Mayo-Lewis

This allows elimination of the radical concentrations from the above equation and the copolymer composition equation (eq. 5),14-16 also known as the Mayo-Lewis equation, can now be derived. 

It is also possible to derive reactivity ratios by analyzing the monomer (or polymer) feed composition v.v conversion and solving the integrated form of the Mayo Lewis equation.10 123 The following expression (eq. 44) was derived by Meyer and Lowry 12j... 

The Mayo-Lewis equation expressing the copolymer composition can be derived from these four elementary reactions. It reads... 

Several important assumptions are involved in the derivation of the Mayo-Lewis equation and care must be taken when it is applied to ionic copolymerization systems. In ring-opening polymerizations, depolymerization and equilibration of the heterochain copolymers may become important in some cases. In such cases, the copolymer composition is no longer determined by die four propagation reactions. 

Polymerization equilibria frequently observed in the polymerization of cyclic monomers may become important in copolymerization systems. The four propagation reactions assumed to be irreversible in the derivation of the Mayo-Lewis equation must be modified to include reversible processes. Lowry114,11S first derived a copolymer composition equation for the case in which some of the propagation reactions are reversible and it was applied to ring-opening copalymerization systems1 16, m. In the case of equilibrium copolymerization with complete reversibility, the following reactions must be considered. 

A general method has been developed for the estimation of model parameters from experimental observations when the model relating the parameters and input variables to the output responses is a Monte Carlo simulation. The method provides point estimates as well as joint probability regions of the parameters. In comparison to methods based on analytical models, this approach can prove to be more flexible and gives the investigator a more quantitative insight into the effects of parameter values on the model. The parameter estimation technique has been applied to three examples in polymer science, all of which concern sequence distributions in polymer chains. The first is the estimation of binary reactivity ratios for the terminal or Mayo-Lewis copolymerization model from both composition and sequence distribution data. Next a procedure for discriminating between the penultimate and the terminal copolymerization models on the basis of sequence distribution data is described. Finally, the estimation of a parameter required to model the epimerization of isotactic polystyrene is discussed. 

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 ( ) ... 

The copolymer composition produced by these two catalysts can be estimated using the Mayo-Lewis equation [38] and these values of i and r2. Figure 10 depicts the hypothetical comonomer content in the polymer (F2) as a function of the mole fraction of comonomer in the reactor (f2). The good incorporator produces a material with higher F2 as f2 increases. In contrast, the composition from the poor incorporator is relatively flat across a broad range and increases only at very high values of/2. The F2 required to render the copolymer amorphous is comonomer-dependent for 1-octene, this value is near 0.19. In this hypothetical system, the good incorporator produces that composition at f2 = 0.57, at which the poor incorporator incorporates very little comonomer (F2 = 0.01). 

Table 9 Reactivity ratios determined for 2-oxazoline copolymerizations utilizing both the Mayo-Lewis terminal model (MLTD) and the extended Kelen-Tiidds (KT) method. Initial defines - 20% conversion and final defines >50% conversion...

The experimental composition data are unequally weighted by the Mayo-Lewis and Fineman-Ross plots with the data for the high or low compositions (depending on the equation used) having the greatest effect on the calculated values of r and r2 [Tidwell and Mortimer, 1965, 1970]. This often manifests itself by different values of r and r2 depending on which monomer is indexed as Mi. 

The composition of the copolymer was determined by either NMR analysis at 90 MHz according to the equations derived by Mochel (21) or by infrared. (22) The agreement of these methods was 2% when applied to copolymer taken to 100% conversion. The reactivity ratios were calculated according to the Mayo-Lewis Plot (13,15), the Fineman-Ross Method (14), or by the Kelen-Tudos equation.(16,17,18) The statistical variations recently noted by 0 Driscoll (23), were also considered. 

Monomer" Mayo- Lewis Kelen- Tudos Fineman- (F-G) -Ross (1/F)- (G/F) ... 

A number of copolymerizations involving macromonomer(s) have been studied and almost invariably treated according to the terminal model, Mayo-Lewis equation, or its simplified model [39]. The Mayo-Lewis equation relates the instantaneous compositions of the monomer mixture to the copolymer composition ... 

The effect of polarity on vinyl monomer copolymerization has long been recognized and is a major factor in the Q, e scheme and copolymerization theory. Mayo, Lewis, and Walling tabulated a number of vinyl monomers into an average activity series and an electron donor-acceptor series (62). The activity series showed the effect of substituents on the ease with which an ethylene derivative reacted with an average radical and on stabilizing the radical which was formed thereby. The electron donor-acceptor series indicated the ability of the substituents to serve as donors or acceptors in radical-monomer interactions. It is significant that in both series the dominant factor is the radical-monomer interaction. 

The intersection procedure put forth in the pioneer report by Mayo-Lewis (ML) [19] assumes expression (6.1) to adopt the following form ... 

The values of these coefficients for the system described within the framework of Mayo-Lewis scheme are to be equal to unity according to relations (6.13) at any monomer feed composition. This condition holds strictly under the copolymerization of styrene with methyl methacrylate (see Table 6.5) while it holds considerably worse under the copolymerization of styrene with acrylonitrile (see Table 6.7). The... 

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. 

Fig. 18. Determination of copolymerization parameters by the Mayo-Lewis method. See p. 255 of ref. 170.

Both the Mayo-Lewis and the Fineman-Ross methods rely on linearizing the copolymer equation. It has been shown that... 

Equation 17 is known as the copolymerization or Mayo Lewis equation. 

Schuller [150] and Guillot [98] both observed that the copolymer compositions obtained from emulsion polymerization reactions did not agree with the Mayo Lewis equation, where the reactivity ratios were obtained from homogeneous polymerization experiments. They concluded that this is due to the fact that the copolymerization equation can be used only for the exact monomer concentrations at the site of polymerization. Therefore, Schuller defined new reactivity ratios, TI and T2, to account for the fact that the monomer concentrations in a latex particle are dependent on the monomer partition coefficients (fCj and K2) and the monomer-to-water ratio (xp) ... 

The Mayo Lewis equation, using reactivity ratios computed from Eq. 18, will give very different results from the homogenous Mayo Lewis equation for mini-or macroemulsion polymerization when one of the comonomers is substantially water-soluble. Guillot [151] observed this behavior experimentally for the common comonomer pairs of styrene/acrylonitrile and butyl acrylate/vinyl acetate. Both acrylonitrile and vinyl acetate are relatively water-soluble (8.5 and 2.5%wt, respectively) whereas styrene and butyl acrylate are relatively water-insoluble (0.1 and 0.14%wt, respectively). However, in spite of the fact that styrene and butyl acrylate are relatively water-insoluble, monomer transport across the aqueous phase is normally fast enough to maintain equilibrium swelling in the growing polymer particle, and so we can use the monomer partition coefficient. 

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