Mechanical properties of penta-graphene, hydrogenated penta-graphene, and penta-CN2 sheets

Computational Materials Science 136 (2017) 181–190

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Mechanical properties of penta-graphene, hydrogenated penta-graphene, and penta-CN2 sheets Minh-Quy Le Department of Mechanics of Materials and Structures, School of Mechanical Engineering, Hanoi University of Science and Technology, No. 1, Dai Co Viet Road, Hanoi, Viet Nam

a r t i c l e

i n f o

Article history: Received 18 August 2016 Received in revised form 3 May 2017 Accepted 4 May 2017

Keywords: 2D material Hydrogenation Mechanical properties Molecular dynamics simulation Penta-graphene

a b s t r a c t The present work investigates through reactive molecular dynamics simulations the mechanical properties of penta-graphene (PG), hydrogenated PG (HPG), and penta-CN2 at 300, 500, 700, and 900 K. Results reveal the higher temperature, the easier and faster transition under tension of PG to a structure, which is similar to a defective graphene. When increasing temperature from 300 to 900 K, the Young’s modulus of PG reduces by 10%, but its yield stress (maximal tensile stress in the first stage before transition) and yield strain decrease considerably. The yield strain and yield in-plane stress of PG are about 10.75% and 3%; and 22.4 N/m and 7.9 N/m at 300 and 900 K, respectively. In the second stage with transition, the axial tensile stress increases, reaches a maximal value then decreases slowly. The maximal inplane stress of PG in the second stage falls within 17–20 N/m when temperature varies in the range of 300–900 K, and the strain at this maximal stress is 40.2, 35.0, 35.4 and 35.6% at 300, 500, 700, and 900 K, respectively. The mechanical properties of HPG and penta-CN2 sheets decrease slightly with an increase of temperature. Hydrogenation reduces the Young’s modulus of PG by 25–28%, but increases the maximal tensile stress because HPG does not undergo phase transformation under tension, while PG does. Among three examined sheets, the penta-CN2 sheet exhibits the highest Young’s modulus, but the lowest tensile strength and fracture strain. At 300 K, the two-dimensional (2D) Young’s modulus, 2D tensile strength and strain at tensile strength of HPG and penta-CN2 sheets are 210 and 314 N/m; 28.7 and 14.4 N/m; and 19.6 and 9.6%, respectively. Ó 2017 Elsevier B.V. All rights reserved.

1. Introduction Recently, penta-graphene (PG), a new two-dimensional (2D) carbon allotrope, which is entirely composed of carbon pentagons, was theoretically predicted [1]. In PG, carbon atoms are arranged in three planes. The middle plane contains sp3-hybridized carbon atoms (denoted as C1), and the upper and the lower planes are formed by sp2-hybridized carbon atoms (denoted as C2), see Fig. 1. While graphene is known as a zero band gap semimetal, PG is a semiconductor with a large band gap [1–4]. The band gap of few-layer PG can be tuned by stacking misalignment [5]. The large band gap of PG and the robust tunable of the band gap make it a potential candidate for optoelectronic and photovoltaic applications. PG has been theoretically predicted to be a promising anode material in Li-ion and Na-ion batteries with high storage capacity and fast charge/discharge rate [6]. Further, hydrogenation changes significantly the mechanical, electronic [7] and thermal properties [8] of PG. Hydrogenation of PG changes the Poisson’s

E-mail address: [email protected] http://dx.doi.org/10.1016/j.commatsci.2017.05.004 0927-0256/Ó 2017 Elsevier B.V. All rights reserved.

ratio from negative to positive, turns the sheet from a semiconductor to an insulator [7], and leads to a significantly increase in thermal conductivity [8]. Both PG and hydrogenated penta-graphene (HPG) can withstand temperatures as high as 1000 K [1,7]. PG exhibits relatively low thermal conductivity [8–10] as compared to graphene. Following the penta-graphene, the prediction of penta-CN2 was also reported [11]. The atomic structure of penta-CN2 is similar to that of PG with the arrangement of N atoms at the place of C2 atoms (see Fig. 1a). Penta-CN2 exhibits larger band gap and higher Young’s modulus than those of PG, and is an insulator [11]. Therefore, beside graphene with superior thermal conductivity and zero band gap, PG, HPG and penta-CN2 provide additional features for more fascinating applications of carbon materials. Due to the very recent predictions of these three sheets (PG, HPG and penta-CN2), researches have just focused on their optoelectronic [2,4] and thermal [8,10,12] properties. In mechanical aspect, their elastic constants have been just predicted by DFT calculations [1,3,7,11]. Molecular dynamics (MD) simulations by Cranford [13] indicated in detail the transition of a free- standing,

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y x H

C2 C1

C1

a)

C2

b)

Fig. 1. Top and side views of the structures of: (a) PG (the atomic structure of penta-CN2 is similar to that of PG with the arrangement of N atoms at the place of C2 atoms); and (b) HPG. The squares marked by green dashed lines denote the unit cells. Red, blue and green spheres represent C1, C2 and H atoms, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

finite-size PG sheet under uniaxial tension at 300 K to a structure containing 5, 6, and 7-rings, which could be viewed as a defective graphene sheet. The broad applications of pentagonal sheets need a comprehension of their mechanical properties up to rupture at various temperatures. Hence, several issues should be further investigated as follows: first, the mechanical properties of HPG and penta-CN2 are explored and compared with those of PG; second, the effects of temperature on the mechanical properties of PG, HPG and penta-CN2 and on the transition process of PG under tension are considered; finally, their fracture mechanism is analyzed. The present work aims at addressing the above-mentioned issues within the context of reactive MD simulations at 300, 500, 700, and 900 K.

2. Computational method ReaxFF potentials have been used to model CAC [14], CAH [15], and CAN [16] interactions. MD simulations were carried out using LAMMPS code [17]. VMD package is used for visualization [18]. The standard Newton equations of motion are integrated in time using the Verlet algorithm [19] with a time step of 0.25 femto second (fs). Periodic boundary condition (PBC) is applied in the planar directions (x- and y-directions in this work) to remove the finite-length effect. Because single-layer PG and HPG are here investigated, a 50 Å vacuum along the thickness of the sheets (z-direction) was used by defining a fixed simulation box size. Thus, the stress is zero in the sheet thickness (z-direction). MD simulations were carried out at 300, 500, 700 and 900 K. The structures were relaxed to zero stress using the Nosé-Hoover barostat and thermostat (NPT) method for 5 pico second (ps). Because atomic arrangements are identical along the x- and y-directions for PG and HPG as shown in Fig. 1, the mechanical behavior along these 2 directions should be the same for each sheet. Therefore, uniaxial tensions were only carried out in the y-direction. In order to guarantee uniaxial stress conditions, zero stress condition in the edge parallel to the tensile direction was achieved by altering the size of simulation box in the direction perpendicular to the

tensile direction using the NPT method. A constant engineering strain rate of 2.5
108 s1 was applied in the tensile direction. Besides uniaxial tension, equi-biaxial tension was also performed for PG in canonical (NVT) ensemble at 300 K for a further examination of the transition of PG. A large (10
10) supercell is used. PG and penta-CN2 sheets contain 600 atoms, HPG sheet contains 1000 atoms. Additional relaxation under fixed boundary conditions (FPC) was also simulated to check the temperature-induced transition of PG. Macroscopic stress tensor is estimated by using the virial theorem [20–22]:

r¼

" # 1X 1 X ðabÞ ðabÞ : mðaÞ v ðaÞ  v ðaÞ þ r f V a2V 2 a–b

ð1Þ

mðaÞ and v ðaÞ are the mass and velocity vector of atom a, respectively. The symbol  denotes the tensor product of two vectors. rðaÞ denotes the position of atom a. rðabÞ ¼ rðbÞ  rðaÞ is the position vector of atom b relative to that of atom a, and f

ðabÞ

is the inter-

ðabÞ

ðabÞ

r ¼ @r@E atomic force exerted on atom a by atom b, where f ðabÞ : r ðabÞ , E is the potential energy of the atomic ensemble. V is the volume of the structure. V ¼ A:t, where A is the surface area of the sheet, and t is the sheet’s thickness. Young’s modulus Y is determined from the first derivative of the stress-strain curve with 0 6 e 6 4% at axial strain e ¼ 0. Without specifying the sheet’s thickness, Yt and rt are the 2D Young’s modulus (or in plane-stiffness) and 2D stress (or in-plane stress), respectively.

3. Results and discussion The potential energy per atom at optimized structure at zero strain was estimated by DFT calculations [1] at 9.22 eV for graphene, 8.57 eV for graphyne, 8.30 eV for PG, 8.28 eV for a-graphyne, and 8.23 eV for (3, 12) – carbon sheet. Hence, PG is more stable than a-graphyne and (3, 12) – carbon sheet, but less stable than graphene and graphyne. The present work estimates

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-7.2

Potential energy/atom, eV

the potential energy per atom at 300 K at 7.56, 6.24, and 5.55 eV for PG, HPG, and penta-CN2, respectively. Therefore, the stability decreases in the order PG, HPG, and penta-CN2 sheets. Deviations in the potential energy per atom between our results and DFT calculations is less than 9% for PG (our result is 7.56 eV and DFT calculations provided a value of 8.30 eV [1]) and falls within 1% for penta-CN2 (our result is 5.55 eV and DFT calculations provided a value of about 5.5 eV [11]). Cranford [13] showed that the critical transition temperature is 600 K for a free-standing finite-size PG. Fig. 2 shows once again here the transition of PG without external mechanical load during the relaxation under FBC at 600 K and higher temperature. Fig. 3 shows an overall decrease in the potential energy per atom of PG in the first stage of the relaxation under FBC. The transition and

-7.3

600K

-7.4

900K

-7.5 -7.6

Relaxation of PG with FBC

-7.7 -7.8 -7.9 0

10

Potential energy/atom, eV

-7.2

20 30 Time, pc

40

50

-7.3 -7.4

-7.3

-7.5 -7.6

-7.4

0

-7.5

1

2

3

Relaxation of PG with PBC

4

5

300K 900K

-7.6 0

10

20

30 40 Time, pc

50

60

Fig. 3. Evolution of the potential energy per atom versus the relaxation time for penta-graphene under: (top) fixed boundary conditions; and (bottom) periodic boundary conditions.

Fig. 2. Snapshots of PG at the relaxation of 50 ps with FBC at: (top) 600 K and (bottom) 900 K. The deformation and transition of PG at 900 K is more severe than at 600 K. Initial C1 and C2 atoms are denoted by pink and cyan balls, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 4. Snapshot of PG at the relaxation of 60 ps with PBC at 900 K without transition. C1 and C2 atoms are denoted by pink and cyan balls, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Table 1 Optimized structure parameters of PG, HPG, and penta-CN2 sheets at zero strain at 0 K. a is the lattice constant; h is the distance between the top and bottom layers of the carbon (or nitrogen) atoms; rC1C2 (rC1N), rC2C2 (rNN), and rC2H are C1AC2 (C1AN), C2AC2 (NAN) and C2AH bond lengths in Å, respectively. Sheets

References

rC1C2 (or rC1N)

rC2C2 (or rNN)

a

h

PG

Present study DFT calculations by Zhang et al. [1]

1.552 1.55

1.323 1.34

3.621 3.64

1.243 1.2

HPG

Present study DFT calculations by Li et al. [7]

1.544 1.55

1.635 1.55

3.496

1.650 1.62

Penta-CN2

Present study DFT calculations by Zhang et al. [11]

1.477 1.47

1.492 1.45

3.521 3.31

1.472 1.52

rC2H

1.111 1.10

320

a) 25 Penta-graphene

280

Axial stresss σ t, N/m

Young's modulus Yt, N/m

300

260

Penta-graphene 240

Penta-CN2

220

Hydrogenated penta-graphene

200

20

300K 900K

15 10 5

180 300

500

700

0

900

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Axial strain

Temperature, K Fig. 5. Effects of temperature on the Young’s modulus.

b) 30

Hydrogenated penta-graphene

Axial stresss σ t, N/m

25 300K 20

900K

15 10 5 0 0

0.05

0.1

0.15

0.2

Axial strain

c) 15 Penta-CN2 12 Axial Stresss σ t, N/m

the reconstruction of the structure occur in this first stage of the relaxation. After this first stage of the relaxation, the potential energy fluctuates around a stable value. The deformation and transition of PG under relaxation with FBC at 900 K is more severe than at 600 K. Initial pentagons still exist at 600 K after a long relaxation time when the sheet is in a stable state, whereas a complete transition occurs almost at the relaxation with FBC at 900 K. In contrast, the PG sheet remains almost flat without transition under relaxation with PBC as shown in Fig. 4 for 900 K. The potential energy per atom (thus the potential energy of the structure) converges almost after 1 ps during the relaxation as shown in Fig. 3 for the relaxation of PG at 300 and 900 K. Therefore, the relaxation time is taken as 5 pc before all tensile tests under PBC. It is clear that PBC is able to reduce the size and edge effects. Edge effects are significant in the transition of PG as seen in the cases with FBC. Table 1 indicates in general a good agreement between the optimized structure parameters of PG, HPG, and penta-CN2 sheets at zero strain at 0 K revealed in the present study and those from DFT calculations [1,7,11]. The C1AC1 bond length is nearly unchanged in PG (1.55 Å) and HPG (1.54 Å), and approximates to the CAC bond lengths in the hydrogenated graphene (graphane) [23–25]. In the other hand, the hydrogenation causes a longer C2AC2 bond length (1.32 Å in PG and 1.64 Å in HPG) and larger distance h between the top and bottom layers of the carbon atoms (h = 1.24 Å in PG and h = 1.65 Å in HPG) as shown in Table 1. Therefore, the hydrogenation leads to a reduction in Young’s modulus of PG as shown in Fig. 5 due to a pre-stretch of C2AC2 bonds. Hydrogenation reduces the Young’s modulus of PG by 25–28% at the temperature in the range of 300–900 K. It is noted that hydrogenation reduces the Young’s modulus of graphene PG by 30% at 300 K [26]. At 300 K, the 2D Young’s modulus of PG, HPG, and penta-CN2 is

300K 900K

9 6 3 0 0

0.02

0.04 0.06 Axial Strain

0.08

0.1

Fig. 6. Evolution of the axial tensile stress versus the axial tensile strain at 300 and 900 K of the: (a) PG sheet; (b) HPG sheet; and (c) penta-CN2 sheet.

M.-Q. Le / Computational Materials Science 136 (2017) 181–190

-7.2

Penta-graphene

Potential energy per atom, eV

300K

-7.4

about 282, 210, and 314 N/m, respectively. These values at 300 K are very close to the predictions by DFT calculations: 263.8 N/m [1] and 267.7 N/m [3] for PG, 205.5 N/m for HPG [7], and 318.5 N/m for penta-CN2 [11]. The 2D Young’s modulus of PG, HPG and penta-CN2 is about 268, 260, and 254 N/m; 195, 186, and 183 N/m; and 303, 299, and 296 N/m at 500, 700 and 900 K, respectively. When the temperature increases from 300 to 900 K, the Young’s modulus of PG and HPG reduces about 10–13%, whereas the Young’s modulus of penta-CN2 decreases less than 6%. The Young’s modulus of PG is comparable to that of h-BN monolayer (267 N/m) [27], the Young’s modulus of HPG is higher than that of hexagonal SiC monolayer (166 N/m) [27] and lower than that of graphane [24,25], and penta-CN2 is stiffer than h-BN sheet. At all examined temperatures, the stress-strain curves and the evolution of the potential energy per carbon atom versus the axial tensile strain of PG include two stages as shown in Fig. 6a and 7. In the first stage without transition, the axial stress and the potential energy per atom of the PG increases monotonously with an

-7.2 -7.3

500K 700K

-7.6

900K

-7.4 -7.5 -7.6 0

-7.8

0.03 0.06 0.09 0.12 0.15

-8

-8.2

0

0.2

0.4

0.6

0.8

185

1

Axial strain Fig. 7. Evolution of the potential energy per carbon atom versus the axial tensile strain of the PG.

8

8 5 8 6 8 6 8 8 9 7 5 9 4

5 5 55 5

a)

b)

c) Fig. 8. Snapshots of the PG sheet under uniaxial tension at 300 K: (a) transition of a half PG sheet at e = 14.06%; (b) initial pentagonal rings exist still at e = 32.81%; and (c) complete transition at e = 33.44%. Initial C1 and C2 atoms are denoted by pink and cyan balls, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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8 6 9 6 8 5

4

7 3

8 8

8 9

Fig. 9. Snapshots of the PG sheet under uniaxial tension at 500 K: (a) partial transition at e = 10.63%; and (b) complete transition at e = 11.25%. Initial C1 and C2 atoms are denoted by pink and cyan balls, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

increase of axial tensile strain. The stress and the potential energy drop at a yield point at the end of the first stage. The stress and the strain at the yield point denote as the yield stress (maximal tensile stress before transition) and the yield strain, respectively. In the second stage, the stress increase and reaches a maximal value then decreases slowly. Just after the maximal stress in the first stage, the PG sheet transitions in the second stage. Figs. 8-10 show 3, 4, 5, 6, 7, 8, and 9-rings during transition. These carbon membered rings form various new 2D carbon allotropes [28–52]. The potential energy fluctuates due to the local relaxation, transition between

different carbon rings and reconstruction of the system with a continuous tensile loading during the second stage. The transition from original pentagons to other rings takes a longer period at 300 K than at 500, 700, and 900 K. At 300 K, the yield strain of PG is 10.75%. The transition does not take place until an axial strain of 13.75%. About a half PG sheet transitions at an axial tensile strain of 14.06% as indicated in Fig. 8a. Fig. 8b shows the existence of the pentagons from the initial structure at an axial tensile strain of 32.81%. The transition from pentagons to other rings almost complete at an axial tensile strain of 33.44% as seen in Fig. 8c. In contrast, the transition from pentagons to other rings happens quickly at 500, 700, and 900 K. The yield strain of PG is 10.38% at 500 K. Just after the yield point, local transition occupies about one third of the PG sheet at 10.63% as shown in Fig. 9a. Initial pentagons are not seen in Fig. 9b at an axial strain of 11.25%, indicating a complete transition. Fig. 10a shows the early transition of PG at 900 K at a tensile strain of 3.13%, just after the yield strain (e  3%). Initial pentagons are not seen at e = 3.75% as shown in Fig. 10b, exhibiting a quick transition process. After a complete transition from pentagons to other rings, the PG sheet exhibits different structures during the tension as indicated in Fig. 10b at e = 3.75% and in Fig. 10c at e  35.6% at the maximal stress of the second stage. Hence, the local relaxation, transition between different carbon rings and reconstruction of the system still repeat with further tensile loading during the second stage. It should be noted that PG can withstand temperatures as high as 1000 K according to ab initio molecular dynamics (AIMD) predictions [1,7], the PG sheet transitions at a very small uniaxial tensile strain (e  3.13%) at 900 K. We see the higher temperature, the easier and faster transition of PG under tension. This issue is opposite to the transition of MoS2, which is a hexagonal structure with Mo atoms in the middle plane and S atoms in the two outer parallel planes. The transition under tension of MoS2 have been observed only at low temperatures below 40 K [53]. Common defects in graphene [54,55] include 5-7-7-5 defects (or Stone-wales defect); single vacancies that form 5-9 defects; double vacancies that can form 5-8-5, 555-777, and 5555-6-7777 defects. Under tension, PG transitions to a sheet with 3, 4, 5, 6, 7, 8, and 9-membered rings. The sheet in the second phase called ‘‘defective graphene” because besides 3, 4, 5, 7, 8, and 9-rings, 6rings are dominant in this sheet. With 3, 4, 5, 6, 7, 8 and 9-rings, the sheet in the second phase is different from commons defective graphene sheets [54,55] and other 2D carbon allotropes. These 2D carbon allotropes include graphynes and graphdiyne [32–35], pentaheptites with 5 and 7-rings [36,37], haeckelites with 5-6-7 rings [38], squarographenes [39], (4, 8) carbon sheet structured by 4 and 8-rings [40–44], octite (8-rings surrounded by 6 and 5-rings) [45], OP-graphene with 5 and 8-rings [46,47], W sheet (4-rings surrounded by 6 and 8-rings [48], pharagrahene (7-rings surrounded by 5 and 6-rings) [28], S-, D-, and E-graphene [49], HOP graphene [50] and pentahexoctite with 5-6-8 rings [29], 2D carbon sheet of fused pentagons [51], biphenylene sheets with 4-6-8 rings [52]. Due to the pre-strained state (transition under tension) and the difference in the atomic structure, the stress-strain response of the second phase is not similar to the above-mentioned 2D carbon allotropes and defective graphene sheets [56–59]. Previous MD simulations [56–58] and DFT calculations [59] showed that the decrease in the tensile strength of graphene due to defects depends largely on the defect type, the defect location and the defect ratio. A Stone-wales defect (5-7-7-5 defect) could reduce 5–14% in the tensile strength of graphene, depending on the tensile direction and defect location, according to DFT calculations by Ren and Cao [59]. A reduction of 54% in the fracture strength of graphene along the zigzag direction due to a linear arrangement of repeat unit 5–8–5 defect was found in MD simulations with Airebo potential by Wang et al. [58]. The maximal in-plane stress in the second

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M.-Q. Le / Computational Materials Science 136 (2017) 181–190

6 7 7 6

a)

c)

b) Fig. 10. Snapshots of the PG sheet under uniaxial tension at 900 K: (a) local transition (at the top right corner of the sheet) at a very early stage of the tension at e = 3.13%; (b) complete transition at e = 3.75%; and (c) e  35.6% at post-transition maximal stress. Initial C1 and C2 atoms are denoted by pink and cyan balls, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

stage of PG at 300 K (18.2 N/m) is about 45% of the in-plane tensile strength of graphene along the zigzag direction (40 N/m) [60–62]. Hence, defects reduce about 55% in the tensile strength if the sheet in the second stage is considered as a defective graphene. It should be emphasized that the sheet in the second stage is formed by 3, 4, 5, 6, 7, 8, and 9-membered rings under a pre-strained state. The yield in-plane stress (22.4 N/m) and the maximal in-plane stress of the second stage (18.2 N/m) of PG at 300 K are lower than the in-plane tensile strength of (4, 8) carbon sheet (34.4, 27.3 N/m) predicted by DFT calculations [44], and than the in-plane tensile strength of phagraphene with 5-6-7 carbon rings (28.5 N/m) predicted by MD simulations with Tersoff potential [63]. It is noted that differences between MD simulations could be associated with the potential formulation, and not necessarily structural in nature. The fracture of HPG and penta-CN2 sheets occurs suddenly with a drop in their stress–strain curves as shown in Fig. 6b and c. HPG and penta-CN2 sheets exhibit a brittle fracture without phase transformation at all examined temperatures. For brittle fracture, the maximal axial stress and the strain at maximal stress refer to fracture stress (or tensile strength) and fracture strain, respec-

tively. Fig. 11 shows the fracture shapes of HPG and penta-CN2 sheets at 900 K at an axial tensile strain just after the strain at maximal stress. Fig. 12 pertains to the effects of temperature on the maximal stress and the strain at maximal stress of PG, HPG, and pentaCN2 sheets. Before transition, the maximal stress (or yield stress) and yield strain of PG decrease rapidly with an increase of temperature. The yield in-plane stress of PG is about 22.4, 21.4, 17.3, and 7.9 N/m at 300, 500, 700, and 900 K, respectively. The maximal inplane stress of PG in the second stage falls within 17–20 N/m when temperature varies in the range of 300–900 K. The strain at this maximal stress in the second stage is about 40.2, 35.0, 35.4 and 35.6% at 300, 500, 700, and 900 K, respectively as shown in Fig. 12. The fracture stress (or tensile strength) and fracture strain of HPG decrease slightly with an increase of temperature from 300 to 900 K. The in-plane tensile strength and fracture strain of HPG are about 28.7, 27.0, 25.5, and 23.8 N/m; and 19.6, 18.4, 16.1, and 14.4% at 300, 500, 700, and 900 K, respectively. MD simulations with AIREBO potential [26] estimated the in-plane tensile strength and fracture strain of graphane at 11.9 and 19.3 N/m; and 5.9 and 11.4% in the armchair and zigzag directions, respectively. Hence,

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Maximal stress σ t, N/m

30 25 20 15 Penta-graphene, before transition Penta-graphene, post-transition Penta-CN2 Hydrogenated penta-graphene

10 5 300

900

500 700 Temperature, K

0.45

Strain at maximal stress

0.4 0.35 Penta-graphene, before transition Penta-graphene, post-transition Penta-CN2 Hydrogenated penta-graphene

0.3 0.25 0.2 0.15

0.1 0.05 0 300

900

500 700 Temperature, K

Fig. 12. Effects of temperature on the: (top) maximal stress, and (bottom) strain at maximal stress.

30 Along x-axis Axial Stress σ t, N/m

25

Fig. 11. Fracture shapes under uniaxial tension at 900 K of: (top) HPG sheet at e = 14.44% (the strain at maximal stress is 14.38%); and (bottom) penta-CN2 sheet at e = 5.63% (the strain at maximal stress is 5.44%). Black, green and pink balls represent C, H, and N atoms, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Along y-axis

20 15 10

Equi-biaxial tension of PG at 300K

5 0 0

0.05

0.1 0.15 Axial Strain

0.2

0.25

Fig. 13. Stress-strain curve of the PG sheet under equi-biaxial tension at 300 K.

HPG exhibits a higher tensile strength and fracture strain than those of graphane. The tensile strength of HPG is higher than that of PG, and the fracture strain of HPG is higher than the yield strain of PG as shown in Fig. 12 because HPG does not undergo phase transformation under tension, while PG does. It should be noted that hydrogenation of graphene could reduce the tensile strength and fracture strain by  65% [26]. The in-plane tensile strength and fracture strain of the pentaCN2 sheet are about 14.4, 13.7, 12.1, and 10.6 N/m; and 9.56,

8.38, 6.97 and 5.44% at 300, 500, 700, and 900 K, respectively. Tensile strength of the penta-CN2 sheet is well lower than that of HPG, and lower than the maximal stress of PG in the second stage. At 300, 500, and 700 K, tensile strength of the penta-CN2 sheet is lower than the yield stress of PG. At 900 K, tensile strength of the penta-CN2 sheet is higher than the yield stress of PG because PG transitions at low tensile strain. Fracture strain of HPG is 2 times higher than that of penta-CN2 sheet. Fracture strain of the pentaCN2 sheet approximates to yield strain of PG.

M.-Q. Le / Computational Materials Science 136 (2017) 181–190

189

strain curves along x- and y-directions are almost indentical because atomic arrangements are identical along these two directions as shown in Fig. 1a. The maximal stress and the strain at maximal stress in the first stage are about 20.4 N/m and 4.75%, respectively. The yield strain under equi-biaxial tension (4.75%) is well lower than that under uniaxial tension (10.75%). In the second stage, the stress-strain curves along x- and y-directions are slightly different because atomic arrangements during transition under equi-biaxial tension are different along these two directions as shown in Fig. 14. The maximal stress and the strain at the maximal stress in the second stage are about 26–27 N/m and 14.1%, respectively.

4. Conclusion The mechanical properties of PG, HPG, and penta-CN2 were examined at 300, 500, 700, and 900 K through reactive MD simulations. Main findings are summarized below.  At all examined temperatures, the uniaxial tensile stress-strain curves of PG include two stages. In the first stage before transition, the axial stress of the PG sheet increases monotonously with an increase of axial tensile strain then drops from the yield point at the end of the first stage. In the second stage with transition of PG to 3, 4, 5, 6, 7, 8, and 9-rings, the axial tensile stress fluctuates but increases, reaches a maximal value, and then decreases slowly.  The transition of PG under tension takes a longer period at 300 K than at 500, 700, and 900 K. This transition process becomes easier and faster with higher temperature. After a complete transition from initial pentagons to other rings, the local relaxation, transition between different carbon rings and reconstruction of the system still repeat with further tensile loading.  An increase of temperature reduces slightly the Young’s modulus of PG and decreases considerably its yield stress and yield strain due to the transition. Temperature has also slight effects on its maximal stress in the second stage and on the strain at this maximal stress.  Hydrogenation of PG reduces the Young’s modulus by 25–28% due to an increase of the C2-C2 bond length. However, hydrogenation increases the maximal tensile stress of PG because HPG does not undergo phase transformation under tension, while PG does.  The mechanical properties of HPG and penta-CN2 decrease slightly with an increase of temperature. Among three examined sheets, the penta-CN2 sheet exhibits the highest Young’s modulus, but the lowest tensile strength and fracture strain.

Acknowledgements

Fig. 14. Snapshots of the PG sheet under equi-biaxial tension at 300 K: (a) e  5.04%; (b) e  14.1% at post-transition maximal stress; (c) e  23.8%. Initial C1 and C2 atoms are denoted by pink and cyan balls, respectively. The strain at maximal stress in the first stage before transition is about 4.75%. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

To further examine the different conditions for the transition of PG, equi-biaxial tension test of PG at 300 K was carried out. Fig. 13 shows the stress-strain curve of the PG sheet under equi-biaxial tension at 300 K. In the first stage before transition, the stress-

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