Polyelectrolytes: Theory, Properties and Applications

22 F. Benmouna, R. Kaci and M. Benmouna

while in the crew-cut configuration, the scaling is like

2 3 2 1 2 1 

ln cmc b (^) a
  N /vN /v. (21)
Figure 11. Schematic representation of micelles with a) star like and b) crew cut
configurations.
Here, we disregard the other parameters of the micelle focusing on scaling
trends in terms of Na and Nb only. The minimization of the free energy with
respect to Nag gives the equilibrium aggregation number which according to
[28, 33], and keeping only terms showing Na and Nb, may be written as
2 1 /5 6/5
ab
ag b 35
/
/
b
45 ln
NNv v
NN
N

 


. (22)

Here, the critical exponent v is related to the solvent quality towards
monomer A. It is 1/2 for a theta solvent and 3/5 for a good solvent. These
results indicate that the micelle properties are much more sensitive to Nb than
Na even though Na is much higher than Nb. This is due to the fact that
equilibrium properties of star like micelles depend logarithmically on the
corona size Na and the continuous decrease of the concentration of A
monomers as we move farther away from the core. In the crew cut
configuration, the core size is dominant and its radius much larger than the
corona thickness. Minimization of the free energy gives the following scaling
for the equilibrium aggregation number