Capturing Polymer Network Statistics using Branching Random Walks*

Recent studies have established a connection between the macroscopic mechanical response of polymeric materials and the statistics of the shortest path (SP) length between distant nodes in the polymer network. Since these statistics can be costly to compute and difficult to study theoretically, we introduce a branching random walk (BRW) model to describe the shortest path statistics from the coarse-grained molecular dynamics (CGMD) simulations of polymer networks. We postulate that the first passage time (FPT) of the BRW to a given termination site can be used to approximate the statistics of the SP between distant nodes in the polymer network. 

We develop a theoretical framework for studying the FPT of spatial branching processes and obtain an analytical expression for estimating the FPT distribution as a function of the cross-link density. 

We demonstrate by extensive numerical calculations that the distribution of the FPT of the BRW model agrees well with the SP distribution from the CGMD simulations. The theoretical estimate and the corresponding numerical implementations of BRW provide an efficient way of approximating the SP distribution in a polymer network.  

Our results have the physical meaning that by accounting for the realistic topology of polymer networks, extensive bond-breaking is expected to occur at a much smaller stretch than that expected from idealized models assuming periodic network structures.

Our work presents the first analysis of polymer networks as a BRW and sets the framework for developing a generalizable spatial branching model for studying the macroscopic evolution of polymeric systems.