Chain‐length dependent termination in rotating sector polymerization, 1. The evaluation of the rate constant of chain propagation kp

The chain‐length distributions (CLDs) of polymers prepared by rotating‐sector (RS) techniques under pseudostationary conditions were simulated for the case of chain‐length dependent termination and analysed for their suitability of determining the rate constant of chain propagation k_p from the positions of their points of inflection. The tendency to underestimate k_p is a little more pronounced than in pulsed‐laser polymerization (PLP) but, interestingly, the situation improves in the presence of chain‐length dependent termination. The estimates also were found to be more precise a) for smaller rates of initiation, b) for higher order points of inflection, c) if termination is by combination, d) if the role played by the shorter one of the two chains becomes less dominant. Taken in all, the determination of k_p from the points of inflection in the CLD of RS‐prepared polymers may well compete with the more famous PLP method, especially if some care is taken with respect to the choice of experimental conditions.     

Absolute propagation rate coefficients in radical polymerization from gel permeation chromatography of polymers produced by intermittent initiation

The pulsed laser polymerization technique is now a well accepted method to determine propagation rate coefficients for radical polymerization from molar mass distributions resulting from intermittent initiation. A simplified apparatus for the periodic photoinitiation is used which is much less expensive than the laser equipment. The usefulness of the simplified equipment was proved by the determination of k_p for styrene at technically relevant temperatures up to 130°C for the first time. Furthermore, careful inspection of the molar mass distribution (mmd) reveals that depending on the reaction conditions, inflection points (L_i) can not only be found at integer multiples of k_p • t_o • [M] but also at 0.5ⁱ • k_p • t_o • [M], i = 1, 2, 3, … . A rule to find the inflection points leading to correct values for k_p is proposed. It is shown that the shape of the mmd inter alia depends on the amount of primary radical termination compared to the termination reaction between growing chains. With dominant primary termination, the maxima of the distribution will give the correct k_p, whereas in the absence of primary termination the inflection points should be used. Experimental conditions like initiator concentration, light intensity etc. may influence the position of the L_i at least to some extent, and so may give a small but principal error or uncertainty in k_p. A new mathematical method for the time-dependent simulation of the resulting mmd is presented which allows the calculations being performed on a PC within an acceptable time.     

Simulation of pseudostationary radical polymerization, 1. Chain-length distributions for arbitrary initiation profiles

A procedure is developed which allows to treat arbitrary periodic initiation profiles (asymmetric and symmetric triangle profiles, sinusoidal profiles, Gaussian profiles etc.) in pseudostationary radical polymerization. Using an iterative method these profiles are transformed into the (likewise periodic) radical profiles and into the chain-length distributions of the resulting polymer in case of termination by disproportionation. These distributions are analysed for the position of their inflection points which may be used for experimental determination of the elementary rate constant of chain propagation k_p. It turned out that for all profiles that have at least one discontinuity (e.g. asymmetric triangle profiles) the position of the point of inflection is a correct measure of k_p for a conveniently wide range of experimental parameters. In case of profiles without discontinuity (symmetric triangle profiles, sinusoidal and Gaussian profiles) the position of the inflection point is shifted to lower values which means that the k_p values determined on this basis will be a little too small. In most cases, however, the error introduced by this fact will not exceed the overall error of the experiment so that in practice the method of determining k_p in pseudostationary polymerization is not restricted to those profiles which exhibit discontinuities.     

Pseudostationary Pulsed Laser Polymerization: A Simple Method for the Direct Determination of Axial Dispersion as Occurring in Size Exclusion Chromatography

Theoretical distribution curves were calculated for several values of pulse separation t₀ and concentrations of initiating radicals ρ formed by each pulse for both types of termination (disproportionation and combination). The absolute and relative peak widths of the additional peaks for the number, molar mass, and hyperdistribution were determined and compared. In all cases, the peak widths of these different distributions became the same with increasing values of C = k_tρt₀ and/or L₀ = k_p[M]t₀, with [M] = monomer concentration and k_t and k_p are the rate constants of termination and propagation, respectively. This is similar to the behavior of Poisson distributions if only the degrees of polymerization are fairly high (P̄_n ≥ 50). The analogy is further supported by the finding that under the same conditions the peak widths themselves approach the theoretical ones for Poisson distributions. Thus, the fulfilment of these two criteria suggests that the various peaks in multimodal distributions should be treated in the same formal way as Poissonian peaks although they clearly originate from a superposition of adjacent Poissonian peaks of different amplitude and, on the whole have a peak width somewhat in excess of the theoretical one of true Poisson peaks of the same P̄_n. The point of inflection can be used as a measure of L₀ without reservation only if the first criterion is not fulfilled. The influence of axial dispersion on the location of the extrema was calculated by use of standard deviations σ_ad,k (0.05 and 0.025) to give “experimental curves” for which a pronounced increase in the absolute and relative peak widths was observed. Based on an assumed additivity of the peak variances, a simple procedure was tested for the direct determination of σ_ad,k. The accuracy increased with increasing peak chain lengths and C values for those values determined from the first peak. The values determined from the second peak in the chain length region between 400–700 were closest to the input value.

The points of inflection on the low and high molecular weight side of the additional first three peaks are given as a function of the peak maximum. The lower and upper lines were calculated with Equation ( 11 ) and ( 12 ), respectively.     

Improving kp Data Originating From Higher Order PLP‐SEC Peaks

Summary: Based on certain features, especially the width of the so‐called extra peaks in the simulated chain length distribution (CLD) of polymers prepared by pulsed laser polymerization (PLP), it is calculated by which factor the positions of the higher order points of inflections and maxima deviate from the theoretical L₀ data that are to be used for the evaluation of k_p. These corrections, which can be put into the form of master equations, are for slightly chain length dependent termination by disproportionation or combination and cover a wide range of chain lengths and primary radical production and a reasonable range of axial dispersion σ_ad,k, caused by the chromatographic device used in the evaluation of the chain length distribution. They can be applied either to the point of inflection on the low molecular weight side of the extra peaks as well as to the peak maximum. For usual extents of column broadening (σ_ad,k ≈ 0.05) the mean error that is about 7% for uncorrected data from second order points of inflection is reduced to the order of 1.5% even if no assumption concerning the mode of termination is made. The situation is a little less satisfactory for the correction of the positions of the second order peak maxima. Third order peak data are a priori less falsified and yield still better results after correction. Thus the proper treatment of higher order peaks helps to extend the range of chain lengths for which highly reliable k_p data can be gained from PLP experiments followed by chromatographic analysis.

Plots of lequation/tex2gif-stack-1.gif/(nL₀) versus lg(L₀) obtained from first order (circles), second order (triangles) and third order (squares) peaks showing uncorrected values in the left diagram and corrected values using correction functions X in the right one, both calculated for σ_ad,k = 0.05. (+) and (×) represent ill‐defined peaks.     

Investigation of the Chain Length Dependence of kp: New Results Obtained with Homogeneous and Heterogeneous Polymerization

New experimental results were collected for the free radical polymerization of styrene by pulsed laser polymerization in solution or in microemulsion. The location of the point of inflection (on the low molecular weight side) and the maximum of the first peak in the chromatograms (measured by size-exclusion chromatography) was used to extract k_p data. The extent of band broadening was determined with narrow polystyrene standards with an assumed Poisson chain length distribution. For a given experiment both k_p values (obtained via the point of inflection and the maximum) were corrected and thus became identical in most cases. Even after the correction, the effect of chain length dependence persists to a higher chain length.     

A Method of Improving the Theoretical Basis of kp Determination from PLP‐SEC Measurements

Making use of hitherto ignored features (such as the peak width) contained in the chain‐length distributions of polymers prepared by pulsed‐laser polymerization (PLP), corrections are calculated from simulated chain‐length distributions for improving the accuracy of the “characteristic chain length” L₀ data on which the evaluation of the propagation rate constant k_p is based. These corrections refer to a wide range of chain lengths and primary radical production, slightly chain‐length‐dependent termination by disproportionation or combination, and a reasonable extent of axial dispersion introduced by the chromatographic device used in the evaluation of the chain‐length distribution. They can be applied to the point of inflection on the low‐molecular‐weight side of the extra peaks as well as to the peak maximum. The remaining mean error which, of course, concerns the evaluation of L₀ only, is shown to be of the order of 1.0–1.5%, if the mode of termination is unknown, and comes down to about half that value if information on the mode of termination is available. Although all the other errors inherent in the size exclusion chromatography (SEC) method are still present, this method constitutes substantial progress with respect to the accuracy of determining k_p data from PLP experiments followed by chromatographic analysis.

Hyper mass distributions calculated for L₀ = 200, C = 5 and b = 0.16 for termination by disproportionation considering Poissonian and Gaussian broadening.     

Monte Carlo simulation of chain length distribution in radical polymerization with transfer reaction

In this paper, the basic principle and a Monte Carlo method are described for numerically simulating the chain-length distribution in radical polymerization with transfer reaction to monomer. The agreement between the simulated and analytical results shows that our algorithm is suitable for systems with transfer reaction. With the simulation algorithm, we confirm that transfer reaction has a similar effect as disproportionation on the molecular weight distribution in radical polymerization with continuous initiation. In the pulsed laser (PL) initiated radical polymerization with transfer reaction, the ‘waves’ on the chain-length distribution profile become weaker as the ratio of transfer reaction rate constant, k_tr, to the propagation rate constant, k_p, is increased in the case with either combination-type or disproportionation-type termination. Moreover, it seems that the combination termination has a broadening effect on the waves. Therefore, k_p can also be determined by precisely locating the inflection point L_o on the chain-length distribution profile for radical polymerization with transfer reaction, unless k_tr is large enough to smear out the waves on the chain-length distribution.     

Termination rate coefficients derived from PLP-SEC data

A novel procedure is outlined by which the termination rate coefficient, k_t, may be deduced from molecular weight and monomer conversion data of pulsed laser polymerization (PLP) – size exclusion chromatography (SEC) experiments. For this k_t analysis only the central part of the molecular weight distribution (MWD) between the first point of inflection (POI), that is also used for k_p analysis, and the third such POI is taken into account. Within this region a characteristic ratio of areas under the MWD is fitted either by using PREDICI or by applying a lumping scheme method. The success of the lumping scheme procedure is demonstrated for the bulk polymerization of butyl methacrylate. The k_t values derived by this method refer to small initial degrees of monomer conversion as are typical of PLP-SEC investigations. The relatively fast and efficient lumping scheme technique is restricted to situations where k_t may be considered independent of chain length and where chain transfer processes are not important.     

Alternating copolymerization of α-methylstyrene and maleic anhydride

Alternating copolymers of α-methylstyrene (α-MeSt) and maleic anhydride (MAn) were prepared by free-radical-initiated polymerization in bulk, benzene, or butanone as solvents. By applying the generalized model described by Shirota and co-workers, the reactivity ratios k_1c/k₁₂ and k_2c/k₂₁ were calculated from the change of copolymerization rate with monomer feed at constant total monomer concentration. From the equation R_p = R_p(f) + R_p(CT) were calculated R_p(f) and R_p(CT), and it was found that in benzene the reaction proceeds predominantly by the addition of CT-complex monomers, while in butanone, cross propagation of free monomers predominates. Termination occurs predominantly by homotermination of α-MeSt macro free radicals, k_t22, although the cross termination k_t21 is also operative.     

Alternating copolymerization of β-methylstyrene and maleic anhydride

Alternating copolymers of β-methylstyrene and maleic anhydride were prepared by free-radical-initiated polymerization in bulk and in toluene as a solvent. The reactivity ratios k_1c/k₁₂ and k_2c/k₂₁ were calculated from the change of copolymerization rate with a monomer feed at a constant total monomer concentration according to the generalized model of Shirota and coworkers. From the equation R_p = R_p(f) + R_p(CT) were calculated R_p(f) and R_p(CT), and it was found that in toluene the copolymerization proceeds predominantly by the addition of CT-complex monomers. Termination occurs predominantly by homotermination of β-methyl-styrene macro free radicals, k_t22, but the cross termination k_t21 is also operative.     

Effect of photoinitiator on the molar mass distribution obtained from a pulsed laser polymerization

The choice of the photoinitiator is not usually regarded as being as important consideration in attempting to determine the value of the propagation rate coefficient k_p from a pulsed laser-initiated polymerization (PLP). It is shown that in fact the choice of the photoinitiator can profoundly influence the success of such experiments. Specially, for a number of monomers it was found that the successful determination of k_p is essentially impossible when 1,1-azodicyclohexanecarbonitrile is employed as a photoinitiator. The likely reason for this phenomenon is discussed from which it would seem appropriate to be wary in using any azo compound as a PLP photoinitiator. From PLP experiments with a non-azo photoinitiator, ambient pressure k_p has been determined for bulk polymerization of methyl methacrylate over an extended temperature range.     

Free Radical Bulk Polymerization of Styrene: Simulation of Molecular Weight Distributions to High Conversion Using Experimentally Obtained Rate Coefficients

Previously obtained experimental conversion‐dependences of the propagation rate coefficient (k_p), the termination rate coefficient (k_t) and the initiator efficiency (f) for the free‐radical bulk polymerization of styrene at 70 °C have been used to simulate the full molecular weight distributions (MWD) to high conversion using the software package PREDICI, providing a robust test of the kinetic model adopted. Satisfactory agreement with the experimental MWD's (GPC) was obtained up to approximately 70% conversion. Beyond 70% conversion, the high MW shoulder that appears was correctly predicted, although the amount of such polymer was somewhat underestimated. This discrepancy is believed to probably have its origin in experimental error in the conversion‐dependences of k_p, k_t and f, in particular k_t, that were employed in the simulations, rather than indicate a more fundamental short‐coming of the model employed.

     

Pseudostationary Polymerization: Improving the Determination of the Rate Constant of Propagation

Pulsed-laser initiated polymerization (PLP) leads to chain length distributions with characteristic extrapeaks. The low molecular weight side points of inflection L_LPI are located near to (multiples) of a specific chain length L₀ which is equal to the product of pulse separation t₀ and propagation frequency k_p[M], i.e. rate constant of propagation k_p times monomer concentration [M], allowing a direct determination of k_p. However, Poissonian broadening inherent in the polymerization process as well as Gaussian broadening due to axial dispersion caused by the size exclusion chromatographic (SEC) device leads to a shift of L_LPI as compared to L₀ – its extent depending on the experimental parameters chosen – which in turn causes an error up to 10–20% in the rate constants evaluated. Fortunately, comparison of the experimental peak width with some sort of theoretical peak width yields several types of correction factors and furthermore master-correction functions which are able to reduce the remaining error on average by at least a factor of 10.     

Emulsion polymerization. V. Lowest theoretical limits of the ratio kt/kp

It is assumed that the propagating polymeric radicals have no diffusive mobility in the very highly viscous medium within latex particles. Chain growth takes place on a lattice when the polymeric radical reacts with a monomer present on a lattice site adjacent to that occupied by the reactive chain end. For termination, two radical chain ends must be positioned on adjacent lattice sites at the same instant. The k_t/k_p ratios calculated with this model are either similar to or somewhat lower than the values determined in emulsion polymerization experiments. A minimum value of k_t/k_p can be calculated with the aid of the rate equation of Part III by assuming that only “living” polymer is produced during emulsion polymerization. This value of k_t/k_p is significantly lower than that calculated by the lattice model. Since the value corresponding to the lattice model gives the slowest practically achievable termination rate, it is concluded from these calculations that emulsion polymerization cannot be carried out under conditions in which chain termination is completely suppressed.     

Variations of Rate Coefficients and Termination Mechanism During the After‐Effects of a Light‐ Induced Polymerization of a Dimethacrylate Monomer

This work was aimed at studying variations in the termination mechanism occurring during the after‐effects of a light‐induced polymerization of a dimethacrylate monomer after the irradiation had been discontinued. The experimental method was based on differential scanning calorimetry. The initiation was stopped at various moments of the reaction corresponding to different degrees of double‐bond conversion (starting conversions). Three termination models: monomolecular, bimolecular, and mixed were used to calculate the ratio of the bimolecular termination and propagation rate coefficients k_t^b/k_p and/or the monomolecular termination rate coefficient k_t^m. The models were determined over short time intervals (conversion increments) of the dark reaction giving different values of rate coefficients for each time interval (interval approximation method). Two‐stage statistical analysis was used to find the model that best reproduced the experimental data obtained for each conversion increment. This enabled variations in the termination mechanism during the after‐effects to be followed. It was found that the termination mechanism changed with the time of the dark reaction from the bimolecular reaction to the mixed reaction when the light was cut off at low and medium double‐bond conversions. At higher starting conversions a monomolecular termination mechanism dominated from the beginning of the dark reaction. The mixed termination model was the only model to describe correctly the variations of rate coefficients in the dark, i. e., the increase in k_t^m and the decreasein the k_t^b/k_p ratio.     

Pulsed‐laser polymerization‐gel permeation chromatographic determination of the propagation‐rate coefficient for the methyl acrylate dimer: A sterically hindered monomer

The methyl acrylate dimer (MAD) is a sterically hindered macromonomer, and the propagating radical can fragment to an unsaturated end group. The propagation‐rate coefficient (k_p) for MAD was obtained by pulsed‐laser polymerization (PLP). The Mark–Houwink–Sakaruda parameters required for the analysis of the molecular weight distributions (MWDs) were obtained by multiple‐detector gel permeation chromatography (GPC) with on‐line viscometry. The small radical created by the fragmentation results in a short‐chain polymer that means the MWD may no longer be given by that expected for “ideal” PLP conditions; simulations suggest that the degree of polymerization required for “ideal” PLP conditions can be obtained from the primary point of inflection provided the GPC traces also show a clear secondary inflection point (radicals terminated by the second, rather than the first, pulse subsequent to initiation). Over the temperature range of 40–75 °C, the data can be best fitted by k_p/dm³ mol^?1 s^?1 = 10^6.1 exp(?29.5 kJ mol^?1), with a moderately large joint confidence interval for the Arrhenius parameters. The data are consistent with an increased activation energy and reduced frequency factor as compared with acrylate or methacrylate; both of these changes can be ascribed to hindrance. © 2001 John Wiley & Sons, Inc. J Polym Sci Part A: Polym Chem 39: 3902–3915, 2001     

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.     

Free‐radical copolymerization of styrene with butyl acrylate. II. Elemental kinetic copolymerization step predictions from homopolymerization data

The free‐radical copolymerization of styrene and butyl acrylate has been carried out in benzene at 50 °C. The lumped k _p/k parameter (where k _p and k _t are the average copolymerization propagation and termination rate constants, respectively) has been determined. Applying the implicit penultimate unit model for the overall copolymerization propagation rate coefficient and the terminal unit effect for the overall copolymerization termination rate coefficient and using the homopolymerization kinetic coefficients, we have found good qualitative agreement between the experimental and theoretical k _p/k values. The variation of the copolymerization rate in solution with respect to the values previously found in bulk has been ascribed to a chain length effect on the copolymerization termination rate coefficient. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 42: 130–136, 2004     

Increase of the average reactivity of active species due to a shift to the more reactive species in ionic propagation accompanied by termination

The presented simulations demonstrate that in polymerizations proceeding on two kinds of species, differing in reactivity and being in equilibrium, the expected decrease of the rate of polymerization due to termination may happen to be compensated by the relative increase of concentration of the more reactive species. This takes place, for instance, in the polymerization proceeding simultaneously on ions and ion pairs if ions are more reactive. Because of termination the total concentration of ionic species during the course of polymerization decreases while the proportion of ions increases due to increasing dilution. The maximum compensation is observed when simultaneously k_(ions)/k_{(ion pairs)} → and K_d/[I]₀ → 0, where k are the propagation rate constants, K_d is the equilibrium constant of dissociation and [I]₀ is the starting concentration of initiator. Then, the degree of compensation (the ratio of the rate with compensation to the rate without termination) is becoming equal to ([P*]/[P*]₀)^1/2, where [P*] is the actual, total concentration of the growing species and [P*]₀ is the initial total concentration (before any termination has taken place).