Large-/! limit 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. 

It must be remembered, however, that the Leibler-type mean field theory [197] is believed to be accurate for the limit of infinite chain length, N—for finite N effects of order parameter fluctuations are important and change the character of the transition from second order to first order even for symmetric composition [185,186,192,210,211]. With a self-consistent Hartree approximation that one believes to be valid for large N, Eq. (41) gets replaced by [234]... 

The one-component lattice gas of 5.3 may also be treated in the Bethe-Guggenheim approximation, which is a generalization and improvement upon, the simple mean-field theory. The latter follows from the former in Uie limit of large c and small e. The resulting mean-field theory is then necessarily thermodynamically consistent, because the Bethe-Guggenheim approximation is consistent for all c and e. In the present two-component model, in which the only interactions are infinitely strong repulsions, there is no simplification we can make beyond the Bethe-Guggenheim approximation and still retain thermodynamic consistency there is no parameter e, and, while the coordination number c is at our disposal, there is no limit to whidi we can usefully take h. 

The potential V that enters the above band equation is expected to be reduced from the LDA result by the correlation effect of C/jf. Based on a mean-field argument Varma and Yafet (1976) proposed that is scaled down by the factor 1 — Hj in the limit of very large U f. The results of a computer simulation by Blankenbecler et al. (1987) shows that large C/ff tends to increase the f electron localization and cuts down the hybridization matrix element, in qualitative agreement with the mean-field theory. 

Assuming a mean force field around each particle rather than treating local force field fluctuations (mean field theory), the multi-component approach is expected to yield meaningful results in the low q-limit, i.e., on large length scales, where local fluctuations are averaged to a mean value. 

The predictions of mean-field theory for phase equilibrium in pores are equivalent, in the large pore limit, to the thermodynamic model. However, it provides a more redistic representation of the fluid behavior as the pores become smaller. In particular, it predicts the thickening of the adsorbed layers on the pore walls, and the change from capillary condensation to pore Ailing at the critical pore size. 

It is suggestive that the narrow Kondo resonance states of individual 4f impurities will form heavy quasiparticle bands in a periodic lattice of 4f ions. A satisfactory microscopic theory of heavy-band formation has yet to be developed. The Hamiltonian of eq. (107) can be generalized to the lattice by introducing a Bose field h,- at every lattice site. However, in this model it is no longer practicable to restrict to physical states with = 1 at every site. The most successful approach so far consists in a mean-field approximation for the Bose field (Coleman 1985, 1987, Newns and Read 1987) that is valid for large N and r < It can be applied both for the impurity and the lattice model. It starts from the observation that in the limit with QJN= fixed, the rescaled... 

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