Molecular weight plays a critical role defining the physical properties of a polymer, like mechanical strength, degradation rate, and solubility. The two methods most commonly used method for determining molecular weight are size exclusion chromatography (SEC) and end group analysis, which are frequently employed to measure the number-averaged molecular weight (Mn) of a polymer. However, both methods have their limitations. SEC is restricted by the choice of solvent, polymer absorption to the column, and the consumption of large amounts of solvent. On the other hand, NMR end group analysis is typically limited to relatively small molecular weights.

To overcome these limitations, one can also use pulsed field gradient NMR, which offers the possibility to measure the self-diffusion coefficients of molecules. To learn more about the principles of pulsed field gradients, you can read through our blog post series titled “Gradients in NMR Spectroscopy” [1]. According to the Stokes-Einstein equation, the self-diffusion coefficient depends on the hydrodynamic radius of a particle as

where D is the diffusion coefficient, *k* is the Boltzmann constant, *T* is the absolute temperature, Î· is the solvent viscosity, *R*_{H} is the hydrodynamic radius. The hydrodynamic radius can be related to the molecular weight by using the empirical Rouse-Zimm model, which utilizes the equation (with arbitrary parameters b and Î½):

This enables the rewriting of the Stokes-Einstein equation (where b’ and Î½ are arbitrary parameters):

The equation can be transformed into a linear form as follows:

By plotting log(D) against log(M), a linear function can be obtained with the slope v and the *y*-axis intercept log(bâ€™). This type of plot can be used for linear calibration to define molecular weight, which is valid for the given polymer at a specific concentration and temperature, as shown in Figure 1.

Figure 1: graphical representation of the equation log(D)=log(bâ€™)-n log(M)

To demonstrate the feasibility of using benchtop NMR systems, we performed calibrations for polystyrene (PS) and polymethylmethacrylate (PMMA) standards at a concentration of 5 mg/mL of CDCl_{3} (0.3 wt%). The Spinsolve software available with the instrument was used to automatically determine D, as shown in Figure 2 for a PMMA sample. The values of the parameters Î´, Î”, and the power of the gradient were adjusted depending on the MW of the polymers. For polymers with high MW, it was necessary to use strong gradients to keep Î´ and Î” as short as possible, in order to minimize the effects of T_{1} and T_{2} relaxation. The number of scans was kept constant at 8 for each measurement, as the mass of polymer per volume of solvent was the same for each polymer solution. Regardless of the set of parameters used, the total measurement time was less than 10 minutes, which is competitive with SEC where a run typically takes around 40 minutes. The table displays the MW and D for each polymer measured on a Spinsolve 80. As shown in Figure 3, D values as low as 1.2 10^{-11} mÂ²/s were measured without the need to use the full range of gradient strength.

Figure 2: Screenshot of the Spinsolve software displaying the selected parameters and the plot of the 8 steps of the PGSTE pulse sequence for PMMA with a molecular weight (MW) of 90000 g/mol at a concentration of 5 mg/mL in CDCl3. The graph on the right displays the Stejskal-Tanner plot, depicting ln(integral) as a function of Î³Â² gÂ² Î´Â² (Î” – Î´/3). The data points are fitted with a linear function, from which the slope is used to obtain the displayed diffusion coefficient.

The log(D) vs log(M) plot shown in Figure 3 shows a robust linear correlation between molecular weight and diffusion coefficient for PS and PMMA, supported by high correlation coefficients that attest to the data’s reliability. The coefficients log(b’) and -Î½ obtained from the linear fits can be used to determine molecular weights of unknown samples based on their diffusion coefficients. The good agreement with literature values [2] verifies that the calibration curve can be transfered to benchtop NMR instruments. The availability of strong gradients on Spinsolve instruments allows for the measurement of diffusion coefficients for high molecular weight polymers, which tend to diffuse slowly. These findings highlight the potential of benchtop NMR systems for polymer analysis and characterization.

Figure 3: Diffusion coefficient of PS and PMMA for different molecular weights. The measurements were performed on a Spinsolve 80 delivering a maximum pulsed gradient of 0.5 T/m. The plot on the right shows log(D) vs log(M) for PMMA (blue) and PS (orange). The coefficient log(bâ€™) and â€“ Î½ were determined as -7.407 and -0.5954 for PS and -7.438 and -0.5777 for PMMA with for both polymers excellent correlation coefficient: 0.998 and 0.998 for PS and PMMA, respectively.

The independence of a polymer’s diffusion coefficient on the instrument or frequency makes the calibration curve universal and transferable from one lab to another, as reported in a previous publication by the group of Prof Junkers [3]. This simplifies the analysis process in comparison to SEC, which is column-dependent. To demonstrate the validity of the calibration curve of the paper [3], a standard polyethylene glycol (PEG) sample with a MW of 41300 and Mn of 31700 g/mol was analyzed on the Spinsolve at a concentration of 1 mg/mL in D_{2}O. The resulting diffusion coefficient was determined to be 3.45 10^{-11} mÂ²/s. By using the calibration values of -8.23 and 0.49 for log(bâ€™) and Î½, respectively, from the publication [3], a MW of 36200 g/mol was extracted, which is close to the expected value.

The results presented in this blog post indicate a good agreement between the diffusion coefficient obtained from the high-field instrument and the benchtop NMR Spinsolve for the polymers PS, PMMA, and PEG, suggesting low interlab variability. In contrast to SEC, PGSTE measurements are faster, and the calibration curves are transferrable across different labs, moreover, it can be applied to any solvent and polymer without limitations. The availability of strong gradients on the Spinsolve spectrometers (0.5 T/m) enables measurement of very low diffusion coefficients, making it possible to analyze high molecular weight polymers.

**References:**

[1] technical notes explaining gradient in NMR Spectroscopy with benchtop NMR Spinsolve:

Part 1: The Basics: https://magritek.com/2016/06/19/gradients-in-nmr-spectroscopy-part-1-the-basics/

Part 2: Pulsed Gradients: https://magritek.com/2016/06/27/gradients-in-nmr-spectroscopy-part-2-pulsed-gradients/

Part 3: Encoding for Displacement using Pairs of Pulsed Gradients: https://magritek.com/2016/07/03/gradients-in-nmr-spectroscopy-part-3-encoding-for-displacement-using-pairs-of-pulsed-gradients/

Part 4: A short Intermezzo on Diffusion: https://magritek.com/2016/07/10/gradients-in-nmr-spectroscopy-part-4-a-short-intermezzo-on-diffusion/

Part 5: The Pulsed Gradient Spin Echo (PGSE) Experiment: https://magritek.com/2016/07/18/gradients-in-nmr-spectroscopy-part-5-the-pulsed-gradient-spin-echo-pgse-experiment/

Part 6: Mixture Analysis by Diffusion Ordered Spectroscopy (DOSY): https://magritek.com/2016/07/25/gradients-in-nmr-spectroscopy-part-6-mixture-analysis-by-diffusion-ordered-spectroscopy-dosy/

[2] Diffusion in Polymer Solutions: Molecular Weight Distribution by PFG-NMR and Relation to SEC; X. Guo, E. Laryea, M. Wilhelm, B. Luy, H. Nirschl, G. Guthausen; Macromol. Chem. Phys.; Volume 218, Issue 1 (2017) 1600440 (https://doi.org/10.1002/macp.201600440)

[3] Solvent-Independent Molecular Weight Determination of Polymers Based on a Truly Universal Calibration; P.-J. Voorter, A. McKay, J. Dai, O. Paravagna, N. R. Cameron, T. Junkers; Angew. Chem. Int. Ed.; Volume 134, Issue 5 (2022) e202114536 (https://doi.org/10.1002/ange.202114536