Elastic colloidal monopoles and reconfigurable self-assembly in liquid crystals

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Abstract

Monopole-like electrostatic interactions are ubiquitous in biology1 and condensed matter2,3,4, but they are often screened by counter-ions and cannot be switched from attractive to repulsive. In colloidal science, where the main goal is to develop colloidal particles2,3 that mimic and exceed the diversity and length scales of atomic and molecular assembly, electrostatically charged particles cannot change the sign of their surface charge or transform from monopoles to higher-order multipoles4. In liquid-crystal colloids5,6,7, elastic interactions between particles arise to minimize the free energy associated with elastic distortions in the long-range alignment of rod-like molecules around the particles5. In dipolar6,8, quadrupolar8,9,10,11,12 and hexadecapolar13 nematic colloids, the symmetries of such elastic distortions mimic both electrostatic multipoles14 and the outermost occupied electron shells of atoms7,15,16. Electric and magnetic switching17,18, spontaneous transformations19 and optical control20 of elastic multipoles, as well as their interactions with topological defects and surface boundary conditions, have been demonstrated in such colloids21,22,23. However, it has long been understood5,24 that elastic monopoles should relax to uniform or higher-order multipole states because of the elastic torques that they induce5,7. Here we develop nematic colloids with strong elastic monopole moments and with elastic torques balanced by the optical torques induced by ambient light. We demonstrate the monopole-to-quadrupole reconfiguration of these colloidal particles by unstructured light, which resembles the driving of atoms between the ground state and various excited states. We show that the sign of the elastic monopoles can be switched, and that like-charged monopoles attract whereas oppositely charged ones repel, unlike in electrostatics14. We also demonstrate the out-of-equilibrium dynamic assembly of these colloidal particles. This diverse and surprising behaviour is explained using a model that considers the balance of the optical and elastic torques that are responsible for the excited-state elastic monopoles and may lead to light-powered active-matter systems and self-assembled nanomachines.

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Fig. 1: Oppositely charged elastic colloidal monopoles.
Fig. 2: Colloidal interactions between elastic monopoles.
Fig. 3: Optical switching of the signs and strengths of interactions of the elastic monopoles.
Fig. 4: Out-of-equilibrium colloidal assembly.

Data availability

All data generated or analysed during this study are included in the published article and its Supplementary Information and are available from the corresponding author upon reasonable request.

Code availability

The codes used in this study for the numerical simulation and calculation are available upon request.

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Acknowledgements

We thank P. Ackerman, A. Hess, S. Park, J.-S. Tai and M. Tasinkevych for technical assistance and discussions. This research was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, under Award ER46921, contract DE-SC0019293 with the University of Colorado at Boulder.

Reviewer information

Nature thanks Dong Ki Yoon and the other anonymous reviewer(s) for their contribution to the peer review of this work.

Author information

Y.Y. and Q.L. synthesized the azobenzene-containing dye molecules. Q.L. fabricated the colloidal particles. Y.Y. and B.S. performed the experiments and Y.Y. performed the numerical modelling. Y.Y., B.S. and I.I.S. analysed the data. I.I.S. conceived the idea and directed the project, designed the experiments, provided funding and wrote the manuscript, with the input from all authors.

Correspondence to Ivan I. Smalyukh.

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The authors declare no competing interests.

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Extended data figures and tables

Extended Data Fig. 1 Hexagonal colloidal platelet with azobenzene monolayers dispersed in a nematic LC.

a, Three-photon excitation fluorescence microscopy images of a platelet in an LC cell, obtained at the cross-sectional plane (xz plane) passing through the middle of the platelet, and at the vertical plane (xy plane) that is orthogonal to the plane of the cell and the large-area faces of the platelet. Both planes pass through the centre of mass of the particle. b, Scanning electron microscopy image of an individual platelet placed on a substrate. c, Schematic of a platelet suspended in a nematic LC. Green rods indicate the director field n(r); red rods indicate the orientation of the azobenzene molecules on the surface of the platelet; the black semi-sphere on one of the vertices of the hexagon represents the surface point defects called ‘boojums’. The optical torque that rotates the platelet away from its equilibrium state is balanced by the counteracting elastic torque caused by director twisting, as shown by the blue and red curved arrows.

Extended Data Fig. 2 Microdisplay-based illumination control system setup.

a, Schematic of the optical illumination setup. White light from the lamp is split in half and directed into two optical paths using a beam splitter, filtered by two separate dichroic mirrors to supply blue light to two of the LC microdisplays (LCDs). After passing through the microdisplays, which define the spatial patterns of illumination, the two light beams are recombined by a dichroic prism and then separated again on the basis of their polarization. The two separated light paths correspond to patterns generated on the two LCDs and their polarization is further controlled by the half-wave plates. Additional mirrors enable fine-tuning of the position of the projected patterns after the light is re-combined and coupled into the microscope. In the schematic, ‘50/50’ denotes the plate beam splitters (BSW10R, Thorlabs); ‘DM-B’ indicates the dichroic mirrors that reflect blue light; L1 and L2 are convex lenses; ‘HWP’ denotes half-wave plates; ‘P’ represents the polarizer; ‘DCP’ is a dichroic cross-prism; ‘PBS’ is a cube polarizing beam splitter (CCM1-PB251, Thorlabs); M1 and M2 are silver mirrors. b, Spectra of the red imaging light obtained by filtering the light of the microscope lamp, and of the blue excitation light from the microdisplay control system. Solid and dashed blue lines are the spectra of the two blue channels when turned on separately.

Extended Data Fig. 3 Optical response of azobenzene-containing molecules and monopole-like platelets in a nematic LC.

a, Absorbance spectrum of the photosensitive molecules of derivative methyl red in toluene at a concentration of 5 × 10−5 M. The spectrum was obtained using a 1-cm-thick cuvette with a spectrometer (Cary 500, Varian). b, Time dependence of the azimuthal orientation angle of the platelet upon switching on the blue excitation light at time t ≈ 1.2 s, showing how the monopole moment is turned on upon light exposure. c, Time dependence of the azimuthal orientation angle of the platelet relative to its equilibrium position upon switching off the blue excitation light at t = 0 s.

Extended Data Fig. 4 Computer simulated director configuration of an elastic monopole.

a, Cross-sections of the director structure in the xz, yz and xy planes passing through the centre of the structure. The local director orientation is shown by grey cylinders with blue and red ends. b, Zoomed-in view of the cross-sections shown in a. c, Distance dependence of the director deviation nx along the z axis. The simulated results (black squares) are fitted with the function nx −1/r anticipated for monopoles (red lines). R represents the effective particle size.

Extended Data Fig. 5 Characterization of interactions between elastic monopoles.

The strength of the elastic monopoles considered here differs from that in the main-text figures. a, c, Separation distance versus time of monopolar attraction (a) and repulsion (c). Red lines in a and c are the best fits of the experimental data with the function rc(t) = (r03 − 3αt)1/3 and with fitting coefficients r0 = 16.7 μm, α = 21.1 μm3 s-1 (a) and r0 = 16.2 μm, α = −10.6 μm3 s−1 (c). b, d, Dependence of the corresponding potential and force (insets) on the distance. Blue lines are the best fits of the experimental data with a potential proportional to ±1/rc. The elastic charge estimated from the fitting parameters is 0.51 μm (a, b) and 0.36 μm (c, d).

Extended Data Fig. 6 Angular dependence of quadrupolar interactions.

Trajectories are colour-coded with the time elapsed since the release of the platelets from the laser traps, and the duration of interaction is tmax − tmin ≈ 100 s. The insets are micrographs of platelets repelling each other when the separation vector that points from the centre of one particle to that of the other is roughly parallel or perpendicular to n0. The images were taken using parallel polarizers shown by the white double arrows. By contrast, the platelets attract when the separation vector is at an angle with respect to n0, as shown by the trajectory in the middle; the corresponding distance-versus-time dependence is shown in Fig. 4e.

Extended Data Fig. 7 Computer-simulated polarization states of the red imaging light after passing through the sample.

The angle ϕ between the long axis of the polarization ellipse and n0 and the corresponding ellipticity are shown with red dots and black squares, respectively. The horizontal axis (z − R)/R represents the distance from the edge of the particle along the z direction relative to the particle radius R.

Supplementary information

Video 1: Optical switching of the elastic monopole signs and interactions

Elastic monopoles of opposite signs are first induced by exposure to linearly blue light beams with different linear polarizations. Then, changing the polarization of the blue light that excites the upper particle flips the sign of the induced monopole to be the same as that of the lower particle. Following this, the direction of the interaction also switches from repulsion to attraction. “P” and “A” mark parallel polarizer and analyzer of the microscope with a quarter-wave plate inserted in-between (fast axis of the waveplate is shown with an orange double arrow). Regions exposed to polarized blue light are shown in dashed boxes, with the corresponding linear polarization directions marked with blue double arrows. The direction of interaction force is shown with a pair of white arrows on the left

Video 2: Out-of-equilibrium colloidal interaction

Two colloidal platelets attract while spinning and later assemble while continuing to spin in the same direction. Polarization of the blue excitation light (blue double arrow) is parallel to n0

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