Continuum mechanics modeling and simulation of carbon nanotubes

Continuum mechanics modelling and simulation of carbon nanotubes. Marino Arroyo Laboratori de C`alcul Num`eric (LaC`aN), Departament de Matem`atica Aplicada III, Universitat Polit`ecnica de Catalunya, E-08034 Barcelona, Spain Ted Belytschko Department of Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA July 5, 2005 Abstract. The understanding of the mechanics of atomistic systems greatly benefits from continuum mechanics. One appealing approach aims at deductively constructing continuum theories starting from models of the interatomic interactions. This viewpoint has become extremely popular with the quasicontinuum method. The application of these ideas to carbon nanotubes presents a peculiarity with respect to usual crystalline materials: their structure relies on a two-dimensional curved lattice. This renders the cornerstone of crystal elasticity, the Cauchy-Born rule, insufficient to describe the effect of curvature. We discuss the application of a theory which corrects this deficiency to the mechanics of carbon nanotubes. We review recent developments of this theory, which include the study of the convergence characteristics of the proposed continuum models to the parent atomistic models, as well as large scale simulations based on this theory. The latter have unveiled the complex nonlinear elastic response of thick multiwalled carbon nanotubes, with an anomalous elastic regime following an almost absent harmonic range. Keywords: Continuum mechanics, carbon nanotubes, finite elasticity, atomistic models, nanotube-based devices 1. Introduction Carbon nanotubes (CNTs) can be viewed as graphene sheets (a twodimensional hexagonal lattice of carbon atoms) wrapped into cylinders one atom thick of specific chirality and diameter. Although irradiation or the synthesis process can produce defects into the crystal structure, these nanostructures are remarkably perfect. Since their discovery in 1991 [18], carbon nanotubes have attracted much attention due to their unique structure and mechanical, chemical and electronic properties [11]. These properties have made of carbon nanotubes the central element in an array of nano-structured materials, as well as nano-sensors and devices. Mechanics plays a central role in nanotube-based nanotechnology. On the one hand, the exceptional mechanical properties (a nominal Young’s modulus of 1 TPa, ideal strength of over 100 MPa) Arroyo, M. and Belytschko, T., Continuum mechanics modeling and simulation of carbon nanotubes, Meccanica, Vol. 40, Issues 4-6, pp. 455-469, 2005 2Marino Arroyo and Ted Belytschko ultrasonic bath, the conditions of our experiments appear to lead to much greater bending angles. In order to reproduce the HREM images and to analyze bent structures in atomistic detail, we have performed computer simulations of bending of single and double-walled nanotubes. The interaction between the carbon atoms was modeled by the many-body potential due to Tersoff,13 as parameterized by Brenner.14 This potential is short-ranged, with a cut-off ,3 Å, and the total energy of the tube is written as a sum over the atomic sites in the local form Eb5( iEi,~1! where the difference between Eiand the cohesive energy per atom of a flat, infinite graphene sheet14 can be interpreted as an excess ‘‘strain energy’’ localized at the site idue to the local curvature of the bent tube. Each term Eidepends on the pairwise distance of the neighbors relative to the atom iand on the angles enclosed by the nearest neighbor triplets that include this atom. The parameters of this potential have been fitted to the experimental atomic densities and binding energies of the bulk phases of carbon, i.e., diamond and graphite,14 and have been used to study both the energetics15 and the growth properties16,17 of isolated nanotubes. Tersofftype potentials have also been used to study the deformation in nanotube bundles.18 In addition to the Tersoff-Brenner interaction, we have also included an additional van der Waals interaction19 between the adjacent shells of a multi-walled tube and between the top and the bottom sides of the same shell when they are in close proximity, e.g., just above a kink. The simulation for a tube of a given length, diameter and helicity was started by first relaxing the tube in the straight ~horizontal!geometry. The bending angle was increased in small steps and the tube was relaxed at each position by the method of conjugate gradients. The presence of the bending force was simulated by keeping the end atoms fixed for each configuration. Up to 3000 atoms were included in the simulations. We have carried out bending simulations with the above setup for a large number of single-walled tubes of varying diameters and helicities. In all simulations the same generic features appear: for bending below a sharp critical curvature, the tube undergoes a simple compression on the inner ~bending!side, and a stretching on the outer side. This is the harmonic regime in which Hooke’s law is obeyed, and the energy increases quadratically as a function of the bending angle, see Fig. 3. In the harmonic regime, the hexagonal rings on the tube surface are only slightly more strained than FIG. 1. HREM images of kink structures formed in nanotubes under mechanical duress. ~a!,~b!Single kinks in the middle of single-walled nanotubes with diameters of 0.8 nm and 1.2 nm, respectively. The gap between the tip of the kink and the upper wall is about 0.4 nm in ~b!.~c!,~d!A multi-walled tube ~diameter ;8nm!showing a single kink and a two-kink complex, respectively. The two kinks in ~d!are separated by about 3.5 nm. FIG. 2. Atomic structure of a single kink obtained in the computer simulation of bending of the single-walled tube in Fig. 2 ~diameter ;1.2 nm!. The coding denotes the local strain energy at the various atoms, measured relative to a relaxed atom in an infinite graphene sheet ~Ref. 14!. The strain energy scale ranges from 0 to 1.2 eV/atom, from left to right. 2090 Iijima et al. : Structural flexibility of carbon nanotubes J. Chem. Phys., Vol. 104, No. 5, 1 February 1996 Downloaded¬20¬Jul¬2005¬to¬147.83.70.202.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://jcp.aip.org/jcp/copyright.jsp Iijima et al. (1996) Figure 1. Buckling at the nanoscale: experimental observation, atomistic calculation (Reprinted with permission from Iijima, S., C. Brabec, A. Maiti, and J. Bernholc: 1996, Structural flexibility of carbon nanotubes. Journal of Chemical Physics 104(5), 2089-2092. Copyright 1996, American Institute of Physics.) and exponential Cauchy-Born rule-based continuum calculation has prompted intense research in nanostructured materials such as nanotube-based nanocomposites. On the other hand, the strong function of other properties such as electrical conductance [37] or chemical reactivity [33] with deformation has deep implications in the design of nanotube-based devices and sensors. From a mechanics viewpoint, the strong crystalline network together with the hollow cylindrical geometry confers carbon nanotubes with features of strong crystalline solids as well as features of flexible macromolecules, with a very rich nonlinear elasticity. Indeed, similarly to bulk crystalline solids, deformation and failure is mediated by plasticity and fracture when the in-plane behavior of graphene is probed. Theoretical studies suggest that strain relaxation in deformed nanotubes can occur through plastic activity–a Stone-Wales defect acts as a dislocation dipole that splits into two dislocations that can glide [26]. More importantly, fracture of nanotubes subject to tension has been reported experimentally [40], and theoretical work has highlighted the importance of preexisting defects in the fracture behavior of nanotubes [25]. The mechanical failure of nanotubes bears particular relevance with regards to nanostructured materials that try to exploit the strength of carbon nanotubes. On the other hand, the three-dimensional film structure makes of nanotubes extremely flexible nanostructures; they can sustain very large distortions in a reversible manner [13, 37]. Transmission electron micrographs as well as atomistic simulations show that when severely deformed, carbon nanotubes tend to buckle displaying morphologies remarkably reminiscent of macroscopic buckled shells [19], Fig. 1. This flexibility and resilience offers opportunities for devices and sensors. The combination of super-high stiffness and strength and high flexibility makes of nanotubes unique mechanical systems. Since the pioneering work of Yakobson and co-workers [39], who first explicitly acknowledged the analogy between the mechanics of carbon Continuum mechanics of carbon nanotubes 3 nanotubes and the continuum elasticity of shells, the use of continuum mechanics concepts in nanotube mechanics has become common practice. Continuum mechanics rationalizes the mechanics of materials in terms of irreducible sets of parameters (e.g. elastic moduli, Peierls stress), is amenable to analytical calculations, and affords very efficient simulations (e.g. finite elements). The most classic approach for continuum modelling of atomistic systems adopts a standard continuum theory (e.g. linear elasticity), and adjusts the material parameters to the data available. This phenomenological approach is simple and useful, particularly when the mechanics are well understood a priori, and the phenomena are simple enough to be encoded in the model. This approach has enjoyed great popularity for nanotubes, which have been modelled as thin elastic shells [32, 39], classical elastic beams [30], and non-local elastic beams that intend to capture the wave dispersion behavior of nanotubes [38] to name a few . While this type of methods have been extended to the finite deformation regime [27], the reduced number of fitted parameters raises doubts on the appropriateness of these models when material nonlinearities are important. This point is further developed in the paper. Another approach to continuum modelling of crystalline materials aims at deductively constructing continuum theories starting from models of the interatomic interactions as realistic as possible, with a minimal set of assumptions [34, 35]. In the present work, we will follow this path. The paper is organized as follows. We first briefly outline the bottomup approach to crystal elasticity based on the Cauchy-Born hypothesis, and the extension proposed to capture the curvature of lattices of reduced dimensionality. The simple yet nontrivial example of an atomic chain deforming in 2D is developed in detailed, and for this model, an asymptotic and numerical study of the relationship between continuum and atomistic models is reported. Then, examples of the application of the theory to the simulation of carbon nanotubes are presented, which include large scale simulations which help understand the complex mechanics of thick multi-walled carbon nanotubes (MWCNTs). Finally the conclusions and outlook are summarized. 2. Continuum modelling of curved lattices 2.1. The Cauchy-Born hypothesis The essential idea behind the present theory, as in classical finite crystal elasticity, is to use the Cauchy-Born rule [12] (here a generalized CB rule) to express the geometric quantities characterizing the deformation of the lattice in terms of the deformation of the continuum. For 4Marino Arroyo and Ted Belytschko ϕ F Aa Figure 2. The standard Cauchy-Born rule for space-filling crystals: the undeformed body is subjected to a deformation map ϕ, whose linear part, the deformation gradient Ftransforms locally the underlying lattice space-filling crystals, the central hypothesis behind molecular theories of nonlinear elasticity is that, at the scale of the atomic spacing, the deformation of the crystal is homogeneous. Therefore, locally, the crystal is subjected to an homogeneous deformation characterized by the deformation gradient F, which transforms the undeformed lattice vector Ainto a=FA (see Fig. 2). This relation allows us to express geometric quantities such as bond lengths and bond angles, which enter as arguments in the inter-atomic potentials, in terms of the right Cauchy- Green deformation tensor. By averaging over the crystal unit cell, it is straightforward to formulate a continuum hyper-elastic constitutive relation explicitly in terms of the atomistic potential. This systematic approach produces constitutive models which inherit the symmetries of the underlying lattice, and can be used to assess the onset of material instability [17] or as the basis for efficient finite element simulations [35, 36]. Some authors interested in the mechanics of carbon nanotubes have used the standard crystal elasticity theories to describe the inplane behavior of the wall (i.e. the mechanics of graphene), and study the elasticity and onset of failure [41]. The Cauchy-Born rule is a kinematic hypothesis, whose validity can be checked a posteriori, for instance by comparing with direct atomistic calculations. In the absence of defects, or emergence of microstructure, this hypothesis has been found to accurately predict the nonlinear elasticity of bulk crystals. The treatment of complex lattices requires some extra care [10, 36, 2]. Also recently, there has been work on the mathematical analysis of this hypothesis, focusing on interesting cases in which it fails (see for instance [16]). 2.2. Exponential Cauchy-Born rule Now consider a curved crystalline solid of reduced dimensionality, such as the graphene monolayer. It is natural to treat the equivalent continuum as a solid of reduced dimensionality, e.g. a surface without thickness in the case of a crystalline film. Indeed, the two-dimensional nature of the lattice does not suggest any mechanically meaningful Continuum mechanics of carbon nanotubes 5 W W T x 0 W X w x a A F=T F expx Figure 3. Illustration of the surface kinematics and the exponential Cauchy-Born rule. The exponential map transforms the vector w=FA tangent to the surface into a chord of the surface a. thickness [4]. As illustrated in Fig. 3, in this kinematic setting the standard Cauchy-Born rule produces deformed lattice vectors which are tangent to the surface. However, deformed lattice vectors should be chords of the surface, i.e. connecting two atoms on the surface. In [2] it was recognized that the standard Cauchy-Born rule does not capture the curvature of the film, and a generalized kinematic rule was proposed exploiting the natural map between the tangent of a manifold and the manifold itself provided by the exponential map. By composing the exponential map with the standard Cauchy-Born rule a= exp ◦FA (1) we define the so-called exponential Cauchy-Born (ECB) rule, which leads to expressions of the deformed lattice measures in terms of the stretch (first fundamental form) and the curvature (second fundamental form) of the surface, has been shown to correct the deficiencies of the standard rule for curved lattices, and to lead to accurate models [5]. For the sake of clarity, let us describe the resulting continuum theory in the simplest nontrivial example of an atomic chain containing N atomic bonds and deforming in two dimensions (see Fig. 4). The discrete atomistic potential for this system has a bond stretch contribution and a three-body contribution: Eatom =X bonds Vs(ai) + X angles Vθ(θj) (2) =A X bonds e Vsai A+X angles e Vθθj−π A .(3) 6Marino Arroyo and Ted Belytschko ϕ F A FA λ, κ θ a Figure 4. Illustration of the continuum and discrete models for the toy example of an atomic chain in 2D The potential energy has been rewritten in the last part of the equation for convenience. We consider two- and threebody potentials that need not be harmonic. The local approximation of the exponential Cauchy-Born rule in this simple setting produces the following relations between the discrete deformation measures, namely the bond length aand the angle between adjacent bonds θ, and the continuum strain measures, i.e. the stretch λand the curvature κof the deformed curve with respect to the undeformed initially straight body [1]: a(λ, κ) = 2 κsin λκA 2;θ(λ, κ) = π−λκA (4) where Adenotes the undeformed bond length (see Fig. 4). By plugging these expressions into the atomistic potential, and averaging over a unit cell of length A, the equivalent continuum is a curve (initially straight and of length L=NA) whose energy is Econt =ZL 0 WECB(λ(X), κ(X)) dX (5) with the strain energy density WECB(λ, κ) = e Vsa(λ, κ) A+e Vθθ(λ, κ)−π A.(6) 2.3. Relationship between discrete and continuum The methods of crystal elasticity bridge atomistic descriptions of matter and finite deformation elasticity. If the crystal is simple, stable, free of defects or microstructure, and is homogeneously deformed, then the methods based on the Cauchy-Born rule are exact. In other situations, for instance when solving a boundary value problem (minimizing the Continuum mechanics of carbon nanotubes 7 Aε=εA, AεNε=L, e Vθ,ε (·) = εe Vθ· ε,e Vs,ε (·) = εe Vs· ε Figure 5. Sequence of atomistic problems used to assess numerically the convergence of continuum models. potential energy of a crystal subject to loads and boundary conditions), the relationship between the atomistic solution and that provided by the continuum model with the Cauchy-Born constitutive relation is not so clear. The mathematical community has addresses this issue by investigating in what sense continuum models arise as asymptotic limits of discrete models (see for instance [6, 7]). Typically a sequence of re-scaled atomistic problems is considered, and the emergence of a continuum model in the limit is established. The re-scaling of the problems can be understood as the though experiment of making the atomic spacing smaller and smaller at fixed size of the body (large body limit). In [6], this limit was performed assuming that the continuum and the atomistic deformations are equal wherever both are defined, while in [7] it was proven that the solutions of the sequence of discrete problems (minimizers of the relevant discrete energies) converge to the solution of a continuum problem through Γ-convergence techniques. In the present section, we provide the asymptotic continuum limit of the chain model described above, extending part of the analysis of [6] to treat the curved lattice. Then, the convergence of the minimizers for a given example is established numerically, which provides information on the convergence of the discrete model to different continuum models. More details about this analysis will be provided elsewhere. We first consider the sequence of re-scaled problems depicted in Fig.5; a circular chain of fixed extended length Lis stretched by 50% laterally by pulling two atoms in a diameter. Starting from a reference model with atom spacing Aand N=L/A atoms, a sequence of models with atomic spacing Aε=εA and Nε=N/ε atoms is considered, with energy Eatom,ε =εA  X bonds e Vsai εA+X angles e Vθθj−π εA  . 8Marino Arroyo and Ted Belytschko Adapting the ideas in [6] to the present setting, and by noting that ai A=λ+O(A) and |θj−π| A=λκ +O(A), a hyper-elastic continuum model for a line with strain energy Wasymp(λ, κ) = e Vs(λ) + e Vθ(λκ) (7) arises in the limit as ε→0 under the hypothesis that the atomistic and the continuum deformations coincide wherever they are both defined, and some regularity hypothesis. We can now ask ourselves whether the actual solutions of the sequence of atomistic problems converge to the solution of a continuum problem with the strain energy density of Eq. (7). Mathematically, this is answered through Γ-convergence techniques [7]. Here, we investigate this convergence numerically, by actually solving the sequence of atomistic problems, and the continuum problem with an overkill discretization. The left plot in Fig. 6 shows the convergence of the relative difference between the atomistic equilibrium energy for several values of εand the continuum equilibrium energy for the asymptotic model. It can be observed that indeed the solutions of the re-scaled atomistic problems converge to the solution of the asymptotic continuum model. Note that the continuum model based on the exponential Cauchy-Born hypothesis, with the strain energy density in Eq. (6), seems to be out of the picture. Unlike the asymptotic model, this model does depend on the atomic spacing, and therefore for different values of εthe resulting continuum problem is different. The left plot of Fig. 6 shows that as ε→0, the continuum model based on the exponential Cauchy-Born rule converges to the asymptotic model. While this analysis provides useful information (it shows that, in the limit of vanishing atomic spacing, the models agree), it does not address a more interesting question from a practical viewpoint: how does a continuum model reproduce the parent atomistic model at finite, or even large, values of the atomic spacing (relative to the body size)? To address numerically this issue, we plot the relative difference of the energy provided by the continuum models (the asymptotic limit and the ECB model) and that provided by the atomistic model (right plot of Fig. 6). This plot shows for both continuum models, the relative error with respect to the atomistic solution decreases as ε→0 with a rate of convergence of 2. It can also be observed that the ECB model always provides more accurate representations of the atomistic model as compared with the asymptotic model, particularly for large values of ε. Continuum mechanics of carbon nanotubes 9 10−2 10−1 10−2 10−1 100 ε Relative energy error w.r.t. limit 10−2 10−1 10−2 10−1 100 ε Relative energy error w.r.t. atomistic Figure 6. Numerical study of the convergence of several continuum models for an anharmonic potential. Left: convergence of the atomistic model (•) and the ECB model () to the limit asymptotic model. Right: convergence of the asymptotic (•), the ECB (), and WLC (H) continuum models to the atomistic model. We also consider a third continuum model characterized by the strain energy density typical of the worm-like chain (WLC) model WW LC (λ, κ) = 1 2ks(λ−1)2+1 2kθκ2,(8) where the elastic moduli ksand kθare matched to the infinitesimal elastic response of the atomistic chain. This model, while harmonic in the strain measures, is (geometrically) nonlinear. This kind of phenomenological model has the same spirit of the continuum models for carbon nanotubes based on geometrically nonlinear shell models and a parameter fit for the elastic response (see e.g. [27]). As can be observed in the right plot, this model does not converge to the atomistic model. Thus, while qualitatively correct and capturing the geometric nonlinearities, this model is unable to correctly capture the material nonlinearities arising from the atomistic potentials. Thus, we have seen that given an atomistic model, there are several appropriate continuum models which can be considered. These include the asymptotic limit model Wasymp, but also the model based on the ECB hypothesis WECB and possibly other models that we have not considered. We have provided a procedure to compare the relative performance of the different continuum models with respect to parent atomistic model, and have seen that for the examples considered, the ECB model outperforms the asymptotic model. The results also show that a naive yet reasonable phenomenological model based on parameter fitting does not converge to the atomistic model. To conclude this section, it should be carefully noted that while for the simple example of the atomic chain the asymptotic model can be easily obtained, it is not straightforward to obtain an asymptotic model for a system like 16 Marino Arroyo and Ted Belytschko outlining some avenues of current and future research in the field of continuum and multiscale modelling of nanotubes. Coupling with atomistics. Despite we have seen that atomistic-based nonlinear elasticity goes a long way in describing the mechanics of carbon nanotubes, some phenomena such as fracture require the explicit tracking of individual atoms, at least in some region. Coupled ECB continuum-atomistic modelling of fracture of carbon nanotubes has been reported in [42]. Multiphysics analysis. As we have mentioned, the relevance of mechanics in nanotube science and technology is largely due to the strong function of other properties such as electrical conductance or chemical reactivity on deformation. Coupling continuum mechanics with atomistic methods capable of resolving the electronic structure (at fixed geometry) is an obvious opportunity [1] to further exploit the efficiency of continuum simulations. See [21, 28] for recent work along these lines. Mesoscopic models. The effective analysis and design of nano-devices ultimately needs simple yet accurate models amenable to the engineering practice. The atomistic-based continuum mechanics constitute a first step towards such mesoscopic models. Some efforts in formulating mesoscopic descriptions can be found in [9, 43]. Theoretical underpinnings of the continuum-discrete connection. We have seen that a number of justified and even convergent continuum models can be associated with a given atomistic model. 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