BIMOLECULAR FREE-RADICAL TERMINATION AT LOW CONVERSION

Click to increase image sizeClick to decrease image size

ACKNOWLEDGMENTS

J. d. K. gratefully acknowledges the financial support of the Netherlands Organization for Scientific Research. Furthermore, Dr. Gregory Russell, Prof. Dr. Michael Buback, Dr. Johannes Schweer, and Prof. Dr. Bob Gilbert are acknowledged for their fruitful remarks and stimulating discussion on all that has to do with bimolecular free-radical termination (and much more).

Notes

^†When two radicals combine, for instance, the reaction energy must be released within one vibration period, or the radicals will simply dissociate again. For simple molecules in the gas phase at low pressure, collision with a third body to dissipate this energy might not readily occur, and termination becomes a third-order reaction [3]. For polymeric species in solution, this behavior is not observed; either the collision frequency with solvent molecules is high enough, or the energy is dissipated over a vibrational, rotational, or any conformational mode of the polymer chain. Besides, because a solvent cage surrounds the locus of reaction, some combinations and subsequent redissociations may occur prior to a “successful” termination event. In some specific cases, firstorder termination processes also may occur (e.g., by means of isomerization of radicals to unreactive species) [4].

^†It should be noted that, in the literature, still conflicting statements are made about this “general recognition” (e.g. Ref. [9]).

^†A detailed discussion of the dynamics of polymer solutions (also) at higher conversions is presented in Refs. [42] and [46]. Some interesting contributions on the termination kinetics in these regimes can be found in Refs. 47-60.

^† D_s represents the diffusion of a polymer chain due to Brownian or thermal motion, while D_m is a coefficient that characterizes the relaxation of a concentration gradient present in a solution. At 0% conversion, these two diffusion coefficients are identical.

^†Recall that in poor solvents both k_q and D_s decrease with increasing polymer concentration.

^†The excluded volume effect is the effect that a polymer segment repels or excludes the presence of all other polymer segments in the volume that it is occupying itself.

^†Although tempting to conclude, constant C is not identical with k ¹ _t ^,1 as Eq. 11 was derived for the long-chain limit [150].

^†For sake of simplicity, “small” may be considered as a “polymer” of chain length 1.

^†Note that, assuming the Smoluchowski equation with Stokes-Einstein diffusion dynamics, k_q should scale as T/η. Care has to be taken when determining activation energies for free-radical termination reactions. A temperature increase is usually accompanied by a change in the radical chain-length distribution, resulting in misleading apparent activation energies [130].

^†Strictly speaking, the Smoluchowski model is not applicable to free-radical termination reactions. The derivation of this model comprises the assumption of an A + B reaction in which one of the reactants is present in large excess, a so-called single-sink model. Termination, however, represents a “multisink” problem in which all A and B species can react with each other. Nevertheless, the Smoluchowski equation has proved to be an accurate description for the termination kinetics of small radicals [20], [31].

^†Note, however, that these authors only consider a reaction to be “diffusion controlled” if a system cannot reach an equilibrium on the typical timescale of reaction and local depletion of reactants occurs. For termination reactions, the typical timescale of reaction is severely decreased by intermolecular excluded volume effects, the mean time needed for reaction is larger than typical polymer relaxation times, and consequently, this reaction is not diffusion controlled. It is important to realize that this definition differs from the one used in this article.

^†The spin multiplicity parameter p _spin takes into account that only radicals encountering each other in a singlet state (antiparallel electron spins) can terminate, as opposed to those in a threefold degenerated triplet state [21], [229] [Russell-Saunders coupling [230]]. For small radicals in (dilute) solution, a value of p _spin = 0.25 seems to be justified [31]. Some additional discussion on the value of p _spin can be found in Ref. [213].

^†In Eqs. 27 and 28, an additional factor 2 has been introduced to correct for differences in the definition of the rate of termination.