Relative and Absolute Measures of Heterogeneity of Polymer Sizes
Jack B. Carmichael · Journal of Macromolecular Science Part A - Chemistry · 1968
The common measure of relative heterogeneity of a polymer sample is the ratio Mw/Mn, where Mw is the weight-average molecular weight and Mn is the number-average molecular weight [1]. A convenient measure of the absolute heterogeneity of polymer sizes is the variance of the molecular weight distribution. The variance [Var(M)] is formulated in terms of Mw and Mn as follows: where E(M) and E(M2) are, respectively, the first and second moments of the polymer molecular weight distribution taken about the origin. The rth moment of the molecular weight distribution about the origin is defined as where Mi represents the molecular weight of polymer containing i units and P(Mi) represents the fraction of polymer molecules with molecular weight Mi. It is easy to show that Mw and Mn can be represented in terms of moments as follows [1,2]: Substitution of E(M2) and E(M) in terms of Mw and Mn into Eq. (1)yields The positive square root of the variance is defined as the standard deviation, δ The relative utilities of MY/Mn and u as measures of breadth of the molecular weight distribution will be illustrated for the Poisson distribution. The concentration distribution for species produced in a living polymerization described by a single rate constant with monomer concentration not held constant is given by [3] and, for second-order disappearance of monomer, where M0 and P0 represent initial concentrations of monomer and initiator, respectively, k is the rate constant, and t is time.