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Chapter 7: Size Exclusion Chromatography

Chapter 7: Size Exclusion Chromatography

Last updated

Jan 28, 2025
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- Chapter 6: Osmometry
- Chapter 8: Light Scattering

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Size Exclusion/Gel Permeation Chromatography (SEC/GPC)

Size exclusion chromatography (SEC), also called gel permeation chromatography (GPC) allows the measurement of an entire molecular weight distribution, which is more favorable than the measurement of a single molecular weight average from either intrinsic viscosity or osmometry measurements.

The experimental apparatus for SEC consists of a long column packed with porous polymer beads of varying diameters and pore sizes, typically on the order of 50-100,000 ˚A. Additionally people will typically utilize multiple columns with different sized beads to increase the resolution. The beads themselves are crosslinked polymers, or gels, leading to the term gel permeation chromatography.

A polymer solution is then pushed through the column slowly (such that the shear rate is not high enough to bias the polymer conformations), and as it flows the solution interacts with the porous beads. As the solution reaches the bottom of the column, eluted volume is collected as a function of time and a concentration detector (which is generally some sort of light scattering device, UV detector, IR detector, or refractive index (RI) detector) is used to determine the concentration of a given sample. Also the solvent is typically chosen to be a good solvent in order have a different refractive index from the polymer.

Figure $\PageIndex{1}$ : SEC/GPC Schematic.

Now the polymers will have a wide distribution of molecular sizes and the beads will also have a distribution of pore sizes. So the polymers will be able to penetrate into some of the porous beads depending on the size of the polymers and the size of the pores. Thus depending on how many beads each polymer interacts with (how many porous beads the polymer can penetrate) the time taken to reach the body of the column will change, leading to a difference in polymer sizes eluting at different times. The larger polymers will typically elute first as they will penetrate the smallest number of beads while the smaller polymers will come out much later. Thus we are able to separate by size (SEC) based on this residence time, the time spent inside the beads. Quick note, this is not an exact technique and really typically only useful as a relative technique that must be calibrated to a reference sample.

Figure $\PageIndex{2}$ : SEC/GPC Experimental Method.

Thermodynamic Principles of SEC:

SEC is a unique case and will depend on the competition between to entropic contributions this time instead of an entropic and enthalpic contribution. The competition is between

Favorable entropy increase upon mixing of polymer and solvent inside the pore of the bead
Unfavorable loss of configurational entropy of large size polymers when they enter bead pore size smaller than polymer’s unperturbed size

For the unfavorable term we remember that for polymer deformation the change in entropy for compression is given by

$\Delta S \approx \bigg(\frac{R_0}{R}\bigg)^2 \nonumber$

So we can write the change configurational entropy term for larger polymers being compressed when attempting to enter small pores as

$\Delta S_{conf} \approx \frac{R_{polymer}}{R_{pore}} \nonumber$

where R_poreis approximated as the ratio of the average pore volume to pore surface area. The entropy increase is the same that we have seen previously from Flory (see how important that lecture was). Having identified these two basic entropic considerations, we can now fully flesh out our understanding of SEC in the next section.

Eluted Volumes of SEC

We have our physical understanding but now we need to relate this to what will actually happen in the experiment which is measuring some eluted volume of polymeric materials. To start let’s divide this volume that is eluted from the column into two parts

V_othe void volume external to the beads in the column
V_ithe pore volume internal to the beads

A polymer will sample some part of this entire volume, depending on the size of the polymer, but the solvent will sample the entire volume so

$\text{pure solvent eluted volume} = V_{es} = V_o + V_i \nonumber$

$\text{polymer solution eluted volume} = V_{ep} = V_o + V_iK_{se} \nonumber$

The difference between the two eluted volumes lies in the amount of pore volume sampled, which depends on the parameter K_se, our size exclusion equilibrium coefficient which like similar equilibrium constants can be defined by

$K_{se} = \frac{\text{polymer concentration inside bead}}{\text{polymer concentration outside bead}} = \frac{c_{2i}}{c_{2o}} \nonumber$

We can relate this to an entropy for pore permeation (just like chemical equilibria constants). Assuming no enthalpy change upon permeation

$\Delta G_{pp} = -RT\ln K_{se} = -T\Delta S_{pp} \nonumber$

$K_{se} = e^{\Delta S_{pp}/R} \nonumber$

In order to find K_sewe need to approximate this entropy. Well that will be the expression that we started with for the change in conformational entropy upon entering the bead which is the size of the unperturbed polymer divided by the size of the pore. Well pore sizes are difficult to measure so instead we will go with another more easily measured parameter which is A_sis the pore surface area:volume ratio (and hence a characteristic size for the pores). So now the change in entropy is

$\Delta S_{pp} = -RA_s \langle r^2 \rangle ^{\frac{1}{2}} \nonumber$

Plugging this back in for K

$V_{ep} = V_o +V_i \exp (-A_s \langle r^2 \rangle ^{\frac{1}{2}}) \nonumber$

Here the physical insight is that the elution volume is proportional to an exponential relation of the polymer radius, which we know is related to molecular weight. Thus, we can relate molecular weight to elution volume.

Unfortunately, with this technique, as previously mentioned, we first need to determine our constants by utilizing a known sample with a known molecular weight (calibration). Then we can yield a complete molecular weight distribution and polydispersity index.

So we have a method for deriving the complete molecular mass distribution based on the relationship derived above between molecular mass and elution volume. We can thus relate the detector response as a function of elution volume (i.e. time) to molecular mass, since a greater detector response corresponds to a higher concentration of polymer at a given volume.

Summary: Viscosity, Membrane Osmometry, and SEC/GPC

We have now studied size exclusion/gel permeation chromatography. In this experimental setup, columns are packed with a dense mixture of porous beads, with differing pore sizes. A polymer sample is then poured through the column. Because of the presence of the beads, polymer coils of different sizes will attempt to either flow directly through the column without interacting with the beads or spend some time permeating into the porous beads themselves. Smaller polymers will be able to get into more pores and thus will spend more time in the column, while larger beads will tend to pass directly through the column without passing through any pores. As a result, we can extract samples from the elution volume of the column as a function of time, assuming that each sample will be relatively monodisperse, and with larger coils emerging from the column earlier. Further characterizing each volume will thus reveal a complete size distribution of the sample.

Gel permeation chromatography

Key Ideas: Fill column with porous beads of varying pore size. Pour solution through column; larger polymer coils will pass through column first since they do not spend time in pores. Can obtain specific molecular weight distribution by measuring elution volume, concentration of polymer as function of time.
Key equations: : $V_{ep} = V_o +V_i \exp (-A_s \langle r^2 \rangle ^{\frac{1}{2}})$ . $V_{ep}$ is elution volume, V_ois volume of solution outside of pores, V_iis volume inside pores, A_sis pore surface:volume ratio, r²is polymer coil radius. Note that this equation just serves to illustrate that the volume eluted is related to coil size.
Key Experimental Method: First calibrate GPC using known molecular weight sample to obtain elution volumes - in other words obtain molecular weight vs. elution volume curve. Can then add new sample and compare to previously calibrated result to gain molecular weight distribution. Most modern GPCs will be precalibrated and can give size distribution easily. Will generally need some other means (light scattering, viscosity) to obtain polymer concentration.
Key Insights: obtaining full molecular weight distribution of polymer sample, rather than just averages (as obtained from other characterization techniques).

Size Exclusion/Gel Permeation Chromatography (SEC/GPC)

Thermodynamic Principles of SEC:

Eluted Volumes of SEC

Summary: Viscosity, Membrane Osmometry, and SEC/GPC

Gel permeation chromatography

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