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Theory Answers
How does separation work in Gel Permeation Chromatography?
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Tetra Detector Array
A revolutionary, integrated multiple-detector device designed for the characterization of natural and synthetic polymers and copolymers, proteins, protein conjugates, excipients and other macromolecules. Comprised of modular Refractive Index, Ultra-Violet, Low Angle Light Scattering and Viscometer detectors.
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GPC Theory: Viscometer Detector |
Design
The most common DV design is the 4-capillary
bridge design invented by Dr. Max Haney, founder of Viscotek. Four capillary tubes
R1–R4 with internal diameters of approximately 0.25
mm are arranged in a balanced bridge configuration,
analogous to the Wheatstone bridge common in electrical
circuits (Figure 1). Differential pressure transducers
measure the pressure difference DP across the
midpoint of the bridge and the pressure difference IP
from inlet to outlet. A delay volume is inserted in the
circuit before capillary R4, in order to provide a reference
flow of solvent through R4 during elution of the
polymer sample. The requirements of the delay volume
are:
- It must have internal volume larger than the net
elution volume of the GPC column.
- The flow resistance must be negligible compared to
the capillary resistances.
The capillary tubes are chosen so that the flow
resistances are almost equal. In this case, the DP
output signal will be nearly zero and most of the pump
pulsations will be cancelled out in the differential
bridge measurement. DP will respond to the viscosity of the sample as it elutes from the GPC, as shown in
Figure 2. The first peak corresponds to the sample as it
elutes into capillaries R1, R2 and R3, while solvent
flows through capillary R4. The second, negative peak
is the breakthrough in the delay volume. At this point
in time, R4 contains the sample and R1, R2 and R3
contain solvent. The breakthrough peak is not
required for the calculation and is simply an artifact of
the measurement. A clever innovation in the viscometer
design that eliminates this breakthrough peak
has been recently patented.
Theory
Poiseuille’s Law of flow through a tube relates the
pressure drop P to the flow rate Q, the viscosity η and
the resistivity R of the tube:
Referring to Figure 1, the DP signal is equal to the difference between the pressure
drop across R3 and the pressure drop across R4 + Delay. During the elution of sample
(first peak in Figure 2) the following equation expresses DP in terms of Poiseuille’s
Law.
Q+ is the flow rate through the positive flow circuit, Q– is the flow rate through
the negative flow circuit, η is the viscosity of the sample, and η0 is the viscosity
of the solvent. Likewise, IP can be expressed as follows:
Dividing equation 2 by equation 3 yields:
The ratio of flow rates through the parallel flow circuits can be calculated in
terms of the relative resistances of each circuit.
At this point we apply the assumption that the capillary tube resistances are equal.
Then equations 4 and 5 combine to give the following expression relating viscosity
of sample η and viscosity of solvent η0.
The specific viscosity of the solution is defined as:
Insertion of this definition into equation 7 yields
the basic equation of the DV:
Viscosity Functions
The DV permits the accurate and sensitive calculation of specific viscosity of the
eluting polymer sample. However, the function of primary interest is the intrinsic
viscosity, which is defined as the ratio of specific viscosity to concentration
in infinitely dilute solution.
Classically the intrinsic viscosity is determined by extrapolation of the ratio
ηsp/C through various concentrations to zero concentration. This is impractical
for chromatographic detection and it also turns out to be unnecessary. For the low
concentrations in the range where GPC is practical, the single point estimation
of intrinsic viscosity due to Solomon and Ciuta is sufficiently accurate.
In refractive index, it was shown how the refractive
index detector allows the calculation of the concentration profile across the chromatogram,
Ci. In light scattering, it was shown how adding the light scattering detector to the RI detector
allows the calculation of the molecular weight profile across the chromatogram.
Equation 9 yields the specific viscosity profile, ηsp,i. The intrinsic viscosity
profile can then be calculated as follows:
The specific viscosity profile is offset by an amount σ, corresponding to the offset
of the viscometer detector relative to the RI detector. The weight-average intrinsic
viscosity is of particular importance because it corresponds to the bulk intrinsic
viscosity of the sample ie., that which would be measured in a conventional glass
tube viscometer. This is proven by the following derivation:
Applications: Universal Calibration
The earliest use of the viscometer detector was for Universal
Calibration, a column calibration method of determining molecular weight
distribution that does not require the standards and samples to have identical structures.
Universal Calibration still has utility in certain applications, particularly with
samples having low molecular weights and/or low values of dn/dc. However, for the
majority of polymers, light scattering is preferred for determining molecular weight.
The viscometer detector is still very useful for measuring other polymer properties;
particularly those related to size or structure,
hence the popularity of the triple
detector system. Two triple detector application areas will now be discussed:
Applications: Hydrodynamic Radius
Einstein showed that the viscosity of a solution is related to the hydrodynamic
radius of the particles in the solution.
φ is the volume fraction of particles in the total volume of the suspension. By
converting φ to concentration units, it can be shown that equation 14 actually relates
the intrinsic viscosity, molecular weight and hydrodynamic radius as follows:
The triple detector SEC measures [η] and M7 absolutely, so Rh is thereby absolutely
derived. This measurement is useful for polymers, where the distribution of Rh can
be determined. In Figure 3 the measured Rh is overlaid with the weight fraction
distribution. It can be seen that the Rh distribution is measurable, with good precision
over the entire distribution of this polymer sample.
Rh measurement is especially useful for proteins, as shown in Figure 4, where the
Rh can be computed accurately for the dimer and trimer, as well as the monomer unit.
Applications: Branching
Intrinsic viscosity is related to the degree of long chain
branching in polymers through the following factor:
[η]M,br denotes the intrinsic viscosity of the branched polymer at molecular weight
M, and [η]M,lin is the intrinsic viscosity of the corresponding linear polymer at
the same molecular weight M. ε is a structure value having an average value of approximately
0.8. An example of branching calculations is shown in Figures 5 and 6. Figure 5
shows the overlay of intrinsic viscosity versus molecular weight plots (‘Mark-Houwink’
plots) for linear and branched PVA polymers. Even though
the amount of branching in this type of polymer is quite small, the Mark-Houwink
plots shows clearly the difference in structure when compared to the linear molecule.
The ratio for g’ in equation 16 is calculated from this data. Figure 6 shows the
branching distribution overlaid with the weight fraction distribution of the branched
PVA sample. The number of branches is shown on the left hand axis.
Conclusion
The viscometer detector provides the final important part of the triple detection
system. We have seen that the RI detector provides an accurate concentration profile
and the light scattering provides accurate molecular weight. The viscometer now
provides the all important structural data, allowing GPC/SEC to be used to determine
such parameters as branching in polymers or hydrodynamic size differences in proteins.
No single or dual detector combination can so easily provide these important parameters
— the power of triple detection lies in the combination of complementary information
provided by all three detectors.
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