The Application Of Fractional Calculus In Study Of Dielectric Relaxation

The history of dielectric relaxation goes back to the 18 th century, and it still attracts extensive research in recent years. The basic reason may be that the dielectric relaxation is a dissipative process in nature and the dissipation dynamics is an important research subject of physics. Based on a wide range of applications of the fractional calculus theory in many fields such as physical and chemical engineering in recent years, we are aware of the dynamic process represented by the fractional calculus is also in nature dissipative, and the fractional constitutive equations have been successfully applied in describing the viscoelastic mechanical phenomena. Fractional constitutive equations can be solved easily in frequency domain and they are suitable for studying the dielectric relaxation. There were some tentative works on using fractional calculus to study dielectric relaxation; however, simply replacing integer-order calculus by fractional-order operators can’t guarantee the dynamical stability. The constitutive equations are stable when the relaxation models are constructed using the fractional element.In this paper, the microscopic mechanism of dielectric relaxation is discussed in detail, and the cause of the dissipation is analyzed firstly. Such study help us understanding the relationship between the dielectric properties and structure and important theoretical value and practical significance. Then the fractional element for the dielectric relaxation, the ―cap-resistor‖, is introduced, and the basic fractional models of dielectric relaxation are constructed using cap-resistors. The constitutive equations of the fractional models are derived, and their complex permittivities are presented. Analysis on the relaxation characteristics show that these fractional models can offer relaxation processes with diverse frequency dependence.Comparing with the process establishing the classical Cole- Cole equation, we find the concept of complex impedances used by Cole brothers is completely consistent with the cap-resistor. And the classical Cole- Cole equation can be regarded as a special case of fractional Maxwell model, so the fractional Maxwell model is referred to as the generalized Cole- Cole equation. Finally, we simulate the relaxation process of propyl alcohol and liquid crystal cells embedded with Pd nanoparticles by using fractional models, we fit the model parameters by using conjugate gradient optimization and genetic algorithm. The fitting results show that the fractional Poynting-Thomson model can delineate the dielectric relaxation behavior of glycerol very well. The fractional Maxwell and Zener model can simulate the main characteristics of the relaxation of liquid crystal cells embedded with Pd nanoparticles, and the fractional Zener model offers an excellent description for the experiment data.