epoxy

Nano-scale fracture toughness and behavior of graphene/epoxy interface Hossein Salahshoor and Nima Rahbar Citation: Journal of Applied Physics 112, 023510 (2012); doi: 10.1063/1.4737776 View online: http://dx.doi.org/10.1063/1.4737776 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/112/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Mode I fracture toughness behavior of hydro-thermally aged carbon fibre reinforced DGEBA-HHPA-PES systems AIP Conf. Proc. 1459, 117 (2912); 10.1063/1.4738416 Mechanical behavior and fracture of graphene nanomeshes J. Appl. Phys. 117, 024302 (2015); 10.1063/1.4905583 Preparation of graphene oxide/epoxy nanocomposites with significantly improved mechanical properties J. Appl. Phys. 116, 053518 (2014); 10.1063/1.4892089 Fracture toughness of the sidewall fluorinated carbon nanotube-epoxy interface J. Appl. Phys. 115, 224305 (2014); 10.1063/1.4881882 Nanofracture in graphene under complex mechanical stresses Appl. Phys. Lett. 101, 121915 (2012); 10.1063/1.4754115

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JOURNAL OF APPLIED PHYSICS 112, 023510 (2012)

Nano-scale fracture toughness and behavior of graphene/epoxy interface Hossein Salahshoor and Nima Rahbara) Department of Civil and Environmental Engineering, University of Massachusetts Dartmouth, North Dartmouth, Massachusetts 02747, USA

(Received 11 May 2012; accepted 20 June 2012; published online 19 July 2012) Atomistic simulations are performed to investigate the nano-scale interfacial fracture toughness between graphene and epoxy. Nano-mechanical properties of graphene and epoxy are initially studied using molecular dynamics simulations. A novel method is suggested to accurately model the behavior of the graphen/epoxy interface during the curing process of the epoxy as a function of temperature. The computed interfacial fracture energy is computed at about 0.203 J/m2, which is in good agreement with available experimental data. It is also shown that the adhesion between cured epoxy and graphene layer increases the pre-existing waviness of the 2-dimensional ˚. graphene sheet in a 3-dimensional space. The waviness amplitude is computed to be about 3.23 A C V 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4737776] I. INTRODUCTION

Graphene is a flat monolayer of carbon atoms tightly packed into a two-dimensional (2D) honeycomb lattice and is a basic building block for graphitic materials of all other dimensionalities.1 Their unique mechanical, thermal, and electronic properties proposed their wide potential applications. Graphene is the basic structural element in carbon nanotubes. Their high surface area, Young’s modulus, and strength make them great fiber materials for composites. This paper is focused on understanding the adhesion mechanisms and interfacial fracture energies between a layer of graphene and epoxy matrix. The main assumption here is that the interaction energy is positively correlated with the adhesion energy. The governing energy term in computing the interaction energy for interfaces is Van der Waals (VdW) interactions. This has led to development of different methods by different researchers to investigate adhesion mechanisms and interfaces. Odegard et al. presented a method for nanocomposites in which the carbon nanotube, the local polymer around the nanotube, and the nanotube/polymer interface is modeled as an effective continuum fiber.2 Later, cohesive laws of separation have been evaluated for graphene planes using VdW potentials by different researchers.3–5 In all of these studies, the dynamics of atomistic structure near the interface were ignored. Wei studied the adhesion between CNT (Carbon Nano-Tube) and a polymer by studying the interfacial shear strength and change of interfacial VdW energy using molecular dynamics.6 He found that interfacial shear strength and VdW energy is increased by increasing the applied strain. Interfacial sliding tests are modeled using atomistic simulations to evaluate shear strength of interfaces by Chowdhury et al.,7 Awasthi et al.8 and Pavia and Curtin.9 These pull out tests have provided the force versus displacement response. Ab-initio molecular dynamics establishes a more accurate framework for atomistic modeling of interfaces. Hence, this paper is focus a)

Author to whom correspondence should be addressed. Electronic mail: [email protected].

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to study the interaction energy between a graphene sheet and a crosslinked polymer using ab-initio molecular dynamics. II. MATERIALS AND METHODS

The first step in the presented atomistic modeling was the simulation of the mechanical properties of polymer (epoxy) and graphene. The resulting elastic properties were then verified with the reported experimental values. This was followed by the simulations of the adhesion and interface formation between polymer and graphene. Molecular modeling was performed using the MATERIAL STUDIO 6.0 software package.10 A. Epoxy

The epoxy system used in this study is based on diglycidyl ether of bisphenol-a (DGEBA), which is commercially known as EPON 828. The curing agent used with the epoxy (DGEBA) is aminoethyl piperazine (AEP), commercially known as EPICURE 3200, Figure 1. Since polymers are long chain molecules, long-range interactions and non-bonded terms play an important role in computing the overall energy of the system. Non bonded energy terms are mostly related to Van der Waals forces being dipole-dipole interactions and electrostatic forces, which are ionic interactions. In our simulations, the VdW and electrostatic interactions were implemented using Lennard-Jones and Coulombs law, respectively. Kinetic energy and bonded terms are the remaining parts of the total energy. Bonded terms consist of bond stretching, bending, torsion and out of plane angle terms, and their interaction. Similar reactivity was considered for the end groups to model the crosslinking process. The charge distribution was calculated by Qeq method, after building the molecular structure in Figure 1.11 In order to provide reactive sites to establish the crosslinks between nitrogen and carbon atoms, the end active hydrogens were removed. An amorphous epoxy model was constructed by using self-avoiding random walk method of Theodorou and Suter.12 The 3D unit cell was built with 15 resin molecules and 10 agents. The molecules were packed into the cell with a density of 1.16 g/cm3.

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FIG. 1. The chemical structure of: (a) DGEBA; (b) AEP.

Crosslinking process was established by performing cyclic NVT (canonical ensemble) and NPT (isothermal-isobaric ensemble) simulations. The Nose thermostat algorithm was used for NVT simulations and Berendsen thermostat and barostat algorithms for NPT simulations. The details of the crosslinking process were presented earlier by Wu and Xu.13 The time step was chosen to be 1 fs for all simulations. The dynamic simulations were performed for curing at T ¼ 500 K. The system was further cooled down by a rate of 100 K/20 ps. The time length of both NVT and NPT simulation was 20 ps. The simulations for 10 ps, 50 ps, and 100 ps were also performed. It was observed that the results would not undergo significant changes. The Young’s Modulus of the polymer was computed by applying a strain amplitude of 0.003 to the cell. This force was exerted to the system by applying a strain amplitude of 0.003 to the cell. Dreiding14 and condensed-phased optimized molecular potential (Compass)15 force fields are among the most common force fields in simulations of epoxy polymers. They have been previously used to study the mechanical properties of epoxy polymers.13,18 Both force fields were also used in this study, and the results are presented and compared in Sec. III. B. Graphene

An armchair graphene was modeled here. A unit cell of graphene with periodic boundary conditions is constructed. The Compass force field was used as the potential, since this potential has been successfully used to model carbon systems.16,17 The Young’s modulus of the graphene sheet was calculated by applying a strain to the system as expressed in Sec. II A.

total energy of the system and interface were computed in each step. Also, the roughness of the graphene was measured in the process of crosslinking in each step. III. RESULTS AND DISCUSSION

In order to verify the validity of the presented method, the mechanical properties of crosslinked polymers and graphene were computed in this section and compared with the experimental values. A. Mechanical properties of polymer

Molecular modeling of the polymer was performed by the methodology prescribed earlier. The temperature of the system was initially raised to 500 K and then, the system was cooled down to room temperature with a rate of 100 K/20 ps. The cyclic NVT and NPT simulations were performed at each cooling step, which helped the system to relax and reach to its minimum energy configuration. Using this procedure, the bonds between the epoxy and the agents were established through the reactive sites. Compass and Dreiding force fields have been previously used to study mechanical properties of epoxies.13,18 In this paper, we have used both force fields to model the DGEBA/AEP polymer and derived the mechanical properties. The results in each case are compared with experimental values. Once the molecules have been placed within the super cell, a series of alternating MD (Molecular Dynamics)

C. Interface

A new epoxy model was developed with a ratio of 10:1 (molecular weight). For this simulation, the epoxy and the graphene were modeled in a 3D unit cell of the size ˚ . Similar crosslinking procedure was applied 83 24 70 A here through different curing steps. The epoxy and graphene were cured together. Curing simulation was started at the room temperature and then, the temperature was raised up to 358 K. First, an energy minimization and equilibration dynamic simulation was performed at 298 K and the temperature was then raised to 358 K. The system was then cooled down by a rate of 20 K/1000 ps. The curing process was followed by cyclic NVT and NPT simulations. Total simulation time 5 ns having 5 106 timesteps equal to 1 fs. Both Compass and Dreiding force fields were used for the interface study. Accordingly, the

FIG. 2. Variation of epoxy Young’s modulus as the simulations of curing process progresses using Drieding and Compass force fields. The thermal step for cross-linking is 100 K from 298 raised to 500 and cooled back to 298 again.


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simulations and static energy minimizations (molecular mechanics simulations) were used to establish the equilibrated molecular structure at the corresponding density. The molecular structures were gradually equilibrated to minimize any residual stresses in the model. After constructing the super cell and a preliminary dynamics simulation, the 2nd step was started to simulate the crosslinking procedure. 20,000 time steps of NVT simulations were performed at room temperature (298 K). Afterwards, the system’s temperature was gradually raised to 500 K. It can be observed that after relaxing the system in the 500 K (3rd simulation), the Young’s Modulus values decreases, Figure 2. This process was repeated for the temperatures at 400 K and 300 K. 0

9:5541 B 5:7585 B B 4:6630 Cij ¼ B B 1:3357 B @ 0:9755 0:1614

5:7585 4:6630 1:2595 3:2319 3:2319 5:5935 1:6605 0:0736 0:7725 0:3619 0:8707 1:4038

The calculated stiffness matrix is symmetrical, which further verifies the validity of the suggested modeling framework. Lame’s constants can also be calculated using the above elastic matrix 1 2 k ¼ ðC11 þ C22 þ C33 Þ ðC44 þ C55 þ C66 Þ; 3 3

(2)

1 l ¼ ðC44 þ C55 þ C66 Þ 3

(3)

to be about k ¼ 3:57 GPa and l ¼ 0:95 GPa. Figure 2 presents the results for Compass force field and Dreiding force field. The glass transition temperature, Tg, of the DGEBA/AEP polymer is about 388 K.19 Hence, the sudden raise of the temperature beyond the polymers Tg will impose different mechanical behavior. The 4th step is dynamic and relaxation simulations at 400 K. Accordingly, some fluctuations in total energy and Young’s Modulus of the polymer (in comparison with experiment) were observed in the 4th step of the simulations. Generally, at temperatures close to the Tg, the behavior of the system needs to be simulated more accurately with a different approach. In terms of mechanical behavior, the material is in a transition state, which leads to these fluctuations. Hence, these fluctuations were expected and were not considered in this study. In the last step, the temperature was set back to the room temperature. There is a remarkable difference observed between the Young’s modulus in 2nd and 5th step, which were both performed at the room temperature. This difference clearly shows the effect of the crosslinking process in the mechanical behavior of the polymer. Figure 2 also presents the computed Young’s modulus as a function of the simulations step for both Compass and Dreiding force field. The Dreiding force field shows fewer

The first step to verify the mechanical properties is to compute the elastic properties at nano-scale. In order to calculate the components of the elastic stiffness matrix, Cij , predefined strain was applied to the polymer and virial stresses rij were computed. At each step of the model, cyclic NVT and NPT simulations at different temperatures were performed and the elastic constants, Cij , were derived using the second derivative of potential energy with respect to strain. Cij ¼

1 @2u @ri ¼ : V @i @j @j

(1)

The stiffness matrix is derived as

1:3357 1:6605 0:0736 1:4437 0:3329 0:5144

1 0:9755 0:1614 0:7725 0:8707 C C 0:3619 1:4038 C C: 0:3329 0:5144 C C 0:5757 1:0205 A 1:0205 0:8290

fluctuations, but the results are significantly different from the experimental data. Meanwhile, the computed Young’s modulus using the Compass force field approached to the experimental values as the simulation progressed. It is also clear from these results that the Dreiding force field is not suitable for modeling cross-linking process in the crosslinked epoxy studied here, since the Young’s modulus does not significantly change as the simulation progresses. Using Compass force field, the final computed Young’s modulus is about 2.64 GPa. The experimental data show the Young’s modulus of DGEBA/AEP polymer is in the range of 2.7–2.9 GPa.18 Hence, the suggested simulation process is capable of modeling the mechanical properties of crosslinked polymers.

B. Mechanical properties of graphene

Mechanical behavior of graphene has been extensively studied by numerous scientists.20–24 The Young’s modulus of graphene is previously measured to be about 1 TPa.22–24 In this work, graphene is modeled with 540 atoms. The hexagon carbon cores have aromatic bonds within themselves and with each other. Since the model is a 3D periodic unit cell and we need a 2D plate of carbon atoms, a vacuum slab is placed above the graphene atoms. By this technique, the interaction between sheets on the axis normal to the graphene is almost vanished. In this case, the non-bonded terms, which can significantly contribute to the total energy, is almost vanished from the energy of the graphene by making the vacuum slab thick enough. As described earlier, in order to calculate the mechanical properties, strain was applied to the system and the Young’s modulus was calculated for the armchair carbon sheet. The calculated Young’s modulus is about 1049 GPa. It is


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observed that the simulation results have excellent compatibility with the presented values in the literature, both analytically and experimentally.14–16 C. Interface of the epoxy and graphene

The interfacial fracture energies between the crosslinked polymer and graphene were computed in the next step. Periodic boundary conditions were applied to the unit cell. The vacuum slab technique was applied to remove the non-bonded terms between different cells along the direction normal to the graphene and epoxy layers, Figure 3. Unit cells ˚ is in the nor˚ 3, in which 70 A dimensions are 83 24 70 A mal direction and establishes the large vacuum slab. Crosslinking process was applied in several steps of curing. In this way, the curing could be done with sufficient accuracy, while the glass transition temperature (Tg) is not surpassed. Simulations of curing are started at the room temperature, going to 358 K and cooling down to room temperature again. The whole dynamics simulation is 5 ns with a time-step of 1 fs. The curing was performed by cooling down the system at a rate of 20 K/1000 ps. Figure 3 shows the curing process for crosslinking. It is widely known that 2-dimensional perfect crystal does not exist in a 2-dimensional space.25 However, existence of a 2-dimensional crystal in a 3-dimensional space is possible. This is possible through creation of corrugation and waviness in graphene sheet.26–29 The thermodynamic stability of the sheet leads to two different modes of buckling or bending. The buckling mode requires generation of dislocations and since the graphene is a perfect crystal, it will not happen. The bending mechanism assumes no defects and requires out-of-plane bending which is observed in the deformation mechanism of interface. This waviness of a graphene sheet has been studied by previous researchers. Kirilenko et al. predicted the roughness ˚ and 2.7 A ˚ withof a single graphene sheet to be around 1.7 A out and with an amorphous carbon substrate, respectively.28 Consequently, adhesion of the graphene sheet to its substrate causes higher roughness and corrugation in the graphene sheet. Figure 4 clearly shows the waviness of the graphene sheet while interacting with the epoxy.

FIG. 3. Variation of temperature as a function of time in the simulation of cross-linking process.

FIG. 4. Simulation results for different stages after relaxation and thermal step of 15 K: (a) initial model; (b) first curing in 298 K, 1000 ps; (c) second curing started in 358 K, 1000 ps; (d) cooled down system to 338 K, 1000 ps; (e) 3rd step of cooling down to 318 K, 1000 ps; (f) last step of post-curing again in 298 K, 1000 ps of simulation.

The roughness of the graphene sheet was computed in each step of the simulation. The results are presented in Figure 5. In higher temperatures, atoms have higher kinetic energy and move faster. This leads to larger waviness in the graphene sheet. As the system is cooled down, the amplitude of the graphene sheet decreases until the temperature is reached to room temperature. To the knowledge of authors, the roughness of graphene interacting with epoxy is studied for the first time here. This value is calculated to be about ˚. about 3.23 A An interesting observation in the simulation with longer time length is the interaction of the polymer with the graphene layer of the upper unit cell. While the vacuum slab makes the effect of the unit cell in the normal direction minimal, in longer simulations, some parts of polymer were eventually attracted to the upper graphene, Figure 4. It should be noticed that cell parameters change during cyclic dynamic simulations of NPT and NVT ensembles. The final and initial cell parameters and their change due to dynamic simulations using Compass force field are presented in the Table I. Compass and Dreiding were used for the adhesion simulations of polymer and organic systems. Although, Dreiding force field results in a reasonable estimate for the interfacial fracture energy, it does not show the correct mechanism of adhesion between polymer and graphene. The results show


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FIG. 5. Left: The amplitude of graphene during the curing process. Right: the amplitude in different temperatures.

TABLE I. The initial and the final equilibrated simulation cell parameters. Length a b c

˚) Initial length (A

˚) Final lengths (A

83.1 24.0 69.8

74.4 21.5 62.5

that Compass force fields leads to a better description of the adhesion between polymer and organics. This is due to the fact that Compass is a potential derived for systems with condensed phase and high changes in densities. Also, in Compass force field, non-bonded terms are reparametrized, so this potential models non-bonded interaction more accurately.17 Hence, non-bonded energy terms, which have the largest contribution to the interaction energy in studying the adhesion phenomenon, are estimated more precisely. Computing the interface toughness is essential in designing tough composites. Here, the assumption is that the interfacial fracture energy is positively correlated with work of adhesion at nanoscale. Work of adhesion, estimated by the interaction energy per unit area, describes how strong the

adhesion between two materials is. Hence, in order to estimate the work of adhesion for the desired interface, the interaction energy should be computed. The total energy of the system is calculated by summing the energy of each layer, the polymer and the graphene. The total energy consists of three main parts: potential energy, Epotential , non-bonded energy, Enonbonded , and kinetic energy, Ekinetic ET ¼ Epotential þ Enonbonded þ Ekinetic :

(4)

Generally, the governing part in the energy terms is the potential part, which is related to the stretch ratios, bending angels, torsional (dihedral) angles, inversions, and hydrogen bonds. However, the non-bonded interactions play the main role in the adhesion of the interfaces. The non-bonded interactions consist of two parts, Van der Waals enrgy, EVdW and electrostatic energy, Eelec , Enonbonded ¼ EVdW þ Eelec :

(5)

Finally, the kinetic energy is referred to the vibrational behavior of atoms and their velocities. This term does not play an important role in solids, especially amorphous solids

FIG. 6. Left: The computed total energy and the energy of the graphene and epoxy atoms. Right: the computed interaction energy between epoxy/graphene using Eq. (6).


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TABLE II. Energy values for the parts present in the final equilibrated system. Energy terms

Energy (kCal/mol) 87.8 103 76.3 103 12.3 103 0.9 103

Total energy Polymer energy Graphene energy Interaction energy

polymer and graphene are also verified using the suggested framework. The results of mechanical properties of materials and the interfacial energies are in good agreement with available experimental data. The calculated work of adhesion shows a weak interface between graphene and DGEBA/AEP. This suggests that non-bonded interactions, especially Van der Waals forces and polymer chain wrapping, do not contribute significantly to the interfacial fracture toughness. 1

like polymers and organics. The total energy in the graphene/epoxy system is calculated as the sum of the energies of the adhesive and graphene sheet Einteraction ¼ ET ðEgraphene þ Eepoxy Þ:

(6)

Therefore, in order to calculate the interaction energy, first, the energy of the whole system was computed. The interaction energies were then computed by subtracting the summation of the energies of each layer, which was computed separately, Egraphene and Eepoxy , from the total energy. The variations of these energies as a function of simulation steps are presented the Figure 6. Table II also shows the final values of the energies, after the dynamics simulations. The negative sign of the interaction energy clearly states that the two layers are adhering to each other. Finally, the work of adhesion, Wadhesion , was calculated by dividing the interaction energy by the contact area. Wadhesion ¼ Einteraction =A:

(7)

The calculated interfacial fracture energy is about 0.203 J/m2. The results are in agreement with the experimental results.29 Ganesan et al.29 have experimentally measured the interfacial fracture toughness between the Multi-Walled carbon Nano-Tube (MWNT) and similar epoxies. They have reported the interfacial fracture energy between MWNT and Epon828 interface to be in the range of 0.05–0.25 J/m2. Atomic force microscopy method can also be used to further verify the computed interfacial fracture energy using contact mechanics models.30 IV. CONCLUSIONS

In this study, the interfacial fracture energy between crosslinked DGEBA and AEP polymers and graphene is studied. A novel approach to crosslinking is suggested and Compass and Dreiding force fields are examined. It is shown that the Compass force field models the crosslinking process more accurately than Dreiding force field. Mechanical properties of

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