Scattering from Polymers

About the Cover

The art was provided by N . S. Murthy, W. Tang, and K. Zero of AlliedSignal Inc. from a fully drawn fiber of a copolymer of PET obtained at the Advanced Polymers Beamline X27C in the National Synchrotron Light Source, Brookhaven National Laboratory.

A C S S Y M P O S I U M S E R I E S 739

Scattering from Polymers Characterization by X-rays, Neutrons,

and Light

Peggy Cebe, EDITOR Tufts University

Benjamin S. Hsiao, EDITOR University of New York at Stony Brook

David J. Lohse, EDITOR Exxon Research and Engineering Company

American Chemical Society. Washington, DC

Library of Congi

Scattering from ρ Cebe, editor, Benjamin S. Hsiao, editor, uavid J . Lonse, editor.

p. cm.—(ACS symposium series, ISSN 0097-6156 ; 739)

Developed from a symposium sponsored by the Division of Polymeric Materials: Science and Engineering, at the 216th National Meeting of the American Chemical Society, Boston, Mass., August 21-27,1998.

Includes bibliographical references and index.

ISBN 0-8412-3644-5

1. Polymers—Analysis Congresses. 2. Scattering (Physics)—Congresses.

I. Cebe, Peggy. II. Hsiao, Benjamin S. III. Lohse, David J. IV. American Chemical Society. Division of Polymeric Materials: Science and Engineering. V. American Chemical Society. Meet­ing (216th : 1998 : Boston, Mass.) VI. Series.

QD139.P6S4 1999 547'.7046—dc21 99-35339

CIP

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Foreword

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Table of Contents

Preface xi INTRODUCTION 1. Introduction to Scattering from Polymers

Peggy Cebe 2 SCATTERING METHODS 2. Characteristics of Small-Angle Diffraction Data from Semicrystalline Polymers and Their

Analysis in Elliptical Coordinates N. S. Murthy, D. T. Grubb, and K. Zero 24

3. Analysis of SAXS Fiber Patterns by Means of Projections

N. Stribeck 41 4. Studying Polymer Interfaces Using Neutron Reflection

D. G. Bucknall, S. A. Butler, and J. S. Higgins 57 5. Neutron Diffraction by Crystalline Polymers

Yasuhiro Takahashi 74 6. Simulation of Melting Transitions in Crystalline Polymers

Lucio Toma and Juan A. Subirana 93 7. Neutron Spin Echo Spectroscopy at the NIST Center for Neutron Research

N. Rosov, S. Rathgeber, and M. Monkenbusch 103 POLYMER CRYSTALLIZATION AND MORPHOLOGY 8. Isothermal Thickening and Thinning Processes in Low Molecular Weight Poly(ethylene

oxide) Fractions Crystallized from the Melt: Effects of Molecular Configurational Defects on Crystallization, Melting, and Annealing Er-Qiang Chen et al. 118

9. Investigating the Mechanisms of Polymer Crystallization by SAXS Experiments

G. Hauser et al. 140 10. Simultaneous In-Situ SAXS and WAXS Study of Crystallization and Melting Behavior of

Metallocene Isotactic Poly(propylene) Patrick S. Dai et al. 152

11. Lamellar Morphology of Narrow PEEK Fractions Crystallized from the Glassy State and

from the Melt M. Dosière, C. Fougnies, M. H. J. Koch, and J. Roovers 166

12. Real-Time Crystallization and Melting Study of Ethylene-Based Copolymers by SAXS, WAXD, and DSC Techniques Weidong Liu et al. 187

13. A Scattering Study of Nucleation Phenomena in Homopolymer Melts

Anthony J. Ryan, Nicholas J. Terrill, and J. Patrick A. Fairclough 201 14. Crystallization and Solid-State Structure of Model Poly(ethylene oxide) Blends

James Runt 218 15. Transient Rotator Phase Induced Nucleation in n-Alkanes

E. B. Sirota and A. B. Herhold 232 COMPLEX FLUIDS AND BIOPOLYMERS 16. Highly Ordered Supramolecular Structures from Self-Assembly of Ionic Surfactants in

Oppositely Charged Polyelectrolyte Gels Shuiqin Zhou et al. 244

17. Some Thermodynamic Considerations of the Lower Disorder-to-Order Transition of

Diblock Copolymers M. Pollard et al. 261

18. Analysis of the Structure, Interaction, and Viscosity of Pluronic Micelles in Aqueous

Solutions by Combined Neutron and Light Scatterings Yingchun Liu and S.-H. Chen 270

19. Optical Probe Study of Solutionlike and Meltlike Solutions of High Molecular Weight

Hydroxypropylcellulose Kiril A. Streletzky and George D. J. Phillies 297

20. SANS Studies of Polymers in Organic Solvents and Supercritical Fluids in the Poor

Theta, and Good Solvent Domains, Y. B. Melnichenko et al. 317 21. Destruction of Short-Range Order in Polycarbonate-Ionomer Blends

Ryan Tucker et al. 328 22. Scattering from Magnetically Oriented Microtubule Biopolymers

Wim Bras et al. 341 POLYMERS UNDER FLOW 23. What Is a Model Liquid Crystalline Polymer Solution?: Solvent Effects on the Flow

Behavior of LCP Solutions S. Chidambaram et al. 356

24. X-ray Scattering Measurements of Molecular Orientation in Thermotropic Liquid Crystalline Polymers under Flow, Wesley R. Burghardt Victor M. Ugaz, and David K. Cinader, Jr. 374

25. X-ray Rheology of Structured Polymer Melts

Geoffrey R. Mitchell and Elke M. Andresen 390 26. Phase Separation Kinetics during Shear in Compatibilized Polymer Blends

Alan I. Nakatani 405

BLOCK COPOLYMERS 27. Ultra-Small-Angle X-ray Scattering and Transmission Electron Microscopy Studies

Probing Grain Size of Lamellar Sytrene-Butadiene Block Copolymers Randall T. Myers et al. 436

28. Block Crystallization in Model Triarm Star Block Copolymers with Two Crystallizable

Blocks: A Time-Resolved SAXS-WAXD Study G. Floudas et al. 448

29. Temperature- and Pressure-Induced Microphase Separation Transitions of a Polystyrene-

block-Butadiene Copolymer Melt W. De Odorico, H. Ladynski, and M. Stamm 456

30. Ordering Kinetics between HEX and BCC Microdomains for SI and SIS Block

Copolymers Hee Hyun Lee and Jin Kon Kim 470

31. SAXS and Rheological Studies on the Order-Disorder and Order-Order Transitions in

Mixtures of Polystyrene-b-Polyisoprene-b-Polystyrene and Low Molecular Weight PS Seung-Heon Lee and Kookheon Char 496

32. Thermoreversible Order-Order Transition between Spherical and Cylindrical Microdomain

Structures of Block Copolymer Kohtaro Kimishima et al. 514

INDEXES Author Index 532 Subject Index 534

Preface

This volume consists mainly of chapters based on presentations made at a symposium on "Scattering from Polymers", which was held at the American Chemical Society (ACS) national meeting in Boston on August 23-27,1998, and was sponsored by the Division of Polymeric Materials: Science and Engi­neering, Inc. This three-day symposium was well attended, reflecting the in­tense current scientific activity and large degree of practical interest in this field. The symposium covered all aspects of scattering from polymeric materials, but the emphasis was on scattering from block copolymers, crystalline polymers, complex fluids, multicomponent systems, polymeric surfaces, and the applica­tion of scattering to processing. Both the experimental and theoretical aspects of scattering were addressed, with a focus on the applications of scattering to polymer technology.

This book is organized slightly differently than the symposium, reflecting the selection of the topics and the fact that several chapters were solicited from authors who had not given talks during the meeting. Following an intro­duction on polymer scattering, there are sections on scattering methods, on crystallization and morphology of polymers, on polymer solutions and biopoly-mers, on block copolymers, on liquid crystalline polymers, and on blends and flow. The chapters are divided this way simply to focus the attention of the reader. As one reads through them, it is clear that several of them could have fit into two or more of the sections, but this is the structure that makes the most sense to us.

We are very grateful to the organizations that sponsored this symposium: Exxon Research and Engineering Company, the Petroleum Research Fund of ACS, and the ACS Division of Polymeric Materials: Science and Engi­neering, Inc. We also thank Anne Wilson and Kelly Dennis of ACS Books

xi

for their help in making this volume a reality. Our thanks also go to Pat Kocian of Exxon for her secretarial help with the symposium and this book.

PEGGY CEBE Department of Physics and Astronomy Tufts University 4 Colby Street Medford, MA 02155 peggy@cebe.phy.tufts.edu

BENJAMIN S. HSIAO Chemistry Department State University of New York at Stony Brook Stony Brook, N Y 11794-3400 bhsiao@notes.cc.sunysb.edu

DAVID J . LOHSE Corporate Research Laboratory Exxon Research and Engineering Company Route 22 East Annandale, NJ 08801 djlohse@erenj.com

xii

INTRODUCTION

Chapter 1

Introduction to Scattering from Polymers Peggy Cebe

Department of Physics and Astronomy, Tufts University, 4 Colby Street, Medford, MA 02155

This chapter is presented as an introduction to scattering for the non-specialist. The underlying physics will be presented in simple form so the reader will become familiar with the language used to describe scattering, the important mathematical tools for analysis, and the most common structural parameters deduced from the data. We have made heavy use of a number of very excellent references, and we recommend them to the reader who desires further study. These include Alexander's early work from 1969 X-ray Diffraction Methods in Polymer Science (1). Though recent books improve on the descriptions of instrumentation and data analysis presented by Alexander, this book is still a worthwhile reference for the beginner in scattering. The treatments of crystallinity and orientation, lattice distortion, and wide angle scattering are still useful today.

Fava's book Methods of Experimental Physics contains two chapters on X-ray scattering, covering the unit cell and crystallinity, and crystallite size and lamellar thickness (2). Two books on small angle scattering are recommended. Glatter and Kratky's edited collection (3) treats experimental methods and data analysis, and contains a large section on applications to polymers, inorganics, and bio-macromolecules. The relationship between small angle X-ray scattering and neutron scattering is described by Feigin and Svergun (4), including instrumentation for both types of scattering.

The most up-to-date addition to the reader's library should be the book by Balta-Calleja and Vonk (5) titled X-ray Scattering of Synthetic Polymers. This work was published nearly a decade ago but constitutes the most important reference on this subject available. Measurement of lattice constants, line breadth, crystallinity and orientation are described in detail along with a modern description of small angle scattering and analysis of intensity through the correlation function.

Several references were used extensively in the mathematical treatments shown below. These are: Introduction to Solid State Physics by C. Kittel (6), Solid State Physics by Ashcroft and Mermin (7), Diffraction Physics by Cowley (8) and Fourier Optics: an Introduction by Steward (9). While these books do not consider polymers specifically, they have excellent general treatments of scattering from crystals. Cowley's

2 © 2000 American Chemical Society

3

book especially is recommended for its unified handling of diffraction of electromagnetic radiation (X-rays and visible light) and particles (electrons and neutrons).

In the following sections we describe the interaction cf monochromatic, coherent radiation with scattering centers, which results in spherical scattered waves. Interference among these waves creates the intensity pattern sensed by a detector. The connection between the observed scattered wave intensity and the structure of matter is ultimately sought. We are especially interested to find the spatial periodicities within our material that lead to interference. To this end, the mathematical techniques of Fourier transformation and convolution are presented. We end the chapter with sections on the small angle scattering from lamellar systems, and neutron scattering.

Physics of Scattering

The phenomena we call "scattering" refers to the interaction of waves with atoms in which the wave is redirected ("elastically" scattered) without any change in its wavelength. The wave may constitute electromagnetic (EM) radiation or particles, such as neutrons or electrons. For electromagnetic radiation, the energy, ε, is related to the wavelength, λ, through ε =hc/X =1ι(ω/2π), where h is Planck's constant (h =6.63xl0"34

J-s), c is the speed of electromagnetic wave propagation in a vacuum (c =3xl08m/s), and ω is the angular frequency in radians/s. Therefore, in direct analogy with mechanical collisions, elastic scattering is an energy conserving process. Processes which are non-energy conserving, for example, absorption of radiation followed by re-emission at a different wavelength, will not be considered in our discussion of scattering. However, analysis cf the inelastic scattering provides information about the dynamics of the scatterers, and is a growing area of research for the SANS (small angle neutron scattering) community studying diffusion.

The incident electromagnetic radiation travels in a direction characterized by the wavevector k whose magnitude is |k|=k=2n/X. The E M radiation is a transverse wave with electric (E) and magnetic (H) fields mutually perpendicular and in-phase, and their cross product points in the direction defined by k. The E M wave is a solution to the Helmholtz wave equation, ν 2 Ψ- ν 2 (9 2 Ψ/3ί) = 0 where ν is the speed of the propagation of the wave, ν and k are related through v=œ/k.

Whether EM radiation or particles are of interest for scattering, the solution to the Helmholtz equation has the same form. Including the time dependent portion, this solution is E(r,t)=Eoexp{i[k- r-cut]} for the electric field component of the wave. This is the equation of a plane wave with amplitude |Eo|, characterized at a given instant in time by surfaces of constant phase which are planes (i.e., for which k* r=constant).

In a collection of atoms irradiated by unpolarized E M radiation, scattered photons will be emitted in all directions, and the atoms will appear to be acting (approximately) as secondary sources of spherical waves. The spherical wave solution to the Helmholtz equation gives an electric field, Es(r,t), dependent upon the distance, r, between the point of observation, P, and a source located at the origin, according to:

E s(r,t) = E 0(b/r)exp{i[kr-œt]} (1)

where b is called the "scattering length." Since the direction of the scattered radiation for a spherical wave is radial, k* r has been replaced in the argument of the exponential by

4

Figure 1. Plane wave with temporal and spatial coherence lengths, w and t, moving in direction kt toward a scatterer of radius, a. Scattered waves will be detected at P.

kr. The factor 1/r is required for energy conservation, so that the intensity, I=ESES*, falls off as 1/r2 on the surface of the sphere.

Now we consider some restrictions on the dimensions of the wave, the scatterer and the distance to the observation point (detector), following the treatment of Messiah (10). In Figure 1 (drawn after Ref. 10) several planes of constant phase travel in the direction kj toward a scattering center characterized by radius, a. The planes are shown in projec­tion as lines, representing wave crests separated by λ. The wave has finite longitudinal and transverse dimensions, w and t. A detector is placed at Ρ a distance R away from the scatterer. The transverse dimension, t, represents the distance over which the wave is spatially coherent and depends upon the variability of the direction of travel, kj. The longitudinal dimension, w, represents the distance over which the wave is temporally coherent and depends upon the variability of the wavelength, λ. For a well- defined, quasi- monochromatic plane wave, both w and t must be much greater than λ, or t >> λ and w >> λ. In order for the incident electric field to be considered uniform over the volume occupied by the scatterer, we also require that the dimension cf the scatterer be less than the dimensions over which the wave is coherent, or a << t and a << w.

Furthermore, so that the electric field amplitude will vary slowly over the volume containing the scatterer, the size of the scatterer must be less than the wavelength, or a<<λ. This condition is easily met for X-rays (λ~10-8cm) scattering from electrons (a~10-13cm), and for neutrons scattering from atomic nuclei (λ~10 - 1 3cm). In the latter case, the wavelength is given by λ=h/mv for neutrons of mass, m, and velocity, v. The neutron wavelength is λ~10 - 8-10 - 7cm and depends upon the velocity given by the experimental apparatus (4).

A final consideration of distances concerns the placement of the detector. The detector must be far enough from the scatterer so that the electric and magnetic fields are transverse to the radius vector, a condition not met close to the scatterer. Thus, the detector must be in the "far field" for which R>>λ. The detector must also be positioned so that it senses only the scattered wave, and not the incident wave transmitted in the forward direction. If 20 is the angle between ki and R in Figure 1 (the customary angle designation in scattering), then the detector distance must be such that sin2θ >> (t/2)/R.

5

The special case cf scattering cf EM radiation by a charged particle is relevant for the interaction of X-rays with crystals. The scattering of E M radiation by an electron produces a spherical wave, modified by a geometrical factor associated with the incident beam's polarization state. The amplitude scattered by an electron in an atom is (8):

| E S I = - Eo {(e^/me2) sin α } /r (2)

where the minus sign in front of EQ reflects a 180° phase shift between the incident EM radiation and the scattered radiation. Here, the scattering length (from equation 1) is Ib l^ /mc 2 , the classical electron radius. This arises according to Thompson scattering theory (11) from the strength of scattering of a single charged particle of mass, m, and charge, e. The reciprocal dependence on particle mass means that X-ray scattering from the nucleus is negligibly small, the nuclear mass being 103 times smaller than the electron mass. The geometrical factor since is dependent upon the polarization of the incident E M radiation, as described below.

Figure 2 shows the geometry used to describe the polarization effects. The incident E M radiation is shown traveling in the direction ζ toward an electron at the origin. Scattered radiation is emitted in the direction indicated by ks which lies in the x-z plane. It is convenient to consider two orthogonal polarization states for the incident radiation, since any arbitrary polarization state can be written as a linear superposition of these two states. The two polarization states, |E s | |j and | E S | ^ are defined to be parallel and perpendicular, respectively, to the plane containing kj and 1%. Figure 2 shows the incident radiation polarized in the χ-direction, α is the angle between the direction of acceleration of the electron and 1%, the scattered EM wave direction. 2Θ is the angle between lq and kg, referred to as the "scattering angle".

Figure 2. Electric field of a plane wave, Eq, traveling along the direction k*t and incident on an electron. The wave is polarized along x. Scattered wave has wavevector, ks, and makes an angle of a with x, and 2 Θ with h

6

For the incident wave polarized in the χ -direction, parallel to the plane cf the page, the electron accelerates along χ, and α = 90°- 2Θ. |E S | | j becomes (8):

| E S | | | = - E 0 {(e^mc2) cos 2Θ } Ix (3)

For an incident wave polarized in the y-direction, perpendicular to the plane of the page, the electron accelerates along y, and α = 90°. |E S | χ becomes (8):

|E s | i = -E 0 ( e 2 /mc 2 ) / r (4)

For unpolarized incident radiation, the scattered intensity will be the average cf the intensities contributed by the parallel and perpendicular fields. This yields a well known result for scattered intensity:

I s= Io(e^/mc 2) 2[(l+cos 220)/2r 2] (5)

When small angle scattering is considered (i.e., 2Θ very small and a~90°), the polarization effects of the incident and scattered radiation may be neglected (3).

Scattering from a Collection of Atoms

In a collection of atoms containing many electrons, the scattered radiation collected at a distant point Ρ shown in Figure 3 (drawn after Ref. 6) will depend upon the phase relationship among the waves scattered from each atom. A large intensity will be measured at Ρ provided that there is constructive interference among the scattered waves emitted by the atoms. On the other hand, f crests from some scattered waves overlap

Figure 3. Geometry of scattering from two point scatterer s at A and B. Spherical scattered waves are detected at P.

7

with troughs from other scattered waves at the time they arrive at P, then partial or complete destructive interference will occur. The intensity at Ρ will be very weak or zero. We now derive the conditions leading to constructive interference.

If the incident beam of EM radiation is spatially coherent across its width, t, and that width is much greater than the volume occupied by the scatterers, then waves scattered from different atoms within the volume will likewise be coherent. That means that there is a fixed phase relationship among the waves scattered by the electrons associated with these atoms. The phase relationship is expressed by a factor of e1^ which multiplies the scattered wave amplitude. The phase φ =(2π/λ)(Δ)^Δ where Δ is the path difference. Since all waves scattered by the electrons have the same amplitude (given by equation 2), they differ only in their phases, relative to a reference point.

Figure 3 shows an ordered array of scatterers (6). Two electrons at A and Β a vector r apart, have been singled out. Whether or not these two electrons produce constructive interference at the distant location of the detector depends upon the instantaneous phase of the incident electric field at A and Β. An approximation is made in considering the scattered wave amplitude. The First Born Approximation (8), or single-scattering approximation, assumes that the amplitude of the spherical scattered wave is very small compared to the incident wave. When the scattered wave encounters another electron, it is not scattered a second time. Thus, only the incident wave scatters from the electrons.

In Figure 4a, electrons at A and Β are shown in relation to the incident and scattered wave crests. For simplicity, the figure depicts the incident electric field at point A as a crest (maximum) and the scattered electric field at point A also as a maximum. The 180° phase shift between incident and scattered waves is omitted in the figure, and is understood to occur at both points A and B, and therefore will not change the relative phase. The condition for constructive interference between the waves scattered from A and from Β is that the path difference Δ must be an integral multiple cf the wavelength, or equivalently, the phase φ must be an integral multiple of 2π. Following the treatment from Ashcroft and Mermin(7), in Figure 4a the path difference is:

Δ = r cos ψΐ + r cos ψ 5 = r - nj - r · n s = τηλ (6)

where η is a unit vector defined by k = 2πη/λ, and m is an integer. The constructive interference condition for the phase is written φ = r · (kj - kg) =2πηι which can be stated using the exponential phase factor as:

e i ( k s - k i ) . r = e i K . r = h ( 7 )

Κ (= kg - kj) is called the "scattering vector". Figure 4b shows the geometry in terms of scattering angle, 2Θ, from which |K| =2|^8Ϊηθ=4π8Ϊηθ/λ. Different authors in this book may use either K, q, or s (where 8=ς/2π=Κ/2π) interchangeably for the scattering vector.

Only the component of the vector r in the direction of Κ makes any contribution to the phase. Thus, all scatterers have the same phase if they lie on planes having their surface normals parallel to Κ (i.e., the planes themselves are perpendicular to K) . One such plane, perpendicular to the page, is indicated by the dashed line in Figure 4b. Planes parallel to this one, separated from it by a distance d, will yield constructive interference. For such planes of scatterers, dK=d^sin9/À)=27tm resulting in the well known expression of Bragg's Law: 2dsin0 = mX.

8

Now the conditions for constructive interference expressed by equation 7 allow us to calculate the net amplitude of scattered waves. The amplitude F of the electric field at Ρ will be the sum of the amplitudes scattered from A and B, or:

F = | E s | ( e

i 0 + e i K ' r > (8)

To generalize to an array of scatterers at positions rj, we sum over all vectors i*j (5):

F = | E S J Sj ( e i K' rJ ) (9)

Figure 4. a.) Geometry for constructive interference of waves scattered from A and B. The path length difference, rcosi#+reas%, must be an integral multiple of λ. b.) Geometry of scattering where A/ and ks each make an angle Θ with respect to the dashed line. Bragg's law is satisfied for a set of planes perpendicular to the plane of the page having Κ as their surface normal and containing the dashed line.

9

The exact locations of the many electrons in a bulk sample are unknown, so it is usual to characterize a collection of atoms by the electron density, p(r) (8):

i X r ) = Z i P i ( r ) * δ ( r - r i ) (10)

Here pi(r) is the electron density of the atom located at ri, and δ(r-r) is the delta function. (The symbol * represents mathematical convolution, which will be treated in the next section.) The volume, dV, around an atom contains Pi(r)dV electrons. If the volume is small compared to the wavelength of radiation (a <<λ in Fig. 1), the electric field of the incident wave may be taken as uniform over the volume. Then all electrons within the volume see the same incident electric field, and contribute equally to the scattered field. The amplitude of the scattered field due to all volume elements is:

F(K) = | E S | J p ( r ) e i K ' r dV r (U) Vr

For the sake of clarity in expressions to follow, we make several simplifications in equation 11. The scattering amplitude for one electron, |E S | , will from now on be taken as unity, and will be understood to apply to all equations dealing with scattered wave amplitude. The integral in equation 11 will be treated, without loss of generality, in one dimension. Scaling to three dimensions is straightforward. Finally, we replace Κ by u= Κ/2π. Then u has units of reciprocal length, |u|=2sinO/λ. The scattering amplitude is:

+00

F(u) = J p(x) e 2 7 l i u x dx (12) -00

The experimentally measured quantity is the intensity of scattered radiation, I(u), which is related to the scattered wave amplitude by: I(u)=F(u)F*(u). If the electron density function is known, the intensity of the scattered wave can be calculated. Most often it is the reverse problem that is of interest: we want to deduce the structure (electron density profile in the sample) from the measured intensity. In the next section we present the mathematical tools that are used in solving this problem.

Fourier Transforms, Convolution, and Correlation

The expression in equation 12 is recognizable as a Fourier transform. F(u) and p(x) constitute a Fourier transform pair, with p(x) written as:

+00

p(x) = J F(u) e - 2 7 î i u x du (13) -00

Now we consider the effect of replacing x by x-a in equation 12, which constitutes a translation of scatterers in real space by an amount, a. This yields F (p(x-a))= F(u)exp(27riua). The amplitude of scattering from the translated scatterers is the original amplitude, F(u), multiplied by a phase factor (which is ultimately eliminated upon squaring to obtain the intensity.) Thus, translating the scatterers in real space introduces a phase factor which does not affect the intensity distribution in the scattering pattern. This results in ambiguity in interpretation of the intensity, called the "phase problem".

10

The forward process, going from p(x) to intensity, is mathematically unique; the reverse process, going from intensity to p(x) is not unique. From the measured intensity, we can not deduce p(x) except to within an arbitrary phase factor.

Nonetheless, by using the Fourier transform and two other mathematical operations, convolution and correlation, we can obtain several important structural parameters from the experimentally measured scattered intensity. The usual assumption is that the system can be represented by two-phases, with either sharp or diffuse boundaries between the phases. Some structural parameters of interest include: average separation distance between the phases; specific surface, O s ; average thicknesses cf the two phases; and width of the boundary between phases, if diffuse. These parameters are obtained from the "auto-correlation" function of p(x), which is a specific type of convolution.

Convolution and Correlation Functions

For the non-specialist in scattering, we provide in this section a very detailed look at the meaning and use of mathematical convolution. Using simple model profiles for the electron density distribution, we introduce the major structural parameters that are obtained from the correlation function.

First, we will illustrate the convolution operation. A function, h(x) is a convolution of two functions f(x) and g(x) provided that:

+ 00

h(x)=/f(x')g(x-x')dx' (14) -00

Mathematically we write the convolution in shorthand as h(x)=f(x)*g(x)=g(x)*f(x). The convolution is a scheme by which one function, g(x), is positioned according to a rule described by the other function, f(x) (9). The value of h at a particular point τ , η(τ), is weighted according to the degree of overlap between the two functions. Function g is first inverted, and then shifted to a new origin at τ. The product between f(x') and g(T-x') is calculated, and then integrated over all values of the dummy variable, x\ To map out the entire function h(x), the value of τ is made to range from -<» to +».

The convolution operation is illustrated in Figure 5a-f. In Figure 5a (drawn after Ref. 9), function 9x) is a series of delta functions equally spaced along the x-axis. Function g(x) is a one-dimensional "rectangle" function of width d and height unity, centered about the origin. For |x| < d/2, g(x) =1; for |x| > d/2, g(x) =0. The delta function placement of the rectangles results in non-zero overlap between flx) and g(x-x) only at positions Xj where Xj-d/2 < τ < Xj+d/2. The Xj are the positions of the delta function spikes. The resulting convolution product h(x) is shown on the right of Figure 5a. The special mathematical properties of the delta-function (infinite height, zero width, and unit area) cause the width of the rectangles in h(x) to be d, the same as the width of g(x). In general, when both fix) and g(x) have finite widths, the width of h(x) will be greater than the width of either of the convolved functions.

h(x), depicted in Figure 5a, represents uniform positioning of a series of rectangles. There are many important physical systems for which this model serves as an ideal. For example, f(x) might represent the locations of atom centers along the direction χ within a crystal and g(x) might represent the electron density profile, Pi(x), as in equation 10. Then h(x) shows the positions of the electron dense regions within the crystal. Another example pertinent to small angle scattering would have g(x) as the electron density difference between components A and Β in a lamellar diblock copolymer. The height of

11

the g(x) rectangle would range between the densities of the components, PA and ρβ. Function f(x) would represent the locations of the centers of the copolymer lamellae along a direction parallel to their surface normals. The convolution product, h(x), then depicts a series of equally spaced A-B diblock lamellae of width d.

Figure 5b-e illustrates successive stages in the convolution of a function with itself, called the "auto-convolution", given by h(x)=f(x)*f(x). In this example, the function f(x) comprises two identical rectangles each of height, y, width, d, a distance, L, apart. These are shown as the unfilled rectangles located symmetrically about the vertical axis. For the convolution, the function f(x-x) (replacing g(x-x') in equation 14) is shown shaded. In each of the frames 5b-5e, f(x-x) is moved to different locations, as indicated by the progressively increasing value of τ. Note that since in this example f(x) has a center of symmetry, the convolution operation requiring the reversal of f(x) results in the shaded function being just exactly a shifted version of f(x). Other examples of the convolution of non-symmetric functions may be found in the references (5,8,9).

In Figure 5b, for τ< -(L+d) there is no overlap between flx) and f(t-x), so h(x) (Figure 5f) will be zero. In Figure 5c, τ reaches -(L+d) and overlap of f(x) and f(x-x) begins. Increasing τ to -L (Figure 5d) results in complete overlap of one rectangle from f(x) and f(x-x), giving a large contribution to the convolution product, so h(-L) = y 2 d. Further increase to τ =-d begins the region of maximum overlap between i(x) and f(x-x). The maximum of h(x) will occur at τ = 0 when both rectangles overlap. As τ increases further, the shaded rectangles will eventually pass beyond the empty ones, and h(f) will be zero for all x>L+d. In this example, from examination of h(t) alone we can deduce some aspects of the structure of f(x). The greatest value of h(x) will always occur at τ =0 where there is complete overlap. The first minimum of h(x) occurs at x=d, giving the width of the rectangles. The value of h(0) is 2y2d (from equation 14). The subsidiary maxima in h(x) occur at τ = ±L, giving the separation distance between the rectangles.

Figure 5 illustrates the procedure for determining the convolution product. A related mathematical process is that of correlation, written as h(x) = fix) ο g(x). This is a special kind of convolution and is defined as: fix) ο g(x) = flx)*g(-x). The correlation between two functions f(x) and g(x) is similar to convolution but g(x) is not reversed. The specific case of autocorrelation is defined as f(x)*f(-x). If function f(x) has a center of symmetry, then the autoconvolution and the autocorrelation are identical operations, as was the case for the f(x) in Figure 5b-f. In that case, f(x) ο f(x) = f(x)*f(-x).

An important theorem relates the Fourier transform and convolution operations. The Convolution Theorem (8,9) states that the Fourier transform of a convolution is the. product of the Fourier transforms, or F (f*g) = F(u)G(u). Applying this to the autocorre­lation yields F [f(x)*f(-x)] = F(u)F(-u). If f(x) is real, F(u)F(-u)=F(u)F*(u)=|F(u)|2. Thus, "the Fourier transform of the autocorrelation of a function f(x) is the squared modulus of its transform" (Ref. 9, p. 81). Application to scattering replaces f(x) with the electron density profile, p(x). We have then the important result that the Fourier transform of the autocorrelation of the electron density profile is exactly equal to the intensity in reciprocal space, \F(u)\2. The autocorrelation function of the electron density has a special name: it is called the generalized Patterson function(8), P(x), given by:

+ 00

P(x)=J" P(x') P(x+x') dxf (15) .00

P(x) provides the means by which structural information is obtained about the electron

12

Figure 5. a.) The auto-convolution between f(x), a set of delta functions, and a rectangle function, g(x), resulting in the convolution product, h(x). b.)-e.) The auto-convolution between a two-rectangle function, f(x) (unshaded), and itself (shaded), using equation 21, illustrating the degree of overlap for: b.) \i\>\L+d\; c.) τ =(L+d); d) τ = -L; e.) τ =-d. f) The resulting convolution product, h(t).

13

density: P(x), like h(x) in Figure 5f, will have peaks at specific distances separating scatterers with large electron densities.

The Convolution Theorem is illustrated in Figure 6a-d for a simple electron density profile. Figure 6a shows p(x), the two-rectangle function, with rectangles of width, d, and separation, L. The height of the rectangles is pc-pa- The arrows show the Fourier transform operations connecting the left and right hand sides of the figure. F(u), is obtained by Fourier transformation of p(x), as F(u)~[sin(ndu/2)/(Kdu/2)]cosLu/2), which is the well-known interference amplitude from two slits. The corresponding intensity is shown in Figure 6d, obtained from the squared modulus |F(u)|2. In Figure 6c, the positive portion of the autocorrelation function of p(x) is shown, arrived at by analytical convolution of the density profile in Figure 6a. The Convolution Theorem connects the measured scattered intensity directly to the electron density correlation function, shown in Figure 6c, by reverse Fourier transformation. In the correlation function, the region extending from x=0 to x=d is called the "self-correlation triangle". Together with the first subsidiary maximum, the self-correlation triangle allows structural parameters of p(x) to be determined. For a more complete discussion of the shape and properties of the correlation function, the reader is referred to the work of Strobl and Schneider (12).

Scattering From Polymer Lamellar Systems

Many types of polymers self-organize into lamellar (sheet-like) morphologies. This includes, among others, semicrystalline polymers, symmetric linear A B diblock copolymers, and smectic main-chain liquid crystalline polymers. The characteristic of the lamellar morphology is that there is a variation of the electron density in the direction normal to the lamellae. The spatial period of the variation can be large, of the order 10-100nm, so the scattering occurs principally at small scattering vectors.

Figure 7a shows the experimental geometry for scattering from an isotropic sample (drawn after Ref. 2). The incident beam direction, kj, is fixed. Radiation is scattered by the sample and collected by a detector, shown here as a two-dimensional detector. The lamellar stacks are arranged in domains; the domains themselves are randomly oriented. Within a domain, the lamellae are characterized by a one-dimensional variation of electron density along the direction of the lamellar surface normal. The electron density alternates between the density of the more dense region, pc, and the less dense regions, pa. The subscripts c and a are chosen to reflect "crystal" and "amorphous" regions, as might be found in a semicrystalline polymer. This scattering entity is equivalent to one in which one phase is empty (containing vacuum) while the second phase has a density equal to the average difference in electron densities, <pc-pa>. The diameter of the incident X-ray beam (not depicted to scale in Figure 7a) is much greater than the size scale of the collection of lamellar domains.

Only a small fraction of domains will be in scattering position at angle 20 relative to kj. Domains with lamellar surface normals parallel to Κ in Figure 4b will scatter at 20. However, since these can have any orientation around the main beam, the scattered intensity will be in the form of a cone, which intersects the detector in a ring. Circum­ferential integration is used to collapse the ring pattern into a radial one-dimensional pattern, thereby increasing the signal-to-noise ratio. If the cross-section of the incident

14

Figure 6. illustration of the Convolution Theorem, a.) The two-rectangle function, p(x), representing a simplified electron density distribution from two scatterers; and b.) its Fourier transform, \F(u)\, the amplitude of scattered radiation; c.) the auto­correlation function, p(x) ο p(x); and d) its Fourier transform, \F(u)\2.

X-ray beam is rectangular, which occurs when slits are used to collimate the beam, the resulting pattern will show smearing of intensity due to the incident beam shape. These effects must be removed by mathematically "desmearing" the pattern to obtain the corrected intensity. The topic of desmearing is extensively covered in the references (3-5). Least distortion of the measured intensity occurs when pinhole collimation is used.

Corrections to Raw Data

The raw intensity must undergo several corrections prior to data analysis to find the structural parameters (1-5). The data must be corrected for: background transmission due to air scattering and/or effects of the sample holder; possible changes in incident beam intensity during real-time data collection (for example, during crystallization or melting studies); and, sample absorption effects. The data may be smoothed to eliminate statistically random noise. Averaging of repetitions of the same experiment is used to verify the sample behavior, and eliminate "bad" data caused by equipment malfunction.

15

Detector

Scattering Vector, s

c) ,γ(ι)

χ

Figure 7. a.) Experimental geometry showing X-rays incident on an isotropic sample comprising stacks of lamellae (shown striped), randomly arranged. A ring pattern will be seen on a 2-D detector, b.) Lorentz corrected intensity, Is2, vs. scattering vector, s, in the small angle range, c.) One dimensional electron density correlation function, y(x) vs. χ for the intensity shown in (b), using equation 21.

The distance from the sample to the detector must be calibrated by reference to standards having sharp scattering patterns, in order accurately to determine the scattering vector.

The intensity is further corrected by a Lorentz weighting factor dependent upon the geometry of the sample(13). In an isotropic sample, intensity is scattered in reciprocal space over spheres of surface area Απ£, where s=2sin6/X. To compare measured intensities spread over spheres of different radii, a Lorentz weighting factor of 4π8 2 is used. In Figure 7b, Lorentz corrected intensity, I c o r rocls 2, is plotted vs. scattering vector, s. (The An scaling factor is suppressed.) The choice about whether to use s or q ^=2π8) as the scattering vector on the abscissa is a matter of taste. One advantage for use of s is that it directly represents the reciprocal space periodicity, having the same meaning as u in equations 12 and 13. Data analysis of the Lorentz corrected intensity profile may involve simple application of Bragg's Law to find LB (the spatial periodicity corresponding to the location of the intensity ring) or calculation of the one-dimensional electron density correlation function. Here, λ =2 LB sin θ ~ 2LB9.

16

In small angle scattering the correlation function, γ(χ), is related to the corrected intensity, I c o r r , by reverse Fourier transformation (see Figure 6) through (14):

+ 00 +00

Y(x) = I Icorr(s) e-2nisx ds=f 4ns2 I(s) e" 2 7 l i s x ds (16) -00 -00

As shown in Figure 7b, the intensity data cover a finite range of s. No intensity data are available below smj„, which is determined by the size cf the beam stop. The width of the detector and the sample-to-detector distance limit sm a x , the largest value of s attain­able. These intensity cut-offs have implications for calculation of γ(χ), since the integration limits extend from -oo to +oo. Extrapolation to low-s may be performed by using a Guinier plot by which I is plotted vs. s2, then extended from s m j n to s =0. Simple extension by linear extrapolation of the plot cf I vs. s below smin is also used. These extrapolation methods change the total area under the scattering curve, but the Lorentz correction factor reduces the ultimate effect of low-s extrapolations on curve area.

Extrapolation to high-s is performed by means of Porod's Law, which predicts that the fall off in the intensity should go as 1/s4. For an isotropic two-phase system, the limiting value of intensity is found from (4):

lim I(s->oo) = (p c-p a)2 [0 8 /8π 3 ] s"4 = Kp/ s4 (17)

where O s is the specific inner surface. Kp is called the Porod's Law constant. If Porod's Law is obeyed at high s, Is4 vs. s will show a high-s region which is independent of s. Empirically it may be that Porod's Law is not obeyed, for example because of thermal density fluctuations within the phases, or diffuse rather than sharp interfaces between the phases. In either case, the high-s data should be corrected to bring the slope into conformity. Correction for thermal density fluctuation is made by plotting Is4 vs. s4. ff the high s region is linear, the data are corrected by a constant equal to the slope of the line. After correction, intensity is made to go to zero with the correct functional dependence upon s, so the integration in equation 16 can be carried out.

In the case that the boundary between the phases is very diffuse, the corrected intensity at large s decreases more quickly than Porod's law, and is modeled as (15):

I'corKs) = C exp(-7i2t2s2)/s4 (18)

In equation 18, I'corr(s) is the intensity after correction for thermal density fluctuation, and t is the thickness of the interphase (a region possessing an intermediate electron density between p c and p a). Plotting ln(I'Corr(s) s4) vs. s2, the scattering at large s is used to find the slope, 7t2t2, from which the interphase thickness is found (15).

An alternative approach to the integral expression of the correlation function in equation 16 is to use a discrete Fourier transformation method. Then the integral expression for the correlation function is replaced by a summation (16):

Ν γ(χ) = Σ 4ns21(s) CUN(H)(X-1) (1 9 )

j=i where ω Ν =βχρ(-2πί/Ν) and Ν is the actual number of data points collected in the interval from smin and s m a x . The correlation function starts off with a spacing along χ cf l / s m a x , and a cubic spline interpolation routine fills in the missing values in the region of interest. This type of algorithm is commonly used in computerized mathematical data

17

analysis and signal processing (16). Both these approaches, extrapolating the intensity regions or interpolating within the correlation function, have the same effect on the Fourier transform. In the Porod's Law extrapolation, data are extended to s-values greater than s m a x , causing the points in the Fourier transform, γ(χ), to be more closely spaced. In the discrete Fourier transform, the filling in of points in γ(χ) occurs by cubic spline fitting. Both are means artificially to create "data" in regions where no experimental data exist. In actual fact, we only have experimental knowledge about the scattering behavior between smin and s m a x . Interpretations cf structural parameters that rely on these extrapolations must be made cautiously, especially since data tend to become very noisy in the high-s region.

Structural Parameters from the Correlation Function

The intensity shown in Figure 7b yields a correlation function, γ(χ), depicted in schematic form in Figure 7c. Variations in average spacing between the phases, average thickness of the phases, and number cf coherently scattering lamellae, all cause the correlation function to take the shape indicated in Figure 7c(12). Following the treatment of Strobl and Schneider, the main structural parameters cf the "corresponding ideal two-phase system" (12) of lamellae are deduced. Many similarities between the observed correlation function and h(x) (see Figures 5f and 6c) are preserved.

The correlation function is defined as: + 00

γ(χ) = J An s2 I(s) COS(2TCSX) ds (20) 0

In this expression, the cosine transform of the complex exponential has been used. The value of γ(χ) at x=0 is called the "invariant", Q, defined as:

+ 00

Q = y(0) = f 4rcs2I(s)ds (21) ο

In some definitions of the correlation function (5,17), Q is used as a normalization factor, dividing equation 20. This normalization factor does not change the shape of the correlation function, affecting only the scaling cf the y-axis: if Q is used as a normalization factor, the correlation function will range between -1 and +1. The invariant is the total area under the scattering curve.

As shown in Figure 7c, the average separation, L, between phases of the same density (called the long period), is given by the location of the first maximum of γ(χ) for x>0. The horizontal line tangent to the first minimum of γ(χ) is called the "experimental baseline". Its coordinate is -A where A= %c2(Pci3a)2. Here, %c (=d/L) is the linear fraction of the phase characterized by electron density, p c. hi a semicrystalline polymer, χ 0 would represent the linear degree cf crystallinity within a lamellar stack. The "true baseline" of an ideal two-phase system would actually be flat, similar to that shown in Figure 6c. This is rarely observed in practice. Therefore, the experimental baseline is an approximation only, and so are parameters determined from it.

The height of the self-correlation triangle, representing the maximum overlap area in the autocorrelation, is:

A+Q = (dTL) (pc-pa)2 = %c (Pc^a)2 (22)

18

The invariant is found from:

Q = Xs%c(l-%c)(Pc-pa)2 (23)

where %s is the spherulite volume filling fraction, ff the sample is uniformly filled with spherulites, χ$ =1. On this assumption, the linear degree of crystallinity will be equal to the ratio A/(A+Q). The average lamellar thickness, d, is obtained (see Figure 7c) as the x coordinate of the baseline intercept with the linear portion of γ(χ) at small χ values.

Finally, the specific inner surface, O s can be determined from the slope of γ(χ) at low x. This slope is defined as (12):

d γ(χ) fax = -(Os/2) (pc-pa)2 = L (pc-pa)2 (24)

from which O s = 2/L. This allows the difference in electron densities, (pc-pa), to be directly determined from equation 17.

In the above treatment, we arbitrarily associated the phase of density p c with length, d. In fact, we can not know which length, d or L-d, is associated with the phase of greater density. The auto-correlation function would be the same if the lengths cf the two phases were reversed, a principle called reciprocity, or Babinet's principle. Other experimental evidence is required to remove the ambiguity in the phase assignment.

Another function used to obtain structure information is the one-dimensional inter­face distribution function, g(x) (18,19). This is simply the second derivative cf the one-dimensional correlation function, or g(x) = γ"(χ). This function gives the probability that two interfaces will be separated by a distance, x. In an ideal two-phase system, the phases would have constant d and L throughout the scattering volume. The interface distribution function would be a series of delta functions. Real polymer systems have a spread of values cf d and L. This causes g(x) to be a smooth curve with broad peaks located at d, L-d, L, L+d, etc. The peak locations and their breadths can be analyzed, and it has been shown(18) that g(x) provides a more reliable estimation of d and L than γ(χ), when the material contains broad distributions of thicknesses.

Finally, we mention real-time approaches to study kinetic processes in polymers, which are becoming more common with the availability of high intensity synchrotron sources cf X-radiation, better detectors, and high speed computers for data analysis. These experiments often use simultaneous detection cf wide and small angle scattering intensities. Some of the problems investigated by this approach include: mechanisms of erystallization(20,21), nucleation phenomena from the melt(22), molecular orientation during flow of liquid crystalline polymers(23), lamella to cylinder transitions in diblock copolymers(24), and structure development during deformation(25). Many of these works will be presented in the later chapters of this volume.

Neutron Scattering

Neutron scattering is being used in a wide variety cf applications in the study cf polymer structure. These include studies of: dimensions of polymer chains in solution, conformation cf chains in networks to test theories cf rubber elasticity, miscibility of blends, structure of block copolymers and semicrystalline polymers, and size and shape ofbio-macromolecules. Adsorbed polymer layers can be studied by neutron reflectivity.

19

Several excellent reviews about neutron scattering in polymers are recommended to the reader (26-31). Here, we introduce the basic principles for the non-specialist.

For elastic small angle neutron scattering (SANS), expressions analogous to those used in X-ray scattering can be written, with some changes in notation common to this field. First, in equation 1 the scattering length, b, depends on the details of the interaction between the neutron and the nucleus (4,26,28). The interaction is controlled by the nuclear and spin density distributions, and in magnetic materials, by the magnetic moment of the incoming neutron and the nucleus. For neutron scattering, b can be either negative (indicating a 180° phase shift upon scattering) as for hydrogen, or positive as for deuterium (4,28). Polymers often consist cf atoms cf low atomic mass which scatter X-rays weakly, therefore one cf the great benefits of SANS is that after deuteration a sample can exhibit excellent contrast for neutron scattering (28).

The average neutron scattering length density (scattering length per unit volume), p n , for a particle (here, considered to be a polymer molecule) comprising a collection of atoms of type i with different scattering lengths, b|, is (30):

p n = ( p m /m ) Σ {ηί bi (25)

where p m is the mass density of the molecule, m is the molecular mass, and n[ is the number of atoms of type i per molecule. If the particle is surrounded by a matrix with a different scattering length density, there will be a contrast factor given by the difference in the densities (pnP - pnm), where ρ stands for "particle" and m for "matrix".

The "scattering cross section" άΣ/άΩ, is the SANS analogue of the intensity, and is the cross-section per unit volume cf sample. άΣ/άΩ is found from equation 12 by squaring, and associating p(r) with the contrast factor for neutron scattering. A collection of identical and widely separated particles in a matrix has scattering cross-section (30):

άΣ/άΩ = C (pnP - pnm)2 (N p /V)|F P(K)| 2 (26)

where C is a constant containing sample and instrument correction factors, and N p / V is the number of particles per unit volume. Fp(K), the "single particle structure factor," depends upon the shape of the particles. Fp(K) is unity at K=0, and decreases to zero as Κ increases, in a manner determined by the shape of the particles. (If there is interparticle interference, another structure term would be needed in equation 26 to describe this effect.) For spherical particles of volume V p = 4m3/3, the structure factor has the form:

Fp(K) = 3 V p {sin(Ka)-Ka cos(Ka)} /(Ka) 3 (27)

There are three regimes of interest in the scattering curve, which determine various parameters cf the particles. At low values of Κ (K->0) Guinier's Law (4) gives the scattering cross-section as:

άΣ(Κ)/άΩ = {dE(K=0)/d^}{exp(-K2Rg

2/3)} (28)

where Rg is the radius cf gyration cf the particle. Rg provides information about the distribution of mass about the particle's center cf mass. A plot cf \η(άΣ/άΩ) vs. K 2

should be linear, with slope proportional to Rg.

20

The different regimes of interest are illustrated in Figure 8 (drawn after Ref. 30) for spherical particles. The cross-section has been normalized to its value at K=0. The oscillations only occur when the particle radius, a, is well-defined, and become damped as the radius takes on a spread of values(26). Figure 8a shows the Guinier region at low K. From equation 26, the intercept at K=0 is equal to C (pnP - pn

mY (N p /V). Thus, the initial portion cf the scattering curve gives information about the particle size and volume fraction, provided the scattering contrast and instrumentation factors are known.

Scattering cross-section vs. Κ is shown in Figure 8b for a larger Κ range. The limiting value of the slope (~K - 4) is given by Porod's Law, illustrated by the straight line of slope -4 in Figure 8b. For neutron scattering Porod's Law is written (cf., equation 17 and recall K=2ns) as(30):

lim άΣ/άΩ (K-»«>) = C(p„P - p ^ ) 2 (N p/V) [2πΑδ] Κ"4 (29)

where A s is 4m2, the area of the spherical particle. For non-spherical particles, such as needles or ellipsoids, it has been shown the slope in the intermediate Κ regime can be used to determined the characteristic shape of the particle (26,31).

Neutron scattering is often performed as a function cf particle concentration to determine specific interactions between components. Blends of metallocene synthesized polyolefins (32) and effects of solvent on dendrimer size (33) have been studied using SANS. Interface widths between two polymer blend components in the melt have been studied by neutron reflectivity (34). Dynamic studies have been undertaken on bimodal melts (35) using neutron spin echo techniques. Some of these recent developments will be reported in the following chapters.

References

1. Alexander, L. E. X-ray Diffraction Methods in Polymer Science; John Wiley & Sons: New York, NY, 1969.

2. Methods of Experimental Physics; Editor, Fava, R.; Academic Press: New York, NY, 1983; Vol 16B.

3. Small Angle X-ray Scattering; Glatter, O.; Kratky O., Eds.; Academic Press: London, 1982.

4. Feigin, L. Α.; Svergun, D. I. Structure Analysis by Small-Angle X-ray and Neutron Scattering; Plenum Press: New York, NY, 1987.

5. Balta-Calleja, F. J.; Vonk, C. G. X-ray Scattering of Synthetic Polymers; Elsevier: Amsterdam, 1989.

6. Kittel, C. Introduction to Solid State Physics; Wiley: New York, NY, 1971. 7. Ashcroft, N . W.; Mermin, N . D. Solid State Physics; Holt, Reinhart, and

Winston: New York, NY 1976. 8. Cowley Diffraction Physics, 2nd Ed.; North Holland: Amsterdam, 1981. 9. Steward Fourier Optics: an Introduction; Ellis Horwood Limited: Chichester,

1983.

21

I

Ο 10 , 20 30 40

Figure 8. Log of normalized neutron scattering cross-section for spherical particles vs. a.) & in the region of low K; b.) Log K. At large K, the slope is determined by Porod's Law and has a Kr4 dependence.

10. Messiah, A. Quantum Mechanics; John Wiley & Sons, Inc: New York, NY, 1958; Vol 1.

11. Kakudo, M. ; Kasai, N . X-ray Diffraction by Polymers; Kodansha Ltd., Elsevier Publishing Co: Tokyo, 1972.

12. Strobl, G.; Schneider, M. J. Polym. Sci., Polym. Phys. Ed 1980, 18, 1343. 13. Crist, B.; Morosoff, N . J. Polym. Sci., Polym. Phys. Ed 1973, 11, 1023. 14. Debye, P.; Anderson, H. R.; Brumberger H. J. Appl. Physics 1957, 28(6), 679. 15. Koberstein, J.; Morra, B.; Stein, R. S. J. Appl. Cryst 1980, 13, 34. 16. MATLAB™ (The Mathworks, Framingham, MA)

22

17. Vonk, C. G.; Kortleve, G. Kolloid-Zeitschrift und Zeitschrift fur Polymere 1967, 220(1), 19.

18. Santa Cruz, C.; Stribeck, N.; Zachmann, H.; Balta Calleja, F. Macromol. 1991, 24,5980.

19. Hsiao, B.; Verma, R.; Sauer, B.; J. Macromol. Sci. 1998, B37(3), 365. 20. Hauser, G.; Schmidtke, J.; Stobl, G.; Thurn-Albrecht, T. American Chemical

Society: PMSE Proceedings, 1998, 79, 344. 21. Dai, P.; Cebe, P.; Alamo, R.; Mandlekern, L . ; Capel, M . op. cit., p.322. 22. Ryan, Α.; Terrill, N. ; Fairclough, P. op. cit., p.358. 23. Ugaz, V.; Burghardt, W. op. cit., p.369. 24. Lai, C.; Russell, W.; Register, R. op. cit., p.380. 25. Hsiao, B.; Fu, X.; Mather, P.; Chaffee, K.; Jeon, H.; White, H. ; Rafailovich,

M . ; Lichtenhen, J.; Schwab, J op. cit., p.389. 26. Windsor, C. J. Appl. Cryst 1988, 21, 582. 27. Lohse, D. Polymer News 1986, 12, 8. 28. Lohse, D. Rubber Chemistry and Technology 1994, 67(3), 367. 29. Higgins, J. S.; Benoit, H. C. Polymers and Neutron Scattering; Clarendon

Press: Oxford, 1994. 30. Fleer, G.; Cohen, M . ; Scheutjens, J.; Cosgrove, T.; Vincent, B.; Polymers at

Interfaces; Chapman & Hall: London, 1993. 31. Methods of Experimental Physics; Eds. Skold, K.; Price, D.; Academic Press:

Orlando, FL, 1987; Vol 23C. 32. Reichart, G.;Graessley, W.; Register, R.; Lohse, D. American Chemical Society:

PMSE Proceedings, 1998, 79, 293. 33. Bauer, Β.; Topp, Α.; Tomalia, D.; Amis, E. op. cit., p.312. 34. Buchnall, D.; Bulter, S.; Hermes, H.; Higgins, J. S. op. cit., p.291. 35. Rathgeber, S.; Willner, L . ; Richter, D.; Appel, M . ; Fleischer, G.; Brulet, Α.;

Farago, B.; Schleger, P. op. cit., p.299.

SCATTERING METHODS

Chapter 2

Characteristics of Small-Angle Diffraction Data from Semicrystalline Polymers and Their Analysis

in Elliptical Coordinates N . S. Murthy1, D. T. Grubb2, and K. Zero1

1Allied Signal Inc., P.O. Box 1021, Morristown, NJ 07962 2Department of Materials Science and Engineering, Cornell University,

Ithaca, NY 14853

The use of small-angle scattering (SAS) of x-rays or neutron from oriented semicrystalline polymers to characterize the lamellar structures is discussed. The features of the SAS data that are commonly analyzed are the lamellar spacing L, angle φ between the lamellar reflection and fiber-axis, the widths of the lamellar reflections along (z) and perpendicular (x) to the fiber-axis, and the integrated intensity of the lamellar peak. In addition, we find that the variations with x of the maxima and the longitudinal width of the lamellar reflection are also related to important aspects of the structure. The intensity maxima of the lamellar reflections in small angle scattering patterns from polymer fibers do not fall on a straight layer line, or on a circular arc. The shape of this arc is analyzed by measuring the periodicity Lφ of the lamellar planes measured parallel to z as a function of φ. A straight line fit to a plot of Lφ

2 vs. tan2

φ shows the elliptical shape of the reflection. This provides a basis for describing the intensity distribution in SAS patterns in an elliptical coordinate system using a minimum number of parameters. It is proposed that a combination of lamellar rotation and shear could cause the lamellar reflection to lie on an ellipse. The increase in the longitudinal width of the lamellar peak with x is attributed to misorientation of the lamellar stacks.

© 2000 American Chemical Society

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INTRODUCTION

Small-angle scattering (SAXS) data from semicrystalline polymers is routinely used to determine the lamellar spacings. However, SAXS data contains a large amount of additional significant information about the lamellar and fibrillar structures,1 and possibly about the connectivity between the amorphous and the crystalline regions. More importantly, it is possible that SAXS data obtained from fibers under strain can be used to relate macroscopic deformation to microscopic structural changes such as slippage of chains, rearrangement of the tie molecules, and the shear between the lamellar and fibrillar structures. Such understanding is useful for example in developing fibers with higher tenacity and modulus and increased dimensional stability.2 This requires careful analysis of the various aspects of the data. Also, with the increased use of synchrotron sources, the limiting step is now the analysis of the data.3 We will discuss efforts over the past two years in analyzing two-dimensional (2-D) SAS data from oriented polymers. The methods discussed here are limited to a specific q-range (< 2.5 nm"1; q=(4nsmθ)/λ) but not to X-rays, though small angle X-ray scattering (SAXS) is the most common technique. The term SAS - for small angle scattering - will be used to include neutron scattering (SANS), which gives results analogous to SAXS, 4 and electron scattering (SAES), which should give the same results if it were used.

The lamellar reflections in small-angle scattering patterns from polymer fibers are often spread onto a curve symmetrical about the fiber axis. These are usually referred to as two-or four-point patterns, the latter sometimes resembling the butterfly pattern frequently found in light scattering. We recently showed that these 2-D patterns could be best analyzed if we describe the intensity distribution in elliptical coordinates because the intensity maxima of the lamellar reflections from oriented polymers fall on an elliptical curve5. We now present new analysis to support this assertion. We will also discuss the physical basis for some of the features in the SAXS pattern in terms of misorientation of the lamellar stacks, deformation of the lamellae, and possible correlation between the lamellar spacing and the orientation of the lamellae.

EXPERIMENTAL

Data were collected at the Cornell High Energy Synchrotron Source (CHESS) on a Fuji image plate using a wavelength of 0.908 A and a sample-to-detector (camera length, C) of 743 mm at F1 beam line. The data from a spun-drawn and heat-treated nylon 6 fiber used in industrial yarns will be discussed in this paper. The data used in this paper have been reported on in our earlier publication.6 The central region of each image was extracted and adjacent pixels binned together to give a 300 x 300 image with 0.2 mm per pixel. For peak fitting, the image was further binned into a 100 x 300 image, and single columns - averages over 0.6 mm in the equatorial or χ direction - were used in the program PeakFit (Jandell). Each 1-D

26

intensity profile was fitted by the sum of four modified Lorentzian peaks, which provided the best fit to the data with least number of parameters.

ANALYSIS AND RESULTS

The SAXS pattern from the fiber used to illustrate the analyses here is shown in Figure 1. The two features obvious in the data are the lamellar reflections and the equatorial streak. Less obvious is an interfibrillar interference peak along the equator at about 50 A. We will here focus only on the lamellar reflections. The various parameters determined by analyzing the data in Figure 1 are listed in Table I and are further elaborated below.

Characteristics of Lamellar Reflections

Lamellar Spacing Figure 2 shows a scan along the fiber-axis through the lamellar reflections.

The lamellar spacing (L) is usually the Bragg spacing corresponding to the position of the lamellar peaks seen in such scans using the expression

L = 1/s (1) where s = (2sin9)M,. However, since the lamellar peak is not quite a Bragg reflection, the correlation function, γ(x), is often analyzed to calculate L.

γ(x) = [ j œ I(sz) cobras,) dsJ/J001(sz) dsz (2) where KsJ = J001(sx sz) sx dsx (3) Subscripts x and z denotes equatorial and meridional directions. L is given by the position of the first maximum in a plot γ(x) vs. x (Figure 3). The curvature in the

Table I. Characteristics that describe the lamellar reflections (Figures 2-8)

Parameter Value Lamellar spacing (Bragg), L 10.3 nm Average lamellar thickness, d 6.0 nm Thickness of the transition Layer, dtr 1.3 nm Integrated intensity, I 218 counts nm-1

Angle between lamellar reflections, x 60° Height of the lamellar stacks, l s 92.4 nm Diameter of the lamellar stacks, D 7.7 nm Misorientation of the lamellar stacks, β 24.2° Ellipticity of the lamellar reflections, ε 0.616

27

1. Two-dimensional (2-D) SAXS pattern from a spun, drawn and heat-set nylon fiber. Fiber-axis (z-axis) is vertical and x-axis is horizontal. Various 1-D scans that follow were generated from this figure.

-0.4 -0.2 0 0.2 0.4 s along ζ (nm"1)

2. A longitudinal slice (i.e., parallel to the fiber axis; z-axis) through the lamellar reflections showing the lamellar peaks.

28

3. Correlation function calculated from the lamellar intensity projected onto the z-axis. The thickness of the transition zone, the thickness of the interlamellar amorphous layers and the lamellar spacing are shown in the figure.

29

plot before the first minimum in these correlation function plots can be analyzed to determine the width of the interphase or the thickness ά\Γ of the transition zone between the crystalline lamellae and the interlamellar amorphous domain, and the thickness d of either the crystalline or the amorphous layers as shown in Figure 3.7 A similar analysis can also be carried out analytically.8

Integrated Intensity The only intensity that is relevant is the one that is integrated over all three

dimensions of reciprocal space.1 Otherwise it is possible to arrive at erroneous conclusions. For instance, peak height or intensity integrated along ζ might show a decrease in lamellar intensity with draw ratio, whereas in fact, a three-dimensionally integrated intensity will indeed show increase in the lamellar intensity. In specimens with a high degree of uniaxial orientation, the integrated intensity can be conveniently obtained as

I = S M J s m a x I (s x )s x ds x (4) where I(sx) = S M J S M A X I(sxsz) dsz (5) In some instances, it is possible to replace the above integrations by multiplying the intensity with appropriate factors analogous to Lorentz factors.1 Absolute intensity measurements is necessary if the intensity from a single sample is to be of any use. However, relative integrated intensities such as the one given in Table I can be of immense value in comparing the data from samples within a set.

Angle Between the Lamellar Reflections The four-point pattern in Figure 1 is attributed either to the organization of

the lamellar stacks in a regular checker-board like lattice,9 or to the oblique angle that the lamellar fold surface makes with the chain or the fiber axis.10 The angle between the two reflections in the four-point pattern is evaluated from an azimuthal scan of the type shown in Figure 4. This angle is sometimes referred to as the tilt-angle of the lamellar plane, i.e., the angle between the normals to the lamellar planes, regardless of whether these planes refer to the fold surfaces or to Bragg planes running through lamellar stacks in adjacent columns.

Widths of the Lamellar Peaks There are two widths to the lamellar peak, the axial width in scans parallel to

the fiber-axis and the transverse width in scans perpendicular to the fiber-axis. The axial width of the peak in longitudinal scans (Figure 2) is used to evaluate the coherence length or height of the lamellar stack (ls) using the Scherrer equation11

l s = λ/[(Δ(2θ) cosG] (6) where Δ(2Θ) is the integral breadth of the lamellar peak along the fiber-axis. Only the first order of the Bragg reflection is seen in many semicrystalline polymers, suggesting that there is a large degree of inhomogeneity and disorder. Hence, the size obtained from the width is only a lower limit, and is always much smaller than that obtained by transmission electron micrographs.

30

2000

1000

-0.4 -0.2 0 0.2

s along x (nm-1)

0.4

4. Azimuthal scan (slice parallel to the equatorial plane; x-axis) through the lamellar reflections for calculating the tilt-angle of the lamellar planes.

31

The transverse width of the lamellar peak in azimuthal scans (Figure 4) is used to calculate the size of the lamellae in the equatorial plane using the above Scherrer equation. Alternatively, the intensity distribution in Figure 4 can be plotted as a Guinier plot (Figure 5), 1 2 and the slope of this curve is used to evaluate the diameter of the lamellae according to the equation 1 3

I(q) = I(0)exp(-q2R2/5) (7) where R is the radius of the lamellar stack and q = (4π8ΐηθ)/λ. Both methods give similar values within the sensitivity of our analysis.

Variations in the Axial Widths of the Lamellar Peak with χ If all the lamellar stacks are perfectly oriented parallel to the fiber-axis, then

the lamellar peak would be streak or a layer line of constant axial width. But in a typical SAS pattern, the width of the layer line increases with the distance from the meridian, as shown in Figure 6. Whole body rotation of the lamellar stack (misorientation) causes the width Δζ, of the layer line to increase with the distance from the meridian.6,14 The rate of such increase in the width is determined by the average angle that the lamellar stack makes with the fiber-axis. This orientation angle (β) of the lamellar stacks is calculated using the expression

[Δζ, «>δ2θ]/(λΡ) = f (sx-s0) + f(sx+s0) (8) where f(s) = l/(21s) + [l/(21s)2 +(s βίηβ)2]"2 (9) in which sc is the position of the maximum of the lamellar peak in the azimuthal scans, i.e., along χ (Figure 4), λ is the wavelength and F is the camera length. l s , the length of the lamellar stack, is same as the coherence length determined from the longitudinal width of the lamellar peak. Whereas coherence length discussed in the previous paragraph is a single-point measurement from the width of the peak in the longitudinal scan through the maximum of the lamellar peak, l s is evaluated in Eqn. 8 by extrapolating the widths of the lamellar peak to sx = 0. Figure 6 shows the reasonably good agreement between Eqn.9 and the raw data. The slope of this curve obtained by least-squares fitting gives the misorientation β of the lamellar stacks.

Variations in the Axial Positions of the Lamellar Peak with χ The lamellar reflections are not flat, but are curved; i.e., there is a continuous

shift in the z-position of the maxima (z0) in the lamellar peaks as a function of χ (Figure 7).2 Because of this curvature, the two-dimensional (2-D) data could not be fitted in Cartesian coordinates. But they are not curved enough to be a circle, hence the polar coordinates ordinarily used in analyzing the wide-angle x-ray diffraction patterns cannot be used either. It appears that the description in elliptical coordinates provides the best fit to the data. This feature of the scattering curve will be analyzed in detail in this paper.

32

Ο 0.5 1 1.5 q2(nm-2)

5. Guinier plot of the intensity of the lamellar reflection measured as function of x.

12.5 ι i — |

0 0 Κ ., ι I . ι ι ι l ι ι ι ι I i i ι ι I ι M t I I I » I I I M I I » » » ' I

-0.02 -0.01 0.00 0.01 0.02

2Θ Χ (Radians)

6. Variations in the axial-width of the lamellar reflections with x.

-15 -10 10 15

X in mm

7. Variations in the peak maxima (zo) of the lamellar reflections with x.

34

Elliptical Characteristics of the SAS Reflections

By analyzing patterns that extend to large scattering angles we have found that the elliptical fit is most likely the best fit to the data. The standard form for this ellipse is

(x/a)2 +(zo/b)2 = 1 (10) where a is the semi-major axis and b is the semi-minor axis. z<, is the position of the intensity maximum along the vertical axis. This equation is linearized by dividing by z 0

2. If φ is the angle between the scattering vector and the z-axis, then tm§ - x/z0. Now we have

(l/Zo)2 = (l/b)2 + (l/a) 2tan^ (11) A plot of (1/ZQ) VS . tan^ would be a straight line with a slope of (1/a)2. The intercept of (1/b)2 is used to calculate a precise and accurate lamellar spacing, which is otherwise a single point measurement. The shape of the ellipse is defined by a dimensionless number, the ellipticity, ε = (1- b/a).

We can further simplify eqn. 11 by substituting tan(20 ) for 2sin9 in Bragg's law. If C is the distance between the sample and the detector, and r is the radial distance from the origin to the point (x, zo) in the SAXS photograph, then we have,

2sin0 = tan(29) = >/(x2+z0

2)/C = r/C (12) So, s = r/(CX ) = 1/LN (13) i.e., (l/r)= (LN/CX) (14)

Here, L N is the periodicity of the lamellae measured in the direction of r. On the meridian, χ = 0, z 0 = + b and

sM = b / (O0=l /L M (15) i.e., l /b=(L M /CX) (16) where L M is the lamellar spacing measured along the meridian ( fiber-axis, φ = 0°) At any arbitrary angle φ, let the periodicity of the tilted lamellar planes measured parallel to the fiber axis be L$, which is given by

l/z 0= (iyCX) (17) Just as L M is the periodicity along the meridian, let L E be the fictitious periodicity along the equator and we have

(l/a) = (L E/Oi) (18) In terms of these spacings, ί φ , L M L E eqn. 11 becomes

L%= L 2

M + L 2

E t a n ^ (19) This is a straight line of slope L 2

E on a plot of L$2 vs. tan% ε = (1 - L E /L M ) . Thus, the shape of the arc can be analyzed by measuring the lamellar spacing L$ at each value of φ, and plotting this as a function of the angle φ from the fiber-axis.

A plot of ί φ

2 vs. tan^ is shown in Figure 8. There is an excellent straight line fit out to tan^ = 25, φ = 66°, confirming the elliptical nature of the shape of the reflection. The intercept is 102.3 ± 0.2 nm2 and the slope is 15.1 ± 0.1 nm2. This gives L M = 10.13 ± 0.01 nm, L E = 3.89 ± 0.03 nm, ellipticity ε = 0.616 ± 0.003.

35

8. The data in Figure 7 in the form of a L§ vs. tan2 φ plot.

36

2-D Fit in Elliptical Coordinates Evaluation of the parameters in Table I by using various 1-D scans of the 2-

D data as discussed above is tedious. An obvious solution is to fit the 2-D data to the various parameters. Such a 2-D fit is best carried out in elliptical coordinates because

of the curvature in the lamellar reflections mentioned earlier.5 Each peak in the elliptical coordinates (u,v) is described as a product of two orthogonal functions, one along u and the other along v. The u-axis and v-axis are analogous, respectively, to the radial and azimuthal directions in the polar coordinates. The Cartesian coordinates χ and y are related to u and ν by the expression

x = V(A 2 + u2)cos(v) (13) y = usin(v) (14)

where A is the distance between the two focii of the ellipse. The intensity at any point (u,v) is expressed as a product of two functions f(u) and g(v). The f and g's can be any of the commonly used functions, we used a Pearson VII function with the shape factor set to 2. The four quadrants are related to each other by symmetry.

The 2-D fit to the data is shown in Figure 9. In addition to a single lamellar peak used to characterize the 4-point pattern in the figure, two additional peaks were used to completely fit the data. One accounts for the equatorial streak related to the scattering from the fibrils and the other for the diffuse scattering in the equatorial plane from the interference between the fibrils. The five parameters -the intensity, position along u and v, and widths along u and v- that describe the lamellar peak are listed in Table II. These parameters can be used to evaluate the features that describe the lamellar structure. The results of the 2-D analysis were found to be in good agreement with the results of 1-D fits shown in Table I.

Table II. Parameters that describe the 2-D fit in elliptical coordinates (Figure 9)

Parameter Value A 62.1 pixels

x-center 149.6 pixels y-center 149.4 pixels Base-line 38.9 counts

Amplitude 6772 counts u-position 33.5 pixels v-position 73.5° u-width 17.8pixels v-width 30.0°

Residuals 2.4 counts

37

9. Elliptical fit to the data show in Figure 1. (a) Raw data, (b) Fit to three functions, (c) The residue between observed and the fitted intensities. The contours are drawn intervals of 200 counts in (a) and (b) and at 20 counts in (c).

38

DISCUSSION

The parameters in Table I fully account for the various features of the lamellar reflections. Many of these have been used in the past. We have introduced two additional parameters, misorientation of the lamellar stacks β and ellipticity ε. These parameters can be derived from a least-squares fit to 2-D data, which is best carried out in elliptical coordinates. The ellipticity is clearly seen in the ί φ

2 vs. tan2<|> (Figure 8). Brandt and Ruland have found a similar characteristics in their SAXS pattern from deformed microdomain structures of block copolymers.15 As we pointed out in our earlier publication,5 a wide variety of SAS patterns found in the literature can be simulated in elliptical coordinates.

The relation between misorientation and the increase in the width of the equatorial streaks and wide-angle reflections have been discussed in detail by Grubb and Prasad.14 We have extended this analysis to determine the degree of orientation of the lamellar stacks. The increase in the longitudinal width of the lamellar reflection with distance from the meridian (Figure 6) clearly shows that lamellar stacks are tilted away from the fiber axis by an angle β. The same misorientation could also affect the curvature of the lamellar reflections. The lamellar reflection from a perfectly oriented stack is a straight layer line, and randomly orientated lamellar stacks would give a circular ring. Therefore, one would expect that some intermediate orientation would cause the layer line to curve, whose shape in between that of a straight line and a circular arc. Our preliminary analysis shows that misorientation cannot fully account for the observed curvature.

A second explanation for the observed curvature is that the ellipticity is due to deformation characteristics of the lamellae or the lamellar stacks as illustrated in Figure 10. The diffraction pattern from a lamellar structure in which the lamellar planes are aligned normal to the fiber-axis is a reflection on the meridian. The lamellar plane could be the locus of the surface of several lamellae in neighboring stacks or fibrils,9 or the surface of a single coherent lamellar crystal.10 Imperfections in the real system would weaken and spread all but the first order points. Suppose the deformation occurs by pure chain slip. Then, during deformation, the lamellar surfaces rotate with no change in the lattice spacing, and the lamellar reflections would spread along a horizontal straight line (Figure 10a) If the rotation of the lamellar planes is associated with a change in the 1-D lattice spacing, then the locus of the reflection is no longer a straight line. An increase of the lattice spacing as the lamellar surfaces rotate away from perpendicular to the fiber axis produces a line with negative curvature (Figure 10b); a reduction in the lattice spacing and the same rotation produces a line with positive curvature (Figure 10c). The extent to which such deformation characteristics contribute to the curvature of the lamellar reflections needs to be explored.

39

0°

10. (a) A straight trace resulting from the tilting of the lamellae with no change in the lamellar spacing, (b) An elliptical trace due to the expansion of the lattice with the tilting of the lamellae, (c) A hyperbolic trace as a result of the contraction of the lattice with the titling of the lamellae.

40

CONCLUSIONS

Two new parameters for describing the SAS data from semicrystalline polymers are introduced. These are the ellipticity of the trace of the lamellar peak-maxima, and the orientation parameter determined from the increase in the longitudinal width of the lamellar peak with the distance from the meridional axis. These two parameters along with the lamellar spacing, tilt-angle of the lamellar plane, the diameter and the coherence length of the lamellar stack, and the lamellar intensity completely describe the SAS data from oriented semicrystalline polymers. These parameters can be obtained by fitting the 2-D SAS from uniaxially oriented semicrystalline polymers in elliptical coordinates.

REFERENCES

1. Matyi, R. J.; Crist, Jr., B. J. Mat. Sci., 1978, 16, 1329. 2. Murthy, N . S.; Grubb, D. T.; Zero, K.; Nelson, C. J.; Chen, G. J. Appl.

Polym. Sci. 1998, 70, 2527. 3. Rule, R. J.; MacKerron, D. H.; Mahendrasingam, Α.; Martin, C.; Nye, T. M .

W. Macromolecules, 1995, 28, 8517. 4. Murthy, N . S.; Orts, W. J. J. Polym. Sci. Polym. Phys. 1994, 32, 2695. 5. Murthy, N . S.; Zero, K.; Grubb, D.T. Polymer 1997, 38, 1021. 6. Murthy, N . S. ;Bednarczyk, C.; Moore, R. A. F.; Grubb, D. T. J. Polym. Sci.

Polym. Phys. 1996, 34, 821. 7. Strobl G. R.; Schneider, M . J. Polym. Sci. Polym. Phys. Ed., 1980, 18, 1343. 8. Verma, R. K.; Velikov, V.; Kander, R. G.; Marand, H.; Chu Β.; Hsiao, B. S.

Polymer 1996, 37, 5357. 9. Statton, W.O. J. Polym. Sci., 1959, 41, 143. 10. Pope D. D.; Keller, A. J. Polym. Sci. Polym. Phys. Ed, 1975, 13, 533. 11. Guinier, A.X-ray Diffraction, Freeman: San Francisco, 1963, p. 124. 12. Crist Jr., B. J. Appl. Cryst., 1979, 12, 27. 13. Guinier, A. X-ray Diffraction, Freeman: San Francisco, 1963, p.328. 14. Grubb, D. T.; Prasad, K. Macromolecules, 1992, 25, 4575. 15. Brandt, M. ; Ruland, W. Acta Polymer. 1996, 47, 498.

Chapter 3

Analysis of SAXS Fiber Patterns by Means of Projections

N. Stribeck

Institut für Technische und Makromolekulare Chemie, Universitat Hamburg, 20146 Hamburg, Germany

Results of two methods for the quantitative analysis of two-dimensio­nal (2D) small-angle X-ray scattering (SAXS) patterns with fiber symmetry are presented. Experimental data originate from studies of poly(ether ester) (PEE) thermoplastic elastomer materials recorded during straining experiments at a synchrotron beamline. The first steps of both methods are similar. By suitable projection of the 2D image data onto a line (longitudinal scattering, first method) or onto a plane (transverse scattering, second method) scattering curves are extracted, which finally can be analyzed in terms of two-phase structural mod­els considering soft domains and hard domains inside the PEE. The studied longitudinal scattering in principal is one-dimensional and originates from chords crossing the soft and hard domains parallel to the direction of strain only. From these curves interface distribution functions (IDF) are computed and analyzed using an advanced stack­ing model. Not only the average domain heights, but four parame­ters characterizing each of two height distributions (hard and soft do­mains) are determined as a function of elongation. With several PEE materials strong equatorial scattering is observed during elongation. The equatorial scattering is similar to the frequently discussed void scattering but originates from an ensemble of rodlike soft domains (needles) in the sample, oriented parallel to the direction of strain. It can be studied using the second method. From the transverse scatter­ing the 2D chord distribution is computed, from which the diameter distribution of the soft needles can be extracted. It is investigated as a function of strain.

© 2000 American Chemical Society 41

42

Introduction If structural transformations shall be investigated, which occur during manufacturing

or processing of polymer materials, one can only use those few methods which are non­destructive and do not interfere with the process itself. In this field scattering methods belong to the most powerful ones. Scattering theory promises the experimenter that he should be able to quantify a wide range of structural parameters, if only the data are recorded with sufficient accuracy. Among these parameters are size and distribution of domains in a multi-phase material, which can be studied using small-angle X-ray scattering (SAXS). Utilizing a powerful source (synchrotron beam) and a high resolution two-dimensional detector, nowadays it becomes possible to record series of detailed scattering patterns with high accuracy during short exposure time.

This chapter presents an overview of work that has been performed by the author aiming to develop adapted (i. e. a structural model is chosen as late as possible) evalu­ation methods for SAXS diagrams with fiber symmetry and the results obtained so far [1-4]. The basic principle of the methods to be presented is the extraction of curves from the 2D data in the scattering patterns by certain kinds of integrations ("projec­tions"). The importance of such projections has early been recognized [5]. Novel is the analysis of the extracted curves in terms of ID and 2D structural models. Applicabil­ity is assessed by comparison of the results with the obvious features of the scattering images.

Experimental Materials

Poly(ether ester)s (PEE) are common thermoplastic elastomers exhibiting a two-phase structure of hard domains in a soft matrix. They can frequently be found in au­tomobiles and as tube materials. Two kinds of multiblock PEEs are investigated. In the first group the soft segment blocks are made from poly(ethylene glycol) (PEG). The second group of samples is made from material in which the soft segment blocks consist of poly(tetrahydrofurane) (PTHF). Hard segment blocks of all samples are made from poly(butylene terephthalate) (PBT). Soft segment block lengths are in the order of mag­nitude of 1000 g/mole. Experiments have been carried out for materials with the hard segment ratio ranging from 35 % to 60 %. Evaluated data presented here originate from experiments on materials with a hard segment content close to 60 %.

When such materials are quenched from the melt, they undergo phase separation forming hard and soft domains. Because phase separation is imperfect, it is common notion that considerable amounts of hard segments reside in soft domains and even the hard domains may contain some soft segments ( fig 1). Otherwise one would not be able to explain the observed domain sizes.

43

Figure 1. Discrimination between first, hard segment and hard domain and, second, between soft segment and soft domain (matrix).

The Experiment

Straining and relaxation experiments are carried out in the synchrotron beam at HA-SYLAB in Hamburg, beamline A2. A typical maximum elongation e = (£ — £Q) /£O « 3 can be reached before the sample slips out of the clamps. During straining small-angle X-ray scattering patterns are recorded on image plates placed 1.8 m behind the sample. After exposing for 1 min the maximum recorded scattering intensity amounts to 60,000 counts. The geometry of the chosen semi transparent beam stop ensures low scattering background at the expense of a relatively large central blind area.

Data analysis is carried out using published computer programs[7], which are freely available[6].

Evaluation Methods Basic Definitions and General Concept

Because all the samples exhibit a scattering pattern with fiber symmetry, it is con­venient to write the intensity I (s) = I ($12, «3) in cylindrical coordinates, with su — \Js\ -f s\ and the component 53 defining the symmetry axis of the pattern. Let the magnitude of the scattering vector be defined by |s| = (2/A)sin0, with λ being the wavelength of radiation and 20 the scattering angle. Because the validity of the tangent plane approximation can be assumed, the complete information of SAXS is in a two-dimensional (2D) pattern. Such patterns may exhibit many reflections, which moreover may vary considerably as a function of tunable parameters (fig 3). Thus the analysis of a 2D pattern appears to be cumbersome because of the wealth of data. But for a 2D image there do exist several methods to extract specific information. Curves gained by such a procedure only reflect certain aspects of the morphology, and thus hopefully can be described by a simple structural model.

First steps towards this goal have been undertaken by Bonart [5]. Based upon the

44

mathematical relation between structure and scattering, he has proposed to analyze pro­jections, which can be extracted from the scattering patterns and reflect a longitudinal and a transverse structure, respectively. Following a concept of Ruland [7,8], the re­sulting curves are not fitted by a complex model, but analyzed step by step in order to "peel off'* information. Finally a chord distribution[9] or an interface distribution[i0] is computed. Only now, based upon the observed properties of the resultant distribution, an adapted model is chosen.

Projections of Fiber Patterns

Definitions of Special Projections Projections are integral operators which map functions onto subspaces of their def­

inition domain. They shall be denoted by a pair of curly parentheses. The best known projection in the field of scattering theory maps the scattering intensity I (s) onto a zero-dimensional subspace, which is the number Q, known as "invariant" or "scattering power"

Bonart's longitudinal structure is obtained by not integrating over the whole reciprocal space, but only over planes normal to the fiber axis yielding a curve

which is a function of S3 only. Thus we identify this curve as a projection onto a one-dimensional subspace and indicate this by subscripting to the pair of braces. Analo­gously Bonart's transverse structure

is computed by integrating along lines running parallel to the direction of strain. In the case of fiber pattern symmetry, this projection can be displayed as a curve. Nevertheless we should bear in mind that it is defined over a two-dimensional domain, the si2-plane. Because, in principal, all these integrals have to be extended to infinity, one should take care to register the scattering intensity over a wide angular range.

Projections and Sections Because of a general theorem of Fourier transformation theory, projections in recip­

rocal space are equivalent to sections in physical space (fig 2). But what is the definition of a section? In this review it may be allowed to explain the idea of a section intuitively.

Every scientist who has investigated fiber patterns, has placed sections in measured images. E. g., he studies the scattering intensity along a straight line extending from the center of the pattern out through the maximum of a reflection. Exactly this is a section of the scattering intensity. Not rarely such section is placed in a manner that it includes

(1)

(2)

(3)

45

Figure 2. Relations between some intensity projections and corresponding structural features in physical space for the case of SAXS from fibers. (Reproduced with permission from reference [2] © 1999 ACS)

an angle φ with the fiber direction. Intuitive reasoning now may lead to the erroneous conclusion that the variation of scattering intensity along the chosen line would describe the correlations (of, e. g. domains) in the chosen direction. But on account of the above mentioned Fourier theorem, the shape of the curve only describes the projection of deliberately oriented correlations onto the chosen direction of the cut. This projection is carried out not in the reciprocal space depicted by the fiber pattern, but in the physical space of the materials structure. Projection in physical space is hard to imagine and, consequently, it can hardly be described by a structural model. Only in the special case of a structure generated from stacks of flat and extended lamellae [8,9,11-17], the intensity found in a section can easily be related to structural notions. Then the well-known "Lorentz correction" is practically applied to isotropic scattering patterns, and it performs similar transformation as does a projection to fiber axis in the case of fiber patterns: It extracts a scattering curve related to a section through a one-dimensional structure in physical space from a scattering pattern.

Objections against an analysis of intensity projections are heard from time to time. Argumentation is based on intuitive reasoning and avoids to make use of scattering the­ory. The most frequent objection tells that a projection of the intensity "doubtlessly smears the details" of information which are present in the 2D scattering pattern. Be­hind this reasoning is the feeling that information gets lost. But the same, of course, is valid when a section of the scattering pattern is analyzed. Moreover, by common under­standing materials structure is preferably described in physical space, and only intensity

46

projection guarantees that no "smearing of structural details" in physical space must be taken into account when the structural model is built.

A second objection starts from the correct finding that different fiber diagrams ("four-and two-point diagrams") may result in identical projected intensities. This objection is not well-founded, since it is based on the erroneous implication that projection analysis would claim to fully characterize oriented structures. By contrast, intensity projection analysis studies intersections of the correlation function, and if two structures result in the same projected intensity with respect to fiber axis, they indeed share the same inter­section of their correlation functions with respect to the fiber direction. Consequently, after projection it is allowed to describe a four-point pattern and a two-point pattern by the same one-dimensional mathematical model of alternating domains.

Although misorientation of lamellar stacks or fibrils will have significant effects on scattering pattern and structure, such effects do not affect the applicability of the method of projection analysis, as long as one accepts that the aim of analysis is not the deter­mination of layer thickness distributions, but the determination of distributions of chord lengths passing through layers in fiber direction only. And just this structural aspect of fiber structure, as to my belief, is the most important one to be related to materials performance. Falling back to the problem of misorientation, inclined layers will simply increase the fraction of longer chords in the resulting chord length distribution, because fiber direction cuts through such layers under an oblique angle. And if the domains are not shaped like layers or needles, the discussion of orientation is obsolete anyway. But now let us start to discuss the most important projections in SAXS fiber patterns.

Invariant It is well-known that the invariant Q contains all information on the non-topological

character of the structure. Thus, if the electron density difference Apei between hard domains and soft matrix is the predominant one, it may be possible to gain information on the volume fraction of hard domains in the sample.

Longitudinal Structure {I}1 (s3) is a one-dimensional scattering intensity which is related to the section

Μ ι (^3) of the correlation function 7 (x) in X3 direction. Thus it contains information only on those chords passing the two-phase system in the direction parallel to the fiber axis, £3· On the right hand side of the middle row in fig 2 we may represent such a chord by a deliberate vertical line. Traveling along such a line, we will alternately move through the hard and the soft domain phase, from time to time crossing a phase boundary. Thus structural parameters of physical interest are J/», the average travel distance inside the hard domain phase, ds, the average travel distance inside the soft domain phase as well as the variances (σ&, and as) of their distributions, which shall be called "domain height distributions" hh (#3) and h$ (x3).

Thus by computing the longitudinal structure we eliminate all information on trans­verse correlation of domains from the scattering pattern and reduce the problem to the case of one-dimensional scattering curves, which is well-known from the theory of lamellar two-phase systems. For the solution of the ID problem appropriate data anal­ysis methods are at hand [2, 8,10,15,16]. As has been mentioned above, analysis of

47

isotropic samples with a lamellar domain structure and the analysis of longitudinal in­tensity projections are closely related to each other. In the first case a Lorentz correction (h (s3) = 27rs| / (S3) ; S3 Ξ s) and in the second case a projection (eq 2) both result in an intensity which is related to a one-dimensional structure in physical space. Thus af­ter the extraction of the "ID intensity" analysis proceeds in the same way for both cases. While from a mathematical point of view the first method is only applicable for isotropic samples with a lamellar structure, the second is generally applicable. From the physical point of view the second method is favorably applied to well-oriented systems. Both kinds of ID intensities share the same ID Porod's law falling off with s j 2 , because they both are related to sections in physical space. Generally comparing projection analysis to an analysis of intensity sections, there is no dilemma concerning the expected fall-off of scattering intensity in the Porod region with projections even in the case of varying misorientation. Finally, the well-known dilemma with intensity sections is only a result of the unknown effect of projection in the space where the structure is.

Transverse Structure If the pattern reveals equatorial scattering, the equatorial streak may be extracted

from the pattern and analyzed separately. In certain cases (If the streak is clearly sepa­rated from other reflections and does not "fan out") it should suffice to mask the mea­sured pattern

In (* i2, S3) = I (S12, S3) Yb (53) (4)

in order to extract a "needle scattering", IN (s 1 2, S3), in particular if this pattern shall only be interpreted after projecting it. Y& (S3) is a shape function, which gives a value of 1 for | 31 < 6/2 and vanishes elsewhere. Thus b is the height of the equatorial band.

When dealing with the equatorial streak of a fiber pattern, it appears suitable to extract the transverse structure {/}2 (S12). Through Fourier transformation relation this projection is linked to a two-dimensional two-phase system made from needle cross sections in a matrix, as indicated in the bottom row of fig 2.

Evaluation of the Transverse Structure An adapted method for the evaluation of the transverse structure had to be devel­

oped!.?]. {I} 2 (s i 2) is generated by a projection process identical to the one carried out by a Kratky camera. In particular {I}2 (s i 2) exhibits Porod's law with the scatter­ing falling off with sj~2

3. Small deviations from the predicted fall off are accounted to the non-ideal structure of the real two-phase system[7,18] and corrected accordingly, leading to the 2D interference function G2 («12) of an ideal two-phase system

G2 (su) = ({J} 2 («12) ~ hi) 5?2/exp J i r V ^ ) - ^2 · (5)

Ap2, Porod's asymptote of the projected SAXS intensity, is the constant governing Porod's law. The non-ideal structure of the real two-phase system is described by IFI and wt. Fluctuations of the electron density are considered by IFU the density fluctua­tion background. wt is the width of the transition zone at the domain boundary.

48

From G 2 ($i2) the 2D chord distribution [19,20] g2 (#12) is computed by poo

92(χη) = π / (Jo(27T^i 2si 2) - J 2 (2TOI 2 SI 2 ) ) 0 2 ( § ι 2 ) dsn. (6) Jo

Here Jo and J 2 denote Bessel functions of the first kind. In general, g2 (xi2) shows the distribution of chords from needle and matrix cross sections and their correlations in the plane normal to the fiber direction.

During the analysis of the experimental data it turned out that g2 (x\2) is positive everywhere. Thus the correlations among the "disks in the plane" are negligible and 92 (^12) represents the chord distribution of an ensemble of uncorrected disks in the (xi, £ 2)-plane (fig 2). Following the principle of late modeling, now it appears rea­sonable to model the cross section of every needle by a circular disk and to ask for the properties of a needle diameter distribution, ho (D). g2 (x\2), can be expressed in terms of ho [D) and the intrinsic chord distribution gc (xi2) of a disk with unit diameter

92(*n) = fhD(D)gc(^)?§ (7)

is simply the superposition of compressed and expanded images from gc weighted by the value of the diameter distribution, ho (£>)> which shall be studied. Eq 7 is the defini­tion of the Mellin convolution [8,21]. ho (D) can be computed by numerical inversion of eq 7 utilizing an iterative van Cittert algorithm similar to the one proposed by Glat-ter[22]. Beyond that one can take advantage of special properties of the Mellin convo­lution and compute parameters characterizing the needle diameter distribution directly from the measured chord distribution utilizing moment arithmetics[3].

Results and Discussion Principal SAXS Patterns of PEE

Images of many different grades of PEE have been recorded. Quantitative evaluation of the full set of data is still in progress. Four different classes of scattering patterns can be observed with these samples, which correspond to different basic structure and can be arranged in the order of increasing elongation (fig 3). None of the materials passes through all four states. E. g. the commercial material Arnitel E2000/60 (DSM, The Netherlands) only shows the "microfibrillar state" and the "soft needle state". For this material the evaluation of the transverse structure has been carried out. The PEE 1000/57 from the laboratory of Fakirov, University of Sofia, on the other hand, starts from the "macro lattice state", transforms into a "microfibrillar structure" and slips from the clamps thereafter. Data from this sample is used to demonstrate the evaluation of the longitudinal structure.

In the uppermost image of fig 3 one observes a scattering pattern with narrow beams of intensity, inclined with respect to the fiber axis. On these beams first and second order of the long period reflection can clearly be detected. Thus the structure can appropriately be described by stacks of tilted lamellae.

49

Figure 3. Typical SAXS patterns from various PEE samples under strain. Elongation increases from top to bottom.

The second state shows indented layer line reflections ("4-point-pattern"), the most appropriate model for which is that of a macro lattice[25,24] from arranged microfibrils. Frequently one observes the shoulder of a second long period, which is not indented at the meridian (5th and 6th point), and has been attributed to slack microfibrils which do not respond to external strain any more[25] (fig 4). Both structural features are collected in a single intensity projection. This superposition may be considered a "smearing". It will be demonstrated that nevertheless the chosen structural model is able to discriminate between both components in the projected data and to return structural data of each when used for fitting. Here the question of how to avoid superposition of multiple components in the projection arises, but let us discuss this issue in the Conclusions section.

Analysis of the Longitudinal Structure

General Evaluation Steps Projections of the SAXS pattern data from sample PEE 1000/57 according to eq 2

yield the scattering curves shown in fig 5(a). From these curves interface distribution functions (IDF) (fig 5(b)) are computed by use of eqs 5 and 6.

50

Figure 4. SAXS pattern of PEE 1000/57 at an elongation c = 0.88. The two different long periods are indicated by arrows emerging from two different labels.(Reproduced with permission from reference [2J © 1999 ACS)

In the final step of data evaluation these curves are fitted using a one-dimensional model function, which unifies[/4] the two most frequently discussed models for one-dimensional statistics, namely stacking statistics[26] and "homogeneous long period distributionM[27, 28}. It turns out that whenever in the original scattering patterns a second long period is observed (fig 4), a fit is only possible if a two-component model is used. This fact is demonstrated in fig 6.

Thus in the end the longitudinal scattering of every one-component sample is de­scribed by two domain height distributions, hh (x$) and hs (x^). Altogether both do­main height distributions, in turn, are defined by a set of six parameters, namely the integral of the IDF, W\ the average domain heights of the hard and the soft domain phase, respectively, dh and ds\ the relative variances of the domain height distributions, O'h/dh and as/ds\ and finally the heterogeneity or skewing parameter, σ//. Conse­quently a fit of a two-component sample yields values for 12 parameters. Now suitable visualizations of these parameters as a function of elongation e should help to gain better understanding of the structural changes during sample straining.

Domain Heights Determined from Longitudinal Structure A plot of the average domain heights as a function of elongation e is shown in

fig 7(a). Although in principal the assignment of soft and hard domain heights is am­biguous, the interaction of the different domain height distributions with external strain gives enough information to identify each distribution uniquely. After this identification, as a first result the volume fraction of hard domains in the original sample,

(8)

51

s3 [nm"1] x3 [nm]

(a) Projections (b) Interface distributions

Figure 5. PEE 1000/57. (a) ID scattering curves (s3) obtained by projection of scattering patterns onto the fiber direction. Curves are labeled with the elongation e. (b) IDFs gi (x3) from longitudinal scattering curves (s 3). (Reproduced with permission from reference [2] © 1999 ACS)

can be computed from its average hard domain height, dh = 7.4 nm, and its average soft domain height, ds = 5.2 nm. The computed value of φ h = 0.59 is close to the hard-to-soft segment ratio of the polymer. Such agreement cannot be expected, since there are several structural features which have been discussed in literature in order to predict or to explain deviations. Nevertheless, data from an IDF analysis frequently result in such an agreement and this fact may become interesting to be studied.

At medium elongation we observe two kinds of microfibrils. The taut component (top in fig 7(a)) elastically interacts with the external strain, while the slack component (bottom in fig 7(a)) collects "garbage" from microfibrils, which are no longer connected to the surrounding elastic network and remain in the relaxed state.

At low elongations in the taut microfibrillar component soft domain heights start to grow continuously as a function of external elongation. The internal elongation of these soft domains is much higher than the externally applied elongation, which is a necessity because of the rigid nature of the hard domains filling the elastic network. Nevertheless, the average long period

L = dh + ds (9)

of the taut component increases slower than external elongation e. This finding reflects the known fact that PEE polymers are far from an ideal elastic material. Pull out of taut tie-molecules from hard domains and a collapse of the hard domains observable at e = 0.8 are characteristic for the straining process of such polymers. The quantitative data now gained from SAXS data analysis may serve to model the filled elastic network

52

1.2

- / \

\ / I comp., ε=0.00

A \ 2 comp., ε=0.88

* \ V^X***" 1 comp., Est.84

, ι 1 . 1 1 I , ι , ι . I 0 10 20 30

x3 [nm]

Figure 6. Model function fits to IDFs g\ (x$). Symbols marks represent data. Solid lines show the best fit. At medium elongation one observes two reflections in the scattering pattern and fitting requires a two-component model. (Reproduced with permission from reference [2] © 1999 ACS)

and finally to describe the material properties of both hard and soft domain phases. Since not only average domain heights have been considered in the model function,

the domain height distributions themselves can be reconstructed from the parameter values of the best fit. Examples for these height distributions are shown in fig 7(b). Here the height distributions h h (χ 3) of the hard domains in the taut microfibrillar component are presented. In the plot of the average domain heights we could hardly observe any change during the initial phase of straining. Watching the height distributions as a whole we now observe a narrowing, indicating that the domains with medium height are the most stable ones. By loss of tall and tiny hard domains to the slack component the height distribution narrows considerably. Thus a material with a narrow distribution of hard domain heights would probably be the more perfect elastomer.

Comprehensive discussion both of theoretical background and results can be found in the original paper[2].

Analysis of the Transverse Structure

General Evaluation Steps For a straining series of the material Arnitel E2000/60 the equatorial scattering has

been extracted and projected onto the transverse plane as discussed in the section "Eval­uation Methods". The resulting curves of {I}2 (su) are presented in fig 8(a). After proper consideration of the non-ideal two-phase structure of the samplefi] the 2D chord distribution g % (χ 12) is computed by means of eq 6. The corresponding curves are shown in fig 8(b). When such equatorial scattering was first observed [29], it was attributed to

53

Figure 7. PEE 1000/57. (a) Average domain heights of taut (top, left scale) and slack (bottom, right scale) microfibrillar components as determined from fits, (b) Height dis­tributions of the hard domains in the taut microfibrillar component as reconstructed from structural parameters determined in fits. (Reproduced with permission from refer­ence [2] ©1999 ACS)

the existence of elongated microvoids in the fiber. Here, with the PEE samples, the formation of these elongated domains is preceded by the destruction of hard domains and growth of a long period reflection which merges with the primary beam before the equatorial scattering emerges. This peculiarity led to the interpretation that the observed equatorial scattering most likely is caused from elongated soft domains, which are a product of the destruction of hard domains. Those elongated soft domains thus should contain almost equal parts of soft and hard segments, according to the chemical compo­sition of the poly(ester ether).

Because in good approximation the curves are positive everywhere, we are allowed to consider the structure to be built from an uncorrelated ensemble of domains, which shall be addressed as "soft needles". Thus by application of moment arithmetics we compute the average needle diameter, Â the relative width of the needle diameter dis­tribution, σ/D and the total cross section of the needles with respect to the total cross section of the fiber.

Data are presented in fig 9(a). From fig 9(a) it becomes obvious that the mean diameter of the needle shaped domains decreases almost linearly with increasing elon­gation, while for rubber elastic behavior, one would have expected a decrease according to D (e) = Do/y/e -f 1. As is shown in fig 9(b), the reason is that the disk diameter distribution alters its shape. With increasing elongation more and more thin needles are emerging, which cause the average diameter to decrease considerably. A different ex­planation for this effect can be given by an increasing raggedness of the needle cross sections circumferences. Extrapolating linearly towards e = 0, one finds a hypothetic average initial diameter D0 = 4.8 nm of the soft domain needles. σ/D (e), the relative

54

(a) Projections (b) Chord distribu­tions

Figure 8. a) Projections {I}2 (sn) of the equatorial scattering onto the plane normal to straining direction. Arnitel E2000/60 . b) Chord distributions g2 (#12) of Arnitel E2000/60 computed from the curves in (a). (Reproduced with permission from refer­ence [3]© 1999 John Wiley.)

width parameter of the disk diameter distribution, hardly increases. The total needle cross section per fiber cross section, becomes constant for elongations e > 2.5. On the other hand, in the observable region of elongations 1.7 < e < 2.5a considerable de­crease is observed. This decrease indicates a strain hardening process of the soft needles: During the straining process the soft material of the needles is compressed in transverse direction with respect to the surrounding matrix material. An assumed increase of the needle density during this process would amplify the observed effect. The measured values could be compared with measurements of Young's modulus, and stress-induced polymorphic transitions[30] could be discussed in conjunction with the presented result. More comprehensive discussion of the theoretical background and the results can be found in the original paper [3].

Conclusions It has been demonstrated, how the utilization of the concept of projections can help

to analyze two-dimensional SAXS patterns with fiber symmetry quantitatively. Because the computation of long ranging integrals is involved in this method, a prerequisite for successful analysis is the careful choice of a small beam stop, a wide vacuum tube, a rather short distance between sample and detector and a fast and linear detector. By do­ing so one can hope to register both all the reflections and also the important background scattering with required accuracy and spatial resolution.

Comparison of the determined structural parameters with the obvious features in a

55

2 " Λ 2.5 " ' 3 0.2 ο, 11 ! I I I L — L

0 2 4 6 D [nm]

1.5 8

(a) Structural parametes (b) Diameter distributions

Figure 9. (a) Characterization of the ensemble of needle-shaped soft domains in Arnitel E2000/60 as a function of elongation e. (b) Diameter distributions, h Β (D), of soft do­main needles computed by numerical Mellin deconvolution of curves shown in fig 8(b). (Reproduced with permission from reference [3] © 1999 John Wiley. )

series of scattering images shows good agreement. Therefore projection analysis ap­pears to be a suitable method, if 2D scattering patterns with fiber symmetry shall be evaluated quantitatively. Both for the identification of structural components and for the purpose of testing the stability of the multi stage evaluation method, it helps to analyze a comprehensive data set, in which an external parameter like the elongation e is slowly varied, Thus one can obtain accurate quantitative data not only on domain size aver­ages, but also on their statistics. All in all application of these methods results in a more detailed description of the domain structure from polymers in the oriented state.

If multiple structural components are obvious in the scattering patterns, one is tempt­ed to separate the contributions directly in the scattering pattern. For the general case this idea is a considerable challenge, and it remains questionable if such an attempt would be successful without the necessity to sacrifice the principle of late structure modeling. A more promising approach, I believe, will be to transform the fiber diagram as a whole into a 3D chord distribution with fiber symmetry. By this operation no structural infor­mation is lost, but the information content of the scattering image will be transformed into physical space. In fact, the two kinds of chord distributions discussed in this work are nothing else but the two principal sections through this 3D chord distribution (in meridional and equatorial direction, respectively).

This investigation has been supported by the Bilateral Cooperation Program between the Uni­versity of Hamburg and the University of Sofia, which is funded by the DAAD. SAXS investi­gations have been supported by HASYLAB Hamburg under project 1-97-06. The study of the

56

Arnitel material has kindly been suggested by Professor Kricheldorf, University of Hamburg. Sample material has been supplied by courtesy of DSM Corp., The Netherlands.

L i t era ture C i t e d

[1] Stribeck, N . Fibre Diffraction Rev. 1997; 6, 20-24. [2] Stribeck, N.; Fakirov, S.; Sapoundjieva, D. Macromolecules 1999; in print. [3] Stribeck, N . J. Polym. Sri., Part B: Polym. Phys. 1999; in print. [4] Stribeck, N . Proc. Am. Chem. Soc. PMSE Prepr. 1998; 79, 393-394. [5] Bonart, R. Kolloid Z. u. Z. Polymere 1966; 211, 14-33. [6] Stribeck, N . Web Page. www.chemie.uni-hamburg.de/tmc/stribeck/. [7] Ruland, W. J. Appl. Cryst. 1971; 4, 71-73. [8] Stribeck, N . Colloid Polym. Sci. 1993; 271, 1007-1023. [9] Méring, J.; Tchoubar-Vallat, D. C. R. Acad. Sc. Paris 1965; 261, 3096-3099.

[10] Ruland, W. Colloid Polym. Sci. 1977; 255, 417-427. [11] Vonk, C. G . Colloid Polym. Sci. 1979; 257, 1021-1032. [12] Vonk, C. G . ; Kortleve, G . Kolloid-Z. u. Z. Polymere 1967; 220, 19-24. [13] Ruland, W. Colloid Polym. Sci. 1978; 256, 932-936. [14] Stribeck, N . J. Phys. IV 1993; 3, 507-510. [15] Verma, R. K.; Velikov, V.; Kander, R. G. ; Marand, H . ; Chu, Β.; Hsiao, Β. S. Polymer 1996;

37, 5357-5365. [16] Verma, R.; Marand, H . ; Hsiao, B. Macromolecules 1996; 29, 7767-7775. [17] Baltà Calleja, F. J.; Vonk, C . G . X-Ray Scattering of Synthetic Polymers. Elsevier, Amster­

dam, 1989. [18] Stribeck, N.; Reimers, C.; Ghioca, P.; Buzdugan, E . J. Polym. Sci., Part B: Polym. Phys.

1998; 36, 1423-1432. [19] Tchoubar,D.; Méring, J. J. Appl. Cryst. 1969; 2, 128-138. [20] Schmidt, P. W. J. Math. Phys. 1967; 8, 475-477.

[21] Marichev, Ο. I. Handbook of Integral Transforms of Higher Transcendental Functions. Ellis Horwood Ltd., Chichester, 1983.

[22] Glatter, O. J. Appl. Cryst. 1974; 7, 147-153. [23] Fronk, W.; Wilke, W. Colloid Polym. Sci. 1983; 261, 1010-1021. [24] Fronk, W.; Wilke, W. Colloid Polym. Sci. 1985; 263, 97-108. [25] Stribeck, N.; Sapoundjieva, D.; Denchev, Z. ; Apostolov, Α. Α.; Zachmann, H . G. ; Stamm,

M . ; Fakirov, S. Macromolecules 1997; 30, 1329-1339. [26] Hermans, J. J. Rec. Trav. Chim. Pays-Bas 1944; 63, 211-218. [27] Stribeck, N.; Ruland, W. J. Appl. Cryst. 1978; 11, 535-539. [28] Strobl, G . R.; Müller, Ν . J. Polym. Sci., Part B: Polym. Phys. 1973; 11, 1219- 1233. [29] Statton, W. O. J. Polym. Sci. 1962; 58, 205-220.

[30] Apostolov, Α . Α.; Boneva, D.; Baltà Calleja, F. J.; Krumova, M.;Fakirov, S. J. Macromol. Sci. - Phys. 1998; 37, 543-555.

Chapter 4

Studying Polymer Interfaces Using Neutron Reflection

D. G. Bucknall1, S. A. Butler2, and J. S. Higgins2

1ISIS Facility, Rutherford Appleton Laboratory, Chilton, Oxon OX11 OQX, United Kingdom

2Department of Chemical Engineering, Imperial College, London SW7 2BY, United Kingdom

Neutron reflection has been used to study the interfaces between the melt phases of crystallisable polymers as well as real time interdiffusion of polymers and oligomers. Both systems are experimentally demanding and have required the use of specialised cells and data collection procedures. The interfacial widths for a number of polymer systems have been determined and the Flory Huggins interaction parameters obtained. In addition, the interdiffusion process has been followed for a polystyrene-polystyrene system above its T g and also for a polystyrene-oligostyrene in-situ in real time using very rapid reflectivity scans.

The interfacial behaviour between the component polymers in blends to a large extent controls the bulk polymer blend characteristics. The understanding of these interfaces is therefore of vital importance. The neutron reflection (NR) technique is ideally suited to study polymer interfaces since it provides a composition profile perpendicular to the interface with a resolution on a sub-nanometer length scale. From self consistent mean-field theory for immiscible homopolymers of infinite molecular weight the interfacial width (w) is is related to the Flory Huggins interaction parameter (χ) by w=(2a)/^[βχ {a is the segment length) [1]. Therefore, simply by measuring the interfacial width it should be possible to extract directly the Flory-Huggins χ parameter. In reality the situation is not so simple because thermally

© 2000 American Chemical Society 57

58

excited capillary waves broaden the interface so that the observed width as determined by NR contains a contribution from two superimposed values, which need to be separated [2]. The broadening effect of these capillary waves is however well understood so that their effect can be calculated and a true estimate of the χ parameter obtained. The interfacial width between crystalline polymers has been measured using NR since many of the most important industrial commodity polymers are semi-crystalline in nature. The surface of crystalline polymers are by their very nature macroscopically rough at room temperature and this has forced measurements to be made in-situ at temperatures above the melt temperatures.

For miscible systems the interface between initially separated materials will develop by diffusion mechanisms. The distance over which this diffusion occurs can often mean that several techniques are required to follow the whole process. At longer annealing times the dominant mechanism is often Fickian diffusion, and this has been investigated by techniques able to observe long length scales such as dynamic secondary ion mass spectroscopy (DSIMS) [3-5], nuclear reaction analysis [6-8] or Rutherford back scattering [9,10], These techniques typically have a resolution of approximately 10 nm at best and are therefore not normally suitable for studying the initial stages of the diffusion process where high resolution techniques are required. NR has been widely utilised for the detailed study of interfacial diffusion and a number of studies have tested the early time regime predictions of the reptation theory [11,12]. Most of the NR studies reported in the literature have been performed using high T g polymers where the samples are annealed and then quenched to room temperature so as to freeze in the structure and enable a full analysis to be performed. However this method does have limitations for studies involving small molecules where diffusion may occur at room temperature and the interface is no longer static, thus preventing this classical approach to NR measurements. This limitation has to a large extent been overcome by making reflectivity in-situ and in real time [13,14]

Polymer Interfaces in the Melt

Neutron reflection (NR) has been successfully applied to the study of interfaces formed between high molar mass amorphous polymers. Such experiments are normally conducted at room temperature (RT) where the interfacial structure of the samples, after annealing, is frozen in (see for example [15] or [16]). There is considerable interest in the interfacial behaviour of crystalline polymers but, until recently, it has not been possible to study such materials by NR. The difficulty lies in the sample required for reflectivity. Neutron reflectivity samples must be microscopically as well macroscopically flat since at a glancing incident angle a large area is sampled and averaged over by the neutron beam. Whereas thin films of amorphous polymers are naturally flat once spin coated from solution onto optically polished substrates, the formation of crystallites causes such thin films of semi-

59

crystalline polymers to become macroscopically rough (see Figure 1). At RT this surface roughness causes drastic loss of specular reflection, making NR measurements at best very difficult but more likely impossible. The problem is not relieved by simply heating the films above the crystalline melt temperature [17]. Although this has the effect of removing molecular roughness associated with the polymer crystallinity, the surface still suffers from residual long range waviness, which adversely affects the reflectivity profile.

Figure 1; Surface of for a thin layer of d-iPP measured using an α-step profiling instrument at RT, showing the macroscopic roughness of a semi-crystalline polymer film (reproduced with permission from reference 18.

One solution to overcome these problems is to observe the polymers in the molten state when sandwiched between a polished silicon substrate and a heated trough which contains a bulk layer of one of the polymers [13,19-21]. This produces both molecularly smooth and macroscopically flat samples allowing accurate determination of the interfacial width and profile. A simple heating cell, shown in Figure 2, has been developed which allows such experiments to be carried out. This cell consists of a brass trough into which a plug of hydrogenous polymer is moulded.

60

A silicon wafer, coated with a layer of deuterated polymer, is clamped face down on to the surface of the polymer plug. The cell is heated by means of thermostated cartridge heaters in the brass base and has been designed such that the neutron beam passes through the silicon wafer, with approximately 90 per cent transmission. Thin films of the deuterated semi-crystalline polymers are prepared by spin casting hot solutions of the polymer directly onto the polished silicon substrate.

Figure 2: Schematic illustration showing a section through the polymer heating cell used for studying crystalline polymer interfaces (reproduced with permission from reference [19]).

The assembled cell is heated to the desired temperature, and aligned before data collection occurs. Although there may be thermal loss from the top surfaces of the cell, it has been found that the polymer within the cell attains and holds the required experimental temperature within approximately 10 minutes of heating. To prevent thermal degradation occurring during the experiment, the whole cell is housed inside an inert atmosphere box with neutron transparent quartz windows front and back. Alignment of the cell is achieved using the neutron beam, so there is always a delay of at least 30 minutes from the point at which the heating is started to when reflectivity data collection begins.

Al l the data reported here have been collected using either the CRISP or SURF reflectometers on the ISIS pulsed neutron source at the Rutherford Appleton Laboratory. These instruments view a liquid hydrogen moderator providing neutrons in a white beam with a wavelength of 0.5-6.5 A. To achieve a large Q range (Q=(4πsin0) /λ ) three different angles are measured for each sample, with the pre-sample collimation slits varied so that at each angle a constant resolution and

61

illuminated area is maintained. Cell calibration, required for the data reduction normalisation factors, is obtained by taking transmission measurements of the Si block. A combination of both non-linear least squares fitting routines and maximum entropy methods have been applied in the reflectivity data analysis [16,22].

Reflectivity measurements using the cell were first performed on a sample of amorphous deuterated polystyrene (dPS) in contact with low density polyethylene (LDPE) at 150C [19]. The characteristics of the polymers are given in Table I. This system had been chosen since it is one of only a few that contain semi-crystalline polymers which have been reported in the literature.

Table I. The characteristics of polymer used in melt interface studies.

Polymer Mw MJMn

deuterated polystyrene dPS 190k 1.08

polyethylene - low density LDPE 44k 5.6 - high density HDPEi 156k 8.2

HDPE 2 58k 6.8 - linear low density LLDPE 150k 1.02

deuterated polypropylene - isotactic d-iPP 274k 5.5 - atactic d-aPP 111k 1.5

The interfacial width obtained from fitting the dPS-LDPE NR profiles is given in Table II. This measured interfacial width is described by a self-consistent mean-field theory, which is broadened by thermally excited capillary waves [2]. The measured interfacial width is given by Gaussian quadrature addition of the intrinsic (unbroadened by capillary waves) interfacial width, Δ0, and the interfacial width associated with the capillary waves, A c. This gives:-

where

and

Δ 2

Π + Δ 2

0 c Δ 0

ΔΙ •Μ {2π/Α0)2

{ΐπ /Ο* +{ΐπ Α , ) * )

(1)

(2)

(3)

62

The capillary wave broadening is therefore dependent on the in-plane coherence length of the neutron, λ£ « 20μηι, the dispersive capillary length, ad = 7cyQct A1,

the Hamaker constant, A, and the interfacial tension γΌ = avkgT^j χ/β (α is the segment length and v*1 is the monomer volume), and d is the layer thickness. This approach has been applied to evaluate the effects of capillary wave broadening on the dPS-LDPE system [19]. The values of χ and γ0 obtained (see Table II) have been shown to agree well with the spread of data found in the literature and give considerable confidence in the use of the cell for applications of this method to obtain accurate values of χ and γ0 for other polymer pairs.

Table II. NR measured interfacial width, w, Flory Huggins parameter, χ, and interfacial tension, γ0.

Polymer pair Τ (°C)

w (nm) X y0 (mJm2)

dPS-LDPE 150 3.0 ± 0.9 0.14±0.09 9.0 ±2.0

diPP-HDPEi 175 6.0 ± 0.6 (1.6±0.1) x 10 2 2.68 ± 0.08 200 7.0 ± 0.7 (1.0±0.2)x ΙΟ"2 2.28 ± 0.03 225 5.7 ±0.1 (1.9±0.1)x 10"2 3.23 ±0.05

daPP (4.4±0.1)x 10"3 - LLDPEi 175 9.9 ±0.1 (4.4±0.1)x 10"3 1,42 ±0.02

- HDPE 2 9.1 ±0.2 (5.1±0.2)x 10"3 1.52 ±0.03

A majority of the data collected using this cell has been with a polypropylene (PP) - polyethylene (PE) system, using a combination of two PP tacticities and a number of PE's of varying densities from linear low density PE (LLDPE) to high density PE (HDPE) [18,20,21]. Deuterated isotactic PP (diPP) is semi crystalline with a melt temperature of 157°C, while deuterated atactic PP (daPP) is amorphous. The melt temperatures of the LLDPE and HDPE used are 102 and 131 °C, respectively. Reflectivity profiles from all the systems were obtained at temperatures well above the melt and glass transition temperatures of the polymers. Data from the diPP-HDPEj system have been collected at 175, 200 and 225 °C and the interfaciai widths obtained. The fitting revealed that in some cases the diPP does not fully wet the silicon [21], but the diPP layer was sufficiently thick to prevent complete dewetting

63

from the substrate, and the film homogeneity and interface with the HDPE1 were believed to be unaffected by this process.

By applying Equations 1-3 to the results of the fitted reflectivity data, the values of the measured interfacial width together with the evaluated values of the χ parameter and γ0 given in Table II were obtained. The interfacial widths measured for daPP-HDPE2 and daPP-LLDPE are seen to be significantly larger than that for diPP-HDPE1, indicating that daPP-PE polymers are more miscible as indicated by the reduction in the calculated χ value. The reason for this increased miscibility is unclear, and is currently under investigation. The similarity in miscibility between daPP with HDPE or LLDPE is expected since the LLDPE is only lightly branched (18 branches per 1000 backbone carbon atoms compared to 3 for the HDPE) and therefore similar to the extent of branching observed in HDPE. Based on the limited number of points measured there does not appear to be any systematic temperature dependence of the interfacial parameters for diPP-HDPEi in the range 175 ≤ Τ (°C) ≤ 225, as may have been expected.

Diffusion Studies

Until recently the use of NR to study miscible polymers has been limited to the investigation of diffusion processes that take place well above ambient temperature. Using the reflectometers at ISIS a full reflectivity profile covering a momentum transfer, Q, from approximately 0.007 - 0.6 A-1 requires measurements at 3 incident angles. This typically takes between one to two hours depending on the sample. Any change in the interface or sample composition profile during this measurement time can corrupt the data. Therefore, to measure diffusion processes has conventionally required the polymer structure to be frozen in. For amorphous polymers, where the glass transition temperature (Tg) lies well above ambient, this is easily achieved by annealing for a given time above T g and then rapidly quenching the sample to RT, before a full reflection profile is collected. This procedure is repeated successively at various annealing times to build up the time dependence of the diffusion process. However, for systems where the interface remains mobile at RT this approach is not adequate and either a quench to below RT must be made to immobilise the system, or in-situ real time reflectivity measurements must be performed.

By utilising the high flux available on the SURF reflectometer, a methodology for conducting neutron reflection experiments in real time has been developed in order to study the initial stages of the interdiffusion of polymers. This is achieved by taking reflectivity measurements at the annealing temperature, but restricting the data collection to a limited Q range from one angle [14,18]. Using a white beam of neutron wavelengths, a partial reflectivity profile for the specified incident angle is obtained. Data collection is therefore restricted at the lower limit to the time required to reduce the error in the data points to be adequate enough to enable the features to be observed and be fitted. Rapid data collection is therefore possible and the time taken to collect such partial reflectivity profiles, with adequate statistics to extract interfacial widths,

64

has been reduced to six minutes. By decreasing the incident angle during the experiment the restricted measurement window is progressively moved to low Q ranges, enabling collection of data from different portions of the full reflection profile. This allows only the most significant regions of the reflectivity profile to be measured.

The viability of this technique has been demonstrated by investigating the interdiffusion of high molar mass deuterated and hydrogenous polystyrene (hPS and dPSi). Subsequent work has extended this methodology to study the ingress of oligomeric polystyrene (OSt) into deuterated high molar mass polystyrene (dPS2). The characteristics of the polymers used are given in Table III.

Table III. The characteristics of the polymers used in real time neutron reflection studies.

Polymer M w MJMn Tg/x:

deuterated polystyrene dPS! 40k 1.02 84 dPS2 101k 1.02 103

oligomeric styrene OSt 1.11k 1.1 27 hydrogenated polystyrene hPS 49k 1.03 104

Polymer-Polymer Interdiffusion

The interdiffùsion of polystyrene hPS and dPSi has been studied at 115°C. Since these polymers are immobile at RT it is possible to prepare a bilayer, and measure a full three angle reflectivity profile of the sample before any interdiffùsion occurs. The sample is then placed onto a preheated sample stage and partial reflectivity profiles are measured every six minutes using fixed angle restricted Q window measurements. By incrementally decreasing the angle of incidence it is possible to monitor the decrease of the interference fringe amplitude, which is associated with interdiffusional processes, within the Q window. After annealing for almost four hours, the sample was removed from the heating stage and quenched to RT before finally collecting a full three angle NR profile. A limited set of all the reflectivity profiles are shown in Figure 3.

65

Figure 3: A selection of reflectivity profiles for the hPS-dPSi bilayer measured in real time (solid symbols) at 115X1 and at RT (open squares) before and after annealing (a and j respectively). The real time reflectivity profiles were measured at a fixed angle with 6 minute count times. The mean time for the profiles plotted are 5 (b), 32 (c), 63 (d), 93 (e), 124 (f), 155 (g), 186 (h) and 212 (i) minutes. The solid lines are fits to data assuming a model as described in the text (reproduced from reference [14]).

66

Due to the restricted Q range in the data from the real time measurements, interpretation of the NR data is limited to specific models and functional form fits based on the fits to the 'as made' sample. The interfacial widths were determined from the reflectivity data by fitting the profiles with a standard two layer model and assuming a simple Gaussian interfacial profile between the dPS1 and the hPS. The interfacial profile is seen to be symmetric as may be expected for a system where the polymers are of approximately equal molecular weight.

Figure 4: A log-log plot of time dependence of interfacial width of hPS-dPS1 (reproducedfrom reference [14]).

The width of the polymer interface as a function of annealing time is shown in Figure 4 and the magnitude of the interfacial widths especially for the longer annealing times have been confirmed using DSIMS. For times approximately t > 6000s the interfacial width, w, increases with a t dependence on as indicated by the solid line in Figure 4. Assuming that the diffusion coefficient, D, is given by the relationship w= faDt) [11], a value of the diffusion coefficient for this system of D = (1.7±0.2) χ 10-17 cmV1 is obtained from the gradient of the linear region of a plot of w versus tA [14]. This compares favourably with published literature values for PS-

67

dPS interdiffusion coefficients, although is perhaps small given the comparatively moderate molecular weights used here. Given this value of D, the reptation time for these polymers can be calculated using the formula τr = Nb Π 1]> where b is

the segment length (= 6.7 A), and Ν is the degree of polymerisation. This gives xr(dPSi) = 3223+363 s and Tr(hPS) = 4333±489 s. The time behaviour of the interfacial width for t > τΓ shows the expected t/2 dependence predicted by Fickian diffusion. The expected tA dependence that has been predicted for times in the regime xr<t<xr (where xR is the Rouse time [23]) is observed and shown by the solid line in

figure for t < 7000s. The calculated xR for these polymers ( τR = dr/d,9 where dT

is the polymer tube diameter ( dT= 57 A [11])) is 215+23 s. The expected change in time dependence at the Rouse time therefore will not be observed in these experiments due to the time resolution of the reflectivity measurements.

Small Molecule Diffusion

The study of small molecule diffusion into polymers using NR requires a slightly different technical approach compared to polymer-polymer systems (described above), since such diffusion processes can take place at RT. To prevent diffusion occurring before data collection can be instigated the polymer and small molecule penetrant have to remain apart. This has been achieved by designing a cell in which an inverted silicon wafer, coated with the deuterated polymer, is held above but separate from a container filled with the penetrant (see Figure 5). The cell can be heated to the experimental temperature and aligned, before remotely raising the container into contact with the silicon and immediately beginning data collection [14,18].

Using such a cell the interdiffusion of oligomeric styrene (OSt) into high molar mass deuterated polystyrene (dPS2) at 65°C has been studied. The polymer characteristics are given in Table III. As before, fixed angle partial reflectivity profiles have been collected every 6 minutes. As for the real time hPS-dPSi measurements, the incident angle was incrementally reduced with time from 0.8-0.5°, in order to progressively move the window in Q to lower ranges. Like the hPS-dPSi NR data, interpretation has been carried out using constrained models and functional form fits because of the restricted Q range. However, the interfacial profile between OSt and dPS2 is more complex than the simple polymer-polymer case and could only be modelled using a highly asymmetric interfacial profile, which is qualitatively similar to predictions from Case II diffusion theory [23-26]. The asymmetry in the interfacial profile is modelled by two discontinuous error functions so that the neutron scattering length density is given by:

68

+p2 (4)

where erfc(x)-1 - erf(x) and all other variables are defined graphically in Figure 6. This functional form has been confirmed by modelling the data using a bilayer with an interface made up of a number of thin layers. This discontinuous error-functional form is applicable for the early and intermediate annealing times, but begins to break down for longer times (t > 140 minutes) where such a simple model of the diffusion process cannot fully describe the data.

// x< xQ then p(x) = |^ - ^

//ΛxQ then p(x) = [p^-f κ -

Figure 5: Schematic illustration of the heatable cell used to study small molecule ingress into high molecular weight polymers (reproduced from reference [18]).

69

Figure 6: Schematic illustration of the discontinuous error functional form used to model the 0St-dPS2 data (reproducedfrom reference [18]).

These asymmetric interfaciai profiles show that the interfacial width on the dPS2

side of the interface is much larger and more extended than that on the OSt layer side, creating a profile with a diffuse tail and an otherwise sharp interface. This confirms that the oligomer diffusion into the polymer is much faster than the polymer diffusion in the opposite direction as may be expected from Case II diffusion theory. [27,28] The time dependent behaviour of the error function widths used to describe the scattering length density profile are shown in Figure 7. The width, w/, on the dPS2

side of the interface, which describes the OSt diffusion into the dPS2, appears to form instantaneously in the time resolution of the reflectivity measurements and then gradually increases with time. The error function width, w2, which describes the dPS2

diffusion into the oligomer, is initially much smaller, but also shows an increase with time. The position of the interface is seen to move towards the polymer suggesting that the system has fully swollen before the first measurement, and what is observed in the time scale of these measurements is the dissolution of the polymer. The change in interfacial position x0(t = 0)-x0(t) shows a t dependence as shown in Figure 8.

The total width of the interface at longer times for such asymmetric systems is expected to show a Φ dependence [23], that is related to the small molecule diffusion coefficient by w=Dst. In this case the oligomer is the smallest molecule, so taking w= w1 + w2, gives a value of Ds for the OSt in this system at 65°C of 8 (± 1) χ 10-17

cmV1 [14]. This time dependence relationship holds for the case where time is larger than the small molecule reptation time, Tr(S), and the polymer chains behave like a transient network swollen by the oligomer chains. This value of the diffusion coefficient would give xT(S) equal to 20 seconds. Under these circumstances it is not expected to see the change to different time dependent behaviour at t < xr(S) as predicted by theory [23].

70

Figure 7: The variation of the interfacial width as a function of annealing time, t, obtained from the discontinuous error functional form of the interfacial profde model used to fit a selection of the reflectivity data obtained for the OSt-dPS2 system (reproducedfrom reference [14]).

71

Figure 8: Plot of shift in interface, x0, as function of square root of time, showing t/2

dependence indicated by the solid line through the data (reproduced from reference [18]).

Conclusions

NR has been demonstrated to be widely applicable technique for detailed studies of polymer interfaces. The requirement for flat and smooth samples has often been thought of as a limitation to the applicability of the technique. This chapter has described some of the work which has been carried out on systems which up to now have not been studied using the NR technique due to problems associated with obtaining adequate samples.

The interfacial widths from semi-crystalline polymers have been determined allowing Flory-Huggins χ parameters and interfacial tension values to be extracted by measuring these samples at elevated temperatures well above the crystalline melt temperature. Further to these studies of immiscible systems the kinetics of the early stages of diffusion process has been observed in-situ in real time on miscible systems. Rapid data collection procedures with reflectivity curves obtained every six minutes

72

have been demonstrated on the test system of polystyrene-polystyrene and a square root time dependence on the interfacial width observed. This has allowed a diffusion coefficient to be extracted which is in close agreement with published values for this system. Oligomer-polymer diffusion using this in-situ real time technique has also been observed and the interfacial profile measured as function of time. The asymmetric interfacial profile is in qualitative agreement with Case II diffusion theories where swelling and dissolution of the polymer by the small molecule penetrant occurs.

With current neutron sources the times of data collection for each partial reflectivity scan will remain of the order of minutes. However, with the advent of the next generation of high intensity neutron sources being planned or built around the world significant advances can be predicted for these real time measurements. 'Time will telP as the saying goes.

References

1. Helfand, E.; Sapse, A. M . J. Chem. Phys. 1975, 62(4), 1327-1331. 2. Sferrazza, M.; Xiao, C.; Jones, R. A. L.; Bucknall, D. G.; Webster, J.; Penfold,

J. Phys. Rev. Lett. 1997, 78(19), 3693-3696. 3. Whitlow, S. J.; Wool, R. P. Macromolecules 1989, 22(6), 2648-2652. 4. Whitlow, S. J.; Wool, R. P. Macromolecules 1991, 24(22), 5926-5938. 5. Agrawal, G.; Wool, R. P.; Dozier, W. D.; Felcher, G. P.; Zhou, J.; Pispas, S.;

Mays, J. W.; Russell, T. P. J. Polym. Sci, Part B: Polym. Phys. 1996, 34(17), 2919-2940.

6. Payne, R. S.; Clough, A. S.; Murphy, P.; Mills, P. J. Nucl. Instrum. Methods Phys. Res., Sect. Β 1989, 42(1), 130-134.

7. Reiter, G.; Steiner, U. Journal De Physique II 1991, 1(6), 659-671. 8. Shearmur, T. E.; Clough, A. S.; Drew, D. W.; Vandergrinten, M . G. D.; Jones,

R. A. L. Physical Review Ε 1997, 55(4), R3840-R3843. 9. Composto, R. J.; Kramer, Ε. J. J. Mater. Sci. 1991, 26(10), 2815-2822. 10. Green, P. F.; Palmstrom, C. J.; Mayer, J. W.; Kramer, E. J. Macromolecules

1985, 18(3), 501-507. 11. Karim, Α.; Mansour, Α.; Felcher, G. P.; Russell, T. P. Phys. Rev. B: Condens.

Matter 1990, 42(10), 6846-6849. 12. Kunz, K.; Stamm, M . Macromolecules 1996, 29(7), 2548-2554. 13. Bucknall, D. G.; Butler, S. Α.; Hermes, H. E.; Higgins, J. S. Physica Β 1998,

241, 1071-1073. 14. Bucknall, D. G.; Butler, S. Α.; Higgins, J. S. Macromolecules submitted. 15. Russell, T. P. Material Science Reports 1990, 5, 171. 16. Bucknall, D. G.; Higgins, J. S. Polymers and Surfaces - A Versatile

Combination; Hommell, H., Ed.; Recent Research Developments in Polymer Science; Research Signpost: Trivandrum, India, In press.

17. Higgins, J. S.; Oiarzabel, L.; Fernandez, M . L. ISIS Annual Report 1993, II, A156.

18. Bucknall, D. G.; Butler, S. Α.; Higgins, J. S. J. Phys. Chem. Solids submitted.

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19. Hermes, H. E.; Higgins, J. S.; Bucknall, D. G. Polymer 1997, 38(4), 985-989. 20. Butler, S.; Hermes, H.; Bucknall, D.; Higgins, J. S. Abstracts Of Papers Of The

American Chemical Society 1997, 213(Pt2), 476-POLY. 21. Hermes, H. E.; Bucknall, D. G.; Butler, S. Α.; Higgins, J. S. Macromolecular

Symposia 1997, 126, 331-342. 22. Sivia, D. S. Data Analysis - A Bayesian Tutorial, Clarendon Press: Oxford,

1996. 23. Brochard-Wyart, F.; de Germes, P. G. Makromol. Chem., Macromol. Symp.

1990, 40(Dec), 167-177. 24. Hui, C. Y.; Wu, K. C.; Lasky, R. C.; Kramer, E. J. J. Appl. Phys. 1987, 61(11),

5129-5136. 25. Hui, C. Y.; Wu, K. C.; Lasky, R. C.; Kramer, E. J. J. Appl. Phys. 1987, 61( 11),

5137-5149. 26. Rossi, G.; Pincus, P. Α.; de Gennes, P. G. Europhys. Lett. 1995, 32(5), 391-

396. 27. Fernandez, M. L.; Higgins, J. S.; Penfold, J.; Shackleton, C. J. Chem. Soc.,

Faraday Trans. 1991, 87(13), 2055-2061. 28. Bucknall, D. G.; Fernandez, M . L.; Higgins, J. S. Faraday Discussions 1994,

98, 19-30.

Chapter 5

Neutron Diffraction by Crystalline Polymers Yasuhiro Takahashi

Department of Macromolecular Science, Graduate School of Science, Osaka University, Toyonaka, Osaka 560, Japan

Neutron diffraction has the several advantages in comparison with X-ray diffraction. Accordingly, new informations about the crystal structure could be obtained differing from the X-ray work. In the present study, two crystalline polymers, poly(vinyl alcohol) and polyethylene-d4 were studied by neutron diffraction. Different crystal structure models have been proposed for atactic poly(vinyl alcohol) by Bunn and Sakurada et al. The models differ principally in the azimuthal angle of the planar zigzag backbone and the hydrogen bonding network. In the present study, the reexamination of the crystal structure analysis was carried out by using both the X-ray and neutron diffraction methods. The crystal structure model proposed by Bunn is found to be correct. The (F 0 - F c ) synthesis was made for the neutron data (100K), in which the hydrogen atoms to be associated with the hydrogen bonds are not incorporated into F c calculation. On the map, three peaks were found, which may be attributed to the hydrogen atoms to be associated with the intramolecular hydrogen bonds between OH groups in an isotactic sequence and the two kinds of intermolecular hydrogen bonds. Neutron structure analysis of polyethylene-d4 was carried out for the data (equator) measured at 10K, 100K, 200K, and 300K by using the rigid body least-squares method, where the translational and librational displacements of the molecular chain were estimated by using the rigid body temperature factor reported by Pawley. The temperature

© 2000 American Chemical Society

75

dependences of the lattice parameters, a and b, were also estimated by the least-squares method. The φ value, the azimuthal angle of the molecular plane with respect to the b-axis, was estimated as 45° within the accuracy of the standard deviation 1° independent of temperature. From the translational and librational displacements at 10K, the static disorder of polyethylene was concluded to be mainly of the translational displacements.

INTRODUCTION

Neutron diffraction has the several advantages in comparison with X-ray diffraction. (i> 2) Neutron is diffracted by atomic nucleus, while X-ray is diffracted by electron. Accordingly, the scattering length of an atom by neutron is independent of the atomic number. Hydrogen and deuterium have the same atomic number, i . e., the same chemical properties, although they have somewhat different physical properties,(3, 4) and have the same scattering length for X-ray diffraction, but they have the different scattering lengths for neutron (Table I).

Table I. Scattering lengths of atoms by Neutron and X-ray.

Neutron X-raya

Coherent Incoherent Η -3.74 25.22 1 D 6.67 4.03 1 C 6.65 0 6 Ν 9.37 1.98 7 Ο 5.80 0 8

aThe values at θ = 0°.

The scattering length by neutron is further independent of the scattering angle 0. The intensities of the reflections with large θ values can be observed strongly and can be measured accurately. This is especially the advantage for crystalline polymers, in which the intensities become weak with the Bragg angle θ because of the disorder contained in the crystalline region and low degree of orientation. Furthermore, absorption of neutron by most elements, for example, A l , is very small. Therefore, the apparatus for low- and high-temperature measurements can be designed easily and the measurements at low and high temperature are easy. Energy of neutron is small because of de Broglie wave. Therefore, it interacts with phonon, i . e., molecular

76

vibration (inelastic scattering) and also interacts with molecular motion (quasielastic scattering).

Examples of the crystal structure analyses of crystalline polymers by neutron diffraction are limited so far.(5,6) This may be attributed to the incoherent scattering by hydrogen atom and to the weak scattering power of crystalline polymer. Incoherent scattering length by hydrogen atom is very large in comparison with the coherent scattering length (Table I). Therefore, it has been considered that the deuterated derivatives of polymers needed for neutron diffraction measurements. Crystalline polymers consist of both crystalline and amorphous regions, and the crystalline region contains a considerable amount of disorder. Diffraction intensity by crystalline polymer are, generally said, so weak and the power of neutron source is not so strong. Therefore, it has been considered to be difficult to measure the sufficient number of reflections accurately. In the present paper, neutron structure analyses of atactic polyvinyl alcohol) and polyethylene^ are successfully carried out. Especially, the former is the protonated polymer and includes statistical disorder. This suggests that neutron diffraction can be applied to a wider range of crystalline polymers.

NEUTRON STRUCTURE ANALYSIS OF POLY (VINYL ALCOHOL)( ?)

In 1948, C. W. Bunn(8) briefly reported the crystal structure of atactic polyvinyl

Figure 1. Crystal structure models of atactic polyvinyl alcohol) proposed by (a) Bunni8) and (b) Sakurada et al (9) Broken lines show hydrogen bonds. (Reproduced with permission from reference 7. Copyright 1997, Wiley-Inter Science)

7 7

Figure 2. Neutron intensity distributions on the equator of poly (vinyl alcohol) (a)before and (b) after smoothing, which are measured at 100K. Al denotes the reflections due to aluminum foil. (Reproduced with permission from reference 7. Copyright 1997, Wiley-InterScience)

78

alcohol) in which OH groups are statistically located on both sides of the molecular plane. Thereafter, Sakurada et al.(9 - π ) briefly reported the crystal structure model, which is different in the azimuthal angle of the molecular plane and hydrogen bonding network from Bunn's model. Figure 1 shows the two crystal structure models. The detailed X-ray structure analysis (Fourier method) of atactic polyvinyl alcohol) was made and briefly reported by Nitta et al., (12) where they supported Bunn's model. Neutron diffraction has an advantage that the hydrogen atoms contribute to the diffraction intensity more than X-ray diffraction. Therefore, it may be possible to determine the azimuthal angle more accurately than X-ray diffraction and to clarify the position of the hydrogen atoms to be associated with hydrogen bonds. In the present analysis, the structure analysis was carried out by using both the X-ray data by Nitta et al.(i2) and the neutron diffraction data newly collected.

Experimental

The commercially supplied atactic polyvinyl alcohol) fiber (Unitika Co., Ltd.) was used as the sample. The specimen for X-ray diffraction measurements was made by arranging the fibers in a cylindrical bundle about 0.5 mm in diameter. X-ray measurements were made by CuKa radiation monochromatized by pyrolite graphite. Neutron diffraction experiments were carried out by high resolution powder diffractometer (HRPD) equipped on JRR-3M installed on Japan Atomic Energy Research Institute (JAERI) using λ = 1.8232 A. The specimen was made by arranging the fibers in a cylindrical bundle about 10 mm in diameter, covered by aluminum foil, and set into the aluminum sample tube with 10 cm diameter. Intensity distribution on the equator was measured every 0.05° from 2Θ = 5° to 165° at 100K, 200K, and room temperature. Due to the limited long range order of the sample, 1(29) data were recorded over the internal 5°<20<100°. Furthermore, on the original distribution, the background intensity was so high and the signal-to-noise ratio was not so good that the distributions were smoothed by averaging the intensities of neighboring seven points, which corresponds to the slit width of 0.35°. The neutron intensity distributions on the equator before and after smoothing, which were measured at 100K, are shown in Figure 2. High background and peaks not to be indexed should be attributed to the incoherent and inelastic contributions. Integral intensities were estimated after the indexing and served to the structure analyses after the correction by Lorentz factor. Temperature dependences of the unit cell parameters a, c, and β, estimated from the peak positions of 200, 002, and 202 reflections of neutron data, are shown in Figure 3.

79

ο 93 -

92 -7.9;

ο

7.7

0 <

ο

5.5

5.4

Ο ,300 Temp./K

Figure 3. Temperature dependence of the unit cell parameters of atactic poly (vinyl alcohol). (Reproduced with permission from reference 7. Copyright 1997, Wiley-InterScience)

Rigid-Body Least-Squares Refinement

The rigid-body least-squares method was first reported by Scheringer,(13) in which well-defined groups of atoms within the structure are treated as rigid bodies. Atactic polyvinyl alcohol) assumes the planar zigzag conformation. On the projection through the fiber axis, the molecule can be treated as a rigid body and assumes the structure shown in Figure 4. Here, O l , 02, H3, and H4 atoms statistically exist on both sides of the molecular plane formed by CI and C2 atoms and possess the probability 0.5. The hydrogen atom in the OH group was ignored during the refinements because the position cannot be specified. The contribution of the hydrogen atom in the OH group to the structure factor is very small, 1/24 for X-ray and 3.739/34.051 for neutron (the ratio of the atomic scattering factor of the hydrogen atom to the sum of the atomic scattering factors of the atoms) and therefore it can be ignored in the analysis. The constrained least-squares method^4) was used for the rigid-body

80

least squares refinements by fixing the bond lengths and bond angles, as shown in Figure 4. Here, the H1 atom was chosen as the origin atom. Accordingly, the variable coordinates are the x and z parameters of the H1 atom and Eulerian angle θ around the fiber axis.

H 1

Figure 4. The molecular framework of atactic polyvinyl alcohol) on the b-projection used in the rigid-body least -squares refinements. (Reproduced with permission from reference 7. Copyright 1997, Wiley-Inter Science)

Structure Refinement by Using X-Ray Data

The observed reflections could be indexed by the Bunn's unit cell parameters with a = 7.81 Å, b = 2.52 A (fiber period), c = 5.51 Å and β = 91.7° and the space group P2 1/m-C 2h 2. Nitta et al.(H) reported the accurate X-ray diffraction intensities of the equatorial reflections (the number of the observed reflections, 21), which were measured by a microdensitometer. The rigid-body least-squares refinement was carried out by using Nitta's intensity data. The values assumed for the molecular framework of polyvinyl alcohol) are shown in Figure 4. During the refinement, the temperature factor for the hydrogen atoms was fixed on 7.0 Å2. The parameters to be refined are Eulerian angle θ, the fractional coordinates of the hydrogen atoms chosen as the origin

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x(Hl) and z(Hl), and the isotropic temperature parameters B(C1), B(C2), B(01), and B(02). The refinement converged to the discrepancy factor 11.1 %, which is slightly different from the results reported by Nitta et al. (10.8 %). This should be attributed to the assumed rigidity of the molecular structure. The atomic positions are essentially the same as those reported by Nitta et al.(12), which support Bunn's model. (8) The hydrogen bonding networks proposed by Bunn are shown in Figure 5. The networks are constructed by the hydrogen bonds (2.84 Â) between to molecules facing each other in the unit cell and the hydrogen bonds (2.81 Â) between neighboring unit cells.

FigureS. The hydrogen bonding networks of atactic polyvinyl alcohol). The scheme is the same as one reported by Bunn. (8) (Reproduced with permission from reference 7. Copyright 1997, Wiley-Inter Science)

Structure Analysis by Using Neutron Data

During the analysis, Bunn's unit cell parameters were adopted. The number of the observed reflections is 27 for 100K, 23 for 200K, and 23 for 300K. The refinement

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was made under the assumption of the rigid body (Figure 4) in the same way as the X-ray structure refinement. The hydrogen atoms to be associated with the hydrogen bonds were ignored during the refinement, because their positions could not be specified. The isotropic temperature parameters of the hydrogen atoms were refined. The parameters to be refined are the Eulerian angle Θ, the fractional coordinates of the hydrogen atom chosen as the origin, x(Hl) and z(Hl), and the isotropic temperature parameters B(H1), B(H2), B(H3), B(H4), B(C1), B(C2), B(01), and B(02). Refinements for the intensity data at 100K, 200K, and room temperature converged to the discrepancy factors R = 26.8 %, 24.8 %, and 20.3 %, respectively. The converged structures are shown in Figure 6 in comparison with the structure obtained by X-ray data (at room temperature).

— X-ray structure

Figure 6. Crystal structures of atactic polyvinyl alcohol) at 100K, 200K, and room temperature obtained by the least-squares refinements. Broken lines indicate X-ray structure at room temperature. (Reproduced with permission from reference 7. Copyright 1997, Wiley-Inter Science)

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The molecule rotates clockwise slightly as the temperature increases. The structures obtained by neutron data are essentially the same as the structure obtained by X-ray data, i . e., Bunn's model. The Eulerian angle θ and hydrogen bond distances systematically change as the temperature increases. The results are consistent with each other. Namely, the hydrogen bond distances increase as the molecule rotates. The slight rotation of the molecule may be caused by the anharmonic vibration of the hydrogen bonds and the motion of the hydrogen atoms to be associated with the hydrogen bonds. This may be supported by the large values of the temperature factors of the oxygen atoms. From the standard deviations of the variable parameters, the accuracy of the structure obtained by the neutron data is comparable to that obtained by X-ray data, in spite of the fact that the R-factors obtained by neutron data are worse than that obtained by X-ray data. This suggests that the incoherent and inelastic contributions to the Laue spots are not significant in the region of 2 θ = 5 - 100°. The rather large discrepancy factors obtained for the neutron data should be attributed to the fact that the hydrogen atoms to be associated with hydrogen bonds were ignored during the refinements. Therefore, the difference synthesis (F 0 - F c synthesis) was made for the 100K intensity data. Here, only independent reflections were taken into

Figure 7. Neutron difference synthesis of atactic poly (vinyl alcohol). Broken and solid contour lines denote the heights with 0.0 and -1.0, respectively. (Reproduced with permission from reference 7. Copyright 1997, Wiley-Inter Science)

84

S o===|>==o II

Figure 8. Intramolecular hydrogen bonds in an isotactic sequence of atactic poly (vinyl alcohol). (Reproduced with permission from reference 7. Copyright 1997, Wiley-Inter Science)

calculation. In Figure 7, the difference synthesis for 100K data is shown. It is possible to find three peaks lower than -1.0, denoted by peaks A, B, and C, which show the hydrogen atoms to be associated with the hydrogen bonds (Figure 7). The minus peaks are due to the minus value of the scattering length of hydrogen atom. These peaks should not be attributed to the anisotropy of the temperature factor of oxygen atom, because the all peaks locate on the asymmetric position with respect to the oxygen atoms. Peak A should be assigned to the hydrogen atom to be associated with the hydrogen bonds between two molecules in the unit cell. On the other hand, Murahashi et al.(is) proposed the intramolecular hydrogen bonds for isotactic polyvinyl alcohol) from the infrared study, and recently, NMR experiments(iO, 16,17) suggest that the intramolecular hydrogen bonds exist in an isotactic sequence of atactic polyvinyl alcohol). Peak Β in the map should be assigned to the hydrogen atom to be associated with the intramolecular hydrogen bonds, which are schematically shown in Figure 8. Accordingly, peak C should be assigned to the hydrogen atoms to be

85

associated with both the intramolecular hydrogen bonds and the intermolecular hydrogen bonds between the neighboring unit cells.

NEUTRON STRUCTURE ANALYSIS OF POLYETHYLENE-D 4( 1 8)

The crystal structure of polyethylene was reported by Bunn in 1939. (19) Thereafter, many works have been done on the polyethylene. During these works, the accurate crystal structure, the accurate angle φ: the azimuthal angle of the molecular plane with respect to the b-axis, was required. By X-ray diffraction measurements, it is very difficult to determine accurately the crystal structure of polyethylene because of the small radius of carbon atom and the small scattering length of hydrogen atom. Neutron diffraction has the advantage that hydrogen atom contributes to the diffraction intensity stronger than X-ray diffraction.(i, 2, 5 - 7) Furthermore, the scattering lengths in the neutron diffraction do not depend on the Bragg angles θ. The reflections with large θ values can be observed and the intensities can be measured accurately. Accordingly, it is possible to determine the accurate crystal structure. On the other hand, the librational motion of the polymer having the long chain structure is an interesting problem. In the previous paper,(20) the librational motion of polyoxymethylene was estimated. In the case of polyethylene, the librational motion can be estimated by using neutron diffraction because of the large contribution to the diffraction intensity of the hydrogen atoms having the large radius.

In the present study, the neutron structure analysis of polyethylene-d4, in which deuterium (scattering length: 6.67) has the scattering length larger than hydrogen (scattering length: -3.74), is carried out on the basis of the rigid body postulate. And the rigid body temperature factor of the molecule is estimated and discussed.

Experimental

The commercially supplied sample of polyethylene-d4 ( M w = 91000, M n = 37000), linear high density polyethylene, was used. The film was made by a hot-press and stretched in the boiling water. The films were stacked and the sample with size 15 χ 10 χ 20 mm3 was prepared. The orientation and crystallinity of the sample were examined by X-ray diffraction.

Neutron diffraction experiments were carried out by Triple Axis Spectrometer TAS-2 in addition to HRPD spectrometers equipped on JRR-3M installed on Japan Atomic Energy Research Institute (JAERI). Intensity distributions on the equator were measured at 10K, 100K, 200K, and 300K by using λ = 1.823 Å. The ranges of

86

measurements by TAS-2 and HRPD were 2Θ < 100° and 2Θ < 150°, respectively. As the intensity data, the integrated reflection intensities of 2 θ < 100° measured by TAS-2 and of 100° < 2Θ < 150° measured by HRPD were adopted. Here, the smoothing of the intensity distributions measured by HRPD are made in the same way as polyvinyl alcohol).!?) The numbers of the observed reflections were 17, 17, 17, and 18 for 10K, 100K, 200K, and 300K, respectively. In Figure 9, the intensity distribution of the equator measured by TAS-2 at 10K is shown.

6000

_

= 4000

β _ m 2000

ο 0 50 100

2θ/°

Figure 9. The intensity distribution on the equator of polyethylene-d4 measured by TAS-2 at 10K. (Reproduced from reference 18. Copyright 1998 American Chemical Society.)

Unit Ce l l Dimensions.

The accurate unit cell dimensions, a and b, were estimated by the least-squares method, where only the independent reflections were used. The temperature dependences of the unit cell dimensions, a and b, are shown in Figure 10. The values above 100K correspond well to the values reported by Swan.(2i) The values at 10K and 100K correspond well to the values at 4K and 90K reported by Avitabile et al.(5), respectively. The temperature dependence of the value a is larger than that of the value b.

87

Figure 10. Temperature dependences of the cell dimensions a and b of polyethylene-d4. (Reproduced from reference 18. Copyright 1998 American Chemical Society.)

Rigid-Body Least-Squares Refinements.

The rigid-body least-squares method was first reported by Scheringer,(13) in which well-defined groups of atoms within the structure are treated as rigid bodies. Thereafter, Pawley(22) reported the rigid-body temperature factor as follows,

Τ = exp{-h · A" 1 -(T+ Vj ·ω·y . ) Â" 1 -h}

V; = Ό , - ( Z r 0 , (Υ,-η) (Ζ,-ζ), 0, -(Xi-ξ)

-(Yi-η), (Xi-ξ), 0 Χ

where h is the reflection indices, A is the transformation matrix from the fractional coordinates Xi to the orthogonal coordinates Xi, Τ is the mean-square translational

tensor, w is the mean-square rotational tensor, and (ξ, ζ, η) is the position of the center of the gravity of the molecule.

Polyethylene assumes the planar zigzag conformation. On the projection through

88

the fiber axis, the molecule assumes the structure shown in Figure 11. The molecular orientation in the unit cell can be defined by the angle Θ, i.e., the complement of the angle φ. In the mean-square translational tensor Τ , the elements,Tn, T 22, and T 1 2 , are significant, and in the mean-square rotational tensor Ο), ω 3 3 is significant. Actually, the constrained least-squares method(i4) was used for the rigid-body least-squares refinements by fixing the bond lengths and bond angles (Figure 11), where the rigid-body temperature factor(22) was taken into consideration. Consequently, the

b

Figure 11. The molecular framework of polyethylene-d 4 on the c-projection. (Reproduced from reference 18. Copyright 1998American Chemical Society.)

variable parameters to be refined are the scale factor, the angle Θ, the elements of the mean-square translational tensor, T a a , T b b , and T a b , and the element of the mean-square rotational tensor ω 3 3 . By the refinements, the R-factors reduced to 11.7 %, 10.1 %, 10.9 %, and 13.3 % for the intensity data at 10K, 100K, 200K, and 300K, respectively. The temperature dependence of the angle θ is given in Figure 12. The angle Θ, the complement of the angle φ, is estimated as 45°, independent of temperature, within the accuracy of the standard deviation 1°. This value 45° corresponds well to the value reported by Stamm et al.(6)

89

100 200 Temperature/K

300

Figure 12. Temperature dependence of angle the azimuthal angle of the molecule with respect to the α-axis of polyethylene-d^ (Reproduced from reference 18. Copyright 1998American Chemical Society.)

Temperature Parameters.

The elements in the mean-square translational tensor and the mean-square rotational tensor are related to the mean-square displacements by following equations,

Tjj = 2jt2<Ujj2> ω 3 3 =2π2<ω2>

The temperature dependences of mean-square translational and librational displacements are shown in Figures 13 and 14, respectively. In Figure 13, the aa and bb elements of the translational displacement tensor are almost the same. This shows that the translational displacements on the c-projection are approximately represented by the circle, i . e., isotropic. The mean-square translational displacement at 10K is about a half of that at 300K (Figure 13), while the mean-square librational displacement at 10K is about one fourth of that at 300K (Figure 14). This suggests that the static disorder of polyethylene is mainly of the translational displacements. The root-mean-square amplitude <ω2>ι/2 was estimated as 13.2° at 300K, which corresponds well to the

90

Figure 13. Temperature dependences of translational displacements estimated by the rigid-body temperature factor. (Reproduced from reference 18. Copyright 1998 American Chemical Society.)

Figure 14. Temperture dependence of the librational displacement estimated by the rigid-body temperature factor. The solid curve show the calculated values. (Reproduced from reference 18. Copyright 1998 American Chemical Society.)

91

values of the carbon (15.8°) and oxygen (13.8°) atoms of polyoxymethylene.(20) This may suggest that the librational motion of polymers having long chain structure is generally large. Cruickshank reported that the mean-square translational and librational motion can be calculated by the Debye temperature and lattice vibrations.(23) The Debye temperature of polyethylene-d4 is not known, but the optical active lattice vibrations are calculated by Tasumi and Shimanouchi. (24) The frequencies of two modes of librational motions, A g and _3g j are reported to be 134 and 109 cm-i, respectively- The mean-square librational motion was estimated by the following equation,(2i)

where h is the Planck constant, I is the moment of inertia, v1 and ν 2 are the frequencies of the librational motions. The calculated curve is shown in Figure 14. The difference between the observed and calculated values shows the static disorder. The observed and calculated values are almost parallel. This suggests that the static librational disorder is independent of the temperature.

The difference between usual (protonated) polymer and deuterated polymer was discussed.(3,4) The crystal structure of polyethylene-d4 is considered to be essentially the same as that of the usual polyethylene, although the temperature factors are somewhat different from the usual polyethylene.

Neutron structure analyses were successfully carried out on two crystalline polymers, atactic polyvinyl alcohol) and polyethylene-d4 Neutron structure analysis supports Bunn's model for atactic polyvinyl alcohol). Three peaks found on the difference synthesis were assigned to the hydrogen atoms to be associated with the intra molecular hydrogen bond in an isotactic sequence of atactic polymer and the intermolecular hydrogen bonds. Neutron diffraction study clarified the detailed crystal structure of polyethylene, in which the azimuthal angle φ of the molecular plane with respect to the b-axis is 45° within the accuracy of the standard deviation Γ. Furthermore, translational and librational motions of the molecule were estimated. The nature of the static disorder in polyethylene was also clarified.

Acknowledgment. I thank Dr. Y. Morii of JAERI for neutron diffraction measurements.

CONCLUSION

92 R E F E R E N C E S

(1) Bacon, G. Ε., "Neutron Diffraction", 1975, Claredon Press, Oxford. (2) Higgins, J. S. ; Benoit, H. C., "Polymers and Neutron Scattering", 1994, Claredon Press, Oxford. (3) Bates, F. S.; Wignall, G. D., Physical Rev. Lett., 1986, 57, 1429. (4) Bates, F. S.; Keith, H. D.; McWhan,D. B., Macromolecules, 1987, 20, 3065. (5) Avitabile, G.; Napolitano, R.; Pirozzi, B.; Rouse, K. D.; Thomas, M . W.; Willis, Β. T. M . , J. Polym. Sci. Polym. Let. Ed., 1975, 13, 351. (6) Stamm, M. ; Fisher, E. W.; Detteinmaier, M.; Convert, P., Discuss. Faraday Soc., 1979, 68, 263. (7) Takahashi, Y., J. Polym. Sci.: Part B: Polym. Phys., 1997, 35, 193. (8) Bunn, C. W., Nature, 1948, 161, 929. (9) Sakurada, I; Futino, K.; Okada, A, Bull. Inst. Chem. Res., Kyoto Univ., 1950, 23, 78. (10) Hu, S.; Horii, F.; Odani, H, Bull. Inst. Chem. Res., Kyoto Univ., 1991, 69, 165. (11) Nakame, K.; Kameyama, M. ; Matsumoto, T., Polym. Eng. Sci., 1979, 19, 572. (12) Nitta, I.; Taniguchi, I.; Chatani, Y. , Annu. Rep. Inst. Fiber Sci., 1957, 10, 1. (13) Scheringer, C., Acta Cryst., 1963, 16, 546. (14) Takahashi, Y.; Sato, T.; Tadokoro, H., ; Tanaka, Y., J. Polym. Sci., Polym. Phys. Ed., 1973, 11, 233. (15) Murahashi, S.; Yuki, H.; Sano, T.; Tadokoro, H.; Chatani, Y., J. Polym. Sci., 1962, 62, S77. (16) Terao, T.; Maeda, S.; Saika, Α., Macromolecules, 1983, 16, 1535. (17) Horii, F.; Hu. S.; Odani, H.; Kitamaru, R., Polymer, 1992, 33, 2299. (18) Takahashi, Y. , Macromolecules, 1998, 31, 3868. 19) Bunn, C. W., Trans. Faraday Soc., 1939, 35, 482. (20) Takahashi, Y.; Tadokoro, H., J. Polym. Sci. Polym. Phys. Ed., 1979, 17, 123. (21) Swan, P. R., J. Polym. Sci., 1962, 56, 403. (22) Pawley, G. S., Acta Cryst., 1964, 17, 457. (23) Cruickshank, D. W. J., Acta Cryst., 1956, 9, 1005. (24) Tasumi, M.; Shimanouchi, T., J. Chem. Phys., 1965, 43, 1245.

Chapter 6

Simulation of Melting Transitions in Crystalline Polymers

Lucio Toma1 and Juan A. Subirana2

1Dipartimento di Chimica Organica, Università di Pavia, Via Taramelli 10, 27100 Pavia, Italy

2Departament d'Enginyeria Quimica, Universitat Politècnica de Catalunya, Av. Diagonal 647, 08028 Barcelona, Spain

We have used a very simple dynamic Monte Carlo approach to simulate polymer behavior in a 2-dimensional lattice. In this paper we show that the model is able to simulate all the basic features of crystallization and melting of real polymer crystals. Crystals of large polymer molecules can be obtained with a thickness which depends on the temperature of crystallization. We also demonstrate that the crystals melt in agreement with the Gibbs-Thompson equation, showing a strong influence of the rate of heating on the melting process. Furthermore thin crystals may thicken upon annealing in adequate conditions. The results obtained may be used in the interpretation of scattering experiments on the crystallization of polymers.

We have recently proposed (1) a simulation model of polymer crystallization through a dynamic Monte Carlo approach in a 2-dimensional lattice. Similar results are obtained in a three dimensional cubic lattice (2). We have found that this method can predict the compaction of polymer chains as folded molecules as they are found in polymer lamellar crystals. The model correctly predicts a gradual thickening of the crystals as the temperature is increased. In this paper we want to test further our model in order to determine if it is able to predict other experimental features of polymer crystallization such as:

• How do larger molecules behave? • Do the simulated crystals follow the Gibbs-Thompson equation? • Do thin crystals become thicker by annealing at higher temperatures?

© 2000 American Chemical Society 93

94

Methods

The algorithm CI/LL for dynamic simulations is the same already described (1,3). The polymer chain is modeled as a self-avoiding walk of a sequence of hydrophobic monomers on a square lattice. The force field is represented by an energy function associated to each couple of non-bonded monomers that assumes the value -1 when two conditions are satisfied: i) a "contact" between monomers (i.e. a distance of one lattice unit) and ii) their parallel (linear-linear) alignment; if only one or none of the conditions are satisfied the energy value is zero. Two residues in contact define an LL-loop that makes all the residues in the loop "cooler" than those outside it; thus, to each residue is associated a cooling factor XO that determines its mobility. Dynamic simulations are performed through a Monte Carlo approach governed by the cooling factors; the approach uses pivot moves actually possible only when is satisfied the condition rnd < Qxp(f(i)/T) where rnd is a random number between 0 and 1 and Τ is the temperature. In our previous papers (1,3) we used the symbol ck in order to represent the temperature. Simulations at proper temperatures result in the formation of lamellar structures whose average thickness L is calculated by the formula L = (N - 1 -f1)/(Nc + 1 -f1), where Ν is the number of monomers in the chain, f1 is the number of rods of length 1, and Nc is the number of bends.

Results

Simulations of longer chains.

The dynamic simulations have been applied to a polymer model chain with a number of monomers TV = 1000, P1000, in addition to the previously reported chains with Ν = 100 and Ν = 300, P100 and P300. P1000 has a sixfold degenerate energy global minimum with £=-912: the six conformations differ in their thickness as L ranges from 39 to 49. We run simulations with the CI/LL algorithm on ensembles of ten independent molecules of Ρ1000 at several temperatures with the same duration (108 time steps) independently from the temperature. After a pre-equilibration period at high temperature (F=100) to yield random conformations, the temperature was instantaneously quenched to the value chosen for each simulation. In Figure 1 are reported the plots of the normalized energy E/Emin and thickness L averaged in the last 10% of the simulations, as a function of T. The same plots also report the corresponding data already obtained for P100 and P300 (1).

95

_1__ι I ι I ι I ι I—ι—I 0.2 0.4 0.6 0.8 1.0 1.2

Τ

Figure 1, Results from simulations on the square lattice with the CI/LL algorithm on the homopolymer chain model P1000 (full squares) compared with the results obtained for P100 (open circles, from ref 1) and P300 (open triangles, from ref 1): normalized energy (E/Em-J and thickness (L) as a function of temperature (T) averaged over 10 independent molecules in the last 10% of the simulations 10 time steps long.

96

Figure 2. Some examples of the final conformations oftPie chain model PI 000 obtained at different temperatures after simulations 10 time steps long.

Al l the plots appear quite similar independently from the chain length also for the very long chain PI000. Figure 2 reports some final lamellar conformation of single molecules obtained at the end of the simulations (at T=QA, 0.6, 0.8). The lamellar behavior appears evident as well as the effect of temperature on the lamellar thickness. It is worth pointing out that all these structures have Z<10 and appear quite different from the global minima of P1000 (1=39-49). The limiting L value only shows a small increase as Ν increases, whereas the value of L at the global energy minima varies significantly as a function of the chain length N: 1=24 for P300 and 1=13-16 for P100. This behavior indicates that the lamellar thickness actually obtained does not depend significantly on molecular weight, in agreement with the experimental observations which show that the lamellar thickness of polymer crystals mainly depends on the undercooling.

97

It appears that the simulation method we are using only allows the molecules to reach a final energy which invariably is about 80% of the global energy minimum, as indicated by the lowest value of Ε which can be reached as shown in Figure 1.

Melting simulations.

The CI/LL algorithm was then used to simulate the melting of lamellar crystals. All these studies have been performed on P300 and we chose as starting conformations regular lamellae with different thickness, some of which are reported in Figure 3. Al l the simulations started at 7=0.6 and temperature was progressively increased to 7=1.63. We adopted three different heating rates (A=3, Λ=1, h=0.3) corresponding to the following heating schemes: i) temperature was increased each time step by a factor 1.000001 in simulations 106 t.s. long (£=3); ii) temperature was increased each time step by a factor 1.000000333 in simulations 3*106 t.s. long 0=1); temperature was increased each time step by a factor 1.0000001 in simulations 107 t.s. long (k=03). In Figure 3 are also presented the plots of energy as a function of temperature for the heating rate k=\ as averages over 20 independent molecules. The increase of the melting temperature by increasing the lamellar thickness of the starting structure is clearly evident. The melting temperature (Tm) was taken as the value of Τ at which the energy becomes one half of the starting energy. Thus, for the crystal with Z=7 the melting temperature turned out to be 7^=0.91, for L=9 Tm

= 1.03, and so on. Similar results were obtained for heating rates &=3 and £=0.3, but the curves moved to higher and lower values of temperature, respectively. Figure 4 illustrates this effect of heating rate on Tm for the crystal with Z,= 14. The change in energy as a function of temperature is shown for four different heating rates. At very low rates (&=0.03) a very sharp transition is found, which indicates that we are very close to thermodynamic equilibrium. As the rate is increased, the polymer releases its energy gradually and maintains part of its energy above its true melting point.

It is known (4) that the melting temperature of a crystal can be expressed by the Gibbs-Thompson equation:

Tm = Tm°[\-2Gj(AHfxl)]

where Tm° is the melting temperature of a crystal of infinite thickness, AHf is the heat of fusion per unit volume of crystal, / is the crystal thickness and σ ε is the surface free energy per unit surface.

A similar equation should apply to our two dimensional model, the only difference being that AHf should be the heat of fusion per (unit of length)2 and σ β

should be expressed as energy per unit of length where one unit of length is one lattice unit; / should correspond to our L. The values of AHfwiW be equal to unity for an infinite crystal. In Figure 5 are reported the plots of Tm (as obtained from the plots in Figure 3 and similar plots not shown for £=3 and ^=0.3) as a function of ML. The extrapolation follows straight lines and allows the determination of Tm° and σ β by linear regression. They turned out to be 1.28, 1.41, 1.61 and 1.16, 1.24, 1.30 for A=0.3, &=1, h=3, respectively.

1=14 £=-247 L=24 E=-253

Β

0.6 0.8 1.0 1.2 1.4 1.6

Τ

Figure 3. a) Examples of the starting geometries for melting simulations of the chain model P300. b) Results of the melting simulations: energy (E) averaged over 20 independent molecules as a function of temperature (Y) for different starting thicknesses. All these simulations were carried out with the same heating rate k-1.

99

Ο

0.6 0.8 1.0 1.2 1.4 1.6

Γ

Figure 4. Results from melting simulations on the chain model P300: energy (E) averaged over 20 independent molecules of starting thickness L=14 as a function of temperature (I). The plots refer to four different heating rates k.

Figure 5. Plots of the melting temperature (TJ as a function of the reciprocal lamellar thickness (1/L) for three different heating rates: k=0.3 (triangles), k=l (squares), andk=3 (circles).

100

The plots given in Figure 5 show that the simulated lamellar crystals do obey the Gibbs-Thompson equation. It is also clear that the slope of the lines depends on the rate of heating, a question which has not received much experimental attention. Furthermore it should be pointed out that the values of σ β calculated from the slopes in Figure 5 are not equivalent to those obtained for three dimensional lamellar crystals. As the value of L is increased, there is an increasing contribution to the surface energy of the straight, lateral dimension of the crystals. This feature is clear by inspection of the folded crystals shown in Figure 3 a. Thus σ β includes contributions to the surface energy of the four sides of the crystal, both the folded and straight sides. We are planning to correct for this effect in our 3D simulations, which are presently in progress.

Crystal thickening.

It is known that thin crystals are metastable from the thermodynamic point of view and spontaneously increase in thickness as the melting point is approached. We tried to reproduce this behavior and performed on P300 the same heating procedures described above starting from a crystal with L=4. The results obtained are illustrated in Figure 6 by the plots of energy and thickness. It is clear that for &=0.3 (and to a lesser extent for h=l) the thickening effect is clearly reproduced by our model. The crystals show first a rapid increase in energy due to partial melting and afterwards a slower decrease which corresponds to the growth of thicker crystals (up to L=6 approximately). When the rate of heating is too large (k=3), the crystals melt directly without any intermediate thickening. Similar results were obtained for P100.

Discussion

The simulations reported in this paper clearly demonstrate that single polymer molecules allowed to crystallize by our method already show the main features of lamellar crystals. In agreement with the results reported in a previous paper (1), in our simulations crystal thickness diminishes as the temperature decreases below the melting point. In Figure 1 we show that this behavior is also found for higher molecular weights. Furthermore the folded molecules melt in agreement with the Gibbs-Thompson equation (Figure 5) and they also thicken as the temperature is slowly increased (Figure 6, k=0.3). These features reproduce the experimental behavior and encourage us to continue our studies in order to model in detail the behavior of single polymers and crystalline lamellae in three dimensions, as an extension of the preliminary results which we have recently reported (2). Our model, though of extreme simplicity in both definition of energy and set of moves, appears able to reproduce in all its aspects the experimental processes of lamellar crystal formation and melting. Several other models have been recently proposed (5-11) that

101

Figure 6. Results from melting simulations on the chain model P300: energy (E) and thickness (L) averaged over 20 independent molecules of starting thickness L=4 as a function of temperature (I); the three plots refer to three different heating rates k.

102

can also reproduce one or more features of homopolymer behavior based on a variable number of energy parameters and relying on a more or less complex set of moves.

Acknowledgments.

This work has been supported in part by Grant PB93-1067 from the DGICYT and in part by Ministero deH'Universita e della Ricerca Scientifica e Tecnologica (Roma).

References

1. Toma, L.; Toma, S.; Subirana, J. A. Macromolecules 1998, 31, 2328. 2. Toma, L.; Subirana, J. A. Polym. Mat. Sci. & Eng. 1998, 79, 367. 3. Toma, L.; Toma, S. Protein Sci. 1996, 5, 147. 4. Hoffman, J.D.; Davis, G.T; Lauritzen, J.I., Jr. In Treatise on Solid State

Chemistry; Hannay, N.B., Ed.; Plenum Press: New York, 1976; Vol. 3, pp. 497-614,.

5. Zhdanov, V. P.; Kasemo, B. Proteins 1997, 29, 508. 6. Doye, J. P. K.; Sear, R. P.; Frenkel, D. J. Chem. Phys. 1998, 108, 2134. 7. Kuznetsov, Υ. Α.; Timoshenko, E. G.; Dawson, K. A. J. Chem. Phys. 1996, 104,

336. 8. Chen, C.-M.; Higgs, P. G. J. Chem. Phys. 1998, 108, 4305. 9. Sundararajan, P. R.; Kavassalis, T. A. J. Chem. Soc. Faraday Trans. 1995, 91,

2541. 10. Yamamoto, T. J. Chem. Phys. 1997, 107, 2653. 11. Liu, C.; Muthukumar, M . J. Chem. Phys. 1998, 109, 2536.

Chapter 7

Neutron Spin Echo Spectroscopy at the NIST Center for Neutron Research

N. Rosov1, S. Rathgeber1,2, and M. Monkenbusch3

1NIST Center for Neutron Research, NCNR, Building 235/E151, 100 Bureau Drive, Stop 8562, Gaithersburg, MD 20899-8562

2University of Maryland, College Park, MD 20742 3IFF, Forschungszentrum Jülich, Jülich D-52425, Germany

A Neutron Spin Echo (NSE) spectrometer is nearing completion at the NIST Center for Neutron Research. NSE spectroscopy measures the pair correlation function with respect to wave-vector transfer and time, unlike other neutron spectrometers, which measure the scattering as a function of wave-vector and energy transfer. The NIST NSE spectrometer can measure wave vector transfers from 0.01 Å-1 to 2Å-1. The accessible time range for a particular setting of the spectrometer extends to nearly 10-7 s at the longest accessible wavelengths. We discuss here the expected properties of the NIST NSE spectrometer, provide a description of the strengths of NSE spectroscopy, and give some detail of the considerations necessary for designing a successful NSE experiment.

Three high-resolution inelastic instruments are nearing completion at the NIST Center for Neutron Research (NCNR) (1): a disk chopper time-of-flight spectrometer (2), a backscattering spectrometer (3), and a neutron spin echo (NSE) spectrometer. Al l three instruments will be available to researchers through reviewed proposals (4)- Both the NSE and Backscattering spectrome­ters will be operational in early 1999 and the Disk Chopper Spectrometer shortly thereafter.

The Disk Chopper and Backscattering spectrometers, like most other inelas­tic neutron spectrometers, measure the scattering function 5(Q,w), where Q is the wave-vector transfer and w is the energy transfer of the scattering. The scattering function is the Fourier transform (in both time and length) of the

© 2000 American Chemical Society 103

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real space pair correlation function. On the other hand, the NSE spectrometer measures the intermediate scattering function I(Q,t) (the cosine transform of 5(Q,w)). This is the same quantity that is measured by dynamic light scatter­ing, albeit over a different time and Q range.

This paper provides a brief introduction to NSE, tailored to the particular details of the NIST NSE spectrometer, which has been optimized for the study of soft condensed matter systems. We will discuss how am NSE experiment is performed and outline the considerations required for choosing experimental systems.

Figure 1: Phase space diagram indicating the region of accessibility for each of the three new inelastic instruments at the NIST Center for Neutron Research. The unshaded region around the NSE region indicates the effect of several planned improvements to the NSE instrument.

Experimental Realization of the Spectrometer

The design of the NCNR NSE spectrometer is based on that of the NSE spec­trometer at the Forschungszentrum Julich (5). Exxon and the Forschungszen-trum Julich are additional members of the participating research team for con­structing the spectrometer.

The NCNR reactor is a 20 MW heavy water reactor with a split core. A guide network is fed from a liquid H2 cold source with an effective Maxwellian

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Table I: N C N R NSE operating characteristics at a glance

> Velocity Selector

• (A) > 4.5A • 8% < Δ λ / λ < 20% (FWHM)

> Polarizer

• 30c Fe/Si supermirror "V" in guide. • Maximum polarization for λ > 5 A

> Main Coils

• Imax - 440 A • Jmax — 0-5 Τ . ΙΠ • 0.1 ns < t < 30ns (at 8A; tm%n, tmax oc λ 3 )

> Sample region

• Active area 5 x 5 cm 2

• Neutron flux > 10 6n/cm 2/s at 8A and Δ λ / λ = 10% • Useful flux between 4.5 A and 12 A

>- Detector

• 25 χ 25 cm 2 area detector • Qmin = OmA-1 at 8 A • Maximum scattering angle 100° (Qmax = 2.1 -"1 at 4.5 A)

temperature of 45 K. The spectrometer is located at the end of guide NG-5, which is coated with 5 8 N i . An optical filter (6) moves the end of the guide out of the direct line-of-sight of the reactor core, removing fast neutrons and core gammas from the beam, while allowing the transmission of neutrons with λ > 4A. A velocity selector roughly monochromates the neutron beam with 8% < Δ λ / λ < 20% (FWHM) for mean wavelengths λ > 4 A. The beam is polarized in the longitudinal direction by a Mezei cavity (7), with maximum polarization for λ > 5 A.

We have performed preliminary measurements of the neutron flux at the end of the guide with a gold-foil technique. Measurements were made with the ve­locity selector set at three wavelengths, λ = 6, 9, and 12 A with Δ λ / λ = 10% F W H M ; the results are shown in Figure 2, where the agreement between the measured results and the calculations is good at 9 A; however, the difference be-

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Figure 2: Calculated neutron fluxes (solid lines) at vanous points along neutron guide NG-5 to the NSE spectrometer with α Δλ/λ = 10% (FWHM) wavelength distribution: Just before taper; just after taper; at end of guide; and at sample position. The three large circles indicate the results of gold-foil measurements at the end of the guide.

tween the results and calculations is around 30% at 6 Å and 12 Å. This difference may be attributed to neglect of wavelength-dependent absorption effects in the windows of the guides and the cold source.

The entire instrument is of amagnetic construction, and the computer pro­gram that operates the spectrometer contains an accurate description of the current distributions, thereby allowing the total field distribution of the more than twenty coils to be calculated with sufficient accuracy to considerably re­duce the time spent tuning the spectrometer. The main coils that provide a precession field contain compensation loops to rapidly reduce the on-axis field outside the solenoid. The inhomogeneities in the field are greatly reduced as a result, which decreases polarization losses, and so there is almost no decrease in the maximum accessible time at higher scattering angles.

The active sample area is 5 χ 5 cm 2 (typical sample sizes for polymer studies are on the order of 3 χ 3 χ 0.3 cm3). This value is constrained by the flipper win­dows and the sample environment. These can be extended somewhat; however, the sample size is ultimately constrained to 10 cm diameter by the active area of Presnel-like correction elements inserted in the main coils. These correction elements allow off-axis and divergent neutrons to satisfy the echo condition (see Eq. 7) and thus allow the use of an area detector. This effectively increases

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the signal strength by a factor of twenty. The form of the sample is not con­strained in principle by the NSE technique; however, the sample environment must not produce stray magnetic fields—e.g., from heating currents or construc­tion materials—that might disrupt the measurement of the echo.

Principles of Neutron Spin Echo

The manipulation of the neutron spins through the spectrometer (8) is shown in Figure 3. After the beam is polarized, the neutron spins are rotated 90° by a n/2 flipper, which begins the precession of the neutron in the main solenoidal field.

In a perpendicular magnetic field B\, a neutron spin will undergo precessions at a frequency UL = — 7ΐ,bι, where 7χ,/(2n) = 29.16MHz/T. If a neutron is polarized perpendicularly to the axis of a solenoidal field, it will precess through an angle

where Ji is the field integral along solenoid % and h is the wavelength of the neutron (h/(m\) is the neutron velocity v). Note that φ can only be determined to mod(27r).

For a beam of neutrons with an incident wavelength distribution /(h) and (h) = (hι), each neutron undergoes a spin precession of φι(l) in the first arm of the spectrometer. The neutron beam, with its broad band of wavelengths, will completely depolarize in this first precession field.

After scattering from the sample, a neutron passes through a π-flipper, thereby changing its phase angle from φ mod (2π) to -(φ mod (2s)) (see Fig­ure 3b). Then, on passing through the second precession field, if the scattering is elastic and the two field integrals are the same, the beam recovers its full polarization at the second π/2 flipper, which rotates the spins back to the lon­gitudinal direction, thereby stopping the precessions. The beam polarization is then analyzed and those neutrons with the correct spin state impinge on an area detector.

If the neutrons are scattered quasi-elastically from the sample (as is almost always the case for soft condensed matter research), changing wavelength by £ l , they will undergo a spin precession with phase angle φι in the second arm of the spectrometer. The phase difference between the two arms of the spectrometer

can be separated into two terms, one due solely to the inelasticity, the other due solely to the difference in the field integral of the two arms of the spectrometer,

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Figure 3: (a) Schematic of a Neutron Spin Echo spectrometer. The various components are discussed in the text, (b) Motion of the neutron spins in the precession region, looking downstream along the beam direction, (c) Motion of the neutron spins looking perpendicular to the beam direction. (Reproduced with permission from Reference (9).)

where Αφ(λ)=φ1(\)-φ2(\). (4)

To first order in δλ and Αφ, the phase shift is composed of a term from the inelasticity and a term from the difference in the field integrals (recall from Eq. 1 that φ α λ):

ψ = ^ ( ( Λ 1 > ) ^ - , 1 ( ( λ ι > ) ^ + Δ ^ λ 1 ) ) ^

= (φ1((Χ1))δΧ + Αφ((Χ1))Χ)/(Χ1). (5)

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The average over the beam, in our quasi-elastic approximation, gives (δλ) = 0, and so

(φ) = Δ ψ ( λ ι ) ) . (6) The spectrometer is therefore in the echo condition when there is no difference between the spin precession angles of the two arms of the spectrometer, i.e., (φ) = 0, which only holds when

Ji = J 2 - (7)

The inelasticity of the scattering can be written as a change in wavelength

where, to first order in δλ,

hw = (9) m λ3

Due to the quantum nature of the neutron spin, only one component of the spin, call it z, can be determined. The polarization of the scattered beam is

(Pz) = <(*#)

= Jf(X)dXx

Js(Q,u)cos ( 1 ( { λ 1 ) ) α ; + Δ 0 ( { λ 1 ) λ ) / { λ 1 ) du. (10)

Since we have assumed 5(Q, ω) is a quasi-elastic scattering law, which is essen­tially an even function of ω, the polarization of the beam is

(Pz) = (Χ)οο8(Αφ({Χ1))Χ/(Χ1))dΧχ

= Jf(\)coa(A<l>((\i))\/{Xx))d\x Js(Q,u)cos(ut)du (11)

where

t = ( 1 ) 2 Π & > - l ( τ ) " ε * ( 1 2 )

is the Fourier time. By adjusting the field so that the echo condition (Eq. 7) is met, the beam

polarization is

<P,) = Jf(X)dXx Js(Q,u)cos(ut)diu = Jf(X)I(Q,t)dX. (13)

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In many experimental cases I(Q,t) varies very slowly with λ and so can be removed from the integrand; in these cases, Neutron Spin Echo spectroscopy directly measures the intermediate scattering function, the energy cosine trans­form of the scattering function. In these situations, it is not even necessary to determine the wavelength distribution; however, if the wavelength distribution is strongly affected by the scattering process, for example by absorption, then the wavelength distribution must be determined.

Even if the condition of small energy transfers does not hold, a linear re­lationship between the phase angle shift φ and the energy transfer δω can be derived for the non-quasi-elastic case, viz.

Here ω = ω((hι), ( λ 2 ) ) is the average energy transfer and φ = ^( (h ι ) , ( λ 2 ) ) the average phase angle shift associated with the average wavelengths {X{) (i = 1,2) before and after the scattering process. The assumption necessary for derivating Eq. 14 is that the wavelength distribution of the neutrons contributing to the echo must be limited:

The echo condition becomes

and in Eq. 5, φ must be replaced by (φ - φ) as given in Eq. 14. To fulfill Eq. 15, the transmission function of the NSE spectrometer has to

be limited to a certain wavelength range. This is mainly done by the velocity selector in the incoming beam and by the limited wavelength acceptance of the analyzer for processes causing a broadening of the wavelength distribution in the scattered beam. For more detailed information see the article by Mezei in Ref. (8).

We will see below, however, that the most powerful application of NSE lies in the field of quasi-elastic scattering with ω = 0, where the first derivation suitably describes the function of the NSE spectrometer.

Applications

There are two simple ways to perform NSE scans: In one, not so frequently used, the time parameter (that is, the field integral) is set as small as possible (although the field must be larger than any stray fields that might depolarize the beam) and so the second term in Eq. 11 reduces to 5(Q),

I l l

"S5 g

1 S "δ t g δ

53 Î2 ^ te «

δ ο ^ ο §

·*{ 111-!, *! I £ 1 r* St *5 ·* ·Ι* k»

"8 Ι « J 6 Ι .s s -S -S j ? ι I ε &

1 2 Λ 3 ^

5 a ^ β ,

"8 ° < ^ 1 s»,

ω ω <ϋ β S •g.» ι β ^ &

e i ~ j S ?

•g g s- u ω e

u g « Ë S* afK

8 υ s £ ' S § s g

.s> . & J g I pq « g a <υ ^ ^

1 b § I 6 i s ι ^ ι

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By varying the difference in the field integrals, one can measure the cosine trans­form of 5(Q)/(A), which, given a constant or slowly varying S(Q), is a cosine function, with a period proportional to (λχ)" 1 , modulated by the cosine trans­form of the wavelength distribution. This scan is often used to "prove" that an NSE instrument is working (see Figure 4) and, more important, to correct either for the effect of strong absorption (or multiple scattering) from the sample or for a strong Q-dependence in the scattering.

The other scan, commonly used for measurements in polymer or similar systems, is taken by setting the spectrometer to the echo condition, which gives Eq. 13, and so the intermediate scattering function. In practice (see Figure 4), the counting rate is measured for several phase current values near the echo position, then fitted to the modulated cosine of Eq. 17 to give the echo amplitude A . The effect of the less-than-perfect efficiency of the flippers, the polarizer, and the analyzer is removed by measuring the count rates with the π/2-flippers off and the π-flipper both off and on, giving ATup and iVdown, respectively. The measured value of the polarization of the scattered beam at the echo point {PZ)M is then determined by

ΊΑ ™ * ~ N -NA • ( 1 8 )

i V up iVdown Inhomogeneities in the magnetic field may further reduce the polarization. As these inhomogeneities are not correlated with S(Q, ω) or / (λ ) , their effect may be simply divided out by measuring the polarization from a purely elastic scatterer (Pz)M- The value of I(Q,t) is then given by

HQyt) {Pz)m \E IM T m - ™ - m ( I 9 )

The time is varied, according to Eq. 12, by changing the field in the main coils. This procedure is particularly useful for measuring quasi-elastic processes or

processes at small energies where discriminating the scattering of interest from any elastic scattering is important. On a conventional spectrometer, deconvolv­ing the instrumental energy lineshape from the physical lineshape can be quite difficult. However, a convolution in ω-space becomes a simple product in the time domain. Rather than performing highly elaborate lineshape analyses in the energy domain to determine, for instance, if a process is elastic or quasi-elastic, one can merely note if the NSE signal is constant or decays with time. Figure 5 shows several scenarios where the inelastic scattering might be determined in the time domain by NSE.

Typical spectra observed by incoherent neutron scattering for a vibrational process show in the energy domain as two inelastic lines with width Γ centered at UQ and — u;o, as well as an elastic feature that comes from the center of mass. A Fourier transformation into the time domain yields a damped cosine oscillating around a nonzero average value. Rotational motions, which show broader quasi-elastic features around the elastic line of the center of mass in the u>-domam, are

Figure 5: A comparison of typical scattering spectra obtained by inelastic neutron scattering in the energy and time domains for (a) vibrational, (b) rotational, (c) translational, and (d) relaxational motion.

114

also observed by incoherent scattering. For a simple Lorentzian-line shape in time-space a single exponential decay to a nonzero plateau value for long-time limit would be observed.

It is possible in principle to measure the incoherent intermediate structure factor Jincoh(Q,£) with the classical NSE spectrometer design (such as that at NCNR); however, in practice it can be difficult to measure I'mcoh(Q,t) in & reasonable measuring time. There are three sources of incoherent scattering:

• isotopic variation in the nuclear cross-section;

• uncorrected motions, e.g., noninteracting rotators; and

• variation in the nuclear cross section due to the nuclear spin.

Al l of these reduce the signal intensity since all incoherent scattering is spread out isotropically in a solid angle of 4π.

The consequences of spin-incoherent scattering, which can have a signifi­cantly deleterious effect on the NSE signal, are shown in Figure 4. Spin-incoherent scattering causes, with a nucleus-dependent probability, a spin flip of both the inelastically and elastically scattered neutrons, e.g., 2/3 of the neutrons scattered from Η undergo a spin flip, whereas deuterium, which has no nuclear spin, has no influence on the neutron spin.

Note that this spin flip caused by spin-incoherent scattering is not the same as that occurring in the π flipper in the sample region. The π flipper changes the phase of the neutron from φ + 2ηπ to -φ + 2ηπ, so that, for instance, on elastically scattering, with the first and second field integrals equal, the neutron ends up at the second π/2 flipper with phase - 0 + 2 η π 4 - 0 + 2 η π = 4ηπ, that is, the original phase. In spin-incoherent scattering precesses, some fraction of the spins (2/3 in the case of Η-scattering) are flipped by 180°, that is, from φ + 2ηπ to φ + (2n -h 1)π and then to —φ + (2n + 1)π by the π-flipper. These spins end up at the second π/2 flipper with phase -φ + (2n 4- 1)π + φ + 2ηπ = π + 4ηπ, and produce an echo that is reversed with respect to the non-spin-incoherent case. The overall echo amplitude is a superposition of the signals with opposite sign from the spin-flipped and non-spin-flipped neutrons, and so is reduced. For scattering from protons, the final signal is -1/3 of the signal from the non-spin-incoherent scattering case; the background is strongly increased as well, reducing the signal-to-noise ratio considerably. If, in addition, there is some coherent scattering present, the spin-incoherent and coherent cross sections have opposite signs for the echo signal and so reduce the echo signal in a way that cannot be decomposed.

The main application of NSE spectroscopy is therefore to measure the inter­mediate coherent scattering function JCoh(Q> £)> the coherent density fluctuations that correspond to some SANS intensity pattern. This type of scattering may be orders of magnitude more intense than the incoherent contributions. In gen­eral, however, measurements on polymeric and biological systems require that the some part of the sample be deuterated. The degree of deuteration is an practical issue that involves optimizing contrast against incoherent background

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and depends in part on the amount of beam time available for a particular experiment.

Translational and relaxational motions that can be observed in coherent scat­tering result in a broad quasi-elastic feature around w = 0, which gives, in the case of a Lorentzian lineshape, a simple exponential decay in the time domain. If the motion is localized in space, the final plateau value of the I c o h (Q,t) is nonzero.

We can summarize that the great advantage of NSE spectroscopy lies in in­vestigations of aperiodic relaxation dynamics. On mesoscopic time scales well separated from atomic time scales, these processes show broad quasi-elastic fea­tures in frequency space, but a featureless decaying structure in the time domain.

Within these constraints, NSE spectroscopy covers a wide range of applica­tions. In classical solid state physics, critical scattering in the fields of mag­netism (10, 11) and structural phase transitions (12, 13) mainly have been investigated. But NSE spectroscopy has also shown to be an extremely useful tool for probing the dynamics of soft matter. In the case of polymer systems, the single chain relaxation of polymers of various structures (stars, cyclic, diblock-copolymers, linear polymers, micelles) were investigated (14, 15, 16). NSE also plays an outstanding role in the investigation of the relaxation processes in glassy systems (17). Beyond this, studies have been performed on complex fluids—shape and size fluctuations of microemulsions (18, 19) and on systems with biological importance (20).

References

1. see J. Res. Nat. Inst. Stand. Technology 1993, 98.

2. Altorfer, F. B.; Cook, J. C.; Copley, J . R. D. Mat. Res. Soc. Symp. Proc. 1995, 376, 119.

3. Gehring, P. M . ; Neumann, D. A. Physica Β 1998, 241-243, 64-70.

4. Information on submitting proposals may be obtained on the NCNR web site <http://www.ncnr.nist.gov/> or by contacting one of the authors: nrosov@nist.gov or silke.rathgeber@nist.gov.

5. Monkenbusch, M . ; Schätzler, R.; Richter, D. Nucl. Instr. Meth. Phys. Res. A 1997, 399, 301-323.

6. Copley, J. R. D. J. Neutron Res. 1994, 2, 95.

7. Krist, T.; Lartigue, C.; Mezei, F. Physica B, 1992, 180, 1005-1006.

8. A much more complete description of NSE can be be found in Neutron Spin Echo; Editor, F. Mezei, Lecture Notes in Physics, vol. 128; Springer-Verlag: Berlin, 1980. The present outline follows that given in the article by J. B. Hayter, pp 53-65.

9. Rathgeber, S. Ph. D. Thesis, University of Aachen, Nordrhein-Westfalen, Germany, 1997.

10. Bewley, R. I.; Stewart, J . R.; Ritter, C.; Schleger, P.; Cywinski, R. Physica Β 1997, 234, 762-763.

11. Sarkissian, Β. V. B. Philos. Mag. Β 1996, 74, 211-217.

12. Kakurai, K. ; Sakaguchi, T.; Nishi, M . ; Zeyen, C. M . E.; Kashida, S.; Ya-mada, Y . Phys. Rev. Β 1996, 53, R5974-R5977.

13. Durand, D.; Papoular, R.; Currat, R.; Lambert, M . ; Legrand, J. F.; Mezei F. Phys. Rev. Β 1991, 43, 10690-10696.

14. Ewen, B.; Richter, D. Adv. Polym. Sci. 1997 134, 1-129.

15. Richter, D.; Willner, L.; Zirkel, Α.; Farago, B.; Fetters, L. J.; Huang, J . S. Macromolecules 1994, 27, 7437-7446.

16. Farago, B.; Monkenbusch, M . ; Richter, D.; Huang, J. S.; Fetters, L. J.; Gast, A. P. Phys. Rev. Lett. 1993, 71, 1015-1018.

17. Arbe, Α.; Richter, D.; Colmenero, J.; Farago, B. Phys. Rev. Ε 1996, 54, 3853-3869.

18. Farago, B. Physica Β 1996, 226, 51-55.

19. Farago, B.; Richter, D.; Huang, J . S.; Safran, S. Α.; Milner, S. T. Phys. Rev. Lett. 1990, 65, 3348-3351.

20. Pfeiffer, W.; Konig, S.; Legrand, J . F.; Bayerl, T.; Richter, D.; Sackmann, E. Europhys. Lett. 1993, 23, 457-462.

POLYMER CRYSTALLIZATION

AND MORPHOLOGY

Chapter 8

Isothermal Thickening and Thinning Processes in Low Molecular Weight Poly(ethylene oxide)

Fractions Crystallized from the Melt Effects of Molecular Configurational Defects

on Crystallization, Melting, and Annealing

Er-Qiang Chen1, Song-Wook Lee1, Anqiu Zhang1, Bon-Suk Moon1, Ian Mann1, Frank W. Harris1, Stephen Z. D. Cheng1,4,

Benjamin S. Hsiao2, Fengji Yen2, and Ernst D. von Meerwall3

1Maurice Morton Institute and Department of Polymer Science, The University of Akron, Akron, OH 44325-3909

2Department of Chemistry, The State University of New York at Stony Brook, Stony Brook, NY 11794-3400

3Department of Physics, The University of Akron, Akron, OH 44325

Three two-arm poly(ethylene oxide) fractions (PEOs) with molecular weights of 2220 g/mol for each arm have been prepared by a coupling reaction using 1,4-, 1,3-, and 1,2-benzene dicarbonyl dichloride. The two arms at the coupling agents thus form angles of 180°, 120°, and 60°, respectively. Self-diffusion coefficients of these two-arm PEOs in the melt are surprisingly different. Wide angle X-ray diffraction patterns reveal that these PEOs possess the identical crystal structure to that of linear PEO. Observations of time-resolved synchrotron small angle X-ray scattering (SAXS) indicate that the samples crystallized below 38°C forming crystals with non-integrally folded (NIF) overall molecular conformations (OMCs). Crystals with integrally folded (IF) OMCs form when the crystallization tempearture (Tc) is above 44°C,. Two different crystal populations with extended and once-folded OMCs are observed in both the 1,4- and 1,3-two-arm PEOs. Only one crystal population with mixed IF OMCs is found for the 1,2-two-arm PEO. The dependence of long period as a function of Tc for these two-arm PEOs is remarkably similar to the melting temperature response to Tc. The annealing effect is examined for samples crystallized at 32°C, subsequently heated to 50°C, and isothermally annealed for various periods of time. A partial melting upon heating and recrystallization during annealing can be identified.

4Corresponding author.

118 © 2000 American Chemical Society

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In the past three decades, low-molecular-weight (LMW) poly(ethylene oxide) fractions (PEOs) have played an important role in understanding polymer crystallization (1-8), In our efforts, we have reported that an initial transient state during the LMW PEO crystallization can be recognized as non-integral folding chain (NIF) crystals which form prior to the final state of integral folding chain (IF) crystals (9-15). In order to investigate the molecular architecture effects on LMW PEO crystallization behavior, several experiments have been designed: molecular weight dependence (15), end group effects (16), and defects at the center of chains (17). Recently, we have focused on the crystallization and melting behavior of three different two-arm PEOs. These PEOs possess an identical MW of 2220 g/mol for each arm (Ma = 2220) and the coupling agents used are 1,4-, 1,3-, and 1,2-benzene dicarbonyl dichloride. The two arms at the coupling agents thus form angles of 180°, 120°, and 60°, respectively (18). It has been found that configurational defects at the center of each of the two-arm PEO chains substantially affect their overall molecular conformation (OMC) in the crystalline state. Wide angle X-ray diffraction (WAXD) expriments indicate that these PEOs exhibit the same crystal structure as that of pure PEO. Upon crystallization at low undercooling (AT), such as at a crystallization temperature (Tc) of 48°C, differential scanning calorimetry (DSC) results reveal two crystal populations. Small angle X-ray scattering (SAXS) experiments also identifies two different long periods. It is speculated that one of the crystal populations possesses an extended OMC in these two-arm PEOs, and thus, one layer of defects is present in between two neighboring lamellae and the long period is smaller. The second crystal population consists of a once-folded OMC. Two layer defects are thus included in between the neighboring lamellae. The crystals with once-folded OMC represent the more stable form compared to those containing the extended OMC. In varying the linkage from the 1,4- to 1,2-positions at the coupling agents, the once-folded OMC population increases under the same crystallization conditions (18).

In this publication, we attempt to understand the crystallization, melting, and annealing behaviors of these three two-arm PEOs using the combined experimental techniques of time-resolved simultaneous synchrotron WAXD, SAXS, and DSC.

Experimental Section

Materials synthesis and characterization. The synthesis of two-arm PEOs has been described in our previous publications (17,18). In brief, 1,4-, 1,3-, and 1,2-benzene dicarbonyl dichlorides were used as coupling agents in conjunction with a LMW PEO fraction (2220 g/mol) [a,œ-methoxy-hydroxy-poly(ethylene oxide), HO-(CH 2CH 2-0) n-CH 3]. Further fractionation was also performed. Gel permeation chromatography (GPC) experiments using tetrahydrofuran at 30°C were carried out to measure the number average MW (Mn) and polydispersity. The GPC was calibrated using standard linear PEOs over a MW range of 500 to 600,000 g/mol. Fourier transform infrared spectroscopy (FTIR, Mattson Galaxy 5020) measurements were carried out between 500 and 4000 cm"1 in order to identify the end groups and

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the coupling agents of the PEOs. A Knauer vapor pressure osmometer (VPO) was used to determine the Mn in toluene at 40°C. Sucrose octaacetate was used in the same solvent and concentration to calibrate the VPO. Finally, light scattering (LS, Wyatt Dawn F) experiments were conducted to insure the accuracy of the polydispersities by measuring the weight average MWs. Self-diffusion coefficients of these two-arm PEOs in the melt were measured via a pulsed-gradient spin-echo (PGSE) nuclear magnetic resonance (NMR) method (19-21). The principle echo on-resonance without Fourier transform was measured using radio frequency phase-sensitive detection. Its attenuation in the presence of a pair of applied magnetic field gradient pulses was detected. The samples were measured at 60.5, 80.5, and 100.5°C. The PEO critical entanglement MW in a monodisperse melt is 4400 g/mol (22).

Equipment and experiments. DSC (TA2000 system) experiments were carried out to study crystallization, melting, and annealing behaviors of these PEOs. The DSC was calibrated with standard materials. Isothermal crystallization was conducted by quenching the samples from the melt to a preset Tc and held for various crystallization times (tc). In the low arrange, a self-seeding technique was employed for the isothermal crystallization as described previously (4). The crystallized samples were then heated above the melting temperature (Tm) at a rate of 5°C/min. Annealing experiments were conducted after quenching the PEO samples from the melt and held for 30 min at 32°C for isothermal crystallization. The samples were then heated to 50°C and isothermally annealed at that temperature for different periods of times (ta). The melting traces of the annealed crystals were recorded by DSC.

Time-resolved synchrotron WAXD and SAXS experiments were carried out on the synchrotron X-ray beam line X27C of the National Synchrotron Light Source at Brookhaven National Laboratories. The wavelength of the X-ray beam was 0.1307 nm. Isothermal crystallizations were carried out on a customized two-chamber hot stage. Temperature control precision was ±0.5°C. Annealing experiments were conducted using the identical thermal history to that of the DSC experiments. Position sensitive proportional counters were used to record the diffraction and scattering data. The diffraction counter was calibrated using silicon crystals of known size. The scattering counter was calibrated with duck tendon scattering peaks at q values of 0.109 nm"1, 0.22 nm 1, 0.33 nm"1, etc. (q= 4nsin®X, where λ is the wavelength of X -ray radiation). The Lorentz correction was performed by multiplying the intensity / (counts per second) by q1. The relative invariant g'was calculated based on f (I-Ib)q2dq which covers a q range between 0.08 and 2 nm*1 (where l0 is the intensity of PEO liquid scattering obtained using the Porod's law extrapolation) (23).

Results and Discussion

Molecular analyses and characterizations. Following the coupling reactions, there remains a mixture of the linear "parent" and two-arm PEOs in the samples. It is necessary to carefully fractionate the mixture. Table 1 lists the analytical results for the MW and MW distribution as measured by GPC, VPO, and LS upon fractionation.

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The samples possess narrow MW distributions. FTIR results indicate that these two-arm PEOs do not possess an absorption band at 3500 cm"1 which originates from hydrogen bonding and OH stretching vibration. The vibration band of the ketone group present in these PEOs is also observed at 1720 cm"1 (17,18). WAXD patterns of the samples crystallized at different Tcs show that the two-arm and linear PEO crystals possess identical structures. Therefore, the defects do not appear to change the crystal structure of the PEOs (17,18).

Table 1. Molecular characteristics of linear and two-arm PEO fractions

Mn

a Mnb My/My? Ma

d Average length (nm)e

Linear PEO 2200 2220 1.02 — 14.0 1,4-two-arm 4490 4550 1.05 2220 29.0 1,3-two-arm 4500 4550 1.05 2220 29.0 1,2-two-arm 4500 4550 1.05 2220 29.0

a Number average molecular weight from GPC. D Number average molecular weight from VPO. c Polydispersities from GPC, and independently checked via LS by measuring the weight average MWs. d Number average molecular weight of each arm. e The average chain lengths were calculated from the equation / = Mn/d, d = 158.2 (4) for the linear fractions. For the two-arm PEO fractions, the average chain lengths were calculated by doubling the linear fraction length and adding the size of the coupling agent.

Self-diffusion coefficients (Ds) measured at three different temperatures using PGSE-NMR for the linear and two-arm PEOs are shown in Figure 1. The Ds of an a,co-methoxy-poly(ethylene oxide) (MPEO) with a MW of 4250 g/mol are larger (i.e. faster) than those of the two-arm PEO(M a = 2220)s at the same temperature despite both fractions having almost identical overall molecular lengths. Furthermore, the D of the 1,4-two-arm PEO is approximately 15% higher than those of the 1,3- and 1,2-two-arm PEOs. This indicates that the molecular dynamics in the melt may be different due to the locations of the two PEO arms at the coupling agents. The activation energies of the linear and two-arm PEOs can be calculated using the Arrhenius equation. A value of 30 ± 2 kJ/mol was determined for these fractions which is in good agreement with the data reported in our earlier work (20).

Isothermal crystallization behavior of the two-arm PEOs. Isothermal crystal­lization processes of these three two-arm PEOs were monitored by simultaneous measurements of synchrotron WAXD and SAXS as shown in Figures 2a and 2b respectively, for the 1,4-two-arm PEO crystallized at 32°C. The WAXD patterns in Figure 2a demonstrate the development of crystallinity as tc increases. The overall crystallization is complete within approximately 4 min, after which the crystallinity reaches a maximum. The SAXS patterns shown in Figure 2b have taken into account the Lorentz correction. An apparent thinning process can be observed during the onset of crystallization. A broad scattering peak is initially observed, with a maximum corresponding to a long period of 18.6 nm at tc = 1 min. The peak intensity gradually increases with increasing tCt and the peak width at the half-maximum

122

Figure 1. Relationship between the self-diffusion coefficients and temperature for the linear and two-arm PEOs.

123

Figure 2. Set of WAXD (a) and SAXS (b) time resolved synchrotron data for 1,4-two-arm PEO crystallized at 32 X.

124

narrows, indicating that the long period becomes more uniform and the layer correlation improves. During the progression of crystallization, the first-order SAXS maximum shifts to a larger q value and the long period reaches 15.3 nm at tc = 4 min (the apparent thinning process). Within 3 min, the total decrease of the long period is 3.3 nm.

A similar phenomenon can be observed for the 1,3-two-arm PEO crystallized at 32°C. Its long period decreases from 18 to 16.7 nm after tc = 4 min. Therefore, the long period change of the 1,3-two-arm PEO is 1.3 nm, which is less than that of the 1,4-two-arm PEO. In addition, the initial scattering peak in the crystallization is asymmetric with a broad shoulder on the low q side.

Both sets of WAXD and SAXS patterns of the 1,2-two-arm PEO crystallized at 32°C are shown in Figures 3a and 3b, respectively. Once again, the maximum crystallinity is reached within 4 min (Figure 3a). However, unlike the 1,4- and 1,3-two-arm PEOs, the long period of the 1,2-two-arm PEO does not appear to thin. The asymmetric SAXS peak with a long period of 16.4 nm gradually increases in scattering intensity without changing the peak shape (Figure 3b). This peak is broader than those of the 1,4- and 1,3-two-arm PEOs, indicating that the lamellar layer correlation length is lower than the other two-arm PEOs.

Upon increasing Tc to 40°C, the overall crystallization rates of these two-arm PEOs decrease. Based on the WAXD patterns in Figures 4a and 5a for the 1,4- and 1,2-two-arm PEOs, respectively, the time needed for completing the crystallization at 40°C is approximately 10 min. In Figure 4b, the SAXS patterns manifest the apparent thinning process of the 1,4-two-arm PEO. The initial long period is 20.6 nm and it decreases to 16.0 nm after tc = 14 min. For the 1,3-two-arm PEO crystallized at 40°C, the crystallization also starts with a broad SAXS peak. The apparent thinning process can be identified by the long period changing from 19.5 to 18.1 nm within 14 min. Figure 5b shows that the SAXS peak of the 1,2-two-arm PEO is rather broad having a maximum at 18.1 nm. This value remains practically unchanged. In all three two-arm PEOs, the SAXS peak widths at the half-heights at Tc = 40°C are narrower than those corresponding to Tc = 32°C.

Upon further increase in Tc, a self-seeding process must be utilized in order to accelerate the crystallization. Figures 6 and 7 are two sets of WAXD and SAXS patterns of 1,4- and 1,2- two-arm PEOs, respectively, crystallized at 48 °C after self-seeding. In all three two-arm PEOs, the crystallinity plateaus at tc > 30 min (Figures 6a and 7a). During the development of crystallinity, these two-arm PEOs also exhibit an apparent thinning behavior which ceases when the crystallinity reaches its maximum. From Figure 6b, the initial long period of the 1,4-two-arm PEO is found to be 24.0 nm. The broad SAXS peak increases in intensity with increasing tc, whereas the peak position shifts to higher q values. More importantly, this peak gradually evolves into two separate scattering peaks. Although their peak positions are in close proximity, these two peaks can be identified at tc = 8 min. The corresponding long periods of these two peaks decrease slightly with prolonged tc, reaching 19.8 and 17.3 nm after tc = 30 min. A similar observation in the 1,3-two-arm PEO is also observed. The long period at initial crystallization is approximately 26.0 nm. After tc

= 10 min, a separate shoulder on the low q side of the scattering peak becomes

125

i ! I . I . I ι I ι I • I • I . 1 . 1

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

q (nm"1)

Figure 3. Set of WAXD (a) and SAXS (b) time resolved synchrotron data for 1,2-two-arm PEO crystallized at 32 °C.

126

Ι

(a) 1,4-two-arm PEO Τ = 40 °C

17 19 21 23 25

2Θ (degree, λ = 0.1307 nm)

27 29

1,4-two-arm PEO Τ =40 °C

0.0 0.2 0.4 0.6 0.8 1.0 1.2 l.<

q (nm1)

1.6 1.8

Figure 4. Set of WAXD (a) and SAXS (b) time resolved synchrotron data for 1,4-two-arm PEO crystallized at 40 X.

127

1,2-two-arm PEO Τ = 40 °C

1

17 19 21 23 25

2Θ (degree, λ = 0.1307 nm)

Figure 5. Set of WAXD (a) and SAXS (b) time resolved synchrotron data for 1,2-two-arm PEO crystallized at 40 *€.

128

1,4-two-arm PEO C

40.0 min 32.0 min 25.0 min 20.0 min 17.5 min 15.0 min 12.5 min 10.5 min 9.5 min 8.5 min 7.5 min 6.5 min 4.3 min 2.7 min 1.3 min

0.2 0.4 0.6 0.8

qCnm"1)

Figure 6. Set of WAXD (a) and SAXS (b) time resolved synchrotron data for 1,4-two-arm PEO crystallized at 48 °C.

129

(a) 1,2-two-arm PEO Τ = 48 °C

12 14 16 18 20 22 24 26 28 30

2Θ (degree, λ = 0.1307 nm)

0.4 0.6 0.8

q (nm"1)

Figure 7. Set of WAXD (a) and SAXS (b) time resolved synchrotron data for 1,2-two-arm PEO crystallized at 48

130

evident. The long periods of these two peaks reach 20.2 nm and 17.5 nm after tc = 30 min. This observation indicates the existence of two populations of lamellae with different long periods formed at 48°C, corresponding to crystals with the extended and once-folded OMCs (18). In addition, the peak widths at the half-maximum of these individual peaks are very narrow in the 1,4- and 1,3-two-arm PEOs, corresponding to a lamellar layer correlation length of approximately 170 nm. However, the 1,2-two-arm PEO exhibits a single SAXS peak and a relatively broad width at the end of crystallization (Figure 7b). This illustrates that the lamellar crystals of the 1,2-two-arm PEO contain a single dominant long period with a relatively broad size distribution. The initial long period of 26.0 nm shifts towards 20.4 nm after tç — 30 min.

Figures 8a - 8c represent the relationships between the final long periods after completing the crystalliztion and Tcs for the three two-arm PEOs. When Tc > 44°C, the 1,4- and 1,3-two-arm PEOs possess two long periods, e.g., nearly at 20.0 nm and 17.5 nm respectively at 48°C, indicating the coexistence of two separate crystal populations (Figures 8a and 8b). As described previously (18), the two-arm PEO crystals with once-folded OMC possess a thicker long period than that of the extended OMC. The long period of 20.0 nm is therefore be associated with the crystals having the once-folded OMC, and the long period of 17.5 nm is attributed to the crystals with the extended OMC. Furthermore, compared to the long period of 14.8 nm for the extended chain crystals of the linear "parent" PEO, one may estimate the thickness of the configurational defect layer to be approximately 2.5 nm. On the other hand, only one scattering maximum can be recongnized for the 1,2-two-arm PEO crystallized at Tc > 44 °C (Figure 8c). The corresponding long period is almost identical to that of crystals with once-folded OMC of the 1,3-two-arm PEO. This implies that the 1,2-two-arm PEO possesses only one population of crystals which have predominantly the once-folded OMC when crystallized at low AT. At Tc < 38 °C, only one long period appears, ranging between 15.5 and 17.0 nm. Note that this long period is usually thinner than that of the extended or once-folded OMCs. It can be speculated that these crystals contain irregular NIF OMCs. Furthermore, the changes of long periods with respect to Tcs for the 1,2- and 1,3-two-arm PEOs are similar, even though the 1,2-two-arm PEO has a single long period at Tc > 44°C. In summary, the crystallization behavior and lamellar morphology of two-arm PEOs are affected by the types of defect linkages at the center of the two PEO arms.

Melting behavior of the linear and two-arm PEOs. Detailed melting behavior of the 1,4-two-arm PEOs has been discussed in reference 18. Figure 9 shows a set of DSC heating diagrams for the 1,3-two-arm PEO following crystallization at different Tcs. It is evident that at relatively low Tcs (< 38°C), the peak temperatures of melting endotherm are almost constant and the lowest compared to that of crystals grown at Tcs above 38 °C. Between 38 and 44°C, the Tm increases about 2°C, and for Tc > 44°C, two endothermie peaks can be found. Upon further increases in Tc, the Tms increase slightly. Figures 10a - 10c represent the summary of the melting behavior for these two-arm PEOs at different isothermal Tcs. Common features can be noted in the other two PEOs. The crystals formed at Tc < 38°C are speculated to contain NIF

131

Figure 8. Relationships between the long periods at the final stage of crystallization and Tcs for three two-arm PEOs: (a) 1,4-two arm, (b) 1,3-two-arm, and (c) 1,2-two-arm PEOs.

132

1,3-two-arm PEO

Τ =32 °C

45 50 55

Temperature (°C)

60

34 °C

36 °C

38 °C

40 °C

42 °C

44 °C

46 °C

48 °C

65

Figure 9. Set of DSC heating diagrams for 1,3-two-arm PEO at a heating rate of 5 °C/min at different Tcs.

133

OMC. These constant Tms observed may be due to annealing during heating (10,11). With further increases in Tc, the Tm starts to increase. Above Tc = 44°C, two separate melting processes can be observed for the 1,4- and 1,3-two-arm PEOs; the lower Tm

most likely represents the crystals with the extended OMC, and the higher one reprents the once-folded OMC (Figures 10a and 10b). This identification has been confirmed by the results obtained from time-resolved SAXS data.

It is interesting to note that Figures 8a and 8b are remarkably similar to Figures 10a and 10b with respect to the long periods and Tms changing with Tc. The only apparent difference is in the 1,2-two-arm PEO (Figure 10c), wherein, two melting endotherms appear above Tc > 44 °C from DSC, but in Figure 8c, only one long period can be observed. However, the lower temperature endotherm of the 1,2-two-arm PEO possesses less than 25% of the overall heat of fusion. Several possibilities may explain this apparent difference (Figures 8c and 10c). The most likely possibility is that the crystal having the lower Tm consists of a mixture of both OMCs, although it is uncertain whether they represent a eutectic system or a solid solution. As long as the crystals possess mixed OMCs on a nanoscopic scale, only a relatively broad scattering peak can be found in the 1,2-two-arm PEO. Further studies are necessary to understand this observed difference.

Annealing behavior in the two-arm PEOs. It has been found that for the two-arm PEO crystals formed at Tc < 38°C, even for a prolonged tc (such as days), cannot change either their long periods (in SAXS) or apparent Tms (in DSC heating at 5°C/min). The question is whether the crystals formed at high ATs (such as at 32°C) can be annealed at high temperatures (such as at 50°C) to form two separate crystals with different OMCs, similar to observations of isothermal crystallization directly from the melt. The annealing experiments show that after the 1,4-two-arm PEO is completely crystallized at 32°C, heated to 50°C, and annealed for different tas, the Tm increases from 54.6°C to 55.2°C at ta = 120 min. When ta is further increased to 900 min, the Tm increases to 55.4°C. Similar behavior of the annealing process can also be observed in the 1,3- and 1,2-two-arm PEOs. Figure 11 describes the Tm

changes during annealing at different tas. It is evident that the initial increase of Tm is rather quick, followed by a slow development. In the 1,4-two-arm PEO, the increment of Tm for the crystals formed at 32°C before versus after annealing at 900 min at 50°C is 0.8°C (55.4°C versus 54.6°C). For the 1,3- and 1,2-two-arm PEOs, the increments are 1.3°C (56.4°C versus 55.1°C) and 1.9°C (56.8QC versus 54.9°C), respectively. This indicates that annealing at higher temperatures can improve the thermodynamic stability of the crystals (increasing Tm). The annealing effect is more predominant in the 1,2-two-arm PEO compared to the 1,4-two-arm PEO.

Interestingly, the annealing experiments lead to only one melting endotherm at 55.4°C for the 1,4-two-arm PEO. When the crystals are grown at 50°C directly from the melt, two Tms of 54.8°C (the Tm of extended OMC crystals) and 55.8°C (the Tm

of once-folded OMC crystals) are observed. Therefore, the Tm of 55.4°C associated with the annealed crystal is between the Tms of the crystals with the extended and once-folded OMCs. Similar observations can be found in the 1,3-two-arm PEO; 56.4°C for the annealed crystals compared with 55.8°C and 57.3°C for the extended

134

υ ο

ε Η

U ο

57

56

55

54

53

58

57

56

55

54

59 58 57 56 55 54 53

(a)

Ο Ο ο ? ο

ο ο • • • • •

I ι I • I ! 1,4-two-arm PEO

ι I ι 1 ι i ι I ι I ι I ι

(b)

- ο ο ο c

ι I ι I i l l

i

j 1,3-two-arm PEO ι I ι I ι i ι I ι I « L _ i ~

I

- ο ο ο ι

ι I ι I . I ι

ι • ° C

t t I ι I ι

ο — ···· 1,2-two-arm PEO . 1 . 1 . 1 .

30 32 34 36 38 40 42 44 46 48 50 52 T C (°C)

Figure 10. Relationships between Tms and Tcs of three two-arm PEOs: (a) 1,4-two-arm, (b) 1,3-two-arm, and (c) 1,2-two-arm PEOs. The filled symbols are results of peak separation.

135

Figure 11. Relationship between Tms and tas for three two-arm PEOs after crystalli­zation at 32 °C and heated to 50 V for annealing at tas.

136

and once-folded OMC crystals respectively. In the 1,2-two-arm PEO, it is 56.8°C for the annealed crystals compared with 56.6°C and 57.6°C for the crystals grown isothermally at 50°C. The results indicate that the annealed crystals may not possess the purely extended or once-folded OMC on a nanoscopic scale. On the contrary, during annealing the most probable transformation of the NIF crystals formed at Tc = 32°C is to form a mixture containing extended and once-folded OMCs.

The annealing behavior of these two-arm PEOs crystallized at low Tcs is also investigated by time-resolved synchrotron SAXS and WAXD. For example, Figures 12a - 12c describe the long period (Figure 12a), crystallinity (Figure 12b), and relative invariant g'(Figure 12c) changes for the 1,4-two-arm PEO during heating and annealing. The temperature profile of the experiment is also included in Figure 12c. Figure 12a demonstrates an initial long period of approximately 14.5 nm. The long period starts to increase at a 44°C, reaching 16.4 nm at 50°C. Annealing isothermally at 50°C leads to a further increase of the long period to 17.8 nm after ta

= 4 min. Little increase in the long period can be found for prolonged tas. It is noted that 44°C is about 10°C below the peak temperature of the melting endotherm, and is even 4 °C lower than the starting Tm (48.0°C). Similar behavior can be observed in the 1,3- and 1,2-two-arm PEOs.

The heating event clearly involves a thickening process of the original thin NIF long period formed at 32°C, which may be achieved by motion of the chain segments. During the thickening, the defects must diffuse to and concentrate on the crystal surfaces. Either the extended or once-folded OMC may thus form in the crystals (depending on the local free energy barriers of motion). The crystals therefore contain a mixture of both IF OMCs showing a single long period with an intermediate spacing and a relatively broad scattering peak. For instance, the annealed crystals of the 1,4-two-arm PEO result in a long period of 17.8 nm at 50°C, while the long periods are 17.3 nm and 19.8 nm for the extended and once-folded OMC crystals isothermally crystallized at 48°C directly from the melt, respectively. Since the long periods after annealing are closer to that of the extended OMC crystal rather than the once-folded OMC crystal, the annealed crystals most likely possess a predominantly extended OMC. Furthermore, since the thickening starts at 44°C, the Tm$ of the crystals grown at Tc < 38°C observed in DSC should be representative of the already thickened crystals during heating.

The crystallinity (Figure 12b) initially remains constant at 0,79 before it starts to decrease (when the long period starts to increase) dropping to 0.68 at 50°C, which is followed by a slow increase. The development of the relative invariant g'with time can be recognized as a two-step process (Figure 12c); an initial β'increases during heating, reaching a maximum at 50°C, followed by a continuous but gentle decrease. In order to explain the Qf change with time in this annealing process, it should be noted that Q' χ Κ<η2>φ0(1 - φ0), where Κ is a constant, <rp-> is the mean square of the electron density difference between the crystalline and amorphous polymer, and φα is the volume crystallinity. Therefore, dQ'/dt χΚ<η2>(1 -2φ0)άφ0/άΙ. Based on the WAXD results, the weight crystallinity (and therefore, φ€) of each the two-arm PEOs is always higher than 0.5 during heating and annealing. This gives rise to a

137

Figure 12. Real-time long period (a), crystallinity (b), and relative invariant Q' (c) changes for 1,4-two-arm PEO crystallized at 32 °C and annealed at 50 <C for various times, tas.

138

negative (1 - 2φ0) term. During heating from 44°C to 50°C, g'increases and thus, dQ'/dt is positive, which requires that awc/dt be negative, i.e. a decrease in the volume crystallinity with time. Hence, it can be concluded that in this heating process, the long period increase is mainly attributed to a partial crystal melting which increases the thickness of the amorphous layer. After g ' reaches a maximum at 50°C, it gradually decreases during annealing while the long period remains nearly constant. The decrease in g'results in a negative dQ'/dt, which makes das/dt positive, namely, increasing the crystallinity with increasing ta, as evidenced by the WAXD results. Therefore, the crystal lamellar thickness increases in the expanses of a reduction of the amorphous layer thickness.

It is observed that in these two-arm PEOs the original long period increases only after the temperature reaches 44°C during heating. Using DSC, a small endotherm at around 44°C for these two-arm PEOs is observed upon crystallization at 32°C. However, this endotherm is not present in the linear PEO. It is speculated that this small endotherm is associated with the cooperative motion of PEO chains with the conflgurational defects. A small but sudden increase of the d-spacing of the (120) planes in these PEO crystals has been also observed in the vicinity of 44°C (24). Further studies are necessary to understand this cooperative motion in the crystals.

Conclusions

In summary, three two-arm PEOs with equal arm lengths (Ma = 2220) have been synthesized. Due to different reaction sites in the coupling agents, these two arms at the center of PEO molecules form angles of 180°, 120°, and 60°, respectively. The different conflgurational defects not only substantially affect the PEO chain motion in the melt, leading to different self-diffusion coefficients, but also influences the two-arm arrangements during crystallization and subsequent melting and annealing. The two-arm PEOs crystallized at Tc < 38°C possess the NIF OMC. Little annealing effect on the crystals can be observed in this temperature region even after prolonged tas. Upon crystallization at relatively high Tc > 44°C, an apparent thinning process of the long period is observed. This process results in two separate crystal populations with different OMCs, namely, the extended and once-folded OMCs for both the 1,4-and 1,3-two-arm PEOs. Only one relatively broad scattering peak is observed for the 1,2-two-arm PEO. It has also been found that the crystals containing the pure once-folded OMC possesses a higher Tm than that of the extended OMC. The changes of Tm with Tc show remarkable similarities compared to those of the long period with Tc. Annealing experiments show that for the crystals formed at 32°C, their stability can be improved during heating and annealing. This process is associated with partial crystal melting, long period increasing, and recrystallization. However, the annealed crystals can only possess a mixture of two IF OMCs.

Acknowledgments. The work was supported by the NSF DMR (96-17030). Research was carried out in part at the National Synchrotron Light Source at Brookhaven National Laboratories.

139

Literature Cited

1. Arlie, J. P.; Spegt, P. Α.; Skoulios, A. E. Makromol. Chem. 1966, 99, 170. 2. Arlie, J. P.; Spegt, P. Α.; Skoulios, A. E. Makromol. Chem. 1967, 104, 212. 3. Spegt, P. Makromol. Chem. 1970, 139, 139. 4. Kovacs, A. J.; Gonthier, A. Colloid & Polym. Sci. 1972, 250, 530. 5. Kovacs, A. J.; Gonthier, Α.; Straupe, C. J. Polym. Sci. Polym. Symp. 1975, 50,

283. 6. Kovacs, A. J.; Straupe, C.; Gonthier, A. J. Polym. Sci. Polym. Symp. 1977, 59,

31. 7. Kovacs, A. J.; Straupe, C. J. Crystal Growth 1980, 48, 210. 8. Kovacs, A. J.; Straupe, C. Faraday Discuss. Chem. Soc. 1979, 68, 225. 9. Cheng, S. Z. D.; Zhang, A.-Q.; Chen, J.-H.; J. Polym. Sci. Polym. Polym. Lett.

Ed. 1990, 28, 233. 10. Cheng, S. Z. D.; Zhang, A.-Q.; Chen, J.-H.; Heberer, D. P. J. Polym. Sci. Polym.

Phys. Ed. 1991, 29, 287. 11. Cheng, S. Z. D.; Chen, J.-H.; Zhang, A.-Q.; Heberer, D. P. J. Polym. Sci. Polym.

Phys. Ed. 1991, 29, 299. 12. Cheng, S. Z. D.; Chen, J.-H. J. Polym. Sci. Polym. Phys. Ed. 1991, 29, 311. 13. Cheng, S. Z. D.; Zhang, A.-Q.; Barley, J. S.; Chen, J.-H.; Habenschuss, Α.;

Zschack, P. R. Macromolecules 1991, 24, 3937. 14. Cheng, S. Z. D.; Chen, J.-H; Zhang, A.-Q.; Barley, J. S.; Habenschuss, Α.;

Zschack, P. R. Polymer 1992, 33, 1140. 15. Cheng, S. Z. D.; Chen, J.-H.; Barley, J. S.; Zhang, A.-Q.; Habenschuss, Α.;

Zschack, P. R. Macromolecules 1992, 25, 1453. 16. Cheng, S. Z. D.; Wu, S. S.; Chen, J.-H.; Zhuo, Q.; Quirk, R. P.; von Meerwall, E.

D.; Hsiao, B. S.; Habenschuss, Α.; Zschack, P. R. Macromolecules 1993, 26, 5105.

17. Lee, S.-W.; Chen. E.; Zhang, Α.; Yoon, Y.; Moon, B. S.; Lee, S.; Harris, F. W.; von Meerwall, E. D.; Hsiao, B. S.; Verma, R.; Lando, J. B. Macromolecules 1996, 29, 8816.

18. Chen. E.; Lee, S.-W.; Zhang, Α.; Moon, B. S.; Lee, S.; Harris, F. W.; Cheng, S. Z. D.; Hsiao, B. S.; Yei, F. Polymer 1999, in press.

19. von Meerwall, E. D. Adv. Polym. Sci. 1983, 54, 1; see also, Rubber Chem. Tech. 1985, 58, 527.

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Akron, Akron, Ohio, 1998.

Chapter 9

Investigating the Mechanisms of Polymer Crystallization by SAXS Experiments

G. Hauser1, J. Schmidtke1, G. Strobl1, and T. Thurn-Albrecht2

1Fakultät für Physik der Albert-Ludwigs-Universität, 79104 Freiburg, Germany

2Department of Polymer Science and Engineering, University of Massachusetts, Amherst, MA 01003

The importance of SAXS experiments for studies of the basic mecha­nisms of polymer crystallization and melting is demonstrated with four ex­amples. (1) Time and temperature dependent measurements on s-PP dur­ing isothermal crystallization and melting yield the dependencies between crystallization temperature, crystal thickness, crystallization rate and melt­ing point. Results show that the lamellar crystallites form in a two-step process. (2) Time dependent experiments on crystallizing PE enable the crystal thickening to be followed from the beginning and confirm the log­arithmic law. (3) Temperature dependent studies on PE during cooling or heating prove that secondary crystallization here is based on a surface crystallization and melting mechanism. (4) The cross-hatching in i-PP has a characteristic signature in the SAXS-curve. Data evaluation yields the total length per unit volume of the edges along the intersections of the crystallites.

1 Introduction Crystallization of polymers generally leads to the formation of layer-like crys­tallites with typical thicknesses in the range of 3-50 nm. Normally they are arranged in stacks, being separated by amorphous layers. Electron microscopy and small angle X-ray scattering are the primary tools used for studies of these nano-structures. Both in combination are required for a reliable structure deter­mination. Electron microscopy directly shows the basic structural elements, and SAXS has to be employed for quantitative measurements, and in particular, when studying the changes with time, for example during a crystallization, or on cooling and heating. This contribution deals with some recent SAXS studies carried out

140 © 2000 American Chemical Society

141

on isotactic and syndiotactic polypropylenes and polyethylene. They provided new insights into the kinetics and mechanisms of structure change. Here, we present the main results of the experiments in their context; the full description can be found elsewhere, as indicated in the text.

2 SAXS-data analysis The majority of partially crystalline systems are set up of stacks of thin laterally extended crystallites. Here, the scattering intensity can be related to the one-dimensional electron density correlation function K(z) defined as

It can be directly obtained by a Fourier transformation of the scattering intensity. If we use the scattering cross-section per unit volume Σ(q),Κ(z) follows as

q denotes the scattering vector, being related to the Bragg-scattering angle vB by

(r e: classical electron radius) [1]. In the case of partially crystalline polymers, a trajectory along the surface

normal passes through amorphous regions with an electron density ρ β ) & and crys­tallites with a core density /9 e , c - As shown by Ruland [2], for such a layer system the second derivative of the correlation function, gives the distribution of distances between interfaces, in the form

The expression between the brackets is set up of a series of distribution func­tions, whereby the subscripts indicate which phases, amorphous and crystalline ones, are to be traversed while going from one interface to the other. Of main importance are the first three contributions. They give the distributions of the thickness of the amorphous and the crystalline layers respectively (h&, hc) and of the sum of both (h&c) which is identical with the long spacing L . Rather than by calculating the second derivative of the correlation function, K" (z) can also be deduced directly from the measuring data, by use of

142

If the boundaries between the crystalline and the amorphous regions are sharp within the resolution limit of SAXS experiments, the asymptotic behavior of in the limit of large scattering angles is given by Porod's law

Hereby Ρ denotes the Porod coefficient which is directly related to the specific in­ternal surface O a c (the area per unit volume of the interface separating crystalline and amorphous regions), by

3 Instrumentation The SAXS experiments discussed in the following were conducted at a conven­tional X-ray source with the aid of a Kratky camera, which was equipped with a temperature controlled sample holder. The changes of the structure during isothermal crystallization and the melting on heating were followed measuring the scattering curves with a position sensitive metal wire detector. A few min­utes counting time were usually sufficient for a curve registration. Deconvolution of the slit-smeared data was achieved applying an algorithm developed in our group [3].

4 Kinetics of primary crystallization

4.1 The s-PP scenario Experiments were carried out on a syndiotactic polypropylene [4] and two sam­ples of syndiotactic poly(propene-eo-octene) [5], synthesized by S. Jungling in the Institute of Macromolecular Chemistry of our university using a metallocene catalyst. The chemical properties of the two samples are given in Table 1 (co-unit contents were determined by NMR spectroscopy).

Table 1: s-Polypropylene and s-poly(propene-co-octene). Properties of samples.

sample octene-units meso-diads M n M w / M n

% (weight) %(mole) % s-PP 0 0 3 104.000 1.7 s-P(P-co-0)4 4 1.7 (3) 73.000 2.1 s-P(P-eo-0)15 15 6.4 (3) 94.000 1.7

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Figure 1: Functions Κ" (ζ) giving the distribution of crystal thicknesses, ob­tained by time- and temperature-dependent SAXS experiments: Changes during isothermal crystallization of s-P(P-co-0)4 at 116 °C (top left) and of s-P(P-co-0)15 at 70 °C (bottom left) and during the subsequent melting processes (right hand sides).

The crystallite formation during isothermal crystallization was studied in time-dependent SAXS experiments. Subsequent to the isothermal crystallization the melting was monitored in temperature dependent SAXS measurements up to the melting point. Fig. 1 displays as an example the curves K"(z) for s-P(P-eo-0)4 and s-P(P-co-0)15 during and subsequent to isothermal crystallization processes.

The formation and melting of the crystallites shows up in the changing height of the ridge dominating Kn(z). The location of the ridge gives the crystal thick­ness. As is observed, this thickness remains constant through both the time of isothermal crystallization and the subsequent melting. Curves indicate thick­nesses of 5.7 nm and 3.5 nm for s-P(P-co-0)4 and s-P(P-co-0)15 respectively. Observations were similar for s-PP and all crystallization temperatures; always the thickness remained constant until melting. From the measurements we ob­tained for the three samples the dependencies of the crystallite thickness (dc), the (Avrami-) time of crystallization (r) and the melting peak (Tf) on the crys­tallization temperature (Tc). The relationships are displayed in Fig. 3, showing τ as a function of Tc and Fig. 2, which depicts the relations between T c , dc and Tf in the form suggested by the Gibbs-Thomson relation, Tf and Tc versus d~l.

144

Figure 2: s-PP and s-P(P-co-O): Relations between the inverse crystallite thick­ness, the crystallization temperature and the melting peak. The dashed line represents the Tf (d~1 )~dependence of a perfect s-PP as obtained by an extrapo­lation procedure. The arrows indicate the limiting temperatures with zero growth rate.

10 6 7 Ί • 1 • 1 « 1 1 1 1 Γ s-P(P-co-0)15

10s i · s-P(P-co-0)4 1 A s-PP

104-

i o 2 - i

101

10ϋ j , J , , , , , , , 1—— 40 60 80 100 120 140 160

r c[°C]

Figure 3: s-PP and s-P(P-co-O): Avrami times of crystallization in dependence on the crystallization temperature.

145

Ο 10 20 30 40 50 z/nm

Figure 4: PE: Interface distribution function at the beginning and the end of an isothermal crystallization at 121 °C.

The outcome can be summarized as follows. With increasing content of non-cry stallizable units (octene-units or meso-diads) one observes, as expected, a systematic shift of the melting points to lower temperatures and similar shifts of the crystallization time versus temperature curves, but surprisingly, no effect at all on the crystal thickness. The thicknesses of all three samples show a common temperature dependence, being inversely proportional to the supercooling below the equilibrium melting point of perfect syndiotactic polypropylene. The latter is located at 196 °C, as determined by an extrapolation based on measured melting points. Crystal thicknesses (being unaffected by the presence of non-cry stallizable units) and growth rates (being strongly affected) are independent properties.

Results may be understood as indicating that crystallization from the melt takes place in two steps. The first step is the formation of 'native' crystallites which, as shown by the slope of the 'crystallization' line Tc versus d" 1 , have a high surface free energy. They change into the final form by structural relaxation processes associated with the surfaces only (the line starts from the equilibrium melting point!). The difference between the temperatures of crystallization and melting is due to this stabilization process, which leaves the crystallite thickness unchanged. There is a further conclusion: Data demonstrate that the popu­lar Hoffman-Weeks plot when applied to random copolymers does not yield the respective equilibrium melting points; it can only be used for perfect homopoly-mers.

146

OJr

0.8

0.7

0.6

0.5

0.4

0.3

0.2

04

°4r

• ο.-' ° α ο · "

10*

o.o* O'O

• •

Te»121*C • Porod constant Ρ Ο Lamellar thicknee* dQ

10» V m i n

I T

Figure 5: Evolution with time of the crystal thickness and the inner surface.

4.2 PE: Kinetics of crystal thickening Different from s-PP, crystallites of polyethylene thicken with time. This was clearly demonstrated by time-dependent SAXS experiments carried out during an isothermal crystallization [6]. Fig. 4 shows a typical result, in a comparison of the interface distribution function K"(z) obtained at the beginning and the final stage of the crystallization process.

At the beginning Kn(z) is dominated by the crystallite contribution (dc), which is indicative for isolated crystallites. During further development the con­tribution associated with the amorphous layers (da) grows and finally reaches the same amplitude as that of the crystallites. Simultaneously the latter is shifted to higher values of z. Observations thus show the change from individual lamellae to the final stack which occurs together with a crystal thickening. For the thick­ening we found, as expected, a logarithmic time dependence. Data are given in Fig. 5, together with the time dependence of the Porod coefficient Ρ Oac-

5 Mechanisms of secondary crystallization 5.1 PE: Evidence for surface crystallization and melting Cooling a sample after completion of an isothermal crystallization at elevated temperatures generally results in a further increase of the crystallinity. In the

147

25r

Ό 5 10 15 20 25 30 35 40 z/nm

Figure 6: PE: Interface distribution functions obtained after an isothermal crys­tallization at 124 °C and a subsequent cooling to 78 °C.

148

-δ Ο 10 20 30 40 50

z/nm

Figure 8: ΡΕ: Interface distribution functions at T- 88 °C obtained for samples crystallized at different temperatures.

discussion of the physical processes contributing to this 'secondary crystallization' two mechanisms are considered. First, it can be connected with the formation of new crystallites, being inserted into the intervening spaces, and second, a reversible thickening of the existing lamellae can be envisaged. Carrying out SAXS experiments, a discrimination is possible. Fig. 6 shows a result which is typical for linear polyethylene [7].

Interface distribution functions were obtained after an isothermal crystalliza­tion at 124 °C and a subsequent cooling to 78 °C. The peak at low ζ is due to the amorphous layers, the other one relates to the crystallites. Cooling leads to a shift of the amorphous and crystalline contribution to lower and higher values respectively. This behaviour is indicative for a surface crystallization, here asso­ciated with a decrease of da from 10 to 5nm and a corresponding increase of dc. Fig. 7 depicts the dependence of da on temperature.

Further experiments demonstrated that this dependence is a unique one, da(T) being independent of the thermal history. Fig. 8 demonstrates this fact. The interface distribution functions of three samples, crystallized at different T c 's and subsequently cooled to 88 °C, show clearly different long periods and different thicknesses of the crystalline layer, whereas the contributions of the amorphous layers are identical.

The dependence da(T) can be quantitatively explained by a model which accounts for the entanglements in the amorphous phase and the change in their

149

Figure 9: i-PP: Scattering curves I(s)sA (s = q/4n) measured at 153 °C and 31 °C.

150

density with da. Surface crystallization and melting can be understood as being driven by the changing entropy of the subchains between the entanglement points.

5.2 i-PP: Formation of the cross-hatched structure Secondary crystallization in isotactic polypropylene occurs in a peculiar way. On cooling additional crystallites are formed, however, many of them are oriented oblique to the primary lamellae rather than parallel as is normally observed. As explained by Lotz and Wittmann [8], this 'cross-hatching' is caused by an epitaxial growth mechanism. If cross-hatching occurs the one-dimensional model used so far in the analysis of SAXS data becomes invalid. There is a simple check which shows the deviations. For a stack of crystallites one expects

K"(z) = 0 since 0 = /ia(0) = hc(0) = hAC(0) etc.

(compare Eq.(4)) and therefore, according to Eq.(5)

oo

0 = / [ l i m q4I(q)-q4I(q)]dq (8) 0

Hence, I(q)q4 must fluctuate about the average given by the limiting value lim(g -> oo)qAI(q). This is indeed observed for P E and s-PP at all temperatures. For i-PP the condition is fulfilled during the primary isothermal crystallization, but then, on cooling, it is violated. Fig. 9 shows the result of a measurement [9],

Cross-hatching produces 'edges' along the lines where the oblique secondary lamellae touch the primary crystallites. Exactly these edges can be directly re­lated to the observed deviation. A theoretical analysis gives the following result

oo

f[ lim q*I(q) - q4I(q)]dq = 2nL8Ap2w{a) (9) J q-*oo ο

Here, the parameter Ls denotes the total length per unit volume of edges at the boundaries of the inner surfaces, i.e. the 'specific inner edge length'; w(a) is a function depending on the angle a enclosed by the two surfaces. Evaluation of the SAXS data obtained for i-PP during cooling yielded the temperature depen­dence of Lsw(a) displayed in Fig. 10. Obviously secondary crystallization here is associated with a growing number of edges, simultaneous with the increases of the crystallinity and the specific inner surface.

For the given cross-hatched structure it is no longer possible to employ the one-dimensional electron density correlation function and interface distance dis­tribution function. Still applicable, because generally valid for two-phase systems, are the Porod-law (eq.(6)) and the equation for the 'invariant'

oo

Φ„{ΐ - ΦΜ = 4τΛΐ / 4 7 r« 2ziiïb (10)

151

Figure 10: i-PP: Temperature dependence of the specific length of edges.

where φ0 describes the crystallinity. They yield the specific inner surface 08

and </>c, provided that the electron density difference Ap% or the overall electron density decrease <t>cApe is known. A dilatometric measurement can give these lacking quantities.

References [1] G. Strobl. The Physics of Polymers, page 408. Springer, 1997.

[2] W. Ruland. Colloid Polym.Sci., 255:417, 1977.

[3] G. Strobl. Acta Crystallogr., A26:367, 1970.

[4] J. Schmidtke, G. Strobl, and T. Thurn-Albrecht. Macromolecules, 30:5804, 1997.

[5] G. Hauser, J. Schmidtke, and G. Strobl. Macromolecules, 31:6250, 1998.

[6] T. Albrecht and G.R. Strobl. Macromolecules, 29:783, 1996.

[7] T. Albrecht and G.R. Strobl. Macromolecules, 28:5827, 1995.

[8] B. Lotz and J.C. Wittmann. J.Polym.Sci., Polym.Phys.Ed., 24:1541, 1986.

[9] T. Albrecht and G. Strobl. Macromolecules, 28:5267, 1995.

Chapter 10

Simultaneous In-Situ SAXS and WAXS Study of Crystallization and Melting Behavior

of Metallocene Isotactic Poly(propylene) Patrick S. Dai1, Peggy Cebe1,5, Malcolm Capel2,

Rufina G. Alamo3, and Leo Mandelkern4

1Department of Physics and Astronomy, Tufts University, Medford, MA 02155 2Department of Biology, Brookhaven National Laboratory, Upton, NY 11973

3Department of Chemical Engineering, Florida Agricultural and Mechanical and Florida State University, Tallahassee, FL 32310

4Institute of Molecular Biophysics and Department of Chemistry, Florida State University, Tallahassee, FL 32306

The isothermal crystallization and subsequent melting behavior of a metallocene isotactic poly(propylene) (m-iPP) was studied by simultaneous in-situ wide-angle X-ray scattering (WAXS) and small-angle X-ray scattering (SAXS), and differential scanning calorimetry (DSC). The m-iPP chosen was one which is known to produce large amounts of the γ modification under normal isothermal crystallization conditions. Both DSC and WAXS data show that during crystallization at 117°C, α and γ modifications appear at about the same time. Thermal analysis by immediate rescan after partial crystallization shows that m-iPP exhibits dual melting endotherms. WAXS scans show that during melting, the γ modification melts first at lower temperature, followed by α modification at higher temperature. Once γ crystals begin to melt, they do not undergo recrystallization, nor do they convert into α phase crystals. We also report changes in the SAXS parameters corresponding to the events observed in DSC and WAXS. Systematic changes in the scattering invariant and Bragg long period are seen during isothermal crystallization and melting of m-iPP.

5Corresponding author.

152 © 2000 American Chemical Society

153

Introduction

Isotactic poly (propylene), iPP, exists in several different crystal structures, depending upon the packing of the helical chains [1-6]. The three modifications are monoclinic (a) [2], trigonal (β) [3], and orthorhombic (γ) [4-6]. The formation of these modifications depends on several factors, such as the thermal treatment conditions, mechanical conditions, the molecular weight, and content of molecular chain defects. Of all the structures, the α modification is the most common, observed in melt crystallization at atmospheric pressure, in commercial-type poly(propylenes).

The β and γ modifications in Ziegler-Natta synthesized iPP usually can only be formed under highly specific conditions. The β modification usually forms under conditions of high cooling rates [7,8], high crystallization temperature [8,9], or stress applied under melt conditions [10], and only in very small amounts. In the presence of selective β-nucleating agents [11-13], the content of β can be significantly increased. The β and α modifications may occur together forming complex spherulitic structures, and at certain temperatures the growth rate of β is larger than that of α [14]. The γ modification occurs even more rarely, but may form in degraded, low molecular weight iPP, or under high pressure conditions [15-18]. The relevance of the γ phase of iPP is that it is "the first and so far unique example of a polymer structure with non-parallel chain stems."[l]

Here we report a study of the isothermal melt crystallization and subsequent melting of an isotactic poly(propylene) (m-iPP), synthesized with metallocene-type catalysts [19]. A material with relatively low isotacticity has been chosen for this study. Characteristics of this material, and several prior studies which include this material, have been reported [20-23]. In this m-iPP, the γ modification occurrence does not require any special crystallization conditions as previously found for conventional iPP, and it can form a significant fraction of the total crystal population [21,23]. Formation of γ-iPP may be attributed to the existence of high contents of stereo and regio defects in the chain [23]. α modification is favored by rapid cooling and low crystallization temperature, while γ is favored by higher crystallization temperature. This m-iPP offers the chance to study the crystallization and melting behaviour of γ phase crystals developed under ambient pressure conditions, in an undegraded material. Using simultaneous SAXS and WAXS we follow structure development during isothermal crystallization and subsequent melting.

SAXS is a widely used method for the investigation of lamellar structure in the two phase systems. To obtain structural parameters, such as the average crystal separation and crystal thickness, the one dimensional electron density correlation function is often used. In syndiotactic poly(propylene) the one dimensional model calculation [24] can be applied since the amorphous phase and crystalline phase form one-dimensional stacks of crystalline lamellae. But in the case of monoclinic iPP, electron microscopy reveals the existence of unique cross-hatched lamellar structure [25-30], and the applicability of the one dimensional model has been questioned by Albrecht and Strobl [31]. These researchers used SAXS and dilatometry to study structure development in PP, and presented a scheme to check for the failure of the

154

one dimensional model. In the present work, we calculate SAXS parameters that depend only on the position of the intensity maximum, reserving the correlation function analysis for a later publication [32].

The origin of the multiple endothermic response in iPP is also an important issue. Contributions to multiple endotherms may arise for many different reasons including melting and recrystallization during DSC scanning [33-35], different levels of perfection, such as primary and secondary crystals [36,37], or different crystal modifications [30,35]. In iPP with a low concentration of defects in the chain, the double melting of monoclinic crystals has also been directly associated with the presence of cross-hatching [38]. Here, we show that the α and γ phase melt at quite different temperatures, and create the dual endothermic response seen in this m-iPP. Contrary to previous findings [39] once the γ crystals have melted, they do not recrystallize and they do not transform into α phase crystals.

Experimental Section

The metallocene isotactic poly(propylene) (m-iPP) used in this study is an experimental product of Hoechst. Characterization of the material shows that the fractional content of isotactic pentads (mmmm) is low at 0.908 mol-%, the M w is 335,500 g/mol, and the polydispersity is 2.3 [20]. Defect content was assessed by 1 3 C-NMR [20] and the stereo and regio defects are 1.68mol% and 0.67mol%, respectively. The material, received as pellets, was formed into films by compression molding at 200°C, then was quenched to room temperature in cold water.

Differential scanning calorimetry (DSC) study was carried out on the TA instruments model 2920-DSC. The heat flow and temperature were calibrated using Indium as a standard. Nitrogen was used as a protection gas (30 ml/min) and no thermal degradation was detected. Thin films were encapsulated in A l pans and the sample mass was about 9 mg. The m-iPP was melted at 200°C for 1 minute, then cooled to 117°C at 10C°/min for isothermal crystallization. The crystallization time at 117°C varied from 1 to 40 minutes, and the sample was immediately re-heated at 5°C/min (without cooling) to observe the development of melting endotherms after partially crystallizing the material.

Real-time small angle X-ray scattering (SAXS) and wide angle X-ray scattering (WAXS) were performed at beam line X12B of National Synchrotron Light Source at Brookhaven National Laboratory. Monochromatic X-radiation with a wavelength λ = 1.54Å was used. The sample was located inside the Mettler FP80 hot stage between two layers of Kapton™ tape. The SAXS data were collected with a two-dimensional position sensitive histogramming detector. The sample to detector distance was 180.0 cm. SAXS data were taken continuously during the experiment, and each scan was of 20 seconds or 60 seconds duration.

Al l SAXS data were first corrected for absorption, beam line intensity fluctuation and background. Then the Lorentz corrected intensity, I(s)s2 (where s=2sinθ/λ),

155

passed a low pass zero-phase digital filter, which was used to eliminate the high frequency noise (comprising more than 90% of the frequency distribution). The filter was run in both forward and reverse directions, to ensure there was no phase shift introduced in the process. The filtered Lorentz corrected intensity was smoothed by an area conserving four-point smoothing function, and finally interpolated by Fourier transformation methods, and extrapolated to scattering vector s = oo by Porod's law.

In this paper, we calculate quantities which are independent of the one dimensional model assumption. Here, the scattering invariant, Q, is found from:

Q is also proportional to:

where x0 is the linear stack crystallinity, x5 is the spherulite volume fraction, and Δρ is the electron density difference between crystals and amorphous phase. The average Bragg spacing, L B , is found from the peak of the Lorentz corrected intensity.

The WAXS data were collected with a Braun 7cm one-dimensional position sensitive wire detector. The detector operated at 3 kV, with Argon/Methane (90/10) gas flowing at 1 ml/min. The d-spacings were calibrated by reference to NaCl and KC1 powders. The 20 angular range covered by the wide angle detector was 9.4° to 41.2°. Scans were collected for 20 seconds or 60 seconds simultaneously with the SAXS scans. Al l WAXS data were first corrected for background. The resulting WAXS data contain high frequency noise, especially during the initial stage of isothermal crystallization, when the crystal content is low. We use a low pass zero-phase digital filter to remove the high frequency noise. Finally, for calculation of the relative area under the α and y peaks, the amorphous scattering curve from the completely melted state was scaled and subtracted from the WAXS data.

In the WAXS diffractograms of iPP [18], many peaks of α, β, and γ crystals are in similar 20 locations. However, each modification has a distinctive reflection peak, which is well defined in our experiment. The α and γ modifications are distinguished by their own characteristic scattering angle 20 and Miller indices (hkl), at 18.5° (130) for a, and 20.2°C (117) for y. No β phase was observed in our study. To quantify the relative amount of α and y crystals, we calculate the area under the characteristic peak for the i - crystal type, Sj (i=l, 2 for α and y, respectively). The ratio:

reflects the relative proportion of the ill* modification. This calculation was performed for both the WAXS peak areas, and the endothermic heat flow areas. The heat of fusion for the α and y forms have been reported to be similar [40].

156

Results and Discussion

DSC and WAXS Results

Compared to the conventional Ziegler-Natta synthesized iPP, the m-iPP used in this study has a lower crystallization temperature and a low melting temperature. Figure 1 shows the DSC exothermic heat flow during isothermal crystallization at 117°C. At this temperature, the crystals develop fairly quickly, with a crystallization half-time of about 4 minutes. Little exothermic heat flow is seen past 15 minutes.

Figure 2a shows the DSC melting curves, immediately after crystallization at 117°C for 1, 2, 3, 4 and 5 minutes. In the case of the 1 minute scan, only one very tiny endotherm at T u = 142°C is observed. In the case of the 2 minutes scan, two endotherms appear: one is at the same position of T u, and the other appears at lower temperature Tj = 132°C. The curve is nearly flat, and its area compared to the higher temperature one is quite small. For the 3, 4 and 5 minutes scans, there are clearly two growing endotherms. The locations of the first and second endotherms are in about the same positions, but the ratio of area of the lower Tj endotherm to that of the higher T u endotherm is significantly larger than in the case of 2 minutes.

When the crystallization time is prolonged beyond the half time to 10, 20 and 40 minutes, as shown in Figure 2b, the double endotherms are barely changed. Both melting temperatures are shifted in the higher temperature direction. This suggests that there are two populations of crystals developed in the m-iPP sample during isothermal crystallization. At the initial stage of crystallization, for the times less than the half time, the two populations grow nearly simultaneously. When the crystallization continues beyond the half-time, both populations remain relatively stable, and the ratio of endothermic peak heights is nearly constant.

Using the data from Figures 2a,b, we calculate the areas underneath the lower and upper endothermic peaks. A flat baseline was drawn underneath the two peaks, and the total area calculated. The peaks were then simply divided by a straight line through the valley between the peaks. The relative area under the lower peak is shown in Figure 3 (solid squares) as a function of original crystallization time. The crystal population forming the lower endotherm, which later on will be shown to be γ phase, increases rapidly at first, and then levels off after 10 min. The γ phase at the end of crystallization represents 0.65 of the total.

To verify the origin of these two endotherms, we must look at the WAXS data. Figure 4 is a sequence of WAXS intensity curves taken during the initial 8 minutes of isothermal crystallization at 117°C. Each scan is taken for 1 minute. The distinctive scattering peak positions for α and γ phase are shown by arrows. Both types of crystals begin to grow within the first 2 minutes. Then follows the rapid growth of γ crystals. By the end of 4 minutes, the γ peak is nearly as high as α peak, and by the end of 5 minutes, the γ peak becomes even higher than the α peak. The small peak at 16° is a combination of a γ reflection and a small amount of noise. At the initial stage, the amount of α crystals is significantly larger than the γ crystals. As time goes on, the γ crystals catch up, and eventually form an even larger amount of

t exo

H — n - r— • • 1 · 1—' ' 1 ' 1 0 5 10 15 20 25 30

Time (min)

Figure 1. DSC exothermic heat flow of m-iPP during isothermal crystallization 117°C.

158

Figure 2. DSC heat flow melting endotherms, immediately after m-iPP crystallized at 117°C for the times indicated. Heating rate of 5°C/min. a.) 1, 2, 3, 4 and 5 min.; b.) 5, 10, 20 and 40 min.

159

0.7

0.6 i

0.5

0.4

0.3

0.2 i

0.1

o o"

o 0 0 o o •

o •

• o

•

o

•

o

•

•

o WAXS • DSC

1 3 10

1 20 30

1 40

Crystallization Time (min)

Figure 3. The time development of γ crystals of m-iPP during isothermal crystallization at 117°C from equation 3. Relative γ fraction using data taken from area of the immediate rescan endotherms shown in Figure 2 ( • ), and from area under the WAXS scattering curves ( Ο ).

160

31 1 ι ι ι I

10 15 20 25 30 35 Scattering Angle 2Θ (°)

Figure 4. WAXS intensity vs. scattering angle taken during the initial eight minutes of m-iPP isothermal crystallization at 117°C.

161

crystals. Using equation 3, the relative area under the γ characteristic WAXS peak is plotted in Figure 3 (open circles) where it may be directly compared to the lower endothermic peak area. The agreement between the two data sets is very close, though the similarity of time scales is a qualitative one.

Figure 5a is a sequence of WAXS intensity curves taken during melting, in the temperature range from 121°C to 140°C. After isothermal crystallization at 117°C for 60 minutes, we quenched the m-iPP sample to room temperature, and then heated at l°C/min from 100°C to 170°C. We took each scan for one minute, so the scans in Figure 5a are separated by 1°C intervals. The crystal scattering intensity begins to drop at about 124°C, and this trend continues until finally γ completely disappeared by 140°C. Even after complete γ melting, a small noise peak remains in the WAXS spectrum at 16°. The intensity of α crystals remains nearly constant within this temperature range. But the subsequent melting of α crystals happens at a faster pace, as indicated in Figure 5b which shows the WAXS intensity curves in the temperature range of 141 °C to 150°C. The α crystal intensity begins to drop at 141°C, and by 148°C, has completely disappeared.

Comparing the above DSC and WAXS observations, during the initial isothermal crystallization at 117°C, the α and γ crystals seem to form simultaneously. During subsequent melting, the γ crystals melt first at Ti, and the α crystals melt second at T u. There is no recrystallization of γ crystals, nor do they convert into α crystals, once they have melted, as indicated by the constant intensity of the α phase reflection during the melting of the γ phase.

SAXS Results

Figure 6a shows the time development of the scattering invariant Q and Bragg long spacing L B during isothermal crystallization. During the initial stage of crystallization, the invariant develops very fast, and after 6 minutes, the increase becomes smaller, and then holds steady for the remainder of crystallization. The long spacing L B decreases from an initial value of 20.9 nm, then decreases steadily with time at a much slower pace. After 60 minutes, L B is reduced to 17.1 nm.

Figure 6b shows the temperature variation of the scattering invariant Q and long spacing L B during subsequent melting after crystallization. In this test, the sample was first quenched from 117°C to room temperature and then reheated at l°C/min. From 100°C to about 124°C, both Q and L B increase together. The change in long period is far greater than can be accounted for on the basis of thermal expansion. The linear coefficient of thermal expansion of PP is given as 14.6 χ 10*5/°C (from 30-60°C) [41] while here the change in L B is 5.3 χ 10"3/°C (from 100°C to 124°C).

The changes in Q and L B can be explained on the basis of crystals perfecting and/or melting during heating. The quenched sample contains a small population of very imperfect crystals which can become perfected through mechanisms such as melt-recrystallization, fold surface smoothing, or rejection of defects from the crystal. These would tend to increase the electron density difference between the crystal

162

A

Scattering Angle 2Θ (°) Scattering Angle 2Θ (°)

Figure 5. WAXS intensity vs. scattering angle taken during melting of m-iPP after isothermal crystallization at 117°C for 60 minutes. Heating rate of 1 °C/min. a.) 121°C to 140°C; b.) 141°C to 150°C

163

Figure 6. Scattering invariant, Q, and Bragg long period, L B , of m-iPP: a.) vs. time during isothermal crystallization at 117°C; b.) vs. temperature during subsequent melting at heating rate of 1 °C/min.

164

phase and the amorphous interlayer in the temperature interval between 100°C and 120°C. Since Q is proportional to the square of the electron density difference (see eqn. 2), small changes in Δρ are amplified. Regarding changes in L B , crystal perfecting should leave L B unchanged. The fact that L B increases further suggests that some imperfect crystals melt at these lower temperatures.

Once the main population of y phase crystals begins to melt, at about 124°C, Q decreases while L B increases. The slope of either Q or L B vs. Τ smoothly changes as γ melting ends and α melting begins. At 145°C, L B seems to decrease, a result of the inability to properly calculate these Bragg long periods: the Bragg peak is moving into the beam stop region at this point, and the intensity is quite weak.

Conclusions

We have performed simultaneous SAXS and WAXS experiments on an m-iPP known to crystallize in α and y modifications. Our results show that:

1. During isothermal crystallization at 117°C, α and y crystals grow simultaneously, within the time resolution of our experiment. The fraction of y crystals is larger than α at the end of the crystallization.

2. Dual endotherms in m-iPP are caused by the melting of different crystal modifications, γ crystals melt at a lower temperature than α crystals.

3. Once γ crystals have melted, they do not recrystallize, nor do they transform into α crystals.

Acknowledgments

Research was supported by the U.S. Army Research Office, Grant DAAH04-96-1-0009. The work performed at Florida State University was supported by NSF Polymer Program (DHR-94-19508).

References

1. Lotz, B.; Wittmann, J. C.; Lovinger, A. J. Polymer 1996, 37, 4979-4992. 2. Natta, G.; Corradini, P. Nuovo Cimento Suppl. 1960, 15, 40. 3. Meille, S. V.; Ferro, D. R.; Bruckner, S.; Lovinger, A. J.; Padden, F. J.

Macromol. 1994, 27, 2615-2622. 4. Bruckner, S.; Meille, S. V. Nature 1989, 340, 455. 5. Lotz, B.; Graff, S.; Wittmann, J. C. J. Polym. Sci. Phys. Ed. 1986, 24, 2017. 6. Meille, S. V.; Bruckner, S.; Porzio, W. Macromol. 1990, 23, 4114-4121. 7. Turner-Jones, Α.; Aizlewood, J.; Beckett D. R. Makromol. Chem. 1964, 75, 134.

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8. Padden, F. J.; Keith, H. D. J. Appl. Phys. 1959, 30, 1479-1484. 9. Keith, H. D ., Padden, F. J.; Walter, N . M. ; Wyckoff, H. W. J. Appl. Phys . 1959,

30, 1485-1488. 10. Devaux, E.; Chabert, B. Polym. Commun. 1991, 32, 464-468. 11. Morrow, D. R. J. Macromol. Sci. Phys. 1969, B3, 53-65. 12. Varga, J. J. Material Sci. 1992, 27, 2557-2579. 13. Vleeshouwers, S. Polymer 1997, 32, 3213. 14. Fillon, B.; Thierry, Α.; Wittmann, J. C.; Lotz, B. J. Polym. Sci. Phys. Ed. 1993,

31, 1407. 15. Morrow, D. R.; Newman, B. A. J. Appl. Phys. 1968, 39, 4944-4950. 16. Sauer, J. Α.; Pae, K. D. J. Appl. Phys. 1968, 39, 4959-4968. 17. Turner-Jones, A. Polymer 1971, 12, 487. 18. Mezghzni, K.; Phillips P. J. Polymer 1997, 38, 5723-5733. 19. Kaminsky, W.; Kulper, K.; Britzinger, Η. H.; Wild, F. R. W. P. Angew. Chem.,

Int. ed. Engl. 1985, 24, 507-508. 20. Isasi, J. R.; Alamo, R. G.; Mandelkern, L. J. Poly. Sci. Phys. Ed. 1997, 35, 2511. 21. Alamo, R. G.; Galente, M . J.; Lucas, J. C.; Mandelkern, L. Polym. Preprint 1995,

36, 285-286. 22. Isasi, J. R.; Alamo, R. G.; Mandelkern, L. J. Poly. Sci. Phys. Ed. in press. 23. Alamo, R. G.; Kim, M . H.; Galente, M . J.; Isasi, J. R ; Mandelkern, L. (to

appear) 24. Strobl, G. R. ; M . Schneider. J. Polym. Sci. Polym. Phys. Ed. 1980, 18, 1340. 25. Norton, D. R.; Keller, A. Polymer 1985, 26, 704-716. 26. Lotz, B.; Wittmann, J. C. J. Poly. Sci. Phys. Ed. 1986, 24, 1541-1558. 27. Khoury, F. J. Res. Natl. Bur Stand. 1966, 70A, 29. 28. Padden, F. J.; Keith, H. D. J. Appl. Phys. 1966, 37, 4013. 29. Padden, F. J.; Keith, H. D. J. Appl. Phys. 1973, 44, 1217. 30. Binsbergen, F. L.; De Lange, B. G. M . Polymer 1968, 9, 23. 31. Albrecht, T.; Strobl, G. Macromol 1995, 28, 5267-5273. 32. Dai, P. S.; Cebe, P.; Capel, M. ; Alamo, R. G.; Mandelkern, L. (to appear) 33. Yadav, Y. S.; Jain, P. C. Polymer 1986, 27, 721. 34. Petraccone, V.; Guerra, G.; de Rosa, C.; Tuzi, A. Polymer 1985, 28, 143. 35. Varga, J. J. Thermal Anal. 1989, 35, 1891. 36. Guerra, G.; Petraccone, V.; Corradini, P.; De Rosa, C.; Napolitano, R.; Pirozzi,

B.; Giunchi, G. J. Polym. Sci. Phys. Ed. 1984, 22 1029. 37. Samuels, R. J. J. Polym. Sci. Phys. Ed. 1975, 13, 1417. 38. Alamo, R. G.;. Brown, G. M. ; Mandelkern, L.; Lehtinen, Α.; Paukkeri, R.

Polymer (in press). 39. Laihonen, S.; Geddie, U. W.; Werner P. E.; Martinez Salazar, J. Polymer 1997,

38, 361. 40. Phillips, P. J. ; Mezghani, K. Polymer 1998, 39, 3735. 41. Brandrup, J.; Immergut, E. H. Polymer Handbook, Third Edition, Wiley

Interscience: New York, 1989, p. V27.

Chapter 11

Lamellar Morphology of Narrow PEEK Fractions Crystallized from the Glassy State

and from the Melt M. Dosière1, C. Fougnies1, M. H. J. Koch2, and J. Roovers3

1Université de Mons-Hainaut, Laboratoire de Physicochimie des Polymères, Place du Parc, 20, B-7000 Mons, Belgium

2European Molecular Biology Laboratory, Hamburg Outstation, EMBL do DESY, Notkestrasse, 85, D-22603 Hamburg, Germany

3Division of Chemistry, National Research Council of Canada, Ottawa, Ontario K1A OR9, Canada

The morphology of five semicrystalline poly(aryl ether ether ketone) or (PEEK) samples with narrow molecular weight distributions has been investigated by differential scanning calorimetry (DSC), wide angle X-ray diffraction (WAXD) and small angle X-ray scattering (SAXS). The values of the degree of crystallinity estimated by WAXD and DSC are in good agreement and range from ~10 to ~53 % for the highest and lowest molecular weight samples, respectively. Most of the samples exhibit the well-known double melting behavior which is interpreted in terms of a melting-recrystallization-melting mechanism. At constant annealing conditions, the crystal thickness (given by the smallest length obtained from the correlation function) does not depend on molecular weight. In contrast, the thickness of the amorphous layer strongly increases with increasing chain length . This leads to a decrease in the linear degree of crystallinity within the lamellar stacks which is consistent with the behavior of the degree of crystallinity obtained from WAXD and DSC measurements. Simultaneous time-resolved SAXS and WAXD data show that the long spacing decreases in the first stage of crystallization for PEEK fractions. A densification of the crystal cores proved by an increase of the crystal density and a slight modification of the distribution of the lamellar thicknesses in the first stage of crystallization are proposed to explain the decrease of 2-3 nm of the long spacing.

© 2000 American Chemical Society

167

Introduction

Poly(aryl-ether-ether-ketone) or PEEK is a high temperature aromatic polymer combining very good thermal and mechanical properties with excellent chemical resistance^-2). It can be obtained either in a fully amorphous state by quenching from the melt or in a semicrystalline state by annealing from the glassy state or crystallization from the melt. Crystallization and melting as well as the morphology of semicrystalline PEEK samples of industrial grades with broad molecular weight distributions have been widely investigated. Most of these studies were made by differential scanning calorimetry (DSC)(J-/2), wide-angle X-ray diffraction (WAXD) (13-19) and small-angle X-ray scattering (SAXS) including time-resolved experiments (20-28), thennomechanical analysis (29-32), transmission electron microscopy (TEM) (33-36) and optical microscopy (OM) (37-40).

The melting curves of semicrystalline PEEK samples generally contain two endotherms. Two hypotheses have been formulated concerning their origin : i) they arise from the presence of two distinct lamellar populations present in the sample before the DSC scan and having different melting temperatures (5,10,21,22); ii) the sample contains only one population of crystals starting to melt at the temperature of the first endotherm and then continuously reorganizing during the scan, the high temperature endotherm representing the final melting of the recrystallized lamellar crystals (3,6,25,26). A compromise between these two extreme hypotheses has been proposed (8). It is admitted that PEEK crystallizes in lamellae which form stacks where crystal cores of limited lateral size alternate with amorphous regions. In approximation, this also justifies the use of a two phase model with amorphous and crystalline regions. With such a model, the correlation function of the SAXS intensity data yields two thicknesses L, and L 2 with for example L, > L 2 and their sum (Lj + L 2) is the long period L p (41). Assignment of the crystal thickness (Lc) to Lj or to L 2

has to be made on the basis of additional information. The crystal thickness L c has been attributed in some cases to the shortest (L2) (24-26) and in other cases to the largest length (Lj) (20-23,27,28). Despite several studies devoted to this topic, the attribution was not settled. Narrow molecular weight PEEK fractions as prepared by Roovers et al. (43). provide the possibility to solve this problem. The kinetics of crystallization of narrow molecular weight fractions of PEEK covering weight average molecular weights from 4000 to 79000 with a degree of polydispersity ranging between 1.19 and 1.49 was studied by differential scanning calorimetry and optical microscopy (44,45). The spherulitic growth rate data were analyzed with the standard theory of crystallization. The range of measurements of spherulitic growth rates on the PEEK fractions was extended to both sides of the temperature range analyzed by Deslandes et al (45) A detailled analysis of these data coupled to long spacing measurements is reported elsewhere (46).

The morphology of semicrystalline PEEK fractions in relation to their molecular weight has been recently investigated (46). The PEEK fractions were crystallized from the glassy state between 250 to 340 °C. The main results of this study will be

168

commented on below. The modifications of the morphology of these narrow molecular weight PEEK fractions during crystallization from the glassy state and from the melt were recently investigated by simultaneous time-resolved WAXD and SAXS synchrotron measurements (48,49). The results of the crystallization of the 8k PEEK fraction from the melt at 300 °C will be taken as example for crystallization of low molecular weight PEEK fractions. The choice of crystallization temperatures equal or above 300 °C is based on previous works indicating that important morphological modifications are observed in industrial grade PEEK in this range of crystallization temperatures (25, 26,35,50).

Experimental Section

Information about the synthesis of the PEEK fractions, annealing experiments from the glassy state, crystallization from the melt, analysis by differential scanning calorimetry, WAXD and SAXS experimental setup for static and time-resolved measurements and the treatment of the experimental data can be found in previous papers (25,26,47).

Results and Discussion

Influence of the crystallization temperature on the melting behavior of PEEK fractions

The melting curves of semicrystalline PEEK fractions annealed at 250, 300 and 330 °C from the glassy state are shown in Figures la-c, respectively. The PEEK fractions exhibit the well known double melting behavior with a low (I) and a high (II) endotherm characterized by peak temperatures Tj and Tm", respectively as described in detail for industrial grade of ?ΈΈΚ(3-12). At least for the low annealing temperatures, this behavior can be attributed to a reorganization of the lamellar crystals. At high annealing temperatures, however, a two step crystallization process takes place : an isothermal crystallization during annealing and a non-isothermal crystallization during cooling from the annealing temperature to room temperature. The latter thermal treatment also leads to two endotherms resulting from the melting of two lamellar populations as illustrated for example for the highest molecular weight PEEK fraction (79k) annealed at 330 °C (Figure lc). For the PEEK fractions, the minimum annealing temperature where there is significant crystallization during cooling following annealing, is lower than 340 °C as determined for industrial grade PEEK (Stabar 200). The temperature range, where annealing only occurs, becomes restricted with increasing molecular weight. This is illustrated by the 79k fraction for which crystallization occurs during cooling after isothermal annealing at 330 °C

169

ο Τ 3

Χ <

220 240 260 280 300 320 340 360 T/°C

Β

ο •α c

χ <

220 240 260 280 300 320 340 360 T/°C

Figure J. DSC heating curves of PEEK fractions crystallizedfrom the glassy state during 1 hr at 250°C (a) ; 300°C (b) and 330°C (c). The values of the weight average molecular weight of the fractions are indicated on the figures. The curves are displaced for better visualization. The bar represents 0.25 W/g

Continued on next page.

79k

32k J \ I

18k A \ I

8k

4k r 220 240 260 280 300 320 340 360

T/°C

Figure 1. Continued.

171

(Figure lc). Such a behavior results from a partial crystallization during the isotherm due to the reduced rate of crystallization. Molecular weight segregation cannot easily be invoked for the origin of the non-isothermal crystallization of these PEEK fractions. The ratio of the lower endotherm to the higher endotherm enthalpy of fusion increases with increasing molecular weight at a fixed annealing temperature. High molecular weight PEEK fractions have significantly lower apparent melting temperatures. Moreover, low molecular weight fractions reorganize more easily than the high molecular weight fractions.

The melting temperatures of both endotherms are plotted in Figure 2 as a function of annealing temperature Ta. The temperature of the second melting peak strongly depends on the molecular weight and seems to be almost independent of the annealing temperature except at high annealing temperatures (Figure 1). The shape of endotherm II is identical to that of initially amorphous samples when heated through their melting in the DSC. Endotherm II thus corresponds to the melting of identical populations of lamellae with a given average molecular weight. This supports the hypothesis of a reorganization mechanism proposed earlier (25,26).

In contrast, the position of the low temperature endotherm I seems to be almost independent on the chain length for a given annealing temperature (Figures 1). This suggests that the lamellar crystals formed at a constant annealing temperature T a start to melt and reorganize approximately at the same temperature independently of their molecular weight, giving rise to the low temperature endotherm peaking around 15 °C above Ta. The behavior of T m

l as a function of annealing temperature is different for low (4k, 8k, 18k) and high molecular weight fractions (32k and 79k). The small increase of T m

! observed for the low molecular weight fractions can be rationalized as follows. As recrystallization is more difficult in the high molecular weight samples (32k and 79k), the exothermic contribution is much less important, leading to a small shift of the first melting peak to higher temperatures. At the end of the DSC scan, as in the case of the initially amorphous samples, the reorganization process stops at a lower temperature in the high molecular weight fractions because the number of entanglements is larger and endotherm II is shifted to lower temperatures.

For the high annealing temperatures, i.e. 310 °C < T a < 340 °C, reorganization of the lamellae is no longer possible at this rather high heating rate (10 °C/min). The remaining sharp peak is thus characteristic of the melting of isothermally crystallized lamellar crystals, its peak temperature corresponds to the melting point of the lamellae in the sample in absence of any reorganization effects. The broad low temperature melting peak is, as already mentioned, due to the melting of lamellar crystals non-isothermally crystallized during the cooling subsequent to the isothermal annealing. These results on narrow PEEK fractions are in agreement with previous conclusions on industrial grades of PEEK(25,26) and a recent study on the crystallization of poly(phenylene sulfide) from the glassy state (51).

To estimate the equilibrium melting temperature Tm° of these PEEK fractions, the melting temperature Tm* was plotted against the annealing temperature T a (Figure 2) according to the Hoffman-Weeks procedure (52). For the low molecular weight fractions, this yields values between 400 and 417 °C in agreement with the value usually quoted (3,8,11). In contrast, the extrapolated values appear to be meaningless

172

2 3 0 2 5 0 270 290 310 330 350 370

T a / ° C

Figure 2. Peak temperature of the first melting endotherm (filled symbols) and of the second melting endotherm (open symbols) against annealing temperature for the various PEEK fractions. The values of the molecular weights are given in the figure.

173

for the two high molecular weight samples. The Hoffmann-Weeks method seems thus not fully adequate to obtain accurate values of the equilibrium melting temperature Tm° for PEEK, probably because reorganization precludes the determination of the true melting temperature.

Degree of crystallinity of PEEK fractions.

The calculation of the degree of crystallinity from DSC measurement requires the knowledge of the melting heat of the PEEK crystal. The value of 130 J/g estimated by Blundell et al. (3) is often used. This value results from an extrapolation of the heat of melting at a density of 1.401 g/cm3. However, Lee et al. (53) have obtained 165 J/g also from an extrapolation of the heat of melting for a crystal density of 1.415 g/cm3, value proposed by Rueda et ai. 15 for the density of the PEEK crystal. A value of 161 ± 20 J/g has been obtained from the Clapeyron equation (54). We made the same choice as Séguéla (55), i.e. a melting heat of 160 J/g for the PEEK crystal taking two independent determinations (161 and 165 J/g) into account. The PEEK fractions cover a particularly wide range of degrees of crystallinity from 0.10 to 0.53 (Figure 3). W C

D S C increases over the whole range of annealing temperatures for the three low molecular weight samples but reaches a maximum for the two high molecular weight samples for annealing temperatures around 300-310 °C. Values of the weight degree of crystallinity W C

W A X D computed from the WAXD diffraction profile by subtracting a scaled diffraction profile obtained from an amorphous sample are in good agreement with those obtained by DSC (Figure 3).

Crystal density of PEEK fractions

The Bragg spacings of the WAXD reflections of all fractions are characteristic of the orthorhombic unit cell of PEEK. The crystal density of the PEEK fractions increases with annealing temperature as reported earlier for industrial grade PEEK crystallized from the glassy state or from the melt (Figure 4)(56,57). For the 4k and 8k fractions, the corresponding crystal densities are relatively high, reaching values of 1.405 g/cm3 for the highest annealing temperatures. This is due to the closer packing of the chains and the better alignment of the ether and ketone bridges for these oligomers.

Lamellar morphology of PEEK fractions

The long period Lp of all PEEK fractions increases with annealing temperature as expected (Figure 5a). At a given annealing temperature, the long spacing increases with the average molecular weight, as already reported for other polymers (42,58). It could be related to an increase of either the amorphous (La) or the crystalline layer thickness (Lc) or even of the two spacings L c and L a , simultaneously. The

174

230 250 270 290 310 330 350 370 390

T a / °C

Figure 3. Weight degree of crystallinity of PEEK samples calculated from WAXD (open symbols) and DSC (fûled symbols) data versus annealing temperature. The values of the molecular weights are given in the figure.

2 4 0 2 6 0 2 8 0 300 320 340 3 6 0

T a / ° C

Figure 4. Crystal density of the various PEEK samples against annealing temperature. The values of the molecular weights are given in the figure.

175

Β

ε c

240 260 2 8 0 300 320 340 360

T a / ° c

Figure 5. Lamellar long spacings for PEEK fractions versus crystallization temperature. The long period Lp is calculated from the Lorentz corrected SAXS intensity curves), L / and L2 are obtained from the correlation function. The values of the molecular weights are given in the figure.

Continued on next page.

176

4k 8k 18k 32k 79k

2 ^ , , , , , 1

2 4 0 2 6 0 280 300 320 3 4 0 360

T a / ° C

Figure 5. Continued.

177

morphological parameters can be obtained from the linear correlation function calculated from the SAXS data assuming a two phase model. This procedure gives the values of the two lengths L 1 and L 2 with Ll > L 2 as already mentioned (Figures 5b-c). The smallest value L 2 was assigned to the crystalline thickness Lc.25,26 This choice implies, as discussed below, that the linear degree of crystallinity in the stack of lamellae v c

l i n = Lc/(LC+La) would generally be smaller than 0.5. This assignment, also made by Jonas et ύ.(24), is based on the following arguments : i) for "physical reasons", some authors claimed that thin lamellae including only a few chemical repeating units along the c-axis could not exist in semicrystalline PEEK (20-22). As the analysis of the correlation function usually yields values of Lx and L 2 around 7.5 and 3.5 nm, respectively, they assigned the larger length (L, ) to the crystalline layer thickness L c . In the present case, the long periods L p obtained for the two lowest molecular weight fractions range from -7.0 to ~9.0 nm for annealing temperatures between 250 and 300 °C. The analysis of the correlation functions gives L, = 4-5 nm and L2 = 3-4 nm (Figures 5b and 5c). As one of these two values must correspond to the crystalline layer, this gives clear evidence that lamellar thicknesses around 4 nm (i.e. - 3 chemical repeating units) can occur in semicrystalline PEEK samples. For these low molecular weight fractions with relatively high degrees of crystallinity, the correlation functions were computed without difficulty and with a good precision since the SAXS peak was well resolved, ii) At constant annealing temperature L 2 is almost independent on the molecular weight of the PEEK fraction while the value of L, strongly increases with increasing molecular weight (Figure 5b). For experiments carried out at a given annealing temperature and characterized by the same degree of supercooling, the crystalline thickness L c is expected to be independent of the molecular weight. The fact that L, increases by as much as 100% at constant annealing temperature over the molecular weight range is obviously inconsistent with its assignment to the crystal thickness. The small variation of L 2 with the molecular weight at different Ta, if any, is compatible with L c = L 2 . iii) It has been pointed out that the value of v c

l m could not be lower than that of the weight degree of crystallinity Wc(57). It should, however, be noted that these quantities, although related, are not strictly equivalent and some errors can occur in their determination. Our data cover a much wider range of degrees of crystallinity (0.10-0.53) than in previous comparisons of v c

i i n and W c (61). The comparison between W C

D S C and v 2

l i n = L2/(L,+L2) is displayed in Figure 6. Similar results are obtained using the values of W C

W A X D . The linear degree of crystallinity v c

l i n and the macroscopic crystallinity may differ but must follow the same trend, which is consistent with the assignment of L c to the smallest length L 2 obtained from the correlation function. Assuming that L, = L c

and hence taking the corresponding values of ν j 1 , 1 1 as representing the linear degree of crystallinity, one would reach the conclusion that the latter increases with increasing molecular weight in obvious contradiction to the WAXD and DSC results.

The major morphological change induced with an increase of the molecular weight is an increase of the thickness of the amorphous regions while the crystalline thickness seems to be only governed by the annealing temperature Ta. It was shown for other semicrystalline polymers (polyethylene, polypropylene, poly(ethylene terephthalate),...) that the thickness of the amorphous regions, depends linearly on

178

Figure 6. Comparison between vjin and Wc^sc for different PEEK samples. The values of the molecular weights are given in the figure. The dashed line is the linear regression through all data.

179

the average dimension of the polymer chain rw = Σ w; M/ 7 2 assimilated to an unperturbed random coil before its crystallization^. For low polydispersity samples, this average dimension rvv is proportional to the square root of weight average molecular weight. The thickness of the amorphous regions following our assignment is proportional to M w

l / 2 for annealing temperatures between 250 and 330 °C (Figure 7). It must be recalled that the analysis of the linear correlation function used throughout this work is based on the assumption that the lamellar stacks have a infinite lateral size. Electron microscopy data of PEEK and analysis of the width of the WAXD peaks reveal that this is far from being the case.

Simultaneous time-resolved WAXD and SAXS investigation of the crystallization of a low molecular weight PEEK fraction from the melt

The simultaneous time-resolved WAXD and SAXS investigation of the crystallization of the five narrow molecular weight PEEK fractions from the glassy state and from the melt is reported elsewhere (48,49). The results of the isothermal crystallization of the 8k PEEK fraction from the melt at 300 °C and its subsequent cooling at -10 °C to 100 °C are typical for the low molecular weight PEEK fractions and will be presented below. Analysis of the SAXS and WAXD intensity curves yielded the following parameters : a) the lamellar dimensions, i.e. the long spacing L p , the crystalline L c and the amorphous L a layer thicknesses obtained from the Lorentz corrected SAXS curves and the correlation functions, respectively (Figure 8); b) the SAXS integrated intensity or the invariant Q (Figure 8) ; c) the weight degree of crystallinity obtained from the WAXD curves (Figure 9) ; d) the crystal density from the Bragg spacings of the WAXD patterns (Figure 9).

Isothermal crystallization During the first stages of crystallization the long spacing, the amorphous and the

crystal thicknesses decrease by 1.9, 1.3 and 0.6 nm, respectively. After a delay of about 500 s, the lamellar dimensions reach constant values of 15.8, 9.7 and 5.0 nm, respectively. The sum of L c and L a , equals to 14.7 nm and is lower than the long period obtained from the Lorentz corrected SAXS curves (15.8 nm). Differences between 5 and 20 % have been already reported between the values of the long period calculated from the Lorentz corrected SAXS curve and the correlation function, the first one being the largest (21). A decrease of the long period by about 2-3 and up to 7 nm during the first stages of the isothermal crystallization from the melt for commercial grade PEEK has already been reported (20,21,23,27). As the degree of polydispersity of these industrial grades is around 3, a segregation of molecular weights could be invoked to explain this decrease of the long period in the first stage of crystallization from the melt. The results obtained with the 8k PEEK fraction, which has a low polydispersity, seems to contradict this hypothesis. Note, however, that molecular weight segregation during crystallization has been experimentally proven for narrow fractions of linear polyethylene (56). Heat transfer considerations can be also invoked when the temperature of the sample is lowered by several tens of

180

14 ι

• 2 5 0 13 o 2 8 0

12 ψ 300 12 310

11 m 3 2 0

Ε 10 D 330

c 9 9

m _j 8

y^ 7

6

'S* 5

4 50 100 150 200 250 300

( M w ) 0 5

Figure 7. Thickness of the amorphous regions La against (Mw)^2 for PEEK fractions annealed during 1 hr from the glassy state at various annealing temperatures given in the figure.

4 \ , , , , , μ o.o 2 0 0 0 2 1 0 0 2 2 0 0 2 3 0 0 2 4 0 0 2 5 0 0 2 6 0 0

Time Is

Figure 8. Lamellar dimensions (long spacing Lp, amorphous thickness L\ and crystal thickness Lc) and invariant Q during isothermal crystallization from the melt at 300 °C of an 8k PEEK fraction.

181

Figure 9. Degree of crystallinity obtained from WAXD data and crystal density during isothermal crystallization from the melt at 300 °C of an 8k PEEK fraction.

182

°C. Numerical simulations show that the temperature of a thin polymer sample with a thickness around 0.1 to 0.3 mm can be changed by 50 to 100 °C in a few seconds assuming perfect thermal contact with the surroundings (60). Lamellar insertion models have been proposed to explain the decrease in long period during the first stage of crystallization (20-22,27,28). These models are, however, inconsistent as recently also recognized by their authors (28).

The change of the invariant during the first stage of the crystallization of the 8k PEEK fraction is similar to that observed with industrial grade PEEK. Note that the invariant increases in the same time interval and remains constant during almost the entire crystallization period. The evolution of the degree of crystallinity and of the crystal density during the first stages of the crystallization is particularly interesting (Figure 9). The crystal density has a particularly low initial value ( 1.304 g/cm3) and it increases to reach at the end of the isothermal crystallization the value of 1.308 g/cm3. The degree of crystallinity has an initial value of 0.29 and reaches a value of 0.41 after a delay of 500 s. Our assignment of the crystal thickness to the smallest value obtained from the correlation function (L2) suggests that the decrease in long spacing is mainly due to a decrease of the thickness of the amorphous regions in the lamellar stacks. At such high temperatures, the mobility of the chains is indeed sufficiently high to allow a diffusion of the chains resulting in a contraction, i.e. a closer packing of the amorphous regions. The macroscopic degree of crystallinity increases by 30 % between the first and the last measurements during the isothermal crystallization. The number of lamellar crystals and/or stacks of lamellae is therefore modified during the isothermal crystallization. A slight modification of the distribution function of lamellar thicknesses could result in a decrease of 2-3 nm of the long spacing. Moreover, the slight increase of the crystal density from 1.304 to 1.310 g/cm3 during the first stage of crystallization supports a densification of the crystal cores of the lamellae.

Cooling from the crystallization temperature to 100 °C at -10 °C/min. The lamellar dimensions L p , L a and Lc continously decrease during cooling from

300 to 100 °C. At 300 and 100 °C, the values of L p , L a and L c are 15.9 and 13.4 nm, 9.5 and 8.1 nm and 5.0 and 4.2 nm, respectively (Figure 10). Such decreases result from the difference between the thermal contraction of the crystalline and amorphous regions on temperature. The invariant also decreases between 300 to 100 °C (Figure 10) due to a difference between the dependences of the thermal coefficients of the crystalline and amorphous regions against temperature. The apparent increase of crystallinity during cooling results from a reduction of the thermal motion at lower temperature (Figure 11). The crystal density increases from 1.312 g.cm3 at 300 °C to 1.38 g.cm3 at 100 °C as a result of thermal contraction (Figure 11). A mean value of the crystal thermal expansion coefficient otc of 2.42 10"4 IC1 can be estimated from Figure 11. This value is in agreement with values previously reported for industrial grades (1.72, 2.22, 2.36, 1.80 and 231 .W*Y;{)(25,51,54,57,61,62).

183

Figure 10. Lamellar dimensions (long spacing Lp, amorphous thickness Lj and crystal thickness Lc) and invariant Q during cooling at -10 °C/min following isothermal crystallization of an 8k PEEK fraction from the melt at 300 °C.

Figure 11. Degree of crystallinity obtained from WAXD (o) and crystal density (+) during cooling at -10 °C/min following isothermal crystallization at 300 °C for a 8k PEEK fraction.

184 Conclusions

The lamellar morphology of five narrow fractions of PEEK synthesized by the ketimide procedure, has been investigated by SAXS, WAXD and DSC. The weight average molecular weights and the degree of polydispersity of these five fractions range from 4000 to 79000 and 1.19 to 1.49, respectively. The weight degree of crystallinity of samples of these five PEEK fractions crystallized from the glassy state ranges between 0.10 and 0.53. The fractions with the lowest weight average molecular weight have the highest degrees of crystallinity. For all fractions, the long spacing increases as expected, with the crystallization temperature. At a given crystallization temperature, the long spacing increases with the weight average molecular weight of the fraction. From the two spacings L, and L 2 (with L, + L 2 = L p

and L, > L 2) obtained from the correlation function of the SAXS intensity curve, only the lowest spacing, i.e. L 2 , gives a linear degree of crystallinity v c

l i n = L 2 / L p which linearly increases with the weight degree of crystallinity W c. The crystal and the amorphous thicknesses are therefore identified to the smallest L 2 and to the largest L, spacings, respectively. The linear relationship between the thickness of the amorphous regions and the square root of the weight average molecular weight for the lamellae crystallized between 250 and 340 °C, is an additional proof of our assignment of the crystal and amorphous thicknesses. Time-resolved simultaneous SAXS and WAXD data show that, even with narrow molecular weight PEEK fractions, the long spacing decreases in the first stage of crystallization. This small decrease of the long spacing (2 - 3 nm) can be explained by a densification of the crystal cores and a progressive filling in the stacks of lamellae. This results in a slight modification of the distribution function of the lamellar thicknesses.

Acknowledgements

This work was supported by the Belgian National Funds for Scientific Research and the European Union through the HCMP Access to Large Installations Project, Contract CHGE-CT93-0040 to the EMBL.

References

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2. Stening ,T.C.; Smith, C .P.; Kimber, P.J. J. Mod. Plast. 1981,86. 3. Blundell, D.J.; Osborn, B.N. Polymer 1983, 24, 953. 4. Kumar, S.; Anderson, D.P.; Adams, W.W. Polymer 1986, 27, 329. 5. Cheng, S.Z.D.; Cao, M.-Y.; Wunderlich, B. Macromolecules 1986, 19,1868. 6. Blundell, D.J. Polymer 1987, 28, 2248. 7. Cebe, P.; Chung, S.; Hong, S.D. J. Appl. Polym. Sci. 1987, 33,487.

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8. Lee, Y.; Porter, R.S. Macromolecules 1987, 20, 1336. 9. Cebe, P., J. Mater. Sci. 1988 , 23, 3721. 10. Bassett, D.C.; Olley, R.J.; Al Raheil,I.A.M. Polymer 1988, 29, 1745. 11. Lee, Y.; Porter, R.S.; Lin, J.S. Macromolecules 1989, 22, 1756. 12. Mehmet-Alkan, A.A.; Hay, J.N. Polymer 1992, 33, 3527. 13. Fratini, A.V.; Cross, E.M.; Whitaker, R.B.; Adams, W.W. Polymer 1986,27,861. 14. Wakelyn, N.T., Polymer Comm.1984, 25, 306. 15. Rueda, D.R.; Ania,F.; Richardson, Α.; Ward, I.M.; BaltaCalleja, F.J. Polymer

Comm. 1983, 24, 258. 16. Hay, J.N.; Kemmisch, D.J.; Langford, J.I.; Rae, A.I.M. Polymer Comm. 1984,

25,175. 17. Wakelyn, N.T. J. Polym. Sci., Polym. Let. 1987, 25, 25. 18. Hay, J.N.; Langford, J.I.; Lloyd, J.R. Polymer 1989, 30, 489. 19. Iannelli, P. Macromolecules 1993, 26, 239. 20. Wang, J.; Alvarez,M.; Zhang,W,.; Wu, Z.; L i , Y.; Chu,B. Macromolecules 1992,

25, 6943. 21. Hsiao, B.S.; Gardner, K.H.; Wu, D.Q. ; Chu, Β. Polymer 1993, 34, 3986. 22. Hsiao, B.S.; Gardner, K.H.; Wu, D.Q. Polymer 1993, 34, 3996. 23. Krüger, K.-N.; Zachmann, H.G. Macromolecules 1993, 26, 5202. 24. Jonas, A.M.; Russell, T.P.; Yoon, Do Y. Macromolecules 1996,28, 8491. 25. Fougnies,C.; Damman,P.; Villers,D.; Dosière,M.; Koch, M.H.J. Macromolecules

1997,30,1385. 26. Fougnies, C.; Damman, P.; Dosière, M. ; Koch, M.H.J. Macromolecules

1997,30,1392. 27. Verma, R.K.; Hsiao, B.S. Trends in Polymer Sci. 1996, 4, 312. 28. Wang, W.; Schultz, J.M. ; Hsiao, B.S. J. Macromol. Sci.,Phys., 1998, B37,667. 29. Hinkley, J.A.; Eftekhari, Α.; Crook, R.A.; Jenson, B.J.; Singh, J.J. J. Polym. Sci.

Part Β 1992, 30, 1195. 30. Nishino, T. ; Tada, K. ; Nakamae, K. Polymer 1992, 33, 736. 31. Jones, D.P.; Leach, D.C. ; Moore, D.R. Polymer 1985, 26, 1385. 32. Chivers, R.A. ; Moore, D.R. Polymer 1994, 35 ,110. 33. Lovinger A.J., Davis D.D. Polymer Comm. 1985, 26, 322. 34. Lovinger A.J., Davis D.D. J. Appl. Phys. 1985, 58, 2843. 35. Lovinger A.J., Davis D.D. Macromolecules 1986, 19,1861. 36. Olley, R. H.; Bassett, D.C. ; Blundell, D.J. Polymer 1986, 26, 344. 37. Chung, C.T.; Chen,M. Polym..Prepr.,A.C.S.San Francisco Meeting,1992,33,420 38. Zhang, Z.; Zeng, H. Makromol. Chem. 1992, 193, 1745. 39. Zhang, Z.; Zeng, H. Polymer 1993, 34, 1551. 40. Zhang, Z.; Zeng, H. Polymer 1993, 34, 4032. 41. Strobl, G.R.; Schneider, M.J. J.Polym.Sci., Polym. Phys. Ed. 1980, 18, 1343. 42. Rault, J.; Robelin-Souffaché, Ε. J. Polym. Sci., Polym. Phys. Ed., 1989,27,1349. 43. Roovers, J.;Cooney, J.D.; Toporowski, P.M. Macromolecules 1990, 23, 1611. 44. Day, M.; Deslandes ,Y.; Roovers, J. ; Suprunchuk, T. Polymer 1991, 32, 1258. 45. Deslandes ,Y.; Sabir, F.-N.; Roovers, J. Polymer 1991, 32, 1267.

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47. Fougnies, C.;Dosière, M.;Koch, M.;Roovers, J. Macromolecules, 1998, 31, 6266. 48. Fougnies, C.; Dosière, M.; Koch, M.; Roovers, J., Macromolecules, submitted. 49. Fougnies, C.; Dosière, M.; Koch, M. ; Roovers, J., Polymer, submitted. 50. Damman, P.; Fougnies, C.; Moulin, J.-F.; Dosière, M . Macromolecules, 1994,

27,1582. 51. Xin Lu, S. ; Cebe, P. ; Capel, M. Macromolecules 1997, 30, 6243. 52. Hoffmann, J.D.; Weeks, J.J. J. Res. Nat. Bur. Std.(U.S.) 1962, 66A,13. 53. Lee, Y., Porter R.S., Lin, J.S., Macromolecules, 1989,22,1756. 54. Zoller, P.; Hehl, T.; Starkweather, H.W.; Jones, G.A. J. Polym. Sci., Part B,

Polym. Phys. 1989,27, 993. 55. Séguéla, R., Polymer Comm., 1993, 34, 1761. 56. Zimmermann, H.J.; Könnecke, K. Polymer 1991, 32, 3162. 57. Chen, F.C.; Choy, C.L.; Wong, S.P.; Young, K. J. Polym. Sci., Polym. Phys. Ed.

1981,19, 971. 58. Mandelkern, L. ; Alamo, R.G. ; Kennedy, M.A. Macromolecules 1990, 23, 4721. 59. Colet, M. -C .; Point, J.-J.; Dosière, M . J. Polym. Sci., Polym. Phys. Ed.,1986, 24,

357. 60. Jakob, M., Heat Transfer, Wiley, New York, 1949, Vol. 1,Chap.13, 251. 61. Choy,C.L.; Leung,W.P.; Nakafuku,C. J. Polym. Sci. Polym. Phys. Ed. 1990,28,

1965. 62. Blundell, D.J.; D'Mello, J. Polymer, 1991, 32,304.

Chapter 12

Real-Time Crystallization and Melting Study of Ethylene-Based Copolymers by SAXS, WAXD,

and DSC Techniques Weidong Liu1, Henglin Yang1, Benjamin S. Hsiao1,4,

Richard S. Stein2, Shengsheng Liu3, and Baotong Huang3

1Chemistry Department, State University of New York at Stony Brook, Stony Brook, NY 11794-3400

2Chemistry Department, University of Massachusetts, Amherst, MA 01003 3Polymer Chemistry Laboratory, Changchun Institute of Applied Chemistry,

Chinese Academy of Sciences 130022, China

Isothermal crystallization and subsequent melting experiments of metallocene-based polyethylenes with different degrees of octene branching were carried out as a function of crystallization temperature using real-time simultaneous synchrotron small-angle x-ray scattering (SAXS) and wide-angle x-ray diffraction (WAXD) techniques and differential scanning calorimetry (DSC). The variation of crystallinity was determined by WAXD; the crystal long period, lamellar thickness, interlayer amorphous thickness and invariant were determined by SAXS. Results from these studies were interpreted using the model of branch exclusion which affects the ability of the chain-reentry into the crystal phase and retards the ability to undergo a lamellar isothermal thickening process during prolonged annealing. The short chain branching turns out to have little effect on the equilibrium melting temperature (Tm°=144.1°C), which is determined by the Thomson-Gibbs equation. The surface free energy of the fold lamellar surface is found to increase with the branch content. Furthermore, our results indicate that a samll fraction of chain branch may be incorporated in the crystals lowering the heat of fusion.

The use of metallocene catalyst has opened up possibilities for producing tailored polyolefms with controlled molecular weights, homogeneous comonomer contents,

Corresponding author.

© 2000 American Chemical Society 187

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narrow chain branching, narrow chain branching distribution, and desired tacticity pattern, etc. Rapid commercialization of metallocene-based polyolefins has led to many new applications of these polymers as well as improvements over existing applications1"6.

The issues of thermodynamics and kinetics factors affecting the structure and morphology development in the metallocene-based polymers are of practical interest. They can be studied using a wide range of time-resolved techniques such as differential scanning calorimetry (DSC), dilatometry, small-angle light scattering (SALS), small-angle x-ray scattering (SAXS) and wide-angle x-ray diffraction (WAXD). In many cases, the ability to study the kinetics in real time has been limited by the capability of experimental techniques to follow the rapid changes in structure. These problems are alleviated in the studies presented here through the use of synchrotron x-ray radiation by simultaneously following WAXD and SAXS7'9 during the isothermal crystallization and subsequent melting in real time. Samples studied were copolymers of ethylene with α-olefin short-chain branch (octene) prepared by metallocene polymerization. The variations of the crystal unit cell parameters and crystallinity can be determined by WAXD; the crystal long period, lamellar thickness, interlayer amorphous thickness and invariant can be determined by SAXS10*12. Results from these studies can be interpreted using the model of branch exclusion which affects the ability of chain-reentry into the crystal phase, the segregation of branched chains near the crystal and amorphous interfaces, and the ability to undergo a lamellar isothermal thickening process during prolonged annealing. Such information can allow us to further correlate structural and morphological variations with the mechanical properties as functions of branch compositions, branch types and processing conditions in the future.

There are several interesting issues to be addresses in this study. (1) It is believed that the branched chains longer than methyl groups are excluded from the crystal because they do not fit into the crystal lattice. The aggregation of these branches at the crystal surface probably prevents the thickening of the crystal lamella. However, this subject is still unclear today. (2) As the Avrami model postulates that crystals arise from a nucleus, probably heterogeneous for polymers such as polyethylene, and that spherulites usually grow with a constant radial growth rate until impingement takes place. However, a secondary crystallization process is often observed during and after the initial spherulite growth. This process occurs probably because of the heterogeniety in molecular weight, the effect of chain branching, and the space-filling process of crystallization. The use of metallocene-based polyethylene (m-PE) provides a good control of molecular heterogeneity. With these polymers, together with real time WAXD and SAXS techniques permits the clarification of some outstanding issues about secondary crystallization. (3) The issue of determining the equilibrium melting temperature for PE using Hoffman-Weeks method13 has always been controversial. In this study, we have adopted the Thomson-Gibbs equation14,15

using the lamellar thickness data extracted from SAXS and the melting temperature from DSC. (4) It has also been postulated that molecule ordering in the amorphous phase may occur prior to crystallization. Evidence for this comes from the earlier appearance of SAXS signals than WAXD crystalline intensities. In this work, a

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quantitative examination of signals from SAXS and WAXD is carried out to address this matter.

Experimental

All ethylene copolymers were synthesized with a soluble metallocene catalyst. The synthetic conditions for the samples were as follows: [Zr] = lxlO^mol, Al/Zr = 1000, temperature = 40°C, reaction time = 0.5 hr, toluene solution =100 ml, ethylene pressure = 0.4 arm. The degrees of branching for ethylene-octene copolymers, determined by C 1 3 NMR as branches per 1000 carbon atoms, are shown in Table 1. Their molecular weights are 1.25 (Sample 1), 1.09 (Sample 2), 0.77 (Sample 3), 0.61 (Sample 4), 0.54 (Sample 5) xl05g/mol measured by gel permeation chromotography (GPC), respectively. Al l samples were purified prior to the experiment usually using the following procedures. They were dissolved in p-xylene by rigorously stirring at 160°C for 1 hour. The polymer concentration was 1% (w/v), and antioxidant 246 was added (0.1% w/w on polymer) when the polymers were precipitated into methanol. The recovered polymers were then dried in vacuum oven at 70°C for 72 hr.

The isothermal crystallization and melting study of m-PE using simultaneous SAXS and WAXD techniques were carried out at the Advanced Polymers Beamline (X27C), in National Synchrotron Light Source (NSLS), Brookhaven National Laboratory (BNL). Two position sensitive detectors (European Molecular Biology Laboratory, EMBL) were used to detect the SAXS and WAXD signals simultaneously. The chosen wavelength was 0.1307 nm, the sample to detector distance for SAXS was 1618 mm and for WAXD was 102 mm, and the data acquisition time was 20 seconds per scan. The specimen was first equilibrated above the melting temperature (> 150°C) for 5 min to erase the thermal history and residual crystallinity, and then rapidly jumped to the desired temperatures for measurements using a dual-chamber temperature jump apparatus. Two modes of experiments, isothermal crystallization (hold for 30 min) and subsequent melting with a heating rate of 2.5°C/min, were carried out.

DSC measurements were carried out with a Perkin-Elmer DSC-7 station. The samples were first heated at a rate of 80°C/min from room temperature to 180°C. After equilibrated at 180°C for 4 min, the sample was cooled at a rate of 2.5°C/min from 180 to 40°C, and then heated at the same rate from 40 to 180°C. DSC measurements were also carried out with the same thermal conditions as in SAXS and WAXS measurements. The heat of fusion was converted to the degree of crystallinity using the value of289 J/g for 100% crystallinity in PE 1 6 .

Results and Discussion

Figure 1 (a) illustrates DSC scans of samples with different degree of short chain branches at a heating rate of 2.5°C/min. Figure 1 (b) shows DSC scans of the samples cooled from 180°C at the same rate. The melting and crystallization temperatures (Tm,

190

Tc), the degree of crystallinity (χ ς, wt %), the heat of fusion (AHm) and the heat of crystallization (AHC) are listed in Table 1. It is seen that both values of T m and %c

decrease with the increase of branch content, indicating that the formation of crystals becomes probably more defective and thinner with higher branch content.

Figure 1. DSC thermograms obtained from m-PE samples with different branching degrees: (a) heating with a rate of 2.5 °C/min; (b) cooling with a rate of2.5°C/min.

Table 1. Thermal Parameters of Ethylene/Octene copolymers Cooled and Heated at 2.5°C/min

Samples Tm AHm Ze AHC Branch

(°C) (J/g) (%) CQ m /1000C SI 131.708 134.787 46.639 118.408 -167.447 0.00 S2 116.958 85.993 29.755 106.491 -93.101 9.65 S3 108.508 69.067 23.899 96.450 -80.901 14.1 S4 107.908 66.509 23.013 96.200 -76.290 20.6 S5 99.258 29.936 10.358 85.908 -48.510 28.6

Isothermal crystallization and subsequent melting experiments by DSC were carried out in the following manner. Samples were first heated up to the temperature above their nominal melting points. Then, they were rapidly cooled to the desired crystallization temperature at a rate of 200°C/min. After isothermally crystallized for 30 min, the samples were subsequently heated up to 150°C at a rate of 2.5°C/min. DSC heating scans of samples under this thermal condition are shown in Figures 2(a)-(e). With the increase in branching concentration, a small second melting peak at lower temperature appears, which suggests that a second population of defective crystals occurs during isothermal crystallization in highly branched samples. Our explanation for this observation is as follows. It is known that crystallization under

191

100 110 120 130 140 150 160 90 100 110 120 130 140

Torçerature^ Temr*3ature(<€)

70 80 90 100 110 120 130 80 90 100 110 120 130 140

Temperature (<€) Tenperature^

Figure 2. DSC melting thermograms for (a) linear, (b) 9.65, (c) 14.1, (d) 20.6, (e) 28.6 branches/1000C PE crystallized at different temperatures for 30 min. The melting scans were initiated at the crys­tallization temperatures with a rate of 2.fC/min.

60 70 80 90 100 110 120 130 140

Température (°Q

192

isothermal condition always proceeds in two stages for most polymers. In the primary stage, crystallization takes place in the virgin melt and produces relatively thicker lamellae; in the secondary stage, crystallization occurs in the restrained melt (imposed by the primary crystals and thus with lower entropy) and produces thinner lamellae. It has been shown that more secondary crystallization occurs at lower isothermal temperatures. In Figure 2, we notice the lack of the annealing-induced melting peak in polymers with less branch concentration. Perhaps, this behavior is due to the increased mobility for reorganization in the crystal with chains of less branch content. In other words, as the linear PE is able to reorganize leading to isothermal thickening, the branched PE probably loses this ability due to the segregation of the branched chains near the crystal surface. As a result, a secondary population of crystal thickness may prevail in order to increase the crystallinity.

Figure 3 shows melting temperature changes with crystallization temperature as observed by DSC. It is seen that melting temperature increases with isothermal crystallization temperature. This is the conventional Hoffman-Weeks plot, which has been used to extrapolate the value of equilibrium melting temperature (Tm°). However, we note that the reorganization ability in the polymer crystal is quite different between the linear and the branched polymer, such a plot does not yield the correct extrapolation of Tm°. Instead, we have adopted the method of Thomson-Gibbs to extrapolate Tm° using the data from time-resolved SAXS and DSC together, which will be illustrated later. Figure 4 shows the degree of crystallinity (DSC) in samples

Figure 3. Relationships between Tm and Tc in samples with various degrees of branching

Figure 4. Crystallinity xc ( by DSC) vs Tc for samples crystallized at various temperatures for 30 min

as a function of crystallization temperature. It is clear that the degree of crystallinity decreases with crystallization temperature in all samples when annealed isothermally. The rate of the crystallinity decrease is about the same for different polymers. The

193

degree of crystallinity also increases with the decrease in the branch content of PE, especially at higher crystallization temperatures. This can be explained by the argument that the reorganization of the chains with higher branch content becomes more difficult as the averaging sequence length decreases.

Isothermal crystallization and subsequent melting behavior in m-PEs were also investigated as a function of crystallization temperature by time-resolved x-ray techniques. Figures 5 (a) and (b) show typical time-resolved SAXS and WAXD

Figure 5. Simultaneously collected (a) SAXS and (b) WAXD profiles of sample S2

profiles of m-PE sample during isothermal crystallization and subsequent melting at a rate of 2.5°C/min. In these figures, a scattering maximum in SAXS (except for linear PE, where the scattering maximum is not seen — an indication that the long period >

194

600A), and two crystal reflections (110 and 200) in WAXD are seen during isothermal crystallization and then disappear upon melting.

The invariant Q, long period I, crystal lamelar thickness lc and interlamelar amorphous thickness la were calculated from the SAXS data using the correlation function method17, which are shown in Figures 6 (a) and (b) for the S2 and S3 samples

Figure 6. Morphological variables Q, L, lc and la of (a) S2 and (b) S3 determined by the correlation function analysis of SAXS data

respectively. It is seen that both values of L and lc decrease with time during the isothermal crystallization period. Initially, the decrease is relatively large; at the later stage, the decrease becomes small. Upon heating, a reversible process (L and le

increase with temperature) is seen. The decreases in L and lc with time can be

195

attributed to secondary crystallization forming thinner lamellae, which is consistent with the slight increase of crystallinity obtained by WAXD. We further observe that the variation of the amorphous layer thickness is relatively small, which suggests that the secondary crystallization process probably proceeds in the form of lamellar stack insertion8. Coinciding with the appearance of annealing-induced endotherm in DSC, it is found that Q9 L, 4 of the S3 sample in Figure 6 (b) show a step change during heating, which can be attributed to the melting of secondary crystallization. The samples S4, S5 show very similar treads of g, L, lc, la change as S3 in Figure 6 (b).

In Figure 6, we can determine several useful variables to explore the effects of crystallization temperatures and branch compositions on the lamellar morphology and the crystallization kinetics. These variables include 4*, the time to reach the plateau value of g, the plateau values of long period (I*), lamellar thickness (4*) and interlamellar amorphous thickness (4*)· The physical meaning of 4* probably indicates the completion of the primary crystallization process. Figure 7 illustrates the values of L*9 4* and 4* as functions of crystallization temperature and branch composition. Al l values increase with the increase in T c as expected; they are also shifted to lower T c values (left) with the increase in branch content. This clearly indicates that the increase in the branch content significantly changes T c (or Tm) but without affecting the crystal thickness. However, the corresponding plateau crystallinity (x0 from WAXD) plot in Figure 8 indicates that the crystallinity decreases with branch concentration (In WAXD, from the relative amount of the integrated intensity of the crystal reflections, the degree of crystallinity %c can be determined as, XcJo00 les2 às I f0°° Is2âs, where / is the total scattering intensity, Ic the scattering intensity of the crystal reflections and s is the scattering vector, s= (2/A)sin(0), 2 0 being the scattering angle.). It is seen that tc* is also functions of both branch concentration and crystallization temperature (Figure 9). With the increase in branch concentration, the time to reach the plateau value of Q increases probably because the averaging sequence length in ethylene is decreased and the reorganization ability of the chain with more branches becomes difficult.

As stated earlier, that the melting temperature can be related to the crystal lamellar thickness 4 by the Thomson-Gibbs equation1415, Tm=Tm° (U2GJAU{° 4) = Tm

0-2aeTm°/AHf0 4, where Tm° is the equilibrium melting temperature, oβ is the surface free energy of the mature lamellae, AHf° the heat of fusion per cubic centimeter, 4 is the lamelar thickness, and T m is the melting temperature. The above equation gives a direct method to extrapolate two parameters, σβ/ΔΗ and Tm°, by plotting the T m (from DSC) vs l / / c * (from SAXS) (Figure 10). The assumption of this plot is that there is no further lamelar thickening in the crystal thickness upon heating, and the melting point is directly proportional to the equilibrated lamellar thickness (4*)· It is seen that linear relationships between T m (the peak melting temperature) and 1/4* are found (see Table 2) for various branch concentrations (black symbols). It appears that these different linear relationships can intercept the T m axis to almost a single point, yielding a common equilibrium melting temperature (Tm°) of 144.1°C (as the average of four different intercepts). The extrapolated 144.1°C is higher than the typical ranges

196

400

350

^ 300

^ 250

200

150

80

* ~3

50

40

(a) (a) δ S2 • Δ • S3 •

• S4 9 Δ

• S5 • • Δ

• • Δ • Δ

9 • Φ •

60 80 100 120 T C ( ° Q

(c) Δ S2 70 • S3

• S4

60 • S5 Δ Δ Η

π 9 • D J »

60 80 100 T C ( ° Q

120

350

300

~ 250

*

- 0 200

150

100

Φ) Δ S2 • S3 • S4

S5 • Δ •

ι *

60 80 100 T C ( ° Q

120

Figure 7. (a) L* (b) le* and (c) la* vs Tc of samples with various degrees of branching

0 • •

• • Δ S2

• • • S3 • S4

• S5

80 90 T ( ° Q

100 110 120 130 50 60

T C ( ° Q Figure 8. χα (obtained by WAXD) vs Tc of Figure 9. tc* vs Tc ofm-PEs with samples with various degrees of branching various degrees of branching

197

of extrapolated values (136.9-142.9°C) determined by thermal analysis methods18 and is slightly lower than the value (145,5°C) determined by Flory and Vrij 1 9 . Recently, Marand et al. have reviewed the subject of the equilibrium melting temperature for PE 1 8 . They have concluded that the value determined by Flory and Vrij perhaps is closest to the true value of Tm°. Typical thermal analysis probably underestimates the equilibrium melting temperature because the assumption of the Holfman-Weeks approach does not truly account for the crystal reorganization during heating. We thus confirm that with the use of the Thomson-Gibbs equation, the equilibrium melting temperature can be estimated correctly. This conclusion somewhat contradicts the report by Mandelkern et al, who stated that the restrictions imposed by copolymer melting do not allow the use of the Thomson-Gibbs equation because the composition of the copolymer during melting may change continuously.20 We feel that their view point rings true when the true value of lamelar thickness and corresponding melting temperature are not obtained correctly, as by most indirect characterization methods. For example, the values for the lamelar thickness used in most studies (by SAXS or Raman spectroscopy — LAM) were determined at room temperature, which was lower than the values determined at crystallization temperature as in this study. The lower value of lamellar thickness would result in a low extrapolated value for equilibrium melting temperature. We believe our study has produced a more correct correlation between the melting temperature and the lamelar thickness, as the sample did not undergo major reorganization during the heating process.

ο

A S2 S2 Regression

Τ S3 S3 Regression

• S4 S4 Regression

• S5 S5 Regression

V S3 0 S4 0 S5

0.000 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 l/'c* °A'1

Figure 10. The Thomson-Gibbs plot (Tm vs l/lc*) of samples with various degrees of branching

198

Table 2. The slope, interception and error of PE samples Samples S2 JS3 S4 S5 interception 144.18 144.32 145.19 142.54 slope -8561.41 -9054.32 -11385.31 -12715.32 r 2 0.963 0.981 1.000 1.000

In Figure 10, however, we notice that the values at lower melting temperature (open symbols) deviate slightly from the linear line. These data were not used for the extrapolation. The main reason for this deviation is that higher heterogeneity in the crystal thickness occurs due to greater secondary crystallization at lower crystallization temperatures. The change of the slope indicates that the ratio of aJAH0

increases, which suggests that either the surface free energy of the lamellar crystals increases or the heat of fusion decreases with the increase of degree of branching. Both cases are expected as we believe that: (1) the majority of the branched chains are probably segregated on the crystal surface, leading to the increase of the crystal surface energy; (2) a small inclusion of some branched chains in the crystals is also likely, which would decrease the heat of fusion for the branched PE. The second viewpoint is consistent with the observations of larger unit cell parameters and broader peak width of the crystal reflection in the WAXD profiles of PE with higher branching.

Both time-resolved SAXS and WAXD techniques are capable to follow the crystallization kinetics. In SAXS, the scattering power Q can be determined as, Ο=4n/ο00 Is2as, where / is the scattering intensity in electron units divided by the volume of the sample and the intensity of the primary beam. As we have collected SAXS and WAXD signals simultaneously, the kinetics results between the two techniques can be compared with each other. In Figure 11, the values of Q and x0 of the m-PE sample with 9.65 per 1000 C branch concentration are plotted together during crystallization and subsequent melting processes. The results show a small time lag between Q (from SAXS) and xc (from WAXD). The invariant Q appears to occur slightly prior to x0 during crystallization and disappears later upon melting. However, this behavior is only seen in the samples with higher branch content and under a low degree of supercooling. Otherwise, the kinetics between SAXS and WAXD is almost identical. The value of Q can be interpreted by a model of three-phase system during crystallizaton: core crystal lamelar phase, an amorphous interlamelar phase, and an interfacial layer 2 1 , 2 2. The invariant for this three-phase system can be defined by Q=Xai(Xc+Xa2)(Ap)2, where %c is crystalline fraction, amorphous fraction and interfacial fraction. Δρ is the density difference between the two phases (crystal and amorphous) measured in electrons per unit volume. The delayed appearance of WAXD may be due to the initial formation of the very imperfect crystals, which do not give sufficient diffraction signals in WAXD but may exhibit large density contract from the amorphous and crystalline phases in SAXS. This is consistent with our vision of defective crystals formed in the higher branched polyethylene samples. Therefore, density fluctuations (manifest the SAXS signals) are probably present prior to crystallization (manifest the WAXD signals) in the sample

199

Figure U. Comparesion of invariant Qfrom SAXS and crystallinity (χε) from WAXD during isothermal crystallization

of high branch content, which also persists upon melting. We intend to further explore this issue later with a higher intensity synchrotron X-ray source.

Conclusion

A series of metallocene-based polyethylenes with different degrees of octene branching have been studied during isothermal crystallization and subsequent melting processes using simultaneous synchrotron SAXS/WAXD and differential scanning calorimetry in real time. It was found that a second population of defective crystals occurs in samples with high branching concentration after isothermal crystallization. In addition, we observed that secondary crystallization dominates at lower isothermal temperature. The decreases in long period and crystal lamelar thickness with time during isothermal crystallization can be attributed to secondary crystallization forming thinner lamellae, which is consistent with the slight increase of crystallinity from WAXD. The formation of thinner lamellae is probably favored by the restrained melt during secondary crystallization and the higher branch density of the sample which reduces the mobility of reorganization in the crystals. The smaller variation of the amorphous layer thickness suggests that the secondary crystallization process probably proceed in the form of lamellar stack. In this study, we confirm that the equilibrium melting temperature (Tm°) cannot be extrapolated using the conventional Hoffman-Week approach because the reorganization behavior of the crystal chains is different between the linear and the branched polymer. However, we demonstrate that Tm° can be extrapolted using the Thomson-Gibbs equation with results from real time SAXS and DSC, which has the value of 144.1°C that agrees reasonably well with the value determined by Flory and Vrij. Furthermore, we found that the surface free energy of the lamellae probably increases with the degree of branching and the heat of

200

fusion for the branched polyethylene may be also decreased. Finally, by comparing the kinetics results of SAXS and WAXD, we observed that the SAXS signals occur prior to WAXD only in the highly branched polymer under low degrees of supercooling. We plan to explore this subject in greater details later.

Acknowledgment

The assistance from Dr. F. J. Yeh, Dr. Z. G. Wang, S. Kim and J. L. Lopez during synchrotron measurements is gratefully acknowledged. The financial support of this project is by a grant from the National Science Foundation (DMR 9732653).

References

1. Wu, P. C. MetCon 94 Proceedings, USA, May 1994. 2. Montagna Α. Α.; Floyd, J. C. Hydrocarbon Processing 1994, 3, 57. 3. Walton, A. L. SPO 196, Proc. Int. Bus. Forum Spec. Polyolefins, 6th 1996. 4. Anon. Res. Discl. 1997, 402 (oct.), pp745-748. 5. Pinchard, G. Metallocenes '95, Int. Congr. Metallecene Polym. 1995, pp 145-157. 6. Reichard, G. C. Ph. D. Thesis, Princeton University, 1997. 7. Ryan, A. J.; Bras, W.; Mant, G. R.; Derbyshire, G. E. Polymer 1994, 35, 4537. 8. Verma, R. K.; Hsiao, B. S. Trends in Polymer Science, 1996, 4(9), 312 9. Kruger, Κ. N.; Zachmann, H.G. Macromolecules 1993, 26, 5202. 10. Hsiao, B. S. and Verma, R. K. J. Synchrotron Radiation, 1998, 11. Hu, S. R.; Kyu, T.; Stein, R. S. J. Polym. Sci.: Part B: Polym. Phys. 1987, 25, 71. 12. Kyu, T.; Hu, S. R.; Stein, R. S. J. Polym. Sci.: Part B: Polym. Phys. 1987, 25, 89. 13. Hoffman, J. D.; Weeks, J. J. J. Res. Natl. Bur. Stand, Part A 1962, 66, 13. 14. Thomson, J. J. Applications of Dynamics; Macmillan, London, 1988. 15. Gibbs, J. W. The Scientific Work of J. Willard Gibbs; V. I. Longmans Green:

New York, 1906. 16. Quinn, F. Α., Jr.; Mandelkern, L. J. J. Am. Chem. Soc. 1958, 80, 3178. 17. Strobl, G. R.; Schneider, M. J. Polym. Sci.: Polym. Phys. Ed., 1980, 18, 1343. 18. Flory, P. J.; Vrij, A. J. Am. Chem. Soc. 1963, 85, 3548. 19. Marand, H.; Xu, J.; Srinivas, S. Macromolecules 1998, 31, 8219. 20. Lu, L.; Alamo, R. G.; Mandelkern, L. Macromolecules 1994, 27, 6571. 21. Hoffman, J. D. Treatise on Solid State Chemistry, Vol. 3. Plenum Press, New

York, 1976. 22. Mandelkern, L. Acc. Chem. Res. 1990, 23, 380.

Chapter 13

A Scattering Study of Nucleation Phenomena in Homopolymer Melts

Anthony J. Ryan1,2, Nicholas J. Terrill2, and J. Patrick A. Fairclough1

1Department of Chemistry, University of Sheffield, Sheffield S3 7HF, United Kingdom

2CCLRC Daresbury Laboratory, Warrington WA4 4AD, United Kingdom

The mechanism of primary nucleation in polymer crystallization has been investigated experimentally and theoretically. Two types of experiments nave been performed on polypropylene. Crystallization with long induction times, studied by small and wide angle X-ray scattering (SAXS and WAXS), reveal the onset of large scale ordering prior to crystal growth. Rapid crystallizations studied by melt extrusion indicate the development of well resolved oriented SAXS patterns associated with large scale order before the development of crystalline peaks in the WAXS region. The results suggest pre-nucleation density fluctuations play an integral role in polymer crystallization.

Processing of semicrystalline thermoplastics relies on the shaping of molten material in either moulds or dies and the stabilization of the shape produced by crystallization [7]. During crystallization a microstructure develops which can control the mechanical and aesthetic properties of the polymer. To produce useful materials it is essential to understand and predict this process.

Polymers in solutions and melts can be regarded as random objects whose size and shape are governed by inter- and intra-molecular interactions and dominated by entropy. In the crystal this is no longer true and the behavior of the chain is now influenced by the proximity of the neighboring chains and the van der Waals forces which act between them. The Gibbs free energy, G, is a balance between entropy, S, and the enthalpy, H=U + pV, where U is the internal energy,ρ the pressure and Vthe volume of the system, thus

G{T,p)={U + pV)-TS (1)

In the melt, entropy dominates and the polymer has a Gaussian (random) structure. Crystallization is a process involving the regular arrangement of chains and

© 2000 American Chemical Society 201

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is consequently associated with a large negative entropy change. For the free energy change upon crystallization to be favorable there must also be a large negative enthalpy change, generally associated with an increase in density and a reduction in internal energy.

The creation of a stable 3-D structure from a disordered state (i.e. a polymer melt) is generally considered a two step process. The first step is called nucleation and involves the creation of a stable nucleus from the entangled polymer melt. Most technical processes involve secondary or heterogeneous nucleation either from specially added surfaces, nucleating agents, or adventitious surfaces, such as dust particles.

In primary (homogeneous) nucleation, creation of a stable nucleus is brought about by the ordering of chains in a parallel array stimulated by intermolecular forces. As a melt is cooled there is a tendency for the molecules to move toward their lowest energy conformation, and this will favor the formation of co-operatively ordered chains and thus nuclei. However two factors impede the ordering required for nucleation: cooling, which reduces diffusion coefficients, and chain entanglements. In fact the thermal motion needed for diffusion may be enough to cause the incipient nuclei to melt. The second step is growth of the crystalline region by the addition of other chain segments to the nucleus. This growth is impeded by low diffusion coefficients at low temperatures and thermal redispersion of the chains at the crystal-melt interface at high temperatures. Thus, the crystallization process is limited to a range of temperatures between the glass transition temperature, Tg, and the melting point, Tm. The alignment of polymer chains at specific distances from one another to form crystalline nuclei will be favored when intermolecular forces are strong. The greater the interaction between chains and the easier they can pack the greater the energy change will be. Thus symmetrical chains and strongly interacting chains are more likely to form stable crystals.

Despite the maturity and market penetration of the polymer industry one aspect of polymer processing, primary nucleation, is little understood. Whereas the growth of polymer crystals is well established in literature, and there are usable theories to predict the kinetics of crystallization, understanding of the initiation or nucleation step remains somewhat a mystery. The available theories are complicated and somewhat unphysical [2]. There is good reason for this. Experimental access to the nucleation step is very difficult, whereas studies of crystal growth are simple enough to be used in undergraduate laboratory classes [3].

Upon cooling a crystallizable polymer melt a hierarchy of ordered structures emerges. Elegant experimental and theoretical work (on melt and solution crystallized materials, especially single crystals) in the 1960s and 70s allowed useful models to be developed [4,5], First there are crystalline 'lamellae', comprising regularly packed polymer chains, each of which is ordered into a specific helical conformation. These lamellae interleave with amorphous layers to form 'sheaves', which in turn organize to form superstructures e.g. spherulites. These structures may be probed by various techniques: wide-angle X ray scattering (WAXS) is sensitive to atomic order within lamellae (giving rise to Bragg peaks), while small-angle X ray scattering (SAXS) probes lamellae and their stacking. Electron microscopy is useful for visualizing the crystal structure (by selected area electron diffraction), the lamellae morphology (by transmission microscopy of surface replicas, or direct scanning electron microscopy) and spherulites (by scanning electron microscopy) and this has been reviewed in some detail [6]. The growth of crystals by secondary nucleation has been explained by various versions of "regime" theory [2].

In the classical picture of polymer melt crystallization we expect and observe Bragg peaks in WAXS after an induction period η. SAXS accompanies the WAXS, corresponding to interleaved crystal lamellae and amorphous regions [7]. No SAXS is expected during η. However, recent experiments [8-12] have reported SAXS peaks

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during the induction period and before the emergence of Bragg peaks. Initially the SAXS peak intensity grows exponentially, and it may be reasonably fitted to Cahn-Hilliard (CH) theory [13] for spinodal decomposition: that is, the spontaneous growth of fluctuations indicative of thermodynamic instability. The peak moves to smaller angles in time, stopping when Bragg peaks emerge. By fitting to CH theory an extrapolated spinodal temperature (at which the melt first becomes unstable towards local density fluctuations) can be obtained [12]. Spinodal kinetics have been reported in different polymer melts: polyethylene terphthalate) (PET) [8-10], poly(ether ketone ketone) (PEKK) [77], polyethylene (PE) and isotactic polypropylene (iPP) [72]. An argument has been made against the spinodal crystallization mode, by Janeschitz-Kriegl [14], who invoked a variety of liquid-gas analogies and consideration of surface tensions, in spite of considerable experimental evidence for spinodal modes. To explain recent results, a spinodally assisted crystallization (SAC) model has been developed [75], which deals with the contributions to the free-energy in terms of coupled order parameters of density and chain conformation and predicts a liquid-liquid phase diagram buried in the liquid-crystal coexistence region.

Recent rheological studies [16] show that gelation occurs in crystallizing melts at very low degrees of crystallinity, typically < 3%, shortly after η. This means that there is a percolating network formed that is observed in the dynamic mechanical response of the system. If impingement of spherulites were the reason for gelation then the gelation would be observed at much higher degrees of crystallinity (typically >40%). There are also anomalies in the measurements of spherulite growth rates [77]. Heat capacity, volume change and X-ray experiments observe an induction time, whereas a plot of the spherulite radius with time is seen to pass through the origin, indicating that spherulites started to grow without an induction period being observed. Both pieces of evidence indicate that large-scale structure is formed prior to the appearance of Bragg peaks or evolution of appreciable enthalpy.

In this paper we present new experimental evidence for precrystallization density fluctuations in a range of polymer melts. In order to separate nucleation from growth, two types of experiments have been performed on polypropylene. Rapid crystallization studied by melt extrusion indicate the development of well resolved oriented SAXS patterns associated with large scale order before the development of crystalline peaks in the WAXS region. Crystallizations with long induction times, studied by small and wide angle X-ray scattering (SAXS and WAXS), reveal the onset of large scale ordering prior to crystal growth.

Experimental

The polypropylene used was a commercial grade, S-30-S (ex. DSM), which was free from any additives. The number-average molar mass (GPC) and the polydispersity were 52 kg/mol and 2, respectively, and the melting point by differential scanning calorimetry (DSC) was 165 ± 2 °C.

Simultaneous SAXS/WAXS/extrusion measurements were made on beamline 16.1 of the SRS at the CCLRC Daresbury Laboratory, Warrington, UK. The details of the storage ring, radiation and camera geometry and data collection electronics have been given in detail elsewhere [18]. An extruder above the x-ray position was used to provide a steady stream of crystallizing polymer past the x-ray beam. Tape extrusion is a steady-state process which shows post-die plug flow, therefore the distance down the spinline where the observation was made correlates with the time since the material left the extruder die. The material in the x-ray beam is continuously replaced by material with the same shear and temperature history. A tape of polymer melt was extruded from a die (of dimensions 0.5 χ 3mm) at 205 °C and collected via a

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wind up mechanism below the x-ray beam. The extruder used was an AXON BX18 that operated in starve-feed mode to minimize the time the polymer spent in the melt. The experimental set up is shown schematically in Figure 1. The distance from the die head to the beam could be varied between 0.3 and 1.8 m. The camera is equipped with a multi-wire area detector (SAXS) located between 3 and 8 m from the sample position with flight path between the WAXS and SAXS detector being under vacuum. Two types of area detector were employed for the WAXS. In order to observe the development of structure during an extrusion experiment a second multi-wire area detector was used which was offset from the center line of the beam and located approximately 20 cm from the extruded tape. Time resolved SAXS/WAXS measurements were made with the multi-wire area detector intersecting either the meridian or the equator. The spatial resolution of the electronic area detectors is 400 pm and they can handle up to -500 000 counts s"1. To probe a wide range of reciprocal space an image plate with a hole was used to record flat plate WAXS patterns contemporaneously with SAXS. The exposure time was set a 30s using a mechanical shutter. A4 size Fuji and Kodak image plates were read with a Molecular Dynamics image plate reader with a spatial resolution of 176 pm.

The scattering pattern from an oriented specimen of wet collagen (rat-tail tendon) was used to calibrate the SAXS detector and HDPE, aluminum and an NBS silicon standard were used to calibrate the WAXS detectors [19], Parallel-plate ionization detectors, placed before and after the sample, recorded the incident and transmitted intensities. The experimental data were corrected for background scattering for sample thickness and transmission, and for the positional alinearity of the detectors.

Simultaneous SAXS/WAXS/DSC measurements were made on beamline 8.2 of the SRS at the CCLRC Daresbury Laboratory, Warrington, UK. The details of the storage ring, radiation and camera geometry and data collection electronics have been given in detail elsewhere [20]. The camera is equipped with a multi-wire quadrant detector (SAXS) located 3.5 m from the sample position and a curved knife-edge detector (WAXS) that covers 120 ° of arc at a radius of 0.2 m. A vacuum chamber is placed between the sample and detectors in order to reduce air scattering and absorption. The WAXS detector has a spatial resolution of 100 urn and can monitor up to -50 000 counts/s. Only 60° of arc are active in these experiments with the rest of the detector being shielded with lead. A beam-stop is mounted just before the SAXS exit window to prevent the direct beam from hitting the SAXS detector which measures intensity in the radial direction (over an opening angle of 70 ° and an active length of 0.2 m) and is suitable only for isomorphous scatterers. The spatial resolution of the SAXS detector is 400 pm and it can handle up to -500 000 counts/s. Disk specimens of polymer (thickness - 1 mm, diameter - 8 mm) were cut from pre-molded sheet. And encapsulated in a DSC pan fitted with mica windows (thickness -25 pm, diameter - 7 mm), and the pan was inserted into a Linkam DSC apparatus of the single-pan design that has been described in detail elsewhere [20]. The cell comprises a silver furnace around a heat-flux plate containing a 3 χ 0.5 mm slot, for X-ray access, and the sample is held in contact with the plate by a spring of low thermal-mass. The temperature was calibrated using the melting points of high purity indium and tin. In the present study, a multi-stage temperature program was used as follows: (i) heat to 200 °C at 50 °C min"1; (ii) hold at 200 °C for 1 min; (iii) cool at 60 °C min"1 to the crystallization temperature and hold for 10-120 minutes depending on the temperature. The crystallization time at each temperature was chosen so that at least half the primary crystallization kinetics were observed. To limit beam damage on longer crystallizations the samples were only exposed to the x-ray beam for 10% of the time after crystallization had started. Data were reduced to intensity versus scattering vector using the CCP13 program xotoko [21]. The peak intensities and areas were calculated using the CCP13 program fit [22]. For the SAXS data a Gaussian peak (whose position was a variable) was fitted on top of a Porod

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Figure 1. A schematic diagram of the SAXS/WAXS/extrus ion experimental set-up.

background. For the WAXS data Gaussian peaks were fitted on top of a Gaussian background, the positions of the 4 Gaussian peaks were set according to the positions of the iPP reflections at M/values of 110,040,130 and (111,131,041).

Results

The crystallization curve shown in Figure 2 was obtained in a SAXS/WAXS/DSC experiment from iPP [23] and shows the classic features of primary crystallization. The detailed molecular structure of the polymer, the specific nature of the nucleation processes and the degree of under-cooling, determines the magnitude of the lamellar thickness and the degree of crystallinity within the lamellar stacks. The crystallization kinetics are analyzed using the Avrami model [24], expressed in terms of the equation

\-Xs =exp(*rw) (2)

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where Xs is the fractional crystallinity, t is time, k is a rate constant and η is an integer that is sometimes interpreted in terms of the growth dimension. Figure 2 shows a plot of Xs versus t from SAXS & WAXS in which the experimental data have been fitted with the Avrami expression over the whole of the primary crystallization process. The fit to the data gives a value for the exponent, n, of 3.0 ± 0.1. The value of η obtained is consistent with random nucleation of spherulites, and is in good agreement with the crystallization kinetics data obtained from dilatometric and calorimetric studies on iPP [25]. It is interesting to note that there are significant deviations of the model from the data at the extremes of crystallization and that the evolution of the structure probed by SAXS & WAXS is also different at the beginning and end of the primary crystallization process. The long time differences have been reviewed in some detailed [26] whereas the initial differences have been largely ignored.

It is difficult to separate nucleation from growth in due to the low concentration of nuclei, which gives rise to poor counting statistics in a scattering experiment. One potential method is to borrow from elementary chemical kinetics [27] and use a flow apparatus. An extruder operating at steady state provides such a set-up (see Figure 1). Polymer above the melting point is extruded from a die, the tape or fiber cools in the air (in our case in a column of chilled nitrogen) prior to being wound up as a solid. Extrusion of tape or fiber is a steady-state process where the crystallization time increases down the spin-line.

0 50 100 150 200 250 300 350 400 t/s

Figure 2. The degree of crystallinity, X, versus crystallization time for Ρ Ρ at 110 °C. The solid circles are SAXS and the open squares are WAXS measurements. The solid line is a fit of the Avrami equation to the average degree of crystallinity. The inset shows a conventional Avrami plot ofln(ln(l-X)) versus In t.

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Figure 3. Data taken during extrusion of iPP with a low wind-up speed using a 2-D wire chamber to collect the SAXS data and an image plate to collect the WAXS data, (a) Close to the die head tear shaped scattering features are observed in 2-D SAXS with only an amorphous halo in 2-D WAXS. (b) At the furthest position from the die head, when crystallization starts, isotropic rings from crystals can be seen in SAXS and in the WAXS.

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This allowed long data collection times (minutes) for very early stages of crystallization (milli-seconds).

Prior to the development of crystals, well resolved, oriented small-angle patterns could be observed with length scales (50-200 A) and intensities that grew down the spin-line. The corresponding WAXS showed no Bragg peaks due to crystals. Figure 3 shows the scattering patterns collected during extrusion of iPP. At short times these early patterns have two SAXS peaks at finite q and no WAXS peaks. We interpret this as a signature of density fluctuations. The orientation observed in the scattering is caused by coupling of density fluctuations with the slight elongational flow-field (the take-up speed was approximately twice the extrusion speed). Once crystallization had been observed in the wide-angle region, the shape of the small-angle pattern changed to that typical of lamellar crystals. Since the elongational flow was weak the crystallization process dominated and only weakly anisotropic crystals were produced. The SAXS peaks shown at early times in Figures 3a were approximately 100 times weaker than the diffraction ring observed in SAXS once spherulitic crystallization had occurred (Figure 3b).

There are a number of other possible interpretations of the data. The SAXS peak could be due to the formation of oriented nuclei, the precursor to the "shish" in "shish-kebab" crystals and row nucleation [28], but this should also lead to orientation in the WAXS which is not observed. Furthermore, the elongation is less than a factor of 2 which is very low for formation of such structures [1,2].

Nucleation could have formed poorly ordered crystals that do not diffract. This would account for the lack of Bragg peaks in the WAXS (peak broadening due to small crystallites), but does not account for the peak in SAXS, the scattering from a low concentration of randomly oriented objects would give a peak at q=0 from the shape factor of the scatterers[2P]. Guinier-Preston zones, which form in systems with conserved order parameters, [30] have nuclei surrounded by a depletion layer. These would give a peak in SAXS at low concentration due to the shape factor. However, in this system with a non-conserved order parameter, the nuclei are not surrounded by a depletion layer. Regions of high density grow from a background of low density with an overall increase in density and the electron density profile does not have a peak at finite q.

The most obvious alternative explanation of the observation is that there is an outer skin of crystalline material, due to the temperature gradient across the tape, which is giving rise to the SAXS. In this situation the crystals formed would be well ordered and one would expect them to diffract at wide angles, which is obviously not the case. Under the current experimental conditions, it is possible to observe diffraction from 1% by volume of a crystalline olefin dispersed in oil [57]. In order to check the temperature of the melt in the scattering volume a number of techniques were tested, an optical pyrometer indicated that the data taken close to the x-ray position (in figure 3a) had a temperature in excess of 120 °C. Whilst it is undoubtedly true that at long times crystallization proceeds from the exterior of the tape, is unlikely that it has occurred in the data presented here.

A caveat to the interpretation of the data concerns the intensity (solid angle) on the Ewald sphere. Assuming peaks of equal strength, the measured intensities scale as \/dF, where d is the Bragg spacing and is typically 100 and 5 A for SAXS and WAXS respectively. The SAXS detector is 5 times further away from the sample so, for the same detector efficiencies, the intensities scale as the reciprocal of distance squared. Overall the measured WAXS intensity will be approximately 0.06 (-52 x 52/1002) of the SAXS intensity for peaks of equal strength. For semicrystalline polymers, not withstanding the argument above, when compared to the background intensity, WAXS peaks are generally very strong and SAXS peaks are often quite weak. Previous SAXS/WAXS studies on polymer extrusion have concentrated on the growth and orientation of crystals [32,33]. Interestingly both studies, Cakmak et. al. [52] on

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PVDF tape (using synchrotron radiation) and Katayama [33] on PET fibers (using a sealed tube source and film as a detector) showed SAXS before WAXS down the spin line, but made no comment on its significance.

Slow crystallizations, with long induction times, have been studied by simultaneous SAXS and WAXS. These experiments, on quiescent samples, show a clear development of a SAXS peak prior to the presence of crystals identified by WAXS. The peak area versus time data in Figure 4 for iPP at 137 °C show, unequivocally, that the SAXS peak grows before the WAXS peak. In the first 200 s of the experiment there is no scattering above the background intensity. Between 200 and 400 s there is a measurable SAXS intensity with no WAXS above the background. After 400 s Bragg peaks are observed and after 800 s the growth in SAXS and WAXS map onto each other. The crystallinity at 1200 s is = 0.1. The logarithm of the peak intensity versus time, for the period where there is SAXS without WAXS, gives a good straight line. Similar behavior has been reported previously for semi-rigid polymers crystallized by devitrifying a glass [8- 1 1] and the kinetics of crystallization after devitrification were analyzed in terms of the Cahn-Hilliard [13] theory for spinodal decomposition.

Figure 4. Integrated intensity data for a crystallization at 128°C in the SAXSWAXS/DSC

It has been shown that the general form of the variation in scattered intensity, 7( ,0, following a quench is given by:

The variation in I(q) at a given time interval is determined by the scattering law for the objects and R(q) is termed the growth rate constant and is given by:

* (?)=-Mtf 2 |^ + 2^ 2j (4)

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Here M is the mobility term, G is the Gibbs free-energy, and κ is a gradient free-energy term. Modifications to equation (4) have been discussed previously by Cook [34] which take into account random thermal fluctuations (inclusion of a Brownian motion term). In employing equation (4) to analyze the data, the extremums are not strictly correct. The q dependence of the Onsager coefficient relating the diffusive flux of polymer molecules to the local chemical potential has been neglected. This may be valid for the early stages of phase separation and a shallow quench. It should be noted that the Onsager coefficient generally does have a q dependence, and according to Pincus [35] is q2 for a polymer blend. Neglecting the Onsager coefficient, R(q)/q2 can be taken as a measure of the dynamic driving force for the growth of the concentration fluctuation with wave vector q/2%. There is a region of q in which R(q)9 and thus R(q)/q2

9 are positive and the concentration fluctuations do not decay but grow and give rise to phase separation. These growing concentration fluctuations have upper and lower critical boundaries to their wave numbers. Outside these limits, the concentration fluctuations decay and do not contribute to the phase separation dynamics. As originally published, the Cahn-Hilliard theory [13] of spinodal decomposition is a macroscopic description and has no direct relation to events at molecular level.

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035 0.004

q2/A"2

Figure 5. A Cahn-Hilliard plot to estimate the effective diffusion coefficient, DM from the q=0 intercept ofRfqj/q2 versus q2 for iPP at 120, 125, 130, 135 and 140 °C. The solid lines are a linear fit to the data.

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The thermodynamic driving force for the growth of the concentration fluctuation with wave vector q/2n, R(q)/<£9 becomes a maximum at q = qm = •(<? 7κ). Thus, the wavelength, q/2n, of the dominant Fourier component of the growing fluctuations in the early stages of phase separation is determined by the maximum dynamic driving force. qm is time independent in the early stages of phase separation and is controlled by thermodynamics [56]. R(q) is further controlled by the transport properties and is related to the flux of molecules where the effective diffusion coefficient, Deff, can be determined from an extrapolation to q=0 of the straight-line portion of R(q)lq2 during phase separation using

and the linearity holds for qm

<q<^2qm

Values of the amplification factor R(q\ for the early stage of crystallization where we observe SAXS but no WAXS, were determined by plotting In/ versus t for discrete wave vectors and finding the slope [36]. R(q)/q2VQrsus q2p\ots were constructed at a range of temperatures and are shown in Figure 5, the data have been truncated at low q [36]. As the crystallization temperature is increased it should be noted that the data get less noisy (as the kinetics slow down and counting statistics improve) and the linear part of the graph is reduced, this is because qm moves to lower values.

Figure 6 shows a plot of £) e f f versus 1/F, the spinodal temperature is determined by the £>eff = 0 intercept. For example, in polypropylene at 410 K, we could estimate

(5)

60

50

40

20 f

30 f

10

0 0.0023 0.0024 0.0025 0.0026

1/(T/K)

Figure 6. Λ plot of Deff versus 1/T (for iPP) to allow calculation of the spinodal temperature from extrapolation to &eff~ 0-

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both the dominant length scale L = 175 Å and the effective co-operative diffusion coefficient Deff = -4.5 AV 1 . By conducting these experiments at a series of temperatures the stability limit could be found at 415 ± 5 K [57] by extrapolation of D e f f to zero. The stability limits obtained in this way are compared to the melting point of an infinite crystal in Table 1 for iPP, PE and PET.

The stability limit is the temperature below which the polymer spontaneously separates into two phases. One of the phases is rich in polymer segments of the appropriate chain conformation to crystallize (trans-gauche arrangement of the carbon backbone in isotactic polypropylene [38]) and the other is concentrated in sequences near entanglements and other defects which cannot crystallize. The stability limit is 7 Κ below the measured melting point for polypropylene with a long spacing of 175 Å and 35 Κ below the thermodynamic melting point of isotactic polypropylene. [39]. Once WAXS from crystals (atomic order on the 1 A scale) was observed, the kinetics reverted to those of nucleation and growth, that is Avrami kinetics with an exponent n=3 (see Figure 2). Similar behavior has also been observed in devitrified glasses of PET, by Imai and co-workers [8-10], and PEEK, by Ezquerra and co-workers [11] however, as the measurements are made close to the glass transition and the dynamics are dominated by the viscosity, estimation of the stability limit is not possible as Deff increases with temperature.

Table 1 Stability limits and thermodynamic melting points [59].

T s T° HDPE 408 417 0.98

iPP 415 459 0.90 PET 499 573 0.87

The quiescent time-resolved SAXS/WAXS and extrusion suggest that a process that strongly resembles spinodal decomposition of chain segments with different average conformations is the nucleation step in polymer crystallization. That polymer crystallization occurs with phase separation is in no doubt, since at the end of the process regions of well ordered crystalline polymer coexist with regions of disordered polymer in a layered morphology (lamellae) with a spherulitic super-structure. Sequences that can be oriented with the right conformation and incorporated into the crystal separate from sequences near entanglements and other defects that can not crystallize and can only be part of the amorphous regions. The transformation from the disordered phase to the better ordered partially crystalline phase proceeds continuously passing through a sequence of slightly more ordered states rather than building up a crystalline state instantaneously. This is consistent with the evolution of SAXS before WAXS. At some stage secondary nucleation must form crystals directly from the melt and a mechanism of continuous transformation could be consistent with a fast homogeneous nucleation process. It is difficult, however, to make a clear distinction between spinodal decomposition and nucleation and growth with nucleation barriers smaller than k&T. Polymer crystallization, like any other phase separation, is kinetically controlled. The structure formed is the one with the highest growth rate. Once a crystallite is formed, its lateral growth rate is much higher than that of the fluctuations and so dominates. In this case the growth mechanism of semi-crystalline polymer lamellae, in the form of spherulites, takes over because the lateral

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growth rate of crystals (typically μιτι s"1) is much faster than the growth rate of the fluctuations (typically A s ) . Thus the combination of the steady-state extrusion and the high intensity, synchrotron X-ray source allows nucleation to be observed.

Discussion

To understand these observations a 'minimalist' phenomenological model has been developed which captures the physics involved. The essential observation is the existence of spinodal-like behavior in a super-cooled melt. By analogy with similar observations from metallurgy [40,41] and recent experiments in colloid-polymer mixtures [42], as well as super-cooled water [43], it is proposed that a metastable liquid-liquid (LL) phase coexistence curve (or 'binodaP) lies buried inside the equilibrium liquid-crystal coexistence region, as shown in Figure 7 [15]. Quenching sufficiently below the equilibrium melting point Ts, we may cross the spinodal associated with the buried LL binodal at temperature Ts < Tm.

In order to crystallize, polymer chains must adopt the correct conformation. For example the chains in crystalline iPP are alternating trans-gauche, but in the melt the conformation is

pw

Figure 7. A generic phase diagram for a polymer melt with a buried liquid-liquid predicted by the SAC model [15]. Tm and 7$ are the melting and spinodal temperatures encountered along the constant density quench path (dotted line). Inset shows the measured induction time as a function of temperature for iPP.

randomly trans or gauche. Furthermore, the radius of gyration of a (very long) chain changes little during crystallization, suggesting [44] that neighboring segments adopt the correct conformation and crystallize 'in situ'. While it is commonly assumed that conformational and crystalline ordering occur simultaneously, we suggest that these

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processes can occur sequentially. Moreover, chains with different conformations have different densities, and therefore also different energy barriers for reorientation between rotational isomeric states (RIS) [45]. Such conformation-density coupling can induce a LL phase transition. At low enough temperatures, the system gives up conformational entropy to relieve packing frustration, and separates into a dense, more conformationally homogeneous liquid and a less dense and more conformationally disordered liquid. In practice, this happens only at appreciable rates by spinodal decomposition, giving rise to two coexisting liquids, with a coarsening interconnected domain texture, as shown in Figure 7. The dense liquid is closer in density and conformation to the crystal phase than the original melt, with a lower energy barrier Δ to crystallization. We expect Δ to decrease with increasing quench depth below F s. The induction time r, is then a sum of the time to coarsen into an intermediate spinodal texture, and an exponential activation time determined by Δ. The strong temperature dependence of η should change over to a much weaker dependence at some Γ ρ < Ts, where Δ < kBTp. This has been found in iPP (inset, Figure 7) [12].

Our arguments so far have been based on conformation-density coupling. An analogous argument may be made in terms of a liquid-crystalline coupling, by which density-orientation effects become more important as the polymer stiffens upon cooling into the preferred helical conformation. This approach was adopted by Imai and coworkers [10], and probed by light scattering. Indeed, the two mechanisms have, in the main, the same physical content.

Until recently, spinodal scattering was mainly observed in polymer melts crystallizing under shear [7,32,49]. This may be understood in a natural way within the present framework. Shear (and extensional) flow couples principally to the orientation of polymer segments, hence straightening chains and biasing the tendency towards LL separation. The LL spinodal line can also be modified by pressure. In particular, it may be possible to access the LL critical point, Fc: recent simulations suggest a massive enhancement of the nucleation rate in the vicinity of Tc [50]. The effect of strain on the crystallization of PET close to the F g has been studied, using elegant time resolved scattering experiments, by Blundell et. al. [57]. Crystallization (determined by WAXS) occurred after the extension of 4:1 and followed first order kinetics (i.e. Avrami Λ=1 which is equivalent to spinodal). Close to the f g the rate of the transformation was temperature insensitive as a reduction in temperature caused an increase in the dynamic driving force but a reduction in mobility. We interpret this data as the LL energy barrier being reduced by chain orientation in extensional flow.

More generally, the coupling of density to (molecular) structural order parameters is an emerging generic theme in the study of super-cooled liquids (water - amorphous ice [43]; polymer melts near the glass transition [52]). Balsara and coworkers [53] have described interesting behavior in hexagonal rod forming block copolymers subjected to a deep quench. The microstructure formation in the liquid and crystal directions is not correlated, the growth of crystalline order occurs before the development of a coherent structure along the liquid directions and they argue that this may be a signal of spinodal decomposition in liquid crystals. The ratio of Ts/Tm

observed is 0.977, which is in the range described here for polymer crystallization. The analogy for the case presented here would be chains pack locally, in straight sections, prior to becoming oriented along their length into lamellae.

Summary

It is obviously very difficult to make a clear distinction between spinodal decomposition and nucleation and growth with nucleation barriers smaller than k^T. Crystallizations with long induction times, studied by small and wide angle X-ray

215

scattering (SAXS and WAXS), reveal the onset of large scale ordering prior to crystal growth. Rapid crystallizations studied by melt extrusion indicate the development of well resolved oriented SAXS patterns associated with large scale order before the development of crystalline peaks in the WAXS region. The results suggest pre­mutation density fluctuations play an integral role in polymer crystallization. It is tempting to consider setting up an instantaneous structure that in the conformation and density that comprises isotropic sine waves with random phase and direction but fixed wavelength. This sets the subsequent lamellar thickness or long spacing. The crystallization process which forms the perfected crystals which show Bragg peaks is a perfection and formation of grains, the growth of which we see as spherulites. Rheology [16\ indicate that a network structure is present at very early stages in crystallisation.

In a technical context, Gahlietner and coworkers have shown [54] that nucleating agents are most effective in the "metastable super-cooling zone". Nucleating agents cease to work in iPP at around 140 °C which is close to the measured Ts and that homogeneous nucleation takes over at temperatures below 110 °C which is close to Γρ. This is in general agreement with our theoretical model and experimental results, the crystallization rate is vanishing small above Ts and homogeneous nucleation is very fast below Tp.

Acknowledgement

The experimental work originated from an EPSRC ROPA grant to Professor RJ Young and AJR at UMIST. The theoretical support of Peter Olmsted, Wilson Poon and Tom Mcleish is most appreciated. The constructive critism of Frank Bates, John Blackwell, Gerhard Eder, Julia Higgins, Ben Hsiao, Andrew Keller (now sadly deceased), Herve Marand, Paul Phillips, Gert Strobl and an anonymous referee are most appreciated.

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4. Keller, Α.; Kolnaar, H.W.H. in Polymer Processing, Meijer, H.E.H., Ed.; Materials Science and Technology, Wiley-VCH, Weinheim, DE, 1997, Vol. 18 pp 189. P. Barham, in Structure property Relations in Polymers, Thomas, E.L., Ed.; Materials Science and Technology,Wiley-VCH, Weinheim, DE, 1993, Vol. 12 pp 153.

5. Eder, G.;Janeschitz-Kriegel, H. in Polymer Processing, Meijer, H.E.H., Ed.; Materials Science and Technology, Wiley-VCH, Weinheim, DE, 1997, Vol. 18 pp 269

6. D. C. Basset, D.C., Principles of Polymer Morphology, Cambridge University Press, Cambridge, UK, 1981.

7. Strobl, G. The Physics of Polymers, Springer-Verlag, Berlin, DE, 1996.

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ibid, 1992, 33, 4457. 10. Imai, M.; Kaji, K.; Kanaya, T. Macromolecules, 1994, 27, 7103. 11. Ezquerra, T.A.; López-Cabarcos, E.; Hsiao, B.S.; Baltà-Calleja, F.J. Phys. Rev.

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Young, R.J.; Ryan, A.J. Polymer, 1998, 39, 2381. 13. Cahn, J.W.; Hilliard, J.E. J.Chem.Phys., 1958, 28, 25. 14. Janeschitz-Kriegel, H. Coll. Polym. Sci, 1997, 81, 1121. 15. Olmsted, P.D.; Poon, W.C.K.; McLeish, T.C.B.; Terrill, N.J.; Ryan, A.J. Phys.

Rev. Lett.,1998, 81, 373. 16. Pogodina, N.V.; Winter, H.H. Macromolecules, 1998, 31, 7103 17. Ratajski, E.; Janeschitz-Kriegel, H. Coll. Polym. Sci., 1996, 274, 938. 18. Hamley, I.W.; Fairclough, J.P.A.; Terrill, N.J.; Ryan, A.J.; Lipic, P.; Bates, F.S.;

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Appl. Polym. Sci., 1996, 61, 649.

Chapter 14

Crystallization and Solid-State Structure of Model Poly(ethylene oxide) Blends

James Runt

Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA 16802

This paper presents an overview of several studies whose overall goal is to elucidate the solid state microstructure and crystallization kinetics of melt-miscible polymer blends. Model blends were prepared with poly(ethylene oxide) and four amorphous polymer diluents: two exhibiting relatively weak intermolecular interactions with PEO and two exhibiting strong interactions. Small-angle x-ray scattering experiments showed that the introduction of strong intermolecular interactions promoted diluent segregation over relatively large length scales, regardless of diluent mobility at the crystallization temperature. In addition, at a given T c, spherulitic growth rates for blends with the strongly interacting polymers are considerably lower than those with weakly interacting polymers with comparable Tgs. To a first approximation, growth rates for PEO and the blends can be superposed when normalized by the degree of supercooling and Τ - T g. Finally, the chapter concludes with a brief review of recent time-resolved small and wide-angle x-ray experiments conducted on PEO and the same blends.

Polymer blends containing crystalline polymers are commonplace and a growing number of commercial materials are polymer mixtures in which at least one of the components is semi-crystalline. Crystallization and solid state microstructure of neat crystalline materials have generally been studied in some depth. However, these features of semi-crystalline blends are less well understood, due at least partly to the greater complexity of these systems. Important considerations in the case of melt-miscible blends are the influence of the polymeric diluent on crystallization kinetics and its ultimate locations) in the lamellar microstructure. Diluent molecules can reside between lamellae in lamellar stacks, between fibrils and/or in interspherulitic regions, yielding different microstructures, which in turn give rise to different material properties. It is important, therefore, to develop an understanding of the

218 © 2000 American Chemical Society

219

factors that influence the extent of diluent segregation. Strong intermolecular interactions are well known to play a critical role in determining melt-miscibility, yet their impact on crystallization and microstructure formation have been rarely examined in depth.

In the past several years, we have undertaken a series of experimental studies focusing on the investigation of the factors responsible for controlling the nature of the solid state microstructure (and the development thereof) and the crystallization kinetics of a series of melt-miscible blends (1-4). Poly (ethylene oxide) [PEO] was chosen as the semi-crystalline component since it exhibits miscibility in the melt with a relatively broad range of amorphous polymers. The current paper presents an overview of several aspects this work: (1) initial small-angle x-ray scattering (SAXS) experiments to elucidate the final microstructure of PEO and the blends; (2) spherulitic growth rate experiments over a range of crystallization temperatures and concentrations and; (3) a real-time small- and wide-angle x-ray scattering study of microstructure development and crystallization of these systems.

Experimental

Materials

Throughout this work, we focused on four melt-miscible poly(ethylene oxide) blend systems. Two exhibit relatively weak intermolecular interactions [PEO with poly(methylmethacrylate) (PMMA) and poly(vinylacetate) (PVAc)] and two exhibit relatively strong interactions (PEO with ethylene - methacrylic acid (EMAA) and styrene - hydoxystyrene (SHS) copolymers). The PEO used in these studies has a viscosity-average molecular weight of 1.44 χ 105. The EMAA copolymer contains 55% by weight methacrylic acid units while the SHS copolymer consists of 50 weight percent styrene and p-hydroxystyrene. The second, diluent polymer is non­crystalline in all cases. In each series, one of the amorphous polymers was chosen to have a relatively low glass transition temperature (Tg) [PVAc and EMAA] and the other a relatively high T g [PMMA and SHS]. The two weakly interacting polymers, PMMA and PVAc, had measured Tgs of 113 °C and 31 °C, while the Tgs of EMAA and SHS were 36 °C and 150 °C, respectively. Molecular weights and polydispersities of the amorphous polymers can be found in refs. / and 5. Various compositions of each blend were prepared by solution casting as described in refs. / and 5 and solvent removal was accomplished at elevated temperature under vacuum.

Small-angle x-ray scattering

'Static' SAXS experiments were performed on the Oak Ridge National Laboratory 10-m pinhole collimated SAXS camera using C u K a radiation (λ = 0.154 nm) and a 20 χ 20 cm2 position-sensitive area detector. The scattering from each sample was determined at two sample-to-detector distances: 2.119 and 5.119 m. The data were azimuthally averaged and converted to an absolute differential cross-section by means of precalibrated secondary standards (6): a high density polyethylene, PES-3, was used for the low q data and a vitreous carbon standard was employed for the

220

high q data. Here, q is the scattering vector defined as (4π/λ) sin(θ/2), where θ is the scattering angle and λ is the x-ray wavelength.

Time-resolved SAXS experiments were conducted on two different beamlines at the National Synchrotron Light Source (NSLS) at Brookhaven National Laboratories. Initial experiments were conducted on Beamline X3A2 at a wavelength of 0.154 nm (2). The scattered intensity was collected by a linear position sensitive photodiode detector coupled to an optical multichannel analyzer. In these early experiments, real-time SAXS scattering during crystallization at 45 °C was analyzed for one composition of each of the model blends, as well as the scattering from neat PEO crystallized at two temperatures.

A much more comprehensive series of experiments have been conducted more recently on the Advanced Polymer Beamline at NSLS, X27C (4). The beamline setup provided access to simultaneous small- and wide-angle x-ray (WAXS) scattering (7). The x-ray wavelength in this series of experiments was 0.1307 nm and pinhole collimation was used in conjunction with a sample to detector distance of 1.190 m. We utilized a specially designed sample holder to allow for a rapid temperature jump between the crystallization temperature, T c, and the temperature in the melt (7). The scattering pattern from a duck tendon was used to calibrate the SAXS detector in the q range from 0 to 0.15 Å-1. A Lupolen standard was used to calibrate the WAXS detector in the 2Θ range of approximately 11° to 38°. Time-dependent bulk crystallinities were determined by resolving the WAXS profiles into crystalline reflections and an amorphous halo, then calculating the area fraction of the crystalline reflections.

Details of the data processing in the different studies can be found in the original publications (1,2,4). The one-dimensional correlation function was calculated for all experimental SAXS curves (8):

where r is the correlation distance. The pseudo two-phase model was applied to the correlation functions of neat PEO and the blends to extract the experimental invariant, Q e x p t . Note that in the case of the 'static' SAXS experiments, Q e x p t is in absolute units, whereas for the time-resolved experiments the Q e x ptare relative values. The invariants in the time-resolved experiments are generally depicted as Q e x p t /Qmax, where Q m a x is the maximum value obtained during the experimental run. Q e x p t is determined from the ordinate of the linear fit to the self-correlation portion of the correlation function:

where w c is the linear crystallinity (given by the ratio of the average lamellar thickness, l c , to the average long period, L), v s the volume fraction of lamellar stacks (given by the ratio of the bulk crystallinity, φ ς, to the linear crystallinity, wc) and Δη1

the linear electron density difference (defined as Δη1 = ηc - η i n t , where ηc and ηint are the electron densities of the crystal and interlamellar amorphous layers, respectively).

In the case of complete incorporation of uncrystallized material in interlamellar regions, the linear crystallinity and electron density difference in eqn. 2 are replaced by their bulk values, φc and Δη, and the invariant becomes (9)

G(r) (1)

Qexpt = V s W c ( l-Wc)Δη 1

2 (2)

221

Q c a i c = φ ο (1 - φ ο )Δη 2 (3)

Δη = T|c - η , where η is the electron density of the amorphous layer when all of the amorphous diluent resides in the interlamellar regions. One can therefore calculate an 'all-interlamellar' invariant ( Q c a i c ) and comparison with Q e x p t indicates the extent of diluent polymer incorporation in interlamellar regions.

The average long period can be written as:

L = ax\x

2i(dGi(k) (4)

where (dGldx) is the slope of a linear fit to the self-correlation portion of G(r). For w c

> 0.5, the intersection of the linear fit with G(r) = 0 is:

r0 = lc(l-w c) = l a w c (5)

Q e x p t , (dGlaY) and ro are obtained directly from the fit to the self-correlation portion of G(r). Together with the long period (derived from the first correlation maximum), this permits quantitative analysis of the lamellar microstructure.

Spherulitic growth rates

Dried films of PEO and the blends were heated to 100 °C (for 3 min.) in a hot stage to erase previous thermal history, then rapidly transferred to a second hot stage set at the desired crystallization temperature. The development of the spherulitic superstructure was then viewed with an Olympus BHSP-300 microscope and the crystallization event recorded with a video camera and VCR. In the case of non-linear growth, spherulite growth rates (G) were determined at the onset of growth. Average growth rates were determined on from 1 - 5 spherulites.

Results and Discussion

'Static' SAXS experiments (1)

The two PEO blends that exhibit weak intermolecular interactions display quite different SAXS behavior after crystallization at T c = 45 °C. For the PEO/PMMA blends, the scattering peaks move to significantly lower q (larger long periods), with increasing PMMA content. The correspondence between experimental and calculated 'all interlamellar5 invariants, as well as experimental and calculated 'all interlamellar' long periods, support complete incorporation of PMMA within lamellar stacks. This is in agreement with the results of other authors (10). However, for the PEO/PVAc blends, the scattering maxima shift only slightly to lower q with increasing PVAc content. In addition, the experimental and calculated invariants do not exhibit good correspondence. These findings, along with optical microscopy observations, suggest at least partial exclusion of the relatively mobile PVAc to interfibrillar regions during crystallization at 45 °C. Unfortunately, quantitative assessment of the extent of PVAc migration was not possible due to the relatively low electron density contrast between amorphous PEO and PVAc.

222

Lorentz-corrected SAXS intensities vs. q for several PEO/EMAA blends are shown in Figure 1. A dramatic increase in long period (from -24 to 45 nm) is observed when EMAA content is increased to only 20%. That this is not primarily associated with incorporation of EMAA in lamellar stacks, as demonstrated in Figure 2. Figure 2 is a plot of the experimental invariant and the invariant calculated assuming all interlamellar incorporation of the EMAA chains. EMAA has an electron density that is considerably smaller than those of neat amorphous and crystalline PEO so that even modest inclusion of EMAA in lamellar stacks would be expected to lead to a significant increase in the invariant, as shown by the solid line in Figure 2. The data therefore indicate very significant displacement of EMAA to extralamellar regions. The microstructural parameters extracted from analysis of the correlation functions show clearly that the increase in long period with increasing EMAA content is due primarily to an increase in lamellar thickness, which is consistent with a lower degree of supercooling (ΔΤ). Similar conclusions were drawn for the SHS blends except that the observed increases in lamellar thickness are more modest, suggesting that the equilibrium melting point depression is not as pronounced for this blend.

Equilibrium melting points of the EMAA and SHS blends were estimated by using the following approach. Using the Lauritzen, Hoffman, Miller theory of crystallization (//) and assuming that the thickening factor and end surface free energy are independent of blending then: l c = ζ · 1/ΔΤ, where ζ is a constant for PEO and the blends. This then permits an estimation of the degree of supercooling at which the EMAA (and SHS) blends were crystallized by comparing the experimental crystal thicknesses with those determined by Arlie et al. for neat PEO (12). Further details of this analysis are described in references 1 and 3. For example, it was estimated that the addition of EMAA to PEO depresses the equilibrium melting point by as much as 5 -10 °C for as little as 10 - 20% EMAA. This seems plausible in light of the dramatic reduction in the crystallization rates of these blends with increasing EMAA content (see the next sections).

The PEO/EMAA and PEO/SHS blends exhibit volume-filling spherulites for all compositions examined in this part of the study (i.e., < 20%). These observations, in concert with SAXS results, indicate at least partial exclusion of the diluent into regions between lamellar stacks at these compositions. The distribution of the second polymer between the interlamellar and interfibrillar regions can be estimated using measured bulk crystallinities and correlation function parameters as follows. The volume fraction of lamellar stacks is determined from the bulk and linear crystallinities (vs = φο/Wc) and the average electron density difference between crystalline and interlamellar amorphous regions can be determined using eqn. 2. Since r|c is known (0.676 mol e-/cc (13)% the average electron density of the interlamellar regions ( η ΐ η 0 can be obtained, and the concentration of amorphous PEO in the interlamellar regions can be calculated as:

ΦΙΕ = (η ΐ ι« -η<ι ) / (η» -η<0 (6)

where Τ[ά and η 8 are the electron densities of EMAA (or SHS) and amorphous PEO (= 0.614 mol e-/cc (13)), respectively. From φ ] 3, the overall volume fraction of amorphous PEO in the interlamellar regions (ILa) can be determined by:

IL a = 0ia(l-W c)V s

223

Figure 1: Lorentz-corrected SAXS intensities as a function of scattering vector for PEO and selected PEO/EMAA blends crystallized at 45 °C: ο - neat PEO; + - 90/10 EMAA; ∆ - 85/15; x - 80/20. From ref L

Figure 2: Experimental and calculated invariants for PEO/EMAA blends crystallized at 45 °C. · - calculated 'all-interlamellar ' invariant; ο - experimental invariant. From ref 1.

224

and the overall volume fraction of amorphous PEO in interfibrillar gaps is simply given by

IFa = 0a~ILa (8)

The corresponding volume fractions of the diluent polymer in interlamellar and interfibrillar regions (i.e., IL d and IFd) can be calculated from

IL d = <|>d-IFd (9)

and

IFd = l - v s - I F a (10)

Table 1 presents the amorphous PEO and diluent polymer distributions for the PEO/EMAA and PEO/SHS blends crystallized at 45 °C. The overall volume fractions of crystalline PEO (φ0), amorphous PEO (φ&) and diluent polymer (φ<0 were obtained from measured DSC bulk crystallinities. At 10% diluent content, the bulk and linear crystallinities are comparable within experimental error and it is not possible to quantitatively determine the diluent distribution. However, comparison of the experimental and calculated invariants does indicate some exclusion from lamellar stacks. As diluent content is increased to 20%, both blends exhibit an increase in the fraction of material between lamellar stacks, l-v s, and a significant reduction in spherulitic growth rates. In addition, at diluent contents up to 20% (15% for PEO/SHS), effectively all of the amorphous PEO resides in interlamellar regions, i.e., IFa « 0 and the interfibrillar regions are composed mainly of the diluent polymer (i.e., 'liquid' gaps) at these compositions. However, -30% (by volume) of the amorphous PEO is included in the interfibrillar regions for the 20% SHS blend, yielding a value for the T g of the regions between stacks (using the Fox-Flory equation and PEO T g — 55 °C) near the crystallization temperature.

The behavior of the weakly-interacting PEO blends seems to suggest that the mobility of the amorphous component at T c determines diluent location: relatively high T g PMMA is trapped between crystal lamellae whereas low T g PVAc is able to diffuse from the growth front and is located partially in interfibrillar regions. However, an examination of the microstructures of the PEO/PMMA and PEO/SHS blends clearly shows that diluent mobility is not the sole criterion for segregation length scale. Despite the reduced mobility of SHS compared to PMMA (due to its significantly higher T g and molecular weight), SHS is located in interfibrillar regions at blend compositions up to 20%, becoming at least partially interspherulitic at higher concentrations (3) whereas PMMA is completely interlamellar at all blend compositions studied. The significantly lower crystal growth rates for the SHS blends allow more time for SHS diffusion over larger distances. This relationship between the length scale of diluent segregation and relative diffusion and growth rates is in good agreement with ideas proposed previously by Keith and Padden (14,15). The segregation behavior of EMAA and SHS further suggests that, given the opportunity, the crystalline microstructure prefers to exclude polymeric diluents.

Tab

le 1

. Am

orph

ous

PEO

and

dilu

ent d

istri

butio

n in

PE

O/E

MA

A a

nd P

EO

/SH

S bl

ends

(1)

Ble

nd

wt.%

di

luen

t V s

Ti

l IL

a IF

a (p

d IL

d IF

d

PEO

/EM

AA

10

«

1 0.

613

(±0.

002)

0.19

(±0.

02)

—

0.12

(±0.

02)

*

15

0.90

(±0.

04)

0.59

5

(±0.

003)

0.12

(±0.

02)

0.16

(±0.

02)

«0

0.19

(±0.

02)

0.09

(±0.

04)

0.10

(±0.

04)

20

0.83

(±0.

03)

0.58

1

(±0.

004)

0.12

(±0.

02)

0.12

(±0.

01)

«0

0.25

(±0.

02)

0.08

(±0.

01)

0.17

(±0.

03)

PEO

/ SH

S 10

0.

95 (±

0.05

)

(-1)

0.61

0

(±0.

003)

0.25

(±0.

02)

—

0.11

(±0.

02)

*

15

0.90

(±0.

05)

0.61

0

(±0.

003)

0.23

(±0.

02)

0.27

(±0.

04)

«0

0.17

(±0.

02)

0.07

(±0.

05)

0.10

(±0.

05)

20

0.78

(±0.

04)

0.60

1

(±0.

003)

0.24

(±0.

02)

0.16

(±0.

03)

0.08

(±0.

04)

0.22

(±0.

02)

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(±0.

02)

0.14

(±0.

03)

valu

e un

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in (s

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226

Spherulitic growth (J)

For PEO and blends crystallized at most of the Tcs examined, typical linear spherulite radius (R) vs. time behavior was observed and hence the growth rate remained constant during the course of spherulite growth. The implication is that the composition at the crystal growth front is effectively unchanged during the crystallization process. This is consistent with the view that, in these situations, the crystallization rate outstrips the rate of diffusion of the second polymer away from the growth front, the second polymer becoming trapped in the crystalline microstructure (14). However, distinctly non-linear growth was observed for some blends, particularly those with strongly-interacting polymers crystallized at relatively low degrees of supercooling. Similar behavior has been observed previously for crystalline polymer - oligomer mixtures (14,16,17). The implication is that there is an increasing concentration of diluent polymer in the liquid mother phase as crystallization proceeds (due to radial diffusion of the diluent), and hence slower spherulite growth as crystallization continues. For blends with strongly-interacting (or low molecular weight) miscible diluents, the degree of supercooling will also be continually depressed during crystallization. At relatively short crystallization times, R ©c t1 and G is constant, whereas there is a crossover at longer times to R °c \ m i.e., G oc t" i /2.

As alluded to earlier, growth rates of PEO spherulites in blends with strongly-interacting polymers are very significantly depressed compared to neat PEO at the same T c. Figure 3 shows the experimental growth rates for neat PEO and the blends with the two lower T g amorphous polymers, PVAc and EMAA. There is a very significant drop in growth rate for the EMAA blends: at comparable T c, G decreases by more than three orders of magnitude for the 30% EMAA blend compared to that of PEO. An important part of this decrease is related to the depression of the equilibrium melting point (Tm°) of the PEO in these mixtures. As before, the degrees of supercooling at which the EMAA (and SHS) blends were crystallized were estimated by comparing experimental crystal thicknesses from SAXS experiments with those derived by Arlie et al for neat PEO (12). The PEO/PVAc blend is relatively weakly-interacting and changes in Tm° with blend composition are assumed to be negligible for the present purposes.

The spread in the growth rate data in Figure 3 collapses down to within about an order of magnitude of that of PEO when plotted vs. ΔΤ (= Tm° - Tc), verifying the relative importance of the Tm° depression for these blends. If the data are further normalized by Τ - T g in a first order attempt to account for transport in the melt, Figure 4 results. Considering the uncertainty in the estimated Tm°s and Tgs and the absence of details concerning mutual diffusion of the polymers, Figure 4 represents reasonable 'master curve' behavior.

Figure 5 summarizes the growth rate vs. T c relationship for blends containing the higher T g diluent polymers. Again, the strongly-interacting blends, i.e., those with SHS, exhibit a very significant reduction in G compared to PEO at comparable T c (a decrease of about four orders of magnitude near T c = 45 °C). These slow rates permit even high Tg, high molecular weight SHS to diffuse over relatively long distances during the course of crystallization. We used the same approach to determine Tm° for the SHS blends as noted above and assumed that changes in Tm° with blend composition for the PEO/PMMA blend are negligible. The corresponding growth rates plotted against ΔΤ are not well superposed, as anticipated since the transport

227

Figure 3: Log spherulite growth rates vs. Tc for PEO and blends with low Tg

amorphous polymers: x-PEO; # - 90 PEO/WPVAc; · - 80/20 PVAc; Δ - 70/30 PVAc; Φ - 90/10 EMAA; EE-80/20 EMAA; Δ - 70/30 EMAA. From ref. 3.

Figure 4: Log spherulite growth rates vs. ΔT • (T-Tg) for PEO and blends with low Tg amorphous polymers: x-PEO; * - 90 PEO/lOPVAc; • - 80/20 PVAc; + - 70/30 PVAc; Φ - 90/10 EMAA; M - 80/20 EMAA; A - 70/30 EMAA. From ref. 3.

228

term is more significant for the higher T g blends. However, as above, normalizing on the basis of ΔΤ · (T - Tg) provides a first order superposition of the experimental growth rate data (Figure 6).

Time-resolved SAXS/WAXS experiments (-0

In this final section, the results of our recent time-resolved SAXS/WAXS study of PEO and model blends will be briefly summarized. Crystallization and microstructure formation were followed for PEO crystallized at four different temperatures and for PEO blends (containing up to 20% PMMA, SHS and EMAA) crystallized at 45 °C and/or 48 °C. For all systems and Tcs investigated, the positions of the wide-angle diffraction maxima remained constant during the course of crystallization. Generally speaking, the rate of development of the bulk crystallinity for the blends followed expectations from the spherulite growth rate experiments: crystallization rates are very significantly depressed for blends with the strongly-interacting diluents. The bulk crystallinity - time profiles required fitting with more than one Avrami expression and the fitted exponents were generally inconsistent with spherulitic growth and instantaneous nucleation (i.e., observations from optical microscopy experiments). Analysis is continuing as of this writing.

As in the static SAXS experiments, final crystal thicknesses (and long periods) were considerably larger for the strongly-interacting blends at a given T c, due to the decrease in the equilibrium melting point of the SHS and EMAA blends. There was a modest increase in final long period for blends with PMMA, consistent with interlamellar placement of the diluent. Average long periods and crystal thicknesses decreased at early times during the course of crystallization for neat PEO and all blends and then remained constant throughout the duration of crystallization (see, for example, the behavior of the 80/20 PEO/SHS blend at T c = 45 °C in Figure 7). A decrease in average L and l c early in the crystallization process has been observed for a number of different polymers (18,19) and it has been proposed that this arises from insertion of lamella within or between lamellar stacks (e.g., 20). The results of our real-time SAXS/WAXS crystallization and melting experiments are generally consistent with an insertion mechanism. In addition, the magnitude of the decrease in average crystal thickness (and long period) early in the crystallization process is often considerably larger (-9 nm) for blends with higher T g diluents (see Figure 7) than for those with lower Tgs (2-3 nm). The origin of this difference is unclear at present but some of our very early work on model mixtures of polymer single crystals with amorphous diluents snowed that reorganization (lamellar thickening) during heating could be inhibited by the presence of higher T g amorphous polymers (21,22). A similar effect can be envisioned during crystallization. In addition, one can also envision modification of infilling processes.

In the preceding paragraphs, data were only considered after the appearance of both wide- and small-angle scattering peaks. However, we noted the appearance of a small-angle scattering maximum that grew in amplitude with time, prior to the appearance of wide-angle diffraction peaks. This has been corroborated for PEO in a recent paper (23). Such electron density fluctuations (on length scales of the order of ca. 15 nm) prior to the appearance of wide-angle scattering during a crystallization experiment have now been observed for a growing number of polymers (e.g., 24-26) and have been associated with a spinodal decomposition-like mechanism. This

Tc(°C)

Figure 5: Log spherulite growth rates vs. Tcfor PEO and blends with high T8

amorphous polymers: x - PEO; A - 90 PEO/10 PMMA; # - 80/20 PMMA; · - 70/30 PMMA; Φ - 90/10 SHS; M-80/20 SHS; A - 70/30 SHS. From ref. 3.

e 1

1000 1500 ΔΤ (T - Tg)

Figure 6: Log spherulite growth rates vs. AT - (T- Tg) for PEO and blends with high Tg amorphous polymers: x - PEO; * - 90 PEO/10 PMMA; · - 80/20 PMMA; + -70/30 PMMA; Φ - 90/10 SHS; m- 80/20 SHS; A - 70/30 SHS. From ref. 3.

230

450 -ι 400 -

350 -

xc 300 -250 -

O). 200 -e

3 150 -100 -

50 -

0 - j 1 1

0 1000 2000 3000 Time (sec)

Figure 7: Plot of average long period (A), lamellar thickness (φ) and amorphous layer thickness (M) versus crystallization time for 80/20 PEO/SHS blend crystallized at 45 °C(4).

process has been described in stressed melts as 'defect clustering' (27) while others have argued that the process can be explained by the presence of a metastable liquid-liquid binodal within the liquid-crystal coexistence curve (26). However, such pre-crystallization behavior has not been universally observed in quiescent melts (e.g., 28). Analysis of our early time SAXS data is in progress at the present time.

Acknowledgments

The author would like express his appreciation to colleagues who contributed to this work: Dr. Catherine Barron, Ms. Melissa Lisowski, Dr. Qiang Liu, Dr. Sapna Talibuddin and Dr. Limin Wu (former and current students and post-docs who worked with the author); Dr. J. S. Lin at Oak Ridge National Laboratory; and Prof. Benjamin Chu, Prof. Benjamin Hsiao, Dr. Li-Zhi Liu and Dr. Fengyi Yeh at SUNY Stony Brook. The research was conducted principally with support from the Petroleum Research Fund, administered by the American Chemical Society.

Literature Cited

1. Talibuddin, S.; Wu, L.; Runt, J.; Lin J. S. Macromolecules 1996, 29, 7527. 2. Talibuddin, S.; Runt, J.; Liu, L.-Z.; Chu, Β. Macromolecules 1998, 31, 1627.

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3. Wu. L.; Lisowski, M. S.; Talibuddin, S.; Runt, J. Macromolecules 1999, In press.

4. Lisowski, M . S.; Liu, Q.; Runt, J.; Yeh, F.; Hsiao, B. Manuscript in preparation.

5. Barron, C. Α., Ph. D. thesis, The Pennsylvania State University, University Park, PA, 1994.

6. Russell, T. P.; Lin, J. S.; Spooner, S.; Wignall, G. D. J. Appl. Cryst. 1988, 21, 629.

7. Hsiao, B. S.; Chu, B.; Yeh, F. http://bnlx27c.nsls.bnl.gov (15 Apr. 1997). 8. Strobl, G. R.; Schneider, M. ; Voigt-Martin, I. J. Polym. Sci., Polym. Phys. Ed.

1980, 19, 1361. 9. Strobl, G. R.; Schneider, M . J. Polym. Sci., Polym. Phys. Ed. 1980, 18, 1343. 10. Alfonso, G. C.; Russell, T. P. Macromolecules 1986, 19, 1143. 11. Hoffman, J. D.; Miller, R. L. Polymer 1997, 38, 3151. 12. Arlie, J. P.; Spegt, P.; Skoulios, A. Makromol. Chem. 1967, 104, 212. 13. Wunderlich, B., Macromolecular Physics, Vol. 3, Academic Press: New York,

1980. 14. Keith, H. D.; Padden, F. J. J. Appl. Phys. 1964, 35, 1286. 15. Keith, H. D.; Padden, F. J. J. Polymer Sci., Polym. Phys. Ed. 1987, 25, 2297. 16. Lee, C. H. Polymer 1998, 39, 5197. 17. Okada, T.; Saito, H.; Inoue, T. Macromolecules 1990, 23, 3865. 18. Hsiao, B.S.; Gardner, K.H.; Wu, D.Q.; Chu, Β. Polymer 1993, 34, 3996. 19. Jonas, A.M. ; Russell, T.P.; Yoon, D.Y. Macromolecules 1995, 28, 8491. 20. Lee, C. H.; Saito, H.; Inoue, T.; Nojima, S. Macromolecules 1996, 29, 7034. 21. Harrison, I. R.; Runt, J. J. Polym. Sci. Polym. Phys. Ed. 1980, 18, 2257. 22. Runt, J. Macromolecules 1981, 14, 420. 23. Ryan, A. J.; Terrill, N . J.; Fairclough, P. A. ACS-PMSE Preprints 1998, 79,

358. 24. Ezquerra, Τ. Α.; Lopez-Cabarcos, E.; Hsiao, B. S.; Balta-Calleja, F. J. Phys.

Rev. Ε 1996, 54, 989. 25. Terrill, N . J.; Fairclough, J. P. Α.; Komanschek, B. U.; Young, R. J.; Towns-

Andres, E.; Ryan, A. J. Polymer 1998, 39, 2381. 26. Olmsted, P. D.; Poon, W. C. K.; McLeish, T. C. B.; Terrill, N . J.; Ryan, A. J.

Phys. Rev. Lett. 1998, 81, 373. 27. Strobl, G. The Physics of Polymers, Springer-Verlag, Berlin, 1996. 28. Liu, W.; Yang, H.; Hsiao, B.S.; Stein, R. S. ACS-PMSE Preprints 1998, 79,

350.

Chapter 15

Transient Rotator Phase Induced Nucleation in n-Alkanes

E. B. Sirota and A. B. Herhold

Exxon Research and Engineering Company, Corporate Research Science Laboratory, Route 22 East, Annandale, NJ 08801

We show that transient metastable rotator phases occur on crystallization of octadecane ( C H 3 - ( C H 2 ) 1 6 - C H 3 ) and other even chain-length alkanes into their triclinic phase from the supercooled melt. This was directly observed using time resolved x-ray scattering. The crystallization temperature (i.e. the limit of supercooling) is determined by the thermodynamic stability of the transient phase with respect to the liquid. We explain the crystallization kinetics of the homologous series of n-alkanes in terms of a crossover from stability to "long-lived" metastability to transient metastability. Such metastable and transient phases are likely to have analogues in polymeric systems.

Introduction Polymers often crystallize into metastable phases when cooling from the melt.

This subject has been recently reviewed by Keller and Cheng. (1) Metastable phases are usually considered to be those which are stable either indefinitely or at least long enough to measure. A transient phase would be a metastable phase which remains for a limited time and may therefore not be readily observed. Clearly the distinction between the two depends on the measurement time required by the specific observation technique. Transient phases are intrinsically difficult to observe and have therefore been the subject of limited study in polymeric(l-3) and other condensed matter systems.(4-6) However, transient phases are intermediate forms which occur on the path to the final state, (7) and therefore study of these phases is key to understanding the process of crystallization. For example, the temperature at which

232 © 2000 American Chemical Society

233

the transient state can nucleate may determine the apparent temperature of crystallization for the stable phase. The properties of the metastable phase can also determine the growth morphology of the crystal. Keller et al. have argued that enhanced chain mobility in the metastable hexagonal phase causes lamellar thickening in polyethylene.(l-3)

The homologous series of n-alkanes (CnH2n+2> abbrv. C n ) are the principle component of petroleum waxes and the building block of many derivative molecules whose properties have been shown to be strongly influenced by their alkyl component. (8-10) They are also the series of oligomers leading up to polyethylene at large n. It is well known that the n-alkanes show an even-odd effect in their crystal structures and melting points.(8,11,12) The kinetics and hysteresis, however, had remained a mystery. Reported here are time dependent x-ray scattering measurements of Cjg which show a transient rotator phase. By comparing this behavior to the chain-length dependent phase diagram of the n-alkanes where the stability of the transient phase is varied, the kinetic behavior of n-alkane crystallization can be explained in terms of a crossover from long-lived to transient metastability. (13) We have shown that the observation that the temperature at which the supercooled liquid transforms to the triclinic phase lies on the same curve as where the rotator phase would be stable with respect to the liquid is not a coincidence, but rather, the crystallization of the triclinic phase is mediated by a transient rotator phase. (13)

Short n-alkanes exhibit various equilibrium rotator phases below their melting point.(9,10,14-21) The rotator phases are lamellar crystals which lack long-range order in the rotational degree of freedom of the molecules about their long axes. In the equilibrium phase diagram, the rotator phases are squeezed out for w-even at lower n by a triclinic non-rotator crystal phase. Bulk melts of the w-odd alkanes (15<w<29) exhibit negligible supercooling at the liquid-rotator (L->R) transitional 1,12,21) The negligible supercooling of bulk n-alkanes has been argued( 12,22) to be related to the presence of an equilibrium surface monolayer rotator phase occurring at the liquid-vapor interface, a few degrees above the crystallization temperature,(23-27) which would serve as an ideal nucleation site for the bulk rotator phase. Ubiquitous tiny bubbles in most bulk situations could serve as such heterogeneous sites. The equilibrium melting points exhibit a significant even-odd effect associated with whether melting is from the triclinic crystal (w-even) or from the rotator phase (n-odd).(l 1,12) The even-odd effect disappears when considering the observed freezing temperatures,(12,13) because the n-even with high melting points supercool, while the w-odd which crystallize into the rotator phase do not. We have previously shown, using time resolved synchrotron x-ray scattering on Ci6, that it is not a coincidence that the amount of supercooling for n-even is equal and opposite to the even-odd difference in the melting temperatures.(13) Ci 5 exhibits large supercooling from the melt and is far from showing a stable bulk rotator phase. In contrast, C22 shows a stable rotator phase on heating and cooling, and C20> only on cooling. C\g shows some supercooling from the melt and the rotator phase is transient (Figure 1). Here we report the kinetic data on Ci g.

234

Experimental

Time dependent x-ray scattering was performed on a 18kW Rigaku x-ray generator and a Huber goniometer. C u K a radiation was used and the beam was defined with a bent graphite monochromator and slits, giving a resolution of Aq=0.0lA'\ A Bicron® single channel detector was used. The sample was contained in a 2mm thick copper sample holder with Be windows in a temperature controlled oven. The C i g was >99% pure from Aldrich and used as obtained. Temperature gradients in the oven lead to absolute uncertainties of <0.1°C.

Figure 1. Phase diagram of the n-alkanes: The dots are the freezing temperatures Tf. The open squares are the melting temperature (Tf^jJ for those η which melt from the triclinic phase. The melting temperatures for the other η coincide with their freezing temperatures. The open circles are the triclinic-rotator phase transformation temperatures on heating (Tj_^g) and the crosses are the average rotator-triclinic transition temperatures (TR^f) on cooling at 0.05 °C/minute. The dashed lines are lines Τχ_+γ=37.7l+2.47(n-20) and TT__>R=31.96+2.60(n-20) and the solid curves are fits to the empirical form(28) yielding 7y= 131.0(n-13.66)/(v& 3.15) and ^T—>lT 134· 7(VL-13. 13)/(n+ 5.39). The regions labeled "stable "metastable " and "transient" refer to the stability of the rotator phase for n-even.

235

Results

The stable low-temperature form of Cjg is the triclinic crystal phase in which the molecules are tilted with respect to the layers by -20°. (29,30) In Figure 2(a) we show high angle scans corresponding to the in-plane packing of Ci g in the stable triclinic crystal phase. By performing repeated scans we were able to determine the primary peak positions in the metastable phase to be ^=1.496A_1 and 1.609À'1 (Figure 2(b)). A peak at ^=1.536A'l is a higher order mixed reflection which occurs in the Rj rotator phase. The stochastic nature of the lifetime of the transient phase allowed us to catch a full scan in that phase by repeating the measurement until a cooling run occurred with a lifetime long enough to measure the spectrum. As will be seen in the histograms of the lifetimes, for C\g this was quite feasible.

Figure 2. (a) A q-scan in the triclinic phase. There is little diffuse scattering and peah occur at well definedpositions q=1.366A", 1.399A-1, 1.487A'K 1.569A'^, and 1.65OA-1. (h) A q-scan caught in the transient rotator phase, showing peaks at q=1.496A-\ 1.536A-1 and L609A'1.

The alkanes with n-odd (n<21) exhibit an equilibrium orthorhombic Ri rotator phase where the molecules are untilted with respect to the layers. (14-20) To clarify the nature of the transient phase, we plot in Figure 3 the peak positions of the rotator phase just below the freezing temperature Tf for w-odd alkanes, along with the positions observed here for Cjg. Also included are the peak positions reported (13) for a transient phase of C\g which has a much shorter lifetime and was obtained using time resolved synchrotron x-ray scattering with a Braun position sensitive detector, at Exxon's beamline X10A at the N.S.L.S. It is clear from this data that the peak positions observed for the transient phase correspond to the interpolated peak position expected for the rotator phase. Also depicted are the peak positions of the untilted, orthorhombic, non-rotator, herringbone crystal phase of the n-odd alkanes which clearly differ from the transient phase.

236

1.65 h

— 1.60 Φ α

—O— Rj Rotator Phase (near Tm e | t) —•—Orthorhombic Crystal

• Observed Metastable Phase °" 1.55

1.50 ο -ο ο

15 16 17 18 19 Carbon number (η)

20 21

Figure 3. The peak positions in the stable R j rotator phase just below the melting temperature (circles) and in the low temperature herringbone crystal phase (squares). The observed positions of the peaks in the metastable phase ofCjg and C jfi (13) (diamonds) are consistent with the rotator phase.

To investigate the transformation kinetics of the transient phase of C i g , we performed "slow quenches" to fixed temperatures. These isothermal runs were performed as follows: The sample was first entirely melted (>10°C above the melting temperature for at least 15 minutes) and then cooled across the freezing point at 0.056°C/minute to a fixed temperature slightly below the highest temperature where any crystallization was observed. The scattering intensity was monitored at #=1.496Α"1, a position at which there was a peak in the R phase but no peak in the triclinic phase. This measurement was repeated many times for 3 different isothermal temperatures: 0.10°C, 0.145°C, and 0.19°C below the crystallization temperature Tf.

The transformation from the transient to the stable phase was stochastic, so each run was different. In Figure 4 we show 3 typical isothermal runs, all at an undercooling of 0.19°C. Since the detector was fixed at a peak position of the rotator phase, the appearance of the R phase can be associated with an increase in scattering above that in the liquid phase. The finite intensity observed at that position in the liquid phase is due to the broad liquid peak centered at q=\35kr^. Since there is not much diffuse scattering in the stable triclinic crystal phase, the intensity at ς'=1.496Α"1 drops below that in the liquid phase. Thus, the lifetime of transient rotator phase can be unambiguously determined from such scans. Also included in Figure 4 is the temperature profile of the quench.

237

Figure 4. Intensity at q= I.496Â"^ for 3 typical quenches obtained at cooling to 0.19°C below the highest observed crystallization temperature. At early times, the sample is in the liquid phase, and at long times, the sample is in the triclinic crystal phase. The scattering level of the liquid phase is greater than that of the final crystal phase. The high intensity at this exposition is associated with the transient rotator phase, which exhibits a peak at that position. Also depicted is the temperature as a function of time for this series of runs.

The histograms for the observed lifetimes of the transient phase are shown in Figure 5. For the 0.10°C subcooling data (Figure 5(a)) the spike at 800 seconds represents the maximum time waited in the scans. Therefore, the 3 points at 800 were actually lifetimes > 800 seconds. For 0.145°C subcooling (Figure 5(b)), the 3 points at 1750 seconds represent similarly lifetimes >1750 seconds. For the 0.19°C subcooling (Figure 5(c)), 620 seconds was the longest lifetime observed. We wish to compare average lifetimes. However, since some of the lifetimes were beyond the experimental measurement, and the length of the experiment was not constant for all subcoolings, we computed an effective average lifetime by counting all lifetimes > 800 seconds as 800 seconds. For subcoolings of 0.10°C, 0.145°C, and 0.19°C we obtained "average" lifetimes of 209s, 166s, and 101s for 30, 68, and 43 data points, respectively. The true lifetimes for the first two are actually longer due to the truncation of the long time data. Another type of measurement, cooling continuously through the transition at -0.07°C/minute, gave an average lifetime of 97 seconds for 38 runs. This corresponds to an average observed subcooling of 0.091°C.

238

Ο 300 600 900 1200 1500 1800 Metastable Lifetime [seconds]

Figure 5. Histograms of the lifetime of the metastable rotator phase in Cjg when quenching at 0.056°C/minute to (a) 0.10°C (b) 0A45°C and (c) 0.19°C below the temperature of the onset of crystallization.

„ 2 Ο

Η- · 2

-4

-6

-8

Transient V , Stable ^Metastable* V

12 14 16 18 2 0 22 24 2 6 η

Figure 6. The transition temperatures, as in Figure 1, but with temperature plotted with respect to Tjin order to enhance the features.

239

The above results show clearly that for C i g, a transient rotator phase occurs upon crystallizing from the supercooled melt. The distribution of transient phase lifetimes raises the issue of what determines the time for conversion into the stable triclinic phase. Systematic measurements of the chain-length dependence of the various transitions were performed using x-ray scattering.(13) These are shown in Figure 1 and as scaled to the freezing point in Figure 6. Temperature was scanned at rates as slow as 0.007°C/minute to systematically obtain transition temperatures. Tf denotes the temperature where crystallization is first observed (cooling slowly at 0.03°C/min). For the chain-lengths where the L<- R transition can be observed both on heating and cooling (i.e. even n>20 and odd-/? ), T R _ ^ L a n c * Tf agree to within -0.1 °C, consistent with the negligible supercooling reported for this transitional 1,12,21)

Discussion and Conclusions

T J ^ R is the temperature where the triclinic phase transforms to the rotator phase on heating (even w>22). T J ^ J is the average temperature at which a rotator phase transforms to the triclinic phase when cooling at a rate of -0.07°C/min. Since adiabatic scanning calorimetry confirmed that the T->R transition does not superheat(21), T ^ ^ R can be considered the equilibrium temperature of the T<->R transition. T R _ ^ J can be considered the temperature below which the R-»T transition is likely to occur on a "reasonable" time scale. T R ^ T and T f are virtually parallel, showing that the supercooling of the R-»T transition is ~6°C.

Nucleation into the rotator phase from the melt occurs at Tf. For Cig , this is well below the extrapolated T R _ » T line and the R->T transformation proceeds rather rapidly. For C i g, the extrapolated Τ ρ ^ γ line is only slightly higher than Tf, thus while the R—>T transformation will certainly occur on laboratory time scales, it occurs much more slowly.

We thus see that the phase diagram for the «-even alkanes contains a crossover from a stable rotator phase (for n>24) to a metastable rotator phase (for w=20, 22) to a transient rotator phase (for «<18). The rotator phase in C i g is just barely transient and is thus long lived on the time scale of a laboratory rotating-anode x-ray scattering measurement. For C\the rotator phase could be observed using time dependent synchrotron x-ray scattering.(13) For C14, where Tf is 20°C below T J ^ R , it is hard to directly observe the transient phase, even though it appears to determine the crystallization temperature.

The ability to recognize the importance of such transient phase induced nucleation may help in understanding crystallization from solutions as well as crystallization from the melt. In addition to the fact that the transient phase determines the crystallization temperatures, it is known that growth morphologies depend on the crystal structure(31-33) so that whether or not growth occurs in a transient phase may impact crystal morphology.

Recent studies of crystallization in both polymer(34-37) and other systems(5,6,38) suggest the presence of precursor phases with densities higher than that of the liquid, but where the symmetry of the ordered phase is not fully developed.

240

The rotator phases are similar to the proposed precursor phases in that their density is higher than the liquid and their symmetry is higher than the crystal, with both quantities being intermediate between the liquid and the low-temperature crystal.

Acknowledgements : We gratefully acknowledge discussions with H. E. King Jr., Moshe Deutsch, Ben

Hsiao, and J. Hutter and the technical assistance of S. Bennett and W. A. Gordon. The NSLS at BNL is supported by the U.S. Dept. of Energy, Divisions, of Materials & Chemical Science.

* Corresponding author: ebsirot@erenj.com

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1993, 70, 958. 25. Wu, X. Z.; Ocko, B. M . ; Sirota, Ε. B.; Sinha, S. K.; Deutsch, M . ; Cao, B. H.;

Kim, M . W. Science 1993, 261, 1018. 26. Ocko, B. M . ; Wu, X . Z.; Sirota, Ε. B.; Sinha, S. K.; Gang, O.; Deutsch, M.

Phys. Rev. E 1997, 55, 3164. 27. Gang, H.; Patel, J.; Wu, X . Z.; Deutsch, M. ; Gang, O.; Ocko, B. M. ; Sirota, E.

B. Europhys. Lett. 1998, 43, 314. 28. Broadhurst, M . G. J. Chem. Phys. 1962, 36, 2578. 29. Nyburg, S. C.; Lüth, H. Acta Cryst. 1972, B28, 2992. 30. Nyburg, S. C.; Pickard, F. M. ; Norman, N. Acta Cryst. 1976, B32, 2289. 31. Bennema, P.; Liu, X . Y. ; Lewtas, K.; Tack, R. D.; Rijpkema, J. J. M. ; Roberts,

K. J. J. Cryst. Growth 1992, 121, 679. 32. Liu, X . Y. ; Bennema, P. J. Appl. Cryst. 1993, 26, 229. 33. Liu, X. Y. ; Bennema, P. J. Cryst. Growth 1994, 135, 209. 34. Imai, M. ; Kaji, K.; Kanaya, T.; Sakai, Y. Phys. Rev. B 1995, 52, 12696. 35. Ezquerra, T. Α.; Lopez-Cabarcos, E.; Hsiao, B. S.; Balta-Calleja, F. J. Phys. Rev.

E 1996, 54, 989. 36. Terrill, N . J.; Fairclough, P. Α.; Towns-Andrews, E.; Komanschek, B. U.;

Young, R. J.; Ryan, A. J. Polymer Comm. 1998, 39, 2381. 37. Olmsted, P. D.; Poon, W. C. K.; McLeish, T. C. B.; Terrill, N . J.; Ryan, A. J.

Phys. Rev. Lett. 1998, 81, 373. 38. Schatzel, K.; Ackerson, B. Phys. Rev. Lett. 1992, 68, 337.

COMPLEX FLUIDS AND BIOPOLYMERS

Chapter 16

Highly Ordered Supramolecular Structures from Self-Assembly of Ionic Surfactants

in Oppositely Charged Polyelectrolyte Gels Shuiqin Zhou1, Fengji Yeh1, Christian Burger2, and Benjamin Chu1,3

1Department of Chemistry, State University of New York at Stony Brook, Long Island, NY 11794-3400

2Max-Planck-Institut für Kolloid and Grenzflächenforschung, Kantstrasse 55, D-14513 Teltow-Seehof, Germany

Small-angle x-ray scattering was used to investigate the nanostructures of complexes formed by interactions of ionic surfactants with oppositely charged polyelectrolyte gels at room temperature (~ 23°C). Highly ordered supramolecular structures of Pm3n space group cubic, face-centered cubic close packing of spheres, hexagonal close packing of spheres, hexagonal close packing of cylinders, bilayer lamellar, and Ia3d space group cubic were, respectively, determined under different experimental conditions. The structural elements of spheres and/or rods were shown to be spherical and/or cylindrical micelles formed by the self-assembly of ionic surfactants driven by both electrostatic and hydrophobic interactions of the charged copolymer chains/surfactants and of the surfactants/surfactants inside the polyelectrolyte gel matrix. The charge density, the chain flexibility and the crosslinker density of the polyelectrolyte gel network chains, and the properties of the ionic surfactants had significant effects on the shape, the aggregation number, and the size of the formed micelles, thus determined the type and the order of the packed structures of those complexes.

The interactions of polyelectrolytes with oppositely charged surfactants can induce complex formation with highly ordered structures. These ordered structures have a host of possible applications ranging from improved mechanical behavior to unusual optical, electrical, and biological properties.1 For example, the use of cationic liposomes in the transfer of genes has become a popular method of gene therapy. M The two-dimensional (2D) hexagonal columnar phase has been shown to

3Correpsonding author.

244 © 2000 American Chemical Society

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be significantly more efficient at transfecting mammalian cells in culture in comparison with the lamellar structure of DNA-cationic liposome complexes.6 Both electrostatic interactions between charged components as well as hydrophobic interactions between the polymer backbone and the alkyl tail of the surfactant are important in driving the self-assembly of surfactant molecules to form ordered structures inside the complexes.7"11

Surfactants in water exhibit very rich and complex self-organized structures which are the basis of the structure-forming abilities of the polyelectrolyte-surfactant complexes.1'12"17 However, little is known about the phase structures of the polyelectrolyte-surfactant complexes. It could be expected that both the charge density and the flexibility of the polyelectrolyte chains as well as the properties of surfactants would significantly affect the structures of the formed complexes.

Substantial efforts have been made to clarify the structures of polyelectrolyte-surfactant complexes in the last five years.1 0 , 1 7"2 5 The systems studied so far included poly(sodium acrylate) gel,1 8 poly(sodium methacrylate),1 poly(2-acrylamido-2-methylpropanesulfonic acid) gel, 1 0 polystyrenesulfonate20"21 or their copolymers22

interacting with alkyltrimethylammonium bromide (CnTAB) and alkylpyridinium chloride (CJPCl) with η being the number of carbon atoms in the tail of the surfactant. The lamellar, cylindrical, and undulating layered structures were most commonly observed.

In this paper, we use the synchrotron small angle X-ray scattering (SAXS) to investigate the supramolecular structures of two kinds of complexes: (1) formed by anionic gels of poly(sodium methacrylate) (PMAA) or poly(sodium methacrylate-co-N-isopropylacrylamide) (PMAA/NLPAM) interacting with cationic surfactants of alkyltrimethylammonium bromide (QTAB) or didodecyldimethylammonium bromide (2Ci2DAB); (2) formed by cationic gels of poly(diallyldimethylammonium chloride)(PDADMACl) interacting with anionic surfactants of sodium alkyl-sulfate (SC„S); The charge density, flexibility, and crosslinking density of polyelectrolyte chains and the properties of surfactant molecules were shown to be critical factors in forming the ordered nanostructures of polyelectrolyte gel-surfactant complexes. Very rich supramolecular structures of Pm3n space group cubic, face-centered cubic close packing of spheres, hexagonal close packing of spheres, bilayer lamellar, Ia3d space group cubic, and 2D hexagonal close packing of cylinders were, respectively, observed in water-equilibrated polyelectrolyte-surfactant complexes.

Experimental

Materials Methacrylic acid (MAA) monomer (Aldrich, 98.5%) was vacuum distilled. N-isopropylacrylamide monomer (NIPAM, ARCOS, 99%) was purified by recrystalization in a toluene/hexane mixture. Diallydimethylammonium chloride monomer (DADMAC1, Fluka Chemicka Biochemika Corp.), Ν, N'-methylenebisacrylamide (BIS) (Ultragrade, Pharmacia LKB) as a crosslinker, N,N, N', N'-tetramethylethylenediamine (TEMED) (Sigma, 99+%) as an accelerator, and

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ammonium persulfate (APS) (Aldrich, 98+%) as an initiator, were used as received. The surfactants of QTAB (Fluka, £98%), 2C l2DAB (Fluka, >98%), and SCnS (Lancaster, 99%) were used without further purification. Deionized water was distilled before use.

Gel preparation Gels were prepared by free radical copolymerization. For PMAA or P(MAA/NIPAM) gels, a 15 wt% aqueous solution (5-8 mL) of reaction mixture with a desired comonomer molar ratio and a crosslinker density of 1 mol% (of total monomers) was bubbled with nitrogen for 15 min to remove oxygen in the reaction mixture. 35 pL of 10 wt% APS solution was added to the mixture. The final solutions were filtered through a 0.22 μπι Millipore filter and injected into the space between two glass plates which were separated by two spacers with a thickness of 0.63 ± 0.04 mm. Gelation was carried out at 65 °C for 24 h. One portion of the resulting gels were washed in distilled deionized water to remove the unreacted monomers, and were titrated with aqueous sodium hydroxide solution to determine the composition of the copolymer. Another portion of the gels was washed in a large amount of dilute sodium hydroxide solution for three weeks in order to neutralize the poly(methacrylic acid) and to remove the unreacted monomers. The dilute sodium hydroxide solution was changed every 1-2 days, and the final pH value was kept at ~ 8.2.

For PDADMAC1 gel, 5 mL 40 wt% aqueous solution of the reaction mixture with a crosslinker density of 0.5 mol% was degassed and bubbled with nitrogen. After 5 pL of TEMED and 25 pL of 10 wt% APS solution were added and well mixed, the mixture was filtered and injected into the space between two glass plates with a thickness of 0.63 ± 0.04 mm. Gelation was carried out at 22 °C for 24 h. The formed gels were then washed in a large amount of distilled water for 3 weeks to remove the soluble polymer and unreacted monomers. The distilled water was changed every 1-2 days.

Gel-surfactant complexation Complexes were prepared by immersing known amounts of neutralized and water swollen PMAA or P(MAA/NIPAM) gel discs (10 mm diameter, 2-3 mm thick) in very dilute aqueous sodium hydroxide solution of surfactants (pH - 8-9) and by immersing known amount of PDADMAC1 gel discs in aqueous solutions of surfactants. The surfactant concentration was approximately 1/3 of the critical micelle concentration (CMC) for the corresponding surfactant in the external solution phase. The total volume of the surfactant solution was controlled at such a level that the ratio r, defined by <number of surfactant molecules in solution>/<number of charged groups in the polymer networks>, was in the range of r £ 2. It means that the number of surfactant molecules was always in excess of the number of charged groups in the copolymer chains for complexation. The gel discs were equilibrated in the surfactant solution for about four weeks before being taken out for SAXS measurements. In addition, the equilibrated PMAA gel-surfactant complexes were also washed on the surface by water, and vacuum dried to determine the compositions of the dried complexes by elemental analysis (M-H-W Lab, AZ). No bromine was detectable inside the anionic polyelectrolyte gel-surfactant complexes.

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X-ray scattering measurements SAXS measurements were performed at the X3A2 State University of New York Beam Line, National Synchrotron Light Source at Brookhaven National Laboratory, using a laser-aided prealigned pinhole collimator.26 The incident beam wavelength (λ) was tuned at 0.154 nm. A 2D detector imaging plate was used in conjunction with an image scanner manufactured by Fujitsu Co. as the detection system. The sample to detector distance was 787 and 490 mm, corresponding to a q range of 0.2 £ q £ 4.8 and 0.2 £ q £ 7.0 nm"1 for PMAA or P(MAA/NTPAM) gel-CnTA complexes and for PDADMAC1 gel-SCS complexes, respectively, where q = (47c/X)sin(9/2) and θ being the scattering angle between the incident and the scattered X-rays. The experimental data were corrected for background scattering and sample transmission. The smearing effect was negligible for this setup.

Results and Discussion

The polyelectrolyte gel underwent a volume contraction when placed in the oppositely charged surfactant solutions. The complexation was a slow diffusion process of surfactant molecules into the polyelectrolyte gels, thus the gels showed different collapse kinetics and various degree of collapse depending on the charge density of polyelectrolyte chains, the surfactant solution concentration, and the tail properties of the surfactant. The elemental analysis results for PMAA gel-C„TAB complexes showed that the complex formation is essentially a stoichiometric binding reaction of the surfactant molecules with the copolymer network chains in terms of charges. Accordingly, there could only be slightly crosslinked polyelectrolyte chains, oppositely charged surfactant ions, and water inside the complexes after the gels reached the equilibrium of collapse. By knowing the weight of the dried gel before complexation, the weight of complexes equilibrated with water (the water on the surface of complex discs was adsorbed before weighing), and the weight of the dried complexes, we could determine the weight fractions of polyelectrolyte chains, surfactant ions, and water inside the complexes.

L Charge Density Effects of Polyelectrolyte Chains on Structures of Complexes. 1. P(MAA/NIPAM) gel-CnTA systems. Figure 1 shows a typical SAXS profile of PMAA gel-Ci2TA complex formed with fully charged PMAA chains. More than ten sharp scattering peaks were observed, which manifested that highly ordered supramolecular structures was formed within the collapsed gel. The location of these peaks had a ratio of 21/2: 41/2: 51/2: 61/2: 81/2: 101/2: 121/2: 141/2: 161/2: 201/2: 211/2, suggesting a cubic phase structure of Pm3n space group which has the extinction rules compatible with the observed reflections. The proposed Miller indices of the corresponding diffraction planes are 110, 200, 210, 211, 220, 310, 222, 321, 400, 420, 421. By decreasing the charge density of the P(MAA/NLPAM) chains to 91%, a similar scattering curve with the same spacing ratio of the scattering peaks was observed. However, the corresponding peak position shifted slightly toward higher q value, suggesting a shrinking of the unit cell. The lattice

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Figure 1. Typical SAXS profile of PMAA gel -Cl2TA complex. The scattered peaks can be indexed according to a cubic structure of Pm3n space group. The dotted line represents the calculated scattering curve based on the connected unit cell model shown in the inset.

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parameter of the unit cell was calculated to be a = 8.54 and 8.21 nm for complexes formed by P(MAA/NIPAM) chains with charge density (mol%) of 100% and 91%, respectively.

The Pm3n cubic phase structure has been observed between the aqueous micellar solution and the hexagonal phase (Ha) in many surfactant/water binary or surfactant/water/oil ternary systems.13*15,27"28 However, it was, for the first time, observed in polyelectrolyte-surfactant complexes. So far, there was still no final judgement on the structure model for Pm3n cubic. The Luzzati group proposed a connected model, built up of a 3D cage-like network of rods with a finite length of a/81/2 and a body-centered cubic (BCC) arrangement of spheres.13 The rods were joined three by three at one end, four by four at the other, forming the bars of the cage in which a spherical micelle was enclosed. Each unit cell contained 24 rods and 2 spheres. The interior of the rods and spheres was occupied by the paraffin chains of surfactants. The Lindblom group27,28 proposed a disconnected structural model on the basis of NMR investigation with the absence of long-range diffusion, which consisted of rod-shaped micelles with an axial ratio of about 2.1. Each unitcell contained eight micelles. Two micelles, occupied at each corner and in the center of the unitcell, showed an isotropic rotational disorder; while the other three pairs of micelles, located at each face of the unitcell in the positions of (1/2, 1/4,0), (1/2, 3/4, 0), and so forth, showed a 2D rotational disorder. However, the overall symmetry for the Pm3n space group needs be restored in the average over all unitcells via a dynamic disorder. It should be noted that for x-ray scattering, the disorder does not need to be dynamic. A frozen disorder may lead to the same results and cannot be distinguished by SAXS. Usually, only few small-angle reflections could be observed because the Pm3n cubic phase structure occurred at high water content. The use of synchrotron X-ray source for our systems produced very detailed scattering curves with a high resolution and ten more scattering peaks, which enabled us to simulate the scattering curve on the basis of the two different models and the known chemical compositions of the complex.

From the Luzzati's wireframe model,13 We have viewed all the spheres as points, and all the rods being infinitely thin. The several most intense peaks observed in experimental scattering curve were retained, and the corresponding superposition of their density waves in real space was used to plot an isosurface of the structure. This truncation of the Fourier series, corresponding to a convolution in real space, put flesh to the wireframe model. If the phase and relative intensities of the first few peaks are correct, we view the resulting model to be a good approximation of the real structure. The resulting space-filled connected unit cell model for the Pm3n cubic structure with φοΐ2ΤΑ

= 0.4 is shown in the inset of Figure 1, which consists of a 3D cage-like continuous network and a BCC arrangement of spheres, both imbedded in a polar matrix consisting of water and copolymer chains. The dotted line in Figure 1 is calculated from this model. It should be mentioned that all the simulated scattering curves in this paper did not yet try to quantitatively match the experimental ones, but only tried to match all the peak positions and the extinction rules and to provide a first order approximation to the intensities. Thus, the

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simulated peaks were usually sharper than the experimental ones, and their intensities were higher in the high q range. Introducing disorder or smooth density transitions at the domain boundaries would then lead the calculated curves to approach the experimental ones more closely. However, at the present stage with the wireframe models, such a refinement would be excessive.

For the Lindblom model, the symmetry of the micelles was lower than that required by the Pm3n space group. They had assumed that the overall symmetry could be restored in the average over all unit cells via dynamic disorder. It means that we need to consider the average of the electron density over the unit cells which will smear out the cylindrical micelles to spheres and oblate disks, respectively. Thus, the part of the scattering curve connected with the hkl peaks will look very similar to that of a non-disordered model consisting of real spheres and disks, or to a disordered model of oblate disks where the spheres in the corners are produced by the rotational disorder of disks. Such a disconnected structural model is shown in the insert of Figure 2. The dotted line in Figure 2 was the calculated scattering curve from this model. The agreement with the experimental curve is worse than that from the connected model. However, presently, we cannot distinguish whether the structure is connected or disconnected from the SAXS results alone.

Figure 3 shows the SAXS profiles of the P(MAA/NIPAM) gel-d 2TA complexes at charge contents of 75% and 67%, respectively. The scattering peaks could be indexed according to the structure of hexagonal close packing of spheres (HCP). To simplify some calculations for simulating the scattering curve, an orthorhombic unitcell, as shown in the insert, was chosen. Putting one sphere at the origin [0,0,0] and one sphere at the center of the base plane [a/2,6/2,0] produced a single 2D closed-packed layer of spheres. The next layer at ζ - c/2 was identical to the base layer but shifted by 6/3 in the y direction. The layer at ζ = c was identical to the one at ζ = 0, resulting in an A-B-A stacking order. Totally, there were four spheres per unitcell. Similarly, for complexes with the same type of structures, the corresponding scattering peaks shifted slightly toward the higher q values with decreasing charge density in the P(MAA/NIPAM) chains. The observed scattering peaks from the complex formed by 67% charge content of P(MAA/NIPAM) chains could be indexed as: 110 and 020 for 1.61 nm 1 , 002 for 1.71 nm 1 , 111 and 021 for 1.82 nm 4 , 112 and 022 for 2.35 nm'1, 113 and 023 for 2.78 nm"1. The dimension of the orthorhombic unit cell was calculated to be a = 4.5, b = 7.8, and c = 7.4 nm for complex with 75% charge content in P(MAA/NIPAM) chains, and a = 4.4, b = 7.7, and c = 7.3 nm for complex with 67% charge content in P(MAA/NIPAM) chains, respectively. The a value corresponds to the sphere-to-sphere center distance.

The dotted line in Figure 3 shows the calculated scattering curve from the orthorhombic unit cell model. In comparison with the calculated scattering curve, the relative intensities of the experimental scattering peaks from the complex formed with 75% charge content in the P(MAA/NIPAM) chains were different, namely, the intensity of the second peak (1.70 nm*1) was much lower than that of the calculated scattering curve. It is known that the scattered intensity can be strongly affected by the form factor of a sphere F(q,R) = {3/(qR)3[sin(qR)-qRcos(qR)]}2 with R being the sphere radius. The low intensity of the second scattering peak suggested that the first

251

Figure 2. Typical SAXS profile of PMAA gel -CnTA complex. The dotted line represents the calculated scattering curve based on the disconnected unit cell model shown in the inset.

Figure 3. SAXS profiles of P(MAA/N1PAM) geUCnTA complexes formed with charge content of 75% and 67% in the copolymer chains, respectively. The dotted line represents the calculated scattering curve based on the inserted orthorhombic unitcell model for the HCP structure.

252

minimum of the form factor function nearly coincided with the position of this peak. This coincidence required that the diameter of the spheres must be about 14% larger than the inter-distance between the spheres. It means that the spheres inside the complexes touched each other and were deformed slightly. Therefore, the structure of the complexes could be described as hexagonal close packing of large deformed spherical micelles. The relative intensities of the scattering peaks from complex with 67% charge content in the P(MAA/NTPAM) chains were more close to those of the calculated curve from the plain HCP structural model. The reason was that the spherical micelles became smaller when the charge density of the copolymer chains was decreased so that the spherical micelles did not deform too much. By further decreasing the charge content of the copolymer chains to 50%, only one broad peak was observed, indicating a less-ordered structure inside the complexes.

In sequencing with decreasing the charge density of the P(MAA/NIPAM) chains, the structures of P(MAA/NIPAM) gel-Ci2TA complexes changed from Pm3n cubic to HCP, finally became less ordered. It should be mentioned that for P(MAA/NIPAM) gel-d4TA and P(MAA/NIPAM) gel-C]6TA complexes, a structure of face-centered cubic close packing of spheres (FCC) was observed between Pm3n cubic and HCP phase structures when decreasing the charge density of the P(MAA/NIPAM) copolymer chains.29 Figure 4 shows the SAXS profiles of P(MAA/NIPAM)-Ci6TA and -Q 4 TA complexes formed with a charge content of 67% in the copolymer chains. Five sharp diffraction peaks with a spacing ratio of 3i/2 . 4i/2 . 8i/2 . n i / 2 : 1 2 i / 2 w e r e o b s e r v e d ? indicating a plain FCC structure. The corresponding diffraction planes can be indexed as 111,200, 220, 311, 222.

L 2. P(MAÀ/NIPAM) gel-2C12DA systems Figure 5 shows the SAXS profiles of the P(MAA/NIPAM) gel-2Ci2DA complexes formed with charge content of the copolymer chains 75%. Two scattering peaks at q = 1.78 and 3.60 nm"1 were observed, corresponding to the 001 and 002 planes of a layered structure. A d spacing (2^qmax) of 3.53 nm was obtained. This thickness is about twice the length of two surfactant molecules, indicating a bilayer arrangment of double-tailed 2C]2DA ions inside the complexes. Such a bilayer structure has been commonly observed in polyelectrolyte-surfactant complexes.1 The ionic headgroup of the surfactant interacted with the oppositely charged copolymer chains by electrostatic interactions, while the hydrophobic tails of the surfactants coupled together through hydrophobic interactions, thus stabilizing the bilayer structures.

Figure 6 shows two SAXS profiles of P(MAA/NIPAM) gel-2Ci2DA complexes at charge content of 67%, and 50% in P(MAA/NIPAM) chains, respectively. At a charge content of 67%, the scattering curve showed three diffraction peaks at q = 1.73, 2.00, and 2.66 nm"1, respectively, with a spacing ratio of 31/2: 41/2: 71/2. At a charge content of 50%, the scattering curves also showed three peaks, but at q = 1.78, 2.06, and 3.24 nm*1, respectively, with a spacing ratio of 31/2: 4 1 / 2: 101/2. The combination of these two scattering curves suggested a cubic structure of Ia3d space group which typically gives diffraction with a spacing ratio of 31/2: 41/2: 71/2: 81/2: JQI/2. j|i/2 Ahj ugk ia3^ C U O Tc i s o n e 0 f t n e m o s t commonly observed phase structures in lipid/water binary systems at phase areas intermediate between the

253

Figure 4. SAXS profiles of Ρ (MAA/NIPAM) gel-Cl6TA and-C14TA complexes formed with 67% charge content of the copolymer chains. The scattering peaks can be indexed according to a FCC structure.

Figure 5. SAXS profiles of Ρ (MAA/NIPAM) gel-2C12TA complexes at charge contents of 100% and 75% in the P(MAA/NIPAM) chains, respectively. The scattering peaks can be indexed as a bilayer lamellar structure.

254

Figure 6. SAXS profiles of PfMAA/NIPAM) gel-2C12TA complexes at charge contents of 67% and 50% in the Ρ (MAA/NIPAM) chains, respectively. The scattering peaks can be indexed as a cubic structure of Ia3d space group.

255

lamellar and the hexagonal phases,30 it was, for the first time, observed in polyelectrolyte-surfactant complexes. The observed four diffraction peaks with a spacing ratio of 31/2: 4m: lm: \0m can be indexed as 211, 220, 321, and 420. Furthermore, the parameters of the Ia3d cubic unit cell of 8.90, and 8.65 nm can be calculated for the two complexes at charge content of 67%, and 50%, respectively. It should be noted that the FCC structure would also produce the first two peaks with the observed ratio of 31/2: 41/2, but the higher order peaks corresponding to 71 / 2 and I0m should be extinct there. The inset of Figure 6 shows the unit cell model of Ia3d cubic structure consisting of two infinite 3D networks, mutually interwined and unconnected with the rods joined coplanarly three by three.16 In surfactants/water binary systems, the single-tailed surfactants usually form the type I structures with paraffin chains inside the rods, while the double-tailed surfactants form the type II structures with the polar moieties inside the rods. For P(MAA/N"PAM)-2Ci2DA complexes at charge content of 67% - 50%, the interior of the rods should be occupied by the water phase with polar head of 2C i 2DA ions and negatively charged copolymer chains, in terms of the compositions of the complexes and the area ratio of head-to-tail of 2Q2DA cations.

IL Flexibility Effect of Polyelectrolyte Chains on Structures of Complexes. Figure 7 shows the SAXS profiles of PDADMAC1 gel-SCS complexes with η = 14 and 12, respectively. The sharp diffraction peaks with a spacing ratio of 1: 31/2: 4m: lm indicated a structure of 2D hexagonal close packing of cylinders. This structure was quite different from the 3D Pm3n cubic structure observed in PMAA gel-QTA complexes,29 namely, Pm3n cubic showed structural elements of both spheres and short rods, while the 2D hexagonal structure showed only the structural element of rods. The structure difference between the two complexes could be ascribed to the difference in flexibility of the PDADMAC1 chains and the PMAA chains. The flexible PMAA chains were relatively easy to bend. Thus, the Ο,ΤΑ ions bound by PMAA chains could form spherical micelles inside the resulting complexes. In contrast, the PDADMAC1 chains were far stiffer due to the five-member ring on the backbone chains. The arrangement of the QS ions (n £ 12) bound by PDADMA chains could only follow the rigid polymer backbone to form the structural element of cylinders. III. Tail Length Effect of Surfactants on Structures of Complexes Figure 8 shows the SAXS profiles of PDADMAC1 gel-SC„S with η =10 and 8, respectively. Nearly ten scattering peaks with a spacing ratio of of 21/2: 4m: 5m: 6m: 81/2: 101/2: 121/2: 141/2: I6m: 2lm were observed, indicating a Pm3n cubic structure formed inside the complexes. In comparison with the 2D hexagonal structure of PDADMAC1 gel-SC„S at η *> 12, the decrease in the hydrophobic tail length of surfactant did induce the structural transition of the formed polyelectrolyte gel-surfactant complexes. As discussed above, the 2D hexagonal structure consisted of only infinitely long rod-like structural elements, while 3D Pm3n cubic structures contained structural elements of both spheres and short rods. For the same PDADMAC1 gel, the decrease in the tail length of surfactants would lead to an increase in the area-per-chain at the polar/apolar interface (SCh) of the CnS ions.

256

Figure 7. SAXS profiles of PDADMACl gel~SC14S and -SC12S complexes. The scattering peaks can be indexed according to a structure of 2D hexagonal close packing of cylinders.

257

Eventually, the interfacial curvature of the C„S ions became so large that they could also self-assemble to micelles with spherical shape inside the PDADMAC1 gel. The conflict between the increase in SCh and the stiffness of the PDADMAC1 chains may be reconciled by a structure containing both rods and spheres. The "relative bulkiness" effects of the polar and the apolar moieties of the surfactant molecules on the phase behavior of lipid/water systems has been discussed elsewhere.14 It should be noted that the SAXS profiles of PDADMAC1 gel-SQoS complex in Figure 8 also exhibited a sharp scattering peak between 121/2 and 141/2. This peak could not be ascribed to the diffraction from the Pm3n cubic structure. Possibly, there was a small amount of 2D hexagonal structure mixed in the complexes during the structural transition from 2D hexagonal to 3D Pm3n cubic. The peak might be from the second-order peak of 2D hexagonal structure. IV. Crosslinker Density Effect of Polyelectrolyte Gels on Structures of Complexes. Figure 9 shows the SAXS profiles of PMAA gel-d2TA complexes formed with different crosslinker density of the polyelectrolyte gel. As expected, an increase in the crosslinker density of the polymer network chains disturbed thé highly ordered structures of complexes. Instead of sharp scattering peaks in the scattering curve at 1.0 mol% crosslinker density, only one broad peak was observed at a high crosslinker density of 7.0 mol%. In contrast, in the PDADMAC1 gel-SCi2S complexes, the ordered structures of 2D hexagonal did not break down as the crosslinker density was as high as 7.7 mol%, and even at a crosslinker density of 12.0 mol%, the disturbance for the SAXS curve only showed a relative lower intensity and less clear high order peaks.31 After analyzing the chemical compositions of the two polyelectrolyte gels of PMAA and PDADMAC1, the dramatic difference of the crosslinker density effects on the ordered structures of the formed complexes is not surprising. The repeating unit length of PMADMAC1 gel are nearly twice that of the PMAA gel, while the definition of the crosslinker density here'represents the molar ratio of the BIS crosslinker to MADMAC1 monomer or to MAA monomer. It means that with the same crosslinker density value, the average mesh size of the PDADMAC1 gel would be nearly twice as large as that of the PMAA gel. Consequently, the disturbance of the crosslinker density on the ordered structures in the PDADMAC1 gel-SCî2S complexes was observed at a much higher crosslinker density value. As a whole, the supramolecular structures were formed inside the 3D gel network and the decrease in mesh size of the polylectrolyte network eventually would hinder the ordered arrangement of the surfactant molecules inside the gel.

Finally, we like to mention that after having analyzed the compositions and the structures of the complexes, we can calculate the size and the aggregation number of the micelles packed inside the complexes. The detail calculation procedure can be found elsewhere.29 Both the aggregation number and the size of micelles packed in the complexes depend on the charge density of polyelectrolyte chains, the flexibility of polyelectrolyte backbone chains, and the hydrophobic tail length of surfactants. Nevertheless, the size and the aggregation number of the micelles formed inside the

258

1 2 3 4 5 q/nm"1

Figure 8. SAXS profiles of PDADMACl gel-SC10S and -SCsS complexes. The scattering peaks can be indexed according to a structure ofPm3n cubic.

1 2 3 4

q/nm"1

Figure 9. SAXS profiles of PMAA gel-C12TA complexes formed at different crosslinker density of the PMAA gel network.

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polyelectrolyte gel were comparable with those of the micelles formed outside the gel (in aqueous solution).29

Conclusions

The interactions of slightly crosslinked polyelectrolyte gels of PMAA or P(MAA/NIPAM) and PDADMAC1 with oppositely charged surfactants could produce complexes with very rich highly ordered nanostructures. Both electrostatic and hydrophobic interactions are very important to drive the complex formation inside the polyelectrolyte gel. The charge density of polyelectrolyte chains, the relative bulkiness of the polar and the apolar moieties of surfactant molecules and the flexibility of polyelectrolyte chains could significantly affect the self-assembly behavior of surfactant molecules inside the gels, and thus determine the shape of structural elements. When the charge density of polyelectrolyte chains is decreased to some value, the structures of the resulting complexes become less ordered. High crosslinker density of the polyelectrolyte network chains could also hinder the formation of highly ordered structure of surfactant molecules inside the complexes. The aggregation number and the size of the micelles packed inside the complexes could be calculated after knowing the compositions and detail structures of the formed polyelectrolyte-surfactant complexes.

Acknowledgement BC gratefully acknowledges the support of this work by the National Science Foundation (DMR 9612386) and the U.S. Army Research Office, Durham (DAAG559710022). The use of the SUNY Beam Line (with support from the Department of Energy) at the National Synchrotron Light Source, Brookhaven National Laboratory is also gratefully acknowledged.

References

1. Ober, C. K.; Wegner, G. Adv. Mater. 1997, 9, 17. 2. Zhu, N.; Liggitt, D.; Liu, Y.; Debs, R. Science 1993, 261, 209. 3. Gershon, H.; Ghirlando, R.; Guttman, S. B.; Minsky, A. Biochemistry 1993, 32,

7143. 4. Crowell, K. J.; Macdonald, P. M. J. Phys. Chem. Β 1997, 101, 1105. 5. Radler, J.O.; Koltover, I.; Saldit, T.; Safinya, C.R. Science 1997, 275, 810. 6. Koltover, I.; Salditt, T.; Radler, J.O.; Safinya, C.R. Science 1998, 281, 78. 7. Interactions of Surfactants with Polymers and Proteins (Eds: Goddard, E. D.;

Ananthapadmanabham, K. P.), CRC Press, Boca Raton, FL 1993. 8. Fundin, J.; Brown, W.; Vethamuthu, M. S. Macromolecules 1996, 29, 1195. 9. Khokhlov, A. R.; Kramarenko, E. Y.; Makhaeva, E. E.; Starodubtzev, S. G.

Macromolecules 1992, 25, 4779. 10. Okuzaki, H.; Osada, Y. Macromolecules 1995, 28, 380.

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11. Chu, B.; Yeh, F.; Sokolov, E. L.; Starodoubtsev, S. G.; Khokhlov, A. R. Macromolecules 1995, 28, 8447.

12. Balmbra, R. R.; Clunie, J. S.; Goodman, J. F. Nature 1969, 222, 11. 13. Tardieu, Α.; Luzzati, V. Biochem. Biophys. Acta 1970, 219, 11. 14. Mariani, P.; Luzzati, V.; Delacroix, H. J. Mol. Biol. 1988, 204, 165. 15. Lindblom, G.; Rilfors, L. Biochim. Biophys. Acta, 1989, 988, 221. 16. Auvray, X.; Petipas, C.; Anthore, R.; Rico, I.; Lattes, A. J. Phys. Chem. 1989,

93, 7458. 17. Seddon, J. M. Biochim. Biophys. Acta 1990, 1031, 1. 18. Khandurina, Yu, V.; Dembo, A. T.; Rogacheva, V. B.; Zezin, A. B.; Kabanov,

V. A. Polym. Sci. 1994, 36, 189.

19. Yeh, F.; Solokov, E. L.; Khokhlov, A. R.; Chu, B. J. Am. Chem. Soc. 1996, 118, 6615.

20. Antonietti, M.; Conrad, J.; Thunemann, A. Macromolecules 1994, 27, 6007. 21. Antonietti, M.; Kaul, Α.; Thunemann, A. Langmuir 1995, 11, 2633. 22. Antonietti, M.; Maskos, M. Macromolecules 1996, 29, 4199. 23. Antonietti, M.; Burger, C.; Effing, J. Adv. Mater. 1995, 7, 751. 24. Antonietti, M.; Radloff, D.; Wiesner, U.; Spiess, H. W. Macromol. Chem. Phys.

1996, 197, 2713. 25. Sokolov, E. L.; Yeh, F.; Khokhlov, A. R.; Chu, B. Langmuir 1996, 12, 6229. 26. Chu, B.; Harney, P. J.; Li, Y.; Linliu, K.; Yeh, F.; Hsiao, B. S. Rev. Sci.

Instrum. 1994, 65, 597. 27. Eriksson, P.; Lindblom, G. J. Phys. Chem. 1985, 89, 1050. 28. Eriksson, P.; Lindblom, G.; Arvidson, G. J. Phys. Chem. 1987, 91, 846. 29. Zhou, S.Q.; Burger, C.; Yeh, F.; Chu, B. Macromolecules 1998, 31, 8157. 30. Luzzati, V.; Tardieu, Α.; Gulik-krzywicki, T.; Rivas, E.; Reiss-Husson, F.

Nature 1968, 220, 485. 31. Yeh, F.; Sokolov, E.L.; Walter, T.; Chu, B. Langmuir 1998, 14, 4350.

Chapter 17

Some Thermodynamic Considerations of the Lower Disorder-to-Order Transition of Diblock Copolymers

M. Pollard1, Ο. K. C. Tsui1, T. P. Russell1, Α. V. Ruzette2, A. M. Mayes2, and Y. Gallot3

1Polymer Science and Engineering Department, University of Massachusetts, Amherst, MA 01003

2Department of Materials Science, Massachusetts Institute of Technology, Cambridge, MA 02139

3Institut Charles Sadron, Strasbourg, France

The lower critical ordering of diblock copolymers is an entropically driven phase transition that is accompanied by a negative volume change on mixing. Small angle neutron scattering (SANS) studies of the phase transition under hydrostatic pressure has shown a very large pressure coefficient δΤ/δΡ = 147°C/kbar. Differential scanning calorimetry studies of the phase transition show that the transition from the disordered to the ordered state is endothermic with an enthalpy, ΔΗ ~ 0.2 J/g. X-ray reflectivity studies of thin copolymer films as a function of temperature exhibits the characteristic thermal expansion of the copolymer film with a discrete change in the film thickness at the transition that corresponds to a 0.35 % volume change. This agrees, within the same order of magnitude, with what would be predicted from the Clapeyron equation.

The order-disorder transition (ODT) in diblock copolymers is a fluctuation induced first-order phase transition that separates a high temperature phase mixed state from a low-temperature microphase separated state. At temperatures below the ODT, also called the upper critical ordering transition (UCOT), repulsive interactions between the monomeric segments of the two blocks drive a segmental demixing that forms a microphase separated structure comprised of nanometer-sized domains. At temperatures above the ODT, thermal energy is sufficient to overcome the unfavorable interactions and segmental mixing occurs. In general, the ODT is determined physically by a balance between energetic and entropie factors unique to block copolymers (1). The coupling between the dynamical behavior of macromolecules and these molecular-level drivers for self-assembly at the ODT has produced a sustained academic interest in the subject of diblock copolymer ordering. Industrial interest in the understanding of the ODT stems from the desire to control

© 2000 American Chemical Society 261

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the processability of block copolymers that are used as thermoplastic elastomers, viscosity stabilizers, and pressure sensitive adhesives (2).

Very few studies, however, have been devoted to the more fundamental aspects of the phase transition, in particular to the measurement of first-order thermodynamic parameters, such as volume and enthalpy, which change discontinuously at the ordering transition. The importance of these parameters to our experimental and theoretical knowledge of block copolymer phase behavior was recently demonstrated by Hajduk et al (3), who examined the ODT phase behavior of a poly(styrene-&/<?cfc-isoprene) (P(S-M)) diblock copolymer under hydrostatic pressure. The primary factor that governed the phase behavior in this study was the coupling between the applied hydrostatic pressure, enthalpy and volume changes at the transition. Calorimetric measurements for P(S-M) indicated an exothermic process in going from the disordered to the ordered state (4,5). The volume reduction upon ordering measured by PVT experiments was consistent with the reported positive pressure coefficient of 20 °C/kbar based on a thermodynamic analysis using the Clapeyron equation. Stiihn and coworkers subsequently reported a volume expansion upon ordering at lower pressures for P(S-M), resulting in a negative pressure coefficient from 1 to 25 bar (6). At higher pressures they observed a positive pressure coefficient that approached the value reported by Hajduk and coworkers. These results indicated the presence of a minimum in the order-disorder phase boundary in T-P plane for this copolymer. Independently, Schwahn and coworkers also observed a minimum in the T-P phase boundary for a block copolymer of ρ ο 1 y (ethylenepropylene) and poly(dimethylsiloxane) (7,8). Taken together, these results suggest a nontrivial pressure dependence of the relative chain packing efficiency between the ordered and disordered states.

Recently, the pressure dependence of the ordering transition in diblock copolymers of perdeuterated styrene and η-butyl methacrylate, denoted P(d-S-b-nBMA) from atmospheric pressure to 1.02 °C/kbar was reported (9). Microphase separation in these materials occurs on heating the disordered, phase mixed copolymer. This transition is termed the lower critical ordering transition (LCOT), since the order-disorder phase behavior is inverted from the usual case (10). The LCOT temperature as a function of hydrostatic pressure was investigated by in-situ small-angle neutron scattering (SANS), yielding a pressure coefficient, dT L C 0 T/dP, of 147 °C/kbar for this transition. The fact that this order-disorder phase boundary rose more steeply with pressure (an order of magnitude higher than the corresponding value found for the ODT studies mentioned above) suggested that a more detailed analysis of changes in the latent heat and volume changes be undertaken for this material. In this brief report, high-pressure SANS experiments and recent measurements to quantify the discontinuous changes in these first-order parameters are discussed.

The changes in enthalpy that occur at the LCOT of a P(d-S-fc-nBMA) copolymer (Mw=85K) were measured using differential scanning calorimetry (DSC). The volume change of the transition was determined by specular x-ray reflectivity measurements of the copolymer as a function of temperature. From the Kiessig fringes in the reflectivity profile, the thickness of the sample can be determined very

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precisely, independent of the film density, affording an alternate means by which the volume change of the transition can be measured. In addition, since the film is confined to a substrate, volume changes only occur normal to the substrate and, consequently, changes in the thickness directly reflect changes in the volume of the film.

Experimental

The symmetric diblock copolymer sample, P(d-S-b-nBMA), used in this study was labeled with deuterium on the styrene block for SANS studies. The weight average molecular weight was 8.5 χ 104 g/mol with a polydispersity index of 1.05 and a styrene fraction of 0.49. The SANS measurements were performed at the Cold Neutron Research Facility at the National Institute of Standards and Technology on beamline NG-3. The LCOT transition was fully reversible in pressure and temperature; GPC showed no significant degradation. DSC measurements were made on a Perkin-Elmer DSC-3 at a scanning rate of +20 °C/min under a helium purge using samples annealed in-situ at 135 °C for 2 hrs (well below the LCOT of 153 °C). The low heat flow expected from this transition (~1 mW) is well above the peak-to-peak noise (-20 μ\¥) observed on this instrument. X-ray reflectivity measurements were performed on a 2430 À thick sample spin coated onto a pre-cleaned 50mm diameter x 5mm thick silicon wafer. The reflectometer was equipped with an 18kW Cu rotating anode generator and a channel-cut Si monochromator that delivered 1.54 Â Cu K a radiation onto the sample (άλ/λ ~ 1.5 χ 10~4, where λ denotes the wavelength). Reflectivity profiles were obtained by rotating the sample θ and the detector 20 so as to measure the specularly reflected radiation as a function of the wavevector, k z (=2π8ΐηθ/λ) up to 0.032 À'1. Above the critical wavevector for Si (kc

~ 0.016 Â"1) the reflectivity decreases rapidly and oscillates with a period ~ n/d, where d is the thickness of the polymer film. The reflectivity profiles were fit using the Parrat formalism assuming a three-layer model consisting of P(d-S-b-nBMA), S1O2, and Si as described elsewhere (12), where the thickness of the oxide layer could be measured independently.

Results and Discussion

Shown in Figure 1 is a series of SANS profiles of the Mw=85K P(d-S-b-nBMA) sample as a function of the momentum transfer, Q (=4nsin0/X, where 29 is the scattering angle) taken over a pressure range from 0.01 kbar to 1.02 kbar at 170 °C. At ambient pressures, the LCOT temperature of this material is 153 °C. At low applied hydrostatic pressures (P < 0.17 kbar), the scattering profiles are sharply peaked at Q M A X ~ 0.021 À"1, with a second order reflection discernable at 0.042 A ( ~

2QMAX)- This is characteristic of a microphase separated sample with randomly oriented lamellar microdomains having a repeat period of 300 À. As the hydrostatic pressure is increased, the scattering profiles become more diffuse, the peak position

264

shifts to higher values of Q, and the second order reflection is lost. The shift in the peak position is greater than expected solely from volume reduction through the material's low compressibility, and is therefore attributed to changes in the amount of chain stretching as the ODT is traversed. At higher pressures (P > 0.17kbar) the scattering profiles are characteristic of a disordered diblock copolymer melt with pre-transitional fluctuations (11). Based on these observations, it is evident that the J>(d-S-b-nBMA) sample has been driven from the microphase separated to the phase mixed state upon application of pressure. Isobaric cooling experiments at moderate pressures also show similar characteristic changes in the scattering profiles as a result of traversing the transition from the ordered to the disordered state. In fact, it has been shown previously that there appears to be an equivalence between temperature and pressure in the thermodynamic behavior of systems that exhibit the LCOT in the vicinity of the transition (9). Data obtained at various hydrostatic pressures, including the temperature dependence of the peak height at Q - 0.02 Â"1, its full-width-at-half-maximum (FWHM), and the peak position QM Ax(T), could be superimposed to produce master curves by shifting the data along the temperature axis. The magnitude of the shift used for each pressure generates master curves that yield a pressure coefficient for the LCOT, (ôTL C 0 T/6P), of 147 °C/kbar for this sample.

10000

m l . . . . I 0 0.01 0.02 0.03 0.04 0.05

Q (A*1)

Figure 1. Small angle neutron scattering profiles of 85Κ P(d-S-b-nBMA) as a function of the momentum transfer Q at a series of hydrostatic pressures ranging from 0.01 kbar to 1.02 kbar.

The DSC thermogram obtained for the P(d-S-b-nBMA) copolymer during heating at 20 °C/min is shown in Figure 2. In going from the disordered to the ordered state, a weak endothermic peak is observed at ~ 193 °C, which we associate with the enthalpy of LCOT. The fact that there is a latent heat associated with this transition

265

indicates that the transition is first order. The amount of sample used in this experiment was 30 mg. The latent heat estimated based on the area under the endothermic peak is ~ 0.2 J/g and is comparable with the numbers typically found in UCOT systems. The apparent increase in the transition temperature compared to the value obtained in SANS, viz. 153 °C, is due to the slow kinetics of block copolymer ordering and the high heating rate required to detect the low heat flow. No exotherm was observed upon cooling. This is attributed to the kinetic dominance of the transition which prevents rapid disordering within the time scale allowed by the high cooling rate and the proximity to the polystyrene glass transition. In addition, we note that a low rate of repeatability was observed over different sample runs for this material. This suggests that the ordering time scale depends sensitively on the pre-annealing times and thermal history. This is under further investigation. Systematic increases in the ODT temperature have also been found in DSC thermograms at elevated heating rates by Kim et al. on a poly(styrene-M0c&-isoprene-Moc£-styrene) copolymer (5).

-205 i ι ' ' ' I ι ι ι ' I I I I I 1 I I I I I I I I I I I I I I I I I I I I I I I I I 180 190 200 210 220

τ r e )

Figure 2. Differential scanning calorimeter scan of 85Κ P(d-S-b-nBMA) taken at a scanning rate of 20 "C/min. The LCOT, observed here at ~ 195 °C, is endothermic and is characterized by a latent heat of - 0.2J/g.

The pressure coefficient of a first order phase transition is related to the enthalpy or latent ΔΗ of the transition and the associated volume change, AV, through the Clapeyron equation as follows (13):

δΤ/δΡ = ΤΔΥ/ΔΗ (1)

266

If the pressure coefficient and the enthalpy of the transition are known, the volume change of the transition can be calculated. From the value obtained above for the latent heat, Equation (1) predicts a volume increase, ÔV7V, of order 0.1 % at the LCOT for the Mw=85K P(i/-S-6-nBMA) at atmospheric pressure. To confirm the volume change experimentally, dilatometric techniques are usually employed, where mercury displacement in a fine bore capillary provides the high precision necessary to detect such small changes. Although dilatometry studies are in progress, x-ray reflectivity was employed as an alternative route to access bV/V experimentally where a high precision in the measurement of thickness can be achieved.

Shown in Figure 3 is a subset of four x-ray reflectivity profiles as a function of scattering wavevector k z measured at different temperatures. The data have been offset vertically for clarity. Below the critical wavevector for the silicon substrate (kc

-0.016 Â"1), there is total external reflection of the x-rays. Above the critical edge at 0.016 Â"1, the x-rays penetrate the sample and the reflectivity signal drops precipitously. The periodic oscillations (Kiessig fringes) in the data arise from interference between x-rays reflected at the air-polymer and polymer-substrate interfaces. The thickness of the sample, d, can, in principle, be determined directly from the fringe separation, though in these studies the data were fit to a three-layered model as described previously. Typically, a 0.3 % variation in the thickness parameter about its best-fit value will give rise to greater than 10 % increase in the <χ 2> parameter. In practice, we find that the error bar of a measurement is better than ±0.1 % according to its reproducibility. As is evident in the data, the fringe separation changes substantially between Τ = 145 and 149 °C, indicating a discontinuous jump in the film thickness.

Figure 3. X-ray reflec­tivity profiles as a function of kz at the temperatures indicated. The reflectivity profiles have been offset vertically for clarity. The oscillations in the data, the Kiessig fringes, character­ize the sample thickness independent of the film density. The line through the data were calculated by a simple three layer model as described in the text.

0.015 0.020 0.025 0.030

k2(Â"1)

267

The film thickness as a function of temperature for the 2430 À thick P(d-$-b-nBMA) film is shown in Figure 4. As expected from thermal expansion, the film thickness increases with increasing temperature. At Τ ~ 148 °C, there is a jump in the film thickness. Independent neutron reflectivity studies on the same sample show that this is the LCOT for the thin film. The apparent decrease in the transition temperature from the bulk value is due to the wetting layers at the substrate-polymer and polymer-air interfaces. These density fluctuations induced at the interfaces stabilize the ordered state. Detailed studies of the effect of film thickness on the ODT have been made by Menelle et al. for the UCOT of a symmetric diblock copolymer of perdeuterated styrene and methyl methacrylate (14) and, more recently, by Mansky et al. on P(d-S-&-nBMA) (15). Using linear extrapolations of the data below and above T- 148 °C in Figure 4, a discontinuity of ~ 7 Â is observed and hence a dV/V of + 0.3 % at the transition to the ordered state. This result is three times larger than the value we estimated above using the Clapeyron equation. It is unclear at present whether this disagreement results from a surface topography that may exist for the copolymer in the ordered state or from an additional contribution to the volume change due to surface effects in thin films that have not been accounted for in the above calculation where bulk parameters were used. Dilatometric measurements are currently in progress to determine the volume change of a bulk sample at the transition.

It is interesting to note that the slope of the curve, which gives the volume thermal expansion coefficient α of the sample according to a= (\ld)(àdléT), is notably larger above the LCOT temperature than below. In the phase mixed state, the thermal expansion coefficient, denoted otmj x, was found to be 3.9 x 10"4 °C'\ whereas the corresponding parameter in the ordered state, denoted oc o r cj e r, is found to be 7.9 χ 1 0 " 4 o C I . According to the literature, the thermal expansion coefficient for polystyrene (15) and poly(n-butyl methacrylate) (16) at similar temperatures are 5.1 x 10" 4 oC~ l and 6.1 χ 10" 4 oC" 1, respectively. Therefore, the thermal expansion coefficients for the homopolymers are between a m j x and otor(jer- I n t n e ordered state, the thermal expansion coefficient is expected to be the weighted average of the coefficients of the two homopolymers i.e. 5.61 x 10 4 °C"1. The discrepancy observed with the measured value of ot o r c j e r arises from a confinement of the copolymer multilayer parallel to the surface of the substrate requiring that the copolymer expand only normal to the film surface under the constraint of having the junction points of the copolymer confined to the microdomain interface. This was discussed by Bassereau et al. previously (18). By contrast, a m | x is ~ 35 % smaller than the weighted average of the thermal expansion coefficients of the two homopolymers, implying that there is a free volume deficit (and hence entropy) in the phase mixed system with respect to the ordered state. Since this deficit increases with temperature, this will eventually drive the system into the ordered state upon heating. It is precisely this mechanism that distinguishes the LCOT from the ODT, which is known to be enthalpically driven.

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2470

2410 125 135 145 155

Τ (°C)

Figure 4. The thickness of a P(d-S-b-nBMA) film determined from x-ray reflectivity as a function of temperature. The bulk LCOT is indicated. The transition in the thin film occurs at ~ 148 °C, lower than the bulk LCOT due to confinement effects, and is characterized by a 7 λ increase in the film thickness. The slopes of the lines were used to determined the thermal expansion coefficient of the film in the disordered and ordered state.

In summary, we have measured the latent heat associated with the LCOT for a Mw=85K P(d-S-&-nBMA) diblock copolymer using standard DSC methods. Coupled with the pressure coefficient obtained previously from SANS measurements, a volume change of - 0.1 % would be expected according to the Clayperon equation. X-ray reflectivity measurements, while sensitive enough to measure this volume change, yielded a much larger value of 0.3 %. It is not clear whether this large volume change reflects an additional contribution coming from confinement or surface effects that had not been taken into account in our application of the Clayperon equation for a bulk sample. Dilatometry studies are currently in progress to address this issue.

The authors wish to acknowledge the support of the US Department of Energy, Office of Basic Energy Science under contract DE-FG02-96ER45612 and the National Science Foundation Materials Research Science and Engineering Center on Polymers at the University of Massachusetts (DMR-9809365).

Acknowledgments

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References

1. Bates, F. Science 1991, 251, 898. 2. Hashimoto, T. in Thermoplastic Elastomers, Legge, N.; Holden, G., Schroeder,

H.; Eds. Hanser Publishers, 1987. 3. Hajduk, D.; Gruner, S.; Erramilli, S.; Register, R.; Fetters, L. Macromolecules

1996, 29, 1473. 4. Kasten, H.; Stühn, B. Macromolecules 1995, 28, 4777. 5. Kim, J. K.; Lee, H. H.; Gu, Q.; Chang, T.; Jeong, Y. Macromolecules 1998, 31,

4045. 6. Steinhoff, B.; Rüllmann, M. ; Wenzel, M. ; Junker, M. ; Alig, I.; Oser, R.; Stühn,

B.; Meier, G.; Diat, O.; Bösecke, P.; Stanley, H. Macromolecules 1998, 31, 36. 7. Schwahn, D.; Frielinghaus, H.; Mortensen, K.; Almdal, K. Phys. Rev. Lett. 1996,

77, 3153 8 Frielinghaus, H.; Mortensen, K.; Almdal, K. Physica Β 1997, 241-243, 1029. 9. Pollard, M ; Russell, T. P.; Ruzette, Α. V.; Mayes, A. M., Gallot, Y .

Macromolecules 1998, 31, 6493. 10. Russell, T. P.; Karis, T. E.; Gallot, Y.; Mayes, A. M . Nature 1994, 368, 729. 11. Bates, F.; Hartney, M . Macromolecules 1985, 18, 2478. 12. Russell, T. P.; Mater. Sci. Rep. 1990, 5, 171. 13. Atkins, P. Physical Chemistry, 4th ed. 141. 14. Menelle, Α.; Russell, T. P.; Anastasiadis, S. H.; Satija, S. K.; Majkrzak, C. F.

Phys. Rev. Lett. 1992, 68, 67 15. Mansky, P.; Tsui, O.K.C.; Russell, T.P. Phys. Rev. Lett., submitted. 16. Polymer Handbook 3rd Ed. Brandrup, J.; Immergut, Ε. H.; Eds. Wiley (New

York) 1989. 17. Rogers, S.; Mandelkern, L. J. Chem. Phys. 1957, 61, 985. 18. Bassereau, P.; Russell, T.P. Israel J. Chem. 1995, 35, 13.

Chapter 18

Analysis of the Structure, Interaction, and Viscosity of Pluronic Micelles in Aqueous Solutions

by Combined Neutron and Light Scatterings Yingchun Liu1 and S-H Chen2

Departments of 1Materials Science and Engineering and 2NucIear Engineering, 24-209, Massachusetts Institute of Technology, Cambridge, MA 02139

We discuss results of analyses of an extensive set of neutron (SANS) and light scattering (LS) intensity distributions from a class of tri-block copolymer micelles in aqueous solutions. We investigated three Pluronics, L44 (ΡΕΟ10 P P O 2 3 P E O 1 0 ) , P84 ( P E O 1 9 P P O 4 3 P E O 1 9 ) and P104 ( P E O 2 7 P P 0 6 1 P E O 2 7 ) in their entire range of disordered micellar phase in temperature and concentration. At room temperature, both polyethylene oxide (PEO) and polypropylene oxide (PPO) are hydrophilic, but at elevated temperatures PPO becomes significantly less hydrophilic than PEO, thus creating a thermodynamic driving force for micellization. It has been known that the resultant micelles are spherical, each consisting of a hydrophobic core and a hydrophilic corona region having a significant hydration. We proposed a "cap-and-gown" model for the microstructure of the micelle, incorporating the polymer segmental distribution and water penetration profile in the core and corona regions. We took into account the inter-micellar correlations using an adhesive hard sphere model. With this combined model, we were able to fit satisfactorily all SANS intensities for the entire micellar range in an absolute scale. We obtained consistent trends for important parameters such as the aggregation number, hydration number per polymer in both the core and corona regions, and surface stickiness. The structure and interaction of micelle stay essentially the same as a function of concentration, but the aggregation number and level of hydration change with temperature. The aggregation number and surface stickiness increase, and the micelle becomes drier with increasing temperature. Micellar core is not completely dry but contains up to 20% (volume fraction) of solvent molecules at lower temperatures. We also proposed a new method for data treatment of LS intensities at all concentrations which is complementary to SANS technique.

270 © 2000 American Chemical Society

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This analysis allows us to extract the average aggregation number of polymer chains in a micelle, the average hydrated volume fraction of a micelle or the volume fraction of polymers in a micelle. These parameters obtained from SANS and LS agree to within experimental error bars. Finally, from the analytical structure factor obtained from the scattering experiments, we calculated the zero-shear viscosity of the micellar solution using the theory developed recently by Verberg, deSchepper and Cohen. We obtained excellent agreement between the theoretically calculated and the measured viscosities as functions of temperature and volume fractions up to 40%.

Introduction

Polyethylene oxide and polypropylene oxide containing block copolymers represent a class of polymers that associate spontaneously in aqueous solution. The self-association is characterized by sensitivity to the temperature (/). In particular, many tri-block copolymers composed of two symmetric end-block polyethylene oxides and a middle-block polypropylene oxide have been synthesized and used industrially as polymer surfactants. Tri-block copolymer surfactants we studied are commercially available under the trade name Pluronic from BASF (2). The total molecular weight and the composition, EO-to-PO molecular weight ratio, of the block copolymer can be adjusted during the synthesis. The copolymer surfactants are water soluble to large weight percentages and in a broad temperature range. Pluronic polymer surfactants find widespread industrial applications as detergents, wetting, foaming/defoaming, emulsification, lubrication and solubilization agents. The are also used in cosmetics, bioprocessing and pharmaceutical products (2,3,4).

The most interesting feature of the Pluronic polymer is its ability to self-associate in an aqueous environment and the resultant rich phase behavior of the solution (3, 5, 6, 7, 8, 9, 10, 11, 12). At low polymer concentrations and low temperatures, the tri-block copolymers in water exist as single-coils, called unimers. At moderately high concentrations or temperatures, the copolymer molecules self-associate to form thermodynamically stable aggregates - micelles. The micelles exist in a disordered phase within substantial temperature and concentration ranges. At even higher concentrations and temperatures, the copolymer chain and micelles can form ordered phases, such as cubic, hexagonal, lamellar phases (5, / / , 12). The self-assembly and phase behavior of the co-polymer solution depends on the total molecular weight and composition of the copolymer.

Micellization occurs above a certain critical micellization concentration and temperature (called c.m.c and c.m.t.). The c.m.c. and c.m.t. curves of the pluronic surfactants have been studied extensively using various techniques (6, 7, 13, 14). Theoretical developments also allow systematic prediction of the unimer-to-micelle transition for a copolymer surfactant with given composition and molecular weight (15, 16, 17, 18). Micellization have no sharp boundary but there is a rather broad coexistence region of large aggregates and single chained polymers. Dynamic light scattering shows that Pluronic micellar solutions exhibit significant polydispersity at low temperatures but become monodisperse at high temperatures (6). Batch-to-batch

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variations of the surfactant supplies with composition heterogeneities, such as di­block copolymer impurities, may considerably affect micelle formation and surface tension, but have little effect on the micellar structure and inter-micellar interaction for concentrations beyond the c.m.c.-c.m.t. line. We therefore limit our study of the microstructure and interaction of micelles far beyond the micellization boundary where we can ignore the effects of impurities and heterogeneities.

In this paper, we focus our attention on a particular Pluronic surfactant family, i.e. copolymers containing 40 to 60 molecular weight ratio of polyethylene oxides and polypropylene oxide. This Pluronic family has several members, namely, Ρ104, P94, P84, L64 and L44, with decreasing molecular weights. Ρ104 has the highest and L44 has the lowest molecular weights. Out of these five, we selected Ρ104 and P84 in particular for a more detailed study. To clarify the effects of copolymer molecular weight, we also made some studies of the system made of L44 for comparison. Molecular weights and molecular volumes of L44, P84 and Ρ104 are given in Table I.

In order to determine the range of the disordered micelle phase, it is important to measure its two boundaries, i.e., the micellization at low concentration and gelation at high concentration. The experimental c.m.c. - c.m.t. boundaries of the Pluronic Ρ104, P84 and L44 in aqueous solutions are determined using light scattering method (19). Micelles are formed as temperature or concentration increases beyond the c.m.c.-c.m.t. curve. This causes an abrupt increase of the scattering signals. The effect of molecular weight of the block copolymer on micellization is significant. For surfactants with the same EO-PO composition, higher molecular weight polymers tend to form micelles at lower concentration and/or temperature. Gelation of Pluronic solutions occurs at high polymer concentrations. For the 40% EO Pluronic family, concentration of the sol-gel transition is only slightly dependent on temperature. The micellization and sol-gel transition is correlated: higher c.m.c. tends to correspond to higher gelation concentration. However, the sol-gel boundary is nearly vertical on the phase diagram. For stable micellar solutions at temperature higher than 30° C, the variation of gelation concentration is normally at most 1 or 2 wt% within a large temperature range. The gelation concentrations for Pluronic Ρ104, P84 and L44 are 20, 22 and 24 wt%, respectively. These gelation concentrations set the upper limit for the concentration ranges of the disordered micellar phases. The disordered micellar phase has an upper temperature boundary called cloud points. The cloud points of micellar solutions ofP84 and P104 are 75 and 85 °C respectively, showing little concentration dependence.

The aim of this paper is to summarize the results obtained on the microstructure and positional correlations of the polymeric micelles at concentrations far beyond the micellization boundary. Polymer concentration at the upper boundary of the disordered micellar phase is about 20 wt%. When the hydration effect of the copolymer is taken into account, the volume fraction of the micelles can be as high as 40 to 50%. The traditional methods in surface science, such as surface tension measurements, are useful in detecting the unimer-to-micelle transition boundary, but are very limited in higher concentration dispersions. On the other hand, the scattering techniques have been proven to be powerful even at higher concentrations (4, 20, 21). We have developed practical methods for analyzing absolute intensity data from both

Tab

le I.

Mol

ecul

ar v

olum

es, s

catte

ring

leng

ths

and

scat

teri

ng le

ngth

den

sitie

s of

pol

ymer

and

solv

ent

Che

mic

al fo

rmul

a M

olec

ular

w

eigh

t M

olec

ular

vol

ume

(A3)

Sc

att.

leng

ths

2>

f (fm

) Sc

att.

leng

th d

ensi

ty

(10"

6Â

2)

EO

-(C

H2) 2

0-44

72

.4

4.14

0.

572

EO

-(C

H2) 3

0-58

95

.4

3.31

0.

347

PO so

lven

t D

20

20

30.3

19

.153

6.

321

P104

P8

4 L4

4

PE0 27

PP0 6l

PE0 27

PEO

l9PP

0 43PE

Ol9

PEO

lQPP

O23PE

Ol0

5900

42

00

2200

9706

69

20

3619

424.

1 30

2.1

158.

1

0.43

7 0.

437

0.43

7

274

SANS and LS experiments. In general, SANS intensity distribution function can be decomposed into an intra-particle structure factor and an inter-particle structure factor. An explicit polymer segment distribution profile, called a Cap-and-Gown model, has been proposed for the calculation of the intra-particle structure factor and an analytical expression for the inter-particle structure factor based on an adhesive hard sphere model is given.

Mortensen (22) has previously given an alternative model for analysis of SANS data for a micellar system formed from P85. This model is based on a core-shell structure of a micelle and a hard sphere inter-micellar interaction. In our experience, if one takes a simple core-shell model of the micelle for the particle structure factor, one needs to introduce a considerable degree of polydispersity of size to fit the large k (k is magnitude of the scattering vector) portion of the scattering intensity distribution. The advantage of our cap-and-gown model of the micellar structure to be described later is that we do not need to postulate a polydispersity to account for the large k intensity distribution. Furthermore, we use a hard sphere with a surface adhesion to model the intermicellar interaction. This is shown to be a necessary ingredient to account for a substantial increase of zero shear viscosity as a function of temperature (23).

We also proposed a novel light scattering intensity analysis method, which is, valid up to about 50% hydrated micellar volume fraction. The new light scattering intensity analysis method significantly extends the concentration range of the traditional Zimm Plot (24). Its simplicity makes it useful also as a fingerprint to detect phase transitions.

Experiment

Small-Angle Neutron Scattering Measurements. SANS experiments were performed at NIST using the 30-m SANS spectrometer (NG7). We used a neutron beam of an average wave length λ of 7 Â and sample to detector distances (SDD) of 2m/15m, to cover a k (magnitude of the wave vector transfer) range of 0.001-0.230 À"1; and a λ of 5 À and SDD of 1.3m/10m to cover another k range of 0.002 to 0.560 À" 1, both with Δλ/λ = 11%. 1 mm path length flat quartz cells were used. Pluronic polymer samples were obtained from BASF. The same batch of polymer was used to carry out all experiments. Polymer solutions were tested by dynamic light scattering for cmc-cmt line before SANS measurements were made. SANS experiments used deuterated water as the solvent to enhance the contrast between the micelle, which are made up largely of protonated polymers and some hydrated solvent molecules, and the solvent. The magnitude of scattering length densities of the polymer segments and the solvent are listed in Table I.

Static Light Scattering Measurements. A Brookhaven Instrument photon correlator was used for the measurements of both static and dynamic light scattering of the Pluronic solutions. The light scattering apparatus uses a He-Ne laser, of wavelength 632.8 nm at a power level of 15 mW, mounted on a goniometer covering an angular range of 25° to 135°. A Pyrex glass sample cell was placed at the center of a brass thermostated block. Filtered toluene was used as an index matching fluid surrounding the cell. An absolute scattering cross section of the micellar solution was determined by using toluene of known Rayleigh scattering cross section per unit volume

275

(Rtoluene=l-4°6 cm"1). The refractive index increments dn/dc for Pluronic polymer in water were measured using a differential refractometer at temperatures ranging from 25 to 60 °C (19).

Theoretical Background

Cap-and-gown model of the micellar structure. We previously proposed a schematic structural model (25), which specifies the polymer segmental distribution and the solvent penetration profile in the copolymer micelle. This model combines two characteristic features of the copolymer micelle: a diffuse distribution of hydrophilic PEO chains of the copolymers in the outer region of the micelle and a compact hydrophobic core consisting largely of the PPO segments. Thus statistically speaking, we assume a uniform compact core but a Gaussian distribution of the polymer segments in the corona region. There is no abrupt boundary between the polymer chains in the corona region and the solvent. Figure 1 illustrates the cap-and-gown model of micellar structure. The micellar core is described as the cap and the diffuse outer layer as the gown. The relative sizes of the cap and the gown are related by the PEO to PPO volume ratio. The figure shows the radial distribution of polymer segment volume fraction Φρ(^ in a micelle.

The radial segmental distribution in the core region starts out uniform within a radius a. It is joined to a Gaussian distribution with a width σ. Denoting by 0 w ( r ) the volume fraction occupied by solvent molecules at a radial distance r, the volume fraction occupied by the polymer segments Ψρ^Γ) is then 1 - The cap-and-gown model assumes that the polymer volume fraction in a micelle is given as

Pp(r)'- r2

exp( s-) for a < r < ©ο, a1

for 0 < r < a,

(1)

Neutron scattering length density profile can be calculated in terms of the known scattering length densities of polymer, pp, and water, pw, as

/ K r ) = ^ ( ' ' ) p w ^Φρ^Ρρ- The difference between the scattering length density of the polymer and that of solvent (the contrast) determines the particle structure factor,

Ap(r) = p ( r ) - p w = [pp - p w ] ^ ( r ) . (2)

The core radius a, the Gaussian width σ and the polymer volume fraction in the core φ ρ are related by a geometrical constraint. We assume that all PPO blocks reside in the core and PEO chains are distributed outside the core as a Gaussian function. The volume of polymer segments in the core, V™ 9 is related to the aggregation number Ν and the volume of PPO segment in a polymer chain by, Vp = (4π/3)α3φρ = NV

PP0. Similarly, the volume of polymer segments outside the core is

276

Figure 1. Illustration of a cap-and-gown model for the radial distribution of polymer segment volume fraction in a micelle. The micellar core is described as the cap and the diffuse outer layer as the gown. The relative sizes of the cap (radius a) and the gown (Gaussian width σ) are related by the PEO to PPO volume ratio. The water penetration profile is complementary to the polymer segment distrubution.

277

V°ut = 4π^ exp^-x2 / σ2^2άχ = NvpEQ. From these two equations, we obtain the

following non-linear equation,

^ + £ ^ 0 - f ^ (3) ' 2 T 3 PVPPO

where t = al σ and erfc(x) is the complementary error function. The right hand side of the above equation contains the known block polymer composition ratio and thus gives a relation between t and Ψρ. As an example, to the first approximation, one may set φ ρ = 1 , assuming no solvent penetration in the core. For Pluronic with 40 % PEO content, the right hand side of the above equation equals 0.444. Solving the non-linear equation, we have t= 1.03. Thus for Pluronic Ρ104, P84 and L44 the core radius-to-Gaussian width ratio is approximately unity.

The cap-and-gown model of the polymer segmental distribution proposed here is qualitatively consistent with theoretical predictions based on a lattice model (17,18). It captures the most important features of the overall segmental distribution and is simple enough to allow an analytical evaluation of the particle structure factor. A merit of the cap-and-gown model is that it describes the complex distribution of the copolymer segments without introducing many parameters. For a given copolymer composition, the core radius a and the Gaussian width σ are known once the aggregation number Ν is known, assuming no water penetration in the core. If the water volume fraction inside the core increases from zero, the core radius and a/σ ratio will increase accordingly. Thus higher water penetration in the core has the similar effects as lower hydrophile-hydrophobe ratio, V P E O / V P P O , of the block copolymer. This makes possible a way to determine the water penetration inside the core by SANS data analysis.

It is straightforward to calculate the particle structure factor from a Fourier transform of the radial distribution of neutron scattering length density, Eq. 2. It is

F(k) = NÇ£b.-p ν )F(k)

F(k) = - PPO ν + ν

PEO PPO

m*") 3 7sin(toc) , 2 2 w — i + — ι — e x p ( - r * )xdx ka ka

(4)

where Ϊχ(χ) is the first order spherical Bessel function. The normalized particle

I l2 F(k)\ .

The normalized form factor of the cap-and-gown model may be considered equivalent to that of a system of polydisperse spheres where boundaries between the spheres and the solvent are smeared out by a size distribution. Determination of the details of the microstructure depends on the micellar compositions and is ultimately limited by the experimental k resolution. The details of the micellar structure cannot be determined by SANS to a length scale smaller than the neutron wavelength

278

without ambiguity. It is likely that the boundary of the micellar core is also dimjse rather than sharp, and the distribution of polymer segments outside the core may deviate from the Gaussian form. But as far as SANS data analysis is concerned, a spatial deviation of the scattering length density profile within a few À is hard to distinguish. The adequacy of the microstructure model can be judged only by the quality of the fits to SANS data.

Inter-micellar interaction and the structure factor. The most important micelle-micelle interaction comes from the excluded volume effect. Volume fraction of the micelles φ characterizes the strength of the interaction. To a first approximation, micelles can be regarded as hard spheres. Micelles are, however, swollen by solvent molecules, mostly in the corona region. Polymer chains and the hydrated water molecules both contribute to the excluded volume. Intermicellar distance is comparable with micellar diameter at high volume fractions. The real volume of a micelle is significantly larger than the volume of the dry polymer chains alone. It is the real volume of the micelle rather than the polymer volume that determines thermodynamical quantities such as the osmotic compressibility and light scattering intensity. Obtaining the real volume of a micelle in a self-consistent way is a non-trivial but important task. It requires a knowledge of the detailed microstructure such as the hydration number per polymer chain. This information on the associated solvent molecules has not been available from theoretical predictions and is hard to extract accurately from scattering intensities at dilute concentrations.

For a more accurate description, polymeric micelles cannot be regarded simply as smooth-surfaced hard spheres because of possible additional interaction at their surface. The origin of the surface adhesion is probably the inter-penetration of polymer chains and the consequent depletion of solvent molecules. When two micelles come into contact, there is an overlapping region where polymer chains can inter-penetrate, squeezing out some solvent molecules. The inter-penetration gives rise to an effective attraction at the micellar surface in addition to the excluded volume repulsion.

Structure factor of the sticky hard sphere model. The structure factor of a system of interacting spheres is determined by the inter-particle interaction potential u(r). We consider a system of hard spheres with adhesive surfaces. The pair-wise inter-particle interaction potential is written as:

oo for 0 < r < R' Ω forR»<r<R (5) 0 for R < r

Φ(τ) k Τ

where R and R' are respectively the outer and inner diameters of the sphere and (R -R*)/2 the thickness of the adhesive surface layer. Baxter (26) gave an analytical solution of the Ornstein-Zernike (OZ) (27) equation in Percus-Yevic (PY) approximation (28) in the limit that the thickness of the adhesive layer goes to zero but the adhesive potential Ω tends to infinity in such a way that one can introduce a finite stickiness parameter l/τ defined as l / t^Oexp^XR-R'yR (29). The second

279

virial coefficient of the system in this limit has an additional negative term, besides the well-known hard sphere term, proportional to the stickiness 1/τ,

v " * H > ( 6 )

The solution is expressed in terms of following set of parameters for a given volume fraction φ (defined in terms of the inner diameter R' of the hard sphere) and the stickiness l/t,

φ(1 + φ/2) φ 6 ( Δ - Λ / Δ 2 - Γ ) _

3 ( ΐ - ^ ι - ^ * ^ ι -< (7)

α _(1 + 2 0 - μ ) 2 30(2 + 0 ) 2 - 2 μ ( 1 + 70 + 02) + μ 2 (2 + 0) (1-0)4 ' Ρ 2(1-0)4

We use a dimensionless parameter Q&kR, the product of the magnitude of the wavevector transfer k and the outer particle diameter R. The inter-particle structure factor S(Q) of the sticky hard sphere system is then given analytically as (25):

1 -1 = 2401 S(Q)

af2(Q) + ft3(Q) + j<l>(xf5(Q) + 20 2 λ 2 / , (β)-20λ/ · . (β) (9)

where various functions are defined as: fo(x) =sin(x)/x; fi(x) = (l-cos(x))/x2; f2(x) = (sin(x) -xcos(x))/x3; f3(x) = (2xsin(x)-(x2-2)cos(x) -2)/x4; f5(x) =(4x3-24x)sin(x) -(x4-12x2+24)cos(x) +24)/ x 6 . The hard sphere limit is obtained by setting 1 / τ -» 0, λ -> 0, μ -» 0. The structure factor given by Eq. 9 is quite adequate for calculating SANS and LS intensities.

Structure factor for an adhesive hard sphere model. Baxter also indicated the possibility of using the PY approximation to solve the OZ equation to the first order in the fractional surface layer thickness ε = (R-R')/R. We gave the closed form structure factor in this case previously (23).

1 s(Q)

ρ2λ2ε2

-1 = 24φ\

/ , (εβ)- -Λ (02) 2 J 3 V •2φ2λ2[φ)-ε2^(εΰ)

- 2 4 ψ 2 ( β ) - ( 1 - ε ) 3 / 2 ( ( 1 - ε ) β ) ] (10)

280

It can be shown that Eq. 10 reduces to Eq. 9 in the limit ε -» 0. The structure factor at finite ε is essential for calculating zero shear viscosity of micellar solutions as we shall discuss next.

The relative viscosity of a colloidal solution (ratio of the solution viscosity to the solvent viscosity) at zero frequency limit ω 0 due to the inter-particle interactions is given by a numerical integral in Q space (Q = kcr) of the following expression,

where # = Α/12ε is the pair correlation function at contact and

A heuristic derivation of the above equation for the case of a hard sphere system was made by Verberg, de Schepper and Cohen (VDC) (30) recently based on a mode coupling theory. The mode coupling theory of dynamics of dense liquids takes into account the additional "cage effect" which is the single most important long-time relaxational mechanism known to be operational in dense fluids. The most important physical insight that guided VDC's work was the recognition of the fact that the cage effect contribution to the shear viscosity dominates all other contributions, including the hydrodynamic interaction which is the most important one at low volume fractions, at higher volume fractions.

Eq. 11 clearly indicates that shear viscosity at high volume fractions depends sensitively on the pair correlation function at contact and on the analytical form of the structure factor at high Q. In evaluating RHS of Eq. 11, one needs to perform a numerical integration to the order of Q = 200 while in analyzing SANS data, one needs to consider Q range of only up to 20. It can be shown (19) that the integrand in the RHS of Eq. 23 oscillates rapidly at high Q, the period of it depends on the value of ε. Thus the precise value of the integral depends considerably on certain non-zero value οίε . One then needs to use Eq. 10 rather than Eq. 9 for the S(Q) and S'(Q) in the integrand of Eq. 11.

A new method for static light scattering intensity analysis The classical procedure of polymer characterization using static light scattering involves extrapolation of the scattered light intensity (excess Rayleigh scattering) to zero concentration limit. In the dilute limit, product of the inverse light scattering cross-section per unit volume (Rayleigh ratio) and concentration is a linear function of the concentration. The two coefficients in the linear equation are the weight average molecular weight and a quantity A2 proportional to the second viral coefficient

B2 = NAm \ ° f t n e solution, where Ν Α is the Avogadro's number and m the mass of the particle. The well-known Zimm Plot, when applied to a micellar solution, uses the following formula (24%

[S'(Q)f s(Q)

(Π)

d(Q) = l-J0(Q) + 2J2(Q).

R„ MP(k) 2 (12)

where C= c-c.m.c. and c and c.m.c. are the polymer concentration and critical micellization concentration in unit of grams of solute per ml, M the weight average

281

molecular weight of the micelle in (g / Mol), A 2 in unit of (ml-Mol / g2), & θ the excess Rayleigh ratio in unit of cm"1 and k = (4 π /XQ) no sin(0/2), the magnitude of the scattering wave vector in the solvent. The optical constant Κ is defined as, K=4 π 2 ηο2^ηΜο)2/ΝΑλο4, where no is the refractive index of water and λο the laser wavelength in air. The essential quantity is C / / ^ . The plot of C/ RQ vs. C deviates from linearity significantly at higher concentrations. However this approach is somewhat ambiguous for treatment of data from polymer micelles. Firstly, there are few experimental data points in the low concentration region in practice. A slight variation in the choice of concentration range can result in a large difference in the slope and intercept of the Zimm Plot. Secondly, a slight uncertainty in the background subtraction can result in a large error in the inverse Rayleigh ratio due to the relatively small signal-to-noise ratio. This problem is particularly severe for polymeric micellar solutions because of the inaccuracy in determination of the background scattering from unknown impurities and heterogeneity's and from the unimers. For these reasons, the traditional zero-concentration extrapolation method is not appropriate for polymer micellar solutions. A better treatment needs to take advantage of the data at higher concentrations where scattering signals are stronger and less affected by the background. Proper consideration for the inter-particle correlation in terms of the structure factor is essential for the theoretical formulation of static light scattering intensity applicable for a broad concentration range.

The Rayleigh ratio can be written in terms of the normalized particle structure factor P(k) and the inter-particle structure factor S(k) as R

e = KCMP(k)S(k). \ n the case of polymeric micelles having diameters in the range of 10-20 nm, the zero k limit can be taken safely (i.e. P(Jt)=l). The Rayleigh ratio can then be written as, Re = KCMS(0), Thus for high concentrations,

The traditional method of viral expansion can be considered as an expansion of 1/S(0) to the first order in concentration. The structure factor at zero wave vector transfer S(0) is a thermodynamic quantity that can be obtained from the equation of state, and is proportional to the osmotic compressibility of the solution.

The new plot We proposed a new plot (31% which treats the light scattering intensity from zero to high concentrations on equal footing. Analogous to the well-known Zimm plot, the central quantity is KC/Re- It is based on an observation that a plot of l n ( K C / R e ) vs. C yields approximately a straight line in the entire concentration range, up to volume fractions of about 50 %. We rewrite the previous equation as,

KC J 1_ M 5(0)

(13)

(14)

282

Analogous to the traditional plots, the zero intercept gives the weight average molecular weight of the micelle. Logarithm of 1/S(0) has nearly linear relationships with the volume fraction for both hard sphere and sticky hard sphere systems as illustrated in Figure 2. The difference between 1/S(0) and βχρ(8φ) is less than 4%. On the logarithmic scale, the differences between ln(l/S(0)) and 8φ are at most 2% for the volume fraction in a range from 0 to 0.45. This level of difference is usually within experimental error bars. To the first approximation, the new plot can be adopted as a simple way of obtaining volume fraction from the slope. The linear shape of the new plot holds also for a sticky hard sphere system. In the sticky hard sphere model, the slope is 8-2/τ. As stickiness increases, the slope gets smaller. This plot is useful if the association and interactions are independent of polymer concentration. The self-assembling polymeric micelles is exactly such a case.

Results

SANS scattering intensity I(k) (in a unit of cm"*) is proportional to the product of the normalized particle structure factor P(k) and the interparticle structure factor S(k), both dimensionless. The constant factor in front of it is product of the polymer concentration C = c-cmc (c = grams of polymer molecules per ml), the contrast factor square, and the aggregation number of a micelle Ν (32). In the contrast factor within the square bracket, the first term is the sum of coherent scattering lengths of atoms comprised of a polymer molecule, p w the scattering length density of the solvent and v m the polymer molecular volume. The complete formula for scattering intensity from a system of monodisperse micelles is

C N A I(k) = à. M

Σ ο . - ρ ν NP(k)S(k) (15)

where N A is the Avogadro number and M p the molecular weight of the polymer. It is important to note that the absolute intensity is directly proportional to the aggregation number N . The polymer concentration and the contrast factor are known quantities.

In this model, the four primary fitting parameters are: the aggregation number of a micelle N , the outer micellar diameter R (or alternatively, the volume fraction of micelles φ), the stickiness parameter l/τ, and the polymer volume fraction in the micellar core φρ. Other parameters, such as the total hydration number Η (no. of water molecules attached to a polymer in a micelle), the volume fraction of micelles φ (or alternatively, the outer diameter of the micelle R), core radius a, Gaussian tail width σ, and the hydration numbers in the core H c , are uniquely related to the four primary parameters. The more sensitive parameters of the fit are the aggregation number N , the volume fraction φ and the micellar diameter R. The aggregation number determines the overall amplitude of the scattering intensity and can be obtained by fitting SANS data in an absolute intensity scale. The volume fraction of micelles controls the peak height of the structure factor S(k). The micellar diameter determines the peak position of S(k). These three parameters basically decide the

283

Figure 2. A demonstration that Log[l/S(0)] vs φ plot exhibits an almost linear relationship for both hard sphere and sticky hard sphere systems.

284

general shape and amplitude of I(k). Two less sensitive parameters determine the detailed rise and fall of the scattering peak and small and large k behaviors of the scattering intensity. These parameters are the stickiness and the polymer volume fraction in the core.

A Fortran code based on a gradient searching non-linear least square fitting method (33) was developed and used to fit SANS intensities in an absolute scale. Quality of the fits is uniformly excellent for Pluronic micellar systems in their entire range of the disordered micellar phases. As an illustration, the calculated particle structure factor P(k), the inter-particle structure factor S(k) and the absolute intensity distribution I(k) from a typical fit is shown in Figure 3 together with the measured one.

Parameters characterizing microstructure of the micelle and interaction between two micelles are thus obtained for Ρ104 and P84 micellar systems. To show the goodness of the fits, a set of concentration series taken for P84 at a temperature 40° C and their fits are shown in Figure 4. The parameters extracted from the fittings are shown in Table II.

Listings on the above table show that at a given temperature, ratio between the volume fraction of micelles to the volume fraction of dry surfactant molecules at different concentrations is a constant. The stickiness parameter characterizing the interaction is also a constant. Furthermore, the aggregation number and the structural parameters of the micelle are independent of concentration. This leads us to conclude that, in contrast to typical ionic micellar systems, the microstructure and interaction of non-ionic micelles formed by tri-block copolymers of PEO and PPO are determined only by temperature, independent of the polymer concentration. We can thus summarize the temperature dependence of a number of the more interesting parameters, averaged over a range of concentrations, for P84 and Ρ104 micellar systems in Table III and Table IV.

The hydration number decreases with temperature as the micelles become more compact at high temperatures. The hydrated water molecules take up a large fraction of volume in a micelle. For example, for Pluronic P84 micelle at an intermediate temperature, the hydration number for each polymer chain is 240. Knowing the specific volume of a water molecule (30 Â 3) and the molecular volume of P84 (6920 Â^), we obtain the volume ratio of hydrated water to polymer in a P84 micelle to be about 1.05. This means that more than 50 % of the micellar volume is occupied by the solvent molecules. The effect of hydration thus contributes significantly to the properties of the micellar solution. This number is hard to obtain from theoretical models for polymer segmental distribution. However, with a proper liquid theory, it can be readily extracted from SANS intensity profile of a concentrated micellar solution. SANS analysis also gives the stickiness parameter. The trend of the stickiness parameter indicates that at low temperatures, the micelles are close to hard spheres. The surface of the micelles becomes stickier as micelles grow at elevated temperatures. This effect is consistent with increased hydrophobicity of the polymers at elevated temperatures.

Water penetration profile is an interesting result. The micelle has regions of core and shell with distinctly different polymer and solvent volume fractions. The core of a Pluronic micelle is not completely dry, in agreement with theoretical prediction (17). The polymer volume fraction in the core is about 92 to 97 % at higher temperatures, independent of temperature and concentration. This suggests that only

285

80

0.00 0.05 0.10

Figure 3. A SANS intensity distribution and its model fit (solid line) as decomposed into the inter- and intra- particle structure factors S(k) and P(k) for a 16 % Pluronic P84 solution in D2O at 40 °C. The inter-particle structure factor is calculated by solving the 02 equation with an inter-micellar potential of an sticky hard sphere system. The intra-particle structure factor is calculated using the cap-and-gown model as the polymer segmental distribution in a micelle.

286

0.00 0.05 0.10 0.00 0.05 0.10 k Â"

Figure 4. A series of SANS data and their fits for Pluronic P84 micellar solutions at 40 °C The fits in absolute intensity scale use an sticky hard sphere model for the inter-particle structure factor and a cap-and-gown model for the form factor. The same stickiness and polymer segmental distribution are used to fit the entire series of SANS data in this graph simutaneously by varying only the polymer concentrations. This indicates that the microstructure of self-associated micelles is independent of the polymer concentration.

Tab

le I

I. P

aram

eter

s fo

r th

e m

icro

stru

ctur

e an

d in

tera

ctio

n of

P84

pol

ymer

ic m

icel

les

at 4

0° C

ext

ract

ed f

rom

SA

NS

inte

nsit

ies.

Con

e.

Agg

rega

tion

no.

Hyd

ratio

n Ν

D

iam

eter

C

ore

radi

us

Mic

elle

VF

(wt%

) Ν

Η

1/

τ R(

Â)

Κ

a (A

) Φ

6.1

52

201

0.28

13

0 0.

94

39

0.13

8.

8 53

22

0 0.

26

133

0.93

38

0.

19

12.5

54

21

2 0.

27

129

0.96

38

0.

26

14.3

55

21

6 0.

24

128

0.97

38

0.

30

16.0

51

20

6 0.

29

127

0.96

37

0.

33

17.9

54

21

1 0.

25

127

0.95

37

0.

35

20.2

52

19

8 0.

27

126

0.97

37

0.

40

Tab

le I

II.

Para

met

ers

of t

he m

icro

stru

ctur

e an

d in

tera

ctio

n of

P84

pol

ymer

ic m

icel

les

extr

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.

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0 38

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8 0.

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66

190

40

115

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13

0 43

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V.

Para

met

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of t

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ctio

n of

P10

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lym

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elle

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from

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Tem

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(°

C)

Agg

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ella

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ratio

n Η

D

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R

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a

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84

62

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6 49

.7

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0.

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106

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171

53.7

10

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0.29

55

12

0 59

7 18

5 56

.0

10.2

0.

50

289

at most 8 % of volume inside the core is occupied by the solvent. However, at low temperatures, the water penetration seems to depend on the concentration. For P84 at 35 0 C, polymer volume fraction in the core goes from 80% at concentration of 2.6 wt %, up to 96% at concentration of 18 wt%. It can be concluded that at high temperatures, a polymeric micelle formed by Pluronic tri-block copolymers becomes more compact. It has a larger association number but smaller hydration number.

We now discuss briefly the analysis of light scattering intensity using the new plot. The new plot extends the working range of static light scattering up to about 50 % of volume fraction, equivalent to 10 to 100 times the concentration range used in the classical treatment. We plot In (KC/R) vs C (polymer concentration in weight %). The plot will be very closely linear. Intercept of the line with the ordinate gives the logarithm of the inverse molecular weight of the micelle or the aggregation number times molecular weight of the monomer. The slope of the line contains information on the hydrated volume fraction in a micelle, a quantity difficult to obtain with conventional light scattering technique.

The slope of this plot yields directly the ratio of volume fraction of hydrated micelles to the polymer concentration. The slope is (8-2/τ)φ/0. Since according to values of the molecular volumes of D 2 0 , EO and PO groups listed in Table I, densities of both the polymers and the solvent are effectively 1.00 to within 1%. The slope can now be expressed as the volume ratio of the bound solvent molecules to the polymer in a micelle, i.e.,

slope γ I solvent

^polymer

(16)

When the excluded volume repulsion dominates, 21% « 8 (τ is of the order of 5), Eq. 16 simplifies to

slope = 8 1 + -solvent

\ polymer y φ

(17) polymer

where 0polymer is given by Vp 0 iy m e r /(Vp 0 i y m e r +V s o ivent)- The number of bound solvent molecules per polymer chain can then be calculated. This new plot offers a convenient way to evaluate the solvation and overall polymer volume fraction in a micelle from static light scattering intensity taken at 90 degree at series of polymer concentrations.

To see an example of the use of Eq.14 for light scattering intensity analysis, we show the new plots in Figure 5 for P84 micellar solutions at a series of temperatures ranging from 30eC to 55°C. The extracted parameters are listed in Table V.

It is straight forward to obtain a reasonably accurate effective weight average molecular weight of a micellar aggregate from the y-intercept in the new plot. Since the intercept is obtained from an extrapolation from high concentration micellar solutions where micelles are rather monodisperse, the aggregation number derived from it should agree with that obtained from SANS analysis. Inspection of listings in Table III and V verifies that it is indeed the case.

290

0.0 0.1 0.2 C (Pluronic P84) [wt%]

0.1 0.2 C [wt%]

Figure 5. The new plots of P84 micellar solutions at a series of temperatures ranging from 30° C to 550 C.

291

Table V. Characterization of Pluronic P84 micellar solutions using the new plot.

Temperature °C

Slope (8-2/TW/C

Molecular weight Μ HO*]

Aggregation no. Ν

30 17.9 0.138 33 35 17.6 0.182 43 40 17.4 0.222 53 45 17.0 0.272 65 50 16.8 0.291 69

55 16.4 0.327 78

At moderate temperature (30-45 0 C), the slopes are 20.2 for Pluronic Ρ104 (Mp=5900), 17.5 for Pluronic P84 (Mp=4200), and 12 for Pluronic L44 (Mp=2200). These values indicate that the volume ratio of the hydrated water molecules to the polymer chains are 2.5, 2.1 and 1.5 for P104, P84 and L44, respectively. This means that the overall polymer volume fractions in a micelle are about 40%, 50% and 70%, for Ρ104, P84 and L44 block copolymer surfactants. The trend of the ratios clearly indicates that the ability of carrying solvent molecules increases with the molecular weight of the polymer surfactant.

The low shear viscosity at a finite frequency ω is expressed as a sum of infinite frequency viscosity and the Brownian part (34).

η(φ, ω) = η(φ, <*>) + ηβ(φ, ω) (18)

The hydrodynamic interactions are essential in calculating the former. Direct pair-wise interparticle interactions have a small effect on the high frequency viscosity, but strongly affect the Brownian contribution. The hydrodynamic interaction dominates the low shear viscosity at low volume fractions, but the inter-particle interaction dominates at high volume fractions. The relative viscosity of the adhesive hard sphere colloidal solution at ω = 0 is likewise calculated according to a sum Άτ = Άι

where (23)

ηοο=1 + 2.5φ + (5.0 + -)φ2 + 6 Αφ3, (19)

and r \ I is given by Eq. 11. The relative viscosity and its two contributions as a function of φ are illustrated in Figure 6, for a system with the surface potential Ω = -1 and the fractional surface layer thickness ε = 0.01. For this case, the interaction part starts to dominate the relative viscosity when φ is about 0.28. At higher φ, the interaction contribution is much larger than the high frequency part, suggesting that precise form of the higher order virial corrections in Eq. 19 become less important.

292

Φ

Figure 6. Theoretically calculated relative viscosity T| r (higher solid line) decomposed into the high frequency part T\oa (lower solid line) and the interaction part (dash line) as a function of φ for an adhesive hard sphere system with the fractional surface layer thickness ε = 0.01 and surface potential Ω = —1.

293

Figure 7. Comparison of theoretically calculated (solid line) and experimentally measured (symbols) viscosities as a function of micellar volume fraction for Ρ104 micellar solutions at a range of temperatures.

294

The low shear viscosity of an adhesive hard sphere system is determined by three parameters: the volume fraction 0, the surface adhesive potential Ω and the dimensionless surface layer thickness ε. Larger surface adhesion potential increases the relative viscosity significantly. The high frequency contributions are less affected by it, but the interaction contribution Vj for an adhesive hard sphere system starts to dominate at a lower φ than in the case of a hard sphere system. The fractional surface layer thickness ε also plays a role in the magnitude of *77. The effect of layer thickness is significant when the surface potential is very attractive. Figure 7 gives an example of the comparison of the theoretical calculation described above and experimentally measured viscosities of Ρ104 micellar solutions in a range of temperatures spanning the disordered micellar phase.

Conclusion

We have shown that the cap-and-gown model for polymer segmental distribution inside a micelle coupled with the sticky hard sphere inter-micellar interaction gives a satisfactory description of the structure and thermodynamics of micellar systems made of Pluronic polymers in aqueous solution. One can also achieve a moderately good fit to the scattering intensity distribution by using an uniform core-shell model for the particle structure factor but in this case one needs to introduce a substantial degree of polydispersity of sizes. The cap-and-gown model, besides being more realistic by taking into account the polymer segmental distribution in the micelle, has a diffuse scattering length density distribution in the corona region so that one can achieve a satisfactory fit to the intensity at large k without having to introduce a polydispersity. This latter feature has an added advantage that one can now construct a reasonable model for the inter-particle structure factor without having to take into account distribution of sphere sizes. We find that as temperature increases, the pure hard sphere interaction is no longer suffice to account for the inter-micellar interaction. Certain amount of attraction between micelles is needed. This point is further confirmed by its necessity to account for the increase of zero shear viscosity as temperature rises. In fact, in order to account for quantitatively the increase of the viscosity as a function of temperature at high volume fractions, a finite fractional range of surface adhesion layer, ε, must be introduced. In this sense, temperature dependence of the viscosity is useful for providing further information on the interparticle interaction. It is well known that the temperature dependence of shear viscosities of gases (rare gases and molecular gases) have already been extensively used as a source for refining their inter-atomic (or molecular) potential functions (55). The ability to determine the correct level of hydration plays a vital role in giving the right volume fraction of the micellar system without which the excluded volume effect can never be properly accounted for. We have shown that our new method for static light scattering intensity analysis enables us to extend the traditional Zimm plot to much higher polymer concentrations and thus allowing us to extract the hydrated volume fraction of a micellar system directly.

295

Acknowledgement This research is supported by a grant to SHC from Materials Chemistry Division

of US DOE. SANS measurements were made with NG7 SANS Station at NIST. The beam time was obtained through PRT arrangement of Exxon Research and Engineering Co., Annandale, NJ with NIST Center for Neutron Research. YCL is grateful for financial assistance from Exxon Research and Engineering Co. In particular, technical support of Jim Sung in carrying out light scattering and viscosity measurements and Dr. John Huang for technical assistance in using the SANS instrument is deeply appreciated.

Literature Cited 1. Lindman, B.; Carlsson, Α., Karlstrom, G.; Malmsten, M . Science 1990, 32,

183. 2. Pluronic and Technical Brochure For Tetronic Surfactants.; BASF Corp.,

Parsipanny, NJ, 1989. 3. Alexandridis, P.; Hatton, T.A. Colloids & Surf. 1995, 96, 1. 4. Chu, B. Langmuir , 1995, 11, 414. 5. Wanka, G.; Hoffmann, H .; Ulbricht W. Macromolecules, 1994, 27, 4145. 6. Zhou, Z.; Chu, Β. J. Colloid & Interfacial Sci.. 1988, 126, 171. 7. Brown, W.; Schillen, K.; Almgren, M ; Hvidt,.S.; Bahadur, P. J. Phys. Chem.

1991, 95, 1850. 8. Schillen, K. ; Brown W.; Johnsen, R. M . Macromolecules, 1994, 27, 4825. 9. Mortensen, K. Prog. in Colloid & Polymer Sci. 1993, 91, 69. 10. Almgren, M . ; Alsins, J.; Bahadur, P. Langmuir, 1991, 7, 446. 11. Almgren, M. ; Brown, W.; Hvidt, S. Colloid Polym.Sci.. 1995, 273, 2. 12. Mortensen, K.; Brown W.; Jorgensen, E. Macromolecules 1994 27, 5654. 13. Alexandridis, P.; Athanassiou, V.; Fukuda, S.; Hatton, T. A. Langmuir

1994, 10, 2604. 14. Grieser, F.; Drummond, C. J. J. Phys. Chem.. 1988, 92, 5580. 15. Karlstrom, J. Phys. Chem. 1985, 89, 4962. 16. Linse, P.; Malmsten, M . Macromolecules 1992, 25, 5434. 17. Linse, P. Macromolecules 1994, 27, 2685. 18. Hurter, P. N. ; Scheutjens, J. M . H. M . ; Hatton, T. A. Macromolecules

1993, 21, 5592. 19. Liu, Y.C., PhD thesis, Massachusetts Institute of Technology, Cambridge,

MA, 1997. 20. Mortensen, K.; Schwahn, D.; Janse, S. Phys. Rev. Lett. 1993, 71, 1728. 21. Mortensen, K.; Pedersen, J. S. Macromolecules 1993, 26, 805. 22. Mortensen, K. J. Phys. Condens. Matter, 1996, 8, A103-A124. 23. Liu, Y . C.; Chen, S. H.; Huang, J. S. Phys. Rev. Ε 1996, 54, 1698. 24. See for example, Flory, P.J. Principles of Polymer Chemistry, Cornell

University Press, 1953. 25. Liu, Y.C.; Chen, S.H.; Huang, J.S. Macromolecules 1998, 31, 2236-2244. 26. Baxter, R. J. J. Chem. Phys. 1968, 49, 2770. 27. Ornstein, L.S. Zernike, F. Proc. Aca. Sci, Amsterdam, 1914, 17, 793.

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28. Percus, J.K.; Yevick, G.J. Phys. Rev. 110, 1 (1958). 29. Ku, C. Y.; Chen, S. H.; Rouch, J.; Tartaglia, P. Int. J. Thermophys 1995,

16, 1119. 30. Verberg, R.; deSchepper, I.M.; Cohen, E.G.D. Phys. Rev. Ε 1997, 55, 3143-

3158. 31. Liu, Y.C.; Chen, S.H.; Huang, J.S. Macromolecules, 1998. 32. Kotlarchyk, M . ; Chen, S. H. J. Chem. Phys. 1983, 79, 2461. 33. Bevington, P. R. Data Reduction and Error Analysis for the Physical Sciences,

McGraw-Hill, New York, 1969. 34. Brady, J.F. J. Chem. Phys. 1993, 99, 567. 35. Hirschfelder, J.O.; Curtiss, G.F.; Bird, R.B. Molecular Theory of Gases and

Liquids, Wiley, New York, 1954.

Chapter 19

Optical Probe Study of Solutionlike and Meltlike Solutions of High Molecular Weight

Hydroxypropylcellulose Kiril A. Streletzky and George D. J. Phillies

Physics Department, Worcester Polytechnic Institute, Worcester, MA 01609

Quasi-elastic light scattering spectroscopy spectra of optical probes diffusing in hydroxypropylcellulose (HPC):water (1,2) are well-described as a sum of two modes, here described as 'fast' and 'slow'. There are correlations between the experimental parameters that characterize spectra and system properties including probe radius and polymer concentration; these correlations separate probe behavior into distinct small-probe and large-probe regimes. For the two probe-size regimes, the fast and slow modes can be grouped into three physical regimes, namely a long time scale regime for the large-probe slow mode, an intermediate time scale regime for the small-probe slow mode and the large-probe fast mode, and a broad time scale regime for the small-probe fast mode. Physical interpretations for the three physical regimes are proposed. The probe diameter separating small-probe and large-probe behavior is the same at all polymer concentrations, consistent with hydrodynamic models for polymer dynamics, but inconsistent with models that assume the existence of a transient gel pseudolattice in solution. It has previously been reported that the zero-shear viscosity of HPC solutions shows a sharp change, at transition concentration c+, in the functional form of its concentration dependence, from a stretched exponential at lower concentration to a power law at higher concentrations. We observe a sharp transition in the concentration dependence of mode parameters at the same concentration c+, supporting the interpretation that the transition is physically real, and not an artifact of the fitting process.

© 2000 American Chemical Society 297

298

Introduction

The nature of polymer motion in semidilute and concentrated solutions remains a major question of macromolecular science. Extant models describe polymer dynamics very differently (3-11). Many experimental methods have been used to study polymer dynamics (12). One method is probe diffusion, in which inferences about polymer dynamics are made by observing the motions of dilute mesoscopic probe particles diffusing in the polymer solution of interest. Probe diffusion can be observed by several experimental techniques, for example, quasi-elastic light scattering spectroscopy (QELSS), fluorescence recovery after photobleaching (FRAP), and forced Rayleigh scattering (FRS).

We use QELSS, which measures the dynamic structure factor (spectrum) S(q,t) of visible light scattered by the system. When the probe particles dominate scattering by the ternary polymer:probe:solvent mixture, QELSS selectively gives information about probe motion. If the spectral lineshape S(q,t) were nearly exponential, one could calculate the diffusion coefficient of probes D p from the initial slope of Log(S(q,t)) (13,14). Near-exponential spectra are generally found for probes in solutions of low-molecular-weight polymers. For probes in solutions of many high-molecular-weight polymers, S(q,t) is also nearly exponential, and D p can again be obtained (15,16). However, for probes in solutions of some high molecular weight polymers, S(q,t) is highly non-exponential, posing the question of how a single diffusion coefficient D p obtained from the initial slope can usefully describe both short- and long-time behavior. With highly non-exponential spectra, a systematic analysis of the lineshape of S(q,t) can reveal information about probe motion more detailed than an average diffusion coefficient.

This paper treats an extensive study (1,2,17,18) of probe diffusion in aqueous solutions of the high-molecular-weight polymer hydroxypropylcellulose (HPC), which is rodlike, semiflexible, and uncharged. Several factors motivated our choice of this system: 1) Earlier studies (19,20,21) of probe diffusion in HPC revealed the bimodality of probe spectra. However, the physical properties and detailed parameterization of the lineshapes for the two modes were non-trivial and incompletely characterized. An objective of our work was to provide a more thorough study of observable lineshape parameters. 2) The viscosity η of aqueous HPC solutions shows, at elevated concentration, an apparent change in the functional form of its concentration dependence. An objective of this paper is to resolve between possible interpretations of this change.

Phenomenologically (22,23), at lower polymer concentrations c, η of HPC solutions has a stretched exponential concentration dependence η = η 0 β χ ρ ( α ο ν ) (1) Here α is a scaling pre-factor and ν and γ are scaling exponents. However, at

elevated concentrations η has a power-law concentration dependence

299

η = η ι ο χ (2)

with exponent χ, η! being η at unit concentration. This transition between the two behaviors is sharp (22), with a well-defined transition concentration c+, and no indication of a crossover region in which neither eq. 1 nor eq. 2 applies.

Ref. (22) denoted these two behaviors of r\(c) as the (lower-concentration) 'solutionlike' regime (eq. 1) and (higher-concentration) 'meltlike' regime (eq. 2), c+

marking the solutionlike-meltlike transition (22). Literature reviews (6,23) find solutionlike-meltlike transitions in r\(c) of solutions of some but not all other polymers. Solutionlike-meltlike transitions are (6) also found in some systems in the concentration dependencies of the steady state compliance and the characteristic shear rate y. When transitions are encountered for multiple viscoelastic parameters, they share a common transition concentration c+. It is interesting to ask whether the apparent transition is real, or whether it is a fitting artifact apparent in some sorts of data. Relative to solutions of these other polymers, c+ and rj(c+) of HPC solutions (22) are low, simplifying sample handling questions at c > c+, and further motivating our selection of HPC: water as the system of interest.

This article summarizes our experiments on probe diffusion in the solutionlike and meltlike regimes of HPC:water. We determined how the parameters which characterize the observed bimodal probe spectra depend on solution properties such as polymer concentration and probe size. To examine the reality of the solutionlike-meltlike transition, we measured the optical probe spectrum S(q,t) over a range of concentrations spanning c+. S(q,t) and η are independent physical quantities, both reflecting transport in polymer solutions. If the solutionlike-meltlike transition manifested itself in the concentration dependence of S(q,t), we would have evidence that the soltionlike-meltlike transition is physically real.

The next section gives our experimental methods and spectral fitting procedures. Further sections describe our results and data interpretation in both concentration regimes. A discussion closes the paper.

Experimental Methods

We studied solutions of hydroxypropylcellulose (Scientific Polymer), nominal molecular weight lMDa. HPC is a semiflexible chemically-modified biopolymer. The polymer is rodlike, very stiff, and has persistence length ~10nm (24). Based on Yang and Jamieson (25) and extending to our molecular weight, lMDa HPC has hydrodynamic radius Rh«55nm and radius of gyration Rg«100nm. HPC is uncharged, so electrostatic contributions to chain dynamics are not an issue. At room temperature HPC is soluble and stable in water. Stock solutions of HPC were prepared using water

300

purified with Millipore Milli-RO and Milli-Q filtration systems to a resistivity of 14-18MO/cm. Other solutions were prepared by serial dilution of the stock.

Our optical probes were carboxylate-modified polystyrene latex spheres (PSL). In the concentration range 0 < c < 7g/L, we used PSL with nominal diameters 14, 21, 38, 50, 67, 87, 189, 282, and 455 nm (Interfacial Dynamics). For concentrations 7 < c < 15g/L, we used PSL with nominal diameters 50, 87, and 189nm. A successful probe diffusion experiment using QELSS requires three conditions: First, the probes must be stable in polymer solution and neither aggregate nor bind the polymer. To stabilize the PSL probes and prevent HPC binding, we followed Phillies et al (21) and included 0.2wt\% of the surfactant TX-100 (Aldrich) in our solutions. Second, the observed light scattering spectrum must correspond to scattering by the probes, polymer scattering being practically unseen. Latex spheres are very efficient scatterers, so very small PSL concentrations (< 0.05 vol %) are enough for the probes to dominate the scattering. Third, for a successful QELSS measurement multiple scattering must be unimportant, a condition satisfied at our probe concentrations.

To confirm that the second requirement was satisfied, we measured under identical conditions the scattering intensity from probe :polymer: solvent solutions and the scattering intensity of the matching probe-free polymer solution. In most cases, scattering by the probe-free polymer solution was less than 1% of scattering by the ternary probe:polymer:solvent mixtures. In the worst case (small probes in concentrated solutions), scattering by the probe-free polymer solution was still <4% of the probe-containing solutions. As a further test that polymer scattering was not distorting our spectra, we subtracted (at the field correlation function level) S(q,t) of probe-free polymer solutions from S(q,t) of the corresponding probe:polymer:solvent mixtures. These difference spectra were analyzed by our standard methods. To within experimental error, difference spectra and the original, unsubtracted, probe:polymer:solvent spectra have (/) the same shapes. Polymer scattering was thus shown not to influence our spectra significantly. To high accuracy our experiment therefore monitors the motions of probes through an unseen polymer solution. The unseen underlying spectrum of HPC was beyond the interest of this study.

Probe diffusion was determined using quasi-elastic light scattering spectroscopy. QELSS monitors the temporal evolution of concentration fluctuations by measuring the intensity I(q,t) of the light scattered at time t, and calculating the intensity-intensity correlation function

Here Τ is the experiment's duration, τ is the delay time, and q is the magnitude of the scattering vector. Samples were illuminated with a CW Ar + laser (Spectra-Physics, 2020-03) having maximum power 2W at 514.5nm. Scattered light was detected by a photometer-goniometer (Brookhaven Instruments, BI-200SM). Most of our experiments were performed at a scattering angle of 90°. However, we also studied q-dependence of S(q,t). Sample cells were placed in a decalin-filled index-

T (3)

ο

301

matching vat and maintained at 25+0.1°C by a water circulator/temperature regulator (Neslab, RTE-110). A digital correlator (Brookhaven Instruments, BI2030AT (270 channels)) operated in the multitau mode, supported by spectral splicing procedures, determined S(q,t). Details of the multitau mode and splicing procedures, and further information on experimental apparatus and sample preparation, appear in refs. (1,17).

To quantitate the spectral lineshape, we applied (/) non-linear least squares methods based on the simplex algorithm. The intensity-intensity correlation function g{2)(q,t) is expressed in terms of the field-correlation function g(1)(q,t) via

g ( 2 ) (q, t) = S(q, t) - Β - A(g ( 1 ) (q, t))2 (4)

where A is the scattering amplitude and Β is the baseline. Spectral analysis consisted of fitting g(1)(q,t) to specific functional forms. We identified expressions for g(1)(q,t) that minimize E=[g(2),(q,t)-g(2)

e(q,t)]2/[g(2)

e(q,t)]2, where g ( 2 )

f and g ( 2 )

e are the fitted and experimentally-measured forms of g(2)(q,t), respectively.

In agreement with earlier studies (19,20,21), our experiments (7) on probes:HPC solutions revealed bimodal spectra. g(1)(q,t) is (1) represented to high accuracy by

g ( 1 ) (q, t) = A f exp(-9 f ) + (1 - A f )exp(-9t p) (5)

Here subscript "f refers to the "fast' mode; the other mode is the "slow' mode. 0 and 0 f are relaxation pseudorates; β and β f are stretching exponents; A f is the amplitude fraction of the fast mode. In some systems, pf <0.5; the Laplace transform of the fast mode then contains a wide range of decay times, including very short times, justifying the appellation "fast' for this mode. The breadth of the fast mode is such that it may finish decaying only at times later than the times at which the slow mode decays.

g(,)(q,t) was represented as a sum of two stretched exponentials. This representation is phenomenological. Spectra of small probes were found (1) to have two shoulders in plots of Ln(S(q,t)) vs. t. Both shoulders to a good approximation were found to be a stretched exponentials. Similar bimodal lineshape was found (1) to describe well the spectra of large probes. Within the limits of the QELSS technique we cannot distinguish a stretched exponential from a sum of many pure exponentials. However, our spectra are bimodal in the sense that they can be described empirically to high precision as a sum of two stretched exponentials, but cannot be described with a single stretched exponential or a sum of two pure exponentials (/).

Results

This Section presents our (1,2,17,18) major experimental results on probe diffusion in HPC solutions. To anticipate the conclusions, large and small probes

302

have qualitatively different spectral properties. A well-defined length scale separates large-probe and small-probe regimes. Within each probe size regime, spectra exhibit characteristic features that are relatively independent of probe radius. Spectra are bimodal. The fast and slow modes of probes in the two size regimes can be grouped into three physical classes, namely 1) the slow mode of the large probes, 2) the large-probe fast mode and the small-probe slow mode, which share a common time scale and have similar physical properties, and 3) the fast mode of the small probes.

We now consider how the dependences of optical probe spectra on polymer concentration, scattering vector, and probe diameter d lead to the above conclusions. We first treat data on the solutionlike regime (c < 7 g/L) and then consider results at larger c (the meltlike regime).

Solutionlike Regime

Refs (1,2) report QELSS spectra of probe:polymer:solvent mixtures using probes with a wide range of sizes (14 < d < 455nm). Differences between small-probe and large-probe behavior are already apparent from a qualitative inspection of g(l)(q,t). On a Log(S(q,t)) vs. Log (t) plot, spectra of small probes show two readily-apparent, visually distinct shoulders. Each shoulder corresponds to a mode. The same plot for large probes appears to show only a single decay, but at long time S(q,t) decays much more slowly than does a single stretched exponential that fits the short-time data. Numerical analysis confirms (1,2) that large-probe spectra are bimodal. The slow-mode stretching exponent depends (/) on probe size, namely β « 1 for large probes and β€(0.6-0.96) for small probes. For the large probes, within the accuracy of our data we were unable to tell if β is exactly unity, or only close to unity; the following reflects fits with β =1.

Quantitative analysis of spectral fitting parameters Θ, 0f, β, β ί 5 and A f reveals their dependences on d, c, and q and reinforces the interpretation that we see distinctive small-probe and large-probe regimes. Small-probe behavior is uniformly seen for probes with d < 50nm; large-probe behavior is uniformly seen for probes with d > 87nm. One notes that small probes have d < R h while large probes have d > R g. Spectra of 67nm probes (for which R h < d < Rg) are transitional between the small- and large-probe spectral forms.

Figure 1 shows how the slow relaxation pseudorate depends on probe diameter. Each point style corresponds to a different polymer concentration, θ falls with increasing d. From this Figure, θ of probes smaller than « 60 nm is approximately independent of c. Probes larger than « 100 nm have θ that decreases markedly with

303

d(nm)

Figure 1. 0(eq. 5) against probe diameter for various polymer concentrations. Rh and Rg are the polymer radius of gyration and hydrodynamic radius, ξ is the estimated correlation length at 0.5, 3, and 7 g/L.

304

increasing c. The transition from small-probe to large-probe regime involves probes with diameters comparable to R , or R g.

The fast relaxation pseudorate 0 f also shows (1,2) a crossover between small- and large-particle behaviors. The crossover diameter for 0 f is the same as the crossover diameter for 0. For small (d < Rh) probes, 0 f is nearly independent of d; for large (d >Rg) probes, 0 f falls as d is increased. For all probes, large and small, 0 f is nearly independent of c.

We now turn to the stretching exponents. Figure 2 shows typical c-dependence of β and βΓ for several probe sizes. $(c) and β ^ ) for other probe sizes can be found in ref. (26). Three distinct characteristic behaviors for β and $ f are apparent. First, for large probes β is a concentration-independent constant, namely β = 1. Second, β from the small-probe slow mode and ${ from the large-probe fast mode both decrease from « 0.9-1 at low c to « 0.6-0.7 in 7 g/L HPC. Third, for small probes β f decreases from « 0.6 at low concentration to « 0.2 in 6g/L HPC. At each concentration, the small-probe β and the large-probe βΓ are substantially larger than the small-probe β^

The three characteristic behaviors for β and $ f correspond to three characteristic magnitudes of θ and 0f. First, the large-probe slow mode has 0e(103, 5xl0"6). Second, the large-probe fast mode and the small-probe slow mode, which as noted above share common values and behaviors for their stretching exponents, have Θ, 0 f

e(10"2, 3*10-3). Third, the small-probe fast mode has 0 f~5xl0"2. This aggregation (2) of the four mode/probe-size combinations into three physical regimes is consistent with the remaining analysis.

The stretched-exponential form exp(-0 tp) can also be parameterized exp(- (ί/τ)β), thereby introducing relaxation times τ=θ"1/β and tf=0f*1/pf as replacements for the relaxation pseudorates θ and 0f. For the various combinations of mode and probe size, one finds 1) teOO^xlO '^s for the large-probe slow mode, 2) τ ^ ί ο χ Ι Ο Λ ί Ο " 1 ^ for the large-probe fast mode, 3) t€(10"4, 8xl0"4)s for the small-probe slow mode, and 4) te(3xl0"4, 5xl0'2)s for the small-probe fast mode. At every concentration, τ of the large-probe slow mode is larger than other relaxation times. The ranges of other relaxation times overlap. The small-probe fast and slow mode relaxation times usually satisfy xf >τ, even though 0 f >0. The difference in ordering between τ and θ arises because the fast mode of small probes is very broad, namely βΓ « 0.2-0.6. Because βΓ « β, the slowest Laplace components of the small-probe fast mode decay at later times than do the slowest components of the small-probe slow mode. Did we correctly label the small-probe modes as 'fast' and 'slow'? Interchanging the 'fast' and 'slow' mode labels for the small spheres would eliminate the smooth concentration and probe size dependences of θ and 0 f seen in Figures 1-3, supporting our choice as to which small-probe modes are to be identified with which large-probe modes as being "fast' or 'slow*.

305

c(g/L)

Figure 2. Stretching exponents (eq. 5) β and Pf against polymer concentration for probes of various diameters (legend).

306

Figure 3a shows θ as a function of c for ail probes. The large-probe and small-probe slow modes clearly have very different concentration dependences. For large probes θ has a strong c-dependence, decreasing as θοβχρί-οχ*). For small probes, θ is c-independent. Fitting parameters are in ref. (1).

Figure 3b shows the concentration dependence of 0f. In the solutionlike regime (c < 7 g/L), 0 f of each probe species is nearly independent of c. The large-probe 0 f and the small-probe θ thus have similar concentration dependences, consistent with our proposed incorporation of the large-probe fast mode and the small-probe slow mode into a single physical regime.

Refs. (1,2) present the dependences of probe spectra on wave vector. For all probes, the slow mode shows diffusive behavior with θ ~ aq2. For all probes, the fast mode has a complicated q-dependence, namely 0 f ~ a,q2+b, at small angles and 0f ~ a2q2+b2 at large angles. The fast modes of large and small probes differ in that a, «0 for large but not small probes.

One recalls that the spectral modes here are stretched exponentials with very different stretching exponents. Unlike pairs of pure-exponential modes, one cannot simply identify a mode as "fast" or "slow" because it decays sooner or later than another mode; stretched-exponential modes decay to some extent on a multiplicity of time scales. However, the observations that (i) the slow modes have a common, diffusive, q-dependence, the same for large and small probes, and (ii) the fast modes have a different, non-diffusive (b„ b2 Φ 0) but also common q-dependence, nearly the same for large and small probes, supports our grouping of modes as "fast" or "slow". It would have been equally possible to refer to the modes as 'sharp', with β « 1, and 'broad', withp f<p.

Meltlike Regime

This section describes probe diffusion in the meltlike regime c >7 g/L. We studied (17) only intermediate and large probes (d « 50, 87, 189 nm). The lineshape S(q,t) does not change at c+; for the large probes S(q,t) remains at all concentrations the sum of a stretched exponential and a pure exponential. While the qualitative lineshape did not change at c+, there are dramatic changes in the concentration dependences of 0 f and % and a less radical change in the concentration dependence of Θ, all occurring near the transition concentration c+. For these probes β remains close to unity at all c.

We start with the parameters showing the most dramatic transitions at c+. Figure 3b shows the concentration dependence of 0f. Below c+, 0 f is independent of c. A radical change is seen near c+. For c>6 g/L, 0 f of the large probes decreases rapidly with increasing c. 0 f of the smaller 50nm probes remains nearly independent of c for c > c+. Above the transition concentration, 0f for each probe species is described well

307

A

c ( g / l )

Figure 3. Relaxation pseudorates (1,17) Θ and 9f(eq. 5) against c for PSL probes (legend, Figure 3 b). a) Slow-mode relaxation pseudorate and fits to 9Qexp(-acv). Dashed lines: fit to all measurements. Solid lines: separate fits for c < c+ and c > c+. b) Fast mode relaxation pseudorate. Dashed lines are simple exponentials. Solid lines are: 1) pure exponentials (50, 87nm probes) or 2) a power law (I89nm probes). Units ofefare(pS)'#

308

by a stretched exponential; for 189nm probes, 0 f could also be described to within experimental error by a power law. For fitting parameters, see ref.(/7). In the meitlike regime, the slope of 0 f increases with increasing d.

As seen in Figure 4, near the transition concentration p f also has a radical change in its concentration dependence. Below c+, pf falls linearly with increasing c. Above c+, βΓ of the larger 87 and 189nm probes instead rises with increasing c. For 50nm probes, p f is independent of concentration for c>c+.

From Figure 3a, θ monotonically decreases with increasing c at all concentrations. At the transition concentration, θ only has a modest change in its concentration dependence. In Figure 3a, dashed lines represent fits of 0(c) over the whole concentration range to a single stretched exponential. For the 87 and 189nm probes, near c+ the data deviates from the best fit stretched-exponential; θ is less than expected from the fit line. Fits of the data to separate stretched exponentials for high and low concentrations, as indicated by the dark lines, do a much better job of representing our measurements. Even though there is no sharp change in the concentration dependence of θ near c+, some change in 0(c) can be seen near c+. Furthermore, from Figure 3a, θ decreases with increasing d. However, the d-dependence of θ changes at c+. Below c+, the slope of 0(c) increases with increasing d. Above c+, the slope in Figure 4a of 0(c) is independent of d.

We determined (17) the q-dependences of the mode parameters for large probes in the meltlike regime. The q-dependences of these parameters are the same above c+

as they were below c+; there is no transition in the wavevector dependence at c+. For large probes, A f is nearly independent of q, and has no substantial dependence of c, over the concentration range studied. For small probes, A f has a very strong q dependence, with A f « 1 being the plausible low-q limit, falling to A f » 0.2 at the largest q studied; A f increases weakly with increasing c. The behavior of A f of large probes does not change significantly between the solutionlike and meltlike regimes.

Relationship Between Viscosity and Relaxation Rates

The diffusion of mesoscopic spheres in simple fluids that are not highly viscous is governed by the shear viscosity η. Relationships between η, θ, and 0f appear in Figures 5a and 5b, which plot θη and θΓη as functions of c. Viscosities were taken from an independent study by Phillies and Quinlan (22).

Figure 5a displays the slow mode behavior. In the solutionlike regime c<c+, for the largest (d > 189 nm) probes θη is constant to within a factor of two. For smaller spheres, 0fr| increases with increasing c, the degree of increase being largest for the smallest spheres, ranging from a 15-fold increase for 50nm spheres up to a 400-fold increase for 14nm spheres. In the meltlike regime, the larger the probe the more nearly θ tracks the solution fluidity η"1. For the 189nm probes there is no visible

309

o.2 y

o.o 12 16

c ( g / l )

Figure 4. Stretching exponent pf(eq. 5) against HPC concentration for probes of diameter 50 (·),87 ( +), and 189 ( *)nm. Lines are linear fits.

310

311

change near c+ in the concentration dependence of θη; θη continues its gradual increase with increasing c. Particularly for smaller spheres, dθη/dc declines with increasing polymer concentration. For the 50 and 87nm probes, θη increases by 50-or 5-fold over the solutionlike regime, but above c+ out to the highest concentrations studied θη only increases by a further factor of three.

Figure 5b displays the fast mode behavior. For all probes, small and large, across the solutionlike regime θ ^ increases by a factor of several hundred. There is a radical change at c+ in the concentration dependence of θ^ . Above c+, θΓη for 189nm probes becomes independent of c; for the 50 and 87nm probes Qfr\ continues to increase, but only by a factor of 50 or 5, respectively over the observed concentration range. In the meltlike regime, η ~ cx. The observed behavior of θΓη implies Qf ~ c*x

for the 189nm spheres in the meltlike regime, which is consistent within experimental error with our data.

Discussion

We now consider major questions raised by our measurements. These include: What physical interpretation may be assigned to the observed spectral modes? What do our measurements reveal about the physical reality of the solutionlike-meltlike transition in η ? Do our results speak to the validity of the coupling model? What does the observed probe crossover imply for the validity of various models of polymer dynamics in solution?

Physical Interpretation of the Mode Structure: The observed spectral modes are (2) rationally grouped into three physical regimes, namely a long time scale regime, an intermediate time scale regime, and a broad time scale regime. Correlations between the solution properties and the spectral fitting parameters suggest physical interpretations of each regime. We discuss a physical interpretation based on the solutionlike regime data; above c+, we do not yet have the needed measurements on truly small probes.

The long time scale regime refers to the slow mode of the large probes. Characteristic properties of this regime are: 1) Spectra are pure exponentials; 2) From its q-dependence, θ - q 2 describes simple diffusive transport; 3) The relaxation pseudorate falls with increasing polymer concentration; and 4) The Stokes-Einstein prediction θ ~ η"1 works to within a factor of 2. The first two properties signify probe transport resembling simple Brownian motion. We infer that the long time scale regime involves time scales sufficiently long that all internal polymer modes have decayed so that probes sample solutions which approximate simple viscous fluids. Because β =1, one could meaningfully write θ = Dq 2; the concentration dependence of this D parallels the concentration dependence of the self-diffusion coefficient of colloid particles with hydrodynamic (27) interactions.

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The intermediate time scale regime refers to the small-probe slow mode and the large-probe fast mode. Modes in this regime have common properties: 1) The mode relaxation is a stretched exponential in time with pe(l.0-0.6), β decreasing with increasing c; and 2) The decay pseudorate, which is nearly independent of c, is 10"2-10"3 for both modes. Furthermore, 3) Ngai and Phillies (10) proposed an extension of the Ngai-Rendell (8,9,11) coupling model to treat polymer transport and probe diffusion. The model reduces the mode time dependence, the q and c dependences of the decay pseudorate, and the concentration dependence of the zero-shear viscosity to a series of concentration-dependent coupling coefficients η and corresponding exponents β = 1-n. The model predicts that β from four disparate paths should agree, and that the β should all decrease with increasing c. These predictions are correct for modes in the proposed intermediate time scale regime, namely the small-probe slow mode and the large-particle fast mode (2,18). The other modes do not follow coupling-model predictions, β obtained from the large-particle slow mode or, separately, from the small-particle fast mode, do not agree with each other or with the coupling coefficient calculated from r|(c).

The small-probe slow mode and the large-probe fast mode do differ in their q-dependence (2). One finds θ - q 2 -» 0 in the low-q limit, while 0 f does not go to zero in this limit.

We have inferred (2) that the intermediate time scale regime corresponds to probe motions that are heavily coupled to incompletely relaxed polymer modes. Mode properties supporting this inference include the highly non-exponential nature of the relaxations, the increase of τ and xf with increasing c, and the increased concentration dependences of τ and xf with increasing probe diameter, expected because larger probes can couple strongly to more polymer chains than small probes do. The detailed nature of the coupling or the identity of the polymer modes in question have not been completely clarified by the present analysis.

The broad time scale regime refers to the small-probe fast mode. Because the mode is so broad (βΓ « 0.6-0.2), for small probes the terminal observable decay of S(q,t) corresponds to the longest-time part of the this mode. Small probes are smaller (d < Rh) than our polymer chains. For c < c+ a probe thus is unlikely to be in contact with more than one or two chains. We infer that this regime plausibly corresponds to probe motion determined by internal modes of a single neighboring polymer chain. Our inference is based in the first instance on the finding A f -» 0 as c -> 0, i.e., the mode arises from the presence of polymers. However, 0 f of the mode is largely independent of polymer c, ruling out effects additive in the number of chains with which the probe is interacting. The identification with internal chain modes is made because internal modes of single chains are the only obvious chain motions whose time scale is substantially concentration-independent at low concentration.

Our results to date lack spectra of small probes in the meltlike regime, so it is difficult to improve the interpretation by using the phenomenology at higher

313

concentrations. The q-dependence of the fitting parameters does not change at c+. Only the intermediate time scale mode is seen to change its behavior markedly in the meltlike regime. Reference to known results (4) on polymer statics suggests a possible explanation. At low concentrations, polymer chains do not interpenetrate. There is a concentration above which chains begin to interpenetrate substantially. Chain interpénétration might reasonably be expected to modify polymer internal modes in ways that the close physical juxtaposition of non-overlapping chains does not. If the intermediate-time-scale mode was in part coupled to incompletely relaxed polymer internal modes, then above the interpénétration concentration there would be a marked change in the relaxation of this mode, consistent with what is observed experimentally.

Reality of the Solutionlike-Meltlike Transition: A solutionlike-meltlike transition in the concentration dependence of the zero-shear viscosity of HPC solutions was reported by Phillies and Quinlan (22). Is this transition real, or is it a fitting artifact? While it is possible to interpret Phillies and Quinlan's viscosity measurements as showing a real, sharp dynamical transition at c+, one could also interpret their c<c+

data as being composed of a series of power laws and crossovers, the final crossover region ending at c+. In this case, the apparent stretched-exponential concentration dependence would only be an empirical approximant. We now ask how the solutionlike-meltlike transition of r\(c) is reflected in our optical probe spectra.

From Figures 3b and 4, the fast mode parameters 6f and pf radically change their c-dependences near c+. Below the transition, Bf is nearly independent of c; above c+, 9 f of the large probes falls sharply. On the other hand, βΓ falls with increasing c for c < c+, but is constant or increases above c+. There is weaker evidence that the slope dLn(Of)/dc changes appreciably at a concentration near c+.

For both 9 f and βΓ, there is one behavior at c < c+, a crossover at approximately the same concentration for all parameters, and a distinct behavior at c > c+. The phenomenology is entirely consistent with a system that has two fundamentally different dynamical behaviors separated by a dynamic crossover at c+. There is no indication in the region c < c+ that one has a series of different behaviors linked by crossovers. Our experimental results based on optical probe diffusion confirm the interpretation of Phillies and Quinlan that r\(c) of HPC solutions shows a sharp dynamic transition at a concentration c+ « 6 g/L.

The Ngai-Rendell coupling model (8-11) indicates that an increase in the polymer concentration should lead to an increase in the coupling parameter η and thence to a decrease in the exponent β. Above the transition concentration c+, the stretching exponent actually increases with increasing c. The observed concentration dependence of β in the meltlike regime is inconsistent with straightforward extensions of the coupling model for this concentration regime.

Implications of the probe-size crossover diameter: Figure 1 reveals a crossover between small-probe and large-probe behavior. As seen in Figure 1, the crossover

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occurs for probes with R h < d < Rg. A similar crossover is seen (1) for 0 f as a function of d. For both θ and 0f, the crossover length scale is independent of c for concentrations up to c[n] «5. (While our large-probe data reach c[r|]=10, without small probe-data we cannot say if the crossover is unchanged for 5<c[rj]<10.) Figure 1 has implications for the acceptability of some polymer dynamics models in solution.

A broadly-accepted family (3-5) of models of polymer dynamics is based on the hypothesis that non-dilute solutions of high-molecular-weight polymers may be described as containing a transient pseudogel lattice, so that at short times non-dilute polymer molecules behave as though they are crosslinked, while at longer times the same molecules are free to move with respect to each other. The short-lived cross­links, whose physical nature is not specified by the models, are entanglement points. In these models, a polymer entanglement is said to be transient because polymer molecules are free to move along their contours until the entanglement unravels. The natural length scale for the solution is the mean distance ξ between entanglements. The model (4) indicates that spheres with d > ξ are trapped by the transient lattice and only free to move when the solution moves; large-sphere motions are therefore governed by the macroscopic shear viscosity. Spheres with d < ξ can move through gaps in the transient lattice, a mode of motion barred to large spheres, so small spheres can move rapidly through solution. The model thus predicts that polymer solutions selectively retard motions of spheres with d > ξ.

From the model, crossover occurs in nondilute solution at some fixed d/ξ not fully specified by the model. The length ξ depends on concentration as ξ - c"3M. Vertical arrows in Figure 1 are estimates of ξ for HPC concentrations 0.5, 3, and 7 g/L. The crossover probe diameter at which probes are trapped by the pseudolattice, being at fixed d/ξ, thus depends strongly on c. Over the observed range of nondilute concentrations the crossover probe diameter should change substantially. In fact, for both θ and 0 f the probe diameter separating small and large probe behavior is R h d R g

independent of concentration. The observed concentration independence of the crossover diameter is inconsistent with transient gel models (4) in which a dimensionless quantity d/ξ separates small from large probes.

On the other hand, the existence of a concentration-independent crossover probe diameter d « (Rh, Rg) is consistent with polymer solution models based on an assumed dominance of hydrodynamic interactions in nondilute solution. Models such as the hydrodynamic scaling model (6,7) identify the chain radius as the primary solution length scale at all concentrations at which the model applies. With this identification, a crossover from small-probe to large-probe behavior, perhaps correlated with differential ability to interact with internal chain modes, would at all concentrations occur over the same range of d/Rg or d/Rh, precisely as observed experimentally.

Literature Cited

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1. Streletzky, Κ. Α.; Phillies, G. D. J. J. Chem. Phys. 1998, 108, 2975-2988.

2. Streletzky, Κ. Α.; Phillies, G. D. J. J. Polym. Sci. B. 1998, 36, 3087-3100.

3. de Germes, P. -G. J. Chem. Phys. 1971, 55, 572-582.

4. de Germes, P.-G. Scaling Concepts in Polymer Physics, Cornell University Press: Ithaca, NY, 1988.

5. Doi, M. ; Edwards, S. F. The Theory of Polymer Dynamics, Oxford University

Press: Oxford, 1986.

6. Phillies, G. D. J. Macromolecules 1995, 28, 8198-8208.

7. Phillies, G. D. J. J. Phys. Chem. 1989, 93, 5029-5039.

8. Tsang K.-Y.; Ngai, K. L. Phys. Rev. Ε 1997, 54, R3067-R3070.

9. Tsang K.-Y.; Ngai, K. L. Phys. Rev. Ε 1997, 56, R17-R21.

10. Ngai, K. L.; Phillies, G. D. J. J. Chem. Phys. 1996, 105, 8385-8397.

11. Ngai, K. L.; Rendell, R. W. Phil. Mag. Β 1998, 77, 621-633.

12. Lodge, T. P.; Rotstein, N.; Prager, S. Adv. Chem. Phys. 1990, 79, 1-132.

13. Phillies, G. D. J. J. Chem. Phys. 1974, 60, 983-989.

14. Phillies, G. D. J. Biopolymers 1975, 14, 499-508. 15. Phillies, G. D. J.; Gong, J.; L i , L.; Rau, Α.; Zhang, K.; Yu, L.-P.; Rollings, J. J.

Phys. Chem. 1989, 93, 6219-6223.

16. Phillies, G. D. J.; Quinlan, C. A. Macromolecules 1992, 25, 3310-3316.

17. Streletzky, Κ. Α.; Phillies, G. D. J. Macromolecules 1999, 32, 145-152.

18. Streletzky, Κ. Α.; Phillies, G. D. J. J. Phys. Chem. Β 1999, in press.

19. Brown, W.; Rymden, R. Macromolecules 1986, 19, 2942-2952.

20. Mustafa, M . B.; Russo, P. S. J. Colloid Interf. Sci. 1989, 129, 240-253.

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21. Phillies, G. D. J.; Richardson, C.; Quinlan, C. Α.; Ren, S.-Z. Macromolecules 1993, 26, 6849-6858.

22. Phillies, G. D. J.; Quinlan, C. Macromolecules 1995, 28, 160-164.

23. Phillies, G. D. J. J. Phys. Chem. 1992, 96, 10061-10066.

24. Bu, Z.; Russo, P. S. Macromolecules 1994, 27, 1187-1194.

25. Yang, T.; Jamieson, A. M . J. Colloid Interface Sci. 1988, 126, 220-230.

26. Streletzky, Κ. Α.; Ph. D. thesis, WPI, Worcester, MA, 1998.

27. Mazur, P.; van Saarloos, W.; Physica 1982, 115A, 21-27.

Chapter 20

SANS Studies of Polymers in Organic Solvents and Supercritical Fluids in the Poor, Theta,

and Good Solvent Domains Y. B. Melnichenko1, E. Kiran2, K. Heath1, S. Salaniwal1, H. D. Cochran1,

M. Stamm3, W. A. Van Hook4, and G. D. Wignall1

1Solid State Division, Oak Ridge National Laboratory*, Oak Ridge, TN 37831 2Department of Chemical Engineering, University of Maine, Orono, ME 04468

3Max Planck Institut für Polymerforschung, 55021 Mainz, Germany 4Chemistry Department, University of Tennessee, Knoxville, TN 37996

We demonstrate that semidilute polymer solutions in supercritical fluids (SCFs) reproduce all main features of the polymer behavior in organic solvents which is indicative of the intrinsic similarity between the thermodynamic properties of polymers in SCFs and in the far sub-critical liquids. Using small-angle neutron scattering, we studied the effect of temperature and pressure on the phase behavior of polystyrene (PS) in organic solvents as well as of poly(dimethylsiloxane) (PDMS) in supercritical carbon dioxide (SC C0 2). The radius of gyration R g of polymer chains in both organic solvents and in SC C 0 2 is invariant during both temperature and pressure quenches down to the critical point of demixing. The limit of infinite polymer miscibility (the Θ condition) of PDMS - SC C 0 2

solutions may be reached by varying the pressure (PΘ=52±4 MPa at the density of SC C 0 2 pco2 = 0.95 g/cm3) and or the temperature (Τ Θ = 65±5 °C). A sharp crossover between the critical and mean field behavior in PDMS - SC C 0 2 is similar to that in solutions of PS in cyclohexane, and at T> Τ Θ , P>PΘ the solutions reach the good solvent domain where R g expands beyond the unperturbed dimensions Rg(Θ) at the Θ condition.

* Managed by Lockheed Martin Energy Research Corporation under contract DE-AC05-96OR-22464 for the U. S. Department of Energy.

© 2000 American Chemical Society 317

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Introduction

In organic solvents, it is well known that the radius of gyration, R g (i.e. the r.m.s. distance of scattering elements from the center of gravity) of polymer molecules depends on the sign and magnitude of the interactions between the chain segments and the molecules of the surrounding liquid. In "good" solvents, the dominating repulsive forces between the segments (excluded volume effects) work to expand the R g, and the second virial coefficient (A2) is positive. In less favorable solvents, the pairwise attractive and repulsive segment interactions may compensate at the "Flory" or "theta temperature" (ΤΘ), where A2=Q, and R g corresponds to the dimension of a volume-less polymer coils, "unperturbed" by the excluded volume effects. By definition, Τ Θ is the critical solution temperature (Tc) for a polymer with the molecular weight Mw = 0 0

and thus corresponds to the threshold of unlimited polymer - solvent miscibility. One of the first significant applications of SANS was to confirm Flory's prediction that polymer chains would adopt such random-walk configurations in the condensed (amorphous) state [1].

In the poor solvent regime (Τ<ΤΘ, A2<0), dominating attractive interactions between the segments work to collapse polymer chains into compact polymer globules. This phenomenon is well understood for dilute solutions of separated polymers which collapse as the temperature approaches T c [2]. SANS has recently been used to extend the experimental observations to "semidilute" solutions of strongly interacting macromolecules. In accord with the de Germes' concept [3], in the critical region T~TC of the solutions polymer chains do not interpenetrate significantly and thus should be collapsed [i.e. Rg(Tc)< Rg(T©)] as in dilute concentration regime. However, experiments on PS in cyclohexane (CH) [4,5] and acetone (AC) [6] have demonstrated that the predicted decrease in R g is not observed as T=>TC. Instead, diverging concentration fluctuations near T c lead to the formation of distinct microdomains of strongly interpenetrating molecules, which prevent the expected collapse [4-6]. Measurements have also been made in supercritical fluids to contrast such systems with organic solvents and to test the prediction [7] that the molecules will adopt the unperturbed dimensions at a critical "theta pressure"(P©) as they do in polymer solutions at the theta temperature. A direct comparison of the phase behavior of PDMS in SC C 0 2 with that of PS in CH and in AC is indicative of the intrinsic similarity between the structure and thermodynamic properties of polymers in SCFs and in sub-critical organic liquids.

Experimental

The SANS data were collected on the W. C. Koehler 30m SANS facility [8] at Oak Ridge National Laboratory (ORNL). The neutron wavelength was λ - 4.75 A (Δλ/λ ~ 5%) and the 64 χ 64 cm2 area detector with cell size ~ 1 cm2 was placed at various sample-detector distances to give an overall range of momentum transfer of 0.005 < Q « 4π λ"1 sin 0 < 0.1 A 1 , where 20 is the angle of scattering. The data were

319

corrected for instrumental backgrounds and detector efficiency on a cell-by-cell basis, prior to radial (azimuthal) averaging and the net intensities were converted to an absolute (± 3%) differential cross section per unit solid angle, per unit sample volume \dI(Q)ldQ in units of cm"1] by comparison with pre-calibrated secondary standards [9]. The experiments in supercritical CO2 were conducted in a high-pressure cell similar to the one used previously for polymer synthesis [10] and the SANS studies of block copolymer amphiphiles in supercritical C 0 2 [11]. The cell's stainless body was fitted with sapphire windows with virtually no attenuation (cell transmission ~ 93%) or parasitic scattering. The sample cross sections were obtained after subtracting the intensities of the cell, and the signal from the C 0 2 amounted to a virtually flat background (~ 0.04 cm"1), which formed only a minor correction to the scattering from the homopolymer solutions.

To obtain R g and the correlation length (ξ) of the concentration fluctuations we make use of the SANS high concentration isotope labeling method [12-14], which allows the determination of both parameters for semidilute polymer solutions. The coherent scattering cross section (άΣΙάΩ in units of cm"1) of an incompressible mixture of identical protonated and deuterated polymer chains dissolved in a solvent is given by [4,5,14]:

/ (β ,*) = / , ( β ,* ) + / , (β ,χ ) , (1)

Is(Q,x) = KnN2Ss(Q) , (2)

lt(Q,x) = LnN2St(Q) . (3)

The subscripts "s" and "t" correspond to scattering from a single chain and total scattering, respectively and thus SS(Q) is the single-chain structure factor, containing information on the intramolecular correlations (and hence R g). Similarly, the total scattering structure factor, St(Q) embodies information on the total (both intra- and intermolecular) correlations between monomer units and is related to the correlation length of the concentration fluctuations, ξ. The structure factors are normalized so that Ss - 1 at (Q = 0) and St = Ss at infinite dilution. Also, χ is the mole fraction of protonated chains in the solvent, and η and Ν are the number density of the polymer molecules and the degree of polymerization, respectively. The prefactors Κ and L are:

Κ = (bH - bDf x(l - x); L = [bHx+(l - x)bD - bs'f , (4)

where bH and bD are the scattering lengths of the protonated and deuterated monomers and bs' is the scattering length of a solvent molecule, normalized to the ratio of specific volumes of the monomer and the solvent molecule.

The prefactor, L, in equation (4) controls the "total" scattering contribution and it has been shown [4] that for isotopic PS mixtures dissolved in fully deuterated acetone (AC-d), L - 0 at χ - 0.214. Similarly for PDMS in C0 2 , an isotopic ratio of χ • 0.512 gives L = 0 at the density of the solvent pco2=0-95 g/cm3. Thus, the

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intramolecular scattering function may be obtained directly from the measured cross section at all temperatures and equation (1) gives the R g directly [6], For PS in CH-d, however, there is no isotopic ratio, 0 <x < 1, which satisfies the condition L(x) » 0, and dX/dQ(Q,x) always contains a minor (< 10%) contribution from the total (intermolecular) scattering, which must be subtracted to extract R g [6].

If all chains are all protonated (JC=1), the prefactor, Κ = 0, and άΣ/dQ ~ St(Q). Thus, the size of the concentration fluctuations may be measured via the Ornstein-

Zernike (O-Z) formalism [9]: dL/dQ(Q) = dL/dQ(0)/(l + β 2 § 2 ) , where ξ is the composition fluctuation correlation length, which may be obtained from the slope of an O-Z plot of [dI/dQ(0)]'! vs. Q2.

Samples of PS-h, defined by their weight (w) and number (n) averaged molecular weights ( M w « 10,200, 11,600 and 533,000 Dalton) and PS-d ( M w -10,500,11,200, and 520,000 Dalton) of polydispersity M w / M n < 1.06 were purchased from Polymer Laboratories. Deuterated solvents AC-d and CH-d (D/(H+D)«0.995) were purchased from Sigma Chemical and were dried over a molecular sieve prior to preparing solutions near the critical concentration C(PS/CH-d)=4.4 wt% [4] and C(PS/AC-d)~20.3wt% [15]. Solutions of (h+d) PS in AC-d and CH-d were prepared at *=0.214 and JC=0.20, respectively. The latter condition provides a reasonably high signal-to-noise ratio, which minimizes the contribution from the intermolecular "total scattering" term in Eq.l .

Samples of protonated PDMS-h with M w - 22,500 47,700, and 79,900 and polydispersity M w / M n < 1.03 were synthesized and characterized at Max Planck Institut fur Polymerforschung, Germany, and PDMS-d ( M w - 27600 and 75600, M w / M n < 1.11) was purchased from Polymer Standards Service GmbH, Mainz, Germany. All solutions were prepared at the volume fraction of the polymer equal to the concentration of polymer coil overlap (φ* - 0.1372,0.0941, and 0.0727 for M w « 22,500 47,700, and 79,900, respectively) which is practically indistinguishable from the critical concentration of phase demixing [3].

Samples were either run at atmospheric pressure in quartz cells as a function of temperature (e.g. PS-h in CH-d [4,5] or in AC-d [6]) or were loaded into the pressure cell with the appropriate isotopic ratio to eliminate (x « 0.512 for PDMS in C0 2 ) or minimize (e.g. χ « 0.2 for PS-h in CH-d) the contribution of the total scattering term [Eq.(l)]. The maximum area accessible to the neutron beam is - 2 cm2

and the path lengths of the quartz or pressure cells may be adjusted over the range, 0.1 to 2.0 cm. to optimize the transmission. The temperature was controlled by circulating fluids (± 0.1 °K) and pressure was applied and measured using a screw-type pressure generator and a precision digital pressure indicator, respectively.

Results and Discussion

Fig.l shows the temperature variation of the correlation length ξ for PS-h in CH-d at the critical concentration of the polymer and it may be seen that the chains do not collapse as T=»TC as observed in dilute solutions [2]. Instead, they maintain their

321

300

Figure 1. Rg(T) and ^(T)for the solution of PS with Mw =115,000 in CH-d between ΤΘ = 40 °C and the critical temperature of phase demixing (Tc).

unperturbed dimensions in accordance with the theoretical predictions of Muthukumar [16] and Raos and Alegra [17]. The size of the concentration fluctuations is very small in the Θ region, but increases dramatically near the critical point where it exceeds the chain dimensions [| » Rg(T©)]. These findings indicate that critical polymer solutions can no longer be considered as an ensemble of collapsed, non-interpenetrating chains. Instead, the diverging thermodynamic fluctuations in the critical region lead to the formation of distinct microdomains, representing unperturbed, strongly interpenetrating macromolecules.

Fig. 2 shows values of ξ and R g over a wide range of pressure and temperature extending from the Θ domain to the poor solvent condition near T c . It may be seen that all isobars for |(T) merge at the theta temperature, 0=40 °C, where the average correlation length ξ(Θ) * 109 ± 5 Â. According to Fujita [18] and Des Cloizeaux and Jannink [19], |(0)=Rg(0)/31 / 2 at the theta condition, so ξ(θ) = 114 Â for M w = 533000 which agrees well with the experimental value.

Unlike the PS-CH system, which may undergo the transition from the poor solvent to the theta solvent by adjusting the temperature, the solvent quality is much poorer for PS - AC. Fig.3 shows the effect of pressure on the thermodynamic state of PS/AC-d solutions and the chain dimensions [Rg = 29 A] are independent of pressure and temperature and remain close to the unperturbed dimensions for Gaussian chains with M w - 11,600 [i.e. R g - 0.27 M w

1 / 2 ~ 29 A]. Conversely, the concentration fluctuations as monitored by the correlation length, ξ, diverge near the critical point and fall when Ρ » P c . However, they do not decay sufficiently to reach ξ(0) *

322

Figure 2. %(T)for PS-h (Mw = 533000) in CH-d at different pressures shown in the inset. Rg(P,T) of PS-h chains in the solution (h+d) PS in CH-d is also shown (open squares).

400

300

o< 200

100

A

ISOTHERMS, 9C: • 51 • 44.9 • 36.6 • 30 • 23.6 * 13.5

10 20 30

P, MPa 40 50

Figure 3. £$P) of PS-h (Mw = 11,600) in AC-d at the different temperatures shown in the inset. Rg(P,T) of labeled PS-h chains in the solution of (h+d) PS in AC-d is also plotted (open squares).

323

R g(0)/3 1 / 2 = 17 À, and thus, the system never leaves the poor solvent domain. Despite this, the macromolecules always exhibit their unperturbed (theta) dimensions, even though they never reach the theta domain. It is remarkable that the diverging concentration fluctuations which prevent chain collapse in PS-CH, also operate in PS-AC. So effective is this mechanism, that even for solutions which never leave the poor solvent domain, the macromolecules are always "stabilized" to exhibit the theta dimensions over wide ranges of pressure and temperature (Fig. (3).

The SANS experiments on PDMS in C 0 2 were designed to compare the chain dimensions with those in organic solvents and to test the prediction [7] that they will adopt "ideal" configurations, unperturbed by excluded volume effects, at a critical "theta pressure"(P©) as they do in polymer solutions at the theta temperature. The effect of temperature on the thermodynamic state of PDMS - C 0 2 solution is illustrated in Fig. 4. The concentration fluctuations diverge as the temperature approaches the phase boundaries, which are themselves a function of pressure. When the correlation length decreases to the theoretical value |(Θ) = R g(©)/3 1 / 2 = 41 A, the Θ condition is reached and this demonstrates that, unlike PS-AC, the solvent quality may be changed from the "poor" to theta solvent by varying the temperature, as in PS-

125

Figure 4. ξ(Τ) of PDMS in SC C02 at different density of the supercritical solvent shown in the inset. The value of the Θ temperature depends crucially on the density of SC CO2, e.g. T& = 65±5°C at pC02 = 0.95 g/cm3, however Te = 80±5°C at pC02 = 0.87 g/cm3. The Θ temperature may hardly be reached at pC02 ^0.84 g/cm3.

324

CH solutions. A typical variation of ξ vs. (T - T c ) in PDMS-SC C 0 2 solutions is shown in Fig.5. As is seen, the critical index ν in a scaling law for the correlation length ξ ~ (T -Tc) V exhibits a sharp crossover from the mean-field value (v=0.5) in the Θ region to the Ising model value (v-1.24) in the critical region around T c . The crossover takes place when the correlation length becomes equal to the radius of gyration of the polymer and thus reproduces the main features of the crossover of ξ observed in solutions of PS in CH-d (see the inset in Fig.5) [4,5], These observations delineate an intrinsic analogy between the temperature behavior of polymers in SCFs and in organic solvents [20] and show that polymer solutions in SCFs belong to the universality class of Ising model.

Figure 5. Variation of ξ as a function of (T-Tc) for solution of PDMS (Mw =22500) in SC C02. The slope gives the value of the critical index v. The inset shows ξ vs. (T-Tc) for a solution of PS with Mw =28000 in CH-d [4,5],

325

Fig.6 shows that PDMS may also enter the "good solvent" domain at a "theta" temperature Τ Θ « 65 ±5 °C, as observed in PS-CH solutions [21]. However, unlike organic solvents, this transition can also be made to occur at a critical "theta pressure" (Ρ Θ * 52 ± 4 MPa) in C0 2 , as illustrated in the inset in Fig.6. To our knowledge, this is the first time that the existence of a "theta pressure" has been demonstrated experimentally and a more detailed description of this phenomenon is given in reference [22].

80

70 μ

60 Η

° < 50

40

30

2 <έθ 40 60 80 Ρ, MPa

R (θ)=41 Λ

20 0 20 40 60

T , ° C

80 100

Figure 6. Expansion of PDMS chains in SC C02 above the Θ temperature Τθ ~ 65°C. Below T&, diverging concentration fluctuations prevent the coil collapsing as observed in organic solvents. The inset illustrates first observation of α θ pressure in solutions of PDMS in SC C02. As observed in organic solvents, the polymer chains do not collapse below P&

326

For Ρ > Ρ Θ and Τ > Τ Θ , the system exhibits a "good solvent" domain, where the polymer molecules expand beyond the unperturbed R g, in a good agreement with results of computer simulations [23]. However, for Τ < Τ Θ , and Ρ < Pe, the chains do not collapse and maintain their unperturbed dimensions, as observed in organic solvents [4-6], where the size of the concentration fluctuations, ξ diverges as T=>TC. Thus, the deterioration of the solvent quality again leads to the formation of microdomains, consisting of interpenetrating polymer coils, as in organic solvents. Near the critical point, the growth of the polymer concentration fluctuations brings together the initially non-overlapping chains, and the macromolecules adopt the unperturbed dimensions, as in highly concentrated systems and in the condensed state. Thus, the stabilization of the molecular dimensions in the poor solvent domain by diverging concentration fluctuations is a universal phenomenon, observed not only in "classical" polymer solutions in organic solvents (e.g. PS in CH), but also in supercritical fluids (e.g. PDMS in C0 2).

A unique attribute of SCFs is that the solvent strength is easily tunable with changes in the system density, offering exceptional control over the solubility. Thus, for PDMS, C 0 2 becomes a "theta" solvent at Ρ Θ -52 MPa and Τ Θ ~65°C, whereas it behaves as a "good" solvent for Ρ > P© and Τ > Τ Θ . Below this transition, the chain dimensions never fall below the "theta" R g, as observed in organic solvents, though for supercritical C0 2 , the system may be driven through this transition as a function of pressure in addition to temperature. Understanding the solubility mechanisms is a necessary condition for the development of C02-based technologies and SANS promises to give the same level of insight into polymers in supercritical media that it has provided in the condensed state, organic solvents and in aqueous systems.

Acknowledgments

Y B M and GDW wish to acknowledge stimulating discussions with M . Muthukumar. The research was supported the Divisions of Advanced Energy Projects and Materials Sciences, U. S. Department of Energy under contract No. DE-AC05-96OR-22464 (with Lockheed Martin Energy Research Corporation) and Grant No. DE-FG05-88ER-45374 (with the University of Tennessee).

References

1. Wignall, G. D. in: Encyclopedia of Polymer Science and Engineering; John Wiley & Sons, Inc.: New York, 1987; Vol. 10, p. 112.

2. Chu, Β; Ying, Q; Grosberg, A. Macromolecules 1995, 28, 180, and references therein

3. De Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, 1979.

4. Melnichenko, Y. B.; Wignall, G. D. Phys. Rev. Lett. 1977, 78, 686.

327

5. Melnichenko, Y. B.; Anisimov, Μ. Α.; Povodyrev, Α. Α.; Wignall, G. D.; Sengers, J. V.; Van Hook, W. A. Phys. Rev. Lett. 1997, 79, 5266.

6. Melnichenko, Y. B.; Wignall, G. D.; Van Hook, W. Α.; Szydlowski, J.; Rebello, L. P.; Wilczura, H. Macromolecules 1998, 31, 8436.

7. Kiran, E.; Sen, Y. L. in: Supercritical Fluid and Engineering Science; ACS Symposium Series 1993.

8. Koehler, W. C., Physica 1986, 137B, 320. 9. Wignall, G. D.; Bates, F. S. J. Appl. Cryst. 1986, 20, 28. 10. DeSimone, J. M. ; Guan, Z.; Elsbernd, C. S. Science, 1992, 257, 945. 11. McClain, J. B.; Betts, D. E.; Canelas, D. Α.; Samulski, E. T.; DeSimone, J.

M. ; Londono, J. D.; Cochran, H. D.; Wignall, G. D.; Chillura-Martino, D.; Triolo, R. Science 1996, 274, 2049.

12. Williams, C. E.; Nierlich, M.; Cotton, J. P.; Jannink, G.; Boue, F.; Daoud, M ; Farnoux, B., Picot, C.; de Gennes, P.-G.; Rinaudo, M. ; Moan, M.; Wolf, C. J. Polym. Sci., Polym. Lett. Ed. 1979, 17, 379.

13. Akasu, A. Z.; Summerfield, G. C.; Jahansan, S. N.; Han C. C.; Kim, C. Y.; Yu, H. J.: Polym. Sci., Polym. Phys. Ed. 1980, 18, 863.

14. King, J. S.; Boyer, W.; Wignall, G. D.; Ullman, R. Macromolecules 1985, 18, 709.

15. Szydlowski, J.; Van Hook, W. A. Macromolecules 1991, 24, 4883. 16. Muthukumar, M . J. Chem. Phys. 1986, 85, 4722. 17. Raos, G.; Allegra, G. J. Chem. Phys. 1996, 104, 1626. 18. Fujita, H. Polymer Solutions; Elsevier: Amsterdam, Oxford, NY, Tokyo;

1990, p. 192. 19. Des Cloizeaux, J.; Jannink, G. Polymers in Solution. Their Modeling and

Structure; Claredon Press: Oxford; 1990, page 815. 20. This conclusion is strongly supported by recent theoretical calculations

which demonstrated a close similarity between the scaling behavior of grafted polymer chains in supercritical and in liquid solvents [Meredith, J. C. and Johnston, K. P., Macromolecules 1998, 31, 5518].

21. Cotton, J. P.; Nierlich, M.; Boue, F.; Daoud, M. ; Farnoux, Β.; Jannink, G.; Duplessix, R.; Picot, C. J. Chem. Phys. 1976, 65, 1101.

22. Melnichenko, Y. B.; Kiran, E.; Wignall, G. D.; Heath, K.; Salaniwal, S.; Cochran, H. D.; Stamm, M. , submitted to Phys.Rev.Lett.

23. Luna-Barcenas, G.; Meredith, J. C.; Sanchez, I. C.; Johnson, K. P.; Gromov, D. G.; de Pablo, J. J., J. Chem. Phys. 1997, 107, 10782; Gromov, D. G.; de Pablo, J. J.; Luna-Barcenas, G.; Sanchez, I. C.; Johnson, K. P., J. Chem. Phys. 1998, 108, 4647.

Chapter 21

Destruction of Short-Range Order in Polycarbonate-Ionomer Blends

Ryan Tucker1, Barbara Gabrýs2, Wojciech Zajac3, Ken Andersen4, M. S. Kalhoro2, and R. A. Weiss1

1Department of Chemical Engineering, University of Connecticut, 91 North Eagleville Road, U-136, Storrs, CT 06269

2Physics Department, Brunei University of West London, 4, Toynbee Close, North Hinksey, Oxford 0X2 9HW, United Kingdom

3The Henryk Hiewodniczanski Institute of Nuclear Physics, Radzikowskiego 152, 31-342 Krakow, Poland institute Laue-Langevin, Grenoble, France

Spin polarized neutron scattering was used to study the short-range order in partially miscible blends of bisphenol A polycarbonate (PC) and lightly sulfonated polystyrene ionomers (SPS). The blends exhibited upper critical solution temperature phase behavior. In the two-phase region of the polymer blend, the short-range order for PC persisted. However, in the miscible, one-phase region, there was a significant reduction of the short-range order of the PC as a result of intimate mixing of the polymers.

Order in amorphous polymers has been debated for over 25 years. One school of thought advocates that local order in polymers is not possible if the chains assume a random Gaussian coil conformation; the other argues that a short-range is present even in melts (/). Both camps give interpretation of experiments as the ultimate proof, and in this respect, the scattering techniques are considered to provide the most direct evidence for either case. The random coil-conformation of high molecular weight bulk polymers has been demonstrated by small angle neutron" scattering (SANS) experiments (2, 3, 4).

Notwithstanding the conclusions from SANS experiments, evidence for local order in amorphous polymers has also been advanced. For example, Geil, Yeh, and their coworkers (5, 6, 7) reported observations of granular or nodular structures with sizes

328 © 2000 American Chemical Society

329

on the order of 10-100 Â on surfaces of PC, polyethylene terephthalate (PET) and polystyrene glasses (7). Frank et. al. (8) observed 50-100 Â granular structures in PC. The evidence for short-range order from electron microscopy studies has been questioned, because those studies dealt with thin films or surfaces, and may not be representative of the bulk polymer.

Harget and Siegmann (9), however, used small angle x-ray scattering (SAXS) measurements, which do sample the bulk polymer, to identify nodular structures in amorphous PET that were similar to the local structures seen in microscopy studies. Gabrys et. al. reported the existence of short-range order in amoprphous poly (methyl methacrylate) using wide angle neutron (WANS) with spin polarization analysis (10). Lin and Kramer (//) performed SAXS experiments on amorphous PC, and they reported large-scale electron density fluctuations that were also consistent with the nodular structures observed by electron microscopy. Amorphous PC has also been investigated by several groups using wide angle x-ray and neutron scattering (12, 13, 14, 15, 16). Cervinka et. al. (13) used WANS to study hydrogenous and deuterated derivatives of PC, and they found reasonable agreement between their experimental data and an "amorphous-cell" model developed by Suter (17), in which the polymer chains take on a trans-trans conformation and lay parallel to one another. Lamers et. al. (12) used WANS with spin polarization analysis to characterize short-range order in bulk PC and three chemically modified polycarbonates. Their results agreed with the model of Cervinka et. al.; they found that the distance between neighboring chains was 4.95 Â, and the correlation length (i.e., the distance over which the short-range order exists) was 28 Â. Recently, Eilhard et. al. (16) combined experimental and theoretical approaches in order to determine structural properties of glassy PC. The neutron scattering experiments with spin polarization yielded a purely coherent part of the scattering, ideally suited for comparison with structure simulation models using newly developed mapping procedures. This approach gave the best agreement between simulated and measured data of all the attempts to determine the structure of PC.

PC is often blended with other polymers (18). Weiss and coworkers (19, 20) reported that lightly sulfonated polystyrene ionomers (SPS) were miscible with PC as a result of the "copolymer effect" (21, 22, 23). That is, strong repulsive interactions between the ionic and nonionic segments within the ionomer favor mixing of the ionomer with PC even though there are no strong, intermolecular, attractive interactions. These blends exhibit upper critical solution temperature (UCST) phase behavior. Typical critical temperatures range from 170°C to 260°C, depending on the sulfonation level of the ionomer and the molecular weight of the polymers. The system also exhibits a miscibility window with respect to the degree of sulfonation, so that at a given composition and temperature, there is a finite range of sulfonation levels that produce miscibility. Changing the counterion used in the ionomer also affects the UCST and the miscibility window (79).

330

The objective of the work reported herein was to assess the effect of the addition of SPS on the short-range order in PC. Short-range order was determined using wide angle neutron scattering with spin polarization analysis.

Experimental Details

Materials

SPS was prepared by solution sulfonation of polystyrene with acetyl sulfate following the procedure of Makowski et. al. (24). This method substitutes a sulfonic acid group at the para-position of the phenyl ring, randomly along the chain. The weight-average molecular weight of the starting polystyrene as determined by gel permeation chromatography (GPC) was Mw= 280,000 and the polydispersity was 2.8. The sulfonation level was determined by titration of the sulfonic acid derivative, HSPS, in a mixed solvent of toluene/methanol (90/10 v/v) with methanolic sodium hydroxide. SPS with lithium and sodium counterions were prepared by neutralizing a solution of HSPS with the corresponding metal hydroxide. The nomenclature used for the ionomers was xj/MSPS, where x.y, and M denote the sulfonation level in mol % of styrene substituted and the counterion, respectively. Bisphenol A polycarbonate was obtained from General Electric Co. and had a Mn= 48,000.

Blend Preparation

Blends were prepared by adding a 3% (w/v) PC solution in tetrahydrofuran (THF) dropwise to a stirred 4% solution of MSPS in THF. The blend solution was then cast into Teflon dishes at 60°C and dried under vacuum.

WANS with Spin Polarization Analysis

Neutron scattering experiments were performed on the D7 instrument at the high flux reactor of the Institute Laue-Langevin (Grenoble, France). In hydrogen-rich samples, strong incoherent scattering is a dominant effect, therefore, the incoherent scattering contribution must be treated and accounted for in some fashion. Typically, incoherent scattering is treated by means of arbitrary "background corrections" which can be unreliable. Spin polarization analysis allows one to separate coherent and incoherent scattering contributions, which provides a means of obtaining a reliable intensity calibration. Thus, the measured structure factors can be directly compared with model calculations without arbitrary adjustments (25). To measure the coherent and incoherent scattering intensity, scattering from non-flipped and spin-flipped

331

neutrons were detected and measured. The non-flip (NF) scattering intensity refers to scattered neutrons that maintain their spin direction. Spin-flip (SF) scattering refers to scattered neutrons that change their magnetic spin. The relation between non-flip and spin-flip intensities determine the coherent and incoherent scattering contributions (//):

X NF = 1 c o h + -Mn

% = | I i n c

(1)

(2)

The coherent structure factor S(q), in absolute units, can be derived from equations. 2 and 3;

S(q) = dcj

άΩ

^NF 2 ^SF

I (3)

SF

where (da/dO)inc is calculated from the chemical composition of the sample. The scattering wavevector range of interest was 0.5 < q (Â*1) < 2.5, with q = (47i/X)sin(0/2). The sample holder was equipped with a furnace to control temperature.

Results and Discussion

It is now widely accepted that the presence of the short-range order in an amorphous polymer produces a peak in the WANS profile, such as peaks appearing at q = 0.6 Â"1

and q = 1.26 Â"1 shown in Figure 1. The first peak is interpreted as due to the correlation between the carbonate groups, with added influence of the isopropylidene groups (16), The more prominent, amorphous halo peak at q = 1.26 Â"1 is due to the correlation between the adjacent chains (12, 13, 16). The spacing between chains (D) and the correlation length (ξ) can be calculated from the peak position (qp) and the width (Aq) of a Gaussian fit to the scattering data using the following two equations.

D = 2%

q P

Y 4% Aq

(4)

(5)

332

For the second peak, analysis of Figure 1 yields for PC, D = 4.99 Â and ξ = 35.25 Â, which are in excellent agreement with the values of D = 4.95 Â and ξ = 28 À reported by Lamers et. al. (12) and those reported by Eilhard et. al. (16): D = 4.93 Â and ξ = 27.6 Â. The simulated values by Eilhard are D = 5.19 Â and ξ = 22.43 Â. (16). The scattering profile of 9%LiSPS is also included in Figure 1. In contrast to the PC data, the 9%LiSPS did not produce a sharp, intense peak, which indicates that the ionomer had little short-range order (27).

20-I A • \ — P C

/ \ 9%LISPS

/ · ·

—,—,—,—,—,—,—,—,—,—,— 0.5 1.0 1.5 2.0 2.5

q(AM)

Figure 1. WANS profiles of PC and 9%LiSPS.

Blends of polystyrene (PS) and PC are immiscible at all temperatures. Figures 2 and 3 show the WANS from PS/PC blends of several compositions. The weak peak at % = 0.6 Â*1 disappeared, but the peak at % = 1.26 Â"1 persisted and increased in intensity with increasing PC blend content, indicating that the short-range order of the PC was unaffected by the addition of the immiscible PS.

The 9%LiSPS ionomer forms a UCST with PC with a critical temperature of ~170°C (19). WANS curves of various blend compositions of 9%LiSPS and PC at 25°C, which is in the two-phase region of the phase diagram, are shown in Figures 4 and 5. The scattering from the two-phase 9%LiSPS/PC blends was similar to that for the immiscible PS/PC system. The characteristic, short-range order peak of PC was present at each blend composition, and the intensity of the peak scaled linearly with the PC content (Figure 6). Those results support the conclusion that the short-range order in PC is unaffected in a two-phase blend. WANS of blends of 5.1%NaSPS, 2%ZnSPS, and 8.45%HSPS with PC in the two-phase region were also measured, and for all compositions the peak characteristic of the short-range order in PC was present and depended linearly on the blend composition.

333

1 61

H 1 • 1 • 1 • 1 ' 1 0.0 0.5 1.0 1.5 2.0 2.5

q (AM)

Figure 2. WANS of PS/PC blends: A, 75/25; B, 50/50.

334

CO

35-^

i

30-

25-

20-

15-

10-

5-

0 -

/ 1

o.o 1.0 1.5

q (AM)

Figure 3. WANS of (25/75) PS/PC blend.

18-

16-

14-

12 A 10-

CO 8 -

6 -

4 -

2 -

1.0 1.5

q(AA-1)

Figure 4. Scattering of9%LiSPS/PC blend in two-phase region, 75/25 composition.

335

- S 1 5 J

!\ I \

ι

1.0 1.5

q f A M ) 2.0 2.5

B

io-|

5 -

0 -0.0

w /i

0.5 1.0 1.5 2.0 2.5

q (AA-1)

Figure 5. Scattering of 9%LiSPS/PC blends in two-phase region: A, 50/50; B, 25/75.

336

& 25

c

2 0 3 0 4 0 5 0 6 0 70 8 0 9 0 100

Weight Percent (%)

Figure 6. Peak intensity vs. PC weight percent.

The 9%LiSPS ionomer was miscible with PC above 170°C. Samples with different compositions were heated to 210°C, which is well within the one-phase region, and held at that temperature for 40 min. before the scattering was measured in order to allow sufficient time for the polymers to mix. Figure 7 compares the WANS data for a (50/50) 9%LiSPS/PC blend in the two-phase and one-phase regions. A clear and significant change in the scattering profile occurred between two-phase blend and the

30-,

H , , , , , , , , . r-0.0 0.5 1.0 1.5 2.0 2.5

q (AA-1)

Figure 7. (50/50) 9%LiSPS/PC blend. Scattering in one- and two-phase regions.

337

one-phase blend. The intensity of the PC short-ranger order peak at % = 1.27 Â'1

decreased by a factor of almost 2 and the peak increased in width. Figure 8 shows similar results for 75/25 and 25/75 blends, and in each case the short-range order peak decreased in intensity and broadened when the blend was moved to the miscible region of the phase diagram. Note that for the (75/25) 9%LiSPS/PC blend, the reduction of the peak intensity upon moving from the two-phase to the one-phase region is small. This small reduction is probably just a consequence of the low concentration of PC, but in may also be that the high concentration of the LiSPS disrupts the ability of the PC to form domains with short-range order. The results in Figures 7 and 8 indicate a decrease in the short-range order of the PC in the one-phase region of the blends.

\ , , . , . , • , r-0.0 0.5 1.0 1.5 2.0 2.5

q (AM)

Figure 8. 9%LiSPS/PC scattering, 1- and 2-phase region: A, 25/75; B, 75/25.

338

The width of the WANS peak for the blends provides a measure of the level of mixing between the ionomer and PC, and more importantly, the level or "amount" of order in PC. The correlation length, ξ, is the measure of the length scale over which the short-range order exists in the blend. Table 1 shows the correlation length values for the 9%LiSPS/PC system in both the two-phase and one-phase region. For each composition, the correlation length decreases from the two- to the one-phase region, indicating that the length scale over which the PC short-range order exists is reduced when the polymer components are miscible. The 50/50 and 25/75 blends, which have the sharpest short-range order peak in the two-phase region, have the largest correlation length values, as well as the largest decrease in correlation length when the blends are taken into the miscible region. This significant change in correlation length for the 50/50 and 25/75 9%LiSPS/PC blends shows the extent of mixing that is occurring between the polymer components.

Table 1 : Correlation lengths for the packing of PC neighbored chains from experiment.

9%LiSPS/PC, ξ (Â)

composition 75/25 50/50 25/75

2-phase region 20.79 49.61 55.11

1-phase region 16.86 11.48 7.62

To ensure that the results described above were indeed due to changes in the short-range order of the PC and not simply temperature effects, e.g., thermal density fluctuations, scattering data were obtained on an immiscible blend, a (25/75) 5.1%NaSPS/PC blend, at 25°C and 200°C (see Figure 9). In this case, the change in the morphology of the blend upon heating is minor compared with that when a phase change takes place The short-range order peaks for the two temperatures in Figure 9 were nearly identical, which indicates that the phase changes and not temperature were responsible for the changes in the short-range order peak in Figures 7 and 8. That is, the reduction of short-range order of the PC occurred when the ionomer and the PC were miscible. Based upon the picture for the short-range order perfected by Eilhard et. al. (16), i.e., PC chain segments aligned in parallel trans-trans conformations, the ionomer appears to disrupt the parallel packing, which is consistent with intimate mixing of the two polymers.

339

35 η

— , , , , , , , , , , 1— 0.0 0.5 1.0 1.5 2.0 2.5

q (AM)

Figure 9. 25/75 5.1 %NaSPS/PC at temperatures of 25 °C and 200 °C.

Conclusions

Amorphous PC exhibits short-range order, which is evident from a WANS peak at % = 1.27 Â*1. In two-phase blends of PC with either polystyrene or sulfonated polystyrene ionomers, the short-range order is retained by the PC. However, in the one-phase region, the PC order is diminished, as evident from a reduction in the WANS peak intensity and increase in the peak broadness. The reduction of the PC short-range order in the miscible blends is believed to result from the intimate mixing of the two polymers, which disrupts the parallel packing of PC chain segments with trans-trans conformations.

Acknowledgment

This work was supported by a grant from the Polymers Program of the National Science Foundation (DMR 97-12194).

Literature Cited

1. Gabrýs, B; TRIP. 1994, 1, 2. 2. Kirste, R.G; Kruse, W.A.; Schelten, J; Makromol. Chem. 1972, 162, 299. 3. Benoit, H; Nature. 1973, 245, 13. 4. Wignall, G.D; Schelten, J; Ballard, D.G.H.; Eur. Polym. J. 1973, 9, 965.

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5. Yeh, G.S.Y.; Geil, P.H; J. Macromol. Sci. 1967, 1, 235. 6. Carr, S.H.; Geil, P.H.; Baer, E; J. Macromol. Sci. 1968, 2, 13. 7. Siegmann, A; Geil, P.H.; J. Macromol. Sci. 1970, 4, 239. 8. Frank, W; Goddar, H; Stuart, H.A.; J. Polym. Sci., Part B: Polym. Lett. 1967, 5,

711. 9. Harget, P.J.; Siegmann, A; J. Appl. Phys. 1972, 43, 4357. 10. Gabrýs, B; Higgins, J.S.; Schärpf, O; J. Chem. Soc. Far. Trans. I. 1986, 82,

1929. 11. Lin, W; Kramer, E.J.; J. Appl. Phys. 1973, 44, 4288. 12. Lamers, C; Schärpf, Ο; Schweika, W; Batoulis, J; Sommer, Κ; Richter, D;

Physica B. 1992, 180, 515. 13. Červinka, L; Fischer, E.W.; Hahn, K; Jiang, B.Z.; Hellman, G.P.; Kuhn, K.J.;

Polymer. 1987, 28, 1287. 14. Mitchell, G.R.; Windle, A.H.; Colloid Polym. Sci. 1985, 263, 280. 15. Schubach, H.R.; Heise, B; Colloid Polym. Sci. 1986, 264, 335. 16. Eilhard, J; Zirkel, A; Tschoep, W; Hahn, O; Kremer, K; Schärpf, Ο; Richter, D;

Buchenau, U; J. Chem. Phys. 1999, 110, 1819. 17. Suter, U.W.; Theodoru, D.N.; Macromolecules. 1985, 18, 1467. 18. Lu, X ; Weiss, R.A.; Proc. Annu. Tech. Conf., Soc. Plast. Eng. 1993, 684 - 686. 19. Lu, X ; Weiss, R.A.; Macromolecules. 1996, 29, 1216. 20. Xie, R; Weiss, R.A.; Polymer. 1998, 39, 2851. 21. Kambour, R.P.; Bendler, J.T.; Macromolecules. 1983, 16, 753. 22. Paul, D.R.; Barlow, J.W.; Polymer. 1984, 25, 487. 23. ten Bricke, G; Karasz, F.E.; Macknight, W.J.; Macromolecules. 1983, 16, 1827. 24. Makowski, H.S.; Lundberg, R.D.; Singhal, G.H.; U.S. Patent 3,870,841, 1975. 25. Schärpf, Ο; Gabrýs, B; Peiffer, D.G.; ILL Report 90SC26T (PTA), Institute

Laue-Langevin. 1990. 26. Gabrys, B, Schärpf, Ο; Peiffer, D. G.; J.Polym.Sci.:Part B: Polymer Physics,

1993, 31, 1891.

Chapter 22

Scattering from Magnetically Oriented Microtubule Biopolymers

Wim Bras1, Gregory P. Diakuir2, Richard C. Denny2, Anthony Gleeson2, Claudio Ferrero3, Yehudi Κ. Levine4, and J. Fernando Díaz5

1Netherlands Organisation for Scientific Research (NWO), DUBBLE CRG/ESRF, BP 220 F38043 Grenoble Cedex, France

2CCLRC Daresbury Laboratory, Warrington WA4 4AD, United Kingdom 3European Synchrotron Radiation Facility, BP 220 F38043 Grenoble

Cedex, France 4Utrecht University, Debye Institute, Section for Computational Biophysics,

P.O. Box 80000, 3508 TA Utrecht, Netherlands 5 KU Leuven, Laboratory for Chemical and Biological Dynamics,

Celestijnenlaan, 200 D, Leuven, Belgium

Microtubules are biopolymers with a large number of functions in all eukaryotic cells. The study of their physical behaviour is complex due to their dynamic self-assembly and disassembly. Attempts to orient the hydrated, and therefore biological viable form, by conventional methods like shearing are hampered by their fragility and their extreme length which allows them only to be oriented in small domains inside a larger disoriented surrounding. For experiments like X-ray fibre diffraction this is not satisfactory. We have developed a method to align these long molecules by assembling them in a controlled way inside a strong magnetic field (7-10 Tesla). This method can possibly be applied to other molecules which resist other methods of alignment but which have sufficient sources of diamagnetism built into their structure. Sources of diamagnetism are for instance regular arrays of peptide bonds (like α-helices and β-sheets), aromatic rings or double and triple C-bonds. Successful fibre diffraction experiments to 18 Å resolution have been obtained with this method. The rather gentle forces applied to the magnetic field and the slow disorientation makes this method extremely suitable for experiments where one wants to study the interaction between molecules and reduce the number of degrees of orientational freedom. This method can also be applied to other bio- or synthetic polymers in solution.

© 2000 American Chemical Society 341

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Introduction

Microtubules play several important roles in the cell. They are involved in intra­cellular transport, motility, mitosis and form an important part of the cytoskeieton [1]. They consist of long chains of tubulin protein dimers, called protofilaments, which connect laterally to form a hollow cylindrical structure with an external diameter of approximately 30 nm and a length which can extend to micrometers. The number of protofilaments can vary between 11 and 17 but the pre-dominant number in a normal functioning cell is 13. The persistence length of these polymers is reported to be between 2000 - 5200 μπι [2,3] so that for most experiments they can be considered to be a rigid rod system. The length distribution is influenced by the biochemical conditions and fluctuates due to dynamic assembly/disassembly properties. With the biochemical conditions in these experiments the average length is approximately 5 μηι [4] giving an average molecular weight of 109 Dalton. Amino acid sequencing shows that the homology in the tubulin protein between organisms as wildly varying as yeast to pigs and humans is at least 75% [5]. The structure of the basic building block, the tubulin dimer, has recently been solved [6] but the structure of the assembled molecule is still not known in detail. Structural properties are important parameters for the study of the (self)assembly mechanism and the interactions with other ligands like for instance anti-cancer drugs.

We have reported previously on Small Angle X-ray Scattering experiments (SAXS) aimed at the elucidation of the structure. The preparation of hydrated, and therefore biologically interesting, samples suitable for small angle X-ray fibre diffraction is not trivial. The only reasonable successful method has been centrifugation over extended lengths of time (>24 hours) and subsequent rehydration [7]. With this method one is not guaranteed that the brute force approach has not changed the structure nor that the rehydration is complete. With other methods like Couette and flow shearing it is possible to create small oriented domains in an otherwise isotropic sample. These domains, however, do not retain their orientation very long and even small temperature gradients are sufficient to distort the system [8,9]. An additional problem is that the molecules only scatter weakly so that, even using a synchrotron radiation source, many hours of data collection are needed in order to obtain a satisfactory statistical data quality.

Diamagnetism can be found in many molecules. The principle sources are aromatic groups, double and triple C-bonds and peptide bonds. In α-helices and β-sheets all the amino acids are in a single plane and can be added vectorially thus creating a possible source of relatively strong diamagnetism. We have investigated the possibility to orient microtubules making use of the interaction between their small diamagnetic moments and high magnetic field. For some polymers this interaction is sufficient to create fully aligned samples, notably for rigid polymers with a low molecular weight containing aromatic groups which are rigidly bound to the backbone. However, this is exception and in general this method is not usable due to a combination of too low a diamagnetic moment and steric hindering which prevents the alignment. For self assembling (bio)polymers like actin and fibrinogen it has been shown that the assembly inside the magnetic field will result in the production of nearly fully aligned samples [10, 11]. This method can also be applied to synthetic polymers in which case it becomes possible to study the aligned samples in solution

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but which can also be thought of as a possible method to create highly aligned solid samples by first assembling the molecules and subsequently evaporating the solvent. With the relatively low prices for modern strong superconducting magnets this method, pioneered several decades ago, might nowadays be a feasible method for creating highly aligned speciality polymers, like for instance electroluminescent ones, which are difficult to align in solution, on an industrial scale.

On theoretical grounds and using the average molecular weight value and assuming that the main contribution to the diamagnetic moment is due to α-helices with an orientation parallel to the tubulin dimer axis, which make up roughly 25% of the total amount of amount of amino acids [6], and thus to the microtubule axis, the diamagnetic moment of the microtubule can be calculated to be approximately to be ΑχΜΤ » / .07 x I0~28 m3 for which it can be calculated that a minimum magnetic field strength of « 6 Τ is required to obtain some degree of alignment [12]. Experimentally it has been found that in fields of approximately 11 Τ the maximum degree of alignment is obtained [12].

From real time magnetic birefringence experiments it was found that the reaction rate is an important parameter for the achievement of full alignment [12]. In these experiments the sample is placed in a strong magnetic field and the birefringence, Δη, is measured either as function of the variable magnetic field strength or as function of other sample parameters, like for instance temperature, whilst the magnetic field is kept constant. For low concentration samples, in a limited orientation range, the measured birefringence has a linear relation with the degree of orientation of the molecules. Contrary to intuition, which predicts that the best alignment will be obtained when the reaction rates are slow so that the molecules have time to align in the field we have found that fast reactions are the best method, see figure 1. This can be understood when one looks at the number of nucleation sites for elongation in the solution. When this number is high, which is generally the case for fast reactions, the overall polymer length will be relatively short. However, these polymers can be sufficiently large to overcome the potential barrier for alignment and be the aligned 'seeds' from which further polymerisation can take place. With less nucleation sites and thus longer molecules the elongated polymers will be locked in their orientation due to steric hindering effects. For polymers with a long persistence length it is even possible that the field can be switched off and that they still retain this orientation even in solutions or in some cases that the orientation even increases with time. This has been previously observed with other self assembling macromolecules [10, 13]. This property has enabled us to perform fibre diffraction studies on these molecules by aligning them off-line and then transporting them to a Synchrotron Radiation beamline, see figure 2.

For rigid rod molecules with an axial ration of 50 - 200 without any constraint applied to them, Flory predicted that the angular distribution of the long axis would be at most 10.2° - 11° [14, 15]. In the case of microtubules this prediction has been confirmed by experiments in which it was shown that the maximum obtainable degree of alignment of microtubules aligned in a high magnetic field but then taken out of the field so that the constraint was removed, was roughly 10° [12]. The analysis of the fibre diffraction pattern is therefore rather difficult since the degree of alignment is such that at higher q-values the reflections belonging to the layer lines can start to overlap with the diffraction arcs from the equator or other layer lines. For the low resolution data this proved to be not too much of a problem. In figure 3 the

344

Δη (xl (T 5 )

4.5

3.5

2.5

1.5

0.5

-0.5

Γ " ' Η ι ι ι ι ι ι ' ι ι I ' • ' » I 1 1 I 1 1 1 1 I 1 1 1 1 .

; / / — ο — f a s t ;

• / / Δ S l o w J

Ί I Ι I I I I I I I I I I . L t , . . . ι . . . . ι . . . . ι . . .

100 200 300 400 500

Time (seconds) 600 700

Figure 1 Magnetic birefringence experiments on microtubules assembled inside a 12 Τ magnetic field. The birefringence, An, is a measure of the degree of alignment. Initially the polymers are unassembled and give no birefringence. Once the assembly starts the signal increases. Fast assembly rates give the highest birefringence and orientation. Figure taken from [13].

direct beam

Figure 2 Three dimensional representation of a low angle X-ray fibre diffraction pattern of magnetically oriented microtubules in which the X-ray beam was at right angles with the polymer long axis. The pattern can be analysed in helical diffraction terms. The orders of the Bessel functions assigned to each peak are indicated. The peaks indicated with Jo>n

a r e due to the cylindrical structure, the peak indicated by JJS is predominantly due the modulations of the outside cylinder wall. Total data collection time was longer than 2 hours.

345

0.20

0.15

Vl(s)

0.10

0.05

0.05

1 Vl(s)

0.25

Figure 3 Equatorial and first layer line intensities of the microtubule fibre diffraction pattern shown in figure 2. The vertical intensity axis is in arbitrary intensity units. The layer line intensity is very weak compared to the equatorial intensity which makes the mathematical correction for the spread in the angular orientation vety difficult. For this reason an alternative method of data collection and interpretation is use as described in the text.

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diffracted intensity on the equator and the first layer line is shown. To obtain these curves it was necessary to correct for the disorientation and several geometrical parameters. For this the CCP13 suite of software was used [12, 16, 17]. Due to the longitudinal staggering of the different protofilaments the surface of microtubules have shallow helical grooves running over them so that the diffraction pattern has to be explained in terms of helical diffraction theory [18, 19].

Algorithms, used for the mathematical correction of the disorientation effects on the diffraction peaks in general requires a high signal to noise ratio in order to render reliable results. In order to increase the accuracy of the determination of the real diffraction intensities and to avoid software induced systematic errors for the higher order diffraction peaks, where the signal to noise ratio is low, a different method has been tried. In this case the microtubules were aligned with their axis parallel to the X-ray beam. This means that one only observes the scattering pattern from the equatorial plane and thus avoid the overlap between the different reflections. An additional advantage is that it is also possible in this case to perform a radial integration over the full area of the two dimensional detector, thus increasing the statistics considerably. This allows one to study the low angle range where the interparticle scattering contributes considerably to the scattering intensity. Biologically this is interesting since the interdistance between microtubules in part determines their function in the cell.

Materials and Methods

The protein tubulin was purified from pig brains and prepared for experiments as described elsewhere [20]. The protein concentration was determined spectrophotometrically. For assembly the tubulin was equilibrated with 10 mM sodium phosphate, 3.4 M glycerol, 1 mM EDTA, 0.1 mM GTP buffer (pH 7.0) after which 1 mM GTP and the desired concentration of MgCl2 were added. The sample were centrifuged for 10 minutes at 50,000 rpm at 4° C to eliminate aggregates. Assembly could be initiated by raising the temperature from 4° C to 37° C. The samples were assembled and aligned in a 9 Τ magnetic field before being transported to the SAXS beamline.

For these experiments the Small Angle X-ray Scattering beamline 2.1 of the SRS at the CLRC Daresbury laboratory was used. This beamline has been described in detail [20] and only a short description is given. For these experiments a gas filled proportional counter with an active area of 200 χ 200 mm2 was used. This photon counting device has been shown to be the most useful detector for this type of work since the low intrinsic background and the absence of read-out noise allow the proper subtraction of the scattering background due to the buffer solution, air scatter and contributions from the cell windows. The electron density difference of the microtubules with respect to the buffer solution is rather small and therefore the proper subtraction is of crucial importance especially in the high q-range. The sample to detector distance was chosen to be 2 and 9 meters covering a scattering vector range from 2 χ 10"2 < q < 1.8 χ 10"1 A " 1 and 4 χ 10"3 < q < 5 χ 10" 2 Â " 1

respectively, where the former experiment was performed to see what the effect of interparticle scattering was.

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Radiation damage was prevented by using elongated cells which were slowly translated through the beam. Optical inspection showed that the damaged areas remain localised and that no transport of material through the cell took place on the time scales of the experiments.

Results

As mentioned in the introduction the accurate determination of the scattered intensity in a sample with a low degree of alignment is a non-trivial exercise especially when this intensity is small with respect to the intrinsic background, due to the buffer solution and unassembled material. For equatorial data there is the possibility to determine these intensities from samples aligned with their long axis parallel to the X-ray beam so that we only probe the equatorial plane which is a projection of the molecule on its basal plane. This will now be a circular symmetric pattern. See Figure 4.

Figure 5 shows the low angle scattering pattern from three different concentrations of microtubules in glycerol assembly buffer. The long microtubule axis was coincident with the X-ray beam so that we are only observing the intensity from the equatorial plane of the molecule. The insert shows the very low angle data, where the effects of interparticle scattering are visible. For this concentration range the (theoretical) interdistance has been calculated to be varying from 1800 - 2600 A. It was calculated that the system is in the dilute regime.

In the very low q-region, where effects due to interparticle scatter will be most pronounced, we indeed find a concentration dependent effect in as such that the scattered intensity, corrected for the different protein concentration, is stronger for the lower concentrations. To gain a better insight some simulations were performed.

The interference effects arising from the packing of the microtubules in a sample were estimated by considering the sample to be a two-dimensional disordered fluid. The microtubules were represented as hard, hollow discs with an inner radius of 86 Â outer radius of 146 À. These discs were placed at random in a square box of side 10 000 Â. The average distance between the centres of the discs was varied by changing the number of discs in the box. The packing of Ν discs into the box was carried out under the constraint that the distance r n m between the centres of any two discs η and m in the box was r n m > 286Â, in order to avoid unphysical overlap. The initial configuration of the discs in the box was changed by allowing each disc to undergo a random displacement with maximum amplitude of 16 Â. The disc to be moved was chosen at random and its displacement was accepted provided it did not overlap any other discs in the box. A new configuration was produced after attempting a displacement of every disc in the box. The simulation proceeded by generating up to 16 000 configurations and evaluating the diffracted intensity I I using the positions of the centres of the discs for every fifth configuration. iLwas calculated using the expression j (q) = I (q)^exp[-iq r] w n e r e *D l s m e scattered intensity from

n,m single hollow discs and q is the wave vector. In practice, only I I (qx) was calculated, as the scattered intensity was found to be symmetric in the {qx,qy} plane as expected. The average distance between the centres of the discs in the box was calculated simultaneously. We evaluated the scattered intensities from configurations containing

348

a

ι . ι 1 0 0.025 0.050

b

ι ι . 1 0 0.025 0 050

Figure 4 Small angle scattering patterns from microtubules aligned with the long axis at right angles to the X-ray beam (panel a) and with the long axis parallel with the X-ray beam (panel b). The scale is in q=2n/d (Â'1).

349

ΙΟ2

Ο 0.01 0.02 0.03 0.04 0.05

q=27i/d (Â"1)

Figure 5 The low angle scattering pattern from microtubules aligned with their long axis parallel to the incident beam for 3 different concentrations. The insert is a magnification of the lowest q-range (with a linear abscissa) showing the effects of interparticle scattering. This effect is much stronger than found in simulated data. This possibly can be due to a non-random distribution of polymers in the samples through clustering in bundles. This will give rise to a locally higher concentration.

350

between 50 and 700 discs, covering an average nearest centre-to-centre distances between 1000 Â and 330 Â. The computational time ranged from 30 minutes at the dilute end to 24 hours for a system of 700 discs. The number of configurations used was chosen judiciously so as to yield statistical fluctuations of less than 3% in the diffracted intensity. The differences in the simulated intensities between different concentration are approximately a factor of 5 less than the experimental values shown in the insert of figure 5, although the tendency for higher central scatter with lower protein concentration remains. This can indicate that the average distance distribution is deviating from that of a complete random distribution and that actually bundles of microtubule polymers are formed. In such a scenario the interparticle scatter will be dominated by the distance distribution inside such a bundle apparently giving a much stronger concentration effect.

The radius of gyration of the cross section, Rc, can be calculated from the Guinier approximation [21].

q

2R*

/{q) = /{o)e"~^~ For cylindrical scattering objects the Rc is related to the dimensions of the cylinder according to:

D2 ι n2 ι 2 j^2 outer ' "inner _j_

1 2 12 Experimentally it was determined that this value ranged between 167 - 160 Â. Since the length, A, is not in the observed scattering range so it can be assumed that this length will not contribute to the scattering and only the first term remains. If we now take the dimer dimension in the radial direction, i.e. the cylinder wall thickness, to be 65 Â [1] and thus substituting R'inner = Pouter ~ 65 Â, we can calculate that the external radius is between 170 - 180 A. This is larger than expected on the basis of electron microscopy and earlier scattering results. These results can be influenced by the large degree of length polydispersity. However, these values have been used as the starting parameters in the fitting procedure described below.

Helical diffraction theory [18] tells us that the scattering pattern on the equator of the fibre diffraction pattern can be explained as a combination of a Fourier transform of the basic hollow cylinder modulated by cylindrical Bessel functions of the order corresponding with the symmetry in the basal plane. This can be approached in two equivalent ways. The first one uses a Jo Bessel function to describe the cylinder and the convolutes this with an unknown function H(q) which represents the shape of the cylinder wall. Another approach is to use the Fourier transform (FT) of a basic hollow cylinder and then add the Bessel function describing these modulations on the bases of intelligent guesses. This has clear analytical advantages. The FT of a smooth walled hollow cylinder is given by [22]:

Ρ Ι Λ - Ρ Jk^Ro^er) inner) t\^)- Komr Kinner

q q The walls of this cylinder will be modulated by electron density grooves between neighbouring protofilaments, both on the inner and outer wall. Due to the 13 fold symmetry this can be expressed as J13 Bessel functions with arguments Jl3(qRinner+Ainner) and Jl3(qRouterA0uter). The magnitude of the Ajnner and Aouter reflects the depth of the grooves on the cylinder wall. From a preliminary

351

analysis it can be seen that the contribution due to the modulation of the inner wall falls outside the scattering range observed in this study but that the modulation due to the outer wall can add scattered intensity in the observed scattering range at q > 0.1 Â" 1. An initial fit to the experimental data with an expression based on equation (1) was therefore limited to the range q < 0.1 A" 1 where is certain that no other contributions will be present. As a starting parameter for the fitting procedure the values obtained via the calculations based on the R c were used. This resulted in a best fit to the data of a cylinder with an Router = 146+5 Â and an dinner = 86+5 Â, see figure 6. The subsequent addition and fit of a contribution of a J13 Bessel function relating to the outside wall over a data range that covers the region q > 0.1 À"1 shows that the grooves between the protofilaments are roughly 21 A deep. The subsequent addition of contributions of the grooves on the inside wall has not been performed yet since at this moment no reliable data obtained from samples aligned with their axis parallel to the X-ray beam is available yet for that data range. Once this data is available the next step will be to reconstruct the equatorial intensity trace and compare this with solution scattering and fibre diffraction patterns in order to be able to deconvolute the overlap between the higher q-range of the equator and the layer lines.

The first layer line can be found on a line with a meridional q-value of «0.157 Â"l. As mentioned in the introduction the degree of alignment is not extremely high and this means that in the equatorial q-range above 0.157 Â"1 it is possible that reflections will start to overlap. However, by using the method of first determining the accurate intensities at lower q on the equator and then fitting these with the appropriate analytical functions it is possible to deconvolute the intensities coming from layer lines. In fact it might be better, once the equatorial diffraction pattern is accurately determined, to revert back to scattering from randomly oriented samples for the deconvolution procedure. This method is being investigated at the moment.

Conclusions

Although not extensively discussed in this work magnetic birefringence has shown that magnetic fields only have to be applied for a limited time during the onset of polymerisation in order to create aligned samples in solution. On the basis of these results it can be predicted that this method of sample preparation might be applied to suitable synthetic polymers as well. This can provide a way to study the molecular transform without interference due to inter-molecular interactions. With a fast reaction the sample only has to remain in the field for several seconds and thus provides the possibility to develop this method into a polymer processing technique.

The application of helical diffraction theory in combination with experiments on suitably oriented molecules allows us to introduce a step wise fitting procedure with which we first can use knowledge of the basic cylindrical structure of the molecule, then add the Bessel function terms relating to the modulations on the cylindrical surface. The next step is to compare these data with solution scattering patterns in order to be able to deconvolute the equatorial and layer line intensities. Once this model is completed the positions of the dimers in the microtubule wall will be known and the scattering parameters obtained from the known dimer structure can

352

A ίο 1

ΙΟ*2

ΙΟ"3

ί ο 4 L — — · — · — · — « — : — · « . — . I . . — . —

Ο 0.05 0.1 0.15 0.2

q(Â"!)

Β

0 0.05 0.1 0.15 0.2 qCÂ"1)

Figure 6 Scattered intensity from microtubules aligned with their long axis parallel to the X-ray beam (panel a). This is equivalent to the equatorial data obtained from a fibre diffraction pattern but one is certain that neither systematic errors due to mathematical procedures used for correction of the angular spread of the molecules are seen nor intensity due to overlap from diffraction arcs of the layer lines. Panel b shows the best fits (χ2 = 0.05) to the data using the fitting procedure described in the text. The dotted line indicates the maximum extend of the experimental data at present. The line indicated with 'cylinder' is the contribution of two Jj functions describing the basic cylinder. The 'modulated' curve is the real fit taking into account the modulations on the cylinder sutface.

353

be used to refine the model with respect to fibre diffraction data. The last step to describe the system is to calculate the interaction between microtubules in solution. Modelling studies describing the interparticle interference effects are being carried out at the moment and indeed show qualitative agreement with the experimental data but give a strong indication that the microtubules are not randomly positioned in the samples but are forming larger bundles.

Hansgerd Kramer and Georg Maret are gratefully acknowledged for their help in the birefringence experiments. Liz Towns-Andrews and Sue Slawson of CCLRC Daresbury Laboratory have been very helpful with the fibre diffraction experiments. Eric v.d. Zee has assisted in some of the diffraction experiments.

Literature Cited

1. Structure of microtubules ed. K.Roberts and J.S Hyams Academic Press 1979

2. F.Gittes, B.Mickey, J.Nettleton, J.Howard J.of Cell Biol., 1993, 120(4), 923-934

3. P.Venier, A.C.Maggs, M.F.Carlier, D.Pantaloni J.Biol.Chem., 1994, 269(18), 13353-13360

4. F.Pirollet, D.Job, R.Margolis, J.Garel EMBO journal 6(11), 1987, 3247-3252

5. P.Valanzuela, M . Quiroga, J.Zaldivar, W.J.Rutter, M.W. Kirschner, D. W.Cleveland Nature 289,1981, 650-655

6. E.Nogales, S. Grayer-Wolf, I.A.Khan, R.F.Luduena, K.H.Downing Nature 375, 1995, 424-427

7. E. Mandelkow Methods in Enzymology 134, 1986, 149-168 8. J.Nordh, J.Deinum, B.Norden Eur.Biophys.J., 14, 1986, 113-122 9. R.E.Buxbaum, T.Dennerl, S.Weiss, S.R.Heidemann Science 235, 1987,

1511-1514 10. J.Torbet, M . Ronzière Biochem.J.219(1984) 1057-1059 11. J.Torbet Biochemistry 25, 1986, 5309-5314 12. W.Bras, G.P. Diakun, J.F.Díaz, G. Maret, H.Kramer, J.Bordas, F.J.

Medrano Biophysical Journal 74(3), 1998, 1509 13. W. Bras PhD thesis Liverpool John Moores University 1995 14. A.Suzuki, T.Maeda, T.Ito Biophys. J., 1991, 59, 25-30) 15. P.J. Flory Proc. Royal Soc. London A234, 1956, 73-89 16. http://www.dl.ac.uk/SRS/CCP13/main.html 17. R.C.Denny Private communication 18. A.Klug, F.Crick, H.Wyckoff Acta.Cryst., 11, 1958, 199-213 19. L.A. Amos , A.Klug, 1974, J.Cell.Sci. 14, 523-549 20. J.M. Andreu, J.Bordas, J.F. Díaz, J. Garcia de Ancos, R.Gil, F.J.Medrano,

E,Nogales, E.Pamtos, E.Towns-Andrews J.Mol. Biol. 226,1992, 169-184 21. Towns-Andrews E., A.Berry, J.Bordas, G.R.Mant, P.K.Murray, K.Roberts,

I.S.Sumner, J.S.Worgan, R.Lewis. Rev.Sci.Instr.60(7), 1989, 2346 - 2349 22. Ο. Glatter in Small Angle X-ray scattering ed. O.Glatter, O.Kratky Academic

Press 1982 23. Diffraction of X-rays by chain molecules B.K. Vainshtein Elsevier 1966

POLYMERS UNDER FLOW

Chapter 23

What Is a Model Liquid Crystalline Polymer Solution?: Solvent Effects on the Flow Behavior

of LCP Solutions S. Chidambaram1, P. D. Butler2, W. A. Hamilton2, and M. D. Dadmun1,3

1Chemistry Department, University of Tennessee, Knoxville, TN 37996 2Solid State Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831

The flow-induced alignment of liquid crystalline solutions of poly (benzyl L-glutamate) (PBLG) in deuterated benzyl alcohol (DBA) and deuterated m-cresol (DMC) is determined using small angle neutron scattering. Surprisingly, the similar solutions show marked differences in their steady state and relaxation response to shear. During shear, the two solutions behave similarly at high shear rates (> 1 s-1), however, at low shear rates the PBLG in DBA shows an increase in orientation with shear rate which is absent in DMC. Upon shear cessation, PBLG in DMC retains flow induced alignment for long times (> 6 hours) while the orientation of PBLG in DBA dissipates quickly (5-10min.). The results are particularly unexpected, as DBA and DMC are isotopes. Possible explanations for this anomalous behavior are discussed. The results exemplify the need for a more complete understanding of the important parameters that affect the flow of LCP solutions so that a more universal theory can be developed which can predict flow behavior of non-model LCP solutions.

Introduction

Liquid Crystalline Polymers (LCPs) are of considerable interest inasmuch as their inherent molecular ordering has dramatic consequences on their macroscopic properties. LCPs are utilized in many high performance applications due to their superior strength and stiffness, excellent solvent resistance, low coefficient of thermal expansion, and low viscosity. Since molecular alignment affects the macroscopic properties and is easily achieved during flow, the rheology and relation between an applied flow field and molecular orientation of LCPs are an area of great interest.2*16

Over twenty years ago, Kiss and Porter2 discovered the unique phenomenon of a

Corresponding author.

356 © 2000 American Chemical Society

357

negative fjrst normal stress difference (Ni) in LCPs. Other anomalies including low die swell, strong dej>£ndence of transient rheological behavior and structure on shear and thermal history, band structures on cessation of flow,6 and decrease in viscosities with increased shear rate at low shear rates (shear thinning)7 have also been observed. Subsequently, there has been considerable interest in understanding the flow of LCP in hopes of explaining the fundamental reasons for this unique behavior. Examples of experimental techniques that have examined the flow of liquid crystalline polymers include rheology,8 flow birefringence9 and scattering techniques (X-ray10, light1 and neutron12). In particular, it has been demonstrated that birefringence, x-ray scattering and small angle neutron scattering (SANS) can be used to accurately characterize the extent of molecular ordering of an LCP, though each has it's advantages and disadvantages.13 Models and theories have also been proposed which account for many of the novel aspects of the rheology of LCPs. 1 4" 1 6

However, a universal model that fully describes the flow characteristics of all LCPs is yet to be developed.

For example, a theoretical model developed by Marrucci and coworkers14 has been successful in explaining the anomalous sign changes in Ni and has been extended to three dimensions by Larson and coworkers.16 In addition to confirming the predictions of Marrucci, Larson's model is successful in understanding the changes in the second normal stress difference, N 2 . However, these theories do not capture all of the complexity of the flow of LCPs. One major deficiency is the inability to account for the existence of a shear thinning regime at very low shear rates, i.e. "Region I" behavior, which occurs in many LCP solutions. These theories predict a low shear rate Newtonian plateau.

PBLG/m-cresol is known as a LCP model system for the study of the flow behavior of LCP solutions, primarily due to it's unusual normal stress behavior. Indeed, Burghardt recently defmed a model LCP solution as one that exhibits behavior predicted by the Doi model, among other factors. Thus, most of the experimental evidence which is utilized to illustrate the flow behavior of LCP comes from the poly(y-benzyl L-glutamate) (PBLG)/m-cresol solution.2'8*913,17*19,22'23 Though there seems to be very good correlation between theory and PBLG/m-cresol solution behavior including the appearance of a negative first normal stress difference, the flow behavior of many LCP solutions does not mimic that of PBLG/m-cresol. Some important aspects that are found in many LCP solutions but not PBLG/m-cresol is a regime at low shear rate where the solution is shear thinning at moderate concentrations, the absence of a negative Ni , and the expedient relaxation of shear-induced alignment after the removal of shear. Moreover, current theory does not allow for this behavior. Therefore, in the drive to develop a broad theoretical description of the flow behavior of LCP which can account for all of the behavior that is observed experimentally, a more specific understanding of the fundamental reasons for the differences between the flow of PBLG/m-cresol and other LCP solutions is needed. If the reasons for this variety of flow behavior can be determined, they can then be accounted for theoretically, and this will lead to a more complete theoretical description of the flow of LCP solutions. Thus, by this argument, a model LCP solutions should be one that allows the systematic study of all the factors which influence the flow of LCP solutions.

Pursuant to this, small angle neutron scattering and rheology have been used to investigate the flow and alignment behavior of two very similar LCP solutions. The solutions are poly(y-benzyl L-glutamate) in deuterated m-cresol (DMC) and

358

deuterated benzyl alcohol (DBA). Both m-cresol and benzyl alcohol are helicogenic solvents in which PBLG forms a cholesteric phase above approximately 12-13 wt% as determined by optical microscopy (for the molecular weights utilized in these experiments; 250,000-320,000). These solutions were chosen as the solvents are isomers, and therefore, it is not expected that there will be significantly different polymer/solvent interactions between the two solutions, yet it is known that they exhibit different alignment behavior.12,13 It is hoped that the results of this systematic examination will provide an explanation for the difference in flow behavior that is observed in these two similar solutions. This, in turn, may provide guidance regarding the choice of important parameters which influence the flow of LCP and must be accounted for in a universal theory.

Experimental

PBLG was purchased from Sigma Chemical Company, stored in a freezer, and dried in vacuum prior to use. The deuterated benzyl alcohol (DBA) was bought from MSD Isotopes while deuterated m-cresol (DMC) was obtained from CDN Isotopes. A 20 wt% solution of PBLG (MW-288000) in DBA and an 18 wt% solution of PBLG (MW-318000) in DMC were used, well into the cholesteric phase in both solvents. The solutions were left to equilibrate for at least a fortnight, often longer, above the gel temperature before being used in the experiments.

The rheology experiments were performed on a Bohlin VOR (dynamic modulus and viscosity measurements) and a Rheometrics Dynamic Stress rheometer (viscosity measurements) using cone and plate fixtures with 2.5° cone. To insure that the flow transient had been surpassed, the sample was pre-sheared for 200 strain units prior to collecting steady viscosity data, or prior to flow cessation in those experiments where evolution of the moduli was monitored. The sample chamber was sealed to prevent solvent loss at long times. The dynamic modulus was obtained using a 1.3% strain at 1 Hz.

Al l SANS experiments were conducted at the High Flux Isotope Reactor (HFIR) at the Oak Ridge National Laboratory in Oak Ridge, TN. In-situ shear experiments were performed on samples contained in a specially designed couette cell with provision for sample heating. The inner component of the couette cell (stator) is temperature controlled. The cup that holds the sample (rotor) is not directly heated but was kept enclosed within a heating jacket to minimize thermal gradients across the sample. With this apparatus, the temperature of the sample was maintained constant throughout the experiment with a precision of ±1°C. The wavelength of neutrons used was 4.75À and the sample to detector distance was 2.008m. Al l solutions were sheared for 300 strain units before a scattering pattern was recorded to ensure that the transients did not influence the data. The experimental setup is such that the neutron beam is incident perpendicular to the flow of the solution and along the shear gradient direction. The scattered pattern is thus recorded in the flow-vorticity plane.

Steady state scattering experiments were performed at different shear rates. At each shear rate, scattering patterns were also recorded after cessation of flow at different time intervals to study the relaxation of the flow induced alignment. Temperature effects in a narrow temperature window (60 - 75 °C) were investigated

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in the case of PBLG/DBA, however all experiments on PBLG/DMC were performed at room temperature. These temperatures were chosen as they represent very similar thermodynamic temperatures. PBLG/DBA forms a gel at ca. 45 °C while the gel temperature for the PBLG/m-cresol solution is approximately -10 °C (as determined gravimetrically). Thus, the conditions for these experiments are approximately 30 ° above the gel temperature. The collected scattering patterns were corrected for sample transmission, detector sensitivity, scattering due to the cell, and background and normalized to absolute units. The corrected data were then converted from 2D patterns to ID plots of I(q, φ) cm"1 vs. φ, where φ is the azimuthal angle around the detector.

A quantitative measure of the degree of ordering was obtained by calculating the alignment factor, Af(q), as proposed by Walker et al. 2 0

2π

J I(q, φ) οο${2φ)άφ

A f ( l ) = - 2 - δ

J Kq, Φ)άφ ο

In the analysis by Walker, it was shown that Af(q) asymptotes to a constant value at q values greater than 0.07 A' 1 . They denote this asymptotic value as the macroscopic alignment factor Af°° which is related to the effective order parameter Sm

by the relation - A f = Sm. In the current analysis, the intensity was averaged over q = 0.06-0.12 A"1, not the whole q range. Therefore, the effective alignment factor reported for these experiments is not a measure of the order parameter of the systems, but is a measure of the orientation of the sample. However, as the relative changes in alignment that occur due to shear rates, temperature, and solvent are of interest, this approximation is justified and does not influence the interpretation of the data.

Results

Steady State

The changes in alignment factor, and therefore molecular orientation, as a function of shear rate for PBLG/DBA (70 °C) and PBLG/DMC (room temperature) are shown in Figure 1. These results clearly demonstrate that there is a marked difference in the response of the two solutions to an applied shear flow. PBLG/DBA exhibits three distinct regions. At low shear rates, the alignment increases with shear rate until it reaches a critical shear rate after which it remains constant and then increases with shear rate again at higher shear rates. This behavior has been observed previously.12 On the other hand, PBLG/DMC exhibits two regions similar to the high shear rate behavior of PBLG/DBA , with the absence of the increase in alignment at low shear rates that is seen in PBLG/DBA. At low and intermediate shear rates (below 10.0 s"1) the solution shows substantial orientation, which is independent of

360

0.5

0.1 1 10 100

Shear Rate γ (s1)

Figure 1. Shear rate dependence of the steady state molecular alignment of PBLG in deuterated benzyl alcohol at 70 °C and deuterated m-cresol at room temperature.

361

shear rate. At shear rates above 10.0 s*1 the alignment increases with shear rate, as in the PBLG/DBA solution. This also is in agreement with previous results.9,13

Upon examination of this data, a first thought is that the shear rates that were examined for the PBLG/m-cresol sample were not low enough to observe this increase in alignment at low shear rate, but that it does exist. To estimate where this behavior should be observed, the data of PBLG/m-cresol can be shifted to lower shear rates so that the transition from Newtonian to shear thinning behavior overlaps that of PBLG/DBA. Completion of this shifting shows that if PBLG/m-cresol were to exhibit behavior similar to PBLG/DBA, it should feature an increase in alignment with shear rate at shear rates below 0.1 s"1. The lowest two shear rates in Figure 1 are below this limit, implying that this regime does not exist in PBLG/m-cresol. Additionally, previous results which examined the shear rate dependence of the alignment of PBLG in m-cresol9 show that the alignment does not decrease with a decrease in shear rate down to 0.01 s"1. More importantly, however, is that the current molecular theories14"16 which describe the flow of LCP solutions do not predict that this behavior should occur.

It is interesting to note that the extent of alignment is larger in m-cresol than in benzyl alcohol for all shear rates, suggesting an easier path to alignment for the tri­cresol solution. Superficially, this is counter intuitive, as one would expect the sample that it is at higher temperature (PBLG/DBA) to align more readily than the one at lower temperature (PBLG/m-cresol). This indicates that the observed differences are due to effects which are subtler than a mere temperature shift.

Correlation of Steady State Alignment to Viscosity

The viscosity as a function of shear rate for PBLG/DMC at room temperature and PBLG/DBA at 70 °C, the conditions of the scattering experiments, are shown in Figure 2. The difference in the magnitude of viscosity of the two solutions can be accounted for by temperature effects on the viscosity of the solvents and is therefore not related to the relative alignment of the two samples. In other words, if the viscosity of PBLG/DBA at 25 °C is estimated by adjusting the viscosity of the solvent with an Arrhenius factor, the viscosity of the solution is shifted up to he dotted line showing qualitatively similar behavior to the PBLG/m-cresol data. It is interesting that the shear dependence of the viscosity of the two solutions is similar, though they exhibit different alignments at low shear rates. In particular, it is curious that the increased molecular alignment with shear rate that PBLG/DBA exhibits at low shear rate does not correlate to shear thinning behavior in the viscosity. Thus, the qualitative difference in molecular alignment that is observed by scattering does not manifest itself as different macroscopic (viscosity) behavior. This will be considered further in the discussion section.

Orientation Relaxation

The change in molecular alignment at different time intervals after cessation of shear is presented in Figure 3 for the PBLG/DMC solution and Figure 4 for PBLG/DBA at 75 °C respectively. The data obtained at 60 °C and 65 °C are very

362

Figure 2. Viscosity behavior of PBLG/DMC and PBLG/DBA under the same conditions as Figure 1.

0.5

^ 0.4

0.3

0.2

0.1

0 0.001 0.01 0.1 1 10

S c a l e d t i m e (γ t)

100 io3

Figure 3. Relaxation of the flow induced alignment in the PBLG/DMC solution after removal of shear for four shear rates.

363

Time after removal of shear (min)

Figure 4. The relaxation of shear induced alignment in PBLG/DBA at 75 °C.

364

similar to those at 75 °C, and thus there is no temperature dependence of this behavior in this temperature window. These results show that there is a notable difference in the relaxation behavior of the two solutions. PBLG/DMC solutions possess significant orientation even after long relaxation times (~ 6 hours). The relaxation behavior for all steady state shear rates depicts the same trend; an initial increase in orientation followed by a slight decrease and a subsequent increase to a steady value. The post-shear alignment also scales with previously applied shear rate and exhibits a minimum in alignment at the same post shear reduced time (« 500). The degree of alignment that occurs after 6 hours of relaxation time also increases with previously applied shear rate.

In contrast, the alignment in PBLG/DBA solutions is not as long lived as that of PBLG/DMC. Al l experiments show the same qualitative behavior; an initial increase in orientation that lasts for about 5-10 minutes and then a relatively sharp decrease that asymptotes to a small positive alignment. The orientation of the solutions sheared at 0.1 s'1 show a final alignment that is almost zero, suggesting no residual alignment. Additionally, during the relaxation following all shear rates except 0.1 s"1, there exists a small peak in orientation at long times. This response has not been observed in the case of PBLG/DMC or in other SANS studies20 involving PBLG/deuterated dimethyl formamide. It must also be noted that the relaxation kinetics do not scale with previous shear rate as has been observed in PBLG/MC solutions.9'21

Dynamic Modulus after Shear Cessation

To correlate the observed alignment relaxation behavior of the solutions to a macroscopic Theological property, the evolution of the dynamic modulus after shear cessation 2 3 of the PBLG/DBA solutions were performed under similar shear rates as the neutron scattering experiment, though the temperature was limited to 65 °C. The results of the mechanical measurements at two shear rates have been plotted with the corresponding alignment factors (SANS data) in Figure 5 to show the correspondence between the changes in the two parameters with time. Logically, the dynamic modulus decreases with an increase in molecular orientation as there is a decrease in resistance to strain with increasing alignment. This correlation is found to be true in the case of PBLG/DBA solutions with a decrease in moduli corresponding very well to the increase in molecular alignment during relaxation. Similarly, the decrease in alignment at relaxation times greater than 10 minutes is found to agree well with the increase in modulus on the same time scale. Similar experiments on the PBLG/m-cresol solution show good correspondence between the dynamic moduli and the evolving alignment structure of the solution after shear cessation.9,21

Discussion

Clearly, these results show that the flow behavior of PBLG in deuterated tri­cresol shows remarkably different behavior than a similar solution in deuterated benzyl alcohol. This is particularly surprising given the similarity in the chemical structure of the two solvents; they are isomers. For the steady state results, the

365

Time (min)

Figure 5. The evolution of the dynamic modulus of PBLG in DBA after the flow field has been removed as well as the corresponding molecular alignment for previous shear rates of 1.0 s'1 and 10 s'1.

366

alignment behavior differs for these two solutions at low shear rates. This is the region where the current understanding of the flow behavior of LCP solutions suggests that the defect structure of the solution is very important in defining the response of the solution to shear flow.25"27 Therefore, the difference that is observed in the low shear rate regime may be a manifest of texture variations (i.e. the structure of a defect in the DBA solution may be inherently different than the structure of a defect in the m-cresol solution). However, the fact that there is no difference in the viscosity behavior of the two solutions is puzzling as one would expect that texture differences would manifest themselves here also. Interestingly, in liquid crystalline hydroxypropylcellulose/D20 solutions, an increase in alignment with shear rate is exhibited at low shear rates, correlating very well to region I shear thinning viscosity behavior. This suggests that region I shear thinning in some LCP solutions does correlate to molecular alignment.28

These results demonstrate that the correlation between viscosity and low shear rate molecular alignment is complex. In particular, it must be emphasized that the viscosity behavior of an LCP solution at low shear rates does not correlate directly to, nor can be utilized to predict, the molecular alignment. At higher shear rates, the correlation between molecular orientation and viscosity appears to be much more robust.

The relaxation of the alignment between the two solutions also differs. The alignment of the DBA solution relaxes back to a near isotropic state while the m-cresol solution retains the shear induced alignment for very long times. However, the dynamic modulus and alignment responses agree very well for both solutions, with a decrease in alignment correlating very well to an increase in the dynamic modulus. This suggests that the dynamic modulus can be directly correlated to the molecular alignment of LCP solutions for the shear rates studied here.

But what is the origin of the differences in the response of the two solutions to an applied shear flow? The first thought is to ascribe the alignment differences to mere temperature effects. As the temperature is increased, the Frank Elastic constants of the solution texture will substantially decrease.24 This, in turn, should result in improved alignment of the sample that is at a higher temperature (PBLG/DBA) over that of the sample that is at a lower temperature (PBLG/m-cresol). This is exactly the opposite of the observed experimental results; the PBLG/DBA is aligned to a lesser extent in the steady state than PBLG/m-cresol. Additionally, as mentioned in the experimental section, the differences in the magnitude of the viscosity of the two samples can be accounted for with temperature effects on the solvent. Finally, the two temperatures that are utilized in the experiments are thermodynamically similar. The combination of all of these factors leads to the conclusion that the alignment effects observed in these experiments are due to more subtle factors than a change in temperature. Three other possibilities are also being examined. First, the PBLG molecule may be more stiff (rod-like) in m-cresol than in benzyl alcohol. If the PBLG molecule is more (semi)flexible in benzyl alcohol, it should be more difficult to align and this could account for the shear rate dependent alignment at low shear rates. Similarly, after the shear field is removed, the fluctuations in the rigidity of the polymer chain in benzyl alcohol could serve as the impetus for the relaxation of the shear induced alignment. Alternatively, the less rigid molecule could merely be more difficult to align and this would account for both the low shear rate and relaxation behavior of PBLG in benzyl alcohol.

367

Second, the differences that are seen in the flow behavior may be due to phase behavior differences between the two samples. Though the temperature of the two sets of experiments is approximately 30° C above the gelation temperature of the two solutions, it is possible that PBLG in DBA may not be as molecularly dispersed as in the PBLG/m-cresol solution. If this is true, it may be that the low shear rate alignment changes are a result of the shear field breaking up aggregates. Similarly, the relaxation back to an isotropic alignment state could be the re-aggregation of aligned molecules after the shear field is removed.

It is possible to test both of these predictions as both the polymer flexibility and aggregation state dramatically affect the phase diagram of LCP solutions. Liquid crystalline polymers exhibit a phase behavior in solution that can be approximated by Figure 6. At high temperature and low concentration, the solution is isotropic. As the concentration is increased, the solution passes through a narrow biphasic region (known as the chimney) to form a liquid crystalline phase. The concentration where the biphasic region is entered (isotropic -> biphasic) is often denoted as C* and where it is exited (biphasic -> liquid crystalline) is often denoted C**. As the temperature is lowered from the isotropic or liquid crystalline phase, the solution enters into a broad biphasic regime. As the flexibility of the polymer chain increases, the chimney region shifts to higher concentrations. In the case of aggregation, if it occurs side-by-side, aggregation will result in a decrease in the aspect ratio while end-to-end aggregation results in an increase in the aspect ratio. In either case, the position of the chimney region (i.e. C* and C**) will depend intimately on the aspect ratio of chain. Thus, if PBLG in BA is more flexible or aggregated side-by-side, then the chimney region of the phase diagram should be shifted dramatically to higher concentrations from that of the PBLG/m-cresol system. Alternatively, if PBLG in BA is aggregated end-to-end, then the chimney region of the phase diagram should be measurably shifted to lower concentrations from that of the PBLG/m-cresol system

Early calculations by Flory2 9 have quantified how flexibility and aspect ratio alters this phase diagram. For this study, the most important change in the phase diagram is that the chimney region shifts in a fairly dramatic fashion. Thus, Flory's original equations29 were utilized to calculate the phase diagram of a semiflexible polymer in solution near the chimney region. From this calculation, C* and C** as a function of polymer flexibility and aspect ratio were determined. The calculated phase diagrams for three rigidities are shown in Figure 7. To evaluate the data in this study, plots which quantify how C* and C** changes with flexibility and aspect ratio were created. This effect is shown in Figure 8 which plots C*(/)/C*(rod) and C**(/)/C**(rod) as a function of/ C*(rod) and C**(rod) are the critical concentrations for a rigid, rodlike molecule, while C*(f) and C**(/) are similar concentrations for a semiflexible polymer, and/is a measure of the flexibility of a chain. One way to view/is as a fraction of bonds along the polymer chain that are flexible. Figure 8 shows that the chimney region shifts to measurably higher concentrations with a very small change in flexibility. Quantitatively, by increasing the amount of flexibility by only 1%, the chimney region shifts up in concentration by a factor of 1.5. Thus, if PBLG in benzyl alcohol is only 1% more flexible than in m-cresol, (from the data below) the C* of PBLG/BA should occur at ca. 14-15%. Similar calculations were completed to demonstrate the affect of aggregation or aspect ratio on the chimney region of the phase diagram. Figure 9 shows quantitatively how altering the aspect ratio of the polymer chain alters the chimney

368

Figure 6. Diagrammatic representation of the phase diagram of a rodlike liquid crystalline polymer in solution.

-o.i

-0.05

0.05

o.i

......... f_oi f=.04

\

\

\ Liquid Crystalline \ Ν

\ \

s

Biphasic

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Φ

Figure 7. The LCP solution phase diagram for polymers with various rigidities near the chimney region.

369

ο u Ο 9 ο

-C*(f)/C*(rod) -C**(f)/C**(rod)

Figure 8. Plot that shows the shift in the chimney region as a result of polymer flexibility.

3 ο 3 ο

side-by-side aggregation

-C*(x)/C*(x=80) -C**(x)/C**(x=80)

end-to-end aggregation

20 40 60 80 100 120 140 160

x = aspect ratio

Figure 9. Effect of polymer aspect ratio (which will depend on molecular aggregation) on C* and C**.

370

region of the phase diagram. Again, a slight change in the aspect ratio will result in a dramatic shift in the chimney region in either direction.

Thus, these data show that if PBLG is slightly more flexible in DBA than in m-cresol or aggregated end-to-end or aggregated side-to-side in DBA, for the conditions of the scattering experiments, the chimney region of PBLG/DBA should occur at a much different concentration regime than in m-cresol.

To test this assumption, the chimney region of the phase diagram of 20 wt.% PBLG (MW = 287,000) in benzyl alcohol and in m-cresol were determined by polarized optical microscopy. For this study, both BA and m-cresol were vacuum distilled immediately prior to sample preparation to minimize water contamination. Poiydispersity effects on the chimney region were eliminated in this analysis by using PBLG that came from the same batch. Figure 10 shows the resultant plot of the phase behavior of these two solutions. For PBLG in BA, C* « 8-9 %; C** « 13-14% while for PBLG in m-cresol, C* « 9-10 %; C** « 12-13%. This shows that the chimney is not shifted relative to PBLG/m-cresol, therefore, PBLG in BA can not be more flexible nor less molecularly dispersed than in m-cresol. Therefore, it is highly unlikely that the anomalous alignment behavior described above is due to either of these two factors. It should be emphasized here that these experiments do not quantify flexibility and thus do not unequivocally eliminate polymer flexibility as a cause for the observed behavior. To more fully quantify the role of polymer flexibility on the results of this study, neutron scattering experiments are underway to quantify the persistence length of PBLG in these two solvents.

The last possible explanation is that there may be an inherent solvent dependence on the structure of the solutions defects or disclinations. It may be that the defects in PBLG/DBA resist alignment or annihilation more than those in m-cresol. If this is the case, one could envision a process where the * stronger5 texture of DBA limits the extent of molecular alignment at low shear rate. In other words, the defects are more long lived, thus the LCP in the immediate vicinity of the defect is not aligned by the shear flow at low shear rates. This would explain the shear rate dependent alignment that is observed in the PBLG/DBA solution. A 'stronger' texture would also explain the alignment relaxation behavior, as surviving defects could act as nucleation points for post shear isotropization. The stronger texture should manifest itself as increase in the Frank elastic constants of PBLG/BA over those of PBLG/m-cresol even at the increased temperature. Qualitative differences between the defect structure of PBLG/BA and PBLG/m-cresol during and after shear are indeed observed by in-situ light scattering. To verify and quantify this effect, the correlation between defect structure and molecular alignment for these two structures is currently under investigation using in-situ shear small angle light scattering and polarized optical microscopy.

One additional possibility to explain these results must also be mentioned. Recently,21,30 Burghardt et. al. have suggested that differences between the alignment behavior of HPC and PBLG may be due to a 'stronger' tendency to form the cholesteric phase in HPC. This cholesteric structure limits the ability of the solution to align and remain aligned after the shear field has been removed. The completion of similar experiments to those described in this publication with a nematic PBG solution (formed from equimolar amounts of PBLG and PBDG) would provide substantial evidence regarding the importance of the cholesteric nature of PBLG in BA on its alignment behavior.

371

Ο

a 40

-

ο ; • • • •

Benzyl Alcohol

• ο

-

ο Isotropic • Biphasic ο LC

-

ο ο /

Χ_ , I I t ι . I . ι

m-cresol • • • ; Ο ο

7 8 9 10 11 12 13 14 15

Φ LCP

Figure 10. Phase diagram of PBLG in benzyl alcohol at 70 °C and of PBLG in tri­cresol at room temperature as determined by polarized optical microscopy. The temperatures of this study coincide with the temperatures of the scattering experiments.

372

It is well known that many parameters can affect the flow of liquid crystalline polymers such as polymer rigidity, polymer-solvent interactions, and polymer concentration. The results presented in this article demonstrate that there exist other, less obvious, parameters which also can dramatically affect the alignment of LCP under shear flow. The current understanding of the reported results suggest that the defect structure of an LCP solution is one such parameter.

Finally, this article documents the need for a more thorough understanding of all of the molecular parameters which influence the flow behavior of liquid crystalline polymers. A universal theory which describes the flow behavior of all LCP will only be developed when all characteristics are defined and accounted for which impact the rheology of LCP solutions. Moreover, a 'model' liquid crystalline polymer solution should be one in which these factors can be systematically studied.

Conclusion

The orientation of PBLG induced by flow in two solvents, deuterated benzyl alcohol (DBA) and deuterated m-cresol (DMC), has been studied by small angle neutron scattering. Unexpectedly, results show marked differences in the steady state and relaxation responses of the solutions to shear. Steady state results show that the solutions behave similarly at high shear rates (> 1 s"1) with an initial nonlinear region followed by a linear increase in alignment. However, at low shear rates the PBLG in DBA shows an increase in orientation with shear rate which is absent in DMC. Relaxation results further contrast the difference in ordering of the two solutions. The PBLG/DMC sample retains the flow induced ordering of molecules even after long periods of time («6 hours). Conversely, the ordering of PBLG in DBA by shear decays to very low orientation rather quickly.

The results also show that the correlation between molecular alignment and rheological parameters is not trivial. The shear rate dependence of the viscosity is not readily related to the molecular orientation particularly at low shear rates, however, the relation between molecular alignment and dynamic modulus is more robust.

These differences are surprising as benzyl alcohol and m-cresol are isomers. Clearly, there exist fundamental distinctions that account for their diverse behavior. A difference in the defect structure of PBLG in BA versus PBLG in m-cresol may prove to be the underlying cause of this alignment behavior. Regardless, the results exemplify the need for a broader choice of 'model' liquid crystalline polymer solutions when examining the flow-induced structure in liquid crystalline polymer solutions. Moreover, a more complete understanding of the important parameters that affect the flow of LCP solutions is needed so that a more universal theory can be developed which can predict flow behavior of non-model LCP solutions.

Acknowledgment

We would like to thank Wes Burghardt for obtaining the dynamic modulus data and for useful discussions. The Division of Materials Sciences, U. S. Department of Energy, supports the research at Oak Ridge under contract No. DE-AC05-960R22464 with Lockheed Martin Energy Research Corp.

373

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1370. 12. Dadmun, M . D.; Han, C. C. Macromolecules 1994, 27, 7522. 13. Hongladarom, K.; Ugaz, V. M. ; Cinader, D. K.; Burghardt, W. R.; Quintana, J.

P.; Hsiao, B. S.; Dadmun, M. D.; Hamilton, W. Α.; Butler, P. D. Macromolecules 1996, 29, 5346.

14. Marrucci, G.; Maffettone, P. L. Macromolecules 1989, 22, 4076. 15. Onogi, S.; Asada, T. in Rheology; Astarita, G.; Marrucci, G.; Nicolais, L.; Eds.;

Plenum: New York 1980, 3, 947. 16. Larson, R. Macromolecules 1990, 23, 3983. 17. Larson, R. G.; Mead, D. W. J. Rheol. 1989. 33. 1251. 18. Larson, R. G.; Prominslow, J.; Baek, S. -G.; Magda, J. J. Ordering in

Macromolecular Systems; Springer-Verlag: Berlin 1994, p. 191. 19. Walker, L. M. ; Kernick, W. A. III.; Wagner, N . J. Macromolecules 1997, 30, 508. 20. Walker, L. M. ; Wagner, N . J. Macromolecules 1996, 29, 2298. 21. Burghardt, W.; Bedford, B.; Hongladarom, K.; Mahoney, M . in Flow Induced

Structures in Polymers; Nakatani, A.I., Dadmun, M.D., Eds., ACS Symposium Series 597, American Chemical Society: Washington, D.C., 1995.

22. Moldanaers, P.; Mewis, J. J. Rheol. 1986, 30, 567. 23. Larson, R.G.; Mead, D.W. J. Rheol. 1989, 33, 1251. 24. DeGennes, P.G.; Prost, J.The Physics of Liquid Crystals; Oxford Science

Publications: Oxford, 1993, p. 103. 25. Walker, L.M.; Wagner, N.J. J. Rheol. 1994, 38, 1525 and references therein. 26. Marrucci, G. in Proc. IX International Conference on Rheology; Mena, Β.,

Garcia-Rejon, Α., Rangle-Nafaile, C. Eds.; Universidad Nacional Autonoma de Mexico: 1984; Vol. 1.

27. Larson, R.G. Rheol. Acta 1996, 35, 150. 28. Dadmun, M.D. in Flow Induced Structures in Polymers; Nakatani, A.I., Dadmun,

M.D., Eds., ACS Symposium Series 597, American Chemical Society: Washington, D.C., 1995.

29. Flory, P.J. Proc. Roy. Soc. (London) 1956, A234, 60. 30. Hongladarom, K.; Secakusuma, V.; Burghardt, W. J. Rheol. 1994, 38, 1505.

Chapter 24

X-ray Scattering Measurements of Molecular Orientation in Thermotropic Liquid Crystalline Polymers under Flow

Wesley R. Burghardt, Victor M. Ugaz, and David K. Cinader, Jr.

Department of Chemical Engineering, Northwestern University, Evanston, IL 60208

In situ synchtrotron x-ray scattering is used to make quantitative measurements of molecular orientation in thermotropic liquid crystalline polymers (TLCPs) in both simple and complex flows. Two different model TLCPs consisting of mesogenic units separated by flexible spacers are studied in simple shear flow as a function of temperature and shear rate. Both samples exhibit significant net molecular orientation under steady shear within the nematic phase. One of the samples undergoes a flipping transition from alignment along the flow direction to alignment along the vorticity direction at low temperatures. This is associated with a subtle phase transition detected by DSC. At low temperatures, it is possible to induce a transition back to orientation along the flow direction by application of high shear rates. A commercial TLCP fully aromatic copolyester was studied in a diverging channel flow. In straight channel sections, significant net orientation along the flow direction is observed. In the expansion region, however, extensional gradients are capaple of inducing alignment transverse to the prevailing flow direction. Both the degree and direction of orientation have been measured as a function of position in this complex flow.

Liquid crystalline polymers have considerable promise as high strength materials. Since the molecules are spontaneously ordered in the liquid state, processing flows can readily induce high degrees of molecular orientation, leading to solid products of remarkable strength and stiffness.1 This potential has been realized in high performance fibers spun from lyotropic solutions of rodlike polymers. In fiber spinning, uniaxial extension effectively promotes molecular orientation along the

374 © 2000 American Chemical Society

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fiber axis. There has also been considerable interest in thermotropic liquid crystalline polymers (TLCPs), which may be processed in the melt phase using such techniques as extrusion and injection molding. Unlike fiber spinning, these processing operations typically involve complex flow fields involving both shear and extension. This complexity makes it difficult to anticipate the final orientation state.

Although TLCPs have the potential for broader technological applications than lyotropes, fundamental understanding of the dynamics of LCP melts under flow is not nearly as advanced as for solutions. In solutions, a consistent theoretical framework built around molecular models for the dynamics of rodlike polymers successfully describes many characteristics of lyotrope rheology.2 Within the linear continuum model of LC hydrodynamics,3 nematics may be classified as either tumbling or shear-aligning depending on how shear flow influences the liquid crystal director. At low shear rates, most lyotropic solutions appear to exhibit tumbing, where hydrodynamic torques act to promote indefinite rotation of the director. In transient shear flows, tumbling results in well-defined oscillatory stress responses (that are however damped by distortional elastic effects acting within the typical polydomain texture4). At higher rates, nonlinear molecular viscoelastic effects become important, as shear flow is able to perturb both the degree and the symmetry of the local distribution of molecular orientation.2 These nonlinear effects lead to a transition from tumbling to flow-alignment. This transition is accompanied by sign changes in the first and second normal stress differences that are the most striking characteristic of lyotropic LCP rheology.5

These dynamical processes may be expected to have profound consequences on the degree to which shear flow generates net macroscopic orientation in lyotropes. Using both flow birefringence and x-ray scattering, our group has made extensive quantitative measurements of molecular orientation in model lyotropic solutions, which have allowed detailed testing of the structural predictions of molecular and mesoscopic models, and provide direct insights into the structural origins of bulk rheological behavior.6 In addition, we have extended our studies on lyotropes beyond simple shear flow to include inhomogeneous shear7,8 and mixed shear-extensional flows9,10 in complex channel geometries. Here, the superposition of even modest amounts of extension can strongly influence molecular orientation by either supression of tumbling,9 or drift of tumbling orbits into or away from the flow direction.10

At present, thermotropic LCPs are poorly understood in comparison to lyotropes. Definitive classification in terms of tumbling vs aligning behavior has been published for only one material- and this material exhibits either behavior depending on the temperature!11 Well-defined rheological data are difficult to obtain on thermotropes, since available commercial materials are subject to many experimental complications such as recrystallization or transesterification in the melt. Recent experiments on idealized model TLCPs by Han and coworkers have shown that reliable mechanical data may be more readily acquired for such systems.12 It is logical, then, to pursue in situ measurements of fluid structure in such materials, that could play a useful role in complementing bulk rheological data. In the first part of this paper, we summarize measurements of molecular orientation as a function of temperature and shear rate in two different model materials. We then turn to more complex flow fields, which are

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of more direct commercial importance. Studies of a commercial TLCP resin in a diverging channel flow clearly demonstrate how superimposed extension can profoundly influence the molecular orientation state.

Model TCLPs in Shear Flow

Experimental

Two model main-chain TLCPs have been studied using x-ray scattering in simple shear flow: (a) DHMS-7,9, and (b) PSHQ-6,12. The molecular structure of these two materials is given here:

Both materials consist of rigid mesogenic units separated by flexible hydrocarbon spacers. DHMS-7,9 is a random co-polyether that has been previously studied by Gillmor and coworkers.13 The sample used here was synthesized by Weijun Zhou at Caltech, with a weight-average molecular weight of 28,000. DSC reveals a nematic-isotropic transition at 190°C, and a crystal melting point of 90eC. An additional, weak thermal transition is found at 120eC. While the sample is nematic for temperatures 120eC < Τ < 190°C, the nature of the phase existing from 90°C < Τ < 120°C has not been clearly established. PSHQ6-12 is a copolyester synthesized by Chang and Han, for which extensive rheological data have been published.14 The sample used here has weight-average molecular weight of 24,300, and was provided by C D . Han. Because of the widely different spacer lengths, PSHQ-6,12 does not crystallize, but rather exhibits a glass transition at 92°C. Between 92°C and 190°C, the sample is nematic; above 190°C the sample is isotropic. The ability to access the isotropic phase in both materials is crucial, since is allows a reproducible intial condition to be established in which previous thermal and shear history is erased. The

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nematic phase occurs at much lower temperatures than for commercial TLCPs, and these materials thus exhibit much greater thermal stability in the melt.

Synchrotron X-ray scattering measurements were performed at DND-CAT (Sector 5) of the Advanced Photon Source at Argonne National Laboratory. Photon energies of either 20 or 25 keV were used; these relatively high energies confine the WAXS features of interest to smaller angles, facilitating the exit of scattered radiation from the shear cell. Shear was generated in a Linkam CSS-450 shearing stage, modified for x-ray experiments.15 Figure 1 provides a schematic illustration of the experimental setup. Flow is generated between two parallel disks, and is interrogated by sending the x-ray beam parallel to, but displaced away from, the axis of rotation. Thin sheets of mica or Kapton serve as the x-ray windows, and are supported by steel plates.

Figure 1. Schematic illustration of the x-ray shearing cell setup. The exploded view depicts the shearing geometry and window design which allows a 1mm diameter x-ray beam to pass through the sample during shear. Reproduced with permission from Reference 15. Copyright 1998 American Chemical Society.

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Results and Discussion

Figure 2 presents representative 2-D WAXS patterns measured during shear flow of DHMS-7,9 at two temperatures: 150°C within the nematic phase, and 110°C within the intermediate phase. The diffuse scattering peak arises from correlations in the lateral packing of mesogenic units; the peak scattering vector provides an estimate of the correlation length d ~ 2n/q* of 5 Â. When a sample is cooled from the isotropic to the nematic phase, there is no prefered orientation, and a polydomain sample emerges in which the director orientation is randomly distributed. Under these circumstances, the scattering takes the form of an isotropic ring. Upon application of shear flow, a bias in orientation is generated, and the scattering becomes anisotropic. Since scattering from an elongated object is concentrated pependicular to its axis, nematics under shear typically exhibit two diffuse peaks along the equator, as seen in Figure 2a.

Within the nematic phase, both materials exhibit scattering patterns qualitatively similar to that in Figure 2a. However, DHMS-7,9 shows a transition in orientation state when the temperature drops below the transition temperature of 120°C into the intermediate phase. Here, diffuse peaks are observed along the meridian, rather than the equator, indicating that the mesogen orientation has flipped by 90° (see Figure 2b) Such an orientation state is frequently refered to as "log-rolling", since the mesogens are on average oriented along the vorticity axis of the shear flow. This type of vorticity alignment has been observed in LCPs on other occasions,16,17 but seems only to appear when there is some possibility for additional fluid ordering in the nematic phase: residual gel phase in a lyotropic system,16 and smectic-like melt ordering in a thermotropic material.17 In the present case, it seems clear that this transition in orientation is induced by the transition detected by DSC, even though the nature of the intermediate phase is poorly defined.

Figure 2. Scattering patterns from DHMS 7,9 in steady shear (a) at 150 °C, 3.0 s'1, and(b) at 110 °C, 0.1 s1.

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A quantitative measure of macroscopic alignment is often extracted from 2-D x-ray scattering patterns in terms of an orientation parameter, S. This is computed by taking the average of the 2 n d Legendre polynomial, P 2, weighted by an azimuthal scan of intensity, 1(a) taken at the peak scattering vector, q*. We adopt the procedure outlined by Mitchell and Windle,18 and compute S using:

π 2

J / fa)P 2 ( C 0 S a) sin vd®

2 jl(a)sinada ο

5 = (P2(cosa)) (P 2(cosa)) 0

Here, (···) represents an average weighted by the experimental azimuthal intensity distribution, while (· · ) o represents an average weighted by the intensity distribution that would be realized if the mesogen alignment were perfect. Following Mitchell and Windle,18 we take this reference state to be a perfectly sharp concentration of scattering at the equator (a = TC/2). While this is appropriate for long, rigid rodlike molecules, it is unlikely that the types of LCP molecules under investigation here would exhibit such idealized scattering with perfect mesogen alignment; for this reason, our measurements of S are likely to understate the actual degree of mesogen ordering (defined in this way, S ranges from 1 for the case of perfect alignment to 0 for the case of an isotropic scattering pattern). This analysis assumes a uniaxial distribution of orientation around the director, which may be only approximately true in shear flow.6

In addition, it assumes that a = 0 corresponds to the average mesogen orientation direction within the sample. For this reason, we have rotated the azimuthal coordinate by 90° for those DHMS-7,9 patterns where, as in Figure 2b, the orientation direction has flipped to the vorticity direction.

Figure 3 presents orientation parameters measured in steady shear flow as a function of shear rate and temperature for the two materials under consideration. Figure 3a shows that orientation is at most a weak function of shear rate in PSHQ-6,12, except at 180*C. Inspection of steady viscosity data for this material suggests that the material may be biphasic at this temperature, explaining the substantial drop in orientation.15 At temperatures fully within the nematic phase, the degree of orientation is observed to increase slowly with decreasing temperature, as would be expected for the molecular order parameter in the nematic phase. The lack of a significant dependence of orientation on shear rate contrasts with most lyotropic systems,6 and suggests the possibility of simpler dynamics. Ugaz and Burghardt have also examined PHSQ-6,12 in transient flows,15 and find that the development of orientation upon inception of shear flow from a random initial condition may be well represented using the assumption that this material exhibits flow-aligning dynamics. Such an interpretation is consistent with the behavior in Figure 3a.

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(a) 0.8-

0.6H

S 0.4H

0.2H

0.01

(b) 0.8-

0.6-

0.4-J

S 0.2-

0-J

-0.2-

-0.4-

^ • • • • é Φ φ

ο ο ο ο

• •

ι ι 1111—

0.1 ι ι ι ι ι 1111

1 .-1ι Shear Rate (s"1)

• t

•

ο •

ο 8 D

0.01 -, ,—ι ι ι ι 11 j—

0.1 τ—ι—ι ι ι 1 1 1 {

1 Shear Rate (s**1)

M i l l

10

•

• •

ο

10

Figure 3. Molecular orientation in steady shear flow, (a) PS HQ 6-12 at 140 °C (ÊÊ), 150 °C (Φ), 160 °C (A), 170 °C (Φ), and 180 °C (O). (b) DHMS 7,9 (28k) at 170 °C (M), 150 °C (Θ), 140 °C (A), 120 °C (Φ), 110 °C (D) and 100 °C (O). Negative values of S are used to represent data with shifted azimuthal coordinate.

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Figure 3b presents similar data obtained for DHMS-7,9. At temperatures well within the nematic phase, the behavior is quite similar to that seen in PSHQ-6,12: there is little change in orientation with shear rate, and small increases in orientation with increased subcooling. Zhou has recently presented direct evidence of flow aligning behavior in a lower molecular weight sample of DHMS-7,9. 1 9 Flow aligning dynamics would again explain the simplicity of orientation behavior during steady shear flow within the nematic phase. The situtation changes dramatically near and below the transition to the intermediate phase at 120°C. A detailed description of the three-dimensional orientation state would require a more complete examination of reciprocal space than is possible in these in situ measurements where only a single projection of the fluid structure is possible. Any analysis using Eq. (1) makes the implicit assumption of a uniaxial orientation state, and while the actual symmetry of the sample is unkown, it seems more plausible to compute a positive value of S relative to the vorticity direction in Figure 2b than to continue the assumption of uniaxial symmetry about the flow direction. For convencient, then, negative values are used in Figure 3b to designate S measurements extracted from patterns in which the orientation direction has flipped. These negative S values thus do not imply a negative uniaxial material. Of particular interest is the fact that shear flow at high rates is able to induce a transition back to orientation along the flow direction within this intermediate phase (represented in Figure 3b by the sign change in S). These flipping transitions in orientation with both temperature and shear rate are in good agreement with neutron scattering observations of chain anisotropy in DHMS-7,9 by Dadmun and coworkers.20

Commercial TLCP in a Diverging Channel Flow

Experimental

We have previously utilized channel flows with contraction or expansion in cross section to study the orientation state of lyotropic LCPs in the presence of mixed shear and extension.9,10 To translate this effort to thermotropic LCPs, we have constructed a channel flow extrusion die, illustrated schematically in Figure 4. The die consists of two steel blocks, heated by cartridge heaters, that sandwich a 1 mm thick spacer that defines the channel flow geometry. In this paper we concentrate on slit flow with a 1:4 expansion in cross sectional area, as illustrated in Figure 4. The upstream channel has a width of 5 mm, while the downstream channel has a width of 20 mm. The entire channel length is 110 mm from the entrance manifold. The steel blocks have trenches along the centerline and across the channel at 3 axial locations to allow access to incident and scattered x-rays. 1 mm thick aluminum "windows" lie recessed at the bottom of the trenches to contain the polymer within the channel defined by the spacer. Polymer pellets were melted and pumped using a 1/2" diameter single screw extruder manufactured by Randcastle.

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Figure 4. Schematic illustration of extrusion die for in situ x-ray scattering from polymer melts in channel flows. Also shown is representative spacer used for 1:4 slit-expansion flow.

X-ray scattering measurements were again performed at DND-CAT of the Advanced Photon Source, using a photon energy of 25 keV. In addition to confining the WAXS features of interest to small angles, the high photon energy helps reduce absorption losses as the x-rays pass through the two aluminum windows. In these measurements, the incident x-ray beam was square, with a cross section of 1 mm2.

Due to the large volume of material needed to maintain steady channel flow with this extrusion setup, experiments on custom synthesized model TLCPs are not possible. Instead, we have studied a commercial LCP resin, Xydar SRT 900, provided by Amoco. This is a fully aromatic random copolyester comprised of four different monomers. Following the suggestion of the manufacturer, we have extruded the polymer at a temperature of 350°C, just above its nominal melting point.

383

Results and Discussion

Figure 5 presents representative 2D x-ray scattering patterns obtained at 3 locations in a 1:4 slit expansion flow of the SRT 900. All channel flow data were taken under steady state conditions. As in Figure 2, correlations in intermolecular packing lead to a broad peak in scattered intensity at a wave vector q* that corresponds to a correlation length d ~ 2n/q* of 5.51 Â. Within the upstream straight channel (Figure 5a), the two nematic peaks located at right angles to the flow direction indicate molecular orientation predominantly in the downstream direction. At this location, the x-ray beam passes through fluid that experiences only shear, although the shear rate is inhomogeneous, ranging from 0 at the midplane to a maximum value at the walls. The scattering pattern thus provides a representation of fluid structure averaged through the slit flow thickness (and, of course, over the finite cross sectional area of the beam). While this situation is more complicated than for simple shear flow, it is still possible to perform a detailed analysis of the orientation state under such circumstances.

Figure 5. 2-D x-ray scattering patterms measured at 3 positions in 1:4 slit-expansion flow ofXydar SRT 900 at a mass flow rate of 5.1 g/min. Incident beam is 1 mm2.

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The situation is more complex in the expansion region, as shown in Figure 5b and 5c. As reported by Cinader and Burghardt,21 a bimodal orientation texture is observed along the centerline in the expansion region. The pattern indicates two discrete populations of orientation, one along the prevailing flow direction and the other transverse to the flow direction. We hypothesize that stretching transverse to the flow direction induces a 90e flipping of orientation near the midplane of the slit-expansion flow, while shear continues to promote orientation along the flow direction near the surfaces of the two windows. The scattering pattern reflects the contributions from the entire slit flow thickness. As seen in Figure 5c, the existence of two orientation populations is also in evidence away from the centerline, complicated by an average orientation angle that is twisted relative to the flow or transverse direction.

The type of orientation parameter analysis embodied in Equation (1) is no longer appropriate given the complexity of the orientation state sampled by the x-ray beam. For instance, while the orientation state near the windows or near the midplane might be locally uniaxial, the overall orientation state contributing the the 2-D pattern clearly is not. In the presence of mixed populations of orientation as evident in Figures 5b and 5c, it is similarly impossible to rationally assign a = 0 to any particular direction. Instead, we elect to develop a measurement of anisotropy of the scattering pattern itself, computed from an azimuthal scan of the intensity taken at

A point on the azimuthal scan may be represented by a unit vector, u, such that «! = cos/3 and w2 = sin/? (β is the azimuthal angle). A weighted average of the second moment tensor of u provides a simple representation of anisotropy in the scattering pattern:21

In this expression, {· · ·) represents an average weighted by the azimuthal intensity

<uu> = (2)

is given as:

2/r Jcos 2 j3/(/})^3

\ΐ(β)άβ (3)

ο

The degree of anisotropy is evaluated by computing the difference in the eigenvalues of the 2nd moment tensor, which we refer to as an anisotropy factor:

385

while the average direction of orientation is determined by the eigenvectors. For centerline measurements, symmetry dictates that the off-diagonal terms in Eq. (2) should be zero. Under these conditions, anisotropy may be characterized by the simpler quantity - {"2^2) · Recognizing that scattering from rodlike molecules is concentrated normal to the molecular axis, we adopt a coordinate system in which the 1-axis is oriented transverse to the flow direction (see Figure 5a). This leads to the expedient result that this quantity approaches a value of +1 for perfect orientation in the flow direction.

While the trench along the centerline of the extrusion die allows orientation state to measured continuously as a function of axial position, measurements away from the centerline are only possible at a few locations dictated by the die design. In order to map out the evolution of average orientation in two dimensions in the expansion region, it was necessary to repeat experiments with a series of 1:4 expansion spacers in which the position of the expansion was shifted relative to the fixed slot locations in the die. Figure 6 illustrates the strategy used, showing the flow field outline for two spacers overlaid with the location of the measurement slots. Using this strategy, the length of straight channel flow upstream of the expansion also changes. Centerline measurements taken with these different spacers, reported as anisotropy factor as a function of axial distance in Figure 6, demonstrate that the evolution of average molecular orientation downstream of the expansion is unaffected by the varying length of the upstream slit flow section.

As discussed by Cinader and Burghardt,21 the expansion in the slit flow has a profound effect on the centerline orientation, as would already be anticipated from Figure 5b. In fact, the anisotropy factor takes on negative values for a substantial region of the flow, indicating that the tranverse orientation state is more prevelant than orientation along the flow direction. Measurements of orientation away from the centerline are summarized in Figure 7. A vector plot is used to represent both the direction and the degree (vector length) of the average orientation state at each position in the expansion region, as characterized by the 2nd moment description in Equations (2) - (4). Along the centerline just downstream of the expansion, note that the direction of average orientation has flipped 90°, in accord with the sign change in anisotropy factor along the centerline in Figure 6 above.

Figure 7 provides a more complete picture of how the expansion influences the evolution in macroscopic orientation. While a substantial drop in average orientation is seen downstream of the expansion, is is largely confined to the middle portion of the channel. Indeed, the orientation is quite strong near the edges of the flow throughout the entire expansion region. The edges are also seen to have an influence on the average orientation direction: even well downstream of the expansion, the average molecular orientation makes a small but finite angle with respect to the edges. In the expansion region, the streamlines are expected to bend outwards; however, the average orientation direction does not track the streamlines, but instead bends in the opposite direction. This issue is obscured to some extent by the fact that the second moment description lumps multiple populations in orientation into an average representation of degree and direction of orientation. Deeper insights into this

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behavior may be obtained by considering the independent evolution of the two discrete populations in orientation that are generated in the expansion region (Figures 5b and 5c); the results of such an analysis will be published elsewhere.22

-0.2 -50 -40 -30 -20 -10 0 10 20 30 40 50 Distance Downstream of Expansion (mm)

Figure 6. Anisotropy factor measured as a function of axial position along centerline in 1:4 slit-expansion flow of Xydar SRT-900 at a flow rate of 5.1 g/min. The symbols represent measurements taken using two different spacers, in which the position of the expansion was shifted in the extrusion die.

387

15

10 : Ξ

5 'ζ Ζ

0 ~z

-5 1 1

•10 - Ξ

-15 Γ ι ι ι ι I » ι ι ι Ι ι ι ι ι I i ι ι ι I r ι r ι I ι ι ι ι l ι ι ι ι > ι ι ι ι I ι ι ι ι I 0 5 10 15 20 25 30 35 40 45

Distance Downstream of Expansion (mm)

Figure 7. Vector plot representing average orientation state in the expansion region, measured in 1:4 slit-expansion flow of Xydar SRT-900, at a flow rate of 5.1 g/min. The length of each vector is proportional to the anisotropy factor, while the direction gives the average molecular orientation.

Summary

These data illustrate how in situ x-ray scattering allows detailed, quantitative measurements of the orientation state in thermotropic liquid crystalline polymers under flow. Unlike model lyotropic LCPs, the model thermotropic PSHQ-6,12 and

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DHMS-7,9 samples studied here in steady simple shear flow exhibit fairly simple orientation behavior as a function of temperature and shear rate. It appears that, within the nematic phase, main chain thermotropes with the mesogen-flexible spacer architecture may generally exhibit flow aligning dynamics. At the same time, DHMS-7,9 exhibits an unexpected flipping transition in orientation, apparently due to a transition to an intermediate phase at temperatures between 90 and 120eC. Within this phase, shear at high rates is able to reverse the flipping, and restore average orientation along the flow direction. In channel flows of commercial thermotropes, we have found that the superposition of extension and shear leads to complex, mixed orientation states. In a slit-expansion flow, extension transverse to the prevailing flow direction leads to a sharp reduction in orientation, and a field of average orientation that varies in a complex way with position.

Acknowledgments

Financial support was provided by a MURI grant on Liquid Crystals, sponsored by the Air Force Office of Scientific Research. We gratefully acknowledge C. D. Han for providing the PSHQ-6,12 polymer, W. Zhou and J. Kornfield for providing the DHMS-7,9 polymer, and B. Dean of Amoco for providing the Xydar resin. We also thank Franklin Caputo for assistance with the experiments. X-ray scattering measurements were performed at the DuPont-Northwestern-Dow Collaborative Access Team (DND-CAT) Synchrotron Research Center located at Sector 5 of the Advanced Photon Source. DND-CAT is supported by the E.I. DuPont de Nemours & Co., the Dow Chemical Company, and the National Science Foundation through Grant DMR-9304725 and the State of Illinois through the Department of Commerce and the Board of Higher Education Grant IB HE HECA NWU 96. Use of the Advanced Photon Source was supported by the U.S. Department of Energy, Basic Energy Sciences, Office of Energy Research under Contract No. W-3M02-Eng-38.

References

1. Donald, A. M. ; Windle, A. H. Liquid Crystalline Polymers; Cambridge University Press: Cambridge, 1992.

2. Marrucci, G.; Greco, F. Adv. Chem. Phys. 1993, 86, 331. 3. Leslie, F.M. Adv. Liq. Cryst. 1979, 4, 1. 4. Larson, R.G.; Doi, M . J . Rheol. 1991, 35, 539. 5. Magda, J. J.; Baek, S.-G.; DeVries, K. L.; Larson, R. G. Macromolecules

1991, 24, 4460. 6. Burghardt, W.R. Macrom. Chem. Phys. 1998, 199, 471. 7. Bedford, B.D.; Burghardt, W.R. J. Rheol. 1994, 38, 1657. 8. Bedford, B.D.; Cinader, D.K. Jr.; Burghardt, W.R. Rheol. Acta 1997, 36, 384. 9. Bedford, B.D.; Burghardt, W.R. J. Rheol. 1996, 40, 235. 10. Cinader, D.K. Jr.; Burghardt, W.R. Polymer, 1999, in press.

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11. Srinivasarao, M . ; Garay, R.O.; Winter, H.H.; Stein, R.S. Mol. Cryst. Liq. Cryst. 1992, 223, 29.

12. Han, C.D.; Chang, S.; Kim, S.S. Mol. Cryst. Liq. Cryst. 1994, 254, 335. 13. Gillmor, J.R.; Colby, R.H.; Hall, E.; Ober, C.K. J. Rheol. 1994, 38, 1623. 14. Chang, S.; Han, C.D. Macromolecules 1997, 30, 1656. 15. Ugaz, V.M. ; Burghardt, W. R. Macromolecules 1998, 31, 8474. 16. Dadmun, M.D; Han, C.C. Macromolecules 1994, 27, 7522. 17. Romo-Uribe, A; Windle, A.H. Macromolecules 1996, 29, 6246. 18. Mitchell, G.R.; Windle, A.H. in Developments in Crystalline Polymers - 2,

Bassett, D.C., Ed.; Chapter 3; Elsevier: London, 1988. 19. Kornfield, J.A.; Zhou, W.; Burghardt, W.R. Paper 127j, AIChE National

Meeting, November, 1998. 20. Dadmun, M.D.; Clingman, S.; Ober, C.K.; Nakatani, A.I. J. Polym. Sci., Part

B: Polym. Phys. 1998, 36, 3017. 21. Cinader, D.K. Jr.; Burghardt, W.R. Macromolecules, 1998, 31, 9099. 22. Cinader, D.K. Jr.; Burghardt, W.R. to be published.

Chapter 25

X-ray Rheology of Structured Polymer Melts Geoffrey R. Mitchell and Elke M. Andresen

Polymer Science Center, J. J. Thomson Physical Laboratory, University of Reading, Whiteknights, Reading RG6 6AF, United Kingdom

X-ray rheology involves the use of in-situ time-resolving x-ray scattering procedures to evaluate the changes in molecular organization, which accompany the imposition of flow and the cessation of flow. Experimental procedures are introduced which facilitate such measurements in which particular emphasis is placed on the relationship between the three-dimensional nature of the flow field and the scattering geometry. These procedures are used to quantify the changes in global preferred orientation for various liquid crystal polymer systems based on hydroxypropylcellulose when subjected to shear flow. These results reveal that although the molecular weight distribution is identical for all three materials studied, there is considerable variation in the response between thermotropic and lyotropic systems. These variations are attributed to the changing balance between viscous and elastic stresses.

X-ray rheology involves the in-situ study of soft materials subjected to flow fields using time-resolving X-ray scattering procedures. Such procedures enable the structural reorganization of complex fluids both during flow and following the cessation of flow to be quantitatively evaluated and related to mechanical rheological measurements. X-ray scattering provides a quantitative probe of many length scales through the use small-angle and wide-angle scattering techniques. These can provide quantitative structural parameters such as global orientation parameters, crystallinity, correlation lengths as well as specific molecular and crystal structure details. A particular advantage of x-ray scattering procedures over the more conventional probes used in rheo-optic experiments, is the ability to evaluate multiphase systems which are strongly light scattering.

390 © 2000 American Chemical Society

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In this article we consider the exploitation of x-ray rheology procedures in the study of structured polymer melts and in particular in the comparison of the response of lyotropic and thermotropic liquid crystal polymers to shear flow. Since such materials have relatively long relaxation times, we expect that considerable structural reorganization will take place at modest shear rates. In particular we explore the influence of the phase type on the response of liquid crystal polymers based on hydroxypropylcellulose (HPC) to steady state shear flow. Liquid crystal polymers are characterized by a significant level of long range orientational order. Molecular segments cluster about a common direction termed the 'director' as shown in Figure 1, the level of orientational clustering reflecting the microscopic order parameters <S2n>. In a polydomain system, there will be a distribution of the directors to give a globally isotropic sample, despite the high level of local anisotropy as shown schematically in Figure 2. Such a sample will contain many orientation-based defects.

Figure 1. A representation of the orientational order in a nematic liquid crystal structure, the rods represent chain segments, the arrow represents the common ordering direction, the director n.

The application of a flow field may lead to the generation of single director distribution and the result is referred to as a monodomain (Figure 2). In x-ray scattering measurement studies, we are only able to access the global or total level of preferred anisotropy described by the normalised amplitudes of teh speherical harmonics, <P2n>> which are a convolution of the microscopic order with the director distribution, <D2n>. It is expected that, in general, the microscopic order parameter is largely unchanged by the flow field except close to defects. As a consequence the measurement of the global preferred orientation will reflect the changes in the director distribution. We obtain the parameters, <P2n>, from the intensity 1(a) at a constant value of IQI, where \Q\=4ns'mQ/X, 2Θ is the scattering angle and λ the incident wavelength, after removal of the background scattering using the following relationship:

η

J I(oc)P2n (cosa) sinada 0 0

Till

P2

m

n \I{a)smada 0

392

where P 2 n

m are coefficients which are related to the scattering for a perfectly aligned system. For the rod-like systems considered here P 2

m = -0.5 and P4

m=0.375. The reader is directed elsewhere for a more detailed discussion of the use of x-ray scattering procedures to evaluate the level of preferred orientation in liquid crystal polymers [1].

Section of a Section of a polydomain texture monodomain texture

Figure 2. A representation of the different director distributions in polydomain and monodomain samples of a nematic liquid crystal system

Materials

α

I 1 1 1 • 1 1 • • • 1 1 ι ι ι ι 1——' -i 1

0 1 2 3 4 Q /A"'

Figure 3. A plot of the scattering intensity I (Q) as a function of \Q\ for a polydomain sample ofPPC in the liquid crystal state at 20°C.

We have selected three closely related liquid crystal polymer systems for these studies. Each is based on HPC and in particular on Aqualon Klucel Grade Ε with a nominal molecular weight of 80,000. Aqueous solutions of HPC exhibit liquid crystal phases for concentrations greater than 45% w/v (2) at room temperature. Under

393

quiescent conditions, these solutions are chiral nematic although under shear flow, it is commonly assumed that the chiral helix unwinds in the flow field (3). The same material exhibits a mobile liquid crystal phase for temperatures above 160°C with a clearing temperature of ~ 205°C (4). We have also studied the propanoate ester of HPC prepared by the reaction of propionyl chloride with Klucel Ε (5). This material (PPC) exhibits lyotropic phases with acetone as the solvent and a thermotropic phase with a clearing temperature of ~166°C, although a strong biphasic region develops at ~ 144°C. It has been suggested that the additional side-chain structures in such HPC derivatives act as an internal plastisizer and in this work we are able to study this material at temperatures deep in the nematic phase at 55 - 110°C. Each of these materials provides a similar broad Q x-ray scattering curve; that for a polydomain sample of PPC in the thermotropic phase is shown in Figure 3. There are diffuse peaks at IQI ~ 0.44, 1.36 and 3À"1 typical of a disordered polymeric material. The diffuse peak at IQI0~ 0.4Â"1 arises from correlations between neighbouring main-chain segments. The precise peak position depends on the chemical configuration of the side-chains and/or the concentration of the solution (6). The azimuthual dependence of the intensity of this peak at constant IQI will be used to evaluate the global orientation parameters <P2>, <P4> using standard procedures described earlier and elsewhere(l).

Experimental Procedures

y Velocity gradient

L ^

ζ

Flow direction

Vorticity vector

Figure 4. The shear flow geometry between one fixed and one moving parallel plates.

The design of shear flow cells for in-situ x-ray scattering must take account of both the three-dimensional nature of the shear flow field and the scattering geometry. Figure 4 shows a schematic of shear flow between parallel plates. The simplest scattering geometry is for the incident beam to lie parallel to the velocity gradient (VV) using relatively x-ray transparent material such as mica for the parallel plates. In this situation, the scattering vector Q is tilted out of the plane containing the flow vector ν and the vorticity vector. In fact, Q lies on the surface of a cone with a semi-angle of 90-Θ about the incident beam as shown in Figure 5. Thus, for small values of Θ, the x-ray beam probes the structure in the plane defined by ν and the vorticity vector. However, when peaks in the wide-angle scattering regime are of interest, such as from semi-crystalline polymers, this out-of-plane tilt may make interpretation of

394

the scattering with respect to the flow geometry a rather complex matter. For the materials studied here 2Θ ~ 5° and hence the x-ray scattering effectively probes the structure in a plane parallel to the parallel plates.

2

] Flow direction

1 Scat te red beams Figure 5. The relationship between the scattering vector, Q, and the flow geometry

We utilize the azimuthual distribution of intensity I (a) to evaluate the level of preferred orientation. Each point on the circle, at constant 2Θ value but with varying a, corresponds to a specific orientation of the scattering vector Q. Inspection of Figure 5 shows that as the scattering vector is slightly tilted out of the plane, it is no longer possible to access the situation where the angle between the flow direction and Q is zero. This is a basic problem with fixed flat detectors such as image plates, film and CCD based detectors. For small scattering angles, as is the case here, the missing data can be safely ignored. However, in many cases, this issue must be addressed if reliable orientational data is to be obtained.

Figure 6 shows a schematic of a shear flow x-ray cell developed at the University of Reading to facilitate the geometry described above (7). The system consists of two stainless steel parallel plates; one rotating and one fixed in which the separation is defined with a phosphor-bronze ring. The rotating plate has slotted windows covered in mica which allow transmission of the incident beam for ~ 90% of each revolution. The fixed plate has a single aperture also covered with a mica sheet which allows the scattered intensity to exit the shear cell for scattering angles < 40 degrees. Mica exhibits modest absorption at wavelengths ~ 1.5À, but has the particular advantage that the scattering from the mica is localized in single crystal type diffraction spots. These can be easily identified in the patterns. If they arise from the window on the moving plate, the intensity in the region of the diffraction spots must be obtained by extrapolation. It is not sufficient to subtract a "empty cell", since the diffraction spots will rotate with the window. The parallel plate arrangement results in a linear variation of the shear rate across the radius of the shear cell. The shear rate resolution of the x-ray scattering procedures, depends upon the effective beam size to the plate radius at the incident beam. In this cell, that particular radius is - 9mm and with a laboratory source the shear rate

3 9 5

resolution is ~ 3% while with the focused beam available at a synchrotron source, the resolution is - 1%. The rotating plate is driven by a stepper motor and encoder system and provides for continuous, stepped and oscillatory flow fields. In previous work on lyotropic liquid crystal polymers (8) we utilized a similar arrangement in which the plate radius was ~ 32mm with an accompanying greater shear rate resolution. Similar rotating shear cells have been used by Romo-Uribe et al (9) and by Hongladarom et al (10). Picken developed an oscillating linear motion shear cell, which has advantages in terms of a uniform shear rate and single global flow direction, although there is an obvious limitation in the shear strain which can be applied before reversing the direction of flow (11).

Figure 6. Schematic of the shear flow cell used for x-ray rheology studies.

Alternative geometries, which evaluate the structure in the planes defined by either y and VV, or W and the vorticity vector, are possible but are not so convenient to implement, particularly in the wide-angle scattering regime. One approach is to use a Couette cell as shown in Figure 7. In this geometry the sample is sandwiched between a fixed inner cylinder and a rotating outer cylinder. If the incident beam is directed along the centre line (Figure 7), the scattering probes the structure in the plane defined by y and the vorticity vector in a similar manner to the parallel plate geometry. However, if the incident beam is located at the tangent to the midpoint between the cylinders as shown in Figure 7, the scattering probes the plane defined by VV and the vorticity vector. Such an arrangement can be used to probe the three dimensional nature of the molecular organization under flow. For example it can confirm or exclude the possibility of a near uniaxial preferred orientation at high shear rates. We have developed such a cell for the study of structured melts in the wide-angle regime (12). The tangential configuration has the complexity of

396

anisotropic absorption corrections, if quantitative evaluation of the intensity data is to be made (12). We have also explored the use of a parallel plate system in which the incident beam lies in the plane of the plates (Figure 8) (13). By placing the incident beam along the center line, the scattering vector lies largely in the plane containing the VV and v, whilst directing the incident beam along the tangential line results in the scattering vector lying in the plane defined by VV and the vorticity vector as shown in Figure 8. This geometry has the particular advantage that by placing the incident beam at intervals between these two limiting configurations the complete three-dimensional nature of the molecular organization can be evaluated. However, a major disadvantage is one of absorption, either in terms of the path length or the anisotropy of the absorption correction as with the Couette cell.

rotating cylinder Figure 7. The flow and scattering geometry for a Couette style flow cell

In this contribution, we focus on results obtained using the parallel plate geometry shown in Figure 6 in which the structure probed is that which lies in the plane defined by the flow vector and the vorticity vector.

To record the x-ray scattering patterns in a time-resolving manner we have utilized a Photonic Science CCD based detector system coupled with a Data Translation frame grabber system. The detector was part of a larger integrated system which facilitated specific programmed sequence of flow and synchronized data collection (14). This enables data collection cycles of less than Is, although in this work, cycle times of ~ 10s were employed. The detector has an active face of ~ 50mm in diameter which allows the full azimuthual dependence of the scattering to be obtained with the incident beam centered on the detector face, although of course the IQI range was somewhat limited. In this work we only recorded the diffuse peak at IQI ~ 0.4 Â"1. Some steady state measurements were made using a laboratory based x-ray source; namely a Cu targeted sealed tube running at 1.6kW with an incident beam monochromator and pinhole collimation. Time-resolving measurements exploited the intense x-ray beam available at the beamline 16.1 of the CCLRC Daresbury Synchrotron. This is a fixed wavelength beam-line mounted on a 6T wiggler with a bent Germanium focusing monochromator and a Quartz mirror.

397

Figure 8. An alternative flow cell geometry to explore the three-dimensional consequences of the flow field

Results

0.30

0.25

0.20

0.15 1 0.10

0.05

0.00

Shear

0

Relaxation

2000

t i m e /s

4000

Figure 9 Values of <P2> obtained for a sample of PPC at 65 °C and at particular times after the imposition of a flow field with a shear rate of 2.5s'1 and after the cessation of the flow field as indicate in the figure.

Figures 9 and 10 show the results from one experiment on PPC deep in the thermotropic phase at 65°C. A steady state shear flow of 2.5s"1 is applied and the level of global preferred orientation is evaluated from a series of x-ray scattering patterns recorded with a cycle time of ~15s over a period of ~ 1000s. Using the azimuthual variation of intensity of the diffuse peak at IQI ~ 0.4 A 1 , values of the global orientation parameters <P2> and <P4> are obtained as a function of time. A value of <P2>, <P4> = 0 indicates a completely isotropic state, while <P2>, <P4> = 1 indicates a fully aligned system. A nematic liquid crystal systems exhibits microscopic order

398

parameters in the range of 0.3 to 0.7 and hence if there is a single director alignment, the global value should not exceed those values.

As Figures 9 and 10 show, a steady level of preferred orientation is achieved within ~ 100s and this is maintained over some 1000s. In general it is found that a steady state is reached within 100-200 strain units (i.e. shear rate χ time) in line with mechanical rheological measurements. At a particular point, the shear flow is halted and the state of preferred orientation is monitored as a function of time following the cessation shear flow. The level of preferred orientation drops rapidly with time, exhibiting a single relaxation time behavior with τ ~ 100s. The values of <P4> which are obtained during steady state shear are ~ 1/3 of the equivalent <P2> values in line with simple models of nematic ordering. <P2> and <P4> represent two components of the orientation distribution and care must be taken not to draw specific conclusions from a single component. However, it is interesting to note that the value of <P4> changes sign during the relaxation stage, possibly reflecting a change in the shape of the orientation distribution. The availability of higher order orientational parameters allows for the reconstruction of the orientational distribution (1) and for a more sensitive test of molecular theories of liquid crystal flow. Although we have made use of a uniaxial description of preferred orientation, we do not know how correct it is to make such an assumption about the molecular organization. To answer such issues we must make use of the alternative flow/scattering geometries shown in Figures 7 & 8. Preliminary measurements using the geometry shown in Figure 8 and lyotropic solutions of HPC suggest that the preferred orientation is uniaxial at high shear rates in that similar values of <P2> are recorded when the orthogonal plane is evaluated (13). Figure 9 Values of <P2> obtained for s sample of PPC at 65°C and at particular times after the imposition of a flow field with a shear rate of 2.5s"1 and after the cessation of the flow field as indicate in the figure

0.15 ι

« .10

0.05 -J.

0.00

-0.05

-0.10 ·

Shear Relaxation

2000

time /s

Figure 10 Values of <P4> obtained for a sample of PPC at 65°C and at particular times after the imposition of a flow field with a shear rate of 2.5s'1 and after the cessation of the flow field as indicate in the figure.

399

By performing a series of measurements as in Figures 9 & 10 but for different shear rates and differing temperatures we can build up a map of the flow behavior. Figure 11 shows a plot of the values of <P2> obtained in steady state flow for the shear rates shown at temperatures of 65, 85 and 110°C. Note that the shear rate is plotted on a logarithmic scale.

Figure 11 Valuesof<P2> obtained for PPC at 65 °C ( O), 85°C ( Φ) and 110°C ( V) during steady state shear

0.6 A

Q_ V 0.3 A

0.1

10 100 1000

Shear rate /s"1

Figure 12 Values of<P2> obtained for a 55% w/v aqueous solution of HPC at 25°C during steady state shear flow ( 16)

400

These curves show three regimes. At low shear rates, there is almost no preferred orientation, whilst at high shear rates, there appears to be a plateau level of <P2> ~ 0.4. Between these two points there is a more or less linear increase of <P2> with the logarithm of the shear rate. The data sets for three temperatures appear rather similar apart from an offset along the shear rate axis.

V

0.6

0.4

0.2

0.0

170°C 180°C 185°C 190°C 195°C

0

V •

0.1

9

é

•

i s

4 V •

1

shear rate /s

•π—

10 ι

100

Figure 13 Values of<P2> obtained for HPC in the thermotropic liquid crystal range during steady state shear flow (17). Temperatures corresponding to the different symbols are given in the inset.

Such data has similarities with earlier studies on lyotropic aqueous solutions of HPC (15,16). The equivalent orientational behaviour in steady state shear flow for a 55% w/v aqueous HPC solution is shown in Figure 12. These data also show three regimes although the level of preferred orientation at low shear rates is clearly greater for the aqueous solution, and the distinction between regimes 2 & 3 are blurred. Lyotropic HPC and thermotropic PPC have considerably different concentrations of liquid crystal forming molecule (55% and 100%), yet their responses to the flow field are similar. It is emphasized that we can make direct comparisons between these two materials since the derivative PPC and the lyotropic solutions were prepared from the polymer; Klucel Ε which was used to prepare the aqueous solutions.

Figure 13 shows the equivalent data obtained for Klucel Ε in the thermotropic phase range (17). The level of preferred orientation is strongly dependent on the temperature. There appears to be no low orientation regime at low shear rates as observed for both the lyotropic HPC and the thermotropic PPC. Since the experiments on thermotropic HPC were performed close to the clearing temperature, the microscopic order parameters will change rapidly with temperature. At high shear

401

rates the material may be close to a monodomain texture and hence the recorded orientation parameters <P2> are essentially the microscopic order parameters. Consideration of the <P2> values shown in Figure 13 show that they do indeed reflect the changes expected from simple models of nematic ordering. Inspection of Figures 12 and 13 shows that the high shear rate values of <P2> for lyotropic and thermotropic HPC are similar, it is the low shear rate behaviour which is different.

We have considered the response to shear flow of three different polydomain textured liquid crystal polymers which have the same molecular length characteristics and might be expected to have similar molecular stiffness. Despite these molecular similarities, the response to shear flow is rather varied. From the viewpoint of mechanical rheology, the lyotropic HPC system differs from the thermotropic HPC and the thermotropic PPC, in that it exhibits two sign changes in the first normal stress difference (18). Such negative first normal stress differences are thought to be associated with the transition from a director tumbling regime to a director-wagging regime. The orientational behaviour of the lyotropic HPC is entirely consistent with such a model. At low shear rates there is extensive tumbling and an increasing level of defects which eventually lead the inhibition of tumbling (19) and the adoption of a flow-aligning regime. Baek et al (18) have speculated that the absence of negative first normal stress differences in thermotropic HPC may be due to increased polymer-polymer interactions which are inevitable in a concentrated system.

Director tumbling is predicted from molecular theories of rigid rod systems subject to shear flow and depends critically on the characteristics of the microscopic order parameters S 2 and S4. The equation below shows the relationship for the tumbling parameter, β, proposed by Larson (20).

When β > 1 the system is flow aligning and if β < 1 the system is tumbling. Using the high shear rate experimental orientation parameters as indicative of the microscopic order parameters we derive values of β ~ 0.9 for the lyotropic HPC, β~1.2 for the thermotropic HPC and β~1.04 for a flow aligning low molar mass nematic system (21). The thermotropic PPC system does not seem to approach a monodomain texture at high shear rates, as may be deduced from the relatively low values of <P2> and from the shear history independent relaxation behaviour (5), and hence we have not attempted to calculate a value for β from the data presented. We must treat these values of β with some caution, since we can not be certain that the director distribution is uniform. Moreover, the expression given for β is itself a simplification based on the first two components of a series representing the microscopic orientational order; values of S6 and S8 must be significant as indeed the x-ray scattering data shows.

There are many factors which can influence the response of a liquid crystal polydomain texture to shear flow. Rey (22) has recently calculated a comprehensive map showing the variation in orientation modes for an initially monodomain sample,

402

subjected to shear flow, as a function of the Ericksen number, which is the balance between viscous and elastic stresses and a parameter describing the ratio of short-range to long-range elasticity. The application of this model or other molecular theories, to the current experimental data, is hampered by the unavailability of a rigorous approach to realistic polydomain systems. We assume that, for the three materials considered here, the ratio of short-range to long-range elasticity is similar since the underlying molecular architectures are identical. As a consequence, variation in the response to shear flow for each material must arise from variations in the Ericksen number.

We can obtain a measure of the relative viscosities by consideration of the relaxation behaviour following the cessation of shear flow. Figure 14 shows the variation in <P2> for a sample of PPC at 65°C following the cessation of a shear rate of 2.5s"1 fitted with a single time constant exponential decay. The time constants obtained are independent of the previous shear rate but strongly depends on the temperature (5). We find that the ratios of these time constants for different temperatures are qualitatively in accord with the critical shear rates for the same temperatures (Figure 11) at which substantial global orientation development develops. This suggests that the similar but shear rate sifted orientation data for PPC (Figure 11) arise from variations in the relaxation times due to differences in temperature. It seems reasonable to view the major differences between lyotropic and thermotropic HPC in a similar manner. The low shear rate variation is not simply a matter of concentration, but reflects the differences in the balance between viscous and elastic stresses.

0.35 -i

0.30 i 0.25 ί

αΓ' o.i5 \ λ ° - 2 0 !

0.00 i 0.05 4

0.10

0 1000 2000 3000

time /s

Figure 14 Values of<P2> obtained for a 55% w/v aqueous solution of HPC at 25°C following the cessation of shear flow (5). The solid line represents a single time constant exponential decay.

403

Summary

We have shown that X-ray rheology procedures can provide useful structural information on the influence of shear flow on the textures of liquid crystal polymers. It is clear that such reorganization is complex and there will be considerable value in developing techniques, which can more directly yield information on the three-dimensional nature of the flow and the changes in molecular organization. The time-responses available using a synchrotron source coupled with a CCD based detector are ideally suited to this type of measurement. Although the three material systems studied here were based on the same underlying hydroxypropylcellulose architecture, the response to shear flow is varied. These variations are not simply related to the whether the material is lyotropic or thermotropic and we attribute the major part of these variations to the changing balance between viscous and elastic stresses.

Acknowledgements

This work was supported by the Engineering and Physical Research Council. The synchrotron studies were performed at the CCLRC Daresbury SRS. We thank Jim Woodcock and Clark Balague for the essential design and construction of the flow stages and Aqualon Ltd for the provision of the samples of Klucel E.

References

1. Mitchell G.R. and Windle A.H. in Developments in Crystalline Polymers-2 1988, Editor D.C.Bassett Elsevier:London Chapter 3

2. Werbowwji R.S. and Gray D.G. Macromolecules 1980 13 69 3. Onogi A. and Asada T. in Rheology Vol 1 edited Astarita G., Marrucci G. and

Nicolais L. Plenum Press New York 1980 p127 4. Shimamura K., White J.L and Fellers J.F J.Appl. Polym. Sci. 1981 26 2165 5. E.M.Andresen and G.R.Mitchell submitted to Macromolecules 6. Keates P.A., Peuvrel E. and Mitchell G.R. Polymer Communications 1992 33

3298 7. G.R.Mitchell, J.A.Pople, E.M.Andresen and P.G.Brownsey Advances in X-Ray

Analysis 1997 41 in press 8. Keates P., Mitchell G.R., Peuvrel-Disdier E., Riti J.B., and Navard P. J. Non-

Newtonian Fluid Mechanics 1994 52 197 9. Romo-Uribe A. and Windle A.H. Macromolecules 1996 29 6246 10. Hongladsrom K., Ugaz V.M. , Cinader D.K., Burghardt W.R., Quintana J.P.,

Hsiao B.S., Dadmun M.D., Hamilton W.A. and Butler P.D. Macromolecules 1996 29 5346

11. Picken S.J., Aerts J., Visser R. and Northolt Macromolecules 1990 23 3849 12. Mitchell G.R. Rheologica Acta 1999 13. Andresen E.M., Pople, J.A. and Mitchell G.R. unpublished work 14. Pople J.A., Keates P.A. and Mitchell G.R. J. Synchrotron Rad. 1997 4 267

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15. Keates P.A., Mitchell G.R., Peuvrel-Disdier E., and Navard P., Polymer 1993 34, 1316

16. Keates P.A., Mitchell G.R., Peuvrel-Disdier E., and Navard P., Polymer 1996 37, 893

17. Andresen E.M. and Mitchell G.R. Europhysics Letters 1998 43 296 18. Baek S.-G., Magda J.J., Larson R.G. and Hudson S.D. J Rheology 1994 38 1473 19. Marrucci G. and Greco F Adv. Chem,. Phys. 1993 86 331 20. Larson R.G. and Mather P.T. in Theoretical Challenges in the Dynamics of

Complex Fluids edited by McLeish T.M. Kluwer Dordrecht 1997 21. Pople J.A. and Mitchell G.R. Liquid Crystals 1997 23 467 22. Rey A.D. Phys Rev Ε 1998 57 5609

Chapter 26

Phase Separation Kinetics during Shear in Compatibilized Polymer Blends

Alan I. Nakatani

Polymers Division, National Institute of Standards and Technology, Gaithersburg, MD 20899

The effect of a diblock copolymer on the phase separation behavior of a polymer blend during shear was examined by light scattering. The phase separation process in the mixtures was initiated by a slow cooling of the sample from the one-phase region into the two-phase region and various shear rates were applied to the samples simultaneously. These studies were designed to imitate behavior which may be encountered during injection molding or other manufacturing processes where mechanical deformation and phase separation may be superimposed on one another, thereby influencing the resultant blend morphology. Results were obtained for a model polystyrene and polybutadiene blend as a function of shear rate for two different final temperatures (shallow and deep quenches) in the two-phase region. Similar experiments were performed on a blend with a mass fraction of 2.5 percent of a symmetric diblock copolymer added. For the shallow quench, the pure blend exhibited an enhancement of the coarsening rate during shear over the quiescent coarsening rate while the modified blend showed a suppressed coarsening rate during shear compared to the zero shear coarsening behavior. For the deeper quench, the pure blend exhibited a crossover in the coarsening from shear induced droplet breakup at low shear rates to shear enhanced coalescence at higher shear rates. The opposite behavior was observed for the modified blend, namely, shear enhanced coalescence at low shear rates, and shear induced droplet breakup (or suppressed coalescence) at higher shear rates.

© U.S. government work. Published 2000 American Chemical Society 405

406

Introduction

Polymer blends and compatibilizing agents historically have been the subject of a wide variety of studies and an extensive body of literature on these materials exists. Without specific chemical interactions between dissimilar polymers, most polymer mixtures tend to phase separate due to the unfavorable entropy of mixing between the polymer chains. Efforts to control or retard the phase separation process have led to the research and development of compatibilizing agents for polymer blends. For a variety of systems the dispersed phase particle size has been found to decrease with increasing copolymer concentration.1 Above a critical concentration of copolymer, the size of the dispersed phase remains constant.

An analogy is often drawn between block copolymers as compatibilizing agents in polymer blends and amphiphilic molecules utilized for stabilization of emulsions. A considerable amount of research on employing block copolymers to modify the phase behavior and morphology of polymer blends has been conducted over the last twenty years. Block copolymers have also been found to improve mechanical properties of polymer blends by improving the interfacial adhesion between phases. Majumdar and coworkers2,3 have examined the effects of compatibilizing agents on the morphology of polymer blends as well as the mechanical properties of the compatibilized blends. Optimization of processing conditions to produce the best material properties have also been examined.4,5

Recent studies in polymer blend behavior have focussed on the influence of simple shear fields on polymer blend morphology. The final morphology resulting from processing of polymer blends has a direct influence on the bulk mechanical properties of the materials. Similarly, extensive research concerning the effect of block copolymers on the resultant size of the domain morphology in polymer blends during processing has been conducted. Polymer blend behavior during shear and the effect of compatibilizers have been examined most commonly by utilizing extruders or other mechanical mixing devices and quenching the extrudate to stabilize the blend morphology. Through subsequent microscopic examination of the extrudate, the particle size distribution within the blend can be measured. These quenching studies are limited to the final morphology of the extrudate and the in-situ behavior in the mixing chamber, where the interplay of droplet breakup in the high shear region and droplet coalescence in the low shear region, has not been studied. Shear light scattering is a technique which has been developed in recent years which may be able to address the issue of temporal evolution of polymer blend morphology during simple shear. The work cited below has particular relevance to the experiments conducted in this report. For more general reviews of the area of polymer blends and block copolymers as compatibilizing agents the interested reader is directed to the cited work and the references listed therein.

Most studies of the behavior of block copolymers as compatibilizing agents consider two opposing effects during deformation: a reduction in critical droplet size due to a reduction in the interfacial tension (droplet breakup) proposed by Taylor, and an increase in droplet size due to increased collision frequency between droplets (droplet coalescence) studied by Smoluchowski. The problem of droplet breakup in a

407

flow field was addressed by Taylor in the 1930's.6,7 Taylor predicted a critical droplet size, Rc, as a function of shear rate, γ, the interfacial tension, σ, the viscosity of the medium, η, and the critical capillary number, Ca*:

Rc = C a V y T | ) . (1)

From Equation 1, it is apparent that by reducing the interfacial tension, or by increasing the shear rate, the critical droplet size decreases. Since the Taylor approach is only applicable to a single droplet suspended in a Newtonian fluid, multiple droplet interactions and non-Newtonian viscosities are not considered. Despite these drawbacks, the Taylor approach remains one of the most commonly used models for predicting the sizes of dispersed droplets during shear.

According to Smoluchowski,8'9 the collision frequency between droplets, J^, ignoring effects due to Brownian motion of the droplets and density driven diffusion is given as:

J 1 2 = (4/3)n1n2y(a1+a2)3 (2)

where nj are the droplet concentrations of a given size, aj. Therefore, the collision frequency between droplets increases as the shear rate, concentrations, or the sizes of the droplets increase. Milner and X i 1 0 have addressed the balance between droplet breakup and collision induced coalescence in a twin screw extruder for a polymer blend with block copolymer formed by a grafting reaction between the two homopolymers. They concluded that the principal mechanism for copolymers to promote compatibilization in polymer blends is suppression of droplet coalescence during collisions as opposed to droplet breakup due to a reduction in the interfacial tension. Milner and X i also concluded that droplet breakup was prevalent in the high shear regime and droplet coalescence dominated in the low shear regime of the extruder. S0ndergaard and Lyngaae-J0rgensenn have utilized a rheometer equipped with light scattering instrumentation to examine the behavior of polystyrene and poly(methyl methacrylate) blended with a diblock copolymer of the two homopolymers. They found that coalescence increases with shear rate up to a critical shear rate, which is in agreement with the predictions of Milner and X i .

Experimental work by Sundararaj and Macosko,12 nicely contrasted the competing effects of droplet breakup and droplet coalescence in both Newtonian and non-Newtonian mixtures. They concluded that the extent of interfacial tension reduction due to the presence of block copolymer was insufficient to be the primary reason for the reduction of the droplet size, and the primary effect of the copolymer was to prevent droplet coalescence through steric stabilization of the droplets. Sundararaj and Macosko also noted that the droplet size as a function of shear rate for a pure blend decreased to a minimum value, then increased at higher shear rates. No data was given for the compatibilized blend and they referred to this shear rate dependence of the dispersed phase size for the pure blend as "anomalous". Sundararaj and Macosko noted that this anomalous behavior has been observed previously by

408

other researchers.13,14,15 The work by Favis and Chalifoux14 suggested that the viscosity ratio between the two phases was a factor. This result is opposite to the results and predictions for the compatibilized blends discussed in the previous paragraph. A study on polypropylene/ polycarbonate blends in twin screw extrusion by Favis and Therrien16 found no effect of screw speed on the size of the dispersed phases. This result also appears to conflict with the results described above, however the range of screw speeds utilized by Favis and Therrien only varied by a factor of two, and it may be possible that even wider variations in the screw speed would result in a variation of the dispersed phase size.

Most commercial polymer blends are considered to be completely immiscible systems, therefore, the small shifts in the phase boundary due to the addition of the copolymer and the subsequent narrowing of the temperature gap between the phase boundary and typical processing temperatures are not very important. However, in some reactive polymer blends, such as the polycarbonate/polyester and polycarbonate/poly(methyl methacrylate) blends studied by Yoon et al.,1 7 , 1 8 the phase boundary and reaction temperatures are separated by a few degrees. The relationship between the phase separation kinetics and the mechanical deformation during processing becomes much more important in such cases. If a sample is undergoing phase separation and shear is applied during the process, what is the interplay between the driving force for phase separation and the mechanical force balance between droplet breakup and collision induced coalescence? The influence of the copolymer on the dynamics of morphological development under these conditions is also of interest. Scott and Macosko19 have addressed the development of morphology of immiscible and reactive polymer blends in the initial stages of mixing. The phase boundaries of the blends were not specified. The critical time scale for observation of morphological changes occurred during the first one or two minutes of mixing. Favis20 has drawn similar conclusions examining the effects of processing parameters on polypropylene/polycarbonate blends. Virtually all of the reports cited indicate very little change in the size of the dispersed phase with longer mixing times. Work by Shih et al.21 which incorporated temperature variations during processing examined the effects of softening and melting, however, the behavior relative to the phase boundary was not discussed.

In this work, we address the issue of competition between phase separation and mechanical mixing utilizing shear light scattering techniques. We have examined the phase separation behavior of a polymer blend during shear and compared the behavior to the same blend with a symmetric diblock copolymer added as a compatibilizing agent. The component polymers of the block copolymer were the same as the blend homopolymers, and the concentrations of copolymer utilized were sufficiently low that aggregates of the copolymer did not form independently. The phase separation process was initiated by changing the temperature of the samples from the one-phase region to the two-phase region. Concurrent with the change in temperature, a simple shear field was applied to the sample and the light scattering patterns were monitored as a function of time. From the intensities and scattering patterns, the relative coarsening rates in the samples as a function of shear rate and final temperature were determined.

409

Materials

The model blend for these experiments consisted of low molecular weight22

polystyrene (PS) ( M w = 2.0x103 g/mol, M w / M n = 1.19) and polybutadiene (PB) (M n = 2.8xl03 g/mol, Mw/M„ = 1.08). For all blends a fixed mass ratio of PS:PB of 60:40 was utilized. The model compatibilizing agent was a symmetric diblock copolymer of perdeuterated polystyrene (PSD) and PB. The diblock was synthesized by coupling the homopolymer precursors (M n of each precursor = 5.0xl03 g/mol) which were prepared using high vacuum, anionic polymerization techniques by Dr. J. W. Mays of the University of Alabama, Birmingham. Similar methods were used to synthesize the PS, while the PB was purchased commercially from Goodyear Tire and Rubber Co. 2 3

A single diblock copolymer concentration (copolymer mass fraction of 2.5%) was used in these experiments for comparison against the pure blend behavior. The mass ratio of the PS:PB remained constant at 60:40 for the modified blend. The samples were prepared by melting the polymers together at 120 °C in an oven with occasional stirring. Due to the low molecular weights of the polymers, the low glass transition temperatures (Tg) of the polymers, and the low cloudpoint temperature (Tc = 113 °C for the pure blend, 107 °C for the modified blend), the oven conditions were sufficient to have the samples in the one phase region and mechanical mixing was sufficient to produce a homogeneous sample.

Experimental

Al l the shear experiments were conducted on a shear light scattering apparatus with transparent cone and plate fixtures. Details of the apparatus have been published previously.24 The experiments were conducted as shown schematically in Figure 1. Steady shear with cooling is a dynamic, non-equilibrium experiment which is a simple attempt to model manufacturing behavior such as injection molding, where a well mixed sample is injected into a cooler cavity, so that deformation and phase separation are occurring concurrently. In these experiments, the sample was allowed to equilibrate at an initial temperature, T i t in the one phase region of the phase diagram for 30 minutes. The oven controls for the shear light scattering apparatus were then set to a final temperature, T f l in the two phase region of the phase diagram. Simultaneously, a shear rate was applied to the sample as the sample cooled and began to phase separate. During the shearing and change in temperature, light scattering images were taken at uniformly spaced intervals for the duration of the mechanical deformation. After cessation of the shear rate, the oven was set back to Tj and the sample was allowed to remix in the one-phase region before the process was repeated with the next desired shear rate. Due to the large thermal mass of the light scattering instrumentation, the change in temperature was not instantaneous as in temperature jump light scattering experiments typically used to study phase separation kinetics.25 Therefore, instead of a sharp, temperature quench, the sample experiences a slow cooling profile. The temperature profile is reproducible as demonstrated in

410

Time

Figure L Schematic of experimental procedure. The top trace indicates the time dependence of the temperature profile while the bottom trace indicates the time dependence of the shear rate profile. The initial temperature, Tu is in the one phase region of the phase diagram. To start the experiment the oven settings are changed to a final temperature, Tp in the two phase region and shear rate is simultaneously applied to the sample. The arrows schematically represent the data acquisition points during the experiment.

140

600 800

Time (s) 1400

Figure 2, Example of the oven cooling reproducibility. Three separate cooling profiles are shown indicating good reproducibility of the time dependence of the temperature profile. Temperatures were monitored at the wall (upper trace) and the middle of the oven (lower trace).

412

Figure 3. Light scattering patterns for the pure blend at a shallow quench depth (AT - 0.5 °C). Data are in column format with each applied shear rate (0, 5.22, and 52.2 s'1) at the top of each column. The elapsed time of the experiments are given to the

413

left. The flow direction in all patterns is along the vertical axis. Grayscale has been adjusted to provide the best contrast. Within each figure the same grayscale lookup table has been utilized for ease of comparison.

414

Figure 2, showing data from three separate cooling runs between the initial (Tj =125 °C) and final (Tf = 112 °C) temperatures. The temperature was monitored on the interior surface of the oven walls as well as in the interior of the oven next to the cone and plate fixtures.

Results

The steady shear during cooling experiments were conducted for both the pure and modified blends at two different final temperatures, to examine the effect of quench depth. As mentioned in the Experimental section, the sample cooling was not instantaneous, however, the term "quench depth" (ΔΤ) will be used here and defined as the difference between the cloudpoint temperature, T c, and the final experimental temperature, T f. The final temperatures were selected so that approximately equivalent quench depths could be compared for the pure and modified blends.

For the pure blend, at a very shallow quench depth (ΔΤ = 0.5 °C), the time dependence of the scattering at zero shear and two shear rates are shown in Figure 3. The images are organized into columns, with the shear rates (0, 5.22 and 52.2 s l) listed across the top of the figure. The elapsed times are given along the left of the figure, so that the time evolution of the scattering pattern at each shear rate can be followed down each column in the figure. The quench depth is very shallow, as evidenced from the zero shear data, showing a very slight increase in the low angle scattering with increasing time indicating domain growth. The zero shear time dependence shows no evidence of a spinodal-like peak developing in the scattering intensity as observed typically in temperature jump light scattering (TJLS) experiments. This is attributed to the slow cooling rate of these experiments compared to the instantaneous temperature change of the TJLS experiments. At 5.22 s"\ a greater degree of low angle scattering is observed at equivalent times after initiation of the experiment compared to the zero shear case. At 52.2 s*1, an even greater degree of low angle scattering is observed at each time. This enhanced low angle scattering with shear, is attributed to the presence of large size scale droplets. The fact that the scattering intensity during shear is greater at equivalent elapsed times is indicative of a higher rate of phase separation (coarsening rate) during shear. Hence, as phase separation proceeds, the applied shear rate appears to promote coalescence due to the increased collision frequency of the droplets.

Since the changes in the scattering patterns shown are often subtle, sector averages (± 15°) of the scattering images were obtained along the axis normal to the flow direction (the horizontal plane in the images) from which the intensity as a function of scattering angle is determined. The growth rate could be determined as a function of scattering vector, q = (4fcn/X)sin(0/2), where η is the refractive index of the sample, λ is the wavelength of the incident radiation (0.6328 μπι), and θ is the scattering angle. From the theories of Cahn-Hilliard and Cook for phase separation in binary mixtures, the q-dependent growth rate can be obtained from the slopes of plots of ln[I(q,t)] versus time. For the sector averaged data of these experiments, the data was corrected for a baseline contribution to the scattering intensity and normalized to

415

the scattering intensity at t=0. As noted previously, the scattering patterns during these experiments showed no evidence of a spinodal-like ring, and with the gradual temperature change used here instead of a sharp temperature quench, it is not clear that mis type of analysis is entirely appropriate. Also the Cahn-Hilliard and Cook analysis is only strictly applicable to early stages of spinodal decomposition. However, the analysis provides a more quantitative evaluation of the scattering patterns and is capable of providing some revealing insight into the later stages of phase separation during the experiments. Of particular interest are the slope values between the last two data points of the ln{[I(q,t)-Baseline]/[I(q,t=0)-Baseline]} versus time plots, which are referred to as the terminal slopes.

The ln{[I(q,t)-Baseline]/[I(q,t=0)-Baseline]} versus time plots for the conditions shown in Figure 3 are shown in Figure 4. Data for every twentieth point are plotted on the figures. The difference in q varies between points due to the sine dependence of q with scattering angle. It is interesting to note that there is very little q dependence in the growth rates, in contrast to the strong q dependence which is typically observed during spinodal decomposition. The greatest changes are seen in the data for the highest q value plotted, indicated by the filled symbols for ease of comparison. For the pure blend, at the shallow quench depth, the quiescent growth is very slight. However, the coarsening rate increases with shear rate as indicated by the greater slope at longer elapsed times.

The results for the pure blend, at a ΔΤ of 2 °C, are shown in Figures 5 and 6. At this deeper quench depth, the quiescent coarsening proceeds at a much faster rate, demonstrated by the higher scattering intensities at equal times after the start of the experiment compared with the shallow quench. The data taken at 5.22 s"1 shows a lower total scattering intensity compared with the zero shear case, as well as some anisotropy in the scattering at the longest times. This relative decrease in scattering intensity is ascribed to fewer and smaller phase separated domains being present. Thus for the deeper quench, the applied shear rate appears to suppress the coarsening of the sample. By comparison, the data at 52.2 s show a greater amount of total scattering at equivalent times than the 5.22 s"1 data, but the scattering is still less than the scattering observed without shear. The degree of anisotropy in the scattering at 52.2 s*1 also appears at much earlier times than in the 5.22 s"1 case. Hence, with increasing shear rate, a crossover in behavior appears to occur, from droplet breakup (or shear inhibited coalescence) to shear enhanced coalescence (or droplet growth). However, higher shear rates are necessary to confirm this crossover behavior. The corresponding ln{[I(q,t)-Baseline]/[I(q,t=0)-Baseline]} versus time plot for the scattering shown in Figure 5 is shown in Figure 6. For the pure blend at this deeper quench depth, the terminal slope is the smallest for an applied shear of 5.22 s*1 and greater for an applied shear rate of 52.2 s"1. The highest slope is observed under quiescent conditions. Application of shear rate also appears to delay the onset of growth under all conditions compared to the quiescent growth.

The behavior for the modified blend is shown in Figures 7 and 9. The corresponding ln{ [I(q,t)-Baseline]/[I(q,t=0)-Baseline]} versus time plots are shown in Figures 8 and 10, respectively. Figure 7 shows the data for a shallow quench (AT = 2.5 °C). The elapsed time necessary for the sample to coarsen at zero shear is much

416

Figure 4. Time dependence of the scattering intensities at various q values for pure blend at a shallow quench depth (AT = 0.5 °C) corresponding to Figure 3 Filled symbols are the data for the highest q value (4.72 pml).

417

418

Figure 5. Light scattering patterns for the pure blend at a deeper quench depth (AT = 2.0 °C). The band of "specks" on the images at 5.22 and 52.2 s"1 are due to ice

419

formation on the active surface of the charge couple device detector and not part of the scattering patterns.

420

5.22 β"1

600 800 1000 1200 1400 1600 Time (s)

Figure 6. Time dependence of the scattering intensities at various q values for the pure blend at a deeper quench depth (AT = 2.0 °C) corresponding to Figure 5.

52.2 s"1

1 1 1 1 1 1 Γ

- • - 0 . 5 1 μπι"1

- ~ β ~ 1.44 μηι"

2.33 μπι"

- —*~3.17 μπι"

3.98 μπι"

- " " · — 4.72 μπι"

0 200 400 600 800 1000 1200 1400 1600 Time (s)

Figure 6. Continued.

422

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re 7

. Li

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the

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.5 °

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.093

, 0.9

3 an

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3 s1

.

424

AT = 2.5°C Os"1

τ 1 Γ

Time (s)

Figure 8. Time dependence of the scattering intensities at various q values for the modified blend at a shallow quench depth (AT = 2.5 °C) corresponding to Figure 7.

425

0.093 s 1

2.5 -W 1

W 1 2 •

g 1.5 -

1

« 1 -

0.5 -

i i 0 •

-0.5 -

200 400 600 800 1000 1200 1400 1600 Time (s)

9.3 s 1

"τ 1 1 r -*~0.51 μπι"1

1.44 μπι"1

"^"2.33 μηι'1

"^"3.17 μιη"1

~^-3.98μιη" 1

~*~ 4.72 μπι"1

200 400 600 800 1000 1200 1400 1600 Time (s)

Figure 8. Continued.

426

Figure 9. Light scattering patterns for the modified blend at a deeper quench depth (AT = 5.0 °C)for applied shear rates ofO, 2.93, 29.3 and 52.2 s1.

427

Figure 9. Continued.

428

AT = 5°C Os

.q # 5 I ι ι ι ι ι 1 1 1 Ο 200 400 600 800 1000 1200 1400 1600

Time (s) Figure 10. Time dependence of the scattering intensities at various q values for the modified blend at a deeper quench depth (AT = 5.0 °C) corresponding to Figure 9.

429

52.2 s"1

3.5 ι 1 1 1 1 1 1 τ

_0.5 I ι ι I 1 1 1 1 1 Ο 200 400 600 800 1000 1200 1400 1600

Time (s)

Figure 10. Continued.

430

longer for the modified blend compared to the pure blend, even for a deeper quench depth. To demostrate these long coarsening times, Figure 7 contains two matrices of scattering images as a function of time after quench and shear rate. The left hand portion of Figure 7 shows the behavior over early times as in Figures 3 and 5, while the right hand side of Figure 7 shows the behavior for much longer times after the temperature quench. The quiescent behavior is shown in both halves of the Figure for comparison.

The decrease in the rate of phase separation is qualitatively in agreement with the TJLS results of Sung and Han showing that the kinetics of phase separation in a diblock copolymer modified polymer blend are much slower than the pure blend at equivalent quench depths. The shear rates utilized to obtain Figure 7 were 0.093,0.93 and 9.3 s"1, which are all lower shear rates than were utilized for the pure blend. At all shear rates, the scattering intensity is less than the zero shear scattering intensity. The effect is most obvious at the longest elapsed times (1513 and 1597 s). For the data taken at 9.3 s"\ the growth in scattering is completely suppressed over the entire time scale of the experiment. The results are indicative of suppression of the phase separation by shear in the modified blend. This is the opposite behavior observed in the pure blend at shallow quench depths.

For a deeper quench (ΔΤ = 5 °C), the modified blend again shows very different behavior than the pure blend. The shear rates utilized in Figure 9 are 2.93, 29.3 and 52.2 s"1. As was the case for the pure blend, without shear being applied to the sample, the phase separation is much faster in the modified blend at the deeper quench depth. When a low shear rate (2.93 s"1) is applied to the sample, the scattering intensity at all elapsed times is much higher than the quiescent scattering intensity. Similar behavior is observed at 29.3 s*1. However, at 52.2 s'\ the scattering intensity is lower at equivalent elapsed times relative to the unsheared sample. Therefore, a crossover in behavior is also observed in the modified blend at deep quench depths, from shear enhancement of coarsening at low shear rates, to a suppression of coarsening at high shear rates. This behavior is also opposite to the results obtained on the pure blend at a deeper quench depth.

For a shallow quench of the modified blend (Figure 8), applied shear rates of 0.093 and 0.93 s*! also demonstrate a delay in the onset of coarsening compared with the quiescent case. The difference in the terminal slope values for applied shear rates of 0.093 and 0.93 s"1 are difficult to distinguish, however, the terminal slopes appear to be greater than for the quiescent case. In contrast, the slopes for an applied shear rate of 9.3 s*1 are much less than the quiescent slopes. For the deeper quench of the modified blend (Figure 10), the applied shear rates of 2.93 and 29.3 s"\ demonstrate a higher growth rate than the quiescent case, while the highest shear rate of 52.2 s"1

demonstrates a slower growth rate than the sample at rest.

Discussion

A summary of the experimental observations from these studies is shown by the bar chart in Figure 11. For various temperature ranges and shear rates, the bars

431

Pure Blend Blend with Block

Enhancement Suppression

s u p p i ^ m i ^ m i

Shear Rate Shear Rate

Figure 11. Summary of coarsening behavior as a function of shear rate and temperature for the pure blend and the blend with block copolymer.

432

designate the observed behavior with the shading indicative of the degree of shear enhanced coalescence (droplet growth) (black) or shear induced droplet breakup (white). For shallow quench depths, the steady shear experiments during cooling demonstrated a shear enhancement of droplet growth in the pure blend, while shear suppression of droplet growth was observed in the modified blend. We therefore conclude that at shallow quench depths the principal effect of the copolymer is to prevent shear enhanced droplet coalescence. For the deeper quench depths, the behavior in both samples showed a transition with increasing shear rate. In the pure blend, shear suppression of droplet growth was observed at low shear rates, while shear enhanced droplet growth was indicated at higher shear rates in agreement with the work on two component blends reported by Sundararaj and Macosko.12 The opposite behavior was observed in the modified blend, where shear enhancement of droplet growth was noted at low shear rates, and shear suppression of droplet growth was indicated at high shear rates. This behavior is the same as predicted by Milner and X i 1 0 and observed by S0ndergaard and Lyngaae-J0rgensenu for three component mixtures. The bar chart emphasizes the striking contrast in behavior between the pure and modified blends. For each category of behavior studied, the exact opposite behavior is observed in the modified blend relative to the pure blend.

The suppression of coarsening in the pure blend followed by an increase in coarsening has been addressed by Sundararaj and Macosko.12 They attributed the behavior to the balance between coalescence at high shear rates predicted by Smoluchowski and the critical droplet size of Taylor which can exist at low shear rates. The arguments presented for the critical droplet size were based on a viscoelastic fluid, while the sample we have examined displays Newtonian viscosity behavior, indicating the presence of viscoelasticity is not necessary to observe the crossover from droplet breakup to coalescence.

The model of Milner and X i 1 0 invokes the relationship between the interdroplet layer draining time and the applied shear rate as an explanation for the modified blend behavior as a function of shear rate. At low shear rates, there is sufficient time for the layer to drain and fusion of the droplets can occur, while at higher shear rates, the interdroplet layer cannot drain away fast enough for fusion to occur. Milner and X i also proposed that the copolymer must migrate toward the backs of the droplets before coalescence can occur and suggest that this migration time also plays a role in the shear rate dependence of the coalescence. We have no evidence from this study to support that hypothesis.

It is important to note that the shear rates applied at these quench depths are insufficient to completely suppress coarsening, since for all shear rates, the scattering intensity is varying with time. Therefore, one should not confuse the final equilibrium and steady shear morphologies with these non-equilibrium, transient structures. Certainly one does not expect the equilibrium phase size during mixing of a pure blend to be larger than the phase sizes in an unmixed blend, since at extremely long times, the unmixed blend will form two distinct layers. The experiments must be extended to longer times to ascertain if a steady state scattering pattern is ever achieved. Similar studies of the steady shear droplet sizes in the two phase region would also help to determine the effects of copolymer on the equilibrium droplet size.

433

Acknowledgments

I would like to acknowledge Mr. D. S. Johnsonbaugh for his assistance in the sample preparation and determination the cloudpoint temperatures. I would also like to acknowledge Drs. L. Sung, J. F. Douglas, C. L. Jackson, and C. C. Han for their enlightening discussions of this work.

REFERENCES

1. Matos, M.; Favis, B.D.; Lomellini, P. Polymer 1995, 36, 3899-3907. 2. Majumdar, B.; Keskkula, H.; Paul, D. R.; Harvey, N. G.; Polymer 1994, 35,

4263-4279. 3. Majumdar, B.; Keskkula, H.; Paul, D. R. Polymer 1994, 35, 3164-3172. 4. Majumdar, B.; Keskkula, H.; Paul, D. R. J. Appl. Polym. Sci. 1994, 54, 339-354. 5. Majumdar, B.; Keskkula, H.; Paul, D. R. Polymer 1994, 35, 5453-5467. 6. Taylor, G. I. Proc. Roy. Soc. London 1932, A138, 41-48. 7. Taylor, G. I. Proc. Roy. Soc. London 1934, A146, 501-523. 8. Smoluchowski, M. Phys. Z. 1916, 17, 557-571, 585-599. 9. Smoluchowski, Μ. Z. Phys. Chem. 1917, 92:129-168. 10. Milner, S. T.; Xi, H. J. Rheol. 1996, 40, 663-687. 11. Søndergaard, K., Lyngaae-Jørgensen, J. In Flow Induced Structure in Polymers;

Nakatani, A. I.; Dadmun, M. D., Eds.; ACS Symposium Series 597; American Chemical Society: Washington, DC, 1995; pp. 169-187.

12. Sundararaj, U.; Macosko, C. W. Macromolecules 1995, 28, 2647-2657. 13. Plochocki, A. P.; Dagli, S. S.; Andrews, R. D. Polym. Eng. Sci. 1990, 30, 741-

752. 14. Favis, B. D.; Chalifoux, J. P. Polym. Eng. Sci. 1987, 27, 1591-1600. 15. Huneault, Μ. Α.; Shi, Z. H.; Utracki, L. A. Polym. Eng. Sci. 1995, 35, 115-127. 16. Favis, B. D.; Therrien, D. Polymer 1991, 32, 1474-1481. 17. Yoon, H.; Feng, Y.; Qiu, Y.; Han, C. C. J. Polym. Sci.: Part B.: Polym. Phys.

1994, 32, 1485-1492. 18. Yoon, H.; Han, C. C. Polym. Sci. Eng. 1995, 35, 1476-1480. 19. Scott C. E.; Macosko, C. W. Polymer 1995, 36, 461-470. 20. Favis, B. D. J. Appl. Polym. Sci. 1990 39, 285-300. 21. Shih, C.-K.; Tynan, D. G.; Denelsbeck, D. A. Polym. Eng. Sci. 1991, 31, 1670-

1673. 22. According to ISO 31-8, the term "Molecular Weight" has been replaced by

"Relative Molecular Mass", symbol Mr. Thus, if this nomenclature and notation were to be followed in this publication, one would write Mr,n instead of the historically conventional M n for the number average molecular weight, with similar changes for Mw, Mz, and Mv, and it would be called the "Number Average

434

Relative Molecular Mass". The conventional notation, rather than the ISO notation has been employed for this publication.

23. Certain equipment and instruments or materials are identified in this paper in order to adequately specify the experimental details. Such identification does not imply recommendation by the National Institute of Standards and Technology nor does it imply the materials are the best available for the purpose.

24. Nakatani, A. I.; Waldow, D. Α.; Han, C. C. Rev. Sci. lnstrum. 1992, 63, 3590-3598.

25. Sato, T.; Han, C. C. J. Chem. Phys. 1988, 88, 2057-2065. 26. Sung, L.; Han, C. C. J. Polym. Sci.: Part B.: Polym. Phys. 1995, 33, 2405-2412.

BLOCK COPOLYMERS

Chapter 27

Ultra-Small-Angle X-ray Scattering and Transmission Electron Microscopy Studies

Probing Grain Size of Lamellar Styrene-Butadiene Block Copolymers Randall T. Myers1, Alexander Karbach2, Anuj Bellare3,

and Robert E. Cohen1,4

1Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139

2Bayer A.G., ZF-PPP R52, 4150 Krefeld 11, Germany 3Department of Orthopedic Surgery, Brigham and Women's Hospital,

Harvard Medical School, Boston, MA 02115

Simultaneous determination of the lamellar morphological length scale and the grain size of two low molecular weight heterogeneous styrene - butadiene block copolymers was accomplished through the use of ultra small angle x-ray scattering measurements. NIST's X23A3 Ultra SAXS beamline of the National Synchotron Light Source in the Brookhaven National Laboratory provided a range of scattering vector, q, from 0.0004 to 0.1 Å-1. Al l of the block copolymer specimens display a clearly resolvable peak in the Ultra SAXS region, and grain size was determined using the spherical form factor. Determination of Porod's law constant and the value of the scattering invariant provided verification of the proposed scattering mechanism by solving for the contrast factor and the volume fraction of the grain boundaries in these specimens. Grain size in a given polymer was a function of annealing temperature and time. Transmission Electron Microscopy validated Ultra SAXS grain size measurements for one of the block copolymers.

It is known that appropriate processing techniques can produce essentially perfectly ordered block copolymer morphologies with a single texture extending throughout the macroscopic dimensions of a specimen (1). The characteristic repeating length scale,

4Corresponding author.

436 © 2000 American Chemical Society

437

d, of these morphologies is dictated by the molecular weights of the constituent block sequences and is on the order of 100 Â. In the absence of extraordinary processing procedures, a second important length scale appears in the block copolymer. The perfection of the morphology is broken up into grains, each of which contains the ordered morphology of length scale d but with essentially random orientation relative to the specimen boundaries. The kinetics of grain growth in block copolymers has been examined extensively by Balsara et al (2-6). These grains typically exhibit a characteristic size, D, which is one or more orders of magnitude larger than the morphological length scale, d.

The grainy structure can influence physical properties. For example gas transport in a grainy lamellar SB diblock is significantly different from that observed in specimens specially processed for series or parallel permeation (7,8). It is believed that the size of the grains may affect physical properties and thus it is important to have a robust technique for grain size measurement.

Conventional small angle x-ray scattering (SAXS) techniques have been employed for decades to characterize block copolymers at the morphological length scale, d (9,10). Recently Ultra SAXS beamlines have been constructed to probe significantly larger morphological features (11). The direct and non-destructive examination of grains in bulk, three-dimensional specimens via Ultra SAXS is advantageous in our ongoing effort to connect mechanical behavior with grain structure in block copolymers. It is the purpose of this paper to demonstrate that Ultra SAXS can be used to characterize bulk specimens of block copolymers simultaneously at both the lamellar morphological length scale, d, and the grain length scale, D.

Experimental

For this study, two low-molecular weight styrene-butadiene block copolymers were used: 9900/9700 and 5400/5350. These specimens were synthesized and sold by Polymer Source, Inc. and have polydispersities of 1.02 and 1.03. The 9900/9700 contains about 90% 1,2-butadiene in the rubber block and will be referred to as S12B10 for the rest of the paper, while the 5400/5350 contains 90% 1,4-butadiene in the rubber block and will be referred to as SB5 for the rest of the paper.

The polymers were dissolved in various solvents at concentrations in the range of 10 weight % and static cast. In this paper we report results for specimens cast from chloroform, a solvent of relatively high volatility (vapor pressure at 25°C is 26 kPa). Annealing at elevated temperatures leads to growth of grains (2). For the low molecular weight SB diblock colpolymers examined here, annealing at 75°C and 50°C for the S12B10 and SB5 respectively produced measureable increases in grain size over the period of several hours. We have attempted to validate grain size results obtained from the Ultra SAXS scattering curves by using Transmission Electron Microscopy for the 9900/9700 S12B10 specimen series.

The Ultra SAXS experiments were performed at the Brookhaven National Laboratory on the X23A3 beamline operated by the National Institute of Standards and

438

Technology. The available range of scattering vector, q = (47c/X)sinO, was 0.1 A 1

to 0.0004 À*1, where 0 is one half the scattering angle and λ = 1.299 Â is the x-ray wavelength (11). The scattering data were desmeared to account for the geometry of the X23A3 beamline using software provided by Dr. Gabrielle Long of the National Institute of Standards and Technology and designed for this specific beamline. The program incorporated the methodology of Lake (12). Although desmearing altered to a small extent the shapes, locations and magnitudes of the peaks in the scattering curves, there was no case in which desmearing caused a peak to appear when none was present in the smeared data. Absolute intensities (units are cm"1) were determined by calibration of the Ultra SAXS instrument at 4.966, 5.989 amd 8.989 keV using transmission absorption-edge measurements on the Κ edges of Ti, Cr, and Cu metal foils

Results

Figure 1 is a set of double logarithmic plots of absolute intensity, I, vs scattering vector, q, for sample S12B10 (9900/9700). Two peaks are observed in the scattering curves. The peak at higher q appears in the conventional SAXS regime and corresponds to the periodic lamellar morphology of the SB diblock copolymer. The

2π lamellar spacing, d = = 170 A, is essentially unchanged by the annealing

Q m a x

protocol described in the figure. The position of the peak at the left varies with annealing time, spanning a range corresponding to a spacing of about 1 μπι. As discussed in detail below, we assosciate this low-q peak with the presence of grains in the materials. For the present we note that continued annealing shifts the peak to lower values of q, corresponding to a larger material length scale. We also note that all of the data below about q=0.01 were accesible only through the use of the Ultra SAXS beamline. Conventional point collimation 2D SAXS experiments in our own laboratory faithfully reproduce the peak shapes and locations above q=0.01. Also in Figure 2 there is a clearly discernable straight line regime of all the scattering curves to the right of the low-q peak. The log-log slope in this region of the scattering curves is -4. Deviations from the slope of -4 occur near q=0.01 where the high-q peak overlaps the descending data from the low-q peak.

Figure 2 presents similar results for the sample SB5 (5400/5350). Again the lamellar spacing (d=100À for this polymer) remains essentially unchanged with time while the low-q peak shifts with annealing by an amount which corresponds to about a factor of 2 in morphological length scale. The data at the lowest values of q in both Figure 1 and 2 fall off in the direction of zero intensity; this trend, coupled with the exceedingly low value of q at the low end of the Ultra SAXS resolution facilitates calculation of the Ultra SAXS invariant, discussed below, with insignificant low-q truncation error.

Figures 3-5 show Transmission Electron Microscopy (TEM) micrographs for the first three S12B10 (9900/9700) samples tested using Ultra SAXS. These figures

439

10°

I 10°

10'

0.1 0.001

no annealing annealed 75C 5 min annealed 75C 1 hour annealed 75C 4 hours

0.01 0.1

Figure 1. Logarithm of absolute intensity vs log q at various annealing times for the 9900/9700 styrene -1,2 butadiene block copolymer (SI 2B10) cast from chloroform. The right peaks correspond to the interlamellar spacing, d, and the left peaks refer to the grain spacing, D.

10*

1(T

no annealing annealed 50C 5 min annealed 50C 1 hour annealed 50C 2 hours annealed 50C 4 hours

0.01 0.001 0.01 0.1

Figure 2. Logarithm of absolute intensity vs log q at various annealing times for the 5400/5350 styrene -1,4 butadiene block copolymer (SB5) cast from chloroform. The right peaks correspond to the interlamellar spacing, d, and the left peaks refer to the grain spacing, D.

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442

Figure 5. Transmission electron micrograph of the 9900/9700 styrene -1,2 butadiene block copolymer (S12B10) cast from chloroform, annealed at 75°Cfor 1 hour, ultramicrotomed and stained with Os04.

d

Jl Pb

Figure 6. Schematic representation of the proposed mechanism. The dark lines represent the electron density differences represented in the Ultra SAXS region corresponding to the grain size. The electron density differences relating to interlamellar spacing are ghosted in.

JL . pGB

443

demonstrate the existence of grains in the block copolymer samples and point to the difficulty encountered in attempting to quantify grain size from TEM data.

The results presented above are representative of, but only a minor fraction of, the numerous data sets we have obtained at the Brookhaven facility. More of our results appear elsewhere (13). Repeated analysis of individual specimens shows excellent repeatablity and reproducibility of all the features of the scattering curves noted above. Thus the experimental facts are clearly depicted in Figures 1-2. In order to determine grain size from Ultra SAXS data, we must propose a mechanism to account for the scattering observed in the low-q regions of our scattering curves.

Mechanism of Scattering

There have been studies focused on the detailed structure of grain boundary morphologies in styrene - butadiene block copolymers (14-18). In these detailed microscopic observations there are clear suggestions that the local composition in the grain boundaries is different from the overall mean composition of the material. It is also known that a free surface leads to an altered local composition in block copolymers (19). We make the assumption that similar, although perhaps smaller, compositions fluctuations arise at the grain boundaries. We will test the internal consistency of this assumption later in the analysis, recognizing that the assumed contrast factor, (p G B - p m )'2, must always lie between zero and either (ps - p m ) or (pb ~ Pm)2- See Figure 6 for definition of the various subscripts on the densities, p, which have units of electrons/Â3. We assosciate the low-q peak in our data with the spherical form factor originally used by Stein et al to determine the size of impinged spherulites in low angle light scattering experiments (20). We also compare grain sizes obtained from the spherical form factor analysis with two alternative methods: a quasi-Bragg approach and a correlation function analysis (21).

Grain size, D, and lamellar spacing, d

The spherical form factor:

Discussion

(D

444

exhibits a peak at a value U=4.0 (20). We use this result to obtain the grain size from 8

the relation D = , where q M A X is the location of the maximum of the low-q peak Q M A X

in the Ultra SAXS scattering curves. We have tabulated grain sizes of the various annealed and unannealed specimens in Tables 1-2. Table 1 also shows grain sizes obtained from the other methods of analysis. While the numerical values differ somewhat, the trends are unaltered. Discussion of the other entries follows.

Table 1: Ultra SAXS Results for S12B10 Annealing Time D(pm) DQB (pm) DCF(pm) lap)2

at75°C none 0.51 0.40 0.26 1.90E-6 0.0995 5 minutes 0.78 0.61 0.38 3.11E-6 0.0991 1 hour 1.03 0.81 0.60 1.36E-5 0.1038 4 hours 1.53 1.20 0.77 1.78E-5 0.0987

NOTE: D Q B : Grain size from quasi-Bragg analysis in which qMAX=27t/DQB

D C F : Grain size from correlation function analysis. See reference 21.

Table 2: Ultra SAXS Results for SB5 Annealing Time

at 75 °C D(pm) (Apf Φ

none 0.52 1.48E-6 0.0975 5 minutes 0.61 1.73E-6 0.0991 1 hour 0.75 2.29E-6 0.1057 2 hours 0.87 2.51E-5 0.0994 4 hours 0.95 3.36E-5 0.1000

The Contrast Factor and Grain Boundary Volume Fraction

There are features in the Ultra SAXS data which enable us to make certain internal consistency checks to support the proposed mechanism of scattering. In particular, the cartoon of Figure 6 indicates that Δρ 2 should have an upper bound of (ps - Pm) 2 ° r (Pb " Pm)2» either of which to a good approximation for our polymers

is equal to ^ s ^ h j . Extracting the contrast factor from our data would therefore

provide one method to support or discredit the proposed scattering mechanism. Also Figure 6 suggests that the volume fraction of the grain boundary is small compared to the volume of material in the grain with mean density, pm. If the analysis of the data indicates otherwise, the mechanism suggested in Figure 6 is in doubt. We use

445

Porod's Law and the scattering invariant,both of which are readily accessible characteristics of the scattering curves in the Ultra SAXS region, to test our mechanism. Porod's Law constant, Cj, is obtained from the region to the right of the low-q scattering peak where intensities decrease with a q*4 dependence.

C l S ^ - l i m ( q 4 i ) = ( S / V ) ( p G B - p m ) 2 (2)

where i is the absolute desmeared intensity and (S/V) is the surface to volume ratio, which for the case of a sphere is equal to 6/D. The invariant in this case is defined as the total area under the Iq2 vs q plot associated with grain scattering and can be expressed as follows:

C 2 • i(q)q2dq = φ(ΐ - φ)(ρ 0 Β - P m ) 2 (3) ζπ η

where φ is the volume fraction of grain boundary material. Values of φ are given in Tables 1-2. Although it would be logical to assume that

the amount of grain boundary material would decrease as grain size increased, there is no such trend in the results in Tables 1-2. Over the range of grain sizes in the materials studied (spanning a factor of 3 from about 0.5 to 1.5 μπι), the volume fraction of the grain boundary material remains essentially constant at φ=0.1. This value is, however, very consistent with the proposed model of Figure 6. A grain size of 0.78 μπι for polymer S12B10 (9900/9700) corresponds to a grain boundary thickness of about 135 À, less than one repeat distance of the block copolymer morphology. Thus the scattering results are consistent with the intuitive view and TEM Figures 3-5 that grain boundary thickness is about the same order of magnitude as the lamellar periodicity. A second consistency check involves the magnitude of the contrast factor (Δρ) 2 which appears in both equations (2) and (3). Using the experimentally determined values of the Porod constant, C,, and the surface to volume ratio from the spherical form factor (S/V = 6/D), we determined the values of (Δρ)2

which appear in Tables 1-2. For both copolymers, this contrast factor tended to increase with annealing time. In all cases, the value is considerably less than the value of (ps - p m ) 2 o r (pb - p m ) 2 , which is 8.4 x 10'5 electrons/Â3 for the styrene-1,2 butadiene copolymer and approximately 4 χ 10"4 electrons/Â3 for the styrene-1,4 butadiene copolymer. Once again, these results are entirely consistent with the mechanism suggested by Figure 6.

Grain Size Determination from Transmission Electron Microscopy

Underwood proposes a method for determining sizes from micrographs for certain particle geometries taking into account the inherent stereology (22). We can calculate

446

a mean intercept length, which for an aggregate containing grains is the average diameter. The surface to volume ratio is S/V = 2 PL, where P L is the number of grain boundaries per unit length, and the diameter is D=2/(S/V). Employing this on Figures 3-5, we can get an average grain size and these are summarized in Table 3 along with the diameters found by Ultra SAXS. As can be seen, the grain sizes are closely correlated.

Table 3. Values of Grain Size found from Ultra SAXS and from TEM micrographs for S12B10

Annealing Time DV (pm) DTEM (m) at75°C

none 0.51 0.45 5 minutes 0.78 0.55 1 hour 1.03 0.83

NOTE: D y is the grain size from Ultra SAXS analysis D T E M is the grain size obtained from TEM micrographs

Summary

While more data and analysis are presented elsewhere (13), we have demonstrated here that grain size can be measured by Ultra SAXS. The findings are consistent with results found elsewhere, that annealing at elevated temperatures causes grain growth (2). The mechanism proposed for scattering in the Ultra SAXS region is validated with a consistent phase fraction, φ, and electron density difference, (Δρ)2. Grain size measurements from TEM micrographs of the same samples are also consistent with Ultra SAXS results.

Acknowledgments

We would like to thank Gabrielle Long and Zugen Fu from the National Institute of Standards and Technology for their assistance in the use of the Ultra SAXS beamline at Brookhaven. We acknowledge NSF/MRSEC for partial support of this project.

References

1. Keller, Α.; Pedemonte, E.; Willmouth, F.M. Nature 1970, 225, 538. 2. Garetz, B.A.; Balsara, N.P.; Dai, H.J.; Wang, Z.; Newstein, M.C.

Macromolecules 1996, 29, 4675. 3. Balsara, N.P.; Dai, H.J.; Watanabe, H.; Sato, T.; Osaki, K. Macromolecules

1996, 29, 3507

447

4. Balsara, N.P.; Dai, H.J.; Kesani, P.K.; Garetz, B.A.; Hammouda, B. Macromolecules 1994, 27, 7406.

5. Garetz, B.A.; Newstein, M.C.; Dai, H.J.; Jonnalagadda, S.V.; Balsara, N.P. Macromolecules 1993, 26, 3151.

6. Balsara, N.P.; Garetz, B.A.; Dai, H.J. Macromolecules 1992, 25, 6072. 7. Csernica, J.; Baddour, R.F.; Cohen, R.E. Macromolecules 1987, 20, 2468. 8. Csernica, J.; Baddour, R.F.; Cohen, R.E. Macromolecules 1989, 22, 1493. 9. Hashimoto, T.; Nagatoshi, K.; Todo, Α.; Hasegawa, H.; Kawai, H.

Macromolecules 1974, 7, 364. 10. Hashimoto, T.; Shibayama, M. ; Kawai, H. Macromolecules 1980, 13, 1237. 11. Long, G.G.; Jemian, J.R.; Weertman, J.R.; Black, D.R.; Burdette, H.E.; Spal,

R. J. Appl. Cryst. 1991, 24, 30. 12. Lake, J.A. Acta Cryst. 1967, 23, 191. 13. Myers, R.T.; Cohen, R.E.; Bellare, A. Macromolecules 1999, in press. 14. Nishikawa, Y.; Kawada, H.; Hasegawa, H.; Hashimoto, T. Acta Polymer. 1993,

44, 247. 15. Gido, S.P.; Gunther, J.; Thomas, E.L.; Hoffman, D. Macromolecules 1993, 26,

4506. 16. Gido, S.P.; Thomas, E.L. Macromolecules 1994, 27, 849. 17. Gido, S.P.; Thomas, E.L. Macromolecules 1994, 27, 6137. 18. Gido, S.P.; Thomas, E.L. Macromolecules 1997, 30, 3739. 19. Mayes, A.M. ; Kumar, S.K. MRS Bulletin 1997, 22, 43. 20. Stein, R.S.; Rhodes, M.B. J Appl Physics 1960, 31, 1873. 21. Porod, G. In Small Angle X-ray Scattering; Glatter, O.; Kratky, O., Eds.

Academic Press: New York, NY, 1982; 18-50. 22. Underwood, E.E. Quantative Stereology, Addison Wesley: Reading, MA, 1970;

80-95.

Chapter 28

Block Crystallization in Model Triarm Star Block Copolymers with Two Crystallizable Blocks:

A Time-Resolved SAXS-WAXD Study G. Floudas1, G. Reiter2, O. Lambert2, P. Dumas2, F.-J. Yen3, and B. Chu3

1Foundation for Research and Technology-Hellas (FORTH), Institute of Electronic Structure and Laser, P.O. Box 1527, 711 10 Heraklion

Crete, Greece 2Institut de Chimie des Surfaces et Interfaces, 15 rue Jean Starcky, B.P. 2488,

68057 Mulhouse Cedex, France 3Department of Chemistry, State University of New York at Stony Brook,

Stony Brook, NY 11794-3400

The kinetics of crystallization in model triarm star block copolymers of two crystallizable blocks (poly(ethylene oxide) (PEO) and poly(ε-caprolactone) (PCL)) and an amorphous block (polystyrene (PS)) have been investigated with time resolved synchrotron wide-angle X-ray diffraction (WAXD) and small-angle X-ray scattering (SAXS). These model block copolymers form a homogeneous melt and the crystallization of the longer block drives the microphase separation. We have explored the block crystallization in the stars as a function of (i) the length of the two crystallizable blocks and (ii) the crystallization temperature. We found "paths" of delaying or even inhibiting individual block crystallization.

Block copolymers composed of crystallizable and amorphous blocks develop structure over different length scales, from the unit cell structure of the crystalline block to the microdomain length scale to the spherulitic superstructure. Such combination of the amorphous and crystalline phases can result in materials with enhanced mechanical properties. The ability to form structures over various length scales (1)(2), the control over the mechanical response (3)(4), and the possibility to template crystallization by fluid mesophases (5) have been the driving force for investigating such materials. We have recenlty reported (6) on the structure of model triarm star block copolymers of two crystallizable blocks (PEO and PCL) and an amorphous block (PS) using X-ray scattering, differential scanning calorimetry

448 © 2000 American Chemical Society

449

(DSC), optical microscopy (OM) and atomic force microscopy (AFM). Our aim is to trigger the thermodynamic interactions by combining two crystallizable blocks with an amorphous block and to develop an understanding of the effect of polymer architecture on the crystallization process. The main results from the earlier study can be summarized as follows. Both PEO and PCL blocks can crystallize in the stars provided that their length ratio is below 3. For the more asymmetric stars, the longer block suppresses the crystallization of the shorter one. The crystallinity, long period and crystalline lamellar thickness are reduced in the stars as compared to the PEO and PCL homopolymers. There is a reduction in the equilibrium melting temperature in the stars (as compared to the values of 343 and 347 Κ for PEO and PCL, respectively), which is caused primarily by the amorphous PS block. The Avrami analysis of the isothermal crystallization calorimetrie experiments indicated a disc-like growth from heterogeneous nuclei, independent of the nature of the crystallized block. However, the crystallization times were sensitive to the block nature. Lastly, different superstructures were formed (spherulites/axialites) depending on the type of the crystallizable block (PEO/PCL). The nucleation sites of the two superstructures were completely independent suggesting a heterogeneous distribution of star molecules with PEO or PCL crystals.

In the present study we use the different unit cells of PEO (monoclinic) and PCL (orthorhombic) of the two crystallizable blocks as a probe of crystallization and we investigate the crystallization kinetics by making temperature jumps - from the homogeneous phase to different crystallization temperatures - as a function of composition and temperature using time-resolved synchrotron SAXS/WAXD.

Experimental

The synthesis of the model triarm star block copolymers has been reported elsewhere (7). Table I gives the molecular characteristics of the three copolymers used in the present study.

Table I. Molecular Characteristics of the Triarm Star Block Copolymers

Sample Mn(PS) Mn(PEO) X103

M„(PCL) X103

M n( total) X103

MJMn

SEL-4.7/20/1.8

4.7 20 1.8 27 1.15

SEL-4.7/20/45

4.7 20 45 70 1.19

SEL-4.7/20/87

4.7 20 87 112 1.29

450

The PEO and PCL weight fraction in the three stars are: 0.88 and 0.03 (SEL-4.7/20/1.8); 0.51 and 0.44 (SEL-4.7/20/45); 0.36 and 0.6 (SEL-4.7/20/87), respectively. Real time-resolved synchrotron SAXS/WAXD measurements have been performed at the X27C beamline of the National Synchrotron Light Source at Brookhaven National Laboratory. The wavelength λ, of the X-rays was 0.1307 nm. Two position sensitive detectors were used; one at small-angle at a distance of 1.2 m from the sample and the other at a wide-angle setting of about 45° angle to the sample. The samples used in this study were prepared directly from the melt and upon crystallization had a thickness of about 1mm. First all samples were heated to 383 Κ for 5 min and subsequently brought to another temperature unit at the preset final temperature. Temperature jumps to different final temperatures in the range 312-316 Κ have been investigated. The Lorentz correction was performed to the data by multiplying the scattered intensity I by q 2 (q=(4jcM,)sinO, q is the scattering wavevector and θ is the scattering angle). Knowledge of the absolute intensity, of the contribution from pure density fluctuations (thermal background) together with an extrapolation to q=0 and to q=°o can result in the absolute invariant. The relative invariant, Q, was calculated here from

where Ib„d is the background scattering, by integrating the (I-Ibgd)q2 curve from q=0.15 to 1 nmwhere the above quantity becomes independent of q.

Typical SAXS/WAXD spectra are shown in Figure 1 for the SEL-4.7/20/1.8 copolymers for a crystallization temperature of 313 K. The SAXS spectrum taken immediately before the T-jump indicated only a minor contribution from concentration fluctuations at 383 K. This is a result of the compatibility of PS and PCL in the melt state (8) and of the star architecture with PEO which further increases the intrinsic compatibility of the copolymer (9). After about 500 s the WAXD patterns develop peaks corresponding to the monoclinic unit cell of PEO. Notice that the crystallization of the shorter block is suppressed in the triarm star. At the same time the SAXS peaks develop signifying the formation of the PEO crystalline lamellar.

The result of the corresponding T-jump for the SEL-4.7/20/87 triarm star block copolymer from 383 to 313 Κ is shown in Figure 2. Notice the distinctly different WAXD Bragg peaks with positions corresponding to the (110) and (200) reflections from the orthorhombic unit cell of PCL. Now, it is the crystallization of the PEO block which is suppressed, notwithstanding the high molecular weight of the PEO block (Mw=2xl04). Simultaneous with the WAXD reflections, a peak in the SAXS spectra appears reflecting the characteristics of the PCL crystals. We have shown earlier (6) that the crystallinity, the long period and the crystal thickness are reduced in the stars and that the magnitude of the reduction depends on the ratio of the amorphous to the crystallized block lengths. The calculated invariant from the

(1)

Results and Discussion

451

SEL 4.7/20/1.8

q (nm'1) Time (s) Channel

Figure L SAXS (left) and WAXD (right) from the SEL-4.7/20/1.8 star block

copolymer taken in increments of 60s following a temperature jump from 383 Κ to

313 K. The SAXS invariant Q and the WAXD intensity of the most intense reflection of

the PEO are also compared.

SEL 4.7/20/87

0,5 1,0 102 1(f 200 300 400 500 600

q (nm'1) Time (s) Channel

Figure 2. SAXS (left) and WAXD (right ) spectra for the SEL-4.7/20/87 triarm star

block copolymer taken at increments of 60 s following a T-jump from 383 to 313 K.

The WAXD pattern after about 400 s shows peaks corresponding to the (110) and

(200) reflections from the orthorhombic unit cell of PCL At the same time a SAXS

peak develops characteristic of the PCL crystals. The SAXS invariant and the

intensity of the most intense WAXD reflections are compared.

452

SAXS spectra and the intensity of the most intense WAXD reflection (110) increase simultaneously, indicating that the PCL crystallization drives the microphase separation in this system as well.

The situation with the third block copolymer, SEL-4.7/20/45, is very different. Earlier conventional X-ray measurements (6) have shown that both blocks are capable of undergoing crystallization, however, not within the same molecule. The distinctly different WAXD patterns corresponding to the PEO and PCL unit cells facilitate the investigation of the crystallization kinetics for both blocks. The result of the T-jump from 383 to 312 Κ is shown in Figure 3. The WAXD spectra at long times exhibit three main reflections: a peak at low q corresponding to the Bragg reflection from the monoclinic unit cell of PEO, an intermediate peak corresponding to the (110) reflection from the orthorhombic unit cell of PCL and a third peak which is an overlapping reflection from both unit cells. The existence of mixed reflections clealry shows that in SEL-4.7/20/45, both blocks can crystallize. However the results from OM have shown that (i) there is a heterogeneous distribution of star molecules with PEO or PCL crystals and (ii) crystallization of the two blocks is not simultaneous. The spectra shown in Figure 3 support the OM conclusion. In the WAXD spectra, first the intermediate reflection appears corresponding to the (110) reflection from the orthorhombic unit cell of PCL and at some later time PEO starts to crystallize.

Figure 3. SAXS (left) and WAXD (right ) spectra of the SEL-4 J/20/45 triarm star block copolymers following a T-jump from 383 Κ to 312 K. Notice the mixed reflections at the WAXD spectra indicating that both PEO and PCL crystallize in their monoclinic and orthorhombic unit cells.

4 5 3

The SAXS invariant and the WAXD peak intensities corresponding to pure PEO, pure PCL (110) and mixed PEO+PCL reflections are compared in Figure 4. The intensity of the mixed reflection follows the increase of Q indicating again that microphase separation is driven by block crystallization. The two pure reflections have a different time-evolution. The PCL block starts to crystallize first and after the PCL crystallization is completed, the PEO block starts to crystallize.

SEL 4.7/20/45

0 500 1000 1500 2000 2500

Time (s)

Figure 4. Time-evolution of the SAXS invariant QSAXS compared to the intensity of the

three most intense WAXD peaks for the SEL-4.7/20/45 at T=312 K.

We have investigated the dependence of the crystallization times - which are operationally defined here as the time required for the first WAXD reflection to appear - of PEO and PCL on the crystallization temperature T c , and the results are shown in Figure 5. The background intensity (Ibgd) was evaluated by monitoring the time-evolution of an area away from the Bragg reflections and analyzed by fitting to Ibgd=A+Bexp(-t/x), where A, Β and τ depend on the T c . Then the QSAXS a n d the three peak intensities were evaluated. The PCL-crystallization is always faster and for smaller undercoolings the difference in the crystallization rates increases. For example, at 316 K, the PEO crystallization is slower by a factor of 4 whereas at 312 Κ the two times are comparable. OM results on the same star block copolymer have shown a higher nucleation probability for PCL crystals than for PEO, in agreement with the data of Fig.4. Furthermore, in OM, the growth rate of PCL crystals (with an

454

axialitic superstructure) was higher than for PEO (with a spherulitie superstructure) which is in agreement with the steeper slope of the IPCL reflection in Fig. 4.

The results shown in Figures 1,2 3 and 5 demonstrate that it is possible to control the nature of the crystallizing block in the stars not only at the synthesis level (by synthesizing asymmetric block copolymers) but also by delaying the crystallization of a particular block (in this case the PEO block) in cases where both blocks can crystallize (i.e., the crystallizable blocks are of comparable length).

SEL-4.7/20/45

/ a """"

/

_ o — 5

313 314 315 316 317

Tc (K)

Figure 5. Dependence of the crystallization times of PEO (squares) and PCL

(circles) on the crystallization temperature in the star SEL-4.7/20/45.

Conclusions

Star block copolymers composed of two crystallizable blocks and an amorphous block offer new possibilities to control the block crystallization at the synthesis level and in the lab by changing the crystallization conditions. We have shown that in asymmetric stars only the longer block will crystallize. In more symmetric stars both blocks can crystallize but not within the same molecule. Furthermore, we have shown here with the use of time-resolved synchrotron SAXS/WAXD, that in stars with comparable block lengths of the crystallizable blocks, it is possible to delay the crystallization of a particular block for long times with a suitable choice of temperature. These features could be important in the design of new materials where a selectivity in block crystallization is required.

455

Acknowledgment

This work was supported by the NATO-CRG 970551 to G.F. and B.C., by a French-Greek collaborative grant (CNRS-NHRF 1998) to G.R. and G.F and by a NSF grant (DMR9612386) to B.C.

References

1. Wunderlich, B. Macromolecular Physics 2. Crystal Nucleation, Growth,

Annealing; Academic Press: New York, 1978. 2. Strobl, G. The Physics of Polymers; Springer-Verlag: Berlin, 1996; Chapter4. 3. Floudas, G.; Tsitsilianis, C. Macromolecules 1997, 30, 4381. 4. Alig, I. Tadjbakhsch, S.; Floudas, G.; Tsitsilianis, C. Macromolecules 1998, 31,

6917. 5. Quiram, D.J.; Register, R.A.; Marchand, G.R.; Ryan, A.J. Macromolecules

1997, 30, 8338. 6. Floudas, G.; Reiter. G.; Lambert, O.; Dumas, P. Macromolecules 1998, 31,

7279. 7. Lambert, O.; Dumas, P.; Hurtez, G.; Riesss, G. Macromol. Rapid Commun.

1997, 18, 343. 8. Li, Y.; Jungnickel, B.-J. Polymer 1993, 34, 9 9. Floudas, G.; N. Hadjichristiidis, Y. Tselikas, I. Erukhimovich, Macromolecules

1997, 30, 3090.

Chapter 29

Temperature- and Pressure-Induced Microphase Separation Transitions

of a Polystyrene-block-Butadiene Copolymer Melt W. De Odorico1,2, H. Ladynski1, and M . Stamm1,3

lMax Planck-Institut für Polymerforschung, Postfach 3148 55021 Mainz, Germany

2Institut für Kernphysik der Universität Frankfurt, August Euler Str. 6, 60486 Frankfurt am Main, Germany

The microphase separation transition (MST) of a nearly symmetric styrene-butadiene diblock copolymer melt has been investigated by small angle x-ray scattering. The scattering profiles change with temperature in intensity, shape and position, and the MST is located at TM S T = 98,5°C at atmospheric pressure. While the peak position shifts linearly in the explored temperature region, peak height and width change abruptly in a temperature region of approximately 1 degree. No hysteresis was observed in the temperature dependent measurements. A similar transition is observed by changing the pressure at constant temperature, and with pressure a distinct hysteresis is observed. A temperature-pressure superposition with a coefficient of +54bar/K was calculated and verified experimentally for some examples. Pressure jump experiments from the disordered to the ordered regime show an incubation time of several hundred seconds. The time development can be described by an Avrami model, and we assume a nucleation and growth process predominant in one dimension for the transition.

Diblock copolymers consist of two blocks of polymer (A and B), covalently bonded together. Besides the technical importance of block copolymers as new materials and as compatibilizing agents in polymer blends, they are of particular scientific interest. Leibler [1] described the phase transition of copolymers in mean field theory, which gives already the main aspects of this phenomenon.

Corresponding author.

456 © 2000 American Chemical Society

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Derived from a Landau-type mean field theory, he provides an analytical function directly comparable with scattering data in the homogeneous state. In contrast to polymer blends no macroscopic separation is possible during demixing due to the connectivity of the blocks and the material exhibits phase separation on the scale of the molecules, the microphase separation transition, MST. Several microphase separated structures of A respectively Β rich domains have been observed [2],[3] building various lattices. The phenomena can be characterized by/and Λ/χ, where / is the composition, Ν the degree of polymerization and χ the temperature dependent Flory-Huggins segment-segment interaction parameter. For nearly symmetric diblock copolymers (f -0.5) a lamellar microstructure is expected from theory, which has been verified with various systems. In the special case (f = 0.5), the transition is predicted by Leibler to be of second order, but the introduction of thermally induced fluctuations by Fredrickson and Helfand [4] show a weakly first order phase transition even for symmetric systems (Brazovskii effect [5]). This was also found in several experiments using small angle x-ray scattering (SAXS) or small angle neutron scattering (SANS) techniques [6],[7],[8].

Fredrickson and Binder [9] further improved this theory to describe the kinetics of the ordering process. Their concentration dependent free energy potential shows two side minima, which have the same depth as the middle one at the microphase separation transition temperature T M ST- Therefore, they presume a coexistence of the disordered and the lamellar phase at T M ST- A S the temperature is further lowered, these side minima become dominant and the transition comes to completion. For a supercooled material, they expect after a "completion time" an Avrami-type ordering transformation with an exponent of 4 equivalent to spherically growing droplets of ordered material. This characteristic time corresponds to the time to form stable droplets of ordered material plus the time needed for the structures to grow to a size that they can be detected by the used technique.

While/and Ν are fixed for a certain material, the interaction parameter can be varied due to its dependency on temperature, which may be assumed as χ ~ αΤ _ 1+β, with a and β as constants. Systems like ours with a positive α exhibit a transition from order to disorder during the raise of temperature. Experiments on polymer blends exhibit a dependence of χ also on pressure [10]. Kasten and Stuhn [11] found a small density discontinuity at T M ST for a comparable material and presumed a linear shift of T M S T with applied pressure derived from the latent heat of the transition (see also [12]) and the Clausius-Clapeyron equation.

Instead, all theories mentioned above presume incompressibility of the material and therefore cannot describe a pressure dependence of the phase transition. A recent theory for the disordered state including compressibility [13] shows only a weak influence of pressure on χ, but cannot describe the transition.

458

Experimental

Sample preparation A poly(styrene-bloek-butadiene) copolymer, P(S-b-B), containing f =0.52 of

deuterated styrene was used in this study. The composition f in this definition is given by f = N P S /N , where N P S and Ν are the degrees of polymerisation of the PS block and the whole chain, respectively. The polymer was prepared via anionic polymerization in our laboratory. For reasons not important for the present study deuterated styrene was used for the PS block. In order to determine the composition of the sample, the polymer was dissolved in deuterated dichloromethane and 1,1,2,2-tetrabromoethane was added as an external standard. A Bruker 500MHz NMR was employed to determine the amount of 1 Η of 1,1,2,2-tetrabromoethane as well as the butadiene content of the copolymer. The molecular weight of the PS(D) block and its polydispersity were determined by gel permeation chromatography (GPC) with polystyrene as a standard before the butadiene component was added. Knowing the PB component by ^ - N M R and the molecular weight of the PS block by GPC, one can calculate the total molecular weight. As obtained from NMR the 1,4-content of PB is >99%. The microstructure is 60% trans-1,4 and 40% cis-1,4 vinyl. The glass transition temperatures Tg of the blocks were obtained by differential scanning calorimetry at a heating rate of 10 K/min. The molecular characteristics of this copolymer are:

Table 1 Characteristic data of the investigated sample

Polymer Sample

Mw(PS) (g/mol-1)

Mw(total) (g/mol1)

M W / M N Tg(PB) (°C)

Tg(PS) (°C)

f=N(PS)/ N(total)

P(S-b-B)-4 8503 12428 1.075 -89 74 0.52

For a better thermal stability 0.1%wt of 2,6,-di-tert-butyl-4-methylphenol as antioxidant was added in solution to the polymer and the sample was dried one week at room temperature, which is a common technique for P(S-b-B) and P(S-b-I) systems [14],[15J. Samples, consisting of disks of 1mm thickness each were prepared in a mechanical press, which was evacuated to ρ « Imbar and heated to Τ = 100°C.

Measurements For the measurements at the x-ray setup in Mainz, the samples were mounted

into a furnace consisting of a heatable copper block with 2mm pinholes. For stability two thin Aluminum windows are used. The SAXS instrument at the institute consists of a Rigaku Rotaflex 18kW rotating anode x-ray source with a graphite double monochromator at a wavelength of λ=0.154 nm (CuKot). Three pinholes collimate the beam over 1.5m to a diameter of about 1mm at the sample position. A Siemens area detector with 512 χ 512 pixels of 0.2mm was applied. The sample - detector distance was 1.2m, yielding an angular resolution of Δ2Θ =

459

0.07°or Aq « 0.05 nm*1. The scattering vector q is defined as q = 4π/λ sinO with 2Θ being the scattering angle and λ the wavelength. The measurements where corrected for non-uniformities in detector efficiency and spatial distortion. Temperature sweeps were performed from low to high temperatures and back by a PID-temperature controller with an accuracy of about 0.2K. Each temperature was held for at least 5min after stabilization to achieve uniform temperature conditions in the sample. In the beginning, each sample was annealed above T M S T to eliminate morphology or orientational effects due to preparation as described previously [17].

The pressure dependent measurements were carried out at the high brilliance synchrotron beamline BL4/ID2 of the ESRF in Grenoble, France. For absorption reasons a photon wavelength of λ=0.1 nm was used [20],[21],[22]. The experimental setup of the hydrostatic pressure cell is similar to the one described in detail elsewhere [23],[24]. It consists of a serial assembly of a hand pump, a pressure gauge, the pressure cell and another pressure gauge connected by flexible tubes and designed for pressures from 50 bar to 2,5 kbar. The pressure is measured before and behind the cell to monitor the applicated pressure at the sample. A change of pressure can be performed in 10 to 20 seconds with an accuracy of 5 bar. The pressure medium is a silicon oil API50 from Wackersilicone, windows are two 1 mm thick diamond single crystals. The sample is wrapped into aluminum foil to avoid contact between the sample and the silicon oil.

Data Analysis The angular distribution of the X-rays originates from the coherent scattering

on the electrons in the penetrated material. If both polymers A and Β differ in their electron densities and if characteristic dimensions are of the magnitude of the photon wavelength, their spatial electron distribution can be measured. For symmetrical diblock copolymers in the phase separated state those conditions are fulfilled and a relatively sharp scattering intensity maximum at small angles originating from the lamellar spacing is observed. According to Leiblers theory [1], the scattering maximum in the disordered state can be interpreted as an effect of volume exclusion (correlation hole). Since blocks A and Β are grafted together, the probability to find another block A in the vicinity of a block A is reduced. Even if this theory does not describe correctly all the details of the transition, one can adapt the theoretical Leibler profile to the data in the disordered regime and thus obtain a good approximation of the Flory-Huggins segment-segment interaction parameter χ knowing f and N.

For the SAXS profiles measured at Mainz, the peak maximum was fitted using a Gauss function below T M S T and a Leibler function above T M S T , respectively. For examination of the second and third order peaks, shear oriented and non-oriented samples were investigated. The main peak was fitted with a Gauss function with constant background and the graphs were normalized in q-values and intensity. At double and triple q-values, a gauss function with exponential background were used as fit functions. The SAXS profiles measured at the ESRF were fitted by Lorentz functions as the profiles are not significantly broadened by the instrumental resolution. It should be kept in mind that the functional form of the peak depends both on the phase state of the sample and the

460

resolution function of the instrument. Therefore different fit functions are used for the different samples and instruments.

Results and Discussion

/. Temperature dependent measurements The temperature dependence of the scattering properties was explored in

several heating and cooling cycles with various step widths. The material did not show any detectable degradation due to the exposure to radiation, light or oxygen. This was checked by comparing T M ST for the different cycles (even after an extended exposure to X-rays) and by a comparison of GPC spectra of used and unused material. The accessible temperature region is limited by the glass transition of the polystyrene block to lower temperatures and by degradation of the polybutadiene block to higher temperatures. The scattering behavior was found not to be affected by the antioxidant additive, especially no rise in intensity at q->0 was observed. The accessible temperature range could be extended by 20 Κ by adding the antioxidant.

Figure 1 shows the first order morphology peaks of SAXS profiles measured at different temperatures. For clarity, only part of the data is depicted. Since the temperature difference between subsequent curves is always 10K in Fig.l, one can clearly identify the transition to be located between 90 and 100°C where a big jump in the intensity occurs. The exact position is obtained from a detailed temperature scan and is T M S T = 98.5°C. At low temperatures, the peaks are of Gaussian shape due to the resolution function of the SAXS instrument (instead of a Lorentz type as measured at other instruments, see below). Near the phase transition the main peak becomes increasingly asymmetric and at T M S T a distinct change in peak height I* and full width at half maximum Aq is observed. The position of the peak q* instead shows an almost continuous shift across the explored temperature region. Beyond T M S T , the profile has a Leibler shape. For the lowest temperature in the disordered state, Τ = 99°C, a value of Νχ = 10.45 was found. This is in good agreement with the theoretical prediction of 10.5 from Leiblers theory. The smallest value for Νχ in the accessible temperature range at Τ = 150°C is Νχ = 10.32, obtained the same way is not much smaller and exhibits the weak dependence of χ on temperature. Even if the estimation of this parameter using this theory is not very accurate, one can suppose, that the tendency is correctly reproduced . In Figure 2 the fit parameters of the SAXS profiles for a heating and a cooling cycle are plotted. In the disordered state, the normalized inverse maximum intensity I V I * is not linearly connected to the inverse temperature 1/T as predicted in the weak segregation limit (WSL) of mean field theory. This indicates the influence of thermally induced fluctuations, which can be seen as some kind of entropie penalty to separation and which suppress the transition.

The data point in brackets corresponds to a measurement directly after the sample preparation. It illustrates the influence of the sample preparation process. The peak position as given by q* shifts linearly across the phase transition; with Rg°c q*"1 [1],[4], no chain stretching at the transition can be confirmed.

461

Figure 1: SAXS profiles ofP(S-b-B) at different temperatures (see box) in the accessible thermal range. The microphase separation transition occurs between 90 and 100°C. Typical counting statistics are 10s total photons per measurement.

462

4,0 3,5

*HH 3,0 ^ 2,5

* ° 2 , 0 ^ 1,5

1,0 0,5

, , 0,41

'g 0,40

£ j 0,39

* 0,38

0,37 0,14

%3 ° ' 1 2

, 5 , 0,10

5" 0,08

0,06

— • — increasing — • — decreasing

temperature

I 1 1 1 1 I 1 1 1 1 I 1

IQB—O—g B

Μ » M I M M 1 I

I M M I " " I

MST

log—o—g— • ι . . . . ι . . . . ι .

2,4 2,5 2,6 2,7 2,8 2,9

1000/T [K"1]

Figure 2; Temperature dependent characteristic parameters ofP(S-b-B) during an upward and downward temperature scan, inverse of the normalized scattering peak maximum, 1*</I* (I*o is the peak intensity at the transition), maximum position, q*t and peak width, Aq, as obtained from fits of a Lorentz function to the q dependent scattering intensity. The solid lines are guides to the eyes. The data point in brackets represents the peak intensity of the sample directly after the preparation. The dotted line indicates the position of the microphase separation temperature TMST-

463

Close to the microphase separation transition, a temperature resolution of IK was obtained. Within this accuracy, no hysteresis was found for heating and cooling cycles. We cannot exclude that there is still a time dependent process similar to effects reported by Sakamoto et al. and Floudas et al. on comparable systems [6],[25]. At a time scale larger than 15 minutes, kinetic processes can be excluded. The curves shown in Fig.2 therefore can be expected to represent the equilibrium behavior of the sample.

The second order reflections (not shown here) are not wholly suppressed as expected theoretically for symmetrical diblock copolymers. This effect can be explained by the slight asymmetry of the block length (in particular when the volume fraction is taken into account) leading to different thicknesses of the PS and PB rich domains, respectively. Both, second and third order reflections are at double and triple values of q* within an error smaller than 2% confirming that the system is lamellar. The higher order maxima appear only in the phase separated state and vanish close to T M S T . Their existence indicates that the spatial concentration profile differs from a sinusoidal shape, which is expected to be true in particular in the strong segregation limit (SSL).

Also in dynamic mechanical measurements of this system [18] T M S T is observed by a clear change in the frequency dependent mechanical behavior. Measured at an appropriate frequency a jump in the shear modulus can be identified as a function of temperature.

2. Pressure dependent measurements Isothermal and isobar scans were carried out at two temperatures (110°C and

120°C) and at one pressure (50bar), respectively. The data were analyzed as described above. Figure 3 shows the fit parameters for the isothermal scan at 110°C. It was found that the phase transition behavior is different for the isothermal scans as compared to the isobar scans. In particular the materials response is not instantaneous on change of the pressure. A detailed exploration of the kinetics is described elsewhere [24]and can be seen in Fig. 5 for a typical example. Therefore, in the transition region the time evolution was taken into account waiting about 1 hour for equilibrium conditions. In figure 3 we show both, the fit parameters for an upward and a downward pressure scan. The transition occurs over a range of about 400bar, but is shifted by about 200 bar between the upward and downward scan. The maximum intensity I* and the full width at half maximum Aq show a quite similar behavior with a distinct hysteresis. The peak position q* also exhibits a small discontinuity and hysteresis, while it is linearly changing with no hysteresis in the temperature dependent measurements. For an estimation of a temperature-pressure superposition a linear regression of pi T M S T pairs (see table 2) was performed.

Table 2: Pressure dependence of TMST The different values for TMST are obtained from experiments at the MPI-P, Mainz and at the ESRF, Grenoble. Some

of the differences can be explained by different calibration of the system.

MPI-P ESRF Ρ/bar 0 50±5 600 + 50 1200 + 50 Τ / °C

MST ^ 98,5 ±1 97 ±1 110 ±1 120 ±1

464

0,7 0,6

A oil P M S T & 0,3

cr o,o4 < 0,02

• B - — ο — § — o -

_o— increasing - • _ decreasing

pressure

ο 500 1000 1500 2000 p [ b a r ]

Figure 3: Pressure dependent characteristic peak parameters ofP(S-b-B) during an upward and downward pressure scan, morphology peak maximum, /*, maximum position, q*, and peak width, Aq, as obtained from a fit of a Lorentz function to the q dependent scattering intensity. The solid lines are guides to the eyes. The dotted line indicates the mean position of the microphase separation pressure pMST.

465

The fit to those values yields: 1°C

T M S T (P) = TMST,O + r r — * Δρ (1) 54bar

The pressure-temperature coefficient of 54 bar/K value is close to the prediction by Kasten et al. [11]. Using the pressure-temperature superposition, we obtain for the pressure dependent transition (shown in figure 3) that the transition region (400bar) corresponds to a temperature region of AT=8°C, while we have obtained a ΔΤ of at most 1°C from the actual temperature scan at atmospheric pressure. Figure 4 shows the superposition of 3 corresponding profiles, i.e. sets of temperature and pressure with the same χ parameter close to the phase transition according to the regression of equ.(l). For comparison, also typical profiles both further in the disordered state as well as in the ordered state are plotted. The small discrepancy at small q values originates from different background subtraction of the empty cell plus the pressure transmitting silicon oil. The measurement at 120°C was taken at a different ESRF beam time with different adjustments and set-ups. The pressure region up to ρ « 50 bar has not been explored and therefore we cannot exclude a non-linearity in this region similar to [26]. Unfortunately, measurements in this pressure regime cannot be performed with our experimental setup.

3. Pressure jump experiments As already mentioned before, the materials response to an increase of

pressure across T M S T is not instantaneous. An example is shown in figure 5, where the time development after a pressure jump form the disordered state into the transition regime is given. The solid lines are Avrami-fits (functional form given in the insert) with an exponent of 2. This exponent is in agreement with previous investigations [6], where also a non-isotropic growth of the phase separated grains has been found from SAXS and TEM investigations, while theory [4] predicts a 3-dimensional spherical growth. The Avrami model together with the characteristic q-dependence during the development of the phase separated structure is indicative for a nucleation and growth mechanism. It should also be noted that time constants are relatively large (e.g. 870s in Fig.5), which might be due to the proximity of the glass transition of PS. A quite large incubation period is observed (approximately 200s in Fig.5), where hardly any changes in the scattering behavior are recorded, which is in full agreement with theoretical predictions [4].

Summary

We have investigated various aspects of the microphase separation transition of diblock copolymers P(S-b-B) by small angle x-ray scattering. The temperature dependence of the scattering behavior of P(S-b-B) agrees well with recent theories. It shows a distinct deviation from mean field behavior in the vicinity of TMST, which may be explained by the influence of thermally induced fluctuations. Static pressure dependent behavior on the other hand seems to be in accordance with the temperature experiments and it seems possible for instance to use a

466

ο,ι

π p=50bar , T = 1 0 0 ° C

ο p = 5 5 0 b a r ? T = 1 1 0 ° C

δ p = 1 1 0 0 b a r , T = 1 2 0 ° C

p = 5 0 b a r ? T = 1 2 0 û C

- p=50bar , T = 7 0 ° C

1 1, I, .1, ι • • ,1 · ι ,, ι ι ι ι ι ι — L u : ι , • • , ι , , , , ι , , , ΙζΐΛ 0,3 0,4 0 ,5 0 ,6

q [nm 4 ]

Figure 4: Superposition of three scattering curves recorded at different temperatures and pressures in the vicinity of ΤΜ$τ according to the calculated pressure-temperature superposition. For comparison 2 data sets further in the disordered and ordered state are also plotted (dashed and solid lines). The latter is not displayed in its original height. A typical count rate is 51(f total photons per second.

467

Γ ι— ,,ι • , ,,,,,,ι ,1 1 10 100 1000

t[s] Figure 5: Time dependence of characteristic parameters of SAXS profdes ofP(S-b-B) after a pressure jump from 600 to 800 bar, inverse of the normalized scattering peak maximum, I*(/I* (I%is the peak intensity at the start of the experiment), maximum position, q*, and peak width, Aq, as obtained from fits to the time dependent scattering intensity. The MST of the system is situated at approximately 600 bar with a „ transition range " of ±200 bar. The lines are Avramifits with an exponent of two as given in the insert and explained in the text.

468

pressure-temperature superposition scheme. This allows to replace a pressure by a temperature change and vice versa in the explored pressure-temperature regime. This statement is, however, already not completely true, since in upward and downward pressure scans a hysteresis is observed which is not present in corresponding temperature scans. In addition, chain stretching at the transition seems to be differently involved, since q* changes smoothly with temperature, while a jump is present during pressure variation. In particular also differences are evident, when the time evolution during pressure jump experiments is considered. This concerns for instance the incubation period and time scales involved. In conclusion, there are many similarities between the two transitions, driven either by a change in temperature or pressure, but there are also distinct differences in the behavior, which still have to be explored further.

Acknowledgments

We greatly acknowledge the help of T. Wagner (Mainz) during the sample preparation and characterization as well as the technical help of M . Bach (Mainz) and Dr. O. Diat, Dr. P. Bôsecke and Dr. M . Lorenzen (Grenoble) during the x-ray experiments at the ESRF. W.D. is grateful to Prof. H. Schmidt-Bôcking (Frankfurt) for the helpful discussions and the general interest as well as to Land Hessen for financial support during those studies.

References

[1] Leibler, L. Macromolecules 1980, 13, 1602

[2] Bates, F.S.; Fredrickson, G.H. Ann. Rev. Phys. Chem., 1990, 41, 525

[3] Bates, F.S. Science, 1991, 251, 898

[4] Fredrickson, G.H.; Helfand, E. J. Chem. Phys., 1987, 87(1), 697

[5] Brazovskii, A. Sov. Phys. JETP, 1975, 41, 85

[6] Sakamoto, N.; Hashimoto, T.; Macromolecules 1998, 31, 3292 and 3815.

[7] Bates, F.S.; Rosendale, J.H.; Fredrickson, G.H. J. Chem. Phys. 1992, 92, 6255

[8] Stühn, B.; Mutter, R.; Albrecht, T. Europhys. Lett. 1992, 18, 427

[9] Fredrickson, G.H.; Binder, K., J. Chem. Phys. 1989, 91, 7265

[10] Janssen, S.; Schwahn, D.; Mortensen, K.; Springer, T. Macromolecules 1993, 26, 5587-5591

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[11] Kasten, H. and Stühn, Β. Macromolecules 1995, 28, 4777-4778

[12] Floudas, G.; Hadjichristidis, N.; Stamm, M. ; Likhtman, A.E.; Semenov, A.N. J. Chem. Phys. 1997, 106, 3318

[13] Vilgis, T .A. , private communication

[14] Förster, L.; Khandpur, A.K.; Zhao, J.; Bates F.S.; Hamley, I.W.; Ryan, A.J.; Bras, W. Macromolecules 1994, 27, 6922

[15] Pan, L.H.; Singh, M.A.; Salomons, G.J.; Gupta, J.Α.; Capel, M.S. J. Macromol. Sci.-Phys., 1997, B36(1), 137-151

[16] Bates, F.S.; Schulz, M.F.; Khandpur, A.K.; Förster, S.; Rosendale, J.H., Faraday Discuss. 1994, 98, 7-18

[17] Bartels, V.T.; Stamm, M. ; Mortensen, K. Pol. Bulletin 1996, 36, 103-110

[18] Wang, S.; De Odorico, W.; Pakula, T.; Stamm, M. , to be published

[19] Tang, H.; Freed, K.F., J. Chem. Phys. 1992, 96 , 8621

[20] Bösecke, P.; Diat, O.J Appl. Cryst., 1997, 30, 867-871

[21] Bösecke, P.; Diat, O. ESRF Annual Report 1995/1996: ID2, http://www.esrf.fr/info/scienee/

[22] Diat, O., ID 2 HIGH BRILLIANCE, ESRF, Handbook ID2A, http://www.esrf.fr/exp_facilities/BLHB.html

[23] Ladynski, H; De Odorico, W.; Stamm, M . J. Non-Cyst. Sol. 1998, 235-237, 491-495

[24] Ladynski, H; De Odorico, W.; Stamm, M . to be published

[25] Floudas, G.; Pakula, T.; Velis, G.; Sioula, S.; Hadjichristidis, N. J. Chem. Phys. 1998, 108, 6498

[26] Steinhoff, B.; Rüllmann, M. ; Wenzel, M. ; Junker, M. ; Alig, I.; Oser, R.; Stühn, B.; Meier, G.; Diat, O.; Bösecke, P.; Stanley, H. B. Macromolecules; 1998; 37(1); 36-40

[27] Avrami, M.J. J. Chem. Phys. 1941, 9, 177

Chapter 30

Ordering Kinetics between HEX and BCC Microdomains for SI and SIS Block Copolymers

Hee Hyun Lee and Jin Kon Kim1

Department of Chemical Engineering and Polymer Research Institute, Pohang University of Science and Technology, Pohang,

Kyungbuk 790-784, Korea

Order to order transitions (OOT) from hexagonally packed cylindrical (HEX) microdomains to body centered cubic (BCC) microdomains and vice versa, for a polystyrene-block-polyisoprene (SI-C1) and a polystyrene-block-polyisoprene-block-polystyrene (Vector 4111) were investigated by rheology and synchrotron small angle X-ray scattering experiments using one dimensional detector (1-D SAXS) and an imaging plate (2-D SAXS). The weight fraction of PS block in both block copolymers was almost the same (0.18), and the molecular weight of SI-C1 was almost half that of Vector 4111. The cylindrical axis of the HEX microdomains in the as-cast samples of both block copolymers was aligned toward the flow direction in a macroscopic scale when an oscillatory shearing with a large strain amplitude and a low frequency was applied to the samples.

Interestingly, we found, via 2-D SAXS and rheology, that the initial cylinder axis of the HEX microdomains in the aligned sample was significantly changed once the sample was heated to the region corresponding to BCC microdomains and cooled to the temperature in HEX region without a shearing. Because of the existence of bridge (tail) conformation of PI block, the transition from HEX to BCC microdomains for Vector 4111 took place slowly compared with SI-C1. The OOT from HEX to BCC microdomains was found to be much easier process than that from the reverse transition of BCC to HEX microdomains.

'Corresponding author.

470 © 2000 American Chemical Society

471

Block copolymers consisting of immiscible block chains form microdomains on the order of 10 nm at temperatures below the order-disorder transition temperature (T 0 D T), because of a chemical junction between the constituent blocks (/). One of the most fascinating features of block copolymers is their ability to self-assemble into various microdomains, such as body-centered cubic spheres (BCC), hexagonally packed cylinders (HEX) and lamellae layers depending on the volume fraction of one block as well as temperature. It is known today that besides these classical structures, a complex structure, e.g. Gyroid (Ia3d), was found for some block copolymers (2).

Although in 1980 Leibler (3) theoretically predicted that for a diblock the microdomain existing at one temperature could be thermoreversibly changed to another, experimental evidence was not reported until 1992 {4-13). Many research groups showed that the transformation from one ordered (microdomain) structure to another in a block copolymer could be described as an epitaxial growth, where structural elements of the resulting phase grow from the original microdomains preserving the orientation of the reflection plane without a long-range transport of one microdomain. For instance, Koppi et al. (6) reported an epitaxial relationship between HEX and BCC microdomains for poiy(ethylene propylene)-6/ocA;-poly(ethyl ethylene) (PEP-PEE) copolymer, where the BCC phase grows epitaxially with the f i l l ] direction coincident with the original cylinder axis in HEX microdomains. Sakurai et al. (11) also reported thermoreversible transition between BCC and HEX microdomains as an epitaxial change for the mixture of polystyrene-6/oc£-polyisoprene (SI) copolymer and dioctyl phthalate. The ordering kinetics from the disordered state to an ordered state, and from one ordered state to another ordered state, have been very important in determining the final morphological control of microdomain structures in block copolymers. This is similar to the polymer blends whose the final mechanical properties strongly depend upon blend morphology which is in turn affected by phase-separation kinetics. However, to date few experimental studies have been investigated on the ordering kinetics between different ordered states (13,14). But, theoretical approaches for predicting this kinetics are available at the present time. Investigating the kinetics of the order-order transition (OOT) for a diblock copolymer in a weak segregation limit regime using a time-dependent Ginzburg-Landau approach, Qi and Wang (75) showed the existence of undulating cylinders as a kinetic pathway during the OOT from HEX to BCC. Using an anisotropic fluctuation theory, Laradji et al. (16) reported similar results to those predicted by Qi and Wang (15).

Recently, we reported the ordering kinetics between HEX and BCC microdomains for polystyrene-6/od:-poryisoprene-&/0c& polystyrene (SIS) copolymer (13). In ref (13), we employed an as-cast sample that was not aligned in a macroscopic scale. In this situation a small angle X-ray scattering with one-dimensional detector (1-D SAXS) could be employed to study the ordering kinetics because of the random orientations of grains in the block copolymer. Also, in a separate paper we have shown that during the transition

472

from HEX to BCC microdomains of the SIS copolymer, an exothermic peak was observed in a thermogram measured by a differential scanning calorimeter during heating run (17). We ascribed this behavior to the existence of undulating cylinders having large packing entropy before transforming into BCC microdomains. Very recently, Ryu et al. (18) showed experimentally the existence of the undulating cylinders during the OOT from HEX to BCC microdomains.

In this study, we investigated the ordering kinetics between HEX and BCC microdomains for SI and SIS copolymers by rheology and synchrotron SAXS experiments with a one-dimensional detector and an imaging plate (2-D SAXS). In order to minimize the effect of different grains in HEX microdomains, the as-cast samples were aligned macroscopically using an oscillatory shearing with a large strain amplitude. The emphasis was placed upon the effect of the chain structure (tri versus di block) on the ordering kinetics during the OOT. We found that the existence of bridge conformation of PI chains in the SIS significantly affected the ordering kinetics from HEX to BCC microdomains. In this study, we report the highlights of our findings.

Experimental

Materials

The SI diblock copolymer (SI-C1) was prepared by anionic polymerization in cyclohexane with s-BuLi as an initiator. The SIS triblock copolymer was a commercial grade (Vector 4111, Dow-Exxon Polymer Co.). The weight average molecular weight (Mw) and the molecular weight distribution (M^/MJ were measured by a low-angle laser-light scattering (LALLS) device with measuring value of dn/dc, and gel permeation chromatography (GPC, Waters Co.), respectively. The microstructures of the PI were determined by Ή nuclear magnetic resonance spectroscope (NMR, Brucker DRX500). The molecular characteristics of SI-C1 and Vector 4111 copolymers are summarized in Table 1. We found that the M w of the latter is almost twice that of the former, and the weight (or volume) fraction of the PS block in both block copolymers is almost same. Samples for rheology and SAXS experiments were prepared by first dissolving a predetermined amount of the block copolymer in toluene (10 wt % in solid) in the presence of an antioxidant (Irganox 1010; Ciba-Geigy Group), and then slow evaporating the solvent at room temperature in a hood for a week and 3 days in a vacuum oven. After a trace of the solvent was completely removed, the specimen was finally annealed at 130 °C for 48 h in a vacuum oven. This sample is referred to as the as-cast sample.

473

Table 1. The Molecular Characteristics of Block Copolymers employed in this study.

Sample code

Total Mwa)

Wt%of PS blockb) MJMn

c> -

Microstructure (mole%)a)

1,2 1,4 3,4

Vector 143,000 18.3 1.11 90.3 9.7 4111 SI-C1 67,200 18.5 1.02 - 92.9 7.1

a ) Determined by a low angle laser light scattering (LALLS ) device b ) Determined by a ! H nuclear magnetic resonance (NMR) spectrometer c ) Determined by gel permeation chromatography (GPC) measurement

474

Rheological Properties

Using an Advanced Rheometric Expansion System (ARES) with parallel plates of 25 mm diameter, we performed dynamic temperature sweep experiments under isochronal conditions with increasing temperature as well as decreasing temperature. The heating and cooling rate of these experiments was 0.5 °C/min. The strain amplitude (γ0) and the angular frequency (ω) were low enough to satisfy a linear viscoelasticity.

Applying a large strain with a low dynamic shear rate at a temperature corresponding to HEX microdomains further aligned the random orientation of HEX microdomains in the as-cast samples of the SI and SIS copolymers toward the flow direction in a macroscopic scale. This process was quite efficient in removing the grain boundary between HEX microdomains in the as-cast samples (19,20). After the samples were quenched to room temperature, the specimen for 2-D SAXS was prepared by cutting the samples near the edge of the parallel plates into a rectangular bar shape. In this study, we will use x-, y-, and z-axes as the flow, the velocity gradient, and the neutral (or vorticity) directions, respectively.

Synchrotron Small Angle X-ray Scattering

Time resolved SAXS experiments were conducted using the beam line (3C2) at the Pohang Light Source (PLS), Korea (21). The incident beam was focused with a toroidal mirror and monochromatized using a double crystal S i ( l l l ) monochromator at a wavelength (λ) of 0.1598 nm, and scattering intensity (I(q)) was detected by a image plate for 2-D SAXS patterns and by a position sensitive diode-array detector (ST-120; Princeton Instruments Inc.; namely 1-D detector) allowing various wave vectors (q = (4π/λ)8ΐη(θ/2) where θ is the scattered angle). Since the block copolymers were well aligned on a macroscopic scale, the imaging plate allowed one to study the microstructure of aligned sample. However, the imaging plate was not appropriate for investigating the ordering kinetics of the block copolymers employed in this study since the ordering took place fast. Thus, in this situation we employed the 1-D detector rather than the imaging plate. But, when the 1-D detector was used, the scanning direction of the specimen (or the tilting of the specimen) was very important because the scattering peak might be absent when the wrong direction is chosen for the aligned samples. SAXS profiles measured by the 1-D detector were obtained continuously during the particular kinetics experiment with an exposure time of 30 s. Two heating blocks were used for the temperature jumping experiment. The sample holder soaked for certain times at initial temperature in the first heating block (annealing block) was manually placed into the second heating block (x-ray passing block) maintained at the setting temperature. After the sample was placed into the second heating block, the specimen reached the setting temperature within 1 min.

Results and Discussion

Figure 1 gives the overall SAXS profiles at several temperatures between

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476

170 and 200 °C for the as-cast samples of Vector 4111 and SI-C1. These profiles were obtained during heating at a rate of 1 °C/min after annealing at 170 °C for 30 min and arbitrarily shifted to avoid overlaps. It can be seen in Figure 1 that theV2qm, where q m is the maximum peak position, was observed at temperatures greater than 190 °C for Vector 4111 and at temperatures greater than 184 °C for SI-C1. When a block copolymer shows SAXS profiles with diffraction peaks appearing at the ratios of 1:V2 :V3:>/4 ... in the reciprocal space, it has the BCC microdomains, while it has hexagonally packed cylindrical (HEX) microdomains when diffraction peaks appear at 1:V3 :VÎ :V7 ... in reciprocal space. On the basis of the results, the Τ ο ο γ of Vector 4111 was 185 ~ 190 °C, while that of SI-C1 was 181 ~ 184 °C.

Figures 2 and 3 give the changes of the maximum scattering intensity, I (qm) and domain spacing, D given by 2n/qm9 with temperature for the as-cast samples of Vector 4111 and SI-C1. It is seen that I (qm) and D changed dramatically near the order to order transition temperature (T 0 0 T ) . When the T 0 0 T is defined as the temperature where the 1 (qm) exhibits a minimum, the T 0 0 T for Vector 4111 and SI-C1 was 186 ± 1 °C and 183 ± 1 °C, respectively. Furthermore, we have the essentially the same values of T 0 0 T for these two block copolymers from the change in the D with temperature as shown in Figure 3. However, the transition at the TQQT of SI-C1 was very sharp (less than ± 1 °C), whereas that of the Vector 4111 was not very sharp (less than ± 4 °C). This implies that the transition from HEX to BCC for SI-C1 took place faster than that for Vector 4111. However, we expect that when the heating rate is reduced to very small one (say 0.1 °C/min), one might obtain a sharp discontinuity in the D for Vector 4111 similar to that for SI-C1.

One interesting thing is that at the T 0 0 T , the D of HEX microdomain is not the exact equal to that of BCC microdomain; rather, the former is slightly larger than the latter (about 4 - 5 %). This led us to conclude that when HEX was transformed to BCC, the interdomain spacing was not exactly maintained but was slightly reduced. By employing a self-consistent field theory in a strong segregation limit, Matsen (22) showed the difference in the domain spacing during the OOT and explained this behavior due to a conformational asymmetry between constituent two blocks. The decrease in the D near the TQQT during the OOT from HEX to BCC microdomains was also reported by Ryu et al (23). However, they reported a somewhat large change in the D (about 7 %). This might be due to the fact that the difference in measuring temperatures (30 °C) was much larger than that employed in this study. As seen in Figure 3, if two different temperatures corresponding to HEX and BCC microdomain are farther away from the T 0 0 T , the change in D would be larger.

Figure 4 gives a temperature sweep experiment of the storage modulus (G') during heating for the as-cast samples of Vector 4111 and SI-C1. The behavior of G' for Vector 4111 was reported in the previous paper (/ 7). When the OOT is taken as the temperature where a minimum in G' appears, the values of the T 0 0 T

for Vector 4111 and SI-C1 are estimated to be 184 ± 1 °C and 179 ± 1 °C, respectively. Although these values are slightly lower than those obtained from SAXS results, we consider this difference to be negligible since two different methods are employed and an oscillatory shear, although it is small, was employed. One thing to note from Figure 4 is that order to disorder temperature (T 0DT) o f Vector 4111 is - 20 °C lower than that of SI-C1 when we consider the

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Temperature (°C)

Figure 2. The changes of the first order peak intensity, I(qJ with temperature obtained during heating at a rate ofl °C/minfor the as-cast samples of (a) Vector 4111 and φ) SI-CI

478

Temperature (°C)

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Temperature ( C)

Figure 3. The changes of the interdomain spacing (D) with temperatures obtained during heating at a rate of 1 °C/min for the as-cast samples of (a) Vector 4111 and (b) SI-C1.

479

Figure 4. Temperature sweep of G'obtained during heating at a rate of 1 °C/min for the as-cast sample of Vector 4111 at 1 rad/s and γ0 = 0.05, and for Sl-Cl at ω = 0.05 rad/s and γ0 = 0.03.

480

temperature where G' drop precipitously as a T 0 D X . This is contradictory to the result of Ryu et al. (23) who showed that the T 0 D T of an SIS was higher (~ 25 °C) than that of the SI. Notice that in ref (23) the molecular weight of SIS was exactly twice that the SI and the volume fraction of PS block in the two was the same. But, the molecular weight of Vector 4111 is not exactly twice that of SI-C1. A somewhat broad polydispersity of Vector 4111 compared with that of SI-C l might affect the rheological properties. It should be mentioned, however, that the temperature sweep of G' might not be applicable to determine T 0 D T of asymmetric block copolymers. For instance, Sakamoto et al. (24) estimated the T 0 D T of Vector 4111 to be ~ 280 °C from the T 0 D T ' s of the mixtures of Vector 4111/dioctyl phthalate, which is much higher than the temperature where G' drops precipitously. For an asymmetric block copolymer, the temperature where G' drops precipitously is defined by the lattice disordering temperature not the T 0 DT (24,25). As mentioned in experimental section, as-cast samples usually have many grains and these grains are oriented randomly on a macroscopic scale. We found that as-cast samples of Vector 4111 and SI-C1 have random orientation of HEX microdomains, which was confirmed by 2-D SAXS results showing a ring pattern of the first order peak.

Figure 5 gives the change in the G' with time for Vector 4111 at 170 °C with γ 0 = 1 and ω = 0.1 rad/s, and for SI-C1 at 160 °C with γ 0 = 1 and ω = 1 rad/s during the alignment, respectively. We could not use the same temperature for both block copolymers, because SI-C1 has a transition close to 170 °C. It might be suggested that when the dimensionless quantity ωτ (τ is the longest relaxation time) of Vector 4111 is similar to that of SI-C1, the orientation degree in both block copolymers would be similar. If both block copolymers are assumed to be in homogeneous state, τ of Vector 4111 would be -10 times larger than that of SI-C l because of τ - M w

3 4 . However, this speculation should be carefully checked because both block copolymers at the shearing temperature have HEX microdomains. It was also found that even when ω was increased to 1 rad/s for Vector 4111, the change in G' with time did not change much. According to Chen and Kornfield (26), these two parameters (γ0 and ω) strongly affected the ordering mechanism, although they only considered the block copolymers with lamellar microdomains.

It is seen in Figure 5 that G' for Vector 4111 decreased at short times (less than 10 min) and reached steady values. But, G' for SI-C1 decreased rapidly at short times, then decreased gradually at longer times, but G' did not level off even at very long times (say 1 h). Thus, we consider that the alignment of HEX microdomains in Vector 4111 might be better than that in SI-CI. Risse et al. (27) reported that the alignment of an SI diblock was poorer than that of an SIS triblock and ascribed this to the lack of bridging chain conformation of the SI. Therefore, the different behavior in the alignment between Vector 4111 and SI-C1 might be attributed to the existence of bridge conformations in the triblock copolymer, although the reason is not clear at the present time.

The 2-D SAXS patterns for the aligned samples of Vector 4111 and SI-C1 at various temperatures during heating and cooling are given in Figures 6 and 7, respectively. Al l the SAXS patterns are taken after annealing for 15 min during heating and 30 min at BCC region. Also, SAXS patterns for a cooled HEX sample from BCC was obtained after annealing for 40min in order to develop complete HEX phase. The alignment was done at 170 °C for the former and at

481

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Ο 20 40 60 80 100 120

Time (min)

Figure 5. The change in G' with time during the alignment by an oscillatory shearing with y0 = 1 and ω-ΟΛ rad/s for the as-cast sample of Vector 4111 and γ0- 1 and ω - 1 rad/s for the as-cast sample of SI-C1. The annealing temperatures for Vector 4111 andSI-Cl were 17(PC and 160°C, respectively, which corresponded to the HEX microdomains in both block copolymers.

482

Figure 6. 2-D SAXS patterns for the aligned sample of Vector 4111 at various temperatures: (a) 170 °C; (b) 185 °C; (c) 200 °C; (d) 170 °C (cooling); (e) 200 °C (reheating)

483

Figure 7. 2-D SAXS patterns for the aligned sample ofSI-Clat various temperatures:(a) 160 °C;(b) 180 °C; (c) 195 °C; (d) 160 °C (cooling); (e) 195 °C (reheating)

484

160 °C for the latter as shown in Figure 5. In this study, only the SAXS patterns on the qx-qy plane are considered. The SAXS patterns on two other planes (qx-qz

and qx-qy planes) will be reported elsewhere (28). The 2-fold diffraction peaks are clearly seen at 170 °C for Vector 4111 and at 160 °C for SI-C1, although the peaks for the latter are somewhat broad. This implies that the cylinders in the HEX microdomains for both samples are well aligned toward the flow directions and the alignment of cylinders in Vector 4111 would be better than that in SI-C1, which is consistent with the rheological behaviors shown in Figure 5.

At 185 °C, the intensities of 90 and 270 0 start to decrease and four new shoulders appear near the main peak, although intensity levels of these shoulders are much lower than main peaks. These shoulders are due to the existence of undulating cylinders at this temperature that would be the precursors of a twinned BCC. Namely, the anisotropic fluctuation in the HEX phase is on a twinned BCC predicted by Qi and Wang (15). When temperature was raised to 200 °C, the undulating cylinders split into BCC spheres. The appearance of new four peaks at V2 qm on the qx-qy plane suggests that the transformation into BCC was completed at this temperature. Also, four other peaks of q^ at 30, 150, 210, 330 0

are clearly seen although the intensity level of these peaks is smaller than that of qm along the qy. The existence of four peaks on the qx-qy plane is different from the results by Almdal et al. (29) that only two-fold qm peaks along the qy direction was shown. But, they examined a twinned BCC structure prepared by a dynamically oscillatory shearing at a temperature corresponding to the BCC phase for a PEP-PEE diblock copolymer, whereas we investigated a twinned BCC structure formed from HEX. As mentioned by Qi and Wang (30), they might observe the fluctuation from the HEX phase formed due to the destruction of the BCC by a shear, rather than Bragg peaks from a twinned BCC structure.

Very interesting phenomena were found when the temperature was lowered from 200 °C to 170 °C, since SAXS pattern is completely different from that of 170 °C before heating. This suggests that during the transition from BCC to HEX some HEX cylinders having different orientations from the original shear direction occurred. However, the positions of the qm became very similar to those at 200 °C although the positions of higher order peaks are significantly changed during BCC to HEX transition. These behaviors imply that the reflection planes corresponding to the qm did not change during this transition. In other words, the direction of {100} plane of HEX lattice became the same as the direction of {110} plane of BCC lattice with the spheres in <111> direction of BCC lattice forming the cylinder axis of the HEX lattice. This is a so-called epitaxial relationship from BCC to HEX. Because twinned BCC has total seven different <111> directions, the newly degenerated HEX cylinders formed from this twinned BCC would also have seven different HEX grains. The cylinder axis of each grain, of course, becomes one of seven <111> directions of the twinned BCC.

In the case of the SI-C1, one notes from Figure 7 that the changes in 2-D SAXS patterns with temperature are very similar to Vector 4111, except that the diffraction peaks are relatively broader compared with Vector 4111. This broadness might be due to less alignment of cylindrical microdomains for SI-C1. On the basis of these results we concluded that the transition mechanism between HEX and BCC microdomains for SI diblock and SIS triblock copolymers would be similar.

485

Figure 8 gives temperature variation in G' for the aligned samples of Vector 4111 and SI-C1. It is noted in Figure 8(a) that with increasing temperature, G' increased twice before leveling off. The initial increase near 175 °C resulted from an appearance of undulating cylinders from well-aligned cylinders. This is consistent with the 2-D SAXS pattern at 185 °C given in Figure 6(b) where four shoulders appeared near qm peaks along qy direction. The second increase in G' just above 187 °C is attributed to the complete splitting of undulating cylinders into BCC spheres. The 3-D microdomain structure of BCC has more elastic properties than the aligned cylinders and thus has higher G' value. This two step increase in G' was also reported by Ryu et al (23). Very interestingly, once the aligned sample of Vector 4111 was annealed at 200 °C for 30 min in order to develop complete BCC microdomains, the G' change with temperature during cooling was completely different from that obtained during the first heating. This is due to the formation of seven degenerated cylinders from the twinned BCC. Among the seven degenerated HEX cylinders, all cylinders except one cylinder formed from [Til] prevented PS chains in HEX domain from relaxing toward the shear direction because the axis of six cylinders is not parallel to the shear direction, thus high G' is obtained.

Furthermore, it is seen that G' obtained during the reheating process after annealing at 170 °C for 40 min is very similar to that obtained during the cooling process. This is also consistent with 2-D SAXS patterns. Thus, thermoreversibility between the HEX and BCC microdomains is expected when degenerated cylinders are obtained from twinned BCC.

It is also shown that G' for the aligned sample of SI-C1 increases twice with increasing temperature, which is similar to the behavior for the aligned sample of Vector 4111: the first increase near 165 °C corresponding to the appearance of undulated cylinders, and the second at 180 °C corresponding to the formation of BCC microdomains. Once the aligned SI-C1 was annealed at 195 °C for 30 min, the temperature dependence of G' during the cooling process was completely different from that obtained during the first heating run. This is because of the degenerated HEX cylinders formed from the BCC. Also, the behavior of G' with temperature during the reheating process is similar to that during the cooling process.

It should be mentioned that the above behavior of G' is not consistent with the result investigated by Koppi et al (6). They showed that the path of G r with temperature obtained during heating for the PEP-PEE copolymer with a pre-shear alignment was similar to that obtained during cooling except some thermal hysteresis between heating and cooling. Although the reason of the different behaviors is not clear at the present time, the modulus measured in PEP-PEE block copolymer seemed to be associated with the distortion of chain conformation rather than microstructure variation during the transition from BCC to HEX transition. It is noted that the viscoelastic contrast of the constituent blocks between PEP-PEE and SI block copolymers is quite different even if the similar oscillatory shearing conditions are employed. In other words, PEP block consisting of HEX microdomains in a PEP-PEE block copolymer is much softer than PS block consisting of HEX microdomains in an SI block copolymer. Thus, even when degenerated HEX microdomains of PEP block were formed from the twinned BCC (or BCC) of the PEP-PEE block copolymer during cooling process, these do not contribute much to the modulus. The viscoelastic contrast might be

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varied with the amplitude and frequency of the oscillatory shearing which can change the relaxation behavior of an aligned sample during the OOT between HEX and BCC microdomains. According to Chen and Kornfield (26), these two parameters are strongly related to the elastic properties between chain conformation and microstructure.

From Figure 8(b), one notes that the OOT from BCC to HEX obtained during the cooling process occurred near 167 °C, while the OOT from HEX to BCC obtained during reheating process after annealing at 160 °C for 40 min took place near 178 °C. This difference is because of the kinetics effect, namely the heating and cooling rates of 1 °C/min are not so slow.

Although the general behavior of the temperature dependence of G' for Vector 4111 and SI-C1 is similar, there exist some differences between the two. First, the second increase in G f of SI-C1 between the first plateau and G' of BCC microdomains was larger than that of Vector 4111, and the OOT transition for ΒΙ­Ο 1 was sharper than that for Vector 4111. This might be attributed to the bridge conformation of PI chains in Vector 4111 that would prevent free motion of PS chains in the PS blocks from being transformed into BCC microdomains. We consider that a complete transformation (or development) into BCC microdomains would be a slow process compared with the breaking process from HEX cylinders (or undulating cylinders) into spheres. Therefore, the development of BCC microdomains for SI-C1 would be easily achieved compared with Vector 4111, which makes the difference in G' of SI-C1 between the first plateau and BCC spheres larger than that of Vector 4111. This argument is consistent with the fact that once BCC microdomains were formed from HEX microdomains, high order peaks of Λ/2 qm and V3 q m for SI-C1 were more visible than those for Vector 4111, which is shown in Figure 1. Furthermore, even if the total molecular weight of SI-C1 was almost half of that of Vector 4111 and T o o x

of SI-C1 was smaller than that of Vector 4111, the temperature range corresponding to BCC microdomains was about twice of that for Vector 4111, as shown in Figure 4. This is attributed to the fact that compared with Vector 4111 more perfectly formed BCC microdomains in SI-C1 would resist disruption or dissolution even at higher temperature. But, poorly organized BCC microdomains in Vector 4111 might be easily transformed into spheres with the liquid-like short-range order (LSO) with increasing temperature (24) since the packing entropy at LSO is much greater than that at BCC microdomains. Although the transition from BCC to LSO is another interesting phenomenon, this is discussed elsewhere (25). However, it would be interesting to investigate whether a long time annealed Vector 4111 at a temperature corresponding to BCC microdomains, say at 200 °C, has a broad temperature range of BCC before disrupting to spheres with LSO.

Figure 9 gives I(qm) variations with temperature obtained during heating at a rate of 1 °C/min for the aligned samples of Vector 4111 and SI-C1. This I(qm) was obtained using the 1-D detector. As well aligned HEX cylinders transformed into twinned BCC spheres, the I(qm) began to decrease near the T 0 0 T . When the T 0 0 T was taken as the temperature that the slope in plots of I(qm) versus temperature changed for the first time, it was 187 °C for Vector 4111 and 182.5 °C for SI-C1. The T 0 0 T s of two block copolymers determined by this method are very similar to those obtained by rheology as shown in Figure 8.

Figure 10 gives the overall SAXS profiles and the variations of the first order

Figure 9. The first order peak intensity variations with temperature obtained during heating at a rate of 1 °C/min for aligned samples of (a) Vector 4111 and (b) SI-C1.

489

1000 f-0 b _ j ! » ι . ι . ι • ι , ι , I , ( • tl

200 400 600 800 1000 1200 1400 1600 1800

200 400 600 800 1000 1200 1400 1600 1800

Time (sec)

Figure 10. The changes of (a) SAXS profiles, (b) the first order peak intensity and, (c) the domain spacing with time when temperature was jumped from HEX microdomains (TOOR - 7.5 °C) to BCC microdomains (TQQT + 7.5 °C) for aligned sample of Vector 4111.

490

peak intensity and the domain spacing with time when temperature was jumped from the HEX microdomains ( T 0 0 T - 7.5 °C) to the BCC microdomains ( T 0 0 T + 7.5 °C) for the aligned sample of Vector 4111. The intensities of high order peaks decreased rapidly, the first order peak intensity and the domain spacing dropped quickly within ~ 2 min. However, a high order peak of Λ/2 qm

corresponding to BCC microdomains did not appear within 30 min, although a weak high order peak of Λ/3 qm is shown. This is because Λ/2 qm peaks are at the 45° from the qy direction.

Figure 11 gives the overall SAXS profiles and the variations of the first order peak intensity and the domain spacing with time when temperature was quenched from BCC microdomains ( T 0 0 T + 7.5 °C) to HEX microdomains (T 0 0 T - 7.5 °C) for Vector 4111. interestingly, the first order peak intensity and the domain spacing did not change during an incubation time of - 10 min before transforming into HEX microdomains. Furthermore, high order peaks of Λ/3 q m

and V4 qm corresponding to HEX microdomains appeared at longer times. This result is in a good agreement with the rheological result in Figure 8(a) that the hysteresis of G' was shown between the cooling and reheating processes, and the T 0OT obtained from cooling was about 12 °C lower than that obtained during heating. Therefore, the transition from HEX to BCC microdomains would be a much easier process than the reverse transition from BCC to HEX microdomains.

Figures 12 and 13 give the overall SAXS profiles and the variations of the first order peak intensity and domain spacing with time for SI-C1 when temperature was jumped from the HEX microdomains (T 0 0 T - 7.5 °C) to BCC microdomains ( Τ ο ο τ + 7.5 °C) and vice versa, respectively. In the case of SI-C1, it was easy to find 1-D detector direction so as to see both the first and higher order peaks. It can be seen in Figure 12 that the intensity of the first order peak and the domain spacing dropped rapidly within ~ 1 min, which is faster than the results of Vector 4111. The high order peak of V2 qm was clearly observed at times longer than 65 s. These results also support our speculation described above that the transformation of HEX microdomains into BCC microdomains for SI-C1 is much faster compared with Vector 4111. We consider that the SAXS profile at 65 s might correspond to the transition state from HEX to BCC microdomains, and two microdomains coexisted due to the superposition of higher order peaks of Λ/2, Λ/3 and V4 qm. The easy detection of the coexistence in SI-C1 compared with Vector 4111 implied that the transition from HEX to BCC microdomains for the former took place faster than that for the latter. This coexistence was not attributed to the long exposure time since the exposure time for each SAXS profile was only 30 s. However, when temperature was quenched from BCC to HEX microdomains, the Λ/2 qm peak was visible up to ~ 30 min in Figure 13. This is because of the slow coalescence of spheres in BCC into HEX cylinders. The D would not change during an incubation time up to 400 s and the increase in D took place very slowly compared with the transition from HEX to BCC microdomains. The results given in Figures 10-13 led us to conclude that the transformations from HEX to BCC microdomains took place much faster than the reverse transformation from BCC to HEX microdomains for bothSI-Cl and Vector 4111.

491

0.2 0.3 0.4 0.5 0.6 0.7 0.8

q (nm*1)

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Time (sec)

Figure 11. The changes of (a) SAXS profiles, (b) the first order peak intensity and, (c) the domain spacing with time when temperature was jumped from BCC microdomains (T00T + 7.5 °C) to HEX microdomains (T00T - 7.5 °C) for Vector 4111.

492

O H 1 I ι I ι ι . ι . » . I . I . 1 . I 1 200 400 600 800 1000 1200 1400 1600 1800

2801- I , I , ι , l , ι , ι , ι , ι , 1 I 200 400 600 800 1000 1200 1400 1600 1800

Time (sec)

Figure 12. The changes of (a) SAXS profiles, (b) the first order peak intensity and, (c) the domain spacing with time when temperature was jumped from HEX microdomains (TQQT - 7.5 °C) to BCC microdomains (T0OT + 7.5 °C) for aligned sample ofSI-Cl.

493

200 400 600 800 1000 1200 1400 1600

Time (sec)

Figure 13. The changes of (a) SAXS profiles, (b) the first order peak intensity and, (c) the domain spacing with time when temperature was jumped from BCC microdomains (T00T + 7.5 °C) to HEX microdomains (T00T- 7.5 °C) for SI-CI.

494

Conclusion

In this study we have shown that the order to order transition kinetics from HEX to BCC microdomains for the Vector 4111 copolymer is much slower than that for the SI-C1 copolymer when the weight fraction of PS in both block copolymer is very similar. This is due to the existence of bridge conformation of PI block in the Vector 4111. Also, the temperature range corresponding to BCC microdomains, which was determined from a plateau in G' during heating, for the SI-C1 was much wider than that for the Vector 4111. This implies that the bridge conformation prevents the perfect ordering of BCC.

It was also found that the initial cylinder axis of HEX microdomains in the aligned sample was significantly changed once the sample was heated to the region corresponding to BCC microdomains and cooled to the HEX region again without a shearing. This result was consistent with the change in G' with temperature.

Finally, the transformations from HEX to BCC microdomains took place much faster than the reverse transformation from BCC to HEX microdomains for bothSI-Cl and Vector 4111.

Acknowledgment

This work was supported by a POSTECH/BSRI special fund (1998) and the Korean Foundation of Science and Engineering (# 97-05-02-03-01-3). Synchrotron SAXS experiments at the PLS (3C2 beam line) were supported by the Ministry of Science and Technology (MOST) and Pohang Iron & Steel Co. (POSCO).

References and Notes

1. Bates, F. S.; Fredrickson, G. H. Annu. Rev. Phys. Chem. 1990, 41, 525. 2. Fredrickson, G. H.; Bates, F. S. Annu. Rev. Phys. Chem. 1996, 26, 501. 3. Leibler, L. Macromolecules 1980, 13, 1602. 4. Almdal, K.; Koppi, Κ. Α.; Bates, F. S.; Mortensen, K. Macromolecules

1992, 25, 1743. 5. Forster, S.; Khandpur, A. K.; Zhao, J.; Bates, F. S.; Hamley, I. W.;

Ryan, A. J.; Bras, W. Macromolecules 1994, 27, 6922. 6. Koppi, Κ. Α.; Tirrell. M. ; Bates, F. S. J. Rheology 1994, 38, 999. 7. Hajduk, D. Α.; Gruner, S. M. ; Rangarajan, P.; Register, R. Α.; Fetters, L. J.;

Honeker, C.; Albarak, R. J.; Thomas, E. L. Macromolecules 1994, 27, 490. 8. Khandpur, A. K.; Förster, S.; Bates, F. S.; Hamley, I. W.; Ryan, A.

J.; Bras, W.;Almdal, K.; Mortensen, K. Macromolecules 1995. 28, 796. 9. Hillmyer, Μ. Α.; Bates, F. S.; Almdal, K.; Mortensen, Κ Ryan, A. J.;

Fairclough, J. P. Science 1996, 271, 976. 10. Sakurai, S.; Kawada, H.; Hashimoto, T.; Fetters, L. J. Macromolecules

1993, 26, 5796. 11. Sakurai, S.; Hashimoto, T; Fetters, L. J. Macromolecules 1996, 29, 40. 12. Sakurai, S. Trends in Polymer Science 1995, 3, 90.

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13. Kim, J. K.; Lee, H. H.; Ree, M. ; Lee, Κ. Β.; Park, Y. Macromol. Chem. Phys. 1998, 199, 641.

14. Vigild, M . E.; Almdal, K.; Mortensen, K.; Hamley, I . W.; Fairclough, J.P. Α.; Ryan, A. J. Macromolecules, 1998, 31, 5702.

15. Qi, S.; Wang, Z. G. Phys. Rev. Lett. 1996, 76, 1670. 16. Laradji, M. ; Shi, A. C.; Noolandi, J.; Dessai, R. C. Macromolecules

1997, 30, 3242. 17. Kim, J. K.; Lee, H. H.; Gu, Q. J.; Chang, T.; Jeong, Y. H.

Macromolecules, 1998, 31, 4045. 18. Ryu, C. Y.; Vigild, M . E.; Lodge, T. P. Phys. Rev. Lett 1998,

81, 5354. 19. Morrison, F. Α.; Winter, H. H.; Gronski, W.; Barnes, J. D.

Macromolecules 1990, 23, 4200. 20. Scott, D. B.; Waddon, A. J.; Lin, Y. G.; Karasz, F. E.; Winter, H. H.

Macromolecules 1992, 25, 4175. 21. Park, B. J.; Rah, S. Y.; Park, Y. J.; Lee, Κ. B. Rev. Sci. Instrum. 1995,

66, 1722. 22. Matsen, M . W.; Bates, F. S. J. Chem. Phys. 1997, 106, 2436. 23. Ryu, C. Y.; Lee, M. S.; Hajduk, D. Α.; Lodge, T. P. J. Polym. Sci. Phys.

Ed. 1997, 35, 2811. 24. Sakamoto, N. ; Hashimoto, T.; Han, C. D.; Kim, D.; Vaidya, N . Y.

Macromolecules, 1997, 30, 5321. 25. Kim, J. K.; Lee, H. H.; Sakurai, S.; Aida, S.; Masamoto, J.; and

Nomura, S.; Kitagawa,Y.; Suda, Y. submitted to Macromolecules (1999). 26. Chen, Ζ. R.; Kornfield, J. A. Polymer 1998, 39, 4679. 27. Riise, B. L.; Fredrickson, G. H.; Larson, R. G.; Pearson, D. S.

Macromolecules, 1995, 28, 7653. 28. Lee, H. H.; Kim, J. K. Manuscript in preparation. 29. Almdal, K.; Koppi, Κ. Α.; Bates, F. S. Macromolecules, 1993, 26,

4058. 30. Qi, S and Wang, Z.-G., Polymer, 1998, 39, 4639.

Chapter 31

SAXS and Rheological Studies on the Order-Disorder and Order-Order Transitions

in Mixtures of Polystyrene-b-Polyisoprene-b-Polystyrene and Low Molecular Weight PS

Seung-Heon Lee and Kookheon Char

Department of Chemical Engineering, Seoul National University, Seoul 151-742, Korea

The order-disorder and order-order transitions (ODT and OOT) of a series of mixtures of polystyrene-block-polyisoprene-block-poiystyrene copolymers (SIS), V4411D (fPS = 0.377) and V4113 (fPS = 0.131), and low molecular weight PS homopolymers with different molecular weights have been investigated by using both small-angle X-ray scattering (SAXS) and rheological measurements. ODT temperatures determined from the discontinuous change in 1/Imax and σ q

2 vs 1/T plots were found to be in reasonable proximity to the ones determined from the discontinuous change in log G' vs Τ plots. OOT was also identified for pure V4113 and a few of its mixtures with PS homopolymers, as evidenced from SAXS measurements. While the phase behavior including ODT of V4411D/PS mixtures was in good agreement with theoretical predictions, that of V4113/PS mixtures showed unusual behavior. This was attributed to the change in PS homopolymer distribution in the microdomain with the addition of low molecular weight PS homopolymer to the SIS triblock copolymer.

Introduction

Block copolymer/homopolymer mixtures have been of considerable interest because they can be easily modified to yield desired properties in polymeric materials such as pressure sensitive adhesives. Phase behavior of these mixtures is generally quite complicated because two different natures of transition can occur at the same time: macrophase and microphase separation. If the molecular weight of a

496 © 2000 American Chemical Society

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homopolymer is sufficiently low, it however tends to be solubilized into one of the microdomains.1"4 In this case, the macrophase separation is suppressed and the microphase separation (or order-disorder transition; ODT) becomes dominant.

There have been many theoretical and experimental works on the ODT of pure block copolymers.5"11 A few researchers have also investigated the ODT of block copolymer/homopolymer mixtures. Whitmore and Noolandi1 calculated theoretical phase diagrams of AB diblock copolymer/A homopolymer mixtures using functional integral formalism. They assumed that the ordered phase has a lamellar structure, so that the third-order term in the Landau-Ginzburg expansion of the free energy vanishes. They found that the spinodal ODT temperature decreases with the addition of the homopolymer if the degree of polymerization of A homopolymer is smaller than the half of that of A block of symmetric AB diblock copolymer, de la Cruz and Sanchez12 also investigated theoretically the effect of the relative molecular weight and composition of the homopolymer on the spinodal ODT by modifying the scattering theory originally proposed by Leibler5 to AB diblock copolymer/A homopolymer mixtures, and obtained similar results with those of Whitmore and Noolandi. Recently, some researchers extended the self-consistent mean-field calculations to block copolymer/homopolymer mixtures having other ordered phases rather than lamellae. Matsen13 found that the addition of homopolymer stabilizes complex ordered phases in the weak segregation regime. Janert and Schick14

investigated the effect of relative homopolymer length on the phase behavior in the weak to intermediate segregation regime considering lamellae, hexagonal, and body centered cubic phases. ODT behavior in previous authors' works, however, showed similar trend to that obtained from the calculations just assuming lamellar phase.

Experimentally, Roe and Zin 1 5 investigated the phase behavior of mixtures of styrene-butadiene diblock copolymer and polystyrene or polybutadiene homopolymers with different molecular weights using small-angle X-ray scattering (SAXS) and turbidity measurements. Baek et al.16 investigated the ODT and phase behavior of mixtures of SIS triblock copolymer/PS homopolymer mixtures by using turbidity and rheological measurements. To the best of our knowledge, however, systematic studies on the order-disorder and order-order transitions in mixtures of block copolymer and low molecular weight homopolymers have yet to been done. Since relatively high molecular weight homopolymers were used in the previous works, the ODT temperature just increased as the homopolymer weight fraction was increased and eventually macrophase separation occurred above a certain value of composition that was quite small.

In present study, we employed both SAXS and rheological measurements to investigate the order-disorder and order-order transitions in a series of SIS triblock copolymer/low molecular weight PS homopolymer mixtures, which did not show macrophase separation in the whole temperature and composition range covered in this experiment. Phase diagrams obtained from both measurements were compared with the predictions based on the Whitmore-Noolandi theory1. The difference between the theory and the experiment was discussed in terms of the change in homopolymer distribution and microdomain morphology by the addition of homopolymer to block copolymer.

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Experimental

Two commercial grades of polystyrene-6/oc£-polyisoprene-6/0eA-polystyrene (PS-PI-PS) copolymers, Vector 4113 and Vector 441 ID (DEXCO Polymers) were used as received. They will be denoted, hereafter, as V4113 and V441 ID, respectively. Four polystyrenes (PS) with different molecular weights were synthesized by anionic polymerization using η-butyl lithium as an initiator and benzene containing a trace amount of tetrahydrofuran as a solvent. The characteristics of these polymers are summarized in Table I. The weight-average molecular weight, M w , and the polydispersity index, M w / M n , were determined by gel permeation chromatography (GPC) using PS standards, and the PS weight percent of the block copolymers was determined by proton nuclear magnetic resonance (H-NMR). The microstructure of PI blocks of the block copolymers was determined to be 93% 1,4 and 7% 3,4 addition by H-NMR.

Table I. Molecular Characteristics of the Polymers Used in This Study

Sample MwxW'3 (g/mol) MJMn PSwt% V4113 13.2-6-117.4-6-13.2 1.19" 15.1

V4411D 17.1-6-80.0-6-17.1 1.08 41.7 PS2 1.9 1.09 100 PS3 2.9 1.06 100 PS6 5.9 1.04 100 PS11 11.4 1.03 100

a 18 wt% of uncoupled diblocks is present in V4113.

Mixture samples were prepared by solution casting method with toluene as a solvent. First, predetermined amounts of block copolymers and PS homopolymers were dissolved in toluene with the addition of 0.3 wt% antioxidant (Irganox 1010, Ciba-Geigy Group). The solvent was slowly evaporated at ambient condition for 3 weeks, and then in a vacuum oven at 56 °C for 3 days. Finally, all the samples were annealed in a vacuum oven at 130°C for 3 days.

Two different methods were used to determine the order-disorder transition temperature (T 0DT) of each mixture. RMS 800 (Rheometrics Inc.) in parallel plate geometry (25 mm diameter and 1.5 mm gap) was used to measure the dynamic viscoelastic storage and loss moduli, G' and G", of the mixtures as a function of temperature. A heating rate of 1 °C/min was used and the strain amplitude was small enough to ensure linear viscoelasticity (typically smaller than 5%) for all the measurements.

Synchrotron small-angle X-ray scattering (SAXS) measurements were carried out at 4C2 SAXS beamline in Pohang Light Source (PLS), which consisted of 2 GeV LINAC accelerator, storage ring, S i ( l l l ) double crystal monochromator, ion chambers, and one-dimensional position sensitive detector with 2048 pixels. The

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wavelength (λ) of the synchrotron beam was 1.59 À and the energy resolution (Δλ/λ) was 5xl0"4, Typical beam size was smaller than l x l mm2. Sample-to-detector distance was 103.7 cm. Scattering profiles were obtained as a function of temperature and then corrected for absorption, air and imide film scattering.

In order to prevent oxidative degradation of the samples, all the experiments were carried out under the nitrogen blanket.

Results and Discussion

Solubility of PS in the Microdomains of SIS Triblock Copolymer

It was found from the hot stage microscopy that V4411D/PS mixtures did not show any macrophase separation up to 40 wt% of PS regardless of PS molecular weight. V4113/PS mixtures with high molecular weight PS, however, showed macrophase separation when PS weight fraction exceeded a certain value. This is due to the fact that V4113 is more asymmetric and hence has more space-restricted microdomain morphology, e.g., cylindrical morphology, than V441 ID, thus being less capable of solubilizing PS homopolymer into its microdomain. It should be noted here that in present work we focussed only on the order-disorder and order-order transition behavior of the mixtures in a composition range showing no macrophase separation.

Changes in Equilibrium Microdomain Morphology of Block Copolymers with the Addition of PS

It is well known that a block copolymer shows multiple X-ray scattering peaks due to its periodic microdomain structure having a long-range order.2 Information on the equilibrium microdomain morphology of block copolymers can be obtained from the relative positions of these multiple peaks, since they exhibit different arrays depending on the shape of the microdomain structure, e.g., 1, 2, 3, 4, ... for lamellae, 1, S , Λ/4 , Λ/7 , Λ /9 , ... for cylinders in hexagonal array, 1, yfl , Λ / 3 , Λ/4 , Λ / 5 , ... for spheres in body centered cubic array, and so on.

Synchrotron SAXS measurements were carried out at room temperature in order to obtain information on the change in the equilibrium microdomain morphology of SIS/PS mixtures with the addition of PS. In Figures 1 and 2, the results for V4411D/PS2 and V4113/PS2 mixtures are presented, respectively, in a semi logarithmic plot of scattered intensity as a function of scattering wave number q. q is given by

(7 = (4;r//l)sin(0/2) (1) where θ is the scattering angle. Al l the measurements were taken after annealing the samples at 130°C for 3 days. Each data set was vertically shifted for clarification. As shown in Figure 1, the relative positions of scattering peaks of V4411D/PS2 mixtures are 1, 2, and 3 up to 30 wt% of PS2, while 1, (4/3)î/2, and 2 at 40 wt% of PS2. It is

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Figure 1. Synchrotron SAXS profiles ofV44JW/PS2 mixtures with different composition at room temperature after annealing the samples at 130 Vfor 3 days. Each data set was vertically shifted for clarification.

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Figure 2, Synchrotron SAXS profiles ofV4113/PS2 mixtures with different composition at room temperature after annealing the samples at 130 V for 3 days. Each data set was vertically shifted for clarification.

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interesting to note that higher order peaks of 60/40 (w/w) V4411D/PS2 mixture appear at the relative positions of (4/3)1/2 and 2. Since the former is the unique characteristic of the gyroid phase and the latter probably results from the lamellar phase, it can be inferred that lamellae and gyroid phases coexist in the 60/40 (w/w) V4411D/PS2 mixture. This further implies that the transition from lamellae to gyroid phase occurs around w P S 2 ~ 0.40, where wP S2 is the weight fraction of PS2 in the mixture. Transition from gyroid to polyisoprene cylinders in hexagonal packing occurs at 0.40 < w P S 2 < 0.50, since the relative positions of the peaks change to 1, V J ,

VÏ , and 4Ï , which is the characteristic of cylinders in hexagonal array, at 50 wt% of PS2. If we assume complete solubilization of PS2 in the polystyrene microdomains of SIS triblock copolymers, overall styrene volume fraction Φ Ρ 5 can be calculated. The values of Φ Ρ 5 at the transition points, Φ Ρ 5 ~ 0.614 and 0.614 < Φ Ρ δ < 0.676, are in reasonable agreement with those of PS volume fraction in SI diblock copolymers reported in the literature9. Therefore, it can be concluded that Φ Ρ 5 can be a good measure of the equilibrium microdomain morphology in block copolymer/ homopolymer mixtures. For V4113/PS2 mixtures, the transition from polystyrene cylinders to lamellae was found to occur at 0.20 < w P S 2 < 0.25 (0.288 < Φ Ρ 5 < 0.33). This kind of transition from one microstructure to another by adding homopolymer to the block copolymer has also been reported by many researchers, but will not be discussed here in detail.

Order-Disorder and Order-Order Transitions Determined from Rheology

In order to determine the order-disorder transition temperature (T0DT)> two different measurements were employed in present study. Rheological measurements were first carried out. Figures 3 and 4 show log G' vs Τ plots of V4411D/PS2 and V4113/PS2 mixtures, respectively, with different composition. Angular frequency ω was fixed to 0.5 rad/sec for all the samples except for V4113 (0.1 rad/sec) and the heating rate was 1 °C/min. As is evident in the figures, G' shows a precipitous drop at a certain temperature for each mixture. Rosdale and Bates7 investigated the order-disorder transition behavior of a series of PEP-PEE block copolymers having lamellar microdomains, and found that both G' and G" drop precipitously at a certain temperature with angular frequencies below critical angular frequencies (coc' and coc" respectively). They also reported that the drop in G* is more pronounced since the critical angular frequency for G' is larger than that for G" (i.e., coc' > coc

M). Following their criterion, the temperature at which there is an abrupt change in G' was chosen as TODT of each mixture. While T 0 D T of V4411D/PS2 mixtures decreases as the PS2 weight fraction was increased in the mixture (see Figure 3), T 0 D T of V4113/PS2 mixtures first decreases up to 10 wt% and increases again (see Figure 4). This unusual behavior will be discussed later.

It should be mentioned here that some researchers argued Bates' criterion for the ODT of highly asymmetric block copolymers and proposed another method. Han and coworkers11,20 used log G' vs log G" plots to determine the ODT temperatures of

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Figure 3. Dynamic storage modulus (G') as a function of temperature for V4411D/PS2 mixtures with different composition.

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Τ 1 I 1 I · ι 1 Γ

Ε 1 1 1 1 . ι , ι , ι , 3

140 160 180 200 220 240 260

Temperature (°C)

Figure 4. Dynamic storage modulus (G') as a function of temperature for V4113/PS2 mixtures with different composition. Each data set was vertically shifted by multiplying JO4, 10s, JO2, JO1, and J0°, respectively, for clarification.

505

asymmetric SI diblock and SIS triblock copolymers, and proposed that the temperature at and above which log G' vs log G" plot does not change any more should be regarded as T 0 D T - Their criterion was not checked for our V4113/PS mixture system, however, since the values of T 0 D T determined from log G' vs log G" plots are usually much larger than those determined from log G' vs Τ plot and exceed the degradation limit for our system (-250 °C).

Order-Disorder and Order-Order Transitions Determined from SAXS

Figure 5 presents typical change in the synchrotron SAXS profiles for 70/30 (w/w) V4113/PS2 mixture with temperature. Scattering intensity is plotted in logarithmic scale in order to see more clearly the change in multiple scattering peaks. Maximum scattered intensity I m a x shows a sharp decrease between 230°C and 235 °C. This discontinuity is more pronounced when the inverse of scattering maximum 1/Imax

and the square of the half-width at half-maximum a q

2 were plotted against inverse temperature 1/T as shown in Figure 6a. Sakamoto and Hashimoto10 investigated the ODT of near symmetric SI diblock copolymers (fPS=0.45 and 0.54) using SAXS, and reported that there exists a clear discontinuity at ODT in 1/Imax vs 1/T plot. They also reported that σ ς

2 shows the same trend as 1/Imax. As can be seen in Figure 6a, σ ς

2

follows exactly the same trend as 1/Imax. From Figure 6a, the temperature at which the discontinuity occurs in 1/Imax and σ ς

2 vs 1/T was chosen as the T 0 D T of the mixture. The value of the T 0 D T was in good agreement with the one determined from the precipitous decrease in log G' vs Τ plot shown in Figure 4.

In Figure 6b, domain periodicity D (= 2n/qmax) was plotted against 1/T. It is interesting to note that there exists a discontinuity in D at ODT. This is in contrary to the results of Sakomoto and Hashimoto10 that D is continuous at ODT. Recently, however, Floudas and coworkers17, 1 8 examined the ODT of SI2 simple graft copolymer, SIB terpolymer, (SI)4 star-block copolymers and asymmetric SI diblock copolymer using SAXS, and reported the discontinuity in D. Ogawa et αι.19 also reported similar results to our study for an asymmetric SI diblock copolymer (fPS=0.303).

It can be clearly seen in Figure 5 that the position of the second order scattering peak relative to the first order scattering peak changes from 2 to 3 1 / 2 between 190°C and 195°C. Local maxima were also found between these temperatures in both 1/Imax

vs 1/T and σ ς

2 vs 1/T plots shown in Figure 6a. In addition to this, a discontinuous change in D occurs in the same temperature range as can be seen in Figure 6b. Above results indicate that order-order transition from lamellae to cylinders in hexagonal array occurs between these temperatures. Similar results on OOT were also observed for asymmetric SIS triblock copolymer (wPS=0.183) by Sakamoto et ai.20 It can be concluded from our results that local maxima in both 1/Imax and σ ς

2 against 1/T and the discontinuous change in D can be attributed to OOT.

I/Imax, s q

2 and D vs 1/T plots for 70/30 (w/w) V4411D/PS2 mixture are presented in Figure 7. Similarly, the discontinuity in D vs 1/T plot as well as 1/Imax and σ ς

2 vs 1/T plots was found at ODT. The T 0 D T value was again in good agreement with the

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507

Figure 6. Scattering parameters as a function of inverse temperature for 70/30 (w/w) V4113/PS2 mixture: (a) inverse of the scattering maximum (1/Ima.x) and square of half-width at half maximum (σ2) plotted against 1/T, and (b) domain spacing (D = 2ïï/qmax) against 1/T. A heating rate of 1 V/min was used.

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T(°C) 300 280 260 240 220 200 180

1.9 2.0 2.1 1000/T(K"1)

Figure 7. Scattering parameters as a function of inverse temperature for 70/30 (w/w) V44Î1D/PS2 mixture: (a) inverse of the scattering maximum (1/Imax) and square of half-width at half maximum (aq

2) plotted against 1/T, and (h) domain spacing (D = 27t/qmay) against 1/T. A heating rate of I V/min was used.

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one determined from the precipitous decrease in log G' vs Τ plot shown in Figure 3. OOT was not observed in this mixture. Pure V4113, 75/25 (w/w) V4113/PS2 mixture and 60/40 (w/w) V4411D/PS2 mixture were also found to show OOT (not shown here). The details of ODT and OOT of these mixtures will be discussed in future publication.

Phase Diagrams of Mixtures of SIS Triblock Copolymers and PS Homopolymers

Phase diagrams of V4411D/PS and V4113/PS mixtures with different PS molecular weight obtained from both measurements are constructed in Figures 8 and 9, respectively. PS homopolymer weight fraction of the mixture was converted to the overall styrene volume fraction by assuming complete solubilization of PS in the polystyrene microdomains of SIS triblock copolymers and shown in the upper x-axis. Theoretical phase diagrams were also constructed in the insets of Figures 8 and 9 based on Whitmore-Noolandi theory1 for comparison with the experimental results. The following expression was used in the calculations for the temperature-dependent Flory-Huggins interaction parameter between styrene and isoprene.21

j = -0.0149 + 38.54/Γ (2) It should be noted here that no adjustable parameter was used in the above

calculations. As shown in Figure 8, TQDT'S of V4411D/PS mixtures were found to either

decrease or increase depending on PS homopolymer molecular weight as PS homopolymer weight fraction in the mixture was increased; T 0 D T decreases with the addition of PS2 and PS3, while it increases with the addition of PSI 1. This behavior is in good agreement with the calculated phase diagram based on the Whitmore-Noolandi theory (see inset of Figure 8). Phase diagram of V4113/PS2 mixtures plotted in Figure 9, however, shows different behavior; T 0 D T first decreases as PS2 weight fraction is increased up to 10 wt% and then increases again. This unusual trend was not predicted from the Whitmore-Noolandi theory (see inset of Figure 9). This can be attributed to the change in homopolymer distribution in the microdomain as PS2 homopolymer is added. If a small amount of low molecular weight homopolymer is added to the block copolymer, it tends to be solubilized uniformly into the microdomains.2 As a large amount of homopolymer is added, however, it is localized in the center of the microdomain. This finally results in the change in microdomain morphology, e.g., from cylinders to lamellae, if homopolymer is further added, as evidenced from the change in relative position of multiple SAXS peaks with increasing PS2 weight fraction in the mixture (see Figure 2). This effect of homopolymer localization on ODT would be more pronounced in mixtures with asymmetric copolymer (V4113) than those with symmetric copolymer (V441 ID).

Conclusion

Triblock copolymer/homopolymer mixtures showing no macrophase separation have been prepared by adding a small amount of a series of low molecular weight

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Figure 8. Phase diagram ofV44UD/PS mixtures. Filled and open symbols are TQDT'S

obtained from rheological and SAXS measurements, respectively. Calculations based on the Whitmore-Noolandi theory are also shown in the inset.

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Figure 9. Phase diagram ofV4113/PS mixtures. Filled and open symbols are T0DT'S

obtained from rheological and SAXS measurements, respectively. Calculations based on the Whitmore-Noolandi theory are also shown in the inset.

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polystyrenes (PS) to near symmetric (V4411D, fPS = 0.377) and highly asymmetric (V4113, fpS = 0.131) polystyrene-6/oc£-polyisoprene-6/oc£-polystyrene copolymers. The order-disorder and order-order transitions (ODT and OOT) of these mixtures were investigated by using both small-angle X-ray scattering (SAXS) and rheological measurements.

ODT temperatures of the mixtures were determined from the discontinuous change in 1/Imax and σ ς

2 vs 1/T plots using SAXS. They were found to be in good agreement with the ones determined from the discontinuous change in log G' vs Τ plots using rheology. OOT was also identified for pure V4113 and a few of its mixtures with PS homopolymers, as evidenced from the change in relative position of higher order scattering peaks as well as the discontinuity in D and local maxima in 1/Imax and o q

2atOOT. Phase diagrams of V4411D/PS and V4113/PS mixtures were constructed from

the ODT measurements using both SAXS and rheology, and compared with theoretical predictions based on the Whitmore-Noolandi theory. ODT of V4411D/PS mixtures were found to either decrease or increase depending on the molecular weight of PS homopolymer added, which is in good agreement with the Whitmore-Noolandi theory. ODT of V4113/PS2 mixtures, however, first decreased and increased again with the addition of PS2, which was not predicted by the theory. We attributed this abnormal phase behavior to the change in the homopolymer distribution in the microdomains with the addition of PS homopolymer.

Acknowledgment

Experiments performed at Pohang Light Source (PLS) were supported in part by MOST and POSCO. We are very grateful to Y. J. Park for his assistance during SAXS experiments at PLS. K. Char is also very grateful to the LG-Yonam Foundation for its financial support during a sabbatical visit (1997 ~ 1998) to Cornell University.

References

1. Whitmore, M. D.; Noolandi, J. Macromolecules 1985, 18, 2486. 2. Tanaka, H.; Hasegawa, H.; Hashimoto, T. Macromolecules 1991, 24, 240. 3. Mayes, A. M.; Russell, T. P.; Satija, S. K.; Majkrzak, C. F. Macromolecules 1992,

25, 6523. 4. Jeon, K.-J.; Roe, R.-J. Macromolecules 1994, 27, 2439. 5. Leibler, L. Macromolecules 1980, 13, 1602. 6. Fredrickson, G. H.; Helfand, E. J. Chem. Phys. 1987, 87, 697. 7. Rosedale, J. H.; Bates, F. S. Macromolecules 1990, 23, 2329. 8. Bates, F. S.; Rosedale, J. H.; Fredrickson, G. H. J. Chem. Phys. 1990, 92, 6255. 9. Khandpur, A. K.; Förster, S.; Bates, F. S.; Hamley, I. W.; Ryan, A. J.; Bras, W.;

Almdal, K.; Mortensen, K. Macromolecules 1995, 28, 8796. 10. Sakamoto, N. ; Hashimoto, T. Macromolecules 1995, 28, 6825.

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11. Han, C. D.; Baek, D. M.; Kim, J. K.; Ogawa, T.; Sakamoto, N. ; Hashimoto, T. Macromolecules 1995, 28, 5043.

12. de la Cruz, M. O.; Sanchez, I. C. Macromolecules 1987, 20, 440. 13. Matsen, M. W. Macromolecules 1995, 28, 5765. 14. Janert, P. K.; Schick, M. Macromolecules 1998, 31, 1109. 15. Roe, R.-J.; Zin, W.-C. Macromolecules 1984, 17, 189. 16. Baek, D. M. ; Han, C. D.; Kim, J. K. Polymer 1992, 33, 4821. 17. Floudas, G.; Hadjichristidis, N. ; Iatrou, H.; Pakula, T.; Fischer, E. W.

Macromolecules 1994, 27, 7735. 18. Floudas, G.; Pispas, S.; Hadjichristidis, N.; Pakula, T.; Erukhimovich, I.

Macromolecules 1996, 29, 4142. 19. Ogawa, T.; Sakamoto, N.; Hashimoto, T.; Han, C. D.; Baek, D. Macromolecules

1996, 29, 2113. 20. Sakamoto, N.; Hashimoto, T.; Han, C. D.; Kim, D.; Vaidya, N. Y.

Macromolecules 1997, 30, 1621. 21. Hashimoto, T.; Ijichi, Y.; Fetters, L. J. J. Chem. Phys. 1989, 89, 2463.

Chapter 32

Thermoreversible Order-Order Transition between Spherical and Cylindrical Microdomain

Structures of Block Copolymer Kohtaro Kimishima1, Tadanori Koga1, Yuko Kanazawa1,

and Takeji Hashimoto1-3

1Hashimoto Polymer Phasing Project, ERATO, Japan Science and Technology Corporation, 15 Shimogamo-Morimoto, Kyoto 606-0805, Japan

2Department of Polymer Chemistry, Graduate School of Engineering, Kyoto University, Kyoto 606-8501, Japan

The mechanism and dynamics of the order-order transition (OOT) in a block copolymer were investigated by means of small-angle X-ray scattering (SAXS), ultra-small-angle X-ray scattering (USAXS) and polarized-light optical microscopy. At high temperatures the spherical microdomains in bcc lattice (bcc-sphere) exist, while at low temperatures the cylindrical microdomains in hexagonal lattice (hex-cylinder) exist. The transition between these ordered phases thermoreversibly occurred with conservation of the grain structures and with conservation of a memory of the lattice orientation. By examining the SAXS pattern from a single grain, it was found that the cylinders are transformed into a series of spheres with their cylindrical axes corresponding to the [111] direction of the bcc-spheres: orientation of cylinders and orientation of the [111] direction of the bcc lattice are conserved in the OOT process from the hex-cylinder phase to the bcc-sphere phase and its reverse transition process, or in a repeated OOT process between these two phases (the conservation of a memory of the lattice orientation). The process of the OOT studied here was found to be spinodal-like, which was contrary to the nucleation and growth process of the order-disorder transition in asymmetric block copolymers.

Introduction

Diblock copolymers in ordered phase form various unique nano-patterns such as alternating lamellar, cylindrical and spherical microdomains, depending on the volume fraction of one of the constituents [1-3]. Recently fascinating patterns such as lamellar

Corresponding author.

514 © 2000 American Chemical Society

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catenoid and gyroid were found as new nano-patterns for diblock copolymer [4-7]. Besides the investigation on the patters in the ordered phase, the phase transition of the block copolymer has been also explored. One typical transition is the order-disorder transition (ODT). The transition between an ordered phase with some nano-patterns and a disordered phase where two blocks are mixed in the molecular levels is called as ODT and the nature of ODT is quite extensively explored as a fundamental problem in physical science from both theoretical [8-11] and experimental viewpoints [12-15].

Another transition for the block copolymer is the order-order transition (OOT) between different kinds of ordered phases. Although the existence of OOT has been theoretically predicted [8,9], experimental reports on thermoreversible OOT have appeared in the literature only recently. Sakurai et al. [16,17] reported the thermoreversible OOT between spherical microdomains packed on a body-centered cubic lattice (hereafter denoted as bcc-sphere) and cylindrical microdomain packed on a hexagonal lattice (hereafter denoted as hex-cylinder) for a polystyrene-6/ocfc-polyisoprene (PS-PI) diblock copolymer [16] and its diocthylphtalate (DOP) solutions [17] with varied polymer concentrations based on small-angle X-ray scattering (SAXS) and/or transmission electron microscopy (TEM) studies.

Bates and co-workers first presented the complex phase behavior in the ordered phase near ODT using a series of poly(ethylene-a//-propylene)-6/oc£-poly(ethylethylene) (PEP-PEE) [4] and PS-PI diblock copolymers [6,7]. They identified new morphology such as a hexagonally perforated layers and a bicontinuous cubic phase having an la 3d space group symmetry (gyroid) by using various technical methods such as dynamic mechanical measurements, TEM, SAXS, and small-angle neutron scattering (SANS). The gyroid structure was also identified by Hajduk et al. [5], using SAXS, TEM, and computer simulations. They observed lamellar microdomains in a PS-PI diblock copolymer consisting of 37wt% of PS at 115°C. After annealing the sample at 150°C, which was approximately 50°C below the ODT temperature, they observed the microdomains were transformed into gyroid. The thermoreversibility between lamella and gyroid was also examined. The complete transition from gyroid into lamella did not occur even after long-time annealing, over 20 hours at 100°C. However, a partial transformation was observed by both TEM and SAXS, showing the nature of the thermoreversibility between the gyroid and lamellar morphologies. The authors also reported the thermoreversible OOT between lamellar and cylindrical microdomains in a polystyrene-Z>/ocA;-poly(ethene-co-butene) diblock copolymer by SAXS and TEM [18].

Now there is no doubt that OOT occurs in the block copolymer. However, the detailed mechanism, process, and dynamics of OOT have not been thoroughly understood compared with those of the ODT. Here we are concerned with the OOT between bcc-sphere and hex-cylinder. Though this type of OOT itself has already been reported, the nature of the transition has not yet been fully understood. Therefore we investigated the transition mechanism and dynamics of this OOT. In this paper we report highlights of our experimental results, full details of which will be described elsewhere [19,20].

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Experimental Section

The sample used in this study was a PS-PI prepared by living anionic polymerization with sec-butyllithium as an initiator and cyclohexane as a solvent. The number-averaged molecular weight, M„, determined by GPC (HLC-8020, THOSO, Co. Ltd.) is 4.4x104 and the polydispersity index, MJM,,, is 1.02 with Mw being the weight-averaged molecular weight. The weight fraction of polystyrene block (PS) determined by 1 H-NMR (JMR-400, JEOL, Co. Ltd.) is 0.20410.005. The diblock copolymer was dissolved into toluene with a small amount of anti-oxidant (BHT) and cast into film specimens from a 5% polymer solution. The film specimens thus obtained were then dried under vacuum until no further weight loss was observed.

SAXS measurements were conducted with an apparatus consisting of a !8kW rotating-anode X-ray generator (M18XHF-SRA, MAC Science Co. Ltd., Yokohama, Japan) with a graphite crystal monochromator, a collimator, and a vacuum chamber for the incident-beam path and scattered-beam path, and a detector. The sample was put in an evacuated chamber to reduce possible thermal degradation. The temperature stability for a static measurement was controlled within ±0.1 Κ fluctuation. The SAXS profiles were measured using a one-dimensional position sensitive proportional counter (PSPC) with line focus optics and were corrected for the absorption of the sample, background scattering, and thermal diffuse scattering arising from the acoustic phonons. The SAXS patterns were also measured with a two-dimensional (2D) imaging plate (IP) [21] (DIP-220, MAC Science, Co. Ltd.) with a point focus optics.

USAXS measurement was conducted using Bonse-Hart type USAXS apparatus consisting of a 18kW rotating-anode X-ray generator (M18XHF-SRA, MAC Science Co., Ltd.) and two germanium channel-cut crystals (one is to monochromatize and collimate the incident beam and the other is to analyze the scattered beam), and a detector. The details of the USAXS camera have been described elsewhere [22]. The measurements were also carried out with specimens in a vacuum chamber to avoid the oxidative degradation of the specimens. In a whole temperature range covered in this USAXS, the sample temperature was calibrated using the same standard platinum resistance (PtlOO) with an ohmmeter as that used for the temperature calibration for the SAXS apparatus and was controlled within ±0.1 Κ fluctuation.

The grain structure of the sample was observed by polarized-light optical microscopy, using a cooled CCD (C5985, Hamamatsu Photonics, Co., Ltd.) for a detector. To minimize an image overlapping in the thickness direction, thin specimens having a thickness of 0.1 mm were used. The specimen was placed on a heating stage (THMS-600, LINKAM Scientific, Co., Ltd.) controlled at various temperatures. Observation was carried out with the specimen enclosed under nitrogen atmosphere to avoid thermal degradation of the sample.

The specific volume of the sample, vsp, was measured using a conventional pycnometer. A small amount of sample (about 0.3g) was placed into a vessel (about 0.5cc in volume), and the bubbles inside the sample were removed in the vacuum oven at 130°C. The rest of the space in the vessel was filled with ethylene glycol (EG) which is a non-solvent for both PS and polyisoprene (PI). The vessel was sealed with a cap equipped with a capillary having the diameter of 0.2mm. The assembly was

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placed into oil bath and the entire system was controlled at various temperatures within O.IK accuracy. The meniscus of EG in the capillary was measured as a function of temperature. The volume of the vessel and capillary was calibrated using EG. The density of EG at 20 °C is 1.109 g/cm3 and the thermal expansion of EG was corrected using following equation [23],

Thermoreversible OOT and OOT Temperature

Figure 1 shows typical SAXS profiles of the block copolymer measured at various temperatures. The intensity, Imeas(q), are plotted in a semi-logarithmic scale against the magnitude of the scattering vector, q, defined by q Ξ (4;r/A)sin(0/2) with λ and θ being the wavelength of the incident X-ray and the scattering angle, respectively. Measurements were carried out in a cooling process from 177.4°C. Before we measured the SAXS profile at a given temperature, we waited for 40 min after attaining that temperature. Then the intensity distribution was accumulated for 30 min and the sample temperature was lowered stepwisely to the next one and kept at that temperature for 40 min before the next SAXS measurement. An average cooling rate corresponding to this cooling scheme was about 15 K/hr. The profiles a and b show a broad first order scattering maximum (marked by a black arrow) and a broad shoulder at around #=0.45 nm"1 (marked by a white thick arrow). The broad shoulder indicates that the block copolymer at these temperatures forms microdomains (may be spherical microdomains) with only short-range liquid-like order [24-26]. This state is different from the disordered state where the PS and PI block chains are mixed in a molecular level and form the dynamical thermal concentration fluctuations [14,15]. The domain­like structures, which may be generated by thermal fluctuations in the disordered state, are dynamical objects and are expected to have no clear interface.

At temperatures at 138.2 and 118.7°C (profiles c and d) the higher-order scattering maxima were observed at the positions of V2 and V3 relative to that of the first-order scattering maximum, qm. Detailed analyses of them indicates the formation of spherical microdomains packed in a body-centered cubic lattice (denoted as bcc-sphere) [19]. Leibler's theory [8] predicts that the disordered state is transformed into the bcc-sphere when the interaction parameter between the constituents, χ, is increased. In this sense the structure at 157.7 and 177.4°C (profiles a and b) is expected to exhibit the disordered state since χ between PS and PI, %Sb

decreases with increasing temperature. However the profiles a and b show evidence of the microdomains (possibly spherical microdomains) as mentioned before. Thus we

= 0.565 χ 10"3 +1.7074 χ 10" 6Γ + 0.293 χ 10"8 Γ 2 , (1)

with Γ and vsp0 being absolute temperature and vsp at 0 °C, respectively.

Result and Discussion

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considered the structure is a "lattice-disordered sphere" (LDS) [25,26], where the bcc lattice was distorted by thermal fluctuations whose scattering profiles may be best interpreted in terms of a bcc-sphere with a large paracrystalline distortion [27-29] or spheres with a spatial distribution as predicted by the Perçus-Yevick type theory [30].

The positions of the higher-order scattering maxima with respect to qm appeared at -J3qm and 4^qm below 99.0°C (profiles e and f)- With further decreasing temperature the higher-order scattering maxima became more pronounced, which is due to the fact that χ8Ι gets larger. At 39.3°C (profile f) the higher-order peaks at least up to *JÏ3qm appeared. This observation indicates the microdomain structure developed is hex-cylinder. Since the measurement was carried out in a cooling process from a high temperature, this morphological transition was not considered to be a non-equilibrium effect encountered by the solution casting process to prepare the film specimens. Any non-equilibrium effects on the morphology in the as-cast film should be relaxed by annealing the film at the high temperatures.

The transition between bcc-sphere and hex-cylinder was observed between 118.7 and 99.0 °C (profiles d and e). To determine the OOT temperature and confirm the thermoreversibility of the transition, the SAXS measurement with a smaller temperature increment and decrement was carried out in heating and cooling processes, respectively. The profiles at each temperature were measured for 30 min after waiting for 40 min since the present temperature had been attained. Then the temperature was increased or decreased to the next one and the measurement was started according to the same scheme as described above. An average cooling rate corresponding to this process was 0.8 K/hr.

Figure 2 shows the temperature dependence of the area under the first-order scattering maximum, Alm calculated by Alm = \q"lmeas(q)dq. Here qu=Q3nm~l

and qt = O.lnm are the upper and lower bound of q, respectively; outside this q domain I,neas{q) becomes significantly smaller than the first-order scattering maximum. Circle and square markers were for AIm values measured in heating and cooling processes, respectively. In this time-scale of the observation, the curves for heating and cooling cycles have different temperature dependence; i.e., hysterisis exists. The OOT temperature, T00T, is defined in this work to be the center of the hyperbolic-tangent-like curve in heating process (116.7°C). The positions of higher order scattering maxima in the SAXS profile (V2gmand 4ïqm for bcc-sphere and 4^qm and 4^qm for hex-cylinder) and the intensity levels ofAIm changed across T00T, which confirms the thermoreversibility of the morphological transition.

Conservation of the Grain Structure

The thermoreversibility was also confirmed by POM observation. Figure 3 shows POM micrographs for the block copolymer taken under crossed-polarizers. The block copolymer specimen sandwiched between two glass slides and having a thickness of 0.1mm was first annealed at 150°C where the LDS structure exists. With lowering the

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χ ι ν"

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ο Heating • Cooling .5 S g

J I I I I ι I I 108 110 112 114 116 118 120 122 124 126

Temperature ( °C )

Figure 2. Temperature dependence ofAIm during the heating (circles) and cooling (squares) processes across T00T.

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Figure 3. POM images under crossed polarizers taken at 112.7°C (parts a and c)for hex-cylinder phase and at 118.6°C (part b) for bcc-sphere phase.

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temperature at a rate of O.lK/min, the sample was cooled down below T00T where the hex-cylinder exists in the sample (Figure 3a at 112.7°C). Since the size of the cylinders is on the order of nm, one cannot observe them by light microscopy. Figure 3a shows an image comprised of many grains of cylinders whose transmitted intensity depends on an orientation of hex-cylinders in the grains and their intrinsic birefringence, primarily due to the form birefringence of cylinders [31]. In each grain observed in Figure 3a the cylindrical microdomains have a coherent orientation. When the axes of cylinders in the grains (corresponding to the principal optical axis of the grain) are inclined with respect to the electric vector axis of the polarizer (P) or analyzer (A), the grains appear bright, while when they are parallel to Ρ or A, the grains appear dark. The brightness of each grain is different due to the difference of the orientation of its optical axis with respect to Ρ or A. Above T00T no clear image was observed as shown in Figure 3b which was taken at 118.6°C above T00T. This is because there is no form birefringence in the spherical microdomain structure. Upon lowering the temperature below TœT, the essentially identical image to Figure 3a, in terms of grain size and brightness of each grains, was recovered (Figure 3c). This result reveals that (i) the grain structure is conserved and (ii) the orientation of the cylinders in each grain is also conserved before/after the two consecutive OOT processes, i.e., the first OOT from hex-cylinder to bcc-sphere and the second OOT from bcc-sphere to hex-cylinder. The result obtained by POM was also confirmed by SAXS measurement with a 2D detector (IP) [19].

When the sample was slowly cooled at a rate of 2.4K/hr from a temperature, where the LDS structure exists, to a temperature where bcc-sphere exists, the two-dimensional (2D) SAXS pattern taken in-situ at a low temperature (Tsphere) shows spot­like diffraction peaks at qm, -J2qm9 etc. This implies that the grains composed of bcc-spheres become large enough so that their size becomes comparable to the size of the incident beam on the sample (1mm in diameter). If the grains are small, the scattering pattern is observed as rings at qm, -Jïqm and etc. From the size of the diffraction spots, the grains were estimated to have a size on the order of mm [19], which is consistent with the POM observation (see Figure 3). Without changing the sample position with respect to the incident beam, the temperature was quickly changed from ^sphereto a temperature, Tcyh where the hex-cylinder phase exists. Then the positions of the second-order diffraction spots were changed from Λ/2 qm to V3#OT, confirming the OOT from bcc-sphere to hex-cylinder. However, the first-order and second-order scattering peaks were also observed as many spots at qm «j3qm, etc, though the diffraction spots on the 2D detector were located in different positions, elucidating the large grain structure which existed at Tcy, even after the OOT into the hex-cylinder phase. When the temperature is raised back up to Tsphere, the original diffraction spots at Tsphere were recovered even after passing through the OOT processes. Similarly when the temperature is lowered down to Tcyh the original spots at Tcyl were again recovered. Even after repeated OOT processes between Tsphere and Tcyh the diffraction spots in the scattering patterns observed at the given temperature (Tsphere or Tcyj) were essentially identical in terms of their spot size and their position on the 2D detector: the diffraction patterns depend only on temperature. This fact reveals that the grain structure did not change before and after the OOT in terms of not only the size but

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also the orientation of the microdomains in each grain, confirming POM results on the conservation of a memory in terms of the grains and the orientation of their internal structure.

Mechanism of the Transition

To investigate the OOT mechanism, we analyzed a scattering pattern from a single grain. For this purpose the sample with a thin thickness (0.3mm) and an X-ray beam of a small size (0.5mm in diameter) were used to minimize the number of grains irradiated by the incident X-ray beam. By scanning the sample with respect to the incident X-ray beam, we found a 6-fold pattern at 111.8°C where the hex-cylinder was observed. Figure 4a shows the scattering pattern, composed of the iso-intensity lines measured with the 2D detector, from almost a single grain of the hex-cylinder. There are a few diffraction spots coming from other grains, though the 6-fold pattern is a main feature of the pattern. The first-order peaks appeared in a hexagonal pattern (connected by dashed lines) at qm and the second-order peaks also appeared in a hexagonal pattern at «J?>qm at different azimuthal angles (also connected by dashed lines). The former is the Bragg reflection from the (10) plane of the hexagonal lattice and the latter is that from (11) plane. In this situation the incident X-ray beam was parallel to the cylindrical axis as illustrated in the schematic in the Figure 4a. Next we changed the temperature to 119.6°C, where the bcc-sphere was observed, without changing the sample position. The hexagonal pattern of the first-order peaks was kept (see Figure 4b).

To observe such a pattern from the bcc lattice, the incident X-ray beam should be parallel to the [111] direction of the bcc lattice as schematically shown in Figure 4b. At this point we can conclude that the cylinders are broken into a series of spheres with the cylindrical axes parallel to [111] direction of the bcc lattice without changing the grain structure. As shown previously this scheme thermoreversibly occurred; i.e., when bcc-spheres are transformed into hex-cylinders, spheres are connected into cylinders along the [111] direction of bcc-sphere. We should note here that there are 4 possible ways in connecting the spheres into the cylinders, because the [111], [Til] , [ i l l ] , and [111] directions are physically equivalent. However nature selects only one of those directions and recovers an original orientation of the cylinders. This means some memory of the lattice orientation exists, which will be discussed later.

Consequently the OOT mechanism does not involve a long-range rearrangement of PS-PI molecules, or a large length-scale translational diffusion of block copolymer molecules but rather involves a transformation induced by instability of the interface curvature brought by the temperature change. When temperature is raised from Tcy, to Tsphere* the system tends to increase interfacial area, causing the cylinders to undulate and eventually break into a series of spheres. When temperature is lowered from Tsphere

to Tcyh the system tends to decrease interfacial area, causing the spheres to deform into ellipsoids that are eventually connected along [111] directions of bcc-sphere and transformed into hex-cylinders.

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Figure 4. SAXS patterns, taken with a 2D detector without changing sample position, from a single grain at (a) 111.8 °C for hex-cylinder and at (b)U9.6°Cfor bcc-sphere.

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Volume Change in Macroscopic and Microscopic Dimensions at OOT

One of the hints to explore the nature of a transition is the change in volume at the transition that is related to the first-order derivative of the free energy with respect to pressure. Here we present the result of the volume change in microscopic dimension as observed by USAXS and that in macroscopic dimension as observed by pycnometry.

One of the merits of using USAXS is the higher angular resolution compared with conventional SAXS. At 117.5°C in the USAXS profile a single peak at qmbcc was observed. Upon cooling the sample, the peak was split into two peaks; one from the bcc-sphere and the other from the hex-cylinder. Upon further cooling the sample, the single peak at qm>cynnd from hex-cylinder was observed. The value of qmtCyijnd is smaller than that of qmibcc9 i.e., the lattice spacing of (10) plane of hex-cylinder (dl0) is larger than that of (110) plane of bcc-sphere, indicating a lattice expansion by 2.7% in length with the transformation from bcc-sphere into hex-cylinder. If this expansion could occur in all directions, it amounts to about 8% volume gain.

The temperature dependence of the specific volume of the sample was measured by pycnometry. The results show no discontinuity at OOT. If the OOT is a first-order phase transition like the liquid-gas phase transition, one may expect discontinuity in volume at OOT. However, even if the OOT is a first-order phase transition, it may not necessarily involve a discontinuity in volume because the transition involves only a symmetry change in the liquid-liquid microphase-separated phases.

We note here that the hysterisis at OOT may be indicative of a first-order phase transition, though the extent of the hysterisis depends on the time-scale of the observations.

Dynamics of OOT as Observed by SAXS and POM

The dynamics of the OOT was further investigated. First we describe the result as observed by time-resolved SAXS. The time-resolved SAXS experiments were conducted after a temperature jump (T-jump) and a temperature drop (T-drop) across T0OT w i *h varying the quench depth, AT, defined by AT = \TQUENCH - T 0 0 T \ . Here TQUENCH

is the temperature to which the sample was quenched at t=0 with / being the time after T-drop or T-jump. Note the area under the first order scattering maximum, A I M , has different intensity levels for the bcc-sphere and hex-cylinder phases (see Figure 2). The value of A I M increased with time after the temperature drop where the transition from bcc-sphere to hex-cylinder occurs. On the other hand, A I M decreased with time after the T-jump from hex-cylinder to bcc-sphere. The change in ALM with time, Alm(t), in both T-jump and T-drop were found to be approximated with good accuracy by a single exponential curve as described by the following equation,

AIm(t) = {A„„(0) - A / mH}exp(-f/T) + A / M H , (2)

with τ being the relaxation time. Note that no appreciable incubation time was

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observed in the change of A[m with / in both transitions from hex-cylinder to bcc-sphere and that from bcc-sphere to hex-cylinder.

Figure 5 shows the relaxation time, τ, plotted against the magnitude of AT in a double logarithmic scale. Open and closed markers are for the T-jump (from hex-cylinder to bcc-sphere) and T-drop (from bcc-sphere to hex-cylinder) processes, respectively. The following observations are noteworthy to point out: (i) the rate of the transition from hex-cylinder to bcc-sphere is much faster than that from bcc-sphere to hex-cylinder, (ii) the larger the ΔΤ, the faster the transition rate, and (iii) both processes show the linear relationships between log(r) and \og(AT) as shown by the solid lines and differ only in terms of the prefactor. This relation maybe explained by a scaling law, τ ~ £2/D~ (AT)"1, where £ and D are the characteristic length relevant to the OOT process and apparent diffusion coefficient, respectively. The mean-field theory predicts D is proportional to the quench depth, D~ D0AT, where D0 is the mutual diffusivity. D 0 and £ = 2n/qm are nearly constant before and after the T-jump in the temperature range covered in this work. As far as eq 2 is concerned, the OOT processes is apparently analogous to the relaxation of concentration fluctuations of single-phase disordered block copolymer melts induced by a temperature change inside the single-phase region.

The process of OOT was also studied by POM. Figure 6 shows the time-resolved POM micrographs after the T-jump from 112.7°C (hex-cylinder phase) to 118.6°C (bcc-sphere phase). The image due to the form birefringence of the cylinders disappeared as a whole gradually with time. On the other hand in the T-drop experiment from 118.6°C to 112.7°C, the image gradually appeared as a whole and the original image at 112.7°C was completely recovered. No signature of a nucleation process was observed and essentially the relaxation times in both T-drop and T-jump were identical to those observed by SAXS.

Time-Resolved USAXS Study at OOT

We investigated the process by a time-resolved USAXS measurement during OOT from hex-cylinder to bcc-sphere. The change in the USAXS profile around the first-order scattering maximum was obtained as a function of time after T-jump from hex-cylinder to bcc-sphere and the transition is found to be classified into 3 stages.

In the first stage, a continuous shift of the peak position toward a larger q was observed, indicating a decrease of the lattice spacing. The shift is very small by about 0.6% and may be due to a cooperative deformation of the grains: the cylinders are elongated and contracted parallel and perpendicular to the cylindrical axes, respectively. Actually we obtained a master curve for the USAXS profiles measured at different time after the T-jump by scaling q with qm, and scattered intensity by maximum intensity, Im. This means that the scaled structure factor is invariant with time and equal to the structure factor for the hex-cylinder at time zero after the T-jump in the first stage. Thus, there occurs a uniform deformation of the grains containing the hex-cylinders with the lattice spacing from dcylind to d'cylind and the length of the cylinder probably changing from icylind to £' c y l i n d, while keeping the

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Figure 5. Relaxation time rasa function of AT in double-logarithmic scale for. Open and closed markers are for the T-jump (from hex-cylinder to bcc-sphere) and T-drop (from bcc-sphere to hex-cylinder) processes, respectively.

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relation (dcylind) ^cylind (^cylind^ ^cylind* though the change from £Cy\ind tO £CyUnd

was not directly observed. At the second stage of the transition, the maximum intensity due to the hex-

cylinder at qm = 27tld'cylind was observed to decrease, while a new maximum corresponding to the bcc-sphere phase appeared at a different characteristic wave number, qm = 2njd'hcc with d'hcc being the spacing of (110) plane of bcc-sphere, and its intensity increased with time. This process reflects the break-up of the cylinders into a series of spheres with their axes parallel to the [111] direction of the bcc lattice. However the existence of two characteristic wave numbers does not necessarily mean that the volume changed. The cubic phase appearing here may be somewhat distorted, as will be clarified immediately below.

Finally at the third stage, a further shift of the peak position toward a larger q was observed, indicating a lattice contraction into a more perfect lattice with the lattice spacing from d'bcc to dbcc. This stage may involve relaxation in the shape of spheres and orientation of block chains as well as that in deformed bcc lattice. These processes may generate defects within the grain or grain boundary. Here we note that those defects in the bcc lattice can provide a memory for the mode selection at OOT from bcc-sphere to hex-cylinder, i.e., the bcc-spheres are connected into the cylinders, while recovering the original orientation of the cylinders.

Concluding Remarks

In this paper we presented the thermoreversible OOT between the bcc-sphere and the hex-cylinder for a PS-PI diblock copolymer using SAXS, USAXS and POM. To investigate the relation between the spatial arrangement of bcc-sphere and hex-cylinder, we prepared a sample having relatively large grains which were developed by cooling the sample from the lattice-disordered spheres with a slow cooling rate. An analysis of the scattering patterns from an effectively single grain revealed that the cylindrical microdomains are burst into spherical microdomains in such a way that the cylindrical axes correspond to the [111] direction of the spheres in bcc lattice. In this thermoreversible transition the grain structure and orientation of the lattice in the grain are conserved. The transition behavior of the OOT induced by a temperature change shows a uniform and gradual change. No incubation time on the OOT process was observed, revealing that the symmetry break induced by the OOT process studied in this work is spinodal-like. This is quite contrary to the nucleation-growth process in the ODT process as the fluctuation-induced first-order phase transition [9-15,32]. By examining the OOT process closely by USAXS, it was found that the OOT process involves the cooperative deformation of the grain structure. The details of the transition mechanism [19] and transition dynamics [20] will be reported elsewhere.

Acknowledgments

We gratefully acknowledge Dr. Tsuyoshi Koga for his useful comments. We thank also to Dr. Jeffrey Bodycomb for useful suggestions on SAXS measurements.

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Almdal, K.; Mortensen, K. Macromolecules 1995, 28, 8796. 8. Leibler, L. Macromolecules 1980, 13, 1602. 9. Fredrickson, G. H.; Helfand, E. J. Chem. Phys. 1987, 87, 697. 10. Fredrickson, G. H.; Binder, K. J. Chem. Phys. 1989, 91, 7265. 11. Hohenberg, P. C.; Swift, J. B. Phys. Rev. E. 1995, 52, 1828. 12. Bates, F. S.; Rosedale, J. H.; Fredrickson, G. H. J. Chem. Phys. 1990, 92, 6255. 13. Stühn, B.; Mutter, R.; Albrecht, T. Europys. Lett. 1992, 18, 427. 14. Sakamoto, N.; Hashimoto, T. Macromolecules 1995, 28, 6825. 15. Hashimoto, T.; Sakamoto, N. ; Koga, T. Phys. Rev. E. 1996, 54, 5832. 16. Sakurai, S.; Kawada, H.; Hashimoto, T.; Fetters, L. J. Proc. Jpn. Acad. 1993, 69

(Ser. B), 13.; Macromolecules 1993, 26, 5796. 17. Sakurai, S.; Hashimoto, T.; Fetters, L. J. Macromolecules 1996, 29, 740. 18. Hajduk, D. Α.; Gruner, S. M. ; Rangarajan, P.; Register, R. Α.; Fetters, L. J.;

Honeker, C.; Albalak, R. J.; Thomas, E. L. Macromolecules 1994, 27, 490. 19. Kimishima, K.; Koga, T.; Kanazawa, Y.; Hashimoto, T., in preparation. 20. Kimishima, K.; Koga, T.; Hashimoto, T., in preparation. 21. Hashimoto, T.; Okamoto, S.; Saijo, K.; Kimishima, K.; Kume, T. Acta Polymer

1995, 46, 463. 22. Koga, T.; Hart, M. ; Hashimoto, T. J. Appl. Cryst. 1996, 29, 318. 23. Polymer Handbook 3rd edition; Brandrup, J.; Immergut, Ε. H., Eds.; John Wiley

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INDEXES

Author Index

Alamo, Rufina G. , 152 Andersen, Ken, 328 Andresen, Elke M , 390 Bellare, Anuj, 436 Bras, Wim, 341 Bucknall, D. G. , 57 Burger, Christian, 244 Burghardt, Wesley R., 374 Butler, P. D., 356 Butler, S. Α., 57 Capel, Malcolm, 152 Cebe, Peggy, 2, 152 Char, Kookheon, 496 Chen, Er-Qiang, 118 Chen, S-H s 270 Cheng, Stephen Z. D., 118 Chidambaram, S., 356 Chu, Benjamin, 244, 448 Cinader, David K., Jr., 374 Cochran, H . D., 317 Cohen, Robert E. , 436 Dadmun, M . D., 356 Dai, Patrick S., 152 De Odorieo, W., 456 Denny, Richard C , 341 Diakun, Gregory P., 341 Diaz, J. Fernando, 341 Dosière M. , 166 Dumas, P., 448 Fairclough, J. Patrick Α., 201 Ferrero, Claudio, 341 Floudas, G., 448 Fougnies, C , 166 Gabrys, Barbara, 328 Gallot, Y., 261 Gleeson, Anthony, 341 Grubb, D. T., 24 Hamilton, W. Α., 356 Harris, Frank W., 118 Hashimoto, Takeji, 514 Hauser, G. , 140 Heath, K., 317 Herhold, A. B., 232

Higgins, J. S., 57 Hsiao, Benjamin S., 118, 187 Huang, Baotong, 187 Kalhoro, M . S., 328 Kanazawa, Yuko, 514 Karbach, Alexander, 436 Kim, Jin Kon, 470 Kimishima, Kohtaro, 514 Kiran, E . , 317 Koch, M . H . J., 166 Koga, Tadanori, 514 Ladynski, H . , 456 Lambert, O., 448 Lee, Hyun Hee, 470 Lee, Seung-Heon, 496 Lee, Song-Wook, 118 Levine, Yehudi Κ., 341 Liu, Shengsheng, 187 Liu, Weidong, 187 Liu, Yingchun, 270 Mandelkern, Leo, 152 Mann, Ian, 118 Mayes, A . M. , 261 Meinichenko, Y. B., 317 Mitchell, Geoffrey R., 390 Monkenbusch, M. , 103 Moon, Bon-Suk, 118 Murthy, N. S., 24 Myers, Randall T., 436 Nakatani, Alan I., 405 Phillies, George D. J., 297 Pollard, M. , 261 Rathgeber, S., 103 Reiter, G. , 448 Roovers, J., 166 Rosov, N., 103 Runt, James, 218 Russell, T. P., 261 Ruzette, Α. V., 261 Ryan, Anthony J., 201 Salaniwal, S., 317 Schmidtke, J., 140 Sirota, Ε. Β., 232

532

Stamm, M. , 317, 456 Stein, Richard S., 187 Streletzky, Kiril Α., 297 Stribeck, N., 41 Strobl, G. , 140 Subirana, Juan Α., 93 Takahashi, Yasuhiro, 74 Terrill, Nicholas J., 201 Thurn-Albrecht, T., 140 Toma, Lucio, 93 Tsui, O. K. C , 261 Tucker, Ryan, 328

Ugaz, Victor M. , 374 Van Hook, W. Α., 317 von Meerwall, Ernst D., 118 Weiss, R, Α., 328 Wignall, G . D., 317 Yang, Henglin, 187 Yeh, Fengji, 118, 244 Yeh, F.-J., 448 Zajac, Wojciech, 328 Zero, K., 24 Zhang, Anqiu, 118 Zhou, Shuiqin, 244

533

Subject Index

«-Alkanes even-odd effect in crystal structures

and melting point, 233 See also Transient rotator phase in­

duced nucleation in n-alkanes Amorphous polymer interfaces, neutron

reflection, 58 Annealing. See Isothermal thickening and

thinning processes in poly(ethylene ox­ide) (PEO)

Auto-convolution, successive stages in convolution of function, 11, 12/

Auto-correlation function, specific type of convolution, 10

Β

Backscattering spectrometer inelastic neutron spectrometer,

103-104 phase space diagram indicating region

of accessibility, 104/ Biopolymers. See Magnetically oriented

microtubule biopolymers Blends. See Phase separation kinetics dur­

ing shear in polymer blends; Polycarbo-nate-ionomer blends; Poly(ethylene ox­ide) (PEO) blends

Block copolymers crystallizable and amorphous blocks,

448 driving force for studying, 448-449 phase behavior of mixtures with homo­

polymers, 496-497 See also Crystallization of block copoly­

mers; Grain size of lamellar styrene-butadiene block copolymers; Micro-phase separation transitions of polystyrene-co-butadiene copolymer; Ordering kinetics for block copoly­mers; Thermoreversible order-order transition of block copolymer

Body centered cubic (BCC) microdo­mains. See Ordering kinetics for block copolymers; Thermoreversible order-order transition of block copolymer

Bonart's longitudinal structure, definition, 44

Bonart's transverse structure, definition, 44

Bragg spacing (L B) changes due to crystals perfecting or

melting, 161, 164 L B versus temperature during subse­

quent melting, 161, 163/ LB versus time during isothermal crys­

tallization, 161, 163/ See also Crystallization and melting be­

havior of metallocene isotactic poly(propylene) (m-iPP)

Butadiene. See Polystyrene-^/ocfc-butadi-ene copolymers

Cahn-Hilliard theory kinetics of crystallization after devitrifi­

cation, 209 plot estimating effective diffusion coef­

ficient (D e f f), 210/ spinodal decomposition, 210-211 See also Nucleation phenomena in ho­

mopolymer melts Chord distributions, definition, 48 Clayperon equation, pressure coefficient

relation to enthalpy and volume change, 265, 268

Collection of atoms conditions for constructive interfer­

ence, 6-7, 8/ electron density characterizing, 8-9 scattering, 6-9

Compatibilizing agents for polymer blends. See Phase separation kinetics during shear in polymer blends

534

Convolution correlation functions, 10-13 illustrating operation, 10-11, 12/ procedure for determining product, 11 structural parameters from measured

scattering intensity, 10 Convolution Theorem

illustration, 14/ relating Fourier transform and convolu­

tion operations, 11, 13 Copolymer effect, miscibility of polycar­

bonate (PC) and sulfonated polysty­rene (SPS), 329

Copolymers. See Ethylene-based copoly­mers; Grain size of lamellar styrene-butadiene block copolymers; Micelles in aqueous solutions; Poly(ethylene ox­ide) (PEO) blends

Correlation, structural parameters from measured scattering intensity, 10

Correlation functions convolution and, 10-13 definition, 17 Q as normalization factor, 17 small-angle scattering relating cor­

rected intensity, 16 structural parameters from, 17-18

Correlation length using small-angle neutron scattering

high-concentration isotope labeling method, 319

See also Polymers in organic solvents and supercritical fluids

Correlation lengths. See Polycarbonate-ionomer blends

Crystalline polymer interfaces illustration showing section through

polymer heating cell, 60/ neutron reflection, 58

Crystalline polymers neutron structure analysis of polyethyl­

ene^, 85-91 neutron structure analysis of poly(vinyl

alcohol), 76-85 See also Neutron diffraction; Simula­

tions of melting transitions Crystallization. See Ethylene-based copol­

ymers; Isothermal thickening and thin­ning processes in poly(ethylene oxide) (PEO); Poly(aryl ether ether ketone) (PEEK); Polyethylene oxide) (PEO) blends

Crystallization and melting behavior of metallocene isotactic poly(propylene)

(m-iPP) differential scanning calorime-try (DSC) study, 154 DSC and wide-angle X-ray scattering

(WAXS) results, 156, 161 DSC exothermic heat flow of m-iPP

during isothermal crystallization, 157/ DSC heat flow melting endotherms im­

mediately after m-iPP crystallization at 117°C for indicated time, 156, 158/

experimental m-iPP sample, 154 explanation for changes in scattering in­

variant (Q) and Bragg long spacing (LB), 161, 164

modifications of WAXS diffractograms of iPP, 155

real-time small-angle X-ray scattering (SAXS) and WAXS, 154

SAXS data correction, 154-155 SAXS results, 161, 164 scattering invariant (g), 155 scattering invariant (Z) and Bragg long

period of m-iPP versus temperature during subsequent melting, 161,163/

scattering invariant (Z) and Bragg long period of m-iPP versus time during isothermal crystallization, 161, 163/

time development of γ crystals of m-iPP during isothermal crystallization at 117°C, 156, 159/

WAXS data collection, 155 WAXS intensity versus scattering angle

during initial isothermal crystalliza­tion, 156, 160/ 161

WAXS intensity versus scattering angle during melting of m-iPP after isother­mal crystallization, 161, 162/

Crystallization from glassy state and melt. See Poly(aryl ether ether ketone) (PEEK)

Crystallization mechanisms of polymers changes during isothermal crystalliza­

tion and subsequent melting of syndi-otactic poly(propene-co-octene) sP(P-co-O) samples, 143/

electron microscopy, 140 evidence for surface crystallization and

melting in PE, 146-148, 150 evolution with time of crystal thickness

and inner surface, 146/ formation of cross-hatched structure in

isotactic polypropylene (iPP), 150-151

iPP scattering curves at 153°C and 31°C, 149/

535

iPP temperature dependence of specific length of edges, 151/

kinetics of crystal thickening for poly­ethylene (PE), 146

kinetics of primary crystallization, 142-146

mechanisms of secondary crystalliza­tion, 146-151

P E change of amorphous layer thick­ness with temperature, 147/

P E interface distribution function at be­ginning and end of isothermal crystal­lization, 145/

P E interface distribution functions after isothermal crystallization and subsequent cooling, 147/

PE interface distribution functions at Τ = 88°C for samples crystallized at different temperatures, 148/

primary tools for study, 140-141 SAXS data analysis, 141-142 SAXS experiments and instrumenta­

tion, 142 scenario with syndiotactic polypropyl­

ene (sPP) and sP(P-co-O), 142-145 small-angle X-ray scattering (SAXS),

140-141 sPP and sP(P-co-O) Avrami times of

crystallization in dependence on crys­tallization temperature, 144/

sPP and sP(P-co-O) relations between inverse crystallite thickness, crystalli­zation temperature, and melting peak, 144/

sPP and sP(P-co-O) sample properties, 142r

Crystallization of block copolymers dependence of crystallization times of

poly(ethylene oxide) (PEO) and poly(e-caprolactone) (PCL) on crys­tallization temperature, 453-454

experimental samples of polystyrene (PS), PEO, and PCL, 449-450

molecular characteristics of triarm star block copolymers, 449f

SAXS and W A X D spectra for star block copolymer SEL-4.7/20/1.8, 450, 451/

SAXS and W A X D spectra for star block copolymer SEL-4.7/20/87, 450, 451/

SAXS and W A X D spectra for third block copolymer SEL-4.7/20/45, 452

time-resolved S A X S / W A X D measure­ments, 450

Crystal structures, atactic polyvinyl alco­hol), 82

Crystal thickening, simulation results, 100, 101/

Detector placement, 4 Diamagnetism, magnetically oriented bio­

polymers, 342-343 Diblock copolymers. See Order-disorder

transition (ODT) in diblock copolymers

Differential scanning calorimetry. See Nu­cleation phenomena in homopolymer melts

Differential scanning calorimetry (DSC). See Crystallization and melting behav­ior of metallocene isotactic polypropyl­ene) (m-iPP); Ethylene-based copoly­mers; Poly(aryl ether ether ketone) (PEEK)

Diffusion, small-molecule by neutron re­flection, 67-69

Diffusion coefficient Cahn-Hilliard plot estimating effective,

210/ 211 determination, 66-67

Diffusion studies, neutron reflection ex­periments in real time, 63-64

Disk chopper time-of-flight spectrometer inelastic neutron spectrometer,

103-104 phase-space diagram indicating region

of accessibility, 104/ Dynamic secondary ion mass spectros­

copy (DSIMS), interface between ini­tially separated miscible materials, 58

Electromagnetic (EM) radiation, direc­tion of travel, 3

Electron density, characterizing collection of atoms, 8-9

Enthalpy thermodynamic parameter, 262 See also Order-disorder transition

(ODT) in diblock copolymers Ethylene-based copolymers

comparison of invariant (Q) from small-angle X-ray scattering (SAXS) and crystallinity from wide-angle X-ray scattering (WAXD) during iso­thermal crystallization, 198, 199/

536

copolymer synthesis, 189 crystallinity by WAXD versus crystalli­

zation temperature for samples with various degrees of branching, 195, 196/

degree of crystallinity in samples as function of crystallization tempera­ture, 192/

differential scanning calorimetry (DSC) measurements, 189

DSC melting thermograms for various polyethylene (PE) crystallized at dif­ferent temperatures, 191/

DSC thermograms from metallocene-PE (m-PE) samples with different branching degrees, 190/

effect of branching on crystallization and melting, 199-200

equilibrium melting temperature using Thomson-Gibbs equation, 199

experimental, 189 invariant (β) , long period (L), crystal

lamellar thickness (/c), and interlamel­lar amorphous thickness (/a) from SAXS data, 194-195

isothermal crystallization and melting experiments by DSC, 190, 192

isothermal crystallization and melting study of m-PE using simultaneous SAXS/WAXD techniques, 189

isothermal crystallization and subse­quent melting behavior of m-PE as function of crystallization tempera­ture by time-resolved X-ray tech­niques, 193-194

issues in study, 188-189 melting temperature changes with crys­

tallization temperature, 192/ morphological variables from correla­

tion function analysis of SAXS data, 194-195

simultaneous SAXS and WAXD pro­files, 193/

slope, interception and error of PE samples, 198i

tailored polyolefins with metallocene catalysts, 187-188

thermal parameters of ethylene-octene copolymers, 190i

thermodynamics and kinetic factors af­fecting structure and morphology de­velopment in metallocene-based poly­mers, 188

Thomson-Gibbs equation, 195, 197 Thomson-Gibbs plot of samples with

various degrees of branching, 195, 197/

time-resolved SAXS and WAXD fol­lowing crystallization kinetics, 198-199

time to reach plateau (β) versus crys­tallization temperature of m-PE with various degree of branching, 195, 196/

values of L, /c, and /a as functions of crystallization temperature and branch composition, 195,196/

Ethylene-methacrylic acid (EMAA) co­polymers. See Poly(ethylene oxide) (PEO) blends

Extrusion. See Nucleation phenomena in homopolymer melts

Fiber analysis. See Poly(ether ester) (PEE); Small-angle X-ray scattering (SAXS)

Flow behavior of liquid crystalline poly­mer (LCP) solutions chimney region of phase diagram by

polarized optical microscopy, 370 complex correlation between viscosity

and shear rate molecular alignment, 366

correlation of steady-state alignment to viscosity, 361

degree of ordering by calculating align­ment factor (A), 359

diagrammatic representation of phase diagram of rod-like LCP solution, 367, 368/

dynamic modulus after shear cessation, 364

effect of polymer aspect ratio on con­centrations entering and exiting bi­phasic region, 369/

evolution of dynamic modulus of poly(y-benzyl) L-glutamate (PBLG) in deuterated benzyl alcohol (DBA) after flow field removal and corre­sponding molecular alignment, 365/

experimental, 358-359 flow behavior of PBLG in deuterated

m-cresol (DMC) versus DBA, 364, 366

LCP solution phase diagram for poly­mers with various rigidities near chimney region, 368/

materials, 358

537

negative first normal stress (Nx) differ­ent in LCPs, 356-357

orientation relaxation, 361, 364 origin of differences in response of two

solutions, 366-367, 370 parameters affecting flow, 372 PBLG/m-cresol as LCP model system,

357 phase diagram of PBLG in benzyl alco­

hol and m-cresol by polarized optical microscopy, 371/

possible inherent solvent dependence on structure of solution defects, 370

quantifying flexibility and aspect ratio altering phase diagram, 367, 370

relaxation of flow-induced alignment in PBLG-DBA solution after removal of shear, 362/

relaxation of shear-induced alignment in PBLG-DBA, 363/

rheology experiments, 358 SANS experiments, 358 shear rate dependence of steady-state

molecular alignment of PBLG in DBA and DMC, 360/

shift in chimney region as result of polymer flexibility, 369/

small-angle neutron scattering (SANS) and rheology of PBLG in DMC and DBA, 357-358

steady state, 359-361 steady-state scattering experiments,

358-359 stronger tendency to form cholesteric

phase in hydroxypropylcellulose (HPC) than in PBLG, 370

theoretical model explaining anoma­lous sign changes in Nu 357

viscosity behavior of PBLG-DMC and PBLG-DBA, 362/

Fluorescence recovery after photobleach-ing (FRAP), probe diffusion technique, 298

Force Rayleigh scattering (FRS), probe diffusion technique, 298

Fourier transforms equation, 9-10 structural parameters from measured

scattering intensity, 10

Gibbs-Thompson equation expressing melting temperature of crys­

tal, 97 simulated lamellar crystals obeying, 100

Good solvent domains poly(dimethylsiloxane) (PDMS) in su­

percritical C0 2 , 325 systems with, 326 See also Polymers in organic solvents

and supercritical fluids Grain size of lamellar styrene-butadiene

block copolymers contrast factor and grain boundary vol­

ume fraction, 444-445 double logarithmic plots of absolute in­

tensity versus scattering vector (q) for sample S12B10, 438, 439/

double logarithmic plots of absolute in­tensity versus scattering vector (q) for sample SB5, 438, 439/

experimental block copolymers, 437 grain size (D) and lamellar spacing

(d), 443-444 grain size determination from transmis­

sion electron microscopy (TEM), 445-446

grain size values by ultrasmall-angle X-ray scattering (USAXS) and TEM,446r

mechanism of scattering, 443 proposed mechanism, 442/ 443 TEM micrograph for annealed (1 hr)

S12B10, 442/ TEM micrograph for annealed (5 min)

S12B10, 441/ TEM micrograph for unannealed

S12B10, 440/ USAXS experiments, 437-438 USAXS results for S12B10 and SB5, 444* See also Thermoreversible order-order

transition of block copolymer Grain structure, conservation in thermore­

versible transitions of block copoly­mers, 519, 522-523

Guinier Law low values of scattering vector K, 19-20 plotting intensity of lamellar reflection

as function of χ in semicrystalline polymer, 31, 32/

Η

G Helmholtz equation solution same for electromagnetic radia-

Gibbs free energy, balance between en- tion or particles, 3 tropy and enthalpy, 201-202 spherical wave solution, 3-4

538

Hexagonally packed cylindrical (HEX) microdomains. See Ordering kinetics for block copolymers; Thermorevers­ible order-order transition of block co­polymer

Homopolymer melts. See Nucleation phe­nomena in homopolymer melts

Hydrogen bonds, intramolecular in isotac­tic sequence of atactic polyvinyl alco­hol), 84-85

Hydroxypropylcellulose (HPC) influence of phase type on response of

liquid crystal polymers, 391 tendency to form cholesteric phase, 370 See also Flow behavior of liquid-crystal­

line polymer (LCP) solutions; Solu­tionlike and meltlike solutions of hy­droxypropylcellulose (HPC); X-ray rheology of structured polymer melts

I Interaction. See Micelles in aqueous solu­

tions Interdiffusion

polymer-polymer by neutron reflec­tion, 64-67

See also Neutron reflection (NR) Interface distribution functions (IDFs)

computation, 49-50 model function fits to IDFs, 52/ poly(ether ester) (PEE), 51/

Intrinsic chord distribution, equation, 48 Invariant

definition, 44 information on volume fraction of hard

domains in sample, 46 Ionic surfactants. See Self-assembly of

ionic surfactants in oppositely charged polyelectrolyte gels

Ionomers. See Polycarbonate-ionomer blends

Isotactic polypropylene (iPP) β and γ modifications of Ziegler-Natta

synthesized iPP, 153 formation of cross-hatched structure,

150-151 scattering curves at 153°C and 31°C,

149/ temperature dependence of specific

length of edges, 151/ See also Crystallization and melting be­

havior of metallocene isotactic poly(propylene) (m-iPP); Crystalliza­tion mechanisms of polymers; Nucle­ation phenomena in homopolymer melts

Isothermal thickening and thinning pro­cesses in poly(ethylene oxide) (PEO) annealing behavior of two-arm PEOs,

133, 136-138 differential scanning calorimetry (DSC)

experiments, 120 DSC heating diagrams for 1,3-two-arm

PEO following crystallization at dif­ferent crystallization temperatures (Tc), 130,132/

equipment and experiments, 120 isothermal crystallization behavior of

two-arm PEOs, 121, 124, 130 isothermal crystallization processes for

three two-arm PEOs by simultane­ous synchrotron wide-angle X-ray dif­fraction (WAXD) and small-angle X-ray scattering (SAXS), 121, 123/

long period, crystallinity, and relative invariant Q' changes for 1,4-two-arm PEO during heating and annealing, 136, 137/ 138

low molecular weight PEO fractions in understanding polymer crystalliza­tion, 119

materials synthesis and characteriza­tion, 119-120

melting behavior for two-arm PEOs at different isothermal Tcs, 130, 133, 134/

melting behavior of linear and two-arm PEOs, 130, 133

melting temperature changes during an­nealing at different annealing times, 133, 135/

molecular analyses and characteriza­tions, 120-121

molecular characteristics of linear and two-arm PEO fractions, 121f

relationship between final long periods after completing crystallization and crystallization temperatures for two-arm PEOs, 130, 131/

self-diffusion coefficients by pulsed-gra-dient spin-echo (PGSE) nuclear mag­netic resonance (NMR) for linear and two-arm PEOs, 121, 122/

time-resolved synchrotron WAXD and SAXS experiments, 120

WAXD and SAXS patterns of 1,2-two-arm PEO crystallized at 32°C, 124, 125/

WAXD and SAXS patterns of 1,2-two-arm PEO crystallized at 40°C, 124, 127/

539

WAXD and SAXS patterns of 1,2-two-arm PEO crystallized at 48°C after self-seeding, 124, 129/

WAXD and SAXS patterns of 1,4-two-arm PEO crystallized at 40°C, 124, 126/

WAXD and SAXS patterns of 1,4-two-arm PEO crystallized at 48°C after self-seeding, 124, 128/

Kinetics of phase separation. See Phase separation kinetics during shear in poly­mer blends

Lamellar peaks variations in axial positions with χ for

semicrystalline polymers, 31, 33/ variations in axial widths with χ for

semicrystalline polymers, 31, 32/ widths of, 29, 31

Lamellar reflections angle between, 29, 30/ characteristics, 26r characteristics in semicrystalline poly­

mers, 26-31 See also Semicrystalline polymers

Lamellar spacing semicrystalline polymers, 26, 29 styrene-butadiene block copolymers,

443-444 See also Grain size of lamellar sty­

rene-butadiene block copolymers Linear poly(ethylene oxide). See Isother­

mal thickening and thinning processes in poly(ethylene oxide) (PEO)

Liquid-crystalline polymers (LCPs) long-range orientational order, 391 molecular alignment affecting macro­

scopic properties, 356-357 orientational order in nematic liquid

crystal structure, 391/ See also Flow behavior of liquid-crystal­

line polymer (LCP) solutions; Molec­ular orientation of thermotropic liq­uid-crystalline polymers (TLCPs) under flow

Longitudinal structure domain height determination, 50-52 evaluation, 46-47 general evaluation steps, 49-50 See also Poly(ether ester) (PEE);

Small-angle X-ray scattering (SAXS)

Lorentz weighting factor, correction to in­tensity, 15

M

Magnetically oriented microtubule bio­polymers algorithms correcting disorientation ef­

fects on diffraction peaks, 346 aligning molecules off-line before dif­

fraction studies, 343, 344/ application of helical diffraction theory,

351, 353 best alignment with fast reactions, 343,

344/ best fit to data of cylinder, 351, 352/ calculating diamagnetic moment of mi­

crotubules, 343 diffraction intensity on equator and

first layer line, 343, 345/ 346 elucidation of structure by small-angle

X-ray scattering (SAXS), 342 helical diffraction theory of scattering

pattern on equator of fiber diffrac­tion pattern, 350-351

interference effects from microtubule packing, 347, 350

long axis of samples parallel to X-ray beam, 347, 348/

low-angle scattering pattern from differ­ent concentrations of microtubules in glycerol assembly buffer, 347, 349/

magnetic birefringence work, 351 materials and methods, 346-347 radius of gyration of cross section (Rc),

350 reaction rate in achievement of full

alignment, 343 roles of microtubules in cell, 342 self-assembling by diamagnetism,

342-343 Mean field theory, phase transition of co­

polymers, 456-457 Mechanisms. See Crystallization mecha­

nisms of polymers Mellin convolution, definition, 48 Melting. See Ethylene-based copolymers;

Isothermal thickening and thinning pro­cesses in poly(ethylene oxide) (PEO); Nucleation phenomena in homopoly­mer melts; Poly(aryl ether ether ke­tone) (PEEK)

Melting transitions. See Simulations of melting transitions

540

Meltlike solutions. See Solutionlike and meltlike solutions of hydroxypropylcel­lulose (HPC)

Melts. See X-ray rheology of structured polymer melts

Metallocene isotactic poly(propylene) (m-iPP) experimental sample, 154 studying isothermal melt crystallization

and subsequent melting, 153 See also Crystallization and melting be­

havior of metallocene isotactic poly(propylene) (m-iPP)

Metallocene polyethylene (m-PE). See Ethylene-based copolymers

Metastable phases. See Transient rotator phase induced nucleation in w-alkanes

Micelles in aqueous solutions ability of Pluronic polymer to self-asso­

ciate in aqueous environment, 271 analysis of light-scattering intensity us­

ing new plot, 289-291, 294 block copolymer self-association in

aqueous solutions, 271 calculating particle structure factor, 277 cap-and-gown model as satisfactory de­

scription, 294 cap-and-gown model consistent with

predictions based on lattice model, 277

cap-and-gown model illustration, 276/ cap-and-gown model of micellar struc­

ture, 275-278 characterization of Pluronic P84 micel­

lar solutions using new plot, 291i comparison of theoretical and experi­

mentally measured viscosity as func­tion of micellar volume fraction for PI04 at range of temperatures, 293/ 294

core-shell model structure of micelle and hard sphere intermicellar interac­tion, 274

determining range of disordered mi­celle phase, 272

first-order spherical Bessel function, 277

Fortran code based on gradient search­ing nonlinear least square fitting method, 284

hydration number versus temperature, 284

intermicellar interaction and structure factor, 278-280

low shear viscosity at finite frequency, 291

low shear viscosity of adhesive hard-sphere system determination, 294

micellization above critical concentra­tion and temperature, 271-272

microstructure and positional correla­tions of polymeric micelles at concen­trations above micellization bound­ary, 272, 274

molecular volumes, scattering lengths and scattering length densities of polymer and solvent, 273r

neutron scattering length density pro­file calculation, 275, 277

new method for static light-scattering intensity analysis, 280-281

new plot, 281-282, 283/ new plots of P84 micellar solutions at

series of temperatures, 290/ parameters characterizing microstruc­

ture of micelle and interaction, 284 parameters for microstructure and in­

teraction of P84 polymeric micelles at 40°C from small-angle neutron scattering (SANS) intensities, 287i

parameters of microstructure and inter­action of P104 polymeric micelles from SANS intensities, 288i

parameters of microstructure and inter­action of P84 polymeric micelles from SANS intensities, 287i

Pluronic surfactants of polyethylene ox­ide and polypropylene oxide, 272

primary fitting parameters in new model, 282, 284

radial distribution of polymer segment volume fraction in micelle, 275

Rayleigh ratio, 281 relative viscosity and two contributions

as function of volume fraction, 291, 292/ 294

relative viscosity of colloidal solution at zero frequency, 280

SANS intensity distribution and model fit as decomposed into inter- and in-traparticle structure factors, 284, 285/

SANS measurements, 274 SANS scattering intensity proportional

to product of normalized particle structure factor and interparticle structure factor, 282

series of SANS data and fits for Pluro­nic P84 micellar solutions, 284, 286/

static light-scattering measurements, 274-275

stickiness parameter characterizing in­teraction, 284

541

structure factor for adhesive hard sphere model, 279-280

structure factor of sticky hard sphere model, 278-279

theoretical background, 275-282 volume fraction of micelles characteriz­

ing interaction strength, 278 water penetration profile, 284, 289 Zimm plot formula for micellar solu­

tion, 280 Micellization, critical concentration and

temperature, 271-272 Microphase separation transitions of poly-

styrene-b/ocfc-butadiene copolymer characteristic data of investigated sam­

ple, 458i data analysis, 459-460 first-order morphology peaks of SAXS

profiles of P(S-6-B) at different tem­peratures, 461/

fit parameters of SAXS profiles of P(S-6-B) for heating and cooling cy­cle, 462/

kinetics of ordering process, 457 mean field theory describing phase tran­

sition of copolymers, 456-457 measurements, 458-459 press-temperature superposition

scheme possible, 465, 468 pressure dependence of Γ Μ 5 Τ (micro-

phase separation transition tempera­ture), 463/

pressure-dependent characteristic peak parameters of P(S-6-B) during up­ward and downward pressure scan, 464/

pressure-dependent measurements, 463, 465

pressure jump experiments, 465 sample preparation, 458 superposition of three scattering curves

at different temperatures and pres­sures in vicinity of Γ Μ 5 τ , 465, 466/

temperature dependence agreeing with recent theories, 465

temperature-dependent measurements, 460, 463

time dependence of characteristic parameters of SAXS profiles of P(S-fe-B) after pressure jump, 465, 467/

Microtubule biopolymers. See Magneti­cally oriented microtubule biopolymers

Models. See Simulations of melting transi­tions

Molecular orientation in thermotropic liq­uid-crystalline polymers (TLCPs) un­der flow anisotropy factor, 384-385 anisotropy factor as function of axial

position along centerline in Xydar SRT-900, 386/

commercial TLCP in diverging channel flow, 381-382

expansion influencing evolution in mac­roscopic orientation, 385-386

log-rolling orientation state, 378 mapping out evolution of average ori­

entation in two dimensions in expan­sion region, 385, 386/

model main-chain TLCPs, random co-polyether DHMS-7,9 and copolyester PSHQ-6,12, 376-377

model TLCPs in shear flow, 376-381 orientation parameter (S) for macro­

scopic alignment, 379 orientation parameters in steady-state

flow as function of shear rate and temperature for model TLCPs, 379, 380/ 381

representative 2D X-ray scattering pat­terns in expansion flow of SRT 900, 383-384

schematic illustration of X-ray shearing cell setup, 377/

schematic of extrusion die for in situ X-ray scattering from polymer melts in channel flows, 382/

studying commercial LCP resin, Xydar SRT 900, 382

synchrotron X-ray scattering experi­ments, 377

vector plot representing average orien­tation state in expansion region, 387/

WAXS patterns for DHMS-7,9 during shear flow, 378/

Monte Carlo approach to simulation model of poly­

mer crystallization, 93 dynamic simulation method for melt­

ing, 94

Ν

Neutron diffraction advantages over X-ray diffraction, 75 atactic polyvinyl alcohol), 76-85 examples of crystal structure analysis

of crystalline polymers, 76

542

experimental, 78, 85-86 polyethylene-d4, 85-91 reflection intensities in crystalline poly­

mers, 75-76 rigid-body least-squares refinements,

79-80, 87-89 scattering lengths of atoms, 75f structure analysis for isotactic polyvi­

nyl alcohol), 81-85 structure refinement by X-ray data for

polyvinyl alcohol), 80-81 temperature parameters for polyethyl­

ene^, 89-91 unit cell dimensions for polyethylene-

d4, 86 See also Polyethylene-d4; Polyvinyl al­

cohol), atactic Neutron reflection (NR)

capillary wave broadening, 62 characteristics of polymer in melt-inter­

face studies, 61t characteristics of polymers in real-time

NR studies, 64t data collection, 60-61 diffusion studies, 63-64 interdiffusion of oligomeric styrene

into high molar mass deuterated polystyrene (dPS), 67, 69/

interfacial behavior of crystalline poly­mers, 58-59

interfacial width from fitting dPS-low density polyethylene (dPS-LDPE) NR profiles, 61

interfacial widths from semicrystalline polymers, 71-72

neutron scattering length density, 67-68

NR determined interfacial width, Flory-Huggins parameter, and in­terfacial tension, 62/

obtaining diffusion coefficient, 66-67 plot of shift in interface as function of

square root of time, 71/ polymer interface width as function of

annealing time, 66/ polymer-polymer interdiffusion, 64-67 polypropylene (PP)-polyethylene (PE)

system, 62-63 providing composition profile perpen­

dicular to interface, 57-58 reflectivity measurements, 61 reflectivity profiles for hydrogenated

polystyrene-dPS (hPS-dPS) bilayer in real time, 65/

sample preparation, 59-60 schematic illustration showing section

through polymer heating cell for studying crystalline polymer inter­faces, 60/

schematic of heatable cell used to study small-molecule ingress into high molecular weight polymers, 68/

small-molecule diffusion, 67-69 surface of thin layer d-iPP (deuterated

isotactic polypropylene) by α-step profiling instrument, 59/

total width of interface at longer times, 69

variation of interfacial width as func­tion of annealing time, 70/

Neutron scattering analogous expressions to X-ray scatter­

ing, 19 applications in polymer structure study,

18-19 average neutron scattering length den­

sity, 19 function of particle concentration de­

termining specific interactions be­tween components, 20

order in amorphous polymers, 329 regimes of interest for spherical parti­

cles, 20 scattering cross section, 19

Neutron scattering length density calculation, 67-68 time-dependent behavior of error func­

tion, 69, 70/ Neutron spin echo spectroscopy (NSE)

active sample area, 106-107 advantage in investigations of aperiodic

relaxation dynamics, 115 applications, 110, 112-115 beam polarization, 110 calculated neutron fluxes at various

points along neutron guide, 106/ comparison of typical scattering spectra

by inelastic neutron scattering in en­ergy and time domains, 113/

consequences of spin-incoherent scatter­ing, 114

design of NIST Center for Neutron Re­search (NCNR) NSE spectrometer, 104-107

inelastic neutron spectrometer, 103-104

main application measuring intermedi­ate coherent scattering function, 115

543

measuring intermediate scattering func­tion, 110

measuring quasi-elastic processes or processes at small energies, 112

motion of neutron spins in precession region in beam direction, 108/

motion of neutron spins perpendicular to beam direction, 108/

NCNR, 103-104 NCNR NSE operating characteristics,

105/ NSE signal as function of phase differ­

ence between incident and scattered beams, 111/

phase difference between two arms of spectrometer, 107-109

phase space diagram indicating region of accessibility, 104/

polarization of scattered beam, 109 polarization of scattered beam at echo

point, 112 preliminary measurements of neutron

flux at end of guide with gold-foil technique, 105-106

principles, 107-110 quasi-elastic scattering law, 109 schematic, 108/ sources of incoherent scattering, 114 typical spectra by incoherent neutron

scattering for vibrational process, 114 NIST Center for Neutron Research

(NCNR). See Neutron spin echo spec­troscopy (NSE)

Nuclear reaction analysis, interface be­tween initially separated miscible mate­rials, 58

Nucleation definition, 202 ordered structures in cooling crystalliz­

able polymer, 202 rheological studies, 203 study by small-angle X-ray scattering

(SAXS) and wide-angle X-ray scat­tering (WAXS), 202-203

See also Transient rotator phase in­duced nucleation in At-alkanes

Nucleation phenomena in homopolymer melts Cahn-Hilliard plot estimating effective

diffusion coefficient, (£>eff), 210/ conformation requirements for crystalli­

zation, 213-214 coupling density to molecular struc­

tural order parameters, 214

degree of crystallinity versus crystalliza­tion time for poly(propylene) (PP) at 110°C, 205-206

difficulty distinguishing between spino­dal decomposition, nucleation, and growth, 214-215

difficulty separating nucleation from growth, 206

effective diffusion coefficient, 211 experimental, 203-205 generic phase diagram for polymer

melt with buried liquid-liquid pre­dicted by SAC model, 213/

growth rate constant [R(q)], 209-210 integrated intensity data for crystalliza­

tion in small-angle X-ray scattering/ wide-angle X-ray scattering/differen­tial scanning calorimetry (SAXS/ WAXS/DSC), 209/

minimalist phenomenological model, 213-214

nucleating agents effective in metasta-ble super-cooling zone, 215

plot of D e f f versus 1/Γ allowing calcula­tion of spinodal temperature, 211/ 212

possible interpretations of SAXS/ WAXS data, 208-209

PP sample, 203 SAXS/WAXS/extrusion suggesting pro­

cess resembling spinodal decomposi­tion, 212-213

SAXS/WAXS scattering patterns dur­ing extrusion of iPP, 207/

schematic of SAXS/WAXS/extrusion experimental set-up, 205/

simultaneous SAXS/WAXS/DSC mea­surements, 204-205

simultaneous SAXS/WAXS/extrusion measurements, 203-204

stability limits and thermodynamic melting points, 212/

thermodynamic driving force for growth, 211

variation in scattered intensity follow­ing a quench, 209

Octadecane. See Transient rotator phase induced nucleation in w-alkanes

Oligomeric styrene real-time neutron reflection studies, 64/ small-molecule diffusion, 67-69

544

Optical probe study. See Solutionlike and meltlike solutions of hydroxypropylcel­lulose (HPC)

Order, short-range methods for detection, 328-329 See also Polycarbonate-ionomer

blends Order-disorder and order-order transi­

tions changes in equilibrium microdomain

morphology of block copolymers with addition of polystyrene (PS), 499-502

commercial grades of polystyrene-fr-polyisoprene-6-polystyrene copoly­mers (V4113 and V4411D), 498

dynamic storage modulus (G') as func­tion of temperature for V4411D/PS2 and V4113/PS2 mixtures, 503/, 504/

methods to determine order-disorder transition temperature (Τ0ΌΤ), 498

molecular characteristics of polymers in study, 498i

order-disorder transition of pure block copolymers, 497

phase behavior of styrene-butadiene diblock copolymers with homopoly­mers, 497

phase diagram of V4113/PS mixtures, 511/

phase diagram of V4411D/PS mixtures, 510/

phase diagrams of mixtures of SIS triblock copolymers and PS homo­polymers, 509

sample preparation, 498 scattering parameters as function of in­

verse temperature for 70/30 V4113/ PS2 mixture, 505, 507/

scattering parameters as function of in­verse temperature for 70/30 V4411D/ PS2 mixture, 505, 508/ 509

solubility of PS in microdomains of SIS triblock copolymer, 499

synchrotron small-angle X-ray scatter­ing (SAXS) profiles of 70/30 V4113/ PS2 mixture as function of tempera­ture, 506/

synchrotron SAXS profiles of V4113/ PS2 samples, 501/

synchrotron SAXS profiles of V4411D/ PS2 samples, 500/

synchrotron SAXS measurements, 498-499

transitions from rheology, 502-505 transitions from SAXS, 505-509 triblock copolymer-homopolymer mix­

tures showing no macrophase separa­tion, 509, 512

Order-disorder transition (ODT) in diblock copolymers changes in enthalpy at LCOT by differ­

ential scanning calorimetry (DSC), 262-263

Clapeyron equation, 265, 268 dependence of ordering on pre-anneal-

ing times and thermal history, 265 diblock P(d-S-fc-nBMA) with deute­

rium-labeled styrene and n-butyl methacrylate, 263

DSC scan of 85 Κ P(d-S-fc-nBMA) as 20°C/min scan rate, 265/

experimental, 263 film thickness as function of Τ for

P(d-S-fc-nBMA) film, 267, 268/ lower critical ordering transition

(LCOT) temperature as function of hydrostatic pressure by small-angle neutron scattering (SANS), 262

measurement of first-order thermody­namic parameters, 262

pressure dependence of ordering transi­tion, 262

repulsive interactions at temperatures below ODT, 261

SANS profiles of P(d-S-ft-nBMA) co­polymer as function of momentum transfer, 264/

segmental mixing at temperatures above ODT, 261

volume thermal expansion coefficient, 267 X-ray reflectivity profiles as function of

scattering wavevector at different temperatures, 266/

Ordering kinetics for block copolymers between hexagonally packed cylindrical

(HEX) and body centered cubic (BCC) microdomains for polysty-rene-fr/ocfc-polyisoprene-fr/oc/c-polystyrene (Vector 4111), 471-472

change in storage modulus (Gr) with time for samples, 480, 481/

changes of first order peak intensity with temperature during heating, 476, 477/

changes of interdomain spacing (D) with temperature during heating, 476, 478/

545

changes of small-angle X-ray scattering (SAXS) profiles, first-order peak in­tensity, and domain spacing with time when temperature jumped from BCC to HEX microdomains for aligned polystyrene-6/ocA>polyiso-prene (SI-C1), 490, 493/

changes of SAXS profiles, first-order peak intensity, and domain spacing with time when temperature jumped from BCC to HEX microdomains for aligned Vector 4111, 490, 491/

changes of SAXS profiles, first-order peak intensity, and domain spacing with time when temperature jumped from HEX to BCC microdomains for aligned SI-C1, 490, 492/

changes of SAXS profiles, first-order peak intensity, and domain spacing with time when temperature jumped from HEX to BCC microdomains for aligned Vector 4111, 487, 489/ 490

2D-SAXS patterns for aligned sample of SI-C1 at various temperatures, 480, 483/

2D-SAXS patterns for aligned sample of Vector 4111 at various tempera­tures, 480, 482/

differences in temperature dependence of G' for Vector 4111 and SI-C1, 487

experimental materials, Vector 4111 and SI-C1, 472

first-order peak intensity variations with temperature for aligned sam­ples, 487, 488/

molecular characteristics of block copolymers in study, 473?

ordering kinetics by rheology and syn­chrotron SAXS, 472

order-to-order transition kinetics, 494 order-to-order transition temperature

(TOOT), 476 rheological properties, 474 SAXS profiles between 170 and 200°C

for as-cast Vector 4111 and SI-C1, 475/

synchrotron SAXS, 474 temperature sweep experiment of G'

during heating, 476, 479/ 480 temperature variation in G' for aligned

samples of Vector 4111 and SI-C1, 485, 486/ 487

transformation from one ordered struc­ture to another, 471

transition from BCC to HEX while lowering temperature different from original, 484

transition mechanism between HEX and BCC for styrene-isoprene (SI) diblock and SIS triblock, 484

Order-order transition. See Thermorevers­ible order-order transition of block co­polymer

Organic solvents. See Polymers in organic solvents and supercritical fluids

Oriented polymers. See Magnetically ori­ented microtubule biopolymers

Phase diagrams chimney region determination by polar­

ized optical microscopy, 370, 371/ mixtures of styrene-isoprene-styrene

triblock copolymers and polystyrene homopolymers, 509, 510/ 511/

See also Flow behavior of liquid-crystal­line polymer (LCP) solutions

Phase separation kinetics during shear in polymer blends behavior of modified blend at shallow

and deeper quench depth, 415, 430 block copolymers modifying phase be­

havior and morphology, 406 collision frequency between droplets,

407 compatibilizing agents, 406 competing effects of droplet breakup

and coalescence, 406-408 immiscibility of commercial polymer

blends, 408 influence of simple shear fields on

blend morphology, 406 light-scattering patterns for modified

blend at deeper quench depth, 426/ 427/

light-scattering patterns for modified blend at shallow quench depth, 422/ 423/

light-scattering patterns for pure blend at deeper quench depth, 418/ 419/

light-scattering patterns for pure blend at shallow quench depth, 412/ 413/

model blend of polystyrene (PS) and polybutadiene (PB), 409

obtaining scattering vector -depen­dent growth rate, 414-415

oven cooling reproducibility, 411/ perdeuterated PS (PSD) as compatibil­

izing agent, 409 546

procedures for shear experiments, 409, 414

pure blend at deeper quench depth, 415

pure blend at very shallow quench depth, 414-415

rate of phase separation for diblock co­polymer modified blend, 430

relationship between interdroplet layer draining time and applied shear rate in modified blend, 432

relationship between phase separation kinetics and mechanical deformation during processing, 408

schematic of experimental procedure, 410/

shear rates insufficient to completely suppress coarsening, 432

summary of experimental observations, 430-432

suppression of and then increase of coarsening in pure blend, 432

time dependence of scattering intensit­ies at various q values for modified blend at deeper quench depth, 428/ 429/

time dependence of scattering intensit­ies at various q values for modified blend at shallow quench depth, 424/ 425/

time dependence of scattering intensit­ies at various q values for pure blend at deeper quench depth, 418/419/

time dependence of scattering intensit­ies at various q values for pure blend at shallow quench depth, 416/ 417/

Physics incident wave polarized in χ and y di­

rection, 5 scattering, 3-5 scattering of electromagnetic radiation

by charged particle, 5 Pluronic micelles. See Micelles in aque­

ous solutions Polarization effects, geometry describing,

6/ Polarized-light optical microscopy (POM)

conservation of grain structure in block copolymer, 519, 522-523

dynamics of order-order transition by POM, 525-526

grain structure measurement, 516 POM images under crossed polarizers

for hex-cylinder and bcc-sphere phase, 521/

See also Thermoreversible order-order transition of block copolymer

Poly(aryl ether ether ketone) (PEEK) comparison between linear degree of

crystallinity (i>2,in) and weight degree of crystallinity (WC

DSC), 177, 178/ cooling from crystallization tempera­

ture to 100°C at -10°C/min, 182 crystal density of PEEK fractions, 173,

174/ degree of crystallinity from wide-angle

X-ray diffraction (WAXD) and crys­tal density during cooling, 183/

degree of crystallinity from WAXD data and crystal density during iso­thermal crystallization from melt, 181/

degree of crystallinity of PEEK frac­tions, 173

degree of crystallinity versus molecular weight of PEEK fractions, 184

differential scanning calorimetry (DSC) heating curves of PEEK fractions crystallized from glassy state, 168, 169/ 170/ 171

experimental, 168 high-temperature aromatic polymer,

167 hypotheses for endotherms in melting

curves of semicrystalline PEEK sam­ples, 167

influence of crystallization temperature on melting behavior of PEEK frac­tions, 168-173

isothermal crystallization, 179, 182 lamellar dimensions and invariant Q

during cooling, 183/ lamellar dimensions and invariant Q

during isothermal crystallization from melt, 180/

lamellar long spacings versus crystalliza­tion temperature, 173, 175/ 176/ 177

lamellar morphology of PEEK frac­tions, 173, 177, 179

melting temperatures of two endo­therms as function of annealing tem­perature, 171, 172/

morphology of semicrystalline PEEK fractions, 167-168

simultaneous time-resolved WAXD and small-angle X-ray scattering (SAXS) investigation of crystalliza­tion of low molecular weight PEEK from melt, 179, 182

547

thickness of amorphous regions versus (Mw)1/2 for annealing temperatures, 179, 180/

weight degree of crystallinity from WAXD and DSC data versus anneal­ing temperature, 174/

Polybutadiene (PB). See Phase separation kinetics during shear in polymer blends

Poly(E-caprolactone) (PCL) crystallizable in star block copolymers,

448-449 See also Crystallization of block

copolymers Polycarbonate-ionomer blends

blend preparation, 330 calculating spacing between chains and

correlation length, 331-332 coherent and incoherent scattering con­

tributions, 331 comparing WANS data for 50/50, 75/

25, and 25/75 blends of 9% LiSPS/PC (sulfonated polystyrene/polycarbo­nate) in two-phase and one-phase re­gions, 336-337

correlation length for packing of PC neighbored chains from experiment, 338i

experimental, 330-331 immiscibility of polystyrene (PS) and

PC at all temperatures, 332 materials, 330 methods to identify order in amor­

phous polymers, 328-329 miscibility of PC and SPS by copoly­

mer effect, 329 peak intensity, wide-angle neutron scat­

tering (WANS) versus PC weight percent, 332, 336/

testing thermal density fluctuations, 338, 339/

WANS curves of various blend compo­sitions of 9% LiSPS and PC in two-phase region of phase diagram, 332, 334/ 335/

WANS of 25/75 PS/PC blend, 334/ WANS of 75/25 and 50/50 PS/PC

blends, 333/ WANS profile for short-range order in

amorphous polymer, 331, 332/ WANS with spin polarization analysis,

330-331 width of WANS peak for measure of

level of component mixing, 338 Poly(diallyldimethylammonium chloride)

(PDADMAC1). See Self-assembly of

ionic surfactants in oppositely charged polyelectrolyte gels

Polyelectrolyte gels. See Self-assembly of ionic surfactants in oppositely charged polyelectrolyte gels

Poly(ether ester) (PEE) analysis of longitudinal structure,

49-52 analysis of transverse structure, 52-54 basic small-angle X-ray scattering

(SAXS) definitions and concept, 43-44

2D chord distribution, 48 definitions of special projections, 44 discrimination between hard segment

and hard domain, and soft segment and soft domain, 43/

domain heights from longitudinal struc­ture, 50-52

evaluation of transverse structure, 47-48

general longitudinal evaluation steps, 49-50

general transverse evaluation steps, 52-54

interface distribution functions (IDFs), 49-50

invariant, 44, 46 longitudinal structure, 46-47 model function fits to IDFs, 52/ PEE 1000/57 ID scattering curves and

IDFs, 51/ PEE 1000/57 average domain heights

and height distributions, 53/ principal SAXS patterns, 48-49 projections and chord distributions, 54/ projections and sections, 44-46 projections of fiber patterns, 44-48 relations between intensity projections

and corresponding structural features in physical space from fibers, 45/

SAXS pattern of PEE 1000/57 at elon­gation e = 0.88, 50/

straining and relaxation experiments, 43 structural parameters and diameter dis­

tributions of soft domain needles, 55/ thermoplastic elastomers, 42 transverse structure, 47 typical SAXS patterns for samples un­

der strain, 49/ Polyethylene (PE)

change of amorphous layer thickness with temperature, 147/

evidence for surface crystallization and melting, 146-148, 150

548

evolution with time of crystal thickness and inner surface, 146/

interface distribution function at begin­ning and end of isothermal crystalli­zation at 121°C, 145/

interface distribution function at Τ = 88°C for samples crystallized at dif­ferent temperatures, 148/

interface distribution functions after iso­thermal crystallization and subse­quent cooling, 147/

kinetics of crystal thickening, 146 melt interface studies, 61/ neutron reflection (NR) of polypropyl­

ene (PP)-PE system, 62-63 NR measured interfacial width, Flory-

Huggins parameter, and interfacial tension, 62/

See also Crystallization mechanisms of polymers; Ethylene-based copoly­mers

Polyethylene-d4

experimental, 85-86 intensity distribution on equator, 86/ mean-square librational motion, 91 molecular framework on c-projection,

88/ neutron diffraction experiments, 85-86 rigid-body least-squares refinements,

87-89 temperature dependence of angle 0,

the azimuthal angle of molecule with respect to α-axis, 89/

temperature dependences of cell dimen­sions a and b, 87/

temperature dependences of librational displacement by rigid-body tempera­ture factor, 90/

temperature dependences of transla-tional displacements by rigid-body temperature factor, 90/

temperature parameters, 89-91 unit cell dimensions, 86 See also Neutron diffraction

Polyethylene oxide) (PEO) crystallizable in star block copolymers,

448-449 See also Crystallization of block copoly­

mers; Isothermal thickening and thin­ning processes in poly(ethylene ox­ide) (PEO); Micelles in aqueous solutions

Poly(ethylene oxide) (PEO) blends amorphous PEO and diluent distribu­

tions in PEO/EMAA (ethylene-meth-

acrylic acid) and PEO/SHS (styrene-hydroxystyrene) blends, 225/

average long period, 221 crystallization and solid-state micro-

structure of semicrystalline blends, 218-219

experimental and calculated invariant (β) , 220-221

experimental and calculated β for PEO/EMAA blends, 223/

experimental growth rates for neat PEO and blends with poly(vinylace-tate) (PVAc) and EMAA, 227/

experimental materials, 219 growth rate versus TC relationship for

blends with high TG diluent poly­mers, 229/

Lorentz-corrected SAXS intensities as function of scattering vector for PEO and PEO/EMAA blends, 223/

mobility of amorphous component at Tc, 224

one-dimensional correlation function, 220

PEO/EMAA and PEO/SHS blends, 222, 224

PEO/PMMA (poly(methyl methacry­late)) blends, 221

PEO/PVAc blends, 221 plot of average long period, lamellar

thickness, and amorphous layer thick­ness versus crystallization time for PEO/SHS blend, 230/

small-angle X-ray scattering (SAXS) calculations, 220-221

spherulitic growth, 226-228 spherulitic growth rate procedure, 221 static and time-resolved SAXS experi­

ments, 219-220 time-resolved SAXS/WAXS experi­

ments, 228, 230 Polymer blends. See Phase separation ki­

netics during shear in polymer blends; Poly(ethylene oxide) (PEO) blends

Polymer interfaces. See Neutron reflec­tion (NR)

Polymer lamellar systems corrections to raw data, 14-17 domains with lamellar surface normals

parallel to scattering vector, 13-14 experimental geometry for scattering

from isotropic sample, 13, 15/ scattering, 13-18

Polymer-polymer interdiffusion diffusion coefficient value, 66-67

549

neutron reflection studies, 64-67 reflectivity profiles for hydrogenated

polystyrene (hPS)-deuterated poly­styrene (dPS) bilayer, 65/

width of polymer interface as function of annealing time, 66/

Polymers in organic solvents and super­critical fluids correlation length and radius of gyra­

tion over wide range of pressure and temperature, 322/

dominating attractive interactions in poor solvent regime, 318

effect of pressure on thermodynamic state of PS/AC-d (deuterated ace­tone) solutions, 322/

effect of temperature on thermody­namic state of PDMS-C02 solution, 323/

experimental, 318-320 obtaining radius of gyration and corre­

lation length of concentration fluctu­ations, 319

PDMS/CO2 in good solvent domain at theta temperature, 325/

polystyrene (PS-h) samples, 320 protonated poly(dimethylsiloxane)

(PDMS) samples, 320 SANS experiments, 320 SANS experiments on PDMS in C0 2

for comparison to chain dimensions in organic solvents, 323-324

scattering structure factor, 319 small-angle neutron scattering (SANS)

data collection, 318-319 solvent strength of supercritical fluids

tunable with system density, 326 systems in good solvent domains, 326 temperature variation of correlation

length for PS-h in deuterated cyclo-hexane (CH-d), 321/

variation of correlation length as func­tion of (T-Tc) for PDMS/CO2 and PS/CH-d, 324/

Poly(methacrylic acid) (PMAA). See Self-assembly of ionic surfactants in op­positely charged polyelectrolyte gels

Poly(methacrylic acid)/poly(iV-isopropyl-acrylamide) (P(MAA/NIPAM)). See Self-assembly of ionic surfactants in op­positely charged polyelectrolyte gels

Poly(methyl methacrylate) (PMMA). See Poly(ethylene oxide) (PEO) blends

Polypropylene (PP) neutron reflection (NR) of PP-

polyethylene (PE) system, 62-63 See also Nucleation phenomena in ho­

mopolymer melts Polypropylene, deuterated

melt interface studies, 61/ NR measured interfacial width, Flory-

Huggins parameter, and interfacial tension, 62/

Polypropylene, isotactic (iPP). See Crys­tallization and melting behavior of met­allocene isotactic poly(propylene) (m-iPP); Crystallization mechanisms of polymers; Isotactic polypropylene (iPP)

Polypropylene, syndiotactic. See Crystalli­zation mechanisms of polymers; Syndio­tactic polypropylene (sPP)

Poly(propylene oxide) (PPO). See Mi­celles in aqueous solutions

Polystyrene (PS) amorphous block in triarm star block

copolymer, 449-450 See also Crystallization of block copoly­

mers; Order-disorder and order-order transitions; Phase separation ki­netics during shear in polymer blends; Polycarbonate-ionomer blends

Polystyrene, deuterated melt interface studies, 61/ NR measured interfacial width, Flory-

Huggins parameter, and interfacial tension, 62/

real-time neutron reflection studies, 64/

reflectivity measurements, 61 small-molecule diffusion, 67-69

Polystyrene, hydrogenated (hPS), real­time neutron reflection studies, 64/

Polystyrene-6/oc/c-butadiene copolymers. See Grain size of lamellar styrene-butadiene block copolymers; Micro-phase separation transitions of polysty-rene-co-butadiene copolymer

Polystyrene-fr/ocfc-polyisoprene. See Or­dering kinetics for block copolymers; Thermoreversible order-order transi­tion of block copolymer

Polystyrene-b/oc/c-polyisoprene-b/oc/c-polystyrene. See Order-disorder and or­der-order transitions; Ordering kinet­ics for block copolymers

Polystyrene ionomers, lightly sulfonated. See Polycarbonate-ionomer blends

550

Poly(vinylacetate) (PVAc). See Polyeth­ylene oxide) (PEO) blends

Polyvinyl alcohol), atactic crystal structure models, 76/ crystal structures at 100 K, 200 K, and

room temperature, 82/ experimental, 78 hydrogen bonding networks, 81/ intramolecular hydrogen bonds in iso­

tactic sequence, 84/ molecular framework on b-projection

used in rigid-body least-squares re­finements, 80/

neutron difference synthesis, 83/ neutron diffraction experiments, 78

neutron intensity distributions on equa­tor, 77/ rigid-body least-squares refinement,

79- 80 structure analysis using neutron data,

81-85 structure refinement by X-ray data,

80- 81 temperature dependence of unit cell pa­

rameters, 79/ See also Neutron diffraction

Poor solvent regime dominating attractive interactions, 318 See also Polymers in organic solvents

and supercritical fluids Porod's Law

asymptotic behavior of scattering cross-section in limit of large scattering angles, 142

evaluation of transverse structure, 47 extrapolation, 17 predicting fall off in intensity, 16 scattering cross-section versus Κ for

larger Κ range, 20, 21/ Pressure-induced transitions. See Micro-

phase separation transitions of polysty-rene-6/oc/:-butadiene copolymer

Primary nucleation, 202 Projections

concept for analyzing 2D-small-angle X-ray scattering (SAXS) patterns with fiber symmetry, 54-55

definitions of special, 44 fiber patterns, 44-48 invariant, 46 longitudinal structure, 46-47 sections, 44-46 transverse structure, 47-48

Pycnometer, specific volume determina­tion, 516-517

Q Quasi-elastic light scattering spectroscopy

(QELSS) probe diffusion technique, 298 See also Solutionlike and meltlike solu­

tions of hydroxypropylcellulose (HPC)

Radius of gyration organic solvents, 318 using small-angle neutron scattering

(SANS) high-concentration isotope labeling method, 319

See also Polymers in organic solvents and supercritical fluids

Relaxation rates relationship to viscosity, 308, 311 See also Solutionlike and meltlike solu­

tions of hydroxypropylcellulose (HPC)

Rheology order-disorder and order-order transi­

tions, 502-505 See also Flow behavior of liquid-crystal­

line polymer (LCP) solutions; X-ray rheology of structured polymer melts

Rigid-body least-squares refinement polyethylene-d4, 87-89 polyvinyl alcohol), 79-80

Rutherford back scattering, interface be­tween initially separated miscible mate­rials, 58

S Scattering

collection of atoms, 6-9 corrections to raw data, 14-17 Fourier transforms, convolution, and

correlation, 9-13 geometry for constructive interference,

8/ geometry from two point scatterers, If geometry where kx and ks making angle

θ with dashed line, 8/ net amplitude of scattered waves, 7-8 neutron, 18-20 phenomenon, 3 physics, 3-5 polymer lamellar systems, 13-18 recommended references, 2-3 several planes of constant phase travel­

ing in direction toward scattering cen­ter, 4/

551

structural parameters from correlation function, 17-18

Scattering invariant (Q) changes due to perfecting or melting,

161, 164 equation, 155 Q versus temperature during melting,

161, 163/ Q versus time during isothermal crystal­

lization, 161, 163/ See also Crystallization and melting

behavior of metallocene isotactic poly(propylene) (m-iPP)

Scattering length, definition, 3-4 Scattering power, definition, 44 Scattering vector (K) description, 7 Self-assembly of ionic surfactants in oppo­

sitely charged polyelectrolyte gels charge density effects of polyelectro­

lyte chains on complex structures, 247-255

charge density of P(MAA/NIPAM) (methacrylic acid/AMsopropylacry-lamide) gel-2C12DA (surfactant) sys­tems, 252-255

charge density of P(MAAZNIPAM) gel-C12TA (surfactant) systems, 247-252

complex formation with highly ordered structures, 244-245

connected model by Luzzati group, 240-250

crosslinker density effect of polyelectro­lyte gels on structures of complexes, 257, 259

disconnected model by Lindblom group, 249, 250

experimental materials, 245-246 flexibility effect of polyelectrolyte

chains on structures of complexes, 255, 256/

gel preparation, 246 gel-surfactant complexation, 246 SAXS profiles of PDADMAC1 (diallyl-

dimethylammonium chloride) gel-SCmS and -SQS (surfactants) com­plexes, Pm3n cubic structure, 255, 258/

SAXS profiles of PDADMAC1 gel-SCUS and -SC,2S complexes, 2D hex­agonal close packing of cylinders, 256/

SAXS profiles of PMAA gel-C12TA complex based on connected unit cell model, 247, 248/

SAXS profiles of PMAA gel-C12TA complex based on disconnected unit cell model, 250, 251/

SAXS profiles of PMAA gel-C12TA complexes with different cross-linker density of PMAA gel network, 257, 258/

SAXS profiles of P(MAA/NIPAM) gel-C12TA complexes at 100% and 75% charge content in P(MAA/NI-PAM) chains, 252, 253/

SAXS profiles of P(MAA/NIPAM) gel-C12TA complexes at 67% and 50% charge content in P(MAA/NI-PAM) chains, 252, 254/ 255

SAXS profiles of P(MAA/NIPAM) gel-Ci2TA complexes in inserted or­thorhombic unit cell model, 250, 251/ 252

SAXS'profiles of P(MAA/NIPAM) gel-C16TA and -Q 4 TA complexes with 67% charge content of copoly­mer chains, 252, 253/

tail length effect of surfactants on struc­tures of complexes, 255, 257

volume contraction of polyelectrolyte gel in surfactant, 247

X-ray scattering measurements, 247 Semicrystalline polymers

angle between lamellar reflections, 29, 30/

azimuthal scan through lamellar reflec­tions for calculating tilt-angle of la­mellar planes, 30/

characteristics describing lamellar re­flections, 26i

characteristics of lamellar reflections, 26-31

correlation function calculated from la­mellar intensity projected onto z-axis, 28f

elliptical characteristics of small-angle-scattering (SAS) reflections, 34, 36

experimental, 25-26 explanations for observed curvature,

38, 39/ Guinier plot of intensity of lamellar re­

flection as function of x, 32/ integrated intensity, 29 lamellar reflections in SAS patterns, 25 lamellar spacing, 26, 29 macroscopic roughness, 59/ neutron reflection for interfacial

widths, 71-72 neutron reflection (NR), 58-59

552

parameters describing 2D fit in ellip­tical coordinates, 36r

parameters describing SAS data, 40 plot of L(0)2 vs. tan2(0) where theta is

angle between scattering vector and ζ-axis and L is periodicity of tilted la­mellar planes parallel to fiber axis, 35/

SAS including neutron scattering, 25 scan along fiber-axis through lamellar

reflections, 27/ two-dimensional fit in elliptical coordi­

nates, 36, 37/ two-dimensional small-angle X-ray scat­

tering (SAXS) pattern from spun, drawn, and heat-set nylon fiber, 27/

uses of SAXS, 25 variations in axial positions of lamellar

peak with JC , 31, 33/ variations in axial widths of lamellar

peak with JC , 31, 32/ widths of lamellar peaks, 29, 31

Separation transitions. See Microphase separation transitions of polystyrene-6/oc/:-butadiene copolymer

Shear light scattering. See Phase separa­tion kinetics during shear in polymer blends

Short-range order. See Polycarbonate-ionomer blends

Simulations of melting transitions crystal thickening, 100, 101/ dynamic Monte Carlo approach, 93 energy as function of temperature over

20 independent molecules, 98/ examples of final conformations of

chain model P1000 at different tem­peratures, 96/

examples of starting geometries for melting simulations of chain model P300, 98/

melting simulations, 97-100 melting simulations of chain model

P300, 99/ methods, 94 Monte Carlo approach, 94 normalized energy and thickness as

function of temperature for homo­polymer chain model P1000, 95/

plots of melting temperature as func­tion of reciprocal lamellar thickness (L) for three heating rates, 99/

reproducing experimental behavior, 100, 102

simulations of longer chains, 94-97

Small-angle neutron scattering (SANS). See Flow behavior of liquid-crystalline polymer (LCP) solutions; Micelles in aqueous solutions; Order-disorder tran­sition (ODT) in diblock copolymers; Polymers in organic solvents and super­critical fluids

Small-angle scattering (SAS) elliptical characteristics, 34-36 experimental for semicrystalline poly­

mers, 25-26 explanations for observed curvature for

semicrystalline polymers, 38, 39/ including neutron scattering, 25 parameters describing semicrystalline

polymers, 40 two-dimensional fit in elliptical coordi­

nates, 36, 37/ See also Semicrystalline polymers

Small-angle X-ray scattering (SAXS) analysis of longitudinal structure,

49-52 analysis of transverse structure, 52-54 basic definitions and general concept,

43-44 2D chord distribution, 48 discriminating between hard segment

and hard domain, and soft segment and soft domain, 43/

dynamics of order-order transition by SAXS, 525-526

evaluation of transverse structure, 47- 48

invariant, 44, 46 investigating lamellar structure in two

phase systems, 153-154 longitudinal structure, 46-47 order-disorder and order-order transi­

tions, 505-509 order in amorphous polymers, 329 pattern from spun, drawn, and heat-set

nylon fiber, 27/ poly(ether ester) (PEE) materials, 42 principle SAXS patterns of PEE,

48- 49 projections of fiber patterns, 44-48 straining and relaxation experiments,

43 transverse structure, 47 typical patterns from PEE samples un­

der strain, 49/ uses in semicrystalline polymers, 25 See also Crystallization and melting

behavior of metallocene isotactic poly (propylene) (m-iPP); Crystalliza-

553

tion mechanisms of polymers; Crys­tallization of block copolymers; Eth­ylene-based copolymers; Nucleation phenomena in homopolymer melts; Order-disorder and order-order transitions; Poly(aryl ether ether ke­tone) (PEEK); Self-assembly of ionic surfactants in oppositely charged po­lyelectrolyte gels; Thermoreversible order-order transition of block co­polymer

Small molecule diffusion, neutron reflec­tion, 67-69

Solutionlike and meltlike solutions of hy­droxypropylcellulose (HPC) behaviors of viscosity in solutionlike re­

gime and meltlike regime, 299 broadly accepted family of polymer dy­

namics models, 314 broad time scale regime, small-probe

fast mode, 312 characteristic behaviors for stretching

components, 304 characteristic properties of slow mode

regime, 311 common properties of intermediate

time scale regime, 312 conditions for probe diffusion experi­

ment using quasi-elastic light-scatter­ing spectroscopy (QELSS), 300

dependences of optical probe spectra on polymer concentration, scattering vector, and probe diameter, 301-314

dependences of probe spectra on wave vector, 306

existence of concentration-independent crossover probe diameter, 314

experimental methods, 299-301 fast and slow modes, 301 field-correlation function, 301 HPC solutions, 299-300 implications of probe-size crossover di­

ameter, 313-314 meltlike regime, 306-308 nature of polymer motion in semidilute

and concentrated solutions, 298 nonlinear least-squares method quanti-

tating spectral lineshape, 301 optical probes, 300 parameters showing most dramatic tran­

sitions at transition concentration, 306-308

physical interpretation of mode struc­ture, 311-313

probe diffusion by QELSS and calcula­tions, 300-301

probe diffusion of aqueous high molec­ular weight HPC solutions, 298

reality of solutionlike-meltlike transi­tion, 313

relationship between viscosity and re­laxation rates, 308, 311

relationship between viscosity and re­laxation rates, fast mode behavior, 310/, 311

relationship between viscosity and re­laxation rates, slow mode behavior, 310/

relaxation times as replacements for re­laxation pseudorates, 304

slow and fast mode relaxation pseu­dorates against concentration for probes, 307/

slow relaxation pseudorate dependence on probe diameter, 303/

solutionlike regime, 302-306 stretched exponential concentration de­

pendence of viscosity of HPC solu­tions, 298-299

stretching exponent against HPC con­centration for probes of various di­ameters, 309/

stretching exponents against polymer concentration for probes of various diameters, 305/

Solvents. See Polymers in organic sol­vents and supercritical fluids

Spinodal decomposition, plot of effective diffusion coefficient versus 1/Γ, 211-212

Star block copolymers. See Crystallization of block copolymers

Structure. See Micelles in aqueous solu­tions

Structure analysis poly(vinyl alcohol) using neutron data,

81-85 See also Neutron diffraction

Structured polymer melts. See X-ray rhe­ology of structured polymer melts

Styrene-butadiene block copolymers. See Grain size of lamellar styrene-butadiene block copolymers

Styrene-hydroxystyrene (SHS) copoly­mers. See Poly(ethylene oxide) (PEO) blends

Styrene oligomers real time neutron reflection studies, 64/ small molecule diffusion, 67-69

554

Sulfonated polystyrene (SPS) ionomers. See Polycarbonate-ionomer blends

Supercritical fluids. See Polymers in or­ganic solvents and supercritical fluids

Supramolecular structures. See Self-assem­bly of ionic surfactants in oppositely charged polyelectrolyte gels

Surfactants. See Micelles in aqueous solu­tions

Surfactants, ionic. See Self-assembly of ionic surfactants in oppositely charged polyelectrolyte gels

SURF reflectometer, neutron reflection experiments in real time, 63-64

Syndiotactic poly(propene-co-octene) (sP(P-co-O)) Avrami times of crystallization in de­

pendence on crystallization tempera­ture, 144/

changes during isothermal crystalliza­tion and subsequent cooling, 143/

kinetics of primary crystallization, 142-145

relations between inverse crystallite thickness, crystallization tempera­ture, and melting peak, 144/

See also Crystallization mechanisms in polymers

Syndiotactic polypropylene (sPP) Avrami times of crystallization in de­

pendence on crystallization tempera­ture, 144/

kinetics of primary crystallization, 142-145

relations between inverse crystallite thickness, crystallization tempera­ture, and melting peak, 144/

See also Crystallization mechanisms of polymers

Temperature-induced separation. See Mi­crophase separation transitions of poly-styrene-6/oc/c-butadiene copolymer

Temperature jump light scattering (TJLS), 414

Temperature parameters dependence of mean-square transla-

tional and librational displacements of polyethylene, 89-91

mean-square librational motion, 91 Thermoreversible order-order transition

of block copolymer conservation of grain structure, 519,

522-523

dynamics of order-order transition (OOT) as observed by small-angle X-ray scattering (SAXS) and polar-ized-light optical microscopy (POM), 525-526

grain structure by polarized-light opti­cal microscopy, 516

mechanism of transition, 523 OOT from bcc-sphere to hex-cylinder,

522 POM images for block copolymer un­

der crossed polarizers, 521/ preparation of polystyrene-b/ocfc-poly-

isoprene (PS-PI) sample, 516 relaxation time as function of Δ Γ for

T-jump and T-drop processes, 526, 527/

SAXS patterns with 2D detector for single grain of hex-cylinder and bcc-sphere, 523, 524/

SAXS profiles of block copolymer in cooling process, 518/

small-angle X-ray scattering (SAXS) measurements, 516

specific volume by conventional pyc-nometer, 516-517

temperature dependence of area under first-order scattering maximum, 519, 520/

thermoreversible OOT and OOT tem­perature, 517-519

time-resolved POM micrographs after T-jump from hex-cylinder to bcc-sphere phase, 526, 528/

time-resolved USAXS study at OOT, 526, 529

ultra-small-angle X-ray scattering (USAXS) measurement, 516

volume change in macroscopic and mi­croscopic dimensions at OOT, 525

Thermotropic liquid-crystalline polymers (TLCP) commercial TLCP in diverging channel

flow, 381-382 interest in, 374-375 model TLCPs in shear flow, 376-381 poorly understood in comparison to ly­

otropes, 375-376 potential for broader technological ap­

plications, 375 transition from tumbling to flow-align­

ment, 375 See also Molecular orientation of ther­

motropic liquid-crystalline polymers (TLCP) under flow

555

Theta solvent domain. See Polymers in or­ganic solvents and supercritical fluids

Thickening and thinning processes, iso­thermal. See Isothermal thickening and thinning processes in poly(ethylene ox­ide) (PEO)

Transient rotator phase induced nucle­ation in n-alkanes crossover from stable to metastable to

transient rotator phase, 239 equilibrium orthorhombic rotator

phase for alkanes with n-odd, 235 equilibrium rotator phases below melt­

ing point, 233 even-odd effect in crystal structures

and melting points, 233 experimental, 234 histograms of lifetime of metastable ro­

tator phase in C 1 8, 237, 238/ metastable phases, 232-233 peak positions of rotator phase just be­

low freezing temperature for n-odd alkanes and C t 8 , 236/

phase diagram of n-alkanes, 234/ </-scans in triclinic phase and transient

rotator phase for d 8 , 235/ quenches for investigating transforma­

tion kinetics of C 1 8 transient phase, 236, 237/

time-dependent X-ray scattering, 234

transformation of triclinic phase to rota­tor phase on heating, 239

transient phase description, 232-233 transient phase determining crystalliza­

tion temperature, 239-240 transition temperatures with respect to

freezing temperature, 238/ 239 Transitions. See Order-disorder and or­

der-order transitions; Order-disorder transition (ODT) in diblock copoly­mers; Ordering kinetics for block copolymers

Transmission electron microscopy (TEM). See Grain size of lamellar sty­rene-butadiene block copolymers

Transverse structure evaluation, 47-48 general evaluation steps, 52-54 See also Poly(ether ester) (PEE);

Small-angle X-ray scattering (SAXS) Triarm star block copolymers

molecular characteristics, 449f See also Crystallization of block

copolymers

Triblock copolymers. See Order-disorder and order-order transitions

Triblock copolymer surfactants. See Mi­celles in aqueous solutions

Two-arm poly(ethylene oxide). See Iso­thermal thickening and thinning pro­cesses in poly(ethylene oxide) (PEO)

Two-dimensional chord distribution, defi­nition, 48

Ultra-small-angle X-ray scattering (USAXS) time-resolved study at order-order

transition of block copolymer, 526, 529

USAXS experiments, 437-438 USAXS results for styrene-butadiene

block copolymers S12B10 and SB5, 444i

See also Grain size of lamellar sty­rene-butadiene block copolymers; Thermoreversible order-order transi­tion of block copolymer

Viscosity relationship to relaxation rates, 308,

311 See also Flow behavior of liquid-crystal­

line polymer (LCP) solutions; Mi­celles in aqueous solutions; Solu­tionlike and meltlike solutions of hydroxypropylcellulose (HPC)

Volume change in macroscopic and microscopic

dimensions at order-order transition, 525

thermodynamic parameter, 262 See also Order-disorder transition

(ODT) in diblock copolymers; Ther­moreversible order-order transition of block copolymer

W

Wide-angle neutron scattering (WANS) experimental with spin polarization

analysis, 330-331 peak intensity versus polycarbonate

content, 332, 336/ presence of short-range order in amor­

phous polymer, 331, 332f

556

profiles for PC/PS (polycarbonate/poly­styrene) blends, 333/, 334/

profiles for PC/sulfonated polystyrene (SPS) blends, 332/ 334/ 335/ 336/

short-range order of poly(methyl meth­acrylate), 329

width of peak for measure of mixing between blend components, 338

See also Polycarbonate/ionomer blends Wide-angle X-ray diffraction (WAXD)

comparison of invariant Q from small-angle X-ray scattering (SAXS) and crystallinity from WAXD during iso­thermal crystallization of ethylene-based copolymers, 198,199/

crystallinity (WAXD) versus crystalliza­tion temperature for ethylene-based copolymers with various degrees of branching, 195,196/

degree of crystallinity from WAXD and crystal density during cooling for poly(aryl ether ether ketone) (PEEK), 183/

degree of crystallinity from WAXD and crystal density during isothermal crystallization of PEEK from melt, 181/

isothermal crystallization and melting of metailocene-polyethylene (m-PE) using simultaneous SAXS and WAXD techniques, 189

isothermal crystallization processes for three two-arm PEOs by simultane­ous synchrotron WAXD and SAXS, 121, 123/

SAXS and WAXD spectra for star block copolymers, 450-452

simultaneous time-resolved WAXD and SAXS investigation of crystalliza­tion of low molecular weight PEEK from melt, 179, 182

WAXD and SAXS patterns of two-arm PEO samples, 124-129

See also Crystallization of block copoly­mers; Ethylene-based copolymers; Isothermal thickening and thinning processes in poly(ethylene oxide) (PEO); Poly(aryl ether ether ketone) (PEEK)

Wide-angle X-ray scattering (WAXS) modifications in WAXS diffractograms

of isotactic poly(propylene) (iPP), 155 order in amorphous polymers, 329 real-time small-angle X-ray scattering

(SAXS) and WAXS, 154

time development of gamma crystals of metallocene iPP (m-iPP) during iso­thermal crystallization, 156, 159/

WAXS data collection, 155 WAXS intensity versus scattering angle

during initial eight minutes of m-iPP isothermal crystallization, 156,160/, 161

WAXS intensity versus scattering angle during melting of m-iPP after isother­mal crystallization, 161, 162/

See also Crystallization and melting behavior of metallocene isotactic poly (propylene) (m-iPP); Nucleation phenomena in homopolymer melts

X

X-ray diffraction (XRD) comparison to neutron diffraction, 75 difficulty determining polyethylene crys­

tal structure, 85 scattering lengths of atoms, 75i structure refinement of poly(vinyl alco­

hol), 80-81 X-ray rheology of structured polymer

melts achieving steady level of preferred ori­

entation, 398 alternative flow cell geometry explor­

ing three-dimensional consequences of flow field, 397/

alternative geometries, 395-396 building up map of flow behavior,

399-400 design of shear flow cells for in situ X-

ray scattering, 393-395 detector system recording X-ray scatter­

ing patterns in time-resolving man­ner, 396

experimental procedures, 393-396 factors influencing response of liquid

crystal polydomain texture to shear flow, 401-402

flow and scattering geometry for Couette style flow cell, 396/

generation of single director distribu­tion, monodomain, 391-392

materials, 392-393 measuring relative viscosities by consid­

eration of relaxation behavior follow­ing cessation of shear flow, 402

molecular theories predicting director tumbling, 401

557

normalized amplitudes of spherical har­monics (P2n): 391-392

plot of scattering intensity as function of Q for polydomain sample of pro-panoate ester of hydroxypropylcellu-lose (PPC) in liquid crystal state, 392/

PPC, 393 relationship between scattering vector

(Q) and flow geometry, 394/ representation of different director dis­

tributions in polydomain and mono-domain samples of nematic liquid crystal system, 392/

representation of orientationai order in nematic liquid crystal structure, 391/

response to shear flow of three differ­ent polydomain textured liquid crys­tal polymers (LCPs), 401

schematic of shear flow between paral­lel plates, 393/

schematic of shear flow X-ray cell by University of Reading, 395/

three regimes of (Ρ2 η) versus shear rate curves, 400

values of (P2n) for aqueous hydroxypro-pylcellulose (HPC) during cessation of shear flow, 402/

values of (P2n) for PPC and HPC dur­ing steady-state shear flow, 399/, 400/

values of (P2n) for PPC sample in ther-motropic phase, 397/, 398/

558