Rotational Diffusion of Spherical-Top Molecules in Liquids

Abstract

The extension of the classical rotational diffusion model proposed by Gordon in his study of linear molecules is applied to the evaluation of the reorientational correlation functions, their Fourier transforms, and correlation times for fluids composed of spherical‐top moelcules. The reorientational correlation times ${\tau }_{\theta }^{\mathrm{(j)}}$ , describing the time correlation of a jth‐order spherical harmonic, are expressed as functions of the correlation time ${\tau }_J$ for the angular momentum of the molecules. These expressions are valid for all values of ${\mathrm{\omega ̄\tau }}_J$ , where $\mathrm{\omega ̄}$ is the root‐mean‐square angular frequency ${\mathrm{(kT\hspace{0.17em}/\hspace{0.17em}I)}}^{\text{1\hspace{0.17em}/\hspace{0.17em}2}}$ for a spherical‐top molecule with moment of inertia $I$ . In the Debye limit ${\mathrm{(\omega ̄\tau }}_J{\text{\hspace{0.17em}≪\hspace{0.17em}1), τ}}_{\theta }^{\mathrm{(j)}}{\mathrm{\hspace{0.17em}\gg \hspace{0.17em}\tau }}_J$ , and ${\tau }_{\theta }^{\mathrm{(j)}}$ are proportional to ${\tau }_J^{\text{−1}}{\mathrm{\hspace{0.17em}/\hspace{0.17em}\omega ̄}}^2$ ; in the perturbed‐free‐rotor limit ${\mathrm{(\omega ̄\tau }}_J{\text{\hspace{0.17em}≫\hspace{0.17em}1), τ}}_{\theta }^{\mathrm{(j)}}$ is proportional to ${\tau }_J$ and is of comparable magnitude. ${\tau }_{\theta }^{\mathrm{(j)}}$ goes through a minimum value when ${\tau }_{\theta }^{\mathrm{(j)}}{\mathrm{,\; \tau }}_J$ , and ${\mathrm{\omega ̄}}^{\text{−1}}$ are approximately equal. Magnetic spin relaxation due to motional modulation of anisotropic spin–rotational interactions is also considered in the framework of this extended diffusion model. The resulting expression for the spin relaxation times involves a correlation function containing both the time correlation of the orientation of the molecule and the angular momentum of the molecule simultaneously. This correlation function and hence the spin‐relaxation times, can be calculated on the basis of the extended diffusion model for all values of ${\mathrm{\omega ̄\tau }}_J$ without assuming any separability or independence of the correlations of the molecular orientation and angular momentum. In the Debye limit, the expression for the spin‐relaxation times reduces to that of Hubbard, and in the perturbed‐free‐rotor limit, it reduces to the result previously obtained by Bloom et al.