Monitoring polydispersity by NMR diffusometry with tailored norm regularisation and moving-frame processing.

Nuclear magnetic resonance (NMR) is currently one of the main analytical techniques applied in numerous branches of chemistry. Furthermore, NMR has been proven to be useful to follow in situ reactions occurring on a time scale of hours and days. For complicated mixtures, NMR experiments providing diffusion coefficients are particularly advantageous. However, the inverse Laplace transform (ILT) that is used to extract the distribution of diffusion coefficients from an NMR signal is known to be unstable and vulnerable to noise. Numerous regularisation techniques to circumvent this problem have been proposed. In our recent study, we proposed a method based on sparsity-enforcing l1-norm minimisation. This approach, which is referred to as ITAMeD, has been successful but limited to samples with a 'discrete' distribution of diffusion coefficients. In this paper, we propose a generalisation of ITAMeD using a tailored lp-norm (1 ≤ p ≤ 2) to process, in particular, signals arising from 'polydisperse' samples. The performance of our method was tested on simulations and experimental datasets of polyethylene oxides with varying polydispersity indices. Finally, we applied our new method to monitor diffusion coefficient and polydispersity changes of heparin undergoing enzymatic degradation in real time.