Soft quasicrystals–Why are they stable?

Abstract

In the last two years we have witnessed the exciting experimental discovery of soft matter with nontrivial quasiperiodic long-range order–a new form of matter termed a soft quasicrystal. Two groups have independently discovered such order in soft matter: Zeng et al. in a system of dendrimer liquid crystals; and Takano et al. in a system of ABC star-shaped polymers. These newly discovered soft quasicrystals not only provide exciting platforms for the fundamental study of both quasicrystals and of soft matter, but also hold the promise for new applications based on self-assembled nanomaterials with unique physical properties that take advantage of the quasiperiodicity, such as complete and isotropic photonic band-gap materials. Here we provide a concise review of the emerging field of soft quasicrystals, suggesting that the existence of two natural length-scales, along with three-body interactions, may constitute the underlying source of their stability.

Acknowledgments

One of the authors (RL) thanks X. Zeng and T. Dotera for fruitful discussions while attending ‘Aperiodic 2006’ in Zao, Japan. This research is funded by the Israel Science Foundation through grant number 684/06.

Notes

^§Early papers in the field were collected by Steinhardt and Ostlund 3; for an introductory textbook on quasicrystals, see 4; for collections of reviews on the physical properties of quasicrystals, see 5, 6; for an introductory textbook on the mathematics of quasicrystals, see 7; for detailed reviews on the mathematics of quasicrystals, see 8, 9; for a detailed overview on the applications of quasicrystals, see 10.

^†Some older texts require crystals to possess so-called ‘forbidden symmetries’ in order to be regarded as quasicrystals. It is now understood that such a requirement is inappropriate. See Ref. 12 for details and Ref. 34 for examples of square and cubic quasicrystals.

^†For the sake of historical accuracy, it should be noted that at some point the blue phase III of liquid crystals, also known as the ‘blue fog’, was thought to have icosahedral quasicrystalline order 70, 71, but this eventually turned out not to be the case 72, 73. Also, incommensurate helical twist-grain-boundary phases are known to exist in smectic liquid crystals 74 75, but the quasiperiodic order in this case is essentially only along the one-dimensional screw axis.

^†Indeed, coarse-grained free energies previously used for amphiphilic self-assembly 43 involve more than one characteristic length scale due to the asymmetry of the molecules and the resulting tendency to form curved interfaces.

^†Models with two order parameters were suggested also for hard quasicrystals 83 and pattern-forming systems 84, yielding additional ground-state symmetries.