Polymerization reactions

A polymer is nothing more than the same small molecule joined to itself, over and over, until a chain long enough to be plastic, fibre, or rubber emerges from what started as a simple liquid or gas. What decides whether that chain grows in a sudden rush or a slow, statistical crawl — and whether its backbone comes out neat and crystalline or tangled and rubbery — is entirely a question of mechanism. Two families cover essentially all synthetic polymers: chain-growth (addition) polymerization, where a reactive chain end adds one monomer at a time in a radical, cationic, or anionic chain reaction, and step-growth (condensation) polymerization, where any two complementary end-groups react and expel a small molecule at every step. Layered on top of mechanism is stereochemistry: when the monomer leaves a stereocentre on the backbone, the pattern of substituent orientation — tacticity — controls whether the polymer is ordered and strong or amorphous and soft. This article ties the two together and adds the one quantitative handle, the Carothers equation, that turns "how far did the reaction go" into "how long is the average chain."

The core insight

Chain-growth polymerization builds a few chains rapidly to full length through a radical, cationic, or anionic chain reaction, while step-growth polymerization builds molar mass slowly across an entire mixture by letting any two complementary end-groups react and lose a small molecule, and in both families the stereochemistry set at each backbone stereocentre — tacticity — is controlled by mechanism and catalyst, not by the monomer alone.

The core idea

Chain-growth (addition) polymerization. A reactive centre — radical, cation, or anion — adds across an alkene monomer's π\pi bond, regenerating the same type of reactive centre on the new chain end, so the process repeats as a genuine chain reaction with three stages:

Initiation: I⋅+M→IM⋅Propagation: IMn⋅+M→IMn+1⋅Termination: chain ends combine or disproportionate\text{Initiation: } \text{I}\cdot + \text{M} \to \text{IM}\cdot \qquad \text{Propagation: } \text{IM}_n\cdot + \text{M} \to \text{IM}_{n+1}\cdot \qquad \text{Termination: chain ends combine or disproportionate}

where: I⋅\text{I}\cdot is the initiating species, M\text{M} is one monomer unit, and IMn⋅\text{IM}_n\cdot is a growing chain of nn monomers still carrying the reactive end. The three mechanistic flavours differ only in what that reactive centre is:

  • Radical: initiator homolysis (e.g. peroxide O–O bond cleavage) generates a radical that adds to the alkene; propagation is fast and largely non-stereoselective, so uncontrolled radical polymerization tends to give a disordered backbone.
  • Cationic: an electrophile adds to the alkene to give a carbocation — an SN1S_N1-type chain — favoured by electron-rich monomers such as isobutylene, since the alkyl substituents stabilise the intermediate cation.
  • Anionic: a nucleophile or base adds to give a carbanion, propagated by further conjugate-addition steps — favoured by electron-poor monomers such as acrylonitrile, where the electron-withdrawing group stabilises the carbanion.

Step-growth (condensation) polymerization. Any two functional end-groups, whether on monomers or on growing oligomers, react together — usually as a nucleophilic acyl substitution — expelling a small molecule (water, HCl, methanol) at each linking step. A diacid and a diol give a polyester plus water; a diamine and a diacid give a polyamide (nylon) plus water. Because reaction can occur between any two species carrying complementary ends, not just between a fixed chain end and fresh monomer, the average chain length across the whole pot climbs only gradually, and — this is the point the Carothers equation below makes precise — very high molar mass demands the reaction be pushed nearly to completion.

Tacticity — stereochemistry of the backbone. When the monomer contributes a stereocentre to the chain (propylene →\to polypropylene, where every other backbone carbon bears a methyl group), the relative configuration at successive stereocentres is described by three patterns:

  • Isotactic — every substituent on the same relative side: regular, crystallisable, and typically packs into an ordered helix.
  • Syndiotactic — substituents alternate sides at each successive centre: also regular and crystallisable, though the packing geometry differs from the isotactic form.
  • Atactic — substituent orientation is random along the chain: irregular, amorphous, and mechanically softer or rubbery.

Tacticity is not a fixed property of the monomer; it is set by the mechanism and catalyst. Ziegler–Natta catalysts coordinate the incoming monomer in a fixed orientation and deliver isotactic or syndiotactic polypropylene, while uncontrolled radical polymerization of a similar monomer gives predominantly atactic material, because the planar, sp²-like radical intermediate has no strong facial bias.

Figure
Three polymer backbones compared showing methyl groups all on one side for isotactic all alternating for syndiotactic and randomly placed for atactic
Tacticity — same monomer, three backbone patterns isotactic all methyls same side syndiotactic methyls alternate sides atactic methyls placed randomly Same monomer, same backbone connectivity — only the stereocentre pattern differs Isotactic and syndiotactic are regular and can crystallise; atactic is irregular and stays amorphous Pattern is set by the polymerization mechanism and catalyst, not by the monomer alone
Derivation ACarothers Equation: Average Chain Length from Extent of Reaction

The goal is to show, using nothing beyond counting end-groups and simple algebra, why step-growth polymerization must be pushed to near-total conversion before it delivers usefully long chains — a fact that explains why condensation polymers are harder to make at high molar mass than addition polymers, and why nylon manufacture insists on near-complete reaction. Step 1. Start with N0N_0 monomer molecules, each carrying one reactive end-group (a simple A–B monomer such as an amino acid has one COOH\text{COOH} and one NH2\text{NH}_2). Every linking step consumes one end-group pair and joins two fragments into one longer fragment, so the number of molecules present, NN, equals the original monomer count minus the number of linking reactions completed. Step 2. Define the extent of reaction pp as the fraction of functional groups already reacted; one group reacts per linking step, so pN0pN_0 linking steps have occurred, leaving N=N0−pN0=N0(1−p)N = N_0 - pN_0 = N_0(1-p) molecules. Step 3. The average degree of polymerization, X‾n\overline{X}_n, is the average number of monomer units per surviving molecule — the original monomer count divided by how many molecules it is now spread across:

X‾n=N0N=N0N0(1−p)=11−p.\overline{X}_n = \frac{N_0}{N} = \frac{N_0}{N_0(1-p)} = \frac{1}{1-p}.

where: X‾n\overline{X}_n is the number-average degree of polymerization and pp is the extent of reaction, 0≤p<10 \le p < 1. Step 4. Because X‾n\overline{X}_n blows up only as p→1p \to 1, chain length stays modest for most of the reaction and rises sharply in the last few per cent of conversion — at p=0.90p = 0.90, X‾n=10\overline{X}_n = 10; at p=0.98p = 0.98, X‾n=50\overline{X}_n = 50; at p=0.99p = 0.99, X‾n=100\overline{X}_n = 100. This is why step-growth polymers need the reaction driven almost to completion (removing the by-product, such as water, to push equilibrium forward), unlike chain-growth polymerization, where each surviving chain is already long by the time it terminates.

Worked example

Nylon-6,6 is made by reacting hexanedioic acid (adipic acid) with hexane-1,6-diamine. If the polycondensation is carried out to an extent of reaction p=0.995p = 0.995, estimate the average degree of polymerization, and contrast this with the radical chain-growth polymerization of styrene initiated by benzoyl peroxide.

Step 1. Apply the Carothers equation with p=0.995p = 0.995: X‾n=11−p=11−0.995=10.005=200\overline{X}_n = \dfrac{1}{1-p} = \dfrac{1}{1-0.995} = \dfrac{1}{0.005} = 200. So on average 200200 monomer units are strung together per chain — and this required the reaction to be carried to 99.5%99.5\% completion; stopping at p=0.98p = 0.98 would have given only X‾n=50\overline{X}_n = 50, a four-fold drop for what looks like a small difference in conversion.

Step 2. Mechanistically, this chain grows one amide bond at a time by nucleophilic acyl substitution: the amine nitrogen lone pair attacks the electrophilic carbonyl carbon of a –COOH\text{–COOH} end, a tetrahedral intermediate collapses expelling water, R–COOH+H2N–R′→R–CONH–R′+H2O\text{R–COOH} + \text{H}_2\text{N–R}' \to \text{R–CONH–R}' + \text{H}_2\text{O}, and the product still carries one free –COOH\text{–COOH} and one free –NH2\text{–NH}_2, each free to react with any other monomer or oligomer bearing a complementary end — exactly the statistical, any-pair-can-react picture the Carothers equation quantifies.

Step 3. Contrast with styrene (PhCH=CH2\text{PhCH=CH}_2) initiated by benzoyl peroxide: initiation gives a radical R⋅\text{R}\cdot that adds to the terminal CH2\text{CH}_2, placing the new radical on the stabilised benzylic carbon, R⋅+CH2=CHPh→R–CH2–C˙HPh\text{R}\cdot + \text{CH}_2=\text{CHPh} \to \text{R–CH}_2\text{–}\dot{\text{C}}\text{HPh}; propagation repeats this addition thousands of times on the same chain before termination (radical–radical coupling or disproportionation) ends it. Unlike nylon's 200200-unit chains built up gradually across the whole pot, a single styrene chain reaches thousands of units almost as soon as it starts, since chain-growth builds molar mass instantly on a few active chains rather than statistically across the mixture.

Step 4. Because propagation proceeds through a planar, sp²-like benzylic radical with no strong facial preference, the configuration at each new stereocentre is essentially random relative to its neighbour, so uncontrolled radical polymerization of styrene-type monomers gives predominantly atactic polymer — the stereochemical counterpart to nylon-6,6 having no stereocentres on its backbone at all (both monomers are achiral and symmetric), so tacticity does not arise for this condensation polymer.

Common traps

  • Calling every polymerization "addition polymerization" — condensation (step-growth) polymers such as nylon and polyester do not proceed by a chain mechanism, and they lose a small molecule at each linking step.
  • Misreading the Carothers equation as linear in pp — X‾n=1/(1−p)\overline{X}_n = 1/(1-p) is sharply nonlinear near p=1p = 1, so the last fraction of a per cent of conversion matters far more for chain length than the first ninety per cent.
  • Assuming tacticity is fixed by the monomer alone — it is set by the mechanism and catalyst of polymerization, not an intrinsic property of the monomer, and it does not arise at all for monomers (like most simple diacids and diamines) that leave no stereocentre on the backbone.
  • Forgetting that radical addition to an unsymmetrical alkene is regioselective (attack at the less hindered carbon, radical ends up at the more stabilised position) — getting this backwards misassigns which carbon becomes the new stereocentre.

Ask the tutor: if you want the full curly-arrow mechanism for cationic or anionic chain-growth polymerization, or a Carothers-equation calculation with a different extent of reaction worked through step by step, ask the AI tutor to walk through it with you.

Why does the Carothers equation only apply to step-growth polymers, not chain-growth ones?

The equation assumes molar mass builds up statistically, by any two end-groups anywhere in the mixture linking together — exactly how step-growth (condensation) polymerization works. Chain-growth polymerization is different: a handful of chains grow rapidly, one monomer at a time, at a reactive centre, while most monomer molecules haven't reacted at all yet. There, chain length depends on kinetics — how many monomer additions happen before termination — not on the overall extent of reaction of the whole pot, so a different (kinetic) treatment is needed instead.

If p = 0.99 already gives Xn = 100, why does nylon manufacture push all the way to p = 0.995 or higher?

Because the relationship is steeply nonlinear near complete conversion: going from p=0.99p = 0.99 to p=0.995p = 0.995 doubles the chain length from 100100 to 200200, and pushing further toward p=0.999p = 0.999 would give 10001000. Useful fibre-grade nylon needs chains long enough to entangle and give mechanical strength, so manufacturers actively remove the condensation by-product (water) to drive the equilibrium as far toward completion as practically possible — the last fraction of a percent of conversion is doing most of the work.

Does every chain-growth polymer end up atactic like radical polystyrene?

No — atacticity is a consequence of the radical mechanism specifically, because the planar radical intermediate has little preference for which face of the alkene reacts next. Cationic and anionic chain-growth can show more stereoselectivity depending on conditions, and when a coordination catalyst (such as Ziegler–Natta systems) controls monomer approach, chain-growth polymerization of propylene-type monomers gives highly isotactic or syndiotactic product instead. Tacticity outcome depends on mechanism and catalyst, not on "addition polymerization" as a blanket label.

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