From Notices of the AMS
Toward a Long-Range Theory of Capillarity
by Enrico Valdinoci
Communicated by Daniela De Silva
We recall the classical theory of capillarity, describing the shape of a liquid droplet in a container, and present a recent approach which aims at accounting for long-range particle interactions.
This nonlocal setting recovers the classical notion of surface tension in the limit. We provide some regularity results and the determination of the contact angle, supplied with various asymptotics.
1. The Classical Theory of Capillarity
1.1. The surface tension as an average of long-range molecular forces
The classical capillarity theory aims at understanding the displacement of a liquid droplet in a container in view of surface tension.
In spite of this elementary formulation, capillarity is a very delicate issue and its study connects a number of fields, including the calculus of variations, geometric measure theory, partial differential equations, mathematical physics, material sciences, biology and chemistry (for a comprehensive introduction to the theory of capillarity, including detailed historical accounts, exhaustive mathematical explanations, and comparisons with real-life experiments, see [Fin86] [Emm87]. A complete understanding of the intriguing patterns related to capillarity will require the virtous blend of different ideas coming both from mathematics and from the applied sciences.
Indeed, surface tension is a complex phenomenon arising as the average outcome of the attractive forces between molecules (such as cohesion and adhesion) and accounts for the interfaces between the droplet, the air, and the container.
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