Diffusion-controlled termination

Termination by self-reaction of propagating radicals is a diffusion-controlled process even at very low conversion. The evidence for this includes the following  

Center of mass or translational diffusion is believed to be the rate-detennining step for small radicals and may also be important for larger spceics. However, other diffusion mechanisms are operative and are required to bring the chain ends together and these will often be the major term in the termination rate coefficient for the case of macromolecular species. These include  

An understanding of this behavior requires that we appreciate termination is a diffusion-controlled reaction best described as proceeding by the three-step process [Mahabadi and O Driscoll, 1977a,b North, 1974]  

Translational diffusion of two propagating radicals (i.e., movement of the whole radicals) until they are in close proximity to each other  

Rearrangement of the two chains so that the two radical ends are sufficiently close for chemical reaction, which occurs by segmental diffusion of the chains, that is, by the movement of segments of a polymer chain relative to other segments 

Theoretical considerations indicate that kc would be very large, about 8 x 109 L mol-1 s 1, in low-viscosity media (such as bulk monomer) for the reaction between two radicals. The rate constants for reactions of small radicals (e.g., methyl, ethyl, propyl) are close to this value (being about 2 x 109 L mol s 1) [Ingold, 1973]. Experimentally determined kt values for radical polymerizations, however, are considerably lower, usually by two orders of magnitude or more (see Table 3-11). Thus diffusion is the rate-determining process for termination, kc 3 fct, and one obtains 

Thus the experimentally observed termination rate constant kt corresponds to k and kff jkx, respectively, for the two limiting situations. 

---

More complex models for diffusion-controlled termination in copolymerization have appeared.1 tM7j Russo and Munari171 still assumed a terminal model for propagation but introduced a penultimate model to describe termination. There are ten termination reactions to consider (Scheme 7.1 1). The model was based on the hypothesis that the type of penultimate unit defined the segmental motion of the chain ends and their rate of diffusion. 

A new rate model for free radical homopolymerization which accounts for diffusion-controlled termination and propagation, and which gives a limiting conversion, has been developed based on ft ee-volume theory concepts. The model gives excellent agreement with measured rate data for bulk and solution polymerization of MMA over wide ranges of temperature and initiator and solvent concentrations. 

Abstract. Auto-accelerated polymerization is known to occur in viscous reaction media ("gel-effect") and also when the polymer precipitates as it forms. It is generally assumed that the cause of auto-acceleration is the arising of non-steady-state kinetics created by a diffusion controlled termination step. Recent work has shown that the polymerization of acrylic acid in bulk and in solution proceeds under steady or auto-accelered conditions irrespective of the precipitation of the polymer. On the other hand, a close correlation is established between auto-acceleration and the type of H-bonded molecular association involving acrylic acid in the system. On the basis of numerous data it is concluded that auto-acceleration is determined by the formation of an oriented monomer-polymer association complex which favors an ultra-fast propagation process. Similar conclusions are derived for the polymerization of methacrylic acid and acrylonitrile based on studies of polymerization kinetics in bulk and in solution and on evidence of molecular associations. In the case of acrylonitrile a dipole-dipole complex involving the nitrile groups is assumed to be responsible for the observed auto-acceleration. 

If ki and k.i are much larger than kj, the reaction Is controlled by kj. If however, ki and k.i are larger than or comparable to kz, the reaction rate becomes controlled by the translational diffusion determining the probability of collisions which Is typical for specific diffusion control. The latter case Is operative for fast reactions like fluorescence quenching or free-radical chain reactions. The acceleration of free-radical polymerization due to the diffusion-controlled termination by recombination of macroradicals (Trommsdorff effect) can serve as an example. 

The diffusion-controlled termination model considers the termination reaction as the reaction... 

When reaction diffusion controlled termination, the ratio of k,/kp[m] was found to be nearly the same for all monomers of the same type of functionality (methacrylate or acrylate) and appeared to be independent of the reaction conditions (i.e., temperature and light intensity). The reported values for this ratio was approximately 3 for the methacrylates and between 6 and 8 for the acrylates. 

The rate of dispersion (co)polymerization of PEO macromonomers passes through a maximum at a certain conversion. No constant rate interval was observed and it was attributed to the decreasing monomer concentration. At the beginning of polymerization, the abrupt increase in the rate was attributed to a certain compartmentalization of reaction loci, the diffusion controlled termination, gel effect, and pseudo-bulk kinetics. A dispersion copolymerization of PEO macromonomers in polar media is used to prepare monodisperse polymer particles in micron and submicron range as a result of the very short nucleation period, the high nucleation activity of macromonomer or its graft copolymer formed, and the location of surface active group of stabilizer at the particle surface (chemically bound at the particle surface). Under such conditions a small amount of stabilizer promotes the formation of stable and monodisperse polymer particles. 

A valid kinetic model of stage 3 emulsion polymerization must account for diffusion-controlled termination and propagation reactions. Marten and Hamielec (J) have proposed such a model based on a free-volune theory and have confirmed its validity for the bulk polymerization of methyl methacrylate (7). Herein is reported an evaluation of this model for the emulsion... 

A comprehensive model which is based on the free-volume theory and which accounts for the effect of molecular weight and solvent on chain entanglements and glassy-state transition has been recently developed by Marten and Hamielec (7 ). This model accounts for diffusion-controlled termination and propagation... 

Diffusion-Controlled Termination and Propagation Model Parameters... 

Because of diffusion-controlled termination and propagation with concomitant at a particular conversion, it is possible to operate a CSTR at considerably higher production rates. Because of the additional heoeficial effects of cold monomer and water feeds on heat removal, much higher production rates are possible than with a batch reactor of the same volume. It should he remembered that polymer production rates are usually limited by heat removal capacity. 

This form predicts that k approaches zero when either k or k approaches zero. Considering that the beginning of the gel effect is the result of translational diffusion controlled termination (1), it is reasonable to assume that k may become very small if the translational diffusivity of the i-mer approaches zero. However, if i-mer radicals whose k = 0 are mixed with j-mer radicals whose k i 0, it must be realized that there can be termination due to the mobility of the j-mer. Equation (3) would predict no reaction and on this basis the authors do not accept the geometric mixing rule as having general applicability. 

In practice, (f) can be calculated by inserting experimental copolymerization rates into Eq. (7.64). The values of (j> thus obtained are frequently greater than unity, and these deviations are ascribed to polar effects that favor cross-termination over homotermination. However, this is not always unambiguous, since the apparent cross-termination factor may vary with monomer feed composition in a given system [25,26]. It is clear also that termination reactions are at least partially diffusion controlled [27,28]. A dependence of segmental diffusivity on the structure of macroradicals is to be expected and dependence of diffusion controlled termination on copolymer composition seems reasonable. It is therefore plausible that the value of the overall termination rate constant ku in copolymerizations should be functions of fractions F and Fi) of the comonomers incorporated in the copolymer. An empirical expression for ku has thus been proposed [27] ... 

---