Rotational and Translational Diffusion in Complex Liquids

Abstract

In two earlier papers, a theoretical and experimental study of the dynamics of molecules in liquids or solids by the neutron-scattering technique was reported. Special attention was given to the quasi-elastically scattered neutron line, the shape and width of which contains considerable information regarding the details of the diffusive process, its origin and nature. It was shown that the origin and formation of the quasielastic peak may be understood if the molecular motion is divided into two partial motions: (1) the motion of the center of gravity, which might be diffusive or vibratory; (2) the motion relative to the center of gravity performed by the molecular constituents, of which the protons are the most important for the case of neutron scattering (this relative motion may be of rotational, hindered rotational, or vibrational nature). In the present work the basic ideas developed in the earlier papers are evolved further. A model for the relative motion of the molecule is defined, and a neutron cross section based on this model is worked out and applied to the neutron-scattering results on glycerol and $n$-propanol. It is assumed that the molecule performs a partially hindered rotation around an effective center of gravity so that each proton may be visualized as performing a vibrational motion around a center which diffusing on the surface of a sphere. The diffusive motion might occur in small steps or in large jumplike steps. The resulting cross section for quasielastic scattering may approximately be described as a Lorentzian line with a full width at half-maximum that is a function of various parameters, such as the true self-diffusion coefficient ${D}_{\mathrm{e}.\mathrm{g}.}$, describing the motion of the molecular center of gravity, and a relative diffusion coefficient ${D}_{\mathrm{rel}}$, describing the proton motion on the surface of the sphere. The relative diffusion coefficient has three components: one each due to small and to large displacements of the protons, corresponding to small and large changes of the molecular orientational angles, respectively; and one due to vibration. It is possible to fit the calculated widths to the observed ones over the entire temperature range investigated for the two liquids, and for the momentum ranges covered in the present studies. This leads to numerical values for the various parameters entering the theory. The values obtained for the various mean lifetimes for the motions are compared with relaxation times obtained from dielectric and ultrasonic data, and certain connections are established. The neutron experiments indicate that the true molecular diffusive motion is a mixed translational-rotational-vibrational displacement. The present work is limited to a study of such systems (complex liquids of not too low viscosity, or solids) in which the rotations are more or less strongly hindered. The consequence of the various neutron observations and other evidences is that the velocity autocorrelation for translational as well as rotational motion is strongly nonexponential for shorter times. The consequences of this for the interpretation of nuclear-magnetic-resonance, dielectric, and ultrasonic data are discussed.