A finite deformation membrane based on inter-atomic potentials for single atomic layer films Application to the mechanics of carbon nanotubes

Transcription

1 A finite deformation membrane based on inter-atomic potentials for single atomic layer films Application to the mechanics of carbon nanotubes Marino Arroyo and Ted Belytschko Department of Mechanical Engineering Northwestern University August 1, 2001 USNCCM-VI, Dearborn

2 Scope and background Goal: Formulate a continuum membrane model for crystalline films one atom thick, in the finite deformation regime.

3 Scope and background Goal: Formulate a continuum membrane model for crystalline films one atom thick, in the finite deformation regime. Established theories for space filling crystals: Crystal elasticity: Elastic constants obtained from inter-atomic potentials. Theory also developed for finite deformations.

4 Scope and background Goal: Formulate a continuum membrane model for crystalline films one atom thick, in the finite deformation regime. Established theories for space filling crystals: Crystal elasticity: Elastic constants obtained from inter-atomic potentials. Theory also developed for finite deformations. To deal with inhomogeneities or defects, the quasicontinuum method has been proposed (Tadmor, Ortiz & Phillips 1996).

5 Scope and background Goal: Formulate a continuum membrane model for crystalline films one atom thick, in the finite deformation regime. Established theories for space filling crystals: Crystal elasticity: Elastic constants obtained from inter-atomic potentials. Theory also developed for finite deformations. To deal with inhomogeneities or defects, the quasicontinuum method has been proposed (Tadmor, Ortiz & Phillips 1996). A key concept is the Born rule (or method of the homogeneous deformations), a kinematic assumption that links the atomic and continuum deformations.

6 Scope and background Appeal of this approach: The resulting hyper-elastic potential is based on the nano-scale physics, rather than in the phenomenology of the material. The model inherits naturally properties of the atomic system such as the crystal symmetries.

7 Scope and background Appeal of this approach: The resulting hyper-elastic potential is based on the nano-scale physics, rather than in the phenomenology of the material. The model inherits naturally properties of the atomic system such as the crystal symmetries. Here, the nano-scale physics are introduced through the inter-atomic potentials, based on experimental data and/or quantum mechanics.

8 lf wth. simowth. le-wall le-wall ture of ture of Scope outer to the inner and tube. background having parablethe to, same the time diameter neededas tothe close inner thedouble-walled single-walledtube. The having closure same time for diameter the double-walled the innertubes is longer, simply Thebecause closureit time takes forsome the double-walled time for atomstubes transfer is longer, fromsim- ply because it takes some time for atoms to transfer from the the outer While to the these inner simulations tube. show that the lip lip interaction alone While cannot these stabilize simulations open-ended show that nanotubes, the lip lip it isinteraction still possible alonethat cannot these stabilize interactions open-ended exert ananotubes, stabilizingitinfluence is still possible act that in conjunction these interactions with other exert effects, a stabilizing such as with influence a large if An application: The mechanics of carbon nanotubes. they if they act in conjunction with other effects, such as with a large Shell-like behavior tubes. d r (inner) tubes. r a (inner) small a small Fig. 4. a High-resolution transmission electron microscope image of a bent tube. Fig. 4. b a A High-resolution computer-generated transmission image electron of a bent microscope tube. The image network of a bent of hexagons tube. b A is computer-generated not disturbed; consequently, image of the a tube bent can tube. unbend The without network any of damage hexagons is not disturbed; consequently, the tube can unbend without any damage Bernholc et al., 1998

9 lf wth. simowth. le-wall le-wall ture of ture of Scope outer to the inner and tube. background having parablethe to, same the time diameter neededas tothe close inner thedouble-walled single-walledtube. The having closure same time for diameter the double-walled the innertubes is longer, simply Thebecause closureit time takes forsome the double-walled time for atomstubes transfer is longer, fromsim- ply because it takes some time for atoms to transfer from the the outer While to the these inner simulations tube. show that the lip lip interaction alone While cannot these stabilize simulations open-ended show that nanotubes, the lip lip it isinteraction still possible alonethat cannot these stabilize interactions open-ended exert ananotubes, stabilizingitinfluence is still possible act that in conjunction these interactions with other exert effects, a stabilizing such as with influence a large if An application: The mechanics of carbon nanotubes. they if they act in conjunction with other effects, such as with a large Shell-like behavior Very large elastic deformations without topological defects of the bond network (Maiti, 2000) tubes. d r (inner) tubes. r a (inner) small a small Fig. 4. a High-resolution transmission electron microscope image of a bent tube. Fig. 4. b a A High-resolution computer-generated transmission image electron of a bent microscope tube. The image network of a bent of hexagons tube. b A is computer-generated not disturbed; consequently, image of the a tube bent can tube. unbend The without network any of damage hexagons is not disturbed; consequently, the tube can unbend without any damage Bernholc et al., 1998 Nanoscience Research Group,UNC-Chapel-Hill

10 Scope and background The continuum models available in the literature are phenomenological, restricted to small strains or to simple situations.

11 Scope and background The continuum models available in the literature are phenomenological, restricted to small strains or to simple situations. Our aim is to follow the path of crystal elasticity to obtain a finite deformation continuum model based on the nano-scale physics for crystalline films one atom thick.

12 Scope and background The continuum models available in the literature are phenomenological, restricted to small strains or to simple situations. Our aim is to follow the path of crystal elasticity to obtain a finite deformation continuum model based on the nano-scale physics for crystalline films one atom thick. Unfortunately, the Born rule does not extend directly to this case. Main challenge: extend the Born rule to this more general situation.

13 Structure of the talk 1. The Born rule for bulk materials 2. Proposed extension to the Rule 3. Local hyper-elastic potential 4. Application to the mechanics of CNT s 4.1 Transverse mechanics 4.2 General 3D deformation 5. Conclusions

14 1. The Born rule for bulk materials: kinematics T XΦ = X Φ (X) Ω0 Φ Ω For bulk materials, let Φ be the configuration that maps the undeformed body Ω 0, an open set of R n, into R n, n being either 1, 2 or 3. The deformed body is denoted as Ω = Φ(Ω 0 ).

15 1. The Born rule for bulk materials: kinematics T XΦ = F X dx dx=fdx Φ (X) Ω0 Φ Ω Classical continuum mechanics: the deformation gradient F maps infinitesimal vectors from the undeformed body into the deformed one: dx = F dx,

16 1. The Born rule for bulk materials: kinematics T XΦ = F X dx dx=fdx Φ (X) Ω0 Φ Ω The Born rule views the lattice vectors of the crystal as infinitesimal vectors, and therefore a = F A, where A and a denote an undeformed and the corresponding deformed lattice vectors.

17 Structure of the talk 1. The Born rule for bulk materials 2. Proposed extension to the Rule 3. Local hyper-elastic potential 4. Application to the mechanics of CNT s 4.1 Transverse mechanics 4.2 General 3D deformation 5. Conclusions

18 2. Proposed extension to the Rule The continuum model for the crystalline film is a membrane without thickness, geometrically, a surface in 3D. The atomic sites are assumed to lie on the surface and the bonds are viewed as chords of the surface.

19 2. Proposed extension to the Rule The continuum model for the crystalline film is a membrane without thickness, geometrically, a surface in 3D. The atomic sites are assumed to lie on the surface and the bonds are viewed as chords of the surface. Objective: Express the deformed geometry of the bonds in terms of the deformation of the membrane. Requirement: Extend the Born rule to surfaces.

20 2. Proposed extension to the Rule X W TΦ = F T X Ω0 Φ(X) w T Φ(X) Ω Ω0 ϕ 0 Φ = ϕ O ϕ 1 0 ξ ϕ Ω Ω Continuum mechanics in manifolds: The deformation gradient F is called the tangent map. It maps T Ω 0 into T Ω. In this setting, the body and its tangent cannot be identified.

21 2. Proposed extension to the Rule: the exponential map The exponential map sends a vector W of the tangent to a point in the surface a distance W along the geodesic from p in the direction W. TpΩΩ p w exp w p Ω The exponential map maps naturally the tangent of a surface into the surface itself. It is a diffeomorphism in a neighborhood of each regular point.

22 2. Proposed extension to the Rule F? W 0 X A Y x a z W Born Rule: a = F A

23 2. Proposed extension to the Rule T X W 0 W 0 X A Y T x W x W Born Rule: a = F A Proposed Rule:

24 2. Proposed extension to the Rule T X W 0 W 0 X A W Y T x W x exp -1 W Born Rule: a = F A Proposed Rule: exp 1 X A

25 2. Proposed extension to the Rule F W 0 T X W 0 X A W Y T x W x w exp -1 W Born Rule: a = F A Proposed Rule: F exp 1 X A

26 2. Proposed extension to the Rule F exp W 0 T X W 0 X A W Y T x W x w z exp -1 W Born Rule: a = F A Proposed Rule: exp x F exp 1 X A

27 2. Proposed extension to the Rule F exp W 0 T X W 0 X A W Y T x W x w a z exp -1 W Born Rule: a = F A Proposed Rule: a = exp x F exp 1 X A

28 2. Proposed extension to the Rule F exp W 0 T X W 0 X A W Y T x W x w a z exp -1 W Born Rule: a = F A Proposed Rule: a = exp x F exp 1 X A a = F Φ A

29 Structure of the talk 1. The Born rule for bulk materials 2. Proposed extension to the Rule 3. Local hyper-elastic potential 4. Application to the mechanics of CNT s 4.1 Transverse mechanics 4.2 General 3D deformation 5. Conclusions

30 3. Local hyper-elastic potential We have proposed the kinematic relation: a = F ΦA Evaluating the exponential map requires the integration of the geodesic differential equations.

31 3. Local hyper-elastic potential We have proposed the kinematic relation: a = F ΦA Evaluating the exponential map requires the integration of the geodesic differential equations. To render the model local and computationally feasible, the exponential map needs to be approximated.

32 3. Local hyper-elastic potential We have proposed the kinematic relation: a = F ΦA Evaluating the exponential map requires the integration of the geodesic differential equations. To render the model local and computationally feasible, the exponential map needs to be approximated. By approximating locally the exponential map, we can express the geometry of the deformed bonds in terms of the continuum deformation: a k = f(c, K); θ kl = g(c, K).

33 3. Local hyper-elastic potential Given the inter-atomic potentials, for instance in the form of a two-body/three body expansion: E = V s (a i ) + V θ (θ kl, a k, a l ), recalling that: a i = f(c, K); θ kl = g(c, K). we can formulate the hyper-elastic potential, considering a representative cell: W (C, K) = 1 S 0 E Cell (a i, θ kl ).

34 3. Local hyper-elastic potential The honeycomb structure of graphene sheets is a Bravais multi-lattice = additional internal elastic variables appear, the inner displacements η (Cousins, 1978). They can be eliminated locally: W = W (C, K; η) Ŵ (C, K) = min η W (C, K; η).

35 Structure of the talk 1. The Born rule for bulk materials 2. Proposed extension to the Rule 3. Local hyper-elastic potential 4. Application to the mechanics of CNT s 4.1 Transverse mechanics 4.2 General 3D deformation 5. Conclusions

36 4. Application to the mechanics of CNT s The undeformed body Ω 0 is a plane graphene sheet.

37 4. Application to the mechanics of CNT s The undeformed body Ω 0 is a plane graphene sheet. A continuum version of the non-bonded interactions is also formulated.

38 4. Application to the mechanics of CNT s The undeformed body Ω 0 is a plane graphene sheet. A continuum version of the non-bonded interactions is also formulated. The continuum membrane is discretized using finite elements. The strain energy density depends on the curvature of the membrane the finite element space needs to be H 2.

39 4.1 Transverse mechanics of CNT s In many situations of interest, only the transverse deformation of NT s is relevant. However, since molecular simulations are intrinsically three dimensional, the symmetry of the problem cannot be fully exploited.

40 4.1 Transverse mechanics of CNT s In many situations of interest, only the transverse deformation of NT s is relevant. However, since molecular simulations are intrinsically three dimensional, the symmetry of the problem cannot be fully exploited. By considering the appropriate kinematic restriction in the continuum membrane, a model of reduced dimensionality, analogous to plane strain/plane stress, can be formulated.

41 4.1 Transverse mechanics of CNT s In many situations of interest, only the transverse deformation of NT s is relevant. However, since molecular simulations are intrinsically three dimensional, the symmetry of the problem cannot be fully exploited. By considering the appropriate kinematic restriction in the continuum membrane, a model of reduced dimensionality, analogous to plane strain/plane stress, can be formulated. In this case, the continuum model is discretized using C 1 Hermite finite elements, and the BFGS Quasi-Newton technique is used to obtain equilibrium configurations.

42 4.1 Transverse mechanics of CNT s Validation of the model: Comparison with molecular mechanics model of a [32,0] CNT in a plane strain situation. X Z Y

43 4.1 Transverse mechanics of CNT s Validation of the model: Comparison with molecular mechanics model of a [32,0] CNT in a plane strain situation. X Y Z Molecular Mechanics Continuum + FE Error in energy under 1.5%

44 4.1 Transverse mechanics of CNT s Model with Standard Born Rule: Molecular Mechanics Standard Born Rule (fine) Standard Born Rule (coarse) The strain energy density is non-convex no bending stiffness

45 4.1 Transverse mechanics of CNT s Transversal stability: [20,0],[20,0] collapsed tube; is E unstable = 0 [26,0], [26,0] collapsed tube; is E local < 0 minimum [32,0] tube; E > 0 [32,0], collapsed is global minimum [42,0] and [32,0] nested tubes; E > 0 [32,0] [42,0], collapsed is global minimum

46 4.1 Transverse mechanics of CNT s NT s in contact: Two [20,0] nanotubes a AFM image of two flattened MWCNT ribbons as indicated by b Scanned close-up image of the MWCNT ribbon marked with the the right-side of a. c Height plot across the line marked in b. e-dimensional rendering of the same flattened ribbon. e width as defined above, namely, the distance bemaxima on the pillow Two contour [24,0] nanotubes from the height plot m. The height obtained from the highest point along d see the arrow is 4.8 nm, slightly larger than twice ght of the flattened MWCNT, and much less than the width of 30 nm, which means that the radius of curof the fold should be less than 2.4 nm, half of the of the fold. This demonstrates the great flexibility of nd of MWCNT ribbon. Another good example is in Figs. 2 b and 2 c, which are AFM images of a T ribbon crossingtwo over[32,0] two individual nanotubes MWCNTs, ving a diameter of 9 nm, and the corresponding threeional representation of the same ribbon, respectively. ight of this ribbon is 1.6 nm, and the width is 30 nm. Yu, Dyer & Ruoff, 2001 FIG. 2. a Twisted and then folded MWCNT ribbon. Note the apparent fold indicated by an arrow along the length of the MWCNT. b MWCNT ribbon crossing over underlying MWCNTs. c Three-dimensional image of b showing how the ribbon conforms remarkably closely to any change in the underlying surface, such as when crossing other MWCNTs, showing its mechanical flexibility orthogonal to the flat of the ribbon. its uncollapsed state, where d is the thickness assigned to each layer of the MWCNT, set equal to the interlayer distance of graphite, 0.34 nm. In the estimation above, each bulb on left and right in the cross section is defined as a half circle having a radius of (H d)/2. The small error introduced by this simple approximation in defining the exact IBM Research

47 4.1 Transverse mechanics of CNT s: Nanorope Thess et al., 1996

48 4.2 General 3D deformation of CNT s Discretization using subdivision surfaces based on Loop s scheme (Cirak, Ortiz & Schröder, 2000). In this technique, a control mesh of triangular elements is used to construct a smooth (H 2 ) surface. The CG method is used to obtain equilibrium configurations.

49 Y 4.2 General 3D deformation of CNT s rame Jul 2001 X Z Bending of a [5,5] nanotube (MM by Shaoping Xiao). θ = 10 0 ame Jul 2001 Y θ = 20 0 X Z e Jul 2001 Y θ = 30 0 X Z

50 4.2 General 3D deformation of CNT s Bending of a [5,5] nanotube.

51 4.2 General 3D deformation of CNT s Bending of a [5,5] nanotube.

52 4.2 General 3D deformation of CNT s Bending of a [10,10] nanotube.

53 4.2 General 3D deformation of CNT s Bending of a [10,10] nanotube.

54 5. Conclusions A finite deformation membrane model for crystalline films one atom thick has been presented.

55 5. Conclusions A finite deformation membrane model for crystalline films one atom thick has been presented. This model has no phenomenological input and is based on the underlying atomical model.

56 5. Conclusions A finite deformation membrane model for crystalline films one atom thick has been presented. This model has no phenomenological input and is based on the underlying atomical model. The key contribution is the extension of the Born Rule to deal with manifolds by means of the exponential map.

57 5. Conclusions A finite deformation membrane model for crystalline films one atom thick has been presented. This model has no phenomenological input and is based on the underlying atomical model. The key contribution is the extension of the Born Rule to deal with manifolds by means of the exponential map. The approximation of the exponential map leads to an elastic potential that depends on C and K.

58 5. Conclusions A finite deformation membrane model for crystalline films one atom thick has been presented. This model has no phenomenological input and is based on the underlying atomical model. The key contribution is the extension of the Born Rule to deal with manifolds by means of the exponential map. The approximation of the exponential map leads to an elastic potential that depends on C and K. The implementation of the model using finite elements shows excellent agreement with molecular mechanics at a fraction of the cost.

59 A finite deformation membrane based on inter-atomic potentials for single atomic layer films Application to the mechanics of carbon nanotubes Marino Arroyo and Ted Belytschko Department of Mechanical Engineering Northwestern University August 1, 2001 USNCCM-VI, Dearborn

60 Scope and background Continuum mechanics seems to work even in the nano-scale mechanics of nanotubes: elastic constants of nanotubes (Lu, 1997), simplified cross-section continuum models to interpret experimental data (Chopra et al., 1995, Yu, Dyer & Ruoff, 2001), linearized bifurcation analysis of classical elastic shells to study buckling patterns of nanotubes (Yakobson, Brabec & Bernholc, 1996).

61 1. The Born rule for bulk materials: kinematics T XΦ = F X dx dx=fdx Φ (X) Ω0 Φ Ω This allows to obtain the geometry of the deformed lattice vectors: a k = where C = F T F. A k C A k and cos θ kl = A k C A l a k a l,

62 Inner displacements A 1 = B 2 + P B 1 P A1 A2 A3 A 2 = B 1 + P A 3 = P B 2

63 Inner displacements p = P + η A 1 = B 2 + p A1 A2 A3 A 2 = B 1 + p A 3 = p B 1 B 2 P h W (C, η)