Inverse-problem-based interdiffusion quantification with NMR

Nuclear magnetic resonance (NMR) is a non-invasive technique that can be used to measure concentration profiles and diffusion coefficient during an interdiffusion process of liquids. Due to their different permeabilities, the liquids induce inhomogeneities in the magnetic field. In the case where the inhomogeneities are of the same order as the difference of their chemical shift, the line-width increases lead to important overlapping effect. Moreover, the line broadening is asymmetric relative to its peak and the line shape is unknown. Since classical deconvolution techniques cannot resolve this overlapping, an inverse-problem-based approach is proposed. A constrained nonlinear least-squares optimization problem with respect to concentration emerges from the error minimization between the simulated and measured spectra. For the spectra calculation, the magnetic field distribution is neccessary, which is described by an elliptic PDE and solved with the finite element method. The representation of the concentration is implemented using parametric curves.