Hexadecapolar colloids

Outermost occupied electron shells of chemical elements can have symmetries resembling that of monopoles, dipoles, quadrupoles and octupoles corresponding to filled s-, p-, d- and f-orbitals. Theoretically, elements with hexadecapolar outer shells could also exist, but none of the known elements have filled g-orbitals. On the other hand, the research paradigm of ‘colloidal atoms' displays complexity of particle behaviour exceeding that of atomic counterparts, which is driven by DNA functionalization, geometric shape and topology and weak external stimuli. Here we describe elastic hexadecapoles formed by polymer microspheres dispersed in a liquid crystal, a nematic fluid of orientationally ordered molecular rods. Because of conically degenerate boundary conditions, the solid microspheres locally perturb the alignment of the nematic host, inducing hexadecapolar distortions that drive anisotropic colloidal interactions. We uncover physical underpinnings of formation of colloidal elastic hexadecapoles and describe the ensuing bonding inaccessible to elastic dipoles, quadrupoles and other nematic colloids studied previously.

In this work, we describe colloidal elastic hexadecapoles that spontaneously form around solid microspheres immersed in a uniformly aligned NLC. The unusual symmetry of elastic distortions arises from conically degenerate 18 boundary conditions for n(r) at the colloidal surfaces, which induce surface point and line defects at the same time. Using a combination of holographic optical tweezers (HOT) 19 , nonlinear optical imaging 20 and polarizing optical microscopy (POM) 1,6 , we probe the n(r)-distortions and quantify colloidal pair interactions by measuring distance and angular dependencies of elastic potentials, demonstrating relations between the director structure and medium-mediated inter-particle forces. Finally, we explain our findings using a model based on elastic multipole expansion and discuss how the experimental framework we have developed may enable colloidal self-assembly into novel forms of tunable pre-engineered matter without known atomic analogues.

Results
Hexadecapolar elastic multipole. When dispersed in a uniformly aligned NLC fluid host, polystyrene microspheres (PSMs) of a radius r 0 locally distort n(r), which is manifested by eight bright lobes around the particle perimeter seen in POM between the crossed polarizer and analyser (Fig. 1a). These bright lobes are separated by eight dark regions within which n(r) at the particle's perimeter is parallel to polarizer or analyser. Using a phase retardation plate and interference of polarized light propagating through the particle-distorted structure, we reveal that n(r) tilting away from the far-field director n 0 switches between clockwise and counterclockwise directions (corresponding to the blue and yellow colours in the micrograph) eight times as one circumnavigates the sphere (Fig. 1b). Bright-field micrographs obtained at different depth of focus reveal presence of weakly scattering surface point defects (boojums) at the particle poles along n 0 as well as a circular loop of a defect line (often called 'Saturn ring') 9,10 at the particle's equator (Fig. 1c,e). Based on POM and three-dimensional nonlinear optical imaging (Fig. 1a-c,e and Supplementary Figs 1-3), we uncover the structure of n(r) distortions schematically shown in Fig. 1d. This structure is consistent with conically degenerate surface boundary conditions for n(r) with respect to PSM's local surface normals s, which were previously demonstrated for NLCs at flat polystyrene-coated surfaces 18 . The director's easy axis orientation lies on a cone of equilibrium polar angle c e (Fig. 1f). To minimize the free-energy cost of bulk elastic distortions, interaction of conically degenerate surface boundary conditions on the microsphere with the uniform n 0 lifts this conical degeneracy and yields an axially symmetric n(r) depicted in Fig. 1d. Extending this analysis to three dimensions, the projection n x of n(r) onto the x-axis orthogonal to n 0 can be visualized around PSM using colours that highlight positive, near zero and negative n x (Fig. 1g). The black points at the poles and ring at the equator of the sphere are regions of discontinuity of n(r) at the NLC-PSM interface and correspond to the boojum and Saturn ring topological defects, respectively. Away from the particle surface and these topological singularities, the experimentally reconstructed n(r) is continuous (Fig. 1d) and consistent with the theoretical model for a hexadecapolar director distortions presented in a similar way in Fig. 1h.
Elastic interactions between hexadecapolar colloids. Elasticitymediated interactions between PSMs (Fig. 2) differ from all NLC colloids studied so far. To get insights into the strength and direction-dependence of these interactions, we utilize HOT to optically trap one 'stationary' PSM and then release another particle at different centre-to-centre vector R orientations with respect to n 0 as well as at different separation distances. Using video-microscopy, we track the ensuing particle motions, which result from a combination of Brownian jiggling and elasticitymediated interactions. The submicron waist and relatively low power (B10 mW) of a focused trapping beam allow us to avoid the influence of the trapping on the measurements 21 . Furthermore, the laser tweezers are used only to bring the particles to the desired initial conditions and are turned off within the time when the pair-interaction is probed with video microscopy, allowing us to avoid possible artefacts associated with the complex effects of the laser trapping light at small interparticle distances. The colour-coded time-coordinate trajectories of particles released from optical traps at different R are shown in Fig. 2a. Unlike in the case of dipolar and quadrupolar NLC colloids 7,8,15,16 , elastic forces are relatively short-ranged and exceed the strength of thermal fluctuations only at distances of four-to-five particle radii r 0 . However, the angular dependence of these forces is very rich, with eight angular segments of inter-particle attractions separated by eight angular segments of repulsions, with the intermediate angular sectors within which particles move sideways as the inter-particle elastic forces are orthogonal or at large angles to R (Fig. 2a). These angular sectors of attraction and repulsion correlate with the bright and dark regions of POM micrographs (Fig. 1a) as well as with the predictions of our model based on elastic multipole expansion (Fig. 2).
To quantify elastic interactions, we first probe the centre-tocentre separation R ¼ 7R7 versus time t for particles released at different angles between R and n 0 within the angular sectors of attraction ( Fig. 3a) and then calculate particle velocities _ R ¼ dR/dt. Because the system is highly over-damped and the inertia effects can be neglected 22 , the experimentally measured R(t) and the simplified equation of motion 0E À x _ R þ F int yield the pair interaction potential U int (inset of Fig. 3a), where x is a drag coefficient measured separately by characterizing Brownian motion of PSMs (Methods and Supplementary Fig. 4) and F int ¼ À rU int is the elastic interaction force. The attractively interacting particles stop short of physically touching each other, instead forming stable dimer assemblies with typical RE(2.05-2.2)r 0 and stable R orientations with respect to n 0 within one of the two angular sectors of assembly in each quadrant dependent on c e : y 1 E22°-26°or y 2 E64°-75°( Fig. 3b-f). Multi-particle assemblies with different combinations of angles y 1 and y 2 are also observed (Fig. 3d). The inter-particle binding energies are measured to be in the range of hundreds of k B T (inset of Fig. 3a), making them robust with respect to thermal fluctuations. Although U int versus y has eight minima, only four of them can be occupied simultaneously in one plane by particles of the same size because of the 'excluded volume' effects, yielding two-dimensional colloidal crystals with rhombic elementary cells ( Supplementary Fig. 5 and Supplementary Note 1). Following similar considerations, a large number of low-symmetry threedimensional colloidal structures can be envisaged too. Since the elastic interactions potential is hundreds of k B T and the particle assemblies can be entrapped in metastable states, the assembly of two-and three-dimensional colloidal lattices from micrometersized particles requires the use of optical tweezers for guiding colloidal particles. Alternatively, the elastic interaction potentials between colloidal particles of smaller size or with weaker surface anchoring can be brought to the order of 10 k B T and lower, so that the crystal self-assembly can occur without the assistance of optical tweezers, which will be explored elsewhere.  Figure 2 | Elastic interactions of colloidal hexadecapoles. (a) Angular dependence of interactions is probed by tracking motion of a particle released from the optical trap and moving with respect to the 'stationary' trapped particle in the centre depending on the orientation of R with respect to n 0 . The elapsed time is coded according to the colour scale (inset) and the maximum elapsed times t max À t 0 are marked next to the corresponding trajectories. R || and R ? denote the centre-to-centre distance R components along and perpendicular to n 0 , respectively. The PSM of radius r 0 is surrounded by the spherical volume of radius 2r 0 that is excluded for centres of other PSMs. The nonlinear zone is shown by a dashed circle at R ¼ 2.4r 0 . (b) Pair-potential U int versus y for two particles at R ¼ 2.4r 0 . Angular zones of repulsion and attraction are highlighted using magenta and green arrows and colouring, respectively.
To model experiments, we exploit the electrostatic analogy of the far-field director distortions because of a colloidal sphere that can be represented as elastic multipoles 23 , albeit our colloids dramatically differ from elastic dipoles and quadrupoles studied so far [7][8][9][10][11][12][23][24][25][26] . Far from the colloidal sphere, the director deviations n m (m ¼ x; y) from n 0 ¼ (0, 0, 1) are small. Assuming n(r)E(n x , n y , 1), the NLC elastic free-energy reads 7,8,23 where K is an average Frank elastic constant 1 . Euler-Lagrange equations arising from the functional (1) are of Laplace type, Dn m ¼ 0, with solutions expanded into multipoles where a l ¼ b l r 0 l þ 1 is the elastic multipole moment of the lth order (2 l -pole) [22][23][24][25][26] and one can find coefficients b l from exact solutions for n(r) or from relevant experiments (Figs 2a and 3a). Odd moments vanish because n(r) is symmetric about the particle centre, similar to analogous electrostatic charge distributions that have no dipole or octupole electric moments [22][23][24][25][26] . The multipole expansion of the induced n(r) shown in Fig. 1d is characterized  by coefficients b 2 , b 4 and b 6 , which also determine the colloidal pair-interaction energy 24 where P l þ l 0 (cosy) are the Legendre polynomials. For colloidal quadrupoles, b 2 dominates and b 4 and b 6 can play a role only at small 25 R. For our particles (Fig. 1), the induced n(r) can be qualitatively understood as a superposition of configurations of two separate quadrupoles, one with the Saturn-ring and one with boojums 9-11,16,22-26 , having opposite signs of quadrupole moments (compare Fig. 4a,d,g,j and 4b,e,h,k). Therefore, the net quadrupole moment is small and the high-order multipoles manifest themselves in a wide range of R. Fitting experimental R(t) within different angular sectors of pair interaction with the corresponding theoretical predictions yields a unique set of parameters b 2 , b 4 and b 6 (Fig. 3a). The quadrupole moment a 2 ¼ À 0.017r 0 3 is about two orders of magnitude smaller than that of elastic quadrupoles 16,24 , consistent with the hexadecapolar symmetry of n(r) that is found playing a dominant role, and even the higherorder term (64-pole) plays a detectable role at relatively small R (Supplementary Fig. 6 and Supplementary Note 2). The rich angular dependence of elastic pair-interactions predicted by equation 3 for coefficients b 2 , b 4 and b 6 obtained from fitting is consistent with our experimental characterization of the hexadecapolar nature of PSM colloids in the NLC host (Fig. 2).

Discussion
It is instructive to compare hexadecapolar NLC colloids to other known elastic multipoles and their electrostatic analogues. Colloidal spheres with strong tangential anchoring at their surface form n(r)-distortions of quadrupolar configuration 16,24,27,28 and are commonly called elastic quadrupoles (Fig. 4a,d,g,j). They have two surface point defects at the two poles of the spherical particle, which are called 'boojums'. Spherical particles with homeotropic surface anchoring can induce two different elastic multipoles, a dipole 23,28 (Fig. 4c,f,l) and a quadrupole 16,23,[27][28][29][30][31][32] (Fig. 4b,e,h,k). The elastic colloidal dipole has a bulk point defect called 'hedgehog' near one of the poles and lacks mirror symmetry with respect to the plane going through its equator orthogonally to n 0 (Fig. 4f,l). Such a colloidal dipole has a small octupole moment as well 33 . The elastic quadrupole around the particle with a strong homeotropic anchoring has a closed disclination loop around the equator called 'Saturn ring' and, in all four quadrants, has an opposite tilt of the deformed director with respect to n 0 as compared with the tangential elastic quadrupole 16 (compare Fig. 4g,j,h,k). In the case when anchoring is weak, an elastic multipole with weak distortions of quadrupolar configuration can be formed somewhat similar to the elastic quadrupole with 'Saturn ring', but its disclination loop is 'virtual' (within particle's volume) as the director is allowed to deviate away from the easy axis orientation 31,32 . The maps of x-component of the director, n x , calculated using methods described in refs 23,24 and plotted on spherical surfaces encompassing the particles and the defects, clearly illustrate the multipolar nature of elastic distortions similar to that in the electrostatics analogues 34,35 . A direct comparison of the symmetry of the director distortions around elastic dipoles and quadrupoles with that of hexadecapolar PSMs in NLCs helps to highlight the very different nature of these highorder NLC colloidal multipoles studied in our work (compare Figs 4j-l and 1h). On the other hand, the hexadecapolar NLC colloids further expand the analogy between electrostatic and elastic colloidal multipoles. In fact, the colour presentations of elastic distortions induced by colloidal dipoles (Fig. 4l), two different quadrupoles (Fig. 4j,k) and our hexadecapoles (Fig. 1g,h) resemble very closely the corresponding dipolar, quadrupolar and hexadecapolar electrostatic charge distributions on a sphere described by a spherical harmonic function 35 s l m (Y, f) ¼ Ncos(mf)P l m (cosY) with (l, m) ¼ (1, 1), (2, ±1) and (4, 1), respectively, where N is a normalization constant, Y is a polar and f is an azimuthal angles, P l m (cosY) is the associated Legendre polynomial; the constant l determines the order of a multipole (that is, 2 l -th pole) and -lrmrl. This close analogy between electrostatic and elastic colloidal multipoles may help devising approaches for self-assembly of colloidal mesostructured composite materials.
To conclude, we describe hexadecapolar NLC colloids with unusual field configurations, highly anisotropic elastic interactions, and versatile forms of self-assembly. Our findings pose a challenge of realizing pure NLC colloidal octupoles, lower order multipoles that have not been observed so far, and provide the means of probing the role of hexadecapolar moments in inter-molecular interactions using colloidal model systems. Self-assembly of hexadecapolar NLC colloids is expected to yield a diverse family of two-and three-dimensional low-symmetry crystal lattices. Beyond the rich experimental platform for fundamental studies, our colloids have potential uses in designing and realizing reconfigurable 'soft' photonic crystals 36 and other NLC-based mesostructured composites 37 with properties that can be pre-engineered through controlling tilted boundary conditions on nanoparticle surfaces 38-40 , particle shape, composition and topology, 2,6,8,41 as well as the NLC host properties, particle active behaviour 42 and the use of external stimuli 12,17 .

Methods
Preparation of nematic colloidal dispersions. Colloidal PSM particles of radii r 0 E2.05, 2.65 and 4 mm and with conically degenerate surface boundary conditions were prepared using one of two different methods as follows 43,44 . In the first method, asymmetric colloidal polystyrene dimers 43,45 were synthesized using a modified seeded polymerization technique [44][45][46] (Supplementary Fig. 1a). Resulting dimers in ethanol were sonicated for tens of minutes using an ultrasonic bath 8891 (Cole-Parmer) to break apart large and small spherical lobes forming the asymmetric colloidal dimer (Supplementary Fig. 1b,c). Colloidal spheres originating from these dimers were then re-dispersed in pentyl-cyano biphenyl (5CB) obtained from Frinton Laboratories, Inc. The larger (diameter D ¼ 2r 0 E4.1 mm) colloidal spheres, which were used as PSMs in our experiments, intrinsically provide conic degenerate surface boundary conditions for the NLC director. In the second preparation method, we used polystyrene divinylbenzene spherical particles PC06N (Bang Laboratories, Inc.) of DE8 mm and DC-05 (Thermo Scientific) of DE5.3 mm dispersed in 5CB. As dispersed, these particles provide tangential alignment for 5CB molecules and the director field n(r). However, after keeping these colloidal dispersions in 5CB at the elevated temperature of E100°C for about 12 h, the alignment at the surface of these polymer particles changed to a conical degenerate ( Supplementary Fig. 1d-f), similar to that reported in ref. 18 for flat surfaces coated by polystyrene. The tilt angle of the easy axis c e was varying approximately within 20°-75°, depending on details of preparation. The comparison of the surface free-energy density F s ¼ W c (cos 2 c À cos 2 c e ) 2 /2 costs (ref. 18) for deviating n(r) to a polar angle c away from the easy axis at c e relative to the bulk elastic energy density is characterized by the so-called conical anchoring extrapolation length K/W c B(50-500) nm o or 0 , where 18 W c B10 À 5 J m À 2 . Since K/W c o or 0 , one can assume that the tilted conically degenerate boundary conditions are rigid with respect to changing c. To prepare samples with NLC dispersions in a uniformly aligned host, dry PSMs were added to 5CB. The dilute NLC colloidal dispersions were then filled into the cells comprised of two glass plates separated by glass spacers defining the cell gap in the range 15-30 mm, which were sealed at all edges using fast-setting epoxy glue after the filling process. The inner surfaces of cell substrates were treated with polyimide PI2555 (obtained from HD Microsystems) for in-plane homogeneous alignment of the far-field director n 0 defined by unidirectional rubbing.
Imaging techniques and data acquisition. POM ( Fig. 1 and Supplementary  Figs 1,2), nonlinear optical imaging (Supplementary Fig. 3) and laser manipulations of PSM particles were performed using a single integrated setup capable of simultaneous conventional optical and nonlinear three-photon excitation fluorescence polarizing microscopy (3PEF-PM) 20   three-photon excitation of 5CB molecules and the 3PEF-PM signals were collected in epi-detection mode with a photomultiplier tube (H5784-20, Hamamatsu). An Olympus Â 100 oil-immersion objective with a high numerical aperture of 1.4 was used for both optical imaging and laser trapping. This experimental setup is described in details elsewhere 19,20 . Motion of colloidal particles was recorded with a charge-coupled camera (Flea, PointGrey) at a rate of 15 fps and their lateral positions versus time were determined from captured sequences of images using motion tracking plugins of ImageJ software (obtained from National Institutes of Health) with the accuracy 8 of 7-10 nm.
Measuring anisotropic diffusion of elastic NLC hexadecapoles. PSMs with conic anchoring distort the homogeneous n(r) and form elastic hexadecapoles in the aligned NLC (Fig. 1a-e and Supplementary Figs 1b,d-f, 2,3b). These hexadecapoles stay suspended in the bulk of NLC ( Supplementary Fig. 3b), which is facilitated by elastic repulsion of PSMs from the confining substrates and their Brownian motion because of thermal fluctuations. Using bright field microscopy and video tracking, one can determine a position of the particle within each frame and then analyse its translational displacements at regular time steps corresponding to the video frame rates. Following the well-established methodology 29 and using the histograms of displacements shown in Supplementary Fig. 4, it is possible to experimentally determine two independent diffusion coefficients D || ¼ 1.63 Â 10 À 3 mm 2 s À 1 and D ? ¼ 1.05 Â 10 À 3 mm 2 s À 1 , which characterize, respectively, the diffusion of elastic hexadecapoles along and perpendicular to the far-field director n 0 . As D || 4D ? , it is easier for the NLC hexadecapoles to move along n 0 than in other directions. The ratio D || /D ? ¼ 1.55 is not much different from values determined for other elastic multipoles studied previously 29,30 (Fig. 4). These coefficients are also used to determine friction coefficients from the Stokes-Einstein relation, which are then subsequently used in the calculation of anisotropic interaction forces 27 between hexadecapoles.