Asymptotic expansions of the contact angle in nonlocal capillarity problems

TL;DR

This paper investigates how fractional nonlocal models of capillarity approximate classical contact angles, revealing asymptotic behaviors as the fractional parameter approaches 1 and 0, with implications for understanding surface interactions.

Contribution

It refines the asymptotic analysis of fractional Young's law, showing how contact angles differ from classical predictions near the limits s→1 and s→0.

Findings

Fractional contact angle is smaller than classical when adhesion coefficient is negative.

Fractional contact angle is larger than classical when adhesion coefficient is positive.

Near s=0, the contact angle depends linearly on the adhesion coefficient.

Abstract

We consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting the family of fractional interaction kernels $|z|^{-n-s}$, with $s\in(0,1)$ and $n$ the dimension of the ambient space. The fractional Young's law (contact angle condition) predicted by these models coincides, in the limit as $s\to 1^-$, with the classical Young's law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for $s$ close to $1$, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient $\sigma$ is negative, and larger if $\sigma$ is positive. In addition, we address the asymptotics of the fractional Young's law in the limit case $s\to 0^+$ of interaction kernels with heavy tails. Interestingly, near $s=0$, the dependence of the contact angle from the relative adhesion coefficient becomes linear.