Behavior of chemically powered Janus colloids in lyotropic chromonic liquid crystal

Abstract

Platinum-coated Janus colloids exhibit self-propelled motion in aqueous solution via the catalytic decomposition of hydrogen peroxide. Here, we report their motion in a uniformly aligned nematic phase of lyotropic chromonic liquid crystal, disodium cromoglycate (DSCG). When active Janus colloids are placed in DSCG, we find that the anisotropy of the liquid crystal imposes a strong sense of direction to their motion; the Janus colloids tend to move parallel to the nematic director. Motion analysis over a range of timescales reveals a crossover from ballistic to anomalous diffusive behavior on timescales below the relaxation time for liquid crystal elastic distortions. Surprisingly we observe that smaller particles roll during ballistic motion, whereas larger particles do not. This result highlights the complexity of phoretically-driven particle motion, especially in an anisotropic fluid environment.

I. INTRODUCTION

Active fluids are out-of-equilibrium systems composed of active elements (particles, rods etc.) suspended in a fluid medium. These active elements transduce energy from their environment into autonomous motion, causing the fluid to exhibit rich dynamic behavior. Such behavior can include collective motion [1-3], motility-induced phase separation [4,5], flow instabilities, and pattern formation [6-8]. Several recent studies have focused on the behavior of active colloids dispersed in isotropic fluids, including self-assembly and their collective dynamics [9-11]. A relatively unexplored direction in this field, concerns the replacement of the suspending fluid by a non-Newtonian anisotropic fluid, a nematic liquid crystal. Liquid crystals (LCs) are elastic fluids with some degree of anisotropy. They have long-range molecular orientational order, described by a director, $\widehat{\mathbf{n}}$.

In this paper, we report on the behavior of individual active Janus colloids in a nematic liquid crystal. A spherical colloid immersed in a uniformly aligned nematic LC will locally perturb the nematic director field as determined by the particle surface anchoring [12-14]. The anisotropic nematic phase is an interesting environment for studying active colloids, since it exhibits both anisotropic viscosity and Frank elasticity [14-16].

Our work complements previously described modes of colloidal motion in a liquid crystal as reviewed recently [16]. The propulsion can be triggered by an externally applied electric field, which causes (i) a realignment of the director around a colloidal sphere and a backflow which displaces the particles [17-19]; (ii) Quincke rotation that converts into translation by interactions with the bounding walls [20,21]; (iii) liquid crystal-enabled electro-osmosis in which a patterned director field is set into motion by an ac electric field [22]; and (iv) liquid crystal-enabled electrophoresis, which requires fore-aft asymmetry of the director field around a colloid in order to separate the space charge in the applied electric field [23-31]. In the latter case, the fore-aft asymmetry can be created by homeotropic alignment of the director [23,24], but Hernàndez-Navarro et al. [28] and Devika et al. [29] demonstrated that snowman-shaped colloidal particles with tangential anchoring create sufficient asymmetry to propel in an external electric field. For a Janus particle, a possible slight fore-aft asymmetry of the director can also be caused by different anchoring conditions at the two hemispheres. Furthermore, Sahu et al. [30] uncovered an electrically powered propulsion of metal-dielectric Janus spheres in a thermotropic liquid crystal which sets quadrupolar rather than dipolar director deformations; the Janus asymmetry is sensed only by the electrostatics of the medium. From the point of view of symmetry of the director and the colloid, our work is similar to the work of Sahu et al. [30] as there is no clear fore-aft asymmetry of the director field. However, there are two differences: (i) the Janus sphere in our study is self-propelled rather than driven by an external electric field, and (ii) the medium is a lyotropic rather than a thermotropic liquid crystal, which can potentially facilitate interface with systems of biological origin.

For this work, we selected chemically powered synthetic active colloids driven by a surface chemical reaction. Common propulsion mechanisms for active Janus colloids include self-diffusiophoresis or self-electrophoresis. Such mechanisms result from asymmetric catalytic reactions on their surfaces that generate gradient-driven fluid flows around the colloids, propelling them forward [32,33]. We used Janus colloids fabricated with platinum (Pt)-capped polystyrene (PS) spheres, that exhibit self-propelled motion in aqueous hydrogen peroxide (H₂O₂) solution via catalytic decomposition into water and oxygen. These Janus colloids typically propel with the PS side leading the way. The Pt-Janus particles were originally thought to propel by means of self-diffusiophoresis [34], whereby a local concentration gradient is established through the catalytic reaction on the Pt surface. The precise mechanism has however been a recent subject of debate. Observations of ionic effects influencing propulsion direction have indicated electrophoretic contributions to the motion [35,36]. The liquid crystal host in our experiments, was carefully selected as disodium cromoglycate (DSCG), a chromonic lyotropic liquid crystal with a nematic phase at room temperature [37]. This material provides an aqueous environment compatible with the hydrogen peroxide-powered Janus particles. DSCG was recently used to investigate the motion of bacteria in liquid crystal, where experiments showed the preferred direction of motion as parallel to the director [38]. Remarkably, at very low bacterial concentrations, Zhou et al. [38] introduced a novel class of active matter—the living liquid crystal—induced by the activity of bacteria moving in passive liquid crystal. These living liquid crystals, exhibit chaotic dynamics and motile topological defects, strikingly similar to the dynamic characteristics of microtubule-based active nematics [39]. Inspired by these results, we focused on the dynamics of individual active Janus colloids in nematic liquid crystal as a synthetic analog to the bacterial system [38]. Elucidating the fundamentals of Janus particle motion in a liquid crystal environment represents the first step towards studying their collective motion.

In our experiments, DSCG was aligned by treating the bounding glass substrates with polyimide SE-7511 to give a planar director orientation [40]. This configuration allowed easy observation of particle position and orientation using fluorescence microscopy. The liquid crystal phase was visualized using crossed polarizers on an optical microscope, revealing liquid crystal distortion profiles around the particles. Tracking the motion of Janus colloids revealed that liquid crystal anisotropy significantly influenced their trajectories. Particles tended to move parallel to the direction of average molecular ordering defined by the director. Our analyses reveal that motion profiles are characterized by a combination of anomalous diffusion and ballistic behavior. We also observed the Janus colloids rolling while moving in DSCG, despite no significant change in their direction of motion.

II. EXPERIMENTAL METHODS

A. Janus colloid preparation

Janus colloids were prepared by coating polystyrene microspheres with platinum (Pt) using electron beam (e-beam) deposition. Polystyrene fluorescent microspheres [FPM-4056-2 (4 μm spheres, Nile red) and FCM-10052-2 (11.8 μm spheres, Yellow), Spherotech, Inc.] were selected as the base particles. Before coating the colloids with a metal layer, they were first diluted in ethanol and then spread evenly on a glass slide. Once dried, the colloids were coated by e-beam deposition with 3 nm of Nickel (Ni) followed by 30 nm of Pt. The Ni layer was added to make the colloids magnetic in order to potentially explore how magnetic torques would affect the colloids’ behavior, although such experiments were not part of the current study. After coating, the non-metallic hemisphere of the particle remained fluorescent. The colloids were then removed from the slide using a small piece of wet lens tissue gently dragged over the slide to pick up the colloids. The tissue was then placed in a plastic 1.5 mL tube containing approximately 0.5 mL of de-ionized water and the tube was shaken and vortexed before removing the tissue. The finished particles were characterized using scanning electron microscopy (SEM) with a ZEISS Gemini SEM 500 instrument. Figure 1(a) shows an SEM image of two particles; the conductive hemisphere (lighter) is clearly visible for each (scale bar = 2 μm).

FIG. 1.

Materials and methods. (a) SEM image, captured with backscattered electron detection mode, of 4 μm-sized fluorescent microspheres (excitation 510 nm, emission 550 nm) half-coated with Pt. Scale bar = 2 μm. (b) Polarized microscopy image of uniformly aligned DSCG inside a pre-assembled cell of thickness 15 μm. Imaged using a full wave ($\lambda$) retardation plate, as indicated by the green line. Polarizer (P) and Analyser (A) directions are also indicated. Scale bar = 2 μm. (c) Cross-sectional schematic of the experimental cell containing DSCG, H₂O₂, and a Janus particle.

B. Liquid crystal cell preparation

We worked with two different Janus colloid diameters: 4.0 μm and 11.8 μm. From a stock solution containing colloids of each respective size, 5 μL was pipetted and resuspended in 30 wt% H₂O₂ aqueous solution to a final concentration of 20 wt% H₂O₂. This solution was then sonicated before mixing with liquid crystal. The Pt coating on the particle is quite reactive, generating significant O₂ bubbles. This problem is especially acute for the larger particles. The bubbles make colloid visualization difficult and form many air-liquid interfaces that influence particle trajectories, therefore, we aimed to minimize their production. To do so for each particle size, we optimized the wait time after the initial addition of H₂O₂ solution to the particle suspension: 1 h for the 4.0 μm Janus colloids and 3 h for the 11.8 μm Janus colloids.

After these respective wait times, 1 μL of a control particle solution (polystyrene divinylbenzene, DVB spheres, 6 μm diameter, 0.5 mg/mL) was also added to the Janus/H₂O₂ solutions. A dispersion of DSCG (Sigma Aldrich) at 25 wt% in de-ionized water was prepared, then subsequently diluted to 15.38 wt% (to be in the nematic phase) by adding 5 μL of the prepared Janus/H₂O₂ solution. This method produced a final 7.21 wt% concentration of H₂O₂, enough for activity, while minimizing oxygen bubbles. To further minimize the effect of bubbles causing advective flows, the Janus + DSCG + fuel solution was loaded into the pre-assembled cell after an additional waiting time. One hour for the 4 μm Janus colloids and 6 h for the 11.8 μm colloids. The cell was constructed from two parallel glass plates (20 mm x 10 mm) separated by 15 μm spacers. The interior surfaces of the glass plates were pre-treated with polyimide SE-7511 [40] and rubbed with a velvet cloth to produce unidirectional planar anchoring. Figure 1(b) shows an optical microscope image of an uniformly aligned cell filled with DSCG. After loading, the cell was sealed with epoxy glue and placed flat on the microscope stage to minimize gravitational effects due to tilt. Proper cell sealing and placement on the stage were important to minimize drift during imaging. The absence of drift was verified using DVB control spheres. The loaded cells were imaged with polarized optical microscopy (POM) using a Leica DM2500P upright microscope with a HI PLAN 40x/0.65 objective lens. Videos were recorded at 1 frame per second over a period of 8 min using a color camera [Basler Ace 2.3 MP]. Figure 1(c) shows a diagram of the experimental cell.

C. Particle tracking and image analysis

Janus colloids were tracked spatially from fluorescence microscopy images using Trackpy—a python package which locates Gaussian bloblike features in video images, follows them through time, and records their trajectories [41]. Tracking yields $x$, $y$ coordinates for each colloid as a function of $i$, the frame number. The liquid crystal was imaged between crossed polarizers with a full wave retardation plate. Figure 1(b), shows an example image of an aligned DSCG nematic phase with the director, polarizer, and retardation plate directions indicated.

III. RESULTS AND DISCUSSION

A. Director distortions around Janus colloids

When a colloidal particle with well-defined surface anchoring is introduced in a uniform director field, the interplay of elastic and surface anchoring forces defines the director configuration surrounding the colloid [42].The inherent asymmetry and distinct surface properties of Janus colloids result in unique interactions with surrounding nematic, leading to novel and varied distortions in the director field. We used POM imaging to investigate the nature and scale of nematic distortions around the colloid. The different Janus configurations discussed here are (i) full moon—where the Pt metallic cap on one side is facing down, maximum brightness; (ii) half-moon—where both sides of the Janus are showing, half bright, and (iii) new-moon—where the metallic cap is facing up, minimum brightness. Figures 2(a)-2(c) show a 4 μm diameter Janus colloid in the full-moon configuration imaged under three different modes. The POM image [Fig. 2(b)] shows a weakly distorted nematic around the colloid. But close observation by inserting a full-wave retardation plate between the crossed polarizers [Fig. 2(c)] reveal slight yellow tinges around the particle, signaling that the nematic distortions on each side of the colloid are locally chiral. This chiral nature becomes a little more obvious in the half-moon configuration [Figs. 2(d)-2(f)]. As large colloids with similar anchoring cause longer-range distortions in the nematic director, making optical observations easier, we see the same local chirality with 11.8 μm diameter Janus colloid [Figs. 2(g)-2(o)]. Similar chiral distortions were observed by A. T. Martínez et al. [43] with passive DVB spheres having tangential surface anchoring in the same liquid crystal. These distortions are characterized by ‘twisted tails’ that extend away from the particle parallel to the far-field nematic director. This behavior can be understood because DSCG has an unusually small twist elastic modulus making twist distortions energetically cheap [44]. Unlike in conventional LCs, where spherical particles with tangential anchoring give quadrupolar distortions with large splay and bend distortions in the surrounding nematic. In DSCG the director twists, reducing the reducing the energies of splay and bend. This twist has handedness rendering chirality to the distortion. The type of twisted tail director configuration we see around the colloid (Fig. 2) is called class-1. In a class-1 configuration symmetry, the director spirals around the central axis in the same sense on opposite sides of the colloid. The director configuration resembles a chiral dipole, characterized as an elastic multipole with both quadrupolar and chiral dipolar terms. These results suggest that, despite the Janus particle’s inherently different hemispherical sides, anchoring around the colloid is overall tangential, albeit with different anchoring strengths between hemispheres.

FIG. 2.

Distortion around inactive Janus colloid inside an aligned cell at different Janus configurations. Microscope images of 4 μm Janus particle inside 18 μm thick cell in full-moon [(a)–(c)] and half-moon [(d)–(f)] configurations under different imaging modes. [(a), (d)] Fluorescence microscopy images. [(b), (e)] Crossed-polarized microscopy images. [(c), (f)] Crossed-polarized microscopy images with an inserted full wave retardation plate (green arrow). Scale bar = 5 μm. Microscopy images of 11.8 μm Janus particle inside 30 μm thick cell in full-moon [(g)–(i)], half-moon [(j)–(l)], and new-moon [(m)–(o)] configurations under different imaging modes. [(g)–(m)] Fluorescence microscopy images. [(h)–(n)] Crossed-polarized microscopy images. [(i)–(o)] Crossed-polarized microscopy images with an inserted full wave retardation plate (green arrow). Scale bar = 10 μm. Magnification 63x water immersion, NA = 0.9.

We quantified the twisted tail lengths for the particles in Fig. 2 using the method described in [43]. For the 4 μm Janus particle in full moon, the distortions were too weak to identify the intensity spread along the equatorial plane. Comparatively, for the half-moon, tail lengths are asymmetric on the sides and approximately 5.65 μm. Similarly in the case of the 11.8 μm Janus particle, the tail lengths are asymmetric on the sides and are approximately 30.00 μm, 43.51 μm, and 41.84 μm for the full-moon, half-moon, and new-moon configurations, respectively. The tail length can increase with particle size (consistent with [43]) and vary with the orientation of the Janus particle.

B. Motion of active Janus colloids in DSCG

To understand the role of liquid crystal in dictating the motion of active Janus particles we performed a series of experiments. First as controls, we studied their behavior in water under two different scenarios, 1) without any fuel, and 2) with fuel (7.21 wt% peroxide solution). With no fuel, the Janus colloids exhibit characteristic Brownian motion [Fig. 3(a)]. In the presence of fuel, the Janus colloids become active and exhibit enhanced Brownian motion with a directional component to their motion, as seen in Fig. 3(b) and in video S1 of the Supplemental Material [55]. These control results were compared with motion in the nematic phase of DSCG in the presence of fuel (7.21 wt% peroxide concentration), [Fig. 3(c)]. When introduced to a uniaxially aligned liquid crystal in the presence of fuel, the Janus colloids show ballistic behavior. The anisotropy of the liquid crystal phase guides the Janus particles to swim along the nematic director. In Fig. 3(c) and video S2 of the SM [55], the DSCG nematic director is oriented at 45° with respect to the analyzer, and we see the active Janus colloid self-propels in a direction close to that of the nematic director.

FIG. 3.

Motion analysis of 4 μm-diameter individual Janus particles. (a) Trajectory of an inactive Janus colloid in water recorded for 5 min at 50 fps. Scale bar = 5 μm. (b) Trajectory of an active Janus colloid in 7.21 wt% H₂O₂ solution captured for a duration of 10 min at 0.2 fps. Scale bar = 5 μm. (c) Trajectory of an active Janus colloid in DSCG inside an aligned pre-assembled cell recorded for 8 min at 1 fps. Scale bar = 10 μm. (d) Mean squared displacement (MSD) plots on a logarithmic scale of individual active Janus colloids moving in uniformly aligned DSCG inside a cell of thickness 15 μm and captured at 1 fps for 8 min. The solid magenta line shows the average (e) MSD plots calculated parallel and perpendicular to the director, averaged over 10 different active Janus colloids moving in DSCG inside an aligned cell of thickness 15 μm, captured for 8 min at 1 fps. (f) MSD plots of active Janus colloids in the presence and absence of DSCG recorded for 1 min at 20 fps.

The overall motion of active Janus particle in DSCG can be explained by considering the interplay between the particle’s intrinsic propulsion mechanisms, anisotropic viscous interactions with the director field, and hydrodynamic effects. To understand this, let’s consider an axis $\mathbf{J}$ which is the polar axis of the Janus particle and an axis $\mathbf{N}$ which is the axis of the director field around the colloid (a line connecting the “tails”).

Propulsion along the overall director is expected because of the viscous anisotropy of DSCG. Measurements of the diffusion constants for from Brownian motion of colloidal spheres [45] show that the ratio ${\sigma }_{\parallel }∕{\sigma }_{\perp }$ of the effective viscosity ${\sigma }_{\parallel }$ for displacements along the director to the viscosity ${\sigma }_{\perp }$ for perpendicular displacements is significantly smaller than 1, ${\sigma }_{\parallel }∕{\sigma }_{\perp }$ = 0.66. In our case, we observe ballistic motion in a direction slightly different to the liquid crystal director. This is because, as the $\mathbf{J}$ axis rotates away from the $\mathbf{N}$ axis due to the propulsion forces generated by the particle and the hydrodynamic flows around the particle, lateral displacements from the director occur. The restoring elastic force arising from the distorted director field tends to align the particle’s motion with the director. The nature of the active Janus particle trajectory is discussed in more detail in Sec. III D.

Motion profiles of the Janus particles were analyzed by plotting the mean square displacement (MSD) of the particles as a function of time (see Appendix B). Figure 3(d) shows MSD plots for 10 different 4 μm-diameter active Janus particles (represented by dotted lines) moving in DSCG. The particles were each tracked for 8 min at 1 frame per second (fps) and their motion plotted on a log-log scale. The solid magenta line represents the average of all 10 trajectories. A linear fit to this average produces a slope of 1.98 ± 0.01 indicating that motion is predominantly ballistic throughout the time of observation. Within a lag time of 480 s (period of observation), 4 μm diameter Janus particles were observed to travel distances ranging from 24 μm to 90 μm. These differences are likely due to variation in the amount of fuel around different particles. Once the Pt + H₂O₂ reaction fuel fully depletes, the Janus particles behave like passive colloids.

C. Anisotropic and anomalous motion of active Janus colloids

To investigate the directional preference of a single active Janus colloid, we calculated MSDs parallel and perpendicular to the director. MSDs, averaged over 10 active Janus colloids, clearly show that motion parallel to the far-field director is substantially longer than motion perpendicular to the director [Fig. 3(e)]. We can understand this behavior by considering the anchoring around the colloids which causes asymmetric director field distortions (Fig. 2) and an effective anisotropic viscosity. When ‘dressed’ by the surrounding director field, the Janus particle suspended in DSCG effectively becomes an anisotropic twisted composite object, with different effective viscosities for motion parallel and perpendicular to the director.

Such viscoelastic effects have been shown to favor particle displacement along the director [45]. In our case, due to misalignment of the $\mathbf{J}$ and $\mathbf{N}$ axes, we did observe some translation perpendicular to the director, but ${\text{MSD}}_{\parallel }$ is significantly larger than ${\text{MSD}}_{\perp }$.

In addition to their anisotropic motion over long timescales, we also observed that the colloids exhibit anomalous diffusion when the time step between measurements of particle position was 0.05s. In this experiment, five different active Janus colloids (4 μm in diameter) moving in DSCG inside a cell of thickness 15 μm were recorded for 5 min. Trajectories for analysis were obtained by splitting the 5 min long video into tracks of 1 min durations. A time-averaged MSD calculation was performed on each 1 min track. Then, an ensemble average was taken by averaging over these 1 min trajectories. As Janus particles are moving in an anisotropic environment, motion parallel and perpendicular to the director were calculated separately [Fig. 3(f)]. The MSD curves reveal two regimes of motion—a regime with a smaller slope at short lag times that slowly transitions into a steeper slope at longer lag times. The crossover time for ${\text{MSD}}_{\parallel }$ corresponds to ${\mathrm{t}}_{\parallel }$ = 0.79 s with a slope of 1.14 ± 0.02 at $\mathrm{t}<{\mathrm{t}}_{\parallel }$ and a slope of 1.95 ± 0.01 at $\mathrm{t}>{\mathrm{t}}_{\parallel }$. The crossover time for ${\text{MSD}}_{\perp }$ is at ${\mathrm{t}}_{\perp }$ = 1.39 s with a slope of 1.09 ± 0.01 at $\mathrm{t}<{\mathrm{t}}_{\perp }$ and 1.85 ± 0.01 at $\mathrm{t}>{\mathrm{t}}_{\perp }$. However, the same MSD curve calculated in water (with H₂O₂ but no liquid crystal), showed no crossover between slopes [Fig. 3(f), black crosses].

When a particle moves in a nematic liquid crystal, it distorts the director field and the associated velocity field. The coupling between the particle’s motion and these fields introduces an intrinsic memory effect, where past disturbances influence the particle’s subsequent motion. As the particle moves, it creates perturbations in the director which relax over characteristic time, $\tau \sim \frac{\eta l^2}{K}$, where $l$ is the characteristic length of the director distortion relaxing back to equilibrium, $\eta$ is the rotational viscosity, which we estimated to be 0.2 Pa.s [46] and $K$ is the Frank elastic constant. These dynamic interactions can result in anomalous diffusion. Assuming $l$ = 1 μm for the class 1 case, an active Pt-coated Janus sphere with a configuration-specific distortion profile and a DSCG twist elastic constant at 23 °C [44], $\tau$ was found to be 0.29 s. This $\tau$ value is slightly less than the crossover time, below which anomalous diffusive behavior was observed. Turiv et al. [45] reported sub-diffusive behavior for tangentially anchored silica spheres in DSCG with crossover times 0.37 s and 0.46 s for motion parallel and perpendicular to the director, respectively. In our case, we observe a slight super-diffusive behavior. This is because of activity affecting particle dynamics over short time scales. Our report of a crossover time for motion perpendicular to the director differs from that reported by Martínez et al. [43] in that their particles were in the bulk, whereas in our case they are close to a substrate. When particles are close to a substrate, the relaxation time of deformations in the surrounding nematic is strongly affected by cell thickness and can be written as ${\tau }_h\sim \frac{{\eta }_{\text{eff}}{h}^2}{K\pi }$, where $h$ is the thickness of the cell. For our cell of thickness 15 μm, ${\tau }_h$ is approximately 3 s, which is slightly above the time we observe anomalous behavior for motion parallel and perpendicular to the director.

D. Rolling

Video analysis reveals the rolling behavior of 4 μm active Janus particles as they move in DSCG (Fig. 4). In this figure we contrast the typical behavior of a 4 μm particle with a larger, 11.8 μm diameter particle. The 4 μm Janus particle undergoes rolling, exhibiting different moon phases [Fig. 4(a)], with the intensity graph showing peaks and valleys. A peak corresponds to a full-moon configuration and a valley indicate a new-moon configuration. The 11.8 μm particle, on the other hand, does not roll, with a constant intensity profile [Fig. 4(b)]. Before discussing these results, it is important to consider possible mechanisms for particle rolling.

FIG. 4.

Rolling of small Janus particles while moving in a uniformly aligned DSCG. (a) Fluorescence microscopy images capturing the moon phases of a 4 μm active Janus particle inside a 18 μm cell for a duration of 5 min and corresponding intensity plots of the of the same Janus particle demonstrating rolling. Scale bar = 5 μm. (b) Fluorescence microscopy images of 11.8 μm active Janus particle inside a 30 μm cell recorded over a duration of 20 min and corresponding intensity plot demonstrating the absence of rolling over the period of observation. Scale bar = 10 μm. (c) Composite image showing the curved trajectory of a rolling 4 μm Janus particle inside a 15 μm DSCG cell, captured every 10 s for 220 s with the $\mathbf{J}$ vector indicated as a red arrow. Scale bar = 10 μm. (d) Graph of $\Delta \theta$, the deviation in direction of motion with respect to the nematic director orientation, and (e) speed, with time for the particle depicted in (c), (f) Histogram of $\Delta \theta$ values that active 4 μm Janus particles take while moving in DSCG observed for a duration of 220 s at every 10 s interval, calculated for seven different particles.

Active particles move via self-generated phoretic mechanisms in the presence of fuel and the motion occurs, in general, due to the interaction of self-generated fields with the interfacial boundary region of the particle. The particle/liquid crystal interfacial boundary layer is a very thin region of the surrounding fluid much less than the size of the particle. A self-generated field around the particle can consist of a solute concentration gradient, set up by the catalytic chemical reaction on the Pt side, or an electric field generated by proton currents at the Pt end that emanate from the vicinity of the equator and end near the pole [35]. These fields create a local pressure imbalance leading to an effective slip velocity along the interfacial boundary layer, driving fluid flows [33,47-49]. The self-phoretic effective slip velocity, $v_s$, induced by the solute concentration field, $C$, along the interface can be written as $v_s=\mu {\nabla }_{\parallel }{C}_s$ [47]. Here, $\mu$ is the characteristic diffusiophoretic mobility and ${\nabla }_{\parallel }{C}_s$ is the tangential gradient of $C$ at the outer limit of the boundary layer. While considering self-propulsion through electrokinetic effects, $\mu$ is replaced with the zeta potential, $\zeta$, and $C$ with a self-generated electric field, $\psi$ [49]. Any spatial nonuniformity in the mobility constant caused by fabrication defects can lead to a rotational component of the effective slip velocity. The particle velocity will be proportional to the sum of the mobilities, and angular velocity is proportional to the difference between the mobilities on the two sides [50]. We expect the particle to turn and align its axis of symmetry along the field gradient. This would mean that after adopting this configuration (e.g., PS side first), rolling should cease. In our case we observe something different. The particles continue to rotate throughout their tracked trajectories and do not reach an orientational equilibrium [Fig. 4(c)]. To try to explain this phenomenon, we can consider the effects of a spatially varying fuel concentration field. Such a variation would add polar and azimuthal components to the local slip velocities at the particle surface, greatly complicating determination of the particle mobility [49,50].

In water [Fig. 3(b)], Janus particles can exhibit spiral trajectories, due to their non-uniform surface properties [51,52]. When active Janus particles are placed in an aligned liquid crystal their hydrodynamic interactions with the nematic are likely to be of the pusher type, as follows from the theoretical model of Lintuvouri et al. [53]. On top of this, the off-center propulsive force and the elastic restoring force from the liquid crystal environment can lead to a slightly curved trajectory, or linear propulsion in a direction slightly different to the nematic director. In Fig. 4(c), we can see a Janus particle moving in aligned DSCG, with the nematic director indicated by a double headed arrow. The particle begins its path in the half-moon configuration, then rolls to adopt a full-moon configuration and continues forward. The corresponding deviation in particle trajectory with respect to the nematic director ($\Delta \theta$) is plotted in Fig. 4(d). While in the half-moon configuration, $\Delta \theta$ is high, then the particle rolls, and $\Delta \theta$ drops to approximately 10°. We observe a rise in velocity [Fig. 4(e)] as the Janus particle trajectory comes more in alignment with the nematic director. Figure 4(f) shows a histogram of $\Delta \theta$s calculated for 7 different active Janus particle trajectories. The most common trajectory direction, $\Delta \theta$ was in the range 4.4°–8.8°.

This behavior is very non-intuitive, based on the simplified picture of Janus particle propulsion. It should be noted that the behavior of each Janus particle moving in DSCG is, to some extent, unique. Although general behaviors can be obtained, the surface chemical properties of each particle can vary from spot to spot. Adding to this, possible spatial non-uniformity of the fuel environment around the colloid, makes each Janus particle different. In addition to these local effects, global inhomogeneities in the fuel concentration field further complicate the behavior. Under these conditions, rolling behavior can be considered stochastic over long timescales. In Fig. 4(c) and in video S3 of the SM [55], we see a fascinating result: the particle rolls with no change in its direction of motion. Considering that particle propulsion occurs, in general, due to catalytic breakdown of H₂O₂ on the particle’s Pt side [33-36], there are two important things to note. First, surface chemical activity—how quickly a concentration gradient can be set-up by the chemical reaction—and second, the surface phoretic mobility—how effectively the surface interacts with this gradient to generate fluid flows [52]. S. Ebbens et al. [35] reported particle motion as a combination of neutral and ionic diffusiophoresis, as well as electrophoretic effects whose interplay can be varied through ionic effects such as pH and salt concentration. DSCG is a disodium salt and high salt concentrations are known to reduce the turnover rate of H₂O₂ [35]. The characteristic diffusion time of reaction products (O₂,H₃O⁺) around the Janus colloid (4 μm) is ${\tau }_d=\frac{R^2}{D}$, where $D=\frac{{\mathrm{k}}_{B}T}{6\pi \eta a}$ depends on the rotational viscosity of DSCG and the radius of the product particle, $a$ [54]. ${\tau }_d$ for O₂ and H₃O⁺ is approximately equal to 0.55 s and 0.37 s, respectively. These, and our results, suggest that while the rotational configuration of the Janus takes time to react to the fuel and set-up a gradient, the existing gradient may give enough thrust to push the particle forward.

Finally, we observed that the large colloids (11.8 μm in diameter) don’t roll while moving in DSCG [Fig. 4(b)] and tend to move in a direction closer to the director. This can be understood as they create a larger elastic distortion in the surrounding liquid crystal director field (Fig. 2) and in general move much more slowly. In addition, because their distortion profiles are sensitive to particle orientation, rolling for larger particles becomes increasingly energetically expensive.

This behavior might also be related to elastic levitation, as described by O. P. Pishnyak et al. [17], where elastic repulsion from the bounding substrates keeps the colloid in the nematic bulk, mediated by director distortions. A large colloid is repelled more strongly from the substrates than a small colloid, thus a larger colloid is often levitating near the midplane of the cell. This leads to just a small difference in shear stresses above and below the colloid and therefore suppresses rolling [55]. Colloidal particles in liquid crystal cells are known to interact elastically with the bounding plates [17,56]. The gap between the spherical surface of a Janus sphere and the flat plates above and below form a double-wedge geometry and thus impose director gradients. In the case of tangential anchoring at the colloid and planar anchoring at the plate, the only region where the director is not distorted is along the line that connects the bottom and the top of the Janus sphere to the neighboring plate; other regions elastically adjust to the tilt of the colloidal surface with respect to the plates. The director gradients become stronger as the particle approaches a flat plate, which means that the sphere is elastically repelled from the plate [17,56]. The repulsion is stronger for larger particles [56]. Experiments with two homeotropically anchored spheres, one large and one small, demonstrate that the larger particle “levitates” closer to the midplane of the cell as compared to a smaller particle, which finds itself closer to the bounding plate [56]. The reason is that the elastic repulsion force grows as r⁴ with the radius r of the colloid [17,56], which is faster than the gravity force proportional to r³. We expect a similar effect for the tangentially anchored colloids, since the repulsive force between two quadrupolar colloids scales as r⁶ [57-59] and the sphere-plate repulsion can be modeled as an interaction of a colloid with its mirror image [17]. Since the smaller colloid is likely to be displaced from the midplane of the cell by gravity, one expects an inverse of the Quincke effect already described for colloids moving in a liquid crystal [20,21]. Namely, in the Quincke effect, an electrically driven rotation of a sphere results in its translation when it is not strictly in the midplane of the cell because of the hydrodynamic interactions with the boundaries [20,21]. In our case, the self-propelled translating colloid is expected to rotate because of the difference between the hydrodynamic interaction with the top and bottom plates, thus, the inverse-Quincke effect.

IV. CONCLUSIONS

In this paper, we report the directed movement of synthetic, chemically powered active Janus particles in a liquid crystal nematic phase. Examination of individual particle trajectories reveals some surprising behaviors. Particle motion is governed by liquid crystal anisotropy, but is also strongly influenced by asymmetries in the individual particles. The particle motion, while close to ballistic over longer timescales, is influenced by the significant, orientation-dependent director distortions around the particle. In addition, nonuniformities in metallic cap coatings and fuel concentration have non-trivial effects on particle propulsion direction and anomalous rolling behaviors. Achieving passive control over simple synthetic active particles is an important goal in the field of active matter, and one that might lead to new ways of achieving collective control. By studying the basic behaviors of anisotropic fluid on single Janus colloids our work helps to lay the foundation for these future directions.

APPENDIX A: TWISTED TAIL LENGTH

We used crossed-polarized microscopy images to calculate the twisted tail lengths associated with the 4 μm and 11.8 μm Janus colloids (Fig. 2). Prior to calculation, the images were converted into 8-bit. As demonstrated in the Supplemental Material from Ref. [43], an intensity profile was plotted for a line along the equatorial plane. The resulting decaying intensity tails were fitted with the Gaussian peak equation,

$$I=I_0+\frac{A}{w}\sqrt{\frac{2}{\pi }}\text{exp}\left(-2{\left(\frac{x-x_c}{w}\right)}^2\right),$$

and the length was estimated using the Full Width at Half Maximum (FWHM) concept. In the above equation, the width parameter $w$ is related to the standard deviation $\sigma$ by $\sigma =\frac{w}{2}$. This gives a FWHM = $\sqrt{2\text{ln}2}w$ representing the length of intensity spread around the colloid. We take the average of the decaying intensity tails on the two sides to find the tail length.

APPENDIX B: MEAN SQUARED DISPLACEMENT (MSD)

MSD curves for each individual particle track defined by {$\mathrm{x}(\mathrm{i})$, $\mathrm{y}(\mathrm{i})$} were calculated using the method described in Ref. [60]. The MSD, for a given time lag $t$, is defined as the average over all pairs of points separated by that time lag using

$$\langle {\Delta }^2(t)\rangle =\frac{{\sum }_{i=1}^{N_i}[{(x(t_i+t)-x(t_i))}^2+{(y(t_i+t)-y(t_i))}^2]}{N_i}.$$

In this definition, $t_i$ is the time for the $i^{th}$ image in the track and $N_i$ represents the number of images over which the average is calculated.

APPENDIX C: INTENSITY ANALYSIS

The fluorescence microscopy videos show a bright feature (the fluorescent side of the Janus colloid) moving in time against a black background. To characterize the orientation of the colloid we took advantage of the fact that only one hemisphere (the nonmetallic side) is fluorescent. By observing a single particle, the total integrated intensity in every video frame was obtained using Trackpy [41]. Maximum intensity in this case corresponds to a full-moon configuration—when the metallic cap is facing down and the fluorescently labeled polystyrene half is facing up. A minimum intensity corresponds to a new-moon configuration—when the metallic cap is facing up and the fluorescently labeled polystyrene half is facing down. To normalize these integrated intensities, the intensity of each frame in a video was divided by ($I_{\max }$), the maximum-observed integrated intensity.

APPENDIX D: $\Delta \theta$ ANALYSIS

To measure how much the direction of motion of active colloid deviates from the nematic director orientation, we calculated $\Delta \theta =\Delta {\theta }_{LC}-{\tan }^{-1}(\frac{dy}{dx})$, where $\Delta {\theta }_{LC}$ is the nematic director orientation (in our case 45°) and $dx$ and $dy$ are the displacements of active Janus moving in liquid crystal along $x$ and $y$ directions, respectively.