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.     

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.     

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.     

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     

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.     

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.     

The kinetics of free radical polymerizing systems at low conversion, 1. On the rate determining step of the bimolecular termination reaction

Using a styrene bulk system as a model, this paper examines rates of termination at very low conversions in bulk and solution polymerizations. No definitive answer to the question of what determines such rates of termination is arrived at. Indeed, it is argued that on the basis of existing kinetic information, no such definitive answer is possible. However several things may be said with conviction. To begin with, it is rigorously shown that low conversion rates of termination cannot be explained by assuming that all radical chain end encounters result in termination, and then using center-of-mass diffusion coefficients of polymer in free solution to calculate rates of chain end encounter. However this does not mean that rates of center-of-mass diffusion do not determine rates of low conversion termination, as is shown; the idea that it may be the case that not all chain end encounters result in termination, this a manifestation of a spin multiplicity effect, is especially worthy of mention. It is also possible to explain low conversion rates of termination, as has traditionally been done, in terms of chain end motions being hindered by the presence of another polymer chain. However in concentrating on interactions between overlapping long chain macroradical coils, this traditional picture is certainly inaccurate, for it is shown that most termination interactions must involve at least one radical of shorter than expected degree of polymerization. This has the important consequence that an understanding of overall rates of dilute solution termination must be founded on an understanding of the diffusional behavior of the ends of short and intermediate length polymer chains.     

Statistical study of the effect of variation of bimolecular rate constant in condensation polymerization

We investigate the effect of variation of bimolecular rate constant with chain length on condensation polymerization by a statistical approach. The variation of reactivity with chain length is simulated in the present study by assigning appropriate weight factors to different species. Formal expressions for the size distribution and various averages of interest have been obtained. Some simple cases are then analyzed and comparisons with the experimental results of Taylor and Howard are carried out.     

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.     

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.     

Effect of solvent on the termination rate constant in initial stages of free radical polymerization

The effect of solvent on the termination rate constant K_t, in the initial stages of free radical polymerizations has been estimated by considering its effect on the viscosity of the medium and on the overall dimensions of the macroradicals. The expression derived predicts that K_t is inversely proportional to the viscosity of the reaction medium, η₀, and that K_t, increases as the overall dimensions of the radicals decrease in poorer solvents. The effect of solvent on the η₀K_t, product depends on the average size and concentration of polymer in the system. For relatively high concentrations of high-molecular-weight polymers η₀K_t can be greater in a better solvent than in a poorer solvent. This trend would be reversed for low concentrations and/or low polymer molecular weights. Good agreement has been found between experimental and estimated values of η₀K_t.     

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.     

Relationships for determining the rate constant of the polymerization growth reaction by means of the emulsion polymerization method

Some simple equations for the emulsion polymerization system were derived on the basis of the Smith-Ewart theory. These are used in calculating the molar monomer concentration and rate of polymerization in units of moles per liter per second for the zero-order region with respect to the monomer as well as in calculating the rate constant of the growth reaction.