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.     

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 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.     

Analysis of polymerization rates in radical polymerization with primary radical termination of some methacrylates

Polymerization rates in polymerizations with primary radical termination of ethyl methacrylate, β-phenylethyl methacrylate, β-methoxyethyl methacrylate, and phenyl methacrylate initiated by 2,2'-azobis-(2,4-dimethylvaleronitrile) at 60°C were analyzed by using a simple linear equation. The values obtained of k_ti/k_ik_p (where k_ti is the primary radical termination rate constant, k_i is the rate constant of addition on to monomer of primary radical, and k_p is the propagation rate constant) on these analyses are discussed on the theoretical base.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Effect of diffusion on rates and molecular weights in graft polymerization

A theoretical analysis has been made of the graft polymerization process in terms of the quantitative interrelationship between the initiation rate R_i, the k_p/k_t^1/ ratio of the monomer, the equilibrium solubility M of the monomer in the polymer, the polymer film thickness L, and the diffusivity D of the monomer in the polymer. It is shown how the values of these parameters in any grafting system interact to lead to diffusion-controlled graft polymerization. Whether graft polymerization is diffusion-free or diffusion-controlled depends on the values of K_p, d, k_p/k_p^1/2, and L as gathered in the parameter A = [(K_p/k_t^1/2)R_i, D,/^1/2] L/2. When the values of the various terms are such that A is less than 0.1 (i.e., D is large while R_i, k_p, and L are small), the reaction is diffusion-free. When A is greater than 3 (i.e., D is small while R_i, k_p, and L are large), the reaction is diffusion-controlled. The derived equations showing the relationship between kinetic and diffusional parameters are theoretically applicable to all grafting systems, i.e., for all monomer-polymer combinations under all conditions of reaction temperature, radiation intensity and polymer film thickness. The theoretical analysis has been verified for the rate and degree of polymerization for the radiation-induced graft polymerization of styrene to polyethylene.     

Diffusion control in radiation graft polymerization with varying dependence of rate on monomer concentration

The quantitative effect of diffusion control on the rate of radiation-initiated graft polymerization has been studied theoretically for systems in which the diffusion-free reaction may show various dependencies of rate on monomer concentration other than the usual first-order dependence. The study is also very general in that it can be applied to systems involving a variety of different modes of initiation and termination. Whether the grafting process is diffusion-free or diffusion-controlled has been analyzed in terms of the interaction of the initiation rate R_i, the propagation and termination rate constants k_p and k_t, the equilibrium solubility M of the monomer in the polymer, the polymer film thickness L, the diffusivity D of the monomer in the polymer, and the diffusion-free kinetic order of dependence v of the grafting rate on monomer concentration. The dependence of the grafting rate for both the diffusion-free and diffusion-controlled reactions on these parameters is expressed both by mathematical experssions and graphically. Diffusion control is shown to occur at a critical value of the parameter A which is proportional to L(k_pR_i^w/k_t^zD)^1/2M^(ε?1)/2 where w, z, and v have different values depending on the specific modes of initiation, propagation and termination in a particular grafting system. The grafting rate is shown to vary with the value of A according to specific mathematical expressions. In comparing diffusion-free to diffusion-controlled reaction, it is shown that the former is independent of L and D while the latter is directly dependent on L and inversely on D^1/2. Further, the change from diffusion-free to diffusion-controlled reaction involves a change in the dependence of rate on monomer from v-order to [(v ? 1)/2]-order. The nonsteady-state as well as the steady-state reaction rates have been analyzed.     

Solvent effect on radical polymerization of butyl acrylate

The propagation and termination rate constants k_p and k_t for the radical polymerization of butyl acrylate initiated by biacetyl have been measured by using the rotating-sector method, in various solvents at 30°C. The value of k_p and initiation rate R_i varied with solvents, while the value of k_t did not change with solvents except for benzonitrile. The variation of k_p with aromatic solvents has a trend against Hammett σ_p of the solvent substituents similar to that for methyl methacrylate or phenyl methacrylate except for the value in benzonitrile, when it is larger than the variation for methyl methacrylate or phenyl methacrylate. The larger variation of k_p for butyl acrylate is compatible with the view that the origin of the solvent effect lies in complex formation between the propagating radical and aromatic solvent molecules. The exceptional decrease in k_p and k_t in benzonitrile is explained by a contraction of the poly(butyl acrylate) chain in the poor solvent.     

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.     

Solvent effect on the radical polymerization of di-n-butyl itaconate

Solvent effect on the polymerization of di-n-butyl itaconate (DBI) with dimethyl azobisisobutyrate (MAIB) was investigated at 50 and 61°C. The solvents used were found to affect significantly the polymerization. The polymerization rate (R_p) and the molecular weight of the resulting polymer are lower in more polar solvents. The initiation rate (R_i) by MAIB, however, shows a trend of being rather higher in polar solvents. The stationary state concentration of propagating poly(DBI) radical was determined by ESR in seven solvents. The rate constants of propagation (k_p) and termination (k_t) were evaluated by using R_p, R_i, and the polymer radical concentration observed. The k_p value decreases fairly with increasing polarity of the solvent used, whereas k_t is not so influenced by the solvents. The solvent effect on k_p is explained in terms of a difference in the environment around the terminal radical center of the growing chain. Copolymerization of DBI with styrene (St) was also examined in three solvents with different physical properties. The poly(DBI) radical shows a lower reactivity toward St in a more polar solvent.     

A comparison of the absolute reactivity of vinyl monomers. I. Kinetic studies on radical polymerization of vinyl monomers initiated by a Mn3+–diglycolic acid redox system

The kinetics of polymerization of acrylamide (AM), acrylic acid (AA), and acrylonitrile (AN) initiated by the redox system Mn³⁺–diglycolic acid (DGA) was studied. All three systems followed the same mechanism; namely, initiation by an organic free radical arising from the oxidation of diglycolic acid and termination by the interaction of polymer radicals with Mn³⁺ ion. The rate coefficients k_i/k₀ and k_p/k_t were related to monomer and polymer radical reactivity, respectively. An inverse relation between monomer and polymer radical reactivity was observed. Monomers with higher Q values gave higher k_i/k₀ values but lower k_p/k_t values. The e values of the monomers were important in determining the reactivities of monomers with nearly the same Q values.     

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.