Abstract
Fundamental relationships are believed to exist between the symmetries of building blocks and the condensed matter phases that they form1. For example, constituent molecular and colloidal rods and disks impart their uniaxial symmetry onto nematic liquid crystals, such as those used in displays1,2. Low-symmetry organizations could form in mixtures of rods and disks3,4,5, but entropy tends to phase-separate them at the molecular and colloidal scales, whereas strong elasticity-mediated interactions drive the formation of chains and crystals in nematic colloids6,7,8,9,10,11. To have a structure with few or no symmetry operations apart from trivial ones has so far been demonstrated to be a property of solids alone1, but not of their fully fluid condensed matter counterparts, even though such symmetries have been considered theoretically12,13,14,15 and observed in magnetic colloids16. Here we show that dispersing highly anisotropic charged colloidal disks in a nematic host composed of molecular rods provides a platform for observing many low-symmetry phases. Depending on the temperature, concentration and surface charge of the disks, we find nematic, smectic and columnar organizations with symmetries ranging from uniaxial1,2 to orthorhombic17,18,19,20,21 and monoclinic12,13,14,15. With increasing temperature, we observe unusual transitions from less- to more-ordered states and re-entrant22 phases. Most importantly, we demonstrate the presence of reconfigurable monoclinic colloidal nematic order, as well as the possibility of thermal and magnetic control of low-symmetry self-assembly2,23,24. Our experimental findings are supported by theoretical modelling of the colloidal interactions between disks in the nematic host and may provide a route towards realizing many low-symmetry condensed matter phases in systems with building blocks of dissimilar shapes and sizes, as well as their technological applications.
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Data availability
Additional relevant data, such as details of the experimental phase diagram, are provided in the Supplementary Information. Additional raw data that further support the findings of this study are available from the corresponding author on reasonable request. Source data are provided with this paper.
Code availability
The custom codes used to generate plots and theoretical phase diagrams, as well as the Landau–de Gennes modelling and polarizing microscopy image simulation codes, are all provided in the Supplementary Information.
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Acknowledgements
We thank M. Almansouri, B. Fleury, J. van de Lagemaat, Q. Liu, L. Longa, S. Park, B. Senyuk and J.-S. Tai for discussions and technical assistance. This research was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DE-SC0019293 with the University of Colorado at Boulder.
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H.M. synthesized disks and performed experiments and J.-S.W. performed Landau–de Gennes modelling of disk-induced distortions of the molecular order, under the supervision of I.I.S. H.H.W. designed the theoretical model and performed the numerical calculations of colloidal ordering of disks, with input from I.I.S. H.M. and I.I.S. analysed data. I.I.S. conceived, designed and directed the research, provided funding and wrote the manuscript, with input from all authors.
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Extended data figures and tables
Extended Data Fig. 1 Characterization of the building blocks of nematic colloids.
a, Three-photon excitation fluorescence spectra of 5CB obtained for polarizations (P) of excitation light parallel and perpendicular to nm. The inset shows the chemical structure and length of a 5CB molecule. b, Scanning electron micrographs of silica-coated disks. c, Transmission electron micrograph image of a single disk on a copper grid. d, A zoomed-in view of the region in the red square in c, showing the silica layer. e, Schematic of a disk with the NaYF4 core, silica layer and polyethylene glycol functionalization. f, Photon-upconverting luminescence spectrum of disks used in confocal imaging. g–i, Simplified schematics of an effective building block of a molecular–colloidal LC formed by a disk (blue) in a fluid host of molecular rods (grey), illustrating its orthorhombic (g), monoclinic (h) and uniaxial (i) symmetries at different orientations. j, Orientational probability versus θ at different T values marked next to the distributions. The peaks of the curves correspond to θe, as depicted with a black dashed line at θe = 32° ± 1° for T = 33.0 ± 0.1 °C. The error in measuring θ is ±1°. k, Schematics of the main notations. l, Asymmetry of the colloidal surface anchoring potential Ue(θ) ≈ ε2(θ − θe)2 + ε3(θ − θe)3 + ε4(θ − θe)4, plotted for ε2 = 88, ε3 = −550 and ε4 = 1,608. m, Time-lapse bright-field micrographs of a disk in 5CB, showing its field-induced motion when subjected to an electric field EDC (black arrow) generated by a d.c. voltage of 5 V applied to in-plane electrodes 1 mm apart. The elapsed time and nm orientations are marked on the micrographs. n, Displacement versus time for disks with different charges in 5CB when subjected to 5 V between electrodes 1 mm apart. o, Histograms of the disk Brownian displacement probed along and perpendicular to nm using video microscopy, with the inset showing a bright-field micrograph of a disk in 5CB. Solid curves are Gaussian fits to experimental data. Errors in measuring the displacements in n, o are ±10 nm. a.u., arbitrary units.
Extended Data Fig. 2 Comparison and visualization of perturbed nematic order at different disk orientations.
a, Comparison of colloidal particles with tangential surface boundary conditions shaped as thin disks with different orientations (top) and a microring (bottom) in 5CB imaged by bright-field microscopy under similar conditions, with the latter inducing four boojum surface point defects visible as dark, light-scattering spots. No light-scattering singular defects are detected around the disks. b–g, Numerical modelling of nm distortions around a disk at θe = 90°. b–d, Visualization of energy-minimizing nm distortions around the disk, where the green isosurface-enclosed region indicates deviation of nm by >2° from its uniform far field and the black isosurface-enclosed region shows where Sm is reduced by >5% below its bulk equilibrium value. Colours of streamlines depict the opposite directions of local nm tilting relative to the midplane of the disk. e–g, nm distortions near the same disk at 0 nm (e), 10 nm (f) and 50 nm (g) away from its midplane. h–m, Numerical modelling of a disk-induced nm corona at θe = 75°. h–j, Visualization of energy-minimizing nm distortions around the disk, where the green isosurface-enclosed region indicates deviation of nm by >2° from its uniform far-field background. The grey pane in i, j that intersects the green isosurfaces helps to show the monoclinic symmetry of nm distortions relative to the midplane of the disk. k–m, Isosurfaces enclosing small regions with Sm reduced by >5% below its bulk value (black surfaces) and colour-coded local nm distortions around the disk. Right-side insets in k depict details of nematic order perturbations at the disk edges. Field lines in l are shown 250 nm away from the disk centre and the single mirror-symmetry plane. n–r, Visualizations similar to h–m, but for θe = 30°. Note the opposite nm tilt at disk edges compared to that in h–m, as highlighted by orange/blue colouring.
Extended Data Fig. 3 Uniaxial nematic dispersions and magnetic field alignment of disks.
a–c, Uniaxial nematic phase characterized by upconverting luminescence confocal images (a, c) and numerical visualization (b) for in-plane (a) and out-of-plane (c) nm, showing the random orientations of the disk normals ωc in the plane perpendicular to nm. The orientation of the cross-sectional confocal image shown in a is depicted in b with a grey plane. d, Confocal image of a sample within a uniaxial–orthorhombic co-existence region with local correlations of ωc orientations. The disk number density is ρ = 0.16 μm−3 (a), ρ = 0.11 μm−3 (c) and ρ = 0.21 μm−3 (d). e–h, Magnetic alignment of initially randomly oriented disk normals ωc (e, f) to point perpendicular to both nm and B at magnetic field amplitude B = 100 mT (g, h). Visualizations in f, h correspond to the images in e, g, respectively; ρ = 0.09 μm−3 and Z*e ≈ +80e in e, g. i, j, Confocal image (i) and numerical visualization (j) for a uniaxial nematic with disks oriented obliquely to nm at T = 31.5 ± 0.1 °C and ρ = 0.08 μm−3. k, l, Confocal image (k) and numerical visualization (l) of a uniaxial nematic at ρ = 0.19 μm−3 and T = 34.0 ± 0.1 °C.
Extended Data Fig. 4 Temperature-dependent orientations and magnetic control of disks.
a–h, Upconversion-based luminescence confocal images (a–c, e–g) and corresponding numerical visualizations (d, h) of disk dispersions at ρ = 0.19 μm−3 with in-plane nm and ωc at T = 27.0 ± 0.1 °C (a), T = 28.0 ± 0.1 °C (b), T = 30.0 ± 0.1 °C (c), T = 31.0 ± 0.1 °C (e), T = 32.0 ± 0.1 °C (f) and T = 33.0 ± 0.1 °C (g), showing the temperature-dependent disk orientations in the presence of a magnetic field. With increasing T at a field of B = 100 mT applied along a normal to the sample, a domain with a magnetically induced unidirectional alignment of ωc (a–d) transforms into a polydomain state at higher temperatures owing to the random ±θe tilting directionality of ωc relative to nm. Image planes are depicted in grey in f–h. The orientations of nm, magnetic field and scale bars shown between a, b and e, f are the same for all confocal images presented. The disk surface charge is Z*e ≈ +80e.
Extended Data Fig. 5 Coarse-grained visualization of LC phases and background alignment.
a–i, Simplified schematics of different hybrid phases of 5CB molecular rods (grey) and colloidal disks (blue) for the orthorhombic nematic (a), monoclinic nematic (b), monoclinic smectic (c), columnar (d), orthorhombic columnar nematic (e), uniaxial nematic (f), uniaxial nematic with colloidal columns (g), isotropic (h) and colloidal nematic (i) states. An elementary columnar cell in d is depicted in red, with details shown in Fig. 5h. j, Phase diagram upon variation of ρ, T and Z*e, with ‘Co-ex’ referring to co-existence regions. Disks aggregate at Z*e < +10e and form Wigner-type crystals at Z*e > +100e (ref. 10). T is measured with an error of ±0.1°; relative errors for ρ and Z*e are ±5% and ±1%, respectively. k, 3PEFPM image of a 5CB dispersion of disks at ρ = 0.32 μm−3 and T = 27.0 ± 0.1 °C for linear polarization of the excitation light parallel to nm. l, 3PEFPM intensity, averaged over the field of view, versus the angle between the linear polarization of the excitation light and nm (measured with an error of less than ±1°) for 5CB with colloidal disks and pure 5CB under the same conditions. m, n, Visualization of nm between two disks at θe = 75° (m) and θe = 30° (n), with black isosurfaces enclosing spatial regions in which Sm is reduced by >5% relative to its equilibrium bulk value. Regions of distorted nm are highlighted by coloured streamlines, with blue and yellow depicting opposite tilt directions.
Extended Data Fig. 6 Colloidal director rotation and nematic ordering versus temperature.
a–i, Numerical visualizations (a, e, i) and upconversion-based luminescence confocal images (b–d, f–h) of disks in the monoclinic and orthorhombic nematic phases with out-of-plane nm at disk number density ρ = 0.31 μm−3 and at T = 27.0 ± 0.1 °C (b), T = 28.0 ± 0.1 °C (c), T = 30.0 ± 0.1 °C (d), T = 31.0 ± 0.1 °C (f), T = 32.5 ± 0.1 °C (g) and T = 33.0 ± 0.1 °C (h). Note the pretransitional smectic correlations in the vicinity of the smectic phase temperature range in f, g. nc tilts out of the sample plane owing to the temperature-dependent variation of θne, consistent with the elliptical cross-section of disks, revealing oblique orientations of ωc and nc relative to nm at T = 32.5 ± 0.1 °C (g) and T = 33.0 ± 0.1 °C (h). Grey planes in a, e, i depict the orientation of confocal images relative to the directors. Scale bars and nm orientation shown between panels b, c and f, g are the same for all images. j, fc(θ, φ) versus azimuthal angle φ of ωc at ρ = 0.34 μm−3 and different T values. k, fc(θ, φ) versus φ at T = 27.0 ± 0.1 °C for different ρ values. l, Δ (red squares) and S (black circles) versus ρ at T = 27.0 ± 0.1 °C. The disk surface charge is Z*e ≈ +80e.
Extended Data Fig. 7 Characterization of colloidal interactions within different phases.
a–c, Distribution of the angle ϑr between nm and the centre-to-centre pair separation vector, defined in the right-side inset of a, in the monoclinic nematic at ρ = 0.31 μm−3 and T = 30.7 ± 0.1 °C (a), orthorhombic nematic at ρ = 0.34 μm−3 and T = 27.0 ± 0.1 °C (b) and uniaxial nematic at ρ = 0.1 μm−3 and T = 27 ± 0.1 °C (c). The error in measuring ϑr is ±2°. d–f, Scatter plots of the nearest-neighbour disk positions in the monoclinic (d), orthorhombic (e) and uniaxial (f) nematic phases at ρ and T values corresponding to a–c. Anisotropic distributions in d and e correlate with the ordering of disks. The disk surface charge is Z*e ≈ +80e.
Extended Data Fig. 8 Monoclinic and magnetic-field-induced triclinic smectic order.
a–d, Upconversion-based luminescence confocal microscopy images (a, b) and numerical visualizations (c, d) of disks in the monoclinic smectic phase at ρ ≈ 0.3 μm−3, T = 32.3 ± 0.1 °C and θne ≈ 45° (a, c) and for the magnetically induced triclinic smectic state under otherwise similar conditions (b, d). The smectic layer normal (yellow double arrow), nc and nm are initially within the image plane (a, c), but then nc is rotated out of the image plane by a magnetic field of B = 100 mT (b), thus switching the smectic order from monoclinic (a, c) to triclinic (b, d). This switching is driven by the tendency of ωc and nc to align perpendicular to B, where the disk normals also tend to reside on a cone of anchoring-defined easy-angle ωc orientations. Right-side insets of a, b show the layer normal, nm, nc and n⊥ and the applied magnetic field B for the monoclinic and the triclinic smectic states, respectively. The layer normal, nc and n⊥ are all mutually orthogonal and the layer normal, nc and nm are all co-planar in a but not in b, which lacks nontrivial symmetry operations and represents triclinic symmetry.
Extended Data Fig. 9 Nematic and columnar phases and their thermal melting.
a–c, Upconversion-based luminescence confocal micrographs (a, c) and numerical visualization (b) showing a uniaxial nematic with differently oriented columns at ρ = 0.08 μm−3 and T = 27.0 ± 0.1 °C (a, b) and the colloidal structure of the biphasic nematic–columnar co-existence region at ρ = 0.17 μm−3 and T = 27.0 ± 0.1 °C (c). d–g, Monoclinic columnar phase with a two-dimensional oblique lattice characterized experimentally (f, g) and numerically (d–f). The inset in f shows the ordering of columns in a sample with out-of-plane nc and in-plane nm, revealing the oblique lattice of colloidal columns. Probing the distribution of the oblique lattice parameters defined in e across multiple sample regions yields angles of χ = 110° ± 5°; d1 = 1.67 ± 0.15 μm, d2 = 1.75 ± 0.15 μm. The angle between the diagonal of the oblique primitive cell and nm varies within 5°–7°. The measuring error is ±1° for angles and ±50 nm for d1 and d2. Coloured nm director streamlines in e depict nm distortions right above the disks. Orange/blue isosurfaces in f enclose regions in which local nm orientation departs from its uniform far-field background by >2°, with colours corresponding to the tilt directionality of the streamlines in e. g, Confocal cross-sections of the columnar phase showing disks at sample depths shifted by 2 μm sequentially, corresponding to the grey planes labelled in d; ρ = 0.32 μm−3 and T = 27.0 ± 0.1 °C. h, i, Confocal photon-upconverting images of disks at ρ = 0.19 μm−3 that show melting of the colloidal columns upon increasing the temperature from T = 27.0 ± 0.1 °C (h) to T = 33.0 ± 0.1 °C (i). j, k, Columns re-emerge upon decreasing T, as revealed by confocal images at T = 30.0 ± 0.1 °C (j) and T = 27.0 ± 0.1 °C (k). The disk charge is +20e.
Extended Data Fig. 10 Stability analysis of phases with partial positional order.
a, Schematic showing the principal reference frames, vectors and angles. b, Phase diagrams with coordinate axes θne versus ρ (left column) and T versus ρ (right column) of the molecular–colloidal LC for higher (0.002; top row) and lower (0.001; bottom row) values of the elastic energy Ξel (renormalized in units of thermal energy) that quantifies the relative strength of elastic attractions with respect to steric repulsion (Supplementary Information). Red-shaded regions represent the monoclinic smectic phase, consistent with the experimental diagram (Fig. 2a). For lower Ξel (weak elastic interactions), the phases with partial positional order appear at higher disk concentrations owing to stronger electrostatic repulsions between the disks. c, Theoretical estimation of the smectic layer spacing σsm versus T in the monoclinic smectic phase.
Supplementary information
Supplementary Information
This file contains the following sections: Supplementary Methods I: Details of theoretical modelling; Supplementary Methods II: Details of computer simulations; Supplementary Methods III: Parameter specifications for the experimental phase diagram; Supplementary References; and Appendix: Numerical codes.
Video 1: Comparison of confocal and brightfield imaging of a uniaxial LC
Confocal photon-upconverting (top left) and brightfield (top right) imaging videos show edge-on discs diffusing in a rectangular 0.2 × 2 mm capillary with out-of-plane nm. Confocal video tracks colloids through a series of depth-resolved scans, each revealing discs’ positions and orientations, but brightfield images show discs at different depths, impeding such characterisation. Elapsed times are marked; Z*e ≈ +80e. On the bottom-left diagram, a circle marks ρ = 0.1 µm−3 and T = 27.0 ± 0.1 °C. Bottom-right visualisation shows discs and the confocal scanning plane; green isosurfaces enclose regions where nm departs from its far-field orientation by >3°.
Video 2: Discs within an orthorhombic nematic
Time-sequential confocal scanning video shows orientations and diffusion of discs at T = 27.0 ± 0.1 °C, ρ = 0.34 µm−3 (first part of the video, out-of-plane nm) and ρ = 0.31µm−3 (second part, in-plane nm). Discs self-align with nc⊥nm. Inset in the last part of the video is a zoomed-in view within the same sample. Elapsed time within each movie fragment (from different samples) is marked on the frames; ρ and T are shown by a filled circle in the bottom-right diagram. Top-right visualisation shows disc ordering and the scanning plane; green isosurfaces enclose regions where nm departs from its far-field by >3°. Z*e ≈ +80e.
Video 3: Discs in a monoclinic nematic LC
Confocal video shows diffusion and ordering of discs at different ρ and T (moving circle in the top-right diagram). Images have monodomain nm and polydomain (beginning of the movie) or monodomain nc oriented obliquely to nm, as shown by double arrows. Note the ellipsoidal appearance of discs in the frames where discs orient obliquely to the image plane (last part of the video). Elapsed time within each movie fragment (from different samples) is marked on the frames. Bottom-right visualisation of monoclinic order also shows cross-sectional planes corresponding to confocal images. Z*e ≈ +80e.
Video 4: Numerical visualisation of perturbed nematic order of the LC host around a disc
Video shows energy-minimising nm-distortions around a disc obtained from Landau-de Gennes numerical modelling. Green isosurface-enclosed regions (left) indicate deviations of nm by >2° from its uniform far-field and the black isosurface-enclosed regions (right) show where Sm is reduced by >5% below its bulk equilibrium value. Orientation of nm and the associated field lines (pink) stay vertical with changing θe; orientation of \({{\boldsymbol{\omega }}}_{{\rm{c}}}\) is shown with a white double arrow.
Video 5: Visualisation of orthorhombic, monoclinic and uniaxial colloidal nematic states
Video shows ordering of discs in nematic phases at different θne. Green isosurface-enclosed regions (left) indicate deviations of nm by >2° from its far-field; black isosurfaces enclose regions (right) where Sm reduces by >5% below its bulk equilibrium value. Pink lines/arrows indicate the far-field nm; blue and green double arrows depict nc and n⊥, respectively. θe = 90° corresponds to the orthorhombic phase, 0° < θe < 90° to the monoclinic state and θe = 0° to the uniaxial phase. For simplicity, the same disc positions are used at different θne; no smectic correlations are considered at intermediate θne.
Video 6: Smectic order and magnetic control
First part of video shows smectic order at ρ and T indicated by a filled circle in the top-right diagram, with disc orientations/positions and directors depicted in the bottom right. The second part compares the monoclinic smectic (top-left) to the smectic with induced triclinic symmetry when nc tilts out of the sample plane in response to |B| = 100 mT (top-right) while discs satisfy the anchoring conditions and tend orienting with normals orthogonal to B; visualisations beneath show orientations of the layer normal (yellow double arrow), B and directors. Elapsed time within each movie fragment is marked; Z*e ≈ +80e.
Video 7: Uniaxial colloidal nematic order of discs within 5CB in nematic and isotropic states
The first part of the video shows the colloidal orientational order of discs dispersed within 5CB in uniaxial nematic state whereas the second half corresponds to 5CB being in the isotropic phase. The corresponding parameters ρ and T of the phase diagram and schematics of ordering are depicted in the top-right and bottom-right, respectively. The elapsed time is marked on the frames within each movie fragment. The disc surface charge is Z*e ≈ +80e.
Video 8: Out-of-equilibrium thermal reconfiguration
Video shows control of colloidal order upon heating from 27 °C to 32.2 °C and then cooling within 990s. Synchronously with video, black filled circle shows ρ and T (top-left diagram); bottom-left visualisation shows disc ordering. Reconfigurations emerge from tendency to form temperature-dependent equilibrium structures, including orthorhombic/monoclinic nematics and monoclinic smectic. Discs align with nc orthogonal to frames at T = 27.0 °C, then gradually tilt to have in-plane nc rotating to θne ≈ 45° relative to nm, exhibiting first nematic and then smectic monoclinic order; smectic layers dissociate and nc rotates back upon decreasing T. Elapsed time is marked on frames; ρ = 0.30 µm−3; Z*e ≈ +80e.
Video 9: Columnar orthorhombic nematic order
Video shows colloidal columns formed by discs, representing a columnar D2h nematic at T = 27.0 °C and ρ = 0.23 µm−3 (marked in the bottom-right diagram). Numerical visualisation (top right) shows colloidal structures, with coloured isosurfaces depicting regions where nm-departures away from its far-field background are >3° in opposite directions (different colours). Individual discs self-organise into columns as a result of competing electrostatic and anisotropic elastic interactions, which spontaneously align and define nc⊥nm. Video was obtained by time-sequential confocal scanning in a 100-μm-thick glass cell ~10-μm away from one of the inner surfaces. Elapsed time is marked on the frames; Z*e ≈ +20e.
Video 10: Orthorhombic nematic order emerging from melting colloidal columns
Video of confocal (left) and brightfield (right) sequential images showing flow and Brownian motion of orientationally-ordered edge-on discs within the orthorhombic nematic phase formed by melting of colloidal columns at T = 34.0 °C and ρ = 0.29µm-3 (marked in the bottom-left diagram) in a rectangular glass capillary with a 200 μm by 2 mm cross-section. Disc order is shown in the bottom-right visualisation, with green isosurfaces enclosing regions where nm departs from its far-field orientation by >3°. Elapsed time is marked on confocal and brightfield video frames; Z*e ≈ +20e.
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Mundoor, H., Wu, JS., Wensink, H.H. et al. Thermally reconfigurable monoclinic nematic colloidal fluids. Nature 590, 268–274 (2021). https://doi.org/10.1038/s41586-021-03249-0
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DOI: https://doi.org/10.1038/s41586-021-03249-0
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