Mayo–Lewis equation
Mayo–Lewis equation
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.[1]
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.[2]
![{\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)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/885a5c41d70a3248f1694d97f6956cff59a8462f)
with the concentrations of the components in square brackets. The equation gives the relative instantaneous rates of incorporation of the two monomers.[4]
Equation derivation
Monomer 1 is consumed with reaction rate:[5]
![{\displaystyle {\frac {-d[M_{1}]}{dt}}=k_{11}[M_{1}]\sum [M_{1}^{*}]+k_{21}[M_{1}]\sum [M_{2}^{*}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b89e87c7c2e3337b43db34931f7e91baec9524ac)
![{\displaystyle \sum [M_{1}^{*}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9733a22de39d62e832d96b2f141bc21a51095910)
![{\displaystyle \sum [M_{2}^{*}]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14e6cd337606d950e25df7907e50d8f73e9307e0)
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}^{*}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/251db64fd536ea6d8b5e6f22270fc15db31e9c03)
![{\displaystyle \sum [M_{2}^{*}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8e7a6a95c9675bcdb49162d8897db7452dbe8a7)
![{\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)\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59c4509d25e4ad69215000be3bb62d8c6d8d537d)
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.
![{\displaystyle {\frac {d\sum [M_{1}^{*}]}{dt}}={\frac {d\sum [M_{2}^{*}]}{dt}}\approx 0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49f4b77a5ec2c2ce37d3cfc8c0c03502dc91d2b3)
![{\displaystyle k_{21}[M_{1}]\sum [M_{2}^{*}]=k_{12}[M_{2}]\sum [M_{1}^{*}]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b932b589370de66ea9c487a34de30c41e2735a44)
or
![{\displaystyle {\frac {\sum [M_{1}^{*}]}{\sum [M_{2}^{*}]}}={\frac {k_{21}[M_{1}]}{k_{12}[M_{2}]}}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21e151fe785d6077ee89435cb6a2befc5da233b2)
Substituting into the ratio of monomer consumption rates yields the Mayo-Lewis equation after rearrangement:[4]
![{\displaystyle {\frac {d[M_{1}]}{d[M_{2}]}}={\frac {[M_{1}]}{[M_{2}]}}\left({\frac {k_{11}{\frac {k_{21}[M_{1}]}{k_{12}[M_{2}]}}+k_{21}}{k_{12}{\frac {k_{21}[M_{1}]}{k_{12}[M_{2}]}}+k_{22}}}\right)={\frac {[M_{1}]}{[M_{2}]}}\left({\frac {{\frac {k_{11}[M_{1}]}{k_{12}[M_{2}]}}+1}{{\frac {[M_{1}]}{[M_{2}]}}+{\frac {k_{22}}{k_{21}}}}}\right)={\frac {[M_{1}]}{[M_{2}]}}{\frac {\left(r_{1}\left[M_{1}\right]+\left[M_{2}\right]\right)}{\left(\left[M_{1}\right]+r_{2}\left[M_{2}\right]\right)}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5280734c9a5a192d935bd4ab0c67f9b8075fe44d)
Mole fraction form
This equation gives the composition of copolymer formed at each instant. However the feed and copolymer compositions can change as polymerization proceeds.
Limiting cases
If both reactivity ratios are very high, the two monomers only react with themselves and not with each other. This leads to a mixture of two homopolymers.
. If both ratios are larger than 1, homopolymerization of each monomer is favored. However, in the event of crosspolymerization adding the other monomer, the chain-end will continue to add the new monomer and form a block copolymer.
. If both ratios are near 1, a given monomer will add the two monomers with comparable speeds and a statistical or random copolymer is formed.
with both values approaching 0 the monomers are unable to homopolymerize. Each can add only the other resulting in an alternating polymer. For example, the copolymerization of maleic anhydride and styrene has reactivity ratios = 0.01 for maleic anhydride and = 0.02 for styrene.[7] Maleic acid in fact does not homopolymerize in free radical polymerization, but will form an almost exclusively alternating copolymer with styrene.[8]
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.
When both , the system has an azeotrope, where feed and copolymer composition are the same.[9]
Calculation of reactivity ratios
Curve Fitting
Mayo-Lewis Method
![{\displaystyle r_{2}={\frac {f_{1}}{f_{2}}}\left[{\frac {F_{2}}{F_{1}}}(1+{\frac {f_{1}r_{1}}{f_{2}}})-1\right]\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d82f6294de9b57d579c0efd25b5f82ba3320653)
Fineman-Ross Method
Fineman and Ross rearranged the copolymer equation into a linear form:[10]
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,
![{\displaystyle \eta =\left[r_{1}+{\frac {r_{2}}{\alpha }}\right]\mu -{\frac {r_{2}}{\alpha }}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7202ada2bbe2b9f2d4fd274182f5f010fd51b601)
