Effect of α-stable sorptive waiting times on microbial transport in microflow cells

The interaction of bacteria in the fluid phase with pore walls of a porous material involves a wide range of effective reaction times which obey a diversity of substrate-bacteria adhesion conditions, and adhesive mechanisms. For a transported species, this heterogeneity in sorption conditions occurs both in time and space. Modern experimental methods allow one to measure adhesive reaction times of individual bacteria. This detailed information may be incorporated into nonequilibrium transport-sorption models that capture the heterogeneity in reaction times caused by varying chemical conditions. We have carried out particle (Brownian dynamic) simulations of adhesive, self-motile bacteria convected between two infinite plates as a model for a microflow cell. The adhesive heterogeneity is included by introducing adhesive reaction time (understood as time spent at a solid boundary once the particle collides against it) as a random variable that can be infinite (irreversible sorption) or vary over a wide range of values. This is made possible by treating this reaction time random variable as having an α-stable probability distribution whose properties (e.g., infinite moments and long tails) are distinctive from the standard exponential distribution commonly used to model reversible sorption. In addition, the α-stable distribution is renormalizable and hence upscalable to complex porous media. Simulations are performed in a pressure-driven microflow cell. Bacteria motility (driven by an effective Brownian force) acts as a dispersive component in the convective field. Upon collision with the pore wall, bacteria attachment or detachment occurs. The time bacteria spend at the wall varies over a wide range of time scales. This model has the advantage of being parsimonious, that is, involving very few parameters to model complex irreversible or reversible adhesion in heterogeneous environments. It is shown that, as in Taylor dispersion, the ratio of the channel half width b to the Brownian bacteria motility coefficient ${(D}_0$ or dispersion coefficient) $t_b{=b}^2{/D}_0$ controls the different adhesion regimes along with the value of α. Universal scalings (with respect to dimensionless time $t_{*}{=t/t}_b)$ for the mean position, $〈X〉{=V}_{\mathrm{eff}}^{*}{t}_{*}^{\theta },$ and mean-square displacement, $〈\Delta X^2〉{=D}_{\mathrm{eff}}^{*}{t}_{*}^{\gamma }$ exist for long-time dispersion and the coefficients were obtained. The model can account for a great many sorptive processes including reversible and irreversible sorption, and sub- and superdispersive regimes with just a few parameters.