Intermolecular Forces, Adhesion, and the Elastic Foundation

A model for the elastic contact between a rigid sphere and an ideal elastic foundation with adhesion has been developed. The model was derived by integrating the full Lennard-Jones potential to arrive at a closed-form equilibrium condition that balances surface energy with strain energy. It was found that the separation height is not a function of the penetration. Using this energy criterion for separation of contact in an elastic foundation, a model for the force displacement relationship was then developed. In this derivation there exists a tensile zone of deformation along the perimeter of the contact. The model also reveals a number of unique aspects of the adhesive contact, including: the maximum adhesion occurs when the apex of the sphere is tangent to the plane of the undeformed surface, the maximum adhesion force \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ F_{\text{adh}} = - 2\pi R\Updelta \gamma $$\end{document}, and the contact area is linearly dependent on penetration. The ability to fit high fidelity indentation data from finite-element analysis and molecular dynamics simulation for thin films was demonstrated. Additionally, experiments were performed on thin films (~40 μm) of PDMS using a custom-built microtribometer with in situ optical interferometry that enabled simultaneous measurements of contact area, penetration depths, externally applied force, and the detailed measurements of the free-surface deformations, which include the predicted tensile zone along the perimeter of contact.