Polymeric solutions theory

Configurational energy for clathrates, 12 Configurations, superposition of, 258 Conformal solution theory, 137 Coordination polymerization, 148, 162, 170... 

The development of a scientific understanding of diffusion in liquid-phase polymeric systems has been largely due to Duda et al. (1982), Ju et al. (1981), and Vrentas and Duda (1977a,b, 1979) whose work in this area has been signal. In their most recent work, Duda et al. (1982) have developed a theory which successfiilly predicts the strong dependence of the diffusion coefficient on temperature and concentration in polymeric solutions. The parameters in this theory are relatively easy to obtain, and in view of its predictive capability this theory would seem to be most appropriate for incorporating concentration-dependent diffusion coefficients in the diffusion equation. 

The results of experimental studies aimed at assessing the validity of Eq. (32) have, for the most part, been inconclusive. Although the exact reasons for this are not entirely clear, certain differences between the experimental conditions employed and those upon which the model is based can be readily identified. In most studies the diffusion coefficient for the particular polymeric solution used was not known and was necessarily treated as an adjustable parameter in the theory. A subjective judgment was thus required as to whether the value of the diffusivity which described the data best was a reasonable one. Generally speaking, the resulting values were unrealistically high. 

Finally, a number of experimental studies have been conducted in a pressure range where the polymeric solution could boil. The vapor bubbles thus created would provide a much larger surface area for mass transfer than the surface area of the wiped film alone. And therefore, for fixed values of the diffusivity and the driving force, predicted values for mass transfer rates would be substantially lower than the measured values. Conversely, for a fixed mass transfer rate and driving force, use of the wiped film surface area alone would require unusually high values of the diffusivity in order to obtain agreement between theory and experiment. 

The main variables in the operation of atomizers are feed pressure, orifice diameter, flow rate and motive pressure for nozzles and geometry and rotation speed of wheels. Enough is known about these factors to enable prediction of size distribution and throw of droplets in specific equipment. Effects of some atomizer characteristics and other operating variables on spray dryer performance are summarized in Table 9.18. A detailed survey of theory, design and performance of atomizers is made by Masters (1976), but the conclusion is that experience and pilot plant work still are essential guides to selection of atomizers. A clear choice between nozzles and spray wheels is rarely possible and may be arbitrary. Milk dryers in the United States, for example, are equipped with nozzles, but those in Europe usually with spray wheels. Pneumatic nozzles may be favored for polymeric solutions, although data for PVC emulsions in Table 9.16(a) show that spray wheels and pressure nozzles also are used. Both pressure nozzles and spray wheels are shown to be in use for several of the applications of Table 9.16(a). 

The concept of free-volume appeared to be very useful and was applied for the theoretical description of many processes in liquids, including polymeric solutions and melts. Taking the free-volume concept as a basis, theories were developed for the diffusion of low-molecular-weight compounds into polymers14,1S, thermal conductivity16, solution and solubility of polymers17, etc. 

Summary The classical treatment of the physicochemical behavior of polymers is presented in such a way that the chapter will meet the requirements of a beginner in the study of polymeric systems in solution. This chapter is an introduction to the classical conformational and thermodynamic analysis of polymeric solutions where the different theories that describe these behaviors of polymers are analyzed. Owing to the importance of the basic knowledge of the solution properties of polymers, the description of the conformational and thermodynamic behavior of polymers is presented in a classical way. The basic concepts like theta condition, excluded volume, good and poor solvents, critical phenomena, concentration regime, cosolvent effect of polymers in binary solvents, preferential adsorption are analyzed in an intelligible way. The thermodynamic theory of association equilibria which is capable to describe quantitatively the preferential adsorption of polymers by polar binary solvents is also analyzed. 

In all but the most basic cases of very dilute systems, with microstructural elements such as rigid particles whose properties can be described simply, the development of a theory in a continuum context to describe the dynamical interactions between structure and flow must involve some degree of modeling. For some systems, such as polymeric solutions, we require modeling to describe both polymer-solvent and polymer-polymer interactions, whereas for suspensions or emulsions we may have an exact basis for describing particle-fluid interactions but require modeling via averaging to describe particle-particle interactions. In any case, the successful development of useful theories of microstructured fluids clearly requires experimental input and a comparison between experimental data and model... 

Let me close with two more familiar examples. We all know that Flory s theory of polymeric solutions is not in agreement with experiment. However, the insight on polymer compatibility gained from the theory was enormous. The same thing may be said about the van der Waals equation of state. We all know that this equation is very approximate, but something remains in our intuition that is extremely helpful. 

Up to now no rigorous quantitative theory describing the kinetics of homopolymerization even of one monomer is available within the framework of modern theory of polymeric solutions. The theory concerning copolymerization will become more complex since the rate constant of a pair of the macroradical recombination could depend not only on their degrees of polymerization but also on the composition and chemical structure of these macroradicals. 

In these complicated extraction systems containing the polymeric and/or hydroxo species, one would expect the solvent used as a diluent to exert a considerable effect on the extraction equilibrium. In the extraction of gallium (III) with decanoic acid it has been found that the less polar the solvent, the more polymerized the extracted species (150). More recently, the solvent effect on the extraction (156) and dimerization (151, 153) of copper(II) decanoate has been interpreted according to regular solution theory (141,142). 

Another limitation of the regular solution theory is the assumption diat is negligible. While this assumption may be valid for solutions in which all components (solute and solvent) are of similar sizes, it breaks down when the molar voinmes of the components are significantly different, i.e., in the case of high molecular weight (polymeric) solvents and low molecular weight solutes. For such cases, more rigorous models that include entropic considerations, such as... 

The principal equations for elastic LS have been reviewed recently [270] and will be only summarized herein. In brief, following the Zimm theory [277], the intensity of the light scattering by a polymeric solution is in relation with the molar mass of the sample according to the following general equation ... 

For non-polymeric mixtures with Nfi, — N = l, this equation was developed earlier by Hildebrand and is called regular solution theory ... 

Knowing the thermal stability of clathrates permits the prediction of experimental conditions for polymerization (8). A detailed analysis of this problem requires the examination of all the involved phases, particularly the solid and liquid phases. Equations for phase equilibria were derived from within the framework of the regular solution theory they contain an interaction parameter W, (whose value is always positive or zero for ideal solutions), which measures the tendency of host and guest to segregate in the liquid phase. The melting or decomposition point is very sensitive to the value of W, especially when it exceeds 2 RT, i.e. when a miscibility gap is observed in the liquid phase. For this reason the PHTP-hydrocarbon clathrates melt congruently between 115 and 180 C, whereas the urea-hydrocarbon... 

In the above considerations, the hydrophobic portions of both the membrane polymer and the small molecules that enter the membrane are expected to interact in the hydrophobic microphases in the membrane. It therefore becomes useful to find a numerical measure of the miscibility of these hydrophobic portions of molecules. In the case of complete molecules, both small and polymeric, the solubility parameter concept has been useful in the past. This concept is related to the enthalpy change which occurs on mixing in regular solution theory as developed by Hildebrand and coworkers (10) and as used for polymer solution theory by Flory (11). The Hildebrand solubility parameter is a measure of the attraction between molecules of the same kind, including dispersion forces, polar forces, and hydrogen bonding... 

Our work addresses these two problems by adopting the solution theory of Hill (10, 11) and by adapting ideas from the Group Renormalization (12, 15) theory of polymer solutions to the prediction of the model parameters from the degree of polymerization of the phase forming polymers. 

One of the reasons for the failure of the ideal solution law is the assumption that a large polymeric solute molecule is interchangeable with the smaller solvent molecule. The law also neglects intermolec-ular forces since the heat of mixing (AH x) assumed to be zero. The Flory-Huggins theory attempted to remedy these shortcomings in the ideal solution law. > ... 

Finally, in addition to predictions for the increments in Miesowicz and Leslie viscosities, the Brochard theory predicts [Brochard, 1979] the increment Syi in the viscosity associated with the twist distortion of a nematic solvent on dissolution of a polymeric solute ... 

Biopolymers are both polar and ionic in character. They exhibit features characteristic of both polymeric solutes and of colloidal particles, giving complex behaviors difficult to describe simply. Enormous responses can occur at low frequencies. Minakata, Imai, and Oosawa have studied theory and experiment for solutions of the polyelectrolyte, tetra-JV-butylammonium polyacrylate (BU4NPA). They observed two low-frequency dielectric dispersion peaks, one at about 100,000 Hz, the other at about 1000 Hz. They suggest that the former is due to a bulk-bulk, Maxwell-Wagner process and that the later and slower process is... 

It was shown that water-soluble polymeric solution such as Separan AP30, NPIO, and Keltrol with polymer concentration less than approximately 1,500, 5,000, and 500 ppm, respectively, is independent of polymer concentration [20]. These observations were consistent with the molecular theories such as the Zimm theory, which states that at infinite dilution, the relaxation time is independent of polymer concentration and is given by... 

For mbbery polymeric membranes, the process of mixing gas A into a polymer P can be described by Hory-Huggins regtrlar solution theory for mixing components within polymer solutions [14] ... 

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