Abstract
We perform largescale quasicontinuum simulations to determine the stable crosssectional configurations of freestanding multiwalled carbon nanotubes (MWCNTs). We show that at an interwall spacing larger than the equilibrium distance set by the interwall van der Waals (vdW) interactions, the initial circular crosssections of the MWCNTs are transformed into symmetric polygonal shapes or asymmetric waterdroplike shapes. Our simulations also show that removing several innermost walls causes even more drastic crosssectional polygonization of the MWCNTs. The predicted crosssectional configurations agree with prior experimental observations. We attribute the radial corrugations to the compressive stresses induced by the excessive interwall vdW energy release of the MWCNTs. The stable crosssectional configurations provide fundamental guidance to the design of single MWCNTbased devices and shed lights on the mechanical control of electrical properties.
Keywords:
Carbon nanotubes; Radial corrugation; Quasicontinuum; vdW interactionsIntroduction
The unique combination of mechanical, electronic, and biochemical properties of carbon nanotubes (CNTs) has found a wide range of applications as building blocks in micro(nano)electromechanical systems (MEMS/NEMS) [16]. It has been experimentally demonstrated that mechanical deformation of CNTs is generally coupled with significant changes in the electronic and magnetic properties [4,7,8]. In addition, the structural stability of CNTbased devices has become a major concern in their applications. These have motivated continuing experimental [911] and numerical studies [1220] on the stable morphologies of CNTs.
Because of their extremely large inplane rigidity compared to their outofplane bending rigidity [21], graphene shells were frequently observed to undergo isometric deformation, featuring local folds [12,1518,2226]. Previous analyses showed that singlewalled carbon nanotubes (SWCNTs) may undergo beam or shell buckling under bending [23,25], compression [23], or twisting [23,24], depending on the aspect ratio of the tube. However, due to the presence of interwall van der Waals (vdW) interactions, the physical mechanisms governing the instability of multiwalled carbon nanotubes (MWCNTs), composed of concentric graphene shells, appear to be more complex [12,1518,27]. For example, wavelike periodic ripplings appear in twisted MWCNT [18], and Yoshimura patterns are present in a bent MWCNT [15,17]. It has been elucidated that these unique deformation patterns are driven by the inplane strain energy release, penalized by the interwall vdW energies [16,18,27,28]. Under pure bending or compression, MWCNTs with interwall covalent bridges exhibit evolving morphologies [17]. More recently, we have showed that helically arranged diamond pattern appears in thick, uniaxially compressed MWCNTs [16]. The helically arranged diamond pattern appears to be a coordinated deformation morphology of the rigid inner walls and compliant outer walls in the MWCNTs.
To date, theoretical analyses of the single MWCNTs have been largely based upon the assumption of perfect circular crosssections [29,30]. Whereas numerical studies [18,28,31,32] predicted that noncircular crosssectional shapes may be energetically favorable in certain conditions. For example, when bringing two CNTs into close contact, the contact region of the CNTs is fattened [19,31,33,34] to favor intertube adhesion energy. Under hydrostatic pressure, crosssectional shape transformation of MWCNTs from circular to polygonal configurations was observed [28]. Recent experiments reported that MWCNTs synthesized in the presence of nitrogen are constituted of walls with uniform chirality and their crosssections are of polygonal shapes rather than circular shapes [35,36]. It has been also observed in the experiments that a narrow inner core remains circular while a wide inner core is elongated with facets. It was argued that the polygonal shapes were stabilized in favor of the interwall adhesion energy due to the increased interwall commensuration. It was also suspected that polygonization may result from the interlayer thermal contraction upon cooling from the synthesis temperature [35].
Motivated by the experiments, in this article, we employ the quasicontinuum method [14,37] to determine the stable crosssectional configurations of freestanding MWCNTs with different interwall spacings and different radii of innermost walls. We show that both factors play significant roles in regulating the crosssectional configurations of the MWCNTs. Our simulations show that the crosssections of the freestanding MWCNTs deviate from the normal circular shape and may be stabilized at polygonal or waterdroplike shapes, depending on the interwall spacings. In addition, when the radius of the innermost wall in an MWCNT is beyond a critical value, the crosssections of the MWCNTs may also be stabilized at polygonal shapes and the extent of crosssection polygonization is even greater. These modeling results agree with prior experimental observations. We attribute the stable corrugated crosssections to the compressive stress induced by the release of the excessive interwall vdW energies in the MWCNTs.
The rest of the paper is organized as follows. "Methodology" section briefly introduces the quasicontinuum method. In "Simulation Results" section, we present our simulation results and elucidate the deformation mechanisms. Conclusion remarks are presented in the last section.
Methodology
We adopt the modified secondgeneration Brenner potential [3840], denoted by MTBG2, to describe the shortrange covalent interactions in MWCNTs, which takes the following form:
where r_{ij} is the distance between atoms i and j, V^{R} and V^{A} are the pairwise repulsive and attractive interactions, respectively, B_{ij} is the bondorder function that has a complicated dependence on the bond angles and bond lengths involving atoms i and j. The interwall vdW interaction is described by a Lennard–Jones (LJ) potential with the parameters given by Girifalco et al. [41], as
where r is the interatomic distance, κ = 2.7 is a dimensionless constant, r_{0} = 1.42 Å is the equilibrium bond length and ∈ = 15.2 eV Å^{6}.
Allatom simulations with empirical interatomic potentials have been widely used to study the deformation of CNTs [42]. However, for the study of thick MWCNTs, fully atomistic simulations are computationally very expensive because of the large number of degrees of freedom involved. To improve the computational affordability, fully atomistic models are here coarsegrained via a quasicontinuum method based on the finite crystal elasticity theory for curved crystalline monolayers [14,15]. Within the theoretical framework, the exponential CauchyBorn rule was proposed to link the kinematics at the atomic and continuum scales:
where ζ is an exponential map [14,43] that transforms the undeformed lattice vector A into a deformed one a. Through a local approximation of the exponential map [14], the deformed lattice vectors and the angles between two lattice vectors can be analytically represented in terms of the continuum deformation measures of the surface, i.e., C and K, the stretch and curvature tensors, respectively. Considering a representative unit cell of area in the graphene lattice, the kinematic link allows analytically determining the hyperelastic strain energy from the underlying interatomic potentials, as
where a and α represent generic lattice vectors and angles between lattice vectors, respectively. The continuum representation of the covalent binding energy for the walls in an MWCNT subject to the deformation map φ that maps from the undeformed to deformed configurations is
where X is a material point in the undeformed configuration, ω^{i} is the surface area of ith wall in an nwalled MWCNT.
Homogenization of the discrete interwall vdW energy density between two unit cells gives rise to the vdW energy density, as
The factor of two on the righthand side of Eq. 6 comes from the fact that each unit cell contains two nuclei. The nonbonded energy between two neighboring shells is then
where X_{i} and X_{i+1} are the two material points that are on the ith and (i + 1)th shells, respectively, in the MWCNT; and are the surfaces of the ith and (i + 1)th shells, respectively.
Based on the coarsegrained constitutive relations for both the bonding and vdW interactions, the constituent shells of the MWCNTs are discretized by finite elements. As extensively tested [12,1418,43], the coarsegrained model accurately reproduces atomistic simulations; the computational efficiency is improved by about two orders of magnitude when compared to its atomistic counterpart. It should be pointed out that the quasicontinuum method described here is incapable of studying the deformation of defected CNTs, which has been a topic of active research [38,4450] for the last decade.
Simulation Results
In the experiments of Ducati et al. [35], the constituent graphene shells of the synthesized MWCNTs are isochiral and likely either zigzag or armchair walls. The measured interwall spacings are 0.355 ± 0.009 nm. To determine the effect of interwall spacings on the stable crosssectional configurations of freestanding MWCNTs, the choice of the MWCNTs to be studied is guided by the experimental settings. Three sets of MWCNTs with isochiral walls, indexed by (5,5)/(10,10)/.../(5n,5n), (9,0)/(18,0)/.../(9n,0), and (2,8)/(4,16)/.../(2n,8n), are chosen for our studies, where n is the number of walls in the MWCNTs. All the MWCNTs are 20 nm long. Based on the tube chirality, the three sets of MWCNTs are characterized as armchair (AC), zigzag (ZG), and chiral (CH) MWCNTs, respectively. For each set, five MWCNTs are chosen for our studies with the numbers of walls being 5, 10, 15, 20, and 25. It is useful to remember that the diameter of an (a, b) SWCNT is given approximately by nm. Prior to structural optimization, the initial crosssections of all the MWCNTs are of circular shape and the interwall spacings are 0.339, 0.352, and 0.359 nm for the AC, ZG, and CH MWCNTs, respectively. One notes that the initial interwall spacings of ZG and CH MWCNTs are 0.12 and 0.19 nm, respectively, larger than the equilibrium spacing of two graphene sheets (0.34 nm) set by the interwall vdW interaction potential described by Eq. 2. To obtain the equilibrium configurations, the MWCNTs are fully relaxed free of any constraints using a limitedmemory BroydenFletcherGoldfarbShanno (BFGS) algorithm [51].
Figure 1 depicts the crosssectional configurations of relaxed MWCNTs. From top to bottom, the three rows correspond to the AC, ZG, and CH MWCNTs, respectively. Figures on each row from left to right represent the relaxed configurations of 5, 10, 15, 20, and 25walled MWCNTs, respectively. It is observed that in the relaxed state, the crosssections of the AC MWCNTs remain circular shape with graphically invisible morphological change. The crosssections of the relaxed 5 and 10walled ZG MWCNT (Figure 1f, g) also remain circular shapes, whereas the 15walled ZG MWCNT (Figure 1h) is stabilized at a polygonal crosssectional configuration with 6 rounded corners, which agrees with the experimental observations [35]. Compared to the 15walled ZG MWCNT, the extent of polygonization of the crosssections is greater for the 20walled ZG MWCNT (Figure 1i). For the 25walled ZG MWCNT, the stabilized crosssection becomes asymmetric, featuring a waterdroplike morphology. For the CH MWCNTs, the crosssectional shape transformation is even more drastic. Similar trends seen in ZG MWCNTs are observed in CH MWCNTs, except that the polygonized and waterdroplike morphologies occur earlier. From these observations, a general trend of the relaxed crosssection morphologies can be obtained: increasing interwall spacings or the number of walls in the MWCNTs drives circularpolygonalwaterdroplike shape transition.
Figure 1. Crosssectional views of relaxed MWCNTs. From left to right on each row, the wall numbers are 5, 10, 15, 20, and 25. Top AC MWCNTs; middle ZG MWCNTs; bottom CH MWCNTs.
We next elucidate the deformation energetics of the MWCNTs. Regarding the walls as thin shells, the total energy of an MWCNT is the sum of the interwall vdW interaction energy, the outofplane bending energy, and the inplane stretching energy. Because of the much higher inplane stiffness of the graphene walls, inplane stretching is not an energetically favorable mode. The deformation morphology of an MWCNT free of external loads is a result of the competition between the interwall vdW interaction energy and the outofplane bending energy. Since the unrelaxed interwall spacings for both ZG and CH MWCNTs are higher than the equilibrium interwall spacing (0.34 nm) set by the vdW interactions, the excessive interwall vdW interaction imposes a compressive stress on the walls. As a result, the MWCNTs are bent to release the interwall vdW interaction energy, which drives the polygonization of the crosssections of the MWCNTs. For the CH MWCNTs, the interwall spacing is the largest and so is the driving force for the shape transformation. As a result, the large driving force leads to shape symmetry breaking, and the crosssections are stabilized at a waterdroplike shape. It should be noted that for the simulations in Figure 1, because of the inner walls are very rigid, it is energetically very costly even to bend these walls. Thus, these inner walls remain circular shapes.
Figure 2 shows the average interwall spacings of the relaxed configuration of the ZG MWCNTs shown in Figure 1 (from f to j). It should be noted that the spacing of the innermost wall is taken as its radius. Except for the 25walled MWCNT, the interwall spacings of all the other MWCNTs monotonically decrease from inner to outer walls. This is because the outer walls are more compliant, easier to bend, and thus more effective in releasing the interwall vdW interaction energy. In contrast, the interwall spacings of the 25walled MWCNT undulate, which may be attributed to the asymmetric crosssectional corrugation. We also note that for the innermost two walls (except for the 25walls MWCNT), the relaxed spacings are larger than the original spacing (0.352 nm). This is because of the curvature effect in rolling a graphene layer into the cylindrical shape. Due to the rolling, the bonds become the chords on the curved surface and are shortened. In the relaxed configuration, these bonds are stretched to their equilibrium length in order to minimize the inplane stretching energy. Thus, the radii of the innermost walls increase. For the rest of the walls, the curvature effects are negligible. Due to the compressive stress generated by the excessive vdW interaction energy release, the interwall spacings are reduced to the values lower than the original spacing. One notices from Figure 1f to j that the rounded corners in the relaxed configurations are increasingly more appreciable graphically when the number of walls increases, indicating increasingly higher bending energy. Our analysis thus concludes that the relaxed configuration is a result of interwall vdW interaction energy release, penalized by the outofplane bending energy.
Figure 2. The interwall spacing of the relaxed ZG MWCNTs shown in Figure 1 (from (f) to (j)).
Previous simulations have demonstrated that the innermost walls in thick MWCNTs act as a hard core that plays an important role in the deformation morphologies of MWCNTs [1618,28]. This explains well the experimental observations that an inner core of small radius remains circular. In order to study the effect of the innermost walls on the stable crosssectional configurations of freestanding MWCNTs, we considered three sets of ZG MWCNTs with removed innermost walls, thereby varying the radius of the innermost wall of the remaining MWCNTs, as shown in Figure 3. In the three rows from top to bottom, 5, 10, and 15 innermost walls, respectively, are removed from the corresponding ZG MWCNTs (9,0)/(18,0)/.../(9n,0). In each row from left to right, the numbers of walls left in the MWCNTs are 5, 10, 15, 20, and 25.
Figure 3. Crosssectional views of ZG MWCNTs with different radii of the innermost walls. From left to right on each row, the wall numbers are 5, 10, 15, 20, and 25. Top ZG MWCNTs with initial, unrelaxed innermost wall radius of 2.1 nm; middle ZG MWCNTs with initial, unrelaxed innermost wall radius of 3.8 nm; bottom ZG MWCNTs with initial, unrelaxed innermost wall radius of 5.6 nm.
Prior to structural optimization, the radii of the innermost walls are 2.1, 3.8, and 5.6 nm for 5wall, 10wall, and 15wall removed MWCNTs, respectively. As shown in Figure 3, upon energy relaxation, the crosssections of all the MWCNTs are transformed from circular to polygonal shapes with flattened sides. The crosssectional shapes in Figure 3h and 3m agree well with reported experimental results [35,52]. A general trend is observed: along each column from top to bottom or along each row from left to right, the configurations of the innermost walls deviate progressively further from the circular shape. These morphological evolutions are generally due to the competition of the interwall vdW energy release and the outofplane bending penalty, as discussed earlier. Along each column from top to bottom, with increasing radius of the innermost walls, the innermost walls and the whole MWCNTs are increasingly more complaint, and thus easier to bend to facilitate interwall vdW energy release. While along each row from left to right, with increasing wall numbers, the increasingly more excessive interwall vdW energy imposes progressively higher compressive stress onto the innermost walls, thereby inducing larger deformation from their circular shape.
Conclusions
We have performed largescale quasicontinuum simulations on the stable crosssectional configurations of MWCNTs. Our simulations show that both the interwall spacing and the radius of the innermost wall play important roles in regulating the stable crosssectional configurations of the MWCNTs. The relaxed crosssectional configurations agree well with prior experimental observations. We attribute the shape transformations to the interwall vdW interaction energy release, penalized by the increase of the wall bending energy. This deformation mechanism is opposite to that in bent, twisted, and compressed MWCNTs, where the deformation morphologies such as wavelike periodic rippling and Yoshimura pattern are driven by the inplane strain energy release, but penalized by the increase of the interwall vdW interaction energy [1618].
It should be pointed out that from our simulations, the crosssectional corrugations occur only for MWCNTs for which interwall vdW interactions are present. This may explain that crosssectional corrugations were rarely reported for MWCNTs in previous allatom simulations since such thick MWCNTs are typically beyond the access of allatom simulations. Given the intimate coupling between the deformation morphologies and the electrical/magnetic properties of CNTs, the predicted stable crosssectional configurations provide fundamental guidance to tailor the electrical properties of MWCNTs.
Acknowledgements
We gratefully acknowledge the grant support from the National Science Foundation grant under award No. 0600661.
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