Biaxial ferromagnetic liquid crystal colloids

Significance

We introduce a soft-matter system with fluidity coexisting with long-range biaxial and ferromagnetic ordering of anisotropic molecular and magnetic colloidal building blocks. The facile polar switching of this complex fluid promises technological applications and rich physical behavior arising from the properties of solid magnetic nanoparticles and their long-range ordering prompted by interactions with the host medium.

Abstract

The design and practical realization of composite materials that combine fluidity and different forms of ordering at the mesoscopic scale are among the grand fundamental science challenges. These composites also hold a great potential for technological applications, ranging from information displays to metamaterials. Here we introduce a fluid with coexisting polar and biaxial ordering of organic molecular and magnetic colloidal building blocks exhibiting the lowest symmetry orientational order. Guided by interactions at different length scales, rod-like organic molecules of this fluid spontaneously orient along a direction dubbed “director,” whereas magnetic colloidal nanoplates order with their dipole moments parallel to each other but pointing at an angle to the director, yielding macroscopic magnetization at no external fields. Facile magnetic switching of such fluids is consistent with predictions of a model based on competing actions of elastic and magnetic torques, enabling previously inaccessible control of light.

Liquid crystals (LCs) that combine fluidity with many forms of orientational and partial positional order are ubiquitous (1, 2). Fluids with polar ordering were envisaged by Born a century ago (3–5), with their study recently guided by prescient theories of Brochard and de Gennes (6–12). An experimental search for small-molecule biaxial nematic fluids has gone on for decades (2, 13). Many types of low-symmetry ordering have been found in smectic and columnar systems (14, 15) with fluidity in only two and one dimensions, respectively (1). However, nematic LCs with 3D fluidity and no positional order tend to be nonpolar, although phases with polar and biaxial structure have been considered (15–17). In colloids, such as aqueous suspensions of rods and platelets, nonpolar uniaxial ordering is also predominant (1, 18). At the same time, there is a great potential for guiding low-symmetry assembly in hybrid LC-colloidal systems, in which the molecular LC is a fluid host for colloidal particles (18). Different types of LC-mediated ordering of anisotropic particles can emerge from elastic and surface-anchoring-based interactions and can lead to the spontaneous polar alignment of magnetic inclusions (6), although the orientations of the magnetic dipoles of colloidal particles were always slave to the LC director n, orienting either parallel or perpendicular to it without breaking uniaxial symmetry (6–12).

In this work, by controlling surface anchoring of colloidal magnetic nanoplates in a nematic host, we decouple the polar ordering of magnetic dipole moments described by macroscopic magnetization M from the nonpolar director n describing the orientational ordering of the LC host molecules. The ensuing biaxial ferromagnetic LC colloids (BFLCCs) possess 3D fluidity and simultaneous polar ferromagnetic and biaxial order. Direct imaging of nanoplates and their magnetic moment orientations relative to n and holonomic control of fields that strongly couple to M and reveal their orientations, as well as numerical modeling and optical characterization, provide the details of molecular and colloidal self-organization and unambiguously establish that BFLCCs have C_s (also denoted C_1h) symmetry. This symmetry, which has three distinct axes and is thus biaxial, is lower than the orthorhombic D_2h symmetry of conventional biaxial nematics (13) and other partially ordered molecular and colloidal fluids (1, 2, 17). We explore polar switching of this system and describe its unusual domain structures. We discuss potential applications and foresee exciting science emerging from the new soft matter framework that the BFLCCs introduce.

Results and Discussion

Our experiments use ferromagnetic nanoplates (FNPs) with average lateral size 140 nm and thickness 7 nm (12) coated with thin (<6 nm) layers of silica and surface-functionalized with polymer (19) to yield conically degenerate surface boundary conditions for n (20) (Fig. 1 A and B and Fig. S1). FNPs spontaneously orient with their magnetic dipole moments m, which are normal to their large-area faces, tilted by an angle θ_me with respect to the far-field director n₀ (Fig. 1C), as dictated by minimization of the surface anchoring free energy arising from the conical boundary conditions imposed by polymer surface functionalization (20). By inducing electrophoretic motion of FNPs in response to an electric field, with the electrophoretic and the viscous drag forces balanced, and by independently estimating the viscous drag coefficient through characterization of Brownian motion (Fig. S2), we find that FNPs have negative surface charge in the range of 100–500e per particle, where e ≈ 1.6 × 10⁻¹⁹ C (Supporting Information) (21). Stabilized against aggregation by weakly screened electrostatic repulsions in the LC due to nanoplate charging, FNPs uniformly disperse and form BFLCCs at concentrations ∼1 wt % (∼0.2 vol %) and higher (Fig. 1D). Salient properties of BFLCCs include magnetic hysteresis and threshold-free polar switching (Figs. 1 E and F and 2). Hysteresis loops are observed for both homeotropic (Fig. 2 A, C, and D) and planar (Fig. 2E) cells with n₀ perpendicular and parallel to confining substrates, respectively, as well as for orientations of applied field B and measured M-components parallel and perpendicular to n₀. Resisted by elastic energy costs of director distortions, magnetic switching of single-domain BFLCCs is threshold-free for fields applied in directions parallel and perpendicular to n₀ and is different from that of uniaxial ferromagnetic LC colloids (6–12). The field B‖n₀ applied to a BFLCC with n₀ along the cell normal z, with a tilted M spontaneously along one of the orientations on the up-cone, rotates M toward B and thereby tilts the director away from the confining surface normal, leading to light transmission through the sample placed between crossed polarizers (Fig. 2B). The angle θ_n between the BFLCC director n and z in the middle of a homeotropic cell increases up to θ_me (Fig. 2 F and G). Reversal of B causes a much more dramatic response of the BFLCC between crossed polarizers mediated by a strong reorientation of M and n (Fig. 2B), with θ_n increasing above 90° at strong fields rather than saturating at θ_me (Fig. 2 F and G). For a BFLCC with the down-cone orientation of M with respect to n₀ this behavior is completely reversed but consistent from the standpoint of the mutual orientations of B, M, and n₀, showing that switching is polar (Fig. 2B) but different from that of uniaxial ferromagnetic LC colloids (9–12). BFLCCs exhibit hysteresis for M-components parallel and perpendicular to n₀ (Fig. 2 A and C–E).

Fig. 1.

BFLCCs formed by collectively tilted FNPs in an LC. (A) Schematic of the FNP. (B) Scanning TEM image of FNPs and a zoomed-in TEM image of a nanoplate’s edge revealing ∼5-nm-thick silica shell (Inset). (C) Schematic of an FNP in LC. (D) BFLCC with M tilted with respect to n₀. (E and F) Experimental (symbols) and theoretical (solid curves) polarized absorption spectra for linear polarizations P, revealing alignment of FNPs without and with B = 2 mT in (E) homeotropic and (F) planar cells. Pure absorbances α_⊥ and α_‖ of FNPs were calculated from experimentally measured values described in ref. 10 for P⊥m and P‖m, respectively.

Fig. S1.

TEM images of as-synthesized FNPs with polygonal (hexagonal and triangular) shapes and with a broad size distribution (A and B) before surface functionalization and (C) after coating with thin silica layers and subsequent PEG functionalization. A, Inset shows a side view of an FNP with the thickness of 9 nm. C, Inset shows a zoomed-in view of the TEM image of an individual FNP with the silica shell visible at its edge, from which the thickness of the silica shell is determined.

Fig. S2.

Magnetic control and dynamics of FNPs in LC hosts. (A) Integrated magnetic holonomic control and optical imaging setup composed of electromagnetic solenoids arranged on an aluminum frame and mounted on an inverted optical microscope (not shown). The solenoids are driven by a computer-controlled DAQ and by three amplified power supply units. This magnetic control system is integrated with an optical imaging system capable of dark-field and bright-field optical microscopy, POM, and confocal fluorescence imaging. The control of phase of ac currents in the three coils, arranged with their axes to match the Cartesian coordinate system axes, allows us to apply a uniform field B in the sample volume above the objective in any desired direction. (B) A histogram showing distribution of lateral displacements Δr of FNPs in a homeotropic glass cell of thickness 30 μm measured using video microscopy for elapsed time steps of 67 ms at no applied fields. (C) A distribution of lateral displacements characterizing diffusion of a single FNP in a homeotropic cell without and with an applied magnetic field B‖x (B = 5 mT). (D) A distribution of lateral displacements characterizing diffusion of a single FNP in a planar cell (n₀‖y) without magnetic field. (E) A distribution of lateral displacements characterizing diffusion of FNP in isotropic phase without and with in-plane magnetic field (B‖x, B = 5 mT). C–E, Insets schematically illustrate geometry of the experiment by depicting the vertical cross-section of the glass cell and the orientations of coordinate axes, m, n₀, and applied fields.

Fig. 2.

Magnetic hysteresis and switching of BFLCCs. (A) Experimental hysteresis loop measured along n₀ in a homeotropic cell. Schematics show orientations of M and n. (B) Experimental (symbols) and computer-simulated (black solid curve) light transmission of an aligned single-domain BFLCC between crossed polarizers in a cell with n₀ and H normal to substrates. (C) Computer-simulated hysteresis loop for a BFLCC with domains (top left inset) fitted to experimental data (triangles) by varying the color-coded (right-side inset) lateral size to cell thickness ratio from 0.5 to 2. (D) Hysteresis loop probed for the same cell as in A but for H⊥n₀. (E) Hysteresis in a planar BFLCC cell for M along x, y, and z axes (Insets). (F and G) Computer-simulated (F) depth profiles of |θ_n| in a homeotropic cell of thickness d = 60 μm at different fields (note that, due to strong boundary conditions for n at the confining substrates, BFLCC cells with smaller d require stronger fields for switching) and (G) field dependencies of the maximum-tilt |θ_n| in the cell midplane for different θ_me. Computer simulations are described in Supporting Information.

We explore FNP-LC dispersions starting from individual particles. The surface anchoring energy per FNP as a function of angle θ_m between m and n₀ can be found by integrating the energy density W_s(θ_m), characterized by the conical anchoring coefficient A (20), over the surface area σ of FNP with radius R while neglecting contributions of side faces:

$$F_s={\displaystyle {\int }_{\sigma }{W}_s}\left({\theta }_m\right)dS=\pi A{R}^2{({\cos }^2{\theta }_m-{\cos }^2{\theta }_{me})}^2/2.$$ [1]

Magnetic fields can rotate FNPs and m away from the minimum-energy orientation at θ_m = θ_me, as discussed by Brochard and de Gennes (6) in the one-elastic-constant (K) approximation while accounting for the energetic costs of rotation-induced elastic distortions for infinitely strong anchoring. By extending this model to the case of finite-strength conically degenerate boundary conditions (Fig. 3A), we find the total elastic and surface anchoring energy cost of rotating the FNP away from the equilibrium orientation for small θ_m − θ_me:

$$F_{se}\approx 4\pi KA{R}^2{\sin }^2\left(2{\theta }_{me}\right){({\theta }_m-{\theta }_{me})}^2/[8K+\pi AR{\sin }^2\left(2{\theta }_{me}\right)].$$ [2]

We were able to vary θ_me between 10–65° by adjusting details of the silica coating and polyethylene glycol (PEG) functionalization that alter the density of the polymer brushes grafted on the FNP surfaces (19, 20). This control of θ_me is consistent with the fact that direct surface functionalization of FNPs without silica coating yields perpendicular boundary conditions (10, 12) whereas a dense PEG functionalization of silica plates yields nearly tangential anchoring (Fig. S3). In the presence of B, the response of individual nanoplates is described by the corresponding energy $F_H=-\mathbf{m}\cdot \mathbf{B}$. In dilute FNP dispersions, the distribution of m and nanoplate orientations due to the total potential energy is then $f\left({\theta }_m\right)=C\exp \left[-(F_{se}+F_H)/k_{\text{B}}T\right]$, where k_B is the Boltzmann constant, T is absolute temperature, and the coefficient C is found from ensuring ${\displaystyle {\int }_0^{\pi }C\exp \left[-(F_{se}+F_H)/k_{\text{B}}T\right]}\sin {\theta }_{m}d{\theta }_m=1$. This field-dependent angular distribution, along with measured material parameters (Supporting Information and Fig. S4), allows us to model experimental absorbance spectra (Fig. 1 E and F). At fields ∼1 mT perpendicular to n₀, the individual FNPs first rotate on the cone of easy orientations to lower F_H while keeping θ_m close to θ_me and F_se near its minimum (Fig. 3 B and C), with the departure θ_m − θ_me ≈ 4° determined by a balance of elastic, surface anchoring, and magnetic torques originating from the angular dependencies of F_se and F_H. These tilted orientations of individual FNPs are consistent with self-diffusion of nanoplates probed by dark-field video microscopy (Fig. S2 and Movie S1).

Fig. 3.

Alignment, rotation, and elastic interactions between magnetically torqued FNPs in LCs. (A) Free-energy minimization for an FNP with finite conically degenerate boundary conditions, with n(r) distorted around the nanoplate and deviating by an angle Δθ from the easy axis orientation at its surface. (B) Equilibrium alignment of representative FNPs at B = 0 in a homeotropic cell. (C) Response of the original four FNPs to a very weak field B₁ ∼0.1 mT, at which the nanoplates rotate mostly on the cone of low-energy orientations. (D) Rotation of FNPs in field B₂ ∼10 mT that induces monopole-type elastic distortions with the n(r) tilt determined by the initial orientation of m on the up- or down-cone. (E–H) Minimization of elastic energy due to FNP-induced distortions prompts long-range interactions (E and F) attractive for nanoplates with like-tilted n(r) and (G and H) repulsive for FNPs with oppositely tilted n(r). Elastic energy of director distortions (dashed ellipsoid in E) lowers with decreasing distance between like-tilted nanoplates (F) and is relieved (G and H) with increasing distance between oppositely tilted FNPs due to incompatible distortions they induce (dashed ellipsoid in H).

Fig. S3.

Optical micrographs showing silane-PEG (5k Da) capped SiO₂ hexagonal colloidal platelets in (A–C) a planar cell and (D–F) a homeotropic cell. The normals to the large-area platelet surfaces of the individual particles point nearly orthogonally to the far-field director n₀ and freely rotate around it, which is different from the alignment of the platelet normal parallel to n₀ that was achieved in our previous work (10, 12). These images, along with refs. 10 and 12, and representative images showing tilted orientation of nanoplates with respect to n₀ in Fig. 6 I and J in the main text demonstrate that the alignment of platelets in the nematic host can be controlled from parallel to orthogonal by varying the surface functionalization with PEG. POM images in C and F were obtained using a 530-nm phase retardation plate inserted between the crossed polarizer (P) and analyzer (A), with the slow axis (γ) marked by a yellow double arrow. The inset between A and D shows a top view of the hexagonal platelet.

Fig. S4.

Distribution of ${\theta }_m$ at B = 0 (black curve), B = 2 mT (red curve), B = 5 mT (green curve), and B = 10 mT (blue curve). The modeling was performed assuming that the nanoplates in a dilute dispersion in a nematic LC respond on an individual basis and for the following parameters: 2R = 140 nm, K = 6 × 10⁻¹² N, A = 10⁻⁵ J/m², ${\theta }_{me}$ = 48.5°, m = 3 × 10⁻¹⁷ Am², ${\rho }_N$= 1.6 × 10¹⁹ m⁻³, and ξ = 1.2 J/m³.

Applied fields alter the distribution of FNP orientations (Fig. 1 E and F and Fig. S4) in a dilute dispersion, prompting additional distortions of the director around individual FNPs. The response of the composite to B both along and perpendicular n₀ is paramagnetic-like and thresholdless (Fig. S5). For example, the field-induced birefringence and phase retardation ∼π in homeotropic cells with n₀ orthogonal to substrates (Fig. S5) is a result of the superposition of weak director distortions prompted by small rotations of individual FNPs in the dilute dispersion. Even the Earth’s magnetic field of ∼0.05 mT can rotate such nanoplates in LC to θ_m − θ_me ≈ 0.3°. At strong fields ∼20 mT, however, the individual FNPs rotate to large angles, so that their moments m approach the orientation of B and rotation-induced n(r) distortions slowly decay with distance away from them (Fig. 3D). The distorted n(r) can have two mutually opposite local tilts induced by rotations of nanoplates dependent on the initial alignments of m on the up- or down-cones (Fig. 3 C–H). These distortions prompt elastic interactions between the nanoplates, attractive for the same tilts and repulsive for the opposite ones (Fig. 3 E–H). Elastic interactions thus separate the nanoplates into domains with magnetic moments m that have the same up- or down-cone orientations (Fig. 4). For example, strong fields B⊥n₀ (∼20 mT) rotate nanoplates and local n(r) in a cell with initial n₀ perpendicular to substrates, causing elastic interactions and formation of ferromagnetic “drops” (localized regions with an increased density of FNPs with the same up-cone or down-cone orientations) when starting from low initial concentrations of nanoplates <0.5 wt % (Fig. 4 A–C). Spatially continuous ferromagnetic domains with M on up- or down-cones emerge in response to the same fields when starting from initial concentrations >0.5 wt % (Fig. 4 D–F). High-resolution dark-field video microscopy monitors kinetics of changes of the local number density of nanoplates in response to B (Fig. 4 G–I), until the interparticle separation becomes smaller than the optical resolution (Fig. 4G). FNP dispersions remain stable after prolonged application of strong fields.

Fig. S5.

(A–J) POM micrographs showing paramagnetic response of a dilute dispersion of FNPs in 5CB in a homeotropic cell. The response to the field along the far-field director n₀ indicates that the platelets spontaneously align with their normal and m tilted (or orthogonal) with respect to n₀ (perpendicular to confining substrates and the POM image plane) and then reorient in response to B while distorting n(r), to which they are mechanically coupled through surface anchoring. The POM micrographs were obtained with a 530-nm phase retardation plate with a slow axis γ (marked by a yellow double arrow) inserted between the crossed polarizer (P) and analyzer (A). The cell thickness is 30 μm. The direction and magnitude of B are marked on the images.

Fig. 4.

Concentration-dependent behavior of FNP-LC dispersions. (A–F) POM micrographs showing a cell with initial homeotropic n₀ for the following FNP concentrations and applied fields: (A) 0.07 wt %, B = 20 mT, (B) 0.13 wt %, B = 20 mT, (C) 0.27 wt %, B = 2 mT, (D) 0.53 wt %, B = 2 mT, (E) 1.07 wt %, B = 2 mT, and (F) 1.54 wt %, B = 2 mT. (G) Changes of local concentration with time in a field of 20 mT applied orthogonally to n₀ of a homeotropic cell. (H and I) Dark-field micrographs corresponding to 0-s and 63.3-s data points marked by red circles in G. The concentration keeps changing at higher fields and longer elapsed times but tracking FNPs at internanoplate distances <400 nm is limited by optical resolution. Orientations of B, polarizer (P), and analyzer (A) are marked on images.

Electrostatic charging of nanoplates in the LC with large Debye screening length λ_D = 0.3–0.5 μm (21) leads to long-range screened electrostatic repulsions. This agrees with video microscopy observations that individual nanoplates rarely approach each other to distances smaller than 0.5–1 μm, even in fields ∼5 mT applied in different directions (Fig. S6). The pair potential U_elect due to the screened Coulomb electrostatic repulsion between FNPs modeled as spheres of equivalent radius R is

$$U_{elect}\left(r_{cc}\right)=(A_1/r_{cc})\exp (-r_{cc}/{\lambda }_D),$$ [3]

where r_cc is the center-to-center pair-separation distance, λ_D = (εε₀k_BT/2N_Ae²I)^−1/2, ε is an average dielectric constant of the LC, ε₀ is vacuum permittivity, N_A is the Avogadro’s number, I is the ionic strength, $A_1={\left(Z^{\ast }e\right)}^2\exp (2R/{\lambda }_D)/\left[{\epsilon }_0\epsilon {(1+R/{\lambda }_D)}^2\right]$, and Z* is the number of elementary charges on a single FNP. For 2R ≈ 140 nm, Z* ≈ 500, ε ≈ 11.1, and λ_D ≈ 378 nm (21), one finds A₁ ≈ 6.8 × 10⁻²³ J/m. Minimization of free energy of the elastic distortions induced by FNPs leads to an elastic pair-interaction potential that contains monopole and highly anisotropic dipolar and quadrupolar terms dependent on magnetic field intensity H:

$$U_{elast}\left(r_{cc}\right)=A_2/r_{cc}+A_3\left(\varphi \right)/{r_{cc}}^3+A_4\left(\varphi \right)/{r_{cc}}^5,$$ [4]

where A₂, A₃, and A₄ are coefficients describing the elastic monopole, dipole, and quadrupole and ϕ is an angle between the center-to-center pair-separation vector r_cc and n₀ (22). The magnetic pair potential due to moments m₁ and m₂ of FNPs is

$$U_m\left(r_{cc}\right)=\frac{{\mu }_0}{4\pi }\frac{1}{{r_{cc}}^3}[{\mathbf{m}}_1\cdot {\mathbf{m}}_2-\frac{3({\mathbf{m}}_1\cdot {\mathbf{r}}_{cc})({\mathbf{m}}_2\cdot {\mathbf{r}}_{cc})}{{r_{cc}}^2}],$$ [5]

where ${\mu }_0$ is vacuum permeability. Superposition of Eqs. 3–5 gives the total interaction potential:

$$U\left(r_{cc}\right)=U_{elect}\left(r_{cc}\right)+U_m\left(r_{cc}\right)+U_{elast}\left(r_{cc}\right).$$ [6]

At r_cc ≥400 nm, corresponding to FNP dispersions up to 0.8 wt % (close to the initial concentration yielding continuous magnetic domains), magnetic pair interactions between the 140- × 7-nm nanoplates with dipoles ∼4 × 10⁻¹⁷ Am² are weak, with U_m ≤1 k_BT. For larger FNPs (Supporting Information and Fig. S1) with magnetic moments up to ∼17 × 10⁻¹⁷ Am² and at higher concentrations of FNPs, U_m including many-body effects overcomes the strength of thermal fluctuations, producing spatial patterns of domains. In Eq. 4, the first monopole term is nonzero only at B pointing away from orientations of m that minimize F_se. The dipolar and quadrupolar terms are always present due to symmetry of elastic distortions induced by the geometrically complex FNPs (Fig. 1B and Fig. S1) tilted with respect to n₀. The elastic dipole and quadrupole terms help maintain correlated orientations of FNPs and their magnetic moments upon formation of BFLCCs in concentrated dispersions. When B rotates the nanoplates, the dominant elastic interactions are of the monopole type, mediating the formation of ferromagnetic “drops” as the local density of nanoplates is increased starting from low initial volume fractions (Fig. 4 A–C) and of continuous domains when starting from higher initial concentrations >0.5 wt % (Fig. 4 D–I). Aggregation of nanoplates is prevented by weakly screened repulsive U_elect. Short-term 5- to 50-s application of a field ∼20 mT or, alternatively, prolonged application of weak fields <1 mT separates the FNPs in concentrated dispersions into domains that exhibit polar switching.

Fig. S6.

Center-to-center separation distance r_cc distributions for a pair of FNPs studied by means of dark-field video microscopy. The distributions were characterized with or without the fields applied in the in-plane directions orthogonal to n₀ (A and B) in a homeotropic cell with n₀ orthogonal to the confining glass plates at (A) B = 0 and (B) B = 4 mT, as well as in a planar cell for (C) B = 0 and (D) B = 4 mT. The insets show the corresponding representative dark-field images of a pair of FNPs in the LC host.

As in uniaxial ferromagnetic LC dispersions (7, 8, 9–12), preparation of BFLCCs can involve quenching of the LC host from the isotropic phase in an external B, which can produce up-cone or down-cone orientation of M within the entire sample. Alternatively, polydomain BFLCCs are obtained from a paramagnetic colloidal dispersion with initially random up- and down-cone orientations of m of FNPs by applying B to separate them into a sample of multidomains of opposite cone orientation (Fig. 5 and Movie S2), as shown using a sequence of micrographs in Fig. S7. The paramagnetic dispersion with random up- and down-cone orientations of FNPs is long-term unstable with respect to formation of ferromagnetic domains due to Earth’s and ambient magnetic fields. By applying B∼20 mT, a polydomain sample with up- and down-cone domains can be transformed to a monodomain BFLCC with the M-cone orientation matching the direction of applied field B‖n₀, as well as reversed by then reversing B to the opposite (Fig. S8). Switching M between up- and down-cones involves singular defects in n(r) visible in polarized optical microscopy (POM) (Fig. S8). Interestingly, even for M oriented on the same up- or down-cone within the entire sample, one observes spontaneous spatial variations of M(r), leading to a structure (Fig. 6) dubbed “left-right domains” with different tilts of M selected from the degeneracy of states of the same up- or down-cone.

Fig. 5.

Up-down domains in a homeotropic cell. (A–C) POM micrographs obtained (A) without B and (B and C) for B‖n₀ (B = 2 mT) (B) without and (C) with a 530-nm phase retardation plate inserted between crossed P and A, with its slow axis γ shown using a yellow double arrow. The POM micrograph in A appears dark because of the light propagation along n₀ and weak optical biaxiality of BFLCCs. (D) Schematic of up-down domains and walls. (E and F) POM images of domains at B in the opposite direction (B = −2 mT) (E) without and (F) with the retardation plate. Complementary POM micrographs B and C and E and F originate from highly asymmetric response of the up- and down-cone domains to opposite vertical fields. (G–I) POM micrographs of domains at B⊥n₀ (B = 2 mT) (G) without and (H and I) with the retardation plate.

Fig. S7.

(A–F) POM micrographs showing the process of initiation of the domain structure in a homeotropic cell with n₀ orthogonal to the confining glass plates by applying magnetic field B = 30 mT in the in-plane direction marked on the micrographs (B⊥n₀). The elapsed time is marked in the bottom-right corners of corresponding micrographs. Orientations of crossed polarizer (P) and analyzer (A) are shown using double arrows.

Fig. S8.

BFLCC domains and their switching. (A–F) POM micrographs showing switching of a BFLCC sample with the up-down domains, with the magnetization cone in some of the domains flipping starting from the initial orientation in the upward direction in response to field B applied in the downward direction. The initial homeotropic structure of the director in A is accompanied by the spatial variations of the magnetization that can live on opposite cones with respect to the n₀; this is evident from nonuniform realignment that becomes noticeable from the POM micrographs (B–F) in the applied field. The domain switching occurs through propagation of defects, the interdomain walls that move to modify the domain structure. The applied field magnitude was 20 mT and its direction normal to substrates is depicted on micrographs. (G–L) Magnetic switching and domain structure of the left-right domain initially aligned by magnetic field B‖n₀ (B = 20 mT) in a homeotropic cell. The elapsed time and the B direction are marked in the bottom-right corners of micrographs. Orientations of crossed polarizer (P) and analyzer (A) are shown using double arrows.

Fig. 6.

Left-right domains in a homeotropic cell. (A) Schematic of the domains and walls. (B–D) POM images at (B) B = 0 and (C and D) at B‖n₀ (B = 2 mT). (E–H) POM images of domains at B⊥n₀ (B = 2 mT) with a 530-nm retardation plate with a slow axis γ (yellow double arrow) inserted between the crossed P and A. The elapsed time is marked on images. (I and J) TEM images of FNPs in a polymerized BFLCC for two microtome cutting planes parallel to n₀, with the image in I containing M and that in J orthogonal to I. The inset in I shows coloring of the domains with differently tilted m (brown arrows) with respect to n₀ (black double arrows). The cross-sections of obliquely sliced FNPs in J reveal their tilt with respect to the image plane and n₀ (Inset). (K) Distribution of orientations of m measured using TEM images.

BFLCCs are modeled using a continuum description invoking minimization of the total free energy composed of elastic, magnetic, and coupling terms [assuming that the boundary conditions for n(r) on confining cell substrates are infinitely strong and neglecting spatial gradients of the FNPs density]:

$$F=F_{elast}+F_{mag}+F_{coupl}.$$ [7]

The elastic energy due to the spatial gradients of n(r) is

$$F_{elast}=\frac{1}{2}{\displaystyle \int \{K_{11}{(\nabla \cdot \mathbf{n})}^2+K_{22}{(\mathbf{n}\cdot \nabla \times \mathbf{n})}^2+K_{33}{[\mathbf{n}\times (\nabla \times \mathbf{n})]}^2\}}dV,$$ [8]

where K₁₁, K₂₂, and K₃₃ are the Frank elastic constants (Table S1) corresponding to splay, twist, and bend deformations, respectively. The magnetic term $F_{mag}=-{\displaystyle \int {\mu }_0}\mathbf{M}\cdot {\mathbf{H}}_{total}dV$ describes the response of the BFLCC to external field H altered by the demagnetizing field, ${\mathbf{H}}_{total}=\mathbf{H}-D\mathbf{M}$, where we neglect the diamagnetic term and the direct interaction of n(r) and H, and D is the demagnetization factor dependent on the sample and field geometry as well as on the domains. The free energy term describing the coupling between n(r) and M(r) reads

$$F_{coupl}=\xi {{\displaystyle \int ({\cos }^2{\theta }_m-{\cos }^2{\theta }_{me})}}^{2}dV\approx \xi {\sin }^2\left(2{\theta }_{me}\right){{\displaystyle \int ({\theta }_m-{\theta }_{me})}}^{2}dV,$$ [9]

where the coupling coefficient ξ originates from the mechanical coupling of individual FNP orientations to n, enhanced by their collective response in concentrated dispersions.

Table S1.

Material parameters of the LC host 5CB of the studied BFLCCs

K11, pN	6.4
K22, pN	3.0
K33, pN	10
ε‖	18.5
ε⊥	7.0
Δε	11.5
ne	1.725
no	1.534
Δn	0.191

Similar to the previous studies involving dispersions of FNPs in 5CB (9–12), we assume that the relatively low volume fractions of the nanoplates have only a small effect on the composite parameters and use the parameters of 5CB listed in our analytical and numerical modeling of BFLCCs.

Different free energy terms often compete, with the elastic term tending to minimize n(r) distortions, the magnetic term rotating M toward B while also prompting formation of domains due to the demagnetizing factor, and the coupling term tending to keep relative orientations of n(r) and M(r) at θ_m = θ_me. Numerical minimization of the free energy given by Eq. 7 yields equilibrium n(r) and M(r) at different fields consistent with the experimental hysteresis and switching data (Fig. 2 A–C). Allowing the magnetic domain size to be a fitting parameter, we model fine details of experimental hysteresis loops, such as the shoulder-like features in the vicinity of B = 0 (Fig. 2C) and domain size behavior (Figs. 5 and 6). This modeling shows that BFLCC domains are governed by the competition between the demagnetizing and elastic free energy terms that exhibit rich behavior when the direction and strength of B are varied. The facile threshold-free polar switching of light transmission through a single-domain BFLCC between crossed polarizers (Fig. 2B) is consistent with the highly asymmetric tilting of n at different θ_me (Fig. 2 F and G).

To understand the richness of BFLCC domain structures, we carried out optical studies (Figs. 4–8 and Figs. S7–S9) and direct imaging of FNP orientations within domains with transmission electron microscopy (TEM) of polymerized and microtome-sliced BFLCCs (Fig. 6 I and J). The up-down domains, in which M lives on two opposite cones θ_m = θ_me, can be observed in homeotropic cells with n₀ orthogonal to substrates (Fig. 5) and also in planar cells with in-plane n₀ (Fig. 7). A magnetic holonomic control system (Fig. S2A), integrated with an optical microscope, allows us to apply B in arbitrary directions, at different tilts with respect to confining plates and different azimuthal orientations, and thus to probe the nature of BFLCC domains (Figs. 5–7). The response of coexisting domains is always present, except when B‖M on the θ_m = θ_me cones, consistent with the C_s symmetry of BFLCCs. The switching of up- and down-cone domains by B‖n₀ is thresholdless (similar to that shown in Fig. 2B), highly asymmetric (polar), and complementary for the two antiparallel directions of B, so that the different domains can be distinguished (Figs. 5 A–F and 7). Up- and down-cone domains in homeotropic cells respond equally strongly to in-plane B (Fig. 5 G–I), although the director within neighboring domains tilts in opposite directions, with homeotropic n(r) in the walls in between. In planar cells, rotations of the in-plane B and the sample between crossed polarizers in POM reveal distorted n(r) and M(r) within the domains (Fig. 7).

Fig. 8.

Three-dimensional structure and dynamics of up-down domains. (A and B) Fluorescence confocal images of the BFLCC with FNPs labeled with dye at B = 0 obtained (A) for the cell midplane and (B) in a cross-section along the yellow line in A. (C–E) Domain interactions and merging (within dashed squares) in a homeotropic cell at B⊥n₀ (B = 30 mT) probed with dark-field microscopy and scattering from FNPs. Elapsed time is marked on images.

Fig. S9.

Absorbance-based bright-field optical micrographs. (A–D) Imaging of the FNP dispersion in 5CB in a homeotropic cell (A and B) before applying the magnetic field right after preparation and (C and D) after applying B = 5 mT field for ∼60 s in the in-plane direction shown on the images and then turning it off. The contrast in images arises from orientation and concentration spatial patterns associated with the domain structure of BFLCCs. (E and F) Polarization-dependent bright-field optical micrographs of a BFLCC with ${\theta }_{me}$∼10° in a homeotropic cell at the applied field B = 5 mT in the in-plane direction shown using arrows. Orientations of the polarizer (P) are shown using white double arrows.

Fig. 7.

Up-down domains in a planar cell. POM images obtained with the 530-nm plate (γ) inserted between crossed P and A for B (B = 2 mT) orthogonal to the rubbing direction n₀ at (A) 0°, (B) 45°, (C) 90°, (D) 135°, and (E) 180° with respect to P, (F and G) before and after reversing B and (H) at B = 0. Dashed cyan lines in insets show n(r). White lines in H depict walls between domains with uniform n₀ and different M-orientations marked by arrows.

BFLCCs prepared to have M on the up- or down-cone within the entire sample slowly develop the left-right domains with different azimuthal orientations of M on the same cones (Fig. 6), separated by analogs of Bloch walls (23) across which M continuously rotates. The presence of left-right domains becomes apparent with B applied at angles to n₀ different from θ_me, including that normal to substrates of a homeotropic cell (Fig. 6 B–D), revealing domains due to their different tilting and then making the sample appear uniform again in B that aligns M roughly along the cell normal. Reversing or applying in-plane B makes this “left-right” domain structure visible again due to different rotations of M within the domains (Fig. 6 E–H). Ferromagnetic domains of both up-down and left-right types are also probed by polymerizing BFLCCs at B = 0 and then directly imaging orientations of nanoplates with TEM (Fig. 6 I–K), revealing θ_me of individual FNPs and M tilted relative to n₀.

Three-dimensional confocal fluorescence (Fig. 8 A and B) and dark-field microscopies (Fig. 8 C–E) and bright-field imaging in a transmission mode that derives contrast from spatially varying absorption of BFLCCs (Fig. S9) provide insights into the spatial changes of local number density of nanoplates. Upon formation of up-down domains, the concentration of nanoplates is depleted in the interdomain walls and increased within the domain regions (Fig. 8 and Fig. S9), becoming more homogeneous again when B is turned off. The ensuing walls (Figs. 5, 7, and 8) between the up-down domains with decreased magnitude of M and an abrupt change of its orientation differ from the common Bloch and Néel walls with a solitonic continuous change of M-orientation (23). The Bloch-like walls between the left-right domains with M on up- or down-cone with respect to n₀ (Fig. 6) have uniform number density of FNPs and localized changes of M-orientation (23).

To conclude, we have introduced a soft-matter system of BFLCCs with the C_s symmetry that combines 3D fluidity and biaxial orientational ordering of constituent molecular and colloidal building blocks. We have identified diverse domain structures and unusual polar switching of BFLCCs. We envisage a rich variety of new fundamental behavior that remains to be probed, such as formation of different topological defects. We also foresee practical uses enabled by threshold-free response of BFLCCs to weak magnetic fields.

Materials and Methods

Barium hexaferrite BaFe_11.5Cr_0.5O₁₉ FNPs were synthesized by the hydrothermal method and then coated with SiO₂ (Supporting Information). These nanoplates were surface-functionalized by trimethoxysilane-PEG (JemKem Technology). Some FNPs were fluorescently labeled with fluorescein isothiocyanate (Sigma-Aldrich). To disperse FNPs in LCs, pentylcyanobiphenyl (5CB; Chengzhi Yonghua Display Materials Co. Ltd.) was mixed with 0.01–20 wt % FNPs in methanol, followed by solvent evaporation at 90 °C for 3 h. The sample was rapidly cooled to the nematic phase of 5CB while vigorously stirring it. The ensuing composite was centrifuged at 500 × g for 5 min to remove residual aggregates and leave only well-dispersed FNPs (10). For fluorescence confocal microscopy, FNPs labeled with the dye were mixed with unlabeled ones in a 1:50 ratio, so the individual labeled FNPs could be resolved. We used TEM CM100 (Philips) for nanoscale imaging. BFLCCs were controlled by a three-axis electromagnetic holonomic manipulation apparatus mounted on a microscope (Fig. S2A). POM of BFLCCs used microscopes BX-51 and IX-81 (Olympus) equipped with 10×, 20×, and 50× dry objectives with N.A.s of 0.3–0.9 and a CCD camera (Spot 14.2 Color Mosaic; Diagnostic Instruments, Inc.). Dark-field imaging additionally used an oil-immersion dark-field condenser (N.A. ≈1.4) and a 100× air objective (N.A. ≈0.6). Video microscopy used a Point Gray camera FMVU-13S2C-CS. Particle dynamics was analyzed by ImageJ software (NIH). Absorbance spectra were obtained using a spectrometer USB2000-FLG (Ocean Optics) integrated with a microscope. Fluorescence confocal imaging used the inverted IX-81 microscope, the Olympus FV300 laser-scanning unit, and a 488-nm excitation laser (Melles Griot). A 100× oil objective with N.A. of 1.42 was used for epidetection of the confocal fluorescence within a 515- to 535-nm spectral range by a photomultiplier tube. Magnetic hysteresis was characterized in 4- × 4- × 0.06-mm homeotropic and planar glass cells (Fig. 2) using an alternating gradient magnetometer (MicroMag 2900; Princeton Measurement Corp.) and a vibrating sample magnetometer (PPMS 6000; Quantum Design).

Details of Sample Preparation

Barium hexaferrite (BaHF) BaFe_11.5Cr_0.5O₁₉ FNPs were synthesized by the hydrothermal method (9); 0.01 M of Ba(NO₃)₂, 0.045 M of Fe(NO₃)₃, and 0.005 M of Cr(NO₃)₃ were dissolved in deionized water and coprecipitated by 2.72 M of NaOH aqueous solution (all ingredients from Alfa Aesar) in a 25-mL Teflon-lined autoclave. The solution was heated to 220 °C at 3 °C/min, held for 1 h, and then cooled down to room temperature. Precipitated powders were washed with 10 wt % nitric acid and acetone and redispersed in 1 mL of water, yielding magnetically monodomain plates (Fig. 1B and Fig. S1) with average diameter of 140 nm and average magnetic moments 2.2 × 10⁻¹⁷ Am² orthogonal to their large-area faces at the maximum magnetic field of 800 kA/m. Then, the BaHF plates were covered with a thin layer of SiO₂ (24). Briefly, 250 mg of 40-kDa poly(vinylpyrrolidone) (PVP-40; Sigma-Aldrich) was dissolved in water by ultrasonication of the solution for 15 min. Subsequently, the PVP-40 solution and 250 μL of 3 wt % of BaHF were mixed together under stirring for 24 h to ensure the adsorption of PVP-40 on the surfaces of BaHF plates. Following this, the dispersions were centrifuged at 20,500 × g for 1 h and the PVP-stabilized particles (achieved using PVP at 0.077 wt %) were redispersed into 10 mL of ethanol. Then, 350 μL of 28 wt % ammonia was added to the dispersion, which was immediately followed by adding 12.2 μL of tetraethyl orthosilicate (TEOS; Sigma-Aldrich) solution while continuously stirring the dispersion. The reaction mixture was then stirred for another 12 h. The silica-coated BaHF (BaHF@SiO₂) were then centrifuged at 15,000 × g for 30 min and redispersed in 6 mL ethanol. The pH was adjusted to 12 by adding 28 wt % ammonia. Then, 25 mg of 5-kDa trimethoxysilane-PEG (Silane-PEG; JemKem Technology) was dissolved in 1 mL hot ethanol and added into BaHF@SiO₂ solution. The solution was kept at 35 °C with stirring for 12 h. The BaHF@SiO₂@PEG was then centrifuged at 9,600 × g for 15 min and washed by methanol two times. To fluorescently label the ferromagnetic plates, the dye FITC (Sigma-Aldrich) was covalently attached to the coupling agent 3-aminopropyltriethoxysilane (APS; Sigma-Aldrich) by an addition reaction of the amine group with the thioisocyanate group (25). The reaction was allowed to proceed for 12 h in a dark environment by slowly stirring a solution containing 4.53 mg of FITC and 5.29 mg of APS in 2 mL of anhydrous ethanol. Then 10 μL of FITC-APS solution was added together with TEOS when BaHF was capped with silica, followed by surface functionalization of silane-PEG described above.

To disperse the ferromagnetic plates in a nematic host, 15 μL of 5CB (Chengzhi Yonghua Display Materials Co. Ltd.) was mixed with 15 μL of a 0.01–20 wt % FNP dispersion in methanol, followed by methanol evaporation at elevated temperature of 90 °C for 3 h, yielding an excellent dispersion in the isotropic phase at no fields. Then, the sample was rapidly cooled to the nematic phase of 5CB while vigorously stirring the dispersion. The ensuing BaHF@SiO₂@PEG-5CB composites were centrifuged at 500 × g for 5 min to remove residual aggregates, so that the final composite contained only well-dispersed particles (10, 12). For fluorescent confocal microscopy, FNPs labeled with the fluorescent dye FITC were mixed with unlabeled FNPs in the ratio of 1:50, so the individual FITC-labeled FNPs could be resolved based on their fluorescence using optical imaging. To polymerize BFLCCs, the FNP-LC dispersion was first prepared using UV-curable LC reactive mesogens (RM23:RM257:RM82 = 11 wt %:22 wt %:33 wt %, all obtained from EM Chemicals) mixed with 1 wt % FNP-doped 5CB (34 wt %). A photo initiator Irgacure 369 (Ciba) was added to the mixture. The mixture was sandwiched by two glass plates with homeotropic surface alignment then activated by magnetic field and photopolymerized using a UV lamp illumination. The sample was then washed by isopropanol, mounted in thermal curable resin, and cut to 200-nm-thick films by microtome at different orientations. The alignment of FNPs in the polymerized LC was then visualized by TEM (using a CM100 instrument from Philips).

The concentration of FNPs in LC was determined by using the Beer–Lambert law (1). The absorbance of the aggregation-free FNP-LC was measured in a 60-μm-thick planar LC cell. The corresponding solid content of FNP-LC sample was weighted after centrifuging, washing by ethanol, and drying the FNP-LC sample. The weight percentage as well as molar concentration of FNP was then determined based on the Beer–Lambert law taking account of the density of FNPs and 5CB is 5.26 g/cm³ and 1.008 g/cm³ at room temperature. Similar to the previous studies (10, 12) involving dispersions of FNPs in 5CB and other LC host fluids, we have determined that the relatively low volume fractions of the nanoplates have only a small effect on the order–disorder nematic–isotropic transition temperature (<1 °C) and the material parameters such as elastic constants. We therefore use material parameters of 5CB in our analytical calculations and in numerical modeling (Table S1).

The FNP’s surface charge was estimated from the balance of the Stokes drag force Φ_s = Ω_viscυ and the electrostatic force Φ_e = (Z*e) E_DC, where Ω_visc is the viscous friction coefficient, υ is the FNP’s moving speed in a direct current (DC) electric field E_DC in LC host in a homeotropic cell, e = 1.6 × 10⁻¹⁹ C is an elementary charge, and Z* is an effective number of elementary charges on the FNP’s surface (21). Taking account of the Einstein relation Ω_viscδ = k_BT (δ is a particle’s diffusion constant at a temperature T), we obtained Z*e = k_BTυ/(δ E_DC). The diffusion constant δ was determined in a separate experiment from the distribution of spatial displacements for the elapsed time interval during the Brownian motion of a single particle in the LC host in a homeotropic cell (Fig. S2A) (21).

Magnetic Holonomic Control and Imaging

Magnetic manipulation of BFLCC was achieved by integrating a microscope with a three-axis electromagnetic apparatus (Fig. S2A) formed by three home-built electromagnets containing custom machined cast-iron cores based on the solenoids (S52051; Fisher Scientific International, Inc.) and one Helmholtz coil (for applying fields along the z axis parallel to the microscope’s axis). The solenoids were mounted directly onto the microscope body. Each electromagnet (Fig. S2A) is independently driven via an amplified power supply (BOP20-5M; Kepco), which was controlled using a computer-driven data acquisition card (DAQ) (USB-6259; National Instruments) via our in-house LabVIEW-based software (National Instruments) (10, 12). These electromagnets can produce magnetic fields up to ∼30 mT (typically much smaller fields were used for BFLCC switching). These relatively low magnetic fields were found to have no significant direct coupling with the LC director (10). POM of BFLCCs was performed using an optical microscope BX-51 and IX-81 (Olympus) equipped with 10×, 20×, and 50× dry Olympus objectives with N.A.s of 0.3–0.9, integrated with a CCD camera (Spot 14.2 Color Mosaic; Diagnostic Instruments, Inc.), polarizers, and a 530-nm full-wave phase retardation plate. Optical imaging in the dark-field mode was performed using the same microscope equipped with an oil-immersion dark-field condenser (NA ∼1.4) and a 100× air objective (N.A. ∼0.6). Video microscopy used a Point Gray camera FMVU-13S2C-CS; the particles were tracked and their dynamics was analyzed by the ImageJ software (NIH). Absorbance spectra were obtained using a spectrometer (USB2000-FLG; Ocean Optics) mounted on the BX-51 Olympus microscope. Fluorescent confocal imaging of BFLCC used the inverted microscope IX-81, a laser-scanning unit (Olympus FV300), and the linearly polarized excitation by a 488-nm laser (Melles Griot). A 100× oil objective (N.A. 1.42) was used for epidetection of the confocal fluoresence signals within a 515- to 535-nm spectral range by a photomultiplier tube.

Mechanical Coupling of Director and Magnetization Through Conical Surface Anchoring on FNPs

In the studied LC dispersion of FNPs, the surfaces of the nanoplates were treated with polymer so that LC molecules and director tend to exhibit tilted equilibrium orientation at ${\theta }_m={\theta }_{me}$ with respect to the surface normal of large-area faces and m. The anchoring energy density $W_s\left({\theta }_m\right)$ only depends on the angle ${\theta }_m$ of magnetic moment m orientation with respect to the director and is independent of the azimuthal angle (20). Because of the inversion symmetry of the nematic and the revolution symmetry of the easy cone (20), $W_s\left({\theta }_m\right)$ is symmetric around both ${\theta }_m=0$ and ${\theta }_m=\pi /2$ and has extrema in these two points. ${\theta }_m=\pi /2$ has a single minimum for ${\theta }_m={\theta }_{me}$ and two maxima at ${\theta }_m=0$ and π/2. Therefore, in lowest approximation, the anchoring energy density can be used in the form derived in ref. 20:

$$W_s\left({\theta }_m\right)=\frac{1}{4}A{({\cos }^2{\theta }_m-{\cos }^2{\theta }_{me})}^2\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\left(A>0\right).$$ [S1]

The total anchoring energy per particle as a function of the angle ${\theta }_m$ can be found by integrating the surface anchoring energy density over the whole surface area σ of the FNP of radius R while neglecting the contribution of side faces, giving Eq. 1 in the main text. However, the lateral dimensions of nanoplates are comparable to the so-called surface anchoring extrapolation length, the length scale that characterizes the interplay of surface anchoring and elastic forces, indicating that rotation of a nanoplate in the LC causes elastic distortions around it as well as the deviation of the director from the easy axis at its surface. Therefore, considering the finite-strength surface anchoring boundary conditions (Fig. 3A), one needs to account for both the surface anchoring energy and elastic energy corresponding to the n(r) distortions. The elastic energy due to the distortion per FNP with lateral dimensions much larger than the nanoplate thickness can be found by extending the result of Brochard and de Gennes obtained for nanoplates with large aspect ratio to the case of the finite-strength conically degenerate surface anchoring conditions (6):

$$F_e=4KR{({\theta }_m-{\theta }_{me}-\mathrm{\Delta }\theta )}^2,$$ [S2]

where $\mathrm{\Delta }\theta$ is the angle to which the director deviates away from the equilibrium ${\theta }_{me}$ at the nanoplate’s surface, as defined in Fig. 3A. The surface anchoring energy per particle expressed in terms of $\mathrm{\Delta }\theta$ then reads

$$F_s=\frac{\pi A{R}^2}{2}{\left[{\cos }^2({\theta }_{me}+\mathrm{\Delta }\theta )-{\cos }^2{\theta }_{me}\right]}^2.$$ [S3]

To find the equilibrium director structure at a given orientation of the nanoplate away from the equilibrium orientation, the $\mathrm{\Delta }\theta$ characterizing the ensuing director structure is determined through minimization of the total energy:

$$\frac{\partial (F_e+F_s)}{\partial \mathrm{\Delta }\theta }=0.$$ [S4]

To obtain an analytical solution, we consider the limit of small deviations $\mathrm{\Delta }\theta$ away from the equilibrium orientations and neglect the higher-order terms. We obtain

$$\mathrm{\Delta }\theta \approx \frac{8K({\theta }_m-{\theta }_{me})}{8K+\pi AR{\sin }^2\left(2{\theta }_{me}\right)}.$$ [S5]

By substituting this result into the expression for the total free energy, we obtain

$$F_{se}\equiv F_s+F_e\approx 4\pi KA{R}^2{\sin }^2\left(2{\theta }_{me}\right){({\theta }_m-{\theta }_{me})}^2/[8K+\pi AR{\sin }^2\left(2{\theta }_{me}\right)],$$ [S6]

which is Eq. 2 in the main text.

The magnetization and director free energy coupling term that favors the tilted mutual orientations of n and M, in analogy with the description of conical surface anchoring energy (20), can be written in the form

$$F_{coupl}={\displaystyle \int \left[-\frac{1}{2}{\beta }_1{\mu }_0{(\mathbf{n}\cdot \mathbf{M})}^2+\frac{1}{4}{\beta }_2{\mu }_0{(\mathbf{n}\cdot \mathbf{M})}^4\right]}dV,$$ [S7]

where β₁ and β₂ are coefficients describing the strength of coupling of the director and the magnetization. This energy of coupling between the director and the magnetization can be expressed in terms of the angle between them by taking into account that $\mathbf{n}\cdot \mathbf{M}=M\cos {\theta }_m$, so that one obtains

$$F_{coupl}={\displaystyle \int \left[\frac{{\mu }_0{\beta }_2{M}^4}{4}{\left({\cos }^2{\theta }_m-\frac{{\beta }_1}{{\beta }_2{M}^2}\right)}^2-\frac{{\beta }_1^2{\mu }_0}{4{\beta }_2}\right]}dV.$$ [S8]

This energy has a minimum at ${\theta }_m={\theta }_{me}$, which implies ${\beta }_2{M}^2$>${\beta }_1$>0 and ${\beta }_1/\left({\beta }_2{M}^2\right)={\cos }^2{\theta }_{me}$, so that the coupling free energy term becomes

$$F_{coupl}=\frac{{\mu }_0{\beta }_1{M}^2}{4{\cos }^2{\theta }_{me}}{\displaystyle \int {\left({\cos }^2{\theta }_m-{\cos }^2{\theta }_{me}\right)}^2}dV+\text{const}.=\xi {\displaystyle \int {\left({\cos }^2{\theta }_m-{\cos }^2{\theta }_{me}\right)}^2}dV+\text{const}.,$$ [S9]

where $\xi ={\mu }_0{\beta }_1{M}^2/(4{\cos }^2{\theta }_{me})$ is called the “coupling coefficient” defined in a form similar to that of Eq. 9 in the main text. For relatively weak surface anchoring and highly dilute FNP dispersions, when rotation of a nanoplate does not produce elastic distortions of the director field around it, the strength of surface-anchoring-mediated coupling between n(r) and M(r) in dilute dispersions can be calculated based on mechanical coupling of individual nanoplates described by Eq. 1 and the number density of the nanoplates:

$$F_{coupl}={\displaystyle \int {\rho }_N{F}_s}dV=\frac{{\rho }_N\pi A{R}^2}{2}{\displaystyle \int {\left({\cos }^2{\theta }_m-{\cos }^2{\theta }_{me}\right)}^2}dV,$$ [S10]

where one can redefine the coupling coefficient as $\xi ={\rho }_N\pi A{R}^2/2$ and ${\rho }_N$ is the number density of FNPs, as introduced in the main text. This regime of coupling of n and M purely through the surface anchoring without involving LC elasticity at highly dilute FNP concentrations is, however, very different from that of our BFLCCs with ferromagnetic properties, which show that even individual nanoplate rotations do produce elastic distortions in the LC host, so accounting for these distortions is important.

For the case of coupling of n and M mediated by both the surface anchoring and the elastic energy we consider the case of small angular departure Δθ of the director from easy axis at the FNP’s surface, as schematically depicted in Fig. 3A, obtaining an approximate expression:

$$\begin{array}{l}{F}_{coupl}=\xi {\displaystyle \int {\left({\cos }^2{\theta }_m-{\cos }^2{\theta }_{me}\right)}^2}dV+\text{const}.\\ \approx -\xi {\text{sin}}^2\left(2{\theta }_{me}\right){\displaystyle \int {\cos }^2({\theta }_m-{\theta }_{me})}dV+\text{const}.\\ \approx \xi {\text{sin}}^2\left(2{\theta }_{me}\right){\displaystyle \int {({\theta }_m-{\theta }_{me})}^2}dV+\text{const}.\end{array}$$ [S11]

corresponding to Eq. 9 in the main text, which we also used in the numerical modeling described below. By comparing this result with Eq. 2 in the main text, we obtain an estimate of the coupling coefficient in the form

$$\xi =\frac{4\pi {\rho }_{N}KA{R}^2}{8K+\pi AR{\sin }^2\left(2{\theta }_{me}\right)}.$$ [S12]

For typical experimental parameters ${\rho }_N$= 1.6 × 10¹⁹ m⁻³, 2R = 140 nm, K = 6 × 10⁻¹² N, A = 10⁻⁵ J/m², ${\theta }_{me}$= 48.5° we obtain ξ = 1.2 J/m³. This allows us to conclude that the elasticity- and anchoring-mediated coupling between the director field n(r) and the magnetization field M(r) is relatively strong, but not infinite in strength. One should note here that this estimate does not account for the collective behavior of the FNPs and can be used to describe only the cases of relatively dilute dispersions. To improve the model further, the collective behavior and elastic, electrostatic, and magnetic interactions between the FNPs in the LC host as they respond to applied magnetic field will need to be taken into account. The estimated value ξ = 1.2 J/m³ is comparable to that derived from experiments for both BFLCCs and uniaxial FNP dispersions in LCs for dilute dispersions (9), but the modeling of ferromagnetic switching of intensity of light transmitted through a high-concentration BFLCC sample between crossed polarizers described below and shown in Fig. 2B provides an estimate of ξ = 10–20 J/m³, which is consistent with the fact that accounting for the internanoplate interactions is important at high FNP concentration and that these interactions act to enhance ξ.

Modeling of Polarized Absorbance Spectra Based on Orientational Ordering of FNPs

In an external magnetic field, the FNP is rotated by the field to minimize the associated magnetic free energy (here we again neglect the direct diamagnetic coupling between the LC director and the applied relatively weak magnetic fields):

$$F_H=-\mathbf{m}\cdot \mathbf{B}.$$ [S13]

For example, for the field B oriented orthogonally to the far-field director n₀, this yields $F_H=-{\mu }_{0}mH\sin {\theta }_m$. For individual nanoplates and very dilute FNP dispersions, the response of the nanoplates to external fields is governed by the competition between the elastic and surface anchoring energies on one hand, which tend to keep the FNP orientation at ${\theta }_m={\theta }_{me}$, and the magnetic free energy term, which tends to align m parallel to the applied field B. Based on the Boltzmann statistics, the distribution of FNP orientations due to the total energy is

$$f\left({\theta }_m\right)=C{e}^{-\frac{F_{se}+F_H}{k_{\text{B}}T}},$$ [S14]

where the normalization coefficient C is determined by ${\displaystyle {\int }_0^{\pi }C\exp \left[-(F_{se}+F_H)/k_{\text{B}}T\right]}\sin {\theta }_{m}d{\theta }_m=1$. On the other hand, this angular distribution of m orientations can be determined from the experimental absorbance spectra. The scalar order parameter S_m describes the orientational ordering of surface normals and magnetic moments m of nanoplates, which is defined as $S_m=\langle 3{\cos }^2{\theta }_m-1\rangle /2$, where ${\theta }_m$ is the angle between the surface normal of FNPs and n₀ and the brackets indicate averaging over all nanoplates. By using the measured experimental absorbance (1, 18) for nanoplates with perpendicular boundary conditions described in detail in refs. 10 and 12 and by taking the surface anchoring strength of the PEG-coated surface for the LC director as 2 × 10⁻⁵ J/m², the scalar order parameter S_m is calculated to be ≈0.54. The pure absorbance of FNPs for the two orthogonal polarizations can be then calculated from the measured absorbance for finite order parameter (1, 18):

$${\alpha }_{\left|\right|}=\frac{2}{3S_m}(A_{\left|\right|}-A_{\perp })+\frac{1}{3}(A_{\left|\right|}+2A_{\perp })$$ [S15]

$${\alpha }_{\perp }=-\frac{1}{3S_m}(A_{\left|\right|}-A_{\perp })+\frac{1}{3}(A_{\left|\right|}+2A_{\perp }),$$ [S16]

where ${\alpha }_{\left|\right|}$ and ${\alpha }_{\perp }$ are the pure absorbances of FNPs for linear polarizations P‖m and P⊥m, respectively. $A_{\left|\right|}$ and $A_{\perp }$ are the measured absorbances of FNPs for finite order parameter and for polarizations of light P‖m and P⊥m, respectively. By using the experimentally determined values of pure absorbances ${\alpha }_{\left|\right|}$, ${\alpha }_{\perp }$ and the distribution of FNP orientations given by Eq. S14, we model the experimental absorbances as follows:

$$A_{\left|\right|}={\displaystyle {\int }_0^{\pi }f\left({\theta }_m\right)}({\alpha }_{\left|\right|}{\cos }^2{\theta }_{me}+{\alpha }_{\perp }{\sin }^2{\theta }_{me})\sin {\theta }_{m}d{\theta }_m$$ [S17]

$$A_{\perp }={\displaystyle {\int }_0^{\pi }f\left({\theta }_m\right)}({\alpha }_{\left|\right|}{\sin }^2{\theta }_{me}+{\alpha }_{\perp }{\cos }^2{\theta }_{me})\sin {\theta }_{m}d{\theta }_m.$$ [S18]

To fit the experimental absorbance spectra (Fig. 1 E and F), we take experimental values m = 3 × 10⁻¹⁷ Am² and B = 5 × 10⁻³ T, calculate the angular distribution with and without the applied field (Fig. S4), and adjust the value of the surface anchoring coefficient A to be 10⁻⁵ J/m² and ${\theta }_{me}$= 48.5° to obtain a good fit of experimental data (Fig. 1 E and F), where these values are consistent with independently estimated parameters based on separate experiments, as described in the main text.

Fabrication of Silica Platelets

In some of the test experiments (Fig. S3) we have used silica platelets coated with PEG that were fabricated as follows. First, a 1-μm-thick SiO₂ layer was deposited on a silicon wafer using plasma-enhanced chemical vapor deposition. Second, photoresist AZ5214 (Clariant AG) was spin-coated on the silica layer. The pattern of hexagonal platelets was defined in the photoresist by illumination at 405 nm with a direct laser-writing system (DWL66FS; Heidelberg Instruments) and then in the silica layer by inductively coupled plasma etching. Finally, the photoresist was removed with acetone and the silicon substrate was wet-etched with 3 wt % sodium hydroxide aqueous solution so that the silica platelets were released and then redispersed in deionized water. To surface-functionalize the platelets and define boundary conditions for n(r) on the surface of these particles, they were treated with an aqueous solution (0.05 wt %) of silane-PEG. The solution was stirred constantly for 12 h. The silica platelets were then centrifuged at 900 × g for 5 min, washed by methanol two times, and finally redispersed in 5CB. Upon infiltration of these colloidal platelet dispersions in 5CB into planar and homeotropic cells using capillary action, the POM micrographs shown in Fig. S3 were obtained using optical microscopy.

Demagnetizing Factor

The magnetization of BFLCC gives rise to a demagnetizing field. For ferromagnetic samples with a simple geometry such as a sphere, an ellipsoid, or a rectangular prism, the demagnetization field is linearly related to the magnetization M by the geometry-dependent constant called the “demagnetizing factor.” The total internal magnetic field is then a sum of the demagnetizing field of the ferromagnetic material and the external magnetic field:

$${\mathbf{H}}_{total}=\mathbf{H}-D\mathbf{M},$$ [S19]

where D is the demagnetization factor dependent on the sample and field geometry, but not on the volume/size of the ferromagnetic sample. Our experimental BFLCC samples have the geometry of rectangular prisms (typically rectangular glass cells of uniform gap thickness in the range of 10–60 μm and lateral dimensions varied within 2–20 mm). The direction-dependent components D_x, D_y, and D_z for such samples along the three orthogonal Cartesian axes can be determined by analytical formulas for a ferromagnetic rectangular prism derived in ref. 26:

$$\begin{array}{l}\pi D_x=\frac{c^2-a^2}{2ac}\ln \left(\frac{\sqrt{a^2+b^2+c^2}-b}{\sqrt{a^2+b^2+c^2}+b}\right)+\frac{b^2-a^2}{2bc}\ln \left(\frac{\sqrt{a^2+b^2+c^2}-c}{\sqrt{a^2+b^2+c^2}+c}\right)\\ +\frac{c}{2a}\ln \left(\frac{\sqrt{b^2+c^2}+b}{\sqrt{b^2+c^2}-b}\right)+\frac{b}{2a}\ln \left(\frac{\sqrt{b^2+c^2}+c}{\sqrt{b^2+c^2}-c}\right)\\ +\frac{a}{2b}\ln \left(\frac{\sqrt{a^2+c^2}-c}{\sqrt{a^2+c^2}+c}\right)+\frac{a}{2c}\ln \left(\frac{\sqrt{a^2+b^2}-b}{\sqrt{b^2+c^2}+b}\right)\\ +2\arctan \left(\frac{bc}{a\sqrt{a^2+b^2+c^2}}\right)+\frac{b^3+c^3-2a^3}{3abc}\\ +\frac{b^2+c^2-2a^2}{3abc}\sqrt{a^2+b^2+c^2}+\frac{a}{bc}(\sqrt{a^2+b^2}+\sqrt{a^2+c^2})\\ -\frac{{(a^2+b^2)}^{3/2}+{(b^2+c^2)}^{3/2}+{(c^2+a^2)}^{3/2}}{3abc}\end{array}$$ [S20]

$$\begin{array}{l}\pi D_y=\frac{a^2-b^2}{2ab}\ln \left(\frac{\sqrt{a^2+b^2+c^2}-c}{\sqrt{a^2+b^2+c^2}+c}\right)+\frac{c^2-b^2}{2ac}\ln \left(\frac{\sqrt{a^2+b^2+c^2}-a}{\sqrt{a^2+b^2+c^2}+a}\right)\\ +\frac{a}{2b}\ln \left(\frac{\sqrt{a^2+c^2}+c}{\sqrt{a^2+c^2}-c}\right)+\frac{c}{2b}\ln \left(\frac{\sqrt{a^2+c^2}+a}{\sqrt{a^2+c^2}-a}\right)\\ +\frac{b}{2c}\ln \left(\frac{\sqrt{a^2+b^2}-a}{\sqrt{a^2+b^2}+a}\right)+\frac{b}{2a}\ln \left(\frac{\sqrt{b^2+c^2}-c}{\sqrt{a^2+c^2}+c}\right)\\ +2\arctan \left(\frac{ac}{b\sqrt{a^2+b^2+c^2}}\right)+\frac{a^3+c^3-2b^3}{3abc}\\ +\frac{a^2+c^2-2b^2}{3abc}\sqrt{a^2+b^2+c^2}+\frac{b}{ac}(\sqrt{b^2+c^2}+\sqrt{a^2+b^2})\\ -\frac{{(a^2+b^2)}^{3/2}+{(b^2+c^2)}^{3/2}+{(c^2+a^2)}^{3/2}}{3abc}\end{array}$$ [S21]

$$\begin{array}{l}\pi D_z=\frac{b^2-c^2}{2bc}\ln \left(\frac{\sqrt{a^2+b^2+c^2}-a}{\sqrt{a^2+b^2+c^2}+a}\right)+\frac{a^2-c^2}{2ab}\ln \left(\frac{\sqrt{a^2+b^2+c^2}-b}{\sqrt{a^2+b^2+c^2}+b}\right)\\ +\frac{b}{2c}\ln \left(\frac{\sqrt{a^2+b^2}+a}{\sqrt{a^2+b^2}-a}\right)+\frac{a}{2c}\ln \left(\frac{\sqrt{a^2+b^2}+b}{\sqrt{a^2+b^2}-b}\right)\\ +\frac{c}{2a}\ln \left(\frac{\sqrt{b^2+c^2}-b}{\sqrt{b^2+c^2}+b}\right)+\frac{c}{2b}\ln \left(\frac{\sqrt{a^2+c^2}-a}{\sqrt{a^2+b^2}+a}\right)\\ +2\arctan \left(\frac{ab}{c\sqrt{a^2+b^2+c^2}}\right)+\frac{a^3+b^3-2c^3}{3abc}\\ +\frac{a^2+b^2-2c^2}{3abc}\sqrt{a^2+b^2+c^2}+\frac{c}{ab}(\sqrt{a^2+c^2}+\sqrt{b^2+c^2})\\ -\frac{{(a^2+b^2)}^{3/2}+{(b^2+c^2)}^{3/2}+{(c^2+a^2)}^{3/2}}{3abc},\end{array}$$ [S22]

where a and b are the lateral dimensions of the rectangular-prism ferromagnetic sample, c is its thickness, and the values of D_x, D_y, and D_z always satisfy the equality condition:

$$D_x+D_y+D_z=1.$$ [S23]

For a uniformly aligned BFLCC sample (for example, the sample with spatially uniform M perpendicular to substrates in a homeotropic cell at a high field of 5 mT and larger, see the magnetometer data in Fig. 2A), the values of D_x, D_y, and D_z can be calculated using equations above, where D_z is the demagnetizing factor for the case of an external field applied along the z axis normal to substrates and D_x and D_y are the corresponding x and y components of D measured in the lateral directions. At low magnetic fields, however, we observed formation of ferromagnetic domains (Figs. 5–8), which reduced the demagnetizing term because of reducing the sample-averaged magnetization in Eq. S19. This formation of domains plays an important role in determining the behavior of the experimental magnetic hysteresis loops (Fig. 2). To account for the fact that the magnetization M(r) changes as a function of coordinates, we split the experimental BFLCC sample into a grid of rectangular prisms with dimensions small enough to describe spatial variations of M(r) and n(r) in a continuum model when minimizing the total free energy. Interestingly, the components of D_x, D_y, and D_z stay the same when discretization maintains the rectangular grid geometry consistent with the geometry of the BFLCC sample used in experiments, as described below. This approach can be applied to the discrete grid in our numerical simulation by calculating the sample-averaged demagnetizing term to obtain the resulting internal magnetic field while accounting for the magnetic domain structure. The demagnetizing factor and its contribution to the free energy through a modified internal magnetic field given by Eq. S19 favor formation of the domains in the geometry of a homeotropic cell with vertical n₀ orthogonal to substrates. In our numerical model described below, domains changed their lateral size to thickness ratio from ∼0.5 to ∼2 for a 60-μm-thick cell, in agreement with experimental observations (Figs. 2C and 4–8). To ensure that the demagnetizing field and energy are calculated properly, we have performed benchmarking tests by splitting a sample with uniform magnetization into a rectangular grid of varying dimensions preserving ratios between a, b, and c, finding the same results for different used rectangular grids. We have also ensured that splitting the sample into ferromagnetic domains with simple geometry and different orientations of M yields expected outcomes in reducing the demagnetizing field and altering the corresponding free energy term. We have incorporated this simple yet robust approach into the free energy minimization described below, which allows considering the effect of spatial variations of M(r) on the sample-averaged demagnetizing field and energy while describing orientations of M(r) within different ferromagnetic domains and regions in between them.

Numerical Modeling of Magnetic Hysteresis Loops and Magneto-Optic Switching

Numerical simulations of minimum-energy structures of BFLCCs formed by the FNP dispersions in LCs used a relaxation routine to minimize total free energy F given by Eq. 7 in the main text (10). Our relaxation routine calculated spatial derivatives of n(r) using a second-order finite difference scheme and evaluated the total free energy density on a rectangular computational grid. Periodic boundaries were implemented along lateral directions whereas fixed homeotropic boundaries were applied at substrate surfaces (we restricted our modeling to the samples treated for homeotropic boundary conditions for n₀). At each time step Δt, evaluation of the functional derivatives of F gives the Lagrange equation $\delta F/\delta Q_i=0$ and the resulting elementary displacement $\delta Q_i=-\mathrm{\Delta }t\left(\delta F/\delta Q_i\right)$ computed for the director n(r) and magnetization M(r), where the subscript $i$ denotes orientations along coordinate axes. The maximum stable time step used in the relaxation routine was determined as $\mathrm{\Delta }t=\min {\left(h_i\right)}^2/2\max \left(K\right)$, where $\min \left(h_i\right)$ was the smallest computational grid spacing and $\max \left(K\right)$ was the largest elastic constant. The steady-state stopping condition was determined through monitoring the change with respect to time of the spatially averaged functional derivative. When this value asymptotically approached zero, the system was assumed to be in the equilibrium and relaxation was complete. All calculations used material parameters of the nematic host 5CB provided in Table S1 while assuming that they do not change upon dispersion of nanoplates in the LC (9–12).

The structure and size of ferromagnetic domains with like-aligned nanoplates and uniform M result from a competition between field-dependent elastic and magnetic energies, with the latter having the demagnetizing field contribution discussed above in Demagnetizing Factor. This competition prompts the domain size to grow or shrink as the strength and orientation of the applied magnetic field are varied. The total-free-energy-minimizing configuration of a polydomain sample for continually varied applied magnetic fields was used to theoretically recreate the hysteresis loops measured experimentally (Fig. 2). Even the fine details, such as the “shoulders” in the experimental data at small fields close to B = 0 (Fig. 2 A and C), were recreated through using domain size as a fitting parameter (Fig. 2C). The lateral dimensions of these domains are found to be 0.5–2 times the cell thickness, consistent with experimental observations (Figs. 4–8). The changes of size of domains can be explained by the interplay of demagnetizing-factor contribution of the magnetic energy and elastic energy of the BFLCC at different applied fields. The minimum-energy n(r) and M(r) configurations in vertical cross-sections are obtained in the form of arrays of azimuthal and polar angles as functions of Cartesian coordinates, which are then used in modeling of transmitted light intensity changes during magnetic switching, as described below.

To model the transmission of light through the BFLCC sample between crossed polarizers (Fig. 2B), we used the Jones-matrix method. We split the BFLCC cell into a stack of thin sublayers while assuming that the orientation of n(r) is constant across the thickness of each of these sublayers. The corresponding coordinate-dependent Jones matrices then have optical axis orientations defined by the orientation of n(r) and the phase retardation defined by the optical anisotropy of the LC and the polar angle of n(r). The ensuing transmission intensity through BFLCC between crossed polarizers was obtained as a result of successive multiplication of Jones matrices corresponding to a polarizer, a series of thin BFLCC slabs each equivalent to a phase retardation plate, and an analyzer. Similar to experimental intensity measurements, we performed calculations for the wavelength of 650 nm. By performing these calculations for the equilibrium director structures of the BFLCC at different magnitudes and directions of the applied field B, we have recreated the experimentally measured results shown in Fig. 2B for independently determined LC host material parameters presented in Table S1.

Numerical modeling provides additional insights into understanding of polar threshold-free switching characterized by means of measuring light transmission through a BFLCC slab placed between crossed polarizers (Fig. 2B). The light transmission in these experiments arises because of magnetic-field-induced tilting of the director away from the original homeotropic orientation (prompted by rotation of M toward B) that increases birefringence and thus also light transmission (Fig. 2B). The strong asymmetry is caused by the fact that originally (before applying fields) the magnetization M is tilted on a cone as shown in Fig. 2B, Inset. As we apply a positive field along the positive +z axis to the sample that has M on the orange up-cone (magenta curve) making a small angle with +z, M slightly tilts toward +z while director slightly tilts away from the cell normal +z‖B, causing a modest effective birefringence and thus also modest light transmission that increases slowly with increasing μ₀H (Fig. 2 B, F, and G). When the applied field is negative (along the negative −z‖B), the magnetization M on the same original cone is forced to rotate dramatically to have orientation closer to that of the applied field (Fig. 2 F and G). This, in turn, causes much stronger field-induced B-dependent birefringence and, thus, also much more dramatic change of intensity of light transmitted through the BFLCC between crossed polarizers with changing the field magnitude B. The switching of another BFLCC sample with the down-cone original orientation of M (blue cone in Fig. 2B, Inset and cyan experimental dependence) is precisely opposite in nature: A strong change of intensity is observed for fields applied along the positive +z but a modest one for μ₀H along the −z. The experimental and computer-simulated dependencies closely match (Fig. 2B). In these dependencies, we plot the curves for asymmetric relatively small range of values of μ₀H. This is because a strong field μ₀H in the direction opposite from the orientation of the cone changes the up-cone BFLCC to a down-cone BFLCC sample through nucleation of domains and propagation of defects, as we describe above and show in Fig. S8 A–F. The nucleation and dynamics of defects cause erratic fluctuations of the transmitted light intensity with time, which are difficult to analyze and model, as well as further complicated by the fact that the phase retardation of light traversing through such a sample with highly distorted director can approach and exceed 2π. Considering this, we omit the high-field range of applied fields and focus on the range of applied fields at which BFLCCs exhibit no defects but just translationally invariant uniform changes of the n(r) and M(r) configurations (Fig. 2B). This experimental characterization of transmitted light intensity is different from the case of magnetometer data (Fig. 2A), which are taken slowly (∼30 min per experimental point), over a large, symmetric range of the μ₀H values, and that also could be modeled using computer simulations by accounting for the structure of domains (Fig. 2C). We also note that the final relaxed structures in very high negative and very high positive magnetic fields applied orthogonally to the BFLCC cell substrates and of the same magnitude, corresponding to extreme tip points of Fig. 2A, exhibit the same light transmission when placed between crossed polarizers, as expected.