Unified Theory of Orientational Relaxation

Abstract

A mathematically simple model for orientational relaxation in liquids is presented. This theory, expressed in terms of Mori's formalism for generalized hydrodynamics, is developed with three orientational variables interrelated by three coupled linear transport equations. The three variables are $\mathcal{D}\mathrm{,d}\mathcal{D}\mathrm{/dt,}\mathrm{and}{d}^2\mathcal{D}{\mathrm{/dt}}^2,$ where $\mathcal{D}$ is a relevant Wigner rotational transformation function. In appropriate limits the theory reduces to that for rotational diffusion, gas‐like extended rotational diffusion (GLED) and solid‐like oscillatory rotations (SLOR) or cell model motions. This theory, which we call the pseudo‐GLED‐SLOR theory, therefore gives a unified picture, encompassing all limits, of molecular rotations in liquids. The assumptions, and consequently the results, differ slightly from those usually introduced in the GLED and SLOR theories; the various limiting forms of molecular reorientation are obtained in the present theory via restrictions on the quantities, $\mathcal{D}\mathrm{,d}\mathcal{D}{\mathrm{/dt,\; d}}^2\mathcal{D}{\mathrm{/dt}}^2,\mathrm{and}{d}^3\mathcal{D}{\mathrm{/dt}}^3,$ whereas the usual quantities of interest are $\mathcal{D},$ the molecular angular velocity, the intermolecular torques, and the rate of change of torques. It is for this reason that we have introduced the term ``pseudo'' in describing the theory. The only restriction that must be placed on the theory is that $d^3\mathcal{D}{\mathrm{/dt}}^3$ changes impulsively, but of course, different assumptions concerning the magnitudes of the other variables lead to different limiting physical situations. In appropriate limits, the present theory exhibits most of the interesting features of the rotational diffusion theory, Gordon's extended rotational diffusion theory, Steele's inertial effects and Ivanov's jump theory; furthermore, it predicts rotational or torsional side bands in the frequency spectrum of the molecular orientational autocorrelation function. In the far wings it predicts an ${\omega }^{\text{−6}}$ frequency dependence. The theory treats $\mathcal{D}\mathrm{,d}\mathcal{D}\mathrm{/dt,}$ and $d^2\mathcal{D}{\mathrm{/dt}}^2$ as dynamical quantities and only $d^3\mathcal{D}{\mathrm{/dt}}^3$ is treated as a Markoffian variable. A careful analysis is presented of the differences between the correlation time $${\tau }_{\theta }={\displaystyle {\int }_0^{\infty }}〈\mathcal{DD}\mathrm{(t)〉\; dt〈}\mathcal{DD}{〉}^{\text{−1}},$$ where $〈\mathcal{DD}\mathrm{(t)〉}$ is the autocorrelation function of $\mathcal{D},$ and the spectral half‐width ${\mathrm{\Delta \; \omega }}_{\text{1/2}}.$