Localization of polymers in random media: Analogy with quantum particles in disorder

Abstract

This chapter reviews the rich behavior of polymer chains embedded in a quenched random environment. It first considers the problem of a Gaussian chain free to move in random potential with short-ranged correlations. The equilibrium conformation of the chain is derived using a replica variational ansatz, and the crucial role of the system's volume is highlighted. 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 Schrodinger equation. This is followed by a physical interpretation of the 1-step replica-symmetry-breaking solution, and the elucidation of the connection with the statistics of localized tail states. The chapter also discusses the more realistic case of a chain embedded in a sea of hard obstacles, thereby showing that the chain size exhibits a rich scaling behavior, which depends critically on the volume of the system. 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. The case of a polymer with self-avoiding (excluded volume) interactions is also considered where it is found that when disorder is present, the polymer attains a shape like that of a pearl necklace, consisting of blobs connected by straight segments. Through the use of Flory type free energy arguments, the statistics of these conformational shapes are analyzed, and the existence of a localization-delocalization transition as a function of the strength of the self-avoiding interaction is shown. The numerical results in the chapter reveal that a polymer in a random potential can be viewed as a glassy system. The work described in this chapter is concerned with static (equilibrium) properties of polymers in random media.