External control strategies for self-propelled particles: Optimizing navigational efficiency in the presence of limited resources

We experimentally and numerically study the dependence of different navigation strategies regarding the effectivity of an active particle to reach a predefined target area. As the only control parameter, we vary the particle's propulsion velocity depending on its position and orientation relative to the target site. By introducing different figures of merit, e.g., the time to target or the total consumed propulsion energy, we are able to quantify and compare the efficiency of different strategies. Our results suggest that each strategy to navigate towards a target has its strengths and weaknesses, and none of them outperforms the other in all regards. Accordingly, the choice of an ideal navigation strategy will strongly depend on the specific conditions and the figure of merit which should be optimized.

Figure 1

(a) Experimentally determined phase diagram of a water-2,6-lutidine mixture with a lower critical point (LCP) at $T_{\text{C}}=307\text{K}$. (b) Dependence of the propulsion velocity $v$ on the illumination intensity $I$. The offset corresponds to the threshold intensity below which no demixing of the fluid is induced by laser illumination. Experimental data is shown as symbols and the solid line corresponds to a linear fit above the threshold.

Figure 2

(a) Original image of a Janus particle acquired by a CCD camera. (b) Image of the particle with increased contrast and calculated barycenter (circle) and intensity centroid (asterisk). The displacement between them points away from the coating and gives the particle orientation $\mathbf{p}$ (arrow). In this image, the position of the intensity centroid is shifted (length of $\mathbf{p}$ increased) to improve visibility. (c) Probability distribution of the angle $\mathrm{\Delta }\phi$ between $\mathbf{p}$ and the particle velocity $\mathbf{v}$ [light bars: without propulsion; dark (red) bars: with propulsion]. (d) Dependence of the particle velocity $v$ on $\left|\mathbf{p}\right|$ (linear fit: solid line).

Figure 3

Schematic description of the different control strategies based on snapshots of simulations (cf. Sec. 2d). The particles start at the initial position ${\mathbf{r}}_0$ and are navigated towards the target area (bullet) around ${\mathbf{r}}_{\text{T}}$. The direct connection, i.e., ideal path, is marked by the horizontal line. If the particle orientation $\mathbf{p}$ points towards the shaded area, the propulsion is set to $v=v_{\text{max}}$ [OnOff (a), MinDev (c)] or to a value between zero and $v_{\text{max}}$ [LinScl (b)].

Figure 4

Simulated time duration to target $T$ with OnOff ${63}^{\circ }$ strategy as a function of the time resolution $\mathrm{\Delta }t$. With increasing $\mathrm{\Delta }t, T$ strongly increases. In our experiments and simulations, we have chosen $\mathrm{\Delta }t=0.5\text{s}$ (vertical dotted line). In this regime, no influence of $\mathrm{\Delta }t$ on $T$ and the other figures of merit is observed. The vertical dashed line corresponds to the particle's rotational diffusion time.

Figure 5

Arrival probability of uncontrolled particles (simulation) within 6 hours for different target distances $l$. With pure Brownian diffusion (dotted line), the probability is only about $15\%$ for $l=30\mu \text{m}$ and below $1/600$ for $l=100\mu \text{m}$. The probability rises for active particles with permanent homogeneous illumination (solid line) to about $25\%$ for $l=30\mu \text{m}$ and approximately $10\%$ for $l=100\mu \text{m}$. With active control (OnOff strategy, dashed line), all particles reach the target area.

Figure 6

Performance of the OnOff strategy depending on the angle ${\alpha }_0$ [see Eq. (2)] for two of the figures of merit: (a) time duration to target $T$ and (b) path deviation $D$. Shown are the simulation results for a target distance of $l=30\mu \text{m}$. The global minima (dashed lines) are at ${\alpha }_0={63}^{\circ }$ for $T$ and ${\alpha }_0={32}^{\circ }$ for $D$.

Figure 7

Experimental trajectories for (a) OnOff ${63}^{\circ }$, (b) OnOff ${13}^{\circ }$, and (c) MinDev strategy. The ideal path from start to target is marked as a solid straight line with the circular target area in real proportion.

Figure 8

(a) Experimental (closed symbols) and numerically (open symbols) determined time duration to target $T$ for different control strategies. (b) Summary of the performance of different control strategies for a target distance of $l=30\mu \text{m}$ (experimental and numerical data as filled and open bars, respectively). The figures of merit are normalized by the mean value of the respective criterion.

Figure 9

Simulations demonstrating the influence of $l$ on $D$ for the control with MinDev (solid) and OnOff ${\alpha }_0$ (dashed). The shaded areas indicate the standard deviation. In the OnOff strategy ${\alpha }_0$ depends on $l$ and is chosen to minimize $D$ at each distance $l$. For small values of $l$ both strategies do not differ much in $D$, whereas for large $l$ the advantage of MinDev starts to gain importance, resulting in a smaller increase than for OnOff.

Figure 10

Experimental trajectories for (a) OnOff ${76}^{\circ }$, (b) OnOff ${35}^{\circ }$, and (c) MinDev and a target distance $l=100\mu \text{m}$.

Figure 11

(a) Experimental (closed symbols) and numerically (open symbols) determined path deviation $D$ for different control strategies. (b) Summary of the performance of different control strategies for a target distance of $l=100\mu \text{m}$ (experimental and numerical data as filled and open bars, respectively). The figures of merit are normalized by the mean value of the respective criterion.

Figure 12

Influence of the diffusional enhancement factor $\epsilon$ for OnOff ${63}^{\circ }$ and $l=30\mu \text{m}$ on the time duration to target $T$. Increasing the random reorientation rate improves $T$ by more than a factor of 2. The inset schematically illustrates the strategy: (a) when the particle is oriented towards the target, the particle exhibits the Stokes-Einstein rotational diffusion $D_R$ and is propelled with $v=v_{max}$; (b) when the particle is not oriented towards the target, the rotational diffusion is increased by a factor $\epsilon$ and the propulsion is deactivated ($v=0$).