Journal of the Korean Physical Society, Vol. 72, No. 6, March 2018, pp. 681∼686
Size Effect of Defects on the Mechanical Properties of Graphene Youngho Park and Sangil Hyun∗ Simulation Team, Korea Institute of Ceramic Engineering and Technology, Jinju 52851, Korea (Received 21 September 2017, in final form 8 November 2017) Graphene, a two-dimensional material, has been studied and utilized for its excellent material properties. In reality, achieving a pure single-crystalline structure in graphene is difficult, so usually graphene may have various types of defects in it. Vacancies, Stone-Wales defects, and grain boundaries can drastically change the material properties of graphene. Graphene with vacancy defects has been of interest because it is a two-dimensional analogy of three-dimensional porous materials. It has efficient material properties, and can function as a part of modern devices. The mechanical properties have been studied by using molecular dynamics for either a single vacancy defect with various sizes or multiple vacancy defects with same defect ratios. However, it is not clear which one has more influence on the mechanical properties between the size of the defects and the defect ratio. Therefore, we investigated the hole-size effect on the mechanical properties of single-crystalline graphene at various defect ratios. A void defect with large size can have a rather high tensile modulus with a low fracture strain compared to a void defect with small size. We numerically found that the tensile properties of scattered single vacancies is similar to that of amorphous graphene. We suspect that this is due to the local orbital change of the carbon atoms near the boundary of the void defects, so-called the interfacial phase. PACS numbers: 61.48.Gh, 61.72.−y, 62.25.−g, 81.40.Jj Keywords: Graphene, Hole defects, Molecular dynamics, Mechanical properties DOI: 10.3938/jkps.72.681
I. INTRODUCTION
Carbon materials are well-known for their superior material properties, such as mechanical, electrical and thermal properties [1]. They can exist in the form of various nanomaterials for all dimensionalities from quasi-zero dimension to three dimensions, that is, fullerene (0D), CNT(1D), graphene(2D) and diamond(3D). Among these, graphene has attracted much interest as a functional material for its 2D nature [2]. In general, synthesizing pure single-crystalline graphene is difficult because graphene usually has various types of defects, such as the vacancies, the topological defects of the Stone-Wales (SW) type, and the line defects such as grain boundaries observable in polycrystalline materials [3]. Although those defects degrade the superior material properties of single-crystalline graphene, graphene with defects can still be applied to new functional devices, if the formation mechanisms of these defects and the correlations of the characteristics of the defects can be fully understood [4]. The mechanical properties of single-crystalline graphene have been numerically studied in pervious works [5, 6], and later the mechanical properties of ∗ E-mail:
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pISSN:0374-4884/eISSN:1976-8524
graphene with defects [7] along with its thermal properties [8, 9] were also investigated by using molecular dynamics (MD) calculations. As the number of vacancies or SW defects increases, Young’s modulus decreases [8] and the fracture strength decreases [9]. Many MD calculations also have been performed on only vacancy defects [10–14]. For two vacancies separated beyond a certain correlation distance, an increase in the distance enhances mechanical properties such as Young’ modulus and the fracture strain [10]. The tensile strength and Young’s modulus decrease as the size of the vacancy defect increases from a single to a double and a sextuple or as the density of vacancies, that is, the defect ratio, increases [11]. On the other hand, the fracture strain behaves differently; it decreases up to a certain value after which it increases [12,13]. In reality, atomic defects can exist as holes (circular voids) with certain finite sizes rather than atomic point vacancies [14]. They are usually generated during the fabrication process and can be formed by electron beam irradiation, for example. Because both the defect size and the defect ratio affect the mechanical properties of graphene with defects, the two geometrical factors, defect size and defect ratio that play roles in determining the characteristics of the materials must be analyzed. For example, if the defect ratio for graphene is fixed, the defect can be a large single hole defect or separated
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©2018 The Korean Physical Society
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Journal of the Korean Physical Society, Vol. 72, No. 6, March 2018
(a)
Fig. 1. (Color online) Modeling of single-crystalline graphene with hole defects with a radius 0.51 nm.
single vacancies or somewhere in between. In all these cases, graphene can have different mechanical properties. Therefore, a systematic study on the effect of the size of the defect at various fixed defect ratios is needed on the properties of graphene. In this study, we investigated mechanical properties, such as Young’s modulus, the fracture strain and ultimate tensile strength (UTS), of graphene with hole defects. First, we fixed the defect ratio and used various random models with various hole defect sizes. We performed MD calculations at room temperature to calculate the mechanical properties. We also compared our results with an analytic prediction based on the homogenization theory for Young’s modulus, which is known to be less strain-rate dependent and more robust under statistical geometric variations.
II. SIMULATION METHODS For modeling single-crystalline graphene with hole defects, we calculated the number of holes for a given defect ratio, that is, the number of atoms removed from the original single-crystalline model, and for holes with certain radii. First, we make a model of pure singlecrystalline graphene for a given size of atomic model. One size of the model is 21 nm × 25 nm with 20,000 carbon atoms. Then, we randomly choose the center positions of the holes and remove atoms within a distance equal to the radius from the center points. We then selected the values of the radius with an equal distance on a logarithmic scale. For statistical stability, we set the distance between the two holes to be more than twice the
(b)
Fig. 2. (Color online) Atomic configurations of singlecrystalline graphene with a hole defect ratio of 10% near the moment of fracture under tensile deformation: (a) a hole defect radius of 4 nm at a tensile strain of 16% and (b) a hole defect radius of 0.08 nm at a tensile strain of 35%.
radius of the holes. We show the case of 64 hole defects, each with a radius of 0.51 nm, in Fig. 1. In the inset figure, the model is an armchair-type model along the x-direction. For our study, we conducted molecular dynamics calculations by using a large-scale atomic/molecular massively parallel simulator (LAMMPS) [15]. We applied tensile deformations on the models in the x-direction at a strain-rate 109 sec−1 and with strains up to 1.3. We used the NPT (isothermal-isobaric) ensemble at a fixed temperature, T = 300 K, and we used the adaptive intermolecular reactive bond order (AIREBO) potential, which can describe carbon interactions more accurately than the other empirical potentials, such as the Tersoff type potentials, as the interatomic potential between the carbon atoms. For defect ratios of 5 ∼ 15%, the radius of the hole defect was in the range of 0.08 ∼ 5 nm. To ob-
Size Effect of Defects on the Mechanical Properties of Graphene – Youngho Park and Sangil Hyun
Fig. 3. (Color online) Stress-strain curves of graphene with defects of holes with various radii (or numbers of holes, Nh ) for a defect ratio of 10%.
tain the stress-strain curves of graphene under tension, we take the average stress over all atoms in the atomic models at each strain value.
III. RESULTS 1. Tensile Deformations
Figure 2 shows the atomic configuration near the fracture under the tensile deformation of single-crystalline graphene with a hole defects ratio of 10%: (a) a hole radius of 4 nm at a tensile strain of 16% and (b) a hole radius of 0.08 nm at a tensile strain of 35%. Here, the inset figure for each case shows the initial state (zero strain). The former case, for which initially a single large hole exists, clearly shows a straight crack path through the hole, as shown in Fig. 2(a). No other part is damaged; this is a simple brittle case just as the pristine graphene without any defect. The latter case in Fig. 2(b), in which initially many small holes exist, that is, single vacancies that are often neighbored, shows some thin connected paths of the atoms just before the fracture. This looks similar to a ductile case where the fracture strain is larger than it is for the brittle case. Figure 3 shows stress-strain curves for the tensile deformation of single-crystalline graphene with hole defects of various radii for a defect ratio of 10%. Instead of the hole radius, we used in this case the number of holes, Nh . In most cases, the stress-strain curves have the same forms as they do in the case of a single crystal. However, quantities, such as Young’s modulus, are found to be different. For the smallest hole-defect cases, that is, single atom vacancies, the form of the stress-strain curve is
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Fig. 4. (Color online) Young’s modulus vs. the radius of the hole defects, for various defect ratios. The dotted line shows Young’s modulus of single-crystalline graphene.
more like that for the case of amorphous graphene than that for the single-crystal case. As Nh increases or the defect radius decreases, the Young’s modulus decreases while the fracture strain increases, except for Nh = 4. The UTS values fluctuate and become larger around Nh = 64 ∼ 256. However, all these quantities seem to have statistical deviations, and because we selected one sample of the stress-strain curve for each case, we performed statistical calculations by taking a few samples for each case. Therefore, we performed all calculations for five random samples.
2. Young’s Modulus, Fracture Strain, and UTS
Young’s modulus vs. hole defect radius for various defect ratios are shown in Fig. 4. As the hole defect radius increases, the Young’s modulus increases for all defect ratios. Here, we calculated the Young’s modulus as the slope in the strain range where the slope of the curve is steepest. That is, the Young’s modulus was calculated mostly in the strain range of 0.0 ∼ 0.05 except for a few small-radius cases where the Young’s modulus was calculated in the strain range of 0.05 ∼ 0.10. As the defect ratio increases, the Young’s modulus decreases, which seems obvious considering the change from perfect single-crystalline graphene (dotted line) to graphene with many defects with much a lower atomic density per unit area. Figure 5 shows the curves of fracture strain vs. hole defect radius for various defect ratios. As the hole-defect radius increases, the fracture strain decreases for all defect ratios. For radii larger than 1.5 nm, the fracture
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Journal of the Korean Physical Society, Vol. 72, No. 6, March 2018
Fig. 5. (Color online) Fracture strain vs. the radius of the hole defects, for various defect ratios. The dotted line shows the fracture strain of single-crystalline graphene.
Fig. 6. (Color online) UTS vs. the radius of the hole defects, for various defect ratios. The dotted line shows the UTS of single-crystalline graphene.
the Young’s modulus case in Fig. 4. strain saturates to the same value for all defect ratios. This may be explained by the fact that the fracture strain is determined by the largest defect. That is, the defects larger than a certain size can become seed points initiating a crack path. Interestingly for small defect radius, the fracture strain is larger for higher defect ratio. This may be because carbon orbitals transitions from sp2 to sp3 bond rotations occur in this case. In this limit of small radius of the hole defect, the fracture strain increases because the material behaves as if it were amorphous. We conjecture that this amorphous property is enhanced as the defect ratio becomes higher. In Fig. 5, we present the fracture strain of single-crystalline graphene as a dotted line. The fracture strain for a small defect ratio of 5% (close to single crystalline) is shown to approach the single-crystalline limit (dotted line) in the limit of zero defect radius. However, for higher defect ratio (e.g., 10% and 15%), the fracture strain for a small defect radius can exceed that of a single-crystalline sample due to the enhanced amorphous property. The ultimate tensile strength in Fig. 6 stays around a similar value except for the region with hole defects of small radius (amorphous-like region) and the region with hole defects of large radius. In the amorphous-like region, for the hole defects with a small radius, this represents an increase in the amorphous property. In the region with hole defects of large radius, the UTS is higher for the larger radius because the stress vs. strain curve has the same form as that for pristine graphene and the Young’s modulus is higher for the larger radius while the fracture strain stays almost the same. As the defect ratio increases, the UTS decreases, and the curves at different defect ratios seem to be parallel to each other, just as in
IV. DISCUSSION Material properties such as stiffness depend on a part of the device such as the surface or the bulk [16]. Then, an effective property G of a composite material consisting of a bulk phase and a surface phase can be expressed by an average of the value of the surface phase GS and the value of the bulk phase GB : G=
GS + GB VB /VS G S V S + GB VB = . VS + VB 1 + VB /VS
(1)
Here, VS is the volume of the surface phase, and VB is the volume of the bulk phase of the composite material. If the material is in the form of a two-dimensional sheet, the volumes in the above equation are replaced by the areas, that is, VB /VS = AB /AS . Then Eq. (1) becomes G=
GS + GB AB /AS . 1 + AB /AS
(2)
In the current model of graphene sheets with defects, the surface phase is interpreted as atoms near the hole defects, and the bulk phase is interpreted as atoms elsewhere. For a given defect ratio, the surface part, including many sp2 -to-sp3 rotated bonds, is increased as the number of holes increases or the radius of a hole defects, r, decreases. That is, AS = 2πrtN,
(3)
where t is the constant width of a ring-shaped region around each defect shown in the inset of Fig. 1. The
Size Effect of Defects on the Mechanical Properties of Graphene – Youngho Park and Sangil Hyun
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(correspondingly defect ratio) of the void (defect) phase is zero, the gap between the upper bound and the lower bound becomes narrow and approaches zero. On the other limit (when the area fraction of the void is one, corresponding to a fully void phase), the gap between the bounds becomes narrow again. However, in the middle range of the area fraction of the void phase, the gap opens wide so that the effective properties depend highly on the microstructures of the composite materials. In the current case, the geometric distribution of the circular defects may contribute to the microstructural variations at higher defect ratios (e.g., 15%). We can observe more fluctuations around the theoretical curve for larger defect ratios. The theoretical prediction curves are seen to become quite off-set from the numerical results as the void phase increases, as shown in Fig. 7.
Fig. 7. (Color online) Young’s modulus for various defect ratios and the theoretically fitted curves. The dotted line shows Young’s modulus of single-crystalline graphene.
surface atoms belong to this ring-shaped region. N is the number of hole defects and N ∼ 1/r2 , hence, AS ∼ 1/r. Because for a small defect ratio, AB ∼ AB + AS and AB is almost a constant, AB /AS is proportional to r and can be expressed as Cr. Then the Eq. (2) becomes G=
GS + GB Cr . 1 + Cr
(4)
Young’s modulus is a robust quantity for describing a material and, unlike UTS and fracture strain, does not depend very much on testing conditions such as the strain rate. Figure 7 shows Young’s modulus vs. the radius of the hole defect and the fitted curves. All fitted curves have the form of Eq. (4) with different values for the fitting parameters GS , GB and C. Here, we have GS = 200, 100, 0(GP A), GB = 800, 700, 600(GP A) and C = 4, 2, 1 for defect ratios 5%, 10% and 15%, respectively. For the case of a hole defect ratio of 5%, the fitted curve and the numerical data agree well with each other. For the other cases, the data points are shown to fluctuate much around the fitted curves. We think that the theoretical predictions in the Eq. (4) are based on the assumption that the distances between two hole defects are large enough so that no geometric correlation exist between them. However, for higher defect ratios (10 ∼ 15%) in our models, that assumption is not true because the correlation between the hole defects is no longer negligible. In some cases, the hole defects can be observed even to overlap. Theoretically, predicted bounds of mechanical properties are known as Hashin-Shtrikman bounds for twophase composite materials [17]. We note that Eq. (1) is an upper bound on the Young’s modulus of graphene with hole defects. In the limit when the area fraction
V. CONCLUSION In this work, we numerically studied the dependence of the mechanical properties of graphene with defects on the various radii of the hole defects. We determined the Young’s modulus, fracture strain and UTS for various sizes of hole defects and various ratios of the hole defect over the domain. As the radius of the hole defect increases, the Young’s modulus was shown to increase and the fracture strain to decrease for fixed defect ratio. Therefore, the case of smaller-radius defects shows a lower Young’s modulus and a higher fracture strain. The case of the smallest-radius defects, that is, single vacancy defects shows a gradually increasing slope of its stress-strain curve, just as seen in amorphous graphene. The behavior of the Young’s modulus can be explained by the homogenization theory where the physical property of the entire system is described by a combination of two phases, atoms near hole defects and atoms elsewhere in the bulk phase. For a fixed radius of hole defects, as the defect ratio increases, the Young’s modulus and the UTS decrease, which represents a mechanical weakening as expected. On the other hand, the fracture strain shows different behaviors, and in the small-radius region, the fracture strain increases as the defect ratio increases, which represents an enhancement of the amorphous characteristics.
ACKNOWLEDGMENTS The authors acknowledge the financial support from the International Cooperative R&D Program of the Ministry of Trade, Industry and Energy (MOTIE) of Korea (grant number: N0001711).
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