Molecular Weight Distributions

On the previous page, we observe a very important phenomenon. When we perform polymerization reactions, we get chains of many different sizes. Indeed, we can control the sizes of those chains by (for example) limiting the number of initiators. This is one way that folks that work with polymers control the distributions of weights of the polymer chains. This is useful, remember, because this allows us to control properties. For example, polyethelene with a bunch of small molecular weights will not stretch much or entangle much. It will be a weak material polymer. Large molecular weights will entangle more - strengthening the material.

So, we better be able to control the size of the molecules, as well as communicate about it! We saw these distributions developing in our simulations - especially NetLogo model 9.7.2. They seemed to look - in general - like the one below in which we had few short polymer chains, a fair amount in some middle molecular weight range, and then few again at high molecular weight. Figure 9.8.1 shows a typical distribution - with a few ways to quantify this distribution - number-average and weight-average molecular weights, discussed below.

When we talk about molecular weight distributions in polymers, we often use a metric called "molecular weight", denoted $M$. This is a number that tells us about the average weight of the polymer chains. We can have high molecular weight polymers (like in the simulation where we had few initiators) and low molecular weight polymers (when we had lots of initiators. There's a few ways to measure this, and it is important to differentiate them. For example - we can measure the number-average molecular weight, $\bar{M}_n$ which is simply the total weight of all the polymer chains divided by the number of polymer chains:

where $n_i$ is the number of chains in some range $i$, and $M_i$ is the mean molecular weight in that range. One can also write this as

Where $x_i$ is the number fraction of polymer chains in range $i$.

The weight-average molecular weight $\bar{M}_w$ is when, instead of looking at the number fraction (i.e. 10 of 100 chains have a length within range $i$, you look at weight fraction $w_i$ (i.e. the fraction of the weight in the polymer in range $i$ is 0.2 of all the weight):

or,

To reach this expression we simple replaced the $n_i$ term in Eq. 9.8.1 - the number of molecules in range $i$ - with $N_i M_i$, which accounts for not just the number of molecules in range $i$, but their weights.

Molecular Weights in Materials Science

It may make sense in basketball to be able compute a number that might help you predict rebounding performance, but how does this help in materials science?

It turns out that certain materials behavior are more dependent on the weight fractions of the large molecule. Some of these you might already be able to predict: mechanical properties like strength (resistance to permanent deformation) and stiffness (resistance to elastic deformation) are highly dependent on the fraction of polymer chains with large molecular weights because those are the ones that entangle with each other. So, when thinking about mechanical properties we need to consider weight-average molecular weights $\bar{M}_n$. Alternatively, the number average molecular weight is more important when we're considering properties that depend on the number of polymer chains, but not the weight - specifically the behavior of polymers in melt and during during reaction (reactions depend on number of reacting chains, not their masses).

Importantly, different characterization methods give different molecular weight values - chromatography is sensitive to $\bar{M}_n$ while light scattering techniques are dependent on $\bar{M}_w$.

Degree of Polymerization Dispersity

While the full molecular weight distribution is the most complete piece of information, we can often only access $\bar{M}_n$ or $\bar{M}_w$ experimentally. So, there's a few other values of interest to report that give us a bit more intuition about the polyer system we're working with.

Degree of Polymerization

If we know the number-average molecular weight $\bar{M}_n$ and the repeat unit of the polymer we can compute the average number of repeat units in a typical polymer chain:

This is sometimes a more powerful way to describe the average chain length because we don't have much context for $\bar{M}_n$, which has units of g/mol. However, if we divide the number-average molecular weight by the mass of the repeat unit, we simply get the number of repeat units in a typical chain!

For example, let's say I have a PVC polymer that has a $\bar{M}_n = 21.050 \times 10^3$ g/mol, and I unit the repeat unit is $\ce{C2H3Cl}$. I can look up the atomic weights of each of the constituent elements: $A_r^{\circ}(\ce{C}) = 12.01$ g/mol, $A_r^{\circ}(\ce{H}) = 1.01$ g/mol, and $A_r^{\circ}(\ce{Cl}) = 35.45.01$ g/mol, so $m = 62.50$ g/mol.

The degree of polymerization is then $DP = \frac{21.050 \times 10^3 \text{g/mol}}{62.50 \text{g/mol}} = 338$. The average chain length is 338 repeat units long - this gives me a much better idea of what's going on!

Dispersity

Again - the values of $\bar{M}_n$ and $\bar{M}_w$ are just ways of measuring average chain size in a polymer system. Alone don't tell you how disperse the molecular weight distribution is. However, the more disperse the distribution, the greater the difference between $\bar{M}_n$ and $\bar{M}_w$. The only time that $\bar{M}_n = \bar{M}_w$ is when the distribution of molecular weights in non-disperse. This is because any dispersity in a specimen leads to us weighting the heavier polymer chains more, causing $\bar{M}_n < \bar{M}_w$.

However, if all chains are the same length (non-disperse), then weighting for larger chain lengths does not matter - all the chains are the same length! We can then define dipersity $Đ = 1 = \frac{\bar{M}_w}{\bar{M}_n}$ for a non-disperse (or uniform) polymer system, and dispersity in general as $Đ = \frac{\bar{M}_w}{\bar{M}_n}$. Note that since $\bar{M}_n < \bar{M}_w$ in all cases where some dispersity exists $Đ \geq 1$. Nearly all synthetic polymers have some dispersity, and values are typically around 1.5-2.0 for most synthetic polymers.

In Figure 9.8.2, we show representations of plots of $x_i$ and $w_i$ to further explore this idea. In the figure, we've taken some molecular weight distribution of chains (like those you observed on previous simulations) and 5inned into $5\times 10^3$ g/mol size bins. On the left side in Figure 9.8.2 we're plotting the number fraction $x_i$: i.e., fraction of the number of chains in the $5\times 10^3-10\times 10^3$ g/mol size range, the $10\times 10^3-15\times 10^3$ g/mol size range, etc.

On the right-hand side we instead plot the weight fractions $w_i$ in each range for the same distribution. This means for each binning region I measure the fraction of the weight in that range.

You can see that the weight-fraction plot is biased towards larger fractions at higher weights. This makes sense because the same number of molecules of high molecule weight take up a larger fraction of the weight of the molecules.

We concede this can get a bit confusing, but as with all things - you need to do it before you understand it. So, let's try something familiar:

Let's consider a classroom of 10 students of varying weights (Section 9.8.14). I want to calculate the number-average "student weights" and weight-average "students weights". The weight-average molecular weight will essentially account for the weight fraction of students - so if we need to make a prediction of some performance that might be heavily dependent on having a few heavier individuals (i.e., likelihood of winning a tug-of-war competition), we have a number that might help us.

\begin{array} .\mathrm{Student} & \text{Weight (lb)}\\ \hline 1 & 104\\ 2 & 116\\ 3 & 140\\ 4 & 143\\ 5 & 180\\ 6 & 182\\ 7 & 191\\ 8 & 220\\ 9 & 225\\ 10 & 380\\ \hline \end{array}

Number-average "Student Weight" Calculation

To calculate the average values, we'll define 40-lb intervalsand bin the students:

\begin{array} .\text{Weight Range (lbs)} &\text{Number of students}\,(n_i) & \text{Mean weight}\,(M_i) \text{(lbs)} & \text{Number fraction}\,(x_i) & \text{Weight fraction}\,(w_i) \\ \hline 81-120 & 2 & 110 & 0.2 & 0.117\\ 121-160 & 2 & 142 & 0.2 & 0.150\\ 161-200 & 3 & 184 & 0.3 & 0.294\\ 201-240 & 2 & 223 & 0.2 & 0.237\\ 241-280 & 0 & - & 0& 0\\ 281-320 & 0 & - & 0& 0\\ 321-360 & 0 & - & 0& 0\\ 361-400 & 1 & 380 & 0.1 & 0.202\\ \hline & \sum n_i = 10 & \sum n_i M_i = 1881 & &\\ \hline \end{array}

After binning, we simply computed $n_i$ and $M_i$ for each defined weight range. We also summed up the total number of students $\sum n_i$ and the total weight of student $\sum n_i M_i$. This allows us to computer the number fraction for each range and the weight fraction for each range in the last two columns.

Two example calculations for number fraction are

$$x_{81-120} = \frac{n_i}{\sum n_i} = \frac{2}{10} = 0.20$$

and

$$w_{81-120} = \frac{n_i M_i}{\sum n_i M_i} = \frac{2 \times 110}{1881} = 0.117$$

With these data, we can compute the full number-average "student weight" and the weight average "student weight" from Eq. 9.8.2 and Eq. 9.8.3, respectively.

$\bar{M}_n = \sum x_i M_i = (0.2 \times 110)+(0.2 \times 142)+(0.3 \times 184)+(0.2 \times 223)+(0.1 \times 380)$

$\bar{M}_n $ =188 lbs

$\bar{M}_w = \sum w_i M_i = (0.117 \times 110)+(0.150 \times 142)+(0.294 \times 184)+(0.237 \times 223)+(0.202 \times 380)$

$\bar{M}_w $ =218 lbs

A Basketball Analog

Before we dig into the details of what we're talking about with Eq. 9.8.1 and Eq. 9.8.3, lets think about this in analog. Let's say we have a sports team - let's say basketball. In basketball it is important to score points and to get rebounds. These days, scoring is done by both smaller players who can shoot well and larger players, while larger, taller players are better at getting rebounds. Often, rebounding is dominated by one or two very tall players, and the other players' height does not matter so much So, let's look at two "height distributions" of players and consider if they can tell us something about various "properties" of the basketball team.

Let's say we've cloned four Jon Emerys (I'm 5'7") and put them on a basketball team with Northwestern's Matthew Nicholson (who is 7'0"). Now, if we want to predict how much the team will score. Let's assume, I'm not a bad shooter (I am), so maybe I average 4 pts a game, compared like Matthew Nicholson's 5 pts a game. It seems like scoring is not highly dependent on height, here, and so we might be able to predict the team's points per game just by looking at the average height of the team. This is the number average.

However, let's say that rebounding is highly dependent on height. If you have a few very tall players, they can have an great influence on the number of rebounds. So, when trying to predict how the team will do in rebounding, we want to count the tall players more - we want to bias our calculation towards the tallest players in order to make predictions. This is what we do with weight-average molecular weights in polymers. Sometimes the larger molecules are much more important for the behavior of a material (for example strength, instead of rebounding) so we essentially count those higher molecular weights more. Essentially, we're more interested in what fraction of the total height is in the tallest player. One can plot such a distribution (for polymers) like in Figure 9.8.2.