[PPT] - Pressure-Induced Structural Transitions in Multiwall Carbon PowerPoint Presentation

SLIDE 1

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

Graduate School of Engineering, Hokkaido University, Japan (shima@eng.hokudai.ac.jp)

Hiroyuki Shima

Pressure-Induced Structural Transitions in Multiwall Carbon Nanotubes

1/11

See H.Shima and M.Sato, Nanotechnology in press

SLIDE 2

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

Hydrostatic pressure > 1GPa

0. Main Finding

= Pressure-induced Radial Corrugation of MWNT

The cross-sectional shape

changes from circular to radially corrugated one. Elastic deformation 2/11

SLIDE 3

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

Radial collapse occurs at 1.0GPa

1. Radial collapse of SWNTs

3/11 Carbon nanotubes are

extraordinarily stiff in the axial direction,
highly flexible in the radial direction.

Results of MD simulations

but

Sun et al., PRB 70 (2004) 165417

S. Zhang et al., PRB 73 (2006) 075423.

SLIDE 4

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

Venkateswaran et al., PRB 59(1999) 10928

Raman spectroscopy:

Tang et al., PRL 85 (2000) 1887

Vanishing a radial breezing mode X-ray diffraction:

1. Radial collapse of SWNTs

Polygonization of SWNT-bundle

X

4/11

SLIDE 5

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

2. Motivation

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. 5/11 What takes place in Multi-walled nanotubes? Atomic-scale simulations for MWNTs = Very challenging! Alternative approach: Continuum elastic approximation Inner walls are relatively stiff in the radial direction so that they can push back the surrounding outer walls.

SLIDE 6

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

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

3. Model and Method

( )

p u , θ

Displacements of a surface element: radial circumferential

( )

p v , θ

6/11

Obtaining the mechanical energy of the deformed MWNT, and then its stable cross-sectional shape under high hydrostatic pressure

SLIDE 7

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain θ

π

d v u v u v u u r p

N N N N N N N N

∫

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − + + = Ω

2 2 2

2 ' '

The mechanical energy of MWNT:

3. Model and Method

Deformation energy: ( ) ( )

∑∫

=

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ − − + ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − + + − =

N i i i i i i i i i i D

d r v u Eh r v u v u r Eh U

1 2 3 2 2 3 2 2 2

' ' ' ) 1 ( 24 2 ' ' ) 1 ( 2 θ ν ν

π

( ) ( ) [ ]

Ω + + =

I D i i

U U p v p u p U , ,

( ) ( )

∑ ∫ ∑ ∫

= − − − = + +

− + − =

N i i i i i i N i i i i i i I

d u u r c d u u r c U

2 2 2 1 1 , 1 1 2 2 1 1 ,

2 2

π π

θ θ

Inter-wall vdW energy Pressure-induced energy

7/11

Apply a variational method to in terms of in order to determine the stable cross- section under high pressure

U

i i v

u , p

See H.Shima and M.Sato, Nanotechnology in press

SLIDE 8

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

10 20 30 40 50 0.0 0.5 1.0 1.5 2.0 2.5 3.0 n=6 n=5 n=2

D = 5.0 nm

Critical pressure: pC [GPa] Number of concentric walls: N

4. Results

[1] Critical pressure curves 8/11

= n 2 = n

For given and , radial deformation occurs just above

N D

C

p

When exceeds 30, radial corrugation is

bserved

N

The wave number of corrugation mode depends

n and

n N D

SLIDE 9

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

10 20 30 40 50 0.0 0.5 1.0 1.5 2.0 2.5 3.0 n=6 n=6 n=5 n=2 n=5 n=2 n=2 n=4

D = 4.0 nm 5.0 nm 8.0 nm

Critical pressure: pC [GPa] Number of concentric walls: N

4. Results

[1] Critical pressure curves

2 8/11

= n 5 = n

(= peculiar to MWNT with )

1 > > N

: Elliptic mode : Corrugation mode

SLIDE 10

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

4. Results

4.0 4.5 5.0 5.5 6.0 10 20 30 40 50

n=6 n=5 n=4 n=2

Number of concentric walls: N Innermost tube diameter: D [nm]

[2] Phase diagram 9/11 nm 4 . 5 = D 36 = N 37 = N 38 = N Increasing with fixed , then… N

SLIDE 11

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

4. Results

[3] Geometric persistence of the innermost tube 10/11

1 5 10 50 10

7

10

6

10

5

10

4

10

3

10

2

10

1

10 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 Index of walls: i

n = 6 n=2

Normalized deformation amplitudes Index of walls: i D=4.0 [nm] 6.0 8.0

2 = n

: Elliptic mode

5 = n

: Corrugation mode In all corrugation modes, the circular shape of the innermost tube persists even under high pressure.

Oval Circular

SLIDE 12

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

The main findings of this study are:

(1) Pressure-induced radial corrugation in the cross-section of MWNT

5. Summary

5 = n 6 = n

Elliptic deformation Radial Corrugation

Oval Circular

(3) Persistent cylindrical geometry

f the innermost tube of MWNT

2 = n

(2) Mode index depend on: i) the tube diameter and ii) the number of concentric wall

Circular shape

C

p p >

C

p p > D N n

Elastic deformation

11/11

(Contact to: shima@eng.hokudai.ac.jp)

SLIDE 13

第13回計算工学講演会（5/21（水）仙台市民会館） 1-5 September 2008, Trend in NanoTechnology 2008 (TNT2008) in Oviedo, Spain

5. Summary

(1) Critical pressure curves

10 20 30 40 50 0.0 0.5 1.0 1.5 2.0 2.5 3.0 n=6 n=6 n=5 n=2 n=5 n=2 n=2 n=4

D = 4.0 nm 5.0 nm 8.0 nm

Critical pressure: pC [GPa] Number of concentric walls: N

(2) Phase diagram of radial deformation

4.0 4.5 5.0 5.5 6.0 10 20 30 40 50

n=6 n=5 n=4 n=2

Number of concentric walls: N Innermost tube diameter: D [nm]

For multi-walled carbon nanotubes, we have demonstrated … (3) Geometric persistence of the innermost tube

1 5 10 50 10

7

10

6

10

5

10

4

10

3

10

2

10

1

10 10 20 30 40 50 0.0 0.2 0.4 0.6 0.8 1.0 Index of walls: i

n = 6 n=2

Normalized deformation amplitudes Index of walls: i D=4.0 [nm] 6.0 8.0

Graduate School of Engineering, Hokkaido University, Japan (shima@eng.hokudai.ac.jp)

Hiroyuki Shima

Pressure-Induced Structural Transitions in Multiwall Carbon Nanotubes

1/11

See H.Shima and M.Sato, Nanotechnology in press

Hydrostatic pressure > 1GPa

= Pressure-induced Radial Corrugation of MWNT

changes from circular to radially corrugated one. Elastic deformation 2/11

Radial collapse occurs at 1.0GPa

3/11 Carbon nanotubes are

Results of MD simulations

but

Sun et al., PRB 70 (2004) 165417

Venkateswaran et al., PRB 59(1999) 10928

Raman spectroscopy:

Tang et al., PRL 85 (2000) 1887

Vanishing a radial breezing mode X-ray diffraction:

Polygonization of SWNT-bundle

4/11

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

( )

p u , θ

Displacements of a surface element: radial circumferential

( )

p v , θ

6/11

Obtaining the mechanical energy of the deformed MWNT, and then its stable cross-sectional shape under high hydrostatic pressure

∫

The mechanical energy of MWNT:

Deformation energy: ( ) ( )

∑∫

( ) ( ) [ ]

Ω + + =

U U p v p u p U , ,

( ) ( )

∑ ∫ ∑ ∫

Inter-wall vdW energy Pressure-induced energy

7/11

Apply a variational method to in terms of in order to determine the stable cross- section under high pressure

U

u , p

[1] Critical pressure curves 8/11

= n 2 = n

For given and , radial deformation occurs just above

N D

p

When exceeds 30, radial corrugation is

N

The wave number of corrugation mode depends

n N D

[1] Critical pressure curves

2

8/11

= n 5 = n

1 > > N

: Elliptic mode : Corrugation mode

n=6 n=5 n=4 n=2

[2] Phase diagram 9/11 nm 4 . 5 = D 36 = N 37 = N 38 = N Increasing with fixed , then… N

[3] Geometric persistence of the innermost tube 10/11

2 = n

: Elliptic mode

5 = n

: Corrugation mode In all corrugation modes, the circular shape of the innermost tube persists even under high pressure.

Oval Circular

The main findings of this study are:

(1) Pressure-induced radial corrugation in the cross-section of MWNT

5 = n 6 = n

Oval Circular

(3) Persistent cylindrical geometry

2 = n

(2) Mode index depend on: i) the tube diameter and ii) the number of concentric wall

p p >

p p > D N n

11/11

(Contact to: shima@eng.hokudai.ac.jp)

(1) Critical pressure curves

(2) Phase diagram of radial deformation

For multi-walled carbon nanotubes, we have demonstrated … (3) Geometric persistence of the innermost tube

(Contact to: shima@eng.hokudai.ac.jp) See H.Shima and M.Sato, Nanotechnology in press