Phase transitions in pressurized semiflexible polymer rings

Abstract

We propose and study a model for the equilibrium statistical mechanics of a pressurized semiflexible polymer ring in two dimensions. The Hamiltonian has a term which couples to the algebraic or signed area of the ring and a term which accounts for bending (semiflexibility). The model allows for self-intersections. Using a combination of Monte Carlo simulations, Flory-type scaling theory, mean-field approximations, and lattice enumeration techniques, we obtain a phase diagram in which collapsed and inflated phases are separated by a continuous transition. The scaling properties of the averaged area as a function of the number of units of the ring are derived. For large pressures, the asymptotic behavior of the area is calculated for both continuum and lattice versions of the model. For small pressures, the area is obtained through a known mapping onto the quantum mechanical problem of an electron moving in a magnetic field. The simulation data agree well with the analytic and mean-field results.

Asymptotic behaviour of convex and column-convex lattice polygons with fixed area and varying perimeter

“…In an earlier paper [18], we determined the mean area of inflated convex and columnconvex polygons of fixed perimeter as a function of the pressure and bending rigidity. This case relates to the problem of two-dimensional vesicles, or equivalently, pressurised ring polymers [1,19,20,21,22] on a square lattice. We showed that in the limit of large pressure, the expression for the average area was the same for convex and columnconvex polygons.…”

Collapse and folding of pressurized rings in two dimensions

“…Different aspects of the LSF closed 2-d random walk problem in the inflated and deflated regime have been discussed in the literature, using a number of analytical and computational techniques (see e.g. [15,16,17,18,19,20]).…”

A Method for Estimating the Location of the Drill-Bit During Horizontal Directional Drilling Using a Giant-Magnetostrictive Vibrator

“…This case relates to the problem of two-dimensional vesicles, or equivalently, pressurized ring polymers [1], [21]- [24] on a square lattice. We showed that in the limit of large pressure, the expression for the average area was the same for convex and column-convex polygons.…”