2.3.2 Free Energy Landscape Formalism
The previous “brute force” method is rather ineffective as all “reactions” and species
are treated equally, regardless of the probability of formation or the stability. It may
very well be that most of the pathways are so improbable or so energy-costly that they
can be completely ignored.
In another approach, the micellization can be seen as a journey on a multidimensional energy landscape (phase space, i.e., a space spanned by all parameters) towards
an equilibrium state that represents a global minimum on the landscape. One can
imagine that the most probable path corresponds to the path of minimal energy,
which naturally introduces a constraint. Such an approach was recently used by
Diamant and coworkers for surfactant micelles [67]. Here, we will briefly review a
generic model for block copolymers.
The Theory of Nyrkova and Semenov
For block copolymer micelles, a conceptual model was developed by Nyrkova and
Semenov [68] who considered the difference between the total free energy of a
micelle compared with that of unimers:
FðP; ϕ 1 Þ ¼ F micelle ðPÞ À P Á F 1 À ðP À 1Þ lnðϕ 1 Þ
(47)
The concentration of a given aggregate is thereby determined by: ϕ P ¼ ϕ 0 exp
[F(P,ϕ 0 )], where ϕ 1 is the concentration of unimers (at equilibrium ϕ 1 ¼ cmc).
In Fig. 5, the potential is plotted for some representative values (for details
see [68]).
Nyrkova and Semenov further assumed that only unimer exchange is active (i.e.,
dominant). By considering the aggregation number as representative of the mean
micellar size and the unimer concentration, the free energy landscape is reduced
and solely spanned by these two variables. In this way, the path from single chains
(unimers) to equilibrium micelles is naturally the path with minimal energy. The
local minima along the path thus represent metastable micelles. These metastable
micelles considerably slow down the growth to the equilibrium micellar state and
can, in some cases, completely deplete unimers and arrest further growth. This
makes micellization an activated process in the sense that there is a (collective)
entropic and enthalpic barrier to overcome in order to form micelles from unimers.
In the words of Nyrkova and Semenov: “The main point is that micelle formation
and their relaxation are activation processes involving collective energy barriers
which can be high enough to considerably slow down or even to virtually suppress
certain channels of relaxation.” These ideas are very similar to those presented
earlier by Aniansson and Wall [54–56]. A very important contribution is thus the
entropic barrier for micellization.
Kinetics of Block Copolymer Micelles Studied by Small-Angle Scattering Methods
79
The previous “brute force” method is rather ineffective as all “reactions” and species
are treated equally, regardless of the probability of formation or the stability. It may
very well be that most of the pathways are so improbable or so energy-costly that they
can be completely ignored.
In another approach, the micellization can be seen as a journey on a multidimensional energy landscape (phase space, i.e., a space spanned by all parameters) towards
an equilibrium state that represents a global minimum on the landscape. One can
imagine that the most probable path corresponds to the path of minimal energy,
which naturally introduces a constraint. Such an approach was recently used by
Diamant and coworkers for surfactant micelles [67]. Here, we will briefly review a
generic model for block copolymers.
The Theory of Nyrkova and Semenov
For block copolymer micelles, a conceptual model was developed by Nyrkova and
Semenov [68] who considered the difference between the total free energy of a
micelle compared with that of unimers:
FðP; ϕ 1 Þ ¼ F micelle ðPÞ À P Á F 1 À ðP À 1Þ lnðϕ 1 Þ
(47)
The concentration of a given aggregate is thereby determined by: ϕ P ¼ ϕ 0 exp
[F(P,ϕ 0 )], where ϕ 1 is the concentration of unimers (at equilibrium ϕ 1 ¼ cmc).
In Fig. 5, the potential is plotted for some representative values (for details
see [68]).
Nyrkova and Semenov further assumed that only unimer exchange is active (i.e.,
dominant). By considering the aggregation number as representative of the mean
micellar size and the unimer concentration, the free energy landscape is reduced
and solely spanned by these two variables. In this way, the path from single chains
(unimers) to equilibrium micelles is naturally the path with minimal energy. The
local minima along the path thus represent metastable micelles. These metastable
micelles considerably slow down the growth to the equilibrium micellar state and
can, in some cases, completely deplete unimers and arrest further growth. This
makes micellization an activated process in the sense that there is a (collective)
entropic and enthalpic barrier to overcome in order to form micelles from unimers.
In the words of Nyrkova and Semenov: “The main point is that micelle formation
and their relaxation are activation processes involving collective energy barriers
which can be high enough to considerably slow down or even to virtually suppress
certain channels of relaxation.” These ideas are very similar to those presented
earlier by Aniansson and Wall [54–56]. A very important contribution is thus the
entropic barrier for micellization.
Kinetics of Block Copolymer Micelles Studied by Small-Angle Scattering Methods
79