Equation of State Theories for Polymers

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-... 

Equation of State Theories for Polymers At constant temperature. Equation (2.76) becomes... 

This is a three-part book with the first part devoted to polymer blends, the second to copolymers and glass transition tanperatme and to reversible polymerization. Separate chapters are devoted to blends Chapter 1, Introduction to Polymer Blends Chapter 2, Equations of State Theories for polymers Chapter 3, Binary Interaction Model Chapter 4, Keesome Forces and Group Solubility Parameter Approach Chapter 5, Phase Behavior Chapter 6, Partially Miscible Blends. The second group of chapters discusses copolymers Chapter 7, Polymer Nanocomposites Chapter 8, Polymer Alloys Chapter 9, Binary Diffusion in Polymer Blends Chapter 10, Copolymer Composition Chapter 11, Sequence Distribution of Copolymers Chapter 12, Reversible Polymerization. 

The first successful theoretical approach of an equation of state model for polymer solutions was the Prigogine-Flory-Patterson theory. It became popular in the version by Flory, Orwoll and Vrij and is a van-der-Waals-like theory based on the corresponding-states principle. Details of its derivation can be found in numerous papers and books and need not be repeated here. The equation of state is usually expressed in reduced form and reads ... 

This equation is not particularly useful in practice, since it is difficult to quantify the relationship between concentration and ac tivity. The Floiy-Huggins theory does not work well with the cross-linked semi-ciystaUine 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. 

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 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. 

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 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. 

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. 

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 ... 

From the historical point of view and also from the number of applications in the literature, the common method is to use activity coefficients for the liquid phase, i.e., the polymer solution, and a separate equation-of-state for the solvent vapor phase, in many cases the truncated virial equation of state as for the data reduction of experimental measurements explained above. To this group of theories and models also free-volume models and lattice-fluid models will be added in this paper because they are usually applied within this approach. The approach where fugacity coefficients are calculated from one equation of state for both phases was applied to polymer solutions more recently, but it is the more promising method if one has to extrapolate over larger temperature and pressure ranges. 

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