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

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

Kinetic model of living radical polymerization

This work studies the kinetics of living radical polymerization by means of both the nonsteady state approach and the quasi-stationary state method. Expressions for the numberand weight-average degress of polymerization and the polydispersity index were derived. Numerical results show that the concentration of residual initiator seriously influences the polydispersity index of the resulting polymer. The calculated outcomes of the non-steady state approach are evidently different from those of the quasi-stationary state method when the magnitude of the rate constant of termination is comparable with that of the propagation rate constant, and the difference becomes negligible if the rate constant of the termination (k_t) is much larger than that of propagation (k_p). The polydispersity index of the resulting polymer increases with decreasing ratios of k_t to k_p or M_O to I_O (initial concentrations of monomer and initiator).     

Kinetic Study of Radical Polymerization VIII. A Comprehensive Study of Solution Copolymerization of Vinyl Acetate and Methyl Acrylate by 1H‐NMR Spectroscopy

Radical copolymerization reaction of vinyl acetate (VA) and methyl acrylate (MA) was performed in a solution of benzene‐d₆ using benzoyl peroxide (BPO) as the initiator at 60°C. Kinetic studies of this copolymerization reaction were investigated by on‐line ¹H‐NMR spectroscopy. Individual monomer conversions vs. reaction time, which was followed by this technique, were used to calculate the overall monomer conversion, as well as the monomer mixture and the copolymer compositions as a function of time. Monomer reactivity ratios were calculated by various linear and nonlinear terminal models and also by simplified penultimate model with r _2(VA)=0 at low and medium/high conversions. Overall rate coefficient of copolymerization was calculated from the overall monomer conversion vs. time data and k _p  . k _t ^?0.5 was then estimated. It was observed that k _p  . k _t ^?0.5 increases with increasing the mole fraction of MA in the initial feed, indicating the increase in the polymerization rate with increasing MA concentration in the initial monomer mixture. The effect of mole fraction of MA in the initial monomer mixture on the drifts in the monomer mixture and copolymer compositions with reaction progress was also evaluated experimentally and theoretically.     

Graft polymers: Chain transfer and branching

The vinyl monomers, methyl methacrylate, ethyl methacrylate, and methyl acrylate were polymerized in the presence of chlorinated rubber or poly(vinyl chloride) in homogeneous solution with benzoyl peroxide as catalyst. A graft polymer was formed by a chain-transfer reaction involving the growing polymer radicals to the backbone of chlorinated rubber or poly(vinyl chloride), in addition to homopolymer from the monomer. The homopolymer was isolated from the polymer mixture by fractional precipitation from methyl ethyl ketone solution with methanol as precipitant. The chain-transfer constants for the branching reactions were evaluated. The ratios k_p/(k_t)^1/2 for the grafting reactions were obtained by a correlation of chain-transfer constants with the extent of branching. The chain-transfer data were correlated on the basis of an extension of the Q–e scheme of Alfrey and Price to polymer–polymer transfer reactions. Specific effects due to the backbone are found to have considerable influence on the course of the chaintransfer reactions and k_p/(k_t)^1/2 of the grafting reactions.     

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 polymerization of acrylic acid in aqueous solution

Aqueous acrylic acid in the presence of cupric chloride has been subjected to γ-irradiation under various reaction conditions and the molecular weights of the resultant poly(acrylic acid) measured. The results, taken in conjunction with previous findings on the dependence of the rate of polymerization on intensity, monomer concentration, and cupric chloride concentration, indicate chain termination solely by cupric ion (rate constant k_tCu) and chain transfer to polymer (rate constant kf). Values have been obtained for k_tCu/k_p, k_f/k_p and G(radical) of acrylic acid. On the basis of these data a theoretical chain-length distribution has been derived which agrees well with distribution measured by gel-permeation chromatography.     

Effect of pyridine on the kinetics of polymerization of methyl methacrylate initiated by benzoyl peroxide and azobisisobutyronitrile at 60°C

Solution polymerization of MMA, with pyridine as the solvent and BZ₂O₂ and AIBN as thermal initiators, was studied kinetically at 60°C. The monomer exponent varied from 0.45 to 0.91 as [BZ₂O₂] was increased from 1 × 10^?2 to 30 × 10^?2 mole/liter in a concentration range of 8.3-4.6 mole/liter for MMA. For AIBN-initiated polymerization the monomer exponent remained constant at 0.69 as [AIBN] varied from 0.4 × 10^?2 to 1.0 × 10^?2 mole/liter in the same concentration range for MMA. The k²_p/k_t Value increased in both cases with an increase in pyridine concentration in the system. This was explained in terms of an increase in the k_p value, which was due presumably to the increased reactivity of the chain radicals by donor-acceptor interaction between the molecules of solvent pyridine and propagating PMMA radicals and in terms of lowering the k_t value for the diffusion-controlled termination reaction due to an increase in the medium viscosity and pyridine content.     

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.     

Radiation-induced grafting of styrene vapor to polyethylene

The radiation-induced grafting of styrene vapor to low-density polyethylene film of 0.063 mm thickness was studied at 23°C at a dose rate of 1.98 × 10⁴ rad/hr. The concentration C of monomer in the film was measured as a function of pre-irradiation exposure time to monomer vapor. The concentration-dependent diffusion coefficient of styrene in polyethylene was calculated to be 4.9 × 10^?9 exp {2.0C/C₀} cm²/sec, where C₀ is the saturation concentration of styrene in the film, and a linear boundary diffusion coefficient for styrene vapor into polyethylene film was found to be 2.0 × 10^?7 cm/sec. The rate of grafting was determined as a function of the concentration of styrene absorbed in the film. The maximum graft yield was obtained with an initial styrene concentration in the film of 4 wt-%. Under conditions of low initial monomer concentration, the grafting rate increases with irradiation time. The results are compared with previously published data on grafting of polyethylene from methanol–styrene solutions. They are explained in terms of the viscosity of the amorphous region as a function of styrene content and the resistance to the diffusion of monomer at the film–vapor interface.     

Photoinitiated polymerization of methacrylic monomers in a polystyrene matrix: Kinetic,mechanistic, and structural aspects

The kinetics and mechanism of the photoinitiated polymerization of tetrafunctional and difunctional methacrylic monomers [1,6‐hexanediol dimethacrylate (HDDMA) and 2‐ethylhexyl methacrylate (EHMA)] in a polystyrene (PS) matrix were studied. The aggregation state, vitreous or rubbery, of the monomer/matrix system and the intermolecular strength of attraction in the monomer/matrix and growing macroradical/matrix systems are the principal factors influencing the kinetics and mechanism. For the PS/HDDMA system, where a relatively high intermolecular force of attraction between monomer and matrix and between growing macroradical and matrix occurs, a reaction‐diffusion mechanism takes place at low monomer concentrations (<30–40%) from the beginning of the polymerization. For the PS/EHMA system, which presents low intermolecular attraction between monomer and matrix and between growing macroradical and matrix, the reaction‐diffusion termination is not clear, and a combination of reaction‐diffusion and diffusion‐controlled mechanisms explains better the polymerization for monomer concentrations below 30–40%. For both systems, for which a change from a vitreous state to a rubbery state occurs when the monomer concentration changes from 10 to 20%, the intrinsic reactivity and k_p/k_t^1/2 ratio (where k_p is the propagation kinetic constant and k_t is the termination kinetic constant) increase as a result of a greater mobility of the monomer in the matrix (a greater k_p value). The PS matrix participates in the polymerization process through the formation of benzylic radical, which is bonded to some extent by radical–radical coupling with the growing methacrylic radica, producing grafting on the PS matrix. © 2001 John Wiley & Sons, Inc. J Polym Sci Part A: Polym Chem 39: 2049–2057, 2001     

Polymerization initiated by an electron donor–acceptor complex. Part I. Polymerization of methyl methacrylate initiated by liquid sulfur dioxide–pyridine complex in the presence of carbon tetrachloride

The polymerization of methyl methacrylate can be initiated by a charge-transfer complex of liquid sulfur dioxide and pyridine in the presence of carbon tetrachloride. The molar ratio of sulfur dioxide and pyridine which participated in the complex was found from a spectrophotometric study to be 2:1. The polymerization proceeds through free-radical intermediates. The overall rate of polymerization is proportional to the square root of the concentration of the complex, and the values of k_p/k_t^1/2 under the various polymerization conditions were satisfactorily consistent with the literature value. For the activation energy of the overall reaction, 8.2 kcal./mole was obtained, and for initiation, 9.7 kcal./mole was evaluated from the values of k_p/k_t^1/2. It was deduced from a kinetic mechanism for the initiation that a primary radical may be produced from the reduction of carbon tetrachloride by an associated complex consisting of liquid sulfur dioxide–pyridine complex and the monomer.     

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

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

Polymerization Kinetics of Water-Soluble N-Vinyl Monomers in Aqueous and Organic Solution

Summary: Free-radical batch polymerization (FRP) of N-vinyl pyrrolidone (NVP) and N-vinyl formamide (NVF) monomers in aqueous solution as well as NVP polymerization in organic (n-butanol) solution has been studied. The differences found in rate of monomer conversion with monomer and solvent choice correlates well with the differences in values of the propagation rate coefficients (k_p) and their variation with monomer concentration measured in independent pulsed-laser polymerization studies, a result demonstrating that a generalized understanding of water-soluble vinyl monomers can be obtained once their k_p differences have been accounted for. A reasonable representation of polymer molecular mass averages and the complete molecular mass distributions for the three systems was obtained by assuming that the rate coefficient for transfer to monomer, polymer, and organic solvent also vary as a function of monomer concentration.     

Graft polymerization kinetics of acrylamide initiated by ceric nitrate–dextran redox systems

The kinetics of the graft polymerization of acrylamide initiated by ceric nitrate—dextran polymeric redox systems was studied primarily at 25°C. Following an initial period of relatively fast reaction, the rate of polymerization is first-order with respect to the concentrations of monomer and dextran and independent of the ceric ion concentration. The equilibrium constant for ceric ion—dextran complexation K is 3.0 ± 1.6 l./mole, the specific rate of dissociation of the complex, k_d, is 3.0 ± 1.2 × 10^?4 sec.^?1, and the ratio of polymerization rate constants, k_p/k_t, is 0.44 ± 0.15. The number-average degree of polymerization is directly proportional to the ratio of the initial concentrations of monomer and ceric ion and increases exponentially with increasing extent of conversion. The initial rapid rate of polymerization is accounted for by the high reactivity of ceric ion with cis-glycol groups on the ends of the dextran chains. The polymerization in the slower period that follows is initiated by the breakdown of coordination complexes of ceric ions with secondary alcohols on the dextran chain and terminated by redox reaction with uncomplexed ceric ions.     

Synthesis and characterization of graft copolymers of methacrylonitrile/methacrylate mixtures onto amylomaize by the ceric ion method

Graft polymerizations of mixtures of methacrylonitrile with n-alkyl methacrylales onto amylomaize were carried out. The graft copolymers were characterized by both IR and ¹³C-NMR spectroscopies. The influence of the monomer feed on the grafting parameters has been studied. The variation of these parameters with the mole fraction of methacrylate in the feed for the first three systems studied, MAN/MMA, MAN/EMA and MAN/BMA, was similar: thus, percent grafting (%G, percent weight of grafted polymer with respect to grafted amylomaize), percent grafted amylomaize (%GA, percent weight of grafted amylomaize with respect to initial amylomaize), percent grafting conversion (%C_g, percent weight of grafted polymer with respect to initial monomer), and percent total conversion (%C_t, percent weight of total acrylic polymer with respect to initial monomer) were increased, but percent grafting efficiency (%GE, percent weight of graft copolymer with respect to total polymer) decreased. The system MAN/HMA presented values of grafting parameters lower than those of the previous systems. The optimum values were obtained at 0.6 HMA mole fraction in the monomer feed. When the number of carbon atoms of the n-alkyl group rises from 1 to 4, the increase of the n-alkyl group length gives rise to increases of the %G %C_g and %C_t values and decreases of the %GE and %GA values. For the largest methacrylate, the grafting reaction appears to be controlled by the lesser accessibility of the monomer to the active sites of the carbohydrate. © 1992 John Wiley & Sons, Inc.     

Kinetics of Aqueous Polymerization of Acrylamide Initiated by Ceric Ammonium Sulfate/2-Mercaptoethanol Redox System

Polymerization of acrylamide monomer, initiated by the redox system involving acidified ceric ammonium sulfate and 2-mercaptoethanol (2-ME) was carried out in an aqueous medium at 25° C. White, rigid polyacrylamide, isolated under controlled experimental conditions, showed a molecular weight of 1.5 × 10⁴ from viscosity measurements. The rate of monomer (M) conversion to polymer was found to be proportional to [M]^1.5, [2-ME]^0.5, and [Ce(IV)]^0.4. Further, the rate of disappearance of ceric ion was observed to be directly proportional to [2-ME] and independent of [M] in the range of 0.16–0.48 mole/liter. The explanation of the above proportionalities is given in terms of a proposed reaction mechanism. Values of the usual rate constants, k_r, k₀/k_t and k_t./k_p ^½ have been computed.     

Estimation of the Characteristic Ratio kp(f/kt)½ and Its Temperature Dependence in Bulk Polymerization of Vinyl Chloride

According to a reaction scheme which as its main features assumes that polymerization is predominantly in the interior of the monomer swollen polyvinyl chloride) particles and that all the decaying initiator finally contributes to the polymerization within the polymer particles, the ratio k_p (f/k_t)^½ = K (where k_p, k_t are rate constants for chain propagation and chain termination, respectively, within the particles and f is initiator efficiency) has been calculated for bulk polymerization of vinyl chloride at three temperatures. K is found to be markedly larger than the corresponding quantity for homogeneous solution polymerization, e.g., at 50°C it is seven times this latter quantity. The characteristic ratio K shows a marked negative temperature dependence, which corresponds to approximately -4.5 kcal/mole for E_p - (E_t/2), when f is assumed to be independent of temperature. This behavior is quite consistent with a strong gel effect being operative at the site of reaction, i.e., the swollen polymer particles can be taken as equivalent to a homogeneous polymerization system at high conversion.     

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 (k_pM)^–1, where c is a dimensionless chain‐transfer constant, k_p the propagation rate coefficient and M the monomer concentration, an analytical expression for k_t 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 k_t 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 k_t.     

Radical polymerization behavior of N,N-disubstituted acrylamide: Absolute rate constants for polymerization and copolymerization of N-acryloylpiperidine

The absolute rate constants of propagation (k_p) and of termination (k_t) of N-acryloylpiperidine (NAPi) were determined by the rotating sector method in bulk; k_p = 273 and k_t = 1.79 × 10⁷ L/mol s at 30°C. It was noted that k_p for NAPi was 100 times smaller than that for N,N-dimethylacrylamide (DMAcAm). The absolute rate constants of cross-propagations for copolymerizations with common monomers were evaluated by combination of the k_p value and the monomer reactivity ratios. Quantitative comparison of the rate constants with those of DMAcAm and poly(DMAcAm) radical shows that NAPi is as reactive as DMAcAm and the smaller k_p value for NAPi is ascribable to much the lower reactivity of the poly(NAPi) radical. The large difference in reactivity of the polymer radicals is discussed in relation to the steric factor of the piperadino and the dimethylamino groups which seems to affect the capability of the carboxamide group to stabilize the polymer radical.     

ESR studies of the radiation-induced multiple graft polymerization

The radiation-induced multiple-graft polymerization was studied by an ESR method. When methyl methacrylate vapor was introduced onto preirradiated polyethylene already grafted with styrene, the second step of grafting of methyl methacrylate occurred mainly in the polyethylene portion. The kinetic treatment proved that the termination rate constant k_t of methyl methacrylate decreased with the amount of styrene grafted in advance. On the other hand, when styrene vapor was introduced onto polyethylene grafted with methyl methacrylate, only radicals of poly(methyl methacrylate) decreased. In this case, the second step of grafting of styrene occurred in the poly-(methyl methacrylate) portion which covered the whole surface of the polyethylene powder. When monomer vapors were alternately introduced onto preirradiated polyethylene powder, the second step of grafting occurred at the growing chain end of the first monomer.