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Thermal relaxation and deformation of indented graphene
Computational Materials Science 79 (2013) 105–109
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Thermal relaxation and deformation of indented graphene Yu-Cheng Fan, Cheng-Da Wu, Te-Hua Fang ⇑, Tao-Hsing Chen Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung 807, Taiwan
a r t i c l e
i n f o
Article history: Received 7 May 2013 Accepted 5 June 2013 Available online 4 July 2013 Keywords: Graphene Temperature Indentation velocity Strain Holding process
a b s t r a c t The nanomechanical properties of graphene under nanoindentation are studied using molecular dynamics simulations based on the Tersoff–Brenner many-body potential and Lennard-Jones potential. The effects of the indentation temperature, indentation velocity, and indenter size are evaluated in terms of atomic trajectories, deformation velocity, indentation force, and strain field. The simulation results show that graphene deformation increases with increasing indentation depth, indentation velocity, temperature, and indenter size. During the holding process, a slight deformation between the center and the edges of the graphene remains due to relaxation, which increases with increasing temperature and indentation velocity. Cracks easily form with high-velocity indentation due to large strain energy accumulation in the material. The area of the deformation region increases with decreasing indentation velocity. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Graphene comprises a single-atom-thick sp2 bonded hexagonal structure of carbon atoms [1,2]. Graphene has attracted great interest in nanodevices due to its excellent mechanical [3,4], electrical [5,6], optical [7,8], thermal [9,10], and chemical [10,11] properties. For example, the electron transfer rate of graphene at room temperature is much higher than that of other known conductor materials. Graphene has a high Young’s modulus (1.0 TPa) and a high thermal conductivity (3000–5000 W/mK) [12]. Graphene properties have been investigated experimentally [13–17] and theoretically using molecular dynamics (MD) [18– 21] and density functional theory [22–24]. Kim et al. [3] grew and transferred graphene films onto nickel layers using chemical vapor deposition (CVD). The patterned graphene film was easily transferred to stretchable substrates with a very low sheet resistance of 280 X/square and a 80% optical transparency. Gao and Hao [25] found that the electronic properties of graphene varied with the external mechanical load on it using quantum mechanics and quantum MD methods. Duan et al. [26] simulated the deformations of a single layer and a circular graphene sheet under a central point load using molecular mechanics simulations and found that the von Karman plate theory can provide accurate predictions for a graphene sheet under linear and nonlinear bending and stretching. Bu et al. [27] analyzed the stress–strain, Young’s modulus, and elastic modulus of graphene nanoribbons using MD
⇑ Corresponding author. Tel.: +886 7 381 4526x5336. E-mail address: [email protected] (T.-H. Fang). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.06.012
simulations. They found that the deformed graphene nanoribbons required an applied strain of about 4–5% to become nearly flat. The present work investigates the mechanical response of graphene under nanoindentation using MD simulation. The effects of the indentation temperature, indentation temperature, and indenter size are discussed.
2. Methodology The MD model of nanoindentation consists of a single-layer graphene substrate and an indenter, as shown in Fig. 1. The substrate and indenter are both made of carbon atoms. The indenter is assumed to be a rigid carbon nanotube with an included angle of 60°. The substrate is circular. To describe the deformation behavior of the substrate under nanoindentation, the edges are set as the boundary atoms to support the whole model, and the internal atoms are set as thermostat atoms. A substrate with a diameter of 10 nm and made up of 3681 atoms is considered. A Cartesian coordinate system is used in the proposed system without any periodic boundary condition (PBC). The origin is set at the center of the substrate. Before indentation, a separation distance of 0.5 nm was set between the indenter and the substrate. The Tersoff–Brenner many-body potential function is adopted to describe the C–C (inside graphene) interaction. The non-bonding interaction between substrate–indenter is described using the Lennard-Jones (L-J) potential model. The parameters for the Tersoff– Brenner many-body potential and the L-J potential are listed in Table 1 [28,29].
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Y.-C. Fan et al. / Computational Materials Science 79 (2013) 105–109
Fig. 1. MD model of nanoindentation. The model consists of a single-layer graphene substrate and an indenter.
Table 1 Parameters for Tersoff–Brenner potential and Lennard-Jones potential. Tersoff–Brenner potential [28]
A (eV) B (eV) k (Å 1)
1.3936 103 3.467 102 3.4879
b n c
1.5724 10 7 7.2751 10 1 3.8049 104
l
2.2119
d
4.384 100
(Å 1) R (Å) S (Å)
1.8 2.1
h
58.2
r
5.7058 10
Fig. 3. Variation of deformation velocity of substrate in Z direction during indentation process (deformation velocity was obtained by averaging over 50 ps). 1
The simulation was performed at an indentation velocity of 10 m/s and a system temperature of 300 K. The maximum indentation depth was set to 2 nm. Fig. 2 shows the variation of the cross section of the substrate with time during the indentation. The indentation simulation includes loading (0–250 ps) and holding (250–270 ps) processes. As the indenter approaches the substrate, the substrate deforms upward due to van der Waals attractive forces between the indenter and the substrate atoms and then deforms downward due to the transfer of interactive force from attractive forces to repulsive forces. The deformation and tension [17] gradually increase with indentation depth. During the indentation, the center of the substrate has the largest amount of defor-
mation due to the direct load applied to it by the indenter. For a given indentation depth, the deformation of the substrate gradually decreases along the radial direction. During the holding process (from 250 to 270 ps), a slight deformation between the center and the edges of the substrate remained due to relaxation. Fig. 3 shows the variation of the deformation velocity of the substrate in the Z direction during the indentation process (the deformation velocity was obtained by averaging over 50 ps). The deformation velocity at the center of the substrate, which is always the largest, is independent of the indentation depth. The deformation velocity gradually decays along the radial direction. Huang and Zhang [30] found that large local tension on a substrate leads to high energy concentration around the indenter. Fig. 4 shows the variation of deformation velocity of the substrate in the Z direction during the holding process (the deformation velocity was obtained by averaging over 5 ps). During the holding process, a large deformation velocity appears in the region between the center and the edges of the substrate, indicating a large elastic recovery due to relaxation. However, the deformation velocity was zero at the center and the edges of the substrate due to the direct load applied to the former and the boundary assumption, respectively.
Fig. 2. Variation of deformation amount of substrate in Z direction with time during indentation.
Fig. 4. Variation of relaxation velocity of substrate in Z direction during holding (relaxation velocity was obtained by averaging over 5 ps).
Lennard-Jones potential [29]
e/kB (K)
3.851
(Å)
3. Results and discussion 3.1. Simulations of nanoindentation process
Y.-C. Fan et al. / Computational Materials Science 79 (2013) 105–109
107
Fig. 5. Variation of strain field of substrate with time during nanoindentation. (a) 50, (b) 100, (c) 150, (d) 200, (e) 250 (maximum indentation depth), and (f) 270 ps (end of holding).
Fig. 5 shows the variation of the strain field of the substrate with time during nanoindentation. The atoms are colored according to the magnitude of their strain. The strain field records the variation of atomic position from the initial time step (after thermal equilibrium) to the specific time step. The amount of strain and the area of the main strain region increase with increasing time. The largest strain occurs at the center, and the strain decays along the radical direction, as shown in Fig. 5b–d. At 250 ps (Fig. 5e), a crack appears at the center because the tension at this point has exceeded the tensile strength of the substrate. During the holding process, the strain slightly increases and extends outward in order to relax the accumulated strain energy, as shown in Fig. 5f. In order to understand the substrate relaxation in detail during the holding process, Fig. 6 shows the development of the strain field of the substrate with holding time. The strain field was obtained by evaluating the difference in atomic position between 250 ps (the maximum indentation depth) and the time step of relaxation. The maximum amount of relaxation occurs in the first 5 ps (Fig. 6a). The relaxation mainly appears in the region near the center, with a maximum relaxation of 0.32 nm. The relaxation progresses with time, as shown in Fig. 6b–d.
3.2. Effect of temperature An indentation simulation was performed at temperatures of 150, 300, 450, and 600 K, respectively, to study the effect of temperature. Fig. 7 shows the variation of indentation force with indentation depth for these temperatures. The indentation force significantly increases with increasing indentation depth and decreases with increasing temperature during the loading process. For a given indentation force, the material at a higher temperature has a larger deformation due to an increase of atomic kinetic energy. During the holding process, the relaxation in the region between the center and the edges of the substrate increases with
Fig. 7. Variation of indentation force with indenter displacement for temperatures of 150, 300, 450, and 600 K.
Fig. 8. Variation of indentation force with indenter displacement for indentation velocities of 2, 5, 10, and 20 m/s. Snapshots show strain field of substrate at the maximum indentation depth for these indentation velocities.
temperature, which indicates that an increase in temperature helps material relaxation.
3.3. Effect of indentation velocity During the indentation process, indentation velocities of 2, 5, 10, and 20 m/s were used, respectively. Fig. 8 shows the variation
Fig. 6. Development of strain field of substrate with holding time. (a) 255, (b) 260, (c) 265, and (d) 270 ps.
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3.4. Effect of indenter size The effect of the indenter size was investigated by varying the indenter length to 0.76, 1.01, 1.35, and 1.66 nm, respectively, with corresponding diameters of 0.76, 1.01, 1.35, and 1.66 nm, respectively. An increase in the indenter length results in increases in the diameter and number of atoms. Fig. 10 shows the variation of indentation force with indentation depth for these indenter lengths. As the indenter approaches the substrate, a larger attractive force appears for a longer indenter due to an increase in the van der Waals attractive force. The interaction force between the indenter and the substrate then gradually transfers from an attractive force into a repulsive force after reaching an indentation displacement of 0.5 nm. The indentation force (repulsive force) and the amount of substrate deformation increase with increasing indenter length due to an increase in the contact area.
Fig. 9. Variation of relaxation velocity of substrate in Z direction during holding. Snapshots show strain field of substrate at end of holding for these indentation velocities.
of indentation force with indentation depth for these indentation velocities. The indentation force significantly increases with increasing indentation velocity. This is because the substrate has a larger amount of deformation at every time step for a higher indentation velocity; therefore, a larger load is needed. Moreover, the area of the deformation region increases with decreasing indentation velocity because the atoms have more time to relax the strain energy through adjustment of atomic positions. Cracks form more easily with a higher indentation velocity because the substrate has insufficient time to release the strain energy and rearrange the atomic configuration. The variation of the relaxation velocity of the substrate in the Z direction during the holding process is shown in Fig. 9. The main relaxation appears in the region between the center and the edges of the substrate. The atoms near the center have the most relaxation. For the whole substrate, the relaxation increases with increasing indentation velocity due to more accumulated strain energy during the loading process, which indicates that the substrate has larger elastic recovery under higher-velocity deformation.
Fig. 10. Variation of indentation force with indenter displacement for indenter lengths of 0.76, 1.01, 1.35, and 1.66 nm.
4. Conclusion This work investigated the mechanical properties of graphene by nanoindentation simulations. Graphene deformation increases with increasing indentation depth, indentation velocity, temperature, and indenter size. During the holding process, a slight deformation between the center and the edges of the graphene remains due to relaxation, which increases with increasing temperature and indentation velocity. Cracks easily form with high-velocity indentation due to large accumulated strain energy. The area of the deformation region increases with decreasing indentation velocity. Acknowledgment This work was supported by the National Science Council of Taiwan under Grants NSC 100-2628-E-151-003-MY3 and NSC 1002221-E-151-018-MY3. References [1] P.Y. Huang, C.S. Ruiz-Vargas, A.M. van der Zande, W.S. Whitney, M.P. Levendorf, J.W. Kevek, S. Garg, J.S. Alden, C.J. Hustedt, Y. Zhu, J. Park, P.L. McEuen, D.A. Muller, Nature 469 (2011) 389–393. [2] W. Regan, N. Alem, B. Aleman, B. Geng, C. Girit, L. Maserati, F. Wang, M. Crommie, A. Zettl, Appl. Phys. Lett. 96 (2010) 113102. [3] K.S. Kim, Y. Zhao, H. Jang, S.Y. Lee, J.M. Kim, K.S. Kim, J.H. Ahn, P. Kim, J.Y. Choi, B.H. Hong, Nature 457 (2009) 706–710. [4] C. Lee, X.D. Wei, J.W. Kysar, J. Hone, Science 321 (2008) 385–388. [5] M.D. Stoller, S. Parl, Y. Zhu, J. An, R.S. Ruoff, Nano Lett. 8 (2008) 34983502. [6] X. Li, Y. Zhu, W. Cai, M. Borysiak, B. Han, D. Chen, R.D. Piner, L. Colombo, R.S. Ruoff, Nano Lett. 8 (2009) 4359–4363. [7] F. Wang, Y. Zhang, C. Tian, C. Giri, A. Zettl, M. Crommie, Y.R. Shen, Science 320 (2008) 206–209. [8] F. Xia, T. Mueller, Y.M. Lin, A. Valdes-Garcia, P. Avouris, Nat. Nanotechnol. 4 (2009) 839–843. [9] A.A. Balandin, S. Ghosh, W. Bao, I. Calizo, D. Teweldebrhan, F. Miao, C.N. Lau, Nano Lett. 8 (2008) 902–907. [10] X. Wang, L.J. Zhi, K. Müllen, Nano Lett. 8 (2008) 323–327. [11] D. Li, M.B. Muller, S. Gilje, R.B. Kaner, G.G. Wallace, Nat. Nanotechnol. 3 (2008) 101–105. [12] X. Huang, X.Y. Qi, F. Boey, H. Zhang, Chem. Soc. Rev. 41 (2012) 666–686. [13] X. Li, W. Cai, J. An, S. Kim, J. Nah, D. Yang, R. Piner, A. Velamakanni, I. Jung, E. Tutuc, S.K. Banerjee, L. Colombo, R.S. Ruoff, Science 324 (2009) 1312–1314. [14] I.F. Cheng, Y.Q. Xie, R.A. Gonzales, P.R. Gonzales, J.P. Sundararajan, B.A.F. Kengne, D.E. Aston, D.N. Mcllroy, J.D. Foutch, P.R. Griffiths, Carbon 49 (2011) 2852–2861. [15] W.W. Cai, Y.W. Zhu, X.S. Li, R.D. Piner, R.S. Ruoff, Appl. Phys. Lett. 95 (2009) 123115. [16] A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M.S. Dresselhaus, J. Kong, Nano Lett. 9 (2009) 30–35. [17] C.S. Ruiz-Vargas, H.L. Zhuang, P.Y. Huang, A.M. van der Zande, S. Garg, P.L. McEuenk, D.A. Muller, R.G. Hennig, J. Park, Nano Lett. 11 (2011) 2259–2263. [18] Q. Lu, W. Gao, R. Huang, Modell. Simul. Mater. Sci. Eng. 19 (2011) 054006. [19] Z. Wei, Z. Ni, K. Bi, M. Chen, Y. Chen, Carbon 49 (2011) 2653–2658. [20] S. Cranford, M.J. Buehler, Model. Simul. Mater. Sci. Eng. 19 (2011) 054003. [21] T.H. Fang, W.J. Cheng, Y.C. Fan, Micro Nano Lett. 6 (2011) 214–216.
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Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Thermal relaxation and deformation of indented graphene Yu-Cheng Fan, Cheng-Da Wu, Te-Hua Fang ⇑, Tao-Hsing Chen Department of Mechanical Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung 807, Taiwan
a r t i c l e
i n f o
Article history: Received 7 May 2013 Accepted 5 June 2013 Available online 4 July 2013 Keywords: Graphene Temperature Indentation velocity Strain Holding process
a b s t r a c t The nanomechanical properties of graphene under nanoindentation are studied using molecular dynamics simulations based on the Tersoff–Brenner many-body potential and Lennard-Jones potential. The effects of the indentation temperature, indentation velocity, and indenter size are evaluated in terms of atomic trajectories, deformation velocity, indentation force, and strain field. The simulation results show that graphene deformation increases with increasing indentation depth, indentation velocity, temperature, and indenter size. During the holding process, a slight deformation between the center and the edges of the graphene remains due to relaxation, which increases with increasing temperature and indentation velocity. Cracks easily form with high-velocity indentation due to large strain energy accumulation in the material. The area of the deformation region increases with decreasing indentation velocity. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Graphene comprises a single-atom-thick sp2 bonded hexagonal structure of carbon atoms [1,2]. Graphene has attracted great interest in nanodevices due to its excellent mechanical [3,4], electrical [5,6], optical [7,8], thermal [9,10], and chemical [10,11] properties. For example, the electron transfer rate of graphene at room temperature is much higher than that of other known conductor materials. Graphene has a high Young’s modulus (1.0 TPa) and a high thermal conductivity (3000–5000 W/mK) [12]. Graphene properties have been investigated experimentally [13–17] and theoretically using molecular dynamics (MD) [18– 21] and density functional theory [22–24]. Kim et al. [3] grew and transferred graphene films onto nickel layers using chemical vapor deposition (CVD). The patterned graphene film was easily transferred to stretchable substrates with a very low sheet resistance of 280 X/square and a 80% optical transparency. Gao and Hao [25] found that the electronic properties of graphene varied with the external mechanical load on it using quantum mechanics and quantum MD methods. Duan et al. [26] simulated the deformations of a single layer and a circular graphene sheet under a central point load using molecular mechanics simulations and found that the von Karman plate theory can provide accurate predictions for a graphene sheet under linear and nonlinear bending and stretching. Bu et al. [27] analyzed the stress–strain, Young’s modulus, and elastic modulus of graphene nanoribbons using MD
⇑ Corresponding author. Tel.: +886 7 381 4526x5336. E-mail address: [email protected] (T.-H. Fang). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.06.012
simulations. They found that the deformed graphene nanoribbons required an applied strain of about 4–5% to become nearly flat. The present work investigates the mechanical response of graphene under nanoindentation using MD simulation. The effects of the indentation temperature, indentation temperature, and indenter size are discussed.
2. Methodology The MD model of nanoindentation consists of a single-layer graphene substrate and an indenter, as shown in Fig. 1. The substrate and indenter are both made of carbon atoms. The indenter is assumed to be a rigid carbon nanotube with an included angle of 60°. The substrate is circular. To describe the deformation behavior of the substrate under nanoindentation, the edges are set as the boundary atoms to support the whole model, and the internal atoms are set as thermostat atoms. A substrate with a diameter of 10 nm and made up of 3681 atoms is considered. A Cartesian coordinate system is used in the proposed system without any periodic boundary condition (PBC). The origin is set at the center of the substrate. Before indentation, a separation distance of 0.5 nm was set between the indenter and the substrate. The Tersoff–Brenner many-body potential function is adopted to describe the C–C (inside graphene) interaction. The non-bonding interaction between substrate–indenter is described using the Lennard-Jones (L-J) potential model. The parameters for the Tersoff– Brenner many-body potential and the L-J potential are listed in Table 1 [28,29].
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Y.-C. Fan et al. / Computational Materials Science 79 (2013) 105–109
Fig. 1. MD model of nanoindentation. The model consists of a single-layer graphene substrate and an indenter.
Table 1 Parameters for Tersoff–Brenner potential and Lennard-Jones potential. Tersoff–Brenner potential [28]
A (eV) B (eV) k (Å 1)
1.3936 103 3.467 102 3.4879
b n c
1.5724 10 7 7.2751 10 1 3.8049 104
l
2.2119
d
4.384 100
(Å 1) R (Å) S (Å)
1.8 2.1
h
58.2
r
5.7058 10
Fig. 3. Variation of deformation velocity of substrate in Z direction during indentation process (deformation velocity was obtained by averaging over 50 ps). 1
The simulation was performed at an indentation velocity of 10 m/s and a system temperature of 300 K. The maximum indentation depth was set to 2 nm. Fig. 2 shows the variation of the cross section of the substrate with time during the indentation. The indentation simulation includes loading (0–250 ps) and holding (250–270 ps) processes. As the indenter approaches the substrate, the substrate deforms upward due to van der Waals attractive forces between the indenter and the substrate atoms and then deforms downward due to the transfer of interactive force from attractive forces to repulsive forces. The deformation and tension [17] gradually increase with indentation depth. During the indentation, the center of the substrate has the largest amount of defor-
mation due to the direct load applied to it by the indenter. For a given indentation depth, the deformation of the substrate gradually decreases along the radial direction. During the holding process (from 250 to 270 ps), a slight deformation between the center and the edges of the substrate remained due to relaxation. Fig. 3 shows the variation of the deformation velocity of the substrate in the Z direction during the indentation process (the deformation velocity was obtained by averaging over 50 ps). The deformation velocity at the center of the substrate, which is always the largest, is independent of the indentation depth. The deformation velocity gradually decays along the radial direction. Huang and Zhang [30] found that large local tension on a substrate leads to high energy concentration around the indenter. Fig. 4 shows the variation of deformation velocity of the substrate in the Z direction during the holding process (the deformation velocity was obtained by averaging over 5 ps). During the holding process, a large deformation velocity appears in the region between the center and the edges of the substrate, indicating a large elastic recovery due to relaxation. However, the deformation velocity was zero at the center and the edges of the substrate due to the direct load applied to the former and the boundary assumption, respectively.
Fig. 2. Variation of deformation amount of substrate in Z direction with time during indentation.
Fig. 4. Variation of relaxation velocity of substrate in Z direction during holding (relaxation velocity was obtained by averaging over 5 ps).
Lennard-Jones potential [29]
e/kB (K)
3.851
(Å)
3. Results and discussion 3.1. Simulations of nanoindentation process
Y.-C. Fan et al. / Computational Materials Science 79 (2013) 105–109
107
Fig. 5. Variation of strain field of substrate with time during nanoindentation. (a) 50, (b) 100, (c) 150, (d) 200, (e) 250 (maximum indentation depth), and (f) 270 ps (end of holding).
Fig. 5 shows the variation of the strain field of the substrate with time during nanoindentation. The atoms are colored according to the magnitude of their strain. The strain field records the variation of atomic position from the initial time step (after thermal equilibrium) to the specific time step. The amount of strain and the area of the main strain region increase with increasing time. The largest strain occurs at the center, and the strain decays along the radical direction, as shown in Fig. 5b–d. At 250 ps (Fig. 5e), a crack appears at the center because the tension at this point has exceeded the tensile strength of the substrate. During the holding process, the strain slightly increases and extends outward in order to relax the accumulated strain energy, as shown in Fig. 5f. In order to understand the substrate relaxation in detail during the holding process, Fig. 6 shows the development of the strain field of the substrate with holding time. The strain field was obtained by evaluating the difference in atomic position between 250 ps (the maximum indentation depth) and the time step of relaxation. The maximum amount of relaxation occurs in the first 5 ps (Fig. 6a). The relaxation mainly appears in the region near the center, with a maximum relaxation of 0.32 nm. The relaxation progresses with time, as shown in Fig. 6b–d.
3.2. Effect of temperature An indentation simulation was performed at temperatures of 150, 300, 450, and 600 K, respectively, to study the effect of temperature. Fig. 7 shows the variation of indentation force with indentation depth for these temperatures. The indentation force significantly increases with increasing indentation depth and decreases with increasing temperature during the loading process. For a given indentation force, the material at a higher temperature has a larger deformation due to an increase of atomic kinetic energy. During the holding process, the relaxation in the region between the center and the edges of the substrate increases with
Fig. 7. Variation of indentation force with indenter displacement for temperatures of 150, 300, 450, and 600 K.
Fig. 8. Variation of indentation force with indenter displacement for indentation velocities of 2, 5, 10, and 20 m/s. Snapshots show strain field of substrate at the maximum indentation depth for these indentation velocities.
temperature, which indicates that an increase in temperature helps material relaxation.
3.3. Effect of indentation velocity During the indentation process, indentation velocities of 2, 5, 10, and 20 m/s were used, respectively. Fig. 8 shows the variation
Fig. 6. Development of strain field of substrate with holding time. (a) 255, (b) 260, (c) 265, and (d) 270 ps.
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Y.-C. Fan et al. / Computational Materials Science 79 (2013) 105–109
3.4. Effect of indenter size The effect of the indenter size was investigated by varying the indenter length to 0.76, 1.01, 1.35, and 1.66 nm, respectively, with corresponding diameters of 0.76, 1.01, 1.35, and 1.66 nm, respectively. An increase in the indenter length results in increases in the diameter and number of atoms. Fig. 10 shows the variation of indentation force with indentation depth for these indenter lengths. As the indenter approaches the substrate, a larger attractive force appears for a longer indenter due to an increase in the van der Waals attractive force. The interaction force between the indenter and the substrate then gradually transfers from an attractive force into a repulsive force after reaching an indentation displacement of 0.5 nm. The indentation force (repulsive force) and the amount of substrate deformation increase with increasing indenter length due to an increase in the contact area.
Fig. 9. Variation of relaxation velocity of substrate in Z direction during holding. Snapshots show strain field of substrate at end of holding for these indentation velocities.
of indentation force with indentation depth for these indentation velocities. The indentation force significantly increases with increasing indentation velocity. This is because the substrate has a larger amount of deformation at every time step for a higher indentation velocity; therefore, a larger load is needed. Moreover, the area of the deformation region increases with decreasing indentation velocity because the atoms have more time to relax the strain energy through adjustment of atomic positions. Cracks form more easily with a higher indentation velocity because the substrate has insufficient time to release the strain energy and rearrange the atomic configuration. The variation of the relaxation velocity of the substrate in the Z direction during the holding process is shown in Fig. 9. The main relaxation appears in the region between the center and the edges of the substrate. The atoms near the center have the most relaxation. For the whole substrate, the relaxation increases with increasing indentation velocity due to more accumulated strain energy during the loading process, which indicates that the substrate has larger elastic recovery under higher-velocity deformation.
Fig. 10. Variation of indentation force with indenter displacement for indenter lengths of 0.76, 1.01, 1.35, and 1.66 nm.
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