Equation of state theory

Blends of poly(vinyl chloride) (PVC) and a-methylstyrene—acrylonitrile copolymers (a-MSAN) exhibit a miscibiUty window that stems from an LCST-type phase diagram. Figure 3 shows how the phase-separation temperature of 50% PVC blends varies with the AN content of the copolymer (96). This behavior can be described by an appropriate equation-of-state theory and interaction energy of the form given by equation 9. 

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. 

Fig. 18. Application of rate theory and equation-of-state theory to Wismer s data for ether superheated in glass. Horizontal displacements represent superheating vertical displacements represent the liquid in tension. Wismer s original Van der Waals plot was different above is the corrected form as given by Volmer (VI).

The maximum superheat which can be achieved with a nonboiling liquid is definitely pressure sensitive. This is evident in Fig. 18 and also in Fig. 28. Both plots show that the possible superheat (and therefore the possible values of ATc) decreases to zero as the critical pressure is approached. This is in agreement with the equation-of-state theory and also the nucleation-rate theory. 

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 some polymer-nonpolar solvent systems, % has been calculated as a function of concentration on the basis of the statistical-thermodynamical theory called the equation of state theory [13,14]. This semiempirical theory takes into account not only the interaction between solute and solvent, but also the characteristics of pure substances through the equations of state of each component. At present, however, we cannot apply this approach to such a complex case as the NIPA-water system. Thus, at the present stage, we must regard % as an empirical parameter to be determined through a comparison between calculated and experimental results. The empirical estimation of % for the NIPA-water system will be described in the next section. 

Chao, K. C., and Robinson, R. L. (eds.) (1986). Equations of State. Theories and Applications. Washington American Chemical Society. 

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. 

To discuss the phase stability of polymer blends in more detail one has to specify the free-energy parameter X. This can be done in terms of an equation-of-state theory [8]. Theories that take into account the compressible nature of the pure components as well as that of the mixture are called equation-of-state theories. As basic quantities characterizing the thermodynamic state of a system the reduced temperature (T), volume (V) and pressure (P) are employed and defined by... 

Fig. 7. Miscibility door for 50/50 blends of PB and SB as a function of copolymer composition. The circles refer to experimentally determined LCSTs and UCSTs. The curve was calculated using the equation-of-state theory discussed in Sect. 2.1. Miscibility occurs to the left of the curve. Inside the dashed area, solution cast films are transparent [39]...

Other unfavourable entropic contributions to the free energy exist which make the entropy of mixing negative. The TAS term in Eq. (1) is thus more unfavourable at higher temperatures. Such terms could possibly arise out of the specific interactions which themselves infer an ordering of the system. Such terms involving an empirical parameter have been included in modified versions of the Equation-of-state theory... 

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. 

Interaction parameters can also be calculated from values of the e qiansion coefficients of polymer blends using Equation-of-state theories, or from values of the isothermal compressibility of the mixture They can also be obtained from measures... 

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... 

Finally there is the problem that the theory was not intended to fuUy describe systems with a specific interaction. If a specific interaction varies with temperature then Xj2 will not be constant. An ideal theory and model would predict this behaviour. The inclusion of yet another adjustable parameter to describe the temperature dependence of Xj2 would not be desirable. In systems where the specific interaction does not vary over the temperature of study the Equation-of-state theory may give a satisfactory description of the system. This underlines the importance of experimental evidence which gives direct information about the specific interactions. 

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. 

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. 

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