Conventional size exclusion chromatography (SEC) relies upon the use of calibration standards of peak molecular weight (Mp), which are used to produce a plot of log molecular weight (MW)
versus retention volume (VR)1-2. This is fitted with a polynomial function that can be used to
assign a MW to any retention volume within the range of the calibrants used. Due to its simplicity, this is by far the most commonly used form of SEC calibration, but conventional calibration has several limitations that will affect its application to analysis of more complex architectures, particularly copolymers and non-linear structures.
h
molecular weight.3 Therefore, in order for an accurate MW to be calculated, the calibration
standards should be of the same topology and repeat unit as the analyte, since any change in functionality or architecture would be expected to have some effect on the hydrodynamic volume.1 Clearly, even if only a few samples of different monomers or architecture are
synthesized, a specific calibration for each system would require an extensive set of known MW standards, which may or may not be commercially available, or well defined enough to accurately be used as a calibrant. Therefore, the majority of molecular weight averages quoted in literature will be an apparent value, based on PS, PMMA or PEO standards, giving an approximation of the real MW for the polymers analysed.
Crucially, for the work presented here, different non-linear structures of identical molecular weight and repeat unit will have different hydrodynamic volumes, and therefore different retention volumes3-6. The same will also be true for polymers with differing degrees of
branching or different types of branching (for example, a star polymer compared to a long chain branched polymer). Therefore a different approach to calibration utilizing two or more detectors, which will be relevant for a larger number of cases, is often necessary for polymers with non-linear structures. Two widely used multi-detector calibrations strategies are Universal Calibration6-7, which combines a concentration-sensitive detector with a viscometry
detector, and Triple Detection2, 4, 8, which uses concentration, viscometry and light scattering
detectors. Due to the insensitivity of light scattering detectors to low molecular weight polymers9-10, applicability of triple detection to such species is limited. For this reason triple
detection will not be discussed in relation to this work.
2.1.2 Universal Calibration
Universal calibration uses a viscometry (VISC) detector along with a concentration-sensitive detector (usually a differential-refractive index, DRI) to give a calibration that is more relevant for a range of topologies, architectures and repeat unit functionality. This method assumes that separation in SEC is dependent only on hydrodynamic volume, which is related to intrinsic
Equation 2.1: Where [η] is intrinsic viscosity, Vhis hydrodynamic volume, M is molecular weight and K is a
constant whose value is independent of polymer structure.6
This relationship, and the assumption that SEC separates molecules only by their
hydrodynamic volume, suggests that plotting log [η].M versus elution volume will be
equivalent to a plot of log Vhversus VR, and the calibration curve obtained will be relevant for
polymers of different architecture and functionality. The success of universal calibration
(while still far from being universal) can be seen in Figure 2.1, showing a plot of log [η].M
against VR, in which samples of differing composition, architecture and degrees of branching
are fitted well by a single calibration. This technique has proved valid for non-linear polymer topologies ranging from long chain branching seen in polyethylene11 to hyper-branched
poly(methyl methacrylate)12.
Figure 2.1: An example of universal calibration, plotting the product of intrinsic viscosity and molecular weight against retention volume for polymer samples of various architectures. Adapted from reference.6
In addition to giving a calibration more relevant to branched structures, viscometry detection can be used to generate Mark-Houwink plots. The Mark-Houwink-Sakurada equation is shown in Equation 2.2:
Equation 2.2: Where M is molecular weight, [η] is intrinsic viscosity and K and αare the Mark-Houwink constants (note that K is not the same constant as appears in Equation 2.1).2
A Mark-Houwink plot is constructed by plotting intrinsic viscosity against MW, on logarithmic scales. This provides a comparison of the intrinsic viscosity across the entire molecular weight distribution (MWD) of the polymer sample. This results in a simple but qualitative method for judging the extent of branching in polymer systems, if a linear standard of similar MW can be used for comparison: a branched polymer will have smaller hydrodynamic volume, and a lower IV than a linear polymer of the same MW, therefore a polymer with consistent branching would be expected to have a lower IV across its entire molecular weight distribution than its linear counterpart.
The slope, α (commonly known as the Mark-Houwink exponent), of a Mark-Houwink plot will
give some information on the conformation of the polymer in dilute solution, with Mark-
Houwink theory linking α values to different architectures in solution.2 These are tabulated in
Table 2.1, with the relationship between slope of the Mark-Houwink plot and polymer architecture demonstrated in Figure 2.2.
Architecture α
Rigid rod -
Linear random coil
(good solvent) 0.5 < α < 0.8 Linear random coil
(θ conditions) 0.5 Random branching (good solvent) 0.33 < α < 0.5 Random branching (θ conditions) 0.33 Hard sphere 0
0 << 0.33 0.33 << 0.5 lo g IV log M 0.5 << 0.8 [] ~M
Figure 2.2: The relationship between the slope of a plot of intrinsic viscosity (IV) against molecular weight (M) to polymer architecture. Adapted from reference.2
The conformation a polymer assumes in dilute solution will be linked to its degree of branching. Linear polymers would be expected to exist as random coils in good solvents,
which is indicated by an α value greater than 0.5, implying a constant increase in intrinsic
viscosity with molecular weight. With increasing branching in polymer samples with the same
molecular weight distribution, the value of α would be expected to decrease, as increasing
molecular weight has less influence on hydrodynamic volume, and therefore IV. If branching
increases to the point that a polymer system becomes crosslinked, α values tending to zero
would be expected, as the molecule becomes more similar to a hard sphere. Generally
speaking, a reduction in α for a given molecular weight is indicative of a decrease in the
intrinsic viscosity and hydrodynamic volume.
2.1.3 Semi-quantitative descriptions of branching by SEC with
viscometry detection
Zimm and Stockmayer used the theory that mean-squared radius of gyration (Rg2) will be
decreased as a result of branching13-14 – a theory that provides the basis for the triple
detection method of column calibration – and defined this reduction with the contraction factor, g (Equation 2.3):
Equation 2.3: The contraction factor g, where subscripts B and L denote the mean-square radius of gyration for branched and linear samples, respectively. Subscript M refers to values of the same molecular weight.13
Since the radius of gyration can only be measured using a multi-angle light scattering (MALS) experiment, which will be difficult to measure at low MWs, a contraction factor can also be measured using viscometry detection. This is defined by the ratios of intrinic viscosities of linear and branched samples, and denotedg’(Equation 2.4):
Equation 2.4: The contraction factorg’. Note that the ratio of intrinsic viscosities of linear and branched samples with the same molecular weight (subscript M) isnotequal to the ratio of intrinsic viscosities of linear and branched samples with the same retention volume (subscript VR).15
This method gives an indication of the extent of branching through comparison of a range of polymers to a linear standard, rather than an absolute branching number. While linear polymers should have a g’of 1, increasing branching will lead to greater deviation from the viscosity of the linear standard, and therefore a lowerg’value.