Steady rotation and wall-mediated dynamics of magnetic Janus particles in oscillating fields

Abstract

Active colloids are synthetic particles inspired by the self-propulsion and adaptability of microorganisms. These particles hold promise to perform intelligent microscale tasks, yet their behavior near boundaries and substrates remains poorly understood. Here we identify how the particle–substrate separation, and applied field frequency dictate the dynamics of magnetic Janus microparticles actuated by an oscillating magnetic field. First, we show that the continuous rotation of the non-inertial Janus particle arises from the coupling between its permanently aligned and field-induced magnetic moments. Second, by varying the particle's height above the substrate, we identify distinct motion regimes, namely, rolling, hybrid, and rotational, where each is defined by characteristic changes in the in-plane rotation, translational dynamics, and trajectory. We reveal the emergence of quasi-steady states at intermediate heights and dynamic in- and out-of-plane rotations that give rise to field frequency-dependent linear and alternating arcs and loops i.e. trochoidal trajectories. The insights gained here provide a framework for designing surface microrollers with programmable kinematics driven by time-varying magnetic fields.

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1 Introduction

Active colloids are microscopic particles that convert energy from their surroundings into persistent motion. With careful design and functionalization, these colloidal particles have the potential to perform tasks at microscale, including targeted drug delivery,^1–3 environmental remediation,^4–6 cargo transport,^7,8 and navigation through complex environments.^9–11 Much of the work in the field of active colloids draws inspiration from nature, where microorganisms have evolved sophisticated strategies to sense their surroundings, communicate, and move through complex environments. They can effectively navigate non-uniform spaces by responding to geometric constraints, mechanical cues, and external gradients.¹² However, maneuvering synthetic colloidal particles in fluids presents significant challenges due to the unique hydrodynamic constraints at these length scales. Microscale swimming operates in a regime dominated by viscous forces, where inertial effects are negligible.¹³ Additionally, Brownian motion introduces unpredictable fluctuations in particle trajectories, making organized motion challenging to achieve. As a result, achieving controlled and efficient motion at the microscale requires not only careful design of propulsion mechanisms, but also a deeper understanding of the environmental factors that influence particle dynamics.¹⁴

External electromagnetic fields enable contactless propulsion of active colloids in fluid media without a priori need to modify the physicochemical properties of the surrounding medium.^15–20 Magnetic fields are of great interest because of their ability to remotely actuate magnetic colloids in both in vitro and in vivo environments. Magnetic propulsion can be achieved using two types of fields: time-varying magnetic fields and spatially varying magnetic fields. Spatially varying (i.e., non-uniform) magnetic fields create a net force on ferromagnetic particles, pulling them toward regions of higher field strength, which is known as positive magnetophoresis.²¹ A major advantage of this method is its simplicity in implementation, as it requires only a permanent magnet. However, the magnetophoretic force decays rapidly with the distance, limiting the effective penetration depth.²² Additionally, the trajectory of the particle animated via magnetophoresis is generally limited to be linear, which is less effective in traversing complex environments compared to curvilinear trajectories.¹⁰

Spatially uniform, time-varying magnetic fields, in which the field strength and polarity vary periodically, can induce continuous rotation of ferromagnetic beads. When these rotating particles are near a substrate, hydrodynamic coupling between the particle and surface breaks symmetry in the viscous Stokes flow, generating net translational motion.^13,23–26 This substrate-mediated mechanism efficiently propels particles through viscous biological fluids and tissue-like micro-environments. Recent experimental studies highlight the importance of surface proximity^27,28 and surface curvature in modulating this translation.^29–33 For example, rotating colloids on curved surfaces exhibit strong interparticle coupling leading to complex collective states and transient assemblies.³¹ However, a key unresolved question remains: how does the height of the rotating particle above the substrate influence its translation and rotational dynamics? Quantifying this dependence is a focus of this study.

The dynamics of active colloids are known to be influenced by nearby boundaries, which modulate their motion through altering hydrodynamic and steric interactions.^{7,21,23,24,31,34–39} However, direct experimental studies that isolate and quantify this effect remain scarce. Addressing this challenge requires not only precise control over particle–substrate separation but also the ability to maintain consistent driving conditions. To meet this need, we designed an experimental system, in which we vary the height of a near neutrally buoyant particle above a flat substrate systematically while applying controlled magnetic actuation. Our model system consists of magnetic Janus particles, composed of polystyrene spheres coated on one hemisphere with an iron cap, suspended in water above a glass substrate, and actuated by an out-of-plane time-varying magnetic field. First, we discuss the origin of the continuous rotation of the non-inertial magnetic Janus particle in the time-varying magnetic field. Second, we explore how the height of the particle above the substrate and the particle anisotropy together dictate translational and rotational dynamics. This experimental study identifies how varying the height of the particle above a substrate governs the dynamics of an active particle (torque driven), and how the Janus character of the ferromagnetic patch leads to non-linear particle trajectories. Specifically, we show that varying the height of the particle above a substrate, together with the applied magnetic field frequency, programs in-plane particle rotation, as well as the translational speeds. These conditions give rise to new propulsion types, with trajectories spanning from linear to alternating arcs and loops (i.e., trochoidal). These results provide new insights into boundary-driven motion of Janus colloids, while highlighting the broader role of particle-level design for engineering intelligent microrobots in confined environments.

2 Results & discussion

2.1 Experimental details

In our experiments, nonmagnetic polystyrene microspheres (radius a = 2.5 µm, zeta potential ∼−30 mV) were used as model particles. These microparticles were immobilized as a close-packed monolayer onto a glass substrate via convective assembly and subsequently transferred to a vacuum metal vapor deposition chamber.⁴⁰ Microparticles were coated with a 20 nm chromium adhesion layer, onto which a 22 nm thick layer of iron was subsequently deposited. The formation of hemispherical iron patch on the microspheres was confirmed using backscattered scanning electron microscopy (Fig. 1a). The deposition parameters were kept constant across all samples to minimize batch-to-batch variability (see Section 4). The magnetic properties of the Janus particles were determined using vibrating sample magnetometry (VSM). The VSM measurements showed a hysteresis loop under cyclic magnetizing and demagnetizing fields, confirming the ferromagnetic nature of the iron patch (see SI Section S1). After metal deposition, Janus particles were scraped and redispersed in deionized water by sonication for 120 minutes. A typical experimental cell was assembled by sandwiching 2 µL of this dispersion between a glass slide and a cover slip separated by a spacer of thickness ∼300 µm, thus approximating a semi-infinite domain above the flat glass substrate.

Fig. 1 Height above the substrate affects dynamics of a rotating Janus particle. (a) SEM image of the iron-capped Janus particles where the brighter regions represent the ferromagnetic patch. Scale bar = 5 µm. (b) Schematic of the setup used in our experiments, and the time varying field applied to the dispersion of the Janus microparticles. The iron patch is represented in gray and the polystyrene core in green. (c)–(e) Overlay of top-view optical microscopy images showing rolling (Δt = 2 s), hybrid (Δt = 6 s), and rotational motions (Δt = 2 s). Red and yellow arrows indicate out-of-plane and in-plane rotations, respectively, and Δt is the total duration of particle motion used to generate each overlay. The hybrid motion reflects a coupled rotational–rolling behavior, characterized by concurrent in-plane and out-of-plane rotation modes. The applied field was H₀ = 8 mT at 35 Hz; scale bar = 5 µm. (f) Schematic side view of the three types of particle motion—rolling, hybrid, and rotation—induced by the vertical sinusoidal field (as illustrated in (b)). (g) Mean square displacement (MSD) of particles undergoing rolling (black, h ∼ 2.5 µm), tumbling (red, at h ∼ 18 µm), and rotating (blue, h ∼ 40 µm) motions. The distinct slopes of MSD vs. τ show the field-driven active motion (∝τ²) and diffusive behavior (∝τ¹) far away from the substrate. (h) The translation speed of a Janus particle decreases with increasing height above the substrate. Error bars represent the standard deviation of mean speeds measured across three experiments for each data point.

The experimental cell was placed at the center of a Helmholtz coil and transferred to the stage of an optical microscope (Fig. 1b). Janus particles were imaged using bright-field microscopy, and their height above the substrate was determined by translating the microscope stage by a known distance to shift the focal plane from the glass substrate to the particle center (see Methods). Independent sedimentation measurements show that the Janus particles settle slowly at a rate of ∼5 µm h⁻¹ (10⁻³ µm s⁻¹; see SI Sections S2 and S3), indicating that their average mass density is closely matched to that of the surrounding fluid. As a result, the particle height h remains effectively constant over the duration of individual experiments (ca. 20 seconds). Different particle heights were accessed by identifying particles at distinct stages of sedimentation toward the substrate. An out-of-plane, time-varying magnetic field was applied using a function generator and power amplifier. The field had the form H(t) = H₀ sin(ωt)e_z where H₀ is the field amplitude, ω is the angular frequency, and e_z is the unit vector normal to the substrate. The resulting field-driven particle dynamics were recorded using a Leica DFC 9000GTC camera mounted on a Leica DM6 microscope.

2.2 Height above a substrate governs particle dynamics

The height of a rotating Janus particle above a substrate influences its motion. We observed three different types of motion by varying the particle height above the substrate as shown for the duration Δt in Fig. 1c–e. For small surface separations (h ∼ 2.5 µm), Janus particles roll across the substrate due to out-of-plane rotation in the oscillating field, with translation arising from hydrodynamic coupling to the nearby wall. At larger heights (here, h > 36 µm), out-of-plane rotation was accompanied by slow in-plane rotation and minimal translation due to weak hydrodynamic coupling with the wall (Fig. 1c–f, and Movie S1). At intermediate heights (2.5 µm < h < 36 µm), the combination of in-plane and out-of-plane rotation, together with stronger rotation–translation coupling (than at h > 36 µm), produced more complex trajectories, including linear and trochoidal paths (discussed later).

The influence of particle height above the substrate on its translational dynamics at fixed field parameters (H₀ = 8 mT at 35 Hz) is quantified by plotting the mean squared displacement (MSD) as a function of lag time τ (Fig. 1g). The MSD scales as ∝τ² when the particle is near the substrate, and the scaling decreases to ∝τ¹ when the particle is far from the substrate, indicating active translational motion close to the surface and nearly diffusive behavior at large separations. This change in MSD scaling with τ highlights two effects: (1) the translational dynamics are inherently coupled to the particle's height above the substrate, and (2) translation is suppressed as the rotating particle moves away from the interface. These trends are further supported by the measured translational speed (Fig. 1h), which shows that speed decreases monotonically with increasing height under constant magnetic field conditions. While these experiments clearly demonstrate the critical dependence of particle speed on height above the substrate, a fundamental question remains: Why do micron-sized Janus particles with negligible inertia rotate at all? To address this, we first focus on the rolling behavior of the particle on the substrate and its dependence on the field frequency.

2.3 Particles roll at small surface separations

At small heights above the substrate (h ∼ a), Janus particles exhibit steady rolling motion along linear trajectories (Movies S1 and S2). The rolling is not constrained to a preferred axis: particles can translate in any direction within the plane, maintaining approximately constant speed (Fig. 2a). To better understand the orientational dynamics, we define a Janus director d as the vector from the particle center to the geometric centroid of the hemispherical metal shell. As detailed below, the director d lies nearly perpendicular both to the substrate normal e_z and to the direction of motion Û, such that rolling proceeds along the particle's Janus equator. Individual particles adopt one of two chiral states: “right-handed” when Û × d ≈ e_z, or “left-handed” when Û × d ≈ −e_z. This handedness appears to be an intrinsic property of each particle, persisting stably across many revolutions and remaining unchanged when the field is switched on and off.

Fig. 2 Rolling at small heights above the substrate (h ∼ a). (a) Linear trajectories showing field-driven rolling along different directions. Scale bar = 5 µm. (b) Average particle speed U vs. driving field frequency ω. Points are the experimental measurements; the solid curve shows model predictions. (c) Schematic of a Janus sphere showing the lab frame (e_y, e_y, e_z) and the particle frame (b_y, b_y, b_z). The director d of the Janus cap (grey) points along the b_y direction. The permanent magnetic moment m_p lies in the b_xb_z plane of the Janus equator. (d) Proposed mechanism of field-driven rotation (about the e_y direction) in an oscillating field. The driving torque e_y·T is proportional to the external field e_z·H = H₀ sin ωt and the e_x component of the magnetic moment, which includes a permanent moment m_p and a field-induced moment α·H.

The translation speed U of the particle depends on the driving frequency ω of the applied field (Fig. 2b). Below a critical value ω_c, the speed increases linearly with frequency as U = κaω, with a fitted traction of κ = 0.034. Above this threshold, however, the translation speed decreases sharply with increasing frequency. Similar behavior is well known for ferromagnetic particles driven to roll by rotating magnetic fields.¹¹ By contrast, the observation of steady rolling in an oscillating field is unexpected, and its physical origins have not been clearly established. Previous studies of rolling in oscillating magnetic fields typically used larger particles, where inertial effects enabled the symmetry breaking required for sustained rotation.³¹ That mechanism cannot account for the present observations, as inertia is negligible on the micron scale (refer to SI Section S4–S6).

We propose an alternative mechanism for steady rolling that considers the torque acting on the particle's magnetic moment, which includes both a permanent contribution^41–44 and an anisotropic field-induced contribution.^45–47 VSM measurements indicate that the particles possess a permanent moment m_p together with a field-induced moment α·H, where α is the magnetic polarization tensor (SI Sections S1 and S4). Owing to the geometry of the Janus cap, this tensor is anisotropic and can be expressed in terms of the particle's principal axes as α = α_xb_xb_x + α_yb_yb_y + α_zb_zb_z (Fig. 2c). If the tensor shares the axial symmetry of an ideal Janus cap with director d = b_y, then one expects α_z = α_x > α_y. In practice, small polarizability differences within the Janus equator (0 < α_z − α_x ≪ α_x) may arise due to imperfections in the metal cap (Fig. 1a) or asymmetries in its magnetization.⁴⁸ By incorporating these small deviations, we develop a mathematical model that explains particle rolling in an oscillating magnetic field without inertial effects.

We consider a single Janus sphere of radius a immersed in a fluid of viscosity η at a height h above a flat substrate. The particle experiences a magnetic torque T = μ₀(m × H), where the total magnetic moment is given by m = m_p + α·H, and μ₀ is the vacuum permeability. Other forces and torques—for example, those due to gravity or Brownian motion—are assumed negligible by comparison. The permanent moment m_p is assumed to lie in the plane of the Janus equator and is parameterized by the angle δ, such that m_p = m_p(sin δb_x + cos δb_z) (Fig. 2c). When the polarizability satisfies α_y < α_x ≈ α_z, the particle orients the director perpendicular to the surface normal; its rotation about the director is described by an angle θ that evolves as

Here, ζ_r = 6πηa³/Y_c is friction coefficient for out-of-plane rotation or, more precisely, the reciprocal of the rotational mobility²³ for a force-free sphere (refer to SI Section S4). The mobility coefficient Y_c for the sphere-plane geometry increases from Y_c = 0.339 at surface separation ξ = (h − a)/a = 0.01 to 3/4 as ξ → ∞ (Fig. S5).²³

Analysis of this simple model reveals that particle rotation in an oscillating field is governed by three dimensionless parameters: the permanent moment angle δ, the small polarizability difference Δ_xz = 1/2(α_z − α_x)H₀/m_p ≪ 1, and the driving frequency ω scaled by the magnetic relaxation rate ω₀ = μ₀m_pH₀/ζ_r. Considering the average speed is related to the time-averaged rotation rate as U = κa〈[small theta, Greek, dot above]〉, the model reproduces the observed dependence of particle translation on the driving frequency (Fig. 2b, solid curve). It further predicts “right-handed” rotation for 0 < δ < π/2 and “left-handed” rotation for −π/2 < δ < 0, with the critical frequency maximized at δ = ±π/4 (SI Fig. S6). While steady particle rotation requires a finite polarizability difference Δ_xz ≠ 0, the critical frequency depends only weakly (logarithmically) on this small parameter (SI Fig. S7).

The observed value ω_c/2π ≈ 47 Hz in Fig. 2b is reproduced using physically realistic parameter estimates: field strength μ₀H₀ = 8 mT, particle radius a = 2.5 µm, fluid viscosity η = 1 mPa s, permanent moment strength m_p ≈ 8.3 × 10⁻¹⁴ A m² (from VSM), surface separation ξ = (h − a)/a ≈ 0.19 implying κ = 0.034 and Y_c = 0.61 (from rolling experiments),⁴⁹ permanent moment angle δ = π/4, and polarizability difference Δ_xz = 0.043 (fitting parameter). Many of these parameters—namely, the surface separation, the moment angle, and the polarizability difference—are expected to vary from one particle to the next, resulting in differences in the critical frequency ω_c. Based on frequency-dependent transitions in the particle dynamics (discussed below), this value was observed to vary from ω_c/2π = 35 to 50 Hz across the different particles analyzed.

Physically, the small polarizability difference ensures that the particle's permanent moment rotates past the vertical direction of the external field, thereby sustaining continuous rotation as the field reverses (ω < ω_c). This behavior is illustrated in Fig. 2d, which shows the time evolution of the relevant quantities over a single oscillation cycle (here with Δ_xz = 0.1 to exaggerate the effect). For rotation in the e_y direction, the magnetic torque T_y = −1/2μ₀m_xH_z (red curve) depends on the e_x-component of the moment and the e_z-component of the field (green curve). The anisotropic induced moment (blue curve) prevents the permanent moment (black curve) from aligning with the external field, allowing it to overshoot with each reversal and thereby sustain continuous rolling motion (see Movie S2). Consistent with this mechanism, the measured particle velocity exhibits a dominant oscillation at twice the driving frequency 2f (see SI Section S8).

2.4 Frequency shapes trajectories at intermediate heights

Janus particles translate along linear trajectories when in close proximity to the substrate and exhibit negligible motion when far from it (Fig. 3a). Consistent with the model presented above, this decrease in translation speed U with increasing height h is attributed to a weaker rotation-translation coupling, as described by the height-dependent traction parameter κ(h). At intermediate heights (2.5 µm < h < 36 µm), we observe new types of particle trajectories in which out-of-plane rolling combines with in-plane rotation to produce trochoidal paths (Fig. 3a). These nonlinear trajectories emerge only above a critical frequency, coinciding with the transition from phase-locked to asynchronous rotation described in the previous section (Fig. 3b). For the particle shown in Fig. 3c, a modest increase in frequency—from 30 to 40 Hz—is sufficient to trigger this qualitative shift from linear to nonlinear motion (Movie S3). Because particle sedimentation is negligible on the experimental timescale (ca. 20 s), this transition cannot be attributed to changes in the particle height above the substrate.

Fig. 3 Particle trajectories as a function of height and frequency. (a) Overlays of the bright field microscopy images showing the trajectories of Janus particles below and above the critical frequency at three heights above the substrate (as indicated). At intermediate heights, the hybrid trajectory transitions from linear to alternating arcs and loops i.e. trochoidal upon increasing the field frequency from 35 Hz to 40 Hz. (b) Plot of particle speed vs field frequency (at H₀ = 8 mT) an intermediate height (h ∼ 18 µm) and on the substrate (h ∼ 2.5 µm, reproduced from Fig. 2). The error bars in (b) represent standard deviation of the mean particle speed measured across triplicate experiments for each data point. The curves are guides to the eyes. (c) The trajectory of a Janus particle at h ∼ 25 µm changes from linear to trochoidal as the field frequency increases from 30 to 40 Hz at t = 6 s (red circle). Scale bar = 5 µm. (d) Overlays of bright field microscopy images during trochoidal motion at h ∼ 25 µm and f = 40 Hz. The top image shows a representative experimental frame with the particle positions overlaid for duration Δt = 5 s. The middle image overlays the same microscopy frame with the in-plane rotation vector IPR (red), drawn from the particle centroid (blue) toward the patch centroid (yellow). The bottom image provides a simplified digital rendering of the same overlay: particles are represented as hollow circles, with centroids and the IPR vector drawn for improved visibility. Inset (right): schematic definition of the IPR vector and the angle β relative to the x-axis, with the patch orientation indicated. The vector (in red), directed toward the metal patch, tends to orient toward the local center of curvature along the trajectory.

Above the critical frequency, the average translation speed decreases nonlinearly with further increases in frequency, following the same qualitative trend observed for particles rolling directly on the substrate (Fig. 3b). We therefore attribute the transition from linear to nonlinear motion to the shift from phase-locked to asynchronous rotation identified for rolling particles at small surface separations. The reduction in translation speed is again attributed to weakened hydrodynamic interactions with the substrate at larger heights.

To characterize the in-plane rotation of the Janus sphere along its trochoidal trajectory, we define the in-plane rotation vector IPR = d − (e_z·d)e_z, which connects the particle centroid (tail) to the center of the metal patch (head) (Fig. 3d and Movie S3). As the particle translates due to rapid out-of-plane rotation, the IPR vector rotates slowly at a variable rate, thereby steering the particle along its trochoidal path. We also quantified the polar tilt of the metal patch (SI Section S9), confirming that the Janus director remains nearly perpendicular to the surface normal, corresponding to a sideways orientation of the patch. Notably, particles always turn toward their metal patch, producing clockwise or counterclockwise rotation for left- and right-handed particles, respectively, when viewed from above. This in-plane particle rotation at intermediate heights is not predicted by our simple rolling model, and its mechanistic origins remain unclear. We therefore sought to experimentally quantify the rotational dynamics of the particle as a function of its height above the substrate.

2.5 Height impacts in-plane rotational and translation

The time evolution of the in-plane orientation of the Janus particle depends on its height above the substrate and the applied field frequency (Fig. 4). We first discuss the dynamics at low frequencies, ω < ω_c, when the particle rotates synchronously with the external field (Fig. 4a–g). At small surface separations (h ∼ a), the in-plane orientation angle β—defined as the angle between the IPR vector and the x-axis (inset Fig. 3d)—is approximately constant, corresponding to the linear rolling trajectories. Due to strong hydrodynamic coupling to the substrate, this steady rolling produces large translational speeds of 16 µm s⁻¹ on average, with relatively small temporal variations (∼25%; Fig. 4e). The reported speed is obtained by linear regression of the particle position over a moving window of 0.1 s, which smooths over oscillations on the shorter time scale, ω⁻¹ = 5 ms. By contrast, at intermediate heights, both the orientation angle β and the translation speed U exhibit larger temporal fluctuations (Fig. 4c and f), which suggest that proximity to the substrate has a stabilizing effect on linear rolling.

Fig. 4 Particle orientation and dynamics. (a) and (h) Vector plot of IPR for rolling (h ∼ 2.5 µm), hybrid (h ∼ 18 µm), and rotating (h ∼ 40 µm) motions, showing distinct propulsion behaviors at f = 35 Hz and f = 40 Hz (at H₀ = 8 mT). Vector magnitudes are normalized to 0.5 µm for clarity. (b)–(d) and (i)–(k) Temporal evolution of the in-plane orientation β for particles at three heights above the substrate at f = 35 Hz and f = 40 Hz, corresponding to rolling, hybrid, and rotating modes. Arrows in (d) and (k) mark spontaneous reversals in the direction of in-plane rotation. (e)–(g) and (l)–(n) Instantaneous particle speed at the corresponding heights for f = 35 Hz and f = 40 Hz. Near the substrate, rolling produces steady translation; far from it, particles rotate without measurable translation; at intermediate heights, hybrid motion exhibits strong speed fluctuations.

Far from the substrate, the particle exhibits continuous in-plane rotation, with β decreasing approximately linearly in time, apart from occasional disruptions (Fig. 4d). The rate of in-plane rotation is roughly 50 times slower than out-of-plane rotation at the applied frequency. As discussed below, this secondary in-plane rotation may originate from perturbations to the rolling mechanism—for example, due to addition of a small gravitational torque or a displacement of the permanent moment from the Janus equator. At these larger heights, however, both in-plane and out-of-plane rotations produce negligible translation as hydrodynamic coupling to the wall is weak (Fig. 4g).

Above the critical frequency, we observe qualitative changes to in-plane particle rotation away from the substrate. Far from the wall, the particle rotates in the plane, but its direction is not fixed: β increases steadily for several revolutions before reversing and continuing in the opposite direction (arrows in Fig. 4k), also predicted by the model (SI Fig. S9). The rate of this in-plane rotation remains much slower than the out-of-plane rolling driven by the field. At intermediate heights, asynchronous rotation produces trochoidal trajectories characterized by large variability in the in-plane rotation rate (Fig. 4h and j). During short “loop” segments, β increases rapidly, whereas during the longer “arc” segments it changes only slowly while the translational speed U remains approximately constant (Fig. 4m). Despite these qualitative changes away from the substrate, the dynamics of linear rolling near the wall remain largely unchanged by the transition to asynchronous rotation (Fig. 4i and l).

The origins of in-plane rotation—and its dependence on particle height and field frequency—remain poorly understood; however, several insights emerge by extending the dynamical model introduced above. First, gravitational torques associated with the added mass of the Janus cap can generate in-plane rotation, even though they are much weaker than the magnetic torques responsible for out-of-plane rolling (SI Section S10). Such gravitational effects cause a rolling particle to “turn” toward its heavier cap, but the resulting in-plane rotation is predicted to occur at a steady rate, in contrast to the time-varying rotation observed in trochoidal trajectories. A second source of in-plane rotation arises when the particle's permanent magnetic moment is slightly tilted relative to the Janus equator (SI Section S7). In the asynchronous regime, this small misalignment produces periodic reversals in the direction of in-plane rotation, reminiscent of those observed experimentally at large heights (Fig. 4k). Although both mechanisms can account for specific features of the dynamics, neither fully reproduces the large fluctuations in β or the loop-arc structure of trochoidal trajectories at intermediate heights.

Importantly, these models rely on the quasi-steady approximation of low Reynolds number hydrodynamics, in which disturbances generated by particle motion are assumed to propagate instantaneously through the viscous fluid. In reality, momentum diffuses over a distance L in a finite time τ = L²/ν, where ν is the kinematic viscosity (SI Section S11). For particles at intermediate heights (L ∼ 20 µm), this viscous diffusion time is τ ∼ 0.4 ms, which is not much smaller than that of the external field: ωτ ∼ 0.1 at 40 Hz. The diffusion time is also similar to the magnetic relaxation time, ω₀τ ∼ 0.6, which sets the maximum angular velocity during particle rolling. Under these conditions, the assumption of instantaneous hydrodynamic response is violated. The resulting hydrodynamic memory effects are likely to influence particle-wall interactions, particularly at intermediate heights where L is the larger while the wall coupling remains significant. Such unsteady hydrodynamic interactions may therefore play a role in producing irregular in-plane rotation rates and the onset of trochoidal trajectories in the asynchronous regime.

Varying the particle's height above the substrate altered the characteristics of the trochoidal trajectory. We observed that at a fixed frequency (higher than the critical frequency), increasing the height of the the particle above a substrate increases the time a particle spends in the loop segment of the trochoidal trajectory (Fig. 5a–c and Movie S4). To quantify the variability of the trochoidal trajectory as a function of particle height, we defined the term prominence of the loop (χ), calculated as the ratio of the time the particle spends in the loop segment of the trajectory to the total duration of its motion, as shown in Fig. 5d. This increase in prominence at fixed field frequency (above the critical value), shows that the particle trajectory becomes increasingly curvilinear as the height from the substrate increases. At a height from the substrate (h > 36 µm), the particle undergoes spontaneous in-plane rotation without measurable translation when driven by the out-of-plane time-varying magnetic field (see Section 2.5).

Fig. 5 Impact of height on trochoidal trajectories (a)–(c) trajectory of the particle center (blue) and the centroid of polystyrene hemisphere (red) at h ∼ 11 µm, ∼18 µm, and ∼25 µm at f = 40 Hz and μ₀H₀ = 8 mT. Scale bar in (a) = 5 µm. (d) The increase in fraction of time spent in the loop form of the trajectory i.e. prominence, χ upon increasing the heights from the substrate, h. Particles spend more time in the loop form as the height of the particle above substrate increases.

3 Conclusions

We have demonstrated that a Janus particle with a ferromagnetic hemispherical patch undergoes rotation in response to an out-of-plane oscillating magnetic field. The continuous rotation of the microparticles with negligible inertia arises from the coupling between their permanent magnetic moment and a small anisotropy in the field-induced moment, which generates a net torque each time the oscillating field reverses. This rotation leads to distinct particle dynamics based on the height of the particle above the substrate. The emergence of in-plane rotation of the particle in the absence of directional control gives rise to unusual particle trajectories. Far from the substrate, the particle exhibits steady in- and out-of-plane rotations, where latter are driven purely by the external magnetic field. The emergence of a transient regime at intermediate separations, combined with whether the particle undergoes phase-locked or phase-slip motion, gives rise to a unique self-propelling behavior.

Our experiments show that the Janus particle's motion becomes organized and periodic when substrate proximity and magnetic-field frequency are appropriately tuned. Yet, precisely capturing and identifying the origin of the types of particle motion remains a significant challenge. Several open questions persist, including: What field and particle parameters govern the radii of the loop and arc sections of the trochoidal trajectory? What determines the prominence of the loop? and What is the influence of nanoscale variations and non-uniformity in the patch properties⁴⁸ on the trajectories and the prominence of the loop? Further experimental and theoretical work is needed to address these questions. These efforts may enable the development of ferromagnetic Janus particles with tunable trajectories and interparticle interactions. Such precise control could facilitate motion-based communication in 3D, a key step toward self-guided active and responsive magnetic swarms.

4 Methods

Our experimental chamber consisted of a dispersion of 5 µm iron Janus particles in water which was sealed between a glass substrate and a glass cover slip. The particle dispersion droplet was sealed with a hydrophobic barrier pen to a liquid film of thickness ∼300 µm. The experimental chamber was designed to create a semi-infinite fluid domain around a Janus particle. The glass substrate acted as a stationary substrate below the particle at a certain height while the fluid above the particle can be considered unbound.

To fabricate iron Janus particles, a monolayer of polystyrene spheres (radius, a = 2.5 µm) was prepared via convective assembly,⁵⁰ followed by deposition of a 20 nm chromium layer (deposition rate, 0.05 nm s⁻¹) and a 22 nm iron layer (deposition rate, 0.05 nm s⁻¹) using physical vapor deposition under vacuum (10⁻⁶ Torr) in a thermal evaporator.^51,52 This experimental chamber was transferred to an electromagnetic Helmholtz coil to apply out-of-plane sinusoidal magnetic field H(t) = H₀ sin(ωt)e_z, where H₀ is the amplitude, and ω is the angular frequency.

Before applying the magnetic field, we determined the vertical distance between the particle center and the substrate called height h. To determine the precise location of the substrate, we used optical microscopy to focus on the gold mask on the substrate (see SI Section S2). The gold masks were made by depositing a 10 nm gold layer in a thermal evaporator on the glass substrate. We adjusted the microscope stage along the Z-axis to focus on the particle and then noted down the distance between the objective and the microscope stage. The same process was repeated by focusing on the gold mask at the substrate. The difference between these distances measured was the particle height, h from the substrate.

During the particle motion, the position of the particles was monitored using a Leica DM6 microscope equipped with a DFC9000 GTC high-speed camera. We tracked the centroid of the particle and the patch separately throughout the experiment using ImageJ TrackMate.⁵³ Camera exposure = 1 ms was used to acquire all experimental data. The data acquired for Fig. 3a and 4 have the following frame rates, fr: (a) f = 35 Hz, h ∼ 2.5 µm, rolling, fr = 82 fps, (b) f = 35 Hz, h ∼ 18 µm, linear hybrid, fr = 160 fps, (c) f = 35 Hz, h ∼ 36 µm, rotating, fr = 212 fps (d) f = 40 Hz, h ∼ 2.5 µm, rolling, fr = 90 fps, (e) f = 40 Hz, h ∼ 18 µm, trochoidal hybrid, fr = 130 fps, (f) f = 40 Hz, h ∼ µm, rotating, fr = 285 fps.

Conflicts of interest

There are no conflicts to declare.

Data availability

The supporting data has been provided as part of the supplementary information (SI). Supplementary information: details on magnetic properties of the particles, particle height and sedimentation rate measurement, dynamical model development, 1D and 3D rotation of the particles, spectral analysis of the particle speed, polar tilt of the particles, potential contributions of inertia, gravitational and hydrodynamics (memory) effects on the Janus particle dynamics. All data reported in this article has been provided as SI. See DOI: https://doi.org/10.1039/d5sm01257k.

Acknowledgements

Authors thank Mr Wenhao Chang for assistance with the sedimentation experiments and Prof. Krishnaswamy Nandakumar for useful discussions. AR, and BB acknowledge the financial support by the National Science Foundation (NSF) under grant numbers CBET-1943986 (NSF-CAREER) and DMR-2522727. KJMB also acknowledges the NSF support via grant DMR-2522726.

Notes and references

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