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Geometric frustration in buckled colloidal monolayers

Abstract

Geometric frustration arises when lattice structure prevents simultaneous minimization of local interaction energies. It leads to highly degenerate ground states and, subsequently, to complex phases of matter, such as water ice, spin ice, and frustrated magnetic materials. Here we report a simple geometrically frustrated system composed of closely packed colloidal spheres confined between parallel walls. Diameter-tunable microgel spheres are self-assembled into a buckled triangular lattice with either up or down displacements, analogous to an antiferromagnetic Ising model on a triangular lattice. Experiment and theory reveal single-particle dynamics governed by in-plane lattice distortions that partially relieve frustration and produce ground states with zigzagging stripes and subextensive entropy, rather than the more random configurations and extensive entropy of the antiferromagnetic Ising model. This tunable soft-matter system provides a means to directly visualize the dynamics of frustration, thermal excitations and defects.

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Figure 1: Ising ground state.
Figure 2: Buckled monolayer of colloidal spheres.
Figure 3: Fluctuation in the number of frustrated bonds per particle as a function of its average.
Figure 4: Tiling the plane with isosceles triangles.
Figure 5: Single-particle dynamics.

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Acknowledgements

We thank B. Chakraborty, R. D. Kamien, D. Li, A. J. Liu, C. D. Modes, T.-K. Ng, S. A. Rice, Y. Snir, T. A. Witten and Y. Zhou for discussions. This work was supported primarily by the NSF through MRSEC grant DMR-0520020 and partially by DMR-0804881 (NSF) and by NAG-2939 (NASA).

Author Contributions Y.H. and A.M.A. initialized the project. A.M.A. synthesized the particles. Y.H. conducted the experiments. Y.S. performed the simulations and provided the tiling model. Y.H. and Y.S. analysed and explained the experimental data. P.Y. characterized the particles. T.C.L. provided theoretical guidance. A.G.Y. provided experimental guidance. Y.H., Y.S., T.C.L. and A.G.Y. wrote the paper.

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Correspondence to Yilong Han or Yair Shokef.

Supplementary information

Supplementary Information

This file contains a Supplementary Method, Supplementary Data, Supplementary References, Supplementary Figures S1-S6 with Legends and Supplementary Table 1. (PDF 276 kb)

Supplementary Movie S1

This file contains Supplementary Movie S1: Raw experimental video at T = 24.7°C, corresponding to Fig. 2A. (MOV 485 kb)

Supplementary Movie S2

This file contains Supplementary Movie S2: Labyrinth pattern at T = 24.7°C, corresponding to Fig. 2B. (MOV 1485 kb)

Supplementary Movie S3

This file contains Supplementary Movie S3: Thermal excitations and defects at T = 24.7°C, corresponding to Fig. 2C. (AVI 4111 kb)

Supplementary Movie S4

This file contains Supplementary Movie S4: Raw experimental video at T = 27.1°C, corresponding to Fig. 2D. (MOV 600 kb)

Supplementary Movie S5

This file contains Supplementary Movie S5: Labyrinth pattern at T = 27.1°C, corresponding to Fig. 2E. (MOV 3138 kb)

Supplementary Movie S6

This file contains Supplementary Movie S6: Thermal excitations and defects at T = 27.1°C, corresponding to Fig. 2F. (AVI 6185 kb)

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Han, Y., Shokef, Y., Alsayed, A. et al. Geometric frustration in buckled colloidal monolayers. Nature 456, 898–903 (2008). https://doi.org/10.1038/nature07595

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