The local geometry of aggregates

Since the suggestion of Hartley in the 30 s that surfactants can self-assemble to form globular aggregates - micelles - in which the hydrophobic chains are essentially molten, it has been clear that in order for surfactant molecules to pack into aggregates, the molecular dimensions must be compatible. For 

These examples allow us to describe tiie structure of surfactant aggregates in terms of the value of the surfactant parameter. Indeed, this is the case for simple closed surfaces, where the interior contains the hydrophobic fraction (v/al l). However, in general this analysis is not sufficient to determine uniquely the interfacial geometry. 

In fact, the molecular shape determines only the local shape of the interface, i.e. the interfacial curvatures. This can be demonstrated using simple equations from differential geometry describing the area of parallel surfaces. (Parallel surfaces are discussed in section 1.13.) If we trace out surface patcties that are parallel to a patch of area z(0) on an interface, the area of a corresponding patch at a distance d from the interface is  

The volume of space occupied by a foliation of parallel surfaces up to a distance I is obtained by integrating eq. (4.1) with respect to d  

Setting v(l) equal to the chain volume, v, I to the chain length and (0) to the head-group area, a, leads to a simple general expression for the surfactant parameter in terms of the curvatures of the interface between hydrophilic and hydrophobic regions, scaled by the characteristic distance, 1  

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