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PHYSICS RESEARCH AND TECHNOLOGY
RECENT PROGRESS IN NEUTRON
SCATTERING RESEARCH
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PHYSICS RESEARCH AND TECHNOLOGY
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PHYSICS RESEARCH AND TECHNOLOGY
RECENT PROGRESS IN NEUTRON
SCATTERING RESEARCH
AGUSTIN VIDAL
AND
MATTHEW CARRIZO
EDITORS
New York
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Copyright © 2013 by Nova Science Publishers, Inc.
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Additional color graphics may be available in the e-book version of this book.
Library of Congress Cataloging-in-Publication Data
Recent progress in neutron scattering research / editors, Agustin Vidal and Matthew
Carrizo.
pages cm
Includes bibliographical references and index.
1. Neutrons--Scattering--Research. I. Vidal, Agustin. II. Carrizo, Matthew.
QC793.5.N4628R43 2013
539.7'58--dc23
2013034687
Published by Nova Science Publishers, Inc. † New York
ISBN: 978-1-62948-100-5 (eBook)
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CONTENTS
Preface vii
Chapter 1 Current Status and Activities of Small-Angle
Neutron Scattering Instrument, SANS-U 1 Hiroki Iwase, Hitoshi Endo and
Mitsuhiro Shibayama
Chapter 2 Analyzing the Phase Behavior of Polymer Blends
by Small-Angle Neutron Scattering 37 Nitin Kumar and Jitendra Sharma
Chapter 3 Small-Angle Neutron Scattering and the Study
of Nanoscopic Lipid Membranes 77 Jianjun Pan, Frederick A. Heberle and
John Katsaras
Chapter 4 Evaluation of the Thermodynamic Properties
of Hydrated Metal Oxide Nanoparticles by
INS Techniques 105 Elinor C. Spencer, Nancy L. Ross,
Stewart F. Parker and Alexander I. Kolesnikov
Chapter 5 Ultra-Low Temperature Sample Environment for
Neutron Scattering Experiments 133 O. Kirichek and R. B. E. Down
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Contents vi
Chapter 6 Strengths of the Interactions in Y Ba2Cu3O6.7 and
La2-xSrxCuO4(x = 0:16) Superconductors Obtained
by Combining the Angle-Resolved Photoemission
Spectroscopy and the Inelastic Neutron Scattering
Resonance Measurements 149 Z. G. Koinov
Index 177
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PREFACE
In this book the authors present current research in the study of neutron
scattering. Topics discussed in this compilation include the current status and
activities of small-angle neutron scattering instrument-SANS-U; analyzing the
phase behavior of polymer blends by small-angle neutron scattering; small-
angle neutron scattering and the study of nanoscopic lipid membranes;
evaluation of the thermodynamic properties of hydrated metal oxide
nanoparticles by INS techniques; ultra-low temperature sample environment
for neutron scattering experiments; and strengths of the interactions in Y
Ba2Cu3O6:7 and La2−xSrxCuO4(x = 0:16) superconductors obtained by
combining the angle-resolved photoemission spectroscopy and the inelastic
neutron scattering resonance measurements.
Chapter 1 – The authors review the current status and activities of the
small-angle neutron scattering instrument (SANS-U) owned by the Institute
for Solid State Physics, The University of Tokyo, at the research reactor (JRR-
3) of the Japan Atomic Energy Agency (JAEA), Tokai, Japan. There have
been two major upgrades of the SANS-U instrument. The second major
upgrade was performed in the last few years. This upgrade allowed an increase
of 3.16 times in the intensity of incident neutrons passing through a sample
and that of the accessible low Q limit to 3.8 × 10−4
Å−1
using focusing lenses
(MgF2 biconcave lenses) and a high-resolution position sensitive detector
(HR-PSD). The HR-PSD consists of a cross-wired position sensitive
photomultiplier tube (PSPMT) combined with a ZnS/6LiF scintillator. The
authors used a high-performance ZnS/6LiF scintillator developed at JAEA
with an optimized ZnS/6LiF scintillator thickness to improve the experimental
efficiency in focusing small-angle neutron scattering experiments. The
versatility of the SANS-U instrument is demonstrated by the variety of
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Agustin Vidal and Matthew Carrizo viii
accessories available and the range of scientific ―hot topics‖ it has been used
in, which have included high-pressure experiments, Rheo-SANS studies of
worm-like micelles, and deformation studies on tough gels.
Chapter 2 - Novel property of neutrons to exploit the inherent differences
in the scattering capabilities of two hydrogen isotopes (namely hydrogen and
deuteron) offers a unique tool for probing soft matter systems. The contrast
achieved through deuteron labeling of a component (as compared to the
hydrogenated ones) provides an opportunity to either hide or highlight one
unit/s or subunit/s of the blend and hence identify and investigate the structure
of the desired-only component in a neutron scattering experiment. The
technique of small-angle neutron scattering (SANS) has particularly been
successful in obtaining the very first signature of phase separation in polymer
blends/mixtures that may begin to occur over a length scale of 10-100 Å.
Through illustrative examples, the chapter presents an overview of the theory
establishing a simple relationship between obtained scattering profile of
systems and the thermodynamic parameters of interest describing their phase
behavior.
Chapter 3 - Recent advances in structural, molecular and cellular biology
have shown that lipid membranes play pivotal roles in mediating biological
processes, such as cell signaling and trafficking. Like proteins and nucleic
acids, lipids in membranes carry out these roles by modifying their transverse
and lateral structures. Along the bilayer normal direction, membranes can
assume different hydrophobic thicknesses by adjusting their lipid chemical
composition, which can in turn lead to altered protein function through
hydrophobic mismatching. In the two-dimensional lateral direction, lipid
membranes can self-compartmentalize into functional domains (rafts) for
organizing membrane-associated biomolecules. In this chapter the authors
focus on the use of small-angle neutron scattering (SANS) to resolve
nanoscopic structure, both normal to and in the plane of the membrane. The
authors present structural parameters of phospholipids with different head
moieties (i.e., choline and glycerol) using the neutron scattering density profile
model, which is guided by molecular dynamics simulations. The authors also
show the role of the glycerol backbone in mediating different modes of
interaction between phospholipids and their neighboring cholesterol
molecules. The authors conclude the chapter by showing how SANS is used to
reveal membrane lateral heterogeneity, specifically by describing their recent
findings which showed that raft size varies as a function of the difference in
membrane thickness between raft and non-raft domains.
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Preface ix
Chapter 4 - In this contribution the authors will present a detailed
methodology for the elucidation of the following aspects of the
thermodynamic properties of hydrated metal oxide nanoparticles from high-
resolution, low-temperature inelastic neutron scattering (INS) data: (i) the
isochoric heat capacity and entropy of the hydration layers both chemi- and
physisorbed to the particle surface; (ii) the magnetic contribution to the heat
capacity of the nanoparticles. This will include the calculation of the
vibrational density of states (VDOS) from the raw INS spectra, and the
subsequent extraction of the thermodynamic data from the VDOS. This
technique will be described in terms of a worked example namely, cobalt
oxide (Co3O4 and CoO).
To complement this evaluation of the physical properties of metal oxide
nanoparticle systems, the authors will emphasise the importance of high-
resolution, high-energy INS for the determination of the structure and
dynamics of the water species, namely molecular (H2O) and dissociated water
(OH, hydroxyl), confined to the oxide surfaces. For this component of the
chapter the authors will focus on INS investigations of hydrated isostructural
rutile (α-TiO2) and cassiterite (SnO2) nanoparticles.
The authors will complete this discussion of nanoparticle analysis by
including an appraisal of the INS instrumentation employed in such studies
with particular focus on TOSCA [ISIS, Rutherford Appleton Laboratory
(RAL), U.K.] and the newly developed spectrometer SEQUOIA [SNS, Oak
Ridge National Laboratory (ORNL), U.S.A].
Chapter 5 - Today almost a quarter of all neutron scattering experiments
performed at neutron scattering facilities require sample temperatures below
1K. A global shortage of helium gas can seriously jeopardize the low
temperature experimental programs of neutron scattering laboratories.
However the progress in cryo-cooler technology offers a new generation of
cryogenic systems with significantly reduced consumption and in some cases a
complete elimination of cryogens. Here the authors discuss new cryogen free
dilution refrigerators developed by the ISIS facility in collaboration with
Oxford Instruments. The authors also discuss a new approach which makes
standard dilution refrigerator inserts cryogen-free if they are used with
cryogen-free cryostats such as the 1.5K top-loader or re-condensing cryostat
with a variable temperature insert. The first scientific results obtained from the
neutron scattering experiments with these refrigerators are also going to be
discussed.
Chapter 6 - The t−U−V −J model predicts phase instabilities which give rise
to a divergence of the charge density and spin response functions. This model
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Agustin Vidal and Matthew Carrizo x
has been the focus of particular interest as model for high-temperature
superconductivity in cuprite compounds because it fits together three major
parts of the superconductivity puzzle of the cuprite compounds: (i) it describes
the opening of a d-wave pairing gap, (ii) it is consistent with the fact that the
basic pairing mechanism arises rom the antiferromagnetic exchange
correlations, and (iii) it takes into account the charge fluctuations associated
with double occupancy of a site which play an essential role in doped systems.
Based on the conventional idea of particle-hole excitations around the Fermi
surface, the authors have performed weak-coupling calculations of the
strengths of the effective interactions in the t−U−V −J model which are
consistent with the antinodal gap obtained by the angle-resolved
photoemission spectroscopy (ARPES), and the well-defined incommensurate
peaks, which have been observed by the inelastic neutron scattering resonance
(INSR) experiments in Y Ba2Cu3O6.7 and La2-xSrxCuO4 (x = 0.16) samples.
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In: Recent Progress in Neutron Scattering … ISBN: 978-1-62948-099-2 Editors: A. Vidal and M. Carrizo © 2013 Nova Science Publishers, Inc.
Chapter 1
CURRENT STATUS AND ACTIVITIES OF SMALL-ANGLE NEUTRON
SCATTERING INSTRUMENT, SANS-U
Hiroki Iwase1*, Hitoshi Endo2 and Mitsuhiro Shibayama3*
1Comprehensive Research Organization for Science and Society, Tokai, Ibaraki, Japan
2Institute of Materials Structure Science, High Energy Accelerator Research Organization, Tsukuba, Ibaraki, Japan
3Neutron Science Laboratory, Institute for Solid State Physics, The University of Tokyo, Tokai, Ibaraki, Japan
ABSTRACT
We review the current status and activities of the small-angle neutron scattering instrument (SANS-U) owned by the Institute for Solid State Physics, The University of Tokyo, at the research reactor (JRR-3) of the Japan Atomic Energy Agency (JAEA), Tokai, Japan. There have been two major upgrades of the SANS-U instrument. The second major upgrade was performed in the last few years. This upgrade allowed an increase of 3.16 times in the intensity of incident neutrons passing
* Corresponding author: Email: Hiroki Iwase ([email protected]), Mitsuhiro Shibayama
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Hiroki Iwase, Hitoshi Endo and Mitsuhiro Shibayama 2
through a sample and that of the accessible low Q limit to 3.8 × 10−4 Å−1 using focusing lenses (MgF2 biconcave lenses) and a high-resolution position sensitive detector (HR-PSD). The HR-PSD consists of a cross-wired position sensitive photomultiplier tube (PSPMT) combined with a ZnS/6LiF scintillator. We used a high-performance ZnS/6LiF scintillator developed at JAEA with an optimized ZnS/6LiF scintillator thickness to improve the experimental efficiency in focusing small-angle neutron scattering experiments. The versatility of the SANS-U instrument is demonstrated by the variety of accessories available and the range of scientific “hot topics” it has been used in, which have included high-pressure experiments, Rheo-SANS studies of worm-like micelles, and deformation studies on tough gels.
INTRODUCTION The small-angle neutron scattering instrument (SANS-U) owned by the
Institute for Solid State Physics, The University of Tokyo, is one of several conventional steady-state pinhole small-angle neutron scattering (PSANS) instruments that are operated in major research-reactor facilities around the world (http://www.issp.u-tokyo.ac.jp/labs/neutron/inst/sans-u/index_e.html). First commissioned in 1991, the SANS-U instrument was installed on the C1-2 cold neutron beamline in the guide hall of the JRR-3 research reactor of the Japan Atomic Energy Agency (JAEA; Tokai, Japan) by Yuji Ito (Ito et al., 1995). The total spectrometer length is 32 m.
The first major SANS-U instrument upgrade was performed between 2002 and 2004, and involved (i) renewing the neutron velocity selector, (ii) replacing the multi-wire-type position-sensitive detector with a detector having an effective area of 645 × 645 mm and a spatial resolution of 5 mm, (iii) replacing the data acquisition system, and (iv) installing MgF2 biconcave lenses. The specifications of the parts that were upgraded are described in previous reports (Okabe et al., 2005, 2007). A number of sample accessories, such as a high-pressure cell (Shibayama et al., 2004; Osaka et al., 2006), real-time stretching cells (Karino et al., 2005; Shibayama et al., 2005; Nishida et al., 2009; Nishida et al., 2012), and shear cells (Shibayama et al., 2007; Matsunaga et al., 2010; Takeda et al., 2011), were developed after the first major upgrade, and as a result, the number of experimental proposals and publications relating to this instrument have steadily increased (http://quasi.issp.u-tokyo.ac.jp/db/index.php).
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Current Status and Activities of Small-Angle Neutron Scattering … 3
The accessible low Q limit (Qmin) for PSANS instruments in research-reactor facilities around the world have typically been in the order of 10−3 Å−1 (Q is the magnitude of the scattering vector defined by Q = (4π/λ)sinθ, where λ and 2θ are the wavelength and the scattering angle, respectively). The SANS-U instrument Qmin is 2.5 × 10−3 Å−1 obtained using a wavelength of 7 Å at a sample-to-detector length of 12 m. A number of focusing small-angle neutron scattering (FSANS) instruments have been successfully constructed over the past few years (from around the millennium) to extend the Qmin to a value in the order of 10−4 Å−1 (Alefeld et al., 1989, Alefeld et al., 1997, Oku et al., 2005, Koizumi et al., 2006, Grunzweig et al., 2007, Koizumi et al., 2007, Brulet et al., 2008, Frielinghaus et al., 2009, Furusaka et al., 2011, Goerigk et al., 2011, Radulescu et al., 2012). Observations of hierarchical structures with a wide range of lengths from one nanometer to a few micrometers have recently played important roles in various fields, such as the biological and soft-materials sciences, engineering, and noncrystalline and solid-state physics. It has been necessary to improve the performance of the SANS-U instrument to meet this increased demand.
A second major upgrade was therefore performed between 2008 and 2011, involving changing the SANS-U instrument from a PSANS instrument into an FSANS instrument by installing a high-resolution position-sensitive detector (HR-PSD) (Iwase et al., 2011). The HR-PSD comprises a cross-wired position-sensitive photomultiplier tube (PSPMT) combined with a ZnS/6LiF scintillator. Using a focusing lens and an HR-PSD, the SANS-U instrument can cover a wide Q range, from 3.8 × 10−4 to 0.35 Å−1 by using a wavelength of 7 Å. The high-intensity FSANS mode allows us to increase the intensity of the neutrons passing through the sample in the conventional Q range from 1.8 × 10-3 to 0.35 Å-1.
In this review, we describe the current specification and the details of the second major upgrade of the SANS-U instrument and recent activities using FSANS on the instrument.
THE SANS-U SPECIFICATIONS Figure 1 shows a schematic of the SANS-U, which is installed on the C1-2
cold neutron beamline at the JRR-3. The SANS-U instrument is composed of (A) renewed attenuators, (B) a mechanical velocity selector (Dornier–Astrium, Immenstaad, Germany), (C) a low-efficiency N2 beam monitor (MR-65, MIRROTRON, Budapest, Hungary), (D) five source apertures, (E) three types
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Hiroki Iwase, Hitoshi Endo and Mitsuhiro Shibayama 4
of collimators (collimation tubes coated on the inside with B4C, Ni guide tubes 20 mm wide and 50 mm high, and (F) neutron focusing lenses (55 MgF2 lenses), (G) a sample aperture, (H) a sample table, (I) a high-resolution position-sensitive detector (HR-PSD), and (J) a main 3He position-sensitive detector (Model 2660N, ORDELA Inc., Oak Ridge, TN, USA). The incident neutron wavelength () is usually set to 7.0 Å, at which the accessible Q range is 3.8 × 10−4 to 0.35 Å−1 (i.e., approximately three orders of magnitude).
New Attenuator System The attenuator system is located in front of the neutron velocity selector
(inside the shield room) and is mainly used for measuring transmission. Polyacrylate slabs of different thicknesses (3, 5, and 7 mm) were previously used. However, we detected a radiation dose of over 20 Sv outside the shield room, which is a working area for users, during inserting the attenuator to measure the transmission. This was caused by -rays being produced by scattered neutrons from the polyacrylate slabs, which further collided with the shield (which is iron or stainless steel).
Figure 1. Schematic drawing of the SANS-U instrument installed on the C1-2 cold neutron beamline at the research reactor (JRR-3). (A) Attenuators (a stack of B4C-coated polyester sheets), (B) the mechanical neutron velocity selector, (C) the beam monitor, (D) five source apertures located at collimation lengths of 16, 12, 8, 4 and 2 m, (E) three kinds of collimators, namely B4C-inner-coated collimation tubes, Ni guide tubes and (F) the neutron focusing lens (55 MgF2 lenses), (G) the sample aperture, (H) the sample table, (I) the high-resolution position-sensitive detector (HR-PSD), (J) the main 3He position-sensitive detector.
We, therefore, installed a stack of B4C-coated polyester films (65 × 50 mm). Figures 2a and 2b show the B4C-coated polyester film and the new attenuator system. Figure 2c shows the transmission of neutrons with = 7.0
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Current Status and Activities of Small-Angle Neutron Scattering … 5
Å against the number of B4C-coated polyester film. Stacks of B4C-coated polyester film containing 12, 20, and 28 films were selected to match the attenuation rates achieved with the old attenuators. The B4C-coated polyester film stacks were sandwiched between aluminum plates. By installing the new attenuator, the radiation dose was dramatically decreased to approximately 5 Sv outside the shield-room.
Figure 2. Attenuator systems installed on the SANS-U instrument. (a) B4C-coated polyester film. (b) Attenuator changer. (c) Transmission of neutrons ( = 7.0 Å) against the number of B4C-coated polyester film. The value of transmission with old attenuators (Polyacrylate slabs with thicknesses of 3, 5, and 7 mm).
Neutron Velocity Selector The incident neutron beam from the cold source is monochromatized by a
mechanical velocity selector with helical slots (Dornier-Astrium, Immenstaad, Germany) behind the main shutter. The neutron wavelength () is related to the rotation frequency () and the selector to rotate the angle () by equation 1 as follows.
(1)
where h and m are Planck’s constant and the neutron mass, respectively, and L is the overall length of the inner rotating part. The selector to rotate the angle is given by
h
Lm
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Hiroki Iwase, Hitoshi Endo and Mitsuhiro Shibayama 6
(2)
where α0 is related to the screw angle of the blades and R and are the distance between the selector rotation axis and the neutron beams and the tilt angle, respectively.
(3)
For the SANS-U instrument, α0 = 48.3°, L = 250 mm, and R =115 mm, so
(4)
The wavelength is calibrated using Bragg reflections from silver behanate (AgBE) powder. Figure 3a shows the Bragg peak from AgBE observed at various NVS rotation frequencies. A plot of the estimated value versus (1/), shown in Figure 3b, allows the for the NVS to be determined using equation 4, and it is usually set to 0°. The typical wavelength and wavelength distribution selected in the SANS-U instrument are = 7 Å and / = 0.1, respectively.
Figure 3. (a) SANS results for silver behanate (AgBE) observed at various NVS rotation frequencies. (b) The values estimated from 1st Bragg peak for AgBE against inverse of the frequency (1/).
0 L R
6.591050 L R
L
2.636103 48.3 2.174
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Current Status and Activities of Small-Angle Neutron Scattering … 7
Beam Monitor An incident neutron beam monitor (with an effective diameter (Ø) of 25
mm), used to normalize the incident neutron intensity, was previously installed in front of the sample aperture. However, scattering from the beam monitor windows led to background scattering in small-angle scattering observations in the lower Q range, Q < 10−3 Å−1. Therefore, we installed a new beam monitor with a large effective area behind the NVS because the size of the neutron beams passing through the beam monitor, which is not collimated, is 20 × 50 mm. The newly installed beam monitor (MR-65), which was purchased from MIRROTRON (Budapest, Hungary) is a low efficiency nitrogen beam monitor with an effective area of 65 × 65 mm. It is filled with N2 and CF4 at 105 Pa, and according to the supplier, the detector efficiency is 8.7 × 10−5 at = 2.8 Å.
Beam Aperture System The X–Z removable source apertures installed in the SANS-U instrument
at a collimation lengths of 16, 12, 8, 4, and 2 m. The apertures can be used to select two or three circular pinhole sizes (Ø 20 mm, Ø 5 mm, and Ø 3 mm or a rectangular open slit (20 mm wide and 50 mm high). All of the positions are precisely determined (to < 0.05 mm). The sample aperture (which is removable), the size of which is usually set to be Ø 15 mm, Ø 10 mm, or Ø 5 mm. The pinholes have, up to now, been made of 1 mm thick cadmium, but the reflectivity and the parasitic scattering are both higher from cadmium than from B4C. This affects the scattering profile background in the Q range of Q < 10−3 Å−1. To reduce the amount of background scattering from the pinholes, we replaced them with new tapered pinholes made from a B4C sintered plate.
Neutron Focusing Lens Several neutron devices have been developed around the world to extend
the Qmin to a value in the order of 10−4 Å−1. In particular, a spherical biconcave MgF2 lens, with a curvature radius, central thickness, and outer diameter of Ø 25 mm, 0.7 mm, and Ø 30 mm, respectively, is used in several SANS instruments (Eskildsen et al., 1998, Choi et al., 2000). On the other hand, Oku and co-workers have uniquely developed two magnetic neutron lenses, a super-conducting magnet lens (Oku et al., 2005), and an extended Halbach-
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Hiroki Iwase, Hitoshi Endo and Mitsuhiro Shibayama 8
type sextupole magnet lens (Oku et al., 2006, 2007). Magnetic lenses, however, require polarized incident neutrons, and thus, MgF2 lenses are appropriate for an unpolarized neutron instrument. Therefore, we used a spherical biconcave MgF2 lens in the SANS-U instrument.
Figure 4 shows photographs of the stack of 55 MgF2 biconcave lenses installed in the most downstream part of the collimator chamber, approximately 1.3 m upstream of the sample position. The MgF2 biconcave lenses were purchased from Ohyo Koken Kogyo Co., Ltd., (Tokyo, Japan). To suppress neutron reflection from the inner surface of the lens holder, 0.5 mm thick Cd rings, each with an inner diameter of 25 mm, were introduced at 15-lens intervals. The edges of the Cd rings were coated with GdO2 paint.
Figure 4. Photographs of a stack of 55 MgF2 spherical biconcave lenses installed in the most downstream region of the collimator chamber, about 1.3 m upstream of the sample position.
Figure 5. Photographs of (1) the main 3He position-sensitive detector (PSD) and (2) the high-resolution position-sensitive detector (HR-PSD). The HR-PSD is mounted on an X–Z movable bench in front of the main 3He PSD. The distance between the HR-PSD and the main 3He PSD is 0.7 m.
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Current Status and Activities of Small-Angle Neutron Scattering … 9
High-Resolution Position Sensitive Detector An HR-PSD with a spatial resolution of less than 1 mm is required to
extend Qmin to a value in the order of 10−4 Å−1, as mentioned earlier (Koizumi et al., 2006). A cross-wired PS PMT-based scintillation detector has been successfully used as a high-resolution detector in the other SANS instruments [NOP (Oku et al., 2005), SANS-J-II (Koizumi et al., 2006, 2007) and mf-SANS (Furusaka et al., 2011)] at the JRR-3.
The HR-PSD assembled by the J-NOP Company Ltd, Japan. The HR-PSD comprises a cross-wired PSPMT (R3239; Hamamatsu Photonics Co. Ltd, Japan) combined with a ZnS/6LiF scintillator. The HR-PSD is packed in an aluminum vessel. According to the PSPMT specifications, the size of the effective PSPMT area and its spatial resolution are approximately Ø 100 mm and 0.45 mm, respectively. ZnS/6LiF scintillation detectors are known to be less efficient than conventional 3He detectors at detecting cold neutrons (by approximately 20%). We used the ZnS/6LiF scintillator developed at the JAEA and determined the optimum scintillator thickness to increase the HR-PSD performance. This will be discussed in more detail in following section.
Figure 5 shows the HR-PSD installed inside the evacuated flight tube of the SANS-U instrument. The HR-PSD is mounted on an X–Z movable bench in front of the main 3He PSD. The data-acquisition system, which was provided by the J-NOP Company, is a modified PSD2K, and it was built into the existing operating and data-acquisition systems.
Main 3He Position Sensitive Detector The multi-wired two-dimensional position sensitive detector (Model
2660N; ORDELA Inc., Oak Ridge, TN, USA) in the flight tube (shown in Figure 5) was used for conventional PSANS measurements. This detector was installed in the SANS-U instrument during the first major upgrade (2002–2004) (Okabe et al., 2005), and has an effective area of 645 × 645 mm and a spatial resolution of 5 mm. The counting gas is 77% 3He and 23% CF4 at 260 kPa. According to the supplier, the neutron detector is 80% efficient at a wavelength of 5 Å. The main 3He PSD can move to change the sample-to-detector distance from 1.03 m to 16 m. The data acquisition system is DAS100 (ORDELA Inc.), which is an integrated board developed for data accumulation and detector configurations.
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Hiroki Iwase, Hitoshi Endo and Mitsuhiro Shibayama 10
OPTIMIZATION OF THE HIGH-RESOLUTION POSITION
SENSITIVE DETECTOR ON THE SANS-U Observing a SANS profile in the Q range of the order of 10−4 Å−1 using
FSANS requires a longer measurement time because a high-resolution SANS measurement, so measurement times usually have a higher proportion of user machine time for FSANS than for PSANS experiments. We need to improve an experimental efficiency of FSANS. We focus that ZnS/6LiF scintillation detectors are well known for having lower detection efficiencies for neutrons than conventional 3He detectors. It is therefore very important to optimize the efficiency of the scintillator without losing spatial resolution. To improve the HR-PSD, we replaced the commercial scintillator [purchased from Applied Scintillation Technologies (AST) Ltd, UK] with a high-performance scintillator developed by Katagiri and co-workers (Katagiri et al., 2004; Kojima et al., 2004).
Scintillation process used a ZnS/6LiF scintillator, in which neutrons are detected through the reaction given as follows:
6Li + n → T + + 4.8MeV Captured neutrons cause visible scintillation light to be generated by the
ZnS/6LiF scintillator, and the light is detected by the PMT. The number of captured neutrons is proportional to the thickness of the scintillator. It is expected that the capture of cold neutrons occurs near the surface of the scintillator (opposite the PMT). In a thick scintillator, part of the generated visible light generated will fail to reach the PSPMT because visible light is obstructed by polycrystalline ZnS particles. However, reducing the thickness of the scintillator proportionately decreases the capture efficiency, thereby decreasing the neutron counts. Therefore, we optimized the thickness of the ZnS/6LiF scintillator under FSANS experimental conditions (i.e. = 7 Å).
Optimization of Thickness of the ZnS/6LiF Scintillator We prepared high-performance ZnS/6LiF scintillators of different
thicknesses by using a procedure described elsewhere (Kojima et al., 2004). The ZnS/6LiF weight ratio in the scintillator was 2:1, so the amount of 6Li was
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higher in the prepared scintillator than in a commercially available scintillator(which has a ZnS/6LiF ratio of 4:1 and a thickness of 0.25 mm).
First, to determine the optimum thickness of the ZnS/6LiF scintillator, the HR-PSD was installed on the sample table, shielded against visible light, and the HR-PSD PMT bias voltage was set to −1000 V. Figure 6 shows the pulse-height spectra for high-performance ZnS/6LiF scintillators of different thicknesses and for a commercial ZnS/6LiF scintillator. The total count increased in all of the channels with scintillator thickness; however, the position of the maximum neutron count peak shifted to a lower value.
Figure 7a shows the total counts, C(t), of the pulse-height spectra recorded using different scintillator thicknesses. Increasing the thickness from 0.180 to 0.433 mm caused the total counts to markedly increase, whereas increasing the scintillator thickness from 0.433 to 0.644 mm caused the total counts to increase only slightly. According to the literature (Price, 1964, Knoll, 1999), the total counts can be expressed using equation 5
(5)
where A and n are an instrumental constant factor and a neutron absorption coefficient, respectively, and n being given by
n = NAtotw/Mw (6) where NA, , tot, w, and Mw are the Avogadro’s number, the density, the total cross section, the weight fraction, and the molecular weight, respectively. For t ≤ 0.433 mm, the experimental results agreed well with the estimates calculated using equations 5 and 6. However, for t > 0.433 mm, the experimental results deviated from the calculated C(t), clearly indicating that the total counts are influenced by the scintillator thickness. The proportion of ZnS (which is polycrystalline) particles increases with the proportion of 6Li, so part of the visible scintillation light generated could not reach the PMT because it was obstructed by ZnS particles.
Figure 7b shows the peak positions of the pulse-height spectra for different scintillator thicknesses. The peak position for the commercial scintillator was estimated to be 25.54 ch. In contrast with the total count behavior, the peak position exponentially decreased from 86.45 to 12.76 ch when the scintillator thickness was increased from 0.180 to 0.644 mm, i.e., the peak position approached the lower discrimination level of approximately 7.0
C t 1 Aexp(nt)
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ch. A low peak position generally leads to poor counting stability because of the difficulties involved in electrical discrimination, whereas higher peak positions give better count stability. A thin scintillator is, therefore, most appropriate in terms of the peak position, and optimizing the scintillator thickness involves a trade-off between the total count and peak position.
We determined the optimum thickness from the results of the total count and the peak position using different scintillator thicknesses. The HR-PSD requires high detection efficiency and approximately the same peak position as that given by the commercial scintillator, but the latter condition is more important for the SANS-U instrument because the background level affects the Qmin (Iwase et al., 2011). The optimum scintillator thickness was determined to be 0.433 mm (Iwase et al., 2012).
Figure 8 shows a comparison of the pulse-height distributions obtained using the optimum ZnS/6LiF scintillator and the commercial scintillator. The estimated total counts were estimated 3.18 105 for the optimum scintillator and 2.27 105 for the commercial scintillator, indicating that the optimum ZnS/6LiF scintillator gave a total neutron count 1.39 times higher than that obtained using the commercial scintillator without giving a different peak position.
Figure 6. Pulse-height spectra for the high-resolution position-sensitive detector (HR-PSD) with a high performance ZnS/6LiF (2:1) scintillator (developed by Katagiri et al.) of different thicknesses and with a commercial ZnS/6LiF scintillator. The scintillator thickness was estimated using a portable coating thickness tester (LZ2000J, Kett Electric Laboratory, Tokyo, Japan). Reproduced with permission of the International Union of Crystallography (J. Appl. Cryst. 2012, 45, 507).
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Figure 7. Scintillator thickness dependencies of (a) total count and (b) peak position on a pulse-height spectrum for the high performance ZnS/6LiF scintillator. Reproduced with permission of the International Union of Crystallography (J. Appl. Cryst. 2012, 45, 507).
Figure 8. A comparison of the pulse-height spectra of the HR-PSD for the optimum ZnS/6LiF scintillator and a commercial scintillator. Reproduced with permission of the International Union of Crystallography (J. Appl. Cryst. 2012, 45, 507).
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Performance of the High-Resolution Detector with Optimum Scintillator
We assessed the performance (including the spatial resolution, image
distortion, and noise levels) of the HR-PSD fitted with the optimized ZnS/6LiF scintillator. The HR-PSD was located in the flight tube at L2 = 11.3 m. The HR-PSD was positioned in the flight tube at L2 = 11.3 m and the HR-PSD PMT bias voltage was set to −1080 V.
We measured the intrinsic noise counts originating from electrical noise and rays by closing the main shutter, and found 8 10−3 cps for the HR-PSD, which was lower than intrinsic noise counts of the main 3He PSD (6.14 cps) at L2 = 12 m.
PSPMT-based scintillation detectors distort the image in the peripheral parts of the effective area, and analytical methods have been proposed to correct this distortion (Hirota et al., 2005). To check the image distortion for the HR-PSD, we measured the incoherent scattering from a low-density polyethylene (LDPE) slab by using a cadmium grid mask filter (4 mm pitch, spot size Ø 1 mm) placed in front of the HR-PSD. Figure 9a shows the grid pattern observed using the HR-PSD. Strong distortion was clearly observed outside the Ø 74 mm area, but the image was negligibly distorted within the Ø 74 mm area. We also confirm the uniformity of the HR-PSD by measuring the incoherent scattering from the LDPE slab without the grid mask filter.
Figure 9. (a) Two-dimensional contour map of the grid pattern detected by the HR-PSD using a grid mask filter made of cadmium (4 mm pitch, spot size Ø1 mm). (1) Active area (Ø74 mm) of the HR-PSD. (2) Beam-stopper position, with a diameter of 2 mm. (b) The circularly averaged scattering profile for a low-density polyethylene (LDPE) slab (t2.03mm), obtained by assuming that the beam centre is irradiated at the beam-stop position. Reproduced with permission of the International Union of Crystallography (J. Appl. Cryst. 2012, 45, 507).
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The incident neutron beam is irradiated to the beam-stop position when scattering from a sample is isotropic to cover a wide Q range. Assuming that the beam center was at the beam-stop position, we obtained a circularly averaged scattering profile, as shown in Figure 9b. The scattering profile was almost flat in the Q range 2 10−4 to 5 10−3 Å−1. We confirmed that the HR-PSD, which has an effective area of Ø 74 mm, can be used without image distortion correction.
FOCUSING SANS IN THE SANS-U INSTRUMENT
Basic Equations The focal length (f) for an FSANS instrument is given by
(7)
where R, NL, and n are the curvature radius, the number of lenses, and the refraction index, respectively. The refraction index (n) is determined from the atomic number density (), the bound coherent scattering length (bc), and the wavelength () using equation 8.
(8)
For a biconcave MgF2 lens with R = 25 mm (Eskildsen et al., 1998, Choi
et al., 2000), bc / = 1.632 10−6 Å−2 (9) However, if the lens thickness is negligible compared to the source-to-lens
distance (D1) and the lens-to-detector distance (D2), f is given by
(10)
f R
2NL 1 n
n 1bc
2
2
1
f
1
D1
1
D2
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The distances D1 and D2 in the SANS-U instrument are 10.7 and 12.7 m, respectively, and f was estimated to be 5.81 m. A wavelength of 7 Å is typically used. Therefore, considering the lens-to-sample distance of 1.3 m, 55 MgF2 lenses are needed to achieve FSANS measurements at = 7 Å and near L2 = 12 m.
FSANS Setups Figure 10 shows the SANS-U experimental setups used for two types of
FSANS. Before the second upgrade, setting a source aperture A0 of Ø 20 mm, a sample aperture As of Ø 7 mm, and a sample-to-detector distance L2 of 12 m, gave an accessible Qmin of 2.5 10−3 Å−1. To improve Qmin (to a value in the order of 10−4 Å−1) without changing L2 from 12 m, the sizes A0 and As have to be set to Ø 2 mm and Ø 0.7 mm, respectively, leading to a drastically decreased neutron intensity through the sample. For FSANS, the beam spot on the detector plane is ideally controlled by the source aperture size, and it is independent of the sample aperture size. Using a smaller source aperture and a larger sample aperture, with focusing collimation, will allow a smaller beam size on the detector to be obtained, with a neutron intensity through the sample higher than what is achieved with the PSANS with A0 = Ø 2 mm and AS = Ø 0.7 mm. The FSANS conditions determined for the SANS-U instrument are shown in Figure 10a. The A0, AS, and L2 values selected were Ø 5 mm, Ø 15 mm, and 12 m, respectively, from optimizing the incident neutron intensity and the Q resolution in terms of experimental efficiency (Iwase et al., 2011).
Figure 10b shows another FSANS experimental setup, in which the lengths L1 and L2 can be varied but kept similar, L1 ≈ L2. Looking at the FSANS geometry required for measuring the data with a Q range in the order of 10−4 Å−1, expanding A0 from Ø 5 to Ø 20 mm and fixing As at Ø 15 mm allowed the intensity to be increased while maintaining a Q-resolution as good as that in conventional PSANS. Hereafter, we refer to this way of using FSANS as the “high-intensity FSANS mode” (Iwase et al., 2011).
Focused Beam Measurements Figure 11 shows the circular averaged focused beam profiles as a function
of Q, which were obtained by using a stack of 55 MgF2 lenses with A0 = Ø 5 mm, As = Ø 15 mm, and L2 = 11.3 m. The focused beams were detected using
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the newly installed HR-PSD (curve 1 in Figure 11) and the main 3He PSD (curve 2 in Figure 11). A two-dimensional counter map for the focused beam detected using the HR-PSD is shown in Figure 11. The focused beam was irradiated at the beam-stopper (off-center) position of the HR-PSD (see Figure 9). We also observed the direct beam profile (curve 3 in Figure 11) collimated using the conventional PSANS instrument, with an A0 of Ø 20 mm, AS of Ø 7 mm, and L2 of 12 m (whose setup was the best possible before the second major upgrade). The focused beam detected using the HR-PSD was much narrower than the direct beam collimated using the PSANS setup. In contrast, the width of the focused beam detected by the main 3He PSD was wider than that detected using the HR-PSD because of the poor spatial resolution of the main 3He detector. The number of data points at Q < 2 10−3 Å−1 was low because the spatial resolution of the main 3He PSD clearly did not match the Q resolution (around 1 10−3 Å−1), suggesting that the beam size of the focused beam is dominated by the spatial resolution of the detector, and that the HR-PSD was indispensable for improving the Qmin to 10−4 Å−1.
Figure 10. Experimental setups for (a) FSANS and (b) high-intensity FSANS.
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Figure 11. Circular averaged focused beam profiles as a function of Q by using (1) the FSANS collimation with the HR-PSD, (2) the FSANS collimation with the main 3He PSD PSD and (3) the PSANS collimation with the main 3He PSD. The sample-to- detector distances are 11.3 m for FSANS and 12 m for PSANS. Reproduced with permission of the International Union of Crystallography (J. Appl. Cryst. 2011, 44, 558).
The focused beam profile can be divided into two regions: (A) an intrinsic
focused beam profile when Q < 7 10−4 and (B) background scattering when 7 10−4 Å−1 < Q < −5 10−3 Å−1. The beam profile in region (A) depends on the width of the source aperture and the chromatic aberration of the lens, the latter of which increases the beam width particularly strongly. However, the background scattering in region B, which follows power-law behavior, Q−3.6, is attributed to parasitic scattering from the pinhole edge, small-angle scattering from the window of the past-sample flight tube, and other artifacts. Background scattering is a serious problem in FSANS, and this is discussed in more detail in following chapter.
The Q resolution (∆Q/Q) of a SANS spectrometer is calculated using equation 11.
(11)
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where ∆/ is the wavelength resolution and ∆ is the angular distribution. The value of ∆/ was set to 0.11 in the SANS-U instrument. ∆ was experimentally determined from the direct-beam intensity distribution. ∆ was estimated from the half-width at half-maximum direct beam profile, the estimated ∆ values being 0.157 mrad for FSANS using the HR-PSD, 0.230 mrad for FSANS using the main PSD, and 0.714 mrad for PSANS. Figure 12 shows the Q dependence for each ∆Q/Q. The ∆Q/Q value was dramatically lower for FSANS using the HR-PSD than that for PSANS with the same L2.
FSANS RESULTS FOR STANDARD SAMPLES
Results for Polystyrene Latex We conducted FSANS measurements on standard samples to confirm the
reliability and the validity of using the SANS-U instrument for FSANS measurements. The standard sample was polystyrene latex (PSL) particles in H2O. The average particle radius (Rav) in the standard was 2980 Å, the standard deviation was 77 Å, and the PSL concentration was 1 wt%. Figure 13 shows the FSANS and PSANS (L2 = 8 and 1.03 m) profiles for the PSL. The PSANS profile at L2 = 12 m is also plotted to show its vertical shift. The scattering intensity was flat in the PSANS profiles in the higher-Q region, Q > 1 10−2 Å−1 because of incoherent scattering by H2O. Qmin was estimated to be 2.5 10−3 Å−1 for PSANS at L2 = 12 m and 3.8 10−4 Å−1 for FSANS, indicating that Qmin was successfully extended by approximately one order of magnitude by using FSANS. This also indicates that by using FSANS (one condition) and conventional PSANS (two conditions: L2 = 8 m and 1.03 m) at a wavelength of 7 Å, the SANS-U instrument can continuously cover a wide Q range of approximately three orders of magnitude, from 3.8 10−4 Å−1.
The theoretical scattering curve for PSL was calculated using reference values and considering the resolution function R(Q). PSL can be characterized using a polydispersed spherical model (Shibayama et al., 2002). The scattering curves observed were represented using the instrumental resolution function shown in equation 12.
(12)
Iobs(Q) Imodel (Q)R(Q Q')dQ'0
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which is approximated using the Gaussian equation
(13)
where
(14)
The dashed line is the theoretical curve for the polydispersed spheres
using the Schulz polydispersity distribution, simulated using reference values (R = 2960 Å). The solid line is the theoretical curve, and smeared because of the instrumental resolution. The theoretical scattering curve agreed with the FSANS profile in the Q range 3.8 10−4 Å−1 < Q <−5.0 10−3 Å−1. The FSANS profile had two maxima at Q = 1.92 10−3 Å−1 and Q = 3.09 10−3 Å−1, whereas the PSANS profile showed Q−4 behavior for 2.5 10−3 Å−1 < Q < 8 10−3 Å−1. These results indicate that FSANS allows a more precise structural analysis in the conventional PSANS Q range (Q > 1 10−3 Å−1) than is possible using PSANS. This is an important advantage of FSANS.
Figure 12. Q dependences of Q resolution (Q/Q) of (1) the FSANS collimation with the HR-PSD, (2) the FSANS collimation with the main 3He PSD and (3) the PSANS collimation with the main 3He PSD.
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Figure 13. FSANS (open circles) and PSANS [L2 = 8 m (open squares) and 1.03 m (open triangles)] profiles for polystyrene latex with an average particle radius of 2980 Å in H2O. The PSANS profile (filled circles) obtained at L2 = 12 m is vertically shifted to avoid overlaps; these are the highest-resolution data obtained before the second major upgrade. The dashed and solid lines are theoretical curves of the polydispersed sphere with the Schulz polydispersity distribution, simulated with catalog values: (solid line) the theoretical curve smeared by the instrumental resolution; (dashed line) the unsmeared curve. Reproduced with permission of the International Union of Crystallography (J. Appl. Cryst. 2011, 44, 558).
Figure 14. (a) FSANS profiles for polystyrene latex with an average particle radius (Rav); Rav = 250 Å (PSL250), 510 Å (PSL500), 995 Å (PSL1000), 2020 Å (PSL2000) and 2980 Å (PSL3000). The dashed line indicates the background of the instrument. The arrows indicate the accessible low-Q limit (Qmin) of each FSANS profiles. Reproduced with permission of the International Union of Crystallography (J. Appl. Cryst. 2011, 44, 558).
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Next, we measured the FSANS for PSL with different average particle radii (Rav) from 250 to 2980 Å. Figure 14 shows the FSANS profiles for the PSL solutions (in H2O) with Rav = 250 Å (PSL250), 510 Å (PSL500), 995 Å (PSL1000), 2020 Å (PSL2000), and 2980 Å (PSL3000). The theoretical scattering curves are also plotted. The FSANS profile for PSL3000 was similar to that shown in Figure 13. The sample thickness and concentrations were 1 mm and 1 wt%, respectively. The Qmin limit systematically improved with increasing average particle radius (volume), with estimated Qmin values being 2.0 10−2 Å−1 for PSL250, 1.1 10−3 Å−1 for PSL500, 6.0 10−4 Å−1 for PSL1000, 5.2 10−4 Å−1 for PSL2000, and 3.8 10−4 Å−1 for PSL3000. (arrows in Figure 14). Qmin evidently depends on the instrument background, but ideally, it also depends on the beam size, which is related to the chromatic aberration of the lens, the source aperture size, the collimation length, the sample-to-detector distance, and the spatial resolution of the detector (Littrell, 2004). Actually, Qmin depended on the beam width for PSL1000, PSL2000, and PSL3000. However, the Qmin values for PSL250 and PSL500 were limited by the background scattering in region B in Figure11. These values of scattering cross-section [d/d (cm-1) were lower than the background intensity (3 103 cm−1) at Q = 110−3 Å−1. In particular, the Qmin for PSL250 (d/d ≈ 4 102 cm−1) was similar to that obtained using PSANS at L2 = 16 m. The importance of minimizing background scattering in an FSANS instrument therefore cannot be overemphasized.
High-intensity FSANS and PSANS measurements were performed using PSL250 solutions, varying both collimation length and sample-to-detector distance (from 16 to 4 m) under symmetrical geometric conditions (L1 ≈ L2) to confirm the capabilities of the high-intensity FSANS mode. The experimental setup for high-intensity FSANS is shown in Figure 10b. Figure 15 shows high-intensity FSANS and PSANS profiles for PSL250 observed at collimation lengths (L1) and sample-to-detector distances (L2) of (a) 16 m, (b) 12 m, (c) 8 m, and (d) 4 m. The SANS intensities were normalized for the measurement time. The standard PSANS settings from before the upgrade (A0 = Ø 20 mm and As = Ø 7 mm) were used. The SANS intensities were collected using the main 3He PSD. If the geometric conditions (both L1 and L2) were the same for high-intensity FSANS as for PSANS, the high-intensity FSANS scattering intensities were 3.16 times higher than the PSANS scattering intensities under all conditions. The gain factors were 3.02 for 16 m, 3.24 for 12 m, 3.29 for 8 m, and 3.09 for 4 m. These results indicate that using FSANS successfully increased the incident intensity by a factor of 3.16 compared to the intensity using conventional PSANS, without depending on the collimation length and
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sample-to-detector distance being varied between 16 and 4 m, when L1 ≈ L2, and that the FSANS Q resolutions are acceptably close to the PSANS Q resolutions. High-intensity FSANS is especially useful for contrast variation SANS and time-resolved SANS experiments.
RECENT SANS-U INSTRUMENT ACTIVITIES USING FSANS
Observing Spatial Inhomogeneity in Phenolic Resins Using Wide-Q Observations
Izumi and co-workers investigated the structures of cured phenolic resins,
which were prepared by compression molding deuterated phenolic resin oligomers and hexamethylenetetramine (HMTA; the curing agent) using a combination of FSANS and PSANS with a wide-Q range, from 4 × 10−4 to 0.2 Å−1. (Izumi et al., 2012). In addition, small-angle X-ray scattering (SAXS) and scanning electron microscopy (SEM) were also performed on the resins. It is considered that cured thermosetting resins, such as phenolic and epoxy resins have an inherent inhomogeneity of the cross-links with sizes ranging from 10 to 100 nm (based on SEM observations). This spatial inhomogeneity in phenolic resins has not yet been directly observed using small-angle scattering techniques, and hence, wide-Q observation of SANS and SAXS were performed on phenolic resins to characterize this inhomogeneity.
Figure 16a shows the SANS profiles for different HMTA/highly deuterated novolac-type phenolic resin oligomer (NVD) mixtures. The weight ratios of HMTA/NVD are 0/1 for NVD00, 0.0627/1 for NVD05, and 0.125/1 for NVD10. The incoherent scattering intensities from HMTA were subtracted from the observed scattering profiles, and and the SANS results clearly showed power-law (Q−3.5) behavior extending over approximately three orders of magnitude, from Q = 4 × 10−4 Å−1 to Q = 0.2 Å−1. This power-law behavior in the scattering profile was independent of the amount of HMTA. The Q−3.5 behavior indicates that the phenolic resins have an inhomogeneity associated with internal fractal interfaces between voids and phenolic resins, with a fractal dimension equal to approximately 2.5 in the range 3–1600 nm.
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Figure 15. Comparisons of SANS profiles for polystyrene latex with an average particle radius of 250 Å in H2O between PSANS (filled circles) and high-intensity FSANS (open diamonds). The PSANS and the high-intensity FSANS profiles were measured at both collimation lengths (L1) and sample-to-detector distances (L2) of 16 m, 12 m, 8 m and 4 m. The insets are the SANS profiles on an absolute scale. Reproduced with permission of the International Union of Crystallography (J. Appl. Cryst. 2011, 44, 558).
Figure 16. (a) SANS profiles for the phenolic resins. (b) SANS (open symbols) and SAXS profiles (filled symbols) normalized by scattering contrast () for the phenolic resins. Circles, NVD00; squares, NVD05; triangles, NVD10. Reproduced by permission of The Royal Society of Chemistry (Soft Matter, 2012, 8, 8438).
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Figure 16b shows the normalized SANS and SAXS profiles for the HMTA/NVD mixtures. The scattering profiles were normalized to the scattering length density difference between the voids (i.e., air) and the phenolic resins (scattering contrast). Q−3.5 behavior was seen in the SAXS scattering profiles when 0.01 nm−1 < Q < 0.2 nm−1. The normalized SANS and SAXS profiles for each sample almost superimpose onto a single curve in that range, clearly indicating that the scattering profiles for the phenolic resins were primarily caused by the scattering contrast between the voids and the phenolic resins. The presence of voids in phenolic resins was confirmed by evaluating the difference in scattering length densities between the SANS and SAXS.
Yoshimura and co-workers have also analyzed the aggregate structures formed by newly synthesized amphiphilic dendrimers, which consist of poly(amidoamine) dendrons and a single alkyl-chain, in aqueous solutions using the FSANS technique (Yoshimura et al., 2013). They found that the aggregates formed by high-generation amphiphilic dendrimers formed hierarchical structures over a wide range (from 1 to 30 nm) of real space. Observing SANS over a wide Q range is a powerful way of studying self-assembled molecular structures. It is worth noting that the new time-of-flight SANS instrument (TAIKAN), which is located on BL15 in the Materials and Life Science Experimental Facility (MLF) in the J-PARC, Japan, can cover a wide Q range, 5 10−4 Å−1 < Q < 20 Å−1 using a magnetic lens and large area detectors systems, and its user program began in March 2012 (http://j-parc.jp/researcher/MatLife/en/instrumentation/ns.html).
Miniemulsion Polymerization Process Observed Using Time-Resolved FSANS
Motokawa and co-workers performed time-resolved FSANS and PSANS
measurements to observe structural changes over a large range of lengths during the miniemulsion polymerization of styrene (St) with the polymerizable surfactant N-n-dodecyl-N-2-methacryloylox-yethyl-N,N-dimethylammonium bromide (C12DMAEMA). (Motokawa et al., 2012). Miniemulsion polymerization has attracted a great deal of interest owing to its unique particle nucleation process, and also because it is an established polymerization method for producing particles with narrow and well-defined size distributions. However, bimodal peaks appeared during the gel permeation chromatographic analysis of the reaction mixture in the early stages of the
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polymerization of C12DMAEMA and 2,2′-azobis(2-amidinopropane) dihydrochloride (V50). Time-resolved FSANS and PSANS allowed the origin of the bimodal peaks and the specific polymerization loci in the system to be determined.
Figures 17a and 17b show the time-resolved PSANS and FSANS profiles for a St/C12DMAEMA/V50/poly-St solution (solution I) and a St/CTAB/V50/poly-St solution (solution II), respectively. When the polymerization time (t) was less than 20 min, the SANS profiles were substantially different for solution I and solution II in the Q range of Q < 0.04 nm−1. For solution I, at t = 0 min, the SANS profile showed power-law behavior (Q−1.5) at low Q range (Q < 0.04 nm−1), indicating that a large aggregate was formed by a number of small droplets. After polymerization was initiated, at t = 10 min, the SANS profile contained a shoulder peak at Q = 0.06 nm−1, and at t > 10 min, the Q dependences of the SANS profiles for solutions I and II were found to be almost identical in the observed Q region.
The solid lines in Figure 17 represent the best-fit theoretical scattering functions for polydisperse hard spheres. The structure factor S(Q) was evaluated using Percus–Yevick approximations (Percus and Yevick, 1958). Except for the solution I results at t = 0 and t = 10 min, the theoretical scattering curves agreed with the experimental results. The average radius (Rav) at t > 10 min, which was almost constant in each solution, being 47–48 nm in solution I and 38–26 nm in solution II. As a result, the droplets formed in the St/C12DMAEMA/V50/poly-St solution in the initial stages of polymerization were not fully stabilized by C12DMAEMA and poly(C12DMAEMA-ran-St) and were found to form the large aggregates. The spherical droplets were stabilized later in the polymerization process (t > 20 min) and a homogeneous dispersion formed in the water phase. In contrast, the droplets stabilized by CTAB in the St/CTAB/V50/poly-St mixture maintained their size and shape throughout the polymerization process. The particle size was found to strongly depend on the size of the droplets formed in the early stages of polymerization.
The SANS intensity in the high Q region depended on the interfacial structure on the surface of the droplet (Porod, 1951). The interfacial thickness was estimated according to the Porod law, and suggested that specific polymerization loci appeared in the early stages of the polymerization of St/C12DMAEMA/V50/poly-St on the surface of the droplet, and that this was the origin of the bimodal peaks in the gel permeation chromatogram.
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Figure 17. Time-resolved SANS (PSANS and FSANS) profiles obtained for (a) solution I and (b) solution II. These SANS profiles were shifted down by a factor, ft (see vertical axis), of 0.1, where ft at t = 0, 10, 20, 60, and 360 min is 10−4, 10−3, 10−2, 10−1, and 1, respectively. The SANS profile at t = 360 is shown as a net absolute intensity scale. Reprinted with permission from Macromolecules 2012, 45, 9435. Copyright 2012 American Institute of Physics.
Double-crystal ultra small angle neutron scattering (DC-SANS)
measurements need to be mentioned when discussing SANS observations at Q values in the order of 10−4 Å−1. In general, using thermal neutrons, DC-USANS can be used to observe SANS profiles in the Q range 10−5 Å−1 < Q < 10−3 Å−1. However, DC-USANS observations on data in the Q range Q > 1 10−4 Å−1 usually require much longer times. DC-USANS are also hardly able to observe anisotropic 2D patterns under certain conditions, such as share flow and stretching. It is also difficult using DC-USANS to observe a small scattering cross section (<103 cm−1). FSANS, in contrast, can easily be used to obtain 2D scattering patterns. As mentioned above, background scattering prevents the Qmin from being improved, so it is necessary to continue to reduce the amount of background scattering.
Structural Analysis of Aggregates Formed by Novel Surfactant in an Aqueous Solution Using High-Intensity FSANS Mode
Kusano and co-workers used FSANS and rheological measurements to
investigate the growth mechanisms of wormlike micelles formed by a newly
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synthesized star-type trimeric surfactant (3C12trisQ), with a hydrocarbon chain length of 12, in aqueous solution (Kusano et al., 2012). A 3CntrisQ molecule consists of three hydrocarbon chains and three hydrophilic groups connected by spacer chains, where n is the number of carbon atoms in the hydrocarbon chain. The critical micelle concentration has been found using electrical conductivity measurements to be an order of magnitude lower for 3CntrisQ surfactants than for the corresponding monomeric surfactants (Yoshimura et al., 2012). SANS results showed a variety of structures depending on the number of carbon atoms in the hydrocarbon chains, with ellipsoidal micelles being found when n = 10, spherical (ellipsoidal) and rod-like (worm-like) micelles at low and high sample concentrations, respectively, when n = 12, and rod-like (worm-like) micelles when n = 14. It was also found that the aggregates formed by 3C12trisQ exhibited sphere-to-rod transitions and worm-like micelle growth in aqueous solution. More SANS measurements were then conducted to elucidate the formation and growth mechanisms of the wormlike micelles formed by 3C12trisQ in aqueous solution at different surfactant volume fractions (φ). The measured φ range was from 0.0007 to 0.0272. The high-intensity FSANS mode was used to obtain a reasonable signal-to-noise ratio at the lowest concentration samples.
Figure 18. (a) Variation of SANS profiles for a star-type trimeric surfactant (3C12trisQ) in aqueous solution with varying volume fractions (φ) from 0.0034 to 0.0272. (b) φ-dependence of the peak-positions (Qm) observed in the Q-range of 0.02 Å–1 < Q < 0.05 Å–1. Reprinted with permission from Langmuir, 2012, 28, 16798. Copyright 2012 American Institute of Physics.
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Current Status and Activities of Small-Angle Neutron Scattering … 29
Figure 19. (a) Comparisons of experimental scattering profiles (symbols) for a star-type trimeric surfactant (3C12trisQ) solutions and the best-fit scattering curves (solid lines). The profiles are vertically shifted to avoid overlapping. (b) Volume fraction (φ) dependence of the number of water molecules per surfactant molecule inside the micelles (nw). Reprinted with permission from Langmuir, 2012, 28, 16798. Copyright 2012 American Institute of Physics.
Figure 18a shows SANS profiles for 3C12trisQ at different volume
fractions in aqueous solution. All of the SANS profiles indicated peak-profiles in the Q range of 0.02 Å−1 < Q < 0.05 Å−1, and these peaks were attributed to repulsive interparticle interactions between the micelles. The sphere-to-rod micelle transition could be easily elucidated by analyzing the φ-dependence of the SANS peak position (Qm) (In et al., 2010). Figure 18b shows the φ-dependence of Qm. Increasing φ from 0.0034 to 0.0102 caused the peak positions to shift to higher Q values, but the Q value decreased when φ increased between 0.0102 and 0.0170. This change in Qm behavior was related to sphere-to-rod micelle transitions. The 3C12trisQ surfactant undergoes sphere-to-rod transitions and can produce wormlike micelles in the absence of salt. Further increases in φ from 0.0170 to 0.0272 caused the peak positions to shift to higher Q values, indicating micellar growth.
Model-fitting analysis was performed using a charged cylindrical or charged ellipsoidal particle scattering function to quantitatively determine the structural parameters. The structure factor S(Q) for a charged micellar system is calculated by applying a rescaled mean spherical approximation as proposed by Hayter and Penfold (Hayter and Penfold, 1981). The radius, length, and scattering contrast (Δρ) were determined by the model fitting analysis. In
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Hiroki Iwase, Hitoshi Endo and Mitsuhiro Shibayama 30
addition, the number of water molecules per surfactant molecule (nw) inside the micelles were assessed from the Δρ value (Kusano et al., 2012). Figure 19 shows the φ-dependence of the nw value for 3C12trisQ micelles. The nw values decreased from 51.4 to 17.1 with increasing φ; thus, approximately 34 water molecules were excluded before the sphere-to-rod transition and micellar growth occurred. The behavior of water molecules inside micelles during sphere-to-rod transition and micellar growth has not been extensively studied. The behavior of water molecules inside micelles during sphere-to-rod transitions and micellar growth was successfully determined in this study.
Other measurements have also been performed; for example, Nishida and co-workers used time-resolved SANS measurements to investigate stress relaxation phenomena in a nanocomposite gel after uniaxial elongation (Nishida et al., 2012). A series of SANS patterns were collected every 30 s without intervals—a measurement time that would be impossible other than the case of high-intensity FSANS mode on the SANS-U instrument.
Chronically insufficient beam time was a major problem before the second major upgrade. The first priority for the second major upgrade was therefore to shorten the standard measurement time to ameliorate this problem. Using the high-intensity FSANS mode successfully increases the intensity of incident neutrons passing through a sample by a factor of 3.16 compared to using conventional PSANS, which suggests that the measurement time can be shortened to 1/3.16 of the time taken using conventional PSANS conditions. We believe that using high-intensity FSANS alleviates the problem of having chronically insufficient beam time.
CONCLUDING REMARKS We successfully upgraded the SANS-U instrument from a conventional
PSANS instrument to a FSANS instrument. Instrumental performance was improved using a focusing lens (a stack of 55 MgF2 biconcave lenses) in terms of both accessible low Q limit (Qmin) and the incident-neutrons intensity passing through the sample. We installed a high-resolution position-sensitive detector (HR-PSD) and optimized the thickness of the high-performance ZnS/6LiF scintillator, which was developed by Katagiri et al., which allowed us to extend Qmin to a value in the order of 10 Å−4. We successfully improved Qmin from 2.5 10−3 to 3.8 10−4 Å−1 by using FSANS. As a result of these changes, the SANS-U instrument can now cover a wide Q range, from 3.8
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Current Status and Activities of Small-Angle Neutron Scattering … 31
10−4 to 0.35 Å−1 (i.e., three orders of magnitude). FSANS can also be used to increase the intensity of incident neutrons passing through a sample compared to the conventional PSANS Q range (2.5 10−3 to 0. 0.35 Å−1). We increased the incident neutron intensity by a factor of 3.16 by using the high-intensity FSANS mode, without changing the collimation length (L1) or the sample-to-detector distance (L2), and by using symmetrical conditions (L1 = L2).
We have summarized recent studies above in which FSANS has been used after the second major upgrade. A number of sample accessories have been developed for the SANS-U instrument, and we have improved the sample accessories after the SANS-U instrument was upgraded. The accessories include a standard sample changer, a high-pressure cell, and real-time stretching cells (because FSANS requires a large beam size, Ø 15 mm). We believe that the SANS-U instrument will be useful in various fields such as investigating nanostructures.
ACKNOWLEDGMENT We would like to thank Prof. Katagiri (JAEA) for discussion of the
scintillator, S. Satoh (KEK) and K. Hirota (RIKEN/J-NOP) for the construction of the high-resolution position sensitive-detector, T. Asami, R. Sugiura and Y. Kawamura [Institute for Solid State Physics (ISSP), The University of Tokyo] for helpful technical support, T. Matsunaga, T. Nishida and T. Kusano and M. Takeda for test measurements of FSANS, T. Oku (JAEA) and S. Okabe (Kyusyu University) for valuable discussions, H. Sakurai (Techno AP) for the improvements of the SANS-U control and data-acquisition systems. Partially of the presented second major upgrade was carried out with the support of the Atomic Energy Initiative, MEXT, Japan.
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In: Recent Progress in Neutron Scattering … ISBN: 978-1-62948-099-2
Editors: A. Vidal and M. Carrizo © 2013 Nova Science Publishers, Inc.
Chapter 2
ANALYZING THE PHASE BEHAVIOR
OF POLYMER BLENDS BY SMALL-ANGLE
NEUTRON SCATTERING
Nitin Kumar and Jitendra Sharma
School of Physics, Shri Mata Vaishno Devi University, Kakryal
Katra, Reasi, India
ABSTRACT
Novel property of neutrons to exploit the inherent differences in the
scattering capabilities of two hydrogen isotopes (namely hydrogen and
deuteron) offers a unique tool for probing soft matter systems. The
contrast achieved through deuteron labeling of a component (as compared
to the hydrogenated ones) provides an opportunity to either hide or
highlight one unit/s or subunit/s of the blend and hence identify and
investigate the structure of the desired-only component in a neutron
scattering experiment. The technique of small-angle neutron scattering
(SANS) has particularly been successful in obtaining the very first
signature of phase separation in polymer blends/mixtures that may begin
to occur over a length scale of 10-100 Å. Through illustrative examples,
the chapter presents an overview of the theory establishing a simple
relationship between obtained scattering profile of systems and the
thermodynamic parameters of interest describing their phase behavior.
E-mail: [email protected]
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Nitin Kumar and Jitendra Sharma 38
1. INTRODUCTION
A variety of phenomena like absorption, reflection, transmission,
luminescence and scattering etc., may occur when a beam of radiation
penetrates matter. Interaction of radiation with matter causes physical,
chemical and biological changes having significant impact on our everyday
life. A systematic study of such interactions readily gives parameters of
interest that are closely linked to the physical properties of matter. In this
chapter, we start by considering the interaction of neutrons with soft-matter
system i.e., polymers. Novel experimental technique of small-angle neutron
scattering (SANS) on polymers – consisting of homopolymer blends – has
been described. SANS is an ideal tool that offers to probe and imprint the very
first signature of a system approaching phase boundary i.e., beginning of phase
separation in polymer blends/mixtures, over a length scale of 10-100 Å. A
better understanding of the intricacies involved in interpreting the
experimentally obtained SANS data from relevant systems helps one to predict
accurately the phase behavior (i.e., miscibility with associated phase
boundaries) and resultant nano-scale structure and morphology of the system.
An overview of the theoretical treatment elucidating the thermodynamic
parameters (linked to the phase behavior of the system) of interest from the
SANS experimental data is therefore presented in detail.
The method described here is based primarily on the postulate of a
dominant wave-like character of radiations. A radiation may consist of
electromagnetic beam of light or X-rays with oscillating electric and magnetic
field components or charged particles such as alpha and beta radiations, beams
of charged particles created by accelerating machines and also include beams
of neutral particles such as neutrons. It is the de Broglie wave character of the
particles that becomes an important measure in the latter case. Due to these
underlying characteristics the technique of neutron scattering has attracted
scientific community the most for its wide-range applicability in studying
elementary particles physics, solid state physics and chemistry, material
science, and biology, etc. As a matter of fact, neutrons interact with matter via
strong, weak, electromagnetic and gravitational interactions. As such insight
into the structural details of soft matter system using scattering techniques
appears as a result of interference pattern produced by radiations scattered
from different sites of the sample. The simplest example of interference
produced by two superimposing waves is the Young‘s double slit experiment.
As shown in figure 1, a radiation propagating through two equidistant slits,
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Analyzing the Phase Behavior of Polymer Blends … 39
results in interference pattern that apart from the wavelength of the radiation
used depends on the distance of separation between the slits.
Figure 1. A typical Young‘s double slit experimental set-up exhibiting the interference
of waves.
2. SMALL-ANGLE NEUTRON SCATTERING (SANS)
Small-angle scattering (SAS) is a technique based on the deflection of
collimated ray away from a straight-line trajectory as it interacts with
structural entities that are larger but comparable within few orders of
magnitude to the wavelength of radiation used. Since the deflections involved
is small (0.1-100) it is generally identified by the prefixing the radiation-type
used with small-angle. Small-angle scattering (SAS) is the combined name
given to the techniques such as small-angle light (SALS), neutron (SANS) and
X-ray (SAXS) scattering, etc. Small-angle neutron scattering (SANS) is a well
established technique capable of probing structures at length scales from
around 1 nm to over 200 nm. It is widely applicable that ranges from the
studies of polymers and biological molecules to nanoparticles, micro-
emulsions and liposome.
The basic principle of operation for small-angle neutron scattering
(SANS) is illustrated in figure 2. The beam of radiation obtained from a
reactor-source here is focused on the sample with the help of a monochromator
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Nitin Kumar and Jitendra Sharma 40
and a collimation line. The monochromator is a rotating cylinder with tilted
lamellae, and selects neutrons of a certain wave length. The collimation
consists of two apertures, which define the maximum divergence of the
incident beam. Right after the second aperture the sample is placed. Usually,
many neutrons pass the sample un-scattered. This high intensity has to be
blocked by a beam-stop close to the detector due to saturation problems of the
detector. Only the scattered neutrons are detected by the detector. The
scattered intensity is recorded as a function of the scattering angle θ. The
scattering vector q is related to θ by
(1)
Neutron scattering is a powerful tool for both basic and applied research in
materials science. Thermal neutron has proved itself a very formidable probe
and consequently a very attractive choice in the study of a wide variety of
materials. For SANS experiments, neutron sources may either be continuous
or pulsed. In case of pulsed sources, the monochromatic beam can be achieved
by the time profile of the elastic scattered beam.
Figure 2. Schematic diagram of a small angle scattering diffractometer. The neutrons
pass from the left to the right. The incident beam is monochromatized and collimated
before it hits the sample. Non-scattered neutrons are absorbed by the beam stopper in
the centre of the detector. The scattered neutron intensity is detected as a function of
the scattering angle θ.
The behavior of the scattered beam from the sample depends on trivial
factors like incoming flux, transmission and geometrical factors and also
proportional to two quantities viz. (i) a contrast factor reflecting the ability of
individual atoms to interact with the radiation and (ii) structure factor obtained
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Analyzing the Phase Behavior of Polymer Blends … 41
as a result of interference pattern formed by the scattered beams from different
sites of the sample, giving information on structural properties.
The small-angle neutron scattering technique can be used to perform
structural studies on colloids, inhomogeneities in alloys and biological
molecules. Some of the typical materials science and biological problems
which can be studied using SANS are: (i) copper cluster in steels (ii) porosity
of coal, oil-bearing rocks, and cement etc., (iii) surfactant structure (iv) size
and shape of biological macromolecules, and (v) location of hydrogen atoms
in large molecules etc.
2.1. Why Neutrons?
An essential characteristic of neutron scattering or SANS which makes it a
useful probe for studying synthetic macromolecules and biomaterials is its
unique ability to offer scattering contrasts between two different isotopes (1H
and 2H, the latter being heavier among the two is identified by the name
―deuteron‖ and expressed by the symbol ‗D‘) of molecular hydrogen. Since
the molecules affected by H/D exchange chemically remains the same, the
optical properties gets modified subjected to different incident radiations. In
order to study the structural behavior of mixtures (polymer blends), there must
exist a significant variation in the scattering abilities of the characteristic
constituent elements to be identified as different scattering centers. One of the
main applications found in multi-component systems is to match a component
or a part of the aggregate with a specific isotopic mixture of the solvent. This
technique called ‗‗contrast match‘‘ has been used effectively to illuminate and
identify the structure of particles composed of different layers (e.g., in micro-
emulsions), to study surfactant layers adsorbed around mineral particles [1],
and for characterization of complex systems [2].
Neutrons are likely the best known elementary particles contained in the
nuclei of every atom. As such most of the lighter nuclei contain almost similar
number of protons and neutrons. The de Broglie wavelength associated
with neutron of mass moving with velocity can be written as
(2)
where h is the Plank‘s constant. Thus, at ambient temperature say, ,
the calculated value for the wavelength of thermal neutrons is about 1.4 Å. It is
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Nitin Kumar and Jitendra Sharma 42
of order similar to that of X-rays and thus implies neutron scattering to be an
effective probe for structural studies of nano-scale materials.
Neutrons are heavy-neutral particles and the interaction between neutrons and
matter is complex. They interact weakly with electrons due to the magnetic
moment present in both electrons and neutrons. Collisions between neutrons
and atomic nuclei are rare events, as both are tiny compared to atoms. Such
collisions can either be elastic or inelastic. In collisions with heavy nuclides,
neutrons lose little energy. On the other hand, collisions with lighter nuclides
such as H, D, He, and C, results in a loss of significant portion of energy for
the neutrons. A neutron can even lose all its kinetic energy in a single collision
with a proton. Thus, lighter nuclides are effective moderators as opposed to the
heavier ones. Capture of neutron by proton and deuterium lead to the
formation of deuterium and tritium, respectively. The cross sections are small,
and many neutrons decay if not captured. The interaction between neutron and
nuclei can generally not be calculated ab-initio, but are estimated based on the
experimental values of the scattering length density and cross-section of the
elements (listed in table1). Due to the magnetic interaction of neutrons with
matter, neutron scattering offers to be a powerful tool for investigating
magnetic structures and fluctuations.
Magnetic scattering from the nuclei of non-magnetic materials is an
incoherent phenomenon as the magnetic moments for such nuclei are totally
uncorrelated which in turn gives rise to an isotropic incoherent background.
This background may be significantly large for some materials such as
hydrogen (1H) whereas oxygen (
16O) and carbon (
12C) are the examples of
elements showing very low incoherent background due to the negligibly small
magnetic moment of these nuclei.
To a large extent, the non-magnetic interactions contribute towards
coherent scattering wherein interference effects are dominant. Due to such
important characteristics of scattering involving non-magnetic materials, the
technique enables one to extract detailed information about the nanoscale
structural features and morphology of the system. As a matter of fact, different
isotopes of the same atom may also have significantly different abilities to
scatter neutrons, and may even possess an opposite sign (positive or negative)
for the scattering length, b. The most important example could be cited for the
case of hydrogen which in its most common isotopic form 1H and the heavier
counterpart, 2H (i.e., also called deuterium, D), have significantly different
scattering lengths: and (see
table 1).
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Analyzing the Phase Behavior of Polymer Blends … 43
Although the two isotopes of the hydrogen 1H and
2H are chemically
similar they differ significantly in terms of optical properties especially when
subjected to neutron as radiation. Specific structural entities (e.g., solute or
solvent) can be highlighted by replacing H with D at particular chemical sites,
leading to the necessary contrast-enhancement required for neutron scattering
measurements. One is thus prompted to select a sample composition with
fewer hydrogen molecules to reduce the incoherent background. In general,
one chooses deuterated solvents because many of the solutes (with attached
hydrogen atoms) could still be cheaper to purchase than the deuterated
solvents. On the other hand, deuterated molecules (solutes) are less commonly
used and require a good knowledge of chemistry to get them synthesized.
Apart from them, in dilute or semi-dilute solutions, the main source of
hydrogen comes from the solvent itself.
For binary mixtures/blends, a scattering signal is obtained only because of
the difference in the individual scattering powers of the two mixing entities
which largely comes due to the contrast generated for neutrons through the
substitution of hydrogen by the deuterium isotope. Even a mixture of identical
polymeric entities that differ only in terms of protonation or deuteration of one
polymer-component can be studied by neutron scattering. Similar techniques
of achieving contrast have been applied for the study of unperturbed chain
conformations [3] and chain dynamics [4] in polymers.
Let us quantify the arguments presented here. The wavelength of a
neutron typically is around 6 to 7 Å and the maximum Q-range (~ 0.3 )
available generally with small-angle scattering measurements does not allow
the resolution of single bond between carbon atoms. Due to such limitations of
resolution, it becomes desirable to replace the nuclear scattering lengths, b,
with a continuous scattering length density function, ρ that averages the b-
values over an appropriate volume V. Thus, the scattering amplitude is
proportional to the contrast which is given by the difference in the scattering
length densities. For neutrons one can write
∑
∑ (3)
where NA is Avogadro‘s number, MV is the molar mass corresponding to the
chosen volume and , the mass density.
To help one understand the robustness of neutrons in achieving the
contrast for scattering measurements in polymer blends, a comparative picture
for the X-ray and light scattering contrast and is discussed and presented here.
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Nitin Kumar and Jitendra Sharma 44
The scattering length density function for X-rays is calculated by taking into
account the electron density difference and is given by
∑ (4)
where is the number of electrons in the ith
atom, and the classical electron
radius, re, is given by
∑ (5)
From the pre-factor in eq. (5), the scattering length for the electron can
be estimated by
(6)
Eq. (6) represents the scattering length for single electron. So in order to
calculate the scattering length of an atom, one must sum up all the scattering
lengths obtained for individual electrons contained in an atom.
Most of the polymers possess similar electron densities, rendering a poor
X-ray contrast. On the other hand, through specific deuterium labeling,
neutron scattering offers a robust and more useful tool for studying the
thermodynamics of polymer blends/mixtures.
Table 1. Coherent and incoherent neutron scattering lengths (bc and bi)
and cross sections (σc and σi) as well as absorption cross section (σa) for
atoms commonly found in polymers [5]
Element bc (fermi) bi (fermi) σc (barn) σi (barn) σa (barn)
H-1 -3.739 25.274 1.757 80.26 0.333
D-2 6.671 4.0400 5.592 2.050 0.000
C-14 6.646 0.0000 5.550 0.001 0.003
N-14 9.360 2.0000 11.01 0.500 1.900
O-16 5.803 0.0000 4.232 0.000 0.000
F-19 5.654 0.0000 4.232 0.001 0.000
Na-23 3.630 3.5900 1.660 1.620 0.530
Si-28 4.149 0.0000 2.163 0.004 0.171
P-31 5.130 0.2000 3.307 0.005 0.172
S-32 2.847 0.0000 1.017 0.007 0.530
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Analyzing the Phase Behavior of Polymer Blends … 45
2.2. Scattering Geometry
In an elastic neutron scattering event, a momentum transfer of
takes place from neutron to the sample. This leads to a minute
translation of the entire sample, but the internal state of the sample remains
unchanged whereas inelastic scattering of neutrons creates or annihilates an
excitation inside the sample so that both the energy of the neutron and the
internal state of the sample is modified. Experimentally, one has to keep track
of not only the flight direction of the scattered neutron but also of its energy.
The scattering of the radiation from two different sites of the sample, ra
and rb has been shown in figure 3. The incident beam of radiation has been
characterized by wavelength λ and direction given by wave vector ki. The
elastically scattered beam of radiation has the same wavelength as that of
incident beam and describes an angle of scattering, 2θ with scattering wave
vector . From the figure 3 it is obvious that the phase difference between the
radiations scattered from two different scattering sites of the sample ra and rb
has been obtained by multiplying the path difference by ⁄ , which may be
expressed as
Figure 3. Scattering geometry and the phase difference for scattering from different
sites of the sample.
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Nitin Kumar and Jitendra Sharma 46
( ) (7)
where , and is the scattering vector. The
magnitude of q is given by
| |
(8)
The wave motion for electromagnetic waves, may be expressed as plane
parallel wave having amplitude oscillating with time (t) and space (r)
(9)
The amplitude of the radiation at a site r and time t scattered from a point
through an angle of 2θ, depends on the capability to scatter at the point
and the phase given by the specific scattering site
[ ( )] (10)
[ ( )] (11)
The phase-factor explicitly gives the phase relative to the
non-interacting beam. Now the total amplitude of the scattered radiation over a
given scattering vector can simply be obtained by taking sum over all
sites in the sample
∑
∑ [ ( )] (12)
Now we can calculate the intensity of the scattered beam only without
taking into account the oscillating wave in time and space. The intensity may
be taken as product of | | along with its complex conjugate . Thus,
one can write the ensemble-averaged intensity as
∑ ∑ ⟨ ⟩ (13)
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Analyzing the Phase Behavior of Polymer Blends … 47
Here is intensity of incident radiation. The normalization of the
scattered intensity with respect to , gives the scattering function as
∑ ∑ ⟨ ⟩ (14)
Since is a continuous function, eq. (14) can be re-written in the
integral form as
⟨ ⟩
⟨ ⟩ [-i(q. )] (15)
where and are replaced with and , respectively. For an isotropic
system the averaged correlation function ⟨ ⟩ depends only on the
distance and remain independent of the specific sites and . So we
can eliminate the r-integral from eq. (15), to write
⟨ ⟩
(16)
where, V is the polymer volume and ⟨ ⟩ is the
correlation function.
Eq. (16) indicates that scattering function is simply the Fourier transform
of the ensemble averaged correlation function correlating densities
separated by distance . For an isolated or individual polymer chain this
correlation function represents the probability of finding segments of chain
separated by a distance of . The correlation function thus measured provides
a measure of the spatial correlation of concentration fluctuations in polymer
blends.
2.3. Structure Factor for a Gaussian Chain
Polymers are long linear macromolecules made up of a large number of
chemical repeat units or monomers, which are linked together through
covalent bonds. The number of monomers in each polymer chain may vary
from hundred to several thousands. It is supposed that the conformation of an
ideal long length polymer chain in the melt state consists of several random
walk steps of monomer segments. The distribution of segment lengths within a
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Nitin Kumar and Jitendra Sharma 48
chain obeys Gaussian statistics [6] and the conformation is often referred to as
ideal or the Gaussian chain conformation.
One can assume that the spatial arrangement of monomers within a
polymer chain follows the random walk statistics. The movement of the
random walker is such that beginning at r0 it moves by , ,
… …… ….. to arrive finally at rN in a total of N steps (as shown in
figure 4). The mean-squared distance between monomers i and j for such a
chain is given by
⟨ ⟩ ⟨
⟩ | | (17)
where and denoted the inter-monomer distance and characteristic
monomer length or Kuhn-length, respectively.
Figure 4. Trajectory of two-dimensional random walk with N-steps having a fixed step
displacement, .
Since (for cases involving larger distances) is sum of many random
variables, application of central limit theorem yields a Gaussian distribution of
the distance given by
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Analyzing the Phase Behavior of Polymer Blends … 49
( ) (
⟨ ⟩) ⁄
(
⟨ ⟩) (18)
Such a Gaussian chain is often characterized as a system of mechanical
beads connected by harmonic springs, as shown in figure 5.
Figure 5. A Gaussian chain in the spring-bead model consisting of N monomer
segments represented as beads connected via N harmonic springs.
The single-chain structure factor or the form factor is given by
∑ ⟨ ⟩
(19)
The configuration average must be taken over the probability distribution
(20)
which, may be simplified further to give an expression in terms of second
moment of the principal axes as
∑ (
⟨ ⟩
⟨ ⟩
⟨ ⟩ )
(21)
where the components of along the three axes are labeled, , and ,
respectively. For the isotropic state, ⟨ ⟩ ⟨
⟩ ⟨ ⟩
⟨
⟩,
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Nitin Kumar and Jitendra Sharma 50
∑ (
⟨
⟩) (22)
∑ *
| |
+
(23)
Using the identity
∑ | | ∑
(24)
where a double sum has been converted to a single sum over all differences,
| |. Therefore, the form factor can be written as
, ∑ *
+
-
,
∑ *
+
- (25)
Assuming N (the degree of polymerization) to be large compared to unity,
eq. (25), the sum can be converted to integral form
(
)
(
)
(26)
Solving the above integral gives the final expression for the form factor
as
(27)
Where, (
); and is called the Debye function. Thus, eq. (27)
gives the scattering of a labeled (isotopic) single coil in polymer melt.
If the end-to-end distance vector is measured for a large number of
polymers in a melt, one will find that the distribution of end-to-end distance
vectors is a Gaussian and that the root mean squared end-to-end distance
scales with the square root of the number of bonds, √| | √ ,
irrespective of the chemical details, and is given by
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Analyzing the Phase Behavior of Polymer Blends … 51
(28)
The size of polymer chains, often expressed in terms of radius of gyration,
, is an experimentally accessible parameter obtained from neutron scattering
measurements and is given by the expression
(29)
Using eq. (29) in eq. (27), the form factor of a Gaussian polymer coil
(represented by Debye function, can be re-written in terms of the radius of
gyration as
[ (
) ] (30)
Eq. (27) may be re-written in terms of the radius of gyration, :
(31)
where, ( ).
In a polymer blend, the chains are expected to follow a random walk
statistics and are therefore, referred to as a Gaussian chain. Polymer chains
also show a similar behavior in polymer solutions at the theta temperature.
3. THERMODYNAMICS OF POLYMER BLENDS
Polymer blends contain chains of at least two different polymers mixed
together and are often called binary (2-component) and ternary (3-component)
etc., depending on the number of components involved. The simplest blend
contains two different homopolymer chains e.g., polymers of polystyrene (PS)
and poly(vinyl methyl ether) (PVME). Each of them contains just one type of
monomer. Homogenous blends may just be formed by simple mechanical-
mixing of the constituent polymers. Scientifically, one of the most interesting
features of a polymer blend is its phase behavior which depends primarily on
the temperature and composition of the components involved. Polymer blends
(like low-molecular weight solvents) may exhibit complete miscibility or
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Nitin Kumar and Jitendra Sharma 52
phase-separation and even various levels of mixing in-between these two
extremes (i.e., partial miscibility). The phase behavior of polymer blends and
solutions is governed by their thermodynamics [6].
From the point of view thermodynamics, the condition of miscibility of
polymer blends/mixtures is determined by the free energy of mixing ( )
which is comprised of two main contributions namely, the entropy
(combinatorial mixing of the monomers) of mixing ( ) and the enthalpy
(interactions between monomers) of mixing ( ). For a stable phase, the
condition of miscibility that must be satisfied is ;
(32)
Therefore, for mixing to take place must be less than zero.
However, eq. (32) i.e., , remains the necessary but not sufficient
condition for miscibility. If the plot of versus (the molar fraction of
component-1) is concave with no inflection point, miscibility is complete over
the entire composition range. However, if the said results in two or more
inflection points, miscibility will be limited to compositions outside the two
compositions with the common tangent, the so-called binodal points. And a
binary mixture/blend having intermediate composition tends to phase-separate
into two phases.
3.1. Flory-Huggins Theory
Flory and Huggins formulated a simple lattice theory to calculate the free
energy [7]. The fundamental lattice theoretical model based on the mean field
theory assumes that the interaction between two segments of chains in a
polymer blend is due to the interaction between the segments itself and an
average field, caused by all other segments present in the system.
Now let us consider that we have an ensemble composed of Ω lattice sites,
of which n1 are occupied by polymer chains of type-1, each having number of
segments N1, and also n2 sites are occupied by polymer chains of type-2 having
number of segments as N2 . The volume fractions for polymers of type 1 and
type 2 can be expressed as ⁄ and ⁄ , respectively. For
an incompressible mixture, . The partition function for the system
can thus be written as
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Analyzing the Phase Behavior of Polymer Blends … 53
∑ *
+ (33)
where Ei is the energy associated with the i-th configuration of polymers 1 and
2, respectively. In the Flory-Huggins lattice model is calculated via mean
field approach where the site dependent energy is replaced by an average
value, . The number of configurations available for and polymers
chains are taken into account via a pre-factor W. Thus, may be re-writtem as
*
+ (34)
Figure 6. Pictorial depiction for the lattice model of polymer blends. The lines
connecting the sites (with dark circles) represent a segment of a typical polymer chain.
For a polymer blend, the average enthalpy change of mixing for
polymer chains 1 and 2 depends on the average number of neighbouring sites
of the polymer segments 1 and 2 (as shown in fig. 6). The calculation of the
enthalpy of mixing is carried out in a manner similar to that of the regular
solution theory using the lattice model. Upon mixing, some 1-1 and 2-2
contacts are replaced by 1-2 contacts. The enthalpy of mixing reflects the
change in the number of interactions (1-1, 1-2, 2-2) and formation of each 1-2
contact may be put as:
1/2 (1-1) + 1/2 (2-2) (1-2)
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The change in the energy of interaction upon breaking a 1-1 contact and a
2-2 contact to make a 1-2 contact is given by
(35)
Since the lattice model does not account for changes in volume, we can
take the enthalpy change upon mixing as
(36)
where , represents the total number of polymer chains,
represents the number of polymer chains 1 and 2, z is the coordination
number, (z-2) represents number of lattice positions available for interaction,
while represents the degree of polymerization for polymer chain 1 and 2,
and indicates the volume fraction of polymer 1 and 2. Since
(37)
Eq. (36) can be re-written as
(38)
where is polymer-polymer interaction parameter known as Flory-Huggins
interaction parameter and is given as
(39)
Let us assume that interaction has no influence on the entropy of mixing.
Now the entropy S is given by the number of possible configurations Ω of the
particles on the lattice i.e.,
(40)
where,
.
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Analyzing the Phase Behavior of Polymer Blends … 55
Let us assume that each chain has been placed randomly on the lattice site
independent of others. In the homogeneous state, the ensemble configuration
term, W, for each polymer chain will be
(41)
possible positions for the center of mass. However, for the de-mixed state, the
two polymer chains will have
(42)
possible states. Therefore, entropy change due to mixing is given by
And, for each polymer of type 1 and 2, the change in entropy will be
(43)
where , (i = 1 or 2) represents the volume fractions for each of the two
different component of polymers having values always less than 1. Eq. (43)
indicates that the entropy change in polymer blends is always positive. Now,
in order to calculate the entropy involved in the system, the entropy
contribution by individual polymer chains and is added to give
(44)
Substituting ⁄ and ⁄ , and dividing by the
number of lattice sites, one gets the change in entropy per unit volume as
*
+ (45)
As a polymer blend normally comprises n polymer chains, eq. (40) can be
re-written as
*
+ (46)
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Now the Helmholtz free energy of mixing is given by
(47)
By substituting eq. (38) and eq. (46), in eq. (47) we get the most
significant Flory-Huggins formula given below:
*
+ (48)
Now since the pressure (P) and temperature (T) are the typical laboratory
parameters, it will be more relevant to take Gibbs free energy function instead
of the free energy only. Thus, Gibbs free energy function for mixing of two
polymers (per mole) is:
*
+ (49)
The entropy of mixing for the blends which is proportional to the chain
length N of the constituent polymers, is relatively small. As such, the entropy
of mixing is strongly influenced by the chain connectivity of the polymers (for
each polymer component). Entropy of the system thus supports mixing
however; the system must overcome the enthalpy barrier for obtaining a
homogenous blend/mixture of polymers.
The ϕ-dependence of the free energy function or the Gibbs free
energy tells whether a system consisting of two polymers 1 and 2
remains phase-separated or in the mixed-up homogeneous state. Now, if the
blend goes for de-mixing such that polymer 1(or 2) is divided into
concentrations of and having a de-mixed free energy, expressed in
terms of the weighted average of the concentrations and (as shown in
figure 7), then
, and (50)
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Analyzing the Phase Behavior of Polymer Blends … 57
Figure 7. The composition ( of polymer 1(or 2) in the binary mixture (polymer
blend) may be expressed as sum of its concentrations and with volume fractions
of α and β, such that and α +β = 1.
It is worth mentioning here that for an adiabatic system in equilibrium, the
free energy will be the minimum. And, if no phase separation occurs, it
satisfies the stability criteria
( ) (51)
Case-I: Let us assume that the free energy function has a -shaped
ϕ-dependence (as shown in figure 8a). The de-mixed state free energy
will be larger than the mixed one, which implies that overall equilibrium
lies with the mixed state (as shown in figure 8a).
Case-II: On the other hand if ϕ-dependence of the free energy function
has a -shape, the free energy of the phase-separated state will
be lower than that of the mixed state, and the thermodynamic equilibrium
shifts to the state of phase-separation (see figure 8b).
Hence the stability of a polymer blend depends on the free energy which
in turn describes the local stability of a homogeneous mixture of composition,
. One can therefore, conclude that thermodynamically, if the free energy has
-shaped ϕ-dependence of between any two binodal points (say,
and ), it satisfies the relation
|
|
(52)
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Figure 8a. Convex shape of the
curve; phase separation increases the
free energy.
Figure 8b. Concave shape of the
curve; phase separation lowers
the free energy.
which implies that for a phase-separated system the concentrations and
of the two thermodynamically stable phases is uniquely determined by the
points of common tangent. The inflection points separating -shaped from -
shaped represents an instability point, known as spinodal point, is given
by the condition
|
|
(53)
The miscibility of polymer mixtures is characterized by two curves say,
the binodal and the spinodal (as shown in figure 9). For regions outside the
binodal, compositional fluctuations are not large and the blend remains in the
stable one phase region. In the concentration region between the binodal and
spinodal the system will be metastable (represented by M in figure 9) and will
decompose via a nucleation-and-growth mechanism. For concentration within
the spinodal region, the fluctuations are large hence rendering the system to
become totally unstable (represented by U in figure 9) towards formation of a
bi-continuos type morphology of phase-separated domains.
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Analyzing the Phase Behavior of Polymer Blends … 59
Figure 9. Typical phase diagram of a binary mixture (polymer blend) clearly showing
the location of phase boundaries (i.e., the binodal and spinodal curves).
Now the free energy function depends mainly on temperature, the binodal
and spinodal temperatures also vary with composition. Owing to temperature
dependence of the free energy function, the binodal and the spinodal curves
meet at a common point in the concentration-temperature phase-diagram and
is often referred as critical point satisfying the condition
(54)
3.2. Miscibility of Polymer Blends
The critical point is the point above/below which a multi-component
solution/mixture is always in the stable homogeneous/mixed state for any
composition. It is defined mathematically as the point where the second and
third derivatives of the Gibbs free energy with respect to the polymer volume
fraction are zero.
Neutron Scattering is a valuable tool to measure the Flory-Huggins
interaction parameter and hence examine the polymer miscibility phase
diagram leading to the very first signature of phase-separation in polymer
blends and mixtures. The phase separation occurs either via heating or cooling
and the polymer blends/mixtures involved are regarded as lower critical
solution temperature (LCST) and upper critical solution temperature (UCST),
respectively. A number of polymeric systems undergo phase separation
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Nitin Kumar and Jitendra Sharma 60
achieved via both heating and cooling and are known to exhibit miscibility
gap. Some others undergo phase separation achieved from either side of the
phase diagram only within a specific temperature region and thus exhibit
closed loop Immiscibility-Island (see figure 10).
Figure 10. Illustrative diagram of typical phase separation behaviors observed in
polymer blends.
Phase diagrams may have either an Upper Critical Solution Temperature
(UCST) or a Lower Critical Solution Temperature (LCST) behavior. UCST
behavior is obtained generally in systems wherein phase separation occurs
upon cooling. In case of UCST, the interaction parameter shows a
behavior, where A is negative and B is positive whereas phase diagrams with
LCST behavior have positive A and negative B values. In both cases, phase
separation occurs when the interaction parameter exceeds its critical value,
which in turn depends only upon the degree of polymerization of the polymer
chain.
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Analyzing the Phase Behavior of Polymer Blends … 61
4. ANALYSIS OF SANS DATA
In the ensuing subsection, the theoretical basis for linking the
thermodynamic parameters of polymer blends and the experimentally obtained
SANS data will presented with illustrative examples demonstrating their
appropriate analysis methodology.
4.1. Experimental Scattering and the Theoretical Correlation
Function
According to Flory-Huggins theory, Gibbs free energy of mixing two
polymers is given by eq. (49):
[
]
The miscible polymer mixtures and suspensions show spatial
concentration fluctuations even in the mixed single phase system. These
concentration fluctuations provide thermodynamics parameters for the Gibbs
free energy, as described already in the last section. For a real polymer, two
types of fluctuations were analyzed (i) chemical composition fluctuations and
(ii) thermal density fluctuations. The study of fluctuations, as a function of
time, leads to the concept of correlation functions which play an important role
in relating the dissipation properties of a system, with the microscopic
properties of the system in a state of the equilibrium. This relationship
(between irreversible processes on one-hand and equilibrium properties on the
other) manifests itself in the so-called fluctuation-dissipation theorem.
According to the fluctuation-dissipation theorem these fluctuations are
characterized by the material compressibility ⁄ where V is the volume
and P the pressure [8]. In scattering experiments with labeled chains one can
neglect the effects of thermal density fluctuations, as they have negligible
contribution compared to composition fluctuations. So, for simplicity we
consider only the concentration fluctuations and assume a constant segmental
density i.e., no thermal density fluctuations. Let us assume that each site in the
lattice is occupied by the polymer segments (figure 6) and thus fulfills the
constraint
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(55)
These fluctuations are measured readily in a scattering experiment. Thus,
scattering methods offers to give valuable insights into the thermodynamics of
polymer blends. The ensemble averaged concentrations ⟨ ⟩ and
⟨ ⟩ must fulfill the relation . The fluctuating
terms are measured by calculating the deviation of the segment density from
average at each lattice site
(56)
(57)
For two different lattice positions with concentration at position and
at position , the product of the fluctuations being an important
measure of the thermodynamics involved, may be written as
(58)
Thus product (58) describes the correlation between segment i at position
with the one indexed j at position, .
Now, one can describe the thermodynamics of polymeric systems by
taking the ensemble average, of the spatial correlated fluctuations:
⟨ ⟩ (59)
The density constraint implies
(60)
which implies, . And, therefore, we can write
(61)
which, leads to a simple and significant result which for an incompressible
two-component mixture describes the concentration fluctuations through a
characteristic single correlation function linked to the self-correlation
function via
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Analyzing the Phase Behavior of Polymer Blends … 63
(62)
Thus, spatial correlation function which contains important information
about the thermodynamics of the system is also accessible experimentally via
the measurement of its Fourier transform in neutron scattering experiments.
4.2. Phase Boundary from the Forward Scattering
It is also important to obtain relationships between essential elements or
parameters of blend thermodynamics and the correlation function or its Fourier
transform measured experimentally. The derivation uses de Gennes [9]
approach based on the linear response theory to calculate the correlation
function. At the onset, perturbation is introduced by subjecting individual
segments to weak external fields and its response measured on the total
energy. Let us also assume that the two segments marked as 1 and 2 at site
are subjected to fields of potential energy and , respectively. The
resultant change in the potential energy of the system will be
(63)
The external potentials and acting on individual positions will
cause local deviation from average composition. Therefore,
where (64)
At equilibrium, the fluctuations in the average composition can be
expressed as
(65)
where is the internal energy of the system. Eq. (65) simplifies to
⟨ ⟩
⟨ ⟩ (66)
For weak external field, one can make the approximation
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Nitin Kumar and Jitendra Sharma 64
(
)
(67)
Using eq. (67) in eq. (66), we have
( ) ⟨ ( )⟩ ( ⟨
⟩) ⟨ ⟩ (68)
As the equilibrium average of ( ) by definition is zero in the absence
of any external field, which implied that the first term in eq. (67) vanish. So,
we have
( )
⟨ ( ) ⟩ (69)
which after using the expression for may be re-written as
( )
⟨ ( ) ∫ ( ) ( )]
( )∫ ( ) ( ) ( ) ( )] ( )⟩ (70)
Assuming a negligible variation in the potentials ( ) and ( ) relative
to the length scale of fluctuations, the first term will vanish (since the
equilibrium average of ( ) is zero). Therefore,
( )
∫ [
⟨ ( ) ( )⟩ ( )
⟨ ( ) ( )⟩ ( ) ] (71)
The equation derived here, and obtained first by de Gennes [9], is
significant and has important consequences as it states the fundamental linear
response theory applicable to the thermodynamics of polymeric systems.
Accordingly, the thermally averaged local concentration fluctuations has linear
dependence on the fields acting on any other site multiplied by a factor equal
to the spatial correlation function, ( ) and (
) (as defined in eq. (59)).
As ( ) ( ), eq. (71) can be re-written as
( )
∫⟨ ( ) ( )⟩ ( ) (
)] (72)
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Analyzing the Phase Behavior of Polymer Blends … 65
Now using eq. (62) i.e., , one can write
(73)
Since the correlation function is of short range, and may be
considered constant over this range. Therefore, eq. (73) can be approximated
by
(74)
Now the Fourier transform of the spatial correlation function can be
measured directly in scattering experiments and is essentially the structure
factor, given by (eq. (16))
(75)
For , we have , so that the integral in eq. (74) can
be written as
(76)
Since the potentials in eq. (76) are assumed almost constant, fluctuation
may be determined using the conditions of thermodynamic
equilibrium expressed in terms of chemical potentials and . In the
absence of external fields, for a system in thermodynamic equilibrium:
(
)|
(77)
where C is a constant and has no dependence on the concentration. Due to
weak perturbations, the local change in chemical potentials and
will also be small. Therefore, change in the free energy derivative can be
expressed in terms of change in the linked concentration as
(
)|
(
)|
(78)
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For small fluctuations (in ). Therefore, conditions of
thermodynamic equilibrium must satisfy the relation
(
) (79)
which, implies
(
)
(80)
Comparing eq. (80) with eq. (76), one can write
(
)
(81)
The above equation clearly shows that structure factor, , is directly
linked to the thermodynamics of the polymer mixtures/blends. More
importantly
i.e., the spinodal point of the blend can thus be
obtained experimentally by calculating the structure factor at q = 0, and
estimating (through extrapolation) the temperature where goes to zero.
4.3. Applying the Random Phase Approximation (RPA)
Based on the linear response theory, random phase approximation (RPA)
applies further the mean field calculations to take into account the excluded
volume effect, the interactions between the chains (segments) and the density
constraint through perturbations (via the potential energy changes).
Let us consider a binary mixture composed of two polymers marked 1 and
2, respectively and placed randomly on the lattice sites, without excluded
volume effects or interaction energies. For such cases, the spatial correlation
between two polymers satisfies
⟨ ⟩ (82)
Due to the constraint that monomers are joined together
⟨ ⟩ (83)
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Analyzing the Phase Behavior of Polymer Blends … 67
⟨ ⟩ (84)
In the presence of external potential fields of energies and ,
changes in the concentration responds through fluctuations (eq. (71))
expressed as
(85)
The response function here is similar to the pair correlation function
expressed in eq. (73). Now, the enthalpic interactions and the volume
constraint both will be taken into account as perturbations using mean field
theory.
If the concentrations of polymer segments in position are
and
respectively, the enthalpic fields acting on
the segments 1 and 2 are given, respectively by
[
] [
] (86)
The condition of volume conservation i.e., , may
thermodynamically be conceived as a force acting on each site, represented by
the potential, . Now the total potential energy representing the
excluded volume effect is then calculated by integrating over volume
(87)
Since , so we can write it as a factor equivalent to
unity. In the linear response theory the thermal averaged local concentration
fluctuations depend linearly on the fields acting on any other sites, with
proportionality constants equal to the spatial correlation functions and
. Assuming that fields of and are acting on the
segments 1 and 2 respectively, the two averages and
thus may be
expressed as
(88)
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Nitin Kumar and Jitendra Sharma 68
(89)
Figure 11:.Experimental determination of the spinodal curve in a binary phase diagram
using the forward scattering data, , recorded from a SANS measurement.
The constrain , implies
(90)
Equations ((88)-(90)) form a set of simultaneous equations for the
unknowns ,
and . In order to solve these equations, we
will take the Fourier transform of and . The Fourier transform of
, can be written as
∫
∫∫
(91)
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Analyzing the Phase Behavior of Polymer Blends … 69
where the definition of eq. (75) has been used for the structure factor, .
Without loss of generality, eq. (75) may also be used to write an expression for
the unperturbed structure factor of non-interacting polymer chains via
(92)
The Fourier transform for the potential can be written as
[
] (93)
The corresponding Fourier transform for may be written as
[
] (94)
The sum of and
is zero as per
[
] (95)
Solving the above three equations ((93)-(95)) for three unknowns, ,
and , yields
*
+
(96)
Here χ is Flory-Huggins parameter. Comparing eq. (96) and eq. (91) the
structure factor can be written as
(97)
The above equation may be rewritten into more general RPA form
(98)
For polymer blends the function has the form
*
+
(99)
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Nitin Kumar and Jitendra Sharma 70
With average degree of polymerization, N, defined as
(100)
As the chains obey Gaussian statistics with a linked Debye function (as in
eq. (31)) the unperturbed non-interacting polymer structure factor will be
(101)
The Debye function in eq. (31) can be approximated by
; with (
)
(102)
Using the approximation of eq. (102) for the Debye function, the RPA
expression in eq. (97) may be re-written as
(103)
Rearranging the terms one can write
*
+ [
] (104)
which indicates that the scattering function has well known form of Ornstein-
Zernike function given by
(105)
Re-writing eq. (105)
(106)
where
(107)
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Analyzing the Phase Behavior of Polymer Blends … 71
represents the structure for forward scattering, and is linked directly to the
thermodynamics of the blends, and the correlation length ξ of the fluctuations
is given by
[
]
√ (108)
Application of RPA offers the means to measure experimentally the
essential parameters needed to understand the thermodynamics of polymer
blends, as already discussed in Flory-Huggins mean field model. Such
measurement of the scattering function as function of temperature and
composition enables one to determine both the phase boundary and the Flory-
Huggins interaction parameter (χ) of the system. And, the mean-field theory
predicts the following scaling relations for the forward scattering and
correlation length:
and (109)
As the spinodal is approached the zero wave vector structure factor
diverges and so is the correlation length as evident from eq. (109).
Figure 12 shows a typical SANS experimental data obtained at six
different temperatures of a 50:50 polymer blend of deuterated polystyrene
(PSD) and poly(vinyl methyl ether) (PVME). The zero wave vector scattered
intensity is estimated from the extrapolation of versus plot in the
low-q region of the fitted curve, i.e., the Ornstein-Zernike function (eq. (106)).
The slope/intercept ratio obtained from the fit of the Ornstein-Zernike function
readily gives the value of the correlation length which could be used to
estimate the spinodal temperature using eq. (109).
Figure 13 shows such an estimate of the spinodal temperature using a plot
of versus for two different polymer blends of PSD and PVME having
compositions of 50:50 with the molecular weight ratios of 300K/90K (value of
estimated spinodal temperature ~ 158 0C) and 500K/90K (estimated spinodal
temperature ~ 173 0C).
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Nitin Kumar and Jitendra Sharma 72
Figure 12. SANS Profile of scattered intensity at different angles for a 50:50 (% wt/wt)
blend of deuterated polystyrene (PSD) and poly(vinyl methyl ether) PVME of
approximate molecular weights 500K (MW = 507,695) and 90K (MW = 90,000),
respectively at six different temperatures. The data presented are fitted to Ornstein-
Zernike function described in eq. (105). Experimental data reproduced from Sharma et
al. (© 2010 Elsevier) [10].
Figure 13. Estimation of the spinodal temperature from plots of versus
(according to eq. (109)) for the blends of PSD-500K/PVME-90K (described in figure
12) and also a yet another blend of PSD-300K/PVME-90K. Experimental data
reproduced from Sharma et al. (© 2010 Elsevier)[10]. The estimated values of the
spinodal temperatures are ~ 173 and ~ 158 0C, respectively.
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Analyzing the Phase Behavior of Polymer Blends … 73
Some general comments about the RPA approach and the parameters
obtained are important and need to be outlined here. The interaction parameter
χ is purely enthalpic and of short range. It only takes into account the
interactions that occur between the segments of the polymer chains, which are
mostly dominated by dipole-dipole interactions. The actual value of χ obtained
experimentally will perhaps be more complex. However, a more precise value
of χ is obtained for high molar mass polymers, where the end effects are
negligible and the temperature dependence occur as scaling only. In spite
of such limitations, the Flory-Huggins model and mean field random phase
approximations are considered to be more effective for studying the
thermodynamics of polymer melts and blends.
4.4. The Mean-Field Cross-Over
Mean field theories have offered robust tools for studying the dynamics of
a network of coupled systems. They are convenient because they are simple
and relatively easier to analyze. However, classical mean field theory neglects
the effects of fluctuations and correlations. As such the random phase
approximation based on the mean-field theory considers a completely random
distribution of the polymer chains where the approach of perturbation is
adopted to study the effects of the resultant interactions. According to
Ginzburg criteria, these types of calculations are only valid as long as the
length scale of fluctuations is smaller compared to the characteristic lengths of
the system.
As such in the small-fluctuation limit, the validity of the subtle link
between rheology and thermodynamics of polymer blends is also established
giving precise estimate of both the binodal and spinodal temperatures of an
LCST polymer blend [11]. Larger thermal fluctuations close to the critical
point, renormalizes the thermodynamics and the assumption of random
organization of polymers is longer valid as the correlation length of
fluctuation, becomes larger than the segmental length, ℓ. So, a more precise
theory that includes the effects of thermal composition fluctuations in a self-
consistent manner (as the phase boundary is approached) is need to explain
this change in the mean field to non mean field behavior of the system.
Taking into account the temperature-dependence of the susceptibility,
, a mean-field critical temperature can be estimated by extrapolating
versus data of the high temperature region. As per Ginzburg
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Nitin Kumar and Jitendra Sharma 74
criterion [12, 15], the relationship between the mean field critical temperature
and the critical temperature , can be written as
⁄ (110)
where is the Ginzburg number. Figure 14 shows the data of forward
scattering versus plot for a polymer mixture of polyisopropylene and
poly(ethylene-alt-propylene) where a deviation from the linear relationship is
clearly visible. The entire data can be fitted to a power law behavior of
(111)
where, (
) is the reduced temperature and is the critical exponent. In
the linear regime has a value of unity while the non-linear regime had a
critical exponent value of . It marks the cross-over from a mean field
( ) to classical fluid (ordinary liquid mixture) behavior exhibiting a 3-d
Ising type scaling [14, 15].
Figure 14. Plot of the forward scattering and the estimated inverse
correlation length as function of temperature for the mixture polyisoprene and
poly(ethylene-alt-propylene) having molecular weights of 2000 and 5000, respectively.
While is the stability limit, the mean-field to non-classical crossover temperature is
represented by , and the dashed lines correspond to Ising theory described in the text
[16]. Experimental data reproduced from Ńtěpánek et al. (© 2002 Elsevier).
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Analyzing the Phase Behavior of Polymer Blends … 75
Experimentally obtained scattering intensity, versus , shown in
figure 14 can be analyzed effectively by applying a single function describing
within the entire one-phase regime. The parameters of such crossover
function are the Ginzburg number, the critical temperature, and the critical
exponents [12].
The limitations of the Flory-Huggins mean field approach is evident from
figure 14 as rather than the region of mean field validity it is the crossover
range of temperature that becomes more relevant for miscible polymer blends.
Experimental studies performed on polymer blends showed that the variation
from the mean-field characteristics, as expressed by Ginzburg number, is
dependent on the degree of polymerization, N. For non-meanfield
characteristics the temperature scale range can be expressed as
| | (112)
where, the exponent exhibits a value of the order of 1 to 2 [17]. Thus
Ginzburg number quantifies the temperature range where thermal fluctuation
affects the Gibb‘s free energy and the 3-d Ising behavior becomes visible.
REFERENCES
[1] Oberdisse J., Adsorption grafting on colloidal interfaces studied by
scattering techniques, Current Opinion in Colloid & Interface Science,
12, 3–8 (2007).
[2] Boué F., Cousin F., Gummel J., Carrot G., Oberdisse J., El Harrak
A., Small angle scattering from soft matter - application to complex
mixed systems, C.R. Physique, 8, 821–844 (2007).
[3] Frielinghaus H., 35th Spring School - Physics meets Biology,
Forschungszentrum Jülich GmbH, Jülich (2004).
[4] Monkenbusch M., 38th Spring School - Probing the Nanoworld,
Forschungszentrum Jülich GmbH, Jülich (2007).
[5] Koester L., Rauch H., Seymann E., Neutron scattering lengths: a survey
of experimental data and methods, Atomic Data and Nuclear Data
Tables, 49, 65-120 (1991).
[6] Doi M., Introduction to polymer physics, Clarendon Press, Oxford
(1996).
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Nitin Kumar and Jitendra Sharma 76
[7] Flory P., Principles of polymer chemistry, Cornell University Press, New
York (1953).
[8] Higgins J.S., Benoit H.C., Polymers and Neutron Scattering, Clarendron
Press, Oxford (1994).
[9] de Gennes, P., Theory of x-ray scattering by liquid macromolecules with
heavy atom label. Le Journal de Physique, 31, 235–238 (1970).
[10] Sharma J., King S.M., Bohlin L., Clarke N., Apparatus for simultaneous
rheology and small-angle neutron scattering from high-viscosity
polymer melts and blends, Nucl. Instrum. Meth. A, 620, 437-444 (2010).
[11] Sharma J., Clarke N., Miscibility determination of a lower critical
solution temperature polymer blend by rheology, J. Phys. Chem. B, 108,
13220-13230 (2004).
[12] Anisimov M.A., Kiselev S.B., Sengers J.V., Tang S., Crossover
approach to global critical phenomena in fluids, Physica A, 188, 487-525
(1992).
[13] Sengers J.V., Supercritical Fluids: Fundamentals for Application: Effect
of Critical Fluctuations on Thermodynamic and Transport Properties of
Supercritical Fluids, ed. by Kiran E. and Sengers J.M.H. Levelt, Kluwer
Academic, Dordrecht (1994).
[14] Schwahn D., Mortensen K., and Yee-Madeira H., Mean-field and Ising
critical behavior of a polymer blend, Physical Review letters, 58, 1544-
1546 (1987).
[15] Bates F.S., Rosedale J.H., Stepanek P., Lodge T.P., Wiltzius P.,
Fredrickson G.H., Hjelm Jr. R.P., Static and dynamic crossover in a
critical polymer mixture, Phys. Rev. Lett., 65, 1893-1896 (1990).
[16] Ńtěpánek P., Morkved T.L., Krishnan K., Lodge T.P., Bates F.S.,
Critical phenomena in binary and ternary polymer blends, Physica A,
314, 411 – 418 (2002).
[17] Schwahn D., Meier G., Mortensen K., Janßen S., On the n-scaling of the
Ginzburg number and the critical amplitudes in various compatible
polymer blends, J. Phys. II (France), 4, 837-848 (1994).
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In: Recent Progress in Neutron Scattering … ISBN: 978-1-62948-099-2
Editors: A. Vidal and M. Carrizo © 2013 Nova Science Publishers, Inc.
Chapter 3
SMALL-ANGLE NEUTRON SCATTERING
AND THE STUDY OF NANOSCOPIC
LIPID MEMBRANES
Jianjun Pan1,2*
, Frederick A. Heberle2
and John Katsaras2,3,4*
1Department of Physics, University of South Florida,
Tampa, Florida, US 2Biology and Soft Matter Division, Neutron Sciences Directorate,
Oak Ridge National Laboratory, Oak Ridge, Tennessee, US 3 Department of Physics and Astronomy, University of Tennessee,
Knoxville, Tennessee, US 4Joint Institute for Neutron Sciences, Oak Ridge National
Laboratory, Oak Ridge, Tennessee, US
ABSTRACT
Recent advances in structural, molecular and cellular biology have
shown that lipid membranes play pivotal roles in mediating biological
processes, such as cell signaling and trafficking. Like proteins and nucleic
acids, lipids in membranes carry out these roles by modifying their
transverse and lateral structures. Along the bilayer normal direction,
* Corresponding author: Jianjun Pan, Ph.D. E-mail: [email protected]
* Corresponding author: John Katsaras, Ph.D. E-mail: [email protected]
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Jianjun Pan, Frederick A. Heberle and John Katsaras 78
membranes can assume different hydrophobic thicknesses by adjusting
their lipid chemical composition, which can in turn lead to altered protein
function through hydrophobic mismatching. In the two-dimensional
lateral direction, lipid membranes can self-compartmentalize into
functional domains (rafts) for organizing membrane-associated
biomolecules. In this chapter we focus on the use of small-angle neutron
scattering (SANS) to resolve nanoscopic structure, both normal to and in
the plane of the membrane. We present structural parameters of
phospholipids with different head moieties (i.e., choline and glycerol)
using the neutron scattering density profile model, which is guided by
molecular dynamics simulations. We also show the role of the glycerol
backbone in mediating different modes of interaction between
phospholipids and their neighboring cholesterol molecules. We conclude
the chapter by showing how SANS is used to reveal membrane lateral
heterogeneity, specifically by describing our recent findings which
showed that raft size varies as a function of the difference in membrane
thickness between raft and non-raft domains.
1. INTRODUCTION
Since the first neutron scattering experiments in the 1940s, substantial
progress has been made in the use of neutron scattering techniques to study a
wide variety of materials. Neutrons are electrically neutral, subatomic
elementary particles, found in all atomic nuclei except hydrogen (H), and have
a nuclear spin of 1/2. Moreover, unlike photons (visible and X-rays), neutrons
are scattered by atomic nuclei, enabling them to differentiate between an
element and its isotopes. These properties make neutrons a suitable probe for
studying both bulk and magnetic properties of materials. In addition, state-of-
the-art neutron sources, including reactor- and accelerator-based (Dura et al.,
2006; Pan et al., 2012b), are being developed and commissioned worldwide.
The greater availability of neutron facilities, coupled with enhanced
instrumentation (e.g., higher flux, increased resolution and improved sample
environments), make neutron scattering an indispensable and powerful
technique for materials research.
In this chapter, we describe applications of small-angle neutron scattering
(SANS), which has been used extensively to explore particle size, shape, inter-
and intra-particle nanostructures in studies of mesoporous and soft (biological
and polymeric) materials, magnetic films, etc., to name a few (Hollamby,
2013; Melnichenko and Wignall, 2007). In the case of biological materials, a
powerful method used by many SANS experiments is the so-called isotopic
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Small-Angle Neutron Scattering and the Study of Nanoscopic … 79
substitution method, whereby one exploits the differences in scattering
between hydrogen and its isotope, deuterium (D). Therefore, by selectively
substituting hydrogen atoms (abundant in biological samples) with deuterium
atoms, different contrast scattering spectra can be obtained. Data analysis is
then used to reveal nanoscopic structural information of the system in
question. Along the same line, varying the D2O/H2O ratio results in different
contrast conditions between the sample and solvent, which give rise to
different neutron scattering length density (NSLD) spectra. The different
methods of creating contrast (i.e., selective labeling or exchange of H2O for
D2O) can also be combined, such that portions of the biomolecule can be made
―invisible‖ to neutrons, while other parts of the molecule can have their
neutron contrast enhanced. We will discuss, in detail, the use of these
techniques as they pertain to structural studies of lipid membranes.
Ever since the inception of the fluid mosaic model of biological
membranes (Singer and Nicolson, 1972), significant developments have been
made with regard to their structure and function. Today it is well recognized
that biological membranes are complex, mesoscopic assemblies that possess
functions which are far more elaborate than a simple permeability barrier in
which proteins reside and carry out their associated functions. Rather,
biomembranes are highly functional dynamic machines that are central to a
host of biological processes, such as the transport of ions and biomolecules,
and signaling, to name a few. In this chapter we focus on the structural aspects
of lipid membranes (i.e., transverse and lateral structures) using the powerful
capabilities offered by the SANS technique, either alone or in combination
with techniques, such as small-angle X-ray scattering (SAXS). By doing so,
we hope to provide some insight into the evolution of the chemical complexity
found in cell membranes.
This chapter is divided into two parts. The first part introduces the
recently developed scattering density profile (SDP) model, and describes how
it is used to jointly refine SANS and SAXS data. In the second part, we
discuss how to extract membrane lateral structure from contrast-matched
samples, using SANS. In particular, we describe a numerical method that has
been successfully applied to obtain domain size in model lipid membranes. We
conclude the chapter by describing how the relationship between line tension
and domain size was elucidated through a series of SANS experiments.
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2. TRANSVERSE MEMBRANE STRUCTURE
2.1. Membranes Are Flexible and Disordered Entities
Conformational flexibility of membranes is an essential feature for many
biological processes. For example, highly curved membranes are formed
during cell endocytosis and cell division. Because the energy cost associated
with membrane deformation increases quadratically as a function of increased
membrane stiffness (Pan et al., 2008a; Rawicz et al., 2000), a flexible
membrane is preferred as it is easier to deform. Aside from the mean field
description of membrane flexibility/fluidity, the spatial and temporal
distribution of individual lipids in a membrane environment confers more
complexity. At any moment, the spatial distribution of each lipid is unique,
i.e., the coordinate of one lipid cannot be transformed into that of another lipid
by simple mathematical translational and rotational operations. Moreover,
even within one lipid, atoms are constantly undergoing vibrational and
rotational movements in the vicinity of their equilibrium positions. As a result,
individual atoms within a membrane cannot be accurately described in three-
dimensional space, something which is commonly done in atomic resolution
protein and nucleic acid crystallography. Rather, the ―fluid structure‖ is
inherently of lower resolution: neighboring lipid atoms must be grouped into
discrete components, depending on their intrinsic physical characteristics such
as scattering power, hydrophobicity, etc. The probability distributions of these
components serve to describe the time-averaged structure of the membrane
(Heberle et al., 2012). Stated another way, due to the fact that biological
membranes are intrinsically disordered entities, their structures are best
described by probability distributions (i.e., the probability of finding one atom
or a group of atoms at a specific position over an extended period of time).
2.2. Small-Angle Neutron and X-Ray Scattering
Small-angle scattering (SAS) is a technique commonly used at
synchrotron (X-rays) and neutron facilities to resolve lipid bilayer structure. In
SAS experiments the scattered signal depends on the difference in scattering
power between membrane components and the aqueous solvent. In the case of
elastic small-angle X-ray scattering, electrons in the lipid bilayer interact with
the incident beam, giving rise to an X-ray signal. The X-ray signal is then used
to determine the lipid bilayer one-dimensional electron density (ED)
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Small-Angle Neutron Scattering and the Study of Nanoscopic … 81
distribution. Components such as the electron-rich phosphate in the polar
headgroup scatter X-rays more strongly–i.e., X-ray scattered intensity
increases as the square of the number of electrons.
The situation, however, is different in elastic small-angle neutron
scattering experiments, where incident neutrons interact with atomic nuclei
(described above). Similarly to X-rays, the neutron scattering power of each
atom or component is characterized by its NSLD. It is well known that
hydrogen and its isotope deuterium have similar coherent neutron scattering
powers, but scatter 180o out-of-phase with each other (i.e., hydrogen‘s
coherent scattering length is negative). Importantly, all of the other atoms
commonly found in biological molecules possess similar neutron scattering
characteristics as those of deuterium. Therefore, in the case of lipid bilayers
surrounded by water (H2O), the SANS signal is most sensitive to hydrogen
deficient moieties, such as the headgroup phosphate and glycerol-carbonyl
backbone. To take full advantage of the scattering power difference between
hydrogen and deuterium, as well as to enhance the mean NSLD difference
between the membrane and solvent, deuterated water (D2O) is often used. In
this case, the lipid bilayer can be approximated as a slab with low NSLD
surrounded by D2O, which has an elevated NSLD. Under these experimental
conditions, the overall bilayer thickness (i.e., the width of the slab) can be
accurately determined (Kučerka et al., 2004). However, when the ratio of
D2O/H2O decreases, the slab model is no longer an accurate description of
bilayer structure, and a more sophisticated bilayer model is required to
correctly interpret the detected SANS signal.
Even though SAXS or SANS are routinely used as standalone techniques
to extract lipid bilayer structure, the scattering power distribution of a lipid
bilayer limits the capability of both techniques in their ability to reveal detailed
structural information. For example, SAXS is suitable for determining the
phosphate headgroup position, but is less sensitive to hydrocarbon chain
thickness. Similarly, SANS is sensitive to the overall lipid bilayer thickness
and the hydrocarbon chain thickness—especially when aided by independently
obtained volumetric information—at high D2O concentration. Moreover,
SANS data is usually of lower resolution than SAXS data (q < 0.3 Å-1
). The
poorer resolution of neutron data also limits the use of low D2O concentrations
for determining transverse lipid bilayer structure. However, by combining the
two techniques, the resulting membrane structure is not only more detailed,
but also more robust than structures obtained from standalone SAXS or SANS
data.
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2.3. Scattering Density Profile Model
The recently developed scattering density profile (SDP) model uses both
SAXS and SANS data to accurately determine lipid bilayer structure (Kučerka
et al., 2012; Kučerka et al., 2008; Kučerka et al., 2011; Pan et al., 2012c). By
properly parsing the lipid bilayer (Figure 1A), the component ED (Figure 2A)
and NSLD (Figure 2B) profiles have the same functional form as the
component volume probability distribution profile (Figure 1B) (Heberle et al.,
2012). As a result, the total ED and NSLD profiles can be represented by a set
of volume probability distribution functions, multiplied by the corresponding
X-ray and neutron scattering power (i.e., number of electrons and neutron
scattering length, respectively). Through Fourier transformation the theoretical
ED and NSLD profiles are used to obtain the X-ray and neutron scattering
form factors (FF), which can be directly compared to the experimental X-ray
and neutron form factors (calculated as the square root of the scattered
intensities). The bilayer structure—that is, the component volume probability
profile and corresponding ED and NSLD profiles—is obtained through
conventional non-linear least-squares minimization.
Figure 1. (A) A snapshot of a POPG lipid bilayer obtained from MD simulations. The
bilayer is divided into distinct components, depending on their scattering power and
volume probability distribution. Specifically, the headgroup (i.e., G1, G2 and G3) and
hydrocarbon chain regions (i.e., CH3, CH2 and CH) are each described by three
distinct components. (B) Volume probability distributions of the lipid components and
water. The color scheme for lipid components follows that of the molecular
representation in (A).
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Small-Angle Neutron Scattering and the Study of Nanoscopic … 83
Figure 2. Component ED (A) and NSLD (B) profiles calculated from component
volume probability distributions in Figure 1B. The NSLD profile of the 100% D2O (B,
gray line) has been divided by 4 for better visualization of the lipid components. The
theoretical X-ray (C, solid line) and neutron (D, solid lines) form factors were jointly
refined against the experimental form factors (symbols). For the neutron form factors,
three D2O concentrations were used in order to increase the experimental data sets
(100, 70 and 50% for green, red and marine, respectively). Notice that the resolution in
reciprocal space is ~ 0.6 Å-1
for SAXS and 0.2 Å-1
for SANS before the scattering
signal was overwhelmed by the incoherent background scattering signal. For
component ED and NSLD profiles, the same color scheme is used as in Figure 1.
An example of SDP model analysis for the fluid 1-palmitoyl-2-oleoyl-sn-
glycero-3-phosphatidylglycerol (POPG) bilayer is shown in Figures 1 and 2
(Pan et al., 2012c). The bilayer was parsed into several structural components
based on information obtained from zero surface tension molecular dynamics
(MD) simulations (Figure 1A). In particular, the hydrated phosphorylcholine
headgroup was parsed into three components: the glycerol-carbonyl backbone
(G1, brown), the phosphate group (G2, magenta), and the terminal glycerol
group (G3, green). The hydrocarbon chain region was parsed into terminal
methyl (CH3, marine), methylene (CH2, cyan), and methine (CH, orange)
groups. The volume probability distributions of these components are shown
in Figure 1B. The G1, G2, G3, CH3 and CH components are represented by
Gaussian functions, and the CH2 component is represented by two error
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Jianjun Pan, Frederick A. Heberle and John Katsaras 84
functions from which the CH3 and CH distributions are subtracted. Last to be
considered is the water distribution: It is obtained from the assumption
(supported by MD simulations) that no persistent vacancy exists in the lipid
bilayer-water system. The volume probability of water molecules (gray line in
Figure 1B) is then obtained by subtracting the total lipid volume probability
from unity (Kučerka et al., 2008).
Importantly, lipid bilayer structural parameters such as lipid area A,
overall bilayer thickness DB, and hydrocarbon chain thickness 2DC are
determined from the component volume probability distribution shown in
Figure 1B. For example, the Gibbs dividing surface of the water/lipid
interface, which is located at the midpoint of the overall bilayer thickness
(DB/2), is defined as a point where the integrated water distribution on one
side and its deficiency on the other side, are equal (areas colored in yellow and
separated by vertical dashed line in Figure 1B). The lipid area is related to the
total bilayer thickness through the relationship A = 2VL/DB, where VL is the
lipid volume, which is determined by a separate density measurement. The
hydrocarbon chain thickness 2DC is represented by the width of the two error
functions, which are the sum of the volume probabilities of CH3, CH2 and
CH.
Figure 2 shows the result from SDP model analysis. The one-dimensional
ED (Figure 2A) and NSLD (Figure 2B) profiles are obtained by multiplying
the component volume probability distribution (Figure 1B) with the
corresponding component‘s total electron number and neutron scattering
length (NSL), respectively. The obtained ED and NSLD profiles are then
jointly refined against the experimental X-ray (Figure 2C) and neutron (Figure
2D) form factors. The best-fit structural parameters representing the
component volume probability distributions are obtained when the difference
between the theoretical from factors (solid lines in Figure 2C and 2D)—which
are calculated from the corresponding ED and NSLD profiles—and the
experimental form factors, is minimized.
2.4. Lipid Headgroup Moiety Influences Lipid Bilayer Structure
To optimize water contact at the headgroup/water interface and to prevent
water from penetrating beyond the headgroup/chain interface, the lipid
headgroup assumes a polarization or charge, depending on its chemical
composition and the environmental conditions (e.g., pH and salt concentration)
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in which it resides. The amphiphilic nature of the lipid headgroup makes it a
preferred binding partner for many important biomolecules. For example, the
lipid headgroup has been implicated to play an essential role in proton lateral
diffusion, enzymatic activity of protein kinase C, the orientation and topology
of membrane proteins, the activation of ion channels, and the activation of
membrane peptides.
Cell membranes consist of a plethora of different lipid headgroup
moieties, with the zwitterionic phosphatidylcholine (PC) headgroup being the
most prominent type (found in both prokaryotic and eukaryotic membranes).
In contrast, the mono-anionic phosphatidylglycerol (PG) headgroup is rare in
mammalian cells, but common in bacterial membranes. To examine the effect
of the headgroup on lipid bilayer structure, Pan and coworkers applied
the aforementioned SDP analysis on SAXS and SANS data collected
from water-suspended PC and PG unilamellar vesicles (ULVs) in their
biologically relevant fluid phase (Kučerka et al., 2011; Pan et al., 2012c).
A significant result from studies of these two commonly used lipids is shown
in Figure 3. Surprisingly, despite POPC‘s (1-palmitoyl-2-oleoyl-sn-glycero-3-
phosphatidylcholine) bulkier headgroup (Figure 3A), it was found that at the
same temperature POPG possesses a larger area per lipid. This implies that the
long-range electrostatic interaction between charged PG headgroups plays a
prominent role in the lateral packing of lipid molecules. This finding also
challenged previous MD simulations performed using the Gromacs force
fields. Those simulations obtained a smaller area for PG, compared to PC,
indicating an inadequate parameterization of inter-lipid interactions. The
newly obtained lipid areas for charged PG lipids are consistent with the
finding that the addition of anionic lipids reduces the membrane rupture
pressure (Shoemaker and Vanderlick, 2002).
From the data above-mentioned analysis of the data, Pan and coworkers
also found that POPC has a larger hydrocarbon chain thickness than POPG
(Figure 3C). Many membrane proteins (e.g., ion channels and GPCR proteins)
control their active-inactive state by altering the conformation of their
transmembrane segments (e.g., helices). Protein structural rearrangement is
affected, for example, by the difference between the membrane hydrophobic
thickness and the thickness of protein transmembrane segments. The different
hydrocarbon chain thicknesses of POPC and POPG may account for the fact
that mammalian and bacterial membranes use homologous proteins with
different amino acid sequence and modulated structures (e.g., hydrophobic
thickness) to achieve the same biological function.
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Figure 3. (A) The POPC headgroup has a negatively charged phosphate group (gold)
and a positively charged ammonium group (blue), while the POPG headgroup has a
negatively charged phosphate group (gold) and a neutral terminal glycerol. Panels (B)
and (C) show temperature-dependent structural parameters (i.e., lipid area A and
bilayer hydrocarbon chain thickness 2DC) of POPC and POPG obtained from SDP
model analysis.
2.5. Lipid Backbone Structure Mediates Cholesterol Location
Not unlike lipid headgroup diversity, the linkage connecting the lipid
headgroup to its hydrocarbon chains also exhibits substantial heterogeneity.
Specifically, lipids whose hydrocarbon chains are attached to the glycerol
backbone through an ether linkage (Guler et al., 2009), instead of the more
common ester linkage (Figure 4A), make up an important class of lipids. Ether
lipids have been implicated in reducing oxidative stress, regulating lipid
metabolism and killing tumor cells (Paltauf, 1994). Moreover, ether lipids are
thought to help maintain the cholesterol gradient between the endoplasmic
reticulum and the plasma membrane by mediating intracellular cholesterol
trafficking (Gorgas et al., 2006; Munn et al., 2003).
To better understand the intermolecular interactions used by ether lipids to
facilitate cholesterol trafficking, Pan et al. used SAXS and SANS experiments,
complemented with all-atom MD simulations, to study the bilayer structure
formed by cholesterol and the ether lipid 1,2-di-O-hexadecyl-sn-glycero-3-
phosphocholine (DHPC) (Pan et al., 2012a). Initial MD simulations were
carried out to formulate an SDP model. Membrane structure was then
determined by analyzing SAXS and SANS data using the SDP model, as
described in section 2.3. Finally, a set of refined MD simulations were
performed using experimentally determined bilayer structural parameters.
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Because the refined simulations were able to reproduce the correct lipid
bilayer structure, it was suggested that more accurate intermolecular
interactions could be inferred. Indeed, the authors found that in the absence of
cholesterol, the ether DHPC lipid has a larger lipid area than its ester
counterpart DPPC (1,2-dipalmitoyl-sn-glycero-3-phosphatidylcholine). This
finding indicates that the two hydrogen atoms at the ether linkage impart a
greater steric repulsive force than the corresponding carbonyl oxygen in the
ester lipid. It was therefore suggested that the chemical composition at the
linkage position plays an important role in determining membrane lateral
packing.
Figure 4. (A) The backbone of the ester DPPC lipid has a carbonyl oxygen atom (red
spheres) attached to each chain, while the oxygen atom is replaced by two hydrogen
atoms (white spheres) in each chain of the ether lipid, DHPC. (B) Lipid apparent areas
ALapp of DPPC and DHPC as a function of cholesterol concentration at 60oC. ALapp is
defined as the lipid hydrocarbon chain volume VHL divided by half of the hydrocarbon
chain thickness DC. DPPC data were taken from Mills et al. (Mills et al., 2008). (C)
DHPC hydrocarbon chain (cyan) arrangement surrounding a cholesterol molecule
(magenta). The snapshot was obtained using lipid area-constrained MD simulations.
Addition of cholesterol to an ether bilayer condenses lipid packing, as
indicated by the smaller apparent lipid area ALapp (Figure 4B), which is defined
as the ratio of lipid hydrocarbon chain volume VHL and half the lipid bilayer
hydrocarbon chain thickness DC. This well-known cholesterol ―condensation‖
effect is comparable between ether and ester lipids, as indicated by their
similar slope shown in Figure 4B. Inputting the experimentally determined
lipid bilayer structure into MD simulations reveals interesting molecular
features. Figure 4C shows a snapshot of lipid chain arrangement surrounding a
cholesterol molecule. The six chains of five neighboring DHPC lipids form a
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pseudo-hexagonal cage, accommodating one cholesterol molecule in its center.
However, a closer examination reveals that hydrocarbon chain packing is not
symmetrical around the cholesterol. The packing near cholesterol‘s rough face
(characterized by the two ring methyl groups) is ―looser‖ than near its smooth
face (Figure 4C). This asymmetrical chain packing near a cholesterol molecule
may explain the distribution of cholesterol tilt angles, which is centered at
approximately 20o from the bilayer normal (Pan et al., 2012a).
More importantly, the refined MD simulations revealed that cholesterol‘s
hydroxyl group (Chol-OH) interacts preferentially with the phosphate non-
ether oxygen atoms. This contrasts with ester lipids, where Chol-OH
predominantly forms hydrogen bonds with the backbone carbonyl oxygen
(Chiu et al., 2002; Pasenkiewicz-Gierula et al., 2000; Smondyrev and
Berkowitz, 1999). As a result, cholesterol is positioned closer to the
membrane/water interface in an ether bilayer than in an ester bilayer. The
unique location of cholesterol in the ether lipid bilayer may provide an
explanation for ether lipid assisted cholesterol trafficking. It is conceivable
that cholesterol‘s elevated position in the ether membrane allows it to be more
readily extracted from the membrane, thereby accelerating its trafficking.
3. MEMBRANE LATERAL STRUCTURE
3.1. Introduction to Membrane Rafts
The cell plasma membrane (PM) is a complex mixture composed of
functional proteins, sterols, and a variety of lipids that act either as cofactors or
general modulators of bulk membrane properties. From the perspective of
thermodynamics, there is a tendency (or potential) for the various membrane
components to randomize within the plane of the bilayer in order to maximize
the translational degrees of freedom of the system, i.e., entropy-driven mixing.
However, like many problems in physics, the energy landscape describing the
pairwise interactions of mixture components (or enthalpy) contains hills and
valleys, and is not the same for different pairs within the membrane system. A
consequence of this enthalpy consideration is that membrane components may
possess a tendency to cluster in order to lower the total energy. The
competition between entropy and enthalpy dictates the extent of non-random
mixing of bilayer components, from small clusters up to complete phase
separation.
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Consistent with the thermodynamic view of cellular membranes,
researchers have long postulated the existence of functional micro-domains, or
membrane rafts, which are enriched in membrane machinery proteins through
the preferential association with raft lipids and sterols. The current consensus
pertaining to membrane lateral heterogeneity is that rafts, which are enriched
in sphingomyelin (SM) and cholesterol, effectively compartmentalize the cell
PM into ordered and disordered regions. These compartments serve to separate
proteins based on their preferential interaction with certain types of lipids, and
control protein functions by influencing protein diffusion and local
concentration (Lingwood and Simons, 2010). An intriguing consequence of
the thermodynamic perspective is that small perturbations to thermodynamic
variables including pressure, temperature, or mixture composition may
dramatically affect the phase state of the mixture, in turn causing significant
redistribution of membrane proteins, and thereby allowing for some degree of
cellular control over raft-mediated processes.
Over the years, studies using high-resolution techniques have suggested
that nanoscopic assemblies exist in biological membranes, but their stability
and lifetimes are presently not known. In addition, various cellular functions
have been linked to lipid rafts, including cell signaling pathways (Foster et al.,
2003; Simons and Toomre, 2000), protein sorting (Anderson and Jacobson,
2002), protein activity modulation (Jensen and Mouritsen, 2004) and
cytoskeletal connections.
Due to the enormous chemical complexity of cell membranes, and the
experimental difficulties associated with intact cells, a fruitful alternative for
elucidating the properties of lipid rafts is to study chemically simplified model
systems with well-controlled lipid compositions. In this respect, lipid mixtures
mimicking the composition of the outer leaflet of the mammalian PM have
received substantial attention. Indeed, macroscopic liquid-liquid phase
separation has been reported in a variety of such mixtures. A typical ternary
lipid mixture capable of exhibiting lateral heterogeneity contains a high-
melting (high-TM) lipid (e.g., SM or saturated long chain phosphatidylcholine
such as DPPC or DSPC), a low-melting (low-TM) lipid with unsaturated (e.g.,
1,2-dioleoyl-sn-glycero-3-phosphatidylcholine, DOPC) or branched
(DiPhytanoylPC) chains, and a sterol molecule, which has a different affinity
for the high-TM and low-TM lipids. By tailoring lipid composition and
temperature, a broad spectrum of lateral segregation between the liquid-
ordered (Lo) and liquid-disordered (Ld) phases can be obtained (Marsh, 2009,
2010). Importantly, the phase diagrams of these model systems provide critical
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information for understanding lipid-lipid interactions governing raft formation
and stability.
3.2. The Principle of SANS in Studying Nanoscopic Rafts
Rafts are currently thought to be ordered domains of nanoscopic
dimensions that can coalesce to form larger, stable platforms upon cell
stimulation (Lingwood and Simons, 2010). In this respect, despite the wide use
of conventional fluorescence microscopy in studying micron-sized phase
separation, light microscopy is fundamentally insufficient for studying the
physiologically relevant nanometer-sized rafts due to the physical limits of
spatial resolution imposed by the diffraction limit (i.e., light microscopy
cannot resolve structures smaller than ~ 200 nm). Moreover, it is well
documented that extrinsic fluorescence probes may perturb the native phase
behavior of a lipid mixture, to an extent that depends not only on the identity
and concentration of the probe, but also on the properties of the mixture itself
(Zhao et al., 2007). Therefore, techniques that are immune to probe-induced
perturbations and have nanometer-scale sensitivity are highly desirable. SANS
is uniquely suited for such a task, and is capable of resolving nanometer-scale
heterogeneity (~ 5-100nm) with minimal perturbation to the native lipid
composition (requiring only partial deuteration of a lipid component).
Importantly, domain sizes obtained from SANS measurements bridge the
micron-sized domains observed by microscopy and the nanometer-sized
domains obtained from high-resolution spectroscopy techniques, such as
Förster resonance energy transfer (FRET) and electron spin resonance (ESR).
SANS therefore offers the unique possibility to delineate the various
physicochemical parameters that control domain size and morphology.
A simple, well-characterized platform for studying lipid domains using
SANS measurements is the ULV system, where single bilayers are suspended
in water. One advantage of the ULV system is that vesicle size can be
precisely controlled by mechanical extrusion, whereby multilamellar vesicles
(MLVs) are pushed through polycarbonate filters of defined pore size. In
addition, the absence of inter-bilayer interference, which is commonly seen in
multilayer vesicles, significantly simplifies data analysis. Finally, the
geometrical constraint imposed by vesicle size yields laterally segregated
domains on the scale of tens of nanometers, a domain size that may be more
relevant to the transient membrane rafts that most likely exist in live cells.
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Similar to the transverse lipid bilayer structure, the SANS signal arising
from membrane lateral heterogeneity is proportional to the square of the SLD
difference between the domain and surrounding bulk phases (Figure 5) (Pan et
al., 2013). In the case of protiated lipids, the SLD difference between saturated
and unsaturated acyl chains is too small to generate a useful scattering signal,
even when large domains are present. The lack of contrast between different
phases also impairs the usefulness of X-ray scattering in studying lipid lateral
segregation. On the other hand, if it is possible to prepare a sample where the
domain and bulk phases contain different concentrations of hydrogen and
deuterium, the SLD difference between the two phases can be significantly
enhanced, enabling the use of SANS to study domain structure (Petruzielo et
al., 2013).
Figure 5. Schematic of contrast matching in a SANS experiment. Lipids phase-separate
into coexisting domains rich in saturated or unsaturated acyl chains. An SLD
difference between the coexisting phases is required to detect domains with SANS.
Neutron contrast is enhanced by isotopic replacement of one lipid‘s protiated
hydrocarbon chain with one that is perdeuterated. The resulting SLD of the phase (Lo
or Ld) that contains the majority of deuterated lipid, will have a larger SLD than the
complementary phase. ULVs are used to depict differences in contrast observed as a
function of temperature. Left, the SLD of the aqueous solvent is matched to the average
SLD of the bilayer such that no contrast is detected at high temperature, where the
lipids are randomly mixed within the plane of the bilayer (ΔSLD = 0). Right, SLD
fluctuations within the plane of the bilayer give rise to an SLD difference between the
aqueous solvent and the acyl chain region of the ULV such that scattering is detected
at lower temperatures, where phase separation of saturated and unsaturated chains
occurs.
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In addition to lateral contrast, the SANS signal also contains contributions
from: (1) SLD differences between the solvent and the spherical vesicle; and
(2) radial SLD fluctuations arising from the different average SLDs of the lipid
headgroup and acyl chain region (Pencer et al., 2005). As we are only
interested in the scattering signal arising from lateral heterogeneity, it is
desirable to minimize these additional contributions. The first contribution can
be minimized by choosing a D2O/H2O ratio for the aqueous solvent that
matches the mean SLD of the vesicle. Although often neglected, the second
contribution can be quite significant due to the marked SLD difference
between the lipid headgroup and hydrocarbon chains (see for example Figure
2B). A remedy for radial SLD fluctuations is to use the proper ratio of chain-
perdeuterated (e.g., DPPC-d62 or DSPC-d70) and chain-protiated (e.g., regular
DPPC or DSPC) di-saturated lipids, such that the mean SLDs of the lipid
headgroup and chains are the same. Because the saturated lipids partition
preferentially into the ordered Lo phase, the lateral SLD contrast is
intrinsically enhanced between the Lo (high SLD) and Ld (low SLD) phases.
In cases where a deuterated high-TM lipid is not available or is prohibitively
expensive to purchase, commercially available sn-1 perdeuterated low-TM
lipids (e.g., POPC-d31 or SOPC-d35) may be used. In this case, the lateral
SLD contrast is achieved by preferential partition of the low-TM lipids into the
Ld phase (high SLD), instead of the Lo phase (low SLD).
3.3. Data Analysis
In analyzing the transverse lipid bilayer structure, an SDP model was
employed to fit SAXS and SANS data simultaneously in order to extract
informative lipid bilayer structure. The same principle is followed here, i.e.,
fitting the SANS signal arising from lipid bilayer lateral heterogeneity to a
model, in order to extract information about the membrane‘s lateral structure,
such as domain size. To achieve this, a suitable model describing phase
coexisting vesicles is a prerequisite. Two approaches have been developed and
successfully applied to analyze phase separated domain sizes, depending on
whether the SANS signal can be described by closed analytical expressions, or
a more computationally intensive, numerical analysis (Pan et al., 2013).
As elaborated in the original methodology paper (Anghel et al., 2007), the
analytical approach can only be used to deal with a simple situation, where a
single round domain is present (the so-called infinite phase separation limit).
In this case, the scattering intensity can be expressed by a series summation
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that depends on domain area fraction, assuming contrast matching between the
vesicle and solvent. The usefulness of this approach has been demonstrated by
studying a 1:1 DPPC:DLPC mixture exhibiting gel/fluid immiscibility over a
range of temperatures (Anghel et al., 2007), and a ternary mixture composed
of DPPC/DOPC/Chol=4/4/2 (Masui et al., 2008).
A more general model that is suitable to describe vesicle phase separation
needs to address multiple domains and interactions between domain pairs. In
this case, a numerical approach that samples all possible configurations of
random domain distributions, or the so-called Monte Carlo (MC) simulation
method, is more practically applied (Hansen, 1990; Henderson, 1996). Figure
6 illustrates how to obtain theoretical scattering intensity for two round
domains floating on a spherical surface. The total area fraction of the round
domains (αd) is usually obtained independently, i.e. from an established phase
diagram. For this example, the total domain area fraction is chosen to be
αd=0.25. The domain-domain interaction is described by a volume excluded
steric interaction. Other types of interactions such as repulsive or attractive
potential can also be incorporated, if parameters representing these interactions
are known, and a Metropolis algorithm can be used (Metropolis et al., 1953).
For the case of volume-excluded interactions, the MC simulation proceeds as
follows (Pan et al., 2013):
1) Generate random coordinates for the domain center, subject to the
constraint that domains do not overlap (the domain radius is fixed by
αd)
2) Generate random points within each object (e.g., two round domains
and the surrounding bulk phase, Figure 6A). The number of points
within each object is proportional to the product of the object‘s
volume, and the SLD difference between the object and aqueous
solvent. A weighting term w is applied to each point to account for the
sign of the object‘s SLD contrast (i.e., w = +1 for positive contrast
and −1 for negative contrast)
3) Calculate the distance between each unique pair of points, weighted
by the product of the pair‘s weighting term (e.g., wi×wj between point
i and point j). The weighted distances are then binned into
probabilities as a function of distance r to construct a total pair-
distance histogram P(r). Note that P(r) can take on negative values
because of the weighting term (Figure 6B)
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4) Repeat steps 1-3 (e.g., >104 times) to generate different domain
configurations, and calculate the corresponding P(r) for each
configuration.
5) Sum the individual P(r) to obtain an ensemble-average P(r). The
scattering intensity (Figure 6C) is then calculated following the
integral Debye formula:
(1)
Figure 6. (A) Schematic of a multi-domain model where two round domains with the
same size float on the surface of a spherical vesicle. (B) Overall pair-distance
probability distribution for the two-domain model. The shell radius is 300 Å and the
shell thickness is 30 Å. The total domain area fraction is fixed at 0.25. The MC
simulation method was used to sample a statistical ensemble of all possible two-
domain configurations. (C) Integration of P(r), using equation (1), generates the
theoretical scattering intensity. The characteristic peak located near 0.008 Å-1
is
correlated with the radius of the round domains.
The power of the MC method lies in its capability to model complex
morphologies whose scattering profiles (or equivalently pair-distance
probability distributions) lack a closed analytical solution. In the case of
multiple domains, the interference, or the so-called structure factor, arising
from multiple, randomly distributed domains on a sphere cannot be (to the best
of our knowledge) expressed analytically. On the other hand, it is
straightforward to account for inter-domain interference numerically using the
MC method (as described in the previous paragraph). To further demonstrate
the use of MC simulation in modeling multiple domains, we calculated the
total pair-distance probabilities and the corresponding scattering intensity
profiles for four scenarios, where the number of domains Nd increased from
one to four, or equivalently decreased in size as illustrated in Figure 7 (the
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total domain area fraction was fixed at αd=0.25 for each model). The scattering
curves were scaled to preserve the Porod invariant , which
does not vary when αd is fixed (Pencer et al., 2006). Polydispersity of vesicle
size was accounted for by averaging an ensemble of vesicles whose sizes
follow the Schulz distribution (Pencer et al., 2005). Of note is that increasing
Nd shifts the position of maximum intensity to larger q values. Furthermore,
the peak width increases systematically and asymmetrically with increasing
Nd, with a marked intensity increase on the right side of the peak. Weakening
of the peak intensity as domain size decreases is consistent with a vanishing
scattering intensity as domain size becomes infinitely small (i.e., random
mixing of lipids), as dictated by the experimental contrast matching
conditions. When the unavoidable background incoherent scattering from
hydrogen is taken into account, the observed trend of the scattering intensity
profile as a function of domain size also sets a size limit (~ 7 nm in our
experience), below which SANS is no longer useful for detecting membrane
lateral heterogeneity (Petruzielo et al., 2013).
Figure 7. Theoretical scattering intensity profiles for the case of 1-4 domains. The total
domain area fraction was fixed at αd=0.25 for each model. The intensity profiles were
obtained by averaging an ensemble of vesicles described by the Schulz distribution,
with an average radius of 300 Å and a polydispersity of 25%. The intensities were
scaled to preserve the Porod invariant, which does not vary when the total domain area
fraction was fixed. For each model the same color scheme was used for the round
domain and calculated scattering intensity profiles.
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3.4. An Application of SANS to the Study of Rafts in
Model Membranes
Ever since the pioneering work by Korlach and coworkers (Korlach et al.,
1999) who first demonstrated that direct observation of micron-sized phases
using conventional fluorescence microscopy (provided that suitable dyes with
different partition coefficients for different domains were added into lipid
mixtures), numerous studies have consistently shown that ternary lipid
mixtures containing lipids with unsaturations in both chains (e.g., DOPC)
exhibit micron-sized phase separation over a wide range of composition and
temperature. In contrast, phase coexistence in ternary mixtures containing an
unsaturation in only one chain (e.g., POPC) can only be inferred from
spectroscopic techniques (e.g., FRET and ESR), indicating that only
nanometer-sized domains are present in these mixtures. We note that although
micron-sized domains have also been reported for mixtures containing singly
unsaturated lipids, they were most likely caused by fluorophore related
artifacts (Ayuyan and Cohen, 2006; Zhao et al., 2007).
Since rafts are thought to function through the coalescence of nanoscopic
domains into larger domains under external stimuli, it is of interest to delineate
what physicochemical parameters determine raft size. To this end, Feigenson
and coworkers have systematically investigated domain size in a four-
component mixture composed of DSPC, DOPC, POPC and cholesterol (Goh
et al., 2013; Heberle et al., 2010; Konyakhina et al., 2011; Konyakhina et al.,
2013). In those experiments, the authors gradually replaced POPC with DOPC
[or changing DOPC fraction ρ=DOPC/(DOPC+POPC)], while keeping the
molar fraction of the saturated lipid DSPC and cholesterol constant. The
results indicate that domain size increases continuously from a few nanometers
(spectroscopy) to microns (microscopy) as ρ increases. Although these
experiments provided useful information pertaining to how domain size may
be changed, the underling mechanism that controls the domain size transition
was not elucidated.
To this end, Heberle and coworkers recently studied phase separation in
the same four-component system using SANS (Heberle et al., 2013). An
example of the SANS data as a function of temperature for ρ=0.1 is shown in
Figure 8A. It is clear that in the high temperature regime (40 and 50oC), no
discernible scattering above background was detected, indicating
homogeneous mixing. As the temperature decreased, an abrupt increase of the
scattering intensity centered near 0.01 Å-1
was observed (10, 20 and 30oC),
suggesting the onset of phase separation. Subsequent analysis using the
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numerical MC method introduced in section 3.3 revealed the presence of
nanometer-sized domains in the low temperature regime. By varying the
degree of lipid mixture unsaturation (or changing ρ), the authors found that
domain radius increased from 6.8 to 16.2 nm as a function of increasing ρ (0 to
0.35).
Figure 8. (A) SANS curves at different temperatures reveal phase coexistence in the
mixture DSPC-d70/DOPC/POPC/Chol=0.39/0.04/0.35/0.22. Under solvent contrast
matching conditions, minimal scattering above background is observed at 50oC (purple
solid line) and 40oC (yellow solid line). However, an abrupt increase in scattering
occurs upon further lowering the temperature to 30C (brown solid line), which
persists at 20oC (green solid line) and 10
oC (cyan solid line). 20
oC data were fitted with
the MC method (smooth black line) using constraints provided by the four-component
phase diagram for this system. The inset shows the total integrated scattered intensity
(Porod invariant) as a function of temperature, further demonstrating the enhanced
scattering observed at lower temperatures. (B) Domain size versus bilayer thickness
mismatch for different mixture compositions in the four-component system (i.e.,
different ρ). Both domain size and thickness mismatch between the coexisting Ld and
Lo phases increased with increasing acyl chain unsaturation (or ρ) in the bilayer.
Because the lipid composition for each phase is known from the
established phase diagram constructed using spectroscopic and microscopic
measurements (Goh et al., 2013; Konyakhina et al., 2011; Konyakhina et al.,
2013), the authors further measured the transverse bilayer structure (e.g.,
bilayer thickness) for each phase at different ρ. It was found that as ρ varies,
the bilayer thickness of the Lo phase remains essentially constant, while the
thickness of the Ld phase decreases continuously as ρ increases. The former
observation is consistent with previous findings that at fractions greater than
20 mol%, cholesterol is capable of straightening saturated lipid chains to their
full extension (Pan et al., 2008b; Pan et al., 2009). The differential responses
of the Ld and Lo phase thicknesses to increasing ρ, coupled with the domain
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size transition as a function of ρ, revealed a positive linear relationship
between domain size and the thickness mismatch (difference) between the Ld
and Lo phases (Figure 8B). This finding is consistent with theoretical
treatments of line tension that predict a quadratic dependence of line tension
on domain thickness mismatch (Kuzmin et al., 2005). It is conceivable that in
order to compensate for increased line tension (or enthalpy), small domains
tend to coalesce into larger ones to reduce the total domain perimeter. This
effect was expected but had not previously been observed in freely suspended
vesicles.
CONCLUSION
SANS is a powerful technique for unveiling the nanoscopic structure of
lipid membranes. In particular, using an SDP model based analysis, SANS can
be integrated with other high-resolution techniques, such as SAXS to yield
robust structural data, including the much sought-after area per lipid and
bilayer hydrophobic thickness. To date, the application of the SDP model to
single lipids, including neutral PC and mono-anionic PG lipids, has revealed
the importance of the lipid headgroup moiety in dictating transverse membrane
structure and packing. Moreover, the unique capability of SDP model analysis
is demonstrated by studying a binary lipid mixture, which showed that
chemical composition of the glycerol linkage between the polar headgroup and
apolar hydrocarbon chains has a profound effect on molecular interaction
modes between phospholipids and nearby cholesterol molecules—compelling
evidence supporting the notion of ether lipid facilitated cholesterol trafficking.
As pointed out, biological membranes are composed of thousands of lipid
species, with each class possessing unique structural features. By combining
neutron and X-ray scattering data, the SDP model presents a tractable
approach for elucidating each lipid‘s fine structure. Along these lines, we are
currently working on structures of several important lipid classes, including
apoptosis related phosphatidylserine (PS) lipids and polyunsaturated lipids.
The structural parameters obtained from the SDP model also provide a metric
for simulators to check the validity of their force fields. In return, it is hoped
that detailed molecular insights can be inferred from fine-tuned simulations,
which are able to reproduce structural quantities determined by experimental
data analysis.
In addition to one-dimensional structural data along the membrane
normal, we have recently exploited the contrast variation offered by neutron
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scattering to study, with unprecedented accuracy, the lateral phase separation
of domain-forming mixtures. It was shown that SANS is sensitive to
nanometer-sized domains, which arguably seem to be of similar size to lipid
rafts thought to exist in resting cells. Moreover, SANS does not require the use
of extrinsic probes, an important feature for systems that are prone to light-
induced artifactual phase separation. As demonstrated by the thickness
mismatch-mediated domain size transition, SANS provides the unique
capability of revealing the heterogeneous lateral structure of biologically
relevant model membranes. We hope that in the near future, this methodology
can be applied to address a question that has vexed biologists and confounded
experimentalists for decades: do membrane rafts exist in vivo?
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In: Recent Progress in Neutron Scattering … ISBN: 978-1-62948-099-2
Editors: A. Vidal and M. Carrizo © 2013 Nova Science Publishers, Inc.
Chapter 4
EVALUATION OF THE THERMODYNAMIC
PROPERTIES OF HYDRATED METAL OXIDE
NANOPARTICLES BY INS TECHNIQUES
Elinor C. Spencer1, Nancy L. Ross
1,, Stewart F. Parker
2
and Alexander I. Kolesnikov3
1Dept. of Geosciences, Virginia Tech, Blacksburg, Virginia, US
2ISIS Facility, STFC Rutherford Appleton Laboratory,
Chilton, Didcot, UK 3Chemical and Engineering Materials Division, Oak Ridge National
Laboratory, Oak Ridge, Tennessee, US
ABSTRACT
In this contribution we will present a detailed methodology for the
elucidation of the following aspects of the thermodynamic properties of
hydrated metal oxide nanoparticles from high-resolution, low-
temperature inelastic neutron scattering (INS) data: (i) the isochoric heat
capacity and entropy of the hydration layers both chemi- and physisorbed
to the particle surface; (ii) the magnetic contribution to the heat capacity
of the nanoparticles. This will include the calculation of the vibrational
density of states (VDOS) from the raw INS spectra, and the subsequent
extraction of the thermodynamic data from the VDOS. This technique
Corresponding author‘s Email: [email protected]
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Elinor C. Spencer, Nancy L. Ross, Stewart F. Parker et al. 106
will be described in terms of a worked example namely, cobalt oxide
(Co3O4 and CoO).
To complement this evaluation of the physical properties of metal
oxide nanoparticle systems, we will emphasise the importance of high-
resolution, high-energy INS for the determination of the structure and
dynamics of the water species, namely molecular (H2O) and dissociated
water (OH, hydroxyl), confined to the oxide surfaces. For this component
of the chapter we will focus on INS investigations of hydrated
isostructural rutile (α-TiO2) and cassiterite (SnO2) nanoparticles.
We will complete this discussion of nanoparticle analysis by
including an appraisal of the INS instrumentation employed in such
studies with particular focus on TOSCA [ISIS, Rutherford Appleton
Laboratory (RAL), U.K.] and the newly developed spectrometer
SEQUOIA [SNS, Oak Ridge National Laboratory (ORNL), U.S.A].
1. INTRODUCTION
Metal oxide nanoparticles are an inherent aspect of the environment in
which we reside. Their unique magnetic, optical, physical and chemical
properties, which often deviate significantly from their bulk counterparts [1-5],
have been exploited in a diverse range of technological and commercially
relevant applications ranging from cosmetics to solar cells. The stability of
metal oxide materials at the nanoscale is provided by the occurrence of water
confined to the surface of the particles [6, 7] that adopts a structure that is
distinct from that of bulk water. These hydration layers have a pronounce
impact on the morphology and polymorphism of the nanoparticles. This aspect
of nanoparticle behaviour has been extensively studied and it has been shown
that the presence of surface water can alter the oxidation-reduction equilibria
and relative phase stabilities of metal oxide particles leading to so-called
‗phase-crossovers‘ whereby the thermodynamically most stable bulk phase no
longer dominate at the nanoscale instead, metastable phases are energetically
favoured [8-16]. This phenomenon is driven primarily by the relative surface
energies of the different phases, which in turn are strongly effected by the
stabilising force provided by surface water. Such size-dependent behaviour has
major implications in the fields of geology, chemistry, and physics.
As surface water is of such critical importance in nanomaterial science it
has been the subject of intense experimental and theoretical investigation.
Multiple density functional theory (DFT) calculations have confirmed the
complex nature of the structure of water confined on the surface of metal
oxide nanoparticles [8, 17-21]. The competition between associative and
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Evaluation of the Thermodynamic Properties … 107
dissociative absorption has been prodigiously documented, and the consensus
is that the surface water achieves a multilayer structure upon the particle
surface; the first water layer comprises a combination of chemically bound
hydroxyl (OH) groups, generated by the dissociation of molecular water upon
interaction with the metal oxide surface, and chemisorbed H2O molecules.
Subsequent layers are typically composed of water molecules hydrogen
bonded to the first water layer and neighbouring H2O molecules.
The ratio of molecular to dissociated water species is believed to be a
consequence of the physical and structural properties of the exposed metal
oxide surface. Moreover, the degree of hydration is dictated by the stability
needs of the particles as well as the synthesis method employed in their
preparation.
A broad range of analytical techniques has provided experimental
evidence for the surface water model proposed by DFT computations. The
importance of hydroxylation to the relaxation, and thus stability, of aluminium
oxide (Al2O3) surfaces at the nanoscale has been demonstrated by multiple
studies utilising surface X-ray diffraction, infrared spectroscopy, and
calorimetric techniques [12, 22-24]. Spectroscopic techniques such as diffuse
reflectance infrared spectroscopy (DRIFTS), FTIR and Raman scattering have
unequivocally verified the presence of chemisorbed OH groups on the surface
of zinc oxide (ZnO) [25], and zirconium oxide (ZrO2) [26], but perhaps the
most conclusive evidence for the multi-layer structural model as well as
information pertaining to the dynamics of surface water has been obtained
from in-depth analysis of metal oxide nanoparticle systems by neutron
scattering methods in particular, quasielastic neutron scattering (QENS) and
INS techniques [27]. These powerful analytical techniques have been
employed to assess the structure and dynamics of the hydration layers of
numerous metal oxide nanoparticle systems including palladium oxide (PdO)
[28], tin oxide (SnO2) [29-31], titanium oxide (TiO2) [30-33], Al2O3 [34], and
ZrO2 [35-38] to name but a few. All of these studies, without exception,
support the multilayer model comprising both dissociated and molecular water
species for the structure of water on the surface of metal oxide nanoparticles.
In complementary INS studies we have focused on evaluating the
thermodynamic properties of the hydration layers confined to the surface of
metal oxide nanoparticles, and it is to this aspect of nanoparticle physics that
we now turn.
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Elinor C. Spencer, Nancy L. Ross, Stewart F. Parker et al. 108
2 .THERMODYNAMICS OF METAL OXIDE NANOPARTICLES
In this section we provide a worked example of the extraction of
thermodynamic data from INS spectra of hydrated metal oxide nanoparticles.
Example calculations of the magnetic contribution to heat capacity of
antiferromagnetic Co3O4 nanoparticles and the heat capacity and entropy of
the surface water on both CoO and Co3O4 particles will be presented.
From a structural perspective the most basic cobalt oxide is CoO. The
bulk structure of this monoxide at room temperature is analogous to that of
rocksalt (cubic, Fm3m), but it transforms below 288 K to a monoclinic phase
(C2/c), a process that is facilitated by a tetragonal distortion of the crystal
structure. This critical transition temperature corresponds to the Néel
temperature (TN) of the material and is associated with the paramagnetic →
antiferromagnetic reordering of the magnetic spins [39, 40]. Interesting, for
CoO the value of TN is dependent on the size of the CoO particles and is
reduced to 265 K for particles < 21 nm in diameter (ΔTN = 23 K) [41].
Moreover, the nature of the absorbed species on the surface of the
nanoparticles can directly influence the state of the surface spins and thus
drastically effect the magnetism of the particles [42].
Whereas the cobalt ions in CoO are exclusively divalent and reside only in
octahedral sites, in the more complex spinel oxide Co3O4 (Fd-3m) the cobalt
ions coexist in both trivalent and divalent oxidation states. In accordance with
a typical spinel structure the diamagnetic low spin Co3+
and high spin Co2+
cations are situated in octahedral and tetrahedral environments, respectively.
The complex interactions between the Co3+
and Co2+
ions leads to bulk Co3O4
having a much lower Néel temperature of 40 K relative to that of CoO [39].
Moreover, consistent with the observations for CoO, TN for Co3O4 is also
particle size-dependent [43, 44]. Although the magnetic transitions in both
CoO and Co3O4 have been evaluated by calorimetric methods [40, 41], the
contribution of the magnetic spinwave to the particle heat capacity has been
determined by INS spectroscopy, which represented the first such
measurement of its kind [45].
2.1. Magnetic Heat Capacity
An INS spectrum for hydrated Co3O4 nanoparticles was collected at 11 K
on the neutron spectrometer TOCSA (vide infra). The low energy region of
this spectrum is displayed in Figure 1.
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Evaluation of the Thermodynamic Properties … 109
Figure 1. The low energy region of the low temperature (11 K) INS spectrum of
hydrated Co3O4 nanoparticles. The pronounced peak at 5 meV is ascribed to a
magnetic spin wave excitation.
The well-defined peak at 5 meV is attributed to the magnetic excitation of
the antiferromagnetic particles and its position in the spectrum was shown to
be independent of the particle size [45]. The exact origin of this magnetic
spinwave is complex, but the electronic environments and spin-orbit coupling
between the Co3+
/Co2+
ions is liable to have an effect on the physical nature of
this excitation. Nonetheless, despite the innate complexity associated with this
type of magnetic excitation, the magnetic contribution to the heat capacity
(Cm) of the Co3O4 particles can be calculated from the peak energy.
To achieve this objective the excitation of the spinwave is modelled as a
simple two-level energy transition. Subsequently, Cm can be expressed in the
following manner: [46]
(1)
where R = universal gas constant (8.3145 J K-1
mol-1
); Emag = magnetic
excitation energy (J); k = Boltzmann‘s constant (1.3807 × 10-23
J K-1
); T =
temperature (K). Substituting in Emag = 5 meV (8.011 × 10-22
J) into this
expression permits Cm for Co3O4 nanoparticles to be evaluated as a function of
temperature; the results from this computation are shown in Figure 2.
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Elinor C. Spencer, Nancy L. Ross, Stewart F. Parker et al. 110
Figure 2. Heat capacity curves for nanoparticulate Co3O4. Cp: isobaric heat capacity
determined by calorimetry (); Cm: magnetic contribution to the heat capacity
calculated from INS data (); CT: contribution of the Néel transition to the total heat
capacity of the Co3O4 particles ().
Also depicted in Figure 2 is the low temperature isobaric heat capacity
(Cp) measured by calorimetry for the Co3O4 nanoparticles employed in the
INS experiment [45]. The peak at 28 K in the Cp curve is assigned to the
combined heat capacity arising from of the paramagnetic/antiferromagnetic
phase transition and the magnetic excitation that is observed in the INS data. It
should be noted that the reduction in the Néel temperature of ΔTN = 12 K
relative to bulk Co3O4 is consistent with the documented effects of finite size
on the TN value of nanoparticluate Co3O4 [43, 44]. By subtracting Cm from Cp
the contribution of the Néel transition to the particle‘s heat capacity (CT) can
be elucidated; the outcome from this calculation is displayed graphically in
Figure 2. It is noteworthy that only INS analysis allows for this type of in-
depth appraisal of the thermodynamic properties of magnetic nanoparticles, no
other analytical technique has the capability of providing such comprehensive
information.
2.2. Thermodynamics of the Confined Surface Water
The use of INS spectroscopy for the determination of the thermodynamic
properties of materials is not a new concept, and such methods have been
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Evaluation of the Thermodynamic Properties … 111
utilized extensively particularly in the field of mineralogy [47]. By combining
ab initio techniques with experimental evidence of the phonon density of
states (PDOS) [or phonon-dispersion relation (PDR)] for a material a diverse
range of macro- and microscopic properties such as the elastic constants,
specific heat capacity, entropy, thermal expansion and lattice dynamics of the
material can be calculated. The most challenging aspect of this research is the
acquisition of the complete PDOS, and this is where INS spectroscopy, with
thermal neutrons, is most readily exploited.
Unlike conventional optical spectroscopic methods e.g. Raman or IR,
neutron spectroscopy is not restricted by selection rules, so theoretically all
phonon modes are accessible. However, although a greater proportion of the
PDOS may be measured by INS some modes may be absent from the spectra
due to the resolution and geometrical limits of the instrumentation.
Nonetheless, this powerful technique has been applied successfully in the
evaluation of a range of geologically relevant minerals such as silicate
perovskites, the polymorphs of Al2SiO5, and the common oxide minerals
MgO, FeO, and SiO2 [47]. Most importantly, the physical properties of these
minerals determined by combined quantum mechanical calculations and
PDOS analysis concur with experimental data obtained from more traditional
laboratory based techniques.
Figure 3. S(E,Q) data for hydrated cobalt oxide nanoparticles. The magnetic peak at
5 meV in the Co3O4·0.40H2O spectrum has been truncated to allow the lower intensity
features of the spectra to be observed more clearly.
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Elinor C. Spencer, Nancy L. Ross, Stewart F. Parker et al. 112
Furthermore, once a detailed model of the phonon distribution within the
material has been extracted from the PDOS, it can be developed further to
allow for calculation of the of structure and dynamics of the material over
wide range of temperatures and pressures that may not be obtainable in an
experimental setting.
Determining the thermodynamic properties of the water confined on the
surface of metal oxide nanoparticles is considerably easier than that of a
complex crystalline material as the vibrational modes of water are restricted to
the 0–120 meV energy range and are thus easily measured, in toto, by INS
spectroscopy. In this section an account will be given of the calculation from
INS data of the surface water isochoric heat capacity and entropy for two
hydrated cobalt nanoparticle systems. The materials under consideration are
the spinel oxide Co3O4·0.40H2O (16 nm) and the simple monoxide
CoO·0.10H2O (10 nm). High-resolution low temperature (11 K) INS spectra
of these oxides were collected on TOSCA (Figure 3). Reminiscent of all metal
oxide nanoparticles those composed of cobalt oxide possess stabilising surface
water layers (vide supra). As a consequence of the extreme differences in the
neutron scattering cross-sections of hydrogen (σH = 82.03 barns) relative to
oxygen (σO = 4.23 barns) and typical transition metals (e.g. σCo = 5.6 barns),
INS spectra of hydrated metal oxide nanoparticles comprise, almost
exclusively, signals arising from scattering by the water species confined to
the surface of the particles (with the exception of magnetic signals). This
affords a unique opportunity to investigate the physical properties of these
water layers without interference from the underlying metal oxide lattice.
As alluded to above, of critical importance for the elucidation of the
thermodynamic behaviour of the hydration layers are the regions of the spectra
related to the translational and librational motions of the surface water species
that span the 0–120 meV energy range (Figure 3). The as-measured INS data,
as represented in Figure 3, correspond to the dynamical structure factor data,
[S(E,Q)] (E = energy transfer, and Q = magnitude of momentum transfer).
After correction for Debye-Waller factor, W(Q),
W(Q) = exp(–BQ2)
(2)
where B = < u2 > i.e. the mean squared atomic displacement, these data must
be transformed to the generalised vibrational density of states [G(E)] prior to
the calculation of the heat capacity. The mathematical expression required for
accomplishing this task is
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Evaluation of the Thermodynamic Properties … 113
(3)
where the n(E, T) term relates to the population Bose factor (Eq. 4).
(4)
The calculation of Q is instrument dependent, but for time-of-flight
spectrometer TOSCA the following equation links Q to the initial (Ei) and
final (Ef) energies:
(5)
where m = neutron mass; ħ = Dirac‘s constant; φ = scattering angle. Ef is a
constant, and for TOSCA the values for the front and back detectors are equal
to 3.35 meV and 3.32 meV, respectively. The incident energy is related to the
transfer energy, E, by the equation Ei = Ef + E.
The low energy region of the INS spectra (< 3 meV) is obscured by the
elastic peak, and in the case of the spectrum for Co3O4 the magnetic peak at
5 meV is nevertheless well resolved. Naturally, these components of the
spectra should not be included in the calculation of the heat capacity of the
hydration layers. To circumvent this issue the low energy regions of the
spectra affected by these features are replaced with simple Debye models of
the form G(E) ≈ AE2, where A is a constant that is inversely proportional to the
cube of the Debye frequency [48]. However, as A also includes an arbitrary
scaling factor (necessary for ensuring that the model is scaled appropriately
with the experimental data) the exact expression for A is inconsequential.
To place the data on an appropriate scale, the areas under the librational
(50–120 meV) and translational (0–40 meV) sections of the spectra are scaled
such that their areas equal three, corresponding to the three degrees of
translational and librational freedom of water. After all data corrections,
conversions and scaling procedures are successfully implemented the
vibrational density of states (VDOS) for the nanoparticle surface water are
obtained (Figure 4). Furthermore, if these manipulations are applied in an
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Elinor C. Spencer, Nancy L. Ross, Stewart F. Parker et al. 114
equivalent manner to all spectra, direct comparison of their intensity profiles is
legitimate.
Figure 4. VDOS for water confined on the surface of cobalt oxide nanoparticles. Also
included is the VDOS for the reference material ice-Ih [49]. The VDOS have been
‗smoothed‘ with a Savitzky-Golay filter so that the features of the VDOS can be
identified [50].
Figure 5. Cv curves for the ice-Ih and the hydration layers confined on the surface of
cobalt oxide nanoparticles: (a) over the 0–300 K temperature range; (b) an expanded
view of the Cv curve within the 0–150 K temperature range. These curves were
calculated from the VDOS of these materials.
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Evaluation of the Thermodynamic Properties … 115
Once the VDOS [v(ω), were ω = phonon angular frequency (s-1
)] have
been extracted from the INS data, the isochoric heat capacity (Cv) of the
nanoparticle surface water can be calculated by application of the following
equation [51]:
(6)
Cp is related to Cv by the expression Cp = TVα2/β + Cv, where V is the
molar volume, α is the volumetric coefficient of thermal expansion, and β is
the isothermal compressibility. However, exact values for these constants are
not readily available for water confined on nanoparticle surfaces, but if we
approximate the water structure as being similar to that of ice-Ih at 100 K then
TVα2/β = 0.025 J K
-1mol
–1 [51]. As the magnitude of this term is negligible,
we can assume that Cp ≈ Cv [52, 53]. Once this relationship has been
established it is possible to evaluate the entropy (S) of the hydration layers
(Eq. 7). The advantage of calculating the entropy is that subtle variations in the
Cv data for different samples are emphasised. Additionally, this relationship
also permits comparison of the results derived from INS measurements with
those obtained from traditional calorimetric techniques [52].
(7)
In the case of the cobalt oxide systems, the room temperature entropy
values are 37.03 and 38.13 J K-1
mol-1
for CoO·0.10H2O and Co3O4·0.40H2O,
respectively, and for ice-Ih S = 36.6813 J K-1
mol-1
. The marginally higher
entropy values for the water confined on the nanoparticle surface relative to
that for the reference material ice-Ih can be trace to the contribution to S from
the high temperature heat capacity of the cobalt oxide systems that exceed that
of ice-Ih at temperatures in excess of ca. 100 K (Figure 5). This rise heat
capacity can in turn be attributed to the additional librational modes in the 45–
60 meV regions of the nanoparticle VDOS that are absent in the ice-Ih VDOS
(Figure 4).
At temperatures below 100 K the Cv curves for both CoO and Co3O3 lie
below that of ice-Ih (Figure 5b). This low temperature region is especially
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Elinor C. Spencer, Nancy L. Ross, Stewart F. Parker et al. 116
sensitive to the energy distribution of translational bands of the VDOS, and
this finding is indicative of redistribution of the translational bands to higher
energy in the cobalt oxide VDOS relative to the equivalent band in the ice-Ih
VDOS, this is consequence of the surface-water interactions influencing the
motion and hydrogen bonding network of the surface water. Moreover, the
low temperature section of the Cv curve for the water on CoO particles is less
than that for the Co3O4 hydration layers, and the first acoustic peak at 7 meV
in the CoO VDOS is strongly suppressed relative to the same peak in the
Co3O4 VDOS. These facts are symptomatic of the CoO water layers
experiencing stronger surface-water interactions than the water confined on
the Co3O4 particles. This discovery is consistent with the correlation of surface
energy with the lattice structure of the oxide. According to this theory, spinel
oxides have lower surface energies than the equivalent rocksalt oxide; the
hydrated surface energies for Co3O4 (spinel) and CoO (rocksalt) are reported
to be 0.92(4) and 2.8(2) J m-2
, respectively [54].
3. HYDRATION LAYER STRUCTURE AND DYNAMICS
In addition to providing unique information pertaining to the
thermodynamic properties of the stabilising water layers on metal oxide
nanoparticles, neutron scattering methods have proved invaluable for
determining both the structure and dynamics of the surface water. Indisputably
the most studied metal oxide systems from this perspective are rutile-TiO2, and
SnO2. In this section it is our objective to discuss briefly INS investigations
concerned with the analysis of the structure and dynamics of water confined
on the surface of metal oxide nanoparticles, focusing primarily on these two
systems. However, it should be appreciated that it is not our intention to
conduct an extensive review of all available literature associated with these
materials, but rather to provide sufficient evidence as to the power of INS
techniques for this type of analysis (for more in-depth information the reader is
directed to refs. 30, 31, 55–57).
TiO2 is a classic example of a metal oxide material that exhibits phase-
crossover at the nanoscale (see Section 1). Rutile is the most stable form of
bulk TiO2 at all temperatures and pressures, yet it is well known that at the
nanoscale both anatase and brookite become thermodynamically stable over a
range of particle sizes. Conversely, the phase diagram for SnO2 is simplistic,
with only a single phase (cassiterite) known to exist irrespective of
temperature, hydrostatic pressure or particle size. As both rutile-TiO2 and
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Evaluation of the Thermodynamic Properties … 117
SnO2 are isostructural, the extreme differences in their phase behaviours
cannot be ascribed to variations in the structure or symmetry of the metal
oxide lattices, but are believed to be associated with disparities in the surface
energies and defects of the dominating crystal faces of the individual phases
[58-60]. As hydration layers act to stabilize metal oxide nanoparticles by
reducing the energy of the exposed crystalline surface (e.g. the surface
energies of anhydrous and hydrous rutile-TiO2 are 2.22(7) and 1.29(8) J m-2
,
respectively) [14, 58] considerable effort has been directed towards
understanding the nature of the water-surface interface, including the structure
and dynamics of the water layers.
Low-temperature (4–10 K) INS spectra (Figure 6) for hydrated SnO2 and
rutile-TiO2 nanoparticles were obtained with the decommissioned High-
Resolution Medium-Energy Chopper Spectrometer (HRMECS) at the now
closed Intense Pulsed Neutron Source (IPNS, Argonne National Laboratory)
[31, 61]
Figure 6. Generalised VDOS measured on HRMECS for hydrated TiO2 and SnO2
nanoparticles: (a) Ei = 50 meV; (b) Ei = 140 meV; (c) Ei = 600 meV.
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Elinor C. Spencer, Nancy L. Ross, Stewart F. Parker et al. 118
The samples employed in this study had the following chemical formulae:
TiO2·0.37H2O and SnO2·1.03H2O. The rutile-TiO2 particles had a rod-like
morphology 7 nm in diameter and 35 nm in length and water coverage of 29
molecules per nm2. The SnO2 particles were truncated rods 4 nm in diameter
and 7 nm in length with a coverage of 29 water molecules per nm2. As these
samples contained a similar number of water molecules per unit area, after
appropriate normalisation and scaling their spectra are directly comparable.
The translational motion of the hydration layer water molecules can be
observed over the 0–40 meV ranges of the INS spectra (Figure 6a). The
translational band in the ice-Ih spectrum has clearly defined peaks that are
replaced by broad featureless bands in the nanoparticle spectra. This is
indicative of the water confined on the metal oxide surfaces experiencing an
isotropic environment, which leads to the energy of translational motion being
independent of the direction of movement relative to the metal oxide lattice.
Overall, there is redistribution to higher energies of the translation bands of the
water situated on the nanoparticles relative to those of ice-Ih, and this is
consistent with the surface water motion being suppressed due to
surface-water interactions relative to the water molecules in ice-Ih. The most
striking evidence for this suppression is the reduction in the first acoustic band
at ca. 7 meV in the nanoparticle spectra with respect to the peak in the ice-Ih
spectrum. This peak is more suppressed in the rutile-TiO2 system than for the
water on the surface of the SnO2 particles. This finding can be rationalized in
terms of the higher surface energy of rutile-TiO2 with respect to SnO2 resulting
in an enhancement in the strength of the surface-water interactions (the
anhydrous surface energies of the SnO2 and rutile-TiO2 particles are 1.72(1)
and 2.22(7) Jm-2
, respectively) [14]. Indeed, the stabilising influence of the
surface water is reflected in the similarity of the hydrated surface energies of
these two systems: SnO2, 1.49(1) Jm-2
; TiO2, 1.29(8) Jm-2
.
The librational modes of water are associated with the rotational motion of
the ‗rigid‘ molecule about three perpendicular axis that pass close to the
oxygen atoms of the H2O molecules; consequently the oxygen atoms
contribute an insignificant amount to the rotational motion of the molecules.
The energies of the librational movements of the water molecules are inversely
proportional to the square root of the non-isotropic moments of inertia of the
molecules and as such are observed in the 50–120 meV regions of the
nanoparticle and ice-Ih INS spectra (Figure 6b). As one might expect, this
motion is sensitive to the hydrogen bond environment of the molecules. The
40–65 meV region on the ice-Ih spectrum is devoid of peaks, yet the
librational bands of the water confined on the surface of the metal oxide
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Evaluation of the Thermodynamic Properties … 119
particles extend over this region. These additional modes are ascribed to the
surface water species riding on the intense optical modes of the oxide lattice
[33]. Unlike the librational band in the ice-Ih spectrum, the corresponding
bands in the nanoparticle spectra are featureless and shifted to lower energy.
This strongly suggests that the hydrogen bond network within the hydration
layers are disrupted and softened relative to the network within the ice-Ih
lattice; this is almost certainly due to the combination of surface-water
interactions and the structural composition of the hydration layers i.e. the
complex combination of associated and dissociated H2O molecules.
The internal vibrations of the water species are discernible in the 150–
500 meV sections of the spectra (Figure 6c). The obvious peaks at ca.
200 meV are associated with the H-O-H bending mode [δ(H2O)], and thus
provides indisputable evidence for the presence of molecular water on the
surface of the metal oxide nanoparticles. The existence of hydroxyl groups
chemically bond to the surface of the particles is confirmed by the broad peaks
at 410–440 meV, which are ascribed to O–H vibrations [υ(H2O)] [20].
The results from this study support a ‗core-shell‘ model for the structure
of hydrated nanoparticles – that is a central core comprising the metal oxide
particle surrounded by a multilayered water shell – and clearly demonstrates
the complexity inherent in the structure of the nanoparticle hydration layers,
which deviates significantly from that of bulk water. In light of the differences
in the structures of bulk water and that of water confined to nanoscale regions,
it is reasonable to expect that these two distinct forms of water will exhibit
unique dynamical behaviour. Given the supremacy of INS techniques for
evaluating the chemical and physical properties of water, it logical to assume
that neutron scattering methods would be ideal for assessing the diffusional
dynamics of the hydration layers bound to the surface of metal oxide
nanoparticles. Indeed, the most effective means of studying the dynamics of
water on the surface of nanoparticles is by the use of QENS. By combining
such experiments with molecular dynamics (MD) simulations a complete
picture of the water transfer dynamics can be acquired. The complexity of the
water dynamics has been confirmed by QENS measurements (at T = 195–
345 K) of water on rutile-TiO2 [56, 57]. In this system the dynamics span a
wide timescale, with the molecular water comprising the outer hydration
layers displaying mobility on the scale of tens of picoseconds to more than a
nanosecond. By evaluating the various timescales over which the diffusion the
water species occur, the many distinct environments encountered by the
surface water can be identified. The translational jumps of the water molecules
of the intermediate water layers are the slowest components observed by
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Elinor C. Spencer, Nancy L. Ross, Stewart F. Parker et al. 120
QENS; this is to be expected as these molecules are restricted by the extensive
hydrogen bond networks in which they participate. The fastest diffusion
components are exhibited by the relatively weakly bound outer water layer,
and in between these two timescale extremes the localized water dynamics
arise. Yet, to achieve a complete assessment of the water behaviour, dynamics
that take place on timescales inaccessible to QENS need to be considered. This
is most evident when analysing the dynamics of the first water layer that is
primarily chemisorbed to the TiO2 surface. Diffusion in this layer typically
happens over tens of nanoseconds, and consequently cannot be observed by
QENS; however, by coupling these neutron investigations with MD
simulations that can access timescales up to 50 ns, this experimental limitation
can be circumvented.
By comparing QENS data and associated MD computations for
isostructural rutile-TiO2 and SnO2 the factors that influence the dynamics of
water confined on the surface of metal oxide nanoparticles can be elucidated.
Such analyses have clearly shown that the diffusion dynamics of the water are
different in these two systems [57]. Although in both cases the water diffusion
in intermediate and outer layers occur on the tens of picoseconds to
nanosecond timescales, the distribution of diffusion processes within this time
frame are notably distinct. Overall, the mobility of the water on the SnO2 is
dampened with respect to water on rutile-TiO2, thus a greater proportion of the
SnO2 QENS signal arises from slow (nanosecond timescale) diffusion process
compared to the TiO2 signal. Moreover, the temperature-dependent behaviour
(Arrhenius vs. non-Arrhenius) of the water layers were remarkably dissimilar
in these two materials, with a fragile-to-strong water transition observed
between 220–210 K for the water on TiO2, but no such transition being
exhibited by the SnO2 hydration layers. This seminal QENS/MD study showed
that the origin of these pronounced differences in the water dynamics for
hydrated SnO2 and rutile-TiO2 nanoparticles can be attributed to the variation
in the spatial arrangements of the water molecules comprising the intermediate
and outer hydration layers.This in turn is strongly influenced by the nature of
the first hydration layer chemically bound to the oxide surface, as the structure
of this layer subsequently dictates the molecular arrangement of the layers
residing upon it. Critically, the first layer on SnO2 comprises a greater number
of dissociated water species (i.e. hydroxyl groups in various bridging modes)
than the corresponding layer on TiO2; DTF calculations indicate that 60% of
water molecules are dissociated in the first layer on SnO2 compared to only
25% on the rutile-TiO2 surface [18, 21] Ultimately, these differences can be
traced to the variations in the electronic structure of the oxide surface, induced
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Evaluation of the Thermodynamic Properties … 121
by the significant differences in the electronic properties of the metal ions,
which strongly effect the proton transfer dynamics occurring at the metal-
water interface.
4. INSTRUMENTATION
In this section we present examples of INS instrumentation ideally suited
for providing the necessary data to perform the types of analysis of metal
oxide nanoparticle detailed in sections 2 and 3.
4.1. SEQUOIA
Vibrational and magnetic excitations can be studied best by INS. The
neutron mass (m) is comparable to atomic masses, the neutron has a spin, and
thermal neutrons have energies comparable to energies of phonon and magnon
excitations in condensed matter, and neutron wavelengths are similar to
interatomic distances in solids and liquids. All of these factors make thermal
neutrons exceptionally favourable for research concerned with the vibrational
and magnetic dynamics of novel materials, and provide comprehensive
information on their dynamical structure factor, S(Q,E), as a function of
momentum and energy transfer (Q and E, respectively). INS experiments at
pulsed neutron sources like the Spallation Neutron Source (SNS, ORNL,
USA), J-PARC (Japan) or ISIS (RAL, UK) are performed mainly with two
types of spectrometers: direct or indirect geometry. In these experiments
neutrons scattered by the sample are collected as a function of neutron energy
and momentum transfer, which are determined by a time-of-flight technique.
Direct geometry spectrometers use monochromatic incident neutrons, where
the desired neutron energy, Ei (typically from a few meV to a few eV), can be
selected by rotating choppers, and the scattering neutrons, Ef, of all energies
are registered by detectors. Indirect geometry spectrometers use ―white‖
incident neutrons (all energies) and the scattering neutrons are detected only
for a fixed final energy (usually ~4 meV).
The collected number of neutrons registered by the detectors as a function
of time-of-flight (t) at scattering angle (θ) is transferred to S(Q,E) function,
where neutron energy and momentum transfer are calculated with the
following simple expressions:
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Elinor C. Spencer, Nancy L. Ross, Stewart F. Parker et al. 122
(8)
(9)
(10)
(11)
(12)
(13)
where ti, Li, vi, ki and tf, Lf, vf, kf are the flight times, flight paths, velocities and
momenta of neutrons before and after scattering by the sample, respectively. Ei
and Ef are the corresponding energies and ET is the energy transfer to the
sample. Due to the very high neutron flux, good energy resolution, and large
detector coverage, direct geometry spectrometers are very efficient for the
study of the vibrational and magnetic excitations in condensed matter,
including liquids, amorphous and polycrystalline solids, and single crystals. In
the latter case, information can be obtained regarding the dispersion curves of
the excitations. Figure 7 shows the fine resolution Fermi chopper spectrometer
SEQUOIA [62,63] at the SNS (ORNL), which is an example of a direct
geometry INS instrument (other similar spectrometers of this type are ARCS
at SNS, HRC at J-PARC, MARI, MAPS and MERLIN at ISIS, HRMECS and
LRMECS at the now closed IPNS). The SNS was designed to have a power of
1.4 MW, and it has already reached a power of 1.1 MW (as of 2012), thus
producing the highest neutron flux among existing pulsed spallation neutron
sources in the world.
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Evaluation of the Thermodynamic Properties … 123
Figure 7. Schematic view of the SEQUOIA spectrometer (taken from ref. [62]).
The neutron beam at SEQUOIA originates from the decoupled H2O
moderator at ambient temperature and travels 20 m to the sample position
along the neutron guide that is utilized to provide a high neutron flux at the
sample (5 × 5 cm2 in size). A T0 chopper, located 9.8 m from the moderator,
blocks the highest energy neutrons generated when the proton pulse hits the
target. It also blocks neutrons that would be transmitted through additional
openings of the high speed Fermi chopper that occur during the 60 Hz time
frame of the source. A Fermi chopper (rotating at a speed of 60 to 600 Hz) is
located 18 m from the moderator and provides the sample with a
monochromatic neutron beam with Ei between 4 meV and 4 eV. There are two
Fermi choppers available on a translation table, which can be set to provide the
desired neutron flux and energy resolution for the experiment. A typical
energy resolution of the SEQUOIA elastic line is about 5% or 2% of the Ei
with the use of the Fermi choppers 1 or 2, respectively, and the resolution is
increased approximately two-fold at energy transfer ~Ei/2 (Figure 8).
Details of the neutron guide and choppers utilized with SEQUOIA are
described elsewhere [63].
The detector array on this instrument consists of ~110,000 pixels (about
10 mm in height and 25.4 mm in width) and is placed along the vertical
cylindrical surface with the shortest sample to detector distance (in the
equatorial plane) of ca. 5.5 m.
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Elinor C. Spencer, Nancy L. Ross, Stewart F. Parker et al. 124
Figure 8. Energy resolution of the direct geometry spectrometer SEQUOIA for Ei=160
meV selected with Fermi chopper 2 rotating at speed 600 Hz (solid line) and the
indirect geometry spectrometer TOSCA (dashed line).
The detectors cover the neutron scattering angles of –30° to 60° in the
horizontal direction, and ±18° in the vertical direction, and the minimum
scattering angle in both directions is 2.5°. To minimize the background signal
the detectors are placed in a vacuum and there is no window between the
sample and the detectors chambers during the INS measurement. A large gate
valve connects the two chambers to isolate them during sample changeover.
Moreover, the interiors of the chambers are covered with boron carbide plates
to further reduce background.
4.2. TOSCA
For INS spectroscopy, the quantities of interest are the energy and
momentum exchanged between the neutron and the sample. From Equations
10 and 12, in order to determine ET and Q, both the incident and final energies
and wavevectors must be known. Indirect geometry spectrometers fix the final
energy to a single value and determine the incident energy by a
monochromator at reactor sources and from the total time-of-flight at a pulsed
spallation source [64]. There are several methods used to fix the final energy
but all depend on the use of beryllium. Beryllium exhibits a very sharp edge at
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Evaluation of the Thermodynamic Properties … 125
~4 meV, this has the result that neutrons with an energy of < 4 meV are
transmitted and those with an energy > 4 meV are Bragg scattered out of the
beam. The result is that beryllium acts as a lon-pass filter for neutrons with < 4
meV energy. The transmission can be improved by a factor of two or so by
cooling the filter below 100 K and all operating spectrometers use cold filters.
Graphite has an edge at ~1.5 meV and by using a sandwich of beryllium and
graphite the bandpass is reduced and the resolution improved, albeit with
reduced flux. FANS (NIST, USA), FDS (LANSCE, USA) and IN1BeF (ILL,
France) all use (or used) such filters in the scattered beam for energy analysis.
Better resolution is obtained by reducing the bandwidth of the energy-
selected neutrons. Figure 9a shows a schematic of an analyser module of
TOSCA at ISIS (a similar arrangement is used for VISION at SNS and
Lagrange at ILL). In this case, the neutrons scattered from the sample impinge
at 45° on a pyrolytic graphite array of crystals, those with the correct energy
(~4 meV) to fulfil the Bragg condition for scattering by the 002 plane are
reflected towards the detectors and those that do not meet the Bragg condition
are transmitted through the graphite and absorbed in the shielding. Bragg‘s law
is:
(14)
Thus neutrons with wavelengths of λ/n (n = 2,3,4…) (corresponding to
energies of 16, 36, 64…meV) will also be scattered towards the detectors, in
addition to the 4 meV neutrons. For time-of-flight (TOF) analysis this is
catastrophic, since the technique requires that the sample to detector flight
time bes known. A neutron of higher energy will have a shorter flight time
(Equation 8), thus will appear as if the incident energy is larger than is the fact.
These higher order neutrons are removed by the beryllium filter (since they
have an energy > 4 meV cut-off).
Figure 9b shows a schematic diagram of TOSCA at ISIS [65]. The
instrument is located 17 m from an ambient temperature water moderator.
There are five modules of the type shown in Figure 9a, in each backscattering
(scattering angle of 135°) and forward scattering (scattering angle of 45°).
Usually a flat sample is placed perpendicular to the neutron beam, and the
detectors lie in the plane of the sample and while it is conceivable to use the
same detectors for both forward and backscattering it is advantageous to have
separate detector banks for forward and backscattering, isolated from each
other by shielding.
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Elinor C. Spencer, Nancy L. Ross, Stewart F. Parker et al. 126
Figure 9. (a) Cut-away view of an analyser module on TOSCA; (b) Schematic view of
TOSCA [66].
ISIS operates at 50 Hz thus there is a neutron pulse every 20 ms, but it is
unique in that it runs two target stations, TS1 and TS2. Four consecutive
pulses go to TS1 (where TOSCA is situated) and the fifth to TS2. The missing
pulse means that there is a one time frame that is 40 ms long, and this is
utilised on TOSCA to extend the energy transfer range down to –3 meV, to
give a very wide spectral range of –3 to > 1000 meV. This is recorded from
every pulse. Thus in contrast to the direct geometry instruments, there are no
user defined instrument parameters such as choice of chopper, its speed and
the incident energy, and TOSCA is very easy to operate.
The use of a white incident beam and a small fixed final energy has
several consequences. For most energies, Ei>>Ef, thus from Equation 12, and
it follows that there is a single value of Q for each value of ET and the
instrument follows a line in (Q,E) space. This means that the spectra, such as
those shown in Figures 1, 3 and 4, resemble those obtained by conventional
vibrational spectroscopy.
The resolution characteristics of indirect geometry instruments are
opposite to those of the direct geometry spectrometers, as shown in Figure 8.
For TOSCA, the resolution is a fraction of ET and degrades with increasing
energy transfer, while for the direct geometry instruments is a fraction of Ei
and improves with increasing energy transfer (note that larger Ei means poorer
resolution). Thus the two types of instrument are highly complementary and
together can provide a complete view across the entire ‗mid-infrared‘ spectral
range 0–500 meV with good energy resolution.
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Evaluation of the Thermodynamic Properties … 127
CONCLUSION
In this chapter we have demonstrated the unique ability of INS techniques
for performing a comprehensive assessment of the physical and structural
properties of metal oxide nanoparticles. By way of a worked example we have
shown how thermodynamic information pertaining to the water confined on
the surface of such particles can be extracted from INS data – information that
cannot be obtained accurately by any other means. Furthermore, by evaluating
cobalt oxide particles we have revealed the immense power of INS methods
for investigating the magnetic properties of nanoparticle systems, with
particular focus on the contribution of the magnetic spinwaves to the heat
capacity of the metal oxide particle. Again, this example is a representation of
the type of information available through INS that is inaccessible by other
analytical techniques.
In section 3 we highlighted the use of INS and QENS for investigating
both the structure and dynamics of the nanoparticle hydration layers. This was
achieved by the presentation of a concise review of the recent analysis of both
hydrated rutile-TiO2 and SnO2 nanoparticle systems. This research has
provided unambiguous evidence that the multilayered water structures on the
nanoparticle surfaces are distinct from that of bulk water due to the extensive
interactions at the water-oxide interface. Subsequently the dynamics of the
water species bound to the nanoparticles is complex, and strongly dependent
on the electronic structure of the underlying metal oxide lattice.
We completed this chapter with a discussion of the INS instrumentation
that is so critical for this type of analysis. A detailed description of both
sequoia and TOSCA was given, and the characteristics of these instruments
that make them so suitable for the analysis of metal oxide nanoparticles were
emphasised.
ACKNOWLEDGMENTS
E. C. S. and N. L. R acknowledge support by the U.S. Department of
Energy, Office of Basic Energy Sciences, Chemical Sciences, Geosciences,
and Biosciences Division under Award DE FG03 01ER15237. Work at ORNL
was supported by the DOE-BES and was managed by UT-Battelle, LLC, for
DOE under Contract DE-AC05-00OR22725.
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In: Recent Progress in Neutron Scattering … ISBN: 978-1-62948-099-2
Editors: A. Vidal and M. Carrizo © 2013 Nova Science Publishers, Inc.
Chapter 5
ULTRA-LOW TEMPERATURE SAMPLE
ENVIRONMENT FOR NEUTRON SCATTERING
EXPERIMENTS
O. Kirichek and R. B. E. Down
ISIS, STFC, Rutherford Appleton Laboratory,
Harwell Science and Innovation Campus, Didcot, UK
ABSTRACT
Today almost a quarter of all neutron scattering experiments
performed at neutron scattering facilities require sample temperatures
below 1K. A global shortage of helium gas can seriously jeopardize the
low temperature experimental programs of neutron scattering
laboratories. However the progress in cryo-cooler technology offers a
new generation of cryogenic systems with significantly reduced
consumption and in some cases a complete elimination of cryogens. Here
we discuss new cryogen free dilution refrigerators developed by the ISIS
facility in collaboration with Oxford Instruments. We also discuss a new
approach which makes standard dilution refrigerator inserts cryogen-free
if they are used with cryogen-free cryostats such as the 1.5K top-loader or
re-condensing cryostat with a variable temperature insert. The first
scientific results obtained from the neutron scattering experiments with
these refrigerators are also going to be discussed.
E-mail: [email protected]
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O. Kirichek and R. B. E. Down 134
Keywords: Dilution refrigerators, neutron scattering, sample environment
INTRODUCTION
According to ISIS1
beam-line proposal statistics almost 25% of all
experiments require sample temperature below 1K [1]. Such sample
environment is usually provided by dilution and 3He evaporation refrigerators.
There are fundamental reasons for this. Firstly, the thermal motion of atoms is
drastically reduced at ultra-low temperature (ULT), significantly improving
the precision of structural measurements. Secondly, the ULT range allows the
study of phase transitions at temperatures below 1K. In recent years, the
experiments where the sample needs to be exposed to a high magnetic field,
low temperature and a neutron beam simultaneously has gained remarkable
high popularity [2]. This sample environment SE is usually achieved by a
combination of dilution refrigerator with superconducting magnet specially
designed for neutron scattering experiments [3].
Conventionally the ULT sample environment in neutron scattering
experiments is provided by dilution refrigerators. For many years the Grenoble
design incorporating sintered silver heat exchangers [4, 5] used to be the
standard of conventional dilution refrigerators. Usually this kind of refrigerator
is built in a liquid helium bath cryostat with all the consequences typical for
liquid cryogen based cryogenics. Hence the extensive use of liquid cryogens
based equipment requires significant resources. It creates a number of
logistical issues including the considerable cost of the cryogens and can also
pose health and safety problems. For example the cost of liquid helium in the
UK has doubled in the last five years and continues to grow. There is also very
strong environmental aspect of this problem. Helium gas is a limited natural
resource, which in case of release rises through and escapes from our
atmosphere [6].
For neutron facilities this problem is of special importance because during
the last two decades they are permanently in the top-ten list of liquid helium
users in the scientific research world somewhere after large accelerators,
fusion reactors and MRI and NMR laboratories. This situation has posed one
of the major risks on sustainability of a neutron facility‘s operation.
1 ISIS – pulsed neutron and muon source at the Rutherford Appleton Laboratory in Oxfordshire,
UK, operated by STFC.
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Ultra-Low Temperature Sample Environment for Neutron … 135
Fortunately the progress in closed cycle refrigerator technology offers a
new generation of cryogenic systems with significantly reduced consumption
and, in some cases, complete elimination of cryogens [1]. This breakthrough
became possible due to a new generation of commercial cryo-coolers
developed during the last decade. The most successful representative is the
pulse tube refrigerator (PTR). The unique feature of the PTR is the absence of
cold moving parts, which considerably reduces the noise and vibrations
generated by the cooler [7, 8]. It also increases the reliability of the cold head
because no expensive high-precision seals are required and the cold head can
be operated without any service inspection.
There are two different approaches to minimize or, in some cases,
completely eliminate the use of cryogens. So-called ―dry‖ systems do not
contain liquid cryogens at all. They are built around a cryo-cooler that utilizes
the cooling power of the cold head [7, 9]. Another approach is realized in re-
condensing systems in which evaporating helium is cooled and returned back
to the cryostat by a cryo-cooler [10]. Here we discuss new cryogen-free
dilution refrigerators based on both methods.
POWERFUL CRYOGEN FREE “DRY” DILUTION
REFRIGERATOR
The development of the cryogen-free dilution refrigerator started at the
very beginning of the 21st century alongside the development of cryogen-free
technology in general [11-13]. The design of modern powerful cryogen-free
dilution refrigerators in most cases is based on the prototype developed by
Kurt Uhlig [14], where the PTR is used to pre-cool the dilution unit. These
powerful dilution refrigerators are usually used in experiments with heavy and
large sample cells like those used for high pressure measurements. The photo
of a similar powerful cryogen-free dilution refrigerator (E-18) developed by
Oxford Instruments in collaboration with ISIS [1] is presented in Figure 1a.
The fridge is capable of cooling large (diameter 200mm; height 250mm) and
heavy (up to 20 kg) samples and provides access for the neutron beam through
a set of thin aluminium alloy windows. The base temperature of the
refrigerator is 15 mK and cooling power is approximately 370 μW at 100 mK.
In experiments where E-18 has been used to cool down 110 cubic-centimeter
solid and liquid 4He sample cells Figure 1b (also attached to the mixing
chamber in Figure 1a) the base temperatures of 35 mK for the solid sample
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O. Kirichek and R. B. E. Down 136
and 50 mK for the liquid sample (up to 25 bar) have been held for weeks [15 -
17]. In these experiments the Bose-Einstein condensation has been studied in
solid and liquid 4He under pressure. In Figure 2(a) we present full 2D S(q,w)
of 4He recoil on the Mari chopper spectrometer and in Figure 2(b) a cut
through the 4He recoil line (courtesy of Jon Taylor, ISIS). These
measurements have been performed at 50 mK. The condensate fraction at 50
mK temperature is found to decrease from 7.25 ± 0 .75% at saturated vapor
pressure to 3.2 ± 0.75% at 24 bar pressure [16, 17]. No evidence for Bose-
Einstein condensation has been found in the solid helium sample [15]. To our
knowledge, the paper published in [15] became the first publication where a
powerful cryogen-free dilution refrigerator has been used in a neutron
scattering experiment.
Today dilution refrigerators of this type are commercially available from a
number of companies such as Leiden Cryogenics [18], Air Liquide [19],
Oxford Instruments [20], LTLab [21], Cryoconcept, Janis, BlueFors and Ice
Oxford.
Figure 1. (a) Powerful cryogen-free dilution refrigerator E-18; (b) 110 cm3 solid
4He
sample cell (also shown attached to the mixing chamber in (a)).
(a) (b)
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Ultra-Low Temperature Sample Environment for Neutron … 137
Figure 2. (a) 4He sample‘s energy transfer on Mari chopper spectrometer; (b) a cut
through the 4He recoil line. Measurements have been performed at 50 mK. (courtesy of
Jon Taylor, ISIS).
TOP-LOADING DILUTION INSERTS IN CRYOGEN
FREE CRYOSTATS
Dilution and 3He refrigerator inserts which can be used with the variable
temperature insert (VTI) of a conventional cryostat or superconducting magnet
are very popular at neutron facilities around the world. These systems
consume cooling power produced at the VTI heat exchanger and can be easily
made cryogen-free if used with cryogen-free cryostats. This can be achieved
by inserting the refrigerator either in the top-loading system based on a cryo-
cooler [22] or in a re-condensing cryostat with a VTI [3]. However the range
of applications of conventional ultra-low temperature inserts is limited by their
relatively low cooling power (~30 μW at 100mK) and small sample space
(outer diameter ≤ 37 mm and height ≤80 mm).
The ISIS facility has developed a top-loading cryogen free system based
on the PTR [23] which can be used as a substitute for conventional liquid
cryogens cryostats. The initial success of this project has led to a joint ISIS –
Oxford Instrument collaboration aimed at developing a 50 mm diameter top-
loading cryogen-free cryostat which provides sample environment for neutron
scattering experiment in the temperature range 1.4 – 300K [24]. The high
cooling power of the cryostat‘s VTI: 0.23W @ 1.9K is sufficient for extending
the sample temperature range below 1K by employing a standard dilution
refrigerator insert KelvinoxVT® in the continuous regime.
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The design of the system is based on the idea of combining a top-loading
cryogen-free system described in Ref. [7] with a helium condensation loop
[25], which has been successfully realized in a first prototype described in
[23]. The PTR selected for the top-loading system is a Sumitomo RP082 B
with cooling power of 1 W @ 4.2 K and an electrical input power to the
compressor of 7 kW.
A similar set of heat exchangers as in Ref. [23] has been used in the
second cryostat design [24] except the heat exchanger at the first stage of the
PTR that is based on sinter rather than coils of copper capillary. This almost
doubled the surface area of 2nd
stage liquid helium heat-exchanger. The sinter
in the 1st stage heat-exchanger also plays a role of an additional 50K filter for
incoming helium gas.
The fixed capillary impedance used in the previous design [23] has been
replaced by a standard Oxford Instruments needle valve that has made
changing the helium circulation rate in a broad range possible.
The major change in comparison with the design of the previous system is
the replacement of the cryostat‘s top-loading insert with the standard Oxford
Instruments VTI. The annulus design of the VTI significantly improves the
recuperation of cooling power stored in the cold helium vapor pumped out of
the VTI‘s 1K pot heat-exchanger. The helium vapor is evacuated through the
annulus space of the VTI by a hermetically sealed rotary pump. After that the
helium vapor goes through a liquid nitrogen cooled trap and returns to the
condensation line of the cryostat. If the system is not operational, the helium
gas is stored in 50 L dump. Two pressure relief valves together with two
standard pressure gauges have been fitted at the inlet of the circulation loop
and at the pumping line in order to prevent over-pressurizing of the system due
to the blockage of the cryogenic part of the circulation loop.
The method of system operation is as follows: the PTR is activated and
the system begins to cool down. Soon after activating the PTR the Helium gas
is released to the inlet of the circulation loop. The pressure of helium gas at the
inlet is set at 0.1 MPa. After reaching equilibrium within few minutes the
circulation rate is settled at 2 · 10-3
Moles per second. At lower temperatures
the circulation rate increases reaching a maximum 10-2
Moles per second. The
total time taken by the sample holder to cool down from room temperature to
base temperature (below 2K) is less than 7 hours. Once cooled, and with a
continuous flow of helium, the cryostat can remain endlessly at base
temperature.
The procedure of sample changing in the cryogen-free system is very
similar to that employed in conventional cryostats: the VTI can be heated
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Ultra-Low Temperature Sample Environment for Neutron … 139
quickly, the sample tube filled with helium gas at just above atmospheric
pressure, the sample stick withdrawn and the sample changed. Once the
sample stick is loaded back into the VTI the heater is turned off and the
sample tube evacuated but for a small volume of exchange gas (~ 5 kPa)
required for sample cooling. The system is then left to re-cool to base
temperature. The total time required for a sample change, starting and
finishing at the sample base temperature, is less than 2 hours.
The procedure of loading and cooling a standard KelvinoxVT® dilution
insert is also similar to the one employed in conventional cryostats. The cool
down procedure consists of three stages. Stage one is pre-cooling of the insert
with heat-exchange helium gas in the refrigerator‘s vacuum can from room
temperature down to below 2K.
After that the VTI is warmed up to 5K and exchange gas is pumped out
for approximately one hour. Once good vacuum in the vacuum can is
achieved, the VTI heater is switched off and dilution refrigerator insert starts
helium mixture condensation and, eventually, circulation in an automated
regime.
The condensation of the 3He/
4He mixture, initiating of
3He circulation and
cooling the refrigerator down to base temperature is shown in Figure 3, where
(a) represents the behavior of the mixing chamber temperature; (b) time
dependence of temperature at the 1K pot heat exchanger of the dilution insert;
and (c) time dependence of the PTR second stage temperature. The refrigerator
operates in fully automated mode.
The time taken from the beginning of the condensation to base
temperature is approximately an hour. The total cool down time of the dilution
insert starting from inserting the refrigerator in the VTI is less than 7 hours,
which is typical of the time taken for conventional cryogen based top-loading
cryostats.
In the test the base temperature of the fridge was 55mK a little higher than
the technical specification 25mK. This can be explained by a massive sample
attached to the mixing chamber, which has not been removed after a previous
neutron scattering experiment, conducted in a standard Variox® cryostat. In
this experiment a similar base temperature of 60mK has been achieved.
All the parameters of the new system are very close to ones of
conventional cryogen based cryostat, such as the Orange cryostat for example.
However the new system does not require cryogenic liquid top-ups which are
challenging from the logistical, economical, operational and safety points of
views.
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O. Kirichek and R. B. E. Down 140
Figure 3. The condensation of the 3He/
4He mixture, initiating of
3He circulation and
cooling refrigerator down to the base temperature, where (a) represents behavior of the
mixing chamber temperature; (b) time dependence of temperature at 1K pot heat
exchanger; and (c) time dependence of PTR second stage temperature.
OPERATING OF TOP-LOADING DILUTION INSERTS
IN VTI OF RE-CONDENSING MAGNET
IN ZERO BOIL-OFF REGIME
A growing number of neutron scattering experiments now require a
combination of extreme conditions such as high magnetic field with ultra-low
temperatures. This sample environment can be achieved by operating a
standard dilution refrigerator insert in the VTI of a re-condensing magnet
cryostat [3]. The design of re-condensing magnet cryostats is usually based on
similar helium bath cryostats [2, 10]. The superconducting magnet is
immersed in the liquid helium. The radiation shield is cooled by the cooler‘s
first stage and the second stage re-condenses helium directly in the helium
vessel. Thus the re-condensing magnet cryostat does not consume any liquid
helium in normal operation. The main advantage of this system is that all
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Ultra-Low Temperature Sample Environment for Neutron … 141
magnet operating procedures, such as cooling, running up to field and
quenching are identical to those of a standard magnet in a bath cryostat. All
standard magnet system accessories like current leads, power supply and
helium transfer siphon can be used. This system also provides a homogeneous
temperature distribution, which is crucial for optimum magnet performance.
Another important advantage is the ability of the magnet, in the case of a cryo-
cooler power failure, to stay at field for more than 48 hours before quenching.
Zero helium boil-off operation of the re-condensing 9T superconducting
magnet cryostat with a standard KelvinoxVT® insert (presented in Figure 4)
running at a base temperature of 63mK during a 24 hour period has been
demonstrated at ISIS [3].
Figure 4. The re-condensing 9T superconducting magnet cryostat with standard
dilution refrigerator insert KelvinoxVT®.
Kelvinox VT®
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O. Kirichek and R. B. E. Down 142
The cooling power for helium re-condensing is provided by a CryoMech
PT410 pulse tube refrigerator. This was designed to re-condense the helium
consumed by the VTI during the operation of a dilution refrigerator insert. The
PTR is inserted into a separate 'sock' which essentially forms an additional
cryostat neck. Both the 1st and 2
nd stages of the PTR are equipped with
specially designed heat exchangers. The first stage heat exchanger is dual
purpose and is intended to transfer the heat load from the radiation shield to
the PTR 1st stage and pre-cool helium gas injected into the cryostat via the
'sock'. The second stage heat exchanger is intended mainly for re-condensing
the helium evaporated from the helium reservoir due to the various heat loads
and condensing externally fed helium gas. There are no special features aiding
the heat exchange between helium gas and the PTR second stage regenerator.
Instead, this heat exchange is provided by helium gas flow driven by
convection along the surface of the tube which contains the regenerator
material. While testing the system it was demonstrated that temperature,
pressure and liquid He level can be kept constant in a steady-state zero boil-off
operation with approximately 600-650mW of excess cooling power on the
second stage of the PTR [2].
To stop the ingress of air into the system the helium reservoir must remain
above atmospheric pressure. A heater was installed in the main helium
reservoir to maintain the pressure at 3 kPa above atmospheric. A ~ 14 kPa
relief valve was installed on the helium exhaust to stop any helium from
leaving the system due to the pressure of the main bath. The VTI that serves as
the 1.5K platform for the dilution refrigerator was cooled by pumping with an
Edwards XDS 35i dry scroll pump. To reclaim the gas from the pump and to
ensure its purity the pump exhaust was connected via NW10 tubing to a
charcoal trap which was cooled to 77K in liquid nitrogen. The inlet and outlet
of the charcoal trap and the entry to the magnet cryostat were fitted with 10µm
sintered filters. The filters stop solid particles and oil droplets from the pump
entering the magnets liquid helium reservoir. The NW10 flexible tube from the
outlet of the charcoal trap was connected to an inlet above the 1st stage of the
pulse tube on the magnet where all helium exhaust gas from the pump was
observed to be reclaimed back to the helium reservoir of the magnet.
The dilution insert cool-down from room temperature to the base
temperature of refrigerator at around 60mK took less than six hours. After the
VTI flow and dilution fridge circulation were stabilized the exhaust of the
pump was connected to the entry of the magnet cryostat. The helium level and
the base temperature of the dilution refrigerator were fairly stable during a 24
hour run at 58% and 63 mK respectively. The pressure in the cryostat is slowly
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Ultra-Low Temperature Sample Environment for Neutron … 143
drifting down during the first ten hours and after that stabilizes around 2.5 kPa
above atmospheric. The temperature of the VTI heat-exchanger demonstrates
similar behavior, stabilizing around 1.7K. During the test the heating power on
the helium reservoir heater stayed at zero. The very slow dynamics observed
in the test allows us to suggest that almost all the excess cooling power of the
PTR‘s 2nd
stage was consumed by condensation of returned helium.
An alternative to the proposed scheme [3] of re-condensing helium in a
superconducting magnet cryostat has been realized in [26]. In this case the
PTR is mounted in vacuum and mechanically attached to the thermal shield
and the wall of the helium can. The excess cooling power has been measured
at around 350 mW, which wouldn‘t be enough to achieve full re-condensing
operation of the magnet cryostat with circulating standard dilution refrigerator
insert.
After commissioning in 2011 both magnets have been released to the ISIS
user program and have been intensively used in neutron scattering experiments
since. Advanced specifications of magnets in combination with the
performance of new cutting-edge TS2 instruments such as the WISH
diffractometer or LET spectrometer have made both magnets extremely
popular within the ISIS user community and has already led to scientific
publications [27, 28]. The outstanding performance of the system is
demonstrated in Figure 5 by the diffraction pattern of the ‗spin ice‘ sample
Ho2Ti2O7 obtained on WISH with the 14T magnet and dilution refrigerator
insert KelvinoxVT®
.
Figure 5. Diffraction pattern of the ‗spin ice‘ sample Ho2Ti2O7 (courtesy of R. Aldus,
UCL) previously used in neutron scattering investigation of the material‘s kagome-ice
plateau [29], obtained on WISH with 14T magnet and dilution refrigerator insert
KelvinoxVT®. The data were obtained at 150 mK and 1.6T.
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O. Kirichek and R. B. E. Down 144
The data was obtained at 150 mK and 1.6T. This data demonstrates very
good angular coverage and excellent data quality (low noise) made possible
when using the collimator and cadmium shielding.
The set-up described above provides cryogen-free high magnetic field and
ultra-low temperature sample environment for neutron scattering experiments.
The system offers a significant reduction in liquid helium consumption as well
as the advantage of operational simplicity and a significant improvement in
user safety. Estimations based on half a year of 14T magnet cryostat operation
statistics show that one re-condensing magnet cryostat should save us ~1800 L
of liquid helium a year.
CONCLUSION AND PERSPECTIVES
Here we have reviewed three different approaches that allow cryogen-free
operation of dilution refrigerators used in neutron scattering ULT sample
environment. In addition to the reduction or complete elimination of
dependence on liquid cryogens the cryogen free approach also offers
operational simplicity and a significant decrease in technical resources
required for DR operation. Cryogen-free technology is also safer. The
involvement of technical personnel regularly handling cryogens raises serious
safety issues associated with asphyxiation due to the lack of oxygen that is
replaced by rapidly evaporating nitrogen or helium. This is a potentially lethal
hazard. Cryogen-free technology significantly reduces and in some cases
completely eradicates all of the cryogenic hazards involved.
There are other cryogen-free options that did not gain popularity in
neutron scattering sample environment due to relative inefficiency or
unsustainability issues.
The first example could be running a dilution refrigerator insert which
incorporates a Joule-Thompson stage such as Cryoconcept TBT®, FRMII style
insert [30] or Oxford Instruments Kelvinox JT® in a standard CCR based top-
loader with base temperature of ~ 3.5K. The operation of such a system has
been recently demonstrated by the FRM II sample environment team [28].
However the time required for pre-cooling, mixture condensation and
circulation is greater than 24 h which is not acceptable in the case of an intense
facility user program.
A few years ago Oxford Instruments developed a lower power cryogen
free dilution refrigerator with cryogenic circulation [31], where the
temperature of the circulating 3He never exceeds 50K. However, this
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Ultra-Low Temperature Sample Environment for Neutron … 145
refrigerator design does require significantly more 3He than a standard one
with similar performance. The recent dramatic increase in the price of 3He [32]
has made the fridge commercially unviable.
ACKNOWLEDGMENTS
We are very grateful to J. Keeping, B. E. Evans, D. J. Bunce, C. R.
Chapman and other members of the ISIS sample environment group involved
in ISIS cryogen free development projects. We are also grateful to Z. A.
Bowden for support and rewarding discussions. We also greatly appreciate the
fruitful collaboration with colleagues from Oxford Instruments.
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In: Recent Progress in Neutron Scattering
Editors: A. Vidal and M. Carrizo
ISBN: 978-1-62948-099-2
c© 2013 Nova Science Publishers, Inc.
Chapter 6
STRENGTHS OF THE INTERACTIONS IN
Y Ba2Cu3O6.7 AND La2−xSrxCuO4(x = 0.16)
SUPERCONDUCTORS OBTAINED BY
COMBINING THE ANGLE-RESOLVED
PHOTOEMISSION SPECTROSCOPY AND THE
INELASTIC NEUTRON SCATTERING
RESONANCE MEASUREMENTS
Z. G. Koinov∗
Department of Physics and Astronomy,
University of Texas at San Antonio,
San Antonio, Texas, US
Abstract
The t − U − V − J model predicts phase instabilities which give
rise to a divergence of the charge density and spin response functions.
This model has been the focus of particular interest as model for high-
temperature superconductivity in cuprite compounds because it fits to-
gether three major parts of the superconductivity puzzle of the cuprite
∗E-mail address: [email protected]
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150 Z. G. Koinov
compounds: (i) it describes the opening of a d-wave pairing gap, (ii) it
is consistent with the fact that the basic pairing mechanism arises rom
the antiferromagnetic exchange correlations, and (iii) it takes into ac-
count the charge fluctuations associated with double occupancy of a site
which play an essential role in doped systems. Based on the conven-
tional idea of particle-hole excitations around the Fermi surface, we have
performed weak-coupling calculations of the strengths of the effective
interactions in the t − U − V − J model which are consistent with the
antinodal gap obtained by the angle-resolved photoemission spectroscopy
(ARPES), and the well-defined incommensurate peaks, which have been
observed by the inelastic neutron scattering resonance (INSR) experi-
ments in Y Ba2Cu3O6.7 and La2−xSrxCuO4(x = 0.16) samples.
1. Introduction
1.1. The Hamiltonian of the t− U − V − J model
The Hamiltonian of the t − U model contains only two terms representing
the hopping of electrons between sites of the lattice and their on-site Hubbard
repulsive interaction U > 0. The t − J model takes into account the spin-
dependent Heisenberg near-neighbor antiferromagnetic (AF) exchange interac-
tion (J > 0). The two models predict phase instabilities which give rise to
a divergence of the charge density and spin response functions, and therefore,
they have been the focus of particular interest as models for high-temperature
superconductivity. In particular, the commensurate and incommensurate peak
structures associated with the neutron cross section have been studied within the
Fermi-liquid-based scenario using the single-band Hubbard t−U model [1]-[3],
or the t−J model [4]-[9]. Common to t−U and t−J models is the expression
for the magnetic susceptibility in the random phase approximation (RPA)
χRPA(Q, ω) =χ(0)(Q, ω)
1 + ϕ(Q)χ(0)(Q, ω),
where χ(0)(Q, ω) is the bare magnetic susceptibility, ϕ(Q) = U in the case of
the t− U model, and ϕ(Q) = −2J(cosQx + cosQy) in the case of a nearest-
neighbor AF interaction.
It is known that in the case when the Hubbard repulsion is large (U/t→ ∞),
the AF exchange J interaction is the consequence of Hubbard repulsion, be-
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Strengths of the Interactions ... 151
cause the t−J model is obtained after projecting out the doubly occupied states
in the Hubbard t−U model, so that J = 2t2/U . Strictly speaking, by projecting
out the doubly occupied states we remove the high-energy degrees of freedom
and replace them with kinematical constraints assuming that the high energy
scale (given by U that in cuprates corresponds to the energy cost to doubly oc-
cupy the same site) is irrelevant. Thus, if the constraint of no double occupancy
is released, we arrive to the conclusion that the susceptibilities should be calcu-
lated using the t− U − J model rather than the t− U or the t− J models (the
corresponding arguments are presented in [10]-[13]).
The t− U − V − J model has also a V term, and the reason to include this
term is that the d-wave superconductivity is due to the near-neighbor interac-
tions, but these could be the near-neighbor AF exchange interaction as well as a
short-range spin independent attractive interaction V between the tight-binding
electrons on the near-neighbor sites of the lattice. The source of the spin inde-
pendent interaction has still not been established with certainty, and therefore,
V should be thought as a phenomenological parameter which stabilizes the d-
wave pairing. Thus, the starting point for our analysis of the commensurate and
incommensurate peak structures associated with the neutron cross section is the
t− U − V − J model:
H = −∑
i,j,σ
tijψ†i,σψj,σ − µ
∑
i,σ
ni,σ + U∑
i
ni,↑ni,↓ − V∑
<i,j>σσ′
ni,σ nj,σ′ + J∑
<i,j>
−→S i.
−→S j ,
(1)
where µ is the chemical potential. The Fermi operator ψ†i,σ (ψi,σ) creates
(destroys) a fermion on the lattice site i with spin projection σ =↑, ↓ along a
specified direction, and ni,σ = ψ†i,σψi,σ is the density operator on site i. The
symbol∑<ij> means sum over nearest-neighbor sites of a 2D square lattice.
The first term in (1) is the usual kinetic energy term in a tight-binding approx-
imation, where tij is the single electron hopping integral. The spin operator is
defined by−→S i = hψ†
i,σ−→σ σσ′ψi,σ′/2, where −→σ is the vector formed by the Pauli
spin matrices (σx, σy, σz). The lattice spacing is assumed to be a = 1 and the
total number of sites is N .
It is expected that a large U value has to be chosen to guarantee that cuprates
are insulators at half-filling. Unfortunately, well controlled solutions in the in-
termediate to strong regime (U ≥ W = 8t, where W is the bandwidth in the
limit U = 0) can be obtained numerically by the Monte Carlo method, limited
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152 Z. G. Koinov
Table 1. Input ARPES parameters and the interaction strengths U , Jand V (units eV) in Y Ba2Cu3O6.7 and La2−xSrxCuO4(x = 0.16) sam-
ples calculated by using the INSR commensurate and incommensurate
peaks. t1 = 0.250 eV, t2 and t3 are the hopping amplitudes for the first
to the third nearest neighbors, µ is the chemical potentia, ∆ is the gap, and
Vψ = 2V + 3J/2
t2 t3 µ ∆ Vψ U J VY BaCuO 0.4t1 0.0444t1 -0.27 0.06 0.265 0.569 0.111 0.049
LaSrCuO 0.148t1 −0.074t1 −0.821t1 0.01 0.0958 0.490 0.030 0.025
due to the exponential increase of computing time with the number of elec-
trons, and by the density matrix renormalization group algorithm, limited by
the two-dimensional entanglement problems. The fact that the phase diagram at
half filling of the extended Hubbard model (J = 0) shows an ”island” in U-V
space where d-wave pairing exists for U < 2t and V < t [14] suggests that
in the weak-coupling regime, where the general random phase approximation
(GRPA) is well justified, V interaction should not be ignored.
Random-phase-approximation expressions for the dynamic spin susceptibil-
ity in the weak-coupling regime have been used to explain the inelastic neutron
scattering resonance (INSR) measurements when doped away from half-filling
[15]-[18]. Recently [19]-[21], the sharp collective mode at the AF wave vec-
tor QAF = (π, π) in underdoped Bi2212 samples has been viewed as a weak-
coupling instability of a Fermi liquid, and the conclusion that in the t−U−V −J
model the Heisenberg effective interaction J is dominant has been put forward.
Since incommensurate peaks in underdoped Bi2212 samples have not been re-
ported, the weak-coupling theory, applied to Bi2212 cuprate family, provides
only a single line in U, J parameter space (see Figure 3 below) which repro-
duces both the angle-resolved photoemission spectroscopy (ARPES) antinodal
gap, and the INSR energy at QAF .
The main goal of the present Chapter is to test whether the INSR measure-
ments of YBaCuO and LaSrCuO compounds can be explained by the weak-
coupling theory. As in the Bi2212 case, we plot lines in U, J parameter space
using the ARPES antinodal gap, the position of a sharp collective excitation at
wave vectors near the AF wave vector QAF , and four well-defined incommen-
surate peaks at wave vectors Qδ = ((1±δ)π, π) and Qδ = (π, (1±δ)π), which
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Strengths of the Interactions ... 153
have been observed in several families of these materials in the INSR studies of
quantum spin fluctuations [22]-[35]. Here, we use a square lattice notation, and
δ is the incommensurability parameter. If the weak-coupling description is cor-
rect, these lines should have only one common point (see Figure 4 and Figure 5)
from which one can estimate the strengths of the interactions U , V and J (see
Table 1).
1.2. The Hubbard-Stratonovich Transformation
and the Bethe-Salpeter Equation
It is known that the single-particle excitations of the Hamiltonian (1) mani-
fest themselves as poles of the single-particle Green’s function, G; while the
two-particle (collective) excitations could be related to the poles of the two-
particle Green’s function, K . The poles of these Green’s functions are defined
by the solutions of the Schwinger-Dyson (SD) equation G−1 = G(0)−1 − Σ,
and the Bethe-Salpeter (BS) equation [K(0)−1−I ]Ψ = 0. Here, G(0) is the free
single-particle propagator, Σ is the electron self-energy, I is the BS kernel, and
the two-particle free propagator K(0) = GG is a product of two fully dressed
single-particle Green’s functions. Since the electron self-energy depends on the
two-particle Green’s function, the positions of the poles have to be obtained by
solving the SD and BS equations self-consistently.
It is widely accepted that the general random phase approximation (GRPA)
is a good approximation for the collective excitations in a weak-coupling
regime, and therefore, it can be used to separate the solutions of the SD and
the BS equations. In this approximation, the single-particle excitations are re-
placed with those obtained by diagonalizing the Hartree-Fock (HF) Hamilto-
nian; while the collective modes are obtained by solving the BS equation in
which the single-particle Green’s functions are calculated in HF approximation,
and the BS kernel is obtained by summing ladder and bubble diagrams.
It should be mentioned that according to the alternative stripes model (see
Ref. [36] and the references therein) the incommensurate peaks are the natu-
ral descendants of the stripes, which are complex patterns formed by electrons
confined to separate linear regions in the crystal. Our approach is based on
the weak-coupling Fermi-liquid-based scenario put forward by various groups
that the experimental peaks associated with incommensurate and commensurate
structure of the magnetic susceptibility probed by neutron scattering in cuprate
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154 Z. G. Koinov
compounds have common origin. The poles of magnetic susceptibility are the
same as the poles of the two-particle Green’s function. The weak-coupling
Fermi-liquid-based scenario allows us to use a unified description of commen-
surate and incommensurate peaks based on the solutions of the corresponding
Bethe-Salpeter (BS) equations in the GRPA. The solutions of the BS equation
manifest themselves as poles of the spin or charge density part of the general
response function. This is due to the fact that the interaction terms in the Hub-
bard Hamiltonian can be separated in two different groups. In the first group
we include interactions which are proportional to the density operator ni,σ, and
therefore, these interactions form the charge density part (or the density chan-
nel) of the general response function. The second group of interactions consists
of interactions proportional to ψ†i,σψi,σ (σ is complimentary to σ). These in-
teractions are responsible for the spin part (or they form the spin channel) of
the general response function. The spin-independent V term belongs to the first
group. Since the U interaction can be rewritten in terms of the spin operators
as −2U/3∑i−→S i.
−→S i + const, the U term in (1) contributes to both the spin
and density channels. The AF interaction contributes to the density and spin
channels because it can be rewritten as a sum of two interactions,
J1 = J4
∑<i,j> [ni,↑nj,↑ + ni,↓nj,↓ − ni,↑nj,↓ − ni,↓nj,↑] and J2 =
J2
∑<i,j>[ψ†
i,↑ψi,↓ψ†j,↓ψj,↑+ψ
†i,↓ψi,↑ψ
†j,↑ψj,↓]. The BS equations in our previous
calculations [43, 44] did not take into account the J2-term in the Hamiltonian
(1). As a result, the Goldstone mode at Q = 0 was not a pole of the two-particle
Green’s function. In the present calculations our BS equations are consistent
with the Goldstones theorem.
Generally speaking, there exist two different GRPA that can be used to cal-
culate the spectrum of the collective excitations of the t− U − V − J Hamil-
tonian. The first approach uses the Green’s function method, while the second
one is based on the Anderson-Rickayzen equations [37]-[39]. According to the
Green’s function method, the collective modes manifest themselves as poles of
both the two-particle Green’s function, K , and the density and spin response
functions. The two response functions can be expressed in terms of K , but it is
very common to obtain the poles of the density response function by following
the Baym and Kadanoff formalism [40], which uses functional derivatives of the
density with respect to the external fields. The second method that can be used
to obtain the collective excitation spectrum of the Hubbard Hamiltonian starts
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Strengths of the Interactions ... 155
from the Anderson-Rickayzen equations, which in the GRPA can be reduced
to a set of coupled equations and the collective-mode spectrum is obtained by
solving the corresponding secular determinant.
Instead of mean-field decoupling the quartic interaction terms, we apply
the idea that we can transform these quartic terms into quadratic form by mak-
ing the Hubbard-Stratonovich transformation for the electron operators. The
attempt to decouple the quartic terms by using two-component Nambu field op-
erators ψ(x) =
(ψ↑(x)ψ†↓(x)
)and ψ(y) =
(ψ†↑(y)ψ↓(y)
)does not work because
the existence of J2 part in the J interaction requires the use of more compli-
cated four-component fermion field operators ( ψ and ψ obey anticommutation
relations):
ψ(x) = 1√2
ψ↑(x)ψ↓(x)ψ†↑(x)
ψ†↓(x)
,
ψ(y) = 1√2
(ψ†↑(y)ψ
†↓(y)ψ↑(y)ψ↓(y)
). (2)
In contrast to the previous approaches, such that after performing the
Hubbard-Stratonovich transformation the fermion degrees of freedom are in-
tegrated out; we decouple the quartic problem by introducing a model system
which consists of a four-component boson field Aα(z) (α = 1, 2, 3, 4) interact-
ing with fermion fields (2).
There are three advantages of keeping both the fermion and the boson de-
grees of freedom. First, the approximation that is used to decouple the self-
consistent relation between the electron self-energy and the two-particle Green’s
function automatically leads to conserving approximations because it relies on
the fact that the BS kernel can be written as functional derivatives of the Fock
ΣF and the Hartree ΣH self-energy I = Id + Iexc = δΣF/δG+ δΣH/δG =δ2Φ/δGδG. As shown by Baym [41], any self-energy approximation is con-
serving whenever: (i) the self-energy can be written as the derivative of a func-
tional Φ[G], i.e. Σ = δΦ[G]/δG, and (ii) the SD equation for G needs to
be solved fully self-consistently for this form of the self-energy. Second, the
collective excitations of the Hubbard model can be calculated in two different
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156 Z. G. Koinov
Figure 1. Diagrammatic representation of the Bethe-Salpeter equation (4)
for the BS amplitude ΨQn2,n1(k; Ω). The single-particle Green’s function
Gn2n1(k,Ω) is denoted by two solid lines, oriented in the direction of electron
propagation. The dashed lines represent the free boson propagator D(0)αβ(ω,Q).
At each of the bare vertices Γ(0)α (n1, n2) (represented by circles) the energy and
momentum are conserved.
ways: as poles of the fermion Green’s function, K , and as poles of the boson
Green’s function,D; or equivalently, as poles of the density and spin parts of the
general response function, Π. Here, the boson Green’s function, D, is defined
by the equation D = D(0) + D(0)ΠD(0) where D(0) is the free boson propa-
gator. Third, the action which describes the interactions in the Hubbard model
is similar to the action ψ†Aψ in quantum electrodynamics. This allows us to
apply the powerful field-theoretical methods, such as the method of Legendre
transforms, to derive the SD and BS equations, as well as the vertex equation for
the vertex function, Γ, and the Dyson equation for the boson Green’s function,
D.
The basic assumption in our BS formalism is that the bound states of two
fermions in the optical lattice at zero temperature are described by the BS wave
functions (BS amplitudes). The BS amplitude determines the probability am-
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Strengths of the Interactions ... 157
plitude to find the first electron at the site i at the moment t1 and the second
electron at the site j at the moment t2. The BS amplitude depends on the rela-
tive internal time t1 − t2 and on the center-of-mass time (t1 + t2)/2 [42]:
ΦQn2n1
(ri, rj ; t1, t2) = exp ı [Q.(ri + rj)/2 − ω(Q)(t1 + t2)/2]ΨQn2n1
(ri − rj , t1 − t2), (3)
where Q and ω(Q) are the collective-mode momentum and the correspond-ing dispersion, respectively. Since n1, n2 = 1, 2, 3, 4, we have to takeinto account the existence of sixteen BS amplitudes. The Fourier transform of
ΨQn2n1(r, t) =
∫dΩ2π
∫ddk
(2π)deı(k.r−Ωt)Ψ
Qn2,n1(k; Ω) satisfies the following BS
equation, presented diagrammatically in Figure 1:
ΨQn2 ,n1
(k;Ω) = Gn1n3(k + Q,Ω + ω(Q))Gn4n2
(k,Ω)∫dΩ′
2π
∫ddp
(2π)d
[Id
(n3 n5
n4 n6|k − p,Ω − Ω′
)+ Iexc
(n3 n5
n4 n6|Q, ω(Q)
)]Ψ
Qn6 ,n5
(p;Ω′). (4)
Here, Gn1n2 (k,Ω) is the single-particle Green’s function, and Id and Iexcare the direct and exchange parts of the BS kernel.
In a similar problem of interacting photons and electrons in electrodynam-
ics, the direct part of the BS kernel does depend on frequency; therefore the
solution of the BS equation is more complicated. As we shall see in the next
Section, in the case of the t− U − V − J model the boson propagatorD(0)αβ(k)
(α, β = 1, 2, 3, 4) is frequency independent, and therefore, the following BS
equation for the equal-time BS amplitude ΨQn2,n1(k) =
∫dΩ2πΨ
Qn2,n1(k; Ω) takes
place:
ΨQn2,n1(k) =
∫dΩ2πGn1n3 (k + Q,Ω + ω(Q))Gn4n2(k,Ω)×
∫ ddp(2π)d
[−Γ(0)α (n3, n5)D
(0)αβ(k − p)Γ
(0)β (n6, n4)
+12Γ
(0)α (n3, n4)D
(0)αβ(Q)Γ
(0)β (n6, n5)]Ψ
Qn6,n5(p). (5)
In the case of t− U − V − J model the interaction in the direct kernel can be
factorized, i.e. D(0)αβ(k − p) = fα(k)gβ(p); therefore it is possible to obtain
the collective excitation spectrum using an 80 × 80 secular determinant. Such
an ambitious task will be left as a subject of future research. Instead, we shall
discuss an approximation in which the dimensions of the secular determinant
becomes 20 × 20.
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158 Z. G. Koinov
2. Spectrum of the Collective Excitations in the GRPA
The single excitations (quasiparticles) of the model we study manifest them-
selves as poles of the fermionic single-particle Green’s function. Because of
the J2 interaction the single-particle Green’s function is a 4 × 4 matrix which
Fourier transform in the mean-field approximation is:
G(k, ıωm) =1
(ıωm)2 −E2(k)
ıωm + ε(k) 0 0 ∆(k)
0 ıωm + ε(k) −∆(k) 00 −∆(k) ıωm − ε(k) 0
∆(k) 0 0 ıωm − ε(k)
.
(6)
Here, ωm is the Matsubara frequency, ∆(k) is the order parameter, and
E(k) =√
∆(k)2 + ε2(k), where ε(k) = ε(k) − µ. For a 2D square lat-
tice the electron energy ε(k) has a tight-binding form ε(k) = t1(coskx +cosky)/2 + t2 cos kx cosky + t3(cos 2kx + cos 2ky)/2 + t4(cos 2kx cosky +
cos 2ky cos kx)/2 + t5 cos 2kx cos 2ky, where ti are the hopping amplitudes for
first to fifth nearest neighbors on a square lattice. The order parameter satisfies
the following gap equation:
∆(k) =1
N
∑
q
(−U + Vψ
[cskcsq + cdkcdq
]) ∆(q)
2E(q)tanh
(βE(q)
2
). (7)
Here, Vψ = (2V + 32J), and we use the following notations: csk = cos(kx) +
cos(ky), cdk = cos(kx) − cos(ky). At a zero temperature the gap equation (7)
has a solution of the form ∆k = ∆0 + ∆1csk + ∆2cdk, where
∆0 = −U
N
∑
q
∆q
2E(q), ∆1 =
VψN
∑
q
csq∆q
2E(q), ∆2 =
VψN
∑
q
cdq∆q
2E(q).
In the case of d-wave pairing one can neglect ∆0 and ∆1. In this approximation
V , J , µ, ti and ∆2 should all be thought of as effective parameters which satisfy
the gap equation:
1 =2Vψ(2π)2
∫dk
g2k√
ε2k
+ ∆2g2k
,
where ∆ = 2∆2 and gk = cdk/2.
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Strengths of the Interactions ... 159
In what follows we will derive the BS equation for the collective modes
ω(Q) in the GRPA. There are sixteen BS amplitudes, but in our version of the
GRPA we take into consideration only four of them:
Ψ =
Ψ↓,↑(k,Q)Ψ↑,↓(k,Q)
Ψ↑,↑(k,Q)Ψ↓,↓(k,Q)
.
Thus, the BS equation in GRPA has the following form Ψ = K(0) [Id + Iexc] Ψ,
where K(0) is a product of two single-particle Green’s functions (6), and Id and
Iexc are the direct and exchange interactions, correspondingly.
Figure 2. Diagrammatic representations of: (a) the BS amplitude, the single-
particle Green’s functions, the direct and exchange interactions; (b) the BS
terms involving just one direct interaction line, and (c) one exchange interac-
tion line.
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160 Z. G. Koinov
In Figure (2a) we have shown the diagrammatic representations of the lead-
ing terms of the direct and exchange interactions. Note, that the direct inter-
actions which are represented by vertical dotted and double dotted lines are
different from the exchange interactions, represented by horizontal doted and
double dotted lines. In Figure (2b) and (2c) we have shown the diagrammatic
representations of the terms involving just one direct interaction line (first four
terms in the righthand side of the BS equation) and just one exchange inter-
action line (last four terms in the righthand side of the BS equation). Thus,
in this approximation we have a set of four coupled BS equations which leads
to a set of 20 homogeneous equations (see the Appendix), and therefore, the
corresponding secular determinant allows us to obtain the collective modes as
a function of Q for a given U/t, V/t and J/t. As can be seen, the elements
of A, B and C blocks are convolutions of conventional two normal GG, two
anomalous FF Green’s functions or FG terms. At the high-symmetry wave
vector, such as QAF , I ia,b and J ia,b with i = 3, 4 involve sine functions, and
therefore, all vanish. I2a,b and J2
a,b also vanish because ε(k,QAF ) is symmetric
with respect to exchange kx ↔ ky. Similarly, the non-diagonal elements of I ija,band J ija,b with i 6= j all vanish. Thus, at QAF each of A and B blocks has 10
different non-zero elements, while block C has 40 non-zero elements. In other
words, the ω dependence of χ (or χ−1) at QAF comes from these 60 non-zero
elements.
The mixing between the spin channel and other channels is neglected in
the RPA, but one can separate m < 20 channels (this is the m−channel re-
sponse theory) and use only m × m secular determinant. To estimate the er-
ror we rewrite the secular determinant as det|χ−1 − V | = det
∣∣∣∣∣P Q
QT R
∣∣∣∣∣ =
det|R|det|P −QR−1QT |. Them−channel response-function theory takes into
account only them×m matrix P, neglecting the mixing with the other 20−m
channels which is represented by them×mmatrixQR−1QT . For example, we
can take into consideration two π channels [45]-[47] with bare π susceptibili-
ties χ011 = I22
ll and χ022 = I22
γγ , respectively. The susceptibilities I2γγ, J2lγ, J22lγ
represent the mixing of the spin and two π channels. Thus, the coupling of the
spin and two π channels (a three-channel response-function theory) leads in the
GRPA to a set of three coupled equations. When the extended spin channel is
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Strengths of the Interactions ... 161
added to the previous three channels, we have a set of four coupled equations (a
four-channel theory) [19, 20].
3. Strengths of Interactions in Cuprates
3.1. Strengths of Interactions in Bi2212 Samples
The corresponding parameters for the electron energy εk = ε(k) − µ are ob-
tained by fitting the ARPES data with a chemical potential µ and hopping am-
plitudes ti for the first three nearest neighbors on a square lattice. For Bi2212
compound with a doping concentration x = 0.12 the corresponding input data
are as follows: t1 = −4t, t2 = 1.2t, µ = −0.94t, ∆ = 35 meV and Vψ = 0.6t,
where t = 0.433 eV [21]. The parameters U, V and J should be adjusted in
such a way that the sharp collective mode of 40 meV which appears at the wave
vector QAF in INSR studies occurs at energy which corresponds to the lowest
collective mode of the corresponding Hamiltonian. In the case of J = 0 the
resonance is determined by the pole of the spin correlation function. Using the
above set of parameters and a resonance energy of 40 meV, we calculate the
RPA value of U of about 1.16 eV. The coupling of the spin channel with other
channels should change the RPA results for U . The coupling of the spin and
two π channels (a three-channel response-function theory) leads in the GRPA
to a set of three coupled equations [21], and the value of U is reduced from 1.16
eV to 0.974 eV. When the extended spin channel is added to the previous three
channels, we have a set of four coupled equations (a four-channel theory), and
according to Ref. [19] U ≈ 300 meV is required in the case when Vψ=0.260 eV
and J = 0.
Within the four-channel theory [19] the collective mode energy has been
calculated by using a 4 × 4 symmetric matrix χ which has only 6 non-zero
elements at QAF : χ11 = Iγγ , χ22 = I11mm, χ33 = I22
γγ , χ44 = I22ll , χ12 = J1
mγ
and χ34 = J22lγ (the other 4 elements χ13 = I2
γγ, χ14 = J2
lγ, χ23 = J12
mγ and
χ24 = I12ml vanish). In Figure 3 we present the results of our calculations of the
set of points in U, J parameter space which reproduce the INSR energy of 40
meV using all twenty channels. We see that the RPA spin correlation function
and the BS equations in GRPA, both provide very similar results for U at point
J = 0. It is worth mentioning that our previous graph (see Figure 2 in Ref.
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162 Z. G. Koinov
Figure 3. Sets of points in U, J parameter space which reproduce the commen-
surate peak in Bi2212 samples at ∼ 40 meV at point QAF = (π/a, π/a). The
V value is V = Vψ/2 − 3J/4 where Vψ = 260 meV. This set is very well
described by the linear function U(J) ≈ −4.002J + 1.157.
[43]) was obtained from the BS equations, but the Goldstone solution leads to
an equation similar to the gap equation but with Vψ = 2V + J/2 instead of
Vψ = 2V + 3J/2 as in the present form of the BS equations. Nevertheless, the
two graphs are quite similar.
Now, we turn our attention to the incommensurate peaks in Bi2212 samples.
Perhaps due to the limited size of single crystals currently available, to the best
of our knowledge, a double-peak structure of the spin response at 38 meV has
been reported only in Ref. [32]. We have used the observed position of this
peak, and by setting the collective mode to be 38 meV we have used the BS
equations to obtain a second set of points in the U, J parameter space, but it
turns out that the second set and the set of points presented in Figure 3 do not
have common point, and therefore, we have concluded that the 38 meV cannot
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Strengths of the Interactions ... 163
be a pole of the spin susceptibility.
3.2. Strengths of Interactions in Y Ba2Cu3O6.7 Samples
from Inelastic Neutron Studies
The commensurate and incommensurate peak structures associated with the
neutron cross section in Y BaCuO have been studied within the Fermi-liquid-
based scenario using the single-band Hubbard t−U model [1]-[3], or the t− J
model [4][9]. The techniques that have been used are based on (i) the Monte
Carlo numerical calculations [1], (ii) the RPA for the magnetic susceptibility
[2, 3], (iii) the mean-field approximation [4, 5], and (iv) the RPA combined
with the slave-boson mean field scheme [6]-[9]. Our approach is based on the
fact that the spin response function and the two-particle Green’s function share
the same poles. Thus, if we assume that the tight-binding form of the electron
energy and the maximum gap can be obtained by fitting the ARPES data, we
can determine the strengths of the interactions from the observed positions of
the commensurate and incommensurate peaks.
In our calculations we use the commensurate peak at ∼ 40 meV and in-
commensurate peaks at ∼ 24 meV and ∼ 32 meV which have been reported in
underdoped Y Ba2Cu3O6.7 [33]. We use the tight-binding form of the energy
and the corresponding chemical potential, both obtained by matching the shape
of the Fermi surface measured by the ARPES. Another input parameter for the
d-wave superconductivity is the maximum gap which usually is assumed to be
equal to the ARPES antinodal gap. Thus, by means of these input parameters
we can plot the graph U vs. J assuming that the energy and the position of
the above commensurate and incommensurate resonances observed by neutron
scattering experiments are described by the solutions of the BS equations.
It is known that the YBaCuO is a two-layer material, but most of the peak
structures associated with the neutron cross section can be captured by one
layer band calculations [14],[48]-[50]. The effects due to the two-layer struc-
ture can, in principle, be incorporated in our approach, but this will make the
corresponding numerical calculations much more complicated. In the single-
layer approximation the mean-field electron energy εk has a tight-binding form
εk = −2t (coskx + cos ky) + 4t′ coskx cosky − 2t′′(cos 2kx + cos 2ky) − µ
obtained by fitting the ARPES data with a chemical potential µ and hopping am-
plitudes ti for first to third nearest neighbors on a square lattice. Using the estab-
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164 Z. G. Koinov
Figure 4. Sets of points in U, J parameter space which reproduce the commen-
surate peak in Y Ba2Cu3O6.7 samples at ∼ 40 meV at point QAF = (π/a, π/a)
and the two incommensurate peaks at points π/a(1 ± δ, 1) and π/a(1, 1 ± δ)at ∼ 24 meV (δ = 0.22) and ∼ 32 meV (δ = 0.19). The V value is
V = Vψ/2 − 3J/4 where Vψ = 264 meV.
lished approximate parabolic relationship [51] Tc/Tc,max = 1−82.6(p−0.16)2,
where Tc,max ∼ 93K is the maximum transition temperature of the system,
Tc = 67K is the transition temperature for underdoped Y Ba2Cu3O6.7, we find
that the hole doping is p = 0.10. At that level of doping the ARPES parameters
are obtained in Refs. [52]: t = 0.25 eV, t′ = 0.4t, t′′ = 0.0444t and µ = −0.27
eV. In the case of d-pairing the gap function is ∆k = ∆dk/2, where the gap
maximum ∆ should agree with ARPES experiments. In the case of underdoped
Y Ba2Cu3O6.7 the gap maximum has to be between the corresponding ∆ = 66meV in Y Ba2Cu3O6.6 and ∆ = 50 meV in Y Ba2Cu3O6.95 [53], so we set
∆ = 60 meV. The numerical solution of the gap equation provides Vψ = 265meV.
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Strengths of the Interactions ... 165
Next, we plot the graph U vs. J by solving numerically the BS equations
assuming that they provide the spectrum of the collective modes ω(QAF ) = 40
meV at the commensurate point QAF , as well as the incommensurate peaks
ω(Qδ) at four incommensurate points π/a(1 ± δ, 1) and π/a(1, 1 ± δ). The
incommensurate 24 meV peak was observed at (Qx, Qy) = π/a(1± δ, 1) and
(Qx, Qy) = π/a(1, 1 ± δ), where δ = 0.22, while the deviation of the 32
meV peak from QAF is about δ = 0.19 [33]. Using these three peaks and the
BS equations we obtain sets of points in the U, J parameter space, which are
presented in Figure 4. It can be seen that the three graphs converge at J ∼
111 meV and U ∼ 569 meV (V ∼ 49 meV). The strength for J should be
comparable to the strength of the superexchange interactions in the underdoped
antiferromagnetic insulator state of the cuprates. The superexchange interaction
in two-dimensional cuprates has been studied by using several experimental
tools, and is now known to be not strongly dependent on materials with the
magnitude of 0.1 − 0.16 eV. Our value of J ∼ 111 meV is in agreement with
the accepted experimental values though some theoretical papers have predicted
different magnitudes. For example, in Ref. [54] the value of J = 0.22 eV has
been predicted using standard cuprate parameters while the calculated value of
the superexchange interaction in Ref. [55] is about J = 0.16 eV. As can be
seen, our weak-coupling Fermi-liquid-based scenario brings the calculated J
closer to the experimental value, and therefore, this fact is a strong support of
the hypothesis that the commensurate resonance and the incommensurate peaks
in cuprates compounds have a common origin.
3.3. Strengths of interactions in La2−xSrxCuO4(x = 0.16) samples
from inelastic neutron studies
To obtain the strengths of the interactions we use three incommensurate peaks
at ∼ 7.1 meV (δ = 0.261), at ∼ 10 meV (δ = 0.255), and at and ∼ 18 meV
(δ = 0.186) which have been reported in Refs. [34, 35]. The tight-binding band
parameters interpolated from values given in Refs. [56]-[58]: t = 0.25 eV, t′ =0.148t, t′′ = −0.5t′ and µ = −0.821t. The maximum energy gap (estimated
according to the prediction of the mean-field theory [59]) is about 10 meV, and
therefore, Vψ = 95.8 meV, which corresponds to maximum value of J about 64
meV. For these three peaks we use the BS equations to obtain three sets of points
in the U, J parameter space, which are presented in Figure 5. It can be seen that
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166 Z. G. Koinov
Figure 5. Sets of points in U, J parameter space which reproduce the incom-
mensurate peaks in La2−xSrxCuO4(x = 0.16) samples at π/a(1 ± δ, 1) and
π/a(1, 1± δ) at ∼ 7.1 meV (δ = 0.261), at ∼ 10 meV (δ = 0.255) and ∼ 18meV (δ = 0.186). The V value is V = Vψ/2 − 3J/4 where Vψ = 95.7 meV.
the three graphs converge at strengths J ∼ 30 meV and U ∼ 490 meV (V ∼
25 meV). By means of these strengths we have calculated the position of the
commensurate peak, which turns out to be at ∼ 40.2 meV. This is in a very good
agreement with the experimental value of 41 ± 2.5 meV [34, 35]. It is worth
mentioning that J ∼ 120 meV in La2CuO4, and therefore, our calculations
support the idea that the antiferromagnetic interaction decreases with doping
[60]. It is worth mentioning that the numerical calculations based on the virtual-
crystal approximation [61] show that the antiferromagnetic interaction is doping
insensitive. This contradicts to the above conclusion that the energy gap of
about 10 meV requires a maximum value of J about 64 meV. Obviously, there
is no consensus about the doping effect on the strength of the antiferromagnetic
interaction.
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Strengths of the Interactions ... 167
4. Conclusion
Based on the conventional idea of particle-hole excitations around the
Fermi surface, we have performed weak-coupling calculations of the
strengths of the effective interactions in the t − U − V − J model which
are consistent with ARPES and INSR experiments in Y Ba2Cu3O6.7 and
La2−xSrxCuO4(x = 0.16) samples. We do not wish to repeat the theoretical
arguments that were advanced against the stripes model, but our unified
description of the INSR peaks strongly supports the idea put forward by various
groups that the commensurate resonance and the incommensurate peaks in
cuprate compounds have a common origin.
Appendix
The BS equation for Ψ↓,↑(k,Q), which involves all diagrams shown inFigure 2 (b) and (c) has the following form (the other three equations for
Ψ↑,↓(k,Q), Ψ↑,↑(k,Q) and Ψ↓,↓(k,Q) are similar):
Ψ↓,↑(k,Q) =∫dΩ2πG↓↓(k + Q, ω + Ω)G↑↑(k,Ω) 1
N
∑q
[Uδk,q − 3J(k − q) − V (k − q)
]Ψ↓,↑(q,Q) +
∫dΩ2πG↓↑(k + Q, ω + Ω)G↓↑(k,Ω) 1
N
∑q
[Uδk,q − 3J(k − q) − V (k − q)
]Ψ↑,↓(q,Q) +
∫dΩ2πG↓↑(k + Q, ω + Ω)G↑↑(k,Ω) 1
N
∑q
[3J(k − q) − V (k − q)]Ψ↑,↑(q,Q) +∫dΩ2πG↓↓(k + Q, ω + Ω)G↓↑(k,Ω) 1
N
∑q
[3J(k − q) − V (k − q)]Ψ↓,↓(q,Q) −∫dΩ2πG↓↑(k + Q, ω + Ω)G↑↑(k,Ω) [U − J(Q) − V (Q)] 1
N
∑q Ψ↓,↓(q,Q) −
∫dΩ2πG↓↓(k + Q, ω + Ω)G↓↑(k,Ω) [U − J(Q) − V (Q)] 1
N
∑q Ψ↑,↑(q,Q) +
∫dΩ2πG↓↑(k + Q, ω + Ω)G↑↑(k,Ω) [J(Q) − V (Q)] 1
N
∑q
Ψ↑,↑(q,Q) +∫dΩ2π
×
G↓↓(k + Q, ω + Ω)G↓↑(k,Ω) [J(Q) − V (Q)] 1N
∑q Ψ↓,↓(q,Q);
J(k) = 2Jcsk, V (k) = 4V csk.
Next, we use the single-particle Green’s functions (6) which at a zero tempera-
ture are as follows:
G↑,↑(k, ωm) = u2(k)
ω−E(k)+ı0++ v2(k)
ω+E(k)−ı0+,
G↓,↓(k, ωm) =v2(k)
ω−E(k)+ı0+ +u2(k)
ω+E(k)−ı0+ ,
G↑,↓(k, ωm) = G↓,↑(k, ωm) = u(k)v(k)[
1ω−E(k)+ı0+ − 1
ω+E(k)−ı0+
],
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168 Z. G. Koinov
where u2k
= [1 + ε(k)/E(k)] /2 and v2k
= 1− u2k
are the coherent factors.
After taking the integration along the frequency we obtain the followingfour equations written in a matrix form:
Ψ(k,Q) = I(1)d
(k) 1N
∑q [3J(k − q) − V (k − q)] Ψ(q,Q) +
I(2)d
(k) 1N
∑q
[Uδk,q − 3J(k − q) − V (k − q)
]Ψ(q,Q) +
I(1)exc(k) [J(Q) − V (Q)] 1
N
∑q
Ψ(q,Q) − I(2)exc(k) [U − J(Q) − V (Q)] 1
N
∑q
Ψ(q,Q),
I(1)d
= I(1)exc =
0 0 M31 M14
0 0 M13 M41
0 0 M33 M11
0 0 M11 M44
, I(2)d
=
M34 M110 0M11 M430 0M31 M130 0M14 M410 0
,
I(2)exc =
0 0 M14 M31
0 0 M41 M13
0 0 M11 M33
0 0 M44 M11
,
M11 := −uvu′v′(
1X
− 1Y
), M13 := u′v′
(v2
X+ u2
Y
),M31 := −uv
(u′2
X+ v′2
Y
),
M33 := u′2v2
X− v′2u2
Y,M44 := v′2u2
X− u′2v2
Y,M14 := u′v′
(u2
X+ v2
Y
),
M41 := −uv(v′2
X+ u′2
Y
),M43 := v′2v2
X− u′2u2
Y, M34 := u′2u2
X− v′2v2
Y,
where u = uk, v = vk, u′ = uk+Q, v′ = vk+Q, X = ω(Q) − ε(k,Q),
Y = ω(Q) + ε(k,Q), and ε(k,Q) ≡ E(k + Q) + E(k).The above set of four equations could be simplified by introducing a
new matrixΦ = T Ψ, where T =
(σx + σz 0
0 σx + σz
). The resulting
equations can be further reduced to a set of two equations for G±(k,Q) =
Φ↓,↑(k,Q)/lk,Q ± Φ↑,↓(k,Q)/γk,Q:
[ω(Q) − ε(k,Q)]G+(k,Q) = U2N
∑q
[γk,Qγq,Q + lk,Q lq,Q
]G+(q,Q)
− U2N
∑q
[γk,Qγq,Q − lk,Q lq,Q
]G−(q,Q) − 1
2N
∑q
[V (k − q) + 3J(k − q)] ×[γk,Qγq,Q + lk,Qlq,Q
]G+(q,Q) − 1
2N
∑q
[V (k − q) − 3J(k − q)][γk,Q γq,Q +mk,Qmq,Q
]G+(q,Q) + 1
2N
∑q
[V (k − q) + 3J(k − q)][γk,Qγq,Q − lk,Q lq,Q
]G−(q,Q + 1
2N
∑q [V (k − q) − 3J(k − q)] ×
[γk,Qγq,Q −mk,Qmq,Q
]G−(q,Q) − U−2J(Q)
2N
∑qγk,Q γq,Q
[G+(q,Q) −G−(q,Q)
]
+U−2V (Q)
2N
∑qmk,Qmq,Q
[G+(q,Q) + G−(q,Q)
],
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Strengths of the Interactions ... 169
[ω(Q) + ε(k,Q)]G+(k,Q) = − U2N
∑q
[γk,Qγq,Q + lk,Qlq,Q
]G−(q,Q)
+ U2N
∑q
[γk,Qγq,Q − lk,Qlq,Q
]G+(q,Q) + 1
2N
∑q [V (k − q) + 3J(k − q)] ×
[γk,Qγq,Q + lk,Q lq,Q
]G−(q,Q) + 1
2N
∑q
[V (k − q) − 3J(k − q)][γk,Qγq,Q +mk,Qmq,Q
]G−(q,Q) − 1
2N
∑q
[V (k − q) + 3J(k − q)][γk,Qγq,Q − lk,Qlq,Q
]G+(q,Q) − 1
2N
∑q
[V (k − q) − 3J(k − q)] ×[γk,Qγq,Q −mk,Qmq,Q
]G+(q,Q) − U−2J(Q)
2N
∑qγk,Qγq,Q
[G+(q,Q) − G−(q,Q)
]
−U−2V (Q)2N
∑qmk,Qmq,Q
[G+(q,Q) + G−(q,Q)
],
where γk,Q = ukuk+Q+vkvk+Q, lk,Q = ukuk+Q−vkvk+Q, γk,Q =ukvk+Q − uk+Qvk, and mk,Q = ukvk+Q + uk+Qvk. The BS equations
are in accordance with the Goldstone theorem. According to this theorem, the
gauge invariance is restored by the existence of the Goldstone mode whose
energy approaches zero at Q = 0. This corresponds to the so-called trivial
solution of our BS equations: G+ = −G− = ∆k/2E(k), and the gap equation
(7) is recovered from our BS equations.
Next, the BS equations can be rewritten in the form:
[G+(k,Q) + G−(k,Q)
]= −
Uωγk,Qω2−ε2(k,Q)
Γ(Q) −Uε(k,Q)lk,Qω2−ε2(k,Q)
L(Q) +
[U−2J(Q)]ωγk,Qω2−ε2(k,Q)
˜Γ(Q) −
[U−2V (Q)]ε(k,Q)mk,Qω2−ε2(k,Q)
M (Q) +∑
i
(2V+ 3J2
)ωγk,Qω2−ε2(k,Q)
λikΓi(Q)
+∑
i
(2V+ 3J2
)ε(k,Q)lk,Qω2−ε2(k,Q)
λikLi(Q) +
∑i
(2V− 3J2
)ωγk,Qω2−ε2(k,Q)
λik˜Γi
(Q)
+∑
i
(2V− 3J2
)ε(k,Q)mk,Q
ω2−ε2(k,Q)λi
kM i(Q),
[G+(k,Q) −G−(k,Q)
]= −
Uε(k,Q)γk,Q
ω2−ε2(k,Q)Γ(Q) −
Uωlk,Q
ω2−ε2(k,Q)L(Q)
+[U−2J(Q)]ε(k,Q)γk,Q
ω2−ε2(k,Q)
Γ(Q) −
[U−2V (Q)]ωmk,Qω2−ε2(k,Q)
M (Q) +∑
i
(2V+ 3J2
)ε(k,Q)γk,Qω2−ε2(k,Q)
λikΓi(Q)
+∑
i
(2V+ 3J2
)ωlk,Qω2−ε2(k,Q)
λikLi(Q) +
∑i
(2V− 3J2
)ε(k,Q)γk,Qω2−ε2(k,Q)
λikΓi
(Q) +
∑i
(2V− 3J2
)ωmk,Qω2−ε2(k,Q)
λikM i(Q).
Here, we have introduced the following variables:
Γ(Q) = − 1N
∑kγk,QZ−(k,Q), L(Q) = − 1
N
∑klk,QZ+(k,Q),˜Γ(Q) = − 1
N
∑kγk,QZ−(k,Q),
Complimentary Contributor Copy
170 Z. G. Koinov
M(Q) = − 1N
∑kmk,QZ+(k,Q), Γi(Q) = − 1
N
∑kγk,Qλ
i
kZ−(k,Q),
Li(Q) = − 1N
∑klk,Qλ
i
kZ+(k,Q),˜Γ
i
(Q) = − 1N
∑kγk,Qλ
i
kZ−(k,Q),
Mi(Q) = − 1N
∑kmk,Qλ
i
kZ+(k,Q),
where Z±(k,Q) = G+(k,Q) ± G−(k,Q) and i = 1, 2, 3, 4. By means of
the last two equations, we can obtain a set of 20 coupled linear homogeneous
equations for Γ(Q), L(Q),Γ(Q), M(Q), Γi(Q), Li(Q),
Γi
(Q) and M i(Q).The existence of a non-trivial solution requires that the secular determinant
det‖χ−1qp − V ‖ is equal to zero, where the interaction is a diagonal 20 × 20
matrix whose elements are
V = diag[U, U,−U + 2J(Q), U − 2V (Q),−(2V + 3J
2
),−(2V + 3J
2
),−(2V + 3J
2
),
−
(2V + 3J
2
),−(2V + 3J
2
),−(2V + 3J
2
),−(2V + 3J
2
),−(2V + 3J
2
),−(2V −
3J2
),
−
(2V −
3J2
),−(2V −
3J2
),−(2V −
3J2
),−(2V −
3J2
),−(2V −
3J2
),−(2V −
3J2
),
−
(2V −
3J2
)].
(8)
The bare response function χqp =
(A BBT C
)where A and B are 4 × 4 and
4 × 16 blocks, while C is 16× 16 block (in what follows i, j = 1, 2, 3, 4):
A =
∣∣∣∣∣∣∣∣∣
Iγ,γ Jγ,l Iγ,γ Jγ,mJγ,l Il,l Jl,γ Il,mIγ,γ Jl,γ Iγ,γ Jγ,mJγ,m Il,m Jγ,m Im,m
∣∣∣∣∣∣∣∣∣
, B =
∣∣∣∣∣∣∣∣∣∣
I iγ,γ J iγ,l I iγ,γ
J iγ,mJ iγ,l I il,l J i
l,γI il,m
I iγ,γ
J il,γ
I iγ,γ
J iγ,m
J iγ,m I il,m J iγ,m
I im,m
∣∣∣∣∣∣∣∣∣∣
,
C =
∣∣∣∣∣∣∣∣∣∣∣
I ijγ,γ J ijγ,l I ijγ,γ
J ijγ,m
J ijγ,l I ijl,l J ijl,γ
I ijl,m
I ijγγ
J ijl,γ
I ijγ,γ
J ijγ,m
J ijγ,m I ijl,m J ijγ,m
I ijm,m
∣∣∣∣∣∣∣∣∣∣∣
.
The quantities Ia,b = Fa,b(ε(k,Q)) and Ja,b = Fa,b(ω), the 1 × 4 ma-
trices I ia,b = F ia,b(ε(k,Q)) and J ia,b = F ia,b(ω), and the 4 × 4 matrices
I ija,b = F ija,b(ε(k,Q)) and J ija,b = F ija,b(ω) are defined as follows (the quanti-
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Strengths of the Interactions ... 171
ties a(k,Q) and b(k,Q) = lk,Q, mk,Q, γk,Q or γk,Q):
Fa,b(x) ≡1N
∑kxa(k,Q)b(k,Q)ω2−ε2(k,Q) , F
ia,b(x) ≡
1N
∑k
xa(k,Q)b(k,Q)λik
ω2−ε2(k,Q) ,
F ija,b(x) ≡1N
∑kxa(k,Q)b(k,Q)ω2−ε2(k,Q)
(λT
kλk
)
ij.
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INDEX
A
accelerator, 78
access, 120, 135
algorithm, 152
aluminium, 107, 135
amino acid, 85
ammonium, 86
amplitude, 43, 46, 156, 157, 159
anatase, 116
ANS, 39
antiferromagnetic, x, 108, 109, 110, 150,
165, 166
antinodal gap, x, 150, 152, 163
apoptosis, 98
aqueous solutions, 25, 35
assessment, 120, 127
atmosphere, 134
atmospheric pressure, 139, 142
atoms, 28, 40, 42, 44, 79, 80, 81, 88, 118,
134
B
backscattering, 125
bandwidth, 125, 151
base, 135, 138, 139, 140, 141, 142, 144
beams, 6, 7, 16, 38, 41
bending, 102, 119
beryllium, 124, 125
bias, 11, 14
biological processes, viii, 77, 79, 80
biological samples, 79
biomaterials, 41
biomolecules, viii, 78, 79, 85
blends, 38, 43, 51, 56, 59, 66, 71, 72, 73, 75,
76
bonding, 116
bonds, 50
boson, 155, 156, 157, 163
C
cadmium, 7, 14, 144
calorimetric method, 108
calorimetry, 110
capillary, 138
carbon, 28, 42, 43
carbon atoms, 28, 43
CCR, 144
cell division, 80
cell membranes, 79, 89, 103
cell signaling, viii, 77, 89
charge density, ix, 149, 150, 154
chemical, viii, 38, 43, 47, 50, 61, 65, 78, 79,
84, 87, 89, 98, 100, 106, 118, 119, 151,
152, 161, 163
chemical properties, 106
cholesterol, viii, 78, 86, 87, 88, 89, 96, 97,
98, 101, 102, 103
Complimentary Contributor Copy
Index 178
choline, viii, 78
circulation, 138, 139, 140, 142, 144
classes, 98
clusters, 88
coal, 41
cobalt, ix, 106, 108, 111, 112, 114, 115,
116, 127
coherent scattering length, 15, 81
collaboration, ix, 133, 135, 137, 145
collisions, 42
color, 82, 83, 95
commercial, 10, 11, 12, 13, 135
communication, 147
community, 38, 143
competition, 88, 106
complement, ix, 106
complex interactions, 108
complexity, 79, 80, 89, 109, 119
composition, viii, 43, 51, 52, 57, 59, 61, 63,
71, 73, 78, 84, 87, 89, 90, 96, 97, 98, 119
compounds, x, 149, 150, 152, 154, 165, 167
compressibility, 61, 115
compression, 23
computation, 109
condensation, 87, 136, 138, 139, 140, 143,
144, 146
configuration, 49, 53, 55, 94
connectivity, 56
consensus, 89, 107, 166
conservation, 67
conserving, 155
construction, 31
consumption, ix, 133, 135, 144
contour, 14
cooling, 59, 60, 125, 135, 137, 138, 139,
140, 141, 142, 143, 144, 146
coordination, 54
copper, 41, 138
correlation, 47, 61, 62, 63, 64, 65, 66, 67,
71, 73, 74, 100, 116, 161
correlation function, 47, 61, 62, 63, 64, 65,
67, 161
correlations, x, 73, 150
cosmetics, 106
cost, 80, 134, 151
covalent bond, 47
critical value, 60
cryogenic, ix, 133, 135, 138, 139, 144, 146
crystal structure, 108
crystalline, 112, 117
crystals, 125
cuprates, 151, 165
cuprite, x, 149
D
data analysis, 90, 98
data set, 83
decay, 42
decoupling, 155
defects, 117
deficiency, 84
deformation, viii, 2, 33, 80
density fluctuations, 61
density functional theory (DFT), 106, 107
Department of Energy, 127
depth, 107, 110, 116
derivatives, 59, 154, 155
detection, 10, 12, 33, 34
deuteron, viii, 37, 41
deviation, 62, 63, 74, 165
diamonds, 24
diffraction, 90, 143, 147
diffuse reflectance, 107
diffusion, 85, 89, 119, 120
diffusion process, 120
direct observation, 96
discrimination, 11
dispersion, 26, 111, 122, 157
displacement, 48, 112
dissociation, 107
distribution, 6, 19, 20, 21, 47, 48, 50, 73, 80,
81, 82, 84, 88, 95, 112, 116, 120, 141,
146
distribution function, 82
divergence, ix, 40, 149, 150
DOI, 129
domain structure, 91
doping, 161, 164, 166
dyes, 96
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Index 179
E
electrical conductivity, 28
electromagnetic, 38, 46
electromagnetic waves, 46
electron, 44, 80, 84, 90, 151, 153, 155, 156,
157, 158, 161, 163
electronic structure, 120, 127
electrons, 42, 44, 80, 82, 150, 151, 153, 157
elementary particle, 38, 41, 78
elongation, 30
elucidation, ix, 105, 112
emulsions, 39, 41
energy, ix, 32, 42, 45, 53, 54, 56, 57, 59, 63,
66, 67, 80, 88, 90, 106, 108, 109, 112,
113, 116, 117, 118, 119, 121, 122, 123,
124, 125, 126, 137, 151, 152, 153, 155,
156, 158, 161, 163, 165, 166, 169
energy transfer, 90, 112, 121, 122, 123, 126,
137
engineering, 3, 146
entropy, ix, 52, 54, 55, 56, 88, 105, 108,
111, 112, 115
environment, vii, 80, 106, 118, 134, 137,
140, 144, 145, 147
environmental conditions, 84
environments, 78, 108, 109, 119
enzymatic activity, 85
epoxy resins, 23
equilibrium, 57, 61, 63, 64, 65, 80, 138
equipment, 134
ESR, 90, 96, 100
ester, 86, 87, 88, 100
ethylene, 33, 35, 74
ethylene oxide, 33, 35
eukaryotic, 85
evaporation, 134
evolution, 79
excitation, 45, 99, 109, 110, 152, 154, 157
experimental condition, 10, 81
extraction, ix, 105, 108
extrusion, 90
F
Fermi surface, x, 150, 163, 167
fermions, 156
field theory, 52, 67, 73, 165
films, 4, 78
filters, 90, 125, 142
flexibility, 80
flight, 9, 14, 18, 25, 45, 102, 113, 121, 122,
124, 125
fluctuations, x, 42, 47, 58, 61, 62, 63, 64,
66, 67, 71, 73, 91, 92, 150, 153
fluid, 74, 79, 80, 83, 85, 93, 101, 102, 103
fluorescence, 90, 96, 99, 100
force, 67, 85, 87, 98, 106
formation, 28, 42, 53, 58, 90, 99, 101
formula, 56, 94
fractal dimension, 23
France, 76, 125
free energy, 52, 56, 57, 58, 59, 61, 65, 75
freedom, 88, 113, 151, 155
FTIR, 107
fusion, 134
G
Gaussian equation, 20
gel, 25, 26, 30, 33, 93
gelation, 35
geology, 106
geometry, 16, 34, 45, 121, 122, 124, 126
Germany, 3, 5
glycerol, viii, 78, 81, 83, 86, 98
graph, 161, 163, 165
graphite, 125
growth, 27, 29, 30, 58
growth mechanism, 27, 58
H
Hamiltonian, 150, 153, 154, 161
hazards, 144
heat capacity, ix, 105, 108, 109, 110, 112,
113, 115, 127
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Index 180
height, 11, 12, 13, 123, 135, 137
helium, ix, 133, 134, 135, 136, 138, 139,
140, 141, 142, 143, 144, 145
helium gas, ix, 133, 138, 139, 142
heterogeneity, viii, 78, 86, 89, 90, 91, 92, 95
histogram, 93
host, 79
Hubbard model, 152, 155, 156
Hungary, 3, 7
hybrid, 146
hydrogen, viii, 37, 41, 42, 43, 78, 79, 81, 87,
88, 91, 95, 107, 112, 116, 118, 120
hydrogen atoms, 41, 43, 79, 87
hydrogen bonds, 88
hydrophobic, viii, 78, 85, 98, 100
hydrophobicity, 80
hydroxyl, ix, 88, 106, 107, 119, 120
hydroxyl groups, 119, 120
I
improvements, 31
in vivo, 99
India, 37
inefficiency, 144
inelastic neutron, vii, ix, x, 105, 150, 152,
165
inertia, 118
infrared spectroscopy, 107
inhomogeneity, 23
INS, vii, ix, 105, 106, 107, 108, 109, 110,
111, 112, 113, 115, 116, 117, 118, 119,
121, 122, 124, 127
insulators, 151
integration, 168
interface, 84, 88, 117, 121, 127
interference, 38, 39, 41, 42, 90, 94, 112
intermolecular interactions, 86
internal time, 157
ion channels, 85
ions, 79, 108, 109
iron, 4
isochoric, ix, 105, 112, 115
isostructural, ix, 106, 117, 120
isotope, 43, 79, 81
J
Japan, vii, 1, 2, 8, 9, 12, 25, 31, 33, 121, 145
L
labeling, viii, 37, 44, 79
landscape, 88
lattices, 117
lead, viii, 42, 78, 160
lens, 3, 5, 7, 8, 15, 16, 18, 22, 25, 30, 34
life sciences, 99
light, 10, 11, 38, 39, 43, 90, 99, 103, 119
light scattering, 43
linear dependence, 64
linear function, 162
linear macromolecule, 47
lipid membranes, viii, 77, 79, 98, 101
lipid metabolism, 86
lipids, viii, 77, 80, 85, 86, 87, 88, 89, 91, 92,
95, 96, 98, 102
liquid phase, 89
liquids, 121, 122
loci, 26
low temperatures, 140
luminescence, 38
Luo, 171
M
machinery, 89
macromolecules, 41, 76
magnet, 7, 34, 134, 137, 140, 141, 142, 143,
144, 145, 146
magnetic, ix, 7, 25, 34, 38, 42, 78, 105, 106,
108, 109, 110, 111, 112, 113, 121, 122,
127, 134, 140, 144, 145, 146, 147, 150,
153, 154, 163
magnetic field, 38, 134, 140, 144, 145
magnetic materials, 42
magnetic moment, 42
magnetic properties, 78, 127
magnetic resonance, 146
magnetic structure, 42
Complimentary Contributor Copy
Index 181
magnetism, 108
magnets, 142, 143, 146
magnitude, 3, 4, 19, 23, 28, 31, 39, 46, 112,
115, 165
mammalian cells, 85
mass, 5, 41, 43, 55, 73, 113, 121, 157
materials, 3, 33, 40, 41, 42, 78, 101, 106,
110, 112, 114, 116, 120, 153, 165
materials science, 3, 40, 41, 101
matrix, 152, 158, 160, 161, 168, 170
matter, 35, 38, 42, 121, 122
mean-field theory, 71, 73
measurement(s), vii, 9, 10, 16, 19, 22, 25,
27, 30, 31, 33, 35, 43, 51, 63, 68, 71, 84,
90, 97, 108, 115, 119, 124, 134, 135,
146, 152
mechanical properties, 102
melt, 47, 50
melting, 89
membranes, vii, viii, 77, 79, 80, 85, 89, 98,
99, 100, 101, 102
metal ions, 121
metal oxide, vii, ix, 105, 106, 107, 108, 112,
116, 118, 119, 120, 121, 127
meter, 101
methodology, ix, 61, 92, 99, 105
methyl groups, 88
Metropolis algorithm, 93
microscopy, 90, 96, 99, 100
mixing, 43, 51, 52, 53, 54, 55, 56, 61, 88,
95, 96, 135, 136, 139, 140, 160
model system, 89, 155
models, 113, 150, 151
moderators, 42
modifications, 99
modules, 125
molar volume, 115
mole, 56
molecular dynamics, viii, 78, 83, 119
molecular structure, 25
molecular weight, 11, 51, 71, 72, 74
molecules, viii, 29, 30, 39, 41, 43, 78, 81,
84, 85, 98, 107, 118, 119, 120
momentum, 45, 112, 121, 124, 146, 156,
157
monomers, 47, 48, 52, 66
Monte Carlo method, 151
Moon, 128
morphology, 38, 42, 58, 90, 106, 118
mosaic, 79, 103
MRI, 134
multi-component systems, 41
N
nanometer(s), 3, 90, 96, 97, 99
nanoparticles, ix, 39, 105, 106, 107, 108,
109, 110, 112, 116, 117, 118, 119, 120,
127
nanoscopic lipid membranes, vii
nanostructures, 31, 78
National Academy of Sciences, 99, 100
Nd, 94
neutral, 38, 42, 78, 86, 98
neutron scattering, vii, viii, ix, 2, 27, 31, 32,
35, 37, 38, 39, 41, 42, 43, 44, 45, 51, 63,
78, 79, 81, 82, 84, 99, 101, 107, 112,
116, 119, 124, 133, 134, 136, 137, 139,
140, 143, 144, 145, 146, 147, 153
neutrons, vii, viii, 1, 3, 4, 5, 8, 9, 10, 27, 30,
32, 37, 38, 40, 41, 42, 43, 45, 78, 79, 81,
111, 121, 122, 123, 125
nitrogen, 7, 138, 142, 144
NMR, 134
novel materials, 121
NSL, 84
nucleation, 58
nuclei, 41, 42, 78, 81
nucleic acid, viii, 77, 80
nuclides, 42
numerical analysis, 92
O
oil, 41, 142
oligomers, 23
operations, 80
optical lattice, 156
optical properties, 41, 43
Complimentary Contributor Copy
Index 182
orbit, 109
overlap, 93
oxidation, 106, 108
oxidative stress, 86
oxide nanoparticles, vii, ix, 105, 106, 107,
108, 111, 112, 114, 116, 117, 119, 120,
127
oxygen, 42, 87, 88, 112, 118, 144
P
pairing, x, 150, 151, 152, 158, 164
palladium, 107
parallel, 46
particle nucleation, 25
partition, 52, 92, 96
pathways, 89
peptides, 85
permeability, 79
permeation, 25, 26
pH, 84
phase boundaries, 38, 59
phase diagram, 59, 60, 68, 89, 93, 97, 101,
116, 152
phase transitions, 134
phenolic resins, 23, 24, 25, 33
phosphate, 81, 83, 86, 88
phosphatidylcholine, 85, 87, 89, 102
phosphatidylserine, 98
phospholipids, viii, 78, 98
photoemission, vii, x, 150, 152
photographs, 8
photomultiplier, vii, 2, 3, 32
photons, 78, 157
physical characteristics, 80
physical properties, ix, 38, 106, 111, 112,
119
physics, 3, 38, 75, 88, 100, 106, 107, 173
pitch, 14
plasma membrane, 86, 88
platform, 90, 142
polar, 81, 98, 102
polarization, 84
polycarbonate, 90
polydispersity, 20, 21, 95
polyisoprene, 74
polymer blends, vii, viii, 37, 38, 41, 43, 44,
47, 52, 53, 55, 59, 60, 61, 62, 69, 71, 73,
75, 76
polymer chain(s), 47, 48, 51, 52, 53, 54, 55,
60, 69, 73
polymer melts, 73, 76
polymer solutions, 51
polymer structure, 70
polymerization, 25, 26, 50, 54, 60, 70, 75
polymerization process, 26
polymerization time, 26
polymer(s),vii, viii, 35, 37, 38, 39, 41,43,
44, 47, 48, 50, 51, 52, 53, 54, 55, 56, 57,
58, 59, 60, 61, 62, 66, 67, 69, 70, 71, 73,
74, 75, 76
polymorphism, 106
polystyrene, 19, 21, 24, 51, 71, 72
polystyrene latex, 19, 21, 24
population, 113
porosity, 41
preparation, 35, 107
pressure gauge, 138
probability, 47, 49, 80, 82, 83, 84, 94, 156
probability distribution, 49, 80, 82, 83, 84,
94
probe, 38, 40, 41, 42, 78, 90
project, 137
propagation, 156
proportionality, 67
propylene, 74
protein kinase C, 85
proteins, viii, 77, 79, 85, 88, 89
proteomics, 99
protons, 41
prototype, 135, 138
purity, 142
pyrolytic graphite, 125
Q
quantum electrodynamics, 156
quasiparticles, 158
Complimentary Contributor Copy
Index 183
R
radiation, 4, 33, 34, 38, 39, 40, 43, 45, 46,
47, 140, 142
radius, 7, 15, 19, 21, 22, 24, 26, 29, 44, 51,
93, 94, 95, 97
radius of gyration, 51
random walk, 47, 48, 51
reactions, 99
redistribution, 89, 116, 118
reflectivity, 7
refraction index, 15
relaxation, 30, 107
reliability, 19, 135
renormalization, 152
repulsion, 150
researchers, 89
resins, 23, 24, 25
resolution, vii, ix, 2, 3, 4, 5, 8, 9, 10, 12, 14,
16, 17, 18, 19, 20, 21, 22, 30, 31, 32, 33,
43, 78, 80, 81, 83, 89, 90, 98, 105, 106,
111, 112, 122, 123, 124, 125, 126
resources, 134, 144
response, ix, 63, 64, 66, 67, 149, 150, 154,
156, 160, 161, 162, 163, 170
reticulum, 86
rheology, 73, 76
rings, 8
risks, 134
rods, 118
room temperature, 108, 115, 138, 139, 142
root, 50, 82, 118
rotation axis, 6
rules, 111
rutile, ix, 106, 116, 117, 118, 119, 120, 127
S
safety, 134, 139, 144
salt concentration, 84
saturation, 40, 102
SAXS, 23, 24, 25, 34, 39, 79, 81, 82, 83, 85,
86, 92, 98
scaling, 71, 73, 74, 76, 113, 118
scaling relations, 71
scanning electron microscopy, 23, 33
scatter, 42, 46, 81
scattering, vii, viii, ix, x, 1, 2, 3, 7, 14, 15,
18, 19, 20, 22, 23, 24, 25, 26, 27, 29, 31,
32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43,
44, 45, 46, 47, 50, 51, 61, 62, 63, 65, 68,
70, 71, 74, 75, 76, 78, 79, 80, 81, 82, 83,
84, 91, 92, 93, 94, 95, 96, 97, 98, 99,
100, 101, 102, 105, 107, 112, 113, 116,
119, 121, 122, 124, 125, 133, 134, 136,
137, 139, 140, 143, 144, 145, 146, 147,
150, 152, 153, 163
scattering intensity, 19, 75, 92, 93, 94, 95,
96
scattering patterns, 27
scintillator, vii, 2, 3, 9, 10, 11, 12, 13, 14,
30, 31, 33
segregation, 89, 91
shape, 26, 41, 57, 58, 78, 163
shear, 2, 35
shortage, ix, 133, 147
showing, viii, 42, 59, 78
signal transduction, 103
signals, 112
signal-to-noise ratio, 28
silver, 6, 134
simulation, 93, 94, 99, 102, 103
simulations, viii, 78, 82, 83, 85, 86, 87, 88,
98, 119
Singapore, 131
single crystals, 122, 162
SiO2, 111
siphon, 141
small-angle neutron (SANS), v, vii, viii, 1,
2, 3, 4, 5, 6, 7, 9, 10, 12, 15, 16, 18, 19,
22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32,
33, 34, 35, 37, 38, 39, 40, 41, 61, 68, 71,
72, 78, 79, 81, 82, 83, 85, 86, 90, 91, 92,
95, 96, 97, 98, 99, 100, 101, 102
small-angle scattering (SAS), 7, 18, 23, 33,
39, 43, 80, 100
SNS, ix, 106, 121, 122, 125
soft matter, viii, 37, 38, 75
solar cells, 106
Complimentary Contributor Copy
Index 184
solid state, 38
solution, 26, 27, 28, 29, 53, 59, 76, 94, 100,
157, 158, 162, 164, 169, 170
solvents, 43, 51
species, ix, 98, 106, 107, 108, 112, 119,
120, 127
specific heat, 111
specifications, 2, 9, 143
spectrometer, ix, 2, 18, 32, 33, 106, 108,
113, 122, 123, 124, 136, 137, 143
spectroscopic techniques, 96
spectroscopy, vii, x, 90, 96, 100, 107, 108,
110, 111, 112, 124, 126, 150, 152
spin, ix, 78, 90, 108, 109, 121, 143, 147,
149, 150, 151, 152, 153, 154, 156, 160,
161, 162, 163
square lattice, 151, 153, 158, 161, 163
stability, 12, 57, 74, 89, 90, 106, 107
standard deviation, 19
state, 2, 3, 45, 47, 49, 55, 56, 57, 59, 61, 78,
85, 89, 108, 142, 147, 165
states, ix, 55, 64, 105, 108, 111, 112, 113,
151, 156
statistics, 48, 51, 70, 134, 144
steel, 4
sterols, 88, 89
stimulation, 90
stress, 30, 146
stretching, 2, 27, 31
structural changes, 25
structure, viii, ix, 26, 29, 37, 38, 40, 41, 49,
65, 66, 69, 71, 78, 79, 80, 81, 82, 85, 86,
87, 91, 92, 94, 97, 98, 99, 100, 101, 102,
103, 106, 107, 108, 112, 116, 117, 119,
120, 121, 127, 153, 162
style, 144
styrene, 25
substitution, 43, 79
superconductivity, x, 149, 150, 151, 163
supplier, 7, 9
suppression, 118
surface area, 138
surface energy, 116, 118
surface tension, 83
surfactant, 25, 28, 29, 30, 41
surfactants, 28
susceptibility, 73, 150, 153, 154, 163
suspensions, 61
sustainability, 134
symmetry, 117, 160
synthesis, 107
T
target, 123, 126
technical support, 31
techniques, vii, 23, 38, 39, 43, 75, 78, 79,
81, 89, 90, 98, 107, 111, 115, 116, 119,
127, 163
technology, ix, 133, 135, 144
temperature, vii, ix, x, 41, 51, 56, 59, 66, 71,
72, 73, 74, 75, 76, 85, 86, 89, 91, 96, 97,
101, 105, 108, 109, 110, 112, 114, 115,
116, 117, 120, 123, 125, 133, 134, 135,
137, 138, 139, 140, 141, 142, 144, 145,
147, 149, 150, 156, 158
temperature dependence, 59, 73
tension, 79, 98, 101
testing, 142
thermal expansion, 111, 115
thermodynamic, vii, viii, ix, 37, 38, 57, 61,
65, 66, 89, 105, 107, 108, 110, 112, 116,
127
thermodynamic equilibrium, 57, 65, 66
thermodynamic parameters, viii, 37, 38, 61
thermodynamic properties, vii, ix, 105, 107,
110, 112, 116
thermodynamics, 44, 52, 61, 62, 63, 64, 66,
71, 73, 88
thin films, 99
three-dimensional space, 80
time frame, 120, 123, 126
tin oxide, 107
titanium, 107
topology, 85
total energy, 63, 88
trafficking, viii, 77, 86, 88, 98
trajectory, 39
transformation, 82, 155
transition metal, 112
Complimentary Contributor Copy
Index 185
transition temperature, 108, 164
translation, 45, 118, 123
transmission, 4, 5, 38, 40, 125
transport, 79, 102
treatment, 38
tumor cells, 86
U
UK, 10, 105, 121, 133, 134
United States (USA), 4, 9, 99, 100, 121, 125
universal gas constant, 109
V
vacuum, 124, 139, 143
valve, 124, 138, 142
vapor, 136, 138
variables, 48, 89, 169
variations, 115, 117, 120
vector, 3, 32, 40, 45, 46, 50, 151, 152, 160,
161
velocity, 2, 3, 4, 5, 41
versatility, vii, 2
vesicle, 90, 92, 93, 94, 95, 99
viscosity, 76
visualization, 83
W
water, ix, 26, 29, 30, 81, 82, 84, 85, 88, 90,
100, 106, 107, 108, 112, 113, 114, 115,
116, 117, 118, 119, 120, 125, 127
water diffusion, 120
water permeability, 100
water structure, 115, 127
wave vector, 45, 71, 152
wavelengths, 121, 125
weight ratio, 10, 23
X
X-ray diffraction, 107
Y
yield, 98
Z
zinc oxide (ZnO), 107
zirconium, 107
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