Kinetics of Chain Transfer Reactions

The simple reaction scheme for vinyl polymerization can be modified to include chain transfer termination steps by which the free-radical site of a growing polymer chain is transferred to another species like a solvent (S) or monomer (M). In the following, P_r and R_r· represent a polymer and a polymer radical comprising r monomer units, respectively.

Initiation:

I → 2 R_I·   (I: Rate of Initiation)
R_I· + M → R₂·

Propagation:

R_r· + M → R_r+1·   (k_p)

Transfer to monomer:

R_r· + M → P_r + M·   (k_fm)

Transfer to solvent (regulator):

R_r· + S → P_r + S·   (k_fs)

Reinitiation by transfer radical:

S· + M → R₁·   (k_ps)
M· + M → R₂·   (k_pm)

Termination:

R_r· + R_s· → P_r+s·   (k_tr)
R_r· + R_s· → P_r + P_s·   (k_td)

Assuming stationary conditions, the rate of reactions of free radicals and regulator is given by:

d[R·] / dt = I - k_fm [M] [R·] - k_fs [S] [R·] + k_ps [S·] [M] + k_pm [M] [R·]
 - (k_tr + k_td) [R·]² = 0                                             

d[S·] / dt = k_fs [S] [R·] - k_ps [S·] [M] = 0

d[M·] / dt = k_fm [M] [R·] - k_pm [M·] [M] = 0

where [R·] is the total polymer radical concentration, and [M], [S] are the concentration of the of monomer and solvent, respectively. If we combine all three equations, we find

I = - (k_tr + k_td) [R·]²

d[S·] / d[M·] ≈ k_fs [S] / k_p [M] = 0

Thus a simple transfer reaction has little or no effect on the overall rate of polymerization. However, chain transfer has a strong effect on the average polymer weight. Furthermore, the amount of chain transfer to monomers is usually quite low because this reaction requires breaking strong carbon-hydrogen bonds.¹

The quatity

d log[S] / d log[M] = k_fs / k_p = C_s

is called the chain transfer constant. It is the slope of the plot of log[S] against log[M].

In the equation above, we neglected reinitiation by radical transfer to polymer chains and assumed that the solvent radicals S· do not participate in termination reactions of the type:

S· + R_r· → P_r+s·   (k_tx)
S· + S· → Q·   (k_ts)

These assumptions are only valid if the monomer concentration is much larger than the polymer concentration and if the concentration of solvent (regulator) radicals is small and/or if their reactivity constant (k_ps) is large, that is, k_ps has to be at least comparable to k_p. If k_ps is much smaller than k_p, which would be the case for comparatively stable radicals, the termination reactions of solvent radicals can not be neglected. For this case, the stationary reactions that involve solvent molecules read

d[R·] / dt = I - k_fm [M] [R·] - k_fs [S] [R·] + k_ps [S·] [M] + k_pm [M] [R·]
 - (k_tr + k_td) [R·]² - k_tx [S·] [R·] = 0                        

d[S·] / dt = k_fs [S] [R·] - k_ps [S·] [M] - k_ts [S·]² - k_tx [S·] [R·] = 0

Unfortunatley, these equations can not be easily verified, because the concentration of the different types of radicals is difficult to measure. Obviously, the additional termination reactions would reduce the rate of polymerization. However, for typical (potent) chain transfer agents it is reasonable to assume that termination reactions involving regulator molecules are negligible. This is the case if the overall reactivity constant for polymerization is independent of the concentration of regulator [S].

Chain transfer depends on the type of transfer agent and on the temperature and polymer type, whereas the viscosity (i.e. polymer concentration, degree of polymerization) and polymerization rate¹ have no or only little effect on the rate of chain transfer. An increase in temperature almost always increases the amount of chain transfer, thus lowering the molecular weight.

Notes

1. There are some exception. For example, vinyl chloride has a rather large chain transfer constant when compared to most other vinyl monomers.