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 equation considers 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 (${\displaystyle M_{1}^{*}\,}$ and ${\displaystyle M_{2}^{*}\,}$) with their reaction rate constants ${\displaystyle k\,}$:

${\displaystyle M_{1}^{*}+M_{1}{\xrightarrow {k_{11}}}M_{1}M_{1}^{*}\,}$

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

${\displaystyle M_{2}^{*}+M_{2}{\xrightarrow {k_{22}}}M_{2}M_{2}^{*}\,}$

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

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]

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

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

The copolymer equation is then:^[3][4][2]

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

with the 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]

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

with ${\displaystyle \sum [M_{1}^{*}]}$ the concentration of all the active chains terminating in monomer 1, summed over chain lengths. ${\displaystyle \sum [M_{2}^{*}]}$ is defined similarly for monomer 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 by ${\displaystyle \sum [M_{2}^{*}]\,}$ followed by division of the first equation by the second 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 using the steady state approximation, meaning that the concentration of each type of active center remains constant.

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

The rate of formation of active centers of monomer 1 (${\displaystyle M_{2}^{*}+M_{1}{\xrightarrow {k_{21}}}M_{2}M_{1}^{*}\,}$) is equal to the rate of their destruction (${\displaystyle M_{1}^{*}+M_{2}{\xrightarrow {k_{12}}}M_{1}M_{2}^{*}\,}$) so that

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

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 yields the Mayo–Lewis equation after rearrangement:^[4]

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

Mole fraction form

It is often useful to alter the copolymer equation by expressing concentrations 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^[6][4]

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

This equation gives the 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 ${\displaystyle r_{1}\,}$ indicates a tendency for ${\displaystyle M_{1}^{*}\,}$ to add ${\displaystyle M_{1}\,}$, while small ${\displaystyle r_{1}\,}$ corresponds to a tendency for ${\displaystyle M_{1}^{*}\,}$ to add ${\displaystyle M_{2}\,}$. Values of ${\displaystyle r_{2}\,}$ describe the tendency of ${\displaystyle M_{2}^{*}\,}$ to add ${\displaystyle M_{2}\,}$ or ${\displaystyle M_{1}\,}$. From the definition of reactivity ratios, several special cases can be derived:

• ${\displaystyle r_{1}\approx r_{2}>>1\,}$ 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.
• ${\displaystyle r_{1}\approx r_{2}>1\,}$. 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.
• ${\displaystyle r_{1}\approx r_{2}\approx 1\,}$. 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.
• ${\displaystyle r_{1}\approx r_{2}\approx 0\,}$ If both values are near 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 ${\displaystyle r_{1}\,}$ = 0.01 for maleic anhydride and ${\displaystyle r_{2}}$ = 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]
• ${\displaystyle r_{1}>>1>>r_{2}\,}$ 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.
• When both ${\displaystyle r<1\,}$, the system has an azeotrope, where feed and copolymer composition are the same.^[9]

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 nonlinear least squares analysis to find the ${\displaystyle r_{1}\,}$, ${\displaystyle r_{2}\,}$ 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.^[10]

Mayo-Lewis Method

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

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

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 in a single point and the area in which most lines 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:^[11]

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

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

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

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,

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

where ${\displaystyle H_{min}\,}$ and ${\displaystyle H_{max}\,}$ are the highest and lowest values of ${\displaystyle H\,}$ from the Fineman-Ross method.^[12] 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 prediction of reactivity ratios is called the Q-e scheme which was proposed by Alfrey and Price in 1947.^[13] 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 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, ${\displaystyle r_{1}}$ 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 of what the monomer pair is in the system.

External links

• copolymers @zeus.plmsc.psu.edu Link

References

1. Copolymerization. I. A Basis for Comparing the Behavior of Monomers in Copolymerization; The Copolymerization of Styrene and Methyl Methacrylate Frank R. Mayo and Frederick M. Lewis J. Am. Chem. Soc.; 1944; 66(9) pp 1594 - 1601; doi:10.1021/ja01237a052
2. Cowie, J.M.G. Polymers: Chemistry & Physics of Modern Materials (2nd ed., Chapman & Hall 1991) p.106 ISBN 0-412-03121-3
3. Fried, Joel R. Polymer Science & Technology (2nd ed., Prentice-Hall 2003) p.42 ISBN 0-13-018168-4
4. Rudin, Alfred The Elements of Polymer Science and Engineering (Academic Press 1982) p.265 ISBN 0-12-601680-1
5. Young, Robert J. (1983). Introduction to polymers ([Reprinted with additional material] ed.). London: Chapman and Hall. ISBN 0-412-22170-5.
6. Fried, Joel R. Polymer Science & Technology (2nd ed., Prentice-Hall 2003) p.44 ISBN 0-13-018168-4
7. Copolymer Reactivity Ratios. Polymer Handbook, 4th ed.; Wiley, 2003; Vol 1, pp 259.
8. Rudin, Alfred The Elements of Polymer Science and Engineering (Academic Press 1982) p.288 ISBN 0-12-601680-1
9. Rudin, Alfred The Elements of Polymer Science and Engineering (Academic Press 1982) p.270 ISBN 0-12-601680-1
10. Tidwell, Paul W.; Mortimer, George A. (1965). "An Improved Method of Calculating Copolymerization Reactivity Ratios". J. Polym. Sci. A. 3: 369–387. doi:10.1002/pol.1965.100030137. Retrieved 14 September 2021.
11. Fineman, M.; Ross, S. D. J. Polymer Sci. 1950, 5, 259.
12. Kelen, T.; Tüdős, F.; Turcsányi, B. Polymer Bull. 1980, 2, 71-76.
13. Seymour, Raymond; Carraher, Charles Jr (1981). Polymer chemistry : an introduction. M. Dekker. p. 326. ISBN 0-8247-6979-1.