The Mayo–Lewis equation or copolymer equation in polymer chemistry describes the distribution of monomers in a copolymer: It is named for Frank R. Mayo and Frederick M. Lewis.
Contents
- Equation derivation
- Instantaneous form
- Limiting cases
- Calculation of reactivity ratios
- Curve Fitting
- Mayo Lewis Method
- Fineman Ross Method
- Kelen Tudos method
- Q e scheme
- References
Taking into consideration a monomer mix of two components
and with reactivity ratios defined as:
the copolymer equation is given as:
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:
with
Likewise the rate of disappearance for monomer 2 is:
Division of both equations yields:
The ratio of active center concentrations can be found assuming steady state with:
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:
or
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
Similarly,
These equations can be combined with the Mayo-Lewis Equation to give
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
An example case is maleic anhydride and styrene, with reactivity ratios:
Neither of these compounds homopolymerize and instead they react together to give almost exclusively alternating copolymer.
When both
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,
Curve Fitting
One of the simplest methods for finding reactivity ratios is plotting the copolymer equation and using least squares analysis to find the
Mayo-Lewis Method
The Mayo-Lewis method uses a form of the copolymer equation relating
For each different monomer composition, a line is generated using arbitrary
Fineman-Ross Method
Fineman and Ross rearranged the copolymer equation into a linear form:
where
Thus, a plot of
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,
where
where
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,
while the reaction of
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,