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Flexoelectricity-driven toroidal polar topology in liquid-matter helielectrics

Nature Physics (2024)Cite this article

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Abstract

Magnetic and electric dipoles have similar symmetry and are therefore expected to exhibit many common structures. However, despite frequent reports of various spin textures composed of magnetic dipoles, investigations on long-range ordered electric dipoles have been scarce, until recently. Here we discover spontaneous toroidal polar topology in an emerging ferroelectric liquid state with polarization helices, dubbed ‘helielectric nematic’. The interplay between liquid-crystal elasticity and polar interactions results in a continuous in-plane rotation of the polarization, producing a periodic toroidal assembly. In partial analogy with magnetism, the neighbouring toroidal domains are separated by circularly shaped domain walls. The local polarization switching enables unique shrinkage and expansion dynamics of the toroidal domains. The discovery provides opportunities for developing designable and switchable ferroelectric-liquid-matter opto-electronics.

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Fig. 1: The emerging polarization LC systems as analogue of the ferromagnetic systems.
Fig. 2: Observation of the toroidal polar topology.
Fig. 3: Determination of the toroidal polar topology in the linear and nonlinear optical regimes.
Fig. 4: Field-induced transformation of the toroidal polar topology.
Fig. 5: Flexoelectricity-driven stabilization of the toroidal polar topology.

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Data availability

All data that support the findings of this study are available in the article and in Supplementary Materials. Source data are deposited in the Open Science Framework (OSF) database of ‘Polar toroidal topology in helielectric nematics’ (https://doi.org/10.17605/OSF.IO/UKHY2).

Code availability

The codes used for the numerical calculations are available from the corresponding author upon reasonable request.

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Acknowledgements

S.A. and M.H. acknowledge the National Key Research and Development Program of China (No. 2022YFA1405000), the Recruitment Program of Guangdong (No. 2016ZT06C322) and the 111 Project (No. B18023). S.A. acknowledges support from the Research Fund for International Excellent Young Scientists (RFIS-II; No. 1231101194), the International Science and Technology Cooperation Program of Guangdong province (No.2022A0505050006) and the Fundamental Research Funds for the Central University (No. 2022ZYGXZR001). M.H. acknowledges support from the National Natural Science Foundation of China (NSFC No. 52273292).

Author information

Author notes

  1. These authors contributed equally: Jidan Yang, Yu Zou.

Authors and Affiliations

  1. South China Advanced Institute for Soft Matter Science and Technology (AISMST), School of Emergent Soft Matter, South China University of Technology, Guangzhou, China

    Jidan Yang, Yu Zou, Jinxing Li, Mingjun Huang & Satoshi Aya

  2. Guangdong Provincial Key Laboratory of Functional and Intelligent Hybrid Materials and Devices, Guangdong Basic Research Center of Excellence for Energy and Information Polymer Materials, South China University of Technology, Guangzhou, China

    Mingjun Huang & Satoshi Aya

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Contributions

S.A. conceived the study and designed the research. J.L. and M.H. synthesized the materials. J.Y., Y.Z. and S.A. made topological analyses. J.Y., Y.Z. and S.A. conducted the measurements and analyses of the material physical properties. Y.Z. and S.A. made the investigation of the analytical solutions and conducted the numerical simulations. J.Y., Y.Z. and S.A. wrote the manuscript. All the authors discussed and amended the manuscript.

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Correspondence to Mingjun Huang or Satoshi Aya.

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Extended data

Extended Data Fig. 1 Numerical calculation of the field-induced dynamics of toroidal polar topology.

a, The first row displays the time evolution of the polarization field and the spatial distribution of the elevation (out-of-plane) angle δ, which aligns well with experimental results. The second row shows the evolution of charge distributions (σb) on DWs and at the droplet’s centre, with corresponding total electric field (E = E0 + Edepol) donated by orange arrows. The direction of the applied d.c. electric field (E0) is shown by a red arrow. b, The evolutions of spatial distributions of the depolarization free energy density fdepol, the total electrostatic energy density felect, the flexoelectric free energy density fflexo, and the total free energy density ftotal, respectively. c, Time evolutions of the total free energy total (black solid line) and the average out-of-plane angle \(\bar\delta\) of the polarization (dark-yellow solid line) in an area with a clear out-of-plane tilting (circled by a green box in the right-bottom corner figure of (b)). The bottom of (c) shows time evolutions of various integrated free energy terms in the circled region, including the splay elastic energy (blue dotted line), twist elastic energy (red dotted line), bend elastic energy (light-yellow dotted line), polarization gradient energy (purple long-dash line), flexoelectric energy (pink dot-dash line and electric) and electrostatic energy (green scattered triangles connected by a long-dash line). The free energies are defined in Eqs. (S53)−(S55). Details about simulation parameters are provided in Methods and Supplementary Discussion 4.

Extended Data Fig. 2 Visualization of the role of the depolarization effect on field-induced dynamics of toroidal polar topology.

Comparisons between the simulation and experimental results on the polarization field reorientation behaviours under d.c. electric fields are shown. a,b, The directionalities of the applied d.c. electric field are shown by red arrows. The first rows of (a) and (b) show simulations of the time evolution of the polarization field under d.c. electric fields by considering the depolarization effect. The results are well consistent with the experimental observations. As a comparison, the second rows of (a) and (b) show the simulation by neglecting the depolarization effect. The results demonstrate complete deviation from the experiments. c,d, (c) and (d) are the corresponding equilibrium polarization fields under the electric fields reconstructed from the experimental FCPM images. Scale bar, 20 μm. The Methods and Supplementary Discussion 4 contain the parameters used in the simulations.

Extended Data Fig. 3 Potential director and polarization fields in the apolar chiral nematic and HN* states.

a-f, Potential director and polarization fields in the apolar chiral nematic (a-c) and HN* (d-f) states under the strong planar anchoring condition. The helical axes of the twisted structures (h) are donated by black arrows. Assuming that the top and bottom directors or polarizations are strongly anchored along the y direction, the possible structures include: the fingerprint-like texture without (a) or with (d) disclinations; Grandjean textures (c and f); the untwisting polarization planar structure without (b) or with (e) disclinations.

Supplementary information

Supplementary Information

Supplementary Figs. 1–17, Materials, Discussions 1–4, List of Videos 1–2 and References.

Supplementary Video 1

PLM observation of the growth process of the toroidal polar topology under crossed polarizers. The video speed is 10× faster than the real observation. The concentration of R811 in RM-OC2 is 0.5 wt%. The thickness of the homemade cell is 3.2 μm. The image size is 200 μm × 200 μm.

Supplementary Video 2

PLM observation of the electric response of the toroidal polar topology. The video speed is 16x faster than the real observation. A square-wave electric field of E = ±4 × 10−3 V μm−1 and field-periodicity of 240 s is applied in the horizontal direction. When switching off the d.c. field, the polarization field returns to equilibrium in the initial state. The image size is 420 μm × 420 μm.

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Yang, J., Zou, Y., Li, J. et al. Flexoelectricity-driven toroidal polar topology in liquid-matter helielectrics. Nat. Phys. (2024). https://doi.org/10.1038/s41567-024-02439-7

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