Pressure-Induced Structural Transitions in Multiwall Carbon Nanotubes Hiroyuki Shima Graduate School of Engineering, Hokkaido University, Japan (shima@eng.hokudai.ac.jp) See H.Shima and M.Sato, Nanotechnology in press 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 1/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
0. Main Finding = Pressure-induced Radial Corrugation of MWNT - The cross-sectional shape Hydrostatic pressure > 1GPa changes from circular to radially corrugated one. Elastic deformation 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 2/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
1. Radial collapse of SWNTs Results of MD simulations Sun et al ., PRB 70 (2004) 165417 Carbon nanotubes are - extraordinarily stiff in the axial direction, but - highly flexible in the radial direction. Radial collapse occurs at 1.0GPa S. Zhang et al ., PRB 73 (2006) 075423. 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 3/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
1. Radial collapse of SWNTs Raman spectroscopy: Vanishing a radial breezing mode Venkateswaran et al., PRB 59 (1999) 10928 X-ray diffraction: Polygonization of SWNT-bundle X Tang et al., PRL 85 (2000) 1887 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 4/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
2. Motivation What takes place in Multi-walled nanotubes? Structural features of multi-walled carbon nanotubes: Multiple concentric walls interact with each other through the intermolecular forces. External pressure leads to a mechanical instability in outside walls due to their large tube diameters. Inner walls are relatively stiff in the radial direction so that they can push back the surrounding outer walls. Atomic-scale simulations for MWNTs = Very challenging! Alternative approach: Continuum elastic approximation 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 5/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
3. Model and Method Continuum elastic-shell model for MWNT Ru, PRB 62 (2000) 16962 Sudak JAP 94 (2003) 7281 Leung PRB 71 (2005) 165415 He, J. Mech. Phys. Solids 53 (2005) 303 Wang JAP 99 (2006) 114317 Displacements of a surface element: ( ) θ u , p radial ( ) θ v , p circumferential Obtaining the mechanical energy of the deformed MWNT, and then its stable cross-sectional shape under high hydrostatic pressure 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 6/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
3. Model and Method The mechanical energy of MWNT: [ ] ( ) ( ) = + + Ω U p , u p , v p U U i i D I U Apply a variational method to in terms of u , i v in order to determine the stable cross- i p section under high pressure ⎧ ⎫ ( ) ( ) 2 ⎡ ⎤ − − ⎪ ⎪ 2 2 3 N π Eh u ' v Eh u ' ' v ' ∑∫ 2 Deformation energy: = + + + θ ⎨ ⎢ ⎥ ⎬ i i i i U u v ' d − ν − ν D i i 2 2 3 ⎪ 2 ( 1 ) r ⎣ 2 r ⎦ 24 ( 1 ) ⎪ 0 r = ⎩ ⎭ i 1 i i i − N 1 c r N c r π π ∑ ( ) ∑ ( ) ∫ 2 ∫ 2 + − = − θ + − θ Inter-wall vdW energy 2 2 i , i 1 i i , i 1 i U u u d u u d + − I i i 1 i i 1 2 2 0 0 = = i 1 i 2 ⎛ ⎞ + − + 2 2 π u v u ' v u v ' ∫ 2 ⎜ ⎟ Ω = + θ N N N N N N Pressure-induced energy p r u d ⎜ ⎟ N N ⎝ ⎠ 2 0 See H.Shima and M.Sato, Nanotechnology in press 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 7/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
4. Results [1] Critical pressure curves 3.0 D D = 5.0 nm For given and , N = n 2 2.5 radial deformation occurs Critical pressure: p C [GPa] p just above C n=2 n=5 2.0 n=6 1.5 When exceeds 30, N radial corrugation is 1.0 observed = 0.5 n 0 n The wave number of 0.0 corrugation mode depends 0 10 20 30 40 50 on and N D Number of concentric walls: N 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 8/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
4. Results [1] Critical pressure curves = : Elliptic mode 2 n 3.0 D = 4.0 nm n=2 n=4 5.0 nm 2.5 Critical pressure: p C [GPa] 8.0 nm n=2 n=5 2.0 n=6 1.5 = n=5 n 5 : Corrugation mode 1.0 n=2 n=6 0.5 0.0 0 10 20 30 40 50 Number of concentric walls: N > > (= peculiar to MWNT with ) N 1 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 8/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
4. Results [2] Phase diagram 50 Increasing with fixed N = nm , then… D 5 . 4 n=6 Number of concentric walls: N 40 = = N 36 N 37 n=5 30 n=4 n=2 20 = N 38 10 4.0 4.5 5.0 5.5 6.0 Innermost tube diameter: D [nm] 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 9/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
4. Results [3] Geometric persistence of the innermost tube In all corrugation modes, the circular shape of the innermost tube persists even under high pressure. = n 2 : Elliptic mode 0 10 1.0 Normalized deformation amplitudes Oval -1 10 n=2 0.8 -2 10 0.6 -3 10 -4 = 10 n = 6 0.4 n 5 : Corrugation mode -5 10 Circular D=4.0 [nm] 0.2 -6 6.0 10 8.0 -7 0.0 10 1 5 10 50 0 10 20 30 40 50 Index of walls: i Index of walls: i 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 10/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
( Contact to : shima@eng.hokudai.ac.jp) 5. Summary The main findings of this study are: n (1) Pressure-induced radial corrugation (2) Mode index depend on: D in the cross-section of MWNT i) the tube diameter and N ii) the number of concentric wall Circular shape p > p > p p (3) Persistent cylindrical geometry C C of the innermost tube of MWNT Elastic deformation Circular Oval Elliptic deformation Radial Corrugation = = = n 6 n 2 n 5 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 11/11 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
( Contact to : shima@eng.hokudai.ac.jp) 5. Summary For multi-walled carbon nanotubes, we have demonstrated … (1) Critical pressure curves (2) Phase diagram of radial deformation (3) Geometric persistence of the innermost tube 50 3.0 0 1.0 10 n=6 D = 4.0 nm n=2 n=4 5.0 nm Normalized deformation amplitudes Number of concentric walls: N 2.5 Critical pressure: p C [GPa] -1 8.0 nm 40 10 n=2 0.8 n=2 n=5 -2 2.0 10 n=5 n=6 0.6 -3 10 30 1.5 -4 n=5 n = 6 10 0.4 1.0 n=2 n=4 -5 n=2 10 20 n=6 0.2 D=4.0 [nm] 0.5 -6 6.0 10 8.0 0.0 -7 0.0 10 10 0 10 20 30 40 50 1 5 10 50 0 10 20 30 40 50 4.0 4.5 5.0 5.5 6.0 Number of concentric walls: N Index of walls: i Index of walls: i Innermost tube diameter: D [nm] See H.Shima and M.Sato, Nanotechnology in press 第 13 回計算工学講演会 1-5 September 2008, Trend in NanoTechnology 2008 （ 5/21 （水）仙台市民会館） (TNT2008) in Oviedo, Spain
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