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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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)
Bindi, L., Steinhardt, P. J., Yao, N., and Lu, P. J. Natural quasicrystals. Science, 324(5932), 1306–1309 (2009)
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)
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)
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)
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)
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)
Fan, T. Y. and Fan, L. Plastic fracture of quasicrystals. Philosophical Magazine, 88(4), 523–535 (2008)
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)
Fan, T. Y., Wang, X. F., Li, W., and Zhu, A. Y. Elasto-hydrodynamics of quasicrystals. Philosophical Magazine, 89(6), 501–512 (2009)
Li, X. F. A general solution of elasto-hydrodynamics of two-dimensional quasicrystals. Philosophical Magazine Letters, 91(4), 313–320 (2011)
Li, X. F. Elastohydrodynamic problems in quasicrystal elasticity theory and wave propagation. Philosophical Magazine, 93(13), 1500–1519 (2013)
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)
Bak, P. Phenomenological theory of icosahedral incommensurate (“quasiperiodic”) order in Mn-Al alloys. Physical Review Letters, 54(14), 1517–1519 (1985)
Bak, P. Symmetry, stability, and elastic properties of icosahedral incommensurate crystals. Physical Review B, 32(9), 5764–5772 (1985)
Lubensky, T. C., Ramaswamy, S., and Toner, J. Hydrodynamics of icosahedral quasicrystals. Physical Review B, 32(11), 7444–7452 (1985)
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)
Lubensky, T. C. Symmetry, elasticity and hydrodynamics in quasiperiodic. Introduction to Quasicrystals, Academic Press, San Diego, 199–273 (1988)
Socolar, J. E., Lubensky, T. C., and Steinhardt, P. J. Phonons, phasons, and dislocations in quasicrystals. Physical Review B, 34(5), 3345–3360 (1986)
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)
Fan, T. Y., Wang, X. F., Li, W., and Zhu, A. Y. Elasto-hydrodynamics of quasicrystals. Philosophical Magazine, 89(6), 501–512 (2009)
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)
Ding, D. H., Yang, W., Hu, C., and Wang, R. Generalized elasticity theory of quasicrystals. Physical Review B, 48(10), 7003–7010 (1993)
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)
Fan, T. Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications, Springer, Beijing (2011)
Dzyaloshinskii, I. E. and Volovick, G. E. Poisson brackets in condensed matter physics. Annals of Physics, 125(1), 67–97 (1980)
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)
Jeong, H. C. and Steinhardt, P. J. Finite-temperature elasticity phase transition in decagonal quasicrystals. Physical Review B, 48(13), 9394–9403 (1993)
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
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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