Viscosity, dilute solution Mark-Houwink equation

The GPC-viscometry with universal calibration provides the unique opportunity to measure the intrinsic viscosity as a function of molecular weight (viscosity law, log [17] (it versus log M) across the polymer distribution (curves 3 and 4 in Fig. 1). This dependence is an important source of information about the macromolecule architecture and conformations in a dilute solution. Thus, the Mark-Houwink equation usually describes this law for linear polymers log[i7] = ogK+ a log M (see the entry Mark-Houwink Relationship). The value of the exponent a is affected by the macromolecule conformations Flexible coils have the values between 0.5 and 0.8, the higher values are typical for stiff anisotropic ( rod -like) molecules, and much lower (even negative) values are associated with dense spherical conformations. 

The viscosity method makes use of the fact that the exponent, a, in the Mark-Houwink equation (see Frictional Properties of Polymer Molecules in Dilute Solution), rj = KM° , is equal to 0.5 for a random coil in a theta-solvent. A series of polymers of the same type with widely different known molecular weights is used to determine intrinsic viscosities [t ] at different temperatures and hence a at different temperatures. The theta-temperature can thus be determined either by direct experiment or, if it is not in the measurable range, by calculation. 

Mattes and coworkers also determined the Mark-Houwink constants for PAn EB in NMP. The Mark-Houwink equation relates the intrinsic viscosity (solution viscosity at infinite dilution) to the polymer molecular weight ... 

The intrinsic viscosities are obtained by making viscosity measurements at different polymer concentrations and by plotting the above expression against the concentration. The limit that this quantity assumes as the infinite dilution is approached is the value desired. Polymer solutions exhibit a Newtonian behavior at low shear rates, changing to a non-Newtonian flow at higher shear rates (6). The intrinsic viscosities should be determined at the low shear rate range. The most commonly used equation that relates the intrinsic viscosity and the molecular weight of a macromolecule is the Mark-Houwink equation ... 

Mark-Houwink equation n. Also referred to as Kuhn-Mark-Houwink-Sakurada equation allows prediction of the viscosity average molecular weight M for a specific polymer in a dilute solution of solvent by [77] = KM, where K is a constant for the respective material and a is a branching coefficient K and a (sometimes a ) can be determined by a plot of log [77] versus logM" and the slope is a and intercept on the Y-axis is K. Kamide K, Dobashi T (2000) Physical chemistry of polymer solutions. Elsevier, New York. Mark JE (ed) (1996) Physical properties of polymers handbook. Springer-Verlag, New York. Ehas HG (1977) Macromolecules, vols 1-2. Plenum Press, New York. 

Although solution viscosity of PLA in solvent is not directly relevant to the processing of molten PLA polymers, this property is often evaluated to determine the molecular weight of resins and processed parts for quality control purposes. The relationship between viscosity and the molecular weight of PLA dissolved in a dilute solution is commonly modeled using the Mark—Houwink equation ... 

This is a standard procedure for molecular weight determinations and involves the use of specially designed viscometers to accurately measure the viscosity of a polymer solution. From this the intrinsic viscosity is determined and hence the molecular weight. The time taken for the polymer solution to pass between two marks on the viscometer is compared to that of pure solvent and the ratio is the viscosity of the solution. Successive dilutions give a range of concentrations and times from which the intrinsic viscosity can be calculated. The value for this is then entered into the Mark-Houwink equation ... 

To perform this analysis, we first prepare a dilute solution of polymer with an accurately known concentration. We then inject an aliquot of this solution into a viscometer that is maintained at a precisely controlled temperature, typically well above room temperature. We calculate the solution s viscosity from the time that it takes a given volume of the solution to flow through a capillary. Replicate measurements are made for several different concentrations, from which the viscosity at infinite dilution is obtained by extrapolation. We calculate the viscosity average molecular weight from the Mark-Houwink-Sakurada equation (Eq. 5.5). 

The specific viscosity )jsp of a dilute solution of spheres is directly related to their hydrodynamic volume VV Nl denotes Avogadro s number. Typically the intrinsic viscosity [tj] follows a scaling law, the so-called Mark-Houwink-Sakurada equation ... 

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 ... 

Dilute Solution Properties. The rheology of dilute polymer solutions has been used extensively to gain insight into the structure and conformation of polymers in solution (11). The intrinsic viscosity provides a measure of the molecular weight of a polymer through a relationship such as the Mark-Houwink-Sakurada equation. Earlier studies of polyacrylamide (PAM) systems and details of the complexity of the characterization of high-molecular-weight water-soluble systems can be found in references 9, 13, and 14. 

In the viscometry method " one measures the viscosity, rf, of a polymer solution at various degrees of dilution. The intrinsic viscosity, [rj], of the polymer-solvent system is obtained by plotting the quantity l (jj - J7o)/ o vs. C (here rjo is the viscosity of the solvent and C is e concentration of the polymer solution) and extrapolating the resulting curve to zero concentration. Then one applies the Mark-Houwink-Sakurada (MHS) equation ... 

Here, a is the exponent in the Mark-Houwink dilute solution viscosity equation... 

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). 

For a solution to abide by the Mark-Houwink type equation, there should be no interaction among the polymer coils, which requires infinite dilution. This is achieved mathematically by defining intrinsic viscosity according to Eq. 5.16. 

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