Mean field scaling theory

The consequences of the excluded volume eflFect are summarized below within the mean field, scaling and renormalization group theories. 

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

Failure of mean-field theories near the critical point has stimulated the development of several alternative theories. One of the more promising is that parameters near a critical point obey scaling laws. 

In practice, the scheme as explained above is not implemented. The consecutive generation of all possible chain conformations is a very expensive step. The reason for this is that there are of the order of ZN number of conformations, where Z is the lattice coordination number. A clever trick is to generate a subset of all possible conformations and to use this set in the SCF scheme. This approach is known in the literature as the single-chain mean-field theory, and has found many applications in surfactant and polymeric systems [96]. The important property of these calculations is that intramolecular excluded-volume correlations are rather accurately accounted for. The intermolecular excluded-volume correlations are of course treated on the mean-field level. The CPU time scales with the size of the set of conformations used. One of the obvious problems of this method is that one should make sure that the relevant conformations are included in the set. Typically, the set of conformations is very large, and, as a consequence, the method remains extremely CPU intensive. 

Scaling theories are restricted to long polymer chains in good solvents and do not include finite chain effects and polymer-solvent interactions. These models should be complemented by more detailed mean-field calculations and molecular simulations. 

Fig. 2. Bead density profiles. Solid line Brushes, mean-field and scaling theory (step function) dashed-dotted line generalization of the Milner et al. theory for brushes in the theta state dashed-double dotted line Milner et al. theory for brushes (EV chains) dashed line EV stars dotted line EV combs. Variable r is scaled to give zero bead density for the smooth curves of brushes at r=l. The brush curves are normalized to show equal areas (same number of units). The comb and star densities are arbitrarily normalized to show similar bead density per volume unit as the step function and EV curves for brushes at the value ol r where these curves intercept...

The second question concerns one particular aspect of general applicability of the simple mean field equations outlined above as opposed to more sophisticated statistical mechanical descriptions. In particular, the equilibrium Poisson-Boltzmann equation (1.24) is often used in treatments of some very short-scale phenomena, e.g., in the theory of polyelectrolytes, with a typical length scale below a few tens of angstroms (1A = 10-8 cm). On the other hand, the Poisson-Boltzmann equation implicitly relies on the assumption of a pointlike ion. Thus a natural question to ask is whether (1.24) could be generalized in a simple manner so as to account for a finite ionic size. The answer to this question is positive, with several mean field modifications of the Poisson-Boltzmann equation to be found in [5], [6] and references therein. Another ultimately simple naive recipe is outlined below. 

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. 

From the theoretical viewpoint, much of the phase behaviour of blends containing block copolymers has been anticipated or accounted for. The primary approaches consist of theories based on polymer brushes (in this case block copolymer chains segregated to an interface), Flory-Huggins or random phase approximation mean field theories and the self-consistent mean field theory. The latter has an unsurpassed predictive capability but requires intensive numerical computations, and does not lead itself to intuitive relationships such as scaling laws. 

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. 

---