Squirmer Model Research Articles

Surface-active microrobots can propel through blood faster than inert microrobots

Abstract Microrobots that can move through a network of blood vessels have promising medical applications. Blood contains a high volume fraction of blood cells, so in order for a microrobot to move through the blood, it must propel itself by rearranging the surrounding blood cells. However, swimming form effective for propulsion in blood is unknown. This study shows numerically that a surface-active microrobot, such as a squirmer, is more efficient in moving through blood than an inert microrobot. This is because the surface velocity of the microrobot steers the blood cells laterally, allowing them to propel themselves into the hole they are digging. When the microrobot size is comparable to a red blood cell or when the microrobot operates under a low Capillary number, the puller microrobot swims faster than the pusher microrobot. The trend reverses under considerably smaller microrobot sizes or high Capillary number scenarios. Additionally, the swimming speed is strongly dependent on the hematocrit and magnetic torque used to control the microrobot orientation. A comparative analysis between the squirmer and Janus squirmer models underscores the extensive applicability of the squirmer model. The obtained results provide new insight into the design of microrobots propelled efficiently through blood, paving the way for innovative medical applications.

Study on the motion characteristics of Janus based on the squirmer model in the flow

The motion characteristics of Janus in the flow are studied numerically using the lattice Boltzmann method based on the squirmer model. The effects of velocity ratio J on the right and left hemisphere surface of Janus, particle Reynolds number Rep, flow Reynolds number Rec, initial orientation angle φ0 on Janus trajectory, and lateral equilibrium position yeq/H are analyzed. The results showed that, for the motion of Janus in stationary power-law fluids in a channel, Janus moves randomly in a small space in shear-thickening fluids when Rep is low and exhibits three motion modes at Rep = 5. The larger the J value, the easier it is for Janus to reach yeq/H. The higher the Rep, the closer the yeq/H is to the lower wall. In shear-thinning fluids, the motion of Janus exhibits significant randomness at Rep = 0.5 and 1, reaches the same yeq/H at Rep = 2 and 3, and tends toward yeq/H near the centerline and along the upper wall, respectively, at Rep = 4 and 5. For the motion of Janus particles in a channel flow of power-law fluids, in shear-thinning fluids, no matter what value J is, Janus reaches yeq/H on the centerline. The lower the Rep, the closer the yeq/H is to the wall. Two particles move toward yeq/H when Rep ≥ 1. The higher the Rep, the closer the yeq/H is to the centerline. The two particles will exhibit the upstream mode at Rep = 2. Two particles eventually reach yeq/H at different Rec. When φ0 > 0°, the two particles first eventually tend toward yeq/H = 0.2 and 0.8. When the value of φ0 is negative, the larger the absolute value of φ0 and higher the Rep, the more likely particles are to exhibit upstream mode.

Efficient navigation of cargo-towing microswimmer in non-uniform flow fields

The vision of deploying miniature vehicles within the human body for intricate tasks holds tremendous promise across engineering and medical domains. Herein, optimal navigation of a cargo-towing swimmer under an applied zig-zag flow is studied by employing direct numerical simulations coupled with a deep reinforcement learning algorithm. Tasks include navigation in flow and shear-gradient directions. We initially explore combinations of state inputs, finding that optimal navigation necessitates swimmers to perceive hydrodynamics and alignment, surpassing reliance solely on hydrodynamic signals while considering their memories. Next, we study combinations of action spaces, allowing dynamic changes in swimming and/or rotational velocities by tuning B1 and C1 parameters of the squirmer model, respectively. By keeping both parameters fixed, cargo-towing swimmers demonstrate superior performance in the flow direction compared to swimmers without load due to tumbling movements influenced by shear flow. In the shear-gradient direction, swimmers without load outperform cargo-towing swimmers, with performance decreasing as load length increases. Across the combination of allowing B1 and C1 to change, the policies from solely dynamic B1 actions demonstrate superior navigation. The policies are then used as a showcase against naive cargo-towing and inert colloidal chains. A t-distributed stochastic neighbor embedding analysis reveals the complex interplay between perceived hydrodynamic signals and swimmer position. In the flow direction, swimmers align effectively with regions of maximum velocity, while in the shear-gradient direction, periodic transitions from minimum to maximum state values occur. Comparing pullers, pushers, and neutral swimmers, cargo-towing swimmers show a reversal in swimming velocity trends, with pullers outpacing neutral and pusher swimmers, irrespective of load lengths. Published by the American Physical Society 2024

Flow structure around a microswimmer at fluid–fluid interface

Living organisms such as bacteria and algae often form biofilms at air–liquid and/or liquid–liquid interfaces. Therefore, it is important to understand the hydrodynamic interaction between the fluid–fluid interface and microorganisms. In the present study, a squirmer model is employed to study the flow field structures around a symmetrically trapped spherical microswimmer at an interface separating two fluids with viscosity contrast. In these simulations, Reynolds ( Re ) and Capillary ( Ca ) numbers are very small, and hence the contribution of inertia and interface deformations are neglected. The squirmer model is validated against the analytical solution obtained by considering a source dipole. It is observed that the flow structure and vorticity distribution around the microswimmers are strongly influenced by the squirmer parameter (β) and viscosity contrast (λ). Furthermore, the interplay between force-dipole and source-dipole along with viscosity contrast leads to a range of flow structures such as symmetric four-lobe and asymmetric quadrupolar flow fields. This flow structure asymmetry around the swimmer is quantified with different steady state orientations (φ). The effect of flow profile on the passive tracer particle transport is explored in terms of trajectories. Specifically, larger values of φ result in more profound curly trajectories.

Sedimentation of a spherical squirmer in a square tube under gravity

In this study, we used a three-dimensional lattice Boltzmann method to simulate the settling motion of a spherical squirmer in a square tube under the effect of gravity. A spherical squirmer model with chirality was chosen to simulate the motion of a real microswimmer in a three-dimensional space and to systematically analyze its kinematic properties. According to the results of this study, we identified seven different motion modes: diagonal plane large-amplitude oscillation, central stable sedimentation, bidirectional spiral motion, rebound motion, unidirectional spiral motion, corner stable motion, and near-wall attraction oscillation. It was shown that the formation of different motion modes is caused by the effects of squirmer-type factor and chirality. squirmer-type factor determines the stable motion position of the squirmer in the channel. Chirality makes the head direction of the squirmer more susceptible to change, thus changing the motion trajectory of the squirmer. In addition, it was found that the self-propelling strength determines the speed of squirmer’s motion, which affects the motion frequency of squirmer’s periodic oscillations.

The effect of particle geometry on squirming in a heterogeneous medium

Biological microorganisms and artificial micro-swimmers often locomote in heterogeneous viscous environments consisting of networks of obstacles embedded into viscous fluid media. In this work, we use the squirmer model and present a numerical investigation of the effects of shape on swimming in a heterogeneous medium. Specifically, we analyse the microorganism's propulsion speed as well as its energetic cost and swimming efficiency. The analysis allows us to probe the general characteristics of swimming in a heterogeneous viscous environment in comparison with the case of a purely viscous fluid. We found that a spheroidal microorganism always propels faster, expends less energy and is more efficient than a spherical microorganism in either a homogeneous fluid or a heterogeneous medium. Moreover, we determined that above a critical eccentricity, a spheroidal microorganism in a heterogeneous medium can swim faster than a spherical microorganism in a homogeneous fluid. Based on an analysis of the forces acting on the squirmer, we offer an explanation for the decrease in the squirmer's speed observed in heterogeneous media compared with homogeneous fluids.

Regularized Stokeslet surfaces

A variation of the Method of Regularized Stokeslets (MRS) in three dimensions is developed for triangulated surfaces with a piecewise linear force density. The work extends the regularized Stokeslet segment methodology used for piecewise linear curves. By using analytic integration of the regularized Stokeslet kernel over the triangles, the regularization parameter ϵ is effectively decoupled from the spatial discretization of the surface. This is in contrast to the usual implementation of the method in which the regularization parameter is chosen for accuracy reasons to be about the same size as the spatial discretization. The validity of the method is demonstrated through several examples, including the flow around a rigidly translating/rotating sphere and in the squirmer model for ciliate self-propulsion. Notably, second order convergence in the spatial discretization for fixed ϵ is demonstrated. Considerations of mesh design and choice of regularization parameter are discussed, and the performance of the method is compared with existing quadrature-based implementations.

The motion of micro-swimmers over a cavity in a micro-channel.

This article combines the lattice Boltzmann method (LBM) with the squirmer model to investigate the motion of micro-swimmers in a channel-cavity system. The study analyses various influential factors, including the value of the squirmer-type factor (β), the swimming Reynolds number (Rep), the size of the cavity, initial position and particle size on the movement of micro-swimmers within the channel-cavity system. We simultaneously studied three types of squirmer models, Puller (β > 0), Pusher (β < 0), and Neutral (β = 0) swimmers. The findings reveal that the motion of micro-swimmers is determined by the value of β and Rep, which can be classified into six distinct motion modes. For Puller and Pusher, when the β value is constant, an increase in Rep will lead to transition in the motion mode. Moreover, the appropriate depth of cavity within the channel-cavity system plays a crucial role in capturing and separating Neutral swimmers. This study, for the first time, explores the effect of complex channel-cavity systems on the behaviour of micro-swimmers and highlights their separation and capture ability. These findings offer novel insights for the design and enhancement of micro-channel structures in achieving efficient separation and capture of micro-swimmers.

Collective behavior of squirmers in thin films.

Bacteria in biofilms form complex structures and can collectively migrate within mobile aggregates, which is referred to as swarming. This behavior is influenced by a combination of various factors, including morphological characteristics and propulsive forces of swimmers, their volume fraction within a confined environment, and hydrodynamic and steric interactions between them. In our study, we employ the squirmer model for microswimmers and the dissipative particle dynamics method for fluid modeling to investigate the collective motion of swimmers in thin films. The film thickness permits a free orientation of non-spherical squirmers, but constraints them to form a two-layered structure at maximum. Structural and dynamic properties of squirmer suspensions confined within the slit are analyzed for different volume fractions of swimmers, motility types (e.g., pusher, neutral squirmer, puller), and the presence of a rotlet dipolar flow field, which mimics the counter-rotating flow generated by flagellated bacteria. Different states are characterized, including a gas-like phase, swarming, and motility-induced phase separation, as a function of increasing volume fraction. Our study highlights the importance of an anisotropic swimmer shape, hydrodynamic interactions between squirmers, and their interaction with the walls for the emergence of different collective behaviors. Interestingly, the formation of collective structures may not be symmetric with respect to the two walls. Furthermore, the presence of a rotlet dipole significantly mitigates differences in the collective behavior between various swimmer types. These results contribute to a better understanding of the formation of bacterial biofilms and the emergence of collective states in confined active matter.

Feeding of planktotrophic squirmers: Effects of mobility and elongation of planktonic particles

Ciliated micro-organisms feed on small planktonic and inorganic particles. Investigating their feeding ability is valuable for understanding corresponding ecodynamics. However, how the feeding ability is affected by the motility and elongation of their food particles remains unknown. In this study, we carry out numerical simulations based on a squirmer model and a point-particle model to represent the ciliated micro-organisms and planktonic particles, respectively. The feeding of the squirmer is accomplished by generating a flow field that attracts particles nearby. This squirmer-induced flow is described by the sum of multiple modes of Legendre polynomials. We adopt a 2-mode model, formed by the first mode and any other one, to investigate the influence of the flow structure on the feeding ability. The latter mode identifies the flow field under the two constraints of the constant maximum surface speed and the fixed vortex size. We find that the dependency of the feeding ability on the flow field varies with the mobility of food particles significantly. For non-motile particles, the feeding ability is little disturbed by the flow mode. While for motile particles, the feeding ability is negatively correlated with the flow mode, which suggests an efficient feeding strategy where the squirmer prioritizes enhancing swimming over attraction to capture more particles. Meanwhile, the elongation of food particles also plays an important role, as more elongated motile particles are more likely to be captured by the flows with high modes. This investigation advances the understanding of feeding on motile and elongated particles.

Hydrodynamic pursuit by cognitive self-steering microswimmers

The properties of biological microswimmers are mostly determined by fluid-mediated interactions, which govern their propulsion, perception of their surrounding, and the steering of their motion for pursuit. Transferring similar functionalities to synthetic microswimmers poses major challenges, and the design of favorable steering and pursuit strategies is fundamental in such an endeavor. Here, we employ a squirmer model to investigate the pursuit of pursuer-target pairs with an implicit sensing mechanism and limited hydrodynamic steering abilities of the pursuer. Two hydrodynamic steering strategies are studied for the pursuer’s propulsion direction by adaptation of its surface flow field: reorientation toward the target with limited maneuverability, and alignment with the target’s propulsion direction combined with speed adaptation. Depending on the nature of the microswimmer propulsion (puller, pusher) and the velocity-adaptation scheme, stable cooperatively moving states can be achieved, where pursuer and target mutually affect their motion and trajectories.

Fluid Dynamics of Squirmers and Ciliated Microorganisms

The fluid dynamics of microswimmers has received attention from the fields of microbiology, microrobotics, and active matter. Microorganisms have evolved organelles termed cilia for propulsion through liquids. Each cilium periodically performs effective and recovery strokes, creating a metachronal wave as a whole and developing a propulsive force. One well-established mathematical model of ciliary swimming is the squirmer model, which focuses on surface squirming velocities. This model is also useful when studying active colloids and droplets. The squirmer model has been recently used to investigate the behaviors of microswimmers in complex environments, their collective dynamics, and the characteristics of active fluids. Efforts have also been made to broaden the range of applications beyond the assortment permitted by the squirmer model, which was established to specifically represent ciliary flow and incorporate biological features. The stress swimmer model imposes stresses above the cell body surface that enforce the no-slip condition. The ciliated swimmer model precisely reproduces the behaviors of each cilium that engages in mutual hydrodynamic interactions. Mathematical models have improved our understanding of various microbial phenomena, including cell–cell and cell–wall interactions and energetics. Here, I review recent advances in the hydrodynamics of ciliary swimming and then discuss future challenges.

Slippery rheotaxis: new regimes for guiding wall-bound microswimmers

The near-surface locomotion of microswimmers under the action of background flows has been studied extensively, whereas the intervening effects of complex surface properties remain hitherto unknown. Intending to delineate the shear-driven dynamics near a planar slippery wall, we adopt the squirmer model of microswimmers and employ a three-dimensional analytical-numerical framework in bispherical coordinates. It is interpreted that both the self-propulsion and the external shear flow are redistributed due to hydrodynamic slippage, followed by modulations in the thrust torque on the microswimmer. Phase portraits of the quasi-steady dynamics indicate that the stable upstream swimming states, known as ‘rheotaxis’, are significantly modulated by the slip length compared with the no-slip case. For puller swimmers, an intricate interplay among the modulated interfacial friction near a slippery surface, velocity gradients of the shear flow and the strength of the squirmer parameter promotes a critical shear rate beyond which a wide range of new rheotactic states exist. Consequently, an escaping microswimmer may exhibit rheotaxis or an existing rheotactic state annihilates due to crashing. Although stable states are absent for pushers without steric interactions, transitions from escaping and undamped oscillations to ‘rheotaxis’ occur in the presence of wall repulsion, but only until the other characteristics are overwhelmed by escape due to enhanced shear. Disclosing the ability of hydrodynamic slippage in broadening the scope of migration against a background flow for a wide range of parameters, the present work paves the way for investigations on the entrapment of microswimmers near complex pathways or sorting using selective rheotaxis.

Dynamics and steady state of squirmer motion in liquid crystal

AbstractWe analyse a nonlinear partial differential equation system describing the motion of a microswimmer in a nematic liquid crystal environment. For the microswimmer’s motility, the squirmer model is used in which self-propulsion enters the model through the slip velocity on the microswimmer’s surface. The liquid crystal is described using the well-established Beris–Edwards formulation. In previous computational studies, it was shown that the squirmer, regardless of its initial configuration, eventually orients itself either parallel or perpendicular to the preferred orientation dictated by the liquid crystal. Furthermore, the corresponding solution of the coupled nonlinear system converges to a steady state. In this work, we rigorously establish the existence of steady state and also the finite-time existence for the time-dependent problem in a periodic domain. Finally, we will use a two-scale asymptotic expansion to derive a homogenised model for the collective swimming of squirmers as they reach their steady-state orientation and speed.

Chemotaxis of two chiral squirmers

External gradients can strongly influence the collective behavior of microswimmers. In this paper, under an external linear chemical gradient, we study the behavior of two hydrodynamically interacting self-propelled chiral swimmers in the low-Reynolds number regime. We use the generalized squirmer model called the chiral squirmer, a spherically shaped body with an asymmetric surface slip velocity, to represent the swimmer. We find that the external gradient favors the attraction between the swimmers and, in some situations, leads to a bounded state in which the swimmers move in a highly synchronous manner. Furthermore, due to this cooperative motion, these swimmers reach the chemical target faster than individual swimmers. This study may help in understanding the collective behavior of chiral swimmers and in designing synthetic microswimmers for targeted drug delivery.

Near- and far-field hydrodynamic interaction of two chiral squirmers.

Hydrodynamic interaction strongly influences the collective behavior of microswimmers. With this work, we study the behavior of two hydrodynamically interacting self-propelled chiral swimmers in the low Reynolds number regime, considering both the near- and far-field interactions. We use the chiral squirmer model [see Burada et al., Phys. Rev. E 105, 024603 (2022)2470-004510.1103/PhysRevE.105.024603], a spherically shaped body with nonaxisymmetric surface slip velocity, which generalizes the well-known squirmer model. The previous work was restricted only to the case, while the far-field hydrodynamic interaction was influential among the swimmers. It did not approach the scenario while both the swimmers are very close and lubrication effects become dominant. We calculate the lubrication force between the swimmers when they are very close. By varying the slip coefficients and the initial configuration of the swimmers, we investigate their hydrodynamic behavior. In the presence of lubrication force, the swimmers either repel each other or exhibit bounded motion where the distance between the swimmers alters periodically. We identify the possible behaviors exhibited by the chiral squirmers, such as monotonic divergence, divergence, and bounded, as was found in the previous study. However, in the current study, we observe that both the monotonic convergence and the convergence states are converted into divergence states due to the arising lubrication effects. The lubrication force favors the bounded motion in some parameter regimes. This study helps to understand the collective behavior of dense suspension of ciliated microorganisms and artificial swimmers.

Active percolation in pusher-type microswimmers

The aggregation of microorganisms in colonies and biofilms underpins a myriad of biological processes, and has crucial implications in ecology and biomedical sciences. While much of our knowledge of microbial motion is based on single-cell mechanisms or cell-cell interactions, the origin of cooperativity in microbial communities is not yet fully understood. Here, we reveal the existence of a continuum percolation transition in two model suspensions of pusher-type microswimmers: an asymmetric dumbbell and a squirmer model. Clusters of swimmers held together by hydrodynamic forces dynamically aggregate and separate. Using simulations with explicit hydrodynamics and theory, we find that as the microswimmers' filling fraction increases, the cluster size distribution approaches a scale-free form and system-spanning clusters emerge.

Chemoreception and chemotaxis of a three-sphere swimmer

The coupled problem of hydrodynamics and solute transport for the Najafi–Golestanian three-sphere swimmer is studied, with the Reynolds number set to zero and Péclet numbers (Pe) ranging from 0.06 to 60. The adopted method is the numerical simulation of the problem with a finite element code based upon the FEniCS library.For the swimmer executing the optimal locomotion gait, we report the Sherwood number as a function of Pe in homogeneous fluids and show that it is quite consistent with that corresponding to the well-known spherical squirmer model. It is also confirmed that little gain in solute flux is achieved by swimming, unless Pe is significantly larger than 10.We also consider the swimmer as a learning agent moving inside a fluid that has a concentration gradient. The outcomes of Q-learning processes show that learning locomotion (with the displacement as reward) is significantly easier than learning chemotaxis (with the increase of solute flux as reward). The chemotaxis learning problem, even at low Pe, has a varying environment that renders learning more difficult. Further, the learning difficulty increases severely with the Péclet number. The results demonstrate the challenges that natural and artificial swimmers need to overcome to migrate efficiently when exposed to chemical inhomogeneities.

Study on the sedimentation and interaction of two squirmers in a vertical channel

We simulated the sedimentation of two self-propelled particles in a two-dimensional (2D) vertical channel using the lattice Boltzmann method. A 2D squirmer model was employed to simulate the microswimmers, and five typical locomotive modes were obtained for a single squirmer, namely, central steady sedimentation, near-wall steady motion, wall-attracted oscillation, large-amplitude oscillation, and small-amplitude oscillation. The locomotive modes of two squirmers are obtained by combinations of different locomotive modes of a single squirmer. It was found that the motion of two squirmers was much more complex than that of a single squirmer, and this complex locomotive mode could be explained by the pressure distribution of the two squirmers. Moreover, we performed a comprehensive analysis of the obtained locomotive modes and determined that the angle at which the two squirmers separated from each other and swimming speed were crucial, which may be the reason for the different locomotive modes of the squirmers that switch from each other.

Simulating Microswimmers Under Confinement With Dissipative Particle (Hydro) Dynamics

In this work we study microwimmers, whether colloids or polymers, embedded in bulk or in confinement. We explicitly consider hydrodynamic interactions and simulate the swimmers via an implementation inspired by the squirmer model. Concerning the surrounding fluid, we employ a Dissipative Particle Dynamics scheme. Differently from the Lattice-Boltzmann technique, on the one side this approach allows us to properly deal not only with hydrodynamics but also with thermal fluctuations. On the other side, this approach enables us to study microwimmers with complex shapes, ranging from spherical colloids to polymers. To start with, we study a simple spherical colloid. We analyze the features of the velocity fields of the surrounding solvent, when the colloid is a pusher, a puller or a neutral swimmer either in bulk or confined in a cylindrical channel. Next, we characterise its dynamical behaviour by computing the mean square displacement and the long time diffusion when the active colloid is in bulk or in a channel (varying its radius) and analyze the orientation autocorrelation function in the latter case. While the three studied squirmer types are characterised by the same bulk diffusion, the cylindrical confinement considerably modulates the diffusion and the orientation autocorrelation function. Finally, we focus our attention on a more complex shape: an active polymer. We first characterise the structural features computing its radius of gyration when in bulk or in cylindrical confinement, and compare to known results obtained without hydrodynamics. Next, we characterise the dynamical behaviour of the active polymer by computing its mean square displacement and the long time diffusion. On the one hand, both diffusion and radius of gyration decrease due to the hydrodynamic interaction when the system is in bulk. On the other hand, the effect of confinement is to decrease the radius of gyration, disturbing the motion of the polymer and thus reducing its diffusion.