Hydrophobically associated networks - Physical Gels from Biological and Synthetic Polymers - page 158

6.2.2 Self-assembly of amphiphilic molecules

The fundamental thermodynamic equations of self-assembly deal with the equilibrium
between aggregated species and single molecules. The association of N amphiphilic
molecules into an aggregate generates a variety of structures in aqueous solution. In most
cases, the micelle structure is a sphere with a hydrocarbon core, the polar or ionic groups
at the surface serving to maintain the solubility in water. The hydrocarbons in micelles are
generally regarded to be in the bulk liquid medium. The core is a small volume of liquid
hydrocarbon. However, parts of the hydrocarbon chains close to the polar surface are
constrained and lie more or less perpendicular to the surface. The effects of such
constraints are nevertheless quite small for simple amphiphilic molecules, as con

rmed

by spectroscopic relaxation methods (Tanford, 1991 ).
Equilibrium thermodynamics requires that, in a system of molecules that form aggre-
gated structures, the chemical potential of all identical molecules in different aggregates
is the same.
If N is the number of molecules in an aggregate, this condition gives

¼

kT
N log

X N
N

0
N

μ ¼ μ N ¼ μ þ const : N ¼ 1 ; 2 ; 3 ; … ð 6 : 1 Þ

Here μ N is the chemical potential of a molecule in an aggregate of aggregation number

N is the standard chemical potential (the mean interaction free energy per molecule)
in a micelle of size N,

N, μ

1 is the standard chemical potential of a single molecule and X N is
the molar fraction or the activity of amphiphilic molecules in micelles of size N
(Israelachvili, 1992 ). Following the law of mass action, one can also write this equili-
brium as a function of the rate of association of molecules, A:

μ

NA 1 $ A N :

For ideal (dilute) solutions without interactions between aggregates, the equilibrium
constant of this reaction is

A ½
A ½

K N ¼

N :

ð 6 : 2 Þ

With

X N
N

½ A N ¼ and A ½¼ X 1 ;

we have

N 1

X N
NX 1 ¼

N ðμ μ Þ

K N ¼

exp : ð 6 : 3 Þ

kT