1. CT – 8 Models for the Gibbs energy:
General form of Gibbs-energy model, temperature and pressure dependences, metastable states, variables for composition dependence

2. Models for thermodynamic properties
Integral Gibbs energy as modeled property (p,T = const.)
„Mean-field approximation“
Mathematical expression is often more general than the physical
model (mathematical description of reality): it is called „formalism“.
Most of models which will be described in next are special cases
of the compound-energy formalism (CEF).
Models for phases can be selected independently (with except
when phases belong to the structure family - e.g. phases with
chemical ordering A2/B2, A1/L12).
Gibbs energy: bonding (ordering/misscibility gap) + configuration.
Selection of model: physical and chemical properties of phase –
crystal structure, bonding type, magnetic properties etc.
Sublattice model – stoichiometric compound is special case of it.

3. Bonding energy of components
EAA + EBB  2.EAB
EAA + EBB  2.EAB
Binary systems – influence of GE

4. Models for thermodynamic properties-cont.
End members:
Sublattices occupied by (A)(B) – stoichiometric compound
Sublattices occupied by (A,B)(A,B) – solution - have
(A)(A), (A)(B), (B)(A), (B)(B) – end members („compounds“)
Parameters (coefficients, variable):
Parameter: quantity, that is part of a model. May be a function of T, p, xi
Each parameter may consist of several coefficients (single numerical value)
(„Variable“ in PARROT optimization software is used instead of „coefficient“.)

5. General form of the Gibbs-energy model:
The total Gibbs energy of a phase :
Gm = srfGm + physGm - T.cnfSm + EGm
„srf“ = surface of reference (unreacted mixture)=  xioGi
“phys“ =Gibbs energy according a physical model (e.g.magnetic transitions)
„cnf“ = configurational entropy (prediction e.g. by CVM), ideal mixing
„E“ = excess Gibbs energy.
srfGm and EGm : there is no attempt to model their physical origin (include, therefore, configurational, vibrational, electronic and other contributions)

6. Surface of reference
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The surface of reference for the Gibbs energy of a phase (A,B)a(D,E)c , according to CEF, plotted above the composition square

7. Excess Gibbs energy in CEF for (A,B)a(D,E)c
EGm = y'Ay'B(y''DLA,B:D + y''ELA,B:E) + y''Dy''E(y'ALA:D,E
+ y'BLB:D,E) + y'Ay'By''Dy''ELA,B:D,E
Each L can be a Redlich-Kister series

8. Phases with fixed composition
Pure element, a stoichiometric compound, or a solution phase (composition kept constant externally) – (some of them are end members). G(T,p) – only.
Temperature dependence: power serie in T:
SER – standard element reference state
Expression is valid only in the interval of temperatures [T1,T2]
How to express the thermodynamic functions using these coefficients?
LFS - CT

9. Thermodynamic models
Pure elements, stoichiometric phases and end-members
of solutions:
G = a + bT + cTlnT + dnTn
Note that the enthalpy, entropy, heat capacity, etc.
can be calculated from G:
H = a – cT – (n-1)dnTn
S = –b – c – clnT – ndnTn-1
Cp = –c – n(n-1)dnTn-1

10. Temperature dependence
LFS - CT
Term T.ln(T) (in expression for G) is from temperature independent heat capacity coefficient
Expressions are valid in limited temperature range above Debye temperature (lower limit of temperature usually is K).
Relations above are empirical. Physical models are rarely used except for feromagnetic transitions.
Decrease of the number of coefficients by using of several temperature ranges is possible.

11. Temperature dependence - example
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12. Pressure dependence
For condensed phases, pressure dependence is needed only at
high pressures (e.g. geochemistry).
Molar volume is important quantity for phase transformations.
For the gas phase – RT ln(P/Po) is enough for description
(but not near the critical- and boiling points).
For condensed phase: Murnaghan model

13. Murnaghan pressure model
Assumption:
bulk modulus depends linearly on pressure.
Compressibility  (inverse of the bulk modulus):
(T,p) = Ko(T)/(1+nKo(T).p)
Ko(T) is compressibility at zero pressure
n is a constant, independent of T,p (usualy  4 for many phases)
Thermal expansivity  depends usually on T
 = o + 1T + 2T-2 + …
avoiding higher powers than one.
For solution phases Gibbs energy is expressed by:
For very high pressures (center of the Earth) – higher-order terms

14. Murnaghan pressure model-example
Phase diagram of pure iron:
T = T(p), T = T(V)
LFS - CT

15. A new pressure model
Lu(2005): Murnaghan model – difficulty in calculating partial Gibbs energy
Empirical relation – Grover (1973):
V(T,p) = Vo(T) – c(T) ln (o(T) / (T,p))
Vo(T), o(T) are volume and compressibility at zero pressure and c(T) is an adjustable function.
By integration:
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16. Expression for the V(T,p)
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Advantages: better suited for modeling of solution phases
can be extended to higher pressures than can Murnaghan model

17. Metastable state, lattice stabilities
Solution phase – temperature extrapolation out of range of stability – necessary for CALPHAD method
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18. Extrapolation of phase boundaries- reconciling Calphad and ab initio „lattice stabilities“
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19. Calculations of „lattice stabilities“ possible:
GCrLiquid – GCrbcc = SmL/bcc (Tmbcc – T )
GCrLiquid – GCrfcc = SmL/fcc (Tmfcc – T )
(m – melting)
GCrfcc - GCrbcc = (SmL/bcc Tmbcc - SmL/fcc Tmfcc) -
T (SmL/bcc - SmL/fcc )
HCrfcc/bcc = (SmL/bcc Tmbcc - SmL/fcc Tmfcc)

20. Temperature extrapolation of Gibbs energy into metastable region
Example:
GHSERFE *T *T*LN(T)
*T** E-08*T** *T**(-1); Y *T-46*T*LN(T) E+31*T**(-9); N SGTE !
GMGLIQ GHSERMG *T E-20*T**7; Y *T *T*LN(T); N SGTE6 !
(Extrapolation of liquid to region of solid phases or solid to region of liquid phases – see next slide.)

21. Metastable state, lattice stabilities
Solution phase – temperature extrapolation out of range of stability – necessary for CALPHAD method
LFS - CT

22. Metastable state, lattice stabilities-cont.
Gibbs energies of compounds (structures), that are never stable for given element, are needed:
Predictions are done by extrapolations or by ab initio calculations.
Unary data – SGTE – Dinsdale A.T.: Calphad 15 (1991) 317, consensus.
Any change in unary Gibbs energy of formation must be connected with check (reassessing) of all parts of the thermodynamic database depending on the previous value.

23. Metastable state, lattice stabilities-cont.
Example of the record of data published in Dinsdale A.T.:Calphad 15(1991) 317
Element: Tantalum

24. Surface of reference of metastable structures Example: Laves phase Cr-Zr
Ab initio calculated values for energy: lattice positions 8a (for Zr) 16d (for Cr)
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) (fitted val.)*T+GHSERZR+ 2*GHSERCR; N 93 !
Guess for energy (old procedure - guessed values):
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) –(fitted val.)-(fitted val.)*T+GHSERZR+2*GHSERCR; N 93 !
(See next slide)

25. Surface of reference
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The surface of reference for the Gibbs energy of a phase (A,B)a(D,E)c , according to CEF, plotted above the composition square

26. Kopp – Neumann rule For compounds with no measured heat capacity:
Heat capacity of compound is equal to the stoichiometric average of heat capacities of the pure elements in their SER state.
Gibbs energy is then:
Gm - i biGiSER = ao + a1.T
(no higher power of T is recommended for this expression
– except of T.ln(T) term in some cases).

27. Variables for composition dependence
Gibbs energy of a phase:
G = N . Gm, where N = i Ni and Gm is molar Gibbs energy
Molar Gibbs energy will be used and corresponding composition
variables will be xi = Ni / N (molar fraction).
(Mass fraction is wi = Mi / M, where Mi is mass of component i and M = i Mi .)

28. Components other than the elements
In oxide systems: use components i (CaO, MgO, O2) instead of elements j (Ca, Mg, O):
In that case:
i = j bij j
Example: Homogeneous reaction 2H2 + O2 = 2H2O
H2O = H O2
Law of mass action (ideal gas): O2 = oGO2 + RT ln (yO2)
H2 = oGH2 + RT ln (yH2)
H2O = oGH2O + RT ln (yH2O),
Where yi is the equilibrium constituent fraction of each molecule i in the gas.
Reaction constant K = oGH2O - oGH2 – 0.5 oGO2 = RT ln (O2 H2 / H2O)
for the reaction H2O = H O2
Reference state of H2O = gas consisting of pure H2O: oH2O

29. Components other than the elements-cont.
Example:
Carbon dissolved interstitially in fcc Fe:
two sublattice model: (Fe)(C,Va) (Va - vacant interstitial sites)
Chemical potential of C - determined indirectly:
GFccFe:Va = GFe
GFccFe:C = GFe + GC
Therefore: C = GC = GFccFe:C - GFccFe:Va

30. Internal degrees of freedom
Example:
Gas formed by the elements H and O.
In equilibrium one may expect to find the constituents H, H2, H2O, O, O2, and O3.
(controlled by internal equilibria)
(H2 and H2O may be used as components rather than as elements)

31. The constituent fraction
The species that constitute the phase are called the „constituents“.
Gas with several species:
consituent fraction, yi, of each specie i, describes the internal equilibrium in the gas.
Example:
H2O above (species-constituents: H, H2, H2O, O, O2, and O3)

32. The constituent fraction-cont.
For crystalline phase with several sublattices:
constituent fraction = „site fraction“
The sum of constituent fraction is unity (on each sublattice)
Mole fraction of the components can be calculated from the constituent fractions („site fractions“):
xi = j bij yj / k j bkj yj
bij are stoichiometric factors.

33. The constituent fraction-cont.
Gas phase:
Partial pressure describes composition of gas phase (Dalton‘s law – ideal gas - constituent i, constituent fraction yi):
pi = p . yi
Phases with sublattices:
Constituent fraction (site fraction) = yi = Ni(s) / N(s)
Ni(s) the number of sites, occupied by the constituent i on sublattice s
N(s) the total number of sites on sublattice s

34. The constituent fraction-vacancies
Vacancy (Va) can be treated as a real component with its chemical potential equal to zero
Structural (constitutional) vacancies are used as real component
Advantageous using of Gibbs energy for the formula unit of the phase (not for the mole of real components)
Example: It is recommended to use Cu2Mg, not Cu Mg (avoid rounding-off errors)
Thermal vacancies stabilize the phase.
Lattice stability parameter for crystalline phase has to be refitted when thermal vacancies are introduced

35. Mole fraction and site fraction
For crystalline phase:
Mole fraction xi of the component i can be calculated from the site fractions yj:
xi = s [(j bij yj(s)) / a(s) k j bkj yj(s)]
ratios a(s) are the set of smallest integers giving the correct ratio between the numbers of sites N(s) on each sublattice. Formula unit of a phase with sublattices is equal to the sum of this site ratios s a(s)
Vacancies must be excluded from this summation
(In the case of more constituents than components one cannot obtain constituent fraction from the mole fraction without a minimization of the Gibbs energy of the phase.)

36. Volume fraction of the constituents
Constituents with very different sizes (chain molecules of polymers) – entropy of mixing is influenced.
Therefore, volume fractions of the constituent are used instead (or directly the volume)– Flory-Huggins model for polymers

37. Flory-Huggins model (1941)
Flory-Huggins solution theory is a mathematical model of the thermodynamics of polymer solutions which takes account of the great dissimilarity in molecular sizes in adapting the usual expression for theentropy of mixing. The result is an equation for the Gibbs energy change ΔGm for mixing a polymer with a solvent:
The result obtained by Flory and Huggins is:
 Gm = RT [n1 ln(1) + n2 ln(2) + n12 1,2] where 1,2 = W12/2kT
and W12 = (Z/2)(2eij – eii – ejj) is „exchange energy“
The right-hand side is a function of the number of moles n1 and volume fraction φ1 of solvent (component 1), the number of moles n2 and volume fraction φ2 of polymer (component 2), with the introduction of a parameter 1,2 to take account of the energy of interdispersing polymer and solvent molecules. The volume fraction is analogous to the mole fraction, but is weighted to take account of the relative sizes of the molecules. (Similarity with regular solution model.)

38. Flory-Huggins model (1941) – cont.
Two-dimensional description of polymer molecule in the solvent lattice

39. Aqueous solutions Molality is used as fraction in the modeling
(number of moles in 1 kg of solvent).
(It can be directly transformed to the
constituent fraction of i. )

40. Questions for learning
1. Explain term „compound energy formalism“ and characterise the models, necessary for mathematical description of Gibbs energy 2. Explain terms „end member“ and „surface of reference“, characterise temperature and pressure dependence of Gibbs energy 3. Explain possibilities for receiving expression for dependence of Gibbs energy of metastable structures of elements on temperature 4.Explain terms „molar fraction“ and „lattice fraction“ 5.Explain the role of vacancies in the models