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How to transfer a chromatogram into the molar mass distribution.
Molar masses cannot be measured directly by gel permeation/size-exclusion chromatography (GPC/SEC). The direct results of a GPC/SEC run are chromatograms or elugrams, which show the detector signal intensity versus the elution volume. These elugrams can be easily compared to each other on the same column set. However, without additional information (such as that from a calibration curve) the molar mass averages and polydispersity are not available. These properties can be determined if the relationship between elution volume and molar mass is known. This article explains more.
Many macroscopic properties of macromolecules can be derived from their molar mass distribution (MMD). In contrast to molar mass averages such as Mn or Mw, which provide reduced information, the MMD describes the complete sample characteristics. Two samples can have the same molar mass averages but have very different molar mass distributions and therefore macroscopic behaviour.
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The difference between an MMD and a chromatogram can be easily understood using the following example: Two laboratories inject the same sample on different instruments. They have a different number of columns with different lengths and inner diameter. This results in two different chromatograms, which is the primary information. Without additional information it is not possible to decide if these chromatograms result from the same sample or not. It is not even possible to tell from the chromatogram if two peaks in the sample correspond to a species with a narrow or a broad molar mass distribution. A broader-looking peak can have a narrower molar mass distribution than a smaller peak, if the broad peak elutes in a column region with high resolution. However, inter-laboratory comparison and distribution information is easy to achieve if MMDs are compared. This process eliminates the experimental conditions. Ultimately only correctly calculated molar mass distributions allow the direct inter-laboratory and long-term comparison of samples and sample properties.
The primary information provided by gel permeation/size-exclusion chromatography (GPC/SEC) measurements is the apparent concentration distribution (chromatogram, h[V]). This is a convolution of sample-related parameters and experimental conditions. The molar mass distribution can be calculated from the signal heights in the elugram by the slice method: Therefore the eluted peak is separated into equidistant time, or volume slices.1,2
For the transformation the retention axis (x-axis, volume or time) will first be changed into a molar mass axis. This is achieved using the information from the molar mass versus elution volume relation.
In the second step the y-axis is converted into mass fractions in one molar mass interval, w(log M). When determining the correct molar mass distribution, the normalized signal height (hi) must be divided by the slope of the calibration curve at every elution volume. This correction can only be neglected in the case of strictly linear calibration curves over the complete separation range. However, this is a feature that even most commercial linear mixed bed columns usually do not exhibit. As soon as a typical GPC/SEC fit function (for example, cubic fit, polynomial 3, or polynomial 5) is used to achieve higher result accuracy, the slope correction is necessary because the data recording occurs linearly, while the molar mass change is not linear. In practical terms this means that for the same measured height (hi) the number of polymer chains on the high molar mass fraction of the elugram is much smaller than on the low molar mass fraction.
The differential distribution, w(M), of the molar mass (M) is defined as w(M) = dm/dM, the mass fraction (m) of the molecule in a dM interval (molar mass).
By simple transformation w(M) can be expressed by measured quantities with h(V) detector signal and σ(V) slope of the calibration curve.
The molecular weight averages can be calculated as:
Number average molecular weight:
Weight average molecular weight:
Please note that the GPC software modules of many high performance liquid chromatography (HPLC) software programmes do not perform the correction with the slope of the calibration curve. This results in incorrect molar mass distributions for all setups with typical non-linear GPC/SEC calibration behaviour. The errors caused by this neglect will increase with the width of the sample and decrease with the data recording frequency. This is dangerous when submitting GPC/SEC results to regulatory organizations such as the US Food and Drug Administration (FDA) or the European Medicines Agency (EMEA) or for Registration, Evaluation, Authorization & restriction of CHemicals (REACH) registration.
Figure 1: Chromatogram versus molar mass distribution. While a chromatogram shows the detector signal height (y-axis) versus the elution volume (x-axis), a MMD displays w(log M) on the y-axis versus log M on the x-axis. Peak shape and breadth can change during transformation depending on the slope of the non-linear calibration curve (displayed in red).
There are several options to determine a molar mass/elution volume relation. From a practical point of view, the methods can be distinguished between those that use reference materials and those which use static light scattering detectors to measure the molar mass on-line for every sample.
Figure 2: Chromatogram versus molar mass distribution. A MMD displays w(log M) on the y-axis versus log M on the x-axis. Graphs that display the signal height or similar on the y-axis for example probably do not show true molar mass distributions.
The most common technique when reference materials are used is to calibrate the system with polymer reference materials with a narrow molar distribution.3 Here different molar masses are measured and the Mp values of the reference materials are plotted versus their measured elution volume. A fit function is then chosen that describes the relation between log M and the elution volume. Because GPC/SEC separates according to hydrodynamic volume and not to molar mass, only apparent molar masses (related to the calibration standards) are obtained if the calibration standards and the samples are chemically or structurally different. The deviation for the molar mass averages and the molar mass distribution can often not be predicted. Still the results for different samples can be compared to each other. Since the method is robust and easy-to-use, many laboratories apply such procedures for quality control and sample comparison as well as for applications where absolute molar masses are not required.
Calibration methods to overcome this limitation use matching reference materials or any of the following techniques:
Another popular approach is the use of on-line static light scattering detectors, such as multi-angle light scattering (MALS), right-angle light scattering (RALS), and low-angle light scattering (LALS).6 The major advantage here is that a combination of a concentration detector with a light scattering detector measures the molar mass at every elution volume on-line with each sample. It is therefore not necessary to establish a calibration curve with molar mass standards.
However, for accurate molar mass determination several sample and instrument parameters need to be known independently on the instrument:
If these parameters are known it is then possible to measure the molar mass at every elution volume and to use this to tranfer the chromatogram into a molar mass distribution.
1. A. Striegel, W.W. Yau, J.J. Kirkland, and D.D. Bly, Modern Size-Exclusion Liquid Chromatography: Practice of Gel Permeation and Gel Filtration Chromatography (John Wiley & Sons, New York, USA, 2009).
2. E. Schröder, G. Müller, and K.-F. Arndt, Polymer Characterization (Hanser, Munich, Germany, 1998).
3. D. Held, The Column4(6), 18–21 (2008).
4. D. Held and P. Kilz, The Column9(14), 11–15 (2013).
5. D. Held and P. Kilz, The Column8(2), 12–16 (2012).
6. D. Held and P. Kilz, The Column5(4), 28–32 (2009).
Daniela Held studied polymer chemistry in Mainz (Germany) and works in the PSS software and instrument department. She is also responsible for education and customer training.
Peter Kilz studied polymer chemistry in Mainz (Germany) and Liverpool (UK). He is one of the founders of PSS and head of the software and instrument department. He is also involved in customer support and training.