Radical Chain Polymerization: Kinetics, Chain Transfer, Gel Effect & Thermodynamics

Chain-growth (addition) polymerization requires an active center—a free radical, carbonium ion, carbanion, or coordination transition-metal complex—capable of rapidly adding monomer molecules sequentially. In free-radical polymerization, chain growth is propagated by an unpaired electron on the terminal carbon atom of the growing macromolecule.

The Three Elementary Kinetic Steps

1. Initiation:

Initiation is a two-step sequence:

• Homolytic Cleavage of Initiator: An initiator molecule $I$ decomposes thermally, photolytically, or via redox transfer into two primary free radicals $R^\bullet$:

• Monomer Addition (Chain Initiation): A primary radical adds to the carbon-carbon double bond of a vinyl monomer $M$ to produce the initial chain radical $M_1^\bullet$:

Because the decomposition rate constant $k_d$ is typically $10^{-6} - 10^{-4}\text{ s}^{-1}$ while $k_i \sim 10^5\text{ L/(mol s)}$, the homolytic decomposition of initiator is the rate-determining step of initiation.

2. Propagation:

The active macroradical adds monomer molecules sequentially in rapid succession:

Addition typically occurs with head-to-tail regioselectivity due to steric repulsion and resonance stabilization of the resulting radical by the pendant substituent $X$ (e.g., phenyl in styrene, ester in acrylates).

3. Termination:

Two growing macroradicals annihilate each other's active unpaired electrons via bimolecular encounter:

• Termination by Combination (Coupling): Two radical chain ends bond covalently to form a single dead macromolecule:

• Termination by Disproportionation: A hydrogen atom is abstracted from the $\beta$-carbon of one radical by the other, producing two dead macromolecules: one with a saturated end and one with a terminal alkene end:

The total termination rate constant is:

Common Free-Radical Initiators

1. Azo Initiators:

• $2,2'$-Azobis(isobutyronitrile) (AIBN): Decomposes thermally with elimination of exceptionally stable nitrogen gas ($\text{N}_2$):

AIBN exhibits first-order decomposition kinetics with minimal induced decomposition from radicals.

2. Peroxide Initiators:

• Benzoyl Peroxide (BPO): Undergoes homolytic cleavage of the weak $O-O$ peroxy bond ($E_a \approx 125\text{ kJ/mol}$):

3. Redox Initiators:

• Potassium persulfate with ferrous ions ($S_2O_8^{2-} + Fe^{2+} \to SO_4^{\bullet-} + SO_4^{2-} + Fe^{3+}$), widely used in aqueous emulsion and suspension polymerizations at room temperature.

Decomposition Kinetics and Half-Life

Initiator decomposition follows strict first-order kinetics:

The initiator half-life $t_{1/2}$ is:

The Initiator Efficiency Factor ($f$) and the Solvent Cage Effect

When an initiator molecule cleaves inside a liquid, the two primary radicals are born inside a surrounding 'cage' of solvent molecules. Before escaping into the bulk solution:

1. They may collide and recombine inside the cage (geminate recombination):

1. They may react with solvent molecules or undergo disproportionation within the cage.

The initiator efficiency $f$ is defined as the fraction of radicals produced by homolysis that successfully escape the solvent cage and initiate polymer chains:

For typical polymerizations in organic solvents, $f$ ranges from $0.30$ to $0.80$. As solvent viscosity increases, radical cage escape is hindered, causing $f$ to decline.

Derivation of the Steady-State Rate Law

To derive the overall rate of polymerization $R_p$, we formulate the rates of the elementary steps:

1. Rate of Initiation ($R_i$):

2. Rate of Propagation ($R_p$):

The rate of monomer consumption occurs primarily via propagation (monomer consumed in initiation is negligible, $< 0.1\%$):

where $[M^\bullet] = \sum_{n=1}^\infty [M_n^\bullet]$ is the total concentration of all growing macroradicals.

3. Rate of Termination ($R_t$):

Because termination is a bimolecular reaction between two macroradicals:

(The factor of 2 represents the consumption of two radical centers per termination event).

The Pseudo-Steady-State Approximation (PSSA)

Within seconds of initiating the reaction, the concentration of active macroradicals reaches a dynamic balance where the rate of radical generation equals the rate of radical termination:

Canceling the factor of 2 and solving for the steady-state radical concentration $[M^\bullet]$:

For typical systems, $[M^\bullet] \approx 10^{-8} - 10^{-7}\text{ mol/L}$, explaining why radical polymerization proceeds smoothly without explosive chain recombination.

The Universal Rate Law of Radical Polymerization

Substituting $[M^\bullet]$ into the propagation rate expression:

This landmark equation reveals the fundamental classical kinetics:

• The rate of polymerization is first-order in monomer concentration: $R_p \propto [M]^1$.
• The rate of polymerization is half-order in initiator concentration: $R_p \propto [I]^{1/2}$.

The square-root dependence on $[I]$ is the experimental hallmark of bimolecular radical termination.

The Kinetic Chain Length ($\nu$)

The kinetic chain length $\nu$ is defined as the average number of monomer molecules polymerized per active radical center initiated:

Substituting $R_p = k_p [M] [M^\bullet]$ and $R_t = 2 k_t [M^\bullet]^2$:

Substituting the steady-state radical concentration $[M^\bullet] = \sqrt{f k_d [I] / k_t}$:

Notice the fundamental kinetic conflict of free-radical polymerization:

• Increasing $[I]$ increases the polymerization rate ($R_p \propto [I]^{1/2}$), but
• Increasing $[I]$ decreases the kinetic chain length and molecular weight ($

u \propto [I]^{-1/2}$). To produce high molecular weight polymer, one must operate at low initiator concentrations and tolerate lower polymerization rates.

Relationship Between $X_n$ and $\nu$

In the absence of chain transfer, the number-average degree of polymerization $X_n$ depends on the termination mode:

1. Pure Disproportionation ($k_{tc} = 0, k_t = k_{td}$):

Each termination event produces two dead polymer molecules from two growing macroradicals. Thus, each dead molecule corresponds to exactly one kinetic chain:

The theoretical dispersity is:

2. Pure Combination ($k_{td} = 0, k_t = k_{tc}$):

Two kinetic chains combine head-to-head to form a single dead polymer molecule:

The theoretical dispersity for combination termination is:

Deconvolution of Mixed Termination Modes

Let $q$ be the fraction of termination events occurring by disproportionation:

Then:

For styrene at $60^\circ\text{C}$, termination occurs almost exclusively by combination ($q \approx 0.05, X_n \approx 1.9 \nu$). For methyl methacrylate (MMA) at $60^\circ\text{C}$, termination is predominantly disproportionation ($q \approx 0.80, X_n \approx 1.1 \nu$).

Chain transfer is an elementary reaction wherein a growing macroradical abstracts an atom (usually hydrogen or halogen) from another molecule $X-Y$ present in the reaction mixture:

The original chain is terminated into a dead macromolecule $M_n-X$, while the newly generated radical $Y^\bullet$ may re-initiate polymerization by adding a new monomer:

Classification of Chain Transfer Agents

1. Normal Chain Transfer ($k_{i,Y} \approx k_p$):

• The new radical $Y^\bullet$ re-initiates rapidly.
• The overall rate of polymerization $R_p$ is unaffected.
• The molecular weight of the polymer is significantly reduced.

2. Retardation ($k_{i,Y} < k_p$):

• The new radical $Y^\bullet$ re-initiates sluggishly.
• Both molecular weight and polymerization rate $R_p$ are decreased.

3. Inhibition ($k_{i,Y} \approx 0$):

• $Y^\bullet$ cannot re-initiate at all (e.g., stable nitroxy radicals like TEMPO, or hydroquinone reacting with oxygen).
• Polymerization completely ceases until the inhibitor is totally consumed.

Transfer Mechanisms in Polymerization Systems

• Transfer to Monomer ($M$): An inevitable physical ceiling on polymer molecular weight. For vinyl chloride, chain transfer to monomer ($C_M \sim 10^{-3}$) is so rapid that molecular weight is determined almost entirely by reaction temperature rather than initiator concentration.
• Transfer to Solvent ($S$): Solvents with weak $C-H$ or $C-Cl$ bonds (e.g., chloroform, carbon tetrachloride, toluene) act as strong chain transfer agents. For $\text{CCl}_4$, chlorine radical abstraction yields trichloromethyl ends ($-\text{CCl}_3$).
• Transfer to Initiator ($I$): Induced decomposition of peroxides.
• Transfer to Chain Transfer Agents (Modifiers / Regulators): Alkyl mercaptans (thiols, such as $n$-dodecyl mercaptan, $R-\text{SH}$) have exceptionally weak $S-H$ bonds ($E_a \approx 365\text{ kJ/mol}$), transferring rapidly to control melt viscosity in industrial rubber and acrylate synthesis.

In 1943, Frank R. Mayo derived the fundamental mathematical expression relating the reciprocal degree of polymerization to chain transfer processes.

Derivation of the Mayo Equation

The total rate of termination of macromolecular chains is the sum of bimolecular termination and all unimolecular chain transfer processes:

The reciprocal number-average degree of polymerization is:

Dividing each term:

where $X_{n,0}$ is the degree of polymerization in the absence of all chain transfer agents ($1/X_{n,0} = \frac{k_t R_p}{k_p^2 [M]^2}$).

Definition of Chain Transfer Constants

We define the dimensionless chain transfer constant $C_X$ as the ratio of the transfer rate constant to the propagation rate constant:

Substituting these constants yields the classical Mayo equation:

The Mayo Plot

To determine the chain transfer constant $C_S$ of a solvent:

1. Polymerizations are conducted at constant $[I]$ and constant temperature while varying the solvent-to-monomer ratio $[S]/[M]$.
2. A plot of $1/X_n$ on the y-axis against $[S]/[M]$ on the x-axis yields a straight line:

• For cyclohexane in styrene: $C_S = 3.1 \times 10^{-6}$ (very inert).
• For toluene in styrene: $C_S = 1.25 \times 10^{-5}$ (benzylic H abstraction).
• For carbon tetrachloride in styrene: $C_S = 1.10 \times 10^{-2}$ (strong transfer agent).
• For $n$-butyl mercaptan: $C_{CTA} = 21.0$ (instantaneous transfer).

In bulk or concentrated solution radical polymerizations (most famously observed with methyl methacrylate), as monomer conversion exceeds approximately $20 - 40\%$, the reaction displays a dramatic, sudden surge in both polymerization rate ($R_p$) and molecular weight ($M_n$), often accompanied by rapid temperature runaway. This phenomenon is known as autoacceleration, the Trommsdorff-Norrish effect, or the gel effect.

Physical Origin: Entanglement and Diffusion-Controlled Termination

Recall the steady-state rate equation:

and degree of polymerization:

Both $R_p$ and $X_n$ scale inversely with $\sqrt{k_t}$.

1. At Low Conversion ($p < 20\%$):

The reaction mixture is a low-viscosity liquid. Both small monomer molecules and large macroradicals diffuse freely. Termination rate constant $k_t \approx 10^7 - 10^8\text{ L/(mol s)}$ is constant.

2. At Intermediate Conversion ($p \approx 20 - 50\%$):

Polymer chains overlap and reach the entanglement threshold ($c^*$). Macroscopic melt viscosity increases by several orders of magnitude ($10^3 - 10^6\text{ cP}$).

• Termination requires two massive macroradicals to diffuse toward each other (center-of-mass translational diffusion, followed by segmental reptation diffusion) until their reactive tips collide.
• Because diffusion of large macromolecules is severely hindered in the viscous entangled gel, the termination rate constant $k_t$ drops catastrophically by 2 to 3 orders of magnitude ($k_t \to 10^4 - 10^5\text{ L/(mol s)}$).

3. Propagation Remains Unaffected:

Unlike macroradicals, small monomer molecules ($M$) diffuse readily through the entangled network mesh. Therefore, the propagation rate constant $k_p$ remains completely unchanged!

4. Kinetic Consequence:

Because $R_p \propto 1/\sqrt{k_t}$ and $X_n \propto 1/\sqrt{k_t}$, as $k_t$ plunges:

• Macroradical concentration $[M^\bullet]$ surges by a factor of 10 to 100.
• The polymerization rate $R_p$ surges dramatically.
• The molecular weight of the polymer formed during this phase is vastly higher than at low conversion.

The Glass Effect at High Conversion ($p > 80\%$)

If the polymerization temperature is below the glass transition temperature of the polymer ($T_{\text{poly}} < T_g$), as conversion approaches $80 - 90\%$, the entire reaction mixture solidifies into a rigid glass. At this stage, even small monomer diffusion is frozen ($k_p$ drops). The reaction halts prematurely at a limiting conversion ($p < 100\%$).

Chain-growth polymerization is a reversible chemical equilibrium between monomer and active polymer chain:

where $k_p$ is the forward propagation rate constant and $k_{\text{dp}}$ is the reverse depolymerization (unzipping) rate constant.

Thermodynamic Enthalpy and Entropy of Polymerization

The Gibbs free energy of polymerization is:

1. Enthalpy of Polymerization ($\Delta H_p < 0$):

Converting one carbon-carbon double bond ($\sigma + \pi$) into two carbon-carbon single bonds ($2\sigma$) releases approximately $60 - 90\text{ kJ/mol}$ of exothermic energy. Thus, almost all polymerizations are exothermic ($\Delta H_p < 0$).

2. Entropy of Polymerization ($\Delta S_p < 0$):

Condensing thousands of independently translating monomer molecules into a single covalently constrained macromolecule results in an enormous loss of translational and rotational degrees of freedom. Thus, $\Delta S_p$ is always strongly negative (typically $-100\text{ to }-130\text{ J/(mol K)}$).

The Ceiling Temperature ($T_c$)

Because $\Delta H_p < 0$ and $\Delta S_p < 0$:

• At low temperatures, the enthalpy term dominates: $\Delta G_p < 0$, and polymerization is thermodynamically favored.
• At high temperatures, the entropy penalty $-T \Delta S_p > 0$ dominates: $\Delta G_p > 0$, and depolymerization (unzipping) is thermodynamically favored!

At the ceiling temperature $T_c$, polymerization and depolymerization are at exact thermodynamic equilibrium ($\Delta G_p = 0$):

Rearranging yields the ceiling temperature formula:

For standard state $[M] = 1.0\text{ M}$:

For pure liquid monomer ($[M] = [M]_0$):

Equilibrium Monomer Concentration $[M]_{\text{eq}}$

At any reaction temperature $T$, there exists an equilibrium monomer concentration $[M]_{\text{eq}}$ below which polymerization cannot proceed:

• For styrene: $\Delta H_p^\circ = -70\text{ kJ/mol}, \Delta S_p^\circ = -105\text{ J/(mol K)} \implies T_c \approx 310^\circ\text{C}$ (far above operating temperatures).
• For $\alpha$-methylstyrene: Steric hindrance between the $\alpha$-methyl and phenyl group weakens the bond ($\Delta H_p^\circ = -35\text{ kJ/mol}, \Delta S_p^\circ = -110\text{ J/(mol K)}$), yielding $T_c = 61^\circ\text{C}$! At room temperature, only low conversion is possible; above $61^\circ\text{C}$, $\alpha$-methylstyrene cannot be polymerized at all!