Abstract
Colloidal interactions in nematic liquid crystals can be described as interactions between elastic multipoles that depend on particle shape, topology, chirality, boundary conditions and induced topological defects. Here, we describe a nematic colloidal system consisting of mesostructures of gold capable of inducing elastic multipoles of different order. Elastic monopoles are formed by relatively large asymmetric mesoflower particles, for which gravity and elastic torque balancing yields monopoletype interactions. Highorder multipoles are instead formed by smaller mesoflowers with a myriad of shapes corresponding to multipoles of different orders, consistent with our computer simulations based on free energy minimization. We reveal unexpected manybody interactions in this colloidal system, ranging from demixing of elastic monopoles to a zoo of unusual colloidal crystals formed by highorder multipoles like hexadecapoles. Our findings show that gold mesoflowers may serve as a designer toolkit for engineering colloidal interaction and selfassembly, potentially exceeding that in atomic and molecular systems.
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Introduction
Introduced by Einstein within a theoretical framework explaining Brownian motion of tiny particles^{1}, colloidal atom paradigm has provided the motivation and means for organizing particles into crystals and other structures, mimicking and even exceeding the diversity of structures in naturally occurring molecular and atomic materials^{2}. Longrange elasticitymediated colloidal interactions between particles^{3} in liquid crystal (LC)^{4} fluids have enabled a host of anisotropic colloidal selfassemblies and composite materials^{5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26}. Colloidal inclusions perturb the uniform background of nematic LC’s groundstate unidirectional molecular orientations, producing distortions in the molecular ordering described by the coordinatedependent director field n(r). These director distortions propagate far beyond the physical extent of the particles themselves^{3}, though confining surfaces with strong boundary conditions can partially localize and limit the extent of spatial propagation of these distortions^{11}. Minimization of director distortions to lower the total free energy cost when colloidal particles are in a close proximity leads to elasticitymediated interactions not present in isotropic host media^{3,11}. Under the oneelasticconstant approximation, the governing EulerLagrange equation, derived from the minimization of free energy, is of Laplace’s type, similar to that of electrostatics, thus allowing for interpreting the nature of longranged n(r) distortions based on the multipole expansions^{3,11,21}. At small interparticle distances, this multipole description is limited by the presence of topological singularities and nonspherical topographic features of the colloidal particles, whereas surface confinement and boundary conditions on sample surfaces effectively limit this description at large interparticle distances^{11}. The confinement effects effectively screen the longrange nature of interactions and can be accounted for analogously to the screening of electrostatic colloidal interactions in presence of counterions and many other types of screening of physical forces described using the mathematical language of multipoles^{27,28,29,30,31}. Deviations of n(r) in opposite directions away from the farfield uniform alignment can be interpreted analogously to opposite charges in electrostatic charge distributions, defining the design principles for achieving diverse types of colloidal interactions and assemblies that mimic the wellunderstood interactions between electrostatic charge distributions^{11}. In addition to theoretical analysis^{5,6,7,12,13,14,15,17}, a number of elastic multipoles have been discovered experimentally^{3,8,9,10,11,16,17,18,19,20,21}. Surface anchoring boundary conditions on the particles and size, shape, topology and chirality are all found to be important factors, defining the behavior of nematic LC colloids^{11}. Besides, colloidal particles can induce different multipoles depending on the types of defects that occur. For example, colloidal spheres can induce elastic dipoles^{3}, quadrupoles^{12}, or hexadecapoles^{17} depending on whether singular point^{3} or “Saturn ring” disclination loop^{32,33} or simultaneously both types of defects^{17} are formed, respectively. However, such ability of achieving different types of elastic multipoles by colloidal inclusions of the same type is limited, consequently limiting the diversity of assemblies and composite materials that can be achieved. Although twophoton polymerization based fabrication of colloidal particles with complex shapes can allow for designing many different types of elastic LC multipoles^{11,16}, it is limited to particles made of polymers and elastomers and cannot be easily scaled, which is a limitation as compared to wet chemical synthesis of colloids.
Here we describe a nematic colloidal system made from mesoflower colloidal particles^{34} capable of inducing elastic multipoles of different order. These mesoflowers are mesostructures of gold with highly diverse shapes and with characteristic sharp spikes of submicron dimensions^{34}. These complex yet diverse particles allow for inducing different elastic multipoles when dispersed in a nematic LC, which for small particles range from dipoles to hexadecapoles, and even higher order multipoles. Larger asymmetric mesoflowers induce elastic monopoles that emerge due to external gravitational torques/forces that are balanced by those originating from the LC’s orientational elasticity; this leads to monopoletype interactions with confining substrates and other colloidal objects, as well as to unusual anisotropy in their Brownian motion. Numerical modeling based on Landaude Gennes free energy minimization, while accounting for the presence of singular defects with corresponding variations of the scalar order parameter in addition to the director configurations, confirms these experimental observations and predicts the existence of other elastic multipoles achieved by systematically varying the distribution of spikes. We reveal changes in physical behavior like pair interactions of colloidal particles stemming from small changes in particle dimensions and shapes, as exemplified by the mesoflower nematic colloidal system. Numerical modeling of interactions in large colloidal systems shows that these effects lead to unusual types of colloidal crystals and transformations between them in the case of highorder multipoles and to demixing of elastic monopoles of opposite signs, very differently from electrostatic interactions in atomic systems. In the spirit of the colloidal atom paradigm, our findings reveal that LC colloids have a great potential of not only expanding the length scales of selfassembly from atomic to colloidal scales^{1,2,3}, but also diversifying the forms of colloidal organization by going beyond what is accessible to atomic systems.
Results
Experimental generation of elastic multipoles
Gold mesoflowers of size ranging from hundreds of nanometers to micrometers are synthesized by a seedmediated growth method (see details in “Methods” section)^{34}. They are then dispersed in a nematic LC, 4cyano4′pentylbiphenyl (5CB) (Fig. 1). The cetyltrimethylammonium bromide (CTAB) coating on the surface of the particles sets perpendicular boundary conditions for the director n(r), while the sharp spikes sticking out in all directions perturb the uniform farfield alignment n_{0} of the LC defined by the rubbing direction of the confining substrates. Although each mesoflower possesses varying number of spikes of different size, the analysis of n(r) around the particles allows for prediction of their colloidal behavior based on electrostatic analogy^{11}. In the optical micrographs taken under crossed polarizers with an additional 530 nm phase retardation plate, director distortions manifest themselves as the colored regions different from the background, indicating that n(r) deviates away from n_{0} around a particle. The direction of director rotations, or rather the rotation of projection of n(r) to the plane of the sample, in blue (yellow) regions of polarizing optical micrographs is extracted on the basis of addition (subtraction) of phase retardation of the waveplate and the birefringent LC sample with the corresponding director orientation patterns. For our experimental geometry, the blue (yellow) polarized interference colors in optical micrographs reveal positive (negative) xcomponents of director and clockwise (counterclockwise) rotations of n(r) away from n_{0} (see the insets in figures showing the details for particular experiments discussed). The diversity of the mesoflowers leads to a variety of polarized interference color patterns, revealing n(r)distortions resembling that of elastic multipoles^{21,26}, depending on the exact patterns of interchanging blue and yellow colored regions distributed around the particles (Fig. 1h–k and Supplementary Fig. 1). In some peculiar cases, the mesoflowers are surrounded by predominately one color (Figs. 1g and 2a, b), implying n(r) rotation to one direction away from n_{0}. Such director distortions can be identified as elastic monopoles, although it has long been believed that they should relax to higher order multipoles^{11,18,26} because rotations of director in one direction would generate elastic torque that would relax such distortions to minimize free energy. In our system, however, gravity prompts this behavior because the density of gold is much higher than that of the LC and can serve as a source of external torques/forces balancing their elastic counterparts. For sufficiently large particles, gravitational forces and torques can compete with the elastic counterparts and the elastic monopoles can be stabilized rather than relax to higher order multipoles. This behavior is very different from that of conventional colloids in isotropic fluid hosts, where the role of gravity is associated with colloidal particle sedimentation, redistribution along the sample height or destabilization. As an order of magnitude assessment of these unusual gravity effects in LCs, we equate effective gravitational potential of the mesoflower ∝Δρ(4πR^{3}/3)gR (here Δρ is the difference between densities of gold 19,320 kg m^{−3} and the LC 1008 kg m^{−3}; g = 9.8 m s^{−2} is the standard gravity; R is an effective radius of the mesoflower) with the elastic energy \(\propto \bar KR\) (the used LC’s average elastic constant \(\bar K\) ~ 6.5 pN). We obtain an estimate of effective threshold radius for particles significantly influenced by gravitational effects, R_{t} ~ 2 μm, well within the range of the studied mesoflower dimensions (Fig. 1). Thus, depending also on particle shape, gravitational effects can be a factor for stabilizing different elastic multipoles when the particle size is comparable or larger than R_{t}. For smaller particles, gravity is not strong enough to compete with elastic forces (Fig. 1d, e), but it can serve as a source of symmetrybreaking torques and forces for particles larger than R_{t}. Particles with intermediate dimensions ~R_{t} may exhibit monopoles as metastable states due to the interplay between the complex shape of the particle, surrounding director distortions and satellite defects, as well as interactions with the confining substrate. Indeed, when we poke, heat or rotate such particles by laser tweezers, different multipolelike color patterns can appear around the same mesoflower of size ~R_{t} (Fig. 2 and Supplementary Fig. 1), including the monopolelike structures.
The intrinsic viscoelastic anisotropy of the LC host leads to anisotropic Brownian motion of colloidal inclusions^{35}, which further depends on the particle’s geometric shape, surface boundary conditions and induced defects^{19,36,37}. For mesoflowers with comparable effective dimensions of spikes extending in directions along and perpendicular to n_{0}, the angular dependence of the mean square displacement (MSD) of individual particles is directly related to n(r) structures that they induce (Fig. 2). In an isotropic phase of the LC, like colloidal spheres, mesoflowers diffuse in all directions with roughly equal probability during a long period of time^{38}, so that the angular dependence of MSD is isotropic. In a nematic phase, even for spherical particles, molecular alignment breaks the symmetry and defines an easy axis for particle diffusion^{35}, which yields dumbbellshaped angular dependencies of MSD, aligned along n_{0} and symmetric with respect to it. However, the mesoflowerinduced n(r) distortions further enrich this behavior (Fig. 2). For example, the dumbbellshaped MSD dependence of elastic monopoles has a long axis tilted away from n_{0} (Fig. 2a, b), with the tilting direction matching the unidirectionally rotated nearby n(r) orientation and correlating with the elastic monopole sign. In contrast, mesoflowers inducing higher order multipoles exhibit MSD angular dependence symmetric with respect to n_{0} (Fig. 2c, d and Supplementary Fig. 2). Interestingly, the diffusion behavior can be altered by switching particleinduced structures between different metastable states associated with reconfiguration of n(r) and elastic multipoles (Fig. 2a, c). Such switching of diffusion anisotropy and mediummediated longdistance correlation of diffusion anisotropy between particles of the same type, cannot be achieved for mesoflowers or other particles dispersed in an isotropic medium, even for anisotropic particles^{38}.
Elasticitymediated colloidal interactions
Dynamics and interactions of mesoflowers dispersed in the LC are probed under polarizing optical microscopy with a 530 nm retardation plate inserted, so that elastic multipoles can be identified by examining the color patterns. Particles are brought to desired initial positions using laser tweezers and then released. In addition to conventional dipoledipole (Supplementary Fig. 3) and quadrupolequadrupole (Fig. 3) interactions that are common in other nematic colloids^{11}, interactions involving elastic monopoles are also observed (Fig. 4a, b). A monopolelike mesoflower surrounded by unidirectionally rotated n(r) (consistent with the blue color in a polarizing micrograph) attracts another mesoflower with a dipolar n(r), surrounded by approximately equal amount of blue and yellow colors within the polarizing micrograph, which is an elastic dipole. The two particles eventually approach each other by sharing bluecolored regions, thus lowering the total energy cost of the ensuing colloidal assembly (Fig. 4a, b). The interaction potential, calculated from the balance of elastic and viscous drag ∝dr_{c}/dt forces, is in the range of hundreds of k_{B}T and its powerlaw distance dependence ∝−r_{c}^{−2} is consistent with that of the monopoledipole interaction (r_{c} is the distance between the centers of interacting colloidal particles). As another example, Fig. 4c, d shows how an assembly of two mesoflowers consisting of one dipole and one monopole attracts another dipolar mesoflower (Fig. 4c, d), with the interactions again consistent with the electrostatic analogy of these nematic colloids. For elastic multipoles of leading orders 2^{l} and 2^{m}, the balance of viscous drag and elastic force \(\propto 1{\mathrm{/}}r_{\mathrm{c}}^{l + m + 2}\) yields the anticipated time dependence of interparticle distance \(r_{\mathrm{c}}\left( t \right) = \left[ {r_0^{l + m + 3}  (l + m + 3)\alpha t} \right]^{1/(l + m + 3)}\), where r_{0} is the initial centertocenter distance and α is a fitting parameter, consistent with experiments for pair interactions for all studied multipoles of the same and different orders (Figs. 3 and 4 and Supplementary Fig. 3).
Computer simulations of elastic multipoles and interactions
Numerical minimization of Landaude Gennes free energy provides insight into the director configurations around mesoflowers with complex shape and different dimensions (Figs. 5 and 6). These particles induce networks of defect lines meandering on their surfaces, typically along the ridges of spikes, with the longrange distortions of n(r) influenced by the spiky topography of particles but not exactly following it (Figs. 5 and 6). For large and strongly asymmetric particles gravitational torques and forces compete with their elastic counterparts, giving the origin to elastic monopoles (Fig. 5), whereas smaller particles tend to induce higher order multipoles (Fig. 6 and Supplementary Fig. 4). For monopolelike particles, the clockwise versus counterclockwise unidirectional rotation of n(r) away from n_{0} corresponds to the opposite signs of monopoles, as confirmed by multipole expansion. The xcomponent of director, n_{x}, which is an effective elastic charge density, has the same sign when plotted on a sphere encompassing the monopoleinducing mesoflowers, though its amplitude is nonuniform, as shown in the Fig. 5a–c, consistent with the leading monopole moment of the elastic charge distribution. Since the twist elastic constant of the LC is the smallest, the bend and splay distortions can also relax through equivalent distortions containing twist, so that the particles orientations can deviate away from n_{0} not only in the plane containing g, the gravitational acceleration, but also in the plane orthogonal to g, as experimentally observed in planar LC cells (Figs. 1–4).
Details of the elasticgravitational torque balance are revealed by simulating an asymmetric particle (similar to lowsymmetry particles that tend to induce elastic monopoles in experiments) placed above a wall in xz plane with tangential boundary conditions (Fig. 5). The xcomponents (see inset in the lower left corner of Fig. 5d for the definition of the reference system) of the elastic \(T_x^{{\mathrm{el}}}\) and gravity \(T_x^g\) torques vary with the angle θ between the particle axis (the line connecting the center of the particle core and the tip of one of the spikes) and n_{0}. The torques are calculated about the center (see Supplementary Fig. 5), and exhibit different signs within the intervals θ ∈ [−180°, −127°] and θ ∈ [−90°, 0°]. The orientations θ = 0°(−180°) and θ ≃ −127° correspond to the minima of the elastic and gravity energies, respectively. For particles with masses \(M \ll \bar K{\mathrm{/}}g\), the elastic energy dominates and the particle adopts one of the two equilibrium orientations either θ_{eq} ≲ 0°, or θ_{eq} ≳ −180°, as shown in Fig. 5f for the case θ_{eq} ≲ 0°. As the particle size increases, the gravity tilts the particle’s axis further away from the purely elastic equilibrium θ = 0° towards negative values of θ_{eq}, the equilibrium orientation behaves according to the upper branch of the curve in Fig. 5f. When starting from the second elastic energy minimum at θ = −180°, the gravity tilts the axis towards less negative θ upon increasing the particle size, towards the minimum of the gravitational energy (lower branch of θ_{eq} curve in Fig. 5f). For sizes larger than R_{t}, the particle would adopt its equilibrium orientation in the vicinity of the purely gravitational equilibrium θ ≃ −127°. The equilibrium orientation θ_{eq} in Fig. 5f was obtained for h = 3.5R_{0}, where R_{0} is the radius of the spherical core of the mesoflower and h is the distance from the particle center to the wall (Supplementary Fig. 6). As the weight of the particle increases the equilibrium particlewall separation h_{eq} decreases, as shown in the inset of Fig. 5e (the gravity acts in the directions towards the wall) and depends on balancing of gravitational and the repulsive elastic forces. The latter also depends on the strength of surface anchoring boundary conditions and is calculated here from the free energy profile versus h for the regime of strong boundary conditions corresponding to the experiments. The corresponding free energy results (Fig. 5e) demonstrate particlewall repulsion, which have similar scaling when using one elastic constant approximation or not, though the elastic torque is almost insensitive to the variation of h for 3.5 ≤ h/R_{0} ≤ 20, justifying the approximations used for calculating θ_{eq} (Fig. 5f). The sizedependent levitation of mesoflowers at different h (inset of Fig. 5e) shows how suspensions of mesoflowers in LCs could be potentially used for separating particles of different dimensions, though the range of particle dimensions that can be effectively separated will depend on the density of particles relative to that of the LC host medium. Therefore, this separation method may be limited to highdensity particles, like the ones made from gold that we use in this study.
The wallmonopole particle repulsive potential ∝ 1/h revealed in Fig. 5e agrees with the predictions of nematostatics^{5,6,14} according to which elastic monopoles with opposite signs repel. Indeed, employing the electrostatic analogy, the interaction with the wall can be modeled using an image elastic monopole, yielding the ∝ 1/h potential. The potentials of mean force between two monopole particles with the same as well as opposite signs of monopole moments (Fig. 5g) also agree with nematostatics:^{9} like (unlike) elastic monopoles attract (repel) each other with an effective potential ∝ 1/r_{c}. Numerical modeling provides insights that formation of elastic monopoles by mesoflowers of gold is facilitated not only by their relatively large size >R_{t}, but also by symmetry breaking often caused by asymmetric distribution and dimensions of individual spikes (Supplementary Fig. 5) within the mesoflower particle (both effects enabled by the competition of gravitational forces and torques with their elastic counterparts, which stabilize elastic monopole configurations).
For particles smaller than R_{t}, varying the number and positions of spikes within mesoflowers generates different multipole series of the director distortions (Fig. 6 and Supplementary Fig. 4). The multipole expansion analysis^{21} for the numerically simulated n(r) shows that certain particles induce stable or metastable structures with strongly pronounced multipole moments of different orders, including octupoles (Fig. 6a–c), hexadecapoles (Fig. 6d, e and Supplementary Fig. 4a–c) and even 64poles (Fig. 6f, g and Supplementary Fig. 4d–f), with the other multipole moments orders of magnitude smaller. The numerically simulated n_{x}distributions on spheres encompassing these multipolar mesoflowers are consistent with the corresponding charge distributions of high leading order electrostatic multipoles (Fig. 6), though they also contain fine features dictated by detailed geometry of the mesoflowers (compare Fig. 6a, c).
Colloidal crystals and demixing of elastic multipoles
A multipolar approximation for an effective interaction between two colloidal particles distance r_{c} apart and with the centertocenter vector forming an angle θ_{c} with the farfield director n_{0} reads:^{17}
where P_{l}(x) is the Legendre polynomial of degree l, Q_{l} characterize the strength of elastic multipole moments, and R_{eff} is the characteristic length scale of the multipole (set in our case by the size of the colloidal particle). The analysis of Eq. (1) shows that a large variety of pair interaction patterns arises for multipoles of different orders (Fig. 7), which can lead to the formation of colloidal crystals and other structures arising from the competition between multipolar elastic and screened electrostatic repulsive interactions^{19}. Because our mesoflower colloids allow for the realization of a large variety of multipoles of different orders, one can systematically explore how such colloidal interaction lead to selforganization depending on the order of the leadingorder multipole.
First, we consider a halfhalf binary mixture of distinct elastic monopoles confined at a plane coplanar to the farfield director. Contrary to electric charges, similar elastic monopoles attract and dissimilar repel^{9,11}, as discussed above for pair interactions. To assess how this behavior impacts collective behavior of many such particles, we assume pairwise additive interaction potential and augment the monopolemonopole elastic interaction with a truncated repulsive Yukawa potential corresponding to the screened electrostatic interactions, which can be tuned by adding counterions through doping LCs with salt and other additives:^{39}
where r_{CO} is the cutoff distance, A and κ are positive constants, with the latter characterizing electrostatic screening effects due to counterions within the LC^{39}. We emphasize that rather strong electrostatic repulsion at short particle separations is needed in order to avoid shortrange steric, elastic and van der Waals interactions leading to aggregation due to the complex surface geometries of particles. Assuming such screened electrostatic repulsions, we perform molecular dynamics simulations using open source Largescale Atomic/Molecular Massively Parallel Simulator (LAMMPS)^{40}. In this simulation as well as in all the subsequent simulations of selforganizations of higher order elastic multipoles, we neglect possible changes in multipole moments, or generation of additional multipoles, as the interparticle distances and orientations vary. We also neglect the shortrange effects that may arise at small interparticle distances and that possibly cannot be fully described within the multipole expansion approach that we consider. Figure 8 shows a series of snapshots along the system trajectory, where a perfect NaCllike crystal order (see Fig. 8a) is assumed as the initial condition. In the course of time, the system of oppositely charged elastic monopoles undergo a spatial segregation, Fig. 8f, which is the direct consequence of the fact that like (opposite) monopoles attract (repel) (see also the Supplementary Movie 1).
Selfassembly of twodimensional nematic colloids with the dominant dipolar or quadrupolar elastic interactions has been previously reported^{22,41,42} and is consistent with the behavior of colloidal dipoles and quadrupoles formed by gold mesoflowers. Driven by the symmetries of the underlying multipole potentials, the dipole colloids tend to assemble headtotail in chains aligned along the farfield director. The chains then assemble in a twodimensional crystalline structure with an antiferromagneticlike alignment of the neighboring chains^{22,41}. Quadrupolar colloids reveal twodimensional crystalline assembly with a rhomboidal unit lattice cell where the colloidcolloid bonds are aligned along the attractive directions of the underlying quadrupole potential^{42}. The sectors of attraction and repulsion are oriented with respect to n_{0}, which defines the longrange orientations for the crystallographic axes of the ensuing colloidal crystals formed by elastic dipoles and quadrupoles, thus precluding the formation of grain boundaries between them. To the best of our knowledge, crystallization of nematic colloids driven by higher order multipole potentials have not been investigated.
Experimental results^{20,21,22,41,42} suggest that the symmetry of the eventual nematic colloids crystals are mainly determined by the distribution of the attractive directions of the corresponding multipole potential. The elastic octupole potential has six directions of attraction forming roughly 60° between each other (Fig. 7a). It is therefore expected that octupole nematic colloids crystalize in two dimensions into a hexagonal lattice, indeed consistent with our LAMMPSbased simulations. The unique alignment of the hexagonal lattice of these elasticoctupolebased crystals with respect to n_{0} in this nematic colloidal system precludes the formation of grain boundaries and, because of this, could potentially be used for the formation of large crystal lattices. The case of hexadecapole colloids is fundamentally more interesting, as in this case the potential has eight directions of attractions (Fig. 7b), which permits two types of quadratic lattice arrangements of hexadecapole colloidal particles, with the corresponding unit lattice cells shown in Fig. 9a, b. In order to verify this hypothesis, we performed molecular dynamic simulations (using LAMMPS) of twodimensional hexadecapole colloids, regularized with repulsive Yukawa potential as defined by Eq. (2). We used constant temperature and constant pressure (NPT) ensemble. Initially, the system was subject to a strong isotropic pressure (Fig. 9c, f). In this initial state the colloids formed a glassy structure, with enhanced local hexagonal ordering revealed by the absolute value q_{6} of the local hexatic bond order parameter (Fig. 9c) and suppressed q_{4} of the quartic bond order parameter (Fig. 9f). In the course of the simulations, the external isotropic pressure was gradually decreased, which resulted in the spontaneous formation of two types of quadratic colloidal lattices and associated grain boundaries between them (Fig. 9d, e, g, h, and Supplementary Movie 2).
Although the hexadecapole elastic interactions tend to form twofold degenerate ground state whose lattice units are depicted in Fig. 9a, b, this tendency can be inconsistent with the efficiency of twodimensional packing at high colloidal concentrations. Colloidal particles are electrostatically charged and mutually repel with screened Coulomb electrostatic potential, where the range and the strength of this potential can be tuned^{39}, which provides additional knobs to control the selfassembly of nematic colloids. We have performed illustrative molecular dynamic simulations of selfassembly behavior of hexadecapole colloids with strong, weakly screened repulsive Yukawa potential. The ensuing selfassembled colloidal configurations are presented on Fig. 10 and Supplementary Fig. 7 (see also Supplementary Movies 3–6). At high concentrations the system reveals glassy (Fig. 10a, d) configurations, while at low concentrations particles assemble into repulsioninduced sparse hexagonal lattices (not shown). Surprisingly, at intermediate values of particle concentrations and depending on the strength of the hexadecapole elastic moment, the system can crystalize into a range of rhomboidal lattices, with some examples shown in Fig. 10c, k and Supplementary Fig. 7c, k, and the corresponding unit lattices in Fig. 10q. Importantly, the symmetry of these lattices differs from the ones of hexadecapolar ground states shown in Fig. 9a, b. Additionally, at some number densities we observe coexisting rhomboidal and hexagonal lattices, see Fig. 10c, k and Supplementary Fig. 7c, k. Many interesting energetically comparable colloidal organizations emerge as a result of dense packing and elastic hexadecapolar interactions favoring different lattices. This unusual crystallization behavior is revealed by analyzing bond orientational order parameters within selfassembled crystallites separated by grain boundaries. These examples illustrate the ability of hexadecapolar nematic colloids to selfassemble into a range of twodimensional lowsymmetry crystal structures, with properties that can be designed through controlling the particle shape, concentration as well as internal properties of the nematic host.
Discussion
In this work, we have developed a nematic colloidal system of mesoflowers where complex and diverse shapes of mesostructures induce elastic multipoles of different order, from monopoles to highorder multipoles like 64poles. By varying the basic building parts of these particles, their colloidal behavior can be effectively controlled. As the shape and size of particles vary, they behave like elastic monopoles that (differently from their electrostatic counterparts) tend to demix into the domains of likecharged elastic monopoles, or as highorder elastic multipoles (e.g., hexadecapoles) forming unusual crystals that can be tuned by packing density and strength of electrostatic repulsions. While here we focused only on the analysis of twodimensional crystal lattices for illustrative purposes, even more complex behavior is expected in three dimensions, where even elastic quadrupoles can form triclinic crystal lattices^{39}. As the symmetries of elastic potentials of highorder multipoles, which form the basis of crystals, become intrinsically incompatible with crystalline lattices, various forms of frustration can arise and lead to complex crystallization behavior that can be potentially tuned through changing concentration of counterions and surface charging. In addition to crystals, quasicrystals and various plastic crystals (with positional order but lacking orientational ordering) could possibly also arise due to diverse multipolar nature of our mesoflower colloids, whereas polydisperse systems are expected to form disordered glassy states. Our findings suggest that these mesostructures may serve as a designer toolkit for engineering predefined colloidal interaction and selfassembly. Access to these structures in large quantities through chemical synthesis may facilitate developments of new composite materials fabricated using colloidal selfassembly. On the other hand, the sharp geometric features of the gold mesoflowers within these colloidal particle assemblies may be of interest for plasmonic enhancement and related applications.
Methods
Sample preparation
Gold mesoflowers are synthesized following an aqueous seedmediated growth procedure^{34}. First, 25 mg of citric acid and 1 mL of 25 mM HAuCl_{4} solution were combined in 35 mL deionized water at 80 °C. Immediately after the color of the solution changed from pale yellow to pink, 100 μL of aniline and 0.5 mL of 25 mM HAuCl_{4} solution were injected with temperature maintained for 5 more minutes. After 5 h, the resultant solution was centrifugated at 4000 rpm to obtain the supernatant as the seed solution for the following steps. The growth solution was prepared by mixing 20 mL of 0.1 M CTAB solution, 335 μL of 25 mM HAuCl_{4} solution, 125 μL of 10 mM AgNO_{3} solution and 135 μL of 0.1 M ascorbic acid solution at 80 °C. 2 mL of the seed solution was added to this growth solution, accompanied with gentle mixing. The mixture was kept at 80 °C for 1 h and cooled down naturally before taking it out for centrifugation at 4000 rpm for 5 min. The residue was collected and washed in deionized water 3 times and then dried in an oven. Liquid crystal suspension of mesoflowers was obtained by sonicating these dried residues within the LC host, after which the suspension was infiltrated between glass substrates with gap defined by monodisperse glass spacers of 10–20 μm in diameter. Prior to being used in cell preparation, the glass substrates were spincoated with polyimide (PI2555, from HD Microsystem) and rubbed unidirectionally to define the bulk LC alignment. The cell edges were sealed using fastsetting epoxy glue.
Optical video microscopy and laser trapping
An inverted optical microscope (IX81, Olympus) with a chargecoupled device (CCD) camera (Flea, PointGrey) and a holographic laser trapping system operating at 1064 nm was used to take polarizing optical micrographs and videos of mesoflowers in the LC, to probe the stable and metastable director configurations around the particles, as well as to set the initial conditions for studying their pair interactions. The laser was turned off during particle diffusion and interaction when videos were being recorded. To assure good optical imaging resolution and robust laser manipulations, we used an oilimmersion objective lens (UPlanSApo 100×, Olympus) with high numerical aperture NA = 1.40 for both imaging and laser trapping. In order to analyze colloidal interactions and diffusion in directions along and perpendicular to n_{0} in the plane of the LC cell, optical videos were analyzed using an imageprocessing software (ImageJ, freeware from the National Institute of Health) to extract positions of the mesoflowers on a framebyframe basis. MSDs that characterize single particle’s Brownian motion were calculated by analyzing particle displacements along directions perpendicular and parallel to n_{0} (denoted Δr_{x} and Δr_{z}, respectively) based on the particle’s positions in each frame of the video. To probe the angular dependence of particle diffusion, a rotation operation was applied
where β is the angle of rotation with respect to n_{0} and \(\left( {\begin{array}{*{20}{c}} {\Delta r_x^\prime } \\ {\Delta r_z^\prime } \end{array}} \right)\) is the displacements in the rotated coordinate frames. MSD was then calculated as <Δr_{x}>^{2} for β varying from 0° to 360°. Diffusion constants D were extracted from MSDs using the relation <Δr_{x}>^{2} = 2Dτ where τ = 1/f and f = 15 Hz is the frame rate of the experimental videos. From StokesEinstein relation, the drag coefficients of diffusing particles were calculated as c_{μ} = k_{B}T/D_{μ} (μ = x, z) and the corresponding drag forces as F_{d} = cv, where k_{B} is the Boltzmann constant, T = 300 K is the room temperature and v is the linear speed of the particle. Because of the highly overdamped nature of our colloidal system, the elastic interaction forces could be then found from their balance with F_{d}. The elastic interaction potential was then calculated by integrating the elastic force over distance.
Simulation of director structures and elastic interactions
The generation of elastic monopole and higherorder multipoles in the n(r) field by mesoflowers is also approached computationally, via numerical minimization of the Landaude Gennes free energy^{26}. The Landaude Gennes free energy functional depends on the tensor order parameter field Q(r) and its spatial derivatives. The functional combines terms which account for variable degree of nematic order, nematic elasticity and biaxiality (at the cores of topological defects), as well as surface anchoring. As such, the Landaude Gennes approach yields theoretical characterization of n(r) and local changes in the degree of nematic ordering that correspond to global or local minima of the free energy functional^{26}. Spiky particles are constructed as a union of a spherical core and a given number of spikes distributed over the surface of the core at predefined locations and orientations (Supplementary Fig. 6). Minimization of the free energy is performed numerically for finite homeotropic surface anchoring at the particle surfaces, and by using variable threedimensional tetrahedral grids^{43}. This minimization yields stable or metastable n(r) structures around the particles, which are then compared to the experimentally reconstructed counterparts^{16,21}.
Nematic configurations around mesoflowers are obtained via numerical minimization of the phenomenological Landaude Gennes free energy functional^{26}
where Q_{ij} = Q_{ji} (i, j = 1,..,3) is a traceless tensor order parameter and summation over repeated indices is assumed. In Eq. (4), the parameter a (unlike the constants b and c) is assumed to depend linearly on temperature T: a(T) = a_{0}(T − T*), where a_{0} is a material dependent constant, and T* is the supercooling temperature of the isotropic phase. Phenomenological parameters L_{1} and L_{2} are related (via an uniaxial Ansatz for Q_{ij}) to the FrankOseen elastic constants. We describe finite homeotropic anchoring, with the strength coefficient W, at the surface of the particles by using \(f_s( {Q_{ij}}) = ({Q_{ij}  Q_{ij}^s})^2\), with \(Q_{ij}^{\mathrm{s}} = \frac{{3Q_b}}{2}\left( {\nu _k\nu _j  \frac{{\delta _{kj}}}{3}} \right)\), where δ_{ij} is the Kronecker delta symbol, and ν is the unit outward normal vector to the particle surface^{44}; \(Q_b = b{\mathrm{/}}8c\left( {a + \sqrt {1  64\,ac{\mathrm{/}}\left( {3b^2} \right)} } \right)\) is the value of the scalar order parameter in the nematic phase, which is thermodynamically favored for 24ac/b^{2} < 1. Minimization of the free energy Eq. (4) is then performed numerically by employing adaptive mesh finite elements method as described in more details in ref. 43. This minimization yields theoretical characterization of n(r) and local changes in the degree of order that correspond to global or local minima of the free energy^{26}.
In our calculations, we use a_{0} = 0.044 × 10^{6} J m^{−3}, b = 0.816 × 10^{6} J m^{−3}, c = 0.45 × 10^{6} J m^{−3}, L_{1} = 6 × 10^{12} J m^{−1}, and L_{2} = 12 × 10^{12} J m^{−1}, which are typical values for 5CB^{45} at T^{*} = 307 K. For these values of the model parameters, the bulk correlation length \(\xi = 2\sqrt {2c(3L_1 + 2L_2)} {\mathrm{/}}b \approx 15\,{\mathrm{nm}}\) at the isotropicnematic coexistence^{46}, 24ac/b^{2} = 1. We use W = 5 × 10^{−4} J m^{−2}, or 10^{−4} J m^{−2}, corresponding to the surface anchoring strengths of interactions for substrates coated with CTAB^{47,48,49}.
Computational geometry of colloidal mesoflowers
We exploit the Open Source Gnu Triangulated Surface (GTS) library^{50} to create triangulated surfaces of the mesoflowers. A spiky mesoflower particle is constructed as a union of a spherical core with the radius R_{0} and a given number of spike particles (Supplementary Fig. 6) decorating the spherical core at predifined locations and orientations. Each spike is chosen as a generalized cylinder with a 5fold symmetry axis, consistent with the experimental electron microscopy images. The lateral surface of each spike whose symmetry axis coincides with the zaxis of a Carthesian coordinates system (x_{i}, y_{i}, z_{i}), i = 1, ..., 5, is parametrized as follows:
where 0 ≤ s < 2π, and 0 ≤ u ≤ 1 are parameters; A(u) = A_{0} + (A_{0} − A_{1})u accounts for the variation of the spike “radius” along the spike symmetry axis with A_{0} > A_{1}, and H is the height of the spike along its symmetry axis.
The triangulated surface of the spike can be manipulated using the functions implemented in the GTS library, including surface translation, rotation, or rescaling. By using these functions together with the GTS function which merges any two surfaces, we generated the spiky mesoflower particles with arbitrary number of spikes placed at predefind relative positions and with predefined orientations. The triangulation of the nematic domain was then performed by using a quality tetrahedral mesh generator^{51}, which supports adaptive mesh refinement. Finally, the discretized functional (5) is minimized numerically using INRIA’s M1QN3 optimization routine^{52}, which implements a limitedmemory quasiNewton method^{53}.
Molecular dynamics simulations
The selfassembly and formation of colloidal crystals are computersimulated using an open source Largescale Atomic/Molecular Massively Parallel Simulator (LAMMPS)^{40}. In the simulations, we use \(\bar KR_{{\mathrm{eff}}}\) as the unit energy, and R_{eff} as the unit length. The total number of particles simulated is N ~ 1000, with the system size L_{x} × L_{y} ranging from 26 × 26 to 37 × 37 R^{2}_{eff}. The potential energy of the system is assumed to be pairwise addative, combining both the elastic energy and Yukawa potential: U = U_{el} + U_{Y}. For the computer simulations on the segregation of monopoles (Fig. 8), the system size is 34 × 34 R^{2}_{eff} and total number of particles N = 1024 with half of them having positive elastic charges and the other half having negative elastic charges. The interaction potential between particles is taken as
where the elastic monopole charges \(Q_1^{\mathrm{a}}\), \(Q_1^{\mathrm{b}}\) = ±0.1, and we set \(\frac{{r_{{\mathrm{CO}}}}}{{R_{{\mathrm{eff}}}}} = 1.5,\,\kappa R_{{\mathrm{eff}}} = 2,\frac{A}{{\bar KR_{{\mathrm{eff}}}^2}} = 0.1\). The simulation is performed using NVT ensemble. Temperature T is fixed by applying a Langevin thermostat with \(\frac{{k_{\mathrm{B}}T}}{{\bar KR_{{\mathrm{eff}}}}} = 10^{  3}.\) For the computer simulations involving hexadecapoles (Figs. 9 and 10, and Supplementary Fig. 7), N = 855 and the effective pair interaction potential is
The elastic hexadecapolar moment Q_{4}, and the relative Yukawa potential strength \(\frac{A}{{\bar KR_{{\mathrm{eff}}}^2}}\) are varied to obtain different colloidal assembly structures. The simulations are performed in NPT ensemble, using a Nose/Hoover temperature thermostat with \(\frac{{k_{\mathrm{B}}T}}{{\bar KR_{{\mathrm{eff}}}}} = 10^{  3},\) and Nose/Hoover pressure barostat. In order to achieve a spontaneous crystallization an isotropic external pressure is gradually released in the course of the simulations leading to the expansion of the system. The local quartic (hexatic) orientational order parameter q_{4}(j) (q_{6}(j)), where j is the index of the particle under consideration, is defined as \(q_n\left( j \right) = \frac{1}{n}{\mathrm{\Sigma }}_{k = 1}^n\exp (in\theta _{jk})\); θ_{jk} is the angle between the particles centertocenter vector r_{jk} and x axis, and the sum runs over n nearest neighbors of particle j.
Data availability
The data that support the findings of this study are available from the corresponding author on request.
Code availability
All codes used in this work are freely available from the authors upon a request.
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Acknowledgements
We thank B. Senyuk, B. Lev, S. Chernyshuk and V. Pergamenshchik for discussions and A. Sanders for technical assistance. This research was supported by the U.S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DESC0019293 with the University of Colorado at Boulder (Y.Y. and I.I.S.). Publication of this article was partially funded by the University of Colorado Boulder Libraries Open Access Fund. M.T. acknowledges financial support from the Portuguese Foundation for Science and Technology (FCT) under Contracts No. IF/00322/2015, UID/FIS/00618/2019 and PTDC/FISMAC/28146/2017 (LISBOA010145FEDER028146).
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Y.Y. and I.I.S. conducted experimental work and analyzed data. M.T. performed numerical modeling. Y.Y., M.T., and I.I.S. wrote the paper. I.I.S. conceived, designed and directed the project.
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Yuan, Y., Tasinkevych, M. & Smalyukh, I.I. Colloidal interactions and unusual crystallization versus demixing of elastic multipoles formed by gold mesoflowers. Nat Commun 11, 188 (2020). https://doi.org/10.1038/s41467019140312
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DOI: https://doi.org/10.1038/s41467019140312
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