# Is it Possible to Detect a Rotating Spherical Colloidal Particle?

Olivier Emile, Janine Emile

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## To cite this version:

Olivier Emile, Janine Emile. Is it Possible to Detect a Rotating Spherical Colloidal Particle?. Advanced Photonics Research, 2024, pp.2400098. 10.1002/adpr.202400098. hal-04870116

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# Is it Possible to Detect a Rotating Spherical Colloidal Particle?

Olivier Emile* and Janine Emile

A single micrometer-size spherical colloid has been set in rotation by transfer of light orbital angular momentum. This particle is floating at an air-water interface. Steady-state rotational frequencies of the order of one hertz have been observed, depending on the topological charge of the beam and on its power, in agreement with expected values. The detection is performed using the rotational Doppler shift of the diffused light. Two time constants have been evidenced in the rotational velocity dynamics. The first one is related to the friction of the colloid with the fluid (air and water), whereas the other one is principally associated with the wall friction of the air-liquid interface with the container. This measurement technique makes it possible to identify dynamic parameters of the rotational movement of any spherical object, which is usually impossible to detect.

particles. Birefringent spherical particle rotation could also be measured using the light polarization change during light particle interaction.[20,21] However, it is often written that it is impossible to detect the rotation of perfect spherical colloids.[1,4] Nevertheless, such detection is sorely needed. We demonstrate here that we can track the rotation of such spherical colloids. We use light carrying orbital angular momentum (OAM), light diffusion, and the so-called rotational Doppler effect, to follow the rotation of a spherical micrometer-size particle at an air-liquid interface and to investigate its rotational dynamics.

## 1. Introduction

The measurement of the rotation of colloidal particles is a major issue in many domains of physics, chemistry, or even in biology.[1-4] In particular, unraveling the rotational motion of colloidal spheres may provide key issues in dense quiescent suspension near glass transition,[5-7] for self-propelling colloids in the vicinity of a confining channel,[8] or in dynamical clustering in denser systems.[9,10] In biology, the tracking of the rotation of ATP and ADP motors may provide valuable information in the biological understanding of ATP synthesis and ADN arrangement or folding.[11-14] This is also true for bacteria's dynamics.[15] Rotation detection is usually difficult to perform. Rotation tracking using imaging processing, sometimes in combination with modified beads in order to introduce asymmetry, is one of the major technologies.[1,16] The other main technique relies on magnetic rotation,[17-19] which is based on the use of magnetic

## 2. Experimental Section

### 2.1. Experimental Set-Up

#### 2.1.1. Spherical Colloid

A 2a = 105 ± 5 µm diameter spherical glass colloid is floating on the free surface of demineralized water, which fills a 16 mm inner diameter glass tube (tube height 7 cm, water height 5 cm). Whereas the existence of a meniscus in such a small tube usually perturbs the experiment, we take advantage of it here to perform observations. Actually, the capillary length is defined as[22] $\lambda_c = \sqrt{\gamma/\rho_w g}$ ($\gamma = 73 \text{ mN m}^{-1}$ at a temperature $T = 20°$ is the surface tension of water, $\rho_w = 10^3 \text{ kg m}^{-3}$ is its density, $g = 10 \text{ m s}^{-2}$ is the gravitational acceleration), leading to $\lambda_c \approx 3 \text{ mm}$. It is only three times less than the radius of the tube. The colloid is still sensitive to the effect of the meniscus due to the contact angle between water and the glass tube. It is pushed toward the center of the tube. The equilibrium position of the sphere is then right in the middle of it. The effect of the meniscus is sufficient to experimentally maintain the sphere in a stationary position, preventing it from randomly moving during the experiment. There is no need for extra confinement nor trapping light.

Is the colloid floating or immersed in the liquid? Let us define the immersion depth *d* (see **Figure 1** for experimental details). For *d/a* = 0, the sphere is floating, for *d/a* = 1, the colloid is half immersed, whereas for *d/a* > 2, the colloid is totally immersed. Let us introduce *α* the angle between the air-water interface and the tangent to the sphere's surface ($\cos \alpha = (a - d)/a$). The sphere is in equilibrium between two forces: its weight and the surface tension (we neglect here the force due to Archimedes' principle). The length of the contact line equals $2\pi a \sin \alpha$. By symmetry, the horizontal components of the

O. Emile
Université de Rennes
F-35000 Rennes, France
E-mail: olivier.emile@univ-rennes.fr

J. Emile
Université de Rennes
CNRS
IPR - UMR 6251, F-35000 Rennes, France

The ORCID identification number(s) for the author(s) of this article can be found under https://doi.org/10.1002/adpr.202400098.

© 2024 The Author(s). Advanced Photonics Research published by Wiley-VCH GmbH. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

DOI: 10.1002/adpr.202400098
<figure><img src="image_3.png" /><figcaption>Figure 1. Experimental arrangement showing, the influence of the capillary length $\lambda_c$, the immersion depth $d$, the radius of the colloid $a$, and the angle $\alpha$.</figcaption></figure>

surface tension sum to zero. The total force is vertical and then equals $F_T = 2\pi a\gamma\sin^2\alpha$. It must be equal to the sphere weight $P = Mg = \frac{4}{3}\pi a^3\rho g$, where $\rho = 2.5 \times 10^3 \text{ kg m}^{-3}$. This then leads to

$$ \sin \alpha = \left( \frac{2 a^2 g \rho}{3 \gamma} \right)^{1/2} \qquad (1) $$

One gets $\sin \alpha = 0.025$ ($\alpha = 1.43^\circ$). Finally the immersion depth is very small ($d/a = 3.1 \times 10^{-4}$). The sphere is thus floating on the air-water interface.

### 2.1.2. Sphere Rotation and Rotation Detection

Before detecting the rotation, the spherical colloid has to be set in motion. We use the OAM carried by the light[23] to transfer angular momentum to the sphere. Briefly, such OAM carrying beams have a non-zero azimuthal phase variation ($\phi = \ell\theta$, $\theta$ being the azimuthal angle, $\ell$ being the so-called topological charge) in a plane perpendicular to the direction of propagation of the beam. Each photon carries $\ell\hbar$ angular momentum, $\hbar$ being the reduced Plank constant. Its transfer has been evidence at the macroscopic scale,[24,25] where a copper ring has been set in rotation using radio waves carrying OAM and at microscopic scale,[26,27] where $1-2 \mu\text{m}$ particles totally immersed in water have been trapped and rotated. It has even been observed for atoms.[28]

In our case, an OAM carrying beam (see Figure 2) is scattered by the spherical colloidal particle. The axis of the rotation of the sphere is aligned with the axis of the beam in order to optimize the OAM transfer. The scattered light doesn't carry OAM anymore. Due to angular momentum conservation, each scattered photon has transferred $\ell\hbar$ to the colloid. It might be that some residual light is scattered into modes carrying AOM. However, this is usually not considered in scattering experiments using OAM. In order to evidence it, the scattering surface has to be embossed with a periodicity corresponding to the desired OAM mode.[29] However, this type of configuration is totally unrealizable for a fluid interface. The available torque $\Gamma_t$ can be then written as

$$ \Gamma_t = \ell\hbar n = \ell\hbar \frac{P\lambda}{2\pi\hbar c} = \ell \frac{P\lambda}{2\pi c} \qquad (2) $$

where $n = P\lambda/(2\pi\hbar c)$ is the number of photon per second in the beam, $P$ is the optical power, $c$ the light velocity in vacuum, and $\lambda$ the light wavelength ($\lambda = 532 \text{ nm}$ in our experimental

<figure><img src="image_4.png" /><figcaption>Figure 2. Principle of the experiment: a) At time $t < t_0$, an OAM carrying beam ($\lambda = 532 \text{ nm}$) with a topological charge $\ell$ is impinging on a spherical colloid particle. During the scattering process, it transfers angular momentum to the colloid due to a torque $\Gamma_t$. In the steady-state situation, the colloid rotates at a constant frequency $f$. b) At time $t > t_0$, the OAM beam is replaced by a superposition of $+\ell$ and $-\ell$. Such a superposition doesn't transfer angular momentum anymore. The dynamics of the particle set in motion is now studied. Due to the rotational Doppler effect, the scattered light is frequency shifted by $\ell f$ for the $+\ell$ component, whereas the $-\ell$ component is shifted by $-\ell f$. This leads to a beat frequency of the scattered photons that equals $2\ell f$, that is, proportional to the rotation frequency. PD: photodiode.</figcaption></figure>

conditions). For $P = 500 \text{ mW}$ measured just before the light impinges on the sphere and $\ell = 5$, this leads to $\Gamma_t = 7 \times 10^{-16} \text{ N m}$. However, the torque relies on a scattering process. Besides, some incident photons are not scattered. Then, the effective torque $\Gamma_t$ should be a little less than this value. Let us call $\eta$ the ratio between the available and effective torques

$$ \eta = \frac{\Gamma_1}{\Gamma_t} < 1 \qquad (3) $$

It has to be noted here that the problem of the interaction of a colloid in the vicinity of a liquid interface in a complicated problem leading to multiple scattering (see for example[30,31]). However, since the colloid is a sphere and because the light propagation axis is perpendicular to the interface and aligned with the sphere, the problem has a rotational symmetry. There are thus no extra forces, neither torques exerted on the colloid.

The equation of motion of the rotating sphere could be expressed as

$$ J\dot{\omega} + 2\beta\omega = \Gamma_1 \qquad (4) $$

where $\omega$ is the angular velocity of the sphere and $J$ is its moment of inertia. It is equal to

$$ J = \frac{2}{5} Ma^2 = \frac{8}{15} \pi a^5 \rho \qquad (5) $$

For $2a = 105 \mu\text{m}$, $J = 1.67 \times 10^{-18} \text{ kg m}^2$. Since it depends on the power $5$ of the particle radius, it is very sensitive to the radius calibration. For example, the $5\%$ relative uncertainty on the colloid radius, leads to a $25\%$ relative uncertainty on the moment of inertia. $2\beta$ is the friction coefficient that accounts for the interaction of the particle with the environment. For a sphere in liquid with infinite extension, it writes[32]
$$2\beta = 8\pi\mu a^3 \qquad (6)$$

$\mu$ is the dynamic viscosity coefficient of the fluid. In air, $\mu_a = 1.85 \times 10^{-5}$ Pa s whereas in water $\mu_w = 1 \times 10^{-3}$ Pa s, that is, nearly two orders of magnitude higher. In our case, although the sphere is not at all immersed, but touches the air/water interface, as it has been shown in Section 2.1.1, when calculating the angle $\alpha$, the effective dynamic viscosity should be close but higher than the value in air $\mu_a$.

We focus on the steady-state response and the damping rotation in the following. In a steady-state situation where the colloid rotates at a constant angular velocity $\omega_0$, according to Equation (4), one gets

$$\omega_0 = \frac{\Gamma_1}{2\beta} \qquad (7)$$

When the torque due to the OAM beam is not applied any more, according to Equation (4), the angular velocity should write

$$\omega(t) = \omega_0 \exp(-t/\tau) \qquad (8)$$

where

$$\tau = J/(2\beta) \qquad (9)$$

is the damping time.

The rotation of the sphere is detected via the rotational Doppler effect.[33] The light diffused by an object rotating at frequency $f$ is frequency shifted by a quantity that equals $\ell f$. Note that such an effect can also be observed for circularly polarized light, the frequency shift would be equal to $\pm f$ depending on the sign of the polarization. In the case of OAM carrying beams, the axis of the rotating colloid and the beam axis should be aligned to optimize the transfer. Here (see Figure 2b), the incoming beam is a superposition of $\ell$ and $-\ell$ beams. They are shifted by $\ell f$ and $-\ell f$, respectively. The scattered light is time modulated. In particular, in the absence of any consideration about the polarization of the light, the detected signal is clearly modulated at a beat frequency that equals $2\ell f$.[34] The same observation has been used to measure the rotation of a rough disk enlightened by a superposition of OAM beams with opposite topological charges.[29,35] The measurement of the beat frequency leads to the knowledge of the rotation frequency. Of course such a superposition doesn't carry OAM, since the angular momentum of the beams cancels. It has to be noted here that such a technique has been successfully applied to the detection of a rotating inhomogeneous sphere that scatters light anisotropically,[36] but never to a perfect sphere, as it is the case here.

The light source is a CW YAG laser ($\lambda = 532$ nm), linearly polarized. The OAM carrying beam is generated by a spatial light modulator (Holoeye Photonics). The light beam is focused on the colloid with a $f = 10$ cm lens, so that the beam size is of the order of the sphere diameter. A XY-translation stage is used to align the water-filled tube axis and the incoming beam. The scattered light is detected by an ordinary photodiode (Thorlabs PDF10A2), analyzed with an oscilloscope, and then registered on a computer.

### 3. Results

We have thus first switched on a $\ell = 5$ OAM carrying beam for 10 s in order to rotate the particle. Assuming we have reached a steady-state situation, we then turn on a $\ell = 5$ and $\ell = -5$ superposition, at $t = t_0$, for a 10 s period. We perform the measurement of the beat pulsation of the scattered light during this period. The results appear on Figure 3.

The beat pulsation has a sinusoidal variation, i.e., in the form $\cos(\omega^{\text{beat}}(t)t + \varphi)$. Also there are very few oscillations, we can measure from Figure 3 the value of the cosine function and the value of the phase $\varphi$ ($\varphi = 53^\circ$). Since we know the value of the corresponding time $t$, we can deduce the value of the beat pulsation versus time $\omega^{\text{beat}}(t)$, including the value at $t = t_0$, $\omega_0^{\text{beat}}$. Besides, from this figure, one can draw several experimental observations. 1) First, since the particle is obviously rotating at $t = t_0$, it means that a $\ell = 5$ OAM carrying beam can transfer OAM to a spherical colloid in order to rotate it. Such scattered light doesn't carry OAM anymore. The topological charge variation is responsible for the OAM transfer; 2) Second, from the beat signal of the scattered light, the rotation velocity of the spherical colloid can be unambiguously deduced. It uses the rotational Doppler effect. Indeed, the rotation velocity $\omega(t)$ equals $\omega(t) = \omega(t)^{\text{beat}}/(2\ell)$. Such a result definitively shows that the detection of the rotation dynamics of perfectly spherical colloids is finally feasible; and 3) Third, the beat frequency decreases with time, meaning that the angular velocity of the sphere is decreasing accordingly. One can note that after 1.5 s, the colloid is not rotating anymore. Then, after a 10 s sequence, the system has indeed reached a steady-state situation.

In order to check whether this rotation is indeed due to the transfer of angular momentum, we have varied the topological charge of the beam, at the same given wavelength, both in the excitation period and in the measurement period. The results appear in Figure 4. $\omega_0^{\text{beat}}$ increases with the topological charge as expected. The solid line corresponds to a parabolic fit of the experimental points.

<figure><img src="image_5.png" /><figcaption>Figure 3. Extracted beat signal between the $\ell = 5$ and $\ell = -5$ scattered OAM beam versus time. The upper graph is a zoom of the beat signal at shorter times. The beat signal of the scattered light measures the rotation of the spherical colloid.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 4. Variation of the beat pulsation at time $t = 0$ versus the topological charge. The solid line corresponds to a fit with a quadratic function.</figcaption></figure>

Actually, the torque and thus the steady-state rotation should vary linearly with the topological charge $ℓ$ according to Equation (2). Besides, the rotational Doppler shift also varies linearly with the topological charge. Then, the beat pulsation that depends both on the rotation velocity and on the topological charge must vary quadratically with $ℓ$, as observed in Figure 4. More precisely, according to Equation (2), (3) and (7), one gets

$$ \omega_0^{\text{beat}} = 2\ell\omega_0 = 2\ell\frac{\Gamma_1}{2\beta} = \ell^2\frac{\eta P\lambda}{\beta 2\pi c} \quad (10) $$

which justifies the quadratic variation of the beat pulsation versus the topological charge. As the charge is increased, the torque is more efficient and the detection becomes apparently easier. However, one cannot increase $ℓ$ at will. In particular, the typical size of the beam increases as the square root of $ℓ$. [37] In the experiment, we had to adapt the focus of the optical beam for each $ℓ$ value. It becomes more difficult to tightly focus the beam as $ℓ$ is increased.

From Figure 3, as already mentioned, one can estimate the beat pulsation $\omega^{\text{beat}}(t)$ and thus the instantaneous rotation velocity $\omega(t)$ of the spherical colloid. This appears in Figure 5. In order to make the variation more explicit, we have used a logarithmic scale. Whereas, from Equation (8), we expect an exponential variation, the figure clearly shows that there are two different time constants. What is their physical origin?

<figure><img src="image_7.png" /><figcaption>Figure 5. Variation of the rotation velocity of the colloid versus time in logarithmic scale. The dotted lines correspond to two exponential decays. After 1.5 s, the angular velocity is below 0.02 rad s⁻¹ and becomes difficult to evaluate precisely. The uncertainty is of the order of 0.005 rad s⁻¹. This may explain the discrepancy at longer times between the exponential theoretical decay and the experimental points on a logarithmic scale.</figcaption></figure>

# 4. Discussion

## 4.1. Smallest Time Constant

Let us focus on the smallest time constant. From Figure 5, we find a damping time $\tau_1 = 22 \pm 2$ ms. According to Equation (9), the friction coefficient equals $2\beta = 7.6 \times 10^{-17}$ kg m² s⁻¹, leading to $\mu_{\text{eff}} = 2.09 \times 10^{-5}$ Pa s, which is consistent with the expected value, i.e., slightly higher than $\mu_a$, the air viscosity. According to Equation (7), for $\omega_0 = 3.0$ rad s⁻¹, the effective torque due to the laser is then equal to $\Gamma_1 = 2.27 \times 10^{-16}$ kg m² s⁻², i.e., three times less ($\eta = 30\%$) than the available torque, which is also in agreement with the expected value. Indeed, only part of the photons participate in the OAM transfer. Note that it is difficult to give a better estimation of the friction coefficient and of the torque since all these parameters depend on the value of the moment of inertia $J$. As already mentioned, for a relative uncertainty on the radius $a$ of the sphere of 5%, the relative uncertainty on $J$ is then 25%, leading to the same type of uncertainty for $\Gamma_1$. The uncertainty on $\mu_{\text{eff}}$ is only 10%, since $\mu_{\text{eff}}$ varies as the square of $a$. Nevertheless, the values found are in reasonable agreement with the friction of the sphere with the surroundings. Thus, the smallest time constant is associated with the damping of the colloid in the air.

## 4.2. Torque Investigation

Since the torque due to the laser depends on the optical power and on the topological charge $ℓ$, according to Equation (2), we have varied these two parameters and investigated the variation of the steady-state rotation ($\omega_0$). The results appear in Figure 6. Obviously, $\omega_0$ varies linearly with the optical power and with the topological charge, as expected. This corroborates the fact that the rotation mechanism of the colloidal sphere relies on the transfer of OAM from the incident beam to the colloid.

## 4.3. Highest Time Constant

What about the higher time constant? From Figure 5, we found $\tau_2 = 0.80 \pm 0.05$ s. What is its physical origin? The water tank has finite dimensions. We thus suspect that this time constant is due to some extra friction. Indeed, the particle rotation causes

<figure><img src="image_8.png" /><figcaption>Figure 6. Variation of $\omega_0$ versus the optical power (left), for a given topological charge $ℓ = 5$, and versus the topological charge for a given optical power (0.5 W) (right). The uncertainty is within the points. After 0.6 W, the departure of the experimental point from a linear variation might be due to some residual heating. We thus restrict our analysis to powers below 0.5 W.</figcaption></figure>
fluid spinning inside the tube that may induce some friction with the tube walls or with the ambient air. In order to check this, we have changed the tube diameter. We have observed that $τ_2$ hardly varies with the tube diameter, indicating that it is mainly the friction between liquid and air, which contributes to this dynamics. The $τ_1$ time constant is little influenced by the tube radius, as long as it remains greater than the capillary length.

Let us return to $τ_2$. More specifically, at a macroscopic scale, the situation of a fluid in a rotating tube has already been studied.[38] The viscous decay of the initial state of motion or equivalently the time required for the interior flow to approach the final state of rigid rotation, with a fluid angular velocity $Ω$ is equal to

$$τ = \frac{L}{\sqrt{ν_KΩ}} \qquad (11)$$

In our case, the final steady state is at rest. L is a characteristic distance corresponding to the length of the tube and $ν_K$ is the kinematic viscosity of the fluid (for air $ν_K = 1.52 \times 10^{-6} m^2 s^{-1}$). Taking for L the radius of the tube (8 mm), and $Ω = ω_0 = 3 rad s^{-1}$ (assuming that the air angular velocity is the particle velocity), this leads to $τ = 1.15 s$, i.e., higher but within the same order of magnitude as the experimental measure of $τ_2$ ($τ_2 = 0.8 s$). This description is here somewhat simplistic. It would also be necessary to consider the shear stresses of the air-liquid interface induced by the particle rotation, the friction of the interface on the tube walls and, possibly, the capillary interaction. However, our measurements do not allow us to refine a hydrodynamic model. Besides, it goes far away from the scope of our work that is dedicated to the detection of a rotating colloid.

## 5. Conclusion

To conclude, we have set into rotation a micrometer-size perfect colloid floating on an air-liquid interface, using the angular orbital momentum of light. The rotation frequency is in agreement with the expected torque. This frequency varies linearly with the optical power and with the topological charge of the beam. The rotation frequency measurement is performed using the rotational Doppler shift, which is shown to be a very powerful technique to access the dynamics of the rotation. This would be of valuable help in the case of perfect spherical particles (as described here), since it is impossible to discriminate between a rotating particle or a particle at rest. Besides, the rotational Doppler technique is not invasive and can be adapted to any situation. We have isolated two damping time constants. The faster one is associated to the friction of the sphere with the surrounding fluid (mostly air), whereas the other one is mostly attributed to the friction at the air-liquid interface.

This kind of measurement can be extended down to the nanometer range up to the molecular level, as soon as there is some light diffused by the sphere.[39–43] The main problem would be to center the light beam axis with the axis of the particle. Besides, the OAM beam would have to be tightly focused, which is not so easy. One alternative could be to use Arago Poisson spots carrying OAM beams that have been shown to be in the nanometer range.[44,45] Such an experiment can then provide valuable

information on the rotational dynamics that lies at the heart of intermolecular, molecule-surface chemistry and cold molecule science. Conversely, this technique could also be extended to macroscopic systems, possibly using radio waves, to track satellites and torque disturbances that could be applied to them.[46–48]

## Acknowledgements

The authors thank S. Boutros and L. Pitois (Université de Rennes) for technical support.

## Conflict of Interest

The authors declare no conflict of interest.

## Author Contributions

**Olivier Emile:** Conceptualization (lead); Data curation (lead); Formal analysis (equal); Investigation (lead); Methodology (equal); Writing—original draft (equal); Writing—review & editing (equal). **Janine Emile:** Conceptualization:Supporting; Data curation:Supporting; Formal analysis:Supporting; Investigation (equal); Methodology (equal); Writing—original draft (equal); Writing—review & editing (equal).

## Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

## Keywords

orbital angular momentum of light, rotation detection, rotational Doppler effect, spherical colloids rotation

Received: June 2, 2024

Revised: November 6, 2024

Published online:

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