Propagation and Termination

Introduction

Once initiation has created radical chain ends, propagation consumes monomer and termination removes radicals. The balance between those events controls both the polymerisation rate and average chain length. The steady-state approximation provides a useful simple rate expression. Its assumptions must be stated before using it quantitatively.

Core explanation

Write a growing radical as P·. During propagation it adds one monomer M: P· + M → PM·. The radical remains at the new end, so the monomer-consumption rate is R p = k p[M][P·] in a simple homogeneous model. Termination can occur by combination: two chain radicals join to form one longer dead chain. Alternatively, disproportionation transfers a hydrogen between two chain radicals, leaving two separate dead chains, commonly one saturated end and one unsaturated end. Both pathways consume two radical ends per event, but they produce different numbers and end structures of dead molecules. If radical production is roughly steady, the rate of initiation balances radical disappearance through termination. With termination event rate k t[P·]² and two radicals lost per event, R i = 2k t[P·]², so [P·] = √(R i/(2k t)) and R p = k p[M]√(R i/(2k t)). Factor-of-two conventions differ across textbooks depending on whether k t denotes event or radical-disappearance rate, so always define it. The steady state applies after a brief start-up interval and before large changes in monomer concentration, viscosity or oxygen content. Near high conversion, diffusion-limited termination can alter the rate strongly; this is sometimes called autoacceleration. Chain transfer also competes with propagation and affects molar mass without necessarily removing all radicals from the system.

Step-by-step reasoning

Write initiation as a source of chain radicals. Write propagation and identify the monomer-consumption rate. Draw combination and disproportionation separately. For a steady state, equate radical production to radical disappearance with an explicit factor-two convention, solve for [P·], then substitute into R p.

Visual explanation

Draw two radical-tipped chains meeting. In the combination branch they become one continuous chain. In the disproportionation branch a hydrogen moves between them, giving two separate chains, one with an alkene end.

Real-world analogy

Two relay runners can link arms and finish as one pair, or pass a token and finish separately. Both actions remove the original active runner state, but the number of final teams differs, as with radical combination and disproportionation.

Real-world example

Industrial radical polymerisations control temperature and initiator supply to regulate chain number and heat generation. When termination slows in a viscous medium, radical concentration can rise and increase the polymerisation rate unexpectedly.

Why?

The radical is regenerated by each propagation step, but termination destroys active ends. Steady state expresses a dynamic balance, not a stop in reaction: radicals are continuously born and consumed while their concentration stays approximately constant.

Common misconception

Disproportionation does not combine two chains into one. It removes two radicals by hydrogen transfer but leaves two separate dead polymer molecules. Also, steady state does not imply that monomer concentration is constant forever.

Worked example

Question: Under an event-rate convention R t,event = k t[P·]², initiation produces radicals at 0.004 mol L⁻¹ s⁻¹. What radical-disappearance rate is needed at steady state? Reasoning: The radical concentration is steady when disappearance balances production, and each termination event removes two radicals. Answer: Radical disappearance is 0.004 mol L⁻¹ s⁻¹, corresponding to 0.002 mol L⁻¹ s⁻¹ termination events.

Quick check

1. Does propagation consume the active radical permanently? Answer: No. Addition moves the radical to the new chain end, allowing further propagation.

Exam focus

Show products for each termination mechanism and define whether k t counts events or radical loss. In rate derivations, the square-root dependence of radical concentration on initiation rate follows from bimolecular termination.

Advanced insight

At high conversion, chain diffusion becomes slow and termination can decrease more sharply than propagation. Radicals then accumulate, causing autoacceleration and potentially a heat-management hazard in bulk polymerisation.

Summary

Propagation adds monomer while preserving a chain radical; combination and disproportionation terminate two radicals by different product routes. A simple steady state balances R i with 2k t[P·]² under an event-rate convention, yielding R p = k p[M][P·].

Practice questions

1. What is the propagation rate in the simple radical model? Answer: R p = k p[M][P·], where [P·] is total active radical-chain concentration.

2. How many dead polymer molecules form when two radicals terminate by combination? Answer: One joined molecule.

3. How many form by disproportionation? Answer: Two separate dead chains after hydrogen transfer.

4. What does steady state mean for radical concentration? Answer: Radical creation and disappearance balance approximately, so [P·] changes slowly during the interval considered.