Three-periodic minimal surfaces

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. 

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. 

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. 

Their quotient surface, under the translation group, is a surface of genus greater or equal to three. Schwarzits, i.e. ( 6,7,8, 3)-maps on surface of genus three, have been used to model 3-periodic minimal surfaces (see [KiOO]). 

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 field of infinite periodic minimal surfaces (IPMS), that was introduced a few decades ago for the analysis of the topology of crystal structures [41], is a different approach to the analysis of nets many common nets are related to the known intersection-free IPMS [42], The IPMS studies have also produced a systematic enumeration of nets that has been recently proposed by the EPINET project (Euclidean Patterns in Non-Euclidean Tilings, see http //epinet.anu.edu.au) instead of working directly in three dimensions, the intrinsic hyperbolic geometry of IPMS is used to map 2D hyperbolic patterns into 3D Euclidean space [43],... 

The three dimensional (3D) cubic V2 phases are arranged as single continuous lipid curved bilayers forming a eomplex network containing two non-intersecting water channels [90]. Three different bicontinuous cubic nanostructures (a family of closely related phases) have been identified in the literature. They have a primitive (P), a gyroid (G), or a diamond (D) infinite periodic minimal surface (IPMS) [88, 89]. 

The most intriguing cases of lower symmetry relatives to known mesophases are anisotropic bicon-tinuous mesophases. These are sponges whose homogeneity lies one rank below the cubic genus-three P, D and gyroid surfaces. The most likely candidates are tetragonal and rhombohedral variants. These include the rPD, tP, tD, tG and rG triply periodic minimal surfaces (18). These surface are deformations of their cubic parent structures, and can be modelled as perturbations of the known bicontinuous cubic mesophases. 

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