# Nano-Level Damage Characterization of Graphene/Polymer Cohesive Interface under Tensile Separation

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## Abstract

**:**

^{−8}(aPa·nm

^{−1}), 9.75 × 10

^{−10}(nm), 2.1 × 10

^{−10}(N·nm

^{−1}) respectively, that is followed by an exponential regressive law with the exponent, α = 7.74. It is shown that the commonly assumed bilinear softening law of the cohesive interface could lead up to 55% error in the predicted separation of the interface.

## 1. Introduction

## 2. Computational Methods

#### 2.1. Molecular Dynamics Simulation

#### 2.1.1. Cross-Linking of the Polymer during the Curing Process

^{3}in order to restrict the formation of unreal configuration and huge forces which occur due to the ring spearing and atoms overlapping [43,44]. The density of the molecular system was increased to the real density through the initial equilibration level by the isothermal-isobaric ensemble (NPT) [23,45,46]. Figure 1b shows the variation of the RVE density with respect to the simulation time after the initial equilibration. The curing process was performed subsequently on this equilibrated molecular system.

- It was assumed that the primary and secondary amines have the same reactivity with the reaction cutoff distance of 5 Ǻ.
- The system was checked for the bond formation at every 50 ps after each equilibration.
- The constants in the equation of the newly-formed bonds were decreased in the first equilibration level and progressively elevated to the real values [47].
- The outputs were examined to update the topology parameters according to the new bonds.
- An annealing process was implemented to release the residual stresses. In this process, the temperature was increased to 450 K and then cooled to 300 K gradually over 2 × 106 time-steps.
- Steps 2 through 5 were iterated to obtain about 80% cross-linking.
- Finally, the last NPT equilibration in an interval time of 3 ns was applied to the RVE.

#### 2.1.2. Tensile Separation Process

^{2}carbon atoms of graphene and all of the polymer atoms in the condensed graphene-polymer nanocomposites [23,54]. In the PCFF force field, the non-bonded Van der Waals interaction between the same type atoms (denoted by ii) is described using the L-J potential as:

^{−7}ns

^{−1}, the other faces were maintained under one atmospheric pressure [56,57].

_{pq}, were calculated using Virial expression as [58,59]:

_{pq}represents the stress components τ

_{xx}, τ

_{yy}, τ

_{zz}, τ

_{xy}, τ

_{xz}, and τ

_{yz}.

#### 2.2. Continuum Mechanics Simulation

#### 2.2.1. Cohesive Zone Model

_{i}are the traction and separation at an interface point, and indices 3, 1, and 2 refer to mode I (normal) load, mode II, and mode III (shear) loadings, respectively. The parameter k

_{i}represents the cohesive stiffness of the interface in the respective load direction. In this study, the pure tensile loading mode was considered, thus the traction-displacement relation could be simplified to ${t}_{3}={k}_{3}{\delta}_{3}$.

#### 2.2.2. Damage Initiation and Propagation Criterion

_{i}, in each loading mode could be calculated by the following equation:

#### 2.2.3. Finite Element Simulation

## 3. Results and Discussion

#### 3.1. Thickness of the Graphene and Graphene/Polymer Interface

^{12}ng/nm

^{3}, as illustrated by the dashed line and without considering the density readings below 0.65 × 10

^{12}ng/nm

^{3}. It is noted that a 0.3% variation of the average density of the epoxy is calculated over the sampling distance of 7 nm, particularly contributed by the higher density gradient next to the graphene/polymer interface. The polymer density diminishes to zero across a small gap to the location of the graphene. This density gradient indicates the non-uniform distribution of the polymer atoms in the nearby graphene/polymer interface.

^{2}(see Table 1) for the RVE. The calculated thickness of the graphene/epoxy interface will be used in the FE model of the RVE with finite-thickness cohesive elements.

#### 3.2. Nanomechanical Behaviors of Graphene/Polymer Interface

^{−10}aPa is achieved. The stress decreases exponentially with the continuous displacement. A similar cohesive stress-displacement response with a nonlinear spring model was predicted for the interface of a pristine graphene/one-layer polymer using the MD approach [10,16]. At the atomic level, the interaction between the neighboring atoms could be described in terms of the interaction energy, as shown in Figure 6b, for two atoms with $\epsilon =3.4\u212b$ and $\sigma =10\mathrm{meV}$. The interaction energy level is at a minimum (also known as the potential well) when the atoms are at their equilibrium distance apart, such that the cohesive (attractive) and repulsive forces between the atoms are balanced. Under the applied separation forces, the interaction energy decreases in an exponential manner, but the energy is restored upon unloading. If the loading continued, the separation energy would diminish to zero, denoting the atomic separation. The inflection point denotes the minimum interaction force as represented by the slope of the energy-distance curve (Figure 6b). Beyond this distance, the required interaction force of atomic separation that diminish exponentially could be interpreted as representing the degradation of the bond strength.

**Graphene/epoxy interface properties:**The elastic properties of the interface were also determined from the MD-calculated stress-displacement curve of Figure 6a. The interface tensile strength, ${T}_{3}^{o}$, is defined by the peak stress level while the corresponding displacement represents the value at the onset of interface damage, ${\delta}_{33}^{o}$. The initial tangential slope of the stress-displacement curve defines the penalty stiffness, k

_{3}, of the interface. The area bounded by the curve represents the critical strain energy release rate, G

_{IC}, for the tensile loading mode, which could be calculated by Equation (7). These properties along with the parameters of the exponential softening law are used to define the graphene/epoxy cohesive interface behavior for the FE simulation. The property values are listed in Table 3.

#### 3.3. Response of the Graphene/Epoxy Interface Model

_{o}. The calculated evolution of the DDE follows the prescribed form of the regressive softening process, as illustrated in the figure. The initial rate of the energy dissipation is faster for the cohesive interface with exponential regressive than that of the linear softening law, as reflected by the initial slope of the curve. The evolving DDE saturates to a constant level when reaching the separation of the graphene/epoxy cohesive interface.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Molecular structure of the polymer matrix used in the molecular dynamics (MD) simulation, (

**b**) variation of RVE density against time after the initial isothermal-isobaric ensemble (NPT) equilibration.

**Figure 2.**The simulation box containing long embedded graphene with the applied displacement of the nanocomposite.

**Figure 4.**(

**a**) Schematic view of the graphene-polymer system, and (

**b**) finite element (FE) model of the half symmetric part of the nanocomposite system.

**Figure 5.**Variation of the epoxy polymer density along the RVE length of the graphene-polymer nanocomposite.

**Figure 6.**(

**a**) The stress-displacement response, as calculated from the MD simulation, (

**b**) interatomic energy-distance curve, and (

**c**) the characteristic exponential decay softening law of the CZM.

**Figure 7.**Comparison of the FE and MD stress-displacement response of the graphene/polymer interface.

**Figure 8.**Comparison of the FE-predicted response of the cohesive interface with different softening laws.

**Figure 9.**Evolution of the total damage dissipation energy (DDE) of the graphene/polymer interface with increasing tensile displacement.

**Table 1.**The equilibrated representative volume element (RVE) specifications of the graphene-epoxy nanocomposite.

Configuration of RVE | Graphene Sheet Length (nm) | Box Volume (nm^{3}) | Number of Epoxy Molecules | Number of Hardener Molecules | Density after Curing Process and Final NPT Equilibration (g/cm^{3}) |
---|---|---|---|---|---|

Long | 4.540 × 4.520 | 143.653 | 258 | 86 | 1.1865 |

Parameter | MD Simulation | Experiments |
---|---|---|

Density, g/cm^{3} | 1.14 | 1.16 |

Poisson’s ratio | 0.39 | 0.3–0.4 |

Young’s modulus, GPa | 2.77 | 2.4–3.4 |

Shear modulus, GPa | 1.03 | 1.0–1.5 |

Parameter | Symbol (Unit) | Value |
---|---|---|

Tensile stiffness | k_{3}, (aPa·nm^{−1}) | 5 × 10^{−8} |

Tensile strength | ${T}_{3}^{0}$, (aPa) | 9.75 × 10^{−10} |

Displacement at damage initiation | ${\delta}_{33}^{o}$, (nm) | 0.0653 |

Displacement at separation | ${\delta}_{33}^{f}$, (nm) | 0.8 |

Exponent for the regressive softening law, (Equation (8)) | α | 7.74 |

Critical Mode I strain energy release rate | G_{IC}, (N·nm^{−1}) | 2.1 × 10^{−10} |

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**MDPI and ACS Style**

Koloor, S.S.R.; Rahimian-Koloor, S.M.; Karimzadeh, A.; Hamdi, M.; Petrů, M.; Tamin, M.N. Nano-Level Damage Characterization of Graphene/Polymer Cohesive Interface under Tensile Separation. *Polymers* **2019**, *11*, 1435.
https://doi.org/10.3390/polym11091435

**AMA Style**

Koloor SSR, Rahimian-Koloor SM, Karimzadeh A, Hamdi M, Petrů M, Tamin MN. Nano-Level Damage Characterization of Graphene/Polymer Cohesive Interface under Tensile Separation. *Polymers*. 2019; 11(9):1435.
https://doi.org/10.3390/polym11091435

**Chicago/Turabian Style**

Koloor, S. S. R., S. M. Rahimian-Koloor, A. Karimzadeh, M. Hamdi, Michal Petrů, and M. N. Tamin. 2019. "Nano-Level Damage Characterization of Graphene/Polymer Cohesive Interface under Tensile Separation" *Polymers* 11, no. 9: 1435.
https://doi.org/10.3390/polym11091435