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the impossibility of writing down a complete Boltzmann-type distribution of all
possible membrane geometries and topologies, appropriately weighted by their
energies and entropies.
An important question in the analysis of mesophase structures is that of the rela-
tive stabilities of intermediate versus bicontinuous mesophases. Most analyses of
the available data suggest that intermediate mesophases of unusual topologies
and geometries fall into three topological classes: noncubic bicontinuous sponges,
branched bilayers, and punctured bilayers. A fourth possible class, ''ribbon'' meso-
phases, consist of geometric distortions of columnar micelles in hexagonal meso-
phases. Experimental studies of surfactant-water phase diagrams suggest that
intermediate structures form in place of bicontinuous cubic mesophases once the
surfactant chains exceed a certain length or their rigidity is enhanced, such as by
replacing hydrocarbon with fluorocarbon chains.
5.5. VESICLES AND BILAYER MEMBRANES
The association of surfactants into relatively simple aggregate structures such as
spheres, ellipses, and disks allows for a reasonably straightforward analysis of
the fundamental aspects of their structure, including their kinetics, thermody-
namics, and geometric considerations. The simplest extension of the simple micel-
lar structures, namely, the rodlike micelles often encountered in systems of ionic
surfactants in solutions of high salt content, presents a number of theoretical diffi-
culties. Such structures are large (relative to spherical systems) and polydisperse,
with no theoretical limit on the length that can be attained. Their average aggrega-
tion number is also very sensitive to the total surfactant concentration, so that the
properties of the system do not always lend themselves to easy analysis. In general,
their unusual properties result from the large dissymmetry in the dimensions of the
structural unit and the effects of the ends of the rods, where the associated mole-
cules are forced to pack into hemispherical caps.
As predicted by the geometric approach to aggregation, amphiphilic materials
that cannot readily pack into neat, closed structures such as simple micelles are
exactly those that are found to produce larger units such as vesicles and extended
bilayers. Such materials will have relatively small head groups, or, as is more com-
mon, their hydrophobic groups will be too bulky to be packed in a manner neces-
sary for normal micelle formation. Such a state of affairs is particularly common for
molecules having more than one hydrocarbon chain, very highly branched chains,
or structural units that produce molecular geometries incompatible with effective
packing into highly curved structures.
Although extended planar bilayers are a thermodynamically favorable option for
the association of some bulky surfactants in aqueous solution, under certain condi-
tions it is more favorable to form closed bilayer systems, leading to the existence of
membranes and vesicles. Such a situation arises from two basic causes: (1) even
large, highly extended planar bilayers possess edges along which the hydrocarbon
core of the structure must be exposed to an aqueous environment, resulting in an
