The Deformation of an Elastic Substrate by a Three-Phase Contact Line

TL;DR

This paper investigates the elastic deformation of a thin substrate caused by a three-phase contact line, revealing that out-of-plane deformation is governed by a linear elastic theory incorporating surface tension effects, independent of substrate stiffness.

Contribution

It provides the first experimental validation of a linear elastic model that accounts for out-of-plane surface tension forces at the contact line.

Findings

Deformation profiles are scale-free and do not depend on substrate elastic modulus.

Out-of-plane deformation is accurately described by a linear elastic theory with surface tension.

Experimental data aligns well with the proposed theoretical model.

Abstract

Young's classic analysis of the equilibrium of a three-phase contact line ignores the out-of-plane component of the liquid-vapor surface tension. While it has long been appreciated that this unresolved force must be balanced by elastic deformation of the solid substrate, a definitive analysis has remained elusive because conventional idealizations of the substrate imply a divergence of stress at the contact line. While a number of theories of have been presented to cut off the divergence, none of them have provided reasonable agreement with experimental data. We measure surface and bulk deformation of a thin elastic film near a three-phase contact line using fluorescence confocal microscopy. The out-of-plane deformation is well fit by a linear elastic theory incorporating an out-of-plane restoring force due to the surface tension of the gel. This theory predicts that the deformation profile near the contact line is scale-free and independent of the substrate elastic modulus.