Mark-Houwink-Sakurada relation

Using the Mark-Houwink-Sakurada relation [rj] — KMa [7] one finds... 

The intrinsic viscosity contains the information on the conformation and molecular motion of each individual polymer chain. It depends on the molecular weight in the power law (the Mark-Houwink-Sakurada relation)... 

As an approximation, assume that the concentration is near zero, and the [jj] = 2.73 X10 nal/g, equation (3.86). This obviates the extrapolation in Rgure 3.14 that is required for more accurate results. Using the Mark-Houwink-Sakurada relation, equation (3.97) and Table 3.11, we have... 

Figure 12-16 shows plots of the intrinsic viscosity [ ] versus the reaction time, r/%, for four water contents, r. [ j] increases with increasing reaction time. Figure 12-17 shows plots of iog[r]] versus log Mn for the fom solutions. It is known that the Mark-Houwink-Sakurada relation holds for [ ] and Mn (Badgley, 1949 Tsuchida, 1975). 

In Pulling-up-sphere Method , we also showed that the value of the exponent a of the Mark-Houwink-Sakurada relation (equation (12-25)) is indicative of the shape of polymers in the solution. The a value estimated from log[ ] vs. log Mn plots in... 

Equation (6164) is known as the Mark-Houwink-Sakurada relation . It generally holds very well, as is also exemplified by the data obtained for two different solutions of poly(isobutylene) presented in Fig. 6.17. 

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

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 ... 

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. 

An appropriate formalism for Mark-Houwink-Sakurada (M-H-S) equations for copolymers and higher multispecies polymers has been developed, with specific equations for copolymers and terpolymers created by addition across single double bonds in the respective monomers. These relate intrinsic viscosity to both polymer MW and composition. Experimentally determined intrinsic viscosities were obtained for poly(styrene-acrylonitrile) in three solvents, DMF, THF, and MEK, and for poly(styrene-maleic anhydride-methyl methacrylate) in MEK as a function of MW and composition, where SEC/LALLS was used for MW characterization. Results demonstrate both the validity of the generalized equations for these systems and the limitations of the specific (numerical) expressions in particular solvents. 

Benoit and co-workers [18] proposed that the hydrodynamic volume, Vr which is proportional to the product of [17] and M, where [17] is the intrinsic viscosity of the polymer in the SEC eluent, may be used as the universal calibration parameter (Fig. 18.3). For linear polymers, interpretation in terms of molecular weight is straightforward. If the Mark-Houwink-Sakurada constants K and a are known, log [t7]M can be written log M1+ + log K, and VT can be directly related to M. The size-average molecular weight, Mz, is defined by this process ... 

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],... 

Equation (3-43) is the Mark-Houwink-Sakurada (MHS) relation. It appeared empirically before the underlying theory which has just been summarized. 

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. 

Molecular weight is related to [r/] by the Mark-Houwink-Sakurada equation given as... 

Even without the use of a UC curve (one must be generated for each series of measurements), measurement of [t7o] is believed by some to yield an intrinsic viscosity-weighted molar mass [2]. Most importantly, there is a historic interest in the relation of [t ] to molar mass and/or size. Indeed, the study and explanation of UC has occupied the theorists for some time and, accordingly, there are various formulations describing such relationships [2]. For linear polymers, the most popular empirical relationship between [77] and molar mass is the Mark-Houwink-Sakurada (MHS) equation... 

The Mark-Houwink-Sakurada (MHS) relation relates the intrinsic viscosity [t ] to the average molecular weight (MW) in the following form ... 

Intrinsic viscosity [t]] of a polymer solution is related to its viscosity average molecular weight by the Mark-Houwink-Sakurada relationship ... 

Universal Calibration In the conventional calibration (described above), there is a problem when a sample that is chemically different from the standards used to calibrate the column is analyzed. However, this is a common situation for instance, a polyethylene sample is run by GPC while the calibration curve is constructed with polystyrene standards. In this case, the MW obtained with the conventional calibration is a MW related to polystyrene, not to polyethylene. On the other hand, it is very expensive to constmct calibration curves of every polymer that is analyzed by GPC. In order to solve this problem, a universal calibration technique, based on the concept of hydrodynamic volume, is used. As mentioned before, the basic principle behind GPC/SEC is that macromolecules are separated on the basis of their hydrodynamic radius or volume. Therefore, in the universal calibration a relationship is made between the hydrodynamic volume and the retention (or, more properly, elution volume) volume, instead of the relationship between MW and elution volume used in the conventional calibration. The universal calibration theory assumes that two different macromolecules will have the same elution volume if they have the same hydrodynamic volume when they are in the same solvent and at the same temperature. Using this principle and the constants K and a from the Mark-Houwink-Sakurada equation (Eq. 17.18), it is possible to obtain the absolute MW of an unknown polymer. The universal calibration principle works well with linear polymers however, it is not applicable to branched polymers. 

Equations (1.28) to (1.30) do not apply to flexible chain molecules, which are not rigid and can exhibit fluctuations in conformation. Here, one approach is to ignore shape anisometry and to make the assumption v = 2.5 in Eq. (1.30). This enables determination of an impermeable sphere-equivalent hydrodynamic volume for flexible chain macromolecules from [ ], provided that M is known. As noted above, the Mark-Houwink-Sakurada equation [Eq. (1.23)] is often used to relate [ j] to M when dealing with flexible coils. 

Simha, R., On the relation between the Mark-Houwink-Sakurada constants. Journal of Polymer Science Part B, Polymer Physics, 37(15), pp. 1947-1948 (1999). 

According to the Mark-Houwink-Sakurada equation, the intrinsic viscosity [jj] might be related to the molecular weight (M) of a linear pol3mier ... 

REPORTED VALUES OF THE MARK-HOUWINK-SAKURADA PARAMETERS RELATING INTRINSIC VISCOSITY TO MOLECULAR WEIGHT FOR PHB... 

Much of the above discussion indicates that, to study excluded volume effects, an accurate determination of unperturbed dimensions is required. For this, a common procedure is to extrapolate intrinsic viscosity of known molecular weight samples to zero molecular weight. Several extrapolations have been used, notably the Stockmayer-Fixman plot. Dondos and Benoit have now introduced a modified version of this, which appears to be linear over a wider range of molecular weights. It introduces a parameter D, which is shown to be linearly related to the exponent of the Mark-Houwink-Sakurada equation. 

In Table II, the large variations in the parameters for the Mark-Houwink-Sakurada equation [Eq. (I)] relating intrinsic viscosity [/y] to molecular weight M (AT and a are parameters which have to be determined for each polymer-solvent system). 

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