Heterogeneity index

The concentrated solution viscosity measurement yields the weight-average degree of association of active chain ends rather than the more conventional number-average (mole fraction) value. However, the calculation of the equilibrium constant for association, K, can be accomplished if Mw and the heterogeneity index of the polymer sample are known. The latter parameter can be determined via postpolymerization characterization. 

In most cases M w is used for general characterization of the molar mass besides I) = Mw/Mn, the heterogeneity index, is of importance as a simple measure for the spread in chain length. In the first example D = 1.67 for A and 3.03 for B. In the second example D = 2.3. For technical polymers D can have values from 1.1 up to 30. 

This simple rule, though based on the assumption that the molar mass distributions are similar, has practical significance because polymers made in one and the same process, do in general not differ much in their heterogeneity index. 

The ratio Mw/ Mn must by definition be greater than unity for a polydisperse polymer and is known as the polydispersity or heterogeneity index. Its value often is used as a measure of the breadth of the molar mass distribution, though it is a poor substitute for knowledge of the complete distribution curve. Typically Mw/ Mn is in the range 1.5-2, though there are many polymers which have smaller or very much larger values of polydispersity index. A perfectly monodisperse polymer would have Mw/ Mn = 1.00. 

The ratio MJM is called the polydispersity index (also known as heterogeneity index and dispersity index) and is an indication of the broadness of molecular weight distribution. As polydispersity index increases, MWD broadens. If the polymer were a single macromolecule, the polydispersity would be 1.0 and the polymer would be said to be monodisperse. 

Before we leave molecular weight discussion mention should be made of the heterogeneity index (HI), or polydispersity index. This is defined as the weight average molecular weight divided by the number average molecular weight ... 

The other unique features of living polymerizations are that M varies directly with conversion (Fig. 22.3, cf. Fig. 22.1) and the heterogeneity index (poly-dispersity) is 1, or very close to it. 

The value for n, however, cannot be obtained from this plot. The Sips (1949) equation (eq. II) takes the heterogeneity of the polyclonal antiserum into consideration (heterogeneity index is oi). 

The Sips graphic representation is then obtained by plotting log r/ n — r) against log c (Fig. 8.1). The intrinsic affinity constant Ko = 1/c is obtained when log rjiji—r) = 0, and the heterogeneity index equals the slope. An inherent drawback of this procedure is the mathematical insensitivity of a double log plot. Assuming normal distribution (with peak at K, 75% of the affinities would be scattered over a range between 0.16 /ifoand 6 A (Nisonoff et al., 1975). 

In addition to the information on the size of molecules given by the molecular weights M and M , their ratio M /M is an indication of just how broad the differences in the chain lengths of the constituent polymer molecules in a given sample are. That is, this ratio is a measure of polydispersity, and consequently it is often referred to as the heterogeneity index. In an ideal polymer such as a protein, all the polymer molecules are of the same size (M, = or M / = 1). This is not true for synthetic polymers -... 

It can be seen that a heterogeneity index for the most probable distri-... 

The number fraction distribution is shown in Figure 4.1 for v = 50, for which the heterogeneity index... 

The breadth of the distribution can often be gauged by establishing the heterogeneity index For many polymerizations, the most probable value is about... 

Hsieh, H.L., McKinney, O.E., 1966. Relationship between heterogeneity index and kinetic ratio of anionicaUy polymerized polymers. J. Polym. Sci. Polym. Lett. Ed. 4(11), 843-847. 

This relationship is truly valid only for a Gaussian type of distribution of binding constants around the mean Ka for the population, and no conclusions can be drawn from the heterogeneity index about the number of antibodies present when the index is less than 1.0. Because of the imprecision of the measurement, even when the index is 1.0 or close to that figure, more than one antibody may be present (Pincus et aL, 1968). 

Hapten Inhibition.— Varying amounts of the strong hapten p-(p-hydroxybenzeneazo)-ben-zenearsonic acid were added to an optimal mixture of purified antibody and antigen XXX at pS. 7.6 before precipitation had begun. This hapten, in 20-fold excess over the added antigen, was able to inhibit the precipitation almost completely, and gave a heterogeneity index d of a = 1.5. A 50-fold excess of the weaker hapten ben-zenearsonic acid had almost no effect on the amount of precipitate obtained with purified antibody and resorcinol-Rj. These results on hapten inhibition with purified antibody are closely similar to those obtained with antiserum. 

In order to determine the tightness of the molecular weight distribution, we calculate the polydispersity index (PDI) or the heterogeneity index (HI), based on the following simple equation ... 

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