Indentation of a Rigid Sphere into an Elastic Substrate with Surface Tension and Adhesion

The surface tension of compliant materials such as gels provides resistance to deformation in addition to and sometimes surpassing that due to elasticity. This article studies how surface tension changes the contact mechanics of a small hard sphere indenting a soft elastic substrate. Previous studies have examined the special case where the external load is zero, so contact is driven by adhesion alone. Here, we tackle the much more complicated problem where, in addition to adhesion, deformation is driven by an indentation force. We present an exact solution based on small strain theory. The relation between indentation force (displacement) and contact radius is found to depend on a single dimensionless parameter: $\omega=\sigma(\mu R)^{-2/3}(9\pi W_{\textrm{ad}}/4)^{-1/3}$, where $\sigma$ and $\mu$ are the surface tension and shear modulus of the substrate, $R$ is the sphere radius, and $W_{\textrm{ad}}$ is the interfacial work of adhesion. Our theory reduces to the Johnson-Kendall-Roberts theory and Young-Dupr\'e equation in the limits of small and large $\omega$ respectively, and compares well with existing experimental data. Our results show that, although surface tension can significantly affect the indentation force, the magnitude of the pull-off load in the partial wetting liquid-like limit is reduced only by 1/3 compared with the JKR limit, and the pull-off behavior is completely determined by $\omega$.