Inertial effects and diffusion

Abstract

Fick’s law of diffusion is equivalent to an equation of motion for an ideal substance in which the thermodynamic driving force is assumed to be balanced by a drag force proportional to drift velocity. Neglect of the inertial force required by Newton’s second law leads to anomalous short-time predictions such as infinite speeds of propagation. Resolution of the propagation anomaly for chemical diffusion is considered for three applications: Knudsen flow in porous materials, interstitial diffusion in elastic solids, and interstitial diffusion in viscoelastic solids. The transients associated with inertial effects for the first two cases occur at very short-times and are typically of little practical significance. Diffusion in viscoelastic solids appears as a possible exception based on observations of linear transport kinetics in ordinary timescales. The constitutive equations for the Maxwell model lead to the prediction that solute-induced dilatations propagate according to the telegraph equation, a widely proposed correction for avoiding the propagation anomaly in standard diffusion theory. The telegraph equation is not universally valid, however. In the case of Knudsen flow, for example, inertial effects give rise to an additional term related to material convection.