Mayo-Lewis model

Mayo-Lewis Binary Copolymeriration Model. In this exeimple we consider the Mayo-Lewis model for describing binary copolymerization. The procedure for estimating the kinetic parameters expressed as reactivity ratios from composition data is discussed in detail in our earlier paper (1 ). Here diad fractions, which are the relative numbers of MjMj, MiMj, M Mj and MjMj sequences as measured by NMR are used. NMR, while extremely useful, cannot distinguish between MiM and M Mi sequences and... 

The diad fractions for the low conversion experiments only are reproduced in Table II. The high conversion data cannot be used since the Mayo-Lewis model does not apply. Again diad fractions have been standardized such that only two independent measurements are available. When the error structure is unknown, as in this case, Duever and Reilly (in preparation) show how the parameter distribution can be evaluated. Several attempts were made to use this solution. However with only five data points there is insufficient information present to allow this approach to be used. 

Penultimate Group Effects Copolymerization Model. This model represents an extension of the Mayo-Lewis model in which the next to last or penultimate group is assumed to affect the reaction rate. Under this assumption the eight reactions represented by the following equations are of importance ( ) ... 

Thus, through the body of the mentioned experimental evidence obtained via different methods that characterize the composition and structure of macromolecules one arrives at a simple conclusion concerning the kinetic model of the binary copolymerization of styrene with methyl methacrylate (I) and with acrylonitrile (II). The former of these systems is obviously described by the terminal model, and the latter one by the penultimate model. However, the latter system characteristics in those cases when high accuracy of the results is not required, may be calculated within the framework of the Mayo-Lewis model. Such a simplified approach was found to be quite acceptable to solve many practical problems. One should note that the trivial terminal model is able to describe a vast majority (at least, 90% according to Harwood [303]) of copolymerization systems which have been already studied. 

The random nature of the copolymerization equilibria can be considered a consequence of two concurrent entropically driven equilibria similar to reactions 2 and 3. These copolymerization equilibria, however, would involve the comonomers interacting reversibly with two different chain ends and the reversible transfer of the different comonomer units between chains. Expressed in another way, the equilibria could be written in a manner similar to the Mayo-Lewis model but with rate constants replaced by equilibrium constants, K, K12, K22, and K21, and comonomer concentrations replaced by the total concentrations of the different siloxane units in the system, M and M2, regardless of their locations in the rings or chains. 

The next step in the protocol answers the question about what is the best method to estimate the reactivity ratios. Historically, because of its simplicity, linearization techniques such as the Fineman-Ross, Kelen-Tudos, and extended Kelen-Tudos methods have been used. Easily performed on a simple calculator, these techniques suffer from inaccuracies due to the linearization of the inherently nonlinear Mayo-Lewis model. Such techniques violate basic assumptions of linear regression and have been repeatedly shown to be invalid [117, 119, 126]. Nonlinear least squares (NLLS) techniques and other more advanced nonlinear techniques such as the error-in-variables-model (EVM) method have been readily available for several decades [119, 120, 126, 127]. 

Tip 13 (related to Tip 12) Copolymerization, copolymer composition, composition drift, azeotropy, semibatch reactor, and copolymer composition control. Most batch copolymerizations exhibit considerable drift in monomer composition because of different reactivities (reactivity ratios) of the two monomers (same ideas apply to ter-polymerizations and multicomponent cases). This leads to copolymers with broad chemical composition distribution. The magnirnde of the composition drift can be appreciated by the vertical distance between two items on the plot of the instantaneous copolymer composition (ICC) or Mayo-Lewis (model) equation item 1, the ICC curve (ICC or mole fraction of Mj incorporated in the copolymer chains, F, vs mole fraction of unreacted Mi,/j) and item 2, the 45° line in the plot of versus/j. 

As early as the 1940s, radical copolymerization models were already developed to describe specific features of the process. Initially, these models were relatively simple models where the reactivity of chain-ends was assumed to depend only on the nature of the terminal monomer unit in the growing chain (Mayo-Lewis model or terminal model (TM)). This model by definition leads to first-order Markov chains. 

It is hence possible to integrate Eqnation 12.70 once the reactivity ratios are known, so that the composition drift can be predicted. The Mayo-Lewis model, or more sophisticated models (e.g., penultimate models), can be extended to the case where one or more reagents is flowed into the reactor in semibatch operation, and the resulting equations numerically integrated to predict the composition drift. 

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