Pseudobinary diffusion coefficients in the Ti-Mo-Ta system
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e (x 2 x )dc 0
1
˜ 311 5 22t ? D
c1 ˜ 312 c2 2 2t ? D x x
˜ 321 5 22t ? D
c1 ˜ 322 c2 2 2t ? D x x
c1
c22
e (x 2 x )dc 0
2
c2
˜ 311 and D ˜ 322 are named direct coefficients; D ˜ 312 and where D 3 ˜ D21, cross ones; t, time of annealing; x, the position relative to the Matano plane x0; and c1 and c2, the atomic concentration of, respectively, elements 1 and 2. Coefficient values are usually represented by plotting constant value curves on the corresponding[2,3] Gibbs diagram. The inaccuracy of this method has already been discussed[4] and is mainly due to the determination of the intersections of the diffusion paths. Other methods have been presented without this determination. The diffusion characteristics are usually described for discrete values of the ternary diagram, using the[5,6] zero flux plane method, the[7,8] effective interdiffusion coefficients determination, or more recently the[9] Kailasam simplified method. Moreover, the presentation of the ternary diffusion characteristics remains less explicit than the binary coefficient one, especially when cross coefficients are significant. Therefore, to improve the readability of ternary diffusion results, we suggest, on the basis of binary and ternary diffusion studies in the system Ti-Mo-Ta, the use of ˜ TiX. This last, named pseudobinary a single coefficient D diffusion coefficient, results from a new extension of the classical Matano method to the ternary diffusion, the Boltzmann–Matano integral equation becoming cTi2
e (x 2 x )dc 0
Ti
˜ TiX 5 22t ? D
cTi
activation energy DE and frequency factor D0 in accordance with a monovacancy diffusion mechanism. Results are in good agreement with Ivanov’s[10] related to the temperature range 1900 8C to 2300 8C. These results will be published later. A ternary diffusion study was also carried out at 1500 8C in the Ti-rich region of the Ti-Mo-Ta diagram. Calculations
(a)
(b)
cTi(x) x
First, the interdiffusion study in the binary system Mo-Ta was carried out between 1500 8C and 1900 8C. Experimental methods were previously described.[10,11] Concentration profiles were obtained by electron probe micro analysis (Camebax SX50) on a polished section parallel to the diffusion direction. Figure 1(a) related to the Mo-Ta couple annealed at 1900 8C for 143 hours presents typical profiles. Both concentration profiles appear symmetric and are characteristic of almost constant coefficients with the composi˜ ) 5 f(1/T ) are associated with tion. Arrhenius plots log (D FABIEN GUILLEMOT, Ph.D. Student, JEAN DEBUIGNE, Emeritus Professor of Metallurgy, and DENIS ANSEL, Professor of Metallurgy and Head, are with the Laboratoire de Me´tallurgie, GRCM, Institut National des Sciences Applique´es, 35043 Rennes Cedex, France. Manuscript submitted March 27, 2000. 3198—VOLUME 31A, DECEMBER 2000
(c) Fig. 1—Concentration profiles of (a) Mo-Ta couple (1900 8C), (b) TiMo65Ta35 couple (1500 8C), and (c) TiMo4.06-TiTa4.06 (1500 8C). METALLURGICAL AND MATERIALS TRANSACTIONS A
˜ TiX with Ternary Diffusion Table I. Compar
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