Kinetics of Isothermal Depolymerization

When vinyl polymers are exposed to high temperatures they often undergo (rapid) depolymerization reactions (also known as chain depropagation). Similar to vinyl polymerization, free radical initiated depropagation can be divided into three stages:

I Initiation
Depolymerization starts with homolytic breaking of bonds either at random positions in the chain backbone or at the chain ends with sequential removal of monomers from:

P_n → R_r· + R_n-r· (Random Scission, k_s)
P_n → R_n-1· + R₁· (End-chain Scission, k_e)

In the case of end initiation, the rate of unzipping (monomer formation) and the rate of consumption of polymer with degree of polymerization n is directly proportional to the number of polymers P_n:¹

R_e = dP_n / dt = - k_e · 2 · P_n

Each initiation reaction produces a radical of length n - 1, thus

dR_n / dt = k_e · 2 · P_n+1

where n is the degree of polymerization, 2 P_n is the number of polymer chain ends of length n and k_e is the rate constant of unzipping.

In the case of random chain scission, the rate of polymer scission is proportional to both the number of polymers, P_n, in the sample and to the number of bonds per polymer (n - 1):

R_s = dP_n / dt = - k_s · (n - 1) · P_n

where k_s is the rate constant for chain scission and R_s is the rate of change in polymer of length n due to chain scission.

Each chain scission leads to the formation of two radicals. Thus, the rate of formation of chain radicals of length n is directly proportional to twice the total number of polymers of length m ≥ n + 1 in the sample because there are two possible ways to cut a chain into two pieces with one of length n

 dR_n / dt = - 2 k_s · ∑_{i > n} P_i

II Depropagation
Once radicals have formed, the polymers will undergo rapid depropagation. There are three types of depropagation reactions:

 R_s· → R_s-n· +  P_n  (Intramolecular H Transfer, k_p1)
R_s· + P_n → P_s + R_n-r· + P_r (Intermolecular H Transfer, k_i)
R_n· → R_n-1· + M  (unzipping or depropagation, k_p2)

The first of these reaction involves the transfer of a hydrogen atom from one side to another side within a single polymer molecule, i.e. intramolecular chain transfer whereas the second reaction involves the exchange of a hydrogen atom between two different polymers, i.e. intermolecular hydrogen transfer. The later reaction is the
the reverse of a propagation, called depropagation or unzipping.

The rate of change in chain radicals due to unzipping is given by

dR_n / dt = k_p · R_n

This process will not change the concentration of polymers since unzipping produces only monomers and shorter chain radicals. However, a radical R_s can also react with a polymer P_n (intermolecular chain transfer). This reaction consumes a radical R_s  and a polymers P_n and creates two new polymers P_r and P_s as well as one new radical R_n-r.

R_s· + P_n → P_s + R_n-r· + P_r

The rate of change in the number of polymers P_n is proportional to the total number of radicals R = ∑_i R_i and to the number of repeat units in the sample n P_n,

 dP_n / dt = - k_i1 · R · n · P_n

Some other intermolecular hydrogen transfer reactions convert radicals of length n into polymers of same length:

R_n· + P_s → P_n + R_s-r· + P_r

The rate of change in the number of polymers P_n due to this reaction is proportional to the number of radicals in the sample R_n and to the total number of repeat units in the polymer molecules per unit volume:

 dP_n / dt =  k_i1 · R_n · ∑_i i · P_i

A third type of intermolecular hydrogen transfer reaction produces polymer chains of length n via chain scission.

R_r· + P_s → P_r + R_s-n· + P_n

The rate of change of polymer P_n is proportional to the total number of radicals R in the sample and to the overall number of polymers in the sample that can form chains of length n:

dP_n / dt =  k_i1 · R · 2 ∑_i>n  P_i

Intramolecular hydrogen transfer produces new polymer chains by chain scission. The rate of change of polymer P_n due to this reaction is proportional to the total number of radicals R that can split into a polymer of length n and another radical: 

dP_n / dt =  k_i2 · 2 ∑_i>n  R_i

III Termination
The classical termination step is the recombination of two chain radicals. This reaction can be considered the reverse
of a random chain scission. The other termination step is dispropotionation which involves the transfer of a hydrogen atom from one chain radical to another chain radical.

R_r· + R_s· → P_r+s   (Recombination, k_tr)
R_n· + R_s· → P_n + P_s   (Disproportionation, (k_td)

Chain termination reactions are usually of second order. In the case of disproportionation, the change in polymer concentration P_n will be proportional to R_n and to the total concentration of radical molecules:

dP_n / dt = - dR_n / dt =  k_td · R_n · R

And in the case of second order recombination

R_t = dP_n / dt = - dR_n / dt = ½ · k_tr · ∑_i+j=n  R_i · R_j

The total change in concentration of polymer P_n is the difference between the rate of formation and that of consumption of P_n. Assuming the termination step is disproportionation and propagation occurs by unzipping and intermolecular hydrogen transfer, the total change in concetration can be written  

dP_n / dt = - k_e 2 P_n - k_s (n - 1) P_n -  k_i1 R (n - 1) P_n +
      k_i1 R_n  ∑_i i P_i + k_i1 R 2 ∑_i>n P_i + k_td R_n R

It can be shown that

ρ / M_R = ∑_i i P_i

where ρ is the density of the polymers and M_R the mass of a repeat unit. Rearranging the equation above, yields

dP_n / dt = - k_e 2 P_n - (k_i1 R + k_s) (n - 1) P_n +
      k_i1 R_n  ρ / M_R + k_i1 R 2 ∑_i>n P_i + k_td R_n R

The rate of weight loss can be calculating by multiplying this expression by n and summing over all chain lengths. Often the depolymerization reaction is dominated by chain scission or unzipping. In the case of unzipping with recombination as the termination step, the total change in the number of reepat units is given by  

dM / dt ≈ - 2 k_e M

where N = ∑_n n P_n is the total number of repeat units and M = M_R N is the total polymer mass. If, on the other hand, random chain scission is the main initiation step, the equation for weight loss reads

 dM / dt ≈ -  k_s DP M

where DP is the number average degree of polymerization which is equal to the average number of repeat units per polymer chain.

Notes & References

1. Both the volume and the total mass change during depolymerization when volatiles are released. Therefore, the remaining weight / moles of polymer (divided by the initial weight / moles) is usually recorded rather than changes in concentration.  

2. The same is true for polymers where all hydrogen atoms have been replaced with atoms that form strong bonds with the backbone carbon atoms. A well known example is polytetrafluoroethylene (PTFE) which decomposes mainly via end-chain scission (unzipping).

3. C.L. Beyler and M.M. Hirschler, Thermal Decomposition of Polymers in SFPE Handbook of Fire Protection Engineering 2, Chapter 7, 110-131 (2002).

4. C.H. Bamford, C.F.H. Tipper, Comprehensive Chemical Kinetiks, Vol. 14, Degradation of Polymers, Elsevier, New York, 1975

5. A. Brzozowsja-Stanuch, S. Rabiej, J. Fabia, J Nowak, Polimery, 59, p. 302-307 (2014)

6. (2) S.D. Lee, Z. Peng, L.X. Kong and J.P. Zhong, J. Nanosci. Nanotech., Vol. 6, No. 2 (2006)