Characterization of polymer molecular weight distributions by ratios of molecular weight averages

The molecular weight (MW) distribution of a polymer is characterized by a hierarchy of average MWs and their appropriate combinations. For example, the ratio of the weight-average to the number-average MW is the most frequently used measure of the polydispersity of a polymer. As is well known the lower bound to this ratio is unity, and it has been shown that the upper bound is (m + 1)²/4m, where m = M_max/M_min is the ratio of the highest to the lowest MW of the MW species present in a given polymer. This upper bound corresponds to an extremely bimodal MW distribution of one half weight fraction with M_min and the other half with M_max. The behavior of the upper bound for two special unimodal distributions is investigated: one is the triangular distribution, the other the quadrilateral. The results suggest that the upper bound for all possible unimodal distributions is considerably less than the corresponding general case, especially for large values of m. For example, the maximum ratios for the quadrilateral distribution and the general upper bound are 1.04 and 1.125 for m = 2; 1.43 and 3.205 for m = 10; 2.56 and 25.5 for m = 100; 3.99 and 250.5 for m = 1000, respectively.