Heterogeneous Equilibrium Systems

Document Type

Book

Publication Date

1-1-2023

Abstract

So far, in Part II we have looked at the homogeneous (one-phase) systems, which can be described by uniform spatial distributions of the OP. Let us now look at a heterogeneous equilibrium system composed of two or more phases. In Sect. 2.4 we looked at the heterogeneous equilibrium states using the classical Gibbsian approach of the theory of capillarity. Unlike the classical one, the field-theoretic approach to the problem of phase equilibrium considers an interface between the coexisting phases as a transition zone of certain thickness with spatial distribution of OPs and possibly other parameters. Hence, for the continuous description of such systems, one must know not only the average values of P, T, and OPs, but the spatial distributions of these parameters also. In this chapter we are discussing only the equilibrium properties of the system; nonequilibrium systems will be considered in Chaps. 10 and 11. In this chapter 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-OP-gradient 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 various 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 sharp-interface (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. In this chapter we are finishing column 2 and start erecting column 3 of the beams and columns FTM configuration in Fig. P.1.

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