Termination in Dilute‐Solution Free‐Radical Polymerization: A Composite Model

Abstract

Literature data are summarized for the chain‐length‐dependence of the termination rate coefficient in dilute solution free‐radical polymerizations. In essence such experiments have yielded two parameter values: the rate coefficient for termination between monomeric free radicals, k, and a power‐law exponent e quantifying how kt values decrease with increasing chain length. All indications are that the value e ≈ 0.16 in good solvent is accurate, however the values of k which have been deduced are considerably lower than well‐established values for small molecule radicals. This seeming impasse is resolved by putting forward a ‘composite’ model of termination: it is proposed that the value e ≈ 0.16 holds only for long chains, with e being higher for small chains – the value 0.5 is used in this paper, although it is not held to dogmatically. It is then investigated whether this model is consistent with experimental data. This is a non‐trivial task, because although the experiments themselves and the ways in which they are analyzed are elegant and not too complicated, the underlying theory is sophisticated, as is outlined. Simulations of steady‐state polymerization experiments are first of all carried out, and it is shown that the composite model of termination both recovers the e values which have been found and beautifully explains why these experiments considerably underestimate the true value of k. Simulations of pulsed‐laser polymerizations find the same, although not quite so strikingly. It is therefore concluded that our new termination model, which retains the virtue of simplicity and in which all parameter values are physically reasonable, is consistent with experimental data. Taking a wider view, it seems likely that the situation of the exponent e varying with chain length will not just be the case in dilute solution, but will be the norm for all conditions, which would give our model and our work a general relevance.Normalized chain length distributions from PLP simulations.imageNormalized chain length distributions from PLP simulations.

The Cutthroat Competition Between Termination and Transfer to Shape the Kinetics of Radical Polymerization

“…intercept of log"k t # vs. can now apply Equation (12) to the data of Figure 3 and thereby procure k t 1,1 ≈ 2 × 10 8 L mol -1 s -1 (this value is only an estimate because of the uncertainty introduced by not knowing the mechanism of cross-termination). Both these values are in excellent agreement with those obtained by other methods, [9] although it is stressed that these values pertain to long chains only, not to short chains, meaning that k t 1,1 is not the true value of this quantity. [9] We additionally point out that Equation (8) confirms that 〈k t 〉 is independent of k trX c X in the termination limit, exactly as seen in Figure 1(b) (values at low k trX c X ).…”

“…Both these values are in excellent agreement with those obtained by other methods, [9] although it is stressed that these values pertain to long chains only, not to short chains, meaning that k t 1,1 is not the true value of this quantity. [9] We additionally point out that Equation (8) confirms that 〈k t 〉 is independent of k trX c X in the termination limit, exactly as seen in Figure 1(b) (values at low k trX c X ). Summarizing this section, it has firstly illustrated the capacity of Figure 1 and Equation (8) to explain trends in RP data.…”

“…Although recent theoretical [9] and experimental [10,11] work has shown that this two-parameter model is an oversimplification of reality, it is a nice model to use for calculations, as it clearly exposes the general effects of CLDT on RP kinetics, [12][13][14] and these trends are essentially the same for more complex homo-termination models. [9] The same also holds for cross-termination models, [12][13][14] and so the simplest one will be employed here unless otherwise stated:…”

“…[9,14,15] Most radical polymerizations are carried out with continuous initiation, which means that to excellent approximation they are in a steady state. Thus the steady-state solutions of…”

“…Making the steady-state assumption and the long-chain approximation, use of Equations (5) and (6) in Equations (2), (3) and (7) for the case of k trX = 0 (i.e., the termination limit) results in [9,14,15] (8) This equation holds strictly only for the geometric mean model, the physical basis of which is dubious for RP. [14] However, the remarkable thing about Equation (8) is that it holds qualitatively and semi-quantitatively for all models of cross-termination.…”