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Boundary condition and pre-strain effects on the free standing indentation response of graphene monolayer

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2013 J. Phys.: Condens. Matter 25 475303 (http://iopscience.iop.org/0953-8984/25/47/475303) View the table of contents for this issue, or go to the journal homepage for more

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IOP PUBLISHING

JOURNAL OF PHYSICS: CONDENSED MATTER

J. Phys.: Condens. Matter 25 (2013) 475303 (7pp)

doi:10.1088/0953-8984/25/47/475303

Boundary condition and pre-strain effects on the free standing indentation response of graphene monolayer Lixin Zhou1 , Yugang Wang1 and Guoxin Cao2 1

School of Physics, Peking University, Beijing 100871, People’s Republic of China HEDPS, Center for Applied Physics and Technology, Department of Mechanics and Engineering Science, College of Engineering, Peking University, Beijing 100871, People’s Republic of China 2

E-mail: [email protected]

Received 17 August 2013, in final form 7 October 2013 Published 31 October 2013 Online at stacks.iop.org/JPhysCM/25/475303 Abstract Using molecular mechanics simulations, we investigated the true pre-stress/pre-strain state of graphene in free standing indentation and the effect of the pre-strain (ε0 ) on the free standing indentation response of graphene is also considered. We found that there is essentially no effective pre-tension in graphene during free standing indentation and the reported pre-tensile stress determined from the indentation tests does not show the true pre-stress state of graphene, which is a ‘fake stress’ caused by the assumption (the indenter tip displacement is equal to the displacement of graphene) typically used in the classic indentation analysis. A negative ε0 will increase the van der Waals (VDW) interaction between the indenter tip and graphene to cause a larger overestimation of both values of the elastic modulus (E) and the nonlinear elastic constant (c) of graphene from the classic indentation analysis. However, applying a positive ε0 in graphene, the VDW effect will be significantly decreased, and a more accurate value of E can be obtained, but the value of c will decrease to zero, which may become an effective way to more accurately obtain the elastic stiffness of graphene from indentation tests. (Some figures may appear in colour only in the online journal)

1. Introduction

In order to further validate the continuum analysis of the deformation of graphene, many theoretical and numerical studies have been performed. It has been reported that a graphene monolayer shows nonlinear elastic behavior under in-plane tension [15, 16], and its elastic stiffness decreases with increasing tensile load (i.e., the strain-soften behavior). The mechanical behavior of graphene monolayers is not sensitive to their chirality, and thus graphene can be considered as an isotropic structure [14]. In a free-standing indentation test, graphene is mounted loosely on a substrate containing cylindrical holes (well) and there is a small part of graphene that bulges into the well and adheres to the vertical wall of well by the van der Waals (VDW) interaction between the wall and the graphene [17]. This VDW interaction is typically considered to introduce a pre-tension into the graphene. Based on AFM measurements and classic analysis, Lee et al reported that there is a pre-tensile stress ranging from 0.07 to 0.74 N m−1 [6]. However, the measured

Due to its particular electrical, thermal, chemical and mechanical properties, graphene has attracted extensive research investigations [1–8]. It has been shown that graphene is one of the strongest materials ever found. Due to its thickness of an atomic level, measuring the elastic modulus of graphene monolayers is highly challenging. On the basis of a free standing indentation technique, Lee et al [6] measured the Young’s modulus (E) and ultimate strength of monolayer graphene using an atomic force microscope (AFM) and reported that E is around 1 TPa (quite close to that of carbon nanotubes). Computational simulations of the graphene indentation response have also been developed [9–14]. In the free standing indentation test [6], graphene is assumed to be an isotropic, elastic thin film with a clamped boundary. The indentation response is considered to be similar to the deformation under in-plane tension (i.e., neglecting the indenter tip effect and the bending stiffness of graphene). 0953-8984/13/475303+07$33.00

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pre-stress is too high—even higher than the fracture strength of many conventional materials—while the VDW attraction is typically not a strong interaction. Therefore, whether the VDW attraction is able to create such a large pre-tension is questioned. In the present work, we will investigate the true pre-stress state of graphene monolayers in free standing indentation as well as the effect of pre-strain on the indentation response of graphene. This work can provide a useful guideline to understand the properties of graphene monolayers determined from free standing indentation tests.

2. Study methods In the present work, molecular mechanics (MM) simulations are employed to study the free standing indentation response of graphene monolayers at the atomic level, and have been used to accurately reveal the deformation mechanisms of graphene [13, 14, 18], nanotubes [19, 20], nanowires [21] and nanofilms [22]. In MM simulations, the atomic structure of the system is optimized by searching for the minimum potential energy of the system, which is calculated from the specific potential energy function (i.e., force field function). Unlike molecular dynamic simulations, MM simulations do not consider the temperature effect. However, it has been reported that the elastic modulus of graphene is not sensitive to temperature [23]. Therefore, similar to other simple structures (e.g., nanotubes or nanofilms), the deformation of graphene monolayers can be effectively investigated by means of MM simulations. The COMPASS forcefield [24] is used to optimize the initial or deformed structures of the free standing graphene monolayer, which is the first ab initio force field and has been widely used in studying carbon-type materials [10, 13, 14, 19, 25–27]. A periodic boundary condition is applied in the lateral direction to remove the lateral boundary effect of the graphene sheet. Since the mechanical properties of graphene monolayers are not sensitive to their chirality [14], the chirality angle of graphene is simply selected as θ = 0◦ . To simplify the interaction between the indenter tip and graphene and show the intrinsic indentation mechanism, we select a cylindrical indenter tip instead of using a pyramidal or spherical tip, which can simplify the free standing indentation response as a uniaxial strain tension problem. The indenter tip size R = 0.67 nm and the graphene sheet length L ≈ 34.0 nm. In the present work, the range of L/R ≈ 50 is close to the range used by Lee et al [6] (e.g., L/R ≥ 35). The indenter tip is modeled by a rigid layer of carbon atoms which are uniformly distributed on a cylindrical surface; the axis of the cylinder is parallel to the lateral direction of graphene. Lee et al reported that the graphene covered on a SiO2 well is actually 0.2–1% longer than the actual diameter of the well [6]. In order to show the true boundary condition of graphene in free standing indentation, we built two rigid SiO2 side walls to model the substrate well, and graphene is folded into contact with the SiO2 wall, as shown in figure 1. After the structure is optimized, the indentation load is applied on the atoms located in the middle of the

Figure 1. Computational model of graphene with the side wall VDW adhesion in free standing indentation. (a) Initial structure; (b) deformed structure with a deflection of δ.

suspended graphene. In addition, in order to further show the pre-strain effect on the indentation response of graphene, a pre-strain (ε0 = −1.0–1.0%) is applied to the graphene sheet. A positive sign respresents pre-tension; a negative strain represents pre-compression. The atoms at the edge of the graphene sheet (not including the lateral edges) are fixed to model the clamped boundary conditions.

3. Results and discussion 3.1. Side wall VDW adhesion analysis The pre-stress in a suspended graphene monolayer created by the VDW interaction between graphene and the SiO2 side wall can be calculated from figure 1 (for simplicity only one layer of SiO2 is displayed in the figure). The in-plane strain of suspended graphene ε0 ≈ 3.0 × 10−4 from our MM simulations, which can be compared with the value reported by Lu et al [28] (ε0 = 8.26 × 10−4 based on the VDW interaction between substrates made by carbon atoms and graphene). The adhesion strength can be described by the VDW adhesion energy density 0. The value of 0 between SiO2 and graphene is about 0.1 J m−2 [17], which is lower than that between carbon and graphene (0 = 0.4 J m−2 ) [28]; thus, our result is lower than the value reported by Lu et al. The pre-strain ε0 is not sensitive to the length of the suspended graphene (L) or the length of the vertically aligned graphene (Lv ). In the present study range (L = 10–30 nm and Lv = 5–10 nm), the value of ε0 varies only slightly. Therefore, the VDW interaction between the substrate side wall and graphene is not strong enough to create an effective pre-tension in the suspended graphene sheet. With the indentation load P applied on the center of graphene, the vertically aligned graphene begins to peel off from the SiO2 side wall. During the peeling procedure, there is no stretching observed for the suspended graphene, and there is also no sliding observed for the vertically aligned graphene. 2

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Figure 3. Schematic of the true pre-strain state of graphene in free standing indentation. (a) With in-plane compression. (b) Buckled and bulged graphene. (c) With the VDW interaction between substrate wall and graphene. (d) Graphene is peeled off by the indentation load. Figure 2. The computational model of graphene with the side wall VDW adhesion under a tensile load along the x-direction. (a) Initial structure; (b) deformed structure.

3.2. True pre-stress state analysis Based on the classic free standing indentation analysis, the relationship between in-plane strain (ε) and the displacement of the graphene center δ with a pre-strain ε0 is: q ε = 4 (δ/L)2 + (1 + ε0 )2 − 1 ≈ k (δ/L)2 + ε0 , (1)

Therefore, a very small indentation load can break the side wall adhesion. In order to further test the side wall VDW adhesion strength, we directly applied a tensile load (controlled by displacement) on one SiO2 side wall (keeping the other one fixed) along the length direction (x-direction) (as shown in figure 2). With the tensile load, the distance between two side walls increases and the length of the suspended graphene (L) increases, but the vertical length (Lv ) decreases. The graphene peeled off from the side wall is converted into the suspended part, and there is still no effective stretching in the suspended graphene and no sliding for the vertically aligned graphene. These results show that the VDW adhesion is very strong to resist the load along the adhesion interface but very weak to resist the load normal to the interface. It is straightforward for the load to peel the graphene off by just needing to break the VDW adhesion between one carbon atom line of the vertically aligned graphene (located at the contact position) and the side wall, while the load to make the graphene slide on the side wall needs to break the VDW adhesion of all carbon atoms at the same time. From the above analysis, the VDW adhesion between the substrate side wall and graphene will not create an effective pre-tension in the suspended graphene. Actually, the phenomenon of a larger size of suspended graphene than the substrate well has been widely reported [6, 8, 17, 28]. The creation mechanism for this phenomenon is as follows (as shown in figure 3): (1) due to a lattice mismatch as well as strong adhesion, the SiO2 substrate produces an in-plane compression of graphene; (2) with the VDW attraction between the side wall and graphene, the graphene suspended over the substrate well is buckled to release the compressive stress and bulged into the substrate well; (3) the bulged graphene adhered to the side wall by the VDW interaction. Therefore, a larger size of graphene than the substrate well is not created by a pre-tension but by a pre-compression.

where L is the suspended membrane length (equal to the diameter of the substrate well) and the parameter k is the fitting parameter. The fitting parameter k varies slightly with ε0 : k = 1.963–1.9ε0 in the range (ε ≤ 5% and −1% ≤ ε0 ≤ 1%). Similar to under in-plane stretching, a thin film only has in-plane strain in indentation based on the classic indentation analysis. The strain energy density u of a film with a pre-strain ε0 under a uniaxial strain load is: 1 1 Es (2) ε 2 + cε 3 + σ0 ε, 2 21−ν 6 where ν is Poisson’s ratio, c is the third-order nonlinear elastic constant, the subscript s represents in-plane stretching and σ0 is the pre-stress corresponding to ε0 . If the nonlinear effect is neglected (c = 0), u can be simplified to: u=

1 Esl ε 2 + σ0 ε, (3) 2 1 − ν2 where the superscript l represents the linear strain–stress assumption. With a pre-strain ε0 , the strain energy density in free standing indentation will be: u=

l 2  1 Eind 2 k + ε , (4) (δ/L) 0 2 1 − ν2 where the subscript ind represents the indentation test. The work done by the indenter tip is equal to the system strain energy U, which leads to Z δ U= P dδ or P = dU/dδ, (5)

u=

0

where P is the indentation force and δ is the deflection of the membrane under the indenter tip. δ is typically assumed to be 3

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Figure 5. The calculated second-order elastic modulus Eind l and the pre-strain ε0 0 . The value of Eind l is normalized by its counterpart determined from in-plane tension with no pre-strain Es0 l . The dashed line is the reference line corresponding to ε0 0 = ε0 .

values of E and ε0 of graphene can be determined by fitting the indentation load–displacement curve as equation (6), which was also used by Lee et al [6]. The simulated P–δt relationship of a graphene monolayer is shown in figure 4(a), where the solid lines in the figure are the fitting curves by equation (6). The fitted value of ε0 is displayed in figure 5. For ε0 ≤ 0, the P–δt relationship does not fit well with equation (6) and the relationship between the indentation displacement (δt ) and the in-plane strain created by the indentation load (ε–ε0 ) does not match with equation (1) (as shown in figure 6(a)). This is caused by the coupling effect between the VDW interaction between the indenter tip and the graphene and the pre-strain in the graphene. This VDW effect will cause a difference between the displacement of the center of the graphene monolayer (δ) and the indenter tip displacement (δt ) (δ > δt ) [13]. A negative ε0 (vertical deflection) will further increase the VDW effect by increasing the contact area between the indenter tip and the graphene (as shown in figure 7). Theoretically, the P–δ relationship of graphene follows equation (6) and the relationship between δ and ε–ε0 matches with equation (1), and have been validated by our MM simulation results (as shown in figures 4(b) and 6(b)). In the classic analysis, δ is assumed to be equal to δt . Based on this assumption, a certain positive value of ε00 can be fitted from equation (6) based on the P–δt relationship of the graphene without pre-strain (ε0 = 0) (as shown in figure 5) and the accompanying pre-stress is:

Figure 4. The free standing indentation response of graphene, (a) the P–δt relationship; (b) the P–δ relationship. P is the indentation load, δt is the indenter tip displacement and δ is the displacement of graphene. Both the values of δt and δ are normalized by the graphene length L.

equal to the indenter tip displacement δt in the classic analysis. Thus, the P–δt relationship of a free standing membrane with a pre-strain ε0 under a cylindrical indenter tip is:     P 2k2 Eind l δt 3 2kEind l ε0 δt = + , (6) W L 1 − ν2 L 1 − ν2 where W is the width of the graphene membrane. It should be noted that for ε0 < 0, the pre-compression will be rapidly released by the vertical deflection of the film (i.e., there is no pre-compressive stress in graphene). Thus, in free standing indentation tests, the film with a negative ε0 is equivalent √ to the suspended film with an initial deflection (δ0 = L ε0 /k) and P can only be measured when δt ≥ δ0 ; the second term in equation (6) does not mean a pre-compressive stress, but rather represents the value of the initial deflection, and the

σ00 =

l Eind k(δ 3 − δt3 ) . 1 − ν2 δt L 2

(7)

This value is a ‘fake tensile stress’ and does not show the true pre-stress state of graphene in the free standing indentation test, which is directly created by the assumption used in the classic analysis (δ = δt ). For the cases with ε0 < 0, the fitted pre-strain ε0 0 from equation (6) is much lower than the 4

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Figure 7. The initial structures of a graphene monolayer in free standing indentation, (a) ε0 < 0 (−1.0%); (b) ε0 = 0; and (c) ε0 > 0 (1.0%).

3.3. Effect of pre-strain on the indentation response of graphene The values of the elastic modulus Eind l of graphene with a pre-strain (−1% ≤ ε0 ≤ 1%) fitted from equation (6) are displayed in figure 5. Eind l is normalized by Es0 l (the second-order elastic modulus determined from in-plane tension with ε0 = 0). For ε0 ≤ 0, there is a larger overestimation of Eind l (up to 30%), and the calculated pre-strain is much lower than the applied one (ε0 0 < ε0 ). However, if there is a positive ε0 , there is a very good estimation of Eind l , e.g., with ε0 = 0.01, Eind l /Esl ≈ 0.97 (only about 3% underestimation of the elastic modulus). This is mainly caused by the coupling effect between the VDW effect and the pre-strain in graphene. Our recent work has already shown that the classic analysis will overestimate the value of E due to neglecting the VDW effect [13]. The VDW effect is highly sensitive to the value of ε0 in graphene since the contact area between the indenter tip and graphene is highly sensitive to the pre-strain, e.g., a positive ε0 can significantly reduce the contact area and thus reduce the VDW effect. With a positive ε0 , the difference between δ and δt is reduced. The P–δt relationship is very close to the P–δ relationship and follows equation (6), while the relationship between δt and ε–ε0 matches with equation (1) (as shown in figures 4 and 6). In addition, these results also provide an effective method to more accurately determine the value of E of graphene by using the classic indentation analysis: adding a pre-tensile stress to graphene in indentation tests. The pre-strain effect on the nonlinear response of graphene can be analyzed by using strain energy analysis. At atomic scale, the strain energy density of graphene with a pre-strain ε0 in free standing indentation can be expressed as:  1 u = u0 + Evdw + Eg ε 2 2 2(1 − ν )  1 + 6 cvdw + cg ε 3 + σ0 ε, (8)

Figure 6. (a) The relationship between the normalized indenter tip displacement δt /L and the in-plane strain created in indentation ε–ε0 ; (b) the relationship between the normalized displacement of graphene δ/L and ε–ε0 .

applied value ε0 . However, using the P–δ relationship, the calculated pre-strain value ε0 0 is very close to ε0 . These results further show the difference between δ and δt can significantly affect the calculated pre-strain. In addition, the nonlinear response of graphene neglected by equation (6) may also have some influence on the fitted pre-strain. Therefore, the classic indentation analysis (i.e., the P–δt relationship) is not able to give the true pre-strain state of graphene. In the experimental work by Lee et al [6], they show a huge discrepancy between the pre-tensile stress (σ0 = 0.07–0.74 N m−1 ) and the pre-strain (ε0 = 0.2–1% corresponding to the pre-stress of 0.7–3.4 N m−1 ) in graphene. From the above pre-stress state analysis, it is clearly shown that the pre-strain represents a larger size of graphene than the substrate well which is created by the bulged graphene with a pre-compression (induced by the lattice mismatch between substrate and graphene); the pre-tensile stress determined is a ‘fake stress’ caused by the assumption (δ = δt ) used by the classic indentation analysis. Actually, graphene is not in the pre-tensile state in free indentation tests.

where u0 is the initial VDW interaction energy density, the second-order elastic stiffness: E = Eg + Evdw , and the third-order nonlinear elastic constant: c = cg + cvdw . The 5

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graphene obtained from in-plane tension with a pre-strain ε0 is also shown in figure 8, which is essentially insensitive to ε0 .

4. Conclusions Using MM simulations, we investigate the true pre-strain state of graphene in free standing indentation tests, and the effect of pre-strain on the indentation response of graphene is also considered. To the best of our knowledge, this is the first treatment showing the true pre-strain state of graphene in a free standing indentation test and analyzing the physical source of the pre-stress/pre-strain in graphene measured in indentation tests. Due to the lattice mismatch and the strong adhesion between the substrate and graphene, a pre-compression is created in graphene. Under the VDW interaction between the substrate side wall and graphene, the pre-compressed graphene (a negative ε0 ) bulged into the substrate well to release the pre-stress and then adhered to the substrate side wall. This side wall adhesion is very weak and no effective pre-tension is essentially created in the suspended graphene. Under the indentation load, the vertically aligned graphene is easily peeled off. Therefore, in free standing indentation tests, the graphene with a negative ε0 is equivalent to a suspended graphene with a certain deflection. The deflection of graphene created by the initial compression (a negative ε0 ) will increase the VDW effect, which increases the difference between the indenter tip displacement δt and the deflection of graphene δ (δt is assumed to be equal to δ in the classic analysis). Thus, the pre-tensile stress of graphene fitted from the P–δt relationship based on the classic analysis does not show the true pre-stress state of graphene, but is a kind of ‘fake stress’. The physical source of the fake stress is neglecting the difference between δt and δ (δt < δ) and the nonlinear response of graphene in the classic analysis (equation (6)). With a negative ε0 , the second-order elastic stiffness E will be overestimated by about 20–30% from the P–δt relationship based on the classic analysis; while the third-order nonlinear constant c can be overestimated by more than 400% since the VDW effect is highly nonlinear. However, a positive ε0 can significantly decrease the VDW effect in the indentation tests of graphene; thus, a quite accurate value of E can be determined from the classic analysis. This actually provides an effective way to quite accurately determine the value of E from free standing indentation analysis: applying a pre-tensile stress to graphene monolayer. In addition, a positive ε0 can significantly reduce the value of c to make graphene behave as a linear material.

Figure 8. The relationship between the third-order nonlinear elastic constant and the pre-strain ε0 . The values of the nonlinear constant are normalized by the nonlinear constant determined from in-plane tension with no pre-strain cs0 .

subscripts g and vdw represent the contributions from the deformation of graphene and the VDW interaction between the indenter tip and graphene. The pre-stress can be also separated into two components: σ0 = σvdw + σg , and the corresponding pre-strain ε0 = σ0 (1 − ν 2 )/E. By fitting the MM simulation results as equation (8), the second-order elastic modulus E and the third-order nonlinear elastic constant (c) can be fitted. The value of c and its components are normalized by the third-order nonlinear elastic constant obtained from the in-plane tension with no pre-strain cs0 , as shown in figure 8. Actually, there is no compressive pre-stress (negative σ0 ) in graphene in free standing indentation. When ε0 < 0, the stress is released by the lateral deflection—thus, σ0 = 0, i.e., u is essentially insensitive to a negative ε0 . The negative ε0 will change the contact area between the indenter tip and graphene; thus, it can still slightly affect u by affecting the values of u0 , Evdw and cvdw in equation (8). For a positive ε0 , both the values of cg and cvdw rapidly decrease to near zero. Thus, a positive ε0 can significantly reduce the nonlinear deformation of a graphene monolayer in free standing indentation, in the other words, with a pre-tension (ε0 > 0), a graphene monolayer can be effectively simplified as a linear material. For a negative ε0 , both the values of cg and cvdw rapidly increase with ε0 . Therefore, the values of c and its components are highly sensitive to the value of ε0 , which is also caused by the dependence of the VDW effect on the value of pre-strain ε0 , since the VDW interaction is highly nonlinear. The pre-strain is hardly avoided for a graphene monolayer mounted on the substrate; thus, the nonlinear property of graphene cannot be accurately determined using indentation analysis due to the coupling effect between the pre-strain and the VDW effect. As a reference, the third-order nonlinear elastic constant (cs ) of

Acknowledgments We acknowledge the financial support provided by the Ministry of Science and Technology of China (2013CB933702 and 2010CB832904) and the National Natural Science Foundation of China (11172002 and 11075005). 6

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Boundary condition and pre-strain effects on the free standing indentation response of graphene monolayer.

Using molecular mechanics simulations, we investigated the true pre-stress/pre-strain state of graphene in free standing indentation and the effect of...
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