Statistical branching of heterochains

Abstract A general theory for the statistical branching of heterochains is proposed on the basis of the random sampling technique. Consider a polymer mixture that consists of N types of chains whose weight fractions are w i ( i = 1, 2, …, N ), and number‐ and weight‐average chain lengths are P̄ np,i and P̄ wp,i , respectively. Suppose the transition probability that a branch point on a chain of type i is connected to a chain end of a type j chain is given by p ij . When the branching density of chains of type i is ρ i , the weight‐average chain length is given by \documentclass{article}\pagestyle{empty}\begin{document}$\bar P_w = W\sum\nolimits_{m = 0}^\infty T ^m \sum\nolimits_{n = 0}^\infty {SU} ^n 1$\end{document} , where S is the diagonal matrix whose elements are \documentclass{article}\pagestyle{empty}\begin{document}$S_{ii} = \bar P_{wp,i}$\end{document} , 1 is the column vector whose elements are all unity, U is the transition matrix whose elements are given by \documentclass{article}\pagestyle{empty}\begin{document}$u_{ij} = \rho _i p_{ij} P_{np,j} ,T$\end{document} is another transition matrix whose elements are given by t ij = ( w j /w i )U ji , and W is the row vector whose elements are w i . Simpler expressions of P̄ w are presented for binary systems. In addition to the multicomponent systems, the present equation could also be used such as for free‐radical polymerization with long‐chain branching, by considering primary chains formed at different times as different types of polymer chains. For the prediction of the full molecular weight distribution, a Monte Carlo simulation method is used to illustrate the resulting distribution profiles.