Carbon-Nanotube Geometries as Optimal Configurations

Mainini, Edoardo; Murakawa, Hideki; Piovano, Paolo; Stefanelli, Ulisse

doi:10.1137/16m1087862

Cited by 11 publications

(24 citation statements)

References 40 publications

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“…Nanotubes with µ = µ us will be referred to as unstretched nanotubes. The aim of [55,56] was to prove that G α us is a local minimizer. This has been illustrated numerically in [55] and checked analytically in [56], for a restricted class of perturbations.…”

Section: Resultsmentioning

confidence: 99%

“…The other classical choice, namely the so-called armchair topology, could be considered as well. The reader is referred to [56] for some results on unstretched armchair geometries. e 1 Figure 2.…”

Section: 2mentioning

confidence: 99%

“…This work is the culmination of a series on the geometry and mechanics of nanotubes [55,56]. The theoretical outcomes of this paper have been predicted computationally in [55], where stability of periodic configurations have been investigated with Monte Carlo techniques, both for zigzag and armchair topologies under moderate displacements.…”

Section: Introductionmentioning

confidence: 99%

“…The theoretical outcomes of this paper have been predicted computationally in [55], where stability of periodic configurations have been investigated with Monte Carlo techniques, both for zigzag and armchair topologies under moderate displacements. A first step toward a rigorous analytical result has been obtained in [56] for both zigzag and armchair topologies under no stretching. In [56], stability is checked against a number of non-periodic perturbations fulfilling a specific structural constraint, which is related to the nonplanarity of the hexagonal cells induced by the local geometry of the nanotube.…”

Section: Introductionmentioning

confidence: 99%

“…A first step toward a rigorous analytical result has been obtained in [56] for both zigzag and armchair topologies under no stretching. In [56], stability is checked against a number of non-periodic perturbations fulfilling a specific structural constraint, which is related to the nonplanarity of the hexagonal cells induced by the local geometry of the nanotube. Here, we remove such constraint and consider all small perturbations, even in presence of stretching.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Characterization of Optimal Carbon Nanotubes Under Stretching and Validation of the Cauchy–Born Rule

Friedrich¹,

Mainini²,

Piovano³

et al. 2018

Arch Rational Mech Anal

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Carbon nanotubes are modeled as point configurations and investigated by minimizing configurational energies including two-and three-body interactions. Optimal configurations are identified with local minima and their fine geometry is fully characterized in terms of lower-dimensional problems. Under moderate tension, we prove the existence of periodic local minimizers, which indeed validates the so-called Cauchy-Born rule in this setting.2010 Mathematics Subject Classification. Primary: 82D25.

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Section: Resultsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Characterization of Optimal Carbon Nanotubes Under Stretching and Validation of the Cauchy–Born Rule

Friedrich¹,

Mainini²,

Piovano³

et al. 2018

Arch Rational Mech Anal

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Graphene ground states

Friedrich¹,

Stefanelli²

2018

Z. Angew. Math. Phys.

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Graphene is locally two-dimensional but not flat. Nanoscale ripples appear in suspended samples and rolling-up often occurs when boundaries are not fixed. We address this variety of graphene geometries by classifying all ground-state deformations of the hexagonal lattice with respect to configurational energies including two-and three-body terms. As a consequence, we prove that all ground-state deformations are either periodic in one direction, as in the case of ripples, or rolled up, as in the case of nanotubes.2010 Mathematics Subject Classification. 70F45, 82D80.

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Ripples in Graphene: A Variational Approach

Friedrich

Stefanelli

2020

Commun. Math. Phys.

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Suspended graphene samples are observed to be gently rippled rather than being flat. In Friedrich et al. (Z Angew Math Phys 69:70, 2018), we have checked that this nonplanarity can be rigorously described within the classical molecular-mechanical frame of configurational-energy minimization. There, we have identified all ground-state configurations with graphene topology with respect to classes of next-to-nearest neighbor interaction energies and classified their fine nonflat geometries. In this second paper on graphene nonflatness, we refine the analysis further and prove the emergence of wave patterning. Moving within the frame of Friedrich et al. (2018), rippling formation in graphene is reduced to a two-dimensional problem for one-dimensional chains. Specifically, we show that almost minimizers of the configurational energy develop waves with specific wavelength, independently of the size of the sample. This corresponds remarkably to experiments and simulations.

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Carbon-Nanotube Geometries as Optimal Configurations

Cited by 11 publications

References 40 publications

Characterization of Optimal Carbon Nanotubes Under Stretching and Validation of the Cauchy–Born Rule

Characterization of Optimal Carbon Nanotubes Under Stretching and Validation of the Cauchy–Born Rule

Graphene ground states

Ripples in Graphene: A Variational Approach

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