Unavoidable emergent biaxiality in chiral molecular-colloidal hybrid liquid crystals

Disk or rod-shaped $\beta-\rm{NaYF_{4}}$:Yb/Er particles are synthesized following the hydrothermal synthesis methods described in detail elsewhere mundoor2021 ; mundoor2018 ; mundoor2019electrostatically ; wang2007controlled ; yang2012one . Precursors and solvents used for the synthesis of colloidal particles are of analytical grade and used without additional purifications, and they are bought from Sigma Aldrich if not specified otherwise. To synthesize nanodisks, 0.7 g of sodium hydroxide (purchased from Alfa Aesar) is dissolved in 10 ml of deionized water and then added with 5 ml of oxalic acid solution (2g, 19.2 mmol) at room temperature to obtain a transparent solution. Under vigorous stirring, we then add 5 ml of sodium fluoride solution (202 mg, 4.8 mmol) to the mixture. After 15 minutes of stirring, 1.1 ml of $\rm{Y(NO_{3})_{3}}$ (0.88 mmol), 0.35 ml of $\rm{Yb(NO_{3})_{3}}$ and 0.05 ml of $\rm{Er(NO_{3})_{3}}$ are added into the mixture while the stirring continues for another 20 minutes at room temperature. Subsequently, the solution is transferred to a 40-ml Teflon chamber (Col-Int. Tech.) and heated to and kept at 200 °C for 12 hours (h). The mixture is then cooled down naturally to room temperature, and the particles precipitated at the bottom are collected by centrifugation, rinsed with deionized water multiple times, and finally dispersed in 10 ml of deionized water. Colloidal rods are prepared using a similar protocol: 1.2 g of NaOH is dissolved in 5 ml of deionized water and mixed with 7 ml of ethanol and 20 ml of oleic acid under stirring, followed by adding 8ml of NaF (1 M), 950 $\rm{\mu l}$ of $\rm{Y(NO_{3})_{3}}$ (0.5 M), 225 $\rm{\mu l}$ of $\rm{Yb(NO_{3})_{3}}$ (0.2 M), and 50 $\rm{\mu l}$ of $\rm{Er(NO_{3})_{3}}$ (0.2 M) and stirring for 20 minutes. The obtained white viscous mixture is transferred into a 50 ml Teflon chamber, kept at 190°C for 24 h, and then cooled down to room temperature. The particles deposited at the bottom of the Teflon chamber are collected and washed with ethanol and deionized water multiple times and finally dispersed in cyclohexane.

In some cases, silica microrods synthesized following an emulsion-templated wet-chemical approach kuijk2011synthesis are also used. To synthesize them, 1 gm of polyvinylpyrrolidone (PVP, molecular weight 40000) is dissolved in 10 ml of 1-pentanol, followed by the addition of 950 $\rm{\mu l}$ of absolute ethanol (Decon labs), 280 $\rm{\mu l}$ of deionized water, 100 $\rm{\mu l}$ of sodium citrate solution (0.18 M), and 130 $\rm{\mu l}$ of ammonia solution (28%). The bottle is shaken vigorously using a vortex mixer after each addition. Then, 100 $\rm{\mu l}$ of tetraethyl orthosilicate (TEOS, 98%) is added under agitation. The bottle is incubated at 25 °C for the next 8 h. The solution becomes milky white after the reaction, and it is centrifuged at 6000 revolutions per minute (RPM) for 10 minutes to separate the as-synthesized rods. The precipitated rods are then washed two times with water followed by another two rounds of washing with ethanol at 3000 RPM for 5 minutes. Finally, to improve the monodispersity and to remove other lightweight impurities, the rods are centrifuged at 500 RPM for 30 minutes and dispersed in ethanol, with the procedure repeated two more times.

Homeotropic surface anchoring boundary conditions for the director and 5CB (pentylcyanobiphenyl or 4-cyano-4’-pentylbiphenyl) molecules on the $\beta$-$\rm{NaYF_{4}}$:Yb/Er disk surfaces is controlled through surface-functionalization with a thin layer of silica and polyethylene glycol. First, 5 ml of hydrogen peroxide ($\rm{H_{2}O_{2}}$) is added to 1 ml of colloidal disk dispersion in deionized water. Then, under vigorous mechanical agitation, 100 $\rm{\mu l}$ of nitric acid is added drop by drop into the solution. After 12 h of agitation, disks are separated from the liquid by centrifugation and transferred into 1 ml of ethanol. The colloidal dispersion is then mixed with 75 mg of polyvinyl pyrrolidone (molecular weight 40,000) in 4 ml of ethanol and kept under continuous mechanical agitation for another 24 h. The particles are collected and redispersed in 5 ml of ethanol, before the addition of 200 $\rm{\mu l}$ of ammonia solution and 6 $\rm{\mu l}$ of tetraethyl orthosilicate under mechanical agitation that lasts 12 h. Disks are collected, washed with ethanol and deionized water, and redispersed in 4 ml of ethanol. The pH value of the mixture is adjusted to 12 by adding ammonia solution (28% in water). Then, under mechanical agitation at 35°C, we add 35 mg of silane-terminated polyethylene glycol (molecular weight 5,000, dissolved in 1 ml of ethanol at 50°C) to the solution. After another 12 h of agitation, the surface-functionalized disks are again collected, washed with ethanol and water, and dispersed in 1 ml of ethanol.

As for the hydrothermal-synthesized rods, the surface chemical treatment not only provides the desired anchoring preference but also controls the cylinder aspect ratio. For this, 4 ml of the nanorod dispersion is added with 200 $\rm{\mu l}$ of HCl in 2 ml of water and kept stirred overnight. The nanorods are then transferred from organic to aqueous phases. The nanorods are collected by centrifugation, washed with deionized water and ethanol three times, dispersed in deionized water, and then finally re-dispersed in ethanol. The process of etching with acid and redispersion is repeated two more times, with HCl treatments of 12 hours and 3 hours, respectively. The aspect ratio of the nanorods is increased during acid treatment to an average value of $L_{\rm c}/D_{\rm c}\approx 60$.

Similarly, the emulsion-templated rods are slowly etched in a mild basic condition hagemans2016synthesis with 0.5 mM NaOH for 24 h, followed by drying at 80°C for another 4 h. After this, the functionalization of silica rods is done by adding 100 $\rm{\mu l}$ of perfluorooctyltriethoxysilane (TCI America) to 0.9 mL ethanol dispersion of the silica rods. The mixture is kept at room temperature for 3 h before being washed and redispersed three times in ethanol. After vacuum-drying inside a desiccator and heating at 60 °C for 1 h, the microrods are immersed in a perfluorocarbon liquid (Fluorinert FC-70, Alfa Aesar) and kept at 60 °C for 1 h before being cooled down to the room temperature and redispersed into ethanol for storage. The fusion of perfluorocarbon oil onto the perfluorosilane functionalized rods results in a fully covered and stable slippery surface layer, giving desired boundary conditions.

A small amount of left-handed chiral dopant cholesterol pelargonate is added into molecular 5CB (Frinton Labs and Chengzhi Yonghua Display Materials Co. Ltd). To obtain the equilibrium pitch $p$ of the molecular chiral mixture, the weight fraction of the used chiral additive is roughly estimated by $c_{d}=\frac{1}{6.25p}$. The actual pitch is later revealed using optical microscopy by observing the periodicity of defect lines in Gradjean-Cano wedge cells smalyukh2002three . The surface-functionalized particles are then dispersed into such molecular chiral LC. In a typical experiment, 20 $\rm{\mu l}$ of colloidal dispersion in ethanol is mixed with 20 $\rm{\mu l}$ of the molecular LC. The mixture is then heated to 75°C and kept for 2 h to completely evaporate the organic solvent. A well-dispersed colloidal-molecular hybrid LC is usually obtained after quenching back to room temperature under mechanical agitation liu2014electrically ; mundoor2015mesostructured ; evans2011alignment . Additional centrifugation can be carried out to remove the particle aggregation formed during the isotropic to chiral nematic phase transition of the molecular LC.

Hybrid LCs containing the colloidal dispersion are infiltrated into glass cells with gap thickness typically chosen to be between $p/2$ and 10$p$, which is experimentally set using Mylar films or silica spheres. To achieve unidirectional planar boundary conditions for 5CB molecules, cell substrates are coated with 1wt.% aqueous polyvinyl alcohol and rubbed unidirectionally. Typically, the geometry and planar boundary conditions of the cell give a sample with its helical axis ${\bf\hat{\chi}}$ perpendicular to the glass substrate and with the helical twist of the cholesteric host LC in compliance with the designed boundary conditions at the confining glass surfaces.

We use different optical microscopy methods to visualize the colloidal orientations inside the hybrid LC, among which are three-photon excitation fluorescence polarizing microscopy (3PEFPM), photon-upconverting confocal microscopy, polarizing optical microscopy and phase contrast microscopy. Using 3PEFPM, optical imaging of director structures of the molecular host medium is performed using a multimodal 3-dimensional (3D) nonlinear imaging system built around a confocal system FV300 (Olympus) and an inverted microscope (Olympus IX-81) mundoor2015mesostructured ; lee2010multimodal . The 3D imaging of the $\beta$-$\rm{NaYF_{4}}$:Yb/Er particles designed to exhibit upconversion luminescence is performed with the same setup when the colloidal dispersions are excited with a laser light at 980 nm; this photon-upconversion-based imaging of colloidal particles minimizes the background signal from the molecular LC, making such a technique ideal for our study. A $100\times$ objective (Olympus UPlanFL, numerical aperture 1.4) and a 980-nm pulsed output from a Ti:Sapphire oscillator (80 MHz, Coherent, Chameleon ultra) are utilized, along with a set of Galvano mirrors on the optical path to achieve sufficient positional accuracy while scanning the sample horizontally. In addition, the vertical re-positioning is achieved by a stepper motor on which the objective could be adjusted to focus at the desired sample depth, enabling 3D scanning with high accuracy. Luminescence signals are epi-collected using the same objective before being sent through a pinhole and detected by a photomultiplier tube. The data obtained from several scanning planes are combined into a 3D tiff image to be analyzed at a later time. The phase contrast images are taken using a $60\times$ objective (Olympus UPlanFL N, variable numerical aperture 0.65-1.25), mounted on another microscope system (Olympus IX-83), at various vertical positions controlled by a motorized sample stage.

The colloidal orientations, representing the normal direction of disks or the long axis of rods, are analyzed on the basis of two-dimensional (2D) slices of a 3D sample using ImageJ software (freeware from the National Institute of Health, schneider2012nih ), with the error in measured colloidal angles is about $\pm 1^{\circ}$. The ensuing statistical data are transferred to Matlab software for visualization, as well as for further analysis. The color thresholds of the images are carefully adjusted to avoid the interference of colloids out of focus. From the 3D stacks of images, the slice plane perpendicular to the helical axis ${\bf\hat{\chi}}$ gives the azimuthal orientational distribution ($\varphi$), whereas the vertical slice plane reveals the polar distribution ($\theta$). Since the colloidal orientations are highly confined, as shown by the high value of uniaxial order $S_{\rm{cc}}$, we assume that the two distributions are independent so that the overall distribution can be written in factorized form $f(\varphi,\theta)=f(\varphi)f(\theta)$. For the same reason, we ignore the effect of the projection from the 3D volume to the slice planes. After the analysis of particles by ImageJ, average azimuthal colloidal orientations are calculated for the data obtained in each ${\bf\hat{n}}-{\bf\hat{\tau}}$ slice plane and plotted against the sample depth ($z$) position of the cross-sectional plane, revealing the helical twist of the colloidal axes. The corresponding helical pitch $p$ of each 3D volume is subsequently calculated from the slope of the linear dependence of the azimuthal angle on the vertical position ($d\varphi/dz=q=360^{\circ}/p$) and is in agreement with the initially designed value mentioned above, confirming the undisturbed molecular helical pitch at low colloidal concentrations. Finally, the colloidal orientation distribution is visualized in the molecular director coordinate system frame. The azimuthal angle in the molecular coordinate is calculated by subtracting the molecular twist from the measured colloidal orientations, $\delta=\varphi-qz$ representing the fluctuation of colloidal orientation around that of a perfect helix. The non-orientable property of the colloidal axis ${\bf\hat{u}}=-{\bf\hat{u}}$ enabled us to express the fluctuation angles, $\delta=\delta+\pi$, for example, within a [-90°,90°] range. Histograms of angular probability distribution with 5° bin width are calculated for each fluctuation angle, and Gaussian fitting $e^{-\sigma*\rm{angle}^{2}}$ is performed to each distribution accordingly with $\sigma$ being the fitting parameter later used to quantify the peak width. The choice of the fitting equation is justified by the analytical prediction of energy dependence on deviation angles (to the leading order), which is detailed in the Results section. In the case of narrow orientational distributions, the visualization is cropped to a smaller angle range after calculation performed in the full [-90°,90°] range. In the case of planar rods and the longer homeotropic rods, it is impractical to do the re-slicing given the limited number of images taken, and the distributions found within achiral nematic LCs are adopted instead as we expect no difference between the horizontal and vertical distributions in such condition with no biaxiality. The same histograms are subsequently utilized in the computation of the colloidal orientation order parameters, which is summarized in Results.

Computer simulations are carried out to study the interplay between molecular LC order near the colloidal surface and the colloidal orientation. The simulations are based on minimizing the mean-field Landau-de Gennes free energy for the molecular LC host mundoor2021 ; ravnik2009landau ; mori1999multidimensional ; tai2020surface ; yuan2018chiral . We consider a thermotropic bulk free energy density describing the isotropic-nematic transition of LCs complemented with elastic contributions associated with LC director distortions occurring in the bulk volume of the LC:

with the 3-by-3 matrix ${\bf Q}^{\rm(m)}$ being the molecular tensorial order parameter describing the local average molecular ordering, $x_{i}$ ($i=1-3$) being cartesian coordinates, and $\epsilon$ the 3D Levi-Civita tensor. Summation over all indices is implied. Among the bulk energy terms, $A$, $B$, and $C$ are thermotropic constants and $L_{i}$ ($i=1-4,6$) are the elastic constants related to the Frank-Oseen elasticities via:

with $K_{11}$, $K_{22}$, $K_{33}$ and $K_{24}$ respectively denoting the splay, twist, bend and saddle-splay elastic moduli, and $S_{\rm{eq}}^{\rm(m)}$ being the equilibrium uniaxial scalar order parameter. In addition to the bulk LC energy there is a contribution due to the boundary condition of the molecular LC at the colloidal surfaces which reads:

with $W_{0}$ the surface anchoring strength, ${\bf P}={\bf\hat{v}}\otimes{\bf\hat{v}}$ the surface projection tensor, ${\bf\hat{v}}$ the surface normal director, and $\tilde{\bf Q}={\bf Q}^{\rm(m)}+\frac{1}{2}S_{\rm{eq}}^{\rm(m)}{\bf I}$. The equilibrium angle $\theta_{\rm e}=0$ corresponds to vertical or homeotropic anchoring at the boundary, and $\theta_{\rm e}=\pi$ leads to planar degenerate anchoring zhou2019degenerate . The dimensions and anchoring forces associated with colloidal particles are represented as boundary conditions inside the numerical volume with the parameters kept constant for each simulation. Specifically, width $D_{\rm{c}}=1\rm{\mu m}$ and thickness $L_{\rm{c}}=10\rm{nm}$ are used for thin disks in all computer simulated results, while $D_{\rm{c}}=28\rm{nm}$ and $L_{\rm{c}}=1.7\rm{\mu m}$ for long rods, if not specified otherwise. The total energy is then given by the integration of Eq. (1) over LC volume and Eq. (3) over colloid-molecule interfaces, with the colloidal volume excluded in the integral of free energy densities.

The total energy is numerically minimized based on the forward Euler method integrating:

with $t$ being the scaled energy-relaxation time of the LC. Adaptive Runge-Kutta method (ARK23) and FIRE, Fast Inertial Relaxation Engine, are adopted to increase numerical efficiency and stability kutta1901beitrag ; sussman2019fast . The steady-state and termination of simulation are determined by the change in total free energy in each numerical iteration, which is usually monotonic decreasing. The values of biaxiality and local orientations of molecular directors are subsequently identified as eigenvalues and eigenvectors of ${\bf Q}^{\rm(m)}$ mottram2014introduction . Alternatively, chirality axes (${\bf\hat{n}}$,${\bf\hat{\chi}}$,${\bf\hat{\tau}}$) are also represented as the eigenpairs of handedness tensor (or chirality tensor, beller2014geometry ; efrati2014orientation ) with results having high consistency with the biaxial approach mentioned above. A more detailed description and comparison of the two approaches are to be given in Discussion.

For a thin homeotropic rod, the anchoring effect at the two ends of the particle is ignored in our simulations by setting the length of the simulation box equal to the rod length $L_{\rm c}$, which hardly changes the energies but greatly improves the numerical stability. The following parameters are used for all computer simulations: $A=-1.72\times 10^{5}\rm{Jm^{-3}}$, $B=-2.12\times 10^{6}\rm{Jm^{-3}}$, $C=1.73\times 10^{6}\rm{Jm^{-3}}$, $K_{11}=6\rm{pN}$, $K_{22}=3\rm{pN}$, $K_{33}=10\rm{pN}$, $K_{24}=3\rm{pN}$ and $S_{\rm{eq}}^{\rm(m)}=0.533$. The simulations are carried out in a Cartesian colloidal frame using equidistant grid sets and are consistent with those based on a radial-basis-function approach performed within the molecular frame fornberg2015solving .

The symmetry-breaking of the nematic colloidal geometry, induced by the twisted alignment of chiral molecules, can be revealed at the single particle level by visualizing the LC distortion field around a single colloidal particle [Fig. 2]. For cylinder-shaped particles dispersed in isotropic solvent, such as thin disks or slender rods in ethanol, a continuous rotational symmetry can be observed locally with the symmetry axis being the disk normal or the long axis of the rod because the other two orthogonal axes are geometrically equivalent. When the cylindrical colloids are dispersed into a chiral nematic, however, the uniaxial symmetry is broken in view of the boundary condition at particle-molecule interfaces and the far-field helical configuration of the LC molecules. Although the realigning effect induced by the homeotropic boundary conditions at the colloidal surfaces is rather weak with $W_{0}=10^{-6}\rm{J/m^{2}}$ and deviation angle $\ll 1^{\circ}$ [Fig. 2a,b], for example, it is evident that the rotational symmetry of the surface-defect-dressed cylindrical colloids becomes discrete (2-fold rotation) once the colloids are immersed in a chiral LC [Fig. 2c]. Clearly, stronger surface anchoring forces and higher chirality (shorter pitch) lead to the significantly stronger molecular LC distortion as well as the ensuing emergent biaxiality as shown by the computer-simulated distortion in nematic director. Also, the single-particle symmetry-breaking is observed even when the helical pitch $p$ is much larger than the particle dimension, with the particle sizes around 1-2 $\rm{\mu m}$ [Fig. 2c]. This demonstrates that the shape biaxiality of the dressed colloidal particle imparted by the molecular chirality of the host is unavoidably developed at all strengths of surface anchoring and values of molecular chirality [Fig. 2c].

To analyze the equilibrium orientation of the cylindrical particles, we perform several sets of simulations at various colloidal orientations and resolve the corresponding free energies. A thin disk with perpendicular boundary condition [Fig. 3], for example, favors alignment in which the disk normal vector orients along the molecular director ${\bf\hat{n}}$. Deviations away from the equilibrium direction give rise to an increase in the overall free energy of the system [Eq. (1) and Eq. (3)]. We emphasize that the free energy profiles are distinct for the two deviation angles $\delta$ and $\zeta$ in Fig. 3 (with lower energy penalty for orientational fluctuation along $\delta$). Though weak, the difference between the two angles and the broken uniaxial symmetry as a consequence of the chirality in the molecular LC host are unambiguous. Furthermore, Fig. 3 (c,d) demonstrate that a stronger surface anchoring force with a higher value of $W_{0}$ leads to more pronounced energetical nondegeneracy of the two deviation angles. Using mean-field numerical simulation of the LC host, we are able to validate the local biaxial symmetry of the orientational probability distribution of an individual colloid, arising from the inequivalence of ${\bf\hat{\chi}}$ and ${\bf\hat{\tau}}$ in the molecular LC host.

In contrast to the case of a disk, a homeotropic rod feels a strong energy penalty when aligned towards ${\bf\hat{n}}$ and reaches a state of minimal surface anchoring energy when the long axis points along the ${\bf\hat{\tau}}$-direction such that the LC director at the rod surface naturally complies with the homeotropic surface anchoring condition [Fig. 4] mundoor2018 ; tkalec2008interactions . The symmetry-breaking of ${\bf\hat{\chi}}$ and ${\bf\hat{\tau}}$, evident from the difference between the two energy landscapes [Fig. 4], is observed again with deviation along the angle $\gamma$ being more energetically favored than that along $\theta$. With the cases of colloidal rods along ${\bf\hat{n}}$, ${\bf\hat{\chi}}$, and ${\bf\hat{\tau}}$ all giving distinct free energies, the biaxiality in the ensuing colloidal orientation probability distribution is explicit, which is contributed solely by the chirality in the 5CB host. The results are in agreement with the empirical evidence shown below.

The alignment of the colloidal particles with respect to the helical director field of the molecular host LC is probed optically in our experiments. For example, Fig. 5. (a) demonstrates confocal fluorescence micrographs of disk-shaped particles immersed in a 5CB liquid crystal doped with a chiral dopant. The molecular orthogonal frame (${\bf\hat{n}}$,${\bf\hat{\chi}}$,${\bf\hat{\tau}}$), which is marked in each micrograph, is robustly controlled by substrates with planar anchoring force (Methods). The average normal direction of the colloidal disks in each vertical slice, which is expected to lie parallel to ${\bf\hat{n}}$, rotates along the sample depth, as clearly shown with the edge-on perspective [Fig. 5. (a)]. Subsequently, the twisted arrangement of the disk direction is analyzed and a quasi-uniform twisting rate is found throughout the sample depth [Fig. 5. (b)], with thermal fluctuation present. We assume that the helix of molecular director ${\bf\hat{n}}$ has a linear trend identical to the one found using colloidal orientations, which is shown by the red line in Fig. 5. (b), since the period of the twisted arrangement of the colloids closely matches the designed molecular pitch $p$. We also ascertain that the colloidal density remains very low such that the molecular LC alignment is not expected to be disturbed by the introduction of colloidal particles. Once the orientational distribution of the thin disks is projected onto the co-rotating molecular frame, we observed Gaussian-like distributions [Fig. 5. (c)].

With a particle number density (volume fraction $\approx$ 0.026%) far below the phase transition threshold mundoor2021 , direct interactions between colloidal disks are negligible and each particle experiences an orientational potential imposed mainly by the surrounding molecular LC, as designed in our numerical simulations discussed above [Fig. 3]. The statistical results of particles can thus be treated as the thermal distribution of a single particle and qualitative agreement is found when compared to the modeling [Fig. 3]. As shown by the numerical modeling earlier, the colloidal director distributions are weakly asymmetric due to the biaxiality imparted by the chiral molecular host, demonstrating a stronger energy barrier of the thin disk fluctuating in the $\zeta$ direction. As a result, the peak widths (full width at half maximum, FWHM) of the colloidal orientation distributions differ along two deviation angles (27.6° in $\delta$ and 23.1° in $\zeta$, Fig. 5. (c)), indicating a biaxial $D_{2}$ symmetry of an individual disk with 2-fold rotations around ${\bf\hat{n}}$ instead of uniaxial $D_{\infty}$ one, even though the cylindrical particles themselves are of uniaxial symmetry.

We further use phase contrast microscopy to determine the colloidal alignment directions for planar rods [Fig. 6. (a)]. In this case, the colloidal thin rods with a planar surface condition are dispersed within the 5CB chiral molecular host. By carefully changing the focal position inside the sample, we obtain a good linear relationship between the average rod direction and depths, representing a helical structure within which the colloidal rods point along ${\bf\hat{n}}$ [Fig. 6. (b)]. Like the case of homeotropic disks, we again clearly observe a biaxial orientational symmetry in the molecular frame revealed by distinct probability distributions along ${\bf\hat{\tau}}$ and ${\bf\hat{\chi}}$ (FWHM=15.6° in $\delta$ and 11.8° in $\zeta$) despite the weak chirality ($p\approx 100\rm{\mu m}$) of the hybrid LC system [Fig. 6. (c)]. The explicit experimental observations of the biaxial symmetry in the colloidal orientation probability distribution, in agreement with the numerical simulation, can only be attributed to the chiral twisting of the surrounding molecular LC.

To see the symmetry-breaking behavior in another type of alignment, such as perpendicular, we adopt the same methods of colloidal orientation analysis but for the colloidal inclusion of thin rods with a perpendicular boundary condition [Fig. 7]. In consistency with the numerical calculation [Fig. 4], the rods aligns towards ${\bf\hat{\tau}}$ axis in thermal equilibrium [Fig. 7 (a,d)]. After measuring the azimuthal angle $\varphi$ of rod long axis in each verticle frame and converting them to 3D distribution in the molecular orthogonal frame, we clearly see a narrower distribution is discovered for the longer rods [Fig. 7 (c,f)]. Due to the larger surface area of the longer particle, to which the surface energy is proportional, a stronger energy barrier develops for the longer colloidal rods to fluctuate away from the energetically ideal configuration along ${\bf\hat{\tau}}$. Furthermore, to our surprise, the orientational distributions of homeotropic rods behave dramatically differently from the non-chiral limit, in which case a degeneracy of alignment along the ${\bf\hat{\chi}}$ and ${\bf\hat{\tau}}$-axes would be expected and the distribution along $\eta$ should be uniform to have the uniaxial symmetry. Instead, we find an exceptionally strong energy barrier for rods deviating along $\eta$ towards the helical axis ${\bf\hat{\chi}}$, with the Gaussian fittings showing peaks even sharper than those along $\gamma$ [Fig. 7 (c,f)]. The symmetry of the hybrid LC system is thus strongly biaxial as experimentally illustrated by the distinct peak widths of the orientational probability distributions. We will demonstrate that the rod orientation distributions can not be interpreted from surface anchoring effects only, but can be explained by considering the elastic distortions generated in the bulk of the molecular host. This is addressed in detail in the following sections using a comprehensive analytical model.

In order to facilitate comparison with experimental results, we define the colloidal orientational order which measures the principal direction of alignment of the colloids along the cholesteric helix. Taking the local molecular LC director ${\bf\hat{n}}$ as a reference frame we define a colloidal uniaxial order parameter as follows:

with ${\langle...\rangle}_{f}$ denoting a thermal average, and a colloidal biaxial nematic order parameter that measures the relative orientational order with respect to the principal directions orthogonal to ${\bf\hat{n}}_{h}$:

Alternatively, we can probe the orientational order from the tensorial order parameter for colloids ${\bf Q}_{\rm{c}}=\frac{3}{2}\langle{\bf\hat{u}}\otimes{\bf\hat{u}}\rangle_{f}-\frac{1}{2}\bf{I}$ which measures orientational order with respect to the principal colloidal alignment direction independently from the chosen reference frame. The corresponding uniaxial and biaxial order parameters defined within the colloidal frame are denoted by $S_{\rm cc}$ and $\Delta_{\rm cc}$, respectively. In case of colloids aligning along the molecular director ${\bf\hat{n}}$, the two frames coincide and the corresponding values of order parameters are identical.

To quantify the symmetry-breaking of the colloidal orientational distribution in experiments, we measured the uniaxial $S_{\rm{cc}}$ and biaxial $\Delta_{\rm{cc}}$ order parameters for both disks and rods (Table  1). The uniaxial order parameter $S_{\rm{cc}}$, as a measure of unidirectional ordering (Eq. (37)), represents the strength of orientational confinement which greatly depends on the synthesized materials and the ensuing surface anchoring effects. Subsequently, the non-equivalence of axes orthogonal to the average colloidal/molecular axis is evaluated using the biaxial order parameter $\Delta_{\rm{cc}}$ (Eq. (38)), with $-1<\Delta_{\rm{cc}}<1$. The values of $S$ are experimentally determined to be $S_{\rm{cc}}=S_{\rm{cm}}=0.66$ for homeotropic disks dispersed in chiral 5CB-based LC [Fig. 5] and $S_{\rm{cc}}=S_{\rm{cm}}=0.94$ for rods with planar boundary condition [Fig. 6]. The values of $\Delta_{\rm{cc}}$ are found to be 0.067 and 0.014, respectively, showing a robust symmetry-breaking among ${\bf\hat{\chi}}$ and ${\bf\hat{\tau}}$ leading to biaxial orientational symmetry.

When the average orientations of the two components differ, however, the choice of reference frame determines the values of order parameters. Stronger orientational fluctuations are found for the shorter rods with homeotropic anchoring (Fig. 7 a-c) with $S_{\rm{cc}}=0.70$ and $\Delta_{\rm{cc}}=0.12$, while the dispersion of the longer rods showed a smaller orientational distribution with higher uniaxial order parameter $S_{\rm{cc}}=0.86$ and lower biaxiality $\Delta_{\rm{cc}}=0.065$, though still higher than that measured for planar rods. The order parameters obtained are in good agreement with the analytical prediction using Eq. (19) and Eq. (35), which give $S_{\rm{cc}}=0.75,\Delta_{\rm{cc}}=0.17$ for the shorter rods, and $S_{\rm{cc}}=0.90,\Delta_{\rm{cc}}=0.058$ for the longer rods using the experimental parameters.

The enhanced biaxiality for homeotropic rods aligning perpendicular to the molecular director ${\bf\hat{n}}$ will be discussed in the following section. Calculated in the molecular reference frame, the negative values of $S_{\rm{cm}}$ and large values of $\Delta_{\rm{cm}}$ simply represent the geometry in which the average colloidal director lies orthogonal to the molecular one ${\bf\hat{n}}$.

For colloidal rods immersed in a chiral LC, the biaxial order developed at the level of the colloids is much more pronounced for rods with homeotropic boundary condition, whose energy-favored orientation is along ${\bf\hat{\tau}}$ and perpendicular to ${\bf\hat{n}}$. As a consequence, the rotational symmetry (with the rotation axis being ${\bf\hat{\tau}}$) is obviously not continuous, with ${\bf\hat{n}}$ being the material axis representing the actual molecular direction and ${\bf\hat{\chi}}$, in contrast, an “imaginary” one. The dissimilarity and the resulting symmetry breaking are thus more pronounced than those in the case of colloids aligned along ${\bf\hat{n}}$, as clearly shown above by biaxial colloidal distribution probabilities in the experimental results.

Analytical theory predicts a similar type of realignment and enhanced biaxiality to occur for homeotropic rods immersed in chiral host LCs where the chiral and biaxial “dressing” around the rod will also be much more pronounced than that of rods with planar or parallel boundary conditions. With the significant contribution from the elastic energy of the background molecular LC to the total free energy [Eq. (35)], we are provided an additional control of this emergent biaxiality by tuning elasticities of the molecular host to boost the biaxiality of the hybrid LC [Eq. (26)]. Likewise, disks with planar anchoring, which could be realized through appropriate surface functionalization mundoor2019electrostatically , exhibit equivalent perpendicular alignment (but this time along the helical axis ${\bf\hat{\chi}}$) which would also give strongly enhanced biaxial order in the disk orientation distribution.

In the weak molecular chirality regime, we may characterize the leading order contribution of chirality to colloidal biaxiality by expanding the biaxial order parameter up to the quadratic order in the inverse pitch $q=2\pi/p$:

where the length scale corresponds to the colloidal dimensions; $a=D_{\rm c}$ for thin disks and $a=L_{\rm c}$ for cylinderical rods. The zeroth order term must be zero given that no intrinsic biaxiality can be expected from purely uniaxial components at zero chirality. Also, the linear term proportional to $qa$ must vanish since the value of biaxiality should not depend on the handedness of the host material. Following the results Eq. (18), Eq. (38), and the free energies for each type of colloids, we computationally verify the quadratic scaling $\Delta\sim\Delta_{0}(qa)^{2}$ within the weak chirality approximation $qa\ll 1$ [Fig. 11(a)]. Interestingly, the quadratic scaling of biaxial order with the parameters $qL_{\rm c}$ or $qD_{\rm c}$ resembles the theoretical prediction by Priest and Lubensky for a single-component molecular LC priest1974biaxial , in which case $L_{\rm c}$ needs to be replaced by $L_{\rm m}$ denoting the size of the molecules. Despite the different derivations of biaxiality from component material(s), the agreement between the results from our hybrid LC system and single-compound LC reveals the underlying physical principle, namely a close relationship between biaxial order and chirality. Most interestingly, the prefactor $\Delta_{0}$ turns out to be very different for each system considered and has a distinct, non-trivial dependence on the surface anchoring strength [Fig. 11(b)].

The molecular biaxial order parameter $\Delta_{\rm{m}}$ measures the broken uniaxial symmetry of the LC host (which is 5CB). The $\Delta_{\rm{m}}$ is associated to the tensorial local mean-field order parameter by mundoor2021 ; mottram2014introduction :

with the molecular director field ${\bf\hat{n}}$ and the biaxial director ${\bf\hat{m}}$ orthogonal to each other. Here, $S_{\rm m}$ is the scalar order parameter measuring the uni-directionality of ${\bf\hat{n}}$, with $S_{\rm m}\geq\Delta_{\rm{m}}\geq 0$. Accordingly, in the numerical computation, the order parameters are determined by the diagonalization of the Q tensor:

where $\lambda_{1}>\lambda_{2}>\lambda_{3}$ are the eigenvalues of ${\bf Q}^{\rm(m)}$. The directors ${\bf\hat{n}}$ and ${\bf\hat{m}}$ are then found by calculating the eigenvectors corresponding to $\lambda_{1}$ and $\lambda_{2}$, respectively. Since eigenvalues are interpreted as the “directionalities” along each orientation (eigenvector), the calculation of $\Delta_{\rm{m}}$ in Eq. (41) corresponds exactly to finding the inequivalence of the two minor axes (${\bf\hat{m}}$ and ${\bf\hat{n}}\times{\bf\hat{m}}$), and the value of biaxiality is a measure of the broken rotational symmetry along ${\bf\hat{n}}$, analogous to the colloidal orientation distributions illustrated above. Using numerical modeling based on the Q-tensor representation of the LC order parameters, we find $\Delta_{\rm{m}}$ at a far-field helical background to be of the order of $10^{-7}$, which is precisely the value predicted using $\Delta_{\rm{m}}\sim(qL_{\rm{m}})^{2}$ with the size of a 5CB molecule being in the nanometer range $L_{\rm{m}}=2\rm{nm}$ priest1974biaxial , showing the intrinsic biaxial order in the molecular chiral liquid crystal.

Interestingly, we also discover that $\Delta_{\rm{m}}$ greatly increases from $10^{-7}$ in the far-field limit to $10^{-4}$ or even $10^{-3}$ near the colloidal surfaces [Fig. 12], being especially prominent at the regions where the surface anchoring force favors a distinct molecular director alignment from the helical far-field. The enhanced biaxiality induced by the colloidal particles is qualitatively interpreted as the mismatch of two axes – the particle surface anchoring orientation ${\bf\hat{v}}$ and background LC aligning direction ${\bf\hat{n}}$. As a quantification of the broken uniaxial rotation symmetry of ${\bf\hat{n}}$, higher values of $\Delta_{\rm{m}}$ can be found at particle surfaces with a greater discrepancy in the two orientations, with maximum $\Delta_{\rm{m}}$ located at regions where surface normal director perpendicular to background far-field [Fig. 12]. Furthermore, within LC regions with a $\Delta_{\rm{m}}$ dominated by particle surface and far exceeding the background value $10^{-7}$, the biaxial director ${\bf\hat{m}}$ is found to coincide with the perpendicular component of the surface anchoring director to the nematic director ${\bf\hat{m}}={\bf\hat{v}}-({\bf\hat{v}}\cdot{\bf\hat{n}}){\bf\hat{n}}$, confirming the idea that the colloidal surface induces molecular biaxiality order by introducing an energy landscape for ${\bf\hat{n}}$ without uniaxial symmetry.

In case of no host chirality, biaxial order stabilized by correlations between colloidal particles immersed in nematic 5CB was reported in mundoor2018 . Furthermore, compared to pure molecular LCs without colloids, the frustrated alignment of ${\bf\hat{n}}$ induced by the presence of colloidal particles also leads to a reduced bulk nematic order parameter $S_{\rm m}$ and the formation of defects in cases with strong surface anchoring. In our systems, though, we expect the two independent contributions to the “biaxialization” of uniaxial 5CB liquid crystal – the introduction of chiral dopant and of colloidal particles – to have negligible effects on the free energies in our analytical model, which is evident by the induced values of $\Delta_{\rm{m}}$ and has been confirmed by the numerical modeling using a tensorial order parameter ${\bf Q}^{\rm(m)}$.

As suggested in the section above, the intrinsic biaxiality of a chiral nematic LC allows us to define local biaxial directors even in absence of colloidal particles. The molecular biaxial order persists $\Delta_{\rm{m}}\sim(qL_{\rm{m}})^{2}$ as long as the chirality $q$, or the helicity in the director alignment, is non-vanishing. To accurately account for this unavoidable biaxiality, we modify and expand the calculation in Ref. efrati2014orientation ; beller2014geometry for uniaxial chiral nematics, in which the chirality-associated directors (${\bf\hat{n}}$,${\bf\hat{\chi}}$,${\bf\hat{\tau}}$) are found by diagonalizing a 3-by-3 handedness tensor $\bf{H}$ defined as:

with summation over indices assumed. The trace $\sum_{i}{\bf H}_{ii}=-{\bf\hat{n}}\cdot(\nabla\times{\bf\hat{n}})$ gives the helicity of the LC director alignment field. Considering the intrinsic biaxial order in chiral LCs, we can similarly construct the handedness tensor using the molecular tensorial order parameter ${\bf Q}^{\rm(m)}$:

The uniaxial definition Eq. (42) can be recovered by expanding the equation using Eq. (40) with $\Delta_{\rm{m}}=0$. Note that the trace of the handedness tensor again represents the helicity and is identical to the chiral part of elastic free energy ($L_{4}$ term in Eq. (1)). Strikingly, in numerical simulation we discovered that the helical director field ${\bf\hat{\chi}}$, which is computed as the eigenvector corresponding to the eigenvalue with the largest absolute value, thoroughly matches the directors calculated by diagonalizing ${\bf Q}^{\rm(m)}$: ${\bf\hat{\chi}}={\bf\hat{n}}\times{\bf\hat{m}}$ [Fig. 13], which also immediately suggests ${\bf\hat{\tau}}={\bf\hat{m}}$ (Note all directors are head-tail symmetric). The excellent overlap of the two orthogonal frames, (${\bf\hat{n}}$, ${\bf\hat{\tau}}$, ${\bf\hat{\chi}}$) originating from chirality and (${\bf\hat{n}}$, ${\bf\hat{m}}$, ${\bf\hat{n}}\times{\bf\hat{m}}$) representing biaxiality, directly demonstrates the biaxial feature in chiral nematic LCs. The energy-minimizing of ${\bf Q}^{\rm(m)}$ automatically incorporates these symmetries once all degrees of freedom beyond those for pure uniaxial nematics are allowed. Consequently, one can straightforwardly identify chirality through the concomitant biaxial properties using $q\sim\sqrt{\Delta_{\rm{m}}}/a$ and ${\bf\hat{\chi}}={\bf\hat{n}}\times{\bf\hat{m}}$ instead of investigating the helical twisting and spatial derivatives of the LC directors. These values are well-defined from the biaxiality calculation even inside LC defects with reduced uniaxial order parameter $S_{\rm m}$.