Assembly and transport of nematic colloidal swarms above photo-patterned defects and surfaces

We investigate the dynamic assembly and swarm translocation of anisometric colloidal particles dispersed in a nematic liquid crystal and driven above a photosensitive surface. We use liquid crystal-enabled electrophoresis to propel these particles via an alternating electric field perpendicular to the sample cell. By manipulating the anchoring conditions on one surface of the experimental cell, we obtain a spatially extended spiral pattern of the liquid crystal orientation that induces the dynamic assembly of a rotating colloidal mill. This structure can be transported by translocating the topological defect above the photosensitive surface. We complement our findings with a theoretical model that captures the basic physics of the process, by formulating an analytic equation for the director field above the surface. Our reconfigurable nematic assemblies may be used as a test bed for complex swarming behaviour in biological and artificial microscale systems.


Introduction
The formation of coherent structures from a disordered array of interacting, self-propelling elements is currently an active research topic in different fields of science [1][2][3][4]. The interest arose from both a fundamental and an applied point of view. On the one hand, monitoring and controlling the transport properties of large populations of artificial micromachines may help understanding the underlying mechanisms that govern spontaneous formation of coherent structures from self-propelled units, as observed in disparate biological systems [5][6][7][8][9][10][11]. On the other hand, these investigations have also direct technological applications related with the field of drug delivery in microfluidic and vein networks. In particular, examples include the controlled release of biochemical cargos attached to functionalized colloidal particles [12][13][14][15], the miniaturization of simple operations in lab-on-a-chip devices [16][17][18][19][20] or the realization of different functional tasks at the microscale [21][22][23].
The formation of particle swarms from ensembles of artificial prototypes driven by an external field has been recently reported by different research groups who use electric [24,25], optic [26] or magnetic fields [27,28]. In most of the cases, the driven particles are dispersed in an isotropic fluid such as water, and are propelled through lithographic structures or assembled along a circular confinement. An alternative approach that is gaining interest in the community is the use of photosensitive surfaces in a nematic liquid crystal matrix, where the dispersed particles can be driven by an electric field along predefined tracks [29][30][31][32][33]. The use of an anisotropic medium for the particle motion may provide different advantages to self-propelling particle systems. For example the possibility of controlling the mean molecular orientation of the dispersing medium via an external field, the ability of switching from an anisotropic to an isotropic medium by varying the system temperature, or finally the absence of hard walls or confinements that may alter the particle motion due to steric constraints.
Motivated by these prospects, in this article we investigate the dynamics of an ensemble of anisometric colloidal particles that are dynamically assembled into a rotating mill organized around a single topological defect of a liquid crystal on a surface that imposes planar alignment. In addition, we study the translocation of the colloidal swarm when the topological defect is relocated on the surface. In this case we observe that the condensed pattern disassembles, forming a polarized phase that moves along the director field. To understand the system dynamics, we theoretically determine the director field in contact with the photosensitive surface, and show that a simple simulation scheme describes well the observed phenomena. Our colloidal model system may be used to explore the rich many-body physics of collective organization and swarming in microscale driven systems, and may also inspire the use of similar strategies to control and guide biological entities along predesigned paths [33,34].
The paper is outlined as follows. After the description of the experimental system, in section 2, we discuss the propulsion of a single colloidal particle, section 3. In section 4 we describe the formation of the experimentally observed mill pattern and quantify its properties, which is then followed in section 5 by a comparison with a simple numerical model. An analytic expression for spiral pattern is derived in the appendix. The process of translocation is discussed in section 6, in which we compare the results of the measurements with the predictions of the numerical model. We summarize the results and draw the conclusions in section 7.

Experimental system
We prepare thin cells enclosing a solution of colloidal particles dispersed in a nematic liquid crystal (MLC-7029, Merck) characterized by a negative dielectric anisotropy Δò=−3.6 (at 1 kHz). As shown in figure 1(a), these cells are composed by two microscope slides with an area of 15×25 mm 2 , 0.7 mm thickness, and coated with a thin layer of indium-tin oxide (ITO) with a sheet resistance of ∼100 Ω sq (VisionTek Systems). The plates are cleaned by sonication in a 1% Micro-90 (Sigma-Aldrich) solution, rinsed with ultrapure water (18.2 M cm W , Millipore Milli-Q), and dried at 80 • C for 30 min. Afterwards, the plates are introduced in a plasma surface treatment equipment (ZEPTO, Diener Electronic). One of the plates is functionalized with a photosensitive surface in a toluene medium (peptide synthesis grade, Scharlau) by mixing two silanes in a ratio 5:1, (3-aminopropyl)triethoxysilane (APTES, Sigma-Adrich) and an azosilane compound, (3)-4-(4-((4-octylphenyl) Figure 1. (a) Schematic of the experimental cell showing one anisometric particle enclosed between two ITO plates and dispersed in a nematic liquid crystal of negative dielectric anisotropy. An AC voltage is applied between both plates. The colloid can be characterized by two lateral dimensions (d x , d y ). (b) Sequence of microscope images showing the controlled propulsion of one particle when subjected to an AC field with amplitude E=0.76 Vμm −1 and frequency f=10 Hz. The corresponding video (Video1) can be found in the supporting information which is available online at stacks.iop.org/NJP/20/075006/mmedia. (c) Speed, v, versus amplitude, E, of the applied electric field for an individual anisometric particle. Scattered points are experimental data, continuous red line is a quadratic fit to the data. diazenyl)phenoxy)-N-(3-(triethoxysilyl)propyl)butanamide (GalChimia). Butylamine is added as catalyzer. The other plate is coated with a thin layer of a polyimide compound (0626 from Nissan Chemical Industries), after plasma surface activation, in order to achieve a strong homeotropic anchoring of the liquid crystal at the surface. Both plates are disposed with the ITO facing inwards and are glued together with a separation of about 23 μm using Mylar spacers (Goodfellow). The anisometric particles used in this work have a pear-like shape with two connected spherical lobes (Magsphere Inc). The particles are made of polystyrene and have lateral dimensions d x =4 μm, d y = 3 μm, see figure 1(a).
The experimental system is composed of an upright optical microscope (Nikon Eclipse 50iPol) containing two collimated epi-illumination LED light sources of wavelength 455 nm (Thorlabs M455L3, 900 mW) and 365 nm (Thorlabs M365L2, 190 mW). Brightfield illumination for microscopy was performed with a red longpass filter (Lambda 645 nm) to avoid perturbing the azosilane coating [31,[35][36][37]. The microscope objective is also used to focus light coming from the LEDs onto the sample cell projecting a Gaussian intensity profile. Images of the particle dynamics are recorded with a CMOS camera (AVT Marlin F-131B). The external, alternating current (AC) is applied with a voltage amplifier (TREK model PZD700) controlled by a function generator (Agilent DSOX2002A).

Single particle transport
We start by describing the transport properties of an individual anisometric particle. Figure 1(b) illustrates different experimental snapshots of a single particle driven in the liquid crystal (LC). To understand the mechanism of motion, one has to consider the experimental geometry, as schematically depicted in figure 1(a). In the absence of an external field, the homeotropic alignment at the surface of the two cells forces the nematic director n to point along the z-axis. Since the LC has a negative dielectric anisotropy, under the applied field n orients perpendicular to the field direction, say along the x-axis. As a consequence, also the main axis of the dispersed particle aligns along n. Inside the LC, the colloidal particle distorts the nematic matrix creating topological defects around its surface [38]. For our pear-shaped colloids we find two point defects located at both apexes of the particle, in the form of a non-symmetric 'double-boojum' [39].
The applied AC electric field induces electroosmotic flows around the particles. The anisometric shape of particles breaks the symmetry of these flows leading to net phoretic propulsion at speed v [29]. Given the confined geometry, the motion is quasi-two-dimensional, and thermal fluctuations of the particle are negligible due to its relative large size and the high viscosity of the LC medium. Most particles move with their large lobe ahead, and exhibit a ballistic-like trajectory as shown by the sequence of images infigure 1(b) (see also Video1 in the supporting information). Further, we note that, since the AC field is applied perpendicularly to the plane of motion, linear (DC) electrophoretic effects resulting from the attraction toward the two electrodes are also negligible.
The particle phoretic speed v can be controlled by varying the amplitude E of the applied AC field, as shown in figure 1(c). Similarly to [29], we detect a quadratic dependence of v on E typical of electroosmotic flows developing around the particle, v ). Note that the propulsion velocity is independent of the polarity of the field, which enables the application of AC fields. From the fit to the experimental data, we determine the threshold field for motion, E 0 =0.16±0.06 Vμm −1 , figure 1(c). Here E 0 represents the minimum field required to generate enough ionic flow around the particle to provide propulsion.
In the collective particle dynamics we will keep the field parameters fixed to E=0.76 Vμm −1 and f=10 Hz, which results in an averaged single particle speed of v=15.1 μms −1 . Finally, we note that at a fixed amplitude, the frequency dependence v=v( f ) is even more complex, due to the nature of the ionic motion within the LC, as reported in previous works [29,40]. We, therefore, keep the frequency constant through all experiments.

Dynamic assembly on a spiral pattern
Controlling the orientation of the director, n, allows steering the trajectory of the colloidal particles that are propelled in the LC by the applied AC field. We investigate the collective dynamics of the anisometric particles above a spiral pattern characterized by a central topological defect, which is shown in figure 2(a) under crossed polarizer and analyzer. This complex pattern was obtained with the following steps. First we use UV light to irradiate a large, circular region of the surface coated with the azosilane (see section 2), which forces the dye molecules there to be in the cis configuration, i.e. to orient parallel to the surface. Thus, the LC director n acquires a planar anchoring on this area. After that, we irradiate a smaller central spot with blue light, forcing the azosilane layer to revert to the trans-configuration. In this situation, the director n acquires a homeotropic alignment inside and outside a circular corona of planar molecules. When the AC field is applied, the LC molecules with planar alignment in this corona extend inwards and outwards, and n forms a spiral pattern with a central topological defect of charge +1, figure 2(a). Indeed, the LC used here is characterized by a bend elastic constant smaller than the splay one, and thus bend distortions are favored during orientation of the LC matrix.
In figure 2(b) we show the dynamic assembly of an ensemble of anisometric particles following this spiral pattern. As shown by the corresponding Video2 in the supporting information, once the external field is applied, particles move towards the topological defect, which appears as a small black spot at the center of the image. Particles follow a spiral trajectory, and once close to the defect they start orbiting around it at a constant distance. As more particles arrive from outside, the density of the rotating cluster increases, until reaching a size of N≈1000 particles after t=113 s, figure 2(b). The cluster rotates as a solid body around the topological defect, and is completely dynamic in nature, since its structure can be easily disassembled by switching off the field or reducing its strength below E 0 (see section 3). The cluster's rotational motion follows the chirality of the underlying pattern, which is randomly selected upon irradiation, while the particles located in the central denser region keep a constant inter-particle distance larger than the particle diameter. This indicates the presence of a long range repulsive interaction that may arise from the electrostatic dipoles induced by the applied AC field. Another interesting feature is the formation of chains at the periphery of the core. This chaining arises from the LC-mediated quadrupolar elastic interaction induced by the double-boojum defects located at both apexes of the particles. The chains tend to spiral at a certain angle to minimize the elastic energy and ensure the relocation of the incoming particles [38].
In order to analyze the dynamic state, we show in figure 2(c) particle image velocimetry analysis of the colloidal pattern. Indeed, we find that, after a short transient, the cluster of particle organizes into a rotating mill pattern, with an almost constant linear velocity everywhere in the cluster and, thus, with an angular velocity that decays with the distance to the center of the cluster.

Theoretical description of the colloidal mill
To model the dynamics of particles assembled in the colloidal mill, we apply the minimal scheme able to capture all the general trends of the dynamic assembly. We consider particles to be spherical with a prescribed phoretic speed and restrict our consideration to the two-dimensional (x, y) plane, which is parallel to the electrodes. The overdamped motion of an ensemble of N colloidal particles with positions r i (i=1, ..., N) in a LC matrix is described by the following equations of motion where the first term on the right hand side of equation (1) describes the phoretic propulsion of individual particles whose direction is along the local LC director, the second one is a force that accounts for interactions with other particles, and z || denotes the friction coefficient of the particle moving along the director field. This model implies that particles propel strictly along the director field n r t , , and the strength of propulsion generally depends on the field parameters (E, f ). Since we are only interested in describing the particle dynamics at fixed f and E, we set the strength of propulsion as v v 1 0.2 , 2 i 0 where v 0 is the constant deterministic mean speed of the particles and ξ is a Gaussian random variable with zero mean and unit variance. The noise distribution has been measured in experiments, and it is about 20% of the mean particle speed, which is reflected on the weight of this term. This distribution resulted from the presence, in ) is shown in figure 3(a). The spiral pattern is characterized by the parameters α 1 =−π/5, α 2 =−π/4 (see the appendix), and the inner and outer radii are set to r 1 =2 μm and r 2 =1000 μm, respectively. We note that although the pattern is derived for the elastic constants K S =16. represents a more accurate solution.
In figure 3(b) we show the result from a numerical simulation, where N=800 particles are assembled above the spiral pattern (see also the corresponding Video3 in the supporting information). In order to reproduce the dynamic features observed in the experimental system, we introduce long-range repulsive interaction between the colloidal particles. We have checked and found that different long-range repulsion potentials seem to work fine to describe the dynamics of the colloidal assembly process. To be specific, in the control simulations we stick to the repulsive potential of the form U r r 1 , which leads to the repulsive force exerted on particle i by The constant C 1 z || is taken as 2000 μm 5 s −1 in the simulations. We note that this law of repulsion is a justified assumption because the particles are polarized in the applied AC field and acquire an electric dipole moment [41,42]. This dipole is always perpendicular to the electrodes and to the plane of particles, and therefore all the dipole moments stay parallel to each other. As a result, they are expected to repel according to the above dipolar interaction force.
To account for a finite size of particles, we also introduce a steep short-range repulsive potential of the form U r r r 1 4 , which is applied at distances r r 2 ij 0 1 24 s < = and is vanishing otherwise. Here, σ is an effective diameter of the particle. At short distances, r ij <r 0 , this potential leads to the strictly repulsive force In our study, we set C 1 m s and σ=5 μm. Note that σ is slightly larger than the actual size of particles, d x , d y . To better replicate the experimental observations, we assume that defects in the liquid crystal in the vicinity of particles prevent the direct contact of particles, leading to a larger effective hard core distance.
Finally, we also account for quadrupolar interactions, which cause a weak chaining of particles depending on the underlying pattern. The corresponding force can be represented as [43] F r where ϑ is the angle between the far-field orientation of the nematic director n and the vector connecting the center of particles, r ij . In the simulations, we set C 500 m s . Note that figure 3(b) reflects the consequence of these interactions leading to spiral chaining at the periphery of the hard core.
The comparison between the experimental and numerical data is shown in figure 4, where we plot different observables related with the dynamics of the colloidal mill as obtained in experiments (figures 4(a), (b)) and simulations (figures 4(c) and (d)). In the former case, we use video microscopy and tracking routines [44] to extract the positions (x i , y i ) of each particle i, and analyze the radial and azimuthal particle velocities averaged over the ensemble, , as measured in polar coordinates from the central defect. Both quantities are shown in figure 4(a) for the rotating mill, and compared with the numerical case in figure 4(c). Both types of data present the same trend, with a vanishing radial velocity when the mill is formed and an azimuthal component that approaches −1. Both quantities are negative since the particles are moving toward the center thus against the radial direction, outwards by definition (v ρ <0), and in a clockwise sense (v θ <0). The main difference between experiments and simulations refer to transient states determined by initial conditions, which are clearly different in both cases. As expected, the convergence to the final state shows similar trends in the evolution of v ρ and v θ .
Apart from the instantaneous velocity, we characterize the dynamics in terms of two other order parameters that describe, respectively, the degree of particle alignment and collective rotation. The first one is the normalized polarity P of the pattern, defined as: with v i being the velocity of particle i at position r i from the center of mass of the ensemble, and the summation is performed over all particles in the ensemble. The second parameter is the normalized angular momentum of the particle ensemble, defined as As described in [45], both order parameters are necessary to quantitatively determine whether the collection of propelling particles is in a coherent flock state (high value of P, low M) or in a single-mill state (low P, high M). As shown in figures 4(b) and (d), we find that in the experiments and simulations both assemblies show a negligible polarity P, while the angular momentum M increases toward unity.

Colloidal translocation: experimental and numerical results
While we just described the formation of a colloidal mill, the photosensitive surface also allows to transport these dynamic states everywhere within the cell. For a given assembly, particle speed due to liquid crystal-enabled electrophoresis (LCEEP) drops down to zero at electric field frequencies of about 50 Hz. For greater stability, we choose 1 kHz as the frequency to freeze the assembly in place, while we pull the topological defect from the initial position (center of the formed colloidal mill) for a certain distance. This translocation is performed through a disclination line that forms as the UV light spot is dragged between the initial and the final destination on the photosensitive surface. We demonstrate this feature in the sequence of images in figure 5(a), where a mill pattern is translocated a distance of 175 μm (see also Video4 in the supporting information). During translocation, the colloidal swarm moves as a polarized pattern developing a leading edge that is followed by all other particles. These dynamic features are reflected in figure 5(b), where we evaluate the corresponding two order parameters M and P. A colloidal mill is initially assembled under f=10 Hz. It is subsequently held in place by switching to f=1 kHz and a UV light spot is dragged from the center of the mill to a destination spot ( figure 5(a)). When the frequency is switched back to 10 Hz, LCEEP becomes active again, dismantling the assembly as particles are driven towards the destination spot. Both order parameters rapidly exchange values raising the value of P and reducing the rotational motion (thus M). The particles move at a constant speed towards the translocated topological defect, raising the value of P to unity, while M vanishes. After 150 s, enough particles have gathered around the destination spot for the mill pattern to form again, thus increasing the corresponding value of M and reducing the degree of alignment of the propelled particles, thus lowering P.
In order to reproduce numerically the experimental situation illustrated in figure 5, we perform a procedure similar to the experimental one. In particular, the operation of transferring one defect from one initial position to a new one at a certain distance is obtained by switching from one spiral pattern to another one with a shifted position after a time t 1 n r n r q n r q t t t t t , , 0 , , where the two patterns are determined by the functions ψ 1 and ψ 2 and q 1 and q 2 specify their locations. The two patterns in equation (9) are generated using equation (A.7) as shown in figure 6(a). We then simulate the translocation process using equation (1) for N=1000 particles in a simulation domain corresponding to a rectangular box, 0xL x , 0yL y with L x =500 μm and L y =675 μm. Spiral patterns 1 and 2 in equation (9)  ), respectively. Other parameters are kept the same, as for the case of mill. Figure 6(b) shows the results of the numerical simulations (see also Video5 in the supporting information) with the corresponding calculated order parameters M and P shown in figure 6(c). At time t=0, the particles are uniformly distributed within a circle of radius 200 μm centered at (200 μm, 200 μm). They start to move along the spiral orbits towards the center of attraction, q 1 , forming a rotating mill. During this process, both parameters, P and M, are non-vanishing and their evolution have opposite tendencies following the previously identified trends.
The switch from pattern 1 to pattern 2 occurs at t=t 1 =50 s, when the behavior of P and M displays an abrupt jump caused by the instant shift of the 'center of attraction' and particle propulsion towards position 2. In contrast to this instantaneous switch, reconstruction of the destination mill is gradual, in analogy with the experiments, as can be seen from the smooth decrease in P and increase in M. Eventually, the center of mass of particles nearly reaches the position of the spiral pattern, q 2 and we observe another rotating mill. This state approximately corresponds to P≈0 and M≈1.

Conclusions
In this article we have demonstrated a method to trap and transport a large ensemble of anisometric colloids dispersed in a nematic liquid crystal matrix and driven by nonlinear electrophoresis. The particles are propelled above a photosensitive surface where a topological defect generating a spiral pattern is created or erased by light. The experiments are combined with a theoretical model that calculates the pattern topology and explains the observed dynamic states and how they depend on the underlying pattern orientation. While our colloidal swarm was transported along a simple track, namely a line connecting two topological defects, in principle more complex patterns may be easily designed to steer the colloidal motion. The advantages of this method over other strategies to propel microscale matter in viscous fluids are the possibility to control the mean molecular orientation of the dispersing medium using an external field, and the independent control of the surface topology through optical means. These two external fields are uncoupled and can be easily interchanged during the experiments, a feature that gives further functionality to the transport in colloidal systems. Appendix. Plane spiral patterns of the director field in a nematic liquid crystal To model mill patterns, here we are interested in finding spiral solutions of the director field that are consistent with the continuum theory of liquid crystals and at the same time can be adjusted to render the experimental conditions. A simple 'magic spiral' solution for a nematic liquid crystal confined by two concentric cylinders of radii r 1 <r 2 was obtained by Parodi, see [46]: showing a monotonic change of the angle made by the director with the radial direction. It is based on three basic assumptions: (i) the director field is restricted to the plane normal to the axes of the cylinders; (ii) the anchoring is homeotropic (normal director) at the inner cylinder and planar (tangential director) at the outer one; (iii) the one-constant approximation, implying that the splay, K S , and bend, K B , constants are equal. Later, Williams went beyond the one-constant approximation and also extended the analysis to an arbitrary alignment at the outer boundary but kept it homeotropic at the inner one [47]. However, to achieve a reasonable agreement with the experiment, it is crucial for us to be able to arbitrarily adjust the alignement of the director field also at the inner border. Furthermore, the radial dependence of the angle ψ(r) can be non-monotonic even at slightly different constants, which may become important for describing spiral patterns at large distances. Therefore, for the sake of generality, we will stick to the case K K S B ¹ and adapt the discussed solutions. We start by formulating the Frank-Oseen free energy of the nematic liquid crystal. For the two-dimensional case as in assumption (i) above (n z =0), the corresponding energy expressed as a function of the director field n n r = ( ) and written per unit length along the cylinder is We now proceed to the polar coordinates (ρ, θ) and represent the director field as (n ρ , n θ )=(cosψ, sinψ) with ψ=ψ(ρ). Using this ansatz in equation (A.2), integrating it over the polar angle θ and introducing the combinations of elastic constants, K ± =(K S ±K B )/2, we obtain where s(r 1 )=0 and s 2 =s(r 2 )=ln(r 2 /r 1 ). Here, the subscripts 'ρ' and 's' denote the corresponding derivatives. Applying a standard procedure to minimize the energy functional (A.3), we arrive at an ordinary differential equation for ψ(s) The values α 1 and α 2 determine the arbitrary alignments of the director field at the inner (ρ=r 1 ) and outer (ρ=r 2 ) borders, respectively. The boundary value problem described by equations (A.4) and (A.5) admits two simple fundamental solutions. One is given by ψ(s)=α 1 =α 2 =0, π. The sink-like pattern with ψ(s)=π corresponds to an aster, n n n , 1 ,0 = = - To make an analytical progress, we resort to the approximation of weak distortion, ψ = 1. By retaining the leading terms only, we end up with the linear ordinary differential equation The partial case of one-constant approximation, K S =K B , follows from equation (A.7) in the limit k → 0,