Swelling of two-dimensional polymer rings by trapped particles
School of Chemistry, Raymond & Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Tel Aviv, Israel
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Accepted: 8 September 2006
Published online: 9 October 2006
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 ∼ 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〉 ∼ N, to a smooth state with 〈A〉 ∼ 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 a 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〉 ∼ 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〉 ∼ 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.
PACS: 36.20.Ey Macromolecules and polymer molecules: Conformation (statistics and dynamics) – / 64.60.Cn Order-disorder transformations; statistical mechanics of model systems – / 61.25.Hq Macromolecular and polymer solutions; polymer melts; swelling –
© EDP Sciences, Società Italiana di Fisica and Springer-Verlag, 2006