Copolymer equation penultimate model

Equation (7.32) shows that pjj is constant for a particular copolymer if the terminal model applies therefore the ratio NmjMi/Nmi also equals this constant. Equation (7.49) shows that Pj u is constant for a particular copolymer if the penultimate model applies therefore the ratio NmiMiMi/NmiMi equals this constant, but the ratio NmjMj/Nmj does not have the same value. 

It has been argued that for a majority of copolymerizations, composition data can be adequately predicted by the terminal model copolymer composition equation (eqs. 5-9). However, in that composition data are not particularly good for model discrimination, any conclusion regarding the widespread applicability of the implicit penultimate model on this basis is premature. 

Following a procedure similar to that used in deriving Eq. (7.11), the instantaneous copolymer composition equation for the penultimate model is then given (Hamielec et al., 1989) by... 

A flrst important question concerns whether the goal is to discriminate between competing models (i.e., terminal vs penultimate model kinetics) or to seek the best parameter estimates. We flrst assume that terminal model kinetics are being considered and later discuss implications regarding the assumption of penultimate model kinetics. As seen in the previous section, for terminal model kinetics, reactivity ratios are typically estimated using the instantaneous copolymer composition equation or the Mayo-Lewis equation, expressed in two common forms. Equations 6.7 and 6.11. 

In eqn [24], fi is the mole fraction of monomer i in the monomer mixture. In the penultimate models, two monomer-derived segments at each radical chain end are taken into account, for example, by the rate coefficient kuy which refers to addition of a monomer molecule j to a radical chain end where both terminal and penultimate units consist of spedes i. Equation [24] is used to derive q and also q =fejj/feji from measured compositions of monomer mixture and copolymer, fi and Fi, respectively. 

While copolymer composition is well-described by the terminal model, the copolymer-averaged propagation rate coefficient (kp. Equation 3.45) for many common systems [10, 26, 27] is not. The measured kp values can be higher or lower than the terminal model predictions, with the deviation substantial in some cases. The implicit penultimate unit effect model, which accounts for the influence of the penultimate monomer-unit of the growing polymer radical on the propagation kinetics [26, 27], provides a good representation of this behavior ... 

Other statistical models. The second-order Markov model is sometimes also applied to copolymers. Here, the probability of addition of a given monomer depends not only on the identity of the chain end monomer, but also on the nature of the preceding or penultimate monomer unit. As there are then four possible types of chain end to consider (namely, -AA, -AB, -BA, and -BB), there are eight addition probabilities which describe addition of the A and B monomers (e.g. Pg g represents the probability of B adding to a -BA chain end). As with the first-order Markov case, only half of these are independent because (Paaa + aab) = ( aba + abb) = Equations for... 

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