Combined refinement of diffusion coefficients applied on the Nb-C and Nb-N systems
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I.
INTRODUCTION
DIFFUSION is an important phenomenon, especially in materials considered for high-temperature applications where it occurs in both production and use. To evaluate diffusivities, to model the concentration profiles arising during the diffusion process, and to simulate the phase band evolution in the Nb-C and Nb-N systems, three approaches were used: (1) an analytical solution of diffusion equations for the concentration profile (Section II–B), (2) an analytical solution of diffusion equations for the phase boundary movement (Section II–C–1), and (3) a numerical solution of diffusion equations that yielded both the development of the phase band structure and the concentration profiles in all phases (Section II–C–2). Diffusion coefficients are usually calculated from the first derivative of the measured concentration profile. This technique is applied predominantly to solid/solid diffusion couples (e.g., References 1 and 2), but it can be modified to be used for gas/solid diffusion. Unfortunately, as this method works with derivatives, it is very sensitive to the scatter in the measured concentrations. Thus, it is not practicable for calculating the diffusivities of light elements. Therefore, as in the first technique (Section II–B), the approximation of the measured concentration profile by a function, which is the analytical solution of the diffusion
DAVID RAFAJA, on leave from the Faculty of Mathematics and Physics, Charles University, CZ-121 16 Prague, Czech Republic, is Research Scientist, Institute for Chemical Technology of Inorganic Materials, Vienna University of Technology, A-1060 Vienna, Austria. WALTER LENGAUER, Professor, and HERBERT WIESENBERGER, Research Fellow, are with the Institute for Chemical Technology of Inorganic Materials, Vienna University of Technology. MANECH JOGUET, Graduate Student, is with the Laboratory of Metallurgy and Physical Chemistry of Materials, F-35043 Rennes, France. Manuscript submitted July 9, 1997. METALLURGICAL AND MATERIALS TRANSACTIONS A
equations for concentration profiles, was applied by taking into account that the diffusion coefficient may depend on the concentration of the in-diffusing species. This method can only be employed for the calculation of diffusion coefficients in the metalloid-rich phases of Nb-C and Nb-N, as the concentration profiles in the intermediate phases Nb2C and Nb2N are very flat[3,4] and the maximum concentrations of carbon and nitrogen in the respective solid solution are low. Diffusion coefficients in the intermediate phases can be obtained from layer growth. For the reaction diffusion, several models of layer growth were reported in the literature. However, not all of them offer a suitable tool for the evaluation of diffusion coefficients. As already stated by Jost,[5] the complete set of diffusion coefficients cannot be obtained from only the layer growth investigated in a series of diffusion experiments, which were carried out with samples having an infinite geometry. An exception is when one diffusion coefficient is kn
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