Composition as a thermodynamic variable Gibbs free energy of binary solutions Entropy of formation and Gibbs free energy of an ideal solution Chemical potential of an ideal solution Regular solutions: Heat of formation of a solution Activity of a component, Henry’s and Raoult’s laws Real solutions: interstitial solid solutions, ordered phases, intermediate phases, compounds Equilibrium in heterogeneous systems Reading: Chapter 1.3 of Porter and Easterling, Chapters 9.5, 9.6, 9.9, 9.10 of Gaskell

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Solid Solutions (terminology)

Solid solutions are made of a host (the solvent or matrix) which dissolves the minor component (solute). The ability to dissolve is called solubility. Solvent: in an alloy, the element or compound present in greater amount Solute: in an alloy, the element or compound present in lesser amount Solid Solution: homogeneous maintains crystal structure contains randomly dispersed impurities (substitutional or interstitial) Second Phase: as solute atoms are added, new compounds / structures are formed, or solute forms local precipitates Solubility Limit of a component in a phase is the maximum amount of the component that can be dissolved in it (e.g. alcohol has unlimited solubility in water, sugar has a limited solubility, oil is virtually insoluble). The same concepts apply to solid phases: Cu and Ni are mutually soluble in any amount (unlimited solid solubility), while C has a limited solubility in Fe. Whether the addition of impurities results in formation of solid solution or second phase depends the nature of the impurities, their concentration and temperature, pressure… MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Composition as a thermodynamic variable

Real materials are almost always mixtures of different elements rather than pure substances: in addition to T and P, composition is also a variable. To understand conditions for equilibrium and phase diagrams (like the one below) we have to understand how the Gibbs free energy of a given phase depends on composition, temperature and pressure. Since many real experiments are performed at fixed pressure of 1 atm, we will focus on G(T, composition).

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Gibbs free energy of a binary solution (I)

Let’s consider a binary solution of A and B atoms that have the same crystal structures in their pure states and can be mixed in any proportions - form a solid solution with the same crystal structure (unlimited solid solubility). Example: Cu and Ni. 1 mol of homogeneous solid solution contains XA mol of A and XB mol of B. XA and XB are the mole fractions of A and B in the alloy.

XA + XB = 1

Let’s consider two steps of mixing: 1. Bring together XA mol of pure A and XB mol of pure B 2. Mix A and B to make a homogeneous solution After step 1 the free energy of the system is

Gstep1 = XAGA + XBGB

G

Gstep1

GA

X AG A GB

Gibbs free energy per mole before mixing

X BG B

0

XB

1

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei

Gibbs free energy of a binary solution (II)

After step 2 the free energy of the system is

G step2 = G step1 + ΔG mix

Where ΔGmix is the change of the Gibbs free energy caused by the mixing.

ΔG mix = ΔH mix − TΔ Smix

ΔH mix = H step2 − H step1 Δ Smix = Sstep2 − Sstep1

- heat of mixing of the components (heat of formation of a solution) - difference in entropy between mixed and unmixed states (entropy of formation of a solution)

Let’s first consider an ideal solution – interactions between atoms A-A, B-B and A-B are identical, and ΔHmix = 0. The free energy change upon mixing is only due to the change in configurational entropy:

ΔGid = −TΔ Smix mix

Δ Smix = Sstep2 − Sstep1

Sstep1 = k Bln1 = 0 - there is only one way the atoms can be arranged before mixing Therefore Δ Smix = Sstep2

MSE 3050, Phase Diagrams and Kinetics, Leonid Zhigilei...