The gibbs-thomson effect in dilute binary systems
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perature dependence of the specific strength for the tested alloys
2␣V m RTr
冢
冣
[1]
where xB␣(r) and xB␣(⬁) are the equilibrium atomic fractions of B in ␣ at a curved interface of a spherical precipitate ( ) of radius r and a planar interface, respectively. The term ␣ is the specific interfacial free energy, which is assumed to be isotropic and independent of precipitate size (r), and V m is the molar volume of . When  is not pure B, e.g., its composition is given by xB, a number of different expressions have been proposed or employed. These include[3–11] 2␣V m
xB␣(r) ⫽ xB␣(⬁) exp
冢 x RTr 冣
[2]
xB␣(r) ⫽ xB␣(⬁) exp
2␣V B RTr
[3]
 B
冢
冣
and xB␣(r) ⫽ xB␣(⬁) exp
Fig. 4—Specific strength of (Ir,Rh)75Nb15Ni10 alloys compared with other high-temperature materials.
alloys based on combining Ir with Rh, which has higher specific strengths compared with Ir-, Rh-, and Ta-based alloys and is comparable to W-based alloys at 1600 ⬚C.
REFERENCES 1. Y. Yamabe-Mitarai, Y. Koizumi, H. Murakami, Y. Ro, T. Maruko, and H. Harada: Scripta Metall., 1997, vol. 36, pp. 393-98. 2. Y. Yamabe-Mitarai, Y. Ro, T. Maruko, and H. Harada: Metall. Mater. Trans. A, 1998, vol. 29A, pp. 537-49. 3. Y.F. Gu, Y. Yamabe-Mitarai, and H. Harada: in Iridium, E.K. Ohriner, R.D. Lanam, P. Panfilov, and H. Harada, eds., TMS, Warrendale, PA, 2000, pp. 73-84. 4. Y.F. Gu, Y. Yamabe-Mitarai, Y. Ro, T. Yokokawa, and H. Harada: Metall. Mater. Trans. A, 1999, vol. 30A, pp. 2629-39. 5. Y.F. Gu, Y. Yamabe-Mitarai, X.H. Yu, and H. Harada: Mater. Lett., 1999, vol. 41, pp. 45-51. 6. N.S. Stoloff and C.T. Sims: in Superalloy II, C.T. Sims, N.S. Stoloff, and W.C. Hagel, eds., John Wiley & Sons, New York, NY, 1987, p. 519. 7. H. Tanaka, Y. Tan, A. Kasama, and R. Tanaka: Proc. Int. Conf. on Advanced Technology in Experimental Mechanics’ 99, Japan Society of Mechanical Engineers, Tokyo, Japan, 1999, p. 539. METALLURGICAL AND MATERIALS TRANSACTIONS A
1 ⫺ xB␣(⬁) 2␣V m xB(⬁) ⫺ xB␣(⬁) RTr
冢
冣
[4]
Equation [3], in which V B is the partial molar volume of B in , is given by Gaskell in a well-known textbook.[6] Unfortunately, this is an incorrect expression as pointed out by Hillert.[12] To illustrate this, one may consider such a case where the partial volume of an interstitial is so small that it can be neglected. Equation [3] says that xB␣(r) should then be independent of precipitate, but from a molar Gibbs energy diagram, one can see that it is not independent.[12] The reason Eq. [3] is incorrect may be elaborated as follows. As can be seen from Reference 6, until Eq. [77], Gaskell is only considering a single phase and Eq. [77] holds under constant composition (i.e., a constant xB). However, it should be noted that one cannot define the partial molar volume or any other partial molar quantities of a component in a single phase with a fixed composition unless the phase under consideration is in equilibrium with another phase.[12] Also, when considering a two-phase equilibrium, Gaskell’s Eq. [78], reproduced as Eq. [3] above, is derived
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