Computational Materials Science 80 (2013) 15–26

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Computational Materials Science

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Fracture and negative Poisson’s ratio of novel spanned-fullerenesnanotube networks under tension

0927-0256/$ - see front matter � 2013 Published by Elsevier B.V.http://dx.doi.org/10.1016/j.commatsci.2013.04.033

⇑ Corresponding author. Tel.: +47 73 59 47 00; fax: +47 73 59 47 01.E-mail address: zhiliang.zhang@ntnu.no (Z.L. Zhang).

J.Y. Wu, J.Y. He, Z.L. Zhang ⇑NTNU Nanomechanical Lab, Norwegian University of Science and Technology (NTNU), Trondheim N-7491, Norway

a r t i c l e i n f o

Article history:Received 14 December 2012Received in revised form 18 April 2013Accepted 19 April 2013Available online 11 May 2013

Keywords:Mechanical propertiesCarbon nanotubeFullereneNano-bambooNano-meshNano-truss structureNegative Poisson’s ratio

a b s t r a c t

Carbon-based nanomaterials have attracted significant attention due to their unique physical properties.In this study, various multi-dimensional graphitic architectures are constructed by spanning fullereneswith carbon nanotube (CNT) super-bonds. The mechanical properties of these novel architectures are sys-tematically investigated by full atomistic simulations. The stress and strain of 1D nano-bamboo struc-tures upon the onset of instability are almost constant, about 1/5 and 1/2, respectively, of those of aperfect CNT. The deformation and fracture behavior of 2D and 3D periodic graphitic nanostructures arelargely dictated by the inter-fullerene distance and loading orientation. Surprising negative Poisson’sratio observed in 2D and 3D networks is revealed to originate as a result of curvature-flattening or rigidmechanical model. The magnitude of Poisson’s ratio is strongly dependent on the level of strain, CNTlength as well as temperature. The insight on the deformation mechanism of these periodic graphiticnanostructures will facilitate the integration of low-dimensional materials towards high-dimensionalorganized structures to realize targeted multi-functional properties.

� 2013 Published by Elsevier B.V.

1. Introduction

The discovery of fullerene and nanotube [1,2] has triggeredenormous interest in carbon-based nanomaterials. Over the lasttwo decades, both experimental and theoretical studies have dem-onstrated that they, as unique zero-/one-dimensional materials,possess predominant physical, chemical, and mechanical proper-ties, in additional to the advantage of light weight, which inspirefurther efforts to utilize their promising properties in practicalnanotechnology applications [3–6]. For example, their electricaland chemical properties enable them as candidates for medicalimaging devices, superconductors, catalysts, diagnostic tools, andetc. [7–17]. Their superior stiffness, strength and resilience whichexceed the prior-art materials make them attractive as new fillerfor advanced reinforced composite materials [18,19].

Beyond the individual fullerene and nanotube, their derivativesare also equally important nanomaterials. The first hybrid carbonnanostructures termed ‘‘nanopeapod’’ – fullerenes encapsulatedin CNTs, has been fabricated, and these nanopeapods have shownto exhibit a variety of intriguing physical and chemical properties[20–23]. Another type of hybrid nanostructures based on fullereneand CNTs, called ‘‘nanobuds’’, hold exceptional field emission prop-erties [24]. Until now, a series of other CNTs-based junctions with

difference morphologies, such as I-, L-, T-, Y-, H-, K- and X-shapedjunctions, have been reported [25–36]. Because of the junctionthese nanoclusters possess special properties which the straightCNTs do not have. For example, a Y-shaped CNT junction displaysrectifying and electrical switching properties, such that it can serveas a switching device, and the origin of this electrical switchingproperty is though not well understood yet [37–39].

Profiting from the ability of synthesizing various shapedCNT-junctions and assembly of CNTs in terms of position andorientation [40–44], complex 2D and 3D CNT architectures canbe fabricated to realize specific functional properties for a varietyof meso- and macro-scale engineering applications [45–52]. Onenovel 3D CNT architecture with tree-like complex hierarchicalbranch, (a primary stem abruptly dividing into 3, 4, 8 and 16branches) has been precisely synthesized through pore designand templated assembly. This 3D complex nanostructure has inter-esting applications in biotechnology such as drug delivery [53]. Byinfiltrating a polymer into a reticulate CNT architecture, CNT-net-work-based reinforced composites are formed to achieve superiormechanical properties [54], originating from strong molecular-le-vel coupling between reticulate CNTs and polymer chains. Very re-cently, covalently bonded 3D CNT networks have also beenproduced at large scale [55], in which an effective stress distribu-tion in the network improves the mechanical strength of the mate-rials. In parallel with the experimental production of CNT-basedhierarchical structures, computer-aided design and simulation of

Table 1Tensile strain ef and the corresponding tensile stress rc as well as the Young’smodulus for the 1D nano-bamboos.

Specimen, N ef rc (N/m2) E (GPa)

0 0.153 25.7 1491 0.134 23.6 1562 0.133 23.4 1603 0.131 23.6 1654 0.130 24.0 1766 0.128 24.2 1868 0.125 24.2 194

10 0.125 24.3 2001 0.280 117 896

Fig. 1. Schematic illustration of the units of CNT-fullerene based architectures. (a) Faceting giant fullerene C372 (octahedral, D2d) with six pentagon–pentagon pairs (top);(6,6) CNT segment. (b) A fullerene hub with 2-terminal for coaxially connecting (6,6) CNT segments (left); 1D nano-bamboo with a CNT length N = 10 (middle); side-view of1D nano-bamboo (right). (c) Two side-views of a hub with coplanar 4-terminal (left and middle); one unit cell of 2D X-shaped nano-mesh (right). (d) A hub with bi-planar 8-terminal (left), analogous to zeolite; a unit cell of bcc-lattice (middle); one representational unit cell of 3D nano-truss matrix (right), similar to a bcc lattice crystal. Coloringatoms of the hub clusters are according to potential energy in the construction. The red color highlights the fullerenes in the nodes of 1D, 2D and 3D architectures for clarity.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2. Stress–strain responses of 1D nano-bamboo with N varying from 0 to 10.N = 0 and 1 respectively represent a linear polymerization of fullerenes and aperfect (6,6) CNT.

16 J.Y. Wu et al. / Computational Materials Science 80 (2013) 15–26

2D and 3D nanostructures consisting of fragments ofCNTs-junctions helps to exploit their potentials [56–64]. Usually,2D polygonal CNT networks, such as super-triangle, super-squareand super-graphene [57,58,62,64,65], are built up by employingY- and X-shaped CNT junctions as building bricks. These 2D CNTnetworks can be utilized to construct hierarchical 0D and 1Dsuper-architectures [56,66,67], such as super-fullerene, super-nanotube and super-nanocone. Similarly, a design of 3D hexagonal,

orthogonal and diamond-like super-architectures has been pro-posed based on various multiplanar CNT-junction clusters andtheir corresponding electrical and mechanical properties havebeen investigated [57,58,68]. Despite the increasing interest inCNT architectures, nano-joining and assembly conforming 2D and3D networks are in very early stages.

In this paper we invoke the concept of hierarchical nano-joiningto construct new multi-dimensional architectures composed ofCNTs and fullerenes. A systematic full atomistic simulation studyon the mechanical properties of selected 1D, 2D and 3D fullerene– single wall CNT architectures is performed. The details of theatomistic models created and the computational method usedare documented in Section 2. The results and discussions are pre-sented in Section 3. Some concluding remarks are made at theend of the paper.

(a) (b)

Fig. 3. Stress–strain relations of 2D nano-mesh with N varying from 0 to 10 under uniaxial tension tests along x and y directions as shown in schematically on top of thefigure, respectively.

J.Y. Wu et al. / Computational Materials Science 80 (2013) 15–26 17

2. Models and methodology

2.1. Models

Unlike most of the nanostructures studied in the literature,which consists of only CNTs, we adopt a CNT (6,6) and a giant ful-lerene C372 with D2d molecular symmetry as parent compounds forconstructing novel covalent bonded 1D nano-bamboo, 2D nano-mesh and 3D nano-truss systems. As illustrated in Fig. 1, the 1D,2D and 3D nanostructures are built from a (6,6) CNT and fullereneC372. Three atomic pairs are added as patches in the face of fuller-ene to create an open-cage cluster with a terminal. This terminal isable to coaxially dock a CNT segment, forming three octagon ringsdispersed at the junction. When the fullerene is opened with twocollinear terminals, a 1D bamboo-like nanostructure is generated.The octahedral fullerene C372 consisting of eight faces allows oneto construct 2D and 3D architectures, with the representative unitof ordered 2D and 3D network shown in Fig. 1c and d, respectively.The truss-like 3D network with an all-sided coalescence of fuller-enes with CNTs resembles a body center cubic (bcc) lattice. Wefound that the binding energies of all these architectures are largerthan that of the fullerene C60. The mechanical responses of as-gen-erated nanostructures under tension are systematically scrutinizedwith respect to the connection CNT length N which varies from 0 to10.

2.2. Computational method

The full atomistic simulations are carried out using LAMMPSsoftware package capable of running on large computing clusters.The adaptive intermolecular reactive empirical bond-order (AIREB-O) potential [69] based on the widely used second-generationBrenner potential is used for carbon atomic interactions in the

simulations [70]. The cutoff parameter for the REBO part of the po-tential is set to be 2.0 Å as described by Shenderova [71] in order toavoid the spuriously high bond forces and nonphysical phenome-non during the fracture process. And the cutoff for the Lennard–Jones part is set as 10.2 Å. The applied potential has been success-fully utilized to model the deformation of carbon-based materialsunder mechanical loading as well as stochastic bond-breaking/reforming process [61,63–65,67]. Periodic boundary condition(PBC) is implemented to preclude edge effects, and the appliedPBC direction depends on the specific dimension of the constructednanostructures. As-constructed architecture models with tripleunit cell in the PBC direction are first quasi-statically relaxed byminimizing the total potential energy through a conjugate gradientmethod. During the relaxation process, all the full atomistic simu-lations are subjected to a NPT ensemble, carried out at zero pres-sure and 1 K temperature with Nosé–Hoover barostat andthermostat up to reach equilibrated state. Then mechanical loads,such as uni-/bi-/tri-axial tension under deformation-control asour previous study [72], are performed on 3D equilibrated struc-tures until complete rupture of each specimen. The samples are de-formed under a reasonable strain rate 0.0001/ps and with strainincrement of every 0.1 ps based on Nosé–Hoover thermostat.Stresses and strain are calculated every 1000 steps.

In order to characterize the mechanical properties, Young’smodulus and Poisson’s ratio as well as the fracture stress and strainare determined. Young’s modulus is calculated as the initial slopeof the stress–strain curve. The nominal strain parallel to the direc-tion of deformation is defined as:

e ¼ L� L0

L0¼ DL

L0; ð1Þ

where L0 and L are the equilibrium and the current length of the en-tire systems in the direction of loading. To determine the force and

Fig. 4. Local atomic structure developments during uniaxial tension. (a) Equilibrated and just before fracture state of 1D nano-bamboo with N = 0 from two side-views. The x-directional viewed configuration changes from hexagon to chimonanthus-like. The corners with curvature undergo small deformation due to the rigid surface in the vicinityof defects. (b and c) Sequential deformation snapshots of 2D networks when N = 0 and 10 in x (upper) and y (lower) directions, respectively. The fracture pattern is governedby the tension direction. It is clearly shown that, when N = 10, the strain-reorientation of CNT segments turns up, resulting in a nonlinear stiffening behavior. Brighter colorsindicate higher potential energy. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

18 J.Y. Wu et al. / Computational Materials Science 80 (2013) 15–26

the tensile strength, the per-atom stress tensor in response to theexternal load is computed by:

raij ¼

1Va

12

mavai v

aj þ

Xb¼1;n

rjabri

ab

!; ð2Þ

where i and j are indices of the coordinate system; a, b are theatomic indices; ma and va are the mass and velocity of atom a;rab is the distance between the two atoms; and Va is the atomic vol-

ume of atom a. The overall stress of the nanostructures is obtainedby averaging all the atoms. The tensile stress and strain are read di-rectly from the peak point of the stress–strain curve.

Under uniaxial tension, the transverse repositioning of nano-structures perpendicular to the load axis gives a measure of thestrain from which the Poisson’s ratio can be calculated as below:

m ¼ � deT

deL; ð3Þ

Fig. 5. (a and b) Deformation dependence of the ratio of the transverse and longitudinal strains for 2D nano-mesh under uniaxial tension along y and x direction, respectively;Transformation of locally atomic motifs evidently shows an (x–y) in-plane expansion of the fullerene hub when respectively undergo strain of 0.12 and 0.10 in y and x axis,which reveals the negative Poisson’s ratio of this 2D nano-mesh.

J.Y. Wu et al. / Computational Materials Science 80 (2013) 15–26 19

where eT and eL are the transverse and longitudinal strains, respec-tively. In a biaxial tension along the x and y directions, the threedirectional strains can be expressed by following equation:

ex ¼rx

E� m

ry

E; ey ¼

ry

E� m

rx

E; ez ¼ �

mEðrx þ ryÞ: ð4Þ

From above, the Poisson’s ratio is given below:

m ¼ ez

ez � ex � ey: ð5Þ

The strain along the three Cartesian coordinates can be calcu-lated by recording the instant dimension of the simulation systemsduring biaxial loading.

For an isotropic system under hydrostatic bi- and tri-axial ten-sion following stress–strain relations are obtained:

rbiaxial ¼ E=ð1� mÞebiaxial;rtriaxial ¼ E=ð1� 2mÞetriaxial; ð6Þ

where E and v are the Young’s modulus and Poisson’s ratio of an iso-tropic material, respectively.

3. Results and discussion

3.1. One- and two-dimensional nanostructures

We first focus on the mechanical properties of as-built 1D archi-tectures made of 0 + 1 D structural units. The stress–strain rela-tions of 1D CNT-fullerene nano-bamboos with varying CNTlength (N) under tension are shown in Fig. 2. The stress–strain rela-tion for the perfect CNT (N =1) is also included for the sake ofcomparison. It is generally known that both the MD with AIREBOpotential and quantum mechanics calculations [73] overestimatethe fracture strain of CNT, while the fracture patterns predictedare consistent with the experimental observation. For the CNT un-der tension, a nonlinear down-swing elastic (softening) relationfollows the linear behavior and is ended by fracture. For the 1D

nano-bamboos (N = 1–10) an initial linear stage is observed, anup-swing (hardening) behavior proceeds before the down-swingbehavior and fracture. To a large extent the bamboo-like structurewith negative curvature at the joints and positive curvature of ful-lerenes contribute to the three-stage elasticity, which is similar tothe initial elastic behavior of nanocrystalline graphene with wrin-kling structure [74]. The range of linear elastic regime of the nano-bamboo decreases with the increase of CNT segment length. Thefailure of 1D nano-bamboo, revealed by stress drop in the stress–strain curves, takes place due to successively irreversible bondbreakings to release the excess strain energy [75]. Further loaddrops indicate the subsequent deformation with growth of nucle-ated crack to complete rupture. The tensile stress, tensile strainand Young’s modulus for the 1D nano-bamboos are listed in Table1 together with the reference of perfect (6,6) nanotube. Except thecase with N = 0, the tensile stress of the nano-bamboo structuresremains almost constant and is approximately 1/5 of that of CNT.The tensile strain on the other hand is about half of the value ofCNT. The length of segment CNT has a minor effect on the tensilestrength and tensile strain. The Young’s modulus of 1D nanostruc-tures, increase slightly with the increase of CNT length. Distin-guished from others, the N = 0 nano-bamboo structure whichcontains dense defects exhibits a higher critical stress and fracturestrain, and a lower stiffness.

In order to study the mechanical properties of the 2D nano-mesh structures, uniaxial tensile tension tests are performed bydeforming 3 � 3 � 1 unit cells along either the x- or y-directions,as illustrated in the inset snapshots of Fig. 3. Due to a rhombus lay-out, anisotropic stress–strain relations are observed in these 2Dnetworks. At a given CNT segment length, the tensile stress ofthe case loaded along x-direction is nearly two times of that ofthe case loaded along the y-direction. On the other hand, the ten-sile strain of the case loaded horizontally is almost half of thatloaded vertically. Upon the onset of instability, a dramatic loaddrop occurs in the case loaded along the x-direction, while the

Fig. 6. Stress–strain curves of 3D nano-truss networks with N varying from 0 to 10under uni-, bi-, and tri-axial tension, respectively.

Fig. 7. Comparison of stress–strain relations of 3D nano-truss network with N = 0under uni-, bi- and tri-axial tensions as well as from the continuum mechanicssolution.

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fracture behavior for the vertical loading case can be characterizedas ‘‘ductile’’ with a small load drop followed by a large ‘‘plastic’’deformation.

Due to a realignment of CNT segments in the 2D networks withlarge N to the tension direction along the x-direction, a stiffeningbehavior turns up, as is the case for the stretching of graphyne’sfamily sheet [76]. As the CNT segment length N increases from 0to 10, the stiffness decreases and the corresponding stress–straincurves of 2D networks become increasingly nonlinear. The smallvalue of stress and stiffness found in large size networks is consis-tent with the sparser layout of CNT-fullerene per area. The changeof CNT segment length plays an insignificant role on the tensilestrain along the x-direction when compared with tension alongthe y-direction.

To gain more insight into the deformation mechanism of these1D and 2D architectures, atomic structural evolutions during the

tension tests are displayed in Fig. 4. All the unstrained nanostruc-tures hold low-energy with higher-energy bonds appeared in thedefects at the junctions and fullerenes. Fig. 4a shows two typicalsnapshots of 1D nano-bamboo structure (N = 0) at the initial stateand just before fracture from z and x side-views. With the increaseof strain, contraction of the fullerenes and extension of junction re-gions as well as increase of bond energy in the whole structure areclearly seen. Interestingly, the hexagon shape changes to chimo-nanthus-like, implying an inhomogeneous contraction. Such struc-tural development is attributed to smaller contraction in thevicinity of defects, which is also observed for others nano-bam-boos. The atomic details of fracture process of 2D networks withN = 0 and 10 are shown in Fig. 4b and c, respectively. It shows thatthe fracture pattern of these 2D networks depends on the tensiondirection and the CNT segment length. As illustrated in Fig. 4b,higher bond energy occurs with linear and zigzag-like distributioncorresponding to tension along the y and x directions. The nano-mesh fails in a brittle manner when stretching along the x direc-tion, while a ductile failure pattern along y direction. In the caseof N = 10 shown in Fig. 4c, the stretching of nano-mesh leads tothe structural change from collinear-CNT segments to non-collin-ear. When subject to sufficiently large strain along the x direction,the fracture in the nano-mesh is homogenously distributed. Inter-estingly, CNT segments of nano-mesh fall in line and adhere to-gether ultimately resulting in a ‘‘seamless’’ appearance. Thestress concentration mainly occurs at the junctions during the ten-sile loads, and the rupture always takes place at topological defectsat the junctions.

Materials with network structures may exhibit a negative Pois-son’s ratio, termed ‘‘auxetic’’ [77]. By the same token, an in-depthinspection of the Poisson’s ratio (�ex/ey or�ey/ex) on these 2D nano-mesh structures would be help to understand their mechanicalproperties. Eq. (3) has been used to calculate the Poisson’s ratio.The deformation dependence of the Poisson’s ratio for selected2D nano-meshes is plotted in Fig. 5. It shows that the in-plane Pois-son’s ratio of 2D nano-meshes is strongly dependent on the CNTsegment length. The 2D nano-mesh with very short CNT length(only for N = 0, 1) exhibits a negative Poisson’s ratio approaching�0.3. The negative Poisson’s ratio arises from a short-gapped in-ter-fullerene distance which facilitates the curvature flattening,allowing an out-of-plane positive Poisson’s ratio in z direction. Thisapparently differs from the mechanism for traditional 2D auxeticmaterials with negative Poisson’s ratio [78–80]. The value of the

(a) (b)

Fig. 8. Deformation dependence of the ratio of the transverse and longitudinal strains for the 3D networks under uniaxial tension. The Poisson’s ratio decreases with thedecrease of the CNTs segment length N. An abrupt change of the Poisson’ ratio indicates the failure of the materials.

Fig. 9. Stress–strain curves of 3D nano-truss networks with varying N from 0 to 10under uniaxial tension along h110i direction.

J.Y. Wu et al. / Computational Materials Science 80 (2013) 15–26 21

Poisson’s ratio of the nano-mesh is strongly strain-dependent.When N = 0, the magnitude of the Poisson’s ratio rises with appliedstrain along the y direction, whereas a constant Poisson’s ratio un-der tension along x direction over a significant range of strain is ob-served. The value of the Poisson’s ratio does not turn to positive upto fracture, which is different to the prior-art auxetic materials. Inthe case of N = 1 and 2, the value of the Poisson’s ratio linearly de-creases with the applied strain. Despite the overestimation of theCNT breaking strain by using the AIREBO potential, the observationof negative Poisson’s ratio in CNT-fullerene nanostructures shouldnot be affected.

3.2. Three-dimensional nano-truss structures

In this section we study the mechanical properties of fullerene3D foam-like nanostructures formed by CNTs and fullerene. Thebasic 3D bcc-lattice nano-truss unit is displayed in Fig. 1d. A setof uni-, bi- and tri-axial tension tests have been carried out by fullatomistic simulation. Fig. 6 compares the three loading cases withCNT length N varying from 0 to 10. Except the uniaxial tension withsmall N where linear elasticity was observed, the 3D nanostruc-tures exhibits nonlinear elasticity and the CNT length plays a keyrole on the mechanical properties in a similar way to the 2Dnano-mesh structures. The dense 3D network (small N) has more

constraints that limit the realignment and bending of CNTs, and re-sults in a linear behavior. A nonlinear up-swing behavior appearsin the sparse network due to the increase of CNTs length, as inFig. 6a and b. Such nonlinear stiffening behavior results from thedifferent deformation mechanisms at different strain level, suchas realignment and bending dominated deformation at smallerstrain, and the stretching dominated deformation at larger strain.Under a hydrostatic triaxial tension, however, a stiffening behavioris already seen in the denser 3D network after the initial linearelastic stage, as shown in Fig. 6c, in contrast to the case of uni-and bi-axial tension. The stiffening becomes more pronouncedwhen the inter-fullerene distance reduces. The stiffening mainlycomes from stretching-governed deformation posterior to thebending-induced curvature-flattening stage. Approaching the crit-ical strain, the stiffness slightly degrades for all the hydrostatic tri-axial tension cases. The degradation is caused by an-harmonicpotential term of carbons. Subsequent sharp stress load drop re-veals a permanent failure due to bond-breaking at the junctionfor all tests.

In order to compare the uni-, bi- and tri-axial tension, we spe-cifically select one case N = 0 as an example, Fig. 7. This 3D networkis able to sustain high stretch under uniaxial straining when com-pared with bi- and tri-axial tension. The difference between bi- andtri-axial tension is insignificant initially. The value of the slope canbe ranked: uniaxial > biaxial > triaxial, which is opposite to con-ventional materials, indicating the existence of negative Poisson’sratio for this novel nanomaterial, similar to what we found forthe 2D nano-meshes.

An in-depth investigation on the Poisson’s ratio is then per-formed. Fig. 8 presents the calculated strain-dependent Poisson’sratio of the 3D networks. It can be seen that the CNTs length dom-inates the Poisson’s ratio of the 3D networks. With N = 0, the 3Dcovalent pack of fullerenes exhibits a negative Poisson’s ratio whensubjected to both uni- and bi-axial tension tests, Fig. 8a. The phys-ical origin of negative Poisson’s ratio observed in the 3D networksdiffers from that of 2D nano-mesh network in the role of the con-tractility in z-direction (out-of-plane) plays. The negative Poisson’sratio of 3D network is attributed to a reduction of absolute curva-ture in fullerene hub due to high constraint from direct covalentconnection between fullerenes without participation of CNTs. Thelinkage of CNTs would actively involve the bending and stretchingdominated deformation, and cause a different stress concentrationon the fullerene hub. The Poisson’s ratio is strikingly strain-depen-dent, changing sign from negative to positive when an extensionalstrain reach around 0.065 and 0.09 for bi- and uniaxial tension,

(a) (b)

(c) (d)

Fig. 10. (a and b) Deformation dependence of the ratio of the transverse and longitudinal strains, �eh�110i/eh110i and �eh001i/eh110i for the 3D networks under uniaxial tension,respectively. (c and d) Ratio of transverse and longitudinal strains as a function of CNT segment length N at eh110i = 0.01.

22 J.Y. Wu et al. / Computational Materials Science 80 (2013) 15–26

respectively, which suggests that the transverse strain begins toact in an opposite sense to the longitudinal strain.

Many cubic metals when stretched along the specific 110 off-axis direction become auxetic [81]. Accordingly, we conduct ten-sion tests on the bcc-lattice 3D nano-truss network along theh110i orientation and the resulting stress–strain relations with Nvarying from 0 to 10 are shown in Fig. 9. We again see clearly thatthe CNT length dominates the mechanical properties. The tensilestress, tensile stain and stiffness all decreases with the increaseof the segment length N for the case under tension along theh110i direction. Compared with Fig. 6a we observe that both thetensile strain eh110i and tensile stress rh110i are smaller than thoseunder h100i direction, resulting from a pure stretching of octagonrings in the fullerened hub. The corresponding Poisson’s ratio isdetermined and plotted in Fig. 10. It can be seen that this bcc-lat-ticed 3D network also exhibits a negative Poisson’s ratio when ten-sion is applied along the h110i direction. Fig. 10c and d comparethe value of �eh�110i/eh110i and �eh001i/eh110i as a function of CNTslength at a specific small strain eh110i = 0.01. It must be noted thatthis 3D network yields a very wide range of Poisson’s ratios thatare both controlled and tunable by virtue of the microstructuralparameter – the length of CNTs. The magnitude of �eh�110i/eh110iand �eh001i/eh110i roughly covers a wide range from �0.4 to 0.3,and from 0.1 to 1.2, respectively. The spectacular change of �eh001i/eh110i appears in this 3D network exceeds 0.5, which is the thresh-old value for isotropic materials in the theory of elasticity. Interest-ingly, a dramatic ‘‘flip’’ of the Poisson’s ratio from

negative ? positive ? negative with increase of CNTs length oc-curs, Fig. 10c. The findings can be utilized on tailor-making auxeticCNT-fullerene networked nanostructures with the wanted proper-ties. Two different mechanisms, involving curvature flattening andrigid mechanical model, are identified to be responsible for the‘‘flipping’’ of Poisson’s ratio. Due to a dilatability in h�110i but acontractility in h001i, the volumetric change during the tensionbecomes an interesting parameter. Fig. 11 presents the deforma-tion dependence of relative volumetric change when strained un-der h100i and h110i direction, together with their derivativesdetermined by a linear fitting in the range of 0–0.01, respectively.

The definition of relative volume change is expressed as:

DVV¼ LxLyLz

Lx0 Ly0Lz0

� 1; ð7Þ

where Lx0 , Ly0, Lz0 , and Lx, Ly, Lz are the equilibrium and the instant

three directional length of the model, respectively.With Eqs. (1) and (3), we obtain the following relationship for

the case of uniaxial tension along the x direction:

DVV¼ vzxvyxex e2

x þ 1� 1vzx� 1

vyx

� �ex �

1vzxþ 1

vyx� 1

vzxvyx

� �� �;

ð8Þ

where vzx and vyx represent the orientation-dependent Poisson’s ra-tio; ex is the strain in the x direction; DV and V are the volumechange after straining and the initial volume, respectively. The rel-

(a) (b)

(c) (d)

Fig. 11. (a and b) Deformation dependence on the relative volume change for the 3D networks under uniaxial tension along h100i and h110i directions, respectively. (c andd) Slope of relative volume change versus deformation as a function of CNT segment length N, linearly fitted in strain range from 0 to 0.01, respectively.

J.Y. Wu et al. / Computational Materials Science 80 (2013) 15–26 23

ative volume change curves in Fig. 11a and b shows a power-lawpattern within the elastic region, well described by Eq. (8). It canbe seen that the CNTs length remarkably dominates the volumetricchange. The rate of relative volume change in a small deformationshows a general decreasing trend with the increase of CNTs length.The positive value of the rate change reveals a total expansion of thesystem volume, irrespective of the sign of the Poisson’s ratio. Underlarge strain, the relative volume change becomes negative for longCNTs length as a result of a large shape change of the 3D networkas well as the increase of Poisson’s ratio.

Visualizing the detailed structural evolutions of this 3D nano-truss provides insights on the structural and mechanical proper-ties, such as the deformation mechanism, the negative Poisson’s ra-tio and fracture modes. Fig. 12 displays straining directiondependence of the representative atomic structural developmentof the 3D nano-truss, together with the schematic drawing of thedeformation mechanisms based on the rigid mechanical model.These atomic configuration evolutions give a justification of theinteresting properties mentioned above. For N = 0, the initialside-view shows the porosity of the 3D nanostructures, analogousto the zeolite structure [82], offering a possibility for molecularstorage and filters. During the deformation, three stages can beidentified: firstly, there is a curvature flattening process with theincrease of the strain, as presented in the zoomed-in local struc-tures of Fig. 12a and c, which results in a negative Poisson’s ratio.In this stage, the curvature-flattening dominates the deformationwith no irreversible structural changes, triggering an expansion

of pore between fullerenes as well as the fullerene cages. Secondly,the subsequent graphitic structure with negative curvature ulti-mately causes a contraction when the strain approaches a criticalvalue. This is confirmed by the change of Poisson’s ratio from neg-ative to positive as shown in Fig. 7a. Finally, an additional strainincrement leads to a fracture at the weak octagon rings at the junc-tion. The fracture propagates in the plane perpendicular to thestrain direction towards complete rupture, in which a number ofmonoatomic carbon chains formed, similar to what is widely ob-served in the stretching of pure CNTs. A longer CNT for spanningthe fullerenes is exploited to tune the properties of this 3D net-works. Particularly, sequential deformation configurations of the3D network with N = 10 are displayed in Fig. 12b and d. It can benoticed that the long length of CNTs brings a sparse 3D network,and orientation dependence of deformation mechanism on the3D network with long CNTs length is also clearly observed. In addi-tion, a different deformation mechanism is revealed by comparingwith that of N = 0. Three deformation stages are also observed fromthe overall tension processes for both loading directions. Strainingalong the h100i direction involves direct stretching of all the CNTsin the network, whereas half of CNTs in the network when de-formed along h110i undergo direct tension because the axis ofother CNTs is perpendicular to the tension direction. For the formercase, straining along h100i, an initial tension of the system inducesthe angles change between CNTs, accompanying with the struc-tural transformation of fullerene hub where the negative curvaturestructures become either flattening or more curved as in the

Fig. 12. Size- and orientation-dependence on the atomic structural development of 3D networks. (a) Sequential deformation snapshots of 3D networks with N = 0 underh100i directional tension. The zoomed-in local atomic structures clearly illustrate the mechanism of negative Poisson’s ratio. (b) Sequential deformation snapshots of 3Dnetworks with N = 10 under h100i directional tension. The zoomed-in local atomic structures distinctly display the buckling of fullerene in the node of 3D network prior tothe fracture. (c) Sequential deformation snapshots of 3D networks when N = 0 under h110i directional tension. The negative Poisson’s ratio is explained by the zoomed-inlocal atomic structures. (d) Sequential deformation snapshots of 3D networks when N = 10 under h110i directional tension. (e) Schematic illustration of deformation of 3Dnetworks under h110i directional tension. The green line denotes the infinite stiff CNT segments connecting the fullerenes, acting as hinges. Ideally, the Poisson’s ratio(vh�110i/h110i = �eh�110i/eh110i) would approach the isotropic lower limit v = �1 based on the rigid mechanical models. The local potential energy shows how the high amount ofstress during uniaxial tension resides on the fullerene nodes. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of thisarticle.)

24 J.Y. Wu et al. / Computational Materials Science 80 (2013) 15–26

zoomed-in local structure of Fig. 12b. As clearly shown in the sec-ond zoomed-in local motif of Fig. 12b, an excessive straining in-duces a striking inversion of a two-pentagon without bondbreaking, which is similar to the inversion of a two-pentagon nano-cone via an indenter compression [83]. Finally, one octagon ringinitially develops to a clean cleavage followed by rest fracturing,and the formation of monoatomic carbon chains. For the latter caseof straining along h110i, an incipient straining also induces thechange of angles as well as the structural change of the fullerenehubs. As schematically illustrated in Fig. 12e, a unit cell consistsof double X-shaped junctions which locate respectively in two per-pendicular planes colored by blue and orange which share one ful-lerene hub, and initial angles a between CNTs would uniformly

change to a0 after an elastic strain based on the rigid mechanicalmodel. As it can be expected, the top-viewed snapshots inFig. 12d shows that the tension in h110i direction increases theview-plane-distance between fullerene hubs as a result of the uni-form change of angles, indicating a negative Poisson’s ratio. Themechanism of this negative Poisson’s ratio differs with the caseof N = 0 in the absence of CNT segments. Ideally, the Poisson’s ratio(vh�110i/h110i = �eh�110i/eh110i) would approach the isotropic lowerlimit v = �1 from the rigid mechanical model. However, a transi-tion of Poisson’s ratio from positive to negative revealed in our cal-culations comes mainly from this rigid mechanical modeldomination. These two mechanisms for the negative Poisson’s ratioin the graphitic nanostructure differ with that of the hypothetical

(a) (b)

Fig. 13. (a) Stress–strain curves of 3D nano-truss networks with a specific length N = 0 under uniaxial tension along h1 0 0i direction at temperature range from 1 to 1000 K.(b) Deformation dependence of the ratio of the transverse and longitudinal strains for the 3D networks under uniaxial tension.

J.Y. Wu et al. / Computational Materials Science 80 (2013) 15–26 25

pure-carbon auxetics crystal structures by Baughman and Galvão[84]. Beyond the tensile limit, excess angle change and the octagonrings damage occurs, generating morphology as displayed in thirdsnapshot of Fig. 12d, similar to a metallic plastic deformation. Fur-ther straining results in compete rupture with a clean cleavage. Asa whole, the strong correlation between deformation mechanismand organizational parameters demonstrates a close structure–property relationship in this 3D nano-truss network.

3.3. Temperature effect

Next we study the effect of temperature on the mechanicalproperties of this 3D nanostructure with N = 0 under tension alongthe h100i direction. Fig. 13 compares the stress–strain relationsand the corresponding strain-dependent Poisson’s ratio of 3D net-works with temperature varying from 1 to 1000 K. As the temper-ature increases from 1 to 1000 K, the fracture strengthmonotonically decreases from 19.03 GPa to 6.31 GPa, and the frac-ture strain from 0.168 to 0.070. The slope of the stress–strain curveat different temperatures fitted from 0 to 0.01 strain respectivelygives the Young’s modulus values: 123.42, 119.56, 114.95,109.76 GPa, which also decreases at higher temperatures due tothe thermal-softening mechanism. Apparently, the feature ofstrain-dependent Poisson’s ratio is not altered with the increaseof temperature and this 3D nanomaterial still exhibit a negativePoisson’s ratio at high temperature. The Poisson’s ratio of mostglass-forming materials increases with increase of temperature[85]. However, Poisson’s ratio of this 3D nanomaterial demon-strates a reverse trend.

4. Conclusions

CNT-based network, as an emerging class of material, attractsextensive research interest in recent years from both the scientificcommunity and industry. In this study we proposed novel cova-lently bonded 1D, 2D and 3D nanostructures (nano-bamboo,nano-mesh, nano-truss) made of fullerenes spanned by CNTs. Themechanical properties of such architectures are systematicallyinvestigated via full atomistic simulation. Our simulations revealthat the CNTs length plays a significant role on the mechanicalproperties. 2D CNT-fullerene nano-mesh structures exhibit an in-plane negative Poisson’s ratio when N = 0, as a result of curva-ture-flattening. 3D bcc-lattice like nano-truss network displaysan orientation-dependent deformation behavior. Under h100i

direction tension, negative Poisson’s ratio is also observed but onlyfor the case of N = 0, while a ‘‘flipping’’ behavior of Poisson’s ratiowith ‘‘Negative ? Positive ? Negative’’ appears when the tensionis applied along h110i direction. The Young’s modulus decreases athigh temperatures because of the thermal-softening mechanism.The findings in our paper would provide guidance to optimal de-sign of CNT-fullerene based structures towards tailor-madeproperties.

Acknowledgements

This work is supported by the Research Council of Norway un-der NANOMAT KMB Project (MS2MP) No. 187269 and the compu-tational resources provided by NOTUR – the NorwegianMetacenter for Computational Science.

Appendix A. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.commatsci.2013.04.033.

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