 Course number: 527 M1700  Designation: Graduate course  Instructor: Chao-Sung Lin, MSE Dept., (office), (lab)  Office hours: 2 ~ 5 pm, Thursday Phase Transformation

Textbook and suggested references  Phase Transformations in Materials [electronic resource available in NTU library], Gernot Kostorz ed., Weinheim, New York Chichester, Wiley-VCH. (Strongly Recommended)  Phase Transformations in Metals and Alloys, D. A. Porter and K. E. Easterling, 2nd ed., Nelson Thornes, UK (Strongly Recommended)  The Theory of Transformations in Metals and Alloys: Equilibrium and general Kinetic Theory, J. W. Christian, 2nd ed., Pergamon Press Ltd.  Solidification Processing, M. C. Flemings, McGRAW-HILL.  Diffusion in Solids, P. G. Shewmon, McGRAW- HILL.

Recommended background  Materials Science and Engineering  Physical Metallurgy Physical Metallurgy Principles, R. E. Read-Hill  Metallurgical Thermodynamics Materials Thermodynamics, D. R. Gaskell Chemical Thermodynamics of Materials, C. H. P. Lupis.

Topics  Thermodynamics and Phase diagrams (4)  Solidification (3)  Diffusion Kinetics (3)  Second Phase Precipitation (4)  Diffusionless Transformations (2) Grading  Mid-Term (50%)  Final (50%)

Thermodynamics and Phase diagrams Thermodynamic properties  External influences P, T, V (any two of them are required)  Thermodynamics properties U, internal energy S, entropy H = U + PV, enthalpy G = H – TS, Gibbs free energy A = U – TS, Hemholtz free energy  Useful parameters c v, constant volume heat capacity c p, constant pressure heat capacity

Gibbs free energy change  Partial molar Gibbs energy of an ideal gas is the standard molar Gibbs energy (at 1 bar) p i is the partial pressure in bar Consider the reaction for the formation of 2 dn moles of H 2 S

Chemical equilibrium  The reaction will proceed as  G < 0  When equilibrium is reached,  G = 0 Where K is the equilibrium constant Note:  G < 0 is a necessary condition for a reaction to occur, but is not a sufficient condition.-  need to consider the rate of the reaction

Predominance diagrams  A particularly simple of phase diagram as calculated by the equilibrium constant at a specific temperature. For the Cu-S-O system at 1000 K, and consider the coexistence of Cu 2 O and Cu 2 SO 4 a straight line with a slope of (-3/2)/2 = -3/4 vertical line

Procedure: 1.Choose Cu as the base element. 2.Consider the formation each solid compound from one mole of the base element, Cu. Say, for CuO 3.Obtain  G 0 from tables of thermodynamics properties  G is calculated for any given p SO2 and p O2. 5.The one with the most negative  G is the stable compound.

Ellingham diagrams For the Cu-O system, choosing temperature and partial pressure of O 2 as the axes

Thermodynamics properties of mixing  The properties change result from the mixing process  Y m = Y after mixing – Y before mixing Gibbs energy of mixing for the mixing of liquid Au and Cu to form 1 mole solution Note: for the solution to be stable,

Chemical potential  The partial molar quantity of a thermodynamics property  The partial molar Gibbs free energy, also known as the chemical potential, is the rate of the Gibbs energy as component i is added. Note:  i =  i (T, n) is an intensive function. 2.When two or more phases are in equilibrium, the chemical potential of any component is the same in all phase. I II dn Cu

Tangent construction divided by n Au + n Cu 1 g Cu - g Au

Gibbs-Duhem equation In general,

Relative partial properties  The difference between the partial Gibbs energy of a component in solution and that in a stand state.  The activity a i of the component relative to the chosen standard state is defined in terms of the relative partial Gibbs energy as For Au-Cu binary system where  h m is the enthalpy of mixing  s m is the entropy of mixing

Ideal Raoultian solution  For ideal solution  Need to consider configuration entropy only, Boltzmann’s equation Appling Stirling’s approximation

Excess properties  y E = y - y ideal

 An equation obtained using tangent constructions  The Gibbs-Duhem equation of excess properties Note: 1.Excess Gibbs energy markedly influences the form of the phase diagram. 2.If g E < 0, the solution is thermodynamically more stable than an ideal solution. 3.g E > 0  positive deviation, the same atoms dislike each other 4.g E < 0  negative deviation, different atoms like each other  Activity coefficient

Binary phase diagram  Complete solid and liquid miscibility  Complete liquid miscibility, with solid miscibility gap Eutectic Peritectic  Liquid miscibility gap  Intermediate phases tie line

latent heat of fusion Note: As T decreases, the g s curve descends relative to g l. Common tangent construction to ensure that the chemical potential of Ge and Si are equal in the solid and liquid phases at equilibrium.