[GPa] p Critical pressure: C

Graduate School of Engineering, Hokkaido University, Japan

([email protected])

Hiroyuki Shima

Pressure-Induced Structural Transitions in Multiwall Carbon Nanotubes

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

Hydrostatic pressure > 1GPa

0. Main Finding

= Pressure-induced Radial Corrugation of MWNT

- The cross-sectional shape changes from circular to

one.

Elastic

deformation

Radial collapse occurs at 1.0GPa

1. Radial collapse of SWNTs

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.

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

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.

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.

Continuum elastic-shell model for MWNT

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

^{v ,} ( ) ^{θ p}

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

π θ

v d u v u v u u

r

p

∫

=

Ω ²

0

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 v u

r u U Eh

1 2

0 3

2 2

2 3 2 2

' '' ) 1 ( 24 2

' ' )

1 (

2 θ

ν ν

π

( ) ( )

[ ^{p u}

^{p v}

^p ] ^{= U}

^{+ U}

^{+ Ω}

U , ,

( ) ∑ ∫ ( )

∑ ∫

= − −

−

= + − + + −

= ^N

i

i i i

i i N

i

i i i

i i

I c r u u d

d u r u

U c

2

2 0

2 1 1

, 1

1

2 0

2 1 1

,

2 2

π

π θ θ

Inter-wall vdW energy

Pressure-induced energy

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

0 10 20 30 40 50 0.0

0.5 1.0 1.5 2.0 2.5 3.0

n=6 n=2 n=5

D = 5.0 nm

Critical pressure: p C [GPa]

Number of concentric walls: N

4. Results

[1] Critical pressure curves

= 0 n

= 2

n For given and ,

radial deformation occurs just above

N D

p

When exceeds 30, radial corrugation is observed

N

The wave number of

corrugation mode depends on and

n

N D

0 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=2 n=5

n=2 n=4 D = 4.0 nm

5.0 nm 8.0 nm

Critical pressure: p C [GPa]

Number of concentric walls: N

4. Results

[1] Critical pressure curves

= 2 n

= 5 n

(= peculiar to MWNT with )N >>1

: Elliptic mode

: Corrugation mode

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

4 nm .

= 5 D

= 36

N N = 37

= 38 N

Increasing with fixed , then…

N

4. Results

[3] Geometric persistence of the innermost tube

1 5 10 50

10^-7 10^-6 10^-5 10^-4 10^-3 10^-2 10^-1 10⁰

0 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

= 5

n

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. Summary

= 5

n n = 6

Elliptic deformation Radial Corrugation

Oval Circular

(3) Persistent cylindrical geometry of 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

p

p >

p

p >

D

N n

Elastic deformation

(Contact to: [email protected])

5. Summary

(1) Critical pressure curves

0 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=2 n=5

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^-1 10⁰

0 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

(Contact to: [email protected])

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