The Quenched Instationary Polymerization System, 4. Combination of Two (or More) Polymerization Periods with Different Rates of Initiation

A generalization of quenched instationary polymerization techniques is presented which includes the combination of at least two periods that differ in the rate of initiation (I₂〈I₁, I₂〉 I₁ /mol·l^–1·s^–1). The radical and polymer chain‐length distributions were calculated (RCLD, PCLD, respectively) and the superimposed distribution curves (RCLD + PCLD) were analyzed with respect to the determination of the rate constant of propagation k_p . In all cases the extrema of the first and second derivatives are related in a simple way to the product k_p[M]t_x . For a combination of n initiation periods n different values for t_x are to be inserted, thus offering the chance of a multiple determination of k_p . The broadness of the appearing peaks is introduced as the difference between the points of inflections and may be useful for the detection of irregularities (e. g., chain‐length‐dependent rate constants).     

The quenched instationary polymerization system, 1. Derivation of the radical and polymer chain length distributions for the case of δ‐pulse like initiation

A new instationary polymerization system is presented including, as an essential element, the complete deactivation of all active radicals by reaction with an inhibitor at a certain time after chain initiation. The complete kinetic scheme, the set of differential equations, as well as the analytical solutions are presented. A proof is presented that the reaction with the inhibitor during the quench period dominates any other possible reaction such as propagation and bimolecular termination. As a result, the radical spectrum present at the beginning of the quench period is converted (almost) completely unchanged and instantaneously into a polymer chain length distribution. The quenched radical spectrum appears as a single additional peak in the experimentally observable total chain length distribution. In the case of δ‐pulse initiation the analytical solutions of the differential equations reduce to a simple poisson distribution for the radical concentrations as a function of time. Theoretical expressions for the maximum and the points of inflections (low and high molecular weight side) were derived and their applicability for the direct determination of k_p was tested. All of them turned out to be equally well suited for this purpose.     

Quenched Instationary Polymerization Systems, 6. The Direct Determination of Axial Dispersion in Size Exclusion Chromatography by Comparison of Theoretical and Ideal Peak Widths

For the direct determination of axial dispersion in size exclusion chromatography a simple method is presented which makes use of the measured and ideal peak widths. The peak width can be defined in two ways: either absolute as the difference of successive points of inflection or relative as the ratio of these points. If the absolute peak width is invariant for the number, molar mass and hyper distribution then this distribution can unambiguously be classified as Poissonian. The relative peak width for such distributions is strictly determined by the experimental parameters. It is demonstrated that axial dispersion only leads to an additive increase in the peak variances for peaks with a relative peak width smaller than 1.25. Thus, it is possible to determine directly the axial dispersion of an experimental size exclusion chromatography set‐up by the use of Poisson distributions prepared by quenched instationary polymerization techniques or any other technique leading to ideal Poisson distributions.     

The Quenched Instationary Polymerization System, 3. Derivation of the Radical and Polymer Chain Length Distributions for Any Combination of Two Periods Differing in Initiation Characteristics

The radical and polymer chain length distributions and their respective convolutes were calculated for any combination of an initiation period followed by a dark period without any initiation. The distributions are characterized by the appearance of an asymmetric (broad) additional peak. The choice of the experimental parameters (i. e. duration of the initiation and the dark period) influences the location of the peak as well as the extension of the complete distribution curve. The first and second derivatives also show a characteristic pattern in the region of the peak. A single experiment yields at least two k_p-values which are determined from different parts of the distribution. These values have to be identical unless k_p is not characterized by a single value. In principle, a single experiment will suffice to display possible irregularities in k_p-constancy.     

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 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 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.     

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.     

New aspects of pseudostationary polymerization and their application

Contrary to the stationary state little thought has been given so far to the general principles of the pseudostationary state. In this discourse an attempt is made to demonstrate that — within wide limits — arbitrary initiation profiles may be used to determine k_p/k_t (k_p = rate constant of chain propagation, k_t = rate constant of chain termination) from the frequency dependence of rate of polymerization (in analogy to the rotating-sector technique) as well as to evaluate k_p from the chain-length distribution (CLD) of samples prepared under pseudostationary conditions. Adverse factors like nonspontaneous transformation of absorbed photons into primary radicals do not invalidate this result. The existence of a universal relationship (independent of the initiation profile) is proved to exist for the second moment of the CLD of samples prepared under pseudostationary initiation conditions for constant (chain-length independent) k_t. Pseudostationarity, however, might be also achieved if not the initiation but the termination is periodically varied. In this case the CLD has a completely different shape but allows determination of k_p likewise. Finally, the case of chain-length dependent k_t is shortly discussed in connection with pulsed-laser initiation. Although the general equation for the second moment of the CLD does not apply any longer for this case some generality appears to exist under these conditions, too.     

The quenched instationary polymerization system, 2. Derivation of the radical and polymer chain length distributions for an incomplete pre‐effect

An instationary polymerization system with constant chain initiation is prevented from attaining a final stationary state by the occurrence of a highly efficient quench reaction which deactivates all radicals present at a certain time t_in. The set of differential equations was used to derive the radical spectrum as a function of polymerization time. These, the numerically calculated polymer chain length distributions and the convolutes of both were analyzed in order to determine the chain lengths of the extrema of the first and the second derivatives. These values were found to be proportional to k_p [M] t_in. The values were recalculated by using already derived correction factors and the deviations from the input parameter were in most cases smaller than 1% but became more pronounced as soon as the total polymer concentration was bigger than the total radical concentration.     

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.     

Dark‐cure studies of cationic photopolymerizations of epoxides: Characterization of the active center lifetime and kinetic rate constants

We have characterized the effective rate constants for termination/trapping (k_t/t) and propagation (k_p) for solvent‐free cationic photopolymerizations of phenyl glycidyl ether for conversions up to 50%. We have performed dark‐cure experiments in which active centers are produced photochemically for a specified period of time until the initiating light is shuttered off, and then the polymerization rate is monitored in the dark. This method is especially well suited for characterizing cationic polymerizations because of the long active center lifetimes. Our analysis provides profiles of the instantaneous kinetic rate constants as functions of conversion (or time). For photopolymerizations of phenyl glycidyl ether initiated with iodonium photoinitiators, k_t/t and k_p remain essentially invariant for conversions up to 50%. For the photoinitiator (tolycumyl) iodonium tetrakis (pentafluorophenyl) borate (IPB), the values of k_t/t at 50 and 60 °C are 0.027 and 0.033 min^?1, respectively. The corresponding values of k_t/t for diaryliodonium hexafluoroantimonate (IHA) are 0.041 and 0.068 min^?1. The values of k_p at 50 °C for IPB and IHA are 0.6 and 0.4 L mol^?1 s^?1, respectively. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 2064–2072, 2003     

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.     

Kinetic and ESR studies on the radical polymerization of α‐n‐(α′‐methylbenzyl) β‐ethyl itaconamates derived from racemic and (s)‐α‐methylbenzylamines

The polymerization of α‐N‐(α′‐methylbenzyl) β‐ethyl itaconamate derived from racemic α‐methylbenzylamine (RS‐MBEI) by initiation with dimethyl 2,2′‐azobisisobutyrate (MAIB) was studied in methanol kinetically and with ESR spectroscopy. The overall activation energy of polymerization was calculated to be 47 kJ/mol, a very low value. The polymerization rate (R_p ) at 60 °C was expressed by R_p = k[MAIB]^0.5±0.05[RS‐MBEI]^2.9±0.1. The rate constants of propagation (k_p ) and termination (k_t ) were determined by ESR. k_p was very low, ranging from 0.3 to 0.8 L/mol s, and increased with the monomer concentration, whereas k_t (4–17 × l0⁴ L/mol s) decreased with the monomer concentration. Such behaviors of k_p and k_t were responsible for the high dependence of R_p on the monomer concentration. R_p depended considerably on the solvent used. S‐MBEI, derived from (S)‐α‐methylbenzylamine, showed somewhat lower homopolymerizability than RS‐MBEI. The k_p value of RS‐MBEI at 60 °C in benzene was 1.5 times that of S‐MBEI. This was explicable in terms of the different molecular associations of RS‐MBEI and S‐MBEI, as analyzed by ¹H NMR. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 4137–4146, 2000     

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.

     

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.     

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.     

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).     

Kilohertz Pulsed‐Laser‐Polymerization: Simultaneous Determination of Backbiting,Secondary, and Tertiary Radical Propagation Rate Coefficients for tert‐Butyl Acrylate

For the first time, a 1000 Hz pulse laser has been applied to determine detailed kinetic rate coefficients from pulsed laser polymerization–size exclusion chromatography experiments. For the monomer tert‐butyl acrylate, apparent propagation rate coefficients k_p^app have been determined in the temperature range of 0–80 °C. k_p^app in the range of few hundreds to close to 50 000 L·mol^–1·s^–1 are determined for low and high pulse frequencies, respectively. The apparent propagation coefficients show a distinct pulse‐frequency dependency, which follows an S‐shape curve. From these curves, rate coefficients for secondary radial propagation (k_p^SPR), backbiting (k_bb), midchain radical propagation (k_p^tert), and the (residual) effective propagation rate (k_p^eff) can be deduced via a herein proposed simple Predici fitting procedure. For k_p^SPR, the activation energy is determined to be (17.9 ± 0.6) kJ·mol^–1 in excellent agreement with literature data. For k_bb, an activation energy of (25.9 ± 2.2) kJ·mol^–1 is deduced.

     

Photoinitiated polymerization of methacrylic monomers in a polystyrene matrix: Kinetic,mechanistic, and structural aspects

The kinetics and mechanism of the photoinitiated polymerization of tetrafunctional and difunctional methacrylic monomers [1,6‐hexanediol dimethacrylate (HDDMA) and 2‐ethylhexyl methacrylate (EHMA)] in a polystyrene (PS) matrix were studied. The aggregation state, vitreous or rubbery, of the monomer/matrix system and the intermolecular strength of attraction in the monomer/matrix and growing macroradical/matrix systems are the principal factors influencing the kinetics and mechanism. For the PS/HDDMA system, where a relatively high intermolecular force of attraction between monomer and matrix and between growing macroradical and matrix occurs, a reaction‐diffusion mechanism takes place at low monomer concentrations (<30–40%) from the beginning of the polymerization. For the PS/EHMA system, which presents low intermolecular attraction between monomer and matrix and between growing macroradical and matrix, the reaction‐diffusion termination is not clear, and a combination of reaction‐diffusion and diffusion‐controlled mechanisms explains better the polymerization for monomer concentrations below 30–40%. For both systems, for which a change from a vitreous state to a rubbery state occurs when the monomer concentration changes from 10 to 20%, the intrinsic reactivity and k_p/k_t^1/2 ratio (where k_p is the propagation kinetic constant and k_t is the termination kinetic constant) increase as a result of a greater mobility of the monomer in the matrix (a greater k_p value). The PS matrix participates in the polymerization process through the formation of benzylic radical, which is bonded to some extent by radical–radical coupling with the growing methacrylic radica, producing grafting on the PS matrix. © 2001 John Wiley & Sons, Inc. J Polym Sci Part A: Polym Chem 39: 2049–2057, 2001