Theory of polydisperse reacting polymer systems

The kinetics of irreversible reactions between polymer chains of different molecular weights are studied, with emphasis on the case of highly reactive end groups. We calculate the rate constant k(N, M) for reaction between chains of lengths N and M respectively, in dilute and semi-dilute solutions and in the melt. In all cases, k(N, M) is dominated by the shortest chain: the limit k(N) ≡ k(N, ∞) is well-defined and scales as if both chains were of length N. In dilute solutions k(N, M) obeys mean field theory, being proportional to the equilibrium reactive group contact probability. For melts and concentrated solutions, k(N, M) follows diffusion-controlled laws: k(N, M) ≈ (R/τ_N)?(M/N) where R_N and τ_N are the coil size and relaxation time of the shortest chain N, and ?(M/N) is a cross-over function describing the approach to the asymptotic form k(N) for M/N ? 1. We calculate the leading contributions to this cross-over function, which has universal forms depending on the concentration regime. The implications of these results for high-conversion free-radical polymerization are discussed.