Further investigation of the determination of solute interaction parameters by analysis of phase equilibria using a line
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[I]
where y~ is the Raoultianactivity coefficient of component 1 and XI is the atom fraction of component 1. They used a composition and temperature dependent representation for otx2 of the form oq2 = (A1 + A 2 / T ) + (As + A4//T)X2
[2]
w h e r e the A j ' s are adjustable constants (the so-called interaction parameters) and T is absolute temperature. This formalism allows In y~ to have an i n v e r s e temperature dependence and a power s e r i e s dependence on composition. Through the use of Eq. [2], Rao et al~ were able to correlate and reduce the experimental data for a given system into this simple orparameter representation. It is possible to extend the a - p a r a m e t e r representation to h i g h e r o r d e r s as shown in Eq. [3] 7/ ~ (A2i+1 + A 2 i , 2 / T ) X ~ [3] i=o w h e r e n is the o r d e r of the a - p a r a m e t e r . The lead-tin system has been reevaluated u s i n g a h i g h e r o r d e r representation than that employed by Rao et a l ) In the present study the simplex technique for the generation of the c~-parameter constants was explored and compared with a l e a s t squares technique. It was found t h a t , although the resulting s - p a r a m e t e r constants generated by these two techniques were s i m i l a r , the least squares technique was superior in many respects t o the simplex method. One distinct advantage of the l e a s t squares method is that it has a much s m a l l e r size for the computational m a t r i x than the simplex technique. If the set of data t o be analyzed is l a r g e , then computer storage of the l a r g e simplex computational m a t r i x may present a problem. Also manipulation of this l a r g e m a t r i x r e q u i r e s much longer computational time with simplex than it does with the l e a s t squares method. Another advantage of the l e a s t squares technique is that negative or-parameter constants can be determined directly. Since simplex can generate only or,2 =
S. K. TARBYand C. J. VAN TYNE are Professor and Graduate Student, respectively, Department of Metallurgy and Materials Science, Lehigh University, Bethlehem, PA 18015. M. L. BOYLE, formerly GraduateStudent,Lehigh University, is now Materials Engineer, United States Nuclear Regulatory Commission, Washington, D.C. 20555. Manuscript submitted September13, 1976. METALLURGICAL TRANSACTIONS B
positive results, r a n g e values1 must be used to offset the problem. By subtracting the r a n g e values from the calculated constants, the true values of the a parameter constants become known, but this procedure is not efficient and is a nuisance. In reexamining the eutectic lead-tin system, it was a s s u m e d that the lattice stability parameters 2 were equal to zero, i . e . , A G ~ ~ ~' = A G ~ ~/3) = 0
[4]
where ot represents the crystal structure of pure lead and [3 the crystal structure of pure tin. This assumption was also implicitly made by Rao et a l .1 It simplifies the analysis of the lead-tin system in that only two phases are c o n s i d e r e d - a solid and a liquid. Howev
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