Smoothening Transition of a Two-Dimensional Pressurized Polymer Ring

TL;DR

This paper investigates the phase transition of a two-dimensional pressurized polymer ring, revealing a second-order transition from crumpled to smooth states with a new critical scaling at the transition point.

Contribution

It introduces a comprehensive analysis combining theoretical and simulation methods to characterize the phase transition and critical behavior of pressurized polymer rings.

Findings

Identifies a critical pressure scaling as p_c ~ N^{-1}.

Shows the transition is second-order and belongs to the mean-field universality class.

Derives exact properties of the smooth state using transfer-matrix methods.

Abstract

We revisit the problem of a two-dimensional polymer ring subject to an inflating pressure differential. The ring is modeled as a freely jointed closed chain of N monomers. Using a Flory argument, mean-field calculation and Monte Carlo simulations, we show that at a critical pressure, $p_c \sim N^{-1}$, the ring undergoes a second-order phase transition from a crumpled, random-walk state, where its mean area scales as $<A> \sim N$, to a smooth state with $<A>\sim N^2$. The transition belongs to the mean-field universality class. At the critical point a new state of polymer statistics is found, in which $<A>\sim N^{3/2}$. For $p>>p_c$ we use a transfer-matrix calculation to derive exact expressions for the properties of the smooth state.