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Comparison of continuum-based and atomistic-based modeling of axial buckling of carbon nanotubes subject to hydrostatic pressure
Computational Materials Science 79 (2013) 619–626
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Comparison of continuum-based and atomistic-based modeling of axial buckling of carbon nanotubes subject to hydrostatic pressure B. Motevalli c,d, A. Montazeri b,d, J.Z. Liu c, H. Rafii-Tabar a,d,⇑ a Department of Medical Physics and Biomedical Engineering, and Research Center for Medical Nanotechnology and Tissue Engineering, Shahid Beheshti University of Medical Sciences, Evin, Tehran, Iran b Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran c Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Australia d Computational Physical Sciences Research Laboratory, School of Nano-Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
a r t i c l e
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Article history: Received 21 January 2013 Received in revised form 16 May 2013 Accepted 21 June 2013
Keywords: Molecular dynamics simulation Aspect ratio Buckling Hydrostatic pressure Carbon nanotube
a b s t r a c t Molecular dynamics simulation (MD) is used to investigate the effect of hydrostatic pressure on the axial buckling of single-walled carbon nanotube (SWCNT). Three different types of nanotube with small, medium and large aspect ratios (ARs) are considered. It is known that continuum-based models for axial buckling of nanotubes subjected to hydrostatic pressure are only applicable to those with small ARs, and that they are incapable of representing nanotubes with large ARs. In this paper, by employing MD simulations, the contribution of hydrostatic pressure to the axial buckling of nanotubes is studied for all ranges of ARs. In the case of small ARs, our MD results prove to be in agreement with those obtained from continuum-based models. However, concerning an SWCNT with a large AR, we found that the effect of hydrostatic pressure on the buckling properties is in total opposition to nanotubes with small ARs. This new trend is in contrast to the general conclusions, previously reported in continuum models, on how hydrostatic pressure affects the buckling properties. We conclude that the AR plays an essential role in determining the observed changes in the critical buckling loads when a nanotube is subjected to both an internal and an external hydrostatic pressure. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Carbon nanotubes (CNTs) are quasi one-dimensional nanostructures that have attracted considerable attention in various branches of nanoscience and nanotechnology [1]. Studying the mechanical characteristics of CNTs is an important research topic, considering their broad domain of applications such as nanosized strain sensors and actuators, nanofluidic devices, drug delivery platforms, reinforcement agents in composites, and as building blocks in many nano-electromechanical systems (NEMS). Since the reliability of many nanodevices depends in a critical way on our understanding of the response characteristics of CNTs to the axial compressive loading, the study of the stability behavior of these exotic nanostructures can be a great help in their potential applications as basic elements in many nanotechnology-based devices. Consequently, investigation into their buckling mode geometries and properties has occupied a centre stage in many
⇑ Corresponding author at: Department of Medical Physics and Biomedical Engineering, and Research Center for Medical Nanotechnology and Tissue Engineering, Shahid Beheshti University of Medical Sciences, Evin, Tehran, Iran. Tel.: +98 2123872566; fax: +98 2122439941. E-mail address: rafi[email protected] (H. Rafii-Tabar). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.06.043
of the basic and applied nanomechanics research programmes. Yakobson et al. [2] used molecular dynamics (MD) simulations to characterize the buckling properties of single-walled carbon nanotubes (SWCNTs), when subjected to an axial compressive load. They compared the results of their atomistic scale modeling for axial compressive buckling with results obtained from a simple continuum-based shell model, and found a remarkable agreement between the two sets of results. The buckling modes of CNTs, with different lengths and radii, when subjected to axial compression have also been studied by several groups [3,4]. For example, using MD simulation, Feliciano et al. [4] showed that the differences in aspect ratios lead to distinct buckling modes in SWCNTs. A small-aspect-ratio SWCNT primarily exhibits a shell-type buckling, whereas an increase in the aspect ratio leads to a columnar-type buckling mode. Their findings also demonstrated that further compression of the already columnar-buckled SWCNT returns to a shell-type buckling. Furthermore, the effects of vacancy defects and dopants on the buckling behavior of nanotubes have also been studied [5,6]. Other MD-based studies of CNTs subject to axial, torsional, and bending loads, as well as internal and external pressures, have also been reported by other workers [7–12]. Although there are many theoretical and experimental studies concerned with the buckling properties of CNTs when subjected
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to simple axial loads, due to computational complexity, the investigation of these properties when combined loads are applied has received scant attention. Recently, we investigated the buckling behavior of SWCNTs under combined torsional and compressive loads using MD simulation [13]. New results on the buckling mode shapes of SWCNTs under different rates of axial and rotational displacement were obtained. In practical applications, such as in composite materials, CNTs often experience a combination of different loads, simultaneously. A special case which may emerge in practical applications is when the CNTs simultaneously experience both axial compression and internal/external hydrostatic pressures, e.g., axially compressed CNTs embedded in an environment that also imposes an external pressure [14]. Also, it has been shown that CNTs can act as smart carriers at nanoscales. The carrier can act as a template in biological applications [15], in hydrogen transport [16], and for the purpose of gas purification [17]. Using a series of MD simulations, Ni et al. [18] demonstrated that regardless of the filling material, filled CNTs can withstand significantly higher buckling forces compared to the unfilled ones. It is thus important to investigate the mechanical buckling of CNTs under such loading conditions. Only few studies are available on the critical buckling load of CNTs when subjected to the combined axial and hydrostatic pressure loads, and all the available studies are based on continuumbased models. Ru [19] used an elastic double-shell model to study the axially compressed buckling of a double-walled carbon nanotube (DWCNT) embedded in an elastic medium. To extend their studies, Ru et al. [20,21] have also developed a multi-shell model to study a similar problem on an individual multi-walled carbon nanotube (MWCNT). Accordingly, they obtained the critical axial stress and the buckling modes for various radial pressures, and they compared the results with those of SWCNTs with the same loading condition. Shen and Zhang [22] also developed a continuum-based shell model and included the effect of defects and thermal environment on the buckling properties of CNTs under combined axial and radial loads. From these studies, it was observed that internal pressure postpones the axial buckling into higher strain, while external pressure prompts the nanotube to collapse earlier at lower strains. However, as demonstrated in the present paper, these conclusions obtained from continuum-based models are only applicable to CNTs that lie within a range of AR wherein the nanotube buckles predominantly in a shell mode. For CNTs with larger ARs, however, an opposite trend is observed. Moreover, it is revealed that for certain ARs, referred to as medium range, the trend of the effect of pressure does not follow the trend of either CNTs with small ARs (predominant buckling in shell mode) or CNTs having large values of AR. The postbuckling behavior of DWCNTs under hydrostatic pressure has also been studied using continuum-based models [22,23], wherein each constituent nanotube in the DWCNT was described as an individual elastic shell and the interlayer van der Waals normal stress was taken into account, and the interlayer friction was assumed to be negligible. The nonlinear prebuckling deformations of the shell and the van der Waals interaction were both accommodated. It was shown that whereas SWCNTs have a stable postbuckling path, DWCNTs have an unstable postbuckling behavior due to the presence of van der Waals interaction. Finite element (FE) models were also used to analyze the effects of radial (hydrostatic) pressure on the axial buckling of these nanostructures [24]. The results of FE-based simulations of SWCNTs were compared with the classical (local) and nonlocal continuum theories. It is worth noting that there exist two different buckling regimes for thin-walled cylinders under compressive axial loading. One is the columnar (global) buckling, in which the cylinder keeps its circular cross section and buckles sideways as a whole, in a
Fig. 1. Transmission electron microscopy images of buckled nanotubes observed in the nanotube-polymer composites: (a) global buckling and (b) local buckling [26].
Table 1 Geometrical parameters of three selected samples of SWCNTs having different ARs. Type
(n, m)
L (Å)
L/R
Small AR Large AR Medium AR
(15, 15) (5, 5) (10, 10)
51.56 100.84 100.84
5.07 29.75 14.87
similar fashion to a beam buckling. The other is the shell (local) buckling, in which the cylinder deforms in a wavy fashion to decrease the compressive energy, while the cylinder’s axis remains straight just as in a thin-shell buckling [25]. Usually, shell buckling happens in cylinders with small ARs, whereas the occurrence of columnar buckling is expected to take place in cylinders that have large ARs. CNTs behave very much like thin-walled cylindrical structures and embrace the shell buckling and columnar buckling regimes under compressive axial loading, as shown in Fig. 1. As mentioned earlier, the effect of large ARs on the buckling behavior of CNTs subjected to hydrostatic pressure has not been addressed in the previous continuum-based studies. Their results demonstrated similar trends for the effect of radial pressure on the critical axial stress of nanotubes having different values of ARs. In general, the results show that the critical load and strain increase almost linearly as the internal pressure is increased, while the external pressure has an opposite effect. This is due to the fact that all of these continuum-based methods implement shell theory to extract the governing equations, a theory which is only applicable to nanotubes with small ARs. It should be noted that CNTs used in composites and other devices, such as NEMS and field emission transistors, must be long enough to show their superior mechanical and electrical properties. Thus, investigation of the effect of internal/external pressure loads on the buckling properties of CNTs with large ARs is a necessary topic of research which cannot be covered by continuum-based models due to the impossibility of applying hydrostatic pressures in the beam theories used for these types of nanotubes. In this paper, three case studies involving SWCNTs with different ARs are investigated. Our results reveal the emergence of a new physical phenomenon during buckling due to the influence of nanotube AR. We have found that the buckling mode of an SWCNT under pure axial compression (shell or columnar buckling) determines how the pressure affects the buckling properties of SWCNTs. In the next section, the geometrical models and simulation details are given. Subsequently, the corresponding results for these samples are represented and discussed in Section 3. Finally, some concluding remarks are summarized. 2. Model The Tersoff–Brenner potential energy function (PEF) [27,28], which has proven its accuracy in modeling the interaction of
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Fig. 2. Buckling deformation of a (15,15) armchair SWCNT (Sample 1 as a typical SWCNT with small AR) under hydrostatic pressure: (a) pure compression, (b) 1 GPa, (c) 3 GPa, (d) 5 GPa, (e) 1 GPa, (f) 2 GPa, (g) 3 GPa, (h) 4 GPa.
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Fig. 3. The effect of hydrostatic pressure on the critical compressive load and critical strain (Sample 1: (15,15) SWCNT with small AR).
covalently bonding carbon atoms, was used to describe the energetics of CNTs. The potential has the general form of
V Br ðr ij Þ ¼ f c ðr ij Þ ½V R ðr ij Þ þ Bij V A ðr ij Þ
ð2:1Þ
where VR represents the repulsive pair-wise potential, such as the core–core interaction, and VA represents the bonding due to the valence electrons. The many-body feature of the potential is represented by the bond-order function Bij between atoms i and j, which depends on the local atomic environment in which a particular bond is located and fc is a cut-off function to account for the nearest neighbors of an atom i. Detailed explanation of the terms of this potential can be found in [25]. The present simulations were conducted in a canonical ensemble at a constant temperature of T = 300 K. To control the temperature, the Nosè–Hoover thermostat [29] was used and the velocity-Verlet integration algorithm [30] was utilized to integrate the equations of motion, with the simulation time-step set at dt = 1 fs. To apply a hydrostatic pressure, equivalent radial forces were imposed on each atom, while the imposition of an axial compressive load was modeled by incremental displacements. In each simulation, the system was initially relaxed for 3500dt, and then the equivalent radial forces were imposed on each atom and the system was relaxed again for 10,000dt. Thereafter, while the equivalent forces were being applied, an incremental displacement was applied and the system was relaxed for 3000dt. Afterwards, the incremental displacements were imposed until the onset of buckling in the structure. 3. Results and discussions Three case studies involving an SWCNT with different ARs were investigated as three samples having small, large, and medium values of AR. The designation of small, medium, and large ARs are according to [1]. The geometrical parameters for these three samples are listed in Table 1. The results, given in this section, proved that, depending on the AR, the hydrostatic pressure has a totally different effect on the critical compressive load of the SWCNT. 3.1. The effect of hydrostatic pressure on SWCNTs with small AR In this case, referred to as Sample 1, a (15,15) armchair SWCNT with a length of 51.56 Å was taken. The buckling deformation of this SWCNT under pure compression is demonstrated in Fig. 2a.
As revealed, the SWCNT deforms in a shell buckling mode with local deformations appearing on its wall. The corresponding critical compressive load and critical strain for this case were obtained as 31.47 nN and 2.5%, respectively. The effect of internal pressure on the buckling properties of nanotube is displayed on the righthand side of Fig. 3. As shown in this figure, the critical load and strain increase almost linearly as the internal pressure increases. Similar trends were also observed in the continuum-based shell theory studies [19–22]. Indeed, due to the buckling mode of this SWCNT under pure compression, such a trend was expected. Actually, the internal pressure imposes stretching forces on the wall of the nanotube, causing resistance against local deformations. As shown in Figs. 2b–d, the internal pressure does not change the buckling deformation of SWCNTs with small AR significantly, and that the shape is similar to that of pure compression. On the other hand, as is shown on the left-hand side of Fig. 3, the external pressure weakens the SWCNT against compressive load. In fact, in this case, the external pressure pushes the atoms inward, resulting in a stronger tendency of the nanotube to collapse under axial compressive load. We note that [19–22] also obtained similar trends on the effect of hydrostatic pressure for SWCNTs. From Figs. 2e–h, it is observed that the external pressure affects the buckling deformation mode, especially, at high pressures. Accordingly, in the last two cases, the nanotube was totally crumpled. In Fig. 3, it is seen that when the external pressure is 4 GPa, the nanotube deforms without the imposition of any compressive axial load. However, this pressure may not be the critical one. As seen in Fig. 3, when the external pressure is 3 GPa, the nanotube buckles when a very small strain is applied. Thus, one can conclude that the critical pressure is between 3 and 4 GPa for this sample. 3.2. The effect of hydrostatic pressure on SWCNTs with large AR As an example of this type of nanotube, a (5,5) armchair SWCNT with a length of 100.84 Å, referred to as Sample 2, was considered. The buckling deformation of this nanotube under pure compression is demonstrated in Fig. 4a. As seen from this figure, the nanotube buckles in a columnar mode with a global deformation geometry. Also, the corresponding critical load and critical strain were obtained to be 21.97 nN and 3.97%, respectively. Fig. 5 displays the variation of buckling properties under different internal and external pressures. Surprisingly, an opposite trend is observed in this case in comparison with the previous case and the cases
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Fig. 4. Buckling deformation of a (5,5) armchair SWCNT (Sample 2 as a typical SWCNT with large AR) under hydrostatic pressure: (a) pure compression, (b) 1 GPa, (c) 2 GPa, (d) 1.5 GPa, (e) 3 GPa, (f) 5 GPa.
reported in the literature [19–22]. In contrast to the nanotubes with small AR, it is seen that the internal pressure weakens the buckling properties of the SWCNT, while, the external pressure improves these properties. We note that the presence of internal pressure increases the stretching stress on the wall of the nanotube. The presence of this stress enhances the instability of nanotube against the columnar-buckling mode. In other words, with the presence of such stress, the tendency of the nanotube towards global buckling increases. The right hand-side of Fig. 5 shows how internal pressure has reduced the critical buckling load and the strain. Moreover, as is shown in Figs. 4b and c, the inter-
nal pressure does not affect the buckling mode of this type of SWCNT. However, in the case of external pressure, the situation is reversed. Note that in general, the imposition of an axial compression causes a lateral expansion of the nanotube. The instability of the nanotube against the global buckling increases with this lateral expansion. Since in the case of an external pressure, the forces acting on the nanotube resist the expansion, the stability of the nanotube against global buckling is enhanced. Fig. 5 reveals that external pressure has a significant effect on the stability of nanotubes having large AR against columnar buckling under
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Fig. 5. The effect of hydrostatic pressure on the critical compressive load and critical strain (Sample 2: (5,5) SWCNT with large AR).
compressive load. However, it is observed that the external pressure enhances the critical load and strain up to a certain pressure (around 2 GPa), while beyond this pressure, the buckling properties decline almost linearly. It is also observed that beyond this point, the buckling properties decrease gradually with a linear variation. It is believed that the decline in the buckling properties is due to the change in the buckling mode shape over this specific pressure. As seen in Fig. 4e, when the external pressure is increased to 1.5 GPa, the buckling deformation is totally changed. It is observed that high external pressure causes the wall of the long nanotube to deform and collapse into a twisted shape under the compressive load, and no column buckling is seen. As the tube buckling mode geometry is changed into this new form, higher external pressure raises the tendency of the tube to deform into a twisted shape with collapsed walls, causing higher degradation in the buckling properties. To be specific, it can be concluded that the external pressure improves the buckling properties of SWCNTs having large AR up to the point that the buckling mode is still columnar. Thereafter, in the transition zone from columnar buckling to collapsing walls (local deformation), a slight change in buckling properties occurs. After crossing this region, the external pressure weakens the buckling properties due to its enforcement of the wrinkling deformation. 3.3. The effect of hydrostatic pressure on SWCNTs with medium AR In this case, it was found that for certain ARs the trend of changes in buckling properties under hydrostatic pressure is not similar to the two previous cases. These ARs are located between small and large ones, and are here referred to as medium ARs. Actually, one may consider the medium AR to lie in a range wherein the buckling mode changes from a predominantly columnar buckling to a predominantly shell buckling mode. As an example for this case, a (10,10) armchair SWCNT with a length of 100.84 Å, which is referred to as Sample 3, was considered. The axial buckling deformation of this SWCNT under pure compression is displayed in Fig. 6a. As can be seen from this figure, the nanotube is deformed locally with a lobe at its middle. Also, the corresponding critical load and critical strain were obtained as 39 nN and 3.72%, respectively. The effect of external and internal pressures on this SWCNT is displayed in Fig. 7. From the right-hand
side of this figure, it can be seen that the internal pressure has a slight effect on the buckling properties of this SWCNT until the pressure reaches 2 GPa. It is observed that up to 1 GPa, the buckling properties slightly improves, while up to 2 GPa these properties slightly decline. After crossing this point, a significant drop is observed in these properties. This significant drop can be explained by the effect of internal pressure on the buckling deformation of the SWCNT. As shown in Fig. 6, the internal pressure has a significant effect on the buckling deformation of this SWCNT, for which, when the pressure is increased, the buckling deformation transforms from a local deformation to a global columnar deformation. Indeed, since at first the wall of the SWCNT tends to deform locally, the insertion of internal pressure resists against this mechanism of deformation, for which a slight improvement is observed in the buckling properties (Fig. 7). However, it should be noted that the internal pressure also decreases the stability of the tube along the axial direction resulting in an increase in the tendency of the nanotube to buckle in a columnar shape. As the internal pressure promotes a dominant tendency in the nanotube to buckle in a columnar shape (after 2 GPa), also seen in the previous case, a higher internal pressure leads to a higher unstable condition of the SWCNT, resulting in a significant drop of the buckling properties. On the other hand, since this SWCNT tends to buckle locally under pure compressive load, exerting external pressure weakens its buckling properties, as described in Section 3.1. Fig. 7 reveals that external pressure affects the critical load and the critical strain, significantly. It is also observed that external pressure has a significant effect on the buckling deformation of SWCNTs with medium ARs. Figs. 6i and h, show that the SWCNT is totally flattened under high external pressure. Regarding our results for the three different ARs, we can conclude that the buckling mode shape plays an important role in indicating how the hydrostatic pressure affects the mechanism of the buckling. Furthermore, the presence of an internal pressure raises the tendency of the global buckling, while the imposition of an external pressure paves the way for a local buckling deformation to take place. We have observed that, the hydrostatic pressure can change the buckling mechanism from a local deformation to a global one (e.g. Fig. 7), and vice versa (Fig. 5). Accordingly, whenever the nanotube buckles predominantly in local mode, the external pressure would decrease the critical buckling load, while the
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Fig. 6. Buckling deformation of a (10,10) armchair SWCNT (Sample 3 as a typical SWCNT with medium AR) under hydrostatic pressure: (a) pure compression, (b) 0.5 GPa, (c) 1 GPa, (d) 1.5 GPa, (e) 3 GPa, (f)1.5 GPa, (g) 2 GPa, (h) 3 GPa, (i) 4 GPa.
internal pressure would postpone the occurrence of buckling. However, in the case of global buckling, the effect of pressure is reversed.
4. Conclusions The stability behavior of SWCNTs when subjected to combined loads of axial compression and hydrostatic pressure has been investigated using a set of MD-based simulations. Continuumbased models can only employ shell theories to obtain the dynamic equations of SWCNTs under hydrostatic pressure and axial compression. These theories are, however, only applicable to nano-
tubes with small ARs. These continuum-based studies have generally observed that the internal pressure postpones the critical axial stress and critical strain, while the external pressure weakens the resistance of a nanotube under axial load. Here, since we have employed atomistic-based simulation, we have been able to explore the effect of internal/external pressure loads on the buckling properties of SWCNTs with all types of ARs. To investigate the role of pressure, three samples of SWCNTs with different ARs were investigated. It has been demonstrated that the results from continuum-based studies are in agreement only with the case of nanotubes having small values of AR, where the nanotube predominantly buckles in a shell mode. Meanwhile, in the case of SWCNTs with large ARs, internal pressure reduces the stability of
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Fig. 7. The effect of hydrostatic pressure on the critical compressive load and critical strain (Sample 3: (10,10) SWCNT with medium AR).
the nanotube against the axial load, while external pressure enhances its stability, leading to a totally opposite trend in comparison with nanotubes having small ARs. Furthermore, it was observed that for a range of ARs, the trend of the variation of buckling properties due to the pressure follows none of these distinct cases. Finally, it is concluded that the buckling deformation has an important role in determining whether increasing the internal/external pressure postpones the buckling or accelerates it.
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Contents lists available at ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Comparison of continuum-based and atomistic-based modeling of axial buckling of carbon nanotubes subject to hydrostatic pressure B. Motevalli c,d, A. Montazeri b,d, J.Z. Liu c, H. Rafii-Tabar a,d,⇑ a Department of Medical Physics and Biomedical Engineering, and Research Center for Medical Nanotechnology and Tissue Engineering, Shahid Beheshti University of Medical Sciences, Evin, Tehran, Iran b Faculty of Mechanical Engineering, K.N. Toosi University of Technology, Tehran, Iran c Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Australia d Computational Physical Sciences Research Laboratory, School of Nano-Science, Institute for Research in Fundamental Sciences (IPM), Tehran, Iran
a r t i c l e
i n f o
Article history: Received 21 January 2013 Received in revised form 16 May 2013 Accepted 21 June 2013
Keywords: Molecular dynamics simulation Aspect ratio Buckling Hydrostatic pressure Carbon nanotube
a b s t r a c t Molecular dynamics simulation (MD) is used to investigate the effect of hydrostatic pressure on the axial buckling of single-walled carbon nanotube (SWCNT). Three different types of nanotube with small, medium and large aspect ratios (ARs) are considered. It is known that continuum-based models for axial buckling of nanotubes subjected to hydrostatic pressure are only applicable to those with small ARs, and that they are incapable of representing nanotubes with large ARs. In this paper, by employing MD simulations, the contribution of hydrostatic pressure to the axial buckling of nanotubes is studied for all ranges of ARs. In the case of small ARs, our MD results prove to be in agreement with those obtained from continuum-based models. However, concerning an SWCNT with a large AR, we found that the effect of hydrostatic pressure on the buckling properties is in total opposition to nanotubes with small ARs. This new trend is in contrast to the general conclusions, previously reported in continuum models, on how hydrostatic pressure affects the buckling properties. We conclude that the AR plays an essential role in determining the observed changes in the critical buckling loads when a nanotube is subjected to both an internal and an external hydrostatic pressure. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction Carbon nanotubes (CNTs) are quasi one-dimensional nanostructures that have attracted considerable attention in various branches of nanoscience and nanotechnology [1]. Studying the mechanical characteristics of CNTs is an important research topic, considering their broad domain of applications such as nanosized strain sensors and actuators, nanofluidic devices, drug delivery platforms, reinforcement agents in composites, and as building blocks in many nano-electromechanical systems (NEMS). Since the reliability of many nanodevices depends in a critical way on our understanding of the response characteristics of CNTs to the axial compressive loading, the study of the stability behavior of these exotic nanostructures can be a great help in their potential applications as basic elements in many nanotechnology-based devices. Consequently, investigation into their buckling mode geometries and properties has occupied a centre stage in many
⇑ Corresponding author at: Department of Medical Physics and Biomedical Engineering, and Research Center for Medical Nanotechnology and Tissue Engineering, Shahid Beheshti University of Medical Sciences, Evin, Tehran, Iran. Tel.: +98 2123872566; fax: +98 2122439941. E-mail address: rafi[email protected] (H. Rafii-Tabar). 0927-0256/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.commatsci.2013.06.043
of the basic and applied nanomechanics research programmes. Yakobson et al. [2] used molecular dynamics (MD) simulations to characterize the buckling properties of single-walled carbon nanotubes (SWCNTs), when subjected to an axial compressive load. They compared the results of their atomistic scale modeling for axial compressive buckling with results obtained from a simple continuum-based shell model, and found a remarkable agreement between the two sets of results. The buckling modes of CNTs, with different lengths and radii, when subjected to axial compression have also been studied by several groups [3,4]. For example, using MD simulation, Feliciano et al. [4] showed that the differences in aspect ratios lead to distinct buckling modes in SWCNTs. A small-aspect-ratio SWCNT primarily exhibits a shell-type buckling, whereas an increase in the aspect ratio leads to a columnar-type buckling mode. Their findings also demonstrated that further compression of the already columnar-buckled SWCNT returns to a shell-type buckling. Furthermore, the effects of vacancy defects and dopants on the buckling behavior of nanotubes have also been studied [5,6]. Other MD-based studies of CNTs subject to axial, torsional, and bending loads, as well as internal and external pressures, have also been reported by other workers [7–12]. Although there are many theoretical and experimental studies concerned with the buckling properties of CNTs when subjected
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to simple axial loads, due to computational complexity, the investigation of these properties when combined loads are applied has received scant attention. Recently, we investigated the buckling behavior of SWCNTs under combined torsional and compressive loads using MD simulation [13]. New results on the buckling mode shapes of SWCNTs under different rates of axial and rotational displacement were obtained. In practical applications, such as in composite materials, CNTs often experience a combination of different loads, simultaneously. A special case which may emerge in practical applications is when the CNTs simultaneously experience both axial compression and internal/external hydrostatic pressures, e.g., axially compressed CNTs embedded in an environment that also imposes an external pressure [14]. Also, it has been shown that CNTs can act as smart carriers at nanoscales. The carrier can act as a template in biological applications [15], in hydrogen transport [16], and for the purpose of gas purification [17]. Using a series of MD simulations, Ni et al. [18] demonstrated that regardless of the filling material, filled CNTs can withstand significantly higher buckling forces compared to the unfilled ones. It is thus important to investigate the mechanical buckling of CNTs under such loading conditions. Only few studies are available on the critical buckling load of CNTs when subjected to the combined axial and hydrostatic pressure loads, and all the available studies are based on continuumbased models. Ru [19] used an elastic double-shell model to study the axially compressed buckling of a double-walled carbon nanotube (DWCNT) embedded in an elastic medium. To extend their studies, Ru et al. [20,21] have also developed a multi-shell model to study a similar problem on an individual multi-walled carbon nanotube (MWCNT). Accordingly, they obtained the critical axial stress and the buckling modes for various radial pressures, and they compared the results with those of SWCNTs with the same loading condition. Shen and Zhang [22] also developed a continuum-based shell model and included the effect of defects and thermal environment on the buckling properties of CNTs under combined axial and radial loads. From these studies, it was observed that internal pressure postpones the axial buckling into higher strain, while external pressure prompts the nanotube to collapse earlier at lower strains. However, as demonstrated in the present paper, these conclusions obtained from continuum-based models are only applicable to CNTs that lie within a range of AR wherein the nanotube buckles predominantly in a shell mode. For CNTs with larger ARs, however, an opposite trend is observed. Moreover, it is revealed that for certain ARs, referred to as medium range, the trend of the effect of pressure does not follow the trend of either CNTs with small ARs (predominant buckling in shell mode) or CNTs having large values of AR. The postbuckling behavior of DWCNTs under hydrostatic pressure has also been studied using continuum-based models [22,23], wherein each constituent nanotube in the DWCNT was described as an individual elastic shell and the interlayer van der Waals normal stress was taken into account, and the interlayer friction was assumed to be negligible. The nonlinear prebuckling deformations of the shell and the van der Waals interaction were both accommodated. It was shown that whereas SWCNTs have a stable postbuckling path, DWCNTs have an unstable postbuckling behavior due to the presence of van der Waals interaction. Finite element (FE) models were also used to analyze the effects of radial (hydrostatic) pressure on the axial buckling of these nanostructures [24]. The results of FE-based simulations of SWCNTs were compared with the classical (local) and nonlocal continuum theories. It is worth noting that there exist two different buckling regimes for thin-walled cylinders under compressive axial loading. One is the columnar (global) buckling, in which the cylinder keeps its circular cross section and buckles sideways as a whole, in a
Fig. 1. Transmission electron microscopy images of buckled nanotubes observed in the nanotube-polymer composites: (a) global buckling and (b) local buckling [26].
Table 1 Geometrical parameters of three selected samples of SWCNTs having different ARs. Type
(n, m)
L (Å)
L/R
Small AR Large AR Medium AR
(15, 15) (5, 5) (10, 10)
51.56 100.84 100.84
5.07 29.75 14.87
similar fashion to a beam buckling. The other is the shell (local) buckling, in which the cylinder deforms in a wavy fashion to decrease the compressive energy, while the cylinder’s axis remains straight just as in a thin-shell buckling [25]. Usually, shell buckling happens in cylinders with small ARs, whereas the occurrence of columnar buckling is expected to take place in cylinders that have large ARs. CNTs behave very much like thin-walled cylindrical structures and embrace the shell buckling and columnar buckling regimes under compressive axial loading, as shown in Fig. 1. As mentioned earlier, the effect of large ARs on the buckling behavior of CNTs subjected to hydrostatic pressure has not been addressed in the previous continuum-based studies. Their results demonstrated similar trends for the effect of radial pressure on the critical axial stress of nanotubes having different values of ARs. In general, the results show that the critical load and strain increase almost linearly as the internal pressure is increased, while the external pressure has an opposite effect. This is due to the fact that all of these continuum-based methods implement shell theory to extract the governing equations, a theory which is only applicable to nanotubes with small ARs. It should be noted that CNTs used in composites and other devices, such as NEMS and field emission transistors, must be long enough to show their superior mechanical and electrical properties. Thus, investigation of the effect of internal/external pressure loads on the buckling properties of CNTs with large ARs is a necessary topic of research which cannot be covered by continuum-based models due to the impossibility of applying hydrostatic pressures in the beam theories used for these types of nanotubes. In this paper, three case studies involving SWCNTs with different ARs are investigated. Our results reveal the emergence of a new physical phenomenon during buckling due to the influence of nanotube AR. We have found that the buckling mode of an SWCNT under pure axial compression (shell or columnar buckling) determines how the pressure affects the buckling properties of SWCNTs. In the next section, the geometrical models and simulation details are given. Subsequently, the corresponding results for these samples are represented and discussed in Section 3. Finally, some concluding remarks are summarized. 2. Model The Tersoff–Brenner potential energy function (PEF) [27,28], which has proven its accuracy in modeling the interaction of
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Fig. 2. Buckling deformation of a (15,15) armchair SWCNT (Sample 1 as a typical SWCNT with small AR) under hydrostatic pressure: (a) pure compression, (b) 1 GPa, (c) 3 GPa, (d) 5 GPa, (e) 1 GPa, (f) 2 GPa, (g) 3 GPa, (h) 4 GPa.
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Fig. 3. The effect of hydrostatic pressure on the critical compressive load and critical strain (Sample 1: (15,15) SWCNT with small AR).
covalently bonding carbon atoms, was used to describe the energetics of CNTs. The potential has the general form of
V Br ðr ij Þ ¼ f c ðr ij Þ ½V R ðr ij Þ þ Bij V A ðr ij Þ
ð2:1Þ
where VR represents the repulsive pair-wise potential, such as the core–core interaction, and VA represents the bonding due to the valence electrons. The many-body feature of the potential is represented by the bond-order function Bij between atoms i and j, which depends on the local atomic environment in which a particular bond is located and fc is a cut-off function to account for the nearest neighbors of an atom i. Detailed explanation of the terms of this potential can be found in [25]. The present simulations were conducted in a canonical ensemble at a constant temperature of T = 300 K. To control the temperature, the Nosè–Hoover thermostat [29] was used and the velocity-Verlet integration algorithm [30] was utilized to integrate the equations of motion, with the simulation time-step set at dt = 1 fs. To apply a hydrostatic pressure, equivalent radial forces were imposed on each atom, while the imposition of an axial compressive load was modeled by incremental displacements. In each simulation, the system was initially relaxed for 3500dt, and then the equivalent radial forces were imposed on each atom and the system was relaxed again for 10,000dt. Thereafter, while the equivalent forces were being applied, an incremental displacement was applied and the system was relaxed for 3000dt. Afterwards, the incremental displacements were imposed until the onset of buckling in the structure. 3. Results and discussions Three case studies involving an SWCNT with different ARs were investigated as three samples having small, large, and medium values of AR. The designation of small, medium, and large ARs are according to [1]. The geometrical parameters for these three samples are listed in Table 1. The results, given in this section, proved that, depending on the AR, the hydrostatic pressure has a totally different effect on the critical compressive load of the SWCNT. 3.1. The effect of hydrostatic pressure on SWCNTs with small AR In this case, referred to as Sample 1, a (15,15) armchair SWCNT with a length of 51.56 Å was taken. The buckling deformation of this SWCNT under pure compression is demonstrated in Fig. 2a.
As revealed, the SWCNT deforms in a shell buckling mode with local deformations appearing on its wall. The corresponding critical compressive load and critical strain for this case were obtained as 31.47 nN and 2.5%, respectively. The effect of internal pressure on the buckling properties of nanotube is displayed on the righthand side of Fig. 3. As shown in this figure, the critical load and strain increase almost linearly as the internal pressure increases. Similar trends were also observed in the continuum-based shell theory studies [19–22]. Indeed, due to the buckling mode of this SWCNT under pure compression, such a trend was expected. Actually, the internal pressure imposes stretching forces on the wall of the nanotube, causing resistance against local deformations. As shown in Figs. 2b–d, the internal pressure does not change the buckling deformation of SWCNTs with small AR significantly, and that the shape is similar to that of pure compression. On the other hand, as is shown on the left-hand side of Fig. 3, the external pressure weakens the SWCNT against compressive load. In fact, in this case, the external pressure pushes the atoms inward, resulting in a stronger tendency of the nanotube to collapse under axial compressive load. We note that [19–22] also obtained similar trends on the effect of hydrostatic pressure for SWCNTs. From Figs. 2e–h, it is observed that the external pressure affects the buckling deformation mode, especially, at high pressures. Accordingly, in the last two cases, the nanotube was totally crumpled. In Fig. 3, it is seen that when the external pressure is 4 GPa, the nanotube deforms without the imposition of any compressive axial load. However, this pressure may not be the critical one. As seen in Fig. 3, when the external pressure is 3 GPa, the nanotube buckles when a very small strain is applied. Thus, one can conclude that the critical pressure is between 3 and 4 GPa for this sample. 3.2. The effect of hydrostatic pressure on SWCNTs with large AR As an example of this type of nanotube, a (5,5) armchair SWCNT with a length of 100.84 Å, referred to as Sample 2, was considered. The buckling deformation of this nanotube under pure compression is demonstrated in Fig. 4a. As seen from this figure, the nanotube buckles in a columnar mode with a global deformation geometry. Also, the corresponding critical load and critical strain were obtained to be 21.97 nN and 3.97%, respectively. Fig. 5 displays the variation of buckling properties under different internal and external pressures. Surprisingly, an opposite trend is observed in this case in comparison with the previous case and the cases
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Fig. 4. Buckling deformation of a (5,5) armchair SWCNT (Sample 2 as a typical SWCNT with large AR) under hydrostatic pressure: (a) pure compression, (b) 1 GPa, (c) 2 GPa, (d) 1.5 GPa, (e) 3 GPa, (f) 5 GPa.
reported in the literature [19–22]. In contrast to the nanotubes with small AR, it is seen that the internal pressure weakens the buckling properties of the SWCNT, while, the external pressure improves these properties. We note that the presence of internal pressure increases the stretching stress on the wall of the nanotube. The presence of this stress enhances the instability of nanotube against the columnar-buckling mode. In other words, with the presence of such stress, the tendency of the nanotube towards global buckling increases. The right hand-side of Fig. 5 shows how internal pressure has reduced the critical buckling load and the strain. Moreover, as is shown in Figs. 4b and c, the inter-
nal pressure does not affect the buckling mode of this type of SWCNT. However, in the case of external pressure, the situation is reversed. Note that in general, the imposition of an axial compression causes a lateral expansion of the nanotube. The instability of the nanotube against the global buckling increases with this lateral expansion. Since in the case of an external pressure, the forces acting on the nanotube resist the expansion, the stability of the nanotube against global buckling is enhanced. Fig. 5 reveals that external pressure has a significant effect on the stability of nanotubes having large AR against columnar buckling under
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Fig. 5. The effect of hydrostatic pressure on the critical compressive load and critical strain (Sample 2: (5,5) SWCNT with large AR).
compressive load. However, it is observed that the external pressure enhances the critical load and strain up to a certain pressure (around 2 GPa), while beyond this pressure, the buckling properties decline almost linearly. It is also observed that beyond this point, the buckling properties decrease gradually with a linear variation. It is believed that the decline in the buckling properties is due to the change in the buckling mode shape over this specific pressure. As seen in Fig. 4e, when the external pressure is increased to 1.5 GPa, the buckling deformation is totally changed. It is observed that high external pressure causes the wall of the long nanotube to deform and collapse into a twisted shape under the compressive load, and no column buckling is seen. As the tube buckling mode geometry is changed into this new form, higher external pressure raises the tendency of the tube to deform into a twisted shape with collapsed walls, causing higher degradation in the buckling properties. To be specific, it can be concluded that the external pressure improves the buckling properties of SWCNTs having large AR up to the point that the buckling mode is still columnar. Thereafter, in the transition zone from columnar buckling to collapsing walls (local deformation), a slight change in buckling properties occurs. After crossing this region, the external pressure weakens the buckling properties due to its enforcement of the wrinkling deformation. 3.3. The effect of hydrostatic pressure on SWCNTs with medium AR In this case, it was found that for certain ARs the trend of changes in buckling properties under hydrostatic pressure is not similar to the two previous cases. These ARs are located between small and large ones, and are here referred to as medium ARs. Actually, one may consider the medium AR to lie in a range wherein the buckling mode changes from a predominantly columnar buckling to a predominantly shell buckling mode. As an example for this case, a (10,10) armchair SWCNT with a length of 100.84 Å, which is referred to as Sample 3, was considered. The axial buckling deformation of this SWCNT under pure compression is displayed in Fig. 6a. As can be seen from this figure, the nanotube is deformed locally with a lobe at its middle. Also, the corresponding critical load and critical strain were obtained as 39 nN and 3.72%, respectively. The effect of external and internal pressures on this SWCNT is displayed in Fig. 7. From the right-hand
side of this figure, it can be seen that the internal pressure has a slight effect on the buckling properties of this SWCNT until the pressure reaches 2 GPa. It is observed that up to 1 GPa, the buckling properties slightly improves, while up to 2 GPa these properties slightly decline. After crossing this point, a significant drop is observed in these properties. This significant drop can be explained by the effect of internal pressure on the buckling deformation of the SWCNT. As shown in Fig. 6, the internal pressure has a significant effect on the buckling deformation of this SWCNT, for which, when the pressure is increased, the buckling deformation transforms from a local deformation to a global columnar deformation. Indeed, since at first the wall of the SWCNT tends to deform locally, the insertion of internal pressure resists against this mechanism of deformation, for which a slight improvement is observed in the buckling properties (Fig. 7). However, it should be noted that the internal pressure also decreases the stability of the tube along the axial direction resulting in an increase in the tendency of the nanotube to buckle in a columnar shape. As the internal pressure promotes a dominant tendency in the nanotube to buckle in a columnar shape (after 2 GPa), also seen in the previous case, a higher internal pressure leads to a higher unstable condition of the SWCNT, resulting in a significant drop of the buckling properties. On the other hand, since this SWCNT tends to buckle locally under pure compressive load, exerting external pressure weakens its buckling properties, as described in Section 3.1. Fig. 7 reveals that external pressure affects the critical load and the critical strain, significantly. It is also observed that external pressure has a significant effect on the buckling deformation of SWCNTs with medium ARs. Figs. 6i and h, show that the SWCNT is totally flattened under high external pressure. Regarding our results for the three different ARs, we can conclude that the buckling mode shape plays an important role in indicating how the hydrostatic pressure affects the mechanism of the buckling. Furthermore, the presence of an internal pressure raises the tendency of the global buckling, while the imposition of an external pressure paves the way for a local buckling deformation to take place. We have observed that, the hydrostatic pressure can change the buckling mechanism from a local deformation to a global one (e.g. Fig. 7), and vice versa (Fig. 5). Accordingly, whenever the nanotube buckles predominantly in local mode, the external pressure would decrease the critical buckling load, while the
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Fig. 6. Buckling deformation of a (10,10) armchair SWCNT (Sample 3 as a typical SWCNT with medium AR) under hydrostatic pressure: (a) pure compression, (b) 0.5 GPa, (c) 1 GPa, (d) 1.5 GPa, (e) 3 GPa, (f)1.5 GPa, (g) 2 GPa, (h) 3 GPa, (i) 4 GPa.
internal pressure would postpone the occurrence of buckling. However, in the case of global buckling, the effect of pressure is reversed.
4. Conclusions The stability behavior of SWCNTs when subjected to combined loads of axial compression and hydrostatic pressure has been investigated using a set of MD-based simulations. Continuumbased models can only employ shell theories to obtain the dynamic equations of SWCNTs under hydrostatic pressure and axial compression. These theories are, however, only applicable to nano-
tubes with small ARs. These continuum-based studies have generally observed that the internal pressure postpones the critical axial stress and critical strain, while the external pressure weakens the resistance of a nanotube under axial load. Here, since we have employed atomistic-based simulation, we have been able to explore the effect of internal/external pressure loads on the buckling properties of SWCNTs with all types of ARs. To investigate the role of pressure, three samples of SWCNTs with different ARs were investigated. It has been demonstrated that the results from continuum-based studies are in agreement only with the case of nanotubes having small values of AR, where the nanotube predominantly buckles in a shell mode. Meanwhile, in the case of SWCNTs with large ARs, internal pressure reduces the stability of
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Fig. 7. The effect of hydrostatic pressure on the critical compressive load and critical strain (Sample 3: (10,10) SWCNT with medium AR).
the nanotube against the axial load, while external pressure enhances its stability, leading to a totally opposite trend in comparison with nanotubes having small ARs. Furthermore, it was observed that for a range of ARs, the trend of the variation of buckling properties due to the pressure follows none of these distinct cases. Finally, it is concluded that the buckling deformation has an important role in determining whether increasing the internal/external pressure postpones the buckling or accelerates it.
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