Flory-Huggins theory mean-field approximation

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. 

A wide variety of theories have been developed for polymer solutions over the later half of the last century. Among them, lattice model is still a convenient starting point. The most widely used and best known is the Flory-Huggins lattice theory (Flory, 1941 Huggins, 1941) based on a mean-field approach. However, it is known that a mean-field approximation cannot correctly describe the coexistence curves near the critical point (Fisher, 1967 Heller, 1967 Sengers and Sengers, 1978). The lattice cluster theory (LCT) developed by Freed and coworkers (Freed, 1985 Pesci and Freed, 1989 Madden et al., 1990 Dudowicz and Freed, 1990 Dudowicz et al., 1990 Dudowicz and Freed, 1992) in 1990s was a landmark. 

For polymers, x is usually defined on a per monomer basis or on the basis of a reference volume of order one monomer in size. However, x is usually not computed from formulas for van der Waals interactions, but is adjusted to obtain the best agreement between the Flory-Huggins theory and experimental data on the scattering or phase behavior of mixtures (Balsara 1996). In this fitting process, inaccuracies and ambiguities in the lattice model, as well as in the mean-field approximations used to obtain Eq. (2-28), are papered over, and contributions to the free energy from sources other than simple van der Waals interactions get lumped into the x parameter. The temperature dependences of x for polymeric mixtures are often fit to... 

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. 

In this section, we mention very briefly some recent theoretical developments, which go far beyond the simple Flory-Huggins theory. As was emphasized above, the Flory-Huggins theory suffers from two basic defects (i) Using a lattice model where polymers are represented as self-avoiding walks is a crude approximation, which neglects the disparity in size and shape of subunits of the two types of chain in a polymer blend, as well as packing constraints, specific interactions etc. (ii) Even within the realm of a lattice model, the statistical mechanics (involving approximations beyond the mean field approximation) is far too crude. 

Distinctly from the mean field approximation in the Flory-Huggins theory (sec Chapter 3), the lines of crossover (i.e. the lines representing the change of the conformational mode of macromolecules) arc readily plottcxl on this diagram in the single-pha.se region (solution). 

Statistical thermodynamic theories provide a powerful tool to bridge between the microscopic chemical structures and the macroscopic properties. Lattice models have been widely used to describe the solution systems (Prigogine 1957). Chang (1939) and Meyer (1939) reported the earliest work related with the lattice model of polymer solution. The lattice model was then successfully established by Flory (1941, 1942) and Huggins (1942) to deal with the solutions of flexible polymers by using a mean-field approximation, and to derive the well-known Flory-Huggins equation. 

Polymer blends. Although it is well known that the mean-field Flory-Huggins theory of the thermodynamics of polymer systems is not a rigorously accurate description, especially for polymer blends, it is sufficiently valid that its use does not incur serious errors. Furthermore, de Gennes [16] used the mean-field random-phase approximation to obtain the scattering law for a binary polymer blend as ... 

The theory presented above for dilute polymer solutions is based upon the Flory-Huggins Equation (3.22) which strictly is not valid for such solutions because of the mean-field approximation. Nevertheless, whilst Equations (3.38) and (3.45) do not accurately predict fix -/ ), they are of the correct functional form, i.e. the relationships... 

When one fits the Flory Huggins theory to experiment [7], nontrivial dependence of Flory Huggins interaction parameters on temperature and volume fractions also result, but might have other reasons than those noted above in particular, it is important to take into account the disparity between size and shape of effective monomers in a blend, and also the effects of variable chain stiffiiess and persistence length [25, 26]). To some extent, such effects can be accounted for by the lattice cluster theories [27 30], but the latter still invokes the mean-field approximations, with the shortcomings noted above. In the present article, we shall focus on another aspect that becomes important for the equation of state for polymer materials containing solvent pressure is an important control parameter, and for a sufficiently accurate description of the equation of state it clearly does not suffice to treat the solvent molecules as vacant sites of a lattice model. In most cases it would be better to use completely different starting points in terms of off-lattice models. 

Being a mean-field approach, the Flory-Huggins theory neglects fluctuations which are important in the vicinity of the critical point or the spinodal (cf. Sect. 2), i.e., just where the square gradient approximation is useful. 

In spite of the deviation from the mean-field type behavior, the coexistence curve of the blend B was well described by the Flory-Huggins theory under the quasibinary approximation with the interaction parameter x that was consistent with those estimated by the molecular weight dependence of cloud points. 

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