The Deformation of an Elastic Substrate by a Three-Phase Contact Line E. R. Jerison

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Introduction

The authors study in detail the contact line around a droplet of water deposited on a gel and surrounded by air. Such a droplet experiences forces due to the three different interfaces: the solid-vapor surface tension, the solid-liquid surface tension, and the liquid-vapor surface tension. In equilibrium, all forces must be balanced. The lateral components of all surface tensions are balanced when the droplet relaxes in its steady-state shape, characterized by a contact angle between it and the solid such that the solid-liquid and solid-vapor tension equal the liquid-vapor tension. However, the liquid-vapor tension has also a component perpendicular to the solid substrate. While it is reasonable to speculate that this force is balanced somehow by an elastic deformation of the solid, an attempt to calculate the magnitude of these forces with pre-existing theories led to pathological results, since relevant quantities such as the vertical component of the liquid-vapor surface tension diverged at the contact line. Existing theories on this system either have avoided this regime, or introduce reasonable cutoffs but do not agree with experimental observation.

To address this lack of understanding, the authors have observed the elastic deformation of a gel solid under a droplet of water in air and developed a linear elastic theory which describes their observations well for displacements out of the plane of the solid substrate. The deformations of the solid were observed by embedding fluorescent beads in the gel at two planes, the interfacial one and 3μm below it, and monitoring their position with microscopy in the presence and absence of the drop.