Models for the excess Gibbs energy: models with three or more sublattices, models for phases with order-disorder transitions, Gibbs energy for phases that never disorder, models for liquids, chemical reactions and their models CT – 12: Models for the excess Gibbs energy:

Models with three or more sublattices Model for phase with three sublattices is (A, B, …) a1 (K, L, …) a2 (U,V,…) a3 Molar fraction x i of components we get from constituent fractions y j by using x i =  j b ij y j / (  k  j b kj y j ) (b ij are stoichiometric factors of component i in constituent j) The formula unit of a phase with sublattices is equal to the sum of the site ratios  s a (s), where the ratios a (s) describe the ratio between the numbers of sites N (s) on each sublattice. Molar fractions x i we get from site fractions y j (s) by using (vacancy excluded) x i =  s [  j b ij y j (s) / (a (s)  k  j b kj y j (s) ) ] More sublattices model can be used only if we have enough reliable experimental data or ab initio calculated data or crystallographic relations exists (ordering on the fcc lattice)

Models with three or more sublattices-cont. The Gibbs-energy expression for a phase with three sublattices is Most important are parameters in the surface of reference, o G i:j:k ! LFS - CT

Models for intermetallic phases - example The  phase has five different crystallographic sublattices Simplification using coordination numbers : Sublattice: 2a 4f 8i 8i´ 8j Coord. number: Simplified model: (2a + 8i´)=10 4f=4 ( 8i + 8j)=16 Similarity: fcc bcc mixture (bcc-type)

Models for intermetallic phases – cont. Simplifications using ab initio calculated (at 0 K) results with CEF: Two sublattice model: (A,B)(Va,…) Vrestal (2001) Five sublattice model: (Re,W) 2 (Re,W) 4 (Re,W) 8 (Re,W) 8 (Re,W) 8 Fries,Sundman (2002), SRO neglected, substitutional disordered regular solution parameter introduced Proposed model: (A,B) 10 (A,B) 20 Joubert (2006) Problem is under discussion yet – no unification of models exist. In databases, the literature available data are used (mainly three sublattice model 10:4:16, coordination number based, mixing in last sublattice only).

Models for intermetallic phases – example LFS - CT

Models for intermetallic phases – example LFS - CT

Models for intermetallic phases – example LFS - CT

Models for metal-non-metal phases Carbide, nitride, boride phases: Crystallographic information M 23 C 6 : (Cr,Fe,…) 21 (Cr,Fe, W, Mo,…) 2 C 6 It does not describe the full composition range. Model: (Cr,Fe,…) 20 (Cr,Fe, W, Mo,…) 3 C 6 is used The wustite phase – constituents are ions: Model: (Fe 2+, Fe 3+, Va) 1 (O 2- ) 1 Sundman (1991) The spinel phase: Model: (Fe 2+, Fe 3+ ) 1 (Fe 2+, Fe 3+, Va) 2 (O 2- ) 4 Four sublattice model is also used for spinel phase: Example:Al 2 MgO 4 (MgO-Al 2 O 3 ) Model: (Mg 2+, Al 3+ ) 1 (Al 3+, Mg 2+, Va) 2 (Va, Mg 2+ ) 2 (O 2- ) 4 Hallstedt(1992)

Models for phases with order-disorder transition Ideal model (without excess parameters): treatment of ordering is identical with Bragg-Williams-Gorsky treatment Excess parameter in CEF: more realistic description: Disordered state: constituents distributed randomly on sublattices Ordered state: constituents have different fractions in different sublattices First order transition often (Cu-Au). Degeneracy enforces several restrictions on the possible parameters in CEF, otherwise disordered state would never been stable – Ansara (1988). Model: two different descriptions for ordered and disordered phase: In literature for D0 22, L2 1, (not generally recommended). Reasonable contribution of the configurational entropy to ideal entropy in sublattice model for ordered phase (modeled in terms of excess Gibbs energy). For future – models using CVM are recommended

Disordered state of an ordered state – partitioning of Gibbs energy expression Single Gibbs energy function for the ordered and disordered states is disadvantageous in influencing the disordedred state by parameters which describe ordered phase and it is also cumbersome for multicomponent system – therefore partitioning: General part – depends only on the composition of the phase (mole fractions x) Ordering part – contribution of LRO only (depends on site fractions y): G m = G m dis (x) +  G m ord (y)  G m ord (y) must be zero when phase is disordered:  G m ord (y) = G m ord (y) - G m ord (y replaced by x) Ordering (LRO) decreases with increasing temperature and at T c dissapears. Some SRO remains even in disordered state (T >T c ).

Quasi-chemical model and LRO Calculate „site fraction“ from the „bond fraction“: y A ‘ = y AA + y AB y B ‘ = y BA + y BB y A ‘‘ = y AA + y BA y B ‘‘ = y AB + y BB Here, it is y AB ≠ y BA for to be able to describe LRO Quasi-chemical model can be formally treated as two-sublattice model with a contribution from SRO (by variable  ): y AA = y A ‘ y A ‘‘ +  y AB = y A ‘ y B ‘‘ -  y BA = y B ‘ y A ‘‘ -  y BB = y B ‘ y B ‘‘ +  Quasi-chemical model is suitable namely for liquids. For large SRO - negative entropies appear (when not allowing LRO). Models for crystalline phases with explicit SRO: CVM based methods

Simultaneous L1 2 and L1 0 ordering in FCC lattice Four sublattice model used for L1 2 and L1 0 ordering on the fcc lattice Disordered A1 phase can be described with the same model as above Restrictions on parameters from the symmetry of the lattice – e.g. L parameters are described in disordered part (not used for ordered part). If bond energy u AB depends only slightly on composition – G parameters can be written: o G A:A:A:B = G A3B = 3u AB +  u 1  u 1,  u 2 are corrections to the experimental data o G A:A:B:B = G A2B2 = 4u AB Numbers 3 and 4 come from the number of AB o G A:B:B:B = G AB3 = 3u AB +  u 2 bonds in each end member.

Approximation of SRO contribution to the Gibbs energy SRO contribution to the Gibbs energy of the fcc phase can be approximated with L A:B:C:D = -  G 2 /(zRT). For four-sublattice model one can have three different such parameters: L A,B:A,B:A:A = L A,B:A:A,B:A = … = L AA L A,B:A,B:A:B = L A,B:A:A,B:B = … = L AB L A,B:A,B:B:B = L A,B:B:A,B:B = … = L BB In the lack of experimental data one may set all of these parameters equal and write L ** = u AB +  u 3 (Abe and Sundman 2003) Examples are shown further.

Approximation of SRO contribution to the Gibbs energy-example. LFS - CT

FCC L1 0 /L1 2 ordering Partitioned single Gibbs-energy function: G m tot = G m A1 (x i ) +  G m ord G m A1 (x i ) =  i=A,B x i o G i + RT  i=A,B x i ln(x i ) + x A x B  4 =0 (x A – x B ). L A1 A,B  G m ord = G m ord (y i ) - E G m ord G m ord (y i ) =  i=A,B  j=A,B  k=A,B  l=A,B y (1) i y (2) j y (3) k y (4) l o G ijkl + + RT  4 s=1  i=A,B y (s) i ln(y (s) i ) + E G m ord E G m ord =  3 s=1  4 t=s+1 y (s) A y (s) B y (t) A y (t) B L ** For the disordered part of Gibbs energy (G m A1 (x i ) ), the contribution of SRO must be included and parameters L will have following values: o L A1 A,B = G A3B + 1.5G A2B2 + G AB L AA L BB + l 0 1 L A1 A,B = 2G A3B – 2G AB L AA – 0.75 L BB + l 1 2 L A1 A,B = G A3B – 1.5G A2B2 + G AB3 – 1.5L AB + l 2 3 L A1 A,B = -0.75L AA L BB 4 L A1 A,B = -0.75L AA + 1.5L AB – 0.75L BB It can be derived in four-sublattice model with (y i = x i ) using substitional model

Approximation of SRO contribution to the Gibbs energy-example c Values of u AB = ,  u 1 = and  u 2 = in J.mol -1. l o is disordered parameter. It can be applied also to ordering in hcp phases. LFS - CT

Transforming a four-sublattice ordered fcc model to the two sublattice model The relation between the parameters for the two-sublattice model L1 2 can be derived from a four-sublattice model in which the site fractions on three sublattices are set equal and related to normal parameters in two sublattice models. Two sublattice model – calculations are significantly faster. Software generates these parameters in equations.

Transforming a four-sublattice ordered fcc model to the two sublattice model - equations New symbols introduced in equations above are defined as: (u AB are from binary system,  u 4 up to  u 7 can be optimized to fit data in the ternary system ) LFS - CT

B32, D0 3, and L2 1 ordering in BCC lattice B32, D0 3, and L2 1 are ordered forms of the A2 structure type (BCC) and they require four sublattices for their modeling BCC ordering requires two bond energies: nearest and next-nearest neighbors. For B2 ordering (BCC), it is sufficient to have two sublattices (central atom and eight corners atoms) Ideal composition of B32 is AB (as for B2), but all nearest neighbors are different Ideal composition of D0 3 is A 3 B (as for L1 2 ), but D0 3 ordering does not have identical surroundings in the three sublattices The L2 1 phase - example: Heusler phase A 2 BC (Cu 2 MnAl)– only in ternary systems (The same arrangement of sites as D0 3, but two sublattices have the same atoms of A, other two have different elements B,C).

Ordered phases, which never disorder but are not stoichiometric compounds Intermetallic phases, like sigma phase, Laves phase etc., can be described by partitioning to disordered substitutional and ordered sublattice description: G m = G m dis (x) – T cnf S m dis +  G m ord  G m ord (y) = G m ord (y) G m dis (x) is described by substitutional model, G m ord (y) includes sublattice according the crystalline structure cnf S m dis is subtracted from disordered part and configurational entropy is calculated for ordered part only. Examples: Laves phase C15: (A,B) 2 (A,B) Sigma phase: (A,B) 10 (A,B) 4 (A,B) 16 or (A) 10 (B) 4 (A,B) 16 (A) 8 (B) 4 (A,B) 18

Example Database for Laves phase C15: PHASE LAVES_C ! CONST LAVES_C15 :CR,ZR:CR,ZR: ! PARAMETER G(LAVES_C15,CR:CR;0) *GHSERCR; N 93 ! PARAMETER G(LAVES_C15,ZR:ZR;0) *GHSERZR; N 93 ! PARAMETER G(LAVES_C15,ZR:CR;0) GHSERCR+2*GHSERZR; N 93 ! PARAMETER G(LAVES_C15,CR:ZR;0) GHSERZR+2*GHSERCR *T; N 93 ! PARAMETER G(LAVES_C15,CR:CR,ZR;0) ; 6000 N 93 ! PARAMETER G(LAVES_C15,ZR:CR,ZR;0) ; 6000 N 93 ! PARAMETER G(LAVES_C15,CR,ZR:CR;0) ; 6000 N 93 ! PARAMETER G(LAVES_C15,CR,ZR:ZR;0) ; 6000 N 93 !

Example Database for sigma-phase PHASE SIGMA I ! CONST SIGMA :FE,MN,NI:CR,MO,V,W:CR,FE,MN,MO,NI,V,W : ! Fe-Cr system: PARAMETER G(SIGMA,FE:CR:CR) *GFEFCC+22*GCRBCC *T; N HIL91,LEE92 ! PARAMETER G(SIGMA,FE:CR:FE) *GFEFCC+4*GCRBCC +18*GFEBCC *T; N HIL91,LEE92 ! Fe-Mo system: PARAMETER G(SIGMA,FE:MO:MO) *GFEFCC+22*GMOBCC *T; 2200 N AND88 ! PARAMETER G(SIGMA,FE:MO:FE) *GFEFCC+18*GFEBCC +4*GMOBCC *T; N AND88 FRI89 !

Models for liquids Models used for liquids: Substitutional-solution model (CT-9) Associate-solution model (for systems with tendency to SRO) (CT-11) Quasi-chemical entropy for liquids (improvement of associate model) The cell model (specially for oxides – cell as constituent) Ionic-liquid two-sublattice model

The modified quasi-chemical model Associate-solution model uses the ideal configurational entropy – improvement: quasi-chemical entropy expression – Hillert (2001) Simple system: (A 1+, B 1+ ) P (C 1-, D 1- ) Q The configurational entropy in ionic-liquid model is generally given by: cnf S m = -R[ P  i y Ci ln(y Ci ) + Q(  j y Aj ln(y Aj ) + y Va ln(y Va ) +  k y Bk ln(y Bk ))], where P and Q are equal to the average charge on the opposite sublattice, and the Gibbs energy in the present system is (P=Q=1): LFS - CT

The modified quasi-chemical model-example LFS - CT

The cell model Kapoor (1974), Guy and Welfringer (1984) Special form of quasi-chemical entropy (cell with one anion and two cations) – originally developed for oxides (CaO – SiO 2 ). Cell is treated like constituent. Entropy expression of the cell model is: where u,v are stoichiometric coefficients in oxide, D i =  n j=i v j x j, first two sums are over all component oxides, last sum over j is for all m constituents. Model is not often used.

The partially ionic-liquid two-sublattice model Representative system (A a+, B b+ ) P (C c-, D d- ) Q Electro-neutrality condition for P  Q: introducing equivalent fractions defined by z A = (N A /a)/ ((N A /a) + (N B /b)) z C = (N C /c)/ ((N C /c) + (N D /d)) Where a+, b+, c-, and d- are the valences of A, B, C, and D, respectively, and P = Q = 1. It is not possible to extend this introduction of equivalent fractions to systems with neutral constituents

The partially ionic-liquid two-sublattice model- cont. Improvement: partially ionic two-sublattice liquid model, Hillert (1985): Model for systems with only cations (C) (metallic systems) and for non-metallic liquids (e.g. liquid sulfur). Model uses constituent fractions as composition variables Hypothetical vacancies (Va) (or neutral species (B)) are introduced on anion sublattice (anion is denoted as (A)). Charge of ions are denoted as, and i,j,k are used to denote a specific constituent Model is: (C i i+ ) P (A j j-, Va, B k 0 ) Q P,Q are numbers of sites on the sublattices (vary with composition to maintain electro-neutrality): P =  j j y Aj + Q y Va Q =  i i y Ci and y i denotes the constituent fraction of constituent i. Mole fractions for cation-like components are x Ci = P y Ci / (P+Q(1-y Va )) and for anion-like and for neutral species are x Di = Q y Di / (P+Q(1-y Va )) (x Va = 0) Gibbs energies and entropy are given by:

The partially ionic-liquid two-sublattice model Where o G Ci:Aj is Gibbs energy of formation for i + j moles of atoms of liquid C i A j, o G Ci and o G Bi are the Gibbs energies of formation per mole of atoms of liquid C i an B i, respectively. Q comes from the variation of the number of sites with composition. G m is defined for a formula unit with (P + Q(1 – y Va )) moles of atoms. ( cnf S is random configurational entropy on each sublattice and E G m excess Gibbs energy ) LFS - CT

Compatibility between different liquid models Substitutional-solution model (Fe, Cr) will be written in ionic-liquid two-sublattice model as (Fe 2+, Cr 3+ ) Q (Va) Q and all parameters for interactions between cations can be used in both models. Substitutional-solution model (Fe, C) will be written in ionic-liquid two-sublattice model as (Fe 2+ ) P (Va, C) Q, valid for higher concentration of carbon. System Cu-S with associate Cu 2 S, modeled in associate-solution model as (Cu, S, Cu 2 S) is in ionic-liquid two-sublattice model written as (Cu 1+ ) P (S 2-,Va, S) Q. Parameters can be identified and used in both models. Different physical models of the system may yield exactly the same mathematical formalism and good results.

The aqueous solutions Parameters for Pitzer model and some other models used for aquous solutions can be evaluated using mentioned models. The Pitzer model evaluates the ionic activities of a solution as a function of solution ionic strength (LRO), interaction terms (SRO), temperature, and pressure. Parameters are not stored in general databases

A model for polymers – the Flory-Huggins model Models proposed for polymer systems – constituents are very different in size ( 1, 2 ) and volume: M G m = RT [ x 1 ln(  1 /x 1 ) + x 2 ln(  2 /x 2 )] + (  1  2 ( 1 x x 2 )  12 )/ 1 where x 1, x 2 are mole fraction,  12 is an interaction parameter, and  1 = 1 x 1 /( 1 x x 2 ),  2 = 2 x 2 /( 1 x x 2 ) are parameters. For the Gibbs energy expression following term is accepted: G m = x 1 o G 1 +x 2 o G 2 +RT[x 1 ln(  1 ) + x 2 ln(  2 )] +x 1 x 2 L 12 /( 1 x x 2 ), where L 12 = 2  12 (BIOSYM – molecular modeling software) CALPHAD 32 (2008) 217 – data for TC

Chemical reactions and thermodynamic models Solubility product Homogeneous reaction: 2H 2 + O 2 = 2H 2 O, it was discussed earlier (CT-8). Next example: Heterogeneous reaction: (Al) + (N) = AlN, in liquid steel. Supposing liquid steel as ideal solution we have for the reaction: o G AlN AlN = o G Al L + RT ln(x Al L ) + o G N L + RT ln(x N L ), where superscripts denote phases (liquid and solid AlN). Rearranging this gives solubility product of AlN in liquid phase: x Al L x N L = exp (( o G AlN AlN – o G Al L – o G N L ) / RT) = K equil

Classification of the different models Possible classification of different models, following Sundman (1990): LFS - CT

Adjustable parameters in the models Most important parameters are those in surface of reference part, srf G m. (they are multiplied by the lowest power of the fractions!) All parameters in srf G m must be referred to the reference state of the element (SER) (never set to zero!) To many parameters and coefficients of the models are not able to improve the fit between descriptions and measurements significantly. It is better to start a calculation with fewer coefficients than with some unnecessary ones. The effect of each coefficient on shape of calculated curves should be known at least qualitatively. Starting calculations „by hand“ is adviceable before the least-square optimisation method starts. It is mandatory, that all possible thermochemical information (even estimated) will be used to obtain realistic values of thermodynamic properties of system by fitting the phase diagram. („Solid point in universe.“ Not mere „curve fitting“.)

Limitations in the models In available thermodynamic databases, the ideal (point) configurational entropy is used and SRO must be modeled as an excess entropy. This may give bad extrapolation to higher-order systems. It may be cured by using quasi-chemical or CVM based models, but the effort needed to change these models is considerable and also calculations using them is slow. Simplifications in crystal structures in modeling are acceptable, but thermodynamic information is necessary (from experiment or from ab initio calculations).

Questions for learning 1.Describe the model for description of Gibbs energy of intermetallic phases 2. Describe the model for description of Gibbs energy of metal-non- metal phases 3. Describe the model for description of Gibbs energy of B32, D0 3, and L2 1 ordering in BCC structure and for simultaneous L1 2 and L1 0 ordering in FCC lattice 4. Describe partially ionic liquid model for liquids and compare it with associate model 5. Describe model for expression of Gibbs energy of polymers