Three-sphere Microswimmer Research Articles

The effect of axisymmetric confinement on propulsion of a three-sphere microswimmer

Swimming at the microscale has recently garnered substantial attention due to the fundamental biological significance of swimming microorganisms and the wide range of biomedical applications for artificial microswimmers. These microswimmers invariably find themselves surrounded by different confining boundaries, which can impact their locomotion in significant and diverse ways. In this work, we employ a widely used three-sphere swimmer model to investigate the effect of confinement on swimming at low Reynolds numbers. We conduct theoretical analysis via the point-particle approximation and numerical simulations based on the finite element method to examine the motion of the swimmer along the centerline in a capillary tube. The axisymmetric configuration reduces the motion to one-dimensional movement, which allows us to quantify how the degree of confinement affects the propulsion speed in a simple manner. Our results show that the confinement does not significantly affect the propulsion speed until the ratio of the radius of the tube to the radius of the sphere is in the range of O(1)−O(10), where the swimmer undergoes substantial reduction in its propulsion speed as the radius of the tube decreases. We provide some physical insights into how reduced hydrodynamic interactions between moving spheres under confinement may hinder the propulsion of the three-sphere swimmer. We also remark that the reduced propulsion performance stands in stark contrast to the enhanced helical propulsion observed in a capillary tube, highlighting how the manifestation of confinement effects can vary qualitatively depending on the propulsion mechanisms employed by the swimmers.

Autonomous elastic microswimmer

A model of an autonomous three-sphere microswimmer is proposed by implementing a coupling effect between the two natural lengths of an elastic microswimmer. Such a coupling mechanism is motivated by the previous models for synchronization phenomena in coupled oscillator systems. We numerically show that a microswimmer can acquire a nonzero steady state velocity and a finite phase difference between the oscillations in the natural lengths. These velocity and phase differences are almost independent of the initial phase difference. There is a finite range of the coupling parameter for which a microswimmer can have an autonomous directed motion. The stability of the phase difference is investigated both numerically and analytically in order to determine its bifurcation structure.

Reciprocal microswimmers in a viscoelastic fluid

We suggest several reciprocal swimming mechanisms that lead to locomotion only in viscoelastic fluids. In the first situation, we consider a three-sphere microswimmer with a difference in oscillation amplitudes for the two arms. In the second situation, we consider a three-sphere microswimmer in which one of the frequencies of the arm motion is twice as large as the other one. In the third situation, we consider a two-sphere microswimmer with a difference in size for the two spheres. In all these three cases, the average velocity is proportional to the imaginary part of the complex shear viscosity of a surrounding viscoelastic medium. We show that it is essential for a micromachine to break its structural symmetry in order to swim in a viscoelastic fluid by performing reciprocal body motions.

Nonequilibrium probability flux of a thermally driven micromachine.

We discuss the nonequilibrium statistical mechanics of a thermally driven micromachine consisting of three spheres and two harmonic springs [Y. Hosaka etal., J. Phys. Soc. Jpn. 86, 113801 (2017)JUPSAU0031-901510.7566/JPSJ.86.113801]. We obtain the nonequilibrium steady state probability distribution function of such a micromachine and calculate its probability flux in the corresponding configuration space. The resulting probability flux can be expressed in terms of a frequency matrix that is used to distinguish between a nonequilibrium steady state and a thermal equilibrium state satisfying detailed balance. The frequency matrix is shown to be proportional to the temperature difference between the spheres. We obtain a linear relation between the eigenvalue of the frequency matrix and the average velocity of a thermally driven micromachine that can undergo a directed motion in a viscous fluid. This relation is consistent with the scallop theorem for a deterministic three-sphere microswimmer.

A three-sphere microswimmer in a structured fluid

We discuss the locomotion of a three-sphere microswimmer in a viscoelastic structured fluid characterized by typical length and time scales. We derive a general expression to link the average swimming velocity to the sphere mobilities. In this relationship, a viscous contribution exists when the time-reversal symmetry is broken, whereas an elastic contribution is present when the structural symmetry of the microswimmer is broken. As an example of a structured fluid, we consider a polymer gel, which is described by a “two-fluid” model. We demonstrate in detail that the competition between the swimmer size and the polymer mesh size gives rise to the rich dynamics of a three-sphere microswimmer.

State diagram of a three-sphere microswimmer in a channel

Geometric confinements are frequently encountered in soft matter systems and in particular significantly alter the dynamics of swimming microorganisms in viscous media. Surface-related effects on the motility of microswimmers can lead to important consequences in a large number of biological systems, such as biofilm formation, bacterial adhesion and microbial activity. On the basis of low-Reynolds-number hydrodynamics, we explore the state diagram of a three-sphere microswimmer under channel confinement in a slit geometry and fully characterize the swimming behavior and trajectories for neutral swimmers, puller- and pusher-type swimmers. While pushers always end up trapped at the channel walls, neutral swimmers and pullers may further perform a gliding motion and maintain a stable navigation along the channel. We find that the resulting dynamical system exhibits a supercritical pitchfork bifurcation in which swimming in the mid-plane becomes unstable beyond a transition channel height while two new stable limit cycles or fixed points that are symmetrically disposed with respect to the channel mid-height emerge. Additionally, we show that an accurate description of the averaged swimming velocity and rotation rate in a channel can be captured analytically using the method of hydrodynamic images, provided that the swimmer size is much smaller than the channel height.

Swimming trajectories of a three-sphere microswimmer near a wall.

The hydrodynamic flow field generated by self-propelled active particles and swimming microorganisms is strongly altered by the presence of nearby boundaries in a viscous flow. Using a simple model three-linked sphere swimmer, we show that the swimming trajectories near a no-slip wall reveal various scenarios of motion depending on the initial orientation and the distance separating the swimmer from the wall. We find that the swimmer can either be trapped by the wall, completely escape, or perform an oscillatory gliding motion at a constant mean height above the wall. Using a far-field approximation, we find that, at leading order, the wall-induced correction has a source-dipolar or quadrupolar flow structure where the translational and angular velocities of the swimmer decay as inverse third and fourth powers with distance from the wall, respectively. The resulting equations of motion for the trajectories and the relevant order parameters fully characterize the transition between the states and allow for an accurate description of the swimming behavior near a wall. We demonstrate that the transition between the trapping and oscillatory gliding states is first order discontinuous, whereas the transition between the trapping and escaping states is continuous, characterized by non-trivial scaling exponents of the order parameters. In order to model the circular motion of flagellated bacteria near solid interfaces, we further assume that the spheres can undergo rotational motion around the swimming axis. We show that the general three-dimensional motion can be mapped onto a quasi-two-dimensional representational model by an appropriate redefinition of the order parameters governing the transition between the swimming states.

Elastic Three-Sphere Microswimmer in a Viscous Fluid

We discuss the dynamics of a generalized three-sphere microswimmer in which the spheres are connected by two elastic springs. The natural length of each spring is assumed to undergo a prescribed cyclic change. We analytically obtain the average swimming velocity as a function of the frequency of the cyclic change in the natural length. In the low-frequency region, the swimming velocity increases with the frequency and its expression reduces to that of the original three-sphere model by Najafi and Golestanian. In the high-frequency region, conversely, the average velocity decreases with increasing the frequency. Such a behavior originates from the intrinsic spring relaxation dynamics of an elastic swimmer moving in a viscous fluid.

Swimmer-Microrheology

We discuss a locomotion of a three-sphere microswimmer in a viscoelastic medium and propose a new type of active microrheology. We derive a relation which connects average swimming velocity and frequency-dependent viscosity of the surrounding medium. In this relation, the viscous contribution can exist only when the time-reversal symmetry is broken, whereas the elastic contribution is present only when the structural symmetry of the swimmer is broken. The Purcell's scallop theorem breaks down for a three-sphere swimmer in a viscoelastic medium.