consumed or expelled, which leads to a faster net growth or reduction of the
micelles towards a new equilibrium. The second process is characterized by τ 2 ;
however, it is expected to have a more complex concentration dependence. This
comes from the fact that since the micelle equilibration only proceeds via unimer
exchange, any intermediate metastable aggregates will consume unimers and
hinder the formation of the final stable micellar state. The time scale thereby
depends on the concentration and kinetics of all improper micelles (P < P eq or
P > P eq ) and thus more than the width of the mean distribution. As seen in
Eqs. 22–23, A-W theory predicts that the τ 2 increases with R, which characterizes
the population and in some sense the lifetime of the improper (out of equilibrium)
micelles as ϕ 0 is the total concentration. The quantity R thus depends on the
thermodynamic stability and, hence, on the lifetime of improper, metastable
micelles. This concentration dependence is thus difficult to predict because
τ 2 depends on the concentration of the different types of micelles of size P, ϕ p .
This highlights the fact that micelle formation is an activated process because
metastable micelles must dissolve by unimer expulsion before the equilibrium
micelles can be formed. This issue will be discussed in more detail later when we
return to non-equilibrium micellization kinetics in section 2.3.
Note that the A-W theory was derived under the assumption that the micellar
size distribution is Gaussian, independent of concentration, and can be taken to be
essentially the equilibrium distribution. Thus, in this linear relaxation process, the
micellar population is imagined to be “shifted” to a new mean aggregation value.
The existence of two relaxation constants and the corresponding concentration
dependence predicted by Aniansson and Wall has been well corroborated in many
experiments, especially for surfactant systems with large hydrophobic tails, where
the two processes can be observed more clearly [54]. It has been shown that for
classical ionic surfactants 1/τ 1 shows a consistent increase with concentration,
whereas the second process, characterized by τ 2 , seems to first increase and then
decrease again upon higher micelle concentration [51, 54]. This complex concentration dependence of τ 2 was later addressed in a work by Lessner and coworkers
[52, 53] and attributed to previously ignored effects of charge screening, which can
promote micellar fusion or fission [57]. This conclusion was based on temperaturejump experiments where the mechanism seemingly changed on going from low to
high concentrations. Screening of the charges at high ionic strengths in ionic
surfactants lowers the repulsion, facilitating fusion as an increasingly important
exchange mechanism. Fusion/fission is in general expected to play more of a role in
micellar systems with low repulsion, such as in nonionic micelles [58]. The
relaxation time for the fusion/fission mechanism is expected to decrease with
concentration, and a tentative description was given by Lessner et al. [53]. However, this does not seem to be strictly necessary to understand the experimental
results because the A-W theory also predicts a complicated concentration dependence of τ 2 simply because metastable micelles restrict micellar growth by consuming and depleting unimers [54]. In any case, a complication for charged
micelles, not considered in the original A-W theory, is the effect of co-solutes
such as counter-ions accompanying the main surfactant chain. These ions will
additionally affect the thermodynamics by lowering the cmc, which in turn leads
Kinetics of Block Copolymer Micelles Studied by Small-Angle Scattering Methods
69
micelles towards a new equilibrium. The second process is characterized by τ 2 ;
however, it is expected to have a more complex concentration dependence. This
comes from the fact that since the micelle equilibration only proceeds via unimer
exchange, any intermediate metastable aggregates will consume unimers and
hinder the formation of the final stable micellar state. The time scale thereby
depends on the concentration and kinetics of all improper micelles (P < P eq or
P > P eq ) and thus more than the width of the mean distribution. As seen in
Eqs. 22–23, A-W theory predicts that the τ 2 increases with R, which characterizes
the population and in some sense the lifetime of the improper (out of equilibrium)
micelles as ϕ 0 is the total concentration. The quantity R thus depends on the
thermodynamic stability and, hence, on the lifetime of improper, metastable
micelles. This concentration dependence is thus difficult to predict because
τ 2 depends on the concentration of the different types of micelles of size P, ϕ p .
This highlights the fact that micelle formation is an activated process because
metastable micelles must dissolve by unimer expulsion before the equilibrium
micelles can be formed. This issue will be discussed in more detail later when we
return to non-equilibrium micellization kinetics in section 2.3.
Note that the A-W theory was derived under the assumption that the micellar
size distribution is Gaussian, independent of concentration, and can be taken to be
essentially the equilibrium distribution. Thus, in this linear relaxation process, the
micellar population is imagined to be “shifted” to a new mean aggregation value.
The existence of two relaxation constants and the corresponding concentration
dependence predicted by Aniansson and Wall has been well corroborated in many
experiments, especially for surfactant systems with large hydrophobic tails, where
the two processes can be observed more clearly [54]. It has been shown that for
classical ionic surfactants 1/τ 1 shows a consistent increase with concentration,
whereas the second process, characterized by τ 2 , seems to first increase and then
decrease again upon higher micelle concentration [51, 54]. This complex concentration dependence of τ 2 was later addressed in a work by Lessner and coworkers
[52, 53] and attributed to previously ignored effects of charge screening, which can
promote micellar fusion or fission [57]. This conclusion was based on temperaturejump experiments where the mechanism seemingly changed on going from low to
high concentrations. Screening of the charges at high ionic strengths in ionic
surfactants lowers the repulsion, facilitating fusion as an increasingly important
exchange mechanism. Fusion/fission is in general expected to play more of a role in
micellar systems with low repulsion, such as in nonionic micelles [58]. The
relaxation time for the fusion/fission mechanism is expected to decrease with
concentration, and a tentative description was given by Lessner et al. [53]. However, this does not seem to be strictly necessary to understand the experimental
results because the A-W theory also predicts a complicated concentration dependence of τ 2 simply because metastable micelles restrict micellar growth by consuming and depleting unimers [54]. In any case, a complication for charged
micelles, not considered in the original A-W theory, is the effect of co-solutes
such as counter-ions accompanying the main surfactant chain. These ions will
additionally affect the thermodynamics by lowering the cmc, which in turn leads
Kinetics of Block Copolymer Micelles Studied by Small-Angle Scattering Methods
69