Radicals number distribution

A crucial distribution for the polymerization rate is the radical number distribution, which describes the number of polymer particles with a given number of radicals (Np(n), n > 0). The average number of free radicals per polymer particle is defined by (Asua, 2007)... 

This expression is plotted in Fig. 6.7 for several large values of p. Although it shows a number distribution of polymers terminated by combination, the distribution looks quite different from Fig. 5.5, which describes the number distribution for termination by disproportionation. In the latter Nj,/N decreases monotonically with increasing n. With combination, however, the curves go through a maximum which reflects the fact that the combination of two very small or two very large radicals is a less probable event than a more random combination. 

Figure 6.11 Comparison of the number distribution of n-mers for polymers prepared from anionic and free-radical active centers, both with f = 50.

Figure 9.2 Calculated (a) number and (b) GPC distributions for three polymers each with =100. The number distributions of chains formed by conventional radical polymerization with termination by disproportionation or chain transfer...

Experimental studies show, however, that these limiting approximations must be used with caution. For example, with some emulsion polymerization systems the mean number of radicals per particle may run from one-half to several depending on the size of the particle (I). Assuming that the polymerization process is stationary with known rates of radical arrival and termination, Stockmayer (6) and O Toole (3) have shown how to calculate not only the mean number of radicals but the entire number distribution as well. Until now, no methods of the same generality seem to exist for calculating the polymer size distribution. 

Lin and Chiu (1979) developed a theory for the MWD in a zero-one system that correctly predicts (where M is continuous rather than discrete) the free-radical lifetime distribution Ni leading to a value of 2 for P. The formalism of Lin and Chiu involves calculating the number of chains containing m monomer units. Their final results involve sequence summations over the range K m < oo. 

FIGURE 4.1 Number distribution curves for a polymer prepared by anionic living polymerization and, for comparison, that generated by free-radical polymerization with termination by combination (both curves calculated for v = 50). 

In an emulsion polymerization system, radicals are distributed among the polymer particles. The size of these particles is so small that there are only a small number of radicals per particle, as an average less than one radical per particle in many cases of practical interest. The compartmentalization of radicals among the particles is the most distinctive kinetic... 

For transfer-dominated systems the shape of the polymer distribution is the same as that of the free radical distribution, [Rj ] [Pp], keeping in mind that the chain length i equals the degree of polymerization, P. It tnms out that this situation is in fact more general Clay and Gilbert (497) have shown that for partly termination controlled distributions also, the following equation holds true, in which the slope of the logarithmic number distribution (Xp = [Pp]/Xlp =o[Rf ]) i correlated with the kinetic parameters ... 

It is important to note at this stage that the use of a truncation chain length in the simulations impedes the use of equation 3.34. This equation requires integration over the complete MWD, which is not available from the simulation output. To deal with this problem, it is necessary to use additional information that is available from the simulations. Beside the number of radicals at time / = 0, also the number of radicals that are still present when the simulation is stopped must be known. As termination is assumed to occur exclusively by combination, this number of residual radicals actually represents twice the number of polymer chains that are missing in the number distribution because of the limited time-span over which the simulations were performed. Hence, with knowledge about the concentration of residual radicals, denoted as [7 ], the integral in equation 3.34 can be replaced by ... 

A mass of polymer will contain a large number of individual molecules which will vary in their molecular size. This will occur in the case, for example, of free-radically polymerised polymers because of the somewhat random occurrence of ehain termination reactions and in the case of condensation polymers because of the random nature of the chain growth. There will thus be a distribution of molecular weights the system is said to be poly disperse. 

If every collision of a chlorine atom with a butane molecule resulted in hydrogen abstraction, the n-butyl/5ec-butyl radical ratio and, therefore, the 1-chloro/2-chlorobutane ratio, would be given by the relative numbers of hydrogens in the two equivalent methyl groups of CH3CH2CH2CH3 (six) compared with those in the two equivalent methylene groups (four). The product distribution expected on a statistical basis would be 60% 1-chloro-butane and 40% 2-chlorobutane. The experimentally observed product distribution, however, is 28% 1-chlorobutane and 72% 2-chlorobutane. 5ec-Butyl radical is therefore formed in greater anounts, and n-butyl radical in lesser anounts, than expected statistically. 

Consider now an encounter (F) radical pair formed from two free radicals. Since there are three components to the triplet state, T+i, To and T j, and only one singlet component, S, the encounter of two free radicals having uncorrelated spins leads to a statistical distribution of T and S radical pairs. However, some of the S radical pairs will react without undergoing T-S mixing, and this has the effect of increasing the relative number of T radical pairs. Consequently the F-pairs will give the sam e type of polarization as the T-pairs, but the degree of polarization will be less. 

Example 13.5 Determine the instantaneous distributions of chain lengths by number and weight before and after termination by combination. Apply the quasi-steady and equal reactivity assumptions to a batch polymerization with free-radical kinetics and chemical initiation. 

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