Overview

You now have a great toolbox of monomers and polymerization mechanisms by which you can make all sorts of exciting polymer materials! You can decide what sorts of monomers and polymerization mechanisms to use, when and how to control tacticity, and how polymerization mechanism impacts skeletal structure for both step growth and chain growth (radical, cationic, anionic, and ring opening). Let’s expand that breadth even more, and think about how we can mix monomers together to create copolymers. In Lesson 1, we learned the difference between homopolymer and copolymer, but most of the polymers we have actually talked about up until now have been homopolymers which incorporate a single type of monomer. In this Lesson, we learn how to predict the ratios in which a monomer will be incorporated into a polymer, and how that composition might drift over the course of a reaction.

Learning Outcomes

By the end of this lesson, you should be able to:

• Explain under what circumstances, and why, the composition of a polymer changes with extent of reaction
• Distinguish between homopropagation and cross propagation reactions
• Describe kinetics of binary chain copolymerization
• Define reactivity ratio
• Apply copolymer composition equations to predict polymer compositions
• Describe, in terms of reactivity ratios, the conditions leading to random copolymerization, azeotropic copolymerization; block copolymers, alternating copolymers, or homopolymers
• Interpret plots of F_A vs f_A
• Explain composition drift using reactivity ratios

Lesson Checklist

Please refer to the Canvas Calendar for specific timeframes.

Questions?

If you have questions, please feel free to post them to the General Questions and Discussion forum. While you are there, feel free to post your own responses if you, too, are able to help a classmate.

n this lesson, we will consider what happens in terms of kinetics and skeletal structure when we polymerize two or more different monomers together to create a copolymer. We already learned in general what a copolymer is in Lesson 1 “what is a polymer?”. Make sure to refresh your memory on the difference between homopolymer and copolymer, and the definitions of alternating copolymer, block copolymer, random copolymer, and statistical copolymer.

Let’s consider a copolymer formed from reacting two monomers together. We consider a system in which the different monomers can react with each other, and they can also react with themselves. An example would be in the case of chain growth polymerization of two different monomers with vinyl groups. (This is an important point, because we saw some examples of step polymerization in which you could only get an alternating copolymer, RA₂+RB₂, just because of the specific functional groups that had to react. We are not considering that situation right now.) Consider the situation shown in Figure 8.1 where we have a purple and green monomer, and the purple happens to be incorporated faster into the polymer than the green one. Well, then at the start of our reaction, the polymer we form is going to contain mostly purple monomers. Later in the course of reaction, we find we have used up most of the purple monomers but have lots of green monomers left, so we begin to incorporate the green monomers more often than purple. As you can see, this leads to a distribution of the purple and green monomers along the polymer that is uneven; we have more repeat units of purple at one end of our polymer and more green at the other. You could also imagine that you stop the reaction before getting to a high extent of reaction, and end up with a polymer that has almost no green monomers at all! So what factors influence this distribution of purple and green along the length of the polymer? Can we predict and describe how various monomers will be incorporated when they are polymerized together?

Figure 8.1: Schematic representation of how two different monomers can be incorporated at different rates into the copolymer and the effect that could have on the polymer composition

Source: Lauren Zarzar

Let’s narrow our focus to 1) chain growth polymerization and 2) binary copolymerization. The chain growth part you already know, and for binary copolymerization we mean that there will only be two different monomers, which we will call A and B. Now because we have two different monomers, that means we can also have two different active centers: -A* and –B*. We designate the active enter with an * because it could be any number of species, such as radical, anion, or cation. Since A can react with A, A can react with B, and B can react with B, we have several different modes of addition that can occur:

$$\begin{array}{c}\text{Homopropagation reactions} \\ \text{ 1) }\sim \text{A*+A}\stackrel{{\text{ k}}_{\text{AA}}}{\to }\sim \text{AA*} \\ \text{ 2) }\sim \text{B*+B}\stackrel{{\text{ k}}_{\text{BB}}}{\to }\sim \text{BB*}\end{array}$$

$$\begin{array}{c}\text{Cross-propagation reactions} \\ \text{ 3) }\sim \text{B*+A}\stackrel{{\text{ k}}_{\text{BA}}}{\to }\sim \text{BA*} \\ \text{ 4) }\sim \text{A*+B}\stackrel{{\text{ k}}_{\text{AB}}}{\to }\sim \text{AB*}\end{array}$$

When an active center on A reacts with another A, or when an active center on B reacts with another B, we call these homopropagation reactions. When an active center on B reacts with A, or when an active center on A reacts with B, these are called cross-propagation reactions. We can describe the rate of monomer consumption for both A and B monomers as the sum of the processes that consume those monomers. For example, A is consumed in reactions (1) and (3) as outlined above. B is consumed in reactions (2) and (4). We then write the rate of consumption for each A and B monomer:

For any point during the reaction, we now have expressions that describe how much A or B is being incorporated. What about the ratio of incorporation? By looking at a ratio of the rates of incorporation of A and B, we can immediately have a sense of which monomer is being preferentially consumed:

This equation for d[A]/d[B] looks a little scary, but if we could just substitute for the concentration of active centers, we could simplify it….. How can we figure out the steady state concentration of active centers?

To start, we can describe the rate of change of [A*] and [B*], which are the active centers, very similarly to how we described the change in [A] and [B]. A* is consumed in reaction (4) but created in reaction (3). (Homopropagation reaction (1) also involves A* but there is no net change in the concentration of A* in reaction (1), thus we do not need to consider it). B* is consumed in reaction (3) and produced in reaction (4). (Again, homopropagation reaction (2) has net zero effect on [B*] so we don’t consider it). Therefore, we can write the change in concentration of active centers:

Let’s apply what we just learned to steady state conditions. Remember our steady state assumption? At steady state, we assume there is no change in the concentration of active centers, so:

$$\frac{d\left[ A* \right]}{dt}=0\text{ and }\frac{d\left[ B* \right]}{dt}=0$$

When we apply this steady state condition and set d[A*]/dt=d[B*]/dt with the equations above, we get:

This equation gives us the steady state ratio of A and B active centers. We can also now substitute this into Equation 8.3 to get:

Where $r_A=\frac{k_{AA}}{k_{AB}}$ and $r_B=\frac{k_{BB}}{k_{BA}}$. We call Equation 8.7 the copolymer composition equation. The copolymer composition equation is another example of an instantaneous equation, because the concentrations of the monomers change over time. These new variables, r_A and r_B , are called the monomer reactivity ratios. (Do not confuse the reactivity ratios with the reactant ratio r that we learned for Carothers theory!)

So what does this copolymer composition equation tell us? For any given instant in time, this equation tells us the molar ratio of A and B that is being incorporated into the copolymer. Sometimes, it can also be helpful to express the incorporation of A and B in terms of a fraction: what fraction of monomer being incorporated is A (=F_A), or what fraction of monomer being incorporated is B (=F_B)? Because we are only considering a binary case with two different monomers, we can express F_A and F_B as:

$$F_A=\frac{d\left[ A \right]}{d\left[ A \right]+d\left[ B \right]}$$
$$F_B=1-F_A$$

To summarize, F_A and F_B represent the mole fraction of A or B being incorporated into the polymer at a particular time.

What about the mole fraction of A and B monomer left in the reaction? Similarly, we can describe those with f_A and f_B (note the lowercase to distinguish from F_A and F_B):

$$f_A=\frac{\left[ A \right]}{\left[ A \right]+\left[ B \right]}$$
$$f_B=1-f_A$$

To summarize, f_A and f_B represent the mole fraction of A and B monomer in the reaction mixture (not in the polymer!)

We can now rewrite our copolymer composition equation (equation 8.7) as a function of f_A, f_B, F_A, F_B:

We can use these relationships to predict copolymer composition from the co-monomer composition and reactivity ratios

This result tells us that 46% incorporation is of A and the rest is B. What is of importance to notice here is that although you start with a 50/50 ratio of monomers, your polymer does NOT have the same monomer ratio! You are incorporating less A than you are B as a result of these reactivity ratios. *side note - Why is “low monomer conversion” specified? It is because as you let the reaction proceed to higher monomer conversions, the ratio of A and B monomer will change! Even though we start with an equimolar ratio of A and B, because here we are using up B faster than A, we are going to deplete our monomer mixture of B. Thus, at higher monomer conversions, the concentration of A will be higher than 0.5, and the concentration of be will be lower than 0.5. By specifying “low monomer conversion” we are assuming that the values of f_A and f_B are not changing significantly.*

It’s worth taking a more in depth look at these reactivity ratios given how important they are; reactivity ratios dictate the composition of the copolymer formed instantaneously, and influence the overall distribution of A and B groups in the copolymer. Let’s look at their definition again. What do they mean?

$$r_A=\frac{k_{AA}}{k_{AB}}{\text{ r}}_{\text{B}}\text{=}\frac{k_{BB}}{k_{BA}}$$

Notice that they are the ratio of rate coefficients for homopropagation to cross-propagation for A and B monomer! Let’s try to translate these mathematical relationships into an intuitive understanding.

A useful way of representing information from the copolymer equation is to plot F_A as a function of f_A. Let’s walk through how to do this using the reactivity ratios from the previous question, r_A = 0.9 and r_B = 1.2. First, we are going to just choose example values of f_A ranging from 0 to 1 (listed in the f_A column of the data Table 8.1 for Figure 8.2). Next, we can find f_B for each data point, because f_B = 1-f_A. Then, we plug the given values of r_A = 0.9 and r_B = 1.2 and the values of f_A and f_B into the expression:

$$F_A=\frac{r_A{f}_A{ }^2+f_A{f}_B}{r_A{f}_A{ }^2+2f_A{f}_B+r_B{f}_B{ }^2}$$

The values we calculated are shown below in Table 8.1. Plotting F_A for each value of f_A yields the graph shown below in Figure 8.2. Notice that the answer to the previous problem, which was F_A=0.46, in fact corresponds to f_A=0.5 as expected.

Figure 8.2. Plotting F_A as a function of f_A for r_A = 0.9 and r_B = 1.2

Source: Lauren Zarzar

Looking at plots of F_A as a function of f_A is very helpful in visualizing the relative incorporation of A and B monomers into the polymer. Figure 9.1 in the textbook (and replicated as Figure 8.3 below) shows such plots for varying combination of reactivity ratios. Can you rationalize why each curve as the shape it does? We will spend a significant amount of time considering this graph!

Figure 8.3: Plotting F_A as a function of f_A

Source: Figure 9.1 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.

Figure 8.3 is a plot of $F_A=\frac{r_A{f}_A{ }^2+f_A{f}_B}{r_A{f}_A{ }^2+2f_A{f}_B+r_B{f}_B{ }^2}$ for various values of reactivity ratios and monomer ratios. This is a key figure illustrating how the reactivity ratios and mole fraction of monomers affect the incorporation of the monomer into the polymer, ultimately determining the polymer composition. The shapes of each of these curves is meaningful, and we will discuss how to interpret this plot.

Take a minute to use the interractive graph below to explore the copolymerization reaction. You can use the sliders to change the values of r_A and r_B. You can also ust the text entry box to directly enter values for r_A and r_B. Try adjusting the values to match some of the curves in Figure 8.3 above.
Note: If you hover your mouse over the plot, the X = N/A and Y = N/A boxes will show the coordinates of you cursor.

Intuitively, when r_B and r_A = 1 then both homopropagation and cross-propagation for ~A* and ~B* happen at equal rates. If k_AA=k_AB (which is the case for r_A=1) then ~A* has no preference for adding to A or B. Similarly, if k_BB=k_BA (which is true for r_B=1) then ~B* also has no preference for adding to A or B. Therefore, the relative amounts of A and B incorporated into the polymer simply reflect the distribution of A and B in the monomer solution. There are relatively few copolymerizations that fulfill this requirement. However, also realize that the reactivities of the 2 active centers (~A* and ~B*) don’t have to be the same for random copolymerization.

So we just learned that a random copolymer forms when f_A=F_A, which can happen when r_A=r_B=1. What happens when this is not the case? For instance, what if f_A<F_A, what does that mean? Look at the conditions highlighted below, where an exemplary line for r_A=4 and r_B=0.1 ( highlighted in blue ). We see that this curve always falls above the line f_A=F_A indicating that the fraction of A being incorporated into the polymer is greater than the fraction of A in the monomer mixture for all molar fractions of A. This is because both active centers (~A* and ~B*) prefer to add to A monomer, which we know because k_AA>k_AB and k­_BA>k_BB, based on the given values of reactivity ratios (i.e. r_A>1 and r_B<1). We can use the same pattern of logic to understand why for the exemplary curve r_A=0.1 and r_B=10 ( highlighted in yellow ), f_A >F_A for all molar ratios of A; both active centers prefer to add to B monomer, so we always incorporate less A into the polymer than there is A in the monomer solution.

Figure 8.4: Plotting F_A as a function of f_A.
r_A=4 and r_B=0.1 ( blue plot )
r_A=0.1 and r_B=10 ( yellow plot )

Source: Modified from Figure 9.1 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.

Another special case of copolymerization is that of azeotropic copolymerization, where r_A and r_B are both less than 1, or, r_A and r_B are both greater than one (this case is rarer). Two examples are highlighted below. Notice that these curves cross the line f_A=F_A at a point called the azeotropic composition (f_A)_azeo. These points have been noted with green dots in figure 8.5. The name “azeotropic” comes from the analogy with phase diagrams of liquid-liquid mixtures, where an azeotropic mixture is characterized by having a vapor phase upon boiling with the same ratio of components as the liquid mixture. Similarly, for copolymerization, at the azeotropic composition, (f_A)_azeo, we find f_A=F_A indicating that monomer A is being incorporated into the polymer in the same proportion in which A exists in the monomer solution. Substituting the fact that f_A=F_A into our copolymer equation (equation 8.8) we can simplify to get an expression for the azeotropic composition:

$${\left( f_A\right)}_{azeo}=\frac{1-r_B}{2-r_A-r_B}$$

Figure 8.5: Highlighted curves are example conditions for azeotropic copolymerizations,
and the green dots indicate the azeotropic composition.

Source: Modified from Figure 9.1 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.

PROBLEM

A special case of azeotropic copolymerization is where both r_A and r_B=0. What would be the structure of this polymer?

Source: Modified from Figure 9.1 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.

1. Poly(A)-block-poly(B)
2. Poly(A)-rand-poly(B)
3. Poly(A)-graft-poly(B)
4. Poly(A)-alt-poly(B)

Click here for the answer.

ANSWER

D. Poly(A)-alt-poly(B)

Because both reactivity ratios are 0

$$\frac{k_{AA}}{k_{AB}}=\frac{k_{BB}}{k_{BA}}=0$$

This means that both k_AA and k_BB must be zero. Since homopropagations can’t occur, the sequence MUST alternate!

For most copolymerizations, f_A ≠ F_A and so, one monomer is preferentially consumed during polymerization. This means that as the reaction proceeds, the overall composition of the comonomer mixtures changes, i.e., f_A and f_B change over the course of the polymerization. And if f_A and f_B change, then F_A and F_B must also change as a function of monomer conversion! This process, which the monomer composition and polymer composition change over the course of the reaction, is called copolymer composition drift. Copolymer composition drift can lead to synthesis of polymers having very different composition over their lengths, and can become especially significant at higher monomer conversions.

Figure 8.1: Schematic representation of how two different monomers can be incorporated at different rates into the copolymer and the effect that could have on the polymer composition

Source: Lauren Zarzar

Given below are some example values of reactivity ratios for pairs of monomers for free radical polymerization. Notice that you need to consider the monomers together, and not individually; the reactivity ratio is not inherent to the monomer, but rather is a function of what you are trying to copolymerize and the conditions of the reaction.

The exact reactivity ratios are often hard to predict, and therefore are mostly determined experimentally. We can get a sense of how well various monomers stabilize the radical active center in comparison to one another based on these reactivity ratios; do the trends match what we predict? Resonance enhances radical stability; if we polymerize a monomer that stabilizes a radical well via resonance with one that doesn’t (such as, styrene with vinyl chloride) we find that the more stabilizing monomer (styrene) is preferentially incorporated (r_styrene>>1, r­_{vinyl chloride} <<1). More alkyl substituents also help to stabilize radicals (i.e., a radical on a tertiary carbon is more stable than a secondary carbon), so let’s compare methyl methacrylate versus methyl acrylate. The methyl methacylate should stabilize the radical better on a tertiary carbon, and thus we find that the methyl methacrylate is more readily incorporated into the polymer, as predicted.