Optimal motility strategies for self-propelled agents to explore porous media

Microrobots for, e.g., biomedical applications, need to be equipped with motility strategies that enable them to navigate through complex environments. Inspired by biological microorganisms we re-create motility patterns such as run-and-reverse, run-and-tumble, or run-reverse-flick applied to active rodlike particles in silico. We investigate their capability to efficiently explore disordered porous environments with various porosities and mean pore sizes ranging down to the scale of the active particle. By calculating the effective diffusivity for the different patterns, we can predict the optimal one for each porous sample geometry. We find that providing the agent with very basic sensing and decision-making capabilities yields a motility pattern outperforming the biologically inspired patterns for all investigated porous samples.

Figure 1

Schematic representation of the geometry of the rigid self-propelled agent.

Figure 2

A two-dimensional slice through a typical randomly generated porous geometry with mean pore radius $r_{\text{p}}\approx 2.6\mu \mathrm{m}$. The colors indicate the local thickness, a measure for the pore size (see Methods).

Figure 3

Two-dimensional projections of example trajectories without confinement for the four biologically inspired motility patterns. The pictograms (not to scale) show the phases involved in the respective pattern, i.e., forward swimming, backward swimming and rotation. The trajectory for reverse-when-stuck is not depicted, as this pattern reduces to straight swimming if there are no pores to get trapped in. For easier distinction, the temperature is reduced by a factor of 60 compared to the simulations used in our analysis.

Figure 4

Mean-squared displacement of the run-and-reverse pattern at various mean pore sizes $r_{\text{p}}$. The black lines indicate different scaling behaviours as a guide to the eye.

Figure 5

Solid, colored lines, left axis: Effective diffusion coefficient $D_{\text{eff}}$ as function of average pore size for all motility strategies. Dashed, gray line, right axis: Connectedness of the accessible void space measured as the fraction of the void space occupied by the largest connected region with local thickness $\tau \left(\mathbf{r}\right)>r_{\text{body}}$. The shaded areas denote one standard error of the mean over $N_{\text{ensemble}}=7$ statistically independent simulations.

Figure 6

Joint probability density of attempted tumble angle ${\mathrm{\Theta }}_{\text{tumble}}$ and actual tumble angle ${\mathrm{\Theta }}_{\text{tumble}}^{*}$ between start and end of a tumble at mean pore radius $r_{\text{p}}\approx 2\mu \mathrm{m}$. The black line indicates ${\mathrm{\Theta }}_{\text{tumble}}^{*}={\mathrm{\Theta }}_{\text{tumble}}$.

Figure 7

Two dimensionless quantities as a function of the mean pore size. Blue: The mean actual tumble angle $\langle {\mathrm{\Theta }}_{\text{tumble}}^{*}\rangle$ normalized by the mean attempted tumble angle $\langle {\mathrm{\Theta }}_{\text{tumble}}\rangle$. Orange: The fraction of pores smaller than the agent size, i.e., with $\tau \left(\mathbf{r}\right)<l_{\text{body}}$. The shaded areas denote one standard error of the mean over $N_{\text{ensemble}}=7$ statistically independent simulations. For further explanations, see the main text.

Figure 8

Effective diffusion coefficient as a function of mean run time. Colors denote the motility stategy. Solid, dashed and dotted lines correspond to different mean pore sizes. The shaded areas denote one standard error of the mean over $N_{\text{ensemble}}=5$ statistically independent simulations.