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Dielectric Relaxation Time, a Non-linear Function of Solvent Viscosity by M D Magee

Offprinted from the Journal of The Chemical Society, Faraday Transactions II, 1974, 70 929-938

Abstract

By extending the Debye and Wirtz treatments to the case where the frictional force opposing rotation of the molecule differs from that of the bulk liquid, the relaxation time (tau) is found to depend non-linearly upon the macroscopic viscosity (eta). The resulting expression is of the form tau = tau’.eta/ (eta + eta’). When eta << eta’ it reduces to a Debye-like equation but when eta increases tau does not increase in proportion but eventually approaches a constant value. Relaxation time data for a large number of solutions are found to conform to this relationship over the viscosity range 0.3 to 200 cP.

South Devon Technical College, Torquay
Received 14th August, 1972

Using a simple hydrodynamic model in which the molecule is regarded as a sphere rotating in a continuous viscous medium, Debye found¹ that the molecular relaxation time (tau) should be linearly dependent upon the viscosity (eta) of the medium. He went on to show that, in general, this is not true. Other workers have proposed modified relationships which fall broadly into two categories, those based upon a theoretical model and those which are purely empirical. Some of the empirical equations² are quite successful in predicting the magnitude of relaxation times under some conditions, but most of them retain the linear dependence of tau upon eta which is not obeyed in practice. Of the theoretical relationships, Hill’s³ is the most fundamental. Beginning with Andrade’s theory of viscosity and extending it to the case of mixtures of liquids, she extracts a local viscosity effective between species at the molecular level. It is upon this “mutual viscosity” that the relaxation time depends.

Hill’s theory satisfactorily explains the observed five-fold increase in relaxation time of chlorobenzene in liquid paraffin solutions compared with benzene solutions, whereas the Debye relationship predicts a 350-fold increase. Evaluation of the mutual viscosity, however, requires a good deal of information not yet readily available and therefore much additional work is needed. Approximations⁴ to Hill’s equations again reduced to linear relationships between tau and eta. Various expressions² predict that the ratio of microscopic to macroscopic viscosity should fall as the volume of the solvent molecule rises relative to that of the solute molecule, but of these only the Wirtz equation is well founded.⁵ For solutes and solvents of similar molecular volume Wirtz’s equation differs from Debye’s equation only by a numerical factor. This gives a closer agreement with the magnitude of observed relaxation times and attributes the low relaxation time in viscous solvents to the difference in molecular size between solvent and solute. But the explanation of the unusually low values in liquid paraffin is unlikely to be sirnply a molecular volume effect, since the paraffin is not the rigid molecule required by theory but a mixture of flexible hydrocarbons, each segment of which is likely to behave like a much smaller molecule. Overall, macroscopic viscosity is discredited as a pointer to the behaviour of relaxation times.⁶

The object of this paper is to show that the macroscopic viscosity is still useful for predicting relaxation times providing that the non-linearity of the relationship is acknowledged.

THE DEBYE APPROACH

That the molecular relaxation time depends upon macroscopic viscosity implies that dielectric relaxation involves regions of the liquid and not just rotation of single molecules. This argument,⁷ seemed to be confirmed experimentally by Schallamach.⁸ Under certain conditions, the separate relaxation regions of the components of a binary mixture of polar liquids coalesce into a single region, even when the separate relaxation times are sufficiently far apart to yield bimodal absorption The tentative explanation is that the volume undergoing relaxation contains at least four, and perhaps many more molecules. When the relaxing region is sufficiently large that its composition is that of the bulk liquid, then a single absorption is obtained, otherwise separate relaxation peaks occur. Later work⁹ contradicted Schallamach’s conclusion, the loss curves of mixtures being separable into relaxation times characteristic of each individual molecule. Schallamach’s results were attributed to the structure-making influence of the low temperatures he used. However, these two approaches are not mutually exclusive. A molecule trapped in a partially structured region of the liquid may lose its orientation by rotation of the whole region as well as by rotation of the individual molecule within the region. In each case resistance to rotation is provided by quasifrictional forces and, neglecting local fields, the externally applied field E may be taken to be that which acts upon the dipoles in the liquid. The orientations (phi_i, theta_i) of the molecules may be represented by points on a sphere, and the rate (J_theta) at which the molecules pass through a unit length of an arbitrary latitude (theta = constant) off the sphere in the direction of increasing theta, yields the rate of the reorientation of the dipoles under the joint influence of the Brownian movement and the applied field. If theta is referred to the direction of the applied field, then J_theta is given by:

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Dielectric Relaxation Time, a Non-linear Function of Solvent Viscosity

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