Calculation of ternary mixing properties of melts and minerals from the bounding binaries

Calphad, Vol. 25, No. 1, pp. 97-108,200l 0 2001 Elsevier Science Ltd All rights reserved 0364-5916/01/$ - see front matter PII:SO364-5916(01)00033-5

CALCULATION

OF TERNARY MIXING PROPERTIES OF MELTS AND MINERALS FROM THE BOUNDING BINARIES

Sung Keun Lee’” and Soo Jin Kim’ 1.Department of Geological Sciences

Seoul National University Seoul, 15l-742, Korea 2.Department of Geological and Environmental Sciences Stanford University Stanford CA 94305 USA

ABSTRACT

Among the several methods that have been used to predict thermodynamic properties of ternary alloys and oxides from three binary data, we used the models proposed by Kohler(l), Toop(Z), and Muggianu et aL(3) in order to evaluate the possibility of the application of these methods to geologically important systems, such as zeolites, clay minerals and silicate glasses. These models can represent the ternary excess Gibbs energy of the NaCl-KCl-Hz0 and Ca-Mg-Fe’+ garnet (Ca~A1&0~2 Mg3AlzSis012- Fe3AlzSi3012)systems relatively well without a ternary correction term. The deviation from the reference data increases in the central region of the composition triangle. Toop’s model with constant mole fractions of NaCl and Mg best simulated the ternary systems among the models. The path independent Muggianu et al. model was applied to diopside (CaMgSi206) -jadeite (NaAlSizOs) acmite (NaFd’Si20a) ternary as an example. Although there exists intrinsic uncertainty in calculation without the ternary interaction term, these models, especially, Muggianu et al. and Kohler can be good approximation methods for prediction of the ternary excess properties of the natural mineral solid solutions, devoid of ternary experimental thermodynamic data. 1. Introduction

A significant number of geologically important silicate mineral assemblages from the lower mantle to earth surfaces -can be described as ternary oxide systems. The ternary thermodynamic properties have an important role in the calculation of phase relations. Therefore, it is of crucial importance to predict ternary phase equilibria and ternary interaction parameters that can usually be obtained from solution calorimetry with thermodynamic modeling. Microscopic approach using multinuclear NMR has been also effective in obtaining thermodynamic mixing properties of ahnninosilicates glasses whose interaction parameter calculated from the oxygen site populations shows remarkable similarity with experimental data.(4,5) However, the experimental preparation is difficult for ternary solid solutions and the appropriate number of experimental setups to estimate thermodynamic properties has often been hampered by the experimental difficulties of minerals and melts and the problems of kinetics. The latter is a particular problem for mineral assemblages with low temperature stability field, such as clay minerals and zeolites. In such cases, it is important to have sophisticated models where three binary data that constitute a specific ternary system are used to approximate the ternary thermodynamic properties of the system.(dQ Received on 27 July 2000 97

98

S. K. LEE AND S. J. KIM

There have been diverse approaches for calculating the properties of ternary regular solutions. The excess Gibbs energy of a ternary regular solution, AG” , may be expressed as a linear combination of three binary interaction parameters: AG”IRT=x,x,L,_, +x,x,L,_,+x,x,L,_, (1) where x,, x2 and xj are the mole fractions of components

1, 2, and 3, respectively,

and Li_j is the temperature

independent interaction parameter of binary i-i. The application of the above equation has been extended to nonregular ternary solutions whose interaction parameters vary with temperature, pressure and composition. Different composition paths have been used to calculate the ternary excess Gibbs energy.( 1-3, 6, 8-11) These methods have been applied to calculate the thermodynamic excess properties of the significant number of ternary systems including silicate melts( 12) and ternary oxide solutions (for example, (10, 13-15) and (16)) and mineral solid soultions.( 16, 17) Among these methods, the excess Gibbs energy based on the Muggianu et al. model is relatively consistent with the experimental data (10, 18), and this method has been commonly applied to the calculation algorithm of phase diagrams(19) Several geologically important systems such as clay minerals, zeolites, and silicate glasses melts are devoid of ternary thermodynamic data and have bounding binaries where the deviation from ideal&y is often not significant.(lZ) Therefore, several models given above can be directly applied to these systems in order to obtain reliable thermodynamic data.(20, 21) For example, the Kohler model was used for aluminosilicate liquids (22) and borosilicate melts.(12) The Muggianu et al mode1 was applied for Ca-Mg-Fe” garnet (CasAlsSisOt2 - MgsAlzSisOtz FesA1zSis0r2)(7) without comparison with the results obtained using other methods. The authors, however, discussed the superiority of Muggianu et al model.(7) On the other hand, it has been suggested that the Toop and the Kohler model can be preferred for the calculation of ternary data near a binary join since the Muggianu model uses binary interaction parameters further from the binary join in the case of concentrated solutions near binary joins.(23) In spite of these diverse efforts at applications, these methods have not been systematically evaluated with rock forming minerals and melts so far. Here, we evaluate seveta\models in two thetmodynamically fairly well-established systems: NaCl-KCI-Hz0 gamet.(25, 26) using previous assessments.(24, 25) We also apply the Muggianu ternary liquid(24) and Ca-Mg-Fe et al. mode1 to infer the phase relations of the diopside(CaMgSizOs)-jadeite(NaAlSizO6)-acmite(NaFe ‘SizO6) ternary system that is complicated by incomplete calorimetric data and the order-disorder transformation along the diopsidejadeite join. We briefly discuss the application of NMR spectroscopy and quantum simulations for estimation of binary or pseudo-binary interaction parameter in silicate glasses. 2. Mathematical Model For Ternary Excess Properties The Gibbs energy of a ternary system is strongly dependent on the behaviors of its three bounding binary systems. Kohler( I), Toop(%), and Muggianu et a1.(3) proposed models for estimating the excess Gibbs energy of any composition point at an isothermal section of a ternary system by using only thermodynamic data of three binary systems. Here, a brief review of each method is presented.(see (6) and (18) for detail discussion) The different composition paths for the calculations of ternary excess properties in each model are illustrated in Fig.l. In the Toop model, the excess Gibbs energy for a composition point 0 is determined by the relationship between the interaction parameters of binary composition points A, B, and C. The excess Gibbs energy can be calculated along composition paths C-B, and 2-A:

AC”

=

&W:2 2

I,,

+5

LAG,:*I,, + (I- xzI*LAG:,lx,/x, 2

where [AGE3 I,,,x, is the excess Gibbs energy of the binary l-3 system at the same relative concentration components

and, [AG,zjlX,

is

the excess Gibbs energy of the i-j system at a composition path of constant x2.

of two

CALCULATION

OF TERNARY MIXING

PROPERTIES

99

The calculated excess Gibbs energy of composition point 0 varies with the path taken. In the Kohler model, the excess Gibbs energies of binary constituents with the same ratios of mole fractions of components such as A’, B’, and C’ are combined to calculate the ternary excess Gibbs energy (Fig. 1b): AG”

=(1-~d2W;“Jx,/x2 +U--X,)~[AC&x, +(1-~,~~W3x,,x,

(3)

where [AG,yjI, ,I, is the excess Gibbs energy of binary i3 B’=B

B”

Fig. 1 Composition paths of mathematical models. Li.1is

interaction parameter of component i and componentj. See text for subscript notations.

j system at

composition path of constant xi /xi. The

excess Gibbs energy of any composition point in ternary system can be calculated using the Muggianu et al. model by the combination of three binary interaction data at composition points, A”, B” and C”:

(4)

where [AGiYj],_, is the excess Gibbs energy of binary system i-j at a composition path of ‘xi - xj = constant’. 3. Sources of Thermodynamic

Data

3.1. The N&I-KCI-Hz0 ternary liquid system. The experimental solubility data of the NaCl-KCl-Hz0 system have been obtained from differential thermal analysis (DTA)(27, 28). The Gibbs energies of mixing for NaCl-KC1 binary solids and liquids as well as for solid saturated NaCl-KCl-Hz0 ternary liquid were modeled using an asymmetric Margules treatment: the coefficients of sub-regular solution in the NaCl-KC1 binary and quasi-regular solution of MCI-H20 in this system were evaluated (M stands for Na or K.)(24) The equation for ternary excess Gibbs energy of this system fitted by Sterner et al. (24)is given below:

AG” = ~w,xn,oLIN.c,-n,o +~~c,~n,oLix,-n,o +x NaClxKC/

(L:,NaC,-KC, +L',,NaC,-KC,xNaC,)

(5)

L!N~CI-KCI-H,O +x N.CIXKCIXH20 where pressure-temperature dependent binary and ternary interaction parameters are as given in Table 1. All the interaction parameters were fitted with following expression: L’=aT+bP+c

(6)

100

S. K. LEE AND S. J. KIM

We use Eq. (5) as a reference for modeling the solution properties at 1200 K, 1 kbar in order to examine contributions from binary terms only. The ternary interaction term of Eq. (5) was eliminated and the combinations of three binary interaction parameters using the methods given above were used to calculate the ternary data. The ternary excess Gibbs energies were also compared with total G” as calculated from the complete Eq. (5). Table. 1 Summary of coefficients of the interaction parameters in NaCI-KCl-Hz0 ternary liquid and Ca-Mg-Fe2’ Garnet system.

System

Solution model

a

b

C

NaCl-KCLH20 ternary (24) NaCl-Hz0 KCl-Hz0 NaCI-KC1

Quasi-regular Quasi-regular Quasi-subregular

LL 4

NaCI-KCl-Hz0

5.0544 3.2839 -1.8908

-1.0742 -1.2842 -0.2580

-4988.0 -6495.4 -2050.0

1.0039

0

-271.9

-5.1158

1.2656

3263.0

Ca-Mg-Fe2+ Garnet (25) Ca -Mg

Quasi-subregular

&,_~a L,.c,

-20.82 -2.49

0.068 0.140

65128 14306

Mg -Fe

Quasi-subregular

LM~_F~

-12.40 22.09

0.050 -0.034

22265 -24166

-14.51 15.51

0.135 0.040

17526 -18113

LFe-hfg

Fe -Ca

Quasi-subregular

L,+_c. &a-Fe

Ca-Mg-Fe

7110 J/mol

3.2. The Ca-Mg-Fe’+ garnet (Ca3A1&012 - M&AI#isOl2 - FepUd33012) ternary system. Due to geologic importance as a geothermometer, thermodynamic mixing properties of garnet has been studied for several decades((25, 26, 29) and references therein) In this study we used the Margules parameters for Ca-Mg-Fe2+ garnet ternary system obtained by Mukhopadhyay et al. (25). The ternary excess Gibbs energy is given below;

where each pressure and temperature dependent binary Margules parameters can be expressed as Eq. (6) and coefficient is given in Table 1. We use the same procedures with NaCl-KCI-Ha0 ternary liquid in calculating Ae of Ca-Mg-Fe2+ garnet ternary system at 900K and Skbar.

CALCULATION OF TERNARY MIXING

PROPERTIES

101

4. Calculation of Thermodynamic Properties 4.1. The NaCl-KCI-Hz0 system. Three different paths for calculation of ternary properties of NaClXCLH20 can be chosen for the Toop model as shown in Table 2. The ternary G” of the system calculated using Toop 1 reproduce that predicted from Eq. 5 with a maximum deviation of not more than 50 J/mol at the central region of the composition triangle as illustrated in Fig. 2a. Compared with the other models, the Toop model corresponding to the combination of paths where the mole fraction of NaCl is held constant exhibits the smallest deviation from the experimental ternary data. The Toop model with composition path of constant Hz0 (Toop 2) can reproduce the ternary data with R2 values about 0.9865 (Table 3). In this case, because the system is composed of two binary regular solutions: NaCl-Hz0 and KCl-H20, and one subregular solution, NaCl-KCl, the calculated result is the same as that calculated using the Kohler model (Fig. 2b). Compared with other methods, Toop3 is subject to the largest error (Fig. 2~). The results of the Muggianu et al. model are compared with the data calculated from Eq. (5). The results are inferior, compared with the Toopl results, but are better than those obtained with Toop 3 (Fig. 2d). Table. 2 Three different composition paths for the Toop model NaCl-KCl-Hz0 ternary liquid and Ca-Mg-Fe2+ Garnet system. (x,=constant, x/x~=constant)

i

j

k

NaCl-KCl-HrO system Toopl Toop2 Toop3

NaCl

Hz0

KC1

KC1 NaCl

Ca Mg Fe

Mg Fe Ca

H20

KC1 NaCl Hz0

Ca-Mg-Fe2+ Garnet Toop(Ca) Toop(Mg) Toop(Fe)

Fe Ca Mg

All methods yield results in good agreement with experimentally derived data near the three binary regions. The difference between the Gibbs energies calculated using these methods and the reference data increases as composition approaches the central region where the relative concentration of three components are similar as manifested in Fig. 3. Differences between the calculated excess Gibbs energies and the reference data at constant mole fraction of NaCl are illustrated in Figs. 3 and 4. As can be seen, at xN&/ = 0.05, the excess Gibbs energies calculated using the Toopl model do not deviate by more than 2 % (20 J/mol) from the reference data. The error increases as we move through the Muggianu, Toop 2 and Kohler, and Toop 3 models. The differences between calculated and reference values also increase with increasing NaCl concentration (Fig. 3 b, c). The percentage errors of the results calculated using the methods given above are concentrated in the HsO-NaCl binary region where the interaction parameter is small and the mixing behavior is similar to ideal solution. Therefore a slight deviation from experimental data makes a relatively considerable error (Fig. 4). Figure 5 illustrates the differences between calculated Gibbs energy of mixing from Muggianu et al. model and the reference(Eq.5). The calculated Gibbs energy of mixing shows good agreement ‘with reference in part because of relatively small deviation from ideal@.

S. K. LEE AND S. J. KIM

102 a

Hz0

C

NaCl

b

H20

NaCl

d

H20

NaCl

KC1

H20

NaCl

KC1

Fig. 2 Differences between excess Gibbs energies calculated using the empirical formula (Eq.5) and the predicting methods. a) Toop 1, b) the Kohler model and Toop 2, c) Toop 3, d) The Muggianu et al. model. Thick and thin solid lines represent the excess Gibbs energy from reference (Eq.5) and that calculated the predicting method, respectively. Contours are drawn from 100 to 1000 level with 100 (JImal) increments. Table. 3 R2 of calculation

Mug

Ko

Toop3 or Toop(Ca)

Toop2 or Toop(Mg)

Toopl or Toop(Fe)

0.9865

0.9919

0.9994

0.9693

NaCl-KCl-Hz0 system RZ

0.9864

0.9865

0.9794

Ca-Mg-Fe2+Garnet R2

0.9959

0.9944

0.9955

Note) Mug, and Ko refer the methods of Muggianu et al., and KohJer, respectively.

CALCULATION OF TERNARY MIXING

w

0

0.1

( -20

40

0.3

0.4

:

ig-60 I 8 k

0.2

XIilO

P fc

103

PROPERTIES

,.:

.~ I.

-80

:-

‘*. . ._..__. I-’

:

.:

C

-100

d

Fig. 3 Comparison of deviations of excess Gibbs energy calculated using the predicting methods from the reference calculated using the empirical formula at various cross section with constant mole fraction of NaCl. a) XNacliS 0.05, b) xN&t iS 0.3, C) ,&ct iS 0.5, d) Contour of difference between excess Gibbs energy calculated using the empirical formula and the method of Toop 1 (mole fraction of NaCl is constant). Unit of the number in figure is J/mol. Gexp represents the reference value calculated by Eq. 5). Gmu, GKo, GTol, GTo2 and GTo3 represent the deviation of excess Gibbs energy calculated using the predicting methods of Muggianu et al., Kohler, Toopl, Toop2, and Toop3 respectively.

H20

KC1

NaCl

Hz0

Hz0

KCI NaCl Fig. 4 Comparison of error of excess Gibbs energies in NaCLKCLH20 system calculated using the predicting methods from the reference calculated using the empirical formula (Eq. 5). Contours are errors of excess Gibbs energy the method of Toop 1 (mole fraction of NaCl is constant). Unit of the number in figure is %.

KC1 NaCl Fig. 5 Comparison of Gibbs energy of mixing NaClKCl-Hz0 system calculated using the empirical formula and the predicting method of Muggianu et al. Thick and thin solid line represent excess Gibbs energy calculated by the empirical formula and predicting method, respectively. Unit of the number in figure is J/mol.

104

S. K. LEE AND S. J. KIM

4.2. The Ca-Mg-Fe*+garnet ternary system. The composition paths for the Toop model are given in Table 2. The Muggianu et al. model and the Kohler model reproduce the ternary excess Gibbs efiergy of the system with R2 of 0.9959 and 0.9944, respectively (Fig 6.a). Toop (Mg) gives best approximation for the ternary properties (Fig. 6b). On the other hand, the results from Toop (Ca) are significantly deviated from the reference with Rz of 0.9693. The comparison between the calculated excess Gibbs energy using the Muggianu et al. model and the combination of binary data without using any model is given in Fig. 6.C, which illustrates the relative magnitude of ternary interaction parameter and manifests the enhancement of reproducibility of ternary data using the models given above. Ca

Ca

a

e

Fe

Ca b

Fe

Mg

Fig. 6 Difference between excess Gibbs energies calculated using the empirical formula (thick line) and the predicting methods in Ca-Mg-Fe’+ garnet. Unit of the number in figure is J/mol a) The Muggianu et al. model vs the Kohler model. Thin line and dashed line refer calculated excess Gibbs energy using the Muggianu et al. model and the Kohler model, respectively. b) The Toop model. Thin and the thinner line denotes Toop(Ca) and Toop(Fe) respectively. Dashed line refers to Toop(Mg). c) The Muggianu et al. model vs the combination of three binary data. Thin line denotes the calculated results using the Muggianu et al. model and the thinnest line is the combination of binary interaction data in Eq.7 without application of models (see text).

5. Discussion 5.1 The diopside (CaMgSizO6) -jadeite (NatiS&&) - acmite (NaFe%izO6) system. Here, as an example, the Muggianu et al. model is applied for estimation of the ternary mixing properties of the system diopside-jadeiteacmite. The purpose of this example is not to obtain the thermodynamic properties of this ternary system but to illustrate the possibility of application of the models. The mixing properties of the binary diopside-jadeite system show ideal mixing behavior above 1150°C (30). On the other hand, the positive enthalpy of mixing has been proposed for the same system.(31-33) The excess Gibbs energy of this binary haa been fitted with a second order Margules equation that was used in our calculation. (33): AG”” =x,x,,,[B+C(x,,,,

-x,)+D(x,,,

-x,)‘]

(8)

CALCULATION OF TERNARY MIXING

105

PROPERTIES

where xjd, and xdiopare the mole fractions of diopside and jadeite, respectively, and B = 19800 + 4.4T, C = 9600 9.lT, D = -8200 + 16.4T From crystal chemical considerations, diopside and acmite are believed to mix ideally (30). Newton and Smith (34) and Popp and Gilbert (35) also suggested ideal mixing between the jadeite and acmite components, which was recently confirmed by Liu and Bohlen (36). Therefore, we assume that there is no contribution from these two binary systems to the total Gibbs energy of the ternary system. Fig. 7 illustrates three pseudobinaries (1, 2 and 3) selected to demonstrate the compositional path dependency of activity-composition relations. Because we assume that there is no excess mixing property between jadeite and acmite or between acmite and diopside, the excess Gibbs energies at the same mole fractions of diopside increase from pseudo binary 1 to 3. Activities along each pseudobinary with the ratio of mole fractions of acmite and jadeite being held constant can be calculated using the equation given below:

Acmite

Jadeite

Diopside

Fig. 7 Pseudo-binaries of ternary system diopsidejadeite-acmite. l), 2) and 3) represent the composition paths -%cmirdxid = 2, &cm&/~ =l and -&mite&d = 0.5, respectively. Unit of the number in figure is J/mol. Dashed lines are calculated excess Gibbs energy at 1OOOKusing the Muggianu et al. model.

c3G” (9)

The analytically derived activity-composition relationship along pseudobinary 1 (Xacm&j~ei~ =2) is given by the following equation:

(10)

where ydiopis the activity coeffkient of dioposide. B, C, and D are given in Eq. (8). The calculated activity coefficient for each psuedobinary as a function of mole fraction of diopside is shown in Fig. 8. 1.4

b

--WOK

1.2

-1IWK

1.0

1.0

--14MK 8

p.0 $ 0.4

0.8 0.6 0.4

0.2 2 0.0

02

0.6

0.4

0.8

1.0

0.6 0.0 rs!d 0.0

0.2

Xdop

0.4

0.6

0.8

1.0

0.0

Xdop

0.2

0.4

0.6

0.8

*m

Fig. 8 Activity-composition relations in the system diopside - jadeite- acmite using the method of Muggianu et al. Pseudo-binary 2 ~1) C) Pseudo-binary 3 a) Pseudo-binary 1 (&c&Jxjd

=

2)

(X~cm&jd

(&cmi&/xjd

4.5)

1.0

106

S. K. LEE AND S. J. KIM The deviation from ideal solution (In ydiop=0) behavior also increases with decreasing temperature for all three

pseudobinaries. The magnitude of the deviation increases from pseudobinary 1 to pseudobinary 3 where the degree of deviation is most significant. The above information can be used in order to calculate ternary phase relations. Acmite

Diopside

Jadeite

Figure 9 shows the phase relations in ternary system at 1200 K determined from the Gibbs energy minimization method using Thermo-Calc. (19) The two miscibility gaps in binary diopside-jadeite join due to second order term in Eq. 8 propagate into the ternary system. The miscibility gap in rather higher temperature is due to the assumption of molecular mixing. On the other hand, the phase relations in this system are further complicated by order-disorder transformation at the intermediate compositions (omphacite). Eq. (8) based on the first order transformation does not make explicit provision for ordering and therefore, the phase boundary ordered P2/n and disordered C2/c for omphacite cannot be determined using this equation.

Fig. 9 Miscibility gap in diopside-jadeite-acmite system

On the other hand, the thermodynamics of cation ordering in this system has been formulated using the BraggWilliams model (37), Landau theory (38) and symmetric formalism.(39). The formulations incorporating these concepts provide improved prospects to the solution modeling of this binary system.( 16) 5.2. Ternary interaction parameters. Strong ternary interaction, such as ordering or phase separation can cause instability of the models presented here. Helffrich and Wood(40) reported that ternary interaction term could not be eliminated in modeling the equation for a ternary subregular solution. Therefore, if the triple interactions are dominant, the ternary interaction parameter that represents interaction involving triplets of different molecular species can not be ignored. On the other hand, as shown in the case of the NaCl-KCI-Hz0 and Ca-Mg-Fe garnet ternary, these models reconstruct the ternary excess properties. The ternary interaction parameter for the system with little deviation from ideality may be expressed as binary interaction terms that simulate the forces between two components. Although the ternary interaction parameter is essential to reproduce the exact thermodynamic value of the system, these methods, including the Kohler and Muggianu et al. model provide a reasonable approximate solution for the thermodynamic properties, The ternary excess Gibbs energy of a specific system that does not contain a ternary complex can be approximated and obtained by the methods given above. On the other hand, several spectroscopic methods can be used for the calculation of interaction parameter. In particular, solid state NMR can provide quantitative site populations of mixing units in ternary aluminosilicate glasses, such as Si-0-Si, Si-O-Al, and Al-O-Al from which the thermodynamic mixing properties were calculated based on quasichemical approximation. (4, 5, 41) Similar method was applied to binary borosilicate glasses, whose interaction parameter (8.9 f 0.5 kJ/mol) shows close similarity with those from calorimetry (12.9 f 1 kJ/mol) and ab initio molecular orbital calculations (11.4 f 3 kJ/mol)(Lee and Stebbins, in preparation). These methods can be extended to ternary systems. 5.3. Evaluation of model. The models presented above are parameter dependent, i.e. the sensitivity and reliability of the models are related to the mixing properties of each binary system expressed as the binary interaction parameters as well as the relative Gibbs energy differences among each binary. The NaCl-KC1 system has a dominant effect on the ternary excess properties because this subsystem is very non-ideal, expressed as a subregular solution. The reason for the relatively high accuracy of the Toop 1 model stems from the fact that the interaction parameter of the NaCIKC1 system decreases with KC1 concentration, which in turn, contributes to the negativity of the system. Because the calculated excess Gibbs energy using the empirical formula is smaller than the one calculated using the methods given above, the model with the composition path with smallest excess Gibbs energy shows the smallest deviation from the experimentally derived data. Although the ternary excess Gibbs energy of the NaCl-KCl-HrO system can be obtained with the smallest error by the method of Toop 1, care must be taken for the application of this model because the result is path dependent. There is no significant difference between the results calculated by the models of Muggianu et al. and Kohler and both models can represent the ternary excess properties for NaCl - KC1 - Hz0 and CaMg-Fe2+ garnet systems relatively well. On the other hand, it has been suggested that the Toop and the Kohler model

CALCULATION

OF TERNARY MIXING

PROPERTIES

can be superior near each binary since the Muggianu model uses binary interaction parameter further from the binary join in case of the concentrated solutions near binary join (23), which is not manifested in the results of our calculation, showing similar results near binary. The ternary phase stability of each phase may be estimated by comparing the Gibbs energies calculated using the methods illustrated above in spite of difficulties in obtaining appropriate three binary data and the fact that the common rock-forming minerals are also multi-component. There exists intrinsic uncertainty in calculation without the ternary interaction term. These models, however, especially of Muggianu et al. and Kohler, were successful in obtaining the ternary mixing properties of the minerals and can be good approximations for the other system, such as clay minerals, zeolites and glasses where the binary interaction parameters are not much deviated from ideality and the experimental data are not sufficient, Acknowledgements

Critical reviews by J. Stebbins and two anonymous reviewers greatly improved the manuscript. We are grateful to M. Cho, T. Fridriksson, J. Ganguly and D.N. Lee for constructive comments on an earlier version of the manuscript. We also thank to D.N Lee and J.H. Shim for helping in calculation using Thermo-Calc. This research was funded by Stanford Graduate Fellowship and BSRJ project provided by the Ministry of Education, Korea. References

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