Introduction

Hubert Aaronson's wide ranging contributions to materials science over the last four decades has focussed repeatedly on three major topics and their applications to phase transformations:[7][6][5][4][3][2][1]

1) solution thermodynamics and multicomponent phase equilibria;[11][10][9][8]

2) equilibrium shapes of surfaces with anisotropic surface energy;[21][20][19][18][17][16][15][14][13][12] and,

3) the morphology of growing precipitates.[25][24][23][22]

In this paper, we hope to honor Hub and his contributions by illustrating how each of these topics derives from a common mathematical basis and that the graphical constructions derived for each separately can shed new insight into the others.

Tangent constructions are powerful graphical methods, widely used for heuristic, computational and theoretical purposes in our field.[2][27][26] We will look at differences in how tangent constructions are used in each of three topics, and explore how these varied methods may be used in the others.

For multicomponent phase equilibria (at constant temperature and pressure) the tangent construction is performed on a plot of the molar Gibbs free energy, , which has to be convex from below at equilibrium. (Here is a short-hand [vector] notation for the molar composition of a multicomponent phase or system.) For first order phase changes, may be multivalued with a different for each phase, usually with a different symmetry. Concavities in this plot represent metastable and unstable phases and compositions for which the free energy is too high. Such concavities lead to ranges in composition (sometimes called miscibility gaps) where single phases are not in equilibrium. These concavities are removed with the common tangent construction, that identifies for a particular average composition whether a single phase is in equilibrium, and if not what mix of phases will have the lowest free energy. This graphical process, termed convexification, leads to a convex hull of . Each point on that convex hull represents the lowest free energy of the system of a given average composition, including permitting the equilibrium to be multiphase. Common tangents are often used to construct phase diagrams from a nonconvex .

A quite similar set of graphical procedures is applied to the orientation dependent surface free energy per unit area , except that the construction is performed on a radial plot of the reciprocal of [28] . Here is the normal to the surface. Note that the components of are the analogs of the concentrations of the chemical species.[29] Concavities are removed by convexification, and the points on the convex hull of represent (the reciprocal of) the lowest free energy a surface with a certain average orientation can achieve. Interface orientations for which lies inside the convex hull have too high an energy and are unstable. The tangent construction identifies which surfaces become corrugated at equilibrium, and specifies which orientations of lower energy coexist to replace a high energy surface, even though there is greater surface area. If there is more than one possible phase state of the surface (facetted, surface melted, wetted, etc.) may be multivalued, just as . The convexification of has many analogies to that for : It identifies the occurrence of orientation gaps and surface phase transitions.

These two examples are based on finding minima in free energy. For the kinetic example we consider the cases of growth rates that are constant in time and may be orientation dependent.[31] This occurs not only with interface controlled growth[32] and massive transformations,[33] but also with such diffusional growth processes as cellular precipitation,[34] eutectic and eutectoid growth,[1] discontinuous coarsening,[36][35] liquid film migration [37] and diffusion induced grain boundary migration.[38] When there is growth anisotropy, certain orientations tend to disappear from the shape. To determine whether a particular orientation will be part of a limiting outward growing shape, the same graphical convexification is performed on a plot of .[36] Those fast growing orientations that will eventually disappear from a growing crystal show up as concavities in this plot. The common tangent construction will determine whether some orientations will disappear into an edge or a corner depending on whether the tangent plane touches at two or more distinct points. No energy minimization is involved, but it is a Huygens principle of least time.[39][36]

In these constructions there are many analogies. Composition gaps have their analogs in orientation gaps; two phase equilibria become edges, three or more phases in equilibrium become corners. There are quite analogous conditions on the curvature of and for stability with respect to undulations in and ; both are called spinodals.[40]

There is another well known graphical construction that confirms the analogy between the orientation dependence of and its kinetic counterpart . The Wulff construction performed on gives the shape with the least surface energy for the volume it contains.[41] The same construction on gives the limiting shape of a growing crystal; it also the shape that will grow most slowly, the one that adds the least volume.[32] The Wulff shape is more basic than the convexified . It contains all the information that is in a convexified , but the converse is not always true, and it is more convenient than for many applications.[43]. The following question suggests itself: Is there an equivalent Wulff-like construction for ? As we will show below, the answer is yes and the construction gives a new method for obtaining a well-known plot in solution thermodynamics.

In this paper we will briefly describe the mathematical basis that links all three topics. A thorough discussion of this topic will appear elsewhere.[44] Because theoretical thoughts about these topics developed quite independently, exploration of these analogies creates opportunities to exploit the various separate methods and discoveries for new uses. We will try to answer how far these analogies can be pushed, and which methods developed for one of these topics can be adapted to the others.

One example has already been suggested and put to use.[29] Information about stable compositions is efficiently stored in the phase diagrams in which simple rules derived from solution thermodynamics and the phase rule play an important role in their construction, interpretation and in many applications. These diagrams identify stable compositions, two-phase regions with tie-lines, three-phase tie-triangles, etc., joining coexisting compositions. The phase rule allows a cataloging of first-order phase changes. Phase diagram extrapolations are a very useful tool for predicting stability, metastable equilibrium compositions, and the order in which phases appear upon cooling. Information about stable orientations can be stored in an analogous diagram, called an -diagram where interface orientation play the same role as composition in a phase diagrams. The orientations that meet at each point on a curved edge are joined by tie lines; tie polygons specify the orientations that meet at corners. First-order surface phase changes, such as wetting and facetting, conform to a modified phase rule.[29]

However, perhaps these analogies are not so direct as they seem as some puzzles should have arisen in the minds of the reader. Namely:

1. Why are the tangent constructions performed on while they are performed on or ?

2. The condition for local stability for a two component systems is , while the equivalent condition, , on two-dimensional crystals is more complex. For more than two components the Hessian, the matrix of second derivatives of , must be positive definite for phase stability, while the condition on for three dimensional surfaces can not be so simply expressed. Are there formulations in which equivalent conditions have the same simple form?

3. Energy is minimized for and ; what is minimized for ?

These conundrums will disappear when these topics are put into a single mathematical framework.