Moving contact line dynamics: from diffuse to sharp interfaces

TL;DR

This paper unifies two scaling laws for slip length in moving contact lines by identifying regimes where diffuse or sharp interface models apply, showing the slip length depends on different variables in each regime.

Contribution

It demonstrates that the slip length scales differently depending on the ratio of microscopic interfacial width to diffusive length, unifying diffuse and sharp interface models.

Findings

Slip length scales as (l_D l)^{1/2} in diffuse regime

Slip length depends only on macroscopic variables in sharp regime

Dynamic contact angle matches Cox's theory with rescaled slip length

Abstract

We reconcile two scaling laws that have been proposed in the literature for the slip length associated with a moving contact line in diffuse interface models, by demonstrating each to apply in a different regime of the ratio of the microscopic interfacial width $l$ and the macroscopic diffusive length $l_D= (M\eta)^{1/2}$, where $\eta$ is the fluid viscosity and $M$ the mobility governing intermolecular diffusion. For small $l_D/l$ we find a diffuse interface regime in which the slip length scales as $\xi \sim(l_Dl)^{1/2}$. For larger $l_D/l>1$ we find a sharp interface regime in which the slip length depends only on the diffusive length, $\xi \sim l_D \sim (M\eta)^{1/2}$, and therefore only on the macroscopic variables $\eta$ and $M$, independent of the microscopic interfacial width $l$. We also give evidence that modifying the microscopic interfacial terms in the model's free energy functional appears to affect the value of the slip length only the diffuse interface regime, consistent with the slip length depending only on macroscopic variables in the sharp interface regime. Finally, we demonstrate the dependence of the dynamic contact angle on the capillary number to be in excellent agreement with the theoretical prediction of \cite{Cox1986}, provided we allow the slip length to be rescaled by a dimensionless prefactor. This prefactor appears to converge to unity in the sharp interface limit, but is smaller in the diffuse interface limit. The excellent agreement of results obtained using three independent numerical methods, across several decades of the relevant dimensionless variables, demonstrates our findings to be free of numerical artifacts.