# 3D Modelling of Mass Transfer into Bio-Composite

^{*}

## Abstract

**:**

## 1. Introduction

_{E}dispersed in the bulk matrix. P

_{E}is identified using a standard analytical formula for binary medium. Same standard analytical formula is then applied to the virtual binary composite containing the pseudo particles. To eliminate limitations imposed by the sphere equations that physically limits the maximal volume fraction investigated (random packing of congruent spheres imposed a non-negligible lattice volume), change in particle shape from spherical to cubic was proposed and found not significant, at least in the polymer-gas permeability area [16]. However, this approach is restricted to regular dispersion of homogenous particle-size and could not take into account heterogeneity in interphase thickness, particles size and distribution into the matrix.

## 2. Materials and Methods

#### 2.1. Experimental Parameters for Water Vapor Transfer

- Effective moisture diffusivity $D\left[{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}\right]$ value of each phase matrix and particle.
- The boundary concentrations of water vapor $\left[\mathrm{mol}\xb7{\mathrm{m}}^{-3}\right]$ in the PHBV matrix in contact with dry air and with humid air (relative humidity of 95%). These concentrations were determined using the experimental water vapor sorption isotherm of PHBV film at 20 °C.
- Water vapor partition coefficient $K=54.52$, calculated as the slope of the linear relation between the water concentration in PHBV matrix and WSF particles, obtained from experimental water vapor sorption isotherm at 20 °C for matrix and particles.

Sample | Diffusivity ^{a} $\times {10}^{-12}$$\left[{m}^{2}\xb7{s}^{-1}\right]$ | Permeability
$\times {10}^{-13}$ $\left[\mathrm{mol}\xb7m\xb7{m}^{-2}\xb7{s}^{-1}\xb7P{a}^{-1}\right]$ | Upper Boundary Concentration ^{a}$\left[\mathrm{mol}\xb7{m}^{-3}\right]$ | Lower Boundary Concentration ^{a}$\left[\mathrm{mol}\xb7{m}^{-3}\right]$ | Partition Coefficient ^{a} |
---|---|---|---|---|---|

PHBV matrix | $2.615\pm 0.56$ | $8.29\pm 3.96$^{b} | $337.14$ | $0$ | $54.52$ |

WSF particle | $18.39\pm 4.93$ | $1664\pm 451$^{c} | - | - |

^{a}Obtained from dynamic sorption experiments (DVS, Dynamical Vapor Sorption system, Surface Measurement System, London, UK) [24];

^{b}Correspond to the average of three experimental sets of measures made in the same laboratory and directly measured from gravimetric experiment (Modified ASTM procedure) [24,25,26].

^{c}Calculated as the product of experimental particle diffusivity by experimental particle solubility obtained from dynamic sorption experiments (DVS, Dynamical Vapor Sorption system, Surface Measurement System, London, UK) [24].

#### 2.2. 2D Image Analysis

#### 2.3. 3D Structure Generation

#### 2.4. Mathematical Modelling and Geometry

#### 2.4.1. 3D Structure Generation

- Step 1. The position (center coordinates ${x}_{p},{y}_{p},{z}_{p}$) and orientation (azimuth ${\theta}_{p1}$ and elevation ${\theta}_{p2}$ angle) are randomly drawn using uniform distributions.
- Step 2. The non-overlapping of the particle with the horizontal faces of the RVE ($z=0$ and $z={L}_{z}$) and with the existing particles is tested.

#### 2.4.2. Governing Equations

#### 2.4.3. Boundary Conditions

#### 2.4.4. Effective Permeability Evaluation

#### 2.5. Numerical Simulations

## 3. Results and Discussion

#### 3.1. From Particle Morphology to 3D Structure Generation

#### 3.2. Simulations of Two-Phase Model

#### 3.2.1. Selection of Mesh and RVE Sizes

^{3}) were generated using experimentally observed particle size distribution: 185 structures for ${\phi}_{p}$ = 5.14 %v/v, 199 structures for ${\phi}_{p}$ = 11.4 %v/v and 158 structures for ${\phi}_{p}$ = 19.52 %v/v.

#### 3.2.2. Selection of the Number of Structures to Analyze

#### 3.2.3. Numerical Results of the 2-Phase Model

- the physical properties of the fiber particle and especially its diffusivity value would be modified once embedded into the polymer matrix compare to the one measure on the native component,
- the diffusivity of the polymer matrix, measured before fiber particles addition, would be modified after fiber particles addition and therefore not well representative of what occurs in the composite material,
- the presence of an interphase, third compartment with its own physical properties, at the interface matrix/particle would influence the overall permeability into the composite.

#### 3.2.4. Modification of Particles Diffusivity Values in the Two-Phase Model

#### 3.2.5. Modification of Matrix Diffusivity Values in the Two-Phase Model

^{−1}), which is also in favor of higher polymer mobility (less entanglements) and thus increased of ${D}_{m}$ value in composite. A modification of the crystal size was also highlighted by the authors: the size of crystals was increased from 1.03 to 1.18 nm in the presence of WSF. This higher crystal size is also in favor of an increase of diffusivity in the continuous phase because of lower tortuosity in the less tight crystal structure.

#### 3.3. Simulations of Three-Phase Model

## 4. Conclusions

## Supplementary Materials

_{m}with calculated ones by using Maxwell-Wagner-Sillar equation. Figure S3. Evolution of the numerical relative permeability of water vapor as a function of the mesh element size for the composite of φ

_{p}= 5.14 %$v/v$. Mesh tests are performed with 10 structures. Figure S4. Evolution of the numerical relative permeability of water vapor as a function of the mesh element size for the composite of φ

_{p}= 11.4 %$v/v$. Mesh tests are performed with 10 structures. Figure S5. Evolution of the numerical relative permeability of water vapor as a function of the mesh element size for the composite of φ

_{p}= 19.52 %$v/v$. Mesh tests are performed with 8 structures. Numerical results corresponding to some mesh element sizes are not available due to mesh errors encountered on some simulations. Figure S6. Evolution of the numerical relative permeability of water vapor as a function of the diffusivity and thickness of the interphase for the composite of φ

_{p}= 5.14 %$v/v$: Comparison between experimental and numerical results. The numerical results (bullets) correspond to the average of the relative permeability of 20 structures (e

_{i}= 1 µm), 23 structures (e

_{i}= 2.5 µm) and 26 structures (e

_{i}= 5 µm). The volume fraction of the interphase was φ

_{i}= 1.85 ± 0.34 %$v/v$ for e

_{i}= 1 µm, φ

_{i}= 5.71 ± 1.24 %$v/v$ for e

_{i}= 2.5 µm and φ

_{i}= 15.19 ± 3.36 %$v/v$ for e

_{i}= 5 µm. Figure S7. Evolution of the numerical relative permeability of water vapor as a function of the diffusivity and thickness of the interphase for the composite of φ

_{p}= 11.4 %$v/v$. The numerical results (bullets) correspond to the average of the relative permeability of 36 structures (e

_{i}= 1 µm), 36 structures (e

_{i}= 2.5 µm) and 21 structures (e

_{i}= 5 µm). The volume fraction of the interphase was φ

_{i}= 3.32 ± 0.9 %$v/v$ for e

_{i}= 1 µm, φ

_{i}= 9.93 ± 2.69 %$v/v$ for e

_{i}= 2.5 µm, and φ

_{i}= 22.47 ± 6.65 %$v/v$ for e

_{i}= 5 µm. Figure S8. Evolution of the numerical relative permeability of water vapor as a function of the diffusivity of the interphase for the composite of φ

_{p}= 19.52 %$v/v$ and e

_{i}= 1 µm: Comparison between experimental and numerical results. The numerical results corresponded to the relative permeability of 14 structures. φ

_{i}and np are the interphase volume fraction and the number of particles respectively. Figure S9. Evolution of the numerical relative permeability of water vapor as a function of the diffusivity of the interphase for the composite of φ

_{p}= 19.52 %$v/v$ and e

_{i}= 2.5 µm: Comparison between experimental and numerical results. The numerical results corresponded to the relative permeability of 19 structures. φ

_{i}and np are the interphase volume fraction and the number of particles respectively. Figure S10. Evolution of the numerical relative permeability of water vapor as a function of the diffusivity of the interphase for the composite of φ

_{p}= 19.52 %$v/v$ and e

_{i}= 5 µm: Comparison between experimental and numerical results. The numerical results corresponded to the relative permeability of 6 structures. φ

_{i}and np are the interphase volume fraction and the number of particles respectively.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Abbreviations | |

RVE | Representative Volume Element |

FEM | Finite Element Method |

PHBV | Poly(3-HydroxyButyrate-co-3-HydroxyValerate) |

WSF | Wheat Straw Fibers |

2D, 3D | Two and Three Dimension |

E | Mean |

SD | Standard Deviation |

Probability Density Function | |

CDF | Cumulative Distribution Function |

cte | Constant value |

Latin symbols | |

${L}_{x},{L}_{y},{L}_{z}$ | RVE length along x-axis, y-axis and z-axis $\left(\mathrm{m}\right)$ |

${a}_{p},{b}_{p},{c}_{p}$ | Major, minor and third axis of the particle $\left(\mathrm{m}\right)$ |

${x}_{p},{y}_{p},{z}_{p}$ | Center coordinates of the particle $\left(\mathrm{m}\right)$ |

${D}_{k}$ | Diffusivity of water vapor in the phase k $\left({\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}\right)$ |

${P}_{k}$ | Permeability of water vapor in the phase k $\left(\mathrm{mol}.\mathrm{m}.{\mathrm{m}}^{-2}\xb7{\mathrm{s}}^{-1}\xb7{\mathrm{Pa}}^{-1}\right)$ |

$P$ | Permeability of water vapor in the composite $\left(\mathrm{mol}\xb7\mathrm{m}\xb7{\mathrm{m}}^{-2}\xb7{\mathrm{s}}^{-1}\xb7{\mathrm{Pa}}^{-1}\right)$ |

${c}_{k}$ | Concentration of water vapor in the phase k $\left(\mathrm{mol}\xb7{\mathrm{m}}^{-3}\right)$ |

${\overrightarrow{J}}_{k}$ | Molar surface flux vector of water vapor in the phase k $\left(\mathrm{mol}\xb7{\mathrm{m}}^{-2}\xb7{\mathrm{s}}^{-1}\right)$ |

${p}_{upper}-{p}_{lower}$ | Water vapor pressure differential across the film $\left(\mathrm{Pa}\right)$ |

K | Partition coefficient: concentration ratio between particles and matrix at equilibrium $\left({\mathrm{c}}_{\mathrm{p}}/{\mathrm{c}}_{\mathrm{m}}\right)$ (-) |

M | Velocity (non-physical property) $\left(\mathrm{m}\xb7{\mathrm{s}}^{-1}\right)$ |

${e}_{i}$ | Interphase thickness $\left(\mathrm{m}\right)$ |

${n}_{p}$ | Number of particles (-) |

Greek symbols | |

${\alpha}_{p}$ | Aspect ratio: ratio between major and minor axis of the particle (${a}_{p}/{b}_{p})$ (-) |

${\phi}_{k}$ | Volume fraction of the phase k: ratio between the volume of the phase k and the composite volume $\left(\%v/v\right)$ |

${\varphi}_{z}$ | Molar flux (along z-axis) of water vapor across a composite face $\left(\mathrm{mol}\xb7{\mathrm{s}}^{-1}\right)$ |

${\theta}_{p1}$ | Azimuth angle: angle between the x-axis and the orthogonal projection of the semi-major axis onto the xy-plane (degree°) |

${\theta}_{p2}$ | Elevation angle: angle between the semi-major axis and its orthogonal projection onto the xy-plane (degree°) |

Subscripts | |

m | Matrix |

p | Particle |

i | Interphase |

## Appendix A

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**Figure 1.**Modelling a particle as a spheroid in 3D space. (

**a**) ${a}_{p}$: major axis, ${b}_{p}$: minor axis, ${c}_{p}$: third axis (

**b**) ${\theta}_{p1}$: Azimuth angle. ${\theta}_{p2}$: elevation angle.

**Figure 2.**Representation of the RVE showing: (

**a**) the boundaries conditions at the six composite faces and at the interface matrix–particle for a two–phase structure; (

**b**) the boundaries conditions at the matrix–interphase and interphase–particle interfaces for a three–phase structure.

**Figure 3.**Comparison between experimental distribution and fitted distributions of (

**a**) major axis and (

**b**) aspect ratio obtained from 2869 particles.

**Figure 4.**Examples of 3D composite structures (two-phase system) generated for particles volume fraction corresponding to true composite materials (

**a**) ${\phi}_{p}=5.14\%v/v$ (122 particles), (

**b**) ${\phi}_{p}=11.4\%v/v$ (214 particles) and (

**c**) ${\phi}_{p}=19.52\%v/v$ (611 particles). Examples of 3D composite structures (three-phase system) built by adding interphase to the two-phase structure (

**b**) with different interphase thicknesses (

**d**) ${e}_{i}=1\mathsf{\mu}\mathrm{m},$ (

**e**) ${e}_{i}=2.5\mathsf{\mu}\mathrm{m}$ and (

**f**) ${e}_{i}=5\mathsf{\mu}\mathrm{m}$.

**Figure 5.**Tetrahedral mesh presentation of 3D composite structures corresponding to particles volume fraction of ${\phi}_{p}=11.4\%v/v$ (240 particles): (

**a**) two–phase system and (

**b**) three–phase system.

**Figure 6.**Comparison between experimental and numerical relative permeability $P/{P}_{m}$ (calculated using the 2-phase model).

**Figure 7.**Effect of particle diffusivity ${D}_{p}$ on numerical relative permeability $P/{P}_{m}$ (calculated using the 2-phase model) into the composite of ${\phi}_{p}=19.52\%v/v$ (hypothesis tested on 10 structures).

**Figure 8.**Comparison between experimental and numerical relative permeability $P/{P}_{m}$ (calculated using the 2-phase model) for different particle diffusivity values varying from ${D}_{p}=1\times {10}^{-14}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ to ${D}_{p}=1\times {10}^{-9}{\mathrm{m}}^{2}\xb7{\mathrm{s}}^{-1}$ (hypothesis tested on 10 structures).

**Figure 9.**Effect of matrix diffusivity ${D}_{m}$ on numerical relative permeability $P/{P}_{m}$ (calculated using the 2-phase model) into the composite of ${\phi}_{p}=19.52\%v/v$ (hypothesis tested on 10 structures).

**Figure 10.**Comparison between experimental and numerical relative permeability $P/{P}_{m}$ (calculated using the 2-phase model): adjustment of numerical model to experimental data by fitting the matrix diffusivity ${D}_{m}$ (fitting performed on 10 structures for each composite).

**Figure 11.**Evolution of the numerical relative permeability of the composite $({\phi}_{p}=19.52\%v/v)$ as a function of the diffusivity of interphase and for different thicknesses of the interphase The numerical results (symbols) correspond to the average of the relative permeability of 14 structures $\left({e}_{i}=1\mathsf{\mu}\mathrm{m}\right)$, 19 structures $\left({e}_{i}=2.5\mathsf{\mu}\mathrm{m}\right)$ and 6 structures $\left({e}_{i}=5\mathsf{\mu}\mathrm{m}\right)$. Lines represent the experimental relative permeability for ${\phi}_{p}=19.52\%v/v$: solid line for mean value, dashed lines for mean ± standard deviation.

**Figure 12.**Comparison between experimental and numerical relative permeability: Effect of the particles volume fraction ${\phi}_{p}$ and effect of the diffusivity ${D}_{i}$ and thickness ${e}_{i}$ of the interphase.

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**MDPI and ACS Style**

Kabbej, M.; Guillard, V.; Angellier-Coussy, H.; Wolf, C.; Gontard, N.; Gaucel, S.
3D Modelling of Mass Transfer into Bio-Composite. *Polymers* **2021**, *13*, 2257.
https://doi.org/10.3390/polym13142257

**AMA Style**

Kabbej M, Guillard V, Angellier-Coussy H, Wolf C, Gontard N, Gaucel S.
3D Modelling of Mass Transfer into Bio-Composite. *Polymers*. 2021; 13(14):2257.
https://doi.org/10.3390/polym13142257

**Chicago/Turabian Style**

Kabbej, Marouane, Valérie Guillard, Hélène Angellier-Coussy, Caroline Wolf, Nathalie Gontard, and Sébastien Gaucel.
2021. "3D Modelling of Mass Transfer into Bio-Composite" *Polymers* 13, no. 14: 2257.
https://doi.org/10.3390/polym13142257