On the thermodynamics of periodic phases

Abstract: "We here present a theory of thermodynamical equilibrium for materials for which the free-energy density [psi](x) depends not only on the concentration (or mass density) u(x) and its gradient u'(x), but also on u"(x), the second gradient of u. We show that a broad class of such materials can exhibit equilibrium states that are periodic in a nontrivial way. It is possible that the theory of such second-order materials may supply insight into periodic layering phenomena observed in various types of mixtures, such as concentrated soap solutions and metallic alloys, but in this paper we do not refer to applications. Instead, we examine the mathematical implications of the theory."