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Molecular Mechanism Responsible for Reentrance to Ia3d Gyroid Phase in Cubic Mesogen BABH(n)
J. Phys. Soc. Jpn. 81 (2012) 094601
Quest for the Gyroid Labyrinth: Geometry and Topology in Soft Matter
by Tomonari Dotera (Department of Physics, Kinki University)
Published August 17, 2012
Over the past two decades, the concepts of geometry and topology have been intensively employed in the interdisciplinary exploration of the “Gyroid Labyrinth” in fields as diverse as biology, chemistry, physics, material science, and mathematics.
Fritz Laves (1906-1978) was a Swiss crystallographer who classified crystal structures on the basis of topological concept and described close-packed crystal structures, now known as Laves phases. In 1932, he described a fundamental three-dimensional lattice, “the Laves graph” [Fig. 1(a)], whose space group is I4132 (No. 214); it is also referred to as the gyroid lattice or network [1-3]. The gyroid lattice is fundamental in the sense that all lattice points and bonds are identical (vertex-transitive and edge-transitive). These properties are shared among the well-known cubic lattices—FCC, BCC, simple cubic, and diamond lattice—whose coordination numbers are 12, 8, 6, and 4, respectively. The coordination number of the Laves graph is 3, the smallest number in the sequence. Even with these characteristics, such a fundamental lattice has surprisingly been missing in Solid-state Physics classrooms and textbooks for many years. The double-gyroid network (mostly known in the Soft Matter community as space group Ia3d, No. 230) consists of the Laves graph and its inversion [Figs. 1(b) and 1(c)].
|Fig. 1: (a) Laves graph. (b) Gyroid network. (c) Double-gyroid networks typically seen in block copolymer systems. These two networks are alternately right-handed and left-handed. Helices can be found in the (100) and (111) directions. (d) Gyroid surface.|
Based on the Laves graph, in 1969, NASA scientist Alan Schoen discovered a triply periodic minimal surface, now known as the gyroid (G) surface  [Fig. 1(d)]. He found that the G surface is a member of the Bonnet-associated family of P (primitive, or “plumber's nightmare”) and D (diamond) surfaces. This type of minimal surface (defined by zero mean curvature) is three-dimensionally periodic and bisects the space into a pair of identical non-intersecting channels, both of which are characterized by the abovementioned double networks. Around the same time in 1967, Luzzati and Spegt found the first gyroid phase in a soap surfactant system . Since then, bicontinuous cubic phases based on G, D, and P minimal surfaces have been found in various soft materials such as biological lipids, surfactants, liquid crystals, inorganic mesoporous materials, and block copolymers.
Significant advances in materials synthesis have led to the emergence of a number of different types of G-related structures. In Ref. , a reentrant lyotropic transition of Ia3d phases in the mesogen BABH(n) system is reported. This system has long been investigated by the authors because of its mysterious reentrant behavior. Here, these novel layered and double-layered structures have been identified using the maximum entropy method, which is well known in crystallography; it should be noted that advanced crystallography is now necessary for the elucidation of mesoscopic self-assembly in soft matter.
Since the gyroid lattice consists of ten-membered rings, it tends to have photonic band gaps, similar to the diamond lattice characterized by six-membered rings. Moreover, it is found to have a negative refractive index. Recently, in the wing scales of several butterfly species, gyroid structures made up of chitin have been found . It has been shown that the gyroid photonic structure reflects a brilliant green color. The gyroid structure is an angle-independent reflector; it is thus more advantageous than multilayered photonic structures. Interestingly, the gyroid lattice is chiral (enantiomorphic), and therefore, the reflectance depends on right or left circular polarization [8,9]. In addition, future technological advances include gyroid structures that can be fabricated experimentally, for example, using block copolymer self-assembly followed by selective etching and subsequent metal deposition.
-  A. H. Schoen: NOTICE OF AMS June/July (2008) 663.
See also: http://schoengeometry.com/e_tpms.html.
-  S. T. Hyde et al.: Angew. Chem. Int. Ed. 47 (2008) 7996.
-  The term “Laves lattice” is known as the dual lattice of the Archimedean tiling, which is different from the Laves graph.
-  A. H. Schoen: Technical Note NASA-TN-D- 5541, NASA, Washington, DC (1970).
-  V. Luzzati and P. A. Spegt: Nature 215 (1967) 701.
-  Y. Nakazawa, et al.: J. Phys. Soc. Jpn. 81 (2012) 094601.
-  K. Michielsen and D. G Stavenga: J. R. Soc. Interface 5 (2008) 85.
-  M. Saba et al.: Phys. Rev. Lett. 106 (2011) 103902.
-  The name “gyroid” stems from these screw (gyroid) rotations.
Copyright © 2012 The Physical Society of Japan.