TABLE OF CONTENTS The main breakthrough achieved by Gibbs some 130 years ago was the understanding that at equilibrium a system of coexisting phases can always be characterized by some appropriate potential functions that achieve minimum (maximum for entropy), and the derivation of many powerful laws (e.g., the Phase Rule, the Gibbs Duhem equation, etc.) that could be applied to understanding the behavior of heterogeneous mixtures. [4] However, neither the classical Gibbsian thermodynamics nor phase diagram research do not pursue the goal of incorporating microstructure of heterogeneous systems into its formalism; that is, the spatial distribution of phases and their temporal evolution toward equilibrium. While the theory can predict the concentrations of components and the relative amounts of the competing phases (using the common tangent construction and the Lever Rule) in a heterogeneous mixture at equilibrium, it cannot predict how coexisting phases will be distributed in space. The main reason is that Gibbsian formalism does not contain intrinsic length scales, and considers all interfaces as mathematical surfaces that are infinitely thin (or sharp ). Of course, the classical Gibbsian thermodynamics is internally consistent, it is just that Gibbs purposefully chose not to deal with interfaces of a certain finite width (e.g., in his theory of absorption), neither can it predict their characteristic length scale(s).
In the asymptotic limit both the sharp interface and the diffuse-interface formalisms coalesce (see below), and that has lead many researchers to believe that there...
