Dynamics of ellipsoidal tracers in swimming algal suspensions

Enhanced diffusion of passive tracers immersed in active fluids is a universal feature of active fluids and has been extensively studied in recent years. Similar to microrheology for equilibrium complex fluids, the unusual enhanced particle dynamics reveal intrinsic properties of active fluids. Nevertheless, previous studies have shown that the translational dynamics of spherical tracers are qualitatively similar, independent of whether active particles are pushers or pullers—the two fundamental classes of active fluids. Is it possible to distinguish pushers from pullers by simply imaging the dynamics of passive tracers? Here, we investigated the diffusion of isolated ellipsoids in algal C. reinhardtii suspensions—a model for puller-type active fluids. In combination with our previous results on pusher-type E. coli suspensions [Peng et al., Phys. Rev. Lett. 116, 068303 (2016)], we showed that the dynamics of asymmetric tracers show a profound difference in pushers and pullers due to their rotational degree of freedom. Although the laboratory-frame translation and rotation of ellipsoids are enhanced in both pushers and pullers, similar to spherical tracers, the anisotropic diffusion in the body frame of ellipsoids shows opposite trends in the two classes of active fluids. An ellipsoid diffuses fastest along its major axis when immersed in pullers, whereas it diffuses slowest along the major axis in pushers. This striking difference can be qualitatively explained using a simple hydrodynamic model. In addition, our study on algal suspensions reveals that the influence of the near-field advection of algal swimming flows on the translation and rotation of ellipsoids shows different ranges and strengths. Our work provides not only new insights into universal organizing principles of active fluids, but also a convenient tool for detecting the class of active particles.

Figure 1

Schematics showing the flow patterns induced by a single pusher (a) and a single puller (b). The color indicates the magnitude of the flow, which decays from the origin following a $r^{-2}$ scaling. The arrows indicates the direction of the flow. E. coli is a typical pusher-type microswimmer [39], whereas C. reinhardtii we study here is a typical puller-type microswimmer.

Figure 2

Experimental setup and method. (a) The left panel shows a schematic of the adjustable wire-frame device. The right panel shows a zoomed view of a region of $130\times 130\mu {\mathrm{m}}^2$ in the center of the free-standing film. The solid red line indicates the trajectory of the ellipsoid over a time interval of 10 s. The dashed lines indicate the trajectories of algae. Scale bar: 20 $\mu \mathrm{m}$. (b) Reference frame transformation. The displacement vector, $\delta {\mathbf{x}}_n$, of an ellipsoid in a small time interval can be decomposed into two orthogonal components ($\delta x_n, \delta y_n$) in the laboratory frame based on the $x-y$ axis or ($\delta {\tilde{x}}_n, \delta {\tilde{y}}_n$) in the body frame based on the $\tilde{x}-\tilde{y}$ axis. The two coordinates are connected through a 2D rotational matrix specified by the orientation of the ellipsoid in the laboratory frame, ${\theta }_n$.

Figure 3

Dynamics of ellipsoids in the laboratory frame. (a) Mean-square translational displacements of ellipsoids (${\mathrm{MSD}}_{\text{T}}$) at different algal concentrations $\varphi$. (b) Mean-square rotational displacements of ellipsoids (${\mathrm{MSD}}_{\text{R}}$) at the same $\varphi$. The dashed lines indicate the slope of the data. (c) Effective translational diffusivity, $D_T$ (the left axis), and effective rotational diffusivity, $D_R$ (the right axis), as a function of $\varphi$. $D_T$ and $D_R$ were extracted from the long-time diffusions of the corresponding MSDs. The dashed line indicates a linear relation between $D_{T,R}$ and $\varphi$.

Figure 4

Probability distribution functions of laboratory-frame translational displacements (${\mathrm{PDF}}_{\text{T}}$) (a) and rotational displacements (${\mathrm{PDF}}_{\text{R}}$) (b) in a time interval $\mathrm{\Delta }t=0.1$ s at different algal concentrations $\varphi$. The dashed lines are the PDFs of Brownian ellipsoids with bare translational and rotational diffusivity given in the text. The solid lines are fits using Eq. (1). (c) The weighting factor, $f$, as a function of $\varphi$ for both ${\mathrm{PDF}}_{\text{T}}$ and ${\mathrm{PDF}}_{\text{R}}$. $f$ indicates the relative contributions of diffusion and advection. The solid lines are visual guides.

Figure 5

Dynamics of ellipsoids in the body frame. (a) Mean-square translational displacements of ellipsoids along the major axis (${\mathrm{MSD}}_{\text{a}}$) and along the minor axis (${\mathrm{MSD}}_{\text{b}}$) at different algal concentrations $\varphi$. (b) Effective diffusivity along the major axis, $D_a$, and along the minor axis, $D_b$, as a function of $\varphi$. $D_a$ and $D_b$ were extracted from the long-time diffusions of ${\mathrm{MSD}}_{\text{a}}$ and ${\mathrm{MSD}}_{\text{b}}$. The dashed line is a linear fit of $D_b$ for guiding the eye. (c) Anisotropic diffusion of ellipsoids, $D_a/D_b$, as a function of $\varphi$.

Figure 6

Comparison of anisotropic diffusions in pullers and pushers. (a) Anisotropic diffusion of ellipsoids, $D_a/D_b$, in puller-type C. reinhardtii, and in pusher-type E. coli. The concentrations of active particles are normalized by the maximal concentration studied in experiments. ${\varphi }_{\text{max}}=3.9\%$ for C. reinhardtii. ${\varphi }_{\text{max}}=40n_0$ for E. coli, where $n_0=8\times {10}^8$ cells/ml. (b) The ratio of the correlation times along the major and minor axis, ${\tau }_a/{\tau }_b$, in C. reinhardtii and in E. coli. Horizontal dashed lines indicate the ratio of 1. Since the maximal algal concentration we can achieve is much smaller than the maximal bacterial concentration due to different microbial physiology of C. reinhardtii and E. coli (see Sec. 2b), it may seem arbitrary to normalize $\varphi$ by ${\varphi }_{max}$. Thus, we also plot $D_a/D_b$ (c) and ${\tau }_a/{\tau }_b$ (d) as a function of the normalized laboratory-frame translational diffusivity $D_T/D_{T,0}$, which show similar increasing and decreasing trends for pullers and pushers, respectively. The data for E. coli are extracted from Ref. [39].

Figure 7

Probability distribution functions of body-frame translational displacements along the major axis (${\mathrm{PDF}}_{\text{a}}$) (a) and along the minor axis (${\mathrm{PDF}}_{\text{b}}$) (b) in a time interval $\mathrm{\Delta }t=0.1$ s at different algal concentrations $\varphi$. The dashed lines are the PDFs of Brownian ellipsoids with bare diffusivity along the major and minor axis given in the text. The solid lines are fits using Eq. (1). (c) The weighting factor, $f$, as a function of $\varphi$ for both ${\mathrm{PDF}}_{\text{a}}$ and ${\mathrm{PDF}}_{\text{b}}$. The solid lines are visual guides.

Figure 8

Origin of the anisotropic diffusion. Black squares are the ratio of the correlation times along the major and minor axes, ${\tau }_a/{\tau }_b$, as a function of algal concentrations $\varphi$. Red circles are the average ratio of the mean-square displacements along the major and minor axes in the superdiffusive regime, ${\langle \langle \mathrm{\Delta }{\tilde{x}}^2\rangle /\langle \mathrm{\Delta }{\tilde{y}}^2\rangle \rangle }_{\mathrm{\Delta }t}$, as a function of $\varphi$. The horizontal dashed lines indicate ${\langle \langle \mathrm{\Delta }{\tilde{x}}^2\rangle /\langle \mathrm{\Delta }{\tilde{y}}^2\rangle \rangle }_{\mathrm{\Delta }t}$ = 1.13 (upper, red) and ${\tau }_a/{\tau }_b=1$ (lower, black), respectively.