The Mayo–Lewis equation or copolymer equation in polymer chemistry describes the distribution of monomers in a copolymer. It was proposed by Frank R. Mayo and Frederick M. Lewis. The equation considers a monomer mix of two components M_1, and M_2, and the four different reactions that can take place at the reactive chain end terminating in either monomer (M_1^, and M_2^,) with their reaction rate constants k,: The reactivity ratio for each propagating chain end is defined as the ratio of the rate constant for addition of a monomer of the species already at the chain end to the rate constant for addition of the other monomer. The copolymer equation is then: with the concentrations of the components in square brackets. The equation gives the relative instantaneous rates of incorporation of the two monomers.

Equation derivation

Monomer 1 is consumed with reaction rate: with \sum[M_1^] the concentration of all the active chains terminating in monomer 1, summed over chain lengths. \sum[M_2^] is defined similarly for monomer 2. Likewise the rate of disappearance for monomer 2 is: Division of both equations by followed by division of the first equation by the second yields: The ratio of active center concentrations can be found using the steady state approximation, meaning that the concentration of each type of active center remains constant. The rate of formation of active centers of monomer 1 is equal to the rate of their destruction so that or Substituting into the ratio of monomer consumption rates yields the Mayo–Lewis equation after rearrangement:

Mole fraction form

It is often useful to alter the copolymer equation by expressing concentrations in terms of mole fractions. Mole fractions of monomers M_1, and M_2, in the feed are defined as f_1, and f_2, where Similarly, F, represents the mole fraction of each monomer in the copolymer: These equations can be combined with the Mayo–Lewis equation to give This equation gives the composition of copolymer formed at each instant. However the feed and copolymer compositions can change as polymerization proceeds.

Limiting cases

Reactivity ratios indicate preference for propagation. Large r_1, indicates a tendency for M_1^, to add M_1,, while small r_1, corresponds to a tendency for M_1^, to add M_2,. Values of r_2, describe the tendency of M_2^*, to add M_2, or M_1,. From the definition of reactivity ratios, several special cases can be derived:

Calculation of reactivity ratios

Calculation of reactivity ratios generally involves carrying out several polymerizations at varying monomer ratios. The copolymer composition can be analysed with methods such as Proton nuclear magnetic resonance, Carbon-13 nuclear magnetic resonance, or Fourier transform infrared spectroscopy. The polymerizations are also carried out at low conversions, so monomer concentrations can be assumed to be constant. With all the other parameters in the copolymer equation known, r_1, and r_2, can be found.

Curve Fitting

One of the simplest methods for finding reactivity ratios is plotting the copolymer equation and using nonlinear least squares analysis to find the r_1,, r_2, pair that gives the best fit curve. This is preferred as methods such as Kelen-Tüdős or Fineman-Ross (see below) that involve linearization of the Mayo–Lewis equation will introduce bias to the results.

Mayo-Lewis Method

The Mayo-Lewis method uses a form of the copolymer equation relating r_1, to r_2,: For each different monomer composition, a line is generated using arbitrary r_1, values. The intersection of these lines is the r_1,, r_2, for the system. More frequently, the lines do not intersect in a single point and the area in which most lines intersect can be given as a range of r_1,, and r_2, values.

Fineman-Ross Method

Fineman and Ross rearranged the copolymer equation into a linear form: where and Thus, a plot of H , versus G , yields a straight line with slope r_1, and intercept -r_2,

Kelen-Tüdős method

The Fineman-Ross method can be biased towards points at low or high monomer concentration, so Kelen and Tüdős introduced an arbitrary constant, where H_{min} , and H_{max} , are the highest and lowest values of H , from the Fineman-Ross method. The data can be plotted in a linear form where and. Plotting \eta against \mu yields a straight line that gives -r_2/\alpha when \mu=0 and r_1 when \mu = 1. This distributes the data more symmetrically and can yield better results.

Q-e scheme

A semi-empirical method for the prediction of reactivity ratios is called the Q-e scheme which was proposed by Alfrey and Price in 1947. This involves using two parameters for each monomer, Q and e. The reaction of M_1 radical with M_2 monomer is written as while the reaction of M_1 radical with M_1 monomer is written as Where P is a proportionality constant, Q is the measure of reactivity of monomer via resonance stabilization, and e is the measure of polarity of monomer (molecule or radical) via the effect of functional groups on vinyl groups. Using these definitions, r_1 and r_2 can be found by the ratio of the terms. An advantage of this system is that reactivity ratios can be found using tabulated Q-e values of monomers regardless of what the monomer pair is in the system.