Non-linear scission/recombination kinetics of living polymerization.

Living polymers are formed by reversible association of primary units (unimers). Generally the chain statistical weight involves a factor sigma < 1 suppressing short chains in comparison with free unimers. Living polymerization is a sharp thermodynamic transition for sigma << 1 which is typically the case. We show that this sharpness has an important effect on the kinetics of living polymerization (one-dimensional association). The kinetic model involves i) the unimer activation step (a transition to an assembly-competent state); ii) the scission/recombination processes providing growth of polymer chains and relaxation of their length distribution. Analyzing the polymerization with no chains but unimers at t = 0, with initial concentration of unimers M greater or approximately M(*) (M(*) is the critical polymerization concentration)), we determine the time evolution of the chain length distribution and find that: 1) for M(*) << M << M(*) /sigma the kinetics is characterized by 5 distinct time stages demarcated by 4 characteristic times t(1), t(2), t(3) and t(*); 2) there are transient regimes (t(1) less or approximately t less or approximately t(3)) when the molecular-weight distribution is strongly non-exponential; 3) the chain scissions are negligible at times shorter than t(2). The chain growth is auto-accelerated for t(1) less or approximately t less or approximately t(2) : the cut-off chain length (= polymerization degree ((n)w) N(1) proportional, variant t(2) in this regime. 4) For t(2) < t < t(3) the length distribution is characterized by essentially 2 non-linear modes; the shorter cut-off length N(1) is decreasing with time in this regime, while the length scale N(2) of the second mode is increasing. (5) The terminal relaxation time of the polymer length distribution, t(*), shows a sharp maximum in the vicinity of M(*); the effective exponent (partial partial differential ln 1/t(*)) divided by (partial partial differential ln M) is as high as approximately sigma(-1/3) just above M(*).