3 Application to curved crystalline monolayers: carbon nanotubes 24. 3.1 Introduction . ..... as crystal elasticity breaks down in the vicinity of defects, and the quasi-continuum ...... crystallography of the underlying lattice, in particular for the chirality of ...... Yu, M., O. Lourie, M. Dyer, K. Moloni, T. Kelly, and R. Ruoff (2000).
NORTHWESTERN UNIVERSITY
Finite Crystal Elasticity for Curved Single Layer Lattices: Applications to Carbon Nanotubes
A DISSERTATION
SUBMITTED TO THE GRADUATE SCHOOL IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
for the degree
DOCTOR OF PHILOSOPHY
Field of Mechanical Engineering
By
Marino Arroyo
EVANSTON, ILLINOIS
June 2003
c Copyright by Marino Arroyo 2003
All Rights Reserved
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ABSTRACT Finite Crystal Elasticity for Curved Single Layer Lattices: Applications to Carbon Nanotubes Marino Arroyo A method for the systematic reduction of degrees of freedom in the static analysis of lattice systems of reduced dimensionality is presented. The traditional methods of crystal elasticity, valid for space-filling crystals, are extended to deal with crystalline films in three dimensions, and chains in two or three dimensions. A generalization of the Cauchy-Born rule, the exponential Cauchy-Born rule, is key to these developments. This methodology allows us to formulate hyperelastic constitutive relations for continua of reduced dimensionality (lines, surfaces) exclusively in terms of the underlying lattice model, and written in closed-form, i.e. they do not involve local or constrained atomistic calculations. These models are shown to very accurately mimic the parent discrete model in the full nonlinear regime. This theory is applied to the mechanics of carbon nanotubes. The continuum model is discretized with finite elements, providing a computationally advantageous alternative to atomistic calculations. Large multi-walled nanotubes containing millions of atoms are efficiently handled in this manner, and unusual experimental observations are reproduced. The symmetry of several deformation modes can be treated analytically, and reduced two and one-dimensional models which encapsulate interesting mechanics of nanotubes are formulated. The linear response of nanotubes is characterized by elastic moduli which are written explicitly in terms of the interatomic potential.
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ACKNOWLEDGEMENTS
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Conventions and notations The conventions and notations follow mostly Marsden and Hughes (1983). For the sake of completeness, the most relevant are summarized below.
Conventions The notation follows in general these guidelines: 1. Points and point mappings are denoted by lightface. 2. Tensors and vectors are denoted by boldface. 3. Objects defined in the undeformed body (material quantities) are denoted with upper case, spacial objects with lower case, and referential objects with Greek letters. 4. Summation on repeated indices is implied for parametric and convected indices (α, β, ...), material indices (A, B, ...), and spacial indices (a, b, ...). Other indices (i, j, k, n, I, ...) do not represent components of vectors and tensors, and summation is not implied.
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Notations Rn Ω0
Euclidean n−space undeformed body (a m−differentiable manifold in R3 or an open set of Rm , m < 3) Ω deformed body (a m−differentiable manifold in R3 , m < 3) Ω referential or parametric body (an open set of Rm ) x, X, ξ points in Ω, Ω0 and Ω, denoted with lightface x, X, ξ corresponding position vectors in Euclidean space Φ deformation map, a point mapping from Ω0 into Ω ϕ0 undeformed configuration, a point mapping from Ω into Ω0 ϕ deformed configuration, a point mapping from Ω into Ω corresponding position vectors in Euclidean space Φ, ϕ, ϕ0 Tx Ω tangent space of Ω at x, i.e. the m−dimensional vector space tangent to Ω at the point x, “centered” at x TΩ Tangent bundle of Ω, i.e. the set {(x, w), x ∈ Ω, w ∈ Tx Ω} TΦ tangent map, a linear transformation from T Ω0 into T Ω ∗ Φg pull-back of a tensor by the mapping Φ Φ∗ C push-forward of a tensor by the mapping Φ ⊗ tensor product N symmetrized tensor product symm × cross product f ◦g composition of maps FA linear transformation acting on a vector A (F is viewed as a linear transformation) F·A contraction of tensors (now F is viewed as a (11 ) tensor and A as a (10 ) tensor) S:C double contraction of tensors < · | · >, k · k Euclidean inner product and norm WA Contravariant components of the vector W QA Covariant components of the one-form Q [F]BB0 Matrix representation of the two-point tensor F in the bases B0 and B [C]B0 Matrix representation of the one-point tensor C in the basis B0
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Contents Abstract
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Acknowledgments
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Conventions and notations
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Contents
vii
List of Figures
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List of Tables
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List of Boxes
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1 Introduction 1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Central idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 The 2.1 2.2 2.3 2.4
exponential Cauchy-Born rule Introduction . . . . . . . . . . . . . . . . . Cauchy-Born rule for space-filling crystals Why films and filaments are more difficult Kinematics of solids of low dimensionality 2.4.1 Deformation map . . . . . . . . . . 2.4.2 Deformation gradient . . . . . . . . 2.4.3 Exponential map . . . . . . . . . . 2.5 The exponential Cauchy-Born rule . . . . 2.6 A toy example: atomic chain in 2D . . . . 2.6.1 Atomistic model . . . . . . . . . . 2.6.2 Continuum model . . . . . . . . . . vii
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2.6.3 Thought example and discussion . . . . . . . . . . . . . . . . 19 2.6.4 Numerical example . . . . . . . . . . . . . . . . . . . . . . . . 21 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . 23
3 Application to curved crystalline monolayers: carbon nanotubes 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Exponential Cauchy-Born rule for 2D lattices . . . . . . . . . . . . 3.2.1 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Local approximation of the exponential Cauchy-Born rule . 3.3 Finite deformation membrane for nanotubes . . . . . . . . . . . . . 3.3.1 Lattice structure of graphene . . . . . . . . . . . . . . . . . 3.3.2 Interatomic potential and constitutive model . . . . . . . . . 3.3.3 Non-bonded interaction and external forces . . . . . . . . . . 3.3.4 Boundary value problem . . . . . . . . . . . . . . . . . . . . 3.4 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Finite element approximation . . . . . . . . . . . . . . . . . 3.4.2 Discrete minimization problem . . . . . . . . . . . . . . . . . 3.5 Numerical validation of the theory . . . . . . . . . . . . . . . . . . . 3.5.1 Compressed (18,0) nanotube . . . . . . . . . . . . . . . . . . 3.5.2 Results with the standard Cauchy-Born rule . . . . . . . . . 3.5.3 Bent (10, 10), (15, 15) two-walled nanotube . . . . . . . . . 3.5.4 Twisted (10,10) nanotube . . . . . . . . . . . . . . . . . . . 3.5.5 Twisted (30,30) nanotube . . . . . . . . . . . . . . . . . . . 3.6 Large-scale examples . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6.1 Bent 5-walled nanotube . . . . . . . . . . . . . . . . . . . . 3.6.2 Bent 34-walled nanotube: rippling . . . . . . . . . . . . . . . 3.7 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . 4 Reduced model for transverse deformations 4.1 Introduction . . . . . . . . . . . . . . . . . . 4.2 Reduced kinematics . . . . . . . . . . . . . . 4.3 Reduced model and variational principle . . 4.4 Numerical examples . . . . . . . . . . . . . . 4.4.1 Validation test . . . . . . . . . . . . 4.4.2 Representative simulations . . . . . . 4.5 Summary and conclusions . . . . . . . . . .
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5 Elastic moduli of carbon nanotubes 89 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.1.1 Elastic moduli from atomistic potentials . . . . . . . . . . . . 89 viii
5.2 5.3
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5.1.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . Elasticity tensors for curved graphene . . . . . . . . . . . . . . Infinitesimal elastic moduli of graphene . . . . . . . . . . . . . 5.3.1 In-plane moduli . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Genuine two-dimensional linear elasticity . . . . . . . . 5.3.3 Bending modulus . . . . . . . . . . . . . . . . . . . . . 5.3.4 Verification of the expressions for the elastic moduli . . 5.3.5 Comparison of Brenner’s potential with ab initio data . 1D finite elasticity for stretched nanotubes . . . . . . . . . . . 5.4.1 Elastic potential for cylindrical nanotubes . . . . . . . 5.4.2 Inner relaxation . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Radial relaxation . . . . . . . . . . . . . . . . . . . . . 5.4.4 Equilibrium state . . . . . . . . . . . . . . . . . . . . . 5.4.5 Stress and stiffness of straight nanotubes . . . . . . . . 5.4.6 Validation of the theory and discussion . . . . . . . . . 5.4.7 Dependence of Ys on stretch: simplified failure analysis 5.4.8 Dependence of elastic moduli on diameter and chirality Summary and conclusions . . . . . . . . . . . . . . . . . . . .
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90 91 93 94 95 96 97 99 100 101 102 102 103 104 106 109 111 114
6 Conclusions
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References
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A Aspects of the constitutive model for carbon nanotubes 125 A.1 Principal curvatures and directions . . . . . . . . . . . . . . . . . . . 125 A.2 Inner relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 A.2.1 Newton’s method iterations . . . . . . . . . . . . . . . . . . . 127 A.2.2 Inner forces and inner elastic constants . . . . . . . . . . . . . 127 A.2.3 Derivatives of pi with respect to the inner displacements . . . 128 A.2.4 Second derivatives of pi with respect to the inner displacements129 A.3 Derivatives of ai and θi with respect to the strain measures . . . . . . 130 A.4 Calculation of Q(x), Q 0 (x) and Q 00 (x) . . . . . . . . . . . . . . . . . 131 B Variations of the strain measures 132 B.1 Continuum problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 B.2 Discrete problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 C Numerical quadrature for subdivision finite elements 134 C.1 Problem description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 C.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 ix
D Derivation of the bending modulus 140 D.1 Kinematic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . 140 D.2 Bending modulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
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List of Figures 2.1 Deformation map and its tangent map for space-filling bodies . . . 2.2 Standard Cauchy-Born rule . . . . . . . . . . . . . . . . . . . . . . 2.3 Illustration of the difficulties encountered when trying to apply the standard Cauchy-Born rule a = FA to surfaces; note that F cannot map A into a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Deformation and its tangent map for surfaces in 3D . . . . . . . . . 2.5 Illustration of the proposed extension of the Cauchy-Born rule . . . 2.6 Illustration of the continuum rope-like model for an atomic chain deforming in 2D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The exponential map of the circle defined at each point of the curve by the unit normal and the normal curvature. . . . . . . . . . . . . 2.8 Comparison between the molecular mechanics (◦) and continuum mechanics () simulations. The atomistic system has 64 atoms and the continuum simulation is performed with (a) 16 Hermite Finite Elements, i.e. the number of degrees of freedom is halved, and (b) 64 Hermite Finite Elements. . . . . . . . . . . . . . . . . . . . . . . . . 3.1 3.2 3.3 3.4 3.5
3.6
Membrane kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . The exponential map transforms the vector w = FA tangent to the surface into a chord of the surface a. . . . . . . . . . . . . . . . . . Principal directions of the curvature tensor in the undeformed and the deformed bodies, and a lattice vector A . . . . . . . . . . . . . Exponential map in fictitious cylinders in each principal direction . Graphene honeycomb multi-lattice: the two simple Bravais lattices, depicted in different colors, are relatively displaced by the inner displacement η, which consequently also affects the bond vectors which are transformed form A0i into Ai . The unit cell of area S0 is also represented. It contains two nuclei and three inequivalent bonds. . . Finite element discretization . . . . . . . . . . . . . . . . . . . . . .
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Two numerical surfaces coming to van der Waals contact, but failing to feel it because of insufficient quadrature points: the finite element nodes are represented by •, the quadrature points for the non-bonded term by ×, and the van der Waals cut-off radius by circles, which here do not overlap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compressed 8.7 nm long (18,0) nanotube: comparison between the full atomistic model and the continuum/finite element model. (a) super-imposed deformed configurations for atomistic (black spheres) and finite element (gray surface) calculations, and (b) strain energy evolution for the atomistic model (—) and for the continuum simulation (•). The strain energy for a continuum model in which the inner displacements are not relaxed also depicted (). . . . . . . . . . . . Compressed 8.7 nm long (18,0) nanotube: unphysical deformations obtained with the standard Cauchy-Born rule for coarse (a), regular (b), and fine (c) meshes. (d) strain energy evolution for the atomistic model (—) and for the continuum simulation based on the standard Cauchy-Born rule for the coarse (), the regular (◦), and the fine (•) meshes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bent 12.56 nm long (10,10), (15,15) two-walled nanotube: (a) superimposed deformed configurations for atomistic (black spheres, and solid lines) and finite element (translucent gray surface) calculations, and (b) strain and non-bonded energy evolution for the atomistic model (—) and for the continuum simulation (•). . . . . . . . . . . Twisted 25.11 nm long (10,10) nanotube: super-imposed deformed configurations at three twisting angles for atomistic calculation (black spheres) and continuum finite element calculation (gray surface). . . Twisted 25.11 nm long (10,10) nanotube: (a) Comparison of the strain energy as a function of the twisting angle for atomistic calculation (—), and continuum/FE calculation (•), and strain energy density evolution if the non-bonded interactions are ignored (- - ). (b) Comparison of the non-bonded energy evolution for atomistic calculation (—) and continuum/FE calculation (•). . . . . . . . . . Twisted 37.67 nm long (30,30) nanotube: comparison between the atomistic model and the continuum/finite element model for two twisting angles. (a) super-imposed deformed configurations for atomistic (black spheres) and finite element (gray surface) calculations, and (b) map of the strain energy density on the finite element computational mesh (red is high, blue is low). . . . . . . . . . . . . . . . . . Bending of a 5-walled carbon nanotube, inspired in the experiments by Iijima et al. (1996) . . . . . . . . . . . . . . . . . . . . . . . . . xii
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3.15 Rippling of a 34-walled carbon nanotube: (a) comparison with experimental observation (reprinted with permission from Kuzumaki et al. (1998), http://www.tandf.co.uk), (b) side-section, (c) crosssections of the simulation, and (d) experiment on larger MWCNT (reprinted with permission from Poncharal et al. (1999). Copyright 1999 American Association for the Advancement of Science). . . . . . 69 3.16 Rippling of a 34-walled carbon nanotube: (a) Top and (b) side views of the simulated deformation. . . . . . . . . . . . . . . . . . . . . . . 70 4.1
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4.5 4.6 4.7 5.1
5.2 5.3
5.4 5.5
Illustration of the kinematics of the nanotube with uniform deformation in the axial direction, and the cylinder used to approximate the exponential map. . . . . . . . . . . . . . . . . . . . . . . . . . . . . The exponential map in the cylinder defined at each point of the surface by the unit normal, the normal curvature and the direction of the axis. The geodesics are either straight lines, circles, or helices. (a) Actual molecular model used in comparison, (b) Comparison of 20 element exponential Cauchy-Born rule continuum model with molecular mechanics and (c) Results obtained with a model constructed from the standard Cauchy-Born rule. . . . . . . . . . . . . . . . . . Which is more stable, circular or collapsed? (answer: for the (20,0) and (26,0) tubes, circular, and for the (32,0) and (40,0) tubes, collapsed.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transverse stability of a Multi-Walled Nanotube. . . . . . . . . . . Equilibrium configurations for pairs of nanotubes in van der Waals contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Equilibrium configuration of a bundle of seven closely packed (22,0) nanotubes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strain energy relative to planar graphene for fully relaxed nanotubes of varying radius plotted versus a quadratic fit of the bending energy with the bending modulus predicted by the continuum theory. . . . Strain energy relative to planar graphene for an axially deformed and otherwise unconstrained (10,10) nanotube. . . . . . . . . . . . . . . Evolution of the surface Young’s modulus as a function of stretch: in the regime labelled “defect nucleation” dislocations not modelled by this theory can take place. . . . . . . . . . . . . . . . . . . . . . . . Dependence of Young’s modulus on diameter (a) and chirality (b). . Dependence of Poisson’s ratio on diameter (a) and chirality (b). . .
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C.1 Computational mesh . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 xiii
C.2 Solution obtained with two integration points (strain energy density map), and evolution of the energy during the minimization iterations for one and two integration points. . . . . . . . . . . . . . . . . . . C.3 Sequence of deformations of the mesh for iterations 160, 170, 180, 200, 300 and 600 of the minimization procedure and one integration point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.4 Map of the Jacobian determinant of the deformation relative to the planar reference configuration —unrolled cylinder— for iterations 170 and 600 evaluated with one and three integration points —the colors represent the value at each element integration point when a single point is used, and the average when three integration points are considered. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.5 Map of the mean curvature for iteration 170 evaluated with one and three integration points —the colors represent the value at each element integration point when a single point is used, and the average when three integration points are considered. . . . . . . . . . . . . . C.6 Map of the Jacobian determinant for a bending test, in a computation with a single integration point. . . . . . . . . . . . . . . . . . . . . .
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List of Tables 3.1
Twisted 25.11 nm long (10,10) nanotube: error of the three considered finite element models at two twisting angles . . . . . . . . . . . . . . 63
4.1
Comparison of 20 element model (C+FE) with molecular (MM): Energy in J/mol. . . . . . . . . . . . . . . . . . . Comparison of 36 element model (C+FE) with molecular (MM): Energy in J/mol. . . . . . . . . . . . . . . . . . .
4.2 5.1 5.2
mechanics . . . . . . . 83 mechanics . . . . . . . 84
Elastic properties of graphene from ab initio calculations, and from Brenner’s potentials (∗ obtained from the closed-form expressions). . . 100 Comparison of strain energy relative to planar graphene for an axially deformed and otherwise unconstrained (10,10) nanotube (eV/atom) . 107
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List of Boxes 3.1
Constitutive model: calculation of the strain energy density and the stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3.2 Calculation of the elemental internal energy and forces . . . . . . . . 52
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Chapter 1 Introduction 1.1
Motivation
Traditionally, continuum mechanics theories have dealt with solids at scales much larger than the atomic spacing. Solid state physics and quantum chemistry have been the basic framework to analyze systems in which individual atoms are distinguishable. Lattice statics, molecular mechanics, first principles calculations are perceived as the natural simulation methods to study such systems. In atomistic simulations, each individual atom of the system needs to be tracked, even though in many situations the deformation of the solid is very smooth and its features can be described in continuum terms. In statics simulations, the large number of unknowns leads to many iterations in the minimization routines, while in molecular dynamics, the critical integration time-step is extremely restrictive. This limits the size and observation time of computable systems to often unrealistically small domains loaded at very high rates. In addition to the computational cost, it is difficult to extract mechanical insight from these calculations. In the context of space-filling crystals, the molecular theories of elasticity have proven to be very fruitful in systematically constructing hyperelastic constitutive models in terms of the atomistic description of the system (Stakgold 1950; Ericksen 1970; Weiner 1983). This approach is the material modelling analog of the “bottom-up” paradigm in nanotechnology. Finite crystal elasticity relies on a central kinematic hypothesis linking the continuum and the atomistic deformations, namely the Cauchy-Born rule (Milstein 1982; Ericksen 1984). This hypothesis has been recently proven to be a consequence of first principles of mechanics for a restricted class of two-dimensional lattices, elevating its status from postulate to theorem (Friesecke and Theil 2002). Continuum models of crystalline systems can be the basis of efficient simulation methods, when combined for example with the finite el1
2 ement method (Tadmor et al. 1999). The straight application of a local theory such as crystal elasticity breaks down in the vicinity of defects, and the quasi-continuum method has been proposed to deal with these situations (Tadmor et al. 1996; Shenoy et al. 1999). The problem which motivates the present work is that of the “crunchy mechanics” of carbon nanotubes (Yakobson and Smalley 1997). Carbon nanotubes are tubular nano-structures that can be viewed as pristine graphene sheets (twodimensional hexagonal lattices) rolled into cylinders. Carbon nanotubes are chemically inert, and display extraordinary mechanical, electrical, and thermal properties, which make them candidates for a wide variety of nano-devices and nano-structured materials. In particular, experiments and atomistic simulations reveal that carbon nanotubes are extremely resilient to mechanical deformation; they can be severely bent, twisted, and compressed reversibly, and the observed morphologies very closely resemble buckled macroscopic shells (Iijima et al. 1996; Yakobson et al. 1996). Given the large elastic deformations of these crystalline defect-less structures, finite crystal elasticity naturally suggests itself for the analysis of the mechanics of carbon nanotubes.
1.2
Central idea
The central idea behind this thesis is the extension of the methods of crystal elasticity to lattices of reduced dimensionality, that is films in 3D, or chains in 2D or 3D. For this purpose, the geometry of the Cauchy-Born rule, corner-stone of finite crystal elasticity, in the context of continuum mechanics on manifolds provides clues as to how to extend this kinematic rule to the case of curved lattices. The Cauchy-Born rule is generalized by means of the differential geometry concept of the exponential map, and the exponential Cauchy-Born rule is proposed. This extended kinematic rule accounts for the fact that the lattice vectors are chords of a curved manifold, whereas the standard Cauchy-Born rule treats them as tangent vectors to the manifold. This methodology leads to hyperelastic constitutive relations for continua of reduced dimensionality (lines, surfaces) based exclusively on the underlying lattice model. However, these models are written in closed-form, and do not involve constrained atomistic calculations, thereby falling within the strict framework of continuum mechanics. The traditional approach to model such reduced dimensionality systems by means of continuum mechanics consists on adopting an existing continuum theory, say thin shell theory (Yakobson et al. 1996; Ru 2000; Ru 2001), or Kirchhoff elastic rod theory (Yang et al. 1993), and fit the parameters of this theory to the particular discrete model of interest. Thus, for instance, the usual continuum idealization of
3 carbon nanotubes is a thin shell. Thin shell theory aims to describe space-filling materials which form thin structures, while carbon nanotubes are formed by a 2D material (graphene is a single layer atomic film). This conceptual drawback is naturally overcome by the special continuum theories which derive from the exponential Cauchy-Born rule.
1.3
Outline
The exponential Cauchy-Born rule is developed in Chapter 2. An abstract exposition of the theory is complemented by its realization in the simplest yet complete example of a one-dimensional atomic chain deforming in two dimensions. Chapter 3 develops in detail the application of the exponential Cauchy-Born rule to curved crystalline two-dimensional monolayers. A continuum hyperelastic constitutive relation for a surface without thickness is formulated for curved graphene, explicitly written in terms of the interatomic potential that describes the atomistic system (here the potential for hydrocarbons by Brenner (1990)). The hyperelastic potential depends on frame-indifferent strain measures, namely the Green deformation tensor and the pull-back of the curvature tensor. Given the fact that graphene is a Bravais multi-lattice, additional internal elastic variables, the inner displacements, must be appropriately treated. The continuum variational statement of the statics of the new model involving a continuum version of the van der Waals interactions is provided, and its numerical discretization by finite elements detailed. The continuum/finite element computational approach to the nonlinear mechanics of carbon nanotubes is numerically tested against a comprehensive series of independently run full atomistic simulations, involving single and multi-walled nanotubes under different loading conditions. These comparisons demonstrate the remarkable accuracy of the continuum theory in the full nonlinear regime, for very large deformations. Simulations of multi-walled nanotubes containing millions of atoms are efficiently handled by the continuum/finite element technique, and unusual features observed in experiments, like the rippling effect, are reproduced. The continuum simulations are estimated to be several orders of magnitude faster than the corresponding atomistic calculations. Since the continuum stain energy potential is written is closed-form, special cases of deformation can be treated (semi-)analytically. Chapter 4 develops a 2D model for the transverse mechanics of nanotubes, and Chapter 5 deals with the elastic moduli of planar graphene and straight nanotubes. In particular, an explicit expression of the bending modulus of graphene in terms of the functional form of the atomistic potential is developed from the proposed theory. This appears to be a new result. A one-dimensional model for uniaxially stretched nanotubes which encapsulates a rich mechanical behavior is developed, and its validity carefully tested
4 against atomistic calculations. The evolution of Young’s modulus with stretch, and the dependence of elastic moduli on diameter and chirality are easily studied with this one-dimensional model. Parts of Chapters 2 and 3 have been published in Arroyo and Belytschko (2002), while Chapter 4 and Section 2.6 appear in Arroyo and Belytschko (2003b). The application of the exponential Cauchy-Born rule to filiform solids in three dimensions is developed elsewhere (Arroyo and Belytschko 2003a).
Chapter 2 The exponential Cauchy-Born rule 2.1
Introduction
This chapter investigates the geometric structure of the standard Cauchy-Born rule for space-filling crystals, which is hidden by the predominant use of Euclidean coordinates. In the context of lattices of reduced dimensionality, it is argued that the continuum object replacing these lattices should also be a continuum of reduced dimensionality, the atomic bonds being chords of the corresponding surface or curve. The shortcomings of the standard Cauchy-Born rule in this setting become apparent from simple geometric arguments. The finite kinematics of deforming surfaces or curves and its geometric structure suggests an extension of the standard Cauchy-Born rule. This extension is based on the differential geometry concept of the exponential map, and is described in abstract terms. Then, this abstract presentation is complemented by its realization in the simplest, yet complete, situation, i.e. an atomic chain deforming in two dimensions. The formulation of the equivalent rope-like continuum object is detailed, and a simple example illustrating the effectiveness of this model in mimicking the parent atomic chain is provided.
2.2
Cauchy-Born rule for space-filling crystals
The formulation of a finite deformation continuum model for space-filling defect-less crystals based on the Cauchy-Born rule is relatively straightforward. The CauchyBorn rule links the atomistic deformation to that of the continuum medium. Then, a representative crystallite is considered, and, for a given continuum deformation, the continuum strain energy density is defined to be the energy of the crystallite subject to the deformation divided by its volume. The details of the procedure 5
6
F=TF
TX W
TF (X) W
0
w=FW
W
F (X)
X
W0
F
W
Figure 2.1: Deformation map and its tangent map for space-filling bodies are presented in several references, and will also be described later in the present thesis. The focus here is on the fundamental kinematic assumption that links the continuum and the atomic deformations, i.e. the Cauchy-Born rule. The details of the atomistic model are deliberately omitted. Assume for the moment that we are dealing with space-filling continuum bodies, i.e. open subsets of the ambient Euclidean space. Let Φ be the deformation that maps the undeformed body Ω0 ⊂ Rn , into Rn , n being either 1, 2 or 3. If X denotes a point in the undeformed body, its image after deformation is x = Φ(X). The deformed body is denoted as Ω = Φ(Ω0 ) and is an open set of Rn . The deformation gradient is the derivative of the vector-valued vectorial function Φ, F = DΦ = ∂Φ/∂X ∈ Rn×n . At each point X, the deformation gradient is a linear transformation from Rn into Rn , which maps “infinitesimal” material vectors, dx = FdX (see Malvern (1969), p. 156). From a differential geometry point of view, the deformation gradient is called the tangent map of Φ, and is denoted as F = T Φ. Let us call the infinitesimal neighborhoods of X and x the tangent spaces of the undeformed body and the deformed one, respectively denoted as TX Ω0 and Tx Ω (see Fig. 2.1 for an illustration). Then, using this language, the deformation Φ maps the undeformed body into the deformed one, and the tangent map F = T Φ maps the tangent space of the undeformed body into the tangent space of the deformed body. The central hypothesis behind molecular theories of elasticity at finite strains is that, at the scale of the atomic spacing, the deformation of the crystal is homogeneous. Consequently, as the crystalline solid deforms, lattice vectors undergo a linear transformation (Stakgold 1950; Ericksen 1970; Weiner 1983). This approach, often
7
F A B
a b
Figure 2.2: Standard Cauchy-Born rule referred to as the method of the homogeneous deformations (Martin 1975; Cousins 1978), is most commonly abstracted through the Cauchy-Born rule (Milstein 1982; Ericksen 1984): a = FA, (2.1) where A denotes an undeformed lattice vector and a the same vector in the deformed crystal (see Fig. 2.2 for an illustration). Multi-lattices, that is lattices with more than one atom in the unit cell, require a special treatment; the homogeneous deformation is assumed to affect each of the simple lattices composing the multi-lattice, while additional kinematic variables describing the relative shifts of the simple lattices must be introduced (Stakgold 1950; Ericksen 1970; Martin 1975; Cousins 1978; Weiner 1983; Tadmor et al. 1999). More details about the treatment of multilattices are provided in Chapter 3. The Cauchy-Born rule is generally assumed to hold as long as the crystal is free of defects, slips, phase transformations, and other inhomogeneities or special crystallographic phenomena. The geometry of the lattice vectors, that is their length and the angles they form with other lattice vectors in the deformed crystal, can therefore be extracted from the continuum deformation through the Green deformation tensor C = FT F using standard continuum mechanics relations: kak =
p < A | CA >
and
cos θ =
< A | CB > , kakkbk
(2.2)
where B and b represent another undeformed and deformed lattice vector, θ is the angle a and b form in the deformed crystal, and < · | · > and k · k denote the standard Euclidean inner product and norm. Once the geometry of the deformed lattice vectors is linked to the continuum deformation, a constitutive model based on the atomistic description can be constructed by identifying the continuum strain energy density with the potential energy of the atomic system for a representative cell divided by its volume. One could argue that the rule expressed by Eq. (2.1) is formally inconsistent,
8 because the lattice vectors A and a, each connecting two atomic positions, are physical entities that lie in the undeformed and deformed body respectively, while the tangent map F = T Φ maps elements of the tangent of the undeformed body into elements of the tangent of the deformed body. This inconsistency can also be viewed from a more classic standpoint: the lattice vectors have finite length while the deformation gradient maps “infinitesimal” material vectors, dx = FdX. These objections are circumvented by noting that for homogeneous deformations, Eq. (2.1) holds exactly, even for material vectors of finite length; in this case the body and its tangent are undistinguishable. Remark 2.1. In the present work, the Cauchy-Born rule is viewed as a fundamental postulate, from which continuum mechanics constitutive models are formulated. In the literature, as well as here, the validity of this hypothesis is tested a posteriori, for instance by comparing the predictions it furnishes with full atomistic calculations. A rigorous mathematical analysis of the validity of the Cauchy-Born rule has only recently been performed for a family of 2D lattices (Friesecke and Theil 2002). It is concluded that for a range of spring parameters, the Cauchy-Born rule is a theorem in the sense that, if a lattice is subject to an affine transformation on its boundary, then the deformation is also affine, i.e. homogeneous, within the lattice.
2.3
Why films and filaments are more difficult
Consider the case in which we have a monolayer crystalline film (such as a graphene sheet) deforming arbitrarily in 3D (the arguments presented in this section also hold for one-dimensional lattices deforming in two or three dimensions). It is natural in this case to treat the continuum solid as a surface, a two-dimensional body without thickness. Indeed, the nature of the lattice, a two-dimensional arrangement of atoms whose energy depends exclusively on the positions of these atoms (BornOppenheimer approximation), does not suggest any meaningful thickness. The sheet is then a two-manifold in R3 —a surface. We postulate that the atoms lie on the surface, and therefore the lattice vectors are chords of the surface. We would like to use a kinematic rule in order to express the geometry of the deformed lattice vectors in terms of a variable characterizing the deformation of the continuum surface, such as the Green deformation tensor in Eq. (2.2). The abstraction of the Cauchy-Born rule a = FA becomes useful in this context. The homogeneous deformations point of view, equivalent to the Cauchy-Born rule for space-filling crystals, becomes inoperative for surfaces, for which it is not clear how to define a homogeneous deformation. Indeed, a uniform metric and uniform curvature are not necessarily compatible (do Carmo 1976), i.e. there may not exist a surface with such uniform properties (this does not happen for curves). Nevertheless,
9
F?
TX W0 X
W0
A
Tx W
x a
W Figure 2.3: Illustration of the difficulties encountered when trying to apply the standard Cauchy-Born rule a = FA to surfaces; note that F cannot map A into a if we try to directly use the Cauchy-Born rule to transform lattice vectors of curved continua through the deformation gradient, we face a fundamental difficulty. In the context of deforming surfaces, the language of differential geometry introduced previously acquires a clear visual meaning, and for instance the tangent space Tx Ω is the tangent plane to the surface Ω at the point x, viewed as the vector space R2 “centered” at x (see Fig. 2.4 for an illustration). Note the essential distinction between the manifold, which is curved, and the tangent space, which is flat (this geometric structure is hidden in the case of space-filling materials described with Euclidean coordinates). As noted previously, the lattice vectors should be viewed as chords of the surface, not as tangent vectors to the surfaces. In the present setting, it is clear that F = T Φ, which maps vectors tangent to Ω0 into vectors tangent to Ω, cannot be used to transform the undeformed lattice vector A into the deformed lattice vector a. This argument is sketched in Fig. 2.3. To illustrate these issues, consider the following situation. Suppose that the undeformed crystal is planar, and is rolled into a cylinder without stretch. If the standard Cauchy-Born rule is used, the deformed lattice vectors emanating from the same point remain coplanar, i.e. they lie in the tangent space to the cylinder at that point. In this tangent space, energetically relevant geometric quantities such as the length of the lattice vectors and the angles they make remain unchanged (the deformation described is an isometry). Therefore, the energy will remain unchanged upon rolling. However, the actual lattice vectors, which are chords of the cylinder, do not remain coplanar and their lengths change. Therefore, we can expect that the
10 energy of the atomic system will change in this deformation. Thus, a continuum model describing only the behavior of the tangent space is blind to the fact that the plane sheet is being rolled, and assigns zero energy to the deformation.
2.4
Kinematics of solids of low dimensionality
The present section describes the continuum kinematics of solids of reduced dimensionality, i.e. surfaces in R3 , or curves in R2 or R3 . Two useful references for this section are Marsden and Hughes (1983) for a mathematical treatment of continuum mechanics, and do Carmo (1976) for the differential geometry of curves and surfaces.
2.4.1
Deformation map
As detailed in further chapters, it is often possible and convenient to consider a flat undeformed body, i.e. Ω0 is an open set of Rm , 1 < m < 3. However, it may be useful to define a curved undeformed configuration, and for this reason the theory is formulated accordingly. Let the undeformed body Ω0 be a m−differential manifold in R3 (a curve if m = 1, a surface if m = 2). Let us assume for simplicity that it can be described by a single chart; there exists a differentiable and invertible parametrization of Ω0 from a referential body Ω, an open set in Rm (which can be called the parametric space). This parametrization is called the undeformed configuration ϕ0 : ϕ0 : Ω ⊂ Rm −→ R3 ξ
(2.3)
7−→ X = ϕ0 (ξ)
where ξ represents any point in Ω and X its image in the deformed body. Similarly, the deformed body Ω is another m−differential manifold in R3 , and it is parametrized on the same referential body Ω by the current or deformed configuration: ϕ : Ω ⊂ Rm −→ R3 (2.4) ξ 7−→ x = ϕ(ξ) with Ω = ϕ(Ω). The deformation map Φ, mapping the undeformed body into the
11
TX W 0
X
Tx W
TF =F
x
w
W
W
F = j o j0-1
W0
j0
x
j w
W Figure 2.4: Deformation and its tangent map for surfaces in 3D deformed body, can be expressed as the composition of two maps: Φ = ϕ ◦ ϕ−1 0 : Ω0 −→ Ω X
(2.5)
7−→ x = Φ(X) = ϕ ϕ−1 0 (X)
The deformation map Φ maps two differential manifolds, and its tangent T Φ (the deformation gradient F) maps the tangent spaces of Ω0 and Ω (see Fig. 2.4). At each point X of Ω0 , TX Φ = F(X) is a linear transformation taking elements of TX Ω0 (the m−dimensional vector space whose elements lie in the tangent plane/line to the surface/curve Ω0 at X and emanate from this point) into TΦ(X) Ω (analogously, the vector space tangent to the deformed body at Φ(X) “centered” at this same point). The following section characterizes the tangent of the deformation.
2.4.2
Deformation gradient
Let us assume for simplicity that the referential body is described by Euclidean coordinates {ξ α } with the corresponding standard basis B = {Ξα }, where the index α ranges from 1 to m. The Euclidean space R3 is described by Euclidean coordinates {x1 , x2 , x3 }, with the standard basis B = {i1 , i2 , i3 }. Let ϕa0 and ϕa denote the a−th Euclidean component of the undeformed and deformed configurations respectively. The natural or convected basis vectors (tangent to Ω0 and Ω respectively) are defined
12 as
∂ϕa ∂ϕa0 Gα = α ia and gα = α ia , α = 1, m, ∂ξ ∂ξ
(2.6)
where summation on the repeated indices is implied. Then, at each point X, C0 = {Gα } is the convected basis of TX Ω0 , and similarly for each x ∈ Ω, C = {gα } is the convected basis of Tx Ω. The corresponding dual bases of the cotangent spaces of Ω0 and Ω are also defined. The cotangent space Tx∗ Ω is the space of one-forms on Tx Ω, i.e. the linear mappings from the tangent into R. The dual basis is defined by the relations gα (gβ ) = δβα (analogous relations hold for the undeformed body). By noting that the image through the tangent of a configuration of the basis vectors Ξα are the convected basis vectors, we can express in a simple way the tangent maps of the configurations as: T ϕ0 = Gα ⊗ Ξα
and
T ϕ = gα ⊗ Ξα ,
(2.7)
∗
where B = {Ξα } is the dual basis of B. We can then characterize the deformation gradient or tangent of the deformation as follows. Recall the definition of the deformation map Φ = ϕ ◦ ϕ−1 0 . Invoking the chain rule and Eq. (2.7), we can write its tangent map T Φ = T ϕ ◦ T ϕ−1 0 , or deformation gradient as: F = gα ⊗ Gα : T Ω0
−→ T Ω α
(2.8) α
W = W Gα 7−→ w = FW = W gα . Therefore, the matrix representation of F in the convected bases of C0 and C is the m × m identity matrix, with the information about the deformation contained in the basis vectors.
2.4.3
Exponential map
A simple definition of the exponential map is given in (Morgan 1993), for a manifold M: “The exponential map expp at a point p in M maps the tangent space Tp M into M by sending a vector v in Tp M to the point in M a distance kvk along the geodesic from p in the direction v.” The exponential map is a diffeomorphism—invertible and differentiable— in a neighborhood of each regular point p of the manifold, that naturally maps the tangent of a manifold into the manifold itself in an intrinsic manner. It can be defined
13 because of the existence and uniqueness of geodesics at any point given a direction in the tangent space. The exponential map is defined here in geometric terms, and its evaluation requires knowledge of the geodesics, which are trivial in the case of curves. For surfaces however, obtaining the geodesics in a particular coordinate system involves solving the geodesic differential equations. These equations are a system of nonlinear ordinary differential equations whose unknowns are the parametric coordinates of the geodesic, and whose coefficients are the Christoffel symbols of the surface. Finding the geodesics, and thus the exponential map, is much simpler in some particular cases, as detailed later. More details about the exponential map in the case of surfaces can be found in do Carmo (1976). It is worth commenting on another important application of the exponential map in mechanics: the transformation of infinitesimal rotations, represented by skew-symmetric matrices or the corresponding spin vector, into finite rotations, represented by proper orthogonal matrices (Simo and Hughes 1998). The underlying idea is very similar to the one used here. Indeed, the infinitesimal rotations are the tangent to the finite rotations. Further geometrical insight is gained by understanding the correspondence between the finite rotations and the unit sphere (Simo and Fox 1989).
2.5
The exponential Cauchy-Born rule
In order to construct a continuum mechanics theory from an atomistic model, we need distribute the deformation energy of the discrete atomic system into the continuum. For this purpose, following the path of crystal elasticity, we need an analog of the standard Cauchy-Born rule, relating the deformation of the lattice vectors to that of the lower dimensionality curved continuum. In Section 2.3, the breakdown of the standard Cauchy-Born rule for curved continua was sketched. We saw that the kinematic rule a = FA presents formal inconsistencies. In this section, these are overcome by introducing a new kinematic rule that exploits the exponential map. The proposed rule is kept at an abstract level, and should be seen as a framework for practical models. Reference to Fig. 2.5 is particularly useful to visualize the ideas presented below. Let A denote an undeformed lattice vector, defined as the chord between points −−→ X, the tail, and Y , the head, i.e. A = XY with X, Y ∈ Ω0 . Recall that in a neighborhood of each regular point the exponential map is invertible. We assume that Y is close enough to X so that the expX has an inverse at Y . If the inverse of the exponential map at X is applied to Y , a vector W ∈ TX Ω0 is obtained. This vector can be legitimately transformed through the deformation gradient F into a vector w ∈ Tx Ω, which in turn can be sent back to a point z ∈ Ω through the
14
exp
F
TX W
0
W0
w
W
X A
Y
TxW
x a
z
exp-1
W Figure 2.5: Illustration of the proposed extension of the Cauchy-Born rule exponential map at x = Φ(X). Then the deformed lattice vector a can be defined as the chord of the surface Ω emanating from x and whose head is z. Let us define the point map FX := expΦ(X) ◦ F(X) ◦ exp−1 X from Ω0 into Ω, that is: FX : Ω0 −→ TX Ω0 Y
−→ TΦ(X) Ω
−→ Ω
(2.9)
7−→ W = exp−1 X (Y ) 7−→ w = FW 7−→ z = expΦ(X) (w).
−−→ Then, given an undeformed lattice vector A = XY emanating from X, the proposed −−−−→ kinematic rule consists on first obtaining z = FX (Y ), and then a = Φ(X)z. In order to keep the notation simple and highlight the fundamental ideas, the following abuse of notation is used. The exponential map at X ∈ Ω0 maps tangent vectors W ∈ TX Ω0 into points in the surface Y = expX (W). Since, with fixed −−→ X, the chord A = XY and the point Y can be identified, we will simply write A = expX (W). Similarly, we write a = expΦ(X) (w), instead of z = expΦ(X) (w) and −−−−→ a = Φ(X)z. With this notation, the extension to the Cauchy-Born rule, which will be called the exponential Cauchy-Born rule, can be written simply as: a = FX (A) := expΦ(X) ◦ F(X) ◦ exp−1 X (A).
(2.10)
15 The above map overcomes the formal inconsistencies of the standard Cauchy-Born rule pointed out in Section 2.3 by exploiting the natural way to map the tangent space and the manifold provided by the exponential map. This map transforms an undeformed lattice vector into a deformed one based on the deformation of the continuum curved object. Although this model provides a theoretical and formal way to overcome the previously mentioned difficulties of the standard Cauchy-Born rule, its practical application is not straightforward. To determine the geodesics, and thus the exponential map, the geodesic differential equations must be integrated. The exact implementation of the proposed model results in a computationally very complex non-local model. Indeed, the deformed lattice vectors would depend not only on the deformation of the surface/curve at a particular point, but also in a neighborhood. However, approximations to the exponential map can be used, rendering the model possibly local and computationally feasible. One can consider the kinematic assumption a = FX (A) as a general framework for a family of extensions to the Cauchy-Born rule in different situations, as developed in the following chapters. Remark 2.2. Note that strictly speaking FX , mapping points in Ω0 into Ω, is intrinsic, i.e. can be performed “from inside” the manifold, without recourse to the ambient Euclidean space. However, the exponential Cauchy-Born rule is necessarily extrinsic, because it ultimately transforms lattice vectors in R3 . The extrinsic part of the map stems from the definition of lattice vectors as chords of the surface/curve, a concept which requires the ambient space R3 . In the extended kinematic rule, the deformation gradient expresses changes in intrinsic length between the atoms (length inside the manifold, defined as the length of the shortest curve between them on the manifold) and in intrinsic angles, but not necessarily the actual Euclidean length and angles of the lattice vectors, which are the quantities of interest in atomistic models. In particular, when applied to surfaces (see Chapter 3), the local approximation of the exponential Cauchy-Born rule depends on the second fundamental form, an extrinsic quantity. As detailed in Arroyo and Belytschko (2003a) for curves in three dimensions, the local approximation of the exponential Cauchy-Born rule depends on the curvature and the torsion of the curve. Remark 2.3. If this model is applied to space-filling bodies, it results exactly in the standard Cauchy-Born rule. Indeed, the geodesics in a subset of the Euclidean space Rn are straight lines and the exponential map is, loosely speaking, an identity map. Remark 2.4. It can be said that this is a higher order model, because, for surfaces, it involves not only the metric but also its derivatives (the Christoffel symbols depend on the metric and its first derivatives). We prefer to think of it in the geometric terms presented. The present approach defined in geometric terms automatically leads to hyperelastic potentials that satisfy frame-indifference. A “higher order” adhoc model will have difficulties in this respect.
16 Remark 2.5. Note that in general z 6= Φ(Y ) (see Fig. 2.5). Actually, the same happens with the standard Cauchy-Born rule, unless the deformation is homogeneous. Similarly, in the present theory there are some special cases where (2.10) is kinematically exact in this sense.
2.6
A toy example: atomic chain in 2D
In this section, we illustrate the exponential Cauchy-Born rule for the simplest case, an atomic chain deforming in 2D. The resulting continuum model is a hyperelastic rope whose strain energy density depends on the stretch and the curvature of the continuum object. This constitutive model is based exclusively on the atomistic description of the chain. In this case the exponential map is approximated at each point by the exponential map of the circle, for which a closed-form expression is straightforward.
2.6.1
Atomistic model
The strain energy of the atomistic system is described by means of bond stretch Vs and bond angle Vθ potentials. The strain potential energy of the atomic chain can be written as a function of the atomic position vectors xi ∈ R2 :
Πchain (x1 , ..., xn ) =
nB X k=1
Vs (ak ) +
mB X
Vθ (θl ),
(2.11)
l=1
where ak denotes the bond lengths, θl denotes the angle that adjacent bonds form, and nB and mB are the number of bonds and adjacent bonds, respectively. This particular atomistic model is chosen for simplicity, but the approach is not restricted to this structure of the interatomic potential by any means. The exponential CauchyBorn rule provides a link between the atomistic and the continuum deformations, and can be combined with any atomistic model of choice.
2.6.2
Continuum model
As illustrated in Fig. 2.6, the undeformed body is considered to be a one-dimensional line segment that is allowed to deform in 2D. In this simple setting, the referential
17
Atomistic system A
F, TF
W 0, T W 0
w x 1/k n
A X
W
Tx W
Figure 2.6: Illustration of the continuum rope-like model for an atomic chain deforming in 2D. and the undeformed body can be identified, Ω ≡ Ω0 . Consequently, the exponential Cauchy-Born rule simplifies to a = expΦ(X) ◦F(X)A in this setting. The deformation map can be described in terms of its Euclidean components 1 Φ (X) and Φ2 (X), with X ∈ Ω0 ⊂ R. In this case, the Euclidean components of the deformation gradient are [F] = [Φ1,X , Φ2,X ]T , where (·),X denotes differentiation with respect to X. The Green deformation tensor C is a scalar, whose square root is the stretch Λ of the deformed rope: √ Λ=
C=
q
Φ1,X
2
2 + Φ2,X .
(2.12)
The curvature κ of the deformed rope can be written as: κ=
1 2 1 1 2 Φ Φ − Φ Φ , ,X ,XX ,X ,XX Λ3
(2.13)
and can be interpreted geometrically as the inverse of the radius of curvature of the curve. As we mentioned in Section 2.5, in order to obtain a practical method the exponential Cauchy-Born rule needs to be approximated. It is desirable that the approximation of the exponential Cauchy-Born rule leads to a local model, i.e. one in which the strain energy depends on the local deformation of the rope. The strategy followed to obtain such an approximation is to consider the exponential map at each point, not of the original curve, but of a circle of radius r = 1/κ with the same
18 normal as the original curve (see Fig. 2.6). Thus, locally, this circle replaces the original curve. The exponential map of the circle is readily available in closed-form. The first part of the exponential Cauchy-Born rule maps the lattice vector A of length A into a vector tangent to the curve whose Euclidean components are [w] = A[F]. Therefore, its length is √ w=
CA.
(2.14)
The exponential map of the circle is illustrated in Fig. 2.7. The length of the tangent vector w is “walked” on the geodesic to obtain expΦ(X) w, and thus the chord a. Since the geodesic of the circle is trivially the circle itself, the length of the arc defined by the ends of a is w. Let θ denote the angle formed by two adjacent deformed lattice vectors. Consider the triangle formed by the ends of a and the center of the circle. This triangle is isosceles, and its equal angles are θ/2. Therefore its third angle, the angle subtended by the arc of length w, is γ = π − θ. Consequently, we can relate the length of the arc, w, to the radius of the circle r = 1/κ and the angle γ: w = γr = (π − θ)/κ.
(2.15)
Since the length of the unequal side of the triangle can be easily computed as γ a = kak = 2r sin , 2 it follows that a=
2 κw sin κ 2
and
θ = π − κw.
(2.16)
(2.17)
Note from Eqs. (2.14) and (2.17) that the quantities a and θ, which are the arguments of the atomistic energy (see Eq. (2.11)), are expressed in terms of the continuum deformation. Thus, we can view a and θ as derived strain measures, expressed in terms of the basic strain measures C and κ; these derived strain measures are adequate for the formulation of atomistic-based continuum models. The next step is to consider a representative crystallite of the atomistic system, which in this case is a cell of length A including a single nucleus in the undeformed crystal. In a homogenization process, the energy of this deformed cell containing one bond and one angle between adjacent bonds is identified to the strain energy density of the continuum multiplied by the undeformed volume of the cell: A · W (Φ) = Vs (a) + Vθ (θ). Since our aim is to formulate a hyperelastic continuum
19
expF(X) w
F(X)
a q/2 1/k n
Figure 2.7: The exponential map of the circle defined at each point of the curve by the unit normal and the normal curvature. model, the elastic potential W (Φ) is a strain energy per undeformed volume, in this case undeformed length. The continuum strain energy density depends on the deformation map Φ through the local strain measures C and kn . Therefore, the hyperelastic potential of the continuum rope can be written as: " 1 Vs W (C, κ) = A
! # √ √ κ CA 2/κ sin + Vθ π − κ CA . 2
(2.18)
The total strain energy of the continuum system approximating the atomistic energy of Eq. (2.11) can then be written as: Z Πrope (Φ) =
W (C, κ) dΩ0 .
(2.19)
Ω0
By taking derivatives of the hyperelastic potential W with respect to the strain measures, Lagrangian stress measures arise: the work conjugate to C is an axial stress analogous to the second Piola-Kirchhoff stress tensor, and the conjugate of κ is a bending moment-like stress.
2.6.3
Thought example and discussion
Suppose an initially rectilinear undeformed rope of length nA is bent into a circle of radius r with uniform stretch. The continuum stretch is Λ = 2πr/(nA), and the curvature is κ = 1/r. Since the atoms are postulated to lie on the continuum surface, the corresponding atomic chain containing n bonds is deformed into a regular
20 polygon of n sides whose circumcircle has a radius r. The bond length and angle predicted by the continuum model (see Eqs. (2.14,2.17)) are: a = 2r sin
π and θ = π − 2π/n. n
(2.20)
It is easy to see that these predictions coincide exactly with the actual bond lengths and angles of the atomic chain deformed into a regular polygon; this is a situation in which z = Φ(Y ) (see Remark 2.5). Therefore, the predicted energetics for this finite deformation are also exact. Of course, for a general deformation with non-constant stretch and curvature, the local approximation of the exponential Cauchy-Born rule will lead to approximate energetics. The examples presented later demonstrate, however, that this approximation is very accurate. The inadequacy of the standard Cauchy-Born rule can be illustrated easily in the present example. The standard Cauchy-Born rule corresponds to taking a = w = FA. Suppose that our rectilinear one-dimensional undeformed body is deformed into a circle without stretch, i.e. Λ = C = 1. The application of the standard Cauchy-Born rule leads to deformed lattice vectors that are tangent to the rope. Consequently, two lattice vectors emanating from the same nucleus remain collinear after deformation, so the angle they form is unchanged irrespective of the bending of the rope. Furthermore, since the rope is bent without stretch, the length of the deformed lattice vectors also remains unchanged (see Eq. (2.14)). Therefore, the energy of such a model will remain unchanged, and the resulting rope has zero bending stiffness. However, the real lattice vectors do not remain coplanar and their length changes due to the curvature even if Λ = C = 1, since from Eq. (2.20) it follows that for this isometric deformation a = nA/π sin
π and θ = π − 2π/n. n
(2.21)
Therefore, the energy of the atomic system will change when deformed in this fashion. Thus, a continuum model based on the standard Cauchy-Born rule is blind to the fact that the rope is being bent, and assigns zero energy change to the deformation, in sharp contrast with the exponential Cauchy-Born rule, which predicts the correct energetics. Although an intuitive approach would associate the continuum stretch to the stretch of the bonds, and the continuum curvature to changes in bond angles, the proposed model couples these deformation modes. Indeed, the continuum bond length a, which is the argument for the interatomic stretch potential, depends both on C and κ in a non-linear fashion. The same applies to the continuum bond angle
21 θ. This feature is essential and makes the continuum model exact for deformations that map an initially straight chain into a circular arc with constant stretch. Thus, as in the case of the standard Cauchy-Born rule for bulk crystalline materials, the resulting continuum model for the rope is exact for such homogeneous deformations. Note that in this thought example we have considered a particular deformation with uniform stretch and curvature, and checked that the energies provided by the continuum and the atomistic models coincide. This “consistency” result does not guarantee that given a boundary value problem, the continuum model will provide a good approximation to the atomistic solution. Such “stability” rigorous analysis has been lacking until very recently even for the case of space-filling crystals. The work of Friesecke and Theil (2002) represents a first step in this direction for a class of simple 2D lattices. The following section provides numerical evidence of the accuracy of the continuum rope-like hyperelastic model in boundary value problems.
2.6.4
Numerical example
In this section, a numerical example that demonstrates the effectiveness of the continuum model in Eq. (2.18) is presented. The continuum model is discretized using C 1 Hermite finite elements, because the formulation requires bounded second derivatives of the deformation. The finite element model is then compared to its atomistic counterpart. In the presented example, a uniform force field leftwards is applied to a circular atomic chain with one fixed point. The continuum counterpart is subject to a body force, and the corresponding exterior potential is: Z B · Φ dΩ0 ,
Πext = Ω0
where B is the body force per unit undeformed length. If the forces applied on the atomic system are a certain constant force f acting on each atom, then B is simply obtained by dividing f by the undeformed interatomic spacing A. Figure 2.8(a) shows the results obtained with both molecular mechanics and continuum mechanics discretized with FE (16 elements). Starting with a ring, the load is applied in eight stages, and then equilibrium configurations are obtained at each stage by minimizing the energy with the conjugate gradients method. For the continuum model, a reference body is also defined. The reference body is a straight segment and corresponds to an absolute minimum of the elastic energy. Then, this segment is wrapped into a ring, and a constraint keeping the ends of the segment together is imposed.
22
(a) 16 elements
(b) 64 elements
Figure 2.8: Comparison between the molecular mechanics (◦) and continuum mechanics () simulations. The atomistic system has 64 atoms and the continuum simulation is performed with (a) 16 Hermite Finite Elements, i.e. the number of degrees of freedom is halved, and (b) 64 Hermite Finite Elements. Very good agreement between the molecular mechanics simulation and the FE one can be observed. The latter has fewer degrees of freedom. The optimized shapes almost coincide even for very large non-homogeneous deformations, although the FE model seems to be slightly stiffer. At the last stage, the difference in elastic energy between the atomistic and the FE model is 0.15%. An advantage of the FE model is that the mesh can be adapted according to some optimality criterion to obtain accurate solutions at optimal cost. Figure 2.8(b) shows how a “converged” finite element simulation based on the continuum model compares with the molecular mechanics simulation. It can be seen that the deformations are even closer than in the previous case. In this case, the FE model has more degrees of freedom and leads to a more expensive simulation (in Hermite finite elements in 2D, each node carries four degrees of freedom). In principle, this fine model would be able to develop sub-lattice features that cannot exist in the molecular system, but the numerical simulations show it does not. The simulation should be understood as an illustration of how the molecular system compares with the exact continuum system. To sum up, the examples presented demonstrate the ability of the continuum hyperelastic rope to accurately mimic the atomistic systems.
23
2.7
Summary and conclusions
The geometric structure of the standard Cauchy-Born rule in the context of finite deformation continuum mechanics of manifolds has been investigated. This analysis highlights some inconsistencies of this standard kinematic assumption linking the continuum and atomistic deformations. These inconsistencies are minor formalities in the case of space-filling crystals, but become crucial for low-dimensionality crystals deforming in R3 . We have shown that the standard Cauchy-Born rule cannot be applied “as is” to the case of curved crystalline sheets or ropes. Physical arguments demonstrate that the standard Cauchy-Born rule produces models blind to the curvature of the lower-dimensionality continuum. The postulate is made that the continuum model replacing a lattice of dimensionality m < 3 should be a m−dimensional manifold, and that the lattice vectors should be viewed as chords of such a surface/curve. Under these circumstances, an extension of the standard Cauchy-Born rule based on the exponential map which overcomes the shortcoming of the standard rule for curved continua is proposed. This extended rule is called exponential Cauchy-Born rule. Essentially, the inverse of the exponential map maps the undeformed lattice vectors (chords of a curved manifold) to the tangent of the undeformed body. This tangent vector can then be transformed by the deformation gradient, producing a vector tangent to the deformed body, which is “brought back” to the deformed body by the exponential map. An illustrative example of an atomic chain deforming in two dimensions has been presented. The resulting simple rope-like continuum encompasses all of the fundamental ideas, and is a model for the more complicated cases discussed in further chapters. This continuum rope-like model is qualitatively analyzed, and its accuracy is verified by comparing atomistic simulations of the original chain system with the solutions provided by the continuum discretized with finite elements.
Chapter 3 Application to curved crystalline monolayers: carbon nanotubes 3.1
Introduction
Motivation Carbon nanotubes hold great potential for applications in nano-structured materials and nano-electro-mechanical devices. These large molecules, or small solids, have been analyzed by atomistic simulations based on analytical potentials (Yakobson et al. 1996; Gao et al. 1998; Cornwell and Willie 1997; Belytschko et al. 2002; Hertel et al. 1998), tight-binding methods (Rochefort et al. 1999; Hern´andez et al. 1999), and ab initio calculations (Zhou et al. 2001; Maiti 2000; Lier et al. 2000; S´anchez-Portal et al. 1999). These simulations have helped in the understanding of experimentally observed phenomena such as buckles in bent nanotubes or collapsed and twisted configurations. Furthermore, given the difficulty of carrying out experiments at this scale, simulations provide information not accessible otherwise. Although atomistic simulations appear as the best suited approach to analyze carbon nanotubes, their computational cost limits them to relatively small systems for very short time intervals. Indeed, for bundles or multi-walled nanotubes, the computational cost becomes prohibitive. The present chapter addresses the space scale issue, and a continuum mechanics approach for nanotubes is proposed. Brief review of continuum modelling of nanotubes Continuum mechanics simulations can model much larger systems than atomistic calculations because they do not need to track every single nucleus, and the dis24
25 cretization is independent of the atomic sites. Furthermore, continuum theories help in the understanding of mechanical phenomena beyond the numerical output of atomistic simulations. Many authors (Chopra et al. 1995; Yu et al. 2001; Sohlberg et al. 1998; Zhong-can et al. 1997; Yakobson et al. 1996) have analyzed to various extents the mechanics of carbon nanotubes through continuum mechanics, although, as noted by Yakobson et al. (1996) “its relevance for a covalent-bonded system of only a few atoms in diameter is far from obvious”. Even without explicitly formulating a continuum theory, recourse to continuum mechanics concepts such as shells, Young’s modulus, or buckling is ubiquitous in the literature. Nevertheless, in our opinion, the current application of continuum mechanics to carbon nanotubes presents two limitations. The first one, pervasive in continuum analysis of solids, stems from the phenomenological character of the constitutive relations. The properties of a postulated material model, e.g. linear elasticity, are fit to data accessible from experiments or atomistic simulations. Several authors (Yakobson et al. 1996; Ru 2000; Ru 2001; Popov et al. 2000; Lourie et al. 1998; Yu et al. 2000) assume linear-elastic isotropic thin shell theory as a natural idealization for the mechanics of carbon nanotubes. In this case, Young’s modulus and Poisson’s ratio define the elastic material. Elastic shell theory requires the definition of a wall thickness which is most commonly assumed to be the equilibrium inter-layer distance in graphite t = 0.34 nm. One can argue that the physical nanotube has a finite thickness, since the electronic clouds occupy a certain volume of space. Nevertheless, once the abstraction of the BornOppenheimer approximation is made, that the energy of the system depends only on the nuclear positions, graphene sheets become two-dimensional crystals. Thus, as noted by Hern´andez et al. (1999), the notion of thickness for a wall one atom thick is quite inappropriate from a purely mechanistic point of view. Furthermore, the selection of this thickness together with the application of elastic thin shell formulas leads to inconsistencies between the axial and the bending moduli of nanotubes. This problem has been formally overcome either by defining a fictitious thickness of t = 0.066 nm for which the thin shell formulas for Young’s modulus and the flexural stiffness are consistent, thereby modifying the value of Young’s modulus (Yakobson et al. 1996), or by defining the bending effective thickness as an independent elastic parameter (Ru 2000). Both ad hoc approaches manifest the inadequacy of thin shell theory for an intrinsically two-dimensional material such as the graphene monolayer. It should be noted in this respect that shell theory is a convenient model for space-filling (bulk) materials that form structures very thin in one dimension. The second limitation in the current applications of continuum mechanics to carbon nanotubes and other monolayer crystalline sheets is that in most cases the proposed models are restricted to infinitesimal deformations. Extensions beyond linearized elasticity are scarce and restricted to simplified settings (Chopra et al. 1995; Yu et al. 2001), and do not represent a general approach to the nonlinear
26 mechanics of nanotubes. For instance, the buckling patterns observed in atomistic simulations have been studied through linearized bifurcation analysis of linear elastic shells by Yakobson et al. (1996), thus going strictly beyond linear theory. The predominance of small deformation elastic analysis of nanotubes contrasts with the fact that nanotubes undergo very large nonlinear deformations elastically, reversibly, with intact bond topology. There is experimental evidence that carbon nanotubes can be dramatically bent and buckled many times with an Atomic Force Microscope (AFM) tip without damaging the crystalline structure (Falvo et al. 1997). Other researchers (Walters et al. 1999) have induced very large elastic deformations on freely suspended nanotube ropes cyclically with an AFM tip, without observing irrecoverable deformations. Kinks and buckles have been observed (Chopra et al. 1995; Falvo et al. 1997) and associated with elastic structural instabilities. Atomistic simulations support these observations (Bernholc et al. 1998; Maiti 2000), and carbon nanotubes have been severely deformed in calculations before fracture or plasticity appear. This celebrated resilience of the bond network together with the often noted shell-like behavior supports the idea of modelling carbon nanotubes by an elastic finite deformation continuum model that fully accounts for the reversible nonlinearities. Finite crystal elasticity Molecular theories of elasticity bridge atomistic descriptions of crystalline solids with finite deformation continuum mechanics (Stakgold 1950; Ericksen 1970; Martin 1975; Cousins 1978; Milstein 1982; Weiner 1983; Ericksen 1984), and thus provide a natural framework to overcome the above mentioned limitations. An alternative asymptotic approach has been recently proposed by Friesecke and James (2000), and Qian et al. (2002) proposed a method based on constrained atomistic calculations in combination with meshfree methods, in the spirit of the non-local quasicontinuum method. The Cauchy-Born rule, also referred as method of the homogeneous deformations, is a fundamental kinematic assumption that links the deformation of lattice vectors to that of the continuous medium, and has been recently proven rigorously to hold under certain conditions (Friesecke and Theil 2002). By means of the Cauchy-Born rule, the continuum elastic potential can be obtained by equating the deformation energy of a representative cell of the lattice to that of an equivalent volume of the continuum. The resulting continuum constitutive model depends only on the atomistic description of the system, without additional phenomenological input. In addition, constitutive models constructed in this fashion inherit the crystal symmetries and anisotropy, and can treat the finite deformations. Finite crystal elasticity has been used to obtain elastic moduli and study the stability of crystals (Stakgold 1950; Hill 1975; Cousins 2001), to study phase transformations in solids (Ericksen 1984; Zanzotto 1996), and recently these ideas have been cast in a
27 computational framework to solve general boundary value problems in combination with finite elements by the quasicontinuum method (Tadmor et al. 1996; Shenoy et al. 1999; Tadmor et al. 1999; Smith et al. 2000). This method has emerged as an efficient and accurate simulation method at the nano-scale. There are several variants of the quasicontinuum method in the literature. Basically, when applied to materials with long-range interactions which easily develop lattice defects, such as metals, constrained atomistic calculations which are adaptively refined down to the atomic spacing are required (Tadmor et al. 1996; Shenoy et al. 1999). On the other hand, when applied to covalently-bonded systems like silicon, a local version relying on the Cauchy-Born rule allows us to write in closedform the continuum constitutive law in terms of the atomistic model, i.e. no local atomistic calculations are needed (Tadmor et al. 1999; Smith et al. 2000). The local quasicontinuum method is particularly advantageous when dealing with crystals with predominant short-range interactions, such as graphene. The method proposed here can be seen as a generalization of this local quasicontinuum to curved crystalline monolayer sheets. Scope and outline Unfortunately, the development of crystal elasticity theories for carbon nanotubes is not straightforward. The basic kinematic law linking atomistic and continuum deformations, the Cauchy-Born rule, cannot be applied “as is” to the case of curved crystalline sheets one atom thick, particularly if curvature effects are to be accounted for. In other words, it is applicable only to space-filling crystals—a graphene sheet deforming in its plane is space-filling in two-dimensions, but a bent graphene sheet must be described in 3D, where it is not space-filling. In Chapter 2, we developed an exponential Cauchy-Born rule which extends the standard Cauchy-Born rule to curved single layer lattices. The resulting special finite deformation continuum mechanics model does not correspond with conventional shell theories, and is original in that it views carbon nanotubes as surfaces without thickness, unlike other continuum mechanics approaches. (Qian et al. (2001) and Zhang et al. (2002) concurrently presented continuum models with crystal elasticity ideas, but viewed the tube wall as a shell with thickness). This is why the term membrane is adopted, as opposed to shell, which carries implicitly the notion of thickness. It should be noted however that by doing this, another terminological convention is violated, since the elastic energy of a classical membrane depends only on its stretch; in the present theory, it depends also on the curvature, and thereby has bending stiffness as well. In the present chapter, the abstract presentation of the exponential Cauchy-Born rule of Chapter 2 is realized for the case of two-dimensional crystalline monolayers deforming in R3 . A local approximation of the exponential Cauchy-Born rule is proposed, which results in hyperelastic constitutive relations which depend on the
28 stretch and the curvature of the surface. The model is particularized to graphene to apply it to the simulation of carbon nanotubes. The finite deformation constitutive relation is constructed in terms of the atomistic description of the system (here, Brenner’s potential), and does not require local atomistic calculations, i.e. it is written in closed-form. Details about its evaluation are provided. A continuum version of the non-bonded interactions is also formulated, and the continuum statement for the statics of the membrane is provided. The finite element implementation of the theory is developed, and a detailed numerical study involving buckling of nanotubes under different loading conditions demonstrates that the continuum/finite element approach very accurately mimics the parent atomistic model in the full nonlinear regime. Furthermore, it is shown that the model predicts many qualitative features of experiments. Our simulations suggest that, in the absence of bond rearrangement or defects, the nonlinear mechanics of carbon nanotubes can be accurately modelled within the strict framework of continuum mechanics, i.e. without any sort of coupled or underlying atomistic calculation. Numerical examples also illustrate the dramatic computational saving which can be achieved for large multi-walled nanotubes containing millions of atoms, and replicate some unusual features observed in experiments. The full atomistic calculations for these examples are only possible at the present time by super-computers.
3.2
Exponential Cauchy-Born rule for 2D lattices
This section presents a concise formulation of the kinematics of the surface replacing the curved single layer two-dimensional lattice. This presentation focuses on the practical computation of the variables of interest for a numerical implementation of the theory. Then, the local approximation of the exponential Cauchy-Born rule for surfaces is presented.
3.2.1
Kinematics
As argued in Chapter 2, the continuum object replacing the graphene monolayer is a surface without thickness. The atoms are assumed to lie on the surface, and therefore, the bonds are chords of the surface. It is convenient to define the undeformed or reference system as the planar graphene sheet. This choice is natural for carbon nanotubes, since this state represents the ground (equilibrium) energy level. The spirit of the notation follows Marsden and Hughes (1983), and do Carmo (1976) is a useful reference for the differential geometry of surfaces. See also Dierkes et al. (1992) for a compact and clear presentation of the differential geometry of
29 x2
j0 A01
A0 2
X2
W
j
x x1
Q0
TxW
A03
x
W X
j0(x)
W0 X1
x3
F = j o j0-1 x1
F (X ) = j (x) x2
Figure 3.1: Membrane kinematics surfaces, particularly with respect to the expression of the fundamental forms in the parametric space.
General setting, coordinate systems, and notation The undeformed body is considered to be two-dimensional, i.e. Ω0 is an open set of R2 , and models a slab of planar graphene sheet. The deformation map Φ maps this undeformed body into the Euclidean space X ∈ Ω0 7−→ x = Φ(X) ∈ R3 . The deformed body Ω = Φ(Ω0 ) is a smooth surface. Let us describe the undeformed body by Euclidean coordinates {X 1 , X 2 }. The corresponding orthonormal basis of the tangent of the undeformed body T Ω0 is B0 = {I1 , I2 }. Analogously, the Euclidean coordinates {x1 , x2 , x3 } describe R3 , and the associated standard basis is B = {i1 , i2 , i3 }. It is convenient in the numerical formulation to define a parametric body Ω ⊂ R2 . Let this parametric body be described by Euclidean coordinates {ξ 1 , ξ 2 }, and the corresponding orthonormal basis is B = {Ξ1 , Ξ2 }. The undeformed configuration is a differentiable and invertible map ξ ∈ Ω 7−→ X = ϕ0 (ξ) ∈ Ω0 ⊂ R2 such that ϕ0 (Ω) = Ω0 . Similarly, the deformed configuration maps smoothly and bijectively the parametric body into the deformed body ξ ∈ Ω 7−→ x = ϕ(ξ) ∈ Ω ⊂ R3 , ϕ(Ω) = Ω. The deformation map is then −1 Φ = ϕ ◦ ϕ−1 0 : X ∈ Ω0 7−→ x = Φ(X) = ϕ(ϕ0 (X)) ∈ Ω.
(3.1)
30 The vector from the origin to the point Φ(X) in R3 is denoted by Φ(X), and coincides with the vector ϕ(ξ) where ξ = ϕ−1 0 (X). The components of these vectors, for instance ϕ, in the standard basis B coincide with the components of the point mappings ϕ in the coordinate system {x1 , x2 , x3 }, and are therefore denoted by the same symbol ϕa , a = 1, 2, 3. Note that boldface is reserved for vectors and tensors, while points (x, X, ξ, . . .) and point mappings (Φ, ϕ0 , ϕ, . . .) are denoted with lightface. This general setting is illustrated in Fig. 3.1. At each point of the surface x ∈ Ω, the tangent space Tx Ω is a linear space which can be viewed as the plane tangent to Ω at x “centered” at this point. The convected basis of the tangent of the deformed body T Ω, C = {g1 , g2 } is defined in terms of the components of ϕ in the coordinate system {x1 , x2 , x3 } and the corresponding Euclidean basis vectors of R3 by gα =
∂ϕ ∂ϕa = ia , α = 1, 2. ∂ξ α ∂ξ α
(3.2)
The following conventions and notations expand on the general guidelines provided at the beginning of the thesis. Components in the convected basis C , as well as in the parametric Euclidean basis B, are denoted by Greek indices (α, β, ...), and run from 1 to 2. Components in the Euclidean coordinate system of Ω0 and the associated basis B0 , are denoted by upper case indices (A, B, ...) which run from 1 to 2, while lower case indices (a, b, ...) denote components in the Euclidean coordinate system of R3 and the basis B, and run from 1 to 3. Summation on these indices when repeated is implied. Other indices (i, j, k, n, I, ...) do not represent components of vectors and tensors in these bases, and summation is not implied by the repetition of these indices. Super-indexes (contravariant indexes) act on forms, while sub-indexes (covariant indexes) act on vectors. Brackets with a basis in the subscript denote the matrix representation of a tensor in that particular basis. The matrix representation of two point tensors requires two bases in the subscript to specify the basis used for each index. To keep the notation simple, we do not distinguish in this notation between the bases and the corresponding dual bases. It is understood that covariant indices are expressed in the dual bases. For one point tensors, if the same basis is used for each index, only one basis in the subscript is sufficient.
Tangents of the configurations and deformation gradient In the following, indicial notation, invariant notation, as well as matrix representation of tensors are provided on many occasions for clarity. The matrix representation of the tangent map of the undeformed configuration in the Euclidean bases B-B0 , is denoted according the our conventions as [T ϕ0 ]B0 B . Its components can be com-
31 α puted as (T ϕ0 )A α = ∂ϕA 0 /∂ξ . The element of area in the undeformed body can be expressed as dΩ0 = det(T ϕ0 )dξ 1 dξ 2 . (3.3)
Since both B and B0 are Euclidean bases, det(T ϕ0 ) can be computed as the matrix determinant det[T ϕ0 ]B0 B . On the other hand, noting that the image of the referential basis vector Ξα through the tangent of the deformed configuration T ϕ is the convected basis vector gα , we have [T ϕ]C B = [Id]2×2 (the information about the deformation is contained in the convected basis vectors). Applying the chain rule, the deformation gradient can be written as F = T Φ = T ϕ ◦ T ϕ−1 0 . For pull-back operations in the following sections, the components of the deformation gradient in the bases B0 -C are needed. Thus, we have −1 −1 [F]C B0 = [T ϕ]C B [T ϕ−1 0 ]BB0 = [Id]2×2 [T ϕ0 ]B B = [T ϕ0 ]B B , 0
0
(3.4)
or in components β −1 α F α A = δ α β (T ϕ−1 0 ) A = (T ϕ0 ) A .
(3.5)
The metric and the Green deformation tensors The matrix representation of the metric tensor of the surface Ω in the convected basis is: g11 g12 [g]C = (3.6) , g21 g22 where the covariant components (the components in C ) are obtained from the convected basis vectors as gαβ =< gα | gβ >= (gα )a (gβ )a ,
(3.7)
where < · | · > denotes the Euclidean inner product. The metric tensor is nothing but the expression of the Euclidean inner product in the tangent of the surface Ω; the first fundamental form I(w) = kwk, w ∈ T Ω can be written in the convected coordinates as I(w) = gαβ wα wβ , where w = wα gα . Note that the matrix in Eq. (3.6) is symmetric. The Green deformation tensor (with lowered indexes) is defined as the pull-back of the metric tensor C[ = Φ∗ g. Therefore, its matrix
32 representation is [C[ ]B0 = [F]TC B0 [g]C [F]C B0 , or CAB = gαβ F α A F β B .
(3.8)
Using the formulas in the previous section, the matrix expression of the Green deformation tensor (with lowered indexes) in the basis B0 is the symmetric matrix:
α −1 β [g]C [T ϕ0 ]−1 , or CAB = gαβ (T ϕ−1 [C[ ]B0 = [T ϕ0 ]−T 0 ) A (T ϕ0 ) B . B B B B 0
0
(3.9)
Since the Green deformation tensor (with lowered indexes) C[ is always expressed in the Euclidean basis B0 in this thesis, we do not need to distinguish between the tensors C[ and C. To simplify the notation, the [ is dropped.
The principal curvatures The unit normal to the deformed body Ω is n=
1 g1 × g2 , kg1 × g2 k
(3.10)
or in components a
n =p
a bc (g1 )b (g2 )c , d ef d gh (g1 )e (g2 )f (g1 )g (g2 )h
(3.11)
where k · k denotes the Euclidean norm, and a bc is the permutation symbol. The covariant components of the curvature tensor, that is the matrix elements of [k]C , can be obtained as kαβ =< n | gα,β >= na (gα,β )a , (3.12) where (·),β denotes ∂(·)/∂ξ β . The second fundamental form of the surface Ω can be expressed in convected coordinates as II(w) = kαβ wα wβ . Similarly to the metric tensor, the pull-back of the curvature tensor K = Φ∗ k can be expressed in the Euclidean basis B0 as the symmetric matrix: [k]C [T ϕ0 ]−1 , [K]B0 = [T ϕ0 ]−T B B B B 0
0
α −1 β or KAB = kαβ (T ϕ−1 0 ) A (T ϕ0 ) B .
(3.13)
33 The principal curvatures k1 and k2 , and the principal directions v1 and v2 of the surface Ω are the eigenvalues and eigenvectors of the Weingarten map, i.e. they are characterized by being the maximum and the minimum of the quotient II(w/I(w), w ∈ T Ω and w 6= 0. Note that the principal directions are tangent to the surface. Using convected coordinates, the principal curvatures and directions are found as solutions of the generalized eigenvalue problem [k]C [v]C = k [g]C [v]C .
(3.14)
Alternatively, it is possible to find the principal curvatures, and the pull-back of the principal directions expressed in B0 solving the generalized eigenvalue problem [K]B0 [V]B0 = k [C]B0 [V]B0 ,
or
KAB V B = k CAC V C ,
(3.15)
where now V1 , V2 ∈ T Ω0 . The eigenvectors, which are C−orthogonal, are normalized with respect to C, so that CAB (Vn )A (Vm )B = δnm .
(3.16)
Details about the solution of this eigenvalue problem, as well as formulas for the derivatives of the principal curvatures and directions with respect to C and K are provided in Appendix A.1.
3.2.2
Local approximation of the exponential Cauchy-Born rule
As detailed in Chapter 2, the standard Cauchy-Born rule postulates that the lattice vectors deform according to the linear transformation a = FA, where A denotes an undeformed lattice vector and a this vector after deformation. This kinematic assumption, which for space-filling crystals has proven to be useful and rigorously valid in some cases, fails to capture the mechanics of curved crystalline monolayers. This is particularly true with regards to the effect of bending a single layer crystalline sheet. Inspired by the geometric structure of the Cauchy-Born rule in the context of the finite deformation kinematics of surfaces, an extension of the Cauchy-Born rule to account for the curvature of the film was proposed in Chapter 2. In the present setting, the undeformed body is considered for convenience to be flat, i.e. Ω0 ⊂ R2 .
34
F =TF expx X
W0
A
Tx W
x W
w a
Figure 3.2: The exponential map transforms the vector w = FA tangent to the surface into a chord of the surface a. Therefore, the exponential Cauchy-Born rule can be summarized with the formula a = exp ◦ FA. The basic idea is that, in the setting of 2D lattices deforming in 3D, the standard Cauchy-Born rule produces a deformed lattice vector which is tangent to the surface, not a chord. The exponential map brings this tangent vector back to the surface (see Fig. 3.2 for an illustration). In general the evaluation of the exponential map, and therefore the application of the extended rule, requires knowledge of the geodesic curves. In a given coordinate system, these are obtained by integration of a system of two nonlinear ODE’s. The coefficients of these equations are the Christoffel symbols. In general, a closed-form solution of these equations is not available, but one can construct approximations based on these equations. For instance, they could be solved numerically, or simplified so that a closed-form solution is available. These approaches have obvious drawbacks. The first one is the computational cost and complexity of the resulting method. Secondly, it would be difficult to guarantee frame-indifference. In Chapter 2, when considering an atomic chain deforming in 2D, a geometric approach was adopted. In this simple kinematic setting, the curve replacing the chain was locally assumed to be a circle with the curvature and the stretch of the actual curve. Then, since the exponential map for the circle is trivial, the deformation of the lattice vectors was written in terms of the local deformation of the continuum rope. We now consider an arbitrary deformation of the crystalline film. Rather than trying to build at each point of the membrane a local representation based on its local deformation, the exponential map is approximated by decoupling the principal directions. The procedure separately considers each principal direction V1 and V2 of the curvature tensor, and two corrections for the tangent deformed lattice vector w are obtained from the exponential map for fictitious cylinders of radius 1/k1 and 1/k2 (the exponential map of the cylinder can be easily obtained in closed-form). Finally, the corrections in each direction are added to w to obtain a. This approach
35 is simple and leads to accurate predictions according to the numerical experiments described later. Basic setup For the planar undeformed crystal, the exponential Cauchy-Born rule simplifies to: a = expΦ(X) ◦ F(X) A,
(3.17)
where we are identifying Ω0 and TX Ω0 . The first part of this map can be readily performed and is equivalent to the standard Cauchy-Born rule: w = FA.
(3.18)
We call this vector the tangent deformed lattice vector, and it can be thought of as the push-forward of A. F
X
2
V2
v2 = F V2
X
V1
v1 = F V1
x b
A
W 0 ,TX W0
X1
w = FA
Tx W
Figure 3.3: Principal directions of the curvature tensor in the undeformed and the deformed bodies, and a lattice vector A Given a deformation map for the originally planar membrane, the local deformation can be characterized by the Green deformation tensor C and the curvature tensor expressed in Ω0 , K. The eigenvalue problem (3.15) defines the principal directions expressed in the undeformed configuration and principal curvatures of the surface Ω. Figure 3.3 illustrates that only the push-forward of V1 and V2 by Φ, that is v1 and v2 , are orthonormal in the Euclidean sense. This figure also shows a generic undeformed lattice vector A. Note that v1 , v2 and w are vectors of the tangent to the surface TΦ(X) Ω. Consider an auxiliary Euclidean coordinate system of R3 , {e x1 , x e2 , x e3 } centered at x = Φ(X) and whose axes are parallel to v1 , v2 , and v1 × v2 . The associated e = {v1 , v2 , v1 × v2 }. Note that B and B e only differ by a orthonormal basis is B
36 rigid body transformation. Consider also the restriction of this coordinate system eT Ω = {v1 , v2 }. to Tx Ω, {e x1 , x e2 } with the basis B x Let us define the angle β (see Fig. 3.3) that v1 and w form in TΦ(X) Ω. Recalling that the principal directions are C−normalized, this angle is characterized by: V1 · C[ · A cos β = √ , A · C[ · A
V2 · C[ · A sin β = √ . A · C[ · A
(3.19)
The tangent deformed lattice vector w can be obtained as kwk = √ length of the √ [ eT Ω can be comA · C · A = CAB AA AB . The components of w in the basis B x puted as: w1 w2
=
< w | v1 > < w | v2 >
=
V1 · C[ · A
=
V2 · C[ · A
CAB AA (V1 )B
.
(3.20)
CAB AA (V2 )B
k2 < 0 v1 Dw1
Tx W
w
x
C1
Tx W
Dw2
w
x
v2
k1 > 0 C2
Figure 3.4: Exponential map in fictitious cylinders in each principal direction
Principal direction 1 Figure 3.4 illustrates the approximation to the exponential map for a surface with negative Gaussian curvature—a hyperbolic point. Consider a fictitious cylinder C1 of radius 1/k1 passing through point x = Φ(X) whose tangent plane is Tx Ω and whose axis is perpendicular to v1 . Using the above defined coordinate systems, the
37 cylinder C1 can be parametrized isometrically from Tx Ω into R3 as 1
2
C1 : f1 (e x ,x e )=
1 1 1 2 1 sin k1 x e; x e; (1 − cos k1 x e) . k1 k1
(3.21)
The geodesic of this cylinder passing through x and tangent to w is c(s) =
1 1 sin [k1 (cos β)s] ; (sin β)s; (1 − cos [k1 (cos β)s]) , k1 k1
(3.22)
where s denotes the arc-length parameter. Note that this curve is a circle if β = 0 mod π, a straight line if β = π/2 mod π, and a helix otherwise. Consequently, by evaluating the above expression at s = kwk, the image of w through exponential map of C1 at x is:
expx,C1 w Be =
1 k1
sin k1 w
1
w2
.
(3.23)
1 1 (1 − cos k w ) 1 k1
Finally, the exponential correction in the first principal direction, that is the difference between the above vector and w, is:
[∆w1 ]Be =
sin k1 w − w . 0 1 1 (1 − cos k w ) 1 k1 1 k1
1
1
(3.24)
Principal direction 2 In this case, the parametrization of the cylinder C2 of radius 1/k2 in the coordinate systems described above is 1
2
C2 : f2 (e x ,x e )=
1 1 2 2 x e; sin k2 x e; (1 − cos k2 x e) , k2 k2 1
(3.25)
38 and similarly we obtain
[∆w2 ]Be =
0 1 k2
sin k2 w2 − w2 . 1 (1 − cos k2 w2 )
(3.26)
k2
Note that all of the deformed geometric quantities are expressed in terms of quantities defined in the undeformed configuration.
Final formula The exponential Cauchy-Born rule a = expΦ(X) ◦ F A is then approximated by the map a = FA + ∆w1 + ∆w2 . By defining Q(x) = sin x/x, the expression for the e is: deformed lattice vector in the orthonormal basis B 1 a 2 [a]Be = a = a3
w Q(k1 w ) 1
1
w2 Q(k2 w2 )
.
(3.27)
k2 (w2 )2 k1 (w1 )2 2 1 2 2 Q (k w /2) + Q (k w /2) 1 2 2 2
Bearing Eq. (3.20) in mind, and the fact that k1,2 and V1,2 are obtained from the eigenvalue problem (3.15), it is clear that [a]Be depends only on the undeformed lattice vector A, the Green deformation tensor C and the pull-back of the curvature tensor K. The length of a deformed bond, and the angle between two deformed bonds a and b obtained following Eq. (3.27) can be computed simply as a = kak =
√
ac ac
and
θ = arccos
ac b c = arccos . ab ab
(3.28)
Thus, it is clear that ultimately the bond lengths and angles have been expressed in terms of the continuum strain measures C and K. Basically, we have obtained the derived strain measures a = f (C, K; A) and θ = g(C, K; A, B). These continuous new strain measures are adequate to formulate continuum models from the atomistic description of the system, i.e. given both the lattice structure and the interatomic
39 potential. Note that, although here the potential depends on the bond lengths and angles, it is straightforward to apply the methodology presented here to atomistic descriptions which include dihedral angles.
3.3
Finite deformation membrane for nanotubes
In this section, the lattice structure of graphene is described, and the need to account for the inner displacements—additional internal elastic variables— is highlighted. Then, given an interatomic potential, the hyperelastic potential for the continuum membrane is formulated. The continuum version of the exterior and the non-bonded potential is then presented. With these ingredients, the variational statement of the continuum boundary value problem is provided for the statics of the membrane.
3.3.1
Lattice structure of graphene
The graphene lattice is defined in the undeformed body Ω0 . This lattice has three inequivalent bonds A0i , i = 1, 2, 3 (see Fig. 3.5 for an illustration). Given an orientation Θ0 which by symmetry considerations takes values in (−π/6, π/6], these undeformed bond vectors can be defined as
, (3.29) sin Θ0 cos(Θ0 − 2π/3) , sin(Θ0 − 2π/3)
[A01 ]B0 = A0
[A02 ]B0 = A0
cos(Θ0 + 2π/3) sin(Θ0 + 2π/3)
, [A03 ]B0 = A0
cos Θ0
where A0 is the equilibrium bond length (see Fig. 3.1). When modelling nanotubes, the initial deformed configuration is a cylindrical surface, and thus the initial deformation map maps the undeformed planar graphene sheet into this cylinder. The chirality in the tube can be specified by selecting the appropriate orientation Θ0 with respect to the rolling orientation. For example, p suppose a (n1 , n2 ) nanotube is modelled. Its ideal radius is given by R0 = A0 3(n21 + n1 n2 + n22 )/2π, and the √ chiral angle by arctan[ 3n2 /(2n1 + n2 )] (Saito et al. 1992). In this situation, the undeformed body for such a nanotube of length L can be defined in {X 1 , X 2 } as Ω0 = (0, L) × (0, 2πR0 ). The initial deformed configuration of the nanotube bring-
40
S0 A1 A2 A3
B1 B2
P
h
A01 A02 A03
Figure 3.5: Graphene honeycomb multi-lattice: the two simple Bravais lattices, depicted in different colors, are relatively displaced by the inner displacement η, which consequently also affects the bond vectors which are transformed form A0i into Ai . The unit cell of area S0 is also represented. It contains two nuclei and three inequivalent bonds. ing this undeformed body into the initial cylinder of length L can be defined in 2 2 {x1 , x2 , x3 } as Φ1 = X 1 , Φ2 = R0 cos X , and Φ3 = R0 sin X . In this situation, the R0 R0 chiral angle coincides with Θ0 . Thus, a zig-zag nanotube is characterized in this setting by Θ0 = 0, while an arm-chair nanotube by Θ0 = π/6. When dealing with graphene, special attention must be paid to the fact that the honeycomb lattice is a Bravais multi-lattice. These lattices have more than one basis nucleus, and can be viewed as a collection of inter-penetrating simple lattices (see Fig. 3.5). Note that one atomic site (say a black one) and the lattice basis vectors B1 and B2 , are not enough to define the entire lattice, in particular the white sites. Either a white site or the shift vector P is also needed. The position vectors of the atomic sites can then be obtained as: Xn = ni Bi + mP,
(3.30)
where summation on the index i is implied, n1 and n2 are integers, and m takes the values 0 (black) or 1 (white). The standard crystal elasticity treatment of multi-lattices is to assume that the homogeneous deformation affects each of the simple lattices. Additional kinematic variables describing the relative shifts of the simple lattices must be introduced to properly describe the configurations of uniformly strained multi-lattices. These relative shifts are called inner displacements (Stakgold 1950; Ericksen 1970; Martin
41 1975; Cousins 1978; Tadmor et al. 1999). The optical modes are the analog of the inner displacements in lattice dynamical theories (Weiner 1983). The relative displacement of the basis nuclei cannot be represented by a homogeneous deformation, and is instead an internal mode of deformation. It is clear from Fig. 3.5 that a perturbation in the shift vector by η leaves the basis vectors unchanged, but changes the configuration of the lattice by perturbing the triplet of bond vectors A0i by the same amount. In the continuum setting, additional kinematic variables must be introduced to account for these rearrangements within the unit cell, which for graphene simply affect its elasticity; for other materials they may describe phase transformations (Tadmor et al. 1999; Tadmor et al. 2002). Let η denote the inner displacements, which following Tadmor et al. (1999), are defined in the undeformed body, previous to the “macroscopic” deformation Φ—η is a vector field in T Ω0 . This guarantees rotational invariance of this kinematic variable. Due to the inner displacements, the undeformed lattice vectors in T Ω0 become: Ai = A0i + η,
i = 1, 2, 3.
(3.31)
In the present theory, a given continuum deformation transforms the triplet of undeformed bond vectors according to the exponential Cauchy-Born rule: ai = FX (Ai ) = FX (A0i + η). Note that this implies that we only allow for inner rearrangements in the undeformed planar lattice. Therefore, the inner displacements are viewed as an “in plane” effect and the local rearrangements of the basis atoms cannot move the atoms “out of the surface”. Through Eqs. (3.27) and (3.28), it is possible to express the derived strain measures in terms of the local deformation of the surface and the inner displacements. Thus, the lengths of these deformed bond vectors we can be written as ai = f (C, K; Ai ) = f¯(C, K, η; A0i ),
i = 1, 2, 3,
(3.32)
and the three angles these inequivalent bonds form after deformation as θi = g(C, K; Aj , Ak ) = g¯(C, K, η; A0j , A0k ),
i = 1, 2, 3,
(3.33)
where {i, j, k} is an even permutation of {1, 2, 3}. The dependence on the inner displacements field through Eq. (3.31) has been emphasized. Remark 3.1. It is easy to check that indeed, in the case of graphene, it is sufficient to consider three inequivalent bonds and angles with regards to the exponential
42 Cauchy-Born rule. It suffices to show, on the one hand, that the lengths of the deformed bonds corresponding to Ai and to −Ai are the same, and on the other hand, that the angle between the bonds Ai and Aj after deformation is equal to the angle between −Ai and −Aj after deformation. To see this, note that the deformed bond a corresponding to an undeformed bond A and the deformed bond a− corresponding to −A are related by a symmetry with respect to the line defined by Φ(X) and n.
3.3.2
Interatomic potential and constitutive model
Once the lattice structure of the crystalline sheet has been described, and characterized in the continuum setting, a model for the potential energy of the atomistic system is needed. One of the interatomic potentials considered in this thesis is the Brenner potential for hydrocarbons (Brenner 1990), which follows the bond-order formalism (Tersoff 1988). This analytical potential has been widely used for carbon nanotubes (Yakobson et al. 1996; Cornwell and Willie 1997), and expresses the energy in terms of bond lengths and angles, as a sum over the bonds: E=
XX ¯ij VA (rij ) , VR (rij ) − B i
(3.34)
j>i
¯ij depends on the lengths of the bonds and angles adjacent to the ij−th where B bond. Another potential employed in the calculations in Chapter 4 follows the many-body expansion formalism of the MM2/MM3 models. In particular, a 2body/3-body expansion is considered: E=
X bonds
Vs (rij ) +
X
Vθ (θijk , rij , rik ).
(3.35)
angles
Note that the present approach is not limited to analytical potentials (Tadmor et al. (1999) presented a quasicontinuum method based on the tight-binding method, while in Tadmor et al. (2002) an ab initio Hamiltonian was considered). By considering a representative cell, which for the graphene honeycomb lattice is hexagonal, contains √ two nuclei, one of each inequivalent bonds, and has a surface area of S0 = (3 3/2)A20 (see Fig. 3.5), the strain energy density (energy per unit area) of the continuum membrane can be written by dividing the energy of this cell
43 by its area. For instance, for Brenner potential it is
W = W (C, K, η) =
3 1 X ¯ j , ak , θj , θk )VA (ai ) , VR (ai ) − B(a S0 i=1
(3.36)
where {i, j, k} is an even permutation of {1, 2, 3}. The dependence of this hyperelastic potential on the stretch C and curvature K of the surface, and on the inner displacement field η can be traced in Eqs. (3.32) and (3.33). The dependence of the energy on the undeformed lattice vectors has been omitted. Note that these strain measures and the definition of the inner displacements in the undeformed body guarantee frame indifference—rotational invariance—of the hyperelastic potential. The inner displacements can be eliminated at the constitutive level. Given a deformation of the surface, the strain energy density can be minimized with respect to η: ∂W b (C, K) = arg min W (C, K, η) η =⇒ = 0. (3.37) η ∂η η =ηb After this inner relaxation, the strain energy density can be written as a function of C and K only c (C, K) = W (C, K, η b (C, K)) . W (3.38) Note that, while a closed-form expression for the hyperelastic potential W is availc (C, K) requires the solution of a bivariate able (see Eq. (3.36)), the evaluation of W minimization problem, which is solved numerically by Newton’s method (see Appendix A.2). A membrane and a bending stress tensors can be defined by taking derivatives of the elastic potential with respect to the strain measures. As noted by Tadmor et al. (1999), in doing so one can benefit from the fact that the inner displacements are in internal equilibrium. Indeed, for the derivative with respect the stretch and using Eq. (3.37), we have c ∂W = ∂C
∂W ∂W ∂b ηA + A ∂C ∂η ∂C
= η =ηb
∂W , ∂C η =ηb
(3.39)
and therefore this derivative can be computed in closed-form from the function W .
44
Given CAB and KBC , 1. Principal curvatures: Solve the eigenvalue problem of Eq. (3.15), and obtain the principal curvatures, the pull-back of the principal directions, and their derivatives with respect to the strain measures (see Appendix A.1): kn ,
∂kn ∂kn , , ∂CAB ∂KAB
(Vn )A ,
∂(Vn )A ∂(Vn )A , , ∂CBC ∂KBC
n = 1, 2.
2. Inner relaxation: Minimize W (C, K, η) with respect to η, and find (see Appendix A.2): b : relaxed inner displacements • η c : relaxed strain energy density • W • (Ai )A = (A0i )A + ηbA ,
i = 1, 2, 3: updated undeformed lattice
3. Exponential Cauchy-Born rule: Compute bond lengths and angles (see Eqs. (3.27), (3.28), (3.32) and (3.33)), and their derivatives with respect to the strain measures (see Appendix A.3): ai ,
∂ai ∂ai , , ∂CAB ∂KAB
θi ,
∂θi ∂θi , ∂CAB ∂KAB
4. Stress tensors: Apply the chain rule to Eq. (3.36), and recall Eqs. (3.40) and (3.41): 3 X ∂W ∂ai ∂W ∂θi AB S =2 + , ∂a ∂C ∂θ ∂C i AB i AB i=1 m
AB
3 X ∂W ∂ai ∂W ∂θi = + , ∂a ∂K ∂θ ∂K i AB i AB i=1
Box 3.1: Constitutive model: calculation of the strain energy density and the stresses
45 Thus the membrane or second Piola-Kirchhoff stress tensor can be defined as
S=2
c ∂W ∂W =2 , ∂C ∂C η =ηb
(3.40)
and similarly, a Lagrangian bending (symmetric) tensor can be defined as c ∂W ∂W m= = . ∂K ∂K η =ηb
(3.41)
Box 3.1, in combination with the appendices, describes the calculation of the strain energy density and the stresses.
3.3.3
Non-bonded interaction and external forces
The non-bonded or van der Waals interactions are generally treated by interatomic potentials that only act between non-bonded pairs of atoms. These diffuse interactions are critical to the mechanics of nanotubes interacting with substrates or packed in bundles, of multi-walled nanotubes, and of nanotubes in their collapsed configurations. The non-bonded energy of the atomistic system can be written as Enb =
X X i
Vnb (rij ) =
j>i,j ∈B / i
1XX Vnb (rij ) 2 i
(3.42)
j ∈B / i
where Vnb is the non-bonded potential, rij is the distance between atoms i and j, and Bi is the set of atoms bonded to atom i. A simple argument involving two representative cells of area S0 each containing n nuclei (n = 2 for graphene, see Fig. 3.5) allows us to write the continuum van der Waals energy double density as Vnb (d) =
n S0
2 Vnb (d)
(3.43)
46 where d is the distance between two points in the deformed body. The continuum counterpart of the total non-bonded energy takes then the form Πnb
1 = 2
Z Ω0
Z
Vnb (kΦ(X) − Φ(Y )k) dΩ0Y dΩ0X ,
(3.44)
Ω0 −BX
where BX is a ball centered at X with a radius that is a function of the potential cut-off radius to account for the fact that this potential does not affect bonded atoms. The classical Lennard-Jones potential has been adopted in this thesis for the non-bonded interactions (Girifalco et al. 2000). This reference also develops continuum non-bonded potentials for graphitic structures of fixed geometry (planar graphene, cylindrical nanotubes, spherical C60 molecules), using similar ideas. When external forces are applied on the nuclei (e.g. electrostatic forces), the continuum counterpart is a body force, and the corresponding total external energy is: Z Πext = B · Φ dΩ0 , (3.45) Ω0
where B is the body force per unit undeformed area. If the forces applied on the atomic system consist of a constant force f acting on each atom, then B is simply given by the expression: n B= f. (3.46) S0
3.3.4
Boundary value problem
From the developments of the previous section, we can express the internal energy of an elastic membrane whose undeformed configuration is a planar body Ω0 , and which is subject to the deformation map Ψ , as Z Πint (Ψ ) =
c (C(Ψ ), K(Ψ )) dΩ0 . W
(3.47)
Ω0
The total potential energy of the system is then Π(Ψ ) = Πint (Ψ ) − Πext (Ψ ) + Πnb (Ψ ).
(3.48)
47 The stable equilibrium deformation maps of the system are given by:
Φ = arg inf Π(Ψ ) , Ψ ∈C
(3.49)
i.e. the equilibrium deformation is a minimizer of Π. C is the appropriate space of deformation maps or trial functions accounting for essential boundary conditions. According to the principle of stationary energy, the equilibrium configurations of the system Φ are stationary points of the potential energy functional, and verify the principle of virtual work:
Z 0 = δΠ(Φ) = Ω0
1 S : δC + m : δK dΩ0 − δΠext + δΠnb , 2
(3.50)
for all δΦ ∈ V, the corresponding space of admissible variations. The expressions of δC and δK in terms of δΦ are described in Appendix B . The variations of the non-bonded and the external energy functionals are δΠnb
1 = 2
Z
Z
Ω0
Ω0 −BX
1 V0 (kΦ(X) − Φ(Y )k) kΦ(X) − Φ(Y )k nb
(3.51)
< Φ(X) − Φ(Y ) | δΦ(X) − δΦ(Y ) > dΩ0Y dΩ0X , and
Z B · δΦ dΩ0 .
δΠext =
(3.52)
Ω0
3.4
Numerical implementation
The present section provides details on the numerical approximation of the above boundary value problem for the hyperelastic membrane. The resulting simulations are intended to mimic the lattice statics, also called molecular mechanics or “zero temperature” equilibrium configurations, of the atomistic system. The configurations are first approximated by a finite element scheme. The present membrane theory places a requirement on the finite element space: it must have bounded second order derivatives. Then, the problem of finding stable configurations is solved by direct minimization of the discretized potential energy.
48
3.4.1
Finite element approximation
Let us define the following notation: the superscript (·)h denotes discretized finite element fields, as well as nodal values in global numbering, while the superscript (·)e denotes the restriction to element e of a finite element field or nodal values in local element numbering. The correspondence between these global and the local numbering schemes is established through the standard scatter and gather operations (Belytschko et al. 2000).
Configurations In the finite element setting, the undeformed domain is triangulated in nel elements S f e e so that Ω0 = nel T e=1 0 and T0 ∩ T0 = ∅ if e 6= f . Following the isoparametric concept, a referential or parametric element T is defined, and takes the role of the parametric body Ω. The parametric element is mapped into each undeformed element T0e through the elemental undeformed configuration ϕe0 ; thus, the finite element undeformed configuration ϕh0 is a parametrization of Ω0 defined element-wise. It can be written in terms of the shape functions NI (ξ 1 , ξ 2 ) and nodal coefficients. For each of the components A of the discrete undeformed configuration ϕh0 in the coordinate system {X 1 , X 2 }, and for each element e we have: (ϕe0 )A (ξ 1 , ξ 2 ) =
X
1 2 1 2 (ϕe0 )A I NI (ξ , ξ ), ∀ (ξ , ξ ) ∈ T ,
(3.53)
I
where (ϕe0 )A I are the nodal coefficients in the local element numbering, while according to the above mentioned notation, the nodal coefficients in the global numbering are (ϕh0 )A J. The finite element deformed configuration ϕh is defined in a similar fashion. Each element, T is mapped into the deformed element T e through ϕe . We can express each of the components a of the deformed configuration in the coordinate system {x1 , x2 , x3 }, and for each element e as follows: (ϕe )a (ξ 1 , ξ 2 ) =
X
(ϕe )aI NI (ξ 1 , ξ 2 ), ∀ (ξ 1 , ξ 2 ) ∈ T .
(3.54)
I
The finite element approximation of the position vectors is trivial from that of the
49 x2
h
j0
jh
T
x1
T
X2
T0
e
e
x3
X1
h
h
h
F = j o (j 0 )-1
x1
x2
Figure 3.6: Finite element discretization point mappings, and for instance we have ϕe (ξ 1 , ξ 2 ) =
X
ϕeI NI (ξ 1 , ξ 2 ), ∀ (ξ 1 , ξ 2 ) ∈ T ,
(3.55)
I
where the components of the vector of nodal coefficients ϕeI ∈ R3 in the basis B are simply (ϕe )aI . The nodal degrees of freedom of the system are ϕhJ , while (ϕh0 )J define the undeformed configuration. Note that the vector nodal coefficients ϕhJ only corresponds to the position vector of node J if the finite element scheme is interpolating; the one used here is not. The finite element deformation map is then Φh = ϕh ◦ (ϕh0 )−1 . This general setting is sketched in Fig. 3.6. Here, subdivision finite elements based on Loop’s scheme are used (Cirak et al. 2000). There are 12 shape functions NI (ξ 1 , ξ 2 ) (the quartic box spline shape functions); the approximated field within a triangular element depends not only on the nodal coefficients of its three nodes, but also on the coefficients of its first neighborhood of nodes. This approximation scheme produces H 2 fields, i.e. fields with up to second square integrable derivatives. This is crucial in the present theory since the strain energy density depends the curvature of the surface K, which therefore needs to be square integrable. The need of finite second derivatives of the shape functions is clear from Eq. (3.57) below. Any other smooth enough discrete parametrization of the surface can be used instead of subdivision finite elements, e.g. the element-free Galerkin approximation, applied to Kirchhoff shells by Krysl and Belytschko (1996).
50 Calculation of the strain measures The following quantities are needed in the numerical calculation of the strain measures. The convected basis vectors can be computed, recalling Eqs. (3.2) and (3.54), for each element e as gα =
X
NI,α ϕeI ,
or (gα )a =
X
I
(ϕe )aI NI,α .
(3.56)
I
Note that these vectors are tangent to the finite element surface T e . For the curvature tensor, the derivatives of the convected basis vectors are needed, and are obtained as gα,β =
X
NI,αβ ϕeI ,
or (gα,β )a =
X
I
(ϕe )aI NI,αβ .
(3.57)
I
For the pull-back operations (see Eqs. (3.9) and (3.13)), the matrix elements of [T ϕh0 ]B0 B are required, and in the finite element context can be calculated in each element e as ∂(ϕe0 )A X e A e A (T ϕ0 ) α = = (ϕ0 )I NI,α . (3.58) ∂ξ α I This 2×2 matrix needs to be inverted once at each integration point at the beginning of the computation, is then stored and retrieved each time a tensor needs to be pulled-back.
3.4.2
Discrete minimization problem
Stable configurations of the discrete system are obtained by direct minimization of the potential energy Π(Φh ) of the discretized system. Numerical methods that only require gradients of the discrete energy with respect to the degrees of freedom have been used, namely the Conjugate Gradients method and the BFGS Quasi-Newton method (Gilbert and Nocedal 1992; Liu and Nocedal 1989). We now describe the numerical calculation of the internal and non-bonded energy and the forces. The external contributions are straightforward.
51 Internal energy Since the finite element undeformed configuration ϕh0 is fixed, the finite element deformation map Φh is determined by the array of nodal coefficients ϕhJ . The internal energy for this deformation map is computed by splitting the integral over the undeformed body into elements, transforming these element-wise integrals to integrals in the referential element, and approximating these integrals by numerical quadrature: Z
h
Πint (Φ ) = = ≈
c (C(Φh ), K(Φh )) dΩ0 W
Ω0 nel XZ e=1 T nel X nint X e=1 i=1
c (C, K) det(T ϕe ) dT W 0 c (C, K) det (T ϕe0 )A ω i , W α ξi ξi | {z }
(3.59)
ωie
where nint is the number of quadrature points for the internal energy, ξi denote the quadrature points and ω i the corresponding weights for the referential element. The definition of the weights ωie for the deformed element, which include the determinants, is convenient for subsequent equations.
Internal forces The internal nodal forces of the discrete system are the derivatives of the internal energy with respect to the nodal degrees of freedom (fint )J =
∂Πint , ∂ϕhJ
(3.60)
where (fint )J corresponds to the global numbering. The application of the chain rule allows to compute the elemental contributions to these forces element-wise:
e (fint )I
! c ∂CAB c ∂KCD ∂W ∂W + det(T ϕe0 ) dT = e e ∂CAB ∂ϕI ∂KCD ∂ϕI T nint X 1 AB ∂CAB CD ∂KCD ωie , ≈ S + m e e 2 ∂ϕI ∂ϕI ξi i=1 Z
(3.61)
52
e a • Initialize Πeint = 0, (fint )I = 0.
• Scatter (ϕe )aI from (ϕh )aJ . • Loop over the quadrature points i = 1, nint (everything evaluated at ξi ): 1. Compute the Green deformation tensor (following Eqs. (3.56), (3.7), (3.9) and (3.58)), and its derivatives with respect to the degrees of freedom (see Appendix B): ∂CAB CAB , . ∂(ϕe )aI 2. Compute the (pull-back of the) curvature tensor (following Eqs. (3.11), (3.57), (3.12), and (3.13)), and its derivatives with respect to the degrees of freedom (see Appendix B): KAB ,
∂KAB . ∂(ϕe )aI
c (C, K) and the stress tensors S AB 3. Compute the strain energy density W and mAB (see Box 3.1). 4. Increment the elemental internal energy and forces: Πeint
←
Πeint
e a (fint )I
←
e a (fint )I
c e + W ωi , 1 AB ∂CAB CD ∂KCD + S + m ωie , 2 ∂(ϕe )a ∂(ϕe )a I
I
e a h a • Gather (fint )I into (fint )J , and add Πeint to the total internal energy.
Box 3.2: Calculation of the elemental internal energy and forces
53 where ωie is defined in Eq. (3.59). These elemental forces are then gathered into the global internal force array (fint )J . The computation of the internal energy and forces in each element is summarized in Box 3.2. The details of the calculation of the derivatives of the strain measures with respect to the nodal degrees of freedom are provided in Appendix B. It is worth mentioning that the symmetry of the strain and the stress tensors is taken advantage of in the computer implementation by means of Voigt notation (Belytschko et al. 2000).
Non-bonded energy For the sake of simplicity, the exclusion of “bonded” integration points in the computation of the non-bonded energy has been overlooked in the following expressions (see Eq. (3.44)). However, it is important not to compute the non-bonded energy of pairs of integration points which, in the undeformed body, lie within the bonded distance. The numerical evaluation of the non-bonded energy entails a double integral, which is performed as a double loop over the elements. Because of the high computational cost of this operation, the computer implementation includes a binning algorithm to search for close (within van der Waals interaction distance) neighbors, and the neighbor lists are updated every few energy evaluations. The numerical approximation of this energy can be written as 1 Πnb (Φ ) = 2 h
=
Z Ω0
nel Z X e=1
=
Z
Vnb kΦh (X) − Φh (Y )k dΩ0Y dΩ0X
Ω0 −BX nel Z X
T f =e+1
T
Vnb kϕe (ξ) − ϕf (ζ)k det(T ϕe0 ) det(T ϕf0 ) dT ζ dT ξ
nel X nnb X nel X nnb X e=1 i=1
Vnb k ϕe (ξi ) − ϕf (ζj ) k ωie ωjf , | {z } f =e+1 j=1 ϕe−f i−j
(3.62)
where ϕe−f i−j denotes the vector defined by the i−th quadrature point of element e and the j−th quadrature point of element f , nnb is the number of quadrature points for the non-bonded term, and ωie is defined in Eq. (3.59). Note that the numerical quadrature for the integration of the non-bonded term needs not be the same of that used for the internal energy term, i.e. in general nnb 6= nint. The latter is dictated by the stability of the finite element discretization and is independent of the element size (see Appendix C), while the former is determined by the van der Waals cut-off distance relative to the element size. The non-bonded potential decays with distance, and a cut-off radius is usually implemented. Large nanotubes display
54
Figure 3.7: Two numerical surfaces coming to van der Waals contact, but failing to feel it because of insufficient quadrature points: the finite element nodes are represented by •, the quadrature points for the non-bonded term by ×, and the van der Waals cut-off radius by circles, which here do not overlap. smooth deformations, which can be accurately represented with very large finite elements relative to this cut-off radius. In this situation, it may be necessary to sample the above integrals with a large number of quadrature points to accurately resolve the scale of the non-bonded interactions. Otherwise, two very close surfaces could fail to “feel” the van der Waals interactions simply because the quadrature points sampling the above integrals on each surface are separated too much (see Fig. 3.7 for an illustration). In the numerical simulations presented later in this chapter, up to 12 Gauss points per element are needed for the integration of the nonbonded term for the largest nanotubes. Note however that the number of integration points for the non-bonded term is generally considerably smaller than the number of nuclei in the atomistic model, resulting in important computational savings in the calculation of the non-bonded interactions.
Non-bonded forces We can define a non-bonded energy between elements e and f > e as:
Πe−f nb =
nnb nnb X X
Vnb kϕe−f k ωie ωjf . i−j
(3.63)
i=1 j=1
This energy results in an elemental force in the nodes of element e nnb nnb
e/f (fnb )I
XX ∂Πe−f 1 e−f 0 nb V kϕ k NI (ξi ) ωie ωjf ϕe−f = = nb i−j i−j , e−f ∂ϕeI kϕ k i−j i=1 j=1
(3.64)
55 as well as a corresponding elemental contribution of the force in element f
f /e
(fnb )I =
∂Πe−f nb ∂ϕfI
=−
nnb X nnb X
1
i=1 j=1
kϕe−f i−j k
V0nb kϕe−f k NI (ζj ) ωie ωjf ϕe−f i−j i−j ,
(3.65)
which are gathered into (fnb )J accordingly.
3.5
Numerical validation of the theory
In order to study the mechanical response of carbon nanotubes under different loading conditions (compression, twisting, or bending), the positions of the nodes at the end of the tubes are incrementally displaced and, in each step, the energy is minimized. The present section describes a set of simulations in which nanotubes are deformed incrementally beyond the load where structural instabilities occur, in the full nonlinear regime. In order to test the accuracy and performance of the continuum/finite element computational scheme, full atomistic calculations for the same nanotubes and loadings are performed independently, and the equilibrium configurations and energies provided by the two methods are compared. The continuum model is constructed to mimic the response of nanotubes obeying the particular atomistic potential of choice, which is assumed to be “true”. Therefore, when we speak of the accuracy or the error of the membrane model, the reader should understand that the atomistic model at hand is taken as the reference. In the following examples, the Brenner potential is adopted (the second parameter set in Brenner (1990)) for the bonded interactions, and the Lennard-Jones potential for the non-bonded interactions corresponds to the graphene-graphene parameter set by Girifalco et al. (2000). It is worth noting that in performing these comparisons, it is important to make sure that the continuum and the atomistic models follow the same equilibrium path. Indeed, for a given problem and model, depending on computational parameters such as the load step-size or the convergence tolerance, different energetically very close deformations are possible, especially for large nanotubes. For instance, in the example of Section 3.5.5, the first buckled configuration observed in most computations displays two-fold symmetry. However, in some cases, both atomistic and continuum simulations show buckled deformations with three-fold symmetry for a range of moderate twisting angles. To facilitate the comparisons, small random perturbations are introduced at the initial guess for each load-step. Our numerical experience shows that this usually rules out the above mentioned meta-stable states, which are of physical significance, but rather inconvenient for our comparisons.
56 As mentioned before, subdivision finite elements based on Loop’s scheme (Cirak et al. 2000) which satisfy the smoothness requirements of the membrane theory have been used. In this method, a control surface mesh whose nodes have only translational degrees of freedom parameterizes a smooth surface. The control or computational mesh is presented in one of the examples below as a faceted surface. In most cases, it is the post-processed smooth surface defined by the computational mesh that is provided. This smooth surface is the actual numerical representation of the deformation of the continuum membrane, while the computational mesh only represents the degrees of freedom of the discrete model. The molecular mechanics and the finite element computer codes are comparable, and actually share many routines like those defining the interatomic potentials and the minimization routine. Therefore, the computational times are also comparable. For a given number of degrees of freedom, the calculation of the energy and the forces for the continuum/finite element model is more expensive than for the atomistic system (the metric and the curvature tensors of the surface, as well as their derivatives with respect to the nodal degrees of freedom must be computed, and the inner relaxation must be performed). Nevertheless, the reduction of degrees of freedom that the continuum model allows for a given problem makes this approach much more efficient above a certain size of nanotubes. This is not only due to the computational savings in each calculation of the energy and the forces for a much smaller system, but also to the fact that the reduction in degrees of freedom of the continuum/finite element systems also makes the minimization routine converge in fewer iterations for given tolerances and convergence criteria. Furthermore, it is possible to take larger load increments with the continuum/finite element model without compromising the convergence of the minimization routine—in the computational time comparisons, however, the number of load steps is the same for the atomistic and the continuum calculations.
3.5.1
Compressed (18,0) nanotube
Figure 3.8 shows the results for a (18,0) zig-zag nanotube compressed beyond the buckling point. In this example, both the atomistic and the finite element model have about 4,300 degrees of freedom. The superimposed final deformed configurations for the continuum/finite element and atomistic calculations are reported for a compression of 10.35%. The characteristic system of “fins” described in the literature (Yakobson et al. 1996), i.e. perpendicular flattenings of the tube, are clearly visible. The morphological agreement between the continuum and the atomistic simulations is remarkable, and the deformed membrane nearly coincides exactly with the positions of the nuclei provided by the atomistic calculation. The right part of the plot presents the evolution of the internal or binding energy of the sys-
57 −7.26
Strain energy (eV/atom)
−7.27 −7.28 −7.29 −7.3 −7.31 −7.32 −7.33 −7.34 −7.35 −7.36 0
2
4
6
8
10
Compression (%)
(a)
(b)
Figure 3.8: Compressed 8.7 nm long (18,0) nanotube: comparison between the full atomistic model and the continuum/finite element model. (a) super-imposed deformed configurations for atomistic (black spheres) and finite element (gray surface) calculations, and (b) strain energy evolution for the atomistic model (—) and for the continuum simulation (•). The strain energy for a continuum model in which the inner displacements are not relaxed also depicted (). tem as a function of the compression. For reference, Brenner’s potential predicts a ground energy for graphene of Egraphene = −7.3756 eV/atom. Before the buckles form, the energy displays a nearly quadratic growth. In this stage, the energies provided by the two methods are undistinguishable (comparable to the minimization tolerance). Note that the energy evolution is matched in absolute terms; for the comparison, neither of the curves has been shifted vertically. The buckling load is correctly predicted by the continuum simulation. After this point, the energy grows nearly linearly. In this regime, characterized by very large local deformations, the continuum approach is also very accurate; the error in the last reported step is 4% (this percentage, as all results reported subsequently, is relative to the strain energy variation, i.e. energy at observation point minus initial energy). This plot also reports the evolution of the strain energy for the continuum model, but without the inner displacement relaxation, in order to illustrate its crucial role in the correct modelling of the elasticity of nanotubes. It can be observed that the errors are considerable; before buckling occurs, the strain energy is over-estimated by 60%, and at the end by about 20%. The buckling point is poorly under-estimated in the absence of the internal relaxation. In some cases, even the deformation branches are not correctly predicted without the internal relaxation.
58
Strain energy (eV/atom)
−7.26
−7.28
−7.3
−7.32
−7.34
−7.36
−7.38 0
2
4
6
8
10
Compression (%)
(a)
(b)
(c)
(d)
Figure 3.9: Compressed 8.7 nm long (18,0) nanotube: unphysical deformations obtained with the standard Cauchy-Born rule for coarse (a), regular (b), and fine (c) meshes. (d) strain energy evolution for the atomistic model (—) and for the continuum simulation based on the standard Cauchy-Born rule for the coarse (), the regular (◦), and the fine (•) meshes. This simulation provides a stringent test for the continuum theory since a very small nanotube, only a few atoms in diameter, is severely deformed with local radii of curvature that approach the bond length. From a practical point of view, the continuum formulation is aimed at larger or longer nanotubes, for which it provides significant computational savings, as illustrated later. Nevertheless, an attractive attribute of this model is its good performance even for such severe deformations, and when the scale of the finite elements is comparable, or even smaller than the scale of the bond lengths.
3.5.2
Results with the standard Cauchy-Born rule
This excellent behavior contrasts with a continuum membrane model directly constructed from the standard Cauchy-Born rule without the proposed exponential extension. As discussed in Section 2.3, the energy of such a model is invariant under isometric deformations (deformations that keep C unchanged, i.e. bending without stretch). Thus, this energy does not depend on the curvature of the membrane, and the model has zero bending stiffness. This non-convexity of the hyperelastic
59 potential manifests itself in numerical simulations as a pathological mesh dependency. Because of the finite dimension of the discrete FE space, and fact that the boundary conditions may not be compatible with an isometric deformation, the numerical method still finds a solution which minimizes the total discrete energy Π(Φh ). Nevertheless, as the mesh is refined, the numerical method picks solutions with increasingly finer features. Figure 3.9 illustrates this behavior for the compressed nanotube of the previous example. The results are provided for three meshes, consisting of 820, 1830, and 3240 nodes. As the mesh is refined, the numerical solution is able to develop finer folds, and accommodate the deformation nearly isometrically. These solutions do not correspond to the behavior of the compressed carbon nanotube depicted in Fig. 3.8. This mesh dependency can also be observed in the energy evolution. It can be observed that as the mesh is refined, the increase of energy during the deformation vanishes. The energy evolution for the atomistic system is provided to highlight these unphysical results. These phenomena are reminiscent of the response of materials for which the strain energy density is physically con-convex (e.g. martensitic materials or nematic elastomers), which develop microstructures with increasingly fine features in the process of energy minimization (Dacorogna 1989; Conti et al. 2002).
3.5.3
Bent (10, 10), (15, 15) two-walled nanotube
Next, we describe an example in which the non-bonded interactions are critical. A two-walled nanotube is bent by rotating each end by 20◦ in opposite directions with respect to an axis perpendicular to the axis of undeformed nanotube, and passing through its center. At an angle near 9◦ , a single buckle forms in the center of the two-walled nanotube. The nanotubes are chosen so that initially, their walls are at approximately the van der Waals equilibrium distance. Figure 3.10 reports the deformed configurations for the atomistic and the continuum/finite element models at the end of the simulation, with a side and a top view of the buckle. To facilitate the visualization, the continuum solution is displayed as a translucent gray surface. The atomistic model is displayed by black spheres for the outer tube and solid lines for the bonds of the inner tube. It can be observed that the outer tube displays a sharper kink than the inner tube, in agreement with reported experimental observations and atomistic calculations (Iijima et al. 1996). Again, the agreement between the two models despite the very large, localized deformations is remarkable. The evolution of the deformation energy is also very well predicted, again in absolute terms, with perfect matching in the quadratic regime, and only slight discrepancies later. At the final stage, the error is about 6%. The evolution of the non-bonded energy as a function of the bending angle is also provided, and the agreement is excellent,
60
−5.4
Non−bonded energy (eV/atom)
−7.34
Strain energy (eV/atom)
−7.342 −7.344 −7.346 −7.348 −7.35 −7.352 −7.354 −7.356 −7.358 −7.36 0
5
10
15
Bending angle (°)
20
−3
x 10 −5.6 −5.8 −6 −6.2 −6.4 −6.6 0
5
10
15
Bending angle (°)
20
Figure 3.10: Bent 12.56 nm long (10,10), (15,15) two-walled nanotube: (a) superimposed deformed configurations for atomistic (black spheres, and solid lines) and finite element (translucent gray surface) calculations, and (b) strain and non-bonded energy evolution for the atomistic model (—) and for the continuum simulation (•).
61 which demonstrates the accuracy of the continuum version of these interactions. The buckling angle can be identified in this plot as a sharp increase of the nonbonded energy, probably due to the compression of the walls of the nanotubes at the buckle. Note that the change of non-bonded energy is much smaller than the change of strain energy. Nevertheless, the non-bonded interactions determine the morphology of the buckle, and inter-penetration of the walls of the two nanotubes will occur in their absence. In this example, both models have around 15,000 degrees of freedom, and the computation time is two times longer for the continuum/finite element approach. The objective of this example is not to prove the computational savings that the continuum model can provide, but rather prove its accuracy for highly strained multi-walled nanotubes.
3.5.4
Twisted (10,10) nanotube
In this example, a (10,10) nanotube 25 nm long is twisted by rotating its ends in opposite orientations with respect to the axis of the tube. Three representative snapshots of the deformation process are shown in Fig. 3.11. The evolution of the strain energy is presented for both the atomistic and the continuum/finite element calculations in Fig. 3.12. The strain energy evolution if the non-bonded interactions are not included is also reported in this figure (dashed line). The evolution of the non-bonded energy is also presented. This example exhibits two structural instabilities. In the first one, a non-uniform deformation mode develops for a twisting angle of each end of about 50◦ . The onset of this instability is evident in the first snapshot of the deformation in Fig. 3.11, and can be identified in the strain energy evolution as the kink that ends the nearly quadratic regime. As loading proceeds, the wall of the tube comes into van der Waals contact with itself and adhesion energy is gained. Then, the van der Waals interactions harden the twisting response of the tube. This can be noticed by observing the deviation between the response with (solid) and without (dashed) non-bonded interactions. The dashed line demonstrates the fundamental effect on the global response of these interactions, despite being less than 2% of the total energy change. The second kink in the strain energy evolution, near 230◦ , indicates the development of a secondary structure. After this point, the flattened twisted ribbon folds onto itself. The snapshots demonstrate that, even for these intricate deformed morphologies, the continuum mechanics theory is surprisingly accurate, and the finite element model remarkably fits the atomic positions. When it comes to the energetics, the agreement is also excellent, both for the strain and the non-bonded energies (note that the discrepancy in the non-bonded interactions at 300◦ is only 0.1% of the total energy variation). This analysis has been performed with four different meshes, a coarse one with
62
Figure 3.11: Twisted 25.11 nm long (10,10) nanotube: super-imposed deformed configurations at three twisting angles for atomistic calculation (black spheres) and continuum finite element calculation (gray surface). 0 −3
Non−bonded energy (eV/atom)
Strain energy (eV/atom)
−7.22 −7.24 −7.26 −7.28 −7.3 −7.32 −7.34 −7.36 0
50
100
150
200
Twisting angle (°)
(a)
250
300
x 10 −0.5
−1
−1.5
−2
−2.5 0
50
100
150
200
250
300
Twisting angle (°)
(b)
Figure 3.12: Twisted 25.11 nm long (10,10) nanotube: (a) Comparison of the strain energy as a function of the twisting angle for atomistic calculation (—), and continuum/FE calculation (•), and strain energy density evolution if the non-bonded interactions are ignored (- - -). (b) Comparison of the non-bonded energy evolution for atomistic calculation (—) and continuum/FE calculation (•).
63 Table 3.1: Twisted 25.11 nm long (10,10) nanotube: error of the three considered finite element models at two twisting angles
◦
220 300◦
Coarse 1.1% 7.4%
Medium Fine 0.39% 0.37% 3.6% 2.3%
Super fine 0.29% 1.5%
6,666 degrees of freedom and 22 elements in the perimeter, a medium one with 10,164 degrees of freedom and 28 elements in the perimeter, a fine one with 16,520 degrees of freedom and 32 elements in the perimeter, and a super-fine mesh with over 30,000 degrees of freedom. The atomistic system has 12,000 degrees of freedom. The results reported above are for the medium mesh. The relative errors of the finite element models before the second instability (220◦ ) and at the end of the analysis (300◦ ) are reported in Table 3.1. Note that this error includes both the finite element approximation error, and the continuum modelling error. The table shows that the finite element solution at 220◦ is excellent, even for the coarse mesh. After the second instability, the severe deformation makes the analysis with the coarse mesh less accurate. It is observed that mesh refining reduces the errors to a very low level, given the very large deformations and the fact that a small atomistic system is being analyzed by a local continuum model. Thus, this simulation suggests that local deformation features do not necessarily require coupling with atomistic calculations, and simple mesh refinements is sufficient to obtain accurate solutions, as indicated in the table. Of course, from a practical point of view, it makes little sense to model an atomistic system with an approximate finite element model with more degrees of freedom. As shown in the next example, accurate computations of larger tubes, which typically display deformation features that are larger relative to the bond length, require finite element models with far fewer degrees of freedom than the atomistic model.
3.5.5
Twisted (30,30) nanotube
In example, a larger (30,30) nanotube is twisted until the tube flattens and folds onto itself. Figure 3.13 shows the superimposed deformations for the atomistic and the continuum/finite element calculations at two twisting angles. As before, the atoms coincide with the continuum membrane. The map of the strain energy density on the computational finite element mesh is also shown for illustration. It can be observed that the severely bent areas display high strain energy. In this case, the atomistic
64
(a)
Y Z X
Z
(b)
X
50◦
Y
75◦
Figure 3.13: Twisted 37.67 nm long (30,30) nanotube: comparison between the atomistic model and the continuum/finite element model for two twisting angles. (a) super-imposed deformed configurations for atomistic (black spheres) and finite element (gray surface) calculations, and (b) map of the strain energy density on the finite element computational mesh (red is high, blue is low). system has 54,000 degrees of freedom while the continuum/finite element model only 5,070. The computational time with the continuum/finite element approach is seven times smaller than with the full atomistic calculation. This fast finite element calculation provides an accurate solution within 0.8% in strain energy for the first reported twisting angle, and within 5% for the final angle. Again, a finite element model with a finer mesh of 9,696 degrees of freedom reduces the error at 75◦ below 1.3%. Remark 3.2. Note that all of the finite element calculations presented here are performed with uniform meshes. As we have seen, these calculations provide very accurate results for moderate deformations and coarse meshes, but finer meshes are required when the deformations are severe. One of the major advantages of the continuum-based simulations is that the mesh can be tailored to the problem under consideration through adaptive mesh refinement. Furthermore, the mesh can be changed during the analysis very easily since the model is hyperelastic. Thus, the proposed continuum model can bring computational technology to carbon nanotubes calculations such as adaptivity, which can further reduce the computational cost by orders of magnitude. Remark 3.3. The possibility of performing the global and the inner minimizations in a staggered fashion has been preliminary explored in this thesis. Although a few
65 more global iterations may be needed to converge, the computational cost of the continuum model may be greatly reduced in this fashion.
3.6
Large-scale examples
This section describes some examples of bending of multi-walled carbon nanotubes (nested tubes in van der Waals contact). In these examples, in which larger nanotubes are considered, the finite element model benefits from the fact that the element sizes can be chosen irrespective of the atomic spacing. Consequently, the computational cost is greatly reduced. These large scale calculations are performed with a parallel implementation of the method. The scalability of the method is excellent, and for the larger examples of Section 3.6.2, up to 20 processors have been used at 80% efficiency. The computational times never exceeded 8 hours on a Beowulf cluster with 1.13 GHz Athlon processors.
3.6.1
Bent 5-walled nanotube
Iijima et al. (1996) presented a series of TEM (Transmission Electron Microscope) images of bent and buckled single and multi-walled carbon nanotubes. They also provided atomistic calculations for small single and two-walled nanotubes, revealing the energetics behind the formation of kinks. These experiments show that 5-walled nanotubes display single kinks, but in addition systems of double kinks are observed. We observed single kinks in Section 3.5.3. In the continuum/finite element calculations, a 5-walled (23,23) (28,28) (33,33) (38,38) (43,43) multi-walled nanotube whose dimensions match those of the experiments is considered. The corresponding atomistic system for a 35 nm long nanotube has about 276,000 degrees of freedom, while the finite element model has 20,000. For this larger nanotube, the first kink forms when both ends of the nanotube are rotated only 5◦ . For moderate bending angles, until 18◦ , this is the only deformation mode observed. For larger bending angles, depending on slight perturbations of computational parameters such as the number of load steps or the minimization tolerances, two equilibrium paths can be distinguished. A series of calculations can be qualitatively classified as following path A, or path B (see Fig. 3.14). Path A is characterized by a system of three buckles, while path B displays only two. Path B breaks the symmetry of the system at a bending angle of 19◦ , and eventually reaches a symmetric configuration. The figure shows three dimensional images of the numerical deformations, as well as longitudinal sections. The sections are particularly useful since they reveal the internal structure of the deformation, and they
66
Figure 3.14: Bending of a 5-walled carbon nanotube, inspired in the experiments by Iijima et al. (1996)
67 are the numerical analogs of the experimental TEM slices. Cross sections are also provided for the last configuration of each path, and are marked by thick lines in the longitudinal sections. It can be observed that the single kink configuration reported by Iijima et al. (1996) agrees very well with the simulation for a bending of 18◦ . On the other hand, there is qualitative agreement between the experimental image displaying a complex kink structure and the deformation pattern observed in the longitudinal section of the final configuration of path B. The three-kink complex observed in the simulations is not reported in these experiments. Systems of consecutive kinks in multi-walled hollow carbon nanotubes have been reported by Ruoff and Lorents (1995). Nevertheless, disagreements between the experiments and the simulations can be observed, which have several likely sources. First, the response of large nanotubes is quite sensitive to the boundary conditions; we do not know the load on the observed nanotubes. Furthermore, as clearly observed in some of the TEM images in Iijima et al. (1996), surrounding material as well as material inside the nanotubes is very likely to have an effect on the nanotube deformation. Finally, the accuracy of the interatomic potential in reproducing the energetics of deformation is limited, as further explored in Chapter 5. From the studies of the continuum/finite element approach in Section 3.5, it can be inferred that the presented simulations faithfully reproduce the corresponding atomistic model, and therefore we do not consider the present modelling approach as a fundamental source of the deviations from experiments. Since the simulations reproduce the major physical phenomena observed in the experiments, it possible to proceed to extract information from the calculations which are not available from the experiments. Let us first analyze the energetics of the process. Note from the strain energy evolution depicted in Fig. 3.14 that, despite the morphological differences between these two paths, they are energetically very close. It can be noticed that path B is slightly energetically more favorable. In Iijima et al. (1996), the transitions between buckling patterns are qualified as “sharp”. While this is true with regards to the deformed configurations, these results show that it is not so in energetic terms. Indeed, our simulations show that large systems of multi-walled nanotubes rarely display sharp transitions in the energy evolution; the characteristic quadratic-linear sudden transition observed for the energy evolution of smaller nanotubes in the previous sections is not present here. Indeed, while for unbuckled structures the energy growth with deformation is roughly quadratic, and for buckled structures roughly linear, the behavior of the buckled structures strongly constrained by van der Waals interactions falls typically in between these two cases. This constraint on the kinks which causes this hardening effect is clearly observed in the sections of the simulations. The increment of nonbonded energy in this experiment is 12% of the total energy increment, a much
68 larger fraction than for single-walled nanotubes. In addition to the energetics, the simulations also provide a three-dimensional picture of the deformation, unlike TEM which provides “slices”. This makes the interpretation of experimental observations much easier and complete. For instance, atomistic simulations of bending found in the literature typically involve small nanotubes, which develop kinks with a simple structure (Iijima et al. 1996). See also Section 3.5.3 for an illustration. As can be observed in Fig. 3.14 from the longitudinal section for 18◦ , one may be led to think that the 5-walled nanotube displays the same simple structure. However, the three-dimensional picture reveals two other systems of pairs of buckles tilted in the transverse direction. Actually, the simulation shows that between two simple buckles there is always a pair of tilted buckles, and vice-versa. The transverse structure of these types of buckles can observed in the cross sections of the final configurations. These cross sections also show that the inner-wall delamination, which can be observed both in the experimental and the numerical slices, can be associated to transverse buckles. This alternation of simple centered buckles and pairs of tilted buckles is even more apparent in the following example.
3.6.2
Bent 34-walled nanotube: rippling
This section presents a computational study of an experimentally observed phenomena called rippling, i.e. the development of periodic wavelike buckles in thick multi-walled nanotubes reported for instance in Poncharal et al. (1999). This deformation mode has been associated with the reduction in the measured bending modulus of nanotubes as their diameter increases. A simplified analysis of a twodimensional anisotropic elastic material by finite elements has been reported in Liu et al. (2001). In experiments reported by Kuzumaki et al. (1998), rippling is also observed. A 34-walled (5,5), ... (170-170) nanotube about 23 nm in diameter and 124 nm in length is considered, matching the dimensions of that reported in Kuzumaki et al. (1998). This system consists of about 5.9 · 106 atoms, that is 17.6 · 106 degrees of freedom. The finite element model used in these calculations has 300,000 degrees of freedom, that is 60 times less than the atomistic system. By considering several finite element meshes, it is concluded that this model provides a “converged” solution. It is clear from these numbers that the memory requirements are greatly reduced, and the computational time is shortened by orders of magnitude. The rippled nanotube reported in Poncharal et al. (1999) is thicker, with a diameter of 31 nm (about 45 walls). Figures 3.15 and 3.16 show the results from these simulations, as well as some experimental TEM images. The image of the nanotube in Kuzumaki et al. (1998)
69
Figure 3.15: Rippling of a 34-walled carbon nanotube: (a) comparison with experimental observation (reprinted with permission from Kuzumaki et al. (1998), http://www.tandf.co.uk), (b) side-section, (c) cross-sections of the simulation, and (d) experiment on larger MWCNT (reprinted with permission from Poncharal et al. (1999). Copyright 1999 American Association for the Advancement of Science).
70
Figure 3.16: Rippling of a 34-walled carbon nanotube: (a) Top and (b) side views of the simulated deformation. has been super-imposed on the numerical slice (a), which is also presented separately (b). The general morphological agreement is remarkable. The simulations reproduce very well the general features of the rippled nanotubes: nearly periodic wavelike distortions, whose amplitude vanishes for the inner tubes and smoothly increases towards the outer layer (see the images from Poncharal et al. (1999) for a more regular rippling). The periodicity of the simulated ripples is very regular, except near the boundaries (periodic boundary conditions are not imposed). A careful look at the experimental images reported in Kuzumaki et al. (1998) reveals that the nanotube is not uniformly bent; the left portion in the reproduced TEM image has a smaller radius of curvature than the right portion, and displays sharper ripples. Actually, Kuzumaki et al. (1998) conjecture that in the left side ripples, the sharply bent layers develop a row of sp3 atoms. Rippling however is generally associated with elastic, reversible deformations (Liu et al. 2001; Suenaga et al. 2001). In the reported simulations, the bond lengths and angles furnished by the continuum theory describe local deformations far away from transitions from the sp2 bond structure (according to Brenner’s potential). Our simulation more closely reproduces the right section of the nanotube in Fig. 3.15 (a), and the wavelength of the ripples is well predicted (around 10 nm). This value is also close to that observed in Poncharal et al. (1999). It should be noted that, as in the example described in Section 3.6.1, the actual loading conditions on the nanotubes are very difficult to estimate. Again, the calculations provide extra information not available from experiments.
71 The three-dimensional images of the deformation in Fig. 3.16 enrich the description of the phenomenon. Similarly to the example in Section 3.6.1, the three-dimensional picture reveals an intercalation of two families of buckles. The cross-sections in Fig. 3.15(c), marked as thick lines in the longitudinal slice in Fig. 3.15(b), provide a new perspective on the deformation, with flat sections and the wedge-like sections, separated by nearly circular sections. Information about the energetics of the rippling deformation can also be extracted from the calculations. The hypothesis made in Poncharal et al. (1999), that the rippling mode is likely to be energetically favorable as compared to uniform bending, is supported by our calculations. For the nanotube in our simulations with Brenner’s potential, the strain energy of the uniformly bent nanotube relative to the ground energy of graphene is 0.052 eV/atom, while for the rippled structure the strain energy is 0.010 eV/atom. For reference, the strain energy of a relaxed (10,10) nanotube, that is the minimum energy required to maintain its curvature, is 0.023 eV/atom. This means that the rippled nanotube in Figures 3.15 and 3.16 has a lower average strain energy than a relaxed (10,10) nanotube. Additional simulations indicate that the chirality of the nanotubes has no effect on the rippling deformation mode, consistent with the observation in Chapter 5 that the anisotropy of deformed graphene only manifests itself for small nanotubes. In other simulations on shorter tubes, the symmetric configuration of Fig. 3.16 is not the most stable structure; if the length of the tube is far apart from a multiple of the wavelength of the ripples, these adopt a helical distribution on the surface of the nanotube to fit an integer number of ripples in the available length between the fixed boundaries. This nonsymmetric rippling may be responsible for TEM images such as that reported in Suenaga et al. (2001), where no clear ripples can be distinguished. Actually, if sufficiently large perturbations are introduced in the equilibrium configuration of Fig. 3.16, a slightly helical distribution of ripples is obtained with almost identical energy and no frustrated ripples near the boundaries.
3.7
Summary and conclusions
A finite deformation continuum mechanics theory for carbon nanotubes in terms of the atomistic description of the system has been developed. This continuum membrane is based on the exponential Cauchy-Born rule presented in Chapter 2. The continuum theory has been discretized by finite elements. It has been shown to mimic very accurately the full non-linear mechanics of the parent atomistic system, at a much lower computational cost than conventional atomistic calculations. This computational efficiency is exploited to analyze and replicate experimental observations involving systems of millions of atoms.
72 Continuum theory With regards to the membrane theory, the following conclusions can be drawn: • The term membrane is chosen to emphasize that, unlike shells, the continuum object replacing curved graphene sheets is a surface without thickness (but with bending stiffness). This is a unique feature of this theory. • A local approximation scheme for the exponential Cauchy-Born rule is proposed. It leads to a hyperelastic membrane whose strain energy density depends on the stretch (Green deformation tensor) and the curvature (pull-back of the curvature tensor) of the surface, is frame indifferent and applicable in the finite deformation regime. • The membrane constitutive model is strictly based on the atomistic description of the system, with no other phenomenological input. The theory is strictly framed within continuum mechanics, with a closed-form expression for the elastic potential in terms of continuum strain measures; it does not require constrained atomistic calculations. The last point is of particular importance, since it allows for (semi-)analytical treatments in some situations of interest, as described in Chapters 4 and 5. Details on the evaluation of the constitutive model are provided. • The proposed theory, like other crystal elasticity theories, accounts for the crystallography of the underlying lattice, in particular for the chirality of nanotubes, and treats consistently the inner elasticity of the non-primitive graphene lattice. • A continuum version of the non-bonded interactions, crucial in the mechanics at the nano-scale, is provided, and its accuracy is tested numerically. • A recent rigorous analysis of the Cauchy-Born rule for a family of 2D lattices concludes that this rule is actually a theorem if certain conditions of interatomic potential are met (Friesecke and Theil 2002). The numerous numerical tests presented in this chapter suggest that an analogous result holds for the exponential Cauchy-Born rule.
Finite element calculations The configurations of the membrane are approximated by finite elements, and equilibrium deformations are found by direct minimization of the discretized potential energy. Details about the computation of the discrete energy and nodal forces are provided. A battery of tests with a membrane based on Brenner’s potential are
73 carried out, for single and multi-walled nanotubes under different loading conditions. Full atomistic calculations are performed independently, and the energetics and deformations are compared with the results of the continuum/finite element approach. These comparisons demonstrate the remarkable accuracy of the continuum theory in the full nonlinear regime, for very large deformations. More specifically, the following conclusions can be reached from these comparisons: • The discrepancies between the atomistic and the continuum simulations have two distinct causes: one is a modelling error, that arises from treating the discrete system as a continuum, and the other is a numerical error, due to the finite element discretization. The presented simulations have shown that the latter can be reduced by mesh refinement, while the former is very small even for severe deformations. These modelling errors are much smaller than the errors associated with current interatomic potentials. • Our simulations suggest that, in the absence of lattice defects or bond rearrangements, it is possible to accurately describe the nonlinear mechanics of carbon nanotubes with exclusively continuum finite element calculations, without any recourse to atomistic calculations. Although this may seem surprising, experimental observations and atomistic simulations of nanotubes in general display smooth deformations (notable exceptions are experimentally observed fracture, and simulated plasticity). These observations can be interpreted as evidence that, as long as the integrity of the lattice is maintained, the mechanical response of carbon nanotubes depends on the detailed atomistic arrangements only through the elasticity of the curved monolayer lattice (the scale of the bond length and that of the overall deformation are well separated). Given the celebrated resilience of the carbon network in graphene, our method can be a valuable tool in the design and analysis of nanotube devices in which only reversible deformations are expected. • The continuum model allows us to exploit the well developed technology of engineering computations to the analysis of carbon nanotubes; for instance, the discretization can be optimized through adaptive mesh refinement, using small elements to capture fine features of the deformation. Thus, the continuum approach enables the choice of the level of accuracy, or the total number of degrees of freedom of a model. For instance, a very coarse mesh suffices to accurately predict the onset of a structural instability. However, to accurately describe severely deformed nanotubes with very localized features, a finer mesh is required, sometimes with fewer degrees of freedom than the original atomistic system. This contrasts with atomistic calculations, which unavoidably need to track each atom.
74 Continuum/finite element calculations of large nanotube systems have been presented. Full atomistic calculations of these systems are beyond our computational reach, while the finite element calculations run times are always below 8 hours. These simulations replicate unusual experimental observations such as debonding of inner nanotubes and rippling, and provide extra information not available from experiments which enrich the description of the observed phenomena. In particular, the three dimensional picture of the deformation reveals that, in addition to buckles observed in Transmission Electron Microscope images, other systems of tilted buckles are also present in large multi-walled nanotubes. Furthermore, valuable information about the energetics behind these deformation processes is available from the simulations.
Future work Several lines of future research suggest themselves, namely: • An analysis of the ranges of applicability and accuracy of the exponential Cauchy-Born rule to complement the numerical studies is desirable. In particular, the local approximation of the exponential Cauchy-Born rule deserves special attention, and alternative approximations could be envisioned. • The results provided by computationally efficient finite element calculations can be used to generate initial guesses for more sophisticated atomistic models, e.g. ab initio calculations. Another possible synergy of the continuum model with atomistic methods is to analyze physical phenomena beyond the capability of continuum mechanics, such as electronic transport, by atomistic methods, but on deformed configurations provided by the hyperelastic membrane. • As detailed in Chapter 5, present interatomic potentials for carbon provide limited accuracy. One exciting avenue is to construct the continuum constitutive model directly from a quantum mechanical model. This would open the possibility of treating another curved crystalline monolayer system, BN nanotubes, for which analytical potentials are not available. • The application of the present continuum formulation to dynamical simulations is not completely justified; in particular, the behavior of high frequencies and their thermalization is not obvious. Furthermore, the treatment of the inner displacements, and how their dynamics interact with the overall dynamics needs to be studied. Although developments in this area are needed, this model should be applicable to the dynamics of nanotube structures in some
75 regimes, thereby providing considerable additional computational savings associated with larger time-steps.
Chapter 4 Reduced model for transverse deformations 4.1
Introduction
Carbon nanotubes have very different longitudinal and transverse behavior. In the longitudinal direction, they are extremely stiff with reported Young’s moduli of the order of 1 TPa, while in the transverse direction, they are very flexible, and the effect of van der Waals interactions becomes crucial. There are many situations in which the transverse behavior is of interest, for example in the study of the transverse stability of collapsed single-walled or multi-walled nanotubes, their interaction with substrates (Ruoff et al. 1993; Chopra et al. 1995; Yu et al. 2001), or the structure of nanoropes, i.e. bundles of nanotubes (Thess et al. 1996; Salvetat et al. 1999; L´opez et al. 2001). A reduced model for nanotubes may also be useful for studying the properties of the recently synthesized crystals of nanotubes—ordered arrays of nanotubes with identical diameter and chirality on the micron scale (Schlittler et al. 2001). In this chapter, a reduced model for the transverse deformation of carbon nanotubes is developed, which exploits analytically the translational symmetry of these deformations. This continuum formulation models three dimensional nanotubes under uniform deformation in the axial direction, i.e. allowing only transverse deformation and uniform axial stretch. Despite modelling the constrained deformation of three-dimensional nanotubes, the continuum model is two-dimensional. This model can be derived either by simplifying the general membrane of Chapter 3, or by approximating the exponential map of the constrained deformation by the exponential map of the cylinder. Such a model of reduced dimensionality—we model a twodimensional object, a surface, with a one-dimensional object, a curve— is possible 76
77
F
X2
x2
L A0 2
A01
Q0
1/k n
W
A03
P
W0
X1
x1
x3
Figure 4.1: Illustration of the kinematics of the nanotube with uniform deformation in the axial direction, and the cylinder used to approximate the exponential map. thanks to the fact the exponential extension of the Cauchy-Born rule allows us to write the constitutive model in closed-form, without local atomistic calculation of any kind. Therefore, constrained deformations can be treated analytically.
4.2
Reduced kinematics
Deformation map We consider deformations that are uniform along the axis of the tube. The undeformed body is a rectangular slab of graphene, Ω0 = (0, L) × (0, P ) as described in Section 3.3.1, where L and P are the undeformed length and perimeter. These deformations can be written in the Euclidean coordinates {X 1 , X 2 } and {x1 , x2 , x3 } introduced in Chapter 3 as Φ(X) = Λ1 X 1 i1 + Φ2 (X 2 ) i2 + Φ3 (X 2 ) i3 ,
(4.1)
where the x1 axis is parallel to the tube axis (see Fig. 4.1). Note that Φ2 and Φ3 only depend on X 2 , therefore (·)0 denotes differentiation with respect to this variable. Along the X 1 axis, Λ1 denotes the uniform stretch. We define the undeformed lattice vectors as in Eq. (3.30); Θ0 is the chiral angle of the nanotube.
78 Green deformation tensor The circumferential stretch is Λ2 =
q
[(Φ2 )0 ]2 + [(Φ3 )0 ]2 ,
(4.2)
which does depend on X 2 , it is not uniform. The matrix representation of the Green deformation tensor is then
2
(Λ1 ) [C]B0 = 0
0 (Λ2 )2
.
(4.3)
Curvature It is easy to see that the only non-zero component of the pull-back of the curvature tensor is K22 . The principal curvatures of the surface are 0 and κ, which can be written as κ=
1 K22 = −(Φ3 )0 (Φ2 )00 + (Φ2 )0 (Φ3 )00 . 3/2 2 (Λ2 ) [(Φ2 )0 ]2 + [(Φ3 )0 ]2
(4.4)
The associated principal directions the basis vectors I1 and I2 . Note that 1/κ is the local radius of curvature of the cylindrical surface.
Exponential Cauchy-Born rule The local approximation of the exponential map can be obtained either by particularizing the general expressions of Section 3.2.2, or by considering a local approximation of the exponential map of the surface Φ(Ω0 ) as the exponential map of a suitably defined cylinder (see Figs. 4.1 and 4.2), which locally replaces the original surface. Let us define the tangent deformed lattice vector w = FA as in Section 3.2.2. Its components in the auxiliary basis of the tangent plane to the cylinder can be written as w1 = Λ1 A1 and w2 = Λ2 A2 . (4.5) The components of the deformed lattice vector resulting from this local approxima-
79
T
F(X)
n
W
w
F (X) a
exp w
Figure 4.2: The exponential map in the cylinder defined at each point of the surface by the unit normal, the normal curvature and the direction of the axis. The geodesics are either straight lines, circles, or helices. tion to the exponential map in the auxiliary Euclidean coordinate system is then 1 w 2 2 [a]Be = . (4.6) w Q(κw ) κ(w2 )2 Q 2 (κw2 /2) 2 Consequently, the bond lengths and angles seen as derived continuum strain measures can be written in terms of the strain measures Λ1 , Λ2 and, κ, and the undeformed lattice vectors. For instance 4 kak = 2 sin2 κ 2
4.3
κw2 2
+ (w1 )2 .
(4.7)
Reduced model and variational principle
Following the same procedure of Section 3.3.2, it is possible to write the continuum strain energy density of the two-dimensional continuum corresponding for instance to the 2-body/3-body expansion potential of Eq. (3.35) as
W = W (Λ1 , Λ2 , κ, η) =
1 S0
"
3 X i=1
Vs (ai ) + 2
3 X i=1
# Vθ (θi , aj , ak ), ,
(4.8)
80 where {i, j, k} is an even permutation of {1, 2, 3}. Recall from Section 3.3.1 that there are three inequivalent bonds and three inequivalent angles, and note that the unit cell contains one bond of each type and two angles of each type. The dependence on the inner displacements η follows from identical arguments as for the general membrane.We similarly define an effective potential with the relaxed inner displacements c (Λ1 , Λ2 , κ) = W (Λ1 , Λ2 , κ, η b (Λ1 , Λ2 , κ)) . W
(4.9)
The corresponding expression of the principle of virtual work is Z 0=L 0
P
! c c c ∂W ∂W ∂W δΛ1 + δΛ2 + δκ dX 2 − δΠext + δΠnb , ∂Λ1 ∂Λ2 ∂κ
(4.10)
c/∂Λ1 is an axial first Piola-Kirchhoff stress component, ∂ W c/∂Λ2 is a where ∂ W c/∂κ is a bending circumferential first Piola-Kirchhoff stress component, and ∂ W stress. The components of the first Piola-Kirchhoff stress, an in plane stress, are expressed in units of force divided by length. Depending on the treatment of the axial stretch Λ1 (see Eq. (4.1)) different situations can be studied: Plane strain. We can consider the situation in which the value of Λ1 is prescribed. In this case, the unknowns of the variational problem (4.10) are Φ2 and Φ3 (see Eq. (4.1)). If Λ1 = 1, a deformation analogous to plane strain conditions is achieved. This applies to axially constrained nanotubes. Λ1 can also be prescribed an arbitrary value to study the transverse behavior of stretched or compressed nanotubes; the behavior will change due to the non-linearity of the model. In this situation, δΛ1 = 0 and the axial component of the in-plane stress does not appear in the variational principle. This means that the axial stress can be computed a posteriori, but does not play a role in the solution of the problem. Plane stress. Alternatively, the axial component of the membrane stress can be prescribed, for instance, to be zero. This would be the case of axially unconstrained nanotubes. In this case, in addition to Φ2 and Φ3 , the axial stretch Λ1 becomes an unknown of the problem.
81
4.4
Numerical examples
In this section, numerical simulations of stable configurations of carbon nanotubes in different situations are reported. The reduced continuum model described in the previous section is discretized by Galerkin finite elements (FE), and the total energy of the discrete system is minimized using the BFGS quasi-Newton technique. Note that only a one-dimensional (beam-like) discretization is needed. Thus, the original discrete molecular system is replaced by a continuum model which is subsequently transformed by the FE method into another discrete system. However, in principle we are free to design the FE discretization so that the FE model has fewer degrees of freedom than the original system. Furthermore, since the continuum model is 2D, while the full atomistic model is 3D, the computational cost is further reduced. First, the exponential Cauchy-Born rule-based continuum model is validated by comparing FE simulations based on it with full atomistic calculations. In these comparisons the interatomic potential used in the molecular mechanics (MM) simulations is used to construct the continuum constitutive equation, and analogous boundary conditions are considered in both calculations. Since the continuum model is intended to mimic the atomistic system, which is viewed as “true”, the term error should be understood as the deviation from the atomistic model. Simulations show that the agreement is excellent with regard to the energetics as well as to the stable configurations. Simulations of a model based on the standard Cauchy-Born rule are also provided, illustrating the deficiencies of such a model. The exponential Cauchy-Born rule simulations also show that, for the transverse mechanics of nanotubes, the relaxation of the inner displacements greatly affects the energetics but has very little impact on the minimum energy configurations. The continuum model is then applied to simulate several situations where the transverse behavior of carbon nanotubes and the effect of van der Waals interactions are important. The interatomic potentials fall into the general form described in Eq. (3.35). The two-body potential Vs is a Morse potential while the three-body potential depends only on the angle Vθ and is harmonic with a sextic correction. The parameters are taken from the MM2 model. The non-bonded interactions are based on the classical Lennard-Jones potential. Warning. The interatomic potential used in these simulations is not realistic. As detailed in Section 5.3.3, a many-body expansion potential without dihedral terms produces graphene sheets without bending stiffness when planar. Therefore, the simulations should be understood as a proof of concept, since continuum results match atomistic calculations; the results on nanotubes are qualitative. The variational principle in Eq. (4.10) imposes restrictions on the finite element interpolation spaces. The virtual internal work term involves variations on the curvature of the test functions, and therefore the finite element space needs to be H 2 , i.e. have up to second square integrable derivatives. This is why C 1 Hermite finite
82 Molecular Mechanics Standard Born Rule (fine) Standard Born Rule (coarse)
X Z
Molecular Mechanics Continuum + FE
Y
(a)
(b)
(c)
Figure 4.3: (a) Actual molecular model used in comparison, (b) Comparison of 20 element exponential Cauchy-Born rule continuum model with molecular mechanics and (c) Results obtained with a model constructed from the standard Cauchy-Born rule. elements are chosen. Note that the discretization of the configuration described in Eq. (4.1) requires the approximation of the scalar functions Φ2 (·) and Φ3 (·), i.e. the curve in R2 described by these functions needs to be parametrized with respect to the finite element degrees of freedom. Each of these functions is approximated by piecewise cubic polynomials; each node I carries four degrees of freedom: Φ2I , Φ3I , (Φ2 )0I and (Φ3 )0I . The internal work term of the variational principle is integrated using 3 Gauss points per element, while the integration of the non-bonded interactions term may require more integration points depending on the size of the finite elements relative to the van der Waals cut-off distance, see in Section 3.4.2. Four integration points are required for this term in some of the simulations.
4.4.1
Validation test
To validate the proposed reduced continuum model, a FE discretized version is compared to a molecular mechanics model. A (32,0) zig-zag carbon nanotube (the standard description of carbon nanotubes in terms of two integers is described by Saito et al. (1992)) is considered (see Fig. 4.3(a)). The molecular model used in the comparison has 384 atoms, that is 1152 degrees of freedom, while the FE model has 20 nodes and consequently 80 degrees of freedom. Note that the discrete FE model reduces the computational cost, not only because large elements relative to the crystal cell size can be used, but also because of its reduced dimensionality. The first configuration studied consists of simply rolling a graphene sheet into a tube in an isometric transformation, without any kind of relaxation. This configura-
83 Table 4.1: Comparison of 20 element model (C+FE) with molecular mechanics (MM): Energy in J/mol. η=0 MM Original tube 14.58 Relaxed tube 10.26 Squeezed A 20.85 Squeezed B 48.56
C+FE 14.69 10.28 21.22 49.17
Error (%) 0.81 0.22 1.8 1.3
Relaxed η MM
C+FE
Error (%)
6.338 12.95 30.68
6.324 13.11 30.46
-0.22 1.2 -0.75
tion is called Original tube in Table 4.1. The table shows the excellent agreement between the energy obtained with the molecular model and that obtained via the continuum model and FE. As argued in Chapter 5, the energy of the continuum model should be exact in this situation. Note however that the continuum membrane is discretized using an approximation space that does not exactly reproduce a circle, and thus introduces small discretization errors. Then several “plane strain” situations are considered. This condition is enforced in the molecular model by setting to zero the nuclear displacements in the direction of the axis of the tube at the nuclei located at both ends of the tube. In the continuum model, we simply enforce Λ1 = 1. Also, two kinds of energy minimization are considered. The first one freezes the inner displacements to those of the graphene sheet in equilibrium, i.e. in the continuum model by prescribing η = 0. In this particular example, invoking symmetry considerations, this constrained minimization can be easily implemented in the molecular model by setting to zero the displacements of all the nuclei in the direction of the tube axis. This incomplete analysis is performed to highlight the effect of the inner relaxation. The other analysis is an unconstrained structural optimization of all the nuclear positions. In the continuum, the inner displacements c at each Gauss point. are relaxed in order to calculate W The situations considered are: Relaxed tube: The Original tube is relaxed without any constraint other than the plane strain conditions; Squeezed A: Displacements at the ends of one diameter of the tube are prescribed so that this diameter of is squeezed to 3/4 of its original size; Squeezed B: Displacements at the ends of one diameter of the tube are prescribed so that this diameter of is squeezed to 1/2 of its original size. Table 4.1 presents the equilibrium energies for both the MM and the Continuum FE simulations, as well as the relative error of the FE calculation with respect to
84 Table 4.2: Comparison of 36 element model (C+FE) with molecular mechanics (MM): Energy in J/mol. η=0 MM Original tube 14.58 Relaxed tube 10.26 Squeezed A 20.85 Squeezed B 48.56
C+FE 14.60 10.28 21.05 49.01
Error (%) 0.14 0.21 0.96 0.93
Relaxed η MM
C+FE
Error (%)
6.338 12.95 30.68
6.324 12.99 30.34
-0.22 0.31 -1.1
MM. A positive value of error means that the MM energy is lower than the FE energy. Note that this error includes contributions not only from the modelling of the discrete atomic system as a membrane, but also from the FE discretization. Fig. 4.3(b) compares the equilibrium configurations for the continuum/FE model and the MM model in the Squeezed B situation. Despite the large deformations to which the tube is subjected, the agreement is excellent. Table 4.1 shows that the equilibrium energies obtained with the continuum model are in all the cases very accurate approximations of the MM energies. The discrepancies are in all the cases below 2 %. The effect of the inner relaxation in the magnitude of the energies is very important. In this table 20 finite elements have been used, while 32 hexagonal cells span the same perimeter in the MM model. Therefore, we expect the FE model to be more constrained and yield higher equilibrium energies. This can be noticed in the columns corresponding to the frozen inner displacements. However, when those are relaxed, the FE model reaches lower energies than the molecular model, still remaining very accurate. Probably the continuum treatment of the inner displacements allows for this extra relaxation. Although the effect of the inner relaxation in the equilibrium energies is very important, in these simulations its effect on the stable configurations is negligible. This can be explained by noting that the in-plane behavior of the model is very stiff, while the flexural behavior is very compliant. Therefore, a slight perturbation of inplane deformation (the inner rearrangements are an in-plane effect) has a dramatic influence on energy, but not in these flexural-dominated optimal deformations. This suggests that in these examples, the inner relaxation is nearly uncoupled from the bending deformation. This is not the case for other types of deformation such as those presented in Chapter 3. Table 4.2 shows the results obtained with 36 finite elements. The errors obtained are smaller in all the cases except in the Squeezed B situation with inner relaxation. This indicates that in general the richer discretization decreases the overall error, but also that the finer mesh allows other modelling errors to manifest themselves. Indeed, the error probably increases in the last case because the continuum model
85
[20,0] tube; ∆E = 0
[26,0] tube; ∆E < 0
[32,0] tube; ∆E > 0
[40,0] tube; ∆E > 0
Figure 4.4: Which is more stable, circular or collapsed? (answer: for the (20,0) and (26,0) tubes, circular, and for the (32,0) and (40,0) tubes, collapsed.) is more compliant than the molecular one with regard to the inner displacements. However, simulations carried out with even finer meshes indicate that the results “converge” to a very accurate result. This very good accuracy of the continuum model based on the exponential Cauchy-Born rule sharply contrasts with the results provided by a continuum model constructed from the standard Cauchy-Born rule. A similar experiment to that reported in Section 3.5.2. As mentioned then, the resulting hyperelastic potential is non-convex, i.e. the model has zero bending stiffness. The pathological mesh dependency in the numerical implementation of such a model is illustrated by Fig. 4.3(c); sharper kinks in the numerical solution are observed as the mesh is refined. The equilibrium energy of the FE solutions is almost zero, which is not realistic.
4.4.2
Representative simulations
The next simulations illustrate the application of the continuum/FE model to the transverse mechanics of nanotubes in different situations. In these applications, the computational cost of analogous molecular mechanics simulations would be much higher than the cost of the presented calculations. The first example studies the stability of the circular and the collapsed configurations of carbon nanotubes. Because of the van der Waals attraction potential, the energy of the system is reduced when two walls adhere. On the other hand, for the wall of a nanotube to come in contact with itself, significant elastic energy
86
[42,0] and [32,0] nested tubes; ∆E > 0
Figure 4.5: Transverse stability of a Multi-Walled Nanotube. is required. This tradeoff is probably responsible for the observation by Gao et al. (1998) that below a certain radius, only the circular configuration is stable. For greater radii, the collapsed configuration is at least meta-stable. Subsequently, another threshold radius separates the nanotubes for which the circular configuration is energetically favorable to those in the collapsed configuration. Figure 4.4 shows the simulations performed for several nanotubes. In this and subsequent figures, the nodes shown are nodes of the finite element mesh; they are not atoms. In these calculations, the fully relaxed circular configuration is deformed so that the wall of the nanotube is brought in contact with itself at the van der Waals equilibrium distance, and then the energy is minimized. The sign of the difference in energy between the circular configuration and the relaxed configuration is also reported, i.e. a positive difference means that the energy of the configuration presented on the right is lower. In some cases, the nanotube goes back to the original configuration (this is the case of the (20,0) nanotube). This implies that the collapsed configuration is not stable. The collapsed configuration is stable for the (26,0) nanotube, but this only constitutes a local minimum of the energy since the circular configuration has lower energy. For the (32,0) and (40,0) nanotubes, the collapsed configuration is the energetically favorable structure. This is expected because larger nanotubes are more flexible and have more wall area to gain adhesion energy. Figure 4.5 displays a similar analysis for a multi-walled nanotube for which the collapsed configuration yields lower energy. A similar competition of elastic and adhesion energy occurs when two nanotubes are brought to the van der Waals equilibrium distance. Figure 4.6 shows the equilibrium configurations obtained when this numerical experiment is performed with nanotubes of different sizes. Again, the larger nanotubes have larger portions of flattened walls. We also report a simulation of a bundle of nanotubes under plane strain. Figure 4.7 shows the equilibrium configuration of the system. A TEM image of such a
87
Two [20,0] nanotubes
Two [24,0] nanotubes
Two [32,0] nanotubes
Figure 4.6: Equilibrium configurations for pairs of nanotubes in van der Waals contact. nanorope has been reported by Salvetat et al. (1999). Carbon nanotubes tend to be closely packed in hexagonal lattices in the nanoropes and crystals of nanotubes (Thess et al. 1996; Schlittler et al. 2001). As can be seen from Fig. 4.7, the equilibrium configuration displays a flattening of the nanotube walls, or partial polygonalization.
4.5
Summary and conclusions
We have further explored a methodology to construct continuum models for oneatom thick crystalline films. The proposed model is a hyperelastic membrane whose elastic potential energy is written in closed-form exclusively in terms of the interatomic potentials that constitute the molecular description of the system. The analysis of the present work is based on the exponential the Cauchy-Born rule presented in Chapter 2. The general methodology then is particularized to analyze the transverse mechanics of carbon nanotubes. This model explicitly exploits the symmetry of such a deformation, and leads to a model of reduced dimensionality—a curve deforming in two-dimensions. The proposed model is discretized using Hermite finite elements,
88
Figure 4.7: Equilibrium configuration of a bundle of seven closely packed (22,0) nanotubes. yielding an alternative simulation method that is faster than atomistic calculations. Several simulations highlighting the relevance of van der Waals interactions in the transverse mechanics of nanotubes are reported. The results show that the continuum model based on the exponential Cauchy-Born rule very well approximates the stable configurations and energies of the corresponding molecular mechanics model. Results agree with molecular mechanics calculations within 2% in the equilibrium energies. This sharply contrasts with the non-physical results obtained from a model based on the standard Cauchy-Born rule. We also show the important effect of the inner rearrangements of the crystal structure of graphene on the equilibrium energies.
Chapter 5 Elastic moduli of carbon nanotubes 5.1 5.1.1
Introduction Elastic moduli from atomistic potentials
The study of the relation between elastic continua and crystalline solids dates back to the origins of elasticity, with the structure theory of elasticity developed by Cauchy in the early nineteenth century, and the so-called Cauchy relations of the elastic moduli, which turned out to be valid only for a particular class of lattices considered by Cauchy—see Note B in Love (1927) for a historical account and a discussion, and Stakgold (1950). The problem of expressing the elastic moduli in terms of the interatomic interactions of the lattice is well established with modern molecular theories of elasticity. There are two equivalent approaches for extracting the elastic moduli from the atomistic description (Weiner 1983), namely the method of the homogeneous deformations (Stakgold 1950; Martin 1975; Cousins 1978), also called Cauchy-Born hypothesis (Milstein 1982; Ericksen 1984), and the method of the long waves within lattice-dynamical theories (Born and Huang 1954). For examples of the application of these classical methods to planar graphene, see Zhang et al. (2002) and Al-Jishi and Dresselhaus (1982), respectively. The method based on the Cauchy-Born rule is purely static and, unlike the latter, allows large deformation nonlinear elastic constitutive laws to be formulated. The idea behind the Cauchy-Born rule, and how it allows hyperelastic potentials to be formulated from the atomistic potentials has been discussed in Chapters 2 and 3. Recall that the resulting continuum constitutive model depends only on the atomistic model in hand, without additional phenomenological input or fitting of 89
90 any kind. In addition, constitutive models constructed in this fashion inherit the crystal symmetries and anisotropy, and are finite deformation models in nature. As will become clearer later in this chapter, it is essential that finite deformations are accounted for, even to obtain infinitesimal elastic properties of nanotubes, since a finite deformation is required to roll a planar graphene sheet into a cylinder of finite radius. Because of the nonlinearity of the model, the elastic moduli depend on the state around which the system is linearized.
5.1.2
Outline
The present chapter deals with the derivation of elastic moduli of carbon nanotubes in terms of the interatomic potential, from the finite deformation membrane theory for curved monolayer lattices presented in Chapter 3. These elastic moduli, which derive naturally from the theory, acknowledge the two-dimensional nature of graphene, and possess unusual units. This agrees with the opinion of Hern´andez et al. (1998) that the notion of thickness for a sheet one atom thick is rather awkward from a purely mechanistic point of view. Assigning a thickness to the wall of nanotubes viewed as shells is very common in the literature (Yakobson et al. 1996; Ru 2000; Ru 2001; Qian et al. 2001; Zhang et al. 2002). The most fundamental flaw of this approach is that shell theory is a convenient model for bulk materials that form structures very thin in one dimension, while nanotubes are intrinsically two-dimensional systems. Since the present membrane theory for nanotubes is written in closed-form in terms of the atomistic potential (no local atomistic calculations are needed), it can be exploited analytically. Section 5.2 presents an overview of the Lagrangian elasticity tensors that arise from the membrane theory, and provides a precis on the crystal symmetries of graphene. Then, the analytically tractable case of planar graphene is developed in Section 5.3. Infinitesimal moduli of planar graphene are provided. In particular, the bending modulus of planar graphene is written explicitly in terms of the functional form of the potential. This, to the best of our knowledge, is an original contribution. These expressions are validated by comparisons with detailed atomistic calculations, and the elastic properties derived from the popular potential developed by Brenner (1990) are compared with recent ab initio data. We also illustrate how the explicit expressions for the elastic moduli in terms of the atomistic potential can be useful in the parameter fitting. Section 5.4 develops a 1D hyperelastic model for uniaxially stretched and otherwise unconstrained nanotubes from the general theory. This model is applicable at finite stretches for nanotubes of finite radius, and accounts for the chirality, for finite radial relaxation, and inner relaxation. Its evaluation requires the numerical solution of univariate and bivariate minimization problems and therefore constitutes a semi-analytical approach. Very
91 careful validation of this model against atomistic calculations reveals that it reproduces the exact energetics of finite stretches up to numerical precision. These findings suggest that the configuration space of uniaxially stretched and otherwise unconstrained nanotubes can be reduced to the four parameter configuration space of our model. Further tests show that fewer than four parameters are insufficient. This one dimensional hyperelastic model is then used to analyze the dependence of the elastic modulus with respect to stretch, yielding a simplified failure analysis, and to study the systematic dependence of elastic properties on chirality and diameter, whose effects can be isolated by the continuum model. The anisotropy induced by finite deformations is also analyzed.
5.2
Elasticity tensors for curved graphene
Recall from Chapter 3 that the strain energy density (energy per unit undeformed area) of the continuum membrane replacing a graphene sheet deforming in three dimensions can be written explicitly in terms of the atomistic potential W = W (C, K, η) ,
(5.1)
where C and K are the Lagrangian expressions of the first and the second fundamental forms of the membrane, and η denotes the inner displacements field, also expressed in the undeformed configuration. The selection of these strain measures and the definition of the inner displacements in the undeformed body guarantee material frame indifference—rotational invariance—of the hyperelastic potential. Furthermore, this hyperelastic potential retains the symmetries of the crystal. This means that if we denote by G ⊂ SO(2) the invariance group of the underlying lattice, W QT CQ, QT KQ, QT η = W (C, K, η) , ∀ Q ∈ G, (5.2) for all symmetric positive definite second order tensors C, and for all symmetric second order tensors K. (Note that if we denote by Q the rotation viewed as a point mapping corresponding to the proper orthogonal tensor Q, then the arguments of the left hand side in the above equation are obtained by the pull-back operations Q∗ C = QT CQ, Q∗ K = QT KQ, and Q∗ η = QT η, see Marsden and Hughes (1983)). It is easily shown by counter-examples that the finite deformation membrane theory is not isotropic, i.e. G 6= SO(2). Instead, it displays the six-fold symmetry of graphene. In particular, as shown later in this chapter, the continuum membrane for nanotubes accounts for the chirality of the tubes. Given a deformation of the surface, the inner displacements are an independent
92 kinematic variable that can be eliminated at the constitutive level by minimizing the strain energy density with respect to η:
b (C, K) = arg min W (C, K, η) η η
=⇒
∂W = 0. ∂η η =ηb
(5.3)
After this inner relaxation, the strain energy density can be written as a function of C and K only c (C, K) = W (C, K, η b (C, K)) . W (5.4) Note that, while a closed-form expression for the hyperelastic potential W is availc (C, K) requires the solution of a bivariate able (see Eq. (3.36)), the evaluation of W minimization problem, which in general must solved numerically. Note also that if the graphene sheet is planar, i.e. K = 0, this theory results exactly in standard finite crystal elasticity. Work conjugate stress measures to the above mentioned strain measures for the surface can be derived from the hyperelastic potential c ∂W ∂W S=2 =2 ∂C ∂C η =ηb
and
c ∂W ∂W m= = . ∂K ∂K η =ηb
(5.5)
c by Note that it is possible to substitute the derivatives of the effective potential W derivatives of the analytically available potential W as long as these are evaluated at the relaxed inner displacements, as argued in Section 3.3.2. The first of these stress measures is an in-plane stress, and corresponds to the Second Piola-Kirchhoff stress (Marsden and Hughes 1983). It has units of force divided by length (surface tension), while the second is a moment-like stress that has units of force. Note that these are not stress resultants; their unusual units for stress tensors follows from the fact that the membrane has no thickness. Similarly, effective (i.e. at the relaxed inner displacements) Lagrangian elasticity tensors can be defined as
Cin−plane = 4
c ∂2W , ∂C2
Cbending =
c ∂2W , ∂K2
and
Cmixed = 2
c ∂2W . ∂C∂K
(5.6)
The first of these elastic tensors is a measure of the purely in-plane stiffness of the membrane and is measured in units of force divided by length. It corresponds to the second elasticity tensor. The second represents the bending stiffness and the third
93 is an in-plane/bending coupling stiffness. Note the unusual surface tension units for the in-plane elastic modulus, a consequence of the two-dimensional nature of a graphene sheet. In the calculation of these elastic moduli, the fact that the inner displacements c are in internal equilibrium does not allow us to replace in these expressions W by W , as in Eq. (5.5); extra terms with crossed derivatives of W with respect to the strain measures and the inner displacements arise (see Tadmor et al. (1999) and Section 5.3). It is also important to emphasize that, although closed-form expressions for these moduli in terms of the atomistic potential can be obtained for a given local finite deformation characterized by C and K, these expressions will b (C, K) is not available in closedalso depend on η. For an arbitrary deformation η form. Therefore, the evaluation of these moduli requires in general the numerical solution of the bivariate minimization problem in Eq. (5.3) (see Appendix A.2 for an algorithm).
5.3
Infinitesimal elastic moduli of graphene
As pointed out above, the proposed membrane theory furnishes closed-form expressions for the elastic moduli in terms of the interatomic potential of choice, provided the relaxed inner displacements are known. The current section develops these closed-form expressions for the analytically tractable situation of planar graphene. In this situation, symmetry considerations allow to conclude that the inner displacements vanish. Although this is a very simplified setting, the moduli can be written in closed-form, and are found to provide good estimates of elastic properties for nanotubes. Potentials following the bond-order formalism (Tersoff 1988) are considered, although explicit expressions can be obtained for any interatomic potential. The popular potential for hydrocarbons developed by Brenner (1990) falls within this category. The precise expressions of the in-plane moduli in terms of the functional form of the potential are provided. The derivation of in-plane moduli of graphene from the interatomic potential, accounting for the effect of the inner relaxation, can be obtained following standard methods of crystal elasticity (Cousins 2001; Weiner 1983), and has been recently reported by Zhang et al. (2002). A brief discussion of the appropriate definition of Young’s modulus and Poisson’s ratio for a genuinely two-dimensional material, for which plane strain and plane stress do not make sense, is then presented. The bending modulus of graphene is derived for an initially planar sheet. This modulus has been frequently analyzed from atomistic calculations (Kudin et al. 2001; Hern´andez et al. 1998), but to our knowledge, this is the first closed-form expression of this material parameter in terms of the atomistic potential.
94 Finally, the closed-form expressions are tested against atomistic calculations, and it is illustrated how they can help in the parameter fitting process for a bond-order potential.
5.3.1
In-plane moduli
By calculations analogous to those presented by Tadmor et al. (1999), the previously defined in-plane second elasticity tensor can be computed at the relaxed inner b as displacements η = η "
Cin−plane
∂2W ∂2W =4 − · ∂C2 ∂C∂η
∂2W ∂η 2
−1
# ∂2W · . ∂η∂C
(5.7)
This tensor in the referential configuration, taken here to be the equilibrium configuration, corresponds to the infinitesimal tensor of elastic moduli. The general form of a bond order potential provides the energy of bond ij as ¯ij VA (rij ), Eij = VR (rij ) − B
(5.8)
¯ij depends only on the angles adjawhere for moderate deformations of graphene, B cent to bond ij (Brenner 1990). In the following, all the expressions are evaluated at the equilibrium configuration of graphene, characterized by equal bond angles of 2π/3 rad, and the equilibrium bond length A0 . As argued previously in Chapter 3, graphene has three inequivalent bonds and angles. We denoted the angles adjacent to bond vector Ai , i = 1, 2, 3 as θj and θj , where {i, j, k} is an even permutation of {1, 2, 3}. Therefore, the strain energy density can be written as (see Eq. (3.36))
W =
3 1 X ¯ j , θk )VA (ai ) , VR (ai ) − B(θ S0 i=1
(5.9)
¯ can be seen as a function with two arguments. Let us denote by V 0 and and B R 00 VR the first and second derivatives of VR with respect to their only argument, and ¯ 0 denote the first derivative of the bond order B ¯ with respect similarly for VA . Let B to any of its arguments (when evaluated at graphene at equilibrium, the choice ¯ 00 denotes the second derivative of B ¯ with of argument does not matter), while B 0 ,0 ¯ ¯ with respect to one of its arguments. By B we denote the second derivative of B
95 respect to the first and the second arguments. We define the following quantities defined at ai = A0 and θi = 2π/3
Crr =
VR00
−
¯ 00 , BV A
Cθθ
3VA ¯ 00 ¯ 0 ,0 ), and −B = 2 (2B A0
Crθ
√ 2 3 0 ¯0 V B. = A0 A
(5.10)
Lengthy but otherwise straightforward calculations lead to the following expression, evaluated at the relaxed planar configuration of graphene
where
and
e + λ Id ⊗ Id, Cin−plane = µ (I + I)
(5.11)
1 (Crr + Cθθ )2 µ = √ Crr − Crθ − Cθθ − Crr + Crθ − Cθθ 4 3
(5.12)
1 (Crr + Cθθ )2 λ = √ Crr + Crθ + Cθθ + Crr + Crθ − Cθθ 4 3
(5.13)
are the Lam´e coefficients. The underlined terms correspond to the effect of the inner displacements, i.e. the second term on the right hand side of Eq. (5.7). The tensor of infinitesimal moduli in Eq. (5.11) has the general form of a two dimensional fourth-order isotropic tensor. Denoting by {I1 , I2 } the Euclidean basis of R2 , the identity tensors in this equation are defined as I = δ AC δ BD IA ⊗ IB ⊗ IC ⊗ e = δ AD δ BC IA ⊗ IB ⊗ IC ⊗ ID , and Id = δ AB IA ⊗ IB (Belytschko et al. 2000). ID , I Thus, it is found from this calculation that the infinitesimal elasticity tensor is isotropic, a well known fact about hexagonal lattices. Note that this does not imply that graphene is isotropic for finite deformations, i.e. G 6= SO(2) (see Eq. (5.2)). The examples provided later demonstrate that nanotubes, which are finitely deformed graphene sheets, exhibit some anisotropy, and the elastic properties show systematic variations with respect to chirality.
5.3.2
Genuine two-dimensional linear elasticity
The expressions relating the conventional Lam´e coefficients with the Young’s modulus Y and Poisson’s ratio ν for bulk isotropic linearly elastic materials are Y =
µ(3λ + 2µ) , λ+µ
and
ν=
λ . 2(λ + µ)
(5.14)
96 Young’s modulus and Poisson’s ratio are defined from the thought experiment of applying uniaxial tension σ11 to a prismatic elastic body, and measuring the strains in each direction. Then Y := σ11 /ε11 and ν := −ε22 /ε11 = −ε33 /ε11 . One may be tempted to use these expressions in Eq. (5.14) for graphene. However, it is important to bear in mind that, although the expression for the infinitesimal elastic tensor in the previous section in terms of the Lam´e coefficients has a very familiar appearance, this fourth-order tensor is defined in a two-dimensional space, the plane in which the graphene sheet deforms. If one subjects a rectangular slab of graphene to uniaxial tension σ11 , measures the strains ε11 and ε22 (ε33 is not defined), and adopts the natural definitions Ys := σ11 /ε11 and νs := −ε22 /ε11 , the resulting expressions in terms of the Lam´e coefficients in Eq. (5.11) are
Ys =
4µ(λ + µ) , λ + 2µ
and
νs =
λ , λ + 2µ
(5.15)
where the subscript emphasizes the fact that they refer to a surface continuum. These modified definitions are adopted in the following. As usual, νs is nondimensional. In this case, Ys has units of surface tension (Hern´andez et al. 1998). The in-plane shear modulus (with units of surface tension) coincides with µ.
5.3.3
Bending modulus
Our extended theory of crystal elasticity for curved lattices of reduced dimensionality allows us to obtain a closed-form expression for the bending stiffness of graphene. We consider an initially planar graphene sheet. We then calculate the second derivative of the strain energy density with respect to the curvature in a given direction. We are interested in the scalar modulus
Cbending =
c ∂2W , ∂κ2
(5.16)
where κ is the only non-vanishing principal curvature of the monolayer. Appendix D describes the derivation of this modulus in terms of the functional form of the inter-atomic potential. Several intermediate results deserve special attention. On the one hand, the fact that the second term of Eq. (D.5) vanishes reveals that, unlike the in-plane moduli, the infinitesimal bending modulus is insensitive to the inner displacements. On the other hand, the it follows from the derivation of Appendix D that for any two-dimensional hexagonal lattice whose interatomic potential depends only on bond lengths and angles (and not on dihedral angles for
97 instance), the infinitesimal bending modulus around the planar state can be written as 3 3 X ∂W ∂ 2 ai X ∂W ∂ 2 θj Cbending = + . (5.17) 2 2 ∂a ∂κ ∂θ ∂κ i j i=1 j=1 Remarkably, this modulus does not depend on second derivatives of the atomistic potential function. This means that if we naively adopt a quadratic two-body/threebody expansion of the energy of graphene along the lines of MM models E=
X
ks (ri − A0 )2 +
bonds
X
kθ (θi − 2π/3)2 ,
(5.18)
angles
at the planar equilibrium configuration we have ∂W/∂ai = 0 and ∂W/∂θi = 0, and therefore such a lattice has zero initial bending stiffness, which is unphysical for graphene. As expected, the infinitesimal bending stiffness of planar graphene does not depend on the direction in which the sheet is being bent, that is planar graphene is isotropic with respect to bending. For bond order potentials, the bending modulus adopts the particularly simple expression Cbending =
5.3.4
1 ¯ 0 (2π/3, 2π/3). VA (A0 )B 2
(5.19)
Verification of the expressions for the elastic moduli
The explicit expressions for Young’s modulus, Poisson’s ratio, and the bending modulus derived in the previous sections are checked against atomistic calculations. The second parameter set of the potential proposed by Brenner (1990) has been adopted in the comparisons. According to Eqs. (5.10), (5.12), (5.13), (5.15), and (5.19), Ys = 235.8 J/m2 ,
νs = 0.4123,
2
and Cbending = 2.177 eV˚ A /atom.
(5.20)
The result for Young’s modulus perfectly matches that reported in Zhang et al. (2002) for the same potential. To compare with values of Young’s modulus provided in units of pressure which assume a thickness t=0.34 nm, simply operate as follows: 694 MPa = 236 J/m2 / 0.34 nm. To extract Young’s modulus and Poisson’s ratio of planar graphene from atomistic calculations, one can subject a slab of graphene to a small uniaxial deformation,
98
Strain energy (eV/atom)
0.2
0.15
Atomistic arm−chair Atomistic zig−zag Quadratic fit
0.1
0.05
0 0
1
2
−1
1/R (nm )
3
4
Figure 5.1: Strain energy relative to planar graphene for fully relaxed nanotubes of varying radius plotted versus a quadratic fit of the bending energy with the bending modulus predicted by the continuum theory. and measure from the simulation the longitudinal forces and the lateral deformations to estimate the moduli. To avoid the issue of the dangling bonds on the sides of the slab or special lateral periodic boundary conditions, it is simpler to consider a very large nanotube, for which the effect of curvature is negligible. We consider a (150,150) nanotube 60 hexagonal cells in length, stretch it a small amount (0.02%), and optimize the structure, allowing in particular for radial relaxation (only degrees of freedom in the axial direction are prescribed). Then, to exclude the effect of the boundaries, the actual stretch of the central part of the tube is measured from the simulation, as well as the forces and the lateral deformation. These careful estimations perfectly match (to four significant digits) with the analytical values for Ys and νs presented in Eq. (5.20). The significance of the inner displacements on Ys and νs can be explored by omitting the second term in the right hand side of Eq. (5.7), that is omitting the underlined terms of Eqs. (5.12) and (5.13). In this case, Ysno inner = 337.8 J/m2 and νsno inner = 1.1580. Thus, the effect of the inner displacements is very important. Although these values more closely agree with accurate ab initio data (see Section 5.3.5), they do not represent the actual behavior of atomistic systems described by Brenner’s potential. To validate the expression for the bending modulus, we compute the strain
99 energies (relative to the ground state of planar graphene) of fully relaxed nanotubes of varying radius. We plot the energy of these nanotubes versus the inverse of their equilibrium radius, and compare it with the simple quadratic expression 1/2 Cbending (1/R)2 . A priori, one could expect this approximation to be valid for very small curvatures, that is very large nanotubes. However, a quadratic approximation to the bending energy has been shown to very accurately describe ab initio results for a wide range of radii (S´anchez-Portal et al. 1999; Kudin et al. 2001). Here, the modulus of Eq. (5.19) predicted by the continuum theory is employed. Figure 5.1 shows that the agreement between the atomistic calculations and the quadratic energy is excellent. For small nanotubes (right side of the plot) some deviations are noticeable. These deviations suggest that for Brenner potential, curvature has the effect of increasing the bending modulus, and zig-zag tubes appear slightly stiffer to curvature. Nevertheless, for nanotubes of diameter larger than 1 nm, the bending modulus provided by the continuum analysis very accurately predicts the bending stiffness of actual atomistic models of nanotubes. The insensitiveness to chirality for large enough tubes is also clear from the figure. From these tests, we conclude that the explicit expressions of the elastic moduli of graphene in terms of the interatomic potential very accurately describe the elastic properties of the actual atomistic system.
5.3.5
Comparison of Brenner’s potential with ab initio data
Table 5.1 reports the values of the elastic moduli of planar graphene Ys , νs , and Cbending derived from the above explicit expressions for second parameter set of Brenner’s potential. The predicted equilibrium bond length for graphene with this potential is 0.145 nm. The second generation bond-order potential by Brenner et al. (2002) is also analyzed, and the associated elastic properties reported in Table 5.1. The equilibrium length predicted by this potential is closer to the widely accepted value of 0.142 nm. Note that the overestimation of the equilibrium bond length with the original potential by Brenner slightly biases the elastic properties. The moduli from empirical potentials are compared with available data from ab initio calculations by Kudin et al. (2001), which very well agree with other published data (S´anchez-Portal et al. 1999). It can be observed that the bond-order potentials deviate significantly from the ab initio data. The second-generation bond order potential slightly improves Young’s modulus and Poisson’s ratio, but provides a worse bending stiffness. These comparisons suggest that the transferability of these potentials is limited with regards to the elasticity of graphene. Further investigation in Section 5.4.8 reveals that the Brenner potential does reproduce qualitative features obtained from quantum mechanical models. Equation (5.19) provides valuable insight into the relation between this elasticity
100 Table 5.1: Elastic properties of graphene from ab initio calculations, and from Brenner’s potentials (∗ obtained from the closed-form expressions). 2
Ys (J/m2 ) νs Cbending (eV˚ A /atom) ∗ Brenner (1990) 236 0.412 2.2 Brenner (2002)∗ 243 0.397 1.8 Kudin et al. (2001) 345 0.149 3.9 and the functional form of the potential. Consider the following exercise. Let us ¯ does not re-scale the bond order function of Brenner’s original potential, so that B 0 ¯ change, but B does in order to fit the ab initio value for Cbending . Note that changing ¯ 0 does not alter the ground energy of graphene, nor the equilibrium the value of B bond length. By doing this, we observe that the obtained value for Young’s modulus is 336 J/m2 , very close to the ab initio value. Poisson’s ratio also dramatically improves to a value of 0.16. This exercise illustrates how the expressions provided in the previous sections can facilitate the fitting of an analytical potential, here a bond-order potential.
5.4
1D finite elasticity for stretched nanotubes
To formulate an elastic rod model from the three-dimensional membrane, we constrain the membrane to cylindrical configurations. We thus assume that, as the nanotube is tested axially, the cylindrical symmetry is not lost. This hypothesis, whose accuracy is checked a posteriori by comparing the theoretical predictions of this analysis with actual atomistic simulations, is supported by ab initio calculations (S´anchez-Portal et al. 1999). The different relaxations processes taking place in uniaxially stretched nanotubes are rationalized in the continuum setting by inner and radial relaxations, which are performed numerically. The appropriate measure of the elastic stiffness of nanotubes seen as one-dimensional rods is defined around the equilibrium state. Comparison with atomistic calculations suggests that the one-dimensional hyperelastic model replicates the exact energetics of uniaxially stretched nanotubes. In conclusion, the influence on the elastic properties of stretch, chirality, and ideal radius are analyzed.
101
5.4.1
Elastic potential for cylindrical nanotubes
√ We consider a (n1 , n2 ) nanotubepof unit length, chirality Θ0 = arctan[ 3n2 /(2n1 + n2 )] and ideal radius R0 = A0 3(n21 + n1 n2 + n22 )/2π, where A0 is the graphene equilibrium bond length (Saito et al. 1992). Similarly to Section 3.3.1, we consider deformations from an undeformed body Ω0 = (0, 1) × (0, 2πR0 ) into a cylinder defined by Φ1 = Λ1 X 1 ,
Φ2 = Λ2 R0 cos
X2 , R0
and
Φ3 = Λ2 R0 sin
X2 , R0
(5.21)
that is, a cylinder with uniform longitudinal stretch Λ1 and uniform circumferential stretch Λ2 . Note that the deformation is parametrized by the pair (Λ1 , Λ2 ). For such a deformation, the strain measures are uniform on Ω0
2
(Λ1 ) [C]B0 = 0
0 (Λ2 )2
and
0 [K]B0 = 0
0 1 Λ2 R0
,
(5.22)
and the principal directions coincide identically with the basis vectors of the Euclidean basis of R2 , B0 . The exponential Cauchy-Born rule maps an undeformed lattice vector A whose Euclidean components are A1 and A2 , into the deformed lattice vector a defined in an Euclidean basis of R3 as (see Chapters 3 and 4)
[a]Be =
Λ1 A1 2
A ) Λ2 A2 Q( R 0 2 2 Λ2 (A ) Q 2 ( A2 ) 2R0 2R0
,
(5.23)
where Q(x) = sin x/x. Since the undeformed lattice vectors depend on the inner displacements, from this expression we can write the three inequivalent lengths and angles of the bonds of graphene as ai = f¯(Λ1 , Λ2 , η), and θi = g¯(Λ1 , Λ2 , η),
i = 1, 2, 3,
(5.24)
102 following the same rationale that led to Eqs. (3.32) and (3.33) (the explicit dependence on the undeformed lattice has been dropped). Then, either by plugging Eq. (5.24) into Eq. (3.36), or by constraining the general potential W (C, K, η) to the above described deformations, the reduced hyperelastic potential W = w(Λ1 , Λ2 , η) (5.25) can be written in closed-form in terms of the atomistic potential. Since the strain measures are uniform on the body, we can view the nanotube as a rectilinear elastic rod whose configuration space is described by (Λ1 , Λ2 , η). Recalling that w represents an energy per unit undeformed area, we can write the energy per unit undeformed length of the nanotube as E(Λ1 , Λ2 , η) = 2πR0 w(Λ1 , Λ2 , η).
5.4.2
(5.26)
Inner relaxation
Given a deformation characterized by the pair (Λ1 , Λ2 ), the inner displacements can be treated as for the general case by relaxing η as follows
b (Λ1 , Λ2 ) = arg min E(Λ1 , Λ2 , η) η η
=⇒
∂E = 0. ∂η η =ηb
(5.27)
b is a constant vector in Ω0 . The Note that, since the deformation is uniform, η following effective energy function for cylindrically constrained deformations follows b 1 , Λ2 ) = E(Λ1 , Λ2 , η b (Λ1 , Λ2 )). E(Λ
(5.28)
Once again, the inner relaxation cannot be performed analytically in general, and the bivariate minimization problem is solved numerically, for instance by simplifying the expressions in Appendix A.2 and using Newton’s method. This procedure is very fast; our experiments with the Brenner potential show that convergence within machine precision is reached in three iterations at most.
5.4.3
Radial relaxation
When nanotubes are tested axially a Poisson effect is observed in atomistic calculations, i.e. changes in the axial stretch result in changes in the equilibrium radius
103 of the nanotube. If we allow for this radial relaxation, the elastic potential can be further simplified to model straight nanotubes deformed axially and otherwise unconstrained. For this purpose, consider the relaxed circumferential stretch e b Λ2 (Λ1 ) = arg min E(Λ1 , Λ2 ) Λ2
=⇒
b ∂E = 0. ∂Λ2 Λ2 =Λe 2
(5.29)
The energy per unit undeformed length of the nanotube relaxed radially, thus acting like a rod, can be written exclusively in terms of the axial stretch as e 1 ) = E(Λ b 1, Λ e 2 (Λ1 )). E(Λ
(5.30)
Note that the kinematic assumptions and these relaxations have reduced the configuration space of the carbon nanotube to the single scalar parameter Λ1 . This function is a hyperelastic potential for the nanotube viewed as a one-dimensional rod, that is an strain energy per unit undeformed length. Despite its trivial appearance, the above one-dimensional hyperelastic potential is derived strictly from the atomistic potential, and accounts for the finite curvature of graphene to make a nanotube, for finite radial relaxation and finite inner rearrangements. Furthermore, it can be evaluated a finite axial stretches. Its evaluation requires the solution of the the bivariate minimization problem in Eq. (5.27), e 1) nested in the univariate minimization problem in Eq. (5.29). As shown later, E(Λ accounts for the chirality of the nanotube. Remark 5.1. Note that, for a sufficiently well behaved potential E(Λ1 , Λ2 , η), the order in which the relaxations are performed is irrelevant; the scalar-valued one die 1 ) can be simply obtained by, for fixed Λ1 , minimizing E with mensional potential E(Λ respect to its other arguments. Splitting the relaxation as described is conceptually clearer, and makes the implementation of the minimization problems easier.
5.4.4
Equilibrium state
In order to meaningfully define infinitesimal elastic moduli as properties of nanotubes, it is essential to consider a stress free equilibrium state. It is observed in atomistic simulations (S´anchez-Portal et al. 1999) that the structure obtained by applying the isometric mapping that wraps the graphene sheet into a cylindrical nanotube (Λ1 = Λ2 = 1) is not in equilibrium, i.e. the energy E0 of this nanotube of unit length and radius R0 is not the minimum energy that the nanotube can reach. The energetics of the fully relaxed nanotube can be easily rationalized with the help
104 of the continuum model. The inner relaxation is automatically accounted for in the b Then, the potential E b can be minimized with respect to both stretches, potential E. yielding the equilibrium configuration of the nanotube (Λ1eq , Λ2eq ). The energy of this optimal configuration Eeq can be written as: e 1eq ) = E(Λ b 1eq , Λ2eq ) > Egraphene , E0 > Eeq = E(Λ
(5.31)
where Egraphene denotes the ground energy of planar graphene.
5.4.5
Stress and stiffness of straight nanotubes
e 1 ) is the hyperelastic potential of the nanotube viewed As pointed out earlier, E(Λ as a hyperelastic 1D rod. As a matter of fact, in this simple case, the axial stretch is the one-dimensional deformation gradient F ≡ Λ1 . From this point on we will rather use this notation for the axial stretch of the nanotube. Thus, F = 1 corresponds to the ideal length of the nanotube, and Feq = Λ1eq characterizes the free-standing nanotube with no axial stress. Different stress measures can be obtained from classical continuum mechanics relations. The first Piola-Kirchhoff stress tensor (a scalar in this case) is e b ∂E ∂E P = = ∂F ∂F
e2 Λ2 =Λ
∂E = , ∂F Λ2 =Λe 2 , η =ηb
(5.32)
where similar arguments as presented in Section 3.3.2 allow to write the last two identities. Note that P (Feq ) = 0. In this one-dimensional setting, the Jacobian determinant coincides with F , and the Cauchy or physical stress coincides with the first Piola-Kirchhoff stress σ = P . These stresses are actually axial forces, the resultant across the nanotube section. Several distinct elastic moduli, or elasticity tensors, can be defined in the Lagrangian description of continuum mechanics. These moduli are functions of the deformation. Therefore one must carefully consider which is the most appropriate modulus to characterize the axial stiffness of the nanotube, and at which configuration this modulus should be considered. The most natural choice for the configuration is the axially unstressed, equilibrium configuration Feq . As for the modulus, two different arguments lead to the same conclusion. On the one side, one may argue that the third (spacial) elasticity modulus a, the push-forward of the first elasticity e 2 F , is the appropriate choice (Marsden and Hughes 1983; Betensor A= ∂ 2 E/∂
105 lytschko et al. 2000). The reason behind this choice is that this modulus appears in the linearization of the theory relating the Cauchy stress with the usual small strain ε. Following standard continuum mechanics relations we have
a|F =Feq = Feq
e ∂2E ∂F 2
.
(5.33)
F =Feq
A somewhat more elegant way to obtain the same result is to change the reference configuration to the equilibrium configuration. A new normalized stretch is then defined F ∗ = F/F eq and the corresponding strain energy density referred to this new parameter is eq e ∗ (F ∗ ) = 1 E(F/F e E ), (5.34) F eq where the original strain energy density has been divided by the equilibrium stretch e ∗ is an energy per unit reference length in order to preserve the property that E and the reference has changed. Once this change of reference state has been made, the question of which elasticity tensor to consider becomes superfluous since, when evaluated at the new stress-free reference state F ∗ = 1, all the elasticity tensors coincide to 2 e∗ ∂ E c∗ |F ∗ =1 = , (5.35) ∂F ∗2 ∗ F =1
which is the infinitesimal tensor of elastic moduli, a scalar in this case. It is straightforward to show from Eq. (5.34), that c∗ |F ∗ =1 = a|F =Feq .
(5.36)
Note that the modulus c∗ has units of force, and has the physical interpretation of a stiffness modulus of the nanotube viewed as a one-dimensional rod. It is therefore a structural property of the nanotube. On the other hand, following Hern´andez et al. (1998), we can define the surface Young’s modulus (with units of surface tension) Ys =
c∗ 2πReq
(5.37)
as a force divided per circumferential length of the fully relaxed nanotube where Req = Λ2eq R0 . This point of view is consistent with our membrane theory. Thus,
106 graphene is viewed as the material that composes the tubular structure and, unlike c∗ , Ys is a material property. However graphene is a rather unusual material, since it is intrinsically two-dimensional and its elastic properties do not have units of pressure. Maybe for this reason, the most common convention for expressing Young’s modulus of nanotubes is to divide the rod’s axial modulus by an effective area Y = c∗ /(2πReq t),
(5.38)
where the effective thickness of the nanotube is usually assumed to be the interlayer distance of graphite t = 0.34 nm. This way of expressing the Young’s modulus emanates from the idealization of the nanotube as an elastic thin shell, although as noted previously this point of view is not mechanically consistent. Remark 5.2. Note that a closed-form expression for the modulus in Eq. (5.38) in terms of the atomistic potential is available by expanding for instance Eq. (5.33). e cannot simply be substituted As argued in Section 5.2, the second derivatives of E by second derivatives of E, and extra cross-derivatives arise. Nevertheless, once the b , the elasequilibrium state is known, that is (Λ1eq , Λ2eq ) and the corresponding η tic modulus can be simply evaluated. This contrasts with the common approximate methods of fitting polynomials or finite difference forms of derivatives (Kudin et al. 2001; Lier et al. 2000; Xiao and Liao 2002). In the work by S´ anchez-Portal et al. (1999), ab initio elastic properties are found in this manner, and the authors estimate an error of about 10% in the calculation of Young’s modulus. Here, the only numerical step is the calculation of the equilibrium state, which involves a bivariate minimization problem. These small problems can be easily solved to machine precision by quadratic methods at insignificant computational cost.
5.4.6
Validation of the theory and discussion
Recall that in Remark 2.1 we noted that in this thesis, kinematic rules such as the standard Cauchy-Born rule, or here the exponential Cauchy-Born rule, are viewed as postulates, whose accuracy are tested a posteriori by comparison with atomistic calculations. The basic hypothesis made in the derivation of the above reduced potentials is that the atoms of an axially deformed and otherwise unconstrained nanotube do not leave the cylindrical surface, and that the axial and the circumferential deformations are homogeneous, as well as the inner rearrangements. Let us consider an example to test the validity of these assumptions. A (10,10) nanotube is considered. An atomistic model of such a nanotube is incrementally stretched in the axial direction, and the structure is optimized at each step. Full radial relaxation of the structure is allowed (only degrees of freedom
107 Table 5.2: Comparison of strain energy relative to planar graphene for an axially deformed and otherwise unconstrained (10,10) nanotube (eV/atom)
Λ1eq
Λ1 0.9604701 0.9751906 0.9801357 = 1.0011929 1.0638382 1.1240572 1.1844136
Atomistic 0.0562190 0.0363382 0.0316380 0.0227191 0.0977064 0.2945486 0.5857545
Continuum 0.0562169 0.0363380 0.0316380 0.0227193 0.0977066 0.2945488 0.5857547
in the axial direction at the ends of the tube are constrained). Since no periodic boundary conditions are adopted in the atomistic calculations, special care must be taken to exclude the boundary effect of dangling bonds. The energies are quite sensitive to any inhomogeneity at the ends (see for instance Lier et al. (2000), where the effect of the end-caps in ab initio calculations of the Young’s modulus is discussed). Thus, for the analysis only the central section spanning 6 hexagonal cells of a tube spanning 30 are considered. The actual stretch experienced by these 6 cells is “measured” from the simulation, and the energy corresponding to these e to obtain the energetic cells is observed. Then, this measured stretch is fed into E prediction of the continuum model (with relaxed radius and inner displacements). The comparison is presented in Table 5.2 and Fig. 5.2. The second parameter set of the potential for hydrocarbons by Brenner (1990) is adopted, and the stretches are relative to an ideal configuration defined by the equilibrium C-C bond length for this model, A0 = 0.145 nm. The reported energies are relative to the ground state for planar graphene with the above mentioned bond length, Egraphene = −7.3756 eV/atom. For the strain energy E0 − Egraphene of the ideal nanotube (simply rolled without stretch from the planar graphene sheet) without any sort of relaxation, the atomistic and the continuum models agree to machine precision to a value of 0.0230195 eV/atom. The comparison for engineering strains ranging from -4% to 18% are reported in Table 5.2. The agreement is excellent even for such large deformations, and the discrepancies are of the order of the numerical tolerance in the optimizations. The only exception is the most compressive case for which the agreement begins to degrade, due to the onset of compressive buckling, see Section 3.5.1. As can be noted, the equilibrium axial stretch is not 1.0, but slightly larger. Similarly, the equilibrium circumferential stretch is Λ2eq = 1.0024170. This comparison is shown graphically in Fig. 5.2, where the anharmonic response
108
Strain energy (eV/atom)
0.7 0.6 0.5
Atomistic Continuum No inner relaxation No radial relaxation
0.4 0.3 0.2 0.1 0 0.95
1
1.05
1.1
1.15
Axial stretch Figure 5.2: Strain energy relative to planar graphene for an axially deformed and otherwise unconstrained (10,10) nanotube. of the model for large stretches is apparent. This figure also shows the energies obtained by the continuum model without the inner relaxation but relaxing the radius on the one hand, and without radial relaxation but relaxing the inner displacements on the other hand. This is done to illustrate the significance of these relaxations. The overestimates of the energy are apparent, as well as the fact that these partially relaxed continuum models lead to stiffer models. It can be observed that the effect of the inner displacements is more important. For this nanotube, the application of our fully relaxed model yields a value for Ys of 233 J/m2 , which agrees with careful estimates from atomistic calculations using the same potential, and finite difference approximation to the derivatives. If the inner displacements are not relaxed, the resulting value is 336 J/m2 , an error of 44%. If, from the nanotube at rest, Young’s modulus is calculated freezing the radius, the value obtained is 282 J/m2 , an error of 21%. Thus, it is clear that although a priori these effects may seem unimportant, they must be carefully considered for accurate results. The above results, particularly those of Table 5.2, suggest that the kinematic postulates at the origin of the present continuum models are fully justified. The exponential Cauchy-Born rule is kinematically exact for the deformations defined by Eq. (5.21), and it appears that the optimized configurations of stretched atomistic models of nanotubes conform to these cylindrical homogeneous deformations,
109 leading to exact energetics of the continuum model. Although the numerical evidence of the correctness of the theory is very strong, it has only been tested on a number of nanotubes and with several interatomic potentials. For this reason, sound theoretical analysis with precise statements on the ranges of validity would be desirable. Pioneering work in this field for the simple case of 2D lattices has recently rigorously shown that analogous hypothesis, namely the Cauchy-Born rule, is a theorem under certain conditions on the interatomic interactions (Friesecke and Theil 2002). The results reported above strongly suggest that a similar result holds in the present setting. From the presented results, it can also be concluded that the deformation of axially stretched and otherwise unconstrained nanotubes can be “exactly” parametrized in terms of (Λ1 , Λ2 , η), that is the configuration space of stretched nanotubes can be reduced to four parameters. We have also shown that not accounting for any of these parameters leads to incorrect over-constrained models. In other words, less than four parameters are insufficient (unless, by symmetry considerations, one can eliminate one of the components of the inner displacements). Numerical evidence shows that these statements hold even for finite stretches, as long as the structure remains stable. In atomistic calculations, by appropriate periodicity conditions and assuming some symmetries, it is also possible to reduce the atomistic configuration space by eliminating redundant degrees of freedom (Kudin et al. 2001). In any case, the present approach reaches an optimal reduction from a completely different prospective.
5.4.7
Dependence of Ys on stretch: simplified failure analysis
e 1 ) come from both the interatomic The nonlinearities in the reduced potential E(Λ potential, and the kinematics. The elastic moduli can vary with stretch in a very nonlinear manner. In this section, the variation of the elastic stiffness of straight nanotubes with deformation is analyzed. Since finite deformations are considered, attention must be paid on the precise definition of the elastic modulus. As argued in Section 5.4.5, the spacial third elasticity tensor is adequate, and therefore we analyze the modulus defined by Ys (F ) =
a , 2πReq
(5.39)
110 250
Ys (J/m2)
200
(10,10) (14,5) (17,0)
Buckling
150
Defect nucleation
100 50 0 0.9
1
1.1
1.2
1.3
Axial stretch Figure 5.3: Evolution of the surface Young’s modulus as a function of stretch: in the regime labelled “defect nucleation” dislocations not modelled by this theory can take place. where F = Λ1 denotes the axial stretch. The results for the bond-order potential by Brenner (1990) are depicted in Fig. 5.3 for an arm-chair, a zig-zag, and a chiral nanotube of similar diameters. Similarly to Belytschko et al. (2002), the cutoff function of the Brenner potential, which introduces an abrupt increase in the interatomic force, has not been included in this analysis. Due to defect nucleation for moderate tensile strains, the validity of the simplified analysis presented here is limited to stretches for which this cut-off function is not active. The general trend is a monotonic reduction of this spacial elastic modulus with tensile stretch. For compressive deformation, the arm-chair tube displays a maximum stiffness for slightly compressive deformation, which then decreases as the tube is further deformed. This behavior, which is also noticeable for chiral tubes, cannot be observed for the zig-zag tube. The results for compressive strains are applicable as long as buckling does not occur. For tensile strains, kinetic analysis predicts defect nucleation for strains of around 17% for this potential (Samsonidze et al. 2002). It can be observed that for nominal strains of around 32% for the (17,0) nanotube, this spacial modulus vanishes. Following Hill (1975), with the notation of Marsden and Hughes (1983), the condition for linear stability of the (1D) solid reads a = c + σ > 0, or equivalently Ys (F ) > 0. Thus, Fig. 5.3 can be used to obtain
111 an estimate of the ultimate tensile strength of stretched nanotubes from the stability of the one-dimensional rod, which agrees with atomistic calculations with the same potential (Yakobson et al. 1997). Of course, this analysis cannot capture the occurrence symmetry-breaking instabilities, inhomogeneous phenomena such as the Stone-Wales transformation, or kinetic effects (Samsonidze et al. 2002; Belytschko et al. 2002). Nevertheless, it provides an upper bound on the tensile strain, which is much tighter than the continuum estimate of 52% based on the same potential reported in Zhang et al. (2002). Again, we should emphasized that these results do not involve fitting of energy data or finite difference approximation of derivatives (Xiao and Liao 2002), but rather simple evaluation of closed-form expressions.
5.4.8
Dependence of elastic moduli on diameter and chirality
As mentioned previously, the continuum membrane, and thus the 1D rod hyperelastic potential, accounts for the chirality of the nanotube. On the other hand, the ideal radius of a nanotube R0 is a parameter in this rod-like model, and characterizes the “amount” of graphene used to form the nanotube, as well as its initial curvature.√ Chirality and ideal radius ofpreal nanotubes obey the relations Θ0 = arctan[ 3n2 /(2n1 + n2 )] and R0 = A0 3(n21 + n1 n2 + n22 )/2π (Saito et al. 1992). However, in the continuum setting, these two parameters can be chosen independently. An interesting consequence is that the continuum model allows us to analyze these two effects independently and continuously, in contrast with atomistic simulations, which are restricted to the numerable set of crystallographically admissible nanotubes. This also means that “virtual” nanotubes that do not exist in reality can be considered. Actually, when analyzing atomistic calculations, researchers have tried implicitly or explicitly to separate the effects of these two parameters (S´anchez-Portal et al. 1999; Hern´andez et al. 1999). Indeed, the functional dependence of elastic and energetic properties of nanotubes with respect to a continuously varying radius has been fitted to atomistic simulations (Cornwell and Willie 1997; Gao et al. 1998). The analysis provided below shows that the effect of these two parameters on the properties of nanotubes can be distinctly established. Systematic dependences can be isolated. The second parameter set of the bond-order potential by Brenner (1990) is used. The effect of diameter and chirality on the surface Young’s modulus Ys as defined in Eq. (5.37), is described in Fig. 5.4. Young’s modulus has been considered to be insensitive to diameter and chirality (Lu 1997). From recently reported ab initio calculations, S´anchez-Portal et al. (1999) concluded that the effects of diameter and
112 chirality are small, since variations fell within the accuracy of their method for evaluating Young’s modulus, 10%. Nevertheless, a dependence on the tube radius was noticed from atomistic simulations with the Brenner potential (Cornwell and Willie 1997), although the predicted affine variation of Young’s modulus with the radius is inconsistent with other calculations (Robertson et al. 1992; Popov et al. 2000; Hern´andez et al. 1999; S´anchez-Portal et al. 1999). Calculations based on empirical potentials (Robertson et al. 1992), tight-binding methods (Hern´andez et al. 1999), and lattice-dynamical models (Popov et al. 2000), predict that small nanotubes are relatively more compliant, and that Young’s modulus increases with the diameter, saturating in a plateau for large diameters. This phenomenon has been explained by the weakening of the bonds due to curvature (Hern´andez et al. 1999). The reduction of Ys relative to graphene for small tubes of about 0.4 nm in diameter has been predicted to be between 5% and 10% (Hern´andez et al. 1999; Popov et al. 2000). A systematic dependence on the chirality has rarely been reported. From the tight-binding calculations by Hern´andez et al. (1999), the deviation between armchair and zig-zag nanotubes cannot be interpreted systematically. Nevertheless, Popov et al. (2000), using a lattice dynamical model together with a long-wave perturbation technique, showed that the modulus of chiral tubes is bounded from above by armchair tubes, slightly stiffer, and from below by zig-zag tubes. These trends were also previously obtained from Brenner potential (Robertson et al. 1992). To summarize, Young’s modulus seems to exhibit a relatively small but systematic dependence on the tube diameter. To a lesser degree, a slight dependence on the chirality has been reported. This dependence is however difficult to establish because it is not possible by atomistic methods to keep the diameter fixed and vary the chirality. Furthermore, generally used approximate fits and finite difference approximations, limit the accuracy of the reported values for elastic moduli, making it impossible to distinguish subtle effects. The previously reported dependence of Young’s modulus on the diameter is also obtained from calculations based on the simple one-dimensional rod model. Figure 5.4(a) shows the variations of Young’s modulus with diameter for tubes with different chirality. As reported in the literature, smaller tubes are softer and a plateau is reached by large tubes, which corresponds to the modulus of plane graphene obtained in Section 5.3.5. Our results also agree with the previously noted fact that armchair nanotubes are the stiffest, and zig-zag the softest (Popov et al. 2000). Figure 5.4(b) shows that the effect of chirality on Ys is small. Nevertheless, thanks to the freedom for choosing independently the diameter and the chirality in the continuum model, the isolated effects can be revealed. The results for several fixed ideal diameters as a function of chirality are reported. This figure clearly pictures the systematic dependence of Young’s modulus with chirality. Consistent with symmetry considerations, the slope of the curves is zero for armchair and zig-
113 240 Planar graphene
236
235
225
Ys (J/m )
Θ=0° Θ=15° Θ=30°
220 215
2
Ys (J/m2)
(18,18)
234
230
232 (16,0)
230
(11,3)
(7,6)
228
210
226 (10,0)
205 200 0
Planar graphene
1
2
3
Tube diameter (nm)
(a)
4
5
224 0
10
20
30
Chiral angle (°)
(b)
Figure 5.4: Dependence of Young’s modulus on diameter (a) and chirality (b). zag tubes. It is clear that from the discrete values of crystallographically consistent combinations of diameter and chirality (denoted with solid circles), it is very difficult to separate the effects of these two parameters and establish precise trends. The effect of chirality is stronger for smaller tubes and, as the diameter is increased, the limit of plane graphene with no dependence on the chirality is reached. This last observation brings up the issue of the isotropy of nanotubes’ walls. The well known fact that graphene is isotropic in the infinitesimal regime was already pointed out and analytically derived in Section 5.3.1. We also pointed out that graphene is not isotropic in finite deformations. The presented chirality-dependent results for nanotubes of finite curvature are a manifestation of the anisotropy induced by the finite deformation. The infinitesimal isotropy is approached asymptotically for large nanotubes. Figure 5.5 shows analogous results for Poisson’s ratio. The systematic variations with chirality and ideal radius are very weak, and the effect on this property is roughly the inverse of the effect on Ys . Note however that the maximum value for νs occurs for chiral tubes. The planar graphene asymptote is apparent from the curves.
114
(10,0)
0.418 0.417
ν
0.43
ν
0.419
Θ=0° Θ=15° Θ=30°
0.435
0.425
0.416
(7,6)
(11,3)
0.415
0.42
0.414
0.415
(16,0)
0.413
0.41 0
1
2
3
4
5
0.412 0
Tube diameter (nm)
(a)
(18,18)
10
20
30
Chiral angle (°)
(b)
Figure 5.5: Dependence of Poisson’s ratio on diameter (a) and chirality (b).
5.5
Summary and conclusions
The main results and conclusions of the present chapter can be summarized as follows: • Explicit expressions of the infinitesimal elastic moduli of planar graphene in terms of the functional form of the atomistic potential are provided. In particular, the expression of the bending modulus derived from our theory cannot be obtained by the standard methods of crystal elasticity. This expression demonstrates that hexagonal 2D lattices whose potential function is a twobody/three-body expansion (no dihedral dependency) from the planar state have zero initial bending stiffness. The elastic properties derived from the bond-order potentials by Brenner (1990) and Brenner et al. (2002) are verified with atomistic calculations, and compared to available ab initio data. It is shown how these expressions can guide the parameter fitting given a functional form, here the bond-order formalism (Tersoff 1988). • A reduced finite deformation 1D model for uniaxially stretched and otherwise unconstrained nanotubes which accounts for the finite curvature, the finite inner and radial relaxations, and for the chirality of the tubes is formulated. Careful comparisons with atomistic calculations suggest that this model exactly captures the energetics of the atomistic system. This implies that the kinematic assumptions that are the basis of the theory hold exactly, similarly to what has been rigorously demonstrated in another context for a class of 2D
115 lattices by Friesecke and Theil (2002). Such a precise analysis of the exponential Cauchy-Born rule would be desirable. The continuum theory provides an optimal parametrization of the configuration space of uniaxially stretched and otherwise unconstrained nanotubes in terms of four parameters. • The dependence of Young’s modulus with respect to finite stretches is analyzed for the potential by Brenner (1990). Upper bounds on the stability limit of stretched nanotubes are obtained, tighter than previous results based on continuum analysis and the same potential (Zhang et al. 2002). In the continuum model, it is possible to vary independently and continuously the chirality and the ideal radius of the analyzed nanotubes. This allows us to isolate these effects, and the systematic dependence of elastic properties on these two parameters is elucidated. This systematic analysis is very difficult or impossible from atomistic calculations. The results are consistent with tight-binding and ab initio results, which indicates that, although the Brenner potential fails to accurately predict the elastic properties of graphene, it correctly reproduces qualitative behavior, like the weakening of in-plane stiffness with curvature. • The quantitative comparisons of the elastic properties predicted by popular analytical potentials with ab initio data highlight their limited transferability with regards to the mechanics of carbon nanotubes. Constructing the continuum model directly from a quantum mechanical model appears thus as a natural development. According to the results in this chapter, an ab initio continuum model for straight nanotubes is straightforward: it suffices to appropriately parametrize the ab initio energies and forces in terms of the reduced configuration space defined by Λ1 , Λ2 , and η. This chapter conveys the concept that carbon nanotubes should be analyzed by elastic models which acknowledge the two-dimensional nature of graphene, e.g. with in-plane elastic moduli measured in units of surface tension and bending moduli with units of force times length. For the analytical treatment of the mechanics of carbon nanotubes, a linearized version of our theory seems more appropriate than commonly used thin shell elastic theory (Ru 2000; Ru 2001).
Chapter 6 Conclusions The main conclusions of the presented work have been drawn at the end of each chapter. The most salient results are summarized below: • Finite crystal elasticity has been generalized to the case of curved lattices of low dimensionality by means of the exponential Cauchy-Born rule, which corrects the inconsistencies of the standard Cauchy-Born rule for curved continua. • The exponential Cauchy-Born rule has been applied to construct a finite deformation membrane model for the mechanics of carbon nanotubes, written explicitly in terms of the atomistic description. There is no recourse to constrained atomistic evaluation of the energy and forces. Instead, the constraints are treated analytically, at the constitutive level, by relating the deformation of a template with continuum strain measures. For this reason the resulting model is strictly speaking a continuum mechanics constitutive theory. • This continuum mechanics theory has been combined with finite elements. Numerous comparisons with full atomistic calculations demonstrate that the continuum model mimics the parent atomistic model with remarkable accuracy, in the full nonlinear regime, and for very large deformations. These simulations suggest that in the absence of lattice defects, continuum simulations suffice to accurately describe the nonlinear mechanics of nanotubes. These simulations benefit from the well developed technology of engineering computations, and for instance adaptive mesh refinement can reduce the computational costs by orders of magnitude relative to the full atomistic calculations. This fact makes the continuum simulations competitive even for nanotubes of small diameter. • Simulations of multi-walled carbon nanotubes containing six million atoms have been performed with 100,000 finite element nodes. These simulations compare very well with experimental observations of rippling of nanotubes, 116
117 and provide valuable information not available from experiments. In particular, the three-dimensional picture of unusual deformation morphologies is accessible, while transmission electron micrographs only provide slices of the deformation. Furthermore, the simulations provide detailed information about the energetics of the deformation processes. For instance, it is shown that the rippled deformation mode is energetically very favorable, and its average strain energy is considerably lower than that of a fully relaxed (10,10) nanotube. • Since the continuum stain energy potential is written is closed-form, special cases of deformation can be treated (semi-)analytically. A 2D model for the transverse mechanics of nanotubes is described, and a one-dimensional model for uniaxially stretched nanotubes which encapsulates a rich mechanical behavior is developed. From the analysis of this one-dimensional model it is concluded that the configuration space of uniaxially stretched and otherwise unconstrained nanotubes can be reduced to exactly four parameters. • The infinitesimal elastic moduli of planar graphene are explicitly written in terms of the functional form of the analytical potential. While the in-plane moduli can be obtained by standard methods, the bending modulus requires our extended theory of crystal elasticity. As for future developments, a few issues have been identified. Although exhaustive numerical validation has been presented, a rigorous analysis of the ranges of applicability of the exponential Cauchy-Born rule is desirable, in the lines of the work by Friesecke and Theil (2002) for the standard Cauchy-Born rule. On the other hand, constructing the continuum theory directly from a quantum mechanical model would highly enhance the practical interest of the method, given the limited accuracy of analytical potentials. Finally, the application of the exponential Cauchy-Born rule to filiform solids in three dimensions is an exciting area that deserves attention. A first step in this direction is presented in Arroyo and Belytschko (2003a).
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Appendix A Aspects of the constitutive model for carbon nanotubes A.1
Principal curvatures and directions
The eigenvalues of the generalized eigenvalue problem of Eq. (3.15), the principal curvatures of the surface Ω, can be easily obtained (do Carmo 1976) from the Gaussian curvature K11 K22 − K212 det[K]B0 K= = , (A.1) 2 det[C]B0 C11 C22 − C12 and the mean curvature H=
1 K11 C22 − 2K12 C12 + K22 C11 1 trace [C]−1 . B0 [K]B0 = 2 2 2 C11 C22 − C12
(A.2)
The principal curvatures can then be written as k1,2 = H ±
√
H 2 − K.
(A.3)
Two C−orthonormal principal directions V1 and V2 are straight-forward to obtain. The derivatives of kn and Vn with respect to C and K can be obtained from standard formulas (Fox and Kapoor 1968), valid for the case in which k1 6= k2 . For
125
126 the principal curvatures, we have ∂kn = Vn ⊗ Vn , ∂K and
or
∂kn = (Vn )A (Vn )B , ∂KAB
∂kn ∂kn = −kn , ∂C ∂K
Introducing the symbol
N
symm
A
O
n = 1, 2,
n = 1, 2.
(A.4)
(A.5)
denoting the symmetrized tensor product, i.e.
B=
symm
1 (A ⊗ B + B ⊗ A) , 2
(A.6)
the derivatives of the principal directions can be obtained as ∂Vn 1 = Vm ⊗ ∂K (kn − km ) or
and
! Vn
O
Vm ,
(A.7)
symm
1 ∂(Vn )A (Vm )A (Vn )B (Vm )C + (Vm )B (Vn )C , = ∂KBC 2(kn − km )
(A.8)
∂Vn 1 ∂Vn = − Vn ⊗ Vn ⊗ Vn − kn , ∂C 2 ∂K
(A.9)
for n = 1, 2 and where {n, m} is a permutation of {1, 2}. For repeated eigenvalues, the procedure for differentiating the eigenvalues and eigenvectors becomes cumbersome (Friswell 1996). In computations, this case rarely occurs, and our experience indicates that numerical differentiation is a simple and robust alternative to compute the stresses at a particular quadrature point for which √ H 2 − K 0 c = W (η k ) and η b = ηk • W Although no result on the convexity of W as a function of η is known to us, numerical experience indicates that for graphene and Brenner’s potential, the inner relaxation always converges to a minimum within machine precision in two or three Newton iterations.
A.2.2
Inner forces and inner elastic constants
The residual r = W,η and the Jacobian J = W,ηη can be interpreted as inner out-of-balance forces and an inner elastic constants (Cousins 2001). This Section provides details for their calculation. To keep the expressions compact, let us define an array of bond lengths and angles for the three inequivalent bonds of graphene p = [a1 , a2 , a3 , θ1 , θ2 , θ3 ], and let pi (i = 1, 6) denote the i−th entry of this array. As for the inner out-of-balance forces, applying the chain rule and recalling Eq. (3.36), we have 6 X ∂W W ,η = pi,η . (A.11) ∂pi i=1
128 On the other hand, the inner elastic constants can be written in the following manner, which highlights the symmetry:
W,ηη =
6 X ∂W
∂2W pi,η ⊗ pi,η ∂pi ∂(pi )2 i=1 X O ∂2W + 2 pi,η pj,η . ∂pi ∂pj symm i
