Mayo–Lewis equation

The Mayo-Lewis equation or copolymer equation in polymer chemistry describes the distribution of monomers in a copolymer [1]:

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^* %2B M_1 \xrightarrow{k_{11}} M_1M_1^* \,
M_1^* %2B M_2 \xrightarrow{k_{12}} M_1M_2^* \,
M_2^* %2B M_1 \xrightarrow{k_{21}} M_2M_1^* \,
M_2^* %2B M_2 \xrightarrow{k_{22}} M_2M_2^* \,

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]%2B\left [M_2\right]\right)}{\left [M_2\right]\left (\left [M_1\right]%2Br_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.

Contents

Limiting cases

From this equation several limiting cases can be derived:

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

Both of these compounds do not 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%2Bf_1 f_2}{r_1 f_1^2%2B2f_1 f_2%2Br_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 %2B M_2)} \,

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

f_1 = 1 - f_2 =  \frac{M_1}{(M_1 %2B 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 feed

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^*] %2B 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_2^*] %2B k_{22}[M_2]\sum[M_1^*] \,

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^*]} %2B k_{21}} {k_{12}\frac{\sum[M_1^*]}{\sum[M_2^*]} %2B 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]}\,

External links

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. ^ Introduction to polymers R.J. Young ISBN 0412221705