Step- and chain-growth polymerisation

Result

Step- and chain-growth polymerisation

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Statement

Two mechanisms for building macromolecules.

Why it matters

Before a polymer's bulk material properties — its glass-transition temperature, or its behaviour in solution as described by flory-huggins theory — can even be discussed, the more basic question of how the chain was actually assembled from monomers must be settled, since the mechanism directly controls the resulting degree-of-polymerisation and molar-mass distribution. Step-growth and chain-growth are the two fundamentally distinct logics by which small molecules are built up into macromolecules, and essentially every synthetic polymer follows one or the other.

The distinction also has an immediate, practical consequence for how a polymerisation is actually run and monitored: the two mechanisms show completely different relationships between reaction time, monomer conversion, and the molar mass of the polymer obtained, a difference with direct implications for reactor design and process control in industrial polymer manufacture.

Hypotheses

Step growth: any two species present — monomer, oligomer, or polymer chain — bearing complementary reactive functional groups may react with each other at any stage of the reaction.There is no special distinction between "growing chain" and "monomer" as in chain growth; a dimer can react with another dimer, a monomer, or a long oligomer with equal ease, provided the two required complementary functional groups are both present and available. Chain growth: only an active, reactive chain end (a radical, cation, or anion) can add a new monomer unit; monomer does not react directly with monomer.Once a chain's active end is deactivated (terminated), that chain stops growing permanently; growth is confined entirely to a comparatively small population of currently active chains at any instant, while the great majority of monomer remains as yet completely unreacted. The Carothers equation (Proof, Step 2) assumes exactly stoichiometric balance between the two complementary functional groups and no competing intramolecular cyclisation; either a slight stoichiometric imbalance or significant cyclisation caps the achievable degree of polymerisation well below the equation's idealised prediction.

Proof

\text{Step growth: monomers with complementary functional groups react stepwise, any-to-any, often releasing a small molecule (condensation).}

A diol and a diacid, for example, react at their end groups to form an ester linkage plus water; the resulting dimer still carries one free hydroxyl and one free carboxylic acid end, so it can itself react further with any other available complementary species, growing the chain incrementally at every step. A

\bar X_n = \frac{1}{1-p}, \qquad p=\text{fractional conversion of functional groups}

The Carothers equation gives the number-average degree of polymerisation as a function purely of overall functional-group conversion \(p\); because \(1/(1-p)\) grows very slowly until \(p\) is extremely close to \(1\), high molar mass in a step-growth polymerisation requires conversion far more complete than is needed in most other reaction contexts. A

\text{Chain growth: initiation (creates a reactive centre)} \to \text{propagation (rapid, repeated monomer addition)} \to \text{termination.}

An initiator generates a reactive centre (e.g. a radical), which then adds monomer units one at a time, very rapidly, to the same single growing chain until termination (for radicals, typically combination of two growing chains, or disproportionation) permanently ends that particular chain's growth. A

\text{High molar mass polymer is present from very early conversion in chain growth, while much monomer remains unreacted.}

Because each individual chain grows from initiation to full length within a very short time relative to the overall reaction, at any given moment during a chain-growth polymerisation the reaction mixture consists mostly of a small number of full-length polymer chains alongside a large amount of still-unreacted monomer — the opposite time-conversion behaviour to step growth, where a broad distribution of oligomer sizes exists at intermediate conversion and only reaches high molar mass as \(p\) approaches \(1\) (Step 2). A

\text{Step growth: } \bar X_n \text{ depends only on overall conversion } p\text{. Chain growth: instantaneous chain length depends on the propagation-to-termination rate ratio.}

These are the two mechanisms' defining kinetic signatures: step growth's degree of polymerisation is set purely by how far the reaction has proceeded overall (Step 2), essentially independent of how fast that conversion was reached, while chain growth's kinetic chain length is set by the competition between propagation and termination rate constants, largely independent of the overall fraction of monomer consumed at any given moment. B

Result

\text{Step growth: } \bar X_n=\frac{1}{1-p} \qquad\qquad \text{Chain growth: initiation}\to\text{propagation (fast, repeated)}\to\text{termination}

Reading. Step growth builds molar mass gradually and uniformly across every species present, reaching high molar mass only very near complete conversion; chain growth builds each chain to (near) full length almost immediately, with high molar mass polymer coexisting alongside a large pool of still-unreacted monomer.

Scope. The Carothers equation requires exact stoichiometric balance of the two complementary functional groups and negligible cyclisation (Hypotheses); chain-growth kinetics as described here requires a genuine, distinguishable active chain end, not merely a formal any-to-any condensation mechanism.

Corollaries & converses

degree-of-polymerisation's resulting molar-mass distribution shape differs characteristically between the two mechanisms — step growth gives the broad "most probable" distribution even at full conversion, while some chain-growth systems can give a much narrower distribution.
glass-transition and other bulk mechanical properties depend sensitively on the molar mass and its distribution, so the choice between step-growth and chain-growth mechanisms has direct, downstream materials consequences.
Step-growth polymers (polyesters, polyamides) are frequently condensation polymers releasing a small molecule such as water; common chain-growth polymers (polyethylene, polystyrene, poly(vinyl chloride)) are typically addition polymers formed with no by-product, though the mechanistic distinction (Hypotheses), not the presence or absence of a by-product, is what actually defines the classification (Common errors).

Fails without

Drop the any-to-any coupling assumption for step growth (Hypotheses): if only monomer, and never oligomer with oligomer, could react, the system would instead show chain-growth-like time-conversion behaviour, and the Carothers equation's prediction that high \(\bar X_n\) requires conversion very close to \(100\%\) would no longer hold.
Drop the single-active-centre restriction for chain growth: if monomer could react directly with monomer with no active centre required, there would be no mechanistic reason for high molar mass polymer to appear early in the reaction while most monomer remains unreacted (Step 4), and the sharp step/chain kinetic contrast disappears entirely.

Common errors

Equating "condensation polymer" with step growth and "addition polymer" with chain growth as though these were strictly synonymous pairs of terms — the true distinguishing criterion is any-to-any functional-group coupling (step) versus single-active-centre sequential monomer addition (chain), a correlation that holds for most common industrial polymers but is not the actual definition.
Assuming high degree of polymerisation is reached early in a step-growth reaction; the Carothers equation (Step 2) shows \(\bar X_n\) remains modest until conversion is extremely close to \(100\%\).
Assuming chain-growth polymerisations require exact stoichiometric balance between two different monomers, a requirement specific to step growth's complementary-functional-group mechanism, not to chain growth's single-monomer (or copolymer) addition scheme.
Forgetting that a slight stoichiometric imbalance in a step-growth reaction, even with conversion pushed to its practical maximum, caps the achievable \(\bar X_n\) well below the idealised Carothers prediction (Hypotheses' \(t3\) caveat).

Discussion

Wallace Carothers, at DuPont in the late 1920s and 1930s, carried out the foundational work establishing the kinetic treatment of step-growth polymerisation used throughout this page, in the course of his research programme that also produced nylon, one of the first major synthetic step-growth polymers. Paul Flory's subsequent statistical treatment, developed over the following decades, generalised the molar-mass distribution analysis for both mechanisms, underlying degree-of-polymerisation's and flory-huggins's more detailed results.

Common misconception: that step-growth and chain-growth polymerisations can always be told apart simply by whether a small molecule is eliminated during the reaction. While this correlation is common and useful as a rough heuristic, the mechanistic classification given in the Hypotheses — any-to-any functional-group coupling versus single-active-centre sequential addition — is the actual, defining distinction, and is what determines the very different time-conversion behaviour described in Step 4.

Worked examples

\text{Carothers equation at } p=0.90,\ 0.99,\ 0.999: \bar X_n = 10,\ 100,\ 1000

Direct substitution into Step 2's formula shows how steeply \(\bar X_n\) rises only in the last fraction of a percent of conversion: reaching a modest degree of polymerisation of \(10\) requires only \(90\%\) conversion, but reaching a technologically useful \(\bar X_n=1000\) requires conversion of \(99.9\%\), an extremely demanding practical target. A

\bar X_n\ \text{rises from}\ 10\ \text{to}\ 1000\ \text{over the narrow conversion range}\ p=0.90\ \text{to}\ p=0.999

Reading. The steep, late-conversion sensitivity of the Carothers equation explains why industrial step-growth polymerisations require driving the reaction to very high, carefully controlled conversion (often by removing the small-molecule by-product to shift the equilibrium) to obtain commercially useful molar masses.

Scope. This same steep late-conversion dependence applies to any step-growth polymerisation obeying the idealised, stoichiometrically balanced Carothers relationship.

Problems

Compute \(\bar X_n\) for a step-growth polymerisation carried to \(p=0.95\) conversion.
Solution
\(\bar X_n=1/(1-0.95)=1/0.05=20\).
Explain, without deriving the modified Carothers equation explicitly, why a slight excess of one bifunctional monomer over its stoichiometric partner caps the maximum achievable degree of polymerisation even as \(p\to1\) for the limiting reagent.
Solution
Once every molecule of the limiting (deficient) functional group has reacted, every remaining chain end necessarily carries only the excess functional group, which has no complementary partner left to react with; further reaction of the limiting group's remaining unreacted molecules is impossible, capping chain growth at a finite maximum \(\bar X_n\) regardless of how close to complete the limiting group's own conversion is driven, unlike the perfectly stoichiometric case where \(\bar X_n\) grows without bound as \(p\to1\).
Contrast, for step growth and chain growth respectively, roughly how much monomer remains unreacted at the point where high molar mass polymer first appears in significant quantity.
Solution
In step growth, high molar mass appears only once conversion \(p\) is extremely close to \(1\) (Worked example), meaning almost no unreacted monomer remains by the time high-molar-mass chains form. In chain growth, by contrast, individual chains reach high molar mass almost immediately after initiation, while the great majority of monomer in the reaction mixture is often still completely unreacted (Step 4) — the two mechanisms show essentially opposite relationships between overall monomer conversion and the appearance of high molar mass polymer.