Viscosity of dilute polymer solutions

Einstein derived the following equation for the viscosity of suspensions of rigid, uncharged, spherical particles which do not interact with the suspension medium  

It may be assumed that under the action of a shear stress the polymer coil with the solvent enclosed by it behaves like an Einsteinian sphere, and hence, the viscosity of polymer solution should obey Eg. (3.141) for noninteracting spherical particles. Noninteraction of the polymer coils requires infinite dilution. Mathematically this is achieved by defining a quantity called the intrinsic viscosity, [77], according to equation (Young and Lovell, 1990)  

If Vh is the hydrodynamic volume of each polymer molecule (it is assumed that aU polymer molecules are of the same molecular weight) then 

A more satisfactory form of this equation to satisfy the condition of noninteraction of polymer coils is given in terms of intrinsic viscosity, [77], defined by Eq. (3.142). 

This equation can be rearranged to give an equation for the hydrodynamic volume of an impermeable (nondraining) polymer molecule in infinitely dilute solution  

The expression in equation (2.31) reduces to that given by the random-walk model when = 1 in equation (2.28). 

Dissolving even a small quantity of a high molecular weight polymer in a good solvent results in a marked increase in solution viscosity. Solution viscosity depends on the nature of polymer, its molecular weight, concentration of the solution, and the temperature. Viscometry is therefore a convenient practical experimental method to determine an average molecular weight (My) of 

The basic notion of solution viscosity is illustrated in Fig. 2.7 of a volume of fluid in a shear field (for instance a film of polymer solution confined between parallel plates where one is stationary and the other is moving in the x-direction at a constant velocity v). Assuming no slippage between the liquid and the plate, the force per unit area applied on the volume, the shear stress t, results in a rate of deformation or a strain rate y where 

The viscosity of water at 20°C is about 1.00 cP whereas that of olive oil is about 10,000 cP at the same temperature. 

For simple low-molecular-weight liquids, viscosity 17 is usually independent of the shear rate (i.e., the linear equation (2.33) applies at constant 

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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. 

Gundert F, Wolf BA (1986) Viscosity of dilute polymer solutions molecular weight dependence of the Huggins coefficient Makromol Chem 187 2969... 

Cooke BJ, Matheson AJ (1976) Dynamic viscosity of dilute polymer solutions at high frequencies of alternating shear stress. J Chem Soc Faraday Trans II 72(3) 679-685 Curtiss CF, Bird RB (1981a) A kinetic theory for polymer melts. I The equation for the single-link orientational distribution function. J Chem Phys 74 2016—2025 Curtiss CF, Bird RB (1981b) A kinetic theory for polymer melts. II The stress tensor and the rheological equation of state. J Chem Phys 74(3) 2026—2033 Daoud M, de Gennes PG (1979) Some remarks on the dynamics of polymer melts. J Polym Sci Polym Phys Ed 17 1971-1981... 

Kalashnikov VN (1994) Shear-rate dependent viscosity of dilute polymer solutions. J Rheol... 

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 study of hydrodynamic properties (sedimentation, diffusion and viscosity) of dilute polymer solutions is the most widely used method permitting the characterization of geometric properties (size and conformation) of polymer molecules. 

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... 

Viscosities of dilute polymer solutions increase as the polymer concentration is increased (21). Several correlations between these two variables have been suggested. A widely used relationship is the Huggins equation which relates the viscosity to concentration in a quadratic functionality as follows ... 

The concentration dependence of the Newtonian viscosity of dilute polymer solutions may be expressed in the form of a polynomial ... 

Brochard, R, Viscosities of dilute polymer solutions in nematic hquids, J. Polym. Sci. Polym. Phys. Ed., 17,1367-1374 (1979). 

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. 

Viscometry Determination of the increase of viscosity of dilute polymer solution resulting from the presence of polymer molecules Mv Thousands to millions... 

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 apparent viscosity of dilute polymer solutions can be represented by the power-law model over a wide range of shear rates [3,4]. For such fluids, the shear rate depends on, among other factors, the power-law index. The shear rate for a power-law fluid in a co-axial rotational viscometer (Couette flow) is ... 

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). 

Til is absolute viscosity of dilute polymer solution and and kj are the proportionality constants. This model was investigated with PEG samples (PEG-20000 and PEG-10000) under the aforementioned experimental conditions using water as solvent. Based on this model, the method was developed and tested with an unknown polyvinyl alcohol sample against the conventional capillary viscosity method. There was good agreement between this method and the conventional method. The method proposed by Mao and co-workers [189] has a number of advantages over the conventional viscosity method it... 

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