On Gyroid Interfaces

Große-Brauckmann, Karsten

doi:10.1006/jcis.1996.4720

Cited by 76 publications

(81 citation statements)

References 14 publications

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“…17; whether this discontinuity corresponds to a bifurcation with two solutions to the problem (such as is observed for the CMC Gyroid surfaces in ref. 19 for high enough values of h 0 ) is to be addressed by future studies. The structural data for 3etc(187 193) phases in Fig.9 to 14 is based on triangulations with 1500 to 6500 points for the patch shown in Fig.…”

Section: A Numerical Methodsmentioning

confidence: 97%

Polycontinuous geometries for inverse lipid phases with more than two aqueous network domains

et al. 2013

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Inverse bicontinuous cubic phases with two aqueous network domains separated by a smooth bilayer are firmly established as equilibrium phases in lipid/water systems. The purpose of this article is to highlight the generalisations of these bicontinuous geometries to polycontinuous geometries, which could be realised as lipid mesophases with three or more network-like aqueous domains separated by a branched bilayer. An analysis of structural homogeneity in terms of bilayer width variations reveals that ordered polycontinuous geometries are likely candidates for lipid mesophase structures, with similar chain packing characteristics to the inverse micellar phases (that once were believed not to exist due to high packing frustration). The average molecular shape required by global geometry to form these multi-network phases is quantified by the surfactant shape parameter, v/(al); we find that it adopts values close to those of the known lipid phases. We specifically analyse the 3etc(187 193) structure of hexagonal symmetry P6 3 /mcm with three aqueous domains, the 3dia(24 220) structure of cubic symmetry I 43d composed of three distorted diamond networks, the cubic chiral 4srs(24 208) with cubic symmetry P4 2 32 and the achiral 4srs(5 133) structure of symmetry P4 2 /nbc, each consisting of four intergrown undistorted copies of the srs net (the same net as in the Q G II gyroid phase). Structural homogeneity is analysed by a medial surface approach assuming that the headgroup interfaces are constant mean curvature surfaces. To facilitate future experimental identification, we provide simulated SAXS scattering patterns that, for the 4srs(24 208) and 3dia(24 220) structures, bear remarkable similarity to those of bicontinuous Q G II -gyroid and Q D II -diamond phases, with comparable lattice parameters and only a single peak that cannot be indexed to the wellestablished structures. While polycontinuous lipid phases have, to date, not been reported, the likelihood of their formation is further indicated by the reported observation of a solid tricontinuous mesoporous silicate structure, termed IBN-9, which formed in the presence of surfactants [Han et al

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Section: A Numerical Methodsmentioning

confidence: 97%

Polycontinuous geometries for inverse lipid phases with more than two aqueous network domains

et al. 2013

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“…The discovery of this morphology in diblock copolymers in the 1990s led the author to study gyroids theoretically, and numerically using Brakke's surface evolver, with the following results. (iii) the entire family was computed with the Surface Evolver; it includes the minimal gyroid, and degenerates in touching spheres (figure 8) [3,11].…”

Section: Constant Mean Curvature Families Of Periodic Surfacesmentioning

confidence: 99%

Triply periodic minimal and constant mean curvature surfaces

Große-Brauckmann

2012

Interface Focus.

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We want to summarize some established results on periodic surfaces which are minimal or have constant mean curvature, along with some recent results. We will do this from a mathematical point of view with a general readership in mind.Keywords: interface; curvature; minimal surface 1. INTRODUCTORY MATERIAL CurvaturesLet us start with the definition of the curvature of a planar curve G, oriented by its normal n. Up to sign, its curvature k at a given point is the inverse of the radius R . 0 of a circle which agrees with the curve at the point up to second order; if the curve coincides to second order with a straight line, then we define k ¼ 0. The curvature is positive provided the circle has its centre to the side of the normal n, and negative otherwise.The standard definition of curvature for a surface uses the notion of curvature for curves as follows. At any given point p of a surface S contained in Euclidean space R 3 , the surface normal n and any tangent direction v span a normal plane. The surface intersects the normal plane in a curve G, which is oriented by the surface normal. The curvature of G at p is called the normal curvature of the surface in direction of the tangent direction v. At any point p [ S, the normal curvature attains a maximal and a minimal value k 1 ð pÞ, k 2 ð pÞ, called principal curvatures for two orthogonal tangent directions: v 1 ð pÞ ? v 2 ð pÞ. In order to understand orthogonality, we write the surface as a graph (x, y, f (x, y)) over its tangent plane, and then use the Taylor expansion for f. Its second-order term involves the symmetric Hessian matrix ð@ ij f Þ, whose minimal and maximal eigenvalues are the principal curvatures, and they are indeed attained at orthogonal eigendirections. The average of the principal curvatures is the mean curvature,moreover K :¼ k 1 k 2 is the Gauss curvature. If H ð pÞ ¼ 0; then the k i have opposite signs, and so p is a saddle point of the surface such that K ð pÞ 0. Parallel surfacesWe present another way to introduce H, which is relevant to our purpose. Let us start with the case of a curve s 7 ! G ðsÞ. Let G t ðsÞ :¼ GðsÞ þ tnðsÞ be the parallel or offset curve at (constant) distance t from G ¼ G 0 . For example, if G is a circle of radius R with inner normal n, then G t is the circle of radius R À t, meaning that the length element scales with ðR À tÞ=R when we go from G to G t . If a general curve G is approximated by a circle of radius R ¼ 1=k; then its parallel curve is approximated by a circle of radius R À t. Therefore, also for a general curve G, the length elements relate asWe view this equation as to say that curvature is the first-order change in length when going from a curve to its parallel curves. In fact, (1.1) says it is the negative of this number, and indeed the parallel curves to a positively curved curve become shorter. Upon integration, we obtainWe apply the same approach to the area element of a parallel surface S t ðu; vÞ :¼ Sðu; vÞ þ tnðu; vÞ:We need two assertions: first, the length elements ds 1 and ds 2 in the two ...

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“…Die beiden Domänen sind bis auf eine Punktspiegelung identisch. Der Gyroid wurde in den 60er Jahren von Alan Schoen erfunden [31] und ist seitdem mathematisch umfassend untersucht worden [9,10]. Die komplexe Form symmetrischer bikontinuierlicher Minimalflächen lässt sich durch ein geometrisches Netzwerk aus Stäben und Knoten, das in den Labyrinthen zentriert liegt, veranschaulichen.…”

Section: Der Gyroid: Eine Dreifach-periodische Minimalflächeunclassified