Polymerization Reactions in Molecular Modeling: Simulated Annealing Based Solution of the Traveling Connoisseur Problem

Polymerization Reactions in Molecular Modeling: Simulated Annealing Based Solution of the Traveling Connoisseur Problem

                A long term goal of material science research is the capability to synthesize materials with exactly predetermined and customizable physical properties at low cost.  As this lofty goal has not yet been attained, the industrial demand for niche materials with novel applications continues to be unfilled.  Combinatorial synthesis can provide large libraries of certain materials, yet at great investment of time and resources.  For applications demanding precisely defined material properties, the combinatorial search may be limited to an appropriately chosen set of candidates by first simulating the full spectrum of combinatorics and predicting physical properties.

                For example, the primary obstacle to economically sustainable alcohol based biofuels is the cost of separation.  The large expense of separating water and bioalcohols with conventional distillation may be circumvented by performing pervaporation with a polyacrylate copolymer membrane.  As the membrane must be selected for desired absorbance and diffusion properties of a particular alcohol-water mixture, this problem is sufficiently multivariate to lend itself well to simulation.  Previous work has created molecular models of linear chains and cross-linked polymers by polymerizing a simulated mixture of monomers.  In real copolymers, certain interspecies connections are extremely unlikely and reactions rarely go to completion.  To address more general systems of copolymers, the mutual reactivity of different monomers and the extent of polymerization are considered in an algorithm for polymerizing an atomistic model.  The long term goal of this work is to produce a general algorithm for producing amorphous polymer models with consideration for the specific reaction probability as well as overall and individual species conversion.

                The first step in creating polymer model structures is to simulate a polymerization reaction by creating bonds between unreacted monomers in a simulation box.  Relaxed model structures are obtained by polymerizing in a sequence to achieve minimum total new bond length, since molecules in a fluid state will be at distances greater than equilibrium bond length.  Thus, the problem becomes to propose new bonds meeting the requirements for allowable reactions and extent of reaction while minimizing overall new bond length.  This multivariable optimization problem is a generalization of the classic Traveling Salesman problem, herein denoted the Traveling Connoisseur.  While the traveling salesman must visit N of N cities, the more discriminating travelling connoisseur needs to visit only n ≤ N of N cities and has no additional cost other than the travel cost.  A solution to the Traveling Connoisseur is developed employing simulated annealing, a Monte Carlo optimization technique wherein a random initial solution is proposed and then modified with stochastic changes.  Each change is evaluated according to the Metropolis Criterion, which always accepts changes which lower the cost and sometimes accepts moves which increase the cost.  The probability with which uphill moves are accepted decreases throughout the simulation, until the system configuration finally ceases to change, yielding the desired optimum solution.

                This algorithm, as currently implemented, allows addition type reactions, such as those occurring in polyacrylates, polystyrene, and polyethylene.  The user may allow or deny any interspecies and intraspecies reactions, thereby restricting chemically impossible combinations.  The method has been used to produce model structures of linear homopolymers such as polybutyl acrylate and poly-2-hydroxy ethyl acrylate, copolymers of butyl acrylate and 2-hydroxy ethyl acrylate, as well as cross-linked polyacrylates.  Ongoing work is focused on generating configurations of multiple disjoint chains in tandem.