Three-bead steering microswimmers

Abstract

The self-propelled microswimmers have recently attracted considerable attention as model systems for biological cell migration as well as artificial micromachines. A simple and well-studied microswimmer model consists of three identical spherical beads joined by two springs in a linear fashion with active oscillatory forces being applied on the beads to generate self-propulsion. We have extended this linear microswimmer configuration to a triangular geometry where the three beads are connected by three identical springs in an equilateral triangular manner. The active forces acting on each spring can lead to autonomous steering motion; i.e., allowing the swimmer to move along arbitrary paths. We explore the microswimmer dynamics analytically and pinpoint its rich character depending on the nature of the active forces. The microswimmers can translate along a straight trajectory, rotate at a fixed location, as well as perform a simultaneous translation and rotation resulting in complex curved trajectories. The sinusoidal active forces on the three springs of the microswimmer contain naturally four operating parameters which are more than required for the steering motion. We identify the minimal operating parameters which are essential for the motion of the microswimmer along any given arbitrary trajectory. Therefore, along with providing insights into the mechanics of the complex motion of the natural and artificial microswimmers, the triangular three-bead microswimmer can be utilized as a model for targeted drug delivery systems and autonomous underwater vehicles where intricate trajectories are involved.

A general perturbative approach for bead-based microswimmers reveals rich self-propulsion phenomena

“…Both, translational and rotational motion were shown to scale with the square of the driving force. Also here, the perturbative approach used in [21] does only hold in the limit n  0. The triangular swimmer is composed of three spherical beads, each of radius a, connected by identical springs of equilibrium length L and spring constant k. All beads are placed in the x-z-plane with orientation θ the angle between the connection of bead 3 to the middle between bead 1 and 2 and the x-axis ( figure 3).…”

“…Furthermore, setting the square of all three driving force amplitudes equal to a constant allows us to keep the power input at the same order of magnitude, as the power input is known to scale with the square of the force for a bead in the viscous regime [20]. Later in this section, we apply our method to the driving protocol used in [21,43] for comparison. Throughout the whole section on triangular swimmers, we make use of the Oseen approximation for the hydrodynamic interactions of the beads.…”

“…The accuracy of the results in terms of the bead separation is tuned by the choice of the mobility matrix, whereas the precision in terms of the actuation strength is controlled by the perturbation order. This is essentially different from previous calculations [28,30,21] as we split the equation of motion by orders of the driving force and systematically solve the long-term limit of each order, while the assumption of very large bead separation is not made within the perturbation scheme, but is imposed by the hydrodynamic model. Consequently, given a sufficiently precise mobility matrix, the presented method provides arbitrarily precise results for the swimming velocity.…”

“…The locomotion of swimmers at small scales has been an active area of research in recent years [1], with a variety of microswimmer models being proposed, both experimental [2][3][4][5][6][7][8][9][10][11] and theoretical [12][13][14][15][16][17][18][19][20][21]. A number of these models aims at understanding the propulsion mechanisms of small organisms such as bacteria or algae cells, or at designing artificial microswimmers.…”

“…The full controllability of a triangular swimmer in the 2D space has recently been shown analytically [21], expanding on the use of the bead-spring model [20]. However, the perturbative calculations in [20,30,21] hold only in the limit of very large bead separations where the swimming velocity becomes extremely small. Especially since this limit is inaccessible in experiments (see [5,9]) as external disturbances become exuberant and can break the swimmer, an investigation of swimmers with smaller bead separations is required and still lacking.In this paper, we fill this gap by presenting a general perturbative framework to calculate the full behavior of arbitrarily shaped bead-spring swimmers, i.e.…”