Mean-field theory of Flory

Early workers already felt that the 2 series for might be of use only in the very vicinity of 2 = 0. It was then natural that a considerable interest arose in deriving closed approximate expressions which may be used to describe aa up to larger values of 2. Various methods have been proposed, and those published by the end of the 1960s are summarized and discussed in detail by Yamakawa [2]. The earliest one is the historic mean-field theory of Flory [14], which gives... 

Though these new theoretical values of 7 are clearly different from 5, the differences are only about 10%, indicating that the mean-field theory of Flory nearly hit the target. This means that eq 1.15 is by no means an unreasonable approximation for a discussion of excluded-volume effects in polymer solutions. The problem is whether 2.60 in it is adequate or not. It is also apparent that the various closed approximate equations of third or fourth-power type derived in the 1960s now have lost their significance. 

Chemical gelation has been extensively investigated by sophisticated experiments (7) and accounted for by different theories from the original mean-field theory of Flory (2) to the concept of fractal geometry and the connectivity transition... 

Consider a single crosslinked cluster, where the crosslinks can be everywhere along the chains. The mean field theory of Flory and Stockmayer calculates for the size of the cluster the typical N law, where N is the total amount of monomers in the cluster. This law is true in ideal noninteracting systems for any kind of branched molecule or lattice animal. " The cluster of size R has then... 

Statistical thermodynamic mean-field theory of polymer solutions, first formulated independently by Flory, Huggins, and Staverman, in which the thermodynamic quantities of the solution are derived from a simple concept of combinatorial entropy of mixing and a reduced Gibbs-energy parameter, the X interaction parameter. 

Statistical network models were first developed by Flory (Flory and Rehner, 1943, Flory, 1953) and Stockmayer (1943, 1944), who developed a gelation theory (sometimes referred to as mean-field theory of network formation) that is used to determine the gel-point conversions in systems with relatively low crosslink densities, by the use of probability to determine network parameters. They developed their classical theory of network development by considering the build-up of thermoset networks following this random, percolation theory. 

Near point B the fluctuations are large and the intensity of light scattering at small angles I tends to diverge. In a mean field theory of the Flory type the divergence is of the form I (v — v,) K This is compatible with Tanaka s data. However, in this problem, since each blob is interacting only with a restricted number of other blobs (P 1), there is no reason to believe that the exponents are of a mean field type. 

Comments on Flory s Theory by de Gennes According to de Gennes, Hory did not realize the existence of the critical point c in the polymer solution. His mean-field theory of or in the dilute polymer solution leads, indeed, to a correct expression of scaling law,... 

It is well-known that the Flory mean-field theory of the polymer self-excluded volume problem yields excellent results. If R is some scalar measure of the polymer chain size, such as the radius of gyration, and N is the number of monomer units in the chain, then R scales with N as... 

The values of the Flory exponent for three dimensions is only approximate, but very close to the rigorous result of 0.5886. It is experimentally impossible to measure differences arising from such a small discrepancy in the exponent. As a result, the mean field theories of the type derived above have been very useful in organizing the conceptual framework for polymer physics. 

The canonical model for hard chains has been the tangent-hard-sphere freely jointed chain model, and several equations of state are available for fluids composed of these molecules. There are two categories of hard chain equations of state (i) those based on the mean field ideas of Flory and Huggins, and (ii) those based on the theory of associating fluids. ... 

Figure A2.5.27. The effective coexistence curve exponent P jj = d In v/d In i for a simple mixture N= 1) as a fimction of the temperature parameter i = t / (1 - t) calculated from crossover theory and compared with the corresponding curve from mean-field theory (i.e. from figure A2.5.15). Reproduced from [30], Povodyrev A A, Anisimov M A and Sengers J V 1999 Crossover Flory model for phase separation in polymer solutions Physica A 264 358, figure 3, by pennission of Elsevier Science.

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. 

To summarize, d = 4 makes a border line, the upper critical dimensionJ for the excluded volume problem. For d < 4 both the cluster expansion and the loop expansion break down term by term in the excluded volume limit. For d i> 4 the expansions are valid, the leading n- or c-dependence of the results being trivial, however. We may state that for d > 4 the random walk model or Flory-Huggins type mean field theories catch the essential physics of the problem. As will be explained more accurately in Chap. 10 the mechanism behind this is the fact that for d > 4 two nncorrelated random walks in general do not cross. 

Both the Flory-Stockmayer mean-field theory and the percolation model provide scaling relations for the divergence of static properties of the polymer species at the gelation threshold. 

In mean field theory, two parameters control the phase behavior of diblock copolymers the volume fraction of the A block /A, and the combined interaction parameter xTak- V. where Xab is the Flory-Huggins parameter that quantifies the interaction between the A and B monomers and N is the polymerization index [30], The block copolymer composition determines the microphase morphology to a great extent. For example, comparable volume fractions of block copolymer components result in lamella structure. Increasing the degree of compositional asymmetry leads to the gyroid, cylindrical, and finally, spherical phases [31]. 

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