A physical hypothesis for Cole-Davidson behavior

It is well-known that for a polar dielectric liquid exhibiting a single relaxation time, graphs of the real and imaginary parts of the permittivity against the logarithm of frequency are of the hyperbolic tangent and hyperbolic secant type, respectively. This was first predicted by Debye, but it was pointed out later by Kenneth and Robert Cole that when /spl epsi/" is plotted against /spl epsi/' for such a liquid, with frequency as parameter, then a semicircle results. However, for many liquids these predictions are not fulfilled. Debye himself only expected them to apply for frequencies lower than the mid-frequency of the dispersion region, and then only for liquids consisting of a weak solution of polar molecules in a solvent consisting of non-polar molecules. In fact, his predictions do hold for some liquids not conforming to those restrictions. The restrictions were felt by Debye to be necessary because he ignored dipolar inertia in his calculations, and he also ignored dipole-dipole interactions. They will also be ignored in this paper. However, a wide range of polar liquids exhibit dielectric characteristics which are not in conformity with Debye behavior, and for which no generally acceptable physical reasons have been put forward. When,/spl epsi/" is plotted against /spl epsi/', sometimes a skewed are results, which at low frequencies is semi-circular, but at high frequencies approximates to a straight line. Various empirical equations have been formulated to give the variation of /spl epsi/" with /spl epsi/' as the frequency is varied, and these equations can in general be analyzed to give a distribution of relaxation times, rather than the single relaxation time of ideal Debye behavior. Nevertheless, the replacement of the geometrical description by an algebraic description of the behavior does not explain the basic physical reason giving rise to it. In this paper, the hypothesis is put forward that the physical causes giving rise to the observation of the skewed arc, in particular to Cole-Davidson behavior, are the temporal and spatial fluctuations continuously occurring in the liquid, which at the molecular level is subject to severe turbulence. A consequence is that at any instant some dipoles will be more able than others to exhibit oscillatory rotary motion, i.e. to vibrate, under the influence of an applied sinusoidal field, and so to execute oscillations of greater magnitude. Even though such dipoles may be very few in number, the effect of their large oscillations could be very significant. The magnitude of the dipole oscillations might be expected to be smoothly distributed over a wide range, but for this preliminary investigation the approximation has been made that the great majority of dipoles will have the small oscillation amplitudes predicted by earlier work, while a small proportion of dipoles will be free to execute much larger oscillations. It is shown that this crude assumption does predict the observation of a skewed are quite similar qualitatively to the Cole-Davidson plot. It is intended to refine the analysis in order to find whether the Cole-Davidson plot can be predicted quantitatively on the basis of instantaneous inhomogeneities in the liquid structure.