Nematic liquid crystal boojums with handles on colloidal handlebodies Qingkun Liua, Bohdan Senyuka,b, Mykola Tasinkevychc,d, and Ivan I. Smalyukha,b,e,f,1 aDepartment of Physics, bLiquid Crystal Materials Research Center, and eDepartment of Electrical, Computer, and Energy Engineering and Materials Science and Engineering Program, University of Colorado, Boulder, CO 80309; cMax-Planck-Institut für Intelligente Systeme, D-70569 Stuttgart, Germany; dInstitut für Theoretische Physik IV, Universität Stuttgart, D-70569 Stuttgart, Germany; and fRenewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder, CO 80309 Edited by Mark J. Bowick, Syracuse University, Syracuse, NY, and accepted by the Editorial Board April 26, 2013 (received for review January 23, 2013) Topological defects that form on surfaces of ordered media, dubbed multistable topology-constrained director patterns that form boojums, are ubiquitous in superfluids, liquid crystals (LCs), Langmuir around these handlebody-shaped particles. We show that 2D monolayers, and Bose–Einstein condensates. They determine supercur- defects in orientation of the director at the LC–colloid interface rents in superfluids, impinge on electrooptical switching in polymer- have a net strength adding to the value of the Euler characteristic dispersed LCs, and mediate chemical response at nematic-isotropic of the handlebody particle χ = 2 − 2g, as expected (17), although fluid interfaces, but the role of surface topology in the appearance, one typically also encounters defects of opposite signs at different stability, and core structure of these defects remains poorly under- locations on handlebody surfaces that self-compensate each other stood. Here, we demonstrate robust generation of boojums by con- and often annihilate with time, providing multiple different ways trolling surface topology of colloidal particles that impose tangential of satisfying topological constraints at multiple stable or meta- boundary conditions for the alignment of LC molecules. To do this, we stable orientations of particles. Topological charges of 3D textures design handlebody-shaped polymer particles with different genus g. of boojums, on the other hand, satisfy the topological charge conservation due to all defects induced in the LC. Numerical When introduced into a nematic LC, these particles distort the nematic modeling based on free energy minimization yields results con- molecular alignment field while obeying topological constraints and sistent with experimental findings and provides insights into dif- induce at least 2g − 2 boojums that allow for topological charge con- ferent handle-shaped core structures of the corresponding SCIENCES servation. We characterize 3D textures of boojums using polarized boojums. We discuss these findings from the standpoint of self- fi APPLIED PHYSICAL nonlinear optical imaging of molecular alignment and explain our nd- assembly–based approaches for fabrication of a unique class of ings by invoking symmetry considerations and numerical modeling of topological composites. experiment-matching director fields, order parameter variations, and nontrivial handle-shaped core structure of defects. Finally, we discuss Results and Discussion how this interplay between the topologies of colloidal surfaces and The lithographically fabricated handlebody colloids (details are boojums may lead to controlled self-assembly of colloidal particles in provided in Materials and Methods and Figs. S1 and S2)had nematic and paranematic hosts, which, in turn, may enable reconfig- a surface Euler characteristic χ = 0, −2, −4, −6, −8 (Table 1); urable topological composites. lateral diameters ranging from 5 to 10 μm; and rounded-square cross-sections of about 1 μm. A dispersion of these particles in colloids | low-dimensional topology | elasticity | topological theorems | a nematic LC, pentyl cyanobiphenyl (5CB), was infiltrated into thin disclinations cells, which, in the absence of inclusions, had a uniform in-plane director n0 in their interior. The handlebody colloids induced di- eing inspired by Lewis Carroll’spoemThe Hunting of the rector distortions around them that are described by the director fi n r n BSnark, Mermin (1, 2) introduced a term “boojum” to name eld ( ) approaching 0 at large distances. To minimize free elusive at the time surface defects in superfluids. This term and the energy due to elastic distortions (18), the particles tend to align n concept of boojums quickly penetrated different fields of physics with their ring planes parallel to 0 (Fig. 1), although metastable fi n and materials science, ranging from liquid crystals (LCs) to con gurations with ring planes perpendicular to 0 are occasion- Langmuir monolayers, and to Bose–Einstein condensates (3–6). ally observed (Fig. 2) and can be reproducibly obtained using local However, unlike in the case of their bulk topological counterparts melting of the LC by laser tweezers, followed by quenching it back (7–10), called “hedgehog” point defects, the appearance of boo- to the nematic state. Although mass density of the ultraviolet- ∼ 3 jums is rarely controlled at will. In LCs, boojums can spontane- sensitive polymer SU-8 ( 1,190 kg/m )comprisingparticlesis ∼ 3 ously appear at their interfaces with isotropic media, such as water. slightly higher than that of the 5CB nematic host ( 1,020 kg/m ), They are commonly associated with the geometries of thin LC repulsive elasticity-mediated interactions between particles and fi fl confining surfaces of cells with strong planar surface anchoring lms on surfaces of isotropic uids (11), LC droplets (3, 4, 12), and n r colloidal inclusions (13, 14). For example, spherical surfaces with of ( ) cause levitation of the colloidal tori in both orientations in tangential boundary conditions for rod-like LC molecules, and the vicinity of cell midplane (Figs. 1 F and G and 2E) while un- n dergoing both translational and rotational thermal fluctuations. director describing their local average orientation, are known to n induce two boojums per LC droplet or per colloidal inclusion in A single colloidal torus with a ring plane parallel to 0 is found to induce four boojums (Fig. 1). A less frequently observed the LC host (3, 4, 13, 14). However, despite the recent progress in fi generating and controlling bulk LC defects using director re- metastable con guration of a torus particle aligned perpendicular alignment with focused laser beams, chirality, and surface topol- ogy of colloidal particles with perpendicular surface anchoring introduced into LCs (7–10), similar control of boojums has not Author contributions: I.I.S. designed research; Q.L., B.S., M.T., and I.I.S. performed re- been demonstrated. search; Q.L., B.S., M.T., and I.I.S. analyzed data; and I.I.S. wrote the paper. In this work, we fabricate polymer microparticles with the to- The authors declare no conflict of interest. pology of a handlebody of genus g ranging from one to five that are This article is a PNAS Direct Submission. M.J.B. is a guest editor invited by the Editorial capable of imposing tangentially degenerate surface boundary Board. conditions for the director n. Using a combination of holographic Freely available online through the PNAS open access option. laser tweezers (HOT), bright-field microscopy, polarizing optical 1To whom correspondence should be addressed. E-mail: [email protected]. fl microscopy (POM), and three-photon excitation uorescence This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. polarizing microscopy (3PEF-PM) (15, 16), we control and deduce 1073/pnas.1301464110/-/DCSupplemental. www.pnas.org/cgi/doi/10.1073/pnas.1301464110 PNAS Early Edition | 1of6 Downloaded by guest on September 30, 2021 Table 1. Disclination strengths and point defect charges due to somewhat deviating from the ones in simulated textures (e.g., g-tori in a nematic LC compare Fig. 3 B, E, and J, showing a stronger asymmetry of No. of defects with strength/charge textures in simulations than in the experiments). Similar con- clusions can be extended to colloids of genus 3, 4, and 5 (Figs. 4 g χ = Σi si = 2Σi Nbi s =+1 s = −1 mb =+1/2* mb = −1/2* Examples and 5 and Table 1). For all colloids, the Euler characteristic measures the total strength of the point disclinations piercing the 1 0 2 2 2 2 Fig. 1C LC–colloid interface, Σi si = χ, in agreement with the Poincaré– 0 0 0 0 Fig. 2 Hopf index theorem (17). Although some boojums of opposite 2 −2 2 4 3 3 Fig. 3D signs annihilate with time and director structures around particles 46 5 5Fig. S3B transform between different metastable and stable states, this 3 −4 2 6 4 4 Fig. 4B relation always prevails. For example, two of the three boojums 4 8 6 6 Fig. 4L shown in the upper part of Fig. 4O (with surfaces of reduced − 4 6 2 8 5 5 Fig. 4I scalar-order parameter depicted using red and green colors) can 4 10 7 7 Fig. 4K annihilate with time, in agreement with the experiments (Fig. 4 A − 5 8 2 10 6 6 Fig. 5B and B). However, defect annihilation typically does not proceed 3 11 7 7 Fig. 5K until a minimum number of topologically required boojums is left, *Characteristic of boojums mb is defined in the main text. because some of these defects help to reduce elastic free energy (Figs. 1–5 and Table 1). Furthermore, colloids in all stable and metastable states undergo anisotropic diffusion (18), exhib- to n0 contains no boojums (Fig. 2) but, instead, a nonsingular iting both rotational and translational thermal motion (Fig. 5 D axially symmetrical n(r) (with bend, splay, and twist distortions) that satisfies tangential boundary conditions on the particle sur- face while approaching n0 at large distances from its surface (Fig. 2 G–I “ ” ns r A B C boojum ). The pattern of the 2D surface nematic director ( ) at the n0 surface of a torus with tangentially degenerate boundary con- ditions contains no defects in the latter case but four 2D defects = A A (point disclinations) in the former case, two of strength s 1and Z` two of strength s = −1, where s is defined as the number of times 5 μm ns(r) rotates by 2π as one circumnavigates the defect core once P P (Fig.