End-growth/evaporation living polymerization kinetics revisited

Abstract

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.

Supramolecular Self-Assembly and Radical Kinetics in Conducting Self-Replicating Nanowires

“…This dependence agrees with the theoretical picture of fibril growth due to the reaction rate controlled concatenation of fibrils. Another slow mechanism of fibril growth by end attachment of free TAA molecules (whose number is expected to be very low as concentration c * ≪ c A ) is less likely to be dominant (this mechanism leads to L ∝ √ t , see ref ).…”

Anisotropic Self-Assembly of Supramolecular Polymers and Plasmonic Nanoparticles at the Liquid–Liquid Interface

“…The fibril scission energy E sc ∼ 24.5 k B T (estimated by all-atomic modeling including solvation effects) 48 is high enough to ensure very low critical aggregation concentration. The length distribution of fibrils is defined by the balance between fibril scission and fusion events (the end evaporation mechanism 55 is subdominant for long fibrils): ( n ) + ( m ) ⇌ ( n + m ), that is, c n + m = Kc n c m , where c n is concentration of n -mers. The scission probability is defined by E sc , whereas fusion is associated with entropy loss (confinement to the effective bond volume, v b ), leading to an equilibrium constant of the reaction K = v b e E sc / k B T .…”

On the kinetics of body versus end evaporation and addition of supramolecular polymers

“…[13] Indeed, the prevalent molecular pathway for self-assembly is dictated by the complex molecular structure of the monomers involved, as well as by the type of bonding between monomers that form a polymer. This results in molecular pathways that are more complex than the simple pathway proposed by Oosawa, which is sometimes also referred to as end evaporation and addition [14][15][16].…”