Kinetics chain-length-dependent

As with the rate of polymerization, we see from Eq. (6.37) that the kinetic chain length depends on the monomer and initiator concentrations and on the constants for the three different kinds of kinetic processes that constitute the mechanism. When the initial monomer and initiator concentrations are used, Eq. (6.37) describes the initial polymer formed. The initial degree of polymerization is a measurable quantity, so Eq. (6.37) provides a second functional relationship, different from Eq. (6.26), between experimentally available quantities-n, [M], and [1]-and theoretically important parameters—kp, k, and k. Note that the mode of termination which establishes the connection between u and hj, and the value of f are both accessible through end group characterization. Thus we have a second equation with three unknowns one more and the evaluation of the individual kinetic constants from experimental results will be feasible. 

Bamford43,59 63 has proposed a general treatment for solving polymerization kinetics with chain length dependent kt and considered in some detail the ramifications with respect to molecular weight distributions and the kinetics of chain transfer, retardation, etc. 

Eqs. (26) and (27) apply irrespective of the nature of the initiation process it is required merely that the propagation and termination processes be of the second order. They emphasize the very general inverse dependence of the kinetic chain length on the radical concentration and therefore on the rate of polymerization. The kinetic chain length may be calculated from the ratios k /kt as given in Table XI and the rate of polymerization. Thus, for pure styrene at 60°C... 

If the degree of polymerization is controlled principally by chain termination so that Xn is proportional to the kinetic chain length, the temperature coefficient of the average molecular weight will depend... 

This decomposition usually shows little dependence on solvent, so if Ed for decomposition in chloroprene is likewise 30.9 kcal. per mole, then since Eox = 25.1 kcal. per mole Ep = 9.6 kcal. per mole, assuming termination to require no energy of activation. This is 1.2 kcal. per mole larger than kp for styrene oxidation (8). Values of e for azobisisobutyronitrile in oxidation systems usually lie in the range 0.6 to 0.8 if e = 0.7, the above equation for the decomposition of the azonitrile and that given earlier for the initiated oxidation of chloroprene permit calculation of kp/kt1/2 for chloroprene and also the kinetic chain lengths of the oxidations (Table IV). 

In MC simulation, any kinetic event can be accounted for, as long as the probability of each kinetic event is represented exphcitly. Chain length dependent kinetics can be accounted for in a straightforward manner if the functional form is provided. In conventional MC simulations of molecular build-up processes, the monomeric units are added to each growing polymer molecule one-by-one therefore, a multitude of random numbers and calculations are required to simulate the formation of each polymer molecule. To get around this problem, a new concept, the competition techniqueyV/as proposed in order to drastically reduce the amount of calculation required for the simulation [263,264]. 

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... 

Fig. 15. Comparison of the conversion dependence of the kinetic chain length L ( ) and the tnomen-taneous degree of polymerization P ( ) of PTS-12...

Computational Aspects of Free Radical Polymerization Kinetics with Chain Length Dependent Termination... 

The kinetic chain length is thus inversely dependent on the radical concentration [Eq. (6.123)] or the polymerization rate [Eq. (6.124)]. This is of great practical significance as it shows that any attempt to increase the rate of polymerization by increasing the radical concentration will be only at the expense of producing smaller size polymer molecules. Equations (6.123) and (6.124) are applicable for all cases of bimolecular termination irrespective of the exact mechanism (combination or disproportionation) and also irrespective of the nature of the initiation process. Thus, for any monomer the Idnetic chain length will be Independent of whether the polymerization is initiated by thermal, redox, or photochemical means, or of the initiator used, if the [M ] or Rp is the same. 

It is obvious from Eq. (6.125) for kinetic chain length (i/) that the temperature dependence of u is determined by the ratio kp / kdktY. Using the Arrhenius expression (6.179) for the individual rate constants one obtains... 

Equation (P6.44.23) shows that the distribution function for the degree of polymerization of polymer formed by a free-radical chain mechanism, in which chain transfer is absent, depends only on the kinetic chain length and the ratio of disproportionation to coupling. The plot of vs. x according to Eq. (P6.44.23) gives the... 

Figure 4 shows the dependence of the time to failure on the rate of bimolecular radical termination which is directly related to diffusion of the polymeric segments in the solid matrix. Again, the log-log plot is linear with a positive slope and this time, again less than unity. Increased diffusion shortens the kinetic chain length and increases the time to failure. 

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