Good’s theory of cascade processes applied to the statistics of polymer distributions

Abstract

Important statistics of polymerization reactions, whether of the condensation or addition type, can be calculated rather simply and in a standardized way, by an adaptation of Good's stochastic theory of cascade processes. Examples of such statistics are the various molecular weight averages, the gel point, and the sol fraction. The use of generating functions in the theory greatly reduces the use of probability theory and it allows the direct calculation of the required statistics without the need of explicit expressions for the distributions concerned, or for the summations required in calculating their moments. The generating functions required are mostly combinations of powers of the basic form (1-$\alpha$) $\theta_1$+$\alpha\theta_2$ where $\theta_1$ and $\theta_2$ are dummy variables (or unity) and $\alpha$ some parameter measuring the conversion of a functionality. Ordinary (non-vectorial) generating functions suffice when the system contains essentially one type of repeat unit, but for copolymers (in a broad sense) generalization to vectorial generating functions is required. Calculations of the former type, useful in describing the principles involved, include the calculation of new sol-fraction equations for simple polycondensation reactions, and a somewhat more exact sol-fraction equation for the vulcanization of chains initially distributed randomly in length. Copolymerization systems are then exemplified by calculating the weight-average molecular weight and sol fraction for the system glycerol/adipic acid from general formulae derived. Exact allowance is made for the statistical effects due to complete or partial elimination of water and to the difference in rate of esterification of primary and secondary hydroxyls. The gel-point condition for a system involving copolymerization of s different units is generally found by equating to zero a determinant of order s. The mechanism of polycondensation reactions could be elucidated by comparing experimental measurements of the statistical parameters with those calculated from postulated kinetic schemes using the unified and comparatively simple theory here presented.