A theoretical framework was developed for treating the diffusion of a single unconstrained atomic species on a crystal lattice. It involved a generalization of the classical theories of atomic diffusion, and diffusion-induced phase separation, to incorporate constitutive non-linearities, external forces and deformation of the lattice. Within this framework, atomic diffusion was regarded as being a microscopic process which was described by 2 independent kinematic variables. One was the atomic flux, which defined the local motion of atoms relative to the motion of the underlying lattice. The other was the time-rate of the atomic density, which took account of non-local interactions between migrating atoms and characterized the kinematics of phase separation. Generalized forces were introduced which were power-conjugate to each of these rates, and it was required that these forces should satisfy ancillary microbalances which were distinct from the conventional balance that involved forces which expended power above the rate at which the lattice deformed. A mechanical version of the second law, which took the form of an energy imbalance and accounted for all of the power expenditures (including those due to atomic diffusion and phase separation), was used to derive restrictions on constitutive equations. Within these limits, the microbalance which involved forces that were conjugate to the atomic flux provided a generalization of the usual constitutive relationship between the atomic flux and the gradient of the diffusion potential. This relationship, in conjunction with the atomic balance, yielded a generalized Cahn-Hilliard equation.

Theory of Atomic Diffusion on Fixed and Deformable Crystal Lattices. E.Fried, S.Sellers: Journal of Elasticity, 2000, 59[1-3], 67-81