Elastic Deformation of Carbon‐Nanotube Nanorings

Zheng, Meng; Ke, Changhong

doi:10.1002/smll.201000337

Cited by 34 publications

(26 citation statements)

References 73 publications

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“…[2][3][4] we consider a surface S with parametric equations r = r(q 1 , q 2 ). The portion of the 3D space in the immediate neighborhood of S can be then parametrized as R(q 1 , q 2 ) = r(q 1 , q 2 ) + q 3N (q 1 , q 2 ) withN (q 1 , q 2 ) the unit vector normal to S. We then find, in agreement with previous studies [2][3][4], the relations among G ij and the covariant components of the 2D surface metric tensor g ij to be…”

Section: Pacs Numbersmentioning

confidence: 99%

“…with α indicating the Weingarten curvature tensor of the surface S [2,4]. We recall that the mean curvature M and the Gaussian curvature K of the surface S are related to the Weingarten curvature tensor by…”

Section: Pacs Numbersmentioning

confidence: 99%

“…Now we can apply the thin-layer procedure introduced by Da Costa [2] and take into account the effect of a confining potential V λ (q 3 ), where λ is a squeezing parameter that controls the strength of the confining potential. When λ is large, the total wavefunction will be localized in a narrow range close to q 3 = 0.…”

Section: Pacs Numbersmentioning

confidence: 99%

“…The second term in the square brackets corresponds to Da Costa's curvature induced scalar potential [2].…”

Section: Pacs Numbersmentioning

confidence: 99%

“…The current theoretical paradigm relies on a thin wall quantization method introduced by Da Costa [2]. In Ref.…”

mentioning

confidence: 99%

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Curvature-induced geometric potential in strain-driven nanostructures

et al. 2011

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The experimental progress in synthesizing low dimensional nanostructures where carriers are confined to bent surfaces has boosted the interest in the theory of quantum mechanics on curved two-dimensional manifolds. The current theoretical paradigm relies on a thin wall quantization method introduced by Da Costa [2]. In Ref.[3] Jensen and Dandoloff find that when employing the thin wall quantization in presence of externally applied electric and magnetic fields, an unphysical orbital magnetic moment appears. This anomalous result is taken as evidence for the breakdown of the thin wall quantization. We show, however, that in a proper treatment within the Da Costa framework: (i) this anomalous contribution is absent and (ii) there is no coupling between external electromagnetic field and surface curvature [1]. PACS numbers:To derive the thin wall quantization in presence of external electromagnetic field we start with the Schrödinger equation that is minimally coupled with the four component vector potential in a generic curved threedimensional space. Adopting Einstein summation convention and tensor covariant and contravariant components, we havewhere Q is the particle charge, G ij are the contravariant components of the metric tensor and A i are the covariant components of the vector potential A with the scalar potential defined by V = −A 0 . The covariant derivative D i is as usual defined bywhere v j are the covariant components of a vector field v and Γ k ij are the Christoffel symbolsFrom Eq. (1) one obtainswhere for convenience we introduced the time gauge covariant derivative D 0 = ∂ t − iQA 0 / . To proceed further, it is useful to define a coordinate system. As in Ref.[2-4] we consider a surface S with parametric equations r = r(q 1 , q 2 ). The portion of the 3D space in the immediate neighborhood of S can be then parametrized as R(q 1 , q 2 ) = r(q 1 , q 2 ) + q 3N (q 1 , q 2 ) withN (q 1 , q 2 ) the unit vector normal to S. We then find, in agreement with previous studies [2][3][4], the relations among G ij and the covariant components of the 2D surface metric tensor g ij to bewith α indicating the Weingarten curvature tensor of the surface S [2, 4]. We recall that the mean curvature M and the Gaussian curvature K of the surface S are related to the Weingarten curvature tensor byNow we can apply the thin-layer procedure introduced by Da Costa [2] and take into account the effect of a confining potential V λ (q 3 ), where λ is a squeezing parameter that controls the strength of the confining potential. When λ is large, the total wavefunction will be localized in a narrow range close to q 3 = 0. This allows one to take the q 3 → 0 limit in the Schrödinger equation. For the sake of clarity, it is better to introduce the indices a, b which can assume the values 1, 2 alone. From the structure of the metric tensor, one finds the following limiting relations:With this, the Schrödinger equation Eq. (2) can be then recast aswhere H S is an effective 2D Hamiltonian that depends on the surface S alone and readsTh...

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Section: Pacs Numbersmentioning

confidence: 99%

Section: Pacs Numbersmentioning

confidence: 99%

Section: Pacs Numbersmentioning

confidence: 99%

“…The second term in the square brackets corresponds to Da Costa's curvature induced scalar potential [2].…”

Section: Pacs Numbersmentioning

confidence: 99%

“…The current theoretical paradigm relies on a thin wall quantization method introduced by Da Costa [2]. In Ref.…”

mentioning

confidence: 99%

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Curvature-induced geometric potential in strain-driven nanostructures

et al. 2011

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The Interesting Influence of Nanosprings on the Viscoelasticity of Elastomeric Polymer Materials: Simulation and Experiment

Tian

et al. 2012

Adv Funct Materials

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Among all carbon nanostructured materials, helical nanosprings or nanocoils have attracted particular interest as a result of their special mechanical behavior. Here, carbon nanosprings are used to adjust the viscoelasticity and reduce the resulting hysteresis loss (HL) of elastomeric polymer materials. Two types of nanospring‐filled elastomer composites are constructed as follows: system I is obtained by directly blending polymer chains with nanosprings; system II is composed of the self‐assembly of a tri‐block structure such as chain‐nanospring‐chain. Coarse‐grained molecular dynamics simulations show that the incorporation of nanosprings can improve the mechanical strength of the elastomer matrix through nanoreinforcement and considerably decrease the hysteresis loss. This finding is significant for reducing fuel consumption and improving fuel efficiency in the automobile tire industry. Furthermore, it is revealed that the spring constant of nanosprings and the interfacial chemical coupling between chains and nanosprings both play crucial roles in adjusting the viscoelasticity of elastomers. It is inferred that elastomer/carbon nanostructured materials with good flexibility and reversible mechanical response (carbon nanosprings, nanocoils, nanorings, and thin graphene sheets) have both excellent mechanical and low HL properties; this may open a new avenue for fabrication of high performance automobile tires and facilitate the large‐scale industrial application of these materials.

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Effects of Nanostructure and Coating on the Mechanics of Carbon Nanotube Arrays

Poelma

Fan

et al. 2016

Adv Funct Materials

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Nanoscale materials are one of the few engineering materials that can be grown from the bottom up in a controlled manner. Here, the effects of nanostructure and nanoscale conformal coating on the mechanical behavior of vertically aligned carbon nanotube (CNT) arrays through experiments and simulation are systematically investigated. A modeling approach is developed and used to quantify the compressive strength and modulus of the CNT array under large deformation. The model accounts for the porous nanostructure, which contains multiple CNTs with random waviness, van der Waals interactions, fracture strain, contacts, and frictional forces. CNT array micropillars are grown and their porous nanostructure is controlled by the infiltration and deposition of thin conformal coatings using chemical vapor deposition. Flat‐punch nanoindentation experiments reveal significant changes in material properties as a function of coating thickness. The simulations explain the experimental results and show the novel failure transition regime that changes from collective CNT buckling toward structural collapse due to fracture. The compressive strength and the elastic modulus increase exponentially as a function of the coating thickness and demonstrate a unique dependency on the CNT waviness. More interestingly, a design rule is identified that predicts the optimum coating thickness for porous materials.

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Elastic Deformation of Carbon‐Nanotube Nanorings

Cited by 34 publications

References 73 publications

Curvature-induced geometric potential in strain-driven nanostructures

Curvature-induced geometric potential in strain-driven nanostructures

The Interesting Influence of Nanosprings on the Viscoelasticity of Elastomeric Polymer Materials: Simulation and Experiment

Effects of Nanostructure and Coating on the Mechanics of Carbon Nanotube Arrays

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