Diffusivity as a Function of Direction

In general, the properties of crystals and other types of materials, such as composites, vary with direction (i.e., macroscopic materials properties such as mass diffusivity and electrical conductivity will generally be anisotropic). It is possible to generalize the isotropic relations between driving forces and fluxes to account for 

D is called the diffusivity tensor and acts as an object that connects one vector to another (e.g., the flux vector with the gradient vector). This connection can be written in matrix form as in Eq. 4.57. The diffusivity tensor D is symmetric (i.e., Dtj = Dji) for any underlying material symmetry. 

The anisotropic form of Fick s law would seem to complicate the diffusion equation greatly. However, in many cases, a simple method for treating anisotropic diffusion allows the diffusion equation to keep its simple form corresponding to isotropic diffusion. Because Dtj is symmetric, it is always possible to find a linear coordinate transformation that will make the Dij diagonal with real components (the eigenvalues of D). Let elements of such a transformed system be identified by a hat. Then 

The diagonal elements of D are the eigenvalues of D, and the coordinate system of D defines the principal axes Xi, 2, 3 (the eigensystem). In the principal axes coordinate system, the diffusion equation then has the relatively simple form 

3The quintessential resource for this topic is Nye s book on crystal properties [7]. 

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