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Chain- and chainmail-like nanostructures from carbon nanotube rings
Computational Materials Science 161 (2019) 76–82
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Chain- and chainmail-like nanostructures from carbon nanotube rings Julian F.R.V. Silveira, Andre R. Muniz
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Department of Chemical Engineering, Universidade Federal do Rio Grande do Sul, Rua Luiz Englert s/n, 90040-040 Porto Alegre, RS, Brazil
A R T I C LE I N FO
A B S T R A C T
Keywords: Nanomaterials Molecular dynamics Mechanical properties 1D materials Nanomeshes
A computational study on the structure, stability and mechanical properties of carbon nanotube (CNT) rings and derived 1D and 2D superstructures is presented. CNT rings were used as building blocks for CNT-based nanochains and nanochainmails, i.e., nanostructures similar to conventional chains and chainmails. Molecular dynamics simulations show that these structures are stable, mechanically strong and highly flexible, reaching strains as high as 0.50 without breaking. The stability of CNT rings was shown to be dependent on both ring and CNT diameters, affecting directly the mechanical behavior of the CNT-based superstructures. The interesting combination of strength, flexibility and lightness exhibited by these materials enable their use in several potential applications, such as reinforced materials, membranes and actuators in miniaturized devices.
1. Introduction Carbon-based nanostructures have been thoroughly studied over the last years due to their outstanding and unique set of physical properties. They are suitable for several practical applications, especially those that require materials with high mechanical strength and low weight [1–5]. The synthesis of nanostructures with characteristic dimensions (length, area) large enough for certain practical applications, without significant loss in performance, is currently one of the greatest challenges in the field [5]. One approach used to circumvent this problem is combining smaller units of carbon nanostructures to form extended superstructures. There are several studies proposing the creation of such superstructures, through the assembly of fullerenes [6,7], carbon nanotubes, [8,9], graphene [10], diamond nanothreads [11] and even mixtures of them [8,12–14], achieved by the establishment of covalent bonds between smaller units of a given material. These materials can potentially overcome some of the limitations of using graphene and carbon nanotubes in certain applications, and enhance the effective use of their mechanical, electronic and thermal properties, enabling new practical applications [8–14]. The synthesis of rings made out of carbon nanotubes (CNTs), also known as CNT rings, carbon nanorings, nanotube tori, among other similar names, has been reported by several authors [15–23], using different methods. In general, CNTs are coiled during the process, and a connection between their extremities is established, being stabilized by either covalent bonds or intermolecular interactions. The controlled synthesis of such structures opens the possibility of creating large superstructures, by arranging them in such way that these are
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interpenetrated forming the nanoscale analog of a one-dimensional chain, on which each individual CNT ring is a link. Expanding this idea to more dimensions allows us to envision 2D chainmail-like structures and even 3D superstructures. The resulting CNT-based nanostructures (as illustrated in Fig. 1), analogously to macroscopic conventional chains and chainmails (similar to those used as medieval armors), would exhibit a high mechanical strength and a low weight. A potential advantage of creating CNT-based nanochains would be the possibility of obtaining large and continuous structures, maintaining to some degree the remarkable mechanical properties of carbon nanotubes [24,25], also introducing some flexibility [19,23]. Carbon nanotube rings have been subject of several theoretical studies, with emphasis on their structural, electronic and magnetic properties [26–36]. The possibility of using these nanostructures for mechanical applications has not been properly investigated. A previous study [37] evaluated the mechanical strength of a CNT ring-based chain-like structure, describing the material using a single link pulled by a pair of straight CNT segments; promising properties are reported, but a more comprehensive description and understanding of mechanical behavior of such nanochains and accurate evaluation of mechanical properties as function of their characteristic dimensions is still elusive. Therefore, the objective of the present work is to carry out a systematic study on the creation of CNT rings-based 1D nanochains and 2D nanochainmails with varied characteristic dimensions (CNT and ring diameters), evaluating their stability and mechanical behavior under tensile strain using classical molecular dynamics simulations.
Corresponding author. E-mail address: [email protected] (A.R. Muniz).
https://doi.org/10.1016/j.commatsci.2019.01.048 Received 4 October 2018; Received in revised form 25 January 2019; Accepted 28 January 2019 0927-0256/ © 2019 Elsevier B.V. All rights reserved.
Computational Materials Science 161 (2019) 76–82
J.F.R.V. Silveira, A.R. Muniz
and 2D stresses are reported for nanochains and nanochainmails respectively (as done before for nanotubes [24] and single-layer graphene [43]), along with the conventionally reported 3D stress. The 1D stress is 1 i i reported in force units, calculated by τnm ,1D = L1D Snm , where L1D is the length of the 1D material; consequently, the 1D tensile strength corresponds to the force required to break a single chain. Analogously, the 1 i i 2D stress is computed by τnm ,2D = A2D Snm , where A2D is the surface area Ω of the 2D material. The volume required for the computation of 3D stresses was calculated considering that the cross-sectional area of a single 1D nanochain At , right before the onset of fracture, can be approximated by the cross-sectional area of two nanotubes, such as Ω = L1D ∙At . Consistently, the thickness of the 2D nanomesh planar structure Lt was taken as twice the nanotube diameter, and then Ω = Lt ∙A2D . The diameter of a nanotube was taken as the sum of the average atomic distance between diametrically opposed atoms in the structure, and the Van der Waals distance (0.340 nm), as done in previous works [24]. Fig. 1. Example of the atomic structure of (a) a (5,5) CNT ring (with a zoomed view depicting the curved nanotube structure), and of CNT ring-based nanostructures, namely (b) a nanochain and (c) a nanochainmail.
3. Results 3.1. Structural stability of individual CNT rings
2. Computational methods The atomic structures of individual CNT rings were formed by rolling up carbon nanotubes and joining their extremities through covalent C–C bonds, in order to establish a continuous honeycomb graphene-like lattice, as illustrated in Figs. 1 and 2. Bending of nanotubes leads to C–C bond straining, resulting in internal stresses on the material. These stresses might induce structural transformations on the rings, leading to other configurations than perfect and symmetric rings as desired. The occurrence and extent of these transformations are expected to depend on the nanotube diameter/chirality, as well as of the ring diameter. In order to evaluate the stability of the CNT rings and how the aforementioned factors affect the atomic structure and morphology of carbon rings, we performed structural relaxations as described in Section 2 for rings of varying diameters (from 15.7 to 125.4 nm), using armchair (n,n) CNTs with diameters from 0.67 to 2.68 nm (equivalent to n varying from 5 to 20), as listed in Table 1. For future reference, each ring presented in this table is labeled as Ri, with i designated by capital letters {A, B, C, …, T}. We observed that the initially perfect nanotube rings (as seen in Fig. 1) undergo distinct morphological transformations upon structural relaxation. Some of the structures retained their characteristic ring shape, and were classified as Ideal Rings – “IR”, as illustrated in Fig. 2(a). The other rings relaxed to configurations different from the
The atomic structure and mechanical properties of CNT rings and derived superstructures were computed by classical molecular dynamics (MD) simulations, using the LAMMPS package [38]. The interatomic interactions were described by the AIREBO potential [39], widely used for studying carbon-based nanostructures, yielding reliable results for mechanical properties [24,25,40]. The C–C cutoff distance in the potential was set to 2.0 Å to avoid spurious high forces at high bond strains, as suggested in previous studies [41]. A timestep of 1 fs was used in the integration of the equations of motion, and temperature and pressure control (the latter necessary when relaxing supercell dimensions) were carried out using the Berendsen thermostat and barostat [42]. Stability tests were conducted for individual rings, consisting of a structural relaxation at 0.1 K for 200 ps, followed by gradual heating up to 300 K, and then thermal equilibration at this temperature (for at least 200 ps). Uniaxial and biaxial tensile strain tests were respectively carried out for 1D nanochains and 2D nanochainmails, using periodic boundary conditions and supercells with 8–10 and 48–75 rings for 1D and 2D structures respectively (containing from 32,000 to 256,000 atoms). The systems were initially subjected to structural relaxation at 0.1 K for 200 ps, and then to a thermal equilibration to 300 K (gradual heating followed by at least 200 ps at final temperature). Finally, a constant strain rate of 0.001 ps−1 was applied in the supercell (along either one or two perpendicular directions), until fracture is observed. Stress-strain curves are obtained by computing the atomic stress along the trajectories, using the virial equation 1 1 1 i i τnm = Ω ⎡−mi vmi vni + 2 ∑j ≠ i rnij Fmij⎤ = Ω Snm , where τni , m is the n, m ⎣ ⎦ component of the atomic stress tensor for atom i , Ω is the volume, mi is i the atomic mass, vk is the k component of the atom’s velocity vector, rkij and Fkij are the k components of the distance and the force (vectors) i between atoms i and j , for each neighbor atom j , and Snm is an auxiliary tensor component (to be used in other definitions later in the text, computed directly in LAMMPS via compute stress/atom). The tensile strength and fracture strain are obtained at the maximum of the curve (at the onset of fracture). Preliminary convergence studies (dependence of results with respect to number of rings on a supercell, strain rate, etc.) were carried out to ensure the quality of reported results. The definition of volume of 1D and 2D structures, required to compute atomic stress in MD simulations, is not straightforward. The interlayer spacing of layers characteristic of graphite (0.335–0.340 nm) has been typically used to define the thickness of the surfaces required to compute stresses in single layer graphene sheets and carbon nanotubes [24,25,40]. In order to present a consistent set of results, both 1D
Fig. 2. Atomic structures of different nanotube rings before (top) and after (bottom) structural relaxation, illustrating the three patterns of ring morphology observed upon structural relaxation: (a) IR (ring RA), (b) PR (ring RM), (c) FR (ring RQ). 77
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Table 1 Structural parameters of CNT rings, indices i = {A, B, …, T} used to designate each Ri ring configuration, and ring morphology observed upon structural relaxation (IR – ideal rings, PR – polygonal rings and FR – flat rings, as described in the text). Nanotube diameter D (nm) and chirality
Ring diameter d (nm) 15.67
31.34
62.68
125.36
0.67 – (5,5)
A IR
B IR
C IR
D IR
0.94 – (7,7)
E IR
F IR
G IR
H IR
1.34 – (10,10)
I PR
J PR
K IR
L IR
2.01 – (15,15)
M PR
N PR
O PR
P PR
2.68 – (20,20)
Q FR
R FR
S PR
T PR
Fig. 3. Rearrangement energy for rings RA, RE, RI, RM, and RQ (all with D = 15.67 nm and varying nanotube diameter).
initial, which can be classified in two groups. In some rings, the nanotube cross-section went from a circular to a flatter shape in some points, inducing folds on the structure, but maintaining the regular nanotube structure in finite sub-domains, as illustrated in Fig. 2(b). The introduction of these folds result in a sort of polygonal structure, not as round as a perfect ring; these were then labeled as Polygonal Rings – “PR”. Other rings relaxed to what we called Flat Rings – “FR”; in this case, the nanotube undergoes a transformation similar to the PR, but the nanotube cross-section relax to a flat shape along the whole ring extension, as depicted in Fig. 2(c). The ring maintains partially its original shape, exhibiting some local folds as in the PRs. Similar structural features exhibited by relaxed CNT rings have been reported by other authors [30–32]. The classification of the resulting structures observed upon structural relaxation for all investigated rings Ri are summarized in Table 1. In general, rings formed by nanotubes of small diameter (< 1 nm) have a tendency to maintain the original ring-like shape (IR) regardless the ring diameter. Rings composed of thicker nanotubes need to be significantly large to maintain the original ring shape, otherwise they relax toward PR structures. Also, when the CNT diameter is larger than 2 nm, the cross section tends to collapse to a flat shape (leading to PR or FR, depending on ring diameter). Only armchair CNTs were considered in this analysis; in order to check if the nature of these transformations depend of nanotube chirality as well, analogous tests were carried out for rings composed by CNTs of other chiralities, with diameters similar to ring RA ((6,4) and (8,1)) and RI ((11,9) and (12,8)). The observed morphological transformations were the same as before, suggesting that chirality has little or no effect on them, compared to the other investigated parameters (nanotube and ring diameters). The relative energy between the initial (perfect rings) and final (IR, PR, FR) relaxed configurations was also analyzed. The rings RA, RE, RI, RM, and RQ (with same diameter, but built of CNTs of varying diameter) were relaxed back to 0.1 K after the stability test (at 300 K), and the energy was computed. Initial structures were relaxed directly at 0.1 K, and in this case they keep the original ring shape. The difference between these energies (normalized by the number of atoms) was defined as the “rearrangement energy” (RE). A negative value means that the final configuration (either IR, PR or FR) is more stable than the initial one (perfect ring). Fig. 3 shows the variation of this property with nanotube diameter for the selected rings. The ideal rings (RA, RE) showed RE = 0, as expected. The polygonal and flat rings (RI, RM, RQ) displayed a negative RE value (the larger the diameter, the higher the RE), showing that the observed morphological transformations occur as a mechanism for stress relief in the structure, and are expected to occur
spontaneously at finite temperatures when energy barriers are overcome. We also analyzed the possibility of using “structural reinforcements”, aiming to help the PR and FR configurations to maintain their original structure (closer to IR). Hindering of the aforementioned structural transformations would be interesting and useful for certain purposes, such as the formation of rings with uniform stress distribution throughout the structure, which would potentially lead to stronger rings for design of nanochains. A straightforward approach toward this objective would consist in filling the CNT interior with some other material. One possibility was tested, by inserting C60 fullerenes within the nanotube, creating a carbon peapod – like ring structure [6,44]. To prepare the carbon-peapod-like structures, rings RI (made out of (10,10) nanotubes) were filled with a varied number of fullerenes (10, 20, 40 and 50), and subjected to the same structural relaxation procedure used before. The initial and final structures are shown in Fig. 4. Results show that the insertion of fullerenes was partially effective or ineffective in helping to maintain the initial ring-like structure. With the lowest number of fullerenes, the nanotube kept a circular ring-like shape, but exhibiting some localized folds, dividing the structure in pockets evenly distributed, containing one or two nanoparticles each. The fullerenes agglomerated when added in higher amounts, resulting in the formation of intercalated full and empty sub-sections, and the relaxed structures exhibited a PR behavior as observed for the original RI ring. Even with higher amounts (close to a possible saturation) the transformations were observed. These results suggest that this strategy (using filled CNTs) is not suitable to avoid the onset of structural
Fig. 4. Atomic configurations before (top) and after (bottom) structural relaxation of peapod-like CNT rings (built from ring RI, made out a (10,10) CNT) with (a) 10, (b) 20, (c) 40 and (d) 50 C60 fullerenes. 78
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Fig. 5. Atomic structures of CNT-based nanochain NCA (a) before and (b, c) after tensile straining. Zoomed views of the structure under increasing strain is given in (d–f), depicting the stress distribution near the inter-ring contacts and fracturing process. Atoms are colored according to the local atomic von Mises stress (scale is given in the right side).
transformations as a mechanism for stress relief. 3.2. Mechanical properties of CNT ring-based 1D nanochains and 2D nanomeshes Some of the CNT rings studied in the previous section were used to create CNT rings-based nanochains (NC), i.e., a nanoscale equivalent of macroscopic chains made of CNT rings, as illustrated in Figs. 1(b) and 5(a). These were relaxed and then subjected to uniaxial tensile strain tests as described in Section 2, in order to investigate their mechanical behavior. Nanochains based on rings Ri with i = {A, B, C, I, J} were created and labeled as NCi for future reference. This set of rings was chosen to analyze the dependence of mechanical properties on both CNT and ring diameters (and consequently on the ring morphology – IR, PR, FR). As expected from the analysis presented in Section 3.1, the NCs built from stable rings (IRs) were stable as well (NCA, NCB, NCC), and behave as perfect chains at finite temperatures, either in their relaxed state or when subjected to uniaxial strain. Snapshots of the structural evolution of the NCA upon tensile straining are shown in Fig. 5. The structure deforms uniformly at lower strains, inducing the rings to reach an elliptical shape at higher strains (with the main axis aligned with the direction of applied strain). This deformation mechanism is reversible, i.e., the original configuration (with circular ring-like units) is restored upon strain release. This reversibility has also been observed in experimental studies on deformation of individual CNT rings [23]. The atomic stress is stored primarily on the contact between two rings, which is the region on which the fracture starts, as seen in Fig. 5(d–f). Stress-strain curves of NCs exhibiting this behavior are shown in Fig. 6, and computed properties are given in Table 2. The results reported in Table 2 show that there is a decrease on the tensile strength of the material (NCs), compared to the pristine CNTs.
Fig. 6. Stress-strain curves for CNT rings-based nanochains NCA, NCB and NCC.
This reduction on strength was expected, considering that the stress generated by application of strain is mainly stored in the contact between two adjacent chain links as discussed in the previous paragraph and seen in Fig. 5(d–f); when straight CNTs are subjected to tensile strain, stress is evenly distributed throughout the structure, and the C–C bonds are strained uniformly throughout the material. Also, the C–C bonds in the CNT rings are intrinsically strained even before application of any tensile deformation, due to imposition of curvature on the material as described in the previous section, which introduces residual stresses in the structure (also observed by the non-zero stress before the application of strain, as seen in Fig. 6). Despite this reduction on
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Table 2 Mechanical properties computed for several CNT ring-based chains (NCx) and pristine CNTs used in their creation. Structure
1D Ultimate Strength (nN)
3D Fracture Stress (GPa)
Fracture Strain
NCA NCB NCC NCI NCJ (5,5) CNT (10,10) CNT
6.2 4.7 3.5 11.6 16.2 73 150
3.8 2.9 2.2 2.6 3.7 101.0 103.5
0.51 0.48 0.44 0.42 0.54 0.21 0.21
strength, the reported values are still quite high compared to conventional engineering materials (steel, ceramics, polymers, etc.). Interestingly, significantly higher fracture strains are achieved by CNT ringsbased nanochains (up to ∼2.5 times the values exhibited by straight CNTs), due to the deformation mechanism observed for the rings (going from a circular to elliptical shape with the main axis aligned with the direction of applied strain). This high flexibility, associated to a relatively high strength, can be explored in several ways toward practical applications. The use of larger rings built out of the same nanotube (comparing NCA, NCB and NCC, from smaller to larger ring diameter), leads to a decrease on the reported properties, considering that the volume of the material increases, but only a small fraction of the material (practically the same in the three cases) is actively storing stress. Also, the stress-strain curves are smoother for nanochains built of rings of smaller diameter, as observed in Fig. 6. These curves exhibit oscillations as strain increases, as a result of thermal fluctuations on ring shape and orientation, and consequently in the effective contact area between adjacent rings. Rings with larger diameters deform more easily due to their larger flexibility, and consequently more fluctuations are observed, especially at lower applied strains. Only chains made of ideal rings (IR) exhibited this reversible behavior, characterized by reaching significant strains without fracturing, and returning to their original state upon strain removal. A different behavior was observed for chains created with rings with a different morphology (PR, namely NCI and NCJ nanochains), affecting the resulting properties. Details of the structure of these nanochains are depicted in Fig. 7(a) and (b). In NCI (rings have the same diameter of those of NCA, but made of a CNT of larger diameter), the presence of the local folds on the rings inhibits the establishment of a smooth and efficient contact between links as seen in the previous case (Fig. 5). As discussed in Section 3.2, this ring morphology is observed for rings of small diameter made of CNTs with larger diameters. Also, the C–C bonds at these folds are initially more strained compared to the others in the material. These factors lead to a significant reduction on the 3D strength and fracture strain comparing NCA to NCI (variations of 32 and 18% from the highest value), as seen in Table 2. The 1D strength in NCI is higher because there are a larger number of C–C bonds within the CNTs cross-section (as also observed for (5,5) and (10,10) CNTs). The chain NCJ is composed of rings of a larger diameter compared to NCI, which allow a better distribution of the folds in the structure when subjected to tensile strain, as seen in Fig. 7(b). In this case, the rings assume a rectangular-like shape upon straining, and folds become localized at the extremities of the links. A more efficient inter-ring contact is established, compared to the previous case (NCI), resulting in a mechanical strength close to that exhibited by the chains made out of perfect rings (IR), as seen in Table 2. However, the aforementioned deformation is not reversible and the original ring shape is not recovered upon removal of strain, i.e., the rectangular-like shape of the constituent rings persists, differently to the previous case. The same concept used to create the CNT rings-based nanochains can be applied to generate two-dimensional nanostructures that we named nanochainmails, i.e., a nanoscale equivalent of macroscopic
Fig. 7. Atomic structure of CNT-based nanochains (a) NCI and (b) NCJ, illustrating differences on ring morphology and resulting effectiveness of contact between neighboring rings. Atoms are colored according to the local atomic von Mises stress (scale is given at the bottom).
chainmails made of CNT rings, as illustrated in Figs. 1(c) and 8(a). This nanostructure was created using RA rings, by linking five independent NCA chains (ten rings each) parallel to each other by introducing five RA rings perpendicularly arranged between each pair of chains, resulting in a structure with a total of 75 rings. Consequently, each ring is either connected to two or four rings on the structure. This nanostructure was relaxed, and subjected to a biaxial tensile strain test as described in Section 2. Atomic configurations of the material under applied strain are depicted in Fig. 8(b–d). As in the previous case (NCi’s), the rings deform from their characteristic circular ring-like shape upon straining; in some of them, the cross-section evolve into an ellipsoid as well, but others (namely, the ones interconnecting two perpendicular chains) assume a rhombus/diamond-like shape, aligned with the directions of applied biaxial strain. In the latter case, some segments of the CNT rings are straightened, and folds arise at the inter-rings contacts, which concentrate most of the resulting stress as in the CNT-based nanochains. Again, this stress concentration induces the onset of fracture at the contacts. The material possesses a tensile strength of 0.23 N/m (2D)// 1.14 GPa (3D), and a biaxial fracture strain of 0.31. Properties are inferior to those exhibited by single layer graphene sheets (∼42 N/m// 130 GPa [43]), as in the previous case (nanochains) when compared to CNTs, but much higher than those possessed by conventional materials with small thicknesses. The strength is in the same order of magnitude of those exhibited by nanochains, but the fracture strain is lower, considering that some rings are deformed along two directions as explained above. Despite of that, the cross-sectional area of the nanochainmails increased by ∼70% prior to fracture, demonstrating their enhanced flexibility compared to conventional carbon-based nanomaterials. Nanochainmails exhibited an interesting combination of strength, flexibility, porosity and lightness, suitable to be used as highly flexible elastic porous membranes; the mass density (computed using the definition of volume described in Section 2) is of 138 kg/m3, providing a specific strength (ratio of tensile strength to density) of 8.2 × 106 N·m/kg, similarly to other carbon nanomaterials [7,11,24,25].
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Fig. 8. (a–d) Zoomed views of the atomic structure of a CNT-based nanochainmail under increasing strain, depicting the stress distribution and fracturing process. Atoms are colored according to the local atomic von Mises stress (scale is given in frame (d)).
4. Conclusions
nanostructures, and we hope that the present study motivates future efforts toward the synthesis of these materials and their use in miniaturized devices and other practical applications.
We presented a comprehensive computational study on the atomic structure and mechanical properties of 1D and 2D nanomaterials built from carbon nanotube rings, that we called CNT-based nanochains and nanochainmails respectively, because of their structural similarity with macroscopic chains and chainmails. These are characterized by a low density (∼138 kg/m3, in the 2D case), a relatively high mechanical strength, in the range of 1–4 GPa (inferior to those exhibited by straight CNTs, but much superior to those of conventional materials), and an enhanced flexibility (reaching strains as high as ∼0.50 (1D, uniaxial) and 0.30 (2D, biaxial), in an elastic manner). The morphology of some CNT rings depart from the ideal one (characterized by a circular crosssection), because of intrinsic residual stresses imposed by the curvature of the CNT surface. Structural changes occur as a mechanism of stress relief on the material, leading to rings with polygonal cross-sections and/or to the flattening of the CNT cross-section. Rings with larger diameters and/or built from CNTs with smaller diameters are the ones likely to preserve their ideal ring-like morphology. Consequently, the mechanical behavior and properties were found to be dependent as well on both ring and nanotube diameters. Ideal rings provided the strongest structures with reversible transformations on their morphology upon application and removal of tensile strain. This set of properties make these materials suitable for applications on which strong, lightweight, flexible and porous materials are required, such as membranes, reinforced composites, actuators in nanoelectromechanical devices (NEMS), among others. The success in the synthesis and stabilization of CNT rings [15–23] brings good expectations toward realization of such
CRediT authorship contribution statement Julian F.R.V. Silveira: Conceptualization, Methodology, Investigation, Writing - review & editing. Andre R. Muniz: Conceptualization, Methodology, Supervision, Writing - review & editing. Acknowledgements The authors acknowledge the National Laboratory for Scientific Computing (SDumont supercomputer, LNCC/MCTI, Brazil) and Centro Nacional de Supercomputação (CESUP/UFRGS) for providing computational resources for the calculations reported in this paper. J.F.R.V.S. acknowledges CNPQ for a PhD scholarship. A.R.M. acknowledges grant #449824/2014-4 Chamada MCTI/CNPQ/UNIVERSAL 14/2014. References [1] H.W. Kroto, J.R. Heath, S.C. O'Brien, R.F. Curl, R.E. Smalley, C 60: buckminsterfullerene, Nature 318 (6042) (1985) 162–163. [2] S. Iijima, Helical microtubules of graphitic carbon, Nature 354 (6348) (1991) 56–58. [3] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, A.A. Firsov, Electric field effect in atomically thin carbon films, Science 306 (5696) (2004) 666–669. [4] R. Van Noorden, The trials of new carbon, Nature 469 (7328) (2011) 14–16.
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Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
Chain- and chainmail-like nanostructures from carbon nanotube rings Julian F.R.V. Silveira, Andre R. Muniz
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T
Department of Chemical Engineering, Universidade Federal do Rio Grande do Sul, Rua Luiz Englert s/n, 90040-040 Porto Alegre, RS, Brazil
A R T I C LE I N FO
A B S T R A C T
Keywords: Nanomaterials Molecular dynamics Mechanical properties 1D materials Nanomeshes
A computational study on the structure, stability and mechanical properties of carbon nanotube (CNT) rings and derived 1D and 2D superstructures is presented. CNT rings were used as building blocks for CNT-based nanochains and nanochainmails, i.e., nanostructures similar to conventional chains and chainmails. Molecular dynamics simulations show that these structures are stable, mechanically strong and highly flexible, reaching strains as high as 0.50 without breaking. The stability of CNT rings was shown to be dependent on both ring and CNT diameters, affecting directly the mechanical behavior of the CNT-based superstructures. The interesting combination of strength, flexibility and lightness exhibited by these materials enable their use in several potential applications, such as reinforced materials, membranes and actuators in miniaturized devices.
1. Introduction Carbon-based nanostructures have been thoroughly studied over the last years due to their outstanding and unique set of physical properties. They are suitable for several practical applications, especially those that require materials with high mechanical strength and low weight [1–5]. The synthesis of nanostructures with characteristic dimensions (length, area) large enough for certain practical applications, without significant loss in performance, is currently one of the greatest challenges in the field [5]. One approach used to circumvent this problem is combining smaller units of carbon nanostructures to form extended superstructures. There are several studies proposing the creation of such superstructures, through the assembly of fullerenes [6,7], carbon nanotubes, [8,9], graphene [10], diamond nanothreads [11] and even mixtures of them [8,12–14], achieved by the establishment of covalent bonds between smaller units of a given material. These materials can potentially overcome some of the limitations of using graphene and carbon nanotubes in certain applications, and enhance the effective use of their mechanical, electronic and thermal properties, enabling new practical applications [8–14]. The synthesis of rings made out of carbon nanotubes (CNTs), also known as CNT rings, carbon nanorings, nanotube tori, among other similar names, has been reported by several authors [15–23], using different methods. In general, CNTs are coiled during the process, and a connection between their extremities is established, being stabilized by either covalent bonds or intermolecular interactions. The controlled synthesis of such structures opens the possibility of creating large superstructures, by arranging them in such way that these are
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interpenetrated forming the nanoscale analog of a one-dimensional chain, on which each individual CNT ring is a link. Expanding this idea to more dimensions allows us to envision 2D chainmail-like structures and even 3D superstructures. The resulting CNT-based nanostructures (as illustrated in Fig. 1), analogously to macroscopic conventional chains and chainmails (similar to those used as medieval armors), would exhibit a high mechanical strength and a low weight. A potential advantage of creating CNT-based nanochains would be the possibility of obtaining large and continuous structures, maintaining to some degree the remarkable mechanical properties of carbon nanotubes [24,25], also introducing some flexibility [19,23]. Carbon nanotube rings have been subject of several theoretical studies, with emphasis on their structural, electronic and magnetic properties [26–36]. The possibility of using these nanostructures for mechanical applications has not been properly investigated. A previous study [37] evaluated the mechanical strength of a CNT ring-based chain-like structure, describing the material using a single link pulled by a pair of straight CNT segments; promising properties are reported, but a more comprehensive description and understanding of mechanical behavior of such nanochains and accurate evaluation of mechanical properties as function of their characteristic dimensions is still elusive. Therefore, the objective of the present work is to carry out a systematic study on the creation of CNT rings-based 1D nanochains and 2D nanochainmails with varied characteristic dimensions (CNT and ring diameters), evaluating their stability and mechanical behavior under tensile strain using classical molecular dynamics simulations.
Corresponding author. E-mail address: [email protected] (A.R. Muniz).
https://doi.org/10.1016/j.commatsci.2019.01.048 Received 4 October 2018; Received in revised form 25 January 2019; Accepted 28 January 2019 0927-0256/ © 2019 Elsevier B.V. All rights reserved.
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and 2D stresses are reported for nanochains and nanochainmails respectively (as done before for nanotubes [24] and single-layer graphene [43]), along with the conventionally reported 3D stress. The 1D stress is 1 i i reported in force units, calculated by τnm ,1D = L1D Snm , where L1D is the length of the 1D material; consequently, the 1D tensile strength corresponds to the force required to break a single chain. Analogously, the 1 i i 2D stress is computed by τnm ,2D = A2D Snm , where A2D is the surface area Ω of the 2D material. The volume required for the computation of 3D stresses was calculated considering that the cross-sectional area of a single 1D nanochain At , right before the onset of fracture, can be approximated by the cross-sectional area of two nanotubes, such as Ω = L1D ∙At . Consistently, the thickness of the 2D nanomesh planar structure Lt was taken as twice the nanotube diameter, and then Ω = Lt ∙A2D . The diameter of a nanotube was taken as the sum of the average atomic distance between diametrically opposed atoms in the structure, and the Van der Waals distance (0.340 nm), as done in previous works [24]. Fig. 1. Example of the atomic structure of (a) a (5,5) CNT ring (with a zoomed view depicting the curved nanotube structure), and of CNT ring-based nanostructures, namely (b) a nanochain and (c) a nanochainmail.
3. Results 3.1. Structural stability of individual CNT rings
2. Computational methods The atomic structures of individual CNT rings were formed by rolling up carbon nanotubes and joining their extremities through covalent C–C bonds, in order to establish a continuous honeycomb graphene-like lattice, as illustrated in Figs. 1 and 2. Bending of nanotubes leads to C–C bond straining, resulting in internal stresses on the material. These stresses might induce structural transformations on the rings, leading to other configurations than perfect and symmetric rings as desired. The occurrence and extent of these transformations are expected to depend on the nanotube diameter/chirality, as well as of the ring diameter. In order to evaluate the stability of the CNT rings and how the aforementioned factors affect the atomic structure and morphology of carbon rings, we performed structural relaxations as described in Section 2 for rings of varying diameters (from 15.7 to 125.4 nm), using armchair (n,n) CNTs with diameters from 0.67 to 2.68 nm (equivalent to n varying from 5 to 20), as listed in Table 1. For future reference, each ring presented in this table is labeled as Ri, with i designated by capital letters {A, B, C, …, T}. We observed that the initially perfect nanotube rings (as seen in Fig. 1) undergo distinct morphological transformations upon structural relaxation. Some of the structures retained their characteristic ring shape, and were classified as Ideal Rings – “IR”, as illustrated in Fig. 2(a). The other rings relaxed to configurations different from the
The atomic structure and mechanical properties of CNT rings and derived superstructures were computed by classical molecular dynamics (MD) simulations, using the LAMMPS package [38]. The interatomic interactions were described by the AIREBO potential [39], widely used for studying carbon-based nanostructures, yielding reliable results for mechanical properties [24,25,40]. The C–C cutoff distance in the potential was set to 2.0 Å to avoid spurious high forces at high bond strains, as suggested in previous studies [41]. A timestep of 1 fs was used in the integration of the equations of motion, and temperature and pressure control (the latter necessary when relaxing supercell dimensions) were carried out using the Berendsen thermostat and barostat [42]. Stability tests were conducted for individual rings, consisting of a structural relaxation at 0.1 K for 200 ps, followed by gradual heating up to 300 K, and then thermal equilibration at this temperature (for at least 200 ps). Uniaxial and biaxial tensile strain tests were respectively carried out for 1D nanochains and 2D nanochainmails, using periodic boundary conditions and supercells with 8–10 and 48–75 rings for 1D and 2D structures respectively (containing from 32,000 to 256,000 atoms). The systems were initially subjected to structural relaxation at 0.1 K for 200 ps, and then to a thermal equilibration to 300 K (gradual heating followed by at least 200 ps at final temperature). Finally, a constant strain rate of 0.001 ps−1 was applied in the supercell (along either one or two perpendicular directions), until fracture is observed. Stress-strain curves are obtained by computing the atomic stress along the trajectories, using the virial equation 1 1 1 i i τnm = Ω ⎡−mi vmi vni + 2 ∑j ≠ i rnij Fmij⎤ = Ω Snm , where τni , m is the n, m ⎣ ⎦ component of the atomic stress tensor for atom i , Ω is the volume, mi is i the atomic mass, vk is the k component of the atom’s velocity vector, rkij and Fkij are the k components of the distance and the force (vectors) i between atoms i and j , for each neighbor atom j , and Snm is an auxiliary tensor component (to be used in other definitions later in the text, computed directly in LAMMPS via compute stress/atom). The tensile strength and fracture strain are obtained at the maximum of the curve (at the onset of fracture). Preliminary convergence studies (dependence of results with respect to number of rings on a supercell, strain rate, etc.) were carried out to ensure the quality of reported results. The definition of volume of 1D and 2D structures, required to compute atomic stress in MD simulations, is not straightforward. The interlayer spacing of layers characteristic of graphite (0.335–0.340 nm) has been typically used to define the thickness of the surfaces required to compute stresses in single layer graphene sheets and carbon nanotubes [24,25,40]. In order to present a consistent set of results, both 1D
Fig. 2. Atomic structures of different nanotube rings before (top) and after (bottom) structural relaxation, illustrating the three patterns of ring morphology observed upon structural relaxation: (a) IR (ring RA), (b) PR (ring RM), (c) FR (ring RQ). 77
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Table 1 Structural parameters of CNT rings, indices i = {A, B, …, T} used to designate each Ri ring configuration, and ring morphology observed upon structural relaxation (IR – ideal rings, PR – polygonal rings and FR – flat rings, as described in the text). Nanotube diameter D (nm) and chirality
Ring diameter d (nm) 15.67
31.34
62.68
125.36
0.67 – (5,5)
A IR
B IR
C IR
D IR
0.94 – (7,7)
E IR
F IR
G IR
H IR
1.34 – (10,10)
I PR
J PR
K IR
L IR
2.01 – (15,15)
M PR
N PR
O PR
P PR
2.68 – (20,20)
Q FR
R FR
S PR
T PR
Fig. 3. Rearrangement energy for rings RA, RE, RI, RM, and RQ (all with D = 15.67 nm and varying nanotube diameter).
initial, which can be classified in two groups. In some rings, the nanotube cross-section went from a circular to a flatter shape in some points, inducing folds on the structure, but maintaining the regular nanotube structure in finite sub-domains, as illustrated in Fig. 2(b). The introduction of these folds result in a sort of polygonal structure, not as round as a perfect ring; these were then labeled as Polygonal Rings – “PR”. Other rings relaxed to what we called Flat Rings – “FR”; in this case, the nanotube undergoes a transformation similar to the PR, but the nanotube cross-section relax to a flat shape along the whole ring extension, as depicted in Fig. 2(c). The ring maintains partially its original shape, exhibiting some local folds as in the PRs. Similar structural features exhibited by relaxed CNT rings have been reported by other authors [30–32]. The classification of the resulting structures observed upon structural relaxation for all investigated rings Ri are summarized in Table 1. In general, rings formed by nanotubes of small diameter (< 1 nm) have a tendency to maintain the original ring-like shape (IR) regardless the ring diameter. Rings composed of thicker nanotubes need to be significantly large to maintain the original ring shape, otherwise they relax toward PR structures. Also, when the CNT diameter is larger than 2 nm, the cross section tends to collapse to a flat shape (leading to PR or FR, depending on ring diameter). Only armchair CNTs were considered in this analysis; in order to check if the nature of these transformations depend of nanotube chirality as well, analogous tests were carried out for rings composed by CNTs of other chiralities, with diameters similar to ring RA ((6,4) and (8,1)) and RI ((11,9) and (12,8)). The observed morphological transformations were the same as before, suggesting that chirality has little or no effect on them, compared to the other investigated parameters (nanotube and ring diameters). The relative energy between the initial (perfect rings) and final (IR, PR, FR) relaxed configurations was also analyzed. The rings RA, RE, RI, RM, and RQ (with same diameter, but built of CNTs of varying diameter) were relaxed back to 0.1 K after the stability test (at 300 K), and the energy was computed. Initial structures were relaxed directly at 0.1 K, and in this case they keep the original ring shape. The difference between these energies (normalized by the number of atoms) was defined as the “rearrangement energy” (RE). A negative value means that the final configuration (either IR, PR or FR) is more stable than the initial one (perfect ring). Fig. 3 shows the variation of this property with nanotube diameter for the selected rings. The ideal rings (RA, RE) showed RE = 0, as expected. The polygonal and flat rings (RI, RM, RQ) displayed a negative RE value (the larger the diameter, the higher the RE), showing that the observed morphological transformations occur as a mechanism for stress relief in the structure, and are expected to occur
spontaneously at finite temperatures when energy barriers are overcome. We also analyzed the possibility of using “structural reinforcements”, aiming to help the PR and FR configurations to maintain their original structure (closer to IR). Hindering of the aforementioned structural transformations would be interesting and useful for certain purposes, such as the formation of rings with uniform stress distribution throughout the structure, which would potentially lead to stronger rings for design of nanochains. A straightforward approach toward this objective would consist in filling the CNT interior with some other material. One possibility was tested, by inserting C60 fullerenes within the nanotube, creating a carbon peapod – like ring structure [6,44]. To prepare the carbon-peapod-like structures, rings RI (made out of (10,10) nanotubes) were filled with a varied number of fullerenes (10, 20, 40 and 50), and subjected to the same structural relaxation procedure used before. The initial and final structures are shown in Fig. 4. Results show that the insertion of fullerenes was partially effective or ineffective in helping to maintain the initial ring-like structure. With the lowest number of fullerenes, the nanotube kept a circular ring-like shape, but exhibiting some localized folds, dividing the structure in pockets evenly distributed, containing one or two nanoparticles each. The fullerenes agglomerated when added in higher amounts, resulting in the formation of intercalated full and empty sub-sections, and the relaxed structures exhibited a PR behavior as observed for the original RI ring. Even with higher amounts (close to a possible saturation) the transformations were observed. These results suggest that this strategy (using filled CNTs) is not suitable to avoid the onset of structural
Fig. 4. Atomic configurations before (top) and after (bottom) structural relaxation of peapod-like CNT rings (built from ring RI, made out a (10,10) CNT) with (a) 10, (b) 20, (c) 40 and (d) 50 C60 fullerenes. 78
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Fig. 5. Atomic structures of CNT-based nanochain NCA (a) before and (b, c) after tensile straining. Zoomed views of the structure under increasing strain is given in (d–f), depicting the stress distribution near the inter-ring contacts and fracturing process. Atoms are colored according to the local atomic von Mises stress (scale is given in the right side).
transformations as a mechanism for stress relief. 3.2. Mechanical properties of CNT ring-based 1D nanochains and 2D nanomeshes Some of the CNT rings studied in the previous section were used to create CNT rings-based nanochains (NC), i.e., a nanoscale equivalent of macroscopic chains made of CNT rings, as illustrated in Figs. 1(b) and 5(a). These were relaxed and then subjected to uniaxial tensile strain tests as described in Section 2, in order to investigate their mechanical behavior. Nanochains based on rings Ri with i = {A, B, C, I, J} were created and labeled as NCi for future reference. This set of rings was chosen to analyze the dependence of mechanical properties on both CNT and ring diameters (and consequently on the ring morphology – IR, PR, FR). As expected from the analysis presented in Section 3.1, the NCs built from stable rings (IRs) were stable as well (NCA, NCB, NCC), and behave as perfect chains at finite temperatures, either in their relaxed state or when subjected to uniaxial strain. Snapshots of the structural evolution of the NCA upon tensile straining are shown in Fig. 5. The structure deforms uniformly at lower strains, inducing the rings to reach an elliptical shape at higher strains (with the main axis aligned with the direction of applied strain). This deformation mechanism is reversible, i.e., the original configuration (with circular ring-like units) is restored upon strain release. This reversibility has also been observed in experimental studies on deformation of individual CNT rings [23]. The atomic stress is stored primarily on the contact between two rings, which is the region on which the fracture starts, as seen in Fig. 5(d–f). Stress-strain curves of NCs exhibiting this behavior are shown in Fig. 6, and computed properties are given in Table 2. The results reported in Table 2 show that there is a decrease on the tensile strength of the material (NCs), compared to the pristine CNTs.
Fig. 6. Stress-strain curves for CNT rings-based nanochains NCA, NCB and NCC.
This reduction on strength was expected, considering that the stress generated by application of strain is mainly stored in the contact between two adjacent chain links as discussed in the previous paragraph and seen in Fig. 5(d–f); when straight CNTs are subjected to tensile strain, stress is evenly distributed throughout the structure, and the C–C bonds are strained uniformly throughout the material. Also, the C–C bonds in the CNT rings are intrinsically strained even before application of any tensile deformation, due to imposition of curvature on the material as described in the previous section, which introduces residual stresses in the structure (also observed by the non-zero stress before the application of strain, as seen in Fig. 6). Despite this reduction on
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Table 2 Mechanical properties computed for several CNT ring-based chains (NCx) and pristine CNTs used in their creation. Structure
1D Ultimate Strength (nN)
3D Fracture Stress (GPa)
Fracture Strain
NCA NCB NCC NCI NCJ (5,5) CNT (10,10) CNT
6.2 4.7 3.5 11.6 16.2 73 150
3.8 2.9 2.2 2.6 3.7 101.0 103.5
0.51 0.48 0.44 0.42 0.54 0.21 0.21
strength, the reported values are still quite high compared to conventional engineering materials (steel, ceramics, polymers, etc.). Interestingly, significantly higher fracture strains are achieved by CNT ringsbased nanochains (up to ∼2.5 times the values exhibited by straight CNTs), due to the deformation mechanism observed for the rings (going from a circular to elliptical shape with the main axis aligned with the direction of applied strain). This high flexibility, associated to a relatively high strength, can be explored in several ways toward practical applications. The use of larger rings built out of the same nanotube (comparing NCA, NCB and NCC, from smaller to larger ring diameter), leads to a decrease on the reported properties, considering that the volume of the material increases, but only a small fraction of the material (practically the same in the three cases) is actively storing stress. Also, the stress-strain curves are smoother for nanochains built of rings of smaller diameter, as observed in Fig. 6. These curves exhibit oscillations as strain increases, as a result of thermal fluctuations on ring shape and orientation, and consequently in the effective contact area between adjacent rings. Rings with larger diameters deform more easily due to their larger flexibility, and consequently more fluctuations are observed, especially at lower applied strains. Only chains made of ideal rings (IR) exhibited this reversible behavior, characterized by reaching significant strains without fracturing, and returning to their original state upon strain removal. A different behavior was observed for chains created with rings with a different morphology (PR, namely NCI and NCJ nanochains), affecting the resulting properties. Details of the structure of these nanochains are depicted in Fig. 7(a) and (b). In NCI (rings have the same diameter of those of NCA, but made of a CNT of larger diameter), the presence of the local folds on the rings inhibits the establishment of a smooth and efficient contact between links as seen in the previous case (Fig. 5). As discussed in Section 3.2, this ring morphology is observed for rings of small diameter made of CNTs with larger diameters. Also, the C–C bonds at these folds are initially more strained compared to the others in the material. These factors lead to a significant reduction on the 3D strength and fracture strain comparing NCA to NCI (variations of 32 and 18% from the highest value), as seen in Table 2. The 1D strength in NCI is higher because there are a larger number of C–C bonds within the CNTs cross-section (as also observed for (5,5) and (10,10) CNTs). The chain NCJ is composed of rings of a larger diameter compared to NCI, which allow a better distribution of the folds in the structure when subjected to tensile strain, as seen in Fig. 7(b). In this case, the rings assume a rectangular-like shape upon straining, and folds become localized at the extremities of the links. A more efficient inter-ring contact is established, compared to the previous case (NCI), resulting in a mechanical strength close to that exhibited by the chains made out of perfect rings (IR), as seen in Table 2. However, the aforementioned deformation is not reversible and the original ring shape is not recovered upon removal of strain, i.e., the rectangular-like shape of the constituent rings persists, differently to the previous case. The same concept used to create the CNT rings-based nanochains can be applied to generate two-dimensional nanostructures that we named nanochainmails, i.e., a nanoscale equivalent of macroscopic
Fig. 7. Atomic structure of CNT-based nanochains (a) NCI and (b) NCJ, illustrating differences on ring morphology and resulting effectiveness of contact between neighboring rings. Atoms are colored according to the local atomic von Mises stress (scale is given at the bottom).
chainmails made of CNT rings, as illustrated in Figs. 1(c) and 8(a). This nanostructure was created using RA rings, by linking five independent NCA chains (ten rings each) parallel to each other by introducing five RA rings perpendicularly arranged between each pair of chains, resulting in a structure with a total of 75 rings. Consequently, each ring is either connected to two or four rings on the structure. This nanostructure was relaxed, and subjected to a biaxial tensile strain test as described in Section 2. Atomic configurations of the material under applied strain are depicted in Fig. 8(b–d). As in the previous case (NCi’s), the rings deform from their characteristic circular ring-like shape upon straining; in some of them, the cross-section evolve into an ellipsoid as well, but others (namely, the ones interconnecting two perpendicular chains) assume a rhombus/diamond-like shape, aligned with the directions of applied biaxial strain. In the latter case, some segments of the CNT rings are straightened, and folds arise at the inter-rings contacts, which concentrate most of the resulting stress as in the CNT-based nanochains. Again, this stress concentration induces the onset of fracture at the contacts. The material possesses a tensile strength of 0.23 N/m (2D)// 1.14 GPa (3D), and a biaxial fracture strain of 0.31. Properties are inferior to those exhibited by single layer graphene sheets (∼42 N/m// 130 GPa [43]), as in the previous case (nanochains) when compared to CNTs, but much higher than those possessed by conventional materials with small thicknesses. The strength is in the same order of magnitude of those exhibited by nanochains, but the fracture strain is lower, considering that some rings are deformed along two directions as explained above. Despite of that, the cross-sectional area of the nanochainmails increased by ∼70% prior to fracture, demonstrating their enhanced flexibility compared to conventional carbon-based nanomaterials. Nanochainmails exhibited an interesting combination of strength, flexibility, porosity and lightness, suitable to be used as highly flexible elastic porous membranes; the mass density (computed using the definition of volume described in Section 2) is of 138 kg/m3, providing a specific strength (ratio of tensile strength to density) of 8.2 × 106 N·m/kg, similarly to other carbon nanomaterials [7,11,24,25].
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Fig. 8. (a–d) Zoomed views of the atomic structure of a CNT-based nanochainmail under increasing strain, depicting the stress distribution and fracturing process. Atoms are colored according to the local atomic von Mises stress (scale is given in frame (d)).
4. Conclusions
nanostructures, and we hope that the present study motivates future efforts toward the synthesis of these materials and their use in miniaturized devices and other practical applications.
We presented a comprehensive computational study on the atomic structure and mechanical properties of 1D and 2D nanomaterials built from carbon nanotube rings, that we called CNT-based nanochains and nanochainmails respectively, because of their structural similarity with macroscopic chains and chainmails. These are characterized by a low density (∼138 kg/m3, in the 2D case), a relatively high mechanical strength, in the range of 1–4 GPa (inferior to those exhibited by straight CNTs, but much superior to those of conventional materials), and an enhanced flexibility (reaching strains as high as ∼0.50 (1D, uniaxial) and 0.30 (2D, biaxial), in an elastic manner). The morphology of some CNT rings depart from the ideal one (characterized by a circular crosssection), because of intrinsic residual stresses imposed by the curvature of the CNT surface. Structural changes occur as a mechanism of stress relief on the material, leading to rings with polygonal cross-sections and/or to the flattening of the CNT cross-section. Rings with larger diameters and/or built from CNTs with smaller diameters are the ones likely to preserve their ideal ring-like morphology. Consequently, the mechanical behavior and properties were found to be dependent as well on both ring and nanotube diameters. Ideal rings provided the strongest structures with reversible transformations on their morphology upon application and removal of tensile strain. This set of properties make these materials suitable for applications on which strong, lightweight, flexible and porous materials are required, such as membranes, reinforced composites, actuators in nanoelectromechanical devices (NEMS), among others. The success in the synthesis and stabilization of CNT rings [15–23] brings good expectations toward realization of such
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