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Important events have recently taken place in structural crystallography. First, the rigid concept of a “perfect crystal” has been relaxed to embrace more general structures, such as quasi-crystals. Second, an advance has been made from the classical geometry of coordination polyhedra to three-dimensional differential geometry. The decisive step in this direction has been the use of curved surfaces in describing a great variety of structures.
The essence of the key concept of a minimal surface is as follows. When a wire frame is dipped into soapy water, a thin film is formed. Surface tension minimizes the energy of the film, which is proportional to its surface area. As a result, the film has the smallest area consistent with the shape of the frame and with the requirement that the mean curvature of the film be zero at every point. If a minimal surface has space group symmetry, it is periodic in three independent directions. Triply periodic minimal surfaces (TPMS) are of special interest, because they appear in a variety of real structures such as silicates, bicontinuous mixtures, lyotropic colloids, detergent films, lipid bilayers and biological formations.[3-11] For example, the interface between single calcite crystals and amorphous organic matter in the skeletal element in sea urchins is described by the P minimal surface.[6,7] TPMS are omnipresent in the natural and man-made worlds, and provide a concise description of many seemingly unrelated structures. They have become of interest not only to the structural chemist, but also the biologist, structural engineer and materials scientist, and are echoed in art and architecture.14 The Schwarz P surface is found in the ternary mixtures of oil, water and surfactant, in the zeolite sodalite and in the perovskite-type structure of. These self-assembled structures are templates in the inorganic and organic polymerisation reactions which lead to mesoporous molecular sieves[15,16] and to contact lens materials. Atoms in numerous zeolites lie on minimal surfaces.[11,18,19] Further structures which are related to minimal surfaces are cristobalite, diamond, quartz, ice, W3Fe3C (cutting steel), starch and N6F15, and many properties of these solids stem from this fact. TPMS may even have applications in cosmology as membranes (or “branes”).
It is imperative that the TPMS be properly examined in the practical context. They offer a unique way of expressing collective laws and strategies for understanding the larger correlations between related structures with potential for applications in molecular and materials design. We are confident that we shall soon be able to describe, quantify and predict transformation pathways in regular structures, such as phase transitions in silicates and catastrophic phase changes in oil-water mixtures.
The application of minimal surfaces to the physical world has so far been descriptive, rather than quantitative. The reason for this is that it was thought that the parameters of minimal surfaces could not be calculated analytically, while numerical calculations were inaccurate and consumed vast amounts of computer time. A surface element of an infinite surface was calculated by numerical integration, and larger portions constructed by combining the elements. We have been able to derive explicit, analytical, general equations for the parameters of triply-periodic minimal surfaces, which led to the discovery of important new properties.[21-32] Accurate calculations can now be easily carried out using a personal computer.
The coordinates of points lying on minimal surfaces are described in terms of integrals involving the (“Weierstrass”) function of the form
It was thought that the Weierstrass integration could not be performed analytically. However, via a rigorous procedure, we were able to express these integrals in terms of standard functions, and gave explicit general equations for the
coordinates.[21-32] The well-known properties of the functions appearing in the solution provide new insights into the features of minimal surfaces, and permit their systematic calculation. The values of the coordinates calculated from the equations are finite everywhere, even at points for which the polynomial under the square root is equal to zero. This has been a major obstacle to numerical calculations, particularly since the most interesting parts of a minimal surface occur in the near vicinity of such singularities. We gave the limiting values of all important quantities characterizing the surfaces, expressed the aspect ratio c/a and the surface-to-volume ratio in terms of the two parameters, and listed the exact coordinates. As a by-product, we have solved important mathematical problems not directly related to minimal surfaces. Some of them relate to optimization theory using the method called Taboo
Search. Others, known as “special functions”, are already being implemented in commercial software packages.
Minimal surfaces are found in soap films, in assemblies of lipid layers in contact with water and in lyotropic colloids.[36-38] However, the structure of such liquid mixtures is difficult to study quantitatively. The breakthrough came with the discovery of a new family of mesoporous silicas synthesised in water-surfactant mixtures. These solid materials,[15,39] the structure of which mimics the liquid-liquid minimal surface, have dramatically expanded the range of crystallographically defined pore sizes from the micropore (< 13 Ĺ) to the mesopore (20 to 100 Ĺ) regime. The synthesis uses ordered arrays of surfactant molecules as a “template” for the three-dimensional polymerization of silicates. The mesoporous materials obtained in this way exhibit several remarkable features: (i) well-defined pore size and shape; (ii) fine adjustability of the pore size; (iii) high thermal and hydrolytic stability; and (iv) a very high degree of pore ordering over micrometer length scales. These properties result directly from the interplay between organized arrays of surfactant molecules and silicate species in the aqueous phase. Since the number of possible surfactants which can be used is virtually limitless, many different minimal surfaces can be readily prepared. The cubic mesophase is the same as the phase found in the water-cetyltrimethylammonium bromide system. The midplane of the silicate wall lies on the G minimal surface. Such a structure can be viewed as a single infinite silicate sheet which separates the surfactant species into two equal and disconnected volumes. It is advantageous for the silicate wall to occupy a periodic minimal surface, because it can then maximize the wall thickness for a given surfactant/SiO2 volume fraction.
et al. have used periodic nodal surfaces, closely related to TPMS, for direct structure determination from powder X-ray diffraction data. Given just a few strong low-index reflections, a periodic nodal surface which divides the unit cell into regions of high and low electron density, is generated. Fourteen known zeolite structures were examined to verify the validity of the procedure, and in all cases the framework atoms were found to lie on just one side of the calculated curved surface. In other words, the surface defines a structure envelope. The same approach was successfully applied to ionic and organic structures. Since these reflections, with large d-spacings, are precisely the ones that are least likely to be involved in overlap in a powder diffraction pattern, such surfaces can also be calculated using powder data. The resulting restriction in the region of the asymmetric unit in which atoms are likely to be found has immediate implications for any of the direct-space methods of structure determination from powder data. The use of this structure envelope as a mask in a straightforward grid search procedure reduced the computer time required to solve several framework structures by as much as two orders of magnitude.
Ordered graphitic foams related to fullerenes and crystalline mesoporous oxides
Mackay and Terrones[43-45] have predicted the existence of a new kind of ordered graphite foams, related to fullerenes, with topologies of triply periodic minimal surfaces likely to be found. We have also shown this to be possible for MCM-type materials, and an example of an energy minimised crystalline form of MCM-48 has been
demonstrated. These new structures, which are of great practical interest, are constructed by introducing 8-membered rings of carbon atoms into a sheet of 6-membered rings, thus giving saddle-shaped surfaces. The existence of such rings has been confirmed by electron
microscopy,[47,48] and curved graphitic foams are present in “fouled” molecular sieve catalysts. The study of ordered periodic foams as 2-D manifolds, not only for graphite but for other materials, will open the possibility of new kinds of structures with novel properties.
Potential applications of periodic minimal surfaces in engineering, as high-strength structural elements, heat exchangers and sound absorbers, can only be assessed if proper solid
three-dimensional models are constructed. In the past, models were made by putting together small sections of the surface, which is unsatisfactory for engineering purposes. Polymer models are being made in Cambridge by stereolithography.