Solution viscosity theory

The development of the solution viscosity theory relates the coil volume to intrinsic viscosity - ... 

The segmental friction factor introduced in the derivation of the Debye viscosity equation is an important quantity. It will continue to play a role in the discussion of entanglement effects in the theory of viscoelasticity in the next chapter, and again in Chap. 9 in connection with solution viscosity. Now that we have an idea of the magnitude of this parameter, let us examine the range of values it takes on. 

The various physical methods in use at present involve measurements, respectively, of osmotic pressure, light scattering, sedimentation equilibrium, sedimentation velocity in conjunction with diffusion, or solution viscosity. All except the last mentioned are absolute methods. Each requires extrapolation to infinite dilution for rigorous fulfillment of the requirements of theory. These various physical methods depend basically on evaluation of the thermodynamic properties of the solution (i.e., the change in free energy due to the presence of polymer molecules) or of the kinetic behavior (i.e., frictional coefficient or viscosity increment), or of a combination of the two. Polymer solutions usually exhibit deviations from their limiting infinite dilution behavior at remarkably low concentrations. Hence one is obliged not only to conduct the experiments at low concentrations but also to extrapolate to infinite dilution from measurements made at the lowest experimentally feasible concentrations. 

The Smoluchowski theory for diffusion-controlled reactions, when combined with the Stokes-Einstein equation for the diffusion coefficient, predicts that the rate constant for a diffusion-controlled reaction will be inversely proportional to the solution viscosity.16 Therefore, the literature values for the bimolecular electron transfer reactions (measured for a solution viscosity of r ) were adjusted by multiplying by the factor r 1/r 2 to obtain the adjusted value of the kinetic constant... 

In support of the association theory, colloid chemists cited non-reproduceable cryoscopic molecular weight determinations (which were eventually shown to be caused by errors in technique) and claimed that the ordinary laws of chemistry were not applicable to matter in the colloid state. The latter claim was based, not completely without merit, on the ascerta-tion that the colloid particles are large aggregates of molecules, and thus not accessible to chemical reactants. After all many natural colloids were shown to form double electrical layers and adsorb ions, thus they were "autoregulative" by action of their "surface field" (29). Furthermore, colloidal solutions were known to have abnormally high solution viscosities and abnormally low osmotic pressures. 

Polyelectrolyte complexation in aqueous solution between PEI and PMAA has been studied through viscometry, conductometry, potentiometry, and IR spectroscopy [90]. Upon addition of increasing concentrations of PMAA to an aqueous PEI solution, viscosity dropped suddenly around a 1 to 4 ratio of PMAA to PEI because of the complexation and subsequent coiling of the complexed chains. Reduced viscosity then rose past this ratio indicating that the stoichiometry of the complex occurs in a 1 4 (PMAA groups PEI groups) formation. Conductance and titration experiments agreed with this theory. The... 

In a free solution, the electrophoretic mobility (i.e., peiec, the particle velocity per unit applied electric field) is a function of the net charge, the hydrodynamic drag on a molecule, and the properties of the solutions (viscosity present ions—their concentration and mobility). It can be expressed as the ratio of its electric charge Z (Z = q-e, with e the charge if an electron and q the valance) to its electrophoretic friction coefficient. Different predictive models have been demonstrated involving the size, flexibility, and permeability of the molecules or particles. Henry s theoretical model of pdcc for colloids (Henry, 1931) can be combined with the Debye-Hiickel theory predicting a linear relation between mobility and the charge Z ... 

Muthukumar[691 described a theory of a fractal polymer possessing solution viscosity. Solutions containing dilute, semidilute, and high concentrations of fractal polymers were examined intra- and interfractal hydrodynamic interactions as well as excluded volume effects were included in the treatment. 

In summarizing the intrinsic viscosity relations presented in this section, it must be admitted that they represent nothing more than rather small semi-empirical refinements of the Flory excluded volume theory and the Flory-Fox viscosity theory. For a large fraction of the existing body of experimental data, they offer merely a slight improvement in curve-fitting. But for polymers in good solvents it is believed that a more transcendental result has been achieved. The new equations permit more reliable assessment of unperturbed chain dimensions in such cases, and in some instances (e. g., certain cellulose derivatives see Section III B) they offer possible explanations of heretofore paradoxical solution behavior. 

Zi is the charge on species i, coefficient A depends on various solute and solvent properties, and the coefficients are specific to the individual ions. Parameters for the Jones-Dole equation at room temperature are tabulated by Marcus [79]. A semiempirical extension of the Jones-Dole equation to higher concentrations, and also a method for extrapolating room-temperature parameters to higher temperatures, are described by Lencka et al. [80]. Jiang and Sandler [81] have developed a different method, based on liquid-state theory, that also appears promising for correlation and limited prediction of electrolyte solution viscosities. 

The exponent v characterizes the swelling of a long polymer chain in very dilute solutions. In theory, it could be measured in several ways. However, can we trust results obtained by the simplest technique which consists in measuring the intrinsic viscosity These measurements produce values for the exponent v which are always lower than those obtained by light scattering measurements of the radius of gyration. It was necessary to explain this discrepancy in order to make a proper comparison between experimental and theoretical values of the exponent v. 

Deriving molecular dimensions in solution from viscosities depends on the model assumed for the conformations of the free molecules. Since any a- or - triple helical sections of our gelatins vrc>uld be melted at 30 C. we assume near randomness for the chains, and a lew ellipticity for the molecular envelopes. Further, the success of Flory s viscosity theory (17) has shown that the hydrodynamically effective volume of randomly coiled (and of many other) chain molecules is not very different from the volume encompassed by the meandering segments. Thus we treated our data as if they pertained to random coil molecules. The measured layer thicknesses then describe the level within the adsorbed interphase below v ich the segmental density is equal to, or larger, than the effective coil density of the free molecules. 

Dissolved particles of density pi travel through a solvent of density pi under the influence of a centrifugal field. They sediment in the direction of the centrifugal field when pi> pi, and move to the center of rotation when P2 < pi. Under otherwise constant conditions, the rate of sedimentation (or flotation) depends on the mass and shape of the particles as well as on the solution viscosity. Therefore all of these quantities can, in theory, be determined from the rate of sedimentation. 

When a size-exclusion chromatograph is calibrated correctly, one can know the molecular weight of a polymer just based on the time it takes to pass, or elute through the column. From Fox and Flory s theory of solution viscosity one can learn that the size of a solvated macromolecular coil is directly correlated with its solution viscosity. The correlation is ... 

The given information affords to define more precisely the developed semiempirical turbulent theory of polymer solutions [4,53. In this case it is necessary to connect the length of the mixing path or of turbulence viscosity with the increased longitudinal solution viscosity in the presence of macromolecules, and the last - with molecular characteristics of a polymer and with its solution concentration. 

In electrochemical measurements using rotating electrode technique and data treatment using Koutecky—Levich theory, the most commonly used solution viscosity is kinematic viscosity. Therefore, in the following sections, we will only focus on this kinematic viscosity. 

In conclusion, both the Graessley and Bueche theories confirm the general experimental observation that the reduced solution viscosity, ti/Doj function of X. Therefore, solution rheological behavior can be described by two material parameters, the zero shear viscosity and polymer response time. In turn, these parameters are functions of macromolecular structure and solvent properties. 

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