Chain-length dependent termination in pulsed-laser polymerization, 1. Interaction with propagation

Using a new simulation procedure in which each individual propagation step is subjected to a Poisson process it was proved that in case of chain-length dependent termination the apparent rate of propagation no longer coincides with the true one. This is caused by the polydispersity of the chain-length distribution of the growing chains: shorter chains are removed preferentially. This effect is comparatively small although significant. The consequences for the determination of the rate constant of chain propagation k_p are nearly negligible.     

The influence of the Poissonian character of chain growth on propagation probabilities in free radical polymerization with chain-length dependent termination

It is shown that the (established) method of deriving chain-length distributions from propagation probabilities is not fully consistent with the Poissonian character of chain propagation if termination is chain-length dependent: the fluctuation of chain propagation leads to somewhat lower radical concentrations (rates of propagation). The deviation is the more prominent the more important is the role assigned to the shorter one of the two chains involved in the termination process by the model applied.     

New aspects of chain-length dependent termination

A survey is given on a selection of recently developed methods for the evaluation of the rate coefficient k_t of termination and its chain-length dependence. In particular these are the time-resolved single-pulse pulsed laser polymerization (TR-SP-PLP), the single pulse pulsed laser polymerization in combination with the analysis of the molecular weight distribution produced (SP-PLP-MWD), the methods yielding an average k_t either from the second moment of the chain-length distribution (CLD) or from the rate of polymerization, and a method focusing on the chain-length dependence of k_t consisting in an analysis of the CLD resulting from PLP experiments carried out at low pulse frequencies (LF-PLP). The results obtained by these methods are compared and discussed. The role of the shielding of the two radical chains by their appendant coils is emphasized.     

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

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

Chain-length dependent termination in pulsed-laser polymerization, 2. On the prospects of determining the bimolecular termination constant kt from the second moment of the chain-length distribution

The correct (event weighted) average of k_t, 〈k_t〉, has been calculated for pseudostationary laser-induced polymerization for a kinetic scheme with chain-length dependent termination and compared to the average k̄_t obtained by formally solving for k_t the expression for the second moment of the chain-length distribution valid for chain-length independent termination (represented by the product of rate of polymerization ν_p and weight average degree of polymerization P̄_w). It is shown that there is a fair agreement between the two quantities. This may be used to recover experimentally the power-law governing the dependence of k_t on chain-length, especially its exponent.     

A comprehensive Monte Carlo simulation of styrene atom transfer radical polymerization

<正>An optimized and high-performance Monte Carlo simulation is developed to take thorough account of four different cases of termination in styrene ATRP.According to the simulation results,the bimolecular termination rate constant sharply drops throughout the polymerization when either chain-length dependency of termination rate constant,gel effect,or both together is applied to the simulation.In addition,as expected,the initiator is quickly decomposed at the early stages of the polymerization.The concentration of the catalyst in lower oxidation state decreases at first and then plateaus at higher conversion;furthermore,the steady concentration of M_t~nY/L in the polymerization is the highest when the chain-length-dependent diffusion-controlled termination rate constant is employed in the simulation.The rates of deactivation and chain end degradation reactions are also smaller in this case.Therefore,the fraction of dormant chains is higher throughout the reaction and consequently the portion of dead polymers decreases.Besides,molecular weight increases linearly with conversion;however,when neither gel effect nor chain-length dependency of termination rate constant is considered,the molecular weight deviates from linearity at the end of the reaction.The peak of chain length distribution shifts toward higher molecular weight too during the reaction.Finally,the molecular weight distribution broadens at higher conversion;however, the chain length distribution of polymers produced under conditions of applying chain-length-dependent diffusion-controlled termination rate constant is narrower.     

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.     

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.     

SP-PLP-EPR study of chain-length-dependent termination in free-radical polymerization of n-dodecyl methacrylate, cyclohexyl methacrylate, and benzyl methacrylate: evidence of "composite" behavior

The chain-length dependence of the termination rate coefficient in n-dodecyl methacrylate (DMA), cyclohexyl methacrylate (CHMA), and benzyl methacrylate (BzMA) bulk free-radical homopolymerizations at ambient pressure and at temperatures from -20 to 0 degrees C is deduced using the recently developed technique of SP-PLP-EPR: pulsed-laser polymerization (PLP) in which time-resolved EPR measurement of radical concentration, cR, is made following each single pulse (SP) of an excimer laser. The decay of cR results from termination of radicals of almost identical size. Their chain length, i, increases linearly with time, t, after applying a SP. The rate coefficient, kt(i,i), for termination of two radicals of size i is determined by fitting the experimental cR vs t data. This process demonstrates that (at least) two power-law exponents are necessary to describe kt(i,i) over the extended chain-length range of i = 1 to 1000. This is consistent with the so-called "composite model" , which uses power-law exponents alpha(S) and alpha(L) to describe termination of radicals either shorter or longer, respectively, than a crossover chain length, ic. The fourth parameter obtained from fitting the SP-PLP-EPR data with this model is kt(1,1), the termination rate coefficient for two radicals of degree of polymerization 1. Previous DMA experiments are reanalyzed while new experimental results are reported and analyzed for CHMA and BzMA. The parameter values for CHMA and BzMA termination at 0 degrees C are almost identical-kt(1,1) approximately 3 x 10(7) L mol(-1) s(-1), alpha(S) approximately 0.50, ic approximately 90, and alpha(L) approximately 0.21-and they are close to those for DMA at 0 degrees C: kt(1,1) approximately 1 x 10(7) L mol(-1) s(-1), alpha(S) approximately 0.64, ic approximately 50, and alpha(L) approximately 0.18. The results fully support the composite model in that the chain-length dependence is more pronounced for shorter than for longer radicals, i.e., alpha(S) > alpha(L). Moreover, the power-law exponent that characterizes termination of long-chain radicals is close to the theoretical value of alpha(L) = 0.16. In fact all parameter values-including the small differences between DMA and CHMA/BzMA-are more-or-less in accord with expectations based on polymer dynamics. Furthermore, our results suggest that termination of methacrylate radicals with large cyclic or long n-alkyl substituents may be affected by steric shielding of the radical functionality.     

The incorporation of side reactions into pseudostationary polymerization, 3. The closed solution of a kinetic scheme for pulsed-laser polymerization comprising concomitant continuous initiation

A procedure is developed that allows the calculation of chain-length distributions of polymers prepared by periodic modulation of the initiation process considering concomitant continuous initiation. For the case of a (pseudostationary) laser-pulse initiated polymerization process a closed solution could be derived for the pseudostationary radical concentration and for the chain-length distribution of dead polymer terminated by disproportionation or stabilized by chain-transfer to monomer or solvent. The analysability of the characteristic peaks appearing in the chain-length distributions of laser-pulse initiated polymers (which is the key for determining the rate constant k_p) is only moderately influenced by continuous thermal radical formation if the extent of this side reaction is not pathologically large, i.e. as long as the amount of primary radicals created by the laser-pulse appreciably exceeds that produced in the dark reaction.     

A new method for evaluation of the characteristic constant for primary radical termination in free radical polymerization

A new model has been proposed for estimation of the characteristic rate constant for primary radical termination, using the radical life time, rate of polymerization, and rate of initiation. The model can be used to estimate the characteristic rate constant for primary radical termination under most conditions of free radical polymerization. By applying this model to high conversion polymerization data, it is possible to compare the conversion dependence of the characteristic rate constant for primary radical termination and initiation rate as well as the conversion dependence of the termination rate constant.The model has applied to various published experimental data and the results compared with literature values.     

Chain-length dependent termination in pulsed-laser polymerization, 6. The evaluation of the rate coefficient of bimolecular termination kt for the reference system methyl methacrylate in bulk at 25°C

The values for the rate coefficient of chain termination k_t in the bulk polymerization of methyl methacrylate at 25°C were formally calculated (i) from the second moment of the chain-length distribution and (ii) from the rate equation for laser-initiated pseudostationary polymerization (both expressions were originally derived for chain-length independent termination) by inserting the appropriate experimental data including the rate constant of chain propagation k_p. These values were treated as average values, k and k , respectively. They exhibited good mutual agreement, even the predicted gradation (k < k by about 20%) was recovered. The log-log plot of k_t vs. the average degree of polymerization of the chains at the moment of their termination v′ yielded exponents b of 0.16–0.17 in the power-law k _t = A · v′^−b, A ranging from 1.1 × 10⁸ to 1.3 × 10⁸ (L · mol⁻¹ · s⁻¹). A 70% contribution of disproportionation to overall termination has been assumed in the calculations.     

Diffusion-controlled rate constant of termination reaction in radical polymerization

Smoluchowski's theory has been modified and the improved theory was applied to diffusion-controlled polymerization. This application proved that the rate-controlling process is not transrational diffusion but the segmental diffusion. The segmental diffusion-controlled rate constant was derived by the collision theory. This rate constant explains the experimental fact that the diffusion-controlled rate constant of bimolecular termination in radical polymerization of alkyl methacrylate is inversely proportional to solution viscosity and independent of the molecular weight of the polymeric free radical.     

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.     

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.     

Collision between polymeric radicals for termination rate constants

The motion of each polymeric radical during a collision between the polymeric radicals with the same radius is treated as completely random motion. The result obtained is: k_t = 0.250k_s (where k_t is the chain-termination rate constant and k_s is the reaction rate constant between radical chain ends). On taking the motion of the primary radical during a collision between a primary radical and a large polymeric radical to be completely random, the result obtained is: k_ti = 0.250k_si (where k_ti is the primary radical termination rate constant and k_si is the reaction rate constant between primary radical and radical chain end). On substituting k_s for k_si in the second equation, the rate constant obtained becomes the chain termination rate constant between the very small polymeric radical and the very large polymeric radical, and identical to the former equation. This identity indicates that the effect of the difference of the size of the polymeric radicals on the collision process relating to the chain termination rate constant should not be large.     

The termination reaction in free radical copolymerization. I. Methyl methacrylate and butyl- or dodecyl methacrylate

Using a spatially intermittent reactor, the absolute rate constant for the termination reaction in free radical copolymerization has been measured for the monomer pair methyl methacrylate (MMA)–butyl methacrylate (BMA). For the pair MMA–dodecyl methacrylate (DMA) the relative rate constant for termination has been measured. In both cases the termination rate constant was a monotonically changing function of the monomer feed composition. This function can be well approximated by a simple calculation of the enchained monomer units' contributions to the average segmental friction coefficient of the copolymer chain. An attempt to apply a previously derived theoretical treatment based on penultimate unit effects produced physically unrealistic results.     

Monte Carlo simulation of kinetics and chain-length distribution in radical polymerization

In this paper, the Monte Carlo method for numerically simulating the kinetics and chain-length distribution in radical polymerization is described. Because the Monte Carlo method is not subject to the assumption of steady-state, it is particularly suitable for studying the kinetic behaviour before the steady-state has been reached and for systems in which the steady-state assumption may be violated. Illustrative applications of the algorithm given in this paper not only demonstrate convincingly both the feasibility and usefulness of the algorithm, but also provide some new insight into the illustrative examples. For the case of pseudostationary radical polymerization such as rotating-sector and pulsed-laser initiations, we have found that the pseudostationary radical concentration can be reached after two or three initiation periods. However, the number-average chain-length x̄_n reaches the pseudostationary value much slower than the radical concentration. It is oscillatively reaching the pseudostationary value, and the amplitudes of the oscillations are decreasing with time. We have also found that the chain-length distribution of the resulting polymer in the case of pseudostationary radical polymerization with termination by combination has stronger periodic modulation. Hence, it should be easier to locate the points of inflection in practice. Therefore, the rate constant of propagation, k_p, can be determined precisely for systems which are dominated by a combination-type of termination.     

Radical polymerization of vinyl acetate with primary radical termination

Vinyl acetate was polymerized at high initiation rate with 2,2′-azobis(2,4-dimethyl valeronitrile) as initiator at 50°C. In this polymerization, the power dependence of polymerization rate on the initiation rate is smaller than at lower concentration of monomer. This dependence was kinetically analyzed at each given concentration of monomer. Average degree of polymerization of polymer formed depends on the concentration of initiator. This dependence was explained by considering chain and primary radical terminations and transfer to monomer of polymer radical, and the initiator efficiency (=0.503) was deduced. It was found that the chain termination is inversely proportional to solvent viscosity, but the primary radical termination is not inversely proportional to solvent viscosity. Further, the value of the primary radical termination rate constant (=1.4 × 10⁹l./mole-sec) was estimated.