Determination of diffusion coefficients through gels with non-negligible finite-volume effects in the compartments of the diffusion cell

Diffusion cells are used to measure diffusion coefficients (D_M) in gels. These measurements are of interest to understand and predict the availability of nutritive or toxic chemical species in waters, soils and sediments. When the diffusive flux from the donor to the acceptor compartment is constant (steady-state regime), D_M is determined from the slope of the linear plot of the acceptor concentration vs time. However, at long enough times, there is a non-negligible concentration depletion in the donor compartment concomitant to a concentration increase in the acceptor compartment. Accordingly, the accumulation plot bends downwards preventing a linear fitting. This is the case of metals whose solubility (especially depending on pH values) limits the concentration in the donor compartment and the time required to reach concentrations above the limit of quantification in the acceptor compartment implies a non-negligible decrease of the concentration in the donor compartment. In this work, a simple linear regression is shown to provide the diffusion coefficient values from experiments exhibiting finite-volume effects. This expression is validated against rigorous numerical simulation as well as reported values in the literature. Diffusion coefficients of Zn, Ni and Pb in agarose cross-linked polyacrylamide (APA) gels (used in Diffusive Gradients in Thin-Film devices, DGT) are determined under finite-volume effects. The resulting values agree with those obtained under the standard linear regime.