Coordinated Buckling of Thick Multi-Walled Carbon Nanotubes Under Uniaxial Compression

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1 Nano Res (00) 3: 3 4 DOI 0.007/s Research Artcle Coordnated Bucklng of Thck Mult-Walled Carbon Nanotubes Under Unaxal Compresson Xu Huang, Hongyan Yuan, K. Jmmy Hsa, and Suln Zhang ( ) Department of Engneerng Scence and Mechancs, The Pennsylvana State Unversty, Unversty Park, PA 680, USA Department of Mechancal Scence and Engneerng, Unversty of Illnos, Urbana, IL 680, USA Receved: 5 October 009 / Revsed: 7 November 009 / Accepted: 8 November 009 The Author(s) 00. Ths artcle s publshed wth open access at Sprngerlnk.com ABSTRACT Usng a generalzed quas-contnuum method, we characterze the post-bucklng morphologes and energetcs of thck mult-walled carbon nanotubes (MWCNTs) under unaxal compresson. Our smulatons dentfy for the frst tme evolvng post-bucklng morphologes, rangng from asymmetrc perodc rpplng to a helcal damond pattern. We attrbute the evolvng morphologes to the coordnated bucklng of the consttuent shells. The post-bucklng morphologes result n sgnfcantly reduced effectve modul that are strongly dependent on the aspect rato. Our smulaton results provde fundamental prncples to gude the future desgn of hgh-performance, MWCNT-based nanodevces. KEYWORDS Carbon nanotubes, bucklng, unaxal compresson, rpplng Introducton Mult-walled carbon nanotubes (MWCNTs) have found a wde range of applcatons as basc buldng blocks n mcro(nano)-electromechancal systems (MEMS/NEMS) [ 5], owng to ther unque combnaton of mechancal, electronc, and bochemcal propertes. Snce carbon nanotubes (CNTs) are slender, thn-walled structures that are susceptble to bucklng under compressve loadng, falure nduced by mechancal nstablty has been a major concern n the performance of such CNT-based nanodevces [5]. The behavor of sngle-walled CNTs (SWCNTs) under dfferent loadng condtons (bendng, twstng, and compresson) has been well characterzed by varous methods, rangng from all-atom molecular dynamcs (MD) smulatons [6 0] to contnuum beam [8, ] and shell theory-based modelng [8, ]. These studes predcted that the bucklng morphologes feature a sngle local knk n SWCNTs beyond a crtcal bendng stran. Ths phenomenon hghlghts the very large energetc cost of stretchng a CNT compared to bendng t. In bent and twsted MWCNTs, the sngle sharp knk s replaced by perodc undulatons [5, ]. Specfcally, the Yoshmura (damond shaped) pattern appears n the compressve secton of bent MWCNTs [5, 3 5, 7, 8,, 3], whlst perodc wave-lke rpples develop n both the length and crcumferental drectons n twsted MWCNTs [3, 4, 6,, 3]. These deformaton morphologes result n a substantally reduced effectve modulus [3, 4, 6]. The mechancal responses of MWCNTs Address correspondence to suz0@psu.edu

2 Nano Res (00) 3: are generally reversble upon unloadng, ndcatng ther nonlnear, purely elastc nature. An MWCNT s composed of a number of concentrc thn shells. The aspect ratos (tube length dvded by tube dameter) of the nnermost and outermost shells of a gven MWCNT may dffer by two orders of magntude. Under unaxal compresson, solated shells wth the nnermost shell aspect rato may exhbt beam (columnar) bucklng, whle those wth the outermost shell aspect rato exhbt shell bucklng. In realty, the nter-shell van der Waals (vdw) nteractons play a crtcal role, constranng the bucklng of these consttuent shells. It s thus nterestng to examne whch bucklng mode an MWCNT wll adopt under the nfluence of nter-shell vdw nteractons. Expermental evdence of the nstablty of unaxally compressed MWCNTs has recently emerged [7 9]. Waters et al. [7, 8] conducted nanondentaton measurements of vertcally algned short, thck MWCNT forests (wth aspect rato rangng from to ). Hysteress was observed n ther load dsplacement curve, whch ndcates the occurrence of plastc deformaton wth two possble sources: defect nucleaton (Stone Wales defects) and propagaton [30 35], or adhesons between neghborng MWCNTs under the compressve loadng. Usng scannng electron mcroscopy (SEM), they found that under compresson the buckled MWCNTs exhbted strongly dstorted cross-sectons and wrnkled sdewalls. Due to the large sze of the nanondentaton tps used n ther experments, several MWCNTs were smultaneously compressed. At a certan stage of the ndentaton, neghborng MWCNTs may come to contact wth each other, whch poses dffcultes n the post-bucklng analyss for a sngle MWCNT. They also found that the measured bucklng stran s nether consstent wth the predcton by contnuum theory usng the elastcty parameters of Yakobson et al. [8], nor wth the theoretcal model developed by Ru [36]. Usng atomc force mcroscopy (AFM), Yap et al. [6] characterzed the nstablty of ndvdual MWCNTs wth dfferent aspect ratos under cyclc compresson. Whle ther experments unambguously demonstrated that nstablty of MWCNTs wth large aspect ratos s governed by Euler bucklng, the post-bucklng deformaton morphology of MWCNTs wth relatvely small aspect ratos remans unknown, largely due to the expermental dffcultes n n stu magng of the deformed MWCNTs wth nanoscale resolutons. Yap et al. hypotheszed that the nstablty of short, thck MWCNTs can be attrbuted to symmetrc shell bucklng. All-atom smulatons wth emprcal nteratomc potentals have been wdely used to study the deformaton of CNTs. However, for the study of thck MWCNTs, fully atomstc smulatons are computatonally prohbtve because of the large number of degrees of freedom nvolved. Ths explans why most prevous atomstc studes have been lmted to SWCNTs or MWCNTs wth only a few layers. In ths artcle, we systematcally characterze the deformaton morphologes and energetcs of thck MWCNTs ( 0 layers) wth relatvely small aspect rato under unaxal compresson through a large-scale, effectve quas-contnuum method, and ratonalze the bucklng mechansms. Our smulatons dentfy evolvng postbucklng morphologes, rangng from asymmetrc perodc rpplng to helcal damond patterns. We attrbute the evolvng morphologes to the coordnated bucklng of the consttuent shells. We further note that the post-bucklng morphologes result n sgnfcantly reduced effectve modul that are strongly dependent on the aspect rato. Our smulaton results may provde fundamental prncples to gude the desgn of future hgh-performance, MWCNT-based nanodevces.. Methodology We adopt the modfed second-generaton Brenner potental [37], denoted by MTB-G, to descrbe the short-range covalent nteractons n MWCNTs; ths takes the form: R A j j j j> V = TB V ( r ) B ( r ) V ( r ) () R V where r j s the dstance between atoms and j, A and V are the parwse repulsve and attractve nteractons, respectvely, and j B s the bond-order functon that has a complcated dependence on the bond angles and bond lengths nvolvng atoms and j. The long-range nonbondng nteracton s descrbed

3 34 Nano Res (00) 3: 3 4 by a Lennard Jones (LJ) potental wth the parameters gven by Grfalco et al. [38], as 6 6 r0 r0 VLJ() r = κ 6 r0 r r where r s an nteratomc dstance, κ =.7 s a dmensonless constant, r0 =.4 Å s the equlbrum bond length, and = 5. ev Å 6. To mprove the computatonal affordablty wthout sacrfcng accuracy, the fully atomstc models employed are here coarse-graned by a quascontnuum method based on the fnte crystal elastcty theory for curved crystallne monolayers [39 4]. Wthn ths theoretcal framework, the exponental Cauchy Born rule was proposed as a way of lnkng the knematcs at the atomc and contnuum scales: () a= ζ ( A) (3) where ζ s an exponental map that transforms the undeformed lattce vector A nto a deformed one a. Through a local approxmaton of the exponental map, the deformed lattce vectors and the angles between two lattce vectors can be analytcally represented n terms of the contnuum deformaton measures of the surface. Consder a representatve unt cell of area S 0 contanng two nequvalent nucle and three nequvalent bonds n the reference, undeformed confguraton: the hyperelastc stran energy densty W can be formulated n terms of the MTB-G potental, W = W( CKη ; ; ) = 3 R A V ( a ) + (,,, θ, θ ) ( ) b a a j ak j k V a S (4) 0 = where C and K are the stretch and curvature tensors, respectvely; a, aj, a k are the three nequvalent bonds, θ j, θk are the angles between these three bonds, and S0 = (3 3 ) A s the area of the unt cell. Note that the hexagonal lattce s a Bravas mult-lattce, so an addtonal knematc varable η must be ntroduced to descrbe the relatve shft between the two basc lattces. Ths nner varable s elmnated by mnmzng the stran-energy densty functon, gvng rse to a stable local lattce arrangement wthn the unt cell. After ths nner relaxaton, the stran energy densty can be wrtten as a functon of only C and K: W ˆ = W ˆ[ CKη ; ; ( CK ; )] (5) Thus, the contnuum representaton of the covalent bndng energy for the shells n an MWCNT subject to the deformaton map φ that maps from the undeformed to deformed confguratons s n E Wˆ C[ φ( X)]; K[ φ( X )] dω (6) = = Ω [ ] where X s a materal pont n the undeformed confguraton, and Ω s the surface area of the -th shell n an n-walled MWCNT. Homogenzaton of the dscrete nonbonded energy densty between two unt cells gves rse to the vdw energy densty, as VvdW() r = VLJ() r S0 The factor of two on the rght-hand sde of Eq. (7) comes from the fact that each unt cell contans two nucle. The nonbonded energy between two neghborng shells s then E V X X n + vdw = φ φ Ω Ω + + Ω vdw Ω ( ) ( ) d 0 d = where X and X + are the two materal ponts that are on the -th and (+)-th shells, respectvely, n the MWCNT; Ω 0 and Ω + 0 are the surfaces of the -th and (+)-th shells, respectvely. Based on the coarse-graned consttutve relatons for both the bondng and nonbondng nteractons, the consttuent shells of the MWCNTs are dscretzed by fnte elements. In our smulatons, an n-walled MWCNT s ndexed by (5,5)/(0,0)/.../(5n,5n), where the nter-shell spacng s roughly 0.34 nm. As the frst step, the MWCNTs are fully relaxed free of any constrants usng a lmted-memory Broyden Fletcher Goldfarb Shanno (BFGS) algorthm [4]. A rng of fnte-element nodes at each end of the MWCNTs are then held fxed and the MWCNTs are (7) (8)

4 Nano Res (00) 3: unaxally compressed by stepwse movement of the fxed nodes to the center n the axal drecton. At each step, the MWCNTs are relaxed under the prescrbed compressve stran; the reacton forces at the ends and the deformaton morphology can be extracted from the equlbrated confguratons. The coarse-graned model reduces the number of degrees of freedom by two orders of magntude as compared to ts atomstc counterpart, yet remans fathful to the nonlnearty of the nteratomc nteractons and the basc characterstcs of hexagonal lattces. For nstance, the coarse-graned model nherts the chralty of the atomstc counterpart, and gves rse to the correct stackng energy between the neghborng shells. Thus, the coarse-graned model reproduces the deformaton morphologes and energetcs of the correspondng atomstc models for defect-free, homogeneously deformed CNTs.. Results and dscusson. Evolvng post-bucklng morphologes We choose a 0-walled CNT as a representatve example to llustrate the deformaton morphologes and energetcs of thck MWCNTs under unaxal compresson. The aspect rato of the MWCNT s ρ = L/ d = 5, where L s the length, and d s the dameter of the outermost wall of the MWCNT (also regarded as the dameter of the MWCNT). Fgure shows the deformaton morphologes at dfferent compressve strans. The deformaton morphologes of four consttuent shells the nnermost, the 5th, the 0th, and the outermost shells are shown from left to rght n each of Fgs. (a) (c). We observed that, upon constrant-free geometry optmzaton as the frst step (Fg. (a)), the shells n the MWCNT undergo rgd rotaton wth respect to ther central axs. From nnermost to outermost layers, the neghborng shells rotate n alternatng drectons (clockwse and counter clockwse). We attrbute ths alternatng rotaton to the regstry-dependent nter-shell nteractons [43]. In partcular, the arrangement of the neghborng shells n the ntal confguraton (the ntal guess) s close to A A stackng, whch s energetcally dsfavored [44]. The rotatons of the neghborng shells n alternatng drectons rearrange the neghborng shells from A A to A B stackng, whch lowers the nter-shell vdw nteractons wthout affectng the n-plane deformaton energetcs. Ths rearrangement shows the capablty of the coarse-graned model to nhert the chralty of ts atomstc counterpart. Fgure Evoluton of the deformaton morphologes of a unaxally compressed 0-walled MWCNT. Each subfgure conssts of four consttuent shells of the 0-walled MWCNT. From left to rght n each subfgure, the shells are the nnermost, the 5th, the 0th, and the outermost shells. (a) Fully relaxed, undeformed confguraton. (b) At a compressve stran of 3 ε = 8.3 0, wave-lke rpples appear along the 0th and the outermost shells, whle rpples are not observable n the other two nner shells. (c) At a compressve stran of ε =.8 0, a helcal damond pattern runs from one end to the other for the outer three shells. The nnermost shell s further bent and twsted throughout ts length As the compressve stran ncreases, all shells of the MWCNT undergo unform shortenng. At a 3 crtcal compressve stran of ε = 8.3 0, wave-lke perodc rpples appear (Fg. (b)) along the tube. The rpplng pattern, however, s graphcally undetectable n the two nnermost shells. Such a rpplng morphology n the nner shells would cause a large deformaton energy penalty snce nner shells are much more rgd. The rpplng ampltude progressvely ncreases from nner to outer shells, and ncreases wth ncreasng compressve stran. The rpplng n all shells has the same rpplng perodcty, manfestng the coordnated nature of the deformaton of the consttuent shells. It s nterestng to note that the rpplng pattern s not exactly perpendcular to the axal drecton (the loadng drecton) of the MWCNT, but has a small tlt. Wth a slghtly ncreased compressve stran, the rpplng pattern evolves nto a damond bucklng pattern (Fg. (c)). Fgure (c) shows

5 36 Nano Res (00) 3: 3 4 that at ε =.8 0 a helcal damond pattern appears n the three outer shells. In contrast, the nnermost shell devates from ts lnear confguraton, but wthout a graphcally vsble damond pattern. The helcal damond patterns predcted by our smulatons have not been observed n ether other numercal smulatons or prevous experments. The constraned deformaton of the nnermost tube merts detaled study. Fgure dsplays the deformaton mode of the nnermost tube n the 0-walled MWCNT at dfferent appled strans. To ad vsualzaton of the deformed confguratons (n green), a reference tube (n blue) s also plotted. The reference tube represents the unformly shortened confguraton wthout any radal dsplacement under the appled stran. The radal dsplacements of the deformed tubes due to unaxal compresson are rescaled by a factor of 0 to further ad vsualzaton. At a compressve stran of 3 ε = at whch wave-lke rpples appear (Fg. (b)), the nnermost tube remans straght (Fg. (a)) so that the deformed and the reference tubes overlap. At the crtcal stran of ε =.8 0, the nnermost tube s deformed nto a spral confguraton around the reference tube (Fg. (b)). At the same crtcal stran, the helcal damond pattern on the outer shells also appears, ndcatng a coordnated bucklng mechansm of the nner and outer shells. A further ncrease n the appled compressve stran results n an ncrease n the spral ampltude (Fgs. (c) and (d)). The nnermost tube s smultaneously bent and twsted, as seen from the twst marks n the spral confguratons and confrmed by the calculated reacton forces at the ends of the tube. or damond pattern formaton throughout the entre loadng process. The two sets of curves are graphcally ndstngushable up to the bfurcaton pont and thereafter devate gradually from each other. Ths bfurcaton pont corresponds to the onset of rpple formaton n the MWCNT. In the pre-bfurcaton regme, the total system energy ncreases quadratcally wth respect to the compressve stran, whle the nter-shell nonbondng energy decreases lnearly due to the unform shortenng of the MWCNT upon compresson. In the post-bfurcaton regme, the nter-shell nonbondng energy stored n the buckled morphology s much hgher than n the dealzed case, whle the total system potental energy s stll lower. Ths clearly suggests that the release of n-plane stran energy acts as the drvng force for the rpplng and damond pattern formaton, and s opposed by the ncrease n the nter-shell nonbondng energy. Snce the n-plane stffness n MWCNTs s much hgher than the transverse stffness, the rpplng deformaton and damond pattern effectvely release the n-plane stran energy, leadng to an energetcally favorable state at suffcently hgh compressve load.. Deformaton energetcs To further reveal the physcal mechansms governng the evoluton of the deformaton morphologes, we plot n Fg. 3(a) the total system energy (n unts of aj) and the nter-shell nonbondng energy (vdw nteracton energy) as a functon of the appled compressve stran. The fully relaxed, undeformed confguraton s taken to be the reference energy state. For comparson, the energy of the MWCNT undergong an dealzed deformaton s also plotted, where the deformaton of the MWCNT s perfectly cylndrcal wthout rpplng Fgure The deformaton confguratons of the nnermost tube n the 0-walled MWCNT. The tube n blue represents the unformly shortened confguraton wthout any radal dsplacement under the appled compressve strans, whle the tube n green represents the deformed confguraton. To ad vsualzaton, the radal dsplacements of the deformed tubes due to unaxal compresson are rescaled by a factor of 0. At zero compressve stran, the confguratons of these two tubes overlap. At fnte appled stran, the deformed tube adopts a spral confguraton. The ampltude of the spral confguraton ncreases wth ncreasng compressve 3 stran. (a) ε = ; (b) ε =.8 0 ; (c) ε =.9 0 ; (d) ε =.47 0

6 Nano Res (00) 3: Fgure 3 Energetcs and mechancal responses of a 0-walled MWCNT under unaxal compresson. For comparson, an dealzed case s also depcted, where the deformaton of the MWCNT s constraned to a perfect cylndrcal shape wthout bucklng throughout the entre loadng process. (a) As compared wth the dealzed deformaton mode, the rpplng deformaton and damond pattern formaton beyond the bucklng pont releases the n-plane stran energy, penalzed by the ncrease of the nterlayer van der Waals (vdw) energy. The undeformed confguraton s taken as the reference energy state. In the dealzed case, the nterlayer vdw energy decreases lnearly due to the unform shortenng of the MWCNT upon compresson. (b) Stress stran relatons. The post-bucklng morphologes result n a substantally reduced effectve modulus as compared to the pre-bucklng regme Fgure 3(b) shows the stress stran response of the MWCNT. For the dealzed case, the compressve stress ncreases lnearly wth compressve stran throughout the entre loadng process (sold black lne). For the mnmum energy path (red open crcles) where the rpplng and damond pattern morphologes develop, the compressve stress frst ncreases lnearly wth respect to the stran untl a sudden drop appears, ndcatng the onset of local bucklng. Beyond the bucklng pont, the compressve stress contnues to ncrease wth the appled stran, but wth a sgnfcantly reduced stffness as compared wth that of the prebucklng regme. If the nter-shell vdw nteracton energy s neglected, an MWCNT n the pre-bucklng regme can be treated as a seres of lnear sprngs, wth the effectve modulus computed by Y = F ( π d ) n ε t = where F s calculated reacton force at the appled compressve stran ε. The term n parenthess n Eq. (9) s the effectve cross-sectonal area of the MWCNT, where d s the radus of the -th shell, and t s the thckness of the shells. For the MTB-G potental, the Young s modulus of the shells s 43/t J/m based on the exponental Cauchy Born (9) rule [40] and t = nm [8]. Wth these geometrcal parameters, the calculated modulus s 5/t J/m, whch s farly close to the analytcal result (the dfference s 3%)..3 Bucklng mechansms and onset bucklng stran The bucklng energetcs and morphologes of a 0-walled MWCNT provde nsghts nto the bucklng mechansms and the post-bucklng responses of MWCNTs. To relate the bucklng mechansms of an MWCNT to that of ts consttuent shells, we note that the bucklng mode of each consttuent shell n the MWCNT depends on ts aspect rato. Bucklng of an SWCNT wth large aspect rato follows the Euler beam theory, where the bucklng stran s gven by ε beam cr = π ( ρ ) (0) In contrast, SWCNTs wth a small aspect rato undergo shell bucklng under compresson wth a bucklng stran gven by ε shell cr = d t 3( ν ) () where ν s Posson s rato, and ν = for the

7 38 Nano Res (00) 3: 3 4 MTB-G potental [40]. Note that the bucklng stran for shell bucklng s ndependent of shell length. For the n-walled MWCNT wth an aspect rato of ρ, the aspect rato of the outermost shell s ρ, whle that of the nnermost shell s n ρ under the condton that the nnermost tube s (5, 5). Thus, for suffcently large n, the nnermost shell s prone to beam bucklng, whle the outermost shell undergoes shell bucklng. The nter-shell nonbondng nteracton has a constranng effect on both bucklng modes, whch delays the bucklng as compared wth solated shells. For MWCNTs wth relatvely small aspect ratos (whch are the focus of ths study), t s concevable that for the most general case there exst three crtcal bucklng strans: the onset bucklng stran for perodc rpplng of the outermost shell ε outer rpplng ; the onset bucklng stran of the symmetrc damond pattern ε outer sym, and the onset bucklng stran of the spral confguraton of the nnermost shell ε nner spral. Analytcal evaluatons of these crtcal strans requre consderaton of the nter-shell nteractons, whch s straghtforward but lengthy. Dependng on the aspect rato and the thckness of the MWCNT, the range of the stran for symmetrc bucklng may be too narrow to be detectable. For example, n our smulatons of 0-walled MWCNTs, the symmetrc damond pattern was not observed durng the compresson, ndcatng that ε outer sym s farly close to ε nner spral. In contrast, for the 5-walled MWCNTs, as wll be dscussed later, there s a detectable stran range wthn whch the symmetrc damond pattern appears. Snce the focus of the present study s on the thck, short MWCNTs, the outermost shell tends to rpple frst as the compressve stran reaches ε outer. rpplng Due to the nter-shell vdw nteractons, the rpplng morphology propagates from the outermost shell all the way to the nnermost shells. As the appled compressve stran ncreases, the rpplng ampltude also ncreases. As the compressve stran reaches ε outer sym, rpplng may no longer be the energetcally favorable deformaton mode, and may be replaced by the symmetrc damond pattern. Snce the nnermost shell s sgnfcantly more rgd n the transverse drecton than the outermost shell, nether the rpple nor the damond pattern s the energetcally favorable deformaton mode for the nnermost shell. Accommodatng the deformaton mode propagated from the outermost shell thus results n sgnfcant stran energy n the nnermost shell. To release ths excessve stran energy, the nnermost shell abandons the deformaton mode propagated from the outermost shell, and tself bends. Due to the constranng effects of the outer shells, the deformaton mode of the nnermost shell adopts a spral confguraton. Such a spral confguraton of the nnermost shell n turn propagates all the way to the outermost shell. In accommodatng ths deformaton mode, the symmetrc damond pattern s dsrupted, and replaced by the helcal damond pattern. The helcal damond pattern represents a hybrd bucklng mode of the MWCNT when beam bucklng of the nnermost shell and shell bucklng of the outermost shell are both actvated. The bucklng response of the MWCNT appears to be a result of the coordnated deformaton of the nnermost and outermost shells. The perodc rpplng s a characterstc shell bucklng mode that has been wdely observed n compressed thn-shell-lke structures. Yet the tltng of the rpplng demands further explanaton. The tlted rpplng pattern observed n the compressed MWCNTs resembles that observed n a twsted MWCNT [3, 4, 6, ], whch ndcates that the MWCNT s nternally twsted. The dfference between these two rpplng patterns s that the tlt angle of rpples n a twsted MWCNT s much larger. Examnaton of the reacton force at the tube ends reveals that the shells are ndeed twsted, whch explans the tltng of the rpplng pattern. Snce the appled stran s purely compressve, the torson must be generated by nternal relaxaton of the shells. The torson, and hence the tltng of the rpplng pattern, may be attrbuted to the nter-shell lattce regstry. The rpplng pattern leads to an ncrease n surface area, whch ntroduces an effectve lattce msmatch of the neghborng shells. The lattce msmatch dsrupts the orgnally establshed n-regstry postons of the neghborng shells, rasng the stackng energy between the shells. As a result, an nter-shell force (torson n ths case) s generated n order to return out-of-regstry atoms to n-regstry postons. The

8 Nano Res (00) 3: nternally generated torson lowers the total system energy, generatng and lockng the twst. A smlar nter-shell regstry drven twst has been observed n all-atom molecular mechancs smulatons of bent MWCNTs [45]. We have systematcally computed the bucklng strans of a seres of -, -, 5-, 0-, 5-, 0-, and 5-walled CNTs. The lengths of all these CNTs were the same, L = 68 nm. Thus, ther aspect ratos were dfferent, rangng from 4 to 00. In Table the numercally determned bucklng strans are compared wth the Euler beam theory (Eq. (0)) and the shell bucklng theory (Eq. ()). We found that these two theores descrbe well the extreme cases. The bucklng stran of the SWCNT ((5, 5) tube) agrees wth the predcton of the Euler beam theory, whle those of the 5-, 0-, and 5-walled CNTs show neglgble dfferences from the predctons of the shell bucklng theory. Note that n applyng the shell bucklng theory, a delay factor s added to Eq. () to take nto account the stffenng effect stemmng from the nter-shell nonbondng nteractons. Specfcally, the predcted bucklng stran s αε shell cr, where α =.4. For moderate aspect ratos, however, both theores fal to predct the correct bucklng stran..4 Dependence of the aspect rato on the postbucklng modulus Prevous studes showed that the post-bucklng responses of bent and twsted thck MWCNTs follow unque scalng laws [3, 4, 6, ]. To examne whether smlar scalng laws exst for unaxally compressed MWCNTs, we have systematcally computed the stress stran relatons for a seres of thck MWCNTs, ncludng 0-, 5-, 0-, and 5-walled MWCNTs. Agan, the length of these MWCNTs was fxed at 68 nm, whle the aspect ratos vared. The stress stran relatons for these MWCNTs are plotted n Fg. 4. In the pre-bucklng regme, all the MWCNTs follow the same stress stran relaton, governed by the lnear elastcty of the MWCNTs. Then, the compressve stress drops abruptly, ndcatng the onset of bucklng. The bucklng stran decreases wth ncreasng thckness of the MWCNTs (and thus wth decreasng aspect rato). For all the MWCNTs, the post-bucklng effectve modul are substantally smaller than that of the lnear elastc regme. In addton, the post-bucklng effectve modulus ncreases monotoncally wth ncreasng thckness of the MWCNTs. For the 0-walled MWCNT, the effectve post-bucklng modulus s negatve. Besdes the helcal Table Comparsons of the computed bucklng strans wth the predctons by the Euler beam theory and the shell bucklng theory. The lengths of all the CNTs consdered here were the same, L = 68 nm. For the SWCNT (n = ), the computed bucklng stran agrees well wth that predcted by the Euler beam theory. For the 5-, 0-, and 5-walled MWCNTs, the computed bucklng strans agree well wth those predcted by the shell bucklng theory. For moderate aspect ratos (n = 0), both theores fal to predct the bucklng strans ε n t/d d/l ε cr Shell bucklng theory Euler beam theory Fgure 4 Stress stran relatons of thck MWCNTs under unaxal compresson. All the MWCNTs share the same modulus n the pre-bucklng regme. The 0-walled MWCNT has a negatve post-bucklng modulus, whch s dstnctly dfferent from other MWCNTs due to the dfference n the bucklng mechansms. The post-bucklng response of 0- and 5-walled MWCNTs can be approxmated by a lnear law

9 40 Nano Res (00) 3: 3 4 damond pattern n the post-bucklng regme, overall the 0-walled MWCNT s clearly bent, whch s dfferent from the other, thcker MWCNTs. Ths hybrd bucklng mode explans why the bucklng stran of the 0-walled CNT can nether be predcted by the Euler beam theory nor the shell bucklng theory. In contrast, we dd not observe global bendng n the 5-, 0-, and 5-walled MWCNTs n ther post-bucklng morphologes wthn the range of the appled stran. Thus, ther post-bucklng mechansms can be consdered to be smlar, as manfested by that fact that ther bucklng strans can all be descrbed by shell bucklng theory. Though the post-bucklng stress stran relatons are ntrnscally nonlnear, we note that for MWCNTs wth smaller aspect ratos (the 0- and 5-walled MWCNTs), ther post-bucklng stress stran relatons are approxmately lnear over a wde range of the appled compressve stran, gvng rse to nearly constant post-bucklng modul. Ths suggests that for thck MWCNTs wth small aspect ratos, ther stress stran responses can be approxmately descrbed by a blnear law, smlar to that for twsted and bent thck MWCNTs [6, ]. The aspect-rato dependent nstablty of the MWCNTs explans well two sets of experments by Yap et al. [6] and Waters et al. [7, 8]. Snce the MWCNTs used n the experments of Yap et al. were of a relatvely large aspect rato, negatve modul were observed, just as for the 0-walled MWCNT studed here. In contrast, the MWCNTs used n the experments of Waters et al. had small aspect ratos,.e., they were very short, thck MWCNTs. The post-bucklng mechancal response was more lke that of the 5-walled MWCNT studed here. Such short, thck MWCNTs do not possess a negatve modulus. We also found that, as the compressve stran ncreases further, the MWCNT wll eventually bend, and the deformaton morphology resembles the descrpton of Waters et al. (.e., hghly dstorted cross-sectons and wrnkled sde-walls). Snce our model cannot address the actvtes of defects, partcularly the nucleaton and moton of the 5/7 dslocatons n CNTs, t s ncapable of reproducng the hysteress observed n the experments of Waters et al. 3. Conclusons Our coarse-graned smulatons predct, for the frst tme, unque evolvng deformaton morphologes n unaxally compressed short, thck MWCNTs, ncludng wave-lke rpplng, and symmetrc and helcal damond patterns. The evoluton of the post-bucklng morphologes of MWCNTs depends on ther aspect ratos, and s a result of coordnated deformatons of the nner and outer shells. Our smulatons show that onset bucklng stran of MWCNTs wth small aspect ratos under axal compresson can be well predcted by the shell bucklng theory, whle that for SWCNTs wth large aspect ratos can be predcted by Euler beam theory. Nether theory can descrbe the bucklng of MWCNTs wth moderate aspect ratos. Due to the stffenng effect of the nter-shell vdw nteractons, the bucklng stran of the outermost shells s delayed by a factor of α.4. The post-bucklng modulus s substantally reduced, and ncreases monotoncally wth ncreasng thckness. The mechancal responses for very short, thck MWCNTs under unaxal compresson can be approxmated by a blnear law for a large range of the appled compressve stran. Our smulaton results agree wth, and explan, prevous expermental data. Acknowledgements We gratefully acknowledge support from the Natonal Scence Foundaton (NSF) grants under Awards Nos and (Clark V. Cooper, program manager), and K. J. H. acknowledges NSF fnancal support under Grant Nos. CMMI and CMMI References [] Baughman, R. H.; Zakhdov, A. A.; de Heer, W.A. Carbon nanotubes The route toward applcatons. Scence 00, 97, [] Zhang, S. L.; Lu, W. K.; Ruoff, R. S. Atomstc smulatons of double-walled carbon nanotubes (DWCNTs) as rotatonal bearngs. Nano Lett. 004, 4, [3] Tombler, T. W.; Zhou, C. W.; Alexseyev, L.; Kong, J.; Da, H. J.; Le, L.; Jayanth, C. S.; Tang, M. J.; Wu, S. Y.

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