Thermal stability of idealized folded carbyne loops
© Cranford; licensee Springer. 2013
Received: 4 October 2013
Accepted: 7 November 2013
Published: 20 November 2013
Self-unfolding items provide a practical convenience, wherein ring-like frames are contorted into a state of equilibrium and subsequently pop up’ or deploy when perturbed from a folded structure. Can the same process be exploited at the molecular scale? At the limiting scale is a closed chain of single atoms, used here to investigate the limits of stability of such folded ring structures via full atomistic molecular dynamics. Carbyne is a one-dimensional carbon allotrope composed of sp-hybridized carbon atoms. Here, we explore the stability of idealized carbyne loops as a function of chain length, curvature, and temperature, and delineate an effective phase diagram between folded and unfolded states. We find that while overall curvature is reduced, in addition to torsional and self-adhesive energy barriers, a local increase in curvature results in the largest impedance to unfolding.
KeywordsCarbyne Molecular dynamics Unfolding Adhesion Torsion Curvature Stability
A clever trick by product designers is self-unfolding structures such as collapsible laundry hampers and pop-up’ tents. These ingenious designs involve a continuous ring structure that unfolds’ to a larger configuration. Similar mechanisms have been proposed for systems ranging from stretchable electronics  to polymer membranes [2, 3] and hollow shell structures . Here, we focus on the smallest possible unfolding system - a closed chain of carbon atoms - to investigate the limits of stability at the atomistic scale. Insights from such structures can then be applied to more complex macromolecular systems, such as responsive polymer [5, 6] or protein-based materials [7–10].
In simplest terms, additional elastic strain energy due to curvature triggers unfolding from the three-loop configuration. However, to completely unfold from an initial coiled state at the molecular scale, both torsional and self-adhesive energetic barriers must be overcome, resulting in a range of stable conditions, depending on initial curvature (κ) and temperature (T). In terms of coiled, looped, or otherwise convoluted geometries (such as knotted molecular structures [24, 25]), a mono-atomistic linear structure provides the simplest and most fundamental platform to explore stability and unfolding. Other systems, such as convoluted protein structures or DNA, would be more complex to analyze (due to kinetic hindrance of side-chain interactions, for example), but similar looped structures exist [26–28] and are also dictated by a balance of thermal and mechanical contributions [29–31].
While linear carbon chains have been experimentally attained, such a closed carbyne has yet to be synthesized. However, recent developments of carbon materials such as annulenes [32–34] and extended porphyrins  suggest that carbon may allow such atomistic control’ and design of such molecular structures. Similar folded/looped atomistic structures include molecular knots [36, 37], foldamers [38, 39], and cyclic heterostructures [39–42]. The use of homogeneous carbon eliminates the effects of more complex structures (such as torsional rigidity or steric interactions). However, while carbyne is used here as an idealized model system, the general behavior can serve as an analog to such systems and reflect the dynamics at a molecular scale.
Full atomistic simulations are implemented using classical MD, utilizing the first-principle-based ReaxFF potential [43, 44], known to provide an accurate account of the chemical/mechanical behavior of carbon nanostructures [21, 45–49]. Due to a bond order-based formulation, ReaxFF can reflect the bond hybridization of the polyyne structure of carbyne, as well as the effect of other valence terms (angle and torsion), without explicit parameterization . It is noted that at such a scale, electron behavior may play a critical role. For example, a previous study demonstrated that in linear carbon chains, a local perturbation through the displacement of a single atom creates atomic force and charge density Friedel-like oscillations . Other electron-dependent effects may include Jahn-Teller distortions  or Möbius topologies [52, 53]. While such complex behavior is incapable of being replicated by MD potentials, it is deemed sufficient for the current scope of length and temperature effects on unfolding. A time step is chosen to be on the order of a fraction of femtoseconds (0.1 × 10-15 s) to ensure the stability and reflect the high vibrational frequency of the acetylene groups of carbyne. All simulations are subject to a canonical (NVT) ensemble, with varying prescribed temperature (10 to 800 K), performed using the massively paralyzed modeling code LAMMPS (http://lammps.sandia.gov/) .
In terms of the ring structure, while linear carbyne chains have been shown to be stable [19, 58], imposing a closed-loop geometry may be energetically unfavorable. To directly assess the stability of looped carbyne here, a linear chain was equilibrated to determine the difference in atomistic energy in comparison with the 54-atom looped structure, resulting in a nominal difference of 0.02 eV atom-1 and suggesting structural stability. For comparison, the energy difference between flat graphene and a fullerene is in the order of 0.2 eV atom-1, while the cohesive energy of carbyne has been found to be in the order of 6.99  to 8.19 eV atom-1, in close agreement with the value of 7.4 eV atom-1 calculated here at a finite temperature of 300 K.
We also wish to assess the stability in comparison with other non-carbyne molecular configurations. Empirically, similar ring-like structures with as few as 20 carbon atoms have been observed in the synthesis of fullerenes , as well as many intermediate bonded chain forms (e.g., so-called bow tie structures or cycloadducts) [60–63]. To explore whether such intermediate forms may be energetically favorable, simple trial structures were equilibrated to assess the potential energy (also depicted in Figure 2), indicating that the looped/ring structure is more favorable than other intermediate forms. As depicted in Figure 2, the most energetically favorable structures are the fullerenes, with rings more favorable than that of the intermediate’ structures (in agreement with previous studies ; absolute relative energy differences would be dependent on the MD potential used). Increasing the bending energy (through folding) could increase the energy such that a transition may occur. That being said, the modeled structure not being the most energetically favorable does not imply that it cannot exist. Such an argument would indicate that fullerenes themselves should not exist, yet C20 fullerenes, bowls, and rings have been observed . Less favorable intermediate structures are proposed pathways to fullerene synthesis [62, 63] and can exhibit interesting properties or result in the synthesis of unique structures . The focus here only involves the stability of a presumed folded structure.
The looped structures are equilibrated at a nominal temperature (10 K) and then subjected to temperature increase to a target temperature (with a rate of approximately 0.001 K fs-1) over 10 ps. As the molecular structures are isolated in a vacuum, the use of temperature as a variable is a direct measure of the kinetic energy of the atoms, independent of any insulating or damping effects an explicit solvent may contribute. Once the target temperature is reached, constant temperature is maintained, and the system is allowed to freely evolve for up to 0.1 ns to assess the stability of the configuration (test trials up to 5.0 ns were also ran to ensure equilibrium; in all cases, if unfolding was initiated, it occurred at a timescale less than 0.1 ns). The critical temperature of unfolding is then determined for each structure. Since the process is stochastic across the chain and the temperature is an ensemble average, the designated unfolding temperature only approximates the magnitude of energy required to trigger unfolding, and thus a range of critical temperatures emerges for the structures across multiple simulations. While the temperature variation was used to induce unfolding, of note is that the carbyne chains do not begin to disassociate until temperatures exceed approximately 3,500 K regardless of size (and a loss of any definitive curvature), defining an accessible temperature range for the ring structures.
Results and discussion
Root mean square deviation
Adhesion and torsional barriers
A recent macroscale investigation has determined that the way these rings behave depends on a single characteristic known as overcurvature  or how much more curved the three-loop configuration is than a flat circle of the same circumference. Here, each structure has the same initial overcurvature (equal to three). However, at the molecular scale, where temperature and self-adhesion effects are on the same energetic scale as strain energy, the relationship between curvature and stability is more complex. Indeed, due to the imposed overcurvature of the three-loop conformation, it could be anticipated that a relaxation of bending strain energy results in the necessary energy to unfold, assuming that the energy is sufficient to overcome the energy barrier due to adhesion and/or torsion (a full twist/rotation is necessary to unfold a looped chain).
For torsion, involving a complete rotation of the carbyne chain about itself, the associated energy barrier would be a function of the initial curvature. A simple five-atom chain was constructed with a set of 14 initial curvatures ranging from 0.016 to 0.395 Å-1 and subjected to incremental twist while tracking the potential energy (representative plots are given in Figure 5a). During the simulation, one terminal atom is fixed, along with the second-to-the-last atom at the opposite end, while the adjacent terminal atom is then rotated about an axis of rotation and constant curvature maintained. The maximum energy barrier was calculated to be approximately 10 kcal mol-1, exhibited at large curvatures (>0.1 Å-1). A recent study quantified the torsional stiffness of carbyne, albeit using ab initio methods, a straight chain configuration, and the rotation of end-groups . The reported energy barrier due to torsion ranged from approximately 0.2 to 0.6 eV, or 5 to 14 kcal mol-1. While the simulation approach and boundary conditions were different, the energy barrier determined here (approximately 10 kcal mol-1) is in the same order of magnitude and in a relatively good agreement. For adhesion, three carbyne chains were brought into contact and incrementally separated to determine the interchain adhesion energy (Figure 5b) of approximately 0.5 kcal mol-1 atom-1. For the worst case scenario (the longest chain of 180 carbons resulting in three adhered 60 carbon rings plus the highest recorded torsional barrier), we calculate a maximum energy barrier of approximately 40 kcal mol-1 - smaller than all but the minimum (n = 54) required energy increase indicated by the unfolding structures (also note that n = 54 unfolds with nominal kinetic energy required, at approximately T ≈ 10 K, representing the smallest possible stable three-loop structure). This indicates an additional contribution that must be overcome to induce unfolding, and we hence turn to the analysis of curvature.
Global and local curvature analysis
Critical unfolding temperatures
where P0 is considered the temperature-independent persistence length. In effect, the apparent bending rigidity increases with temperature, also supported by previous theoretical results; a recent ab initio (temperature-free) investigation reports the bending stiffness to be in the order of 5.3(10-2) nN-nm2, while a finite temperature (300 K) molecular dynamics study reports a stiffness of approximately 13 to 20(10-2) nN-nm2. Here, D0 is the rigidity at zero temperature (as carbyne is not ideally flexible) and thus is approximated as 5.3(10-2) nN-nm2.
Using the fitted slope of 4.2 ± 0.85 K Å-1, the energy barrier to unfolding, Ω, is determined to be approximately 98 to 366 kcal mol-1 (best fit with Ω = 139 kcal mol-1), which agrees well with the magnitude of measured energy barriers (40 to 400 kcal mol-1). This range may be seemingly large as the energy of cohesion for the chains is in the order of 7 eV or approximately 160 kcal mol-1; one may expect the chains to break before unfolding. However, the barrier is due to the bending strain energy required, which, by definition, requires the involvement of numerous atoms (rather than a single cleavage site , for example). In consideration of the relatively large flexural rigidity of carbyne, such bending energy barriers can be quite significant. If we consider the change in curvature for n = 72, from approximately 0.27 Å-1 to local peaks of 0.5 Å-1, then we can approximate the length that undergoes the local increase in curvature by equating the energy barrier, Ω, with the local bending strain energy. For n = 72 at 200 K (for a bending rigidity of D200K = 10.4 nN-nm2 by Equation 5), this results in local curvature increase in approximately 7.4 to 27.5 Å of the loop. This range of length is in good agreement with the size/span of the peaks depicted in Figure 8. Similarly, considering n = 144 at 725 K (for a bending rigidity of D725K = 24.0 nN-nm2), with curvature increases from 0.11 Å-1 to local peaks of 0.3 Å-1, results in local curvature increasing in approximately 7.2 Å to 27.2 Å to develop the determined energy barrier, again in good agreement with Figure 8, which indicated multiple (but short spanning) peaks across the molecular length. It is noted that there is an intrinsic relationship between the magnitude of local curvature and necessary length, i.e., a longer length can develop the equivalent energy barrier with a smaller curvature as Ub ∝ Lк2.
The results confirm that, while global unfolding implies an overall reduction in curvature, continuity of the molecular loop results in local increases in curvature, resulting in a small yet finite energy barrier to surpass. For longer loops (with less stored bending strain energy due to a decrease in curvature), a higher temperature (e.g., kinetic energy) is required to induce unfolding. In contrast, short loops (with high bending energies) unfold at relatively low temperatures. Using carbyne as a platform, the potential for folding can serve to extend the accessible design space of such materials. It is noted that the heterogeneous/local curvature as depicted in the snapshots in Figure 3, as well as plotted in Figure 7, was not explicitly considered in terms of energy contribution. Rather, the limiting cases - the curvature of the three-loop structure and the curvature of an unfolded ring - were used to estimate the necessary energy. Here, all structures begin in an ideal configuration, and the deviations from the ideal curvatures are due to thermal fluctuations; the thermal energy (essentially molecular kinetic energy) must impose overcurvature to trigger the unfolding process. Since the heterogeneous curvatures are stochastic (the results plotted are only representative), temperature is used as a proxy to evaluate the necessary energy to unfold.
It behooves us to note that the looped carbyne structure modeled herein is not attainable experimentally and is intended as an ideal model platform to explore the unfolding phenomena. A similar idealized bead-spring-type’ model could have been constructed but would be subject to the arbitrariness of parameterization. Carbyne provides a compromise - an ideal structure with physical, fundamental, and proven molecular-scale parameterization/behavior through the ReaxFF potential. It is the simplest case from a molecular perspective (a non-reactive homogeneous chain, no solvent, etc.) and is necessary to isolate and observe the thermal contribution to unfolding as well as the local curvature effect. Indeed, understanding the stability and mechanics of folded carbyne loops can be of use in modifying transport properties or triggering mechanisms in active molecular systems.
Finally, understanding the mechanical principles of such geometries will help not only in dissecting existing looped systems but also in designing and constructing commercially useful self-assembling nanostructures. For example, the local curvature increase may be isolated in a particular, flexible molecular hinge’ or activated by an enzyme in biological systems. When one thinks of folding/unfolding at the molecular scale, DNA and similarly protein structures are likely to come to mind. In terms of insights to such structures, the governing folding/unfolding phenomenon is quite different from carbyne loops. However, there are insights even from this simple system; DNA can exhibit looped configurations, which can serve to suppress the formation of gene products, or facilitate compaction of DNA as a whole [26–31, 76, 77]. The size of the loops also affects the mechanical stability [26–28] and has been analyzed via elastic assumptions  and thermodynamic cost . Similar to the carbyne system here, larger loops are shown to be more stable. The observation that local curvature undergoes an increase may shed light into the attainment of such structures. Indeed, for small DNA looped structures to be stable, extensive local curvature is required (which can be potentially controlled by sequence; see  and references therein). While at a different scale, clearly there is an interplay between curvature, local flexibility, and temperature similar to that of the structures observed here. There are no direct insights from carbyne to macromolecules such as DNA, just as the general study of overcurvature in collapsible laundry baskets was not applied at the molecular scale here. But there are indeed potential indirect corollaries.
While carbon chains have been primarily studied as extensions from graphene  or carbon nanotubes [79, 80], isolated carbynes and related structures may inspire an even smaller generation of nanomaterials, with increased functionality due to their intrinsic flexibility and ability to attain exotic topologies. Development of looped systems may lead to novel devices that unfold’ per design with some external event - a potential novel nanoscale trigger - motivated by commercial pop-up tents and collapsible laundry hampers.
S.W.C. acknowledges the generous support from NEU's CEE Department. The calculations and the analysis were carried out using a parallel LINUX cluster at NEU's Laboratory for Nanotechnology In Civil Engineering (NICE).
- Sun YG, Choi WM, Jiang HQ, Huang YGY, Rogers JA: Controlled buckling of semiconductor nanoribbons for stretchable electronics. Nat Nanotechnol 2006, 1: 201–207. 10.1038/nnano.2006.131View ArticleGoogle Scholar
- Klein Y, Efrati E, Sharon E: Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 2007, 315: 1116–1120. 10.1126/science.1135994View ArticleGoogle Scholar
- Kim J, Hanna JA, Byun M, Santangelo CD, Hayward RC: Designing responsive buckled surfaces by halftone gel lithography. Science 2012, 335: 1201–1205. 10.1126/science.1215309View ArticleGoogle Scholar
- Shim J, Perdigou C, Chen ER, Bertoldi K, Reis PM: Buckling-induced encapsulation of structured elastic shells under pressure. Proc Natl Acad Sci USA 2012, 109: 5978–5983. 10.1073/pnas.1115674109View ArticleGoogle Scholar
- Guan JJ, He HY, Lee LJ, Hansford DJ: Fabrication of particulate reservoir-containing, capsulelike, and self-folding polymer microstructures for drug delivery. Small 2007, 3: 412–418. 10.1002/smll.200600240View ArticleGoogle Scholar
- Ionov L: Soft microorigami: self-folding polymer films. Soft Matter 2011, 7: 6786–6791. 10.1039/c1sm05476gView ArticleGoogle Scholar
- Stepanskiy LG: Sonication-induced unfolding proteins. J Theor Biol 2012, 298: 77–81.View ArticleGoogle Scholar
- Neidigh JW, Fesinmeyer RM, Andersen NH: Designing a 20-residue protein. Nat Struct Biol 2002, 9: 425–430. 10.1038/nsb798View ArticleGoogle Scholar
- Sulkowska JI, Noel JK, Onuchic JN: Energy landscape of knotted protein folding. Proc Natl Acad Sci USA 2012, 109: 17783–17788. 10.1073/pnas.1201804109View ArticleGoogle Scholar
- Dean FB, Stasiak A, Koller T, Cozzarelli NR: Duplex DNA knots produced by Escherichia coli topoisomerase I - structure and requirements for formation. J Biol Chem 1985, 260: 4975–4983.Google Scholar
- Kavan L, Kastner J: Carbyne forms of carbon: continuation of the story. Carbon 1994, 32: 1533–1536. 10.1016/0008-6223(94)90149-XView ArticleGoogle Scholar
- Chalifoux WA, Ferguson MJ, McDonald R, Melin F, Echegoyen L, Tykwinski RR: Adamantyl-endcapped polyynes. J Phys Org Chem 2012, 25: 69–76. 10.1002/poc.1874View ArticleGoogle Scholar
- Lin ZZ, Ning XJ: Controlling the electronic properties of monatomic carbon chains. Epl-Europhys Lett 2011, 95: 47012. 10.1209/0295-5075/95/47012View ArticleGoogle Scholar
- Khoo KH, Neaton JB, Son YW, Cohen ML, Louie SG: Negative differential resistance in carbon atomic wire-carbon nanotube junctions. Nano Lett 2008, 8: 2900–2905. 10.1021/nl8017143View ArticleGoogle Scholar
- Tykwinski RR, Chalifoux W, Eisler S, Lucotti A, Tommasini M, Fazzi D, Del Zoppo M, Zerbi G: Toward carbyne: synthesis and stability of really long polyynes. Pure Appl Chem 2010, 82: 891–904. 10.1351/PAC-CON-09-09-04View ArticleGoogle Scholar
- Gibtner T, Hampel F, Gisselbrecht JP, Hirsch A: End-cap stabilized oligoynes: model compounds for the linear sp carbon allotrope carbyne. Chem-Eur J 2002, 8: 408–432. 10.1002/1521-3765(20020118)8:2<408::AID-CHEM408>3.0.CO;2-LView ArticleGoogle Scholar
- Cataldo F: A method for synthesizing polyynes in solution. Carbon 2005, 43: 2792–2800. 10.1016/j.carbon.2005.05.024View ArticleGoogle Scholar
- Eisler S, Slepkov AD, Elliott E, Luu T, McDonald R, Hegmann FA, Tykwinski RR: Polyynes as a model for carbyne: synthesis, physical properties, and nonlinear optical response. J Am Chem Soc 2005, 127: 2666–2676. 10.1021/ja044526lView ArticleGoogle Scholar
- Chalifoux WA, Tykwinski RR: Synthesis of polyynes to model the sp-carbon allotrope carbyne. Nat Chem 2010, 2: 967–971. 10.1038/nchem.828View ArticleGoogle Scholar
- Itzhaki L, Altus E, Basch H, Hoz S: Harder than diamond: determining the cross-sectional area and Young's modulus of molecular rods. Angew Chem Int Edit 2005, 44: 7432–7435. 10.1002/anie.200502448View ArticleGoogle Scholar
- Nair AK, Cranford SW, Buehler MJ: The minimal nanowire: mechanical properties of carbyne. Epl-Europhys Lett 2011, 95: 16002. 10.1209/0295-5075/95/16002View ArticleGoogle Scholar
- Hu YH: Bending effect of sp-hybridized carbon (carbyne) chains on their structures and properties. The Journal of Physical Chemistry C 2011, 115: 1843–1850. 10.1021/jp111851uView ArticleGoogle Scholar
- Castelli IE, Salvestrini P, Manini N: Mechanical properties of carbynes investigated by ab initio total-energy calculations. Phys Rev B 2012, 85: 214110.View ArticleGoogle Scholar
- Dobrowolski JC, Mazurek AP: Model carbyne knots vs ideal knots. J Chem Inf Comp Sci 2003, 43: 861–869. 10.1021/ci020063wView ArticleGoogle Scholar
- Deng WY, Qiu WY: Helical chirality in model mirror-imaged carbyne trefoil knots. J Mol Struct 2008, 875: 515–519. 10.1016/j.molstruc.2007.05.035View ArticleGoogle Scholar
- Becker NA, Kahn JD, Maher LJ: Bacterial repression loops require enhanced DNA flexibility. J Mol Biol 2005, 349: 716–730. 10.1016/j.jmb.2005.04.035View ArticleGoogle Scholar
- Yuann JMP, Tseng WH, Lin HY, Hou MH: The effects of loop size on Sac7d-hairpin DNA interactions. Bba-Proteins Proteom 2012, 1824: 1009–1015. 10.1016/j.bbapap.2012.05.011View ArticleGoogle Scholar
- Vafabakhsh R, Ha T: Extreme bendability of DNA less than 100 base pairs long revealed by single-molecule cyclization. Science 2012, 337: 1097–1101. 10.1126/science.1224139View ArticleGoogle Scholar
- Cherstvy AG: Looping charged elastic rods: applications to protein-induced DNA loop formation. Eur Biophys J Biophy 2011, 40: 69–80. 10.1007/s00249-010-0628-5View ArticleGoogle Scholar
- Levene SD, Giovan SM, Hanke A, Shoura MJ: The thermodynamics of DNA loop formation, from J to Z. Biochem Soc T 2013, 41: 513–518. 10.1042/BST20120324View ArticleGoogle Scholar
- Olson WK, Grosner MA, Czapla L, Swigon D: Structural insights into the role of architectural proteins in DNA looping deduced from computer simulations. Biochem Soc T 2013, 41: 559–564. 10.1042/BST20120341View ArticleGoogle Scholar
- Spitler EL, Johnson CA, Haley MM: Renaissance of annulene chemistry. Chem Rev 2006, 106: 5344–5386. 10.1021/cr050541cView ArticleGoogle Scholar
- Stevenson CD: Annulenylenes, annulynes, and annulenes. Accounts Chem Res 2007, 40: 703–711. 10.1021/ar600067nView ArticleGoogle Scholar
- Castro C, Karney WL: Mechanisms and Mobius strips: understanding dynamic processes in annulenes. J Phys Org Chem 2012, 25: 612–619. 10.1002/poc.2904View ArticleGoogle Scholar
- Saito S, Osuka A: Expanded porphyrins: intriguing structures, electronic properties, and reactivities. Angew Chem Int Edit 2011, 50: 4342–4373. 10.1002/anie.201003909View ArticleGoogle Scholar
- Lukin O, Vogtle F: Knotting and threading of molecules: chemistry and chirality of molecular knots and their assemblies. Angew Chem Int Edit 2005, 44: 1456–1477. 10.1002/anie.200460312View ArticleGoogle Scholar
- Andrae D: Molecular knots, links, and fabrics: prediction of existence and suggestion of a synthetic route. New J Chem 2006, 30: 873–882. 10.1039/b601895eView ArticleGoogle Scholar
- Ghosh K, Moore JS: Foldamer structuring by covalently bound macromolecules. J Am Chem Soc 2011, 133: 19650–19652. 10.1021/ja2087163View ArticleGoogle Scholar
- Yamato K, Kline M, Gong B: Cavity-containing, backbone-rigidified foldamers and macrocycles. Chem Commun 2012, 48: 12142–12158. 10.1039/c2cc36391gView ArticleGoogle Scholar
- Fu HL, Liu Y, Zeng HQ: Shape-persistent H-bonded macrocyclic aromatic pentamers. Chem Commun 2013, 49: 4127–4144. 10.1039/c2cc36698cView ArticleGoogle Scholar
- Sisco SW, Moore JS: Directional cyclooligomers via alkyne metathesis. J Am Chem Soc 2012, 134: 9114–9117. 10.1021/ja303572kView ArticleGoogle Scholar
- Hoger S: Shape-persistent phenylene-acetylene macrocycles: large rings-low yield? Angew Chem Int Edit 2005, 44: 3806–3808. 10.1002/anie.200500681View ArticleGoogle Scholar
- Chenoweth K, Van Duin ACT, Goddard WA: ReaxFF reactive force field for molecular dynamics simulations of hydrocarbon oxidation. J Phys Chem A 2008, 112: 1040–1053. 10.1021/jp709896wView ArticleGoogle Scholar
- Strachan A, Kober EM, Van Duin ACT, Oxgaard J, Goddard WA: Thermal decomposition of RDX from reactive molecular dynamics. J Chem Phys 2005, 122: 054502. 10.1063/1.1831277View ArticleGoogle Scholar
- Van Duin ACT, Dasgupta S, Lorant F, Goddard WA: ReaxFF: a reactive force field for hydrocarbons. J Phys Chem A 2001, 105: 9396–9409. 10.1021/jp004368uView ArticleGoogle Scholar
- Nielson KD, Van Duin ACT, Oxgaard J, Deng WQ, Goddard WA: Development of the ReaxFF reactive force field for describing transition metal catalyzed reactions, with application to the initial stages of the catalytic formation of carbon nanotubes. J Phys Chem A 2005, 109: 493–499. 10.1021/jp046244dView ArticleGoogle Scholar
- Chen N, Lusk MT, VanDuin ACT, Goddard WA: Mechanical properties of connected carbon nanorings via molecular dynamics simulation. Phys Rev B 2005, 72: 085416.View ArticleGoogle Scholar
- Buehler MJ: Mesoscale modeling of mechanics of carbon nanotubes: self-assembly, self-folding, and fracture. J Mater Res 2006, 21: 2855–2869. 10.1557/jmr.2006.0347View ArticleGoogle Scholar
- Cranford SW, Buehler MJ: Mechanical properties of graphyne. Carbon 2011, 49: 4111–4121. 10.1016/j.carbon.2011.05.024View ArticleGoogle Scholar
- Cahangirov S, Topsakal M, Ciraci S: Long-range interactions in carbon atomic chains. Phys Rev B 2010, 82: 195444.View ArticleGoogle Scholar
- Kato T, Yoshizawa K, Yamabe T: Vibronic coupling and Jahn-Teller effects in negatively charged annulene. Chem Phys 1999, 247: 375–386. 10.1016/S0301-0104(99)00226-8View ArticleGoogle Scholar
- Rzepa HS: Mobius aromaticity and delocalization. Chem Rev 2005, 105: 3697–3715. 10.1021/cr030092lView ArticleGoogle Scholar
- Herges R: Topology in chemistry: designing Mobius molecules. Chem Rev 2006, 106: 4820–4842. 10.1021/cr0505425View ArticleGoogle Scholar
- Plimpton S: Fast parallel algorithms for short-range molecular-dynamics. J Comput Phys 1995, 117: 1–19. 10.1006/jcph.1995.1039View ArticleGoogle Scholar
- Kertesz M, Koller J, Azman A: Ab initio Hartree-Fock crystal orbital studies. 2. Energy-bands of an infinite carbon chain. J Chem Phys 1978, 68: 2779–2782. 10.1063/1.436070View ArticleGoogle Scholar
- Liu M, Artyukhov VI, Lee H, Xu F, Yakobson BI: Carbyne from first principles: chain of C atoms, a nanorod or a nanorope. Acs Nano 2013. doi:10.1021/nn404177r doi:10.1021/nn404177rGoogle Scholar
- Artyukhov VI, Liu M, Yakobson BI: Mechanically induced metal-insulator transition in carbyne. arXiv: 1302–7250.
- Lin ZZ, Yu WF, Wang Y, Ning XJ: Predicting the stability of nanodevices. Epl-Europhys Lett 2011, 94: 40002. 10.1209/0295-5075/94/40002View ArticleGoogle Scholar
- Chuvilin A, Kaiser U, Bichoutskaia E, Besley NA, Khlobystov AN: Direct transformation of graphene to fullerene. Nat Chem 2010, 2: 450–453. 10.1038/nchem.644View ArticleGoogle Scholar
- Prinzbach H, Weller A, Landenberger P, Wahl F, Worth J, Scott LT, Gelmont M, Olevano D, Von Issendorff B: Gas-phase production and photoelectron spectroscopy of the smallest fullerene, C-20. Nature 2000, 407: 60–63. 10.1038/35024037View ArticleGoogle Scholar
- Allison C, Beran KA: Energetic analysis of 24 C-20 isomers. J Mol Struc-Theochem 2004, 680: 59–63. 10.1016/j.theochem.2004.04.042View ArticleGoogle Scholar
- Goroff NS: Mechanism of fullerene formation. Accounts Chem Res 1996, 29: 77–83. 10.1021/ar950162dView ArticleGoogle Scholar
- Strout DL, Scuseria GE: A cycloaddition model for fullerene formation. J Phys Chem-Us 1996, 100: 6492–6498. 10.1021/jp9530212View ArticleGoogle Scholar
- Kawasumi K, Zhang Q, Segawa Y, Scott LT, Itami K: A grossly warped nanographene and the consequences of multiple odd-membered-ring defects. Nat Chem 2013, 5: 739–744. 10.1038/nchem.1704View ArticleGoogle Scholar
- Grossfield A, Zuckerman DM: Quantifying uncertainty and sampling quality in biomolecular simulations. Ann Rep Comp Chem 2009, 5: 23–48.View ArticleGoogle Scholar
- Lazaridis T, Karplus M: “New view” of protein folding reconciled with the old through multiple unfolding simulations. Science 1997, 278: 1928–1931. 10.1126/science.278.5345.1928View ArticleGoogle Scholar
- Thielges MC, Fayer MD: Protein dynamics studied with ultrafast two-dimensional infrared vibrational echo spectroscopy. Accounts Chem Res 2012, 45: 1866–1874. 10.1021/ar200275kView ArticleGoogle Scholar
- Mouthuy P-O, Coulombier M, Pardoen T, Raskin J-P, Jonas AM: Overcurvature describes the buckling and folding of rings from curved origami to foldable tents. Nat Commun 2012, 3: 1290.View ArticleGoogle Scholar
- Rutter JW: Geometry of Curves. Boca Raton: Chapman & Hall; 2000.Google Scholar
- Landau LD, Lifshitz EM: Theory of Elasticity. 2nd English edn. Oxford: Pergamon Press; 1970.Google Scholar
- Grosberg AIU, Khokhlov AR: Statistical Physics of Macromolecules. New York: AIP Press; 1994.Google Scholar
- Yamakawa H: Modern Theory of Polymer Solutions. New York: Harper & Row; 1971.Google Scholar
- Hagerman PJ: Flexibility of DNA. Annu Rev Biophys Bio 1988, 17: 265–286. 10.1146/annurev.bb.17.060188.001405View ArticleGoogle Scholar
- Brinkers S, Dietrich HRC, De Groote FH, Young IT, Rieger B: The persistence length of double stranded DNA determined using dark field tethered particle motion. J Chem Phys 2009, 130: 215105. 10.1063/1.3142699View ArticleGoogle Scholar
- Moras G, Pastewka L, Walter M, Schnagl J, Gumbsch P, Moseler M: Progressive shortening of sp-hybridized carbon chains through oxygen-induced cleavage. J Phys Chem C 2011, 115: 24653–24661. 10.1021/jp209198gView ArticleGoogle Scholar
- Semsey S, Virnik K, Adhya S: A gamut of loops: meandering DNA. Trends Biochem Sci 2005, 30: 334–341. 10.1016/j.tibs.2005.04.009View ArticleGoogle Scholar
- Zhang Y, McEwen AE, Crothers DM, Levene SD: Statistical-mechanical theory of DNA looping. Biophys J 2006, 90: 1903–1912. 10.1529/biophysj.105.070490View ArticleGoogle Scholar
- Castelli IE, Ferri N, Onida G, Manini N: Carbon sp chains in graphene nanoholes. J Phys-Condens Mat 2012, 24: 104019. 10.1088/0953-8984/24/10/104019View ArticleGoogle Scholar
- Xu B, Lin JY, Lim SH, Feng YP: Structural and electronic properties of finite carbon chains encapsulated into carbon nanotubes. J Phys Chem C 2009, 113: 21314–21318. 10.1021/jp906980yView ArticleGoogle Scholar
- Zhao XL, Ando Y, Liu Y, Jinno M, Suzuki T: Carbon nanowire made of a long linear carbon chain inserted inside a multiwalled carbon nanotube. Phys Rev Lett 2003, 90: 187401.View ArticleGoogle Scholar
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