The key to theoretical prediction of the behaviour of self-assembling systems is an
understanding of structure. This structure is the global spatial manifestation of the sum
of all local interactions. As a consequence of the subtlety of this connection, prediction
of structure from the basis of a detailed model of the specific molecular interactions is not
feasible, as our mathematical limitations force the imposition of strict geometrical
assumptions. Instead, such physical treatments are sacrificed for the freedom of
description offered by simplified geometrical approaches. In this thesis two examples of
such geometrical motivations, of particular reference to the phases observed in binary
surfactant-water mixtures, are analysed and extended.
Helfrich attributed the energy cost of fluctuation of the surfactant aggregate to the
bending of the interface separating the hydrophilic and hydrophobic regions, and
proposed a simple curvature energy function describing this. Applying simple stability
considerations, we find that this functional form cannot be reconciled with the intuition of
a preferred interfacial curvature, and propose an alternative phenomenological description
which is consistent with this notion and with existing continuum models of the surfactant
film.
For the specific case of preferentially flat films (that is, zero spontaneous
curvatures) the surfactant bilayers have a tendency to form lamellar phases, which exhibit
interesting behaviour attributed to the bending energy. To investigate this, we calculate
this energy directly for the specific case of ionic surfactants in aqueous electrolytes, thus
permitting the inference of formulae for the bilayer bending modulus (characterising its
degree of stiffness).
Perhaps the most striking structural feature of surfactant-water systems is the
observation of bicontinuous phases. The partitioning interface in these formations is
found to be modelled in many cases by the class of triply periodic minimal surfaces.
Here we derive new examples of these special surfaces and present an algorithm for the
parametrisation of this class, thus facilitating a quantitative assessment of the degree to
which these surfaces match the real surfactant interface
