Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals

Some of the most exotic condensed matter phases, such as twist grain boundary and blue phases in liquid crystals and Abrikosov phases in superconductors, contain arrays of topological defects in their ground state. Comprised of a triangular lattice of double-twist tubes of magnetization, the so-called ‘A-phase’ in chiral magnets is an example of a thermodynamically stable phase with topologically nontrivial solitonic field configurations referred to as two-dimensional skyrmions, or baby-skyrmions. Here we report that three-dimensional skyrmions in the form of double-twist tori called ‘hopfions’, or ‘torons’ when accompanied by additional self-compensating defects, self-assemble into periodic arrays and linear chains that exhibit electrostriction. In confined chiral nematic liquid crystals, this self-assembly is similar to that of liquid crystal colloids and originates from long-range elastic interactions between particle-like skyrmionic torus knots of molecular alignment field, which can be tuned from isotropic repulsive to weakly or highly anisotropic attractive by low-voltage electric fields.

topological particles. The intensity in (d-e) increases from blue (lowest) to green, to yellow, and then to red (highest).

Supplementary Figure 3 | Many-body interactions between hopfion-based particles starting
from initial conditions in the forms of arrays subjected to different voltages. a, Laser induced mobile torons (generated at relatively low laser powers ~50 mW) and their interaction at different U starting from the same initial conditions. b, Structural organization of topological particles versus elapsed time (marked on the images) starting from an initial array of torons and upon application of U=2.7V peak to peak, 1 kHz square wave, at which attractive interactions lead to a small-periodicity hexagonal arrangement (Fig. 2c). c, Structural organization of topological particles versus elapsed time (marked on the images) starting from an initial array of optically generated torons and upon application of U=3.1V peak to peak, 1 kHz square wave, at which strongly anisotropic dipolar interactions lead to a linear-chain self-assembly (Fig. 2d); note that one dipolar topological particle disappeared (due to the toron-umbilic annihilation) between frames 4 and 5.
Supplementary Figure 4 | Numerical modeling of the 3D director structure and polarizing optical micrographs of skyrmionic particles in CNLCs. a-e, In-plane cross-sections of a topological particle, whose vertical cross-section is shown in Fig. 1o, at five different sample depths while moving from the cell top (a) to the cell midplane (c) and to (e) the bottom part of the cell. f-i, Details of director structure of a topological particle with cross-section shown in Fig.  1o, which is reproduced here as part (f) with labeling of vertical cross-sections orthogonal to it and shown in (g-i). j,k, Computer-simulated polarizing optical micrographs corresponding to the director configurations shown in Fig. 1n,o, respectively; the smaller insets shown to the right of computer-simulated micrographs are the corresponding experimental polarizing optical micrographs. Figure 5 | Diffraction pattern obtained using a monodomain hexagonal array of self-assembled topological particles, similar to that shown in Fig. 2a. The pattern was obtained using a HeNe laser beam. K further helps to stabilize these skyrmionic field configurations, as we have discussed for torons previously. 3 We therefore set

Supplementary
K while using all other experimentally measured elastic and dielectric constants of the used liquid crystal (Supplementary Table 1).
Minimization of the free energy to find the equilibrium director field is implemented with the relaxation method. 2 Electric free energy is calculated with a static voltage profile. This is a reasonable assumption because the absolute value of the dielectric anisotropy    Table 1.
For all voltages U, the initial director configuration used as a starting point of the minimization procedure described above was a toron director field configuration deduced from the experiments at U=0. Additionally, to obtain the equilibrium n(r)-field configurations at different U, the toron was surrounded by TIC configuration for the corresponding U relaxed separately; 3 the entire sample volume was then relaxed for the respective U. The relaxed director configurations at different voltages and d/p ratios yield a broad range of n(r)-structures discussed in details in the main text of this work, which are found to be in agreement with experiments.

Laser manipulation and 3D optical imaging
We used an integrated system for simultaneous optical manipulation and 3D imaging, which was built around an inverted microscope IX 81 (Olympus). The holographic optical trapping  Supplementary Figures 1-3) and monochromatic light (Fig. 4a) through the sample between crossed polarizers is used in polarizing microscopy studies. The use of scanned monochromatic laser light allowed us to simultaneously obtain co-located polarizing optical micrographs and both 2PEF-PM or 3PEF-PM depth-resolved images, an example of which is shown in Fig. 4a,b. We use oil-immersion objectives with high numerical aperture (NA), 60x (NA=1.42) and 100x (NA=1.4), both from Olympus. The same objectives are used for imaging as well as optical trapping. The n(r) and its azimuthal orientation patterns presented in the main text were obtained through the analysis of Stokes parameters described in details in Ref. 6.

Preparation of polymerized CNLC samples
The partially polymerizable cholesteric LC composite was prepared by first mixing 69% of nonreactive negative-dielectric-anisotropy nematic AMLC-001 (from Alpha Micron Inc.) with 30% of a diacrylate nematic (consisting of 12% of RM 82 and 18% of RM 257 obtained from EM Chemicals) and 1% photoinitiator Irgacure 184 (from CIBA Specialty Chemicals), which was then followed by doping this nematic mixture with CB15 (from EM Chemicals) to obtain a cholesteric of pitch p equal to 15 μm. 7 The ensuing mixture was first dissolved in dichloromethane to homogenize, heated to 80 °C for one day to remove the solvent through slow evaporation, and then cooled down to obtain a room-temperature CNLC mixture. This CNLC mixture was then used to optically generate and subsequently polymerize at different applied voltages solitonic structures of interest. The cholesteric mixture was infiltrated into LC cells with thickness d comparable to p. To fabricate the cells, glass substrates with conductive indium tin oxide coatings were spin coated with polyimide SE1211 (Nissan) at 2700 rpm for 30 s and then baked (5 min at 90 °C followed by 1 h at 180 °C) to set strong vertical surface boundary conditions for the LC director. This particular type of samples was used to obtain depth-resolved images similar to that shown in Fig. 1i.