Average Degree of Polymerization

The average degree of polymerization (DP) is equal to the average number of reacted monomers per (inactive) polymer chain.  Assuming stationary conditions, the DP is given by:

DP = (d[M] / dt) / (d[P] / dt)

where [M] and [P] are the concentration of monomer and polymer. The amount of monomer consumed per time intervall, d[M] / dt, depends on the velocity constant of polymerization k_p and on the (stationary) concentration of chain radicals [R·] and monomer [M]:

d[M] / dt = k_p [M] [R·]

Unreactive polymers are produced via combination-disproportionation of chain radicals, and via radical transfer to a solvent (s) and/or monomer (m) molecule. The rate of formation of unreactive polymer, d[P] / dt, is given by^[1,2]

 d[P] / dt = k_fm [M] [R·] + k_fs [S] [R·] + (k_tr/2 + k_td) [R·]²

where [S] is the solvent concentration, k_t the velocity coefficient of chain termination via recombination (r) or disproportionation (d), and k_f the velocity coefficient of chain transfer to solvent (s) or monomer (m). Using these two relations, we obtain for the mean molecular chain length following expression:

 DP = k_p [M] [R·]  / {k_fm [M] [R·] + k_fs [S] [R·] + (k_tr/2 + k_td) [R·]²}

which on inversion becomes

 1 / DP = (k_tr/2 + k_td) [R·] / k_p [M] + k_fm / k_p +  k_fs [S] / k_p [M]
= K_T [R·] / [M] + C_M + C_S [S] / [M]

where C_M and C_S are the so-called transfer constants, which are defined by

  C_M = k_fm / k_p ;     C_S = k_fs / k_p ;    C_T = (k_tr/2 + k_td) / k_p ;

The average life time of a kinetic chain τ is equal to [R·] / I, where [R·] is the concentration of chain radicals in the stationary state and I is the rate of radical formation which is given by^[2]:

I = (k_tr + k_td) [R·]²

Making the substitution [R·] = {I/(k_tr + k_td)}^1/2, we obtain for the degree of polymerization:

 1 / DP = (k_tr/2 + k_td) I^1/2 / {(k_tr + k_td)^1/2 k_p [M]} + C_M + C_S [S] / [M]

In a cataylyzed polymerization the rate of initiation is often proportional to the concentration of catalyst, to be more specific, to the fraction of catalyst radicals f [Cat] which initiate polymerization:

I = 2f k_d [Cat]

where k_d is the reaction constant for a unimolecular decomposition of catalyst. Making the substitution

[R·] = {I/(k_tr + k_td)}^1/2 = {2f k_d [Cat] / (k_tr + k_td)}^1/2

we obtain another expression for the degree of polymerization:

 1 / DP = K_T {2f C_D [Cat]}^1/2 / [M] + C_M + C_S [S] / [M]

where C_D = k_d / (k_tr + k_td). Thus, the average degree of polymerization depends on the monomer and solvent concentration and on the velocity constants of radical formation, chain termination and chain transfer. In the absence of chain transfer the expression above can be written as

 1 / DP₀ = (k_tr/2 + k_td) I^1/2 / {(k_tr + k_td)^1/2 k_p [M]}
= K_T {2f C_D [Cat]}^1/2 / [M]

Therefore

 1 / DP = 1 / DP₀ + C_M + C_S [S] / [M]

The amount of chain transfer to monomers is usually small because this reaction requires breaking strong carbon-carbon or carbon-hydrogen bonds.¹ Thus

 1 / DP ≈ 1 / DP₀ + C_S [S] / [M]

This equation was first derived by Mayo (1943) and has proven to be very useful in the experimental determination of transfer constants C_S.

Notes

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