On the theory of aggregation and micellization: PEO-PVP copolymer in water.

We develop a theoretical approach to micellization of the PEO-PVP block-copolymer in water. This copolymer is a weak polyelectrolyte due to protonation of VP blocks. The theory accounts for non-linear ion screening, and predicts strong position dependence of both ion concentration and the effective Debye length. We consider both the case when the local Debye length is small compared to the core radius and the case when it is large. We found that the effective (local) pH is not uniform even inside one micellar core, hence non-uniform protonation of the core with higher charge density near the surface. In many cases the core charge is concentrated in a relatively thin surface layer. Considering statistical weights of non-equilibrium micelles and their continuous evolution we show that kinetics of both formation and dissociation of typical block-copolymer or surfactant micelles can be extremely slow. Thus micelle formation at the genuine (equilibrium) critical micelle concentration (c.m.c.) is totally suppressed (involves astronomical time scales) if the micelles are big enough. An 'apparent' critical micelle concentration (c.m.c.*) is introduced to account for this effect. The apparent c.m.c.* could be much higher than the genuine equilibrium c.m.c., i.e. a significant hysteresis is inherent in these systems. We also determine the ranges of meta-stability of micelles depending on the experimental time-scales.