Polymer Conformations in Solution

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.

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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? $\left( 1{\text{mL=10}}^{\text{24}}{\text{Å}}^3\right)$

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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.