Ellipsoidal aggregates

The close-packed-spheron theory8 incorporates some of the features of the shell model, the alpha-particle model, and the liquid-drop model. Nuclei are considered to be close-packed aggregates of spherons (helicons, tritons, and dineutrons), arranged in spherical or ellipsoidal layers, which are called the mantle, the outer core, and the inner core. The assignment of spherons, and hence nucleons, to the layers is made in a straightforward way on... 

ZnO particle morphologies are very complex and diversiform in comparison with Ti02. Thus, monodispersed ZnO particles with well-defined morphological characteristics, such as spherical, ellipsoidal, needle, prismatic, and rod-like shapes, have been obtained. Aggregates composed of these basic shape particles have also been achieved. The methods used for synthesis of these ZnO powders include alkali precipitation [214-216], thermal decomposition [217], hydrothermal synthesis [218], organo-zinc hydrolysis [219], spray pyrolysis [220], and other routes. 

Numerous books and reviews have been published on this subject (e.g. Fendler and Fendler, 1975 Mittal, 1977). Therefore, the structural characteristics of micelles will be presented only to the extent that is necessary for the subsequent discussions. These surfactants form micelles at concentrations above the cmc (critical micelle concentration). Such micelles have average radii of 12-30 A and contain 20-100 surfactant molecules. The hydrophobic part of the aggregate forms the core of the micelle while the polar head groups are located at the micellar surface. Micelles at concentrations close to their cmc are assumed to possess spherical and ellipsoidal structures (Tanford, 1973, 1978). A schematic representation of a spherical ionic micelle is shown in Fig. 1. 

As mentioned earlier, surfactants aggregate to form micelles, which may vary in size (i.e., number of monomers per micelle) from a few to over a thousand monomers. However, surfactants can form, besides simple micellar aggregates (i.e., spherical or ellipsoidal), many other structures also when mixed with other substances. The curved micelle aggregates are known to change to planar interfaces when additives, the so-called cosurfactants, are added. A reported recipe consists of... 

The biochemical structure of a membrane is that of a lipid bilayer composed of phospho- and sphingolipids, as well as cholesterol. These lipids are amphipathic in nature, that is, they each have a polar and a nonpolar end. In water the nonpolar (hydrophobic, lipophilic) ends will seek to avoid the polar solvent and aggregate into a bilayer with the polar (hydrophilic, lipophobic) ends oriented towards the outside of the bilayer. As this structure extends in all directions the exposed nonpolar regions will close up and form a sphere (or ellipsoid) with water trapped inside and excluded outside. See Figures 2a and 2b. 

According to our observations and to literature data (4, 19, 22, 24, 47, 49, 50, 54, 56, 63, 64, 65), the following aggregate shapes occur in the end structures spheres, ellipsoids, long, substantially cylindrical rods in... 

Soft-core reverse micelles are spherical or ellipsoidal aggregates consisting of a water core separated from a continuous apolar phase by a surfactant shell. It is well known that in the absence of water, some surfactants such as sodium bis(2-ethylhexyl) sulfosuccinate (AOT) can form dry aggregates, while others such as sodium dodecyl sulfate (SDS) or hexadecyl-trimethylamonium bromide (CTAB) need a cosurfactant, e.g., a short chain alcohol, to form micelles. 

One of the conceptually simplest representations of a molecule, also used in the construction of hard-sphere space filling models, is obtained by centring spheres of suitable Van der Waals radii at the positions of the atomic nuclei. To convey a feeling of size the aggregate is enclosed in a generating ellipsoid that circumscribes the atom most remote from the centre to define the volume of the Van der Waals body [212]. 

The hydrophobic part of the aggregate molecules forms the core of the micelle while the polar head groups are located at the micelle-water interface in contact with the water molecules. Such micelles usually have average radii of 2... 4 nm and contain 50... 100 monomers in water. Their geometric structure is usually roughly spherical or ellipsoidal. In non-aqueous nonpolar solvents, the micellar structures are generally the inverse of those formed in water. In these solvents, the polar head groups form the interior of the micelle while the hydrocarbon chains of the ions are in contact with the nonpolar solvent. 

Polymers in Solution. Polyacrylamide is soluble in water at all concentrations, temperatures, and pH values. An extrapolated dieta temperature in water is approximately —40°C (17). Insoluble gel fractions are sometimes obtained owing to cross-link formation between chains or to die formation of imide groups along die polymer chains (18). In very dilute solution, polyacrylamide exists as unassociated coils which can have an ellipsoidal or beanlike structure (19). Large aggregates of polymer chains have been observed in hydrolyzed polyacrylamides (20) and in copolymers containing a small amount of hydrophobic groups (21). 

For anisotropic particles, the percolation limit is a function of the aspect ratio. For ellipsoids of revolution, the percolation limit for a simple cubic lattice was studied by Boissonade et al. [85]. They found as the aspect ratio increases from 1 (a sphere) to 15 (a fiber), the percolation limit decreased from a volume fraction of 0.31 to 0.06 and the correlation length (i.e., aggregate size) did not change (i.e., it was the same as that of the sphere). 

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