Statistics of Colloidal Suspensions Stirred by Microswimmers

We present a statistical analysis of the experimental trajectories of colloids in a dilute suspension of the green algae Chlamydomonas reinhardtii. The measured probability density function (pdf) of the displacements of colloids covers 7 orders of magnitude. The pdfs are characterized by non-Gaussian tails for intermediate time intervals, but nevertheless they collapse when scaled with their standard deviation. This diffusive scaling breaks down for longer time intervals and the pdf becomes Gaussian. However, the mean squared displacements of tracer positions are linear over the complete measurement time interval. Experiments are performed for various tracer diameters, swimmer concentrations, and mean swimmer velocities. This allows a rigorous comparison with several theoretical models. We can exclude a description based on an effective temperature and other mean field approaches that describe the irregular motion as a sum of the fluctuating far field of many microswimmers. The data are best described by the microscopic model by J.-L. Thiffeault, Distribution of particle displacements due to swimming microorganisms, Phys. Rev. E 92, 023023 (2015).

Figure 1

Microscope image of a microswimmer, sketches of colloids, and tracer trajectories. Color encodes the time. The sketched microspheres (gray disks) and the swimmer are scaled. (a) Trajectory of a microsphere over 80s and a short part of the swimmer trajectory at around $t\approx 70\mathrm{s}$). At the moment of interaction, the irregular Brownian motion of the tracer is superimposed on a loop. (b) Simulated trajectory with a similar loop due to a spherical squirmer with a speed of $U=35\mu \mathrm{m}/\mathrm{s}$ and a radius $\ell =6\mu \mathrm{m}$ moving along the direction of the arrow at $t\approx 60\mathrm{s}$ [12]. A representative thermal Brownian motion was added. (c) Experimental trajectory of a much bigger sphere interacting with a swimmer (not shown) at $t\approx 40\mathrm{s}$. In this case, the Brownian motion is less pronounced, but the size of the loop is comparable to (a),(b).

Figure 2

(a) Mean squared displacements, MSDs, for various tracer diameters in the suspensions of slow and fast swimmers and without swimmers. (b) Effective diffusion coefficients vs swimmer number density $n$ for various tracer diameters $d$ of slow swimmers, and one example for fast swimmers. For particles with a diameter $d\ge 3\mu \mathrm{m}$, ${\mathrm{D}}_2\mathrm{O}$ was added to the TAP solution to prevent sedimentation (open symbols). The lines correspond to predictions by Eq. (1). Error bars in swimmer densities are only shown for the $2\mu \mathrm{m}$ colloids. Errors bars in $D_{\mathrm{eff}}$ are of the order of the symbol size.

Figure 3

Experimental $\text{pdf}(\widehat{x};t)$ () at $t=0.4\mathrm{s}$ and a number density of $n=1.5\times {10}^6{\mathrm{cm}}^{-3}$ for colloids of $d=3\mu \mathrm{m}$. The small red shadowed region indicates the 95% bootstrap confidence interval [36]. Various characterizations and predictions are shown: () a superposition of Gaussian and exponential, () CTRW model, () the microscopic model, and the asymptotic pdf $\propto {\widehat{x}}^{-4}$ in the inset; see text for parameters. The measured pdf for pure Brownian motion is shown as (). The yellow area indicates the data range presented in Ref. [9]. The black arrow indicates the scaled swimming distance of the microswimmers within 0.4 s.

Figure 4

Experimental pdf at $n=1.5\times {10}^6{\mathrm{cm}}^{-3}$ for various time intervals $t_i$ to visualize the breakdown of the diffusive scaling. The pdf becomes a Gaussian for $t_i>2\mathrm{s}$ as predicted in Ref. [12]. The inset shows the kurtosis as measure to quantify the non-Gaussian tails.