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 surfaces18. The director’s easy axis orientation lies on a cone of equilibrium polar angle ψ 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.

Figure 1: Elastic hexadecapole induced by a colloidal PSM. (a–d) Optical micrographs obtained with (a,b) POM and (c,e) bright field microscopy. P, A and γ mark the crossed polarizer, analyser and a slow axis of a 530 nm retardation plate (aligned at 45° to P and A), respectively. (d) Schematic diagram of induced n(r) (green rods) satisfying the tilted boundary conditions at the PSM surface (red rods), with the ‘easy axis’ at a constant angle ψ e to a local normal s to the surface (black rods). (f) Schematic of conic degenerate surface boundary conditions. (g,h) Three-dimensional visualization of the x-component n x of n(r) (g) at the surface of PSM for ψ e =45° and (h) at a spherical surface of radius 1.2r 0 shown using a dashed red circle in d. Blue, yellow and magenta colours correspond, respectively, to a positive, near-zero and negative n x according to the colour scheme shown in g. Dashed equatorial line in d and a black solid line in g depict the ‘Saturn ring’ surface defect loop at the particle’s equator visible in c. Black hemispheres in d and g show surface point defects boojums at the poles of the particle visible in e. Scale bars, 2 μm. Full size image

Elastic interactions between hexadecapolar colloids

Elasticity-mediated 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 elasticity-mediated interactions. The submicron waist and relatively low power (∼10 mW) of a focused trapping beam allow us to avoid the influence of the trapping on the measurements21. 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 inter-particle 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 colloids7,8,15,16, 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).

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 −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=2.4r 0 . (b) Pair-potential U int versus θ for two particles at R=2.4r 0 . Angular zones of repulsion and attraction are highlighted using magenta and green arrows and colouring, respectively. Full size image

To quantify elastic interactions, we first probe the centre-to-centre separation R=|R| 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 =d R/d t. Because the system is highly over-damped and the inertia effects can be neglected22, the experimentally measured R(t) and the simplified equation of motion 0≈−ξ +F int yield the pair interaction potential U int (inset of Fig. 3a), where ξ is a drag coefficient measured separately by characterizing Brownian motion of PSMs (Methods and Supplementary Fig. 4) and F int =−∇U int is the elastic interaction force. The attractively interacting particles stop short of physically touching each other, instead forming stable dimer assemblies with typical R≈(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 ψ e : θ 1 ≈22°-26° or θ 2 ≈64°-75° (Fig. 3b–f). Multi-particle assemblies with different combinations of angles θ 1 and θ 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 θ 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 three-dimensional 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 micrometer-sized 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 3: Self-assembly of hexadecapolar colloids. (a) Separation R versus time t for attractive interactions at different θ (measured upon the dimer formation). The solid red line shows a least-squares fit with R(t) obtained from the simplified equation of motion with (b 2 , b 4 , b 6 )=(−0.017, −0.092, 0.003); it is impossible to reproduce such a dependence R(t) with the parameter b 2 only since the quadrupole–quadrupole interaction is repulsive along θ≈23°. Inset shows the corresponding U int versus R. (b) and (c) POM micrographs of self-assembled colloidal dimers of hexadecapoles with R aligned at θ 1 and θ 2 to n 0 , respectively. (d) POM micrograph of a kinked colloidal chain of hexadecapoles. (e,f) Schematic diagrams of n(r) (green lines) for colloidal dimers shown in c,d, respectively. Dashed lines and black hemispheres in c and d depict surface defect loops and surface point defects at the poles of PSMs, respectively. Scale bars, 2 μm. Full size image

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 multipoles23, albeit our colloids dramatically differ from elastic dipoles and quadrupoles studied so far7,8,9,10,11,12,23,24,25,26. Far from the colloidal sphere, the director deviations n μ (μ=x; y) from n 0 =(0, 0, 1) are small. Assuming n(r)≈(n x , n y , 1), the NLC elastic free-energy reads7,8,23

where K is an average Frank elastic constant1. Euler–Lagrange equations arising from the functional (1) are of Laplace type, Δn μ =0, with solutions expanded into multipoles

where a l =b l r 0 l+1 is the elastic multipole moment of the lth order (2l-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 moments22,23,24,25,26. 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 energy24

where P l+l ′ (cosθ) are the Legendre polynomials. For colloidal quadrupoles, b 2 dominates and b 4 and b 6 can play a role only at small25 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 boojums9,10,11,16,22,23,24,25,26, 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 =−0.017r 0 3 is about two orders of magnitude smaller than that of elastic quadrupoles16,24, consistent with the hexadecapolar symmetry of n(r) that is found playing a dominant role, and even the higher-order 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).