The contact angle on an elastic substrate. 1. The role of disjoining pressure in the surface mechanics

Abstract

Using the correct physical mechanism for the transmission of the surface tension stress of a free liquid interface to the substrate, viz., the disjoining pressure of the substrate/liquid/vapor system, the macroscopic and microscopic structure of three-phase contact for a liquid drop on a linearly elastic substrate is elucidated. A variational treatment which includes the elastic free energy with the surface energies is shown to yield the augmented Young–Laplace equation for the drop shape and a natural boundary condition at the microscopic contact line, which produces the important result that the microscopic contact angle is zero—a conclusion which is also valid on a rigid substrate within the confines of the Derjaguin approximation for the intersurface forces. The macroscopic contact angle—the angle that the drop appears to make at the contact line when the system is observed on length scales greater than the range of the disjoining pressure in the system—is shown to obey Young's equation with a line-tension-like correction term. The correction depends directly on the height of the substrate deformation at microscopic contact and explicit expressions for the deformation in terms of integrals over the disjoining pressure are exhibited. For typical van der Waals-type disjoining pressures the effect of the elasticity of the substrate is shown to be very large for low Young's modulus materials (hydrogels) but to be essentially negligible for typical inorganic materials with high moduli (oxides and clays).