# Investigating the Influence of Diffusion on the Cohesive Zone Model of the SiC/Al Composite Interface

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}/AlSi12 interfaces. They determined the diffusion constants and interdiffusion coefficients for the ternary system.

## 2. Simulation Results and Discussion

#### 2.1. Crack Growth Process

#### 2.2. Traction–Separation Response

_{z}and σ

_{xz}were obtained by averaging the atomic virial stress at a distance of ±30 Å near the crack (i.e., region a in Figure 2). The separation or crack opening displacement was also determined by calculating the average displacement of atoms within region a in Figure 2. The opening displacements in the normal direction (Δz) and the shear direction (Δx) were defined and measured as the average displacements of atoms in the upper half of the region compared with those in the lower half. The magnitude of the total crack opening displacement was defined as $\Delta r=\sqrt{\Delta {x}^{2}+\Delta {z}^{2}}$.

_{max}indicates the maximum normal stress, δ

_{0}is the corresponding displacement, and the additional parameters α and β control the overall shape and the decay rate of the stress.

_{max}is the critical stress of fracture, E is the elastic modulus of the cracked specimen, a

_{h}is the crack half length, and woa is the work of adhesion. The work of adhesion is defined as [5]

_{SiC}and Γ

_{Al}are the surface energies of the SiC and Al, respectively, and Γ

_{SiC/Al}is the interfacial energy between SiC and Al. The surface and interfacial energies are defined as follows:

_{1}and A

_{2}are the surface area for computing the surface energy and the area of the interface, respectively (e.g., see [29]). Furthermore, E

_{SiC/Al}, E

_{SiC}, and E

_{Al}are the total energies of the SiC/Al interface, bulk SiC, and bulk Al after relaxation, respectively. It is worth noting that the surface and interfacial energies were computed for the 6H-SiC/Al and 3C-SiC/Al interfaces with the orientation relationships $\left(0001\right)[2\overline{1}\overline{1}0{]}_{\mathsf{\alpha}-\mathrm{SiC}}\Vert \hspace{0.33em}\left(111\right)$[110]

_{Al}and ${\left(111\right)\left[01\overline{1}\right]}_{\beta -\mathrm{SiC}}\Vert \hspace{0.33em}{\left(111\right)\left[01\overline{1}\right]}_{\mathrm{Al}}$, respectively (see Section 3.2).

^{2}as evaluated by previous researchers [30,31]. It can also be observed that the work of adhesion was smaller than the work of separation in Mode I, as presented in Table 1, which was an expected result. Based on Young’s modulus values obtained from Table 1 and an initial crack length of 40 Å, Equation (2) predicts the fracture stresses presented in Table 2. The data presented in Table 1, obtained from MD simulations, showed reasonable consistency with the results obtained using Griffith’s theory in Table 2. However, the critical fracture stresses derived from atomistic simulations were slightly lower than those predicted by the analytical Griffith’s theory.

#### 2.3. Cohesive Zone Model

_{max}is the maximum cohesive strength, λ is a non-dimensional parameter defined in Equation (6), and α and β are constants that will be determined using the MD results. Also, in Equation (6), u

_{n}and u

_{t}are the normal and tangential separations, and δ

_{n}and δ

_{t}are the maximum allowable normal and tangential separations, respectively. The cohesive element reaches failure when the value of λ equals 1.

## 3. Simulation Methodology

#### 3.1. Potential Functions

_{ij}represents the bond length between atoms i and j, and f

_{R}(r

_{ij}), f

_{C}(r

_{ij}), and f

_{A}(r

_{ij}) are the repulsive and attractive atomic pair interactions and the optional cutoff function that determines the interaction range, respectively. Also, b

_{ij}is a function that modulates the attractive interaction and incorporates many-body interactions.

_{0}, D

_{0}, and α are the distance between atoms, the equilibrium bond length, the well depth of the potential, and the width of the potential, respectively. The parameters of the Morse potential for the Al–C and Al–Si interactions are provided in Refs. [15,16,38].

#### 3.2. Molecular Dynamics Model

_{Al}was considered for the α-SiC particulate-reinforced Al [41]. On the other hand, the orientation relationship ${\left(111\right)[01\overline{1}]}_{\beta -\mathrm{SiC}}\Vert \hspace{0.33em}{\left(111\right)\left[01\overline{1}\right]}_{\mathrm{Al}}$ was considered for the β-SiC whisker-reinforced Al [42].

^{−10}and a force tolerance of 1 × 10

^{−10}eV/Å. Next, the NVT canonical ensemble at a constant temperature of 300 K was imposed on the sample for 20 ps to adjust the volume and relax the assembled interface system. Two paths were followed depending on whether the systems underwent heat treatment.

#### 3.3. Loading Conditions

^{8}s

^{−1}. A simulation time step of 1 fs was employed to ensure accurate results.

_{z}and -v

_{z}were imposed on the atoms situated on the top and bottom boundaries (region b) in the z-direction, respectively. The imposed global strain rate was equal to

_{z}is the size of the simulation box in the z-direction and t

_{b}is the thickness of the boundary layer (see Figure 2). The atoms between these boundary layers initially had velocities that uniformly increased from −v

_{z}to v

_{z}. The specific velocity assigned to each atom depended on its relative position along the z-direction, ranging from the lowest to the highest coordinate.

_{x}and −v

_{x}, respectively, which was constant in time. The imposed shear strain rate is given by

## 4. Conclusions

- The formation of the diffusion layer did not cause a significant effect on the elastic and shear moduli.
- The tensile and shear strengths of the Si-terminated interfaces were lower than those of their C-terminated counterparts before heat treatment. However, after heat treatment, the strengths of the two interfaces approached each other.
- The formation of a diffusion layer increased the tensile strength of the C- and Si-terminated interfaces by about 20% and 40%, respectively, compared with the interfaces before heat treatment.
- Following heat treatment, the work of separation increased by approximately 30% and 100% for the C- and Si-terminated interfaces, respectively.
- The shear strength was significantly lower than the tensile strength at the 6H-SiC(0001)/Al(111) and 3C-SiC(111)/Al(111) interfaces. Therefore, there was no direct correlation between shear and tensile strengths for these interfaces, unlike isotropic materials, where the shear strength was about half that of the tensile strength.
- The existing continuum-based cohesive zone model was consistent with the proposed traction–separation law based on the MD results.
- The hierarchical multiscale modeling of the interface in finite element software can be done using the cohesive zone model obtained from the MD simulations.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Sample Availability

## References

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**Figure 1.**Snapshots of crack propagation at interface models under pure tensile loading (mode I). The C-terminated 6H-SiC/Al (

**a**) before and (

**b**) after DL formation. The Si-terminated 6H-SiC/Al (

**c**) before and (

**d**) after DL formation.

**Figure 2.**Schematic view of the C-terminated 6H-SiC/Al simulation model, dimensions, and coordinated system.

**Figure 3.**Traction–separation relation for the SiC/Al composite under tensile loading for both cases before and after DL formation. The (

**a**) C- and (

**b**) Si-terminated 6H-SiC/Al interfaces. The (

**c**) C- and (

**d**) Si-terminated 3C-SiC/Al interfaces.

**Figure 4.**Snapshots of the Si-terminated 6H-SiC/Al interface models under shear loading (mode II). (

**a**) Before and (

**b**) after DL formation.

**Figure 5.**Traction–separation curves for the C-terminated 6H-SiC/Al interface under different mixed-mode loadings. Normal stress (

**a**) before and (

**b**) after DL formation. Shear stress (

**c**) before and (

**d**) after DL formation.

**Figure 6.**Traction–separation curves for the Si-terminated 6H-SiC/Al interface under different mixed-mode loadings. Normal stress (

**a**) before and (

**b**) after DL formation. Shear stress (

**c**) before and (

**d**) after DL formation.

**Figure 7.**Traction–separation curves for the C-terminated 3C-SiC/Al interface under different mixed-mode loadings. Normal stress (

**a**) before and (

**b**) after DL formation. Shear stress (

**c**) before and (

**d**) after DL formation.

**Figure 8.**Traction–separation curves for the Si-terminated 3C-SiC/Al interface under different mixed-mode loadings. Normal stress (

**a**) before and (

**b**) after DL formation. Shear stress (

**c**) before and (

**d**) after DL formation.

**Figure 9.**Comparison of the proposed CZM with the present MD results and continuum-based CZM models formulated by Needleman [32]. The (

**a**) C- and (

**b**) Si-terminated 6H-SiC/Al interfaces. The (

**c**) C- and (

**d**) Si-terminated 3C-SiC/Al interfaces.

**Table 1.**The elastic modulus, maximum tensile stress, toughness, and work of complete separation of SiC/Al composites under pure tensile loading (mode I).

Composite Material | Annealing Condition | E (GPa) | σ_{max}(GPa) | Toughness (10 ^{9} J/m^{3}) | Work of Separation (J/m^{2}) |
---|---|---|---|---|---|

C-terminated 6H-SiC/Al | Before DL formation | 161.7 | 4.39 | 0.174 | 4.81 |

After DL formation | 163.5 | 5.17 | 0.231 | 6.41 | |

Si-terminated 6H-SiC/Al | Before DL formation | 162.9 | 3.55 | 0.052 | 2.49 |

After DL formation | 164.7 | 5.02 | 0.217 | 5.59 | |

C-terminated 3C-SiC/Al | Before DL formation | 165.5 | 4.40 | 0.164 | 4.50 |

After DL formation | 167.4 | 5.40 | 0.183 | 5.71 | |

Si-terminated 3C-SiC/Al | Before DL formation | 163.7 | 3.57 | 0.071 | 3.17 |

After DL formation | 166.0 | 5.21 | 0.232 | 6.21 |

**Table 2.**The work of adhesion and critical fracture stress obtained using Griffith’s theory in Equation (2) for the C- and Si-terminated 6H-SiC/Al and 3C-SiC/Al interfaces at 300 K.

6H-SiC/Al | 3C-SiC/Al | |||
---|---|---|---|---|

C-Terminated | Si-Terminated | C-Terminated | Si-Terminated | |

woa (J/m^{2}) | 1.09 | 0.74 | 1.12 | 0.86 |

σ_{max} (GPa) (Equation (2)) | 5.24 | 4.38 | 5.43 | 4.73 |

**Table 3.**The shear modulus and maximum shear stress of SiC/Al composites under shear loading (mode II).

Composite Material | Annealing Condition | G (GPa) | τ_{max} (GPa) |
---|---|---|---|

C-terminated 6H-SiC/Al | Before DL formation | 36.9 | 1.71 |

After DL formation | 36.5 | 1.80 | |

Si-terminated 6H-SiC/Al | Before DL formation | 39.2 | 1.12 |

After DL formation | 35.2 | 1.82 | |

C-terminated 3C-SiC/Al | Before DL formation | 39.8 | 1.72 |

After DL formation | 32.2 | 1.60 | |

Si-terminated 3C-SiC/Al | Before DL formation | 33.1 | 1.39 |

After DL formation | 30.7 | 1.65 |

**Table 4.**The loading angle θ and corresponding boundary velocities v

_{x}and v

_{z}for mixed-mode loading.

θ (°) | v_{x} (Å/ps) | v_{z} (Å/ps) |
---|---|---|

0 | 0 | 0.01225 |

15 | 0.00317 | 0.01183 |

30 | 0.00612 | 0.01060 |

45 | 0.00866 | 0.00866 |

60 | 0.01060 | 0.00612 |

75 | 0.01183 | 0.00317 |

90 | 0.01225 | 0 |

Composite Material | Annealing Condition | σ_{max} | δ_{n} (Å) | α | β |
---|---|---|---|---|---|

C-terminated 6H-SiC/Al | Before DL formation | 4.39 | 27 | 0.759 | 0.663 |

After DL formation | 5.17 | 28 | 0.703 | 0.414 | |

Si-terminated 6H-SiC/Al | Before DL formation | 3.55 | 17 | 0.689 | 0.939 |

After DL formation | 5.02 | 24 | 0.764 | 0.732 | |

C-terminated 3C-SiC/Al | Before DL formation | 4.40 | 28 | 0.783 | 0.816 |

After DL formation | 5.40 | 28 | 0.932 | 0.859 | |

Si-terminated 3C-SiC/Al | Before DL formation | 3.57 | 20 | 0.724 | 0.621 |

After DL formation | 5.21 | 24 | 0.880 | 0.829 |

**Table 6.**The elastic constants C

_{11}, C

_{12}, and C

_{44}, bulk modulus K, Young’s modulus E, shear modulus G, and Poisson’s ratio ν determined through MD simulations using the EAM and Tersoff potential functions and compared with other MD simulations and experimental data.

Material | Method | C_{11}(GPa) | C_{12}(GPa) | C_{44}(GPa) | K (GPa) | E (GPa) | G (GPa) | ν |
---|---|---|---|---|---|---|---|---|

Al | Present ^{a} | 107.03 | 61.06 | 31.05 | 76.38 | 62.67 | 22.99 | 0.363 |

Present ^{b} | 105.09 | 59.46 | 30.66 | 74.67 | 62.12 | 22.82 | 0.361 | |

MD ^{c} | 107.21 | 60.60 | 32.88 | 76.14 | 63.44 | 23.31 | 0.361 | |

Experiment ^{d} | 107.3 | 60.08 | 28.3 | 75.7 | 63.83 | 23.48 | 0.359 | |

3C-SiC | Present | 383.78 | 144.41 | 239.75 | 224.20 | 304.81 | 119.68 | 0.273 |

MD ^{e} | 390.1 | 142.7 | 191.0 | 225.1 | 313.6 | 123.7 | 0.268 | |

Experiment ^{f} | 390 | 142 | 256 | 225 | 314.2 | 124 | 0.267 |

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**MDPI and ACS Style**

Tahani, M.; Postek, E.; Sadowski, T.
Investigating the Influence of Diffusion on the Cohesive Zone Model of the SiC/Al Composite Interface. *Molecules* **2023**, *28*, 6757.
https://doi.org/10.3390/molecules28196757

**AMA Style**

Tahani M, Postek E, Sadowski T.
Investigating the Influence of Diffusion on the Cohesive Zone Model of the SiC/Al Composite Interface. *Molecules*. 2023; 28(19):6757.
https://doi.org/10.3390/molecules28196757

**Chicago/Turabian Style**

Tahani, Masoud, Eligiusz Postek, and Tomasz Sadowski.
2023. "Investigating the Influence of Diffusion on the Cohesive Zone Model of the SiC/Al Composite Interface" *Molecules* 28, no. 19: 6757.
https://doi.org/10.3390/molecules28196757