Biological and artificial microswimmers often self-propel in external flows of vortical nature; relevant examples include algae in small-scale ocean eddies, spermatozoa in uterine peristaltic flows and bacteria in microfluidic devices. A recent experiment has shown that swimming bacteria in model vortices are expelled from the vortex all the way to a well-defined depletion zone (A. Sokolov and I. S. Aranson, Rapid expulsion of microswimmers by a vortical flow. Nat. Commun., 2016, 7, 11114). In this paper, we propose a theoretical model to investigate the dynamics of elongated microswimmers in elementary vortices, namely active particles in two- and three-dimensional rotlets. A deterministic model first reveals the existence of bounded orbits near the centre of the vortex and unbounded orbits elsewhere. We further discover a conserved quantity of motion that allows us to map the phase space according to the type of the orbit (bounded vs unbounded). We next introduce translational and rotational noise into the system. Using a Fokker-Planck formalism, we quantify the quality of trapping near the centre of the vortex by examining the probability of escape and the mean time of escape from the region of deterministically bounded orbits. We finally show how to use these findings to formulate a prediction for the radius of the depletion zone, which compares favourably with the experiments (A. Sokolov and I. S. Aranson, Rapid expulsion of microswimmers by a vortical flow. Nat. Commun., 2016, 7, 11114).
Bacteria
Cilial pumping is a powerful strategy used by biological organisms to control and manipulate fluids at the microscale. However, despite numerous recent advances in optically, magnetically and electrically driven actuation, development of an engineered cilial platform with the potential for applications has remained difficult to realize1-6. Here we report on active metasurfaces of electronically actuated artificial cilia that can create arbitrary flow patterns in liquids near a surface. We first create voltage-actuated cilia that generate non-reciprocal motions to drive surface flows at tens of microns per second at actuation voltages of 1 volt. We then show that a cilia unit cell can locally create a range of elemental flow geometries. By combining these unit cells, we create an active cilia metasurface that can generate and switch between any desired surface flow pattern. Finally, we integrate the cilia with a light-powered complementary metal-oxide-semiconductor (CMOS) clock circuit to demonstrate wireless operation. As a proof of concept, we use this circuit to output voltage pulses with various phase delays to demonstrate improved pumping efficiency using metachronal waves. These powerful results, demonstrated experimentally and confirmed using theoretical computations, illustrate a pathway towards fine-scale microfluidic manipulation, with applications from microfluidic pumping to microrobotic locomotion.
Cellular appendages conferring motility, such as flagella and cilia, are known to synchronise their periodic beats. The origin of synchronization is a combination of long-range hydrodynamic interactions with physical mechanisms allowing the phases of these biological oscillators to evolve. Two of such mechanisms have been identified by previous work, the elastic compliance of the periodic orbit or oscillations driven by phase-dependent biological forcing, both of which can lead generically to stable phase locking. In order to help uncover the physical mechanism for hydrodynamic synchronization most essential overall in biology, we theoretically investigate in this paper the effect of strong confinement on the effectiveness of hydrodynamic synchronization. Following past work, we use minimal models of cilia where appendages are modeled as rigid spheres forced to move along circular trajectories near a rigid surface. Strong confinement is modeled by adding a second nearby surface, parallel to the first one, where the distance between the surfaces is much smaller than the typical distance between the cilia, which results in a qualitative change in the nature of hydrodynamic interactions. We calculate separately the impact of hydrodynamic confinement on the synchronization dynamics of the elastic compliance and the force modulation mechanisms and compare our results to the usual case with a single surface. Applying our results to the biologically relevant situation of nodal cilia, we show that force modulation is a mechanism that leads to phase-locked states under strong confinement that are very similar to those without confinement as a difference with the elastic compliance mechanism. Our results point therefore to the robustness of force modulation for synchronization, an important feature for biological dynamics that therefore suggests it could be the most essential physical mechanism overall in arrays of nodal cilia. We further examine the distinct biologically relevant situation of primary cilia and show in that case that the difference in robustness of the mechanisms is not as pronounced but still favors the force modulation.
