Numerical techniques for the calculation of multicomponent phase diagrams
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a p r e v i o u s p a p e r , 1 a method w a s p r e s e n t e d for the c a l c u l a t i o n of the c o m p o s i t i o n s of two p h a s e s in e q u i l i b r i u m in a b i n a r y s y s t e m . In a t e r n a r y s y s t e m , if t h r e e p h a s e s a r e in e q u i l i b r i u m at a given t e m p e r a t u r e , t h e i r c o m p o s i t i o n s a r e fixed and can be s i m p l y obtained as i n t e r s e c t i o n s of two t i e - l i n e s ( d e t e r m i n e d f r o m the c a l c u l a t i o n of t w o - p h a s e e q u i l i b r i a ) . In a q u a t e r n a r y o r h i g h e r o r d e r s y s t e m , the d o m a i n s c o r r e s p o n d i n g to the e q u i l i b r i u m c o e x i s t e n c e of m o r e than two p h a s e s m a y s t i l l be deduced f r o m the e q u i l i b r i u m between two p h a s e s (e.g., as the i n t e r s e c t i o n of two s u r faces). T h u s a m e t h o d to c a l c u l a t e the c o m p o s i t i o n of two p h a s e s in e q u i l i b r i u m is s u f f i c i e n t to c a l c u l a t e a ternary phase diagram in its entirety, and is necessary for the calculation of higher order phase diagrams. The method presented for binary systems I consists in solving two equations expressing the equality of the chemical potentials of each component in the two phases. A method based on the same principle has been used by Kaufmaa et al. a'a to compute isothermal sections of ternary phase diagrams. It uses a NewtonRaphson iteration technique to solve the system of the three equations expressing the equality of the chemical potentials (of each of the three components in the two phases) and is similar to that devised by Hurle and Pike, 4 also for ternary mixtures. Ansara et al. ~ s t a r t with e s s e n t i a l l y the s a m e e q u a t i o n s but solve t h e m by a d i f f e r e n t technique b a s e d on a w i n n o w ing of c o m p o s i t i o n s . T h e s e methods b e c o m e s o m e w h a t c o m b e r s o m e with a l a r g e n u m b e r of c o m p o n e n t s and a d i r e c t m i n i m i z a t i o n of the Gibbs f r e e e n e r g y of the s y s t e m a p p e a r s p r e f e r a b l e . S p e n c e r et al. 6 u s e d this p r i n c i p l e to adapt a s i m p l e x t e c h n i q u e developed by N e l d e r and Mead. 7 The p r e s e n t c o n t r i b u t i o n adopts the s a m e p r i n c i p l e but develops a d i f f e r e n t technique b a s e d on a q u a d r a t i c r e p r e s e n t a t i o n of the Gibbs f r e e e n e r g y of the s y s t e m . (This technique was b r i e f l y outlined in a p r e v i o u s p u b l i c a t i o n ; 8 it is p r e s e n t e d h e r e in d e t a i l ) .
HENRI GAYE, formerly Graduate Student, Department of Metallurgy and Materials Science, Carnegie-MellonUniversity, is now with IRSID, Maizieres-les-Metz,France. C. H. P. LUPIS is Professor of Metallurgy and Materials Science, Carnegie-MellonUniversity, Pittsburgh, Pa. 15213. Manuscript submitted October 11, 1974. METALLURGICALTRANSACTIONSA
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