Swelling of two-dimensional polymer rings by trapped particles

Area of Science:

• Polymer physics
• Statistical mechanics
• Soft matter

Background:

• The mean area of a two-dimensional Gaussian ring diverges at a critical pressure (p c ~ N -1).
• Previous work established a second-order transition for inextensible freely jointed rings from crumpled to smooth states.

Purpose of the Study:

• To investigate the effect of trapped ideal-gas particles on polymer ring swelling and critical behavior.
• To extend existing models (Gaussian and freely jointed rings) to incorporate internal particle pressure.

Main Methods:

• Exact solution for the Gaussian ring model.
• Flory argument, mean-field theory, and Monte Carlo simulations for the freely jointed ring model.
• Analysis of two ensembles: fixed number of trapped particles (Q) and particles in contact with a reservoir of fixed chemical potential.

Main Results:

• Criticality disappears in both models when swelling is driven by a fixed number of trapped particles (Q) due to the absence of an area constraint.
• In the Gaussian model, mean area scales as [A] ~ NQ, maintaining pressure at p c.
• In the freely jointed model, mean area follows [A] ~ N(2) f (Q/N), with particle pressure always exceeding p c.
• Criticality is retained when particles interact with a reservoir of fixed chemical potential.

Conclusions:

• The presence of trapped ideal-gas particles fundamentally alters the pressure-induced phase transitions in polymer rings.
• The ensemble of fixed particle number and the ensemble of fixed chemical potential are shown to be inequivalent for these swelling phenomena.
• The study provides insights into the statistical mechanics of confined polymers and the role of internal pressure.