## EQUILIBRIUM CRYSTALLIZATION

### Composition X

Since we are examining equilibrium crystallization, the final solid must have the same bulk compostion as the initial liquid. The bulk composition of X, in tems of the three end member components which define the system, is 60% Wo, 20% An, 20% Sp. The compostion of X is determined by drawing a series of lines (the red lines) through X parallel to the sides of the triangle and the corresponding percentage of each component can be calculated.

As these are the only phases present in this ternary system and all three coexist at the ternary eutectic, the liquid must move to the eutectic. The final liquid will be a mixture of An Wo and Sp, in the proportions of the eutectic.

So we know what our initial liquid composition is and we know what our final liquid composition will be - What path will the liquid follow?

The path followed by the liquid is shown as the blue line on the image.
1. To begin, X is at some temperature well above the liquidus and as such is 100% liquid.
As the temperature drops to the liquidus the system still consists of 100% liquid 0% solid.
2. Upon cooling to the liquidus, the bulk composition does not change, and reaches the liquidus at point X, which lies in the field of Wo + L.
At this point Wo begins to crystallize and it is in equilibrium with liquid. Due to the crystallization of Wo, the liquid composition must change, i.e., it is becoming depleted in the Wo component and enriched in the An and Sp components. The result is that the liquid composition moves directly away from the Wo composition point.
3. As the liquid moves away from X towards Z, it is changing in the following way:
• It is becoming depleted in Wo. Wo is being removed from the liquid into solid Wo crystals, and
• It is becoming enriched in An and Sp, it is moving towards the An-Sp side of the diagram.
4. At Point Z
The two phases in equilibrium are Wo and L. Two questions to ask here are:
1. What are the proportions of Wo and L at Z?
2. What are the compostions of Wo and L at Z?
Proportions
Applying the lever rule at Z will provide the proportions of Wo and L.
• % Wo = XZ/WoZ *100 = 24%
• % L = WoX/WoZ * 100 = 76%
Compositions
The composition of the two phases Wo and L at Z, must be expressed in terms of the three end member components of the ternary diagram An, Wo and Sp.
• The solid Wo is pure Wo (100%Wo, 0% An, 0% Sp).
• The liquid composition can be determined by drawing a series of three lines through Z, with each line parallel to one side of the triangle. This gives a composition for the liquid of 47% Wo, 27% Sp and 25% An.
5. As crystallization continues from Z to P, Wo continues to crystallize and the liquid continues to move directly away from Wo.
6. At P, which lies on the cotectic between the field of Wo + L and Sp + L.
Sp begins to crystallize along with the Wo, because the liquid has reached the boundary curve between the fields of Wo + L and Sp + L. The result is that the liquid composition now moves down the boundary curve towards E as both Wo and Sp are removed from the liquid.
7. At Q
Here three phases are in equilibrium, Wo + Sp + L. The proportion of solid (Wo+Sp) to liquid (L) and the composition of the solid and liquid phases can be determined. In order to calculate these values we must first draw a line from Q, the point of interested, through X, our bulk composition, back to intersect the edge of the triangle. This line on the image extends from Q through X to T.

Proportions of Solid to Liquid
• % Solid (Wo+Sp) = QX/QT * 100 = 45%
• % Liquid (L) = TX/QT *100 = 55%
Proportion of Wo to Sp
• % Wo = TSp/WoSp * 100 = 89%
• % Sp = TWo/WoSp * 100 = 11%.

Compositions of Phases
• Solid Wo is pure Wo (100% Wo, 0% Sp, 0% An)
• Solid Sp is pure Sp (0%Wo, 100% Sp, 0% An)
• The liquid composition can be determined by drawing a series of lines through Q, with each line parallel to one side of the triangle.
8. Crystallization of Wo and Sp together continues from Q to E.
9. At E, the ternary eutectic
Wo and Sp continue to crystallize and are joined by An, with all three crystallizing together, in the proportions given by the eutectic.
The system stays at the temperature of the eutectic until the last droplet of liquid crystallizes into the last grains of Wo, Sp and An.
The compostion of the last liquid last liquid to exist in the system is represented by E.
The proportions of the solid phases in the final solid after the last liquid has crystallized is given by X (60% Wo, 20% Sp, 20% An) - OUR INITIAL BULK COMPOSITION!
In order to maintain equilibrium as the solid phases are crystallized the liquid composition is constantly changing as represented by the path from X to P to E, the solid blue line on the diagram.