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CT – 11 : Models for excess Gibbs energy

CT – 11 : Models for excess Gibbs energy. M odels for the excess Gibbs energy: associate-solution model, quasi-chemical model, cluster-variation method, modeling using sublattices: models using two sublattices. The associate-solution model. Systems with SRO:

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CT – 11 : Models for excess Gibbs energy

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  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 18 1995, 356 – associate-solution model Ba – O system J.Houserová, J.Vřešťál: VII.seminarDiffusionandThermodynamicsofMaterials,Brno,1998,p.93 - ionicliquid model Ba – O system xO

  8. The associate-solution model – example (Comparisonwithionicliquid model) J.Houserová, J.Vřešťál: VII.seminarDiffusionandThermodynamicsofMaterials, Brno, 1998, p.93. Example: Ba-O system - parameters (database) Model: PHASE IONIC_LIQUIDBA 2 1 1 ! CONST IONIC_LIQUIDBA :BA+2:O,O-2,VA-2: ! Data: • PARA G(IONIC_LIQUIDBA,BA+2:O;0) 298.15 +2*GHSEROO-2648.9-2648.9 • +31.44*T+31.44*T;,, N ! • PARA G(IONIC_LIQUIDBA,BA+2:VA-2;0) 298.15 +2*GBALIQ;,, N ! • PARA G(IONIC_LIQUIDBA,BA+2:O-2;0) 298.15 +2*GBAO;,, N ! • PARA L(IONIC_LIQUIDBA,BA+2:VA-2,O-2;0) 298.15 -41429.53-6.897814*T;,, N! • PARA L(IONIC_LIQUIDBA,BA+2:O,O-2;0) 298.15 -61973.98;,, N! • (Associate-solution model isbetterfordescriptionoflessorderedsystems, ionicliquid model is more appropriateforsolutionswithhighdegreeofordering.)

  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 maintainelectroneutrality. It is possible to handle the whole range of compositionsfrompure 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. 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 fixedbut 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) 244-249 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. Example: T.Mohri et al.: First-principles calculation of L10-disorder phase equilibria for Fe-Ni system. CALPHAD 33 (2009) 244-249 Cluster variation method – cont.

  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. PartialGibbsenergycannotbecalculatedforcomponentsbutforendmembersonly 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. Surfaceof reference forreciprocalsystem In modelswith more sublattices – major part ofGibbsenergyisdescribed in thesurfaceof reference ! LFS - CT

  30. Reciprocal solutions – cont. Miscibility gap in reciprocalsolution model – inherenttendency to formit in themiddleofthesystem. Sometimesdifficult to suppress. When no data foroneendmemberofreciprocalsystem are available – recommendedassumptionis (e.g. to calculateoGA:C) ∆G = oGA:C + oGB:D - oGA:D - oGB:C = 0 To avoidthemiscibility gap – introductionof (Hillert 1980) EGm = (yAyByCyD)1/2 LA,B:C,D where LA,B:C,D = - ∆G2 / (zRT) , z isthenumberofnearestneighbors (Successfulalsoforcarbidesandnitrides.)

  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 Thesameconstituentscanbe in bothsublattices. Gibbsenergyforthis model is: LFS - CT

  34. Interstitial solutions Most commonapplicationoftwo-sublattice model: C and N in metals. Theyoccupytheinterstitialsites in metallicsublattices. Introducingvacancies (Va): „real“ constituentswithchemicalpotentialequalzero they are excludedfromthesummation to calculate mole fraction Model forcarbide, nitride, boride – B1 structure, (canbetreated as fccstructure), is: (Fe, Cr, Ni, Ti,….)1 (Va, C, N, B)1

  35. Models for phases with metals and non-metals According to crystallography – twometallicsublatticesformetallicelementscanbemodeled: e.g. case M23C6 (Fe, Cr, …)20 (Cr, FeMo, W,…)3 C6 Anotherinteresting case isthe spinel phase – constituents are ions: (Fe2+, Fe3+)1 (Fe2+, Fe3+, Va)2 (O2-)4

  36. The Wagner-Schottky defect model (1930) Idealcompound (ofconstituents A,B) + 2 defects (X,Y) allowingdeviationfromstoichiometry (a,b) on bothsidesoftheidealcomposition: CEF description as reciprocalsolution model is: (A, X)a (B,Y)b • Thedefectscanbe: • Anti-siteatoms • Vacancies • Intersticials • Mixtureofabovedefects

  37. The Wagner-Schottky defect model – cont. Varioustypesofmodels: (A)a (B)b (Va, A, B)c Both A and B prefer to appear as interstitials on thesameinterstitial sublattice (A, B)1 (B,Va)1 Orderedbcc (B2) has twoidenticalsublattices – theyoftenhaveanti-siteatoms on onesideoftheidealcompositionandvacancies on theother (A, B, Va)1 (B, A, Va)1 Defectsincluding on bothsublattices (anti-siteatomsandvacancies)

  38. Mathematicalexpressionsfor Wagner- Schottky model in CEF: The Wagner-Schottky defect model - cont. c LFS - CT nxisthenumberof X on A sites, nynumberof Y on B sites, n isthetotalnumberofsites, oGX:Bisenergyforcreatingdefect X in first sublattice...

  39. The Wagner-Schottky defect model – cont. Colon (:) isused to separateconstituents in differentsublattices Comma (,) isusedbetweeninteractingconstituents in one sublattice Wagner –Schottky model isapplicableforverysmallfractionof non-interactingdefects oGA:AandoGB:B are unary data Itisrecommended 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 parametersisrecommended, as it is usual in CEF

  40. A model for B2/A2 ordering of bcc LFS - CT Ordering = atom in the center ofthe unit cell isdifferentfromthoseatthecorners Examples: Fe-Si, Cu-Znsystems: B2 to A2 issecond-ordertransition– bothphases are treated by thesameGibbsenergyexpression, Fe-Ti: separatephasefromthedisordered A2 (treatedusingWagner-Schottky model)– Itis not recommendedforproblemswithextension to ternarysystems. Extendingintotheternary – B2 mayformcontinuoussolution to anotherbinarysystemwithsecond-order A2/B2 transition (e.g. Al-Fe-Ni).

  41. 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. A model for B2/A2 ordering of bcc – cont. 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 bringscontributionalso in disorderedstate. Sublattices are crystallographicallyequivalent, thereforebinary model is: (A, B, Va)1 (A, B, Va)1 nineendmemberscanbereduced to sixusingrequirement oGA:B = oGB:A oGA:Va = oGVa:A oGB:Va = oGVa:B withoGA:A = oGAbcc oGB:B = oGBbcc ForoGVa:Valarge positive valueispresentlyrecommended – itisfictitiousanyway.

  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 L12orderingmeansthattheatoms in thecorners are ofdifferentkindfromtheatoms on thesides: idealstoichiometryis A3B whichcanbemodeled as (A, B)3 (A, B)1. (Usuallyfirst- ordertransition, twodifferentmodels.) (Constituents in thefirst sublattice haveeightnearestneighbors in thesame sublattice – relationexistsbetweenoGA:Band LA,B:*. Possibleimprovementisfour-sublattice model: Fe-Alsystem. More sublattice model issubstantiatedonlywithenoughexperimetal data orab initio data. Itisstillsubjectofdiscussion.)

  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

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