The Mark-Houwink Equation

An equation finding much application to polysaccharides is the Mark-Hou-wink equation  

Mark-Houwink Constants for Some Dispersed Polysaccharides 

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This relationship with a = 1 was first proposed by Staudinger, but in this more general form it is known as the Mark-Houwink equation. The constants k and a are called the Mark-Houwink coefficients for a system. The numerical values of these constants depend on both the nature of the polymer and the nature of the solvent, as well as the temperature. Extensive tabulations of k and a are available Table 9.2 shows a few examples. Note that the units of k are the same as those of [r ], and hence literature values of k can show the same diversity of units as C2, the polymer concentration. 

By taking the logarithm of both sides of Eq. (9.34), the Mark-Houwink equation is transformed into the equation of a straight line ... 

Equations (9.42) and (9.46) reveal that the range of a values in the Mark-Houwink equation is traceable to differences in the permeability of the coil to the flow streamlines. It is apparent that the extremes of the nondraining and free-draining polymer molecule bracket the range of intermediate permeabilities for the coil. In the next section we examine how these ideas can be refined still further. 

The analysis of the main properties of aqueous solutions of polyacrylamide and copolymers of acrylamide has been reviewed [4,5]. The main characteristics of aqueous solutions of polyacrylamide is viscosity. The viscosity of aqueous solutions increases with concentration and molecular weight of polyacrylamide and decreases with increasing temperature. The relationship between the intrinsic viscosity [q]) in cmVg and the molecular weight for polyacrylamide follows the Mark-Houwink equations ... 

This equation appears to have a number of names, of which the Mark-Houwink equation is the most widely used. In order to use it, the constants K and a must be known. They are independent of the value of M in most cases but they vary with solvent, polymer, and temperature of the system. They are also influenced by the detailed distribution of molecular masses, so that in principle the polydispersity of the unknown polymer should be the same as that of the specimens employed in the calibration step that was used to obtain the Mark-Houwink constants originally. In practice this point is rarely observed polydispersities are rarely evaluated for polymers assigned values of relative molar mass on the basis of viscosity measurements. Representative values of K and a are given in Table 6.4, from which it will be seen that values of K vary widely, while a usually falls in the range 0.6-0.8 in good solvents at the 0 temperature, a = 0.5. 

If changes in the molecular weight distribution can be neglected, substitution of the Mark-Houwink equation into Eq. 7 leads to Eq. 8, where a is the Mark-Houwink exponent. 

In the following four years Mark successively reported on the viscosity and molecular weight of cellulose (40), Staudinger s Law (41), high polymer solutions (42), and the effect of viscosity on polymerization rates (43). Confident of his findings, he proposed (at the same time as R. Houwink) the general viscosity equation now known as the Mark-Houwink Equation (44, 45). 

Mark and Houwink were the first to formulate the equation in the power form and to demonstrate its validity by means of empirical values. In reality, the Mark-Houwink Equation is simply the Einstein viscosity equation, which assumed spheres, transferred to particles with size dependent particle density. 

The molecular weight is normally measured, for convenience sake, by solution viscosity and is often given as the intrinsic viscosity. There is a wide range of solutions used, with the average molecular weight related to the intrinsic viscosity by the Mark-Houwink equation ... 

As a norm, polyester molecular weights are reported by their intrinsic viscosities (IV), [r ]. The two are related by the Mark-Houwink equation, as follows ... 

Viscosity Measurements. A Zimm-Couette type low shear viscometer was used. The intrinsic viscosities were estimated from single concentration viscosity measurements using the equations for the concentration dependence of the specific viscosity (5,6). The Mark-Houwink equation was used to determine My (5,6). 

Staudinger showed that the intrinsic viscosity or LVN of a solution ([tj]) is related to the molecular weight of the polymer. The present form of this relationship was developed by Mark-Houwink (and is known as the Mark Houwink equation), in which the proportionality constant K is characteristic of the polymer and solvent, and the exponential a is a function of the shape of the polymer in a solution. For theta solvents, the value of a is 0.5. This value, which is actually a measure of the interaction of the solvent and polymer, increases as the coil expands, and the value is between 1.8 and 2.0 for rigid polymer chains extended to their full contour length and zero for spheres. When a is 1.0, the Mark Houwink equation (3.26) becomes the Staudinger viscosity equation. 

What is the value of the exponent a in the Mark-Houwink equation for polymers in theta solvents ... 

When is the Flory equation similar to the Mark-Houwink equation ... 

The experimental determination of polymer intrinsic viscosity is done through the measurement of polymer solution viscosity. The connotation of intrinsic viscosity [hi/ however, is very different from the usual sense of fluid viscosity. Intrinsic viscosity, or sometimes called the limiting viscosity number, carries a far more reaching significance of providing the size and MW information of the polymer molecule. Unlike the fluid viscosity, vdiich is commonly reported in the poise or centipoise units, the [h] value is reported in the dimension of inverse concentration xinits of dl/g, for exanple. The value of [hi for a linear polymer in a specific solvent is related to the polymer molecular weight (M) through the Mark-Houwink equation ... 

The Mark-Houwink equation for the linear polystyrene/THF system at 25 C was determined accurately by W. Graessley et al. We used their equation as follows ... 

It is of some interest that the mechanical treatment of poly(vinyI chloride) under conditions of high shear stress in the melt, in a Brabender Plastograph, was found to increase the degree of LCB, as estimated from deviations from the Mark-Houwink equation 204). This is presumably due to chain scission under the influence of the shear stress followed by attack of the radicals produced on other polymer chains, and subsequent recombination reactions. 

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