# Predictive Optimization of Electrical Conductivity of Polycarbonate Composites at Different Concentrations of Carbon Nanotubes: A Valorization of Conductive Nanocomposite Theoretical Models

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Materials Used and Blending Process of PC-CNT Conductive Composites

^{®}OD2015 as an insulating polymer matrix, and as conductive nanofillers multi-walled NC7000 carbon nanotubes (MWCNTs, from NanoCyl SA, Sambreville, Belgium) produced by catalytic chemical vapor deposition (CCVD) method; their physical properties are reported in [15]. The densities of PC and MWCNTs are 1.19 g/cm

^{3}and 1.75 g/cm

^{3}respectively. CNTs are added in an appropriate weight or equivalent volume concentration, as presented in Table 1. The composite compounds are melt-blended at 280 °C for 5 min at 150 RPM in a micro 15 DSM micro-compounder. Composite pellets are twice hot-pressed in Fontijne press under 290 °C, 7.5 T, and 2.5 min to produce films with 140 µm thickness.

#### 2.2. Observation and Characterization

#### 2.3. Hyperparameter Optimization

#### 2.4. Mean Square Root Error and Coefficient of Determination Methods for Model Fitting

^{2}) is used to measure the precision of the fitting of our predicted plots with the experimental values; we have calculated R

^{2}based on the ratio between the sum of squares of the regression (SSR) and the total sum of squares (SST) [20] given by the following equation:

_{i}and $\overline{{\mathrm{y}}_{\mathrm{i}}}$ is the observed and predicted value of electrical conductivity obtained by experimental procedure and Mamunya model, respectively.

## 3. Results and Discussion

#### 3.1. Theoretical Models Background and Applicability

#### 3.1.1. Determination of Percolation Using Power Law Theory

#### 3.1.2. Numerical Method to Predict the Electrical Conductivity

^{2}= 1; from this result, we can present PCHIP as an effective way to predict the electrical conductivity at the whole range of CNTs loading.

#### 3.2. Applicability and Validation of Models to Predict the Electrical Conductivity of PC-CNT Composites

#### 3.2.1. Voet Model

#### 3.2.2. Bueche Model

^{4}–10

^{7}S/m, and the conductivity of pure PC is 10

^{−12}S/m. This model is highly recommended to be applied in composites systems constituted of components that do not differ largely in physical properties, which makes the additive mixing rule theory applicable in this case [7]. For instance, we applied a series of hyperparameter grid-searches in order to optimize the convenient conductivity of CNTs, which satisfies this model; from a series of optimization, its value is estimated to be 2368 S/m. The disruption of conductivity of CNTs can be explained by the tendency of CNTs to break and drastically increase the number of defects during the melt blending process supplied by strong mechanical stresses at hot temperature. The shear stress imparting on the surface of a CNT can induce a pulling effect (a tensile force) on the nanotube inducing fracture and surface damages of CNTs. It results in a decrease of CNTs conductivity compared to pristine MWCNT [33]. From TEM images of PC-CNT, it is observed that below the percolation threshold, CNTs are randomly dispersed in the PC matrix. As we increase the weight concentration of CNTs above the percolative network formation, CNTs make electrical connections between themselves. In addition, we observe that CNTs are aligned in the biaxial flow direction because of the hot pressing [34]. Figure 5 shows the distribution of 1 and 2 wt.% of CNTs in PC from TEM observation.

#### 3.2.3. McCullough Model

#### 3.2.4. Mamunya Model

^{2}= 0.967 implying the feasibility of our presented model as presented in Equation (13) to predict the AC electrical conductivity.

^{2}respectively; it is determined using a contact angle goniometer (model: 100-00-(115/220)-S, Rame–Hart–German [45]).

^{2}is maximized, thus approaching the same fit error, obtained from Equation (14), as shown in a comparative performance of the coefficient of fitting R

^{2}between the two Mamunya algorithms from Figure 9a,b presenting k as an independent exponent factor on one hand (Equation (10)), and in another hand, as a parameter function of various factors, as shown from the integration of Equations (14) and (15) into the Mamunya model. The optimal values of A and B obtained from a series of an hyperparameter optimizations are found to be 0.11 and 0.03 respectively, as shown in Figure 10 depicted by the black mark, resulting in low mean square error value ($\mathrm{MSE}=0$), hence approaching at this optimized values the experimental electrical conductivity; at these values of A and B, it is resulted a linear behavior of electrical conductivity as obtained in Figure 8 approaching by this result the same coefficient of determination R

^{2}as obtained from Equation (13). A and B are generally defined for composites with different polymer matrices charged by a unique nanofiller; in our case, by using a hyperparameter optimization and MSE method for a composite formed by a single polymer matrix. These values do not largely differ from the ones obtained from an experimental approach [46]. The respective values of exponent’s k factors are then recorded in Figure 11. For an optimized design, an extensive Mamunya model can be written as:

#### 3.2.5. Sohi Model Extension

^{−1}(S) and the conductivity of the polymer matrix [36], as follows:

^{−1}(from a data sheet provided by NanoCyl). Figure 13 displays the experimental and the theoretical conductivity of PC-CNT simulated for a geometric factor varying from 1 to 0.001 according to the Sohi model. From these plots, it can be concluded that the Sohi model fails to predict the electrical conductivity of the PC-CNT composites that can be explained by the mathematical nature of this model that is essentially governed by the conductivity of the polymer matrix, as seen by the plots in Figure 13. In the present work, we are using a low volume concentration of carbon nanotubes; the first term of Equation (17) has no significant improvement in the conductivity of the composite.

#### 3.3. Electromagnetic Absorption Performance

_{1}, which refers to the initial electromagnetic absorption index of the insulating polymer matrix which is quite difficult to predict by the SB model due to its lowest value. The value of percolation threshold using SB model is around 0.1 vol.% of CNTs less than the critical volume fraction obtained from the power-law of electrical conductivity model; replacing this value ${\mathrm{x}}_{\mathrm{c}}=0.1\text{}\mathrm{vol}.\%$ in the logarithmic power law scaling line, the magnitude of exponent factor ascribed to the dimensionality of the network system is lowered by 1.3 in magnitude; this type of difference has also reported elsewhere [32,48]. From 0.1 to 2 vol.%, more closed packed conductive pathways of CNTs are formed, interacting with incident microwave signal and leading to an increase in the electromagnetic absorption ratio [13,34]. Its maximal values are 50% and 48% at 15 and 25 GHz respectively. From Figure 16, it is observed that increasing the frequency leads to a shifting in the electromagnetic absorption index towards higher values, inducing a smaller volume fraction of CNTs for the percolation threshold. PC-CNT composites show also frequency dependency and electromagnetic absorption more pronounced at 15 GHz. With the progressive increase in filler concentration above 2 vol.% volume fraction, electromagnetic absorption ratio reaches a plateau of saturation and further CNTs enhancement does not show an increase in absorption which makes concentration 2 vol.% the optimum filler content for electromagnetic absorption. Complementary analysis of larger CNT concentrations in polymer has been realized; it results that the ultimate EMI absorber is targeted in this range of loading [14].

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Electrical conductivity of PC-CNT vs vol.% of CNTs (

**a**), and the plot of $\mathrm{log}(\mathsf{\sigma})$ vs. $\mathrm{log}\left(\mathsf{\varphi}-{\mathsf{\varphi}}_{\mathrm{c}}\right)$ in (

**b**).

**Figure 2.**PCHIP numerical method and experimental data of electrical conductivity ${\mathsf{\sigma}}_{\mathrm{c}}$ for PC-CNT composites at several volume fraction of reinforcement $\mathsf{\varphi}$.

**Figure 3.**Theoretical conductivity against volume fraction of CNTs based on Voet model for PC-CNT conductive composites. (

**a**) low volume fraction, (

**b**) remarkable linearity.

**Figure 4.**Theoretical and experimental plots of conductivity against volume fraction of CNTs based on the Bueche model for PC-CNT conductive composites.

**Figure 5.**TEM observation at 8000× of a PC-1 wt.% CNT blend composite in (

**a**), and a PC-2 wt.% CNT composite in (

**b**).

**Figure 6.**Theoretical and experimental plots of conductivity against volume fraction of CNTs based on McCullough model for PC-CNT conductive composite system.

**Figure 7.**Experimental and Mamunya theoretical electrical conductivity against volume fraction of CNTs loading.

**Figure 8.**Experimental electrical conductivity and Mamunya theoretical conductivity as function of AR against CNTs volume fraction.

**Figure 9.**Comparative performance of MSE and ${R}^{2}$ for Mamunya theoretical model, as described respectively by (

**a**) Equation (10) and (

**b**) Equations (14) and (15). $\mathsf{\varphi}$ (k) values are the experimental concentrations considered in this work, thus corresponding to k samples.

**Figure 11.**Calculated k exponent factors for minimizing the MSE factors according to optimized A and B constants.

**Figure 12.**Comparative theoretical and experimental plots of conductivity against volume fraction of CNTs based on the Mamunya model for different aspect ratio for CNT nanofiller.

**Figure 13.**Theoretical and experimental plots of conductivity against volume fraction of CNTs based on the Sohi model for PC-CNT conductive composite system.

**Figure 16.**Experimental absorption ratio and Steffen–Boltzmann (SB) fit at 15–25 GHz for PC-CNT composites.

CNTs (wt.%) | 0.25 | 0.5 | 1.5 | 2.5 | 3 | 3.5 | 4 | 4.5 |

CNTs (vol.%) | 0.17 | 0.34 | 1.03 | 1.71 | 2.05 | 2.41 | 2.75 | 3.1 |

**Table 2.**Steffen–Boltzmann parameters values obtained from fitting electromagnetic absorption ratio against volume fraction experimental curves at 15 and 25 GHz.

Composition | Frequency [GHz] | ${\mathit{R}}^{\mathbf{2}}$ | ${\mathbf{A}}_{\mathbf{1}}$ | ${\mathbf{A}}_{\mathbf{2}}$ | ${\mathsf{\phi}}_{\mathbf{c}}$ | $\mathsf{\Delta}\mathsf{\phi}$ |
---|---|---|---|---|---|---|

PC-CNT | 15 | 0.975 | $-30.69\pm 107.29$ | $47.90\pm 1.86$ | $0.00156\pm 0.0102$ | $0.00393\pm 0.00252$ |

25 | 0.99 | $-37.02\pm 77.88$ | $46.31\pm 0.84$ | $0.00106\pm 0.0045$ | $0.0026\pm 0.00117$ |

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**MDPI and ACS Style**

Sidi Salah, L.; Ouslimani, N.; Chouai, M.; Danlée, Y.; Huynen, I.; Aksas, H.
Predictive Optimization of Electrical Conductivity of Polycarbonate Composites at Different Concentrations of Carbon Nanotubes: A Valorization of Conductive Nanocomposite Theoretical Models. *Materials* **2021**, *14*, 1687.
https://doi.org/10.3390/ma14071687

**AMA Style**

Sidi Salah L, Ouslimani N, Chouai M, Danlée Y, Huynen I, Aksas H.
Predictive Optimization of Electrical Conductivity of Polycarbonate Composites at Different Concentrations of Carbon Nanotubes: A Valorization of Conductive Nanocomposite Theoretical Models. *Materials*. 2021; 14(7):1687.
https://doi.org/10.3390/ma14071687

**Chicago/Turabian Style**

Sidi Salah, Lakhdar, Nassira Ouslimani, Mohamed Chouai, Yann Danlée, Isabelle Huynen, and Hammouche Aksas.
2021. "Predictive Optimization of Electrical Conductivity of Polycarbonate Composites at Different Concentrations of Carbon Nanotubes: A Valorization of Conductive Nanocomposite Theoretical Models" *Materials* 14, no. 7: 1687.
https://doi.org/10.3390/ma14071687