1. CT – 11 : Models for excess Gibbs energy
Models for the excess Gibbs energy: associate-solution model, quasi-chemical model, cluster-variation method, modeling using sublattices: models using two sublattices

2. The associate-solution model
Systems with SRO:
„associate“ – association between unlike atoms when attractive forces are not strong enough to form stable molecule
(If the molecule is formed (introduced) – it is constituent in the solution)
LFS - CT

3. The associate-solution model – cont.
Associate-solution model introduce formaly fictitious „associate“ („cluster“) as a constituent in the solution as a modeling tool:
it is not stable enough to be isolated, but its life time is longer than the mean time between thermal collisions
„Associate“ is a new constituent (it has to be introduced!) and it creates an internal degree of freedom: Gibbs energy of its formation should fit experimental data – sharp minimum in the enthalpy curve corresponds to enthalpy of formation of associate and its stoichiometry.
Stoichiometric composition: concentration of associate is high, configurational entropy is low.

4. The associate-solution model – cont.
Gibbs-energy expression for substitutional associated solution (i is constituent):
yi is site fraction of constituent (molar fractions of components are not the same as constituent fractions)
EG can be modeled as in the substitutional solution
LFS -CT

5. Example of introducing associates
System Mg-Sn: additional constituent Mg2Sn in the liquid (high degree of SRO) – (see concentration scale)
LFS - CT

6. Example of introducing associates – cont.
Gibbs energy expression:
oGMg2Sn determines the fraction of Mg2Sn in the liquid
Model will behave differently for the model (Mg, Sn, Mg2/3Sn1/3)
LFS -CT

7. The associate-solution model – example
Zimmermann E., Hack K., Neuschütz K.: CALPHAD , 356 – associate-solution model
Ba – O system
J.Houserová, J.Vřešťál: VII.seminar Diffusion and Thermodynamics of Materials,Brno,1998,p.93 - ionic liquid model
Ba – O system xO

8. The associate-solution model – example
(Comparison with ionic liquid model)
J.Houserová, J.Vřešťál: VII.seminar Diffusion and Thermodynamics of Materials, Brno, 1998, p.93.
Example: Ba-O system - parameters (database)
Model:
PHASE IONIC_LIQUIDBA !
CONST IONIC_LIQUIDBA :BA+2:O,O-2,VA-2: !
Data:
PARA G(IONIC_LIQUIDBA,BA+2:O;0) *GHSEROO
+31.44*T+31.44*T;,, N !
PARA G(IONIC_LIQUIDBA,BA+2:VA-2;0) *GBALIQ;,, N !
PARA G(IONIC_LIQUIDBA,BA+2:O-2;0) *GBAO;,, N !
PARA L(IONIC_LIQUIDBA,BA+2:VA-2,O-2;0) *T;,, N!
PARA L(IONIC_LIQUIDBA,BA+2:O,O-2;0) ;,, N!
(Associate-solution model is better for description of less ordered systems, ionic liquid model is more appropriate for solutions with high degree of ordering.)

9. The ionic liquid model
The ionic liquid model is the modified sublattice
model, where the constituents are cations (Mq+),
vacancies (Vaq–), anions (Xp–) and neutral species (Bo).
Assumes separate „sublattices“ for Mq+ and Vaq-,
Xp–, Bo, e.g., (Cu+)P(S2–, Va–, So)Q, (Ca2+, Al3+)P(O2–,
AlO1.5o)Q or (Ca2+)P(O2–, SiO44–, Va2–, SiO2o)Q
The number of „sites“ (P, Q) varies with
composition to maintain electroneutrality.
It is possible to handle the whole range of
compositions from pure metal to pure non-metal.

10. Non-random configurational entropy
Strong interactions:
Random (ideal) entropy of mixing, Sid = -R in yi ln(yi),
is not valid
Long-range ordering (LRO): sublattices model
Short-range ordering (SRO):
Quasi-chemical approach – pairwise bonds
Cluster variation method (CVM) – clusters
= new fictitious constituents in the equation
Sid = -R in yi ln(yi)

11. The quasi-chemical model
Guggenheim (1952):
Assuming the bonds AA, BB and AB
Formation of bonds is treated as chemical reaction:
AA + BB  AB + BA
„fictitious“ constituents AB and BA
(bonds in crystalline solids of different orientation)
Number of bonds per atom for a phase = z,
Gibbs energy expression for the „bonds“:
LFS -CT

12. The quasi-chemical model - cont.
In the expression for surface reference:
oGAB = oGBA (due to symmetry)
If yAB = yBA no LRO – „degeneracy“ in disordered state
It should be added the additional term: RT yAB ln 2
compared with the case on ignoring this degeneracy
In the expression for configurational entropy:
cnfS is overestimated: number of bonds is z/2 times larger than the
number of atoms per mole and, for oGAB = 0 entropy
should be identical to Sid of components A and B.)
Therefore, modified configurational entropy expression (Guggenheim 1952):
LFS -CT

13. The quasi-chemical model - cont.
No SRO - modified configurational entropy expression gives the same configurational entropy as a random mixing of the atoms A and B (first term=0): constituent fractions can be calculated from the mole fractions by the set of equations:
yAA = xA2 , yAB = yBA = xA xB, yBB = xB2
Small SRO - Modified configurational entropy expression is valid!
Strong SRO, z > 2, may be even cnfS < 0
Very strong SRO: situation can be treated as LRO: sublattice model
(Quasichemical and CVM models are alternative models to the regular solution model – Differences in entropy term)

14. Cluster variation method
Kikuchi (1951):
Clusters with 3,4, and more atoms – depending on the crystal structure are independent constituents (similarity with the quasi-chemical formalism)
Corrections to the entropy expression taking into account that clusters are not independent (share corners, edges, surfaces…)
Example:
configurational entropy for FCC lattice in tetrahedron approximation:
No LRO:
clusters A, A0.75B0.25, A0.5B0.5, A0.25B0.75, B are „end members“ of the phase:
Surface of reference is:
The ideal configurational entropy for this system is:
LFS - CT

15. Cluster variation method – cont.
Typical cluster used in tetrahedron approximation in FCC lattice
Saunders N., Miodownik P.: CALPHAD, Pergamon Press, 1998.

16. Typical cluster used in tetrahedron approximation in BCC lattice
Cluster variation method – cont.
Typical cluster used in tetrahedron approximation in BCC lattice
B. Sundman, J. Lacaze: Intermetallics 2009

17. Cluster variation method – cont.
Example: tetrahedron approximation – cont.
Clusters are not independent (share corners…) – it reduces the configurational entropy:
The term degSm is due the fact that the 5 clusters above are degenerate cases of the 16 clusters needed to describe LRO (4 different A0.75B0.25, 6 different A0.5B0.5, and 4 different A0.25B0.75 this means adding term
LFS .- CT

18. Cluster variation method – cont.
The variable pAA is a pair probability that is equal to the bond fraction in the quasi-chemical-entropy expression.
It can be calculated from the cluster fractions as:
and the mole fractions from the pair probabilities are:
xA = pAA + pAB
xB = pBB + pBA
LFS - CT

19. Cluster variation method – cont.
LFS - CT

20. Cluster variation method – cont.
Discussion:
- Associate model overestimates the contribution of SRO to the Gibbs energy
- The CVM extrapolation gives an unphysical negative entropy at low T
- Cluster energies in CVM depend on composition.
In CEF - energies are fixed but the dependence of Gibbs energy on composition is modeled with EG (needs less composition variables than CVM does – see example)
Example:
CVM tetrahedron model for FCC with 8 elements: at least 84=4096 clusters
CEF 4-sublattice requires only 8 x 4 = 32 constituents.

21. Cluster variation method – cont. – example:
Saunders N., Miodownik P.: CALPHAD, Pergamon Press, 1998.

22. Cluster variation method – cont.
Example: T.Mohri et al.: First-principles calculation of L10-disorder phase equilibria for Fe-Ni system. CALPHAD 33 (2009)
N is total number of atoms, xi, yij, wijkl are cluster probabilities of finding atomic configuration specified by subscript (s) on a point, pair and tetrahedron clusters, respectively, and ,  distinguish two sublattices in the L10 ordered phase. Entropy is calculated for disordered, S(dis), and ordered: L10, S(L10), phases, kB is Boltzmann constant.

23. Cluster variation method – cont.
Example: T.Mohri et al.: First-principles calculation of L10-disorder phase equilibria for Fe-Ni system. CALPHAD 33 (2009)

24. Modeling using sublattices
Atoms in crystalline solids – occupy different type of sublattices Sublattices represent LRO – modify entropy and excess Gibbs energy Example: (A,B)m(C,D)n m,n – ratio of sites on the two sublattices (smallest possible integer numbers) A,B,C,D – constituents (in CEF)

25. Modeling using sublattices – cont.
Special cases: stoichiometric phase AmCn Substitutional solution model (A,B)mCn reciprocal solutions (A,B)m(C,D)n

26. Reciprocal solutions Example: NaCl + KBr = NaBr + KCl
Model: (Na,K) (Cl,Br) as (A,B)1(C,D)1
Reciprocal Gibbs energy of reaction:
∆G = oGA:C + oGB:D - oG A:D - oGB:C
Entropy: ideal configurational entropy in each sublattice weighted for total entropy with respect to the number of sites on each sublattice (different from the configurational entropy given by substitutional model!):
LFS - CT

27. Reciprocal solutions – cont.
Partial Gibbs energy cannot be calculated for components but for end members only
LFS - CT

28. Reciprocal solutions – cont.
Excess Gibbs energy: Excess parameters are multiplied with three fractions - two from one sublattice and one from the other. Additionally, reciprocal excess parameter is multiplied by all four fractions having largest influence in the center of the square (A,B,C,D – components, primes – sublattices): Binary L – parameters can be expanded in an Redlich – Kister formula (with concentration dependence – not advisable)
LFS - CT

29. Reciprocal solutions – cont.
Surface of reference for reciprocal system
In models with more sublattices – major part of Gibbs energy is described in the surface of reference !
LFS - CT

30. Reciprocal solutions – cont.
Miscibility gap in reciprocal solution model –
inherent tendency to form it in the middle of the system.
Sometimes difficult to suppress.
When no data for one end member of reciprocal system are available – recommended assumption is (e.g. to calculate oGA:C)
∆G = oGA:C + oGB:D - oG A:D - oGB:C = 0
To avoid the miscibility gap – introduction of (Hillert 1980)
EGm = (yAyByCyD)1/2 LA,B:C,D
where LA,B:C,D = - ∆G2 / (zRT) ,
z is the number of nearest neighbors
(Successful also for carbides and nitrides.)

31. Reciprocal solutions – cont.
Example:
LFS - CT

32. Reciprocal solutions – cont. - example.
Projection of miscibility gap in the HSLA steels – (Nb,Ti) (C,N) system
M. Zinkevich-Sommer school Stuttgart 2003

33. Models using two sublattices
Two-sublattice CEF – generally:
(A,B,….)m(U,V,….)n
The same constituents can be in both sublattices.
Gibbs energy for this model is:
LFS - CT

34. Interstitial solutions
Most common application of two-sublattice model: C and N in metals.
They occupy the interstitial sites in metallic sublattices.
Introducing vacancies (Va):
„real“ constituents with chemical potential equal zero
they are excluded from the summation to calculate mole fraction
Model for carbide, nitride, boride – B1 structure, (can be treated as fcc structure), is:
(Fe, Cr, Ni, Ti,….)1 (Va, C, N, B)1

35. Models for phases with metals and non-metals
According to crystallography – two metallic sublattices for metallic elements can be modeled: e.g. case M23C6
(Fe, Cr, …)20 (Cr, Fe Mo, W,…)3 C6
Another interesting case is the spinel phase – constituents are ions:
(Fe2+, Fe3+)1 (Fe2+, Fe3+, Va)2 (O2-)4

36. The Wagner-Schottky defect model (1930)
Ideal compound (of constituents A,B) + 2 defects (X,Y) allowing deviation from stoichiometry (a,b) on both sides of the ideal composition:
CEF description as reciprocal solution model is:
(A, X)a (B,Y)b
The defects can be:
Anti-site atoms
Vacancies
Intersticials
Mixture of above defects

37. The Wagner-Schottky defect model – cont.
Various types of models:
(A)a (B)b (Va, A, B)c
Both A and B prefer to appear as interstitials on the same interstitial sublattice
(A, B)1 (B,Va)1
Ordered bcc (B2) has two identical sublattices – they often have anti-site atoms on one side of the ideal composition and vacancies on the other
(A, B, Va)1 (B, A, Va)1
Defects including on both sublattices (anti-site atoms and vacancies)

38. The Wagner-Schottky defect model - cont.
Mathematical expressions for Wagner- Schottky model in CEF: c
LFS - CT
nx is the number of X on A sites, ny number of Y on B sites, n is the total number of sites, oGX:B is energy for creating defect X in first sublattice...

39. The Wagner-Schottky defect model – cont.
Colon (:) is used to separate constituents in different sublattices
Comma (,) is used between interacting constituents in one sublattice
Wagner –Schottky model is applicable for very small fraction of non-interacting defects
oGA:A and oGB:B are unary data
It is recommended to set: LA,B:A = LA,B:B = LA,B:*
LB:B,A = LA:B,A = L*:B,A
i.e. interaction in each sublattice is independent on occupation of the other sublattice
For larger defect fraction – Redlich-Kister expansion of interaction parameters is recommended, as it is usual in CEF

40. A model for B2/A2 ordering of bcc
LFS - CT
Ordering = atom in the center of the unit cell is different from those at the corners
Examples:
Fe-Si, Cu-Zn systems: B2 to A2 is second-order transition – both phases are treated by the same Gibbs energy expression,
Fe-Ti: separate phase from the disordered A2 (treated using Wagner-Schottky model)– It is not recommended for problems with extension to ternary systems. Extending into the ternary – B2 may form continuous solution to another binary system with second-order A2/B2 transition (e.g. Al-Fe-Ni).

41. A model for B2/A2 ordering of bcc – cont.
Phase diagram Cu-Zn: A.Dinsdale, A.Watson, A.Kroupa, J. Vrestal, A.Zemanova, J.Vizdal: Atlas pf Phase Diagrams for Lead-Free Soldering. COST 2008, Printed in Brno, Czech Rep.
LFS - CT

42. A model for B2/A2 ordering of bcc – cont.
Symmetry in the model: oGi:j = oGj:i
Li,j:k = Lk:i,j
Li,j,k:l = Ll:i,j,k
brings contribution also in disordered state.
Sublattices are crystallographically equivalent, therefore binary model is:
(A, B, Va)1 (A, B, Va)1
nine end members can be reduced to six using requirement
oGA:B = oGB:A
oGA:Va = oGVa:A
oGB:Va = oGVa:B with oGA:A = oGAbcc
oGB:B = oGBbcc
For oGVa:Va large positive value is presently recommended – it is fictitious anyway.

43. B2 ordering in bcc lattice – example Fe-Al system
B. Sundman, J. Lacaze: Intermetallics 2009

44. A model of L12/A1 ordering of fcc
LFS - CT
The L12 ordering means that the atoms in the corners are of different kind from the atoms on the sides: ideal stoichiometry is A3B which can be modeled as (A, B)3 (A, B)1. (Usually first- order transition, two different models.)
(Constituents in the first sublattice have eight nearest neighbors in the same sublattice – relation exists between oGA:B and LA,B:*. Possible improvement is four-sublattice model: Fe-Al system.
More sublattice model is substantiated only with enough experimetal data or ab initio data. It is still subject of discussion.)

45. L10,L12/A1 ordering in fcc lattice-example: Au-Cu system
LIQ
FCC_A1
L10
L12
L12
B. Sundman, J. Lacaze: Intermetallics 2009

46. Questions for learning
1. Describe associate-solution model
2. Describe quasi-chemical model and cluster variation method
3. Describe model for reciprocal solutions and their tendency to
create miscibility gap
4. Describe Wagner-Schottky model and two sublattice model for
interstitial solutions
5. Describe model for B2/A2 ordering in BCC structure and
L10,L12/A1 ordering in FCC structure