Diffusion with attrition

This article treats the problem of the sharp front observed when a diffusing substance interacts irreversibly with binding sites within the medium. The model consists of two simultaneous partial differential equations that are nonlinear and cannot be solved in closed form. The parameters are the diffusion coefficient D in the direction under consideration (x), the interaction constant k, the binding-site concentration μ and the boundary concentration of the diffusing ion c ₀. Our aim is to develop methods to enable the estimation of these parameters from the experimental data. An analytical solution for the case k → ∞, as found by others, is given first and then a finite element analysis package is used to obtain numerical solutions for the general case. Graphs are presented to illustrate the effects of the various parameters. Simple graphical procedures are described to compute μ and c ₀. The position of the advancing front ξ then provides, together with μ, a way to estimate D. A mathematical identity relating D and x and a second one involving D, k and t help to reduce the complexity of the problem. A new, measurable quantity S(t) is defined as S(t) = max_x(- df(x,t)/dx), where f is the total concentration (free + bound) of the diffusing ion at time t, and detailed plots are furnished that permit the computation of k directly from S(t), μ and D. The accuracy with which such methods can be expected to determine the various parameters of the model is considered at some length. Finally, in a concluding section, we simulate typical experimental data, examine the validity of our methods, and see how their accuracy is affected by controlled amounts of various kinds of noise.