Inertial effects in the theory of dielectric and Kerr effect relaxation of an assembly of non-interacting polar molecules in strong alternating fields

Abstract

The ensemble averages 〈cos θ〉 and 〈cos 2 θ〉 which describe dielectric and Kerr effect relaxation are calculated for alternating fields (i) E cos ω t, (ii) E 1 cos ω 1 t + E 2 cos ω 2 t, (iii) E 1 + E 2 cos ω t dielectric fluids in which polar molecules are constrained to rotate in two dimensions. The ensemble averages 〈 P 1(cos ϑ)〉 and 〈 P 2(cos ϑ)〉 are calculated for the same fields when the polar molecules can rotate in space. Two different techniques for calculating the above ensemble averages, correct to second order in the field strength, are outlined. One of these is based on a perturbation solution of the differential-difference equations which arise from separating variables in the Fokker-Planck equation describing the problem. The second method is based on a modification of the Smoluchowski equation originally described by Sack. Both methods yield the same results to the order in field strength to which the calculation is carried. In general in the 〈cos 2θ〉 and 〈 P 2(cos ϑ)〉 calculation one finds that the sum and difference frequencies of the applied field well as constant and second harmonic terms will occur. Results correct to terms cubic in the field strength for an impressed field E cos ω t are also given for the dielectric response, there the third harmonic of the field occurs. All the formulae reduce to earlier results when inertia is neglected. The formula given earlier by Rocard for the birefringence in an ac field due to permanent dipoles including inertial effects appears to differ from that of the present work, on the other hand his formula for the mean dipole moment in the low-field limit is in agreement. The present formula indicates that the birefringence due to permanent dipoles when inertial effects are included should vanish at a critical frequency, decrease, then increase to zero at high frequencies. If induced moments are taken into account the birefringence contains a frequency- and time-independent part which persists to the highest frequencies in accordance with the observations of Raman and Sirkar.