Hybrid analytical finite element method for dielectric response of PE/TiO2 nanodielectric materials

An accurate three-dimensional (3D) model was developed using hybrid analytical finite element method (H-FEM) for the simulation of frequency-domain dielectric response of low-density polyethylene (PE) filled with titanium dioxide (TiO2) nanoparticles. The input values of dielectric permittivity of nanoparticle and interphase were calculated analytically using the mixture model and adjusted by an optimization procedure. The effective permittivity of PE/TiO2 nanocomposites was then modelled by COMSOL Multiphysics. The model output results agreement with the experimental values indicate that the developed H-FEM 3D model is suitable for use in solving dielectric response problems of different nanodielectric materials in frequency domain. Furthermore, the simulation results also offer further understanding into the effect of the nanoparticle interphase on the final dielectric properties of the nanodielectric materials.


Introduction
Polyethylene is extensively used as dielectric in medium and high voltage components. However, the physical properties of this material must be enhanced in order to expand its engineering use. It has been demonstrated that adding sufficiently well dispersed nano size inorganic fillers to the polymer leads not exclusively to a change of its mechanical and thermal properties, but also to improve of its dielectric properties [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].
The complex effective permittivity of polymeric nanocomposites is dependant upon its microstructure, the volume fraction, the shapes and type of the components, including the matrix itself, the fillers and a possible third phase known as the interphase. Several studies have shown that this interphase plays a significant role in improving the dielectric performance of nanocomposites [4,[17][18][19][20][21][22]. This is could be explained by the fact that below 100 nm the volume fraction of the interphase becomes higher as the particle size decreases. However, the exact dielectric properties of the interphase are still not clearly known in most cases.
For some particular geometry, such as periodic or laminated structures, the effective permittivity of a twophase material can be evaluated from the analytical solution of the field distribution resulting in the various analytical models such as the generally used laws of mixtures and even for some very specific geometries in exact results. However, if the structure is disordered, as it is expected for a real compounded composite, analytical models cannot be directly used for a precise estimation of the material effective permittivity, especially if the information on both real and imaginary parts are needed (which, however, can be determined experimentally). Alternatively, numerical techniques such as finite elements method can be used to calculate the effective dielectric permittivity of composite materials and these techniques have turned to be the most efficient approach to model and predict the physical properties of two or multi-phase materials.
In this paper, numerical simulation of PE/TiO 2 nanodielectric were developed to predict their dielectric behavior and to help designing the nanocomposites material with optimum electrical properties for electrotechnique applications. The frequency-domain complex dielectric permittivity of PE/TiO 2 composites has been calculated by numerical simulation using a commercial FEM software (Comsol Multiphysics). A new Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. approach has been proposed to evaluate the input parameters of the FEM model, including dielectric properties of the nanoparticles and their surrounding interphases. A comparison between the numerical results and the experimental ones is also reported.

Exeperimental
Low density polyethylene powder (PE) with a density of 0.922 g cm −3 (Marplex, Melbourne, Australia) was mixed with TiO 2 nanopowder (P25, Sigma-Aldrich, US) to prepare PE/TiO 2 nanodielectric containing 3 wt% of TiO 2 nanoparticles using ball milling and hot-pressing process. More details on ball milling process can be found in the literature [23]. For subsequent comparison, neat PE was subjected to the same process. Broadband frequency-domain dielectric spectroscopy (BDS) experiments were performed in the frequency range from 10 0 to 10 4 Hz at 23°C. The samples with an average thickness of 0.50 mm were sandwiched between two circular brass electrodes of 40 mm in diameter and the complex dielectric permittivity was measured by taking the average value from three points for each frequency. The measurement setup of dielectric permittivity by BDS is shown in figure 1.

Analytical approaches
In analytical methods for solving single inclusion problems, the effectual properties of heterogeneous media is the way out to such problems where an electric field E o is applied away from inclusion in z-direction [24][25][26][27][28][29][30][31][32][33][34]. For the case of spherical inclusion with radius R, the solution of the Laplace equation in terms of spherical coordinates is given as: Here V (r, AE) denotes the electrical potential, r represents the radial coordinate, AE is the angle between the position vector and the z-axis, e 1 is permittivity of the matrix and e 2 is the permittivity of the inclusion. The electric field along the z-axis inside the inclusion according to (1) is given by e e e = + - For a general case of ellipsoidal inclusion, analogous calculations give Here A 1 is the depolarization factor in the direction of ellipsoid principal axis and parallel to electrical field E o [25]. When particles are spherical than A 1 =A 2 =A 3 =1/3, so (3) becomes similar to (2). According to definition, for a two-component heterogeneous linear material the effective dielectric permittivity given as [35]: where e c denotes effective permittivity, q 1 is volume fraction of matrix, q 2 the volume fractions of the inclusion and brackets 〈 〉 represents an average over phase 1, 2 or over the volume of material. In composite materials the analytical calculation of electric field is possible only when the minority phase exists in regular shape inclusions and in minor concentration. Eventually dealing with several matrix systems and periodic regular inclusion arrangements, the exact solution to these matrixes can be calculated [36,37]. For instance, by assuming á ñ = á ñ = á ñ E E E 1 2 in (4), the equation gives the mixture model.
So, this gives the exact solution for a laminated structure that is applied parallel to the electrical field. The solution of single inclusion problem equation (3) can likely be used for a dilute suspension having a permittivity e 2 in a continuum matrix of permittivity e 1 and ellipsoidal shape inclusion. While dealing with such cases it is assumed that electrical field (E 0 ) and average field in matrix are equivalent, this gives [25]: e e e e e e e = -- In above equation A i denotes the depolarization factor of the ellipsoid for the ith-axis. While, for spherical particles since A1=A2=A3=1/3 as a result equations (7) and (6) are identical. The analytical expressions for depolarization factor (A i ) for spheroids (a=b≠c), oblate spheroids (a=b>c) and prolate spheroids (a=b<c) can be found in literature [24].
Additionally, another method known as effective medium approximation also depends on the solution of single inclusion boundary value problem. Generalizing the Maxwell approximation for ellipsoidal inclusion having two-phase composite, comprising of matrix (phase 1) containing a perfectly orientated ellipsoidal inclusion (phase 2) gives: It can further be verified that equations (6) and (8) are analogous to each other, in case of randomly orientated ellipsoidal inclusion it would be analogous to (7) [38].
This equation also represents the exact solution of the coated spheres model. Moreover, equation (8) represents the multi component composites having various kinds of ellipsoidal inclusions.
In the family of effective medium approximation, Bruggeman introduced self-consistent approximation [39] that leads to minor changes in (8)

Numerical model description
When the electric field vector present inside the material is well-known, the effective permittivity of nanodielectric can be calculated. For an entirely electrostatic case, it can be calculated by solving the Poisson's equation given as: where e r and e 0 are relative permittivity and vacuum permittivity respectively, V is electrical potential and ρ presents charge density. When ρ=0 (neutral condition) and possible conductivity σ and dielectric losses are considered, (12) can generally be written as [42][43][44]: Here e r (the complex permittivity) is given by In above equation (14) e r is the involvement to the imaginary portion because of dielectric relaxation. When the properties of each phase and material microstructure are familiar, we can numerically solve (12) and (13) with the help of finite elements method (FEM). After the numerical evaluation of field distribution, the relative permittivity e ¢ r ( ) and dielectric losses (e r ) can be calculated by several equivalent methods. One such method is the calculation of total current and complex impedance.
where Z′ and Z″ are the real and imaginary part of the complex impedance Z, while d represents the distance between the two electrodes, f is the frequency, S is the area of the plate and ε r ″ is the imaginary part of equation (14), including the contribution of direct conductivity. As a simplified approach, 27 particles with weight fraction of 3 wt% are considered randomly distributed in a block of polyethylene matrix. The inclusions were assumed to have spherical geometry with a diameter of 50 nm and each nanoparticle has an interphase of thickness of 10 nm. A Matlab script was created to generate the random non-overlapping nanoparticles in a unit cubical domain based on dynamic collision algorithm (DCA) as shown in figure 2. The top face of cube was fixed to a constant potential of 1 V and the opposed face was set to ground (V=0), while the other faces of the cube were set to the periodic conditions.
In the first approximation, the interphase was considered has the same value as the nanoparticles. The numerical results of the real and imaginary part of the effective permittivity obtained with the developed 3D model were compared with experimental measurement from BDS and then, the values for the permittivity of interphase will be adjusted to match well the numerical results with the experimental values. Figure 3 presents the dielectric response of PE and PE/TiO 2 nanocomposites over the 10 0 to 10 4 Hz frequency range at room temperature (23°C). From figure 3(a), it can be observed that over the 10 0 to 10 4 Hz frequency range, the relative dielectric permittivity (ε′) of PE/TiO 2 nanocomposites is higher as compared to that of PE. At high frequencies (higher than the relaxation frequency shown in figure 3(b)), this can be attributed to the fact, that the ε′ of TiO 2 is much higher than the ε′ of PE. The additional increase in the permittivity at lower frequencies is related to the relaxation mechanism shown in figure 3(b). In the case of dielectric losses (ε″), it can be seen from figure 3(b) that for non-polar PE no relaxation peak can be observed. However, for PE/TiO 2 , one relaxation peak was observed in the vicinity of 10 Hz. This behavior is known as the Maxwell-Wagner-Sillars polarization, which is mostly related to a difference in conductivity between the two phases [45].

Model validation
In this 3D model, the dielectric permittivity of polyethylene matrix and that of the TiO 2 nanoparticles were frequency dependent, while the conductivities were assumed constants. The electrical conductivity of PE was fixed to s 1 =10 −15 Ω −1 m −1 , as measured by Keithley 6517 electrometer and those of the nanoparticle, s 2 and interphase, s 3 were chosen to be between 10 −8 and 10 −10 Ω −1 m −1 , as often found in the literature. The real, e ¢ 1 and imaginary, e 1 parts of the complex relative permittivity, e 1 assumed to PE matrix are obtained from BDS measurement. However, the real, e ¢ 2 and imaginary, e 2 parts of the complex dielectric permittivity of the nanoparticles, e 2 were calculated separately by using the above mixture's formula (equation (5)).
In the first estimation, the complex relative permittivity of the interphase e 3 was considered has the same value as the nanoparticles,  The surface plots of the electric field distribution in the 3D model were obtained by FEM simulations and are shown in figure 4. As it can be seen, there is a field enhancement at the matrix-nanoparticle interface in the z-direction, and the field is almost constant and smaller within the inclusions. This is due the fact that the two inhomogeneous phases (polymer and nanoparticle) have different permittivities. Figure 5 shows simulation and experimental results about the effective real and imaginary part of PE/TiO 2 nanodielectric. It appears that at low frequency the real relative dielectric permittivity predicted by the developed H-FEM model agree well with the experimental data obtained by BDS. However, at high frequency once noticed that the simulated values are lower than those experimentally measured. In the case of dielectric loss permittivity, a good agreement can be observed between the experimental and simulated results in terms of profile shape, but the experimental values are higher than that the numerical ones.
To reduce errors and optimise the validity of this H-FEM model, the imaginary part of the dielectric permittivity of the interphase that was previously calculated by mixture model, is scaled with a factor of 2.3. Figure 6 shows a comparison of experimental results and the numerical values predicted by the corrected H-FEM model with the adjusted complex dielectric permittivity of the interphase which can be written in this case as e e e = ¢ - j 2.3 18 3 3 3 ( ) It can be noticed that the numerical values for both real and imaginary parts of the effective relative dielectric permittivity of the nanodielectric now agree well with those obtained experimentally by BDS. These results indicate that the corrected model is reliable and confirm the validity of our simulation procedure.

Conclusion
This paper presents H-FEM 3D model for predicting the dielectric properties such as electrical field distribution and dielectric permittivity of PE/TiO 2 nanodielectric materials in frequency domain. The model was built based on FEM combined with the experimental BDS measurement and mixture model.
The numerical results supported by experimental data obtained by BDS indicate the validity of the developed 3D model and reliability of the proposed strategy to calculate the dielectric permittivity of nanoparticle and interphase by using the adjusted mixture model. Finally, this numerical model can be extended to design nanocomposites materials with optimum dielectric properties for electrotechnical applications.