In the case of an isotropic material both surface tension and diffusivity are independent of the crystalline orientation and are given by and , say. This evolution equation becomes
where . No analytic solutions are known for equation (6) (Cahn & Taylor, 1994). Mullins (1957) assumed the surface slope was everywhere small, which gives the linearised equation
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
where C1 and C2 are arbitrary constants and is the solution of the Herring equation (Herring, 1951)
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
where 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)
The constitutive functions show degeneracy and for values of b less than around the evolution equation satisfactorily models an isotropic material.