Grooving of a Bicrystal by Surface Diffusion

Isotropic Materials

In the case of an isotropic material both surface tension and diffusivity are independent of the crystalline orientation and are given by gamma0 and D0, say. This evolution equation becomes

evolution equation (6)

where euqation for B. No analytic solutions are known for equation (6) (Cahn & Taylor, 1994). Mullins (1957) assumed the surface slope was everywhere small, abs(yx) << 1 which gives the linearised equation

linearised equation. (7)

Tritscher (1996a) showed that (7) was also an integrable form of the governing equation (5) for a class of anisotropic materials with behaviour similar to a liquid crystal and constitutive relations

constitutive relations (8)

where C1 and C2 are arbitrary constants and gamma is the solution of the Herring equation (Herring, 1951)

Herring equation (9)

By use of symmetry recursion operators applied to the linearizable nonlinear diffusion equation Broadbridge & Tritscher (1996) derived a further linearizable form of the evolution equation

evolution equation (10)

where D0, gamma0 are weights for the surface diffusivity and surface tension and b is an arbitrary parameter which orients the crystalline lattice relative to the coordinate axis. The constitutive relations for equation (10) are (Tritscher, 1996)

constitutive relations

The constitutive functions show degeneracy and for values of b less than around pi/4 the evolution equation satisfactorily models an isotropic material.





Derivation of the Evolution Equation

Isotropic Materials

New Constitutive Functions

Symmetry Reductions

The Symmetric Grain Boundary Groove Problem