# Multiple Analytical Models to Evaluate the Impact of Carbon Nanotubes on the Electrical Resistivity and Compressive Strength of the Cement Paste

^{1}

^{2}

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^{*}

## Abstract

**:**

## 1. Introduction

_{3}O

_{4}), and carbon nanotube (CNT) [20,21,22,23,24,25,26]. Nanotechnology may be used in a variety of industries, including the construction sector, through nanoparticles in the manufacturing of cement-based products. Additionally, using conductive nanoparticles in cement-based material highly increased the sensitivity of the material by increasing electrical conductivity [10,20,27].

## 2. Research Significance

## 3. Methodology

#### 3.1. Statistical Evaluation

- (i)
- Carbon Nanotubes (CNTs)

^{2}/g. The proportion of carbon nanotubes utilized in the cementitious mixes ranged between 0 and 1.5 percent by weight of cement. Moreover, the standard deviation, variance, skewness, and kurtosis are 0.34, 0.118, 0.48, and 0.61, respectively (Figure 2).

- (ii)
- Water to Cement Ratio (w/c)

- (iii)
- Curing Time (t)

- (iv)
- Electrical Resistivity (ρ)

#### 3.2. Modeling

^{2}value. Therefore, four different models are proposed to test the Effect of the various mixture proportions described above on cement paste adjusted with CNTs, as stated below.

^{2}values should be obtained.

#### 3.2.1. Linear Regression Model

#### 3.2.2. Multi Logistic Regression Model

#### 3.2.3. Nonlinear Regression Model

#### 3.2.4. Artificial Neural Network

^{2}. However, due to a complication of equation of multi hidden layer, the single hidden layer with four neurons was chosen in this study based on trial and error to get minimum RMSE and lowest MAE Figure 6 and higher R

^{2}. Equation (5) shows the ANN for one hidden layer.

_{1,}β

_{2,}β

_{3,}and β

_{4,}can be determined by multiplying the attribution values which were given by software with each variable as shown below.

#### 3.3. Assessment Criteria for Models

^{2}, RMSE, MAE, SI, and OBJ which are specified, have been used to test and evaluate the efficiency of the suggested models.

## 4. Analysis and Output

#### 4.1. Predicted and Measured Electrical Resistivity Relationships

#### 4.1.1. Linear Regression Model

^{2}, RMSE, and MAE for the training dataset, is 0.69, 115.37, and 77.43 Ω.m, respectively Figure 7. In addition, the scatter index for this model is 0.725 for the training dataset.

#### 4.1.2. Multi Logistic Regression Model

^{2}, RMSE, and MAE, are 0.88, 71.24, and 44.79 Ω.m, respectively. Moreover, the SI values for the current model are 0.45 for the training dataset.

#### 4.1.3. Nonlinear Regression Model

^{2}, RMSE, and MAE—are 0.75, 104.84, and 62.45 Ω.m, respectively. Moreover, the SI values for the current model are 0.659 for the training dataset.

#### 4.1.4. Artificial Neural Network Model

^{2}, lower RMSE value, and lower MAE value because it will reduce the error. The predicted electrical resistivity vs. actual is shown in Figure 10, which indicates the main idea of generating data based on the ANN model.

^{2}, RMSE, and MAE, are 0.93, 54.48, and 36.37 Ω.m, respectively. Moreover, the SI values for the current model Equation (14) are 0.34 for the training dataset.

#### 4.1.5. Comparison between Different Models

^{2}, and OBJ, have been placed to determine the efficacy of the proposed models. Figure 11 provides a comparison of the models based on RMSE and MAE, and Figure 12 indicates R

^{2}for training and testing dataset of cement paste modified with CNTs. Among the four main models, the ANN model has higher R

^{2}value and lower RMSE and MAE values relative to the LR, MLR, and NLR models, the previous study also used the ANN model for predicting compressive strength of cement mortar with R

^{2}value slightly less than achieved from this study and the RMSE and MAE values are greater than this study results [50]. Another research used ANN model to predict the mechanical properties such as compressive strength of concrete modified with carbon nanotube which get R

^{2}value slightly higher than achieved from this study and this might be due to the high data numbers used in this study [51]. In addition, Figure 13 indicates that the residual error for all models uses dataset preparation, training, and testing. It can be shown from both figures that the actual and calculated values of electrical resistivity are closer to the ANN model, suggesting the superior efficiency of the ANN compared to other models. The OBJ rates of all proposed models are given in Figure 14. The OBJ values for LR, NLR, MLR, and ANN are 105.06, 82.44, 57.49, and 43.10, respectively. The OBJ value of the ANN model is 59% less than the LR model, 47.7% lower than the NLR model, and 25% less than the MLR model. This also indicates that the ANN model is more efficient in estimating the electrical resistivity of cement paste mixtures modified with CNTs.

#### 4.2. Correlation between Compressive Strength and Electrical Resistivity Based on the Proposed Model and Models from the Literature

^{2}, RMSE, MAE, SI, and residual error, the proposed model gives better performance for predicting the compressive strength of cement paste compared to the models presented in the literature. The R

^{2}value for the proposed model is 0.74, which is 10.80% larger than the logarithm function and 16.22% greater than the linear model Equation (15); the RMSE value is 13.04% lower than logarithm function and 19.35% lower than the linear function. In addition, the proposed model also has a lower MAE value than the other models, 14.77% lower than the logarithm model and 25.37% lower than the linear model Equation (15) as shown in Figure 18. The proposed model also has a lower scatter index value than the models proposed in the literature, which is 14.29% lower than the logarithm model Equation (16) and 20.48% lower than the linear model Equation (15) in Figure 19. The residual error for all three models is indicated in Figure 20. From Figure 20, the proposed model is also giving a better residual error compared to the other models.

## 5. Sensitivity Examination

^{2}, RMSE, and MAE—were determined independently. In the first scenario, all parameters were considered to account for estimated electrical resistivity (scenario no. 1 in Table 1). For the second scenario, w/c was removed in order to see its effect on statistical. In the third scenario, the curing time was removed. Lastly, CNT was removed. From both Table 1 (Scenario 1–4) and Figure 21, it was noticed that the curing time is the most critical and sensitive parameter, which affects the output results by removing curing time, the R

^{2}was reached the lower limit, and both RMSE and MAE have high magnitude. This happens because there is a significant difference between the performance of CNT-based paste at early and long ages [52]. The curing time for the collected data was varied from 1 to 180 days in this analysis. Increasing the curing time significantly improved the electrical resistivity of cement paste mixes containing CNTs. Almost all of the experimental findings listed in Table A1 (Appendix A) support this.

## 6. Conclusions

^{2}, RMSE, MAE, SI, and OBJ. Results indicated that the sequence of models are LR, NLR, MLR, and ANN, which means that the ANN was the best model proposed in this study based on data collected from literature which gives higher R

^{2}and lower RMSE and MAE.

^{2}, MAE, RMSE, SI, and residual error, the model proposed in this study gives better performance than that of the models proposed by literature to predicate the electrical resistivity of CNT-based paste as a function of electrical resistivity.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Effect of Carbon Nanotubes on Cement Paste Resistivity at Different w/c Ratios and Different Curing Age

**Table A1.**Effect of carbon nanotubes on cement paste resistivity at different w/c ratios and different curing ages.

No. of Data | References | w/c | Curing Time (day) | CNT % | Electrical Resistivity (Ω.m) |
---|---|---|---|---|---|

1 | [4] | 0.2 | 28 | 0 | 72.33 |

2 | 0.2 | 28 | 0.1 | 68.39 | |

3 | 0.2 | 28 | 0.5 | 32.54 | |

4 | 0.2 | 28 | 1 | 6.42 | |

5 | [28] | 0.2 | 27 | 0 | 72.53 |

6 | 0.2 | 27 | 0.1 | 68.39 | |

7 | 0.2 | 27 | 0.5 | 32.54 | |

8 | 0.2 | 27 | 1 | 6 | |

9 | [30] | 0.27 | 1 | 0 | 9.67 |

10 | 0.27 | 7 | 0 | 20.92 | |

11 | 0.27 | 28 | 0 | 90.65 | |

12 | 0.27 | 60 | 0 | 330 | |

13 | 0.27 | 90 | 0 | 620.22 | |

14 | 0.27 | 180 | 0 | 1252.23 | |

15 | 0.27 | 1 | 0.05 | 5.64 | |

16 | 0.27 | 7 | 0.05 | 16.75 | |

17 | 0.27 | 28 | 0.05 | 57.8 | |

18 | 0.27 | 60 | 0.05 | 188.57 | |

19 | 0.27 | 90 | 0.05 | 241.61 | |

20 | 0.27 | 180 | 0.05 | 601.56 | |

21 | 0.27 | 1 | 0.15 | 4.97 | |

22 | 0.27 | 7 | 0.15 | 14.78 | |

23 | 0.27 | 28 | 0.15 | 51.17 | |

24 | 0.27 | 60 | 0.15 | 157.14 | |

25 | 0.27 | 90 | 0.15 | 238.66 | |

26 | 0.27 | 180 | 0.15 | 597.14 | |

27 | 0.27 | 1 | 0.25 | 4.15 | |

28 | 0.27 | 7 | 0.25 | 13.01 | |

29 | 0.27 | 28 | 0.25 | 46.11 | |

30 | 0.27 | 60 | 0.25 | 141.92 | |

31 | 0.27 | 90 | 0.25 | 223.44 | |

32 | 0.27 | 180 | 0.25 | 518.57 | |

33 | 0.27 | 1 | 0.35 | 3.59 | |

34 | 0.27 | 7 | 0.35 | 12.95 | |

35 | 0.27 | 28 | 0.35 | 43.12 | |

36 | 0.27 | 60 | 0.35 | 128.66 | |

37 | 0.27 | 90 | 0.35 | 219.51 | |

38 | 0.27 | 180 | 0.35 | 484.69 | |

39 | 0.27 | 1 | 0.45 | 2.94 | |

40 | 0.27 | 7 | 0.45 | 12.87 | |

41 | 0.27 | 28 | 0.45 | 38.21 | |

42 | 0.27 | 60 | 0.45 | 115.89 | |

43 | 0.27 | 90 | 0.45 | 207.23 | |

44 | 0.27 | 180 | 0.45 | 468.97 | |

45 | 0.27 | 1 | 0.55 | 2.87 | |

46 | 0.27 | 7 | 0.55 | 11.79 | |

47 | 0.27 | 28 | 0.55 | 33.93 | |

48 | 0.27 | 60 | 0.55 | 113.44 | |

49 | 0.27 | 90 | 0.55 | 190.04 | |

50 | 0.27 | 180 | 0.55 | 417.9 | |

51 | 0.27 | 1 | 0.65 | 2.06 | |

52 | 0.27 | 7 | 0.65 | 10.7 | |

53 | 0.27 | 28 | 0.65 | 33.17 | |

54 | 0.27 | 60 | 0.65 | 103.13 | |

55 | 0.27 | 90 | 0.65 | 178.75 | |

56 | 0.27 | 180 | 0.65 | 374.2 | |

57 | 0.27 | 1 | 0.75 | 1.57 | |

58 | 0.27 | 7 | 0.75 | 9.7 | |

59 | 0.27 | 28 | 0.75 | 30.08 | |

60 | 0.27 | 60 | 0.75 | 102.83 | |

61 | 0.27 | 90 | 0.75 | 164.02 | |

62 | 0.27 | 180 | 0.75 | 362.41 | |

63 | 0.27 | 1 | 0.85 | 1.42 | |

64 | 0.27 | 7 | 0.85 | 9.3 | |

65 | 0.27 | 28 | 0.85 | 28.24 | |

66 | 0.27 | 60 | 0.85 | 75.97 | |

67 | 0.27 | 90 | 0.85 | 137.5 | |

68 | 0.27 | 180 | 0.85 | 339.82 | |

69 | 0.27 | 1 | 0.95 | 1.37 | |

70 | 0.27 | 7 | 0.95 | 8.72 | |

71 | 0.27 | 28 | 0.95 | 24.01 | |

72 | 0.27 | 60 | 0.95 | 64.13 | |

73 | 0.27 | 90 | 0.95 | 117.86 | |

74 | 0.27 | 180 | 0.95 | 312.81 | |

75 | 0.27 | 1 | 1 | 1.2 | |

76 | 0.27 | 7 | 1 | 8.23 | |

77 | 0.27 | 28 | 1 | 22.25 | |

78 | 0.27 | 60 | 1 | 58.19 | |

79 | 0.27 | 90 | 1 | 102.14 | |

80 | 0.27 | 180 | 1 | 276.47 | |

81 | [31] | 0.27 | 7 | 0 | 218.5 |

82 | 0.27 | 28 | 0 | 268.3 | |

83 | 0.27 | 60 | 0 | 360.8 | |

84 | 0.27 | 90 | 0 | 473.8 | |

85 | 0.27 | 120 | 0 | 536.4 | |

86 | 0.27 | 7 | 0.25 | 111.4 | |

87 | 0.27 | 28 | 0.25 | 167.3 | |

88 | 0.27 | 60 | 0.25 | 211.7 | |

89 | 0.27 | 90 | 0.25 | 250.4 | |

90 | 0.27 | 120 | 0.25 | 302.9 | |

91 | 0.27 | 7 | 0.5 | 84.5 | |

92 | 0.27 | 28 | 0.5 | 134.3 | |

93 | 0.27 | 60 | 0.5 | 178.6 | |

94 | 0.27 | 90 | 0.5 | 203.9 | |

95 | 0.27 | 120 | 0.5 | 245.6 | |

96 | [32] | 0.4 | 7 | 0.05 | 0.798 |

97 | 0.4 | 14 | 0.05 | 1.225 | |

98 | 0.4 | 28 | 0.05 | 3.46 | |

99 | 0.4 | 7 | 0.1 | 0.897 | |

100 | 0.4 | 14 | 0.1 | 1.84 | |

101 | 0.4 | 28 | 0.1 | 5.52 | |

102 | 0.4 | 7 | 0.3 | 1.9 | |

103 | 0.4 | 14 | 0.3 | 2.18 | |

104 | 0.4 | 28 | 0.3 | 5.22 | |

105 | 0.4 | 7 | 0.5 | 3.29 | |

106 | 0.4 | 14 | 0.5 | 3.9 | |

107 | 0.4 | 28 | 0.5 | 5.34 | |

108 | [33] | 0.485 | 28 | 0 | 84 |

109 | 0.485 | 28 | 0.1 | 61 | |

110 | 0.485 | 28 | 0.5 | 71 | |

111 | [36] | 0.4 | 28 | 0 | 500 |

112 | 0.4 | 28 | 0.25 | 350 | |

113 | 0.4 | 28 | 0.5 | 190 | |

114 | 0.4 | 28 | 0.75 | 176 | |

115 | 0.4 | 28 | 1 | 180 | |

116 | 0.4 | 28 | 1.5 | 35 | |

Remarks | Ranged between 0.2–0.485 | varied between 1–180 (days) | Ranged between 0–1.5 (%) | varied between 0.798–1252.23 (Ω.m) |

No. Data | References | Electrical Resistivity (Ω.m) | Compressive (MPa) |
---|---|---|---|

1 | [32] | 536.4 | 73.7 |

2 | 473.8 | 70.1 | |

3 | 360.8 | 65.37 | |

4 | 302.9 | 76.85 | |

5 | 268.3 | 58.51 | |

6 | 250.4 | 72.125 | |

7 | 245.6 | 76.96 | |

8 | 211.7 | 69.425 | |

9 | 203.9 | 74.04 | |

10 | 196.48 | 68.22 | |

11 | 190.04 | 64.74 | |

12 | 178.6 | 71.225 | |

13 | 167.3 | 62.56 | |

14 | 134.3 | 65.71 | |

15 | 111.4 | 48.16 | |

16 | 105.978 | 56.88 | |

17 | 90.65 | 49.23 | |

18 | 84.5 | 52.21 | |

19 | [35] | 20.92 | 27.66 |

20 | 11.79 | 47.72 | |

21 | 1.57 | 32.2 | |

22 | 11.87 | 47.72 | |

23 | 33.93 | 59.53 | |

24 | 190.04 | 64.74 | |

25 | [33] | 84 | 31.7 |

26 | 61 | 33.8 | |

27 | 71 | 35.3 | |

28 | [53] | 2.09 | 18.73 |

29 | 8.29 | 45.95 | |

30 | 2.41 | 25.68 | |

31 | 5.27 | 36.49 | |

32 | 3.73 | 31.27 | |

33 | 1.79 | 14.86 | |

34 | [54] | 3.04 | 45.6 |

35 | 2.94 | 45.1 | |

36 | 2.88 | 44.6 | |

37 | 3.74 | 54.7 | |

38 | 3.74 | 53.8 | |

39 | 7.08 | 37.51 | |

40 | 7.06 | 25.48 | |

41 | 5.63 | 19.7 | |

42 | [55] | 71 | 35.3 |

43 | 61 | 33.8 | |

44 | 60.73 | 49.49 | |

45 | 28.04 | 37.19 | |

46 | [52] | 6.99 | 35.03 |

47 | 6.92 | 34.39 | |

48 | 6.81 | 31.99 | |

49 | 6.54 | 33.53 | |

50 | 6.45 | 30.6 | |

51 | 6.23 | 29.47 | |

52 | 6.18 | 24.9 | |

53 | 6.05 | 28.2 | |

54 | 5.98 | 26.62 | |

55 | 5.79 | 19.38 | |

56 | 5.71 | 21.92 | |

Remarks | Electrical resistivity ranged between (1.57–536.4) Ω.m | Compressive strength varied between (14.86–76.86) MPa |

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**Figure 1.**This study follows the flow chart diagram process (R

^{2}is coefficient of determination; root mean square, RMSE; mean absolute error, MAE; scatter index, SI; and objective, OBJ).

**Figure 5.**Optimal ANN structure based on RMSE and R

^{2}: (

**a**) one hidden layer; (

**b**) two hidden layers; (

**c**) three hidden layers.

**Figure 6.**Choosing best-hidden layer and neurons for Artificial Neural Network model based on lower RMSE and MAE values.

**Figure 7.**Comparison between measured and predicted the electrical resistivity using Linear. Regression (LR) model (

**a**) training dataset, (

**b**) testing dataset.

**Figure 8.**Comparison between measured and predicted the electrical resistivity using Multi Linear Regression (MLR) model (

**a**) training dataset, (

**b**) testing dataset.

**Figure 9.**Comparison between measured and predicted the electrical resistivity using Non Linear Regression (NLR) model (

**a**) training dataset, (

**b**) testing dataset.

**Figure 10.**Comparison between measured and predicted the electrical resistivity using ANN model (

**a**) training dataset, (

**b**) testing dataset.

**Figure 11.**Comparison of the RMSE and MAE performance parameters of different developed models for training data and testing data.

**Figure 12.**Comparison of the R

^{2}performance parameters of different developed models for training data and testing data.

**Figure 15.**Comparison of the SI performance parameter of different developed models for training data and testing data.

**Figure 16.**Variation in predicted and measured values of electrical resistivity based five different approaches using training data set.

Scen. No. | Input Combination | Removed Parameter | R^{2} | MAE (Ω.m) | RMSE (Ω.m) | Ranking |
---|---|---|---|---|---|---|

1 | w/c, CNT, curing time | - | 0.93 | 36.37 | 54.48 | - |

2 | CNT, curing time | w/c | 0.89 | 49.49 | 69.41 | 3 |

3 | w/c, CNT | Curing time | 0.18 | 140.67 | 190.21 | 1 |

4 | w/c, curing time | CNT | 0.63 | 73.01 | 127.53 | 2 |

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**MDPI and ACS Style**

Piro, N.S.; Mohammed, A.S.; Hamad, S.M. Multiple Analytical Models to Evaluate the Impact of Carbon Nanotubes on the Electrical Resistivity and Compressive Strength of the Cement Paste. *Sustainability* **2021**, *13*, 12544.
https://doi.org/10.3390/su132212544

**AMA Style**

Piro NS, Mohammed AS, Hamad SM. Multiple Analytical Models to Evaluate the Impact of Carbon Nanotubes on the Electrical Resistivity and Compressive Strength of the Cement Paste. *Sustainability*. 2021; 13(22):12544.
https://doi.org/10.3390/su132212544

**Chicago/Turabian Style**

Piro, Nzar Shakr, Ahmed Salih Mohammed, and Samir Mustafa Hamad. 2021. "Multiple Analytical Models to Evaluate the Impact of Carbon Nanotubes on the Electrical Resistivity and Compressive Strength of the Cement Paste" *Sustainability* 13, no. 22: 12544.
https://doi.org/10.3390/su132212544