Three-periodic surfaces

In the latter the surfactant monolayer (in oil and water mixture) or bilayer (in water only) forms a periodic surface. A periodic surface is one that repeats itself under a unit translation in one, two, or three coordinate directions similarly to the periodic arrangement of atoms in regular crystals. It is still not clear, however, whether the transition between the bicontinuous microemulsion and the ordered bicontinuous cubic phases occurs in nature. When the volume fractions of oil and water are equal, one finds the cubic phases in a narrow window of surfactant concentration around 0.5 weight fraction. However, it is not known whether these phases are bicontinuous. No experimental evidence has been published that there exist bicontinuous cubic phases with the ordered surfactant monolayer, rather than bilayer, forming the periodic surface. 

The SCL2 structure is composed of three different embedded periodic surfaces. The middle surface is the Schwarz minimal surface P. Similarly, the middle phase surface in GLl (Fig. 8(c)) and GL2 (Fig. 8(d)) structures is the Schoen minimal surface G. 

Crisp and coworkers found that the development of surface crystallinity was related to the speed of set. The faster the reaction, the shorter was the inhibition period before surface crystallization took place. When the setting time of a cement was between two and three minutes, surface crystallinity developed in a few minutes. When it was seven minutes, surface crystallinity was delayed by three hours. The reaction rate was affected by the chemical composition and physical state of the cement components. Well-ignited zinc oxide, the presence of magnesium in the... 

The periodic surface is the surface that moves onto itself under a unit translation in one, two, or three coordinate directions similarly as in the periodic... 

The one hundred year history of froth flotation may be classified into three periods. The earliest stage is from the end of the 19 century to the early 20 century, i.e. surface flotation or bulk oil flotation. The natural hydrophobic sulphide minerals can be collected by the addition of oil. Froth flotation came into practice in 1909 with the use of pine oil, mechanical flotation machine in 1912, and xanthate and aerofloat as collectors in 1924—1925 (Gaudin, 1932 Sutherland and Wark, 1955). 

The International Code Council (ICC) was formed by the consolidation of three formerly separate fire code organizations International Conference of Building Officials (ICBO), which had published the Uniform Fire Code under its fire service arm, the International Fire Code Institute (IFCI) Building Officials and Code Administrators (BOCA), which had published the National Fire Prevention Code and Southern Building Congress Code International (SBCCI), which had published the Standard Fire Prevention Code. When the three groups merged in 2000, in part to develop a common fire code, the individual codes became obsolete however, they are noted above since references to them may periodically surface. The consolidated code is IFC-2006, International Fire Code. 

Table 12-3. Comparison of calculated structural parameters for the kaolinite(O)—H2O system obtained from geometry optimizations of a cluster model (ONIOM(B3LYP/SVP PM3 method) and the periodic DFT(PW91) approach) (static relaxation and MD simulation) [73], Bond lengths and interatomic distances are in A, angles are in degrees. Superscript w stands for water, subscripts distinguish O and H atoms of three different surface OH groups...

Three-periodic hyperbolic surfaces of infinite genus carve space into two intertwined sub-volumes, both resembling three-dimensional arrays of interconnected tubes. They are simple candidates for the interfaces in bicontinuous structures, consisting of two continuous subvolumes [4, 5]. As such they have attracted great interest as models for microstructured complex fluid interfaces, biological membranes, and structures of condensed atomic and molecular systems, to be explored in subsequent Chapters. 

The simplest three-periodic hyperbolic surfaces are "Infinite Periodic Minimal Surfaces" (EPMS, named by Alan Schoen [6]). For these surfaces, the mean curvature is constant on the surface, and everywhere identically zero. This is a defining characteristic of minimal surfaces. For these structures, the sub-volumes can be geometrically identical. This occurs if the IPMS contains straight lines. Such surfaces have been called "balance surfaces" by Koch and Fischer [7]. We focus primarily on IPMS in this book. Some further discussion of general properties of minimal surfaces is in order here, since a number of their geometrical and topological properties will be required for later chapters. 

The work of Weierstrass and Riemann on analytic functions and surface geometry provided the setting for the work of Schwarz who pioneered the study of three-periodic minimal surfaces (IPMS). Schwarz, a student of... 

Weierstrass, worked out the Weierstrass representation (eqs. 18) for two of the simplest IPMS, now called the P- and D-surfaces. (The latter is sometimes called the F-surface.) He also described three further three-periodic minimal surfaces, the CLP-(Crossed Layers of Parallels), H-, and T- surfaces. (The last is also called "Gergonnes surface"). These IPMS are illustrated in the Appendix. For the first time, the extraordinary complexity of these surfaces was revealed [12]. For the next half-century, his work was extended by others. 

Figure 1.17(c) A single node of the three-periodic D-surface. Four funnels (on one side of the surface) meet at each node, at angles of 109.5. Image courtesy of David Anderson, (d) A model of a portion of the D-surface. The surface partitions space into two interpenetrating open labyrinths, each lying on a diamond lattice. 

This product form for the Weierstrass polynomial is readily generalised, and offers a useful route to the discovery and parametrisation of three- periodic minimal surfaces (IPMS). It turns out that for all "regular surfaces" (which are the topologically simplest IPMS), the distribution and character of the flat point images (the location and type of the branch points (of R(,0))) in the complex plane) alone suffice to construct the Weierstrass polynomial, and thus the complete IPMS, using the Weierstrass equations. 

All these surfaces have one important characteristic in common. The flat "points" are not located within any finite portion of the surface. Rather, the surfaces become asymptotically flat (e.g. the trumpet-shaped "ends" in the catenoid). As the number of flat points increases beyond four, the flat points are located at fixed identifiable sites and the surface closes up to b ome periodic in three dimensions. This distinction between one- or two-periodic and three-periodic minimal surfaces is a crucial one, since it implies that the average Gaussian curvature () of one-, and two-periodic minimal surfaces is usually zero, due to the overwhelming contribution from the... 

The simplest, non self-intersecting three-periodic minimal surfaces have a genus of three per unit cell. Six such surfaces are known (plus lower symmetry cases for some). They belong to three distinct isometric families. 

Table 1.3 List of the simpler three-periodic minimal surfaces (IPMS), together with their crystallographic symmetries. Those surfaces that carve space into two interpenetrating open labyrinths are marked with a tick, a cross denotes IFMS that are self-intersecting. In most cases, two space groups are listed for each IPMS, the first is that of the surface ctssuming both sides are equivalent, the second is the s)munetry displayed by the surface assuming inequivalent sides.

Many more examples of interpenetration in inorganic chemistry lead to a recognition of the ubiquity of hyperbolic surfaces of infinite genus -exemplified by three-periodic minimal surfaces - that demands consideration. In the giant structure of Cu4Cd3 the Cu atoms are separated from the Cd atoms by a surface that resembles a minimal surface. In diamond, cubic ice and cristobalite, all the atoms are located on one side of the surface and the space on the other side is empty. If ice is subjected to very high pressure, the same structure appears on both sides of a minimal surface (double ice or ice IX), with almost double the density of ordinary ice (Fig. 2.8). Similarly, diamond is expected to transform to a double-diamond structure with metallic properties at sufficiently high pressure. 

This result suggests a remarkably simple picture of these frameworks as dense packings of flexible discs (each containing a single Si02 group) on periodic hyperbolic surfaces close to three-periodic miiumal surfaces of infinite genus. 

Table 4.1 Geometric properties of some periodic minimal surfaces. The "genus" of each three-periodic minimal surface (IPMS) is the genus of a unit cell of the IPMS (with symmetrically distinct sides). The "symmetry" refers to the crystallographic space group for the surface (assuming equivalent sidesl.liie surfaces are tabulated in order of deviation of the homogeneity index from the "ideal" value of 3/4. ...

Notice that the homogeneity indices of these IPMS are close to those of perfectly homogeneous minimal surfaces. Clearly then, three-periodic minimal surfaces are quasi-homogenous hyperbolic surfaces, in contrast to... 

We have already argued that a self-assembled bilayer composed of equivalent surfactant "blocks" should form a homogeneous minimal surface, tracing the mid-surface of the bilayer. Within the constraints of this simple geometric model, we thus expect the formation of hyperbolic bUayers, wrapped onto three-periodic minimal surfaces of genus three or four per unit cell. 

An intermediate phase of tetragonal syiiunetry - the T phase - has also been detected in a number of systems. A rod structure related to a square mesh surface was foimd to agree well with X-ray and NMR data on a perfluorinated surfactant-water mixture forming the T phase [22], [34]. These examples demonstrate that surfactant or lipid monolayers lining mesh surfaces as well and bilayers wrapped onto three-periodic minimal surfaces (IPMS) are indeed found in these self-assembled systems. 

It is clear from the universal diagram (Fig. 4.11) that a variety of bilayer phases can form only if die surfactant parameter is between about 0.5 and 1.7. For higher values of the surfactant parameter, steric constraints e.g. head-group crowding) preclude the formation of curved hyperbolic bilayers or monolayers. The opportunity for bilayer pol)rmorphism exists for surfactant parameters l3dng between 1.0 and about 1.5. Iliese bilayer phases are expected to adopt cubic or rhombohedral symmetries, corresponding to the most homogeneous three-periodic minimal surfaces. 

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