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.     

The Quenched Instationary Polymerization System, 5. The Peak Broadness as an Invariant Quantity of the Number,Molar Mass and Hyper Distributions

The total number, molar mass and hyper distributions generated by quenched instationary polymerization techniques are dominated by the radical chain length distribution (RCLD) whereas the contribution from the polymer chain length distribution (PCLD) is in most cases negligible. For the determination of the rate constant of propagation (k_p) the location of different extraordinary points of the distribution curves is determined by the use of the first and second derivatives. For the number, molar mass and hyper distributions these points are related in an unambiguous way to k_p[M]t_x and can be used to extract k_p. The choice of t_x (duration of the dark period, or an initiation period, or the sum of different periods) depends on the experimental conditions (δ‐pulse, incomplete pre‐effect, combination of periods differing in initiation extent) and is essential for the proper determination of k_p. The broadness of appearing peaks (introduced as the difference between two successive points of inflections) turned out to remain the same irrespective which type of distribution curve was analyzed. Analytical expressions for the peak broadness were derived for different types of quenched instationary polymerization conditions. For δ‐pulse initiation the broadness of the Poisson peak depends simply on the number of propagation steps that occurred whereas for non‐δ‐pulse initiation conditions the peak broadness is governed by the corresponding duration of the initiation period.     

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.     

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.     

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.     

Pulsed Laser Polymerization at Low Conversions: Broadening and Chain Transfer Effects

Pulsed laser polymerization (PLP) is widely employed to measure propagation rate coefficients k_p in free radical polymerization. Various properties of PLP have been established in previous works, mainly using numerical methods. The objective of this paper is to obtain analytical results. We obtain the most general analytical solution for the dead chain molecular weight distribution (MWD) under low conversion conditions which has been hitherto obtained. Simultaneous disproportionation and combination termination processes are treated. The hallmarks of PLP are the dead MWD discontinuities located at integer multiples of n₀ = k_pt₀C_M, where t₀ is the laser period and C_M is the monomer concentration. We show that chain transfer reduces their amplitude by factors , consistent with numerical results obtained by other workers. Here c_tr is the chain transfer coefficient and Ln₀ (L = integer) are the discontinuity locations. Additionally, transfer generates a small amplitude continuous contribution to the MWD. These results generalize earlier analytical results which were obtained for the case of disproportionation only. We also considered two classes of broadening: (i) Poisson broadening of growing living chains and (ii) intrinsic broadening by the MWD measuring equipment (typically gel permeation chromatography, GPC). Broadening smoothes the MWD discontinuities. Under typical PLP experimental conditions, the associated inflection points are very close to the discontinuities of the unbroadened MWD. Previous numerical works have indicated that the optimal procedure is to use the inflection point to infer k_p. We prove that this is a correct procedure provided the GPC resolution σ is better than nequation/tex2gif-stack-1.gif. Otherwise this underestimates Ln₀ by an amount of order σ²/n₀.

Schematic of a chain transfer reaction with monomer as the transfer agent.     

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.     

Chromocene-based catalysts for ethylene polymerization: Kinetic parameters

The chromocene catalyst for ethylene polymerization shows a high response to hydrogen which leads directly to highly saturated polyethylenes containing methyl groups as the major terminal functionality in the polymers. At a polymerization temperature of 90°C the ratio of termination rate constants for hydrogen (k_H) and ethylene (k_M) is k_H/k_M = 3.60 × 10³. The ratio of k_H to the chain propagation constant (k_p) is k_H/k_p = 4.65 × 10^?1 A simple relation that can be derived from polymerization kinetics and the Quackenbos equation exists between melt index and hydrogen–ethylene ratio. A deuterium isotope effect (k_H/k_D) = 1.2 was calculated for the termination reaction. The overall polymerization process has an apparent activation energy of 10.1 kcal/mole. Oxygen addition studies show catalyst activity is proportional to initial divalent chromium content.     

Monte Carlo simulation of the chain length distribution in pulsed-laser polymerization experiments in microemulsion

The Monte Carlo method has been used for numerically simulating pulsed-laser polymerization (PLP) in microemulsion, in order to establish if a shift from inflection point to peak maximum as the best measure of the propagation rate constant, k_p, will occur theoretically. Termination is assumed to be instantaneous in the simulations as droplet sizes can be very small in microemulsions. From the results of the simulations it is found that instantaneous termination indeed causes the peak maximum to become the best measure of k_p. From these results it can be deduced that in bulk it is not simply the Poisson-broadening that causes the peak maximum to yield an overestimation of k_p. This overestimation is rather caused by the fact that the termination rate is finite leading to an asymmetrical peak in the molecular weight distribution. In combination with broadening this yields the inflection point to be the best measure of k_p in the bulk.     

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.     

Intermolecular chain transfer to polymer with chain scission: general treatment and determination of kp/ktr in L,L-lactide polymerization

A general kinetic treatment of the system with intermolecular chain transfer followed by fast reinitiation is given. It leads to the broadening of the molecular weight distribution (MWD), the number of growing chains being invariable. Thus, this system can be considered as a special case of living polymerization. A general method has been elaborated allowing the determination of the ratio of the rate constant of propagation (k_p) to the rate constant of the bimolecular transfer (k⁽²⁾_tr) from the dependence of the MWD on monomer conversion. Numerical values of k_p/k⁽²⁾_tr equal to ≈ 10² and 25 were thus determined for the polymerization of L , L -lactide (L , L -dilactide) initiated with aluminium tris(isopropoxide) trimer ({Al(OⁱPr)₃}₃) and tributyltin ethoxide (ⁿBu₃SnOEt), respectively.     

Modeling Termination Kinetics of Non‐Stationary Free‐Radical Polymerizations

Pulsed‐laser induced polymerization is modeled via an approach presented in a previous paper.^[1] An equation for the time dependence of free‐radical concentration is derived. It is shown that the termination rate coefficient may vary significantly as a function of time after applying the laser pulse despite of the fact that the change in monomer concentration during one experiment is negligible. For the limiting case of t ≫ c^–1 (k_pM)^–1, where c is a dimensionless chain‐transfer constant, k_p the propagation rate coefficient and M the monomer concentration, an analytical expression for k_t is derived. It is also shown that time‐resolved single pulse‐laser polymerization (SP–PLP) experiments can yield the parameters that allow the modeling of k_t in quasi‐stationary polymerization. The influence of inhibitors is also considered. The conditions are analyzed under which M (t) curves recorded at different extents of laser‐induced photo‐initiator decomposition intersect. It is shown that such type of behavior is associated with a chain‐length dependence of k_t.     

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.     

Is the rate constant of chain propagation kp in radical polymerization really chain-length independent?

A careful investigation of the k_p data obtained from pulsed-laser polymerization at different pulse separations t₀ in a lot of systems has revealed that k_p exhibits a slight but significant decrease when t₀ is increased, corresponding to an about 20% decrease of k_p extending over several hundreds in degree of polymerization. Transformation of this integral effect to individual chain-lengths reduces this range, of course, but still shows more than one hundred propagation steps to be concerned. This is interpreted in terms of a decrease of the monomer concentration at the site of propagation caused by the segments already added to the growing chain.     

Propagation Kinetics of Free-Radical Methacrylic Acid Polymerization in Aqueous Solution. The Effect of Concentration and Degree of Ionization

Propagation rate coefficients, k_p, of free-radical methacrylic acid (MAA) polymerization in aqueous solution are presented and discussed. The data has been obtained via the pulsed laser polymerization – size-exclusion chromatography (PLP-SEC) technique within extended ranges of both monomer concentration, from dilute solution up to bulk MAA polymerization, and of degree of ionic dissociation, from non-ionized to fully ionized MAA. A significant decrease of k_p, by about one order of magnitude, has been observed upon increasing monomer concentration in the polymerization of non-ionized MAA. Approximately the same decrease of k_p occurs upon varying the degree of MAA ionization, α, at low MAA concentration from α = 0 to α = 1. With partially ionized MAA, the decrease of k_p upon increasing MAA concentration is distinctly weaker. For fully ionized MAA, the propagation rate coefficient even increases toward higher MAA concentration. The changes of k_p measured as a function of monomer concentration and degree of ionization may be consistently interpreted via transition state theory. The effects on k_p are essentially changes of the Arrhenius pre-exponential factor, which reflects internal rotational mobility of the transition state (TS) structure for propagation. Friction of internal rotation of the TS structure is induced by ionic and/or hydrogen-bonded intermolecular interaction of the activated state with the molecular environment.     

Improving kp Data Originating from PLP Number Distributions

Summary: Based on certain features, especially the width of the so‐called extra peaks in the simulated number‐chain‐length distribution (CLD) of polymers prepared by pulsed laser polymerization (PLP), it is calculated by which factor the positions of the points of inflections and maxima deviate from the theoretical L₀ data that are to be used for the evaluation of k_p. These corrections are for slightly chain‐length‐dependent termination by disproportionation or combination and cover a wide range of chain‐lengths and primary radical production. They can be applied either to the point of inflection on the low‐molecular weight side of the extra peaks or to the peak maximum. On average, the mean error that is about −2.5% for uncorrected data from first‐order points of inflection is reduced to the order of less than 1% even if no assumption concerning the mode of termination is made. The situation is similar for the positions of the first‐order peak maxima where the mean error of about +7% likewise can be cut down to less than 1% if the proper correction function is chosen. Second‐ and third‐order peaks data, which are a priori less falsified, yield still better results after correction. Mass sensitivity of the detection process has comparatively little effect: it is only for unrealistically high extents of chain‐length dependence in detection that considerable falsifications are to be expected. As an additional result it turned out that correction functions obtained for number distributions are also applicable to mass spectrometry raw distributions and even for mass distributions x(l) · l provided Poissonian broadening is the only broadening process.

Number distribution x_C(l) calculated for termination by combination times attenuation function F₁(l).     

STUDIES ON THE KINETICS OF PROPYLENE POLYMERIZATION WITH THE COMPLEXTYPE TiCl_3 CATALYST

The active center concentration C_p, the rate constant k_p, and the activation energy of chain propagation E_p in the polymerization of propylene with complex-type TiCl_3-(C_2H_5)_2AlCl catalyst system were studied. The Mn was corrected by (?) value determined by GPC. The values thus obtained for C_p, k_p, and E_p at 50℃were 3.01 mol/mol Ti, 6.27 1/mol·sec, and 5.10 Kcal/mol respectively.The kinetic parameters were compared with those obtained from conventional TiCl_3·AlCl_2 catalyst, showing that the higher activity of the complex-type catalyst over the conventional catalyst is not only due to the higher C_p of the former, but to a greater extent due to the increase of the k_p value.     

Molecular Weight Distribution in Living Polymerization Proceeding with Reshuffling of Polymer Segments due to Chain Transfer to Polymer with Chain Scission, 2. Monte Carlo Simulation of Polymer Reshuffling Proceeding with Disproportionation of Chain Functionalities

The method for analyzing the reshuffling of polymer segments developed previously has been extended to systems involving the disproportionation of chain functionalities. The effect of interchain exchange reactions of this type, leading to the redistribution of chain lengths and of the chain functionalities (redistribution of living and dead chain ends), was analyzed by means of the Monte Carlo simulations. In the systems, in which no propagation occurs (monomer concentration equal to zero), a set of polymer chains containing one living and one dead end was taken as an initial material. A series of simulations were performed for systems with differing molecular weight distributions of the starting macromolecules. Uniform (no chain length distribution polymer – all chains are of the same length), Poisson, and the most probable (geometric) distributions were taken into consideration. Although the molecular weight distributions (MWDs) of functionally different chains of the same polymer were different apart from the eventual equilibrium conditions, the overall MWD was very close to that observed in analogous systems without disproportionation. The same was observed concerning MWDs in modeled polymerization systems, in which reshuffling and disproportionation accompanied propagation. Consequently, a method of estimating the ratio of rate constants of propagation and reshuffling (i. e. k_p /k _tr) in the relevant polymerization systems, using the observed polydispersity indexes, was proposed. The extent of disproportionation can be evaluated from the determined relationships of the polydispersity index and of the monofunctional chains fraction as functions of the average number of chain transformations.     

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.