Statistical branching of heterochains

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 P?_np,i and P?_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 $\bar P_w = W\sum\nolimits_{m = 0}^\infty T ^m \sum\nolimits_{n = 0}^\infty {SU} ^n 1$, where S is the diagonal matrix whose elements are $S_{ii} = \bar P_{wp,i}$, 1 is the column vector whose elements are all unity, U is the transition matrix whose elements are given by $u_{ij} = \rho _i p_{ij} P_{np,j} ,T$ 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 P?_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.