Diffusion-controlled reaction. V. Effect of concentration-dependent diffusion coefficient on reaction rate in graft polymerization

The effect of diffusion on radiation-initiated graft polymerization has been studied with emphasis on the single- and two-penetrant cases. When the physical properties of the penetrants are similar, the two-penetrant problem can be reduced to the single-penetrant problem by redefining the characteristic parameters of the system. The diffusion-free graft polymerization rate is assumed to be proportional to the v power of the monomer concentration C, in which the proportionality constant a = k_pR/k, where k_p and k_t are the propagation and termination rate constants, respectively, and R_i is the initiation rate. The values of v, w, and z depend on the particular reaction system. The results of our earlier work were generalized by allowing a non-Fickian diffusion rate, obtained from an extension of the Fujita free-volume theory, which predicts an essentially exponential dependence on the monomer concentration of the diffusion coefficient, D = D₀ [exp(δC/M)], where M is the saturation concentration. It was shown that a reaction system is characterized by the three dimensionless parameters v, δ, and A = (L/2)[aM^(v?1)/D₀]^1/2, where L is the polymer film thickness. Graft polymerization tends to become diffusion controlled as A increases. Larger values of δ and v cause a reaction system to behave closer to the diffusion-free regime. The transition from diffusion-free to diffusion-controlled reaction involves changes in the dependence of the reaction rate on film thickness, initiation rate, and monomer concentration. Although the diffusion-free rate is w order in initiation rate, v order in monomer, and independent of film thickness, the diffusion-controlled rate is w/2 order in initiator rate and inverse first-order in film thickness. The dependence of the diffusion-controlled rate on monomer is dependent in a complex manner on the diffusional characteristics of the reaction system.     

Diffusion-controlled reaction. VI. Radiation graft polymerization of styrene to polyethylene

The radiation graft polymerization of styrene to polyethylene was studied under diffusion-controlled conditions of radiation intensity I, monomer concentration M₁, and polymer sample thickness L. The results of the present study together with previous work under diffusion-free conditions verify our theoretical model for the diffusion-controlled reaction. The grafting rate is inverse first order in L for diffusion-controlled reaction and independent of L for diffusion-free reaction. The order of dependence of grafting rate on radiation intensity for diffusion-controlled reaction is one-half that for diffusion-free reaction. Diffusion control leads to a decrease in the order of dependence of grafting rate on monomer concentration. The decrease is greater than theoretically predicted; possible reasons for this effect are described.     

Diffusion-controlled reaction. VI. Radiation graft polymerization of 4-vinylpyridine to polyethylene

The radiation-initiated graft polymerization of 4-vinylpyridine to high-density polyethylene was studied over a wide range of reaction conditions of radiation intensity I, monomer concentration M₁, and polymer film thickness L. The conditions included both diffusion-free and diffusion-controlled graft polymerizations. The results corroborate our previous theoretical predictions on the effect of I, M₁, and L on the experimental grafting rate. The grafting rate is inverse first order in L for diffusion-controlled reaction and independent of L for diffusion-free reaction. The dependence of grafting rate on radiation intensity decreases from 1/2 to 1/4 order for diffusion-controlled reaction. Diffusion control results in a decrease in the dependence of rate on monomer concentration. The observed decrease is somewhat greater than theoretically predicted.     

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.     

Kinetics of diffusion-free radiation graft polymerization of styrene onto polyethylene

The diffusion-free radiation graft polymerization of styrene onto polyethylene has been studied. The grafting rate shows a dependence on monomer which is far different than what has been assumed. Further, the dependence on monomer changes with increasing dose rate as does the dependence of grafting rate on radiation dose rate. Three different regions of behavior are defined: (1) a region of low dose rate where the grafting rate is 1/2-order in dose rate and 3/2-order in monomer; (2) a region of intermediate dose rate where the grafting rate is intermediate between 1/2-and zero-order in dose rate and 5/2-order in monomer; and (3) a region of high dose rate, where the grafting rate is independent of dose rate and at least 5/2-order in monomer. Various possible mechanisms responsible for these effects are discussed, including the effects of viscosity on the initiation and termination reactions, the possibility of ionic graft polymerization, and energy transfer.     

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.     

Grafting onto wool. V. Kinetics of graft copolymerization of methyl methacrylate in the viscous media of wool fibers

The graft copolymerization of methyl methacrylate in S-carboxymethylated wool fibers was investigated in the aqueous LiBr-K₂S₂O₈ system. The rate of grafting, the degree of polymerization of graft polymer, and the number of grafting sites were determined on varying the thiol content at a constant concentration of monomer. Kinetic considerations lead to the following expression in agreement with the experimental results: Z/DP = {(k_td + k_tc)/k_p²[M]²} R_p, where Z is the number of DNP endgroups of polymer; DP is the average degree of polymerization; k_p, k_td, and k_tc are the rate constants of propagation, termination by disproportionation, and termination by recombination, respectively; [M] is the concentration of monomer in fibers, and R_p is the overall rate of grafting. For wool fibers in media sufficiently high viscosity, the rate constants k_td and k_tc of diffusion-controlled termination are approximately equal and not affected by the change in cross-link density, provided that the thiol and disulfide interchange occurs. The possibility of occurrence of mechanical bond scission through a radical mechanism is involved in systems with extremely small amounts of thiol groups.     

Radiation-induced graft polymerization of styrene to poly(vinyl chloride)

The radiation-induced graft polymerization of styrene to poly(vinyl chloride) (PVC) was investigated. Relations between the rate of grafting and the dose rate when the polymer is irradiated in liquid monomer or in monomer vapor, and between the rate of grafting and monomer concentration absorbed in the polymer have been investigated. The rate of grafting in monomer vapor was found to be far larger than that in liquid monomer. A high rate of grafting in monomer vapor was thought to result from a lower concentration of monomer in PVC during irradiation. An experiment carried out on PVC containing the monomer at various concentrations showed that the rate is largest at a monomer concentration of about 3.5 mole/l. and is smaller for higher and lower concentrations. On the assumption that the theory of homogeneous homopolymerization can be applied to this grafting reaction, the value of k_p²/k_t has been obtained, where k_p and k_t are propagation constant and termination constant, respectively. The value of k_t greatly increases when the monomer concentration exceeds 3.5 mole/l. This increase of k_t can be accounted for if it is assumed that the monomer absorbed in the polymer works as a plasticizer and increases the molecular motion of the polymer. A measurement of the elastic modulus of PVC containing the monomer at various concentrations showed that this is, in fact, the case.     

Synthetic thermally reversible gel systems. IV

The polymerization kinetics in water of acrylylglycinamide (AG) initiated by K₂S₂O₈ was studied over the temperature range 40.0 to 60.0°C. Monomer concentration was varied from 7.8 × 10^?3 to 31.2 × 10^?3M and catalyst from 1.85 × to 11.10 × 10^?5M. The rate expression is ?d[M]/dt = R_p, = k_1.22[K₂S₂O₈]^0.5[M]^1.22, and the overall empirical rate constant, k_1.22 = 1.14 × 10¹¹e^?15,800/RT 1.^0.72 mole^?0.72 min^?1. To explain the dependence on monomer, a kinetic scheme which includes a bimolecular reaction (k₂) between monomer and initiator is suggested. The simplified expression which describes the initial rate of polymerization is: ?d[M]/dt = R_p, = k₄(2[I]/k₅)^1/2[M](k₁ + k₂[M])^1/2, where k₁, k₂, k₄ and k₅ are rate constants for S₂O₈ = decomposition, a bimolecular reaction between monomer and initiator, propagation, and termination, respectively. Individual bimolecular rate constants are expressed in liter/mole-min. The equation predicts a dependence on monomer concentration between 1.0 and 1.5 with 1.5 being approached a t high monomer concentrations. Plots of R_P²/[M]² versus [M] are linear, as predicted by the postulated reaction route and values for k₂ and k₄/k₅^1/2 were obtained from the slopes and intercepts of these plots. The temperature dependence of the bimolecular monomer-initiator reaction is k₂ = 5.19 × 10²¹e^?36,000/RT. Instead of the usual behavior, the k₄/k₅^1/2 ratio was found to decrease with temperature and the difference of activation energies, (E₄ ? E₅/2), is ?1.50 kcal. The temperature dependence of the propagation to square root of the termination rate constant ratio is k₄/k₅^1/2 = 6.16e^1500/RT. These rather unusual results may be related to the ability of AG polymers in water to form thermally reversible gels; even above the gel melting points, the polymers are considerably aggregated in solution. This would tend to make the bimolecular termination reaction more temperature dependent and also account for the high values (59–69) for the k₄/k₅^1/2 ratios. For similar temperatures, the overall rate constants for AG are approximately four times those for acrylamide.     

Variations of Rate Coefficients and Termination Mechanism During the After‐Effects of a Light‐ Induced Polymerization of a Dimethacrylate Monomer

This work was aimed at studying variations in the termination mechanism occurring during the after‐effects of a light‐induced polymerization of a dimethacrylate monomer after the irradiation had been discontinued. The experimental method was based on differential scanning calorimetry. The initiation was stopped at various moments of the reaction corresponding to different degrees of double‐bond conversion (starting conversions). Three termination models: monomolecular, bimolecular, and mixed were used to calculate the ratio of the bimolecular termination and propagation rate coefficients k_t^b/k_p and/or the monomolecular termination rate coefficient k_t^m. The models were determined over short time intervals (conversion increments) of the dark reaction giving different values of rate coefficients for each time interval (interval approximation method). Two‐stage statistical analysis was used to find the model that best reproduced the experimental data obtained for each conversion increment. This enabled variations in the termination mechanism during the after‐effects to be followed. It was found that the termination mechanism changed with the time of the dark reaction from the bimolecular reaction to the mixed reaction when the light was cut off at low and medium double‐bond conversions. At higher starting conversions a monomolecular termination mechanism dominated from the beginning of the dark reaction. The mixed termination model was the only model to describe correctly the variations of rate coefficients in the dark, i. e., the increase in k_t^m and the decreasein the k_t^b/k_p ratio.     

Aqueous Polymerization on Clay Surface. 2. The Polymerization of Methyl Methacrylate on Hydrogen Bentonite: Effect of Monomer Concentration and Temperature

Aqueous free radical polymerizations of methyl methacrylate with the hydrogen bentonite/ethanol system have been accomplished with less transfer to monomer in spite of high monomer concentrations and temperature. The overall initial rate has a first-order dependence on monomer. It is proposed that initiation does not occur in the aqueous phase. The apparent activation energy of 15 kcal/mol corroborates a twofold increase in rate for a 10°C rise in temperature. The frequency of bimolecular termination is quite small as is evident from k_p ² /k_t, values at various temperatures.     

Estimation of molecular weight distribution based on chain length dependence of termination rate

When a chain length dependence of polymer-polymer termination is given by k_t,ns = const. (n^?2a + s^?2a) where n and s are the chain lengths for the polymer radicals and a is parameter, an instantaneous weight fraction of the non-reacting polymers is derived as: where h and k? are the kinetic parameters, p is a parameter depending on a, and p_n is instantaneous number-average chain length. Such a weight fraction corresponds to the experimental one over a wide range of conversion in the polymerization of styrene. On the scope of this correspondence, the polymer-polymer termination rate is estimated as: k?_t = 8πR₀D₁/100 ( = 4πR_sD_s) where R₀ is reaction radius between monomer radicals and D₁ is the diffusion coefficient of the monomer; R_s is reaction radius between segment radicals with n ? 100 and D_s is the diffusion coefficient of the segment. The Fujita-Doolittle theory applies to such a rate. Further, the rate also yields 1.5 × 10⁷1./mole-sec, which is the observable extent at conversions less than 0.2.     

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.     

Free‐radical copolymerization of styrene and m‐isopropenyl‐α,α′‐dimethylbenzyl isocyanate studied by 1H NMR kinetic experiments

The free‐radical copolymerization of m‐isopropenyl‐α,α′‐dimethylbenzyl isocyanate (TMI) and styrene was studied with ¹H NMR kinetic experiments at 70 °C. Monomer conversion vs time data were used to determine the ratio k_p × k_t^?0.5 for various comonomer mixture compositions (where k_p is the propagation rate coefficient and k_t is the termination rate coefficient). The ratio k_p × k_t^?0.5 varied from 25.9 × 10^?3 L^0.5 mol^?0.5 s^?0.5 for pure styrene to 2.03 × 10^?3 L^0.5 mol^?0.5 s^?0.5 for 73 mol % TMI, indicating a significant decrease in the rate of polymerization with increasing TMI content in the reaction mixture. Traces of the individual monomer conversion versus time were used to map out the comonomer mixture composition drift up to overall monomer conversions of 35%. Within this conversion range, a slight but significant depletion of styrene in the monomer feed was observed. This depletion became more pronounced at higher levels of TMI in the initial comonomer mixture. © 2002 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 40: 1064–1074, 2002     

Absolute propagation rate coefficients in radical polymerization from gel permeation chromatography of polymers produced by intermittent initiation

The pulsed laser polymerization technique is now a well accepted method to determine propagation rate coefficients for radical polymerization from molar mass distributions resulting from intermittent initiation. A simplified apparatus for the periodic photoinitiation is used which is much less expensive than the laser equipment. The usefulness of the simplified equipment was proved by the determination of k_p for styrene at technically relevant temperatures up to 130°C for the first time. Furthermore, careful inspection of the molar mass distribution (mmd) reveals that depending on the reaction conditions, inflection points (L_i) can not only be found at integer multiples of k_p • t_o • [M] but also at 0.5ⁱ • k_p • t_o • [M], i = 1, 2, 3, … . A rule to find the inflection points leading to correct values for k_p is proposed. It is shown that the shape of the mmd inter alia depends on the amount of primary radical termination compared to the termination reaction between growing chains. With dominant primary termination, the maxima of the distribution will give the correct k_p, whereas in the absence of primary termination the inflection points should be used. Experimental conditions like initiator concentration, light intensity etc. may influence the position of the L_i at least to some extent, and so may give a small but principal error or uncertainty in k_p. A new mathematical method for the time-dependent simulation of the resulting mmd is presented which allows the calculations being performed on a PC within an acceptable time.     

Influence of the stereochemical configuration on the radical polymerization of methacrylic monomers: cis‐ and trans‐(2‐cyclohexyl‐1,3‐dioxan‐5‐yl) methacrylate

The synthesis of two new isomeric monomers, cis‐(2‐cyclohexyl‐1,3‐dioxan‐5‐yl) methacrylate (CCDM) and trans‐(2‐cyclohexyl‐1,3‐dioxan‐5‐yl) methacrylate (TCDM), starting from the reaction of glycerol and cyclohexanecarbaldehyde, is reported. The process involved the preparation of different alcohol acetals and esterification with methacryloyl chloride of the corresponding cis and trans 5‐hydroxy compounds of 2‐cyclohexyl‐1,3‐dioxane. The radical polymerization reactions of both monomers, under the same conditions of temperature, solvent, monomer, and initiator concentrations, were studied to investigate the influence of the monomer configuration on the values of the propagation and termination rate constants (k_p and k_t ).The values of the ratio k_p /k_t ^1/2 were determined by UV spectroscopy by the measurement of the changes of absorbance with time at several wavelengths in the range 275–285 nm, where an appropriate change in absorbance was observed. Reliable values of the kinetics constants were determined by UV spectroscopy, showing a very good reproducibility of the kinetic experiments. The values of k_p /k_t ^1/2, in the temperature interval 45–65 °C, lay in the range 0.40–0.50 L^1/2/mol^1/2s^1/2 and 0.20–0.30 L^1/2/mol^1/2s^1/2 for CCDM and TCDM, respectively. Measurements of both the radical concentrations and the absolute rate constants k_p and k_t were also carried out with electron paramagnetic resonance techniques. The values of k_p at 60 °C were nearly identical for both the trans and cis monomers, but the termination rate constant of the trans monomer was about three times that of the cis monomer at the same temperature. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 3883–3891, 2000     

Solid-state polymerization of long-chain vinyl compounds. III. Mechanism of γ-ray-initiated postpolymerization in layered structures of octadecyl methacrylate and acrylate

The mechanism and kinetics of the γ-ray-initiated postpolymerization of octadecyl methacrylate and acrylate in lamellar crystals were investigated by a simple model. This model assumes that the initiation points are distributed as in a checkerboard and that polymerization probability of the monomer molecules decreases conically around each initiation point. The two-dimensional polymerization can be characterized in this cone model by two parameters, a and r; a represents the polymerizability of the monomer for a given condition, and r depends on the number of initiation points per unit area. G values for the initiation reaction of octadecyl methacrylate and acrylate were estimated as 0.8 and 1.6, respectively. The two-dimensional postpolymerization of long-chain compounds proceeds in two stages. The rate of polymerization is very high and zero order with respect to monomer concentration in the first stage. It is lower and obeys first-order kinetics in the second stage. The rate constants of the zero-and first-order polymerizations were k_p0 = 1.73 molecule sec^-1 and k_p1 = 0.93 sec^?1, respectively, for octadecyl acrylate at 20°C.     

Polymerization of acrylamide and methacrylamide photoinitiated by azidopentamminecobalt(III) chloride

The kinetics of polymerization of acrylamide and methacrylamide, photoinitiated by azidopentamminecobalt(III) chloride in homogeneous aqueous acid medium was studied systematically. Monochromatic wavelengths 365, 405, and 435 mμ were employed for irradiation. Polymerization proceeded without any induction period, and the reaction was followed by measurements of rate of monomer disappearance (bromometrically), rate of complex disappearance (spectrophotometrically), and the chain lengths of the polymer formed (viscometrically). The dependences of the rate of polymerization on variables like light intensity, light absorption fraction by the complex, wavelength, monomer concentration, hydrogen ion concentration, nature of the acid used (HClO₄, HNO₃, and H₂SO₄), etc., were studied. The rate of polymerization of acrylamide depended on the unit power of monomer concentration and on the square root of light absorption fraction k_ε and light intensity I. The rate of methacrylamide polymerization was proportional to the unit power of monomer concentration and fractional powers of 0.25 and 0.30 of k_ε and I, respectively. A kinetic reaction scheme is proposed and discussed in the light of the experimental results, and it has been concluded that (1) the primary photochemical act is an electron transfer reaction from the azide ion to Co(III) in the complex, (2) initiation of polymerization is by azide radical, (3) termination is by mutual destruction of polymer radicals.     

Synthesis and radical polymerization of hydrophilic methacrylates with oxyethylene units in the pendant chain

The syntheses of methacrylic monomers of the general structure where n is 3, 4, 5, or 6, were performed by the reaction of the corresponding alcohol ethers with methacryloyl chloride. The alcohol ethers were previously prepared by different synthetic procedures involving the monoetherification of the starting glycols. The polymerizations kinetics of the monomers were examined at several temperatures in the bulk and in dioxane solutions. NMR spectroscopy and electron paramagnetic resonance techniques were used to study the kinetics of polymerization. The polymerization rate parameter, expressed as (2f)^1/2k_p/〈k_t〉^1/2, and the values of the propagation rate coefficient k_p and the termination rate coefficient 〈k_t〉/f, where f is the efficiency factor of the initiator, were determined. The reactivity of the monomers depended on the size of the ester residue in such a way that the longer the lateral chain was, the higher the polymerization rate was and the lower the termination rate coefficient was. On the contrary, the dependence of k_p on the chemical structure was very small. In the solution polymerizations of all these monomers (monomer concentration = 1 mol L^?1), the radical concentrations remained almost constant until very high conversions, whereas in the bulk, a different behavior was observed that depended on the number of oxyethylene units in the side chain of the monomer. In this sense, for n = 4, 5, or 6, the radical concentration remained almost invariable with the reaction time, whereas for n = 3, a moderate increase occurred at low conversions, contrasting with the important increase observed at similar conversions for n = 1. This showed that the gel effect in these methacrylic monomers was greatly dependent on the number of bonds of the lateral chain. © 2003 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 41: 1567–1579, 2003     

Mechanism of the reduction of molecular nitrogen to hydrazine by niobium (iii) hydroxide

The effect of the concentration of water on the rate of reduction of molecular nitrogen to hydrazine by niobium(iii) hydroxide in alkaline H₂O−MeOH and D₂O−MeOD mixtures was studied. In both cases, the reaction rate is maximum when [H₂O]=4 mol L⁻¹, and the inverse isotopic effect (K ^D/k ^H>1) is observed when [H₂O]<20 mol L⁻¹. Similar regularity was observed for the reaction of hydrogen elimination. It was found that HD is formed in the H₂O−MeOH system in the presence of D₂. The conclusion was made that the ratedetermining stage in hydrazine formation is the transfer of a hydride ion to the dinitrogen molecule coordinated to the binuclear Nb^III center. A kinetic scheme satisfactorily explaining the effect of the concentration of water ([H₂O]=1.5−49.0 mol L⁻¹) on the reaction rate constant was proposed. Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1600–1604, September, 1997.