Modeling the Effects of Chain Length on the Termination Kinetics in Multivinyl Photopolymerizations

A model is presented that predicts photopolymerization kinetics over several orders of magnitude change in initiation rate. The model incorporates polymerization features that have long been assumed negligible when examining multivinyl photopolymerizations. The assumption that radical termination is chain‐length‐independent is relaxed by incorporating a chain‐length‐dependent termination (CLDT) parameter based on Random‐walk theory into the kinetic model. Experiments and modeling of multivinyl free‐radical photopolymerizations clearly demonstrate that CLDT is important at low conversions, where a deviation from the classical square‐root relationship between polymerization rate (R_p) and initiation rate (R_i) is observed (R_p ∝ R _i^α, α = ¹/₂, classically). At moderate conversions, when reaction diffusion dominates termination, a transition region is observed from a chain‐length‐dependent to a chain‐length‐independent region. During this transition, long chain – long chain termination is reaction diffusion controlled while the short chain – short chain termination event remains translational and segmental diffusion controlled. The scaling exponent, α, gradually increases throughout this region until achieving the classical value, where once attained, a plateau is observed. Chain‐length effects were also examined by including chain‐transfer (CT) reactions into the kinetic expressions. Upon CT agent addition, a transition region is still observed; however, at low conversion, α adheres more closely to the classical predictions. Most importantly, the model clearly demonstrates a transition from a CLDT region at low conversion to reaction diffusion controlled termination region at high conversion, where chain length is unimportant.     

The Importance of Chain-Length Dependent Kinetics in Free-Radical Polymerization: A Preliminary Guide

The effect of chain-length dependent propagation at short chain lengths on the observed kinetics in low-conversion free-radical polymerization (frp) is investigated. It is shown that although the values of individual propagation rate coefficients quickly converge to the high chain length value (at chain lengths, i, of about 10), its effect on the average propagation rate coefficients, 〈k_p〉, in conventional frp may be noticeable in systems with an average degree of polymerization (DP_n) of up to 100. Furthermore it is shown that, unless the system is significantly retarded, the chain-length dependence of the average termination rate coefficient, 〈k_t〉, is not affected by the presence of chain-length dependent propagation and that there exists a simple (fairly general) scaling law between 〈k_t〉 and DP_n. This latter scaling law is a good reflection of the dependence of the termination rate coefficient between two i-meric radicals, k, on i. Although simple expressions seem to exist to describe the dependence of 〈k_p〉 on DP_n, the limited data available to date does not allow the generalization of these expressions.     

Average termination rate coefficients in emulsion polymerization: Effect of compartmentalization on free‐radical lifetimes

A method is presented by which the time‐dependent average termination rate coefficient in an emulsion polymerization may be calculated as an appropriate average of the chain‐length‐dependent termination rate coefficients. The method takes advantage of the fact that the overall termination rate is dominated by terminations between rapidly moving short radicals and much slower long ones. This termination rate coefficient is suitable for use in the Smith–Ewart equations describing the compartmentalization of radicals in an emulsion polymerization. Rate data in emulsion polymerizations can be quantitatively interpreted if the kinetics fall into one of two categories: zero–one (showing compartmentalization; intraparticle termination is not rate‐determining) or pseudo‐bulk (no compartmentalization; intraparticle termination is rate‐determining). The new method can be used to interpret rate data for systems falling between these categories and also can be used to find termination rate coefficients from Monte Carlo simulations of termination kinetics. The latter is especially useful for predicting and understanding kinetics in controlled radical polymerizations in disperse media. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 1076–1089, 2005     

Miniemulsion polymerization of styrene with chain transfer agent as cosurfactant

Miniemulsion polymerization of styrene with the chain transfer agent n-dodecyl mercaptan (DDM) used as cosurfactant was studied. Droplet size and shelf life for unpolymerized miniemulsions were measured and compared with those of equivalent macroemulsions. The miniemulsion monomer droplets with dodecyl mercaptan as cosurfactant were very stable. Shelf lives were from 17 h to 3 months. The kinetics of miniemulsion polymerization were studied. Unlike other miniemulsion systems where the cosurfactant does not act as a chain transfer agent, the polymerization rate falls with cosurfactant level because the chain transfer agent enhances radical desorption from the particles. The polymerization rate in all the miniemulsions was lower than that of the corresponding macroemulsions. Polymerized particles were larger than in the corresponding macroemulsions, but molecular weights were lower. Results indicate that DDM can serve as an effective cosurfactant as well as a chain transfer agent. The fact that the molecular weights are lower in the miniemulsion reactions indicates predominant droplet nucleation. © 1997 John Wiley & Sons, Inc.     

Chain length‐dependent termination in pulsed‐laser polymerization. VIII. The temperature dependence of the rate coefficient of bimolecular termination in the bulk polymerization of styrene

The photosensitized polymerization of styrene in bulk was investigated in the temperature range of 25–70°C with respect to the average rate coefficient of bimolecular chain termination k̄_t, especially its chain length dependence at low conversions, by means of pulsed laser polymerization (PLP). Three methods were applied: two of them were based on equations originally derived for chain length independent termination taking the quantity k_t contained therein as an average k̄_t, while the third one consisted in a nonlinear fit of the experimental chain length distribution (CLD) obtained at very low pulse frequencies (LF‐PLP) to a theoretical equation. The exponent b characterizing the extent of chain length dependence was unanimously found to decrease from about 0.17–0.20 at 25°C to 0.08–0.11 at 70°C, slightly depending on which of the three methods was chosen. This trend toward more “ideal” polymerization kinetics with rise of polymerization temperature is tentatively ascribed to a quite general type of polymer solution behavior that consists in a (slow) approach to a lower critical solution temperature (LCST), which is associated with a decrease of the solvent quality of the monomer toward the polymer, an effect that should be accompanied with a decrease of the parameter b. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 697–705, 2000     

Gel effect in the bulk reversible addition–fragmentation chain transfer polymerization of methyl methacrylate: Modeling and experiments

A model for the evaluation of the kinetics and the chain length distribution in living/controlled radical polymerization mediated by reversible addition–fragmentation chain transfer (RAFT) in bulk is presented. Using the free volume theory, the model accounts for the diffusion limitations over both termination and RAFT exchange reactions. Model predictions are compared to experimental results of methyl methacrylate polymerization with cumyl dithiobenzoate as a RAFT agent. It is shown that the polymerization retardation observed in living systems at large conversions is well predicted. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 1071–1085, 2006     

Scope for Accessing the Chain Length Dependence of the Termination Rate Coefficient for Disparate Length Radicals in Acrylate Free Radical Polymerization

A method that utilizes reversible addition fragmentation chain transfer (RAFT) chemistry is evaluated on a theoretical basis to deduce the termination rate coefficient for disparate length radicals k in acrylate free radical polymerization, where s and l represent the arbitrary yet disparate chain lengths from either a “short” or “long” RAFT distribution. The method is based on a previously developed method for elucidation of k for the model monomer system styrene. The method was expanded to account for intramolecular chain transfer (i.e., the formation of mid-chain radicals via backbiting) and the free radical polymerization kinetic parameters of methyl acrylate. Simulations show that the method's predictive capability is sensitive to the polymerization rate's dependence on monomer concentration, i.e., the virtual monomer reaction order, which varies with the termination rate coefficient's value and chain length dependence. However, attaining the virtual monomer reaction order is a facile process and once known the method developed here that accounts for mid-chain radicals and virtual monomer reaction orders other than one seems robust enough to elucidate the chain length dependence of k for the more complex acrylate free radical polymerization.     

Photopolymerization of methyl methacrylate with azo-containing polydimethylsiloxane as photoinitiator: Effect of siloxane chain length

The kinetics of the free radical photopolymerization of methyl methacrylate (MMA) initiated by azo-containing polydimethylsiloxane (PSMAI) and azobisisobutyronitrile (AIBN) was investigated. The greater polymerization rate R_p in MMA/PSMAI systems may be due to the higher value of the initiation rate R_i and the lower value of the termination rate constant k_t than that in MMA/AIBN system. The reaction orders with respect to initiators PSMAI decreased with an increase in polydimethylsiloxane chain length (SCL) in PSMAI. The observed deviations in polymerization rate from rate equation could be explained in terms of primary radical termination. The photoinitiator efficiency Φ of initiators decreased with increase in SCL, while the ratio of the rate constants for chain termination and chain initiation by primary radical increased with SCL. The fraction β of primary radicals entering into termination in MMA/PSMAI systems were larger than that in MMA/AIBN system. © 1996 John Wiley & Sons, Inc.     

Neutral nickel oligo- and polymerization catalysts: the importance of alkyl phosphine intermediates in chain termination

An unconventional chain termination reaction has been explored for the SHOP (Shell higher olefin process)-type, anilinotropone, and salicylaldiminato nickel-based oligo- and polymerization catalysts by using density functional theory (DFT). Starting from the tetracoordinate alkyl phosphine complex, the termination reaction was found to involve a rearrangement of the alkyl chain to form a pentacoordinate β-agostic complex, β-hydride elimination, and olefinic chain dissociation and to compete with propagation at sufficiently high phosphine concentration and/or basicity. It provides the first complete and convincing mechanistic rationale for the decreasing chain lengths observed upon increasing phosphine concentration and basicity. The unconventional reaction was found to be a major termination pathway for the SHOP-type catalyst and is very unlikely to lead to branching and olefin isomerization, which is critical for explaining why the SHOP catalyst, in contrast to the anilinotropone and salicylaldiminato catalysts, tends to lead to the oligomerization of ethylene to form linear α-olefins. Based on our results we have proposed a new and extended catalytic cycle for the SHOP-type ethylene oligomerization catalyst. Finally, the importance of the new termination reaction for the SHOP-type catalyst suggests that this reaction may also operate with other ethylene oligomerization nickel catalysts. This prediction was confirmed for a pyrazolonatophosphine catalyst, for which the new termination route was found to be even more facile, which explains the short oligomers produced by this catalyst.     

The Influence of Chain Length‐Dependent Propagation on the Evaluation of the Chain Length Dependence of the Rate Coefficient of Bimolecular Termination, 1. PLP–SEC Methods

Chain length distributions have been calculated for polymers prepared by pulsed laser polymerization (PLP) under the condition that not only chain termination but also chain propagation is subject to chain length dependence. The interplay between these two features is analyzed with the chain length dependence of the rate coefficient of termination k_t introduced in the form of a power law and that of propagation k_p modeled by a Langmuir‐type decrease from an initial value for zero chain length to a constant value for infinite chain lengths. The rather complex situation is governed by two important factors: the first is the extent of the decay of radical concentration [R] during one period under pseudostationary conditions, while the second is that termination events are governed by [R]² while the propagation goes directly with [R]. As a consequence there is no general recommendation possible as to which experimental value of k_p is best taken as a substitute for the correct average of k_p characterizing a specific experiment. The second point, however, is apparently responsible for the pleasant effect that the methods used so far for the determination of k_t and its chain length dependence (i.e., plotting some average of k_t versus the mean chain‐length of terminating radicals on a double‐logarithmic scale) are only subtly wrong with regard to a realistic chain length dependence. This is especially so for the quantity k_t* (the average rate coefficient of termination derived from the rate of polymerization in a PLP system) and its chain length dependence.     

RAFT polymerization kinetics: How long are the cross‐terminating oligomers?

We extend a new model for the kinetics of reversible addition‐fragmentation chain transfer (RAFT) polymerization. The essence of this model is that the termination of the radical intermediate formed by the RAFT process occurs only with very short oligomeric radicals. In this work, we consider cross‐termination of oligomers up to two monomers and an initiator fragment. This model accounts for the absence of three‐armed stars in the molecular weight distribution, which are predicted by other cross‐termination models, since the short third arm makes a negligible difference to the polymer's molecular weight. The model is tested against experiments on styrene mediated by cyano‐isopropyl dithiobenzoate, and ESR experiments of the intermediate radical concentration. By comparing our model to experiments, we may determine the significance of cross‐termination in RAFT kinetics. Our model suggests that to agree with the known data on RAFT kinetics, the majority of cross‐terminating chains are dimeric or shorter. If longer chains are considered in cross‐termination reactions, then significant discrepancies with the experiments (distinguishable star polymers in the molecular weight distribution) and quantum calculations will result. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 47: 3455–3466, 2009     

Diffusion-controlled vinyl polymerization. I. The gel effect

It is well known that the reaction rate and molecular weight of vinyl polymers can change markedly during the course of polymerization and that these changes are due to the influence of diffusion on the termination reaction. The chain length dependence of the termination rate constant has been considered in this work and has resulted in a general method of treating the polymerization kinetics and molecular weight distribution. This method is independent of the form of the chain length dependency and is capable of dealing with both disproportionation and recombination modes of termination. A specific model for the termination rate constant with chain length dependence is proposed and is based on free volume theory and entanglement coupling. Master curves for the characteristics of the reaction rate and molecular weight distribution are presented with the application of this model.     

Gel formation in free radical polymerization via chain transfer and terminal branching

Gel formation in free-radical polymerization via chain transfer to polymer, recombination termination, and terminal branching due to either chain transfer to monomer or disproportionation termination is investigated using the method of moments. It is found that no gel can possibly form in the systems consisting of initiation, propagation, and one of the above reactions. However, systems with the following combination of reactions are found to be capable of gelling. They are: chain transfer to polymer + recombination termination; chain transfer to polymer + terminal branching due to disproportionation termination; and terminal branching due to transfer to monomer + recombination termination. Systems with the following combination of reactions are incapable of gelling; transfer to polymer + terminal branching due to transfer to monomer; and terminal branching due to disproportionation termination + recombination termination. An examination of the gelation mechanisms reveals that the formation of multivinyl macromonomers during the course of polymerization is the reason that systems involving terminal branching gel. Sol/gel diagrams are generated to give critical kinetic parameters required for gelation. It is found that terminal branching does not always promote gelation due to the adverse effect on chain length through chain transfer to monomer and termination by disproportionation, reactions which generate terminal double bonds. © 1994 John Wiley & Sons, Inc.     

Kinetics of polymerization rate based on the dependence of termination rate on chain length

When the structure of a primary radical resembles that of the chain end of the polymer radical, the rate of the primary radical termination is approximately the same as the termination rate between the oligomer radical and the polymer radical. The rate constant of termination between polymer radicals of chain length n and s, which involve the primary radicals, is k_t,ns = const.(ns)^?a. In the polymerization of methacrylonitrile initiated by 2,2′-azobisisobutyronitrile in dimethylformamide at 60.0°C, the value of a is found to be 0.091. From data obtained previously in the bulk polymerization of styrene initiated by 1-azobis-2-phenylethane at 60.0°C, the value of a is found to be 0.167. Because such a values are so large that they are not estimated by the excluded volume, the termination rates are discussed by adding the dependence of the diffusion of the segments to that for chain length.     

Determination of the Mode of Free Radical Termination from Pulsed Laser Polymerization Experiments

The molecular‐weight distribution (MWD), obtained by pulsed laser polymerization (PLP) at the high termination rate limit has been considered for investigating termination kinetics. The proposed methodology takes into account both the composite model for termination and the chain‐length dependencies of propagation for short‐chain and long‐chain radicals. Power‐law expressions are used to represent propagation and termination of long‐chain radicals (where k and k represent the maximum “virtual” rate coefficients for monomeric radicals, and α and β capture the chain‐length dependencies for propagation and termination), with the combined value of (β − α) evaluated from the MWD, after correcting for the influence of the kinetics of short‐chain radicals. A novel method is also developed for determining the mode of termination, δ, from MWDs produced by PLP at the high termination rate limit. Simulations for methyl methacrylate (MMA) polymerization at 25 °C confirm that the method can be applied robustly in the presence of complicating factors such as chain transfer to monomer and SEC broadening. The analysis of an experimental MWD obtained for MMA polymerization at 25 °C results in estimates of 0.14 ± 0.03 for (β − α) and 0.75 ± 0.04 for δ.

     

Degradative chain transfer in vinyl acetate polymerizations using toluene as solvent

In this work, we propose that retardation in vinyl acetate polymerization rate in the presence of toluene is due to degradative chain transfer. The transfer constant to toluene (C_trs) determined using the Mayo method is equal to 3.8 × 10^?3, which is remarkably similar to the value calculated from the rate data, assuming degradative chain transfer (2.7 × 10^?3). Simulations, including chain‐length‐dependent termination, were carried out to compare our degradative chain transfer model with experimental results. The conversion–time profiles showed excellent agreement between experiment and simulation. Good agreement was found for the M_n data as a function of conversion. The experimental and simulation data strongly support the postulate that degradative chain transfer is the dominant kinetic mechanism. © 2007 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 45: 3620–3625, 2007     

Polymerization and reptation,an analytical treatment

Free-radical polymerization without crosslinking is considered. By taking the chain length as a continuous variable the reaction kinetics is treated in a simple way for the stationary state. Equations for instantaneous average degrees of polymerization and related quantities are generalized to allow for chain-length dependence of the termination rate constant. For the Ito model, which treats reptation-controlled termination, approximate versions of these equations are derived in the limits of low and high conversion. The observed structure in the peak of the gel permeation chromatograms is shown not to be related to the bimodality in the instantaneous molecular weight distribution. None of the results is very sensitive to the precise reptation behavior.     

Influence of n‐dodecyl mercaptan as chain transfer agent in dispersion polymerization of styrene

Dispersion polymerization of styrene with n‐dodecyl mercaptans (DDM) as the chain transfer agent was investigated. PS particles with various molecular weight, molecular weight distribution (MWD), and particle diameter were prepared by varying the concentration of DDM and also the addition time of DDM before and after the particle nucleation. The average particle diameter was increased, whereas polymerization rate, molecular weight, and MWD were decreased with increasing DDM concentrations from 0 to 10 wt %. The effect of addition of DDM before and after particle nucleation was studied at 0.4, 0.8, and 1.0 wt % DDM. The addition of DDM before particle nucleation produced PS particles of relatively large particle diameter and low molecular weight when compared with the addition of DDM after particle nucleation. This study shows that particle nucleation occurs in about 5–6 min, which corresponds to the 15–16% conversion, 372–378 nm in D_n , and provides a facile way to control the particle size and interesting information about the particle formation using the delayed addition of DDM. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 6612–6620, 2008     

Kinetic model of reversible addition‐fragmentation chain transfer polymerization in microemulsions

A simplified kinetic model for RAFT microemulsion polymerization has been developed to facilitate the investigation of the effects of slow fragmentation of the intermediate macro‐RAFT radical, termination reactions, and diffusion rate of the chain transfer agent to the locus of polymerization on the control of the polymerization and the rate of monomer conversion. This simplified model captures the experimentally observed decrease in the rate of polymerization, and the shift of the rate maximum to conversions less than the 39% conversion predicted by the Morgan model for uncontrolled microemulsion polymerizations. The model shows that the short, but finite, lifetime of the intermediate macro‐RAFT radical (1.3 × 10^?4–1.3 × 10^?2 s) causes the observed rate retardation in RAFT microemulsion polymerizations of butyl acrylate with the chain transfer agent methyl‐2‐(O‐ethylxanthyl)propionate. The calculated magnitude of the fragmentation rate constant (k_f = 4.0 × 10¹–4.0 × 10³ s^?1) is greater than the literature values for bulk RAFT polymerizations that only consider slow fragmentation of the macro‐RAFT radical and not termination (k_f = 10^?2 s^?1). This is consistent with the finding that slow fragmentation promotes biradical termination in RAFT microemulsion polymerizations. © 2009 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 48: 604–613, 2010     

Polymerization processes at conditions of moderately short kinetic chains by calorimetry

A method has been developed for the determination of the heat of the acts of chain growth q_g and of the summary heats of the acts of chain initiation and termination q^** for the radical polymerization of vinyl monomers at the conditions of moderately short kinetic chains, when the length of the kinetic chain v < 100, but the degree of polymerization is ¯P > 20. The analysis of the experimental data, obtained by investigating the kinetics of polymerization of vinyl monomers in solution by calorimetry has led to the conclusion, that at v > 100 an approximation of the long material chains can be used (when the contribution of q^** is negligibly small) with a permissible experimental error; however, in the region of moderately short chains a correction for q^** must be introduced, in order to obtain the correct kinetics of the process.Translated from Teoreticheskaya i Éksperimental'naya Khimiya, Vol. 24, No. 6, pp. 695–701, November–December, 1988.