The pinning effect of dilute defects

TL;DR

This paper investigates how small, periodic inhomogeneities in a Bernoulli free boundary problem cause pinning of solutions, with a focus on asymptotic behavior and contact angle hysteresis.

Contribution

It provides an asymptotic expansion for the range of pinned slopes caused by small defects in the Bernoulli free boundary problem, linking to contact angle hysteresis.

Findings

Derived asymptotic expansion for pinned slopes

Analyzed capacity-like pinning effect of a single defect

Linked pinning phenomena to contact angle hysteresis

Abstract

We consider the Bernoulli free boundary problem with ``defects", inhomogeneities in the coefficients of compact support. When the defects are small and arrayed periodically there exist plane-like solutions with a range of large-scale slopes slightly different from the background field value. This is known as pinning. By studying the capacity-like pinning effect of a single defect in the Bernoulli free boundary problem, we can compute the asymptotic expansion of the interval of pinned slopes as the defect size goes to zero for lattice aligned normal directions. Our work is motivated by the issue of contact angle hysteresis in capillary contact lines.