Swelling of two-dimensional polymer rings by trapped particles

TL;DR

This paper investigates how trapped particles influence the swelling and phase transition behavior of two-dimensional polymer rings, revealing ensemble-dependent critical phenomena and exact solutions for Gaussian models.

Contribution

It extends existing models to include trapped particles, demonstrating the disappearance of criticality in fixed particle number ensembles and providing exact and approximate solutions.

Findings

Gaussian model solved exactly showing swelling proportional to NQ

Freely jointed model exhibits a single scaling law for mean area

Criticality persists in fixed chemical potential ensemble

Abstract

The mean area of a two-dimensional Gaussian ring of $N$ monomers is known to diverge when the ring is subject to a critical pressure differential, $p_c \sim N^{-1}$. In a recent publication [Eur. Phys. J. E 19, 461 (2006)] we have shown that for an inextensible freely jointed ring this divergence turns into a second-order transition from a crumpled state, where the mean area scales as $<A> \sim N$, to a smooth state with $<A> \sim N^2$. In the current work we extend these two models to the case where the swelling of the ring is caused by trapped ideal-gas particles. The Gaussian model is solved exactly, and the freely jointed one is treated using a Flory argument, mean-field theory, and Monte Carlo simulations. For fixed number $Q$ of trapped particles the criticality disappears in both models through an unusual mechanism, arising from the absence of an area constraint. In the Gaussian case the ring swells to such a mean area, $<A> \sim NQ$, that the pressure exerted by the particles is at $p_c$ for any $Q$. In the freely jointed model the mean area is such that the particle pressure is always higher than $p_c$, and $<A>$ consequently follows a single scaling law, $<A> \sim N^2 f(Q/N)$, for any $Q$. By contrast, when the particles are in contact with a reservoir of fixed chemical potential, the criticality is retained. Thus, the two ensembles are manifestly inequivalent in these systems.