-plots

To draw out the analogy to the above discussion of binary phase diagrams, we consider an example of a two dimensional crystal. Discussion of three dimensional crystals can be found elsewhere[54].

A parallel geometric construction is made for an orientation dependent surface tension () in Figure 2. This particular example is a first order expansion of a with square symmetry. It could also be written as . The figure is remarkably similar to the construction for in Figure 1 except that the figures are closed curves since the range of is all of

Increasing values of increase the anisotropy in and tends to create higher energy orientations which disappear from the equilibrium shape. Thus, plays a similar role to temperature, T, on the construction for the regular solution and so is the ordinate for the -diagram illustrated in the center of Figure 2. The critical value of alpha is 4/7. Also, note that we use as the ordinate which is convenient for this case of square symmetry.

Plots of appear on the left side of Figure 2 for three different values of , one above and two below the critical anisotropy. The gradient construction shown on the right column of the figure show that `ears' develop as the anisotropy increases just as in the case for lower temperatures in the gradient construction for . Any orientation on the `ears' is unstable and will break up into orientations given by the crossing in the -plot. Those parts on the concave part of the `ears' (outside the spinodes) are metastable.[40]

Consider the geometrical relations for the gradient construction in Figure 2. According to Equation 2 (for surface energies, these are the Cahn-Hoffman equations [47][46] ) the unit normal to the surface must be the orientation vector. Therefore, for all stable orientations, the surface of must also be the surface of the Wulff shape. In this sense, the interior region of the -plot must be equivalent to that obtained by the Wulff construction (See below).