The Deformation of an Elastic Substrate by a Three-Phase Contact Line E. R. Jerison
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 arising from the existence of 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, previous attempts 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.
Schematic of a liquid droplet on an elastic substrate. The red arrows show the surface tension forces acting on the contact line (a fourth force pointing downwards from the elastic deformation of the substrate completes the force balance diagram). The magnified detail shows the anticipated and confirmed deformation of the elastic substrate due to the upward liquid-vapor surface tension.
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. Their observations are recorded in Figure 2.
In previous work, the authors have solved the linearized displacement equation for an isotropic elastic solid of finite thickness. An infinite thickness solution had been presented before by Boussinesq, which however predicted a divergence of the displacement at infinite distances from the contact angle; introducing a finite thickness in the problem prevents this divergence and provides a more physical result. To this equation they have now additionally incorporated a term that accounts for the energy cost due the creation of additional surface area when the solid is deformed; this amendment resolves an additional peculiarity of the Boussinesq solution, which was associated with the stress in the solid at the contact line. The solution of this newly presented equation provides a theoretical prediction for the displacement profile of the gel substrate around the contact line in the directions tangential to the contact line and perpendicularly to the solid interface. The agreement between this calculation and data on the perpendicular displacement is excellent; deviations from the behavior of the system in the tangential direction are still noticeable. Pinning of the droplet on the gel surface and viscous drag near the contact line are two factors that might account for this discrepancy.
Measured profile of the elastic substrate under and around the droplet. x is the direction along the contact line and z is the direction perpendicular to the contact plane; the droplet lies to the right of the peaks. The plot on the left shows data from the top surface of the substrate and the plot on the right shows data from 3um below it. The theory developed by the authors is plotted with dashed lines (without accounting for surface tension) and with solid lines (including surface tension). The agreement between theory and experiment for displacements along z is remarkable. It appears that more need still be understood about the forces acting along x.