Effect of polymerization and depolymerization reactions on molecular size distribution in living polymers

Auluck, F. C.; Gupta, Gopal; Nanda, Vikas

doi:10.1007/bf02846327

Cited by 4 publications

(3 citation statements)

References 3 publications

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“…The presented plots hinted us that plateau level can be expected or at least not excluded . We found the corresponding solution in the paper by Auluck et al, who studied the effect of depolymerization on CLD in batch reversible polymerizations . They found solution for CLD in the intermediate stage of a batch reversible polymerization, when most of monomer is consumed.…”

Section: Results Of Numerical Simulations and Theoretical Analysissupporting

confidence: 74%

“…They did not notice that their solution is in fact also solution for CLD of polymer formed in open reactor at constant monomer concentration [M] = k d / k p . This solution is as follows

f r (X_{i}^{*}) = ([], I_{i - 1} (2 τ) + I_{i} (2 τ)) \times e^{- 2 τ}

where fr (X i *) is the fraction of living i ‐mers, in the brackets we have the sum of modified Bessel functions of orders i − 1 and i , and τ, the normalized (dimensionless) reaction time, is equal to k d × t , t is the real reaction time (for the considered homopolymerization in open reactor τ is also equal to k p [M] × t ).…”

Section: Results Of Numerical Simulations and Theoretical Analysismentioning

confidence: 99%

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Steady State and Equilibrium in Reversible Copolymerization at Constant Comonomer Concentrations

Szymański

Sosnowski

Cypryk

2017

Macro Theory & Simulations

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Selected aspects of copolymerization processes carried on at constant comonomer concentrations are analyzed theoretically and modeled by Monte Carlo method. It is confirmed that some combinations of initial parameters lead to stationary conditions of copolymer formation for both irreversible and reversible systems which can be regarded as the first‐order Markov chain process. However, this study shows that for many copolymerization systems the stationary conditions are attained only at high number‐average degree of polymerization DPn, and for some reversible copolymerizations, attaining equilibrium, stationary conditions are not observed at all. The analysis shows that the chain length distribution (CLD) for copolymerization carried out under steady state conditions at constant comonomer concentrations, equal to equilibrium concentrations for infinitely high DPn, is approximately described by the modified Bessel and exponential functions. This type of CLD is analytically proved and confirmed by the Monte Carlo simulations for the analogous homopolymerization process.

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Section: Results Of Numerical Simulations and Theoretical Analysissupporting

confidence: 74%

“…They did not notice that their solution is in fact also solution for CLD of polymer formed in open reactor at constant monomer concentration [M] = k d / k p . This solution is as follows

f r (X_{i}^{*}) = ([], I_{i - 1} (2 τ) + I_{i} (2 τ)) \times e^{- 2 τ}

Section: Results Of Numerical Simulations and Theoretical Analysismentioning

confidence: 99%

Steady State and Equilibrium in Reversible Copolymerization at Constant Comonomer Concentrations

Szymański

Sosnowski

Cypryk

2017

Macro Theory & Simulations

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“…Further comments on the intermediate stages were made by Nanda and co-workers. 36,37 In 1984 Taganov 38 addressed anew the issue of the time development of the MWD in reversible polymerization, assuming that the polymer chains are long, the initiation is instantaneous, and the rate constants do not depend on chain length. His analysis also predicts a Poisson distribution in the beginning and a Flory-Schulz distribution at equilibrium, with a Gaussian distribution convoluted with the Poisson distribution for the intermediate stage.…”

Section: Stagementioning

confidence: 99%

Living poly(α-methylstyrene) near the polymerization line. VII. Molecular weight distribution in a good solvent

Das

Zhuang

Andrews

et al. 1999

The Journal of Chemical Physics

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We have measured the molecular weight distribution ͑MWD͒ in a case of equilibrium polymerization. We have studied the time development of the MWD of ''living'' bifunctional poly͑␣-methylstyrene͒ in tetrahydrofuran after a quench to 21 K below the polymerization temperature, T p. We see an intermediate Gaussian distribution evolving toward a final exponential distribution, as expected from theoretical considerations. We see a longer equilibration time for the number average molecular weight (M n) as well as for the weight average molecular weight (M w) than for the monomer concentration ͓͑M͔͒, whereas theories predict that M n and ͓M͔ will relax together and that M w will take much longer. We attribute the delayed equilibration and a second peak at about M n /4 to the effects of ionic aggregation of the living polymers. We have also studied the equilibrium MWD of this system as a function of the temperature below T p , and thus as a function of the number average degree of polymerization ͑L͒. These measurements and the time study discussed above are the first experimental evidence that the equilibrium MWD for an organic polymer in a state of equilibrium polymerization is an exponential/Flory-Schulz distribution, and is consistent with scaling predictions. Near T p and at low L, we observe a deviation from the exponential distribution, which may be evidence of the effect of a chain-length dependence of the equilibrium constant for polymerization, or of the effects of polydispersity on correlations due to excluded volume. In addition, the measured L is about two times less than that expected from the initiator concentration; this could result from ionic aggregation or from chain transfer reactions.

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End-growth/evaporation living polymerization kinetics revisited

Semenov

Nyrkova

2011

The Journal of Chemical Physics

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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.

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Effect of polymerization and depolymerization reactions on molecular size distribution in living polymers

Cited by 4 publications

References 3 publications

Steady State and Equilibrium in Reversible Copolymerization at Constant Comonomer Concentrations

Steady State and Equilibrium in Reversible Copolymerization at Constant Comonomer Concentrations

Living poly(α-methylstyrene) near the polymerization line. VII. Molecular weight distribution in a good solvent

End-growth/evaporation living polymerization kinetics revisited

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