A mathematical formalism for diffusion in crystalline solids incorporating deviations from the Fickian behavior
Diffusion in crystalline solids includes processes such as lattice diffusion, surface diffusion, grain boundary diffusion, triple junction diffusion and diffusion along dislocations. The key parameter that is used to quantify the diffusion processes is the diffusion coefficient, D. In order to calculate the diffusion coefficient from experimental data, several mathematical models have been proposed over the years. These models, by implicitly assuming that the diffusion mechanisms for all the above processes are identical, are based on the same laws, namely, Fick's diffusion laws. However, owing to the differences in their structure, studies have shown that the diffusion mechanisms are different for different diffusion processes. The aim of our research is to analyze these differences in detail and propose a model that describes the various diffusion processes more accurately. An important outcome of the new model is that the diffusion coefficient alone is seen to be inadequate to quantify diffusion processes. A new parameter (in addition to variables such as segregation parameter and grain boundary width) is introduced and its characteristics are described. We also present plots that illustrate how the classical concentration profiles get modified under the new formalism for instantaneous and constant source conditions. A method to calculate the diffusion parameters from experimental data using the new model is described. ^ A further outcome of our analysis is as follows. Fick's second law can be derived in two ways. Firstly, it can be derived by combining Fick's first law and the continuity equation. This method is called the continuum method. Secondly, it can be derived ab-initio by considering the random walk of the diffusing particle. This method is called the statistical method. As a consequence of our analysis, we demonstrate that the statistical method is the physically correct approach to derive Fick's second law and that the continuum approach is based on an incorrect assumption. ^ Finally, a method to design stable triple junctions based on Herring's relation is described and experimental results that demonstrate the occurrence of enhanced diffusion of nickel through triple junctions of copper and diffusion induced triple junction migration are presented. ^
Alexander H. King, Purdue University.
Engineering, Materials Science
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