Dynamics of Enhanced Tracer Diffusion in Suspensions of Swimming Eukaryotic Microorganisms

In contexts such as suspension feeding in marine ecologies there is an interplay between Brownian motion of nonmotile particles and their advection by flows from swimming microorganisms. As a laboratory realization, we study passive tracers in suspensions of eukaryotic swimmers, the alga Chlamydomonas reinhardtii. While the cells behave ballistically over short intervals, the tracers behave diffusively, with a time-dependent but self-similar probability distribution function of displacements consisting of a Gaussian core and robust exponential tails. We emphasize the role of flagellar beating in creating oscillatory flows that exceed Brownian motion far from each swimmer.

Figure 1

Swimming cells and tracer particles. (a) Chlamydomonas trajectories over an interval of $\sim 1\mathrm{s}$. (b) PDF of swimming speed obtained at 30 fps, averaged over 30 s, and (inset) mean squared displacement of swimmers in the imaging plane. The line $⟨\Delta r^2⟩\sim (\Delta t{)}^2$ shows the motion to be ballistic for intervals up to several $s$. (c) Trajectories of $2.0\mu \mathrm{m}$ microspheres observed at 50 fps. Inset: loopy tracer path, sampled at 500 fps (bright-field), induced by a cell passing within $\sim 8\mu \mathrm{m}$.

Figure 2

Probability distribution functions for tracer displacements. (a) Overlaid PDFs at a fixed time interval $\Delta t=0.12\mathrm{s}$, for various volume fractions, showing the broadening of the Gaussian core and appearance of exponential tails. (b) Diffusive rescaling of PDFs at $\varphi =2.2\%$ and various time intervals, illustrating data collapse.

Figure 3

Diffusional properties. (a) Mean squared tracer displacement at various cell concentrations. Concentration dependence of (b) effective diffusivity, with linear fit, and (c) fractional contribution of PDF tails.

Figure 4

Tracer dynamics. (a) Trajectory of a tracer particle imaged at 500 fps, in a frame of reference moving with a Chlamydomonas cell. Black disk represents the approximate size of the cell. (b) Induced radial motion fades into Brownian motion beyond $\sim 25\mu \mathrm{m}$ from the cell.