- Mayo-Lewis equation
The Mayo-Lewis equation or copolymer equation in
polymer chemistry describes the distribution ofmonomer s in acopolymer ["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. LewisJ. Am. Chem. Soc. ; 1944; 66(9) pp 1594 - 1601; DOI|10.1021/ja01237a052] :Taking into consideration a monomer mix of two components and and the four different reactions that can take place at the reactive chain end terminating in either monomer () with their
reaction rate constant s :::::
and with reactivity ratio's defined as:
::
the copolymer equation is given as:
:
with the
concentration of the components given insquare bracket s. The equation gives the copolymer composition at any instant during the polymerization.Limiting cases
From this equation several limiting cases can be derived:
* with both reactivity ratios very high the two monomers have no inclination to react to each other except with themselves leading to a mixture of twohomopolymer s.
* with both ratio's 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 as such giving rise toblock copolymer
* with both ratio's around 1, monomer 1 will react as fast with another monomer 1 or monomer 2 and arandom copolymer results.
* with both values approaching 0 the monomers are unable to react in homopolymerization and the result is analternating polymer
* 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 is
maleic anhydride andstilbene , with reactivity ratio:
* Maleic anhydride ( = 0.08) & cis-stilbene ( = 0.07)
* Maleic anhydride (( = 0.03) & trans-stilbene ( = 0.03)Both of these compounds do not homopolymerize and instead, they react together to give exclusively alternating copolymer.
Another form of the equation is:
where stands the
mole fraction of each monomer in the copolymer:and the mole fraction of each monomer in the feed:
When the copolymer composition has the same composition as the feed, this composition is called the "azeotrope".
Calculation of reactivity ratios
The reactivity ratio's can be obtained by rewriting the copolymer equation to:
with
in the feed
and
in the copolymer
A number of copolymerization experiments are conducted with varying monomer ratio's and the copolymer composition is analysed at low conversion. A plot of versus gives a straight line with slope and intercept .
A semi-empirical method for the determination of reactivity ratio's is called the
Q-e scheme .Equation derivation
Monomer 1 is consumed with
reaction rate ["Introduction to polymers" R.J. Young ISBN 0412221705] :with the concentration of all the active centers terminating in monomer 1 or 2.
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
External links
* copolymer equation applet @eng.utah.edu [http://www.eng.utah.edu/~nairn/applets/Copoly.html Link]
* copolymers @zeus.plmsc.psu.edu [http://zeus.plmsc.psu.edu/~manias/MatSE443/chapter5.pdf Link]References
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