REVIEW www.rsc.org/softmatter | Soft Matter
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.
This journal is ª The Royal Society of Chemistry 2009
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
This journal is ª The Royal Society of Chemistry 2009
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
This journal is ª The Royal Society of Chemistry 2009
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
This journal is ª The Royal Society of Chemistry 2009
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
This journal is ª The Royal Society of Chemistry 2009
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.
This journal is ª The Royal Society of Chemistry 2009
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).
This journal is ª The Royal Society of Chemistry 2009
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.
This journal is ª The Royal Society of Chemistry 2009
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.
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