Objectives and Motivation
To understand the physical principles of Size Exclusion Chromatography (SEC) and its use in the
determination of the molecular weight distribution of a polymer sample.
Introduction
Gel Permeation Chromatography (GPC), often also known as Size Exclusion Chromatography, is a
separation method for high polymers, similar to but advanced in practice over gel filtration as carried out
by biochemists, that has become a prominent and widely used method for estimating molecular-weight
distributions since its discovery in 1961. The separation takes place in a chromatographic column filled
with beads of a rigid porous “gel”; for polymer molecular-weight determinations, highly crosslinked porous
polystyrene and porous glass are preferred column-packing materials. The pores in these gels are of the
same size as the dimensions of polymer materials.
A sample of a dilute polymer solution is introduced into a solvent stream flowing through the column. As
the dissolved polymer molecules flow past the porous beads, the can diffuse into the internal pore structure
of the gel to an extent depending on their size and the pore size distribution of the gel. Larger molecules
can enter only a small fraction of the internal portion of the gel, or are completely excluded; smaller
polymer molecules penetrate a larger fraction of the interior of the gel. The different molecular species are
eluted from the column in order of their molecular size. Therefore, the larger molecules leave the column
first, and the smaller molecules, latter.
A specific column or set of columns (with gels of different pore sizes) is calibrated empirically to give such
a relationship, by means of which a plot of amount of solute versus retention volume, such as the ones
obtained for this laboratory, can be converted into a molecular-size-distribution curve. For convenience,
commercially available narrow-distribution polystyrenes are often used as standards.
In order to obtain relevant information from the GPC chromatogram, it is necessary to convert it into a
Molar Mass Distribution Curve (MMD) curve. Knowledge of the relationship between molar mass (M) and
elution volume (Ve) – volume of solvent required to elute a particular polymer species from the point of
injection to the detector – is needed to do so. Such relationship comes from the expression for total elution
volume (Ve),
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Ve = V0 + Vi exp(-As * L/2)
(1)
Where Ve = Total Elution Volume
V0 = Void Volume (Volume in the system outside the porous beads)
Vi = Internal Volume (Volume of solvent inside the beads)
As = Surface area per unit pore volume
L = Mean molecular projection of the molecule when free in solution
Since L is proportional to the molar mass (M), this is the elution volume to molar mass relationship needed.
This expression arises from the thermodynamics on which the Gel Permeation Chromatography is based
on. First, it is known that for separations exclusively performed by size exclusion, the enthalpy change
should be zero. Recalling that G = H - TS, if H=0, then G = - TS. It is also known that the
relationship between the Gibbs Free Energy and the equilibrium constant is G = - RT ln K.Then if we
equate this two Gibbs Free Energy expressions we have - TS = - RT ln K. Solving for K we have the
following expression for K as a function of Entropy (S), K = exp (S/R). But what is all this effort in
calculating K? The answer comes from the original expression for the total elution volume, which is the
sum of the void volume (volume in the system outside the porous beads) and the internal volume (volume
of solvent inside the beads). However, the term for internal volume is dependent of how deep the molecule
penetrates inside the pores in the bed. Thus, the term K represents the fraction of internal pore volume
which has been occupied by the molecule, but it is also viewed as an equilibrium contant. Then Ve = V0 +
Vi*K. If the expression for K in terms of entropy is substituted in this equation, we have
Ve = V0 + Vi* exp (S/R).
(2)
Methods of statistical thermodynamics based on models of coils entering pores of different shapes are used
to obtain an alternative expression for S. The result has the general form S = -R As * L/2, which is
inserted in equation (2) to give equation (1) described above.
Now, GPC is not a primary method as usually practiced. As it is mentioned above, it requires calibration in
order to convert raw experimental data into molecular weight distribution. The raw data in gel permeation
chromatography consist of a trace of detector response (e.g. differential refractometer, continuous
viscometer, etc) – which is proportional to the amount of polymer in solution – and the corresponding
elution volumes.
A series of commercially available anionically polymerized polystyrenes is particulary well suited to be
used as standard for calibration, since anionic-polymerized polystyrene can be produced as closely
monodisperse polymers and for a wide range of molecular weights. When such sample is injected into the
GPC column set, the resulting chromatogram is narrower than that of a whole polymer, but is not a simple
spike because of band broadening effects and dispersion along the columns, and because the polymer
standard is itself not exactly monodisperse. Since the distribution is very narrow, no serious error is
committed by assigning the elution volume corresponding to the peak of the chromatogram to the
molecular weight of the particular polystyrene. The peaks appear in decreasing order of molecular weight,
being the larger molecules the first to be eluted. A “cocktail” of several polystyrenes covering different
molecular weights is usually used and result in a peak of each one of them.
Once every molecular weight is assigned to its elution volume, a plot of log(M) versus elution volume can
be constructed. A straight line should result and this is known as the calibration curve.
After having the calibration curve ready, an analysis of the unknown polymer can be performed. Again a
plot of the reading signal at the end of the set of columns versus the elution time is the result of the run.
Elution time can be converted into elution volume if the pump flow is known. Note that the calibration
curve to be used must cover the same range of elution volume as the known sample. It is also important to
consider a baseline to which the detector response will be referred; this will be considered the origin (y=0)
and the deviation will be proportional to the concentration of polymer. These recorder divisions are in fact
the product NiMi, the product of the number of molecules of “i” molecular weight times “i” molecular
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weight. If the unknown sample and the standard are compared at the same elution volume, it is possible to
obtain Mi for such volume, applying the linear fitting expression for log(M) as a function of Ve. Once all
Mi’s are determined, two columns – Ni and NiMi2 - can be obtained for each molecular weight. Therefore,
Mn = Sum (NiMi) / Sum (Ni)
(3)
Mw = Sum (NiMi2) / Sum (NiMi)
(4)
PDI = Mw/Mn
(5)
and
Where Mn stands for Number Average Molecular Weight, Mw for Weight Average Molecular Weight and
PDI is the Polydispersity Index.
Conclusions
GPC is an extensively used technique to determine the molecular weight distribution of polymers.
However, caution should be used during its application due to possible sources of error. Some of them are
due to the PS standards which may not be exactly monodisperse. Other possible source of error comes from
the election of the baseline; different baselines may result in different values of Molecular Weight
Distribution (MWD). Errors may also arise from the precision of the detector and from the presence of
branched polymers, since they are more compact than linear molecules of the same molecular weight and
therefore have smaller hydrodynamic volumes.
Bibliography
Rudin, A. “The Elements of Polymer Science and Engineering”, Academic Press (1998), USA
Billmeyer, B. “Textbook of Polymer Science” , Wiley-Interscience (1984), USA
Young R and Lovell P, “Introduction to Polymers”, Stanley Thornes (2000) UK
Collins, E et al. “Experiments in Polymer Science”
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