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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 }∝4*N* ^{2} for *N* _{0} < 2*N* and *t* _{max }∝*NN* _{0}/(1 − *N*/*N* _{0}) for *N* _{0} > 2*N*. 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 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, , 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** ≈ 0.77*N* _{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.

© 2011 American Institute of Physics

Received 21 December 2010
Accepted 08 February 2011
Published online 15 March 2011

Acknowledgments:
This work was partially supported by the French ANR Grant No. “DYNABLOCKs” (ANR-09-BLAN-0034-01).

Article outline:

I. INTRODUCTION
II. OVERVIEW OF THE END-GROWTH KINETICS
A. The kinetic equations
B. Equilibrium state; parameter jumps
C. The fast relaxation stage
D. The slow stages
III. THE CONTINUOUS APPROXIMATION FOR KINETIC EQUATIONS FOR *N* ≫ 1
IV. RELAXATION AFTER A SMALL PARAMETER JUMP
A. The CLD perturbation analysis
B. Relaxation of the mean lengths *N* _{ n }, *N* _{ w }
C. The fast stage
V. NONLINEAR RELAXATION
A. Relaxation to much longer chains, α = *N*/*N* _{0} ≫ 1
B. Terminal relaxation for α < 0.5 (*N* < *N* _{0}/2)
C. Relaxation to much shorter chains: *N* ≪ *N* _{0} (α ≪ 1)
VI. DISCUSSION
VII. SUMMARY AND CONCLUSIONS

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2011-03-15

2016-02-14

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