Universal hydrodynamic constant

Problem 3.5 Using the universal hydrodynamic constant O = 2.5 x 10 dL cm-3 mol (where dl is deciliters), calculate the intrinsic viscosity for polyethylene of molecular weight M — 10 daltons. Note that for polyethylene, = 7.3.-----------------------------------------... 

Usually the function [Cn) M] (intrinsic viscosity times molecular weight) is used to represent hydrodynamic volume which is plotted versus elution volume. For such a plot the calibration curves of many polymers fall on the same line irrespective of polymer chemical type. Universal calibration methodology usually requires knowledge of Mark-Houwink constants for the polymer/ temperature/solvent system under study. 

From the primary calibration curve based on polystyrene standards and the Mark-Houwink constants for polystyrene (K,a) a universal calibration curve (Z vs. v), based on hydrodynamic volume is constructed. Z is calculated from... 

In summary, the approach outlined here is a straightforward method for determining representative values of viscosity ratios [ n ] MA /[ h ] LB I certainly g values significantly less than 1.0 are expected for such highly branched polymers (33). However, the anomalous dependence of g (v) on M[v1a suggests that 1) the core/shell hydrodynamic configuration and/or chromatographic artifacts invalidate universal calibration, and/or 2) the LB elution behavior does not conform to that of polystyrene in the assumed, constant manner. Further work is necessary to elucidate these points. 

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 ... 

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. 

Empirically, it is known that if different polymers have the same elution volume from a given GPC column, the product [x] M will be constant, where [ ] is determined under the same conditions Tand solvent). This follows from the Einstein-Simha relation, whereby the quantity xi M is proportional to the hydrodynamic volume. The hydrodynamic volume depends on temperature and solvent, and is roughly half the volume calculated from Sg. Using the universal calibration method, molecular weight determinations can be made for a polymer using known standards of a different polymer (Grubisic et al., 1967 Puskas and Hutchinson, 1993 Kuo et al., 1993). These molecular weight standards are commercially available for various polymers. From Eq. (3.13),... 

Yamakawa and co-workers find that a universal scaling relationship exists between and aj , provided that occasional system-specific effects, such as solvent dependence of the unperturbed dimensions [Horita et al., 1993], solvent dependence of the viscosity constant o [Konishi et al., 1991], draining effects in the theta solvent (Konishi et al., 1991], specific solvent interactions, and solvent dependence of the hydrodynamic bead diameter [Tominaga et al., 2002], are taken into account. Thus, in Figure 1.5 we reproduce a plot of log versus log which superimposes values... 

In this equation A and B represent universal constants, R represents a typical linear dimension of the chain (usually the end-to-end distance), and V = hydrodynamic volume of the chain in solution. One may assume that the product [n] M is associated with the volume of the chain. This assumption will be undertaken in a later discussion of MWD utilizing a size separation technique (GPC). 

It should also be noted that two types of interaction contribute to the value of ko- hydrodynamic interactions and static excluded volume interactions. In a good solvent both contributions are of the same order of magnitude, so that, as for hard spheres, we expect positive values for and k. Moreover, some universal ratios can be constructed from these numbers, such as kj[rf. In a 0 solvent, the excluded volume interactions vanish (the second virial coefficient is 2=0) and only the hydrodynamic interactions contribute. In this case, k is negative. At higher concentrations, in the semi-dilute regime, the diffusion becomes cooperative, the diffusion constant goes through a minimum and then increases as a function of concentration. 

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