Similarities in Self-Assembly of Proteins and Surfactants: An Attempt to Bridge the Gap - Food Colloids: Self-Assembly and Material Science - page 59

yields X 1 + X 2 + + X N ¼ X total , the total mole fraction of surfactant in
solution. Equation (1) then yields 1

X N ¼ N{X 1 exp[ m 1 m 0 N /kT]} N .

(2)

Because the total mole fraction cannot exceed unity, there appears automat-
ically an upper boundary for X 1 , also known as a critical aggregation concen-
tration, or critical micelle concentration, X c , given by

X c ¼ exp[ ( m 1 m 0 N /kT)]. (3)

We have not assumed anything about the type of surfactant, nor the morphol-
ogy of the aggregates. So, for that matter, the above would be equally valid for
proteins.
To advance further with the description set out above, one needs to include
molecular information about the surfactant in order to calculate the interaction
potential between surfactants within an aggregate of size n, and subsequently to
arrive at an expression for the chemical potential of a surfactant in that
aggregate of size N, i.e., m 0 N . Such an expression should, in principle, also
contain information on the shape of the aggregate, because aggregate shape
also influences the distances between the hydrophilic and hydrophobic parts of
the surfactants and thus their chemical potential. One should realise at this
point the following general complicating factor. If one wants to arrive at a size
distribution of aggregates of a particular surfactant on the basis of an equilib-
rium approach as above, one must include the shape of the aggregate in order
to calculate the chemical potential. However, it is this shape itself that also
should, in principle, be part of the minimization scheme of the free energy of
the overall system. In fact, one should minimize the free energy on the basis of
both the size distribution and the morphology of the system simultaneously.
Setting aside for a moment this complication in variational calculus, one may
alternatively set an a priori shape first, and put forward hydrophilic and
hydrophobic interactions within such an aggregate, and hence determine the
average size as a function of experimental parameters.
Israelachvili et al. 2 have argued that, despite the intrinsic difficulties of
incorporating all the hydrophilic interaction contributions, the contribution
from the repulsive interaction to the chemical potential per surfactant, within
an aggregate of size N, can be assumed to adopt a simple form:

m 0 N hydrophilic p De 2 / e a (4)

Equation (4) is based on modelling the energy contribution as the energy of a
capacitor with a charge per unit area of e/a, and a separation D of the planes
(resulting from the double-layer of charge with thickness D), where e is the
relative dielectric constant of the medium around the surfactant head-group.
Indeed, Tanford 3 has shown that this 1/a dependence explains satisfactorily the
micellar size and the critical micelle concentration.