# Multiphysics Modeling Framework for Soft PVC Gel Sensors with Experimental Comparisons

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

## 3. Results

#### 3.1. Solid Mechanics Study

#### 3.2. Transport and Migratory Effects

#### 3.3. Model Results with Experimental Validation

## 4. Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Dong, T.-Y.; Zhang, X.-L.; Liu, T. Artificial muscles for wearable assistance and rehabilitation. Front. Inf. Technol. Electron. Eng.
**2018**, 19, 1303–1315. [Google Scholar] [CrossRef] - Maeda, Y.; Hashimoto, M. Lightweight, Soft Variable Stiffness Gel Spats for Walking Assistance. Int. J. Adv. Robot. Syst.
**2015**, 12, 175. [Google Scholar] [CrossRef] - Washington, J.; Kim Neubauer, K.J. Chapter 3—Soft Actuators and Their Potential Applications in Rehabilitative Devices. In Soft Robotics in Rehabilitation; Jafari, A., Ebrahimi, N., Eds.; Academic Press: Cambridge, MA, USA, 2021; pp. 89–110. ISBN 9780128185384. [Google Scholar] [CrossRef]
- Xia, H.; Hirai, T. Electric-Field-Induced Local Layer Structure in Plasticized PVC Actuator. J. Phys. Chem. B
**2010**, 114, 10756–10762. [Google Scholar] [CrossRef] [PubMed] - Asaka, K.; Hashimoto, M. Electrical properties and electromechanical modeling of plasticized PVC gel actuators. Sens. Actuators B Chem.
**2018**, 273, 1246–1256. [Google Scholar] [CrossRef] - Hwang, T.; Frank, Z.; Neubauer, J.; Kim, K.J. High-performance polyvinyl chloride gel artificial muscle actuator with graphene oxide and plasticizer. Sci. Rep.
**2019**, 9, 9658. [Google Scholar] [CrossRef] [PubMed] - Frank, Z.; Olsen, Z.; Hwang, T.; Kim, K.J. Modelling and experimental study for PVC gel actuators. In Proceedings of the ASME 2019 Dynamic Systems and Control Conference, DSCC 2019, Park City, UT, USA, 8–11 October 2019; Volume 3. [Google Scholar] [CrossRef]
- Neubauer, J.; Olsen, Z.J.; Frank, Z.; Hwang, T.; Kim, K.J. A study of mechanoelectrical transduction behavior in polyvinyl chloride (PVC) gel as smart sensors. Smart Mater. Struct.
**2022**, 31, 015010. [Google Scholar] [CrossRef] - Neubauer, J.; Kim, K.J. Tunable polyvinyl chloride (PVC) and thermoplastic polyurethane (TPU)-based soft polymer gel sensors. Smart Mater. Struct.
**2022**, 31, 115025. [Google Scholar] [CrossRef] - Sharif, M.A. PVC, gel smart sensor for robotics sensing applications: An experimental and finite element simulation study. Eng. Res. Express
**2022**, 4, 035029. [Google Scholar] [CrossRef] - Ju, M.; Mezghani, S.; Jmal, H.; Dupuis, R.; Aubry, E. Parameter Estimation of a Hyperelastic Constitutive Model for the Description of Polyurethane Foam in Large Deformation. Cell. Polym.
**2013**, 32, 21–40. [Google Scholar] [CrossRef] - Schrodt, M.; Benderoth, G.; Kühhorn, A.; Silber, G. Hyperelastic description of polymer soft foams at finite deformations. Tech. Mech.-Eur. J. Eng. Mech.
**2005**, 25, 162–173. [Google Scholar] - Liu, C.-H.; Chen, Y.; Yang, S.-Y. Quantification of hyperelastic material parameters for a 3D-Printed thermoplastic elastomer with different infill percentages. Mater. Today Commun.
**2020**, 26, 101895. [Google Scholar] [CrossRef] - Bien-Aimé, L.K.M.; Blaise, B.B.; Beda, T. Characterization of hyperelastic deformation behavior of rubber-like materials. SN Appl. Sci.
**2020**, 2, 648. [Google Scholar] [CrossRef] - Hill, R. Aspects of Invariance in Solid Mechanics. Adv. Appl. Mech.
**1979**, 18, 1–75. [Google Scholar] [CrossRef] - Lin, C.-Y. Alternative Form of Standard Linear Solid Model for Characterizing Stress Relaxation and Creep: Including a Novel Parameter for Quantifying the Ratio of Fluids to Solids of a Viscoelastic Solid. Front. Mater.
**2020**, 7, 11. [Google Scholar] [CrossRef]

**Figure 1.**(

**A**) Generalized Maxwell material model (

**left**) with SLS or Zener model represented in Maxwell (

**center**) and (

**B**) Kelvin–Voigt (

**right**) forms (Lin) [16].

**Figure 2.**Illustration of interfacial polar molecule behavior and preferred orientation (

**left**) and substrate effects such as hydrogen bonding on polar molecules (

**right**).

**Figure 3.**Fully cured PVC DBA P4 gel tray with punched sample to limit irregularities such as edge effects during the curing process.

**Figure 4.**A detailed view of the force application mechanism with associated equipment for mechanoelectrical testing (

**right**) including magnified region of soft polymeric sensing region (

**left**).

**Figure 5.**Dynamic mechanical analyzer (

**left**) with magnified testing region of PVC gel sample (

**right**) to determine stress–strain relationship under compressive loading applications.

**Figure 6.**Flow chart of multiphysics model including the three major physics modules included in this study with plasticizer concentration gradient and electric potential response (blue).

**Figure 7.**COMSOL model initial mesh including increased density near the surface due to modeled adsorptive surface phenomena.

**Figure 8.**DMA results for P4, P5, and P6 samples with one-, two-, and three-term Storakers hyperelastic material model fits (

**top**), with residuals plot (

**bottom**) showing good agreement among experimental results for this compressive hyperelastic material model.

**Figure 9.**Viscoelastic stress relaxation observed in PVC gel samples when undergoing stepped compressive strain input (black) and SLS model fit (blue) for low (

**left**) and high (

**right**) compressive loading applications.

**Figure 10.**Variation of steady-state response parameter ${\mathrm{K}}_{\mathrm{A}}$ (

**left**) in adsorption-based mathematical model and transient response parameters ${\mathrm{k}}_{\mathrm{dyn}}$ and $\mathsf{\tau}$ (

**right**) displaying effects on fractional occupancy of adsorption sites $\mathsf{\Theta}$.

**Figure 11.**Steady-state relative concentration plot of compressed PVC gel sample with 0.1 compressive strain displaying higher plasticizer concentrations in a layer approximately 100 $\mathsf{\mu}$m near the interfacial layer.

**Figure 12.**Obtained experimental data and mathematical model results for transient (

**top**) steady-state (

**bottom**) response characteristics showing ability of the model to reflect experimental results of soft polymer gel sensors.

Variable | Units | Description |
---|---|---|

$\mathsf{\u03f5}$ | $\left[-\right]$ | Strain on polymer gel sample |

$\mathsf{\rho}$ | $\left[{\mathrm{kg}/\mathrm{m}}^{3}\right]$ | Density |

$\mathsf{\Theta}$ | $\left[-\right]$ | Fractional occupancy of adsorptive sites |

${\mathrm{V}}_{\mathrm{resp}}$ | $\left[\mathrm{V}\right]$ | Voltage response |

$\mathrm{c}$ | $\left[{\mathrm{mol}/\mathrm{m}}^{3}\right]$ | Concentration of bulk species |

${\mathrm{K}}_{\mathrm{S}}$ | $\left[-\right]$ | Equilibrium constant of surface layer |

${\mathrm{S}}_{\mathrm{E}}$ | $\left[\mathrm{V}\right]$ | Proportionality constant |

$\mathsf{\tau}$ | $\left[\mathrm{s}\right]$ | Time constant |

${\mathrm{c}}_{\mathrm{sat}}$ | $\left[{\mathrm{mol}/\mathrm{m}}^{2}\right]$ | Saturated surface concentration |

${\mathrm{k}}_{\mathrm{ads}}$ | $\left[{\mathrm{m}}^{3}/\mathrm{mol}\cdot \mathrm{s}\right]$ | Adsorption rate constant |

${\mathrm{k}}_{\mathrm{des}}$ | $\left[1/\mathrm{s}\right]$ | Desorption rate constant |

${\mathrm{N}}_{\mathrm{ads}}$ | $\left[{\mathrm{mol}/\mathrm{m}}^{2}\cdot \mathrm{s}\right]$ | Rate of adsorption |

${\mathrm{N}}_{\mathrm{des}}$ | $\left[{\mathrm{mol}/\mathrm{m}}^{2}\cdot \mathrm{s}\right]$ | Rate of desorption |

${\mathsf{\lambda}}_{1,2,3}$ | $[-]$ | Principal stretches |

${\mathsf{\mu}}_{\mathrm{n}}$ | $\left[{\mathrm{N}/\mathrm{m}}^{2}\right]$ | Shear modulus |

${\mathsf{\alpha}}_{\mathrm{n}}$ | $\left[-\right]$ | Storakers model parameter |

${\mathsf{\beta}}_{\mathrm{n}}$ | $\left[-\right]$ | Storakers model parameter |

$\mathrm{J}$ | $\left[-\right]$ | Volumetric parameter |

$\mathsf{\sigma}$ | $\left[{\mathrm{N}/\mathrm{m}}^{2}\right]$ | Stress |

${\mathrm{k}}_{\mathrm{n}}$ | $\left[{\mathrm{N}/\mathrm{m}}^{2}\right]$ | Bulk modulus |

$\mathrm{W}$ | $\left[{\mathrm{J}/\mathrm{m}}^{3}\right]$ | Strain energy density function |

${\mathrm{J}}_{\mathrm{ads}}$ | $\left[{\mathrm{mol}/\mathrm{m}}^{2}\cdot \mathrm{s}\right]$ | Modeled adsorption rate |

${\mathrm{J}}_{\mathrm{des}}$ | $\left[{\mathrm{mol}/\mathrm{m}}^{2}\cdot \mathrm{s}\right]$ | Modeled desorption rate |

${\mathrm{J}}_{\mathrm{dyn}}$ | $\left[-\right]$ | Modeled dynamic interfacial effect |

${\mathrm{k}}_{\mathrm{dyn}}$ | $\left[-\right]$ | Dynamic interfacial rate constant |

$\mathrm{E}$ | $\left[{\mathrm{N}/\mathrm{m}}^{2}\right]$ | Compressive modulus |

$\mathrm{t}$ | $\left[\mathrm{s}\right]$ | Time |

${\mathrm{D}}_{\mathrm{j}}$ | ${[\mathrm{m}}^{2}/\mathrm{s}]$ | Diffusivity of species j |

${\mathrm{c}}_{\mathrm{j}}$ | ${[\mathrm{mol}/\mathrm{m}}^{3}]$ | Concentration of species j |

${\mathrm{z}}_{\mathrm{j}}$ | $[-]$ | Charge number of species j |

${\mathrm{u}}_{\mathrm{m},\mathrm{j}}$ | ${[\mathrm{m}}^{2}/\mathrm{s}\cdot \mathrm{V}]$ | Mobility of species j in medium m |

$\mathrm{F}$ | $[\mathrm{C}/\mathrm{mol}]$ | Faraday constant |

$\mathrm{V}$ | $\left[\mathrm{V}\right]$ | Electric potential |

Name | P4 | P6 | P8 |
---|---|---|---|

PVC: plasticizer ratio | 1:4 | 1:6 | 1:8 |

Model Fit | P4 Error | P6 Error | P8 Error |
---|---|---|---|

One-term | 3.04% (0.497 kPa) | 6.96% (4.094 kPa) | 7.58% (4.198 kPa) |

Two-term | 3.07% (0.505 kPa) | 1.83% (0.646 kPa) | 3.02% (0.493 kPa) |

Three-term | 3.02% (0.499 kPa) | 1.71% (0.430 kPa) | 3.02% (0.513 kPa) |

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**MDPI and ACS Style**

Neubauer, J.; Kim, K.J.
Multiphysics Modeling Framework for Soft PVC Gel Sensors with Experimental Comparisons. *Polymers* **2023**, *15*, 864.
https://doi.org/10.3390/polym15040864

**AMA Style**

Neubauer J, Kim KJ.
Multiphysics Modeling Framework for Soft PVC Gel Sensors with Experimental Comparisons. *Polymers*. 2023; 15(4):864.
https://doi.org/10.3390/polym15040864

**Chicago/Turabian Style**

Neubauer, Justin, and Kwang J. Kim.
2023. "Multiphysics Modeling Framework for Soft PVC Gel Sensors with Experimental Comparisons" *Polymers* 15, no. 4: 864.
https://doi.org/10.3390/polym15040864