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.     

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.     

The effects of chain length dependent propagation and termination on the kinetics of free-radical polymerization at low chain lengths

The kinetics of the free-radical polymerization of methyl methacrylate at 60 °C in the presence of large amounts of n-dodecanethiol (DDM) were investigated as a test at short chain lengths of our recently proposed composite termination model [Smith GB, Russell GT, Heuts JPA. Macromol Theory Simul 2003;12:299]. The experimental data were found to show an unexpected and therefore very interesting trend: at high concentrations of DDM, the rate actually increases as chain lengths become smaller. No termination model can explain this unusual behaviour. However it was found that these results are very successfully described when, in addition to our composite termination model, chain length dependent propagation is included in modelling of the kinetics. These findings have important ramifications for all kinetic studies involving short polymeric radicals, including in low conversion living radical polymerization systems and studies determining the chain length dependence of the termination rate coefficient.     

On the (im)possibility of evaluating correct individual rate constants of chain propagation kp and chain termination kt by combining kp2/kt and kp/kt data for chain‐length dependent termination

On the basis of simulated data two ways of evaluating individual rate constants by combining k_p²/k_t and k_p /k_t (k_p , k_t = rate constants of chain propagation and termination, respectively) were checked considering the chain‐length dependence of k_t. The first way tried to make use of the fact that pseudostationary polymerization yields data for k_p²/k_t as well as for k_p /k_t referring to the very same experiment, in the second way k_p²/k_t (from steady state experiments) and k_p/k_t data referring to the same mean length of the terminating radical chains were compared. In the first case no meaningful data at all could be obtained because different averages of k_t are operative in the expressions for k_p /k_t and k_p²/k_t. In spite of the comparatively small difference between these two averages (≈15% only) this makes the method collapse. The second way, which can be regarded as an intelligent modification of the “classical” method of determining individual rate constants, at least succeeded in reproducing the correct order of magnitude of the individual rate constants. However, although stationary and pseudostationary experiments independently could be shown to return the same k_t for the same average chain‐length of terminating radicals within extremely narrow limits no reasonable chain‐length dependence of k_t could be derived in this way. The reason is an extreme sensitivity of the pair of equations for k_p/k_t and k_p²/k_t towards small errors and inconsistencies which renders the method unsuccessful even for the high quality simulation data and most probably makes it even collapse for real data. This casts a characteristic light on the unsatisfactory situation with respect to individual rate constants determined in the classical way, regardless of a chain‐length dependence of termination. As a consequence, all efforts of establishing the chain‐length dependence of k_t are recommended to avoid this way and should rather resort to methods based on inserting a directly determined k_p into the equations characteristic of k_p²/k_t or k_p/k_t, properly considering the chain‐length dependent character of k_t.     

Chain‐length dependent termination in rotating sector polymerization, 1. The evaluation of the rate constant of chain propagation kp

The chain‐length distributions (CLDs) of polymers prepared by rotating‐sector (RS) techniques under pseudostationary conditions were simulated for the case of chain‐length dependent termination and analysed for their suitability of determining the rate constant of chain propagation k_p from the positions of their points of inflection. The tendency to underestimate k_p is a little more pronounced than in pulsed‐laser polymerization (PLP) but, interestingly, the situation improves in the presence of chain‐length dependent termination. The estimates also were found to be more precise a) for smaller rates of initiation, b) for higher order points of inflection, c) if termination is by combination, d) if the role played by the shorter one of the two chains becomes less dominant. Taken in all, the determination of k_p from the points of inflection in the CLD of RS‐prepared polymers may well compete with the more famous PLP method, especially if some care is taken with respect to the choice of experimental conditions.     

On the conundrum of deriving exact solutions from approximate master equations

We derive the exact time-evolution for a general quantum system under the influence of a bosonic bath that causes pure phase noise and demonstrate that for a Gaussian initial state of the bath, the exact result can be obtained also within a perturbative time-local master equation approach already in second order of the system–bath coupling strength. We reveal that this equivalence holds if the initial state of the bath can be mapped to a Gaussian phase-space distribution function. Moreover, we discuss the relation to the standard Bloch–Redfield approach.     

Evaluation of polymerization rate by chain length dependence on polymer–polymer termination and primary radical termination in radical polymerization

In order to analyze the polymerization rate at high initiation rate and/or low monomer concentration, the rate equations are derived by a rate formulated previously for polymer–polymer termination and another rate for primary radical termination, which is formulated here (both rates depend on chain length of polymer radical). Such equations would be applicable to the kinetic data in the polymerizations of styrene and methyl methacrylate. This shows that the assumption that both rates are independent of chain length overestimates the rate of primary radical termination.     

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.     

On the use of exact state counting methods in RRKM rate calculations

The use of direct counts of molecular energy level degeneracies, P(E), t internal energy, E, RRKM unimolecular reaction rate calculations is clarified. Highly accurate densities of states for use in practical rate calculations can be obtained by utilizing an energy level “grain” size of 1–5 cm^?1 and by suitable averaging of P(E) over small energy intervals. An easily programmed algorithm for rounding energy levels to multiples of this “grain” size is given and applied to sum and density calculations.     

Effect of primary radical termination on chain length and initiator efficiency

In polymerization with primary radical termination, when reaction between primary radicals, which escape from solvent cage, is not negligible, a relation between chain length and polymerization rate is found regardless of tractable approximate procedures. Such a relation is applied to the kinetic data obtained in the polymerizations of methyl methacrylate (MMA) and vinyl acetate (VA) initiated by 2,2′-azobis(2,4-valeronitrile) at 50.0°C. Further, when the primary radical termination is high, an initiator efficiency can not be approximated to a ratio of the primary radicals escaping from the cage to the total primary radicals formed in the cage. In the polymerization of MMA, after the primary radicals escapes from the cage, they immediately react with the monomer. Thus, the reaction between the primary radicals is not significant. However, in the polymerization of VA, the rate of reaction between the primary radical and the monomer might be comparable to the rate of reaction between the primary radicals when the initiator concentration is quite high.     

Chain length‐dependent termination in pulsed‐laser polymerization. VIII. The temperature dependence of the rate coefficient of bimolecular termination in the bulk polymerization of styrene

The photosensitized polymerization of styrene in bulk was investigated in the temperature range of 25–70°C with respect to the average rate coefficient of bimolecular chain termination k̄_t, especially its chain length dependence at low conversions, by means of pulsed laser polymerization (PLP). Three methods were applied: two of them were based on equations originally derived for chain length independent termination taking the quantity k_t contained therein as an average k̄_t, while the third one consisted in a nonlinear fit of the experimental chain length distribution (CLD) obtained at very low pulse frequencies (LF‐PLP) to a theoretical equation. The exponent b characterizing the extent of chain length dependence was unanimously found to decrease from about 0.17–0.20 at 25°C to 0.08–0.11 at 70°C, slightly depending on which of the three methods was chosen. This trend toward more “ideal” polymerization kinetics with rise of polymerization temperature is tentatively ascribed to a quite general type of polymer solution behavior that consists in a (slow) approach to a lower critical solution temperature (LCST), which is associated with a decrease of the solvent quality of the monomer toward the polymer, an effect that should be accompanied with a decrease of the parameter b. © 2000 John Wiley & Sons, Inc. J Polym Sci A: Polym Chem 38: 697–705, 2000     

Unusual chain length dependent adsorption of linear and branched alkanes on UiO-66

In this work, the zero coverage adsorption properties of C₅–C₁₀ n- and iso-alkanes on the UiO-66, UiO-66-Me and UiO-66-NO₂ metal–organic frameworks are studied by gas phase pulse chromatography. Analysis of enthalpy values, entropy values, Gibbs free energies and Henry constants reveals unusual chain length dependent adsorption behaviour of linear and branched alkanes, caused by the complex structure of the zirconium metal–organic framework UiO-66. The UiO-66 structure consists of a small, tetrahedral and large, octahedral cage. It is shown that at specific carbon chain lengths (e.g. C₆–C₇ for n-alkanes), distinctive jumps in adsorption enthalpy, entropy values and Henry constants occur. This chain length dependent effect is even more pronounced for 2- and 3-methyl alkanes and double branched alkanes. This distinctive shift in adsorption behaviour occurs at a molecular size that coincides with the cavity dimensions of the smallest, tetrahedral cage. The resulting selective adsorption arises from confinement effects and is function of both the molecular shape and size.     

Chain-length dependent termination in pulsed-laser polymerization, 5. The evaluation of the rate coefficient of bimolecular termination kt for the reference system styrene in bulk at 25°C

Following earlier suggestions the values for the rate coefficient of chain termination k_t in the bulk polymerization of styrene at 25°C were formally calculated (a) from the second moment of the chainlength distribution (CLD) and (b) from the rate equation for laser-initiated pseudostationary polymerization (both expressions 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 number-average degree of polymerization of the chains at the moment of their termination yielded exponents b of 0.16–0.18 in the power-law k_t = A · P_n ^−b, A ranging from 2.3 × 10⁸ to 2.7 × 10⁸ L · mol⁻¹ · s⁻¹. These data are only slightly affected if termination is not assumed to occur by recombination only and a small contribution of disproportionation is allowed for.     

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.     

The segment-cloud model of the second virial coefficient of polymer solutions: effects of chain length and branching

The segment-cloud model for polymer molecules has been used, and the second virial coefficient A₂ obtained as a function of the interaction parameter z for linear and branched chains having different values of n. It is observed that the chain length effect, though much smaller than in the perturbation theory, increases as the degree of branching increases. Also, the branching parameter g is found to be a better correlating parameter than the segment density distribution for A₂. This is in contrast to earlier results for the perturbation theory of the excluded volume.