Free Radical Polymerization
Part 2: Rate of Polymerization

The classical free radical polymerization kinetic is often described using the steady-state method. This method is based on the assumption that the concentration of the radical intermediates stays constant throughout the polymerization. In other words, the rate of generation of free radicals is equal to the rate of radical termination:

Ri = Rt ⇒ d[M*] / dt = 0

This is usually true after a short initiation phase which typically lasts a minute or so. Assuming unimolecular decomposition of intiators and bimolecular termination reactions,1 the total concentration of all chain radicals [M*] is given by

[M*] = (f kd [I] / kt)½

Substitution of this expression into the general rate equation of vinyl polymerization yields5

Rp = kp (f kd / kt)½ [I]½ [M] = K [I]½ [M]

This equation is known as the "square-root law". It predicts that the rate of monomer consumption (polymer growth) in free-radical polymerization is first order in monomer concentration but only half order in initiator concentration. Thus, doubling the amount of initiator will not double the rate of polymerization but only increase it by a factor of √2 ≈ 1.4.

The quantity kp2 (f kd / kt) can be estimated from the rate of polymerization, Rp, and the measured concentration of monomer [M] and initiator [I]:

Rp / [I] [M]2 = kp2 (f kd / kt)

If the efficiency f and rate constant for (spontaneous) decomposition kd are known, kp2/ kt can be claculated from the observed polymerization rate Rp at a given monomer [M] and initiator concentration [I].

The initiator concentration [I] does not change very much during propagation because termination reactions are rare events. Thus, a free radical polymerization follows approximately first order kinetics.2

Polymerization of undiluted vinyl monomers and of monomers in concentrated solution (> 40 - 50%) shows a noticeable deviation from first-order kinetics at a conversion of about 25 percent.2 At this point, the rate of conversion increases markedly which is known as the gel or Trommsdorff effect. The effect is particularly pronounced with (meth)acrylic esters, but has also been observed with other monomers such as styrenes and vinyl acetate, but the effect is less pronounced. Norrish and Smith3, and Trommsdorff4 postulated that the drastic increase in the rate of polymerization is caused by a noticeable decrease in the rate of termination due to a dramatic increase in viscosity which drastically slows down chain termination reactions. Hence, a drastic increase of the ratio kp2/ kt (as much as a hundredfold) is observed as the polymerization proceeds.

An important quantity in free radical polymerization is the kinetic chain length ν, which is defined as the average number of monomers consumed for each radical that initiates polymer growth until chain termination. This quantity is given by the ratio of rate of propagation Rp to rate of initiation Ri. The later is equal to the rate of termination Rt under steady state conditions:

ν = Rp / Ri = Rp / Rt 

ν = kp [M] [M*] / 2 kt [M*]2 = kp [M] / 2 kt [M*] ⇒

or

ν = (kp/ 2kt ) [M] / [M*] = (kp2/ 2kt ) [M]2 / Rp

The kinetic chain length is related to the number-average degree of polymerization (DP), which is the average number of monomers consumed per polymer chain formed. The final length of a polymer chain will depend on side reactions (chain-transfer) and the type of termination reaction. For example, termination by recombination will double the final polymer chain length:

DP = 2 Rp / Ri = 2ν

whereas disproportionation will not change the average degree of polymerization:

DP = Rp / Ri = ν

Chain transfer, on the other hand, will lower the average degree of polymerization:

DP = Rp / (Rt + Rtr) < ν = Rp / Rt

where Rtr is the rate of transfer. Hence, the more effective chain transfer agents (greater Rtr), the shorter the final polymer chain length.

Notes & References

  1. Termination can occur via recombination or disproportionation. Since both reactions follow the same kinetic, the mechanism does not have to be specified.

  2. Paul J. Flory, Principles of Polymer Chemistry, 1st Edition 1953 Cornell University

  3. R. Norrish and R.  Smith, Nature 150, 336 (1951)

  4. E. Trommsdorff, H Koehle, and P Lagally, Makromol. Chem. 1, 169 (1948)

  5. D. Braun, Int. J. Poly. Sci., Vol. 2009, 893234/1-10

  • Summary

    The process of chain initiation involves two steps; the first being the decomposition of the initiator and the second the addition of a monomer to the primary radical R· to yield the chain radical.

  • The free radicals can be produced by thermal, catalytical or photochemical decomposition of organic peroxides, hydroperoxides, azo and diazo compounds.

  • The growth of the polymer chains occur by successive addition of monomers.

  • The general assumption is that the radical reactivity is independent of the chain length.

  • Termination occurs via combination or disproportionation.

  • In the case of combination or coupling, two growing polymer chains react with each other to a single nonreactive polymer chain.

  • In the case of disproportionation, a hydrogen is transferred from one radical to the other resulting in two nonreactive polymer chains.

  • The molecular weight increases rapidly at early stage and remains approx. the same throughout the polymerization.

  • Some residual monomer remains even after long reaction times.

  • Transfer reactions to other polymer chains are usually rare events and transfer reactions to solvent can be avoided by selecting a suitable solvent.

  • With respect to the radical concentration, the propagation is first order, while the radical termination is second order.