### Example 8: Four phases, invariant point in first orientation

The most general kind of invariant point arises whenever three-phase univariants cross each other (if at least one phase is involved in both reactions).

Let's imagine following along a three-phase univariant reaction in our binary system and watch what happens when a fourth phase gets involved. In (a), the fourth phase is everywhere above the interior tangent segment defining the univariant assemblage, so it is not stable. However, in Figure (b), still following the univariant reaction to keep the original three phases lined up on their common tangent, we have reached a point where the fourth phase touches the same tangent, so all four are in mutual equilibrium. This is the invariant point. If we move in any direction, one or more of the phases will move off the common tangent, and at the invariant point the composition of all four phases is fixed: we have no degrees of freedom that allow us to keep this four phase assemblage. Proceeding further in the same direction in P-T space, we find in Figure (c) that as the fourth phase dropped below the common tangent to the other three that our original univariant reaction becomes metastable (indicated by the yellow line segment) because the one- and two-phase assemblages involving the fourth phase now have lower than any combination of the three phases involved in the univariant. The three still share a common tangent (since we kept moving along the Clausius-Clapeyron direction), it just is not part of the envelope of minimum curves and tangent segments anymore. This is a general behavior: at any non-degenerate invariant point, each univariant curve switches from stable to metastable. The locations of the three points selected for Figures (a),(b), and (c) are shown as blue circles in P-T space in Figure (d): we moved along the BCD univariant through the ABCD invariant point and onto the metastable extension of BCD. Now here's the remarkable thing: starting from our invariant point, it is always possible to move out along four different univariant reactions, each defined by a distinct subset of three of the four phases (or, alternatively, by the one phase among the four not participating). That is, in a binary system, a four-phase invariant point marks the mutual intersection of four univariant curves. For each subset of three phases, we can write a balanced reaction between them, and knowing their entropies and volumes we can calculate the Clausius-Clapeyron slope of the univariant curve involving these three as it takes off from the invariant point. Only one arrangement of the stable and metastable univariant curves around the invariant point is allowed in a binary system; the topological restrictions on arrangments of stable and metastable reactions are expressed by Schreinemaker's Rules, which we will not elaborate here except to say that they are a geometric consequence of the upward concavity of all the (X) curves.

T-X and P-X sections through an invariant point can have three different topological appearances. Either the two phases that are intermediate along the composition axis have stability fields on opposite sides of the invariant line, or they have stability fields on the same side of the invariant line, or one of them pinches out at the invariant point and appears there only as a point along the invariant line. Which case manifests depends on the orientation of the stable and metastable reactions relative to the P and T axes. In the example shown in the applet (initialized to example 8), the T-X section has the first behavior and the P-X section has the third behavior.

Since this applet was written, browers have stopped supporting applets for security reasons. You can download a standalone executable that runs all the functionality of the applet on your own platform here. When you run it, select "Example 8" to get this page's content.

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by Paul Asimow and Nizar Almoussa