6.2 Reactivity ratios and copolymer composition

Reactivity Ratios and Copolymer Composition

Reactivity ratios describe how two monomers "compete" during copolymerization. They let you predict whether a copolymer will be random, alternating, or blocky, and they're the key input for calculating what composition you'll actually get in the final polymer. Understanding them connects monomer chemistry to real copolymer structure and properties.

Reactivity Ratios in Copolymers

When two monomers (M1 and M2) copolymerize, a growing chain ending in M1• can either add another M1 or add an M2. The reactivity ratio captures which option the chain end prefers.

• $r_1 = k_{11}/k_{12}$, where $k_{11}$ is the rate constant for M1• adding M1, and $k_{12}$ is the rate constant for M1• adding M2
• $r_2 = k_{22}/k_{21}$, where $k_{22}$ is the rate constant for M2• adding M2, and $k_{21}$ is the rate constant for M2• adding M1

How to read these values:

• $r_1 > 1$: an M1• chain end prefers to add its own monomer (self-propagation favored)
• $r_1 < 1$: an M1• chain end prefers to add M2 (cross-propagation favored)
• $r_1 = 1$: M1• has no preference between M1 and M2

The same logic applies to $r_2$ for the M2• chain end.

The product $r_1 r_2$ tells you the type of copolymer you'll get:

Calculations of Copolymer Composition

The Mayo–Lewis equation (also called the instantaneous copolymer composition equation) relates the mole fraction of M1 in the copolymer ($F_1$) to the mole fraction of M1 in the monomer feed ($f_1$):

$F_1 = \frac{r_1 f_1^2 + f_1 f_2}{r_1 f_1^2 + 2f_1 f_2 + r_2 f_2^2}$

where $f_2 = 1 - f_1$.

This equation gives you the instantaneous composition, meaning the composition of polymer being formed at that specific moment. Here's why the distinction matters:

1. At the start of the reaction, the feed composition is whatever you loaded into the reactor, and the equation directly gives you the copolymer composition being produced.
2. As the reaction proceeds, whichever monomer is consumed faster will become depleted in the feed. This shifts $f_1$, which shifts $F_1$. The result is composition drift: the copolymer formed later in the reaction has a different composition than the copolymer formed early on.
3. At low conversions (below about 10%), the feed hasn't changed much, so the instantaneous composition is a good approximation of the overall average.
4. At higher conversions, you need to integrate the Mayo–Lewis equation over the conversion range to get the true average composition. This is sometimes done numerically or using the Skeist equation for integrated composition.

Interpretation of Composition Diagrams

Copolymer composition diagrams plot $F_1$ (copolymer composition) on the y-axis against $f_1$ (feed composition) on the x-axis. A diagonal line from (0,0) to (1,1) represents the case where the copolymer composition exactly matches the feed.

The shape of the curve depends entirely on $r_1$ and $r_2$:

• Ideal random ($r_1 = r_2 = 1$): The curve sits right on the diagonal. Whatever you put in the feed, you get in the copolymer.
• $r_1 > 1$, $r_2 < 1$: The curve lies above the diagonal. The copolymer is enriched in M1 compared to the feed because both chain ends preferentially incorporate M1. Acrylonitrile–butadiene copolymers show this behavior.
• $r_1 < 1$, $r_2 > 1$: The curve lies below the diagonal. The copolymer is enriched in M2. Ethylene–propylene copolymers are an example.
• Alternating ($r_1 \approx r_2 \approx 0$): The curve flattens sharply toward $F_1 = 0.5$. No matter what the feed ratio is, the copolymer composition stays close to 50:50 because both chain ends strongly prefer cross-propagation. Maleic anhydride–vinyl acetate behaves this way.

One feature to watch for: when the curve crosses the diagonal, that crossing point is called the azeotropic composition. At that feed ratio, the copolymer composition equals the feed composition, so there's no composition drift during the reaction.

Effects on Copolymer Structure

The combination of reactivity ratios and feed composition determines how monomer units are distributed along the chain. That sequence distribution directly controls properties.

1. Random copolymers ($r_1 \approx r_2 \approx 1$) have a statistical distribution of M1 and M2 along the chain. Their properties fall between those of the two homopolymers. Styrene–acrylonitrile (SAN) is a common example: it has a single $T_g$ that depends on composition.

2. Alternating copolymers ($r_1 \approx r_2 \approx 0$) have a regular ABAB pattern. This regularity can give them unique properties not seen in either homopolymer. Maleic anhydride–ethylene copolymers are a classic case.

3. Blocky or gradient copolymers ($r_1 > 1$, $r_2 > 1$) contain long runs of each monomer. These long sequences can phase-separate into distinct domains, similar to polymer blends but connected by covalent bonds. This microphase separation is what gives well-defined block copolymers (like SBS thermoplastic elastomers) their combination of rubbery and glassy behavior.

Specific properties affected by sequence distribution include:

• Thermal properties: A random copolymer typically shows a single $T_g$ between those of the homopolymers. A block copolymer can show two separate $T_g$ values, one for each block.
• Mechanical properties: Block copolymers can combine stiffness from one block with elasticity from another. Random copolymers give averaged mechanical behavior.
• Solubility and surface properties: Alternating hydrophobic–hydrophilic sequences behave very differently from blocky arrangements of the same monomers, which matters in applications like drug delivery carriers, polymer compatibilizers, and adhesive formulations.

By choosing monomers with the right reactivity ratios and controlling the feed composition throughout the reaction, you can tailor copolymer structure for a specific application.