# Microscale Thermal Modelling of Multifunctional Composite Materials Made from Polymer Electrolyte Coated Carbon Fibres Including Homogenization and Model Reduction Strategies

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## Abstract

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## 1. Introduction

## 2. Modelling Methods and Material Properties

#### 2.1. Representative Volume Element of PeCCF Polymer Composite

#### 2.2. Heat Transfer Problem and Virtual Material Testing

_{i}. The symbol $\theta ({\mathbf{x}}_{\mathrm{i}},t)$ is the temperature in a spatial point of domain Ω

_{i}denoted by the vector ${\mathrm{x}}_{\mathrm{i}}$ and at time t. For the discrete description the heat conductivity is assumed to be isotropic for all domains and ideal contact at domain interfaces is assumed. For $\frac{\partial}{\partial t}\theta ({\mathbf{x}}_{\mathrm{i}},t)=0$ the stationary heat tranfser equation results:

- Case 1: Virtual thermal conductivity measurement

- Case 2: Joule heating and heat transfer

#### 2.3. Assumptions and Material Properties

#### 2.4. Homogenization Methods

#### 2.4.1. Spatial Averaging

#### 2.4.2. Series and Parallel Connection

#### 2.4.3. Three-Phase Lewis-Nielsen Model

#### 2.4.4. New Two-Level Lewis-Nielsen Approach

#### 2.5. Principles for Model Reduction

## 3. Results and Discussion

#### 3.1. Stationary Heat Flux Solution and Accuracy of Homogenization

#### 3.2. Transient Heat Flux Solution and Effects of Homogenization

#### 3.3. Accuracy of 2D Reduced Model

## 4. Conclusions

- Based on the discrete RVE representation of the PeCCF composite on the microscale a characteristic temperature gradient in the transversal plane of ${\Delta}\theta =1.0\phantom{\rule{4.pt}{0ex}}\frac{\mathrm{K}}{\mathrm{mm}}$ was indicated for heat convection at the surface. It should be noted, that this number is a linear average. The discrete temperature gradient could be larger. This property is important, as such a thermal gradient in the thickness direction of a typical laminate could lead to additional thermal stresses within the laminate plies. Such temperature gradients have been neglected in earlier studies on PeCCF composite materials. With this new recognition it is expected that the temperature distribution within the material plays an important role for the mechanical performance.
- The influence of the solid polymer electrolyte coating on the transversal heat flux was identified to be small, since the thermal properties are in the range of the matrix material and the volume fraction of this domain is small. Especially in the transversal plane the effective length of the coating (the thickness) is small, which reduces the temperature drop in its domain. However, since the coating represents an additional interphase, the fibre volume fraction is geometrically limited by the presence of the coating layer. This leads to smaller fibre volume fractions, which induces a smaller thermal conductivity in the transversal plane compared to classical carbon fibre—matrix composites. Accordingly, a high fibre volume ratio in PeCCF composites is needed for improved thermal conductivity.
- The homogenization of the thermal conductivity $\lambda $ in the transversal plane was found to be best by the Three-Phase Lewis-Nielsen method and the new Two-Level Lewis-Nielsen approach. The resulting stationary temperature distributions in the transversal plane are compared to those resulting from virtually measured thermal conductivities. Both methods represent narrow lower and upper bounds respectively. The new Two-Level Lewis-Nielsen approach enabled tailoring the effect of the coating by including its geometric appearance on particle level (PeCCF level). Based on the effective thermal conductivity, a suitable and efficient prediction of temperature distributions and temperature gradients are enabled. This new method directly addresses the special geometric composition of the PeCCF compound and is therefore very accurate to predicit the thermal conductivity in the transversal plane.
- The homogenization of specific heat c and density $\rho $ were identified best by the volume average, based on the RVE model, and the parallel ROM method, which are similar approaches. Both quantities showed minor influence on the transient heat up process. The discrete temperature difference at a certain time step during heat up was small compared to the resulting stationary temperature. Accordingly, the application of effective properties in a homogenized material representation has negligible influence on the prediction of the transient heat flux problem.
- With respect to the macroscale (laminate of PeCCF composite), the thickness direction governs the heat flux, since heat convection only appears at the surfaces of the laminate. While joule heating is effective only in longitudinal direction, the width direction is uneffected from both physical effects. In a newly proposed reduced 2D approach for the microscale, a simplified and accurate prediction of the thermal field in transversal direction was demonstrated. The homogenized and reduced 2D model in the ${y}_{1},{y}_{3}$ plane showed a more conservative representation of the temperature gradient of ${\Delta}\theta =1.5\phantom{\rule{4.pt}{0ex}}\frac{\mathrm{K}}{\mathrm{mm}}$. Compared to the discrete 3D model, this difference is reduced to the fact, that in the 3D representation a discrete heat source distribution is given by the fibre distribution. This is vanished in the reduced model and replaced by a continous heat source distribution. This allows no heat exchange in width direction which effects the resulting temperature field. However, in terms of computational cost, the 2D model was up to 97% more efficient compared to the discrete model, which is a huge benefit of this new approach. This is especially important for future coupled multiscale models. Accordingly, the more conservative temperature distribution can be accepted in order to provide an efficient modelling approach for the microscale thermal heat flux problem.
- With respect to applications, the representation of the PeCCF composite material with homogenized material constants in the thermal problem will be of importance. Applications such as structural batteries, thermal management structures and composites with adaptive stiffness will be subjected to significant joule heating during electrical current conduction. The efficient predicition of the thermal field is crucial for the multiphysically coupled modelling of such multifunctional materials. The described approaches and results pave the way for the coupled representation of the thermo-mechanical behaviour of the PeCCF composite.

## 5. Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PeCCF | Polymer electrolyte coated carbon fibre |

SPE | Structural polymer electrolyte |

RVE | Representative volume element |

BC | Boundry condition |

FEM | Finite element method |

DMF | dimethylformamide |

ROM | Rule of mixtures |

## Appendix A. RVE Generation Algorithm

**Figure A2.**Geometry generation example: (

**a**) The resulting coordinate pair distribution and (

**b**) Corresponding 3D RVE.

## Appendix B. Technical Data on FEM Discretization

**Figure A3.**Mesh geometries for FEM calculations: (

**a**) Refined mesh M1, (

**b**) Coarse mesh M2, (

**c**) Homogenized RVE mesh M3 and (

**d**) 2D mesh M4; The coordinate origin is placed at the hidden corner.

**Table A1.**Technical details of FE meshes M1-M4 compare Figure A3.

Properties | Mesh M1 | Mesh M2 | Mesh M3 | Mesh M4 |
---|---|---|---|---|

Element types | Prisma | Triangle | ||

Shape functions | Quadratic serendipity * | |||

Reference fibre volume ratio ${v}_{\mathrm{c}\mathrm{f}}$ | 0.55 | - | - | |

Nr. of elements domain 1 | 15,371 | 3090 | - | - |

Nr. of elements domain 2 | 28,196 | 2670 | - | - |

Nr. of elements domain 3 | 19,057 | 4490 | - | - |

Total nr. of elements | 62,624 | 10,250 | 2560 | 578 |

Total degrees of freedom (DOF) | 168,206 | 29,216 | 7573 | 1217 |

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**Figure 1.**Assumptions for composite material domains, domain interfaces and discussed directions referred to the macroscale (laminate).

**Figure 2.**RVE with fibre volume ratio ${v}_{\mathrm{c}\mathrm{f}}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0.5$, boundary face indicators (${\mathrm{Y}}_{\mathrm{i}}$), and global coordinate system ${y}_{\mathrm{i}}$ with i = 1, 2, 3.

**Figure 3.**Schematic of the Two-level Lewis-Nielsen approach, applying the method first on particle level (level 1) and second on particle filled matrix level (level 2), Equation indices are given with respect to the material phase or mixture.

**Figure 4.**Temperature distribution corresponding to BC case 2; (

**a**) Isotherms in RVE, (

**b**) Temperature distributions along ${y}_{3}$ axis ($({y}_{1},{y}_{2})=(10\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m}$, $10\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m})$), (

**c**) Temperature difference related to respective maximum Temperature ($({y}_{1},{y}_{2})=(10\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m}$, $10\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m})$).

**Figure 5.**Transient evolution of temperature difference in thickness direction ${y}_{3}$ over time based on the discrete RVE model.

**Figure 6.**Transient heat up process of the RVE with volume averaged temperature ${T}_{\mathrm{av}}$; (

**a**) Variation of effective specific heat $\langle c\rangle $, (

**b**) Temperature distribution along ${y}_{3}$ axis ($({y}_{1},{y}_{2})=(10\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m}$, $10\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m})$, $t=10\phantom{\rule{4.pt}{0ex}}\mathrm{s}$), (

**c**) Variation of effective density $\langle \rho \rangle $, (

**d**) Temperature distribution along ${y}_{3}$ axis ($({y}_{1},{y}_{2})=(10\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m}$, $10\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m})$, $t=10\phantom{\rule{4.pt}{0ex}}\mathrm{s}$).

**Figure 7.**Comparison between 3D discrete and 2D homogenized and reduced model; (

**a**) Transient heat up process, (

**b**) Stationary Temperature distribution along ${y}_{3}$ axis (${y}_{1}=10\phantom{\rule{4.pt}{0ex}}\mathsf{\mu}\mathrm{m}$).

Property | Symbol | Value | Reference |
---|---|---|---|

RVE edge length | a | 20 μm | chosen |

Diameter carbon fibres * | ${d}_{\mathrm{f}}$ | 5.0 μm | [28,29] |

SPE coating thickness | ${t}_{\mathrm{c}}$ | 0.5 μm | [28] |

Domain | Sym. | Value | Unit | Explanation | Reference |
---|---|---|---|---|---|

${{\Omega}}_{1}$: Carbon Fibre IMS65 | ${\rho}_{1}$ | 1.78 | $\frac{\mathrm{g}}{{\mathrm{cm}}^{3}}$ | Density | [29] |

${\lambda}_{1}$ | 50 | $\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}$ | Thermal conductivity | assumed based on [21] | |

${c}_{1}$ | 1400 | $\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ | Specific heat capacity | assumed based on [31] | |

${\kappa}_{1}$ | $1.45\xb7{10}^{-3}$ | ${\Omega}\xb7\mathrm{cm}$ | Specific electrical resistance | [9] | |

${{\Omega}}_{2}$: Coating * | ${\rho}_{2}$ | 1.3 | $\frac{\mathrm{g}}{{\mathrm{cm}}^{3}}$ | Density | [10] |

${\lambda}_{2}$ | 0.21 | $\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}$ | Thermal conductivity | [32] | |

${c}_{2}$ | 2090 | $\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ | Specific heat capacity | [32] | |

${\kappa}_{2}$ | $\to \infty $ | ${\Omega}\xb7\mathrm{cm}$ | Specific electrical resistance | ** assumed | |

${{\Omega}}_{3}$: Matrix | ${\rho}_{3}$ | 1.3 | $\frac{\mathrm{g}}{{\mathrm{cm}}^{3}}$ | Density | [21] |

${\lambda}_{3}$ | 0.19 | $\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}$ | Thermal conductivity | [32] | |

${c}_{3}$ | 1050 | $\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ | Specific heat capacity | [32] | |

${\kappa}_{3}$ | $\to \infty $ | ${\Omega}\xb7\mathrm{cm}$ | Specific electrical resistance | ** assumed |

Volume Ratios | Symbol | Value | Reference Volume |
---|---|---|---|

Fibre volume ratio | ${v}_{\mathrm{f}}$ | 0.34 | RVE |

Coating volume ratio | ${v}_{\mathrm{c}}$ | 0.16 | RVE |

Matrix volume ratio | ${v}_{\mathrm{m}}$ | 0.50 | RVE |

Coated fibre volume ratio | ${v}_{\mathrm{c}\mathrm{f}}$ | 0.49 | RVE |

Particle fibre volume ratio | ${v}_{\mathrm{p}\mathrm{f}}$ | 0.69 | PeCCF * |

Particle coating volume ratio | ${v}_{\mathrm{p}\mathrm{c}}$ | 0.31 | PeCCF * |

Method | $k=\langle \lambda \rangle $ in $\left[\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}\right]$ | Input Parameters | Subsection | Comment |
---|---|---|---|---|

Volume average | 17.30 | - | Section 2.4.1 | based on RVE |

Series (ROM) | 0.29 | Table 3 | Section 2.4.2 | |

Parallel (ROM) | 17.30 | Table 3 | Section 2.4.2 | |

Three-Phase | 0.40 | $A=6.6$ | Section 2.4.3 | Maximum square |

Lewis-Nielsen | ${v}_{\mathrm{max}}=0.79$ | package assumed | ||

Two-Level Lewis-Nielsen | 0.43 | $A=6.6$ | Section 2.4.4 | Maximum square package assumed |

${v}_{\mathrm{max}}=0.79$ | ||||

${A}_{\mathrm{p}\mathrm{f}}=0.1$ | ||||

${v}_{\mathrm{p}\mathrm{f},\mathrm{max}}=1$ | ||||

Virtually measured | $\langle \mathbf{\lambda}\rangle ={[{\lambda}_{1},{\lambda}_{2},{\lambda}_{3}]}^{\mathrm{T}}={[17.30,0.41,0.42]}^{\mathrm{T}}$$\left[\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}\right]$; BC Case 1; Section 2.2 |

Method | $k=\langle c\rangle $ in $\left[\frac{\mathbf{J}}{\mathrm{kg}\xb7K}\right]$ | $k=\langle \rho \rangle $ in $\left[\frac{\mathrm{g}}{{\mathrm{cm}}^{3}}\right]$ | Input Para. | Subs. | Comment |
---|---|---|---|---|---|

Volume average | 1327.50 | 1.46 | - | Section 2.4.1 | based on RVE |

Series (ROM) | 1251.70 | 1.43 | Table 3 | Section 2.4.2 | |

Parallel (ROM) | 1327.50 | 1.46 | Table 3 | Section 2.4.2 |

Computation Time in [s] | ||
---|---|---|

Model | Transient Problem | Stationary Problem |

3D discrete | 450 | 137 |

3D homogenized | 17 | 6 |

2D homogenized | 14 | 5 |

Relative reduction of computational cost | ≈97% | ≈96% |

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**MDPI and ACS Style**

Schutzeichel, M.O.H.; Kletschkowski, T.; Monner, H.P. Microscale Thermal Modelling of Multifunctional Composite Materials Made from Polymer Electrolyte Coated Carbon Fibres Including Homogenization and Model Reduction Strategies. *Appl. Mech.* **2021**, *2*, 739-765.
https://doi.org/10.3390/applmech2040043

**AMA Style**

Schutzeichel MOH, Kletschkowski T, Monner HP. Microscale Thermal Modelling of Multifunctional Composite Materials Made from Polymer Electrolyte Coated Carbon Fibres Including Homogenization and Model Reduction Strategies. *Applied Mechanics*. 2021; 2(4):739-765.
https://doi.org/10.3390/applmech2040043

**Chicago/Turabian Style**

Schutzeichel, Maximilian Otto Heinrich, Thomas Kletschkowski, and Hans Peter Monner. 2021. "Microscale Thermal Modelling of Multifunctional Composite Materials Made from Polymer Electrolyte Coated Carbon Fibres Including Homogenization and Model Reduction Strategies" *Applied Mechanics* 2, no. 4: 739-765.
https://doi.org/10.3390/applmech2040043