The Gibbs-Thompson relation within the gradient theory of phase transitions

Abstract

This paper discusses the asymptotic behavior as ɛ → 0+ of the chemical potentials λɛ associated with solutions of variational problems within the Van der Waals-Cahn-Hilliard theory of phase transitions in a fluid with free energy, per unit volume, given by ɛ2¦▽ϱ¦2+ W(ϱ), where ϱ is the density. The main result is that λɛ is asymptotically equal to ɛEλ/d+o(ɛ), with E the interfacial energy, per unit surface area, of the interface between phases, λ the (constant) sum of principal curvatures of the interface, and d the density jump across the interface. This result is in agreement with a formula conjectured by M. Gurtin and corresponds to the Gibbs-Thompson relation for surface tension, proved by G. Caginalp within the context of the phase field model of free boundaries arising from phase transitions.