Statistical Theory

With Carothers theory, we were dealing with number-average quantities. But how do we predict the molar mass distributions? If we treat each reaction between A and B groups in a step of polymerization as a random event, we can start to apply probability and statistics to the reaction and get an estimation of the distribution of the degrees of polymerization of the polymers formed for a given extent of reaction. Essentially, we are trying to find $P (x)$ , which is the probability of existence of a molecule containing exactly x units at time t with extent of reaction p. Without going into the derivation here, we find the probability of finding a sequence of x units:

$P (x) = (1 - p) p^{(x - 1)}$

Since $P (x)$ is the probability of finding an “x-mer”, it must also be representative of the mole fraction of x-mers:

$m o l f r a c t i o n x-mer = \frac{N_{x}}{N} = (1 - p) p^{(x - 1)}$

Where $N_{x}$ is number of x-mers and $N$ is total number of molecules at time $t$ . Often, it’s hard to measure the number of molecules present at any given time $(N)$ , but it’s easy to find $N_{0}$ , which is the number of molecules we initially start with . Can we substitute in $N_{0}$ ? In order to relate $N$ to $N_{0}$ we use a familiar equation:

${\bar{x}}_{n} = \frac{1}{1 - p} = \frac{N_{0}}{N}$

If we solve this above equation for $N$ , we find $N = N_{0} (1 - p)$ . We substitute this relationship into $N_{x} = N (1 - p) p^{(x - 1)}$ and find an expression for the number of “x-mers” at a given extent of reaction:

$N_{x} = N_{0} {(1 - p)}^{2} p^{(x - 1)}$

We may also be interested in the weight fraction of x-mers, in addition to the number fraction. How do we solve for that? We start with the standard definition of weight fraction:

$w_{x} = \frac{m a s s o f m o l e c u l e s w i t h d e g r e e o f p o l y m e r i z a t i o n x}{t o t a l m a s s o f a l l m o l e c u l e s}$

$= \frac{N_{x} (x {\bar{M}}_{0})}{N_{0} {\bar{M}}_{0}} = \frac{x N_{x}}{N_{0}}$

We substitute into this with $\frac{N_{x}}{N_{0}} = {(1 - p)}^{2} p^{(x - 1)}$ and get the weight fraction of x-mers at a given extent of reaction:

$w_{x} = x {(1 - p)}^{2} p^{(x - 1)}$

PROBLEM

You are polymerizing the below monomers (1 mole of each) by step growth polymerization. When you stop the reaction, there are 0.1 moles of acid chloride groups left. What weight fraction of monomer do you have left?

molecular diagrams of Phenolphthalein and terephthaloyl chloride (TCL)

Phenolphthalein (left) and terephthaloyl chloride (TCL) (right)

Source: Lauren Zarzar

Click here for the answer.

ANSWER

First, we find the extent of reaction:

$p = \frac{N_{A}}{N_{B}} = \frac{(2 m o l - 0.1 m o l)}{(2 m o l)} = 0.95$

Then we can find the weight fraction of monomer (because x=1 is monomer!)

$w_{x} = x {(1 - p)}^{2} p^{(x - 1)}$

$= 1 {(1 - 0.95)}^{2} {0.95}^{(1 - 1)}$

$= 0.0025$

PROBLEM 2

You are polymerizing the below monomers (1 mole of each) by step growth polymerization. You stop the reaction when it's 99% complete. What fraction of the mixture is monomer, on a per mole basis?

Phenolphthalein (left) and terephthaloyl chloride (TCL) (right)

Source: Lauren Zarzar

Click here for the answer.

ANSWER 2

We can plug directly into the $P (x)$ equation:

$P (x) = (1 - p) p^{(x - 1)}$

$= (1 - 0.99) {0.99}^{(1 - 1)}$

We can use these relationships we have learned from the statistical theory to plot the most probable distributions for $P (x)$ and $w_{x}$ :

Enter image and alt text here. No sizes!

Figure 3.5: Probable distributions for

P (x)

and

w_{x}

Source: Figure 3.1 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.

What can we learn from these plots? First, consider the plot of $P (x)$ for a few different values of extent of reaction (which is the left graph). Notice that $P (x)$ , even for the high extents of reaction, is always larger for the lower values of $x$ . This is telling us that even if we go to high extents of reaction, polymers with low degrees of polymerization are still more likely to be found than those with high degrees of polymerization. Now look at the second graph, at right, which is a plot of $w_{x}$ . In contrast, this plot has a peak, which is actually very close to $M_{n}$ . As extent of reaction increases, we see that the peak weight fraction moves to high values of $x$ – this makes sense, because the polymers with higher degrees of polymerization are formed at higher extents of reaction. We also notice that the breadth of the curve increases as extent of reaction increases, which tells us that the dispersity of the polymer is increasing.

From these distributions, we can solve for the number average molar mass using this relationship:

${\bar{M}}_{n} = \sum P (x) M_{x}$

Doing a series of substitutions (i.e. substitute in $P (x) = (1 - p) p^{(x - 1)}$ , $M_{x} = x {\bar{M}}_{0}$ , and then simplify) we find:

${\bar{M}}_{n} = \frac{{\bar{M}}_{0}}{1 - p}$

Does this look familiar? It’s the same relationship as we got from Carothers theory!

Without going through the derivation, we can apply a similar analysis to get information about the weight average molar mass as well. We find:

${\bar{M}}_{w} = {\bar{M}}_{0} \frac{1 + p}{1 - p}$ and ${\bar{x}}_{w} = \frac{1 + p}{1 - p}$ and $d i s p e r s i t y = 1 + p$