Triply Periodic Bicontinuous Cubic Microdomain Morphologies by Symmetries

Wohlgemuth, Meinhard; Yufa, Nataliya A.; Hoffman, James T.; Thomas, Edwin L.

doi:10.1021/ma0019499

Cited by 263 publications

(203 citation statements)

References 31 publications

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“…Skeletal frames and IPMS were calculated using the level-set approximation. 48 gyroid models were calculated and the intensity for {110}, {200}, and other reflections agree well with SAXS from the material.…”

Section: Introductionmentioning

confidence: 70%

A Re-Evaluation of the Morphology of a Bicontinuous Block Copolymer−Ceramic Material

et al. 2007

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The structures of a poly(isoprene-block-ethylene oxide) (PI-b-PEO) block copolymer-directed aluminosilicate mesostructure and the resulting ceramic material obtained from calcination were studied via smallangle X-ray scattering (SAXS) and transmission electron microscopy (TEM). The PI minority phase (volume fraction 0.36) formed a continuous network of channels, previously reported 1,2 to be consistent with the plumber's nightmare 3 morphology. The solvent casting process used to form the material caused it to shrink uniaxially by ∼30%, deforming the network structure within it. Calculated structure factors for constant-curvature and constantthickness models of a distorted double gyroid structure are consistent with SAXS from the material, while [100] and [111] projections of the distorted double gyroid structure match the TEM data. Because the structural data from the material is most consistent with a distorted version of the double gyroid morphology, the previous assignment of the plumber's nightmare morphology must be reconsidered. Approaches for structural assignment are also discussed.

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Section: Introductionmentioning

confidence: 70%

A Re-Evaluation of the Morphology of a Bicontinuous Block Copolymer−Ceramic Material

et al. 2007

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“…As shown below, applying different perturbations to this structure will lead us to PhCs with frequency-isolated linear point and line degeneracies (i.e. Weyl points and line nodes).A DG PhC consists of two single gyroids (SG) in a body-centered-cubic(bcc) lattice.An SG surface is a bi-continuous triply-periodic minimal surface defined by g(r) = sin(2πx) cos(2πy) + sin(2πy) cos(2πz) + sin(2πz) cos(2πx) [19]. The red gyroid in Fig.…”

mentioning

confidence: 99%

“…An SG surface is a bi-continuous triply-periodic minimal surface defined by g(r) = sin(2πx) cos(2πy) + sin(2πy) cos(2πz) + sin(2πz) cos(2πx) [19]. The red gyroid in Fig.…”

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confidence: 99%

Weyl points and line nodes in gyroid photonic crystals

et al. 2013

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Weyl points and line nodes are three-dimensional linear point-and line-degeneracies between two bands. In contrast to Dirac points, which are their two-dimensional analogues, Weyl points are stable in the momentum space and the associated surface states are predicted to be topologically non-trivial. However, Weyl points are yet to be discovered in nature. Here, we report photonic crystals, based on the double-gyroid structures, exhibiting frequency-isolated Weyl points with intricate phase diagrams. The surface states associated with the non-zero Chern numbers are demonstrated. Line nodes are also found in similar geometries; the associated surface states are shown to be flat bands. Our results are readily experimentally realizable at both microwave and optical frequencies.Two-dimensional (2D) electrons and photons at the energies and frequencies of Dirac points exhibit extraordinary features [1][2][3][4][5]. As the best example, almost all the remarkable properties of graphene are tied to the massless Dirac fermions at its Fermi level [6,7]. Topologically [8], Dirac cones are not only the critical points for 2D phase transitions but also the unique surface manifestation of a topologically gapped 3D bulk. In a similar way, it is expected that if a material could be found that exhibits a 3D linear dispersion relation, it would also display a wide range of interesting physics phenomena. The associated 3D linear pointdegeneracies are called "Weyl points". In the past year, there have been a few studies of Weyl fermions in electronics [9][10][11][12][13][14]. The associated Fermi-arc surface states, quantumHall-effect [15], novel transport properties [16] and the realization of the Adler-Bell-Jackiw anomaly [17] are also expected. However, no observation of Weyl points has been reported. Here, we present a theoretical discovery and detailed numerical investigation of frequency-isolated Weyl points in perturbed double-gyroid(DG) photonic crystals(PhCs) along with their complete phase diagrams and their topologicallyprotected surface states. PhCs containing frequency-isolated linear line-degeneracies, known as "line nodes", and their flatband surface states are also presented. Unlike the proposed Weyl points in electronic system thus far, our predictions in photonics are readily realizable in experiments.Before proceeding, we first point out one intriguing distinction between the 2D Dirac points and the 3D Weyl points. 2D Dirac cones are not robust; they are only protected by the product of time-reversal-symmetry(T) and parity(P, inversion). In 2D, Dirac cone effective Hamiltonian takes the form of H(k) = v x k x σ x + v z k z σ z ; this form is protected by PT (product of P and T) which requires H(k) to be real. Thus, one can open a gap in this dispersion relation upon introducing a perturbation proportional to σ y = 0 −i i 0 that is imaginary;for example, even an infinitesimal perturbation that breaks just P or just T will open a gap. In contrast, 3D Weyl points are topologically protected gapless dispersions robust...

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“…This FF is determined by a corresponding value of 't' which forms part of the general surface equation of the gyroid. Both the theoretical and replica gyroids were produced using the technique described by Wohlgemuth et al [26] for generating the gyroid's three-dimensional minimal surface. An equation, which defines this surface is sin 2px a…”

Section: Microwave Experimental Methods and Theoretical Modellingmentioning

confidence: 99%

Electromagnetic characterization of millimetre-scale replicas of the gyroid photonic crystal found in the butterfly Parides sesostris

Pouya

Vukusic

2012

Interface Focus.

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We have used three-dimensional stereolithography to synthetically replicate the gyroid photonic crystal (PC) structure that occurs naturally in the butterfly Parides sesostris. We have experimentally characterized the transmission response of this structure in the microwave regime at two azimuthal angles (f ) over a comprehensive range of polar angles (u). We have modelled its electromagnetic response using the finite-element method (FEM) and found excellent agreement with experimental data. Both theory and experiment show a single relatively broad transmission minimum at normal incidence (u ¼ 08) that comprises several narrow band resonances which separate into clearly identifiable stop-bands at higher polar angles. We have identified the specific effective geometric planes within the crystal, and their associated periodicities that give rise to each of these stop-bands. Through extensive theoretical FEM modelling of the gyroid PC structure, using varying filling fractions of material and air, we have shown that a gyroid PC with material volume fraction of 40 per cent is appropriate for optimizing the reflected bandwidth at normal incidence (for a refractive index contrast of 1.56). This is the same gyroid PC material volume fraction used by the butterfly P. sesostris itself to produce its green structurally coloured appearance. This infers further optimization of this biological PC beyond that of its lattice constant alone.

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Triply Periodic Bicontinuous Cubic Microdomain Morphologies by Symmetries

Cited by 263 publications

References 31 publications

A Re-Evaluation of the Morphology of a Bicontinuous Block Copolymer−Ceramic Material

A Re-Evaluation of the Morphology of a Bicontinuous Block Copolymer−Ceramic Material

Weyl points and line nodes in gyroid photonic crystals

Electromagnetic characterization of millimetre-scale replicas of the gyroid photonic crystal found in the butterfly Parides sesostris

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