Asymptotic Expansions of the Contact Angle in Nonlocal Capillarity Problems

We consider a family of nonlocal capillarity models, where surface tension is modeled by exploiting the family of fractional interaction kernels |z|−n−s, with s ∈ (0, 1) and n the dimension of the ambient space. The fractional Young’s law (contact angle condition) predicted by these models coincides, in the limit as s → 1−, with the classical Young’s law determined by the Gauss free energy. Here we refine this asymptotics by showing that, for s close to 1, the fractional contact angle is always smaller than its classical counterpart when the relative adhesion coefficient σ is negative, and larger if σ is positive. In addition, we address the asymptotics of the fractional Young’s law in the limit case s → 0 of interaction kernels with heavy tails. Interestingly, near s = 0, the dependence of the contact angle from the relative adhesion coefficient becomes linear.