Evaluation of Molar Volume Effect for Calculation of Diffusion in Binary Systems
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ssion by Boltzmann[1] and Matano[2] was developed to calculate diffusivity, which is dependent on the concentration composition of the diffusion profile of binary systems. Z c c ¼ 1 @x ðx x0 Þdc ½1 D 2t @c c¼c c where x0 is a point that corresponds to the so-called Matano interface[2] for which it applies as follows: Z cþ ðx x0 Þ dc ¼ 0 ½2 c
The c- and c+ represent the left- and the righthand-side molar fractions of a component at the ends of the diffusion profile. The calculation of diffusivity according to relation [1] primarily requires the determination of the Matano interface. Moreover, relation [1] reflects the concentration diffusion profile in an inverse form. The practical simplification of the Boltzmann-Matano procedure is obtained from the De Broeder modification[3] based on the expression of the relative MAREK VACH and MARTIN SVOJTKA, Scientists, are with the Institute of Geology, Academy of Sciences of the Czech Republic v.v.i., 165 00 Prague 6, Czech Republic. Contact e-mail: [email protected] Manuscript submitted December 2, 2011. Article published online August 14, 2012. 1446—VOLUME 43B, DECEMBER 2012
concentration Y ¼ ðc c Þ=cþ c . The resulting relation no longer requires the determination of the Matano interface, and the diffusion profile is included as a functional dependence of the concentration on the distance c = f(x). 2 Zx 1 4ð1 Y Þ c ¼ ðcðxÞ c Þdx D 2tð@c=@xÞx¼x 1 3 Z1 ðcþ cðxÞÞdx5 ½3 þY x
Relation [3] does not include the change in molar volume in dependence on the concentration of the diffusing components. For the calculation of the diffusivity from the diffusion profiles with an indispensable change in molar volume, a treatment can be applied as proposed by Sauer and Freise[4] and by Wagner.[5] 2 Zx ð c c ÞV Y þ m c ¼ 4ð1 Y Þ D dx Vm 2tð@c=@xÞx¼x 1 3 Z1 ð1 YÞ 5 þY dx ½4 Vm x
The relative concentration Y has the same meaning as in relation [3] Y ¼ ðc c Þ=ðcþ c Þ. METALLURGICAL AND MATERIALS TRANSACTIONS B
The behavior of the changes in molar volume in a binary diffusion profile may be linear, which corresponds to Vegard’s rule. Vm ¼ VA NA þ VB NB
½5
However, Vegard’s rule does not apply to a number of alloys. An alloy does not behave as an ideal mixture, and the course of the changes in its molar volume is nonlinear. The positive deviation from Vegard’s rule represents the swell form of this course, and the negative deviation represents the shrink. The subject of this article is the theoretical evaluation of the effect of molar volume on the concentration dependent diffusivity of binary systems that are calculated using the analytical integration of relation [4] with the substitution of a suitable regression function that represents any form of concentration diffusion profile.
By accounting for Eq. [7], relation [3] takes the form following the substitution of the regression function [6]. " ! # 2 X 1 a i c ¼ @ þ dc þ e =@ D 2t c þ bi i¼1 cc¼c " Z " # 2 cþ X 1 1 Y ai þ dðc c Þ dc c þ bi c þ bi c i¼1 ! 3 Zc X 2 ai a
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