high temperature phase equilibria and phase diagrams || the calculation of phase diagrams

CHAPTER 5

The calculation of phase diagrams

5.1 T H E D E V E L O P M E N T OF PHASE D I A G R A M C A L C U L A T I O N A N D CALCULATION STRATEGY

In dealing with a system containing a number of phases at equilibrium, the following conditions must be satisfied:

1. The total free energy of the system is minimized. 2. Each component has equal chemical potential in the phases which

are in coexistence. 3. The free energy change of the system is zero.

In principle the equilibrium phase assemblage in a system, thus the phase diagram can be obtained by thermodynamic calculation if the free energy or the thermochemical parameters of the phases involved in the equilibrium are known. Actually the computation work on high temperature phase diagrams may stem from the studies by J. J. Van Laar^ in the early part of this century. At that time he defined a series of mathematical expressions for synthesizing binary equilibrium diagrams. The success and validity of this development has been recognized and confirmed by both the experimental and theoretical diagramists. However, the development in the field of phase diagram calculation was still slow some fifty years after Van Laar's elegant papers were published because of lack of thermochemical data and complexity of the equilibria calculation procedures. Subsequently, most of the early work was limited to simple systems which contain a limited number of phases. A substantial amount of progress toward the calculation of high temperature phase diagram has been made since the sixties, and during the past twnety years a number of computer programs and facilities have been established, which allow the plotting of phase diagrams by equilibrium calculation and offer equilibrium data in a much shorter time than experimental investigation would allow. In addition, the calculation of results may also be helpful in evaluating the experimental data from different sources. The main progress that promotes the calculation methods may be summarized:

153

154 High temperature phase equilibria arid phase diagrams

(1) Compilation of high temperature thermochemical data. Under the auspices of the Joint Army-Navy-Air Force Thermochemical Panel, the first edition of JANAF Thermochemical Tables^ were issued in 1965, which contain a great number of free energy and thermochemical parameters of interest in the high temperature systems. Subsequently revised edition and supplements of JANAF Tables and the like collections were continued. These published data formed a necessary and important data base for the heterogeneous equilibria calculations.

( 2 ) Progress in non-ideal solution theory. Several important approximation expressions and structural models were suggested up to 1960s to represent the excess free energy of mixtures, which allowed the phase equihbria calculations beyond the ideal solution restrictions.

(3) Computer application. By the end of sixties, L. Kaufman and coworkers established a series of computer programs^"^ and applied them to calculate a number of binary and ternary equilibrium diagrams of the refractory metallic systems. Later on G. Eriksson published the SOLGAS-M I X program"^"^ which is capable of computing equilibria in gas participating high temperature systems. These and the other computation programs developed meant the speed and reliability of equilibrium calculations were greatly improved.

(4) CALPHAD project. Under the organization and active participation of Kaufman and his colleagues, the C A L P H A D (calculation of phase diagram) project has been operational since 1973. Subsequently, the open yearly meeting started in 1975 and the first volume of the quarterly international journal CALPHAD was issued in 1977.

Two principal approaches are available for phase diagram computation, by either minimizing the total free energy of the system or equilizing the chemical potentials for each of the components in the coexisting phases. A variety of computational procedures have been designed along such lines. Figure 5.1 illustrates one example of a computational route where the equilibrium state is attained by minimizing the total free energy of the system.

5.2 DATA COLLECTION

5.2.1 S e l e c t i o n o f s p e c i e s a n d c o l l e c t i o n o f d a t a

After deciding the ranges of composition, temperature and pressure, the first step towards a phase equilibria calculation is to select the species involved in the equilibria and to collect their thermochemical data. A lot of

The calculation of phase diagrams 155

D e f i n i n g s y s t e m

D e f i n i n g c o m p o s i t i o n , t e m p e r a t u r e a n d p r e s s u r e

1 D e f i n i n g s p e c i e s a n d c o l l e c t i n g t h e i r t h e r m o c h e m i c a l d a t a

1 S e l e c t i n g t e m p e r a t u r e - , p r e s s u r e - a n d

c o n c e n t r a t i o n - d e p e n d e n t e x p r e s s i o n s

f o r t h e r m o d y n a m i c f u n c t i o n s

A d j u s t i n g t h e

c o e f f i c i e n t s

D e t e r m i n i n g t h e c o e f f i c i e n t s o f t h e t h e r m o

d y n a m i c f u n c t i o n e x p r e s s i o n s

M i n i m i z i n g f r e e e n e r g y o r e q u a l i z i n g

c h e m i c a l p o t e n t i a l f o r e a c h c o m p o n e n t

C o m p a r i n g c a l c u l

a t i o n r e s u l t s w i t h

e x p e r i m e n t a l d a t a

Τ

O u t p u t

E q u i l i b r i u m d a t a a n d / o r p h a s e d i a g r a m

FIG. 5.1 Steps of phase diagram calculation.

reference data tables, for instance in refs. 2 and 10-19, can be used for this purpose. Generally speaking, the species should be chosen from as many as possible in order to increase the reliability of the calculation results. If necessary a number of the species may be eliminated if their equilibrium concentrations are extremely low.

The thermodynamic functions can be evaluated resting upon either experimental data or estimation calculations. The following experimental approaches have provided and are providing the most thermochemical and thermodynamic data for the high temperature species.

(1) Calorimetry. Integration of heat capacity gives enthalpy and entropy

\/T+Y^H,, (5.1) H{T) = H{Q) +

5(71 = 5(0) + (5.2)

156 High temperature phase equilibria and phase diagrams

(2) Chemical equilibrium measurement. Free energy is given by

AG''=-RTlnK (5.4)

where AG^ and Κ are the standard free energy and equilibrium constant of chemical reaction. Differentiating equation (5.4) gives

dAG''/dT=-Rd{ln K)/dT

since d AG'^/dT= - AS"" and AG^ = Δ / / ^ - Τ

AH'' = RT^d{\nK)/dT (5.5)

(3) High temperature mass spectrometry. According to section 3.3.4.2, we may express the equilibrium constant of a high temperature mass spectrometric reaction as

A{gHB{g) = C{gHD{g)

f r ^ PCPD ^ Ic ΙρΕρΕοηΛηΒ^Α^ΒίΑΐΒ

PAPB II Iß EAEßncnuCcGjyycyD

Inserting Ä^into equation (5.4) allows calculation of free energy. The free energy function (G?-//^)/ris defined basing on the third law of

thermodynamics. This function often changes more slowly with temperature than does the free energy itself. We can write AG^ for temperature Tas

A G ^ / r = A ( G ^ - / / ? ) / r + AHlIT (5.6)

Inserting equation (5.4) gives /ON

AH', = -RT In K,- TA(^^^^ (5.7)

e

^ ( 5 ? l M ) . ^ ( ^ ) p r o d u c , s - l ( M ;

(4) Electrochemical emf and its temperature coefficient. At constant temperature and constant pressure the emf of a Galvanic cell

E= -AG/ZF (5.8)

where Cp is the heat capacity, Η and S are enthalpy and entropy, and i / (0 ) and 5(0) are the integral constants. Subscript tr denotes phase transition. Different heat capacity expressions are often inserted into equations (5.1) and (5.2) before and after the phase transitions. Gibbs free energy G is calculated from

G = H-TS (5.3)


^^G\ _ _

ST j r dT

äH=-zFE-zFT,\ dE\

dT

(5.9)

(5.10)

In the concentration cell

A\A''*'X' l-^(^-B)

^ = - ^ I n (»MAB)

where a ^ represents the activity of A in the mixture AB. The partial thermodynamic functions are

(5.11)

AH^=-zFE-zFT^^ (5.12)

( 5 ) Statistical mechanics calculation. The expressions given below are for ideal gas state:

Translation entropy:

5„ = Ä ( § l n M + f In r + l n P - 1 . 1 6 4 ) (5.13)

Rotation entropy:

linear molecule S, = Ä[ln Γ+1η( /+10*° ) - l n σ*-2.695] (5.14)

nonlinear molecule

Vibration entropy:

5, = /?[f In r + i ln(/i/2/3 χ 10^")

-1ησ-3 .471 ] (5.15)

1 — expl / hVi

\ kT

hv¡ kf

expl fhv¡\ [ict)

- 1

(5.16)

where Μ is the molecular weight, Ρ the pressure, / the moment of inertia of rotation, σ the symmetry number of the molecule, for the homonuclear diatomic molecular σ = 2, heteronuclear diatomic molecule σ = 1 , linear

where £• is the emf of the cell, Fthe Faraday constant and Ζ the electronic charges of the species responsible for the reaction. Differentiating equation (5.8) yields


TjK l o g ^ ä o

300 -41.5337 -16.9034 -89.6528 1000 -8.5484 -6.8257 -19.6118

Transform the reference state of CIO into Clj and O j

ClO(3) = 0.5Cl2(3) + 0.502(i)

Gäo=/?rin ^ α ο = ΛΓ(1η In / : ° , , - 0 . 5 In

Substituting ΚΡ%, we get

o = 99,031 J/mole at 300 Κ

Gao = 89,404 J/mole at 1000 Κ

and symmetric polyatomic molecule σ = 2 and asymmetric polyatomic molecule σ = 1, ν the vibrational frequency, and h and k are the Planck and Boltzmann constants respectively. The moments of inertia and vibrational frequencies of a molecule can be evaluated from the bond lengths, bond angles and geometric configuration of the molecule and the spectroscopic terms.

It should be noted that if data of the species involved in the equilibria calculation are taken from thermochemical tables which are defined on different reference states, then transformation calculations are required to bring the data into the same base. In conventional tables the data may be defined as atomic gases, stable form of elements or stable oxides. For instance, the specific reference state of CaO. SÍO2 may be the atomic gases Ca, Si and O; the stable forms Ca, Si and O 2 ; or the oxides CaO and SÍO2. Different values of the thermodynamic functions for the same substance will be found if the reference state is different. The following calculation offers an example showing how to transform from the reference state of atomic gases into the molecular forms.

In ref. 12, whose reference state is atomic gases, we have

C\0(g) = C\{g) + 0(g)

f^T=Pc\Po/Pc\o

ClM = Cl{gHCl{g)

02 = 0( ) + 0( ) ^^=Ρθ2/ΡοΡο

The equilibrium constant of the reactions is^^


5 .2 .2 E s t i m a t i o n o f t h e r m o c h e m i c a l a n d t h e r m o d y n a m i c d a t a

Although a large number of thermochemical data have been collected and published so far, the data for some special interest may still be unavailable. A number of empirical and semiempirical methods for estimating thermochemical and thermodynamic properties have been developed.^^'^^ The numerical values thus obtained may be used in phase equihbria calculation or at least suggest an approximate pattern of the phase equilibria.

(1) Heat capacity. It has been known that the heat capacity of gases approximates a constant, Cp = 2\ J/K/mole for the monatomic molecule gases and = 29 J/K/mole for diatomic gases at room temperature, which correspond to the translational and translational plus rotational energy contribution in monatomic and diatomic molecules respectively. The vibrational energy in a diatomic molecule rises with increasing temperatures. The heat capacity of diatomic gases can be expressed as

ς , = 27.852 + 4.157χ (J/K/mole)

in temperature interval 300K-2300K. For heavy molecules whose molecular weight is larger than 100, the heat capacity approximates to 37 J/K/mole in the above temperature range. The geometric configuration of the molecules will also affect the heat capacity of polyatomic gases. The heat capacity can be roughly written as follows:

Linear molecule: 20.92 + 16.74Λ^

(the number of linkages inside the molecule)

Nonlinear molecule: 25.1 + 1 6 , 1 4 X

(the number of linkages inside the molecule)

According to Dulong and Petit's rule, the heat capacity at constant volume is a constant 25 J/K/mole. This rule often holds for the solid elements at room temperature or high temperatures. The heat capacity of solids increases with increasing temperature. At the first transition temperature, polymorphic transition or melting, Cp = 29.3 —30.3 J/K/mole of atoms.

Estimation of heat capacity of solids or liquids: Example At room temperature the heat capacity of CdO is

43.2 J/K/mole, and the melting point is 1385°C, then

ς , = 39.3 + 12.9 X 10"^Γ (J/K/mole)

Additive method:^^


(2) Entropy. The entropy of gases can be approximately evaluated by their molecular weights: the empirical formulas are:

The number of atoms in molecule oO ^298

1 110.88 + 33.05 log M (±6 .7 ) 2 101.25 + 69.2 log Λ / ( + 5.86) 3 37.66+111.71 log M (±7.53)

4 -7.53 + 146.44 log M ( ± 6 . 7 )

5 -131.8 + 207.111 log Μ (±11.3)

Τ. Β. Lindemer et al^^ have provided an additive method for calculation of entropy of polyatomic gases. They consider the polyatomic molecule D^E^O^+^y as a complex molecule composed of x(Z)O), z{EO) and 0.5>^(θ2), where D and Ε may be all the elements except O and H. The entropy of the supposed complex molecule is expressed roughly as

Table 5.1 lists the heat capacity parameters of ionic components used in estimation calculations.

Example The heat capacity of Al2(S04)3 is estimated from the parameters in Table 5.1.

ς,(Αΐ2(8θ4)3, 298K) = 2 X 19.7 + 3 χ 76.5 = 268.9 J/K/mole

where the experimental value is 259.4 J/K/mole. Heat capacity additive rule in chemical reaction: The difference in heat capacities between reactants and products of

chemical reaction is usually small. Then we may suppose

Σ (^p)products Σ (^p)reactants ^

Since and AS are subtracted and vary simultaneously in the chemical reactions, the errors involved in the above equation for free energy calculation should tend to cancel out.


Slgi{cx) = \.5R In M , , + 4 In 298.15-2.349 + / :

+ x i^" - 27.253 -1 .5R In M)oo

+ z(5" - 27.253 - 1 . 5 Ä In M)£o

+ 0.53;(5°-27.253-1.5/?lnM)o, (5.17)

where the 5" term can be obtained from reference tables or estimated empirically. is a constant

K= -1.5/? In 298.15 + 12 (J/K/mole)

TABLE 5.1 Capacity parameters in ionic solids (J/K/mole) at 298 K^^

Cationic part Anionic part

Ag 25.7 Ho 23.0 Sn 23.4 Η 8.8 Al 19.7 In 24.3 Sr 25.5 F 22.8 As 25.1 Ir (24) Ta 23.0 α 24.7 Ba 26.4 Κ 25.9 Th 25.5 Br 26.0 Be (9.6) La 25.5 Ti 21.8 I 26.4 Bi 26.8 Li 19.7 Tl 27.6 O 18.4 Ca 24.7 Mg 19.7 U 26.8 S 24.5 Cd 23.0 Μη 23.4 V 22.2 Se 26.8 Ce 23.4 Na 25.9 Y (25) Te 27.2 Co 28.0 N b 23.0 Zn 21.8 OH 31.0 Cr 23.0 Nd 24.3 Zr 23.8 SO^ 76.5 Cs 26.4 Ni (27.5) N O 3 64.5 Cu 25.1 Ρ 14.2 Ρ (23.5) Fe 25.9 Pb 26.8 C O 3 58.5 Ga (21) Pr 24.3 Si (25) Gd 23.4 Rb 26.4 CrO^ 91 Ge 20.1 Sb 23.8 M 0 O 4 90 Hf 25.5 Se 21.3 W O 4 97.5 Hg 25.1 Sm 25.1 U O 4 111

evaluated from the diatomic molecules whose molecular weight locates between M^^^q^ — M^^q^ . For instance the entropy of the molecule F 4 W O can be estimated

^ 2 9 8 ( F 4 W O = 2F2 + W O ) = 1.5Λ In Μρ^^ο + 4 In 298.15 - 2.349 + Κ

+ 2(S°-27.253-1.5Äln M)^^

+ (5° - 27.253 -1 .5R in Μ ) ^ ο


or

. 298 (F4WO = WF + FO + F2) = 1.5/? In Mp.wo

+ 4 In 298.15-2.349 + / r

+ (5^-27.253-1.5/? In

+ (5^ - 27.253 -1 .5R In M)^^

+ (5^ - 27.253 -1 .5R In Μ ) F .

If a polymerized molecule consists of three or more monomers, an additional term —0.5/?(n —2) In 298.15 must be inserted into equation (5.17), where η is the order of polymerization. For instance, the order of polymerization of BcjOj and BcgOs is three and five respectively.

Two approaches are often used to estimate the entropy of solid substance.

(1) The entropy of a compound is calculated from its atomic or ionic components. Tables 5.2 and 5.3 list the entropy contribution parameters for conventional cations and anions in ionic solids. The calculated and experimental entropies are given in Table 5.4.

(2) Entropy of complex oxides can be approximately calculated by the summation of the component oxides.

(3) Heat and entropy of vaporization. According to Pietel-Trouton, the entropy of vaporization is nearly a constant:

5, = i/,/r, = 92 J/K/mole (5.18)

where subscripts ν and b denote vaporization and boihng. If the boiling

TABLE 5.2 Entropy contribution in solids (J/K/mole) at 298 K^^

Ag 53.6 Eu 59.0 N b 51.0 Sm 59.0 Al 33.5 Fe 43.5 Nd 58.2 Sn 54.8 As 47.9 Ga 46.9 Ni 43.9 Sr 50.2 Au 64.0 Gd 59.8 Os 63.2 Ta 62.3 Β 20.5 Ge 47.3 Pb 64.9 Tb 59.8 Ba 57.3 Hf 61.9 Pd 53.1 Te (56.1) Be 18.0 Hg 64.4 Pr 57.7 Th 66.5 Bi 65.3 Ho 60.7 Pt 63.6 Ti 41.0 C 21.8 In 54.4 Ra 66.1 TI 64.4 Ca 38.9 Ir 63.6 Rb 49.8 υ 66.9 Cd 54.0 Κ 38.5 Re 62.8 ν 42.3 Ce 57.7 La 57.7 Rh 52.3 w 62.8 Co 44.4 Li 14.6 Ru 52.3 Y 50.2 Cr 42.7 Lu 61.9 S (35.6) Yb 61.5 Cs 56.9 Mg 31.8 Sb 55.2 Zn 45.6 Cu 45.2 Mn 43.1 Sc 40.6 Zr 50.6 Dy 60.2 M o 51.5 Se (48.5) Er 60.7 Na 31.4 Si 33.9


TABLE 5,3 Entropy contribution in solids (J/K/mole) at 298

Charge of cations

Anion + 1 + 2 + 3 + 4

F ( 2 3 . 0 ) 19.7 ( 1 6 . 7 ) 20 .9

CI 4 0 . 6 33.9 28 .9 33.9

Br 54.4 45 .6 ( 3 7 . 7 ) ( 4 1 . 8 )

I 61.1 56.9 52.3 54 .4

OH ( 2 0 . 9 ) 18.8 12.6

O 10.0 2.1 2.1 4.2

S 34.3 20 .9 5.4 10.5

SO^ 92 .0 72 .0 57.3 50 .2

Se ( 6 6 . 9 ) 47 .7 ( 3 3 . 5 )

Te ( 6 9 . 0 ) 50.6 ( 3 7 . 7 )

C O 3 63.6 47 .7 ( 3 3 . 5 )

SÍO4 ( 7 9 . 5 ) 57.7 ( 3 7 . 7 ) 33.1

SÍO3 70.3 43 .9 ( 2 9 . 3 )

P O , ( 1 0 0 . 4 ) 71.1 ( 5 0 . 2 )

TABLE 5.4 Comparison of calculated entropy of inorganic solids with literature data (J/K/mole) at 298 Κ

Calculated Literature data^* Calculated Literature data*^

LiF 37.3 35.7 MgS 52.7 50.3

BeFj 57.4 53.4 BaS 78.2 78.2

C0F2 83.8 82 .0 GeS 68 .2 66 .0

BaFj 96.7 96 .4 NdSe 91.7 90 .4

NaCl 72.0 72.1 Ba(OH)2 94.9 100.8

KCl 80.1 82 .6 LaOCl 88.7 82.8

AlBr3 146.6 180.3 LiBeF3 95 .0 89.1

LaBr3 170.8 182.0 LiBeF^ 132.6 124.7

Irl 222 .2 224.7 MgC03 79.5 65.9

ΙΓΙ2 124.7 108.8 LijSO^ 108.7 121.3

BÍI3 177.4 159.0 CaSO^ 105.2 111.8

CaO 41 .0 39.8 CUSO4 109.2 117.2

B2O3 47.3 53.8 MgSiOj 75.7 67.8

SÍO2 42.3 41 .5 MgSiO^ 121.8 95 .2

GeOj 55.7 55.3 CajSiO^ 135.5 120.5

temperature is known, then the heat of vaporization can be simply calculated. This relation satisfies most substances except those having extremely low boihng points.

(4) Heat and entropy of melting. Different from vaporization, the entropy of melting is much smaller and varies from substance to substance, but is characteristic of the structure of solids. For inorganic substances, the melting entropy often decreases as the covalency or disorder of a structure


(5) Heat of formation. In many cases the heat of formation has a more important contribution to free energy than entropy. But, unfortunately, so far a generally usuable method for estimation of the heat of formation is still lacking. Most of the proposed approximation approaches are only applied to some specified or limited areas. Subsequently several such methods are briefly described.

(1) Interpolation from analogous compounds. Heat of formation of analogous compounds, consisting of the same metal and non-metallic elements in the same group in the Periodic Table or of the same non-metallic element with metallic elements of the same group, usually have a smooth plot (not necessarily linear or monotonous). Interpolation from the plot may possibly give an acceptably approximative value of unknown heat of formation. The following gives the interpolation examples:

From CaO(318)-^CaS(238.1)^CaSe(156.5)-^CaTe(x), we have x = 113-142; CaSn(104.6)^Ca3Sb2(145.6)->CaTe(x)^

Cal2(178.2), x = 146-176; Mg2Sn(25.5)^Mg3Sn2(66.1)^ MgTe(104.6)-^Mgl2(120.1), CaTe(x) , x = 126-167

where the numerals in the brackets are negative heats of formation in kJ/mole of atoms. The heat of formation for CaTe is estimated -290,000 + 29,000 J/mole.

(2) Figure 5.2 shows the Born-Haber thermochemical cycle which has been employed to evaluate the lattice energy of crystalline substances. However if the lattice energy and the other heat terms are known, the thermochemical cycle may also be used for calculation of the heat of formation, as demonstrated in Fig. 5.2.

(3) A number of empirical parametric formulas have been proposed for estimation of the heat of formation. However the physical and chemical parameters proposed there are frequently ambiguous and the parameters used may be different from different sources, therefore attention must be paid to the applicability and accuracy of these parameters. Otherwise erroneous results and unreliable conclusions can possibly be reached.

Kapustinskii:^"^

^H%JW=a\ogZ^b

where ζ is the atomic number, W the atomic valence and a and b are constants.

increases. Chemical analogues usually have similar entropy of melting although the melting point may be quite different. The melting entropy of alkali halides was found to be approximately equal to 12.6 J/K/mole of atoms and for binary alloys to be 9.2 J/K/mole of atoms for a disordered lattice and 14.6 J/K/mole of atoms for an ordered lattice.

+ Δ Η

M e ( s ) sub


+ I

165

M e ( g ) Heat o f s u b l i m a t i o n I o n i z a t i o n e n e r g y

X 2 ( g ) ^diss

- > X ( g ) -E

Heat o f d i s s o c i a t i o n

- Δ Η ^

E l e c t r o n a f f i n i t y

l.leX( s ) Heat o f f o r m a t i o n L a t t i c e e n e r g y

- Δ Η ^ . υ - Ι - Δ Η 3 ^ ^ Δ Η U i s s

FIG. 5.2 Born-Haber thermochemical cycle.

Reznitskii:^^

where Ε is the standard electrode potential. Michdlov-Petrosyan and Babushkin:^^

-AH%s=-l + Σ Δ ^ . + Σ Δί/disso

where ε , is the mean bond energy, AH^ the vaporization energy of sohd elements and AU^-,^^^ the dissociation energy of gaseous molecules.

Wen Yuan Kai et alr}^

where is the heat of reaction and Wj the parameter of energy of formation of metallic ions.

Electronegativity parameter formula:^®

Aif598=-96.48

where Ε is the electro negativity of elements. (4) Lindemer et alP have offered simultaneous inequahties for free

energy changes of a series of chemical reactions to calculate heat of formation if the relevant entropies are known. For example, in the binary system A-B there are three intermediate compounds A2B, AB and AB2, the reactions of formation are

AÂBÂ2B

A2B-ÂB2^3AB

ΑΒΛ-Β-^ΑΒ2


Based on = AHÍ-T AS? < 0, we have

Ai/?i - Δ / / ί 1 - TiS¡, ~ S^ - Χ O (5.19)

(5.20)

Δ / / ? 2 - Δ < - r ( S ? 2 - S ¿ ) < 0 (5.21)

where subscripts 11, 12 and 21 refer to the intermediate phases AB, A B 2 and A2B, respectively. For instance AHi2 denotes the standard heat of formation of A B 2 . In the approximation calculation, three parameters a, b and c are inserted into (5.19)-(5.21) to transform them into equahties. If the entropy terms and at least one of the heat of formation terms are known, then equation (5.22) is obtained by eliminating AH^2 ^ ^ 2 1 ·

a + b + c=- A//?i + Τ AS?i (5.22)

The unknown heat of formation can be solved by approximation calculations.

(6) Gibbs free energy. The temperature dependent expression of free energy and free energy function can be written as

AG^ = AH^gs - Τ AS%^ -h Τ A Q 298

tr/J (5.23)

and

fen= Τ

^ Τ " ^ 2 9 8 — {5γ — ^298 ) ~ »^298

ο 298

(5.24)

where Δ represents the difference between products and reactants and Γ„ the transition temperature, where 298< Γ „ < Γ. If taking ACp = 0 , then

G° = Δ / / °98 - Τ ASU + Δ[Σ - TIT,^-\ (5.25)

If no phase transition occurs,

Δ σ ° = Δ / / , % - 7 ' Δ 5 2 % (5.26)


Compound

Carbides Periods 2 and 3 Periods 4 -7

0.97 + 0.09 1.08±0.11

0.97+0.10 1.03 + 0.10

Nitrides Periods 2 and 3 Periods 4 -7

0.97 + 0.09 1.08 + 0.04

0.93 + 0.17 1.04+0.06

Oxides Periods 2 and 3 Periods 4^7

1.05 + 0.11 1.15±0.11

Fluorides 1 .21+0.13 1.05 + 0.14

Suicides 1 .02±0.15 1.16 + 0.24

Borides M e / B ^ l M e / B < l

1.02 + 0.10 1.03 + 0.11 0.96 + 0.11

5 .2 .3 M o d i f i c a t i o n o f t h e r m o c h e m i c a l o r t h e r m o d y n a m i c d a t a

In phase diagram calculation, inconsistency may sometimes exist in the data collected from different sources or in the results obtained. The following approaches may be employed in processing the data for the equilibria calculation.

Two computational methods^^ have been proposed to estimate the molar heat capacity contribution in equation (5.24). The first method defines a parameter as equation (5.27):

^ ~ x / ? ( M ) + z / ? ( y ) ^^"^'^

/?= C ^ ( - - - ] d T „ (5.28) J 298 \ ^ ^ i r /

where is called the integral of heat capacity contribution and evaluated from known free energy functions. The second method defines the parameter R 2 as

(w + ) ; ) / ? ( M , r j ^ ^ - ( x - h z ) / ? ( M . O , ) ^^-^^^

Average and R 2 are calculated (Table 5.5) and used to evaluate the unknown free energy function in analogous compounds.

TABLE 5.5 Rj and R j / o r calculating heat capacity integral (I?)^'


5.3 M A T H E M A T I C A L T R E A T M E N T OF T H E R M O D Y N A M I C F U N C T I O N S

5.3.1 T e m p e r a t u r e d e p e n d e n t e x p r e s s i o n

In principle the relation between heat capacity and temperature can be mathematically approximated with a polynomial equation of temperature. Equation (5.30) is one of such polynomials usually used in the phase diagram and equilibria calculations.

Cp=m^ + m^T-^m^T-^-^m^T^-^m^T^ + · · ·

The enthalpy H, entropy S and Gibbs free energy G are derived

s=s{ToH To

CpdT

Τ

G = H{To)-S{ToHT To

^dT To Τ

Substituting equation (5.30) into (5.31)-(5.33) yields

H=m^+mj^T^m^ Τ^β -m^T'^+m^

S=m2-\-m^ In T-^m^T-m^T'^/l + m^T^/l

+ m 7 r V 3 + - "

G = m^-m2T-{-m^T{i-ln T)-m^T^/2

-m^T-'/2-m^T^/6-mjT^/l2+ · · ·

= mi-(m2-m^)T-^m^Tln T-m^T^/2

-m^T-^/2-m^T^/6-m^T''/\2+ · · ·

where m^ = H{TQ) and m2 = S{TQ) are integral constants.

(5.30)

(5.31)

(5.32)

(5.33)

(5.34)

(5.35)

(5.36)

(1) In evaluating the data collected from different sources a least square method is usually applied to ñtting the data points and issuing the most reliable expression for the points. Unreliable data points may be omitted according to the error theory.

(2) Equilibria are calculated and compared with experimental measurements, if available. This procedure is also used for data modification and extrapolation.


5 .3 .2 C o n c e n t r a t i o n d e p e n d e n t e x p r e s s i o n

The free energy of a multicomponent system at given temperature Tis

G = ¿ xfii-^RT ¿ Xflnxf + ^^G (5.37) 1 = 0 1 = 0

where X; and G¿ represent the mole fraction and molar free energy of component i. The first term at the right side of equation (5.37) corresponds to the free energy of mechanical mixture, the second to the mixing free energy of an ideal solution, where AH^^^=0 and ^5^^^^ = — Λ X χ,· In(Xí), and the last term is the so-called excess free energy characteristic of the non-ideality of the mixture. The former two terms can be directly obtained from the data of pure components and concentrations, but the excess free energy is dependent upon the concentration, which can only be described either by a fitted equation with activity measurements or by an expression deduced from an assumed structural model of mixing.

5.3.2,1 Mathematical approximation expression of

excess free energy

(1) Binary system. The free energy of binary mixture is

G = XiGi+X2G2 (5.38)

where subscripts 1 and 2 represent components 1 and 2, and G the molar free energy, G¿ = Gi + /?rin(a¿), a^=7¿x¿, where a and y are activity and activity coefficient. Substituting Gs into (5.38) gives

G = x^G^-\-X2G2-\-RT{x^ Ina^-f X2 In «2)

= x^Gi-\-X2G2-\-RT{xi In X1 + X2 In X2)

+ ΑΓ(χ ι1η7ι+Χ2ΐη72) (5.39)

A comparison of (5.37) and (5.39) gives

^^G = /?r(xi In y, + X2 In 72) (5.40)

The activity coefficient y depends upon concentration at given temperature T. Because the excess term disappears for the pure components, the polynomial for a binary system has to include the common term χ 1X2 or x(x— 1), where x = Xi or X2. The polynomials commonly used are

Margules expression

- σ = χ . Χ 2 Σ ^1^2 (5.41) ί = 0


Borelius expression:^^ η

i = 0

Redlich and Kister expression:^^-^^

i = 0

Bale and Pelton expression:^'^

i = 0

(5.42)

(5.43)

(5.44)

In these functions A¡, B¡, C¡ and /),· may be a function of temperature. The Legendre polynomial of degree i is defined as

Pnix) = ( 2 « - l ) ( 2 n - 3 ) . . . l

n! X —

« ( n - l )

2(2η-1) · ,n -2

n ( H - l ) ( « - 2 ) ( » - 3 ) •

2 · 4 ( 2 η - 1 ) ( 2 η - 3 )

The Legendre terms PQ-PS are

PoW = l

P i ( x ) = x

1 P 2 W = ^ ( 3 x ' - l )

P3(x) = ^ ( 5 x ^ - 3 x )

P 4 W = | ( 35x* -30x^ + 3)

Ρ5(χ) = ^(63χ ' -70χ3 + 15χ) O

and the recursion term is

The expressions (5.41)-(5.44) can be transformed into each other in a binary system, where x^ + X2 = 1. However they have different behaviour in numerical calculations.


Ternary system ( ' ' ' ( ? 2 ) e x t r a p o i a t i o n + '""i a

i

Quaternary system ( ^G3 ) e x t r a p o i a t i o n + """G^

n-component system _ , ) e x t r a p o i a t i o n + """G^

(2) Ternary and multicomponent systems. The excess free energy in a ternary system is described by polynomials composed of adjacent binary and ternary additional terms with a common factor x^X2X^.^^~^^ The ternary terms are

- G 3 = x , X 2 ^ 3 Σ "Σ {Α,Χ\Χ\ΧΓ-^) (5.45) i = 0 j = 0

which disappear in the binary systems, where x^, X2 or x^ becomes zero. Two different principles are often used to treat binary description in the

ternary system. (1) All three components are treated in the same way, for example in

ref. 35, where the mathematical expression has symmetric form. (2) A couple of the components are considered as similar,^for

example, if component 1 « component 2, then binary system 1 - 3 « binary system 2-3. The expression has an asymmetric form.

Based on these treatments the ternary expressions of excess free energy are

Kohler's expression:

- G = ( l - X 3 ) ' '""GuiîX-îf ^ ' ^ 2 . 3

+ (l-X2)'^'G3,i+ZPofcÛ2^S O'J* ' ) (5-46)

Toop's expression:

X 1 + X 3 X l - h X 3 ^ ' ^

+ ( 1 - X 2 ) ^ ^^G3,i+ZPo-fc i i S (5.47) The excess free energy of an n-component system contains two-, three-, . . . , (n— l)-component terms and n-component additional terms.

-^G = x,G, + /? rx , ln Xf + ^^G i

Binary system ^''G2


5.3.2.2 Structural model

The deviation of measured activities in high temperature mehs from those calculated by ideal solution can be imagined to be caused by interactions between the components of the system. As a result, fragmental molecules (atoms or ions) and/or associates are formed. These must be treated as additional species with respect to the formulae of the mixing entropy, resulting in an activity description which can not be simply described by ideal solution terms of the original system. Structural model treatment aims to explain the activity changes in terms of the occurrence of the additional species and thereafter express their activities.

Consider the binary system A-B, in which the associated molecule Aßj is formed in the homogeneous liquid phase by

iA+jB=AiBj (5.48)

Take the bulk composition as n^ + rig, then the mole fraction x^ = ^Α/(^Α + "β) and = n^Hn^ + n^). If + = 1, then x^ = and = . At equilibrium suppose the number of moles of A, Β and Αβ^ is n^^, n^^ and η^.β., from (5.48) we have

(5.49)

nB,k = nB-JnA,Bj (5.50)

« Α , β + ^Β,β + n^iBj = + + (1 - i -j)n^,Bj

= 1 + (1- í-jX^^^. (5.51)

The equilibrium constant

( n x - ' « A . i . , ) K - ; « ^ , B , y

Suppose pxq species of associated molecules are formed, where

¿=1,2 ,3 , ,p

; = 1 , 2 , 3, ,q

then η m

« A . c = « - Σ Σ í " x i B , (5-53) í= l j= l

« B , e = nß- Σ Σ J"ABJ (5.54) 1 = 1 J = l


The equihbrium constant can be described as

^ = π π f^u

Π π « AiBj 1 - Σ Σ (ΐ-^·-;Κ.,

Σ Σ ^^MBj] ( η Β - Σ ΣJ^Λ,BJ i = l j = l / \ i=lj=í J

(5.55)

In a modelled structure calculation, if the number of molecular, or associated species is greater than the number of components, their concentrations can not all be derived from the bulk composition. Additional parameters are required in order to describe the homogeneous equilibria in the system. For this purpose the following methods have been proposed:

(1) Supposing the reactions for forming the associated molecules have the same value of equilibrium constant."^ " ^ Along this line R. C. Masson deduced the following relations in silicate melts:

s i o r + s i o r = s i , o r + o - K,,J^^^^^

s i o r ^ s i , o r = s i 3 0 ? o - + o - ^ . - [ ^ i S ' o ; : ]

sior+si„of<'',V'-=si„,i0^1V/>- + o ^ -^ [ S i „ , i O i ! . V / ' - ] [ o ^ - ] '^ '" [S io r ] [S i ,o i i , \V»- ]

^11 = ^ 1 2 = ^ 1 3 = * ' '

The distribution of concentration of the polymerized silicate anions is determined by equilibrium constant K. In the MO -SÍO2 binary systems the activity expression is

1 ^ S i O z — 1 ^MO —

3 - a : + + 5 — 1 ~ ^MO ^MO _^ ^

^-^MO

(2) The equilibrium constant of possible chemical reactions is calculated and compared with experimental activity data, and the predominant


5.4 CALCULATION OF PHASE D I A G R A M S

Thermodynamic equilibrium at given constant temperature, pressure and amounts of components is determined by minimizing the Gibbs free

equilibrium is chosen from the temperature dependence of the equilibrium constant and calculated activities."^^

(3) Fitting activity data by Monte Carlo calculation."^"^ (4) All the possible polymerized anions in MY4(Si04 and

BeF4)-containing melts are calculated and fitted to measured activities. The mixing entropy changes are evaluated by the configurational entropy of volume fractions and enthalpy changes by the number of non-bridging anions."^^

(5) In the melt consisting oiA.B and Afi^ species, (refer to equations (5.48)-(5.51), the enthalpy and entropy changes are defined to include the terms representing formation of the associated species and the behaviour of a regular solution,"^^"^^

A//=A//^^^-hAi/^^ (5.56)

A5=A5^^^ + A5-^ (5.57)

where superscripts as and reg denote associated species and regular solution. The enthalpy and entropy changes of the regular solution are

A//^^^ = C-^(n^,n^,)/n +Cr^(n^,,n^..3;/n

+ C r ^ K c « ^ . B , ) / « (5.58)

AH^^ = n^^,^AH%^ (5.59)

AS^^^ = - R{n^^, In x^., + n^,. In x^,, + n^^^^ In x^^^^) (5.60)

AS- = n^^sj^S%^ (5.61)

where C*^ , C[^^ and are the parameters relating to the interactions between the species A and B, A and AiBj, and AiBj and B, respectively. AH^.ß. and A^^.^^. are the enthalpy and entropy of formation.

In view of the structure of melts, the methods (1) and (4) are usually applied to silicate, borate and phosphate systems in which the interactions are strong and stable associated species are formed. Methods (2) and (5) are applicable to the systems in which the interactions are weak and fewer associated species with low stabilities are formed.


5.4.1 C a l c u l a t i o n s b y t h e i d e a l s o l u t i o n m o d e l

A close agreement of calculated equilibrium diagrams by the ideal solution model with experimental liquidus, solidus curves has been observed in a few high temperature systems in which tendencies towards formation of dissociated and associated molecules are low.

For a binary liquid we have

G(/) = n^GaO + n ,G,( / ) + ä t / - ^ I n + \n^-\-n2 n^-\-n2 « 1 + ^ 2

where and «2 are the number of moles of components 1 and 2. The following equations describe the free energy of the phase assemblages of mechanical mixture of solids, pure solid + liquid, and solid solution+ hquid, respectively.

G{s)/n,G,{s)-^n2G2Ís) (5.62)

G(5i + / ) = Gl (s) + (n,-m,)G, (/) + « 2 ^ 2 (/)

^RT( " ^ - ^ ^ In " ^ - ^ ^ ^ « 1 — ^ 1 + ^ 2 « 1 — + « 2

+ ^ In ^ (5.63) ni+m^+n2 « 1 — ^ ι 4 - η 2 /

energy. At the same time, the free energy change of the system is equal to zero and the equahty of the chemical potential of each component holds in the coexisting phases. Principally there are two approaches for the equilibrium calculation. The first approach calculates the free energy of all the possible phase assemblages, regardless of whether they are stable or metastable. Afterwards the most stable state is picked out by comparison of the free energies calculated. The second approach takes the chemicals as variables, then the equihbrium state is determined by equalizing the chemical potentials. So far a number of computer programs^" ' ^"^"^ have been designed and established by which phase equihbria and phase diagrams can be directly constructed on the basis of thermochemical data of the phases involved in the equihbria. In addition, comparisons between the calculation results and experimental measurements are sometimes helpful to modify and evaluate the thermochemical data used in the equilibria calculation as well as to improve the calculation accuracy and reliability. Subsequently, several practical examples will be presented in order to further elucidate the thermodynamic and computational approaches for phase diagram calculation.


G(ss + l) = n'l Gl (s) + n'zGjis)

+ Rli , In , + , In , \n'i + n'2 n'l+ «2 n'l + n'j n'l + n'j

+ (ni-n ' i )Gi( / ) + (n2-n'2)G2(0

+ .RT( " r " ' ^ , l n " r " ' ^ ,

« 1 - n ' l + « 2 - ^ 2 n^ + n\^-n2-n'2

where is the number of moles in solid phase and n\ and η 2 are the number in the solid solution phase. Minimizing the free energy in the above equations reveals the equilibrium phase assemblage. This section deals with the free energy-composition plots by which the equilibrium phase composition can be obtained via the tangent line rule. The analytical computational approach will be described in section 5.4.4, which is most satisfactorily applied to the calculation in multicomponent systems.

Figure 5.3 illustrates the free energy-concentration plots of a binary system in which pure liquid was chosen as the reference state. It may be seen that at temperature all the mixtures of solid have lower free energy than the liquid. Therefore the solid phase is the stable phase. At temperature Γ3 the liquid phase is stable. and Γ3 are the temperatures below the eutectic point and above liquidus respectively. Between the two extremes, for instance, at Γ2, where G^{s)>G^{l) but 0 2 ( 5 ) > G 2 ( / ) , the equihbrium phase composition can be obtained as follows: Firstly draw straight lines through G^{s) and ^2 (5 ) insecting the free energy curve for the hquid mixture, for instance, at L^-L^ and L[-L^. The two end points of these tie lines will represent the composition of a solid phase and a hquid phase. Secondly, we draw a vertical line through the composition χ at which the equilibrium is calculated. The intersection point between the vertical line and the tie lines will give the free energy of the system, where the coexisting phases are the end members of the tie lines. By comparisons we would find that the phase assemblages containing solid 1 are unstable since their free energies are always higher than that of the liquid mixture. On the other hand, the equilibrium composition can be attained on the line G2(s)-L^ where the free energy is a minimum. The basic procedures for constructing phase diagram by the free energy-composition plots are summarized.

(1) If the free energies of both solids 1 and 2 are lower than the minimum of the free energy curve of liquid mixtures or all the sohd mechanical mixtures have lower free energies than the same composition of liquid mixtures, the temperature is below the solidus or eutectic point.


GA

V

s _ ^

V

\ ^

X

Γ

1 1 1 I — 0:2

Mol fraction

FIG. 5.3 Free energy-composition plot of binary system.

(2) Solid 1 or 2 or both are under the free energy curve of the liquid mixture but above the minimum. An equilibrium liquid composition can be obtained by drawing a straight line through the free energy of the solid(s) tangent to the free energy-composition curve for the liquid mixture. The tangent point corresponds to the equilibrium composition of hquid phase. The temperature is between the eutectic point and liquidus temperature of the bulk composition.

(3) If the free energies of both solids 1 and 2 are located above the free energy of the liquid, the temperature is above the liquidus and liquid is the stable phase.

Figures 5.4 and 5.5 show the free energy plots of AI2O3-SÍO2 mixtures at four temperatures and the phase diagram calculated by the above-mentioned approach.


^ ^ ^ - mm _ l 1 1 L .

S i 0 2 ( ) . 2 ( ) . lO.CO.K 3 : 2 0 . 2 0 . 10.(ί().8Αΐ2θ:Η SÍO2O.2O. 1 0.(50.8 3 : 2 0 . 2 0 . U).(i 0 .8A1

Mol fraction 2 O 3

Ι8Γ)0Κ

0.037 i lO.OTi) /

- 3í)0()0 -21500

- 39000 . . 1 , 1

S i ( ) : O . 2 0 . I ( ) . ( ; 0 . 8 3 : 2 0 . 2 0 . I ().(; O .SAbO:^ SÍO2O .20 . I 0 . ( ) 0 . 8 3 : 2 0 . 2 0 . 4 0 . ( ) 0 . 8 A 1

FiG. 5.4 Free energy-composition plot of the binary system A l j O j - S i O j .

.0,

2100,

2000

1800

1000

1100

1200

L

/ L + : Í A I 2 0 : h I X · 2SÍO2

:^Αΐ2θ. · 2SÍO2 + A I 2 O H

:^Αΐ2θ. · 2SÍO2 + A I 2 O H

SiO ^ + - 3 A I 2 O S · 2SÍO2

-

SiOi 1/3 3AI2ÜH V'2 Al.Oa •2SÍO2

Mol fraction

FIG. 5.5 Calculated A l j O j - S i O j phase diagram.

5 .4 .2 C a l c u l a t i o n s u s i n g s t r u c t u r a l m o d e l s o f t h e

l i q u i d

As mentioned above the structural model considerations offer an approach to calculating phase diagrams of systems in which the behaviour of the mixtures deviates from the ideal solution but httle or no activity data are available. Structural models can be proposed based on structural knowledge, activity data or theoretical considerations. The following example will explain how to calculate the rare earth sequioxide-beryllia


" B e O , e = l - 2 x + ^ 3 (5.66)

Σ η _ . ο = 1 - | ^ + ^ (5.68)

In these equations the subscript e denotes the equilibrium state and Σ n_ q indicates the summation of negative charged and neutral species. From the above chemical reaction we have

Substituting the equihbrium constant into (5.65) gives

"^'Ih ,>)--2·0«« ί (5.69) According to the hquidus temperature expression,

where subscripts A and / represent component A {REO^ ^ or BeO) and fusion, Η is the enthalpy and a the activity. In the calculation by using a structure model, one often replaces activity by concentration, i.e. a quasi-ideal solution is assumed. Solving equations (5.69) and (5.70) yields liquidus temperature Γ. Figure 5.6 illustrates the calculation results.^^

systems by the structural model proposed in ref. 43. The structure model calculation for the Yb203-BeO system suggested the following equilibrium reaction:

Yb203 + 2BeO = 2Yb^ ^ + Be20f "

Be20f-=2BeO + 302-

A liquidus temperature fitting calculation gives the equilibrium constant expression

log K= -2.1066+ 1748.9/Γ (5.65)

Equation (5.65) is used to construct the phase diagrams of the heavy rare earths systems, where the similar structure model is assumed.

In the /?£Όι 5-BeO systems, where RE= Dy, Ho, Er or Lu, consider the mole composition REO^^ = x and BeO = l—x, at equilibrium suppose the moles O^" ='ίο^-,β' then


2()()()r

21()0h

2200 h

2000 h iHOOh

KiOO

120( BeO 20 10 00 (mol^c;)

RE=Dy, IIo, Er, Lu

HO Rp:o,.r.

FIG. 5.6 Calculated phase diagrams of the binary systems D y j O j , H 0 2 O 3 , ErjOj and L u 2 0 3 - B e O .

5 .4 .3 K a u f m a n a n d B e r n s t e i n c o m p u t e r programs^-^

The main strategy for calculation of phase diagrams described above is to omit the excess free energy term by assuming ideal or quasi-ideal solution. At the beginning of the seventies Kaufman and Bernstein published the book Computer Calculation of Phase Diagrams, in which around 100 metallic equilibrium diagrams were calculated by assuming a regular solution model. This section will deal with some details of the calculation method.

5 . 4 . 3 . 1 Liquid-liquid equilibrium

From equation (5.37), the free energy of liquid binary mixture can be written as

G' = ( l - x ) G { + x G ^

+ ΛΓ[χ1 ηχ + ( 1 - χ ) 1 η ( 1 - χ ) ] + ^^σ' (5.71)

where χ is the mole fraction of the component 2, G[ and G2 are the free energies of the pure liquid states of components 1 and 2, and ""G' is the


excess free energy of mixing, which is zero when χ = 0 or 1. In regular solution assumption,

"^G^ = L x { l - x ) (5.72)

where L is a constant characteristic of the interaction in the liquid. When L = 0 , the solution is ideal. The partial molar free energy is

G[^G'-a(^^ = G[^RT\n{\-x) + Lx^ (5.73)

= G[-\-RT\xvx + L{\-xY (5.74)

In the two Hquid region including liquids x, and χ, , we have

and

From (5.73) and (5.74)

Ä n n ( l - x , , ) + xf,L = / ? r i n ( l - x , J + x,2,L (5.75)

RT In X,, + (1 - XtfL = RT In x, + (1 -x^fL (5.76)

For the case of a regular solution, a symmetrical miscibility gap is expected by equations (5.75) and (5.76) since x, = 1 — χ, , then

RT l - 2 x , ,

At the critical temperature, ^ = - ^ = - ^ = - ^ = 0,

(5.77)

7; = ^ (5.78)

If L > 0 , r^>0, then a miscibility gap results.

5.4.3.2 Solid-liquid equilibrium

Referring to equations (5.71) and (5.72), the free energy of a regular solid solution phase can be written as

G^ = ( l - x ) G f + xGf

+ RTlx In x + (1 - x ) ln(l - x ) ] + 5x( l - x ) (5.79)


where Β is the interaction constant for sohd solution β. The partial molar free energy of components 1 and 2 is

Gf = Gf + ln(l - x ) + 5x ^ (5.80)

Gi = Gl + RT\nx^B{\-xY (5.81)

At equilibrium between the solid solution and liquid phases,

^ f | . , = G i | „ (5.82)

GiU-Gil, (5.83)

where and x, are the equilibrium compositions of solid solution and liquid phases. From equations (5.73), (5.74) and (5.80)-(5.83),

^G{•'^^-RT\rJ¡^^^^ = x¡B-xfL (5.84)

^Gi-^*-^RT\n(^ = (^-x^fB-(\-x^ΫL (5.85)

^G{^' = G[-G^

AGt' = G^-Gi

Kaufman and Bernstein define the curve Xq[_T] or 7Ό[χ] to represent the equilibrium between the phases β and /.

AG'-'l Γο, X ] = (1 - x ) A G f T o ] + xAGfl To]

+ ( L - 5 ) x ( l - x ) = 0 (5.86)

AG'-r T, x o ] = (1 - x o ) A G f T ] + Xo ΔίϊΓ'[ Τ]

+ ( L - 5 ) x o ( l - X o ) = 0 (5.87)

The XqIT] or Γο[χ] curve defines the locus of points where G^ = G'. The equilibrium relation between composition and temperature is obtained by solving the equation (5.86) or (5.87).

5.4.3.3 Equilibria involving the intermediate

compound

Consider an intermediate compound of composition x. The free energy of formation of phase φ is defined by

G' = (1 - x^)Gi + x^G* + x^(l - Χψ) (L - C) (5.88)


where k is selected as a base phase for the compound and C is a constant. If the phase boundary composition of the liquid phase coexisting with the φ phase is x,^, at the equilibrium point

G\U=G%, = Gtl^^^ (5.89)

G'2U = GtU^ = Gt\^^^^ (5.90) Substituting equations (5.73) and (5.74) into (5.89) and (5.90), and multiplying equation (5.89) by (1 — x^) and (5.90) by x^ gives

= (1 - x ^ ) G { +x^G^ + Λ Γ [ ( 1 - x ^ ) ln(l - x , ^ )

+ x^ In x¿^] + L(x^ — ΙΧφΧΐφ + xjj,) (5.91)

Equations (5.88)-(5.91) define x,^ as a function of temperature. The free energy equation between β and φ is

G^ = (1 - x ^ ) G f + x^G| + /?Γ[(1 - x ^ ) ln(l - x ^ ^ ) + x^ In x^^]

+ fi(x^-2x^x^^ + x^^^) (5.92)

5.4.3.4 Calculation of phase diagrams

Based on binary equihbrium equations, Kaufman and Bernstein built a series of computer programs to solve the equilibrium state in binary systems. The main program is called TRSE (Two Regular-Solution Equilibria). Data input includes the free energy difference parameters, the interaction constants and the temperature interval and increments. The program is used for solving equations (5.84) and (5.85). Program X O T O follows to compute X o [ Γ ] and Γο[χ] curves by using equations (5.86) and (5.87), and then LCRSE (Line Compound Regular-Solution Equihbria) to compute the boundary of the two-phase field between a regular solution phase, for example liquid or β phase, and the intermediate compound φ by means of equations (5.91) or (5.92). The output is fed into an A'-F plotter to plot the phase diagram.

5.4.3.5 Equilibrium calculation in a ternary system

The regular solution model has been extended to equilibrium calculations in the ternary system. The free energy of ternary regular solution is given by

G'lx,y, n=zG'l-^xG^-^yG^

-\-RT(z In z + x In x + y In y)

^xzE\2 +yzE\^ ^xyE\^ (5.93)


where subscripts 1, 2 and 3 represent components 1, 2 and 3, χ and y are the mole fractions of 2 and 3. ζ is the mole fraction o f l , z = l — x — y and the Ε terms are the binary interaction parameters for regular solutions. The high order interactions are ignored here.

Along similar lines as described in the binary system, the equilibria equations for the ternary regular solution system are

AG{-' + RT\J'A + {E\,xi--E{2^^^^

+ {AE^'x^y^ - AE^yj) = 0 (5.94)

Δ σ Γ ' ^ + ΛΓ1η(ίί ;^ + [ £ Ϊ , ( 1 - χ , ) ^ - ^ ί , ( 1 - χ / ]

+ {E\yi-E{yf)-lAE'y,{l-x,)--AE^yj{l--Xj)-] = 0 (5.95)

AGr'^ + /?rin(^^) + ( £ Í , x , 2 - £ { 2 ^ / ) + [ £ Í 3 ( l - 3 ; , ) ^

-EU{l-yj)'li-lAE'x,(l-y,)-AE^Xjil-yj):\ = 0 (5.96)

In these equations the subscripts j and k represent phases ; and k. The ternary analogue of the binary equihbrium equation (5.87) is

A G ^ ^ ' ' [ X o , y o . n = G%Xo,yo, n-G^lxo.yo, Γ ] (5.97)

Computer programs T E R N A R Y , M I G A P and TERCP were built to solve the equilibria, miscibility gap and compound-solution phases equilibria.

Kaufman and Nesor^ have calculated non-metallic phase diagrams using an approach essentially similar to the above description and in Refs. 5 and 3. The excess free energy of mixing for a binary liquid or a solid solution is given by

- G ^ = X ( 1 - X ) [ / Í : , 2 ( 1 - X ) + / Í : 2 I X ] (5.98)

where ^21 functions of temperature (but not of composition). The free energy of compound phase ψ having composition x^ is defined

by equation (5.99)

G' = (1 - x^)Gl^ -f x^G^ + Χφ{1- Χφ)

χίΚ,2{1-ΧφΗΚ2,Χφ-C{ Γ ) ] (5.99)

where Κ refers to the base phase selected for the compound and C( T) is a temperature dependent function, is the compound parameters. The calculated phase diagrams of M g O - F e 3 0 4 - A l 2 0 3 and the subsidiary binary systems are given in Fig. 5.7.


(a) M g , ,0,

C=Cürundum

P = P e r i c l a s e

M8n .On P = P c r i c l a s e

'=^0.429°0.571

FIG. 5.7 Calculated phase diagram of the ternary system M g O - F e j O ^ - A l j O j (a) and the subsidiary binary systems (b)-(d).^


( b )

4 6 1Λ

/ o MgÜ

: i 29 38 48 59 Tl 85 100 PK

{.lijiiid 1 1. -

,ίΟΟΙ)

2()0()

,ίΟΟΙ)

2()0()

-L + Ρ

P t í r i c l a s e

IS71 1940 ( Ρ )

• S p i n e l / S o l i d /

1 ODD Unliit I O l / 1 S 1 / S ^ Ρ

-

^'jS 10 20 30 an 50 60 70 80 90 100

c, 2M0

2315

22?ö"

C

SP

SP . ρ

0 .5 0 .5

ί ' ^ ^ 0 . 2 8 6 ° 0 . 4 2 8 ' - ί % . 1 4 3 % . 1 4 3 ^ = '^'

( d ) / ο A l ^ O j

13 21 29 38 48

600

S = S p i n e l Sol id S o l u t i o n

1140

S p i n e l M i s c i b i l i t y Gap S j + S ^

( S ) S p i n e l • Corundum (CI

1 ( 1 1 i

FIG. 5.7. continued.


(5.100) i = 1 i = 1

where m is the number of gas species, superscripts g and 0 indicate gas phase and standard state. According to Dalton's gas law

p = xP/X (5.101)

where Ρ and X represent the total pressure and total number of moles in gas phase. Substituting (5.101) into (5.100) gives

102) G'=t ^fG^^' ^RT^x, \ln(^) + In p] (5. /=i i=l L \ ^ /

Take the number of pure condensed phases as 5 , their contribution to free energy is

G'= Σ ^ÍGP^' (5.103) í = l

where c indicates pure condensed phase. The total free energy of the system is then given by

i = l

+ Α Γ X X, I n ^ + In Ρ + ¿ G?'^ (5.104)

Based on minimization of the total free energy of a system, G. Eriksson established the computer program SOLGAS for calculating equilibria in multicomponent systems.

Dividing equation (5.104) by Λ Γ gives a dimensionless quantity (G/RT),

RT = i ^ f [ ( ^ ) > > n f + l . f ] + E x ? ( ^ ) ; (5.105)

The mass balance relations of the system are

Σ + Σ «.W = ^- 0 = 1' 2 , . . . , / ) (5.106) i = l i = l

5 .4 .4 M u l t i c o m p o n e n t e q u i l i b r i u m c a l c u l a t i o n s w i t h a p a r t i c i p a t i n g g a s p h a s e

5 .4 .4 .7 Computer program SOLGAS^

Consider an n-component system composed of gases and pure condensed phases and assume the ideal gas law is obeyed. The free energy contribution of the gaseous mixture is


where a^^ represents the number of atoms of the jth element in a molecule of the /th substance, b^ is the total number of moles of the;th element, and / is the total number of elements.

Minimizing equation (5.105), subject to equation (5.106) as subsidiary conditions, we obtain the Lagrange function

- Ι / · ' Ρ ) : - " | - » ' ' ] Λ Σ . ( ^ ) ; + 4t <-^f + Σ ciljx¡-b\ 0 = 1, 2 , . . . , / )

\ i = l f=l /

where λ is Lagrangian multiplier. From ^ = ^ = 0

' | ^ ) ' + l n | + l n / > - ¿ ( i = l , 2 , . . . , m ) (5.107)

- Σ V o = o ( i = i , , . . . , s ) RT

(5.108)

Expanding equations (5.107) and (5.108) in a Taylor series about an arbitrary point {yl, yl,.. •, yi: y{, yl, •. •, y¡) and neglecting terms involving derivatives of second and higher orders yields

Σ <yf+ Σ ^tjyí-bj+ Σ < W - 3 ' f ) + Σ a^ixt-yD-o i=l i=i i=l i=l

0 = 1 , 2 , . . . , / )

í ^ ^ + , „ ^ + l n P - X A X + | - | = 0

(5.109)

RTL

(¿=1,2, . . . , m )

where Y= ¿ yf i= 1

xf is calculated from equation (5.110).

Γ Λ Χ / ( r ° \ » /ν? \ « f = y í [ Σ + γ - [jf)^ - In ( ^ y j - In

(5.110)

( i = l , 2 , . . . ,m) (5.111)


The summation of equation (5.111) over i gives

189

I m m Γ / / 7 Ο

j=l i = l 1=1 RTL

+ ln(4 + l n P (5.112)

Define C -as a correction term in cases where the initial guess of the mole numbers does not satisfy the mass balance relations

Cj= l<^fjyf-bj 0 = 1 , 2 , . . . , / ) (5.113) i = l

Substituting (5.111) and (5.113) into (5.109) gives

' YÍ-. k=i \^ / i=l i = l

+ lnP - ς

α = ι , 2 , . . . , / ) (5.114)

where rj, = r,, = £ {afjaf, )yf ( ; , fc = 1, 2, . . . , / ) . i = l

The equations (5.108), (5.112) and (5.114) constitute a system of (/ + 5H-1) hnear equations, consisting of (/ + sH-l) unknown quantities ( ; = 1, 2 , . . . , / ) , (i = 1, 2 , . . . , s) and (X/Y-l). Therefore the x* values can be directly solved and then x^ values are obtained from equation (5.111).

The conditions which must be met in the equilibria calculation are (1) non-negative x-values, (2) minimal free energy of system, and (3) mass balance relations. Iterative procedures are performed under these conditions to attain the equihbrium phase and species assemblage.

5.4.4.2 Computer programs SOLGASMI)^^ and S 0 L G A S M I X - P V ^ 2

In addition to gas and condensed species, liquid or solid solution phases are considered in the SOLGASMIX program. The total free energy in the system is

G = G^ + G'" + G^ (5.115)

the superscript m represents the mixed phases (liquid or/and solid solutions). Consider a system consisting of one gaseous phase, q liquid and solid mixtures, and 5 pure condensed phases, and assuming ideal conditions in gas and in mixtures, equation (5.115) can be rewritten in the form


G= Σ Σ ^jG^pi-^RTlnP+RTlní'^] p=l í= l L

+ Σ Σ

p = 2 i=l

+ Σ Σ ^ ρ ί ί ^ ρ ί (5.116) p=q+2 í=l

Dividing the equation by Λ Γ yields the dimensionless expression

RT

1 mp

= Σ Σ ^ρ.· » = 1 i = l

l „ ( ^ ) + , „ /

Σ|/"ΡΙ^'"(^)] + Σ Σ (5.117)

where and denote the total amount and total number of substances in pth phase respectively.

The mass balance relations are

q+s+l mp Σ l^pijXpi = bj 0 = 1 , 2 , . . . , / )

Γ = 1 1 = 1 (5.118)

where Up^j refers to the number of atoms of the ;th element in a molecule of the ith substance in the pth phase.

The total free energy of the system is minimized by the Lagrange method with the mass balance relations as subsidiary conditions. Expanding the resulting equations into a Taylor series about an arbitrary point y/pi (p= 1 ,2 , . . . , q-\-s+1; / = 1, 2 , . . . , m^) and neglecting second and high order derivatives, we obtain (/-f qf + s + 1 ) linear equations:

q+l I q + s + 1 nip

Σ Σ Vpífc+ Σ h ^ - i Σ W P Í + Σ Σ S Í ^ Í p=lk=l ρ=1 / i=l ρ = ί + 2 ί = 1

q+l nip

= Σ Σ « ρ ο · σ ρ . - ^ ρ ί ) + ί ' ^ 0 = 1 , 2 , . . . , / ) ρ = 1 ί = 1

(5.119)

Σ h Σ «ρο·>'ρί= Σ fpi ( Ρ = 1 , 2 , . . . , 9 + υ j=l 1 = 1 i=l

(5.120)


where

I, (SI (p = g + 2,g + 3 , . . . , g + s + l ; i = l )

rpik= Σ Kj^pik)ypi

(5.121)

i = l

fpi=ypi + lnP ( P = l )

fpi=ypi + ln ip = 2, 3,...,q+l)

Solution of equations (5.119)-(5.121) gives the values of the (l+q + s+l) unknowns.

λJ (i=i,2,...,l)

^ - 1 ( p = l , 2 , . . . , 4 + l )

Xpi (P = ^ + 2 , ^ - h 3 , . . . , q f - h s + l ; i = l )

TABLE 5.6 Species and their free energy entered in SOLGASMIX-PV computer program

Species G«(J/mole) Species G®(J/mole)

Gaseous species: Ar C H O CO C O 2 C F , C F , C O F C O F 2 C H F 3 C H F O F F2 H F H F O SiF SÍF2 SÍF3 SÍF4 SÍOF2 SÍHF3 Η

H O O H O 2

- 2 0 8 8 3 - 4 0 . 1 4 6 Τ H2O2 - 1 1 6 0 2 0 - 8 4 . 8 9 0 Γ O - 3 9 5 8 5 0 - 0 . 2 6 8 Τ - 1 8 7 4 0 0 + 26.288 Γ Ο3 - 9 3 1 8 8 5 + 150.875 Τ SiO - 1 7 7 1 1 0 - 3 2 . 8 3 8 Τ S iOj - 6 4 2 2 5 0 + 57.699 Τ Condensed mixtures: - 7 0 2 3 8 0 + 1 2 7 . 3 8 0 Τ SiO, ,ΟΗ{\) - 3 8 4 5 0 0 + 41.208 Τ SiOiil)

8 3 1 7 2 - 6 4 . 0 6 0 Γ Ο SiOj 5 0 H ( s )

- 2 7 6 4 9 0 - 2 . 2 1 4 Τ S i 0 2 ( s ) - 1 3 6 1 0 0 + 53.711 Τ Pure condensed phases:

- 4 8 6 0 0 - 7 9 . 1 6 2 Τ C - 6 9 3 9 3 0 - 1 6 . 9 9 7 Τ Si

- 1 1 2 4 1 0 0 - 6 1 . 7 9 8 Τ SiC - 1 6 3 3 0 0 0 + 1 5 8 . 0 0 0 Γ SiO^O)

- 9 9 2 7 9 0 + 77.938 Τ SiOjís) - 1207900+ 135.180 Τ S iO, jOHO)

2 2 5 2 1 - 5 9 . 2 2 9 Τ SiO, ,OU{s) O

- 2 5 0 4 8 0 + 57.489 Τ

3 7 7 2 3 - 1 4 . 0 0 5 Τ 1 6 5 7 4 - 5 0 . 3 4 2 Τ

- 1 4 0 8 4 0 + 1 1 2 . 1 7 Τ 254090 + 66.346 Τ

Ο 144920 + 67.106 Τ

- 1 1 1 3 2 0 - 7 5 . 6 7 3 Τ - 3 5 4 7 2 0 + 29.000 Γ

- 1 0 6 4 3 4 6 + 273.830 Γ - 9 1 8 0 8 3 + 184.012 Τ

- 1 0 6 4 3 4 6 + 273.830 Γ - 9 2 6 0 1 5 + 187.987 Τ

Ο Ο

693950-171 .94 Τ - 9 1 8 0 8 3 + 184.012 Τ - 9 2 6 0 1 5 + 187.987 Τ

- 1 0 6 4 3 4 6 + 273.830 Γ - 1 0 6 4 3 4 6 + 273.830 Γ


Gas

eous

spec

ies:

Ar

0.20

000

x10

-1

CH

O0.

8168

8x

10-19

CO

0.75

510

X10

-7

CO

20.

4999

2x

10-3

CF2

0.12

564

x10

-20

CF 4

0.90

807

x10

-21

CO

F0.

3515

3x

10-1

4

CO

F20.

1291

7x

10-11

CH

F30.

1023

8x

10-26

CH

FO0.

9232

6x

10-1

9

F0.

3121

6x

10

-5

F2

0.15

339

X10

-12

HF

0.10

935

x10

-4

HFO

0.94

208

x10

-12

SiF

0.42

530

x10

-18

SiF2

0.21

603

x10

-11

SiF3

0.13

635

x10

-6

SiF 4

0.49

300

x10

-3

SiO

F 20.

6755

9x

10

-5

SiH

F30.

8827

1x

10

-15

H0.

4410

6X

10-1

0

H2

0.23

073

X10

-13

P20

0.54

332

X10

-9

TA

BL

E5.

7Eq

uilib

rium

data

outp

ut/r

omSO

LG

ASM

IX-P

Vpr

ogra

m

HO

0.63

452

X10

-7

H0

20.

1145

3X

10-9

H20

20.

4402

8X

10

-15

°0.

1124

8X

10-1

0

O2

0.10

040

x10

+1

03

0.16

767

X10

-7

SiO

0.50

448

x10

-10

Si0

20.

2525

5X

10-8

Con

dens

edm

ixtu

re:

Mol

efr

actio

nSi

0l.s

OH

(I)

o.()()(

)()()x

10°

Si0

2(1)

O.()(

)()()(

)x10

°

Si0l

.sO

H(s

)0.

3201

5x

10-9

Si0

2(s)

0.55

001

x10

-2

Pure

cond

ense

dph

ase:

CO.

()()()

()()x

10°

SiO.

()()()

()()x

10°

SiC

O.()(

)()()(

)x10

°S

i02

(1)O.

()()()

()()x

10°

Si0

2(s)

o.()()(

)()()x

10°

Si0l

.sO

H(I

)o.(

)()()()(

)x10

°Si

0l.S

OH

(s)

o.()()(

)()()x

10°

O.()(

)()()(

)x10

°O.

()()()

()()x

10°

O.()(

)()()(

)x10

°O.

()()()

()()x

10°

O.()(

)()()(

)x10

°O.

()()()

()()x

10°

o.()()(

)()()x

10°

o.()()(

)()()x

10°

o.()()(

)()()x

10°

0.62

724

X10

-2

0.11

322

X10

-4

0.43

525

X10

-10

0.11

119

x1

0-s

0.99

248

xlO

s0.

1657

5X

10-2

0.49

869

x10

-s0.

2496

5x

10-3

0.58

262

X10

-7

0.1(

)()()(

)()x

101

Part

ialp

ress

ure

(Pa)

Mol

eSp

ecie

s

0.19

771

X10

4

0.80

751

X10

-14

0.74

643

x10

-20.

4941

9X

102

0.12

429

X1

0-1

5

0.89

765

X10

-16

0.34

749

X10

-9

0.12

769

X10

-6

0.10

121

x10

-21

0.91

267

x10

-14

0.30

858

x10

°0.

1516

2X

10-7

0.10

809

X10

1

0.93

199

X10

-7

0.42

043

X10

-13

0.21

355

X10

-6

0.13

478

X10

-1

0.48

734

X10

2

0.66

783

x10

°0.

8725

8X

10-1

0

0.43

599

x10

-50.

2280

8X

10-8

0.53

906

X10

-4

Parti

alpr

essu

re(P

a)M

ole

Spec

ies


fpi—ypi (p = 2, 3 , . . . , q + i)

The terms represent the excess free energy of mixing. SOLGASMIX-PV program is formed by inserting the relation P V = R T

into the SOLGASMIX program mentioned above. The newly programming calculation offers the possibility of calculating equilibria for either constant pressure or constant volume conditions.

An example calculation concerns the species distribution in the system SÍO2-CH4-H2O-O2-CF4 . The initial mole compositions are: Si 0.0006, C 0.0005, 02.02, Η 2.75x10"^ and F 0.0005. Table 5.6 lists the free energy of the species concerned here. The reference states are the stable form of Ar, F2, H2, O2 , C and Si. Equilibrium data at 1500°C and 1 atm calculated by SOLGASMIX-PV are shown in Table 5.7.

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Activities are inserted into the equations to replace the mole fraction terms for the non-ideal mixtures.

''pi=ypi ( y ^ ) (P = 2. 3 , . . . , q-h 1; i = 1, 2 , . . . , m^)

where y^, is an activity coefficient which can be obtained by mathematical approximation of experimental activity data or by structurally modelled calculation. Then the/^,- in equation (5.119) will be changed into


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241 (1979). 52. T. M. Besmann, Oak Ridge National Laboratory, Oak Ridge Report, ORNL-TM-5775

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Nakamura, Acta Metall, 25, 207 (1977). 63. L Ansara, P. Desre and E. Bonnier, J. Chim. Phys. 66, 297 (1969). 64. L Ansara, CALPHAD VII, Project Meeting, April 10-13 (1978). 65. G. Kaestle and K. Koch, CALPHAD VII, project Meeting, April 10-13 (1978). 66. J. F. Counsel, E. B, Lees and P. J. Spencer, Metal Sei. 5, 210 (1971). 67. Η. Gaye and C. Η. P. Lupis, Metall. Trans. 6A, 1049; 1057 (1975). 68. Kuo Chu Kun and Yen Tung Sheng, Amer. Ceram. Soc. Bull. 59, 365 (1980).

high temperature phase equilibria and phase diagrams || the calculation of phase diagrams

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