Termination Kinetics of Methyl Acrylate and Dodecyl Acrylate Free‐Radical Homopolymerizations up to High Pressure

The single pulse (SP) – pulsed laser polymerization (PLP) technique has been applied to measure k_t /k_p, the ratio of termination to propagation rate coefficients, for free‐radical bulk homopolymerizations of methyl acrylate (MA) and dodecyl acrylate (DA) between 10 and 50°C at pressures from 10 to 2 500 bar. k_t /k_p is obtained from experimental monomer concentration vs. time traces that are determined via time‐resolved (μs) near infrared monitoring of monomer conversion induced by single excimer laser pulses of about 20 ns width. With k_p being known from PLP–SEC experiments, chain‐length averaged k_t is immediately obtained from k_t /k_p. For MA, k_t remains constant up to about 15% monomer conversion and clearly decreases upon further polymerization. For DA at pressures of 100 bar and above, a plateau value of constant k_t is observed up to about 60% monomer conversion whereas at lower pressure, e. g. at 10 bar, k_t slightly increases in the very initial conversion region, but also exhibits a plateau k_t value at moderate and high conversions. The occurrence of such plateau k_t values and their pressure and temperature dependence are consistent with the view that plateau regions of k_t are best understood in terms of diffusion control via segmental mobility.     

Investigation into the Kinetics of n‐Pentyl Methacrylate Radical Polymerization

The precise knowledge of rate coefficients is of key importance for the understanding and the application of radical polymerization processes. Propagation rate coefficients, k_p, of n‐pentyl methacrylate (PnMA) radical polymerization are measured in bulk and partly in toluene solution over an extended temperature range via pulsed laser polymerization (PLP) in conjunction with size‐exclusion chromatography (PLP–SEC). Rate coefficients, k_t^i,i, for termination of two radicals of chain length i are determined as a function of chain length by single‐pulse PLP in conjunction with electron paramagnetic resonance spectroscopy (SP–PLP–EPR). The as‐obtained data that allow for modeling PnMA polymerization kinetics and product properties at moderate degrees of monomer conversion are compared with reported data for several other alkyl methacrylates. A distinct family behavior of this group of monomers is seen.     

Dependence of termination kinetics in methyl acrylate–dodecyl acrylate free‐radical copolymerization on comonomer composition,pressure, temperature and conversion

The termination kinetics of the free‐radical bulk copolymerization of dodecyl acrylate (DA) and methyl acrylate (MA) has been investigated at various monomer mole fractions between 15 and 50 °C and up to 2000 bar. The ratio of termination to propagation rate coefficients, (k_t/k_p)_copo, is measured via the single pulse‐pulsed laser polymerization (SP‐PLP) technique. Chain‐length averaged k_t,copo are deduced from (k_t/k_p)_copo in conjunction with k_p,copo data that are estimated via a simplified version of the implicit penultimate unit effect (IPUE) model. At low and moderate degrees of monomer conversion extended ranges of almost constant k_t are observed where termination is controlled by segmental diffusion with important contributions of steric effects. These plateau k_t values are significantly – by almost two orders of magnitude – different for MA and DA. The increase with MA content of k_t,copo is adequately described by a penultimate unit model which uses the geometric mean approximation for estimating rate coefficients of cross‐termination between radicals of different free‐radical terminus. The model applies within the entire pressure and temperature range of the present study. At high degrees of monomer conversion, at and above 50%, homo‐k_t and k_t,copo are almost insensitive toward the composition of the monomer mixture. The termination rate under these conditions is essentially controlled by reaction diffusion.

Relative monomer conversion, c_M(t)/c_M,0, vs. time, t, plot of a methyl acrylate (MA) – dodecyl acrylate (DA) copolymerization (f_MA,0 = 0.5) induced by a single laser pulse at 30 °C and 10 bar. The difference between measured and fitted (to Equation (4)) data is illustrated by the plot of residuals (res) in the lower part of the figure.     

Termination Kinetics of tert‐Butyl Methacrylate and of n‐Butyl Methacrylate Free‐Radical Bulk Homopolymerizations

Summary: Termination kinetics in tert‐butyl methacrylate (tert‐BMA) and n‐butyl methacrylate (n‐BMA) bulk homopolymerizations has been studied via the single pulse‐pulsed laser polymerization‐near infrared (SP‐PLP‐NIR) method between 40 and 80 °C at pressures from 500 to 2 250 bar. Toward increasing monomer conversion, the chain‐length averaged termination rate coefficient, 〈k_t〉, for both monomers exhibits the methacrylate‐specific sequence of an initial plateau region, assigned to control by segmental diffusion, followed by a steep decrease of 〈k_t〉 at intermediate conversion, which is assigned to translational diffusion control, and a weaker decrease of 〈k_t〉, associated with reaction‐diffusion control, at still higher degrees of monomer conversion. Despite this similarity, the two isomeric monomers clearly differ in absolute size of 〈k_t〉 and in the monomer concentration ranges where the transitions between the different types of diffusion control occur. The differences are assigned to effects of chain mobility which is hindered to a larger extent in tert‐BMA than in n‐BMA. As a consequence, the 〈k_t〉 behavior of tert‐BMA at 80 °C is close to the one of n‐BMA at 40 °C. Investigations into the chain‐length dependence of k_t, in particular into k_t(i,i), the rate coefficient for termination of two radicals of identical size, support the evidence on the different types of diffusion control that operate as a function of monomer conversion. In the initial conversion range, the power‐law exponent which characterizes the chain‐length dependence of larger (entangled) radicals, is found for both monomers to be close to the theoretical value of α = 0.16.

Dependence of log(〈k_t〉/k_p) on monomer conversion, X, for n‐BMA and tert‐BMA bulk homopolymerizations at 2 000 bar and 70 °C. Circles and triangles represent independent data sets obtained from separate experiments.     

Kinetics and Modeling of the Radical Polymerization of Trimethylaminoethyl methacrylate chloride in Aqueous Solution

For the radical polymerization of ionized trimethylaminoethyl methacrylate chloride (TMAEMA) in aqueous solution, two strategies to determine the propagation rate coefficient (k_p) are proposed for systems where the pulsed‐laser polymerization–size‐exclusion chromatography (PLP–SEC) method fails. This problem occurs with some fully ionized or sterically highly hindered monomers, where termination may become too slow. As TMAEMA is a borderline case with k_p being accessible by PLP–SEC and from single‐pulse–pulsed‐laser polymerization with electron paramagnetic resonance (SP–PLP–EPR) spectroscopy, studies into this monomer allow for judging the quality of the suggested alternative approaches of k_p measurement and serve for consistency checks of the previously published k_p and termination rate coefficient (k_t) data. Within both approaches, k_p/〈k_t〉 ^0.5 is measured via chemically initiated polymerization, with 〈k_t〉 referring to chain‐length‐averaged termination. The k_p/〈k_t〉 ^0.5 data are combined either with k_p/〈k_t〉 values from highly time‐resolved near‐infrared detection of monomer conversion induced by a single laser pulse (SP–PLP–NIR) or with Predici modeling on the basis of known chain‐length‐dependent termination kinetics. As coupled rate coefficients are measured, the obtained k_p data also provide 〈k_t〉 for a particular chain‐length distribution. The differences between propagation and termination rates of nonionized and fully ionized monomers are discussed.

     

Termination Kinetics of Dibutyl Itaconate Free‐Radical Polymerization Studied via the SP–PLP–ESR Technique

Summary: The termination kinetics of dibutyl itaconate (DBI) bulk polymerization was studied via SP–PLP–ESR single pulse–pulsed laser polymerization with time‐resolved detection of free‐radical concentration by electron‐spin resonance, at temperatures between 0 and 60 °C. As is characteristic of PLP experiments, termination rate coefficients, k_t(i,i), are measured for radicals of (almost) identical chain length (CL) i. CL‐averaged 〈k_t〉, for chain lengths up to 200 monomer units, and also kequation/tex2gif-stack-1.gif referring to termination of very small‐size radicals are directly deduced from measured DBI radical concentration vs time traces. At 45 °C, 〈k_t〉 is (3.4 ± 0.6) · 10⁵ L · mol^?1 · s^?1 and kequation/tex2gif-stack-2.gif is (7.2 ± 1.0) · 10⁵ L · mol^?1 · s^?1. Both rate coefficients are independent of monomer conversion up to the highest experimental conversion of 18%. The associated activation energies are E_A(〈k_t〉) = 23.0 ± 3.2 kJ · mol^?1 and E_A(kequation/tex2gif-stack-3.gif) = 27.6 ± 2.8 kJ · mol^?1, respectively. “Model‐dependent” and “model‐free” analyses of radical concentration vs time profiles indicate a pronounced CL dependence of k_t(i,i) for DBI radicals of moderate size, 5 < i < 50. The lowering of k_t(i,i) with CL corresponds to an exponent α close to 0.5 in a power‐law expression k_t(i,i) = kequation/tex2gif-stack-4.gif · i^?a. At higher chain lengths, the variation of k_t(i,i) with CL becomes weaker and may be represented by an α value of 0.16 or even below. These results are consistent with models according to which α varies to a larger extent at low CL and to a smaller extent at high CL with the crossover region between the two ranges being located somewhere around i = 100.

Conversion‐dependence of 〈k_t〉 and kequation/tex2gif-stack-5.gif from laser‐induced photopolymerization of DBI.     

Novel Access to Propagation Rate Coefficients of Radical Polymerization by the SP–PLP–EPR Method

A novel method for k_p measurement is presented, in which time‐resolved electron paramagnetic resonance detection of radical concentration after applying a single laser pulse (SP?PLP?EPR) is carried out in conjunction with measuring monomer conversion induced by the laser pulse. The method is particularly useful for the study of fully ionized monomers, as will be outlined by experiments on trimethylaminoethyl methacrylate chloride (TMAEMA, 2‐(methacryloyloxy)‐N,N,N‐trimethylethan‐1‐aminium chloride, 20 wt% in aqueous solution) at 60 °C. In addition to k_p, the method provides termination rate coefficient data. The reliability of the so‐obtained rate coefficients is demonstrated by chemically induced experiments and by measuring TMAEMA conversion after single‐pulse initiation via near‐infrared (NIR) spectroscopy.

     

Online Monitoring of Chain Transfer in Free‐Radical Polymerization

A recently introduced, automated method for online monitoring of polymerization reactions was used to study free‐radical transfer reactions. The persulfate initiated polymerization of acrylamide (AAm) in water was chosen as the test system. Chain transfer properties of ethanol (EtOH) and propanol (PrOH) were investigated. Different methods of computing the transfer constant are compared, including those based on the slope and intercept behavior of the monitored cumulative weight‐average molecular mass as a function of conversion, M̄_w (f), the reduced viscosity, and corresponding size exclusion chromatography analysis of the reaction end products. To a close approximation, the chain transfer agents were found to obey the form expected when ideal free‐radical polymerization takes place and radical transfer from propagating radicals to the chain transfer agent (CTA) is slower than from the CTA to monomer, that is, the polymer molar mass decreases with increasing chain transfer agent, but there is no appreciable effect on the kinetics of monomer conversion. The AAm kinetics were characterized in terms of the ratio k_p² /k_t, where k_p and k_t are the propagation and termination rate constants, respectively.     

Termination Kinetics of 1‐Vinylpyrrolidin‐2‐one Radical Polymerization in Aqueous Solution

Termination kinetics of 1‐vinylpyrrolidin‐2‐one radical polymerization in aqueous solution has been studied at 40 °C between 20 and 100 wt.‐% VP. The <k_t>/k_p values from laser single‐pulse experiments with microsecond time‐resolved NIR detection of monomer conversion, in conjunction with k_p from literature, yield chain‐length‐averaged termination rate coefficients, <k_t>. Because of better signal‐to‐noise quality, experiments were carried out at 2 000 bar, but also at 1 500, 1 000, and 500 bar, thus allowing for estimates of <k_t> at ambient pressure. The dependence of <k_t> on monomer conversion indicates initial control by segmental diffusion followed by translational diffusion and finally reaction diffusion control. To assist the kinetic studies, viscosities of VP–water mixtures at ambient pressure have been determined.

     

Termination kinetics of free-radical polymerization of styrene over an extended temperature and pressure range

The termination rate coefficient k_t of the free radical bulk polymerization of styrene is determined between 30 and 90°C up to a maximum pressure of 2800 bar. The majority of polymerization experiments has been carried out at monomer conversions up to 20 per cent. In this range a single value of k_t is sufficient to describe termination rate at constant pressure and temperature. Toward higher conversion, significant changes in k_t are observed. The data are measured by a pulsed laser polymerization technique and partly by conventional chemically initiated experiments, both with 2,2′-azoisobutyronitrile (AIBN) as the initiator. Online spectroscopy is applied toward measurement of styrene conversion. The experimental termination rate coefficients up to 20 per cent monomer conversion are adequately represented by the expression: Activation volume and activation energy of k_t are very close to the corresponding activation parameters that characterize the pressure and temperature dependence of the inverse of styrene monomer viscosity. Varying laser pulse repetition rate has been used to investigate a potential chain-length dependence of k_t at low conversion. It turns out that effects of this kind are not sufficiently pronounced to be safely established in view of the experimental precision of ±25 per cent that is reached in the k_t determinations.     

Critically Evaluated Rate Coefficients for Free‐Radical Polymerization, 5,

Summary: Propagation rate coefficients, k_p, for free‐radical polymerization of butyl acrylate (BA) previously reported by several groups are critically evaluated. All data were determined by the combination of pulsed‐laser polymerization (PLP) and subsequent polymer analysis by size exclusion (SEC) chromatography. The PLP‐SEC technique has been recommended as the method of choice for the determination of k_p by the IUPAC Working Party on Modeling of Polymerization Kinetics and Processes. Application of the technique to acrylates has proven to be very difficult and, along with other experimental evidence, has led to the conclusion that acrylate chain‐growth kinetics are complicated by intramolecular transfer (backbiting) events to form a mid‐chain radical structure of lower reactivity. These mechanisms have a significant effect on acrylate polymerization rate even at low temperatures, and have limited the PLP‐SEC determination of k_p of chain‐end radicals to low temperatures (<20 °C) using high pulse repetition rates. Nonetheless, the values for BA from six different laboratories, determined at ambient pressure in the temperature range of ?65 to 20 °C mostly for bulk monomer with few data in solution, fulfill consistency criteria and show excellent agreement, and are therefore combined together into a benchmark data set. The data are fitted well by an Arrhenius relation resulting in a pre‐exponential factor of 2.21 × 10⁷ L · mol^?1 · s^?1 and an activation energy of 17.9 kJ · mol^?1. It must be emphasized that these PLP‐determined k_p values are for monomer addition to a chain‐end radical and that, even at low temperatures, it is necessary to consider the presence of two radical structures that have very different reactivity. Studies for other alkyl acrylates do not provide sufficient results to construct benchmark data sets, but indicate that the family behavior previously documented for alkyl methacrylates also holds true within the alkyl acrylate family of monomers.

Arrhenius plot of propagation rate coefficients, k_p, for BA as measured by PLP‐SEC.     

Propagation and Termination Kinetics of Poly(Ethylene Glycol) Methyl Ether Methacrylate in Aqueous Solution

The propagation rate coefficient, k_p, of poly(ethylene glycol) methyl ether methacrylate (M_n ≈500 g mol^?1) has been measured via pulsed‐laser polymerization (PLP)–size‐exclusion‐chromatography in aqueous solution between 5 wt% monomer and bulk polymerization at temperatures from 22 to 77 °C. k_p increases significantly toward higher water content, as is observed for other water‐soluble monomers. This entropy‐motivated effect enhances the pre‐exponential. The activation energy, E_A(k_p), is more or less identical to the characteristic value of methacrylates. The chain‐length‐dependent rate coefficient, k_t^i,i, for termination of two radicals of chain length i has been investigated at low degrees of monomer conversion via the single‐pulse–PLP–electron paramagnetic resonance technique. k_t^i,i turned out to be adequately represented by the composite model designed by the Russell group. The power‐law exponents for the chain‐length dependence of small and long radicals are close to the numbers reported for other monomers. The rate coefficient for termination of two radicals of chain length unity scales with the fluidity of the reaction mixture. Viscosity measurements prior to polymerization thus enable estimates of termination rate.

     

Free‐Radical Bulk Polymerization of Styrene: ESR and Near‐Infrared Spectroscopic Study of the Entire Conversion Range

The free radical polymerization of styrene in bulk was monitored by ESR and FT near‐infrared spectroscopy at 70°C for a series of concentrations of the initiator, dimethyl 2,2′‐azobis(isobutyrate). In order to obtain detailed kinetic information over the entire conversion range, and the gel effect range in particular, conversion and free radical concentration data points were accumulated with exceptionally short time intervals. The polystyrene radical concentration ([St∗︁]) went through a sharp maximum at the gel effect, a feature that has hitherto escaped observation due to the rapid concentration changes in the gel effect range relative to the data point time intervals of previous studies. Temperature measurements throughout the polymerization were employed to calculate that a temperature increase was not the cause of the [St∗︁] maximum, which thus appears to be a genuine feature of the gel effect of this system under isothermal conditions. The propagation rate constant (k_p) as a function of monomer conversion exhibited a marked dependence on initiator concentration at high monomer conversion; the sharp decrease in k_p with increasing conversion was shifted to higher conversions with increasing initiator concentration.     

Critically evaluated rate coefficients for free‐radical polymerization, 3. Propagation rate coefficients for alkyl methacrylates

Pulsed‐laser polymerization (PLP) in conjunction with the analysis of the molecular weight distribution (MWD) via size‐exclusion chromatography (SEC) remains recommended by the IUPAC Working Party on Modeling of polymerisation kinetics and processes as the method of choice for the determination of propagation rate coefficients, k_p, in free‐radical polymerization. k_p data from PLP‐SEC studies in several laboratories for ethyl methacrylate (EMA), butyl methacrylate (BMA) and dodecyl methacrylate (DMA) bulk free‐radical polymerizations at low conversion and ambient pressure are collected. The data fulfill consistency criteria and the agreement among the data is remarkable. These values are therefore recommended as constituting benchmark data sets for each monomer. The results are best fit by the following Arrhenius relations: For the methacrylates under investigation k_p increases with the size of the ester group. For example, in going from MMA to DMA, k_p at 50°C is enhanced by a factor of 1.5.     

PLP‐SEC Study into the Free‐Radical Propagation Rate Coefficients of Partially and Fully Ionized Acrylic Acid in Aqueous Solution

Summary: Propagation rate coefficients, k_p, for acrylic acid (AA) polymerization at 6 °C in aqueous solution were measured via pulsed laser polymerization (PLP) with the degree of ionization, α, varied over the entire range between 0 and 1. These measurements were carried out in conjunction with aqueous‐phase size‐exclusion chromatography (SEC). Strictly speaking, the reported k_p's are “apparent” propagation rate coefficients deduced from the PLP‐SEC data under the assumption that the local monomer concentration at the radical site is identical to overall monomer concentration. At an AA concentration of 0.69 mol · L^?1, the apparent k_p decreases from 111 000 L · mol^?1 · s^?1 at α = 0 to 13 000 L · mol^?1 · s^?1 at α = 1.0. The significant lowering of k_p with higher α is attributed to the repulsion between both monomer molecules and macroradicals becoming negatively charged. Addition of up to 10 mol‐% (with respect to AA) sodium hydroxide to the fully ionized aqueous AA solution leads to an enhancement of k_p up to 57 000 L · mol^?1 · s^?1.

Dependence of apparent k_p values on the degree of ionization of acrylic acid (a) and on pH (b) for aqueous polymerizations of acrylic acid.     

Termination kinetics in free-radical bulk terpolymerization — the systems methyl acrylate – butyl acrylate – dodecyl acrylate and methyl methacrylate – butyl methacrylate – dodecyl methacrylate

Termination rate coefficients, k_t , for the terpolymerization of mixtures containing the three acrylates: methyl, butyl, and dodecyl acrylate or the corresponding members of the methacrylate family: methyl, butyl, and dodecyl methacrylate, have been measured via the single pulse (SP)-PLP technique. In the homopolymerization of each of the six monomers an initial plateau region of almost constant k_t is seen which extends at least up to 15% and, for dodecyl acrylate and dodecyl methacrylate, even up to about 50% monomer conversion. Terpolymerization k_t in this region of low and moderate conversion is remarkably well described by a rather simple correlation which exclusively contains homopolymerization k_t and the composition of the monomer mixture. This correlation turns out to be very useful for the modeling of k_t in mixtures of monomers that exhibit such an initial plateau region of k_t . For the systems under investigation, terpolymerization k_t may be estimated within ± 30%. This finding is very remarkable in view of the enormous differences in homopolymerization k_t within each monomer family, e. g., k_t of methyl acrylate exceeds the corresponding dodecyl acrylate value by a factor of 50 (or 5 000%). The entire set of experiments has been carried out at 40°C and 1 000 bar where the signal to noise ratio of the laser-induced single pulse experiments is very satisfactory. No reason is seen why the conclusions about modeling of terpolymerization k_t should not be valid at other reaction conditions including ambient pressure.     

Kinetics of the early stages of high‐energy radiation initiated polymerization

We describe our method based on pulse radiolysis with optical detection developed for the examination of the kinetics and mechanism of the first steps of high‐energy radiation initiated polymerization. The absorption spectra of the intermediates were obtained in cyclohexane solution of hexanediol diacrylate (HDDA) of different concentrations. In dilute solution (10 mmol·dm^–3) and short time (10 μs) after the pulse, the spectrum of the monomer radicals was observed. On increasing the monomer concentration, the maximum of the spectrum was shifted to longer wavelength indicating the start of the oligomerization reaction. The increase in the time of observation resulted in a similar shift in dilute solution. From the kinetic curves the rate coefficients of termination for the monomer radicals (2·k_t,m) and average rate coefficients of termination for the oligomer radicals of different chain length (2·k_t) were determined. The average rate coefficient of termination was found to decrease in time (that is with increasing chain‐length).     

Low Conversion 4‐Acetoxystyrene Free‐Radical Polymerization Kinetics Determined by Pulsed‐Laser and Thermal Polymerization

Summary: The free‐radical polymerization kinetics of 4‐acetoxystyrene (4‐AcOS) is studied over a wide temperature range. Pulsed‐laser polymerization, in combination with dual detector size‐exclusion chromatography, is used to measure k_p, the propagation rate coefficient, between 20 and 110 °C. Values are roughly 50% higher than those of styrene, while the activation energy of 28.7 kJ · mol⁻¹ is lower than that of styrene by 3–4 kJ · mol⁻¹. With known k_p, conversion and molecular weight data from 4‐AcOS thermal polymerizations conducted at 100, 140, and 170 °C are used to estimate termination and thermal initiation kinetics. The behavior is similar to that previously observed for styrene, with an activation energy of 90.4 kJ · mol⁻¹ estimated for the third‐order thermal initiation mechanism.

Joint confidence (95%) ellipsoids for the frequency factor A and the activation energy E_a from non‐linear fitting of k_p data for 4‐AcOS (black) and styrene (grey).     

A Suitable Photoinitiator for Pulsed Laser‐Induced Free‐Radical Polymerization

Detailed kinetic studies into free‐radical polymerization via pulsed laser experiments ideally require photoinitiators which almost instantaneously dissociate into primary free‐radical fragments that rapidly add to monomer molecules and thus induce macromolecular growth. 2‐Methyl‐4′‐(methylthio)‐2‐morpholinopropiophenone (MMMP) is shown to be such a suitable photoinitiator. Measurement of monomer conversion induced by a single laser pulse, within the so‐called single‐pulse pulsed laser polymerization (SP–PLP) experiment, provides direct information about the chain‐length dependence of the termination rate coefficient if MMMP is used as the photoinitiator.

Relative monomer concentration vs time trace of a methyl acrylate homopolymerization at 40 °C and 2 000 bar where MMMP was used as the initiator. The primary radical concentration from MMMP photo‐decomposition increases from curve (a) to (b) to (c) by the ratios 1:2.2:5.7.     

The laser flash-initiated polymerization as a tool of evaluating (individual) kinetic constants of free radical polymerization, 1. Outline of method and first results

A method is presented which allows the determination of k_p/k_t-values in free radical polymerization. It is based on measurements of the (average) rate of polymerization under pseudostationary conditions, the polymerization being initiated by laser flashes of short duration. For ρk_t t₀ ? 1 (ρ being the additional polymer radical concentration produced by each laser flash, k_t the bimolecular termination constant between polymer radicals, k_p the rate constant of chain propagation, t₀ the time separating two successive laser flashes) k_p/k_t may be obtained as the slope of a linear plot of the fractional conversion per flash vs. ln t₀. Dividing the intercept by the slope yields ln (pk_t). Thus, if p is accessible, separation of k_p/k_t-data into its individual constituents may be accomplished without making any use of stationary polymerization data. Application of this method to the polymerization of styrene sensitized by benzoin or AIBN at 25°C gives k_p/k_t-values of 1,0 · 10^?6 which are in fair agreement with those obtained by other methods.