Polymer-Chain Adsorption Transition at a Cylindrical Boundary

TL;DR

This paper models the adsorption transition of polymer chains near a cylindrical boundary using a reduced random walk approach, revealing exponential decay near criticality and a linear crossover at higher energies.

Contribution

It introduces a solvable model for polymer adsorption on cylindrical boundaries based on a reduced random walk framework.

Findings

Adsorbed monomers fraction vanishes exponentially near critical energy.

Crossover to linear increase in adsorption fraction occurs above a certain energy threshold.

Model captures transition behavior from cylindrical to planar boundary conditions.

Abstract

In a recent letter, a simple method was proposed to generate solvable models that predict the critical properties of statistical systems in hyperspherical geometries. To that end, it was shown how to reduce a random walk in $D$ dimensions to an anisotropic one-dimensional random walk on concentric hyperspheres. Here, I construct such a random walk to model the adsorption-desorption transition of polymer chains growing near an attractive cylindrical boundary such as that of a cell membrane. I find that the fraction of adsorbed monomers on the boundary vanishes exponentially when the adsorption energy decreases towards its critical value. When the adsorption energy rises beyond a certain value above the critical point whose scale is set by the radius of the cell, the adsorption fraction exhibits a crossover to a linear increase characteristic to polymers growing near planar boundaries.