GPC Theory: Viscometer Detectors
The most common differential viscometer design is the 4-capillary bridge design invented by Dr. Max Haney, founder of Viscotek. Four capillary tubes R1 to 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 ηo 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 ηO
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 the refractive index theory pages, 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. 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 GPC/SEC system measures [η] and M 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 BSA (bovine serum albumin) 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 plot 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.
Systems for GPC/SEC:
The Viscotek TDAmax is a complete, temperature controlled, advanced, multidetector GPC/SEC system suitable for all macromolecular applications, particularly research. It consists of three unique and complementary components – The Triple or Tetra Detector Array (TDA), the GPCmax integrated solvent and sample delivery module and the OmniSEC software.
The Viscotek 270max is a modular advanced multi detector detector system that operates at ambient temperature. It is perfect for the routine full characterization of natural and synthetic polymers, copolymers and proteins.
The Viscotek RImax is a modular, conventional calibration system. It offers simple operation and full upgradeability to advanced detection. Designed for routine GPC/SEC and teaching purposes. Operates with the same powerful OmniSEC software as used in the advanced systems.
