Block copolymer micelles, mean-field theory

There is a substantial body of theoretical work on micellization in block copolymers. The simplest approaches are the scaling theories, which account quite successfully for the scaling of block copolymer dimensions with length of the constituent blocks. Rather detailed mean field theories have also been developed, of which the most advanced at present is the self-consistent field theory, in its lattice and continuum guises. These theories are reviewed in depth in Chapter 3. A limited amount of work has been performed on the kinetics of micellization, although this is largely an unexplored field. Micelle formation at the liquid-air interface has been investigated experimentally, and a number of types of surface micelles have been identified. In addition, adsorption of block copolymers at liquid interfaces has attracted considerable attention. This work is also summarized in Chapter 3. 

Semianalytical mean-field theories of block copolymer micellization were formulated by Noolandi et al. [ 197] and by Leibler et al. [198]. In the approach of Noolandi et al., the micellar characteristics were obtained through a minimization of the Gibbs free energy for an isolated micelle. This was applied to PS-PB micelles, and the obtained theoretical values were in good agreement with the experimental ones. 

This chapter is organized as follows. The thermodynamics of the critical micelle concentration are considered in Section 3.2. Section 3.3 is concerned with a summary of experiments characterizing micellization in block copolymers, and tables are used to provide a summary of some of the studies from the vast literature. Theories for dilute block copolymer solutions are described in Section 3.4, including both scaling models and mean field theories. Computer simulations of block copolymer micelles are discussed in Section 3.5. Micellization of ionic block copolymers is described in Section 3.6. Several methods for the study of dynamics in block copolymer solutions are sketched in Section 3.7. Finally, Section 3.8 is concerned with adsorption of block copolymers at the liquid interface. 

Mean field theories for block copolymer micelles... 

Mean-Field Theory of Block Copolymer Micelles Boxlike Model. 81... 

In this section, we give a brief review of important selected theories for surfactant and block copolymer micelles. First, the classical thermodynamic theories covering both mean-field and scaling approaches are briefly reviewed before discussing kinetics. Classical theories for equilibrium and near-equilibrium surfactant and block copolymer micelle kinetics will be briefly reviewed before covering nonequilibrium kinetics in the final part. 

Hurter, RN., Scheutjens, J.M.H.M., and Hatton, T.A., Molecular modeling of micelle formation and solubilization in block copolymer micelles. 1. A self-consistent mean-field lattice theory. Macromolecules, 26, 5592, 1993. 

Keywords Block copolymer Disordered micelles Fluctuation effect Order-disorder transition Self-consistent mean-field theory... 

Among other approaches, a theory for intermolecular interactions in dilute block copolymer solutions was presented by Kimura and Kurata (1981). They considered the association of diblock and triblock copolymers in solvents of varying quality. The second and third virial coefficients were determined using a mean field potential based on the segmental distribution function for a polymer chain in solution. A model for micellization of block copolymers in solution, based on the thermodynamics of associating multicomponent mixtures, was presented by Gao and Eisenberg (1993). The polydispersity of the block copolymer and its influence on micellization was a particular focus of this work. For block copolymers below the cmc, a collapsed spherical conformation was assumed. Interactions of the collapsed spheres were then described by the Hamaker equation, with an interaction energy proportional to the radius of the spheres. 

The mean-field approach provides a convenient framework for the analysis of copolymer self-assembly leading to micellar structures. Combining the mean-field approach with the local electroneutrality approximation (LEA) enables us to generalize the theory for micelles with ionic coronal blocks that feature stimuli-responsive properties. 

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