1. CT – 3: Equilibrium calculations: Minimizing of Gibbs energy, equilibrium conditions as a set of equations, global minimization of Gibbs energy, driving force for a phase

2. Equilibrium conditions dU = T.dS-p.dV +  j  j.dn j (see CT-2) Conjugated properties: T,p,  j – intensive, S,V,n j – extensive At constant entropy, volume, and n j, equilibrium is characterized by minimum of the internal energy. Most proper conditions: p,T,n j dG = V dp – S dT +  j  j dn j (mole fraction: x i = n i /N, N =  j n j ) G may have several minima. That with the most negative value of G is „global minimum“ which corresponds to the „stable equlibrium“ and another ones are „local minima“ and correspond to the „metastable equilibria“

3. Stable and metastable states Metastable state Stable state Unstable state Metastable state Stable state

4. Equilibrium conditions – cont. For determination of phases presented in equilibrium: analytical expression of G needed. Total Gibbs energy: G =   m .G  m (m   0, it is amount of phase  ) Amount of components i: N i = N.x i o Introduce: x i =   m . x  i - lever rule Equilibrium condition: min (G) = min (   m . G  m (T,P, x  i or y (l,  ) k )) (x i  is definite function of y k (l,  ) ; m , x  i are unknowns)

5. 6 Rule 1: If we know T and C o, then we know: --the composition and types of phases present. Example : Cu-Ni phase diagram Adapted from Fig. 9.2(a), Callister 6e. (Fig. 9.2(a) is adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH, 1991). PHASE DIAGRAMS: composition and types of phases

6. 7 Rule 2 : If we know T and C o, then we know: --the composition and amount of each phase. Example : Cu-Ni system Adapted from Fig. 9.2(b), Callister 6e. (Fig. 9.2(b) is adapted from Phase Diagrams of Binary Nickel Alloys, P. Nash (Ed.), ASM International, Materials Park, OH, 1991.) PHASE DIAGRAMS: composition and amount of phases Lever rule: m L /m  = C o C  /C o C L

7. Types of phase diagrams of Cu

8. Equilibrium conditions as a set of equations Equilibrium condition using chemical potential: Constraints relating m , x i  and N i used to eliminate variables m  and N i : G i  (T,P,x i  ) = G i  (T,P,x i  ) (i = 1,…,c,  = 1,…p-1,  =  + 1,…,p) By the definition: G i  (T,P,x i  ) =  i (i = 1,…,c,  = 1,…,p) Nonlinear equations – appropriate iteration algorithm Unknown: x i  and  i (for stoichiometric phases, modifications are necessary)

9. G m  as function of site fractions y k (l,  ) instead of mole fractions x i  Lagrange-multiplier method Constraints: (1) Total amount of each component N i is kept constant (2) Sum of site fractions in each sublattice is equal 1 (3) Sum of charge of ionic species in each phase is equal 0

10. Constraints mathematically LFS - CT

11. Lagrange-multiplier method-cont. Each constraint is multiplied by „Lagrange multiplier“ and added to the total Gibbs energy min (G) = min (   m . G  m (T,P, x  i or y (l,  ) k )) to get a sum L. If all constraints are satisfied, L is equal to G and a minimum of L is equivalent to a minimum of the total Gibbs energy G.

12. Newton‘s method To find x for which y=0: (also for searching the minimum of G m ) (df/dx) x=xi.  x i = -f(x i ), x i+1 = x i +  x i There exists cases, where this method diverges LFS - CT

13. There exists cases, where Newton‘s method diverges Starting with x 1 - diverges Starting with x 3 – solution on the left, Starting with x 4 – solution on the right, x5, x6 - finally on the left – influence of starting values on the result of minimization LFS - CT

14. Newton-Raphson method It is extension of Newton‘s method to more than one variable (n equations for n unknowns). All iterative techniques like the Newton- Raphson one need an initial constitution for each phase in order to find the minimum of Gibbs energy surface for the given conditions.

15. Thermocalc – starting point tools Automatic starting values Set-all-start values

16. Global minimization of the Gibbs energy Miscibility gap problem (solution phases only) LFS - CT

17. Compounds with fixed compositions Equilibrium set of phases is given by the tangent „hyperplane“ defined by the Gibbs energies of a set of compounds constrained by the given overall composition and with no compound with a Gibbs energy below this hyperplane („global“ minimum).

18. Gibbs energies of a set of compounds A x B B G/kJ.mol -1 LFS -CT

19. Minimization techniques to find global equilibrium Gibbs energy surface of all solution phases is approximated with a large number of „compounds“ which Gibbs energy has the same value as the solution phase at the composition of the compound (dense grid about 10 4 (100x100) compounds, for multicomponent system about 10 6 such compounds) Search for hyperplane representing equlibrium for the compounds is then carried out.

20. Minimization techniques to find global equilibrium When minimum for these „compounds“ has been found, the „compound“ in this equilibrium set must be identified with regard to which solution phase they belong to. Each „compound“ --- initial constitution of the solution phase --- is used in a Newton-Raphson calculation to find the equilibrium for the solution phases (correct, not wrong)

21. Limitation of the method to find the global equilibrium T, p and overall composition must be known For other conditions as starting point (e.g. activity of components) –-- indirect procedure: Overall composition calculate first and use it for a new equilibrium calculation.

22. Conditions for a single equilibrium The equilibrium conditions as a set of equations contain fewer equation than unknowns – the difference = number of degrees of freedom „f“ Therefore: „f“ extra conditions (equations) must be added to select definitely single equilibrium „Unknown state variable“ = „constant value“ : Example: For binary system i-j: (f=0) T = 1273, p = 101352, x i = 0.1,  i = -40000 Thermocalc: Fe – W – Cr system set-condition t=1273 x(W)=0.15 x(Cr)=0.35 p=1E5 n=1

23. Conditions for a single equilibrium – cont. For each calculation step: which and how many phases are present (Gibbs energy description exist only for phases). Calculation steps with different sets of phases may be compared The phases set with lowest Gibbs energy describes the stable equilibrium Example: Thermocalc: rej ph * res ph liq bcc fcc sigma Chi R Mu

24. Example Different starting points may give different sets of equilibrium phases for the same overall composition. Check the total Gibbs energy for global minimum (In new codes checked automatically.)

25. Ag-In system

26. Output from POLY-3, equilibrium number = 1, Ag-In system Conditions: T=500, X(IN)=2E-1, P=100000, N=1 DEGREES OF FREEDOM 0 Temperature 500.00, Pressure 1.000000E+05 Number of moles of components 1.00000E+00, Mass 1.09260E+02 Total Gibbs energy -3.15128E+04, Enthalpy -1.91077E+02, Volume 0.00000E+00 Overal composition Component Moles W-Fraction Activity Potential Ref.state AG 8.0000E-01 7.8982E-01 1.6046E-03 -2.6752E+04 SER IN 2.0000E-01 2.1018E-01 5.2290E-06 -5.0558E+04 SER FCC_A1#1 Status ENTERED Driving force 0.0000E+00 Number of moles 5.6253E-01, Mass 6.1400E+01 Mass fractions: AG 8.06153E-01 IN 1.93847E-01 HCP_A3#1 Status ENTERED Driving force 0.0000E+00 Number of moles 4.3747E-01, Mass 4.7860E+01 Mass fractions: AG 7.68872E-01 IN 2.31128E-01

27. Mapping a phase diagram 2 or 3 variables of the conditions are selected as axis variables with lower and upper limit and maximal step. All additional conditions – kept constant throughout the whole diagram Start: „initial equilibrium“ for Newton-Raphson calculation (with all phases „entered“) All results of calculations usually stored – any phase diagram may be displayed at the end of calculations

28. Mapping a phase diagram – cont. Example (in Thermocalc): set-axis-variable 1 x(Ag) 0 1.025 s-a-v 2 t 300 1200 10 map (T in K)

29. „Stepping“ By stepping with small decrements of the temperature (or enthalpy or amount liquid phase-generally one variable) one can determine the new composition of the liquid and then remove the amount of solid phase formed by resetting the overall composition to the new liquid composition before taking the next step (Scheil solidification scheme: no diffusion in solid phase, high diffusion in liquid phase)

30. Example – Scheil-Gulliver solidification scheme x Ni = 0.1

31. Azeotropic points Maxima and minima of binary two-phase fields Setting additional conditions For binary: x  - x  = 0 For ternary: x B  - x B  = 0 x C  - x C  = 0

32. The driving force for a phase LFS - CT

33. Driving force-application Driving force ΔG ,  G FCC : (Fig.2.5) (difference in G of paralel tangents for phases and stability tangent) - theory of nucleation of phases - minimization of G (whether another phases set exists that is more stable than calculated set of phases)

34. Conditions for a single equilibrium – cont. Adding phase to the calculated stable phase set: Positive „driving force“ of the phase – repeat calculation Removing phase from the selected set: Calculation finds negative amount for one of selected phases

35. Conditions for a single equilibrium – cont. Phases with miscibility gaps may have more than one driving force at different compositions – test must be performed for each of these compositions Test by experiment when some phases appear in calculations to be stable but experimentally are found to be not stable

36. Questions for learning 1.What is a difference between stable and metastable states? 2. What is principle of Lagrange-multiplier method? 3. What is principle of Newton – Raphson method? 4. What conditions must be fulfilled for single equilibrium calculation? 5. What means „mapping“ and „stepping“ in calculations of phase equilibria?