Boundary shape engineering for the spatial control of confined active particles

Abstract

Unlike an equilibrium gas, the distribution of active particles can be very sensitive to what happens at the boundaries of their container. Experiments and simulations have previously highlighted the possibility of exploiting this behavior for the geometric control of active particles, although a general theoretical framework is lacking. Here we propose a boundary method based on the flux transfer formalism typical of radiometry problems, where surface elements transmit and receive "rays" of active particles with infinite persistence length. As in the case of blackbody radiation, a Lambert scattering law results in a uniform distribution of active particles within the cavity, while other scattering laws result in specific patterns of particle accumulation in the bulk or over the boundary walls. We validate our method's predictions with numerical simulations and demonstrate its practical utility by spatially controlling swimming microalgae confined in light-defined arenas. The presented boundary method offers a simple and efficient way to predict particle distributions when both the geometry of the boundaries and the scattering law are known. In addition, it provides a general design principle for engineering container shapes optimized for transport, accumulation, and sorting of self-propelled colloids and microorganisms.

Figures (8)

• Figure 1: Geometric description of the boundary model.
• Figure 2: Spatial probability density distributions calculated with the boundary method in four different confinement geometries for the set of scattering angle distributions defined in Table \ref{['fig:ftable']}.
• Figure 3: Validation of the boundary method with active particle simulations. Two dimensional spatial probability density distributions obtained with particle simulations are shown for a generic shape and a set of scattering laws. Results of the two methods are compared on plots of density profiles along two sample lines, with the simulation results represented by gray circles and the boundary model results by black lines.
• Figure 4: Scatter angle distribution of Euglena gracilis cells. a Image of the measurement's light pattern with a sample timelapse of swimming cells superimposed. The enlarged area shows a zoomed view on two cells going through scattering at the light-dark boundary. White arrows depict the local boundary normal. b Polar histogram of the measured scattering angles. Blue line plots the best fit with $~\cos^\alpha(\theta+\theta_0)$, while the red line shows the Lambert cosine law. c Dependence of the outgoing angle $\theta$ from the incoming angle $\theta^\prime$. The white plot shows the mean outgoing angle in respective intervals of the incoming angle.
• Figure 5: Spatial distribution of Euglena gracilis cells confined in a circular light patch. a Bright-field image snapshot of cells during a measurement. The light pattern's edge is marked by the magenta line. b A random sample of persistent cell trajectories. c Boundary model results. d Two-dimensional spatial distribution of the cells (smoothed with a 0.5 bin wide Gaussian). e Radial spatial probability density distributions calculated from measurement data (green crosses) and from simulation results of particles with scattering distributions of $\cos \theta$ (blue circles) and $\cos^8 \theta$ (orange circles). Theoretically calculated density curves are shown as black lines. Scalebars 0.5 mm.