Swelling of two-dimensional polymer rings by trapped particles

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.