The Cutthroat Competition Between Termination and Transfer to Shape the Kinetics of Radical Polymerization

There is a fascinating interplay between termination and transfer that shapes the kinetics of radical polymerization (RP). In one limit all dead-chain formation is by termination, in the other by transfer. Because of chain-length-dependent termination (CLDT), the rate law for RP takes a different form in each limit. However, common behavior is observed if one instead considers how the average termination rate coefficient varies with average degree of polymerization. Examples are given of using these principles to understand trends in actual RP data, and it is also demonstrated how to extract quantitative information on CLDT from simple steady-state experiments.     

The nature of the chain-length dependence of the propagation rate coefficient and its effect on the kinetics of free-radical polymerization. 1. Small-molecule studies

In this paper we summarize and analyze the currently available small-molecule data, both experimental and theoretical, that is relevant to chain-length-dependent propagation in free-radical polymerization (FRP). We do this in order to appreciate the nature of chain-length-dependent propagation, because workers are becoming increasingly cognizant of its necessity in reaching a complete understanding of FRP kinetics. We show that studies of addition in small-molecule (model) systems support a chain-length dependence (at short chain lengths i) which is described by the following functional form, which therefore can be said to be physically realistic: , where the values of C₁ and i_1/2 are of the order of 10 and 1, respectively. These results are supported by transition state theory, which predicts a very similar behavior for the Arrhenius frequency factor. We illustrate that in systems with low number-average degree of polymerization (DP_n), this chain-length dependence can dramatically affect the observed (chain-length-averaged) propagation rate coefficient 〈k_p〉, which can be significantly higher than the long chain value, k_p. However, this effect is only observed if the activation energy for the first radical addition is similar to that for propagation. In the case that the former is significantly higher (e.g., when choosing a less than optimal initiator or in the case of retardative chain transfer), the chain-length-dependent propagation predicted by our model will not be observed, and in fact a significant lowering of 〈k_p〉 can in cases be expected up to relatively high DP_n.     

A comprehensive Monte Carlo simulation of styrene atom transfer radical polymerization

<正>An optimized and high-performance Monte Carlo simulation is developed to take thorough account of four different cases of termination in styrene ATRP.According to the simulation results,the bimolecular termination rate constant sharply drops throughout the polymerization when either chain-length dependency of termination rate constant,gel effect,or both together is applied to the simulation.In addition,as expected,the initiator is quickly decomposed at the early stages of the polymerization.The concentration of the catalyst in lower oxidation state decreases at first and then plateaus at higher conversion;furthermore,the steady concentration of M_t~nY/L in the polymerization is the highest when the chain-length-dependent diffusion-controlled termination rate constant is employed in the simulation.The rates of deactivation and chain end degradation reactions are also smaller in this case.Therefore,the fraction of dormant chains is higher throughout the reaction and consequently the portion of dead polymers decreases.Besides,molecular weight increases linearly with conversion;however,when neither gel effect nor chain-length dependency of termination rate constant is considered,the molecular weight deviates from linearity at the end of the reaction.The peak of chain length distribution shifts toward higher molecular weight too during the reaction.Finally,the molecular weight distribution broadens at higher conversion;however, the chain length distribution of polymers produced under conditions of applying chain-length-dependent diffusion-controlled termination rate constant is narrower.     

Average termination rate coefficients in emulsion polymerization: Effect of compartmentalization on free‐radical lifetimes

A method is presented by which the time‐dependent average termination rate coefficient in an emulsion polymerization may be calculated as an appropriate average of the chain‐length‐dependent termination rate coefficients. The method takes advantage of the fact that the overall termination rate is dominated by terminations between rapidly moving short radicals and much slower long ones. This termination rate coefficient is suitable for use in the Smith–Ewart equations describing the compartmentalization of radicals in an emulsion polymerization. Rate data in emulsion polymerizations can be quantitatively interpreted if the kinetics fall into one of two categories: zero–one (showing compartmentalization; intraparticle termination is not rate‐determining) or pseudo‐bulk (no compartmentalization; intraparticle termination is rate‐determining). The new method can be used to interpret rate data for systems falling between these categories and also can be used to find termination rate coefficients from Monte Carlo simulations of termination kinetics. The latter is especially useful for predicting and understanding kinetics in controlled radical polymerizations in disperse media. © 2005 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 43: 1076–1089, 2005     

Radiation-induced catalytic polymerization. I. Theory

The kinetics of radiation polymerization on a solid catalyst is discussed, under the condition that only linear termination of the chain takes place. All the kinetic equations are balance equations of particles of each type adsorbed by unit mass of the catalyst, and this makes it possible to account for the effect on the kinetics of the time dependence of the magnitude of the part of its surface on which the reactions we are considering may take place. Integro-differential equations are used for calculating the molecular weight distribution of the resulting polymer; this ensures higher accuracy of the formulas obtained than when differential equations are used and makes it possible to eliminate a number of limitations generally involved in the transition to differential equations. An expression has been found for the molecular weight distribution of the polymer product which allows for the possibility of radiation-induced catalytic polymerization on the resulting adsorbed polymer. Expressions have been derived for the average molecular weight and yield (weight and molecular) of the polymer formed. Asymptotic formulas have been obtained (for large irradiation times) for all the above values. The conclusions that can be drawn concerning the mechanism of the process based on a comparison of the formulas obtained with kinetic curves plotted from experimental data are given. It is shown how such a comparison can be utilized for calculating the rate constants for polymerization and chain termination reactions.     

Quasi-steady-state laws in enzyme kinetics

To understand how enzymes work is essential for understanding life processes. And, in enzyme kinetics, a fundamental assumption is the so-called Quasi-Steady-State Assumption, which has the history of more than 80 years and has been proven very fruitful in analyzing the equations of enzyme kinetics. Many experimental results and numerical results have shown the validity of the assumption. So, an important problem is if it is always true. If it is always true, then it should be a law, not only an assumption. In this paper, we prove mathematically rigorously that it is indeed always true. Hence, it is a law, and we name it the Quasi-Steady-State Law. Actually, more precisely, we have two Quasi-Steady-State Laws. In one of them quasi-steady state means that the concentration of the enzyme-substrate complex remains approximately constant, and in the other it means that the change rate of the concentration of enzyme-substrate complex is extremely tiny.     

Bimolecular radical termination: New perspectives and insights

The reversible addition‐fragmentation chain transfer‐chain length dependent termination (RAFT‐CLD‐T) method has allowed us to answer a number of fundamental questions regarding the mechanism of diffusion‐controlled bimolecular termination in free‐radical polymerization (FRP). We carried out RAFT‐mediated polymerizations of methyl acrylate (MA) in the presence of a star matrix to develop an understanding of the effect of polymer matrix architecture on the termination of linear polyMA radicals and compared this to polystyrene, polymethyl methacrylate, and polyvinyl acetate systems. It was found that the matrix architecture had little or no influence on termination in the dilute regime. However, due to the smaller hydrodynamic volumes of the stars in solution compared to linear polymer of the same molecular weight, the gel onset point occurred at greater conversions, and supported the postulate that chain overlap (or c*) is the main cause for the observed autoacceleration observed in FRP. Other theories based on “short–long” termination or free‐volume should be disregarded. Additionally, since our systems are well below the entanglement molecular weight, entanglements should also be disregarded as the cause of the gel onset. The semidilute regime occurs over a small conversion range and is difficult to quantify. However, we obtain accurate dependencies for termination in the concentrated regime, and observed that the star polymers (through the tethering of the arms) provided constriction points in the matrix that significantly slow the diffusion of linear polymeric radicals. Although, this could at first sight be postulated to be due to reptation, the dependencies showed that reptation could be considered only at very high conversions (close to the glass transition regime). In general, we find from our data that the polymer matrix is much more mobile than what is expected if reptation were to dominate. © 2008 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 46: 3155–3173, 2008     

Monte Carlo simulation of pulsed laser polymerization

Pulsed laser polymerization (PLP) has been simulated using a Monte Carlo procedure. From the results of numerous simulations it has been shown that the molecular weight distribution (MWD) consists primarily of two superimposed distributions. One distribution, a relatively broad background, represents the termination reactions during the dark period; the other, a rather sharply peaked distribution, represents the termination reactions occurring as a consequence of the large number of small radicals produced during the laser pulse. The postulate that the inflection point on the sharp peak can be used to calculate that the propagation rate constant was tested and found to be accurate to within 3%. The relative position of the broad and sharp distributions on the chain length scale determines the qualitative appearance of the overall MWD and is in turn governed by the rate of photoinitiation and the relative values of termination and propagation rate constants. This explains the qualitatively different shapes of MWD which have been experimentally observed. Finally, it is shown that the occurrence of chain length dependent termination reactions precludes the use of an analytical expression to deduce quantitative or qualitative information about the termination reaction from PLP data.     

Linear Chain Termination in the Kinetics of Postpolymerization of Dimethacrylates

The kinetics of postpolymerization (after turning off UV irradiation) of various dimethacrylates differing in their nature and molecular weight was studied over a wide range of temperatures. For each temperature, a series of kinetic curves, varying in the initial conversion during a dark period, was obtained. The proposed kinetic model is based on the following assumptions. The process in the interphase layer at the liquid monomer-solid polymer boundary has the most significant contribution to the kinetics of postpolymerization. Chain termination in the interphase layer occurs by the unimolecular reaction, is controlled by the chain growth rate, and presents the act of self-burying of an active radical in a conformational trap. A wide spectrum of characteristic times is inherent in unimolecular chain termination, and the relaxation function is described by the Kohlraush' stretched exponential law. The rate law obtained agrees well with experimental data. This fact made it possible to estimate the rate constants (k _t) and the activation energies of chain termination and to establish the scale dependence of k _t on the molar concentration [M₀] of the monomer in a block. It is suggested that both the stretched exponential law and the scale k _t-[M₀] dependence are due to a wide spectrum of characteristic times of relaxation exhibiting the properties of a fractal set.     

Kinetic models of sorption: a theoretical analysis

The kinetics of sorption from a solution onto an adsorbent has been explored theoretically. The general analytical solution was obtained for two cases. It has been shown that at high initial concentration of solute (sorbate) the general equation converts to a pseudo-first-order model and at lower initial concentration of solute it converts to a pseudo-second-order model. In other words, the sorption process obeys pseudo-first-order kinetics at high initial concentration of solute, while it obeys pseudo-second-order kinetics model at lower initial concentration of solute. The theoretical results (derived equations) show that the observed rate constants of pseudo-first-order and pseudo-second-order models are combinations of adsorption and desorption rate constants and also initial concentration of solute. The obtained theoretical equations are used to correlate experimental data for sorption kinetics of some solutes on various sorbents. The predictions of the theory are in excellent agreement with the experimental data.     

Calorimetric study of photopolymerisation of divinyl monomers

Isothermal differential scanning calorimetry (photocalorimetry) enables real‐time measurements of rates of polymerisation as functions of irradiation time to be made. Since it is possible to stop initiation at any point in a reaction (by cutting off the light) and to monitor the dark reaction, this method has been widely used for determinations of the (apparent) rate constants of polymerisation of multifunctional monomers at various degrees of double bond conversion. The work reported here presents the results of photocalorimetric studies of polymerisation kinetics of six related acrylates and methacrylates. Two main topics are discussed: the effects of sulphide and ether groups present in the monomers on network formation, and the determination of polymerisation rate constants according to three termination models (monomolecular, bimolecular and mixed).     

Kinetics of the gas-phase thermal chlorination of methane

Based on information concerning the rate constants of elementary steps corrected with consideration for experimental data obtained under laboratory, pilot-plant, and industrial conditions, the kinetics of the gas-phase process of thermal chlorination of methane was considered. The form of the rate equation of the process depends on the mode of chain termination. Of four possible variants, cross termination with the participation of a chlorine atom and a hydrocarbon radical and quadratic-law termination on hydrocarbon radicals are significant under industrial conditions. The fractions of the participation of either of these variants at various degrees of chlorine conversion were determined. Rate equations were found to describe the chlorination of methane, methyl chloride, methylene chloride, and chloroform with cross and quadratic-law chain terminations. The overall kinetic order of these equations with respect to reactants was 1.5. Combined equations that imply the simultaneous occurrence of cross and quadratic-law chain terminations were proposed.     

An exclusive kinetic model for the methylcyclohexane dehydrogenation over alumina-supported Pt catalysts

The experimental data of six different types of Pt-containing alumina catalysts are used to study the detailed and rigorous kinetics of the methylcyclohexane dehydrogenation reaction. A large number of kinetic rate equations were formulated using the power law kinetics and the Langmuir-Hinshelwood-Hougen-Watson (LHHW) kinetics, which were tested against the experimental data. For each catalyst, the elementary reaction step “the loss of first molecular hydrogen” in the LHHW single-site surface reaction mechanism was observed to be the rate-determining step. The form of the kinetic rate model developed in the study is believed to be applicable to any Pt-loaded alumina catalyst.     

A Two‐Step Interval and Modified Rosenbrock Method for Kinetic Parameter Estimation in Copolymerization Systems: A Robust and Reliable Approach

Summary: The kinetics of solution free radical copolymerization of isobutyl methacrylate (i‐BMA) and lauryl methacrylate (LMA) in benzene, initiated with 2,2‐azoisobutyronitrile (AIBN) were studied at different monomer feed compositions at low conversion levels. In order to avoid the complications of copolymerization kinetics, the pseudo‐kinetic rate constant method was applied in constant and variable volume polymerization systems. A two‐step procedure based on interval analysis and the modified Rosenbrock method was used to estimate the kinetic parameters of copolymerization. In the first step, initiation, coupled propagation‐termination and transfer rate parameters were determined from steady state kinetic equations using interval analysis. Since the objective function is non‐linear, non‐convex and has multiple local optima, a robust computational technique, based on the Interval Newton/Generalized Bisection (IN/GB) algorithm, was developed to solve this set of non‐linear algebraic equations. This method was used with mathematical and computational guarantees of certainty to find the global optimum. In the second step, the system of mole balance, population balance and moment equations, which are highly stiff ordinary differential equations, were discritized and solved by the modified Rosenbrock method. The results of the first computational step were inserted as an active or equality constraint in the second step to calculate the individual elementary rate parameters of the reaction. Statistical analysis indicated that the copolymer composition is well described by the terminal unit model (TUM), but the implicit penultimate unit effect (IPUE) model of Fukuda and coworkers is more suitable for describing the rate data. In contrast to most previously studied systems, it was found that propagation and coupled rate parameters are greater than those predicted by the TUM.

Variation of number average molecular weight of the copolymer in polymerization system for various initial monomer feed compositions at different reaction times (solid lines are computed results).     

Emulsion polymerization of vinyl acetate. II

The emulsion polymerization of vinyl acetate was investigated at low ionic strengths and has quite unusual kinetics. The rate of polymerization is dependent on the initiator concentration to the first power and independent of soap concentration. In seeded polymerizations, the rate of polymerization depends on initiator to the 0.8 power, particle concentration to the 0.2 power, and monomer volume to 0.35 power. In all cases the rate of polymerization is almost independent of monomer concentration in the particles until 85–90% conversion. These results were rationalized by the following mechanism: (a) polymerization initiates in the aqueous phase because of the solubility of the monomer and is stabilized there by adsorption of ionic soap on the growing polymer molecule; (b) the growing polymer is swept up by a particle at a degree of polymerization (under our conditions) of about 50–200. Growth continues in the particle. This sweep-up is activation-controlled as both particle and polymer are charged. (c) Chain transfer to the acetyl group of monomer gives a new small radical which cyclizes to the water-soluble butyrolactonyl radical, and reinitiates polymerization in the aqueous phase; (d) the main termination step is reaction of an uncharged butyrolactonyl radical with a growing aqueous polymer radical. A secondary reaction at low ionic strength is sweep-up of an aqueous radical by a particle containing a radical. At high ionic strength, this is the major termination step. The unusual kinetic steps are justified by data from the literature. They are combined with the usual mechanisms operating for vinyl acetate polymerization and kinetic equations are derived and integrated. The integral equations were compared with the experimental data and shown to match it almost completely over the whole range of experimental variables.     

The photocopy method: measuring living chain distributions in radical polymerization

We have developed a method to measure living chain molecular weight distributions (MWDs) in free radical polymerization (FRP). By laser photolysis of photoinhibitor molecules included in the polymerizing mixture, the living chains are instantaneously flooded with small molecule radicals carrying fluorescent labels. These radicals react with living chain radical end groups, kinetically freezing growth of living chains and simultaneously end -labelling them: the living chain population has been photocopied. The living MWD is obtained from subsequent analysis by GPC equipped with fluorescence detection. We have measured low conversion thermally initiated PMMA living MWDs, Exponential behaviour is found for large chain length N, in accord with classical Flory-Schultz theory, but at smaller N we establish strong deviations, consistent with the stretched exponential predicted by modern FRP theory incorporating first principles chain length dependencies of termination rate constants. However, this behaviour may derive at least partially from distortions produced by the photocopying technique which can generate power law or logarithmic forms at small N.     

Evaluation of polymerization rate by chain length dependence on polymer–polymer termination and primary radical termination in radical polymerization

In order to analyze the polymerization rate at high initiation rate and/or low monomer concentration, the rate equations are derived by a rate formulated previously for polymer–polymer termination and another rate for primary radical termination, which is formulated here (both rates depend on chain length of polymer radical). Such equations would be applicable to the kinetic data in the polymerizations of styrene and methyl methacrylate. This shows that the assumption that both rates are independent of chain length overestimates the rate of primary radical termination.     

Thermal decomposition of nickel azide

The results on the thermal decomposition of anhydrous nickel azide prepared by reacting nickel azide solution with AnalaR Me₂CO are reported in the temperature range 490–525 K. The sample starts to decompose immediately after it is raised to the decomposition temperature and the rate of decomposition continuously decreases. The decomposition kinetics have been explained in terms of exponential decay law for α?0.19 and contracting volume law for α > 0.19. The domain of exponential decay law corresponds to the initial decomposition pertaining to surface nucleation and two dimensional surface growth of the product phase, while contracting volume law explains the growth of the product phase into the bulk. The role of defects has been explained by electrical conduction studies on the sample.