Interface mechanical properties of graphene

Interface mechanical properties of graphene reinforced copper nanocomposites Ke Duan, Li Li∗, Yujin Hu∗, Xuelin Wang State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Abstract The interface load transfer ability between graphene (Gr) and metal-matrix materials is often the limiting factor in determining the mechanical property of Gr reinforced metalmatrix nanocomposites. Herein, the interfacial properties of graphene reinforce copper nanocomposite (Gr/CNc) are investigated by performing a series of pull-out simulations using molecular dynamics simulation method, considering the effects of Gr dimensions and aggregations. It is found that the interface strength of Gr/CNc is stronger than carbon nanotube reinforced Cu nanocomposite (CNT/CNc) owing to the two dimensional platelet structure of Gr. In particular, we found that an appropriate formulae can be developed to predict the interfacial shear strength for an arbitrary dimensions single layer Gr/CNc system. However, the interface load transfer ability of Gr/CNc reduces significantly if Gr aggregation phenomenon happens. Keywords: Nanonanocomposite, Graphene, Molecular dynamics simulation, Interfacial shear strength

1. Introduction Metal-matrix nanocomposites have received extensively interests for its great potential applications in a variety of fields, such as aerospace industry, electronic packages, and thermal management materials [1–6]. It is well known that the performance of reinforcement material plays a key role on the property of nanocomposites [7]. Several kinds of carbon allotropes and nanostructures: nanofiber, fullerene, carbon nanotube (CNT), and graphene (Gr) are expected to be the promising reinforcements. In particular, carbon nanotube and graphene are receiving the significant attention in the past decades as a novel reinforcement material owing to their superior mechanical properties, good electrical conductivity and exceptional thermal properties [8–20]. ∗

Corresponding author Email addresses: [email protected] (Li Li ), [email protected] (Yujin Hu ), [email protected] (Xuelin Wang) Preprint submitted to Materials Research Express

October 24, 2017

Until recently, carbon nanotube has been the dominant carbon reinforcement material for metal matrix nanocomposites [3–5, 21–25]. For instance, the CNT-reinforced aluminum nanocomposites exhibit high strength and high strain-hardening ability and significant improved the mechanical properties of aluminum matrix [3, 4]. The mechanical strength of CNT/Cu nanocomposite (CNT/CNc) is found to be more than three times greater than that of pure copper [23]. However, the applications of CNT reinforced metal composites are hampered by the challenges such as high fabrication cost and severe CNT agglomerations during fabrication process [26, 27]. Moreover, the entanglement phenomenon due to the tubular structure and high aspect ratio of CNT sometimes results in rough and porous nanocomposites. Compared with carbon nanotube, graphene can be prepared in large quantities by inexpensive methods such as electrochemical and mechanical exfoliation of graphite [28–31]. Besides, graphene may induce much more stronger interfacial bonding with the matrix materials than CNT for its higher surface area [32]. Rafiee et al. show that the mechanical properties of polymer matrix reinforced by a low content of graphene are better over those with CNTs [33]. Enhancement in the thermal and electrical properties of polymer based composites reinforced by a small amounts of graphene has been also reported by other researchers [34–36]. For the graphene/metal nanocomposites, the successful studies in this area are very limited [37–49]. For the particular case of Gr/Cu nanocomposite (Gr/CNc), the elastic modulus and yield strength of Gr/CNc containing 2.5 vol% graphene-oxide obtained by Hwang et al. are 1.3 and 1.8 times higher than those of pure Cu [44]. Moreover, several study also found an increment around 39% in hardness of Gr/Cu composites compare to its pure copper counterpart [45, 46]. Even so, the mechanical improvement of Gr/metal composites is still below the theoretical value [50]. This deviation is mainly caused by the insufficient interfacial bonding and aggregation tendency of graphene due to the large surface areas. Undoubtedly, understanding the interface behavior between graphene and metal matrix is of great significance for the fabricating of high-quality Gr-reinforced metal nanocomposites. However, to our knowledge, surprisingly very little research involved in this topic due to the fact that it is extremely difficult, time-consuming, and expensive to investigate the graphene-metal interface behavior with experiments. Molecular dynamics (MD) simulation method, which is capable of obtaining the unmeasurable microscopic details, has been considered to be the most natural and powerful tool for nanoscale analysis [51–55]. In this work, we attempt to investigate the interface behavior of a particular case: Gr/CNc using MD. First of all, the pull-out process of a Gr and SWCNT from Cu matrix was carried out, aiming at comparing the interfacial strength between Gr/CNc and CNT/CNc. Afterwards, the effects of Gr dimensions including the Gr length and width on the interfacial shear strength are determined. At last, the influence of Gr aggregations to the interface load transfer ability have been explored and accounted for. 2. Computational methods 2.1. Simulation models In this study, the studied Gr/CNc is composed of copper matrix and a single layer graphene, as shown in Figure 1. The graphene is located at the center of copper matrix. 2

The lattice direction along x coordinate axes is [100] for the Cu matrix. The length of Cu matrix and Gr was kept the same, whereas the width of Cu matrix was set as two times that of Gr, and the height of Cu matrix was 4 nm, as shown in Figure 1 (a). As depicted in Figure 1 (b), the initial spacing gap between Cu matrix and Gr was set to be 0.35 nm [56]. We have adopted the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code to perform the pull-out process of Gr from Cu-matrix. Our simulation consists of the following steps:

Cu matrix

H

0.35 nm

a) Equilibration: At the beginning, total potential energy of the simulation structure was minimized using a conjugate gradient algorithm by adjusting their positions. Then, the simulation system was equilibrated using the NVT ensemble [where the number of molecules (N), volume (V), and temperature (T) are kept constant] and Nose-Hoover thermostat for 200 ps at 1 K temperature. The famous Velocity-Verlet algorithm was employed to update the position and velocity of the atoms of simulation system. The length of each time step was 0.5 fs. Periodic boundary conditions were applied to the y and z directions while the free boundary condition was imposed to the longitudinal direction (x direction). b) Pull-out: Subsequent to the equilibration, discrete displacement was applied on the Gr atoms using a displacement controlled method. Three layers of carbon atoms at the right end of the Gr were subjected to discrete displacements of 0.01 nm along the x direction until the Gr was completely pulled out from Cu matrix. The three top and bottom atomic layers of Cu atoms were fixed during the whole pull-out process. After each discrete displacement loading step, the simulation system was then equilibrated using NVT ensemble for 1 ps to ensure that the system has reached a new stable state. Then the total energy and pull-out force at the new equilibration state was extracted for analysing the interfacial behavior of the simulation structure.

Interface Graphene Section perimeter

z y

L

x

(a)

Cu

C

(b)

(c)

Figure 1: Atomic model of Gr/Cu nanocomposite (a) general view, (b) section view, (c) schematic of the interface

2.2. Potentials The adaptive Intermolecular Reactive Empirical Bond order (AIREBO) potential, which is based on the second-generation many-body inter-atomic Brenner potential (REBO), was used to describe the interactions between C-C atoms within the Gr [57]. This potential was 3

extensively utilized in previous studies of simulating the mechanical properties of carbon nanostructures, especially for carbon nanotubes and graphene [58]. The potential of Gr can be expressed as: " # XX 1 XX T orsion E= EijREBO + EijLJ + Ekijl (1) 2 i j6=i k6=i j6=i,k where EijREBO is the hydrocarbon REBO potential developed by Brenner et al.; EijLJ is long range interactions using a form similar to the standard Lennard-Jones (LJ) potential; T orsion term is an explicit 4-body potential that describes various dihedral angle preferences. Ekijl Additionally, the EAM (an embedded atom method) developed by Daw and Baskes was adopted to model the Cu-Cu interactions [59, 60]. The total energy of an atom i is given by: ! X 1X φαβ (rif ) (2) Ei = Fα ρβ(rij ) + 2 j6=i j6=i where Fα denotes the embedding energy, which is a function of the atomic electron density ρ. φ is a pair potential interaction, α and β are the element types of atoms i and j, respectively. The two summations in the equation (2) are over all neighbors i and j within the cutoff distance. Due to the weak interaction between graphene and Cu matrix, the Lennard-Jones (LJ) 12-6 potential was used to model the interaction. This potential well defined the weak van der Waals interactions for carbon-metal systems, and the expression can be given as: h 6 i 12 − σr r < rc (3) ELJ = 4ε σr where ε and σ are the coefficients of the well-depth energy and the equilibrium distance, respectively. rc is the cutoff distance and was set to 1 nm in this study. The LJ parameters are ε=0.02578 eV and σ=0.30825 nm for C-Cu [61]. 3. Results and discussion 3.1. Comparison of Gr and CNT copper nanocomposites To characterize the interface performance of Gr/CNc, the total energy and pull-out force of the simulation system were extracted after every discrete displacement loading step. Figure 2 presents the total energy and pull-out force for Gr/CNc with Gr dimensions of 3.94 × 2 nm2 . It can be observed that the pull-out force (black hollow cycle) can be divided into three stages. The pull-out force increases in the first stage and decreases in the third stage. The length of the first and third stage is about 1.0 nm, which is nearly the same as the cutoff distance of van der Waals (vdW) interaction. In the remaining middle part, the pull-out force oscillates around a stable value. From Figure 2, we can also note that the total energy (red line) of the simulation system increases gradually as the Gr is being pulled 4

out from the Cu matrix. The increase of total energy during the complete pull-out process is equal to the total work done by vdW forces. A two-dimensional schematic of the pull-out situation is presented in Figure 3 (a) to account for the obtained pull-out force and total energy curves. It should be mentioned that only vdW interactions exist between the Gr-Cu interface. The van der Waals interaction between Gr and Cu matrix can divided into three domains. In the left domain (domain I), the cumulative resultant vdW force is nearly normal to the x direction and has no contribution to the counteraction of the applied pull-out force (Fpull ). However, Gr atoms within the domain II have a larger x direction component of the cumulative force. In the domain III, there are no vdW interactions exist because the distance between carbon and Cu atoms are beyond the cutoff distance of the used LJ potential. Thus, only the domain II is responsible for counteracting the pull-out force. Based on the above explanations, the obtained trend of pull-out force presented in Figure 2 can be well understood. In the first stage of pull-out process, more and more carbon atoms enter the section II (Figure 3 (a)) and cause an increase of the pull-out force. In the middle stage of pull-out process, the carbon atoms within the domain II remain the same so that the pull-out force do not vary during pull-out process. The oscillation of the pull-out force is because the repetitive breaking and reforming of vdW interactions during pull-out process. In the third stage of pull-out process, the carbon atoms within the domain II decrease and cause a drop of the pull-out force. 6 .0

S ta g e 1

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-3 0 4 4 0

0 .5 -3 0 4 6 0

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2

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5

P u ll-o u t d is p la c e m e n t (n m )

Figure 2: Total energy and pull-out force as a function of pull-out distance for the case of Gr/CNc.

Dividing the maximum pull-out force by initial interface area has been commonly utilized to determine the maximum interfacial shear strength (τ ) [62]. However, in this work, an average pull-out force was used in calculating τ due to the reason that the oscillating phenomenon of the pull-out force in the middle stage may cause a larger error if we simply use the maximum value of the pull-out force curve [25]. The expression of computing the

5

Ⅰ

Ⅰ

(a)

Ⅲ

Ⅱ

(b)

Ⅱ

Ⅲ

vdW Interactions

vdW Interactions C

C

Fpull

Cu Z

Fpull

CNT

Cu

Cumulative Forces Cumulative Forces

X

Figure 3: A two-dimensional schematic of the cumulative resultant vdW forces. (a) Gr/CNc. (b) CNT/CNc. Domain I: The cumulative forces are almost normal to the x direction. Domain II: The resultant cumulative forces counterbalance Fpull . Domain III: No vdW interactions exist due to the distance is beyond the cutoff distance. -3 9 7 3 0 2 .5

-3 9 7 4 0 -3 9 7 5 0 P u ll-o u t fo r c e T o ta l e n e r g y

-3 9 7 7 0 -3 9 7 8 0

1 .0

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-3 9 8 0 0 -3 9 8 1 0

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Figure 4: Total energy and pull-out force as a function of pull-out distance for the case of CNT/CNc.

maximum interfacial strength is given as:

τ=

Fpm Fpm = A cL

(4)

where Fpm denotes the maximum pull-out force during pull-out process, A is the initial interfacial area and can be computed as A = cL, c and L are the section perimeter and length of the reinforcement, respectively. The value of Fpm is obtained as (Fpm = ∆ELtotal ). Using equation (4), an interfacial shear strength 312 MPa was obtained for the Gr/CNc. As comparison we also conducted pull-out simulation for a (5, 5) CNT/CNc. The reason for the selection of (5, 5) CNT is that the volume fraction and interface area of (5, 5) CNT is nearly the same as the used graphene. Figure 4 shows the pull-out force and total energy as a function of increasing pull-out displacement for the CNT/CNc. It can be noted that the increase of total energy is about 72 eV, which is smaller than the change in total energy of Gr/CNc (120 eV). Besides, the maximum pull-out force (2.3 nN) for CNT/CNc is 6

much less than Gr/CNc (4.9 nN). To account for the difference in pull-out force of CNT/Cu and Gr/CNc, a schematic of the interaction between Cu matrix and CNT is presented in 3 (b). Three distinct domains in Figure 3 (b) are similar to the situation of Figure 3 (a) and therefore no more description is given here. Compared with the interface interaction of Gr/CNc (Figure 3 (a)), it is clear that less vdW interaction is formed for every carbon atoms due to the hollow structure of CNT. Thus, the needed maximum pull-out force is smaller. Furthermore, the calculated interfacial shear strength of CNT/CNc is 238.6 MPa, which is less than the obtained value (312 MPa) for Gr/CNc. Therefore, we can conclude that the interfacial performance of Gr/CNc is better than that of CNT/CNc. 3.2. Effect of Gr length and width Investigating the effect of Gr parameters on properties of nano-materials is another one of the advantages of MD simulations compared with the experimental method due to the extremely difficulty in controlling the dimensions of Gr during the fabricating process. In this part, we first examined the influence of Gr lengths on the interfacial shear strength using four different lengths; 3.94 nm, 5.90 nm, 7.87 nm, and 9.84 nm with a width of 2 nm. Figure 5 shows the increase in total energy with the increasing pull-out displacement of the Gr/CNc and the inset plots the pull-out force. Several noteworthy points can be concluded from Figure 5. First of all, the total energy of the simulation system increases linearly with the increasing pull-out distance. Furthermore, the four ∆Etot curves nearly have the identical slope value (the slope of ∆Etot curve represents Fpm ). This phenomenon indicates that increasing the length of Gr has no effect on the maximum pull-out force but only extends the middle stage of the pull-out profile, which can be clearly seen from the inset in Figure 5. The reason for the length-independence pull-out force is because the domain I and III (Figure 3 (a)) have no contribution to the counteraction of pull-out force. Only the domain II (Figure 3 (a)) is responsible for the counterbalance of pull-out force. Increase the Gr length has no effect on the number of carbon atoms within the domain II, and therefore has no contribution to the pull-out force. The obtained interfacial shear strength of Gr/CNc with different Gr length is shown in Figure 6. It can be observed that τ ) due to the reason that pull-out force is inversely proportional to the length of Gr (τ = 1238 L is independent with Gr length. After the influence of Gr lengths are illuminated, the effects of Gr widths on the interfacial shear strength are also explored and discussed. Four different widths were chosen, i.e., 2 nm, 4.12 nm, 5.82 nm and 10.0 nm. The length of these monolayer Grs is 3.94 nm. Figure 7 shows the change in total energy (Etot ) of Gr/CNc with different Gr width. We can note in Figure 7 that the slope of Etot curves increases linearly with the increasing widths. This trend is also reflected by the enlarging phenomenon of pull-out force shown in the inset of Figure 7. Different from the influence of Gr length, the numbers of carbon atoms within the domain II increase with the increase in Gr width, resulting in larger pull-out force needed to counterbalance the cumulative vdW forces. Similarly, the effect of Gr width on the interfacial shear strength is also different from that of Gr length. It can be seen from Figure 8 that τ converge to a certain value instead of zero if the Gr width increases to very large value. 7

3 2 0 2 8 0

G r e m b e d d e d le n g th 3 .9 4 n m 7 .8 7 n m

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Figure 5: Total energy profiles for Gr/CNc with different embedded lengths. The inset shows the variation of pull-out force during the pull-out process.

I n te r fa c ia l s h e a r s tr e n g th (M P a )

3 2 0

M D d a ta F ittiin g c u rv e

2 8 0 2 4 0

τ=1238/L 2 0 0 1 6 0 1 2 0 4

6

8

1 0

G r le n g th (n m )

Figure 6: Effects of Gr length on the interfacial shear strength of Gr/CNc.

According to the previous explanation for the interface, it can be known that maximum pull-out force (Fpm ) used to calculate the interfacial shear strength is related to the width of Gr and approximately has a linear relationship (Fpm = a + kw). Therefore, from equation (4), the expression for τ can be written as (5). It is easy to see that τ converges to the first term of equation (5) when Gr width increased to a very large value.

τ=

Fpm a + kw k a = = + cL 2wL 2L 2wL

(5) 8

where a and k are the constant coefficients, c is the section perimeter of Gr and is considered as c = 2w in this study, w and L are the width and length of Gr, respectively. Using the obtained results for monolayer Gr/CNc under different Gr width, we concluded a specific relationship between τ and Gr dimensions as: 1 τ= L

199.6 1130.4 + w

(6)

This formula allows one to get the value of τ for a single layer Gr/CNc system that is too large to model and compute due to involved number of degrees of freedom. For instance, according to the formula, interfaical shear strength is about 11.5 MPa for Gr/CNc system with Gr dimensions of 100 × 100 nm2 . 6 4 0 5 6 0

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th .0 .1 2 .8 2 0 .0

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Figure 7: Total energy profiles for Gr/CNc with varies embedded widths. The inset shows the variation of pull-out force during the pull-out process.

3.3. Influence of stacked Gr structure In the previous parts, the interfacial property of single layer Gr/CNc was explored and discussed. However, it is noteworthy that the aggregation of Gr was commonly observed in fabricating process of the Gr/metal nanocomposites. Therefore, it is of great significance to investigate the effects of stacked Gr structure on the interface load transfer capability. For the multilayer Gr/CNc, there are many kinds of possible pull-out situations among an arbitrary pull-out process. In this study, a three layers Gr/CNc system was considered to explore the influence of Gr aggregations on the purpose of reducing simulation costs. The length and width for each piece of Gr are 3.94 nm and 2 nm, respectively. Four typical situations were studied, i.e., Situation 1: only pull-out the middle Gr (Figure 9 (a)); Situation 2: only pull-out the upper or bottom Gr (Figure 9 (b)); Situation 3: simultaneous 9

3 1 5

I n te r fa c ia l s h e a r s tr e n g th (M P a )

M D d a ta F ittin g c u rv e 3 1 0

3 0 5

τ= 2 8 6 . 9 + 5 0 . 6 5 / w

3 0 0

2 9 5

2 9 0 2

4

6

8

1 0

G r w id th (n m )

Figure 8: Effects of Gr width on the interfacial shear strength of Gr/CNc.

pull-out the two adjacent Gr (Figure 9 (c)); Situation 4: pull-out all Grs at the same time (Figure 9 (d)). Figure 10 presents the ∆Eint and pull force curves for these four different situations. From the ∆Eint curves, we can firstly note that the increase in total energy of the Gr/CNc become larger when much more Gr layers were pulled out. Meanwhile, a larger pull-out force is observed with the increasing pull-out Gr layers as depicted in the small figure embedded in Figure 10. However, there is an interesting phenomenon that the curves for situation 2 are almost coincides with that of situation 3, which indicates that the middle layer Gr almost has no effect on the increase in total energy of simulation system. To account for the obtained simulation results, a two-dimensional schematic of the interface interactions between stacked Grs amd Cu matrix for situation 2 has been depicted in Figure 11. The vdW interaction is related to the distance of two atoms and the vdW force will be vanished if the distance is beyond the cutoff distance (1 nm) of the defined LJ potential. From Figure 11, one can found that the carbon atoms in the top layer Gr can not form any vdW interactions with the bottom Cu matrix due to their long distance (1.03 nm). Similarly, the carbon atoms in the bottom layer Gr also can not form any vdW interactions with the top Cu matrix. In regard to the middle layer Gr, the cumulative resultant forces along the positive x direction are mainly caused by the attraction from another two Gr layers. It should be mentioned that the attraction between the middle Gr layer and Cu matrix is very weak and can be negligible for their long distance. Therefore, the middle layer Gr is also pulled out for the situation 2. According to equation (4), the calculated interfacial shear strength for situation 1 to 4 are 99.8, 186.6, 186 and 300.48 MPa, respectively. At first glance one may conclude that the interfacial performance of the three layers Gr/CNc may as good as the single layer Gr/CNc for the reason that τ of situation 4 is 300.48 MPa, which is comparable to that of single layer Gr/CNc (312 MPa ). However, it should be noted that the mass fraction 10

of the graphene for the three layers Gr/CNc is nearly two times larger than that of single layer Gr/CNc. Besides, the interfacial shear strength of another three situations is much less than that of a single layer Gr/CNc. Therefore, we can conclude that the stacked graphene structure severely weaken the interface performance of Gr/CNc. This may accounted for the phenomenon that graphene aggregation greatly reduced the properties of Gr/CNc observed in many experimental researches [50, 63, 64]. (a)

(b)

pull

pull

GNR Cu matrix

(c)

(d)

pull

pull

z x

Figure 9: Schematic of the four studied pull-out situations.

s it s it s it s it

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Figure 10: Total energy profiles for different pull-out situations of the three layer Gr/CNc. The inset shows the variation of pull-out force during the pull-out process.

11

0.35 nm

vdW Interactions

Top Gr Fpull C 0.34 nm

Cu

Middle Gr Bottom Gr

Z Cumulative Forces

X

Figure 11: A two-dimensional schematic of the cumulative resultant vdW forces for the pull-out profile of multi-layer Gr/CNc.

4. Concluding remarks In this article, molecular dynamics (MD) simulations were performed to investigate the interfacial properties of graphene reinforced Cu nanocomposites. According to the obtained results, the concluding remarks can be drawn as follows: • Owing to the 2D platelet structure of Gr, the interfacial strength of Gr reinforced Cu nanocomposite (Gr/CNc) is much better than that of CNT reinforced Cu nanocomposite (CNT/CNc). • For single layer Gr/CNc, there is a specific relationships between interfacial shear strength and the Gr dimensions, which allows one to predict the interfacial shear strength for an arbitrary dimensions single layer Gr/CNc system. • The stacked structure of multilayer Gr greatly weaken the interface strength of multilayer Gr/CNc, which indicates that the Gr aggregation has a negative effect on the interface load transfer. These simulation results are very helpful for clarifying the enhancement mechanism of graphene in the mechanical properties of graphene-reinforced metal-matrix nanocomposites and providing useful guidance for designing graphene reinforced materials with excellent performance. Acknowledgments This work is supported by the National Natural Science Foundation of China (Grant Nos. 51375184 and 51605172), the Natural Science Foundation of Hubei Province (Grant No. 2016CFB191) and the Fundamental Research Funds for the Central Universities (Grant No. 2015MS014).

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