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

Transcription

1 Review Progress in Computer Simulation of Bulk, Confined, and Surface-initiated Polymerizations Erich D. Bain, Salomon Turgman-Cohen, Jan Genzer* In this article we provide a brief summary of computational techniques applied to investigate polymerization reactions in general, with a focus on systems under confinement and initiated from surfaces. We concentrate on two major classes of techniques, i.e., stochastic methods and molecular modeling. We describe the major principles of the two classes of methodologies and point out their strengths and weaknesses. We review a variety of studies from the literature and conclude with an outlook of these two classes of computer simulation approaches as 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 and other processes and thus enable predicting the properties of the final product. Many reviews and monographs have described approaches facilitating the prediction of the characteristics of the final products, including the time evolution of molecular weight, molecular weight E. D. Bain, S. Turgman-Cohen, J. Genzer Department of Chemical & Bimolecular Engineering, North Carolina State University, Raleigh, North Carolina , USA Jan_Genzer@ncsu.edu S. Turgman-Cohen Present address: School of Chemical Engineering, Cornell University, Ithaca, New York , USA 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- CI TM, [8] that can perform those calculations. While modeling 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 8 wileyonlinelibrary.com DOI: /mats

2 Progress in Computer Simulation of Bulk, Confined, and Surface-initiated... reactions in confined geometries and on surfaces. Because recent reviews have dealt in depth with in silico polymerization in confined spaces, [9 11] we will concentrate primarily on the class of macromolecules prepared by direct 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 socalled 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. 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 limitation may 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, computer models 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, termination, 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, reducing the accessibility of the reaction center 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 Erich D. Bain obtained his B.S. degree in chemical engineering from the University of Alabama in 2005, and his PhD in chemical engineering from North Carolina State University, under the direction of Prof. Jan Genzer, in He is currently a contract research assistant in the Genzer research group, focusing on synthesis and characterization of polymer brushes for surface modification applications. Salomon Turgman-Cohen received his B.S. degree in Chemical Engineering from Purdue University in West Lafayette, Indiana in In 2010 he completed a PhD in Chemical Engineering at North Carolina State University under the guidance of Prof. Jan Genzer and Prof. Peter K. Kilpatrick. He is currently a post-doctoral associate in the group of Prof. Fernando Escobedo at Cornell University and is applying computer simulation techniques to environmental and sustainability problems. Jan Genzer received his Dipl.-Ing. in Chemical & Materials Engineering from the Institute of Chemical Technology in Prague, Czech Republic in 1989 and his Ph.D. in 1996 in Materials Science & Engineering from the University of Pennsylvania. After two post-doctoral stints with Prof. Ed Kramer at Cornell University ( ) and UCSB ( ), Genzer joined the faculty of chemical engineering at the NC State University as an Assistant Professor in fall He is currently the Celanese Professor of Chemical & Biomolecular Engineering at NC State University. His group at NC State University pursues research related to the behavior of polymers at surfaces, interfaces, and in confined geometries. 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 polymerization and chain freedom (i.e., confined vs. free). 3. Computer Simulation Approaches for Polymerization in Bulk and in Confined Geometries Figure 2 depicts the various computational approaches utilized for in silico polymerizations. We divide the relevant modeling approaches into five categories, depending on the 9

3 E. D. Bain, S. Turgman-Cohen, J. Genzer Figure 1. A schematic depicting polymerization reactions under various degree of confinement ranging from the bulk all the way to the onedimensional 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 shown with darker color). Technically, the grafted systems can be considered to 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 of optimal length scales (bottom axis) and the amount of localization information that can be modeled (left axis). length scales probed and on the ease to simulate confined environments: quantum mechanical 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 represent 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 macromolecules mainly due to available computation resources. In order to simulate polymerization reactions of longer macromolecules, one has to give up some 10

4 Progress in Computer Simulation of Bulk, Confined, and Surface-initiated... 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. By means of these techniques, one can obtain a reasonably complete picture of the entire process, including the spatio-temporal development of chain growth and depletion of monomers. In most 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 implemented. For instance, in classical MD methods that may implement Lennard-Jones atomistic potential (or other more complex potentials), the polymerization process can only be monitored for relatively short polymerization 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 MD models 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, can model branched macromolecules, 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 selfavoidance of the chains is achieved. Information about rate constants describing the individual reactions is generally not available in the molecular simulations. While the system size that can be treated with the molecular models is much larger than that in the QM models, 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 of weakness for molecular simulations, and it models rigorously the time dependence of reactions, resulting in relatively accurate predictions of reaction kinetics. The GSSA method considers each reacting species independently, allowing calculation of the distributions of molecular weight, sequence distribution, 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 are made, such as the steady-state approximation 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 full molecular weight distribution, particularly at high conversion. However, 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 polymerizations; among these are the method of moments, [23 27] kinetic lumping, [28,29] orthogonal collocation, [30 32] numerical fractionation, [33 35] Galerkin methods, [36 38] and sectional 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 in more accurate predictions of multivariate distributions with fewer assumptions and greater computational efficiency than comparable deterministic 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 simulations. In both RRE and GSSA approaches one may employ hydrodynamics and heat and mass transfer principles to better predict the properties of macromolecules. 11

5 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 the various methods in terms of a variety of 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 copolymerization, 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 limitations of the various computational approaches mentioned earlier. While the QM methods can provide detailed mechanistic information about the polymerization process, the complexity of the computation and limited computational resources prohibit the study of realistic macroscale polymerization reactions and their chemical evolution. Molecular simulations (MD and MC) 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 descriptions 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 require methods by which the monomers may react to form polymers. These methods often involve probabilities of reaction that are unrelated to real rate constants. Nevertheless, 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 polymerizations in confined spaces and from surfaces 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 Table 1. Attributes of various computational methods in describing general polymerization. Quantum mechanics(qm) Molecular dynamics(md) Monte Carlo(MC) Stochastic simulation algorithm(gssa) Reaction rate equations(rre) System size < nm nm nm 100 nm >100 mm Assumptions Non-relativistic Schrödinger equation, values of fundamental physical constants Ergodic hypothesis, potential energy functions, coarse graining System at equilibrium Homogeneous system volume Deterministic formulation of chemical kinetics, steady state approximation (for analytical solution) Kinetic constants Molecular weight distribution Can predict rate constants from first principles Yes, but computationally limited No No Yes Yes Full Full Full Can estimate by some numerical methods Monomer sequence distribution in copolymers Full Full Full Full Can estimate by some numerical methods Polymerization mechanism Full description Some coarse-graining Some coarse-graining Severe coarse-graining Unavailable 12

6 Progress in Computer Simulation of Bulk, Confined, and Surface-initiated... 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. RREbased models of surface-initiated controlled radical polymerization have been developed by Zhu and coworkers [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 description of such systems, simulation methods capable of 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 simulation methods that have traditionally been employed in describing polymerizations in bulk, i.e., the GSSA originally devised by Gillespie, and the molecular models. We will point out cases relevant to polymerization in confined spaces and on surfaces 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 single deterministic trajectory, and (2) the reaction medium 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 Table 2. Attributes of various computational methods in describing polymerization under confinement. Quantum mechanics (QM) Molecular dynamics (MD) Monte Carlo (MC) Stochastic simulation algorithm (GSSA) Reaction rate equations (RRE) Confinement due to solvent quality Implicit only Implicit or explicit Implicit or explicit Volume restriction, diffusion-dependent rate constants In principle an approximate method should be possible Confinement due to presence of impenetrable walls Short length scales Yes Theoretically possible with reaction diffusion master equation Not known Substrate geometry Short length scales Yes Yes Depends on resolution of subvolumes No Grafting density of chains Requires multiple chains Yes Yes Not known No Monomer spatial distribution Not feasible Yes Yes Depends on resolution of subvolumes No 13

7 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 applications. [51] The direct method of the GSSA involves two basic steps. First, the probability a i (x) of each reaction channel i in system state x is calculated as the product of the molecular rate constant (proportional to the bulk reaction rate constants k i ) 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: X j i¼1 a i ðxþ > r 1 a 0 ðxþ (1) where r 1 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 a i (x) of occurring, that reaction has the highest probability of being chosen by the algorithm. Here a 0 is the sum of probabilities for all reaction channels: a 0 ðxþ ¼ XM i¼1 a i ðxþ (2) The second step in the direct method involves calculating the time interval for the chosen reaction. The time step is calculated as: t ¼ lnðr 2Þ a 0 ðxþ where r 2 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. Furthermore a hybrid method known as tau-leaping saves computation time by approximating the GSSA results (3) 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] The GSSA is less computationally expensive than the MD or MC methods, 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 steadystate 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 systems with large numbers of channels, and systems whose differential equations cannot be solved analytically. Since the GSSA takes account of the stochastic trajectory of real reactions, it is ideally suited to simulating systems with 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 reacting molecule, limited computation resources have typically restricted system sizes to picomoles and below. Nevertheless that often can be considered a large enough sample size to obtain statistically significant results. The GSSA has several advantages for modeling polymerization 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- 14

8 Progress in Computer Simulation of Bulk, Confined, and Surface-initiated... 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 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 k r,n, corresponding to the reaction type r (e.g., propagation, termination, 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 polymerization of the individual chains is solved by creating 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 involves a polymer chain, a polymer is chosenfrom thelist bymeans of a third randomly generated number, and the chain length is modified according to the rules of the chosen reaction channel. Lu et al. [56] were among the first to demonstrate that a 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 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 in radical concentration for a continuously initiated FRP, and weight distribution of the resulting set of chains. GSSA models for FRP that include the effect of chain transfer [57,58] give more nuanced and physically realistic results. The GSSA 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-phenylenevinylene) via sulfinyl precursor route, [63] and polymerization of propylene by single and multi-site Ziegler-Natta catalysts. [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 extensions with bisoxazoline [66 69] and telomerization with 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 to high flow rates, 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, polymer chain lengths may be estimated according to 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, Figure 3. (Left) Time evolution of radical concentration for a continuously initiated free-radical polymerization simulated using Gillespie s stochastic simulation algorithm (GSSA). (Right) Molecular weight distribution of chains produced from the same simulation. Reprinted with permission from ref. [56] 15

9 E. D. Bain, S. Turgman-Cohen, J. Genzer 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] The GSSA has been compared directly against the discrete Galerkin method for calculating the weight distributions of free-radical polymerization. [76] The Galerkin method is employed commonly in commercial models of polymerization and in some cases is able to generate accurate molecular weight distributions in only seconds of computation 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 shown to be quite versatile and can give results that are equally or more accurate than the Galerkin model for a variety of mechanism of polymer formation. A polymerization model based on a hybrid of GSSA and the h-p Galerkin method used in the commercial software package PREDICI TM has been demonstrated. [77] The chain length distribution has been solved deterministically by PREDI- CI TM, while additional properties, i.e., copolymer sequence distribution and branching point distribution, are determined 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 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 penultimate termination models, [79] multiblock copolymerization, [80] and gradient copolymerization. [81 85] The bivariate distribution of copolymer composition and molecular weight can be obtained by combining GSSA with simultaneous property accounting algorithms by means of a twodimensional fixed pivot technique. [86] Sequence distribution can be also tracked in conjunction with long chain branching distribution. [87] Reactivity ratios may be determined from a given sequence distribution using a GSSA model of copolymerization. [88] Studies have been performed on modification of cis-1,4-polybutadiene backbones 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 simulated in a straightforward application of the GSSA, often yielding great insight into the results of experimental studies. Mechanisms studied by GSSA include nitroxidemediated, [93 98] atom transfer radical polymerization (ATRP) with varying initiator functionality, [99,100] copolymerization by ATRP, [84,101] length-dependent termination rates in ATRP, [102] the cross reaction between dithioester and alkoxyamine used in reversible addition-fragmentation chain transfer (RAFT), [103] and RAFT polymerization of methyl acrylate mediated by cumyldithiobenzoate. [104] Figure 4. Comparison of hybrid GSSA-Galerkin algorithm output (points) with the average value calculated by the Galerkin method alone (thin lines) for monomer sequence distribution in a copolymerization at early reaction times, i.e., 60 s (left) and at late reaction times, i.e., 600 s (right). The Galerkin solutions match closely with regression averages of the stochastic hybrid results (thick lines). Reprinted with permission from ref. [77] 16

10 Progress in Computer Simulation of Bulk, Confined, and Surface-initiated... Figure 5. Weight fraction distributions for free-radical polymerization (dotted lines) modeled by the GSSA at conversions, from top to bottom, of 9.5, 29.5, and 69.4%, and living radical polymerization (solid lines) at conversions, from left to right, of 6.2, 24.9, and 49.8%, respectively. Reprinted with permission from ref. [94] 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 equilibrium between active and dormant 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 GSSA Approaches for Diffusion Limited Polymerization The approaches discussed so far have dealt with polymerizations in solutions or bulk, or in systems with a continuous distribution of species. Traditionally the GSSA cannot describe polymerization at interfaces and in Figure 6. Weight fraction distribution evolution with time for GSSA simulation of living radical polymerization (squares) and experimental ATRP of styrene (lines). Reprinted with permission from ref. [99] 17

11 E. D. Bain, S. Turgman-Cohen, J. Genzer confined geometries because it assumes that all reactants are small molecules in a perfectly mixed volume. However, several polymerization systems have been studied by GSSA that take account of diffusion limitation, including imperfectly mixed bulk polymerization, emulsion polymerizations, branched polymerizations, and polymerizations 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 freevolume 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 diffusion-limited. For the microemulsion copolymerization of hexyl methacrylate and styrene in microemulsion [111] rate constants for radical entry and desorption were determined by iteration to fit the experimental data. For polymerization 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 phaseseparated polymerizations. [113] Branched and network polymerizations contain spatial effects similar to those found in confined and surfacegrafted 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. The GSSA is Figure 7. Experimental data for free-radical polymerization of methyl methacrylate (points) compared with the output from GSSA featuring volume restricted according to diffusion length to account for imperfect mixing (lines). The top panel uses diffusion parameters from the literature, while the bottom panel adjusts diffusion parameters for a better fit. Reprinted with permission from ref. [107] 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 polymerization with 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 diffusion coefficients were calculated based on Vrentas 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 18

12 Progress in Computer Simulation of Bulk, Confined, and Surface-initiated... which neighboring functional group the radical will react with, again based on the weighted distribution of reaction probabilities. Diffusion was 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 GSSA Approaches for Polymerizations in Confinement and Spatially Varying Systems Figure 8. Reactions considered in GSSA model of inverse emulsion polymerization of acrylamide. Reprinted with permission from ref. [112] developed a GSSA-based algorithm [116] that considers various diffusion limited phenomena according to previously 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 simulate a 3D model of the chain structure, based on 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-based modification 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 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 is modeled by considering discrete jumps between neighboring elements, with each jump treated as a kinetic event associated with a rate constant k ¼ D/l 2, where l is the length scale of a subvolume. In this way the spatial distribution of reactants within a mesoscale volume can be simulated by means of a matrix of smaller homogeneous sub-volumes. Figure 11 illustrates schematically this discretization for a simple one-dimensional space. Often RDME methods based on the GSSA are used to simulate spatial behavior of nonlinear chemically reacting systems such as the Brusselator. [121,122] However, the methods may also be well-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 simulations 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 19

13 E. D. Bain, S. Turgman-Cohen, J. Genzer Figure 9. Chain transfer reaction between branched polymers with topology modeled exactly using GSSA. Reprinted with permission from ref. [116] studied systems. [123] The validity of GSSA for simulating spatially inhomogeneous systems was tested by comparing analytical, numerical, and stochastic (GSSA) solutions of RDME against microscopic MC simulations for nonequilibrium reacting systems. [124] It was found that element size should be on the order of the mean free path 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 Figure 10. Two-dimensional lattice-based GSSA simulations of cross-linking free-radical polymerization with difunctional monomers. Functional group conversion is increased from left to right, (1) 10%, (2) 20%, (3) 31%, (4) 50%, and (5) 75% conversion. The initiation rate constants are 0.1 s 1 in row (a) and 10 s 1 in row (b). Each color represents a separate kinetic chain produced by a single free radical. Reprinted with permission from ref. [119] 20

14 Progress in Computer Simulation of Bulk, Confined, and Surface-initiated... Figure 11. Discretization of one-dimensional space into subvolumes for analysis by RDME. Solid arrows represent allowed jumps. Line graphs are shown for periodic boundaries and hard boundaries. Reprinted with permission from ref. [129] 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 of species from sub-volumes [129] and incorporation of tauleaping 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 geometry have been addressed in various ways by the use of GSSA for modeling biological polymerizations, such as polymerization of lignin, [131] prion aggregation, [132,133] viral capsid self-assembly, [134] and origin of life. [135] In particular, a significant amount of work has focused on the application of GSSA to motility in eukaryotic cells via polymerization/depolymerization of actin. [ ] 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 internal 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. [ ] GSSA simulations have accounted for experimentally observed length fluctuation in propagating filaments due to a complex interplay among different actin monomer states. [ ] Nucleation of actin bundles from a surface-bound network of precursors was modeled using a lattice-based Figure 12. Results of a three-dimensional bistable reaction diffusion system simulated by the NSM, an optimized GSSA for RDME. Part A shows the correlation time of molecules for different system volumes and diffusion coefficients. Insets show the number of A and B molecules with time. Part B shows the time evolution of reactant numbers and positions within the system volume, for different diffusion coefficients. Reprinted with permission from ref. [126] 21

15 E. D. Bain, S. Turgman-Cohen, J. Genzer 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 equilibrium. 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-dimensional 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 polymerization proceeding from an interface found unique structural patterns resulting from chain crowding and competition between alternative branching orientations. [148] The lower portion Figure 13 shows stochastically simulated two-dimensional actin networks with two characteristic distributions of branching angle Monte Carlo and Molecular Dynamics Simulation 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. The Metropolis algorithm is one such MC method 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 ¼minð1; e ðeb EAÞ=kBT Þ; (4) where E i is the energy corresponding to the configuration i, k B is the Boltzmann constant, and T is the absolute temperature. Since many standard texts describe the MC method and its implementation in great detail, [ ] 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) [ ] 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 depolymerization reactions and reaches an equilibrium molecular weight 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 occur with 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 by ensemble averaging. MC simulations have 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-lattice MC algorithm was 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 slower decaying high molecular weight tails than their bulk 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 22

16 Progress in Computer Simulation of Bulk, Confined, and Surface-initiated... Figure 13. (upper left) Result from lattice-based GSSA showing self-assembly of bundles from a mixture of actin, fascin, and Arp2/3 at the surface of a bead coated with Wiskott Aldrich syndrome protein. Regions (b) and (c) correspond to the lower right hand and upper left hand boxes in (a), respectively. Reprinted with permission from ref. [144] (upper right) Polymerization of actin filaments in a filopodium of length h n, 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 pattern illustrated 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] 23

17 E. D. Bain, S. Turgman-Cohen, J. Genzer Figure 14. Molecular weight distribution of EP polymers obtained in Monte Carlo simulation with the bond fluctuation model. The inset shows an attempt to scale the data according to mean-field approximation. Reprinted with permission from ref. [157] 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 [ ] (Figure 16). Early MC simulations of KG consisted of bi- and tetra-functional monomers that reacted randomly on a lattice. In these early models the simulation continued until Figure 15. Schematic of the EP investigated by Milchev et al. In EP the polymers and the free monomers reach thermodynamic equilibrium. 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. [ ] 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 which was 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 surfaceinitiated polymers had broader MWDs than bulk-initiated counterparts (Figure 17). Thus even in the absence of termination reaction, the gradient in monomer concentration 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] 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 dissipative and random pairwise forces are added to the typical MD simulation, has been developed. [174] In DPD, the molecular details of the system are coarse-grained, resulting 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] 24

18 Progress in Computer Simulation of Bulk, Confined, and Surface-initiated... Figure 16. Snapshot of an early kinetic gelation simulation. Dots represent bifunctional monomers and circled dots represent tetrafunctional cross-linkers. The solid lines represent formed bond and the stars represent active centers. Reprinted with permission from ref. [162] hydrodynamics effects may be studied with the technique, something that is not possible with MC or MD. Several problems related to reactive polymer systems have been investigated with MD and DPD. To achieve 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 accomplished 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 three dimensions. [176] The motivation for these studies, apart from understanding the polymerization process, was to devise new methods to generate initial configurations 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). As with MC, a model akin to KG has been investigated by an event-driven MD simulation [178] (Figure 20). Two types of hard-particles shaped as prolate spheroids were simulated. Each particle was either bi-functional or pentafunctional. 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 in which largemulti-functional particles with arbitrary shapes and curvatures act as initiators for the polymerization reaction. 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 density 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 systems. 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 computations; 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 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 that pertains to the latter category has 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 surfacebound 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 25

19 E. D. Bain, S. Turgman-Cohen, J. Genzer Figure 18. Coarse-grained mapping used in ref. [177] to study the polymerization of ethylbenzene into polystyrene. A single bead represents ethylbenzene while bonded ones represent styrene units. The tacticity of the polymer depends on the distribution of R or S beads. Reprinted with permission from ref. [177] Figure 17. Polydispersity index for good (top) and poor (bottom) solvent conditions as a function of monomer conversion for the simulation of surface-initiated living polymerization. The PDI increases with increases grafting density of initiators and the dashed lines represent polymerizations in bulk. Reprinted with permission from ref. [13] such grafting from processes have found application in many important technological areas, including, lubricants, anti-fouling layers in bio-adsorption, matrices for attaching nanoparticles, and other applications. In this article we have provided a succinct overview of strategies for applying 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 for modeling bulk polymerizations due 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. Simulations of bulk polymerizations with imperfect mixing achieve good fits to experimental data by considering diffusion limitations at propagating chain ends. A mathematically similar approach has described emulsion polymerizations with significant mass transfer between phases. 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 confinement by impenetrable surfaces. Lattice-based GSSA and stochastic simulations of the RDME enable the application of Gillespie s method to spatial distribution problems that 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 important 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 attain with the current state of the art experimental techniques. The ability of molecular 26

20 Progress in Computer Simulation of Bulk, Confined, and Surface-initiated... Figure 20. A model of kinetic gelation simulated by the MD technique. The two hard ellipsoids of revolution are either bifunctional or pentafunctional. Reprinted with permission from ref. [178] Figure 19. Number of reacted initiators for homogeneously distributed initiators (top) and for initiators spatially localized in a small area of the simulation. N G is the number of simulation time steps and the initial number of initiators is 80. Reprinted with permission from ref. [177] 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 in which 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 systems and recipes to achieve a target molecular weight distribution, grafting densities, or compositions. Since obtaining information such as the molecular weight distribution or comonomer sequence distribution of grafted polymers is a technically challenging experimental endeavor, data obtained from molecular simulations may also aid in the development of a sound theory of surface grafted polymerization. Such a theory may 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 supporting our work in the area of surface-initiated polymerization. Received: May 16, 2012; Revised: July 24, 2012; Published online: September 19, 2012; DOI: /mats