Computer simulations of polymer chain relaxation via Brownian motion

Abstract

Numerical simulations are employed to study the Brownian motion of a bead-rod polymer chain dissolved in a solvent. An investigation is conducted of the relaxation of the stress for an initially straight chain as it begins to coil.For a numerical time step δt in the simulations, conventional formulae for the stress involve averaging large ±O(1/(δt)1/2) contributions over many realizations, in order to yield an O(1) average. An alternative formula for the stress is derived which only contains O(1) contributions, thereby improving the quality of the statistics.For a chain consisting of n rods in a solvent at temperature T, the component of the bulk stress along the initial chain direction arising from tensions in the rods at the initial instant is $k\hat{T}\times n(\frac{1}{3}n^2 + n +\frac{2}{3})$. Thus the bead-rod model yields results very different from other polymer models, such as the entropic spring of Flory (1969), which would assign an infinite stress to a fully aligned chain. For rods of length l and beads of friction factor $\hat{\zeta}$ the stress decays at first on $O(\hat{\zeta}\hat{l}^2/k\hat{T}\times 1/n^2)$ time scales. On longer time scales, this behaviour gives way to a more gradual stress decay, characterized by an $O(k\hat{T}\times n)$ stress following a simple exponential decay with an $O(k\hat{T}/\hat{\zeta}\hat{l}^2\times 1/n^2)$ rate. Matching these two limiting regimes, a power law decay in time t is found with stress $O(k\hat{T}\times n^2\times (k\hat{T}\hat{t}/\hat{\zeta}\hat{l}^2)^{-1/2})$. The dominant physical processes occurring in these separate short, long and intermediate time regimes are identified.