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Solutions for hydrodynamics of 5- and 10-fold symmetry quasicrystals

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Abstract

This study carries out a complete analysis of time-space solution of hydrodynamics of pentagonal/decagonal quasicrystals. The behaviors of wave propagation for phonons and diffusion for phasons and coupling between phonon-phason fields are explored explicitly. Comprehensive discussion on physical time-space variations of all hydrodynamic field variables of the alloy quasicrystals is given. The computational specimen is simple, convenient in testing computational results, and provides a possibility that is easy to test experimentally. The quantitative results of mass density, viscosity velocities, phonon displacements, phason displacements, phonon stresses, phason stresses, viscosity stresses, and their time-space variations help us understand the motion of solid quasicrystals in a hydrodynamic condition (long-wavelength and low-frequency). The analysis presented in this paper can be used for octagonal and dodecagonal quasicrystals and is easily extended to other two-dimensional quasicrystals and three-dimensional icosahedral quasicrystals. Some problems explored by the computational results are also discussed.

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References

  1. Shechtman, D., Blech, I., Gratias, D., and Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry. Physical Review Letters, 53(20), 1951–1953 (1984)

    Article  Google Scholar 

  2. Bindi, L., Steinhardt, P. J., Yao, N., and Lu, P. J. Natural quasicrystals. Science, 324(5932), 1306–1309 (2009)

    Article  Google Scholar 

  3. Zeng, X., Ungar, G., Liu, Y., Percec, V., Dulcey, A. E., and Hobbs, J. K. Supramolecular dendritic liquid quasicrystals. nature, 428(6979), 157–160 (2004)

    Article  Google Scholar 

  4. Takano, A., Kawashima, W., Noro, A., Isono, Y., Tanaka, N., Dotera, T., and Matsushita, Y. A mesoscopic Archimedean tiling having a new complexity in an ABC star polymer. Journal of Polymer Science Part B: Polymer Physics, 43(18), 2427–2432 (2005)

    Article  Google Scholar 

  5. Hayashida, K., Dotera, T., Takano, A., and Matsushita, Y. Polymeric quasicrystal: mesoscopic quasicrystalline tiling in ABC star polymers. Physical Review Letters, 98(19), 195502 (2007)

    Article  Google Scholar 

  6. Talapin, D. V., Shevchenko, E. V., Bodnarchuk, M. I., Ye, X., Chen, J., and Murray, C. B. Quasicrystalline order in self-assembled binary nanoparticle superlattices. nature, 461(7266), 9649–67 (2009)

    Article  Google Scholar 

  7. Fischer, S., Exner, A., Zielske, K., Perlich, J., Deloudi, S., Steurer, W., Lindner, P., and Förster, S. Colloidal quasicrystals with 12-fold and 18-fold diffraction symmetry. Proceedings of the National Academy of Sciences, 108(5), 1810–1814 (2011)

    Article  Google Scholar 

  8. Fan, T. Y. and Fan, L. Plastic fracture of quasicrystals. Philosophical Magazine, 88(4), 523–535 (2008)

    Article  Google Scholar 

  9. Li, W., Fan, T. Y., and Wu, Y. L. Plastic analysis of crack problems in three-dimensional icosahedral quasicrystalline material. Philosophical Magazine, 89(31), 2823–2831 (2009)

    Article  Google Scholar 

  10. Fan, T. Y., Wang, X. F., Li, W., and Zhu, A. Y. Elasto-hydrodynamics of quasicrystals. Philosophical Magazine, 89(6), 501–512 (2009)

    Article  Google Scholar 

  11. Li, X. F. A general solution of elasto-hydrodynamics of two-dimensional quasicrystals. Philosophical Magazine Letters, 91(4), 313–320 (2011)

    Article  Google Scholar 

  12. Li, X. F. Elastohydrodynamic problems in quasicrystal elasticity theory and wave propagation. Philosophical Magazine, 93(13), 1500–1519 (2013)

    Article  Google Scholar 

  13. Cheng, H., Fan, T. Y., Sun, J. J., and Wei, H. Possible soft-matter quasicrystals of 5- and 10-fold symmetries and hydrodynamics. Computational Materials Science, 105, 47–54 (2015)

    Article  Google Scholar 

  14. Bak, P. Phenomenological theory of icosahedral incommensurate (“quasiperiodic”) order in Mn-Al alloys. Physical Review Letters, 54(14), 1517–1519 (1985)

    Article  Google Scholar 

  15. Bak, P. Symmetry, stability, and elastic properties of icosahedral incommensurate crystals. Physical Review B, 32(9), 5764–5772 (1985)

    Article  MathSciNet  Google Scholar 

  16. Lubensky, T. C., Ramaswamy, S., and Toner, J. Hydrodynamics of icosahedral quasicrystals. Physical Review B, 32(11), 7444–7452 (1985)

    Article  Google Scholar 

  17. Levine, D., Lubensky, T. C., Ostlund, S., Ramaswamy, S., Steinhardt, P. J., and Toner, J. Elasticity and dislocations in pentagonal and icosahedral quasicrystals. Physical Review Letters, 54(14), 1520–1523 (1985)

    Article  Google Scholar 

  18. Lubensky, T. C. Symmetry, elasticity and hydrodynamics in quasiperiodic. Introduction to Quasicrystals, Academic Press, San Diego, 199–273 (1988)

    Chapter  Google Scholar 

  19. Socolar, J. E., Lubensky, T. C., and Steinhardt, P. J. Phonons, phasons, and dislocations in quasicrystals. Physical Review B, 34(5), 3345–3360 (1986)

    Article  Google Scholar 

  20. Rochal, S. B. and Lorman, V. L. Minimal model of the phonon-phason dynamics in icosahedral quasicrystals and its application to the problem of internal friction in the i-AlPdMn alloy. Physical Review B, 66(14), 144204 (2002)

    Article  Google Scholar 

  21. Fan, T. Y., Wang, X. F., Li, W., and Zhu, A. Y. Elasto-hydrodynamics of quasicrystals. Philosophical Magazine, 89(6), 501–512 (2009)

    Article  Google Scholar 

  22. Zhu, A. Y. and Fan, T. Y. Dynamic crack propagation in decagonal Al-Ni-Co quasicrystal. Journal of Physics: Condensed Matter, 20(29), 295217 (2008)

    Google Scholar 

  23. Ding, D. H., Yang, W., Hu, C., and Wang, R. Generalized elasticity theory of quasicrystals. Physical Review B, 48(10), 7003–7010 (1993)

    Article  Google Scholar 

  24. Hu, C., Wang, R., and Ding, D. H. Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals. Reports on Progress in Physics, 63(1), 1–39 (2000)

    Article  MathSciNet  Google Scholar 

  25. Fan, T. Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications, Springer, Beijing (2011)

    Book  Google Scholar 

  26. Dzyaloshinskii, I. E. and Volovick, G. E. Poisson brackets in condensed matter physics. Annals of Physics, 125(1), 67–97 (1980)

    Article  MathSciNet  Google Scholar 

  27. Fan, T. Y. Poisson bracket method and its applications to quasicrystals, liquid crystals and a class of soft matter (in Chinese). Chinese Journal of Theoretical and Applied Mechanics, 45(4), 548–559 (2013)

    Google Scholar 

  28. Jeong, H. C. and Steinhardt, P. J. Finite-temperature elasticity phase transition in decagonal quasicrystals. Physical Review B, 48(13), 9394–9403 (1993)

    Article  Google Scholar 

  29. Guo, L. H. and Fan, T. Y. Solvability on boundary-value problems of elasticity of three-dimensional quasicrystals. Applied Mathematics and Mechanics (English Edition), 28(8), 1061–1070 (2007) DOI 10.1007/s10483-007-0808-y

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. School of Physics, Beijing Institute of Technology, Beijing, 100081, China

    Hui Cheng, Tianyou Fan & Hao Wei

  2. School of Science, Hebei Engineering University, Handan 056038, Hebei Province, China

    Hui Cheng

Authors
  1. Hui Cheng
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  2. Tianyou Fan
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  3. Hao Wei
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Correspondence to Tianyou Fan.

Additional information

Project supported by the National Natural Science Foundation of China (No. 11272053)

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Cheng, H., Fan, T. & Wei, H. Solutions for hydrodynamics of 5- and 10-fold symmetry quasicrystals. Appl. Math. Mech.-Engl. Ed. 37, 1393–1404 (2016). https://doi.org/10.1007/s10483-016-2133-9

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  • DOI: https://doi.org/10.1007/s10483-016-2133-9

Keywords

Chinese Library Classification

2010 Mathematics Subject Classification

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