Aggregate structure in the Euclidean desert

If we confine ourselves to simple geometries, the explanation of the gross phase behaviour indicated is immediate given the maximisation of entropy 

At first sight the multi-molecular, multi-component assemblies of biology occur in dilute solutions where global packing constraints are not a problem. In fact that is not so. Compartmentalisation of components occurs naturally, and within those compartments global packing is a main issue. Even so their transformations in response to biochemical processes require more detailed accounting of the interplay between intra- and inter-molecular forces. 

These ideas can be formalised in terms of statistical mechanics to some extent, and an outline of the main ideas is given in the following section. We remark parenthetically that there are profound difficulties confronting the definition of an aggregate. The nature of the hydrophobic free energy of transfer of a hydrocarbon from water to the hydrophobic core of a micelle can be measured, but its temperature dependence is not understood because it depends on water, an unknown quantity. For the same reasons, solution theory, does not even tell us whether mole fractions or mole volumes are the correct ratios to use to determine entropy. However, provided certain assumptions are allowed [62-65], then simple rules emerge. The rules are if v is the hydrocarbon chain volume, a the head-group area, and 1 of an optimal 

The same progression can be observed if one adds progressively a single-chained surfactant (S) (v/al = 1/3) to a lamellar phase of a double-chained surfactant D v/al=l) of the same head-group area and chain length. Here it is the effective volume which changes according to. 

If curvature so prescribed is the major determinant of self-assembly, then phase diagrams ought to exhibit uiuversality, yet ionic and non-ionic phase diagrams are different (Fig. 3.4). 

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