The Debye theory of rotary diffusion: History, derivation, and generalizations

Motivated by the Debye theory of rotary diffusion in a dipolar fluid, we systematically develop a continuum mechanical theory of rotary diffusion. This theory generalizes classical kinematics to include continuous rotary degrees of freedom and introduces an additional balance law associated with the rotary degrees of freedom. Various constitutive relations are proposed in accordance with standard procedures of nonlinear continuum mechanics. The resulting set of equations provides a properly invariant and thermodynamically consistent theory that allows for constitutive nonlinearities. In particular, the classical Debye theory along with the Nernst-Einstein relations are shown to follow from a special case of linear constitutive relations and an assumption of ideality in which the free energy consists only of a classical entropic contribution. Within our theory, the notion of osmotic pressure arises naturally as a consequence of accounting for forces that act conjugate to the rotary degrees of freedom and serves as the driving force for rotary diffusion.