A Menagerie of Binary Phase Diagrams

The phase diagram in Figure 30-4 is the simplest possible two-component phase diagram at constant pressure.

Figure: The so-called "lens" phase diagram. The upper line is the limit of $\ensuremath{{f}^{\mbox{solid}}} \rightarrow 1$ and is called called the solidus curve. The lower line is called the liquidus curve.

Figure 30-6: A variation on the lens phase diagram.

Consider how the Gibbs phase rule relates to the above phase diagrams.

The Gibbs phase rule is: $D= C + 2 -f$

However, $P$ is constant so we lose one degree of freedom: $D = C +1 -f$

In the two phase region-- $D = 2 + 1 - 2 =1$--so there is one degree of freedom.

Question: What is the degree of freedom? What does it mean?

• If temperature is changed at fixed $\ensuremath{\langle \ensuremath{{X}_\circ} \rangle}$, then the change in volume fraction of phases is determined. In other words there is a relation between $dT$ and $d \ensuremath{{f}^{\mbox{solid}}}$.

• If $\ensuremath{\langle \ensuremath{{X}_\circ} \rangle}$ is changed with fixed phase fractions then $\Delta T$ is determined by the change.

Consider another two-component phase diagram and see if it violates the Gibbs phase rule.

Figure 30-7: Is this a possible phase diagram?

Consider the three-phase region: $D = C +1 - f = 0$

Because there are no degrees of freedom, the three-phase region must shrink to a point in a two component system. This places restrictions on the topology of binary phase diagrams.

The diagrams below illustrate how such an invariant point (i.e., three phase equilibria in a two component system) arises:

Figure 30-8: Liquid is stable at all compositions at this temperature.

Figure 30-9: One of the solid phases becomes stable.

Figure 30-10: The second solid phase becomes stable as well, but not at the same compositions as the first.

Figure 30-11: At one unique temperature (the Eutectic) the two phase regions converge-this is the invariant point.

Figure 30-12: Below the eutectic, the two solid phases are separated by a two-phase region.

This yields the following phase diagram

Figure 30-13: The free curves from Figures 30-8 through 30-12, result in a eutectic phase diagram.