Carbon nanotubes collapse phase diagram with arbitrary number of walls. Collapse modes and macroscopic analog
Y. Magnin, F. Rondepierre, W. Cui, D.J. Dunstan, A. San-Miguel
CCarbon nanotubes collapse phase diagram witharbitrary number of walls. Collapse modes andmacroscopic analog.
Y. Magnin a,b, ∗ , F. Rondepierre c , W. Cui d , D.J. Dunstan e , A. San-Miguel c, ∗∗ a MIT Energy Initiative, Massachusetts Institute of Technology, Cambridge, MA, UnitedStates b Consultant, Total@Saclay NanoInnov, 2 boulevard Thomas Gobert, 91120 PalaiseauCedex, France. c Univ Lyon, Universit´e Claude Bernard Lyon 1, CNRS, Institut Lumi`ere Mati`ere,Campus LyonTech - La Doua, F-69622 LYON, France d School of Physics and Electronic Engineering, Jiangsu Normal University, Xuzhou221116,China e School of Physics and Astronomy, Queen Mary University of London, London, E1 4NS,UK
Abstract
Carbon nanotubes tend to collapse when their diameters exceed a certainthreshold, or when a sufficiently large external pressure is applied on theirwalls. Their radial stability of tubes has been studied in each of these cases,however a general theory able to predict collapse is still lacking. Here, wepropose a simple model predicting stability limits as a function of the tubediameter, the number of walls and the pressure. The model is supportedby atomistic simulations, experiments, and is used to plot collapse phasediagrams. We have identified the most stable carbon nanotube, which can ∗ Corresponding author ∗∗ Corresponding author
Email addresses: [email protected] (Y. Magnin), [email protected] (A. San-Miguel)
Preprint submitted to Carbon February 1, 2021 a r X i v : . [ phy s i c s . c o m p - ph ] J a n upport a maximum pressure of ∼
18 GPa before collapsing. The latter wasidentified as a multiwall tube with an internal tube diameter of ∼ ∼
30 walls. This maximum pressure is lowered depending on the internaltube diameter and the number of walls. We then identify a tube diameterdomain in which the radial mechanical stability can be treated as equivalentto macroscopic tubes, known to be described by the canonical L´evy-Carrierlaw. This multiscale behavior is shown to be in good agreement with exper-iments based on O-ring gaskets collapse, proposed as a simple macroscopicparallel to nanotubes in this domain.
1. Introduction
Low-dimensional carbon structures such as fullerenes, graphene, carbonnanotubes (CNT), nanocones, nano-junctions [1, 2, 3, 4, 5, 6] have deeplychanged fundamental concepts of condensed matter physics during the lastdecades [7]. The many associated technological breakthroughs have openedperspectives in a broad range of applications, ranging from electronics [8, 9],sensor developments [10],energy transport and storage [11, 12, 13] or biol-ogy and medical sciences such as drug delivery technology [14]. While bothgraphene and CNT sp structures have concentrated an important part ofthe recent research efforts, there exist many hybrid structures between thesetwo, which have been much less explored [15]. In the literature, they arereferred as ”collapsed nanotubes”, ”flattened carbon nanotubes”, ”closed-edge graphene nanoribbons” or ”dogbones”. Such structures correspond to ageometrical evolution of the CNT radial cross-section, from circular towardsa continuum of shapes, in which the internal walls become closer, in at leastone radial direction. The terms mentioned above are most frequently usedwhen the distance between the internal walls in the collapse direction tendsto the graphitic interlayer distance, where van der Waals interactions (vdW)have to be considered. For clarity, we will refer to this state as the ”col-lapsed” shape, and the deviations from the circular cross-section leading toit as ”collapse transition” shapes. The latter can be either first-order-like2or large tube diameters, or continuous for smaller ones [16], going throughdifferent geometries including oval, race-track or polygonal [17], all groupedin the ”collapse transition” domain.Characterizing the CNT collapse behavior is greatly motivated by the changein electronic structure from the pristine circular cross-section to the deformedor collapsed geometry [18, 19, 20, 21, 22, 23, 24]. Hence, a geometrical elec-tronic tuning based on a shape modification may offer an interesting alter-native to substitutional doping in nano-engineering design [25, 26, 27]. Soon Figure 1: Scheme of the different physical mechanisms allowing the evolution of a carbonnanotube to a symmetrical collapsed structure (s): s1 External applied pressure; s2 Selfcollapse for large tube diameters; s3 Defective tubes; s4 Through charge injection. Mech-anisms leading to an asymmetrical collapsed structure (a): a1 Through interaction witha substrate; a2 Through interaction with other nanotubes. after the first dedicated study about CNT by Iijima et al. [28], collapsed ge-3metries were evidenced by electron microscopy on large CNT diameters [29].It is generally admitted now that a collapse event is favored for large tubediameters and/or for small numbers of concentric tubes [30, 31, 32, 33, 34].Other collapse parameters at ambient conditions were also identified, includ-ing interactions between the external walls of CNT, interactions with a sub-strate or with molecular adsorbates [35, 36, 37], defect formation by electronbeam irradiation [38], or due to the application of electrostatic fields [39, 40].The different mechanisms that can be responsible for collapse are illustratedin Figure 1.For small tube diameters with a stable circular cross-section at ambient con-ditions, it has been shown that a high external pressure causes collapse. Thiswas first shown for SWCNT, where the collapse pressure has been demon-strated to be a function of CNT diameter [41, 42, 43, 44, 45, 16, 46, 47, 48,49, 50, 51, 52, 53, 54]. Pressure-induced collapse is also observed for double-wall [55, 56, 57, 58], triple-wall [59] and multi-wall carbon nanotubes [60].Any effect of helicity (or chirality) - a geometrical characteristic reported tooriginate from the configurational tube edge entropy [61] - on the pressureresponse of CNT was reported to be small, or to occur only in tubes of verysmall diameters [62].In this work, we explore the stability conditions of CNT from their circu-lar cross-section to collapsed shapes, focusing on the stability domain as afunction of geometrical parameters (diameter and number of tube walls),as well as the effect of external pressure. The instability of the circularcross-section is found to be driven by the competition of the elastic energy,related to the bond-bending energy, and the external and internal forces.External forces include the pressure applied to CNT external surfaces, andinteractions with the surrounding molecular medium, while internal forcesinclude the vdW interactions of the tube walls. While a number of atom-istic calculations and models have tried to predict the CNT stability condi-tions [43, 63, 16, 55, 56, 47, 60, 32, 53, 54, 63], none of them fully coveredstructural tube properties, including tube diameters, number of walls, etc.In addition, we propose a multiscale approach, showing that the collapseonset of a range of nanotubes with diameter about nanometer sizes, arewell described by the canonical L´evy-Carrier law (LC), formulated 150 yearsago [64], and originally developed for macroscopic tube collapse. Our mod-ified LC-based model behaves consistently in this domain with atomisticsimulations, density functional tight-binding (DFTB), molecular dynamics(MD), as well as with a macroscale analog based on O-ring gaskets defor-4ation. Our model is then used to plot collapse phase diagrams for carbonnanotubes, providing a better understanding of the radial stability and col-lapse mechanisms of single- and multi-wall nanotubes (MWCNT), includingnanotube bundles as a function of their diameter from the nanometer to dozenof nanometers. We finally think that such an approach could be adapted formore complex porous materials.
2. Results and discussion
In mechanics, the radial collapse pressure P c for macroscopic tubes with adiameter d , is known to scale as P c ∝ d − , as expressed by L´evy-Carrier [64].At the nanoscale, it has been shown that such a formalism is consistent forSWCNT, when including an additional correction term, β /d . This termemerges both in simulations, as well in experiments, and may be relatedto the large built-in curvature energy for small tube diameters, or to thediscrete nature of the nanotubes [65]. This approach is called the modifiedLC equation [66], and is written, P c = 24 Dd (cid:18) − β d (cid:19) , (1)where D is the bending stiffness of graphene, and β corresponds to the di-ameter of the smallest free-standing stable SWCNT [66]. All parameters aregiven in Table 1. When d (cid:29) β , the LC law is recovered; however, when d < β , P c <
0, corresponding to unfeasibly small unsupported tube diame-ters.Eq.1, originally based on experimental observations [54], has been shown tobe well-suited for SWCNT with diameters in the range d ∼ γ F − C , in order to account for the surrounding PTM (anargon bath in MD simulations) [67]. Doing so, we obtain the enthalpy of aSWCNT of length L as, H = 48 Dd (cid:18) − β d (cid:19) πL d + P πL d + γ F − C πL d , (2)5inimizing Eq.2 as a function of d , we obtain the collapse pressure forSWCNT interacting with the PTM as, P c = 24 Dd (cid:18) − β d (cid:19) − γ F − C d . (3) For a bundle formed of SWCNT, Eq.3 can be modified following Pugnoet al. [67], considering that each individual tube in a bundle interacts withits neighboring tubes, acting as a SWCNT pseudo-fluid, while the outersurface of the bundle interacts with the PTM. Then, P c can be obtained byminimizing the corresponding enthalpy expression, P c = 24 Dd (cid:18) − β d (cid:19) − (cid:18) γ C − C d + γ F − C d B (cid:19) (4)where d B represents the bundle diameter. In Eq.4, γ C − C is the carbon for-mation energy, corresponding to the inter-tube vdW interactions, while thelast term represents the interaction between the PTM and the external sur-face area of the bundle. It is worth to note that following Pugno et al. [32]and consistent with atomistic modeling [59], the bundle will have undergonepolygonization before collapse. To go a step further, we now generalize our model for MWCNT. We followthe methodology proposed by Gadakar et al. [56] in which friction betweentubes is neglected so that the bending stiffnesses are additive. Thus, the netexternal pressure needed to collapse the N walls of a MWCNT is written asthe sum of the pressures P c i needed to collapse the i = 1 to N correspondingindividual SWCNT. The pressure energy needs to be distributed betweenthe different tubes, leading to the additive character of the P c . In MWCNT,the vdW inter-wall interactions are compensated, considering that each in-dividual inner-tube interacts both with its next inner- and next outer-tube, i -1 and i +1 respectively. However, it is necessary to account for interactionof the innermost tube ( i =0) with only its next larger tube neighbor ( i =1),and for the interaction of the outermost tube ( i = N -1) with ( i = N -2), andfinally with the external PTM (the last term in Eq.5). Accounting for all6 able 1: Parameters used in the various L´evy-Carrier models. D β γ F − C ) This work γ C − C ) Ref [69, 70, 71] δ P c = N − (cid:88) i =0 Dd i (cid:18) − β d i (cid:19) − (cid:18) γ C − C d − γ C − C d N − + γ F − C d N − (cid:19) . (5)Inter-tube distances are reported to range in between 0.27 nm and 0.42 nm,however, the most common distances in MWCNT are about 0.32-0.35 nm [68].For the sake of simplicity, we have considered that all tubes in a MWCNTrange at the graphitic interlayer distance δ (see Table.1), and that d i = d + 2 δ · i in Eq.5. We may note here that even if a general MWCNT canno longer be considered a thin tube, our method of evaluation of the col-lapse pressure as contribution of various SWCNT in interaction with theirenvironment allows us to keep using the LC expression, which is in fact onlyvalid for thin-walled tubes. In order to check the accuracy of our model, we compare it with bothnumerical and experimental data. Simulations have been performed withtwo algorithms, the DFTB for small tube diameters [72], and MD based onthe empirical AIREBO bond order potential [73], accounting both for C-Ccovalent bonds and for long-range vdW interactions [74]. All simulationshave been performed for tubes or bundles immersed in an Ar bath at 300K.Ar-Ar and C-Ar interactions have been modeled by a (12-6) Lennard-Jonespotential, using the Lorentz-Berthelot mixing rule (see section Method forsimulation details).In Fig.2.a, we show the evolution of P c as a function of the tube diame-ter from DFTB, MD, and from the theoretical models detailed above. As7an be seen, for small d , the modified LC approach (light green line), isin good agreement with the DFTB simulations (yellow stars). Hence, when d < P c decreases, corresponding to a situation where tube diam-eters are so small that the curvature energy is large enough to make tubesunstable. It is noteworthy that a deviation is observed when it is comparedto the macroscopic LC law (dashed black line), that does not include thesmall-diameter correction term discussed above. When d > P c , corresponding to atube diameter where interactions between the PTM and the tube walls dom-inate, while the curvature energy is negligible for such diameters. SWCNTsimulations (red circles), show a self-collapse diameter of 5.3 nm, in goodagreement with the experimental data reported; about 5.1 nm is cited in thereview of He et al. [15]. This agreement allows then to fit γ F − C (see Table1), in the vdW-LC model for SWCNT. As shown in Fig.2.a, we found anexcellent agreement between simulations and the theoretical model for thefull diameter range simulated. We then compared the vdW-LC model ap-plied on bundles. In Fig.2.a, the gray squares correspond to the results ofsimulations performed on bundles made of 37 SWCNT. As can be seen, theself-collapse diameter is smaller than that of isolated SWCNT, a behavioralready observed in [75]. This behavior could be explained by the contribu-tion of the polygonisation of the tubes to the surface energy which resultsfrom inter-tube interactions and which tends to lower P c . Using Eq.4 andthe γ F − C previously fitted from isolated SWCNT, we see that the vdW-LCmodel applied to the bundle configuration is in good agreement with simu-lations using the C-C interaction energy γ C − C (see Table 1). Note that inthe bundle used in simulations, d B (cid:29) d and the interaction between thePTM and the external bundle surface could be neglected compared to theinter-tube interactions.In Fig.2.b, we compare experiment with the vdW-LC for MWCNT (Eq.5).The tube stability is presented as innermost tube diameter d against thenumber of tube walls N at ambient pressure. The continuous line separat-ing the two domains represents the critical internal diameter for collapse ofMWCNT at ambient pressure. The experimental points (red circles) corre-spond to collapsed tubes observed by electron microscopy, and are extractedfrom different sources summarized in the work of Balima et al [34]. As ex-pected, these points are mainly found in the collapsed domain. We haveunderlined a particular point from the work of He et al [15] (yellow square),which corresponds to the determination of the critical collapse diameter by8 igure 2: a. Collapse pressure P c of SWCNT as a function of the tube diameter d , usingthree models based on the L´evy-Carrier formula: the standard LC (black dashed line),the modified LC (green line) and the vdW-LC developed in this work for SWCNT (redline), and bundles (blue line). The results are compared to numerical simulations, DFTB(yellow stars) and MD simulations with a long-range bond order potentials for SWCNT(red circles), and bundles (blue squares). b. Innermost tube diameters d of MWCNTfor which collapsed configurations have been observed, plotted against the number of tubewalls N . The red circles correspond to experimental observation of collapsed tubes, whilethe yellow square corresponds to an experimental determination of the collapse pressure forSWCNT. The black line corresponds to the prediction from our vdW-LC model. The grayarea corresponds to the phase where tubes are not collapsed (circular cross section), whilethe part of the diagram above corresponds to nanotubes collapsed at ambient pressure. We have demonstrated that the vdW-LC model is a robust, simple andsuitable approach in predicting single- and multi-wall carbon nanotubes sta-bility, as well as for bundles. We now use this model to determine the nan-otube collapse phase diagram. In Fig.3.a, we plot the stability diagram ofMWCNT at ambient pressure, extending beyond the domain explored inFig. 2.b. The stable phase, either circular or collapsed, depends on d and N . There also exists an instability domain, i.e., in which tubes cannot ex-ist, for very small d (red domain) in Fig,3.a,b. Hence, with an increasinglylarge d , the number of tube walls has to be increased in order to stabilize acircular MWCNT. Nevertheless, d has a maximum at d =12nm for N =45walls, corresponding theoretically to the largest possible internal cavity inMWCNT. For larger N , d slightly decreases and becomes asymptoticallyconstant at d =11.3 nm. This behavior results from the competition be-tween the inter-tube vdW and the tube-PTM interactions, corresponding tothe second term in the Eq.5.In Fig.3.b, we see that the smallest possible tube diameter for a free-standingSWCNT corresponding to d =0.44 nm [66] can be slightly reduced when in-creasing the number of tube walls. This result is qualitatively supported inthe literature, where d ∼ d ∼ N =13 MWCNT [78]. Our calculationsdo not allow for internal tube diameters below ∼ igure 3: a . Stability diagram of carbon nanotubes at ambient pressure as a function ofthe internal diameter d , and the number of tube walls, N . Three regions are defined fromthe bottom to the top: (1) the red phase corresponds to the tubes that cannot exist due totoo small an internal diameter, (2) the gray phase, corresponding to the stability domain oftubes with a circular cross-section, and (3) the white phase, corresponding to the stabilitydomain of collapsed tubes. b . Zoom on the red phase in (a) (unstable tubes), for smalltube diameters. c . Stability diagram for MWCNT with d =0.56 nm, corresponding tothe tubes found to show the highest collapse pressure. The red dashed line correspondsto the maximum collapse pressure found in the model. in the model, and going beyond the scope of this work.The maximum pressure needed to collapse any MWCNT can now be foundby searching the maximum collapse pressure from Eq.5, as a function of d and N . The maximum P c value depends on N but is found invariably at d ∼ N . The maximum value of P c evolvesfrom P c ∼ N , con-verging to a maximum at P c =18.2 GPa. As shown in Fig.3.c, we note thequick convergence of P c for N ranging from 1 to 4. The number of walls isthus found to increase the stability pressure by about ∼ P c evolves as a function of d , for N rang-ing from 1 to 100, in the collapse phase diagram, Fig.4.a. As can be seen,for small d , N has no significant effect on P c . So, we may approximate thatall curves collapse in a single one for internal diameters below d ∼ d , N plays a more significant role on tubes stability. Interestingly,in Fig.4.b, the collapse pressures obtained with the LC model are compa-rable with those from the vdW-LC model for certain tubes. We consider a11 igure 4: a. Collapse pressure P c as a function of the innermost tube diameter d fordifferent numbers of tube walls N , ranging from 1 to 100 and determined from the the-oretical model accounting for vdW interactions. The gray area represents the multiscaledomain, i.e. a d range where the original macroscopic LC model agrees with the vdW-LCmodel at the nanoscale (see text and (b) for details and criteria). b. Normalized collapsepressure P c d as function of the internal diameter for tubes with N =1, 2, 3, 5, 10, 20and 100 walls from inner to outer curves. The continuous part of the curves represent thedomain in which within ± c. Domain of validity of the continuummechanics corresponding to the LC model (gray area). We have considered that the LCmodel is valid when it differs by less than 5% from a linear approximation in the P c d representation as function of d . P c d plotted as function of d can be approxi-mated by the LC law corresponding to a constant. We show this behaviorfor N = 1, 2, 3, 5, 10, 20 and 100, where the continuous part of plots canbe assimilated to horizontal lines, assuming an uncertainty on experimentalpressure determinations of the order of ± N to higher diameters, and converging for N = 20, for internal diametersranging from ∼ ∼ ±
3. A macroscopic model for the collapse of carbon nanotubes
We have shown above that Eq.1 is a suitable multiscale equation for com-paring the collapse behavior of macroscopic tubes with certain tubes at thenanoscale. This result is thus consistent in predicting P c for SWCNT withdiameters ranging from d ∼ d =11.25, 13.7, 16.25 and 28.75 mm. The O-rings are placed betweentwo transparent plates to limit movement in the direction of the torus axis. Apressure vessel is made by using a large outer O-ring surrounding the smallerinner one. The vessel is connected to a bicycle pump with a pressure gauge togenerate and monitor the pressure acting on the inner O-ring (Fig.5.a). Thedeformation of this O-ring is measured as a function of the applied pressure,and quantifieded by image analysis (details are given in the section Method.Videos of the experiment are also available in the Supplementary Informa-tion). The O-rings used in this experiment are not strictly speaking tubes,but rings, and we first verify that they follow the LC law. To do so, fourmeasurements per O-ring diameters have been performed, and the averaged13 igure 5: a. Schema and picture of the experiment, in which O-rings are collapsed by afluid pressure medium (air or water). b. Averaged collapse pressure P c of O-rings in theexperiment described in (a), as a function of their diameters d (red circles). Experimentaldata are found to fit P c ∝ d − . ± . (black line). c. Schema and picture of the experimentin which O-rings are collapsed by a solid pressure medium (ball-bearings ). P c = a.D − α with a =1.51 J (a constant), and α =3.0 ± d ranging from ∼ Figure 6: a. Evolution of an O-ring bundle immersed in the ball-bearing pressure-transmitting medium during a compression cycle. b. Collapse behavior of a SWCNTbundle immersed in a solid argon medium. Simulation was performed at pressures rang-ing from 1 to 3 GPa with a pressure increased (from the left to the right). instead of air or water) compared to the previous experiment (Fig.5.a) isnecessary to match simulations. Indeed, the pressure needed to collapse abundle with SWCNT diameter around 1 nm is a few GPa, a pressure atwhich the argon PTM used in simulation and in experiment is solid [82].The ball-bearings tend to form an hexagonal close-packed planar domain,which mimic a macro-crystalline argon PTM. In the simulation, we havebuilt a bundle formed of 37 SWCNT with d =1.3nm, immersed in an argonbath at P ∼ P c - butgoes progressively through ovalisation to collapse shape over the range P c to1.5 P c , see also Fig.3 in [54].
4. Conclusion
We propose a simple theoretical model to determine the stability domainof carbon nanotubes as a function of their diameters and their number ofwalls. The geometrical stability limits at ambient, as well as collapse pres-sures for arbitrary number of walls are characterized from the long range Vander Waals interactions, introduced into the modified L´evy-Carrier equation,formulated for tubes at the nanoscale. The model proposed in this work isvalidated by numerical simulations at the nanoscale, as well as experiments.We have thus found that depending on the number of tube walls, nanotubesshow a maximum collapse pressure ranging from ∼
13 to 18 GPa with aninner-tube diameter of d =0.56nm. It is noteworthy that our model fits ex-perimental data despite neglecting multi-wall nanotubes inter-layer friction.When d is smaller than 0.56nm, the collapse pressure drops right down,due to the strong tube curvature, resulting in unstable nanotubes. For largetube diameters, the collapse pressure decreases, corresponding to the Van derWalls interactions that tend to favor the collapse. From the collapse phasediagrams plotted with the model presented in this work, we have shown thatthe L´evy-Carrier equation (originally established for macroscopic tubes) iscompatible with tube diameters of a few nanometers, and depending on thenumber of tube walls. This is an important result underlying that the collapseprocess of most of the common nanotubes produced by standard experimen-tal techniques takes place as in macroscopic tubes, linking behavior at thenano- to the macroscale. This behavior was verified comparing numericalsimulations with experiments at the nanoscale and at the macroscale, where16anotubes were replaced by polymer O-rings. We think that such an analogymaybe interesting in order to study mechanical deformation of nanotubes un-der pressure more easily, or more complex porous systems as Metal OrganicFrameworks (MOF), zeolites, or even disordered porous materials as kerogen.
5. Methods
Density functional tight-binding calculations were performed using theDFTB software package [72] with the matsci-0-3 parameter [84]. This algo-rithm was used only for small tube diameters ranging from d =0.5 to 1.4nm,due to its expensive computational time. In this approach, the Kohn-Shamdensity-functional theory is approximated with fitted integrals from refer-ence calculations. The method increases simulations efficiency compared todensity functional theory (DFT), while keeping ”a priori” a better accuracycompared to the empirical approaches. The C-C Slater-Koster parametersimplemented in this work have been extensively used for CNT simulationsand can be found elsewhere [84]. In Fig.2.a, we have determined P c for dozenof armchair SWCNT. For each pressure, a random displacement of 0.002nmis applied on each atom, and both atomic positions and cell vectors were op-timized until the magnitude of all forces became smaller than 10 − Ha/Bohr.In this process, P was increased by steps of 0.2 GPa, up to the tube collapse.This phenomenon is generally found to be abrupt, and can be easily identi-fied from a discontinuity in the enthalpy as a function of P , correspondingto the transformation in a collapse shape. In some rare cases, especially forsmall d , the discontinuity is not visible, and P c was determined by eye, i.e.,we assigned P c to the first collapsed geometry found.A second set of simulations using an MD algorithm were performed to studylarger tube diameters. MD simulations were conducted for systems withSWCNT immersed in an argon bath, in order to transmit pressure to thenanotubes. Inter-atomic interactions (Ar-Ar and Ar-C) were modelled by a(12-6) Lennard-Jones potential (LJ), with a cutoff fixed at 2 nm, U = 4 (cid:15) (cid:20)(cid:16) σr (cid:17) − (cid:16) σr (cid:17) (cid:21) , (6)where σ corresponds to the atomic diameter, r ij is the inter-atomic distance,17 able 2: Lennard-Jones parameters for argon and carbon atoms. Atoms σ (nm) (cid:15) (meV)Ar-Ar 0.341 10.3C-C 0.336 2.4Ar-C 3.382 5and (cid:15) ij is the interaction energy between two atoms i and j . The LJ pa-rameters (given in Table 2) for the interactions of the different species aredetermined with the Lorentz-Berthelot mixing rule as, σ ij = (cid:18) σ i + σ j (cid:19) and (cid:15) ij = √ (cid:15) i (cid:15) j , (7)The C-C interactions were modelled with the long range bond order poten-tial AIREBO [73]. This potential models both the carbon covalent bonds,and long range vdW interactions with a cutoff fixed at 2 nm.The first step of simulations consists in adsorbing Ar molecules in a bulkphase with the grand canonical Monte Carlo algorithm GCMC ( µ Ar , V , T ),where µ Ar is the chemical potential of Ar, V the volume of the simulation boxat T =300K. GCMC was thus performed to generate a pure argon bulk config-uration at pressures of about P=2 GPa. A tube (or a bundle) is then insertedinside the bulk configuration, and the inner part of the tube is cleared of ar-gon atoms. MD simulations are then performed in the isothermal-isobaricensemble ( N , P , T ), and P c is determined from enthalpy as a function of P . When tube collapse occurs, the tube energy changes abruptly, and thecollapse pressure can be easily identify. In Fig.7, we show the tube shapeevolution, for a SWCNT with d ∼ ∼ In order to mimic the elastic properties of the radial buckling of SWCNT,we have used toroidal elastomer gaskets (O-ring), usually used for applica-tions as vacuum seals. O-rings were placed between two transparent plates18 igure 7: Shape evolution of a SWCNT with d ∼ ∼ of PMMA (poly(methyl methacrylate)) with 25mm thickness, while a largergasket diameter was used to create a cavity around the smaller one. Wethen used dynamometric keys in order to ensure a uniform and weak pres-sure over the O-ring. Both O-rings and internal plate surfaces were oiled inorder to reduce the friction. Holes were then drilled into the top plate andwere used to connect a bicycle pump and a manometer to vary and monitorthe pressure respectively, Fig.8. A schema and a picture of the experimentis shown in Fig.5.a. Doing so, we was able to observe the effect of a ra-dially applied pressure on O-rings for different diameters ( d =11.25, 13.7,16.25 and 28.75 mm), and taking care that displacements are constrainedinto the plane. In order to validate that O-rings can be compared to simili of a SWCNT in the consistent LC domain, we have first shown that theyverify the macroscopic LC approach itself, i.e., that the collapse pressure isinversely proportional to the cube of the torus diameter, Fig.5.b.The second experiment conducted in this work is devoted to O-rings deforma-tion using a pressure-transmitting medium that mimics the situation whereSWCNT are immersed in a high-pressure ( > igure 8: Determination of the collapse pressure for individual O-rings from video imageanalysis during the compression following the scheme of Fig.5.a. a. Time evolution of O-ring radial sectors, measured in pixels from the image. The color scale represents the radialdistribution of the pixels detected. The higher this value, the more the O-ring presents acircular shape. On the contrary, the lower this value, the more the O-ring deviates froma circular shape, corresponding to a collapse situation. b. Pressure measured from themanometer in image analysis. The correlation with (a) allows to determine the collapsepressure. See videos in the supplementary material.
Figure 9: a. Four-lobe collapse of an O-ring obtained using a mixed medium of ball bear-ings and plastic beads (red colour). The mixed medium leads to the formation of smallercrystalline domains in the pressurized transmitting medium. b. Numerical simulation oftube collapse with a metastable state corresponding to a four-lobe shape (the pressuretransmitting medium is not shown in this snapshot).
Acknowledgments
We acknowledge the platform PLECE of the University de Lyon andiLMTech (CNRS and University Claude Bernard Lyon 1). Y. Magnin grate-fully acknowledges the Computational Center of Cergy-Pontoise University(UCP) for the computational time. A. San-Miguel and D. J. Dunstan ac-knowledge the support of the 2D-PRESTO ANR-19-CE00-0027 project.21 eferences [1] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, R. E. Smalley,C60: Buckminsterfullerene, nature 318 (6042) (1985) 162–163.[2] A. K. Geim, K. S. Novoselov, The rise of graphene, in: Nanoscienceand technology: a collection of reviews from nature journals, WorldScientific, 2010, pp. 11–19.[3] S. Iijima, T. Ichihashi, Single-shell carbon nanotubes of 1-nm diameter,nature 363 (6430) (1993) 603–605.[4] N. Yang, G. Zhang, B. Li, Carbon nanocone: A promising thermalrectifier, Applied Physics Letters 93 (24) (2008) 243111.[5] D. Wei, Y. Liu, The intramolecular junctions of carbon nanotubes, Ad-vanced materials 20 (15) (2008) 2815–2841.[6] S. Nasir, M. Z. Hussein, Z. Zainal, N. A. Yusof, Carbon-based nano-materials/allotropes: A glimpse of their synthesis, properties and someapplications, Materials 11 (2) (2018) 295.[7] T. C. Dinadayalane, J. Leszczynski, Fundamental structural, electronic,and chemical properties of carbon nanostructures: graphene, fullerenes,carbon nanotubes, and their derivatives, Handbook of computationalchemistry (2012) 793–867.[8] L.-M. Peng, Z. Zhang, S. Wang, Carbon nanotube electronics: recentadvances, Materials today 17 (9) (2014) 433–442.[9] M. M. Shulaker, G. Hills, N. Patil, H. Wei, H.-Y. Chen, H.-S. P. Wong,et al., Carbon nanotube computer, Nature 501 (7468) (2013) 526–530.[10] K. Chen, W. Gao, S. Emaminejad, D. Kiriya, H. Ota, H. Y. Y. Nyein,et al., Printed carbon nanotube electronics and sensor systems, Ad-vanced Materials 28 (22) (2016) 4397–4414.[11] E. Halakoo, A. Khademi, M. Ghasemi, M. Yusof, R. J. Gohari, A. F.Ismail, Production of sustainable energy by carbon nanotube/platinumcatalyst in microbial fuel cell, Procedia CIRP 26 (2015) 473–476.2212] H. Kim, K.-Y. Park, J. Hong, K. Kang, All-graphene-battery: bridgingthe gap between supercapacitors and lithium ion batteries, Scientificreports 4 (2014) 5278.[13] T. Saito, T. Yamada, D. Fabris, H. Kitsuki, P. Wilhite, M. Suzuki,et al., Improved contact for thermal and electrical transport in carbonnanofiber interconnects, Applied Physics Letters 93 (10) (2008) 102108.[14] A. Bianco, K. Kostarelos, M. Prato, Applications of carbon nanotubes indrug delivery, Current opinion in chemical biology 9 (6) (2005) 674–679.[15] M. He, J. Dong, H. Wang, H. Xue, Q. Wu, B. Xin, et al., Advancein close-edged graphene nanoribbon: Property investigation andstructure fabrication, Small 15 (29) (2019) 1804473. , doi:10.1002/smll.201804473 .URL [16] P. Tangney, R. B. Capaz, C. D. Spataru, M. L. Cohen, S. G. Louie,Structural transformations of carbon nanotubes under hydrostatic pres-sure, Nano Lett. 5 (11) (2005) 2268–2273. doi:10.1021/nl051637p .[17] S. Rols, I. N. Goncharenko, R. Almairac, J. L. Sauvajol, I. Mirebeau,Polygonization of single-wall carbon nanotube bundles under high pres-sure, Phys. Rev. B 64 (2001) 153401. doi:10.1103/PhysRevB.64.153401 .[18] P. E. Lammert, P. Zhang, V. H. Crespi, Gapping by squashing: Metal-insulator and insulator-metal transitions in collapsed carbon nanotubes,Phys. Rev. Lett. 84 (2000) 2453–2456. doi:10.1103/PhysRevLett.84.2453 .URL https://link.aps.org/doi/10.1103/PhysRevLett.84.2453 [19] M. S. C. Mazzoni, H. Chacham, Bandgap closure of a flattened semi-conductor carbon nanotube: A first-principles study, Applied PhysicsLetters 76 (12) (2000) 1561–1563. arXiv:https://doi.org/10.1063/1.126096 , doi:10.1063/1.126096 .URL https://doi.org/10.1063/1.126096 doi:10.1103/PhysRevLett.85.154 .URL https://link.aps.org/doi/10.1103/PhysRevLett.85.154 [21] O. G¨ulseren, T. Yildirim, S. Ciraci, C¸ . Kılı¸c, Reversible band-gap engi-neering in carbon nanotubes by radial deformation, Physical Review B65 (15) (2002) 155410.[22] A. Impellizzeri, P. Briddon, C. P. Ewels, Stacking- and chirality-dependent collapse of single-walled carbon nanotubes: A large-scaledensity-functional study, Phys. Rev. B 100 (2019) 115410. doi:10.1103/PhysRevB.100.115410 .URL https://link.aps.org/doi/10.1103/PhysRevB.100.115410 [23] C. E. Giusca, Y. Tison, S. R. P. Silva, Atomic and electronic struc-ture in collapsed carbon nanotubes evidenced by scanning tunnelingmicroscopy, Physical Review B 76 (3) (2007) 035429.[24] F. Balima, S. Le Floch, C. Adessi, T. F. Cerqueira, N. Blanchard,R. Arenal, et al., Radial collapse of carbon nanotubes for conductiv-ity optimized polymer composites, Carbon 106 (2016) 64–73.[25] L. Duclaux, Review of the doping of carbon nanotubes (mul-tiwalled and single-walled), Carbon 40 (10) (2002) 1751 –1764, carbon Nanotubes:The Present State. doi:https://doi.org/10.1016/S0008-6223(02)00043-X .URL [26] X. Ma, L. Adamska, H. Yamaguchi, S. E. Yalcin, S. Tretiak, S. K. Doorn,et al., Electronic structure and chemical nature of oxygen dopant statesin carbon nanotubes, ACS nano 8 (10) (2014) 10782–10789.[27] D. Machon, V. Pischedda, S. Le Floch, A. San-Miguel, Perspective:High pressure transformations in nanomaterials and opportunities inmaterial design, Journal of Applied Physics 124 (16) (2018) 160902. doi:10.1063/1.5045563 . 2428] S. Iijima, Helical microtubules of graphitic carbon, Nature 354 (6348)(1991) 56–58. doi:10.1038/354056a0 .URL https://doi.org/10.1038/354056a0 [29] N. G. Chopra, L. X. Benedict, V. H. Crespi, M. L. Cohen, S. G. Louie,A. Zettl, Fully collapsed carbon nanotubes, Nature 377 (6545) (1995)135–138. doi:10.1038/377135a0 .URL [30] L. X. Benedict, N. G. Chopra, M. L. Cohen, A. Zettl, S. G. Louie,V. H. Crespi, Microscopic determination of the interlayer bindingenergy in graphite, Chemical Physics Letters 286 (5) (1998) 490 – 496. doi:https://doi.org/10.1016/S0009-2614(97)01466-8 .URL [31] T. Tang, A. Jagota, C.-Y. Hui, N. J. Glassmaker, Collapse of single-walled carbon nanotubes, J. Appl. Phys. 97 (7) (2005) –. doi:10.1063/1.1883302 .[32] N. M. Pugno, The design of self-collapsed super-strong nan-otube bundles, J. Mech. Phys. Solids 58 (9) (2010) 1397–1410. doi:10.1016/j.jmps.2010.05.007 .URL [33] C. Zhang, K. Bets, S. S. Lee, Z. Sun, F. Mirri, V. L. Colvin, et al.,Closed-edged graphene nanoribbons from large-diameter collapsed nan-otubes, ACS Nano 6 (7) (2012) 6023–6032, pMID: 22676224. arXiv:https://doi.org/10.1021/nn301039v , doi:10.1021/nn301039v .URL https://doi.org/10.1021/nn301039v [34] F. Balima, S. L. Floch, C. Adessi, T. F. Cerqueira, N. Blanchard, R. Are-nal, et al., Radial collapse of carbon nanotubes for conductivity opti-mized polymer composites, Carbon 106 (2016) 64 – 73. doi:https://doi.org/10.1016/j.carbon.2016.05.004 .[35] T. Hertel, R. E. Walkup, P. Avouris, Deformation of carbon nanotubesby surface van der waals forces, Phys. Rev. B 58 (1998) 13870–13873.25 oi:10.1103/PhysRevB.58.13870 .URL https://link.aps.org/doi/10.1103/PhysRevB.58.13870 [36] K. Yan, Q. Xue, Q. Zheng, D. Xia, H. Chen, J. Xie, Radial collapseof single-walled carbon nanotubes induced by the cu2o surface, J. Phys.Chem. C 113 (8) (2009) 3120–3126. arXiv:http://pubs.acs.org/doi/pdf/10.1021/jp808264d , doi:10.1021/jp808264d .[37] J. Xie, Q. Xue, H. Chen, D. Xia, C. Lv, M. Ma, Influence of solidsurface and functional group on the collapse of carbon nanotubes, TheJournal of Physical Chemistry C 114 (5) (2010) 2100–2107. arXiv:https://doi.org/10.1021/jp910630w , doi:10.1021/jp910630w .URL https://doi.org/10.1021/jp910630w [38] N. G. Chopra, F. Ross, A. Zettl, Collapsing carbon nanotubes withan electron beam, Chemical Physics Letters 256 (3) (1996) 241 – 245. doi:https://doi.org/10.1016/0009-2614(96)00475-7 .URL [39] O. E. Shklyaev, E. Mockensturm, V. H. Crespi, Modeling electrostati-cally induced collapse transitions in carbon nanotubes, Phys. Rev. Lett.106 (2011) 155501. doi:10.1103/PhysRevLett.106.155501 .URL https://link.aps.org/doi/10.1103/PhysRevLett.106.155501 [40] H. R. Barzegar, A. Yan, S. Coh, E. Gracia-Espino, G. Dunn,T. W˚agberg, et al., Electrostatically driven nanoballoon actuator, NanoLetters 16 (11) (2016) 6787–6791, pMID: 27704855. doi:10.1021/acs.nanolett.6b02394 .[41] A. Sood, P. Teresdesai, D. Muthu, R. Sen, A. Govindaraj, C. Rao, Pres-sure behaviour of single wall carbon nanotube bundles and fullerenes: Araman study, Phys. Status Solidi B 215 (1) (1999) 393–401, InternationalConference on Solid State Spectroscopy - (ICSSS), Schwabisch Gmund,Germany, Sep 05-07, 1999. doi:{10.1002/(SICI)1521-3951(199909)215:1<393::AID-PSSB393>3.0.CO;2-8} .[42] S.-P. Chan, W.-L. Yim, X. G. Gong, Z.-F. Liu, Carbon nanotube bundlesunder high pressure: transformation to low-symmetry structures, Phys.Rev. B 68 (2003) 075404. doi:10.1103/PhysRevB.68.075404 .2643] R. B. Capaz, C. D. Spataru, P. Tangney, M. L. Cohen, S. G. Louie,Hydrostatic pressure effects on the structural and electronic propertiesof carbon nanotubes, Phys. Status Solidi B 241 (14) (2004) 3352–3359. doi:10.1002/pssb.200490021 .[44] J. A. Elliott, J. K. W. Sandler, A. H. Windle, R. J. Young, M. S. P.Shaffer, Collapse of single-wall carbon nanotubes is diameter dependent,Phys. Rev. Lett. 92 (2004) 095501. doi:10.1103/PhysRevLett.92.095501 .[45] A. Merlen, N. Bendiab, P. Toulemonde, A. Aouizerat, A. San Miguel,J. L. Sauvajol, et al., Resonant raman spectroscopy of single-wall carbonnanotubes under pressure, Phys. Rev. B 72 (2005) 035409. doi:10.1103/PhysRevB.72.035409 .[46] S. Zhang, R. Khare, T. Belytschko, K. J. Hsia, S. L. Mielke, G. C.Schatz, Transition states and minimum energy pathways for the collapseof carbon nanotubes, Phys. Rev. B 73 (2006) 075423. doi:https://doi.org/10.1103/PhysRevB.73.075423 .[47] M. Hasegawa, K. Nishidate, Radial deformation and stability of single-wall carbon nanotubes under hydrostatic pressure, Phys. Rev. B 74(2006) 115401. doi:10.1103/PhysRevB.74.115401 .URL https://link.aps.org/doi/10.1103/PhysRevB.74.115401 [48] C. Caillier, D. Machon, A. San-Miguel, R. Arenal, G. Montagnac,H. Cardon, et al., Probing high-pressure properties of single-wall car-bon nanotubes through fullerene encapsulation, Phys. Rev. B 77 (12)(2008) 125418. doi:10.1103/PhysRevB.77.125418 .[49] M. Yao, Z. Wang, B. Liu, Y. Zou, S. Yu, W. Lin, et al., Ramansignature to identify the structural transition of single-wall carbonnanotubes under high pressure, Phys. Rev. B 78 (20) (2008) 205411. doi:10.1103/PhysRevB.78.205411 .[50] A. J. Ghandour, D. J. Dunstan, A. Sapelkin, G-mode behaviourof closed ended single wall carbon nanotubes under pressure,physica status solidi (b) 246 (3) (2009) 491–495. arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/pssb.200880503 , doi:https://doi.org/10.1002/pssb.200880503 .27RL https://onlinelibrary.wiley.com/doi/abs/10.1002/pssb.200880503 [51] C. A. Kuntscher, A. Abouelsayed, K. Thirunavukkuarasu, F. Hennrich,Pressure-induced phenomena in single-walled carbon nanotubes: Struc-tural phase transitions and the role of pressure transmitting medium,physica status solidi (b) 247 (11-12) (2010) 2789–2792. arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/pssb.201000150 , doi:https://doi.org/10.1002/pssb.201000150 .URL https://onlinelibrary.wiley.com/doi/abs/10.1002/pssb.201000150 [52] Y. Sun, D. Dunstan, M. Hartmann, D. Holec, Nanomechanicsof carbon nanotubes, PAMM 13 (1) (2013) 7–10. arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/pamm.201310003 , doi:https://doi.org/10.1002/pamm.201310003 .URL https://onlinelibrary.wiley.com/doi/abs/10.1002/pamm.201310003 [53] T. F. Cerqueira, S. Botti, A. San-Miguel, M. A. Marques, Density-functional tight-binding study of the collapse of carbon nanotubes underhydrostatic pressure, Carbon 69 (0) (2014) 355–360. doi:10.1016/j.carbon.2013.12.036 .[54] A. C. Torres-Dias, T. F. Cerqueira, W. Cui, M. A. Marques, S. Botti,D. Machon, et al., From mesoscale to nanoscale mechanics in single-wallcarbon nanotubes, Carbon 123 (2017) 145 – 150. doi:http://dx.doi.org/10.1016/j.carbon.2017.07.036 .[55] X. Ye, D. Y. Sun, X. G. Gong, Pressure-induced structural transitionof double-walled carbon nanotubes, Phys. Rev. B 72 (2005) 035454. doi:10.1103/PhysRevB.72.035454 .URL http://link.aps.org/doi/10.1103/PhysRevB.72.035454 [56] V. Gadagkar, P. K. Maiti, Y. Lansac, A. Jagota, A. K. Sood, Collapseof double-walled carbon nanotube bundles under hydrostatic pressure,Phys. Rev. B 73 (2006) 085402. doi:10.1103/PhysRevB.73.085402 .[57] A. L. Aguiar, E. B. Barros, R. B. Capaz, A. G. Souza Filho, P. T. C.Freire, J. Mendes Filho, et al., Pressure-induced collapse in double-28alled carbon nanotubes: Chemical and mechanical screening effects, J.Phys. Chem. C 115 (13) (2011) 5378–5384. doi:10.1021/jp110675e .[58] S. You, M. Mases, I. Dobryden, A. A. Green, M. C. Hersam, A. V. Solda-tov, Probing structural stability of double-walled carbon nanotubes athigh non-hydrostatic pressure by raman spectroscopy, High PressureRes. 31 (1) (2011) 186–190. , doi:10.1080/08957959.2011.562897 .[59] R. S. Alencar, W. Cui, A. C. Torres-Dias, T. F. T. Cerqueira, S. Botti,M. A. L. Marques, et al., Pressure-induced radial collapse in few-wallcarbon nanotubes: A combined theoretical and experimental study, Car-bon 125 (2017) 429 – 436. doi:10.1016/j.carbon.2017.09.044 .[60] H. Shima, M. Sato, Pressure-induced structural transitions in multi-walled carbon nanotubes, physica status solidi (a) 206 (10) (2009)2228–2233. doi:10.1002/pssa.200881706 .URL http://dx.doi.org/10.1002/pssa.200881706 [61] Y. Magnin, H. Amara, F. Ducastelle, A. Loiseau, C. Bichara, Entropy-driven stability of chiral single-walled carbon nanotubes, Science362 (6411) (2018) 212–215.[62] A. C. Torres-Dias, S. Cambr´e, W. Wenseleers, D. Machon, A. San-Miguel, Chirality-dependent mechanical response of empty and water-filled single-wall carbon nanotubes at high pressure, Carbon (2015) 442–541 doi:10.1016/j.carbon.2015.08.032 .[63] M. H. F. Sluiter, Y. Kawazoe, Phase diagram of single-wall carbonnanotube crystals under hydrostatic pressure, Phys. Rev. B 69 (2004)224111. doi:10.1103/PhysRevB.69.224111 .[64] G. Carrier, On the buckling of elastic rings, Journal of Mathematics andPhysics 26 (1-4) (1947) 94–103.[65] D. J. Carter, D. J. Dunstan, W. Just, O. F. Bandtlow, A. S. Miguel,Softening of the euler buckling criterion under discretisation of compli-ance (2020). arXiv:2011.14120 .2966] A. C. Torres-Dias, T. F. Cerqueira, W. Cui, M. A. Marques,S. Botti, D. Machon, et al., From mesoscale to nanoscale mechan-ics in single-wall carbon nanotubes, Carbon 123 (2017) 145 – 150. doi:https://doi.org/10.1016/j.carbon.2017.07.036 .URL [67] N. M. Pugno, J. A. Elliott, Buckling of peapods, fullerenes andnanotubes, Physica E: Low-dimensional Systems and Nanostructures44 (6) (2012) 944 – 948. doi:http://dx.doi.org/10.1016/j.physe.2011.12.024 .URL [68] O. V. Kharissova, B. I. Kharisov, Variations of interlayer spacing incarbon nanotubes, RSC Adv. 4 (2014) 30807–30815. doi:10.1039/C4RA04201H .URL http://dx.doi.org/10.1039/C4RA04201H [69] X. Meng, B. Zhang, H. Li, F. Li, Z. Kang, M. Li, et al., A theoreticalanalysis on self-collapsing of nanotubes, International Journal of Solidsand Structures 160 (2019) 51–58.[70] Y. Han, K. C. Lai, A. Lii-Rosales, M. C. Tringides, J. W. Evans, P. A.Thiel, Surface energies, adhesion energies, and exfoliation energies rele-vant to copper-graphene and copper-graphite systems, Surface Science685 (2019) 48–58.[71] L. Girifalco, R. Lad, Energy of cohesion, compressibility, and the po-tential energy functions of the graphite system, The Journal of chemicalphysics 25 (4) (1956) 693–697.[72] B. Aradi, B. Hourahine, T. Frauenheim, Dftb+, a sparse matrix-basedimplementation of the dftb method, J. Phys. Chem. A 111 (26) (2007)5678–5684.[73] S. J. Stuart, A. B. Tutein, J. A. Harrison, A reactive potential for hy-drocarbons with intermolecular interactions, The Journal of chemicalphysics 112 (14) (2000) 6472–6486.3074] Y. Magnin, G. F¨orster, F. Rabilloud, F. Calvo, A. Zappelli, C. Bichara,Thermal expansion of free-standing graphene: benchmarking semi-empirical potentials, Journal of Physics: Condensed Matter 26 (18)(2014) 185401.[75] M. Motta, A. Moisala, I. A. Kinloch, A. H. Windle, High performancefibres from ’dog bone’ carbon nanotubes, Advanced Materials 19 (21)(2007) 3721–3726.[76] J.-C. Blancon, A. Ayari, L. Marty, N. Bendiab, A. San-Miguel, Elec-tronic transport in individual carbon nanotube bundles under pressure,J. Appl. Phys. 114 (14). doi:{10.1063/1.4824544} .[77] L. Guan, K. Suenaga, S. Iijima, Smallest carbon nanotube assignedwith atomic resolution accuracy, Nano Letters 8 (2) (2008) 459–462,pMID: 18186659. arXiv:https://doi.org/10.1021/nl072396j , doi:10.1021/nl072396j .URL https://doi.org/10.1021/nl072396j [78] X. Zhao, Y. Liu, S. Inoue, T. Suzuki, R. Jones, Y. Ando, Smallestcarbon nanotube is 3 ˚a in diameter, Physical review letters 92 (12)(2004) 125502.[79] K. Syassen, Ruby under pressure, High Pressure Research 28 (2) (2008)75–126. arXiv:https://doi.org/10.1080/08957950802235640 , doi:10.1080/08957950802235640 .URL https://doi.org/10.1080/08957950802235640 [80] J. Zang, A. Treibergs, Y. Han, F. Liu, Geometric constant definingshape transitions of carbon nanotubes under pressure, Phys. Rev. Lett.92 (2004) 105501. doi:10.1103/PhysRevLett.92.105501 .URL https://link.aps.org/doi/10.1103/PhysRevLett.92.105501 [81] J. Prasek, J. Drbohlavova, J. Chomoucka, J. Hubalek, O. Jasek,V. Adam, et al., Methods for carbon nanotubes synthesis—review, J.Mater. Chem. 21 (2011) 15872–15884. doi:10.1039/C1JM12254A .URL http://dx.doi.org/10.1039/C1JM12254Ahttp://dx.doi.org/10.1039/C1JM12254A