A microscopic theory of small-droplet adhesion on solid surfaces

Droplets resting on solid surfaces resist lateral forces before sliding; however, the microscopic origins of the sliding criterion and solid-liquid adhesion remain elusive. Here, we develop a microscopic mean-field theory that incorporates the dependence of solid-liquid interfacial energy on the droplet's body energy and maps the full energy landscape under coupled normal and lateral loads. Within this framework, we propose a sliding criterion within the small-droplet limit, in which the solid-liquid and solid-gas interfacial energies become equal, as a result of the formation of a novel microscopic lubrication-like interfacial state. This criterion explains the observed enhancement of adhesion under increasingly negative normal loads, thereby violating Amontons' first law. Furthermore, we reveal the microscopic origin of pinning force associated with contact angle hysteresis. Our approach eliminates inconsistencies associated with body-force balance in small droplets and achieves quantitative agreement with counterintuitive experimental observations across diverse conditions.