# Statistical Modeling of Photo-Bending Actuation of Hybrid Silicones Mixed with Azobenzene Powder

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

^{2}by changing the controller power 25, 50, 75, and 100%. The bending and unbending behaviors of hybrid silicones (0, 1, 5, and 10 wt%) were observed by turning on UV light for 3 min and turning it off for 4 min. This observation was repeated three times in each condition. Such repetitive observations were performed at each light intensity (150, 300, 450, or 600 mW/cm

^{2}). The movies were recorded by a personal smartphone (iPhone X, Apple). The bending deflection was calculated from the recorded movies by using software ImageJ. The experimental results of deflections were fitted with an exponential function for bending and unbending each by using solver function in excel ver. 16.16.2 (Microsoft), and then modeled by using a statistical software DesignExpert ver. 11.1.2.0 (Stat-Ease). Model constructions are tested by linear, two factor interaction (2FI), quadratic, cubic, and quartic functions. These functions are represented as combinations of two variables x

_{1}and x

_{2}corresponding to the percentage of the powder content and the UV intensity (Table 1).

## 3. Results & Discussions

#### 3.1. Photo-Actuation Behavior

#### 3.2. Bending Evaluation

^{2}). The deflection δ′ in the photo-bending process at each condition was successfully fitted by least square to an exponential model:

^{2}, and increased up to A = 13.2 at the intensity 600 mW/cm

^{2}. This tendency is applied to other hybrid silicones of 0, 1 and 5 wt%. As to the value of ${\tau}_{b}$, it decreased for increasing light intensity, but there was no clear relation with the amount of powder contents.

^{2}and increased up to ${\tau}_{u}$ = 74.4 s after irradiated at 600 mW/cm

^{2}. This tendency was almost observed in the other hybrid silicones (with different powder content), although some conditions of the hybrid silicones of 0 and 1 wt% did not match with the trend. When comparing the samples at the same light intensity, the trend is that higher powder contents lead to higher ${\tau}_{u}$ values, although the maximum appeared at 5 wt%.

#### 3.3. Model Construction

#### 3.3.1. Model Construction for Photo-Bending

^{−6}(significant) and the related lack of fit has a p-value of 0.228 (not significant, which means that the cubic model is adequate enough) as shown in Table 5. For instance, a linear model presents a p-value of 7.49 × 10

^{−19}(also significant), but a lack of fit of 4.86 × 10

^{−15}(significant, which means that this model is not suited to fit the data). A quartic model scores comparable adjusted R

^{2}and predicted R

^{2}, but has a p-value of 0.0871 (not significant), showing this quartic model is aliased.

_{%}is the weight percentage of powder (in %) and P

_{UV}the intensity of UV light (in mW/cm

^{2}). For any parameter to be significant, its F-value should be higher than 0.05 and its p-value should be lower than 0.05. Among variables in the full cubic model, the terms of P

_{%}and P

_{UV}

^{2}are not significant due to p-value larger than 10%, while all the other parameters of the cubic model have a p-value lower than 4.0% (Table 6). This is how the cubic model can slightly be simplified by removing the variables that are not significant. The new proposed model is, therefore:

^{−35}. The two-dimensional (2D) contour plot and 3D response surface of the reduced cubic model is represented in Figure 4. The response surface shows that the value A increases depending on the increase of both the UV intensity and the powder contents.

^{−11}(significant) and the lack of fit has a p-value of 0.107, meaning not significant (Table 7). For instance, the p-value of the linear model is 1.40 × 10

^{−3}(significant), but the lack of fit is 3.84 × 10

^{−10}(significant) and cannot be used. A quartic model also scores comparable adjusted R

^{2}and predicted R

^{2}, but is aliased.

_{UV}the power of UV light and P

_{%}the percentage of powder. This model can be simplified by removing the coefficients that are not significant. The terms of P

_{UV}

^{2}, P

_{%}

^{2}P

_{UV}, P

_{%}P

_{UV}

^{2}, and P

_{UV}

^{3}are not significant (p-value larger than 30%), while all the other parameters of the cubic model have a p-value lower than 4.5% (Table 8). The new proposed model is, therefore:

^{−16}. The response surface of the new model is presented in Figure 5. The 3D surface shows that the value ${\tau}_{b}$ at a certain powder content decreases depending on the increase of the light intensity, but the value ${\tau}_{b}$ is susceptible to the percentage of powder in a not monotonous manner.

#### 3.3.2. Model Construction for Unbending

^{−4}(significant) and the lack of fit has a p-value of 0.263, meaning not significant (Table 9).

_{UV}the power of UV light and P

_{%}the percentage of powder. This model can be simplified by removing the coefficient that are not significant (Table 10). The terms of P

_{%}P

_{UV}, P

_{UV}

^{2}and P

_{UV}

^{3}are not significant (p-value larger than 25%), while all the other parameters of the cubic model have a p-value lower than 3.5%. The new proposed model is, therefore:

^{−21}. The response surface of the new model is presented in Figure 6, showing that the value ${\tau}_{u}$ increases depending on the percentage of powder. Thus, the unbending behavior of hybrid silicones has also been expressed as an experiments-based mathematical model.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Preparation of a hybrid silicone mixed with azobenzene powder. (

**a**) Curing the first layer. (

**b**) Coating uncured silicone containing azobenzene powder as the second layer. (

**c**) Obtained hybrid silicone after curing the second layer. (

**d**) Cutting for bending observations.

**Figure 2.**Shape change of a hybrid silicone with azobenzene powder (10 wt%) before and after ultraviolet (UV) light irradiation. (

**a**) Elongation of the hybrid silicone when free to expand. (

**b**) Bending of the hybrid silicone when constrained by a paper. Dotted lines in panels show the initial position before UV irradiation. In both cases, UV light (365 nm) was irradiated 3 min at 600 mW/cm

^{2}.

**Figure 3.**Time dependence of deflection δ′ of hybrid silicones under UV light illumination at different light intensities and after stopping the irradiation. Panels show the results of hybrid silicones of (

**a**) 0, (

**b**) 1, (

**c**) 5, and (

**d**) 10 wt%. Plots drawn by small circle indicate the average value of deflection repeatedly observed three times, with standard deviations. Solid lines are the results of fitting to exponential models for photo-bending and unbending processes.

**Figure 4.**The value A in the photo-bending process, drawn by the reduced cubic model. (

**a**) Contour plot in two dimensions (2D) and (

**b**) surface plot in three dimensions (3D). The dots represent the experimental points (i.e., the fit of the exponential model) and properly fit with the cubic model proposed in this work.

**Figure 5.**The value ${\tau}_{b}$ in the photo-bending process, drawn by the reduced cubic model. (

**a**) Contour plot in 2D and (

**b**) surface plot in 3D. The dots represent the experimental data (i.e., the fit of the exponential model). The cubic model gives satisfying fit with these experimental data.

**Figure 6.**The value ${\tau}_{u}$ in the unbending process, drawn by the reduced cubic model. (

**a**) Contour plot in 2D and (

**b**) surface plot in 3D. The dark dots correspond to experimental data over the model response and light dots corresponds to experimental data below the model surface.

x_{1} | x_{2} | x_{1}x_{2} | x_{1}^{2} | x_{2}^{2} | x_{1}^{2}x_{2} | x_{1}x_{2}^{2} | x_{1}^{3} | x_{2}^{3} | x_{1}^{2}x_{2}^{2} | x_{1}^{3}x_{2} | x_{1}x_{2}^{3} | x_{1}^{4} | x_{2}^{4} | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Linear | Y | Y | - | - | - | - | - | - | - | - | - | - | - | - |

2FI | Y | Y | Y | - | - | - | - | - | - | - | - | - | - | - |

Quadratic | Y | Y | Y | Y | Y | Y | Y | - | - | - | - | - | - | - |

Cubic | Y | Y | Y | Y | Y | Y | Y | Y | Y | - | - | - | - | - |

Quartic | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y | Y |

0 wt% | 1 wt% | 5 wt% | 10 wt% | |
---|---|---|---|---|

Length from irradiated center to the tip/mm | 20.0 | 18.9 | 18.0 | 16.5 |

Width/mm | 4.9 | 4.5 | 4.9 | 5.5 |

Thickness/mm | 1.4 | 1.5 | 1.7 | 1.5 |

**Table 3.**Summary of A and ${\tau}_{b}$ obtained by fitting to the exponential model for photo-bending.

Light (mW/cm^{2}) | The Value of A | $\mathbf{The}\text{}\mathbf{Value}\text{}\mathbf{of}\text{}{\mathit{\tau}}_{\mathit{b}}$ | ||||||
---|---|---|---|---|---|---|---|---|

0 wt% | 1 wt% | 5 wt% | 10 wt% | 0 wt% | 1 wt% | 5 wt% | 10 wt% | |

150 | 0.57 | 1.78 | 3.41 | 4.88 | 66.7 | 43.6 | 70.7 | 60.1 |

300 | 1.23 | 3.57 | 6.61 | 8.56 | 61.6 | 46.9 | 65.8 | 49.8 |

450 | 2.04 | 5.32 | 9.54 | 12.0 | 58.0 | 45.6 | 62.1 | 47.0 |

600 | 2.48 | 6.07 | 11.5 | 13.2 | 52.9 | 41.8 | 53.6 | 40.2 |

Light (mW/cm^{2}) | The Value of A | $\mathbf{The}\text{}\mathbf{Value}\text{}\mathbf{of}\text{}{\mathit{\tau}}_{\mathit{u}}$ | ||||||
---|---|---|---|---|---|---|---|---|

0 wt% | 1 wt% | 5 wt% | 10 wt% | 0 wt% | 1 wt% | 5 wt% | 10 wt% | |

150 | 0.62 | 1.78 | 3.69 | 5.05 | 48.7 | 52.5 | 70.9 | 69.0 |

300 | 1.28 | 3.59 | 6.97 | 8.60 | 49.9 | 50.9 | 72.4 | 67.6 |

450 | 2.18 | 6.97 | 9.91 | 12.0 | 51.4 | 55.5 | 74.7 | 71.2 |

600 | 2.60 | 8.60 | 11.8 | 13.1 | 44.7 | 52.8 | 85.7 | 74.4 |

Source | Sequential p-Value | Lack of Fit p-Value | Adjusted R^{2} | Predicted R^{2} |
---|---|---|---|---|

Linear | 7.49 × 10^{−19} | 4.86 × 10^{−15} | 0.837 | 0.820 |

2FI | 1.42 × 10^{−4} | 4.62 × 10^{−13} | 0.880 | 0.864 |

Quadratic | 1.09 × 10^{−12} | 4.66 × 10^{−5} | 0.966 | 0.960 |

Cubic | 2.18 × 10^{−6} | 0.228 | 0.983 | 0.977 |

Quartic | 0.0871 | 0.619 | 0.985 | 0.977 |

Source | Sum of Squares | df | Mean Square | F-Value | p-Value |
---|---|---|---|---|---|

Model | 739. 9 | 9 | 82.2 | 311.4 | 9.14 × 10^{−33} |

P_{%} | 0.69 | 1 | 0.69 | 2.6 | 0.113 |

P_{UV} | 60.6 | 1 | 60.6 | 229.4 | 1.09 × 10^{−17} |

P_{%}P_{UV} | 22.7 | 1 | 22.7 | 86.1 | 2.62 × 10^{−11} |

P_{%}^{2} | 35.6 | 1 | 35.6 | 134.8 | 4.60 × 10^{−14} |

P_{UV}^{2} | 0.026 | 1 | 0.026 | 0.099 | 0.755 |

P_{%}^{2}P_{UV} | 5.4 | 1 | 5.4 | 20.4 | 5.98 × 10^{−5} |

P_{%}P_{UV}^{2} | 1.5 | 1 | 1.5 | 5.6 | 0.0229 |

P_{%}^{3} | 4.5 | 1 | 4.5 | 17.0 | 1.93 × 10^{−4} |

P_{UV}^{3} | 1.3 | 1 | 1.3 | 4.8 | 0.0345 |

Residual | 10.0 | 38 | 0.26 | - | - |

Lack of Fit | 2.1 | 6 | 0.36 | 1.45 | 0.228 |

Pure Error | 7.9 | 32 | 0.25 | - | - |

Corrected Total | 749.9 | 47 | - | - | - |

Source | Sequential p-Value | Lack of Fit p-Value | Adjusted R^{2} | Predicted R^{2} |
---|---|---|---|---|

Linear | 0.00140 | 3.84 × 10^{−10} | 0.220 | 0.156 |

2FI | 0.202 | 3.75 × 10^{−10} | 0.232 | 0.157 |

Quadratic | 0.0253 | 1.51 × 10^{−9} | 0.324 | 0.225 |

Cubic | 1.76 × 10^{−11} | 0.107 | 0.824 | 0.761 |

Quartic | 0.0186 | 0.837 | 0.856 | 0.785 |

**Table 8.**ANOVA results for the ${\tau}_{b}$ parameter in the complete cubic model for photo-bending.

Source | Sum of Squares | df | Mean Square | F-Value | p-Value |
---|---|---|---|---|---|

Model | 4075.7 | 9 | 452.9 | 25.5 | 1.77 × 10^{−13} |

P_{%} | 1988.5 | 1 | 1988.5 | 112.1 | 6.84 × 10^{−13} |

P_{UV} | 79.1 | 1 | 79.1 | 4.5 | 0.0413 |

P_{%} P_{UV} | 116.7 | 1 | 116.7 | 6.6 | 0.0144 |

P_{%}^{2} | 508.5 | 1 | 508.5 | 28.7 | 4.38 × 10^{−6} |

P_{UV}^{2} | 9.8 | 1 | 9.76 | 0.55 | 0.463 |

P_{%}^{2}P_{UV} | 0.24 | 1 | 0.24 | 0.01 | 0.907 |

P_{%}P_{UV}^{2} | 18.4 | 1 | 18.4 | 1.0 | 0.315 |

P_{%}^{3} | 2158.7 | 1 | 2158.7 | 121.7 | 2.09 × 10^{−13} |

P_{UV}^{3} | 16.1 | 1 | 16.1 | 0.9 | 0.346 |

Residual | 674.2 | 38 | 17.7 | - | - |

Lack of Fit | 178.7 | 6 | 29. 8 | 1.92 | 0.107 |

Pure Error | 495.5 | 32 | 15.5 | - | - |

Corrected Total | 4749.9 | 47 | - | - | - |

Source | Sequential p-Value | Lack of Fit p-Value | Adjusted R^{2} | Predicted R^{2} |
---|---|---|---|---|

Linear | 2.26 × 10^{−9} | 1.85 × 10^{−10} | 0.569 | 0.531 |

2FI | 0.277 | 1.54 × 10^{−10} | 0.571 | 0.528 |

Quadratic | 5.88 × 10^{−12} | 0.00260 | 0.869 | 0.844 |

Cubic | 3.64 × 10^{−4} | 0.263 | 0.915 | 0.888 |

Quartic | 0.144 | 0.493 | 0.921 | 0.884 |

Source | Sum of Squares | df | Mean Square | F-Value | p-Value |
---|---|---|---|---|---|

Model | 6718.9 | 9 | 746.5 | 57.05 | 2.53 × 10^{−19} |

P_{%} | 311.3 | 1 | 311.3 | 23.79 | 1.95 × 10^{−5} |

P_{UV} | 159.7 | 1 | 159.7 | 12.21 | 0.00123 |

P_{%} P_{UV} | 0.010 | 1 | 0.010 | 0.0 | 0.978 |

P_{%}^{2} | 1313.2 | 1 | 1313.2 | 100.34 | 3.26 × 10^{−12} |

P_{UV}^{2} | 27.3 | 1 | 27.3 | 2.08 | 0.157 |

P_{%}^{2}P_{UV} | 168.3 | 1 | 168.3 | 12.86 | 9.44 × 10^{−4} |

P_{%} P_{UV}^{2} | 64.0 | 1 | 64.0 | 4.89 | 0.0331 |

P_{%}^{3} | 101.8 | 1 | 101.8 | 7.78 | 0.00823 |

P_{UV}^{3} | 14.8 | 1 | 14.8 | 1.13 | 0.294 |

Residual | 497.3 | 38 | 13.1 | - | - |

Lack of Fit | 100.7 | 6 | 16.8 | 1.35 | 0.263 |

Pure Error | 396.6 | 32 | 12.4 | - | - |

Corrected Total | 7216.2 | 47 | - | - | - |

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**MDPI and ACS Style**

Taniguchi, T.; Blanc, L.; Asahi, T.; Koshima, H.; Lambert, P. Statistical Modeling of Photo-Bending Actuation of Hybrid Silicones Mixed with Azobenzene Powder. *Actuators* **2019**, *8*, 68.
https://doi.org/10.3390/act8040068

**AMA Style**

Taniguchi T, Blanc L, Asahi T, Koshima H, Lambert P. Statistical Modeling of Photo-Bending Actuation of Hybrid Silicones Mixed with Azobenzene Powder. *Actuators*. 2019; 8(4):68.
https://doi.org/10.3390/act8040068

**Chicago/Turabian Style**

Taniguchi, Takuya, Loïc Blanc, Toru Asahi, Hideko Koshima, and Pierre Lambert. 2019. "Statistical Modeling of Photo-Bending Actuation of Hybrid Silicones Mixed with Azobenzene Powder" *Actuators* 8, no. 4: 68.
https://doi.org/10.3390/act8040068