1 / 46

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:

tuan
Download Presentation

CT – 11 : Models for excess Gibbs energy

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related