Progress in Computer Simulation of Bulk, Confined, and ... · Progress in Computer Simulation of Bulk, Conﬁned, and Surface-initiated Polymerizations Erich D. Bain, Salomon Turgman-Cohen,

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

Macromol. Theory Simul. 2013, 22, 8–30

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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-

Progress in Computer Simulation of Bulk, Confined, and Surface-initiated . . .

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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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Macromol. Theory Sim

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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




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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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

Macromol. Theory Simu


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




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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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



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]




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

ul. 2013, 22, 8–30


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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]



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

l. 2013, 22, 8–30


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

ul. 2013, 22, 8–30


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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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



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

l. 2013, 22, 8–30


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




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

ul. 2013, 22, 8–30


Figure 9. Chain transfer reaction between branched polymers with topology modeled exactly using GSSA. Reprinted with permission fromref.[116]

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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]



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

l. 2013, 22, 8–30


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]




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

ul. 2013, 22, 8–30


22

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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



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

l. 2013, 22, 8–30


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]



� 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim23


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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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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]



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



� 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinhe

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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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



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

l. 2013, 22, 8–30


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]


www.mts-journal.de

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-




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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Keywords: computer simulation; Gillespie; grafting from; poly-merization; stochastic

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