Application of the McNabb-Foster trapping equations to the diffusion of oxygen in dilute niobium alloys
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I,
INTRODUCTION
IN 1963 McNabb and Foster J proposed two equations which could be used as the basis of a continuum model for diffusion and trapping of interstitials in solids. The equations apparently are similar to some used in other applications but were not used to describe diffusion and trapping prior to the McNabb-Foster effort and, therefore, have become associated with their names. Initially the equations were not used very extensively because it was too difficult to find analytical solutions for boundary conditions comparable to experimental conditions. McNabb and Foster and Oriani, z however, pointed out that for the special case where local equilibrium existed between the diffusible and trapped species. the results could be approximated by the use of an effective or apparent diffusion coefficient. Koiwa 3 in 1974 developed an atomic model which confirmed and extended the Oriani approach. Recently McLellan 4 has also proposed a model which leads to a slightly different expression for the apparent diffusion coefficient. In 1975 Caskey and Pillinger 5 employed a finite difference approach to solve the McNabbFoster equations, and this made it possible to employ them more extensively for a variety of conditions. However, until a short time ago little effort had yet been made to use numerical analysis to apply the McNabb-Foster equations to actual experimental data. Robertson 6 recently reported using a finite difference approach similar to that of Caskey and Pillinger in an analysis of data for hydrogen in thoria dispersed nickel. The work which will be reported here was an effort to determine further how useful and reliable such an approach might be. Results obtained by numerical analysis are often difficult to verify. However, the approach should have a broad range of applicability, including cases involving nonuniform trap distributions, if the method can be shown to be reliable. 7 In these studies the results obtained earlier by Lauf and Altstetter s on the diffusion of oxygen in niobium alloys R. C. FRANK is Director of Research, Augustana Research Foundation, and Professor of Physics, Augustana College, Rock Island, IL 61201, R.J. LAUF is with Metals and Ceramics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37830, and C.J. ALTSTETTER is with the Department of Metallurgy and the Materials Research Laboratory, University of Illinois at Urbana-Champaign, IL 61801. Manuscript submitted December 27, 1979. METALLURGICALTRANSACTIONS A
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using the Oriani approximation were reevaluated using the numerical analysis of the McNabb-Foster equations and computer-assisted curve fitting to obtain ratios of the release to trapping coefficients, p/k, and these were used to calculate trap binding energies.
II.
DATA TREATMENT USING E F F E C T I V E DIFFUSION COEFFICIENTS
As discussed in the earlier paper, 8 the experiment involved measuring the oxygen concentration as a function of time at one surface of a flat slab of niobium alloy while permitting the oxygen to diffuse out from the other surface. Six all
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