Quantitative explanation for uphill diffusion of Sn during reactive diffusion between Cu-Sn alloys and Nb

In a bronze method, uphill diffusion of Sn takes place from a binary Cu-Sn alloy to Nb3Sn during reactive diffusion between the Cu-Sn alloy and Nb at temperatures around 1000 K. In order to account for the occurrence of the uphill diffusion quantitatively, the phase equilibria in the ternary Cu-Nb-Sn system were theoretically analyzed using a thermodynamic model for phases with different sublattices. In this ternary system, there is no ternary compound and the solubility of the third component is very small for all the phases. Consequently, the ternary interaction was assumed negligible, and thus the Gibbs energy of each phase was expressed with the thermodynamic parameters of the relevant binary systems. In an isothermal section calculated at 1053 K, the three-phase equilibrium of Cu + Nb + Nb3Sn appears at an activity of Sn a(Sn)(b) of Sn for the binary Cu-Sn alloy is greater than the activity asn of a(Sn)(c) for the three-phase equilibrium of Cu + Nb + Nb3Sn, Nb3Sn is spontaneously produced owing to the reactive diffusion. As a result, the uphill diffusion of Sn occurs from the Cu-Sn alloy to Nb3Sn. The chemical driving force Delta G(Sn) for the uphill diffusion is evaluated by the equation Delta G(Sn) = RT ln(a(Sn)(b)/a(Sn)(c)). This equation semi-quantitatively explains the growth behavior of Nb3Sn.