Copyright 2011 KIEEME. All rights reserved. http://www.transeem.org175

† Author to whom all correspondence should be addressed:E-mail: eric.david@etsmtl.ca

Copyright ©2014 KIEEME. All rights reserved.This is an open-access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

pISSN: 1229-7607 eISSN: 2092-7592DOI: http://dx.doi.org/10.4313/TEEM.2014.15.4.175

TRANSACTIONS ON ELECTRICAL AND ELECTRONIC MATERIALSVol. 15, No. 4, pp. 175-181, August 25, 2014

Simulation and Modeling of Polyethylene/Clay Nanocomposite for Dielectric Application

Bouchaib Zazoum, Eric David†, and Anh Dung NgôDepartment of Mechanical Engineering, École de technologie supérieure (ÉTS), 1100 Notre-Dame Street West, Montreal QC H3C 1K3, Canada

Received March 31, 2014; Revised April 11, 2014; Accepted June 2, 2014

In this paper, the simulation and modeling of a polyethylene/clay nanocomposite were undertaken to predict the nanocomposite’s dielectric behavior and to help design a nanocomposite material with optimum electrical properties for electrotechnical or electronic applications. A 3-D simulation model using the finite elements method was employed in order to study the effective permittivity and electric field distribution of two-phase nanocomposite materials for ordered and random distributions of inclusions in a low-loss host matrix such as polyethylene. The influence of the dispersion of reinforcing particles, and of the permittivity and radius of the inclusions, was analysed. The simulation results were compared with alternative, known theoretical solutions obtained from classical models, and were found to be in good agreement. The numerical results also indicate that for fixed volume fractions of nanoparticles the effective permittivity of the mixture, for ordered and random distributions, does not vary with the degree of dispersion. The variation of the effective permittivity with the particle radius is shown, using numerical data, to agree with the analytical modules.

Keywords: Dielectric nanocomposites, Effective permittivity, Modeling, Finite elements method

Regular Paper

1. INTRODUCTION

In recent years, nanocomposites have attracted a great deal of attention from many researchers. Several studies have shown that the incorporation of nanometric inclusions in a polymer matrix can often significantly improve its mechanical, dielectric and optical properties when compared to a pure polymer matrix [1-9], provided that the particles are reasonably well dispersed. In the electrical insulation field, polyethylene is extensively used in medium/high voltage electrical cables due to its excellent di-electric properties, very low dielectric losses and high intrinsic breakdown strength. In order to improve these properties and be able to meet new needs, such as those relating to insulation systems in DC power cables, composite materials consisting of a thermoplastic matrix and using nano-filler as a reinforcing filler

can be used; this results in a possible increase in breakdown strength, dielectric endurance, thermal conductivity, and allows the electrical conductivity to be tailored to avoid space charge accumulation in DC applications.

The effective permittivity of composite materials generally depends on the material microstructure, which includes the vol-ume fraction, as well as the shapes and types of component. For some specific conditions (i.e. when the mixture has a periodic well-defined structure), the effective permittivity can be calcu-lated from the analytical solution of the field distribution. This has resulted in various analytical models, such as the commonly used laws of mixtures. However, if the structure is disordered, as may be the case for a real compounded composite, analytical models cannot be used to accurately estimate the material’s ef-fective permittivity. However, it should be noted that in this case, it can still be measured experimentally.

Numerical methods have been developed and calculation capabilities expanded to evaluate the effective dielectric permit-tivity of composite materials. Numerical simulations have in fact grown to represent another method for dealing with a large pro-portion of the physical problems of composite materials.

Trans. Electr. Electron. Mater. 15(4) 175 (2014): B. Zazoum et al.176

In this paper, the effective permittivity of a polyethylene-based nanocomposite is calculated by numerical simulation us-ing the Comsol Multiphysics software, which is based on the fi-nite element method (FEM). The influence of dispersion as well as the variation of the permittivity and radius (or the volume fraction) of inclusions on effective permittivity is studied. The electric field and polarization distribution in the nanocomposite materials are also reported in this paper. A comparison between the numerical results and those of the analytical models is pre-sented.

2. ANALYSIS

The effective dielectric permittivity of a multi-phase material can be estimated either by analytical or numerical methods.

2.1 Analytical models

The starting point of most analytical approaches to estimat-ing the effective properties of heterogeneous media is the solu-tion of the single inclusion problem for which a constant field Eo along the z-direction is applied at a distance from the inclu-sion. This approach has been detailed in many textbooks and review papers [10-20]. In spherical coordinates, the solution of the Laplace equation for a spherical inclusion of radius R is given by:

(1)

with

where Ψ(r,θ) is the electrical potential, r is the radial coor-dinate, θ is the angle between the position vector and the z-coordinate, and ε1 and ε2 are the permittivities of the inclusion and the matrix, respectively. It should be noted that the same equations hold in steady-state AC conditions in which the po-tential would be a phasor and the permittivity can take complex values, including a possible conductivity term. The electrical field along the z-direction inside the inclusion according to (1) is given by:

(2)

A similar calculation can be made for the more general case of an ellipsoidal inclusion leading to:

(3)

where A1 is the depolarization factor along the ellipsoid princi-pal axis parallel to the electrical field [11]. For spherical particles, A1 = A2 = A3 = 1/3 and (3) is identical to (2). By definition, the ef-fective dielectric constant of a two-component heterogeneous linear material can be defined by [21]:

(4)

where εc is the effective permittivity, q1 and q2 are the volume fraction of the matrix and the inclusion, and the brackets denote averages over phase 1, phase 2 and over the material’s volume. An analytical calculation of the electrical field in a composite material can only be done if the minority phase is present in a small concentration and has regular shape inclusions. A number of results can be found in which an exact solution for several ma-trix systems with periodic arrangements of regular inclusions is obtained [22,23]. In the case of a dilute suspension of ellipsoidal shape inclusions with a permittivity ε2 in a continuum matrix of permittivity ε1, it is possible to use the solution of the single-inclusion problem (equation (3)), assuming that the field Eo is equivalent to the average field in the matrix (phase 1). This leads to [11]:

(5)

A similar procedure leads, for randomly oriented inclusions, to:

(6)

where Ai is the depolarization factor for the ith axis of the el-lipsoid. For spherical particles, A1 = A2 = A3 = 1/3. In the case of spheroids for which two axes are equal (a = b ≠ c), the analytical expressions for Ai for oblate spheroids (disk-like spheroids with a = b > c) and prolate spheroids (needle-like spheroids with a = b < c) can be found in literature [10].

Another approach, the effective medium approximation, also relies on the solution of the single-inclusion boundary value problem. The generalization of the Maxwell approximation for ellipsoidal inclusion leads to:

(7)

for a two-phase composite consisting of a perfectly ori-ented ellipsoidal inclusion (phase 2) inside a matrix (phase 1). It can be shown that this equation is equivalent to equation (5), and the randomly oriented case would be equivalent to (6). This is also the exact solution of the coated-spheres model [24].

A different approach in the effective medium approximation family is the self-consistent approximation, which was originally developed by Bruggeman [25]. It leads to a slight modification of (7):

(8)

which in turn leads to a quadratic equation for the effective permittivity. For the case of the randomly oriented ellipsoidal, equation (8) can be written [26] as:

2

0 0 2

0 0

cos( , ) cos

( , ) cos cos

r E r AE r Rr

r E r BE r R

θψ θ θ

ψ θ θ θ

= − − ≥

= − + ≤

3 2 1

2 1

2 1

2 1

2

2

A R

B

ε εε ε

ε εε ε

−= −

+−

=+

1

2 12 0

1

113

E Eε εε

− −

= +

1

2 12 1 0

1

1E A Eε εε

− −

= +

1 1 1 2 2 2c

q E q EE

ε εε < > + < >=

< >

1 2 1 2 2 1 21

1 1 2 2 1

(1 )(1 ) [ (1 )](1 )( )c

q A q A qA q

ε εε εε ε ε

− − + + −=

+ − −

( ) ( )

( )

32 2 1

1 21 1 2 1

32 1

21 1 2 1

13

13

i ic

i i

qqA

qqA

ε εεε ε ε

εε

ε ε ε

=

=

− ++ −

=− +

+ −

∑

∑

1

21 ( ) 1 0j i

j j c jj

q Aε ε

ε εε

−

=

−∑ − + =

12

1 ( ) 1 0j cj j c j

c

q Aε ε

ε εε

−

=

− ∑ − + =

177Trans. Electr. Electron. Mater. 15(4) 175 (2014): B. Zazoum et al.

(9)

Finally, using a symmetric integration technique [27], it can be shown that the Looyenga equation for randomly oriented ellip-soids, independent of their shape, is given by:

(10)

As mentioned previously, all the above equations also hold in steady-state AC conditions. In this case, the electric fields can be replaced by their respective phasors and the permittivity can be replaced by their complex representation. Accordingly, equations (5) to (10) can also be used to predict a composite dielectric re-sponse, i.e., the variation of the effective complex permittivity as a function of frequency, by replacing ε1 and ε2 by their frequency-dependent complex representations.

2.2 Numerical models

The effective permittivity of the dielectric mixture can be cal-culated once the electric field vector is known inside the mate-rial. In a purely electrostatic case, it can be calculated by solving the Poisson’s equation given by:

(11)

where εr, ε0 and Ψ are the relative permittivity, the vacuum per-mittivity and the electrical potential, respectively. ρ is the charge density. For the neutral condition (ρ = 0), and if we take into ac-count a possible conductivity σ and dielectric losses, then (11) can be written in the steady state more generally as [28-30]:

(12)

where the complex permittivity is given by:

(13)

Once the material microstructure and the properties of each phase are known, (11) or (12) can be numerically solved by the finite elements method (FEM) using a commercially available package like COMSOL Multiphysics. After the field distribution is numerically evaluated, the effective permittivity of the dielectric mixture can be calculated in several ways [30-38]. In this paper, the effective dielectric permittivity of the composites was calcu-lated by using the averages of the electric field and dielectric dis-placement values. Therefore, the effective dielectric permittivity can be expressed as follows:

(14)

where <D> denotes the mean value of the electric displace-ment field and <E> is the mean value of the electric field over the

material. Both averages were taken in the direction of the applied field.

3. RESULTS AND DISCUSSION

3.1 Simulation setup

The morphology of polyethylene/clay nanocomposite in the nanometric scale observed by scanning electronic microscopy (SEM) is shown in Fig. 1. As can be seen, two components are presented in the nanocomposite. This two-phase nanocom-posite consists of 5 wt% of nanoclay particles dispersed in a polyethylene matrix. A comparison between the morphology of polyethylene with 5 wt% nanoclay, PE/O-MMT (Fig. 1(a)), and polyethylene with 5 wt% nanoclay and 10 wt% of compatibilizer, PE/O-MMT/PE-MA (Fig. 1(b)), shows that the density and size of aggregates were decreased in the compatibilized nanocompos-ite. This improvement in the nanoclay dispersion is due to the presence of the polar compatibilizer, PE-MA [8]. In this paper, the dielectric permittivity of the polyethylene matrix and that of the nanoclay reinforcing fillers are taken to be as follows: ε1=2.3 and ε2=4.4, for the matrix and the filler, respectively. A single-inclusion 3-D model cell was used as a first approach, as shown in Fig. 2. The inclusion was assumed to have a spherical form to simulate the shape of a nanoclay particle, and the radius of this was later calculated in order to meet the requirement of a 5% volume fraction of the particles. The bottom face of the cube was set to a constant potential (Ψ = 1 V), and the opposite face was

3 1 1 2 21

1 2

( ) ( ) 0( ) ( )

c ck

k c c k c c

q qA A

ε ε ε εε ε ε ε ε ε=

− −∑ + =

− + − +

1/3 1/3 1/31 1 2 2c q qε ε ε= +

Fig. 1. SEM micrograph of (a) PE/O-MMT and (b) PE/O-MMT/PE-MA nanocomposites.

Fig. 2. Unit cell model of a single-inclusion two-component periodic composite material.

0( ) 0rjω ε ε→ ∧ → ∧ ∇• ∇Ψ =

0

' "r j σε ε εωε

∧ = − +

0c

DE

εε

< >=< >

0( )rε ε ρ→ →

∇• ∇Ψ = −

(a) (b)

Trans. Electr. Electron. Mater. 15(4) 175 (2014): B. Zazoum et al.178

Fig. 4. Polarization field distribution in the nanocomposites with (a) 1 particle, (b) 8 particles, (c) 27 particles, and (d) 64 particles (ordered distribution).

Fig. 5. Electric field distribution in the nanocomposites with (a) 1 particle, (b) 8 particles, (c) 27 particles, and (d) 64 particles (random distribution).

set to ground (Ψ = 0). The other faces of the cube were set to pe-riodic conditions.

3.2 Ordered distribution

In order to study the effect of the dispersion of nanoclay particles on dielectric properties, such as electric field distribu-tions and effective permittivity, four geometries (with the same volume fraction q=0.05) were drawn with 1, 8, 27 and 64 spheres corresponding to particle normalized radii of 0.229, 0.114, 0.076 and 0.057 (Figures 3(a)-3(d)).

The surface plots of the electric field distribution in the 3-D model were obtained by FEM simulations, and are shown in Fig. 3.

As can be predicted by the single-inclusion solution, a field enhancement is present at the interface between the particle and the matrix on the bottom and top sides in the z-direction, the field is almost constant and smaller within the inclusion. The maximum value of the electric field Emax increases as the degree of dispersion of the nanoclay particles increases, and the highest value is obtained when the number of spheres is 27.

The polarization vector (in C/m2) distributions for all 4 geom-etries are shown in Figure 4, along with an enhancement at the matrix-particle interfaces in the z-direction. The highest value of polarization was found at the surface of the particles in the cell with 64 spheres. It is evident from the images that the polariza-tion increases as the degree of dispersion of the nanoclay par-ticles increases.

3.3 Random distribution

In the above section, the uniform reinforcing particles are considered to be orderly dispersed in a polyethylene matrix. In real materials, nanoclay particles or any reinforcing filler are distributed more or less randomly in the polymer matrix with ag-glomerate size distributions. Figures 5(a)-5(d) show four geome-tries and electric field distribution plots for different numbers of randomized nanoclay particles. The volume fraction of nanoclay particles was set at 5% and dispersed in a unit cell.

The corresponding effective permittivity and normalized

maximum electric field Emax/E0 (where E0=1 kV/mm) of the nano-composite materials are plotted in Figs. 6 and 7, respectively. As can be seen from Fig. 6, the effective permittivity for ordered distributions does not vary significantly with the number of par-ticles. This is due to the fact that the volume fraction of nanoclay particles was constant and set at 5%. On the other hand, the ef-fective permittivity of the random distribution was found to be higher than that of ordered distributions. This change is due to a decrease of the distance between inclusions in the random dis-tribution.

Figure 7 shows the normalized maximum value of the electric field in the z-direction as a function of the particle dispersion. At a low particle number, it can be seen that the normalized maxi-mum electric field is almost identical for the ordered distribution

Fig. 3. Electric field distribution in the nanocomposite with (a) 1 particle, (b) 8 particles, (c) 27 particles, and (d) 64 particles (ordered distribution).

(a) (b)

(c) (d)

(a) (b)

(c) (d)

(a) (b)

(c) (d)

179Trans. Electr. Electron. Mater. 15(4) 175 (2014): B. Zazoum et al.

as for the random cases, and we can also see it increases as the quality of dispersion is improved. For particle numbers higher than 27, the normalized maximum value of the electric field in the random distribution decreases slightly as the quality of dis-persion increases, while in the ordered distribution, the electric field is found to increase considerably with an increase in the number of particles.

3.4 Effect of the permittivity of the inclusion on effective permittivity

In this section, the two geometries of 64 spheres of the ordered and random distributions presenting the distribution of the electrical potential simulated by FEM are shown in Figs. 8(a) and 8(b) to show the effect of a variation of the dielectric permittiv-ity of inclusions on the effective permittivity of the ordered and random nanocomposites. The resulting effective permittivities obtained from FEM are compared with those calculated from equation (7), the Maxwell-Garnett model, equation (9), the Brug-

geman model, and equation (10), the Looyenga model. Figure 10 shows the effect of varying the permittivity of the

inclusion ε2 on the normalized maximum value of the electrical field in the z-direction. As expected, for ordered and random dis-

Fig. 6. Comparison of effective permittivity for random and ordered nanoparticle distributions.

Fig. 7. Comparison of normalized maximum electric field for random and ordered nanoparticle distributions.

Fig. 8. Electric potential distribution in the nanocomposite with 64 particles: (a) ordered distribution and (b) random distribution. The red lines present the electrical field stream lines.

Fig. 9. Numerical and theoretical results of the effective permittivity as a function of the dielectric permittivity of inclusions.

Fig. 10. Normalized maximum electric field for ordered and random nanoparticle distributions as a function of the dielectric permittivity of inclusions.

(a) (b)

Trans. Electr. Electron. Mater. 15(4) 175 (2014): B. Zazoum et al.180

tributions, the maximum value of the electrical field increases as the value ε2 increases.

3.5 Effect of the radius (and volume fraction) of the inclusion on effective permittivity

The distributions of the electric potential simulated by FEM for the two geometries for the single-inclusion case, of the or-dered and random distributions, are shown in Fig. 11. In the sim-ulation, the permittivity of the polymer matrix ε1 is still assumed to be 2.3, and the permittivity of the inclusion ε2 is set at 4.4. To study the effect of nanoclay loading on the effective permittivity εc, the normalized radius of the inclusion r/r0 (where r0=1 μm) was varied from 0.010 to 0.400, corresponding to a volume frac-tion of nanoparticles ranging from 0.0004% to 26.81%.

The effective permittivity of the nanocomposites as a func-tion of the radius of nanoclay particles is plotted in Fig. 12. It can be observed that the effective permittivity increases with the nanoclay particle radius, and it is evident from this figure that the effective permittivity of the ordered and random distribu-tions obtained from FEM is quite similar to that calculated with the Maxwell-Garnett model, and is also very close to those of the Bruggeman and Looyenga models. These results confirm the va-lidity of our simulation method.

4. CONCLUSIONS

A 3-D simulation model using the finite elements method was developed in order to study the effective permittivity and electric field distribution of polyethylene/clay nanocomposite materials for electrical applications. An enhancement of the electric field and polarization was observed as the degree of dispersion of the nanoclay particles increased; however, the effective permittivity of the nanocomposites was not affected by improving the qual-ity of dispersion of nanoclay particles in the host matrix. The numerical results indicate that the Maxwell-Garnett model is ap-propriate for evaluating the effective permittivity of ordered dis-tributions, while the Bruggeman Symmetry model remains the most suitable for calculating the effective permittivity for ran-dom distributions. It was observed that in both distributions the maximum value of the electrical field and the effective permit-tivity εc increase as the value ε2 increases. Finally, this numerical model can be extended to design nanocomposite materials with optimum dielectric properties for electrotechnical or electronic applications.

ACKNOWLEDGMENTS

The authors are grateful for financial support from the Na-tional Sciences and Engineering Research Council of Canada (NSERC).

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