Theory of intramolecular spin relaxation by translational diffusion in locally ordered fluids

Abstract

The theory of spin relaxation induced by translational diffusion of small molecules or ions in locally ordered fluids, developed in parts I and II of this series, is extended to cylindrical geometry as, for example, in polyelectrolyte solutions or in hexagonal lyotropic liquid crystals. The theory is based on the Smoluchowski diffusion equation with nonuniform potential of mean force and translational diffusivity and on the cylindrical cell model. Formally exact closed-form expressions are derived for the zero-frequency spectral density associated with radial diffusion, while a general numerical algorithm is described for computing the full frequency-dependent spectral density. Several useful approximations, such as the dynamic cell approximation, the steady state approximation and the surface diffusion approximation, are formulated and their accuracy quantitatively assessed. Calculations are reported for a mean-field interaction model based on the nonlinear PoissonBoltzmann equation, with emphasis on applications of the theory to counterion spin relaxation in polyelectrolyte solutions. l. INTRODUCTION In locally ordered fluids, such as macromolecular solutions and lyotropic liquid crystals, nuclear spin relaxation can be induced by modulation, through translational motion of the fluid molecules, of the magnitude and orientation of the residual intramolecular spin-lattice coupling, which is only partially averaged by local molecular motions near an interface. In part I I-1] and part II I-2] of this series of papers, we developed a theoretical framework, based on the so-called continuous diffusion model (CDM), for treating spin relaxation in locally ordered fluids bounded by planar interfaces. In the CDM theory, the geometry of the fluid region and the potential of mean force acting on the spin-bearing species determine the equilibrium distribution of spins as well as their translational dynamics. The latter is characterized by a time-dependent distribution function, or propagator, which, in the CDM theory, is obtained from the Smoluchowski mean-field diffusion equation, containing also the translational diffusivity tensor of the spin-bearing species. These distribution functions are then used to construct the time correlation functions and spectral densities that determine the observable spin relaxation rates. We showed in parts I and II that the predictions of the CDM theory may differ qualitatively from those of the traditional discrete