Hexadecapolar colloids

Abstract

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.

Liquid Crystal Colloids

“…The leading-order multipoles are typically monopoles, dipoles, and quadrupoles, with higher-order multipoles rarely playing dominant roles, though we will see that they can be important for certain types of nematic colloids (2,(8)(9)(10)(11)(12)(13)(20)(21)(22)(23)(24). Much like in the case of electrostatic charge distributions, the far-field distortions of n(r) due to a colloidal particle can be represented as elastic multipoles (2,13,(20)(21)(22)(23)(24). Far from a colloidal particle, the deviations of the director n  (=x; y) from the far-field uniform director n 0 =(0, 0, 1) are small (24,25).…”

“…Much like in the case of electrostatic charge distributions, the far-field distortions of n(r) due to a colloidal particle can be represented as elastic multipoles (2,13,(20)(21)(22)(23)(24). Far from a colloidal particle, the deviations of the director n  (=x; y) from the far-field uniform director n 0 =(0, 0, 1) are small (24,25). Assuming one-elastic-constant approximation and representing the field as n(r)  (n x , n y , 1), the LC elastic free energy is expressed as…”

“…where a l =b l r 0 l+1 is the elastic multipole moment of the l-th order (2 l -pole), r 0 is particle's radius and expansion coefficients b l can be found from experiments (24,25). This electrostatics analogy provides a framework for understanding, predicting, and engineering both the director structure and self-assembly of nematic colloids.…”

“…The tilt of n(r) away from n 0 switches between clockwise and counterclockwise directions eight times as one circumnavigates the hexadecapole (Figure 2l), which is different from dipoles, where it switches two times, and quadrupoles, where it switches four times. Both the surface boojums at the particle's poles along n 0 as well as the Saturn ring at the particle's equator are induced by hexadecapoles (24). To minimize the free energy cost of bulk elastic distortions, interaction of conically degenerate BCs on the microsphere with the uniform n 0 lifts the conical degeneracy of surface anchoring and yields an axially symmetric n(r) that can be thought of as a superposition of the quadrupolar structures with boojums and Saturn ring (Figure 2l) (24).…”

Key-lock colloids in a nematic liquid crystal

“…Recently it was reported that polystyrene colloidal particles can align the LC molecules at an oblique angle (conical degenerate anchoring) giving rise to a new type of LC colloids [31]. Here we restrict our study to homeotropic and planar degenerate anchorings.…”