(299h) Bacteria-Swimming-Induced Hydrodynamic Diffusion of Tracer Particles

We present the first theoretical estimate of the hydrodynamic diffusion of tracer particles in a homogeneous, isotropic suspension of swimming bacteria producing Stokesian hydrodynamic disturbances. While there are many experimental and numerical investigations of enhanced diffusion due to swimming bacteria in the literature, no theoretical estimate for the enhancement in tracer diffusivity has been proposed so far. The theory assumes pair-wise interactions between a bacterium and a colloidal tracer particle with the hydrodynamic disturbance due to the bacteria estimated using slender-body theory. The method of averaged equations is then employed to determine the hydrodynamic diffusivity D_hyd of the beads at long length and time scales when a bead has undergone many interactions with bacteria. The hydrodynamic diffusivity D_hyd depends upon the Peclet number Pe=UL/D and the non-dimensional bacteria concentration nL³  where U is the swimming speed of the cells, L is the total length of cells including the flagella bundle,  D is the Brownian diffusivity of the tracer bead, and n is the number of cells per unit volume.  At high Peclet numbers, the hydrodynamic diffusion has a pure convective origin and D_hyd~ UL(nL³). At low Peclet numbers, the bead can sample the hydrodynamic disturbance of a bacterium diffusively throughout the period τ between tumbles leading to a characteristic radial distance for interactions of r ~ (Dτ)^1/2 and an enhancement to the diffusivity proportional to UL(nL³)U(τ/D)^1/2 . We complement our theory with numerical simulations of bacteria-bead interactions and experiments on moderately dense suspensions of E. coli bacteria with concentrations ranging from nL³ = 0.6 to 14. Numerical simulations of an interacting bead-bacteria pair allow us to assess the importance of excluded volume, which limits the ability of beads to sample near-field hydrodynamic disturbances. Experiments involve time resolved imaging of fluorescent beads in a bacterial suspension confined in a microchannel and tracking the images to obtain the bead diffusivity. Comparison of theory, simulation and experimental data reveals the remarkable fact that the pair-wise interactions model can adequately estimate hydrodynamic diffusion in suspensions with nL³ as large as 14.