Lewis’ equations

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. 

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). 

It is a fortunate coincidence that the product M n Z also appears in the Lu-Lewis equation (equation 6.4). It can be used to assess the effective Z value from the follo A/ing equations ... 

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 ... 

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. 

An investigation of the copolymer composition demonstrated the important effect of monomer transport on the copolymerization. The droplets in the macroemulsion act as monomer reservoirs. In this system, the effect of monomer transport will be predominant when an extremely water-insoluble comonomer, such as DOM, is used. In contrast with the macroemulsion system, the miniemulsion system tends to follow the integrated Mayo Lewis equation more closely, indicating less influence from mass transfer. 

The transfer reactions to the solvent and the initiator have been described for butadiene, isoprene, or vinyl acetate polymerizations using thermally decomposed hydrogen peroxide in methanol or rm-pentanol (Table 3.5)l55). The Mayo-Lewis equation has been applied... 

Using an improved Mayo-Lewis equation, the ratio of termination constant (kt) and propagation constant (kp) can be determined 154). This ratio is 8.3 for polymerization of MMA at 60 °C. 

In studies of the kinetics of copolymerization of cyclic compounds the Mayo—Lewis equations [150] for kinetics of copolymerization have been applied, often with deserved caution. Many monomer reactivity ratios have been derived in this way. A large number of them have been summarized previously [7, 151] and we will not repeat them here nor attempt to update the lists. Instead we shall concentrate on some of the factors that seem to be important in regulating the copolymerizations and on some of the newer approaches that have been suggested for dealing with the complicated kinetics and give only a few examples of individual rate studies. 

Equations (1.198) and (1.199) are also known as Lewis equations. The mass transfer coefficients f3m calculated using this equation are only valid, according to the definition, for insignificant convective currents, fn the event of convection being important they must be corrected. The correction factors C, = /3 n/l3m for transverse flow over a plate, under the boundary layer theory assumptions are shown in Fig. 1.50. They are larger than those in film theory for a convective flow out of the phase, but smaller for a convective flow into the phase. 

If the complexed radical is inactive (k n = kx 2 = k22 = k21 = 0), Eq. (7.8) reduces to the ordinary Mayo-Lewis equation and no solvent effect on the reactivity ratio will be observed. Busfield et al.108) studied the solvent effect on the free radical copolymerization of vinyl acetate and methyl methacrylate. The methyl methacrylate content is unaffected by benzene and ethyl acetate. This result seems to be consistent with our assumption that the complexed radical is inactive in propagation. However, the solvent effect might not be observed in the case in which the reactivity of the complexed radical is proportional to that of the uncomplexed radical, because also in this case Eq. (7.8) reduces to the Mayo-Lewis form. It is difficult, therefore, to expect from the copolymerization experiment some evidence to support the concept of the complex formation. 

Equation 6.7 is known as the copolymerization or the Mayo-Lewis equation. The physical meaning of Equation 6.7 is better appreciated by writing it in terms of mole fractions. If /j is the mole fraction of unreacted monomer i and F is the mole fraction of monomer i in the copolymer formed instantaneously, then... 

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. 

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