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A thermodynamic model for multicomponent melts, with application to the system CAO-MGO-AL₂O₃-SIO₂ Berman, Robert G. 1983

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A THERMODYNAMIC MODEL FOR MULTICOMPONENT MELTS, WITH APPLICATION TO THE SYSTEM CAO-MGO-AL203-SI02 by ROBERT G. BERMAN B.A., Amherst College, 1975 M.Sc, University Of B r i t i s h Columbia, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department Of Geological Sciences We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF May © Robert- G. BRITISH COLUMBIA 1983 Bermaln, 1983 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree that p ermission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of (SeoU^vtaA^c^evNOLT The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Jone. 1,(183 i i Abstract A thermodynamic model i s proposed for c a l c u l a t i o n of liquidus r e l a t i o n s in multicomponent systems of geologic i n t e r e s t . Mineral-melt reactions are written in terms of l i q u i d oxide components, and balanced on the stoichiometry of liquidus phases. In order to account for non-ideality in the l i q u i d , a 'Margules solution' i s derived in a generalized form which can be extended to systems of any number of components and to polynomials of any degree. C a l i b r a t i o n of the model i s achieved using the method of l i n e a r programming. Thermodynamic properties of liquidus minerals and the melt are determined which are consistent with available calorimetric measurements and phase e q u i l i b r i a data concerning liquidus r e l a t i o n s , binary and ternary l i q u i d i m m i s c i b i l i t y gaps, and s o l i d - s o l i d P-T reactions. Application of t h i s model to the ternary CaO-Al 20 3-Si0 2 system shows that a fourth degree polynomial equation for the excess free energy of the l i q u i d solution i s necessary to adequately reproduce phase r e l a t i o n s . Successful reproduction of the l i q u i d i of 24 minerals in the quaternary system CaO-MgO-Al 20 3-Si0 2 requires no additional terms in the expansion for the excess free energy. For 69 invariant or piercing points on the joins used in c a l i b r a t i o n of the quaternary system, the average differences between calculated and experimentally determined invariant points are 0.24±14.5° and 0.52±0.55 oxide weight percent (owp). Calculated liquidus r e l a t i o n s on the .5-35% A1 20 3 and 5-10% MgO planes compare favorably with i i i experimental data, and piercing points on 17 other quaternary joins not used in the c a l i b r a t i o n have average differences of -1.6±18.0° and 0.43±0.64 owp. S i g n i f i c a n t inconsistencies have been i d e n t i f i e d in several experimental studies and are most readily attributed to inhomogenous s t a r t i n g materials. A l l liquidus diagrams are recalculated by computer programs that trace univariant curves and isothermal sections while checking each point on a curve for metastability. Isobaric quaternary univariant curves have been calculated and are presented in stereographic projections. The intersections of these curves define the T, X positions of a l l quaternary invariant points at one atmosphere pressure. The model permits c a l c u l a t i o n of a variety of melt properties which can be used to supplement the limited data avail a b l e from other sources. Calculated l i q u i d a c t i v i t i e s , heats of fusion, and heats of mixing are compared with available experimental data. Table of Contents A b s t r a c t i i L i s t of Tables .v L i s t of F i g u r e s v i Acknowledgements v i i i I. INTRODUCTION 1 I I . A THERMODYNAMIC MODEL FOR MULTI COMPONENT MELTS 6 A. THEORETICAL CONSIDERATIONS 6 B. METHODOLOGY ...15 C. FORMULATION OF CONSTRAINTS 20 1. L i q u i d u s C o n s t r a i n t s . . 2 0 2. I m m i s c i b i l i t y C o n s t r a i n t s 25 I I I . APPLICATION TO THE CAO-AL 20 3-SI0 2 SYSTEM 29 A. THERMOCHEMICAL PROPERTIES 32 B. LIQUIDUS RELATIONS 39 C. DISCUSSION 47 1. Component A c t i v i t i e s 48 2. Heats of Mixing 53 3. E n t h a l p i e s of Fusion 55 IV. APPLICATION TO THE SYSTEM CAO-MGO-AL 20 3-SI0 2 58 A. METHODOLOGY 58 B. RESULTS 70 1. C a l o r i m e t r i c Data 70 2. Phase R e l a t i o n s on C a l i b r a t i o n J o i n s 73 3. Phase R e l a t i o n s on N o n - c a l i b r a t i o n J o i n s 88 C. DISCUSSION 109 1. L i q u i d P r o p e r t i e s 110 2. L i q u i d I m m i s c i b i l i t y 122 3. P e t r o l o g i c C o n s i d e r a t i o n s 124 D. CONCLUSIONS 133 BIBLIOGRAPHY 136 APPENDIX A - C a l c u l a t e d L i q u i d u s Diagrams For The Quaternary C a l i b r a t i o n J o i n s 150 APPENDIX B - C a l c u l a t e d L i q u i d u s Diagrams For Quaternary J o i n s Not Used In C a l i b r a t i o n 158 V L i s t of Tables I. Source of Experimental data f o r the System C a O - A l 2 0 3 - S i 0 2 30 I I . Experimental data f o r s o l i d - s o l i d r e a c t i o n s i n the system C a O - A l 2 0 3 - S i 0 2 31 I I I . M i n e r a l s i n the system C a G ~ A l 2 0 3 - S i 0 2 33 IV. Thermochemical values f o r m i n e r a l s and l i q u i d oxides in the system C a O - A l 2 0 3 - S i 0 2 34 V. Margules parameters f o r C a O - A l 2 0 3 - S i 0 2 l i q u i d s 35 VI. Heats of f u s i o n f o r phases i n the system C a O - A l 2 0 3 - S i 0 2 57 V I I . Experimental data f o r s o l i d - s o l i d r e a c t i o n s i n the system CaO-MgO-Al 20 3-Si0 2 59 V I I I . M i n e r a l s i n the system CaO-MgO-Al 20 3-Si0 2 68 IX. Nonstoichiometry i n l i q u i d u s phases i n the CaO-MgO-Al 20 3-Si0 2 system 69 X. Thermochemical values f o r m i n e r a l s and l i q u i d oxides i n the system CaO-MgO-Al 20 3-Si0 2 71 XI. Margules parameters f o r CaO-MgO-Al 20 3-Si0 2 l i q u i d s .72 X I I . Binary i n v a r i a n t p o i n t s i n the CaO-MgO-Al 20 3-Si0 2 system 74 X I I I . Ternary i n v a r i a n t p o i n t s i n the CaO-MgO-Al 20 3-Si0 2 system 75 XIV. Comparison of c a l c u l a t e d and experimental i n v a r i a n t p o i n t s on c a l i b r a t i o n j o i n s 91 XV. Comparison of c a l c u l a t e d and experimental i n v a r i a n t p o i n t s on n o n - c a l i b r a t i o n quaternary j o i n s 102 XVI. Heats of f u s i o n f o r phases i n the system CaO-MgO-Al 20 3-Si0 2 112 XVII. Quaternary i n v a r i a n t p o i n t s i n the CaO-MgO-Al 20 3-Si0 2 system 129 v i L i s t of f i g u r e s 1. Two-dimensional l i n e a r programming problem 17 2. Binary phase diagram showing d e r i v a t i o n of l i q u i d u s c o n s t r a i n t s 19 3. Binary G-X diagram showing d e r i v a t i o n of i m m i s c i b i l i t y c o n s t r a i n t s 27 4. Heat c a p a c i t i e s of c r y s t a l l i n e and l i q u i d oxides 38 5. C a l c u l a t e d l i q u i d u s diagram f o r the system C a O - A l 2 0 3 - S i 0 2 40 6. C a l c u l a t e d versus e x p e r i m e n t a l l y determined l i q u i d u s diagram f o r the system C a O - A l 2 0 3 - S i 0 2 41 7. C a l c u l a t e d phase diagram f o r the CaO-Si0 2 system 42 8. C a l c u l a t e d phase diagram f o r the A l 2 0 3 - S i 0 2 system ...43 9. C a l c u l a t e d phase diagram f o r the CaO-Al 20 3 system ....44 10. C a l c u l a t e d phase diagram f o r the pseudobinary system SiO ?-CaAl 20„ 49 11. A c t i v i t y contours of S i 0 2 l i q u i d at 1600°C i n the system C a O - A l 2 0 3 - S i 0 2 50 12. C a l c u l a t e d a c t i v i t i e s of S i 0 2 and CaO l i q u i d s at 1500 and 1600°C i n the system CaO-Si0 2 51 13. C a l c u l a t e d heats of mixing i n l i q u i d s on the j o i n Si0 2-CaAl 20„ 54 14. Comparisons of d i f f e r e n t experimental determinations of i n v a r i a n t p o i n t l o c a t i o n s 62 15. L i q u i d u s phases i n the quaternary CaO-MgO-Al 20 3-Si0 2 system 67 16. Phase diagram of the system CaO-Si0 2 76 17. Phase diagram of the system MgO-Si0 2 77 18. Phase diagram of the system A l 2 0 3 - S i 0 2 78 19. Phase diagram of the system Ca0-Al 2O 3 79 20. Phase diagram of the system MgO-Al 20 3 80 21. Phase diagram of the system CaO-MgO 81 22. L i q u i d u s diagram of the system C a O - A l 2 0 3 - S i 0 2 83 23. L i q u i d u s diagram of the system CaO-MgO-Si0 2 84 24. L i q u i d u s diagram of the system CaO-MgO-Al 20 3 85 25. L i q u i d u s diagram of the system M g O - A l 2 0 3 - S i 0 2 ...86 26. L i q u i d u s diagram of the 10 weight percent MgO plane of the CaO-MgO-Al 20 3-Si0 2 system 90 27. L i q u i d u s diagram of the 5 weight percent A l 2 0 3 plane of the CaO-MgO-Al 20 3-Si0 2 system 93 28. L i q u i d u s diagram of the 10 weight percent A l 2 0 3 plane of the CaO-MgO-Al 20 3-Si0 2 system 94 29. L i q u i d u s diagram of the 15 weight percent A l 2 0 3 plane of the CaO-MgO-Al 20 3-Si0 2 system ..95 30. L i q u i d u s diagram of the 20 weight percent A1 20 3 plane of the CaO-MgO-Al 20 3-Si0 2 system 96 31. L i q u i d u s diagram of the 25 weight percent A1 20 3 plane of the CaO-MgO-Al 20 3-Si0 2 system 97 32. L i q u i d u s diagram of the 30 weight percent A l 2 0 3 plane of the CaO-MgO-Al 20 3-Si0 2 system .98 33. L i q u i d u s diagram of the 35 weight percent A1 20 3 plane v i i of the CaO-MgO-Al 20 3-Si0 2 system 99 34. L i q u i d u s diagram f o r the system M g S i 0 3 - A l 2 0 3 - C a S i 0 3 .108 35. Heats of mixing on the C a A l 2 S i 2 0 8 - C a M g S i 2 0 6 j o i n ....114 36. S i 0 2 l i q u i d a c t i v i t i e s at 1600°C i n the system C a O - A l ? 0 3 - S i 0 2 116 37. S i 0 2 l i q u i d a c t i v i t i e s at 1600°C i n the system M g O - A l 2 0 3 - S i 0 2 117 38. S i 0 2 l i q u i d a c t i v i t i e s at 1600°C i n the system CaO-MgO-Si0 2 118 39. S i 0 2 l i q u i d a c t i v i t i e s at 1600°C on the 10 weight percent MgO plane of the system CaO-MgO-Al 20 3-Si0 2 ..119 40. S i 0 2 l i q u i d a c t i v i t i e s at 1600°C on the 20 weight percent MgO plane of the system CaO-MgO-Al 20 3-Si0 2 ..120 41. S i 0 2 l i q u i d a c t i v i t i e s at 1600°C on the 30 weight percent MgO plane of the system CaO-MgO-Al 20 3-Si0 2 ..121 42. L i q u i d u s diagram f o r the system CaO-MgO-Al 20 3-Si0 2 ..126 43. S p i n e l p r o j e c t i o n of l i q u i d u s r e l a t i o n s i n the system CaO-MgO-Al 20 3-Si0 2 132 v i i i Acknowledgement I would l i k e to express my deep a p p r e c i a t i o n to Dr. T.H. Brown, whose guidance and e n t h u s i a s t i c support were of immeasureable h e l p to me at a l l stages of t h i s p r o j e c t . His e a r l i e r work with melt systems p r o v i d e d the s t a r t i n g p o i n t f o r t h i s t h e s i s , and h i s a b i l i t y to weather the f i n a n c i a l storm unleashed through the course of my work allowed f o r i t s complet i o n . I a l s o thank Dr. M a r t i n Engi f o r h i s c o n t i n u a l h e l p f u l advice and f o r h i s c r i t i c a l review of p o r t i o n s of t h i s manuscript. I am indebted to E. H. P e r k i n s f o r a s s i s t a n c e with computer-related problems, and f o r making a v a i l a b l e h i s ' s t a t e of the a r t ' computer-g r a p h i c s package. 1 I . INTRODUCTION The progress which g e o l o g i s t s have made i n d e s c r i b i n g the pe t r o g e n e s i s of igneous rocks i s l a r g e l y a r e s u l t of the i n s i g h t gained through experimental phase e q u i l i b r i u m s t u d i e s undertaken du r i n g the present c e n t u r y . Many of these s t u d i e s , performed on simple systems ( l e s s than f i v e components), served to e l u c i d a t e phase r e l a t i o n s among common igneous m i n e r a l s and p r o v i d e d a general framework f o r p e t r o g e n e t i c models ( T u t t l e and Bowen, 1958; Yoder and T i l l e y , 1962). More r e c e n t l y , experimental s t u d i e s have i n c l u d e d n a t u r a l s t a r t i n g m a t e r i a l s i n order to a s c e r t a i n e f f e c t s of a d d i t i o n a l components on phase r e l a t i o n s , and to answer p e t r o g e n e t i c q u e s t i o n s concerning p a r t i c l u l a r rock compositions (e.g. Green and Ringwood, 1967; Robertson and W y l l i e , 1971; Mysen and Boettcher, 1975). Because melts can form with i n f i n i t e c o m p o s i t i o n a l v a r i a b i l i t y , the task of f i l l i n g the gaps between s t u d i e s on simple and n a t u r a l systems can not r e a l i s t i c a l l y be achieved s o l e l y with a d d i t i o n a l experiments. T h i s c o n s i d e r a t i o n underscores the need f o r a model capable of p r o v i d i n g P, T, X i n t e r p o l a t i o n between the experimental r e s u l t s of these systems. Such a model i s necessary to assess q u a n t i t a t i v e l y the r o l e of d i v e r s e processes such as f r a c t i o n a l c r y s t a l l i z a t i o n , p a r t i a l f u s i o n , or l i q u i d i m m i s c i b i l i t y i n the p e t r o g e n e s i s of the wide range of compositions found i n igneous rocks. The e s s e n t i a l i n g r e d i e n t of a model of t h i s s o r t i s the a b i l i t y to c a l c u l a t e the Gibbs f r e e energy of s i l i c a t e l i q u i d s as a f u n c t i o n of temperature, p r e s s u r e , and composition. The 2 most s a t i s f y i n g approach would base c a l c u l a t i o n s on the d i s t r i b u t i o n and mixing p r o p e r t i e s of a c t u a l s p e c i e s present i n melts. U n f o r t u n a t e l y , the mathematical b a s i s of s p e c i a t i o n c a l c u l a t i o n s i n v o l v i n g polymer d i s t r i b u t i o n s (Masson, 1977; Hess, 1980) has been extended no f u r t h e r than to b i n a r y systems, and treatment of s i l i c a - r i c h systems i s d i f f i c u l t . An a l t e r n a t e approach to t h i s problem i s to d e s c r i b e melt systems i n terms of chemical e q u i l i b r i a . L i m i t e d knowledge re g a r d i n g the p h y s i c a l s t r u c t u r e of s i l i c a t e melts (see B o t t i n g a et a l . , 1981 f o r a b r i e f review) r e q u i r e s that models must be e v a l u a t e d by t h e i r a b i l i t y to reproduce e x p e r i m e n t a l l y determined phase r e l a t i o n s i n multicomponent systems. B r i e f d i s c u s s i o n of s e v e r a l models i l l u s t r a t e s the v a r i e t y of ways i n which t h i s problem has thus f a r been formulated. The model of Burnham (1975; 1979) d e f i n e s 'quasi-c r y s t a l l i n e ' components, which, normalized to an equi-oxygen b a s i s , mix i d e a l l y i n s i l i c a t e melts. T h i s model s u c c e s s f u l l y accounts f o r water s o l u b i l i t y i n s y n t h e t i c and n a t u r a l systems, and adequately reproduces r e l a t i o n s i n s e v e r a l pseudobinary systems as a f u n c t i o n of pressure (Burnham, 1979). A d d i t i o n a l s p e c i e s are hypothesized to form by s p e c i a t i o n r e a c t i o n s when the a c t i v i t y of any of the i n i t i a l s p e c i e s are c a l c u l a t e d to be n o n i d e a l . In i t s present form, the model can not be g e n e r a l i z e d fo r c a l c u l a t i o n of multicomponent phase e q u i l i b r i a because s p e c i a t i o n r e a c t i o n s are proposed only a f t e r i t i s found that the i n i t i a l s p e c i e s do not mix i d e a l l y . An a d d i t i o n a l drawback of t h i s model i s i t s i n a b i l i t y to account f o r i m m i s c i b i l i t y due 3 to i t s assumption of i d e a l a c t i v i t y - c o m p o s i t i o n r e l a t i o n s . B o t t i n g a and Richet (1978) present a model in which l i q u i d f r e e e n e r g i e s are d e f i n e d i n terms of the e q u i l i b r i u m d i s t r i b u t i o n of melt s p e c i e s which have the same s t o i c h i o m e t r y as known l i q u i d u s phases. G e n e r a l i z e d equations are presented, u t i l i z i n g the Flory-Huggins a c t i v i t y approximation, which make i t p o s s i b l e to extend t h i s model to multicomponent systems. Thus f a r t h i s model has been s u c c e s s f u l i n r e p r o d u c t i o n of s e v e r a l pseudoternary phase diagrams ( d i o p s i d e - a n o r t h i t e -w o l l a s t o n i t e , w o l l a s t o n i t e - a k e r m a n i t e - g e h l e n i t e , f o r s t e r i t e -s p i n e l - k a l s i l i t e ) . The only melt model which has so f a r been a p p l i e d to n a t u r a l magma compositions i s that of Ghiorso and Carmichael (1980), who use 'normative m i n e r a l ' compositions as t h e i r l i q u i d components, and a ' r e g u l a r ' s o l u t i o n formalism to d e s c r i b e t h e i r a c t i v i t i e s . T h e i r c a l c u l a t e d mixing parameters allow the c a l i b r a t i o n of u s e f u l geothermometers for metaluminous n a t u r a l systems, but, i n i t s present form, t h e i r model f a i l s to d e s c r i b e phase r e l a t i o n s i n the component sub-spaces. In a d d i t i o n , t h i s model encounters d i f f i c u l t i e s i n rocks which are o u t s i d e of the composition space d e f i n e d by non-negative p r o p o r t i o n s of the end-members. The approach taken i n t h i s work i s s i m i l a r to that of Ghiorso and Carmichael (1980) i n that f i x e d end-member components are used. A d i f f e r e n t c h o i c e of components, however, r e f l e c t s a major d i f f e r e n c e i n o v e r a l l s t r a t e g i e s . The l i q u i d oxides are chosen as components so that the model can be 4 c a l i b r a t e d i n simple oxide systems. The philosophy u n d e r l y i n g t h i s approach i s that a f u l l y c o n s i s t e n t model f o r n a t u r a l magma compositions r e q u i r e s due account f o r the sub-systems of t h i s multicomponent system. I t should be emphasized t h a t t h i s approach does not imply the e x i s t e n c e of oxide s p e c i e s i n the melt, nor does the occurrence of other melt s p e c i e s i n v a l i d a t e the r e s u l t s of t h i s s t o i c h i o m e t r i c approach. T h i s approach makes use of the enormous wealth of experimental data that has been generated i n simple oxide systems(cf. Osborn and Muan, 1960), but i t w i l l , of course, encounter d i f f i c u l t i e s i n p o r t i o n s of the composition space that have not been e x p e r i m e n t a l l y i n v e s t i g a t e d . Although most of the l a t t e r s i t u a t i o n s occur i n g e o l o g i c a l l y unimportant systems, i t i s hoped that mixing p r o p e r t i e s and phase r e l a t i o n s can be p r e d i c t e d i n these systems by e x t r a p o l a t i n g energy s y s t e m a t i c s (such as f r e e energy of mixing v a r i a t i o n s with i o n i z a t i o n p o t e n t i a l of c a t i o n s (Hess, 1980)) d e r i v e d from other oxide systems. Because of the need to span a much l a r g e r c o m p o s i t i o n a l space than t h a t of Ghiorso and Carmichael (1980), t h i s model w i l l n e c e s s a r i l y be more complex. S u i t a b l y formulated with g e n e r a l i z e d equations, however, t h i s complextiy can be r e a d i l y handled by computer. The l a r g e task of c a l i b r a t i n g the model w i l l be j u s t i f i e d by the gain i n knowledge of mixing p r o p e r t i e s of simple systems, and by the a b i l i t y to i n t e g r a t e the thermodynamic p r o p e r t i e s of these sub-systems i n t o one l a r g e system. I n t e r p o l a t i o n between and e x t r a p o l a t i o n from a l l 5 e x p e r i m e n t a l l y determined compositions c o u l d then be achieved with a higher degree of c o n f i d e n c e . The d e t a i l s of the model and the methodology of c a l i b r a t i o n are d e s c r i b e d i n chapter I I . R e s u l t s are presented f o r a t e r n a r y oxide system i n chapter I I I , and f o r the quaternary CaO-M g O - A l 2 0 3 - S i 0 2 system i n chapter IV. Because a p p l i c a t i o n s are made only at one atmosphere, pressure w i l l be ignored as an i n t e n s i v e v a r i a b l e i n the d e r i v a t i o n s presented i n chapter I I . 6 I I . A THERMODYNAMIC MODEL FOR MULTICOMPONENT MELTS A. THEORETICAL CONSIDERATIONS In t h i s formulation of mineral-melt e q u i l i b r i a , a l l r e a c t i o n s are w r i t t e n on the b a s i s of oxide components, balanced on the s t o i c h i o m e t r y of the l i q u i d u s phases. In general nc Mineral « Melt = v±0± (1 ) i=l where v± are the s t o i c h i o m e t r i c c o e f f i c i e n t s of the i t h l i q u i d oxide Oi. For example, e q u i l i b r i u m between l i q u i d and f o r s t e r i t e i s expressed by Forsterite = Melt = 2 MgO ( l i q u i d ) + S i 0 2 ( l i q u i ( J ) < 1 a ) At e q u i l i b r i u m , the Gibbs f r e e energy of the l i q u i d i s equal t o that of s o l i d . Adopting as standard s t a t e s the pure minerals and l i q u i d oxides at the temperature (and pressure) of i n t e r e s t , gives nc (.) + n ( s ) R T l n a ( s ) = vi["± + n i R T l n a i ] <2> where ue and n ±, and a s and a i f represent the standard s t a t e f r e e energies and a c t i v i t i e s of the mineral and l i q u i d oxides, r e s p e c t i v e l y , T i s the temperature i n K e l v i n s , R i s the gas constant, and n i s the number of d i s t i n c t s i t e s upon which mixing occurs. For l i q u i d s , as wi t h gases, n i s equal to u n i t y . 7 I f the l i q u i d u s mineral i s pure, s u b s t i t u t i n g the product of the a c t i v i t y c o e f f i c i e n t , 7 , and the mole f r a c t i o n , X f o r the a c t i v i t y of the i t h oxide (a=yX) g i v e s nc _ v ( s ) = ]C v i y i + R T l n X i + R T l n Y i i=l *-(3) The example of f o r s t e r i t e - m e l t e q u i l i b r i u m i s w r i t t e n Fo 2 w° + RTlnX n + RTlnY 1 LVet) ^*°c« M g 0 C £ > J f L° + RTlnX O J r t + RTlnY (3a) L S i 02(£) S±°2(l) Si02(£)J In order t o use equation 3, the component a c t i v i t y c o e f f i c i e n t s must be evaluated a c c u r a t e l y . These are r e l a t e d t o the excess f r e e energy of a s o l u t i o n , G , by S J ' excess ** nc G „ = / . X.RTlnY. < 4) excess / J i 1 i=l where nc i s the number of components and G i s def i n e d as excess the d i f f e r e n c e between the free energy of the s o l u t i o n and the i d e a l (entropy of mixing) c o n t r i b u t i o n t o the t o t a l free energy, i . e . G = G - G . excess solution ideal In t h i s study, use i s made of the e m p i r i c a l formulation a r i s i n g from the work of Margules(1895) that G can be 3 3 excess approximated by a v a r i a b l e degree polynomial expansion. This method has been used e x t e n s i v e l y i n the geologic l i t e r a t u r e f o r 8 d e s c r i b i n g the d e v i a t i o n s from i d e a l i t y i n s o l i d s o l u t i o n s c o n s i s t i n g of two and three components (e.g. Thompson and Waldbaum, 1969; Wood and N i c h o l l s , 1978; L i n d s l e y et a l . , 1981). For a binary s o l u t i o n , G. e x c e s s = A + BXj+ CX2 + DXJ> + (5) Equation 5 can be s i m p l i f i e d by a p p l i c a t i o n of the boundary c o n d i t i o n s that G e x c e s s approaches zero as each mole f r a c t i o n approaches u n i t y , and recombination such that a l l terms are of the same degree (Thompson, 1967). C o n s i d e r a t i o n of the f i r s t t hree terms of equation 5 leads t o G e x c e s s = ^ ( 6 a ) while c o n s i d e r a t i o n of the f i r s t four terms of equation 5 gives G = W i o X i x J +W 9 1 X*X , ( 6 b ) e x c e s s 12 1 2 11 1 / The W terms are l i n e a r a l g e b r a i c combinations of the o r i g i n a l c o e f f i c i e n t s of equation 5, and are c a l l e d Margules parameters. Equation 6a i s that of a symmetric s o l u t i o n as G _„„__ must be CACCOO symmetrical about the compositional midpoint of the s o l u t i o n . By analogy, equation 6b i s that of an asymmetrical s o l u t i o n . For most authors, 'symmetric* s o l u t i o n has become synonymous wit h ' r e g u l a r ' s o l u t i o n (e.g. P o w e l l , 1974; Grover, 1977), w h i l e Hardy (1953) dubbed the asymmetrical formulation 9 ' s u b r e g u l a r ' . Mar g u l e s parameters can be t r e a t e d i n the same f a s h i o n as the a n a l o g o u s thermodynamic s t a t e f u n c t i o n s , so t h a t , f o r example, /dw G\ ( i n a l l e q u a t i o n s o t h e r than e q u a t i o n 7 , ' W i s used as an a b b r e v i a t i o n f o r ' WQ ' . ) In c o n t r a s t t o the nomenclature r e f e r r e d t o above, C a r m i c h a e l e t a l . ( 1 9 7 7 ) r e q u i r e t h a t ' r e g u l a r ' s o l u t i o n s s a t i s f y t h e c o n d i t i o n t h a t W = 0 , w h i l e Guggenheim ( 1 9 5 2 ) r e f e r s t o such s o l u t i o n s as ' s t r i c t l y r e g u l a r ' . T h i s usage appears t o be more i n k e e p i n g w i t h H i l d e b r a n d ' s ( 1 9 2 9 ) o r i g i n a l sense of t h e term ' r e g u l a r ' i n r e f e r r i n g t o tho s e s o l u t i o n s c h a r a c t e r i z e d "by non-zero h e a t s of m i x i n g but i d e a l e n t r o p i e s of m i x i n g . In an attempt t o a v o i d t h e c o n f u s i o n r e g a r d i n g usage of the term ' r e g u l a r ' , t h e s e s o l u t i o n s w i l l be r e f e r r e d t o as 'Margules s o l u t i o n s ' . The c o m p l e x i t y o f t h e s o l u t i o n model i s d e s i g n a t e d by t h e degree of p o l y n o m i a l used i n t h e i n i t i a l e x p a n s i o n , o r by the number of s u f f i x e s used w i t h the Ma r g u l e s parameters (Wohl, 1 9 4 6 ) . For t h e s p e c i a l c a s e s of second and t h i r d degree p o l y n o m i a l s , the terms symmetric and asymmetric can s t i l l be used, and f o r t h e case of a s o l u t i o n i n which the M a r g u l e s p a r a m e t e r s have no te m p e r a t u r e dependency, the m o d i f i e r ' t e m p e r a t u r e - i n d e p e n d e n t ' can be used. Thompson ( 1 9 6 7 ) a l s o p r e s e n t s an e x p r e s s i o n f o r a t e r n a r y 10 asymmetric s o l u t i o n . The d e r i v a t i o n of t h i s e x p r e s s i o n i s shown i n d e t a i l by Anderson and L i n d s l e y (1981), who p o i n t out the i n c o n s i s t e n c i e s i n the ' l i t e r a t u r e with r e g a r d t o s u b s c r i p t systems f o r the Margules parameters. I f t h e i r W 1 2 i s equal to Thompson's W2,, and v i c e v e r s a , t h e i r e x p r e s s i o n i s the same as Thompson's (although Thompson omitted the t e r n a r y i n t e r a c t i o n term). In an e f f o r t t o r e s o l v e such i n c o n s i s t e n c i e s a g e n e r a l i z e d equation i s pre s e n t e d which can r e a d i l y be a p p l i e d t o s o l u t i o n s of any number of components and can i n c o r p o r a t e any degree po l y n o m i a l i n the i n i t i a l expansion: nc-1 n c n c  G e x c e s s ~ Z Z Z W l . i 9 . . . i ( X i , V * * > (8) h'1 h'h W i 1 2 p 1 2 p where p i s the degree of polynomial used i n the i n i t i a l e x p r e s s i o n f o r G . F i n n e r t y (1977) p r e s e n t s an e x c e s s e s s e n t i a l l y i d e n t i c a l equation f o r the s p e c i a l case of a second degree p o l y n o m i a l . For a t h i r d degree polynomial and a t e r n a r y s o l u t i o n equation 8 reduces to nc -1 n c n c G e x c e s s " Z Z Z W I j k < X i X J V (9) i - l J - i k-J which when expanded g i v e s - W e s s " " m V A + " u A t t + W H 3 X 1 X 1 X 3 + + W 2 2 3 X 2 X 2 X 3 + MJJJXJXJXJ + W 1 2 3 X L X 2 X 3 ( ( O ( 1 1 This i s the same as the expression given by Anderson and L i n d s l e y (1981) i f the obvious s u b s t i t u t i o n s f o r d i f f e r e n t s u b s c r i p t s are made, i . e . Anderson and L i n d s l e y ' s W12 i s equal to W112. This n o t a t i o n seems p r e f e r a b l e as i t i s w e l l s u i t e d to computer manipulation, and the s u b s c r i p t s of the Margules parameters can be extended f o r any degree polynomial. Furthermore, the s u b s c r i p t s serve as reminders of the powers of the mole f r a c t i o n s i n each term. Equations 4 and 8 can be used to d e r i v e a general equation f o r the c a l c u l a t i o n of the a c t i v i t y c o e f f i c i e n t of a l l components. For component m, d i f f e r e n t i a t i o n of equation 4 with respect to component m, holdi n g a l l other components constant except component n (dX n=-dX m) g i v e s : / l ^ c e s s \ . RxinYjjj - RTlnY n + x ± R T l n ( - | ^ ) ( n , \ ° A m /X q=X r..X n c i-1 \ /Xq-Xx-.-Xnc Xq^Xjj.Xn Xq7<XTn,Xn Because a form of the Gibbs-Duhem equation at constant pressure and temperature i s i ^ l \ W TD / A Q xA.x. q m n i t f o l l o w s t h a t / d G e x c e s s \ m m n Y m . m n Y n ( 1 3 ) \ /Xq-Xi-.-Xnc Xq^Xjn,Xn Component m can be considered apart from the other 12 components of a s o l u t i o n and, wi t h rearrangement, equation 4 becomes nc "excess • E W { 1 4 ) n=l S u b s t i t u t i n g f o r R T l n 7 n from equation 13, noting that the mole f r a c t i o n s always sum to u n i t y , and rearranging gives R T l m r m - Gexcess W ^ e x c e s s j ( 1 5 ) n?Hn Xq?Xm,Xn Equation 15 i s independent of any model f o r G e x c e s s • D i f f e r e n t i a t i o n of the g e n e r a l i z e d Margules expansion (equation 8) g i v e s : nc-1 nc nc / a Gexcess\ ^ ]T) .. . ^ w^^.. . ± [ o n ^ X ^ . . .Xi /Xm \ d X * h - X y . . ^ l 2 - i l V V l " QnXijX^-•-Xip/Xjjj (16) where Q and Q are terms which sum how many of the i 's are nt n equal to m and n, r e s p e c t i v e l y . Thus, f o r the s u b s c r i p t m, nc ^ • E 0 ^ • ( 1 7 ) S u b s t i t u t i n g equation (16) and (8) i n t o equation (15) gi v e s nc-1 nc nc RTlnY m - Z Z X V i l ± 2 " - i r V 1 ! 13 + n/m and because nc 2 x n ( V i 1 X i 2 . . . X l p / X m - Q n X i l X l 2 . . . X l p / X n ^ ( 1 8 ) nc m ( 1 9 ) n=l n^ hn i f f o l l o w s that nc-1 nc nc^ » T i n r m - E Z ••• E » i , i 2 . . . i J x i 1 x i 2 - - x i « - E v P i - i 4 - n x X X /X - 0 X. X. . . . X . I (20) Grouping the l a s t term of equation (20 ) w i t h the f i r s t term, and no t i n g that nc XX-P <21) n=l where p i s the degree of polynomial used i n the i n i t i a l Margules expansion, gives nc-1 nc nc RTlnY nc—x _ _ i -i p * 1 ! + (l-p)X, X. ...X, (22 ) Equation 22 i s the general expression r e l a t i n g a c t i v i t y c o e f f i c i e n t s to Margules parameters i n systems of any number of components, using polynomial expansions such as equation ( 5 ) 14 which in c l u d e any number of terms. For example, i n a binary system, using a t h i r d degree polynomial f o r G e x c e s s , equation (22) reduces t o 2 2 2 RTlnYi = W U 2 ( 2 X 1 X 2 - 2XjX 2) + W 1 2 2(X 2 - 2 X i X 2 ) (23a) RTlnY2 = WU2(xJ - 2 ^ ) + W ^ U X ^ -Tkfa) (23b) which can be rearranged t o show t h e i r equivalence expressions given by Thompson(1967): RTlnY x = X2[wx + 2(W2 - W^xJ RTlnY2 - Xj[w2 + 2(Wj - W^xJ Returning t o the example of f o r s t e r i t e - l i q u i d e q u i l i b r i u m , w i t h s u b s t i t u t i o n of equation 22 f o r the a c t i v i t y c o e f f i c i e n t s of MgO and S i 0 2 f equation 3a can be w r i t t e n i n the form 4 o - <*5g0 + VSi0 2> + < 2 R T l n XMgO + R T l n X S i 0 2 > + " ' ^ g O + XSi0 2> ( 2 5) where use i s made of the a b b r e v i a t i o n s of W f o r the sum of the Margules parameters, and X'^ Q , x ' s i o . f o r t h e s u m o f t h e c o e f f i c i e n t s (the bracketed q u a n t i t y i n equation 22) of the Margules parameters f o r the components MgO and S i 0 2 , r e s p e c t i v e l y . Notice t h a t , even f o r multicomponent e q u i l i b r i a , w ith (24a) (24b) 15 tion 25 remains linear in the Margules parameters (W's). equa B. METHODOLOGY In recent years, pe t r o l o g i s t s have increasingly turned to the re s u l t s of phase equilibrium studies to estimate thermochemical data and mixing properties of solutions. Most estimates (e.g. Haas et a l . , 1982) are obtained by linear regression of phase e q u i l i b r i a data expressed in the form of equations si m i l a r to equation 3. This method of treating the data t a c i t l y assumes that the equilibrium conditions (T,X) are known, by producing f i t s designed to be as close to as many of the estimated equilibrium points as possible. Most phase e q u i l i b r i a techniques do not, however, identif y the equilibrium conditions of a reaction, only the r e l a t i v e s t a b i l i t i e s of the product and reactant assemblages. Thus, although the equilibrium condition i s expressed as , (26) reaction " ^products 'reactants experimental r e s u l t s convey information regarding whether the sign of A G r e a c t i o n i s po s i t i v e (reactants stable) or negative (products s t a b l e ) . In recognition of t h i s s i t u a t i o n , Haas et a l . (1982) use weighting factors in order to "reduce(d) the tendency of the regression to s e t t l e on the midpoint of a (experimental) bracket". In considering data for coexisting c l i n o - and orthopyroxenes, Lindsley, et a l . (1981) found that only very i l l ranges of compositions within cert a i n brackets were sma. 16 c o n s i s t e n t w i t h the c o n s t r a i n t s imposed by the r e s t of the data. They performed repeated l e a s t - s q u a r e s r e g r e s s i o n s , " a d j u s t i n g compositions at each c y c l e according to a s t r a t e g y designed to reduce the r e s i d u a l s " (pg. 166). A more rig o r o u s mathematical treatment of phase e q u i l i b r i a data i s obtained with the use of l i n e a r programming, as o u t l i n e d by Gordon (1973; 1977) i n h i s treatment of s o l i d - f l u i d r e a c t i o n s . The technique of l i n e a r programming has already been described i n the geologic l i t e r a t u r e (e.g. Greenwood, 1967; Day and Kumin, 1980). I t c o n s i s t s of an algorithm f o r c o n t i n u a l l y improving an o b j e c t i v e f u n c t i o n while maintaining f e a s i b i l i t y , i . e . consistency with the sense of a l l i n e q u a l i t y c o n s t r a i n t s (Figure 1). In r e l a t i o n to phase e q u i l i b r i a s t u d i e s , i n e q u a l i t y c o n s t r a i n t s can be w r i t t e n f o r each h a l f - b r a c k e t , i . e . each determination of the c o n d i t i o n s where e i t h e r reactants or products are s t a b l e . In c o n t r a s t to l i n e a r r e g r e s s i o n which provides a unique f i t which tends towards the midpoints of experimental brackets while not ensuring c o n s i s t e n c y w i t h a l l b r a c k e t s , l i n e a r programming ensures consistency with a l l experimental data, but provides a range of s o l u t i o n s which can be unique only f o r a given o b j e c t i v e f u n c t i o n (Figure 1). Gordon (1973) uses the A G r e a c t l o n and A S r e a c t i o n as o b j e c t i v e f u n c t i o n s , although any property of a phase or combination of such p r o p e r t i e s c o u l d be used. In a d d i t i o n , some l i n e a r programming computer packages a l l o w use of n o n - l i n e a r o b j e c t i v e f u n c t i o n s , so t h a t a s o l u t i o n s i m i l a r t o that produced by r e g r e s s i o n (but always i n t e r n a l l y c o n s i s t e n t w i t h a l l data) F i g u r e 1; Two-dimensional l i n e a r programming problem, showing f e a s i b l e r e g i o n d e f i n e d by f o u r l i n e a r c o n s t r a i n t s , and o p t i m a l s o l u t i o n s (A and B) f o r two d i f f e r e n t o b j e c t i v e f u n c t i o n s . 18 c o u l d be achieved (Gordon, 1977). In c o n t r a s t to r e g r e s s i o n a n a l y s i s , l i n e a r programming o f f e r s the advantage that a model can be c o n s t r a i n e d by the T, X p o s i t i o n of a n o n - s t o i c h i o m e t r i c l i q u i d u s m i n e r a l , without knowledge of the a c t i v i t y - c o m p o s i t i o n r e l a t i o n s of the m i n e r a l . Reference to F i g u r e 2 shows that the ' l i q u i d s t a b l e ' c o n s t r a i n t r e p r e s e n t s the upper l i m i t f o r the s t a b i l i t y f i e l d of any l i q u i d u s m i n e r a l . Since the reduced a c t i v i t y of a non-s t o i c h i o m e t r i c phase expands i t s s t a b i l i t y f i e l d r e l a t i v e to the s t o i c h i o m e t r i c phase, c a l c u l a t i o n s of the ' l i q u i d s t a b l e ' c o n s t r a i n t assuming s t o i c h i o m e t r y are v a l i d f o r both s t o i c h i o m e t r i c and n o n - s t o i c h i o m e t r i c m i n e r a l s . In t h e i r i n v e s t i g a t i o n of l i q u i d i m m i s c i b i l i t y i n the CaO-S i 0 2 system, Tewhey and Hess (1979) p o i n t out that the numerical valu e s d e r i v e d f o r Margules parameters are extremely s e n s i t i v e to the estimated compositions of the c o e x i s t i n g phases. S l i g h t changes i n the composition of e i t h e r phase cause l a r g e d i f f e r e n c e s i n the c a l c u l a t e d values of the Margules paramters. T h i s o b s e r v a t i o n emphasizes the importance of i n c o r p o r a t i n g experimental u n c e r t a i n t i e s i n t o the thermodynamic e v a l u a t i o n of phase e q u i l i b r i u m data. U n c e r t a i n t i e s are e a s i l y i n c o r p o r a t e d i n t o the l i n e a r programming problem by a d j u s t i n g the temperature and composition of each experimental h a l f - b r a c k e t i n such a way as to produce the maximum bracket width (Figure 2 ) . . Where e q u i l i b r i u m c o n d i t i o n s have been estimated but not bracketed, estimated u n c e r t a i n t i e s can be added so as to c r e a t e a bracket. 19 1 8 0 0 1 7 0 0 H O 1 8 0 0 H Ul or 3 1 5 0 0 or UJ 9| 1 4 0 0 UJ 1 3 0 0 H 1 2 0 0 0.0 A • Liquid stable AG r < 0 • A + A 3 B 2 ~ T ~ 0.2 CM CQ CO < CQ < Solid stable • AG r > 0 AB + B ~ i 0.6 0.4 Mole F r a c t i o n 0.8 1.0 B F i g u r e 2: B i n a r y phase diagram showing T, X c o o r d i n a t e s f o r l i q u i d u s c o n s t r a i n t s . I n c o r p o r a t i o n of e s t i m a t e d e x p e r i m e n t a l e r r o r s i n t e m p e r a t u r e and c o m p o s i t i o n y i e l d the p o s i t i o n of ' l i q u i d s t a b l e ' and ' s o l i d s t a b l e ' c o n s t r a i n t s . The reduced a c t i v i t y of n o n s t o i c h i o m e t r i c AB ca u s e s the e x p e r i m e n t a l l y d e t e r m i n e d l i q u i d u s ( s o l i d l i n e l a b e l l e d 'NS') t o occ u r a t h i g h e r t e m p e r a t u r e s than t h a t c a l c u l a t e d f o r s t o i c h i o m e t r i c AB ( t h i n n e r s o l i d c u r v e l a b e l l e d 'ST'). Note t h a t t h e p o s i t i o n of the ' l i q u i d s t a b l e ' c o n s t r a i n t s a r e c o n s i s t e n t w i t h t h e p o s i t i o n o f t h e s t o i c h i o m e t r i c l i q u i d u s , w h i l e two ' s o l i d s t a b l e ' c o n s t r a i n t s ( s t a r s ) a r e i n c o n s i s t e n t w i t h t h e s t o i c h i o m e t r i c l i q u i d u s . A l s o n o t i c e t h e c i r c l e d i n t e r s e c t i o n s which show t h a t , f o r t h e s t o i c h i o m e t r i c diagram, t h e r e a c t i o n p o i n t o c c u r s a t h i g h e r t e m p e r a t u r e and t h e e u t e c t i c a t lower t e m p e r a t u r e than t h e e x p e r i m e n t a l l y d e t e r m i n e d i n v a r i a n t p o i n t s . 20 C. FORMULATION OF CONSTRAINTS In t h i s treatment of melt systems, two primary sources of data are used as c o n s t r a i n t s f o r the model: s o l i d - l i q u i d ( l i q u i d u s ) and l i q u i d - l i q u i d ( i m m i s c i b i l i t y ) e q u i l i b r i a . Before pr e s e n t i n g r e s u l t s f o r a tern a r y system, the complete c o n s t r a i n t set used i n the c a l c u l a t i o n s w i l l be dis c u s s e d . 1. L i q u i d u s C o n s t r a i n t s For every experimental p o i n t b r a c k e t i n g the l i q u i d u s of a given system, r e l a t i o n 2 6 (with the s u b s t i t u t i o n of the appro p r i a t e i n e q u a l i t y sign) must be s a t i s f i e d . W r i t t e n i n the form of 2 6 , equation 3 becomes nc ( 2 7 ) nc » ^ V i f y i + RTlnX ± + RTlnYij - V°s < 0 i° = A H ° + y*(Cp)dT - T | S 0 + y*(Cp/T)dTJ ( 2 8 ) At any temperature T ( i n K e l v i n s ) , the standard s t a t e chemical p o t e n t i a l - o f any pure phase or component, u°, i s c a l c u l a t e d by the r e l a t i o n T „ .X y J  °  1 2 9 8 - 1 5 " 2 9 8 . 1 5 where AH°,S°, and Cp are the enthalpy at the reference temperature of 298.15 K, the entropy at 298.15 K, and the heat c a p a c i t y (at constant pressure) which i t s e l f i s expressed as a f u n c t i o n of temperature (discussed below). Combining equations 27 and 28, and rea r r a n g i n g , g i v e s 21 nc Z^Vi[AHf ts*+fcv^at " t y^cPi/T>dT+xiwj i = 1 298.15 298.15 /T T nc (Cps)dT - T f (Cps/T)dT > - Z v i(RTlnX.) 2 9 )  J i=l 298.15 298.15 where, s u b s t i t u t i n g f o r RTln7 i, the r i g h t hand s i d e (RHS) of equation 22 has been abbreviated as discussed p r e v i o u s l y i n the d e r i v a t i o n of equation 25. I t should be noted that the ' l i q u i d s t a b l e ' c o n s t r a i n t i n the form of equation 29 a p p l i e s only t o s t a b i l i t y w ith respect to one mi n e r a l . I f the assumption i s made that the exp e r i m e n t a l i s t determined s t a b l e e q u i l i b r i a , i t i s a l s o necessary to s t i p u l a t e that the l i q u i d i s s t a b l e with respect to a l l other minerals as w e l l . In p r a c t i c e i t i s found that such a d d i t i o n a l c o n s t r a i n t s are r a r e l y r e q u i r e d , and co n s e q u e n t i a l l y the s i z e of the problem i s g r e a t l y reduced. R e c a l c u l a t i o n of phase diagrams wi t h a ge n e r a l i z e d technique which considers a l l mineral s t a b i l i t i e s a t a given composition i s necessary to ensure that the above c o n d i t i o n i s s a t i s f i e d . I f a l l e n t h a l p i e s , e n t r o p i e s , and heat c a p a c i t i e s are known, these v a r i a b l e s can be s h i f t e d to the RHS of r e l a t i o n 29, and the set of Margules parameters can be c a l c u l a t e d from ex p e r i m e n t a l l y determined l i q u i d u s c o n d i t i o n s . C a l o r i m e t r i c measurements of e n t h a l p i e s and ent r o p i e s are p r e s e n t l y a v a i l a b l e f o r many of the common igneous minerals. Because u n c e r t a i n t i e s are always a s s o c i a t e d with these measurements, a l l e n t h a l p i e s and e n t r o p i e s of minerals are t r e a t e d as "constrained" v a r i a b l e s , a l l o w i n g them f l e x i b i l i t y w i t h i n the estimated e r r o r 22 b r a c k e t s . This method allows determination of the enthalpy and/or entropy of any other mineral f o r which no data are a v a i l a b l e . Heat c a p a c i t y (Cp) data from d i f f e r e n t i a l scanning c a l o r i m e t r y or drop c a l o r i m e t r y measurements a l s o e x i s t f o r the m a j o r i t y of common igneous m i n e r a l s . Heat c a p a c i t i e s can be expressed by an e m p i r i c a l equation of the form (Haas and F i s h e r , 1976) Cp » A + BT + CT"2 + D T ( - 0 , 5 ) + ET 2 ( 3 0 ) Because Cp measurements u s u a l l y stop at temperatures below that of the l i q u i d i of many minerals i n oxide systems, i t i s extremely important that the Cp f u n c t i o n permit reasonable e x t r a p o l a t i o n s t o high temperatures. As pointed out by Holland (1981), use of equation 30 o f t e n r e s u l t s i n unwarranted and p h y s i c a l l y improbable i n f l e c t i o n s i n the f u n c t i o n Cp(T). C o n s i d e r a t i o n of k i n e t i c theory i n d i c a t e s that Cv (the heat c a p a c i t y at constant volume) approaches a l i m i t i n g value of 3R/gram-atom (R i s the gas constant) at high temperature (see e.g. K i e f f e r , 1979). As Cp i s r e l a t e d to Cv by Cp - Cv + T(Va2/B) (31 ) Cp should become approximately l i n e a r i n T at high temperatures, w i t h the slope equal to the bracketed q u a n t i t y i n equation 31. Holl a n d (1981) r e p o r t s t h a t , using t y p i c a l thermal expansion and 23 compressibility values for s i l i c a t e minerals, Cp's should have a high temperature l i m i t of around 27 Joules/gram-atom-K. The commonly used Maier-Kelley equation uses only the f i r s t three terms of equation 30, which r e s u l t s in considerable overestimations of Cp at high temperature. It has been found (Berman and Brown, 1983a) that the equation Cp + CT"2 + D T ( - ° ' 5 ) + FT" 1 (32) constrained such that A i s p o s i t i v e while C,D, and F are negative, r e s u l t s in adequate reproduction of Cp measurements as well as reasonable extrapolations to high temperatures. The heat capacity c o e f f i c i e n t s of the liqu i d u s minerals could be f i t to the calorimetric measurements at the same time as the phase e q u i l i b r i a constraints are f i t (Haas et a l . , 1982). In an e f f o r t to reduce the size of the f i n a l problem, however, the Cp data for each mineral was f i t independently p r i o r to consideration of the liquidus data (Berman and Brown, 1983a). Consequently, a l l terms in r e l a t i o n 29 involving the Cp of minerals were moved to the RHS. At present there i s a pronounced shortage of thermochemical data for l i q u i d oxides and s i l i c a t e s ; for recent compilations and discussion of such data, see Carmichael et a l . (1977), Bottinga and Richet (1978), and Navrotsky (1981). These authors also point out the problems in extrapolating properties of glasses through the glass t r a n s i t i o n temperature to l i q u i d s of the same composition. At the melting temperature of unary 24 systems, the p r o p e r t i e s of the l i q u i d oxides are l i n k e d to those of the s o l i d oxides by r e l a t i o n 29. In these s p e c i a l cases, the X',W, and lnX terms are a l l zero, so that knowledge of the heat or entropy of f u s i o n p l a c e s c o n s t r a i n t s on the thermochemical p r o p e r t i e s of the l i q u i d o x i d e s . In p r a c t i c e , b r a c k e t s are a c r e a t e d around these values by means of the estimated e r r o r s , and l i q u i d e n t h a l p i e s and e n t r o p i e s are t r e a t e d as v a r i a b l e s i n the f i n a l l i n e a r programming problem. Carmichael et a l . (1977) have d e r i v e d p a r t i a l molar heat c a p a c i t i e s of the l i q u i d oxides from experiments on multicomponent s i l i c a t e melts. They found t h a t , w i t h i n t h e i r experimental u n c e r t a i n t y , there are no excess heat c a p a c i t i e s on mixing, and the heat c a p a c i t i e s of the oxides are temperature independent w i t h i n the range of t h e i r experiments (1200-1650 K). R i c h e t and B o t t i n g a (1980) present Cp data f o r K 2 S i , 0 9 and N a A l S i 3 0 8 l i q u i d s i n the range 752-1493 K and 1158-1486 K, r e s p e c t i v e l y , and review data p e r t a i n i n g to s i l i c a t e l i q u i d heat c a p a c i t i e s . They conclude t h a t a l l data are p r e s e n t l y c o n s i s t e n t with the premise that l i q u i d heat c a p a c i t i e s are constant i n the s t a b l e and supercooled temperature range. The e x p e r i m e n t a l l y i n v e s t i g a t e d temperatures cover, however, only a f r a c t i o n of the range of the l i q u i d u s temperatures commonly encountered w i t h i n simple oxide systems. Because i t i s u n l i k e l y t h a t heat c a p a c i t i e s are constant over the wide temperature ranges i n some of these systems (up to 2000 K), equation 32 i s used f o r the l i q u i d o xides. Richet and B o t t i n g a (1980) a l s o note that the magnitude of s i l i c a t e l i q u i d heat c a p a c i t i e s are 25 q u i t e uniform, 29.2 Joules ± 10% per degree per gram-atom. In accordance w i t h t h i s observation,the c o n s t r a i n t has been imposed t h a t a l l oxide heat c a p a c i t i e s are i n the range 26-32 Joul e s per degree per gram-atom. The C p - c o e f f i c i e n t s of equation 32 are l e f t as v a r i a b l e s i n the f i n a l problem; l i q u i d oxide heat c a p a c i t i e s are compared wi t h some measured values below. 2. I m m i s c i b i l i t y C o n s t r a i n t s Many b i n a r y and t e r n a r y oxide systems d i s p l a y regions of metastable or s t a b l e l i q u i d i m m i s c i b i l i t y ( G r i e g , 1927). The coexistence of immiscible l i q u i d p a i r s imposes t i g h t c o n s t r a i n t s on a s o l u t i o n ' s mixing p r o p e r t i e s through the c o n d i t i o n that the chemical p o t e n t i a l s of a l l components must be equal i n both l i q u i d s . Thus, f o r component 1 i n c o e x i s t i n g phases A and B , v^CA) = WjCB) ( 3 - 3 ) where - U° + RTlnX 1 + RTlnY x (34) Since the standard s t a t e chemical p o t e n t i a l , v°i, depends only on the temperature, M I ( A ) = M ^ ( B ) . Equation 33 can be arranged t h e r e f o r e to give RTlnY^(A) - RTlnY^(B) = RTlnX^B) - RTlnXj (A) ( 3 5 ) The s u b s t i t u t i o n that RTln7=X'W', as above, gi v e s 26 [ x | ( A ) - Xj(B)]w' = RTlnX 1(B) - RTlnX^A ) (36) G i v e n t h e te m p e r a t u r e and c o m p o s i t i o n of c o e x i s t i n g l i q u i d s , e q u a t i o n 36 (and t h e nc-1 e q u a t i o n s f o r the o t h e r components) can be used t o c a l c u l a t e t h e s e t of t h e v a l u e s of t h e Ma r g u l e s p a r a m e t e r s ( W ) . E q u a t i o n 33 i s r e p r e s e n t e d i n F i g u r e 3 by t h e common tan g e n t t o t h e f r e e e n e r g y - c o m p o s i t i o n c u r v e . E q u a t i o n s 33-36 can be adapte d t o t h e l i n e a r programming problem by i n c o r p o r a t i n g e r r o r s i n t h e c o m p o s i t i o n s of t h e c o e x i s t i n g l i q u i d s . R e f e r e n c e t o t h e b i n a r y system i n F i g u r e 3 shows t h a t f o r c o m p o s i t i o n p a i r s (A" and B*) y i e l d i n g a w i d e r s o l v u s than t h e e s t i m a t e d e q u i l i b r i u m c o m p o s i t i o n s , t h e f o l l o w i n g r e l a t i o n s h o l d : V L ( A - ) > P X ( B + ) , v 2 ( A ~ ) < y 2 ( B + ) (37) F o r c o m p o s i t i o n p a i r s (A* and B") y i e l d i n g a narrower s o l v u s t h a n t h e e q u i l i b r i u m c o m p o s i t i o n s , yx(A+) < VjOr) v 2(A +) > y 2 ( B _ ) (38) F o r most s o l u t i o n s , r e l a t i o n s 37 and 38 s u f f i c e t o ensure t h a t i m m i s c i b l e p a i r s l i e w i t h i n e x p e r i m e n t a l b r a c k e t s o r the e s t i m a t e d e r r o r range. F o r systems i n which t h e e x c e s s f r e e energy i s a c u t e l y a s y m m e t r i c , t a n g e n c i e s t o t h e G-X c u r v e can be such t h a t t h e s e r e l a t i o n s do not guar a n t e e c o n s i s t e n c y w i t h the 27 CO _Q) D O L. 0) c LU CD 0) CO j Q 2000 -0 -Mi(B~) -2000 H - 4 0 0 0 H Mi(A)| Mi(A +) - 6 0 0 0 H = - 8 0 0 0 H -10000 Free Energy of Solution M 2( A ) 0.0 0.2 0.4 Component 1 1.0 n 1 0.6 0.8 Component 2 F i q u r e 3: Binary G-X diagram showing d e r i v a t i o n of c o n s t r a i n t s f73S c o e x i s t i n g immiscible l i q u i d s (A and B). Because c o n s i d e r a t i o n i s given to compositional e r r o r s , " ^ J E * equating chemical p o t e n t i a l s i n both l i q u i d s are replaced by the i n e q u a l i t y c o n s t r a i n t s discussed i n the t e x t . 28 experimental data. In such systems i t i s most p r a c t i c a l to e s t a b l i s h the p o s i t i o n of the solvus by c o n s t r a i n i n g the p o s i t i o n of the s p i n o d a l curve, d e f i n e d by the c o n d i t i o n that the second d e r i v a t i v e of the f r e e energy of the s o l u t i o n with respect to a c o m p o s i t i o n a l v a r i a b l e i s equal to zero. Thus, compositions can be c o n s t r a i n e d to l i e o u t s i d e the s p i n o d a l (narrowing the solvus) by f o r c i n g the second d e r i v a t i v e to be p o s i t i v e , or i n s i d e the s p i n o d a l (widening the s o l v u s ) by f o r c i n g the second d e r i v a t i v e to be n e g a t i v e . The same method can be used to i n s u r e s t a b i l i t y of the l i q u i d s o l u t i o n at compositions f o r which no evidence of i m m i s c i b i l i t y has been observed. 29 I I I . APPLICATION TO THE C a O - A l 2 0 3 - S i 0 2 SYSTEM The t e r n a r y system C a O - A l 2 0 3 - S i 0 2 i s w e l l s u i t e d as a t e s t of the s t o i c h i o m e t r i c - M a r g u l e s s o l u t i o n model because t h i s system has been w e l l c h a r a c t e r i z e d e x p e r i m e n t a l l y , and s o l i d s o l u t i o n e f f e c t s are n e g l i g i b l e i n most m i n e r a l s . Most of the experimental work w i t h i n the t e r n a r y system was performed by Rankin and Wright (1915), but many r e v i s i o n s of phase boundaries and phase compositions ( p a r t i c u l a r l y on the C a O - A l 2 0 3 binary) have been proposed s i n c e t h e i r p i o n e e r i n g work. Table I summarizes the sources of the experimental data adopted as primary c o n s t r a i n t s f o r t h i s system. As a s t a r t i n g p o i n t f o r the f i t of l i q u i d u s r e l a t i o n s , the thermochemical p r o p e r t i e s of m i n e r a l s i n t h i s system were c o n s t r a i n e d through a c o n s i s t e n t set a n a l y s i s of s o l i d - s o l i d r e a c t i o n s determined e x p e r i m e n t a l l y as a f u n c t i o n of pressure and temperature. Table II l i s t s the r e a c t i o n s that were c o n s i d e r e d , as w e l l as the u n c e r t a i n t i e s adopted f o r these experimental data. C o n s t r a i n t s d e r i v e d from these r e a c t i o n s by i n c o r p o r a t i o n of estimated e r r o r s (Hammerstrom and Brown, 1982) were added to the melt c o n s t r a i n t s so that c o n s i s t e n c y with both s e t s of data was i n s u r e d . The e a r l i e s t attempts at f i t t i n g the melt data employed a t h i r d degree polynomial f o r the excess f r e e energy of the l i q u i d (two Margules parameters fo r each b i n a r y system, and one t e r n a r y parameter). I t q u i c k l y became apparent that a d d i t i o n a l f l e x i b i l i t y was needed i n order to do j u s t i c e to the l i q u i d u s data while m a i n t a i n i n g c o n s i s t e n c y with the thermodynamic Tab! e I: Source of Experimental Data f o r the System System Unary Systems S i O i - A l I O J CaO-AlzOi Source G r i e g (1927) Chase et a l . (1974) S c h a i r e r and Bowen (1955) Aksay and Pask (1975) Davis and Pask (1972) Nurse et a l . (1965) R o l i n and Thanh (1965) CaO-SiOi Rankin and Wright (1915) C a O - A l ! 0 3 - S i d Osborn and S c h a i r e r (1941) Osborn (1943) Welch and Gutt (1959) Gutt (1968) G r i e g (1927) Tewhey and Hess (1979) Rankin and Wright (1915) Osborn (1942) S c h a i r e r and Bowen (1947) Osborn and S c h a i r e r (1941) G e n t i l e and F o s t e r (1963) F i l o n e n k o and Lavrov (1949) React ions c r i s t o b a l i te=melt corundum=melt; lime=melt c r i s t o b a l i te=mu11i te corundum l i q u i d u s m u l 1 i t e 1 i q u i d u s a l l i n v a r i a n t p o i n t s i n F i g u r e 7 corundum=ca1ciurn hexaluminate c a l c i u m hexa1 urninate = c a l c i u r n d i a l u m i n a t e c r i s t oba 1 i t e = t r i dym'i t e t r i d y m i t e = p s e u d o w o l l a s t o n i t e a l p h a l a r n i t e = m e l t p s e u d o w o l l a s t o n i te=melt pseudowol1 a s t o n i t e = rank i n i t e a l p h a 1 a r n i t e = r a n k i n i t e a l p h a 1 a r n i t e = t r i c a 1 c i u r n s i l i c a t e 1 i m e = t r i c a l c i u m s i l i c a t e two 1i qu i d f i e l d two 1 i q u i d f i e l d a l l i n v a r i a n t p o i n t s i n F i g u r e 5 except f o r the f o l l o w i n g : a n o r t h i te=melt a n o r t h i te=pseudowol1astoni t e t r i dym i t e = a n o r t h i t e g e h l e n i t e = pseudowol1astoni t e a n o r t h i te=corundum=calc i urn hexa1 urni nate a n o r t h i te=corundum=calci um hexa1umi nate Tabl e 11: Experimental Data f o r S o l i d - s o l i d R e a c t i o n s React ion Data source A p p l i ed E r r o r s (1) g r o s s u l a r + q u a r t z = a n o r t h i t e + 2 w o l 1 a s t o n i t e : B o e t t c h e r (1970) 10' 100 b a r s Newton (1966) 10' 400 b a r s Huckenholz et a l . (1975) 10' 5% Windom and B o e t t c h e r (1976) 10' 1000 b a r s (2) 2 g r o s s u l a r = 3 w o l l a s t o n i t e + g e h l e n i t e + a n o r t h i t e : Huckenholz e t a l . (1975) 10' 5% (3) 3 a n o r t h i t e = g r o s s u l a r + 2 k y a n i t e + q u a r t z : H a r i y a and Kennedy (1968) 10' 5%t (4) g r o s s u l a r + corundum = g e h l e n i t e + a n o r t h i t e : Huckenholz et a l . (1975) 10' 5% B o e t t c h e r (1970) 10' 100 b a r s (5) k y a n i t e = s i l l i m a n i t e : R i c h a r d s o n et a l . (1968) 10' 100 b a r s (6 ) a n d a l u s i t e = s i l l i m a n i t e : Holdaway ( 1971) 10' 1 .5% (7) k y a n i t e = a n d a l u s i t e : R i c h a r d s o n et a l . (1969) 10' 100 b a r s Holdaway (1971) 10' 1 .5°/= (8) a l p h a q u a r t z = be t a q u a r t z : K e i t h and T u t t l e (1952) 5' Cohen and Klement (1967) 10' 2% (9) b e t a q u a r t z = t r i d y m i t e : Fenner (1913) 5' O s t r o v s k y (1966) 10' 2% (10) b e t a q u a r t z = be t a c r i s t o b a l i t e : Fenner (1913) 5" - O s t r o v s k y (1966) 10' 2% (11) w o l 1 a s t o n i t e = p s e u d o w o l 1 a s t o n i t e : Osborn and S c h a i r e r (1941) 10' (12) a l p h a 1 a m i te=al pha' l a r n i t e : B r e d i g (1950) 5' (13) a l p h a ' 1 a m i te=gamma l a r n i t e : B r e d i g (1950) 5' (14) a l p h a ' l a r n i t e = b e t a l a r n i t e : B r e d i g (1950) 5' t 29000 bar d a t a p o i n t moved an a d d i t i o n a l 1000 b a r s to a l l o w f o r k y a n i t e s t a b i l i t y w i t h r e s p e c t to corundum + q u a r t z 32 p r o p e r t i e s of minerals d e r i v e d from the s o l i d - s o l i d r e a c t i o n s . Table III l i s t s a l l phases which were c o n s i d e r e d in the f i t of s o l i d - s o l i d and l i q u i d u s data. Tables IV and V present the thermodynamic p r o p e r t i e s of a l l m i n e r a l s and the l i q u i d which r e s u l t from f i t t i n g the data with a f o u r t h degree Margules expansion. A d d i t i o n a l f l e x i b i l i t y r e s u l t s from having twelve mixing parameters f o r the melt phase, three f o r each binary and three w i t h i n the t e r n a r y system. Allowance f o r temperature dependence of the Margules parameters, i . e . W = W T'W , leads to 24 parameters f o r the t e r n a r y system. In the present f o r m u l a t i o n , there i s no excess heat c a p a c i t y (W =0). A. THERMOCHEMICAL PROPERTIES The f i r s t s tep i n s o l v i n g a l i n e a r programming problem i n v o l v e s . f i n d i n g values of a l l v a r i a b l e s that are c o n s i s t e n t with a l l c o n s t r a i n t s . Because i t can be d i f f i c u l t (and consequently expensive) to i d e n t i f y i n c o n s i s t e n c i e s in the c o n s t r a i n t s e t , c o n s e r v a t i v e estimates of u n c e r t a i n t i e s i n the c a l o r i m e t r i c data have been used. T h i s helps to ensure that i n c o n s i s t e n c i e s are avoided not only i n the c a l o r i m e t r i c data, but a l s o i n use of the c a l o r i m e t r i c data i n c o n j u n c t i o n with phase e q u i l i b r i a s t u d i e s of both s o l i d - s o l i d r e a c t i o n s and l i q u i d u s r e l a t i o n s . T h i s l a t t e r c o n s i d e r a t i o n i s e s p e c i a l l y important i f e q u i l i b r i a i n v o l v e phases which may a t t a i n d i f f e r e n t o r d e r i n g s t a t e s . I t must be emphasized that t h i s approach i s adopted i n t h i s study because the primary focus i s on l i q u i d u s r e l a t i o n s , not on d e r i v a t i o n of the t i g h t e s t c o n s i s t e n t set of mineral p r o p e r t i e s . 33 Table I I I : M i n e r a l s i n the System CaO-Al 20 3-Si0 2 M i n e r a l a-quartz 0-quartz 0 - t r i d y m i t e /3-cr i s t o b a l i t e Corundum Lime A n d a l u s i t e Kyanite S i l l i m a n i t e W o l l a s t o n i t e Pseudowollastonite R a n k i n i t e 0 - l a r n i t e 7 - l a r n i t e • ' — l a r n i t e a - l a r n i t e T r i c a l c i u m s i l i c a t e T r i c a l c i u m aluminate Calcium aluminate Calcium dialuminate Calcium hexaluminate A n o r t h i t e G e h l e n i t e Grossular A b b r e v i a t i o n Formula a-qz S i 0 2 0-qz S i 0 2 Tr i d S i 0 2 C r i s S i 0 2 Coru A1 20 3 Lime CaO Anda A l 2 S i O s Kyan A l 2 S i 0 5 S i l l A l 2 S i 0 5 Woll CaSi0 3 PsWo CaSi0 3 Rank C a 2 S i 3 0 8 0 ' l r Ca2SiO« 7 ' l r Ca2SiO« a ' l r Ca2SiO„ Larn Ca2SiO« c 3s, C a 3 S i 0 5 C 3A, C a 3 A l 2 O s C,A, C a A l 2 0 8 C,A2 • CaAl,0 7 C,A6 CaAl1 20, 9 Anor C a A l 2 S i 2 0 8 Gehl C a 2 A l 2 S i 0 7 Gros C a 3 A l 2 S i 3 0 34 Table IV: Thermochemical Values f o r Minerals and Li q u i d Oxides Mineral Beta t n d y m l t e Beta C r i s t o b a l I t e Beta quartz Alpha quartz Lime Corundum Pseudowollastonlte Wollastonlte RankInlte Alpha 1arnIte Alpha' 1arnIte Gamma l a r n l t e Enthalpy' Entropy' -909.46S 42.030 27.414 -907.48842.385 rhf 43.93040.42 rhf 27.41440.02 rbb -90B.522 43.625 27.381 -908.346+2.090 rhf 43.400±0.13 rhf 27.381*0.01 rbb -910.505 bb83 41.713 bb83 23.718 23.718±0.01 rbb -910.800 41.643 -910.70011.000 co78 41.46010.20 co78 -635.09010.900 co78 38.10010.40 co78 16.76410.00 rbb -1675.71111 .300 co78 50.92010. 10 co78 25.57510.01 rbb -1629.65012.594 rhf 87.45010.84 rhf 40.08010.14 rbb -1636.48011 .506 Ch78 82.01010.84 rhf 39.93010.10 rbb -3940.436 213.390 -3958.84712.092 ky62 210.87011.26 kk61 -2289.259 -2309.060 143.296 127.622 96.506 hrh 52.458 52.298 -2320.294 118.B40 59.110 -2318.61412.441 rhf 120.50010.84 rhf 59.11010.18 rbb Beta l a r n l t e T r i c a l c i u m s i l i c a t e -293 T r i c a l c i u m aluminate -3606.151 Calcium aluminate -2330.292 Calcium dlalumlnate -4007.413 Calcium hexalumlnate -10724.605 Andaluslte -2310.140 125.950 51.600 2308.06013.220 rhf 127.61010.84 rhf 51.60010.27 rbb -2943.382 166.020 2931.65440.920 ky62 168.61541.30 kk61 207.950 205.43011.26 kk61 72.742 hrh 115.348 114.22010.84 kk61 175.060 177.82011.30 kk61 -2585.711 93.332 51.590 2586.32512. 100 rhf S3.22010.42 rhf 51.58010.01 wg79 Kyanlte S1111mam te Anorthite Gehlenlte Grossular SIOi l i q u i d A1 101 1 1 qu1d CaO 1Iquld -2590.297 83.907 44.230 -2590.53011.900 rhf 83.76010.34 rhf 44.22010.01 W|j79 -2582.562 96.950 50.000 -2584.56011 .740 rhf 96.11010.42 rhf 50.05010.01 wg79 -4229.953 1B9.900 100.740 -4233.94041.297 nt80 199.30010.30 rhf lOO.79010.05 rbb -3984.014 206.520 90.330 -3980.92911 .297 ch81 209.BOOH.64 rhf 90.24010.09 rbb -6638.086 252.160 125.270 -6641.70942.803 Ch78 254.68041.26 WS79 125.30040.03 rbb -906.608 -1584.090 -534.114 45.535 52. COO 55.375 -Heat Capacity* 0 86.576 -1495314. 339.510 235.676 252.572 241.921 247.882 322.749 304.519 220.693 345.537 973.423 228.016 238.437 -1038737. -3384403. -5517950. -512804. -487461. 253.038 -2292825. 422.521 379.775 532.297 117.432 143.415 60.831 -2189052. -1405905. -B04B5B5. -1769104. -11333.60 -7421.32 -9587.00 -522.58 -1O067.91 -772.17 -12621.25 -1891.23 -2182.967 -1997.54 -2059.37 -672.40 -27883.55 -3135.46 -206.80 -63511. 12 -26116.87 -543.84 ' -23775.99 -1807.07 -3219.94 -2415.33 ' Enthalpy of formation from the elements at 298.15*C. 1 bar (K11OJoulea/mole) • T h i r d law entropy at 298.15'C, 1 bar (Joules/mole) » Volume In cc/mole , ^ „ . • Heat capacity equation (u/mole-K): Cp • A • CT-' • DT-'-' • FT-' (Berman and Brown 1983a) References: bb83 • Berman and Brown ( 1983a) / ch78 - Charlu et al . (1978) / co78 - COOATA Task Group (1978) ch81- Charlu et a l . (1981) / hrh - Haas et a l . (1982) / kk61 - Kelley.and King (1961) ky62 • K e l l e y (1962) / nt80 - Newton et a l . (1980) / rbb - Roble et al . (1965) rhf - Roble et a l . (1978) / w|)79 • Winter and Ghose(1979) / ws79 • Westrum et a). (1979) 35 Table V; Margules Parameters for Si0 2-Al 20 3-CaO Liquids System Parameter Wu(J/mol) n W (J/mol) s Si0 2-CaO SAAA SSAA SSSA (6) (3) (1) 63617.160 1642663.510 -106635.220 23.740 763.870 -28.130 Si0 2-CaO SCCC SSCC SSSC (9) (5) (2) -898692.640 -350208.150 -14081.800 -240.770 48.620 35.490 Al 20 3-CaO ACCC AACC AAAC (12) (11) (10) -455634.210 -725166.290 -240214.840 -2.470 -255.390 -26.700 Si0 2-Al 20 3-CaO SSAC SAAC SACC (4) (7) (8) -2847911.220 -2149042.640 209108.930 -1046.350 -641.840 313.360 S = SiO : A = A1 20 3 CaO Margules parameters are grouped with respect to subsystems of the ternary system. Numbers following each parameter designation show the order in which the parameters would be used i f generalized equations such as equation 8 were used to c a l c u l a t e the ternary excess free energy (for the component order SiO z, A l 2 0 3 f CaO). 36 As d i s c u s s e d i n the p r e v i o u s s e c t i o n , use of l i n e a r programming provides a r e g i o n of f e a s i b l e s o l u t i o n s . A unique s o l u t i o n can be achieved only i n c o n j u n c t i o n with o p t i m i z a t i o n of a given o b j e c t i v e f u n c t i o n . The r e s u l t s presented i n Tables IV and V were obtained by m i n i m i z a t i o n of the standard s t a t e enthalpy of w o l l a s t o n i t e , which i s s l i g h t l y l e s s negative than the c a l o r i m e t r i c value. A l l other e n t h a l p i e s were determined by the l i n e a r programming f i t , except f o r the those of a-quartz, lime, and corundum, which were c o n s t r a i n e d w i t h i n quoted c a l o r i m e t r i c u n c e r t a i n t i e s (Table I V ) . Molar volumes were p e r m i t t e d to vary w i t h i n s t a t e d u n c e r t a i n t i e s of the volume measurements. The volume of s i l l i m a n i t e was moved 0.04 cc o u t s i d e i t s quoted e r r o r bracket i n order t o s a t i s f a c t o r i l y reproduce the a l u m i n o s i l i c a t e phase diagram. Standard s t a t e e n t r o p i e s were t r e a t e d as bounded v a r i a b l e s with the width of b r a c k e t s equal to twice the quoted experimental u n c e r t a i n t i e s . Only the entropy of lime was changed beyond these c a l o r i m e t r i c l i m i t s ; the value adopted by Helgeson et a l . (1978) as being c o n s i s t e n t with low temperature data was found to a l s o improve the f i t of the l i q u i d u s d ata. With the exception of t r i c a l c i u m s i l i c a t e and r a n k i n i t e ( d i s c u s s e d below), heats of formation not c o n s t r a i n e d i n the f i t are i n e x c e l l e n t agreement with c a l o r i m e t r i c measurements. In p a r t i c u l a r , c a l c u l a t e d heats of formation of a n o r t h i t e , g e h l e n i t e , and g r o s s u l a r are very c l o s e to r e c e n t l y r e v i s e d c a l o r i m e t r i c estimates ( C h a r l u et a l . , 1978; Newton et a l . , 37 1980; C h a r l u et a l . , 1981) which are q u i t e d i s c o r d a n t with e a r l i e r v a l u e s ( c f . Robie et a l . , 1978). These r e s u l t s c l e a r l y demonstrate the u t i l i t y of t h i s approach f o r o b t a i n i n g thermodynamic p r o p e r t i e s of l i q u i d u s m i n e r a l s . T h i s c o n s i d e r a t i o n i s e s p e c i a l l y important because i t p r o v i d e s a t o o l to determine thermochemical p r o p e r t i e s of phases f o r which c a l o r i m e t r i c data are l i m i t e d or absent (compounds on the C a O-Al 20 3 b i n a r y , f o r i n s t a n c e ) . Heat c a p a c i t i e s of l i q u i d oxides are expressed i n the form of equation 32 with the sig n of the c o e f f i c i e n t s c o n s t r a i n e d as above (A>0; C, D, F<0). In the present f i t , CaO and A l 2 0 3 l i q u i d s have constant heat c a p a c i t i e s , while S i 0 2 l i q u i d shows a small temperature dependence above 1200 K (Figure 4). These r e s u l t s are compatible with a l l o b s e r v a t i o n s of s i l i c a t e l i q u i d s i n the s t a b l e and supercooled s t a t e (Richet and B o t t i n g a , 1980). Carmichael et a l . (1977) determined Cp f o r S i 0 2 i n multicomponent melts over the temperature range 1200-1650 K. T h e i r value (87 J/mole«K) i s very c l o s e to the average value (84.4 J/mole'K) c a l c u l a t e d f o r t h i s temperature range. Although t h e i r value f o r S i 0 2 l i q u i d i s w e l l bracketed by the composition of t h e i r melts (33-81 mole % S i 0 2 ) , other oxides are not as w e l l c o n s t r a i n e d , and W e i l l et a l . (1980b) p o i n t out d i f f i c u l t i e s with t h e i r Cp's f o r A l 2 0 3 and CaO l i q u i d s . The c a l c u l a t e d Cp f o r A l 2 0 3 (143.4 J/mole-K) i s s u b s t a n t i a l l y higher than the value (103 J/mole«K) determined by Carmichael et a l . (1977), but very c l o s e to the 146 J/mole*K value r e p o r t e d by W e i l l et a l . (1980b). The Cp f o r CaO (61 j/mole«K) i s s i g n i f i c a n t l y 38 160 1 4 0 -o E \ 1 2 0 -CO Q) D 1 0 0 -O 8 0 --t-> *o D Q_ 6 0 -D O D CD 40 20 H 1 I 4 0 0 . . i — i — i — i — i — i — i — i — i — r 8 0 0 1200 1600 2000 2 4 0 0 2 8 0 0 3200 Temperature (Kelvin) Figure 4; Heat c a p a c i t i e s of c r y s t a l l i n e and l i q u i d oxides. C o n s t r a i n t s placed on l i q u i d heat c a p a c i t i e s are discussed i n t e x t . 39 lower than values r e p o r t e d by Carmichael et a l . (1977) and W e i l l et a l . (1980b), 80.8 and 96 J/mole-K, r e s p e c t i v e l y . On an atomic b a s i s , however, these l a t t e r values (40.4 and 48 J/mole'K) are much higher than the Cp's r e p o r t e d f o r s i l i c a t e l i q u i d s , 29.2 J/atom«K ± 10% (Richet and B o t t i n g a , 1980). B. LIQUIDUS RELATIONS L i q u i d u s r e l a t i o n s i n the system C a O - A l 2 0 3 - S i 0 2 are d i s p l a y e d i n F i g u r e s 5-9. These phase diagrams were c a l c u l a t e d from the data i n Tables IV and V by computer programs which t r a c e i s o t h e r m a l s e c t i o n s and c o t e c t i c boundaries, while checking a l l p o i n t s on each curve f o r m e t a s t a b i l i t y (Brown and Perkins,,1983; Berman and Brown, 1983b). The p o s i t i o n of e x p e r i m e n t a l l y determined i n v a r i a n t p o i n t s are p l o t t e d with the c a l c u l a t e d phase r e l a t i o n s . A l l temperatures are based on the 1968 I n t e r n a t i o n a l P r a c t i c a l Temperature Scale (Preston-Thomas, 1976). D i f f e r e n c e s between the c a l c u l a t e d and e x p e r i m e n t a l l y determined temperatures f o r each i n v a r i a n t p o i n t i n F i g u r e 6 are i n d i c a t e d i n parentheses. In f i t t i n g the l i q u i d u s data and i n r e c a l c u l a t i n g the phase diagrams a l l m i n e r a l s were c o n s i d e r e d to be s t o i c h i o m e t r i c . The l i q u i d u s of m u l l i t e c o u l d not be f i t using e i t h e r the 3:2 or 2:1 A l 2 0 3 : S i 0 2 s t r u c t u r a l formula. Attempts to approximate i t s p o s i t i o n were u n s u c c e s s f u l and suggest h i g h l y n o n - i d e a l s o l i d s o l u t i o n p r o p e r t i e s f o r m u l l i t e . An a d d i t i o n a l c o m p l i c a t i o n i s the apparent i n c o n s i s t e n c y caused by the use of d i f f e r e n t experimental techniques f o r c o l l e c t i o n of b i n a r y and t e r n a r y 40 S i0 2 0.2 0 .4 0 .6 0 .8 CaO Weight Fraction A l 2 0 3 Figure 5: Calculated liquidus diagram for the system C a O - A l 2 0 3 - S i 0 2 , showing primary liquidus phases, phase boundaries, and isothermal sections at 100°C i n t e r v a l s . The shaded area represents the experimentally determined f i e l d of nonstoichiometric m u l l i t e . The calculated spinodal and estimated binodal curves are shown* 41 SiO 2 CaO Weight Fraction A l 2 0 3 Figure 6: Calculated liquidus diagram for the system CaO-Al 20 3-Si0 2. Triangles and squares mark compositions of experimentally determined invariant points and maxima, respectively. Temperatures in degrees Centigrade. For invariant points which are well constrained experimentally, the temperature differences (calculated-experimental) are shown in parentheses. The experimentally determined temperatures of the S i 0 2 and CaSiO„ polymorph t r a n s i t i o n s were treated as fixed values. Shaded area as in Figure 5. 42 3200 O 2800 H Ui CD CD ST 2400 CD _2 2000 -| D i _ CD C L Q5 1600 H 1200 Lime + Liquid Cristobalite + Tridymite + Liquid : — ^ — „ , Rankinitd Pseudowollustonitd + Liquid + Liquid 1 1 1 •flricalcium Silicate + Liquid 0.0 —r 0.2 —r 0.4 V 0.6 SiO 0.8 1.0 Weight Fraction CaO Figure 7; Calculated liquidus diagram for the system CaO-Si0 2. Symbols show the position of experimentally determined invariant points. The spinodal curve i s shown within the two-liquid f i e l d . 43 2200 H 2 0 0 0 H o CO CP CP i 1 8 0 0 -CP ~o 1 6 0 0 -CP 1_ D -»-' n l _ 1 4 0 0 -CP Q_ E CP 1— 1 2 0 0 -1000 0.0 A l 2 0 3 0 . 4 Weight Fract ion Figure 8: Calculated liquidus diagram for the system Al,0, SiO,. Symbols mark liquidus points located ^ P ^ i m e n ^ l y ; o s ^ t a s t a b l e f i e l d s of l i q u i d i m m i s c i b i l i t y and spmodal decomposition are shown. 44 3 2 0 0 0 .0 A l 2 ° 3 0 .2 0.4 0 .6 0 .8 Weight Fraction CaO Figure 9: Calculated liquidus diagram for the system CaO-Al 20 3. Symbols as in Figure 7. 45 l i q u i d u s data. Ternary l i q u i d u s data were ob t a i n e d by quenching techniques which favor the 2:1 s t o i c h i o m e t r y (Aramaki and Roy, 1962), while b i n a r y l i q u i d u s data was determined with s o l i d s t a t e d i f f u s i o n couples which favor the 3:2 s t o i c h i o m e t r y (Aksay and Pask, 1975). The e x p e r i m e n t a l l y determined m u l l i t e f i e l d has been shaded i n F i g u r e s 5 and 6 and covers the c a l c u l a t e d metastable e x t e n s i o n of the a n o r t h i t e f i e l d almost to the A l 2 0 3 - S i 0 2 b i n a r y . T h i s e x t e n s i o n of the a n o r t h i t e f i e l d i s due to the low temperature of the metastable b i n a r y e u t e c t i c between corundum and t r i d y m i t e . The r e p r o d u c t i o n of phase r e l a t i o n s i s e x c e l l e n t c o n s i d e r i n g the experimental u n c e r t a i n t i e s i n the l o c a t i o n of phase boundaries and the e f f e c t s of s o l i d s o l u t i o n i n l i q u i d u s m i n e r a l s . The maximum d i f f e r e n c e s between c a l c u l a t e d and ex p e r i m e n t a l l y determined i n v a r i a n t p o i n t s i n v o l v i n g s t o i c h i o m e t r i c m i n e rals are 17°C and 1.5 oxide weight per cent. Greater d i f f e r e n c e s occur at i n v a r i a n t p o i n t s which were ' l o o s e l y ' f i t because of s i g n i f i c a n t experimental u n c e r t a i n t i e s . Extremely high temperatures make the me l t i n g p o i n t s of lime, l a r n i t e , corundum, and t r i c a l c i u m s i l i c a t e d i f f i c u l t to a s c e r t a i n . The p l o t t e d i n v a r i a n t p o i n t compositions on the CaO- A l 2 0 3 b i n a r y are taken from Nurse et a l . (1965), but t h e i r exact l o c a t i o n i s s t i l l i n doubt c o n s i d e r i n g the lack of consensus with other i n v e s t i g a t o r s of t h i s system (Rankin and Wright, 1915; G e n t i l e and F o s t e r , 1963; R o l i n and Thanh, 1965). A s i m i l a r argument a p p l i e s to the p o s i t i o n of the c a l c i u m hexaluminate-corundum-anorthite i n v a r i a n t p o i n t ( G e n t i l e and 46 F o s t e r , 1963; F i l o n e n k o and Lavrov, 1949). Many i n v a r i a n t p o i n t s i n v o l v e minerals which are s l i g h t l y n o n - s t o i c h i o m e t r i c . Because no a c t i v i t y models for these phases have been i n c o r p o r a t e d i n t h i s study, l i q u i d u s s u r f a c e s f o r these phases are c a l c u l a t e d at lower temperatures than those determined e x p e r i m e n t a l l y . The maximum d e v i a t i o n i s 41°C at the t r i d y m i t e - p s e u d o w o l l a s t o n i t e b i n a r y e u t e c t i c . The c a l c u l a t e d lower temperature of t h i s e u t e c t i c , along with that of the p s e u d o w o l l a s t o n i t e - r a n k i n i t e e u t e c t i c , are e x p l i c a b l e by e a r l y work on the CaO-Si0 2 b i n a r y (Day and Shepherd, 1906) which i n d i c a t e d up to 2% excess S i 0 2 and CaO i n p s e u d o w o l l a s t o n i t e . 5 wgt. % C a A l 2 S i 2 0 8 i n c r i s t o b a l i t e , and 10 wgt. % S i 0 2 i n a n o r t h i t e (Longhi and Hays, 1979) can account f o r the higher temperature of the experimental pseudobinary e u t e c t i c , and, along with non-stoichiometry i n p s e u d o w o l l a s t o n i t e , f o r the s h i f t i n temperature and composition of the p s e u d o w o l l a s t o n i t e -t r i d y m i t e - a n o r t h i t e e u t e c t i c i n the t e r n a r y system. Reduction of the a c t i v i t y of a - l a r n i t e produced by 2-3 wgt. % A l 2 0 3 at 1400-1500°C (Lea, 1970) would improve the r e p r o d u c t i o n of the a l a r n i t e - g e h l e n i t e pseudobinary e u t e c t i c . The c a l c u l a t e d higher temperature of ' the a ' - l a r n i t e - r a n k i n i t e - g e h l e n i t e r e a c t i o n p o i n t , and the lower temperature of the a ' - l a r n i t e - c a l c i u m a l u m i n a t e - g e h l e n i t e e u t e c t i c can be r e c o n c i l e d with 1 wgt. % A 1 2 0 3 i n a ' - l a r n i t e at 1300-1350°C (Lea, 1970). A d d i t i o n a l non-s t o i c h i o m e t r y i n t h i s t e r n a r y system has been r e p o r t e d as A l 2 0 3 i n c r i s t o b a l i t e at the m u l l i t e - c r i s t o b a l i t e b i n a r y e u t e c t i c (Aramaki and Roy, 1962), 5-6 atom % S i 0 2 / ( S i 0 2 + A l 2 0 3 ) i n 47 t r i c a l c i u m aluminate (Lea, 1970), and 0.9 wgt. % A l 2 0 3 i n t r i c a l c i u m s i l i c a t e at 1500°C (Midgley and F l e t c h e r , 1962). C. DISCUSSION Although, t o t h i s p o i n t the problem can be c o n s i d e r e d p r i m a r i l y as a f i t t i n g e x e r c i s e , the value of the present approach l i e s i n the f a c t that t h i s f o r m u l a t i o n permits i n t e r p o l a t i o n between, and e x t r a p o l a t i o n beyond the experimental data, as w e l l as the c a l c u l a t i o n of a v a r i e t y of melt p r o p e r t i e s which transcend the nature of the o r i g i n a l phase e q u i l i b r i u m d a ta. F i g u r e 8 shows the complete A l 2 0 3 - S i 0 2 phase diagram which r e s u l t s from f i t t i n g o nly the corundum and c r i s t o b a l i t e l i q u i d i . A consequence of t h i s f i t i s the p r o d u c t i o n of a wide metastable t w o - l i q u i d f i e l d , r e f l e c t e d i n the sig m o i d a l shape of the c a l c u l a t e d corundum l i q u i d u s . T h i s occurrence i s not without experimental b a s i s , f o r s e v e r a l workers (e.g. Risbud and Pask, 1977; Jantzen and Herman, 1979; McPherson, 1980) have recognized evidence of i m m i s c i b i l i t y i n A l 2 0 3 - S i 0 2 g l a s s e s , although no consensus has been reached as to the c o m p o s i t i o n a l l i m i t s or even the number of s o l v i . A d d i t i o n of c a l c i u m oxide causes the r a p i d disappearance of t h i s metastable two l i q u i d f i e l d i n the t e r n a r y system. An i n t e r e s t i n g f e a t u r e r e s u l t i n g from t h i s f i t of l i q u i d u s r e l a t i o n s i s the h i g h l y asymmetric f r e e energy of mixing on the CaO-Si0 2 j o i n . The l a r g e negative excess f r e e energy of mixing i n CaO-rich compositions (compare the Margules parameters i n Table IV) i s suggestive of complexing i n the melt, which m i r r o r s 48 that observed i n the numerous l i q u i d u s compounds i n t h i s system (Hess, 1977). The CaO-rich p o s i t i o n of the t e r n a r y minimum in the f r e e energy of mixing is.due to the dominant e f f e c t of the CaO-Si0 2 mixing. Because the Margules f o r m u l a t i o n permits computation of the l i q u i d f r e e energy at any composition and temperature, phase r e l a t i o n s can be c a l c u l a t e d along any j o i n i n the C a O - A l 2 0 3 - S i 0 2 system. F i g u r e 10 i s an example of a pseudobinary phase diagram between the compositions of S i 0 2 and CaAl 20„. 1. Component A c t i v i t i e s Using equation 22, the a c t i v i t y c o e f f i c i e n t s , and hence the a c t i v i t i e s of a l l components can be c a l c u l a t e d at any temperature and composition. F i g u r e 11 shows contours of S i 0 2 l i q u i d a c t i v i t i e s i n t h i s t e r n a r y system at 1600°C. The i n f l e c t i o n s of contours r e f l e c t the presence of (metastable) immiscible l i q u i d f i e l d s extending i n t o the t e r n a r y system from the A l 2 0 3 - S i 0 2 and CaO~Si0 2 b i n a r i e s . A l s o p l o t t e d i n F i g u r e 11 are a c t i v i t y contours c a l c u l a t e d by Rein and Chipman (1965) on the b a s i s of measurements of the d i s t r i b u t i o n of s i l i c o n between Fe-Si-C a l l o y s and s l a g compositions (Rein and Chipman, 1963). Although no measurements were made at more than 50 weight per cent S i 0 2 at 1600°C, a c t i v i t i e s above 50 weight per cent S i 0 2 were estimated from p o i n t s of s i l i c a s a t u r a t i o n ( u n i t a c t i v i t y f o r c r y s t a l l i n e S i 0 2 ) taken from the c r i s t o b a l i t e 1600°C l i q u i d u s isotherm as drawn by Osborn and Muan (1960). The a c t i v i t i e s c a l c u l a t e d by Rein and Chipman (1965) have been r e f e r e n c e d to l i q u i d S i 0 2 through the 49 1800 1700 - ^ CO CP CD i 1600-cn CD 1500-CD u. D D 1400-CD a . E Te Te 1300-1200-f 0.0 S i0 2 0.4 Weight Fraction Figure 10; Calculated phase diagram for the pseudobinary system Si0 2-CaAl 20,. Ternary relations are approximate. 50 Figure 11; A c t i v i t y contours of S i 0 2 l i q u i d at 1600 C. Discrepancies between calculated ( s o l i d curves) and experimentally estimated (dashed curves) a c t i v i t i e s are discussed in the text. 51 Figure 12: Calculated a c t i v i t i e s of S i 0 2 and CaO l i q u i d s at and 1600UC. Experimentally determined a c t i v i t i e s at 1 (dashed curve) are shown for comparison. 52 calculated equilibrium constant for the reaction S i 0 2 ( c r y s t a l ) =  s i ° 2 ( l i q u i d ) a t 1600°C. Rein and Chipman (1965) quote an uncertainty of 0.13 in log a c t i v i t y values, but t h i s value i s probably an underestimate because of the large uncertainties in the thermodynamic data for SiC. A c t i v i t y contours, however, are derived s o l e l y from the s i l i c o n d i s t r i b u t i o n experiments, and are therefore independent of t h i s source of error. The calculated contours are in excellent agreement with Rein and Chipman's data, p a r t i c u l a r l y below X S i 0 = 0.5. At compositions richer in s i l i c a , the calculated contours are less steep than those drawn by Rein and Chipman. This divergence stems from f i t t i n g the c r i s t o b a l i t e -tridymite boundary as determined by Rankin and Wright (1915), whereas Rein and Chipman based th e i r s i l i c a - r i c h contours on the steeper sloped boundary curve drawn by Osborn and Muan (1960). The greatest discrepancy in calculated a c t i v i t i e s occurs on the CaO-Si0 2 binary, where measured S i 0 2 a c t i v i t i e s (Rein and Chipman, 1965) are systematically lower than those calculated in t h i s study. Expermimentally, a cross-over point between d i f f e r e n t temperature a c t i v i t y curves has been located at X s ± 0 = 0.53 (Rein and Chipman, 1965), very close to the ideal 'a = X' l i n e . The calculated a c t i v i t y curves for 1500°C and 1600°C cross at X S 1 Q = 0.42. This discrepancy suggests that, even though the l i q u i d free energies derived from f i t t i n g the phase e q u i l i b r i a are approximately correct, these data do not, in t h i s instance, adequately constrain the temperature dependence of the excess free energy on t h i s binary j o i n . 53 2. Heats Of Mixing Navrotsky et a l . (1982) have recently obtained heats of mixing along the j o i n Si0 2-Ca,. 5A10 2 from calorimetric measurements of glasses at 985 K. Their data, normalized to the jo i n Si0 2-CaAl 2O f t, shows a minimum around X S i 0^ = 0.55 and a tendency towards unmixing at S i 0 2 - r i c h compositions (Figure 13). Heats of mixing between c r y s t a l l i n e compounds along t h i s j o i n are also plotted and display a minimum around X S i 0 = 0.5. These measurements on glasses are not d i r e c t l y comparable to heats of mixing in l i q u i d s because of the lack of knowledge concerning the systematics of the glass to l i q u i d t r a n s i t i o n in compositions along t h i s j o i n . The f i t of the phase e q u i l i b r i a data in the system Ca0-Al 20 3-Si0 2 did not completely f i x heats of mixing along t h i s j o i n . A l l f i t s produced heats of mixing sim i l a r in magnitude to those measured on the glasses, with p o s i t i v e heats of mixing i n S i 0 2 - r i c h compositions and a minimum between X c, r t « 0.25-0.45. Compositions of Si0 2-poor glasses S1O2 were s l i g h t l y o f f the binary j o i n (Navrotsky et a l . , 1978), but, as shown in Figure 13, t h i s moves the position of the calculated minimum only s l i g h t l y . At present i t i s unclear whether t h i s s h i f t in the position of the minimum res u l t s from the use of a simple configurational entropy model, or whether i t r e f l e c t s a r e a l difference between mixing in glasses and l i q u i d s . As discussed by W e i l l et a l . (1980a), such a difference would imply excess heat ca p a c i t i e s of mixing or variations in the glass t r a n s i t i o n temperatures across the j o i n . No data are presently a v a i l a b l e to assess these p o s s i b i l i t i e s . 54 5 0 0 0 CO D O c 0 4 - 5 0 0 0 H - 1 0 0 0 0 H > N - 1 5 0 0 0 . c . - 2 0 0 0 0 L U - 2 5 0 0 0 L X Crystal + Glass 0 . 0 SiOn CaAI 2Si 2 0 6 CaO/AI20 3=1.25 CaAI 2Si 20 8 T T T 0 .2 0.4 0 .6 Mole Fraction T 0.8 1.0 CaAI 2 0 4 F i g u r e 13: C a l c u l a t e d heats of mixing i n l i q u i d s on the j o i n S i 0 2 - C a A l 2 0 , (CaO/Al 20 3=1.0) and f o r l i q u i d s w i t h CaO/Al 20 3=1.25. Heats of mixing f o r glasses and c r y s t a l l i n e compounds on the j o i n are taken from Navrotsky et a l . (1982). 55 3. E n t h a l p i e s Of Fusion Knowledge of the e n t h a l p i e s of f u s i o n of common minerals i s important i n understanding the energy budgets i n v o l v e d i n magmatic processes (e.g. Yoder, 1976). Heats of f u s i o n are a l s o r e q u i r e d as input data f o r thermodynamic c a l c u l a t i o n s of mineral-melt e q u i l i b r i a ( c f . B o t t i n g a and R i c h e t , 1980; Ghiorso and Carmichael, 1980). To date, very few measurements of heats of f u s i o n have been reported, and commonly heats of v i t r i f i c a t i o n , e x t r a p o l a t e d to the f u s i o n temperature, have been used as approximations. Recent measurements of the heats of f u s i o n of a n o r t h i t e ( W e i l l et a l . , 1980b) and di o p s i d e ( W e i l l et a l . , 1980a) have c l e a r l y demonstrated the inadequacy of t h i s approximation. With the present model the heat of f u s i o n at the f u s i o n temperature (T f) can be c a l c u l a t e d by T f Tf _ HTf "fusion liq u i d mineral (39) where mineral f,mineral J ^ >?!_„.,- + ( C p ) d I (40) and l i q u i d " I] X i K i + / ( C pi ) d TJ + Hexcess (41) i=l "298.15 56 which may be expanded to nc f. T l i q u i d - Z/*rli+ /(4)dT] i-1 ^298.15 nc-1 nc nc + J2 Yu... Z ) w H l i ^ v v - v , A 9 > V 1 V 1 ! v s - i 1 2 " p ( 4 2 > V 1 ! Only the heats of f u s i o n of the oxide components were c o n s t r a i n e d i n the f i t of the l i q u i d u s d a t a . The f i n a l v a l u e s (Table V) f o r S i 0 2 and A l 2 0 3 were kept w i t h i n the u n c e r t a i n t i e s of r e p o r t e d and measured v a l u e s , w h ile that f o r CaO was p e r m i t t e d to be s i g n i f i c a n t l y h i g h e r than the Chase e t a l . (1974) estimate based on the assumed s i m i l a r i t y between the f u s i o n e n t r o p i e s of p e r i c l a s e and l i m e . I f t h i s value were even l a r g e r i t would improve the f i t of l i q u i d u s r e l a t i o n s by c o u n t e r b a l a n c i n g the e f f e c t s of l a r g e n e gative d e v i a t i o n s from i d e a l i t y i n CaO-rich l i q u i d s . T able V shows c a l c u l a t e d heats of f u s i o n f o r other m i n e r a l s i n t h i s system. The c a l c u l a t e d v a l u e s f o r a n o r t h i t e and p s e u d o w o l l a s t o n i t e are i n e x c e l l e n t agreement with heats of f u s i o n r e c e n t l y determined by W e i l l et a l . (1980b) and Adamikovicova e t a l . (1980), r e s p e c t i v e l y . Where c a l o r i m e t r i c data are l a c k i n g , i t i s suggested t h a t t h i s model a f f o r d s a v a l u a b l e a l t e r n a t i v e f o r e s t i m a t i n g heats of f u s i o n of l i q u i d u s m i n e r a l s . T a b l e VI: Heats of F u s i o n Mi n e r a l T ( f u s i o n ) 1 B e t a - c r i s t o b a l i t e 1726 Corundum L i me Pseudowol1astoni t e A n o r t h i te G e h l e n i t e R a n k i n i t e ( * ) L a r n i t e 2032 3101 1550 1555 1590 1493 2205 T r i c a l c i u m s i l i c a t e ( * ) 2197 T r i c a l c i u m a l u m i n a t e ( * ) 1554 C a l c i u m a l u m i n a t e ( * ) 1613 C a l c i u m d i a l u m i n a t e 1776 C a l c i u m h e x a l u m i n a t e ( * ) 1860 H ( f u s i o n ) ' 12590 126855 122077 49852 140440 138369 88692 42514 36179 198020 88600 191544 677277 H ( f u s i o n ) - R e f e r e n c e 7531 Kracek (1930) 15062 Lumsden (1966) 10460 Holm e t a l . (1967) 9581 Chase e t a l . ( 1974) 8159 Robie e t a l . ( 1978) 107529t S h p i 1 ' r a i n e t a l . (1972) 118407f Fomichev (1973) 117152±8.368t Chase e t a l . ( 1974) 79496 Chase e t a l . ( 1974) 57321 Adamkovicova e t a l . (1980) 50208 Adams and Cohen (1966) 82843 Spencer (1973) 27405 Robie et a l . ( 1978) 120081 Adams and Cohen (1966) 166942t F e r r i e r (1969) 122173 Carmichael e t a l . (1975) 135562t Wei 11 et a l . (1980b) 1 C a l c u l a t e d f u s i o n temperature (degrees C e n t i g r a d e ) 1 C a l c u l a t e d heat of f u s i o n ( J o u l e s / m o l e ) 3 E s t i m a t e d heats of f u s i o n i n the l i t e r a t u r e ( J o u l e s / m o l e ) t E x p e r i m e n t a l l y determined v a l u e * I n c o n g r u e n t l y m e l t i n g mineral 58 IV. APPLICATION TO THE SYSTEM CA0-MG0-AL 20 3-SI0 2 The t h e o r e t i c a l b a s i s f o r a thermodynamic model f o r s i l i c a t e melts i s presented i n chapter I I , and i t s a b i l i t y to reproduce l i q u i d - s o l i d and l i q u i d - l i q u i d ( i m m i s c i b i l i t y ) r e l a t i o n s i n the sytem C a O - A l 2 0 3 - S i 0 2 i s demonstrated i n chapter I I I . T h i s chapter r e p o r t s the r e s u l t s of a p p l i c a t i o n of t h i s model to the quaternary system CaO-MgO-Al 20 3-Si0 2. I t w i l l be shown that the model allows the i n c o r p o r a t i o n of a l a r g e percentage of experimental data i n t o a s e l f - c o n s i s t e n t set of l i q u i d u s r e l a t i o n s i n t h i s system. In a d d i t i o n , the technique used to c a l i b r a t e the model permits the d e r i v a t i o n of thermodynamic p r o p e r t i e s of m i n e r a l s and l i q u i d s i n t h i s system, and can thus provide an a l t e r n a t i v e when hig h temperature c a l o r i m e t r i c data are not a v a i l a b l e f o r l i q u i d samples. A. METHODOLOGY As d i s c u s s e d i n chapters II and I I I , three types of data are used to c a l i b r a t e the model. L i q u i d u s data and t h e i r sources f o r the quaternary system CaO-MgO-Al 20 3-Si0 2 are d i s c u s s e d below. Both s o l i d - s o l i d phase e q u i l i b r i u m and c a l o r i m e t r i c data are used to ensure c o n s i s t e n c y of the thermodynamic p r o p e r t i e s of mi n e r a l s i n t h i s system. These data and t h e i r sources are summarized i n Tables VII and X. Each of these data s e t s have experimental u n c e r t a i n t i e s a s s o c i a t e d with them, and, i n the case of l i q u i d u s data, these u n c e r t a i n t i e s can range over an order of magnitude depending on the temperatures i n v o l v e d . Consequently, attempts to f i t these data u t i l i z i n g Table VII: Experimental Data for S o l i d Reaction (1) grossular + quartz • anorthite + 2 wollastonite: (2) 2 grossular • 3 wollastonite + gehlenite + anorthite: (3) 3 anorthite • grossular + 2 kyanite + quartz: (4) grossular + corundum » gehlenite + anorthite: (5) diopside + merwimte • wol lastonite + montlcel 1 i t e : (6) akermanlte • wollastonite + montlcel11te: (7) diopside • mo n t l c e l l i t e » f o r s t e r l t e + akermanlte: (8) akermanlte • diopside + merwlnlte: (9) kyanite • s i l l i m a n i t e : (10) andalusite » s i l l i m a n i t e : (11) kyanite • andalusite: (12) alpha quartz » beta quartz: (13) beta quartz • trldymlte: (14) beta quartz • beta C r i s t o b a l I t e : (15) wollaston1te=pseudowollastonite: (16) alpha 1arnlte«a1pha' l a r n i t e : (17) alpha' 1arn1te=gamma l a r n i t e : •sol id React ions Data source Boettcher (1970) Newton (1966) Huckenholz et a l . (1975) Wlndom and Boettcher (1976) Huckenholz et a l . (1975) Boettcher (1970) Harlya and Kennedy (1968) Goldsmith (1980) Huckenholz et a l . (1975) Boettcher (1970) Yoder (1968) Harker and Tuttle (1956) Yoder (1968) Walter (1963) Yoder (1968) Richardson et a l . (1968) Holdaway (1971) Richardson et a l . (1969) Holdaway (1971) Keith and Tuttle (1952) Cohen and Klement (1967) Fenner (1913) Ostrovsky (1966) Fenner (1913) Ostrovsky (1966) Osborn and Schalrer (1941) Bredig (1950) Bredig (1950) Bredig (1950) Applled Errors 10* 100 bars 10' 400 bars 10* 5% 10' 1000 bars 10* 5% 10' 100 bars 10' 5%t 10* 10% 10" 5% 10* 100 bars 10' 10% 10' 10' 10% 14' 10% 10' 10% 10* 100 bars 10' 1.5% 10* 100 bars 10' 1.5% 5* 10' 2% 5' 10" 2% 5' 10' 2% 10' 5' 5' 5' 60 r e g r e s s i o n techniques would r e q u i r e use of complex weighting f a c t o r s . Ghiorso et a l . (1983) has r e c e n t l y developed a r e g r e s s i o n package designed to circumvent some of these d i f f i c u l t i e s which produced numerical i n s t a b i l i t i e s i n the c a l i b r a t i o n of Ghiorso and Carmichael's (1980) r e g u l a r s o l u t i o n model. An a l t e r n a t i v e to u s i n g r e g r e s s i o n techniques to solve t h i s type of problem i s p r o v i d e d by l i n e a r programming, which i s a mathematical technique f o r s o l v i n g systems of l i n e a r i n e q u a l i t i e s . As d i s c u s s e d p r e v i o u s l y (chapter I I ) , each statement of equation 29 can be expressed as an i n e q u a l i t y , the sign of which depends on whether m i n e r a l or l i q u i d i s s t a b l e at the given set of experimental c o n d i t i o n s . W r i t t e n as i n e q u a l i t i e s , e x p l i c i t account can be taken of u n c e r t a i n t i e s i n c a l o r i m e t r i c data, as w e l l as experimental e r r o r s i n compositions and temperatures of l i q u i d u s data, and p r e s s u r e s and temperatures of s o l i d - s o l i d phase e q u i l i b r i u m d a t a . There are two other d i s t i n c t advantages to f i t t i n g l i q u i d u s r e l a t i o n s with l i n e a r programming. F i r s t , use of polynomials to approximate the excess f r e e energy of the l i q u i d s o l u t i o n a l l o w s f o r the appearance of i n f l e c t i o n s (with respect to composition) i n the f r e e energy s u r f a c e of the l i q u i d . Although i n f l e c t i o n s are necessary to produce l i q u i d i m m i s c i b i l i t y , they o f t e n a r i s e i n c o m p o s i t i o n a l p o r t i o n s of the system where c o n s t r a i n t s are l a c k i n g due to sparse experimental data. In such cases, ' a r t i f i c i a l ' c o n s t r a i n t s can be i n t r o d u c e d into- the l i n e a r programming problem to e l i m i n a t e these i n f l e c t i o n s . 61 The second advantage i s that c o n s t r a i n t s d e r i v e d from the l i q u i d u s p o s i t i o n of a n o n s t o i c h i o m e t r i c mineral can be i n c o r p o r a t e d i n the l i n e a r programming f i t without recourse to an a c t i v i t y model f o r the s o l i d . As d i s c u s s e d i n chapter I I , c a l c u l a t i o n of phase r e l a t i o n s assuming s t o i c h i o m e t r y should r e s u l t i n the l i q u i d u s o c c u r r i n g at lower than e x p e r i m e n t a l l y determined temperatures. The l i q u i d u s temperature can be r a i s e d t o that of the experimental value by accounting f o r the nonstoichiometry with an a c t i v i t y model. Thus, the ' l i q u i d s t a b l e ' c o n s t r a i n t i s s t i l l v a l i d f o r a n o n s t o i c h i o m e t r i c l i q u i d u s phase because the reduced a c t i v i t y of such a phase can o n l y r a i s e the temperature of i t s s t o i c h i o m e t r i c l i q u i d u s , not lower i t . U n l i k e r e g r e s s i o n , which minimizes r e s i d u a l s over an e n t i r e data set but which doesn't ensure c o n s i s t e n c y with estimated e r r o r s f o r each data p o i n t , the i n i t i a l s tep i n the l i n e a r programming a l g o r i t h m i n v o l v e s f i n d i n g a s o l u t i o n which i s c o n s i s t e n t with a l l c o n s t r a i n t s d e r i v e d from the data s e t . A unique s o l u t i o n i s then produced by o p t i m i z a t i o n of a given o b j e c t i v e f u n c t i o n . In the present problem of a n a l y z i n g a l a r g e data base composed of experimental r e s u l t s from d i f f e r e n t l a b o r a t o r i e s u s i n g d i f f e r e n t experimental techniques, there i s no s o l u t i o n c o n s i s t e n t with a l l the data at t h e i r quoted u n c e r t a i n t y l e v e l s . T h i s p o i n t can be a p p r e c i a t e d with r e f e r e n c e to F i g u r e 14, which shows the d i s t r i b u t i o n of v a r i o u s l e v e l s of consensus between d i f f e r e n t experimental determinations of i n v a r i a n t p o i n t 62 Temperature Differences 10 H CO z o CO tr o o u. o oc UJ CO 8H 6 4H 24 0-4»5-9'l0?4'l5-9'20-9'3059'>70 Composition Differences 0- '0.5-'1.0- '2.0->3.0 0.40 0.99 1.99 2.99 Figure 14: D i s t r i b u t i o n of temperature (°C) and composition (oxide wgt. %) differences between d i f f e r e n t experimental determinations of invariant point locations. Three of the four differences greater than 70°C are for binary invariant points greater than 2000°C. Because of inconsistencies discussed in the text, comparisons do not involve the data of Gutt (1963;1964) or El-Shahat and White (1965). 63 l o c a t i o n s . Although the frequency of f a v o r a b l e comparisons ( l e s s than 4°C) i s g r e a t e s t , approximately o n e - t h i r d of the comparisons i n v o l v i n g e x p e r i m e n t a l l y w e l l c o n s t r a i n e d i n v a r i a n t p o i n t s show temperature d i f f e r e n c e s between 10 and 20 degrees and over one oxide weight p e r c e n t . D i f f e r e n c e s of t h i s order of magnitude are i n c o n s i s t e n t with most estimates of experimental u n c e r t a i n t i e s , which might be expected to be 5°C and 0.5 oxide weight percent (e.g. Rankin and Wright, 1915). T h i s d i f f e r e n c e i s of course the d i f f e r e n c e between e s t i m a t i n g experimental p r e c i s i o n and experimental accuracy. I t i s the l a t t e r which i s of c r i t i c a l importance when using l i n e a r programming to achieve c o n s i s t e n c y with a data s e t , because i t can be q u i t e p r o b l e m a t i c , and t h e r e f o r e expensive, to i d e n t i f y the i n c o n s i s t e n t data p o i n t s . In the l i g h t of t h i s d i s c u s s i o n , and because s i m i l a r arguments apply to c a l o r i m e t r i c data and s o l i d - s o l i d phase e q u i l i b r i a data, the f o l l o w i n g procedure f o r c a l i b r a t i n g the model has been adopted. A l l c o n s t r a i n t s d e r i v e d from s o l i d -s o l i d r e a c t i o n s (Table VII) i n c o r p o r a t e d e r r o r s i n pressure and temperature equal to twice that quoted by the e x p e r i m e n t a l i s t s . Standard s t a t e volumes of a l l minerals were c o n s t r a i n e d w i t h i n s t a t e d u n c e r t a i n t i e s , while b r a c k e t s on standard s t a t e e n t r o p i e s were set at twice the s t a t e d u n c e r t a i n t i e s (exceptions are noted below). A l l standard s t a t e e n t h a l p i e s were determined by the l i n e a r programming f i t except f o r those of lime, p e r i c l a s e , corundum, and a-quartz, which were c o n s t r a i n e d w i t h i n the u n c e r t a i n t i e s estimated by CODATA (1978). T h i s procedure was 64 followed i n an attempt to a v o i d adopting thermodynamic p r o p e r t i e s of m inerals which were themselves i n c o n s i s t e n t . T h i s i s extremely important i n f i t t i n g the l i q u i d u s s u r f a c e of a l a r g e number of m i n e r a l s , because i n c o n s i s t e n c i e s between the f r e e e n e r g i e s of the l i q u i d u s phases can t r a n s l a t e i n t o d i s t o r t i o n s i n the l i q u i d ' s f r e e energy s u r f a c e . Melt r e l a t i o n s were f i t by adding c o n s t r a i n t s f o r m i n e r a l l i q u i d i at the l o c a t i o n of e x p e r i m e n t a l l y determined i n v a r i a n t (or p i e r c i n g ) p o i n t s . T h i s g r e a t l y reduced the number of c o n s t r a i n t s i n the problem over the number r e q u i r e d i f a l l l i q u i d u s b r a c k e t s were c o n s i d e r e d . In a d d i t i o n , i n v a r i a n t p o i n t temperatures can o f t e n be a s c e r t a i n e d with r e l a t i v e ease and p r e c i s i o n because experimental charges need not be prepared at i n v a r i a n t p o i n t compositions. As long as an adequate number of experimental charges are examined i n order to l o c a t e i n v a r i a n t p o i n t compositions, these c o n s t r a i n t s form the most e f f i c i e n t and r e l i a b l e data f o r f i t t i n g l i q u i d u s r e l a t i o n s . C onsistency was f i r s t achieved with l i q u i d u s data on b i n a r y and t e r n a r y systems which bound the quaternary composition space. E x p e r i m e n t a l l y w e l l c o n s t r a i n e d data were f i t with e r r o r b rackets of 20°C and 1 oxide weight percent (owp), while l a r g e r b rackets were a p p l i e d to high temperature or otherwise p o o r l y known data. T h i s f i r s t step in the c a l i b r a t i o n of the l i q u i d u s data p r o v i d e d that the model was w e l l behaved at the boundaries of the composition space, but not o v e r l y determined by p o o r l y c o n s t r a i n e d data such as high temperature m e l t i n g p o i n t s . The next c a l i b r a t i o n step i n v o l v e d adding the most r e l i a b l e 65 data from j o i n s w i t h i n the t e t r a h e d r o n , s e l e c t e d on the b a s i s of r e p o r t e d experimental procedures and p r e s e n t a t i o n of data. Only those data were used in which the e x p e r i m e n t a l i s t r e p o r t e d adequate p r e c a u t i o n s i n p r e p a r a t i o n of homogenous s t a r t i n g m a t e r i a l s ( r e q u i r i n g m u l t i p l e f u s i o n s and g r i n d i n g s ) , attainment of e q u i l i b r i u m , and c l o s e b r a c k e t i n g of i n v a r i a n t p o i n t compositions with experimental charges. In a d d i t i o n , only data were used that were presented i n such a way that the width of temperature brac k e t s around the l i q u i d u s c o u l d be assessed. Of the 28 c o m p o s i t i o n a l planes or j o i n s on which l i q u i d u s data had been c o l l e c t e d w i t h i n t h i s quaternary system, only seven, d i s c u s s e d below, met a l l these c r i t e r i a . I t i s the i n t e r s e c t i o n of the l i q u i d i of two or more min e r a l s ( i n a b i n a r y or higher component system) which determines the p o s i t i o n of an i n v a r i a n t p o i n t . Applying 20°C and 1 weight percent composition brackets to the l i q u i d u s of each m i n e r a l does not ensure that the i n v a r i a n t p o i n t w i l l a l s o be w i t h i n 20°C and 1 weight percent of the e x p e r i m e n t a l l y determined p o s i t i o n . For t h i s reason, the f i n a l c a l i b r a t i o n of the model i n v o l v e d f u r t h e r t i g h t e n i n g of l i q u i d u s b r a c k e t s i n order to more c l o s e l y reproduce e x p e r i m e n t a l l y determined i n v a r i a n t p o i n t s , while a l s o adding some ' a r t i f i c i a l ' c o n s t r a i n t s to remove d i s t o r t i o n s i n the l i q u i d u s s u r f a c e where l i t t l e or no data e x i s t s . T h i s l a s t stage of the c a l i b r a t i o n proved to be extremely time consuming, and c o u l d only be achieved by developing computer programs that e f f i c i e n t l y c a l c u l a t e and p l o t l i q u i d u s r e l a t i o n s i n b i n a r y , t e r n a r y , and 66 quaternary composition spaces (Berman and Brown, 1983b). 27 phases occur on the one atmosphere l i q u i d u s i n the quaternary system CaO-MgO-Al 20 3-Si0 2 (Figure 15, Table V I I I ) . The l i q u i d i and thermodynamic p r o p e r t i e s have been f i t f o r a l l phases except w o l l a s t o n i t e , m u l l i t e , and c o r d i e r i t e because of e x t e n s i v e s o l i d s o l u t i o n i n these phases. A p p r e c i a b l e s o l i d s o l u t i o n i n s p i n e l , pyroxenes, m e l i l i t e s , t r i c a l c i u m s i l i c a t e , f o r s t e r i t e , and m o n t i c e l l i t e l e d to c o n s i d e r a t i o n of the ' l i q u i d s t a b l e ' c o n s t r a i n t (as d i s c u s s e d above) f o r some e q u i l i b r i a i n v o l v i n g these phases. A l l other l i q u i d i were f i t assuming s t o i c h i o m e t r y of the l i q u i d u s phases. T h i s appears at present to be a reasonable approximation because the nonstoichiometry r e p o r t e d i n other phases in t h i s system (Table IX) i s very s l i g h t . Before a c t i v i t y models f o r l i q u i d u s phases can be i n c o r p o r a t e d i n t o the m o d e l l i n g of melt r e l a t i o n s i n t h i s system, f u r t h e r work i s necessary to s u b s t a n t i a t e and q u a n t i f y the data summarized in Table IX. In the next s e c t i o n the q u a l i t y of the f i t of t h i s model i s d i s c u s s e d with respect to 1) c a l c u l a t e d thermodynamic p r o p e r t i e s of l i q u i d u s phases, 2) r e p r o d u c t i o n of i n v a r i a n t p o i n t s on the c o m p o s i t i o n a l j o i n s used i n c a l i b r a t i o n , and 3) comparison of c a l c u l a t e d phase r e l a t i o n s on a l l other j o i n s on which experimental l i q u i d u s data has been c o l l e c t e d i n t h i s quaternary system at one atmosphere p r e s s u r e . 67 A l 2 0 3 CaO Weight Percent MgO F i g u r e 15: The q u a t e r n a r y C a O - M g O - A l 2 0 3 - S i 0 2 system showing phases f o r which t h e one atmosphere l i q u i d u s has been f i t , as w e l l as t h o s e phases ( i n p a r e n t h e s e s ) f o r which the l i q u i d u s was not f i t due t o e x t e n s i v e s o l i d s o l u t i o n . M i n e r a l s i n i t a l i c s a r e not s t a b l e on the e x p e r i m e n t a l l y d e t e r m i n e d one atmosphere l i q u i d u s , but apppear a s c o m p o s i t i o n a l end-members i n some e x p e r i m e n t a l s t u d i e s . A b b r e v i a t i o n of phases as i n T a b l e V I I I , and: M u l l = m u l l i t e , C o r d = c o r d i e r i t e , Pyro=pyrope, and CaTs=calcium"Tschermak's m o l e c u l e . 68 Table VIII: Minerals in the System CaO-MgO-Al 20 3-Si0 2 Mineral Abbreviation Formula a-quartz 0-quartz 0-tridymite 0 - c r i s t o b a l i t e Corundum Lime Pericl a s e Andalusite Kyanite S i l l i m a n i t e Wollastonite Pseudowollastonite Rankinite 0 - l a r n i t e 7 - l a r n i t e • » - l a r n i t e a - l a r n i t e Tricalcium s i l i c a t e Tricalcium aluminate Calcium aluminate Calcium dialuminate Calcium hexaluminate Protoenstatite F o r s t e r i t e Spinel Anorthite Gehlenite Grossular M o n t i c e l l i t e Merwinite Akermanite Diopside Sapphirine a-qz Tr i d Cr i s Coru Lime Peri Anda Kyan S i l l Woll PsWo Rank /T l r 7 ' l r a ' l r Larn C 3S, C 3A, C,A-, C,A2 C,A6 PrEn Fors Spin Anor Gehl Gros Mont Merw Aker Diop Saph S i 0 2 S i 0 2 S i 0 2 S i 0 2 A1 20 3 CaO MgO A l 2 S i O s A l 2 S i 0 5 A l 2 S i O s CaSi0 3 CaSi0 3 C a 2 S i 3 0 8 Ca 2SiO, Ca 2SiO, C a 2 S i 0 4 Ca 2SiO, C a 3 S i 0 5 C a 3 A l 2 0 6 CaAl 20, C a A l 4 0 7 C d A 1 1 2 ^ 1 9 MgSi0 3 Mg 2Si0, MgAl 20, C a A l 2 S i 2 0 8 C a 2 A l 2 S i 0 7 C a 3 A l 2 S i 3 0 , 2 CaMgSiO, Ca 3MgSi 20 8 Ca 2MgSi 2 0 7 CaMgSi 2 0 6 M g , A l 1 0 S i 2 0 2 3 Table IX: Nonstolchlometry in Liquidus Phases In the CaO-MgO-Al>0i-S10. System Nonstolchlometry <i wgt% C a A l . S tIOI Mi neral T M d y m l t e « - c n s t o b a l i te System S l O t - C a A l t S l i O i Mul11te (-) L1me P e r l c l a s e W o l l a s t o n i t e (-) a - l a r n l t e a'-larnl te T r i c a l c i u m S i l i c a t e T r i c a l c i u m Aluminate Protoenstat1te F o r s t e r l t e Spine l A n o r t h l t e Gehlen i t e C o r d t e n t e (-) Akermanlte Montlce l11te Merwlnlte 01ops 1de S10>-A1>0i S 1 0 . - C a A l . S 1 . 0 . A l . 0 , - S 1 0 i CaO-MgO CaO-MgO MgO-Mg.S10. MgO-MgAl.0. Ca0-Mg0-S10> CaSIOi-CaMgSIiO. CaO-Al-t0i-S10i C a . S I O . - M g . S l O . CaO-Al1O1-SIO. C a . S I O . - M g . S l O . CaQ-Al .0 , -S10. CaO-MgO-SIO. C a O - A l . O i - S 1 0 i MgSIO,-CaMgSI . 0 . Mg.SlO.-CaMgSI.0 . Mg,S10.-MgO M g . S l O . - M g A l • • . Mg.SIO.-CaMgSIO. Al >0.-MgAl.0. MgAl,0.-MgO MgAl .0 . -Mg.S10 . CaAliSIIO.-SIO. C a . A l . S 1 0 , - C a . M g S I . 0 . MgO-Al .0 . -S10 . C a . A l . S 1 0 . - C a . M g S 1 . 0 . CaMgS10.-Mg.S10. Ca.MgSI.0.-CaMgSIO. CaMgSI>0f-S10> CaMgSI.O.-Mg.SIO. CaMgSI.0.-CaS10. CaMgSI.0 . -Ca.MgSI.0 . CaMgSI .0 . -CaAl .S1 .0 • CaMgSI.0.-MgSIO. excess A1.0 . 5 wgtX C a A l , S I . 0 . excess A 1.0. and S10. 17 wgtX MgO at 2370'C 7.8 wgtX CaO at 2370'C 11 molX Mg.SlO. 18 wgt% A l . 0 . 6 wgtX CaO at 1550'C 22 wgtX CaMgSIiO» 2-3 wgtX A l . 0 . 5 mol% Mg.SlO. at 1630*C 1 wgt% A l . 0 . 3 molX Mg.SlO. at 140O'C 0.9 wgtX A l . 0 . 2 wgtX MgO at 1500'C 5-6 atomX S1/(S1+A1) 2.8 wgtX CaMgSI.0. at 1400'C smal1 X CaMgSI>0> O.S mol% MgO 0.5 mo1% MgAl.O. 30 wgtX CaMgSIO. 86 molX A l . O i 11 wgtX MgO 5 molX Mg.SlO. 10 wgt% S10. 73 wgtX Ca.MgSI.0 . excess SI . A l , Mg 37 wgtX C a . A l . S 1 0 . 30 wgtX Mg.SlO. 10 molX CaMgSIO. at 1485*C 2-3 wgtX S10. 5 wgtX Mg.SlO. smal I X CaSIO, 5 wgtX Ca.MgSI.O. 40 molX C a A l . S I . 0 . 32 wgtX MgSIO> Reference Longhl and Hays (1979) Aramakl and Roy (1962) Longhl and Hays (1979) Aksay and Pask (1975) Doman et a l . (1963) Doman et a l . (1963) Schlaudt and Roy (1965) Alper et a l . (1962) H a t f i e l d et a l . (1970) Scha lrer and Bowen (1942) Lea (1970) Schlaudt and Roy (1966) Lea (1970) Schlaudt and Roy (1966) Mldgley and F le t cher (1962) Mldgley and F l e t c h e r (1962) Lea (1970) Kushlro (1972) Kushlro and Scha lrer (1963) Schlaudt and Roy (1965) Schlaudt and Roy (1965) Rlcker and Osborn (1954) Roy et a l . (1953) Alper et a l . (1962) Schlaudt and Roy (1965) Longhl and Hays (1979) Osborn and Scha lrer (1941) Shreyer and Scha lrer (1961) Osborn and S c h a l r e r (1941) Rlcker and Osborn (1954) Schlaudt and Roy (1966) Schalrer and Kushlro (1964) Kushlro and Scha lrer (1962) Scha lrer and Bowen (1942) Kushlro and Scha lrer (1964) de N e u f v l l l e and S c h a l r e r (1962) Kushlro (1972) ON (-) Liquidus not f i t due to extensive s o l i d s o l u t i o n 70 B. RESULTS 1. C a l o r i m e t r i c Data Phases which were considered i n the f i t of s o l i d - s o l i d and l i q u i d u s data are presented i n Table V I I I . Table X l i s t s the c a l c u l a t e d thermodynamic p r o p e r t i e s of phases in t h i s system along with c a l o r i m e t r i c a l l y measured values f o r comparison. As mentioned above, a l l volumes are w i t h i n quoted u n c e r t a i n t i e s of measured v a l u e s , and e n t r o p i e s w i t h i n twice the quoted u n c e r t a i n t i e s with the f o l l o w i n g e x c e p t i o n s . The entropy of lime was i n c r e a s e d to a value c l o s e r to that found by Helgeson et a l . (1978) to be more c o n s i s t e n t with low temperature e q u i l i b r i a . C a l c u l a t e d e n t h a l p i e s are i n very • good agreement with c a l o r i m e t r i c data, e s p e c i a l l y the recent measurements of the e n t h a l p i e s of a n o r t h i t e (Newton et al.,1980) and akermanite (Charlu et a l . , 1981). General agreement of c a l c u l a t e d and measured val u e s f o r p s e u d o w o l l a s t o n i t e , e n s t a t i t e , and s a p p h i r i n e , phases c o n s t r a i n e d only by l i q u i d u s data, i s p a r t i c u l a r l y encouraging i n so f a r as t h i s demonstrates that t h i s technique can be used to a c c u r a t e l y p r e d i c t the thermodynamic p r o p e r t i e s of minerals from the p o s i t i o n of t h e i r l i q u i d i . A l s o l i s t e d i n Table X are the c a l c u l a t e d thermodynamic p r o p e r t i e s of the l i q u i d oxide components. These v a l u e s , along with the Margules parameters i n "Table XI, allow c a l c u l a t i o n of any p o i n t on the f r e e energy s u r f a c e of the melt phase i n t h i s 71 Table X: Thermodynamic Properties of Minerals and Liquid Mineral Enthalpy Entropy 1 a-quartz -910.800 41.260 -910 .700+1 .000 C078 41 .46+0.20 C078 a-quartz -910.505 41.330 a-trldymite -909.466 41.647 -907 .488±2.385 RHF 43 .9310.42 RHF a - c r l s t o b a l i t e -908.522 43.242 -909 .346+2.090 RHF 43 .4010.13 RHF Corundum -1674.411 51.020 -1675 . 700± 1 . 300 C078 50 .9210.10 C078 Lime -635.990 39.750 -635 .090±0.900 C078 38 .1010.40 C078 P e r i c l a s e -601.200 27.250 -601 .500±0.300 C078 26 .9510.15 C078 Andalusite -2585.738 93.041 -25B6 .325+2.100 RHF 93 .22+0.42 RHF Kyanite -2590.355 83.551 -2591 .844±1.255 SK73 83 .76+0.33 RHF S i l l I m a n i te -2582.407 96.835 -2584 .35411.088 SK73 96 .11+0.42 RHF Pseudowollastonite -1626.774 89.130 -1629 .823±1.297 CH78 87 .4510.84 RHF Wollastonlte -1634.038 83.690 -1636 . 595±1 .506 CH78 82 .0110.84 RHF Rank m i t e -3949.425 213.390 -3958 .685+2.092 KY62 210 .8711.26 KK61 a - l a r n l t e -2295.194 143.302 a'-larnlte -2314.995 127.621 7 - l a r n i t e -2326.229 118.840 -2319 .59611.925 RHF 120 .50+0.84 RHF a - l a r n l t e -2316.075 125.950 -2309 .09410.962 RHF 127 ,6110.84 RHF T r i c a l c i u m S i l i c a t e -2947.307 167.138 -2931 .654+0.920 KY62 168 .5311.26 KK61 T r i c a l c i u m Aluminate -3615.680 202.910 205 .4311.26 KK61 Calcium Aluminate -2336.751 112.540 114 .2210.84 KK61 Calcium Dlalumlnate -4006.668 175.060 177 .6511.30 KK61 Calcium Hexalummate - 10735.301 360.980 Protoenstat1te -1546.097 68.659 -1548 .29010.711 CH75 67 .7810.84 KY43 F o r s t e r i t e -2176.254 93.910 -2175 .86511.130 CH75 94 .1110.10 RB82 Spinel -2307.588 81.420 -2300 .45412.343 SK73 80 .5810.42 KK61 Anorthite -4231.480 199.900 -4231 .39111.297 NTBO 199 .3010.30 RHF Gehlenlte -3988.273 206.520 -3980 .92711.297 CH81 209 ,8311.64 RHF G r o s s u l a r l t e -6644.066 252.160 -6639 .81612.803 CH78 254 ,6811.26 WS79 M o n t l c e l l I t e -2252.623 113.000 -2263 .30110.586 KY62 110.46 HG78 Merwinite -4557.988 248.950 -4569 .34011.590 KY62 253 .1312.09 RHF Akermanlte -3871.440 211.809 -3871 .15511.130 CH81 209 .3312.09 RHF Diopslde -3203.382 144.760 -3205 .39911.715 CH78 143 .0910.84 KK61 Saphirine - 12778.270 443.220 -12785 .86316.527 CH75 SiOt l i q u i d -904.927 45.OOO A1>0> l i q u i d -1548.481 102.785 CaO l i q u i d -554.897 26.228 MgO l i q u i d -491.967 63.685 Oxides in the CaO-MgO-A1.0i-S10. System Volume* Heat Capacity* a c d f 23.464 79 . 166 -229130. -9587 .OOO BB83 23.718 78 . 190 -810399. -7421 . 324 23 .7210.01 RBB 27.414 86 .576 -1495314. -434 . 328 27 .4110.02 RBB 27.381 82 . 196 -11333 .598 27. .3810.01 RBB 25.575 152 .724 -852486. -522 .577 -10067 .910 25. .5710.01 RBB 16.764 57 .076 -403034. -3103 .658 16. .7610.00 RBB 11 .200 59 .682 -136 .496 -4172 .625 11 .2510.00 RBB 51.570 228 .016 -512804. -206 .798 -26116 .871 51 . . 53+0.04 RBB 44.160 238 .437 -487461. -543 .844 -23775 .992 44. .0910.07 RBB 49.856 253 .038 -2292825. -1807 .074 49. .9010.04 RBB 40.240 138 .583 -15533 .477 40. .0810. 14 RBB 40.030 135 .382 -1038737. -11457 .000 39. .9310.10 RBB 339 ,510 -3384403. -772 . 167 -12621 .254 52.458 235 .676 -1891 .230 52.298 252 .572 -2182 .967 59.110 241 .921 -1997 .545 59. .1110.18 RBB 51.600 247 .882 -2059 ,370 51 . .6010.27 RBB 72.742 HRH 322 ,749 -2409 . 159 -3383 .006 304 .519 -28244 .695 220 .693 -1372400. -1461 .313 345 .537 -1199855. -672 .399 -27883 .555 973 .423 -5517950. -3135 .460 -63511 . 117 32.385 156 .998 -943 .737 -6038 .262 32. .3810.08 RBB 43.759 237 .819 -599845. -1954 .035 43. .7910.03 RBB 233 .493 -1674612. -1704 .549 39. 7110.03 RBB 100.740 422 .521 -2189052. -3219 .938 100. .7910.04 RBB 90.150 379 .775 -1405905. -2415. 327 -5271 .992 90. .2410.09 RBB 125.330 532 .297 -8048585. -34185 .965 125. .3010.03 RBB 51.433 197 .586 -1255520. -136 496 -18127 .250 51 . .3610.07 RBB 103.400 453 .374 -1500225. -3180. 631 104 4011.00 RBB 92.722 371. .073 -1728635. -2410. 887 92. 8110.09 RBB 66.190 310. .775 -1356. 450 -19586. 809 66. 0910.10 RBB 1164. .005 -4262430. -3158. 867 -88732. OOO 99. .5010.73 RBB 28.380 121 . .630 -1444. 383 25.575 154. .820 -677. 889 -8071, 355 16.764 76 044 16.764 60. .428 -206. 533 -2674. 372 ' Enthalpy of formation from the elements at 298.15*C, 1 bar (K1lojoules/mole) * T h i r d law entropy at 298.15'C, 1 bar (Joules/mole) ' Volume i n cc/mole * Heat capacity equation (d/mole-K): Cp • A + CT * * + DT-°-» + FT" 1 (Berman and Brown, 1983a) Rererences: BB83 » Berman and Brown (1983a) / CH75 • Charlu et a l . (1978) / CH78 • Charlu et al , (1978) CH81 - Charlu et a l . (1981) / C078 « CODATA Task Group (1978) / HG78 • Helgeson et a l . (1978) HRH » Haas et a l . (1982) / KK61 • Kelley and King (1961) / KY43 • Kelley (1943) KY62 • K e l l e y (1962) / NT80 - Newton et a l . (1980) / RB82 - Roble et a l . (1982) RBB « Roble et a l . (1965) / RHF • Roble et a l . (1978) / SK73 « Shearer and Kleppa (1973) WG79 • Winter and Ghose(1979) / WS79 • Westrum et a l . (1979) 72 Table XI XI: Margules Parameters for the CaO-MgO-Al (1) (4) (10) (2) (7) (16) (3) (9) (19) (20) (22) (25) (21) (24) (28) SSSA SSAA SAAA SSSC SSCC SCCC SSSM SSMM SMMM AAAC AACC ACCC A A AM AAMM AMMM (29) CCCM (30) CCMM (31) CMMM Wi Binary Parameters -161039.81 1803871.61 258911.07 -25525.64 -341962.81 -960867.88 94145.91 -270581.70 -610906.87 -197743.47 -734020.10 -617537.61 -641890.87 727706.30 -691193.17 318144.87 590616.72 114076.93 58.66 ,Oj-Si0 2 System (J/mole) Ternary Parameters -60.41 844.79 110.47 (5) (11) (13) SSAC SAAC SACC -2685775.05 -2833471.13 580678.70 -917.87 -976.80 526.17 34.19 56.62 -247.11 (8) (17) (18) SSCM SCCM SCMM -1143506.91 -2464803.70 -2026666.90 -279.90 -669.00 -555.03 52.77 -59.98 -195.01 (23) (26) (27) AACM ACCM ACMM 343546.79 -2440837.52 -3334297.25 160.77 -526.78 -1148.10 -1.11 -251.94 -76.83 (6) (12) (15) SSAM SAAM SAMM -1828080.99 -3201173.35 -1828080.99 -693.44 -1382.29 -693.44 -224.47 290.88 -227.94 (14) SACM Quaternary 2179011.74 Parameter 1328.50 154.79 322.37 CaO M - MgO A - A1,0, S SiOj system. Numbers preceding «» c J.P? " 3 e 5 ?%; " rJ l i»ed equations such as the component order SiO,, A l 2 0 J r CaO, MgO). 73 system. 2. Phase R e l a t i o n s On C a l i b r a t i o n J o i n s For the 69 e x p e r i m e n t a l l y w e l l d e f i n e d i n v a r i a n t p o i n t s on these j o i n s , the average d i f f e r e n c e s between c a l c u l a t e d and experimental determined i n v a r i a n t p o i n t s i s 0.24±14.5°C and 0.52±0.55 owp (the ± r e f e r s to one standard d e v i a t i o n ) . C a l c u l a t e d i n v a r i a n t p o i n t s , comparisons with e x p e r i m e n t a l l y determined l o c a t i o n s , and the sources of experimental data are l i s t e d i n Tables XII and XIII f o r b i n a r y and t e r n a r y systems, r e s p e c t i v e l y . F i g u r e s 16-25 are c a l c u l a t e d b i n a r y and t e r n a r y phase diagrams which give v i s u a l comparisons of c a l c u l a t e d and e x p e r i m e n t a l l y determined phase r e l a t i o n s . A l l phase diagrams have been c a l c u l a t e d and p l o t t e d by computer programs which t r u n c a t e metastable extensions while f o l l o w i n g c o t e c t i c and i s o t h e r m a l s e c t i o n s (Berman and Brown, 1983b). F i g u r e s 16-21 show phase r e l a t i o n s on the s i x b i n a r y edges of the quaternary system. Symbols mark the p o s i t i o n s of e x p e r i m e n t a l l y determined i n v a r i a n t p o i n t s . Large u n c e r t a i n t i e s are a s s o c i a t e d with the p o s i t i o n of many i n v a r i a n t p o i n t s because of the experimental d i f f i c u l t i e s i n h e r e n t i n c o n t r o l l i n g , measuring, and d e f i n i n g r e f e r e n c e p o i n t s i n extremely high temperature phase e q u i l i b r i u m s t u d i e s . Large temperature b r a c k e t s have thus been allowed at the m e l t i n g p o i n t s of lime, p e r i c l a s e , l a r n i t e , t r i c a l c i u m s i l i c a t e , f o r s t e r i t e , and s p i n e l . In a d d i t i o n , the temperatures and compositions of e u t e c t i c s between s p i n e l and p e r i c l a s e ( F i g . 20) and between p e r i c l a s e and lime ( F i g . 21) are very p o o r l y Table XII : Binary Invariant Points in the CaO-MgO-StO.-Al .0. System Calcu la ted Composition 1 Dif ferences ' Phases Tc Td SIOi A l , Oi CaO MgO S10. Al .0 i CaO MgO Svstem: CaO-Al .Oi C>Ai Lime 1545 5 42. 9 57. 1 1 . 9 -1 . 9 5 0. 1 - 0 . 1 C i A i C A , 1345 -17 50. 0 50.0 - 0 . 7 0. 7 C A . - C i A , 1635 39 63. 3 36.7 - 3 . 2 3. 2 32 - 0 . 7 0. 7 C 1 A 1 - C 1 A . 1764 -13 80. 7 19.2 1 . 7 -1 . 7 0 2. 7 - 2 . 8 C i A . - C o r u 1805 -98 86. 2 13.8 - 4 . 3 4. 3 -27 2. 2 - 2 . 2 System: CaO-SIOi Larn C1S1 2091 -3 30.2 69.7 - 0 . 3 0. 2 Llme -CiS, 2224 110 26.7 73.3 -1. 8 1 . 8 30 -1. 8 1 . 8 Larn-Rank 1474 -4 44.5 55.4 0. 0 - 0 . . 1 7 - 0 . 2 0. , 1 PsWo Rank 1451 -7 45.4 54.6 - 0 . 1 0. . 1 -12 0. 4 - 0 . .4 Tr1d PsWo 1411 -28 61 .9 38. 1 -1. 1 1 . , 1 PsWo 1553 7 51 .7 48.3 Solvus 1686 -23 73.8 26.2 1. 4 -1 . .4 Solvus 1686 -8 73.8 26.2 0. 2 - 0 .2 Solvus 1686 -23 97.7 2.3 -1. 7 1 .7 Solvus 1686 -8 97.7 2.3 - 0 , 6 0 .6 Svstem: MgO-S10i C r l s PrEn 1547 0 63.8 36. 2 -1. . 2 1 . 2 Per! Fors 1901 28 38.2 61 . 8 0. .2 - 0 . 2 Fors -PrEn 1556 -5 60. 1 39. .9 - 0 . .9 0. 9 Fors 1932 17 42.7. 57. .3 Solvus 1675 -31 98.0 2. .0 -1 .2 1 . 2 Solvus 1675 -31 70.4 29. .6 0 .9 - 0 . .9 Svstem: CaO-MgO Lime Per l 2114 -245 53.4 46 .5 13 .6 13 .6 -260 13 .6 13 .5 System: MgO-AI.Oi Coru Spin 1787 -265 92 .6 7 .4 -5 .4 5 .4 Perl Spin 2040 -28 49 .4 50 .6 -5 .6 5 .6 42 -5 .6 5 .6 Spin 2145 -35 71 .7 28 .3 Svstem: A l . O i - S i O i i S i l l Coru 1353 90.0 9 .9 Reference Rankin and Wright (1915) Nurse et a l . (1965) Nurse et a l . (1965) Rankin and Wright (1915) Nurse et al . (1965) R o i m and Pham ( 1965) Nurse et a l . (1965) R o i m and Pham (1965) Nurse et a l . (1965) Welch and Gutt (1959) Welch and Gutt (1959) Gutt (1968) Rankin and Wright (1915) Osborn (1943) Rankin and Wright (1915) Osborn (1943) Rankin and Wright (1915) G r i e g (1927) Tewhey and Hess (1979) Grieg (1927) Tewhey and Hess (1979) Bowen and Anderson (1914) Bowen and Anderson (1914) Bowen and Anderson (1914) Bowen and Anderson (1914) Grieg (1927) Grieg (1927) Rankin and Merwln (1916) Doman et a l . (1963) Rankin and Merwln (1916) Rankin and Merwln (1916) Alper et a l . (1962) Rankin and Merwln (1916) Tc = C a l c u l a t e d Invariant point temperature (*C) Td » Di f ference between c a l c u l a t e d and experimental ly determined Invariant point temperatures (ca lculated-experimental ) 1 Ca lcu la ted Invariant point composition (oxide weight percent) ' Di f ference between c a l c u l a t e d and experimental ly determined Invariant point compositions (ca lculated-experimental ) - Phase ( fo l lowing the dash) 1n reac t ion r e l a t i o n s h i p with l i q u i d 75 Table X I I I : Ternary Invariant Points I n the Ca0-Mg0-S10i-Al1O1 System C a l c u l a t e d Composit ion' D i f f erences ' Phases Tc Td S I O . A1 .0 . CaO MgO S10. A l . 0 . CaO MgO System: C a O - A l i b . - S I O i Lime C S , C A , 1490 16 8. 3 32. 2 59. 4 C1A1 C . S i Larn 1445 -12 9. 9 32. 0 58. 1 C1A1 a' l r -•Gehl 1375 -7 9. 8 41 . 3 49. 0 Gehl C A , CiA< 1512 3 10. 4 51. 8 37. 8 C A , Gehl C A . 1482 3 22. 4 47. 0 30. 6 C A . Gehl Anor 1383 0 32. 0 37. 9 30. 1 Gehl • •a' l r Rank 1371 33 38. 8 1 1 . 3 49. 9 Anor C . A . - •Coru 1432 -64 32. 3 39. 9 27. 8 25 Anor Gehl PsWo 1275 7 41 . 6 20. 9 37. 5 Rank Gehl PsWo 1325 12 40. 4 12. .2 47. 4 PsWo Anor Tr1d 1 174 2 60 .3 15. .3 24. .4 C1A1 Gehl 1537 -19 14 6 51 . .0 34. .4 C A , Gehl 1537 -9 14 .6 51 .0 34 .4 Larn Gehl 1528 -21 26 .7 23 .6 49 .7 Larn Gehl 1528 -21 26 .7 23 .6 49 .7 Coru Anor 1547 -4 41 .6 39 .0 19 .3 PsWo Anor 1316 14 g 47 .4 18 .5 34 . 1 PsWo Gehl 1328 10 40 .5 14 .0 45 .5 Anor Gehl 1384 9 -3 32 .7 36 .9 30 .4 Anor T M d 1366 5 -4 72 .4 17 .8 9 .8 Anor 1550 -8 43 .2 36 .7 20 . 1 Gehl 1588 -7 21 .9 37 .2 40 .9 0. 8 -0 . 6 -0 . 3 1 . 2 -1 . 0 -0 . 2 0. 1 -0 . 7 0. 7 1 . 1 -1 . 5 0. 4 -1 . 9 2. 5 -0 . 6 0. 2 -1 . 1 0. 9 -1 . 1 -0 . 6 1 . 7 -3 . 7 -1 . 1 4 . 8 -0 . 0 -0 . 1 0. 1 -0 . 4 0. 9 -0. 5 -0. 6 0. 4 0. 2 -1 . 7 0. 5 1. . 1 0. 4 0. 2 -0 6 -0. .2 0. 4 -0 .2 -0. .0 -0. 0 0 . 1 0. .5 -1 . 3 0 .8 0 . 2 -0. 3 0 .0 0 . 1 -0. . 1 0 .0 -0 .2 0. .9 -0 .7 -0 .6 0. 8 -0 .2 -0 .3 0 .4 -0 . 1 -0 .3 0 . 1 0 .2 2 .4 -1 .7 -0 .7 0 .4 -0 .2 -0 .2 System: Ca0-Mg0-S10. R a n k - a ' l r Aker Merw-Larn-Per1 Aker Rank PsWo Aker Mont-Per l Aker-Mont-D1op-D1op Aker-C S , Dlop PsWo Fors T M d Merw P e r l -Larn Mont Pen Fors PrEn •Merw P e r l Aker Aker D1op D1op Mont Merw C S , Fors Fors PrEn T M d Larn L1me Fors D1op Aker Aker Merw Aker Dlop v PrEn-Fors Saph Saph Fors-Sp1n Saph PrEn T r l d Saph-Coru-Sp1n Spin Fors Per l Spin Fors T M d Saph-Coru Coru T M d Saph Saph T r l d Coru L1me PeM C A , C s A i PeM C i A i C A , P e M - S p i n Spin C i A i - C i A t C A , Spin C A , C A . Spin Coru 1392 2 43. 9 51 . 2 4. 8 -0.4 10 -0.4 1561 -19 38. 1 44. 0 17. 9 -0 .7 1367 -12 45. 4 50. 3 4. 3 0.9 -11 0.8 1457 18 42. 7 40. 2 17. 0 0.0 1493 -8 40. 1 37. 0 22. 9 -0 .3 1736 -72 28. 4 63. 2 8. 4 -0.1 1444 5 43. 9 33. 8 22. 3 -0 .5 1483 -22 40, .9 31 . 0 28. 1 -0 .6 1360 -25 58. .0 17. 8 24. 2 -0 .5 1343 -30 61 . .2 1B. 0 20. 8 -1 .2 1427 14 43. ,5 50. 3 6. 2 -0.2 1782 -89 26 . 2 63. 6 10. 2 1 .2 1352 -7 50 .5 31 . 3 18. .3 0.5 1394 -8 47 .9 44 . .7 7. .4 0.5 1385 -4 54 .2 23. .5 22. ,3 0.2 1353 -11 61 .7 22 2 16 .0 -0 .9 -20 -0.4 1352 -11 51 .9 30 .7 17 .4 1 . 1 1459 10 43 .3 42 .2 14 .5 0.2 1463 2 44 . 1 41 . 1 14 .8 1390 -3 55 .5 25 .9 16 .6 System: MgO-Al tl 0.-S10. 1318 52 .4 24 .2 23 .3 1335 51 .4 24 .5 24 .0 1245 58 .8 23 . 1 18 . 1 1490 47 .3 35 .5 17 .2 1684 -26 28 .3 18 . 1 53 .6 0.4 2 -28 -0 .5 -1 1699 31 .4 18 .9 49 .7 1307 63 .9 23 .9 12 .2 1312 75 .6 17 .8 6 .6 1312 85 .8 11 .4 2 .7 System: CaO-MgO - A l . 0 . 1461 8 42 .0 51 .2 6 .7 -0 1299 49 . 1 46 .8 4 . 1 1356 -16 50 .0 42 .7 7 .3 -2 1578 23 60 .6 35 .2 4 .2 -2 1726 36 79 .2 17 .7 3 . 1 5 1753 87 .9 7 .2 4 .9 Reference Rankin and Wright (1915) Rankin and Wright (1915) Rankin and WMght (1915) Rankm and WMght (1915) Rankm and WMght (1915) Rankm and WMght (1915) Rankin and WMght (1915) Fllonenko and Lavrov (1949) Gent i l e and Foster (1963) Rankm and WMght (1915) Rankm and Wright (1915) Rankin and WMght (1915) Rankm and WMght (1915) Rao (1968) Rankin and Wright (1915) Pr ince (1951) Rankm and WMght (1915) Rankin and WMght (1915) Osborn (1942) Rankin and WMght (1915) Osborn (1941) Rankm and WMght (1915) Rankin and WMght (1915) Schairer and Bowen (1947) •1.4 Ferguson and Merwln (1919) •0.9 Osborn (1943) •0.3 Osborn (1943) •2.0 Ferguson and Merwln (1919) •1.3 Osborn (1943) •1.3 Ferguson and Merwln (1919) 0.6 Ferguson and Merwln (1919) •5.6 Rlcker and Osborn (1954) -0.0 Ferguson and Merwln (1919) 1.7 Ferguson and Merwln (1919) 0.6 Kushlro (1972) 0.2 Kushlro (1972) -0.6 Osborn (1943) -2.8 Rlcker and Osborn (1954) -2.0 Ferguson and Merwm (1919) -1.0 Schairer and Bowen (1942) -0.7 Bowen (1914) 0.4 Bowen (1914) 0.2 Schairer and Kushlro (1964) 0.4 Kushlro and Schairer (1964) 0. 1 Osborn (1943) Ferguson and Merwm (1919) Ferguson and Merwm (1919) -2.4 Rankin and Merwln (1918) 2.4 Osborn and Muan (1960) 0.2 0.6 Rankin and Merwln (1916) 2.0 0.4 Rankm and Merwm (1916) 1.9 0.7 Rankin and Merwln (1916) -3 .3 -1.8 Rankin and Merwtn (1916) -0.3 5.7 1.5 0.5 0.5 0.4 0.2 -1 .5 -0.2 Tc = C a l c u l a t e d Invariant point temperature C O Td « Di f f erence between c a l c u l a t e d and experimental ly determined i n v a r i a n t po int temperatures (ca lcu la ted-exper imenta l ) • C a l c u l a t e d Invariant point composit ion (oxide weight percent) ' D i f f erence between c a l c u l a t e d and experimental ly determined Invariant point compositions (ca lcu lated-exper imenta l ) - Phase ( f o l l owing the dash) i n r e a c t i o n r e l a t i o n s h i p with l i q u i d 76 3 2 0 0 2 8 0 0 -O o D -»-' D 2 4 0 0 H g_2000-E (D 1 6 0 0 - ^ 1 2 0 0 0 . 2 0 . 4 0 .6 0 .8 S i0 9 Weight Fraction 1.0 CaO Figure 16t Calculated phase diagram of the system CaO-Si0 2. Symbols mark the position of experimentally determined invariant points. Points on the binodal curve determined by Grieg (1927) (crosses) and Tewhey and Hess (1975) ( t r i a n g l e s ) . 77 2 8 0 0 2 6 0 0 H 0 .2 0 .4 0 .6 0 .8 SiOo Weight Fraction MgO Figure 17; Calculated phase diagram of the system MgO-Si0 2. Symbols as in Figure 16. 78 2200 2000 H o o V—s Q) 1800-i _ D -t-' O i _ Q) CL 1600-E Te 1400-1200 A l 2 0 3 0.4 0.6 0.8 Weight Fraction SiO Figure 18: Calculated phase diagram of the system A l 2 0 3 - S i 0 2 . Symbols are experimentally determined liquidus points. 79 3 2 0 0 0 .0 A l 2 0 3 0 .2 0.4 0 .6 0 . 8 Weight Fraction 1.0 CaO Figure 19: C a l c u l a t e d phase diagram of the system CaO-Al 20 3. Symbols as i n Figure 16. I n v a r i a n t p o i n t s determined exp e r i m e n t a l l y by Nurse et a l . (1965) (squares) and R o l i n and Thanh (1965) ( t r i a n g l e s ) . 8Q 2800 Figure 2 0 ; C a l c u l a t e d phase diagram of the system MgO-Al Symbols as i n Figure 16. 81 3 2 0 0 3 0 0 0 H 2 8 0 0 H ¥ 2 6 0 0 2 4 0 0 H CD i _ D •4-* D L - 2 2 0 0 CD D_ E 2 0 0 0 CD ! — 1 8 0 0 Lime + I Peri + Lime Peri + I CaO 0 .8 Weight Fraction 1.0 MgO F i g u r e 21: C a l c u l a t e d phase diagram of t h e system CaO-MgO. Symbols as i n F i g u r e 16. 82 known, the l a t t e r having been l o c a t e d by l a r g e e x t r a p o l a t i o n from a lower temperature CaO-MgO-Al 20 3 i n v a r i a n t p o i n t . The other major experimental u n c e r t a i n t i e s i n the b i n a r y systems occur on the CaO-Al 20 3 j o i n . A consensus seems to have been reached concerning the s t o i c h i o m e t r y of phases on t h i s j o i n , but disagreement s t i l l remains with respect to the s t a b i l i t y f i e l d s of the more aluminous phases on t h i s j o i n (Nurse et a l . , 1965; R o l i n and Thanh, 1965; Rao, 1968). Well documented excess A l 2 0 3 i n s p i n e l accounts f o r the much lower c a l c u l a t e d temperature of the spinel-corundum e u t e c t i c ( F i g . 20). Nonstoichiometry i n m u l l i t e prevented f i t t i n g i t s l i q u i d u s , and the e x p e r i m e n t a l l y determined s t a b i l i t y f i e l d i s shown c o v e r i n g metastable e q u i l b r i a i n v o l v i n g corundum, t r i d y m i t e , and s i l l i m a n i t e i n F i g u r e 18. For the 11* w e l l c o n s t r a i n e d i n v a r i a n t p o i n t s i n these b i n a r y systems, the average d i s c r e p a n c y between c a l c u l a t e d and e x p e r i m e n t a l l y determined i n v a r i a n t p o i n t s i s -2.6±15.9°C and 0.37±.44 owp. These d i s c r e p a n c i e s are w i t h i n the l i m i t s of experimental e r r o r s with the exce p t i o n perhaps of the e u t e c t i c between t r i d y m i t e and p s e u d o w o l l a s t o n i t e ( F i g . 16). Any s o l i d s o l u t i o n i n ps e u d o w o l l a s t o n i t e r e p o r t e d o r i g i n a l l y by Day and Shepard (1906), but l a t e r d i s c r e d i t e d by Bowen et a l . (1933), would improve the r e p r o d u c t i o n of t h i s i n v a r i a n t p o i n t . The t e r n a r y system C a O - A l 2 0 3 - S i 0 2 i s the best s t u d i e d of the t e r n a r y faces of the t e t r a h e d r o n , and the c a l c u l a t i o n s f o r t h i s system have p r e v i o u s l y been presented (chapter I I I ) . The present v e r s i o n of t h i s system, r e v i s e d through c o n s i d e r a t i o n of 83 Figure 22: Calculated liquidus diagram of the system Ca0-Al 20 3-Si02. Lighter l i n e s are liquidus isotherms with a contour i n t e r v a l of 200°, s t a r t i n g with 1300°C. Squares and tri a n g l e s represent the location of experimentally determined invariant points and maxima, respectively. The calculated spinodal and estimated binodal curves are shown. Temperatures of binary invariant points are shown in Table X. The shaded area represents the experimentally determined f i e l d of nonstoichiometric mu l l i t e . The calculated metastable s t a b i l i t y f i e l d of s i l l i m a n i t e i s not shown. 84 Figure 23t C a l c u l a t e d l i q u i d u s diagram of the system CaO-MgO-Si0 2. L i g h t e r l i n e s are l i q u i d u s isotherms with a contour i n t e r v a l of 200°, s t a r t i n g with 1500°C. Symbols as i n Figure 22. The p o s i t i o n s of the s p i n o d a l and b i n o d a l curves w i t h i n the t e r n a r y system are estimated by i n t e r p o l a t i o n between the c a l c u l a t e d b i n a r y p o s i t i o n s . 85 Figure 24; Calculated liquidus diagram of the system CaO-MgO-Al 20 3. Liquidus isotherms as in Figure 23. Symbols as in Figure 22. 86 SiO MgO 0.2 0 .4 0.6 0.8 Weight Fraction A l 2 0 3 Figure 25; Calculated liqu i d u s diagram of the system MgO-Al 20 3-Si0 2. Liquidus isotherms as in Figure 22. Shaded areas represent the experimentally determined f i e l d s of mullite and c o r d i e r i t e , which cover the metastable coexistence of tridymite + corundum, and the lens-shaped f i e l d of saphirine at high s i l i c a contents. The calculated metastable f i e l d of s i l l i m a n i t e i s not shown. The p e r i c l a s e - f o r s t e r i t e - s p i n e l eutectic i s estimated by Rankin and Merwin (1916) (triangle) and Osborn and Muan (i960) (diamond). 87 quaternary data, i s shown i n F i g u r e 22. The agreement with the experimental data i s e x c e l l e n t , with the average d e v i a t i o n of 20 i n v a r i a n t p o i n t s being 2.2±13.1°C and Q..57 + .54 owp. The maximum d e v i a t i o n s are 33°C and 2.5 owp. The MgO-CaO-Si0 2 system i s more d i f f i c u l t to f i t because nonstoichiometry i n many l i q u i d u s phases f o r c e s c o n s i d e r a t i o n of only the ' l i q u i d s t a b l e ' c o n s t r a i n t ( d i s c u s s e d above). The main d i f f e r e n c e s i n t r o d u c e d i n the s t o i c h i o m e t r i c diagram (Figure 23) are the pronounced shrinkage of the s t a b i l i t y f i e l d of t r i c a l c i u m s i l i c a t e , the s h i f t i n the p o s i t i o n of the m o n t i c e l l i t e - f o r s t e r i t e - p e r i c l a s e i n v a r i a n t p o i n t , and the lower temperatures of i n v a r i a n t p o i n t s i n v o l v i n g two pyroxenes. These p o i n t s a s i d e , the average d i s c r e p a n c y f o r the remaining 17 te r n a r y i n v a r i a n t p o i n t s i s -4.7±11.0°C and 0.57±.51 owp, while the maximum di s c r e p a n c y ' i s 20°C and 1.95 owp. The t e r n a r y system CaO-MgO-Al 20 3 (Figure 24) i s dominated by the phase volumes of p e r i c l a s e , s p i n e l , and lime. Only three i n v a r i a n t p o i n t s are w e l l d e f i n e d e x p e r i m e n t a l l y , and the average d e v i a t i o n f o r the c a l c u l a t e d p o i n t s i s 4.8±20.0°C and 1.31±0.9 owp. An e u t e c t i c between 12CaO«7Al 20 3 ( o r i g i n a l l y i d e n t i f i e d as 5 C a O « 3 A 1 2 0 3 ) , lime, and p e r i c l a s e l o c a t e d by Rankin and Merwin (1916), has s i n c e been shown t o be metastable i n m o i s t u r e - f r e e atmospheres (Nurse et a l . , 1965). Rankin and Merwin a l s o estimated the p o s i t i o n of a r e a c t i o n p o i n t between s p i n e l , corundum, and 3CaO*5Al 20 3 by e x t r a p o l a t i o n of bi n a r y phase r e l a t i o n s . The s t o i c h i o m e t r y of ca l c i u m aluminate phases has s i n c e been r e v i s e d , and the c a l c u l a t e d i n v a r i a n t p o i n t at a 88 s i m i l a r temperature but more aluminous composition i n v o l v e s the phases s p i n e l , c a l c i u m d i a l u m i n a t e , and c a l c i u m hexaluminate. Phase r e l a t i o n s i n the M g O - A l 2 0 3 - S i 0 2 system (Figure 25) are v i s u a l l y very d i f f e r e n t from the e x p e r i m e n t a l l y determined phase diagram because e x t e n s i v e s o l i d s o l u t i o n prevented f i t t i n g the l i q u i d i of c o r d i e r i t e and m u l l i t e . The s t a b l e p o r t i o n of the f i e l d of s a p p h i r i n e i s i n agreement with the work of K e i t h and S c h a i r e r (1952); most of i t s l a r g e s t a b i l i t y f i e l d i n the c a l c u l a t e d diagram i s rendered metastable by c o r d i e r i t e , and, at higher s i l i c a compositions, by m u l l i t e . S o l i d s o l u t i o n i n s p i n e l accounts f o r the l a r g e r s t a b i l i t y f i e l d of corundum in the c a l c u l a t e d diagram. Phase diagrams f o r the c a l i b r a t i o n j o i n s w i t h i n the t e t r a h e d r o n are shown in Appendix I. Table XIV summarizes the r e s u l t s of the c a l i b r a t i o n with r e s p e c t to these j o i n s . The average d i f f e r e n c e s f o r the 18 i n v a r i a n t p o i n t s on these j o i n s i s 3.7±7.7°C and 0.37±.28 owp. These r e s u l t s and those presented i n F i g u r e s 16-25 i n d i c a t e r e p r o d u c t i o n of phase r e l a t i o n s i n the bounding b i n a r y and t e r n a r y systems and the seven quaternary c a l i b r a t i o n j o i n s which are w e l l w i t h i n reasonable estimates of experimental u n c e r t a i n t i e s . 3. Phase R e l a t i o n s On N o n - c a l i b r a t i o n J o i n s P r i n c e (1954) presents l i q u i d u s data f o r 102 compositions on the 10 weight percent MgO plane of the system, and estimates the compositions and temperatures of p i e r c i n g p o i n t s on t h i s plane. T h i s data was not i n c l u d e d i n the c a l i b r a t i o n because onl y approximate widths of the l i q u i d u s b r a c k e t s are given, even 89 though they are reported to be small (4°C). Figure 26 shows, however, the excellent agreement between the calculated phase relations and Prince's estimated piercing point compositions. Also shown are the differences between calculated and experimentally determined liquidus temperatures averaged over the number of experimental charges studied within the s t a b i l i t y f i e l d of each phase. These calculations represent the data in the poorest possible l i g h t , because they assume no experimental errors in the composition of the charges. The magnitude of t h i s effect i s proportional to the T-X gradient of the liquidus of each phase. Tridymite, for example, has a large gradient with liquidus temperatures which are approximately 50°C lower for every one weight percent decrease of S i 0 2 . The only major discrepancy in the calculated diagram occurs at the piercing points l a b e l l e d A and B, where, in contrast to the model cal c u l a t i o n s , Prince's data show spinel to be present along with anorthite, mullite, and corundum. These differences are due to the calculated liquidus of spinel being up to 70°C lower than indicated by Prince in several compositions near points A and B. Spinel displays pronounced s o l i d solution towards A l 2 0 3 on the A l 2 0 3 - S i 0 2 binary (Roy et a l . , 1953), but there are no data available to judge whether the present discrepancy can be accounted for by nonstoichiometry at these lower temperatures. A c t i v i t y models for gehlenite and akermanite are needed to raise their liquidus temperatures to those observed experimentally for m e l i l i t e s . The 5 weight percent MgO plane of the quaternary system has 90 SiO # Phase Av +/-0.2 0 .4 0 .6 0.8 CaO Weight Fraction A l 2 0 3 Figure 26: Calculated liquidus diagram of the 10 weight percent MgO plane of the CaO-MgO-Al 20 3-Si0 2 system. Lighter l i n e s are liqui d u s isotherms with a contour i n t e r v a l of 200°, s t a r t i n g with 1300°C. Squares represent experimentally estimated p i e r c i n g point compositions. Triangles are calculated compositions^ of maxima. The f i e l d of l i q u i d i m m i s c i b i l i t y at high S i 0 2 i s not shown. Shown for each experimentally determined l i q u i d u s f i e l d are the number of experimental charges examined, the average differences between calculated and experimentally determined liquidus temperatures (Av), and the standard deviations of these differences ( + ). 91 Table XIV: Comparison of C a l c u l a t e d and Experimental Invariant Points on C a l i b r a t i o n J o i n s Tc C a l c u l a t e d Compositions 1 Td S i 0 2 A1 20, CaO MgO D i f f e r e n c e s * S i 0 2 A l j O , CaO MgO System; Fors Enst Anor 3 Anor Fors Spin* Anor Enst T r i d Spin Anor Spin Fors System: Diop PsWo Anor Diop PsWo Anor System: Diop Fors Anor Anor Fors Spin System: Coru Spin* Coru Anor* Gehl Spin C,A, C,A ( Gehl Anor System: Spin P e r i a ' l r a ' l r Spin System: Spin Gehl a ' l r Spin Gehl C a A l 2 S i 2 0 , - M g 2 S i 0 . - S i 0 2 (Anderson, 1915) 1261 1313 1212 1443 1467 •11 0 -12 -4 -2 54.0 20.0 11.0 14.9 4B.7 22.2 12.2 16.9 61.0 18.8 10.3 43.1 31.1 17.1 43.0 20.4 9.9 8.7 11.2 25.5 -0.1 -0.1 -0.1 -0.0 -0.5 -0.3 -0.9 0.3 0.1 -0.0 -0.4 -0.2 0.0 0.6 0.3 0.3 0.8 0.5 0.6 -0.9 C a S i O j - C a M g S i 2 0 ( - C a A l 2 S i 2 0 | (Osborn, 1942) 1245 -2 49.4 14.6 30.8 5.2 -0.0 0.5 -1.0 0.5 1245 6 49.4 14.6 30.8 5.2 -0.2 0.7 -0.1 -0.4 Ca A l 2 S i 2 0 s - C a M g S i 2 0 ( - M g 2 S i 0 , (Osborn and T a i t , 1952) 1272 -10 49.1 16.0 21.3 13.7 -0.1 0.0 -0.2 0.2 1318 -1 46.2 20.3 17.7 15.8 0.4 -0.9 0.3 0.2 MgAl 2 0 , - C a A l 2 S i j O . - C a j A l j S i O , (DeVries and Osborn, 1957) 1569 11 34.3 43.9 16.0 5.9 1.0 -0.8 0.5 -0.7 1569 1526 1393 1376 7 5 -1 40.5 38.9 18.9 1.8 0.4 -0.3 30.4 38.8 29.3 1.5 -0.4 -0.2 32.2 37.5 29.8 0.5 -0.1 -0.3 0.2 -0.3 0.8 -0.2 0.6 -0.2 Mg0-Ca 2Si0,-MgAl 20« (Prince, 1951) 1406 -13 22.5 24.8 42.0 10.7 -0.4 1407 -13 22.8 24.9 42.5 9.8 -0.1 0.9 0.1 • •0.8 -0.1 M g A l 2 0 . - C a 2 S i 0 . - C a 2 A l 2 S i 0 T (Prince, 1951> 1406 -10 22.8 24.9 42.6 9.6 -0.1 0.1 -0.3 1529 -2 19.0 41.8 35.4 3.8 0.4 -0.7 0.8 0.3 0.1 0.3 -0.6 Aker Anor System: C a 2 M g S i 2 0 7 - C a A l 2 S i 2 0 . (deWys and Foster, 1956) 1221 -14 43.7 16.5 31.7 8.1 0.0 -0.3 0.2 0.1 Tc « C a l c u l a t e d i n v a r i a n t point temperature C O Td - D i f f e r e n c e between c a l c u l a t e d and experimentally determined i n v a r i a n t p o i n t temperatures (calculated-experimental) 1 C a l c u l a t e d i n v a r i a n t point composition (oxide weight percent) * D i f f e r e n c e between c a l c u l a t e d and experimentally determined i n v a r i a n t point compositions (calculated-experimental) * I r v i n e (1975) * Welch (1956) 92 been i n v e s t i g a t e d by C a v a l i e r et a l . (1954) who i n v e s t i g a t e d 37 compositions, 25 of which were i n the primary phase volume of m e l i l i t e . The c a l c u l a t e d l i q u i d u s of g e h l e n i t e l i e s at p r e d i c t a b l y lower temperatures because of the assumption of s t o i c h i o m e t r y . C a l c u l a t e d l i q u i d i f o r the remaining e x p e r i m e n t a l l y i n v e s t i g a t e d compositions are i n reasonably good agreement with the experimental data. The d i f f e r e n c e s between c a l c u l a t e d and e x p e r i m e n t a l l y determined l i q u i d u s temperatures are (1) a n o r t h i t e : 5.5±26.5°, (2) l a r n i t e : 14.5±32.1°, and (3) s p i n e l : 23.2±22.5°. The c a l c u l a t e d l i q u i d u s diagram f o r t h i s plane i s shown in Appendix I I . Osborn et a l . (1954) s t u d i e d l i q u i d u s r e l a t i o n s on planes with constant A l 2 0 3 from 5 to 35 owp. Most experimental mixes were s t u d i e d i n the region of the merwinite f i e l d due to i t s relevance to s l a g compositions, but enough data were c o l l e c t e d to estimate s t a b i l i t y f i e l d s and "quaternary p i e r c i n g p o i n t s on these p l a n e s . Although t h i s data set was . not used i n the c a l i b r a t i o n because of d i s c r e p a n c i e s i n phase boundaries r e p o r t e d by Gutt (1963; 1964), the c a l c u l a t e d l i q u i d i agree very w e l l with t h e i r d a ta. F i g u r e s 27-33 show the c a l c u l a t e d phase diagrams f o r these planes, along with the e x p e r i m e n t a l l y estimated p i e r c i n g p o i n t compositions. A l s o shown are the average d i f f e r e n c e s between c a l c u l a t e d and e x p e r i m e n t a l l y determined l i q u i d u s temperatures f o r a l l compositions s t u d i e d w i t h i n the s t a b i l i t y f i e l d of each phase. The l a r g e s t d i f f e r e n c e s occur f o r p e r i c l a s e , l a r n i t e , and t r i d y m i t e . Because these are a l s o the phases with the 93 Figure 27: Calculated liquidus diagram of the 5 weight percent A l 2 0 3 plane of the CaO-MgO-Al 20 3-Si0 2 system. Symbols as in Figure 26. Lighter l i n e s are liquidus isotherms with a contour i n t e r v a l of 200*, s t a r t i n g with 1400°C. 94 Figure 28: Calculated liqu i d u s diagram of the 10 weight percent A1 20 3 plane of the CaO-MgO-Al 20 3-Si0 2 system. Symbols as in Figure 26. Liquidus isotherms as in Figure 27. 95 Figure 29; Calculated liquidus diagram of the 15 weight percent A1 20 3 plane of the CaO-MgO-Al 20 3-Si0 2 system. Symbols and liquidus isotherms as in Figure 26. 96 S i 0 2 CaO Weight Fraction MgO Figure 30: Calculated liquidus diagram of the 20 weight percent A1 20 3 plane of the CaO-MgO-Al 20 3-Si0 2 system. Symbols and liquidus isotherms as in Figure 26. 97 Figure 31: Calculated liquidus diagram of the 25 weight percent A1 20 3 plane of the CaO-MgO-Al 20 3-Si0 2 system. Symbols as in Figure 26. Liquidus isotherms as in Figure 27. 98 Figure 32; C a l c u l a t e d l i q u i d u s diagram of the 30 weight percent A 1 2 0 3 plane of the CaO-MgO-Al 20 3-Si0 2 system. Symbols as i n Figure 26. L i q u i d u s isotherms as i n Figure 27. 99 SiO 2 CaO Weight Fraction MgO Figure 33; Calculated liquidus diagram of the 35 weight percent A1 20 3 plane of the CaO-MgO-Al 20 3-Si0 2 system. Symbols as in Figure 26. Liquidus isotherms as in Figure 27. 100 steepest l i q u i d i ( p e r i c l a s e : 50°/owp, l a r n : I00°/owp, t r i d y m i t e : 45°/owp), p i e r c i n g p o i n t compositions are not f a r o f f the estimates of Osborn et a l . (1954). D i s c r e p a n c i e s i n the p o s i t i o n of the l a r n i t e - p e r i c l a s e - m e r w i n i t e p i e r c i n g p o i n t s on the 10-15% planes are due to the s y s t e m a t i c a l l y lower temperatures c a l c u l a t e d f o r merwinite. T h i s d i f f e r e n c e can be a t t r i b u t e d to s o l i d s o l u t i o n e f f e c t s i n merwinite which c o n t a i n s approximately 10 mole percent m o n t i c e l l i t e a t 1485°C (Schlaudt and Roy, 1966). Other phase boundaries that would be expected to change s i g n i f i c a n t l y because of s o l i d s o l u t i o n are the m o n t i c e l l i t e - f o r s t e r i t e c o t e c t i c and the m e l i l i t e - p r e s e n t r e a c t i o n s . The only major disagreement between the c a l c u l a t e d and e x p e r i m e n t a l l y determined r e s u l t s occurs i n the s p i n e l s t a b i l i t y f i e l d . C a l c u l a t e d l i q u i d u s temperatures are s y s t e m a t i c a l l y low on the 15% A l 2 0 3 plane, but compare f a v o r a b l y as A l 2 0 3 i n c r e a s e s . The d i f f e r e n c e s are u n l i k e l y to be s o l e l y r e l a t e d t o experimental charges being o f f - c o m p o s i t i o n , as they are c o n s i d e r a b l y g r e a t e r than the maximum gr a d i e n t (25°/owp) of the s p i n e l l i q u i d u s . Osborn et a l . (1954) d i d r e c o g n i z e , however, one anomalously high l i q u i d u s temperature f o r s p i n e l on t h i s plane ( t h e i r Table V I I I , mix 15A-4), so that i t i s p o s s i b l e that they may have had problems r e c o g n i z i n g quench s p i n e l i n some charges. Comparisons between c a l c u l a t e d and e x p e r i m e n t a l l y determined l i q u i d u s r e l a t i o n s are summarized i n Table XV f o r other j o i n s i n the quaternary system f o r which e x p e r i m e n t a l i s t s 101 have attempted to l o c a t e i n v a r i a n t ( p i e r c i n g ) p o i n t s . Phase diagrams f o r a l l systems are presented i n Appendix I I . These r e s u l t s are d i s c u s s e d below, and i t w i l l shown that the main d i s c r e p a n c i e s appear on j o i n s where c o n s i d e r a b l y l e s s care was demonstrated i n ensuring homogeneity of experimental charges. Other d i f f e r e n c e s can be accounted f o r by the l i m i t e d number of experimental charges used to estimate i n v a r i a n t p o i n t s , and by the p r e d i c t e d e f f e c t s of s o l i d s o l u t i o n i n l i q u i d u s phases. C a l c u l a t e d j o i n s i n the high alumina p o r t i o n of the system are i n reasonable agreement with the experimental data at lower temperatures, but show l a r g e r d i f f e r e n c e s at higher temperatures i n more aluminous compositions. The experimental r e s u l t s can be questioned on the b a s i s of the very small number of charges s t u d i e d , p a r t i c u l a r l y near the s p i n e l - c a l c i u m d i a l u m i n a t e -c a l c i u m hexaluminate p i e r c i n g p o i n t (DeVries and Osborn, 1957). C a l c u l a t e d temperatures on the j o i n s CaAl 20<, -MgAl 2O 0 and C a A l 4 0 7 - M g A l 2 0 « are lower than those determined by high temperature microscopy (Rao, 1968). Rao a l s o found c a l c i u m aluminate to be s t a b l e on the l a t t e r j o i n . In t h i s r e s p e c t , h i s r e s u l t s on samples melted i n a i r are at odds with those of Nurse et a l . (1965), who found that c a l c i u m aluminate melts i n c o n g r u e n t l y i n m o i s t u r e - f r e e atmospheres. Temperatures of most MgO-rich i n v a r i a n t p o i n t s are i n agreement with most experimental data, and the l i m i t e d number of experimental charges can be used i n defense of l a r g e c o m p o s i t i o n a l d i f f e r e n c e s i n the l o c a t i o n of i n v a r i a n t p o i n t s , p a r t i c u l a r l y on the j o i n s M g O-CaAl 2Si 20 B-Mg 2SiO a (DeVries and 1Q2 Table XV: Comparison of C a l c u l a t e d and Experimental Invariant Points on n o n - c a l i b r a t i o n Quaternary J o i n s Tc Td S i 0 2 A1 20, CaO MgO SiO, A l 2 0 , CaO MgO A. Alumina-rich J o i n s : Svstem: CaAl 2Si 20.-MgAl 20.-Mg 2SiO. (DeVries and Osborn, 1957) — - ' - - ~ *~ r 0 .6 — 1.5 — 0 . * 1.1 Anor Coru Spin 1481 -1 Gehl C,A 2 Spin Spin System: C,A 2 C,A,« A1 20, 1507 1589 -Ca 2Al -11 -57 C,A, C,A, Spin Gehl C,A2 C,A 2 C,A, C,A 2 Svstem: Gehl i Spin CaAl 2 1507 1565 1672 1499 0,-Ca 2 9 -29 56 -5 Gehl C,A 2 Spin Spin Svstem: C,A 2 CaAl, 1512 1728 0 7-Ca 2 -4 25 Anor Spin Coru Anor Svstem: Spin CaAl, 1481 1465 S i 2 0 t --6 14 39.5 37.6 17.3 5.6 14.0 57.9 26.1 2.0 10.3 51.7 37.5 0.4 6.4 56.5 37.1 0.0 0.0 65.6 30.2 4.3 11.5 50.8 35.5 2.2 14.9 49.8 33.2 2.0 0.0 77.7 19.2 3.2 38.8 37.9 18.1 5.2 40.4 34.3 18.9 6.4 0.8 -1.0 1.6 -1.4 1.0 -1.3 0.2 0.1 6.4 -8.4 3.5 -1.5 0.0 0.2 -0.9 0.7 0.1 -0.3 1.2 -0.9 -0.1 0.0 -0.1 0.1 0.4 0.2 0.8 -1.1 0.5 -0.7 0.5 -0.1 -0.3 0.1 0.0 -0.1 B. Magnesium-rich J o i n s : Aker Spin Mont 5 Alter Spin Anor Mont Aker Mont S p i n 8 Anor Spin Coru* Svstem: C a A l 2 S i 2 0 t - M g A l 2 0 , - C a 2 M g S i 2 0 7 (Schairer and Yoder, 1344 1221 1405 1370 1480 -23 -26 13 -15 -4 37.6 13.333.515.5 0.2 0.8 -0.4 43.2 17.0 31.4 8.4 1.7 -2.4 1.4 39.5 7.5 36.9 16.2 0.5 -0.8 0.5 35.6 13.8 33.2 17.4 -1.0 1.6 -0.9 37.8 38.4 19.2 4.7 0.6 -0.6 0.4 Diop Fors Fors Spin Svstem: Al 20,-CaMgSi 20, (O'Hara and S c h a i r e r , '963) 1322 -5 49.7 10.4 23.2 16.7 -0.5 0.9 -0.2 1312 0 44.9 19.0 21.0 15.1 0.3 -0.5 0.1 System: Fors Spin Fors Spin Anor Anor Fors Diop* Anor Spin Coru* Diop Fors Svstem: Spin Anor Spin Fors P e r i * CaAl 2Si 20,-MgAl 20,-CaMgSi 20. (Schairer and *°cier. 1413" -19" 41.6 17.9 19.4 21.0 1300 -7 45.2 19.1 21.1 14.5 1272 -3 48.1 16.0 22.4 13.5 1481 9 38.8 37.9 18.1 5.2 1356 -3 51.5 5.2 24.0 19.3 0.0 0.0 -0.0 0.8 -0.8 0.4 -0.5 -0.0 -0.2 -0.9 2.4 -0.4 -0.2 0.2 -0.1 1969) •0.5 •0.7 -0.2 0.3 -0.3 -0.2 0.1 1969) -0.0 -0.4 0.7 -1.1 0.0 P e r i Spin Larn P e r i P e r i Spin Svstem: System: CaAl 2Si 20,-MgO-Mg 2SiO, (DeVries and Osborn, 1957) 1465 14 40.4 34.3 18.9 6.4 0.1 0.0 0.0 1602 -14 29.4 17.3 9.5 43.7 -9.5 -1.0 -0.5 CaMgSiO,-MgAl,0,* (El-Shahat and White, 1966a) 1491 -149 28.2 19.0 26.3 26.4 1.1 -2.0 1.0 Ca 1MgSi 2O l-MgAl 20, 2 (El-Shahat and White, 1966b) 1500 -75 30.2 12.5 42.2 15.1 -1.6 3.2 -2.3 1447 -113 25.4 21.8 35.6 17.1 1.1 -2.2 1.6 -0.1 11.0 -0.1 0.7 -0.5 103 Table XV: Comparison of C a l c u l a t e d and Experimental Invariant Points on n o n - c a l i b r a t i o n Quaternary J o i n s (continued) Tc Td S i 0 2 A1 20, CaO MgO SiO, Al,Oj CaO MgO C. Lime-rich J o i n s : System: Ca sAl (0,,-MgO-Ca 2SiO, (Hansen, 1928) o ' l r P e r i CjAT 1 1284 -29 3.9 44.8 48.3 3.0 -2.7 5.2 -0.4 - 2 .0 System: Ca 2MgSi 20,-CaAl 2Si 2O t-CaMgSi 20, (deWys and Foster, 1958) Anor Diop Aker 1221 -7 44.0 16.3 31.5 8.3 -0.7 0.1 0.9 - 0 . 3 Aker Anor 1221 -14 43.7 16.5 31.7 8.1 0.0 -0.3 0.2 0.1 System: CaMgSi 20 (-CaAl 2SiO. (Schairer and Yoder, 1969) C,A § 1532 -45 27.6 46.7 25.7 0.0 0.0 - 0 .0 0.0 0.0 C,A« Spin 1454 20 30.6 41.6 25.7 2.0 -0.8 1.4 - 0 .0 -0 .6 Spin Anor 1295 -15 43.6 19.8 25.8 10.7 1.6 -2.6 0.0 1.0 Diop Anor 1257 -15 45.7 16.3 25.8 12.1 -0.3 0.4 - 0 .0 -0 .2 System: M g 2 A l 2 S i , 0 , 2 - C a 2 A l 2 S i , 0 , 2 (Chinner and S c h a i r e r , 1962) Gehl 1331 -12 40.0 22.6 37.3 0.0 -0.0 -0.0 -0.0 0.0 Gehl Anor 1288 -16 40.4 22.9 33.9 2.8 -0.0 -0.0 0.3 -0.2 Anor Spin 1307 -15 41.0 23.2 29.8 6.0 -0.4 -0.2 3.3 -2.7 Spin 1532 2 44.7 25.3 0.0 30.0 -0.0 -0.0 0.0 -0.0 System: Ca,Al,0,«-MgO-Ca 2SiO,-CaO (McMurdie and In s l e y , 1936) C|A, C»S, Lime P e r i 1439 39 4.9 36.5 53.8 4.8 -2.1 3.0 -0.2 -0.7 C,A, C,S, o ' l r P e r i 1404 20 6.2 36.6 53.0 4.2 -1.3 2.6 0.0 -1.3 System: C.ajMgSijO.-MgAljO.-CajAlaSiOT* (Gutt, 1964) Merw P e r i Larn 1452 -76 29.0 15.4 41.5 14.0 " * * Merw P e r i . S p i n 1403 -19 27.2 19.6 39.6 13.6 Spin Merw Gehl 1399 -19 27.6 19.8 41.3 11.3 Larn Merw Gehl 1401 -33 27.9 19.4 42.1 10.6 Gehl Larn 1444 -34 28.9 19.4 45.8 5.9 System: Ca 2SiO,-Al 20,-Mg 2SiO,* (Gutt, 1963) Coru Spin C,A. 1681 30 12.2 66.2 18.9 2.8 Fors P e r i 1609 -194 40.0 0.0 22.2 37.8 Fors Spin 1690 -1 34.3 19.6 0.0 46.1 C,A, Spin C,A 2 1564 -36 15.8 55.4 26.9 1.9 C,A 2 Spin Gehl 1502 -18 18.2 48.7 31.4 1.8 Gehl Spin Merw 1394 -39 30.3 17.8 40.0 11.9 Merw Gehl Larn 1404 -55 30.0 17.6 43.0 9.4 Larn Merw P e r i 1543 -5 36.1 3.3 43.2 17.4 D. S i l i c a - r i c h J o i n s : System: Si0 2-CaMgSi 20«-CaAl 2SiO, (de N e u f v i l l e and Sc h a i r e r , 1962) Anor Fors T r i d 1195 -26 61.1 15.9 17.1 6.0 -0.1 0.0 0.0 0.1 System: CaMgSi 20,-MgSiO,-CaAl 2Si 20. (Hytonen and S c h a i r e r , 1961) Anor Fors 1302 -6 50.1 21.5 11.9 16.5 0.2 -0.4 -0.2 0.5 Anor Fors Diop 1269 5 50.8 16.0 19.5 13.7 -0.1 0.2 0.3 -0.4 System: Mg,Al 2Si,0, 2-CaMgSi 20. (O'Hara and S c h a i r e r , 1963) Diop Fors 1361 -6 53.6 4.5 21.3 20.7 -0.2 0.5 -0.5 0.2 Fors Spin 1473 9 45.7 23.0 2.4 29.0 -0.3 0.7 -0.7 0.3 Inv a r i a n t p o i n t excluded from average d i f f e r e n c e s reported in text', due to: 1 Acceptance of C,A, incongruent melting (discussed in text) * I n c o n s i s t e n c i e s discussed in t e x t * M e t a s t a b i l i t y of experimental r e s u l t s (discussed in text) * Large experimental u n c e r t a i n t i e s 1 Only compositions excluded due to l i m i t e d number of charges studied -2.2 4.5 -2.9 0.6 -0.2 0.1 -0.6 0.7 0.2 -0.3 0.4 -0.3 -1.0 1.8 -1.7 1.0 0.1 -0.1 0.0 0.1 -2.4 6.2 -2.5 -1.4 -2.1 0.0 17.6-•15.5 0.2 -0.4 0.0 0.2 -0.7 1.6 0.4 -1.3 -1.1 2.9 -0.6 -1.1 -0.4 0.8 0.2 -0.7 0.8 -1.4 -1.9 2.5 1.0 -2.2 -0.8 1.9 1 04 Osborn, 1957) and C a A l 2 S i 2 0 8 - M g A l 2 O u - C a M g S i 2 0 6 ( S c h a i r e r and Yoder, 1969). C o n s i d e r a t i o n of the e f f e c t s of s o l i d s o l u t i o n i n akermanite would improve the r e p r o d u c t i o n of the akermanite-m o n t i c e l l i t e r e a c t i o n p o i n t and a k e r m a n i t e - s p i n e l - a n o r t h i t e p i e r c i n g p o i n t ( S c h a i r e r and Yoder, 1969), although much of the temperature d i s c r e p a n c y i n the l a t t e r i s due to a 16° d i f f e r e n c e i n the experimental d e t e r m i n a t i o n of the a n o r t h i t e - a k e r m a n i t e b i n a r y e u t e c t i c ( S c h a i r e r and Yoder, 1969; deWys and F o s t e r , 1956). The data r e p o r t e d by El-Shahat and White (1966a; 1966b) f o r the j o i n s MgAl 20 4-CaMgSiO f t and M g A l 2 O n - C a 3 M g S i 2 0 8 are h i g h l y d i s c o r d a n t with the model c a l c u l a t i o n s , with temperatures of the s p i n e l and p e r i c l a s e l i q u i d i up to 150°C h i g h e r . For example, El-Shahat and White l o c a t e d the p e r i c l a s e l i q u i d u s at 1760±10°C fo r the m o n t i c e l l i t e composition. The model l i q u i d u s temperature i s 1612°C, which i s c o n s i s t e n t with Ri c k e r and Osborn's (1954) data showing the assemblage p e r i c l a s e + l i q u i d at 1614±3°C and l i q u i d only at 1654±3°C. These d i f f e r e n c e s may be a t t r i b u t e d to inhomogenous charges as El-Shahat and White r e p o r t only one step of g r i n d i n g and mixing unfused s t a r t i n g m a t e r i a l s . In a d d i t i o n , t h e i r l a r g e r s t a b i l i t y f i e l d s of p e r i c l a s e and s p i n e l c o u l d have been produced by the appearance of these phases upon quenching samples i n a i r . Biggar and O'Hara (1970) r e p o r t quench problems with s p i n e l in s i m i l a r compositions even when samples are quenched i n water. S e v e r a l i n c o n s i s t e n c i e s appear between c a l c u l a t e d and ex p e r i m e n t a l l y determined l i q u i d u s r e l a t i o n s i n the high lime 105 p o r t i o n of the quaternary system. A l a r g e d i f f e r e n c e occurs i n the composition of the s p i n e l - a n o r t h i t e r e a c t i o n p o i n t on the j o i n C a 3 A l 2 S i 3 0 , 2 - M g 3 A l 2 S i 3 0 . 2 , which Chinner and S c h a i r e r (1962) l o c a t e d on the b a s i s on mixes prepared at 10% i n t e r v a l s a c r o s s the j o i n . Although the c a l c u l a t e d s p i n e l l i q u i d u s i s on average c o n s i s t e n t with the experimental data (3±12.5°C), the l a r g e s t d i f f e r e n c e (23°C) occurs near t h e i r estimated composition f o r t h i s r e a c t i o n p o i n t . T h i s temperature d i f f e r e n c e , and the low angle of i n t e r s e c t i o n of the s p i n e l and a n o r t h i t e l i q u i d i , combine to produce the rather l a r g e c o m p o s i t i o n a l discepancy i n the p o s i t i o n of t h i s r e a c t i o n p o i n t . S c h a i r e r and Yoder (1969) present the phase diagram C a M g 2 S i 2 0 6 - C a A l 2 S i O s without s u p p o r t i n g l i q u i d u s data. T h e i r l i q u i d u s f o r c a l c i u m hexaluminate i s 45°C hi g h e r than that c a l c u l a t e d at the C a A l 2 S i 0 6 composition. More data i s necessary to a s c e r t a i n l i q u i d u s temperatures i n t h i s composition as t h e i r d e t e r m i n a t i o n appears h i g h when compared to isotherms i n the c a l c i u m hexaluminate f i e l d of the C a O - A l 2 0 3 - S i 0 2 system ( G e n t i l e and F o s t e r , 1963; Osborn and Muan, 1960; Langeman and Chipman, 1956). C a l c u l a t e d phase r e l a t i o n s on the j o i n C a 5 A l 6 0 i 4 - C a 2 S i O a -MgO d i f f e r from those determined by Hansen (1928) because of the m e t a s t a b i 1 i t y i n dry atmospheres of the c a l c i u m aluminate phase, the s t o i c h i o m e t r y of which has been r e v i s e d to 12CaO«7Al 20 3 (Nurse et a l . , 1965). With the e x c l u s i o n of t h i s phase, the pseudoternary phase r e l a t i o n s i n v o l v e the phases t r i c a l c i u m aluminate and c a l c i u m aluminate at temperatures approximately 106 30°C lower than e x p e r i m e n t a l l y determined f o r the 12CaO«7Al 20 3 present e u t e c t i c . T h i s d i f f e r e n c e and the s h i f t i n the composition of t h i s e u t e c t i c are very s i m i l a r i n magnitude to the d i f f e r e n c e s produced by e x c l u s i o n of t h i s phase from l i q u i d u s r e l a t i o n s i n the C a O - A l 2 0 3 - S i 0 2 t e r n a r y system ( F i g u r e 22).. McMurdie and I n s l e y (1936) estimated the l o c a t i o n s of four quaternary i n v a r i a n t p o i n t s i n the h i g h lime and alumina p o r t i o n of the system, but two i n v o l v e d the metastable phase 12CaO-7Al 20 3 ( 5 C a O » 3 A l 2 0 3 ) . The d i f f e r e n c e s i n the remaining two i n v a r i a n t p o i n t s are l a r g e l y due to the higher c a l c u l a t e d l i q u i d u s temperatures f o r p e r i c l a s e (-82±35°C). These d i s c r e p a n c i e s (Table XV) may be caused by inhomogenous experimental charges which were prepared with only one s t e p of mixing, s i n t e r i n g , and g r i n d i n g a f t e r MgO was added to a mixture of the other components. Sample inhomogeneity probably a l s o accounts f o r the l a r g e i n c o n s i s t e n c i e s evident i n the r e s u l t s of Gutt (1963; 1964). He r e p o r t s t h a t samples were prepared with one s t e p of m e l t i n g and g r i n d i n g of powdered m a t e r i a l s , or two steps of s i n t e r i n g and g r i n d i n g f o r compositions with l i q u i d u s temperatures higher than 1550°C. Our c a l c u l a t e d i n v a r i a n t p o i n t s (Table XV) show d i f f e r e n c e s with h i s data which average -38°C and 1.8 owp. Gutt (1963; 1964) r e c o g n i z e d i n c o n s i s t e n c i e s between h i s r e s u l t s and t r a n s l a t i o n s from the constant A l 2 0 3 planes s t u d i e d by Osborn et a l . (1954), p a r t i c u l a r l y the l a r g e r f i e l d s of p e r i c l a s e and merwinite on Gutt's v e r s i o n of the system C a 2 S i O ( 1 - A l 2 0 3 - M g 2 S i 0 l l . 107 In these r e s p e c t s , the model c a l c u l a t i o n s favor the data of Osborn et a l . (1954). Gutt (1963) l o c a t e d the f o r s t e r i t e - p e r i c l a s e e u t e c t i c on the C a 2 S i O « - M g 2 S i O a j o i n at 1806°C and 93 weight percent Mg 2SiOi,. These r e s u l t s can not be r e c o n c i l e d with R i c k e r and Osborn's (1954) data on the CaMgSiO«-Mg 2Si0 4 j o i n , which p l a c e the e u t e c t i c at 1664±20°C and 67 wgt. % M g 2 S i 0 4 . The c a l c u l a t e d e u t e c t i c p o s i t i o n i s 1609°C and 66 wgt. % M g 2 S i O „ . Other apparent temperature anomalies i n Gutt's data have been noted by Biggar and O'Hara (1970). F i n a l l y , Gutt's (1963) r e s u l t s suggest a temperature f o r the l a r n i t e - g e h l e n i t e e u t e c t i c more than 25°C above t h a t independently determined by P r i n c e (1951) and Rankin and Wright (1915). F i g u r e 34 shows the c a l c u l a t e d j o i n C a S i 0 3 - M g S i 0 3 - A l 2 0 3 which was s t u d i e d by Segnit (1956). P i e r c i n g p o i n t temperatures were not estimated by S e g n i t , and f o r each s t a b i l i t y f i e l d , the average d i f f e r e n c e s between c a l c u l a t e d and e x p e r i m e n t a l l y determined l i q u i d u s temperatures are shown. E x c l u d i n g akermanite and g e h l e n i t e which have lower c a l c u l a t e d temperatures- because of s o l i d s o l u t i o n e f f e c t s , the c a l c u l a t e d l i q u i d i are in e x c e l l e n t agreement with the experimental r e s u l t s . The only e x c e p t i o n i s the p s e u d o w o l l a s t o n i t e l i q u i d u s which i s roughly 50°C higher than determined by experiment. The model c a l c u l a t i o n s are supported by the data of Osborn et a l . (1954). For the composition 47.8, 10.0, 36.2, 6.0 ( S i 0 2 , A l 2 0 3 , CaO, MgO owp), Segnit determined a l i q u i d u s temperature of 1290°C, compared with a c a l c u l a t e d value of 1347°C. The same 108 F i g u r e 34; C a l c u l a t e d l i q u i d u s diagram f o r the system MgSi03-Al 20 3-CaSi03. Symbols as i n Figure 26. 109 l i q u i d u s temperature (1347°C) i s c a l c u l a t e d f o r a very s i m i l a r composition (47.0, 10.0, 37.0, 6.0) f o r which Osborn et a l . (1954).located the l i q u i d u s at 1337±6°C. No e x p l a n a t i o n f o r Se g n i t ' s anomalous r e s u l t s c o u l d be found i n h i s d e s c r i p t i o n of h i s experimental procedure. C. DISCUSSION The r e s u l t s presented above demonstrate the a b i l i t y of the s t o i c h i o m e t r i c - M a r g u l e s s o l u t i o n model to reproduce l i q u i d u s r e l a t i o n s i n the quaternary system w i t h i n the u n c e r t a i n t i e s of the experimental data. The average d i f f e r e n c e s f o r non-c a l i b r a t i o n j o i n s (-1.6±18°C and 0.43±.64 owp) are not s i g n i f i c a n t l y worse than those f o r the c a l i b r a t i o n j o i n s themselves. T h i s o v e r a l l agreement i s extremely encouraging both f o r the c o n s i s t e n c y i t p o i n t s out among experiments performed i n many d i f f e r e n t l a b o r a t o r i e s d u r i n g the present century, and f o r the success of t h i s type of model i n reproducing l i q u i d u s r e l a t i o n s over the e n t i r e spectrum of CMAS compositions, and over temperatures ranging from l e s s than 1200°C to near 3000°C. D i s c u s s i o n i s now focussed on some consequences of the f i t of l i q u i d u s r e l a t i o n s as a p p l i e d to c a l c u l a t e d thermodynamic p r o p e r t i e s of the melt, and p r e d i c t i o n of phase r e l a t i o n s which have not been e x p e r i m e n t a l l y i n v e s t i g a t e d . 110 1. L i q u i d P r o p e r t i e s Thermodynamic d e s c r i p t i o n of s i l i c a t e l i q u i d s i s important both f o r c o n t r i b u t i n g to our general understanding of melts, and because c a l c u l a t e d melt p r o p e r t i e s can be u s e f u l l y employed i n much p e t r o l o g i c m o d e l l i n g . T h i s i n f o r m a t i o n i s d i f f i c u l t to o b t a i n d i r e c t l y because of high l i q u i d u s temperatures and the problems i n v o l v e d i n m a i n t a i n i n g the l i q u i d r e f e r e n c e s t a t e . Much work i n the past has been performed on g l a s s y samples, and only r e c e n t l y are these data being u s e f u l l y combined with data on l i q u i d samples so that reasonable e x t r a p o l a t i o n s can be made to g e o l o g i c a l l y r e l e v a n t temperatures. Some c a l c u l a t e d l i q u i d p r o p e r t i e s are b r i e f l y compared with experimental o b s e r v a t i o n s i n order to show the u t i l i t y of thermodynamic mod e l l i n g f o r e s t i m a t i n g these p r o p e r t i e s . Recent data based on r e g r e s s i o n f i t s of multicomponent l i q u i d heat c a p a c i t y data, assuming no excess heat c a p a c i t y of mixing, i n d i c a t e a general i n c r e a s e of l i q u i d oxide heat c a p a c i t i e s with i n c r e a s i n g i o n i c p o t e n t i a l of c a t i o n s (Carmichael and Stebbins, 1982). Values f o r MgO and CaO are re p o r t e d to be as high as 5.7R/atom«K (R i s the gas c o n s t a n t ) . The s i g n i f i c a n c e of these values can be questioned i n the l i g h t of recent data which show excess heat c a p a c i t i e s of mixing i n some s i l i c a t e melt systems (Stebbins et a l . , 1982a). Because these data show l a r g e d e p a r t u r e s , however, from the r e l a t i v e l y narrow range (29 ± 10% Joules/atom«K) of l i q u i d heat c a p a c i t i e s r e p o r t e d by Ric h e t and B o t t i n g a (1980), t h i s range has not been used to c o n s t r a i n l i q u i d heat c a p a c i t i e s as was done i n the 111 e a r l i e r f i t of the system C a O - A l 2 0 3 - S i 0 2 (chapter I I I ) . The only d i r e c t c o n s t r a i n t s p l a c e d on l i q u i d heat c a p a c i t i e s i n the f i t of l i q u i d u s r e l a t i o n s were that they be g r e a t e r than that of the r e s p e c t i v e s o l i d o x i d e s . The l i q u i d oxide heat c a p a c i t i e s d e r i v e d i n the f i t show no systematic trends between the oxides, with Cp(MgO) near 3.4R/atom-K and Cp(CaO) equal to roughly 4.6R/atom«K at high temperatures. Only the heat of f u s i o n of corundum was f o r c e d to be s i m i l a r to i t s e x p e r i m e n t a l l y determined v a l u e . Heats of f u s i o n of the other three components were not f i x e d because of l a r g e u n c e r t a i n t i e s i n estimates f o r these o x i d e s . T h i s r e p r e s e n t s a r e l a x a t i o n of c o n s t r a i n t s p r e v i o u s l y a p p l i e d i n m o d e l l i n g of the C a O - A l 2 0 3 - S i 0 2 system (chapter I I I ) , and a l l o w s f o r c l o s e r r e p r o d u c t i o n of l i q u i d u s r e l a t i o n s i n the quaternary system. Heats of f u s i o n of a l l one atmosphere l i q u i d u s m i n e r a l s are presented i n Table XVI along with l i t e r a t u r e e s t i m a t e s . Comparison of the t a b u l a t e d values shows that the model c a l c u l a t i o n s are w e l l w i t h i n the range of l i t e r a t u r e values estimated from e x t r a p o l a t i o n s of heats of v i t r i f i c a t i o n , and from computations based on f r e e z i n g p o i n t d e p r e s s i o n s or slopes of f u s i o n c u r v e s . The c a l c u l a t e d v a l u e s do show d i f f e r e n c e s with some of the values d e r i v e d i n c a l o r i m e t r i c s t u d i e s i n v o l v i n g l i q u i d samples. The main source of e r r o r i n these measured heats of f u s i o n stem from u n c e r t a i n t i e s i n the l i q u i d heat c a p a c i t i e s used to b r i n g the heat of v i t r i f i c a t i o n up to the m e l t i n g temperature. Previous s o l u t i o n s to the f i t of l i q u i d u s r e l a t i o n s i n d i c a t e s enough f l e x i b i l i t y to be able to Table XVI: Heats Of Fusion Mineral T(fuston)' B e t a - c r l s t o b a l I t e 1736 2036 Lime Pe r i c l a s e Pseudowollaston!te Anorthlte Diopside Forster!te Akermam te Enstat1te(*) 3077 2675 1553 1550 1933 1464 1557 Sp1ne1 Montlcel1!te(*) SaphlMneC*) Gehlenite Rank1n1te(») Larm te Tricalcium S11lcate(») 2226 Tricalcium alum1nate(*) 1588 Calcium alumlnate(*) 1638 Calcium dlalummate 1767 Calcium hexalumlnate(*) 1829 2147 1500 1714 1588 1509 2225 H(fU31on)' 13085 127159 147715 109456 45772 113438 78947 111216 59200 187547 96707 695554 137290 78488 56136 52814 153901 86436 199542 726120 H( fusion) ' 7531 15062 10460 9581 8159 107529t 118407t 117152t 79496 77404 57321t 50208 82843 27405 120081 166942t 135562t 138072t 142611t 128449t 129884 170193 71 128 123900t 81906 75312 61505 1 Calculated fusion temperature (degrees Centigrade) ' Calculated heat of fusion (Joules/mole) ' Estimated heats of fusion In the l i t e r a t u r e (Joules/mole) t Experimentally determined value » Incongruently melting mineral (based on model c a l c u l a t i o n s ) Reference Kracek (1930) Lumsden (1966) Holm et a l . (1967) Chase et a l . (1974) Robie et a l . (1978) Shpl1'rain et a l . (1972) Fomlchev (1973) Chase et a l . (1974) Chase et a l . (1974) Chase et a l . (1974) Adamkovlcova et a l . (1980) Adams and Cohen (1966) Spencer (1973) Robie et a l . (1978) Adams and Cohen (1966) F e r r l e r (1969) Wei 11 et a l . (1980b) Stebblns et a1. (1982c) Wei 11 et a l . (1981) F e r r l e r (1968) Carmichael, et a l . (1977) Ghiorso and Carmichael (1980) S t u l l and Prophet (1971) Proks et a l . (1977) Ghiorso and Carmichael (1980) S t u l l and Prophet (1971) Robie, et a l . (1978) 113 c o n s t r a i n the model w i t h i n the u n c e r t a i n t i e s of s e l e c t e d heats of f u s i o n . Such c o n s t r a i n t s have not been i n c l u d e d , however, because d i s c r e p a n c i e s i n l i q u i d heat c a p a c i t y values (e.g. Stebbins et a l . , 1982b) have not been r e s o l v e d . The model c a l c u l a t i o n s thus y i e l d heats of f u s i o n independent of these c a l o r i m e t r i c data, and these appear to represent reasonable f i r s t order e s t i m a t e s . With the present s o l u t i o n model, heats of mixing can be c a l c u l a t e d on any j o i n w i t h i n the quaternary system. P r e v i o u s l y presented heats of mixing on the j o i n S i 0 2 - C a A l 2 O f t (chapter I I I ) are very s i m i l a r i n magnitude to those measured by drop c a l o r i m e t r y on g l a s s samples, but the c a l c u l a t e d minimum i n the mixing curve i s d i s p l a c e d towards C a A l 2 O a . F i g u r e 35 shows c a l c u l a t e d heats of mixing i n l i q u i d s on the j o i n CaMgSi 2G 6-C a A l 2 S i 2 0 8 . The magnitude of the heat e f f e c t s are again very c l o s e to those determined c a l o r i m e t r i c a l l y on g l a s s e s ( W e i l l et a l . , 1980a), while the minimum occurs approximately 10 mole % c l o s e r to the a n o r t h i t e composition. T h i s general agreement i n c a l c u l a t e d heats of mixing i s very encouraging c o n s i d e r i n g that they are dependent on a simple o n e - s i t e c o n f i g u r a t i o n a l entropy model. Some of the d i f f e r e n c e i n the c a l c u l a t e d heats of mixing may a l s o be r e l a t e d to excess heat c a p a c i t i e s of mixing which Stebbins et a l . (1982a) r e c e n t l y r e p o r t e d i n the Ab-An-Di system, or to d i f f e r e n c e s i n g l a s s t r a n s i t i o n temperatures a c r o s s t h i s j o i n . The data t a b u l a t e d by W e i l l et a l . (1980a) i n d i c a t e , however, that the l a t t e r c o n s i d e r a t i o n has only a small e f f e c t i n the Di-An system. 114 2000 CD Figure 35; Heats of mixing on the CaAl 2Si ?0 ?-CaMgSi 20s j o i n . S o l i d curve represents calculated heats of mixing in I j q u i j s -Dasned curve shows heats of mixing in glasses measured at 970 K (data smoothed by equation 10 of Weill et a l . 1980a;. 1 15 A c t i v i t i e s of the l i q u i d oxides d e r i v e d from the f i t of phase r e l a t i o n s compare q u i t e f a v o r a b l y with a c t i v i t i e s determined through S i d i s t r i b u t i o n experiments e q u i l i b r a t i n g s l a g s with SiC c r u c i b l e s (Rein and Chipman, 1963; 1965). The o r i e n t a t i o n of a c t i v i t y contours i n the three S i 0 2 - b e a r i n g t e r n a r y systems ( F i g u r e s 36-38) are s i m i l a r to those determined e x p e r i m e n t a l l y , and the numerical d i f f e r e n c e s 'in a c t i v i t i e s are g e n e r a l l y l e s s than 0.1. A c t i v i t i e s on the CaO-Si0 2 b i n a r y show the only systematic d i f f e r e n c e s , with c a l c u l a t e d a c t i v i t i e s being d i s p l a c e d to lower S i 0 2 compositions. Rein and Chipman (1965) a l s o determined a c t i v i t i e s of S i 0 2 on the 10, 20, and 30 weight percent MgO planes of the quaternary system. C a l c u l a t e d a c t i v i t i e s on the 10% plane (Figure 39) show the same f e a t u r e s as the C a 0 - A l 2 0 3 - S i 0 2 system, with d i f f e r e n c e s i n t r o d u c e d by d i s c r e p a n c i e s on the CaO-Si0 2 b i n a r y . These d i f f e r e n c e s have l a r g e l y disappeared with a d d i t i o n of 20% MgO (Figure 40), and contours on the 30% plane c l o s e l y p a r a l l e l the experimental contours ( F i g u r e 41), although the numerical d i f f e r e n c e s i n a c t i v i t i e s are l a r g e r (up to 0.23). E r r o r s i n the experimental work and c h o i c e of thermodynamic data i n t h e i r a n a l y s i s of t h e i r d i s t r i b u t i o n experiments can account f o r some of the d i f f e r e n c e s with the c a l c u l a t e d v a l u e s (chapter I I I ) , although the s y s t e m a t i c a l l y higher a c t i v i t i e s c a l c u l a t e d on the 30% plane represent a s i g n i f i c a n t departure from t h e i r data. P o s s i b l e sources of t h i s d i s c r e p a n c y can not be assessed as Rein and Chipman do not present any s u p p o r t i n g data f o r a c t i v i t i e s on the constant MgO p l a n e s . 116 Figure 36: S i 0 2 l i q u i d a c t i v i t i e s at 1600°C in the system CaO-Al 20 3-Si0 2. Calculated l i q u i d a c t i v i t i e s ( s o l i d curves) are compared with experimentally derived a c t i v i t i e s (dashed curves). The l a t t e r have been referenced to l i q u i d S i 0 2 through the calculated equilibrium constant for the reaction S i O 2 ( c r i s t o b a l i t e ) = S i 0 2 ( l i q u i d ) at 1600°C. 117 Figure 37t S i 0 2 l i q u i d a c t i v i t i e s at 1600°C in the system MgG~Al 20 3-Si0 2. See Figure 36. 118 Figure 38: S i 0 2 l i q u i d a c t i v i t i e s at 1600°C i n the system CaO-MgO-Si0 2. See Figure 36. 1 1 9 F i g u r e 39; S i 0 2 l i q u i d a c t i v i t i e s at 1600°C on the 10 weight percent MgO plane of the system Ca0-Mg0-Al 20 3-Si0 2. See Figure 36. 1 2 0 F i g u r e 40: S i 0 2 l i q u i d a c t i v i t i e s at 1600°C on the 20 weight percent MgO plane of the system CaO-MgO-Al 20 3-Si0 2. See Figure 36. 121 Figure 41: S i 0 2 l i q u i d a c t i v i t i e s at 1600°C on the 30 weight percent MgO plane of the system CaO-MgO-Al 20 3-Si0 2. See Figure 36. 122 The d i f f e r e n c e s that do e x i s t , however, do not a l t e r Rein and Chipman's c o n c l u s i o n s based on these a c t i v i t y data. Comparison of F i g u r e s 36-38 c l e a r l y i n d i c a t e 1) the amphoteric nature of alumina (reducing the S i 0 2 a c t i v i t y i n h i g h S i 0 2 compositions, and i n c r e a s i n g i t i n low S i 0 2 compositions i n both the M g O - A l 2 0 3 - S i 0 2 and C a O - A l 2 0 3 - S i 0 2 t e r n a r i e s ) , 2) the s i m i l a r e f f e c t s of CaO and MgO i n systems without A l 2 0 3 , and 3) the g r e a t e r r e d u c t i o n i n S i 0 2 a c t i v i t y caused by MgO or CaO r e l a t i v e to A l 2 0 3 i n low S i 0 2 compositions. These changes i n the S i 0 2 a c t i v i t y can be r a t i o n a l i z e d with the g r e a t e r network modifying a b i l i t y of MgO and CaO r e l a t i v e to A l 2 0 3 , and with the dual r o l e of A l 2 0 3 as network former at low S i 0 2 , and network m o d i f i e r at high S i 0 2 c o n c e n t r a t i o n s . 2. L i q u i d I m m i s c i b i l i t y The a b i l i t y to c a l c u l a t e the e n t i r e f r e e energy s u r f a c e of the melt w i t h i n the quaternary system as a f u n c t i o n of temperature has important p e t r o l o g i c a p p l i c a t i o n s f o r o u t l i n i n g the compositions expected to show l i q u i d i m m i s c i b i l i t y . Our r e s u l t s on the CaO-Si0 2 binary are w i t h i n the estimated 1 owp u n c e r t a i n t i e s estimated for the recent data of Tewhey and Hess (1979), which show a narrower so l v u s than that o r i g i n a l l y determined by G r e i g (1927). The c a l c u l a t e d solvus on the MgO-Si0 2 b i n a r y i s a l s o narrower than G r e i g ' s data by approximately 1 owp on each limb. The lower c a l c u l a t e d temperature of s t a b l e two l i q u i d occurrence on t h i s j o i n p a r a l l e l s the lower temperature found by Tewhey and Hess (1979) on the CaO-Si0 2 j o i n (Table X I I ) . 1 23 Although a technique f o r c a l c u l a t i o n of b i n o d a l s u r f a c e s i n systems of more than two components has not been developed, s o l v i i n t e r n a r y systems can be c l o s e l y estimated by c a l c u l a t i n g the s p i n o d a l s u r f a c e (second d e r i v a t i v e of the f r e e energy of s o l u t i o n equals z e r o ) , and u t i l i z i n g the c o i n c i d e n c e of the b i n o d a l and s p i n o d a l at the c r i t i c a l p o i n t of the s o l v u s . S o l v i estimated by t h i s means are shown in F i g u r e s 22, 23, and 25. The p o s i t i o n of the two l i q u i d f i e l d i n the CaO-MgO-Si0 2 ( F i g . 23) system i s s i m i l a r to that determined by G r e i g (1927), while l i q u i d i n s t a b i l i t y extends to s l i g h t l y higher A l 2 0 3 contents i n both the C a O - A l 2 0 3 - S i 0 2 ( F i g . 22) and M g O - A l 2 0 3 - S i 0 2 ( F i g . 25) systems. Because of the p o s s i b i l i t i e s of c o m p o s i t i o n a l inhomogeneities i n Grei g ' s experimental charges having a r i s e n by the mechanism proposed by Tewhey and Hess, attempts were not made to more c l o s e l y reproduce these i m m i s c i b i l i t y f i e l d s . An i n t e r e s t i n g consequence of the f i t of l i q u i d u s r e l a t i o n s i s the appearance of s e v e r a l metastable two l i q u i d f i e l d s i n p o r t i o n s of the quaternary system other than those r i c h i n S i 0 2 ( F i g u r e s 18, 20, 21). A wide i m m i s c i b i l i t y f i e l d occurs on the A l 2 0 3 - S i 0 2 b i n a r y with a c r i t i c a l temperature of 1830°C, 200° below the l i q u i d u s . Much experimental evidence f o r g l a s s e s i n t h i s system i n d i c a t e s the e x i s t e n c e of an i m m i s c i b i l i t y gap, although, as d i s c u s s e d i n chapter I I I , the a c t u a l p o s i t i o n of the s o l v u s i s s t i l l i n doubt. Another solvus i s c a l c u l a t e d on the MgO-Al 20 3 b i n a r y with a c r i t i c a l temperature of 1980°C, 160° below the s p i n e l l i q u i d u s . There i s no experimental evidence f o r the occurrence of t h i s 1 24 s o l v u s , but arguments presented by Hess (1977) lend credence to i t s e x i s t e n c e . Hess p o i n t s out a c o r r e l a t i o n between s t a b i l i t y of the l i q u i d s o l u t i o n and s t a b i l i t y of l i q u i d u s phases with compositions intermediate to the components of b i n a r y systems. Conversely, i n c o m p o s i t i o n a l regions showing l i q u i d i m m i s c i b i l i t y , no i n termediate compounds form (see F i g u r e s 16 and 17). The i m p l i c a t i o n i s that e n e r g e t i c a l l y f a v o r a b l e bonds that form in s o l i d s w i l l a l s o form and s t a b i l i z e l i q u i d s . The absence of intermediate composition l i q u i d u s phases, r e f l e c t i n g energy b a r r i e r s to formation of chemical bonds, i s manifested i n l i q u i d i n s t a b i l i t y which a t t a i n s i t s lowest energy s t a t e by phase s e p a r a t i o n . In t h i s context, i t i s not s u r p r i s i n g t h at the MgO-Al 20 3 system with s p i n e l as the only intermediate compound, and the A l 2 0 3 - S i 0 2 system with m u l l i t e as the only intermediate compound, both show metastable two l i q u i d f i e l d s . A low temperature i m m i s c i b i l i t y gap i s a l s o c a l c u l a t e d i n the CaO-MgO system (Figure 21), which has only the two end-members as l i q u i d u s phases. The C a O - A l 2 0 3 b i n a r y , with four s t a b l e i n t e r m e d i a t e compounds, i s the only binary system which shows no evidence f o r s t a b l e or metastable l i q u i d i m m i s c i b i l i t y . 3. P e t r o l o g i c C o n s i d e r a t i o n s The components CaO, MgO, A l 2 0 3 , and S i 0 2 account f o r approximately 85% of the chemistry of b a s a l t s , and study of t h i s s i m p l i f i e d b a s a l t system can be p a r t i c u l a r l y i l l u m i n a t i n g because ge o m e t r i c a l r e l a t i o n s h i p s between phases can be e a s i l y a p p r e c i a t e d . Yoder and T i l l e y (1962).brought a t t e n t i o n to the 125 s t r a t e g i c p o s i t i o n of the plane a n o r t h i t e - f o r s t e r i t e - d i o p s i d e which r e p r e s e n t s a thermal b a r r i e r at low pressure s e p a r a t i n g c r y s t a l l i z a t i o n paths of a l k a l i c and t h o l e i i t i c b a s a l t s . R ecently s t u d i e d phase r e l a t i o n s i n t h i s system at e l e v a t e d p r e s s u r e s have rev e a l e d the mechanism by which t h i s thermal b a r r i e r i s broken ( P r e s n a l l et a l . , 1978), and by which mid-ocean r i d g e t h o l e i i t e s may be produced ( P r e s n a l l et a l . , 1979). E x t r a p o l a t i o n of the r e s u l t s from s i m p l i f i e d systems to g e o l o g i c a l l y r e l e v a n t compositions must be c a u t i o u s l y a p p l i e d , of course, because of the u n c e r t a i n e f f e c t s caused by a d d i t i o n a l components. A p p l i c a t i o n of the r e s u l t s of the present work to g e o l o g i c a l problems i s f u r t h e r l i m i t e d because of c o n s i d e r a t i o n of phase r e l a t i o n s i n v o l v i n g only s t o i c h i o m e t r i c phases at one atmosphere. Extension of t h i s work i n these d i r e c t i o n s should prove very v a l u a b l e , because, as d i s c u s s e d below, q u a n t i t a t i v e assessment of phase r e l a t i o n s can be made on the b a s i s of i n t e r p o l a t i o n s between experimental r e s u l t s which may be q u i t e widely separated i n temperature, composition, and p a r t i c u l a r l y i n p r e s s u r e . D i r e c t a p p l i c a t i o n s of t h i s model can p r e s e n t l y be made to problems of i n t e r e s t to the s l a g and ceramic i n d u s t r y , where i t i s important to know l i q u i d u s temperatures, compositions of i n i t i a l melts, and the l o c a t i o n of c o m p o s i t i o n a l r e g i o n s where temperature p l a t e a u s occur (e.g. Osborn et a l . , 1954). Table XVII l i s t s the temperatures and compositions of a l l the c a l c u l a t e d quaternary i n v a r i a n t p o i n t s i n t h i s system at one 126 F i g u r e 42: S t e r e o g r a p h i c p a i r of the l i q u i d u s diagram of the system CaO-MgO-Al 20 3-Si0 2. Temperatures of a l l i n v a r i a n t p o i n t s are l i s t e d i n Tables X I I , X I I I , and XVII. L i q u i d u s phase a b b r e v i a t i o n s are the f i r s t two l e t t e r s of the a b b r e v i a t i o n s used i n Table V I I I . Boundary curves between C a 2 S i O , and S i 0 2 polymorphs, the sma l l s i l l i m a n i t e f i e l d ( F i g u r e 18), and the h i g h - S i 0 2 extension of s a p h i r i n e (Figure 25) have been removed fo r the sake of c l a r i t y . The s t a b l e f i e l d of l i q u i d i m m i s c i b i l i t y at high S i 0 2 i s not shown. 127 A 128 129 Table XVII: Quaternary Invariant Points in the CaO-MgO-Si0 2-Al 20 3 Syst Phases Tc S i 0 2 A l 2 o 3 CaO MgO Anor T r i d Coru Saph 1275 81-48 14. 03 2.31 2.18 Anor Saph T r i d -Coru 1252 63.54 22. 64 5.97 7.86 Diop PrEn Anor T r i d 1191 60.72 16. 23 14.97 8.08 Hytonen and Schairer(1969) 1150110 Diop Anor T r i d PsWo 1 145 59.82 14. 42 23.64 2.12 PrEn Anor Saph T r i d 1 189 59.79 21 . 51 6.28 12.42 -Fors Diop PrEn Anor 1239 54.73 16. 30 16.10 12.87 Hytonnen and Schairer(1969) 1250±10 -Fors PrEn Anor Saph 1252 53. 10 22. 51 7.99 16.40 Fors Anor -Spin Saph 1273 51 .59 22. 84 8.70 16.87 Anor -Spin Saph -Coru 1418 47.57 32. 24 10.25 9.94 Diop Anor PsWo Aker 1217 44.81 15. 47 33.27 6.45 -Fors Diop -Anor Spin 1238 44.26 16. 58 27.81 1 1 .35 Schairer and Yoder (1969) 123813 O'Hara and Biggar (1969) 123412 -Fors Diop Spin Aker 1233 43.71 15. 56 29.56 11.17 Schairer and Yoder (1969) 123813 O'Hara and Biggar (1969) 123213 Diop Spin Aker Anor 1221 43.58 16. 58 30.58 9.26 Schairer and Yoder (1969) 123813 O'Hara and Biggar (1969) 123011 Gehl PsWo Aker Anor 1212 42.77 17. 19 35.17 4.87 -Spin Aker Anor Gehl 1219 42.32 17. 97 33.09 6.62 Rank Gehl PsWo Aker 1274 41 .27 10. 43 44.96 3.34 Rank Gehl Aker - a ' l r 1313 39.46 9. 39 47.00 4.15 -Merw Aker Gehl a'lr 1320 39.04 9. 66 46.70 4.60 -Mont Spin Fors Aker 1341 38.27 12. 93 31.51 17.28 Schairer and Yoder (1969) 134812 Gehl Aker Spin -Mont 1342 36.50 14. 07 37.18 12.25 Mont -Merw Aker Gehl 1345 36.29 13. 48 38.93 1 1 .30 -Spin Mont Gehl -Merw 1357 34.98 14. 45 38.01 12.56 Mont - P e r i Spin Fors 1395 33.30 13. 66 27.78 25.25 Biggar and O'Hara (1970) 1425 Anor Spin -Coru C,A6 141 1 32.82 38. 64 26.59 1 .95 DeVries and Osborn (1957) 1400 35. 36. 26. 3. Spin Anor Gehl -C,A6 1364 32.55 36. 68 29.26 1.51 DeVries and Osborn (1957) 1350 32. 36. 30. 2. Schairer and Yoder (1969) 136015 Spin - P e r i Merw Mont 1392 31.30 15. 50 34.53 18.67 Biggar and O'Hara (1969) 1410 Merw Spin Gehl a'lr 1399 26.91 20. 36 41.65 11 .08 - P e r i Merw a'lr Spin 1402 26.03 20. 77 41 .14 12.06 Osborn et a l . (1969) 1410 Gehl -C,A2 Spin C,A6 1467 22.37 46. 25 29.89 1 .48 DeVries and Osborn (1957) 1450 25. 44. 30. 1 . Gehl C,A, -C,A2 Spin 1485 10.12 51.03 36.65 2.20 Rao (1958) 1475 Spin C,A, a'lr -Gehl 1333 8.46 41 . 98 45.24 4.32 -Spin P e r i C,A, a'lr 1317 6.79 43. 20 45.34 4.68 Peri a'lr C 3A, -C 3S t 1404 6.13 36. 66 52.99 4.23 McMurdie and Insley (1936) 1384 7.5 34. 0 53.0 5.5 P e r i Lime C 3A, -C 3S, 1439 4.85 36. 51 53.77 4.87 McMurdie and Insley (1936) 1400 7.0 33. 5 54.0 5.5 P e r i C,A, a'lr C 3A, 1274 3.81 45. 18 48.13 2.89 - Phase i n re a c t i o n r e l a t i o n s h i p with l i q u i d Experimental i n v a r i a n t point l o c a t i o n s follow c a l c u l a t e d l o c a t i o n s 1 30 atmosphere. These were l o c a t e d by developing computer software which permits c a l c u l a t i o n of the e n t i r e quaternary l i q u i d u s diagram (Figure 42) by f o l l o w i n g each p o i n t on quaternary u n i v a r i a n t curves (Berman and Brown, 1983b). The r e a c t i o n s which occur at i n v a r i a n t p o i n t s , l o c a t e d by i n t e r s e c t i o n of these curves, can be i d e n t i f i e d by s o l v i n g the l i n e a r equation r e p r e s e n t i n g the l i q u i d composition i n terms of the s o l i d phases present at the i n v a r i a n t p o i n t . These r e s u l t s are of course a f f e c t e d by the s o l i d phase compositions, so that the type of r e a c t i o n s can change at i n v a r i a n t p o i n t s i n v o l v i n g n o n s t o i c h i o m e t r i c phases. For example, the p e r i c l a s e - s p i n e l -f o r s t e r i t e - m o n t i c e l l i t e i n v a r i a n t p o i n t changes from a p e r i c l a s e r e a c t i o n p o i n t (Table XVII) to a p e r i c l a s e and f o r s t e r i t e ( b i r e s o r p t i o n a l ) r e a c t i o n p o i n t i f m o n t i c e l l i t e s o l i d s o l u t i o n (Mo 7 8) i s c o n s i d e r e d (Biggar and O'Hara, 1970). Table XVII a l s o shows the few temperatures which have been e x p e r i m e n t a l l y determined or estimated f o r quaternary i n v a r i a n t p o i n t s . The c a l c u l a t e d temperatures are g e n e r a l l y w i t h i n 10°, and many are on the low s i d e of experimental temperatures of i n v a r i a n t p o i n t s with n o n s t o i c h i o m e t r i c m i n e r a l s . T h i s i s c o n s i s t e n t with t h e i r c a l c u l a t i o n f o r s t o i c h i o m e t r i c phases. The En-An-Di-Tr i n v a r i a n t p o i n t i s c a l c u l a t e d 40° higher than that determined by de N e u f v i l l e and S c h a i r e r (1962); t h e i r estimated 10° u n c e r t a i n t y can not be assessed from the data they r e p o r t . The compositions of quaternary i n v a r i a n t p o i n t s have not been l o c a t e d e x p e r i m e n t a l l y , although s e v e r a l i n high alumina compositions have been estimated by DeVries and Osborn (1957), 131 while others i n g e o l o g i c a l l y more a c c e s s i b l e r e g i o n s have been l o c a t e d w i t h i n four-phase s u b s o l i d u s volumes ( S c h a i r e r and Yoder, 1969). Determination of quaternary i n v a r i a n t p o i n t compositions by the c l a s s i c a l method of quenching i s d i f f i c u l t because of the numerous experimental charges necessary to bracket such compositions. T h i s problem underscores one of the primary a s s e t s of t h i s thermodynamic model i n p r o v i d i n g a method of i n t e r p o l a t i o n which allows c a l c u l a t i o n of these i n v a r i a n t p o i n t compositions. I n s p e c t i o n of F i g u r e 42 shows the l a r g e p o r t i o n of the quaternary system which i s occupied by the phase volumes of s p i n e l and p e r i c l a s e . P r o j e c t i o n s provide a u s e f u l way of r e p r e s e n t i n g multicomponent phase r e l a t i o n s , and F i g u r e 43 shows a l l the s p i n e l - p r e s e n t u n i v a r i a n t r e a c t i o n s i n the quaternary system p r o j e c t e d from s p i n e l onto the CaO-MgO-Si0 2 plane. T h i s p r o j e c t i o n c l e a r l y shows the three d i o p s i d e - s p i n e l present i n v a r i a n t p o i n t s which O'Hara and Biggar (1969) demonstrated are more s t a b l e than the assemblage a n o r t h i t e - a k e r m a n i t e - f o r s t e r i t e -s p i n e l suggested by Chinner and S c h a i r e r (1962). The model c a l c u l a t i o n s a l s o agree with O'Hara and Biggar's c o n c l u s i o n that both a n o r t h i t e and f o r s t e r i t e are i n r e a c t i o n r e l a t i o n s h i p with l i q u i d over a small range of temperatures and compositions between the maximum on the f o r s t e r i t e - a n o r t h i t e - s p i n e l curve and the f o r s t e r i t e - a n o r t h i t e - s p i n e l - d i o p s i d e i n v a r i a n t p o i n t . T h i s i s at odds with the i n t e r p r e t a t i o n of S c h a i r e r and Yoder (1969), who i n f e r a temperature maximum on the f o r s t e r i t e - s p i n e l -d i o p s i d e u n i v a r i a n t curve. 132 Figure 43: Spinel projection of liquidus relations in the system CaO-MgO-Al 20 3-Si0 2. Spinel i s present in a l l portions of the projection. Temperatures of invariant points are l i s t e d in Table XVII. Abbreviations as in Table VIII. 133 Use of p r o j e c t i o n s makes i t p o s s i b l e to observe geometric r e l a t i o n s h i p s between the l i q u i d and s o l i d phases, although c a u t i o n must be e x e r c i s e d i n i n t e r p r e t a t i o n s because of d i s t o r t i o n s t hat can a r i s e even with l e g a l p r o j e c t i o n s . P r o j e c t i o n s from other compositions can be used to check f o r such d i s t o r t i o n , but d i r e c t c a l c u l a t i o n of multicomponent phase r e l a t i o n s o f f e r s the only r e l i a b l e technique f o r v e r i f i c a t i o n of observed g e o m e t r i c a l r e l a t i o n s h i p s . One p a r t i c u l a r l y u s e f u l a p p l i c a t i o n would be the s u b s t a n t i a t i o n of p r o j e c t e d phase boundary c u r v a t u r e s which have been used to p r e d i c t c r y s t a l l i z a t i o n sequences i n magma mixing processes (e.g. Walker et a l . , 1979). D. CONCLUSIONS The r e s u l t s presented i n t h i s chapter demonstrate that the s t o i c h i o m e t r i c - M a r g u l e s s o l u t i o n model developed i n chapter II i s s u f f i c i e n t to f i t the CaO-MgO-Al 20 3-Si0 2 l i q u i d u s r e l a t i o n s h i p s , while a l s o s a t i s f y i n g a v a i l a b l e c a l o r i m e t r i c and phase e q u i l i b r i a c o n s t r a i n t s on the thermodynamic p r o p e r t i e s of minerals i n t h i s system. The l i q u i d i of 24 phases s t a b l e on the one atmosphere l i q u i d u s i n t h i s system are reproduced w i t h i n estimated u n c e r t a i n t i e s of the experimental data. I n c o n s i s t e n c i e s i n the experimental data probably r e s u l t from inhomogeneities i n s t a r t i n g m a t e r i a l s . Use of t h i s model r e s u l t s i n s u c c e s s f u l p r e d i c t i o n of the thermodynamic p r o p e r t i e s of the melt ( a c t i v i t i e s , heats of mixing, heats of f u s i o n ) , and e s t i m a t i o n of the thermodynamic p r o p e r t i e s of l i q u i d u s phases f o r which c a l o r i m e t r i c data are not a v a i l a b l e . 134 I t must be emphasized that the thermodynamic p r o p e r t i e s of s o l i d and l i q u i d phases presented i n chapters I I I and IV must be c o n s i d e r e d as s e t s i n which no value can be changed independently of the o t h e r s . I f c o n s i s t e n c y i s to be maintained with a l l a v a i l a b l e data, any r e v i s i o n of phase r e l a t i o n s or thermochemical values can be accomplished only by r e f i t t i n g the c o n s t r a i n t set i n i t s e n t i r e t y . Accurate r e p r o d u c t i o n of l i q u i d u s r e l a t i o n s i n the quaternary system CaO-MgO-Al 20 3-Si0 2 f u r t h e r s the p o s s i b i l i t y that an e x t e n s i o n of t h i s model may be able to reproduce phase r e l a t i o n s i n higher component systems approaching the complexity of n a t u r a l magmas. The disadvantage of r e q u i r i n g a great number of mixing parameters to account f o r n o n - i d e a l i t y i n the l i q u i d i s balanced by the c a p a b i l i t y to a c c u r a t e l y c a l c u l a t e e q u i l i b r i a i n a l l p o r t i o n s of the component sub-spaces. T h i s a b i l i t y to i n t e r p o l a t e between e x i s t i n g experimental data can provide the p e t r o l o g i s t with f r u i t f u l i n s i g h t s f o r f u r t h e r r e s e a r c h . For example, c a l c u l a t e d i n v a r i a n t p o i n t p o s i t i o n s (Table XVII) can serve as guides to the compositions of u s e f u l experimental mixes. A d d i t i o n a l experiments are necessary to i n v e s t i g a t e the i n c o n s i s t e n c i e s which have been p o i n t e d out among the experimental data and the model c a l c u l a t i o n s , as w e l l as check f o r the e x i s t e n c e of the c a l c u l a t e d metastable i m m i s c i b i l i t y gaps on the CaO-MgO and MgO-Al 20 3 b i n a r i e s . Much a d d i t i o n a l experimental work i s a l s o necessary to assess the nature and extent of s o l i d s o l u t i o n s i n l i q u i d u s phases i n t h i s system. Use of the e l e c t r o n microprobe should g r e a t l y f a c i l i t a t e t h i s 1 35 process (e.g. Longhi and Hays, 1979). In a d d i t i o n to o f f e r i n g a technique with which g e o l o g i s t s may u l t i m a t e l y , be abl e to approach p e t r o l o g i c a l problems r e q u i r i n g q u a n t i t a t i v e p r e d i c t i o n of phase r e l a t i o n s , t h i s model allows f o r the d e r i v a t i o n of thermochemical p r o p e r t i e s and heats of f u s i o n of l i q u i d u s m i n e r a l s i n the absence of c a l o r i m e t r i c data, as w e l l as the c a l c u l a t i o n of l i q u i d a c t i v i t i e s and mixing c h a r a c t e r i s t i c s . In t h i s regard, i t should be s t r e s s e d t h a t , although there i s s t i l l a great need f o r much a d d i t i o n a l phase e q u i l i b r i u m data, thermodynamic a n a l y s i s of the a v a i l a b l e data can do much to supplement the present s p a r c i t y of data concerning the p r o p e r t i e s of s i l i c a t e l i q u i d s themselves. 136 BIBLIOGRAPHY 1. Adams, L.H. and Cohen, L.H., 1966. 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The e f f e c t of reduced a c t i v i t y of a n o r t h i t e on the r e a c t i o n g r o s s u l a r + quartz = a n o r t h i t e + w o l l a s t o n i t e ; a model f o r p l a g i o c l a s e in the Earth's lower c r u s t and upper mantle. Amer. M i n e r a l o g i s t 61, 889-896. 169. Winter, J.,K. and Ghose, S., 1979. Thermal expansion and high-temperature c r y s t a l chemistry of the A l 2 S i 0 5 polymorphs: Amer. M i n e r a l o g i s t 64, 573-586. 170. Wohl, K., 1946. Thermodynamic e v a l u a t i o n of b i n a r y and t e r n a r y l i q u i d systems. T r a n s a c t i o n s of the Amer. I n s t . Chem. Eng. 42, 215-249. 171. Wood, B.J. and N i c h o l l s , J . , 1978. The thermodynamic p r o p e r t i e s of r e c i p r o c a l s o l i d s o l u t i o n s . C o n t r i b . M i n e r a l . P e t r o l . 66, 389-400. 172. Yoder, H.S., J r . , 1968. Akermanite and r e l a t e d m e l i l i t e -b e a r i n g assemblages. Carnegie I n s t . Wash. Yb. 66, 471-477. 173. Yoder, H.S., J r . , 1976. Generation of b a s a l t i c magma. N a t i o n a l Academy of Science, Washington, D.C. 174. Yoder, H.S. and T i l l e y , C.E., 1962. O r i g i n of b a s a l t i c magmas: an experimental study of n a t u r a l and s y n t h e t i c rock systems. J . P e t r o l . 3, 342-532. 150 APPENDIX A - C a l c u l a t e d L i q u i d u s Diagrams For The Quaternary C a l i b r a t i o n J o i n s L i q u i d u s diagrams are presented i n the same order as Table 151 152 CaAl 2Si 2° 8 CaSi0 3 Weight Fraction CaMgSi20 153 CaAl 2Si 2° 8 CaMgSI 2 0 6 Weight Fraction Mg 2Si0 4 154 CaAl 2Si 2° 8 Ca 2 AI 2 Si0 7 Weight Fraction MgAI 20 4 155 MgAI 2 0 4 0.2 0.4 0.6 0.8 MgO Weight Fraction Ca2SiO 156 157 1600 -A 1500 H o 5 1400 1300 H Q . E 1200 1 1 0 0 H 1 0 0 0 H I 1 ' Anorthite \ + Liquid \ / Akermanite + Liquid Anorthite + Akermanite - r 1 1 0.0 0.2 0.4 0.6 0.8 ^ 1.0 CaA I 2 S i 2 0 8 Weight Fraction Ca 2 MgS i 2 0 7 1 58 APPENDIX B - C a l c u l a t e d L i q u i d u s Diagrams For Quaternary J o i n s Not Used In C a l i b r a t i o n L i q u i d u s diagrams are presented i n the same order as Table X I I I . Only primary l i q u i d u s r e l a t i o n s are p o r t r a y e d i n pseudobinary s e c t i o n s . 159 MgAI 2 0 4 160 A l 2 0 3 Ca 9 AI 9 Si0 7 Weight Fraction MgAI20 161 162 163 164 M g A I 2 0 4 C a 2 M g S i 2 0 7 Weight Fract ion C a A I 2 S i 0 8 165 2200 -4 2000 H o o 0 ) 1800 H Coru + I g_ 1600 E a> Spin + I 1400 -L Diop + I 1200 0.0 Fors + I 0.2 CaMgSi 2 0 6 0.4 0.6 0.8 Weight Fraction 1.0 A l 2 0 166 MgAI204 CaMgSi 20 6 Weight Fraction CaAI 2Si 20 167 168 1200 H , 1 1 1 T 0.0 0.2 0.4 0.6 0.8 1.0 CaMgSi0 4 Weight Fraction MgAI20 169 2200 - I J 2000 -Spin + I ¥ 1 8 0 0 -— Temperature o o o o \ Larn + I / \ U / Peri + I 1200 -, — • DOH 1 1 1 1 7^ 0.0 0.2 0.4 0.6 0-8 1.0 Ca 3 MgS i 2 0 8 Weight Fraction MgAI 20 4 170 Ca 2SiO 4 MgO Weight Fraction Ca 5 A I 6 0 1 4 171 CaAI2Si 2 0 8 CaMgSi 20 6 Weight Fraction Ca 2 MgSi 2 0 7 172 1 6 0 0 -4 1 5 5 0 H 1 2 5 0 H 1200 -f 1 1 1 1 0.0 0.2 0.4 0.6 0.8 1.0 CaAI 2 Si0 6 Weight Fraction CaMgSi 2 173 1600 - • i i 1550 -1500 -CJ o w 1450 -<D -3 1400-D 0) 9-1350 -Spin + I E 1300-.Gehl + I / D 1250-ionn - i 1 1 0.0 0.2 0.4 0.6 0.8 u C a 3 A I 2 S i 3 0 1 2 Weight Fraction Mg 3 AI 2 Si 3 174 MgAl 2 0 4 0.2 0.4 0.6 0.8 C a 3 M g S i 2 0 8 Weight Fraction Ca 2AI 2SiO 175 176 177 CaAl 2 Si 2 0 8 0.2 0.4 0.6 0.8 MgSi0 3 Weight Fraction CaMgSi20 178 1 2 5 0 H 1200-1 1 1 1 — i r 0.0 0.2 0.4 0.6 0.8 1.0 CaMgSi 2 0 6 Weight Fraction Mg 3AI 2Si 30 P u b l i c a t i o n s 1. Berman, R.G. and Brown, T.H. 1982. Thermodynamic treatment of l i q u i d u s r e l a t i o n s i n multicomponent systems. GSA A b s t r a c t s with Programs, 14, p. 443. 2. Berman, R.G. and Brown, T.H. 1981. A thermodynamic model f o r s i l i c a t e melts. GSA A b s t r a c t s with Programs, 13, p. 409. 3. Berman, R.G. 1981. D i f f e r e n t i a t i o n of c a l c - a l k a l i n e magmas: evidence from the C o q u i h a l l a V o l c a n i c Complex, B r i t i s h Columbia. J o u r n a l of Volcanology and Geothermal Research, 9, p. 151-179. 4. Mathews, W.H., Berman, R.G. and Harakal, J.E. 1981. Mid-T e r t i a r y v o l c a n i c rocks of the Cascade Mountains, southwestern B r i t i s h Columbia, ages and c o r r e l a t i o n s . Canadian J o u r n a l of E a r t h S c i e n c e s , 18, p. 622-664. 5. Berman, R.G., Armstrong, R.L. and Mathews, W.H. 1981. The C o q u i h a l l a V o l c a n i c Complex and Pemberton V o l c a n i c B e l t of southwestern B r i t i s h Columbia. G e o l o g i c a l A s s o c i a t i o n of Canada Programs and A b s t r a c t s ( C o r d i l l e r a n S e c t i o n ) , p. 8. 6. Berman, R. G. and Armstrong, R.L.A. 1980. Geology of the C o q u i h a l l a V o l c a n i c Complex, southwestern B r i t i s h Columbia. Canadian J o u r n a l of E a r t h S c i e n c e s , 17, p. 985-995. 7. Reid, J.B., J r . and Berman, R.B. 1975. Mantle processes beneath Hawaii. EOS, 56, p.1077. 8. Berman, R.B., H a r r i n g t o n , B.G., Woods, J.A. and Reid, J.B., J r . 1975. The p e t r o l o g y of deep-seated cumulate i n c l u s i o n s , West P o t r i l l o Mountains, New Mexico. EOS, 56, p. 1069. 

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