Diffusion of self-propelled particles in complex media

The diffusion of active microscopic organisms in complex environments plays an important role in a wide range of biological phenomena from cell colony growth to single organism transport. Here, we investigate theoretically and computationally the diffusion of a self-propelled particle (the organism) embedded in a complex medium composed of a collection of nonmotile solid particles that mimic soil or other cells. Under such conditions we find that the rotational relaxation time of the swimming direction depends on the swimming velocity and is drastically reduced compared to a pure Newtonian fluid. This leads to a dramatic increase (of several orders of magnitude) in the effective rotational diffusion coefficient of the self-propelled particles, which can lead to “self-trapping” of the active particles in such complex media. An analytical model is put forward that quantitatively captures the computational results. Our work sheds light on the role that the environment plays in the behavior of active systems and can be generalized in a straightforward fashion to understand other synthetic and biological active systems in heterogenous environments.

Figure 1

(a) Schematic representation of a squirmer swimming within a passive monolayer. Velocity field generated by (b) a pusher with $\beta =-5$ and (c) a puller with $\beta =5$.

Figure 2

Rotational diffusion coefficient as a function of Brownian Peclet number, $\mathrm{Pe}=\frac{U}{D_{\theta }^0\sigma }$, for different squirmer types and externally driven particles at ${\varphi }_A=0.5$. The solid lines correspond to the analytical model described in the text. Inset: Phase shift, $\mathrm{\Delta }$, as a function of $\beta$.

Figure 3

Color map of the passive particles area fraction, ${\varphi }_A$, around the different type of active particles in $x\text{−}y$ plane swimming at about Pe = 737: Pushers, pullers, neutral squirmers, and ED particles. Blue regions corresponds to less dense areas, while red regions corresponds to more packed areas.

Figure 4

Translational diffusion coefficient as a function of the Brownian Peclet number for different squirmer types and externally driven particles. The squirmers are on the bottom wall of a channel of height $h=30\mathrm{\Delta }x$. The $D_t$ are normalized by the translational diffusion coefficient of an active Brownian particle, $D_t^{ab}=U_0^2/\left(2D_{\theta }^0\right)$. The solid lines correspond to the analytical model described in the text.

Figure 5

Trajectories of the different types of squirmers swimming at about Pe = 450 within passive monolayers of ${\varphi }_A=0.5$: $\beta =-5$ (black), $\beta =-1$ (red), $\beta =0$ (diamonds), $\beta =1$ (blue), and $\beta =5$ (orange).

Figure 6

Normalized rotational diffusion coefficient as a function of the Pe for squirmers with $\beta =-5$ when embedded in monolayers of passive particles at different particle area fractions.