Star-polyphiles are generalizations of lipids or surfactants [1]: they are small molecules consisting of 3 mutually immiscible chains attached to a common center: a hydrocarbon, a fluorocarbon and a water soluble oligo-ethylene glycol chain [2,3].When brought in contact with water, these selfassemble into mesophases, most of which structurally resemble those found in lipids like monoolein, including lamellar, hexagonal, micellar and bicontinuous cubic mesophases.However, the hydrophobic membrane is split into separate hydrocarbon and fluorocarbon domains [4], which significantly increases structural complexity and hierarchical ordering can occur.Here we give an overview on possible and experimentally determined mesophase geometries, focusing on their experimental investigation by SAXS and SANS in combination with scattering simulations of model structures.Figure 1.Gyroid mesophase based on star-polyphiles with de-mixed hydrocarbon-fluorocarbon domains schematically drawn in the form of stripes.In a binary star-polyphile/water system the demixed domain widths are limited by the geometry of the molecules (A).De-mixed domain widths can be enlarged by replacing part of the star-polyphile with the corresponding double-chain molecules (B)

Nature employs an impressive range of topologically complex ordered nanostructures that occur in various forms in both natural and synthetic materials. A particular class of these exhibits negative curvature and forms periodic saddle-shaped surfaces in three dimensions. Unlike pattern formation on flat or positively curved surfaces like spherical systems, the understanding of patterning on such surfaces is highly complicated due to the structures being intrinsically intertwined in three dimensions. We present a new method for visualisation and analysis of patterns on triply periodic negatively curved surfaces by mapping to two-dimensional hyperbolic space analogous to spherical projections in cartography thus effectively creating a more accessible "hyperbolic map" of the pattern. Specifically, we exemplify the method via the simplest triply periodic minimal surfaces: the Primitive, Diamond, and Gyroid in their universal cover along with decorations from a soft materials, whose structures involve decorations of soft matter on negatively curved surfaces, not necessarily minimal.

The concept of an orbifold is particularly suited to classification and enumeration of crystalline groups in the euclidean (flat) plane and its elliptic and hyperbolic counterparts. Using Conway's orbifold naming scheme, this article explicates conventional point, frieze and plane groups, and describes the advantages of the orbifold approach, which relies on simple rules for calculating the orbifold topology. The article proposes a simple taxonomy of orbifolds into seven classes, distinguished by their underlying topological connectedness, boundedness and orientability. Simpler `crystallographic hyperbolic groups' are listed, namely groups that result from hyperbolic sponge-like sections through three-dimensional euclidean space related to all known genus-three triply periodic minimal surfaces (i.e. the P, D, Gyroid, CLP and H surfaces) as well as the genus-four I-WP surface.

We study the generation of 2D turbulence in Faraday waves by investigating the creation of spatially periodic vortices in this system. Measurements that couple a diffusing light imaging technique and particle tracking algorithms allow the simultaneous observation of the three-dimensional fluid motion and of the temporal changes in the wave-field topography. Quasistanding waves are found to coexist with a spatially extended fluid transport. More specifically, the destruction of regular patterns of oscillons coincides with the emergence of a complex fluid motion whose statistics are similar to that of two-dimensional turbulence. We reveal that a lattice of oscillons generates vorticity at the oscillon scale in the horizontal flow. The interaction of these vortices explains how 2D turbulence is fueled by almost standing waves. Remarkably, the curvature of Lagrangian trajectories reveals a "footprint" of the forcing-scale vortices in fully developed turbulence. Two-dimensional Navier-Stokes turbulence should be considered a source of disorder in Faraday waves. These findings also provide a new paradigm for vorticity creation in 2D flows.

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 I43d composed of three distorted diamond networks, the cubic chiral 4srs(24 208) with cubic symmetry P4232 and the achiral 4srs(5 133) structure of symmetry P42/nbc, each consisting of four intergrown undistorted copies of the srs net (the same net as in the QII(G) 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 QII(G)-gyroid and QII(D)-diamond phases, with comparable lattice parameters and only a single peak that cannot be indexed to the well-established 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., Nat. Chem., 2009, 1, 123].

We construct some examples of finite and infinite crystalline three-dimensional nets derived from symmetric reticulations of homogeneous two-dimensional spaces: elliptic (S-2), Euclidean (E-2) and hyperbolic (H-2) space. Those reticulations are edges and vertices of simple spherical, planar and hyperbolic tilings. We show that various projections of the simplest symmetric tilings of those spaces into three-dimensional Euclidean space lead to topologically and geometrically complex patterns, including multiple interwoven nets and tangled nets that are otherwise difficult to generate ab initio in three dimensions.

Sessions C332The external perturbations can be temperature, light irradiation, pressure, and pulsed magnetic fields all may change the magnetic coupling interaction in transition metal systems.Here, we use pressure as external force to study magneto -structural correlation.We report here the structure and magnetic behavior of one 1-D helical chain iron complex [FeII(μ-bpt)in ambient conditions and under hydrostatic pressure.The complex was crystallized in a tetragonal space group I4 1 /a at ambient pressure, each iron atoms linked by one bpt and one isonicotinic acid legend, which form an infinite helical chain along the screw axis 4 1 at c-axis direction.The iron atoms are iron (2+) high spin state with distorted N4O2 coordinated shell.High-pressure powder X-ray diffraction experiments from 0.03GPa to 2.79GPa pressure were carried out to understand precisely the correlation between the structure and magnetic behavior.The careful analysis of a series high pressure structures showing a pressureinduced shrinking more than 10% along the chain direction, which can describe as spring compressed 10%.This make the iron-iron distance from 4.445(2) Å at ambient pressure change significantly to 4.083(6) Å at around 2GPa.Magneto -structural correlation will be discussed .The coordinated shells of Iron (N4O2) also decrease 0.11 Å about averagely.The spin state of iron and magnetic property of title compound also will be shown in this report

We present a method for geometric construction of periodic three-dimensional Euclidean nets by projecting two-dimensional hyperbolic tilings onto a family of triply periodic minimal surfaces (TPMSs). Our techniques extend the combinatorial tiling theory of Dress, Huson & Delgado-Friedrichs to enumerate simple reticulations of these TPMSs. We include a taxonomy of all networks arising from kaleidoscopic hyperbolic tilings with up to two distinct tile types (and their duals, with two distinct vertices), mapped to three related TPMSs, namely Schwarz's primitive (P) and diamond (D) surfaces, and Schoen's gyroid (G).

During the past decade, interest has grown tremendously in the design and synthesis of crystalline materials constructed from molecular clusters linked by extended groups of atoms. Most notable are metal-organic frameworks (MOFs), in which polyatomic inorganic metal-containing clusters are joined by polytopic linkers. (Although these materials are sometimes referred to as coordination polymers, we prefer to differentiate them, because MOFs are based on strong linkages that yield robust frameworks.) The realization that MOFs could be designed and synthesized in a rational way from molecular building blocks led to the emergence of a discipline that we call reticular chemistry. MOFs can be represented as a special kind of graph called a periodic net. Such descriptions date back to the earliest crystallographic studies but have become much more common recently because thousands of new structures and hundreds of underlying nets have been reported. In the simplest cases (e.g., the structure of diamond), the atoms in the crystal become the vertices of the net, and bonds are the links (edges) that connect them. In the case of MOFs, polyatomic groups act as the vertices and edges of the net. Because of the explosive growth in this area, a need has arisen for a universal system of nomenclature, classification, identification, and retrieval of these topological structures. We have developed a system of symbols for the identification of three periodic nets of interest, and this system is now in wide use. In this Account, we explain the underlying methodology of assigning symbols and describe the Reticular Chemistry Structure Resource (RCSR), in which about 1600 such nets are collected and illustrated in a database that can be searched by symbol, name, keywords, and attributes. The resource also contains searchable data for polyhedra and layers. The database entries come from systematic enumerations or from known chemical compounds or both. In the latter case, references to occurrences are provided. We describe some crystallographic, topological, and other attributes of nets and explain how they are reported in the database. We also describe how the database can be used as a tool for the design and structural analysis of new materials. Associated with each net is a natural tiling, which is a natural partition of space into space-filling tiles. The database allows export of data that can be used to analyze and illustrate such tilings.

We present a method for generating 3D euclidean periodic networks from 2D hyperbolic tilings. We utilize triply-periodic minimal surfaces (TPMS) as a mathematical scaffold to guide this process. These surfaces have an intrinsic hyperbolic geometry as well as an underlying set of discrete hyperbolic symmetries, allowing decoration with tilings of matching symmetry. Hyperbolic tilings of a given symmetry can be enumerated (up to finite complexity), projected onto the surface, and used to derive a labeled quotient graph of nodes and edges, representing a 3D periodic net. Although we have visualized intermediate geometries for this process, it is performed entirely as a topological computation using only neighborhood relationships and labels. The final net geometry is derived from the labeled graph, using Systre [Delgado-Friedrichs 2006], enabling uniqueness and 3D space-group identification. We have published the results of these computations at the EPINET website [Ramsden et al. 2005] for an initial sample of tilings, resulting in an online database of many thousands of 3D nets, each with their own webpage, which can be data mined via geometric and topological properties. Tiling combinatorics can provide a rich palette of both geometric and image data, only a fraction of which has been currently explored.

Crystalline frameworks in 3D Euclidean space can be constructed by projecting tilings of 2D hyperbolic space onto three-periodic minimal surfaces, giving surface reticulations. The technique involves Delaney–Dress tiling theory, group theory, differential and non-Euclidean geometry. Preliminary results of this approach, found at http://epinet.anu.edu.au, are discussed and compared with other approaches.

We propose a general method to study the hierarchical organization of financial data by embedding the structure of their correlations in metric graphs in multi-dimensional spaces. An application to two different sets of interest rates is discussed by constructing triangular embeddings on the sphere. Three-dimensional representations of these embeddings with the correct metric geometry are constructed and visualized. The resulting graphs contain the minimum spanning tree as a sub-graph and they preserve its hierarchical structure. This produces a clear cluster differentiation and allows us to compute new local and global topological quantities.

The bacterial toxin ColE7 bears an H-N-H motif that has been identified in hundreds of prokaryotic and eukaryotic endonucleases, involved in DNA homing, restriction, repair or chromosome degradation.The crystal structures of the nuclease domain of ColE7 in complex with an 8-bp [1], 12-bp and 18-bp duplex DNA have been determined respectively by X-ray diffraction methods.In each of the structure, the H-N-H motif is bound at the minor groove primarily to DNA phosphate groups, with little interaction to ribose groups and bases.This result provides a structural basis for sugar and sequence independent DNA recognition.Structural comparison shows that several families of endonucleases bind and bend DNA in a similar way to that of the H-N-H ColE7, indicating that endonucleases containing a similar His-metal finger fold of active site possess a universal mode for protein-DNA interactions [2].

Our study of the gyroid minimal surface has revealed that there are two distinct covering maps from the hyperbolic plane onto the surface that respect its intrinsic symmetries. We show that if a decoration of \(\mathbb{H}^2\) is chiral, the projection of this pattern via the two covering maps gives rise to distinct structures in \(\mathbb{E}^3\).

We describe a systematic approach to generate nets that arise from decorations of periodic minimal surfaces. Such surfaces are covered by the hyperbolic plane, in the same way that the euclidean plane covers a cylinder. Thus, a symmetric hyperbolic network can be wrapped onto an appropriate minimal surface to obtain a 3D periodic net. This requires symmetries of the hyperbolic net to match the symmetries of the minimal surface. Here, we tabulate all such symmetry groups that are compatible with the H minimal surface.

. Many crystalline networks can be viewed as decorations of triply periodic minimal surfaces. Such surfaces are covered by the hyperbolic plane in the same way that the Euclidean plane covers a cylinder. Thus, a symmetric hyperbolic network can be wrapped onto an appropriate minimal surface to obtain a 3d periodic net. This requires symmetries of the hyperbolic net to match the symmetries of the minimal surface. We describe a systematic algorithm to find all the hyperbolic symmetries that are commensurate with a given minimal surface, and the generation of simple 3d nets from these symmetry groups.

A medial surface (MS) analysis of the rhombohedral infinite periodic minimal surface family rPD is presented. The rPD family of bicontinuous surfaces has been suggested as a pathway for transitions between its two cubic members, the P and the D surface, in mesophases in liquid-crystalline self-assembly. The MS is a representation of a labyrinth as a centered 2D skeleton. By providing a definition of a pointwise channel diameter, the MS allows for an analysis of homogeneity of such surfaces including chain stretching frustration. For the rPD surface, variations of this channel diameter are locally minimal for the D surface, and a horizontal inflection point for the P surface. This may have implications for the phase stability of the corresponding liquid-crystalline mesophases. The MS can be further reduced to a 1D line graph. For the rPD surface, this graph contains curved edges and cannot be deduced from symmetry considerations alone.

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