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 \(s\in (0,1)\) 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 \(s\rightarrow 1^-\), 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 \(\sigma \) is negative, and larger if \(\sigma \) is positive. In addition, we address the asymptotics of the fractional Young’s law in the limit case \(s\rightarrow 0^+\) of interaction kernels with heavy tails. Interestingly, near \(s=0\), the dependence of the contact angle from the relative adhesion coefficient becomes linear.