Mean field approximation, high-concentration

The deficiencies of the Flory-Huggins theory result from the limitations both of the model and of the assumptions employed in its derivation. Thus, the use of a single type of lattice for pure solvent, pure polymer and their mixtures is clearly unrealistic since it requires that there is no volume change upon mixing. The method used in the model to calculate the total number of possible conformations of a polymer molecule in the lattice is also unrealistic since it does not exclude self-intersections of the chain. Moreover, the use of a mean-field approximation to facilitate this calculation, whereby it is assumed that the segments of the previously added polymer molecules are distributed uniformly in the lattice, is satisfactory only when the volume fraction (f>2 of polymer is high, as in relatively concentrated polymer solutions. 

Interstitial impurities (e.g., hydrogen of helium atoms) can easily move over the interstitial cavities in LRC but comparatively large size partial vacancies on the boundaries are traps for small-size atoms. It was established in a series of experiments that amorphous alloys contain traps for hydrogen and helium atoms and that the concentration of these traps is rather high [6.56]. The concentration dependence of interstitial diffusivity in disordered structures was considered phenomenologically [6.56] and by mean-field approximation [6.57],... 

Section IV is devoted to excitons in a disordered lattice. In the first subsection, restricted to the 2D radiant exciton, we study how the coherent emission is hampered by such disorder as thermal fluctuation, static disorder, or surface annihilation by surface-molecule photodimerization. A sharp transition is shown to take place between coherent emission at low temperature (or weak extended disorder) and incoherent emission of small excitonic coherence domains at high temperature (strong extended disorder). Whereas a mean-field theory correctly deals with the long-range forces involved in emission, these approximations are reviewed and tested on a simple model case the nondipolar triplet naphthalene exciton. The very strong disorder then makes the inclusion of aggregates in the theory compulsory. From all this study, our conclusion is that an effective-medium theory needs an effective interaction as well as an effective potential, as shown by the comparison of our theoretical results with exact numerical calculations, with very satisfactory agreement at all concentrations. Lastly, the 3D case of a dipolar exciton with disorder is discussed qualitatively. 

Cellular environments can be modeled as a first approximation as aqueous solvent, corresponding to an implicit solvent model with e = 80. The high concentration of biomolecules and co-solvents inside cells generally attenuates the dielectric response of the remaining water molecules [17]. In a mean field approach, this... 

The approximation becomes poor for the systems in which concentration fluctuations are large. For instance, in dilute polymer solutions, monomers distribute unevenly inside and outside the region occupied by the polymer chains. The spatial variation of the concentration is so high that the mean field assumption cannot be expected to hold. Also, in the region near the critical point of phase separation, where the concentration fluctuation is large, the mean field picture breaks down. 

The first theory of polyelectrolyte complex formation was proposed by Voom and Overbeek [14, 15]. This mean field model was used to describe the binodal compositions, the water content and the critical salt concentration as a function of the polymer chain length. This theoretical description uses the Debye-Hiickel approximation the approximations within the derivation of the electrostatic interaction free energy are therefore only valid at low charge densities. The correlation effects at high concentrations of salt and monomeric units are neglected, and ion pairing effects such as counterion condensation are not taken into account. Despite these limitations, the experimental results could be described reasonably well [13]. 

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