# Thermal Conductivity and Microstructure of Novel Flaxseed-Gum-Filled Epoxy Resin Biocomposite: Analytical Models and X-ray Computed Tomography

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Description of Materials

^{2}sandpaper to obtain parallel surfaces and a height of 20 ± 0.5 mm. For this, the volume of the samples was measured with calipers (4 samples per test) and weighed to an accuracy of 0.01 g.

#### 2.2. Microstructural Characterization of Samples

^{3}. For each sample, a total of four REVs scans were randomly performed, with 8 µm in voxel size. It is important to note that during the image analysis, pores, and flax fibers in contact with the edges of the REV were excluded from the calculation.

#### 2.3. Analytical Approach for Estimating Thermal Conductivity

_{FG}

_{(s)}, with a given FG100/epoxy volume fraction ratio.

#### 2.3.1. Thermal Conductivity of the Fibrous Media

- with ${1-\mathsf{\nu}}_{Fb}=\mathsf{\phi}+{\mathsf{\nu}}_{FG\left(s\right)}$,
- where,
- λ
_{eff}: is the effective thermal conductivity of the fibrous composite. - λ
_{FG}: is the effective thermal conductivity of the porous flaxseed gum (FG) matrix. - λ
_{Fb}_{(s)}: is the local thermal conductivity of the flax fibers (s = solid). - ν
_{Fb}: is the volume fraction of the flax fibers. - φ: is the porosity.
- f
_{s}: is a morphological parameter (fs = 1 for fibers).

_{g}:

- with ${1-\mathsf{\nu}}_{Fb}=\mathsf{\phi}$.
- where,
- λ
_{g}: is the thermal conductivity of the air.

#### 2.3.2. Thermal Conductivity of the Porous Flaxseed Gum Matrix

_{FG}

_{(s)}, three analytical models were selected: Russell, Maxwell, and Bruggeman [33,34]. These models were chosen based on their capability for estimating thermal conductivity for different shapes of pores within the composite material.

- λ
_{FG}: is the effective thermal conductivity of the porous flaxseed gum matrix. - λ
_{FG}_{(s)}: is the local thermal conductivity of the non-porous flaxseed gum matrix. - φ: is the porosity of the flaxseed gum matrix.

#### 2.3.3. Local Thermal Conductivity of the Flaxseed Gum Matrix

_{FG}

_{(s)}, with a given volume fraction ratio of FG and epoxy resin, the analytical models require determination of the intrinsic thermal conductivity of FG and epoxy resin in the matrix. The thermal conductivity of the solid phase can be calculated using the mixing law of series (7) or parallel (8) mixing laws:

- X: is the volume fraction of the pure flaxseed gum (FG100) in the solid matrix.
- λ
_{FG}_{100(s)}: is the intrinsic thermal conductivity of the non-porous FG100. - λ
_{Epoxy}_{(s)}: is the intrinsic thermal conductivity of the non-porous epoxy resin.

## 3. Results and Discussion

#### 3.1. Microstructure Analysis

#### 3.2. Thermal Conductivity Estimation

#### 3.2.1. Thermal Conductivity of the Non-Porous Flaxseed-Gum-Filled Epoxy Matrix

^{−1}·K

^{−1}, respectively.

#### 3.2.2. Local Thermal Conductivity of Chopped Flax Fibers

_{Fb}

_{(s)}, the porosity, and the morphological parameter fs of the solid phase. Considering the experimental thermal conductivity (0.048 W·m

^{−1}·K

^{−1}) and the porosity (76% ± 1.0) obtained from X-ray CT images, the local thermal conductivity of the fibers, λ

_{Fb}

_{(s),}was determined using the following expression:

- λ
_{Fb}: is the effective thermal conductivity of the flax fibers (in air). - λ
_{g}: is the thermal conductivity of the air (λ_{g}= 0.026 W·m^{−1}·K^{−1}).

^{−1}·K

^{−1}), while in the transverse direction, it was 0.17 (W·m

^{−1}·K

^{−1}). These measurements diverge from our results because we used shorter and randomly oriented fibers, compared to their well-oriented fibers.

#### 3.2.3. Intrinsic Thermal Conductivity of the Pure Flaxseed Gum (FG100) and Epoxy Resin

_{Epoxy}

_{(s)}in each sample was determined using Equation (10) [32], where m

_{Epoxy}

_{(s)}represents the mass fraction of the epoxy and ρ

_{matrix}

_{(s)}/ρ

_{Epoxy}

_{(s)}is the ratio of the density of the matrix to that of the epoxy (Table 1).

_{FG(s)}= 1 − ν

_{Epoxy(s)}

_{FG100}and λ

_{Epoxy}(Figure 6).

_{FG100}) were around 0.056 and 0.057 (W·m

^{−1}·K

^{−1}) for both parallel and series models, respectively. Thus, an average value of 0.0565 W·m

^{−1}·K

^{−1}can be considered for FG100. The thermal conductivity of epoxy resin is around 0.28 and 0.288 (W·m

^{−1}·K

^{−1}) for parallel and series models, respectively. Thus, a mean value of 0.284 W·m

^{−1}·K

^{−1}can be considered for epoxy. It can be noted that in the literature, the thermal conductivity of epoxy varies between 0.18 and 0.26 (W·m

^{−1}·K

^{−1}) [39,40]. These values are of the same order of magnitude as that obtained and tend to show the validity of the approach.

#### 3.2.4. Local Thermal Conductivity of Flax Fibers Based on the Effective Conductivity of the Fibrous Composite

_{FG}

_{(s)}was around 0.180 W·m

^{−1}·K

^{−1}. Considering that the porosity of FFG is 65% (Table 2) and by applying Maxwell’s model, the estimated thermal conductivity of the porous matrix, λ

_{FG}, was 0.069 W·m

^{−1}·K

^{−1}.

^{−1}·K

^{−1}) and the experimental effective conductivity of the FFG (λ

_{eff}), the local conductivity of the fibers, λ

_{Fb}

_{(s)}, was deduced from Equation (13). The obtained value was approximately 0.123 (W·m

^{−1}·K

^{−1}).

_{Fb}

_{(s)}= 0.35 (W·m

^{−1}·K

^{−1}) obtained in Section 3.2.2. In [32,41], this overestimation of the thermal conductivity of fibers in air was explained by the potential contribution of radiation inside highly porous Juncus maritimus fibers. Accordingly, the obtained value (0.123 W·m

^{−1}·K

^{−1}) is close to the average thermal conductivity of treated flax fibers 0.1187 (W·m

^{−1}·K

^{−1}), obtained by [19].

#### 3.2.5. Estimating the Thermal Conductivity of the Porous Matrix for Different Volume Fractions of Flaxseed Gum

## 4. Conclusions

^{−1}·K

^{−1}) and pure epoxy resin (0.284 W·m

^{−1}·K

^{−1}) was estimated using both parallel and series mixing laws.

^{−1}·K

^{−1}. However, this value may be overestimated due to the radiation contribution. An alternative estimation was therefore proposed, based on the fibrous composite (FFG). As expected, the fiber’s thermal conductivity obtained was around 0.123 W·m

^{−1}·K

^{−1}. This value is close to the average thermal conductivity of the pure flaxseed gum, FG100 (0.145 W·m

^{−1}·K

^{−1}).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Pictures and SEM images of the cross-section morphology of the three composites: FG20, FG80, and FFG.

**Figure 2.**Example of FG20 sample segmentation using LABKIT plug-in. (

**A**) Original X-ray CT image, and (

**B**) binary image.

**Figure 5.**X-ray CT images of the segmented flax fibers from (

**A**) the fibrous composite (FFG) and (

**B**) the panel’s fibers (in air).

**Figure 6.**The linear trend curves obtained from (

**A**) parallel (Equation (11)) and (

**B**) series mixing laws (Equation (12)).

Sample | m_{FG}(%) | m_{Fibers}(%) | m_{Epoxy}(%) | Bulk Density (g/cm ^{3}) | Thermal Conductivity (W·m ^{−1}·K^{−1}) |
---|---|---|---|---|---|

FG100 | 100.0 | 0.0 | 0.0 | 228.9 ± 9.3 | 0.054 ± 0.001 |

FG80 | 80.0 | 0.0 | 20.0 | 0.231 ± 8.2 | 0.065 ± 0.001 |

FG20 | 20.0 | 0.0 | 80.0 | 0.219 ± 3.0 | 0.057 ± 0.001 |

FFG | 12.0 | 48.0 | 40.0 | 0.194 ± 5.3 | 0.064 ± 0.001 |

Epoxy resin (dense) | 0.0 | 0.0 | 100 | 1.1 | 0.782 ± 0.001 |

Chopped fibers (1 mm in length) | - | 100.00 | - | 0.108 | 0.048 ± 0.001 |

Sample | ν_{FG} (%) | ν_{fibers} (%) | Porosity (%) |
---|---|---|---|

FG100 | 32.0 | - | 68.0 ± 2.4 |

FG80 | 32.0 | - | 68.0 ± 2.5 |

FG20 | 39.0 | - | 61.0 ± 3.0 |

FFG | 19.0 | 16.0 | 65.0 ± 1.1 |

Chopped fibers (1 mm) | - | 24.0 | 76.0 ± 1.0 |

Sample | ν_{FG100} (%) | ν_{Epoxy} (%) |
---|---|---|

FG100 | 100.00 | 0.00 |

FG80 | 80.30 | 19.70 |

FG20 | 59.20 | 40.80 |

FFG | 35.00 | 65.00 |

Epoxy resin | 0.00 | 100.00 |

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**MDPI and ACS Style**

Zaidi, M.; Baillis, D.; Naouar, N.; Depriester, M.; Delattre, F.
Thermal Conductivity and Microstructure of Novel Flaxseed-Gum-Filled Epoxy Resin Biocomposite: Analytical Models and X-ray Computed Tomography. *Materials* **2023**, *16*, 6318.
https://doi.org/10.3390/ma16186318

**AMA Style**

Zaidi M, Baillis D, Naouar N, Depriester M, Delattre F.
Thermal Conductivity and Microstructure of Novel Flaxseed-Gum-Filled Epoxy Resin Biocomposite: Analytical Models and X-ray Computed Tomography. *Materials*. 2023; 16(18):6318.
https://doi.org/10.3390/ma16186318

**Chicago/Turabian Style**

Zaidi, Mohammed, Dominique Baillis, Naim Naouar, Michael Depriester, and François Delattre.
2023. "Thermal Conductivity and Microstructure of Novel Flaxseed-Gum-Filled Epoxy Resin Biocomposite: Analytical Models and X-ray Computed Tomography" *Materials* 16, no. 18: 6318.
https://doi.org/10.3390/ma16186318