Chemical Wulff Shapes

There is a large literature regarding the Wulff shape that minimizes surface energy for a given enclosed volume and the analogous kinetic Wulff shape that give the limiting shape of a crystal growing outwardly under diffusion control that has recently been reviewed.[32] The methods of construction are the same even though one is a minimization problem and the other is a long-time solution of a first-order nonlinear partial differential equation.

The method is one of iterative truncation of space by a set of half planes-each half plane partitions the space into allowed and disallowed half spaces. For each value of a plane is drawn normal to at a distance equal to the value of or respectively and the half space of all the more distant points discarded. When this is done for all , the remaining points form a convex body that is the Wulff shape. The expression for the set of points which survive this construction is given by for ; substituting for gives the kinetic Wulff shape.

The surface of the Wulff shape and the convexified -plot or plot of the characteristics are the same shapes, even though they are obtained by quite different mathematical or graphical operations. In the Wulff constructions there is no restriction to either continuous or differentiable functions. Because no differentiation of date is used, the methods Wulff constructions may be quite superior for noisy data. Even though is expected to be smooth for solutions it is worthwhile to propose a Wulff construction for solution and compound free energy data.

In order to do this we need to convert into the Euclimolar free energy . For the binary regular solution example . The left hand panel of figure 3 shows for which is of the critical temperature, and therefore shows a miscibility gap. The second and third panels show , plotted respectively against and as a radial plot, for this same temperature. The third panel shows in addition one step in the Wulff construction for a single composition. A line for that composition is drawn perpendicular to another line from the origin with slope and length ; all points to the upper right of this line (shown gray) are discarded. Performing the Wulff construction on a finite set of results in a fan of truncation lines shown in the last panel. The clear area at the lower left is the chemical Wulff shape, whose envelope is the -plot. As with the -plot such a figure plots chemical potentials against each other, and compositions are obtained from slopes. It identifies single phases as continuous curves a two-phase equilibrium as the corner.

The inverse Wulff construction, finding the distance of a tangent line from the origin corresponding to a particular composition recovers the Euclimolar free energy. Note that this is equivalent to .

The concavity in the first panel of Fig. 3 shows that at this is not convex. The corner in the Wulff construction in the last panel confirms this. Both are appropriate criteria for nonconvexity of . The convexity of the curves in the middle two panels is of Fig. 3 are of no importance; even though is not convex, both curves are convex. For Herring's tangent sphere construction for finding stable orientation is an alternate test for convexity. But this construction works only for positive functions. As can be seen in the third panel, it does not work for a radial plot of a negative .

Note that the two-phase corner does not touch the Euclimolar free energy plot; the gap is a measure of the reduction in free energy upon phase separation. The chemical Wulff shape does not give metastable phases or their equilibria, except when the entire curve of a stable phase is ignored in the construction. The undiscarded points in the interior of the lower left-hand area are not physically realizable, except possibly as an unknown stabler phase-an ice-9.[56]

Examples of the chemical Wulff construction for the three temperatures in Fig. 1 are illustrated in Fig. 4. They show not only the miscibility gaps at the lower temperature, but also the chemical potentials of the phases at various compositions derived from the normals. Note that in Fig. 4 that the Euclimolar free energy takes on some positive values as the temperature is decreased.