Self-diffusion and impurity diffusion of fee metals using the five-frequency model and the Embedded Atom Method
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The activation energies for self-diffusion of transition metals (Au, Ag, Cu, Ni, Pd, Pt) have been calculated with the Embedded Atom Method (EAM); the results agree well with available experimental data for both mono-vacancy and di-vacancy mechanisms. The EAM was also used to calculate activation energies for vacancy migration near dilute impurities. These energies determine the atomic jump frequencies of the classic "five-frequency formula, which yields the diffusion rates of impurities by a mono-vacancy mechanism. These calculations were found to agree fairly well with experiment and with Neumann and Hirschwald's "T m " model.
I. INTRODUCTION Diffusion in fee metals at medium and high temperatures occurs primarily by a vacancy mechanism. Diffusion is dominated by the contribution of mono-vacancies, but the contribution of di-vacancies is significant at high temperatures. Thus, the total diffusion rate D is given by: D = D]v + D2v = D ° l v e x p ( - e i v / k r ) + £>° 2 v exp(-e 2 v /kr)
(1)
where Dlv is the contribution of mono-vacancies, D2v is the contribution of di-vacancies, £>°lv and D°2v are the preexponential coefficients, QXv and Q2v are the activation energies for the mono-vacancy and di-vacancy mechanism, respectively, k is Boltzmann's constant, and T is the temperature. For the case of self-diffusion, Qu and Q2v are the sum of the formation and migration energies of monovacancies and di-vacancies, respectively. The case of impurity diffusion will be discussed later. Many experiments have been carried out to determine the rate of diffusion by a vacancy mechanism in pure fee metals; the most recent review is by LeClaire and Neumann.59 Most of these experiments were carried out at high temperatures to avoid significant grain boundary effects, but the high temperatures generally resulted in a significant di-vacancy contribution. The di-vacancy contribution has a higher activation energy than the mono-vacancy contribution, so high-temperature experiments yielded higher activation energies than for a mono-vacancy mechanism. The effect of di-vacancies seems to be a major cause of the disagreement among different experimental results. Experiments at moderate temperatures in single crystals largely avoid both di-vacancy and grain boundary effects. Careful theoretical analyses of the best of the experimental data for self-diffusion have been carried out by Neumann and Tolle, 60 Peterson, 61 and Schule. 62 Their analyses yielded values of D° lv , D°2v, Qlv, and Q2v, which are in fairly good agreement with one another, especially for the mono-vacancy terms which play the major role in 102
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diffusion. Neumann and Tolle also carried out similar analyses for several examples of impurity diffusion in Cu, Ag, and Ni.63 The vacancy mechanism for impurity diffusion is well described by the classic "five-frequency formula"; for a recent review, see Le Claire.64 The classic five-frequency formula includes only the effect of single vacancies on the diffusion process, but Mehrer developed
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