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Cite this: Phys. Chem. Chem. Phys., 2015, 17, 12796
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Composition-dependent buckling behaviour of hybrid boron nitride–carbon nanotubes Jin Zhang and S. A. Meguid* The buckling of hybrid boron nitride–carbon nanotubes (BN–CNTs) with various BN compositions and locations of the BN domain is investigated using molecular dynamics. We find that BN–CNTs with large BN composition (438%) only undergo local shell-like buckling in their BN domain. Although similar local shell-like buckling can occur in BN–CNTs with a relatively small BN composition, it can transfer to the global column-like buckling of the whole BN–CNT with increasing strains. The critical strains for local
Received 12th February 2015, Accepted 17th April 2015
shell-like and global column-like buckling decrease with increasing BN composition. In addition, critical
DOI: 10.1039/c5cp00914f
location of their BN domain. As a possible application of the buckling of BN–CNTs, we demonstrate that
strains and buckling modes of the global column-like buckling of BN–CNTs also strongly depend on the the BN–CNT can serve as a water channel integrated with a local natural valve using the local buckling
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of its BN domain.
1. Introduction
fabricate the B- and N-doped CNTs. However, in such nanotubes the B, N, C atoms are usually randomly distributed. To grow the
Carbon nanotubes (CNTs) have been proposed as novel nanomaterials in a wide variety of applications due to their unique properties. For example, CNTs are known to be intrinsically either semiconducting or metallic, depending on their chirality,1 and are thus promising materials for nano-electronic devices.2,3 Analogously, boron-nitride nanotubes (BNNTs), the III–V analogs of CNTs, are electrical insulators with a large band gap (independent of their chirality), but have higher thermal and chemical stability than CNTs.4,5 Thus, BNNTs are promising materials for hydrogen storage applications and as high-temperature ceramics.5 Considering the dramatically contrasting properties of CNTs and BNNTs, many efforts have been expended to explore combinations of these component materials in order to develop hybrid structures with desirable properties. Generally, these hybrid nanotubes can be developed by doping a BN segment into the CNT (i.e., BN–CNTs, as shown in Fig. 1). Such novel hybrid BN–CNTs have been proven to possess properties different from their pristine CNT and BNNT counterparts; the properties of BN–CNTs can be tailored by adjusting the ratio of BN to CNTs (BN composition).6 The unique properties of these BN–CNTs are expected to serve as a novel material platform of next-generation electronic components, sensors, and structural composites. Recently, various synthesis approaches,7 such as chemical vapor deposition and laser vaporization, have been developed to Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada. E-mail: [email protected]
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Fig. 1 Left row: schematic diagram of a BN–CNT after the initial structural relaxation. Here LBN denotes the location of the BN domain, while lBN is the length of the BN domain. Right row (from top to bottom): molecular representation of C–C, C–N, B–N, and C–B bonds in the BN–CNT after the initial structural relaxation.
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heterojunction between two kinds of nanotubes, some promising methods have been proposed. For example, the junctions between CNTs and silicon nanowires have been fabricated by Hu et al.8 using the catalyst to control the growth of the two nanostructures. Most recently, using a continuous CO2 laser vaporization reactor, Enouz et al.9 proposed a possible method to fabricate the heterojunctions between CNTs and BNNTs. At the same time, considerable computational methods, including ab initio simulations10 and first-principles calculations,6,11,12 have been proposed for determining the fundamental geometries, formation energies, and electronic properties of hybrid BN–CNTs. The results by first-principles calculations show that BN–CNTs have the structural stabilities similar to their CNT and BNNT counterparts.6 Concurrently, first-principles calculation results also reveal that the electronic properties of the BN–CNTs can be systematically tuned by changing the relative ratio of CNT to BNNT and/or the nanotube chirality,6 such as incrementally converting a BN–CNT from semiconducting to metallic. Although considerable efforts have been paid to the geometric and electronic properties of such hybrid BN–CNTs, the mechanical properties of this new material remain almost unexplored. The mechanical properties are of particular importance for the application of BN–CNTs in electronics and composite materials. Specifically, it is known that the buckling of such one-atom layer nanotube structures not only substantially reduces their structural rigidity but also leads to significant changes in their electronic properties.13 Thus, the success of BN–CNT-based device applications crucially hinges upon the comprehensive understanding of their resistance to buckling. The individual buckling of pristine CNTs and BNNTs has been widely investigated theoretically and experimentally.14–26 The obtained results illustrate that CNTs and BNNTs display different buckling characters. For example, CNTs and BNNTs have different resistance to buckling instability and buckling mode shapes.14,18,19,21,22,24,25 As a result, the conclusions observed from the individual buckling of CNTs and BNNTs are not applicable for the case of the present hybrid BN–CNT that is a combination of CNTs and BNNTs. This discrepancy has motivated the undertaking of the current study. In this paper, the buckling instability of BN–CNTs has been investigated using molecular dynamics simulations (MDSs). The effect of BN composition and the location of the BN domain are carefully examined. In addition, as a case study, we show that the BN–CNT can serve as a water channel integrated with a local valve naturally located at its BN domain. The results should be instrumental in future structural design of BN–CNT-based applications.
2. Simulation method In the present study, we adopted the classical MDSs to study the mechanical behaviour of BN–CNTs. The basic concept of MDSs is to simulate the time evolution of a system. The atoms in the system are treated as point like masses that interact with one another according to a given potential energy. In the present
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study, the interactions between C atoms were described by Tersoff potentials:27 X
Et ¼
i
Ei ¼
1X Vij ; 2 iaj
Vij = fC(rij)[fR(rij) + bijfA(rij)].
(1)
Here, Et is the total energy of the system, which is decomposed into a site energy Ei and a bond energy Vij. The indices i and j run over the atoms of the system, and rij is the distance from atom i to atom j. fR(rij) and fA(rij) are the repulsive and attractive pair potentials, respectively. The cut-off function fC(rij) is provided to limit the range of the potential, and thus saves the computational resources required in the MDS. Specifically, fR(rij), fA(rij), and fC(rij) have following forms: fR(rij) = Ael1rij,
(2.1)
fA(rij) = Bel2rij,
(2.2)
8 1; > > > > < 1 1 pðrij MÞ fC ðrij Þ ¼ sin ; > 2 2 2N > > > : 0;
rij o M N M N o rij o M þ N rij 4 M þ N (2.3)
Here, A, B, l1, l2, M and N are parameters adjusted to match the material properties. In addition, bij in eqn (1) is the bond order function that determines the strength of the attractive term. It takes the following form, bij ¼ ð1 þ bn znij Þ1=ð2nÞ ; zij ¼
X
m
m
fC ðrij Þgðyijk Þel3 ðrij rik Þ ;
kai;j
gðyijk Þ ¼ 1 þ
a2 a2 2 ; 2 d d þ ðh cos yijk Þ2
(3)
where yijk is the bond angle between bonds ij and ik. In eqn (3), b, l3, m, n, a, d and h are additional parameters adjusted to match the material properties. In the present study, the parameters in Tersoff potentials were adopted from ref. 28 to simulate the C–C interactions and have been proven to be capable of modelling the lattice dynamics and thermal transport properties of CNTs.29,30 Similarly, the interactions for B–N, B–C and N–C were also modelled by Tersoff potentials (eqn (1)). In this case, the parameters in Tersoff potentials of the B–N, B–C and N–C interactions were taken from ref. 31. Such interatomic interactions in the C–B–N system have been successfully employed to evaluate the mechanical and thermal transport properties of hybrid BN–C nanosheets.32,33 In the present study, we considered (8,0) zigzag BN–CNTs (see Fig. 1) with fixed length L about 41 Å but different lengths lBN of the BN domain and different locations of the BN domain denoted by LBN in Fig. 1. Thus, the composition of this hybrid structure is measured by the molar fraction of BN, which is
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approximately denoted as lBN/L. In order to conduct the computational calculations, the whole nanotube has been divided into three sections as shown in Fig. 1. The moving layer at the top section is used to apply displacement boundary condition (compressive strains in the present buckling study), while the fixed layer at the bottom section is used to constrain the nanotube. The middle section is the free layer, where the positions and velocities of the atoms obey Newton’s second law. The atoms of moving and fixed layers are not subject to internal forces. However, they provide internal forces to the atoms in the free layer. Thus, the present MDSs were performed according to the following steps: the atoms were first brought to the positions corresponding to the lowest energy. Subse´–Hoover thermostat34 was applied to thermally quently, a Nose equilibrate the atoms at 1 K to reduce the random thermal fluctuation effect. It is known that the buckling behaviour of nanotubes is sensitive to geometrical perturbations, which are prominent at room temperature. However, the random thermal vibrations associated with thermal fluctuations may cause severe uncertainties in the current effort.35 Therefore, it was decided to conduct the MDS at a temperature near the absolute zero to avoid these uncertainties. In fact, MDS at a temperature near the absolute zero will aid in developing greater understanding of the elastic behaviour of BN–CNTs at a finite temperature. This approach has been widely adopted in many studies concerning the buckling behaviour of CNTs16,18,21 and BN nanocones.36 The velocity Verlet algorithm with the time step of 1 fs was utilized to integrate Hamiltonian equations of motion as determined by Newton’s second law. After the full relaxation with 20 ps, BN–CNTs were quasistatically loaded under compression in the axial direction, i.e., the atoms at the top layer of the BN–CNTs were moved downwards, and meanwhile the atoms at the bottom end layer were fixed. This created a ramp velocity profile where the velocity rises from zero at the fixed layer to its maximum value at the moving layer. To avoid the crystalline defects normally produced due to a high rate of loading in the present simulations, we chose a relatively low strain rate of 0.001 ps1. Finally, the top and bottom layers were kept fixed and the system was relaxed for 1 ps, 1000 time steps, to allow the middle section of the BN–CNTs to reach a new equilibrium state. By repeating the above process, the nanotubes can be compressed continuously until the required strain has been obtained. The total simulation time was different for different structures and between 200 ps and 240 ps. The present MDSs were conducted using large-scale atomic/molecular massively parallel simulator (LAMMPS).37
3. Results and discussion Based on the MDS technique and the chosen interatomic potentials, we display the molecular structure of a BN–CNT after the full structural relaxation. We can see from Fig. 1 that due to the different hybridizations related with the B and N atoms38 the structural relaxation causes each B–N bond to be slightly rotated such that each N atom is rotated out and B atom
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is rotated into a nanotube axis. As a result, in the minimum energy state, all B atoms are arranged in one cylinder and all N atoms in another concentric cylinder with a relatively larger radius. Similar B–N bond rotation effect was also found in pristine BNNTs.38 On the contrary, no significant rotation was found in all C–C, B–C and N–C bonds. In other words, all C atoms together with the B and N atoms at the heterojunctions are arranged in concentric cylinders with almost the same radius (see Fig. 1). After achieving the optimized structure of BN–CNTs, we can examine their buckling behaviour following the methods illustrated above. In this section we will discuss the influence of the length of the BN domain and the location of the BN domain on the elastic properties and buckling instability of the whole BN–CNTs. 3.1
Influence of the length of the BN domain
Firstly, we will study the influence of the length of the BN domain, which measures the BN composition of BN–CNTs, on the mechanical behaviours of the whole BN–CNTs. To this end, we study the elastic properties and buckling instability of BN–CNTs with various lengths of the BN domain in this subsection. Here, the BN domain is placed at the centre of all BN–CNTs. We axially compressed BN–CNTs and recorded the relationship between the axial strain e and the stored strain energy U. In Fig. 2a, we plot the U–e curves for two typical BN–CNTs: BN–CNT-1 with a relatively large molar fraction of BN (lBN = 33.2 Å) and BN–CNT-2 with a relatively small molar fraction of BN (lBN = 7.4 Å). In addition, the results of their pristine CNT and BNNT counterparts with the same geometric size are also illustrated in Fig. 2a. Initially, at the small deformation, U of all nanotubes increases almost quadratically with e. This U–e relationship in the small deformation range offers a means to calculate the effective elastic properties of nanotubes. In order to obtain such effective elastic properties, a continuum model should be introduced. Generally, a nanotube can be most conveniently modelled as a continuum thin shell with an effective wall thickness t and Young’s modulus E. This continuum shell model thus leads to tensile rigidity Y = Et of the nanotube as Y = (q2U/qe2)/(2pRL), where R is the radius of the nanotube. In view of the uncertainty of the effective tube thickness for all nanotubes, we chose the tensile rigidity Y rather than Young’s modulus E to assess the elastic behaviour of nanotubes. The fitted Y of pristine CNTs and BNNTs are respectively 330 N m1 and 231 N m1, which are very close to the values 326 N m1 ((8,0) CNTs) and 253 N m1 ((8,0) BNNTs) obtained by Li and Chou39 and Kudin et al.40 As for BN–CNTs, their Y is found to range between the values of the pristine CNTs and BNNTs and decrease with increasing molar fraction of BN, as shown in Fig. 2b. Such composition-dependent elastic properties of BN–CNTs can be understood by inducing a continuum composite beam (CB) model41 as illustrated by the inset of Fig. 2b. Similar to its molecular representation, the CB model of BN–CNTs is also composed of two parts: CNT and BNNT. Thus, according to the classical composite theory,41 when such CB model of BN–CNTs is subjected to an axial force F, its deformation can be treated as the
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Fig. 2 (a) Variation of the total strain energies of the CNT, BNNT and BN–CNT with their axial strains. (b) The tensile rigidity of BN–CNTs as a function of their molar fraction of BN. The inset shows the continuum CB model of BN–CNTs. (c) Shows the typical tube geometry after BNNT buckling. (d) Shows the typical tube geometry after BN–CNT-1 buckling and (e) shows the typical tube geometry of BN–CNT-1 during the post-buckling. (f) and (g) Respectively show typical tube geometry after the 1st and the 2nd buckling of BN–CNT-2. (h) Shows the typical tube geometry after CNT buckling.
deformation sum of its composite BN and C parts, i.e., FL/(2pRY) = FlBN/(2pRYBN) + F(L lBN)/(2pRYC), where YBN and YC are respectively the tensile rigidity of BNNTs and CNTs. Following that, the tensile rigidity of the BN–CNTs can be expressed as Y¼
YBN YC ; cYC þ ð1 cÞYBN
(4)
where c = lBN/L is the molar fraction of BN characterizing the composition of the hybrid BN–CNT. We calculate Y of BN–CNTs using eqn (4) (Fig. 2b (line)). The results show that the prediction by the present continuum CB model agrees well with the results calculated using MDSs. In addition, from Fig. 2a we can also find that the nanotubes become unstable as the axial strain increases. The strain energy falls down abruptly when the nanotubes buckle. Specifically, we can see from Fig. 2a that there is only one abrupt strain energy drop in pristine BNNTs and CNTs. The buckled shapes of pristine BNNTs and CNTs are shown in Fig. 2c and h, respectively. A global column-like buckling is observed for the CNTs. At the same time, a local shell-like buckling develops at the centre of the column-like buckled CNTs, since one side of the globally buckled CNT is highly compressed. Such buckling modes of CNTs agree well with previous reports.18,19,21 As for pristine BNNTs, a compression-induced global shell-like buckling (displays like flattening ‘‘fins’’) is detected. The different buckled shapes between the CNTs and BNNTs mainly originate from their different material properties and surface patterns.
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The smaller elastic constants and buckled surface of BNNTs both make them more readily amenable to buckle like shells. In Fig. 2a we also find that there exists only one abrupt strain energy drop in BN–CNT-1, corresponding to the occurrence of shell-like buckling as illustrated in Fig. 2d. However, different from its pristine BNNT counterpart, this compression-induced shell-like buckling of BN–CNT-1 exists locally only in its BN domain. A similar local shell-like buckling mode is also observed in BN–CNT-2 (see Fig. 2f), which corresponds to the first abrupt strain energy drop in the U–e curve of BN–CNT-2 as shown in Fig. 2a. The different buckling positions in the BN–CNTs and the BNNTs are due to the difference in the buckling resistance between the BNNTs and the CNTs. From the aforementioned discussion BNNTs are found to have smaller critical buckling strain than the CNTs. As a result, when the strain applied to the BN–CNTs increases, their BN domains will firstly buckle, while, at the same time, the C domains will not. This difference is attributed to the larger buckling resistance of the C domain. On the other hand, when the strain applied to the BNNT goes beyond its critical buckling strain, the entire BNNT will lose its stability, since BNNT is a homogeneous material. The critical strain ecr1 corresponding to such local shell-like buckling of the BN–CNTs is found to decrease with increasing molar fraction of BN (determined by lBN). For example, ecr1 of BN–CNT-2 is 0.12 and is about 30% greater than that of BN–CNT-1. This composition-dependent ecr1 can be qualitatively understood using the continuum mechanics theory (Sanders shell model),
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which gives the critical buckling strain of a tube structure as22 ecr1 pffiffiffiffiffiffiffiffi p (1 0.1l/R t=R). Here l is the length of the structure and equals to lBN for BN–CNTs, since the shell-like buckling occurs locally in the BN domain. From this expression, we can see that ecr1 decreases as lBN increases, which agrees with our MDS results. Finally, we will turn our attention to the post-buckling behaviour of nanotubes. Fig. 2a illustrates that during the post-buckling stage there exists no significant abrupt strain energy drops in the CNT, BNNT and BN–CNT-1, which means that the buckling modes of these nanotubes will not change during their post-buckling stage. Indeed, for example, Fig. 2e shows that during the continued compression of an already buckled BN–CNT-1, it will maintain the shell-like buckling mode in its BN domain, despite the appearance of some localised kinks. However, for BN–CNT with relatively small molar fraction of BN, e.g., BN–CNT-2, when we keep compressing the already buckled BN–CNT, a second drop in strain energy is observed at the strain level of ecr2, which equals to 0.15 for BN–CNT-2 (see Fig. 2a). This indicates that the buckling mode of BN–CNT-2 will topographically change after this critical strain ecr2. Indeed, we can see from Fig. 2g that the buckling mode of BN–CNT-2 has transferred from the local shell-like buckling (Fig. 2f) to the global column-like buckling (Fig. 2g). The different post-buckling behaviours of BN–CNT-1 and BN–CNT-2 are mainly due to their different numbers of the flattening fin following the first buckling mode. According to previous studies on shelllike buckling of CNTs,16 the number of the flattening fin of the buckled zig-zag nanotubes is proportional to the nanotube length. A similar conclusion is found to hold for their BN counterparts. Fig. 2e and f show that after the first buckling mode, the buckled BN domain of BN–CNT-2 has only one fin, while the BN domain of BN–CNT-1 displays four fins. In the post-buckling stage, each fin will serve as a hinge.14 Further compression of the a priori buckled BN–CNT-1 will lead to additional kinks to the shell-like buckling mode (see Fig. 2e). At the same time, the only fin in BN–CNT-2 serves as a hinge connecting two relatively long neighbouring C domains of BN–CNT-2. Thus, when BN–CNT-2 is compressed further, the fin will facilitate the buckling of its C domains, and thus induce the column-like buckling of BN–CNT-2 (see Fig. 2g). The buckling mode transition is observed for BN–CNTs whose molar fraction of BN c is smaller than 0.38; and ecr2 is found to decrease with increasing c due to the weaker elastic stiffness of BN domain compared to its C counterpart. Moreover, in Fig. 2g we find that although there still exists a local shell-like buckling in the BN domain of the globally buckled BN–CNT-2, the mechanism of this local shell-like buckling is completely different from that shown in Fig. 2f. In Fig. 2g, the local shell-like buckling is due to bending, and thus only one side of BN–CNT-2 buckles; in Fig. 2f the local shell-like buckling is induced by the axial compression, as a result both sides buckle. 3.2
Influence of the location of the BN domain
In the above analysis, we have studied BN–CNTs whose BN domains are placed at the centre of nanotubes. Subsequently,
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we shall study the influence of the location of the BN domain on the buckling instability of BN–CNTs. To this end, BN–CNTs with lBN = 7.4 Å, 16.3 Å, and 25.3 Å and varying LBN are considered here. As we depicted in Section 3.1, the buckling mode transition phenomenon can be observed for all BN–CNTs with the length of the BN domain smaller than 16.3 Å (BN composition smaller than 38%), regardless of the location of the BN domain (LBN). The critical strain ecr1 corresponding to the first (local shell-like) buckling and ecr2 corresponding to the second (global column-like) buckling are calculated in Fig. 3a as a function of LBN. We can see from Fig. 3a that ecr1 is almost independent from the location of the BN domain and is around 0.12 for BN–CNTs with lBN = 7.4 Å, around 0.11 for BN–CNTs with lBN = 16.3 Å, and about 0.1 for BN–CNTs with lBN = 25.3 Å. This is because that ecr1 is a parameter corresponding to the local buckling behaviour of the BN domain and thus is almost independent of the global parameters of BN–CNTs, e.g., the location of the BN domain considered here. On the contrary, we find in Fig. 3a that ecr2 strongly depends on the location of the BN domain. It is found that ecr2 holds the minimum value when the BN domain locates at the centre of the BN–CNT. When the BN domain moves from the centre to the ends of the BN–CNT, ecr2 is found to firstly increase and reach a maximum value at LBN = 8.5 Å or 25.6 Å for BN–CNTs with lBN = 7.4 Å and at LBN = 8.5 Å or 17.0 Å for BN–CNTs with lBN = 16.3 Å; after that ecr2 decreases as the BN domain keeps moving to the ends. In this process, the deviation between the minimum and maximum of ecr2 is about 13%. It is known that the global stability of BN–CNTs is determined by their ability to sustain the beam-bending, which is
Fig. 3 (a) The 1st and the 2nd buckling critical strains as a function of the location of the BN domain in BN–CNTs (measured by LBN as shown in Fig. 1). (b) Bending moment diagram of the fixed–fixed buckled nanotube. (c)–(g) Respectively show snapshots of the BN–CNT-2 (lBN = 7.4 Å) with LBN = 8.54 Å, 12.81 Å, 16.96 Å, 21.35 Å, and 25.62 Å after the 2nd buckling.
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measured by the bending stiffness. In Fig. 3b, we plot the bending moment diagram of a buckled fixed–fixed homogeneous beam.42 It can be seen from Fig. 3b that the bending moment is zero between the centre and ends of the beam. Thus, the whole BN–CNT can achieve the largest overall bending stiffness when the BN domain with relatively smaller bending stiffness (compared with its C counterpart) locates in or just near the zero-bending moment area. This theoretical analysis agrees well with our MDS results illustrated in Fig. 3a, where ecr2 holds a maximum value when the BN domain locates between the centre and ends of BN–CNTs. In addition, although the location of the BN domain has no influence on the first buckling modes of BN–CNTs, it may have a significant influence on the configurations of BN–CNTs after their second buckling. As an example, in Fig. 3c–g we display the configurations of BN–CNT-2 (lBN = 7.4 Å) after the second buckling. Comparing Fig. 3c–g, it can be deduced that although all BN–CNTs globally display the column-like deformation, the local patterns of these buckled nanotubes are different. For BN–CNTs with the BN domain at their centre (Fig. 3e), the compression-induced local shell-like buckling mode (induced by the first buckling) changes to the bending-induced local shell-like buckling mode. As for BN–CNTs with the BN domain far from their centre and near their ends (Fig. 3c and g), the compression-induced first buckling mode is preserved after the second buckling. In addition to this preserved compressioninduced first buckling mode, another local shell-like buckling mode develops at the central C domain, which is induced by the bending of the BN–CNTs following the second buckling. For BN–CNTs with the BN domain far from their ends and near their centre, e.g., BN–CNTs with LBN = 12.81 Å and 21.35 Å shown in Fig. 3d and f, it is expected that these two BN–CNTs display the same buckling character, since their BN domains are symmetrical to the centre of the structure. However, we can see from Fig. 3d and f the buckling modes of these two BN–CNTs are quite different actually. The BN–CNT with LBN = 12.81 Å shows a similar buckling behaviour to the BN–CNT with the BN domain at its centre (Fig. 3d), while the BN–CNT with LBN = 21.35 Å shows a similar buckling behaviour to the BN–CNT with the BN domain near its ends (Fig. 3f). This difference originates from the different bond characters between the C–B and C–N bonds. For the BN–CNTs with LBN = 12.81 Å the C–N bonds play a more important role (compared with the C–B bonds) in resisting the buckling since they are located much closer to the centre of the structure (see Fig. 3d) and thus could accommodate much larger bending moment (see Fig. 3b); for those with LBN = 21.35 Å the C–B bonds thus play a more important part in withstanding the buckling for the same reason (see Fig. 3f).
4. A case study of potential applications of the buckling of BN–CNTs The above understanding of the buckling instability of BN–CNTs could be potentially leveraged to enable the design of new functions of nano-devices. Thus, as a demonstration, we report
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Fig. 4 (a) Variation of the total strain energy of the water filled BN–CNT with the axial strain. (b) Initial configurations (without strain) of the water filled BN–CNT. (c) Typical tube geometry before the 1st buckling: open position of the nano-valve. (d) Typical tube geometry after the 1st buckling: closed position of the nano-valve.
a valve-integrated water channel by BN–CNTs. Previous studies have demonstrated that pristine CNTs and BNNTs can both serve as the water channel.43–46 Enthusiasm for utilizing nanotubes for water transport, one crucial but largely unexplored issue for the success of this potential application is the design of a smart nano-valve to control the water flow. In what follows, we will show that the BN–CNT can serve as a water channel integrated with a valve naturally by using the local shell-like buckling of its BN domain. In this simulation, a BN–CNT with the same geometric size of the above BN–CNT-2 is considered. Initially, such BN–CNT was immersed in a water reservoir, and a single-file chain of water molecules inside the BN–CNT was obtained after that (see Fig. 4b). In this study, the non-bonded van der Waals (vdW) interactions of C–water and BN–water systems was simulated by the Lennard-Jones (LJ) atomic pair potential: Vij = 4d[(s/rij)12 (s/rij)6]. Here, the LJ parameters d and s employed in the present simulation were adopted from ref. 47 and 48. Fig. 4a shows the strain energy U-strain e curve of the waterfilled BN–CNT under compression. Similar to the empty BN–CNT case (Fig. 2), U initially quadratically increases with increasing e and then abruptly falls down when e reaches the critical value ecr1 corresponding to the local shell-like buckling of the BN–CNT. Compared with its empty counterpart, the present water-filled BN–CNT is much less stable, since ecr1 = 0.07 measured here is much smaller than 0.12 that obtained from Fig. 2. As indicated in previous studies, the effect of the filler on the buckling strength of nanotubes strongly depends on the size and density of the filled atoms or molecules, the nanotube diameter, length and chirality, among others.49–52 For example, due to the encapsulation of gold atoms49 or crystalline zinc sulphide nanowires,50 the buckling strengths of filled CNTs were greatly enhanced. In the meantime, smaller critical buckling strains similar to those observed in the present study were also found in the water-filled CNTs51 and the hydrogen-filled BNNTs.52 The existence of the water molecules in
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open-ended nanotubes seems to result in extra attractive or repulsive lateral forces imposed on the tube wall. As a consequence, the stress concentration is intensified and the local perturbations present in the walls of the nanotubes lead to a reduction in their buckling strengths and may accelerate their mode of collapse. In Fig. 4c and d we plot the configuration of the water-filled BN–CNT just before and after the buckling. We can see from Fig. 4c that before the BN–CNT buckles the water molecules distribute continually along the BN–CNT just like the case without strain (Fig. 4b). However, after the BN–CNT buckles (Fig. 4d) the water molecules in the local buckled BN domain exit this buckled region due to the change of the vdW force associated with the crimpling of the tube wall. At the same time, the water molecules in the two C domains distribute continually, which is similar to the case before buckling is encountered. In other words, Fig. 4c and d indicate that the water molecules cannot flow through the buckled BN–CNT only due to local buckling of the BN domain. After releasing the external constraint and relaxing the tubes, the BN–CNT recovers to its initial configurations. As a result, in this unloaded case the water molecules can well transport along the BN–CNT again. Based on this analysis, we can see that the local BN domain of the BN–CNT can be treated as a natural local valve in the BN–CNT-based water channel. Finally, it is worth pointing out that the aforementioned calculations were performed at 1 K temperature to exclude the effect of thermal fluctuations. The study was further extended to account for the effect of room temperature (300 K) upon the applicability of our proposed ‘‘nano-valve’’. To this end, we plot U–e curve in Fig. 4a and the buckling mode in Fig. 4d for the same water-filled BN–CNT at room temperature. Analogous to the 1 K results, the room temperature results show that the water molecules are forced to leave the BN domain of BN–CNT due to the increased repulsive vdW forces ensued by the buckling mode of the BN domain (see Fig. 4d). This result indicates that the working mechanism of the present nano-valve at 1 K holds for room temperature. In addition, Fig. 4a shows that the critical buckling strain of the water-filled BN–CNT (the operating strain of the nano-valve) at room temperature is smaller than that observed at 1 K temperature. This is due to the significant effect of the thermal fluctuations at room temperature.
5. Conclusions The buckling instability of BN–CNTs has been studied using MDSs in this paper. The buckling behaviour of BN–CNTs is found to strongly depend on their composition. For BN–CNTs with large BN composition, they only undergo local shell-like buckling in the BN domain; for BN–CNTs with small BN composition, they experience local shell-like buckling firstly but this local shell-like buckling changes to global column-like buckling of the whole BN–CNT with increasing strains. The critical strains for local shell-like and global column-like buckling both decrease with increasing BN composition. In addition,
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the critical strains and buckling modes of the global columnlike buckling of BN–CNTs also strongly depend on the location of the BN domain. As a case study, we demonstrated that the BN domain can serve as a local value for the BN–CNT-based water channel operating by its local buckling character. This analysis suggests that the BN–CNT can be treated as an integrated building block for the nano-device design.
Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Discovery Accelerator Supplement. The authors also wish to offer their thanks for the anonymous reviewers for their constructive remarks.
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Composition-dependent buckling behaviour of hybrid boron nitride–carbon nanotubes Jin Zhang and S. A. Meguid* The buckling of hybrid boron nitride–carbon nanotubes (BN–CNTs) with various BN compositions and locations of the BN domain is investigated using molecular dynamics. We find that BN–CNTs with large BN composition (438%) only undergo local shell-like buckling in their BN domain. Although similar local shell-like buckling can occur in BN–CNTs with a relatively small BN composition, it can transfer to the global column-like buckling of the whole BN–CNT with increasing strains. The critical strains for local
Received 12th February 2015, Accepted 17th April 2015
shell-like and global column-like buckling decrease with increasing BN composition. In addition, critical
DOI: 10.1039/c5cp00914f
location of their BN domain. As a possible application of the buckling of BN–CNTs, we demonstrate that
strains and buckling modes of the global column-like buckling of BN–CNTs also strongly depend on the the BN–CNT can serve as a water channel integrated with a local natural valve using the local buckling
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of its BN domain.
1. Introduction
fabricate the B- and N-doped CNTs. However, in such nanotubes the B, N, C atoms are usually randomly distributed. To grow the
Carbon nanotubes (CNTs) have been proposed as novel nanomaterials in a wide variety of applications due to their unique properties. For example, CNTs are known to be intrinsically either semiconducting or metallic, depending on their chirality,1 and are thus promising materials for nano-electronic devices.2,3 Analogously, boron-nitride nanotubes (BNNTs), the III–V analogs of CNTs, are electrical insulators with a large band gap (independent of their chirality), but have higher thermal and chemical stability than CNTs.4,5 Thus, BNNTs are promising materials for hydrogen storage applications and as high-temperature ceramics.5 Considering the dramatically contrasting properties of CNTs and BNNTs, many efforts have been expended to explore combinations of these component materials in order to develop hybrid structures with desirable properties. Generally, these hybrid nanotubes can be developed by doping a BN segment into the CNT (i.e., BN–CNTs, as shown in Fig. 1). Such novel hybrid BN–CNTs have been proven to possess properties different from their pristine CNT and BNNT counterparts; the properties of BN–CNTs can be tailored by adjusting the ratio of BN to CNTs (BN composition).6 The unique properties of these BN–CNTs are expected to serve as a novel material platform of next-generation electronic components, sensors, and structural composites. Recently, various synthesis approaches,7 such as chemical vapor deposition and laser vaporization, have been developed to Mechanics and Aerospace Design Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Ontario M5S 3G8, Canada. E-mail: [email protected]
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Fig. 1 Left row: schematic diagram of a BN–CNT after the initial structural relaxation. Here LBN denotes the location of the BN domain, while lBN is the length of the BN domain. Right row (from top to bottom): molecular representation of C–C, C–N, B–N, and C–B bonds in the BN–CNT after the initial structural relaxation.
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heterojunction between two kinds of nanotubes, some promising methods have been proposed. For example, the junctions between CNTs and silicon nanowires have been fabricated by Hu et al.8 using the catalyst to control the growth of the two nanostructures. Most recently, using a continuous CO2 laser vaporization reactor, Enouz et al.9 proposed a possible method to fabricate the heterojunctions between CNTs and BNNTs. At the same time, considerable computational methods, including ab initio simulations10 and first-principles calculations,6,11,12 have been proposed for determining the fundamental geometries, formation energies, and electronic properties of hybrid BN–CNTs. The results by first-principles calculations show that BN–CNTs have the structural stabilities similar to their CNT and BNNT counterparts.6 Concurrently, first-principles calculation results also reveal that the electronic properties of the BN–CNTs can be systematically tuned by changing the relative ratio of CNT to BNNT and/or the nanotube chirality,6 such as incrementally converting a BN–CNT from semiconducting to metallic. Although considerable efforts have been paid to the geometric and electronic properties of such hybrid BN–CNTs, the mechanical properties of this new material remain almost unexplored. The mechanical properties are of particular importance for the application of BN–CNTs in electronics and composite materials. Specifically, it is known that the buckling of such one-atom layer nanotube structures not only substantially reduces their structural rigidity but also leads to significant changes in their electronic properties.13 Thus, the success of BN–CNT-based device applications crucially hinges upon the comprehensive understanding of their resistance to buckling. The individual buckling of pristine CNTs and BNNTs has been widely investigated theoretically and experimentally.14–26 The obtained results illustrate that CNTs and BNNTs display different buckling characters. For example, CNTs and BNNTs have different resistance to buckling instability and buckling mode shapes.14,18,19,21,22,24,25 As a result, the conclusions observed from the individual buckling of CNTs and BNNTs are not applicable for the case of the present hybrid BN–CNT that is a combination of CNTs and BNNTs. This discrepancy has motivated the undertaking of the current study. In this paper, the buckling instability of BN–CNTs has been investigated using molecular dynamics simulations (MDSs). The effect of BN composition and the location of the BN domain are carefully examined. In addition, as a case study, we show that the BN–CNT can serve as a water channel integrated with a local valve naturally located at its BN domain. The results should be instrumental in future structural design of BN–CNT-based applications.
2. Simulation method In the present study, we adopted the classical MDSs to study the mechanical behaviour of BN–CNTs. The basic concept of MDSs is to simulate the time evolution of a system. The atoms in the system are treated as point like masses that interact with one another according to a given potential energy. In the present
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study, the interactions between C atoms were described by Tersoff potentials:27 X
Et ¼
i
Ei ¼
1X Vij ; 2 iaj
Vij = fC(rij)[fR(rij) + bijfA(rij)].
(1)
Here, Et is the total energy of the system, which is decomposed into a site energy Ei and a bond energy Vij. The indices i and j run over the atoms of the system, and rij is the distance from atom i to atom j. fR(rij) and fA(rij) are the repulsive and attractive pair potentials, respectively. The cut-off function fC(rij) is provided to limit the range of the potential, and thus saves the computational resources required in the MDS. Specifically, fR(rij), fA(rij), and fC(rij) have following forms: fR(rij) = Ael1rij,
(2.1)
fA(rij) = Bel2rij,
(2.2)
8 1; > > > > < 1 1 pðrij MÞ fC ðrij Þ ¼ sin ; > 2 2 2N > > > : 0;
rij o M N M N o rij o M þ N rij 4 M þ N (2.3)
Here, A, B, l1, l2, M and N are parameters adjusted to match the material properties. In addition, bij in eqn (1) is the bond order function that determines the strength of the attractive term. It takes the following form, bij ¼ ð1 þ bn znij Þ1=ð2nÞ ; zij ¼
X
m
m
fC ðrij Þgðyijk Þel3 ðrij rik Þ ;
kai;j
gðyijk Þ ¼ 1 þ
a2 a2 2 ; 2 d d þ ðh cos yijk Þ2
(3)
where yijk is the bond angle between bonds ij and ik. In eqn (3), b, l3, m, n, a, d and h are additional parameters adjusted to match the material properties. In the present study, the parameters in Tersoff potentials were adopted from ref. 28 to simulate the C–C interactions and have been proven to be capable of modelling the lattice dynamics and thermal transport properties of CNTs.29,30 Similarly, the interactions for B–N, B–C and N–C were also modelled by Tersoff potentials (eqn (1)). In this case, the parameters in Tersoff potentials of the B–N, B–C and N–C interactions were taken from ref. 31. Such interatomic interactions in the C–B–N system have been successfully employed to evaluate the mechanical and thermal transport properties of hybrid BN–C nanosheets.32,33 In the present study, we considered (8,0) zigzag BN–CNTs (see Fig. 1) with fixed length L about 41 Å but different lengths lBN of the BN domain and different locations of the BN domain denoted by LBN in Fig. 1. Thus, the composition of this hybrid structure is measured by the molar fraction of BN, which is
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approximately denoted as lBN/L. In order to conduct the computational calculations, the whole nanotube has been divided into three sections as shown in Fig. 1. The moving layer at the top section is used to apply displacement boundary condition (compressive strains in the present buckling study), while the fixed layer at the bottom section is used to constrain the nanotube. The middle section is the free layer, where the positions and velocities of the atoms obey Newton’s second law. The atoms of moving and fixed layers are not subject to internal forces. However, they provide internal forces to the atoms in the free layer. Thus, the present MDSs were performed according to the following steps: the atoms were first brought to the positions corresponding to the lowest energy. Subse´–Hoover thermostat34 was applied to thermally quently, a Nose equilibrate the atoms at 1 K to reduce the random thermal fluctuation effect. It is known that the buckling behaviour of nanotubes is sensitive to geometrical perturbations, which are prominent at room temperature. However, the random thermal vibrations associated with thermal fluctuations may cause severe uncertainties in the current effort.35 Therefore, it was decided to conduct the MDS at a temperature near the absolute zero to avoid these uncertainties. In fact, MDS at a temperature near the absolute zero will aid in developing greater understanding of the elastic behaviour of BN–CNTs at a finite temperature. This approach has been widely adopted in many studies concerning the buckling behaviour of CNTs16,18,21 and BN nanocones.36 The velocity Verlet algorithm with the time step of 1 fs was utilized to integrate Hamiltonian equations of motion as determined by Newton’s second law. After the full relaxation with 20 ps, BN–CNTs were quasistatically loaded under compression in the axial direction, i.e., the atoms at the top layer of the BN–CNTs were moved downwards, and meanwhile the atoms at the bottom end layer were fixed. This created a ramp velocity profile where the velocity rises from zero at the fixed layer to its maximum value at the moving layer. To avoid the crystalline defects normally produced due to a high rate of loading in the present simulations, we chose a relatively low strain rate of 0.001 ps1. Finally, the top and bottom layers were kept fixed and the system was relaxed for 1 ps, 1000 time steps, to allow the middle section of the BN–CNTs to reach a new equilibrium state. By repeating the above process, the nanotubes can be compressed continuously until the required strain has been obtained. The total simulation time was different for different structures and between 200 ps and 240 ps. The present MDSs were conducted using large-scale atomic/molecular massively parallel simulator (LAMMPS).37
3. Results and discussion Based on the MDS technique and the chosen interatomic potentials, we display the molecular structure of a BN–CNT after the full structural relaxation. We can see from Fig. 1 that due to the different hybridizations related with the B and N atoms38 the structural relaxation causes each B–N bond to be slightly rotated such that each N atom is rotated out and B atom
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is rotated into a nanotube axis. As a result, in the minimum energy state, all B atoms are arranged in one cylinder and all N atoms in another concentric cylinder with a relatively larger radius. Similar B–N bond rotation effect was also found in pristine BNNTs.38 On the contrary, no significant rotation was found in all C–C, B–C and N–C bonds. In other words, all C atoms together with the B and N atoms at the heterojunctions are arranged in concentric cylinders with almost the same radius (see Fig. 1). After achieving the optimized structure of BN–CNTs, we can examine their buckling behaviour following the methods illustrated above. In this section we will discuss the influence of the length of the BN domain and the location of the BN domain on the elastic properties and buckling instability of the whole BN–CNTs. 3.1
Influence of the length of the BN domain
Firstly, we will study the influence of the length of the BN domain, which measures the BN composition of BN–CNTs, on the mechanical behaviours of the whole BN–CNTs. To this end, we study the elastic properties and buckling instability of BN–CNTs with various lengths of the BN domain in this subsection. Here, the BN domain is placed at the centre of all BN–CNTs. We axially compressed BN–CNTs and recorded the relationship between the axial strain e and the stored strain energy U. In Fig. 2a, we plot the U–e curves for two typical BN–CNTs: BN–CNT-1 with a relatively large molar fraction of BN (lBN = 33.2 Å) and BN–CNT-2 with a relatively small molar fraction of BN (lBN = 7.4 Å). In addition, the results of their pristine CNT and BNNT counterparts with the same geometric size are also illustrated in Fig. 2a. Initially, at the small deformation, U of all nanotubes increases almost quadratically with e. This U–e relationship in the small deformation range offers a means to calculate the effective elastic properties of nanotubes. In order to obtain such effective elastic properties, a continuum model should be introduced. Generally, a nanotube can be most conveniently modelled as a continuum thin shell with an effective wall thickness t and Young’s modulus E. This continuum shell model thus leads to tensile rigidity Y = Et of the nanotube as Y = (q2U/qe2)/(2pRL), where R is the radius of the nanotube. In view of the uncertainty of the effective tube thickness for all nanotubes, we chose the tensile rigidity Y rather than Young’s modulus E to assess the elastic behaviour of nanotubes. The fitted Y of pristine CNTs and BNNTs are respectively 330 N m1 and 231 N m1, which are very close to the values 326 N m1 ((8,0) CNTs) and 253 N m1 ((8,0) BNNTs) obtained by Li and Chou39 and Kudin et al.40 As for BN–CNTs, their Y is found to range between the values of the pristine CNTs and BNNTs and decrease with increasing molar fraction of BN, as shown in Fig. 2b. Such composition-dependent elastic properties of BN–CNTs can be understood by inducing a continuum composite beam (CB) model41 as illustrated by the inset of Fig. 2b. Similar to its molecular representation, the CB model of BN–CNTs is also composed of two parts: CNT and BNNT. Thus, according to the classical composite theory,41 when such CB model of BN–CNTs is subjected to an axial force F, its deformation can be treated as the
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Fig. 2 (a) Variation of the total strain energies of the CNT, BNNT and BN–CNT with their axial strains. (b) The tensile rigidity of BN–CNTs as a function of their molar fraction of BN. The inset shows the continuum CB model of BN–CNTs. (c) Shows the typical tube geometry after BNNT buckling. (d) Shows the typical tube geometry after BN–CNT-1 buckling and (e) shows the typical tube geometry of BN–CNT-1 during the post-buckling. (f) and (g) Respectively show typical tube geometry after the 1st and the 2nd buckling of BN–CNT-2. (h) Shows the typical tube geometry after CNT buckling.
deformation sum of its composite BN and C parts, i.e., FL/(2pRY) = FlBN/(2pRYBN) + F(L lBN)/(2pRYC), where YBN and YC are respectively the tensile rigidity of BNNTs and CNTs. Following that, the tensile rigidity of the BN–CNTs can be expressed as Y¼
YBN YC ; cYC þ ð1 cÞYBN
(4)
where c = lBN/L is the molar fraction of BN characterizing the composition of the hybrid BN–CNT. We calculate Y of BN–CNTs using eqn (4) (Fig. 2b (line)). The results show that the prediction by the present continuum CB model agrees well with the results calculated using MDSs. In addition, from Fig. 2a we can also find that the nanotubes become unstable as the axial strain increases. The strain energy falls down abruptly when the nanotubes buckle. Specifically, we can see from Fig. 2a that there is only one abrupt strain energy drop in pristine BNNTs and CNTs. The buckled shapes of pristine BNNTs and CNTs are shown in Fig. 2c and h, respectively. A global column-like buckling is observed for the CNTs. At the same time, a local shell-like buckling develops at the centre of the column-like buckled CNTs, since one side of the globally buckled CNT is highly compressed. Such buckling modes of CNTs agree well with previous reports.18,19,21 As for pristine BNNTs, a compression-induced global shell-like buckling (displays like flattening ‘‘fins’’) is detected. The different buckled shapes between the CNTs and BNNTs mainly originate from their different material properties and surface patterns.
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The smaller elastic constants and buckled surface of BNNTs both make them more readily amenable to buckle like shells. In Fig. 2a we also find that there exists only one abrupt strain energy drop in BN–CNT-1, corresponding to the occurrence of shell-like buckling as illustrated in Fig. 2d. However, different from its pristine BNNT counterpart, this compression-induced shell-like buckling of BN–CNT-1 exists locally only in its BN domain. A similar local shell-like buckling mode is also observed in BN–CNT-2 (see Fig. 2f), which corresponds to the first abrupt strain energy drop in the U–e curve of BN–CNT-2 as shown in Fig. 2a. The different buckling positions in the BN–CNTs and the BNNTs are due to the difference in the buckling resistance between the BNNTs and the CNTs. From the aforementioned discussion BNNTs are found to have smaller critical buckling strain than the CNTs. As a result, when the strain applied to the BN–CNTs increases, their BN domains will firstly buckle, while, at the same time, the C domains will not. This difference is attributed to the larger buckling resistance of the C domain. On the other hand, when the strain applied to the BNNT goes beyond its critical buckling strain, the entire BNNT will lose its stability, since BNNT is a homogeneous material. The critical strain ecr1 corresponding to such local shell-like buckling of the BN–CNTs is found to decrease with increasing molar fraction of BN (determined by lBN). For example, ecr1 of BN–CNT-2 is 0.12 and is about 30% greater than that of BN–CNT-1. This composition-dependent ecr1 can be qualitatively understood using the continuum mechanics theory (Sanders shell model),
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which gives the critical buckling strain of a tube structure as22 ecr1 pffiffiffiffiffiffiffiffi p (1 0.1l/R t=R). Here l is the length of the structure and equals to lBN for BN–CNTs, since the shell-like buckling occurs locally in the BN domain. From this expression, we can see that ecr1 decreases as lBN increases, which agrees with our MDS results. Finally, we will turn our attention to the post-buckling behaviour of nanotubes. Fig. 2a illustrates that during the post-buckling stage there exists no significant abrupt strain energy drops in the CNT, BNNT and BN–CNT-1, which means that the buckling modes of these nanotubes will not change during their post-buckling stage. Indeed, for example, Fig. 2e shows that during the continued compression of an already buckled BN–CNT-1, it will maintain the shell-like buckling mode in its BN domain, despite the appearance of some localised kinks. However, for BN–CNT with relatively small molar fraction of BN, e.g., BN–CNT-2, when we keep compressing the already buckled BN–CNT, a second drop in strain energy is observed at the strain level of ecr2, which equals to 0.15 for BN–CNT-2 (see Fig. 2a). This indicates that the buckling mode of BN–CNT-2 will topographically change after this critical strain ecr2. Indeed, we can see from Fig. 2g that the buckling mode of BN–CNT-2 has transferred from the local shell-like buckling (Fig. 2f) to the global column-like buckling (Fig. 2g). The different post-buckling behaviours of BN–CNT-1 and BN–CNT-2 are mainly due to their different numbers of the flattening fin following the first buckling mode. According to previous studies on shelllike buckling of CNTs,16 the number of the flattening fin of the buckled zig-zag nanotubes is proportional to the nanotube length. A similar conclusion is found to hold for their BN counterparts. Fig. 2e and f show that after the first buckling mode, the buckled BN domain of BN–CNT-2 has only one fin, while the BN domain of BN–CNT-1 displays four fins. In the post-buckling stage, each fin will serve as a hinge.14 Further compression of the a priori buckled BN–CNT-1 will lead to additional kinks to the shell-like buckling mode (see Fig. 2e). At the same time, the only fin in BN–CNT-2 serves as a hinge connecting two relatively long neighbouring C domains of BN–CNT-2. Thus, when BN–CNT-2 is compressed further, the fin will facilitate the buckling of its C domains, and thus induce the column-like buckling of BN–CNT-2 (see Fig. 2g). The buckling mode transition is observed for BN–CNTs whose molar fraction of BN c is smaller than 0.38; and ecr2 is found to decrease with increasing c due to the weaker elastic stiffness of BN domain compared to its C counterpart. Moreover, in Fig. 2g we find that although there still exists a local shell-like buckling in the BN domain of the globally buckled BN–CNT-2, the mechanism of this local shell-like buckling is completely different from that shown in Fig. 2f. In Fig. 2g, the local shell-like buckling is due to bending, and thus only one side of BN–CNT-2 buckles; in Fig. 2f the local shell-like buckling is induced by the axial compression, as a result both sides buckle. 3.2
Influence of the location of the BN domain
In the above analysis, we have studied BN–CNTs whose BN domains are placed at the centre of nanotubes. Subsequently,
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we shall study the influence of the location of the BN domain on the buckling instability of BN–CNTs. To this end, BN–CNTs with lBN = 7.4 Å, 16.3 Å, and 25.3 Å and varying LBN are considered here. As we depicted in Section 3.1, the buckling mode transition phenomenon can be observed for all BN–CNTs with the length of the BN domain smaller than 16.3 Å (BN composition smaller than 38%), regardless of the location of the BN domain (LBN). The critical strain ecr1 corresponding to the first (local shell-like) buckling and ecr2 corresponding to the second (global column-like) buckling are calculated in Fig. 3a as a function of LBN. We can see from Fig. 3a that ecr1 is almost independent from the location of the BN domain and is around 0.12 for BN–CNTs with lBN = 7.4 Å, around 0.11 for BN–CNTs with lBN = 16.3 Å, and about 0.1 for BN–CNTs with lBN = 25.3 Å. This is because that ecr1 is a parameter corresponding to the local buckling behaviour of the BN domain and thus is almost independent of the global parameters of BN–CNTs, e.g., the location of the BN domain considered here. On the contrary, we find in Fig. 3a that ecr2 strongly depends on the location of the BN domain. It is found that ecr2 holds the minimum value when the BN domain locates at the centre of the BN–CNT. When the BN domain moves from the centre to the ends of the BN–CNT, ecr2 is found to firstly increase and reach a maximum value at LBN = 8.5 Å or 25.6 Å for BN–CNTs with lBN = 7.4 Å and at LBN = 8.5 Å or 17.0 Å for BN–CNTs with lBN = 16.3 Å; after that ecr2 decreases as the BN domain keeps moving to the ends. In this process, the deviation between the minimum and maximum of ecr2 is about 13%. It is known that the global stability of BN–CNTs is determined by their ability to sustain the beam-bending, which is
Fig. 3 (a) The 1st and the 2nd buckling critical strains as a function of the location of the BN domain in BN–CNTs (measured by LBN as shown in Fig. 1). (b) Bending moment diagram of the fixed–fixed buckled nanotube. (c)–(g) Respectively show snapshots of the BN–CNT-2 (lBN = 7.4 Å) with LBN = 8.54 Å, 12.81 Å, 16.96 Å, 21.35 Å, and 25.62 Å after the 2nd buckling.
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measured by the bending stiffness. In Fig. 3b, we plot the bending moment diagram of a buckled fixed–fixed homogeneous beam.42 It can be seen from Fig. 3b that the bending moment is zero between the centre and ends of the beam. Thus, the whole BN–CNT can achieve the largest overall bending stiffness when the BN domain with relatively smaller bending stiffness (compared with its C counterpart) locates in or just near the zero-bending moment area. This theoretical analysis agrees well with our MDS results illustrated in Fig. 3a, where ecr2 holds a maximum value when the BN domain locates between the centre and ends of BN–CNTs. In addition, although the location of the BN domain has no influence on the first buckling modes of BN–CNTs, it may have a significant influence on the configurations of BN–CNTs after their second buckling. As an example, in Fig. 3c–g we display the configurations of BN–CNT-2 (lBN = 7.4 Å) after the second buckling. Comparing Fig. 3c–g, it can be deduced that although all BN–CNTs globally display the column-like deformation, the local patterns of these buckled nanotubes are different. For BN–CNTs with the BN domain at their centre (Fig. 3e), the compression-induced local shell-like buckling mode (induced by the first buckling) changes to the bending-induced local shell-like buckling mode. As for BN–CNTs with the BN domain far from their centre and near their ends (Fig. 3c and g), the compression-induced first buckling mode is preserved after the second buckling. In addition to this preserved compressioninduced first buckling mode, another local shell-like buckling mode develops at the central C domain, which is induced by the bending of the BN–CNTs following the second buckling. For BN–CNTs with the BN domain far from their ends and near their centre, e.g., BN–CNTs with LBN = 12.81 Å and 21.35 Å shown in Fig. 3d and f, it is expected that these two BN–CNTs display the same buckling character, since their BN domains are symmetrical to the centre of the structure. However, we can see from Fig. 3d and f the buckling modes of these two BN–CNTs are quite different actually. The BN–CNT with LBN = 12.81 Å shows a similar buckling behaviour to the BN–CNT with the BN domain at its centre (Fig. 3d), while the BN–CNT with LBN = 21.35 Å shows a similar buckling behaviour to the BN–CNT with the BN domain near its ends (Fig. 3f). This difference originates from the different bond characters between the C–B and C–N bonds. For the BN–CNTs with LBN = 12.81 Å the C–N bonds play a more important role (compared with the C–B bonds) in resisting the buckling since they are located much closer to the centre of the structure (see Fig. 3d) and thus could accommodate much larger bending moment (see Fig. 3b); for those with LBN = 21.35 Å the C–B bonds thus play a more important part in withstanding the buckling for the same reason (see Fig. 3f).
4. A case study of potential applications of the buckling of BN–CNTs The above understanding of the buckling instability of BN–CNTs could be potentially leveraged to enable the design of new functions of nano-devices. Thus, as a demonstration, we report
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Fig. 4 (a) Variation of the total strain energy of the water filled BN–CNT with the axial strain. (b) Initial configurations (without strain) of the water filled BN–CNT. (c) Typical tube geometry before the 1st buckling: open position of the nano-valve. (d) Typical tube geometry after the 1st buckling: closed position of the nano-valve.
a valve-integrated water channel by BN–CNTs. Previous studies have demonstrated that pristine CNTs and BNNTs can both serve as the water channel.43–46 Enthusiasm for utilizing nanotubes for water transport, one crucial but largely unexplored issue for the success of this potential application is the design of a smart nano-valve to control the water flow. In what follows, we will show that the BN–CNT can serve as a water channel integrated with a valve naturally by using the local shell-like buckling of its BN domain. In this simulation, a BN–CNT with the same geometric size of the above BN–CNT-2 is considered. Initially, such BN–CNT was immersed in a water reservoir, and a single-file chain of water molecules inside the BN–CNT was obtained after that (see Fig. 4b). In this study, the non-bonded van der Waals (vdW) interactions of C–water and BN–water systems was simulated by the Lennard-Jones (LJ) atomic pair potential: Vij = 4d[(s/rij)12 (s/rij)6]. Here, the LJ parameters d and s employed in the present simulation were adopted from ref. 47 and 48. Fig. 4a shows the strain energy U-strain e curve of the waterfilled BN–CNT under compression. Similar to the empty BN–CNT case (Fig. 2), U initially quadratically increases with increasing e and then abruptly falls down when e reaches the critical value ecr1 corresponding to the local shell-like buckling of the BN–CNT. Compared with its empty counterpart, the present water-filled BN–CNT is much less stable, since ecr1 = 0.07 measured here is much smaller than 0.12 that obtained from Fig. 2. As indicated in previous studies, the effect of the filler on the buckling strength of nanotubes strongly depends on the size and density of the filled atoms or molecules, the nanotube diameter, length and chirality, among others.49–52 For example, due to the encapsulation of gold atoms49 or crystalline zinc sulphide nanowires,50 the buckling strengths of filled CNTs were greatly enhanced. In the meantime, smaller critical buckling strains similar to those observed in the present study were also found in the water-filled CNTs51 and the hydrogen-filled BNNTs.52 The existence of the water molecules in
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open-ended nanotubes seems to result in extra attractive or repulsive lateral forces imposed on the tube wall. As a consequence, the stress concentration is intensified and the local perturbations present in the walls of the nanotubes lead to a reduction in their buckling strengths and may accelerate their mode of collapse. In Fig. 4c and d we plot the configuration of the water-filled BN–CNT just before and after the buckling. We can see from Fig. 4c that before the BN–CNT buckles the water molecules distribute continually along the BN–CNT just like the case without strain (Fig. 4b). However, after the BN–CNT buckles (Fig. 4d) the water molecules in the local buckled BN domain exit this buckled region due to the change of the vdW force associated with the crimpling of the tube wall. At the same time, the water molecules in the two C domains distribute continually, which is similar to the case before buckling is encountered. In other words, Fig. 4c and d indicate that the water molecules cannot flow through the buckled BN–CNT only due to local buckling of the BN domain. After releasing the external constraint and relaxing the tubes, the BN–CNT recovers to its initial configurations. As a result, in this unloaded case the water molecules can well transport along the BN–CNT again. Based on this analysis, we can see that the local BN domain of the BN–CNT can be treated as a natural local valve in the BN–CNT-based water channel. Finally, it is worth pointing out that the aforementioned calculations were performed at 1 K temperature to exclude the effect of thermal fluctuations. The study was further extended to account for the effect of room temperature (300 K) upon the applicability of our proposed ‘‘nano-valve’’. To this end, we plot U–e curve in Fig. 4a and the buckling mode in Fig. 4d for the same water-filled BN–CNT at room temperature. Analogous to the 1 K results, the room temperature results show that the water molecules are forced to leave the BN domain of BN–CNT due to the increased repulsive vdW forces ensued by the buckling mode of the BN domain (see Fig. 4d). This result indicates that the working mechanism of the present nano-valve at 1 K holds for room temperature. In addition, Fig. 4a shows that the critical buckling strain of the water-filled BN–CNT (the operating strain of the nano-valve) at room temperature is smaller than that observed at 1 K temperature. This is due to the significant effect of the thermal fluctuations at room temperature.
5. Conclusions The buckling instability of BN–CNTs has been studied using MDSs in this paper. The buckling behaviour of BN–CNTs is found to strongly depend on their composition. For BN–CNTs with large BN composition, they only undergo local shell-like buckling in the BN domain; for BN–CNTs with small BN composition, they experience local shell-like buckling firstly but this local shell-like buckling changes to global column-like buckling of the whole BN–CNT with increasing strains. The critical strains for local shell-like and global column-like buckling both decrease with increasing BN composition. In addition,
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the critical strains and buckling modes of the global columnlike buckling of BN–CNTs also strongly depend on the location of the BN domain. As a case study, we demonstrated that the BN domain can serve as a local value for the BN–CNT-based water channel operating by its local buckling character. This analysis suggests that the BN–CNT can be treated as an integrated building block for the nano-device design.
Acknowledgements This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Discovery Accelerator Supplement. The authors also wish to offer their thanks for the anonymous reviewers for their constructive remarks.
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