Achiral Planar Microswimmers (APMs)

• APMs are planar, non-chiral microstructures that achieve propulsion via rotation–translation coupling in low-Reynolds-number (Stokes) flows.
• They employ magnetic, phoretic, or viscoelastic actuation to generate controlled trajectories such as straight, circular, or precessional motion.
• In vivo demonstrations, such as with zebrafish embryos, highlight their potential for targeted drug delivery, microscale manipulation, and biomedical imaging.

Searching arXiv for the cited APM and related microswimmer papers to ground the article in the primary literature. Achiral planar microswimmers (APMs) are low-Reynolds-number swimmers whose propulsion does not rely on geometric handedness. In the magnetic microrobotics literature, the term typically denotes planar rigid bodies with nonzero rotation–translation coupling under externally imposed rotation; in related active-matter literature, it also encompasses achiral swimmers whose motion is confined to a plane and acquires persistent translation or circular trajectories through asymmetry, boundary effects, or medium memory. Across these variants, the central question is how an achiral object can generate net motion in Stokes flow, where time-reversible actuation ordinarily forbids swimming. Recent work has extended this subject from in vitro control to the safe injection and clear imaging of a lithographically fabricated APM in live zebrafish embryos, establishing an in vivo platform for follow-up studies of microswimmer locomotion (Wang et al., 25 Aug 2025).

1. Defining features and geometric classes

An APM is, in the strictest sense used in magnetic propulsion studies, a planar body whose structure lies in a single plane and whose geometry is achiral, yet whose hydrodynamic response permits translation under rotation. In the zebrafish study, the representative APM is a rigid, flat two-arm SU-8 microstructure with a $120^\circ$ opening, dimensions $$50\,\mu\mathrm{m}\times 20\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$$, and a width-to-height ratio of $4{:}1$. Its body is planar, its overall shape is superimposable on its mirror image, and its magnetic functionality is provided by a thin Ti–Co–Ti stack deposited on the planar substrate (Wang et al., 25 Aug 2025).

A related minimal magnetic APM is the rigid three-sphere swimmer composed of three identical spheres of radius $a=2.25\,\mu\mathrm{m}$ in a $90^\circ$ bent configuration. That geometry is planar and achiral in 3D, with two mutually perpendicular planes of symmetry but not three, and it exhibits a nonzero translation–rotation coupling block $\mathbf{C}$ in its mobility matrix. In that setting, achirality does not preclude propulsion; rather, propulsion depends on whether the symmetry class allows a nonvanishing coupling between torque-driven rotation and translation (Buzhardt et al., 2019).

The broader literature summarized here uses the term more expansively for effectively planar achiral swimmers. L-shaped phoretic particles confined near a substrate, V-shaped magnetic or electric propellers, and even spherical Janus colloids in quasi-2D viscoelastic media all realize planar achiral dynamics without requiring a helical body. A useful synthesis from pitch theory is that some achiral objects have non-vanishing pitch matrices, with the achiral class characterized by one zero moment of pitch and two equal-magnitude, opposite-sign moments, $(0,\Lambda,-\Lambda)$. This formalizes the statement that geometric chirality is not a necessary condition for rotation-induced translation at low Reynolds number (Duraes et al., 2023).

2. Hydrodynamic basis and symmetry

The hydrodynamic description of APMs is formulated in Stokes flow, where inertia is negligible and force, torque, translation, and rotation are linearly related. In one common representation,

$$\begin{pmatrix} \mathbf{V} \ \boldsymbol{\Omega} \end{pmatrix} = \begin{pmatrix} \mathbf{K} & \mathbf{C} \ \mathbf{C}^{\intercal} & \mathbf{M} \end{pmatrix} \begin{pmatrix} \mathbf{F}^e \ \mathbf{T}^e \end{pmatrix},$$

so propulsion by a rotating field requires a nonzero translation–rotation coupling $\mathbf{C}$. The same content can be recast through the pitch matrix $\mathsf{P}$, defined by

$$50\,\mu\mathrm{m}\times 20\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$$0

which directly links rotation to translation for a force-free rigid body. The pitch formulation is especially useful because it makes symmetry constraints explicit: if the symmetries of the body force $$50\,\mu\mathrm{m}\times 20\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$$1, no torque-driven swimming is possible; if not, achiral swimming can occur even without helicity (Buzhardt et al., 2019, Duraes et al., 2023).

For magnetically driven APMs, the applied torque is the dipole–field torque

$$50\,\mu\mathrm{m}\times 20\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$$2

and the governing regime is characterized by

$$50\,\mu\mathrm{m}\times 20\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$$3

The zebrafish APM work explicitly frames locomotion in this low-Reynolds-number regime and attributes propulsion to symmetry breaking produced by the swimmer shape, magnetic alignment, and hydrodynamics under a conical rotating magnetic field. The object is achiral, but the driven stroke is not kinematically symmetric over a cycle, so steady forward thrust becomes possible (Wang et al., 25 Aug 2025).

Symmetry analysis refines this picture. For geometrically achiral planar objects with a permanent dipole, parity $$50\,\mu\mathrm{m}\times 20\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$$4, charge conjugation $$50\,\mu\mathrm{m}\times 20\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$$5, and combined $$50\,\mu\mathrm{m}\times 20\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$$6 symmetries partition the dynamics into distinct classes. Highly symmetrical $$50\,\mu\mathrm{m}\times 20\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$$7-even objects with permanent dipoles exhibit no net propulsion in synchronous rotation. Less symmetrical $$50\,\mu\mathrm{m}\times 20\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$$8-even objects can propel individually, but symmetry pairs their propulsive states so that the average ensemble velocity vanishes. By contrast, $$50\,\mu\mathrm{m}\times 20\,\mu\mathrm{m}\times 5\,\mu\mathrm{m}$$9-odd magnetization orientations yield unidirectional motion analogous to that usually associated with chiral helices (Sachs et al., 2017).

A distinct hydrodynamic route appears in viscoelastic media. For spherical, rigid, achiral Janus colloids in a polymer solution, the fluid memory generates a delayed propulsion force $4{:}1$0 that is not collinear with the instantaneous orientation $4{:}1$1. The resulting effective internal torque drives a transition from enhanced angular diffusion to persistent circular motion above a critical propulsion speed. In that case, effective chirality is emergent rather than geometric: the swimmer remains achiral, while the fluid’s non-Markovian response creates a persistent phase lag between propulsion and orientation (Narinder et al., 2018).

3. Actuation modes and trajectory phenomenology

The canonical actuation protocol for magnetic APMs is a spatially uniform rotating field. For the three-sphere magnetic swimmer, the field takes the form

$4{:}1$2

or, for steering, a rotated version in which the plane of rotation is tilted by an angle $4{:}1$3. That system exhibits three regimes as the frequency $4{:}1$4 varies: a low-frequency tumbling regime with negligible net translation, an intermediate propulsion regime with precession and net motion perpendicular to the field-rotation plane, and a high-frequency step-out regime in which synchronous rotation fails. For the modeled swimmer, the numerically identified transition frequencies are $4{:}1$5 and $4{:}1$6 (Buzhardt et al., 2019).

The zebrafish APM uses a conical rotating magnetic field generated by a 3D Helmholtz coil system. In the paper’s notation, the field combines a rotating component, a static bias $4{:}1$7, a vertical component $4{:}1$8, and a heading angle $4{:}1$9 that sets the swimming direction. The perpendicular vector to the rotating-field plane,

$a=2.25\,\mu\mathrm{m}$0

sets the swimmer orientation tendency, and modulating $a=2.25\,\mu\mathrm{m}$1 steers the propulsion axis. The same work emphasizes that higher rotation frequency typically yields higher propulsion velocity up to a step-out threshold, although quantitative in vivo velocity curves were not reported there (Wang et al., 25 Aug 2025).

Not all APM trajectories are field-normal translations. In L-shaped phoretic swimmers confined near a substrate, shape asymmetry and an offset propulsion force create a deterministic torque $a=2.25\,\mu\mathrm{m}$2, coupling translation and rotation. The resulting angular velocity is proportional to propulsion strength, $a=2.25\,\mu\mathrm{m}$3, while the orbit radius is approximately

$a=2.25\,\mu\mathrm{m}$4

so $a=2.25\,\mu\mathrm{m}$5 is independent of propulsion strength. Experimentally, these swimmers show $a=2.25\,\mu\mathrm{m}$6 increasing linearly with $a=2.25\,\mu\mathrm{m}$7 and a measured radius $a=2.25\,\mu\mathrm{m}$8, consistent with constant-radius circular motion set by geometry rather than by drive amplitude (Kümmel et al., 2013).

In viscoelastic media, the trajectory class changes discontinuously. For spherical Janus swimmers at $a=2.25\,\mu\mathrm{m}$9 PAAm, the critical propulsion speed is $90^\circ$0; for $90^\circ$1 PAAm, $90^\circ$2. Above threshold, the steady angular velocity follows

$90^\circ$3

and the radius of curvature is $90^\circ$4. The sign of $90^\circ$5 can spontaneously reverse, a behavior that sharply distinguishes memory-induced circular motion from intrinsically chiral swimmers with fixed sense of rotation (Narinder et al., 2018).

4. Materials, fabrication, and representative platforms

The material realization of APMs is heterogeneous, but planar microfabrication is a recurring theme.

Platform	Defining structure	Reported behavior
Zebrafish APM	SU-8 planar two-arm body, Ti–Co–Ti coating, $90^\circ$6, $90^\circ$7 arm angle	Forward thrust under rotating magnetic field; safe injection and clear in vivo imaging
Three-sphere magnetic swimmer	Three rigidly connected spheres, $90^\circ$8 bend, $90^\circ$9	Tumbling, propulsion, and step-out; contactless microparticle manipulation
L-shaped phoretic swimmer	SU-8 planar L, arms $\mathbf{C}$0 and $\mathbf{C}$1, thickness $\mathbf{C}$2, 20 nm Au cap	Constant-radius circular motion; wall sliding or reflection
V-shaped magnetic/electric propeller	Planar V or arc with permanent dipole or induced electric dipole	Symmetry-controlled propulsion under rotating magnetic or electric fields
Janus colloid in viscoelastic fluid	Silica sphere, radius $\mathbf{C}$3, half carbon-coated	Transition from persistent random walk to circular motion

The zebrafish APM is fabricated by standard planar microfabrication. A dextran sacrificial layer is spin-coated on silicon, SU8-2005 is spun at $\mathbf{C}$4 for $\mathbf{C}$5 to obtain $\mathbf{C}$6 thickness, patterned by UV lithography, developed, cleaned by plasma treatment, and coated by physical vapor deposition with a Ti–Co–Ti stack of $\mathbf{C}$7, $\mathbf{C}$8, and $\mathbf{C}$9. Release is achieved by dissolving dextran in water, yielding free-floating APMs in aqueous suspension. The bottom Ti layer improves adhesion of Co to SU-8, while the top Ti layer protects Co from oxidation and enhances biocompatibility for in vivo use (Wang et al., 25 Aug 2025).

L-shaped phoretic APMs are also fabricated lithographically. A $(0,\Lambda,-\Lambda)$0 SU-8 layer is patterned on a silicon wafer, and a $(0,\Lambda,-\Lambda)$1 Au cap is deposited by thermal evaporation with the wafer tilted by approximately $(0,\Lambda,-\Lambda)$2, producing selective coating on one side of the short arm. V-shaped micromachines in the symmetry study are realized by glancing-angle deposition, either as magnetic SiO$(0,\Lambda,-\Lambda)$3 microstructures containing nickel sections or as polarizable SiO$(0,\Lambda,-\Lambda)$4 V-shapes with one Au-coated arm (Kümmel et al., 2013, Sachs et al., 2017).

These fabrication routes reflect a broader distinction within the field. Some APMs rely on body-fixed magnetic moments and external rotational actuation; others rely on internal phoretic forcing, anisotropic polarizability, or medium memory. A plausible implication is that “APM” is best understood as a hydrodynamic and symmetry category rather than a single materials platform.

5. In vivo realization in zebrafish embryos

The most direct in vivo implementation reported so far is the injection and imaging of an APM in a live zebrafish embryo. The model organism is the zebrafish embryo at $(0,\Lambda,-\Lambda)$5 hours post-fertilization, maintained in E2 culture medium in a PDMS chamber. This stage was chosen because embryos are effectively transparent, permitting direct optical observation of microswimmers inside the body, and because the yolk sac provides a relatively large, fluid-filled compartment for microrobot placement and actuation. The heartbeat was clearly observed during experiments, confirming viability and good physiological status (Wang et al., 25 Aug 2025).

The injection protocol is intentionally minimal. The chorion is manually removed with fine tweezers, $(0,\Lambda,-\Lambda)$6 of anesthetic is added, and the embryo is moved to the chamber edge. A microprobe of approximately $(0,\Lambda,-\Lambda)$7 diameter punctures the yolk to create a small wound, and the same probe presses an APM through the opening into the yolk. An important developmental constraint is that embryos younger than $(0,\Lambda,-\Lambda)$8 hpf tend to suffer large outflow of yolk on puncture and die; the $(0,\Lambda,-\Lambda)$9 hpf window avoids that failure mode. Because the injected object is solid, the effective injection volume is only the microscopic displacement associated with the probe and swimmer (Wang et al., 25 Aug 2025).

Imaging is integrated with magnetic actuation. The system comprises three orthogonal pairs of Helmholtz coils, three Kepco BOP20-5M supplies, a National Instruments PCI-6259 DAQ card, and a LabVIEW-based control interface, together with a conventional bright-field microscope using a $$\begin{pmatrix} \mathbf{V} \ \boldsymbol{\Omega} \end{pmatrix} = \begin{pmatrix} \mathbf{K} & \mathbf{C} \ \mathbf{C}^{\intercal} & \mathbf{M} \end{pmatrix} \begin{pmatrix} \mathbf{F}^e \ \mathbf{T}^e \end{pmatrix},$$0 objective and a CMOS camera (BFS-U3-13Y3M-C, $$\begin{pmatrix} \mathbf{V} \ \boldsymbol{\Omega} \end{pmatrix} = \begin{pmatrix} \mathbf{K} & \mathbf{C} \ \mathbf{C}^{\intercal} & \mathbf{M} \end{pmatrix} \begin{pmatrix} \mathbf{F}^e \ \mathbf{T}^e \end{pmatrix},$$1 pixels). Figure-level observations reported in the study show that the APM’s planar geometry is clearly resolvable inside the yolk, providing sufficient spatial resolution to determine body orientation in vivo (Wang et al., 25 Aug 2025).

A common misconception is that this zebrafish work already demonstrated quantitative in vivo swimming. It did not. Its stated contribution was safe injection and clear imaging of an APM in a transparent zebrafish model, thereby demonstrating the possibility for follow-up in-depth studies of swimming motion in vivo. The paper explicitly stops short of reporting in vivo trajectories, speed–frequency curves, or 2D/3D path reconstructions inside zebrafish, even though the actuation and imaging infrastructure is designed to support precisely those measurements in future studies (Wang et al., 25 Aug 2025).

6. Manipulation, collective dynamics, and biomedical significance

APMs are not limited to self-transport; they also function as mobile hydrodynamic manipulators. In the three-sphere magnetic model, the swimmer’s disturbance flow advects a nearby passive cargo without physical attachment. Control is achieved by switching both the plane of field rotation and the actuation frequency above and below the critical frequency. In the tumbling regime, the swimmer acts as a localized stirrer and drives the cargo along circular arcs; in the propulsion regime, it repositions relative to the cargo while still imparting motion through the flow field. The paper concludes that microswimmers can be utilized as mobile manipulators of microparticles in a fluid, and notes that the underlying control strategy is transferable to other geometries and propulsion mechanisms (Buzhardt et al., 2019).

At the suspension level, the multi-species DDFT framework treats achiral planar force-dipole swimmers in planar geometries with steric and hydrodynamic interactions, external traps, and shear-cell boundaries. In binary pusher–puller mixtures, the majority species imposes its behavior on the minority species. For unconfined mixtures, linear stability analysis predicts a threshold for the fraction of pullers and for their propulsion strength beyond which global polar orientational order emerges. In shear cells bounded by driven passive colloids, the confined active swimmers acquire behavior reminiscent of circle swimmers, though with varying swimming radii. None of these effects requires intrinsic swimmer chirality; they arise from collective hydrodynamics, confinement, and externally imposed flow (Hoell et al., 2019).

Several recurring misconceptions are clarified by this body of work. First, circular motion does not imply intrinsic chirality: it can arise from an off-center propulsion force and translation–rotation coupling in asymmetric planar particles, from viscoelastic memory in an otherwise spherical swimmer, or from imposed shear in an achiral planar suspension (Kümmel et al., 2013, Narinder et al., 2018, Hoell et al., 2019). Second, propulsion under rotation does not require a helix: some achiral objects have non-vanishing pitch matrices, and chirality is not essential for propulsion due to rotation–translation coupling at low Reynolds number (Duraes et al., 2023, Sachs et al., 2017). Third, previous in vivo zebrafish studies had visually demonstrated gradient pulling or rolling, but not swimming; the zebrafish APM work is significant precisely because it creates a platform in which true swimming, rather than only transport by pulling or rolling, can be investigated (Wang et al., 25 Aug 2025).

The biomedical motivation is explicit. The cited literature associates microswimmers with targeted drug delivery, cell transport, microsurgical procedures, and imaging-guided interventions. The zebrafish study adds enabling evidence for in vivo feasibility by combining biocompatible coating, minimal mechanical damage, and direct optical observability in a transparent model. At the same time, the same study emphasizes unresolved constraints: biological fluids can exhibit viscosity variations and non-Newtonian behavior; the yolk is a confined cavity with soft boundaries; tissue interactions may hinder planar swimmers by friction or sticking; and embryonic motion and heartbeat can superimpose uncontrolled flows. This suggests that the next stage of APM research is not merely stronger actuation, but systematic comparison between in vitro and in vivo propulsion efficiency, controllability, and biocompatibility under realistic biological confinement (Wang et al., 25 Aug 2025).