Calibration of the Mark-Houwink-Sakurada Equation

The classical method for determining K and a values of the Mark-Houwink-Sakurada (MHS) equation involves fractionation of a whole polymer into subspecies, or fractions, with narrow molecular weight distributions. An average molecular weight can be determined on each such fraction, by osmometry (M ) or light scattering and, if the fractions are narrow enough, the measured 

---

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. 

Fortunately, there is a thorough study available in which the acmal molar mass of PPP 39 (R = methyl, = dodecyl, and R = 3,5-di-tcrt-butylphenyl) was determined by Ught scattering, osmometry, and size exclusion chromatography using universal calibration and compared with the masses obtained from PS calibration. The universal calibration was done on the basis of the Mark-Houwink-Sakurada equation using nine fractions of 39 with molar masses ranging between 27 < < 189 kg/mol. A calibration... 

Molecular weight calibration from a monomer to several million daltons can be carried out by a variety of techniques. Because narrow standards of p(methyl methacrylate) (pMMA) are available, these are often used. Narrow standards of p(styrene) (pSty) are also available and can be used. Using the Mark-Houwink-Sakurada equation and the parameters for pSty and pMMA, a system calibrated with pSty can give pMMA-equivalent values, and vice versa. 

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... 

The conversion of a calibration curve for one polymer (say, polystyrene, as in Fig. 3-10) to that for another polymer can be accomplished directly if the Mark-Houwink-Sakurada equations are known for both species in the GPC solvent. From Eq, (3-43), one can write... 

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. 

Figure 2. GPC elugram of polymer 39 (H = 113 kg/mol) in THF PS calibration (solid line) universal calibration based on the Mark-Houwink-Sakurada equation (dashed line) and on the work-like chain model (dotted line).

Calibration of the SEC trace using PS standards is a very common practice, probably because alternative calibration methods are scarce. The Mark-Houwink-Sakurada (MHS) equation relates the intrinsic viscosity, [ti], with the molar mass ... 

---