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Thermodynamic stability of quasicrystals: From fluid dynamics to soft condensed matter

Banff International Research Station for Mathematical Innovation and Discovery

As early as 1985, Landau free-energy models [1-3] and density-functional mean-field theories [4] were introduced in an attempt to explain the stability of quasicrystals, with only partial success if any. It is only in recent years, that great progress has been made in understanding the thermodynamic stability of quasicrystals in such simple isotropic classical field theories. Much of this has happened thanks to insight from the experimental observation of quasicrystalline order in diverse systems ranging from fluid dynamics to soft condensed matter. The key to unlocking the stability puzzle was in the realization that more than a single length scale was required, but more importantly in figuring out how to introduce these multiple scales into the models, and identifying the remaining requirements [5,6]. We and others have since managed to produce Landau and other mean-field theories with a wide range of quasicrystals as their minimum free-energy states, and have also confirmed some of these theories using molecular dynamics simulations with appropriately designed inter particle potentials [7-12]. I shall give a quick overview of the quasicrystals that can be stabilized in these theoriesâ in systems of one or two types of particles, in two and in three dimensionsâ and attempt to identify a trend that might be emerging in going from Landau theories to more realistic density-functional mean-field theories. It remains an open question whether this trend may eventually lead to understanding the stability of quasicrystals in complex metallic alloys. This research is supported by Grant No. 1667/16 from the Israel Science Foundation. [1] P. Bak, Phys. Rev. Lett. 54, 1517 (1985). [2] N.D. Mermin, S.M. Troian, Phys. Rev. Lett. 54, 1524 [Erratum on p. 2170] (1985). [3] P.A. Kalugin, A.Yu. Kitaev, L.C. Levitov, JETP Lett. 41, 145 (1985). [4] S. Sachdev, D.R. Nelson, Phys. Rev. B 32, 4592 (1985). [5] R. Lifshitz, H. Diamant. Phil. Mag. 87, 3021 (2007). [6] R. Lifshitz, D. Petrich, Phys. Rev. Lett. 79, 1261 (1997). [7] K. Barkan, H. Diamant, R. Lifshitz, Phys. Rev. B 83, 172201 (2011). [8] A.J. Archer, A.M. Rucklidge, E. Knobloch, Phys. Rev. Lett. 111, 165501 (2013). [9] K. Barkan, M. Engel, R. Lifshitz, Phys. Rev. Lett. 113, 098304 (2014). [10] C. V. Achim, M. Schmiedeberg, and H. Löwen, Phys. Rev. Lett. 112, 255501 (2014). [11] P. Subramanian, A.J. Archer, E. Knobloch, A.M. Rucklidge, Phys. Rev. Lett. 117, 075501 (2016). [12] S. Savitz, M. Babadi, R. Lifshitz, IUCrJ 5, 247 (2018). [13] M.C. Walters, P. Subramanian, A.J. Archer, R. Evans, Phys. Rev. E 98, 012606 (2018). [14] S. Savitz, R. Lifshitz, â Self-assembly of body-centered icosahedral cluster crystalsâ , presentation at this conference (2019).

Mathematics; Convex and discrete geometry; Statistical mechanics, structure of matter; Discrete mathematics

video/mp4

Author affiliation: Tel Aviv University

2020-03-30

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/73847

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/