Monte Carlo simulation of random branching in hyperbranched polymers

We study the branching statistics of hyperbranched polymers formed from a one-pot melt polymerization of AB(2) monomers via Monte Carlo simulation. To simulate the reaction ensemble, we use a 3D percolation-type model where each lattice site of a cubic lattice is assumed to be occupied by a single AB(2) monomer and monomers are only allowed to react with their near neighbors (A reacts exclusively with B). We also allow the possibility that the reactivity ratio kappa of free B groups on linear AB(2) units to those on terminal AB(2) units may be different from unity (the so-called "substitution effect"). We study the molecular weight distribution, fractal structure, loop statistics, and degree of branching as a function of both the fraction of reacted A groups p(A) and the reactivity ratio kappa. For p(A) -> 1, we find that the molecular weight distribution of hyperbranched polymers with different p(A) and kappa collapse remarkably well on to a universal curve of the form n(N)N-w(2) = A(N/N-w)(-tau) exp(-BN/N-w), where n(N) is the number density of HBPs with degree of polymerization N and N-w is the weight-average molecular weight (a function of p(A) and kappa) while A, B, and tau are constants independent of p(A) and kappa. Our most accurate determination of tau yields tau = 1.32 +/- 0.01, which is significantly different from the mean-field value of tau = 1.5. This demonstrates the importance of fluctuations in our system. The fractal dimension of HBP chains in the reaction melt is found to be in excellent agreement with the hyperscaling prediction of d(f) = 3 [Buzza, D. M. A. Eur. Phys. J. E 2004, 13, 79] but significantly different from the mean-field result of d(f) = 4 and the percolation result of d(f) = 2.53. We find that the loop distribution obeys the scaling form (m) proportional to m(-alpha)p(A)(m), where (m) is the number density of loops with degree of polymerization m and alpha approximate to 3 for all kappa. Finally, we find excellent agreement between our simulations and the mean-field predictions for the degree of branching.