Heterogeneous Equilibrium Systems

Abstract

In this chapter, we are looking at the heterogeneous equilibrium states using the classical Gibbsian approach—Theory of Capillarity and the field-theoretic one, which considers an interface as a transition region between the phases. To do that we generalize the free energy to a functional of the spatial distributions of the order parameters and introduce a gradient energy contribution into the free energy density. We analyze various forms of the gradient energy and find the square-gradient one to be preferable. Equilibrium conditions in the heterogeneous systems yield the Euler–Lagrange equation, solutions of which are called extremals. We study properties of the extremals in the systems of various physical origins and different sizes and find a bifurcation at the critical size. The results are presented in the form of the free energy landscapes. Analysis of the one-dimensional systems is particularly illuminating; it shows that, using qualitative methods of differential equations, many features of the extremals can be revealed without actually calculating them, based only on the general properties of the free energy. We find the field-theoretic expression for the interfacial energy and study its properties using different Landau potentials as examples. We introduce a concept of an instanton as a critical nucleus and study its properties in systems of different dimensionality. Multidimensional states are analyzed using the drumhead approximation and Fourier method. To analyze stability of the heterogeneous states, we introduce the Hamiltonian operator and find its eigenvalues for the extremals. Importance of the Goldstone modes and capillary waves for the stability analysis of the extremals is revealed.