Modeling and Simulation of the Electromagnetic Properties of Composites Filled by Carbon Nanotubes B. De Vivo, L. Egiziano, P. Lamberti, Member, IEEE, G. Spinelli*, V. Tucci, Senior Member, IEEE Depart. of Inform. Eng., Electr. Eng. and Appl. Math. – DIEM, University of Salerno Via Giovanni Paolo II, 132-84084 Fisciano (SA), Italy, Phone: +39 089 964161 *E-mail address:
[email protected]
Abstract— Low volume percentages of carbon nanotubes are added in insulating resins in order to obtain new lightweight and conductive nanocomposites with improved electromagnetic performances. The properties of such composites are investigated by means of a 3D model in the J-band (5.38-8.18 GHz), X-band (8.2-12.4 GHz) and Pband (12.4-18 GHz). and the numerical data are validated with experimental results. A sensitivity analysis is carried out in order to account the influence of some parameter such as the amount of filler and the dielectric properties of the resin on the electrical behavior of the simulated composites. Keywords- Nanocomposites, Carbon Nanotubes, Modeling, Electromagnetic Properties
I.
INTRODUCTION
Innovative composites containing conducting fillers, such as carbon based particles, are required in several electromagnetic (EM) applications, mainly for the design of radar absorbing materials (RAMs) or shielding enclosures able to ensure an efficient electromagnetic interference (EMI) reduction [1]. Epoxy resin combined with carbon nanotubes (CNTs), principally of the multi-wall (MW) type, is largely adopted for the production of advanced composites that can be adopted in the aerospace industry [2,3]. In fact, driven by the significant weight reduction, such materials may be a valid alternative to the classical metal for the realization of structural parts in new generation aircraft fuselage or cockpit, for which ability to dissipate lightning currents or provide effective shielding again electromagnetic interference is required. In order to design such CNT-filled composites with tailored electromagnetic properties efficient simulation tools are important. For this reason a 3-dimensional model simulating the CNT distribution in a polymer matrix is developed. It allows to investigate, by means of a suitable RC circuit associated to the percolation paths detected in the simulated structures and able to take into account resistive, capacitive and tunneling effects, the EM performances up to the microwave frequency. A sensitivity analysis of the combined effects of the filler concentrations and the dielectric constant of the neat resin on the resulting composite performance is carried out. The numerical results are successfully compared with available experimental data [5] confirming the reliability
of the proposed model and its applicability as an alternative to onerous trial and error experimental investigations needed for material design and as a valuable tool for the performance optimization of EM devices and systems. II.
COMPUTATIONAL PROCEDURE AND MODELING
The main steps of the modeling procedure are the creation of the morphological structures simulating the CNT-based polymer composite, the detection of the percolation paths and the association to them of an equivalent electrical circuit and the evaluation of the electrical properties taking into account the relevant resistive and capacitive effects. Regarding the resistive components, three contributions can be distinguished: RCNT, Rres and Rtun, with reference to the conductive behavior of the filler, of the resin and tunneling effect, respectively, in accordance to the following equations: !", $, %&* = ∙ ; = ∙ ℎ (1) ∙ ≠(≠)
+,-
=
./ 0
/ 123 4
5"6 7
89: 123 4< ;
(2)
where σcnt and Scnt are, respectively, the electrical conductivity and the cross-sectional area of the CNTs, lmn is the distance between two generic points m and n on the filler that corresponds to the effective length involved in the conduction process, h is the Plank’s constant, d the distance between CNTs, e the electron charge, me the mass of electron and λ is the height of barrier typically of few eV, σres is the DC conductivity of the resin. For the capacitive effects, the model take into account the insulator gap between neighboring CNTs (i.e. Cgap) and the dielectric behavior of the resin (i.e. Cres) evaluated as follows: =>?@ = AB ∙ A
∙
0
; =
= AC ∙ A
∙
∙
(3)
where ε0 is the permittivity of free space (8.8542×10-12 F/m) and εres is the relative dielectric constant of the polymer. In order to provide a realistic picture of the material, some physical constrains such as the CNT impenetrability, the minimum distance between two neighbor particles (Van der Waals separation 0.34 nm) and the total containment of the filler in the representative cell are ensured. A 3D model characterized by the above mentioned features is described in Spinelli et al.[6,7,8]. Herein the main features of the model are
summarized for sake of clarity and completeness. The filler particles (CNT) are modeled as straight cylinders to which the desired length L [µm] and diameter W [nm] and therefore a specific aspect ratio (AR =L/W) are assigned. If (xi, yi, zi) and (xf, yf, zf) are respectively the initial and final coordinates of the filler in pending insertion, the cylinders are added one at a time, according to an uniform probability distribution, by adopting the following analytical relations [1]: (4a) " = DEFG1 ∙ I" , $ = DEFG2 ∙ I$ , % = DEFG3 ∙ I% (4b) "L = " + I∅1 cosRS1 T , $L = $ + I∅1 sinRS1 T , %L
(i.e. σ=2[LAB A ^^ ) of CNTs based composites, respectively, as function of the frequency f. Further validation of the model performed by comparing also the imaginary part of the permittivity (i.e. ε’’) and the intrinsic wave impedance is available in a previous work [8]. a 20 0.030 Ref.5 0.030 Sim 0.025 Ref.5 0.025 Sim 0.020 Ref.5 0.020 Sim
18 16
= % + IW1
8
Table 1: Computational parameters Parameter
CNT
Length [µm] Diameter [nm] Conductivity [S/m] Volume fraction of CNT (VF) Min. value [nm] Max. value [nm]
Tunneling distance III.
10
(5)
where randi, for i=1, …, 5 are uniformly distributed random numbers in the interval I= [0, 1]. A Monte Carlo (MC) approach is adopted to handle the random distribution of the coordinates of CNTs. In this way, each MC trial corresponds to the generation of five uniformly distributed parameters. In order to reduce the computational time, a representative cubic cell having side of 5µm is adopted since it appears large enough for an isotropic response and numerical convergence. The results are evaluated as average of a large number (typically >100) of MC trials. The parameters adopted for the numerical analysis are summarized in Table 1.
Subject
12
6
14
16
18
6 4
0
The numerical results are compared with those obtained from an experimental characterization carried out by Micheli et al. for MWCNTs-epoxy composites in the J-band (5.38-8.18 GHz), X-band (8.2-12.4 GHz) and P-band (12.4-18 GHz). In particular, Fig 1a and Fig.1b, show the comparison for the real part (i.e. ε’) of the complex permittivity ε*(ω) = ε^ _ωa − (ε^^ _ωa and for the electrical conductivity
12
f[GHz]
8
1 10 1×106 [2.0, 2.5, 3.0, 3.5]×10-2 0.34 2.10
A. ELECTRICAL PROPERTIES
10
b
10
2
In order to assess the quality of the numerical prediction an experimental validation is carried out either by direct measurements or by considering literature data.
8
0.030 Ref 5 0.030 Sim 0.025 Ref 5 0.025 Sim 0.020 Ref 5 0.020 Sim
12
Value
MODEL VALIDATON AND DISCUSSION
6
14
σ [S/m]
W = 1 − 2 ∙ DEFGY , ∅ = Z1 − W 2 , S = 2[ ∙ DEFG\
ε
'
14
In the above equations Lx ,Ly, Lz are the dimensions of elementary cell along respectively the x, y and z axes while ϑ1, Ø1 and µ 1 are the parameters, related to the angular distribution, according to the following expressions (2):
6
8
10
12
14
16
18
f[GHz] Figure 1. Comparison between numerical results and experimental data [5] for the real part of relative permittivity (a) and for AC electrical conductivity (b) of nanocomposites. A good agreement between the numerical results and the experimental data is observable. The plots show that the real part of the complex effective relative permittivity, which is related to the stored energy within the medium, slightly decrease with increasing frequency. Such frequency behavior is due to the presence of free dipolar functional groups and/or interfacial polarization, dependent to the filler amount, attributable to the presence of conducting impurities. At low frequency the composites show the highest value and, as the frequency increases, the permittivity progressively decreases because both mechanisms become negligible. Moreover, the higher CNTs loading the higher the permittivity of the composites. Instead the electrical conductivity increase with the frequency. This is coherent with the assumption that in such materials the conductivity is mainly governed by the electron tunneling between CNTs. Such mechanism is
In order to have a complete vision of the electromagnetic properties of materials, predictions of the loss tangent (i.e. tanδ= ε’’/ε’) are required. Such parameter accounts for the dielectrics losses inside the composite under test in terms of small fraction of the total energy stored in the dielectric field. This information is particularly useful when CNTs are employed as fillers in composites for EMI shielding applications. The higher is its value, the greater are the losses. In Fig.2 the frequency-dependent behavior of tanδ is reported for nanocomposites with different loading. The results show an increase of the loss effects at higher filler concentrations.
show that skin-depth decreases for composites at higher filler loadings (i.e. VF=0.030) through which the propagation of the electromagnetic field undergoes a strong attenuation. 6
-3
VF=0.030 VF=0.025 VF=0.020
4 3 2 1
0.9
x 10
5
s [m]
strongly influenced by the energy barrier (λ) that is altered with respect to the static condition by the frequency of the perturbing wave. Higher frequencies lead to a decrease of λ which, in turn, modulates the tunneling effect and consequently the electrical conductivity of the composites.
6
8
10
12
14
16
18
f[GHz] Figure 3. Skin-depth as function of frequency for composites filled at different CNTs loadings.
0.8
tan δ
0.7
C. DEBYE RESPONSE
VF=0.030
Useful information regarding the dielectric properties of materials may be provided by the analysis of the complex impedance:
VF=0.025
0.6
VF=0.020
e ∗ _ga = e ^ _ga − (e " _ga
0.5
0.4
6
8
10
12
14
16
18
f[GHz] Figure 2. loss tangent (tan δ) as function of frequency for composites filled at different CNTs loadings.
B. MICROWAVE SKIN-DEPTH The skin-depth (i.e. s) is a frequency dependent parameter indicative of the ability of an electromagnetic field to propagate through a material: (6) b = 1/1[LSB S d where µ0 is the permeability of free space (4π×10-7 H/m) and µr is the relative permeability of the composites, here adopted equal to 1 (although small deviations can be found due to the ferromagnetic behavior of catalyst particles, which may be adopted to produce CNTs). In Fig. 3, the skin-depth of nanostructured composite materials is reported as a function of frequency. It is worth to note, in accordance with eq.6, that the lower is the electrical conductivity (i.e. for VF=0.020, see Fig.v1), the higher is the skin-depth and then the capability of the EM field to propagate within the material. As evident from (6), a Perfect Electric Conductor (PEC), for which the electrical conductivity tends to infinity, is characterized by a zero skin-depth. Moreover, the obtained numerical results
(6)
where e ^ _ga is the real part of the impedance, which is related to the resistive behavior of material and e " _ga is the imaginary part which take into account the capacitive effects due to the insulating resin and capacitive interaction between the metallic nanotubes separated by a thin layer of polymer. A Cole-Cole representation in the complex plane of the imaginary components e " _ga versus the real one e ^ _ga looks like as a half-circle, as shown in Fig. 4. Regardless from the CNTs content, the circle center lies on a line on the real axis while, for each specific concentration, the radius r is strictly dependent on the resistance (R) value exhibited by the material in the low frequency _D ≅ /2a. In fact, the intercepts of the semicircle with the real axis (Z’) returns the value of the bulk resistance of the material. For this reason, the impedance spectra of the composites at different concentrations are characterized by half-circle whose radii gradually decreases with increasing CNTs content. Accordingly, the composite characterized by a lower filler loading (i.e. VF=0.020) presents the widest semicircle (r=3.74×108Ω) implying that it acts, at low frequency, as a resistor of about 7.82×108Ω. The composite filled with high concentration of CNTs (i.e. VF=0.035) shows the smaller half-circle (r= 4.16×107Ω) highlighting the lower resistance (8.57×107Ω) offered by the material due to the greater presence of percolation paths in the network of electrically conductive nanotubes.
IV.
8
4
x 10
VF=0.020[/] VF=0.025[/] VF=0.030[/] VF=0.035[/]
3.5
2.5 2
"
Z [Ω]
3
1.5 1
Frequency increases
0.5 0 0
1
2
3
'
4
Z [Ω]
5
6
7
8 8
x 10
Figure 4. Complex impedance spectrum of the composites. D. RELAXATION TIME In the field of nanocomposites, the relaxation time (τ) typically refers to the time necessary to the polymer chains to return to equilibrium after being disturbed by variable electric field. The characteristic relaxation time, τ, is obtained using the following relationship: (8) j = 1/g = 1/_2[L a 3?k
where fmax is the relaxation frequency, i.e. the value of frequency at which Z” reaches its maximum value. In Fig.5 the relaxation time (τ) vs. the permittivity (εres) of the neat resin, for different volume fraction of the filler is reported. 2
x 10
A 3D simulation model has been developed to predict the EM properties of carbon-based composites in the microwave frequency range. The influence of the filler concentration and the dielectric properties of the employed resin on the EM behavior of the resulting composites have been investigated. The numerical results, successfully validated with experimental data, show an important role played by the permittivity of the resin and therefore the need to carefully select it depending on the applications to which materials are designed. The model working properly with the investigated material parameters can be employed for the design and performance optimization of EM devices and systems based on CNT-filled composites and for the determination of parameters, such as the CNT intrinsic permittivity whose direct measurement is difficult to perform. Improvements of the model will focus on different material parameters ranges, non uniform distributions of CNTs and the effect of their curvature on the electrical properties of the composites. V.
τ = mε
1.5
res
+b
REFERENCES [1]
[2]
[3]
R2=0.999
[4] 2
τ [s]
R =1 -8
1
m=3e
m=2e-8
0.5
[5] R2=0.999
-8
m=1e
m=3e-9
0 2.5
3
3.5
R2=0.999
4
4.5
εres
5
5.5
6
6.5
[6]
Figure 5. Relaxation time (τ) vs. the resin permittivity (εres). [7]
The relaxation time shows a linear dependence on the dielectric constant εres and therefore nanocomposites based on high-permittivity resins are characterized by greater relaxation times. This is coherent with the assumption that τ is correlated to the orientation of dipoles in an oscillating field, whose density increases in the resin with high permittivity.
ACKNOWLEDGMENT
This work has been partly supported by the FARB funds of the University of Salerno and the European Union FP7/2007-2013 under Grant Agreement n° 313978.
-7
VF=0.020[/] VF=0.025[/] VF=0.030[/] VF=0.035[/]
CONCLUSIONS
[8]
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