Aggregation Kinetics in Simulated Telechelic Polymer Gel

1. The Aggregation Kinetics of a Simulated Telechelic Polymer M. Wilson+, A. Baljon+, A. Rabinovitch* + Department of Physics, San Diego State University, San Diego, CA 92128, USA *Department of Physics, Ben-Gurion University of Negev, Beer-Sheva, 84105, Israel Abstract Micelle Transition Reaction Rates Rate - Size Relation • The system exhibits high temperature fluidity (sol state) with a phase transition to a network structure (gel state) with decreasing temperature. • Characteristic temperatures2: • Onset temperature: 0.6<T<0.625 • Micelle transition: T = 0.51 Rates of reactions define the probability of a specific reaction occurring and are determined by observing the frequency of occurrence within simulations. We look for a relation between size distribution and reaction rates. Can we observe the rates in two different phases (sol - gel)? Yes, the simplification of is enough to reproduce all the characteristics of the distribution due to detail balance, In the gel phase, the preferential aggregate size is located at or . Therefore, We investigate the aggregation kinetics of a simulated telechelic polymer gel. In the hybrid Molecular Dynamics (MD)/ Monte Carlo (MC) algorithm, aggregates of associating end groups form and break according to MC rules, while the position of the polymers in space is dictated by MD. As a result, the aggregate sizes change every time step. In order to describe this aggregation process, we employ a master equation. It defines the change in the number of aggregates of a certain size in terms of reaction rates. These reaction rates indicate the likelihood that two aggregates combine to form a large one or that a large aggregate splits in two smaller parts. The reaction rates are obtained from the simulations at several temperatures of the gel. Our results indicate that the rates are not only temperature dependent, but also a function of the sizes of the aggregates involved in the reaction. Using the measured rates, solutions to the master equation are shown to be stable and in agreement with the aggregate size distribution, as obtained directly from simulation data. Furthermore, we show how variations in these rates give rise to the observed changes in the aggregate distribution that characterizes a gel transition. The reaction rates for several temperatures demonstrate the , , and temperature dependence. , (5) The probability of finding an aggregate of size k for several temperatures. . Kinetic Model The reaction rates identify the location of the preferential aggregate size within the gel state. Conclusions Symmetry inversion near characteristic temperature range corresponds with preferential aggregate size. MD / MC Simulation • Aggregation within the simulation can be described in terms of a master equation. • Below the micelle transition temperature aggregates have a preferred size. • At the preferred size, an inversion of symmetry within occurs. • Using the measured rates, the master equation has a solution consistent with the size distribution. • Due to detailed balance, knowledge of the rates provides all information about the size distribution. • We employ a hybrid molecular dynamic (MD) / Monte Carlo (MC) simulation of telechelic polymers1. • Course-grained bead spring model with periodic boundary conditions • 1000 chains that are 8 beads long • Each bead interacts through a Lennard-Jones potential: • MC process creates and destroys junctions between end groups according to Boltzmann factor • Chains and junctions potential: A depiction of the formation and breaking of aggregates through the creation and destruction of junctions. We introduce an aggregation kinetic model, where the formation and destruction of junctions, resulting in a change in aggregate size, are described in terms of a chemical reaction. A change in the size of an aggregate can occur by the following reactions, where we denote an aggregate of size k by . Solutions to the master equation of the kinetic model are in agreement with size distribution. References The time evolution of master equation using measured rates reproduces the size distribution. An unconstrained minimization where , results in equivalent solution. An eigenvalue analysis of the linearized system indicates a stable solution. A master equation describing the aggregation model can be written as [1] K. Kremer and G. S. Grest, Journal of Chemical Physics, 92, 5057 (1990). [2] Baljon A. R. C., Flynn D. and Krawzsenek D., Journal of Chemical Physics, 126 (2007) 044907.