Dependence Of Kt Research Articles

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

AbstractSummary: A novel method for measuring termination rate coefficients, kt, 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 kt 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 kt over the entire chain‐length region (cR = radical concentration).magnified imageTwo straight lines provide a very satisfactory representation of the chain‐length dependence of kt over the entire chain‐length region (cR = radical concentration).

The Respiration Rate of Composting Pig Manure

The rate at which oxygen is consumed during composting is a measure of aerobic microbial activity and is linked to the rate of organic material decomposition. The rate of loss in mass is a function of the mass of the degradable organic fraction and is related to oxygen uptake rate by the reaction rate coefficient, k. The decomposition of a pig manure and straw mix was investigated at temperatures between 10°C and 70°C using respirometric techniques. The oxygen concentrations in the reactor were measured continuously for about 4 days and then converted to hourly oxygen uptake rates for each incubation temperature, T. The specific oxygen uptake rate was used to calculate the reaction rate coefficient at T, kT, for the observed fast and slow stages of decomposition. The effect of the environmental factors was taken into account using a multiplicative approach and a relationship, which expressed kT for each stage as a function of T, was formulated. The maximum measured rate of activity occurred during the fast stage at 60°C where kT fast = 0.31 day−1. Activity increased exponentially with the temperature in both stages up to about 60°C. At higher temperatures, the activity slowed but most noticeably in the fast stage. The dependence of kT on T during each stage was described by a double power expression, which predicted that activity would cease around 73°C. The relationships may be used to improve a compost model that is based on a first order reaction rate kinetics for the decomposition of organic material.

Monomolecular chain termination in the kinetics of dimethacrylate postpolymerization

AbstractThe kinetics of postpolymerization (after ultraviolet illumination was stopped) for a number of dimethacrylates that differed by nature and molecular mass was experimentally studied over a wide range of temperatures. A series of kinetic curves that differed by the starting conversion of the dark period of time was obtained for every temperature. The proposed kinetic model of the process is based on the following main principles: (1) the process at an interface on the liquid monomer–solid polymer (micrograins) boundary takes a main share of the kinetics of postpolymerization; (2) chain termination at an interface is monomolecular, is controlled by the chain propagation rate, and represents by itself the self‐burial act of active radicals in the conformation trap; and (3) monomolecular chain termination is characterized by a wide spectrum of characteristic times and that is why the function of the relaxation is described by Kohlrausch's stretched exponential law. The obtained kinetic equation was in good agreement with all of the sets of experimental data. This permitted us to estimate the rate constant of chain termination (kt) and to determine the scaling dependence of kt on the molar‐volumetric concentration of the monomer in bulk [M0]. We assumed that the stretched exponential law and scaling dependence kt from [M0] were characterized by common peculiarities, namely, a wide range of characteristic times of relaxation possessed by a property of the fractal set. © 2003 Wiley Periodicals, Inc. J Appl Polym Sci 91: 2376–2382, 2004

Anharmonicity in mineral physics: A physical interpretation

Mineral physicists often refer to the quasi‐harmonic approximation. This approximation accepts the intrinsically anharmonic effects of positive thermal expansion coefficient, α, and pressure‐dependent bulk modulus, KT. However, for insulators at high temperatures, the classical specific heat, CV, and the product αKT are assumed to be independent of temperature at constant volume. Departures from this approximation are termed anharmonicity. We interpret this distinction using derivatives of the atomic potential function, ϕ(r), with atomic spacing, r. CV and KT are reasonably explained by harmonic bonds, that is, ϕ ∝ (r − a)2, where a is the equilibrium value of r, so that ϕ″ = d2ϕ/dr2 is the only derivative considered. Thermal pressure or expansion and pressure dependence of KT depend on ϕ‴ (referred to as first‐order anharmonic effects). Temperature dependence of CV requires ϕiv and is a second‐order anharmonic effect. The temperature variation of αKT arises from ϕv, a third‐order effect. By calculating the Grüneisen parameter, γ, as the ratio of thermal pressure to thermal energy, we relate its anharmonicity to a ratio of derivatives of ϕ. For all commonly used finite strain theories, this gives a temperature variation of γ at high temperature and constant volume systematically less than that of CV. Anharmonicity of CV, which has been more comprehensively studied, may be a few percent at 2000 K, but decreases strongly with compression, so that anharmonicity of both γ and CV is negligible (less than 1% in the case of γ under deep‐Earth conditions).

Multipulse Initiation in Pulsed Laser and Quenched Instationary Polymerization: Determination of the Propagation and Termination Rate Coefficients for Dicyclohexyl Itaconate Polymerization

The commonly used pulsed laser polymerization technique and the recently introduced method of quenched instationary polymerization were successfully adapted for the measurement of kinetic coefficients in the bulk polymerization of the highly hindered monomer dicyclohexyl itaconate (DCHI) in the temperature range 20−50 °C. The need for very high free radical concentrations in the polymerization system was met via a multipulse laser initiation, in which a burst of laser pulses substitutes for the commonly used single laser pulse. Thus, two new methods are at hand for the investigation of the kinetic parameters. The chain length distributions that are obtained can be easily analyzed, and the data for the propagation rate coefficients, calculated via these two techniques, show excellent agreement. The activation parameters for kp were calculated as Ea = 22.0 kJ mol-1 and A = 1.74 × 104 L mol-1 s-1 for the first method and as Ea = 22.8 kJ mol-1 and A = 2.49 × 104 L mol-1 s-1 for the second method. The activation energy is comparable with the methacrylate series of monomers. The frequency factor is relatively small and reflects the steric hindrance in the transition state caused by the bulky substitution in the monomer (and/or the radical). The quenched instationary polymerization method was utilized to determine the termination rate coefficient, yielding very low values (200−1000 L mol-1 s-1 in the temperature range 20−50 °C). The low kt may well be related to the monomer viscosity. The temperature dependence of kt shows the same extent as that of the viscosity, indicating the diffusion control of the termination reaction. Modeling of the chain length distributions obtained by the quenched instationary polymerization experiments indicates about 25% termination via combination.

Solution of Certain Problems in the Thermal Diffusion of Binary Gases

Two variants of the experimental method of unique determination of the thermal-diffusion ratio (Kt) of binary gases are described. A formula expressing the temperature and concentration dependence of Kt and obtained by a thermodynamic method is proposed.

Termination Kinetics of Free-Radical Methyl Methacrylate - Dodecyl Acrylate and Dodecyl Methacrylate - Methyl Acrylate Copolymerizations

Copolymerization termination rate coefficients (kt) of the methyl methacrylate–dodecyl acrylate (MMA–DA) and dodecyl methacrylate–methyl acrylate (DMA–MA) systems at 40°C and 1000 bar have been measured using the single pulse (SP)–pulsed laser polymerization (PLP) technique. Plateau regions of kt are observed in the initial polymerization period. The region of constant kt increases with the size of the alkyl ester group, e.g. it extends up to at least 60% monomer conversion in DA and DMA homopolymerizations. The plateau kt values of MA and MMA are significantly above the corresponding DA and DMA values. The kt penultimate unit effect model, which uses the so-called geometric mean approximation, is well suited for representation of the dependence on monomer composition of the plateau kt values. The dependence of kt on monomer composition is quite different at low and at high degrees of monomer conversion. The reason behind this is seen in different types of diffusion control being operative at low and at high degrees of monomer conversion. The plateau-type behaviour is assigned to segmental diffusion control, whereas high-conversion kt is controlled by reaction diffusion.

Photocopying Living Chains. 1. Steady-State

We present the first ever measurements of living chain molecular weight distributions (MWDs), ψo(N), in free radical polymerization (FRP), using a new technique, the “photocopy method”. Though living chains are the fundamental objects in FRP, their MWDs have eluded measurement until now, principally due to their very short lifetimes (≲1 s). In the photocopy method, the living population is converted, essentially instantaneously, to a labeled inert one by “photoinhibitor” molecules activated by a short laser pulse. This floods the FRP with photoinhibitor radicals, which ideally (i) are extremely slow to initiate new living chains yet (ii) couple with existing living chains (and each other) at near diffusion-controlled rates and (iii) carry a fluorescent label. Thus, the living chains are “frozen” and labeled. They are subsequently detected selectively using GPC equipped with a fluorescence detector (a second detector simultaneously detects unlabeled chains). We applied the photocopy method to low conversion methyl methacrylate FRP. Our measured MWDs are exponential as predicted by the classical Flory−Schulz theory (which ignores the chain length dependence of the termination rate constant, kt), but only for chains longer than the mean living chain length N̄o. For N < N̄o, our data are consistent with a stretched exponential as predicted by modern FRP theories accounting for N dependence of kt. However, the small N data may also be accounted for by nonideal effects, initiation of new living chains by photoinhibitors, which lead to power law behavior. Another complication is that thermal initiation persists during the photocopying process in its present form. Thus, post-laser-pulse initiated living chains react with photoinhibitor radicals, distorting the measured MWDs from that of the steady-state living chains. From the measured living and dead MWDs, we infer living and dead chain concentrations and mean lengths and the fraction of living chains terminating via coupling. Finally, using reported values of propagation rate constants, we estimate mean living chain lifetime, polymerization rate, and the average termination rate constant.

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 (kpM)–1, where c is a dimensionless chain-transfer constant, kp the propagation rate coefficient and M the monomer concentration, an analytical expression for kt 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 kt 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 kt.

Modeling Termination Kinetics of Non-Stationary Free-Radical Polymerizations

Pulsed-laser induced polymerization is modeled via an approach presented in a previous paper. 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 (kpM)-1, where c is a dimensionless chain-transfer constant, kp the propagation rate coefficient and M the monomer concentration, an analytical expression for kt is derived. It is also shown that time-resolved single pulsed-laser polymerization (SP-PLP) experiments can yield the parameters that allow the modeling of kt 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 chain-length dependence of kt.

On the (im)possibility of evaluating correct individual rate constants of chain propagationkp and chain terminationkt by combiningkp2/kt andkp/kt data for chain-length dependent termination

On the basis of simulated data two ways of evaluating individual rate constants by combining kp2/kt and kp /kt (kp , kt = rate constants of chain propagation and termination, respectively) were checked considering the chain-length dependence of kt. The first way tried to make use of the fact that pseudostationary polymerization yields data for kp2/kt as well as for kp /kt referring to the very same experiment, in the second way kp2/kt (from steady state experiments) and kp/kt 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 kt are operative in the expressions for kp /kt and kp2/kt. 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 kt for the same average chain-length of terminating radicals within extremely narrow limits no reasonable chain-length dependence of kt could be derived in this way. The reason is an extreme sensitivity of the pair of equations for kp/kt and kp2/kt 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 kt are recommended to avoid this way and should rather resort to methods based on inserting a directly determined kp into the equations characteristic of kp2/kt or kp/kt, properly considering the chain-length dependent character of kt.

Modeling termination kinetics in free-radical polymerization using a reduced number of parameters

An approach for modeling chain-length dependent termination rate coefficients is presented. The method is based on the assumption that free-radical chain length may be considered as a continuous variable. As compared to discrete numerical methods, in continuous modeling the number of independent dimensionless parameters can be significantly reduced. As a consequence, for a wide variety of monomers the conversion dependence of kt can be predicted without extensive numerical calculations. The method may also be used to determine polymerization conditions under which simpler models of kt (which neglect effects arising from the dependence of kt on chain length) may be applied. Calculations for methyl methacrylate, styrene, and butyl acrylate bulk polymerizations up to high degrees of monomer conversion show that the impact of chain length on termination varies with conversion and strongly depends on the type of monomer.

Modeling termination kinetics in free-radical polymerization using a reduced number of parameters

An approach for modeling chain-length dependent termination rate coefficients is presented. The method is based on the assumption that free-radical chain length may be considered as a continuous variable. As compared to discrete numerical methods, in continuous modeling the number of independent dimensionless parameters can be significantly reduced. As a consequence, for a wide variety of monomers the conversion dependence of kt can be predicted without extensive numerical calculations. The method may also be used to determine polymerization conditions under which simpler models of kt (which neglect effects arising from the dependence of kt on chain length) may be applied. Calculations for methyl methacrylate, styrene, and butyl acrylate bulk polymerizations up to high degrees of monomer conversion show that the impact of chain length on termination varies with conversion and strongly depends on the type of monomer.

Chain-length dependent termination in pulsed-laser polymerization, 7. The evaluation of the power-law exponent b from the chain-length distribution in the low frequency (single-pulse) limit for the reference systems styrene and methyl methacrylate in bulk at 25°C

The chain-length distribution (CLD) was examined for polymers prepared by “low frequency pulsed laser polymerization” (LF-PLP), i. e. for very long pulse separations in the so-called “low frequency” or “single pulse limit”. The data were fitted to the theoretical CLD which could be derived in a closed form for a chain-length dependent rate coefficient kt and the parameter b that characterizes this chain-length dependence was determined by this fitting procedure. b values close to 0.2 were obtained for styrene as well as for MMA, indicating a moderate chain-length dependence of kt at low conversions. This result, which is in good agreement with data evaluated by other methods in our laboratory, points to the fact that under these conditions end-segment diffusion is the rate-determining step in bimolecular termination. Factors like moderate chain transfer to monomer and uncertainties with respect to the mechanism of termination (combination or disproportionation) appear to have very little influence on this result.

Modeling Insight into the Diffusion-Limited Cause of the Gel Effect in Free Radical Polymerization

The molecular-level cause of the gel effect in free radical polymerization, associated with a decrease in the termination rate parameter with increasing conversion, is studied by experimental and modeling approaches. The predictive Vrentas−Duda free volume theory serves as a basis for modeling as it handles quantitatively the temperature and polymer concentration dependencies of monomer diffusion. The polymer concentration (c) dependencies of various diffusional processes fit an expression of the form [Dm(c)/Dm(0)]ξx,m, where Dm is the monomer diffusion coefficient and ξx,m expresses the power-law dependence of the diffusional process of interest (x) relative to that of monomer (m). Relevant values of ξx,m vary from ∼ 0.8 for segmental mobility, to ∼1−2 or 3 for oligomeric diffusion, to ∼ 3.0−3.5 for unentangled polymer diffusion, and to ∼ 9.5−10 for entangled polymer diffusion. If the very high conversion regime, where issues unrelated to the origin of the gel effect add unnecessary complications, is avoided, the modeling of conversion−time data requires only that the polymer concentration dependence of the termination reaction, caused by a diffusional process cited above, must be taken into account. Comparison of ξx,m values needed to fit various methyl methacrylate conversion−time data (values of 1.1−5.0 were required, increasing with increasing molecular weight of the polymer produced) with those needed for different diffusion types indicates that termination is governed by diffusion of the shortest radical chains present in significant number. In contrast to some “short−long” termination theories that commonly distinguish “short” and “long” as unentangled and entangled, this study finds that entanglements are irrelevant in distinguishing “short” from “long”. Styrene polymerizations yield ξx,m values substantially below 1. This is ascribed to chain transfer, supported by the fact that methyl methacrylate systems with chain transfer agent show similar suppressed gel effect behavior. The very low ξx,m values result in part from the reduced polymer concentration dependence of kt associated with the highly mobile short radical chains obtained in chain transfer reactions. While systems with significant chain transfer must be studied further, this modeling approach should serve as a robust basis for a more complete description of termination under various conditions.

Kinetics and Mechanism of the Photosensitized Oxidation of Furosemide*

AbstractDetection of O2(1δg) phosphorescence emission, γmax = 1270 nm, following laser excitation and steady‐state competitive methods was employed to measure total rate constants, kT, for the reactions of the diuretic furosemide, 2‐methylfurane and furfurylamine with singlet oxygen in several solvents. Correlation of kT values with solvent parameters and product identification shows that the reaction mechanism is strongly solvent dependent. In aliphatic alcohols, the dependence of kT on solvent parameters is similar to the one observed for triethylamine, suggesting a reaction mechanism involving partial charge transfer from the amino group to the singlet oxygen. In nonprotic solvents, the dependence of kT on solvent parameters resembles the behavior found for 2‐methylfur‐ane and furfurylamine, implying that mostly a 2 + 4 cy‐cloaddition mechanism of singlet oxygen to furane ring of furosemide occurs in these solvents. These mechanistic differences are explained in terms of hydrogen‐bonding interactions between the carboxylic group in the aromatic ring and the amino group of furosemide. Furthermore, direct generation of C2(1δg) by furosemide was detected. Quantum yields of 0.047 ± 0.003 and 0.078 ± 0.004 were determined in acetonitrile and benzene, respectively. This last result may be related, at least partially, to the photodynamic effects of this diuretic drug.

Chain-length dependent termination in pulsed-laser polymerization, 4. The influence of the type of mean involved in the bimolecular termination step

The chain-length distribution (cld) of living and dead chains and their moments have been calculated by an iterative simulation procedure for pseudostationary laser-induced polymerization (without chain-transfer) assuming various types of means between the lenghts of the two chains involved in the termination process. The overall appearance of the cld is dramactically influenced by the type of mean: the stronger the influence of the shorter of the two chains the more prominent are the “extra-peaks” of the cld and the smaller the rate of polymerization. The different types of means are subject to different short-chain effects; in the long-chain limit, however, the correct exponent of the power law governing the chain-length dependence of kt is correctly recovered in all cases. The estimation of kt(1,1) works the better the less prominent is the role of the smaller chain. Because of its close proximity to the “diffusion mean” the geometric mean approximation – although unphysical – is proved to give an excellent qualitative and quantitative description of all effects associated with chain-length dependent termination.

Chain-length dependent termination in pulsed-laser polymerization, 4. The influence of the type of mean involved in the bimolecular termination step

The chain-length distribution (cld) of living and dead chains and their moments have been calculated by an iterative simulation procedure for pseudostationary laser-induced polymerization (without chain-transfer) assuming various types of means between the lenghts of the two chains involved in the termination process. The overall appearance of the cld is dramactically influenced by the type of mean: the stronger the influence of the shorter of the two chains the more prominent are the “extra-peaks” of the cld and the smaller the rate of polymerization. The different types of means are subject to different short-chain effects; in the long-chain limit, however, the correct exponent of the power law governing the chain-length dependence of kt is correctly recovered in all cases. The estimation of kt(1,1) works the better the less prominent is the role of the smaller chain. Because of its close proximity to the “diffusion mean” the geometric mean approximation – although unphysical – is proved to give an excellent qualitative and quantitative description of all effects associated with chain-length dependent termination.

Chain-length dependent termination in pulsed-laser polymerization, 3. On the prospects of determining the bimolecular termination constant kt from rate data

The correct (event-weighted) average of kt, 〈kt〉, has been calculated from simulation data for pseudostationary laser-induced polymerization for a kinetic scheme with chain-length dependent termination and compared to the average k̄t which is obtained by employing the formal procedures, originally designed for the evaluation of individual rate constants from rate data in the case of chain-length independent termination. Satisfactory (and in fact excellent) results are obtained only if the complete equation for the conversion per laser pulse is solved for k̄t. This leads to an almost perfect recovery of the power-law governing the dependence of kt on chain-length, especially the exponent.

Kinetics and Mechanism of the Photosensitized Oxidation of Furosemide

Detection of O2(1δg) phosphorescence emission, γmax = 1270 nm, following laser excitation and steady-state competitive methods was employed to measure total rate constants, kT, for the reactions of the diuretic furosemide, 2-methylfurane and furfurylamine with singlet oxygen in several solvents. Correlation of kT values with solvent parameters and product identification shows that the reaction mechanism is strongly solvent dependent. In aliphatic alcohols, the dependence of kT on solvent parameters is similar to the one observed for triethylamine, suggesting a reaction mechanism involving partial charge transfer from the amino group to the singlet oxygen. In nonprotic solvents, the dependence of kT on solvent parameters resembles the behavior found for 2-methylfur-ane and furfurylamine, implying that mostly a 2 + 4 cy-cloaddition mechanism of singlet oxygen to furane ring of furosemide occurs in these solvents. These mechanistic differences are explained in terms of hydrogen-bonding interactions between the carboxylic group in the aromatic ring and the amino group of furosemide. Furthermore, direct generation of C2(1δg) by furosemide was detected. Quantum yields of 0.047 ± 0.003 and 0.078 ± 0.004 were determined in acetonitrile and benzene, respectively. This last result may be related, at least partially, to the photodynamic effects of this diuretic drug.