Distribution of particle displacements due to swimming microorganisms

The experiments of Leptos et al. [Phys. Rev. Lett. 103, 198103 (2009)] show that the displacements of small particles affected by swimming microorganisms achieve a non-Gaussian distribution, which nevertheless scales diffusively—the “diffusive scaling.” We use a simple model where the particles undergo repeated “kicks” due to the swimmers to explain the shape of the distribution as a function of the volume fraction of swimmers. The net displacement is determined by the inverse Fourier transform of a single-swimmer characteristic function. The only adjustable parameter is the strength of the stresslet term in our spherical squirmer model. We give a criterion for convergence to a Gaussian distribution in terms of moments of the drift function and show that the experimentally observed diffusive scaling is a transient related to the slow crossover of the fourth moment from a ballistic to a linear regime with path length. We also present a simple model, with logarithmic drift function, that can be solved analytically.

Figure 1

The natural log of ${\rho }^2{\mathrm{\Delta }}^2(\rho ,z)$ [integrand of Eq. (4) with $d{V}_{\mathbf{\eta }}=2\pi {\rho }^{2}d(ln\rho )dz$] for (a) a sphere of radius $\ell =1$ in inviscid flow, moving a path length $\lambda =10$ (top) and 100 (bottom), plotted on the same scale. The scale of the integrand doesn't change, only its support. Here $\mathbf{\eta }=(\rho ,z)$ with $z$ the swimming direction and $\rho$ the distance from the $z$ axis. (b) Same as (a) but for a stresslet velocity field. The integral Eq. (4) grows linearly with $\lambda$ for both (a) and (b).

Figure 2

Contour lines for the axisymmetric streamfunction of a squirmer of the form of Eq. (26), with $\beta =0.5$. This swimmer is of the puller type, as for C. reinhardtii.

Figure 3

The function ${\mathrm{\Gamma }}_{\lambda }\left(k\right)$ defined by Eq. (16) for (from broadest to narrowest) $\lambda =12\mu \mathrm{m}, 36\mu \mathrm{m}, 60\mu \mathrm{m}$, and $96\mu \mathrm{m}$.

Figure 4

(a) The pdf of particle displacements after a path length $\lambda =12\mu \mathrm{m}$, for several values of the volume fraction $\varphi$. The data is from Leptos et al. [46], and the figure should be compared to their Fig. 2. The theoretical curves were obtained from Eq. (24) for the model squirmer in Fig. 2, with some noise corresponding to thermal diffusivity as measured in Leptos et al. [46]. Inset: comparison of (from broadest to narrowest) $\beta =2$, 1, $0.5$, and $0.1$, for $\varphi =2.2\%$, showing the sensitivity of the fit $\beta =0.5$. (b) Same as (a) but on a wider scale, also showing the form suggested by Eckhardt and Zammert [47] (dashed lines).

Figure 5

The same distributions as in Fig. 4, but on a log-log plot. The dashed line is the $x^{-4}$ power law predicted by Pushkin and Yeomans [41]. Inset: numerical simulation with only the stresslet far-field displacement included.

Figure 6

(a) For the squirmer model Eq. (26), dependence of the effective diffusivity $D_{\mathrm{eff}}$ on the stresslet strength $\beta$. For small $\beta$, we recover the value for spheres in inviscid flow [22]. An approximate formula is also shown as a solid curve. (b) Comparison of the effective diffusivity data from Leptos et al. [46], showing their fit (solid line). The dashed line is the prediction for $\beta =0.5$, used in this paper.

Figure 7

The minimum volume fraction ${\mathrm{\Phi }}_{\lambda }$ for the threshold of Gaussian behavior [Eq. (41)]. The solid line is the the squirmer model (Sec. 3) with $\beta =0.5$ and radius $\ell =5\mu \mathrm{m}$. The dashed line is for spherical treadmillers (inviscid spheres) of the same radius. The latter require an order of magnitude longer to achieve Gaussianity, due to the short range of their velocity field.

Figure 8

The rate function $I\left(X\right)$ for cylinders (Eq. (43), dashed line) and spheres (solid line, numerical solution of (36)). In both cases we used $C=1$. The linear behavior for large $\left|X\right|$ indicates exponential tails in Eq. (36). When $X$ is small, expanding near the quadratic minimum recovers the Gaussian limit.

Figure 9

(a) The pdfs of particle displacements for squirmers for different times, at a number density $\varphi =2.2\%$. (b) The same pdfs rescaled by their standard deviation exhibit the “diffusive scaling” observed in the experiments of Leptos et al. [46], where the curves collapse onto one despite not being Gaussian. As in the experiments, the scaling is worst for $\lambda =6\mu \mathrm{m}$.

Figure 10

Same as described in the caption of Fig. 9 but for longer times and with a wider scale. In (a) the distributions broaden with time since their standard deviation is increasing; in (b), after rescaling by the standard deviation, the distributions' tails narrow with increasing $\lambda$ as they converge to a Gaussian.

Figure 11

The second and fourth integrated moments of ${\mathrm{\Delta }}_{\lambda }$. These grow ballistically (${\lambda }^q$) for short times and eventually grow linearly with $\lambda$. The slow crossover of ${\mathrm{\Delta }}_{\lambda }^4$ is the origin of the “diffusive scaling” of Leptos et al. [46], since in their narrow range of $\lambda$ the curve is tangent to ${\lambda }^2$.