Corresponding states for polymers

SOL.35. I. Prigogine, A. Bellemans, and C. Naar-Colin, Theorem of corresponding states for polymers, J. Chem. Phys. 26, 710 (1957). 

After a brief consideration of the cell model for polymers ( 4-6) we formulate a theorem of corresponding states for polymers in 7. This theorem permits to correlate in a most remarkable way the thermod3mamic properties of polymers and to show how their size and structure determines these thermodynamic properties. We shall make a constant use of this theorem in Ch. XVTI. 

Equation (16.4.5) is far too cumbersome for actual calculations i s we shall see in 7, it is possible to introduce a theorem of corresponding states for polymers which avoids the use of a detailed cell model. 

I. Introduction 348- 2. Theorem of Corresponding States for Polymer Mixtures 349. 

Note that insoluble mixed thiophene-phosphole conjugated polymers have been characterized by MAS NMR spectroscopy <2006AGE6152>. The chemical shifts recorded in the solid state for polymers having m -phosphole, a -phosphole sulfide, or phosphole-Au(l) moieties compare well with those recorded in solution for the corresponding molecular monomers. 

Theories of Homopolymer Surface Tension. More rigorous expressions for predicting polymer surface tension have been derived from equations of state for polymers. Corresponding states principles have been employed to derive expressions for surface tensions in terms of characteristic equation of state parameters and the associated reduced temperature, pressure, and volume. One proposed expression is of the form (14)... 

Williams, Landel, and Ferry found that C and Ci were similar for many amorphous polymers with Q = 17.44 and C2 = 51.6 in the temperature range between Tg and Tg -i-100 °C. The equation is referred to as the universal WLF equation when Q and C2 assume these values. While this equation is not truly universal, it was developed from a large database for various polymers. When the equation is written in this form, it is clear that Tg serves as a corresponding state for viscoelastic behavior. A plot of log aj versus (T - T) for the data of Fig. 5.15 is shown in Fig. 5.17 here 7] is the inflection point in the modulus curve at Tg. Each DMA vendor has software available that automates the TTS operations of curve shifting, determining values of aj, and fitting the data to the WLF equation. 

To recapitulate, the Flory version of the Prigogine free-volume or corresponding-states polymer solution theory requires three pure-component parameters (p, v, T ) for each component of the solution and one binary parameter (p ) for each pair of components. 

It is often of industrial interest to be able to predict the equilibrium sorption of a gas in a molten polymer (e.g., for devolatilization of polyolefins). Unfortunately, the Prigogine-Flory corresponding-states theory is limited to applications involving relatively dense fluids 3,8). An empirical rule of thumb for the range of applicability is that the solvent should be at a temperature less than 0.85 Tp, where Tp is the absolute temperature reduced by the pure solvent critical temperature. 

Unfortunately, relatively little work has been done on the solution thermodynamics of concentrated polymer solutions with "gathering". The definitive work on the subject Is the article of Yamamoto and White (17). The corresponding-states theory of Flory (11) does not account for gathering. We therefore restrict our consideration here to multicomponent solutions where the solvents and polymer are nonpolar. For such solutions, gathering Is unlikely to occur. 

There Is a large body of experimental literature relating to polymer fractionation In liquid-liquid equilibria. In addition, numerous authors have analyzed polymer fractionation using Flory-Huggins theory. We have considered use of the corresponding states theory to model polymer fractionation for the ethylene/ polyethylene system at reactor conditions (18). Results of the... 

Many further developments can be expected in the use of corresponding-states polymer solution theory in engineering practice. However, the reliability and versatility of this method is now well demonstrated for engineering use. 

The constant K, depends,on the structure of the polymer and on the temperature. The constant A has different values below and above Mr ami is also sensitive to temperature below Af,.. For M M,. the chains are so short that the change in M strongly affects 7 Since properties are best compared at corresponding states, they should be compared at the same / When this is done. A is temperature independent and A = 1... 

The effect of an applied pressure on the UCFT has been investigated for polymer particles that are sterically stabilized by polyisobutylene and dispersed in 2-methy1-butane. It was observed that the UCFT was shifted to a higher temperature as the hydrostatic pressure applied to the system increased. There was also a qualitative correlation between the UCFT as a function of applied pressure and the 6 conditions of PIB + 2-methylbutane in (P,T) space. These results can be rationalized by considering the effect of pressure on the free volume dissimilarity contribution to the free energy of close approach of interacting particles. Application of corresponding states concepts to the theory of steric stabilization enables a qualitative prediction of the observed stability behaviour as a function of temperature and pressure. 

The possible occurrence of a back-skip of the chain for catalytic systems based on C2-symmetric metallocenes would not change the chirality of the transition state for the monomer insertion and hence would not influence the corresponding polymer stereostructure. On the contrary, for catalytic systems based on Cs-symmetric metallocenes, this phenomenon would invert the chirality of the transition state for the monomer insertion, and in fact it has been invoked to rationalize typical stereochemical defects (isolated m diads) in syndiotactic polypropylenes.9 376 60 This mechanism of formation of stereoerrors has been confirmed by their increase in polymerization runs conducted with reduced monomer concentrations.65 In fact, it is reasonable to expect an increase in the frequency of chain back-skip by reducing the monomer concentration and hence the frequency of monomer insertion. 

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