A criterion for the pinning and depinning of an advancing contact line on a cold substrate

TL;DR

This paper develops a theoretical model to predict the pinning and depinning of an advancing contact line on a cold substrate, linking thermal effects, surface shape, and velocity to experimental observations.

Contribution

It introduces a lubrication theory-based model incorporating thermal effects and a Stefan problem to predict contact line behavior during solidification.

Findings

The model predicts a minimal apparent contact angle at a critical velocity.

Transition from continuous motion to pinning is characterized by this critical velocity.

Qualitative agreement with experimental data is achieved, with potential for quantitative improvements.

Abstract

The influence of solidification on the spreading of liquids is addressed in the situation of an advancing liquid wedge on a cold substrate at $T_p < T_f$, of infinite thermal conductivity, where $T_f$ is the melting temperature. We propose a model derived from lubrication theory of contact-line dynamics, where an equilibrium between capillary pressure and viscous stress is at play, adapted here for the geometry of a quadruple line where the vapour, liquid, solidified liquid and basal substrate meet. The Stefan thermal problem is solved in an intermediate region between molecular and mesoscopic scales, allowing to predict the shape of the solidified liquid surface. The apparent contact angle versus advancing velocity $U$ exhibits a minimal value, which is set as the transition from continuous advancing to pinning. We postulate that this transition corresponds to the experimentally observed critical velocity, dependent on undercooling temperature $T_f-T_p$, below which the liquid is pinned and advances with stick-slip dynamics. The analytical solution of the model shows a qualitatively fair agreement with experimental data. We discuss on the way to get better quantitative agreement, which in particular can be obtained when the mesoscopic cut-off length is made temperature-dependent.