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 δ t in the simulations, conventional formulae for the stress involve averaging large ± O (1/(δ t ) 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.