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Confined polymer systems: synergies between simulations and neutronscattering experiments

Ian G. Elliott,a Dennis E. Mulder,a Petra T. Tr€askelin,a John R. Ell,b Timothy E. Patten,c Tonya L. Kuhla

and Roland Faller*a

Received 1st June 2009, Accepted 7th September 2009

First published as an Advance Article on the web 2nd October 2009

DOI: 10.1039/b910693f

Molecular simulations and neutron reflectivity are both extremely valuable tools for determining

the structure of soft matter at interfaces and under confinement. The high resolution structural information

provided by these techniques allows us to obtain a thorough understanding at the molecular level. Here

we present examples of polymer thin films and show the advantages these two techniques offer and how we

can combine the approaches in order to provide a complete structural and thermodynamic picture.

Introduction

In tandem, experiments and computer simulations are a powerful

means of elucidating the structure of polymeric thin-films. It is

becoming increasingly clear that by combining these two tech-

niques much more than the sum of the parts can be obtained.

Experiments obviously explore the physical reality of the

system.1–9 Simulations on the other hand have unprecedented

access to all positions of all particles at all times leading to all

structural and thermodynamic properties, but they are limited to

statements within the boundaries of the underlying models.10–16

Many modern experiments also require a great deal of modeling

to interpret the raw data.17–19 To obtain a high resolution picture

of the structure of materials, the ideal approach is, therefore,

aDepartment of Chemical Engineering and Materials Science, University ofCalifornia, One Shields Avenue, Davis, California, USA. E-mail:[email protected]; [email protected]; [email protected];[email protected]; [email protected]; Fax: +1-530-752-1031; Tel:+1-530-752-5839bPolymer Science and Engineering, University of Massachusetts, Amherst,MA 01003, USA. E-mail: [email protected] of Chemistry, University of California, One Shields Ave,Davis, CA, USA. E-mail: [email protected]

Ian G: Elliott

Ian Gould Elliott was born in

Phoenix, Arizona in 1984 and

grew up in Bend, Oregon. In

2007 he received a BS in chem-

ical engineering and a BS in

chemistry from Oregon State

University. He is currently

starting his third year in the

chemical engineering PhD

program at the University of

California, Davis. His research

involves characterizing polymer

brushes through experimental

methods and simulations as

a student of Tonya Kuhl and

Roland Faller.

4612 | Soft Matter, 2009, 5, 4612–4622

a synergistic combination of simulations and experiments to

explore a broad parameter space, to aid interpretation of raw

experimental data, and to study real, physical systems. The

co-interpretation of simulations and experiments is a powerful

means to obtain high resolution structure on many scales.

In this contribution we focus on the combination of neutron

scattering and molecular dynamics simulations of polymer films

under confinement. We specifically discuss confined polymer

brushes at interfaces as advances in both computational power

and new synthetic strategies have recently allowed investigations

of novel variants of these systems under geometrical confine-

ment.21–24 Polymer brushes have been studied extensively due to

their ability to modify surface properties to prevent colloid

aggregation and to enhance lubrication and adhesion.25–30 They

have been shown to remarkably reduce friction when properly

designed.31 The brush structure can be controlled by appropri-

ately selecting the grafting density, solvent, polymer molecular

weight, and temperature.32

This paper is organized as follows. We start with a brief

discussion of the fundamentals of molecular simulations and of

neutron scattering of confined soft matter. We then describe how

the two techniques combined yield a richer understanding using

Dennis E: Mulder

Dennis Mulder received his

Chemical Engineering BS from

the University of California,

Davis in 1999 and went to work

for NEC Electronics as

a Process Equipment Engineer.

He returned to Davis in 2003 to

start the PhD program in

Chemical Engeering at Univer-

sity of California, Davis as well

as the MBA program. He

earned the MBA in 2005 and is

finishing up his PhD. His

research interests include char-

acterizing polymer brushes using

scattering measurements.

This journal is ª The Royal Society of Chemistry 2009

polymer films as our example. We finish with a discussion and

outlook.

Molecular modeling of confined polymer systems

Molecular modeling is an ideal tool to investigate thin films of

soft materials in general and polymers in particular as many of

the relevant length scales are well suited for molecular

modeling33–35 and the direct integration with experiments is often

possible.18,19,36–38 Analytical theory of such systems is non-trivial

as there is an interplay of system scales with molecular length

scales such that the system in general has different properties

compared to the bulk. It is, thus, crucial to study these systems

computationally to understand new phenomena which may

emerge by this interplay of length scales. Clearly, simulations

alone cannot answer all the fundamental questions as there are

Petra T: Tr€askelin

Petra Tr€askelin received her

PhD in physics at the University

of Helsinki, Finland, in 2006.

She did her PhD thesis on

sticking and erosion at carbon-

containing plasma-facing mate-

rials in fusion reactors. After

graduation she received a schol-

arship from Academy of Finland

for a post-doc position at the

University of California, Davis.

For the past few years she has

been investigating polymer melts

and brushes using atomic scale

modeling. Her other scientific

interests include the develop-

ment of analytical interatomic potentials and hydrocarbon surface

chemistry.

John R: Ell

John R. Ell received his BS in

organic chemistry with a poly-

mer option at Carnegie Mellon

University in 2003 and his PhD

from the University of Cal-

ifornia at Davis in 2008 working

under Timothy E. Patten on

polymer brush coatings. He is

currently a Center for Hierar-

chical Manufacturing post-

doctoral fellow at the University

of Massachusetts Amherst in the

Polymer Science and Engi-

neering Department working

under James J. Watkins and

Kenneth R. Carter. His research

centers on the application of patterned polymeric materials in

nanofluidic devices and he also works with polymer brush surfaces

for block copolymer ordering and drug delivery.


problems associated with model applicability and system equil-

ibration. As experiments explicitly address physical systems, high

resolution experiments which allow simulation model validation

at or close to the length scales of interest are an ideal counterpart.

Molecular dynamics (MD) simulations are fundamentally

based on the assumption that Netwon’s equations are appro-

priate to model the system under study, i.e. relativistic and

quantum effects are neglected.39,40 Here, one uses a potential

energy function, often called a force-field, which assigns every

possible arrangement of atoms a potential energy. These force-

fields typically contain terms for bonds, angles and torsion of the

molecules as well as electrostatic and van der Waals interactions.

The correct choice of force-field is crucial in a simulation as

any error or inaccuracy in the force-field is translated into the

result of the simulation. So one has to ensure that the force-field

reliably represents the system under study; at first glance this

Tonya L: Kuhl

Tonya Kuhl received her BS

from the University of Arizona

and her PhD in Chemical Engi-

neering from University of Cal-

ifornia, Santa Barbara. After

a stay as a postdoctoral research

associate at the UCSB Mate-

rials Research Laboratory, she

joined the faculty of Chemical

Engineering and Materials

Science at UC Davis in 2000

where she is currently

a Professor and the Jeff and

Dianne Childs & Steve Whi-

taker Endowed Chair. Her

research focuses on development

and application of small angle scattering techniques and interaction

force measurements of interfacial thin-films and soft condensed

matter.

Roland Faller

Roland Faller received his

Diploma in physics from the

University of Bayreuth in Ger-

many and his PhD in theoretical

physics from the University of

Mainz, Germany for work per-

formed at the Max-Planck

Institute for Polymer Research.

After a stay as a postdoctoral

research associate at the

University of Wisconsin-Madi-

son he joined in 2002 the faculty

of Chemical Engineering and

Materials Science at UC Davis

where he is currently an Asso-

ciate Professor and the Joe &

Essie Smith Endowed Chair. His research focuses on development

and application of multiscale modeling for soft condensed matter.

Soft Matter, 2009, 5, 4612–4622 | 4613

Fig. 1 (A) Schematic of a neutron reflectivity experiment with a thin,

10 nm layer of polystyrene on a silicon substrate (for specular reflection

qin ¼ qout) and (B) calculated reflectivity curve for system shown in (A)

with either deuterated or hydrogenated polystyrene (dPS or hPS).

would suggest that one should use the most detailed model in

order to achieve the highest possible accuracy. This would nor-

mally be an atomistic model where at least every heavy atom is

represented. Such models can either be obtained from quantum

chemistry calculations of small molecules41 or by optimization

against experimental data.42,43 However, a detailed atomistic

simulation is frequently not possible as the required computer

time is prohibitive. As a result, more computationally efficient,

coarse-grained simulations where typically one interaction site

represents a whole monomer are often used. These coarser

models can either be systematically derived from higher resolu-

tion models44,45 or—more often—chosen as generic models based

on computationally efficient potentials.46 In general the question

under study and the available computer time dictate the force-

field for a particular study. Below, we will show examples using

atomistic as well as coarse-grained simulations.

MD simulations solve Netwon’s equations with the forces

derived from a specified force-field for a particular system. The

result is a so-called trajectory, a time-ordered series of snap-shots

of the system. This trajectory can then be analyzed for structure,

thermodynamics and/or dynamics. For the purpose here let us

explain a few structural properties which can be obtained and

compared to experimental data.

If we are studying a thin film, the thickness is a natural first

variable. In simulations thicknesses can be derived from the

density profile along the film normal. The density profile is easily

obtained by counting the atoms in a certain range of positions

along the desired coordinate, dividing by the appropriate

volume, and multiplying this by the known atomic masses. If an

electron density profile is desired we multiply by the electron

numbers. Clearly, density profiles of parts of the system, e.g.

a certain molecule type can be obtained as well. In order to

obtain the thickness we need to use an appropriate definition

based on the density profile. There are several measures of

thickness which typically lead to slightly different values; one

typical measure is the full width at half maximum in the density

profile.

Simulations can provide information which is inaccessible

from an experiment. For example, if calculating the density

profile of just a portion of the brush or film was desired, this

could be accomplished by reanalyzing an existing simulation

trajectory in a new way. In other words, one does not have to

conduct further simulations to fully explore the system. In

contrast, a physical experiment would require a new experi-

mental run where portions of the brush would have to be selec-

tively deuterated and solvent matched to screen out or highlight

the particular segments. For example, the distribution of chain

ends are often of interest as this indicates if the chains tend to

monotonically extend from the surface or if they fold back down

into the brush with implications in adhesion and lubrication

properties. The distribution of chain ends can be determined

explicitly in simulations and readily compared to the density

profiles of the entire brush.

Another structural quantity is the tilt of the molecules with

respect to the film normal. Again, there are slightly different ways

to define tilt. We can either use the end-to-end vector, i.e. the unit

vector connecting the outermost atoms of the molecule, the

principal axis of the gyration tensor or any other convenient

vector.

4614 | Soft Matter, 2009, 5, 4612–4622

Analagous to crystal diffraction, from the position of the

atoms we can directly calculate structure factors to compare to

scattering experiments as:

SðkÞ ¼X

i

bi expði~k ~riÞ (1)

The sum runs over all particle positions in the system, k is the

wavevector. So in order to compare to neutron scattering

experiments the only necessary input (outside of the simulation

data itself) is the scattering lengths of the different atoms bi. We

can determine static scattering functions as well as the interme-

diate scattering function S(k,t) (by determining the static

structure factor as a function of time instead of averaging over

the simulations) for times of normally up to a few tens of

nanoseconds or correspondingly the high frequency part of the

dynamic structure factor (by Fourier transformations of the

intermediate function).

As simulations have access to all atoms one has direct access to

other structural properties like gyration tensors and radial distri-

bution functions which can be used to characterize the structure of

single molecules as well as the local structure arising from their

packing. A quantity which is often used to describe the structure of

a system based on the position of particles is the pair correlation

function or radial distribution function. It describes the probability

to find a particle at a given distance from another particle

normalized with respect to an ideal gas at the same density.

Simulations have a number of limitations as well. First, as

indicated above the results are always only as good as the model

and representative force-fields. If no model exists for a specific

problem, it is a tedious procedure to develop a new model.

Additionally, one has to ensure that the simulations are long

enough to be equilibrated and sample the relevant phase or

conformation space. Further, the size of the simulation box can

be of concern, especially for long polymer chains, as we have to

avoid self-interaction through periodic boundaries. Still, these

are all known problems which can be avoided by carefully per-

forming the simulations, yielding simulations as a very powerful

partner to experiments.

Neutron reflectivity of confined thin films

Neutron reflectivity of a surface is defined as the ratio of the

number of neutrons elastically and specularly scattered from the


surface to the number of incident particles. Detailed theoretical

descriptions can be found in the literature.47–51 The basic

principles of the technique are illustrated in Fig. 1 for a thin-film

polymer sample in air. Far from the source, the incident neutron

can be treated as a plane wave with wavevector, kI. The magni-

tude of the wavevector in air (superstrate) is given by:

��~kI

�� ¼ kI ¼2p

l¼ mnvi

h(2)

where l is the neutron wavelength, vi is the velocity of the

neutron, and mn is the mass of the neutron. The neutron may be

reflected, transmitted or refracted. For elastically scattered

neutrons |~kI| ¼ |~kout| ¼ kI, and for specular reflection, the

momentum transferred to the neutron in the collision is

perpendicular to the surface and given by:

Qz ¼��~kR � ~kI

�� ¼ 4p sinðqÞl

(3)

The reflectivity is therefore measured as a function of the

wavevector transfer Qz. The curve contains information of

the average neutron scattering length density (SLD or b) of the

sample normal to the surface and can be used to determine

the concentration of atomic species at a particular depth in the

material. The visible fringes in the reflectivity profile arise from

interference between waves being reflected from the top surface

and the buried interfaces above the substrate. The amplitude of

the fringes relates to the SLD contrast between the layers and in

simple systems such as depicted in Fig. 1, the fringes have

a spacing DQ z 2p/Dlayer. The SLD or b of the layer is the

product Nibi, with Ni the atomic number density and bi the

neutron coherent scattering length.52 A key advantage of neutron

reflectivity is that different contrast can be obtained by substi-

tution of deuterium for hydrogen as depicted in Fig. 1b for

a 10 nm thick polystyrene film. Through isotopic substitution the

different SLDs (bdPS¼ 6.46� 10�6 �A�2 versus bhPS¼ 1.41� 10�6

�A�2) enable us to better characterize the structure of the sample.

Another feature is the perfect reflection of neutrons from the

surface when the wavevector is below a critical value,

Qz # Qcritical ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi16pðbsubstrate � bsuperstrateÞ

q. Beyond Qcritical the

reflectivity curve obeys the Fresnel law, R z Qz�4 and decreases

rapidly. From the measured reflectivity profile, the SLD’s, thick-

nesses, and roughness of the layers can be determined by modeling

the SLD profile and iterating to minimize the difference between

the measured reflectivity profile and that obtained from the

modeled SLD profile. However, as the majority of reflectivity

measurements only provide intensity information, the structural

information of interest is indirectly contained within the reflec-

tivity data. The transformation of the data from inverse space to

real space, in the absence of phase information, is mildly ill-posed

and multiple solutions can be obtained. Limiting the possible

solutions through constraints based on the known chemical

identities of layers, expected thickness, etc. is extremely helpful.

Still, the iterative fitting depends critically on starting with an

accurate model for the profile, which clearly benefits from

a synergistic approach between experiments and simulations.47,53,54

In spite of modeling challenges, neutron reflectivity experi-

ments have been very successful in providing detailed density


distribution profiles of polymeric materials at single interfaces

(depth profiling), where the structure of end-grafted polymers in

good, theta, and poor solvents has been investigated as a func-

tion of grafting density.2,3 Probing the structure such polymer

layers adopt under confinement has proven more elusive.

Towards this end, two experimental systems will be discussed

below. In the first case, ultra-high grafting density polystyrene

(PS) brushes are available through new, ‘‘grafting from’’

synthetic schemes based on atom transfer radical polymerization

and allow confinement studies due to lateral crowding of the

chains. Grafting from approaches provide access to the extreme

end of grafting levels, where the equilibrium layer thickness

approaches the length of the strongly stretched polymer chain.

Typically, a monolayer of polymerization initiator is attached to

a surface and polymerization occurs via diffusion of monomer to

the active sites of the growing polymer chains. This growth

process can be contrasted with ‘‘grafting to’’ methods, in which

the layer thickness must increase by diffusion of a polymer chain

through the brush layer to the substrate surface.

In the second case, moderate grafting density PS brushes are

confined between two surfaces to study the compression of the

brushes. In this system, both the experiments and the modeling

are inherently difficult. Experimentally, the biggest challenge is

to obtain uniform confinement between two single crystals of

a suitably large surface area for neutron scattering measure-

ments. The confining surfaces must be kept in close opposition at

separations below the free extension of the brush, typically less

than a few hundred nanometers, and must be kept aligned and

parallel throughout the duration of the measurement in order to

confine the film of interest uniformly. Early work by Cosgrove

and coworkers paved the way for work we present here.55 In their

design, large, optically polished quartz flats were forced to

closely approach using a hydraulic ram, and intersurface

separations of about 100 nm were attained. However, data

interpretation proved difficult because a constant surface sepa-

ration could not be maintained during the course of the neutron

reflectivity measurement.

To overcome this problem experimentally, we have developed

an apparatus that enables single crystal substrates of silicon,

quartz or sapphire with areas up to tens of square centimeters to

be kept parallel at controlled and well-defined separations from

millimeters to less than 1000 A.2,3 To illustrate our approach and

resulting structural information that can be obtained, we

describe work on high density polystyrene (PS) brushes confined

between sapphire substrates. The PS brushes are formed by spin-

coating PS-polyvinylpyridine diblocks to obtain very high

grafting densities. These studies demonstrate that the experi-

mental technique provides a new method for measuring the

density distribution of grafted polymer brush layers under

confined geometries and also highlight the need for a concerted

experimental—simulation modeling approach in order to fully

characterize the system.

Co-analyzing scattering and simulation experiments

Simulations most often determine the radial distribution

function which can be Fourier transformed into the scattering

function (if we assume three-dimensional homogeneity and

isotropy) using56

Soft Matter, 2009, 5, 4612–4622 | 4615

SðkÞ ¼ 1þ 4pr

ðN0

r2 sinðkrÞkrðgðrÞ � 1Þdr (4)

where r is the density of the system and g(r) the radial distri-

bution function. Thus, we can go back and forth between Fourier

space in which scattering lives and direct space in which our

intuition works best. Simulations can therefore provide the link

between powerful scattering techniques to our intuition by

measuring both, radial distribution functions or density profiles

on the one hand and structure factors or scattering profiles on the

other.

In neutron reflectivity experiments, the scattering intensity is

obtained for different values of Q, which depend on the wave-

length or incident angle of the neutron beam and sample.

Because the structural information of interest is indirectly con-

tained within reflectivity data, post-experimental analysis is

required. As commented earlier, this process involves a trans-

formation of the data from inverse space to real space and, in the

absence of phase information, iterative fitting of a model is most

often relied upon to extract the scattering length density (SLD)

profile from the data. As a result, the largest challenge frequently

for physical experiments is generating the appropriate physical

model. Here, especially, a synergistic approach that leverages

simulations and experiments is beneficial. Furthermore, simula-

tions can extend the available parameter space into realms not

accessible directly experimentally. For instance, currently there is

an upper limit on the grafting density of polymer brushes which

can be studied with experiments due to the surface packing of

polymerization initiators. If simulations are anchored to reality

at a few experimentally obtainable grafting densities, a variety of

other grafting densities can be simulated to probe parameter

space more efficiently as well as study regimes currently

unavailable to experiments.

Moreover, research on polymers at interfaces is a perfect

example of a multiscale problem in which processes that take

place on scales ranging from the atomistic (A) to the film

thickness (mm) are all important (Fig. 2). With molecularly-thick

films, the relevant length scales begin to overlap, making a mul-

tiscale approach imperative.29,57 There is a critical need in

computational polymer modeling for addressing this type of

problem with methods that map onto each other a set of models

that represent the same system but are tailored for computational

Fig. 2 A schematic illustration of the wide parameter range and length

scales that can be explored using a coordinated experimental and simu-

lation approach to characterize and predict polymer thin-film structure

under confinement between two surfaces.

4616 | Soft Matter, 2009, 5, 4612–4622

efficiency at one length scale. The mapping process allows a type

of bootstrapping in which the information built up by an

atomistic simulation is retained by and used to tune the next

coarser model. This approach allows information such as the

chemical identity35,58,59 to be retained and facilitates comparison

as well as validation with experimental data.36

Modeling experiments

Computer simulations of fully detailed atomistic models are

limited in their applicability as the length scales reachable are of

the order of a few nanometers.19,60,61 So larger scale coarser-

grained simulations are normally necessary.13

Coarse-grained molecular dynamics simulations have been

performed here to examine systems not amenable to atomistic

simulations. The model we are applying first is based on the

coarse-grained lipid model developed by Marrink.62 Every

interaction site consists of 4–5 heavy atoms and has a mass of

72 amu (e.g. a super atom in Fig. 2). Qualitative information

pertaining to how the system structure changes with polymer

grafting density, chain length, and temperature can be obtained

in this manner. The exact force-field has been presented earlier13

in an application to hydrophilic–hydrophobic copolymers. The

system corresponds to a generic polymer in good solvent such as

PS in toluene or polar polyethylene oxide (PEO) in water.

Four grafting densities (0.174, 0.347, 0.485, and 0.694

chains/nm2) were examined at different temperatures and chain

lengths. The highest two grafting densities correspond to what

can currently be achieved experimentally with atom transfer

radical polymerization (ATRP) method discussed in the next

section on physical experiments. All grafting densities were

characterized at both 300 and 350 K for short chains consisting

of 40 coarse grained monomers. At selected state points longer

chains (up to 150 monomers) were studied as well.

At 350 K the chains extend farther from the surface with

grafting density, as is expected (cf. Fig. 3). Of particular interest

is a characteristic rise of solvent density at the surface observed

for all grafting densities, indicating a polymer depletion layer of

about 2–3 nm. This depletion region occurs because the surface

limits the polymer random walk, forcing the chains to orient

more normal to the surface. This constraint of chain configura-

tions near the surface, reduces the entropy of the system. A

depletion layer is expected and has previously been observed for

lower grafting densities63 (so-called mushroom regime), yet was

unexpected for higher grafting densities as studied here.64

Experiments typically cannot resolve such fine structural details.

In general, at the higher grafting densities the chains extend due

to lateral excluded volume effects, and orient more perpendicular

to the surface. For the 100 monomer systems with the highest

grafting density the polymer density develops a plateau at about

half the bulk density, indicating a saturation limit (not shown).65

The structure factors corresponding to the density profiles in

Fig. 3 are shown in Fig. 4. As discussed above, density profiles

and strutcure factors obtained from simulations aid in devel-

oping models and fitting routines for the experimental data, and

can be validated by experimental results. For this system we

observed that the peak of the chain end (the last few monomers)

density profiles shifts away from the surface with increasing

grafting density. Further, the chain ends on average extend


Fig. 3 Polymer density profiles as a function of distance from surface for

various grafting densities.

Fig. 4 Structure factors for different grafting densities corresponding to

the density profiles in Fig. 3.

Fig. 5 Snapshots of opposing polystyrene brushes without (left) and

with (right) toluene.

Fig. 6 Polystyrene chain aspect ratio in dry and solvated opposing

double brushes.

through the majority of the brush height. This is an example of

simulations providing additional information without extra time

or effort.

In addition to such generic, semiquantitiave simulations one

can also perform high resolution atomistic simulations of smaller

systems. We discuss here an example of short polystyrene chains

grafted to a graphite surface. We studied the behavior of poly-

styrene (PS) brushes in dry (vacuum) conditions and in toluene,

a good solvent. Arrays of 3� 3, 4� 4, 5� 5, and 6� 6 PS chains

were grafted onto a graphite surface with a cross section of

60.4 nm2, corresponding to grafting densities of 0.149, 0.265,

0.413 and 0.595 chains/nm2 Arrays of stretched polystyrene

chains were grafted onto the surface consisting of two graphene

sheets, to construct a polymer brush. The end-grafting was

achieved by restraining the position of one end of the chain

a short distance (2.5–3.0 A) above the graphite layer. These single

brushes were subsequently stacked against each other to create

opposing brushes with a spacing of 12 nm. Toluene molecules

were inserted until the desired density was obtained. All simu-

lations were carried out at 450 K and an united-atom toluene

model, where both the CH and the CH3 units were treated as

pseudo atoms, was applied.66


Equilibrated opposing brushes with and without toluene are

depicted in Fig. 5 which shows that the PS chains in the toluene

solution are more stretched out. The behavior of the polystyrene

chains in the opposing brush configurations follows the same

pattern as for the coarse-grained single brushes discussed above.

An interesting observation is that in all cases studied, toluene

molecules were found to form layers at the graphite surface. The

first toluene enriched layer appears at approximately 0.5 nm

from the surface. This is in excellent agreement with the solvent

enrichment also observed in the coarse-grained model above.

More generally, the effect of grafting density and solvent

conditions on the topology of the polystyrene chains can be

quantified in a variety of ways e.g., via density profiles, the

extension of chains away from the grafting surface, or the prin-

cipal components of the gyration tensor. Specifically, the latter

can be used to compute shape descriptors such as the radius of

gyration, the asphericity, or the acylindricity. The usefulness of

these measures is illustrated in Fig. 6 which shows the aspect

ratio of the PS chains as a function of grafting density. The aspect

ratio is obtained by dividing the smallest by the largest principal

value of the gyration tensor. In the absence of a solvent, this

quantity rises with increasing grafting density from about 0.5 to

about 0.7 indicating a transition from a cigar to a more egg-like

shape. This change is a result of the shorter separation between

the chains leading to stronger chain-chain interactions and

Soft Matter, 2009, 5, 4612–4622 | 4617

Fig. 7 Two dimensional rdfs for 3 � 3 (left) and 6 � 6 (right) atomistic PS opposing brushes.

effectively to a flattening of the chains. The aspect ratio provides

a simple scalar measure to quantify this topological transition. In

solvated brushes this change is almost unnoticeable.

The grafting density and the solvent conditions not only

influence the shape and extent of individual PS chains but also

their mutual interactions. This type of information can be

assessed using radial distribution functions g(r). Owing to the

two-dimensional nature of the systems under investigation

conventional spherically symmetric RDFs are not well suited.

Instead, we have calculated 2D-RDFs which measure the spatial

correlations parallel to the surface (in-plane) and the correlations

perpendicular to the surface (out-of-plane) separately. The result

of this type of analysis is shown for the 3 � 3 and a 6 � 6

opposing brushes in toluene solution in Fig. 7. The chain-chain

separation is reflected in the variations of the RDF in the in-

plane direction which reveals correlations up to the 2nd (in-

plane) neighbor range. In contrast, the RDF in the out-of-plane

direction provides information about the spatial extent of the

chains away from the surface and the interaction between the

chains grafted to the two opposing surfaces. This clearly

demonstrates the importance of separating the in and out-of-

plane correlations. This is a very good example of the strength of

simulations as such an analysis is not available with experimental

data.

As discussed aboved RDFs are not directly accessible experi-

mentally. Scattering experiments can, however, measure struc-

ture factors which are shown in Fig. 8. The most apparent feature

are the periodic ripples which decay in intensity with increasing

Q-vector. Their spacing is inversely proportional to the separa-

tion of the two opposing polymer brushes. The range of the

Fig. 8 Structure factors of atomistically simulated polystyrene opposing

brushes for different grafting densities with and without toluene.

4618 | Soft Matter, 2009, 5, 4612–4622

periodic repetitions as well as their fine structure are directly

related to the structure of the polymer brushes. Comparing Fig. 7

and 8 it is obvious that RDFs provide a much more intuitive

picture than structure factors and the visualizations in Fig. 5

together with the change in aspect ratio (Fig. 6) tell another

important part of the story. The structure factor, however,

establishes a direct connection between simulation and experi-

ment. Successful comparison to experimental data validates

simulations, which subsequently can be exploited to obtain a real

space picture of the physical system that is not accessible by other

means.

Physical experiments

Previously, experimental studies that characterized polymer

brush structure at high grafting densities were rare due to the

challenges associated with forming such layers.67,68 Recently

developed ‘‘grafting from’’ approaches provide access to this

regime by polymerizing monomers directly onto a surface from

solution.69–71 Typically, a monolayer of polymerization initiator

is attached to a surface and polymerization occurs via monomer

diffusion to the active sites of the growing polymer chains. This

growth process can be contrasted to ‘‘grafting to’’ methods, in

which the layer thickness must increase by diffusion of a polymer

Fig. 9 Reflectivity profile for an ATRP grown 20k MW PS film in air.

Solid line through the data is the fit from the inset SLD profile.


Fig. 10 Thickness of dry PS films as a function of MW.

Fig. 11 (A) Reflectivity profile for an ATRP grown 20k MW PS brush in

toluene. Solid line through the data is a fit from the inset SLD profile. (B)

Volume fraction profile showing the parabolic profile of the brush using

Eq. (6).

chain through the brush layer to the substrate surface. In the first

experimental example presented, ultra-high grafting densities of

polystyrene brushes were obtained using atom transfer radical

polymerization under moderate temperatures (see ref. 72 for

details). Neutron reflectivity measurements were used to char-

acterize the films in the dry state (air is a poor solvent for PS) and

under good solvent conditions (toluene). Fig. 9 shows the

reflectivity data and scattering length density (SLD) profile for

a representative PS film in air with Mn ¼ 20k. The data were fit

with nonlinear least-squares regression using the MOTOFIT

reflectivity analysis package.73 The SLD model consisted of

several layers: silicon substrate, 15–25 A native oxide, 15–20 A

initiator with a SLD of 0.4 � 10�6 A�2, polystyrene, and air.

Importantly, the SLD of the polymer layer converged to

1.45 � 10�6 A�2 compared to the expected value of 1.42 � 10�6

A�2 for bulk PS, thus lending credence to the quality of the

polymerization. Note the qualitative agreement between the

initial fringes in Fig. 8 and Fig. 9. The dry film thickness was

found to scale linearly with the molecular weight, demonstrating

that a constant grafting density was obtained as shown in Fig. 10.

The grafting density was calculated by fitting the equation

t ¼�

s

rNA

�Mn

to the data. Using a bulk value PS density,74 r, of 1.05 g/cm3

yields a grafting density of 0.44 chains/nm2 or 2.3 nm2 per chain.

The cross sectional area of a single polystyrene chain in the

crystalline state is 0.7 nm2,75 which establishes a theoretical

maximum grafting density of 1.4 chains/nm2. These grafting

densities are comparable to the atomistic 5 � 5 simulations

above.

Under good solvent conditions, both theory and simulations

(see e.g. Fig. 3) predict a parabolic density profile away from the

surface followed by a long decaying tail over a range of grafting

coverages.20,76–79 Fig. 11 shows the reflectivity data and best fit

based on the SLD profile for Mn ¼ 20k in deuterated toluene

(SLD ¼ 5.66 � 10�6 A�2).


Deuterated toluene is a good solvent for polystyrene and

maintains high neutron contrast to the grafted chains. The region

extending from the initiator was modeled with an additional

layer to account for a possible depletion layer at the anchor

surface (in agreement with both atomistic61 and coarse-grained65

simulation density profiles) followed by a power law profile for

the brush with the end smeared by an error function representing

the decaying tail also found in simulations.

To obtain the polymer brush density distribution, the SLD

profile was converted to a volume fraction profile of PS

extending from the initiator using

SLDfitted ¼ fPS(SLDPS) + (1 � fPS)SLDtoluene (5)

Soft Matter, 2009, 5, 4612–4622 | 4619

Fig. 12 Reflectivity profile for opposing spin coated PS-P2VP layers.

Small Qz interference peaks in the profile result from constructive inter-

ference between the substrates and their overall separation. Higher Qz

peaks are due to the polymer layers. The dashed curve is a fit to the data

based on the scattering length density profile shown in the inset assuming

that the gap between the substrates is uniform. The solid curve assumes

that the gap separation has a Gaussian variation of 60 A.

Fig. 13 (A) Reflectivity profile PS brushes in confinement in deuterated

toluene. The solid curve is a fit to the data based on the scattering length

density profile (inset). A power law profile for the PS portion of the brush at

the P2VP interface was used in the data fitting. (B) Volume fraction profile

of the compressed polymer brushes compared to the profile of an uncon-

strained brush at a single interface (dashed curve adapted from ref. 20).

To compare with theoretical predictions, the main body of the

volume fraction profiles was fitted to a power law:

fðzÞ ¼ f0

1�

�z

h0

�n!

(6)

where f0 is the volume fraction of the brush at the interface and

h0 is a measure of the brush extension.

At lower grafting densities a depletion layer has been observed

at the grafting surface and is also found from our simula-

tions.61,63,65 The effect of a decreased depletion layer with

increasing grafting density has also been observed in Monte

Carlo simulations.64 In the very high grafting density experiments

here, fitting the data did not require a depletion layer. However,

our experimental resolution is much poorer than that of the

simulations above (below 3 nm, see Fig. 3), which precludes

a definitive statement.

To investigate the effect of confinement/compression on the

structure of high density brushes, symmetric polystyrene-poly-2-

vinyl vinylpyridine (PS-P2VP) polymer diblock layers were

prepared by spin coating on sapphire substrates and annealing

above their glass transition temperature, Tg. The crystals were

mounted in an apparatus which is capable of maintaining very

small, uniform separations between the surfaces.3 For simplicity,

we first start with neutron reflectivity measurements of the

system in the dry, non-solvated state, cf. Fig. 12. The high

frequency Keissig fringes at low Q clearly indicate that the gap

spacing between the substrates is quite small, D z 2p/DQfringe

spacing z 1200 �A.30 However, the relatively low visibility of the

fringes indicates there is some variation in separation across the

gap. As the beam footprint is about 1.5 square centimeters, this is

not unexpected as small variations in the intersubstrate separa-

tion have a large impact on the Keissig fringes. A simplified

means to account for these variations is to incoherently average

the representative model reflectivity over a Gaussian distribution

of gap thicknesses.2

R�Qz;Tavg

�¼ 1

sffiffiffiffiffiffi2pp

ðRðQz;TÞ e

�ðT�TavgÞ2

2s2 dT (7)

The same SLD model is used to generate the solid and dashed

curve fits to the reflectivity profile (Fig. 12), but the visibility of

the fringes with the solid curve is reduced by incoherently aver-

aging the gap spacing with a Gaussian standard deviation of 60 A

over the sampled area (neutron beam footprint). This approach

accounts for macroscopic regions of the sampled area that have

small differences in gap spacing. As these regions are larger than

the coherence of the beam, their contributions to the reflectivity

profile add independently to give the total reflected signal. In air

when the polymer brushes are collapsed, this simple approach of

only varying the gap spacing is an accurate reflection of the

system. However, once the brushes are solvated and interacting,

variations in the gap spacing will alter the structure of the brush

layer.

In the case of solvated brushes under confinement, simulations

can enable much more sophisticated and representative models

to be used in fitting the experimental data. The reflectivity profile

for confined PS brushes in deuterated toluene is shown in Fig. 13.

Keissig fringes (D z 1000 �A) are still clearly visible in the

solvated reflectivity profile and indicate that the small gap

4620 | Soft Matter, 2009, 5, 4612–4622 This journal is ª The Royal Society of Chemistry 2009

spacing was maintained under solvation. Under these conditions,

the brushes are compressed to about 75% of their extension

compared to a free, unconfined interface. As a result, their profile

is no longer expected to be that of a simple parabolic profile as

observed at a single surface. Rather, the brushes may now

interdigitate such that the volume fraction between the two

substrates should become more uniform with compression. On

the other hand, interdigitation will not occur if the brushes act as

impenetrable walls. Notably, the PS volume fraction at the

interface increases as the brushes are compressed, rather

than becoming more flattened and uniform with increasing

confinement.

Fitting the data, however, is much more challenging with

solvation. The variability in the intersubstrate separation will

also impact the polymer brush profile as shown schematically in

Fig. 14. In other words, instead of having a 1-dimensional SLD

profile to represent the physical system, the reflectivity data is

comprised of a superposition of reflections from different sepa-

rations, each with slightly different interference fringes and

correspondingly different polymer brush density distributions.

To properly model the data, a 2-dimensional profile that repli-

cates the polymer density distribution as a function of the

intersubstrate separation is now required.2,3 Currently, a single,

1-dimensional profile at the mean substrate spacing is used and

smeared using a Gaussian distribution as in the air example.

Because of the distribution of substrate spacing, each data set,

therefore, contains contributions from different polymer brushes

with varying compression. A more accurate and physically

realistic model would explicitly describe the polymer brush

profile as a function of substrate separation with different

separations contributing to the total reflectivity profile. Impor-

tantly, a series of experiments performed at regular intervals of

mean substrate separation, provides data that contains infor-

mation over a large range of substrate separations. These data

sets could then be fit simultaneously with a 2D model that

describes the profile for all substrate separations. This unique

approach would improve the confidence and reliability of the

structural information tremendously and allow one to robustly

characterize many density profiles simultaneously. Simulations

are crucial for developing such 2-dimensional profiles and

Fig. 14 Variation in the scattering length density profile as a function of

inter-substrate spacing. The dark solid lines represent 1D fitting lines.

With 2D surface models, the profile can be generated continuously and

a more reliable representation of the physical system can be obtained.


providing a tractable analysis model for the experiments, work

that is currently on-going.

Conclusions

We have shown that by judiciously combining simulation and

neutron reflectivity experiments we can elucidate the structural

properties and density profiles of confined polymer systems to

a high degree of detail. We can obtain density profiles along the

film normal and understand the structure of the films in detail.

Both techniques have their own advantages and drawbacks. It

turns out that these are largely complimentary and that by using

them in concert we obtain the advantages of both and minimize

the drawbacks. Simulations are clearly model dependent, but are

extremely valuable due to their high resolution, accessible

parameter space, and ability to discern sample heterogeneities.

Neutron scattering enables the physical structure of soft matter

systems to be determined non-invasively, however significant

effort is needed to extract the results in an unambiguous manner.

This combination of scattering and simulation can also be used

in other areas of soft matter research where local structure

determination is important. A good example is the structure of

supported lipid bilayers.80,81

Acknowledgements

Financial support for this work through the US Department of

Energy, Office of Basic Energy Sciences under grant DE-FG02-

OGER46340 is gratefully acknowledged. PT is additionally

supported through a fellowship by the Academy of Finland.

Computer time at the National Energy Research Supercomputer

Center which is supported by the Office of Science of the

U.S. Department of Energy under Contract No. DE-AC03-

76SF00098 has been used for the simulational part. The SPEAR

reflectometer at the Manual Lujan Jr. Neutron Scattering Center

at Los Alamos National Laboratory, which is supported by

DOE under Contract No. W7405-ENG-36, was used for the

neutron scattering experiments.

Notes and references

1 K. A. Marx, Biomacromolecules, 2003, 4, 1099–1120.2 W. A. Hamilton, G. S. Smith, N. A. Alcantar, J. Majewski,

R. G. Toomey and T. L. Kuhl, J. Polym. Sci., Part B: Polym.Phys., 2004, 42, 3290–3301.

3 T. L. Kuhl, G. S. Smith, J. N. Israelachvili, J. Majewski andW. Hamilton, Rev. Sci. Instrum., 2001, 72, 1715–1720.

4 G. S. Smith, T. L. Kuhl, W. A. Hamilton, D. J. Mulder and S. Satija,Phys. B, 2006, 385–386, 700–702.

5 E. P. K. Currie, M. Wagemaker, M. A. C. Stuart and A. A. van Well,Phys. B, 2000, 283, 17–21.

6 J. B. Field, C. Toprakcioglu, R. C. Ball, H. B. Stanley, L. Dai,W. Barford, J. Penfold, G. Smith and W. Hamilton,Macromolecules, 1992, 25, 434–439.

7 S. Granick, A. L. Demirel, L. L. Cai and J. Peanasky, Israel Journal ofChemistry, 1995, 35, 75–84.

8 S. Granick, S. K. Kumar, E. J. Amis, M. Antonietti, A. C. Balazs,A. K. Chakraborty, G. S. Grest, C. Hawker, P. Janmey,E. J. Kramer, R. Nuzzo, T. P. Russell and C. R. Safinya, J. Polym.Sci., Part B: Polym. Phys., 2003, 41, 2755–2793.

9 T. W. Kelley, P. A. Schorr, K. D. Johnson, M. Tirrell andC. D. Frisbie, Macromolecules, 1998, 31, 4297–4300.

10 K. A. Smith, M. Vladkov and J. L. Barrat, Macromolecules, 2005, 38,571–580.

Soft Matter, 2009, 5, 4612–4622 | 4621

11 S. M. Baker, G. S. Smith, D. L. Anastassopoulos, C. Toprakcioglu,A. A. Vradis and D. G. Bucknall, Macromolecules, 2000, 33,1120–1122.

12 D. R. Heine, G. S. Grest and J. G. Curro, Berlin, 2005.13 M. Hatakeyama and R. Faller, Phys. Chem. Chem. Phys., 2007, 9,

4662–4672.14 V. Vao-soongnern, J. Nanosci. Nanotechnol., 2006, 6, 3977–3980.15 A. Yethiraj, J. Chem. Phys., 1994, 101, 2489–2497.16 A. V. Lyulin, N. K. Balabaev and M. A. J. Michels, Macromolecules,

2002, 35, 9595–9604.17 R. Graf, A. Heuer and H. W. Spiess, Phys. Rev. Lett., 1998, 80,

5738–5741.18 G. D. Smith, W. Paul, M. Monkenbusch and D. Richter, Chem.

Phys., 2000, 261, 61–74.19 M. Doxastakis, D. N. Theodorou, G. Fytas, F. Kremer, R. Faller,

F. M€uller-Plathe and N. Hadjichristidis, J. Chem. Phys., 2003, 119,6883–6894.

20 M. D. Whitmore and J. Noolandi, Macromolecules, 1990, 23,3321–3339.

21 G. L. Cheng, A. A. Boker, M. F. Zhang, G. Krausch andA. H. E. Muller, Macromolecules, 2001, 34, 6883–6888.

22 D. Neugebauer, Y. Zhang, T. Pakula, S. S. Sheiko andK. Matyjaszewski, Macromolecules, 2003, 36, 6746–6755.

23 K. L. Robinson, M. A. Khan, M. V. D. Banez, X. S. Wang andS. P. Armes, Macromolecules, 2001, 34, 3155–3158.

24 J. Ell, T. Patten, D. Mulder and T. Kuhl, ACS Polymer Preprints,2008, 49, 197–198.

25 R. C. Advincula, W. J. Brittain, K. C. Caster and J. Ruhe, PolymerBrushes, Wiley-VCH, Weinheim, 2004.

26 D. H. Napper, Polymeric Stabilization of Colloidal Dispersions,Academic Press, London, 1983.

27 G. S. S. W. A. Hamilton, N. A. Alcantar, J. Majewski, R. G. Toomeyand T. L. Kuhl, J. Polym. Sci., Part B: Polym. Phys., 2004, 42,3290–3301.

28 C. Pastorino, K. Binder, T. Kreer and M. Muller, J. Chem. Phys.,2006, 124, 064902.

29 S. T. Milner, Science, 1991, 251, 905–914.30 R. L. Jones, S. K. Kumar, D. L. Ho, R. M. Briber and T. P. Russell,

Nature, 1999, 400, 146–149.31 J. Klein, E. Kumacheva, D. Mahalu, D. Perahia and L. J. Fetters,

Nature, 1994, 370, 634–636.32 P. Auroy, L. Auvray and L. L�eger, Phys. Rev. Lett., 1991, 66, 719.33 K. Kremer and F. M€uller-Plathe, MRS Bull., 2001, 26, 205–210.34 R. Faller and F. M€uller-Plathe, Polymer, 2002, 43, 621–628.35 J. Baschnagel, K. Binder, P. Doruker, A. A. Gusev, O. Hahn, K. Kremer,

W. L. Mattice, F. M€uller-Plathe, M. Murat, W. Paul, S. Santos,U. W. Suter and V. Tries, Adv. Polym. Sci., 2000, 152, 41–156.

36 F. L. Colhoun, R. C. Armstrong and G. C. Rutledge,Macromolecules, 2002, 35, 6032–6042.

37 R. Faller, F. M€uller-Plathe and A. Heuer, Macromolecules, 2000, 33,6602–6610.

38 G. D. Smith, D. Y. Yoon, W. Zhu and M. D. Ediger, Macromolecules,1994, 27, 5563–5569.

39 M. P. Allen and D. J. Tildesley, Computer Simulation of Liquids,Clarendon Press, Oxford, 1987.

40 D. Frenkel and B. Smit, Understanding Molecular Simulation: FromBasic Algorithms to Applications, Academic Press, San Diego, CA,1996.

41 O. G. Byutner and G. D. Smith, Macromolecules, 2000, 33,4264–4270.

42 R. Faller, H. Schmitz, O. Biermann and F. M€uller-Plathe, J. Comput.Chem., 1999, 20, 1009–1017.

43 E. Bourasseau, M. Haboudou, A. Boutin, A. H. Fuchs andP. Ungerer, J. Chem. Phys., 2003, 118, 3020–3034.

4622 | Soft Matter, 2009, 5, 4612–4622

44 D. Reith, M. P€utz and F. M€uller-Plathe, J. Comput. Chem., 2003, 24,1624–1636.

45 R. Faller, in Reviews in Computational Chemistry 23, ed. K. B.Lipkowitz and T. R. Cundari, Wiley, 2007, pp. 233–262.

46 G. S. Grest and K. Kremer, Phys. Rev. A: At., Mol., Opt. Phys., 1986,33, R3628–R3631.

47 T. P. Russell, Mater. Sci. Rep., 1990, 5, 171–271.48 M. Stamm, in Physics of Polymer Surfaces and Interfaces, ed. I. C.

Sanchez, Butterworth-Heinemann, Boston, MA, 1992, pp.163–201.

49 J. S. Higgins and H. Benoit, Polymers and Neutron Scattering,Clarendon Press, Oxford University Press, Oxford/New York, 1994.

50 I. W. Hamley and J. S. Pedersen, J. Appl. Crystallogr., 1994, 27,29–35.

51 J. S. Pedersen and I. W. Hamley, J. Appl. Crystallogr., 1994, 27,36–49.

52 S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter,Clarendon Press, Oxford Oxfordshire/New York, 1984.

53 C. F. Laub and T. L. Kuhl, J. Chem. Phys., 2006, 125, 244702–244708.

54 A. van der Lee, F. Salah and B. Harzallah, J. Appl. Crystallogr., 2007,40, 820–833.

55 T. Cosgrove, P. F. Luckham, R. M. Richardson, J. R. P. Webster andA. Zarbakhsh, Colloids Surf., A, 1994, 86, 103–110.

56 D. M. Heyes, The Liquid State: Applications of Molecular Simulations,New York, 1998.

57 G. S. Grest, in Polymers in Confined Environments, 1999, vol. 138, pp.149–183.

58 F. M€uller-Plathe, Soft Mater., 2002, 1, 1–31.59 R. Faller, Polymer, 2004, 45, 3869–3876.60 R. Faller, Macromolecules, 2004, 37, 1095–1101.61 P. T. Tr€askelin, T. L. Kuhl and R. Faller, Phys. Chem. Chem. Phys.,

2009, DOI: 10.1039/b911311h.62 S. J. Marrink, A. H. de Vries and A. E. Mark, J. Phys. Chem. B, 2004,

108, 750–760.63 P. Auroy, Y. Mir and L. Auvray, Phys. Rev. Lett., 1992, 69, 93.64 C. M. Chen and Y. A. Fwu, Phys. Rev. E: Stat., Nonlinear, Soft

Matter Phys., 2000, 63, 011506.65 I. G. Elliott, T. L. Kuhl and R. Faller, 2009, submitted.66 A. W. Schuettelkopf and D. M. F. van Aalten, Acta Crystallogr., Sect.

D: Biol. Crystallogr., 2004, 60, 1355–1363.67 C. Devaux, F. Cousin, E. Beyou and J. P. Chapel, Macromolecules,

2005, 38, 4296–4300.68 A. Samadi, S. M. Husson, Y. Liu, I. Luzinov and S. M. Kilbey,

Macromol. Rapid Commun., 2005, 26, 1829–1834.69 K. Matyjaszewski and J. Xia, Chem. Rev., 2001, 101, 2921–2990.70 C. J. Hawker, Acc. Chem. Res., 1997, 30, 373–382.71 Y. Tsujii, K. Ohno, S. Yamamoto, A. Goto and T. Fukuda, Adv.

Polym. Sci., 2006, 197, 1–45.72 J. R. Ell, D. E. Mulder, R. Faller, T. E. Patten and T. L. Kuhl,

Macromolecules, submitted.73 A. Nelson, J. Appl. Crystallogr., 2006, 39, 273–276.74 J. E. Mark, Polymer Data Handbook, Oxford University Press, New

York, 1999.75 V. P. Privalko, Macromolecules, 1980, 13, 370–372.76 S. T. Milner, T. A. Witten and M. E. Cates, Macromolecules, 1988, 21,

2610–2619.77 B. J. Factor, L.-T. Lee, M. S. Kent and F. Rondelez, Phys. Rev.

E: Stat., Nonlinear, Soft Matter Phys., 1993, 48, R2354.78 M. Murat and G. S. Grest, Macromolecules, 1989, 22, 4054–4059.79 A. Chakrabarti and R. Toral, Macromolecules, 1990, 23, 2016–2021.80 E. B. Watkins, C. E. Miller, D. J. Mulder, J. Majewski and

T. L. Kuhl, Phys. Rev. Lett., 2009, 102, 238101.81 C. Xing and R. Faller, J. Phys. Chem. B, 2008, 112, 7086–7094.


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