1. CT – 4: Phase diagrams Phase diagrams: definition and types, mapping a phase diagram, implicitly defined functions and their derivatives. Optimisation methods: the principle of the least- squares method, the weighting factor, Marquardt‘s algorithm

2. Dimensions of phase diagrams Two- or three- dimensional manifold of phase equilibria can be visualized as phase diagram („Higher-order“ phase diagram – usually 2- or 3- dimensional sections) dimensional sections) The largest dimension of phase diagram – number of free variables under the equilibrium conditions (diminishing it introducing x i, H m etc.- recalculating extensive variables to intensive ones) Example: Diagrams in two dimensions: Diagrams in two dimensions: Binary phase diagram: p = const. Binary phase diagram: p = const. Ternary phase diagram: p, T = const. or p, x i = const. Ternary phase diagram: p, T = const. or p, x i = const. Quaternary phase diagram: p, T, x i = const. or: p, x i, x j = const. Quaternary phase diagram: p, T, x i = const. or: p, x i, x j = const. n – components phase diagram: (n-1) – intensive variables constant n – components phase diagram: (n-1) – intensive variables constant

3. Phase diagram and property diagram  Example:  Functional dependence (property diagram): only points on the curve have meaning only points on the curve have meaning V m = V(T), phase , xi, p=const. V m = V(T), phase , xi, p=const. (V m = (V o /T o )T for ideal gas) (V m = (V o /T o )T for ideal gas)  Phase diagram of unary system: p = f(T, V m ) can be drawn at the same coordinates, but T and V m are independent variables, pressure depends on the two coordinates T and V m and coordinate space shows the can be drawn at the same coordinates, but T and V m are independent variables, pressure depends on the two coordinates T and V m and coordinate space shows the „state of the system“ („state“ diagram – phase diagram). „state of the system“ („state“ diagram – phase diagram).

4. Lines in phase diagram can be interpreted as functional dependence Under condition of monovariant equilibrium of a set of specified phases, one coordinate of phase diagram is an implicitly defined function of the other one (functional dependence)

5. Types of phase diagrams Fig.2.6 Three types (Schmalzried and Pelton – 1973): - Axis - two intensive variables (a),(b) - Axis - one intensive variable and quotient of extensive variables (c),(d) - Axis – two quotients of extensive variables (e), (f)

6. Types of phase diagrams LFS - CT

7. Types of phase diagrams –cont. Plane phase diagrams exist for n-component system with (n-1) intensive variables kept constant Lines in phase diagrams represents monovariant equilibria (in Fig.2.6 two-phase equilibria) The intensive variables are the same for all phases in equilibrium, extensive variables differ Coordinates, quotients of extensive variables, represent overall state variable of the whole system as well as individual state variables of the phases

8. Types of phase diagrams –cont. Variables of the individual phases are represented by pair of lines belonging to the two coexisting phases (Fig.2.6, c, d, e, f) Corresponding points on these pairs of lines are connected by stright lines called „tie-lines“ (lines connecting the compositions of phases in equilibrium – isotherms in binary phase diagram, tie-lines or tie-triangles in ternary diagram) tie-lines or tie-triangles in ternary diagram) Intensive coordinate (constant along the „tie-lines“) may be parallel to the axis of the quotient of extensive variables. (Fig.2.6,c,d) The direction of „tie-lines“ is an important part of the information given by phase diagram (Fig.2.6,e,f)

9. Types of phase diagrams –cont. The monovariant equilibria meet in invariant equilibria (where variable of the phases are represented by points – single point in the Fig.2.6.a,b, separate points in other diagrams – in the type of Fig.2.6.c,d three points on a straight line - in the type of Fig.2.6.e,f they form triangle)

10. Reading of binary phase diagram Binary phase equilibria, eutectic, peritectic, eutectoid, peritectoid Example: Fe-C diagram (Callister W.D.: Materials Science and Engineering, John Wiley 1999.)

11. Stable and metastable Fe-C phase diagram (Callister W.D.: Materials Science and Engineering, John Wiley 1999.) and Engineering, John Wiley 1999.)

12. Rules for monovariant lines Thermodynamically possible Thermodynamically impossible

13. Reading ternary phase diagram Ternary phase equilibria, two-phase field, three-phase field Example: Fe-Cr-Ni phase diagram (Raynor G.V., Rivlin V.G. Phase Equilibria in Iron Ternary Alloys. Inst. Of Metals 1988.)

14. Reading ternary phase diagram – cont. Al-Cr-C system: Hallstedt B.:Calphad XXXVIII Prague 2009 - Book of abstracts Two-phase fields with tie-lines Three-phase fields are tie-triangles

15. Reading ternary phase diagram Liquidus projection: eutectic, peritectic, phase transformation Example: Fe - Mo – W Vrestal J. in Landolt-Börnstein Series 2009 in print

16. Phase transformations on liquidus surface E (eutectic) U (phase transformation) P (peritectic)

17. Vertical sections of ternary systems (isopleths) Tie-lines generally are not in the plane of section! The lines do not belong to monovariant equilibria - they show „zero-phase-fraction“ equilibria. Lines show boundaries between an n- and (n-1)-phase field (calculated from equilibrium condition for the n-phase equilibrium with additional condition that amount of one phase is zero, although still present in equilibrium).

18. 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 are usually stored – any phase diagram may be displayed at the end of calculations (In Fig.2.6. diagrams b) – e) could be plotted from single mapping)

19. In non-isoplethal phase diagram calculations: monovariant equilibrium is traced, keeping except one of the axis variable constant and increasing the only variable axis in steps until an invariant equilibrium is found or one of the selected limits of the axis exceeded. Procedure repeated for all monovariant equilibria In isoplethal phase diagrams calculations: „zero-phase-fraction“ line is traced by setting the appropriate conditions: the set of stable phases is constituted by all phases appearing in both areas adjacent to the line, plus the phase appearing only in one area – this phase is assigned the fixed amount of zero. Procedure is continued until an axis limit is reached If in phase diagram a set of lines is completely separated from another set, the mapping procedure must be continued with an additional starting equilibrium leading to the other set (Example: Ag-Pd FCC miscibility gap) Mapping a phase diagram – cont.

20. Example (in Thermocalc - Cu-Sn system): set-axis-variable 1 x(Sn) 0 1.025 s-a-v 2 t 200 1400 10 map Example (in Thermocalc – Bi-Cu-Sn system at 1000 K): set-axis-variable 1 x(Cu) 0 1.025 s-a-v 2 x(Sn) 0 1.025 map -------------------------------------------------------- „sorry, can not continue“ is warning in Thermocalc here, that Newton-Raphson procedure does not converge – may be for bad starting values

21. Types of phase diagrams LFS - CT

22. Implicitly defined functions and their derivatives Fig.2.6.b: T=T(  Mg ) or  Mg =  Mg (T) - implicitly Fig.2.6.c: T implicitly defined function of x Mg MgCu2 as well as the inverse function x Mg C15-MgCu2 (T) for the boundary of two-phase field „C15-Mg 2 Cu“ against „C15-MgCu 2 “ Derivatives of these implicitly defined functions are also defined Gibbs-Konovalov rule: (dT/dx  ) coex = [(x  - x  ) T (  2 G m  /  x  2 )] / [(H 1  - H 1  ) (1- x  ) + (H 2  - H 2  ) x  ] [(H 1  - H 1  ) (1- x  ) + (H 2  - H 2  ) x  ]

23. Optimization methods Set of n-measurable values W i depends on set of unknown coefficients C j via functions F i with values of independent variables x ki : W i = F i (C j, x ki ) i = 1,..., n, j = 1,..., m W i = F i (C j, x ki ) i = 1,..., n, j = 1,..., m (k distinguishes the various independent variables (T, x i...) belonging to measurement i, n  m) W i corresponds to measured value L i W i corresponds to measured value L i Best set of C j : sum of squares of the „errors“ (calculated values by F i – experimental values L i ) (calculated values by F i – experimental values L i ) must be minimal must be minimal „Errors“ are multiplied by weighting factor p i

24. Weighting factor Weighting factor p i is defined by „error equation“: (F i (C j, x ki ) – L i ). p i = i (F i (C j, x ki ) – L i ). p i = i Condition for the best values Cj:  n i=1 i 2 = Min (with respect to the Cj)  n i=1 i 2 = Min (with respect to the Cj)

25. Weighting factor – cont.  n i=1 i 2 = Min (with respect to the Cj) For every coefficient C j, m equations can be derived:  n i=1 i. (  i /  C j ) = 0 j = 1,..., m  n i=1 i. (  i /  C j ) = 0 j = 1,..., m Gauss: i expanded into a Taylor series with only linear terms: i ( C j, x ki )  i o ( C j o, x ki ) +  m l=1. (  i /  C l ).  C l i ( C j, x ki )  i o ( C j o, x ki ) +  m l=1. (  i /  C l ).  C l  C j are corrections to the coefficients C j  C j are corrections to the coefficients C j

26. Weighting factor – cont. Calculation of corrections  C j :  m l=1 (  n i=1 (  i /  C j )(  i /  C l )  C l = -  n i=1 i. (  i /  C j ) j = 1,..., m j = 1,..., m set of m such linear equations for m unknowns  C j : „Gaussian normal equations“

27. Measure of the fit Fit between the resulting coefficients and the measured values can be defined by: Mean square error =  n i=1 ( i 2 /(n-m)) where: n - measurable values, m – unknown coefficients C j, i - defined as (F i (C j, x ki ) – L i ). p i = i, p i weighting factor i - defined as (F i (C j, x ki ) – L i ). p i = i, p i weighting factor

28. Weighting factor – cont. In the optimisation procedures (PARROT etc.): estimated uncertainities of measured quantities are introduced Additionally introduced dimensionless factor „weight“ is left to the responsibility of the user to assign weights that reflect the relative importance of data (default settings of all weights are equal 1)

29. First attempt to optimization Trial and error optimization (by hand): Influence of the change of parameter to the position of phase boundaries – starting value for optimisation program (e.g.PARROT in TC). Example: Cr-Ti Laves phases

30. Cr-Ti-Laves phase C14 PARAMETER G(LAVES_PHASE,CR:TI:CR;0) 298.15 +4*GHSERTI+8*GHSERCR -101605.-3.20*(T-T*ln(T)); 6000.0 N 93 ! PARAMETER G(LAVES_PHASE,CR:TI:CR,TI;0) 298.15 -50000.; 6000.0 N 93 ! Correct value = 3.20Trial value = 2.90 - wrong Pavlů J., Vřešťál J., Šob M.: Calphad 34 (2010) 215

31. Marquardt’s algorithm Combination of Newton-Raphson method with steepest-descent method for equations, nonlinear in coefficients (D.W.Marquardt 1963) Calculation of corrections  C j :  m l=1 (  n i=1 (  i /  C j )(  i /  C l )  C l = -  n i=1 i. (  i /  C j ) j = 1,..., m j = 1,..., m may not converge for  C j, then factor called Marquardt parameter is added to the normalized matrix of equation above. For large Marquardt parameter, the steepest descent method is used, for the small of it, the pure Newton-Raphson technique is in work.

32. Calculation of phase diagram by suspending one phase Example: Metastable Fe-Fe 3 C diagram can be calculated (with cementite) if graphite (and diamond) are suspended

33. Calculation of metastable equilibria Extending this idea: we can calculate any equilibrium for phase, or an assemblage of phases, irrespective of whether these phases represent the stable state of the system

34. Calculation of metastable equilibria – cont. The same rules apply to the metastable equilibria as to the stable one. The total Gibbs energy (at const.T,p) is higher then that of stable equlibrium (could be used for the check). Similarity with stable equilibria: it is still a minimum for the phases included in the calculation and the chemical potentials for all components are the same in all phases.

35. Questions for learning 1. Explain difference between phase diagram and property diagram 2. Find monovariant equilibria and invariant equilibria in binary and ternary phase diagrams 3. Give the rules for extrapolation of monovariant lines in binary phase diagrams 4. Give the rules for direction of monovariant lines in liquidus projection at invariant points at invariant points 5. Define weighting factor and explain its use during optimisation of themodynamic parameters. What is Marquardt‘s algorithm?