Flory’s mean-field theory

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],... 

ATRP was applied to the copolymerization of a monovinyl monomer and a divinyl cross-linker to study the experimental gelation behavior. The fundamental features of ATRP, including fast initiation and reversible deactivation reactions, resulted in a retarded gelation and the formation of a more homogeneous network in the ATRP process compared to gel formation in a conventional radical polymerization. The experimental gel point based on the monomer conversion in the ATRP reaction occurred later than the calculated value based on Flory-Stockmayer s mean-field theory, which was mainly ascribed to intramolecular cyclization reactions. The dependence of the experimental gel points on several parameters was systematically studied, including the ratio of cross-linker to initiator, the concentration of reagents, reactivity of vinyl groups, initiation efficiency of initiators, and polydispersity of primaiy chains. 

Based on Flory-Stockmayer s mean-field theory (F-S theory), the theoretical gel point in a system is reached when the weight-average number of cross-linking unit (v) per primary chain equals unity ... 

Fig. 2 A graphical visualization of the Flory-Huggins mean field theory. The grating positions with white circles illustrate the solvent and the gray circles illustrate the PE. It s assumed that Each field has the same size, no overlap of fields or chains, all field positions are occupied, all polymer-polymer interactions are the same (all chain parts are the same)...

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. 

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]...

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. 

As noted above, Zimm [59] was the first to point out on a theoretical consideration that A2 should depend on M. Flory and Krigbaum [81] in 1950 fonmulated a mean-field theory concluding that A2 should decrease with increasing M. But the predicted M dependence was weaker than that observed in those days. The discrepancy motivated extensive literature on theoretical and experimental studies of A2 which are well summarized in Yamaka wi s book [2]. 

Analysis (Schulz el al., I960, 1963, 1966 Lechner and Schulz, 1973) shows that no mean field theory explains the experimental data however, near the 0 point, the best results are obviously given by Orofino-Flory s (1957) theory and, in the case of good (alhermal)... 

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,... 

In the past several theoretical studies have been concerned with the mutual solubility of linear polyethylene and M-alkanes. In the course of such investigations phase behavior, or pVT relations, of pure n-alkanes has to be dealt with. In the following, three of such models will be discussed briefly Flory s Equation of State theory (EoS), the Mean-Field Lattice Gas (MFLG) model, and the Simha-Somcynsky (SS) theory. 

In the present introduction to Mesodyn we assume the reader has had some exposure to statistical thermodynamics and Flory-Huggins theory, but otherwise we do not suppose familiarity with the typical functional mathematical language of colloid and polymer physics. The introduction chapter to this book contains an extensive list of references to morphologies in block copolymer systems, and mean-field calculations, as in Hamley s book [1]. Here we focus on the work done in our own group [2,3]. General introductory books are those of de Gennes [4] and Doi and Edwards [5]. An excellent recent review paper dealing with dynamical field models is available [6]. 

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