Localization of Polymers in Random Media: Analogy with Quantum Particles in Disorder

TL;DR

This paper reviews the behavior of polymers in random environments, establishing a quantum analogy, analyzing localization phenomena, and exploring how disorder influences polymer conformations and phase transitions.

Contribution

It introduces a novel mapping between polymer localization in random media and quantum particle localization, and analyzes the effects of disorder on polymer conformations and phase transitions.

Findings

Polymer localization is analogous to quantum particle localization in disordered potentials.

System volume critically influences polymer size and behavior in random media.

A localization-delocalization transition occurs depending on self-avoiding interactions.

Abstract

In this chapter we review the rich behavior of polymer chains embedded in a quenched random environment. We first consider the problem of a Gaussian chain free to move in a random potential with short-ranged correlations. We derive the equilibrium conformation of the chain using a replica variational ansatz, and highlight the crucial role of the system's volume. A mapping is established to that of a quantum particle in a random potential, and the phenomenon of localization is explained in terms of the dominance of localized tail states of the Schr\"odinger equation. We also give a physical interpretation of the 1-step replica-symmetry-breaking solution, and elucidate the connection with the statistics of localized tail states. We proceeded to discuss the more realistic case of a chain embedded in a sea of hard obstacles. Here, we show that the chain size exhibits a rich scaling behavior, which depends critically on the volume of the system. In particular, we show that a medium of hard obstacles can be approximated as a Gaussian random potential only for small system sizes. For larger sizes a completely different scaling behavior emerges. Finally we consider the case of a polymer with self-avoiding (excluded volume) interactions. In this case it is found that when disorder is present, the polymer attains a conformation consisting of blobs connected by straight segments. Using Flory type free energy arguments we analyze the statistics of these conformational shapes, and show the existence of a localization-delocalization transition as a function of the strength of the self-avoiding interaction.