Mayo-Lewis copolymer equation

A significant study [13] of the copolymerization in acetonitrile with t-butyl ammonium perchlorate as the electrolyte confirmed that the composition of the copolymer depends on applied potential as well as feed ratio. The authors demonstrated that the copolymer composition obeys the Mayo-Lewis copolymer equation [14] and reported reactivity ratios for the pyrrole/ bithiophene pair for the first time. At a polymerization potential of 1.3 V, r, =4.9 and r2 = 0.04 for pyrrole and bithiophene, respectively, and at 1.5 V, r — 4.3 and a-2 = 0.24. 

A straightforward approach was used in Reference [20] to compute reactivity ratios for two methacrylate-based comonomers and to compare the values obtained with the existing literature. Thus, since the comonomer conversion kinetics is known from ACOMP, the Mayo-Lewis copolymer equation ... 

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. 

Simultaneous polymerization of a mixture of cyclosiloxanes gives polymers whose microstructure is not easily predictable. Typically, the kinetics of ring-opening copolymerization is analyzed in terms of the Mayo-Lewis copolymerization equations (Scheme 5). The aim of such analysis is to determine the Mayo-Lewis reactivity ratios rD = feDD/feox and rx = fexx/fexD, which define the composition of a copolymer." The task is relatively easy when the propagation reactions are irreversible. 

Batch reactors are commonly used in research laboratories because of its simplicity and low cost of operation. The composition of the copolymers produced in batch readors will be dictated by the readivity ratios of the comonomers and the ratio of their concentration in the polymer partides (Mayo-Lewis copolymer composition equation - see eqn [3] in Section 3.14.2.1.2(iv)). Most of the common monomers employed in emulsion polymerization redpes present different reactivities, and a consequence of this is the compositional drift (nonconstant copolymer composition) produced in batch operation. 

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. 

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). 

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. 

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 ... 

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

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) ... 

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. 

Also known as the method of intersections, the method first described by Mayo and Lewis [3] has been widely used for computing reactivity ratios from data fitted to the differential copolymer equation. In this procedure, Eq. (7.11) is recast to the form... 

The copolymer equation, which expresses the composition of growing chains at any reaction time t, was developed in the 1930s by a group of investigators including Wall, Mayo, Simha, Alfrey, Borstal, and Lewis (for instance, and j4-0. 

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. 

The equation of Mayo-Lewis (23) relates the copolymer composition to the monomer composition, and is given by ... 

The copolymerisation of N-cyclohexylmaleimide (CHMI) and methyl methacrylate (MMA) with azobisisobutyronitrile as the initiator is investigated. IR spectroscopy is applied to determine the copolymer compositions of the copolymers synthesised at 60, 70, 80 and 90 deg.C while the conversions of the copolymerisations are controlled to be below 10%. According to the Mayo-Lewis equation, the reactivity ratios of N-cyclohexylmaleimide and methyl methacrylate are calculated. It is proved that N-cyclohexylmaleimide is less reactive, and the optimum temperature of the copolymerisation is 80 deg C. 8 refs. CHINA... 

In Equation 2.108, the copolymer composition distribution is quantified by the variable y, defined in Equation 2.109. The variable y measures the difference between the molar fraction of monomer A in a given polymer chain to the average molar fraction of monomer A in all the chains, already defined in Equation 2.104. The classical Mayo-Lewis equation can... 

The condition to produce a latex with a given copolymer composition is that the ratio of the monomer concentrations in the polymer particles must be kept at the value that ensures the production of the desired composition. This comonomer ratio can be calculated from the Mayo-Lewis equation, Eq. (75), where ri and t2 are the reactivity ratios and yu is the instantaneous composition referred to monomer 1. 

As indicated in Eq. (19), y is the deviation from the average molar fraction of monomer type A in the copolymer, Fa- As usual, the instantaneous value of Fa can be calculated using the molar fraction of monomer A in the reactor, /a, and the reactivity ratios and rs by the Mayo-Lewis equation, Eq. (20) [35]. 

For the foUwing estimation of the copolymerization parameters it is useful to discern between the ovmaU or mixed parameters and the true oopolymerization parameters. First we assume that there are only uniform active centres located on the catalyst sur ce, (i.e., one centre model), and use ethene and comonomer peaks in the NMR spectrum of the polymer mixture for the estimation of the oopolymerization parameters according to the Mayo Lewis equqtion This evaluation, via the r versus diagram, leads to the overall or mixed copolymerization parameters. However, for the estimation of the true copolymerization parameters we now use the following considerations. The Mayo-Lewis equation describes the composition of the copolymer as a function of the initial monomers mixture and the oopolymerization parameters. If we know these and the monomers mixture we can calculate not only the copolymer composition but also, by means of statistical considerations, the sequence length distribution of Mj and M2 sequences in the copolymer... 

Assuming copolymerization between [M1M2] and [M1M3] charge-transfer complexes, the Mayo-Lewis equation for copolymer composition may be modified to obtain Eq. (48). 

The instantaneous copolymer composition Fi is related to the monomer composition/i by Mayo-Lewis equation [232], The cumulative copolymer composition Fi is related to not only the monomer composition/i but also the monomer conversion X. The monomer composition/i is further related to the... 

Note 4 The copolymer composition equation is known also as the Mayo-Lewis equation, counterion (in polymer science)... 

The situation is more complex when a copolymerization is considered because as two or more monomers are involved and the partitioning of the monomer between the phases might be different, this may lead to variations in the copolymer composition. In emulsion copolymerization, the evolution of the copolymer composition depends, in addition to the reactivity ratios, on the partition of the monomer between the aqueous and polymer particle phases (see Section 3.14.2.1.2(iv)). Furthermore, if the monomer is hydrophobic enough, transport limitations through the aqueous phase might control the concentration of the monomer in the polymer particles. On the other hand, in miniemulsion polymerization, the transport of monomer is reduced to such levels that the incorporation of hydrophobic monomers is favored as compared with conventional emulsion polymerization, and the copolymer compositions achieved in batch miniemulsion copolymetiza-tion are closer to those expected from the Mayo-Lewis equation (eqn [3]) under bulk conditions. This trend was experimentally observed by several authors who investigated the copolymer composition produced in batch emulsion and miniemulsion copolymerization using monomers with different water solubilities and reactivity ratios. 

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