Viscosity Mark-Houwink equation

In contrast, Mark maintained that macromolecules could assume many different conformations (shapes) and in collaboration with Guth and Kuhn, proposed a power form for the Staudinger equation i.e., qgp=KM . A similar equation was proposed simultaneously by Roelof Houvdhk and the above equation is now referred to as the Mark-Houwink viscosity equation. 

The Mark-Houwink-Sakurada equation relates tire intrinsic viscosity to tire polymer weight ... 

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). 

Electrostatic repulsion of the anionic carboxylate groups elongates the polymer chain of partially hydrolyzed polyacrylamides increasing the hydrodynamic volume and solution viscosity. The extensional viscosity is responsible for increased resistance to flow at rapid flow rates in high permeability zones (313). The screen factor is primarily a measure of the extensional (elonga-tional) viscosity (314). The solution properties of polyacrylamides have been studied as a function of NaCl concentra-tion and the parameters of the Mark-Houwink-Sakaruda equation calculated... 

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 ... 

Many polymer properties can be expressed as power laws of the molar mass. Some examples for such scaling laws that have already been discussed are the scaling law of the diffusion coefficient (Equation (57)) and the Mark-Houwink-Sakurada equation for the intrinsic viscosity (Equation (36)). Under certain circumstances scaling laws can be employed advantageously for the determination of molar mass distributions, as shown by the following two examples. 

Not only good solubility but also solution behavior differs for hyperbranched polymers compared to linear polymers. For example, hyperbranched aromatic polyesters, described by Turner et al. [71,72], exhibit a very low a-value in the Mark-Houwink-Sakurada equation and low intrinsic viscosities. This is consist-... 

The viscosity average molecular weight is determined through the use of the Mark-Houwink-Sakurada equation [3] using solution viscosity ... 

Intrinsic viscosity is related to the relative viscosity via a logarithmic function and to the specific viscosity by a simple algebraic relationship. Both of these functions can be plotted on the same graph, and when the data are extrapolated to zero concentration they both should predict the same intrinsic viscosity. The specific viscosity function has a positive slope and the relative viscosity function has a negative slope, as shown in Fig. 3.7. The molecular weight of the polymer can be determined from the intrinsic viscosity, the intercept of either function, using the Mark-Houwink-Sakurada equation. 

Since the intrinsic viscosity depends not only on the size of the macromolecule but also on its shape, on the solvent, and on the temperature, there is no simple relationship for the direct calculation of molecular weights from viscosity measurements. However, the Mark-Houwink-Kuhn equation gives a general description of how the molecular weight can be calculated from the intrinsic viscosity ... 

By use of both the appropriate value for in Equation 2 and the Mark-Houwink viscosity expression, one may write... 

The relation between number molecular weight, Mn and intrinsic viscosity, [t ], for poly(penLachlorophenyI methacrylate) (PPCIPh) can be represented by the Mark - Houwink - Sakurada equation [44],... 

Ito et al. [65] investigated the MW dependence of the limiting viscosity for a series of regular polymacromonomers from PEO macromonomers, 26 (m=l) and demonstrated that the universal SEC calibration holds for these polymers. The exponent, a, in the Mark-Houwink-Sakurada equation defined by... 

Intrinsic viscosity is the most useful of the various viscosity expressions because it can be related to molecular weight by the Mark-Houwink-Sakurada equation ... 

Viscosity measurements alone cannot be directly used in the Mark-Houwink-Sakurada equation to relate absolute viscosity and polymer molecular weight, since additional unknowns, K and a must be determined. Therefore, viscometry does not yield absolute molecular weight values it rather gives only a relative measure of polymer s molecular weight. Viscosity measurements based on the principle of mechanical shearing are also employed, most commonly with concentrated polymer solutions or undiluted polymer these methods, however, are more applicable to flow properties of polymers, not molecular weight determinations. 

The Mark-Houwink empirical equation relates intrinsic viscosity and the average molecular weight (M) ... 

Osmotic pressure measurements for the determination of MW were used in 1900 to characterize starch. Twenty years later, the solution viscosity measurements were introduced by Staudinger for this purpose. However, it was Mark and his collaborators who developed the concept of the intrinsic viscosity ([r ]) and demonstrated that it provides information on the volume of individual colloidal particles, thus on MW. For the freely rotating chains the dependence (today known as Mark-Houwink-Sakurada equation) was obtained [Guth and Mark, 1934] ... 

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. 

According to the Mark-Houwink-Sakurada equation, the relationship between the viscosity, molecular weight, and organic polymer type can be formulated as... 

This equation applies for polymeric solutions under theta conditions. Theta conditions are those at which excluded volume effects (expansion of the dimensions of the ideal coil) are exactly compensated by polymer solvent interactions (Chapter 25). The dependence between intrinsic viscosity and MW is given by the Mark-Houwink-Sakurada equation (see also Chapter 1) ... 

Routinely, molecular weights of polymers are conveniently estimated from intrinsic viscosity measurements using the Staudinger (also known as the Mark-Houwink-Sakurada) equation... 

To detemiine the intrinsic viscosity, both inherent and reduced viscosities are plotted against concentration (Q on the same graph paper and extrapolated to zero. If the intercepts coincide, then this is taken as the intrinsic viscosity. If they do not, then the two intercepts are averaged. The relationship of intrinsic viscosity to molecular weight is expressed by the Mark-Houwink-Sakurada equation ... 

Usually, the acid hydrolysis is followed by measuring the decrease in molecular weight, or D.P. with time by intrinsic viscosity. The relationship between intrinsic viscosity [ ] and molecular weight is given by the Mark Houwink Sakarada equation ... 

CV. The intrinsic viscosity [ri] and can be obtained directly from the viscosity distribution, outlined earlier, in connection witli Equation (3). Now, a Mark-Houwink exponent [Equation (6)] can be approximated. The ratio n can then be estimated from the viscosity distribution when the molecular weight distribution is set equal to either a log normal or the even more widely applicable generalized exponential distribution. The parameters characteristic of either assumed molecular weight distribution are easily fit from the moments of viscosity distribution. Once this is done, all average molecular weights can be estimated in principle. Because of analytical uncertainties in the high-molecular-weight tails of the distributions of most synthetic polymers, however, it is wise to confine these estimates to... 

The logarithms of intrinsic viscosities of fractionated samples are plotted against log or log Mn. The constants a and K of the Mark-Houwink-Sakurada equation are the intercept and the slope, respectively, of that plot. Except for the lower molecular weight samples, the plots are linear for linear polymers. Many values of K and a for different linear polymers can be found in the literature [66]. 

Actually, the Mark-Houwink-Sakurada equation applies only to narrow molecular weight distribution polymers. For low molecular weight poly disperse polymers this equation is useful, because the deviations due to chain entanglement are still negligible. On the other hand, chain entanglement in high molecular weight polydisperse polymers affects viscosity and this equation does not really apply. 

Viscosimetry is the mostly utilized method to determine the molecular weight of chitosan due to its simplicity. The method has the disadvantage of not being absolute because it relies on the correlation between the values of intrinsic viscosity with those of molecular weight, as determined hy an absolute method for fractions of a given polymer. This relationship is given by the well-known Mark—Houwink—Sakurada equation. 

The viscosity of polymer solutions depends on both the concentration and the molecular weight of the polymer. Thus, measurement of the viscosity of a solution of polymer can be a way to determine the molecular weight of the dissolved polymer. This can actually be the case providing that the coefficients K and a of the Mark-Houwink-Sakurada equation (Equation 2.5) giving the relationship between the intrinsic viscosity of the polymer, [tj], and the viscosity average molecular weight, M, are known. 

This also means that calibration is required in methods based on viscosity measurements before the method can be applied to characterise the molecular weight of a given polymer. Such calibration is tedious and requires that the molecular weights of a series of the polymer have been characterised by another technique in order to establish the relationship with the intrinsic viscosity. At present, this can be performed using the SEC method coupled with triple detection. It is noteworthy that calibration of the Mark-Houwink-Sakurada equation is valid only for one couple including a polymer and a solvent at a given temperature. 

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