Overview

In this Lesson we are going to begin learning about the various characterization methods that are commonly used to analyze polymers. What methods are useful to analyze the molar mass, the dispersity, the crystallinity, etc.? There are many properties of polymers that we learned about so far, and we have to figure out how to actually measure them! We will begin by getting a general overview of the broad categories of characterization methods and the types of properties they can be used to measure. From there, we will go more in depth on a few select characterization methods: differential scanning calorimetry, size exclusion chromatography, end group analysis, and osmometry.

Figure 11.01: Overview of characterization methods used to analyze polymers

Source: Lauren Zarzar

Figure 11.01 is an overview of some of the key polymer characterization approaches and the properties they are used to measure. Spectroscopy, such as UV-VIS, FTIR, NMR, Raman, and mass spectrometry are used primarily analyze the chemical composition and molecular structure of polymers. For example, we can confirm using these techniques that we in fact made the chemistry of polymer we intended, we can determine the tacticity, or we can figure out end group concentration. Light scattering and diffraction are used to determine the conformations of polymers in solution and crystal structure. To analyze the thermal transitions in polymers, such as the glass transition, melting temperature, or heat capacity, we can use differential scanning calorimetry (DSC). Dilatometry also can provide insight into thermal transitions by measuring changes in free volume. Size exclusion chromatography (SEC) is an important tool that can tell us about the molar mass distributions of polymers. Osmometry and end group analysis are techniques that help us count the “number” of molecules in solution, which yields insight as to the number average molar mass. Osmometry can also be used to measure the Flory-Huggins interaction parameter. Of these techniques, we will not be covering spectroscopy or light scattering or diffraction in much more depth. The others we will discuss how the technique works, what it measures and how, and what are the advantages and limitations of the various approaches.

Learning Outcomes

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

• List the polymer characteristics that that can be analyzed using:
  • osmometry
  • end group analysis
  • size exclusion chromatography
  • differential scanning calorimetry
• Analyze results from:
  • osmometry
  • end group analysis
  • size exclusion chromatography
  • differential scanning calorimetry

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.

The first characterization methods we will address are those that can help us figure out the polymer number average molar mass. $M_n$ was one of the first characteristics of polymers we learned about in this class. Recall this fundamental relationship:

$${\overline{M}}_n={\overline{M}}_0{\overline{x}}_n$$

The number average molar mass is just a function of the molar mass of the repeat unit and how many repeat units there are (i.e., the degree of polymerization). We, of course, know the molar mass of the repeat unit, that just is a function of what polymer we are trying to make. So, how do we figure out $M_n$? Well, if we could just figure out $x_n$ that would be easy….

Now, recall this expression we learned very early on as well, that related the total number of molecules at the start and end of the reaction to degree of polymerization:

$${\overline{x}}_n=\frac{N_0}{N}$$

We can easily find $N_0$ because this is just the total number of monomers we started with. If we could just figure out $N$, the number of molecules at the end of the reaction, then this would allow us to solve for degree of polymerization. And then we could use degree of polymerization to solve for $M_n$!

It turns out, there are a number of measurable physical properties of solutions that depend primarily on the number of solute molecules per unit volume; these are called colligative properties. Examples include boiling point elevation, freezing point depression, osmotic pressure, and vapor pressure. If we can use these techniques to quantify the number of polymer molecules in solution, then we can use these to determine $x_n$ and $M_n$. We will focus on osmotic pressure, because this is the only colligative property that is most accurately measured for polymers of higher molar mass. But there is still a limit – if the polymer is too big and solvent too dilute, it’s hard to measure osmotic pressure, and if the polymer is too small, then the membrane may have a hard time keeping out the polymer solute. But those concerns aside, we will be considering osmometry in depth first!

As a quick review, recall that osmosis is the movement of molecules through a semi-permeable membrane from a region of low solute concentration to a region of high solute concentration. Osmotic pressure is then the minimum pressure that is required to prevent the flow of solvent across the membrane, and in effect is a measure of the tendency of the solution to take up solvent. Review the Wikipedia: Osmotic Pressure page for more information.

Figure 11.02: A simplified osmotic pressure experiment

Source: Figure 11.1 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.

The difference in height between the solutions in the two sides of the chamber, $h$ is related to the osmotic pressure ($\pi$), where $g$ is acceleration due to gravity and ${\rho }_0$ is the solvent density:

$$\pi =h{\rho }_{0}g$$

And we can relate the osmotic pressure to the Flory-Huggins parameter ($\chi$) where $c$ is the polymer concentration (mass/volume) in the mixture, $V_1$ is molar volume of solvent, and ${\rho }_2$ is polymer density:

$$\frac{\pi }{c}=\frac{RT}{{\overline{M}}_n}+\left( \frac{RT}{V_1{\rho }_2^2}\right)\left( \frac{1}{2}-\chi \right)c$$

The quantity $\frac{\pi }{c}$ is called the reduced osmotic pressure.

PROBLEM

Consider the experimental data plotted below and the relationship between reduced osmotic pressure and concentration of polymer. What does the Y intercept of the plot represent?

Source: Figure 11.2 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.
(Data taken from Kamide, K. et al., Br. Polym. J. 15,91, 1983.)

1. The density of polymer
2. χ
3. $\frac{RT}{{\overline{M}}_n}$
4. The molar volume of polymer

Click here for the answer.

ANSWER

C. $\frac{RT}{{\overline{M}}_n}$

The plot has reduced osmotic pressure, $\frac{\pi }{c}$, on the Y axis and $c$ is on the X axis. Looking at the equation:

$$\frac{\pi }{c}=\frac{RT}{{\overline{M}}_n}+\left( \frac{RT}{V_1{\rho }_2^2}\right)\left( \frac{1}{2}-\chi \right)c$$

We see that the quantity $\frac{RT}{{\overline{M}}_n}$ thus corresponds to the Y intercept and $\left( \frac{RT}{V_1{\rho }_2^2}\right)\left( \frac{1}{2}-\chi \right)$ corresponds to the slope. We can use the slope to therefore measure the Flory Huggins parameter and the Y intercept to measure the number average molar mass.

PROBLEM 2

Consider the experimental data plotted below for two different sizes of polystyrene in toluene, and the relationship between osmotic pressure and concentration of polymer. Which sample has higher $M_n$?

Source: Figure 11.2 from Young, Robert J., and Peter A. Lovell.
Introduction to Polymers, Third Edition, CRC Press, 2011.
(Data taken from Kamide, K. et al., Br. Polym. J. 15,91, 1983.)

1. Sample (a)
2. Sample (b)

Click here for the answer.

ANSWER 2

A. Sample (b)

We know that $\frac{RT}{{\overline{M}}_n}$ is the Y intercept. Thus, as $M_n$ increases, the intercept decreases. Sample (b) has the lower intercept, and higher $M_n$ Thus, the intercept helps us measure number average molar mass.

If we know the skeletal structure of a polymer (i.e., how many end groups there are per molecule of polymer) and the end groups are in some way distinguishable from other chemical moieties on the polymer, then end group analysis can be a powerful tool to allow us to “count” the number of molecules in a sample and hence solve for $M_n$.Alternatively, if we know $M_n$ from some other method, then end group analysis can actually help us figure out how many end groups there were on average per molecule. There are a variety of analytical methods that could facilitate quantification of the end groups, such as spectroscopic methods, elemental analysis, and titration. We will not be going into depth regarding spectroscopic analysis, but in this lesson we will go over examples for which acid-base titrations and elemental analysis can be used.

Figure 11.03: Example polymers for end group analysis

Source: Lauren Zarzar

End group analysis would be like counting the stars on these polymers (6) and knowing there are 2 end groups per molecule – we can use that information to figure out that there are 3 polymer molecules.

There are several requirements that must be met in order for end group analysis to be useful:

• There has to be some way to quantify the end group functionality in the first place, usually by titration, elemental analysis, or spectroscopy.
• Other functional groups on the polymer or in the solution can’t interfere with measurement of your target end groups.
• The concentration of end groups has to be sufficient to get an accurate measurement (so may not be good for very high molecular weight samples).
• You have to know how many end groups there are per molecule (to find molar mass), OR you know the molar mass, and want to know the number of end groups per molecule.

Recall from general chemistry laboratory those titration experiments? This analysis method works by neutralizing an unknown concentration of acid or base with a known concentration of acid or base. For example, we may have a polymer that has terminal carboxylic acid groups, which would give an acidic solution when dissolved in water. We could add a strong base, like NaOH, to our sample and neutralize the carboxylic acid groups. If we know how much NaOH is required to neutralize the acid, then we can figure out how many carboxylic end groups there must have been in the first place. Usually, a colorimetric pH indicator is used to determine the end point of the titration. Easy groups to distinguish by acid-base titration are carboxylic acids and amines. Sometimes the polymer as its originally produced does not have an acidic or basic end group, but we can do a reaction to make that functional group amenable to titration. An example would be terminal hydroxyl groups, which are not distinguishable by acid-base titration but can be converted via multiple pathways to carboxylic acids:

Figure 11.4: Terminal hydroxyl groups can be converted to carboxylic acids

Source: Lauren Zarzar

PROBLEM

For end-group analysis, 0.8632 g of a carboxyl terminated polybutadiene sample of the general structure shown below, dissolved in a mixture of ethanol and toluene, consumed 5.2 mL of 0.1242 M alcoholic potassium hydroxide solution in titration using phenolphthalein as the indicator. Calculate the molar mass of the polymer.

Click here for the answer.

ANSWER

First we find the number of carboxylic acid groups, which we know must equal the number of hydroxyl groups:

$$molesCOOHgroups=molKOH$$

$$=5.2ml\ast \frac{1L}{1000mL}\ast 0.1242\frac{mol}{L}$$

$$=6.46\ast {10}^{-4}mol$$

Given the structure of the polymer, we see there are 2 carboxylic acid groups per molecule. Since we know how many carboxylic acids there are, we can figure out how many polymer molecules there are:

$$molespolymer=\frac{6.46\ast {10}^{-4}mol}{2}$$

$$=3.23\ast {10}^{-4}mol$$

To find the number average molar mass, we simply divide:

$${\overline{M}}_n=\frac{0.8632g}{3.23\ast {10}^{-4}mol}$$

$$=2672g/ mol$$

If we can functionalize the end groups of the polymer with an element that is not found elsewhere in the polymer, such as a halogen, then we can use elemental analysis to figure out what mass or atom percentage of that element is in the sample. Working backwards, we can then figure out how many of those atoms there are in total, and hence how many end groups there are in the sample.

PROBLEM

1 g of a sample of polyester polyol of M_n = 3,000 g/mol was treated with bromoacetyl bromide to convert the hydroxyl end groups to bromoacetyl end groups as shown below. The treated polymer was found to contain 4.88% by weight Br by elemental analysis. Estimate the average number of hydroxyl groups on each molecule of the polyol. Bromine = 79.9 g/mol.

Click here for the answer.

ANSWER

For every OH group reacted, the molar mass changes, because the –OH loses the H but gains the two carbons, an oxygen, bromine, and two more hydrogens. Thus, the molar mass of polymer for each hydroxyl group that reacts increase by: (79.9+24+16+2-1)g/mol = 120.9 g/mol

If “x” is the average number of OH groups per polymer molecule that reacted with the bromoacetyl bromide, then:

$$molarmassoftreatedpolymer=\left( 3000+120.9x \right)g/ mol$$

Mass percentage of bromine content found would then be:

$$\frac{79.9x}{3000+120.9x}=0.0488$$

$$x=1.98$$

There are about 2 hydroxyl groups per polymer molecule.

SEC helps us answer the question, what is the molar mass distribution of the polymer? In general, to analyze the sample, you pass a solution (polymer plus solvent) through a column that is packed with porous beads. A detector “watches” when polymer comes out the other end of the column (often by change in refractive index or UV absorption). The software generates a plot of polymer concentration vs. time, which gives you an indication of the molar mass dispersity of the polymer.

Figure 11.5: General description of the principles of SEC

Source: Wikipedia: Size-exclusion chromatography

Importantly, as shown in the figure above, larger polymers exit the column (i.e., elute) faster than smaller polymers. This is a bit counterintuitive for most people. It has to do with the fact that larger molecules have a shorter path length in the column than the smaller polymers. The packing material in the column has a range of pore sizes; the larger polymers cannot fit into the smaller pores, and hence they are more limited in the paths that they can take through the column, and elute faster. Smaller polymers that can go into the smallest pores will ultimately travel greater distances within those pores and hence elute more slowly.

Often, SEC data is not plotted as a function of time, but rather as a function of elution volume. Elution volume (V_e) is the volume of solvent required to move the polymer from the point of injection (one end of the column) to the detector (other end of the column). A small elution volume means that little solvent is required to flush the polymer out of the column (this would correlate to “fast” elution) while a large elution volume means that more solvent is required (correlating with a “slower” elution). Data you collect might look something like this:

Figure 11.6: Generic elution volume plot

Source: Lauren Zarzar

Now even though we say SEC can be used to measure molar mass distributions, in fact, as you may have noticed, it actually is separating polymers by their “size” rather than mass. More precisely, SEC is separating polymers by their hydrodynamic volume or hydrodynamic radius – which is affected by various things, in particular the polymer (chemistry and structure), solvent, solvent/polymer interactions, and temperature.

Figure 11.7: Representation of hydrodynamic volume

Source: Lauren Zarzar

Unfortunately, molar mass does not always correlate with polymer hydrodynamic volume. After all, when we were considering polymer conformations in solution, the polymer “size” could vary widely depending on whether it was coiled or extended. These considerations of polymer conformations in solution are relevant when we try to interpret data from SEC experiments. This means a polymer with the smaller hydrodynamic radius will require a higher elution volume than a polymer with a larger hydrodynamic radius, even if the sample with small hydrodynamic radius has the higher molar mass!

Now that we know that SEC is actually separating polymers by their hydrodynamic volume, how do we go about actually correlating that with the molar mass? The best way to do this is with a calibration curve made using standards which are expected to behave like your sample. To make such a calibration curve, you get a series of low dispersity polymers of known molar mass and run them through the SEC to yield a plot that could look something like the top plot of the figure below, where each peak corresponds to the elution of one of the standards:

Figure 11.8: Elution volume curve correlating with callibraation curve

Source: Lauren Zarzar

Since you know the molar mass of each of your standards, you can correlate each of those peaks with a specific elution volume. This allows you to make the calibration curve (bottom plot of the figure above). Now you can use this calibration curve to correlate the elution volume to molar mass for any sample that you expect to elute similarly in the SEC column (i.e., sample chemistry of polymer, same solvent).

So we can figure out now how to correlate elution volume to molar mass, but how do we get the molar mass distribution of the sample?

Figure 11.9: Correlating an elution plot to a molar mass plot

Source: Lauren Zarzar

First, we normalize the area under the SEC data curve to 1, because presumably all our polymer is somewhere on that plot. Then we can slice it into sections of arbitrarily small widths (v_i). Each slice has a weight fraction (w_i) associated with it, according to the area within that slice. Each slice also has a molar mass which you can figure out using the calibration curves we just learned about:

Figure 11.10: Normalizing the area under the SEC data curve

Source: Lauren Zarzar

Therefore, if you know the weight fraction of polymer with a specific molar mass, we can use that to create our molar mass distribution curve.

PROBLEM 2

Here are two calibration curves for two standards of a polymer with the same chemical composition, same (good) solvent, and same temperature... But different shapes! Which polymer is branched and which one is linear?

Click here for the answer.

ANSWER

2 is branched and 1 is linear.

They are in a good solvent, so the linear polymer will have the larger hydrodynamic volume. (Branching will make the polymer more compact as compared to a linear polymer with the exact same mass, all else equal). The linear polymer will therefore elute faster, with lower V_e, for a given molar mass

Summary of strengths and weakness of SEC

Strengths of SEC

• Can be used for a pretty wide range of polymers and solvents systems – can be used for sensitive biological systems in aqueous solutions as well as synthetic polymers in organic solvents.
• Can get information about molar mass, distribution of molar mass, and dispersity (and also M_n).

Limitation of SEC

• We are using hydrodynamic volume as an analog to molar mass. Depending on your polymer system, you may or may not be able draw an accurate correlation between the two.
• If there is interaction between the column packing material (the stationary phase) and your polymer, this will cause longer elution times and will mimic smaller polymers.

The glass transition temperature (T_g) or melting temperature (T_m) of polymers are important characteristic properties. How do we measure those thermal transitions or phase transitions? Now, we will discuss a common approach to quantifying T_g and T_m, called differential scanning calorimetry (DSC). DSC is a technique that measures heat required to change the temperature of a sample and a reference of known heat capacity. The direction of heat flow is very important in DSC, as that will tell us whether we have an exothermic or endothermic transition, and this is often a point of confusion when analyzing data from DSC.

To prepare for our discussion of DSC, let’s first review heat flow and how it relates to phase transitions, which is something you first learned in introductory chemistry. You likely saw a plot something like this, which depicts the heat associated with temperature transitions for one mole of H₂O:

Figure 11.11: Heat flow during the transitions of water going from ice to liquid water to steam

Source: Aaron Garner

Notice that while the material undergoes a phase transition, the temperature does not change even though heat is being added. For example, look at ΔH_fus in which H₂O goes from solid ice to liquid water. We have to add 6 kJ of heat to cause this transition to happen for 1 mole of water (so it is an endothermic phase transition), but the temperature remains constant at 0°C during this phase transition. Once we get liquid water, as we add heat, the temperature goes up linearly; how much heat you have to add to increase the temperature is a function of the material’s heat capacity.

When we do DSC analysis, we are going to be heating our sample to specific temperatures at a specific rate and looking at how much heat is required to get us to that temperature. In a way, it has a lot of similarities to this plot, except instead of tracking absolute values of heat added, we measure the heat flow. If the heat flow goes up, that means we will be adding more heat to the system to raise it to a given temperature, whereas if heat flow decreases, heat is released from the system to the surroundings. Let’s look at some sample data you might collect from DSC and try to interpret it.

Look at the graph in Figure 11.12. The Y axis is important, and identifying which “direction” is endothermic vs exothermic is key. In this class, I will keep the axis the same, as in what is shown in Figure 11.12 (but be wary that you may see examples in other books or papers that define the axis in the opposite direction). Here, an increase in heat flow means that the DSC machine is putting in extra heat to try to raise the sample temperature – and it has to do this, because the material is undergoing an endothermic phase transition or a change in heat capacity. In the plot in Figure 11.12 below, we see that initially the heat flow is constant while the temperature is increasing. This means the sample has a constant heat capacity. Suddenly, we see a spike in heat flow; this spike corresponds to a phase transition. During a phase transition, heat is either input or released, but there is no change in temperature (recall our phase diagram for water?) This is why you see that sudden increase in the DSC trace – the machine has to put in a lot more heat in order to get the temperature to rise above the transition temperature. In this specific example, the peak is pointing upward, which means we are inputting heat, and thus the phase transition must be endothermic. Thus, if this was a polymer sample, this would be the melting temperature (T_m). (Vaporization is also endothermic, but for polymers, it’s unlikely that we would be vaporizing them).

Figure 11.12: DSC trace showing an endothermic phase transition at T_m

Source: Lauren Zarzar

PROBLEM

A DSC trace is shown below. What is happening at the temperature marked?

Click here for the answer.

ANSWER

The polymer is crystallizing. Notice that this peak is pointing down; at the temperature marked (T), heat flow to the sample is decreased, and that’s because the sample is undergoing an exothermic phase transition. Crystallization is the relevant phase transition for polymers.

Another DSC trace is shown below in Figure 11.13. This one looks slightly different than the melting and crystallization curves we saw before. Here, we notice that the heat flow increased, but never came down again! Therefore, this feature cannot correspond to a phase transition, but must be indicative of something else – such as, a change in the heat capacity (ΔC_p). This change in heat capacity is correlated to the glass transition temperature (T_g), in which the polymer is going from a glassy solid state to a more viscous state.

Figure 11.13: DSC trace showing a change in heat capacity.

Source: Lauren Zarzar

Putting it all together could look something like the DSC trace shown below in Figure 11.14:

Figure 11.14: DSC trace showing multiple phase transitions of a polymer.

Source: Lauren Zarzar

Not all polymers will have all of these transitions, and there are a number of factors that can affect the DSC trace. For example, some polymers do not crystallize, so keep in mind which curves you might expect or not expect to see depending on the polymer chemistry and skeletal structure. It can often be very difficult to define a specific temperature for these transitions, as the peaks may be quite broad or happen over a range of temperatures. The thermal history of the polymer can also affect the measurements of melting (T_m) and crystallization (T_c) and glass transition (T_c) temperatures, because of hysteresis. Heating rates and cooling rates can also affect these measurements. However, in general, DSC is a very powerful technique that helps us to probe the thermal properties of polymers.