Thermodynamics and entanglements of walks under stress

Abstract

We use rigorous arguments and Monte Carlo simulations to study the thermodynamics and the topological properties of self-avoiding walks on the cubic lattice subjected to an external force f. The walks are anchored at one or both endpoints to an impenetrable plane at Z = 0 and the force is applied in the Z-direction. If a force is applied to the free endpoint of an anchored walk, then a model of pulled walks is obtained. If the walk is confined to a slab and a force is applied to the top bounding plane, then a model of stretched walks is obtained. For both models we prove the existence of the limiting free energy for any value of the force and we show that, for compressive forces, the thermodynamic properties of the two models differ substantially. For pulled walks we prove the existence of a phase transition that, by numerical simulation, we estimate to be second order and located at f = 0. By using a pattern theorem for large positive forces we show that almost all sufficiently long stretched walks are knotted. We examine the entanglement complexity of stretched and pulled walks; our numerical results show a sharp reduction with increasing pulling and stretching forces. Finally, we also examine models of pulled and stretched loops. We prove the existence of limiting free energies in these models and consider the knot probability numerically as a function of the applied pulling or stretching force.

Adsorbed self-avoiding walks subject to a force

“…We first recall some earlier results for pulled self-avoiding walks (see section 2.3 in reference [17]). It is known that λ(y) is a non-analytic function of y with a singularity at y = y The number of tails from the origin of length n and height of last vertex h is c + n (0, h).…”

Force-induced desorption of 3-star polymers in two dimensions

“…where this inequality follows from the log convexity of λ(y) [13]. If all three arms have vertices in x 2 = 0 the free energy is bounded above by n (a, y) = log µ 2 and the system is in the free phase.…”

“…λ(y) is singular at y = 1 [2,9,10] and the walk is in a ballistic phase when y > 1. Also λ(y) is a convex function of log y [13].…”

Force-induced desorption of 3-star polymers: a self-avoiding walk model

“…When y = 1 (so that there is no force) ψ(a, 1) = κ(a), the free energy of an adsorbing walk [7]. When a = 1 (so that there is no interaction with the surface) ψ(1, y) = λ(y), the free energy of a pulled walk [1,11,12,15]. There exists a critical value of a, a c > 1, such that κ(a) = log µ d when a ≤ a c and κ(a) > log µ d when a > a c , so that κ(a) is singular at a = a c > 1 [7,13,23].…”