# Analysis of DC Electrical Conductivity Models of Carbon Nanotube-Polymer Composites with Potential Application to Nanometric

Authors: Rafael Vargas-Bernal; Gabriel Herrera-Pérez; Ma. Elena Calixto-Olalde; Margarita Tecpoyotl-Torres
Journal: Journal of Electrical and Computer Engineering
 (2013)

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## Abstract

The design of nanometric electronic devices requires novel materials for improving their electrical performance from stages of design until their fabrication. Until now, several DC electrical conductivity models for composite materials have been proposed. However, these models must be valued to identify main design parameters that more efficiently control the electrical properties of the materials to be developed. In this paper, four different models used for modeling DC electrical conductivity of carbon nanotube-polymer composites are studied with the aim of obtaining a complete list of design parameters that allow guarantying to the designer an increase in electrical properties of the composite by means of carbon nanotubes.


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## Body

## 1. Introduction

In the nanometer era, VLSI circuits must be simulated by novel models that allow us to predict and estimate the electrical behavior waited from their phase of design and during their performance. With the introduction of the nanomaterials as an alternative to silicon, numerous research groups around the world have been developing models to simulate electrical properties of such materials. Within this class of nanomaterials, carbon nanotube-polymer nanocomposites are offering improved electrical properties thanks to inherent electrical properties of the carbon nanotubes either semiconducting or conducting. Previous studies of these materials have predicted successful technological applications [1–3]. However, their design must overcome important barriers involved with the control of phenomena between their interfaces and fabrication methods with the aim of achieving repetitive results.Electrical properties of materials are determined through four fundamental parameters called dielectric constant, tangent of dielectric loss angle, dielectric breakdown, and electrical conductivity. Most polymers exhibit a highly insulating dielectric behavior and a very high electrical resistivity. Thus, it is required adding an electrically conductive filler to polymers for changing their electrical properties. The resulting nanocomposite of such combination can give place to a semiconducting or conducting behavior. Electrical fillers can be ceramic materials, metals, and/or carbon nanotubes. In particular, carbon nanotubes form composite materials which can be applied in gas sensors, biosensors, electromagnetic shielding, antistatic coatings, high-strength low-density corrosion-resistant components, and lightweight energy storage. The use of these nanocomposites will be extended to VLSI circuit design when a complete domain of electrical properties and fabrication methods can be achieved. In addition, fabrication of these materials must be repetitive to be exploited by electronic industry. Several models of DC electrical behavior have been proposed until now; unfortunately, a complete comparative study among these models that determines a strategic methodology of design does not exist.In this paper, an explorative study of the DC electrical models of nanocomposites is realized with the aim of obtaining a complete list of design parameters to achieve electrical behavior well defined. In Section2, the description of four DC electrical models for composites, which will be modeled, is realized. Next, these models are simulated with the aim of identifying main design parameters that increase electrical conductivity in Section 3. In Section 4, results of the simulation and their discussion are analyzed with details. Finally, conclusions highlighting the importance of the study realized are given in Section 5.


## 2. DC Electrical Models for Composites

An electrical model is a mathematical description that represents electrical behavior of an electrical device or system. Direct current (DC) is the unidirectional flow of electric charge. Direct current can flow through metals, semiconductors, insulators, and composites. The contribution to electrical conductivity of the polymer (or matrix) by part of the filler is not continuous and linear, rather it is discrete and nonlinear. There is a critical composition or also called percolation threshold, at which electrical conduction is increased by several orders of magnitude leading the composite from an insulating range to semiconductive and conductive ranges. In this paper, DC electrical models that will be analyzed are: Kirkpatrick’s model, McLachlan’s model, Mamunya’s model, and Sigmoidal function model. The models are associated with an extended basic statistical percolation theory, and the last model makes use of a nonlinear regression.


### 2.1. Kirkpatrick’s Model

This model predicts the DC electrical conductivity based on the likelihood of contact between particles of filler within the composite [3–6]. Its equation is expressed as a power law equation of the form
(1)σm=A(ϕ-Vbc)b,
where σm is the conductivity of the composite, A is the conductivity of the filler, ϕ is the volume fraction of filler, Vbc is the percolation threshold of filler, and b is the critical exponent which depends on the type of space dimension, and it is a characteristic value experimentally obtained. The percolation threshold is the minimum quantity of the filler required to form a continuous network of particles (as shown in Figure 1) to transport electrical charge between ends of the material being electrically polarized by a DC source. As it is illustrated in this figure, there are three main zones in which conductivity of nanocomposites can be found. At low filler concentrations, zone 1 in Figure 1, the conductivity of the composite is very close to the pure polymer matrix. In zone 2, the percolation threshold have been achieved and the electrical conductivity undergoes a drastic increase due to the start of the formation and consolidation of an electrically conductive network through the sample. Finally, in zone 3, the maximum electrical conductivity is obtained when a completely interconnected network is formed.Figure 1
Dependence of electrical conductivity on filler volume fraction.


### 2.2. McLachlan’s Model

Some physical models can describe the macroscopic properties of a medium in accordance with their properties and the relative fractions of its components under the name of effective medium approximations or effective medium theory. The properties under study, in particular, are usually the electrical conductivity or the dielectric constant of the medium. McLachlan et al. [7–11] proposed a statistical model of the DC electrical conductivity of the composite materials based on the effective medium theory by means of the equation
(2)(1-ϕ)(ρm1/p-ρf1/p)ρm1/p+((1-pc)/pc)ρf1/p+ϕ(ρm1/p-rl1/p)ρm1/p+((1-pc)/pc)rl1/p=0,
where ϕ is the volume fraction of filler, pc is the percolation threshold of filler, ρf is the resistivity of the filler, rl is the resistivity of the polymer or matrix, and p is the critical exponent which represents the volume fraction of the phases.


### 2.3. Mamunya’s Model

The model suggested by Mamunya et al. [12–16] makes uses of the surface energy of the polymer and the filler, and it includes the aspect ratio of the filler. In addition, this theory relates the electrical conductivity of composites with the presence of clusters of connected particles. Therefore, there exists a relationship between the electrical conductivity of the composite and the volume fraction of the filler. Their equations are
(3)log(σm)=log(SC)+(log(SF)-log(σc))(ϕ-ϕcF-ϕc)k,
where
(4)k=Kϕc(ϕ-ϕc)3/4,K=A-Bγpf,γpf=γp+γf-2(γpγf)1/2,F=575/(10+AR)+AR.
In this model, σm is the conductivity of the composite, SC is the conductivity at the percolation threshold, SF is the conductivity at the maximum packaging fraction (F), ϕ  is the volume fraction, ϕc is the percolation threshold, γpf is the interfacial tension between polymer and filler, γp is the surface energy of the polymer, γf is the surface energy of the filler, A and B are constants, and AR is the aspect ratio. In Figure 2, the maximum packaging fraction (F) has been calculated to illustrate the effect of the aspect ratio of the filler.Figure 2
Maximum packaging fraction of the filler used in the composite versus aspect ratio of the filler involved.


### 2.4. Sigmoidal Function Model

A sigmoidal function [17–21] is a mathematical logistic function with “S” shape and generically defined as
(5)S(t)=a1+e(-bt+c).
In particular, the model used in the simulations of DC electrical conductivity of nanocomposites was
(6)σm=σp+SF-σp1+exp[-(ϕ-PCP)/w],
where σm is the conductivity of the composite, SF is the conductivity of the filler, σp is the conductivity of the polymer, ϕ is the volume fraction of the filler, PCP  is the volume fraction of the filler in the midpoint of the percolation, and w is the width of the percolation region. Therefore, σ represents the independent variable, and the constants can be identified as  a=SF-σp,b=-1/w,  and  c=PCP/w. This equation is represented in Figure 3.Figure 3
Sigmoidal function model for DC electrical conductivity for nanocomposites.


## 2.1. Kirkpatrick’s Model

This model predicts the DC electrical conductivity based on the likelihood of contact between particles of filler within the composite [3–6]. Its equation is expressed as a power law equation of the form
(1)σm=A(ϕ-Vbc)b,
where σm is the conductivity of the composite, A is the conductivity of the filler, ϕ is the volume fraction of filler, Vbc is the percolation threshold of filler, and b is the critical exponent which depends on the type of space dimension, and it is a characteristic value experimentally obtained. The percolation threshold is the minimum quantity of the filler required to form a continuous network of particles (as shown in Figure 1) to transport electrical charge between ends of the material being electrically polarized by a DC source. As it is illustrated in this figure, there are three main zones in which conductivity of nanocomposites can be found. At low filler concentrations, zone 1 in Figure 1, the conductivity of the composite is very close to the pure polymer matrix. In zone 2, the percolation threshold have been achieved and the electrical conductivity undergoes a drastic increase due to the start of the formation and consolidation of an electrically conductive network through the sample. Finally, in zone 3, the maximum electrical conductivity is obtained when a completely interconnected network is formed.Figure 1
Dependence of electrical conductivity on filler volume fraction.


## 2.2. McLachlan’s Model

Some physical models can describe the macroscopic properties of a medium in accordance with their properties and the relative fractions of its components under the name of effective medium approximations or effective medium theory. The properties under study, in particular, are usually the electrical conductivity or the dielectric constant of the medium. McLachlan et al. [7–11] proposed a statistical model of the DC electrical conductivity of the composite materials based on the effective medium theory by means of the equation
(2)(1-ϕ)(ρm1/p-ρf1/p)ρm1/p+((1-pc)/pc)ρf1/p+ϕ(ρm1/p-rl1/p)ρm1/p+((1-pc)/pc)rl1/p=0,
where ϕ is the volume fraction of filler, pc is the percolation threshold of filler, ρf is the resistivity of the filler, rl is the resistivity of the polymer or matrix, and p is the critical exponent which represents the volume fraction of the phases.


## 2.3. Mamunya’s Model

The model suggested by Mamunya et al. [12–16] makes uses of the surface energy of the polymer and the filler, and it includes the aspect ratio of the filler. In addition, this theory relates the electrical conductivity of composites with the presence of clusters of connected particles. Therefore, there exists a relationship between the electrical conductivity of the composite and the volume fraction of the filler. Their equations are
(3)log(σm)=log(SC)+(log(SF)-log(σc))(ϕ-ϕcF-ϕc)k,
where
(4)k=Kϕc(ϕ-ϕc)3/4,K=A-Bγpf,γpf=γp+γf-2(γpγf)1/2,F=575/(10+AR)+AR.
In this model, σm is the conductivity of the composite, SC is the conductivity at the percolation threshold, SF is the conductivity at the maximum packaging fraction (F), ϕ  is the volume fraction, ϕc is the percolation threshold, γpf is the interfacial tension between polymer and filler, γp is the surface energy of the polymer, γf is the surface energy of the filler, A and B are constants, and AR is the aspect ratio. In Figure 2, the maximum packaging fraction (F) has been calculated to illustrate the effect of the aspect ratio of the filler.Figure 2
Maximum packaging fraction of the filler used in the composite versus aspect ratio of the filler involved.


## 2.4. Sigmoidal Function Model

A sigmoidal function [17–21] is a mathematical logistic function with “S” shape and generically defined as
(5)S(t)=a1+e(-bt+c).
In particular, the model used in the simulations of DC electrical conductivity of nanocomposites was
(6)σm=σp+SF-σp1+exp[-(ϕ-PCP)/w],
where σm is the conductivity of the composite, SF is the conductivity of the filler, σp is the conductivity of the polymer, ϕ is the volume fraction of the filler, PCP  is the volume fraction of the filler in the midpoint of the percolation, and w is the width of the percolation region. Therefore, σ represents the independent variable, and the constants can be identified as  a=SF-σp,b=-1/w,  and  c=PCP/w. This equation is represented in Figure 3.Figure 3
Sigmoidal function model for DC electrical conductivity for nanocomposites.


## 3. Simulation of the DC Conductivity Models for Nanocomposites

The first step taken in the analysis of DC electrical conductivity models was evaluating their behavior by means of the variation of their parameters, with the aim of determining which parameters modify in a great proportion the electrical conductivity of the nanocomposites. The models selected for this study include the models reported by  Kirkpatrick et al. [3–6], McLachlan et al.  [7–11], Mamunya et al. [12–16], and Sigmoidal function [17–21]. The functions were evaluated using Gnuplot 4.6.


### 3.1. Simulations of Kirkpatrick’s Model

The evaluation of the Kirkpatrick’s model was realized for three parameters: critical exponent, conductivity of the filler, and percolation threshold. The volume fraction was restricted to the range of 0.02 to 0.25. The critical exponent of the percolation was varied from 0.1 to 0.8. The conductivity of the filler was varied from5E-20 to 50E-20 S/cm. The percolation threshold was varied from 0.01 to 0.20.


### 3.2. Simulations of McLachlan’s Model

The evaluation of the McLachlan’s model was realized for three parameters: critical exponent, conductivity of the matrix, and percolation threshold. The volume fraction was restricted to the range of 0.02 to 0.25. The critical exponent was varied from 1.68 to 1.90 S/cm. The resistivity of the matrix was varied from4E13 to 5E14 S/cm. The percolation threshold was varied from 0.02 to 0.15.


### 3.3. Simulations of Mamunya’s Model

The evaluation of the Mamunya’s model was realized for three parameters: conductivity at the maximum packaging fraction, conductivity at the percolation threshold, and packaging fraction. The volume fraction was restricted to the range of 0.02 to 0.25. The conductivity at the maximum packaging fraction was varied from1E-10 to 1E-06 S/cm. The resistivity at the percolation threshold was varied from 1E-30 to 1E-22 S/cm. The packaging fraction was varied from 0.055 to 0.64.


### 3.4. Simulations of Sigmoidal Function Model

The evaluation of the Sigmoidal function model was realized for three parameters: midpoint of the percolation region’s width, conductivity of the filler, and width of the percolation region. The volume fraction was restricted to the range of 0.02 to 0.25. The midpoint of the percolation region’s width was varied from 0.05 to 0.20. The conductivity of the filler was varied from1E-15 to 1E-07 S/cm. The width of the percolation region was varied from 5E-03 to 45E-03.


## 3.1. Simulations of Kirkpatrick’s Model

The evaluation of the Kirkpatrick’s model was realized for three parameters: critical exponent, conductivity of the filler, and percolation threshold. The volume fraction was restricted to the range of 0.02 to 0.25. The critical exponent of the percolation was varied from 0.1 to 0.8. The conductivity of the filler was varied from5E-20 to 50E-20 S/cm. The percolation threshold was varied from 0.01 to 0.20.


## 3.2. Simulations of McLachlan’s Model

The evaluation of the McLachlan’s model was realized for three parameters: critical exponent, conductivity of the matrix, and percolation threshold. The volume fraction was restricted to the range of 0.02 to 0.25. The critical exponent was varied from 1.68 to 1.90 S/cm. The resistivity of the matrix was varied from4E13 to 5E14 S/cm. The percolation threshold was varied from 0.02 to 0.15.


## 3.3. Simulations of Mamunya’s Model

The evaluation of the Mamunya’s model was realized for three parameters: conductivity at the maximum packaging fraction, conductivity at the percolation threshold, and packaging fraction. The volume fraction was restricted to the range of 0.02 to 0.25. The conductivity at the maximum packaging fraction was varied from1E-10 to 1E-06 S/cm. The resistivity at the percolation threshold was varied from 1E-30 to 1E-22 S/cm. The packaging fraction was varied from 0.055 to 0.64.


## 3.4. Simulations of Sigmoidal Function Model

The evaluation of the Sigmoidal function model was realized for three parameters: midpoint of the percolation region’s width, conductivity of the filler, and width of the percolation region. The volume fraction was restricted to the range of 0.02 to 0.25. The midpoint of the percolation region’s width was varied from 0.05 to 0.20. The conductivity of the filler was varied from1E-15 to 1E-07 S/cm. The width of the percolation region was varied from 5E-03 to 45E-03.


## 4. Results and Discussion

In this section, the results and discussion of these models is realized. Each model is analyzed separately with the aim of identifying their advantages and drawbacks in the modeling of the nanocomposites. In addition, this analysis allow us determine which are the main parameters that must be considered when a design process related with this type of materials is used to fabricate electronic devices for VLSI circuits and systems.


### 4.1. Analysis of the Kirkpatrick’s Model

In Figure4, the variation of the electrical conductivity of composites with respect to the value of the critical exponent of the percolation is illustrated. Higher electrical conductivities are obtained when the value of the critical exponent is low. Therefore, this parameter must be controlled to manipulate the value of the electrical conductivity of the resulting composite.Figure 4
DC electrical conductivity of the nanocomposite when the critical exponent of the percolation is varied (Kirkpatrick’s model).The value of the conductivity of the filler determines the maximum electrical conductivity obtained for the composite as shown in Figure5. Thus, this parameter can be reinforced as part of a strategy to increase the electrical conductivity of the nanocomposite to be designed.Figure 5
DC electrical conductivity of the nanocomposite at varying the conductivity of the filler (Kirkpatrick’s model).At varying the percolation threshold of the filler in the composite, as shown in Figure6, the electrical conductivity is not changed, but rather the initial value, for which an electrical conductivity is achieved, is displaced in accordance with the value of the percolation threshold.Figure 6
DC electrical conductivity of the nanocomposite when the percolation threshold of the filler is varied (Kirkpatrick’s model).In Figure7, it is depicted that the behavior is obtained when both the critical component of the percolation and the volume fraction of the filler are modified. It was found that when critical exponent is low and volume fraction is high, high electrical conductivities are established. In the case of the conductivity of the filler and the volume fraction of the filler, as shown in Figure 8, the electrical conductivity of the composite is at maximum, if at least the first parameter is great. Finally, the percolation threshold at minimum and the volume fraction at maximum can increase the electrical conductivity of the nanocomposite, as depicted in Figure 9.Figure 7
DC electrical conductivity of the nanocomposite at varying both the critical exponent of the percolation and the volume fraction of the filler (Kirkpatrick’s model).Figure 8
DC electrical conductivity of the nanocomposite when both the conductivity of the filler and the volume fraction of the filler are varied (Kirkpatrick’s Model).Figure 9
DC electrical conductivity of the nanocomposite at varying both the percolation threshold of the filler and the volume fraction of the filler (Kirkpatrick’s model).


### 4.2. Analysis of the McLachlan’s Model

The critical value of the percolation increases the electrical conductivity of the resulting nanocomposite with better intensity for greater values of the volume fraction, as shown in Figure10. Thus, it is feasible considering that this parameter must be favored in the design of this class of materials.Figure 10
DC electrical conductivity of the nanocomposite when the critical exponent of the percolation is varied (McLachlan’s model).The conductivity of the polymeric matrix can be considered as a value to be reinforced during the design of nanocomposites, since it increases the electrical conductivity of them as shown in Figure11. Therefore, a better electrical conductivity of the matrix will lead to better conductive nanocomposites.Figure 11
DC electrical conductivity of the nanocomposite at varying the conductivity of the matrix (McLachlan’s model).In Figure12, at varying the percolation threshold of the filler, it was found that the initial value of the conductivity is restricted for values of the volume fraction greater to itself established in the simulation. Therefore, this parameter holds the increasing trend but the electrical conductivity is not favored.Figure 12
DC electrical conductivity of the nanocomposite when the percolation threshold of the filler is varied (McLachlan’s model).Tridimensional changes in this model for three different parameters can be visualized in Figures13, 14, and 15. In Figure 13, DC electrical conductivity is increased when critical exponent of the percolation and volume fraction of filler are bigger. The conductivity of the polymeric matrix intensifies the electrical conductivity of the composite if both it and the volume fraction are larger. When percolation threshold of the filler and volume fraction are varied, then the electrical conductivity of the nanocomposite is favored if both values are greater.Figure 13
DC electrical conductivity of the nanocomposite at varying both the critical exponent of the percolation and the volume fraction of the filler (McLachlan’s model).Figure 14
DC electrical conductivity of the nanocomposite when both conductivity of the matrix and the volume fraction of the filler are varied (McLachlan’s model).Figure 15
DC electrical conductivity of the nanocomposite at varying both the percolation threshold of the filler and the volume fraction of the filler (McLachlan’s model).


### 4.3. Analysis of the Mamunya’s Model

In Figure16, it is depicted that the change of DC electrical conductivity when the conductivity at the maximum packaging fraction is done. Therefore, it can be concluded that this parameter considerably increases electrical conductivity.Figure 16
DC electrical conductivity of the nanocomposite when the conductivity at the maximum packaging fraction is varied (Mamunya’s model).The conductivity at the percolation threshold uniquely modifies the initial value of the conductivity in the percolation threshold, as shown in Figure17. Thus, an insignificant change in the electrical conductivity for all values of this parameter can be established.Figure 17
DC electrical conductivity of the nanocomposite at varying the conductivity at the percolation threshold (Mamunya’s model).The packaging fraction of the filler only changes the edge of the curve of the conductivity, but does not affect the maximum value of electrical conductivity of the nanocomposite, as shown in Figure18.Figure 18
DC electrical conductivity of the nanocomposite when the packaging fraction of the filler is varied (Mamunya’s model).In Figures19, 20, and 21, the tridimensional changes of these behaviors are illustrated with the aim of emphasizing that the maximum electrical conductivity is achieved when the volume fraction and the conductivity at the maximum packaging fraction are great. Insignificant changes have been found when the conductivity at the percolation threshold of the filler and the packaging fraction are increased, since the maximum conductivity is achieved for almost any value of them.Figure 19
DC electrical conductivity of the nanocomposite at varying both the conductivity at the maximum packaging fraction of the filler and the volume fraction of the filler (Mamunya’s model).Figure 20
DC electrical conductivity of the nanocomposite when both the conductivity at the percolation threshold of the filler and the volume fraction of the filler are varied (Mamunya’s model).Figure 21
DC electrical conductivity of the nanocomposite at varying both the packaging fraction of the filler and the volume fraction of the filler (Mamunya’s model).


### 4.4. Analysis of the Sigmoidal Function Model

In Figure22, it can be observed that varying the midpoint of the percolation region’s width modifies the initially electrical conductivity for small values of volume fraction, but the maximum value can be achieved for any value of the midpoint. Therefore, this value has an insignificant effect on the maximum value of the conductivity.Figure 22
DC electrical conductivity of the nanocomposite when the midpoint of the percolation region’s width is varied (Sigmoidal function model).The conductivity of the filler has an important effect on the maximum value of the electrical conductivity of the nanocomposite, as shown in Figure23. When the value of the conductivity of the filler is great, the conductivity also is increased. Thus, the filler’s conductivity is a very important parameter in the design of these materials.Figure 23
DC electrical conductivity of the nanocomposite at varying the conductivity of the filler (Sigmoidal function model).In the case of the width of the percolation region (Figure24), it has a reduced effect on the range of values of the volume fraction of the filler since the slope of the curve is more abrupt or not, although the maximum value of the conductivity is not affected by this parameter. Therefore, this value has an insignificant effect on the maximum value of the conductivity of the nanocomposite.Figure 24
DC electrical conductivity of the nanocomposite when the width of the percolation region is varied (Sigmoidal function model).In Figures25, 26, and 27, a tridimensional representation of these changes allows the designer of these materials to take into account the regions where electrical conductivity will be maximum, middle, and minimum. In Figure 25, the maximum value of electrical conductivity is achieved when the volume fraction of the filler used was at maximum, and the midpoint of the percolations region’s width is at maximum. The maximum value of conductivity can be achieved when the conductivity of the filler is great, as shown in Figure 26. Finally, in Figure 27 it can be visualized that three different regions are found, but the maximum value is obtained when volume fraction is great.Figure 25
DC electrical conductivity of the nanocomposite at varying both the midpoint of the percolation region’s width and the volume fraction of the filler (Sigmoidal function model).Figure 26
DC electrical conductivity of the nanocomposite when both the conductivity of the filler and the volume fraction of the filler are varied (Sigmoidal function model).Figure 27
DC electrical conductivity of the nanocomposite at varying both the width of the percolation region and the volume fraction of the filler (Sigmoidal function model).


### 4.5. Discussion of the Results

In Table1, it can be found that a recompilation of the effect of each one of the parameters associated with mathematical models is used to determine electrical conductivity of the composite materials based on carbon nanotubes and polymers. It is necessary to remember that volume fraction of carbon nanotubes in the composite is restricted in the range of 0.02 to 0.25.Table 1
Mathematical models
Parameters
Electrical conductivity of the composite materials
Kirkpatrick’s model
Conductivity of the Filler
Critical exponent
Percolation threshold
5
E
-
20–50E-20
1
E
-
20–4.3E-19
0.1–0.8
1.4
E
-
21–4.3E-19
0.01–0.20
2
E
-
20–4.3E-19
McLachlan’s model
Resistivity of the matrix
Critical exponent
Percolation threshold
4
E
13–5E14
0.3
E
-
20–6.2E-20
1.68–1.90
5
E
-
21–2.5E-20
0.02–0.15
0.2
E
-
17–1.63E-16
Mamunya’s model
Conductivity at the percolation threshold
Conductivity at the maximum packaging fraction
Packaging fraction
1
E
-
30–1E-22
6
E
-
15–1E-10
1
E
-
10–1E-06
9
E
-
13–1E-06
0.055–0.64
2
E
-
13–1E-10
Sigmoidal function model
Midpoint of the percolation region’s width
Conductivity of the filler
Width of the percolation region
0.05–0.20
2.8
E
-
12–1E-10
1.5
E
-
15–1E-07
8
E
-
15–1E-19
5
E
-
03–45E-03
1.7
E
-
09–1E-10For Kirkpatrick’s model, an increase in the electrical conductivity can be achieved. In the case of the McLachlan’s model, a similar behavior can be found and specific values must be considered since two solutions can be obtained: (1) an increased behavior or (2) a decreased behavior. In addition, imaginary values in electrical conductivity can be achieved for determined ranges of parameters used in the mathematical model. These details have not been reported previously by different authors using this model. In particular, Mamunya’s model has electrical conductivity restricted to the range where conductivity is at the maximum packaging fraction. Finally, Sigmoidal function model can be used for all values of concentration of the filler, and its behavior can be adjusted only when experimental values have been obtained since it belongs to the group of models based on nonlinear regression.


## 4.1. Analysis of the Kirkpatrick’s Model

In Figure4, the variation of the electrical conductivity of composites with respect to the value of the critical exponent of the percolation is illustrated. Higher electrical conductivities are obtained when the value of the critical exponent is low. Therefore, this parameter must be controlled to manipulate the value of the electrical conductivity of the resulting composite.Figure 4
DC electrical conductivity of the nanocomposite when the critical exponent of the percolation is varied (Kirkpatrick’s model).The value of the conductivity of the filler determines the maximum electrical conductivity obtained for the composite as shown in Figure5. Thus, this parameter can be reinforced as part of a strategy to increase the electrical conductivity of the nanocomposite to be designed.Figure 5
DC electrical conductivity of the nanocomposite at varying the conductivity of the filler (Kirkpatrick’s model).At varying the percolation threshold of the filler in the composite, as shown in Figure6, the electrical conductivity is not changed, but rather the initial value, for which an electrical conductivity is achieved, is displaced in accordance with the value of the percolation threshold.Figure 6
DC electrical conductivity of the nanocomposite when the percolation threshold of the filler is varied (Kirkpatrick’s model).In Figure7, it is depicted that the behavior is obtained when both the critical component of the percolation and the volume fraction of the filler are modified. It was found that when critical exponent is low and volume fraction is high, high electrical conductivities are established. In the case of the conductivity of the filler and the volume fraction of the filler, as shown in Figure 8, the electrical conductivity of the composite is at maximum, if at least the first parameter is great. Finally, the percolation threshold at minimum and the volume fraction at maximum can increase the electrical conductivity of the nanocomposite, as depicted in Figure 9.Figure 7
DC electrical conductivity of the nanocomposite at varying both the critical exponent of the percolation and the volume fraction of the filler (Kirkpatrick’s model).Figure 8
DC electrical conductivity of the nanocomposite when both the conductivity of the filler and the volume fraction of the filler are varied (Kirkpatrick’s Model).Figure 9
DC electrical conductivity of the nanocomposite at varying both the percolation threshold of the filler and the volume fraction of the filler (Kirkpatrick’s model).


## 4.2. Analysis of the McLachlan’s Model

The critical value of the percolation increases the electrical conductivity of the resulting nanocomposite with better intensity for greater values of the volume fraction, as shown in Figure10. Thus, it is feasible considering that this parameter must be favored in the design of this class of materials.Figure 10
DC electrical conductivity of the nanocomposite when the critical exponent of the percolation is varied (McLachlan’s model).The conductivity of the polymeric matrix can be considered as a value to be reinforced during the design of nanocomposites, since it increases the electrical conductivity of them as shown in Figure11. Therefore, a better electrical conductivity of the matrix will lead to better conductive nanocomposites.Figure 11
DC electrical conductivity of the nanocomposite at varying the conductivity of the matrix (McLachlan’s model).In Figure12, at varying the percolation threshold of the filler, it was found that the initial value of the conductivity is restricted for values of the volume fraction greater to itself established in the simulation. Therefore, this parameter holds the increasing trend but the electrical conductivity is not favored.Figure 12
DC electrical conductivity of the nanocomposite when the percolation threshold of the filler is varied (McLachlan’s model).Tridimensional changes in this model for three different parameters can be visualized in Figures13, 14, and 15. In Figure 13, DC electrical conductivity is increased when critical exponent of the percolation and volume fraction of filler are bigger. The conductivity of the polymeric matrix intensifies the electrical conductivity of the composite if both it and the volume fraction are larger. When percolation threshold of the filler and volume fraction are varied, then the electrical conductivity of the nanocomposite is favored if both values are greater.Figure 13
DC electrical conductivity of the nanocomposite at varying both the critical exponent of the percolation and the volume fraction of the filler (McLachlan’s model).Figure 14
DC electrical conductivity of the nanocomposite when both conductivity of the matrix and the volume fraction of the filler are varied (McLachlan’s model).Figure 15
DC electrical conductivity of the nanocomposite at varying both the percolation threshold of the filler and the volume fraction of the filler (McLachlan’s model).


## 4.3. Analysis of the Mamunya’s Model

In Figure16, it is depicted that the change of DC electrical conductivity when the conductivity at the maximum packaging fraction is done. Therefore, it can be concluded that this parameter considerably increases electrical conductivity.Figure 16
DC electrical conductivity of the nanocomposite when the conductivity at the maximum packaging fraction is varied (Mamunya’s model).The conductivity at the percolation threshold uniquely modifies the initial value of the conductivity in the percolation threshold, as shown in Figure17. Thus, an insignificant change in the electrical conductivity for all values of this parameter can be established.Figure 17
DC electrical conductivity of the nanocomposite at varying the conductivity at the percolation threshold (Mamunya’s model).The packaging fraction of the filler only changes the edge of the curve of the conductivity, but does not affect the maximum value of electrical conductivity of the nanocomposite, as shown in Figure18.Figure 18
DC electrical conductivity of the nanocomposite when the packaging fraction of the filler is varied (Mamunya’s model).In Figures19, 20, and 21, the tridimensional changes of these behaviors are illustrated with the aim of emphasizing that the maximum electrical conductivity is achieved when the volume fraction and the conductivity at the maximum packaging fraction are great. Insignificant changes have been found when the conductivity at the percolation threshold of the filler and the packaging fraction are increased, since the maximum conductivity is achieved for almost any value of them.Figure 19
DC electrical conductivity of the nanocomposite at varying both the conductivity at the maximum packaging fraction of the filler and the volume fraction of the filler (Mamunya’s model).Figure 20
DC electrical conductivity of the nanocomposite when both the conductivity at the percolation threshold of the filler and the volume fraction of the filler are varied (Mamunya’s model).Figure 21
DC electrical conductivity of the nanocomposite at varying both the packaging fraction of the filler and the volume fraction of the filler (Mamunya’s model).


## 4.4. Analysis of the Sigmoidal Function Model

In Figure22, it can be observed that varying the midpoint of the percolation region’s width modifies the initially electrical conductivity for small values of volume fraction, but the maximum value can be achieved for any value of the midpoint. Therefore, this value has an insignificant effect on the maximum value of the conductivity.Figure 22
DC electrical conductivity of the nanocomposite when the midpoint of the percolation region’s width is varied (Sigmoidal function model).The conductivity of the filler has an important effect on the maximum value of the electrical conductivity of the nanocomposite, as shown in Figure23. When the value of the conductivity of the filler is great, the conductivity also is increased. Thus, the filler’s conductivity is a very important parameter in the design of these materials.Figure 23
DC electrical conductivity of the nanocomposite at varying the conductivity of the filler (Sigmoidal function model).In the case of the width of the percolation region (Figure24), it has a reduced effect on the range of values of the volume fraction of the filler since the slope of the curve is more abrupt or not, although the maximum value of the conductivity is not affected by this parameter. Therefore, this value has an insignificant effect on the maximum value of the conductivity of the nanocomposite.Figure 24
DC electrical conductivity of the nanocomposite when the width of the percolation region is varied (Sigmoidal function model).In Figures25, 26, and 27, a tridimensional representation of these changes allows the designer of these materials to take into account the regions where electrical conductivity will be maximum, middle, and minimum. In Figure 25, the maximum value of electrical conductivity is achieved when the volume fraction of the filler used was at maximum, and the midpoint of the percolations region’s width is at maximum. The maximum value of conductivity can be achieved when the conductivity of the filler is great, as shown in Figure 26. Finally, in Figure 27 it can be visualized that three different regions are found, but the maximum value is obtained when volume fraction is great.Figure 25
DC electrical conductivity of the nanocomposite at varying both the midpoint of the percolation region’s width and the volume fraction of the filler (Sigmoidal function model).Figure 26
DC electrical conductivity of the nanocomposite when both the conductivity of the filler and the volume fraction of the filler are varied (Sigmoidal function model).Figure 27
DC electrical conductivity of the nanocomposite at varying both the width of the percolation region and the volume fraction of the filler (Sigmoidal function model).


## 4.5. Discussion of the Results

In Table1, it can be found that a recompilation of the effect of each one of the parameters associated with mathematical models is used to determine electrical conductivity of the composite materials based on carbon nanotubes and polymers. It is necessary to remember that volume fraction of carbon nanotubes in the composite is restricted in the range of 0.02 to 0.25.Table 1
Mathematical models
Parameters
Electrical conductivity of the composite materials
Kirkpatrick’s model
Conductivity of the Filler
Critical exponent
Percolation threshold
5
E
-
20–50E-20
1
E
-
20–4.3E-19
0.1–0.8
1.4
E
-
21–4.3E-19
0.01–0.20
2
E
-
20–4.3E-19
McLachlan’s model
Resistivity of the matrix
Critical exponent
Percolation threshold
4
E
13–5E14
0.3
E
-
20–6.2E-20
1.68–1.90
5
E
-
21–2.5E-20
0.02–0.15
0.2
E
-
17–1.63E-16
Mamunya’s model
Conductivity at the percolation threshold
Conductivity at the maximum packaging fraction
Packaging fraction
1
E
-
30–1E-22
6
E
-
15–1E-10
1
E
-
10–1E-06
9
E
-
13–1E-06
0.055–0.64
2
E
-
13–1E-10
Sigmoidal function model
Midpoint of the percolation region’s width
Conductivity of the filler
Width of the percolation region
0.05–0.20
2.8
E
-
12–1E-10
1.5
E
-
15–1E-07
8
E
-
15–1E-19
5
E
-
03–45E-03
1.7
E
-
09–1E-10For Kirkpatrick’s model, an increase in the electrical conductivity can be achieved. In the case of the McLachlan’s model, a similar behavior can be found and specific values must be considered since two solutions can be obtained: (1) an increased behavior or (2) a decreased behavior. In addition, imaginary values in electrical conductivity can be achieved for determined ranges of parameters used in the mathematical model. These details have not been reported previously by different authors using this model. In particular, Mamunya’s model has electrical conductivity restricted to the range where conductivity is at the maximum packaging fraction. Finally, Sigmoidal function model can be used for all values of concentration of the filler, and its behavior can be adjusted only when experimental values have been obtained since it belongs to the group of models based on nonlinear regression.


## 5. Conclusions

The DC electrical conductivity of nanocomposites based on polymers and carbon nanotubes has been studied with the aim of determining which are the main parameters of fabrication that must be considered for truly modifying the electrical behavior and how these change such behavior. This study was realized by computer simulation through four different models at varying three parameters in each one of them. Parameters such as volume fraction of the filler, percolation threshold of the filler, critical exponent of percolation, conductivity of the filler, conductivity of the matrix, conductivity at the maximum packaging fraction of the filler, conductivity at the percolation threshold, packaging fraction of the filler, midpoint of the percolation region’s width, and width of the percolation region were analyzed. The main parameters involved in the increase of electrical conductivity of the nanocomposites are critical exponent of percolation, conductivity of the filler, conductivity of the matrix, conductivity at the maximum packaging fraction, conductivity at the percolation, and width of the percolation region. Unfortunately, a model that includes all these parameters does not exist until now. In McLachlan’s model, specific values must be considered since two solutions can be possible: (1) an increased behavior or (2) a decreased behavior. In addition, imaginary values in electrical conductivity can be achieved for determined ranges of parameters used in the McLachlan’s model. These details have not been reported previously by different authors using this model. The results presented here suggest the development of new electrical conductivity models for nanocomposites, where all these parameters are related with the aim of providing a better approach to the electrical behavior of these materials. A more sophisticated electrical model will allow to lead nanocomposites based on carbon nanotubes and polymers to the electronic industry for fabrication of VLSI circuits and systems.



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*Source: 179538-2013-03-07.xml*