The relative motion of ions in solution. I. Microdynamical models and intermolecular dipolar spin relaxation

This paper is the first in a series involving the theoretical and experimental study of the relative motion of ions in solution. In this paper we describe the general theoretical approach and make explicit calculations for model aqueous electrolytes. In all models the ions are considered to be charged hard spheres and in order to compare with real solutions effective hard sphere diameters must be estimated from crystal radii or space-filling molecular models. Our object is to provide theoretical results for these model solutions which can be tested with nuclear magnetic resonance (NMR) experiments. Therefore, we have calculated the dimensionless spectral density j̄2(ωτ) by solving the Smoluchowski equation including a force term dependent upon the ion–ion potential of mean force. Experimentally, j̄2(ωτ) can be determined from NMR measurements of the interparticle dipole–dipole relaxation rate of the nuclear spins I located on a diamagnetic ion and interacting with the electronic spins S of a paramagnetic species. Theoretically, we examined j̄2(ωτ) as a function of frequency and concentration for pairs of physically realistic ions. The influence of varying the ion size and charge as well as spin eccentricity effects are discussed. It is shown that the ionic charge strongly influences the relative motion giving values of j̄2(ωτ) which are much larger (i.e., by one or two orders of magnitude) for pairs of attractive ions and smaller for the repulsive case than those obtained for neutral particles. Furthermore, for attractive ion pairs the molecular and continuum solvent models yield very different results. The molecular solvent values for j̄2(ωτ) are frequently much larger than those given by the corresponding primitive model and for many physically realistic ion pairs the discrepancy between the models lies well outside the range of uncertainty introduced by the choice of hard sphere diameter. Hence for such systems NMR experiments should easily and clearly distinguish between these different theoretical pictures. For pairs of repulsive ions both models give similar results for dilute solutions, but the discrepancies become larger with increasing ionic strength as Debye screening of the Coulombic interaction increases the importance of the short-range potential.