Micelles mean-field theories

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. 

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. 

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... 

A simple mean field theory for micelle formation by a diblock copolymer in a homopolymeric solvent was developed by Leibler et al. (1983). This model enables the calculation of the size and number of chains in a micelle and its free energy of formation. The fraction of copolymer chains aggregating into micelles can also be obtained. A cmc was found for low copolymer contents even for weak incompatibilities between components. Leibler et al. (1983) emphasize that fora finite aggregation number p, the cmc is a region rather than a well-defined concentration and some arbitrariness is involved in its definition. 

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

Scaling approaches to predicting structural dependencies for micelles are useful to reveal power law behavior, but lack the precision with respect to numerical coefficients. The mean-field theory for self-assembly, as discussed above, unravels the general trends for these complex micellar systems, but implements major approximations. In particular, the boxlike model neglects gradients in the local densities in both the core and corona regions. A more advanced nonlocal mean field model... 

In all models (as for mean-field theories), the reductimi of interfacial area upon micelUzation is considered to be the driving force for micellization. Thus, the interfacial tension is an important parameter. The interfacial free energy per chain of a spherical micelle (Eq. 8) can be written as ... 

Figure 10 Apparent molecular weight Mapp as a function of the concentration c of giant surfactant micelles of sodium sulfbpropyl octadecyl maleate in brine, obtained by means of light scattering at a temperature of 323 K. Triangles experimental data of Berlepsch and coworkers [54], Line theory for self-assembled rigid rods fitted to the data [55]. The link free energy obtained from the fit amounts to G 26 Ab T, which is equivalent to about 65 kJ mor. The micelles have an estimated persistence length of 120 nm and a diameter of 6 nm. Also indicated by the dashed lines are the predictions of mean-field theory in the dilute and semidilute regimes [56].

Regarding semianalytical mean-field theories, Noolandi and Hong [78] and Leibler et al. [79] derived the micellar characteristics by minimising the free energy of both isolated micelles and the whole micellar system. A further development was achieved by Nagarajan and Ganesh [80], who took into account the molar volumes of the solvent and blocks A and B, the interfacial tension between the B block and the solvent and the interaction parameter between the A block and the solvent. This... 

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. 

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