# Dielectric Electroactive Polymers with Chemical Pre-Strain: An Experimentally Validated Model

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}, (nanofillers) can be used to facilitate crystallization of the polymer and have better control of the polymerization process. The addition of such particles resulted in an improved dielectric constant, low dielectric loss, excellent mechanical strength, toughness, and optical transparency. In this research, commercially available industrial grade fluoropolymer materials were adapted through the addition of graphite particles to adjust their electrical properties, and were used to produce dielectric EAPs through a one-step chemical pre-strain process.

## 2. Theoretical Background

_{o}is the permittivity of free space, 8.85 × 10

^{−12}F/m, V represents the applied electrical potential, a

_{1}is an electrostrictive coefficient which is the change in relative permittivity when shape is changed assuming constant volume, and a

_{2}represents a density dependent permittivity [19]. However, the electrostriction effects are small in comparison to the Coulombic charge interaction response [18,19,20]. Therefore, for simplification, the following equations can be used to define the anticipated actuation and corresponding pressures seen by the electroactive material. The corresponding stresses can be seen in Equations (3) and (4).

_{o}is the permittivity of free space which is 8.85 × 10

^{−12}F/m, and V represents the applied electrical potential.

_{o}is the permittivity of free space which is 8.85 × 10

^{−12}F/m, V represents the applied electrical potential, and D is the dielectric thickness.

## 3. Materials and Methods

#### 3.1. Experimental Samples

#### 3.2. Experimental Samples: Chemical Pre-Strain

^{®}from Menards Inc., Eau Claire, WI, USA) for 5 min, removed, and allowed to dry for 12 h at room temperature in a fume hood. According to its MSDS, the evaporation rate of this product is approximately 4.6 per ASTM D3539 [26], which is slower than Acetone (with an evaporation rate of 7.8) but faster than 95% Ethyl alcohol (with an evaporation rate of 1.4) [27]; thus the 12 h period used was sufficient for drying the samples. The solvent interacted with the fluoropolymer material, causing intercalation within the polymer structure, resulting in swelling of the material. However, this process is primarily reversible. Therefore, during the dry time, the material returned to its native state. During the intercalation process, the fluoropolymer material expanded in the x, y, and z directions. These dimensional changes were measured using a standard ruler before, during, and after induced strain [23,24].

#### 3.3. Experimental Samples: Addition of a Conductive Layer

#### 3.4. Experimental Samples: Displacement Measurements

#### 3.5. Model: Construction

^{TM}. Initial inputs to the system included a stress-strain curve of the fluoropolymer material with tensile test data taken at a rate of 20 in/min. The stress-strain data was fit using a 5-order Mooney-Rivlin fit with a root mean square error of 0.272. The model fitting was completed using Creo Elements software for fitting of a hyperelastic model. Multiple forms of strain energy calculations exist including Odgen, Yeoh, and Arruda-Boyce [22]. However, the Mooney-Rivlin model was selected due to the output of the lowest Root Mean Square error with the data fitting. The 5th order Mooney-Rivlin model was constructed using Equation (7) for calculation of strain energy, W [21,30]. The strain energy equation is dependent upon the invariants produced from the left Cauchy Green deformation tensor. The left Cauchy Green tensor is calculated from the deformation tensor, F through FF

^{T}. The C values in the equation are based on material parameters, and d

_{k}is determined based on the shear modulus. The material is assumed to be incompressible and initially isotropic. Therefore, the resulting invariants can be calculated from the eigenvalues of the deformation gradient tensor. These values are generally referred to as the stretch ratios, λ, and are defined in Equations (8)–(10). In the original undeformed state, λ

_{1}= λ

_{2}= λ

_{3}= 1. Therefore, I

_{1}= I

_{2}= 3, which results in Equation (7). Due to incompressibility, J, which is the volume ratio, is set to 1.

_{1}= λ

_{1}

^{2}+ λ

_{2}

^{2}+ λ

_{3}

^{2}

_{3}= (λ

_{1}

^{2})(λ

_{2}

^{2})(λ

_{3}

^{2}) = J

^{2}

#### 3.6. Model: Sensitivity Analysis

^{k}trials (k = 5) was conducted. The variables tested and test ranges can be seen in Table 4. The designed experiments tested the maximum and minimum combinations of all variables. The size correlated with the minimum and maximum geometries used, and the voltage values correlated to achievable ranges with the voltage supply unit maximum (5000 V), which was also under the dielectric breakdown of the fluoropolymer material. The temperature range was selected to be within the working range for the fluoropolymer material. The thickness values were based on minimum and maximum values measured from experimental samples. The dielectric constant was based on experimental measurements and optimal values for electroactive polymer materials.

## 4. Results

#### 4.1. Experimental Samples: Chemical Pre-Strain

#### 4.2. Experimental Samples: Displacement Results

^{2}, and sample 1 had an acceleration of 44.8 mm/s

^{2}. The reason for this variability is unknown, but may be attributed to various factors, such as variation in sample thickness and inconsistencies in the lab-scale manufacturing process.

#### 4.3. Model Results and Experimental Comparison

#### 4.4. Model: Sensitivity Analysis

^{−5}.

## 5. Discussion

^{2}. Sample actuation showed an immediate fast response time, followed by a slow material creep. Material actuation was also voltage, active area, and measurement location dependent. Material displacement was greatest farthest away from the inactive boundaries of the materials. This trend was seen in both experimental and modeled data. Variations in experimental results and discrepancies from model to experimental data were attributed to variation in material thickness and lacking uniformity due to the lab scale manufacturing process. This process will be improved in future work to provide a more uniform response. In conclusion, a novel material and process for the production of industrial grade electroactive polymers was demonstrated, and its effectiveness tested.

## 6. Patents

- Newell, B.A.; Krutz, G.W. Electroactive Actuators, Systems Equipped Therewith, and Methods of Use and Manufacture. U.S. Patent 9683663, 20 June 2017.
- Newell, B.A.; Krutz, G.W. Electroactive Polymers, Methods of Manufacture, and Structures Formed Thereof. U.S. Patent Application 15556696, 10 March 2016.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**(

**A**) Cross-Section of Modeled Actuator (

**B**) Applied Electric Potential at Electrode Surfaces (Sample 4000 V) (

**C**) 2-D Axisymmetric Revolution.

**Figure 4.**Chemical Pre-strain. Left o-ring is without treatment, right o-ring with treatment in solvent for 15 min.

**Figure 9.**Experimental and Model Displacement Data versus Voltage. (

**A**) Stencil 2 Voltage Response; (

**B**) Actuation versus Size.

Set Number | Inactive Region Outer Diameter | Inactive Region Inner Diameter | Active Region Outer Diameter | Active Region Inner Diameter | Active Area | Width of Ring |
---|---|---|---|---|---|---|

1 | 5.7 cm | NA ^{1} | 4.7 cm | NA ^{1} | 17.35 cm^{2} | NA ^{1} |

2 | 5.7 cm | 0.7 cm | 4.7 cm | 1.7 cm | 15.08 cm^{2} | 1.5 cm |

3 | 5.7 cm | 2.7 cm | 4.7 cm | 3.7 cm | 6.60 cm^{2} | 0.5 cm |

^{1}Sample 1 is a circular actuator not a ring structure.

Material Property | Value |
---|---|

Dielectric Constant | 10.7 |

Thickness | 0.2549 mm |

Poisson Ratio | 0.499 |

Bulk Modulus | 2.2986 MPa |

Shear Modulus | 2.29 kPa |

Young’s Modulus | 1.8 MPa |

Electrical Conductivity (Dielectric Layer) | 2.857 × 10^{−14} S/m |

Electrical Conductivity (Electrode Layer) | 10 × 10^{4} S/m |

Relative Permittivity | 2.5 |

Density | 1800 kg/m^{3} |

Thermal Conductivity | 0.2 W/(m·K) |

Heat Capacity at Constant Pressure | 700 J/(kg·K) |

Stencil | Average (mm) | Standard Deviation | Maximum (mm) | Minimum (mm) |
---|---|---|---|---|

1 | 0.234 | 0.054 | 0.305 | 0.152 |

2 | 0.284 | 0.030 | 0.330 | 0.254 |

3 | 0.255 | 0.050 | 0.330 | 0.152 |

Variables | Minimum | Maximum |
---|---|---|

Size (Active Area) | 6.60 cm^{2} | 17.35 cm^{2} |

Voltage | 1000 V | 5000 V |

Temperature | 293 K | 400 K |

Dielectric Constant | 10.7 | 21.4 |

Thickness | 0.329 mm | 0.654 mm |

Initial Dimensions (cm) | 5 min Chemical Strain Dimensions (cm) | 5 min after Removal Dimensions (cm) | % Swell | % Reversal |
---|---|---|---|---|

5.10 | 8.17 | 5.40 | 160.13 | 94.44 |

5.13 | 8.00 | 5.13 | 155.84 | 100.00 |

4.53 | 7.27 | 4.60 | 160.29 | 98.55 |

4.77 | 6.97 | 4.73 | 146.15 | 100.00 |

4.63 | 7.63 | 4.87 | 164.75 | 95.21 |

4.60 | 7.17 | 4.97 | 155.80 | 92.62 |

4.60 | 7.50 | 4.65 | 163.04 | 98.92 |

4.60 | 7.60 | 4.50 | 165.22 | 100.00 |

Set | Active Area | Experimental Displacement |
---|---|---|

1 | 17.35 cm^{2} | 0.9876 mm |

2 | 15.08 cm^{2} | 0.8475 mm |

3 | 6.60 cm^{2} | 0.1891 mm |

Specimen | Active Area | Model Displacement | Experimental Displacement |
---|---|---|---|

1 | 17.35 cm^{2} | 1.1653 mm | 0.9876 mm |

2 | 15.08 cm^{2} | 0.633 mm | 0.8475 mm |

3 | 6.60 cm^{2} | 0.1870 mm | 0.1891 mm |

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**MDPI and ACS Style**

Newell, B.; Garcia, J.; Krutz, G. Dielectric Electroactive Polymers with Chemical Pre-Strain: An Experimentally Validated Model. *Actuators* **2018**, *7*, 50.
https://doi.org/10.3390/act7030050

**AMA Style**

Newell B, Garcia J, Krutz G. Dielectric Electroactive Polymers with Chemical Pre-Strain: An Experimentally Validated Model. *Actuators*. 2018; 7(3):50.
https://doi.org/10.3390/act7030050

**Chicago/Turabian Style**

Newell, Brittany, Jose Garcia, and Gary Krutz. 2018. "Dielectric Electroactive Polymers with Chemical Pre-Strain: An Experimentally Validated Model" *Actuators* 7, no. 3: 50.
https://doi.org/10.3390/act7030050