Theoretical Aspects of Micellisation

Numerous theories, models and mathematical approaches have been developed over the years in order to describe the micellisation process and the dependence of fundamental structural parameters of the micelles, like cmc, aggregation number (Nagg), overall size (Rm), core radius (Rc) and corona thickness (L), on the molecular characteristics of the block copolymer, with respect to the degrees of polymerisation of the constituent blocks (Na and N ), as well as the Flory-Huggins interaction parameters x between the blocks and between the blocks and the solvent. Some of these approaches use the minimisation of the total free energy of the micellar system so as to extract relations between the copolymer and micelle features, while others are based on the scaling concept of Alexander-de Gennes and 

In scaling theories, two limiting cases have to be considered for a monodisperse AB diblock copolymer in a selective solvent for the A block [82], These are the hairy or starlike micelles, with Nj N and accordingly L Rc, and the crew-cut micelles for which Va and Rc L (see Fig. 2). For the case of hairy micelles, based on the star polymer theory of Daoud and Cotton [84], the following scaling relations for N gg and L have been derived  

Zhulina and Birshtein [85] and Halperin [183] have obtained similar results for 

For crew-cut micelles, the theories of Alexander and de Gennes for polymer brushes have been used, assuming micelles with uniformly stretched chains in the core and the corresponding scaling relations are  

However, scaling theories are not able to predict finite chain effects and polymer-solvent interactions, since they are restricted to long polymer chains in good solvents. More detailed mean-field calculations and molecular simulations are needed in order to complement these models. 

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