Macrotransport theory for diffusiophoretic colloids and chemotactic microorganisms

We conduct an asymptotic analysis to derive a macrotransport equation for the long-time transport of a chemotactic/diffusiophoretic colloidal species in a uniform circular tube under a steady, laminar, pressure-driven flow and transient solute gradient. The solute gradient drives a ‘log-sensing’ advective flux of the colloidal species, which competes with Taylor dispersion due to the hydrodynamic flow. We demonstrate excellent agreement between the macrotransport equation and direct numerical solution of the full advection–diffusion equation for the colloidal species transport. In addition to its accuracy, the macrotransport equation requires times less computational runtime than direct numerical solution of the advection–diffusion equation. Via scaling arguments, we identify three regimes of the colloidal species macrotransport, which span from chemotactic/diffusiophoretic-dominated macrotransport to the familiar Taylor dispersion regime, where macrotransport is dominated by the hydrodynamic flow. Finally, we discuss generalization of the macrotransport equation to channels of arbitrary (but constant) cross-section and to incorporate more sophisticated models of chemotactic fluxes. The macrotransport framework developed here will broaden the scope of designing chemotactic/diffusiophoretic transport systems by elucidating the interplay of macrotransport due to chemotaxis/diffusiophoresis and hydrodynamic flow.