Modeling nonlinear dielectric properties of laminated composites

The article is devoted to the development of a method for predicting the nonlinear dielectric properties of layered composites with complex dependences of the dielectric constant on the electric field strength, which consider the effects of a sharp increase and subsequent decrease in the dielectric constant with increasing field strength. The method is based on the use of the theory of asymptotic averaging proposed by N.S. Bakhvalov, E. Sanchez-Palencia, in relation to the nonlinear problem of electrostatics in a layered medium. As a result of applying this method, local nonlinear problems of electrostatics on a periodicity cell are formulated, and an algorithm for calculating effective nonlinear dielectric constants is proposed. It is shown that a layered composite is a transversely isotropic nonlinear dielectric material if its layers are isotropic materials. A numerical example of calculating the nonlinear properties of a 2-layer composite based on barium titanate and ferroelectric ceramic varicond VK4 is considered. Calculations have shown that the developed method for calculating the nonlinear-dielectric ratios of composites is quite effective and makes it possible to predict the dielectric properties of composites at various values of the electric field strength, including the regions where the dielectric permittivities of the layer materials are saturated. The developed technique can serve as a basis for designing new nonlinear dielectric composite materials with unusual properties.

Materials with nonlinear dielectric properties, in particular ferroelectrics, pyroelectrics, etc., are a promising class of materials for creating various devices and electrical devices [1][2][3]. In particular, these materials are used for various types of nonlinear varicond capacitors, which are used "for frequency multiplication and division; for signal detection; to create a modulation frequency in time relay circuits; in memory cells; to create dielectric amplifiers and other devices [4]. One of the promising directions is the creation of computer memory elements for recording and storing information [3,5]. In this regard, the problem of creating methods for predicting the properties of composite materials with nonlinear dielectric properties, which would make it possible to design materials with predetermined dielectric properties, is very urgent. The design of nonlinearly dielectric heterogeneous structures and materials is currently being actively carried out, including using a domain structure at the nanostructural level [5]. However, as a rule, approximate calculation methods are used in this case, which do not always take into account the anisotropy of the properties of the composite in the nonlinear region.

2.
Statement of the quasi-static problem of nonlinear electrostatics for polarizable composites. Let us consider in three-dimensional space a composite (an inhomogeneous medium V with a periodic structure, consisting of N isotropic phases V α , 1,..., N α = , which is under the influence of an alternating electric field. All phases V α of the composite will be assumed to be polarized dielectrics [16] with nonlinear dielectric strength diagrams, and the distribution of the fields of electric strength and electric induction in the composite will be assumed to obey Maxwell's quasi-static equations [16]. x , the formulation of the problem of nonlinear electrostatics for the indicated type of composite will take the form [16,17]: , We seek the solution of problem (7) in the form of asymptotic expansions in the small parameter Substituting (9) into the equations of system (7), we obtain asymptotic expansions for the components of the electric field strength vector { }

Local problems
Substituting expansions (9) -(11) into the system of equations (7), and collecting terms at the same powers of the small parameter, we obtain a recurrent sequence of local problems. For the zero approximation 0 n = , we have the following problem The solution to problem (12) is sought with respect to functions. The input data of this problem is a gradient, so the solution can be presented in a formal form where functions (1)(0) i N depend on the following arguments Substituting (14) into the third equation of system (12), we find the intensity vector in the zero approximation

Average problems
After substituting asymptotic expansions (10) -(11) into the first equation of (11), and averaging it over PC, we obtain the following averaged equation describing the distribution of the average electric field induction in the composite where the notation is introduced Taking into account (15), this relation can be written in the form Here is the averaged composite stress vector is the effective tensor of the dielectric constant of the composite. Substituting asymptotic expansions (10) and (11) into the boundary conditions of problem (7), after averaging, we obtain >= at 1 n > , due to the normalization conditions imposed on local problems (12). The system of equations (16), (19), (20) and conditions (21) forms the statement of the linearized averaged problem of composite electrostatics. This problem is considered with respect to the zero-order potential

Explicit solution to a local problem
Problem (12) is one-dimensional and admits an explicit solution with respect to functions (1) Here we denote the integration operator (15), (20) and (23)

Explicit solution to a local problem
In accordance with the developed methodology, we calculated the effective dielectric constant of a two-layer nonlinear dielectric composite, one of the layers of which was barium titanate 3 BaTiO , and the other was a ferroelectric material -varicond VK4 [4,19,20]. Diagrams of nonlinear dielectric functions (| |) α ε edependences of dielectric permittivities on the electric field strength | | e for 3 BaTiO and VK4 variconds were taken from [19,20]. These experimental functions were approximated (| |) α ε e using analytical functions (6). The values of the constants 0 , , ,