Asymptotic Behavior of Inflated Lattice Polygons

Abstract

We study the inflated phase of two dimensional lattice polygons with fixed perimeter N and variable area, associating a weight exp[pA − Jb] to a polygon with area A and b bends. For convex and column-convex polygons, we show that A /Amax = 1 − K(J)/p 2 + O(ρ −p ), wherep = pN ≫ 1, and ρ < 1. The constant K(J) is found to be the same for both types of polygons. We argue that self-avoiding polygons should exhibit the same asymptotic behavior. For self-avoiding polygons, our predictions are in good agreement with exact enumeration data for J = 0 and Monte Carlo simulations for J = 0. We also study polygons where self-intersections are allowed, verifying numerically that the asymptotic behavior described above continues to hold.

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

“…This is similar to the case for polygons with fixed perimeter, where the asymptotic expressions for area match up to the second term as well [20]. Introducing overhangs in one direction to convert convex polygons to column-convex polygons does not affect the second term in the expansion of equation (49).…”

“…The shapes of convex and row-convex polygons are obtained by minimizing the free energy at fixed area, generalizing the calculation presented in [20]. The equilibrium shape of convex polygons is invariant under rotations by π/2.…”

“…The angle dependent surface tension σ was computed in [20] using simple combinatorial arguments. For convex polygons σ 1 is given by…”

“…Substituting for dλ 1 /da from equation (20) and using equation (21) and simplifying, we obtain the solution…”

“…In an earlier paper [20], 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, pressurized ring polymers [1], [21]- [24] on a square lattice.…”