# 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 ${\displaystyle M_{1}\,}$ and ${\displaystyle M_{2}\,}$ and the four different reactions that can take place at the reactive chain end terminating in either monomer (${\displaystyle M^{*}\,}$) with their reaction rate constants ${\displaystyle k\,}$:

${\displaystyle M_{1}^{*}+M_{1}{\xrightarrow {k_{11}}}M_{1}M_{1}^{*}\,}$
${\displaystyle M_{1}^{*}+M_{2}{\xrightarrow {k_{12}}}M_{1}M_{2}^{*}\,}$
${\displaystyle M_{2}^{*}+M_{2}{\xrightarrow {k_{22}}}M_{2}M_{2}^{*}\,}$
${\displaystyle M_{2}^{*}+M_{1}{\xrightarrow {k_{21}}}M_{2}M_{1}^{*}\,}$

and with reactivity ratios defined as:

${\displaystyle r_{1}={\frac {k_{11}}{k_{12}}}\,}$
${\displaystyle r_{2}={\frac {k_{22}}{k_{21}}}\,}$

the copolymer equation is given as:

${\displaystyle {\frac {d\left[M_{1}\right]}{d\left[M_{2}\right]}}={\frac {\left[M_{1}\right]\left(r_{1}\left[M_{1}\right]+\left[M_{2}\right]\right)}{\left[M_{2}\right]\left(\left[M_{1}\right]+r_{2}\left[M_{2}\right]\right)}}}$

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

## Equation derivation

Monomer 1 is consumed with reaction rate:[2]

${\displaystyle {\frac {-d[M_{1}]}{dt}}=k_{11}[M_{1}]\sum [M_{1}^{*}]+k_{21}[M_{1}]\sum [M_{2}^{*}]\,}$

with ${\displaystyle \sum [M_{x}^{*}]}$ the concentration of all the active centers terminating in monomer 1 or 2.

Likewise the rate of disappearance for monomer 2 is:

${\displaystyle {\frac {-d[M_{2}]}{dt}}=k_{12}[M_{2}]\sum [M_{1}^{*}]+k_{22}[M_{2}]\sum [M_{2}^{*}]\,}$

Division of both equations yields:

${\displaystyle {\frac {d[M_{1}]}{d[M_{2}]}}={\frac {[M_{1}]}{[M_{2}]}}\left({\frac {k_{11}{\frac {\sum [M_{1}^{*}]}{\sum [M_{2}^{*}]}}+k_{21}}{k_{12}{\frac {\sum [M_{1}^{*}]}{\sum [M_{2}^{*}]}}+k_{22}}}\right)\,}$

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

${\displaystyle {\frac {d\sum [M_{1}^{*}]}{dt}}={\frac {d\sum [M_{2}^{*}]}{dt}}\approx 0\,}$

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:

${\displaystyle k_{21}[M_{1}]\sum [M_{2}^{*}]=k_{12}[M_{2}]\sum [M_{1}^{*}]\,}$

or

${\displaystyle {\frac {\sum [M_{1}^{*}]}{\sum [M_{2}^{*}]}}={\frac {k_{21}[M_{1}]}{k_{12}[M_{2}]}}\,}$

Substituting into the ratio of monomer consumption rates eliminates the radical concentrations and yields the Mayo-Lewis equation.

## Instantaneous form

It can often be useful to alter the copolymer equation by expressing concentration in terms of mole fractions. Mole fractions of monomers ${\displaystyle M_{1}\,}$ and ${\displaystyle M_{2}\,}$ in the feed are defined as ${\displaystyle f_{1}\,}$ and ${\displaystyle f_{2}\,}$ where

${\displaystyle f_{1}=1-f_{2}={\frac {M_{1}}{(M_{1}+M_{2})}}\,}$

Similarly, ${\displaystyle F\,}$ represents the mole fraction of each monomer in the copolymer:

${\displaystyle F_{1}=1-F_{2}={\frac {dM_{1}}{d(M_{1}+M_{2})}}\,}$

These equations can be combined with the Mayo-Lewis Equation to give

${\displaystyle F_{1}=1-F_{2}={\frac {r_{1}f_{1}^{2}+f_{1}f_{2}}{r_{1}f_{1}^{2}+2f_{1}f_{2}+r_{2}f_{2}^{2}}}\,}$

This equation gives the instantaneous copolymer composition. It is important to note that the feed and copolymer compositions can change as polymerization proceeds.

## Limiting cases

Reactivity ratios indicate preference for propagation. Large ${\displaystyle r\,}$ indicates a tendency for ${\displaystyle M_{1}^{*}\,}$ to add ${\displaystyle M_{1}\,}$, while small values indicate a tendency for ${\displaystyle M_{1}^{*}\,}$ to add ${\displaystyle M_{2}\,}$. From the definition of reactivity ratios, several special cases can be derived:

• ${\displaystyle r_{1}=r_{2}>>1\,}$ with both reactivity ratios very high the two monomers only react with themselves and not each other leading to a mixture of two homopolymers.
• ${\displaystyle r_{1}=r_{2}>1\,}$ 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 giving rise to block copolymer
• ${\displaystyle r_{1}=r_{2}\approx 1\,}$ with both ratios around 1, monomer 1 will react as fast with another monomer 1 or monomer 2 and a random copolymer is formed.
• ${\displaystyle r_{1}=r_{2}\approx 0\,}$ with both values approaching 0 the monomers are unable to homopolymerize and only add each other resulting in an alternating polymer
• ${\displaystyle r_{1}>>1>>r_{2}\,}$ 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 case is maleic anhydride and styrene, with reactivity ratios:

• Maleic anhydride (${\displaystyle r_{1}\,}$ = 0.01) & styrene (${\displaystyle r_{2},}$ = 0.02)[3]

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

When both ${\displaystyle r<1\,}$, the system has an azeotrope, where feed and copolymer composition are the same.

## 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, ${\displaystyle r_{1}\,}$ and ${\displaystyle r_{2}\,}$ can be found.

### Curve Fitting

One of the simplest methods for finding reactivity ratios is plotting the copolymer equation and using least squares analysis to find the ${\displaystyle r_{1}\,}$, ${\displaystyle r_{2}\,}$ pair that gives the best fit curve.

### Mayo-Lewis Method

The Mayo-Lewis method uses a form of the copolymer equation relating ${\displaystyle r_{1}\,}$ to ${\displaystyle r_{2}\,}$:[4]

${\displaystyle r_{2}={\frac {f_{1}}{f_{2}}}\left[{\frac {F_{2}}{F_{1}}}(1+{\frac {f_{1}r_{1}}{f_{2}}})-1\right]\,}$

For each different monomer composition, a line is generated using arbitrary ${\displaystyle r_{1}\,}$ values. The intersection of these lines is the ${\displaystyle r_{1}\,}$, ${\displaystyle r_{2}\,}$ for the system. More frequently, the lines do not intersect and the area in which most line intersect can be given as a range of ${\displaystyle r_{1}\,}$, and ${\displaystyle r_{2}\,}$ values.

### Fineman-Ross Method

Fineman and Ross rearranged the copolymer equation into a linear form:[5]

${\displaystyle G=Hr_{1}-r_{2}\,}$

where ${\displaystyle G={\frac {f_{1}(2F_{1}-1)}{(1-f_{1})F_{1}}}\,}$ and ${\displaystyle H=\left[{\frac {f_{1}^{2}(1-F_{1})}{(1-f_{1})^{2}F_{1}}}\right]\,}$.

Thus, a plot of ${\displaystyle H\,}$ versus ${\displaystyle G\,}$ yields a straight line with slope ${\displaystyle r_{1}\,}$ and intercept ${\displaystyle -r_{2}\,}$

### Kelen Tudos method

The Fineman-Ross method can be biased towards points at low or high monomer concentration, so Kelen and Tudos introduced and arbitrary constant,

${\displaystyle \alpha =(H_{min}H_{max})^{0.5}\,}$

where ${\displaystyle H_{min}\,}$ and ${\displaystyle H_{max}\,}$ are the highest and lowest values of ${\displaystyle H\,}$ from the Fineman-Ross method.[6] The data can be plotted in a linear form

${\displaystyle \eta =\left[r_{1}+{\frac {r_{2}}{\alpha }}\right]\mu -{\frac {r_{2}}{\alpha }}\,}$

where ${\displaystyle \eta =G/(\alpha +H)\,}$ and ${\displaystyle \mu =H/(\alpha +H)\,}$. Plotting ${\displaystyle \eta }$ against ${\displaystyle \mu }$ yields a straight line that gives ${\displaystyle -r_{2}/\alpha }$ when ${\displaystyle \mu =0}$ and ${\displaystyle r_{1}}$ when ${\displaystyle \mu =1}$. This distributes the data more symmetrically and can yield better results.

### Q-e scheme

A semi-empirical method for the determination of reactivity ratios is called the Q-e scheme. This involves using two parameters for each monomer, ${\displaystyle Q}$ and ${\displaystyle e}$. The reaction of ${\displaystyle M_{1}}$ radical with ${\displaystyle M_{2}}$ monomer is written as

${\displaystyle k_{12}=P_{1}Q_{2}exp(-e_{1}e_{2})}$

while the reaction of ${\displaystyle M_{1}}$ radical with ${\displaystyle M_{1}}$ monomer is written as

${\displaystyle k_{11}=P_{1}Q_{1}exp(-e_{1}e_{1})}$

Where 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, ${\displaystyle r_{1}}$ and ${\displaystyle r_{2}}$ 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 or what the monomer pair is in the system.