Flory-Huggins model theory

Since the Flory-Huggins theory provides us with an analytical expression for AG , in Eq. (8.44), it is not difficult to carry out the differentiations indicated above to consider the critical point for miscibility in terms of the Flory-Huggins model. While not difficult, the mathematical manipulations do take up too much space to include them in detail. Accordingly, we indicate only some intermediate points in the derivation. We begin by recalling that (bAGj Ibn ) j -A/ii, so by differentiating Eq. (8.44) with respect to either Ni or N2, we obtain... 

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. 

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. 

In recent years, there have been substantial efforts to measure the composition dependence of the interaction parameter X12 (Bates et a/., 1988 Han et al, 1988) - see Fig. 4.4, p. 79 - and thus to refine our understanding of this theory. The original and customary derivations of the Flory-Huggins model introduced an interaction parameter that was considered to be independent oi 4>i but it has long been recognized that this is not the typical case (Flory, 1970). We depart here from the customary derivations to lay a groundwork for a basic reconsideration of that composition dependence. For example, we don t insist here that X12 is the traditional Rory-Huggins i2 parameter. 

Our discussion here explores active connections between the potential distribution theorem (PDT) and the theory of polymer solutions. In Chapter 4 we have already derived the Flory-Huggins model in broad form, and discussed its basis in a van der Waals model of solution thermodynamics. That derivation highlighted the origins of composition, temperature, and pressure effects on the Flory-Huggins interaction parameter. We recall that this theory is based upon a van der Waals treatment of solutions with the additional assumptions of zero volume of mixing and more technical approximations such as Eq. (4.45), p. 81. Considering a system of a polymer (p) of polymerization index M dissolved in a solvent (s), the Rory-Huggins model is... 

A comparison of freezing temperature lowering via the mechanisms for the disruption of surface and for the Flory-Huggins model is shown in Fig. 34. Our result, Eq. (27), is similar to that obtained by Solms and Rijke (1971). In their case a polymer system where x < i is used to test the results of the theory. 

The name of this model is an acronym of universal quasichemical, the name of the theory used to drive it. Like the Flory-Huggins model, UNIQUAC separates nonideal contributions to the excess Gibbs free energy into a combinatorial and a residual term ... 

Polymers dissolved in mixed solvents show the phenomenon of Preferential Adsorption. Experimentally, the preferential adsorption eoeffieient, X, is determined. Xis the volume of one of the liquids sorbed in exeess by the polymer (per unit mass of polymer). In general, the Flory-Huggins model of polymer solutions is used to deseribe the Preferential Adsorption. More reeently, equation of state theories have been applied. ... 

The infortnation provided in this chapter can be divided into four parts 1. introduction, 2. thermodynamic theories of polymer blends, 3. characteristic thermodynamic parameters for polymer blends, and 4. experimental methods. The introduction presents the basic principles of the classical equilibrium thermodynamics, describes behavior of the single-component materials, and then focuses on the two-component systems solutions and polymer blends. The main focus of the second part is on the theories (and experimental parameters related to them) for the thermodynamic behavior of polymer blends. Several theoretical approaches are presented, starting with the classical Flory-Huggins lattice theory and, those evolving from it, solubility parameter and analog calorimetry approaches. Also, equation of state (EoS) types of theories were summarized. Finally, descriptions based on the atomistic considerations, in particular the polymer reference interaction site model (PRISM), were briefly outlined. 

The calculation of the entropy of mixing, ASm, of polymers in solution was carried out by Flory (3) and Huggins (4) in 1942. The Flory-Huggins (FH) theory, based on the rigid lattice model developed for solutions of small molecules, leads to the following expression for the free energy of mixing ... 

Recently, phase behaviour of mixtures consisting of a polydisperse polymer (polystyrene) and nematic liquid crystals (p-ethoxy-benzylidene-p-n-butylani-line) was calculated and determined experimentally. The former used a semi-empirical model based on the extended Flory-Huggins model in the framework of continuous thermodynamics and predicted the nematic-isotropic transition. The model was improved with a modified double-lattice model including Maier-Saupe theory for anisotropic ordering and able to describe isotropic mixing. ... 

Like the Flory-Huggins model, the Sanchez-Lacombe lattice fluid theory is based on the assumption that segments of solvent molecules and polymer molecules occupy the lattice sites of a rigid lattice, but vacant lattice sites are also allowed. The number of vacant lattice sites, and as a consequence the total number of lattice sites, are pressure-dependent, and in this way compressibility is introduced. 

The dissolution of polymer blends is generally modelled by the Flory-Huggins solution theory and the Gibbs free energy between the two components can be calculated as AG = AT-TAS [9-11, 16, 21-23, 32-53, 55-75]. 

In the preceding chapter we have, based on the Flory-Huggins theory, discussed the basis for the phase behavior of polymer blends. Miscible polymer blends and polymer solutions have, even in the mixed one-phase system, spatial variations in the polymer concentration. These concentration fluctuations reflect the thermodynamic parameters of the free energy, as described in the Flory-Huggins model. 

Several other equation-of-state models have been proposed The lattice-fluid theory of Sanchez and Lacombe (1978), the gas-lattice model proposed by Koningsveld (1987), the strong interaction model proposed by Walker and Vause (1982), and the group contribution theory proposed by Holten-Anderson (1992), etc. These theories are reviewed by Miles and Rostami (1992) and Boyd and Phillips (1993). The lattice-fluid theory of Sanchez and Lacombe has similarities with the Flory-Huggins theory. It deals with a lattice, but with the difference from the Flory—Huggins model in that it allows vacancies in the lattice. The lattice is compressible. This theory is capable of describing both UCST and LCST behaviour. 

There have been sophisticated calculations of x these systems. In the Flory-Huggins model x is independent of molecular weight and composition. In reality, this parameter has been shown, by scattering experiments and by theoretical calculations to be a function of Af, T and 0. 3,40,49 Theoretical attempts, which are beyond the scope of the mean field predictions of the Flory-Huggins approach which applies stri ct to incompressible systems, have been made to address these questions. 3,4o 49 The theories of Bates and Muthukumar 3 and of Schweizer and Curro °> both have predictions which may be written in the following form... 

The Flory-Huggins (FH) theory is widely u.sed to analyze experimental data. In the FH model the polymer molecules are assumed to occupy the sites of a regular lattice with one monomer per site. For binary blends the FH estimate for the free energy of mixing, AAmi. is ... 

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