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Review
8
Progress in Computer Simulation of Bulk,Confined, and Surface-initiatedPolymerizations
Erich D. Bain, Salomon Turgman-Cohen, Jan Genzer*
In this article we provide a brief summary of computational techniques applied toinvestigate polymerization reactions in general, with a focus on systems under confine-ment and initiated from surfaces. We concentrate on two major classes of techniques,i.e., stochastic methods and molecular model-ing. We describe the major principles of thetwo classes of methodologies and point outtheir strengths and weaknesses. We reviewa variety of studies from the literature andconclude with an outlook of these twoclasses of computer simulation approachesas they are applied to ‘‘grafting from’’polymerizations.
1. Introduction
Computer simulations have emerged as a powerful tool in
predicting the properties of various classes of materials.
When applied to polymerization, computer simulation
methods can be employed in modeling the elementary
reactions andother processes and thus enable predicting the
properties of the final product. Many reviews and mono-
graphshavedescribed approaches facilitating theprediction
of the characteristics of the final products, including the
time evolution of molecular weight, molecular weight
E. D. Bain, S. Turgman-Cohen, J. GenzerDepartment of Chemical & Bimolecular Engineering, NorthCarolina State University, Raleigh, North Carolina 27695-7905,USAE-mail: [email protected]. Turgman-CohenPresent address: School of Chemical Engineering, CornellUniversity, Ithaca, New York 14853-5201, USA
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distribution, copolymer composition, and others.[1–6]
Various techniques have been employed to describe the
polymerization reactions on scales ranging from molecular
to mesoscale employing variants of quantum methods all
the way to the solutions of complex sets of differential
equations; the latter include the effects of hydrodynamics,
and heat and mass transfer.[7] Nowadays, there are even
commercial software packages available, such as PREDI-
CITM,[8] that can perform those calculations.Whilemodeling
and simulation of polymerization processes in bulk has
been covered rather extensively in numerous monographs,
relatively little attention has been paid to situations
involving polymerizations in confined geometries or on
surfaces. Yet, the latter class of polymerization reactions
has received great attention experimentally in recent years
due to either (1) carrying polymerization in chemically
inhomogeneous media or, (2) its prospect of synthesizing
specialty polymers and tailored surfaces.
The purpose of this review is to provide a brief account of
the progress in computer simulations of polymerization
library.com DOI: 10.1002/mats.201200030
Erich D. Bain obtained his B.S. degree in chemicalengineering from the University of Alabama in2005, and his PhD in chemical engineering fromNorth Carolina State University, under the direc-tion of Prof. Jan Genzer, in 2012. He is currently acontract research assistant in the Genzerresearch group, focusing on synthesis andcharacterization of polymer brushes for surfacemodification applications.
Salomon Turgman-Cohen received his B.S.degree in Chemical Engineering from PurdueUniversity in West Lafayette, Indiana in 2005.In 2010 he completed a PhD in Chemical Engin-eering at North Carolina State University underthe guidance of Prof. Jan Genzer and Prof. PeterK. Kilpatrick. He is currently a post-doctoralassociate in the group of Prof. Fernando Esco-
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reactions in confined geometries and on surfaces. Because
recent reviews have dealt in depth with in silico poly-
merization in confined spaces,[9–11] we will concentrate
primarilyon the classofmacromoleculespreparedbydirect
polymerization from surfaces. We will revisit briefly some
aspects of the various methods that have been applied to
describe polymerization reactions in bulk and point out
how some of those approaches can be adopted in the so-
called ‘‘grafting from’’ polymerizations. Given that not
much work has been done in the field of computational
methods applied to ‘‘grafting from’’ polymerization, we
include some general suggestions for researchers to
consider when approaching these problems. We hope that
this work will serve to provide an up-to-date summary of
the field and will stimulate further efforts to apply
molecular simulations to surface-initiated polymerization.
bedo at Cornell University and is applying com-puter simulation techniques to environmentaland sustainability problems.Jan Genzer received his Dipl.-Ing. in Chemical &Materials Engineering from the Institute ofChemical Technology in Prague, Czech Republicin 1989 and his Ph.D. in 1996 in Materials Science& Engineering from the University of Pennsyl-vania. After two post-doctoral stints withProf. Ed Kramer at Cornell University (1996–1997) and UCSB (1997–1998), Genzer joined thefaculty of chemical engineering at the NCState University as an Assistant Professor in fall1998. He is currently the Celanese Professor ofChemical & Biomolecular Engineering at NCState University. His group at NC State Univer-sity pursues research related to the behavior ofpolymers at surfaces, interfaces, and in confinedgeometries.
2. Polymerization in Confined Spaces
The attributes of polymerization processes in confined
geometries, i.e., pores, slits, intercalated layers, capillaries,
or those performed from initiators grafted at interfaces
differ significantly from those of analogous bulk processes.
The physical properties of polymers in confinement, such
as their glass transition temperature and their elastic
modulus, exhibit deviations from bulk behavior.[12]
Furthermore, if the polymerization process occurs under
confinement, altered kinetics and diffusion limitationmay
result in polymers with molecular weights, molecular
weight distribution, topology, and/or composition that
differ significantly from macromolecules synthesized
using identical methods under no confinement (e.g.,
bulk or solution). Since many of these effects are often
challenging to study experimentally, computermodels and
simulations have been a key component of research on
polymerizations in confined geometries. In Figure 1 we
depict various scenarios of polymerization in bulk, in
confined spaces, and from surfaces. While in bulk we can
tailor the polymerization conditions to yield reaction
processes approximately governed by the rates of the
individual chemical steps, i.e., initiation, addition, termina-
tion, and chain transfer, grafting or confining the growing
polymers may affect the rates of these reaction steps. For
instance, the presence of the substrate and its geometry
may limit chain conformational freedom in one or more
dimensions, reducingtheaccessibilityof the reactioncenter
in the growing chain. Hence, polymerizations in pores or
slits experience a higher degree of confinement than those
grafted on planes or spheres. This effect becomes stronger
with increasing the degree of confinement (e.g., decreasing
the size of a pore). Similarly, chain crowding occurs in a
surface-grafted polymerization due to a high density of
grafting points on the substrate. Expanded or collapsed
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chain conformations due to solvent quality have also been
shown to play a role as a confining factor.[13] In addition,
polymerization may depend on diffusion of monomer and
accessibility of chain ends, catalysts, or transfer agents. If
reactions are fast relative to diffusion, it may be necessary
to account for dynamic concentration gradients. All in all,
polymerization in confined spaces is affected by many
environmental parameters that originate from both the
nature of the substrate, the space available for polymeriza-
tion and chain freedom (i.e., confined vs. free).
3. Computer Simulation Approaches forPolymerization in Bulk and in ConfinedGeometries
Figure 2 depicts the various computational approaches
utilized for in silico polymerizations.Wedivide the relevant
modeling approaches intofive categories, dependingon the
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Figure 1. A schematic depicting polymerization reactions under various degree of confinement ranging from the bulk all the way to the one-dimensional space. The cartoons in the top row correspond to polymerizations in physically confined systems, including, (from the left) bulk,small volumes (3D), two closely spaced impenetrable surfaces (2D), and capillaries (1D). The middle row illustrates systems prepared by‘‘grafting from’’ polymerization grafting from flexible objects, from the surfaces of nanoparticles (3D), from flat impenetrable surfaces (2D),and inside concave tubes (1D). The bars below the cartoons depict the effect of curvature (increasing curvature shown with darker color),and degree of confinement (increasing degree of confinement shownwith darker color). Technically, the grafted systems can be consideredto be more confined than the physically confined systems given that the mobility of the chains in the grafted substrates is reduced–this,however, has to be taken with caution since the degree of confinement will also vary with the system size.
Figure 2. Different methods of modeling polymerization compared on the basis ofoptimal length scales (bottom axis) and the amount of localization information thatcan be modeled (left axis).
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E. D. Bain, S. Turgman-Cohen, J. Genzer
length scales probed and on the ease to
simulate confined environments: quan-
tummechanical models (QM), molecular
simulations which typically comprise
either molecular dynamics (MD) or
Monte Carlo (MC) methods, stochastic
methods based on those developed
by Gillespie[14,15] (Gillespie’s stochastic
simulation algorithm, or GSSA) and
finally deterministic models based on
reaction rate equations (RREs).
Quantum mechanical methods repre-
sent a powerful tool for evaluating the
details of the reaction mechanisms on
the atomistic scale. Here, all the reaction
mechanisms present in polymerization
reactions can, in principle, be captured
with high fidelity.[16–19] However, these
techniques, as powerful as they are,
are limited in their ability to model
polymerizations of long chain macro-
molecules mainly due to available
computation resources. In order to simu-
late polymerization reactions of longer
macromolecules, one has to give up some
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chemical information offered by the QM methods and
coarse grain the system. This is precisely what is done in
molecular simulations that involve various variants of MD
and MC methods. In spite of coarse graining, these
molecular approaches represent powerful toolboxes for
predicting the various characteristics of macromolecules
during polymerizations. Bymeans of these techniques, one
can obtain a reasonably complete picture of the entire
process, including the spatio-temporal development of
chaingrowthanddepletionofmonomers. Inmost cases, the
details of the solvent are coarse grained or the solvent
is considered only implicitly. The system size and extent
of polymerization vary, depending on the method imple-
mented. For instance, in classical MD methods that may
implement Lennard-Jones atomistic potential (or other
more complex potentials), the polymerization process can
onlybemonitored for relatively shortpolymerization times
given limited computation resources. This obstacle can be
removed by simplifying the potential considerably, for
instance by implementing square-well[20,21] or hard sphere
potentials. Here, larger system sizes than in the classical
MDmodels may be considered but one has to bear in mind
that the potential may not capture the true interactions
present in the system. Given that even the detailed LJ
potentials are only an approximation of reality and
that the overall aim is to get a qualitative picture of the
process, substantial simplification of the potentials is often
acceptable in this field.
Monte Carlo methods are often used on lattice models.
The choice of model is important since it dictates the
moves possible in the MC algorithm. If the kinetics of the
system are of interest, only moves that preserve realistic
dynamics, such as single bead displacements or reptation,
may be used. The bond fluctuation model (BFM),[22] is
often used when simulating polymers since it exhibits
Rouse dynamics, canmodel branchedmacromolecules, and
allows investigation of dense systems while preserving
integer arithmetic and other advantages of simple lattice
models. If the allowed moves are selected carefully in the
BFM, unrealistic bond-crossing can be avoided and self-
avoidance of the chains is achieved. Information about rate
constants describing the individual reactions is generally
notavailable inthemolecularsimulations.While thesystem
size that can be treated with themolecular models is much
larger than that in theQMmodels, computer resources limit
the maximum polymer length and maximum polymer
number in such simulations. This limitation is mitigated in
techniques that employ GSSA to evaluate a set of reaction
channels involved in polymerization processes.
The GSSA approach is computationally faster than
molecular simulations. It evaluates reaction probabilities
using empirical kinetic parameters, an area ofweakness for
molecular simulations, and it models rigorously the time
dependence of reactions, resulting in relatively accurate
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predictionsof reactionkinetics. TheGSSAmethod considers
each reacting species independently, allowing calculation
of the distributions of molecular weight, sequence dis-
tribution, and branching points for polymerized chains.
In principle, there is no limitation to the number of
reaction channels that can be included in a GSSA model.
A significant level of detail is lost relative to molecular
simulations, however, because the GSSA, as originally
formulated, assumes a homogenous distribution of the
reactive components. This assumption is usually not
applicable in ‘‘grafting from’’ polymerization or in other
confined systems. Recent advances in adapting GSSA for
polymerizations in spatially confined systems will be a
major focus of this review.
The final class of methods we consider are deterministic
models based on RREs. In simple cases, where only the
initiation, propagation, termination, and chain transfer
reactions are considered, a closed analytical solution to the
differential RREs may be available if certain assumptions
aremade, suchas thesteady-stateapproximation involving
the conservation of radical species. While analytical
solutions of the RREs are often useful for describing
reactant concentrations in large scale polymerizations,
they are typically incapable of predicting the fullmolecular
weight distribution, particularly at high conversion. How-
ever, a wide array of more powerful numerical techniques
are employed for dynamic simulation of the deterministic
RREs to describe a variety of polymerization systems.
Kiparissides et al.[3] have presented a helpful summary of
deterministic numerical methods for modeling polymer-
izations; among these are the method of moments,[23–27]
kinetic lumping,[28,29] orthogonal collocation,[30–32] numer-
ical fractionation,[33–35] Galerkin methods,[36–38] and sec-
tional grid methods.[39,40] In most cases deterministic
numerical approaches are not subject to the steady state
approximation (SSA), and they can provide accurate
estimates for distributions of chain length, composition,
and branching points. Often a hybrid approach is used, in
which deterministic methods based on the RRE are
combined with GSSA to give a more robust description of
the system. The GSSA has particular advantages for
calculating distributions of molecular weight and other
parameters, often resulting inmore accurate predictions of
multivariate distributions with fewer assumptions and
greater computational efficiency than comparable deter-
ministic methods.[41] Below system sizes of a few hundred
microns or in the case of very low concentrations of one or
more species such as radicals, random fluctuations in
concentration become important, at which point the GSSA
gives a more realistic description of the variability in a
perfectly mixed system than deterministic RRE simula-
tions. In both RRE and GSSA approaches one may employ
hydrodynamics and heat and mass transfer principles to
better predict the properties of macromolecules.
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E. D. Bain, S. Turgman-Cohen, J. Genzer
The characteristics of the individual methods and the
information obtained are listed in Table 1, where we
compare thevariousmethods in termsofavarietyof factors
relevant to polymerization in general (not necessarily in
confined geometry), including key assumptions, whether
physical rate constants can be used or predicted, and how
much information can be obtained about distributions of
molecular weight, sequence distribution in copolymeriza-
tion, and the mechanism of the reactions making up
polymerization.
Polymerizations in confined geometries have witnessed
enormous growth in the past few years. This has been
motivated by attempts to describe the polymer growth in
heterogeneous systems as well as activities related to
comprehending the polymerization processes in confined
spaces (i.e., in capillaries, or between two parallel slits) and
from surfaces. While the effects of confinement on
polymerizations have entertained a close scrutiny from
the experimental point of view, only a limited amount of
work has been done on modeling and simulation of these
systems.[42–46] This has to do, primarily, with the limita-
tions of the various computational approaches mentioned
earlier. While the QM methods can provide detailed
mechanistic informationabout thepolymerizationprocess,
Table 1. Attributes of various computational methods in describing
Quantum
mechanics(QM)
Molecular
dynamics(MD)
System size <0.1–101nm 1–100nm
Assumptions Non-relativistic
Schrodinger
equation, values
of fundamental
physical constants
Ergodic hypothesis,
potential energy
functions,
coarse graining
Kinetic
constants
Can predict rate
constants from
first principles
No
Molecular
weight
distribution
Yes, but
computationally
limited
Full
Monomer
sequence
distribution in
copolymers
Full Full
Polymerization
mechanism
Full description Some
coarse-graining c
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the complexity of the computation and limited computa-
tional resources prohibit the study of realistic macroscale
polymerization reactions and their chemical evolution.
Molecular simulations (MD andMC) alleviate this problem
by approximating the interaction between atoms and
molecules with empirically derived force fields and can be
further simplified by abandoning fully atomistic descrip-
tions and coarse graining the system. These simplifications
allow for longer times and larger length scales to be probed
and for the distribution of the reactive species to be
monitored during the reaction. A notable disadvantage of
MC and MD in the context of polymerization is that they
requiremethodsbywhich themonomersmay react to form
polymers. These methods often involve probabilities of
reaction that are unrelated to real rate constants. Never-
theless, as will be detailed later in this paper, this class of
approaches has received much attention in the past few
years in describing the growth of polymers in restricted
spaces. The application of GSSA approaches to polymeriza-
tions in confined spaces and fromsurfaces has been limited
severely primarily due to the inability of these techniques
to describe spatial distribution of reacting chains. Some of
those limitations can, in principle, be removed by
incorporating rate constants that account for diffusion
general polymerization.
Monte
Carlo(MC)
Stochastic
simulation
algorithm(GSSA)
Reaction rate
equations(RRE)
1–100nm �100nm >100mm
System at
equilibrium
Homogeneous
system
volume
Deterministic
formulation of
chemical kinetics,
steady state
approximation
(for analytical solution)
No Yes Yes
Full Full Can estimate by some
numerical methods
Full Full Can estimate by some
numerical methods
Some
oarse-graining
Severe
coarse-graining
Unavailable
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limitations, employing lattice-based GSSA methods, or
stochastically simulating a reaction–diffusion master
equation (RDME), as will be discussed below. Deterministic
RRE approaches have been used to model polymerizations
in confined geometries or on surfaces, although no
information about the single chain properties is known
and the effects of confinement on the polymerization
cannot be incorporated at the molecular scale. Hessel and
coworkers[47] used a numerical finite element simulation
package to study the effects of heat and mass transfer on
free radical polymerization in microfluidic devices. RRE-
based models of surface-initiated controlled radical poly-
merization have been developed by Zhu and cowor-
kers[48,49] and Bruening and coworkers.[50] Good agreement
with experimental thickness profiles can be obtained by
allowing the kinetic constant for termination to vary with
catalyst concentration, catalyst ratio, grafting density, and
other parameters. However, the models assume that
concentrations of reactants in the brush layer are
equivalent to those in the bulk, neglecting the effects of
confinement on these quantities. To obtain a more robust
descriptionof such systems, simulationmethods capableof
considering the distribution of polymers and small
molecule reactants within the brush layer are required.
In Table 2 we list several attributes of polymerization
systems under confinement and provide assessment of
how those are treated with the various computational
methods.
In the following sections we describe briefly the
governing principles of two major classes of computer
simulationmethods thathave traditionally been employed
in describing polymerizations in bulk, i.e., the GSSA
Table 2. Attributes of various computational methods in describing
Quantum
mechanics
(QM)
Molecular
dynamics
(MD)
M
C
(
Confinement due
to solvent quality
Implicit only Implicit or
explicit
Imp
ex
Confinement due
to presence of
impenetrable walls
Short length
scales
Substrate geometry Short length
scales
Yes
Grafting density
of chains
Requires
multiple chains
Yes
Monomer spatial
distribution
Not feasible Yes
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originally devised by Gillespie, and the molecular models.
We will point out cases relevant to polymerization in
confined spaces and on surfaces.
3.1. Stochastic Simulation Algorithm
Most polymerization processes consist of a series of
reaction channels. For the case of radical polymerization,
these may include initiator decomposition, initiation,
propagation, reversible termination (e.g., in controlled
‘‘living’’ radical polymerizations), irreversible termination
(by radical combination or disproportionation), and
chain transfer to monomer, solvent, polymer chains, or a
chain transfer agent. The reaction steps of a polymerization
are often formulated as a set of coupled differential
equations. Unfortunately these complex systems of
equations often cannot be solved analytically without
simplifying assumptions, e.g., the SSA, and numerical
solutions are often mathematically and computationally
quite demanding. Furthermore, modeling with a set of
differential equations makes two unrealistic assumptions.
Namely, it assumes that (1) chemical reactions have a
singledeterministic trajectory, and (2) the reactionmedium
is a continuum. While these assumptions work well for
large systems, they are not necessarily valid at the
molecular scale, where a discrete number of molecules
of each species participate in collisions and first-order
reactions (such as decomposition) that are essentially
random. These random events lead to a probability
distribution of reaction trajectories rather than a single
deterministic path for a given set of conditions. While
a coupled set of kinetic events can be modeled exactly
polymerization under confinement.
onte
arlo
MC)
Stochastic
simulation
algorithm (GSSA)
Reaction rate
equations (RRE)
licit or
plicit
Volume restriction,
diffusion-dependent
rate constants
In principle an
approximate method
should be possible
Yes Theoretically possible
with reaction–diffusion
master equation
Not known
Yes Depends on resolution
of subvolumes
No
Yes Not known No
Yes Depends on resolution
of subvolumes
No
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E. D. Bain, S. Turgman-Cohen, J. Genzer
by a so-called chemical master equation, this equation is
difficult if not impossible to solve for many systems.
The GSSA, often referred to as Gillespie’s algorithm,
was devised as a method to stochastically simulate
trajectories of the chemical master equation for coupled
chemical reactions.[14,15] There are several different GSSA
formulations, each similar but suited to different applica-
tions.[51] The ‘‘direct method’’ of the GSSA involves two
basic steps. First, the probability ai(x) of each reaction
channel i in systemstatex is calculatedas theproduct of the
molecular rate constant (proportional to the bulk reaction
rate constants ki) and the number of molecules of each
species participating in the reaction. The reaction step to
occur in a given iteration is chosen stochastically by
choosing the smallest integer j for which:
Xj
i¼1
aiðxÞ > r1a0ðxÞ (1)
where r1 is a randomly generated number on the interval
(0,1). This procedure amounts to a random selection of an
individual reaction channel weighted by the probability of
all available channels. If a certain reaction has the highest
probability ai(x) of occurring, that reaction has the highest
probability of being chosen by the algorithm. Here a0 is the
sum of probabilities for all reaction channels:
a0ðxÞ ¼XM
i¼1
aiðxÞ (2)
The second step in thedirectmethod involves calculating
the time interval for the chosen reaction. The time step is
calculated as:
t ¼ �lnðr2Þa0ðxÞ
(3)
where r2 is a second unit interval random number. The
time step is normalized by the total probability of reaction
in order to provide a physically realistic simulation of
reaction kinetics. The direct method formulation gives
accurate results when iterated for nearly any system
of homogeneous coupled chemical reactions, yet it is
relatively computationally expensive. An alternative
GSSA formulation called the first reaction method
calculates a time interval for each possible reaction
channel, after which the channel with the shortest time
is selected for the given iteration. In both the direct and
first-reaction methods, several hierarchical algorithms
of sorting and selecting the reaction channels have
been developed to improve computational speed. Further-
more a hybrid method known as tau-leaping saves
computation time by approximating the GSSA results
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for long time intervals over which the probability
functions can be expected not to change significantly.[51]
As a side note, the term kinetic Monte Carlo (KMC) is
sometimes used in the literature to refer to a method that
employs random numbers to simulate the dynamic
behavior of non-equilibrium systems. In many cases
KMC methods are equivalent to the GSSA method.[52–54]
TheGSSA is less computationally expensive than theMD
orMCmethods, making it an attractive technique for cases
where its basic assumptions are valid. Since the GSSA is
based on empirically determined reaction rate constants, it
is capable of being quantitatively accurate whereas MC
and MD depend on heuristically determined probability
functions that only provide qualitative results. As opposed
to the standard RRE formulation of chemical kinetics,
the GSSA does not require the implementation of a steady-
state approximation to model free-radical polymerization.
While moments of the molecular weight distribution
can be calculated from the RRE approach, the GSSA
easily allows one to obtain a full molecular weight
distribution of polymers at any point in the reaction, thus
offering a more thorough description of the system.
In principle, the GSSA can provide an exact solution for
nearly any set of discrete reactions, including systemswith
large numbers of channels, and systemswhose differential
equations cannot be solved analytically. Since the GSSA
takes account of the stochastic trajectory of real reactions, it
is ideally suited to simulating systemswith small amounts
of reacting species, i.e., cells and other biochemical systems.
GSSA is also well-suited to model radical polymerization,
where the concentration of active radicals is usually very
small. Since the standard formulations of GSSA explicitly
model each individual reactingmolecule, limited computa-
tion resources have typically restricted system sizes to
picomoles and below. Nevertheless that often can be
considereda largeenoughsample size toobtainstatistically
significant results.
The GSSA has several advantages for modeling poly-
merization systems. Since growing chains are counted
individually, the full molecular weight distribution of the
generated polymers can be obtained at a given conversion.
Non-steady reaction conditions, such as pulsed initiation,
can be considered because the GSSA does not rely on
the SSA. Because it assumes a perfectly mixed system
volume, the originally formulated GSSA is not applicable to
spatially inhomogeneous systems involving, for instance,
diffusion limitation and concentration gradients.However,
refinements such as chain length dependent rate constants
have allowed the GSSA to be applied for diffusion-limited
polymerizations, highly branched polymerizations, and
heterogeneous (i.e., emulsion) polymerizations. More
advanced modifications to adapt GSSA for spatially
varying systems do exist. These techniques can and should
be applied to polymerization reactions in confined geo-
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metries. Here we discuss recent work simulating bulk
polymerization by GSSA, as well as innovations based on
the GSSA that are suited to studying diffusion limitation
and spatially varying systems including geometrically
confined polymerizations.
3.1.1. GSSA Approaches for Modeling Polymerization
Two approaches are available for obtaining chain length
distributions using the GSSA approach. The most obvious
method[55] is to treat each chain length as an independent
chemical species with unique rate constants kr,n, corre-
sponding to the reaction type r (e.g., propagation, termina-
tion, etc.), for an n-length polymer. Hence a system with
maximum chain length N will have a number of reaction
channels proportional to N multiplied by the number of
reactions each chain can participate in.While this approach
has the advantage of allowing a set of size-dependent rate
constants, the computational time increases significantly
since the number of reaction channels considered in each
iteration (cf. Equation 1) grows with N. A more efficient
approach is achieved by assuming that polymer reactions,
i.e., propagation or termination, are independent of chain
length, according to the well-established assumption of
equal reactivity. In this case only a handful of reaction
channels need to be considered for the entire course of the
simulation. The problem of how to track the degree of
polymerizationof the individual chains is solvedbycreating
a list of chain lengths (most efficiently, a list in which the
vector index represents chain length and the value
represents the number of chains of that length). Each time
a reaction channel is chosen that involvesapolymer chain, a
polymer ischosenfromthelistbymeansofathird randomly
generated number, and the chain length is modified
according to the rules of the chosen reaction channel.
Lu et al.[56]were among thefirst todemonstrate thata full
molecular weight distribution could be obtained using
Gillespie’s GSSA to model free radical polymerization. The
reaction was simulated for unsteady conditions including
rotating sector and pulsed laser initiation, demonstrating
Figure 3. (Left) Time evolution of radical concentration for a continuostochastic simulation algorithm (GSSA). (Right) Molecular weight diwith permission from ref.[56]
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conditions for which the steady-state approximation is
valid as well as those for which it is not. Figure 3 illustrates
the lag time of approximately 2 s to establish a steady state
inradicalconcentrationforacontinuously initiatedFRP,and
weight distribution of the resulting set of chains. GSSA
models for FRP that include the effect of chain transfer[57,58]
givemorenuancedandphysically realistic results. TheGSSA
has been used to study the polymerization of butadiene
from the gas phase,[59,60] diacetylene and deuterated
diacetylene 2,4-hexadiynylene bis-(p-toluenesulfonate) in
the solid phase,[61,62] formation of poly(p-phenyleneviny-
lene) via sulfinyl precursor route,[63] and polymerization of
propylene by single and multi-site Ziegler-Natta cata-
lysts.[64,65] The GSSA has been employed as part of a
multiscale model for industrial high pressure low-density
polyethylene (HPLDPE) production.[3] In addition, chain
extensionswithbisoxazoline[66–69] and telomerizationwith
chain transfer agents[70] have been modeled by GSSA.
An important application of the GSSA to polymerization
has involved non-steady state conditions. For example, a
non-steady state GSSA simulation verified an expression
derived analytically for the molecular weight distribution
at very short times of polyolefins produced by coordination
polymerization.[71,72] GSSA has also been employed to
model polymerization in a flow reactor,[73] a case for which
steady-state radical concentration is often not reached at
moderate tohighflowrates, because the residence time in a
section of the tube is on the same order as the startup
time for radical steady state. Used in conjunction with
a kinetic theory for the viscoelasticity of the chains, the
GSSA provided a better fit to experimental data for LDPE
production in a flow reactor than a deterministic model
based on moment equations. To increase the speed of the
GSSA,polymerchain lengthsmaybeestimatedaccordingto
the average number of propagation steps expected for the
lifetime of a given radical.[74] This approach amounts to
solving the deterministic rate equation for propagation,
while initiation and termination are treated stochastically.
A parallelized version of the GSSA has been developed,
usly initiated free-radical polymerization simulated using Gillespie’sstribution of chains produced from the same simulation. Reprinted
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which splits the number of reacting polymers evenly
among processors, reacts them independently for a short
time, updates the global species list via communication
among the processors, then repeats the process.[75]
TheGSSAhasbeencompareddirectlyagainst thediscrete
Galerkin method for calculating the weight distributions
of free-radical polymerization.[76] The Galerkin method is
employed commonly in commercialmodels of polymeriza-
tion and in some cases is able to generate accurate
molecular weight distributions in only seconds of compu-
tation time. However, the Galerkin method is highly
dependent on a priori knowledge about the reacting
system, such as the expected weight distribution, and
hence is applied best in situations where the weight
distribution could be predicted approximately even before
running the computer simulation. Conversely, the GSSA
was showntobequiteversatile andcangive results thatare
equally or more accurate than the Galerkin model for a
variety of mechanism of polymer formation. A polymer-
ization model based on a hybrid of GSSA and the h-p
Galerkinmethod used in the commercial software package
PREDICITM has been demonstrated.[77] The chain length
distribution has been solved deterministically by PREDI-
CITM, while additional properties, i.e., copolymer sequence
distribution and branching point distribution, are deter-
mined in parallel by the GSSA, creating a package that is
both more efficient and gives a more robust set of data
than would be available by either the Galerkin method or
the GSSA approach independently. Figure 4 compares
copolymer sequence distributions calculated by the hybrid
GSSA-Galerkin model with the averages calculated by the
Galerkin method alone.
The GSSA is well suited to studying copolymerization
because the sequence distribution can be estimated or even
Figure 4. Comparison of hybrid GSSA-Galerkin algorithm output (poin(thin lines) for monomer sequence distribution in a copolymerization600 s (right). The Galerkin solutions match closely with regression apermission from ref.[77]
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accounted for exactly for each chain, analogous to the way
in which molecular weight distribution is obtained using
lists. Efficient accounting algorithms[78] are necessary for
this purpose, given the large amount of data processed.
Copolymerization systems studied by the GSSA include
statistical copolymerization with terminal and penulti-
mate termination models,[79] multiblock copolymeriza-
tion,[80] and gradient copolymerization.[81–85] The bivariate
distribution of copolymer composition and molecular
weight can be obtained by combining GSSA with simulta-
neous property accounting algorithms by means of a two-
dimensional fixed pivot technique.[86] Sequence distribu-
tion can be also tracked in conjunction with long chain
branching distribution.[87] Reactivity ratios may be deter-
mined from a given sequence distribution using a GSSA
model of copolymerization.[88] Studies have been per-
formedonmodificationof cis-1,4-polybutadienebackbones
by graft copolymerization with styrene[89] and solid phase
grafting of acrylic acid onto polypropylene (PP).[90] GSSA
was also used to elucidate the mechanism of forming
single monomer or short-chain grafts of maleic anhydride
on PP[91] and PE[92] in the presence of free radicals from
peroxide initiators.
Controlled/‘‘living’’ radical polymerizations are simu-
lated in a straightforward application of the GSSA, often
yielding great insight into the results of experimental
studies. Mechanisms studied by GSSA include nitroxide-
mediated,[93–98] atom transfer radical polymerization
(ATRP) with varying initiator functionality,[99,100] copoly-
merization by ATRP,[84,101] length-dependent termination
rates in ATRP,[102] the cross reaction between dithioester
and alkoxyamine used in reversible addition-fragmenta-
tion chain transfer (RAFT),[103] and RAFT polymerization of
methyl acrylate mediated by cumyldithiobenzoate.[104]
ts) with the average value calculated by the Galerkin method aloneat early reaction times, i.e., 60 s (left) and at late reaction times, i.e.,verages of the stochastic hybrid results (thick lines). Reprinted with
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Figure 5. Weight fraction distributions for free-radical polymeri-zation (dotted lines) modeled by the GSSA at conversions, fromtop to bottom, of 9.5, 29.5, and 69.4%, and living radical polymeri-zation (solid lines) at conversions, from left to right, of 6.2, 24.9,and 49.8%, respectively. Reprinted with permission from ref.[94]
Figure 6.Weight fraction distribution evolution with time for GSSA siATRP of styrene (lines). Reprinted with permission from ref.[99]
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Because the activation/deactivation processes involved in
reversible termination type controlled polymerizations are
typically much faster than the other reaction channels,
these processes occur predominantly and can increase
computation time significantly relative to free-radical
polymerization. He et al.[94] have circumvented this
limitation by incorporating an analytical expression for
the equilibriumbetweenactive anddormant species,while
treating the other reaction pathways stochastically.
Figure 5 compares results for free-radical polymerization
and living radical polymerization, both modeled by GSSA.
Figure 6 depicts the GSSA results of a controlled radical
polymerization and compares them with experimental
data for ATRP of styrene.
3.1.2. GSSA Approaches for Diffusion Limited
Polymerization
The approaches discussed so far have dealt with polymer-
izations in solutions or bulk, or in systems with a
continuous distribution of species. Traditionally the GSSA
cannot describe polymerization at interfaces and in
mulation of living radical polymerization (squares) and experimental
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Figure 7. Experimental data for free-radical polymerization ofmethyl methacrylate (points) compared with the output fromGSSA featuring volume restricted according to diffusion length toaccount for imperfect mixing (lines). The top panel uses diffusionparameters from the literature, while the bottom panel adjustsdiffusion parameters for a better fit. Reprinted with permissionfrom ref.[107]
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E. D. Bain, S. Turgman-Cohen, J. Genzer
confined geometries because it assumes that all reactants
are small molecules in a perfectlymixed volume. However,
several polymerization systems have been studied by
GSSA that take account of diffusion limitation, including
imperfectly mixed bulk polymerization, emulsion poly-
merizations, branched polymerizations, and polymeriza-
tions in biological cells. The simplest means of accounting
for diffusion limitation is by allowing reaction rate
constants to vary with parameters that directly affect
diffusion, such as chain length. For example, rate constants
of chain-end extension reactions have been treated as a
function of chain length,[66,68] and termination rate
constants have been calculated as a function of monomer
conversion.[105] Alternatively, diffusion limitations in
free radical polymerization have been accounted for by
limiting radicals to small volumes or ‘‘microreactors,’’ and
using a chain length-dependent termination rate constant
based on the Smoluchowski equation, which accounts for
macroradical diffusion.[106] A similar approach calculates
the reaction within a ‘‘perfectly mixed volume’’ chosen on
the basis of a diffusion coefficient calculated from free-
volume theory.[107] Figure 7 compares a GSSA simulation
of free-radical polymerization restricted to the perfectly
mixed volume with experimental data.
The limited volume approaches mentioned above are
physically and mathematically very similar to emulsion
polymerization, another diffusion-limited case that has
been modeled by GSSA. Tobita assumes steady state
between entry and desorption of radicals in emulsion
droplets, using either empirical relations,[108] or the more
complex Smith-Ewart equations[109] to estimate the
average number of radicals per particle. In another study,
capture of oligoradicals by micelles is diffusion limited
according to the Smoluchowski equation.[110] Radical
desorption from particles is also considered to be diffu-
sion-limited. For the microemulsion copolymerization of
hexyl methacrylate and styrene in microemulsion[111] rate
constants for radical entryanddesorptionweredetermined
by iteration tofit the experimental data. Forpolymerization
of acrylamide in inverse emulsion[112] diffusion limitations
were neglected altogether by assuming that mass transfer
of monomer to micelles is much faster than propagation,
and the effect of radical desorption on molecular weight
is negligible. Figure 8 depicts the processes considered in
a typical model for emulsion polymerization. A recent
overview covers several multiscale approaches, including
GSSA and others, for interfacial diffusion in phase-
separated polymerizations.[113]
Branched and network polymerizations contain spatial
effects similar to those found in confined and surface-
grafted polymerizations (cf. Figure 1). Besides diffusion
limitations, which become important with increasing
degree of branching, the complex topology can create
confinement-like effects due to chain crowding. TheGSSA is
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able to account for the precise distribution of branching
points using lists, in an analogous manner to accounting
for the distributions of polymer molecular weight and
copolymer sequence mentioned above. An early study[114]
used a GSSA-like approach to model cross-linking poly-
merizationwith a full description of the network structure.
Since diffusion limitation was not considered, a gel point
was determined by a simple cutoff above a fixed number
of branching generations. Another study[115] took into
account not only the full network structure, but also
diffusion dependent rates of propagation, termination, and
radical efficiency factor. Length-dependent polymer diffu-
sion coefficientswere calculated based onVrentas–Vrentas
theory of polymer diffusivity. In principle, one can use the
topological history obtained from a GSSA simulation of
branching polymerization to model the spatial behavior
of the polymer system. Meimaroglou and Kiparissides
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Figure 8. Reactions considered in GSSAmodel of inverse emulsionpolymerization of acrylamide. Reprinted with permission fromref.[112]
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developed a GSSA-based algorithm[116] that considers
various diffusion limited phenomena according to pre-
viously published methods,[117] and models completely
branching structure via a topology array separately from
the chain length array. The researchers then used a random
walk to simulatea3Dmodelof thechain structure, basedon
the stochastically generated topology. Figure 9 provides an
overview of a system to account exactly for topology.
Cross-linking polymerization has also been modeled
using a lattice-basedmodification of GSSA.[118,119] To adopt
GSSA to a lattice simulation, the probability of reaction for
each radical (originating from initiators placed at random
sites on the lattice)was calculated according to the number
of nearest-neighbor unreacted groups for each radical.
A radical was then selected stochastically according to
this weighted probability distribution. Following this step
another random number was generated and used to select
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which neighboring functional group the radical will react
with, again based on the weighted distribution of reaction
probabilities. Diffusionwas not considered in these studies
except through radical propagation; however, the spatial
distribution of polymerizing groups was tracked explicitly
by calculating pair correlation functions for reacted,
unreacted, and branched monomers. Radical trapping
and cyclization were quantified in real space and time
for a variety of conditions relevant to photoinitiated
free-radical polymerization. Figure 10 shows results of
the lattice-based GSSA model for free-radical network
polymerization.
3.1.3. GSSA Approaches for Polymerizations in
Confinement and Spatially Varying Systems
Besides diffusion limitation, an equally important effect in
surface-grafted polymerizations is confinement due to
increased crowding at high grafting density. To the best of
our knowledge this phenomenon has not been addressed
adequately for confined polymerization using GSSA. An
ideal model would take explicit account of local variations
in reactant concentrations, as well as the direction and
rates of diffusion. In recent years, the use of GSSA with
the so-called RDME[120] has been gaining in popularity
for stochastically simulating spatially inhomogeneous
systems. We submit that stochastic simulation of the
RDME is an excellent candidate for application to the
growing fields of polymerizations from surfaces, in
confined geometry, and other spatially varying systems.
In a typical procedure for a reaction–diffusion simulation,
GSSA is used to simulate reactions within each of a
number of small, correlated sub-volumes or elements,
each of which is assumed to possess a homogeneous
distribution of reactants. Diffusion ismodeled by consider-
ing discrete jumps between neighboring elements, with
each jump treated as a kinetic event associated with a rate
constant k¼D/l2, where l is the length scale of a sub-
volume. In this way the spatial distribution of reactants
within amesoscale volume can be simulated bymeans of a
matrix of smaller homogeneous sub-volumes. Figure 11
illustrates schematically this discretization for a simple
one-dimensional space. Often RDMEmethods based on the
GSSA are used to simulate spatial behavior of nonlinear
chemically reacting systems suchas theBrusselator.[121,122]
However, themethodsmayalsobewell-suited for applying
the strengths of GSSA to polymerizations in confined
geometry and grafted at interfaces, because of their
ability to estimate the effects of diffusion limitations and
confinement, while still outperforming molecular simula-
tions in terms of computational efficiency.
Before the RDME was simulated using GSSA, it was
solved analytically[120] or with a stochastic Langevin
equation.[121] As is the case for homogeneous systems,
GSSA is by far the most practical method for commonly
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Figure 9. Chain transfer reaction between branched polymers with topology modeled exactly using GSSA. Reprinted with permission fromref.[116]
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E. D. Bain, S. Turgman-Cohen, J. Genzer
studied systems.[123] The validity of GSSA for simulating
spatially inhomogeneoussystemswastestedbycomparing
analytical, numerical, and stochastic (GSSA) solutions of
RDME against microscopic MC simulations for non-
equilibrium reacting systems.[124] It was found that
element size should be on the order of the mean free path
Figure 10. Two-dimensional lattice-based GSSA simulations of crosFunctional group conversion is increased from left to right, (1) 10%, (constants are 0.1 s�1 in row (a) and 10 s�1 in row (b). Each color represenwith permission from ref.[119]
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between reacting molecules, in order to obtain results
in agreement with the molecular simulations. GSSA
simulations of RDME were also compared against MD
simulation.[125] GSSA was able to reproduce the results of
MD simulation for a bistable reacting system, provided
diffusion was sufficiently fast to ‘‘smooth out’’ local
s-linking free-radical polymerization with difunctional monomers.2) 20%, (3) 31%, (4) 50%, and (5) 75% conversion. The initiation ratets a separate kinetic chain produced by a single free radical. Reprinted
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Figure 11. Discretization of one-dimensional space into sub-volumes for analysis by RDME. Solid arrows represent allowedjumps. Line graphs are shown for periodic boundaries and hardboundaries. Reprinted with permission from ref.[129]
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concentration fluctuations The next sub-volume method
(NSM)[126] is an optimized application of GSSA to reaction–
diffusion systems, allowing for faster calculation by
hierarchically sorting the cells for diffusion. Figure 12
depicts results of the NSM RDME GSSA for a bistable
reaction–diffusion system in three-dimensional space. A
similar method was applied to study chaperone-assisted
protein folding.[127] To speed up computation, hybrid
approaches to RDME have been developed. For example,
reactions in sub-volumes may be treated stochastically
while diffusion is modeled deterministically via finite
volume calculations.[122] An adaptive mesh refinement
algorithm has been devised in which subdivisions of
the system are periodically resized with greater or less
resolution as defined by a refinement criterion based on the
degree of local homogeneity.[128] Other hybrids of GSSA
simulation of RDME include calculating only net diffusion
Figure 12. Results of a three-dimensional bistable reaction–diffusionshows the correlation time of molecules for different system volummolecules with time. Part B shows the time evolution of reactant numcoefficients. Reprinted with permission from ref.[126]
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of species from sub-volumes[129] and incorporation of tau-
leaping and a diffusion propensity function based on
concentration gradients.[130] As algorithm optimizations
combine with continual advances in computation power,
the time is ripe for GSSA simulations of RDME to be applied
to polymerizations at interfaces and in confined geometry.
Some of the above considerations for polymerization in
confined geometryhave been addressed in variousways by
the use of GSSA for modeling biological polymerizations,
such as polymerization of lignin,[131] prion aggrega-
tion,[132,133] viral capsid self-assembly,[134] and origin of
life.[135] In particular, a significant amount of work has
focused on the application ofGSSA tomotility in eukaryotic
cells via polymerization/depolymerization of actin.[136–148]
Since actin filaments polymerize, among other places, in
finger-like projections of a cell’s cytoplasm called filopodia,
they essentially represent polymerizations in a confined
geometry. One has to bear in mind that the comparison
with synthetic polymerizations is not exact since the actin
‘‘monomers’’ themselves are globular proteins with inter-
nal macromolecular structure. Actin filaments are rigid, so
their conformational limitations tend not to be as severe as
that of most flexible polymers in confined space. Diffusion
limitation remains an issue, as are the forces acting on the
filaments from the surrounding cell membrane.[145] Many
factors are relevant in actin filament formation including
nucleotide composition, branching, fragmentation and
annealing, and protein capping.[137–139] GSSA simulations
have accounted for experimentally observed length fluc-
tuation inpropagatingfilamentsduetoacomplex interplay
among different actin monomer states.[140–143]
Nucleation of actin bundles from a surface-bound
network of precursors was modeled using a lattice-based
system simulated by the NSM, an optimized GSSA for RDME. Part Aes and diffusion coefficients. Insets show the number of A and Bbers and positions within the system volume, for different diffusion
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simulation, which models each move between lattice sites
as a stochastic kinetic event.[144] This approach, similar to
the stochastic methods of simulating RDME, illustrates
the advantage of GSSA over traditional lattice-based MC
methods for studying dynamic systems far from equili-
brium. The simulation results are comparable to other
surface-initiated polymerizations as shown in the upper
left portion of Figure 13, but the stiffness of the actin
filaments results in less chain crowding than a typical
polymer brush. A reaction–diffusion approach was used to
split up a filopodium into slices of well-mixed volume in
which filament polymerization could take place.[145] Mass
transfer was considered along the length of the volume,
effectively treating diffusion as hops along a one-dimen-
sional lattice as illustrated in the upper right section of
Figure 13. The same model was extended to include the
effects of capping and anticapping proteins, accounting
for experimentally observed fluctuations in filopodia
length and finite lifetime of filopodia.[146] GSSA was used
to model actin filament growth on the surface of
biomimetic colloidal particles.[147] The results were used
in combination with equilibrium force calculations to
generate a spatial trajectory of actin-propelled colloid
movement. GSSA simulations of actin network polymer-
ization proceeding from an interface found unique
structural patterns resulting from chain crowding and
competition between alternative branching orienta-
tions.[148] The lower portion Figure 13 shows stochastically
simulated two-dimensional actin networks with two
characteristic distributions of branching angle.
3.2. Monte Carlo and Molecular Dynamics Simulation
3.2.1. Monte Carlo Simulation
The MC method in the context of molecular simulation
refers to a technique where the configurational space of a
model is sampled or a system is evolved by generating
random numbers to perform a variety of possible actions.
TheMetropolis algorithm is one suchMCmethod in which
the system changes from one state to another with a set of
probabilities that depend on the change in energy of the
system according to the Boltzmann equation:
PðA ! BÞ ¼ minð1; e�ðEB�EAÞ=kBTÞ; (4)
where Ei is the energy corresponding to the configuration i,
kB is the Boltzmann constant, and T is the absolute
temperature. Since many standard texts describe the MC
method and its implementation in great detail,[149–151] we
just recall briefly a few features. Most MC simulations
are applied to study systems in equilibrium. To this end,
the simulation generates a set of configurations for the
model in question at a specific thermodynamic state. If a
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sufficiently large set of these configurations is generated
and configurational space is sampled appropriately, a
number of ensemble averages and their fluctuations can
be used to compute thermodynamic properties of interest.
Many MC simulations are performed in discretized space
(i.e., on a lattice) although it is also possible to implement
the technique off-lattice. Due to limited computing
resources, it is often necessary to investigate a small
model and use periodic boundary condition (PBC) to extend
the system size to macroscopic scales. The small system
size and use of PBCs sometimes result in ‘‘finite size effect’’
in which the computed averages diverge from the value
obtained if a truly macroscopic system was simulated.
Equilibrium polymerization (EP)[152–154] has previously
been studied by MC simulations. In EP, a set of living
polymers is in equilibrium with a solution of monomers.
The polymer undergoes polymerization and depolymeriza-
tionreactionsandreachesanequilibriummolecularweight
distribution. The equilibrium properties of these systems
depend on temperature, pressure, composition, and the
interactions present in the system (say among monomers
and between monomers and solvent.)[152] One example of
an EP is the polymerization/depolymerization of actin
filaments in eukaryotic cells, which has been studied using
GSSA as described above.
The investigation of EP by means of MC simulations
requires mechanisms by which to move monomers and
polymers and by which monomers and polymers can
polymerize and depolymerize. This is achieved by setting
the probabilities for the various possible reactions. For
example, if the end of a propagating polymer encounters a
free monomer and is within a pre-set reactive distance, a
random number will be generated and the reaction will
occurwith a certain probability. Alternatively, the energies
of the system before and after bond formation/breakage
may be used along with Equation (4) to determine if the
reaction step is accepted. In such a way, the system can
evolve dynamically into an equilibrium state which can be
characterized byensemble averaging.MCsimulationshave
been employed to investigate EPs in solution and in
the melt,[155,156] including the MWD at equilibrium[157]
(Figure 14). The properties of EPs within two impenetrable,
repulsive plates in equilibrium with bulk polymers were
studied by MC.[158,159] It was found, for example, that
the equilibrium molecular weight depended on the
distance between the plates and the overall monomer
density of the system. An off-latticeMC algorithmwas also
used to study EPs in systems tethered to an impenetrable
surface[160] (Figure 15). The simulations showed, for
example, that the MWD of the grafted polymers possess
slowerdecayinghighmolecularweight tails than theirbulk
counterparts. This was due to the development of a free
monomer concentration gradient that favored the growth
of longer chains. Other properties, such as polymer and
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Figure 13. (upper left) Result from lattice-based GSSA showing self-assembly of bundles from a mixture of actin, fascin, and Arp2/3 at thesurface of a bead coatedwithWiskott–Aldrich syndrome protein. Regions (b) and (c) correspond to the lower right hand and upper left handboxes in (a), respectively. Reprinted with permission from ref. [144] (upper right) Polymerization of actin filaments in a filopodium of lengthhn, modeled as a series of discrete subvolumes with reaction and diffusion simulated by GSSA. Reprinted with permission from ref.[145]
(lower) Results of stochastic simulation of actin network formation. Cases A and C demonstrate the þ70/0/�70 degree branching patternillustrated in E, while case B features the� 35 degree branching pattern illustrated in F. The orientation distributions for A-C are shown in D.Reprinted with permission from ref.[148]
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Figure 14. Molecular weight distribution of EP polymers obtainedin Monte Carlo simulation with the bond fluctuation model. Theinset shows an attempt to scale the data according to mean-fieldapproximation. Reprinted with permission from ref.[157]
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E. D. Bain, S. Turgman-Cohen, J. Genzer
monomer concentration profiles and the sizes of the
polymers, were also evaluated.
A similar framework to that of MC simulations of EPs
was used to investigate systems away from equilibrium,
such as irreversible free radical polymerization. One
such example is that of kinetic gelation (KG) in which
bifunctional monomers and polyfunctional cross-linkers
are allowed to react until an ‘‘infinite’’ gel is formed[161–163]
(Figure 16). EarlyMC simulations of KG consisted of bi- and
tetra-functional monomers that reacted randomly on a
lattice. In theseearlymodels thesimulationcontinueduntil
Figure 15. Schematic of the EP investigated by Milchev et al. In EPthe polymers and the free monomers reach thermodynamicequilibrium. Reprinted with permission from ref.[160]
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no more reactions were possible or the gel transition
was reached. The original KG models included no solvents
and no monomer or polymer motions but refinements
throughout the years have incorporated these effects into
the model.[164–169]
The methods used to study KG and EP can be modified
to study controlled radical polymerization,[13,170] which
is the most widely used polymerization technique to
synthesize polymer grafts. Bulk- and surface-initiated
polymerizations were simulated with a MC algorithm
in which the equilibrium between active (propagating)
polymers and inactive (dormant) polymers were included
in an approximate way. Both bulk and surface-initiated
polymers were investigated and the effect of the lifetime
and fraction of living polymers on the broadness of the
MWD was determined.[170] It was observed that the MWD
of surface-initiated polymers was broader than for bulk
initiation due to an early onset of excessive termination
reactions, an effect whichwas enhanced at higher grafting
density of initiators on the surface. Later investigations
probed truly living systems, in which terminations were
excluded.[13] Even without terminations the surface-
initiated polymers had broader MWDs than bulk-initiated
counterparts (Figure 17). Thus even in the absence of
termination reaction, the gradient in monomer concentra-
tion favors the growth of longer polymer chains (similar to
the effect for EP brushes) and results in broader MWDs.
Investigations of similar systems in which bulk and
surface polymers were grown simultaneously allowed
determination of the validity range of the assumption that
these simultaneously grown polymers have equal average
molecular weights and MWDs.[171,172]
3.2.2. Molecular Dynamic Simulation
In MD a model of the chemical entities of interest is
investigated by computing the forces that the particles in
the system exert on each other. The computed forces allow
the numerical solution of Newton’s equations and the
propagation of the system forward in time. A number of
standard texts detail the implementation and theory
behind the MD technique.[149,150,173] In its basic form MD
performs a simulation with the number of particles (N),
volume (V), and energy (E) constant (i.e., NVE ensemble) but
it can be adapted to other ensembles with the aid of a
thermostat, barostat, and/or random particle insertion/
deletions.
To extend MD to longer time- and length-scales, the
method of dissipative particle dynamics, in which dis-
sipative and random pairwise forces are added to the
typicalMD simulation, has been developed.[174] In DPD, the
molecular details of the system are coarse-grained, result-
ing in microscopic particles that represent a fluid element
instead of an atom in a molecule. If one chooses the
conservative, random, and dissipative forces carefully,[175]
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Figure 16. Snapshot of an early kinetic gelation simulation. Dots represent bifunctionalmonomers and circled dots represent tetrafunctional cross-linkers. The solid linesrepresent formed bond and the stars represent active centers. Reprinted with per-mission from ref.[162]
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hydrodynamics effects may be studiedwith the technique,
something that is not possible with MC or MD.
Several problems related to reactive polymer systems
havebeen investigatedwithMDandDPD.Toachieve this, it
is necessary–-just as with the MC method–-to include a
mechanism by which the particles may react to form
polymers (i.e., propagate). In most cases this is accom-
plished by identifying particles within a pre-specified
distance to reactive ends and using MC-style probabilities
to determine if a reaction occurs. Reactive MD has been
applied to the study of irreversible polymerization in two
and threedimensions.[176] Themotivation for these studies,
apart from understanding the polymerization process,
was to devise new methods to generate initial configura-
tions for non-reactive MD simulations. The polymers were
modeled by a bead-spring model and information on
the dimensions and MWD of the polymers was obtained.
A similar study was performed with a coarse grained
model of polystyrene[177] (Figure 18). Besides modeling a
realistic polymer, the latter study also demonstrated the
potential of the technique to simulate polymers growing
in spatially heterogeneous environments by localizing the
initiators within a small portion of the simulation cell
(Figure 19).
AswithMC, amodel akin to KGhas been investigated by
an event-driven MD simulation[178] (Figure 20). Two types
of hard-particles shaped as prolate spheroids were simu-
lated. Each particle was either bi-functional or penta-
functional. Reactions occurred when any of these reactive
patches approached one another within a pre-specified
distance. One can envision a similar system to model
confinement inwhich largemulti-functional particles with
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arbitrary shapes and curvatures act as
initiators for thepolymerizationreaction.
The synthesis of polymer brushes was
investigated by means of a reactive DPD
simulation.[179] Although the authors did
not include a mechanism by which the
polymers may be active/inactive, they
reported narrow MWDs when very slow
reaction rates were employed. The study
noted that increases in the rate of
polymerization and in the grafting den-
sity of the initiators on flat impenetrable
surfaces resulted in broader MWDs, a
result that is in agreement with the
observations of the MC results described
above.[13,170]
Finally we mention the development
of reactive force field models applicable
in MD simulations at the atomistic
level.[180] Conventional force fields used
in MD simulations have a fixed topology
with their bonds, angles, dihedrals and
other interactions defined before the beginning of the
simulation; they cannot therefore describe reactive sys-
tems. Reactive force fields allow for simulating molecules
that can transition from a bonded state to a dissociated
state continuously, thus allowing for chemical reactions
within the MD simulation. These reactive force fields are
normally parameterized against quantum chemical com-
putations; although they are not as accurate as QC
calculations, they allow for larger reactive systems to be
modeled. To our knowledge, reactive force fields have not
yet been applied to polymerizations and might be a useful
tool to include in future studies of polymerization reaction
from surfaces.
3.2.3. Outlook
Computer simulations have emerged as a powerful tool for
studying polymerization processes over the past few years.
While the majority of work in this area has concentrated
primarily on describing polymerizations that take place in
bulk, only a limited number of studies have been devoted
to address polymerizations under confinement. Most work
published thatpertains to the latter categoryhas concerned
on polymerization in confined spaces (i.e., pores or
‘‘nanoreactors’’). Much more work is needed to shed light
on polymerization reactions involving ‘‘grafting from’’
processes, i.e., those that generate polymeric grafts on
surfaces by initiating the polymer growth from surface-
bound centers. While some progress in this area has
occurred during the past few years, our knowledge
regarding the growth of macromolecular chains under
such conditions is rather limited. The motivation for such
studies is clear and sound. Polymer brushes generated by
im25
Figure 17. Polydispersity index for good (top) and poor (bottom)solvent conditions as a function of monomer conversion for thesimulation of surface-initiated living polymerization. The PDIincreases with increases grafting density of initiators and thedashed lines represent polymerizations in bulk. Reprinted withpermission from ref.[13]
Figure 18. Coarse-grained mapping used in ref.[177] to study thepolymerization of ethylbenzene into polystyrene. A single beadrepresents ethylbenzene while bonded ones represent styreneunits. The tacticity of the polymer depends on the distribution ofR or S beads. Reprinted with permission from ref.[177]
26
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E. D. Bain, S. Turgman-Cohen, J. Genzer
such ‘‘grafting from’’ processes have found application in
many important technological areas, including, lubricants,
anti-fouling layers inbio-adsorption,matrices forattaching
nanoparticles, and other applications. In this article we
haveprovidedasuccinctoverviewofstrategies forapplying
two major computation methodologies, i.e., stochastic
methods and molecular modeling, to polymerization
systems in bulk, under confinement, and grafted at
heterogeneous interfaces.
The GSSA has been employed routinely as a powerful
method formodelingbulkpolymerizationsdue to its ability
to model virtually any set of reaction pathways without
need for simplifying assumptions, and its ability to track
the distribution of molecular weight and copolymer
sequence in the individual chains. The primary effects of
Macromol. Theory Simu
� 2013 WILEY-VCH Verlag Gmb
confinement on polymerization, especially the reduction of
available chain conformations due to impenetrable walls
or chain crowding, diffusion limitations of polymers and
monomeric species, and resulting concentration gradients,
have been dealt with in varying degrees by the GSSA.
Simulationsofbulkpolymerizationswith imperfectmixing
achieve good fits to experimental data by considering
diffusion limitations at propagating chain ends. A math-
ematically similar approach has described emulsion poly-
merizationswithsignificantmass transferbetweenphases.
For networks and cross-linked polymerizations, the GSSA
keeps track of branching points, information that can be
used in conjunction with other methods to describe
the conformational limitations faced by each polymeric
branch. The GSSA has proven useful for simulations of
biological polymerizations, frequently involving confine-
ment by impenetrable surfaces. Lattice-based GSSA and
stochastic simulations of the RDME enable the application
ofGillespie’smethod tospatial distributionproblemsthat it
could not accommodate in the past, opening a path for
direct application of this method to polymerizations in
confined geometry and at interfaces. For instance, the
RDME and NSN methodologies, reviewed briefly here,
may provide important new insight into ‘‘grafting from’’
methods of synthetic polymerizations.
Molecular simulations have also emerged as an impor-
tant tool to study polymerization initiated from surfaces
and under confinement. Recent efforts applying these tools
have elucidated many details of polymerizations from
surfaces that are impossible to attainwith the current state
of the art experimental techniques. The ability ofmolecular
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Figure 19. Number of reacted initiators for homogeneously dis-tributed initiators (top) and for initiators spatially localized in asmall area of the simulation. NG is the number of simulation timesteps and the initial number of initiators is 80. Reprinted withpermission from ref.[177]
Figure 20. A model of kinetic gelation simulated by the MDtechnique. The two hard ellipsoids of revolution are eitherbifunctional or pentafunctional. Reprinted with permissionfrom ref.[178]
Progress in Computer Simulation of Bulk, Confined, and Surface-initiated . . .
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simulations to track the position and state of individual
chains and monomers during the polymerization enables
probing these reactions in unprecedented detail. Despite
these advances, there is still much information regarding
polymerization systems that may be extracted through
molecular simulations. One key area in need of attention is
the development of a solid theoretical framework inwhich
the reactive MC and MD techniques may rest. Such a
framework may allow the mapping of the heuristic
probabilities currently used to enable reactions in these
simulations to the kinetic rate constants measured in
experimental work. This kind of information may serve to
guide experimentalists in their selections of appropriate
molecular systemsandrecipes toachieveatargetmolecular
weight distribution, grafting densities, or compositions.
Since obtaining information such as the molecular weight
distributionorcomonomersequencedistributionofgrafted
polymers is a technically challenging experimental endea-
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vor, dataobtained frommolecular simulationsmayalso aid
in the development of a sound theory of surface grafted
polymerization. Such a theorymay relate variables like the
grafting density and reaction rate to the final molecular
weight distribution of the polymers on the surface. Just as
we can use kinetics and probabilistic arguments to model
the condensation and addition polymerizations in bulk,
the development of similar models for surface confined
polymerization would pave the way to the rational design
of macromolecular grafts.
In order to fine-tune the properties of polymeric grafts in
the aforementioned applications, it is important that one
has a good understanding of the process that leads to
the formation of such polymeric scaffolds. Experimental
groups, such as ours, are in desperate need to understand
how the conditions of ‘‘grafting from’’ reactions affect
the final characteristics of the macromolecular grafts.
Those conditions include the effect of confinement (due to
different geometry of the substrate, solvent quality, spatial
distribution of the polymerization centers), reaction type,
and others. It is our hope that this article will stimulate
more discussion on this important topic, which will lead
ultimately to new and more refined insights in the field.
Acknowledgements: We thank the National Science Foundation,Office of Naval Research, and Army Research Office for supportingour work in the area of surface-initiated polymerization.
Received: May 16, 2012; Revised: July 24, 2012; Published online:September 19, 2012; DOI: 10.1002/mats.201200030
ul. 2013, 22, 8–30
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E. D. Bain, S. Turgman-Cohen, J. Genzer
Keywords: computer simulation; Gillespie; grafting from; poly-merization; stochastic
[1] G. Litvinenko, Computation Studies of Polymer Kinetics,Wiley-VCH, Weinheim 2010, pp. 93–126.
[2] M. Ramteke, S. K. Gupta, Int. J. Chem. Reactor Eng. 2011, 9, 1.[3] C. Kiparissides, a. Krallis, D. Meimaroglou, P. Pladis, a. Balt-
sas, Chem. Eng. Technol. 2010, 33, 1754.[4] D. S. Achilias, Macromol. Theory Simul. 2007, 16, 319.[5] T. F. McKenna, J. B. P. Soares, Chem. Eng. Sci. 2001, 56, 3931.[6] E. Vivaldo-Lima, P. E. Wood, A. E. Hamielec, A. Penlidis, Ind.
Eng. Chem. Res. 1997, 36, 939.[7] N. Dotson, R. Galvan, R. Laurence, M. Tirrell, Polymerization
Process Modeling, 1st edition, Wiley-VCH, New York 1995,p. 392.
[8] M. Wulkow, Macromol. React. Eng. 2008, 2, 461.[9] P. B. Zetterlund, Polym. Chem. 2011, 2, 534.[10] P. B. Zetterlund, Macromol. Theory Simul. 2009, 19, 11.[11] H. Tobita, Macromol. Theory Simul. 2007, 16, 810.[12] A. J. Crosby, J.-Y. Lee, Polym. Rev. 2007, 47, 217.[13] S. Turgman-Cohen, J. Genzer, Macromolecules 2010, 43,
9567.[14] D. T. Gillespie, J. Comput. Phys. 1976, 22, 403.[15] D. T. Gillespie, J. Phys. Chem. 1977, 81, 2340.[16] M. L. Coote, Encyclopedia Polym. Sci. Technol. 2006, 7, 1.[17] M. D. Miller, A. J. Holder, J. Phys. Chem. A 2010, 114, 10988.[18] M. Borrelli, V. Busico, R. Cipullo, S. Ronca, P. H. M. Budzelaar,
Macromolecules 2002, 35, 2835.[19] K. Mylvaganam, L. C. Zhang, J. Phys. Chem. B 2004, 108,
15009.[20] B. J. Alder, T. E. Wainwright, J. Chem. Phys. 1959, 31, 459.[21] S. W. Smith, C. K. Hall, B. D. Freeman, J. Comput. Phys. 1997,
134, 16.[22] I. Carmesin, K. Kremer, Macromolecules 1988, 21, 2819.[23] D. S. Achilias, C. Kiparissides, J. Macromol. Sci. Part C: Polym.
Rev. 1992, C32, 183.[24] C. H. Bamford, H. Tompa, Trans. Faraday Soc. 1954, 50,
1097.[25] E. Ginsburger, F. Pla, C. Fonteix, S. Hoppe, S. Massebeuf,
P. Hobbes, P. Swaels, Chem. Eng. Sci. 2003, 58, 4493.[26] M.-J. Park, M. T. Dokucu, F. J. Doyle,Macromol. Theory Simul.
2005, 14, 474.[27] W. H. Ray, J. Macromol. Sci. Rev. 1972, C8, 1.[28] T. J. Crowley, K. Y. Choi, Ind. Eng. Chem. Res. 1997, 36,
1419.[29] W. J. Yoon, J. H. Ryu, C. Cheong, K. Y. Choi,Macromol. Theory
Simul. 1998, 7, 327.[30] M. Nele, C. Sayer, C. Pinto, Macromol. Theory Simul. 1999, 8,
199.[31] A. D. Peklak, A. Butte, G. Storti, M. Morbidelli, Macromol.
Symp. 2004, 206, 481.[32] V. Saliakas, C. Chatzidoukas, A. Krallis, D. Meimaroglou,
C. Kiparissides, Macromol. React. Eng. 2007, 1, 119.[33] G. Papavasiliou, F. Teymour, Macromol. Theory Simul. 2003,
12, 543.[34] P. Pladis, C. Kiparissides, Chem. Eng. Sci. 1998, 53, 3315.[35] F. Teymour, J. D. Campbell, Macromolecules 1994, 27, 2460.[36] M.-Q. Chen, C. Hwang, Y.-P. Shih, Comput. Chem. Eng. 1996,
20, 131.[37] P. D. Iedema, M. Wulkow, H. C. J. Hoefsloot, Macromolecules
2000, 33, 7173.
Macromol. Theory Simu
� 2013 WILEY-VCH Verlag Gmb
[38] M. Wulkow, Macromol. Theory Simul. 1996, 5, 393.[39] A. Butte, G. Storti, M. Morbidelli, Macromol. Theory Simul.
2002, 11, 22.[40] S. Kumar, D. Ramkrishna, Chem. Eng. Sci. 1996, 51, 1311.[41] D. Meimaroglou, P. Pladis, A. Baltsas, C. Kiparissides, Chem.
Eng. Sci. 2011, 66, 1685.[42] P. B. Zetterlund, Y. Kagawa,M. Okubo,Macromolecules 2009,
42, 2488.[43] P. B. Zetterlund, M. Okubo, Macromol. Theory Simul. 2009,
18, 277.[44] P. B. Zetterlund, M. Okubo, Macromolecules 2006, 39, 8959.[45] H. Tobita, Macromol. Theory Simul. 2011, 20, 709.[46] H. Tobita, Macromol. Theory Simul. 2011, 20, 179.[47] C. Serra, N. Sary, G. Schlatter, G. Hadziioannou, V. Hessel, Lab
Chip 2005, 5, 966.[48] X. Gao, W. Feng, S. Zhu, H. Sheardown, J. L. Brash,Macromol.
React. Eng. 2010, 4, 235.[49] D. Zhou, X. Gao, W-jun. Wang, S. Zhu, Macromolecules 2012,
45, 1198.[50] J.-B. Kim, W. Huang, M. D. Miller, G. L. Baker, M. L. Bruening,
J. Polym. Sci. Part A 2003, 41, 386.[51] D. T. Gillespie, Annu. Rev. Phys. Chem. 2007, 58, 35.[52] E. V. Albano, Heterogen. Chem. Rev. 1996, 3, 389.[53] A. Chatterjee, D. G. Vlachos, J. Comput. -Aided Mater. Des.
2007, 14, 253.[54] C. C. Battaile, D. J. Srolovitz, Annu. Rev. Mater. Res. 2002, 32,
297.[55] H. P. Breuer, J. Honerkamp, F. Petruccione, Chem. Phys. Lett.
1992, 190, 199.[56] J. Lu, H. Zhang, Y. Yang, Makromol. Chem. -Theory Simul.
1993, 2, 747.[57] J. He, H. Zhang, Y. Yang, Macromol. Theory Simul. 1995, 4,
811.[58] H. Tobita, Macromolecules 1997, 30, 1693.[59] J. Ling, X. Ni, Y. Zhang, Z. Shen, Polymer 2000, 41, 8703.[60] J. Ling, X. Ni, Y. Zhang, Z. Shen, Polym. Int. 2003, 52, 213.[61] J. Even, M. Bertault, J. Chem. Phys. 1999, 110, 1087.[62] M. Haıdopoulo, M. Bertault, J. Even, L. Toupet, Macromol.
Theory Simul. 2000, 9, 257.[63] P. V. Steenberge, J. Vandenbergh, Macromolecules 2011, 44,
8716.[64] Z-hong. Luo, W. Wang, P. L. Su, J. Appl. Polym. Sci. 2008, 110,
3360.[65] Z.-H. Luo, D.-P. Shi, Y. Zhu, J. Appl. Polym. Sci. 2010, 115, 2962.[66] L. T. Yan, B. H. Guo, J. Xu, X. M. Xie, J. Polym. Sci. Part B 2006,
44, 2902.[67] L. Yan, Z. Qian, B. Guo, J. Xu, X. M. Xie, Polymer 2005, 46,
11918.[68] L.-T. Yan, B.-H. Guo, J. Xu, X. M. Xie, Polymer 2006, 47, 3696.[69] L.-T. Yan, J. Xu, Z.-Y. Qian, B.-H. Guo, X. M. Xie, Macromol.
Theory Simul. 2005, 14, 586.[70] I. Chung, Polymer 2000, 41, 5643.[71] J. B. P. Soares, A. E. Hamielec, Macromol. React. Eng. 2007,
1, 53.[72] J. B. P. Soares, A. E. Hamielec, Macromol. React. Eng. 2008,
2, 115.[73] C. C. Hua, F. Y. Hsu, M. G. Chang, C. J. Kan,Macromol. Theory
Simul. 2004, 13, 550.[74] K. Platkowski, K.-H. Reichert, Polymer 1999, 40, 1057.[75] H. Chaffey-Millar, D. Stewart,M.M. T. Chakravarty, G. Keller,
C. Barner-Kowollik, Macromol. Theory Simul. 2007, 16,575.
[76] M. Seeßelberg, M. Thorn, Macromol. Theory Simul. 1994, 3,825.
l. 2013, 22, 8–30
H & Co. KGaA, Weinheim www.MaterialsViews.com
Progress in Computer Simulation of Bulk, Confined, and Surface-initiated . . .
www.mts-journal.de
[77] C. Schutte, M. Wulkow, Macromol. React. Eng. 2010, 4, 562.[78] L. Wang, L. J. Broadbelt, Macromol. Theory Simul. 2011,
20, 54.[79] A. Habibi, E. Vasheghani-Farahani,Macromol. Theory Simul.
2007, 16, 269.[80] J. Ling, W. Chen, Z. Shen, J. Polym. Sci. Part A 2005, 43,
1787.[81] L. Wang, L. J. Broadbelt, Macromolecules 2009, 42, 7961.[82] L. Wang, L. J. Broadbelt, Macromolecules 2009, 42, 8118.[83] A. Cho, L. Broadbelt, Mol. Simul. 2010, 36, 1219.[84] M. Al-Harthi, M. J. Khan, S. H. Abbasi, J. B. P. Soares, Macro-
mol. React. Eng. 2009, 3, 148.[85] L. Wang, L. J. Broadbelt, Macromol. Theory Simul. 2011, 20,
191.[86] A. Krallis, D. Meimaroglou, C. Kiparissides, Chem. Eng. Sci.
2008, 63, 4342.[87] D. Meimaroglou, A. Krallis, V. Saliakas, C. Kiparissides,
Macromolecules 2007, 40, 2224.[88] R. Szymanski, e-Polymers 2009, Art. No. 044, 1.[89] H. Liang, F. Li, X. He, W. Jiang, Eur. Polym. J. 2000, 36,
1613.[90] Z.-H. Luo, J. Li, X.-L. Zhan, X.-B. Yang, J. Chem. Eng. Jpn. 2004,
37, 737.[91] Y. Zhu, L. An, W. Jiang, Macromolecules 2003, 36, 3714.[92] Y. Zhu, R. Zhang, W. Jiang, J. Polym. Sci. Part A 2004, 42,
5714.[93] J. He, L. Li, Y. Yang, Macromolecular Theory and Simulations
2000, 9, 463.[94] J. He, H. Zhang, J. Chen, Y. Yang, Macromolecules 1997, 30,
8010.[95] M. Drache, Macromol. Symp. 2009, 275–276, 52.[96] J. He, H. Zhang, L. Li, C. Li, J. Cao, Y. Yang, Polym. J. 1999, 31,
585.[97] A. S. Cho, L. Wang, E. Dowuona, H. Zhou, S. B. T. Nguyen, L. J.
Broadbelt, J. Appl. Polym. Sci. 2010, 118, 740.[98] L. Wang, L. J. Broadbelt, Macromolecules 2010, 43, 2228.[99] M. Al-Harthi, L. S. Cheng, J. B. P. Soares, L. C. Simon, J. Polym.
Sci. Part A 2007, 45, 2212.[100] M. Al-Harthi, J. B. P. Soares, L. C. Simon,Macromol. React. Eng.
2007, 1, 95.[101] Ma. Al-Harthi, J. K. Masihullah, S. H. Abbasi, J. B. P. Soares,
Macromol. Theory Simul. 2009, 18, 307.[102] M. Najafi, V. Haddadi-Asl, M. Salami-Kalajahi, H. Roughani,
e-Polymers 2009, Art. No. 030, 1.[103] Y. Ao, J. He, X. Han, Y. Liu, X. Wang, D. Fan, J. Xu, Y. Yang,
J. Polym. Sci. Part A 2007, 45, 374.[104] M. Drache, G. Schmidt-Naake, M. Buback, P. Vana, Polymer
2005, 46, 8483.[105] I. M. Maafa, J. B. P. Soares, A. Elkamel, Macromol. React. Eng.
2007, 1, 364.[106] H. Tobita, Macromolecules 1995, 28, 5119.[107] H. F. Hernandez, K. Tauer, Macromol. Symp. 2008, 271, 64.[108] H. Tobita, Y. Takada, M. Nomura, Macromolecules 1994, 27,
3804.[109] H. Tobita, Y. Takada, M. Nomura, J. Polym. Sci. Part A 1995,
33, 441.[110] S. M. GhafelebashiZarand, S. Pourmahdian, F. AfsharTaromi,
E. Jabbari, B. Dabir, Chem. Eng. Technol. 2003, 26, 969.[111] L. Nie, W. Yang, H. Zhang, S. Fu, Polymer 2005, 46, 3175.[112] K. Platkowski, A. Pross, K.-H. Reichert, Polym. Int. 1998, 45,
229.[113] H. F. Hernandez, K. Tauer, Macromol. React. Eng. 2009, 3,
375.[114] H. Tobita, Macromolecules 1993, 26, 836.
www.MaterialsViews.com
Macromol. Theory Sim
� 2013 WILEY-VCH Verlag Gmb
[115] D. L. Kurdikar, J. Somvarsky, K. Dusek, N. A. Peppas, Macro-molecules 1995, 28, 5910.
[116] D. Meimaroglou, C. Kiparissides, Macromolecules 2010, 43,5820.
[117] A. Keramopoulos, C. Kiparissides, Macromolecules 2002, 35,4155.
[118] M. Wen, L. E. Scriven, A. V. McCormick, Macromolecules2003, 36, 4140.
[119] M. Wen, L. E. Scriven, A. V. McCormick, Macromolecules2003, 36, 4151.
[120] C. W. Gardiner, K. J. McNeil, D. F. Walls, I. S. Matheson, J. Stat.Phys. 1976, 14, 307.
[121] S. Chaturvedi, C. W. Gardiner, I. S. Matheson, D. F. Walls,J. Stat. Phys. 1977, 17, 469.
[122] D. Bernstein, Phys. Rev. E 2005, 71, 1.[123] M. Malek-Mansour, J. Houard, Phys. Lett. A 1979, 70, 366.[124] F. Baras, M. Mansour, Phys. Rev. E, Stat. Phys. Plasmas, Fluids,
Relat. Interdisciplinary Top. 1996, 54, 6139.[125] J. Gorecki, A. L. Kawczynski, B. Nowakowski, J. Phys. Chem. A
1999, 103, 3200.[126] J. Elf, M. Ehrenberg, Syst. Biol. 2004, 1, 230.[127] P. Lecca, L. Dematt, Int. J. Biol. Life Sci. 2008, 4, 211.[128] B. Bayati, P. Chatelain, P. Koumoutsakos, J. Comput. Phys.
2011, 230, 13.[129] S. Lampoudi, D. T. Gillespie, L. R. Petzold, J. Chem. Phys. 2009,
130, 094104.[130] W. Koh, K. T. Blackwell, J. Chem. Phys. 2011, 134, 154103.[131] F. R. D. van Parijs, K. Morreel, J. Ralph, W. Boerjan, R. M. H.
Merks, Plant Physiol. 2010, 153, 1332.[132] T. Poschel, N. V. Brilliantov, C. Frommel, Biophys. J. 2003, 85,
3460.[133] R. Rubenstein, P. C. Gray, T. J. Cleland, M. S. Piltch, W. S.
Hlavacek, R. M. Roberts, J. Ambrosiano, J.-I. Kim, Biophys.Chem. 2007, 125, 360.
[134] M. Hemberg, S. N. Yaliraki, M. Barahona, Biophys. J. 2006, 90,3029.
[135] M. Wu, P. G. Higgs, J. Mol. Evol. 2009, 69, 541.[136] J. Son, G. Orkoulas, A. B. Kolomeisky, J. Chem. Phys. 2005, 123,
124902.[137] A. A. Halavatyi, P. V. Nazarov, S. Medves, M. van Troys,
C. Ampe, M. Yatskou, E. Friederich, Biophys. Chem. 2009,140, 24.
[138] H. Y. Kueh,W.M. Brieher, T. J. Mitchison, Biophys. J. 2010, 99,2153.
[139] A. Matzavinos, H. G. Othmer, J. Theor. Biol. 2007, 249, 723.[140] J. Fass, C. Pak, J. Bamburg, A. Mogilner, J. Theor. Biol. 2008,
252, 173.[141] D. Vavylonis, Q. Yang, B. O’Shaughnessy, Proc. Natl. Acad.
Sci. USA 2005, 102, 8543.[142] X. Li, J. Kierfeld, R. Lipowsky, Phys. Rev. Lett. 2009, 103, 1.[143] A. E. Carlsson, Phys. Biol. 2008, 5, 036002.[144] S. Stewman, A. Dinner, Phys. Rev. E 2007, 76, 1.[145] Y. Lan, G. A. Papoian, Biophys. J. 2008, 94, 3839.[146] P. I. Zhuravlev, Ga. Papoian, Proc. Natl. Acad. Sci. USA 2009,
106, 11570.[147] S. Schmidt, J. van der Gucht, P. M. Biesheuvel, R. Weinkamer,
E. Helfer, A. Fery, Eur. Biophys. J. : EBJ 2008, 37, 1361.[148] J. Weichsel, U. S. Schwarz, Proc. Natl. Acad. Sci. USA 2010,
107, 6304.[149] M. P. Allen, D. J. Tildesley, Computer Simulation of Liquids,
1st edition, Oxford University Press, Oxford 1989, p. 385.[150] D. Frenkel, B. Smit, Understanding Molecular Simulation:
From Algorithms to Applications, 2nd edition AcademicPress, San Diego 2002, p. 638.
ul. 2013, 22, 8–30
H & Co. KGaA, Weinheim29
30
www.mts-journal.de
E. D. Bain, S. Turgman-Cohen, J. Genzer
[151] D. P. Landau, K. Binder, A Guide to Monte Carlo Simulationsin Statistical Physics, 2nd edition Cambridge UniversityPress, Cambridge 2005, p. 432.
[152] S. C. Greer, J. Phys. Chem. B 1998, 102, 5413.[153] S. C. Greer, Annu. Rev. Phys. Chem. 2002, 53, 173.[154] J. Dudowicz, K. F. Freed, J. F. Douglas, J. Chem. Phys. 1999, 111,
7116.[155] A. Milchev, D. Landau, Phys. Rev. E 1995, 52, 6431.[156] A. Milchev, D. P. Landau, J. Chem. Phys. 1996, 104, 9161.[157] Y. Rouault, A. Milchev, Phys. Rev. E 1995, 51, 5905.[158] Y. Rouault, A. Milchev, Macromol. Theory Simul. 1997, 6,
1177.[159] A. Milchev, Eur. Phys. J. E 2002, 8, 531.[160] A. Milchev, J. P. Wittmer, D. P. Landau, J. Chem. Phys. 2000,
112, 1606.[161] P. Manneville, L. de Seze, Numer. Methods Study Crit.
Phenom. 1981, 9, 116.[162] H. Herrmann, D. Landau, D. Stauffer, Phys. Rev. Lett. 1982,
49, 412.[163] R. Bansil, H. J. Herrmann, D. Stauffer, Macromolecules 1984,
17, 998.[164] C. N. Bowman, N. A. Peppas, J. Polym. Sci. Part A 1991, 29,
1575.[165] C. N. Bowman, N. A. Peppas, Chem. Eng. Sci. 1992, 47, 1411.[166] K. S. Anseth, C. N. Bowman, Chem. Eng. Sci. 1994, 49, 2207.
Macromol. Theory Simu
� 2013 WILEY-VCH Verlag Gmb
[167] J. H. Ward, N. a. Peppas , Macromolecules 2000, 33, 5137.[168] M. Nosaka, M. Takasu, K. Katoh, J. Chem. Phys. 2001, 115,
11333.[169] A. K. Poshusta, C. N. Bowman, K. S. Anseth, J. Biomater. Sci.
Polym. Ed. 2002, 13, 797.[170] J. Genzer, Macromolecules 2006, 39, 7157.[171] S. Turgman-Cohen, J. Genzer, J. Am. Chem. Soc. 2011, 133,
17567.[172] S. Turgman-Cohen, J. Genzer, Macromolecules 2012, 45, 2128.[173] D. Rapaport, The Art of Molecular Dynamics Simulation,
2nd edition Cambridge University Press, Cambridge 2004,p. 549.
[174] P. J. Hoogerbrugge, J. M. V. A. Koelman, Europhys. Lett. (EPL)1992, 19, 155.
[175] P. Espanol, P. Warren, Europhys. Lett. (EPL) 1995, 30, 191.[176] R. L. C. Akkermans, S. Toxvaerd, W. J. Briels, J. Chem. Phys.
1998, 109, 2929.[177] K. Farah, H. A. Karimi-Varzaneh, F. Muller-Plathe, M. C.
Bohm, J. Phys. Chem. B 2010, 114, 13656.[178] S. Corezzi, C. DeMichele, E. Zaccarelli, D. Fioretto, F. Sciortino,
Soft Matter 2008, 4, 1173.[179] H. Liu, M. Li, Z.-Y. Z. Lu, Z.-G. Z. Zhang, C.-C. Sun, Macro-
molecules 2009, 42, 2863.[180] A. C. T. van Duin, S. Dasgupta, F. Lorant, W. A. Goddard,
J. Phys. Chem. A 2001, 105, 9396.
l. 2013, 22, 8–30
H & Co. KGaA, Weinheim www.MaterialsViews.com