# Mayo–Lewis equation

The Mayo–Lewis equation or copolymer equation in polymer chemistry describes the distribution of monomers in a copolymer:[1] It is named for Frank R. Mayo and Frederick M. Lewis.

Taking into consideration 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 ($M^*\,$) with their reaction rate constants $k\,$:

$M_1^* + M_1 \xrightarrow{k_{11}} M_1M_1^* \,$
$M_1^* + M_2 \xrightarrow{k_{12}} M_1M_2^* \,$
$M_2^* + M_2 \xrightarrow{k_{22}} M_2M_2^* \,$
$M_2^* + M_1 \xrightarrow{k_{21}} M_2M_1^* \,$

and with reactivity ratios defined as:

$r_1 = \frac{k_{11}}{k_{12}} \,$
$r_2 = \frac{k_{22}}{k_{21}} \,$

the copolymer equation is given as:

$\frac {d\left [M_1 \right]}{d\left [M_2\right]}=\frac{\left [M_1\right]\left (r_1\left[M_1\right]+\left [M_2\right]\right)}{\left [M_2\right]\left (\left [M_1\right]+r_2\left [M_2\right]\right)}$

with the concentration of the components given in square brackets. The equation gives the copolymer composition at any instant during the polymerization.

## Limiting cases

From this equation several special cases can be derived:

• $r_1 = r_2 >> 1 \,$ with both reactivity ratios very high the two monomers have no inclination to react to each other except with themselves leading to a mixture of two homopolymers.
• $r_1 = r_2 > 1 \,$ with both ratios larger than 1, homopolymerization of component M_1 is favored but in the event of a crosspolymerization by M_2 the chain-end will continue as such giving rise to block copolymer
• $r_1 = r_2 \approx 1 \,$ with both ratios around 1, monomer 1 will react as fast with another monomer 1 or monomer 2 and a random copolymer results.
• $r_1 = r_2 \approx 0 \,$ with both values approaching 0 the monomers are unable to react in homopolymerization and the result is an alternating polymer
• $r_1 >> 1 >> r_2 \,$ In the initial stage of the copolymerization monomer 1 is incorporated faster and the copolymer is rich in monomer 1. When this monomer gets depleted, more monomer 2 segments are added. This is called composition drift.

An example is maleic anhydride and stilbene, with reactivity ratio:

• Maleic anhydride ($r_1\,$ = 0.08) & cis-stilbene ($r_2 ,$ = 0.07)
• Maleic anhydride ($r_1\,$ = 0.03) & trans-stilbene ($r_2 ,$ = 0.03)

Neither of these compounds homopolymerize and instead they react together to give exclusively alternating copolymer.

Another form of the equation is:

$F_1=1-F_2=\frac{r_1 f_1^2+f_1 f_2}{r_1 f_1^2+2f_1 f_2+r_2f_2^2}\,$

where $F\,$ stands the mole fraction of each monomer in the copolymer:

$F_1 = 1 - F_2 = \frac{d M_1}{d (M_1 + M_2)} \,$

and $f\,$ the mole fraction of each monomer in the feed:

$f_1 = 1 - f_2 = \frac{M_1}{(M_1 + M_2)} \,$

When the copolymer composition has the same composition as the feed, this composition is called the azeotrope.

## Calculation of reactivity ratios

The reactivity ratios can be obtained by rewriting the copolymer equation to:

$\frac{f(1-F)}{F} = r_2 - r_1\left(\frac{f^2}{F}\right) \,$

with

$f = \frac{[M_1]}{[M_2]} \,$ in the

and

$F = \frac{d[M_1]}{d[M_2]} \,$ in the copolymer

A number of copolymerization experiments are conducted with varying monomer ratios and the copolymer composition is analysed at low conversion. A plot of $\frac{f(1-F)}{F}\,$ versus $\frac{f^2}{F}\,$ gives a straight line with slope $r_1\,$ and intercept $r_2\,$.

A semi-empirical method for the determination of reactivity ratios is called the Q-e scheme.

## Equation derivation

Monomer 1 is consumed with reaction rate:[2]

$\frac{-d[M_1]}{dt} = k_{11}[M_1]\sum[M_1^*] + k_{21}[M_1]\sum[M_2^*] \,$

with $\sum[M_x^*]$ the concentration of all the active centers terminating in monomer 1 or 2.

Likewise the rate of disappearance for monomer 2 is:

$\frac{-d[M_2]}{dt} = k_{12}[M_2]\sum[M_1^*] + k_{22}[M_2]\sum[M_2^*] \,$

Division of both equations yields:

$\frac{d[M_1]}{d[M_2]} = \frac{[M_1]}{[M_2]} \left( \frac{k_{11}\frac{\sum[M_1^*]}{\sum[M_2^*]} + k_{21}} {k_{12}\frac{\sum[M_1^*]}{\sum[M_2^*]} + k_{22}} \right) \,$

The ratio of active center concentrations can be found assuming steady state with:

$\frac{d\sum[M_1^*]}{dt} = \frac{d\sum[M_2^*]}{dt} \approx 0\,$

meaning that the concentration of active centres remains constant, the rate of formation for active center of monomer 1 is equal to the rate of their destruction or:

$k_{21}[M_1]\sum[M_2^*] = k_{12}[M_2]\sum[M_1^*] \,$

or

$\frac{\sum[M_1^*]}{\sum[M_2^*]} = \frac{k_{21}[M_1]}{k_{12}[M_2]}\,$