Surfaces triply periodic

In this section we characterize the minima of the functional (1) which are triply periodic structures. The essential features of these minima are described by the surface (r) = 0 and its properties. In 1976 Scriven [37] hypothesized that triply periodic minimal surfaces (Table 1) could be used for the description of physical interfaces appearing in ternary mixtures of water, oil, and surfactants. Twenty years later it has been discovered, on the basis of the simple model of microemulsion, that the interface formed by surfactants in the symmetric system (oil-water symmetry) is preferably the minimal surface [14,38,39]. 

D. A. Hoffman. Some basic facts, old and new, about triply periodic embedded minimal surfaces. J Physique Colloque 51 C7 197-208, 1990. 

W. Gozdz, R. Holyst. Distribution functions for H nuclear magnetic resonance band shapes for polymerized surfactant molecules forming triply periodic surfaces. J Chem Phys 706 9305-9312, 1997. 

W. Gozdz, R. Hotyst. Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions. Phys Rev E 54 5012-5027, 1996. 

The first periodic (in one direction only) minimal surface [12] discovered in 1776 was a helicoid The surface was swept out by the horizontal line rotating at the constant rate as it moves at a constant speed up a vertical axis. The next example (periodic in two directions) was discovered in 1830 by Herman Scherk. The first triply periodic minimal surface was discovered by Herman Schwarz in 1865. The P and D Schwarz surfaces are shown in Figs. 2 and 3. The revival of interest in periodic surfaces was due to (a) the observation[13-16] that at suitable thermodynamic conditions, bilayers of lipids in water solutions form triply periodic surfaces and (b) the discovery of new triply periodic minimal... 

However, H. A. Schwarz found before 1865 that patches of varying negative gaussian curvature and constant H = 0 could be smoothly joined to give an infinite triply periodic surface of zero mean curvature. About five different types were found by Schwarz and Neovius, but now about 50 more have been described (Schoen 1970 Fischer Koch, 1989 a e). 

Fischer, W. Koch, E. 1989e Genera of minimal balance surfaces. Acta crystallogr. A 45, 726-732. Fogden, A. Hyde, S. T. 1992a Parametrisation of triply periodic minimal surfaces. I. Acta crystallogr. A 48, 442-451. 

Figure 4.7 Images of (left) a portion of a surfactant bilayer wrapped onto the P-surface, a triply-periodic minimal surface, with two interwoven polar labyrinths and (right) a reversed bilayer on the P-surface, with interwoven lipophilic labyrinths.

In his doctoral thesis, Meeks (1976) has given an existence proof of a family of triply periodic embedded minimal surfaces. Each surface corresponds to a placement of four antipodal pairs of points on the unit sphere. 

A model structure of type AA) based on a triply periodic embedded surface S parametrized via the present method as S r(u,u,w (u,y))l(u,u)eD, is in general described (for the purpose of relative intensities) by the following step-change density profile ... 

The I-WP and F-RD minimal surfaces have been shown to provide two counterexamples to a conjecture that has previously been made (Meeks 1978, p. 81, Conjecture 6) A triply periodic minimal surface disconnects into two regions with asymptotically the same volume. The volume fraction of the labyrinth containing the symmetric skeletal graph is 0.5360 0.0002 for the I-WP minimal surface and 0.5319 0.0001 for F-RD. 

Since analytical expressions for only a few continuous triply periodic CMC surfaces are known (e.g. the Enneper-Weierstrass parameterization of the single-gyroid minimal surface with H = 0 and a volume fraction of 50 % [9]), these surfaces are typically modeled with the help of level surfaces. 

In this context, it is interesting to note that it has been argued recently [88] that the (r) = 0 surfaces of order-parameter configurations, which minimize the free energy functional (16), are (often) periodic minimal surfaces. The four best known triply-periodic minimal surfaces G, D, P, and I— WP have been reproduced [88,89], and new candidates for minimal surfaces of high genus have been found [88]. 

Figure 36 Schematic of proposed arrangement of A/B junctions on a flat intermaterial dividing surface (IMDS) for (a) the TB-4 lamellar (all interior blocks form loops) and (b) the SB-2 lamellar structures (interior blocks form bridges and loops), (c) Portion of a triply periodic IMDS of constant mean curvature for the SB-4 structure (interior blocks form bridges and loops). Reprinted from Tselikas, Y. Hadjichristidis, N. Lescanec, R. L. et at. Macromolecules 9S6, 29, 3390. °...

Interestingly, in addition to constituting vesicles and lamellar structures, the bilayer building block also appears in the so-called L3 phase, characterized by a thermally disturbed, triply periodic, minimal surface type of structural organization for which we must have, of course, AP = 0. 

---