Staudinger-Mark-Houwink equation

In this section we have looked at the Staudinger-Mark-Houwink equation as a purely empirical relationship useful for determining the molecular weight of unknown polymeric solutes. A considerable amount of work has been directed toward understanding the theoretical basis for this result. Although a detailed discussion would take us too far afield, we examine certain special cases of Mark-Houwink a values in the next section. 

Solution Intrinsic viscosities are calculated by direct substitution into the Staudinger-Mark-Houwink equation. For cellulose triacetate,... 

Use these data to evaluate the constants k and a in the Staudinger-Mark-Houwink equation for this system. 

Use these data to evaluate the constants in the Staudinger-Mark-Houwink equation. Are the values obtained consistent with the known facts that 35.4°C is the Flory (0) temperature for polystyrene in cyclohexane while benzene is a good solvent for polystyrene at 40°C. 

Viscosity-Average Molecular Mass Molecular mass determined on the basis of viscosity measurements coupled with an empirical equation such as the Staudinger—Mark—Houwink equation. 

As shown in Figure 2, we found a strong concentration dependence of the reduced viscosity even at low concentrations, which in contrast to the findings with the low molecular a-PMMA samples is not linear. It is worthwhile to note that the order with respect to solvent power is obviously the same as found with the other a-PMMA samples. Calculation of the viscometric molecular weight in toluene and chloroform, based on the Staudinger-Mark-Houwink TSMH)-equation is in very good agreement with the My, -value determined by GPC. Values for the constants K and a are taken from the literature. ... 

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. 

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). 

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. 

Fortunately, Staudinger found, in 1950, that for a series of samples of the same polymer in a given solvent and at a constant temperature, the intrinsic viscosity (or the viscosity number) is related to the molar mass of the polymer by the following equation, known as Mark-Houwink equation or Staudinger equation ... 

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] ... 

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

Equation (9-151) is known as the modified Staudinger equation (originally with Or, = 1) or as the Kuhn-Mark-Houwink-Sakurada equation. It was originally found empirically. K and Qrj are empirical constants obtained by calibration (see also Sections 9.9.7 and 9.9.8 and Figure 9-26). In certain special cases, Qr, can also be theoretically calculated (see Table 9-7). 

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. 

On the basis of the assumption that linear macromolecules can also exist as clusters, Hermann Mark (1895-1992, editor s note) co-operated with the Dutch physical chemist Roelof Houwink (1899-1987, editor s note) in Vienna to continue tanpirical development of Staudinger s viscosity equation (Mark-Houwink equation). [...] The corrections/addi-tions to Staudinger s viscosity law made by Mark and Houwink proved to be correct, but they were still being rejected by Staudinger in the 1950s. ([19], p. 410)... 

Mark-Houwink Equation Staudinger (1932) suggested that the molecular weight M of polymers is proportional to the reduced viscosity ... 

The Mark-Houwink-Staudinger (MHS) equation provides a relationship between the intrinsic viscosity (Equation 8.12b) and the (average) molecular weight (M) of (synthetic) polymers ... 

Two of the early theorists that demolished the Staudinger Law were Kuhn and M.L. Huggins. Huggins derived a theoretical form that depends on the nature of the molecules in solution. This equation is now called the Mark-Houwink equation, but like many other historical artifacts, it traces its origin to Huggins. 

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