A semi-analytical method to estimate the slip length of spreading cap-shaped droplets using Cox theory

TL;DR

This paper introduces a semi-analytical approach to estimate slip length in spreading cap-shaped droplets by transforming Cox law, enabling slip length estimation from radius evolution data without needing instantaneous contact angle and speed measurements.

Contribution

It develops a method to estimate slip length from radius-time data using Cox theory, bypassing the need for direct contact angle and speed measurements during spreading.

Findings

The method accurately estimates slip length from radius evolution data.

Numerical simulations validate the approach for partially wetting droplets.

The approach facilitates better modeling of moving contact lines in numerical methods.

Abstract

The Cox-Voinov law on dynamic spreading relates the difference between the cubic values of the apparent contact angle (theta) and the equilibrium contact angle to the instantaneous contact line speed (U). Comparing spreading results with this law requires accurate data of theta and U during the entire process. We consider the case when gravitational forces are negligible and transform the general Cox law in a relationship for the temporal evolution of the spreading radius. For cap-shaped droplets, this enables a comparison of experimental or computational results with Cox theory without the need for instantaneous data of theta and U. The fitting of Cox theory against measured or computed base-radius-over-time curves allows estimating the effective slip length. This is useful for establishing relationships between slip length and parameters in numerical methods for moving contact lines. The procedure is illustrated by numerical simulations for partially wetting droplets employing the coupled level-set volume-of-fluid and phase field methods.