Kinetic Chain Length And Poisson Distribution of Living Polymerization

In carefully controlled systems, an anionic polymerization does not undergo any termination reaction. This type of polymerization is called living polymerization. It results in narrow molecular weight distribution because the rate of polymerization is approximately the same for all active chains and chain growth of all growing chains is initiated at about the same time when the initiator is added.

If RMe is the initiater and M the monomer, then the individual steps of the living  polymerization can be written as follows:

RMe ⇔ Me⁺ + R^-

R^- + M → RM^-

RM_n-1^- + M → RM_n^-

The efficiency of the initiator is usually very high and close to one. Assuming each initiator starts chain growth the final (average) chain length, ⟨x_avg〉, and degree of polymerization, ⟨M_n〉, after all monomer is consumed is then 

⟨x_avg⟩ = 1 + [M₀] / [In₀]

⟨M_n〉 = ⟨x_avg〉 MW₀

where [In] is the total molar concentration of initiator, [M₀] is the initial molar monomer concentration and MW₀ is its molecular weight. The rate of monomer consumption of a living polymerization is given by¹

[M](t) ≈ [M₀] exp(-k_r [In₀] t)

which implies instantaneous initiation of all chains uppon addition of catalyst. The kinetic chain length 〈x(t)〉 is the chain length or average number of repeat units in the growing chains. It can be calculated as follows

〈x(t)〉 ={[M₀] - [M](t)} / [In₀] ≈ [M₀] / [In₀] · {1 - exp(-k_r [In₀] t)}

and

d〈x(t)〉/dt ≈  k_r [M₀] · exp(-k_r [In₀] t) = k_r [M](t)

The conditions at the start of the polymerization are ⟨x_avg〉 = 1 and 〈x(t)〉 = 0 and as the polymerization proceeds, 〈x(t)〉 asymptotically approaches ⟨x_avg〉. To find the distribution function of the chain radicals, we first have to calculate the molar concentration of initiator anion [R^-] and active chain anions [RM_x^-]  as function of time. Assuming all initiator radicals R^- are produced immediately after injected into the reactor at t ≈ 0, the rate of consumption of [R^-] as function of time t is calculated as follows

d[R^-]/dt = d[R^-]/d〈x〉 · d〈x〉/dt ≈ k_r · [M] d[R^-]/d〈x〉 = -k_i [M] [R^-]

 [R^-](t) = [In₀] · exp[-〈x(t)〉]

where we made the assumption k_i ≈  k_r, and [R^-](t = 0) = [In₀] which is not necessarily always the case. In a similar manner we find for the active carbanion concentration

d[RM^-]/dt  ≈ k_r · [M] d[RM^-]/d〈x〉 = k_i [M] [R^-] - k_r [M] [RM^-] ≈ k_r [M] {[R^-] - [RM^-]}

Thus

d[RM^-]/d〈x〉 + [RM^-] ≈ [R^-] = [In₀] · exp(-〈x〉)
exp(〈x〉) {d[RM^-]/d〈x〉 + [RM^-]} = d/d〈x〉 {[RM^-]exp(〈x〉)} ≈ [In₀]
[RM^-] = [In₀] 〈x〉 exp[-〈x〉]

This result can be generalized for any carbanion. For x = 2,..,n the expressions for the carbanions [RM_x^-] read

[RM₂^-] = ½ [In₀] exp[-〈x〉] 〈x〉² 
[RM_n-1^-] = [In₀] exp[-〈x〉] 〈x〉^n-1 / (1-n)!

The mole fraction of chains that consist of n repeat units is the ratio of its molar density [RM_n-1^-] to the molar density of all growing chains:

P(x) = [RM_x-1^-] / ∑_y [RM_y^-] = [RM_x-1^-] / [In₀] = exp[-〈x〉] 〈x〉^x-1 / (1-x)!

This result has the form of a discrete Poisson distribution which is  much narrower than the molar fraction distribution of free radical and condensation polymerization.

Notes

1. This expression is only valid if the kinetic rate constant of initiation is much larger than that of propagtion, k_i » k_r.