Flory’s equation of state theory

Flory s equation-of-state theory is quite complex, requiring as it does the manipulation of many interrelated parameters. Sanchez (Sanchez and Larcombe, 1978 Sanchez, 1980) has therefore developed a somewhat simpler theory based on a lattice fluid or Ising model. This approach also proceeds via... 

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. 

McMaster " demonstrated that LOST behavior arises from free volume effects. In particular, if the pure component thermal expansion coefficients are sufficiently different, LCST behavior becomes more likely. It should be noted again that the normal entropic effects are very small. By using Flory s equation-of-state thermodynamics, it can be demonstrated that LCST behavior should generally be expected for high-molecular-weight mixtures. Although the detailed thermodynamic arguments are beyond the scope of this chapter, the theory has been reviewed recently. " ... 

Parameters related to differences in size and shape and respectively change in internal energy of mixing Entropy correction terms Parameter related to average interaction energy Internal energy of the system Flory s equation of state Flory-Huggins Staverman theory... 

A number of analytical off-lattice approaches are based on the extension of various simple one-component analytical models (such as Flory-Huggins theory, Flory s equation of state model, or Guggenheim theory) to multicomponent systems by the use of some ad hoc combining rules [5-8,15,16]. These approaches likewise neglect the influence of chain connectivity and monomer... 

H-F theory was extended to ternary systems comprising poly disperse polymer by Utracki [1962]. Starting in the early 1960 s considerable effort was made to develop what become known as the equation of state theories [Flory et al., 1964 Eichinger... 

In response to the developing field of polymer blends, two new theories of polymer mixing were developed. The first was the Flory equation of state theory (19,20), and the second was Sanchez s lattice fluid theory (21 4). These theories were expressed in terms of the reduced temperature, T = TIT, ... 

Flory (11) improved the notation and form of Prigogine s expressions, and it is essentially the Flory form of Prigogine s free-volume theory that is of most use for design purposes. The Flory work (11) leads to an equation of state which obeys the corresponding-states principle ... 

Let me close with two more familiar examples. We all know that Flory s theory of polymeric solutions is not in agreement with experiment. However, the insight on polymer compatibility gained from the theory was enormous. The same thing may be said about the van der Waals equation of state. We all know that this equation is very approximate, but something remains in our intuition that is extremely helpful. 

Also, the original Hildebrand approach has been refined to take into account the contribution of polar groups and hydrogen bsolubility parameters. These mndifications of the Flory-Huggins theory and of the solubility parameter concept have made these methods an even more useful tool in the description of solutions, especially of mixtures containing polymer compounds. A comprehensive treatment of these extensions of Flory-Huggins and Hildebrand s theories, as well as the new equation of state approach of Flory (1965), bns re ntly been published (Shinoda, 1978 Olahisi et al 1979). 

By generalizing Flory s dimer theory. Hall and coworkers [22, 23] obtained the GF-D equation of state, which is as follows ... 

About ten years after Flory s development of an equation of state for polymer systems, one began to apply methods of thermodynamic perturbation theory to calculate the thermo-... 

The perturbed-hard-ehain (PHC) theory developed by Prausnitz and coworkers in the late 1970s was the first successful application of thermodynamic perturbation theory to polymer systems. Sinee Wertheim s perturbation theory of polymerization was formulated about 10 years later, PHC theory combines results fi om hard-sphere equations of simple liquids with the eoneept of density-dependent external degrees of fi eedom in the Prigogine-Flory-Patterson model for taking into account the chain character of real polymeric fluids. For the hard-sphere reference equation the result derived by Carnahan and Starling was applied, as this expression is a good approximation for low-molecular hard-sphere fluids. For the attractive perturbation term, a modified Alder s fourth-order perturbation result for square-well fluids was chosen. Its constants were refitted to the thermodynamic equilibrium data of pure methane. The final equation of state reads ... 

The above-mentioned deficiencies of the Flory-Huggins theory can be alleviated, in part, by using the local-composition concept based on Guggenheim s quasichemical theory for the random mixing assumption and replacing lattice theory with an equation-of-state model (Prausnitz et al., 1986). More sophisticated models are available, such as the perturbed hard sphere chain (PHSC) and the statistical associating fluid theory (SAFT) (Caneba and Shi, 2002), but they are too mathematically sophisticated that they are impractical for subsequent computational efforts. 

Wichert, J. M. Gulati, H. S. Hall, C. K. J. (1996). Binary hard chain mixtures. I. Generalized Flory equations of state. /. Chem. Phys., Vol. 105, 7669-7682 Woodward, C. E. Forsman, J. (2008). Density functional theory for polymer fluids with molecular weight polydispersity. Phys. Rev. Lett., Vol. 100,098301 Woodward, C. E. (1991). A density functional theory for polymers Application to hard chain-hard sphere mixtures in slitlike pores. /. Chem. Phys., Vol. 94,3183-3191 Woodward, C. E. Forsman, J. (2009). Interactions between surfaces in polydisperse semiflexible polymer solutions. MacromoL, Vol. 42, 7563-7570 Woodward, C. E. Yethiraj A. (1994). Density functional theory for inhomogeneous polymer solutions. /, Chem. Phys., Vol. 100,3181-3186... 

---