Configurations of polymers attached to probes

TL;DR

This study investigates how polymers attached to curved probes behave differently depending on the size ratio of the polymer to the probe, revealing anisotropic compliance and a crossover in configuration exponents.

Contribution

It introduces a numerical analysis of polymer behavior on curved probes, highlighting the dependence of compliance and configuration exponents on the size ratio.

Findings

Scaled compliance is anisotropic and large when polymer size is comparable to probe radius.

The configuration exponent $b3$ crosses over from boundary-characteristic to shape-dependent values.

Different probe shapes (spherical vs. parabolic) influence the crossover behavior of polymer configurations.

Abstract

We study polymers attached to spherical (circular) or paraboloidal (parabolic) probes in three (two) dimensions. Both self-avoiding and random walks are examined numerically. The behavior of a polymer of size $R_0$ attached to the tip of a probe with radius of curvature $R$, differs qualitatively for large and small values of the ratio $s=R_0/R$. We demonstrate that the scaled compliance (inverse force constant) $S/R_0^2$, and scaled mean position of the polymer end-point $<x_\perp>/R$ can be expressed as a function of $s$. Scaled compliance is anisotropic, and quite large in the direction parallel to the surface when $R_0\sim R$. The exponent $\gamma$, characterizing the number of polymer configurations, crosses over from a value of $\gamma_1$ - characteristic of a planar boundary - at small $s$ to one reflecting the overall shape of the probe at large $s$. For a spherical probe the crossover is to an unencumbered polymer, while for a parabolic probe we cannot rule out a new exponent.