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Materials Thermodynamics Thermodynamic modelling Calculations of phase diagrams using

Materials Thermodynamics Thermodynamic modelling Calculations of phase diagrams using Thermo-Calc software package. Content. Equilibrium condition 2. Gibbs energy for elements and stoichiometric phases 3. Gibbs energy for solution phase: substitutional regular solution

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Materials Thermodynamics Thermodynamic modelling Calculations of phase diagrams using

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  1. Materials Thermodynamics Thermodynamic modelling Calculations of phase diagrams using Thermo-Calc software package Content • Equilibrium condition 2. Gibbs energy for elements and stoichiometric phases 3. Gibbs energy for solution phase: substitutional regular solution 4. Model for ideal gas 5. Sublattice model Interstitials, Wagner-Schottky model, ionic crystalline phases, Order/Disorder 6. Associate solution model 7. Quasichemical model 8. Two-sublattice partially ionic liquid 9. Optimisation

  2. The Gibbs for a system and for a phase Equilibrium is minimum of the Gibbs energy of system xB 1+3+5: G135=n1G1+n3G3+n5G5 1+2+4: G124=n1G1+n2G2+n4G4 1+3+4: G134=n1G1+n3G3+n4G4 2+4+5: G245=n2G2+n4G4+n5G5 2+4+6: G246=n2G2+n4G4+n6G6 2+3+5: G235=n2G2+n3G3+n5G5 3+4+6: G346=n3G3+n4G4+n6G6 3+5+6: G356=n3G3+n5G5+n6G6 xA

  3. Another approach: condition for equilibrium is equality of chemical potentials For equilibrium between three phases s, r and t equilibrium condition can be written as: Chemical potential is partial derivative of the Gibbs energy where Ni is number of moles of component i. T, P and Nj (j≠i)

  4. Stable equilibrium, metastable equilibrium, diffusionless transformation a+b is stable equilibrium (global minimum of G) g b g+b is metastastable equilibrium (local minimum of G) a Diffusionless transformation occurs when Ga=Gb (T0-line) T0 Xbg=b Xgg=b Xaa=b Xba=b T0 a+b b-equil Xbeq Xaeq a-equil. Xa, Xb non-equilibrium

  5. The Gibbs energy for elements and stoichiometric compounds FUNCTION GCORUND 298.15 -1707351.3+448.021092*T-67.4804*T*LN(T) -0.06747*T**2+1.4205433E-05*T**3+938780*T**(-1); 600 Y -1724886.06+754.856573*T-116.258*T*LN(T)-0.0072257*T**2 +2.78532E-07*T**3+2120700*T**(-1); 1500 Y -1772163.19+1053.4548*T-156.058*T*LN(T)+0.00709105*T**2 -6.29402E-07*T**3+12366650*T**(-1); 3000 N !

  6. The Gibbs energy for stoichiometric compounds The Gibbs energy of phase with constant composition at temperature T is expressed by GT = HT - ST ∙T, where HT and ST are enthalpy and entropy at temperature T. The enthalpy and entropy at temperature T are expressed by equations: where DH0f,el298 is enthalpy of formation from elements, S0298 standard entropy and CP is heat capacity, which also depends on temperature: CP= A+B∙T+C/T2 (Maier – Kelly equation) As a result of integration of CP expressed by Maier - Kelly equation a = DH0f298 -298∙A -0.5∙2982∙B +298-1∙C b = -S0298 +A +A∙ln298 +298∙B -0.5∙298-2∙C c= -A, d = -0.5∙B, f = -0.5∙C in the resulting expression of the Gibbs energy of phase

  7. Neumann-Kopp rule Neumann-Kopp rule is used when heat capacity data are not available. is the Gibbs energy of phase  , Gi(T) is the Gibbs energy of pure element i, bi is stoichiometric coefficient This assumption means that the heat capacity of formation of the compound  from elements is equal to zero: Pure oxides can be selected as components to calculate CP for mixed oxide compounds or other oxide species i.e. MgSiO3 and CaSiO3 to calculate heat capacity of diopside CaMgSi2O6 Fe2Al5: CPcomparison of experimental data with DFT (ab-initio) and Neumann-Kopp calculations Disadvantage of using Neumann-Kopp rule is that transformations occurring in the components will be seen on CP curves as kinks at temperatures of transformations which does not have physical reason

  8. Magnetic contribution For element or compound having magnetic ordering GHSER is referred to para-magnetic state and additional term accounting magnetic contribution is included where β is average magnetic moment, t=T/TC, where TC is critical temperature i.e Curie or Neel temperature Pressure contribution Murnaghan equation of state for solid phases where KT is isothermal bulk modulus (KT is inverse compressibility ) K’p is pressure derivative of bulk modulus The molar volume at 1 atm is expressed as function of temperature Volumetric thermal expansion is expressed as function of temperature

  9. Equation of state for gas Ideal gas PVm=RT P0-1 bar Real gas Equation of Van-der-Waals Valid above critical point, for gas at low P and liquid, not valid in two phase region Vm – molar volume, P- pressure, a and b coefficients a - correction for intermolecular forces, b – for finite molecular size f – fugacity, f0 – reference at 1 bar

  10. The Gibbs energy for a solution phase

  11. Lattice stability To calculate G of solution phase the Gibbs energies for both end-members A and B and mixing parameters L (0LA,B,1LA,B, 2LA,B …) should be known For liquid phase the Gibbs energies for both end-members are usually known: 0GLA and 0GLB For continuous solid solutions they are also known Cu-Ni fcc solid solution 0GfccCu and 0GfccNi For limited solid solutions G is knownonly for one end-member, for example, fcc solid solution in the Al-Si system: maximal solubility of Si in fcc is 1.5 at%. How to find 0GfccSi? For compound with homogeneity range the Gibbs for both end-members is not known, for example,TiN fcc. How to find 0GfccTi and 0GfccN? www.sgte.net (temporaryaddresshttp://www.crct.polymtl.ca/sgte/) Unarydatabase: collectionof G(T) for all elements in stableandmetastablestructures

  12. Example: thermodnamic description of Liquid phase in THERMO-CALC format for the Al-Si system FUNCTION GALLIQ 298.15 +3028.879+125.251171*T-24.3671976*T*LN(T) -0.001884662*T**2-8.77664E-07*T**3+74092*T**(-1)+7.9337E-20*T**7; 700 Y -271.21+211.206579*T-38.5844296*T*LN(T)+.018531982*T**2 -5.764227E-06*T**3+74092*T**(-1)+7.9337E-20*T**7; 933.47 Y -795.996+177.430178*T-31.748192*T*LN(T); 2900 N ! FUNCTION GSILIQ 298.15 +42533.751+107.13742*T-22.8317533*T*LN(T) -0.001912904*T**2-3.552E-09*T**3+176667*T**(-1)+2.09307E-21*T**7; 1687 Y +40370.523+137.722298*T-27.196*T*LN(T); 3600 N ! PHASE LIQUID % 1 1.0 ! CONSTITUENT LIQUID :AL,SI : ! PARAMETER G(LIQUID,AL;0) 2.98140E+02 +GALLIQ#; 2900 N 91Din ! PARAMETER G(LIQUID,SI;0) 2.98140E+02 +GSILIQ#; 3600 N 91Din ! PARAMETER L(LIQUID,AL,SI;0) 2.98150E+02 -11340.1-1.23394*T; 6000 N 97Feu ! PARAMETER L(LIQUID,AL,SI;1) 2.98150E+02 -3530.93+1.35993*T; 6000 N 97Feu ! PARAMETER L(LIQUID,AL,SI;2) 2.98150E+02 2265.39; 6000 N 97Feu !

  13. Redlich-Kister parameters The contribution to the enthalpy of mixing for the first five terms (0L-4L) for the same value of the parameter equal to 20000 J/mol in the Redlich-Kister power series Parameters nLij can depend on temperature nLij=naij+nbijT

  14. Non-ideal behavior Eii Eij Ejj Energy difference eij for bonds between unlike atoms Eij and similar atoms Eii and Ejj eij=Eij-0.5(Eii+Ejj) tendency for unlike atoms to be together, forming compounds eij<0 Tendency to phase separation, forming of miscibility gaps eij>0 For substitutional regular solution model z - number of bonds

  15. Application of substitutional model • Substitutional model is used to describe : • liquid phase in metallic systems • Solid metallic phase if atoms are mixing in one single sublattice, the second being empty: i.e. fcc (Fe,Ni)1(Va)1, bcc (Fe,Ni)1(Va)3

  16. Example: Al-Bi system a. – calculated phase diagram of Al-Bi system b. – calculated integral properties at 1400 K c. – calculated activities at 1400 K

  17. Example Al-Y system a. – calculated phase diagram for Al-Y system b. – calculated integral properties of liquid at 1800 K c.- calculated activity of Al and Y in liquid at 1800 K.

  18. Ternary interaction parameters n=3 The excess energy consists from binary interactions (Muggianu extrapolation) Muggianu (1975) For ternary system binary excess contribution is: bin,EGm=x1x2L12+x1x3L13+x2x3L23 bin,EGm=x1x2{0L12+1L12(x1-x2)}+x1x3{0L13+1L(x1-x3)}+x2x3{0L23+1L23(x2-x3)} Ternary excess contribution tern,EGm=x1x2x3L123 L123=x11L123+x22L123+x33L123 For multicomponent system

  19. Other methods of extrapolation into ternary system Geometric presentation: Kohler and Muggianu models are symmetric, Toop is asymmetric Kohler Kohler model assumes that xi/(xi+xj)=const Toop Toop model assumes that x1=const The difference between models exists only L has higher order parameters L1, L2 … (b) Toop Model [1965Too]

  20. Model for multicomponent ideal gas Gas phase usually consists of many constituents (species) which mixed ideally where 0Gi – the Gibbs energy of constituent i, yi – its fraction The Gibbs energy of gas phase depends on pressure where bij is stoichiometric coefficient of gas constituent, p0 – atmospheric pressure, yi=pi/p constitutent fraction equal to ratio of partial pressure pi to total pressure p equation of state for ideal gas V=nRTp-1

  21. Example: system H-O Gas constituents H2, O2, H2O, O, H, O3 T5 There are additional internal degrees of freedom, because constituents can react with each other 2H2+O2=2H2O, 2H=H2, 2O=O2 etc T4 T3 T2 T1 T1<T2<T3<T4<T5

  22. Sublattice model Fluorite Structure 2-sublattice model (Zr+4,Y+3)1(O-2,Va)2 A2/B2 ordering A2: (Fe,Al)1(Va,C)3 Second sublattice is interstitials B2: (Fe,Al)0.5(Fe,Al)0.5(Va,C)3 First sublattice is divided into two for B2 model and the third sublattice is interstitials

  23. Sublattice Model (A,B)a(C,D)c The Gibbs energy of a phase consists from Reference Surface Energy, Configurational Entropy and Excess Energy terms Reference surface energy is expressed as 0Gij is the Gibbs energy of formation of compound iajc, also called end-member, y’i and y’’j are mole fraction of species i and j in the first and second sublattice Configurational entropy is expressed as

  24. Sublattice model For considered case (A,B)a(C,D)c: The excess energy contribution is described as regular solution on each sublattice L – are Redlich-Kister series The end members are related with each others by Reciprocal Reaction: AaDc+BaCc=AaCc+BaDc

  25. Example: modelling of Laves phases in Cr-Ti system Laves phase modelling: a-TiCr2 C15, b-TiCr2 C36, g-TiCr2 C14 (Cr,Ti)2(Ti,Cr)1 Cr2Ti, Cr2Cr, Ti2Ti, Ti2Cr CN12 16d Cr CN16 8a Ti a-TiCr2 - C15 Mixing parameters Cr2Ti+Ti2Cr=Cr2Cr+Ti2Ti DGrec=0

  26. Phase diagram of the Cr-Ti system b-Bcc (Cr,Ti)1(Va)3 a-Hcp (Ti,Cr)1(Va)0.5 a-TiCr2, b-TiCr2, g-TiCr2 (Cr,Ti)2(Ti,Cr)1 Liquid (Cr,Ti)

  27. Sublattice model to describe interstitials Common application of Sublattice model is to describe interstitial solution of C, N, O and other non-metals in metallic phases FCC_A1 (Fe,Cr,Ni)1(Va,C,N,O,B)1 Some carbides and nitrides also forms fcc structure and can be described as single phase which forms a miscibility gap. Example: Fe-N system Liquid (Fe,N)1 Fcc_A1 (Fe)1(Va,N)1 Bcc_A2 (Fe)1(Va,N)3 Hcp_A3 (Fe)2(Va,N)1 Fe4N - stoichiometric

  28. Wagner-Schottky model to describe defects • Wagner-Schottkymodel is to describe narrow homogeneity range in compounds due to various type of defects • (A,X)a(B,Y)b • AaBb is ideal compounds, X and Y are defects. • Defects can be: • anti-site atoms, B in sublattice for A and A in sublattice for B • (A,B)a(B,A)b • 2) Vacancies • (A,B)a(B,Va)b • 3) Interstitials • (A)a(B)b(Va,A,B)c (A,X)a(B,Y)b Wagner-Schottky model consider one defect for each sublattice and contains three parameters 0GA:B, D0GX:B and D0GA:Y (energy of formation of compound, energy of formation of defect on the first and second sublattices). Wagner-Schottky model is applicable when fraction of defects is very small and defects are non-interacting

  29. Wagner-Schottky model – relations with sublattice model (A,X)a(B,Y)b 0GAaBb, D0GA:Y, D0GX:B are parameters of Wagner-Schottky model nX is number of X in A sites, nY number of Y in B sites, n is total number of sites Example: Y-O system Y2O3_R and Y2O3_H are modelled by Wagner-Schottky model (Y+3)2(O-2,Va)3

  30. Ionic crystalline phases Example: Fe-O system, modelling of wüstite FexO phase FexO: (Fe+2,Fe+3,Va)1(O-2)1 Electroneutrality condition: 2Fe+3O-2+VaO-2=Fe2O3 Crystal structure of wüstite

  31. Modelling of Spinel in the MgO-Al2O3 system Spinel Mg1Al2O4 (Mg+2,Al+3)T1(Al+3,Mg+2)O2(Va)I2O4 (Mg+2,Al+3)T1(Al+3,Mg+2,Va)O2(Va)I2O4 (Mg+2,Al+3)T1(Al+3,Mg+2,Va)O2(Mg+2,Va)I2O4 Hallstedt, 1992

  32. Modelling of Spinel phase (Mg+2,Al+3)T1(Al+3,Mg+2)O2(Va)I2O4 AlAl2O4+1+AlMg2O4-1=2MgAl2O4 (Mg+2,Al+3)T1(Al+3,Mg+2,Va)O2(Va)I2O4 AlVa2O4-5+5AlAl2O4+1=8g-Al2O3

  33. Modelling of Spinel phase (Mg+2,Al+3)T1(Al+3,Mg+2)O2(Va,Mg+2)I2O4 MgMg2VaO4-2+MgMg2Mg2O4+2=8g-MgO AlMg2Mg2O4+3+3AlMg2VaO4-1=2Mg5Al2O8

  34. Thermodynamic modelling of pyrochlore structure Zr+4 La+3 (La+3,Zr+4)2(Zr+4,La+3)2(O-2,Va)6(O-2)1(Va,O-2)1

  35. Sublattice model to describe order/disorder Al-Fe system 0GA:B=0GB:A0GA:A=0GbccA0GB:B=0GbccB LA,B:A=LA,B:B=LA,B:* LA:A,B=LB:A,B=L*:A,B LA,B:*=L*:A,B

  36. Order/Disorder In case of disorder y‘i=y“i=xi LA,B:*=L*:A,B + 0GA=0GA:A0GB=0GB:B LA,B=20GA:B-0GA:A-0GB:B+2LA,B:* (y replaced by x)

  37. Example: oder/disorder in fcc phase AuCu L10 L12 (Ni,Si)0.75(Ni,Si)0.25 Au3Cu, AuCu3 L12 (Au,Cu)(Au,Cu)(Au,Cu)(Au,Cu)

  38. Associate solution model Systems with short-range-order (SRO) Unlike atoms stay together, but attractive forces are not strong enough to form a stable chemical molecule New constituent (species) is introduced what creates additional internal degrees of freedom (A, AaBb, B) The reference surface energy and configurational entropy are described where yi is constituent fraction, Gi is the Gibbs energy of constituent i. The excess energy is described the same way as in substitutional model, treating each pair of constituents as independent parameter

  39. Example: Te-Zn system Liquid (Te, ZnTe, Zn) Calculated integral properties and activities at 1900K

  40. Example: Al-N system Liquid (Al, AlN, N) FCC (Al)1(Va)1 - stoichiometric AlN (Al+3)1(N-3)1 - stoichiometric Gas (Al, N, N2, N3)

  41. Example of ceramic system K2O-Al2O3-SiO2 Yazhenskikh, Hack, Müller (2011) List of species in liquid K2O, Al2O3, Si2O4 2/3 K2SiO3 1/2 K2Si2O5 1/3 K2Si4O9 KAlO2 1/3 K2Al4O7 1/4 Al6Si2O13 1/2 KAlSi2O6 2/3 KAlSiO4 List of solid phases K2SiO3, K2Si2O5, K2Si4O9, KAlO2, KAl9O14, K2Al12O19, Al6Si2O13, KAlSiO4, KAlSi3O8 (microcline, K-feldspar, sanidine), KAlSi2O6 (luecite)

  42. Quasi-chemical model Model proposed by Guggenheim for systems with SRO (short-range-order) The formation of the bonds described by chemical reactions AA+BB=AB+BA AB and BA are different in solid phase because orientations of bonds in crystals are important z – is number of bonds per atom xA=yAA+0.5(yAB+yBA) xB=yBB+0.5(yAB+yBA) Mass balance Entropy is overestimated because if z=2 it is identical to entropy of gas Modified entropy In case there is no SRO: yAA=xA2, yBB=xB2, yAB=yBA=xA∙xB

  43. Calculations with modified quasi-chemical model for liquid System Ag-Zr (Kang and Jung, 2010) System MgO-Al2O3-SiO2 Jung, Decterov, Pelton (2004)

  44. The cell model M.L. Kapoor, G.M. Frohberg, H. Gaye and J. Welfringer The model was developed for oxide slag and it has a special form of quasi-chemical entropy. Different cells with one anion and two cations were considered. Cell can have two identical or two different cations. • Cells mix essentially ideally, with equilibria among the cells: [Mg-O-Mg] + [Si-O-Si] = 2 [Mg-O-Si] DG° < 0 • Quite similar to Modified Quasi-chemical Model • Accounts for ionic nature of slags and short-range-ordering. • Has been applied with success to develop databases for multicomponent systems. Di is a sum of component oxides, nj is the oxygen stoichiometry, xjis mole fraction of component oxide, i.e. cells with only one type of cations uimetal stoichiometry, ni is the oxygen stoichiometry in MuOn

  45. Two-sublattice partially ionic liquid model C –cations, A –anions, n-electric charge, Va –vacancies B-neutral species The number of sites P and Q vary with composition in order to maintain electro neutrality The Gibbs energy expression for this model G=srfGm-TcnfSm+EGm

  46. Model parameters 0GCi:Aj is the Gibbs energy of formation of (ni+nj) moles of liquid CiAj, 0GCi and 0GB are the Gibbs energy of liquid metal C and liquid B The excess parameters Li1,i2:jrepresents interaction between two cations, e.g. in the CaO-MgO system LCa+2,Mg+2:O-2 Li1,j2:Va represents interaction between metals in metallic liquid , e.g. in the system Ca-Mg LCa,Mg Li:j1,j2 represents interaction between two anions, e.g. in the system NaCl-NaNO3 LNa+1:Cl-1,NO3-1 Li:j,Va is interaction between metal atom and anion, e.g. system Fe-O LFe+2:O-2,Va Li:j,k is interaction between anion and neutral species, e.g. Fe-S system LFe+2:S-2,S Li;Va,k is interaction between metal atom and neutral species, e.g. Fe-C system LFe+2:Va,C Lk1,k2 is interaction between two neutral species, e.g. Si3N4-SiO2 system LSi3N4,SiO2

  47. Example: Fe-O system (Fe+2)P(O-2,Va,FeO1.5)Q Example: Fe-Si-O system (Fe+2,Si+4)P(O-2,SiO4-4,Va,FeO1.5,SiO2)Q Fe2O3+SiO2 SiO2+Fe3O4 Liq1 Liq Fe2SiO4+Fe3O4 FeO+Fe2SiO4 FeO+Fe Fe+SiO2 Faya=Fe2SiO4 Wüs=FeO1-x Quartz, tri, cri=SiO2 Magnetite=Fe3O4 Hematite=Fe2O3

  48. Compatibility of two sublattice partially ionic liquid model with other models Partially ionic liquid can describe metallic liquids (for this vacancies are introduced in anionic sublattice, non-metallic, polymeric liquids, e.g. liquid sulfur or SiO2 (for this neutral species are introduced in anionic sublattice) and ionic liquids, e.g. liquid salts like NaCl, NaNO3 (cations in the cationic sublattice and anions in the anionic sublattice) Substitutional model Two-sublattice partially ionic liquid (Fe,Cr) (Fe+2,Cr+3)P(Va)Q P=Q (Fe,C) (Fe+2)P(Va,C)Q Associate model (Cu,Cu2S,S) (Cu+1)P(S-2,Va,S)Q (Ca,CaO,Ca2SiO4,SiO2) (Ca+2)P(O-2,SiO4-4,Va,SiO2)Q

  49. Estimates Theory Experiments DTA, Calorimetry, Quantum Mechanics, EMF, Knudsen Effusion, Statistical Metallography, Thermodynamics X-ray Diffractometry, ... Models with adjustable Parameters Ab-initio Calculation Adjustment of Parameters Thermodyn. Functions G, H, S, C p Storage in Databases Equilibria Graphical Phase Diagrams Representation Application Computational Thermodynamics CALPHAD CALculation of PHAse Diagrams Optimization www.calphad.org Equilibrium Calculations Kinetics

  50. Experimental data 1. Phase equilibrium data 1a. Heat treatment to reach equilibrium, XRD for identification of phases, SEM/EDX or microprobe to determine composition of phases Conditions of equilibrium: T and phases in equilibrium = fixed Experimental results: composition of phases 1b. Differential thermal analysis (DTA), XRD before and after DTA Condition of equilibrium: phases in equilibrium =fixed, composition of liquid is fixed=X(bulk) (liquidus), composition of solid is fixed=X(bulk) (solidus) Experimental results: temperature of solidus/liquidus F F B B F F B B B B B F B B B B F B B B XRD,SEM/EDX: At 1600°C in equilibrium are phases: F (0.58, 0.25, 0.17) and B (0.81, 0.03, 0.16) [Sm, Zr, Y] DTA: at fixed composition transformation occurs at 1500°C

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