Relationship between chain termination rate constant and conversion in radical polymerization

At low and high conversions, the chain termination rate constant for bimolecular termination between polymeric radicals given by k_t = A_tD_s, where A_t is a constant and D_s is the diffusion constant of radical chain end, is completely correct. This termination rate constant does not depend on solution viscosity, but conversion.     

Termination rate constants in radical polymerization

A rate constant is generally derived by using Fick's equation corresponding to the spherical interdiffusion of particles. By using this rate constant, chain and primary radical termination rate constants can be approximated to rate constants for the bimolecular reactions between two radical chain ends, and primary radical and radical chain end, respectively. The former is given by k_s = 8πN_LD_sL_s exp { ? L_s/R_s} × 10^?3 1./mole-sec. The latter is given by k_si = 4πN_L(D_s + D_i)L_si exp { ? L_si/R_si} × 10^?3 1./mole-sec. Here, N_L is Avogadro's number; D_s and D_i are the diffusion constants of radical chain end and primary radical, respectively; L_s and L_si are, respectively, the distances between two radical chain ends and between a primary radical and a radical chain end at a thermal energy equal to the coulombic energy of interaction of the net charges; and R_s and R_si are, respectively, the average distances between two radical chain ends and primary radical on a collision.     

Kinetics of polymerization rate based on the dependence of termination rate on chain length

When the structure of a primary radical resembles that of the chain end of the polymer radical, the rate of the primary radical termination is approximately the same as the termination rate between the oligomer radical and the polymer radical. The rate constant of termination between polymer radicals of chain length n and s, which involve the primary radicals, is k_t,ns = const.(ns)^?a. In the polymerization of methacrylonitrile initiated by 2,2′-azobisisobutyronitrile in dimethylformamide at 60.0°C, the value of a is found to be 0.091. From data obtained previously in the bulk polymerization of styrene initiated by 1-azobis-2-phenylethane at 60.0°C, the value of a is found to be 0.167. Because such a values are so large that they are not estimated by the excluded volume, the termination rates are discussed by adding the dependence of the diffusion of the segments to that for chain length.     

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.     

Analysis of polymerization rates in radical polymerization with primary radical termination of some methacrylates

Polymerization rates in polymerizations with primary radical termination of ethyl methacrylate, β-phenylethyl methacrylate, β-methoxyethyl methacrylate, and phenyl methacrylate initiated by 2,2'-azobis-(2,4-dimethylvaleronitrile) at 60°C were analyzed by using a simple linear equation. The values obtained of k_ti/k_ik_p (where k_ti is the primary radical termination rate constant, k_i is the rate constant of addition on to monomer of primary radical, and k_p is the propagation rate constant) on these analyses are discussed on the theoretical base.     

Primary radical termination rate constant at high conversion in radical polymerization

A primary radical termination rate constant given by: k_ti = A_1iD_i, where A_1i is a constant and D_i is the diffusion constant of the primary radical, was examined on the basis of the variation of conversion. It was proved that this rate constant is correct at high conversion. A relationship between primary radical termination rate constant and conversion was derived. The effect of variation of conversion on the gel effect is discussed.     

Effect of viscosity on termination rate constant in radical polymerization at low conversion

By using the expression, k_t = A₁D_s for the chain termination rate constant (where A₁ is a constant and D_s is the diffusion constant of radical chain end), a familiar chain termination rate constant, k_t = A₂/η_s (where A₂ is a constant and η_s is solvent viscosity) was examined with variation of conversion x. It was found that the proportionality of chain termination rate constant and solution viscosity is a valid relation at conversion 0 but is approximate at conversion x_c ≥ x > 0. Here x_c denotes a critical conversion under the average distance around spherical polymers formed in polymerization solution is zero. At conversions above x_c, the inverse relation between chain termination rate constant and solution viscosity is not correct.     

Emulsion polymerization. V. Lowest theoretical limits of the ratio kt/kp

It is assumed that the propagating polymeric radicals have no diffusive mobility in the very highly viscous medium within latex particles. Chain growth takes place on a lattice when the polymeric radical reacts with a monomer present on a lattice site adjacent to that occupied by the reactive chain end. For termination, two radical chain ends must be positioned on adjacent lattice sites at the same instant. The k_t/k_p ratios calculated with this model are either similar to or somewhat lower than the values determined in emulsion polymerization experiments. A minimum value of k_t/k_p can be calculated with the aid of the rate equation of Part III by assuming that only “living” polymer is produced during emulsion polymerization. This value of k_t/k_p is significantly lower than that calculated by the lattice model. Since the value corresponding to the lattice model gives the slowest practically achievable termination rate, it is concluded from these calculations that emulsion polymerization cannot be carried out under conditions in which chain termination is completely suppressed.     

Photopolymerization of methyl methacrylate with azo-containing polydimethylsiloxane as photoinitiator: Effect of siloxane chain length

The kinetics of the free radical photopolymerization of methyl methacrylate (MMA) initiated by azo-containing polydimethylsiloxane (PSMAI) and azobisisobutyronitrile (AIBN) was investigated. The greater polymerization rate R_p in MMA/PSMAI systems may be due to the higher value of the initiation rate R_i and the lower value of the termination rate constant k_t than that in MMA/AIBN system. The reaction orders with respect to initiators PSMAI decreased with an increase in polydimethylsiloxane chain length (SCL) in PSMAI. The observed deviations in polymerization rate from rate equation could be explained in terms of primary radical termination. The photoinitiator efficiency Φ of initiators decreased with increase in SCL, while the ratio of the rate constants for chain termination and chain initiation by primary radical increased with SCL. The fraction β of primary radicals entering into termination in MMA/PSMAI systems were larger than that in MMA/AIBN system. © 1996 John Wiley & Sons, Inc.     

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

Summary: A novel method for measuring termination rate coefficients, k_t, 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 k_t 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 k_t over the entire chain‐length region (c_R = radical concentration).     

The Influence of Chain Length‐Dependent Propagation on the Evaluation of the Chain Length Dependence of the Rate Coefficient of Bimolecular Termination, 1. PLP–SEC Methods

Chain length distributions have been calculated for polymers prepared by pulsed laser polymerization (PLP) under the condition that not only chain termination but also chain propagation is subject to chain length dependence. The interplay between these two features is analyzed with the chain length dependence of the rate coefficient of termination k_t introduced in the form of a power law and that of propagation k_p modeled by a Langmuir‐type decrease from an initial value for zero chain length to a constant value for infinite chain lengths. The rather complex situation is governed by two important factors: the first is the extent of the decay of radical concentration [R] during one period under pseudostationary conditions, while the second is that termination events are governed by [R]² while the propagation goes directly with [R]. As a consequence there is no general recommendation possible as to which experimental value of k_p is best taken as a substitute for the correct average of k_p characterizing a specific experiment. The second point, however, is apparently responsible for the pleasant effect that the methods used so far for the determination of k_t and its chain length dependence (i.e., plotting some average of k_t versus the mean chain‐length of terminating radicals on a double‐logarithmic scale) are only subtly wrong with regard to a realistic chain length dependence. This is especially so for the quantity k_t* (the average rate coefficient of termination derived from the rate of polymerization in a PLP system) and its chain length dependence.     

Gamma radiation-initiated polymerization of styrene at high pressure. II. Chain termination

The pressure dependence of the termination rate constant k_t for the free radical polymerization of monomers such as styrene is a function of polymer chain length, chain stiffness, and monomer viscosity, all of which influence the rate of segmental diffusion of an active radical chain end out of the coiled polymer chain to a position in which it can react with a proximate radical. Although k_t is not sensitive to changes in chain length, the large increase in molecular weight is responsible for a significant reduction in k_t at high pressures. For most of the common vinyl polymers, which exhibit some degree of chain stiffness, k_t is inversely proportional to a fractional power of the monomer viscosity because it depends in part on the resistance of chain segments to movement and in part on the influence of viscosity in controlling diffusion of the chain ends. The fractional exponent appears to increase with pressure and this is interpreted as evidence that the polymer chains become more flexible in a more viscous solvent. Because the fractional exponent is higher for more flexible chains, the value of the activation volume for chain termination is an indication of the degree of flexibility of the polymer chains, provided that the monomer is a good solvent for the polymer and that chain transfer is negligible.     

New aspects of pseudostationary polymerization and their application

Contrary to the stationary state little thought has been given so far to the general principles of the pseudostationary state. In this discourse an attempt is made to demonstrate that — within wide limits — arbitrary initiation profiles may be used to determine k_p/k_t (k_p = rate constant of chain propagation, k_t = rate constant of chain termination) from the frequency dependence of rate of polymerization (in analogy to the rotating-sector technique) as well as to evaluate k_p from the chain-length distribution (CLD) of samples prepared under pseudostationary conditions. Adverse factors like nonspontaneous transformation of absorbed photons into primary radicals do not invalidate this result. The existence of a universal relationship (independent of the initiation profile) is proved to exist for the second moment of the CLD of samples prepared under pseudostationary initiation conditions for constant (chain-length independent) k_t. Pseudostationarity, however, might be also achieved if not the initiation but the termination is periodically varied. In this case the CLD has a completely different shape but allows determination of k_p likewise. Finally, the case of chain-length dependent k_t is shortly discussed in connection with pulsed-laser initiation. Although the general equation for the second moment of the CLD does not apply any longer for this case some generality appears to exist under these conditions, too.     

Radical desorption in the emulsion polymerisation of vinyl monomers

The dependence of emulsion polymerisation rates on a number of important parameters is considered. Attention is paid to the use of seeded emulsion systems for the evaluation of radical desorption coefficients (k _o). Experimental conditions are shown to be important. When the average number of radicals per particle is low, large changes in the rate coefficient for chain termination do not have a large effect on the kinetics. With styrene and methylmethacrylate, radical re-absorption by the polymer particles is shown to be important and radical capture efficiences can be high. Consistency is established between the results of a number of workers and values fork _o are shown to be lower than those calculated from chain transfer rates.     

Updating the Chemiluminescence Oxygen‐Aftereffect Method for Determining the Rate Constant of the Peroxy‐Radical Self‐Reaction: Oxidation of Cyclohexene

Updating the facile chemiluminescence oxygen‐aftereffect method, most suitable for determining the rate constant (k_t) of the peroxy‐radical self‐reaction (main chemiluminescence channel), pertained to considering the sensitivity of such a method toward a disturbing influence of the peroxy radicals of the initiator of the chain oxidation process. Such a disturbance may derive from the side chemiluminescent reaction, which involves peroxy radicals of both hydrocarbon and initiator. To examine the applicability and limitations of the chemiluminescence method under present scrutiny, cyclohexene was used as the model oxidizable hydrocarbon substrate. Computer simulations of the reaction and chemiluminescence kinetics have demonstrated the validity of the considered methodology at the value of the rate constant of the propagation of the overall chain process by peroxy radicals of the initiator higher than 1 m ^?1 s^?1. Despite that the chemiluminescence time profile and the stationary level of the total chemiluminescence intensity depend on the kinetics of the side chemiluminescence channel and on the ratio of the excited‐state generation yields in the mentioned reaction channel and in the main chemiluminescence process, the value of k_t assessed by the oxygen‐aftereffect method has been found independent of variation of these characteristics.     

Investigation of autoacceleration effects during the solution polymerization of styrene

The possibility of obtaining increases in the rate and degree of polymerization through a decrease in the termination rate in nonviscous, homogeneous solution polymerizations of styrene has been investigated. Decreases in the termination rate were achieved through decreasing segmental diffusion of the propagating macroradical by greater occlusion, on the average, of the radical in the coiled polymeric chain. Coiling of the polymeric chain was effected by polymerizing styrene in thermodynamically poor (θ) solvents near the θ temperature for polystyrene. Examples of such systems are diethyl oxalate at 51.5°C. and cyclohexane at 34.6°C. Polymerization under these conditions did lead to a decrease in the k_t/k_p² kinetic ratio; this decrease resulted in increases in the degree of polymerization, but changes in the rate of polymerization, in contrast to the marked increases noted in viscous solution or heterogeneous polymerizations, were not observed. Possible explanations for the latter observations are discussed.     

On exact and approximate methods of calculating an overall termination rate coefficient from chain length dependent termination rate coefficients

In this paper is tackled the problem of calculating the overall termination rate coefficient 〈k_t〉 which follows from values of k_t^{i, j}, by which is denoted the rate coefficient for termination between two free radicals of degrees of polymerization i and j, respectively. The significance of this problem is that polymerization experiments yield 〈k_t〉 values, whereas microscopic models predict K_t^i,j values. An assumption-free method is presented for computing the steady state 〈k_t〉 corresponding to a set of K_t^i,j values. Parallel to this, approximate methods for calculating steady state 〈k_t〉 values are developed: the so-called short-long approximation is used, and coarse graining of the radical chain length distribution is applied. Calculations are firstly carried out using a microscopic termination model which describes an intermediate conversion polymerization system, and then a set of calculations are performed with low conversion conditions in mind. Comparison of the completely exact solutions with the various approximate solutions reveals which of the approximate and therefore more tractable models is suitable for accurate, microscopic modeling of polymerization kinetics at the different physical conditions envisaged. In this respect the credentials of one of the coarse grain approaches are found to be persuasive. Also executed are calculations probing how values of 〈k_t〉 depend on factors, such as the rate of propagation and of chain transfer to monomer, not traditionally regarded as having anything to do with termination rates; interesting results emerge. Because calculations are carried out with real systems very much in mind, the latter (and other) results are felt to be genuinely relevant to the mechanism of actual free radical polymerizations.     

Photopolymerization of Methyl Methacrylate Using the Benzil-Dimethylaniline Combination as the Photoinitiator

Benzil (BZL)-dimethylaniline (DMA) exciplex interaction has been utilized to initiate the photopolymerization of methyl methacrylate at 40°C in bulk and in solution. Depending on the nature of the solvent used, the monomer exponent values varied between 0.47 to 2.76. Initiator exponent values were found to be 0.29 and 0.15 with respect to [BZL] and [DMA], respectively. A low value of k_p ²/k_t and the high initiator transfer constant values indicated significant initiator-dependent termination. The semipinacol radical formed during irradiation is thought to be mainly responsible for primary radical termination while the generated ion radicals are presumed to participate in degradative initiator transfer.     

Kinetics and mechanism of the liquid-phase oxidation of <Emphasis Type="Italic">n</Emphasis>-carboxylic acids

Short reaction chains participate in the oxidation of C₃-C₆, C₈, C₁₀, and C₁₂ n-carboxylic acids. The quadratic-law recombination of peroxy radicals occurs both without and with chain termination. The ratio of the rate constants of these reactions increases to k′/k _t = 4.5 on passing from propanoic acid to pentanoic acid, and then it decreases almost to zero for dodecanoic acid. The anomalous variation of the k′/k _t ratio is explained by the fact that the radicals resulting from carboxylic acid oxidation at the CH bonds nearest to the functional group make different contributions to the recombination process, depending on the carbon chain length. The cross recombination of secondary hydrocarbon peroxy radicals with the HO₂^· radicals resulting from the oxidation of carboxylic acids at the β-C-H bonds proceeds without chain termination.