Translational diffusion of liquids at surfaces of microporous materials: Theoretical analysis of field-cycling magnetic relaxation measurements

We present a theory of nuclear-spin relaxation appropriate to the case of a mobile liquid dipolar spin diffusing in a quasi-two-dimensional model porous system in the presence of rare paramagnetic impurities fixed at the surface of the pores. This theory predicts that the ${}^1\mathrm{H}$ spin-lattice relaxation rate will be linear in two parts when plotted as a function of the logarithm of the magnetic-field strength and the slopes of these distinct linear regions should be in the ratio 10:3. The theory predicts also a typical pore size dependence for such a rate. The theory is tested at several temperatures using acetone, acetonitrile, dimethylformamide, and dimethylsulfoxide on microporous chromatographic glass beads that have paramagnetic ion impurities at the level of 40 ppm. ${}^1\mathrm{H}$ spin-lattice relaxation rates are recorded over magnetic-field strengths corresponding to ${}^1\mathrm{H}$ Larmor frequencies between 0.01 and 30 MHz using a field-switched magnetic relaxation dispersion spectrometer. The data support the theory quantitatively. The diffusion constant $D_{I\perp }$ for the proton-bearing molecule perpendicular to the normal of the pore surface is found to be nearly a factor of 10 smaller than in the bulk solvents. It is characterized by a small activation energy similar to those in the bulk solvent. These results demonstrate that magnetic relaxation dispersion at low magnetic-field strengths in high-surface-area heterogeneous systems may be quantitatively understood in terms of the parameters of the spatial confinement and the local translational dynamics.