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Geometric frustration in buckled colloidal monolayers

Nature volume 456, pages 898–903 (2008)Cite this article

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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References

  1. Moessner, R. & Ramirez, A. R. Geometrical frustration. Phys. Today 59, 24–26 (2006)

    Article  CAS  Google Scholar 

  2. Pauling, L. The structure and entropy of ice and of other crystals with some randomness of atomic arrangement. J. Am. Chem. Soc. 57, 2680–2684 (1935)

    Article  CAS  Google Scholar 

  3. Harris, M. J., Bramwell, S. T., McMorrow, D. F., Zeiske, T. & Godfrey, K. W. Geometrical frustration in the ferromagnetic pyrochlore Ho2Ti2O7 . Phys. Rev. Lett. 79, 2554–2557 (1997)

    Article  ADS  CAS  Google Scholar 

  4. Bramwell, S. T. & Gingras, M. J. P. Spin ice state in frustrated magnetic pyrochlore materials. Science 294, 1495–1501 (2001)

    Article  ADS  CAS  Google Scholar 

  5. Moessner, R. Magnets with strong geometric frustration. Can. J. Phys. 79, 1283–1294 (2001)

    Article  ADS  CAS  Google Scholar 

  6. Ramirez, A. R. Geometric frustration: Magic moments. Nature 421, 483 (2003)

    Article  ADS  CAS  Google Scholar 

  7. Anderson, P. W. The resonating valence bond state in La2CuO4 and superconductivity. Science 235, 1196–1198 (1987)

    Article  ADS  CAS  Google Scholar 

  8. Wannier, G. H. Antiferromagnetism. The triangular Ising net. Phys. Rev. 79, 357–364 (1950); erratum Phys. Rev. B 7, 5017 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  9. Houtappel, R. M. F. Order-disorder in hexagonal lattices. Physica 16, 425–455 (1950)

    Article  ADS  MathSciNet  Google Scholar 

  10. Davidović, D. et al. Correlations and disorder in arrays of magnetically coupled superconducting rings. Phys. Rev. Lett. 76, 815–818 (1996)

    Article  ADS  Google Scholar 

  11. Hilgenkamp, H. et al. Ordering and manipulation of the magnetic moments in large-scale superconducting π-loop arrays. Nature 422, 50–53 (2003)

    Article  ADS  CAS  Google Scholar 

  12. Wang, R. F. et al. Artificial ‘spin ice’ in a geometrically frustrated lattice of nanoscale ferromagnetic islands. Nature 439, 303–306 (2006)

    Article  ADS  CAS  Google Scholar 

  13. Möller, G. & Moessner, R. Artificial square ice and related dipolar nanoarrays. Phys. Rev. Lett. 96, 237202 (2006)

    Article  ADS  Google Scholar 

  14. Nisoli, C. et al. Ground state lost but degeneracy found: The effective thermodynamics of artificial spin ice. Phys. Rev. Lett. 98, 217203 (2007)

    Article  ADS  Google Scholar 

  15. Libál, A., Reichhardt, C. & Reichhardt, C. J. O. Realizing colloidal artificial ice on arrays of optical traps. Phys. Rev. Lett. 97, 228302 (2006)

    Article  ADS  Google Scholar 

  16. Koshikiya, Y. & Hachisu, S. [in Japanese] Lecture at Colloid Symposium of Japan (1982)

    Google Scholar 

  17. Pieranski, P., Strzelecki, L. & Pansu, B. Thin colloidal crystals. Phys. Rev. Lett. 50, 900–903 (1983)

    Article  ADS  CAS  Google Scholar 

  18. Van Winkle, D. H. & Murray, C. A. Experimental observation of two-stage melting in a classical two-dimensional screened Coulomb system. Phys. Rev. Lett. 58, 1200–1203 (1987)

    Article  ADS  Google Scholar 

  19. Weiss, J. A., Oxtoby, D. W., Grier, D. G. & Murray, C. A. Martensitic transition in a confined colloidal suspension. J. Chem. Phys. 103, 1180–1190 (1995)

    Article  ADS  CAS  Google Scholar 

  20. Pansu, B., Pieranski & Pieranski Direct observation of a buckling transition during the formation of thin colloidal crystals. J. Phys. 45, 331–339 (1984)

    Article  CAS  Google Scholar 

  21. Chou, T. & Nelson, D. R. Buckling instabilities of a confined colloid crystal layer. Phys. Rev. E 48, 4611–4621 (1993)

    Article  ADS  CAS  Google Scholar 

  22. Schmidt, M. & Löwen, H. Freezing between two and three dimensions. Phys. Rev. Lett. 76, 4552–4555 (1996)

    Article  ADS  CAS  Google Scholar 

  23. Schmidt, M. & Löwen, H. Phase diagram of hard spheres confined between two parallel plates. Phys. Rev. E 55, 7228–7241 (1997)

    Article  ADS  CAS  Google Scholar 

  24. Zangi, R. & Rice, S. A. Phase transitions in a quasi-two-dimensional system. Phys. Rev. E 58, 7529–7544 (1998)

    Article  ADS  CAS  Google Scholar 

  25. Melby, P. et al. The dynamics of thin vibrated granular layers. J. Phys. Condens. Matter 17, S2689–S2704 (2005)

    Article  CAS  Google Scholar 

  26. Osterman, N., Babič, D., Poberaj, I., Dobnikar, J. & Ziherl, P. Observation of condensed phases of quasiplanar core-softened colloids. Phys. Rev. Lett. 99, 248301 (2007)

    Article  ADS  CAS  Google Scholar 

  27. Alsayed, A. M., Islam, M. F., Zhang, J., Collings, P. J. & Yodh, A. G. Premelting at defects within bulk colloidal crystals. Science 309, 1207–1210 (2005)

    Article  ADS  CAS  Google Scholar 

  28. Han, Y., Ha, N. Y., Alsayed, A. M. & Yodh, A. G. Melting of two-dimensional tunable-diameter colloidal crystals. Phys. Rev. E 77, 041406 (2008)

    Article  ADS  CAS  Google Scholar 

  29. Shokef, Y. & Lubensky, T. C. Stripes, zigzags, and slow dynamics in buckled hard spheres. Preprint at 〈http://arxiv.org/abs/0807.4884〉 (2008)

  30. Ogawa, T. A maze-like pattern in a monodispersive latex system and the frustration problem. J. Phys. Soc. Jpn 52 (Suppl.). 167–170 (1983)

    Google Scholar 

  31. Crocker, J. C. & Grier, D. G. Methods of digital video microscopy for colloidal studies. J. Colloid Interface Sci. 179, 298–310 (1996)

    Article  ADS  CAS  Google Scholar 

  32. Metcalf, B. D. Ground state spin orderings of the triangular Ising model with the nearest and next nearest neighbor interaction. Phys. Lett. A 46, 325–326 (1974)

    Article  ADS  CAS  Google Scholar 

  33. Millane, R. P. & Blakeley, N. D. Boundary conditions and variable ground state entropy for the antiferromagnetic Ising model on a triangular lattice. Phys. Rev. E 70, 057101 (2004)

    Article  ADS  CAS  Google Scholar 

  34. Chen, Z. Y. & Kardar, M. Elastic antiferromagnets on a triangular lattice. J. Phys. C 19, 6825–6831 (1986)

    Article  ADS  Google Scholar 

  35. Gu, L., Chakraborty, B., Garrido, P. L., Phani, M. & Lebowitz, J. L. Monte Carlo study of a compressible Ising antiferromagnet on a triangular lattice. Phys. Rev. B 53, 11985–11992 (1996)

    Article  ADS  CAS  Google Scholar 

  36. Lee, S.-H., Broholm, C., Kim, T. H., Ratcliff, W. & Cheong, S.-W. Local spin resonance and spin-Peierls-like phase transition in a geometrically frustrated antiferromagnet. Phys. Rev. Lett. 84, 3718–3721 (2000)

    Article  ADS  CAS  Google Scholar 

  37. Villain, J., Bidaux, R., Carton, J. P. & Conte, R. Order as an effect of disorder. J. Phys. 41, 1263–1272 (1980)

    Article  MathSciNet  CAS  Google Scholar 

  38. Mau, S. C. & Huse, D. A. Stacking entropy of hard-sphere crystals. Phys. Rev. E 59, 4396–4401 (1999)

    Article  ADS  CAS  Google Scholar 

  39. Liebmann, R. Statistical Mechanics of Periodic Frustrated Ising Systems (Springer, 1986)

    Google Scholar 

  40. Nussinov, Z. Avoided phase transitions and glassy dynamics in geometrically frustrated systems and non-Abelian theories. Phys. Rev. B 69, 014208 (2004)

    Article  ADS  Google Scholar 

  41. Shih, W. Y. & Stroud, D. Two-dimensional superconducting arrays in a magnetic field: Effects of lattice structures. Phys. Rev. B 32, 158–165 (1985)

    Article  ADS  CAS  Google Scholar 

  42. Nussinov, Z. Commensurate and incommensurate O(n) spin systems: novel even-odd effects, a generalized Mermin-Wagner-Coleman theorem, and ground states. Preprint at 〈http://arxiv.org/abs/cond-mat/0105253〉 (2001)

  43. Chakraborty, B., Gu, L. & Yin, H. Glassy dynamics in a frustrated spin system: The role of defects. J. Phys. Condens. Matter 12, 6487–6495 (2000)

    Article  ADS  CAS  Google Scholar 

  44. Yin, H. & Chakraborty, B. Entropy-vanishing transition and glassy dynamics in frustrated spins. Phys. Rev. Lett. 86, 2058–2061 (2001)

    Article  ADS  CAS  Google Scholar 

  45. Ediger, M. D. Spatially heterogeneous dynamics in supercooled liquids. Annu. Rev. Phys. Chem. 51, 99–128 (2000)

    Article  ADS  CAS  Google Scholar 

  46. Blunt, M. O. et al. Random tiling and topological defects in a two-dimensional molecular network. Science 322, 1077–1081 (2008)

    Article  ADS  CAS  Google Scholar 

Download references

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.

Author information

Author notes

  1. Yair Shokef

    Present address: Present address: Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel.,

  2. Yilong Han and Yair Shokef: These authors contributed equally to this work.

Authors and Affiliations

  1. Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104, USA,

    Yilong Han, Yair Shokef, Ahmed M. Alsayed, Peter Yunker, Tom C. Lubensky & Arjun G. Yodh

  2. Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong,

    Yilong Han

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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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