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.     

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).     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Laser Single Pulse Initiated RAFT Polymerization for Assessing Chain‐Length Dependent Radical Termination Kinetics

Summary: A novel method combining RAFT polymerization with pulsed‐laser initiation for determining chain‐length dependent termination rate coefficients, k_t, is presented. Degenerative chain‐transfer in RAFT enables single‐pulse pulsed‐laser polymerization (SP‐PLP) traces to be measured on systems with a narrow radical distribution that remains essentially unchanged during the experiment. SP‐PLP‐RAFT experiments at different polymerization times allow for determining k_t as a function of chain length via classical kinetics assuming chain‐length independent k_t.

Single‐pulse pulsed‐laser polymerization trace for BMPT‐mediated RAFT polymerization of butyl acrylate.     

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     

Free‐Radical Propagation Rate Coefficients via n‐Pulse Periodic Polymerization

Aspects of applying n‐pulse periodic initiation in pulsed laser polymerization/size‐exclusion chromatography (PLP/SEC) experiments are studied via simulation of molecular weight distributions (MWDs). In n‐pulse periodic PLP/SEC, sequences of n laser pulses at successive time intervals Δt₁ up to Δt_n are periodically applied. With the dark time intervals being suitably chosen, n‐modal MWDs with n well separated peaks occur. The n‐pulse periodic PLP/SEC method has the potential for providing accurate propagation rate coefficients, k_p. Among several measures for k_p, the differences in molecular weights at the MWD peak positions yield the best estimate of k_p under conditions of medium and high pulse laser‐induced free‐radical concentration. Deducing k_p from n dark time intervals (corresponding to n regions of free‐radical chain length) within one experiment at otherwise identical PLP/SEC conditions allows addressing in more detail a potential chain‐length dependence of k_p. Simulations are compared with experimental data for 2‐pulse periodic polymerization of methyl methacrylate.

Measured MWD (solid line) and associated first derivative curve (dotted line) for a 2‐pulse periodic bulk polymerization experiment of MMA at 20 °C.     

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.     

Modeling the Effects of Primary Radicals in Free‐Radical Polymerization

The effects of non‐ideal initiator decomposition, i.e., decomposition into two primary radicals of different reactivity toward the monomer, and of primary radical termination, on the kinetics of steady‐state free‐radical polymerization are considered. Analytical expressions for the exponent n in the power‐law dependence of polymerization rate on initiation rate are derived for these two situations. Theory predicts that n should be below the classical value of 1/2. In the case of non‐ideal initiator decomposition, n decreases with the size of the dimensionless parameter α ≡ (k_tz /k_dz) √r_ink_t, where k_tz is the termination rate coefficient for the reaction of a non‐propagating primary radical with a macroradical, k_dz is the first‐order decomposition rate coefficient of non‐propagating (passive) radicals, r_in is initiation rate, and k_t is the termination rate coefficient of two active radicals. In the case of primary radical termination, n decreases with the size of the dimensionless parameter β ≡ k_t,s r_in^1/2/k_p,s M r_t,l^1/2, where k_t,s is the termination rate coefficients for the reaction of a primary (“short”) radical with a macroradical, k_t,l is the termination rate coefficients of two large radicals, k_p,s is the propagation rate coefficient of primary radicals and M is monomer concentration. As k_t is deduced from coupled parameters such as k_t /k_p, the dependence of k_p on chain length is also briefly discussed. This dependence is particularly pronounced at small chain lengths. Moreover, effects of chain transfer to monomer on n are discussed.     

Free‐Radical Termination Kinetics Studied Using a Novel SP‐PLP‐ESR Technique

Summary: A novel method for measuring termination rate coefficients, k_t, in free‐radical polymerization is presented. A single laser pulse is used to instantaneously produce photoinitiator‐derived radicals. During subsequent polymerization, radical concentration is monitored by time‐resolved electron spin resonance (ESR) spectroscopy. The size of the free radicals, which exhibits a narrow distribution increases linearly with time t, which allows the chain‐length dependence of k_t to be deduced. The method will be illustrated using dodecyl methacrylate polymerization as an example.

Two straight lines provide a very satisfactory representation of the chain‐length dependence of k_t over the entire chain‐length region (c_R = radical concentration).     

On the (im)possibility of evaluating correct individual rate constants of chain propagation kp and chain termination kt by combining kp2/kt and kp/kt data for chain‐length dependent termination

On the basis of simulated data two ways of evaluating individual rate constants by combining k_p²/k_t and k_p /k_t (k_p , k_t = rate constants of chain propagation and termination, respectively) were checked considering the chain‐length dependence of k_t. The first way tried to make use of the fact that pseudostationary polymerization yields data for k_p²/k_t as well as for k_p /k_t referring to the very same experiment, in the second way k_p²/k_t (from steady state experiments) and k_p/k_t data referring to the same mean length of the terminating radical chains were compared. In the first case no meaningful data at all could be obtained because different averages of k_t are operative in the expressions for k_p /k_t and k_p²/k_t. In spite of the comparatively small difference between these two averages (≈15% only) this makes the method collapse. The second way, which can be regarded as an intelligent modification of the “classical” method of determining individual rate constants, at least succeeded in reproducing the correct order of magnitude of the individual rate constants. However, although stationary and pseudostationary experiments independently could be shown to return the same k_t for the same average chain‐length of terminating radicals within extremely narrow limits no reasonable chain‐length dependence of k_t could be derived in this way. The reason is an extreme sensitivity of the pair of equations for k_p/k_t and k_p²/k_t towards small errors and inconsistencies which renders the method unsuccessful even for the high quality simulation data and most probably makes it even collapse for real data. This casts a characteristic light on the unsatisfactory situation with respect to individual rate constants determined in the classical way, regardless of a chain‐length dependence of termination. As a consequence, all efforts of establishing the chain‐length dependence of k_t are recommended to avoid this way and should rather resort to methods based on inserting a directly determined k_p into the equations characteristic of k_p²/k_t or k_p/k_t, properly considering the chain‐length dependent character of k_t.     

Collision between polymeric radicals for termination rate constants

The motion of each polymeric radical during a collision between the polymeric radicals with the same radius is treated as completely random motion. The result obtained is: k_t = 0.250k_s (where k_t is the chain-termination rate constant and k_s is the reaction rate constant between radical chain ends). On taking the motion of the primary radical during a collision between a primary radical and a large polymeric radical to be completely random, the result obtained is: k_ti = 0.250k_si (where k_ti is the primary radical termination rate constant and k_si is the reaction rate constant between primary radical and radical chain end). On substituting k_s for k_si in the second equation, the rate constant obtained becomes the chain termination rate constant between the very small polymeric radical and the very large polymeric radical, and identical to the former equation. This identity indicates that the effect of the difference of the size of the polymeric radicals on the collision process relating to the chain termination rate constant should not be large.     

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.