End-growth/evaporation living polymerization kinetics revisited

End-growth/evaporation kinetics in living polymer systems with "association-ready" free unimers (no initiator) is considered theoretically. The study is focused on the systems with long chains (typical aggregation number N ≫ 1) at long times. A closed system of continuous equations is derived and is applied to study the kinetics of the chain length distribution (CLD) following a jump of a parameter (T-jump) inducing a change of the equilibrium mean chain length from N(0) to N. The continuous approach is asymptotically exact for t ≫ t(1), where t(1) is the dimer dissociation time. It yields a number of essentially new analytical results concerning the CLD kinetics in some representative regimes. In particular, we obtained the asymptotically exact CLD response (for N ≫ 1) to a weak T-jump (ε = N(0)∕N - 1 ≪ 1). For arbitrary T-jumps we found that the longest relaxation time t(max ) = 1∕γ is always quadratic in N (γ is the relaxation rate of the slowest normal mode). More precisely t(max )∝4N(2) for N(0) < 2N and t(max )∝NN(0)∕(1 - N∕N(0)) for N(0) > 2N. The mean chain length N(n) is shown to change significantly during the intermediate slow relaxation stage t(1) ≪ t ≪ t(max ). We predict that N(n)(t)-N(n)(0)∝√t in the intermediate regime for weak (or moderate) T-jumps. For a deep T-quench inducing strong increase of the equilibrium N(n) (N ≫ N(0) ≫ 1), the mean chain length follows a similar law, N(n)(t)∝√t, while an opposite T-jump (inducing chain shortening, N(0) ≫ N ≫ 1) leads to a power-law decrease of N(n): N(n)(t)∝t(-1∕3). It is also shown that a living polymer system gets strongly polydisperse in the latter regime, the maximum polydispersity index r = N(w)∕N(n) being r∗ ≈ 0.77N(0)∕N ≫ 1. The concentration of free unimers relaxes mainly during the fast process with the characteristic time t(f) ∼ t(1)N(0)∕N(2). A nonexponential CLD dominated by short chains develops as a result of the fast stage in the case of N(0) = 1 and N ≫ 1. The obtained analytical results are supported, in part, by comparison with numerical results found both previously and in the present paper.