# Emergent dynamics of light-induced active colloids probed by XPCS

Peer Fischer, Thomas Zinn, Theyencheri Narayanan, Sai Nikhilesh Kottapalli, Johannes Sachs, Thomas Sottmann

► To cite this version:

Peer Fischer, Thomas Zinn, Theyencheri Narayanan, Sai Nikhilesh Kottapalli, Johannes Sachs, et al. *Emergent dynamics of light-induced active colloids probed by XPCS*. New Journal of Physics, 2022, 24 (9), pp.093007. 10.1088/1367-2630/ac8a66. hal-05077657

HAL Id: hal-05077657

https://hal.science/hal-05077657v1

Submitted on 21 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
PAPER • OPEN ACCESS

# Emergent dynamics of light-induced active colloids probed by XPCS

To cite this article: Thomas Zinn et al 2022 *New J. Phys.* **24** 093007

View the [article online](#) for updates and enhancements.

## You may also like

- The effect of the spatial sampling rate on quantitative phase information extracted from planar and tomographic edge illumination x-ray phase contrast images
  C K Hagen, P C Diemoz, M Endrizzi et al.

- Edge-illumination x-ray phase-contrast imaging
  Alessandro Olivo

- Boundary-enhancement in propagation-based x-ray phase-contrast tomosynthesis improves depth position characterization
  Huifeng Guan, Qiaofeng Xu, Alfred B Garson et al.
<figure><img src="image_2.png" /></figure>

CrossMark

OPEN ACCESS

RECEIVED

13 March 2022

REVISED

29 July 2022

ACCEPTED FOR PUBLICATION

17 August 2022

PUBLISHED

8 September 2022

Original content from
this work may be used
under the terms of the
Creative Commons
Attribution 4.0 licence.

Any further distribution
of this work must
maintain attribution to
the author(s) and the
title of the work, journal
citation and DOI.

<figure><img src="image_3.png" /></figure>

PAPER

# Emergent dynamics of light-induced active colloids probed by XPCS

Thomas Zinn¹, Theyencheri Narayanan¹,*, Sai Nikhilesh Kottapalli², Johannes Sachs², Thomas Sottmann³ and Peer Fischer²,³

¹ ESRF-The European Synchrotron, F-38043, Grenoble, France

² Max Planck Institute for Intelligent Systems, 70569 Stuttgart, Germany

³ Institute of Physical Chemistry, University of Stuttgart, 70569 Stuttgart, Germany

* Author to whom any correspondence should be addressed.

E-mail: narayan@esrf.fr

**Keywords:** active colloids, active dynamics, velocity fluctuations, x-ray scattering, XPCS, USAXS

Supplementary material for this article is available [online](#).

## Abstract

Self-propelled particulate systems manifest certain collective behavior of living matter, which have been the subject of intense research over the past decades. One of the elegant methods for realizing such active motions is by means of custom synthesized Janus particles suspended in a catalytic medium that can be triggered upon illumination by ultraviolet light. In this work, the evolution of the particle dynamics from passive diffusive to active ballistic behavior upon light illumination was probed by multispeckle x-ray photon correlation spectroscopy (XPCS). This technique enables not only studying the emergence of active motions in three dimensions (3D) but also deciphering different contributions to the overall dynamics. Using a combination of homodyne and heterodyne analysis, the ensemble averaged mean velocity, velocity fluctuations and diffusion coefficient of particles were determined in the thermodynamic limit. Results revealed a gradual transition from diffusive to ballistic dynamics with systematic increase of the catalytic activity. At the intermediate region, the dynamics is dominated by Gaussian velocity fluctuations and an enhanced relaxation rate with a weaker wave vector dependence similar to superdiffusive behavior. For the highest activity, the dynamics became purely ballistic with Lorentzian-like distribution of velocity fluctuations. Presented results demonstrate that different aspects of active dynamics can be investigated in 3D over a broad range of Péclet numbers and other control parameters by means of multispeckle XPCS.

## 1. Introduction

Over the last several decades, active or self-propelled colloids have received significant interest as simple realization of active matter that has been the subject of burgeoning research [1–6]. In contrast to diffusive Brownian particles, active colloids consume energy from the surrounding medium and perform mechanical motion in addition to their Brownian diffusion. Therefore, they mimic living organisms and display certain fascinating features such as directed motion, flocking, swarming, etc [3]. The self-propulsion of particles can be realized in a variety of ways, for instance by chemical gradients resulting from a reaction at the surface of the particles, temperature gradients emanating from selective heating of one side of the particles, response of the particles to gradient of electric or magnetic fields, etc [7–9]. The diffusiophoresis that involves the gradient of concentration is one of the most widely used mechanism for designing active colloids [10]. A common approach uses Janus particles with a catalytic subunit, which when suspended in a catalytic medium such as hydrogen peroxide ($H_2O_2$) solution display self-phoretic motion induced by the chemical gradient around each particle [7, 9]. The active motions of such particles can easily be tracked in
two dimensions (2D) by optical microscopy and therefore they have been widely investigated [4, 9]. Using Janus particles composed of silica spheres with a hemispherical titania (anatase) cap suspended in H₂O₂ solution, the self-phoretic motion can be induced by ultraviolet (UV) light illumination [11, 12].

The motion of active Brownian particles remains ballistic over a time scale defined by the rotational diffusion, beyond which their behavior becomes stochastic [5, 13, 14]. The crossover is via a superdiffusive regime that is characterized by an enhanced effective diffusion coefficient, $D_{\text{eff}}$. In the superdiffusive range, an anomalous wave vector dependence of the relaxation rate is expected [15]. Many of the theoretical predictions concerning active behavior are in the thermodynamic limit with very large number of particles [1, 2], while the vast majority of experiments have been carried out by optical microscopy with a limited system size [4, 16]. In particular, microscope observations often involved the motion near an interface, but it is known that the particles-surface interactions modify the phoretic motions in a variety of ways [5, 12, 17, 18]. For example, the substrate may act as a momentum sink for particles [19] and the propulsion speed of particles near a wall is influenced by the wall slip boundary conditions [20]. Therefore, the bulk dynamics of these particles could be different from their motion near a surface. By employing scattering techniques, active systems can be studied in their thermodynamic limit in three dimensions (3D) [21, 22]. The ensemble averaged dynamic information derived from scattering experiments can complement the results obtained by optical tracking method, which may be subject to limited statistics [23].

In this work, active motions of silica/titania Janus particles were triggered by UV light illumination [11, 22]. The evolution of the active dynamics of these colloids were probed by multispeckle ultra small-angle x-ray photon correlation spectroscopy (UA-XPCS). The time and ensemble averaged static behavior is simultaneously monitored by ultra small-angle x-ray scattering (USAXS). The UA-XPCS provides access to the wave vector dependent ensemble averaged dynamics, which can be decomposed into ballistic and diffusive contributions [21]. Especially, the deviations from the mean velocity or velocity fluctuations of the particles ($\delta v$) and their $D_{\text{eff}}$ can be deduced with high statistics. In particular, the high Péclet number (Pe) range that has been little explored is readily accessible. Furthermore, these photo-active Janus particles may display phototactic behavior enriching their active dynamics [24]. Finally, the activity can be switched off by turning off the UV illumination [11], thereby allowing to probe the gradual return to Brownian dynamics.

Recent advances of the x-ray source (ESRF-EBS) with high degree of coherence, and detector technology (e.g. the Eiger pixel-array detector) in terms of higher spatial and temporal resolution [25], enabled the investigation of shorter time ranges and perform a direction dependent analysis of the particle dynamics. The high frame rate capability of the Eiger 500k detector together with the high coherent photon flux permitted the measurement of ensemble averaged intensity-intensity autocorrelation functions, $g_2(q, t)$, with high signal-to-noise ratio in the sub-millisecond time range. This allowed the decomposition of the emergent particle dynamics into diffusive and propulsive components as a function of time from the onset of active motion upon UV illumination. In addition, the dependencies of parameters on the H₂O₂ concentration and UV power were explored.

## 2. Materials and methods

### 2.1. Sample preparation

Measurements were performed using silica/titania (SiO₂/TiO₂) Janus particles fabricated from SiO₂ beads (SiO₂-F-SC68, microParticles GmbH, Berlin, Germany) with nominal radius $\approx 259 \pm 7$ nm. A close-packed monolayer of SiO₂ particles was formed using the Langmuir-Blodgett technique, which was then transferred onto a silicon wafer. A thin (20 nm) hemispherical layer of TiO₂ was coated onto the silica particles with the wafer held at an oblique angle (85°) and undergoing continuous rotation in a physical vapor deposition chamber [26]. The substrate was then annealed at 450 °C for 2 h for the TiO₂ layer to crystallize into the anatase phase [22]. The coated particles were extracted by sonication and dispersed in Milli-Q deionized water. Typically, $10^{10}$ Janus particles can be fabricated at a time using this approach. When illuminated with UV light of wavelength below 375 nm (corresponding to the bandgap), electron-hole pairs are generated asymmetrically on the anatase side, which initiate reduction reactions in a catalytic medium. The local chemical gradient thereby formed leads to the self-phoretic motion of the particles, which can be tuned with the UV light intensity.

The particles were characterized by scanning electron microscope (SEM) in the dry form and subsequently by USAXS as a dispersion in water. The number density (N) of particles in the dispersion ranged from $2-5 \times 10^6$ mm⁻³. Freshly prepared samples were contained in thin-walled ($\sim 10$ µm) quartz
<figure><img src="image_4.png" /><figcaption>**Figure 1.** (a) Sample stage of the experimental setup used for UV illumination during XPCS measurements. The capillary cell was placed precisely at the center where UV and x-ray beams crossed at 90°. (b) A representative scanning electron microscopy image of the SiO₂/TiO₂ Janus particles used for the experiments. The size bar is 500 nm.</figcaption></figure>

capillaries of diameter 2 mm, with a volume of 25–30 µl. Required amount of H₂O₂ solution (35%) was added into the capillary cell and the concentration was verified by weight. The specified concentration of H₂O₂ in the sample refers to relative volume % of H₂O₂ in the aqueous solvent, though it does not make a significant difference due to the extremely low volume fraction of the particles. Evaporation losses were prevented by a quartz stopper on the open end of the capillary.

## 2.2. X-ray scattering setup

Figure 1(a) depicts the experimental setup used for the UV light illumination during XPCS measurements. The UV light of wavelength 365 nm was produced by means of a light-emitting-diode (LED) (Alonefire SV003 flashlight). The UV light was collected and focused onto the capillary cell using a large lens with diameter (70 mm) and focal length (50 mm). The nominal UV power in the spot (4 mm) varied from 100–200 mW, which was calibrated using a power meter (Power Max 10, Coherent). The results presented here correspond to mainly 200 mW nominal, and the real power in the capillary volume was lower by a factor 2 or more due to the larger spot size with a flatter profile. For this power, the sample temperature increased by up to 6 °C upon illumination but stabilized within about 5 min. The capillary metal holder aided thermal dissipation and fast equilibration. The LED was powered by a variable direct current power supply through a remote controlled relay system (WAGO) for turning on and off the illumination. The UV and x-ray beams crossed the capillary at 90°. While the UV spot covered the whole capillary diameter, the x-ray spot (full width at half maximum size ≈ 25 µm) passed through the center of the capillary.

All experiments were carried out at ID02 beamline of the European Synchrotron Radiation Facility (ESRF) in Grenoble, France [27]. The incident x-ray energy was 12.23 keV, corresponding to a wavelength, $\lambda = 1.014Å$. The sample-to-detector distance was set to 30.7 m for accessing the low $q$ scales relevant to the system. Where $q$ is the magnitude of scattering wave vector, $q$, given by $q = (4\pi/\lambda)\sin(\theta/2)$, with $\theta$ the scattering angle. The coherent beam was selected by a pair of slits of aperture 30 µm (vertical) × 20 µm (horizontal), which provided a photon flux of $10^{12}$ photons/s at the sample position. With these dimensions, the beam was slightly elliptical on the sample and together with the pathlength of 2 mm defined the scattering volume. 2D speckle patterns as illustrated in figure 1 were acquired using an Eiger 500k detector with a pixel size of 75 µm in 8 bit or 4 bit modes [25]. The resulting speckle contrast ($\beta$) was better than 35% at 30.7 m sample-to-detector distance. Further details of the experiment setup for XPCS are described elsewhere [25]. Depending on the dynamics, the acquisition rate varied from 5 to 20 kHz and typically 5000 to 20 000 frames were acquired in order to capture the full decay of $g_2(q, t)$ with sufficient statistics. The acquisition time varied between 0.6 s and 1.2 s. The static scattering profiles were obtained by the azimuthal averaging of the 2D USAXS patterns after applying detector corrections and normalization to an absolute intensity scale [27]. Each pattern was measured with 0.1 s acquisition time and the normalized background subtracted USAXS intensity is denoted by $I(q)$.

## 2.3. Principle of XPCS and data analysis procedure

The XPCS method relies on the coherent properties of the incident x-ray beam and the measured 2D scattering patterns display speckles, which fluctuate due to the spatio-temporal rearrangement of the
particles in the scattering volume [28, 29]. These intensity fluctuations can be related to the dynamics of the suspension via the homodyne intensity-intensity autocorrelation function

$$g_2(\mathbf{q}, t) = \frac{\langle I(\mathbf{q}, t_0)I(\mathbf{q}, t_0 + t) \rangle}{\langle I(\mathbf{q}, t_0) \rangle^2}, \qquad (1)$$

where $I(\mathbf{q}, t)$ is the scattered intensity measured at a given $q, t$ is the lag time from $t_0$, and $\langle \cdot \rangle$ denotes the average over $t$. $g_2(q, t)$ is related to the intermediate scattering function or the first order field-field autocorrelation function, $g_1(q, t)$, through the Siegert relation [30]:

$$g_2(q, t) = 1 + \beta |g_1(q, t)|^2, \qquad (2)$$

where the speckle contrast, $\beta$, depends on the coherence properties of the incident x-ray beam and the resolution of the scattering setup. The dynamic information of the system is contained in $g_1(q, t)$. For the simple case of non-interacting diffusive Brownian particles, $g_1(q, t)$ decays exponentially:

$$g_1(q, t) = \exp[-\Gamma(q)t]. \qquad (3)$$

In the case of spherical isotropic particles, the homodyne relaxation rate, $\Gamma(q)$, contains information about the translational diffusion only [31].

$$\Gamma(q) = D_0 q^2, \qquad (4)$$

where $D_0$ is the translational diffusion coefficient given by $D_0 = k_B T/(6\pi\eta R_H)$, with $k_B$ the Boltzmann constant, $T$ the absolute temperature, $\eta$ the solvent viscosity and $R_H$ the hydrodynamic radius of particles. Corresponding rotational diffusion coefficient, $D_R = k_B T/(8\pi\eta R_H^3)$ [31].

In the case of propulsive particles, the net displacement has contributions from both diffusion and directed motion [32],

$$\mathbf{r}_j(t) - \mathbf{r}_j(t_0) = \Delta\mathbf{r}_{j,D}(t) + \mathbf{v}t + \Delta\mathbf{v}_j t, \qquad (5)$$

where $\Delta\mathbf{r}_{j,D}$ corresponds to the change in position of the particle $j$ due to diffusion, $\mathbf{v}$ is the mean velocity of particles in the scattering volume and $\Delta\mathbf{v}_j$ is the relative velocity of the particle $j$ from the mean velocity or velocity fluctuations. Possible origins for these velocity fluctuations will be discussed later but differences in the catalytic activity of the particles and light illumination certainly contribute to the variance of $\mathbf{v}$ in the present case [22].

For active Brownian particles, there exists an explicit expression for $g_1(q, t)$ based on the solution of Fokker-Planck equation [16, 33]. However, a different approach is used here due to the specificities of XPCS technique. First of all, in the homodyne detection scheme employed here, the intensity fluctuations are modulated by the differences in the Doppler shifts caused by all particle pairs in the scattering volume and it is given by the Fourier cosine transform of the probability distribution of velocity differences, $P(\delta\mathbf{v})$ [32, 34]. As a result, $g_1(q, t)$ is not directly measuring $\mathbf{v}$ and the magnitude of mean propulsion velocity, $\mathbf{v}$, enters the expression via the decorrelation effect when a particle leaves the scattering volume. The resulting $|g_1(q, t)|^2$ for all particles in the scattering volume can be factorized as follows [32, 35]

$$|g_1(q, t)|^2 = |g_{1,D}(q, t)|^2 |g_{1,T}(q, t)|^2 |g_{1,A}(q, t)|^2. \qquad (6)$$

In equation (6), the first term describes the diffusion of particles similar to that in equation (3), the second term accounts the decorrelation effect mentioned above and the third term is the contribution of velocity differences or velocity fluctuations ($\delta\mathbf{v}$) of particles. A constant $\delta\mathbf{v}$ is manifested by the characteristic oscillations in the measured $g_2(q, t)$ functions [36]. Based on the observed features of $g_2(q, t)$, a Gaussian distribution of $\delta\mathbf{v}$ appeared reasonable. For a normal distribution of $\delta\mathbf{v}$ and a Gaussian beam profile, equation (6) leads to [21, 32, 37]

$$g_1(q, t) = \exp[-\Gamma_e(q)t] \times \exp[-(t/t_T)^2] [1 - \alpha + \alpha \exp(-q^2 \delta v^2 t^2/2)], \qquad (7)$$

where $\Gamma_e(q)$ is an enhanced relaxation rate related to the corresponding diffusion constant, $D_e$ and $t_T$ is the transit time of particles across the x-ray beam given by $t_T = \sigma/v$, with $\sigma$ the Gaussian width of the beam. The parameter $\alpha$ was inserted due to the fact that only a fraction ($\alpha$) of particles deviate from $\mathbf{v}$ and contribute to the decay of $g_{1,A}(q, t)$ [21, 36]. When all particles contribute to the advective term, $\alpha = 1$ and this parameter is essential for describing the exponential tail of $g_1(q, t)$, which will be otherwise cut off by the Gaussian terms in equation (7). $D_e$ may deviate from $D_0$ during the propulsion and in addition, the $q$-dependence of $\Gamma_e(q)$ may vary from the expected $D_e q^2$ form.
<figure><img src="image_5.png" /><figcaption>**Figure 2.** Static USAXS profiles of silica/titania Janus particles under different conditions. The oscillations in intensity signify the scattering form factor of particles. The red line displays the modeling in terms of the scattering form factor of polydisperse spheres with a hemispherical cap ($R_s$ = 254 nm, PD ≈ 0.024, $d_{TiO_2}$ = 12 nm and $N \approx 4 \times 10^6$ mm$^{-3}$).</figcaption></figure>

In principle, all three terms in equation (7) can be decomposed from their differences in $q$ and $t$ dependences. However, the decomposition can become delicate due to similar $q$-dependence of diffusive and velocity fluctuation terms. Therefore, $v$ was independently determined from the inadvertent heterodyning effect at the low $q$ region. For a dilute sample, the scattered intensity from the suspension becomes comparable to the parasitic background of the instrument at the lowest $q$ values probed. Therefore, the scattered intensity of the particles mixes with the nonfluctuating background that serves as a reference intensity, resulting in an heterodyne correlation function, which can be described as [38, 39],

$$g_{2h}(q, t) = 1 + \beta[x^2 g_1^2(q, t) + 2x(1-x)g_1(q, t)\cos(q \cdot vt)] + \beta(1-x)^2, \quad (8)$$

where $g_1(q, t)$ is given by equation (7) and $x$ is the ratio of the fluctuating scattered intensity from the suspension to the total scattered intensity including the background. This ratio can be determined from the static measurements with and without the sample. The $\cos(q \cdot vt)$ term in equation (8) leads to periodic oscillations in the measured $g_{2h}(q, t)$ [38], dominated by the $v$ component parallel to $q$. At ultra low angles, the scattered and transmitted beams are nearly parallel, therefore $q$ is nearly perpendicular to the beam direction. As a result, $v$ can be independently estimated from the heterodyning effect at the lowest $q$ value of measurements and the frequency in the oscillations in $g_{2h}(q, t)$ may be used to deduce $v$.

The above treatment is valid for relatively low volume fraction of particles, where the dynamic clustering effect and motility-induced phase separation [6, 40] do not intervene the emergent dynamics. Moreover, at higher volume fraction the $q$-dependence of the diffusive term is expected to be non-monotonic.

From the measured speckle patterns, $g_2(q, t)$ functions were calculated using the python program (Dynamix) developed at the ESRF. Each $g_2(q, t)$ function was calculated pixel-by-pixel for different $q$ values and then azimuthally averaged over all speckles corresponding to a given $q$ bin ($\Delta q = 5 \times 10^{-4}$ nm$^{-1}$) in order to obtain the time and ensemble averaged $g_2(q, t)$. For anisotropic dynamics, the ensemble averaged $g_2(q, t)$ functions were calculated by taking only an azimuthal sector of $\pm 15^\circ$ along the vertical and horizontal directions. The $g_2(q, t)$ is not sensitive to the reversal of the direction up or down and left or right. As a result the averaging for the vertical and horizontal directions were performed using both upper and lower sectors, and left and right sectors, respectively.

# 3. Results and discussion

## 3.1. Static characterization of Janus particles

Figure 1(b) presents an SEM micrograph of the dry Janus particles with the $TiO_2$ hemispherical cap. Prior to each XPCS sequence, the static behavior of the particle suspension was verified by USAXS measurements under the same conditions. Figure 2 displays typical normalized USAXS profiles from the suspensions after the background subtraction. The scattering profiles under different conditions essentially superimpose once the particle concentration difference is corrected. At these dilute concentrations, the USAXS profiles manifest the scattering form factor averaged over the size distribution of particles and the structure factor of
<figure><img src="image_6.png" /><figcaption>**Figure 3.** (a) Typical ensemble averaged $g_2(q, t)$ functions for the Janus particles in water depicting the normal diffusive dynamics characterized by an exponential decay. The continuous lines are fits using equation (2) with $\beta \simeq 0.35$. (b) The relaxation rates, $\Gamma(q)$ vs $q^2$ derived from the analysis shown in (a) as a function of time from the UV power was turned on. When UV power was off, $D_0 \simeq 0.95 \ \mu\text{m}^2 \text{s}^{-1}$ corresponding to $R_H \simeq 260$ nm. $\Gamma(q)$ increased systematically upon UV illumination over the initial several minutes, but the $q^2$-dependence was not altered.</figcaption></figure>

interactions is essentially unity, i.e. $S(q) \simeq 1$. These scattering profiles can be described by a polydisperse sphere (silica) model incorporating the hemispherical cap [41]. The model yielded a mean radius of silica particles ($R_S$) = 254 nm, the polydispersity (PD) in terms a Gaussian size distribution $\simeq 0.024$ and the thickness of the titania cap ($d_{TiO_2}$) = 12 nm. The model scattering length densities of $SiO_2$ and $TiO_2$ were $1.72 \times 10^{-3} \ \text{nm}^{-2}$ and $2.94 \times 10^{-3} \ \text{nm}^{-2}$, respectively which correspond to mass densities 10% lower than the bulk amorphous silica and anatase, that is expected in the colloidal form. The scattering profiles do not show any significant variation in $H_2O_2$ solution and with UV illumination. There was not any indication of dynamic clustering when particles became active, presumably due to the low volume fraction of particles ($\simeq 0.0003$) used in this study. This concentration corresponds to colloidal gas phase in a passive system but in the active case the liquid phase extends to vanishing densities for small values of the noise strength [6].

## 3.2. Passive dynamics of Janus colloids

In this section, the dynamic characterization of the Janus colloidal suspensions by XPCS in the absence of activity (0% $H_2O_2$) is presented. The colloid dynamics without UV illumination showed the typical Brownian behavior characterized by an exponential decay of $g_2(q, t)$ and $q^2$-dependence of the corresponding $\Gamma(q)$ as shown in figure 3. The $R_H$ value derived from this analysis is fully consistent with the static analysis presented in figure 2. As the suspension is continuously illuminated by UV light, $\Gamma(q)$ increased by about 20% over the initial several minutes and then stabilized around that value. This enhancement can be explained in terms of a decrease in viscosity caused by an increase in sample temperature by about 10 °C due to UV absorption. The bulk temperature change measured with a thermocouple was about 6 °C and therefore the local temperature felt by the particles is higher. The onset of convection would have led to a more significant enhancement of the dynamics with a change in the functional form of $g_2(q, t)$. An identical behavior was also observed with similarly sized silica particles as shown in figure S1 of supplemental material (SM) ([https://stacks.iop.org/NJP/24/093007/mmedia](https://stacks.iop.org/NJP/24/093007/mmedia)) [42]. Apart from this heating effect, the dynamics remained Brownian in water and $H_2O_2$ in the absence of UV illumination. For the conditions used in the next section, the enhanced dynamics (20%–25%) due to UV heating effect can be accounted by $D_e$ and the overall active dynamics is an order of magnitude faster.

## 3.3. Active dynamics at different $H_2O_2$ concentrations

This section presents the main results on active dynamics observed with UV illumination. The enhanced dynamics is sensitive to both the $H_2O_2$ concentration and UV power. With $4 \times 10^6 \ \text{mm}^{-3}$ particles, the evolution of dynamics below a nominal UV power of 80 mW and 2% of $H_2O_2$ was less spectacular. Below this concentration, the chemical gradient generated may not be sufficient to induce significant ballistic motion for this range of UV power. Although the trends are similar with increasing UV power or $H_2O_2$ concentration, here the analysis is focused on the latter for the same nominal UV power of 200 mW. The behavior is similar for $H_2O_2$ concentrations varying from 2%–8% (moderate), though the decay of $g_2(q, t)$ is faster at higher $H_2O_2$ content. The dynamics became strikingly different for $H_2O_2$ concentrations exceeding 10% (high) for the UV power used. The time $t_{UV}$ represents the time after UV light illumination is turned on.
<figure><img src="image_7.png" /></figure>

### 3.3.1. Moderate H₂O₂ concentrations

Figure 4 presents representative evolution of the $g_2(q, t)$ functions upon UV illumination for a sample containing 4% of H₂O₂. The enhanced dynamics and a change in the functional form of $g_2(q, t)$ are clearly evident. As in figure 3, $g_2(q, t)$ functions in figure 4(a) can be well described by a simple exponential decay. This expression became inadequate when the dynamics turned faster. In such cases, a straightforward extension of equation (2) analogous to the Kohlrausch-Williams-Watts function, is a compressed exponential function of the following form [15, 29],

$$g_2(q, t) = 1 + \beta \exp[-2(\Gamma(q)t)^\gamma], \quad (9)$$

where $\gamma$ is an exponent describing the compression and mean $\Gamma(q)$ is given by the area under the curve. Figure 5(a) displays the analysis using equation (9) for a given $q$ as a function of $t_{UV}$. The change in functional form is accounted by $\gamma > 1$ and the resulting mean $\Gamma(q)$ at different times are presented in figure 5(b). This approach allowed an empirical description of the overall evolution of the dynamics and extraction of the underlying $q$-dependence. A change from $\Gamma(q) \propto q^2$ to $\Gamma(q) \propto q^\mu$ with $1 \le \mu \le 2$ behavior can be noticed in figure 5(b), which may imply a crossover from diffusive to ballistic dynamics, with an intervening superdiffusive region involving Lévy flight like steps [15].

For a comparison, the corresponding analysis for 8% H₂O₂ sample is shown in figure S2 [42]. Nevertheless, the analysis in terms of equation (9) can only provide limited information about the active dynamics including the $q$-dependence. In order to gain a deeper insight, the analysis was performed using equations (2) and (7), which enabled dissecting the diffusive and ballistic contributions in the observed dynamics. In principle, parameters $D_e$, $v$ and $\delta v$ can be deduced from the $q$-dependence and figure 4(b) shows the description of the data in terms of equation (7). To better constrain the analysis, $v$ parallel to $q$, which is the leading contribution to the transit term in equation (7) was independently derived using the heterodyne analysis.

For the lower $q (=0.0022 \text{ nm}^{-1})$ an additional oscillation can be seen, which is not described by equation (7) and it likely arises from the heterodyning effect mentioned before. Figure 6 presents the heterodyne analysis of $g_2(q, t)$ functions for a sample containing 8% of H₂O₂ following the UV
<figure><img src="image_8.png" /><figcaption>**Figure 5.** Evolution of the ensemble averaged $g_2(q, t)$ functions at a fixed $q$ after turning on the UV light for a sample containing 4% of $H_2O_2$. The continuous lines are fit curves using equation (9). The inset depicts the evolution of the exponent $\gamma$ and the shaded region in gray represents time before UV illumination. (b) The mean $\Gamma(q)$ at different $t_{UV}$ derived from the above analysis demonstrating significant enhancement of the dynamics.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>**Figure 6.** The heterodyne analysis of $g_2(q, t)$ functions at the lowest $q$ (0.0022 nm$^{-1}$) at different times after turning on the UV light for a sample containing 8% of $H_2O_2$. The continuous lines are fit curves using equation (8). For clarity successive $g_2(q, t)$ functions have been shifted by a factor from their baselines.</figcaption></figure>

illumination. The analysis was performed using equation (8) and the oscillation that appeared at later times corresponds to the cosine term. The position of the first minima is an indicative of the minima of the cosine function, which may be used to estimate $v$ parallel to the $q$ direction. The corresponding analysis for 4% $H_2O_2$ sample is provided in figure S3 [42]. The heterodyning effect at this $q$ is also manifested by a reduction in the $\beta$ value from 0.35. For the remaining $q$'s this heterodyning effect is insignificant, and $v$ was fixed at this value in the analysis using equation (7).

Figure 7 presents the analysis for a sample containing 8% of $H_2O_2$. As compared to figure 4, the onset of active motion occurred earlier but the functional form of $g_2(q, t)$ and $q$-dependence are qualitatively similar to 4%. When the particles turned active, the $q$-dependence of $g_2(q, t)$ became squeezed and the functional
<figure><img src="image_10.png" /><figcaption>**Figure 7.** Representative ensemble averaged $g_2(q, t)$ as a function of $q$ for a sample containing 8% of H$_2$O$_2$ (a) before and (b) after 3 min of turning on the UV light. The continuous lines are fit curves using (a) equation (3) and (b) equation (7) as described in the text.</figcaption></figure>

form changed manifesting the dominance of $v$ and $\delta v$ terms, respectively. The dynamics still remained isotropic without significant difference between vertical and horizontal sections of $g_2(q, t)$ functions. The detailed analysis in terms of equation (7) yielded $\delta v$ and $D_e$, while $v$ was fixed at the value derived from the heterodyne analysis.

Figure 8 displays $\Gamma_e(q)$ corresponding to the diffusive contribution obtained using equation (7) for two samples containing 4% and 8% of H$_2$O$_2$. It can be noticed that prior to the onset of active motion, the behavior is that expected for Brownian dynamics [$\Gamma_e(q) \propto q^2$]. When the UV light is turned on, the apparent $q$-dependence has changed to $\Gamma_e(q) \propto q^\mu$, with the exponent, $\mu < 2$. The weaker $q$-dependence of $\Gamma_e(q)$ may signify superdiffusive behavior [15]. Moreover, in the case of asymmetric active particles with net propulsion, propulsive and anomalous diffusive modes coexist over a certain time range below $\tau_R$ [13]. The presented behavior is reminiscent of this scenario involving both propulsion and superdiffusive dynamics. Nevertheless, the possible correlation between $\delta v$ and $\Gamma_e(q)$ terms in equation (7) somewhat excludes a deeper interpretation of the results in figure 8.

The same data are plotted on a linear scale in figure S4 of SM and depict a non-zero intercept when particles became active [42]. This intercept is suggestive of a contribution from the rotational diffusion, which is $q$-independent. However, the homodyne method without polarization analysis employed here is not expected to be sensitive to the weak contribution from the rotational diffusion [31]. This non-zero intercept is more significant for 4% H$_2$O$_2$, possibly arising from an offset in $v$ from that obtained using equation (8).

Figure 9 presents the quantities, $v$, $\delta v$, $D_e$ and $\alpha$ derived from the modeling of $g_2(q, t)$ functions using equation (7) for samples containing 4% and 8% of H$_2$O$_2$. Upon UV illumination, these parameters evolved from their Brownian limit and reached their steady limits in less than 4 min. Subsequently, these parameters gradually decreased likely due to the local depletion of H$_2$O$_2$. But similar values were recovered after stirring the sample. The onset of active dynamics is characterized by a simultaneous evolution of $v$, $\delta v$ and $D_e$. The trend in $D_e$ is consistent with the corresponding increase of sample temperature upon UV illumination. The magnitudes of $v$ and $\delta v$ are larger than those obtained using the heterodyne laser Doppler velocimetry [22], presumably due to the higher concentrations of H$_2$O$_2$.
<figure><img src="image_11.png" /><figcaption>Figure 8. The relaxation rate, $\Gamma_e(q)$, of the diffusive term derived from the analysis using equation (7) for the samples containing 4% (a) and (b) 8% of H$_2$O$_2$. The insets depict the evolution of the exponent, $\mu$ and the shaded region in gray represents time before UV illumination.</figcaption></figure>

For active Brownian particles, the effective diffusion coefficient, $D_{eff}$, is related to the Péclet number (Pe) of the active motion [5],

$$D_{eff} = D_e(1 + 2/9Pe^2), \qquad (10)$$

where Pe is given by the ratio of diffusive and ballistic times, Pe = $2R_s \times v/D_e$, with $R_s$ the mean radius of particles. Here $D_e$ is used instead of $D_0$ to account for the residual changes in the diffusion coefficient with UV illumination. Furthermore, $D_{eff}$ is related to the rotational diffusion time, $\tau_R$ [5],

$$D_{eff} = D_e + \tau_R v^2/3. \qquad (11)$$

From the obtained $v$ and $D_e$ values, the Pe and $D_{eff}$ can be calculated using equation (10). The Pe values reach as high as 22 at the maximum. The ratio $D_{eff}/D_e = T_{eff}/T$, $T_{eff}$ is the effective temperature of the system due to the activity [43]. The $T_{eff}/T$ increased by two orders of magnitude suggesting very strong motor activity. In addition, the parameter $\alpha$ is close to 0.5, which may signify that roughly 50% of particles advect in opposite directions (up or down of the x-ray beam).

A striking feature of the data presented in figure 9 is the relatively large magnitude of velocity fluctuations or $\delta v$. This variance in $v$ may partly arise from the variations in the catalytic activity and UV illumination on different particles. However, a distinguishing feature of aligned active systems as compared to equilibrium systems in the thermodynamic limit is the large number fluctuations [1, 2, 19, 44, 45]. Even spherical active particles display a similar giant number fluctuations [46, 47] attributed to dynamic inelastic clustering or from persistent inhomogeneity in their spatial distribution [1]. The resulting particle number density fluctuations may manifest themselves as the variance of $v$ analogous to colloidal sedimentation [48, 49]. This issue will be further discussed in the following section.

### 3.3.2. High H$_2$O$_2$ concentrations

At high H$_2$O$_2$ concentrations (>10%), the dynamics is much faster manifesting somewhat different features. Figure 10 depicts the evolution of the dynamics for a sample of 12% H$_2$O$_2$ upon illumination of UV light. Prior to turning on the UV, the passive dynamics is very similar to previous samples and
<figure><img src="image_12.png" /><figcaption>**Figure 9.** Main parameters derived from the analysis of $g_2(q, t)$ functions using equation (7) for the samples containing 4% and 8% of H$_2$O$_2$. The parameter $v$ was obtained using equation (8). The shaded region in gray indicates time before UV illumination.</figcaption></figure>

adequately described by pure Brownian behavior with the same $R_H$ value as before. Evidently, the dynamics became rapid immediately upon turning on the UV light, and the $g_2(q, t)$ functions in the steady state decayed two orders of magnitude faster than the pure Brownian motion. Equation (7) still described the evolution of the dynamics during the first minute. But from 2 min onwards, the diffusive part in equation (7) became unimportant and the decay of the $g_2(q, t)$ is fully governed by $v$ and $\delta v$ terms. This is also indicated by the weak $q$-dependence of the $g_2(q, t)$ functions. The faster decay implies much larger amplitudes of $v$ and $\delta v$ than in the moderate H$_2$O$_2$ case, and $Pe$ values reaching close to 900. The feature of the diffusive contribution can be inferred on the decaying tail of $g_2(q, t)$. The resulting $g_2(q, t)$ functions can be sufficiently described by the following simpler form

$$g_2(q, t) = 1 + \beta \exp[-2(t/t_T)^2] \times \exp[-2\Gamma_{\text{app}}(q)t], \quad (12)$$

where $\Gamma_{\text{app}}(q)$ is an apparent relaxation rate. The second term in equation (12) is essentially the same as equation (2), but the corresponding relaxation rates represent different quantities. The $q$-dependence of $\Gamma_{\text{app}}(q)$ became linear as shown in figure 11. This behavior likely stems from a change in the probability distribution of $\delta v$ from the Gaussian as that involved in equation (7) to a Lorentzian function similar to that in turbulent fluids [34]. In that case, $\Gamma_{\text{app}}(q) = q \cdot \delta v$, and the main information that can be derived is the leading contribution of velocity fluctuations. Upon turning on the UV illumination, the rapid increase of $\Gamma_{\text{app}}(q)$ is primarily dominated by velocity fluctuations.

However, the above analysis is incomplete due to the anisotropy in $g_2(q, t)$ functions along the vertical and horizontal directions. Figure 12 presents the horizontal and vertical sections of $g_2(q, t)$ functions averaged over an azimuthal angle of $\pm 15^\circ$. Clearly the decay rates and $q$-dependence are different. Along the vertical direction, $g_2(q, t)$ functions are still described by equation (7) without the diffusive term. On the other hand, the relaxation of $g_2(q, t)$ along the horizontal is adequately traced by equation (12). The continuous production of bubbles by the reduction of H$_2$O$_2$ is likely inducing a vertical flow, leading to the anisotropy in the dynamics. This is also suggested by the value of $\alpha \simeq 1$, implying all particles contribute to the advective term. Along the horizontal direction, the dynamics is dominated by velocity fluctuations and suggest a crossover of the probability distribution function from Gaussian to Lorentzian-like form.
<figure><img src="image_13.png" /><figcaption>**Figure 10.** The azimuthally averaged $g_2(q, t)$ as a function of $q$ for a sample containing 12% of $H_2O_2$ (a) before and after (b) 1 min and (c) 2 min of turning on the UV light. The continuous lines are fit curves using equation (12) with the parameter values shown in the legend.</figcaption></figure>

Furthermore, the dynamics of the sample at these elevated $H_2O_2$ concentrations did not return to pure Brownian limit immediately upon turning off the UV illumination. Figure S5 illustrates this behavior of gradual return to Brownian dynamics after turning off the UV illumination [42]. This finding significantly deviates from microscopy observations, which noticed a cessation of active motion immediately after turning off the UV light [11]. In the absence of traction by a substrate (sink), the momentum conservation requires that particles continue to advect due to the slowly decaying velocity field [19], which is of phoretic origin in the present case. Although the volume fraction of the passive particles is very low, at these high Pe values, the particle positions are correlated via phoretic interactions [40]. As a result, the system may be equivalent to an active liquid state that also exhibits strong number fluctuations [6]. The large particle number fluctuations imply that the corresponding number density has a large variance, which in turn leads to analogous velocity fluctuations [48]. Once the UV light is turned off, the resulting passive system resembles a stratifying colloidal suspension where it is known that velocity fluctuations decay slowly over time [36, 49].
<figure><img src="image_14.png" /><figcaption>**Figure 11.** The apparent relaxation rate, $\Gamma_{\text{app}}(q)$ derived from the analysis using equation (12) for a sample containing 12% of H$_2$O$_2$. The behavior prior to turning on the UV illumination corresponded to Brownian diffusion. Upon turning on the UV light, $\delta v$ contribution dominated.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Figure 12.** The direction dependent analysis of $g_2(q, t)$ for a sample containing 12% of H$_2$O$_2$ before (a) and (d), after 1 min (b) and (e), and 2 min (c) and (f), of turning on the UV light illumination. The upper and lower panels correspond to vertical and horizontal directions, respectively. The continuous lines are fit curves using equation (3) for (a) and (d), equation (7) without the diffusive term for (b) and (c), equation (12) for (e) and (f).</figcaption></figure>

Indeed, the continuous bubbling of the suspension at these H$_2$O$_2$ concentrations may also contribute to the enhanced dynamics and persistent velocity fluctuations. However, when a bubble crosses the scattering volume, the whole integrity of the $g_2(q, t)$ functions is lost as shown in figure S6 [42]. The short acquisition times involved in these measurements enabled avoiding direct impact from the bubbles. Nevertheless, the velocity field of a passing bubble can cause advection of particles in the scattering volume. The features of $g_2(q, t)$ functions and velocity fluctuations at these high H$_2$O$_2$ concentrations are somewhat similar to turbulent convection [50]. Once the flow is initiated, it takes some time before the sample returns from the propulsion range to pure Brownian diffusion. Nevertheless, it is unlikely an effect purely induced by temperature gradient as the control measurements without H$_2$O$_2$ or TiO$_2$ cap did not display such a dramatic enhancement of the dynamics (figures 3 and S1 in SM).

Finally, it should be clarified that the dynamics is sensitive to the UV power, at a lower power (100 mW) the dynamics resembles that at 8% H$_2$O$_2$ concentration as illustrated in figure S7 [42]. At elevated H$_2$O$_2$ concentrations and UV power, XPCS mainly probes the velocity distribution and the information about diffusion lies on the decaying tail of $g_2(q, t)$ functions.
## 4. Summary and conclusion

The results presented in the previous sections illustrate the bulk dynamic behavior of UV-induced active colloids at large Pe values not explored before. The activity is sensitive to both the H₂O₂ concentration and UV power. At moderate H₂O₂ concentrations, the dynamics is dominated by both the propulsion and superdiffusive motions. In this case, the persistence length of propulsion is small enough to capture the superdiffusive dynamics prior to the complete decay of g₂(q, t). The decomposition of measured g₂(q, t) enabled the deduction of parameters v, δv and Dₑ. The observed behavior manifests striking similarity with the dynamics of silica colloids in phase separating critical binary liquid mixtures [21]. At the low concentrations of Janus colloids studied here, there was no indication of dynamic clustering either in static USAXS profiles or g₂(q, t) functions. At higher H₂O₂ concentrations, the observed dynamics is fully dominated by the propulsion and associated velocity fluctuations. The diffusive contribution is at the tail of the decaying g₂(q, t) and therefore Dₑ cannot be determined with sufficient confidence. In addition, the enhanced dynamics persisted even after the UV light was turned off and decayed over tens of minutes, plausibly as a signature of large number fluctuations. The direction dependent analysis showed an anisotropy in the dynamics with the vertical component exhibiting stronger velocity fluctuations. The continuous degassing of the solution may have contributed to this anisotropic dynamics that persisted even in the UV off condition. An analogous behavior as that at the moderate H₂O₂ concentrations can be retrieved at a lower UV power.

USAXS and UA-XPCS enabled probing the microstructure and dynamics of UV triggered active colloids at extremely low volume fractions in 3D. The measured g₂(q, t) functions can be decomposed to propulsive and diffusive components of particle dynamics. A combination of homodyne and heterodyne analysis facilitated independent determination of the propulsion velocity and velocity fluctuations. In addition, the q-dependence of the diffusive dynamics and the enhanced diffusion coefficients were determined simultaneously. Therefore, these scattering methods are suitable for nanoscale investigations of active colloids in 3D over a broad range of volume fractions and Pe values. In particular, different extraneous contributions such as bubbles, convection, etc can be distinguished. For 2D systems, XPCS can be performed in the grazing incidence scattering geometry and the corresponding analysis of g₂(q, t) function is more intricate [28].

## Acknowledgments

The ID02 beamline staff, and Y Chushkin and J Kieffer (ESRF) are thanked for technical support, and dynamix software package, respectively. The ESRF is acknowledged for the provision of synchrotron beam time (SC4996). SNK thanks the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for financial support.

## Data availability statement

The data sets presented in this study can be obtained from the corresponding author upon reasonable request.

## References

[1] Vicsek T and Zafeiris A 2012 *Phys. Rep.* **517** 71–140

[2] Toner J, Tu Y and Ramaswamy S 2005 *Ann. Phys.*, **NY 318** 170–244

[3] Marchetti M C, Joanny J F, Ramaswamy S, Liverpool T B, Prost J, Rao M and Simha R A 2013 *Rev. Mod. Phys.* **85** 1143

[4] Bechinger C, Di Leonardo R, Löwen H, Reichhardt C, Volpe G and Volpe G 2016 *Rev. Mod. Phys.* **88** 045006

[5] Zöttl A and Stark H 2016 *J. Phys.: Condens. Matter.* **28** 253001

[6] Chaté H 2020 *Annu. Rev. Condens. Matter Phys.* **11** 189–212

[7] Ebbens S J and Howse J R 2010 *Soft Matter* **6** 726–38

[8] Kapral R 2013 *J. Chem. Phys.* **138** 020901

[9] Illien P, Golestanian R and Sen A 2017 *Chem. Soc. Rev.* **46** 5508–18

[10] Moran J L and Posner J D 2017 *Annu. Rev. Fluid Mech.* **49** 511–40

[11] Singh D P, Choudhury U, Fischer P and Mark A G 2017 *Adv. Mater.* **29** 1701–328

[12] Gibbs J G 2020 *Langmuir* **36** 6938–47

[13] Golestanian R 2009 *Phys. Rev. Lett.* **102** 188305

[14] Löwen H 2020 *J. Chem. Phys.* **152** 040901

[15] Bouchaud J P 2008 *Anomalous relaxation in complex systems: from stretched to compressed exponentials* (New York: Wiley) ch 11 pp 327–45
[16] Kurzthaler C, Devailly C, Arlt J, Franosch T, Poon W C K, Martinez V A and Brown A T 2018 *Phys. Rev. Lett.* **121** 078001

[17] Das S, Garg A, Campbell A I, Howse J, Sen A, Velegol D, Golestanian R and Ebbens S J 2015 *Nat. Commun.* **6** 8999

[18] Popescu M N, Uspal W E, Dominguez A and Dietrich S 2018 *Acc. Chem. Res.* **51** 2991–7

[19] Ramaswamy S, Simha R A and Toner J 2003 *Europhys. Lett.* **62** 196

[20] Ketzetzi S, de Graaf J, Doherty R P and Kraft D J 2020 *Phys. Rev. Lett.* **124** 048002

[21] Zinn T, Sharpnack L and Narayanan T 2020 *Phys. Rev. Res.* **2** 033177

[22] Sachs J, Kottapalli S N, Fischer P, Botin D and Palberg T 2021 *Colloid Polym. Sci.* **299** 269–80

[23] Novotný F and Pumera M 2019 *Sci. Rep.* **9** 13222

[24] Uspal W E 2019 *J. Chem. Phys.* **150** 114903

[25] Zinn T *et al* 2018 *J. Synchrotron Radiat.* **25** 1753–9

[26] Mark A G, Gibbs J G, Lee T-C and Fischer P 2013 *Nat. Mater.* **12** 802–7

[27] Narayanan T, Sztucki M, Zinn T, Kieffer J, Homs-Puron A, Gorini J, Van Väerenbergh P and Boesecke P 2022 *J. Appl. Crystallogr.* **55** 98–111

[28] Sinha S K, Jiang Z and Lurio L B 2014 *Adv. Mater.* **26** 7764–85

[29] Lehmkuhler F, Roseker W and Grübel G 2021 *Appl. Sci.* **11** 6179

[30] Berne B J and Pecora R 2000 *Dynamic Light Scattering: With Applications to Chemistry, Biology, and Physics* (Dover Books on Physics Series) (New York: Dover)

[31] Degiorgio V, Piazza R and Bellini T 1994 *Adv. Colloid Interface Sci.* **48** 61–91

[32] Chang P H P and Penner S S 1981 *J. Quant. Spectrosc. Radiat. Transfer* **25** 97–104

[33] Kurzthaler C, Leitmann S and Franosch T 2016 *Sci. Rep.* **6** 36702

[34] Tong P, Goldburg W I, Chan C K and Sirivat A 1988 *Phys. Rev. A* **37** 2125–33

[35] Leheny R L, Rogers M C, Chen K, Narayanan S and Harden J L 2015 *Curr. Opin. Colloid Interface Sci.* **20** 261–71

[36] Möller J and Narayanan T 2017 *Phys. Rev. Lett.* **118** 198001

[37] Weber R and Schweiger G 1998 *Appl. Opt.* **37** 4039–50

[38] Pike E R 1977 Photon correlation velocimetry *Photon Correlation Spectroscopy and Velocimetry* vol 23 ed H Z Cummins and E R Pike (New York: Springer) pp 246–343

[39] Lhermitte J R M, Rogers M C, Manet S and Sutton M 2017 *Rev. Sci. Instrum.* **88** 015112

[40] Liebchen B and Löwen H 2019 *J. Chem. Phys.* **150** 061102

[41] Semeraro E F, Dattani R and Narayanan T 2018 *J. Chem. Phys.* **148** 014904

[42] Zinn T, Narayanan T, Kottapalli S N, Sachs J, Sottmann T and Fischer P see supplemental material

[43] Loi D, Mossa S and Cugliandolo L F 2011 *Soft Matter* **7** 3726–9

[44] Narayan V, Ramaswamy S and Menon N 2007 *Science* **317** 105–8

[45] Chaté H, Ginelli F, Grégoire G and Raynaud F 2008 *Phys. Rev. E* **77** 046113

[46] Aranson I S, Snezhko A, Olafsen J S and Urbach J S 2008 *Science* **320** 612

[47] Palacci J, Sacanna S, Steinberg A P, Pine D J and Chaikin P M 2013 *Science* **339** 936–40

[48] Luke J H C 2000 *Phys. Fluids* **12** 1619–21

[49] Mucha P J, Tee S-Y, Weitz D A, Shraiman B I and Brenner M P 2004 *J. Fluid Mech.* **501** 71–104

[50] Shen Y, Xia K-Q and Tong P 1995 *Phys. Rev. Lett.* **75** 437–40