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Question

Consider a random substitutional binary alloy, consisting of Na "A" atoms and Nb "B" atoms. Both types of atoms have Z neighbors. Only nearest neighbor interactions are considered. The bond energies of the AA, BB and AB bonds, respectively, are V AA , V BB , and V AB.

1. Write an expression for the Gibbs free energy of the random alloy, assuming that it is a regular solution and that the change of volume upon mixing is zero. Sketch the shape of G (X) for a number of representative values of c / RT' (positive and negative).
2. For the case of phase separation, make a plot of T vs. XB that shows the phase(s) with lowest free energy. Assume for mathematical convenience that the Gibbs free energies of the standard states of A and B are the same (i.e. V AA = V BB). Define (via equation) the critical temperature Te below which phase separation starts.

Answer 1

The Gibbs free energy of a binary alloy can be expressed as:

G = H - TS

where H is the enthalpy, T is the temperature, and S is the entropy. Assuming that the change in volume upon mixing is zero, the enthalpy of mixing is also zero, and the Gibbs free energy of the alloy is given by:

G = kT[c ln(c) + (1-c)ln(1-c)] + c(1-c)[VAB - VA - VB]

where c is the mole fraction of A in the alloy, k is the Boltzmann constant, and VAB, VA, and VB are the bond energies of AB, AA, and BB bonds, respectively.

The shape of G(X) depends on the values of c/RT'. For positive values of c/RT', G(X) is a minimum at c=0.5, indicating that the alloy is thermodynamically stable in the random state. For negative values of c/RT', G(X) has two minima, one at c=0 and the other at c=1, indicating that the alloy undergoes phase separation into two distinct phases.

To find the critical temperature Te below which phase separation occurs, we need to calculate the second derivative of G with respect to c, and set it equal to zero:

d²G/dc² = 2kT[(1/2c-1/2)+(1/2(1-c)-1/2)] + 2(VAB-VA-VB) = 0

Solving for T, we get:

Te = (VAB - VA - VB)/(k ln(1+sqrt(2)))

To plot T vs. XB for the phase diagram, we need to calculate the free energy of the two phases, assuming that each phase has a different mole fraction of A, denoted by XA and XB (where XA + XB = 1). The free energy of each phase is given by:

G(XA) = kT[XA ln(XA) + (1-XA)ln(1-XA)] + XA(1-XA)[VAB - VA - VB]

G(XB) = kT[XB ln(XB) + (1-XB)ln(1-XB)] + XB(1-XB)[VAB - VA - VB]

The phase(s) with lowest free energy will be the most stable at a given temperature T. We can plot T vs. XB for the case where there is a single phase (i.e. the alloy is homogeneous) and for the case where there are two phases (i.e. the alloy is undergoing phase separation). The critical temperature Te separates the region where the alloy is homogeneous from the region where it undergoes phase separation.