Similarities in Self-Assembly of Proteins and Surfactants: An Attempt to Bridge the Gap - Food Colloids: Self-Assembly and Material Science - page 60

Including hydrophobic interactions lead to an overall chemical potential of
the form 1

m 0 N ¼ m 0 N hydrophobic + m 0 N hydrophilic ¼ g a + c/a + g

(5)

where g denotes the surface tension of, and a is the area between, hydrophobic
groups and water (a constant), and where the term g denotes constant surface
area and/or bulk contributions. Combining Equations (4) and (5) yields an
optimum area, a 0 ¼ (c/ g ) 1/2 , for which the free energy is a minimum. 2 This
relates to a specific number of surfactants per aggregate. Having a larger
number of surfactants per aggregate yields a smaller area per surfactant, and
vice versa. Both cases imply a higher free energy, leading to a size distribution of
micelles.
The optimal area a 0 can be used to provide a practical criterion for the shape
of the micelle versus the sizes of the head-group and tail of the surfactant, 2
assuming that the chemical potential is not influenced too much by the change
of shape, i.e., assuming that a 0 is not much affected by the choice of aggregate
shape. Considering a spherical micelle of radius R, and denoting the volume of
the hydrocarbon tail as v, and the surface area between hydrocarbon and water
as a 0 , we have R ¼ 3v/a 0 . Realising that there is an upper limit to the extension
of the hydrocarbon chain given by the maximum tail length l c , a spherical
micelle will only be formed for 3v/a 0 o l c , leading to the condition for spherical
micelle formation as v/a 0 l c o 1/3. 2 Similarly, we have for cylinders the condi-
tion v/a 0 l c o 1/2, and for planar objects v/a 0 l c o 1.
We may investigate further, now assuming a certain shape, the effect of the
number of surfactants on the chemical potential. Suppose one has an energy of
binding between monomers within an aggregate of magnitude a . In the case of a
rod-like structure, the endpoints are unbound, leading to 2

N m 0 N ¼ N a kT + a kT ¼ (N 1) a kT (6)

or

m 0 N ¼ m 0 1 þ a kT

N ;

ð 7 Þ

where m 0 N is the energy of the surfactant within an infinite aggregate. Similarly,
taking always into account the number of endpoints of the aggregate, we may
derive 2 for planar aggregates:

m 0 N ¼ m 0 1 þ a kT

N 1 = 2 :

ð 8 Þ

And for spherical aggregates we get

m 0 N ¼ m 0 1 þ a kT

N 1 = 3 :

ð 9 Þ