# 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.

## Key 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.