Hostname: page-component-7c8c6479df-hgkh8 Total loading time: 0 Render date: 2024-03-29T01:07:44.729Z Has data issue: false hasContentIssue false

Stability of Soft Quasicrystals in a Coupled-Mode Swift-Hohenberg Model for Three-Component Systems

Published online by Cambridge University Press:  16 March 2016

Kai Jiang
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, P.R. China
Jiajun Tong
Affiliation:
School of Mathematical Sciences, Peking University, Beijing, 100871, P.R. China
Pingwen Zhang*
Affiliation:
LMAM, CAPT and School of Mathematical Sciences, Peking University, Beijing, 100871P.R. China
*
*Corresponding author. Email addresses:kaijiang@xtu.edu.cn (K. Jiang), jiajun.tong@nyu.edu (J. Tong), pzhang@pku.edu.cn (P. Zhang)
Get access

Abstract

In this article, we discuss the stability of soft quasicrystalline phases in a coupled-mode Swift-Hohenberg model for three-component systems, where the characteristic length scales are governed by the positive-definite gradient terms. Classic two-mode approximation method and direct numerical minimization are applied to the model. In the latter approach, we apply the projection method to deal with the potentially quasiperiodic ground states. A variable cell method of optimizing the shape and size of higher-dimensional periodic cell is developed to minimize the free energy with respect to the order parameters. Based on the developed numerical methods, we rediscover decagonal and dodecagonal quasicrystalline phases, and find diverse periodic phases and complex modulated phases. Furthermore, phase diagrams are obtained in various phase spaces by comparing the free energies of different candidate structures. It does show not only the important roles of system parameters, but also the effect of optimizing computational domain. In particular, the optimization of computational cell allows us to capture the ground states and phase behavior with higher fidelity. We also make some discussions on our results and show the potential of applying our numerical methods to a larger class of mean-field free energy functionals.

Type
Research Article
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Shechtman, D., Blech, I., Gratias, D., and Cahn, J. W., Metallic phase with long-range orientational order and no translational symmetry, Phys. Rev. Lett., 53 (1984), 19511953.Google Scholar
[2]He, L. X., Wu, Y., and Kuo, K., Decagonal quasicrystals with different periodicities along the tenfold axis in rapidly solidified Al65Cu20M15 (M=Mn, Fe, Co or Ni), J. Mat. Sci. Lett., 7 (1988), 12841286.CrossRefGoogle Scholar
[3]Tsai, A.-P., Inoue, A., and Masumoto, T., Stable decagonal quasicrystal in the Al-Cu-Co system, Mater. Trans. JIM, 30 (1989), 300304.Google Scholar
[4]Steurer, W. and Deloudi, S., Crystallography of Quasicrystals: Concepts, Methods and Structures, Springer-Verlag Berlin Heidelberg, 2009.Google Scholar
[5]Steurer, W., Twenty years of structure research on quasicrystals. Part 1. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals, Z. Kristallogr. 219 (2004), 391446.Google Scholar
[6]Fujiwara, T. and Ishii, Y., Quasicrystals, Elsevier, 2007.Google Scholar
[7]Tsai, A. P., Icosahedral clusters, icosaheral order and stability of quasicrystals–view of metallurgy, Sci. Technol. Adv. Mater, 9 (2008), 013008.Google Scholar
[8]Bindi, L., Steinhardt, P. J., Yao, N., and Lu, P. J., Natural quasicrystals, Science, 324 (2009), 13061309.Google Scholar
[9]Zeng, X., Ungar, G., Liu, Y., Percec, V., Dulcey, A. E., and Hobbs, J. K., Supramolecular dendritic liquid quasicrystals, Nature, 428 (2004), 157160.Google Scholar
[10]Hayashida, K., Dotera, T., Takano, A., and Matsushita, Y., Polymeric quasicrystal: mesoscopic quasicrystalline tiling in ABC star polymers, Phys. Rev. Lett., 98 (2007), 195502.Google Scholar
[11]Dotera, T., Quasicrystals in soft matter Isr. J. Chem., 51 (2011), 11971205.Google Scholar
[12]Zhang, J. and Bates, F. S., Dodecagonal quasicrystalline morphology in a poly(styrene-bisoprene- b-styrene-b-ethylene oxide) tyetrablock terpolymer, J. Am. Chem. Soc., 134 (2012), 76367639.Google Scholar
[13]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, Proc. Natl. Acad. Sci. USA, 108 (2011), 18101814.Google Scholar
[14]Chaikin, P. M. and Lubensky, T. C., Principles of Condensed Matter Physics, Cambridge University Press, 1995.Google Scholar
[15]Archer, A. J., Rucklidge, A. M., and Knobloch, E., Quasicrystalline order and a crystal-liquid state in a soft-core Fluid, Phys. Rev. Lett., 111 (2013), 165501.Google Scholar
[16]Denton, A. R. and Löwen, H., Stability of colloidal quasicrystals, Phys. Rev. Lett., 81 (1998), 469472.CrossRefGoogle Scholar
[17]Alexander, S. and McTague, J., Should all crystals be bcc ? Landau theory of solidification and crystal nucleation, Phys. Rev. Lett., 41 (1978), 702705.Google Scholar
[18]Bak, P., Phenomenological theory of icosahedral incommensurate (“quasiperiodic”) order in Mn-Al alloys, Phys. Rev. Lett., 54 (1985), 15171519.Google Scholar
[19]Jarić, M. V., Long-range icosahedral orientational order and quasicrystals, Phys. Rev. Lett., 55 (1985), 607610.Google Scholar
[20]Kalugin, P., Kitaev, A. Y., and Levitov, L., Al0.86Mn0.14: a six dimensional crystal, JETP Lett, 41 (1985), 145149.Google Scholar
[21]Gronlund, L. and Mermin, N., Instability of quasicrystalline order in the local Kalugin-Kitaev- Levitov model, Phys. Rev. B, 38 (1988), 36993710.CrossRefGoogle ScholarPubMed
[22]Mermin, N. and Troian, S.M., Mean-field theory of quasicrystalline order, Phys. Rev. Lett., 54 (1985), 15241527.Google Scholar
[23]Swift, J. and Hohenberg, P. C., Hydrodynamic fluctuations at the convective instability, Phys. Rev. A, 15 (1977), 319328.Google Scholar
[24]Müller, H. W., Model equations for two-dimensional quasipatterns, Phys. Rev. E, 49 (1994), 1273-277.Google Scholar
[25]Dotera, T., Mean-field theory of Archimedean and quasicrystalline tilings, Philos. Mag., 87 (2007), 30113019.CrossRefGoogle Scholar
[26]Lifshitz, R. and Petrich, D.M., Theory of color symmetry for periodic and quasiperiodic crystals, Phys. Rev. Lett., 79 (2997), 12611264.Google Scholar
[27]Lifshitz, R. and Diamant, H., Soft quasicrystals –Why are they stable ? Philos. Mag., 87 (2007), 30213030.CrossRefGoogle Scholar
[28]Engel, M. and Trebin, H.-R., Self-assembly of monatomic complex crystals and quasicrystals with a double-well interaction potential, Phys. Rev. Lett., 98 (2007), 225505.Google Scholar
[29]Reinhardt, A., Romano, F., and Doye, J. P. K., Computing phase diagrams for a quasicrystalforming patchy-particle system, Phys. Rev. Lett., 110 (2013), 255503.CrossRefGoogle ScholarPubMed
[30]Janot, C., Quasicrystals: A primer, Oxford Press, 1992.CrossRefGoogle Scholar
[31]Steinhardt, P. J. and DiVincenzo, D. P., Quasicrystals: The State of the Art, World Scientic, Singapore, 1991.Google Scholar
[32]Goldman, A. I. and Kelton, R. F., Quasicrystals and crystalline approximants, Rev. Mod. Phys., 65 (1993), 213230.CrossRefGoogle Scholar
[33]MCarley, J. S. and Ashcroft, N. W., Stability of quasicrystals composed of soft isotropic particles, Phys. Rev. B, 49 (1994), 1560015606.Google Scholar
[34]Jiang, K. and Zhang, P., Numerical methods for quasicrystals, J. Comp. Phys., 256 (2014), 428440.Google Scholar
[35]Jiang, K., Tong, T., Zhang, P., and Shi, A.-C., Stability of two-dimensional soft quasicrystals in systems with two length scales, Phys. Rev. E, 92 (2015), 042159.Google Scholar
[36]Barkan, K., Diamant, H., and Lifshitz, R., Stability of quasicrystals composed of soft isotropic particles, Phys. Rev. B, 83 (2011), 172201.Google Scholar
[37]Rottler, J., Greenwood, M., and Ziebarth, B., Morphology of monolayer films on quasicrystalline surfaces from the phase field crystal model, J. Phys.: Condens. Matt., 24 (2012), 135002.Google Scholar
[38]Dotera, T., Toward the discovery of new soft quasicrystals: from a numerical study viewpoint, J. Polym. Sci. Part B: Polym. Phys., 50 (2012), 155167.Google Scholar
[39]Tang, P., Qiu, F., Zhang, H., and Yang, Y., Morphology and phase diagram of complex block copolymers: ABC star triblock copolymers, J. Phys. Chem. B, 108 (2004), 84348438.Google Scholar
[40]Matsushita, Y., Creation of hierarchically ordered nanophase structures in block polymers having various competing interactions, Macromolecules, 40 (2007), 771776.Google Scholar
[41]Xu, W., Jiang, K., Zhang, P., and Shi, A.-C., A strategy to explore stable andmetastable ordered phases of block copolymers, J. Phys. Chem. B, 117 (2013), 52965305.CrossRefGoogle ScholarPubMed