Florys Mean-Field Theory

Flory pointed out that volume exclusion must cause (R ) to increase. The amount of increase can be expressed by the expansion factor a , defined as 

Consider the segments of a chain, x in number, which pervade a volume V. Let p be the segment density, which is uniform throughout the volume V. Then p = x/V within V and p = 0 outside V. The volume is related to (R ) by 

Equation (5.18) describes the average density corresponding to (R ). (Note This is the major point of mean-field theory see Chapter 9, Appendix E.) The distribution of chain vector R for the unperturbed chain is assumed to be Gaussian in nature, that is, 

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The phase separation in polymer solution is usually described in terms of the Flory mean-field theory. The free energy per unit volume of polymer solution is presented as a function of the average polymer volume fraction cp = ca, where c is the number density of the monomer units and a is the monomer unit size ... 

Incompressible Flory mean-field theory is recovered from Eqs. (5.4) and (5.5) if one assumes the following (i) no excess volume of mixing (ii) a blend composition-independent total packing fraction (iii) the... 

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... 

In critical phenomena there are always at least two independent critical exponents. The Flory mean-field theory yields only one exponent and raises the question can mean-field or self-consistent field arguments yield the missing second exponent The answer to this question is the main subject of this paper. 

Except for some exceptions, the Flory mean field theory accounts well for the observed phenomena. Experimentally measured exponents y vary between 0.59 and... 

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.

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. 

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. 

Flory-Huggins and Landau mean field theories... 

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]. 

Some decades ago Stockmayer [63] first suggested that a flexible polymer chain can transit its conformation from an expanded coil to a collapsed globule on the basis of Flory s mean - field theory [11], Since his prediction, theoretical and experimental studies of this coil - to -globule transition have been extensively conducted [31,64-68],... 

The first mean-field theories, the lattice models, are typified by the Flory-Huggins model. Numerous reviews (see, e.g., de Gennes, 1979 Billmeyer, 1982 Forsman, 1986) describe the assumptions and predictions of the theory extensions to polydisperse and multicomponent systems are summarized in Kurata s monograph (1982). The key results are reiterated here. 

Nb is the number of animals with b bonds), and making use of the aforementioned series expansion (Sect. 6), he could show that the exponent 0 collapses into the classical value, 0 = 5/2, at 8 dimensions, in support of the Lubensky field theory. The Parisi and Sourlas e = 8 -d expansion [34] gave almost identical results for 0, thus supporting dc= 8 as well. The Isaacsson-Lubensky mean-field theory [30] based on the Flory excluded volume theory and the screening concept is more intelligible, which reads... 

Within mean field theory, for a symmetric blend the excess free energy of mixing per volume is given by the Flory-Huggins expression ... 

A quantitative comparison between the mean field prediction and the Monte Carlo results is presented in Fig. 15. The main panel plots the inverse scattering intensity vs. xN. At small incompatibility, the simulation data are compatible with a linear prediction (cf. (48)). From the slope, it is possible to estimate the relation between the Flory-Huggins parameter, x, and the depth of the square well potential, e, in the simulations of the bond fluctuation model. As one approaches the critical point of the mixture, deviations between the predictions of the mean field theory and the simulations become apparent the theory cannot capture the strong universal (3D Ising-like) composition fluctuations at the critical point [64,79,80] and it underestimates the incompatibility necessary to bring about phase separation. If we fitted the behavior of composition fluctuations at criticality to the mean field prediction, we would obtain a quite different estimate for the Flory-Huggins parameter. 

Fig. 15. Inverse maximum of the collective structure factor of composition fluctuations, N/S k 0), as a function of the incompatibility, x - Symbols correspond to Monte Carlo simulations of the bond fluctuation model, the dashed curve presents the results of a finite-size scaling analysis of simulation data in the vicinity of the critical point, and the straight, solid line indicates the prediction of the Flory-Huggins theory. The critical incompatibility, XcN = 2 predicted by the Flory-Huggins theory and that obtained from Monte Carlo simulations of the bond fluctuation model M 240, N = 64, p = 1/16 and = 25.12) are indicated by arrows. The left inset compares the phase diagram obtained from simulations with the prediction of the Flory-Huggins theory (c.f. (47)). The right inset depicts the compositions at coexistence such that the mean field theory predicts them to fall onto a straight line. Prom Muller [78]...

The Flory-Huggins mean-field theory recovers the van t Hoff Law 4Fq. (1-72)] in the dilute limit (as 0—>0). At higher concentrations, the mean-field theory predicts that two-body excluded volume interactions make osmotic pressure proportional to the mean-field probability of monomer-monomer contact (0 ) ... 

K 01 0 4 lhody Tp S, t) X Xb Xc overlap volume iraction. laimensioniessi. d. overlap volume fraction for /9-solvents, [dimensionless], p. 172 semidilute-concentrated crossover volume fraction, [dimensionless], p. 180 crossover volume fraction in mean-field theory, [dimensionless], p. 181 probability for segment s to still be part of the tube at time t, [dimensionless], p. 405 Flory interaction parameter, [dimensionless], p. 142 Flory interaction parameter for a binodal, [dimensionless], p. 150 critical interaction parameter, [dimensionless], p. 152 Florv interaction oarameter for a soinodal. 

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. 

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