Recitation: 12 12/04/03

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Evelyn Carpenter
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1 Recitation: 12 12/4/3 Regular Solution Solution: In an ideal solution, the only contribution to the Gibbs free energy of ing is the configurational entropy due to a random ture: ΔG id G id = x + x µ µ + RT [x ln x + x ln x ] ] [ = G id x µ + x µ = RT [x ln x + x ln x ] When there is a difference in the energetics between, and interactions, it is possible to model the Gibbs free energy of the system using the Regular Solution model: where G R.S. = x µ + x µ + RT [x ln x + x ln x ] + wx x ( ) w + w w = N a z w 2 and w ii corresponds to the energy of the corresponding bond or interaction. For a regular solution, the Gibbs free energy of ing is given by: ΔG R.S. = RT [x ln x + x ln x ] + wx x The enthalpy of ing in a regular solution is just H R.S. = wx x. When H <, Exothermic When H >, Endothermic There is a fundamental flaw in the Regular Solution Model: The key assumption in the R.S. model is that the probability for observing any possible configuration is the same and is independent of the interaction energies between like and unlike atoms. In reality, s T, however, P i P j P i = e E i E j kt P j 1. Therefore, R.S. is a better approximation at high temperatures. Ordering In real materials, it is possible to observed a wide range of ordering behavior, going from a completely random solution to a fully ordered compound.

2 5 4 H TS G 5 H TS G H TS G H? 7 H? w= w= w= Figure 1: Regular Solution Models with different values for w 4 T= w= Figure 2: G R.S. for various T, w =

3 Consider a binary system for which the pure components,, and, exist in the bcc structure. If the interaction parameter w is highly negative, the system will try to maximize the number of bonds in order to minimize its Gibbs free energy, provided the temperature is low enough. t high temperature, entropy dominates. In this system it is possible to consider to sublattice, α and β. n order parameter, η can be defined so: α x 1 = (1 + η) 2 β x 1 (1 + η) = 2 α β For η =, x = x Fully disordered β For η = 1, x = 1, x = Fully ordered α For a system that can undergo order disorder transitions, for a given composition, we have: z RT [ ( ) ( )] 1 η 1 + η F = η 2 w + (1 η) ln + (1 + η) ln To find the value for η at any temperature T it is necessary to minimize the free energy F (η) with respect to η: T zw R η = ( ( ln 1 η ) ( ln 1+η )) η T/Tc Figure 3: η v.s T /T c for x = x =.5

4 Problem: correlation of occupation probabilities. Phase Diagram There is a thermodynamic connection between microscopic behavior of matter and macroscopic observations. Phase Diagrams: i) Expressions for the Gibbs energy of phases can be obtained from microscopic calculations of energies, entropies, etc. ii) y establishing equilibrium conditions, i.e. common tangent construction, it is possible to calculate phase diagrams. iii) In principle, the Gibbs free energy curves and the phase diagrams carry the same information. Example: Temperature, C x(ni) Gmr(*) :X(NI), GMR(LIQUID), 2:X(NI), GMR(FCC_1), -92 E3-93 X(NI) 1 2 (a) Cu Ni Phase Diagram (b) Gibbs free ennergy curves at 125 C Figure 4: Cu Ni System

5 Eutectics, Peritectics L + L +L L (a) Eutectic Transformation (b) Peritectic Transformation Figure 5: Invariant Transformations in inary Systems Example: 11 Temperature, C #1+# Mole Fraction, X(g) Figure 6: g Cu inary Phase Diagram

6 Metastability In some cases, the precipitation of a phase may be limited by kinetic factors, such as nucleation. This usually occurs when the phase transformations occur at low temperatures, where diffusion and nucleation rates are extremely slow. good example is the F e C phase diagram, Why??. C C (a) Stable Equilibrium (b) Metastable Equilibrium Figure 7: Stable vs. Metastable Equilibria Example: Temperature, K 15 bcc hcp gti gti 2 5 x(ti) Figure 8: g Ti Phase Diagram

7 If gt i and/or gt i2 become metastable: Temperature, K 15 bcc hcp Temperature, K 15 bcc hcp gti 5 x(ti) (a) g Ti Phase Diagram, with gt i 2 metastable. 5 x(ti) (b) g Ti Phase Diagram, with gt i 2 and gt i metastable. Figure 9: Metastable g T i Phase Diarams. How do the Gibbs Free energy curves look like???? THE END!!!

Thermodynamic parameters in a binary system

Thermodynamic parameters in a binary system Previously, we considered one element only. Now we consider interaction between two elements. This is not straightforward since elements can interact differently