Surfactants (Soaps and Detergents) and Physicochemical Properties - Surface and Colloid Chemistry: Principles and Applications - page 49

this process; otherwise, very large aggregates would result. Thus, the standard free
energy of a micelle formation, ΔG o mice , could be written as:

ΔG° mice = attractive forces + opposing forces (3.2)

The attractive forces are associated with the hydrophobic interactions between the
alkyl part (alkyl-alkyl chain attraction) of the surfactant molecule, ΔG hydrphobic . The
opposing forces arise from the polar part (charge-charge repulsion, polar group
hydration), ΔG polar , and are of opposite signs. The attractive forces would lead to
larger aggregates. The opposing forces would hinder the aggregation. A micelle with
a definite aggregation number is one in which the value of ΔG o mice is 0. Hence, we
can write ΔG° mice

ΔG° mice = ΔG hydrophobic + ΔG polar (3.3)

The standard free energy of micelle formation will be

ΔG° mice = μ° mice − μ° mono

= RT ln(C mice /C mono ) (3.4)

At CMC, one may neglect C mice , which leads to

ΔG° mice = RT ln(CMC) (3.5)

This relation holds for nonionic surfactants, but will be modified in the case of ionic
surfactants (as shown in the following text). This equilibrium shows that, if we dilute
the system, micelles will break down to monomers to achieve equilibrium. This is a
simple equilibrium for a nonionic surfactant. In the case of ionic surfactants, there
will be charged species.
In the case of ionic surfactants, such as SDS, the micelle with aggregation number
N SD− will consist of counterions, C S+ :

SD− ionic surfactant monomers + C S+ counter = micelle with charge

(N SD− − C S+ ) (3.6)

Since N will be larger than S + , all anionic surfactants are negatively charged.
Similarly, cationic micelles will be positively charged. For instance, the cationic sur-
factant cetyltrimethyl ammonium bromide (CTAB), we have following equilibrium
in micellar solutions:

CTAB—dissociates into CTA + and Br − ions

The micelle with N CTA+ monomers will have C Br− counterions. The positive charge of
the micelle will be the sum of positive and negative ions (N CTA+ + C Br− ). The actual
concentration of each species will vary with the total detergent concentration, as in
the case of SDS (Figure 3.9).