Adhesion of a Rigid Sphere to a Freestanding Elastic Membrane With Pre-Tension

Adhesion between a solid sphere and a thin film is a common but crucial issue in the study of biological membranes and two-dimensional materials. To supplement quantitative knowledge of membrane adhesion, this work addresses the axisymmetric adhesive contact between a rigid sphere and a circular freestanding elastic membrane clamped at the perimeter. For the membranes following linear stretching elasticity with initial radial tension, both the Johnson-Kendall-Roberts (JKR)- and Derjaguin-Muller-Toporov (DMT)-type adhesion as well as the transition regime in-between are considered. The dependency of contact radius and displacement on external force is studied analytically. In essence, the general solution is governed by three dimensionless parameters, reflecting the effects of membrane stretching elasticity, the range of adhesion force, and the membrane size. It is interestingly found that the membrane size does not affect the contact radius and displacement at zero external force at all and has minor influence on the value of pull-off force. The presented closed form solutions might be useful for the understanding of adhesion behaviors of sphere-membrane systems.