Polymer chain in a quenched random medium: slow dynamics and ergodicity breaking

TL;DR

This paper investigates how quenched disorder affects polymer chain dynamics, revealing anomalous diffusion at short times and ergodicity breaking at longer times, with modes freezing at a critical disorder strength independent of solvent quality.

Contribution

It introduces a theoretical framework for understanding slow dynamics and ergodicity breaking in polymers within quenched random media, including explicit calculations of subdiffusive exponents and mode freezing conditions.

Findings

Anomalous diffusion occurs within the Rouse time scale.

Disorder induces a transition to a frozen state for the chain's center of mass.

Rouse modes freeze at a critical disorder strength independent of solvent quality.

Abstract

The Langevin dynamics of a self - interacting chain embedded in a quenched random medium is investigated by making use of the generating functional method and one - loop (Hartree) approximation. We have shown how this intrinsic disorder causes different dynamical regimes. Namely, within the Rouse characteristic time interval the anomalous diffusion shows up. The corresponding subdiffusional dynamical exponents have been explicitly calculated and thoroughly discussed. For the larger time interval the disorder drives the center of mass of the chain to a trap or frozen state provided that the Harris parameter, $(\Delta/b^d) N^{2 - \nu d} \ge 1$, where $\Delta$ is a disorder strength, $b$ is a Kuhnian segment length, $N$ is a chain length and $\nu$ is the Flory exponent. We have derived the general equation for the non - ergodicity function $f(p)$ which characterizes the amplitude of frozen Rouse modes with an index $p = 2\pi j/N$. The numerical solution of this equation has been implemented and shown that the different Rouse modes freeze up at the same critical disorder strength $\Delta_c \sim N^{-\gamma}$ where the exponent $\gamma \approx 0.25$ and does not depend from the solvent quality.