The viscosity of dilute polymer solutions

A continuous fluid is characterized by a local velocity The velocity 

For a Newtonian fluid, this linear relationship is obeyed over a wide range of shear rates (velocity gradients). 

The viscosity of a dilute polymer solution can be represented by a virial expansion  

As noted in Chapter 1, the intrinsic viscosity for a macromolecular solute can often be expressed as a power law in the molecular weight  

Because the exponent a is observed to vary over a wide range from 0.0 to 2.0, a theory that explains this behavior would be helpful. The Kirk-wood-Riseman treatment given in Section 5.10 is tiie basis of a successful understanding of the intrinsic viscosity of Gaussian coils in dilute solution. 

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Viscosity Measurements. Although in typical polymer-plasticizer systems, the polymer is the major component, it is possible to use the viscosity of dilute polymer solutions as a measure of the solvent power of the liquid for the polymer. Thus, liquids with high solvent power for the polymer cause a stretching out of the chain molecules, whereas a liquid of poor solvent power causes the chains to coil up. This is because, in the liquid with poor solvent power, the segments of the polymer chain (the monomer units) prefer to stay close to each other, while in a good solvent, interaction between polymer segments and solvent molecules is preferred. 

The DTO model ignores the overall translations and rotations of the molecule as a whole and refers only to internal vibrational modes. It is therefore incapable of explaining on its own the viscosity of dilute polymer solutions. The enhanced viscosity of dilute polymer solutions is undoubtedly due to a hydrodynamic damping of the polymer as a whole as it translates and rotates in the shear field. This was very well described by Debye (21). We should point out that the Debye viscosity is alternatively derivable from the RB theory. 

The viscosity of dilute polymer solutions can be estimated with fair accuracy. 

In order to overcome this difficulty, Rudin and Strathdee (1974) developed a semi empirical method for predicting the viscosity of dilute polymer solutions. The method is based on an empirical equation proposed by Ford (1960) for the viscosity of a suspension of solid spheres ... 

Rudin and Strathdee (1974) remarked that the equations presented for the viscosity of dilute polymer solutions were valid approximately up to the critical concentration. This leads to a more general definition of a concentrated polymer solution, viz. a solution for which c > ccr. 

The viscosity of dilute polymer solutions is considerably higher than that of the pure solvent. The viscosity increase depends on the temperature, the nature of the solvent and polymer, the polymer concentration, and the sizes of the polymer molecules. This last dependence permits estimation of an average molecular weight from solution viscosity. The average molecular weight which is measured is the viscosity average A/v, which differs from those described so far in this text. Before viscosity increase data are used to calculate Afv of the solute it is necessary, however, to eliminate the effects of solvent viscosity and polymer concentration. The methods whereby this is achieved are described in this section. 

The theory of Zimm (7) uses the assumptions of the Rouse theory and in addition considers hydrodynamic interactions between the moving submolecules and the solvent. The theory also makes use of the method formulated by Kirkwood and Riseman for the evaluation of the viscosity of dilute polymer solutions. A parameter h = where r is the... 

The molecular weight of polymer molecules can be determined by the measurement of the viscosity of dilute polymer solutions [1], The relationship used is the so-called Mark-Houwink (MH) empirical equation ... 

Experimentally, the viscosity of dilute polymer solutions is, in most cases, determined with glass capil- [17] =... 

In fact experimentally the viscosity of dilute polymer solutions follows... 

H. van Oene, Measurement of the Viscosity of Dilute Polymer Solutions, in Characterization of Macromolecular Structure, Natl. Acad. Sci. U.S. Publ. 1573, Washington, D.C., 1968. 

Experimentally, the viscosity of dilute polymer solutions is, in most cases, determined with glass capillary viscometers, making application of the Hagen-Poiseuille s law for laminar flow of liquids. The time required for a specific volume of a liquid to flow through a capillary of... 

The lattice theory of entropy of mixing, derived independently and contemporaneously by Huggins and Flory and the "Huggins constant K ", relating the concentration dependence of the viscosity of dilute polymer solutions are listed among Huggins major contributions to polymer science. He developed new procedures for more quantitative productions of solubiHty. 

Viscosity-Average Moiecuiar Weight. The viscosity of dilute polymer solutions may be related to the molecular weight of the polymer by the appropriate calibration (see Viscometry). The polymer is usually separated into narrow molecular weight distribution fractions, which are characterized by absolute molecular weight methods. The molecular weight is related to the intrinsic viscosity [ j] by the Mark-Houwink relationship (eq. 6). 

The limiting viscosity number depends on the polymer, solvent, and temperature, but imder a given set of conditions it is related to the molecular weight by the Mark-Houwink relation, [ >] = where K and a are constants and M is the molecular weight of the polymer. Tables of K and a are available for a large number of polymers and solvents (31,32). Excellent summaries of equations, techniques, and references relating to the viscosity of dilute polymer solutions are also available (33,34), as is information on dilute polymer solutions that are shear thinning (35). 

S1 Schoff, C.K. Concentration dependence of the viscosity of dilute polymer solutions Huggins and Schulz-Blaschke constants. Polymer Handbook, Brandrap, J., Immergut, E.H., Gralke, E.A. (eds.), 4th Ed., p. VII/265-289, J. Wiley Sons, Inc., New York 1999... 

If the shear rate is higher than the time for the first normal mode, the chain does not have time to respond to the applied perturbation, and only the higher modes are able to be activated. In other words, at times which are shorter than Tj the first normal mode is frozen out and hence cannot contribute to the observed viscosity. Further increase in the rate of shear will progressively remove further modes until the viscosity falls to a value which corresponds to that of the solvent. This simple description, with minor modifications, describes the behaviour of most polymer molecules in dilute solution. Because in solution the backbone motions are effectively liberated, so that the chains are fuUy flexible, the description of the viscosity of dilute polymer solutions is essentially independent of the chemical nature of the molecules. The modes are purely defined by the end to end length of the polymer chains and hence by the molar mass of the polymer. 

A characteristic feature of a dilute polymer solution is that its viscosity is considerably higher than that of either the pure solvent or similarly dilute solutions of small molecules. This arises because of the large differences in size between polymer and solvent molecules, and the magnitude of the viscosity increase is related to the dimensions of the polymer molecules in solution. Therefore, measurements of the viscosities of dilute polymer solutions can be used to provide information concerning the effects upon chain dimensions of polymer structure (chemical and skeletal), molecular shape, degree of polymerization (hence molar mass) and polymer-solvent interactions. Most commonly, however, such measurements are used to determine the molar mass of a polymer. 

The viscosities of dilute polymer solutions most commonly are measured using capillary viscometers of which there are two general classes, namely U-tube viscometers and suspended-level viscometers (Fig. 3.18). A common feature of these viscometers is that a measuring bulb, with upper and lower etched marks, is attached directly above the capillary tube. The solution is either drawn or forced into the measuring bulb from a reservoir bulb attached to the bottom of the capillary tube, and the time required for it to flow back between the two etched marks is recorded. 

Concentration Dependence of the Viscosity of Dilute Polymer Solutions Huggins and Schulz-Blaschke Constants... 

VII / 266 CONCENTRATION DEPENDENCE OF THE VISCOSITY OF DILUTE POLYMER SOLUTIONS... 

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