Mayo–Lewis equation

Monomer 1 is consumed with reaction rate:{{cite book|last=Young|first=Robert J.|title=Introduction to polymers|year=1983|publisher=Chapman and Hall|location=London|isbn=0-412-22170-5|edition=[Reprinted with additional material]|url-access=registration|url=https://archive.org/details/introductiontopo00youn_0}}

$\backslash frac\{-d[M\_1]\}\{dt\}\; =\; k\_\{11\}[M\_1]\backslash sum[M\_1^*]\; +\; k\_\{21\}[M\_1]\backslash sum[M\_2^*]\; \backslash ,$

with $\backslash sum[M\_1^*]$ the concentration of all the active chains terminating in monomer 1, summed over chain lengths. $\backslash sum[M\_2^*]$ is defined similarly for monomer 2.

Likewise the rate of disappearance for monomer 2 is:

$\backslash frac\{-d[M\_2]\}\{dt\}\; =\; k\_\{12\}[M\_2]\backslash sum[M\_1^*]\; +\; k\_\{22\}[M\_2]\backslash sum[M\_2^*]\; \backslash ,$

Division of both equations by $\backslash sum[M\_2^*]\; \backslash ,$ followed by division of the first equation by the second yields:

$\backslash frac\{d[M\_1]\}\{d[M\_2]\}\; =\; \backslash frac\{[M\_1]\}\{[M\_2]\}\; \backslash left(\; \backslash frac\{k\_\{11\}\backslash frac\{\backslash sum[M\_1^*]\}\{\backslash sum[M\_2^*]\}\; +\; k\_\{21\}\}\; \{k\_\{12\}\backslash frac\{\backslash sum[M\_1^*]\}\{\backslash sum[M\_2^*]\}\; +\; k\_\{22\}\}\; \backslash right)\; \backslash ,$

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.

$\backslash frac\{d\backslash sum[M\_1^*]\}\{dt\}\; =\; \backslash frac\{d\backslash sum[M\_2^*]\}\{dt\}\; \backslash approx\; 0\backslash ,$

The rate of formation of active centers of monomer 1 ($M\_2^*\; +\; M\_1\; \backslash xrightarrow\{k\_\{21\}\}\; M\_2M\_1^*\; \backslash ,$) is equal to the rate of their destruction ($M\_1^*\; +\; M\_2\; \backslash xrightarrow\{k\_\{12\}\}\; M\_1M\_2^*\; \backslash ,$) so that

$k\_\{21\}[M\_1]\backslash sum[M\_2^*]\; =\; k\_\{12\}[M\_2]\backslash sum[M\_1^*]\; \backslash ,$

or

$\backslash frac\{\backslash sum[M\_1^*]\}\{\backslash sum[M\_2^*]\}\; =\; \backslash frac\{k\_\{21\}[M\_1]\}\{k\_\{12\}[M\_2]\}\backslash ,$

Substituting into the ratio of monomer consumption rates yields the Mayo–Lewis equation after rearrangement:

$\backslash frac\{d[M\_1]\}\{d[M\_2]\}\; =\; \backslash frac\{[M\_1]\}\{[M\_2]\}\; \backslash left(\; \backslash frac\{k\_\{11\}\backslash frac\{k\_\{21\}[M\_1]\}\{k\_\{12\}[M\_2]\}\; +\; k\_\{21\}\}\; \{k\_\{12\}\backslash frac\{k\_\{21\}[M\_1]\}\{k\_\{12\}[M\_2]\}+\; k\_\{22\}\}\; \backslash right)\; =\; \backslash frac\{[M\_1]\}\{[M\_2]\}\; \backslash left(\; \backslash frac\{\backslash frac\{k\_\{11\}[M\_1]\}\{k\_\{12\}[M\_2]\}\; +\; 1\}\; \{\backslash frac\{[M\_1]\}\{[M\_2]\}+\; \backslash frac\{k\_\{22\}\}\{k\_\{21\}\}\}\; \backslash right)\; =\; \backslash frac\{[M\_1]\}\{[M\_2]\}\; \backslash frac\{\backslash left\; (r\_1\backslash left[M\_1\backslash right]+\backslash left\; [M\_2\backslash right]\backslash right)\}\{\backslash left\; (\backslash left\; [M\_1\backslash right]+r\_2\backslash left\; [M\_2\backslash right]\backslash right)\}$

It is often useful to alter the copolymer equation by expressing concentrations in terms of mole fractions. Mole fractions of monomers $M\_1\backslash ,$ and $M\_2\backslash ,$ in the feed are defined as $f\_1\backslash ,$ and $f\_2\backslash ,$ where

$f\_1\; =\; 1\; -\; f\_2\; =\; \backslash frac\{M\_1\}\{(M\_1\; +\; M\_2)\}\; \backslash ,$

Similarly, $F\backslash ,$ represents the mole fraction of each monomer in the copolymer:

$F\_1\; =\; 1\; -\; F\_2\; =\; \backslash frac\{d\; M\_1\}\{d\; (M\_1\; +\; M\_2)\}\; \backslash ,$

These equations can be combined with the Mayo–Lewis equation to giveFried, Joel R. Polymer Science & Technology (2nd ed., Prentice-Hall 2003) p.44 {{ISBN|0-13-018168-4}}

$F\_1=1-F\_2=\backslash frac\{r\_1\; f\_1^2+f\_1\; f\_2\}\{r\_1\; f\_1^2+2f\_1\; f\_2+r\_2f\_2^2\}\backslash ,$

This equation gives the composition of copolymer formed at each instant. However the feed and copolymer compositions can change as polymerization proceeds.

Reactivity ratios indicate preference for propagation. Large $r\_1\backslash ,$ indicates a tendency for $M\_1^*\backslash ,$ to add $M\_1\backslash ,$, while small $r\_1\backslash ,$ corresponds to a tendency for $M\_1^*\backslash ,$ to add $M\_2\backslash ,$. Values of $r\_2\backslash ,$ describe the tendency of $M\_2^*\backslash ,$ to add $M\_2\backslash ,$ or $M\_1\backslash ,$. From the definition of reactivity ratios, several special cases can be derived:

• $r\_1\; \backslash approx\; r\_2\; >>\; 1\; \backslash ,$ 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.
• $r\_1\; \backslash approx\; r\_2\; >\; 1\; \backslash ,$. 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.
• $r\_1\; \backslash approx\; r\_2\; \backslash approx\; 1\; \backslash ,$. 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.
• $r\_1\; \backslash approx\; r\_2\; \backslash approx\; 0\; \backslash ,$ 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 $r\_1\backslash ,$ = 0.01 for maleic anhydride and $r\_2$ = 0.02 for styrene.Copolymer Reactivity Ratios. Polymer Handbook, 4th ed.; Wiley, 2003; Vol 1, pp 259. Maleic acid in fact does not homopolymerize in free radical polymerization, but will form an almost exclusively alternating copolymer with styrene.Rudin, Alfred The Elements of Polymer Science and Engineering (Academic Press 1982) p.288 {{ISBN|0-12-601680-1}}
• $r\_1\; >>\; 1\; >>\; r\_2\; \backslash ,$ 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.Rudin, Alfred The Elements of Polymer Science and Engineering (Academic Press 1982) p.270 {{ISBN|0-12-601680-1}}

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, $r\_1\backslash ,$ and $r\_2\backslash ,$ can be found.

One of the simplest methods for finding reactivity ratios is plotting the copolymer equation and using nonlinear least squares analysis to find the $r\_1\backslash ,$, $r\_2\backslash ,$ 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.{{cite journal |last1=Tidwell |first1=Paul W. |last2=Mortimer |first2=George A. |title=An Improved Method of Calculating Copolymerization Reactivity Ratios |journal=J. Polym. Sci. A |date=1965 |volume=3 |pages=369–387 |doi=10.1002/pol.1965.100030137 |url=https://onlinelibrary.wiley.com/doi/10.1002/pol.1965.100030137 |access-date=14 September 2021|url-access=subscription }}

The Mayo-Lewis method uses a form of the copolymer equation relating $r\_1\backslash ,$ to $r\_2\backslash ,$:

$r\_2\; =\; \backslash frac\{f\_1\}\{f\_2\}\backslash left[\backslash frac\{F\_2\}\{F\_1\}(1+\backslash frac\{f\_1r\_1\}\{f\_2\})-1\backslash right]\backslash ,$

For each different monomer composition, a line is generated using arbitrary $r\_1\backslash ,$ values. The intersection of these lines is the $r\_1\backslash ,$, $r\_2\backslash ,$ 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 $r\_1\backslash ,$, and $r\_2\backslash ,$ values.

Fineman and Ross rearranged the copolymer equation into a linear form:Fineman, M.; Ross, S. D. J. Polymer Sci. 1950, 5, 259.

$G=\; Hr\_1-r\_2\; \backslash ,$

where $G\; =\; \backslash frac\{f\_1(2F\_1-1)\}\{(1-f\_1)F\_1\}\; \backslash ,$ and $H\; =\; \backslash frac\{f\_1^2(1-F\_1)\}\{(1-f\_1)^2F\_1\}\backslash$

Thus, a plot of $H\; \backslash ,$ versus $G\; \backslash ,$ yields a straight line with slope $r\_1\backslash ,$ and intercept $-r\_2\backslash ,$

The Fineman-Ross method can be biased towards points at low or high monomer concentration, so Kelen and Tüdős introduced an arbitrary constant,

$\backslash alpha\; =\; (H\_\{min\}H\_\{max\})^\{0.5\}\; \backslash ,$

where $H\_\{min\}\; \backslash ,$ and $H\_\{max\}\; \backslash ,$ are the highest and lowest values of $H\; \backslash ,$ from the Fineman-Ross method.Kelen, T.; Tüdős, F.; Turcsányi, B. Polymer Bull. 1980, 2, 71-76. The data can be plotted in a linear form

$\backslash eta\; =\; \backslash left[r\_1+\backslash frac\{r\_2\}\{\backslash alpha\}\backslash right]\backslash mu\; -\; \backslash frac\{r\_2\}\{\backslash alpha\}\; \backslash ,$

where $\backslash eta=\; G/(\backslash alpha+H)\; \backslash ,$ and $\backslash mu=\; H/(\backslash alpha+H)\; \backslash ,$. Plotting $\backslash eta$ against $\backslash mu$ yields a straight line that gives $-r\_2/\backslash alpha$ when $\backslash mu=0$ and $r\_1$ when $\backslash mu\; =\; 1$. This distributes the data more symmetrically and can yield better results.

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.{{cite book |last1=Seymour |first1=Raymond |last2=Carraher |first2=Charles Jr |title=Polymer chemistry : an introduction |year=1981 |publisher=M. Dekker |isbn=0-8247-6979-1 |pages=326}} This involves using two parameters for each monomer, $Q$ and $e$. The reaction of $M\_1$

radical with $M\_2$ monomer is written as

$k\_\{12\}\; =\; P\_1Q\_2exp(-e\_1e\_2)$

while the reaction of $M\_1$ radical with $M\_1$ monomer is written as

$k\_\{11\}\; =\; P\_1Q\_1exp(-e\_1e\_1)$

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, $r\_1$ and $r\_2$ 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.

• copolymers @zeus.plmsc.psu.edu [http://zeus.plmsc.psu.edu/~manias/MatSE443/chapter5.pdf Link]

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