

Dynamics of Enhanced Tracer Diffusion
in Suspensions of Swimming Microorganisms
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Abstract
We observe and statistically quantify the enhanced transport and mixing properties of passive tracer particles in suspensions of eukaryotic swimmers, the alga Chlamydomonas reinhardtii. The biflagellate cells behave ballistically over short intervals with heterogeneous trajectories and swimming speeds around 100 μm/s. However, the probability distribution function (PDF) of displacements for the tracers are selfsimilar and grow diffusively. As the swimmer concentration increases, the Gaussian core of the PDF broadens and develops robust exponential tails. We emphasize the role of flagellar beating in creating oscillatory flows that exceed Brownian motion far from each swimmer.

Swimming Algae
Mixing dynamics by microorganisms is important for functions such as nutrient uptake. Chlamydomonas reinhardtii (Fig. 1) is a biflagellate, unicellular organism, which swims by pulling its body through the fluid using two 1012 μm long flagella in a synchronized “breaststroke” pattern (5060 Hz).
The swimming trajectories of Chlamydomonas are well documented and exhibit a broad spectrum of trajectories (Fig. 2) including typical helical paths with a speed around 100 μm/s. 

Fig. 1 A single Chlamydomonas reinhardtii cell (courtesy K. Drescher). 

Fig. 2 A representative sample of polydisperse swimmer trajectories. 
Passive Tracers
Fluorescent tracer particles (2 μm diameter) are used to visualize the mixing due to the motile Chlamydomonas cells, where trajectories exhibit both Brownian components and large jumps induced by flows from the passing swimmers (Fig. 3). Higher resolution (500 fps) reveals that these advective motions are loops due to the oscillatory flagellar beating (Fig. 3, inset). 

Fig. 3 A representative sample of tracer particle trajectories in the presence of swimmers. 

Movie 1 Tracer particle motion in the presence of swimmers.

Enhanced Mixing
Probability density functions (PDFs) of tracer displacements, Δx, are observed to be Gaussian in the absence of swimmers and develop dramatically extended, exponential tails as the swimmer volume fraction, φ, increases (Fig. 4). This results in an enhancement of the effective tracer particle diffusivity. 

Fig. 4 Enhanced tracer diffusion due to swimming microorganisms results in extended, exponential tails in the displacement PDF. 
Oscillatory Swimming
As a direct measurement of the advection/diffusion balance, we use highspeed imaging (500 fps) to capture the details of tracerswimmer interactions (Movie 2). Figure 5 shows a tracer trajectory in the reference frame of the swimmer. These flows are intrinsically unsteady, which can contribute to the enhanced diffusivity. 

Movie 2 High resolution (500 fps) tracerswimmer interaction 

Fig. 5 (a) Tracer trajectory in the moving reference frame of a swimmer and (b) the induced radial velocity fades to Brownian motion at ~25 μm. 
References
K.C. Leptos, J.S. Guasto, J.P. Gollub, A.I. Pesci, and R.E. Goldstein, Phys. Rev. Lett. 103, 198103 (2009).

