untitled - KUTprl06.pdf

Transient Deformation Regime in Bending of Single-Walled Carbon Nanotubes

A. Kutana and K. P. Giapis

Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, California 91125, USA

(Received 14 August 2006; published 13 December 2006)

Pure bending of single-walled carbon nanotubes between

;

and

;

is studied using molecular

dynamics based on the reactive bond order potential. Unlike smaller nanotubes, bending of

;

and

larger ones exhibits an intermediate deformation in the transition between the buckled and fully kinked

configurations. This transient bending regime is characterized by a gradual and controllable flattening of

the nanotube cross section at the buckling site. Unbending of a kinked nanotube bypasses the transient

bending regime, exhibiting a hysteresis due to van der Waals attraction between the tube walls at the

kinked site.

DOI:

10.1103/PhysRevLett.97.245501

PACS numbers: 62.25.+g, 02.70.Ns, 46.32.+x, 61.46.Fg

Carbon nanotubes possess remarkable mechanical prop-

erties [

–

], making them ideal candidates for nanoelec-

tromechanical systems. Indeed, new nanodevices based on

carbon nanotubes have been demonstrated, including a

mass sensor [

], an oscillator [

], actuators [

], and a

torsional pendulum [

]. A quantitative description of the

elastic properties of nanotubes is necessary to understand

their superior structural flexibility and expand their nano-

scale applications.

Experimental [

] and theoretical [

–

] studies of

basic loading types in carbon nanotubes have established

that, when deformed, carbon nanotubes behave like elastic

shells. This observation stimulated measurements and cal-

culations of Young’s modulus [

] and the Poisson’s

ratio [

] of nanotubes. Yacobson

et al.

[

] employed mo-

lecular dynamics (MD) simulations to study pure bending

of small (

) nanotubes. Cao and Chen [

] esti-

mated the critical buckling strain and curvature of larger

nanotubes (with radii up to 27 A

) using both MD and finite

elements. These studies identified a single critical discon-

tinuity between bending and buckling of nanotubes, which

was delineated by a change in the functional dependence of

the strain energy of the nanotube on bending angle. The

dependence was quadratic in the prebuckling regime but

became linear beyond the buckling point, which was

marked by the appearance of a distinct kink [

]. For

small nanotubes, there is little difference between the

just-buckled and kinked configurations. For example,

Fig.

1(a)

shows the shape of a

;

single-walled carbon

nanotube just after buckling. The nanotube walls at the

buckling point are so close that further bending alters the

nanotube cross section only slightly. However, this is not

the case for larger diameter nanotubes. The just-buckled

wall of a

;

nanotube, shown in Fig.

1(b)

, is further

from the opposite side. Therefore, more bending is re-

quired to bring the two sides close enough (i.e., to the

equilibrium van der Waals distance) to form the kink.

In this Letter, we present results from MD simulations of

large nanotubes (

), which predict the existence

of an intermediate bending regime in the transition be-

tween the just-buckled and fully buckled (termed

‘‘kinked’’) configurations. This ‘‘transient bending re-

gime’’ (TBR) is characterized by a gradual flattening of

the nanotube cross section at the bending site until the

nanotube walls collapse to form the kink. The strain energy

curve exhibits two critical discontinuity points, clearly

demarcating the TBR. Remarkably, unbending of a kinked

nanotube does not retrace the TBR. As the bending angle is

reduced, the tube walls remain collapsed until they

abruptly spring up to the prebuckled state.

Molecular dynamics simulations of bending deforma-

tions in single-walled nanotubes (SWNTs) were performed

using a code developed by Brenner

et al.

, which imple-

ments a reactive bond order (REBO) potential for hydro-

carbons [

]. The REBO potential has been shown to

correctly reproduce the energy of various structural ar-

rangements of carbon, including the 3-coordinated con-

figuration in graphene tubules [

]. Significant

van der Waals interactions exist between proximal nano-

tube walls as, for example, in the kinked configuration.

Such nonbonding interactions were described with a pair-

wise 6 –12 Lennard-Jones potential, with distance and

energy parameters of

407

and

, respectively, for carbon-carbon atom pairs [

All simulations were quasistatic. Test runs at 300 K using a

FIG. 1. Predicted shape of single-walled carbon nanotubes just

after buckling, based on MD simulations. (a) 15.7-nm-long

;

SWNT at

and (b) 23.6-nm-long

;

SWNT at

. Note the difference in scale.

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Berendsen thermostat indicated that the transition points

between deformation regimes occurred at slightly smaller

bending angles but did not otherwise affect the results.

The initial setup of a relaxed straight nanotube was

prepared by minimizing the potential energy of the entire

nanotube. During bending, the nanotube was divided into

three segments: two fixed ends with a length of

which were externally controlled and positioned to satisfy

the boundary condition for the bending angle, and a central

portion whose atoms were allowed to move.

The bending deformation was carried out by rotating the

fixed ends in steps of 0.5

around a line passing through the

middle of the nanotube and normal to its center axis. After

each step, the potential energy of the central segment was

minimized using the conjugate gradient method. Addition-

ally, the axial strain was removed through a succession of

end displacements and central part relaxations. All nano-

tube lengths reported below refer to the extent of the

central nanotube portion, prior to the onset of bending.

Previous simulations of buckling in carbon nanotubes

have shown that the strain energy of the deformed nano-

tube varies nearly quadratically with the bending angle in

the prebuckling regime and is close to linear in the kinking

regime [

]. Our simulations of bending of

;

and

;

nanotubes agree well with elasticity theory calcu-

lations and reproduce results from previous MD studies for

similar SWNTs [

–

]. Indeed, we find only one disconti-

nuity in the potential energy of these nanotubes as they

transition from the prebuckled to the kinked configuration.

However, as the diameter of the nanotube is increased, a

second discontinuity appears in the potential energy curve

at a larger bending angle than the first one. This behavior is

illustrated in Fig.

, where the strain energy

and the

bending moment

dU=d

are plotted as a function of

bending angle

for a

;

SWNT. Three distinct de-

formation regimes are observed for this nanotube, clearly

separated by discontinuity points at

and 32

[

It is instructive to discuss these regimes in view of the

corresponding nanotube longitudinal shape and cross sec-

tion at the buckling point, shown in Fig.

. In the initial

elastic regime, the strain energy

exhibits a quadratic

dependence on the bending angle, while the cross section

experiences progressive ovalization as the bending angle

increases, culminating to the shape in Fig.

3(a)

. The buck-

ling event is marked by an abrupt transition from the oval

cross section to one with the flat top shown in Fig.

3(b)

. The

bending moment’s rate of growth switches from positive

before buckling to negative after buckling.

As the bending angle increases during the TBR, the flat

portion of the top wall expands continuously across the

nanotube [see Figs.

3(c)

and

3(d)

], while the bottom wall

retains its oval shape, albeit with less curvature. As a result,

the top-to-bottom wall distance decreases gradually, reduc-

ing the tube cross section at the buckling site. The strain

energy dependence on the bending angle is no longer

quadratic — in fact, the exponent becomes less than 1.

When the approaching

opposite walls reach

the

van der Waals equilibrium distance of 3.4 A

[Fig.

3(e)

the cross section collapses, forming the kink, and a second

discontinuity is observed. The third bending regime begins

at this point, with no significant change in the shape of the

cross section upon further bending. This regime is charac-

terized by a relatively constant bending moment.

Three deformation regimes are observed for all SWNTs

with diameters between

;

and the largest studied

;

. Nanotube size determines the range of bending

angles over which the TBR occurs, as well as the magni-

tude of the energy drop when the tube cross section col-

lapses. Interestingly, the TBR is fully reversible: If bending

is stopped before the second discontinuity occurs, unbend-

ing recovers the cross-sectional shapes at the buckling

point. The reversible variation in the area of the cross

section during the TBR may act as a means of restricting

the flow of a liquid in the inner cavity of a SWNT and thus

serve as a nanoconduit valve.

Unlike linear elastic bending, the deformation of nano-

tubes during the TBR is inhomogeneous: The strain energy

is not constant along the tube length. It is instructive to

consider the energetics of the buckled cross section and the

nanotube end portions separately. Figure

compares the

potential energy of a

;

tube segment (

of the

tube length) away from the buckling site to that of a thin

ring consisting of 300 carbon atoms at the buckling site.

The onset of buckling corresponds to a redistribution of

strain: The outer portions relax, while the ring at the

buckling site gains a substantial amount of strain energy.

0 1020304050

-3

-2

-1

(eV)

θ

(

rees)

(10

eV)

(30,30)

FIG. 2.

Strain energy

and bending moment

dU=d

for

a 23.6-nm-long

;

SWNT as a function of the bending

angle

. The letters a – e indicate the points for which the tube

shape and cross section at the buckling point are shown in Fig.

TBR denotes the transient bending regime.

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In comparison, the energy gain of an equal-sized ring near

the nanotube end is insignificant, as shown in Fig.

. Upon

further bending, the strain energy of the end segment

decreases slightly, while that of the buckling region con-

tinues to increase significantly. This behavior may be

qualitatively explained by considering the tube as an elastic

shell. It is well known from the theory of elasticity that the

cross section of a uniformly bent cylindrical shell is oval-

ized by a radial force, which ensues from the variation in

the direction of the tensile stress [

]. A much greater

compressive force is applied to the buckling area cross

section during the TBR, because the change in the stress

direction is more abrupt, causing the progressive collapse

of the cross section. The nanotube ends may be considered

as effective levers that amplify the force exerted on the

compressed

cross

section

the

buckling

site.

van der Waals interactions are of no significance until the

second critical discontinuity.

The kinked configuration is marked by the collapse of

the nanotube walls at the buckling site. MD simulations of

SWNTs have predicted that the strain energies of the

circular and collapsed configurations are equal at

SWNT

’

[

]. When

SWNT

, the col-

lapsed configuration is metastable, and the cylindrical

configuration is more stable, while the opposite is true

for

SWNT

. Our simulations predict that bistable

switching between the collapsed and cylindrical forms

can be observed at the kinking site in SWNTs whose radius

is slightly less than

. Figure

5(a)

compares the strain

energy of bending and unbending of a

;

nanotube

(

). Remarkably, unbending after the occur-

rence of the second discontinuity point bypasses entirely

the intermediate bending regime. Rather, the nanotube

switches directly from the kinking to the linear elastic

regime. The kink is preserved by attractive van der Waals

forces between the collapsed nanotube walls, which are

responsible for the unbending hysteresis.

FIG. 4.

Strain energy of various segments of a 23.6-nm-long

;

SWNT as a function of the bending angle

. (1) Tube

segment (

of SWNT length) away from buckling region,

(2) thin ring at the buckling point, and (3) thin ring away from

buckling point. The van der Waals energy is excluded. The TBR

region is observed between 12

and 32

FIG. 3.

Side views and buckling point cross sections of a 23.6-

nm-long

;

SWNT at various bending angles correspond-

ing to the points a – e marked on the strain energy curve in Fig.

Only the portion near the buckling region [darkened region in

Fig.

1(b)

] is shown for the longitudinal views. Bending occurs in

the plane of the journal page.

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In smaller nanotubes, the mutual wall attraction is not

sufficient to produce enough cohesion for the unbending

hysteresis to be observed. For instance, Fig.

5(b)

shows the

energy of bending and unbending of a 15.7-nm-long

;

SWNT, where no dependence on the direction of

bending is found. Unbending recovers the geometrical

shapes observed during bending, including those seen in

the intermediate TBR. This is also true for the

;

SWNT, when van der Waals forces are neglected (not

shown). In that case, unbending retraces the bending path-

way through the transient bending regime. Thus, we con-

clude that van der Waals interactions play a crucial role in

the manifestation of the unbending hysteresis in nanotubes.

In conclusion, MD simulations of pure bending of

SWNTs reveal that nanotubes with size

;

and up

exhibit a transient bending regime, between the linear

elastic and nonlinear kinking regimes expected for smaller

nanotubes. This intermediate regime is characterized by a

gradual flattening of the compressed side of the nanotube

and a concomitant reduction of the cross section at the

bending site. Unbending of fully kinked nanotubes by-

passes the transient bending regime, exhibiting a hysteresis

due to van der Waals attraction between the collapsed walls

at the kinked site.

This work was based on research supported by NSF

(No. CTS-0508096).

Corresponding author.

Electronic address: giapis@cheme.caltech.edu

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0 1020304050

(eV)

θ

(

rees)

(15,15)

(30,30)

(

)

(b)

FIG. 5.

(a) Strain energy

for a 23.6-nm-long

;

SWNT

during bending and unbending operations. While all three bend-

ing regimes are present during bending, unbending proceeds by a

direct switch from the kinking to the linear elastic regime.

(b) Strain energy of bending and unbending of a

;

SWNT. The energy does not depend on the direction of bending.

TBR denotes the transient bending regime.

PRL

97,

245501 (2006)

PHYSICAL REVIEW LETTERS

week ending

15 DECEMBER 2006

245501-4