Thermodynamics of Metals

In this chapter, we will continue the discussion on phase equilibria started in the introductory chapter on thermodynamics, concentrating on binary systems, such as those that are commonly found in metal alloys. The following topics are presented:
Mixing of Phases
Binary Solid Solution Diagram
Binary Eutectic Phase Diagram
Binary Eutectic with Miscibility Gap
Spinodal Decomposition
Binary Phase Diagram with Intermediate Phases

Ternary systems will be discussed in the chapter on thermodynamics of ceramics, and we will discuss mixing and phase separation in greater detail in the chapter on thermodynamics of polymers.

Mixing of Phases

We saw in the introductory chapter that free energy determines which phases will be stable and which will separate into immiscible compound. This is true for solid-liquid, solid-solid and liquid-liquid equilibria. Mix two phases A and B. The free energy of the solution will be:

G = H - T·S The molar enthalpy and entropy of the solution are given by:

Java Equation

Java Equation where X_A = mole fraction of A in solution
X_B = mole fraction of B in solution = (1-X_A)
H° = standard enthalpies of A and B
S° = standard entropies of A and B
Java Equation = enthalpy of mixing
Java Equation = entropy of mixing

Look at the mixing terms more closely. The enthalpy is a measure of how the two types of species interact. It can be approximated by:

Java Equation where V is the interaction energy. The interaction energy tells us how favorably (or unfavorably) the two types of atoms interact with one another. The following observations about V are important:

V<0, A atoms bond more strongly with B atoms
V>0, A atoms bond more stronly with A atoms
V=0, equal interaction between A and B

The entropy term tells us how much more the system is made random by mixing A and B, and can be approximated by:

Java Equation

Note that Java Equation is greater than one since X_A and X_B < 1.

Once we have the enthalpy and entropy of mixing, the free energy of the solution, then, is given by Equation 1:

Java Equation

We can use this equation to generate a phase diagram by plotting free energy vs. composition at different temperatures.

Binary Solid Solution Diagram

solid and liquid phases each have own free energies, G or GL, described be above equation Handout 1: Figs. 16.19 and 16.4, Cu-Ni G at 0 and 100% Ni are the standard state free energies of components A (Cu) and B (Ni) at 1500 , liquid free energy is lowest at all compositions at 1000 , solid free energy is lowest at all compositions at 1300 , free energy curves cross. This generates two-phase region. tangents to free energy curves determine compositions of two-phase boundary. That is, the liquid (E) is more stable than the solid (F), but a combination of the two, as given by point "G" is even lower in free energy and thus more stable. Two phases will exist.

Binary Eutectic Phase Diagram

When the enthalpy of mixing, or the interaction energy, V, is positive for the solid phase, the free energy curves transform from one minimum above a critical temperature to two minima below a critical temperature. Handout 2: Figure 15.5 (Free energy is represented as "F" in this diagram). T1: liquid has lowest free energy T2: tangent between two phases represents lowest free energy. Two-phase region results. T3: two 2-phase regions; L + 1 and L + 2 T4: three phase region appears, L + 1 + 2. This is the eutectic melting point. T5: solid is now more stable, but it phase separates into two solids. two phase solid-solid equilibrium results tangents to free energy curve determines composition of two-phase boundary.

Binary Eutectic Phase Diagram with Miscibility Gap

In this case, there is a liquid free energy curve, a solid ( ) free energy curve, and a free energy curve for the two phase, solid-solid solution region. Handout 3: Figs. 16.20 and 16.6 liquid and solid form phase diagram similar to solid-solution diagram above (Fig. 16.4) solid ( ) phase separates into two solid phases, 1 and 2. tangent between two minima gives the region of "immiscibility" or "miscilibity gap" also known as the "binodal" C 1 and C 2 map out miscibility gap as temperature changes. Again, the two-phase tangent is lowest in free energy.The miscibility gap is a thermodynamic result and is described by the minima of the free energy vs. concentration curve: G/ X = 0 Handout 4: Figure 10. Minima of free energy curve map out the binodal.E. Spinodal Decomposition In addition to the binodal, we can have a region of metastable equilibrium called the "spinode". Handout 4: Figure 9.3 given by the inflection points of the free energy vs. concentration curve: 2 G/ X2 = 0 Handout 5: Figures 14, 15 metastable equilibrium between binode and spinode Between the spinode and the binode, the solution is stable against small composition fluctuations, but can phase separate by nucleation and growth. Spinodal decomposition is a phase transformation process which competes with nucleation and growth. We'll get to this in a minute.

Binary Diagram with Intermediate Phases

Things pretty much stay the same, even if we have intermediate phases, such as Mg2Ni or MgNi2 in the Mg-Ni system: Handout 6: Figures 16.21, 16.16 free energy curves for intermediate phases tend to be sharp peritectic at 761 C; L, Mg2Ni and MgNi2 all in equilibrium; common tangent

Back to the table of contents