# Predicting the Dielectric Properties of Nanocellulose-Modified Presspaper Based on the Multivariate Analysis Method

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental

#### 2.1. Materials

#### 2.2. Preparation of Presspaper Samples

^{2}for 10 min. Prepared presspaper has a thickness of ~400 μm and a grammage of about 480 g/m

^{2}.

#### 2.3. Morphology Characterization

#### 2.4. Chemical Composition

#### 2.5. Crystallinity, DP, and Total Charge

_{002}− I

_{AM})/I

_{002}× 100%

_{200}is the height of the (002) crystalline peak and I

_{AM}is the height of the amorphous halo.

#### 2.6. Tensile Strength

#### 2.7. Breakdown Behavior

## 3. Modeling

#### 3.1. Variables

_{1}to x

_{13}.

_{1}and y

_{2}, are shown in Table 3. There are significant differences in the mechanical and breakdown properties of the samples. The maximum values of tensile strength DC breakdown strength are 121 MPa and 23.4 kV/mm, respectively; the corresponding minimum values are 39 MPa and 9.4 kV/mm; the maximum value is 2 to 3 times the minimum value.

#### 3.2. Small Sample and Multiple Correlation Problems

_{8}), total lignin(x

_{7}), and DP(x

_{12}), x

_{8}will be eliminated first when conducting the multiple linear regression analysis.

^{2}, adjusted R

^{2}, and Mallows Cp value, the physicochemical parameters set suitable for multiple linear regression modeling could be determined. As the expected number of selected variables is 2~5, the range of given variables is 1~6 when conducting best subset selection.

_{1}) is selected. If the number of physicochemical parameters to be considered is increased to two, the best subset is fiber width(x

_{2}) and CrI(x

_{13}). For a different number of variables, there is a corresponding best subset.

^{2}increases, indicating that the fitting effect of the regression model is improved. However, when the number of independent variables reaches a certain value, the goodness of fit that the newly introduced variable brings is very limited. If the number of variables increases from one to two, R

^{2}increases by 32.4%; when it increases from three to four, R

^{2}is only increased by 2.4%. Adjusted R

^{2}can partly eliminate the increase in goodness of fit due solely to an increase in the number of variables. Only when the new variable has a certain explanatory effect on the dependent variable does adjusted R

^{2}increase. Since fiber morphology parameters, such as fiber length and width, etc., affect the mechanical properties of the presspaper when they are introduced, the adjusted R

^{2}value increases. However, when the number of variables reaches two, increasing variables does not lead to a significant increase in the adjusted R

^{2}value.

^{2}value are relatively high, and no further significant increase occurs with increasing variables, the physicochemical parameters that are suitable for establishing the tensile strength regression model of presspaper are fiber width(x

_{2}) and crystallinity(x

_{13}). Similarly, for the AC breakdown strength regression model, the number of variables selected is three, and the selected parameters are fiber length(x

_{1}), fines(x

_{5}), and total charge(x

_{11}). The values of adjusted R

^{2}and Mallows Cp are 82.5% and 4.1, respectively. For the DC breakdown strength regression model, the number of selected variables is three, and the selected parameters are fiber length(x

_{1}), fines(x

_{5}), and total lignin(x

_{7}). The value of adjusted R

^{2}and Mallows Cp are 89.2% and 4.0, respectively.

## 4. Results and Discussion

#### 4.1. Multiple Linear Regression Model for Mechanical Properties of Presspaper

_{1}) and fiber width(x

_{2}) and CrI(x

_{13}) is shown in Table 5. The goodness of fit of the model reached 87%, and predicted R

^{2}reached 73%. The P value is less than 0.05 for x

_{2}and x

_{13}, indicating their strong relationship with the tensile strength of presspaper, and it is necessary for them to be included in the model. To test the degree of collinearity among the selected variables, the variance inflation factor (VIF) is presented in Table 6. This means that there is no correlation between independent variables, if VIF is equal to 1; if VIF is greater than 10, it means that there is a strong link between independent variables; if VIF is between 1 and 5, it means there is a certain degree of correlation between independent variables, but it will not have a serious impact on the regression coefficient estimation of the model. As shown in Table 6, VIFs of x

_{2}and x

_{13}are very close to 1, so there is no problem of multicollinearity. The above shows that, from a statistical point of view, it is necessary and feasible that the tensile strength regression model includes both x

_{2}and x

_{13}. To further illustrate the rationality of the model, Figure 3a shows the normal distribution of standardized residuals. Since the residuals are standardized, the μ of the normal distribution approaches 0, and σ is close to 1. The standardized residual approximates a linear trend, indicating that it follows the normal distribution, and there is no undetermined variable in the established tensile strength model. All the standardized residuals are in (−2, 2), indicating that none of the data is an abnormal observation point. Comparing the fitting values with the actual values, as designated in Figure 3b, they are very close to each other. Actually, the deviations of the fitting values from the actual values were all within ±12%, among which the deviations of the No. 7 and No. 16 samples were the largest at 9.6% and −11.9%, respectively. This indicates that the model has a good fitting effect. The above analysis indicates that the obtained model is reasonable and credible from a statistical point of view.

#### 4.2. Multiple Linear Regression Model of DC Breakdown Strength

_{2}) and fiber length(x

_{1}), fines(x

_{5}), and total lignin content(x

_{7}) is shown in Table 6. The goodness of fit of the model is as high as 91%, and the predicted R

^{2}value is 81%. All the P values of the independent variables are less than 0.05, and the VIF values are less than 5. This shows that the above variables have a clear explanatory effect on the DC breakdown strength, and there is no strong correlation between them. Figure 4a shows the standardized residual results of the DC breakdown strength regression model. The residual basically presents a linear trend, which illustrates that it has a normal distribution. Since standardized residuals are all located in (−2, 2), none of the data points is abnormal, indicating that this model could explain the experiment results statistically. Figure 4b shows the response characteristics of the DC breakdown strength model. The results illustrate that the fitting and actual values are very close. Specifically, the deviations of the fitting values and the actual values were all within ±12%, among which the deviations of the No. 8 and No. 15 samples were the largest at 9.0% and −11.7%, respectively. In summary, the DC breakdown strength regression model of presspaper is statistically reasonable and reliable.

_{1}and x

_{5}in the DC breakdown model is positive.

_{12}is positive in the model. On the other hand, lignin helps to increase the bonding strength between fibers and restrict the internal discharge. Therefore, the regression coefficient of the total lignin content is positive.

#### 4.3. Multiple Linear Regression Model of Breakdown Strength Considering the Nanocellulose Reinforcing Effect

_{14}, as a physicochemical parameter of micro-fibers. Firstly, a linear regression model is established between the performance of presspaper and the content of nanocellulose. The constant term of the model is the performance of the presspaper without nanocellulose, then the previously obtained multiple linear regression model can be substituted, making the new model contain the reinforcing effect of nanocellulose.

_{2}* = 115.8x

_{14}+ 19.6

^{2}value reached 0.998. The regression coefficient of nanocellulose x

_{14}was 115.8, slightly higher than its coefficient in the AC breakdown regression model (y

_{3}*), which is:

_{3}* = 80.8x

_{14}+ 10.4

^{2}reached 0.999. The constant of the regression model is 10.4, which means that when the presspaper does not contain nanocellulose, the AC breakdown strength is 10.4 kV/mm.

_{2}* = 10.1x

_{1}+ 0.52x

_{5}+ 0.54x

_{7}+ 115.8x

_{14}− 7.7

_{3}* = 7.13x

_{1}+ 0.41x

_{5}+ 0.041x

_{12}+ 80.8x

_{14}− 7.6

_{14}in Equations (4) and (5) should not exceed 2.5%.

## 5. Conclusions

- A multiple linear regression model between tensile strength and fiber width variable and crystallinity variable was obtained. The goodness of fit was 87%, and the prediction accuracy of the test samples reached more than 90%. Multiple linear regression models were established for DC breakdown strength of presspaper. The prediction accuracy of the model for testing samples is more than 95%.
- Multiple linear regression models of AC and DC breakdown strength of presspaper considering the reinforcing effect of nanocellulose were established. The model for AC breakdown strength is y
_{2}* = 7.13x_{1}+ 0.41x_{5}+ 0.041x_{11}+ 80.8x_{14}− 7.6. The model for DC breakdown strength is y_{2}* = 10.1x_{1}+ 0.52x_{5}+ 0.54x_{7}+ 115.8x_{14}− 7.7. Among them, x_{1}, x_{5}, x_{7}, x_{11}, and x_{1}_{4}represent fiber length, fines, total lignin content, total charge, and nanocellulose content, respectively.

## Author Contributions

## Funding

## Conflicts of Interest

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Sample Availability: Samples of the presspaper are available from the authors. |

**Figure 3.**Multiple linear regression model of tensile strength of presspaper: (

**a**) Normal distribution test of standardized residuals; (

**b**) contrast of actual value and fitting value of regression model.

**Figure 4.**Multiple linear regression model of DC breakdown strength of presspaper: (

**a**) Normal distribution test of standardized residuals; (

**b**) comparison of fitting value and actual value.

**Figure 5.**(

**a**) Regression model of AC breakdown strength and CNFC content of presspaper; (

**b**) regression model of DC breakdown strength and CNFC content of presspaper.

Physicochemical Parameters | Unit | Variable | Physicochemical Parameters | Unit | Variable |
---|---|---|---|---|---|

Fiber length | mm | x_{1} | Holocellulose | % | x_{8} |

Fiber width | μm | x_{2} | Hemicellulose | % | x_{9} |

Ratio of length to width | − | x_{3} | Ash | % | x_{10} |

Coarseness | μg/m | x_{4} | Total charge | μmol/g | x_{11} |

Fines | % | x_{5} | DP | − | x_{12} |

Shape coefficient | % | x_{6} | CrI | % | x_{13} |

Lignin | % | x_{7} |

Sample | x_{1} | x_{2} | x_{3} | x_{4} | x_{5} | x_{6} | x_{7} | x_{8} | x_{9} | x_{10} | x_{11} | x_{12} | x_{13} |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2.06 | 31.8 | 65 | 207 | 5 | 84.3 | 10.2 | 89.8 | 8.6 | 0.70 | 52 | 1573 | 90.6 |

2 | 2.33 | 31.2 | 75 | 160 | 3.3 | 83.3 | 9.4 | 90.6 | 8.2 | 0.83 | 60 | 1877 | 91.2 |

3 | 2.52 | 31.3 | 81 | 215 | 2.5 | 84.8 | 7.6 | 92.4 | 10.5 | 0.61 | 34.7 | 1328 | 91.6 |

4 | 2.38 | 34.7 | 69 | 208 | 3.2 | 84.6 | 6.4 | 90.3 | 8.8 | 0.71 | 12 | 2038 | 91.5 |

5 | 2.36 | 31.6 | 75 | 196 | 2.1 | 84.2 | 8.8 | 91.3 | 9.4 | 0.56 | 56 | 1730 | 91.2 |

6 | 2.02 | 31.1 | 65 | 182 | 5 | 84.2 | 7.8 | 92.2 | 9.2 | 0.59 | 58.7 | 1772 | 90.9 |

7 | 2.29 | 31 | 74 | 185 | 4.1 | 82.7 | 8.4 | 91.7 | 10.5 | 0.65 | 74.7 | 1775 | 90.9 |

8 | 2.43 | 30.9 | 79 | 219 | 2.8 | 85 | 5.4 | 91.7 | 8.8 | 0.40 | 20 | 1583 | 91.2 |

9 | 2.24 | 33.5 | 67 | 178 | 3.3 | 85.3 | 10.2 | 89.7 | 10.0 | 0.49 | 42.7 | 2038 | 90.1 |

10 | 1.71 | 28.2 | 61 | 113 | 7.7 | 80.2 | 10.0 | 90.1 | 8.1 | 0.91 | 60 | 1664 | 91.3 |

11 | 2.29 | 31.3 | 73 | 184 | 4.5 | 82.9 | 8.1 | 91.6 | 9.5 | 0.63 | 50.7 | 1877 | 90.6 |

12 | 2.18 | 32.4 | 67 | 170 | 4.6 | 83.6 | 9.7 | 90.3 | 9.0 | 0.32 | 46.7 | 2046 | 90.7 |

13 | 2.20 | 30.9 | 71 | 118 | 4 | 82.2 | 1.5 | 96.4 | 14.2 | 0.11 | 36 | 804 | 91.2 |

14 | 1.51 | 19.4 | 78 | 102 | 8.4 | 82.4 | 2.1 | 94.1 | 12.0 | 0.26 | 53.3 | 1156 | 86.8 |

15 | 1.45 | 23.2 | 62 | 108 | 5.7 | 80.6 | 1.1 | 98.6 | 6.5 | 0.08 | 17.3 | 763 | 96 |

16 | 0.98 | 23.2 | 42 | 105 | 20.7 | 86.7 | 0.6 | 95.8 | 9.9 | 0.08 | 13.3 | 1298 | 89.8 |

17 | 1.74 | 27.2 | 64 | 145 | 5.35 | 82.4 | 4.4 | 95.4 | 7.9 | 0.33 | 38 | 1268 | 93.5 |

18 | 1.67 | 27.4 | 59 | 150 | 11.4 | 85.5 | 4.7 | 93.5 | 9.6 | 0.32 | 34.7 | 1514 | 90.5 |

Sample | Tensile Strength y_{1} (MPa) | DC Breakdown Strength y_{2} (kV/mm) | Sample | Tensile Strength y_{1} (MPa) | DC Breakdown Strength y_{2} (kV/mm) |
---|---|---|---|---|---|

1 | 105 | 21.2 | 10 | 100 | 19.5 |

2 | 113 | 22.1 | 11 | 101 | 23.3 |

3 | 102 | 21.2 | 12 | 107 | 20.9 |

4 | 107 | 19.6 | 13 | 112 | 17.8 |

5 | 118 | 23.3 | 14 | 92 | 13.5 |

6 | 105 | 21.3 | 15 | 39 | 9.4 |

7 | 118 | 21.7 | 16 | 75 | 13.2 |

8 | 109 | 23.4 | 17 | 70 | 15.5 |

9 | 121 | 21.6 | 18 | 101 | 18.0 |

Number of Variables | R^{2} | Adjusted R^{2} | Mallows Cp | Selected Variables |
---|---|---|---|---|

1 | 54.3 | 51 | 30.7 | x_{1} |

2 | 86.7 | 84.6 | 2.5 | x_{2}, x_{14} |

3 | 90.7 | 88.4 | 0.7 | x_{2}, x_{11}, x_{13} |

4 | 93.1 | 90.6 | 0.4 | x_{1}, x_{3}, x_{11}, x_{13} |

5 | 94.4 | 91.7 | 1.2 | x_{1}, x_{3}, x_{4}, x_{11}, x_{13} |

6 | 95.6 | 92.7 | 2.1 | x_{1}, x_{2}, x_{3}, x_{5}, x_{6}, x_{13} |

Model Information | Performance of the Prediction | |||||
---|---|---|---|---|---|---|

Independent variable | P | VIF | No. | Actual value | Predicted value | Predicted interval |

x_{2} | 0.000 | 1.02 | 17 | 69.7 | 74.6 | (55.7, 93.6) |

x_{13} | 0.000 | 1.02 | 18 | 100.7 | 95.8 | (78.1, 113.5) |

y_{1} = 608 + 3.89x_{2} − 6.83x_{13} | ||||||

R^{2} = 87%, predicted R^{2} = 73% |

Model Information | Performance of the Prediction | |||||
---|---|---|---|---|---|---|

Independent variable | P | VIF | No. | Actual value | Predicted value | Predicted interval |

x_{1} | 0.000 | 4.90 | 17 | 15.5 | 15 | (11.8, 18.3) |

x_{5} | 0.008 | 4.26 | 18 | 18 | 17.7 | (14.4, 20.9) |

x_{7} | 0.001 | 1.57 | ||||

y_{2} = −7.7 + 10.1x_{1} + 0.52x_{5} + 0.54x_{7} | ||||||

R^{2} = 91%, predicted R^{2} = 81% |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhou, Y.; Huang, X.; Huang, J.; Zhang, L.; Zhou, Z. Predicting the Dielectric Properties of Nanocellulose-Modified Presspaper Based on the Multivariate Analysis Method. *Molecules* **2018**, *23*, 1507.
https://doi.org/10.3390/molecules23071507

**AMA Style**

Zhou Y, Huang X, Huang J, Zhang L, Zhou Z. Predicting the Dielectric Properties of Nanocellulose-Modified Presspaper Based on the Multivariate Analysis Method. *Molecules*. 2018; 23(7):1507.
https://doi.org/10.3390/molecules23071507

**Chicago/Turabian Style**

Zhou, Yuanxiang, Xin Huang, Jianwen Huang, Ling Zhang, and Zhongliu Zhou. 2018. "Predicting the Dielectric Properties of Nanocellulose-Modified Presspaper Based on the Multivariate Analysis Method" *Molecules* 23, no. 7: 1507.
https://doi.org/10.3390/molecules23071507