Overview

Up until this point, we have largely been focused on understanding the molecular scale structure and reactivity of polymers. Now, let’s think bigger – how do those polymer behave when you dissolve them in solution? No molecule exists in isolation, so we must consider how polymers interact with other polymers as well as solvent. And, perhaps even before getting to that point, we need to think about the conformations the polymer can adopt in the first place – is it coiled? Is it elongated? Is the polymer flexible, or is it stiff? In Lesson 9, we will address the concept of polymer conformations and discuss ways in which we can represent the “size” of a polymer and how stiff it is. These concepts will be important for characterizing and representing the behavior of a polymer in solution, the thermodynamics of which will be covered in Lesson 10.

Learning Outcomes

By the end of this lesson, you should be able to:

• Calculate contour length and fully extended end to end distance
• Calculate root mean square end to end distance using freely jointed chain, valence angle model, and hindered rotation model
• Predict which polymers will be more sterically hindered to ration than others
• Describe chain stiffness using the characteristic ratio

Lesson Checklist

Please refer to the Canvas Calendar for specific timeframes.

Questions?

If you have questions, please feel free to post them to the General Questions and Discussion forum. While you are there, feel free to post your own responses if you, too, are able to help a classmate.

Thus far, we have been primarily considering individual molecules of polymers in isolation. But polymers don’t exist in isolation – they exist amongst other polymers and often dispersed in a solvent. So, how does a polymer behave when it is dispersed in a solution – is the polymer extended? Does it coil up? Is the polymer ordered, or is it randomly oriented? How much volume does the polymer take up in the solution? These are the types of questions that we begin to think about now.

Figure 9.1: Four examples of how polymers behave in solutions

Source: Lauren Zarzar

We have previously discussed polymer size in terms of mass (M_w, M_n), but now let’s think in terms of volume – how much space does it occupy, and what is the conformation in solution? There are a few key characteristic parameters that give us some insight into polymer size and conformation.

Figure 9.2: Contour Length compared to Fully Extended Chain Length

Source: Lauren Zarzar

Contour length is one of the simplest, but also one of the least accurate, descriptors of polymer size. Contour length is defined as the length of the polymer if you were to stretch the polymer out in a line. Disregard bond angles, just pull it as straight as possible, with the constraint being the bond distances. This is of course unrealistic, but it gives the absolute maximum length estimation for polymer size.

Fully extended chain length differs from contour length in that it considers that there are preferred bond angles within the backbone of the polymer. For a backbone consisting of all sp³ hybridized carbons, we expect that angle to be 109.5°. Thus, because we have this zigzag in the backbone, then fully extended chain length for a given polymer will be shorter than the contour length. This fully extended chain model isn’t realistic either, although it’s somewhat better than contour length because at least it includes bond angles.

PROBLEM

For polyethylene with molar mass = 280,000 g/mol, what is the fully extended length? Neglect end groups. Polyethylene is shown below.

Click here for the answer.

ANSWER

The molar mass of the repeat unit of polyethylene (as written above) is 28 g/mol. Therefore we can calculate the degree of polymerization:

$$x_n=\frac{280,000g/ mol }{28g/ mol }=10,000$$

And the fully extended length would be:

$$fullyextendedlength\text{ = }2\ast 1.26\text{Å}\ast 10,000=25,200\text{Å}=2.52microns$$

That’s HUGE! For a point of comparison, the width of a human hair is about a hundred microns – and this 2.5 microns is the length of a single molecule. It would be very unlikely for the polymer to actually exist at this fully extended length.

Now that we have considered one extreme in which the polymer is fully stretched out in a line, what if we now consider the other extreme, which is the polymer coiled up into a sphere? How big would that sphere be?

PROBLEM 2

Consider polyethylene with molar mass = 280,000 g/mol, and that polyethylene has a typical density of 0.9 g/mL. Polyethylene is shown below. If the polyethylene could be packed in a sphere, what would be the radius of one molecule? $\left( 1{\text{mL=10}}^{\text{24}}{\text{Å}}^3\right)$

Click here for the answer.

ANSWER 2

$$volumeof1molecule\text{ = }\frac{280,000g/ mol }{0.9g/ mL }\ast \frac{mol}{6\ast {10}^{23}molecules}\ast \frac{{\text{10}}^{\text{24}}{\text{Å}}^3}{mL}=520,000{\text{Å}}^3$$

$$volumeofsphere\text{ = }\frac{4}{3}\ast \pi \ast r^3=520,000{\text{Å}}^3$$

$$\text{Radius}=50\text{Å for a single molecule}$$

Let’s now compare these two extremes of the perfectly stretched out polyethylene with a fully extended length of 25,200Å and the perfectly coiled polyethylene with a diameter of 100Å. This size difference is a bit like comparing a tennis ball to a football field – several orders of magnitude difference! So, what conformation does the polymer actually take? And how many different conformations are possible? The polymer takes on these different conformations through the rotation of single bonds in the backbone. In order to address these questions, let’s take a step back and consider a far simpler and smaller molecule than polyethylene, butane:

Figure 9.3: Butane

Source: Lauren Zarzar

Butane is not big enough to be considered a polymer, but as you may notice, it looks like a polyethylene dimer! It is a simpler case to consider first, before moving onto polyethylene with hundreds or thousands of repeat units. Butane changes conformation through the rotation around single bonds (Figure 9.3). However, not all conformations are created equal. Newman projections, which you learned in organic chemistry, allow us to visualize how substituents are oriented relative to each other when rotating and “looking” down the bond of a molecule. In Figure 9.4 two examples of Newman projections are shown for the staggered (trans) conformation and the eclipsed conformation of butane. We are “looking down” the center carbon bond between carbons 2 and 3. By drawing the Newman projections shown underneath, it is much easier to visualize how the terminal methyl groups are oriented relative to each other.

Figure 9.4: Examples of Newman Projections

Source: Lauren Zarzar

In terms of energy, it is more favorable (lower energy) to have the methyl groups far away from each other, and to have substituents not overlapping with each other, due to sterics. We can draw a plot to show the relationship between the rotation angle and the potential energy of the molecule. Notice that there are three potential energy minima for butane that correspond to the trans and gauche conformations (Figure 9.5). The molecule is most likely to exist in these conformations that minimize potential energy.

Figure 9.5: Location of Trans and Gauche conformations in potential energy minima

Source: Lauren Zarzar based on figure 10.6 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.

What if you add more carbons now – how many conformations are there of pentane? Hexane? Poly(ethylene)? As you add additional carbons to the chain, each of those additional bonds can now have these various rotation angles and the problem of describing the molecular conformation becomes exponentially more complex. We can still have Newman projections along each bond in polyethylene, and it looks very similar to butane except that instead of methyl groups there are now polymer chains:

Figure 9.6: Newman Projections of Polyethylene

Source: Lauren Zarzar

Consider a polyethylene molecule having 10,000 bonds, and if each of those bonds has 3 preferred orientations – how many different conformations of the polymer would there be? 3^10,000, which is a huge number! The probability that the chain exists in any specific one of these confirmations is incredibly small. We can’t predict exactly what conformation the polymer will have at any given instance. The conformation actually fluctuates, at some intermediate between fully extended and spherical. We can use statistics to think about the polymer conformation and size in terms of averages, and what dimensions might be most likely.

One way to think of polymer conformations is to treat the polymer like a freely jointed chain where every bond is free to rotate to any angle, but the bond distances are fixed (“l”) and the number of bonds in the chain is fixed (“n”). We can then think of the polymer conformation as that of a “random walk”. What do we mean by that? Imagine you stand in one place, representing the end of the polymer, and take a step of length “l” in any direction. This corresponds to one bond distance and the position of the second atom in the polymer backbone. Take another step of length “l”, in any direction. You now stand in the position of the third atom in the polymer backbone. Continue for “n” steps, however many bonds are in the backbone of the polymer. Your path may look something like this:

Figure 9.7: Example of a freely jointed chain polymer conformation

Source: Lauren Zarzar

A good way of characterizing this “walk” is the distance from the start to the finish, which we labeled as “r” and call the “end to end distance”. There are many different possibilities for r given values of l and n, just depending on the specific path you take. Similarly, the polymer is constantly fluctuating in conformation and changing end to end distance. While we cannot specify the exact conformation at any given instance, we can know something about the distribution for r values and can use an “average” value of r reflective of that distribution: root mean square (RMS) end to end distance, which is given by the equation 9.1:

Although RMS end to end distance of a freely jointed chain is a better representation of the polymer conformation than either the contour length or fully extended chain length, it’s still not great – after all, we made an assumption that the bond angles and positions could be anywhere, which is definitely not representative of actual polymer molecules. We can try to alter our model to take that into account.

Figure 9.8: Valence Angle Model

Source: Lauren Zarzar

In the valence angle model, we try to account for the fact that the bonds along the backbone of the polymer will have a preferred bond angle. For a fully singled bonded carbon backbone in which all carbons are sp³ we expect those bonds angles to be 109.5°. In this model, although we fix the bond angle, we will still consider each single bond to freely rotate.

We can alter our formula for the mean square end to end distance with that fixed bond angle:

$${\langle r^2\rangle }_{fa}=n{l}^2\frac{1-\cos \theta }{1+\cos \theta }$$

Notice that the subscript “fa” here is to indicate that this is free rotation around a fixed angle. Also notice that this formula is for the mean square end to end distance, not the root mean square. You’d have to take the square root to find RMS end to end distance.

Figure 9.9: Valence Angle Model with Hindered Rotation

Source: Lauren Zarzar

In the valence angle model, we initially assumed that single bonds can freely rotate. But we know that not all orientations are created equal (recall Newman projections?) How can we account for that? In the valence angle model with hindered rotation, we not only account for the bond angles but also account for the fact that single bonds do not freely rotate to any angle, and depending on the substituents on the polymer, some bonds may be more hindered than others. We thus define the mean square end to end distance with a subscript of “0” meaning hindered rotation round a fixed bond angle:

$${\langle r^2\rangle }_0={\sigma }^{2}n{l}^2\frac{1-\cos \theta }{1+\cos \theta }$$

Notice we have introduced a new variable, σ, which is called the steric parameter. Steric parameters are usually determined experimentally from measured values of ${\langle r^2\rangle }_0$. The steric parameter represents how stiff or flexible the polymer is. A stiffer polymer with more hindered rotation will have a high steric parameter, and a more flexible polymer with freer rotation will have a lower steric parameter.

PROBLEM

Which polymer would you expect to have the highest steric parameter σ at a given temperature?

Click here for the answer.

ANSWER

B. The bulkiest substituent will give the most hindered rotation and largest steric parameter.

We can get a sense of how stiff a polymer chain is from the characteristic ratio, C_∞

$$C_{\infty }=\frac{{\langle r^2\rangle }_0}{n{l}^2}$$

Higher steric parameter corresponds to a higher characteristic ratio (all else equal) because a higher steric parameter will give a higher value of ${\langle r^2\rangle }_0$. So polymers that do not rotate as freely are more stiff and have longer end to end distances. The denominator, $n{l}^2$ is the mean square end to end distance based on the “random walk” with free rotation and no restrictions to bond angles (i.e., the denominator represents the most flexible polymer possible!). The numerator is the valence bond model including hindered rotation; this value will be higher for stiffer polymers. Think of the two extremes: an infinitely stiff polymer is just a rod. It can’t coil at all, and the end to end distance is the same as the contour length. Compare that to a very flexible polymer, which will have the ability to coil, and thus will have a much lower end to end distance.

A table of some characteristic ratios and steric parameters is given below (textbook table 10.3)

As expected, bulkier groups correspond to higher steric parameters and higher characteristic ratios. Temperature also plays a role; at higher temperatures there is more energy available to overcome those barriers to rotation, so it becomes easier for single bonds to rotate. If it’s easier for bonds to rotate, then the polymer is overall more flexible, and we therefore see a decrease in the steric parameters and characteristic ratio as we go to higher temperatures.

Summary

Because polymers are such large molecules, they really can take on nearly infinite dramatically different conformations, which makes it difficult to fully describe their conformation at any given time. But we can use the concepts of end to end distances and chain stiffness to help us describe the conformations of polymers and to compare properties across different chemical structures. These same concepts are going to still be useful in our next lesson, Lesson 10, where we will begin to learn about thermodynamics of polymers in solution. We will need to start to take into account interactions between polymers and solvent, and figure out how those interactions actually affect polymer conformations and the properties of polymer solutions.

Reminder - Complete all of the Lesson 9 tasks!

You have reached the end of Lesson 9! Review the checklist on the Lesson 9 Overview / Checklist page to make sure you have completed all of the activities listed there before you begin Lesson 10.