Obtaining aCalibrationCurve
Polydisperse polymers in solution are fractionated according to size or hydrodynamic
volume during GPC, which is also known as size exclusion chromatography. Molecular weight
is related to the hydrodynamic volume. In GPC a dilute polymer solution is injected into a
solvent stream which then flows through a series of columns packed with porous gel beads.
Smaller molecules pass through and around the beads while larger molecules are excluded from
all but the largest pores. Thus fractionation occurs with the largest molecules eluting first. The
molecular weight of an eluting polymer molecule varies exponentially with eluting volume, the
latter of which is proportional to time under constant flow rate conditions. To obtainmolecular
weight data and convert theGPC chromatogram into amolecular weight distribution, the relation
between molecular weight and elution time is obtained from a series of polymer standards of
known molecular weight. The calibration curve is thus obtained from a plot of the logarithm of
molecular weight versus time. Given that GPC is a comparison of hydrodynamic volumes,
unknown molecular weight determinations will be relative to the calibration standards. For a
good introductory reference to polymer science, see R. J. Young and P. A. Lovell, Introduction
toPolymers.
Using PeakSimple 2.08, the result table for each of the three polystyrene standard
chromatograms was copied using DDE into Excel. The natural logarithm of molecular weight
versus time was plotted and a best fit analytical approximation to the curve was obtained from a
third order polynomial,
) (
e
tP
. This is the calibration curve relating molecular weight to elution
time.
ObtainingMolecularWeightAverages
The most common and convenient way to characterize a distribution of molecular weights
making up a polymer sample is using molecular weight averages such as, number average
molecular weight (
n
M
), and weight average molecular weight (
w
M
), as shown in the following
figure for a typical polymer chromatogram.
n
M
is defined as a sum of products of themolecular
weight of each fractionmultiplied by its mole fraction. That is:
i
i
n
MX M
å=
where
i
X
is the
mole fraction of molecules of molecular weight mass
i
M
. The weight average molar mass is
