Geometric-mean termination

Figure 3.4 Termination rate eoefficient versus ehain length for the geometric mean termination model including Poisson broadening variation of chain-length exponent b. Symbols represent the results of the computer simulations, while the lines are the corresponding input values.

The rate constants for the cross termination terms are approximated as the geometric mean of the corresponding homotermination terms. Thus ... 

Termination scheme 11 applies to the geometric mean and phi factor models and scheme 12 Is required for the penultimate effect model. All the above reaction models were used In attempts to simulate kinetic data. 

An examination of elimination data from 44 pigs exposed to excess selenium as sodium selenite in feed was found to fit a one-compartment model of selenium elimination (Davidson-York et al. 1999). Serum selenium levels were monitored over a period of 46 days beginning 1-14 days after termination of exposure to the feed containing excess selenium. Data were not adequate to depict the initial distribution phase, but a geometric mean elimination half-life of 12 days was calculated. It is likely that the period of elimination included in this study corresponds to the second phase described by Thomson and Stewart (1974). 

Of course Equation 8 specifies only homotermination rate coefficients one requires also cross-termination rate coefficients. A simple model for these is that of the geometric mean ... 

This method has historical and educational relevance, since it highlights the geometrical meaning of conjugate directions. From a practical point of view, however, it is obsolete other methods with the same features require tty onedimensional searches only to have a quadratic termination. 

In defining it is assumed that the rate constant of cross-termination is equal to the geometric mean of the rate constants of termination in homopolymerization. This relation is applicable in the case of gas reactions. In copolymerizations, however, the value of can increase up to about 400 (methyl methacrylate/vinyl acetate). In addition, depends on the composition of the mixture. The cause of this composition effect is unknown, although it is significant that is particularly large when the monomers tend toward alternating copolymerization. 

Model D differs from Model C in that a geometric mean approximation (eq. 30) is applied to reduce the number of penultimate termination rate coefficients from 10 to 4 ... 

Olaj and Zifferer [229] have performed Monte Carlo simulations in which they placed 300 polymer chains in a lattice, considering excluded volume effects. From the obtained configurations they evaluated the shielding factor which describes how severely the presence of a polymer coil retards the termination of its own radical compared to small (unshielded) radicals. For chains of unequal size Olaj and Zifferer showed that the geometric mean model provided a reasonable mathematical description of their results, although the harmonic mean model. 

The line of thought explained above is, however, not new and can be traced back to earlier papers where several people have dealt with the derivation of (approximate) closed expression for chain-length dependent kinetics. Mahabadi [151] was the first to explicitly postulate this simple identity between microscopic and macroscopic termination rate coefficient, later followed by more elaborate analyses by Olaj etal. [161, 162, 168]. Beside the geometric mean relation, Olaj s method has also been shown to hold for other power law relations for kt [174] and has been experimentally applied to styrene [175] and MMA [176]. Unfortunately, the effect of chain transfer reactions was ignored in their simulations. 

Figures 3.4 and 3.5 show the results of the simulations using a geometric mean and Smoluchowski model respectively (termination model 2 and 3, respectively). In both simulations the chain-length dependence was varied in strength by adjusting the parameter b in geometric mean model and the parameters a and b in the Smoluchowski model. Note that different scales have been applied in these graphs. As is directly obvious from inspecting...

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