-Shapes from Solution Thermodynamics

We illustrate the following example from two-component regular solutions but the concepts certainly apply to more components and more sophisticated solution models. Although two components allows us to define a single composition , the comparisons are abetted by our introduction of a vector notation.

For a regular solution model , where is a reduced temperature which scales out the energy of mixing and Boltzmann's constant. Letting , the molar free energy becomes In Figure 1, these molar free energies are plotted in the left column of the figure at reduced temperatures above, just below and well below the critical temperature. The common tangent construction at each temperature was used to draw the phase diagram in the middle. The dashed curve in the phase diagram correspond to the spinodals, where .

Chemical potentials for both components can be obtained for this model by any one of a number of equivalent ways, e.g. by taking the derivatives of with respect to and . Another is taking the intercepts at and of tangents at to . The resulting chemical potentials are plotted against each other in the figures of the right column of Figure 1. Since the coordinates of any point of this curve give the values of the two chemical potentials, these point are the ends of is the gradient of . We wish to focus on this -shape.

At high this shape is smoothly curved. Because of the geometric relation in Equation 3 i.e., the Gibbs-Duhem equation in the case of solution thermodynamics, , the normal to the curve is the composition vector . Thus we can know the composition for each part of the curve. Once that is known we can recover from this curve from , but there are other ways.

Below the critical temperature the -plot becomes self-intersecting and develops `swallow-tails' or `ears.' The crossings are places where two phases (smooth curves) have the same chemical potential. Because of the Gibbs-Duhem equation relating slope to composition, the distinct compositions of each of the phases are given by the normals to the curve at the crossing point. The sharpness the corner at the crossing relates to the difference in composition between the two phases in equilibrium, i.e., the width of the miscibility gap in the phase diagram. This analogy between corners in -shapes and phase diagrams extends to multicomponent phase equilibrium.

The locally convex portions of the ears represent metastable compositions; the concave parts unstable compositions. The metastable and unstable part are separated by a spinode. Eliminating the ears produces a convex figure that is the convexified -shape. It contains all the information that was in the convexified plus a graphic display of all the phase equilibria. This diagram illustrates the geometric nature of Equation 3.

The diagrams on the left and right side of Figure 1 are dual to each other. Each can be used to calculate the phase diagram in the center and any diagram on the right side () can be used to calculate its dual -shape which appears on the right side of Figure 1. Note that even though the phase diagram cannot be used to determine any of the other figures uniquely, the special CALPHAD procedures have had considerable successes.[6][53]