Conformations of Randomly Linked Polymers

TL;DR

This paper investigates how random linkages affect polymer conformations, revealing conditions for expansion or collapse, and provides analytical and numerical insights into the size scaling and phase transition behaviors of such polymers.

Contribution

The study introduces a model of randomly linked polymers, analyzes their conformational properties, and establishes scaling laws for collapse transitions, including exact solutions for directed models.

Findings

Ideal chains remain expanded when M << N

Collapse transition relates to percolation in a 1D model

Self-avoiding polymer size is reduced by links, scaling as (N/M)^nu

Abstract

We consider polymers in which M randomly selected pairs of monomers are restricted to be in contact. Analytical arguments and numerical simulations show that an ideal (Gaussian) chain of N monomers remains expanded as long as M<<N; its mean squared end to end distance growing as r^2 ~ M/N. A possible collapse transition (to a region of order unity) is related to percolation in a one dimensional model with long--ranged connections. A directed version of the model is also solved exactly. Based on these results, we conjecture that the typical size of a self-avoiding polymer is reduced by the links to R > (N/M)^(nu). The number of links needed to collapse a polymer in three dimensions thus scales as N^(phi), with (phi) > 0.43.