Periodic minimal surfaces

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. 

The properties of the periodic surfaces studied in the previous sections do not depend on the discretization procedure in the hmit of small distance between the lattice points. Also, the symmetry of the lattice does not seem to influence the minimization, at least in the limit of large N and small h. In the computer simulations the quantities which vary on the scale larger than the lattice size should have a well-defined value for large N. However, in reality we work with a lattice of a finite size, usually small, and the lattice spacing is rather large. Therefore we find that typical simulations of the same model may give diffferent quantitative results although quahtatively one obtains the same results. Here we compare in detail two different discretization... 

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... 

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 1. A. Computer graphic portion of a periodic surface of constant mean curvature, having the same space group and topological type as the Schwarz D minimal surfhce. This surbce, together with an identical displaced copy, would represent the polar/apolar dividing surface in a cubic phase with space group 224 (Pn3m). The two graphs shown would thread the two aqueous subspaces. B. Computer graphic of a portion of the Schwarz D minimal sur ce (mean curvature identically zero). In the 224 cubic phase structure, this sur ce would bisect the surfactant bilayer.

In general, minimal surfaces display self-intersections. The most usual cases are surfaces that intersect themselves everywhere, and the "surface" wraps onto itself repeatedly, eventually densely filling the embedding space. We are only interested in translationally (or orientationally) periodic minimal surfaces, which are free of self-intersections (thereby generating a bicontinuous geometry) or periodic surfaces with limited self-intersections. Elucidation of these cases of interest requires judicious choice of the complex function R(a>) in the Weierstrass equations (1.18). 

DPMS, PMS, PCS, PNS All abbreviations for periodic hyperbolic surfaces infinite periodic minimal surfaces, periodic minimal surfaces, periodic cubic surfaces and periodic nodal surfaces respectively. (The abbreviations P-, D-and G- prepended to these indicate the topology and symmetry of the periodic surface, corresponding to the relative tuimel arangements and black-white sub-group of the P-surface, the D-surface and the gyroid respectively. 

A rigorous mathematical existence proof for a periodic surface of small, nonzero constant mean curvature can be obtained with the methods of the theory of nonlinear elliptic differential equations. The resulting surface would be a perturbation of a known periodic minimal surface, but the intent of chapter is rather to exhibit numerical solutions that extend over wide ranges in mean curvature. 

Figure 8. A periodic surface with the same topological type as the Schwarz P minimal surface, with sinusoidally varying mean curvature in the vertical direction. Two unit cells are shown side by side.

In the interval 0 < t < 1.413 the DG level surface effectively divides space into three embedded continuous subvolumes. This phase which is located between the two SG surfaces, centered on the minimal gyroid surface, forms the so-caUed matrix phase through which the two networks run. Each one of the three networks is periodic in all three principle directions. The matrix phase volume increases with increasing values of t, and thereby gradually reducing the strut diameters of the two enclosed networks. 

The Monte Carlo (MC) method, used to simulate the properties of liquids, was developed by Metropolis et al. (1953). Without going into any detail, it should be pointed out that there are two important features of this MC method that make it extremely useful for the study of the liquid state. One is the use of periodic boundary conditions, a feature that helps to minimize the surface effects that are likely to be substantial in such a small sample of particles. The second involves the way the sample of configurations are selected. In the authors words Instead of choosing configurations randomly, then weighing them with exp[—/i ], we choose configurations with probability exp[—/6 ] and weight them evenly. ... 

Modem surface crystallographic studies have shown that on the atomic scale, most clean metals tend to minimize their surface energy by two kinds of surface atom rearrangements - relaxation and reconstmction [22-26]. In this review, the term surface reconstruction applies to the case in which there is lateral (i.e. in the surface plane) movement of surface atoms such that the surface layer has a symmetry that is different from that of the underlying bulk of the crystal. Hence, the surface layer has a two-dimensional unit cell that is different from the corresponding two-dimensional unit cell of a layer in the bulk. The periodicity of the surface can be defined by Woods notation for example, an unreconstructed surface would be termed as (1x1), whereas if the surface unit cell size was doubled in one of the primary vector directions, it would be termed as (2 x 1), and so on. On the other hand, surface relaxation apphes to the case in which the surface layer is in a (1 x 1) state but the layer is displaced along the surface normal direction from the position expected for bulk termination of the crystal lattice. In this section, both surface reconstruction and surface relaxation effects are described with specific examples chosen to illustrate the phenomena as they are observed in the electrochemical environment. 

Next we consider the effect of the block copolymer composition /= NfJN on the ordered morphology. In the limit of very strong segregation, that is, zero interface width, the natural idea is to let the stable ordered phase correspond to the phase with the minimal interface surface. To illustrate this principle and to obtain a semiquantitative estimate of the values of/for which the transitions between the three classical stmctures occur, we consider an LxLxL volume of the self-assembled diblock copolymer system. The ordered states that will be compared are the lamellar phase, a square lattice of cylinders, and spheres on a simple cubic (SC) lattice. L is the periodicity length scale of the layers, the square, and the cubic lattice (Figure 19). The LxLxL volirme element contains one cylinder resp. one sphere. Volirme conservation (Figure 20), therefore, requires fL = 7tRcL = 4n/SRs, where Rc and Rs are the radii of the cylinder and the sphere, respectively. 

These local structural rules make it impossible to construct a regular, periodic, polyhedral foam from a single polyhedron. No known polyhedral shape that can be packed to fill space simultaneously satisfies the intersection rules required of both the films and the borders. There is thus no ideal structure that can serve as a convenient mathematical idealization of polyhedral foam structure. Lord Kelvin considered this problem, and his tet-rakaidecahedron is still considered the best periodic structure of identical polyhedra that can nearly satisfy the mechanical constraints, while providing the smallest surface energy (or area). A more efficient structure for minimizing the surface energy, has more recently been proposed (known as the Weaire-Phelan structure (10)), but it consists of bubbles of two different types, and whether it is the optimal structure remains an open question. 

When a bulk crystal is cleaved to form a surface the coordination of the surface atoms is reduced and the atoms close to the surface relax to minimize the surface energy of the crystal. As a result of this, the distances between the first few layers of the crystal will change with respect to their bulk values this process is known as surface relaxation. A more severe change of structure is the phenomenon of surface reconstruction. This involves larger (still on the atomic scale) displacements of atoms compared to surface relaxation and the periodicity parallel to the surface may change with respect to that of the bulk. 

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