Florys equation-of-state theory

Unusual Properties of PDMS. Some of the unusual physical properties exhibited by PDMS are summarized in List I. Atypically low values are exhibited for the characteristic pressure (a corrected internal pressure, which is much used in the study of liquids) (37), the bulk viscosity i, and the temperature coeflScient of y (4). Also, entropies of dilution and excess volumes on mixing PDMS with solvents are much lower than can be accounted for by the Flory equation of state theory (37). Finally, as has already been mentioned, PDMS has a surprisingly high permeability. 

The liquid may be a good or poor solvent for the polymer. For this type of system a theoretical relation can be obtained for K by applying the Flory equation of state theory ( -i) or lattice fluid theory (7-10) of solutions. An important prerequisite for the application of these theories is for the polymer to behave as an equlibrium liquid. This condition is generally valid for a lightly crosslinked, amorphous polymer above its Tg or for the amorphous component of a semi-crystalline polymer above its Tg. 

Roe and Zin analyzed the value of the polymer-polymer interaction energy density and its temperature dependence obtained in their work. Starting from the Flory equation-of-state theory they derived the following expression for A ... 

In the Flory equation of state theory [12, 16—21] the parameter of interaction between the volatile substance and the binary stationary phase is written as [26] ... 

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, ... 

T, P, and v can be determined from the Flory equation of state theory. 

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. 

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. 

All of these three effects will be discussed further in later parts of this review. At this stage we will say more concerning advanced theories of polymer miscibility starting with the Equation-of-state theory of Flory and his co-workers. 

Many variations on these and similar theories have been developed and their relative merits have been discussed We beUeve that most theories suffer because they do not address themselves to the important problem of the specific interaction directly. In our own work we have used the Equation-of-state theory of Flory and coworkers. Although it cannot fully describe systems with specific interactions it does have the merits of being moderately easy to use. of allowing volume changes on mixing, and of using parameters which are mostly obtainable either by experimental measurement or calculation. 

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... 

It is possible to simulate the spinodal curves of the phase diagram of polymer pairs using the Equation-of-state theory developed by Flory and co-workers. It is only, however, possible to do this using the adjustable non-combinatorial entropy parameter, Qjj. Another problem arises in the choice of a value for the interaction parameter Xjj. This is introduced into the theory as a temperature independent constant whereas we know that in many cases the heat of mixing, and hence is strongly temperature dependent. The problem arises because Xj was never intended to describe the interaction between two polymers which are dominated by a temperature dependent specific interaction. 

The lattice fluid equation-of-state theory for polymers, polymer solutions, and polymer mixtures is a useful tool which can provide information on equa-tion-of-state properties, and also allows prediction of surface tension of polymers, phase stability of polymer blends, etc. [17-20]. The theory uses empty lattice sites to account for free volume, and therefore one may treat volume changes upon mixing, which are not possible in the Flory-Huggins theory. As a result, lower critical solution temperature (LCST) behaviors can, in principle, be described in polymer systems which interact chiefly through dispersion forces [17]. The equation-of-state theory involves characteristic parameters, p, v, and T, which have to be determined from experimental data. The least-squares fitting of density data as a function of temperature and pressure yields a set of parameters which best represent the data over the temperature and pressure ranges considered [21]. The method,however,requires tedious experiments to deter-... 

In order to explain phase separation on heating, I.e., LCST behavior, the effect of volume changes on mixing must be considered. This effect Is described by equation-of-state theories such as that developed by Flory and co-workers (30). The free volume contributions to the free energy are unfavorable and increase with temperature. 

Predicting blend miscibility correctly is, however, often considerably more challenging than one might guess by looking at the equations given above or even their somewhat more refined versions. Flory-Huggins interaction parameters, their more elaborate versions, and alternative methods such as the equation-of-state theories [9] discussed in Section 3.E, all provide correct predictions in many cases, but unfortunately provide incorrect predictions in many other cases. 

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... 

Free volume approach to the combinatorial entropy The combinatorial entropy of mixing can be more readily derived by a free volume approach which renders the assumptions inherent in the Flory-Huggins theory more transparently obvious. Anticipating what is to be presented in Section 3.3, vis-d-vis the equation-of-state theory, we present a brief account of this alternative derivation. 

Flory (1965 1970) has developed an elaborate equation-of-state theory that endeavours to incorporate the free volume contribution into the various thermodynamic functions. Because of its inherent complexity, only the barest outline of the theory will be adumbrated at this point. Patterson (1968) has developed a separate, but in many ways similar, corresponding states theory. Both theories are in many respects only satisfactory at the semi-quantitative level for nonpolar polymer molecules in apolar solvents. 

Fig. 3.10. Comparison of the predictions of the equation-of-state theory (full line) with the results ( ) of experiment for the concentration dependence of the interaction parameter of poly(isobutylene) in benzene at 2S °C (after Eichinger and Flory, 1968).

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... 

One further difficulty not touched upon in the foregoing discussion is the absence of a truly quantitative theory describing polymer solution thermodynamics. Even a second generation theory, such as the equation-of-state theory, probably only represents a qualitative or, at best, a semi-quantitative theory of polymer solution thermodynamics (Casassa, 1976). In the absence of a fully quantitative theory, it seems justifiable to make do with the classical Flory-Huggins theory, provided that the cracks that have appeared in its superstructure are papered over. These include using the concentration dependent interaction parameter % that is determined experimentally. Most of the theories of steric stabilization that have been developed to-date have unfortunately been based upon a concentration independent interaction parameter (see Table 10.1), although there are some exceptions (see, e.g. Evans and Napper, 1977). 

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