Contact angles on a soft solid: from Young's law to Neumann's law

TL;DR

This paper develops an elasto-capillary model to understand how contact angles on soft solids transition from Young's law to Neumann's law as the solid's elasticity varies, highlighting the role of molecular-scale effects.

Contribution

It introduces a coupled molecular-elastic model that explains the transition of contact angles from rigid to soft solids, bridging classical laws with elastic effects.

Findings

Contact angles deviate from Young's law on soft substrates.

The model predicts a transition to Neumann's law at low elastic moduli.

The transition occurs when the length scale γ/E is a few molecular sizes.

Abstract

The contact angle that a liquid drop makes on a soft substrate does not obey the classical Young's relation, since the solid is deformed elastically by the action of the capillary forces. The finite elasticity of the solid also renders the contact angles different from that predicted by Neumann's law, which applies when the drop is floating on another liquid. Here we derive an elasto-capillary model for contact angles on a soft solid, by coupling a mean-field model for the molecular interactions to elasticity. We demonstrate that the limit of vanishing elastic modulus yields Neumann's law or a slight variation thereof, depending on the force transmission in the solid surface layer. The change in contact angle from the rigid limit (Young) to the soft limit (Neumann) appears when the length scale defined by the ratio of surface tension to elastic modulus $\gamma/E$ reaches a few molecular sizes.