Polymer length distributions for catalytic polymerization within mesoporous materials: Non-Markovian behavior associated with partial extrusion

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24.

Due to the recurrence of 1D random walks (Ref. 23), the end of the polymer is far more likely to return to the same catalyst site after

n_{h} \sim L_{c}

hops, rather than to reach an adjacent catalyst site after

n_{h} \sim {(L_{c})}^{2}

hops. The mean number of hops to reach any catalyst site satisfies

⟨ n_{h} ⟩ \approx L_{c}

⁠.

25.

The residence time of the end of the long polymer of length L at the catalytic site,

τ_{poly} = 1 / h (L)

far exceeds the characteristic time,

τ_{m} = 1 / h_{m} = 1

⁠, for monomer hopping. Thus, even for small monomer densities, there is usually sufficient time for monomers to reach the catalytic site and react before the polymer hops. As a result, typically one has that

P_{rx} \approx 1

⁠.

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27.

The loss term in

d / {dtL}_{av}

due to extrusion has the form

- P_{ex} L_{av} h (L_{av}) / ⟨ n_{h} ⟩ ∣_{L = Lav}

noting that the length of the polymer is reduced from

L \sim L_{av}

to

L \sim O (1)

upon extrusion.

28.

Using the notation in the Appendix, these Markovian equations have the form

d / dt N_{L} = \sum_{m > 0} p_{m} [P_{ret} (L - m) R_{ret} (L - m) N_{L - m} - P_{et} (L) R_{ret} (L) N_{L}] - P_{ex} (L) R_{ex} (L) N_{L}

⁠, for

L > 2

⁠, and

d / {dtN}_{2} = \sum_{L^{'} > 2} P_{ex} (L^{'}) R_{ex} (L^{'}) N_{L^{'}} - P_{ret} (2) R_{ret} (2) N_{2}

⁠. The equations also involve rates for return,

R_{ret} (L)

⁠, and extrusion,

R_{ex} (L)

⁠, of a polymer of length L.

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31.

From Ref. 30, the behavior of

F_{n}

also follows from that of the “survival probability” of not returning after n steps,

S_{n} \sim n^{- 1 / 2}

and the relation

S_{n} = 1 - \sum_{m \leq n} F_{n}

⁠.

32.

The generating function for

{nF}_{n}

satisfies

{zF}^{'} (z) \sim 2^{- 1 / 2} {(1 - z)}^{- 1 / 2}

as

z \to 1

⁠. Application of the discrete Tauberian Theorem (Ref. 23) implies that

n F_{n} \sim 2^{- 1 / 2} n^{- 1 / 2} / Γ (1 / 2)

⁠, for even

n \to \infty

⁠.

33.

The conditional mean return time sampling only those walks which return in the first n steps diverges like

{(2 n / π)}^{1 / 2}

(Ref. 23) consistent with stated large n-behavior of

F_{n}

⁠.

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