Packing Argument

The maximum effective tail length Ic and the tail volume i of a saturated hydrocarbon chain of carbon atoms are estimated as (lanford 1980) 

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This "packing" argument may seem an unnecessary complication. But its advantage comes now. Consider cubic zirconia, ZrOj, an engineering ceramic of growing importance. The structure (Fig. 16.1c) looks hard to describe, but it isn t. It is simply an f.c.c. packing of zirconium with the ions in the tetrahedral holes. Since there are two tetrahedral holes for each atom of the f.c.c. structure, the formula works out at ZrOj. 

Thkea( 5, b, 3)-sphere or torus that is 5i j. Euler formula reads 12/ = p — (b — 6)ph- There are np = pairs of 5-gons. Every such pair defines AnP pattens b5b and 2nP patterns b55b in the boundary sequences of fe-gons. A packing argument yields ... 

Proof. A cycle of 4-gons cannot exist either in case (i) or in case (ii), due to the exclusion of Prismb. So, all 4-gons are part of triples of 4-gons 4,3 — v. Denote by n, the number of such triples. We have the relations p4 = 3nt. Furthermore, by a packing argument, we obtain the inequality ... 

The values of v/aoic for which one should expect bicontinuous cubic phases can be derived by an extension of the packing argument of Israelachvili et al. (Hyde 1990, 1992 Strom and Anderson 1992). Consider a small patch of area z(0) on a curved surface. If this patch is displaced a small distance in a direction normal to the surface, then the area a( ) of the displaced surface becomes... 

The structural units of surfactant-containing liquids are self-assembled aggregates, such as spherical or cylindrical micelles, or bilayers. These supramolecular structural units can then further self assemble into ordered phases, with cubic, hexagonal, smectic, or other symmetry. Consequently, the structural and flow properties of such liquids are amazingly rich. Laws of mass action, combined with geometric packing arguments, allow rationalization, if not prediction, of the phase behavior of many surfactant solutions. 

In this study, we use the same simulation scheme to study the behavior of these model amphiphiles in the presence of a liquid/vapor interface. This is quite feasible since Lennard-Jones particles are known to exhibit liquid/ vapor coexistence [16]. In short, we observe a highly regulated self-assembly process leading to the formation of one or more well-formed bilayers in the liquid phase amphiphiles not included in the bilayers are dispersed in the vapor phase. In this paper, we report on this phenomenon, and examine how the size and overall eoncentration of amphiphiles govern the development and characteristics of the bilayers formed. We also show that bilayer formation can be suppressed if the geometry of the amphiphiles is altered, as is predicted by simple packing arguments. 

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