Flory-Huggins equation-of-state

The interesting features of the photo-heating phase transition are that irradiation causes the originally continuous transition to become discontinuous and the transition temperature to be lowered. These can be explained using the Flory-Huggins equation of state ... 

The Flory-Huggins equation-of-state (22) provides simple expressions for the SCFs (Helfand, 1975c Helfand and Sapse, 1975, 1976 Scheutjens and Fleer, 1979 Hong and Noolandi, 1981 Ploehn et al, 1988),... 

Dar (Dar, 1999 Dar and Caneba, 2002) calculated Flory-Huggins interaction parameters from the Flory-Prigogine equation-of-state theory (Eq. 1.1.7), which adequately represented the experimental system. From the work of Patterson and coworkers, the following expression for the Flory-Huggins interaction parameter, X, was obtained ... 

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... 

Note 1 The Flory-Huggins theory has often been found to have utility for polymer blends, however, there are many equation-of-state theories that provide more accurate descriptions of polymer-polymer interactions. 

The viewpoint parallels that of many other theories of condensed state behavior. The van der Waals theory develops an equation of state for dense gases from the assumption that each molecule moves in an average field provided by its neighbors and that the molecules contribute additively to the pressure. The Flory-Huggins thermodynamic theory of concentrated polymer solutions proceeds similarly. Chains select configurations on a lattice partially occupied by... 

Current thermodynamic theories for polymer systems are combinations of the Flory -Huggins, Guggenheim, and Equations-of-State approaches. All of these theories make use of empirical parameters and are based on assumptions about the underlying molecular model. 

Equation-of-state theories employ characteristic volume, temperature, and pressure parameters that must be derived from volumetric data for the pure components. Owing to the availability of commercial instruments for such measurements, there is a growing data source for use in these theories (9,11,20). Like the simpler Flory-Huggins theory, these theories contain an interaction parameter that is the principal factor in determining phase behavior in blends of high molecular weight polymers. 

In systems with specific interactions random mixing cannot be assumed. Hence, the thermodynamic theories traditionally used to interpret ternary system properties, such as the Flory - Huggins formalism or the equation of state theory of FI ory, are expected not to apply to such systems. 

In Section 7.1 we discussed the thermodynamic condition for a stable mixture given in the Flory-Huggins equation (Eq. 7.1-6), where AS denotes the increase of entropy due to mixing. This equation is based on Boltzmann s principle stating that the entropy of a... 

Corresponding-states theory (Prigogine, 1957 Flory, 1970) incorporates features of the pure component properties and liquid structure in the mixture equation-of-state, producing nonzero values of AFm and contributing enthalpic and entropic terms beyond those in Flory-Huggins theory. The theory assumes that all pure components and mixtures obey the same universal equation-of-state, e.g., (Flory, 1970)... 

The form derived by Scheutjens and Fleer (1979) depends on the Flory-Huggins expression for the free energy of a solution [Eq. (22)]. The Helfand (1975b) SCF [Eq. (39)] can incorporate any equation-of-state indeed, supplementing Eq. (22) by a term accounting for adsorption energy and substituting into Eq. (39) produces... 

The Flory-Huggins forms of y and nP are simply the continuous space analogs of Eqs. (65) and (66). Ploehn and Russel (1989) use a segment equation-of-state giving greater intersegment repulsion than the Flory-Huggins... 

Sanchez and Lacombe (1976) developed an equation of state for pure fluids that was later extended to mixtures (Lacombe and Sanchez, 1976). The Sanchez-Lacombe equation of state is based on hole theory and uses a random mixing expression for the attractive energy term. Random mixing means that the composition everywhere in the solution is equal to the overall composition, i.e., there are no local composition effects. Hole theory differs from the lattice model used in the Flory-Huggins theory because here the density of the mixture is allowed to vary by increasing the fraction of holes in the lattice. In the Flory-Huggins treatment every site is occupied by a solvent molecule or polymer segment. The Sanchez-Lacombe equation of state takes the form... 

Due to the shortcomings of the classical Flory-Huggins lattice model, Flory and co-workers abandoned the whole concept of a lattice, and characterized each pure component by three equation of state parameters, V, T and P which may be evaluated from the pure component data, density, thermal expansion coefficient and... 

It should be pointed out that since I.G.C. measures the total free energy of the interaction, any value of the Flory-Huggins interaction parameter which is derived will be a total value including combinatorial and residual interaction parameters as well as any residual entropy contributions. Similarly when using Equation-of-state theory one will obtain Xj2 rather than Xj. The interactions are measured at high polymer concentration and are therefore of more direct relevance to interactions in the bulk state but this does not remove problems associated with the disruption of intereactions in a blend by a third component. 

As well as the above quoted studies this method has also been used to study the interaction between poly(vinylidene fluoride) and poly(methyl methacrylate) , between poly(ethylene oxide) and the hydroxy ether of bisphenol A ° , and between poly(ethylene oxide) and a poly(ether sulphone) The above equations have also been reformulated in terms of the equation-of-state theory to obtain the interaction energy, which is concentration independent rather than the Flory-Huggins X parameter which is composition dependent. 

The thermodynamic definition of the spinodal, binodal and critical point were given earlier by Eqs. (9), (7) and (8) respectively. The variation of AG with temperature and composition and the resulting phase diagram for a UCST behaviour were illustrated in Fig. 1. It is well known that the classical Flory-Huggins theory is incapable of predicting an LCST phase boundary. If has, however, been used by several authors to deal with ternary phase diagrams Other workers have extensively used a modified version of the classical model to explain binary UCST or ternary phase boundaries The more advanced equation-of-state theories, such as the theory... 

This equation is not particularly useful in practice, since it is difficult to quantify the relationship between concentration and activity. The Flory-Huggins theory does not work well with the cross-linked semi-crystalline polymers that comprise an important class of pervaporation membranes. Neel (in Noble and Stern, op. cit., pp. 169-176) reviews modifications of the Stefan-Maxwell approach and other equations of state appropriate for the process. 

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