Chain Growth Binary Copolymerization

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.