Periodic surface

LEED is the most powerfiil, most widely used, and most developed technique for the investigation of periodic surface structures. It is a standard tool in the surface analysis of single-crystal surfaces. It is used very commonly as a method to check surface order. The evolution of the technique is toward greater use to investigate surface disorder. Progress in atomic-structure determination is focused on improving calculations for complex molecular surface structures. 

Graphite was tised as substrate for the deposition of carbon vapor. Prior to the tube and cone studies, this substrate was studied by us carefully by STM because it may exhibit anomalotis behavior w ith unusual periodic surface structures[9,10]. In particular, the cluster-substrate interaction w as investigated IJ. At low submonolayer coverages, small clusters and islands are observed. These tend to have linear struc-tures[12j. Much higher coverages are required for the synthesis of nanotubes and nanocones. In addition, the carbon vapor has to be very hot, typically >3000°C. We note that the production of nanotubes by arc discharge occurs also at an intense heat (of the plasma in the arc) of >3000°C. 

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 SCLl surface is particularly interesting. Although the outer and the inner surface look different in Fig. 8(a), they have the same surface area. In fact they are built of the same piece of the surface. The picture of 1/8 of the unit cell, see Fig. 8(b), explains how two different periodic surfaces can be built of the same surface patch. 

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... 

In order to find the effect of broadening of the surface on the structure parameters H and K, we first study the ordered phases with the diffusive interfaces. The ordered phases can be described by the periodic surfaces (0(r)) = 0 and we can compare and with H and K. The numerators in the definitions (76) and (77) in the Fourier representation assume the forms [68]... 

S. T. Hyde. The topology and geometry of infinite periodic surfaces. Z Kris-tallogr 757 165-185, 1989. 

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. 

Regulation of microarchitecture has applications in the production of surface coatings. Again, control of the consistency of pattern offers the prospect of the self-assembly of periodic surface features on a scale that would interact with incident light. Paints could be designed to produce... 

The phase shift, As, between the applied sinusoidal AC and the periodical surface concentration is ... 

In what follows we will discuss systems with internal surfaces, ordered surfaces, topological transformations, and dynamical scaling. In Section II we shall show specific examples of mesoscopic systems with special attention devoted to the surfaces in the system—that is, periodic surfaces in surfactant systems, periodic surfaces in diblock copolymers, bicontinuous disordered interfaces in spinodally decomposing blends, ordered charge density wave patterns in electron liquids, and dissipative structures in reaction-diffusion systems. In Section III we will present the detailed theory of morphological measures the Euler characteristic, the Gaussian and mean curvatures, and so on. In fact, Sections II and III can be read independently because Section II shows specific models while Section III is devoted to the numerical and analytical computations of the surface characteristics. In a sense, Section III is robust that is, the methods presented in Section III apply to a variety of systems, not only the systems shown as examples in Section II. Brief conclusions are presented in Section IV. 

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 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... 

In this simple model characterized by a single scalar order parameter, the structures with periodic surfaces are metastable. It simply means that we need a more complex model including the surfactant degrees of freedom (its polar nature) in order to stabilize structures with P, D, and G surfaces. In the Ciach model [120-122] indeed the introduction of additional degrees of freedom stabilizes such structures. 

MORPHOLOGIES OF PERIODIC SURFACE PROFILES AND SMALL PARTICLES A SOURCE OF STEP AND STEP INTERACTION ENERGIES... 

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