Mesoscale Phenomena in Fluid Systems - American Chemical Society

Chapter 19

Grand Canonical Monte Carlo Simulations of Equilibrium Polymers and Networks

Downloaded by COLUMBIA UNIV on July 17, 2012 | http://pubs.acs.org Publication Date: August 17, 2003 | doi: 10.1021/bk-2003-0861.ch019

James T. Kindt Department of Chemistry and Cherry L . Emerson Center for Scientific Computation, Emory University, Atlanta, G A 30322

Grand canonical Monte Carlo ( G C M C ) simulation is a useful technique for mesoscale modeling o f systems of self— assembled aggregates - i.e., structures at equilibrium with each other and with a pool of constituent monomers. A new biased G C M C algorithm has been developed to efficiently handle the growth and equilibration of polydisperse structures in phenomenological simulations. Its efficiency is demonstrated in a preliminary study of the nematic ordering transition in equilibrium polymers. The adaptability and precision o f the GCMC approach is further shown in simulations of self-assembled networks, in which loop formation is demonstrated to suppress the first-order condensation transition for flexible chains, an effect not anticipated by mean-field theory.

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© 2003 American Chemical Society

In Mesoscale Phenomena in Fluid Systems; Case, F., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 2003.


299 The Metropolis Monte Carlo algorithm (1,2) has been widely used for half a century to simulate systems at thermal equilibrium with an external fixedtemperature bath. The extension of the Metropolis algorithm to the grand canonical ensemble, in which the system's composition reflects an equilibrium with a reservoir of particles at fixed chemical potential μ (in addition to thermal equilibrium) was introduced in 1969 by Norman and Filinov (3) and also has become commonplace. Grand canonical Monte Carlo ( G C M C ) is a useful tool for the understanding of the thermodynamics and mesoscale structure of selfassembled systems, particularly those in which thermal fluctuations and aggregate polydispersity play an important role. It has been used extensively in lattice-based simulation, where it can be implemented very efficiently (4-7). Off-lattice models, however, allow a more realistic representation of curvature elasticity and symmetry-breaking transitions. G C M C , and the related Gibbs ensemble Monte Carlo method, are also effective in applications using atomically detailed force-fields (see chapters by Siepmann and McCormick in the present volume); nevertheless, in simulations of phases of self-assembled aggregates on the scale of 100's of nanometers, atomistic simulation is far from practical. This chapter will specifically address off-lattice coarse-grained simulation models. A n advantage of the grand ensemble is that the automatic generation of the density as a function of μ allows easy identification of first-order phase transitions and allows contact to be made with approximate analytic expressions for the free energy. In terms of efficiency, particle exchanges between the system and an infinite reservoir of particles get around the dynamic bottleneck of monomer diffusion between aggregates during equilibration of the aggregate size distribution. On the other hand, like other Monte Carlo moves designed to circumvent rather than reproduce true dynamics, use of this type of move renders essentially impossible the recovery of even qualitative dynamical information from the results. One of the most significant advantage of Monte Carlo methods in phenomenological modeling is the degree of control they afford over the local structure and energetics of self-assembly. Most dynamics-based methods require smoothly varying potentials, which are difficult to develop for a particular morphological outcome (e.g., aggregation into chains, sheets, ribbons, helices, networks, etc.), and even more so for independently adjustable association constants and elastic constants of the structure. In contrast, nondifferentiable (square-well or delta-function) potentials, which are conveniently and efficiently implemented in biased M C simulations, allow association constants and structural properties to be precisely engineered into the potential. This approach might be criticized for being a "top-down" method, in which the simulation results can only confirm presuppositions about the system behavior. It is perhaps better characterized as a "middle-up" approach: it does not address the question of how molecular structure leads to preferred local geometries, free energies of association, and elastic constants, but treats these as inputs to determine the next higher level of structure. (These inputs may come from experiment, from microscopic theory or simulation, or can be treated as a parameters to vary in order to determine whether and how they influence


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mesoscale structure.) As such, it can play a useful role in a multi-scale approach to understanding and predicting mesoscale phenomena. The remainder of this chapter will describe a new and general "polydisperse insertion" (PDI) algorithm for the efficient simulation of systems with a polydisperse distribution of aggregates, followed by examples of the application of G C M C methods to two related problems in self-assembly: the isotropicnematic transition in semiflexible equilibrium polymer systems, and the gasliquid transition in self-assembled networks with 3-fold junctions.


A G C M C Method for Polydisperse, Self-Assembled Systems G C M C on self-assembled systems can quite naturally be performed through directed addition or removal moves that change the size of an aggregate by one (coarse-grained) monomer at a time. There are several motivations, however, for attempting to improve on this approach by performing compound move in which multiple monomers or entire aggregates are added or removed at once. The first is that in a system with average number M of monomers per aggregate, the number of moves necessary to grow or remove an aggregate is of order (M/af with a the number of particles used in each step. Equilibration of the system composition - the number and size distribution of aggregates - becomes slow for systems with large M, but can be improved with an increased step size. Another motivation is that there may be barriers to aggregate growth, as observed in some surfactant systems where spherical micelles coexist with long wormlike micelles, but short cylindrical micelles are nearly absent. (8) For such systems, the ability to bypass unfavorable aggregate growth stages through direct addition or removal of large aggregates will improve equilibration efficiency. Multiple particle insertions are most likely to be useful for models that lack explicit solvent and for semidilute systems, where the influence of nonbonded interactions is important on the level of entire aggregates but weak on level of individual monomers. The configuration-bias Monte Carlo ( C B M C ) approach of Siepmann and Frenkel (9) has been very successful for the insertion of chain molecules, and even branched structures and rings, (10) into crowded environments. A key step of the C B M C method is the generation of multiple possible configurations, from which one is chosen with a Boltzmann-weighted probability; the move's overall acceptance probability is determined by a Rosenbluth weight W (II) which incorporates the weights of all configurations considered. In the "polydisperse insertion" (PDI) method introduced here, a similar process of generation and selection is used; the configurations used are the sub-aggregates generated in the course of a directed sequence of aggregate growth or disassembly moves, and so contain different numbers of monomers. Several algorithms have been developed for the reversible growth of clusters of associating particles through biased Monte Carlo moves. {12-14) For generality, we assume that the aggregate is defined by some connectivity criterion based on each monomer occupying a position within a volume F nd defined with respect to one or more other monomer positions and orientations. b0


301 A given aggregate contains N( unoccupied bonding regions to which monomer addition moves can be directed, and N monomers that can be removed without affecting the integrity of the aggregate. (To preserve reversibility, no PDI move that splits or merges aggregates w i l l be permitted.) For instance, i f the aggregate is constrained to be an unbranched chain that can grow from either end, JVfree = N = 2 for any «-mer with n>l. To show that the use of biased, multi-particle moves will satisfy detailed balance, it is sufficient to show that the resulting ratio of probabilities of adding and removing an Λ-mer equals the ratio o f the probabilities o f adding or removing the same structure through η unbiased single particle moves. The probability of building an w-mer in η successive unbiased additions depends on the product of the probability

OrfP% οΛ>% ο Λ * ϊ

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S *

8 + β

9*>

Figure 1: Schematic illustration of polydisperse insertion Monte Carlo step to give the following ratio of transition probabilities for the η -> η' resizing and its reverse, when the probabilities of choosing an «-mer or « - m e r for resizing are included: J W Jt,^„

l

m

l

N N; a>,{W-co y n

g

n

(N, + l)N;l

Mesoscale Phenomena in Fluid Systems - American Chemical Society

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