Surface minimal

Resolution at tire atomic level of surfactant packing in micelles is difficult to obtain experimentally. This difficulty is based on tire fundamentally amoriDhous packing tliat is obtained as a result of tire surfactants being driven into a spheroidal assembly in order to minimize surface or interfacial free energy. It is also based upon tire dynamical nature of micelles and tire fact tliat tliey have relatively short lifetimes, often of tire order of microseconds to milliseconds, and tliat individual surfactant monomers are coming and going at relatively rapid rates. 

Luzzati V, Delacroix FI and Gulik A 1996 The micellar cubic phases of lipid-containing systems Analogies with foams, relations with the infinite periodic minimal surfaces, sharpness of the polar/apolar partition J. Physique. II 6 405-18... 

Giusti E. (1984) Minimal surfaces and functions of bounded variations. Birkhauser, Boston, Basel, Stuttgart. 

Increase adhesion tension. Maximize surface tension. Minimize contact angle. Alter surfactant concentration or type to maximize adhesion tension and minimize Marangoni effects. Precoat powder with wettahle monolayers, e.g., coatings or steam. Control impurity levels in particle formation. Alter crystal hahit in particle formation. Minimize surface roughness in milhng. 

To minimize surface damage, static SIMS mass spectrometers should be as efficient as possible for detecting the total yield of secondary ions from a surface. Also, to be able to separate elemental from molecular species, and molecular species from each other, the mass resolution usually given as the mass m divided by the separable mass Am, should be very high. With this in mind, two types of mass spectrometer have been used - in early work mainly quadrupole mass filters and, more recently, time-of-flight mass spectrometers. 

Particle shape (for a given volume, the best shape for minimizing surface area and improving filterability is a sphere), and... 

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]. 

A. Structures with Interface Described by Known Minimal Surfaces... 

TABLE 1 Geometrical Properties of the Known Minimal Surfaces Obtained from the Functional (1), for the Parameters /o = 0.0, = -3.0 ... 

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. 

For ordered periodic structures of a period A a dimensionless ratio between the two lengths, Ryy = Kyy /, provides an additional characteristic of the structure. On the basis of the results of Ref. 39, we can estimate this ratio for the periodic minimal surfaces. For simple minimal surfaces, P, D, or G [39], we find respectively = 0.306, Ryyi = 0.195, and Ryyi = 0.248. For more complicated periodic structures [39] its value can be even smaller than 0.1 for example, for the surface labeled GX5, Ryy = 0.073. 

W. Gozdz, R. Holyst. From the plateau problem to minimal surfaces in lipids, surfactants and diblock copolymer systems. Macromol Theory Simul 5 321-332, 1996. 

A. H. Schoen. Inifinite periodic minimal surfaces without self-intersections. Technical Report TN D-5541, NASA, May 1970. 

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

D. M. Anderson. A new technique for studying microstructures NMR band-shapes of polymerized surfactants and counterions in microstructures described by minimal surfaces. J Physique Colloque 57 1-18, 1990. 

A realistic model of a solution requires at least several hundred solvent molecules. To prevent the outer solvent molecules from boiling off into space, and minimizing surface effects, periodic boundary conditions are normally employed. The solvent molecules are placed in a suitable box, often (but not necessarily) having a cubic geometry (it has been shown that simulation results using any of the five types of space filling polyhedra are equivalent ). This box is then duplicated in all directions, i.e. the central box is suiTounded by 26 identical cubes, which again is surrounded by 98 boxes etc. If a... 

The net attraction of surface molecules to the interior of the liquid indicates that molecules are most stable when attractive forces are maximized by as many neighbor molecules as possible. Consequently, a liquid is most stable when the fewest molecules are at its surface. This occurs when the liquid has minimal surface area. Spheres have less surface area per unit volume than any other shape, so small drops of a liquid tend to be spheres. Large drops are distorted from ideal spheres by the force of gravity. 

In order to show how a theory for the number of nuclei formed per unit time can be built up, we consider the following. First, we assume that a nucleus will be spherical (to minimize surface energy). The volvune will be ... 

Nanoclusters tend to form facets in order to minimize surface energy. Since the surface energy of a face-centered cubic (fee) metal follows the order of (111)<(100) <(1 1 0), most nanoclusters are bounded by (1 1 1) and (1 00) surfaces, Figure 2a and b. In many cases, the ratio of the growth rate along the (100) versus (111)... 

In direct insertion techniques, reproducibility is the main obstacle in developing a reliable analytical technique. One of the many variables to take into account is sample shape. A compact sample with minimal surface area is ideal [64]. Direct mass-spectrometric characterisation in the direct insertion probe is not very quantitative, and, even under optimised conditions, mass discrimination in the analysis of polydisperse polymers and specific oligomer discrimination may occur. For nonvolatile additives that do not evaporate up to 350 °C, direct quantitative analysis by thermal desorption is not possible (e.g. Hostanox 03, MW 794). Good quantitation is also prevented by contamination of the ion source by pyrolysis products of the polymeric matrix. For polymer-based calibration standards, the homogeneity of the samples is of great importance. Hyphenated techniques such as LC-ESI-ToFMS and LC-MALDI-ToFMS have been developed for polymer analyses in which the reliable quantitative features of LC are combined with the identification power and structure analysis of MS. 

Surface Behavior. Most extraction processes deal with several phases. At the boundaries between these phases, an interface exists which can be populated with or depopulated of polymer. Situations in which the polymer should accumulate at the surface of one phase are 1. the flocculation of clays and fines or 2. the formation of foams, while situations in which the polymer should depopulate the surface of the phase boundary are 3 minimizing adsorption in mineral acid leaching or 4. minimizing surface tension with surfactants in oil recovery by miscible flooding.,... 

Those chains of class B in Fig. 1, with a loop in the neighboring cube, will embrace a number of segments that is determined by the minimal surface enclosed by the line segment from , to anc by the space curve of the portion of the chain... 

---