2009

Ph.D

HBNI

This thesis studies the statistical mechanics of pressurized ring polymers. These can be thought of as a simple low-dimensional models for the understanding of vesicle shapes and phase transitions, a classic problem first studied several decades ago in the context of the shapes of red blood cells. The model for the two-dimensional vesicle presents many difficulties for analytic studies, arising principally from the self-avoidance constraint. A related class of models in which the polymer ring is allowed to intersect itself, and the pressure term is conjugate to an algebraic or signed area. The effects of semi-flexibility in the inextensible self-intersecting ring problem is investigated. The flexible chain problem is characterised by a continuous phase transition at a critical value of an appropriately scaled pressure, separating collapsed and inflated regimes of the ring. It is shown that this transition survives for non-zero values of the bending rigidity and an analytic form is obtained for the phase boundary separating the collapsed and inflated phases in the scaled pressure-bending rigidity plane.An analogy with the quantum mechanical problem of an electron moving in a magnetic field applied transverse to the plane of motion, is used to reproduce exact results for the flexible chain. Then incorporated with semi-flexibility in both the continuum and lattice models through scaling arguments, obtains very good agreement with numerics. The numerical data was obtained through the exact enumeration method, which explicitly counts the total number of allowed polygons, and hence the partition function. Also several mean-field approaches to this model are performed. The different mean-field approximations, motivated by physical arguments, model the behaviour of the system in different regimes of parameter space are discussed. The usefulness of these results for more realistic systems lies in the fact that self-intersections are irrelevant in the large pressure limit. The results obtained at large pressures should therefore apply both qualitatively and qualitatively to the more realistic case of a pressurised self-avoiding polymer.