Diffusion controlled free radical termination

Early explanations of auto-acceleration postulated that at the onset point the viscosity of the polymer in monomer solution became sufficiently great that growing polymer radicals could not diffuse together in order to terminate each other. The decreased termination rate then led to the increased overall polymerisation rate. However, it is now known that the radical-radical reaction is diffusion controlled from the very start of the reaction, although the large acceleration in rate does not occur until a considerable amount of polymer has been formed. 

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The termination kinetic constant exhibits a somewhat more complex behavior. From the onset of reaction, termination is diffusion controlled (segmental diffusion controlled). The diffusion of the macroradicals is the controlling step and the primary means of free radical termination. At some later conversion, the termination mechanism changes from segmental to reaction diffusion control. In this region, a plateau in k, occurs. Reaction diffusion is a propagation controlled... 

The polymerization rate is directly proportional to the monomer concentration for ideal free radical polymerization kinetics. Deviations from this first-order kinetics can be caused by a whole series of effects which must be checked by separate kinetic experiments. These effects include cage effects during initiator free radical formation, solvation of or complex formation by the initiator free radicals, termination of the kinetic chain by primary free radicals, diffusion controlled termination reactions, and transfer reactions with reduction in the degree of polymerization. Deviations from the square root dependence on initiator concentration are to be primarily expected for termination by primary free radicals and for transfer reactions with reduction in the degree of polymerization. 

A major milestone in the history of polymer science was the macromolecular hypothesis by Staudinger [1]. The molecular structure of polymers started to emerge and nowadays, almost 80 years later, a knowledge base of respectable size has been built by the contributions of thousands of researchers. Nevertheless, there are still many aspects of free-radical polymerizations that are not fully understood. The bimolecular free-radical termination reaction is one such example. The first scientific papers dealing in some detail with the kinetics of this reaction, can be traced back to the 40 s when the gel-effect was discovered [2-4]. From subsequent research it became apparent that this reaction has a very low activation energy and is diffusion controlled under almost all circumstances. A major consequence of this diffusion-controlled nature is that the termination rate coefficient kt) is governed by the mobility of macroradicals in solution and is thus dependent upon all parameters that can exert an effect on the mobility of these coils. Consequently, kt is a highly system-specific rate coefficient and benchmark values for this coefficient do not exist. 

The discussion of the rate of bimolecular termination has, up to now, been mainly of a qualitative nature. The scaling of average or macroscopic kt values with viscosity, solvent effects and coil dimensions were discussed without much attention for the chain-length dependence of this process. This dependence originates from the simple fact that free-radical termination is a diffusion-controlled process. Consequently, the overall mobility of polymer chains and/or polymer chain ends determine the overall rate of radical loss in a polymerizing system. As small chains are known to be much more mobile than large ones, the chain length of radicals can be expected to have a profound effect on the termination kinetics. 

It should, however, be mentioned that diffusion-control of the termination step could be an important variable which has not been taken into consideration in the previous calculations. Whereas Rabel and Ueberreiter have ai gued that this factor was unimportant in their experiments, an increasing amount of evidence indicates that the termination step in most free-radical pol onerization reactions is diffusion controlled 4S). The possibility that this is also the case with ethylene must be kept in mind, although even the qualitative consequences are unpredictable in the absence of specific models. Extrapolations over Inoad pressure and temperature ranges should therefore be made with caution. This is particularly true where the reaction proceeds heterogeneously. 

A characteristic of free radicals is the bimolecular radical-radical reaction which in many cases proceeds at the diffusion-controlled limit. These radical-radical reactions can occur either between two identical radicals or between unlike radicals, the two processes being known as self-termination and cross-termination reactions, respectively. 

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. 

As the polymerization reaction proceeds, scosity of the system increases, retarding the translational and/ or segmental diffusion of propagating polymer radicals. Bimolecular termination reactions subsequently become diffusion controlled. A reduction in termination results in an increase in free radical population, thus providing more sites for monomer incorporation. The gel effect is assumed not to affect the propagation rate constant since a macroradical can continue to react with the smaller, more mobile monomer molecule. Thus, an increase in the overall rate of polymerization and average degree of polymerization results. 

A characteristic reaction of free radicals is the bimolecular self-reaction which, in many cases, proceeds at the diffusion-controlled limit or close to it, although the reversible coupling of free radicals in solution to yield diamagnetic dimers has been found to be a common feature of several classes of relatively stable organic radicals. Unfortunatly, only the rate constants for self-termination of (CH3)jCSO (6 x 10 M s at 173 K) and (CH3CH2)2NS0 (1.1 X 10 M s at 163K) have been measured up to date by kinetic ESR spectroscopy and consequently not many mechanistic conclusions can be reached. 

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. 

In the case ofn-Ge(lll) substrates, surface states affect electrochemical deposition of Pb [319]. At high cathodic potentials, the deposition occurs by instantaneous nucleation and diffusion-controlled three-dimensional growth of lead clusters. Comparing H- and OH-terminated n-Ge(lll) surfaces, the nucleation is more inhibited at n-Ge(lll)-OH, which can be explained by the different densities of Ge surface free radicals, being nucleation sites. In this case, nucleation site density is about 1 order of magnitude lower than that for n-Ge(lll)-H. 

The gel or Trommsdorff effect (11) is the striking autoacceleration of the vinyl polymerization reaction as the viscosity of the monomer-polymer solution increases. Chain termination involving the recombination of two free radicals becomes diffusion controlled and this results in a decrease in the rate of termination. The concentration of active free radicals therefore increases proportionally. To sum up the gel effect the rate of Vazo catalyst initiation increases with temperature the rate of propagation or polymerization increases with the viscosity and the rate of termination of the growing polymer chains decreases with the viscosity. This of course also results in an increase in the molecular weight of linear polymers, but this has no practical significance when crosslinking is part of the reaction. 

For the average free radical in a vessel which is a distance x from the walls, diffusion will be the rate-controlling step in wall termination when to tew, or since D = Do/P, when... 

Now for most gases Dq/c is approximately equal to the mean free path and is very close to lO cm at STP (Table VIII.3), so that in a 500-cc flask (ro = 5 cm), P must be of the order of 0.002/c mm Hg or higher for diffusion to be important in controlling wall termination. Thus if a radical is captured on every collision (c = 1), diffusion control is important at pressures above 0.002 mm Hg. If, however, e = 10 , then the range is 20 mm Ilg or higher. Below these pressures radicals disappear at the walls, but there are no appreciable gradients present. 

For the linear free radical case, the time to vitrification is affected by the initial initiator concentration (Fig. 21). The vitrification curves are again S-shaped. For this case, the same values of kp, k, and k were used throughout the course of the reaction, although it is well known that the termination reaction becomes diffusion controlled at fairly low degrees of conversion... 

The bimolecular termination reaction in free-radical polymerization is a typical example of a diffusion controlled reaction, and is chain-length-depen-dent [282-288]. When pseudobulk kinetics appUes, the MWD formed can be approximated by that resulting from bulk polymerization, and it can be solved numerically [289-291]. As in the other extreme case where no polymer particle contains more than one radical, the so-caUed zero-one system, the bimolecular termination reactions occur immediately after the entrance of second radical, so unique features of chain-length-dependence cannot be found. Assuming that the average time interval between radical entries is the same for all particles and that the weight contribution from ohgomeric chains formed... 

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