Light-induced selective speed alteration of magnetically rolled semiconductor particles

Summary

Microrobot teams or swarms are promising candidates for many applications, such as micromanipulation, microsurgery, or targeted drug delivery. However, attaining individual control of the microrobots, which is a critical component to many of their applications, remains a significant technical challenge. We introduce a method to control the magnetic rolling speed of hematite semiconductor particles using localized UV light, attributed to light-induced changes in particle-substrate friction. Simulations and theoretical models support our experimental observations, showing how particle-substrate separation influences speed. Additionally, we demonstrate fixed patterning of microparticles via selective UV illumination at lower pH, demonstrating selective immobilization of microrobots, a conceptual step toward applications such as targeted drug delivery or patterned cell stimulation in future studies. Therefore, this work provides a novel approach for independent control of microrobot systems by modulating particle-substrate interactions with light.

Graphical abstract

Highlights

• •Localized UV illumination increases magnetic rolling speed of microparticles
• •Speed differences enable the formation of small particle patterns
• •Non-monotonic dependence of speed with pH is consistent with theoretical predictions
• •Light-induced selective adhesion produces fixed patterns of particles on the surface

Applied sciences; Materials science

Introduction

One of the primary objectives of current microrobotic research is to obtain independent control over a group or swarm of microrobots. Such control would significantly enhance the effectiveness and efficiency of microrobot applications.^1,2 For example, individual microrobot control would provide a greater ability to deliver drugs to multiple locations and improve navigation in complex environments.

Various strategies have been explored to control microrobots in an independent manner. Some approaches used specially constructed substrates that can fix or release microrobots on the surface,³ while others used light-induced convection to manipulate individual spherical particles to create patterns, relying on a fluidic bed with low thermal conductivity.⁴ Individual microrobot actuation has also been achieved using optoelectronic tweezers (OET),⁵ and by illuminating individual TiO₂ micromotors with a small disk of light.⁶ While light actuation offers precise spatial control, it typically produces relatively weak forces compared to magnetic actuation,^7,8 making it less suitable for manipulation or other tasks that require strong mechanical forces. Other approaches have employed electrical fields to enhance rolling speed via particle-surface attraction,⁹ or by taking advantage of differences in size, shape, or resonant frequencies to induce varying responses in acoustic or magnetic fields.^10,11 However, such processes involve complex fabrication, inefficient algorithms, and limited scalability.

To address this challenge, we present a method to selectively control the rolling speed of magnetically actuated hematite microparticles using targeted UV light. UV illumination increases the rolling speed of particles by reducing slip due to a light-induced attraction of the particles with the surface.¹² Using a digital micromirror device (DMD), we project user-defined disks of UV light that can be dynamically positioned to follow specific particles. Furthermore, at lower pH, illuminated hematite particles adhere to the substrate, allowing for an additional mechanism of individual control. We exploit this effect to immobilize selected particles at desired locations, creating patterns of particles on the substrate by magnetically rolling UV light induced fixation. This demonstrates another method of pattern formation by moving many particles in a global fashion but then immobilizing certain particles once they are in the desired location. This light-induced adhesion is attributed to both a diffusion-osmotic attraction of the particles to the surface as well as an electrostatic adhesion of the particles once they are in close proximity to the surface.¹² Together, these mechanisms provide a new strategy for independently modulating the mobility and position of magnetically driven microrobots, addressing a key challenge in microrobotic swarm control.

Results

Magnetic rolling and light-induced surface attraction

In this work, peanut-shaped magnetic hematite particles were magnetically rolled in liquid near a glass surface using rotating magnetic fields created by electromagnetic coils (as depicted in Figure 1). The particles have a magnetic moment that is along their short axis.¹³

Figure 1.

Schematic representation of the experimental workspace and the hematite microparticles

(A) A schematic of the experimental setup.

(B) A cartoon showing the magnetically rolled peanut-shaped microrobots and the orientation of their magnetic moment.

The hematite particles were studied at a range of concentrations of hydrogen peroxide, pH, and at high and low light intensity, which all affect the UV light-induced chemical reaction at their surface and the subsequent attraction/repulsion with the substrate.¹²

In this work, we control the speed of individual rolling hematite microparticles by applying localized light “spotlights” that increase the speed of the illuminated rollers. We attribute the speed increase to an attraction of the particles to the substrate, resulting in increased friction and more efficient rolling. We note that Lin et al.¹⁴ reported self-propulsion of hematite particles along their long axis and explained this as diffusiophoretic motion due to asymmetrically generated concentration fields. This is not the dominant mechanism in our current work as there was negligible directional motion without rolling by the applied magnetic field and the rolling velocity was perpendicular to the long axis of the particle. Self-propulsion could also emerge from or be enhanced by advection-induced asymmetry in the concentration fields^15,16 but this effect is also expected to be negligible as the Péclet number associated with solute transport is of the order of 10⁻³ and the rotational motion tends to homogenize the concentrations around the particle.

We have previously found that at a pH greater than approximately 6.5 and with an application of UV light from below, the hematite particles are attracted to the substrate.¹² This was indicated by a decrease in their diffusivity as well as a quenching of their out-of-plane rotational fluctuations. A theoretical model was able to explain this attraction as arising from a weakening of the double-layer repulsion upon initiation of the light-induced reaction on the particle’s surface. The reaction creates ions which reduce the electrostatic double-layer repulsion of the particles with the negatively charged glass surface, resulting in a decrease in separation between the two. We explain the increase in rolling speed to be due to this decrease in the lubrication layer between the particles and the surface, resulting in more efficient rolling with less slipping. A previous study used electric fields to attract particles to a surface which increased their rolling speed,⁹ and another study employed soft spheres that rolled faster due to increased friction.¹⁷ Here, we use light which provides localized as well as on-demand on/off control. This allows for independent speed control of individual rolling particles.

Independent speed control and pattern formation via localized UV illumination

To attain independent control of the microrobots, we created a light “spotlight” using the DMD system. A disk of light, whose size was set to be small enough to only affect one particle at a time, was shone on the sample. The small spot of light was effective at increasing the rolling speed of the microrobots. We demonstrated the ability to create patterns of microrobots by selectively increasing the speeds of specific microrobots while using magnetic rolling to position a group of them in a pattern. Examples of this are given in Video S1 and Figure 2. As can be seen from the figure, a triangular pattern was created using this approach.

Figure 2.

Formation of a triangular pattern with hematite micropartcles

(A) Cartoon showing the creation of a triangular pattern via selective speed alteration by the application of localized UV light.

(B) The corresponding series of experimental images. The dashed circles indicate the location of the disk of light.

Rolling speed as a function of magnetic frequency with and without light

The peanut microrobot’s speed versus the rotating magnetic field frequency at a pH of 8.25, a hydrogen peroxide concentration of 8%, and a light intensity of 250 mW/cm² is shown in Figure 3. For comparison, the case with identical conditions but without light is also plotted. Video S2 shows an example of magnetic rolling with light off and on. In the video, the median speed of the rolling microparticles with the light on was more than two times that with the light off. Although there is not a large difference between the mean and the median, we plot the median speed because some microrobots stuck to the slide or formed clumps which would decrease or increase their speed, respectively; thus, the median is used since it is less susceptible to outliers.

Figure 3.

Experimental data and fitted model of the rolling speed of the microrobots as a function of the frequency of the rotating magnetic field with light on and light off

For the experiment, the pH was 8.25, the hydrogen peroxide concentration was 8%, and the light intensity was 250 mW/cm². The R² of the fit are 0.842 and 0.814 in the light off and light on cases, respectively (data are represented as median ± SEM).

In both cases, with and without light, the rolling speed increases approximately linearly with the frequency of the applied rotating magnetic field up to a certain point known as the step-out frequency.^18,19 At applied frequencies higher than this, the magnetic torque experienced by the particle is insufficient to keep it rotating at the same frequency as the magnetic field. The average rotational frequency and, hence, the rolling speed of the particle decline as the magnetic field frequency increases.

In the linear regime (below step-out), the slope is steeper for the case with applied light than the case without light. We attribute this to a reduced distance between the glass surface and the particle when the light-induced reaction occurs. As the particle gets closer to the surface, the drag on the bottom and top of the particle becomes asymmetric, resulting in a rolling translational motion.

We performed numerical simulations of a peanut-shaped particle rolling near a wall (with a no-slip boundary condition) to determine the dependence of rolling speed on the wall separation distance (see Figure 4). Least squares fitting of the model to the experimental data shown in Figure 3 (the best fit model functions are also plotted on the same axes) gave estimates for wall-particle separations of 0.034 μm with light and 0.117 μm without light. The fitting also estimated the maximum magnetic torque (which is the product of the magnitudes of the magnetic dipole moment of the particle and the applied magnetic field strength) of 1.11 and 1.26 pNμm with and without light, respectively. This quantity should not depend on whether the light is on or off, so the similarity in values indicates self-consistency. To compare this value to the experimentally applied torque, we estimated the magnetic moment of the particle to be 1–1.6 × 10⁻¹⁶ A.m.² based on a coercivity of 0.07 emu/g.²⁰ For an applied magnetic field of around 5 mT, we estimate the maximum torque on the particle to be 0.5–0.8 pNμm, which is of a similar magnitude to the numerical result. The slightly higher numerical estimate could be due to slight differences in the magnetic properties or geometry of the particles than in ref.²⁰ Also, if the particles roll with their long axes slightly non-parallel to the surface this could reduce their effective drag, and hence also lower the torque needed to remain below the step-out frequency.

Figure 4.

Simulation data of the rolling speed of the microrobots as a function of the distance between the bottom of the particle and the wall below

The microrobot is assumed to be horizontal. In one dataset, the particle rotates about its long axis with a fixed rotational frequency of 60 Hz at all distances and in the other dataset, the particle experiences a torque due to a magnetic field rotating with frequency 60 Hz; this frequency is above the step-out frequency when the particle is close to the wall because the rotational drag coefficient increases.

At the lowest examined frequencies (approximately 10 Hz and lower), we note that the experimental rolling speeds are generally above the linear model fitting. This could be simply due to measurement errors, which are much more significant when the true speeds are low, but we propose another possible explanation. At low magnetic field frequencies, the predicted linear increase in rolling speed is based on the assumption that the long axis of the particle remains aligned with the axis of rotation of the magnetic field. As discussed in a previous study,¹² Brownian rotational motion causes stochastic fluctuations of the orientation, leading to measurable tilt angles with respect to the plane of the substrate in the absence of rotating magnetic fields. Simulations reported in the supporting material show that if the peanut rotates with its long axis at a fixed finite angle (45°) to the axis of rotation, then the rolling speed is significantly higher (see Videos S3 and S4). If instead the particle rotates under the action of a rotating magnetic field, then this angle decreases as the particle rotates due to the hydrodynamic drag (see Video S5). Therefore, we expect that at low rotation rates, the average angle is large due to the dominance of Brownian rotation, and the average angle is close to zero at high rotation rates when hydrodynamic effects are dominant.

We note that the experimental observations seemed to corroborate this assertion. We observed rolling behavior that seems to indicate that the particles roll with their long axis more parallel to the substrate at higher frequencies compared to lower frequencies (see Video S6). It was difficult to determine these tilt angles precisely, however, due to the rolling frequency being comparable or greater than the video capture rate. However, for cases in which the rolling frequency is less than the video frame rate, there does appear to be more tilt present at very low frequencies compared to somewhat higher frequencies (1 Hz vs. 5 Hz, for example).

Rolling speed as a function of light intensity

We also studied the effect of light intensity on their rolling speed at a rotation rate of 60 Hz and a pH of 8.15. The results, shown in Figure 5, show that the rolling speed of the peanut microrobots increases with light intensity. Since a higher light intensity is expected to lead to a greater attraction of the microrobot with the substrate according to our model, we attribute this increase in speed to the increase in hydrodynamic drag asymmetry, and hence more efficient translational motion, as previously mentioned. The plateau of the speed at high light intensity can be explained by saturation in reaction rate, after which there is negligible change in the distance between the particle and the surface. Comparing the rolling speeds with the computational results in Figure 4, we estimate the separation to be around 0.16 μm at 0 mW/cm² light intensity and 0.05 μm at 40 mW/cm² light intensity. We note that the rolling speed at high light intensity is at the peak of the theoretical curve taking into account step-out when the particle is very close to the wall. We expect that under different conditions (e.g., at a higher magnetic field frequency so that step-out occurs farther from the wall), there would be a peak in rolling speed at a particular light intensity, beyond which the rolling speed decreases and eventually plateaus.

Figure 5.

The rolling speed of the microrobots as a function of pH

The hydrogen peroxide concentration was 8%, the light intensity was 250 mW/cm², and the rolling frequency was 60 Hz (data are represented as median ± SEM).

Rolling speed as a function of peroxide concentration

Similar to the effect of light intensity, increasing the hydrogen peroxide concentration also resulted in an increase in rolling speed of the microrobots with the light applied (see Figure 6). We expect the photocatalytic reaction to proceed at higher rates as the hydrogen peroxide concentration is increased so this observation is consistent with our explanation of the particle-wall separation decreasing with reaction rate, thereby enhancing the rolling effect. There was no significant variation of rolling speed with hydrogen peroxide concentration with the light off. At all tested hydrogen peroxide concentrations, including the cases with no hydrogen peroxide, the speed was higher with light on than without light. This could be due to other ion-producing photocatalytic reactions occurring at low rates in the absence of peroxide, as has been reported previously for semiconductor particles that are self-phoretic under UV light even in pure water.²¹

Figure 6.

The rolling speed as a function of hydrogen peroxide concentration

The pH was between 8.15 and 8.55, the light intensity was 250 mW/cm² in the case with light on, and the magnetic field frequency was 60 Hz (data are represented as median ± SEM).

Rolling speed as a function of pH

We also studied the effect of pH on the rolling speed, and the results are presented in Figure 7. The speed increases as the pH is increased from 7 to 9 and decreases as the pH is raised further. We noticed that some particles began to stick to the glass slide at the highest pH, indicating that perhaps there is an attractive interaction between some of the particles and the slide. Simulation results shown in Figure 4 indicate that there could be an optimal distance from the wall where the rolling speed is maximal for a fixed magnetic field frequency. At larger distances, the hydrodynamic coupling between the wall and the rotating particle is weak, so the bottom of the particle slips relative to the surface instead of rolling. Below the optimal distance, the increased hydrodynamic drag due to the wall causes step-out and the particle does not rotate with the magnetic field.

Figure 7.

The median rolling speed of the peanut-microrobots as a function of light intensity

The pH was 8.15, the hydrogen peroxide concentration was 8%, and the magnetic field frequency was 60 Hz (data are represented as median ± SEM).

These experimental and numerical observations can be linked together using a theoretical model from our previous work.¹² By considering how the particle-substrate interactions depend on pH and ionic strength, the model exhibited an equilibrium distance that decreases with pH from around 0.6 μm at pH 7 to 0.05 μm at pH 11 without light-induced reactions. The trend with reaction is similar, depending on the rate of reaction, and the equilibria are closer to the wall with reaction. Hence, the experimental trends in rolling speed with pH can be explained if the distance to the wall decreases, passing through the optimal distance at an intermediate pH around 9.

Pattern formation by light-induced adhesion

At sufficiently low pH of the solution, the peanut particles will stick to the slide when the light is applied.¹² Therefore, we applied localized light in order to only stick the microrobots that were within the illuminated area while allowing other microrobots to continue to roll freely. This allowed for the creation of patterns on the substrate, as shown in Figure 8 and Video S7. Although we applied the light to a rather large area in the video, we found that often even using a small area of light was sufficient (see Video S8).

Figure 8.

Hematite microparticles were adhered to a glass slide via UV illumination to form the letters “UD”

The particles were moved to their locations by magnetic rolling.

We also demonstrated sticking a microrobot near a cell (see Video S9 and Figure 9) which demonstrates that microrobots can be selectively immobilized adjacent to cells, enabling pattern formation on cellular surfaces. While no payload was delivered in this work, such positioning could in principle be extended in future studies toward selective drug delivery or localized biochemical stimulation of cells (e.g., morphogens for differentiation).

Figure 9.

Adhering a microparticle near a cell

Series of images showing a peanut-shaped hematite microrobot magnetically rolled to a cell (A and B), and then adhered to the substrate upon application of UV light (C).

Video S9. Sticking a microparticle to the substrate near a Chinese hamster ovary (CHO) cell, demonstrating the potential for delivering and fixing the location of microparticles for cellular drug delivery

The sticking of the particle is confirmed by its immobility after the light was applied and the rotating field was turned on again.10

Discussion

In summary, we have demonstrated that localized UV illumination can selectively modulate the rolling speed of magnetically actuated hematite microrobots, enabling per-particle control within a globally applied magnetic field. By quantifying the dependence of light-induced speed changes on pH, H₂O₂ concentration, and illumination intensity, and by supporting these observations with theoretical modeling, we establish light as a programmable “knob” for tuning near-wall friction and hydrodynamic coupling. This approach allows for independent immobilization and patterning of microrobots which is not achievable with other global actuation strategies.

While our experiments focus on fundamental control principles rather than translational demonstrations, selective immobilization near cells suggests a path toward future applications in biomedical or bioengineering contexts, including localized drug delivery or spatially patterned biochemical stimulation. Realizing such functions will require integration with payload release strategies and automated feedback control, which we identify as important directions for future work. More broadly, our results highlight how combining magnetic torque with optically tunable friction provides a new framework for independent, parallel control of microrobot collectives at the micron scale.

Limitations of the study

Although localized light illumination effectively increased the rolling speed of individual particles, several challenges remain before this approach can be applied to targeted or patterned drug delivery. The enhancement in speed depends on electrostatic interactions with the substrate; therefore, surfaces with low surface charge may be ineffective. In addition, the use of light restricts operation to open or transparent systems. Finally, hydrogen peroxide is toxic to cells at high concentrations; thus, replacing it with a biocompatible fuel would be necessary for implementation in biological environments.

Resource availability

Lead contact

Requests for further information and resources should be directed to and will be fulfilled by the lead contact, Sambeeta Das (samdas@udel.edu).

Materials availability

This study did not generate original materials.

Data and code availability

• •All data have been deposited at Mendeley Data and are publicly available as of the date of publication at https://doi.org/10.17632/zcygz6hrym.1.
• •All original code has been deposited at Mendeley Data and is publicly available at https://doi.org/10.17632/zcygz6hrym.1 as of the date of publication.
• •Any additional information required to reanalyze the data reported in this paper is available from the lead contact upon request.

Acknowledgments

H.S. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada (funding reference number RGPIN-2018-04418). Cette recherche a été financée par le Conseil de recherches en sciences naturelles et en génie du Canada (CRSNG) (numéro de référence RGPIN-2018-04418). This work was supported by the National Science Foundation under grant GCR 2219101, CPS 309 2234869, and the National Health Institute under grant 1R35GM147451. This project was also supported by a grant from the National Institute of General Medical Sciences – NIGMS (5P20GM109021-07) from the National Institutes of Health and the State of Delaware. The authors thank Fatma Ceren Kirmizitas for culturing and providing cells used in the experiments.

Author contributions

Conceptualization, D.P.R. and S.D.; methodology, D.P.R. and H.S.; investigation, D.P.R. and H.S.; writing – original draft, D.P.R.; writing – review and editing, D.P.R., S.D., and H.S.; funding acquisition, S.D. and H.S.; resources, D.P.R. and Z.H.S.; supervision, S.D.

Declaration of interests

The authors have no competing interests to declare.

Declaration of generative AI and AI-assisted technologies in the writing process

The authors acknowledge the use of ChatGPT for language editing support. All AI-assisted texts were reviewed and revised by the authors. The authors take full responsibility for the contents.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Chemicals, peptides, and recombinant proteins
Iron(III) chloride (FeCl3)	Sigma Aldrich	Cat. #: 236489
Sodium hydroxide (NaOH)	Sigma Aldrich	Cat. #: S5881
Sodium sulfate (Na2SO4)	Sigma Aldrich	Cat. #: 238597
Hydrogen peroxide (H2O2, 30%)	Fisher Chemical	Cat. #: H325
Experimental models: Cell lines
Chinese Hamster Ovary (CHO)	Laboratory of Dr. Ron Weiss (MIT)	N/A
Software and algorithms
Python (v3.x)	Python Software Foundation	https://www.python.org/
Custom Python analysis scripts	This paper	https://doi.org/10.17632/zcygz6hrym.1
Mightex Polygon software	Mightex Systems	https://www.mightexbio.com/
Amscope Camera Software	Amscope	https://amscope.com/pages/software-downloads
Custom Python/Arduino control code	Sokolich et al.22	https://doi.org/10.48550/arXiv.2505.06450
Custom Fortran 90 boundary element method	Shum et al.23	Available upon request
Deposited data
Orginal code used in the paper are available in the database: https://doi.org/10.17632/zcygz6hrym.1

Experimental model and study participant details

Chinese hamster ovary (CHO) cells were used to experiment with sticking of the particles near the cells with light. The concentration of hydrogen peroxide was 0.5%.

Method details

Synthesis of hematite peanuts

The particles were made following previous methodology.^24,25 A 100 mL solution of 56% w/v Fe₃Cl₃ was prepared in a glass bottle. Then, 90 mL of 6 N NaOH was added followed by 0.6 M Na₂SO₄. To mix well, the solution was shaken vigorously for 10 minutes. It was then placed in an oven at 100^∘C for 8 days. Once a clear separation was observed between the particles and the supernatant, the supernatant was removed and the remaining particles were centrifuged to wash in DI water three times. SEM images revealed particles with a length of about 1.6 μm and a diameter of 0.5 μm.

Experimental procedure

The hematite particles were then pipetted into a plastic tube with DI water to a volume of 100 μL. NaOH and 30% H₂O₂ were added to this to create the final pH and peroxide concentration. Approximately 70 μL of this was pipetted onto a glass coverslip under a microscope (Zeiss Axiovert 200). The resulting thickness of the liquid droplet was approximately 1 mm at the center. Experiments were conducted near the center of the droplet to avoid boundary effects. Particle density was approximately 2×10^-3μm^-2. Videos were recorded with an Amscope MU903-65 camera.

Magnetic coils

Magnetic fields were applied using a home-built electromagnetic control system as described in ref.²⁶ Three axes of electromagnetic coils were used to apply magnetic fields along any axis, and a joystick on an x-box controller was employed to set the direction of the rolling fields. The rotating magnetic fields are generated by applying sinusoidal currents to the coils at a desired frequency, as described in the reference.

Theoretical model

The rolling speed was calculated for particles at varying distances from a wall by numerically solving the equations of incompressible Stokes flow,

$$-\nabla p+\mu {\nabla }^2\overrightarrow{u}=\overrightarrow{0},\nabla ·\overrightarrow{u}=0\text{,}$$

subject to the boundary conditions

$$\overrightarrow{u}{|}_{z=0}=\overrightarrow{0},\underset{z\to \infty }{\lim }\overrightarrow{u}=\overrightarrow{0},\overrightarrow{u}\left(\overrightarrow{x}\right){|}_{\overrightarrow{x}\in P}=\overrightarrow{U}+\overrightarrow{\mathrm{\Omega }}\times \left(\overrightarrow{x}-{\overrightarrow{x}}_0\right)\text{,}$$

where p is the pressure field, $\overrightarrow{u}$ is the fluid velocity field, μ is the dynamic viscosity of the fluid, P denotes the surface of the particle, $\overrightarrow{U}$ and $\overrightarrow{\mathrm{\Omega }}$ are the rigid body translational and rotational velocity vectors of the particle, respectively, and ${\overrightarrow{x}}_0$ denotes the center of the particle. Two scenarios are considered:

• 1.The rotational velocity Ω is prescribed and the translational velocity $\overrightarrow{U}$ is computed, and
• 2.The particle is given a magnetic dipole moment $\overrightarrow{m}$ and a rotating external magnetic field $\overrightarrow{B}$ is prescribed, resulting in a magnetic torque $\overrightarrow{\tau }=\overrightarrow{m}\times \overrightarrow{B}$ acting on the particle, from which the rotational velocity $\overrightarrow{\mathrm{\Omega }}$ and the translational velocity $\overrightarrow{U}$ are computed.

In both cases, it is assumed that there is no net external force acting on the particle. A boundary element method is used to discretize the particle and solve the system of equations.^23,27

Geometry of model particle

The peanut-shaped particle is modeled as a volume of revolution. The curve that sweeps out the surface of the particle consists of three circular arcs, as shown in Figure S1. Based on SEM images, we choose the dimensions D₀=0.58 μm, D₁=0.66 μm, and L=1.6 μm.

Numerical methods

Let ${\overrightarrow{x}}_0$ denote the center of the particle and let P denote the surface of the particle. The velocity field $\overrightarrow{u}$ at a point $\overrightarrow{x}=\left(x_1,x_2,x_3\right)$ for Stokes flow around the particle satisfies the boundary-integral equation²⁸

$$\overrightarrow{u}\left(\overrightarrow{x}\right)=-\frac{1}{8\pi \mu }\underset{P}{\int }\mathbf{G}\left(\overrightarrow{x},\overrightarrow{y}\right)\overrightarrow{f}\left(\overrightarrow{y}\right)\mathrm{d}S\left(\overrightarrow{y}\right)\text{,}$$ (Equation 1)

where the ij-components of the Green’s function tensor G for Stokes flow satisfying the boundary condition $\overrightarrow{u}\left(x_1,x_2,x_3=0\right)=\overrightarrow{0}$ are given by²⁹

$$G_{ij}\left(\overrightarrow{x},\overrightarrow{y}\right)=\left[\frac{{\delta }_{ij}}{r}+\frac{r_i{r}_j}{r^3}\right]-\left[\frac{{\delta }_{ij}}{\tilde{r}}+\frac{{\tilde{r}}_i{\tilde{r}}_j}{{\tilde{r}}^3}\right]+2y_3\left({\delta }_{j\alpha }{\delta }_{\alpha l}-{\delta }_{j3}{\delta }_{3l}\right)\frac{\partial }{\partial {\tilde{r}}_l}\left[\frac{y_3{\tilde{r}}_i}{{\tilde{r}}^3}-\left(\frac{{\delta }_{i3}}{\tilde{r}}+\frac{{\tilde{r}}_i{\tilde{r}}_3}{{\tilde{r}}^3}\right)\right]\text{,}$$

where $\overrightarrow{r}=\overrightarrow{x}-\overrightarrow{y}$, $\tilde{\overrightarrow{r}}=\left(x_1-y_1,x_2-y_2,x_3+y_3\right)$, $r=\left|\overrightarrow{r}\right|$, and $\tilde{r}=\left|\tilde{\overrightarrow{r}}\right|$. Repeated indices imply summations over α=1,2, and l=1,2,3.

The vector field $\overrightarrow{f}$ is the traction vector or hydrodynamic stress acting on the surface of the particle. This quantity must be solved for subject to constraints, namely, that the velocity of the fluid matches that of the rigid body motion of the particle on its surface,

$$-\frac{1}{8\pi \mu }\underset{P}{\int }\mathbf{G}\left(\overrightarrow{x},\overrightarrow{y}\right)\overrightarrow{f}\left(\overrightarrow{y}\right)\mathrm{d}S\left(\overrightarrow{y}\right)=\overrightarrow{U}+\mathbf{\Omega }\times \left(\overrightarrow{x}-{\overrightarrow{x}}_0\right),\text{for}\overrightarrow{x}\in P\text{,}$$ (Equation 2)

and the force and torque balance equations,

$$\underset{P}{\int }\overrightarrow{f}\left(\overrightarrow{y}\right)\mathrm{d}S\left(\overrightarrow{y}\right)=\overrightarrow{0}\text{,}$$ (Equation 3)

$$\underset{P}{\int }\overrightarrow{y}\times \overrightarrow{f}\left(\overrightarrow{y}\right)\mathrm{d}S\left(\overrightarrow{y}\right)+\mathit{\tau }=\overrightarrow{0}\text{,}$$ (Equation 4)

where τ is the total non-hydrodynamic torque acting on the particle.

The system of (Equations 2, 3, and 4) relate the vectors $\overrightarrow{U},\mathbf{\Omega },\mathit{\tau }$, and the vector field $\overrightarrow{f}$. These equations are solved numerically using a boundary element method^23,27 in which the surface P is discretized into triangular mesh elements and the surface integrals are approximated by quadrature rules. Two scenarios are considered:

• 1.the rotational velocity $\overrightarrow{\mathrm{\Omega }}=\left(0,-1,0\right)$ is prescribed and the required torque τ is computed along with the translational velocity $\overrightarrow{U}$ and traction $\overrightarrow{f}$; and
• 2.the particle is given a magnetic dipole moment $\overrightarrow{m}$ perpendicular to its long axis and a rotating external magnetic field

$$\overrightarrow{B}\left(t\right)=\left(B\cos \left(\omega t\right),0,B\sin \left(\omega t\right)\right)$$

is prescribed, resulting in a magnetic torque $\mathit{\tau }=\overrightarrow{m}\times \overrightarrow{B}$ acting on the particle, from which the rotational velocity $\overrightarrow{\mathrm{\Omega }}$, the translational velocity $\overrightarrow{U}$, and the traction $\overrightarrow{f}$ are computed.

In the first scenario, if the long axis of the particle is aligned with the y-axis, then the velocity is expected to be constant and in the -x-direction by symmetry; it is sufficient to compute the velocity at one instant.

In the second scenario and if the particle is tilted with respect to the y-direction in the first scenario, the velocities $\overrightarrow{U}$ and $\overrightarrow{\mathrm{\Omega }}$ are time dependent and we use a second-order predictor–corrector time integration scheme²³ to compute the trajectories over a few periods of rotation.

Stepping-out of magnetically driven particles

To investigate the step-out behavior of the particles numerically, we considered a particle rotating about the y-axis due to a magnetic torque $\mathit{\tau }=\overrightarrow{m}\times \overrightarrow{B}$, where the dipole moment $\overrightarrow{m}$ of the particle and the magnetic field $\overrightarrow{B}$ are both confined to the x–z plane. The magnitude of the torque required to rotate the particle is proportional to angular velocity Ω,

$$\tau =C\mathrm{\Omega }\text{,}$$

where the rotational drag coefficient C depends on the size and shape of the particle, the fluid viscosity, and the distance h to the wall. If the vector $\overrightarrow{B}$ rotates about the y-axis with low angular frequency ω, then the angle between $\overrightarrow{m}$ and $\overrightarrow{B}$ will approach a steady value at which the magnetic torque causes the particle to rotate at the same frequency as the magnetic field, Ω=ω. Since the magnitude of the torque cannot exceed ${\tau }_{\max }=\left|\overrightarrow{m}\right|·|\overrightarrow{B}|$, the rotation of the particle cannot keep up with the magnetic field above the step-out angular frequency ${\mathrm{\Omega }}_{\max }=\left|\overrightarrow{m}\right|·|\overrightarrow{B}|/C$. The mean angular frequency ⟨Ω⟩ of the particle depends on the angular frequency of the magnetic field according to the equation¹⁸

$$⟨\mathrm{\Omega }⟩=\{\begin{array}{cc}\omega ,& \omega \le {\mathrm{\Omega }}_{\max },\\ \omega -\sqrt{{\omega }^2-{\mathrm{\Omega }}_{\max }^2},& \omega >{\mathrm{\Omega }}_{\max }.\end{array}$$

The function ⟨Ω⟩ multiplied by a rolling coefficient that depends on h, the distance to the glass substrate, (as presented in Figure 2 of the main text multiplied by a frequency of 60 Hz) gives the rolling speed as a function of magnetic field frequency. This is shown in Figure 4 of the main text after fitting to find estimates for the parameters h and τ_max.

Rolling with tilt

The effect of misalignment between the axis of the particle and the axis of rotation, which could be due to Brownian fluctuations or a magnetic dipole moment that is not precisely transverse to the particle axis, for example, was numerically investigated by computing the mean rolling speed of the particle at various starting distances from the wall (measured to the closest point on the particle) with prescribed angular velocity $\overrightarrow{\mathrm{\Omega }}=\left(0,-20\pi ,0\right)$ and the axis $\overrightarrow{e}$ at a fixed 45° angle to the y-direction. The configuration variables are shown in Figure S4 and the calculated rolling speeds are shown in Figure S3 compared with the case of $\overrightarrow{e}$ being aligned with the y-axis. Videos S3 and S4 show examples of rolling under prescribed rotational velocities with θ=0° and θ=45° respectively.

For the second scenario, we set the initial tilt angle of 45° and an initial distance from the wall of 0.034 μm (corresponding to the estimated distance with light on at pH 8.15). The maximum magnetic torque is set to τ_max=1.1 pN μm. We compute the trajectories over many cycles of the magnetic field at two frequencies: 10 Hz (corresponding to ω=20π), which is below the step-out frequency, and 60 Hz, which is above the step-out frequency. As shown in Figure S4, the angle θ between the particle axis and the rotation axis of the magnetic field rapidly decreases to almost 0° in both cases. As anticipated, the angle ψ between the magnetic dipole moment $\overrightarrow{m}$ and the magnetic field $\overrightarrow{B}$ approaches a steady value below the step-out frequency and fluctuates between 0° and 180° above the step-out frequency. Video S5 shows the motion with a magnetic field frequency of 10 Hz.

Varying salt concentration

The rolling speeds of the particles as a function of NaCl concentration (0–10 mM) is shown in Figure S5. Changing the NaCl concentration tunes the Debye length and strength of electrostatic double-layer repulsion, allowing to further probe the mechanism underlying the light-indued modulation of particle-substrate interactions.

Rolling speed over time

The rolling speed over time was determined by measuring the particle speed in 15-minute intervals over a 1-hour period as shown in Figure S6, indicating fairly consistent behavior over this duration.

Spotlight intensity profile

The distribution of the intensity of the light disk was determined by first generating a circular “spotlight” pattern and then capturing images of the pattern for analysis. The spatial intensity distribution was characterized by subtracting a dark frame and conducting a radial intensity average to obtain the profile shown in Figure S7, showing that the intensity is fairly uniform (80% of max) within ∼80% of the distance from the center to the edge.

Quantification and statistical analysis

The software used for data and statistical analysis was Matlab (R2019a). Details on the statistical analyzes performed are stated in the figure captions.

Published: December 23, 2025