  We develop a theoretical framework for the diffusion of a single
unconstrained species of atoms on a crystal lattice that provides a
generalization of the classical theories of atomic diffusion and
diffusion-induced phase separation to account for constitutive nonlinearities,
external forces, and the deformation of the lattice. In this framework, we
regard atomic diffusion as a microscopic process described by two independent
kinematic variables: (i) the atomic flux, which reckons the local motion of
atoms relative to the motion of the underlying lattice, and (ii) the time-rate
of the atomic density, which encompasses nonlocal interactions between
migrating atoms and characterizes the kinematics of phase separation. We
introduce generalized forces power-conjugate to each of these rates and require
that these forces satisfy ancillary microbalances distinct from the
conventional balance involving the forces that expend power over the rate at
which the lattice deforms. A mechanical version of the second law, which takes
the form of an energy imbalance accounting for all power expenditures
(including those due to the atomic diffusion and phase separation), is used to
derive restrictions on the constitutive equations. With these restrictions, the
microbalance involving the forces conjugate to the atomic flux provides a
generalization of the usual constitutive relation between the atomic flux and
the gradient of the diffusion potential, a relation that in conjunction with
the atomic balance yields a generalized Cahn-Hilliard equation.
