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In 1948, Lawrence Stamper Darken published a paper entitled "Diffusion, Mobility and Their Interrelation through Free Energy in Binary Metallic Systems" in which he derived two equations describing solid-state diffusion in binary solutions. Specifically, the equations Darken created relate “binary chemical diffusion coefficient to the intrinsic and self diffusion coefficients.”〔(), Trimble, L.E., D. Finn, and A. Cosgarea, Jr. "A Mathematical Analysis of Diffusion Coefficients in Binary Systems." ''Acta Metallurgica'' 13.5 (1965): 501-507. Web.〕 The equations apply to cases when a solid solution's two interdiffusing components do not have the same coefficient of diffusion. The result of this paper had a large impact on the understanding of solid state diffusion and as a result the equations have come to be known as “Darken’s equations.” Darken's first equation is: : Darken's first equation is used to calculate marker velocity, given here as , in respect to a binary system where the different components have their own corresponding diffusion coefficients, D1 and D2, as was discussed in the Kirkendall experiment.〔Darken, L. S. "Diffusion, mobility and their interrelation through free energy in binary metallic systems." Trans. Aime 175.1 (1948): 184-194.〕 The marker velocity is in terms of length per unit time and the diffusion coefficients are in terms of length squared per unit time. The variables N1 and N2 represent the atom fraction of the corresponding component. In addition, the variable x is the distance term. It is important to note that this equation only holds in situations where the total concentration remains constant. For a binary system this is defined by C1 + C2 = C, where C is the overall concentration of the system that remains constantand C1 and C2 are the corresponding component's concentration. This is equivalent to saying that the partial molar volumes of the two components are constant and equal.〔(), Sekerka, R.F. "Similarity Solutions for a Binary Diffusion Couple with Diffusivity and Density Dependent on Composition." ''Progress in Materials Science'' 49 (2004): 511-536. Print.〕 In addition, the ends of the system need to be fixed in position for the equation to hold. These constraints will be further analyzed in the derivation. Darken's second equation is: : Darken's second equation is used to calculate the chemical diffusion coefficient (also known as the inter-diffusion coefficient), , for a binary solution.〔 The variables N and D are the same as previously stated for Darken's first equation. In addition, the variable a1 is the activity coefficient for the component one. Similar to the first equation, this equation only holds in situations when the total concentration remains constant. To derive these equations Darken mainly references Kirkendall and Smigelskas’s experiment,〔Smigelskas, A. D., and E. O. Kirkendall. "Zinc diffusion in alpha brass." Trans. Aime 171 (1947): 130-142.〕 and W. A. Johnson’s experiment, along with other findings within the metallurgical community. ==Experimental Methods== In deriving the first equation, Darken referenced Simgelskas and Kirkendall's experiment which tested the mechanisms and rates of diffusion, and gave rise to the concept now known as the Kirkendall effect. For the experiment, inert molybdenum wires were placed at the interface between copper and brass components and the motion of the markers was monitored. The experiment supported the concept that a concentration gradient in a binary alloy would result in the different components having different velocities in the solid solution. The experiment showed that in brass zinc had a faster relative velocity than copper, since the molybdenum wires moved farther into the brass. In establishing the coordinate axes to evaluate the derivation, Darken refers back to Smigelskas and Kirkendall’s experiment which the inert wires were designated as the origin.〔 In respect to the derivation of the second equation, Darken referenced W. A. Johnson’s experiment on a gold-silver system, which was performed to determine the chemical diffusivity. In this experiment radioactive gold and silver isotopes were used to measure the diffusivity of gold and silver, because it was assumed that the radioactive isotopes have relatively the same mobility as the non-radioactive elements. If the gold-silver solution is assumed to behave ideally, it would be expected the diffusivities would also be equivalent. Therefore, the overall diffusion coefficient of the system would be the average each components diffusivity; however, this was found not to be true.〔 This finding led Darken to analyze Johnson's experiment and derive the equation for chemical diffusivity of binary solutions. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Darken's equations」の詳細全文を読む スポンサード リンク
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