The shapes of the phase boundaries of two ideal solution phases in ternary and higher order systems
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The authors are grateful to Dr. S. Banerjee for many helpful discussions.
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~X k,--4
REFERENCES 1. D.F. Paulonis, J.M. Oblak, and D.S. Duvall: Trans. ASM, 1969, vol. 62, pp. 611-22. 2. Ya-fang Han, P. Deb, and M.C. Chaturvedi: Met. Sci., 1982, vol. 16, pp. 555-61. 3. M. Sundararaman, P. Mukhopadhyay, and S. Banerjee: Metall. Trans. A, 1992, vol. 23A, pp. 2015-28. 4. M.C. Chaturvedi and Ya-fang Han: Met. Sci., 1983, vol. 17, pp. 145-49. 5. J.M. Oblak, D.S. Duvail, and D.F. Paulonis: Metall. Trans., 1974, vol. 5, pp. 143-53. 6. M. Sundararaman, P. Mukhopadhyay, and S. Banerjee: Acta Metall., 1988, vol. 36, pp. 847-64. 7. C. Crussard and B. Jaoul: Rev. Met., 1950, vol. 57, p. 589.
The Shapes of the Phase Boundaries of Two Ideal Solution Phases in Ternary and Higher Order Systems
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A o
2o
656--VOLUME 25A, MARCH 1994
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lOOB
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Let x7j and x/aj be the mole fractions of the component i of the points 0[j and/3j, respectively. For these three tielines, according to the conditional equations for the existence of a two-phase equilibrium, there exist nine equations as given subsequently~ Gi"+RTlnxTJ=G]t3+RTlnx~(i,j=
1,2,3)
[1]
where G] ~ and G] a are the lattice stabilities of the component i exhibiting the 0[ and/3 structures, respectively, R is the gas constant, and T is temperature. Equation [I] can also be written in the following form:
xT' x7~ G~ - G 7 = RT In ~ = RT In x~--S x7 = RTln--z: xF'
(i = 1, 2, 3)
[2]
or
xT'
x7
x7
xh
xh
g'
-exp\
(oi ~ - o7] RT
./ ( i = 1 , 2 , 3 )
[3]
Define ( G] tJ Ki-exp\.-RT
-
GT]
-/(i=1,2,3)
[4]
Then, Eq. [3] becomes
x?
x7
xh
xf2 xh
--=
SHUANG-LIN CHEN, Research Assistant, and Y. AUSTIN CHANG, Wisconsin Distinguished Professor, are with the Department of Materials Science and Engineering, University of WisconsinMadison, Madison, WI 53706. Manuscript submitted September 8, 1993.
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Fig. 1 - A schematic isothermal-isobaric diagram for a ternary system A-B-C. The phases a and/3 are ideal solutions, and tz~-/3~, a2-/32, et3-fl3, a'-fl', and t~"-/3" are tie-lines of the a-fl phase equilibrium.
SHUANG-LIN CHEN and Y. AUSTIN CHANG The shapes of the phase boundaries of a two-phase region in a ternary isothermal-isobaric diagram depend on the lattice stabilities and excess Gibbs energy terms of these two phases. These boundaries can be obtained numerically by solving the thermodynamic conditional equations. However, there are no analytical solutions to these conditional equations. Recently, when calculating isothermal-isobaric diagrams of a ternary system, we found that the phase boundaries of a two-phase region are straight lines if the two phases are ideal solutions. The objective of the present study is to show rigorously that this is indeed the case, starting from the conditional equations for the existence of a two-phase equilibrium. Figure 1 shows an isothermal-isobaric diagram for a ternary system A-B-C. Assume that the behaviors of the phases a and/3 are ideal and that Ogl-/31, 0[2-/32, and 013-fl3 are three arbitrary ti
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