Statistical Copolymer Number Average Sequence Length

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For copolymers, the molecular weight and copolymer composition are macro scale characteristics that influence copolymer properties. The number-average sequence length, ${\overline{N}}_{M_1}$, also influences copolymer properties through defining the microstructure. Copolymer composition and number-average sequence length are often determined and/or checked by high-resolution NMR spectroscopy. The definition of ${\overline{N}}_{M_1}$ can be seen in (1).

${\overline{N}}_{M_1}\equiv \frac{totalnumberof{M}_{1}inthecopolymer}{totalnumberof{M}_{1}sequencesinthecopolymer}$

(1)

For a binary copolymer made via free radical polymerization, ${\overline{N}}_{M_1}$ can be shown through Markovian statistics.

The first-order Markovian or terminal model describes this as

$p_{11}$ is the probability of Failed to parse (syntax error): {\displaystyle \ce{ \sim\sim M1.+\ \ M1 ->[{}\atop k_{11}]\ \sim\sim M1. }}

$p_{12}$ is the probability of Failed to parse (syntax error): {\displaystyle \ce{ \sim\sim M1.+\ \ M2 ->[{}\atop k_{12}]\ \sim\sim M2. }}

$p_{21}$ is the probability of Failed to parse (syntax error): {\displaystyle \ce{ \sim\sim M2.+\ \ M1 ->[{}\atop k_{21}]\ \sim\sim M1. }}

$p_{22}$ is the probability of Failed to parse (syntax error): {\displaystyle \ce{ \sim\sim M2.+\ \ M2 ->[{}\atop k_{22}]\ \sim\sim M2. }}

The penultimate model (second-order Markovian) describes this as

$p_{111}$ is the probability of Failed to parse (syntax error): {\displaystyle \ce{ \sim\sim M1M1.+\ \ M1 ->[{}\atop k_{111}]\ \sim\sim M1M1. }}

$p_{211}$ is the probability of Failed to parse (syntax error): {\displaystyle \ce{ \sim\sim M2M1.+\ \ M1 ->[{}\atop k_{211}]\ \sim\sim M1M1. }} etc.

where $M_1$ is monomer 1, $\sim \sim \mathrm{M}{\phantom{A}}_1\cdot$ is a propagating radical, and $k_{11}$ is the rate constant of the propagation of these reactions and the rates of termination and initiation are assumed to be negligible. It is assumed that these polymer chains are infinitely long and therefore the reactivity of these chains with monomer is not dependent on chain length. This is an acceptable assumption for most copolymers but not to be used for those for $D{P}_n<200$.

Small Chain Description

The number-average chain length of the small copolymer below can be found using equation (1).

$M_2\underset{\_}{M_1}{M}_2{M}_2\underset{\_}{M_1{M}_1}{M}_2{M}_2{M}_2\underset{\_}{M_1{M}_1{M}_1}$

There are 6 $M_1$s in this copolymer and 3 sequences therefore the number-average sequence length, ${\overline{N}}_{M_1}$ is 2.

Terminal Binary Copolymerization

In general, the first-order Markovian model is sufficient to understand the behavior of binary copolymers, therefore the second-order influences are neglected in this derivation. To begin, $A_n$ is defined as the probability that an $M_1$ sequence has n $M_1$ units in it within a binary copolymer. Therefore,

$A_2=p_{11}{p}_{12}$

(2)

as $p_{11}$ is the probability that there is a 2 $M_1$ sequence and $p_{12}$ is the probability that that sequence is terminated.

$A_n=p_{11}^{n-1}{p}_{12}$

(3)

for a sequence length of $n$, $p_{11}$ is to the $n-1$ as it is for the probability of each additional $M_1$ after the first $M_1$. There is only one $p_{12}$ for any $n$ length sequence as the sequence will end with the addition of a single $M_2$. As this is a binary copolymer, $p_{11}+p_{12}=1$ therefore

$A_n=p_{11}^{n-1}(1-p_{11})$

(4)

To obtain the ${\overline{N}}_{M_1}$ of the entire copolymer rather than the probability of one sequence:

${\overline{N}}_{M_1}=\sum _{n=1}^{\mathrm{\infty }}{A}_{n}n$

(5)

${\overline{N}}_{M_1}=\sum _{n=1}^{\mathrm{\infty }}{p}_{11}^{n-1}(1-p_{11})n$

(6)

${\overline{N}}_{M_1}=(1-p_{11})\sum _{n=1}^{\mathrm{\infty }}{p}_{11}^{n-1}n$

(7)

where $(1-p_{11})$ may be removed from the summation as it is constant across all $n-1$.

$\sum _{n=1}^{\mathrm{\infty }}{p}_{11}^{n-1}n$ converges to $\frac{1}{(1-p_{11}{)}^2}$ for $p_{11}<1$

(8)

Therefore,

${\overline{N}}_{M_1}=\frac{1}{(1-p_{11})}$

(9)

Previously defined qualitatively in the introduction as $p_{11}$ is the probability that $\sim \sim M_1\cdot$ will propagate with $[M_1]$ rather than $[M_2]$,

$p_{11}=\frac{k_{11}[M_1\cdot ][M_1]}{k_{11}[M_1\cdot ][M_1]+k_{12}[M_1\cdot ][M_2]}$

(10)

$p_{11}=\frac{r_1{f}_1}{r_1{f}_1+f_2}$

(11)

as $r_1=\frac{k_{11}}{k_{12}}$ and $r_2=\frac{k_{22}}{k_{21}}$ where $r_1$ and $r_2$ are the reactivity ratios and $f_1=\frac{[M_1]}{[M_1]+[M_2]}$ and $f_2=\frac{[M_2]}{[M_1]+[M_2]}$ where $f_1$ and $f_2$ are the fractional compositions of the monomer mix. Therefore, by plugging (11) into (9)

${\overline{N}}_{M_1}=\frac{1}{1-\frac{r_1{f}_1}{r_1{f}_1+f_2}}$

(12)

${\overline{N}}_{M_1}=\frac{1}{\frac{r_1{f}_1+f_2-r_1{f}_1}{r_1{f}_1+f_2}}$

(13)

${\overline{N}}_{M_1}=\frac{r_1{f}_1+f_2}{f_2}$

(14)

Therefore,

${\overline{N}}_{M_1}=1+r_1\frac{f_1}{f_2}=1+r_1\frac{[M_1]}{[M_2]}$

(15)

${\overline{N}}_{M_2}$ can be derived in the same manner where $p_{22}=\frac{r_2{f}_2}{f_1+r_2{f}_2}$:

${\overline{N}}_{M_2}=1+r_2\frac{f_2}{f_1}=1+r_2\frac{[M_2]}{[M_1]}$

(16)

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The values of $r_1$,$r_2$, $f_1$ and $f_2$ can be derived experimentally, using NMR for the reactivity ratios, and therefore a number average sequence length can be determined for a copolymer.

Run Number

The run number, $N_R$ is the averaged number of sequences of either monomer per 100 monomer units. A perfect alternating copolymer, where $r_1=r_2=1$, has an $N_R$ of 100. The higher the $N_R$, the greater the alternation in the polymer.

References

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