# Electromechanical interactions in a carbon nanotube based thin film field emitting diode.

**ABSTRACT** Carbon nanotubes (CNTs) have emerged as promising candidates for biomedical x-ray devices and other applications of field emission. CNTs grown/deposited in a thin film are used as cathodes for field emission. In spite of the good performance of such cathodes, the procedure to estimate the device current is not straightforward and the required insight towards design optimization is not well developed. In this paper, we report an analysis aided by a computational model and experiments by which the process of evolution and self-assembly (reorientation) of CNTs is characterized and the device current is estimated. The modeling approach involves two steps: (i) a phenomenological description of the degradation and fragmentation of CNTs and (ii) a mechanics based modeling of electromechanical interaction among CNTs during field emission. A computational scheme is developed by which the states of CNTs are updated in a time incremental manner. Finally, the device current is obtained by using the Fowler-Nordheim equation for field emission and by integrating the current density over computational cells. A detailed analysis of the results reveals the deflected shapes of the CNTs in an ensemble and the extent to which the initial state of geometry and orientation angles affect the device current. Experimental results confirm these effects.

**0**Bookmarks

**·**

**146**Views

- [Show abstract] [Hide abstract]

**ABSTRACT:**We consider field emission from carbon nanotubes and other elongated nanostructures. An exact solution for the electrostatic field between a metallic hemi-ellipsoidal needle on a plate (as a cathode) and a flat anode are presented. The basic idea is to replace the cathode by a linearly charged thread in a uniform electric field and to use a set of "image" charges to reproduce the anode. Exact analytical formulas of the electrical field, field enhancement factor, and electrostatic force are found. Using the Fowler-Nordheim theory we obtain an exact analytical formula for the total current. The field enhancement factor, total force and emission current, as well as their distributions on the top of the needle for a wide range of parameters, have been calculated and analyzed.EPL (Europhysics Letters) 01/2009; 85(1):17001. · 2.26 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Fluctuation of field emission current from carbon nanotubes (CNTs) poses certain difficulties for their use in nano-biomedical X-ray devices and imaging probes. This problem arises due to deformation of the CNTs due to electrodynamic force field and electron–phonon interaction. It is of great importance to have precise control of emitted electron beams very near the CNT tips. In this paper, a new array configuration with stacked array of CNTs is analysed and it is shown that the current density distribution is greatly localised at the middle of the array, that the scatter due to electrodynamic force field is minimised and that the temperature transients are much smaller compared to those in an array with random height distribution.Molecular Simulation 05/2009; 35(6):512-519. · 1.06 Impact Factor - SourceAvailable from: Evgeny G Pogorelov[Show abstract] [Hide abstract]

**ABSTRACT:**We present an exact solution for the electrostatic field between a metallic hemi-ellipsoidal needle on a plate (as a cathode) and a flat anode. The basic idea is to replace the cathode by a linearly charged thread in a uniform electric field and to use a set of “image” charges to reproduce the anode. We calculate the field enhancement factor on the needle surface and ponderomotive force acting on the needle. Using the Fowler–Nordheim theory we obtain an exact analytical formula for the total current.Ultramicroscopy. 03/2009;

Page 1

IOP PUBLISHING

NANOTECHNOLOGY

Nanotechnology 19 (2008) 025701 (12pp)

Electromechanical interactions in a carbon

nanotube based thin film field emitting

diode

doi:10.1088/0957-4484/19/02/025701

N Sinha1, D Roy Mahapatra2, Y Sun1, J T W Yeow1, R V N Melnik3

and D A Jaffray4

1Department of Systems Design Engineering, University of Waterloo, Waterloo, ON,

N2L3G1, Canada

2Department of Aerospace Engineering, Indian Institute of Science, Bangalore 560012, India

3M2NeT Laboratory, Wilfrid Laurier University, Waterloo, ON, N2L3C5, Canada

4Department of Radiation Physics, Princess Margaret Hospital, Toronto, ON, M5G2M9,

Canada

E-mail: jyeow@engmail.uwaterloo.ca

Received 13 September 2007

Published 6 December 2007

Online at stacks.iop.org/Nano/19/025701

Abstract

Carbon nanotubes (CNTs) have emerged as promising candidates for biomedical x-ray devices

and other applications of field emission. CNTs grown/deposited in a thin film are used as

cathodes for field emission. In spite of the good performance of such cathodes, the procedure to

estimate the device current is not straightforward and the required insight towards design

optimization is not well developed. In this paper, we report an analysis aided by a

computational model and experiments by which the process of evolution and self-assembly

(reorientation) of CNTs is characterized and the device current is estimated. The modeling

approach involves two steps: (i) a phenomenological description of the degradation and

fragmentation of CNTs and (ii) a mechanics based modeling of electromechanical interaction

among CNTs during field emission. A computational scheme is developed by which the states

of CNTs are updated in a time incremental manner. Finally, the device current is obtained by

using the Fowler–Nordheim equation for field emission and by integrating the current density

over computational cells. A detailed analysis of the results reveals the deflected shapes of the

CNTs in an ensemble and the extent to which the initial state of geometry and orientation angles

affect the device current. Experimental results confirm these effects.

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Field emission from CNTs was first reported in 1995 by

three research groups [1–3]. With significant improvement in

processing techniques, applications of CNTs in field emission

devices (e.g., field emission displays, x-ray tube sources,

electron microscopes, cathode-ray lamps, nanolithography

systems etc) have been successfully demonstrated [4–6]. The

cathodes in these devices are better electron field emitters than

Spindt-type emitters and diamond structures [7]. The field

emission performance of a single isolated CNT is found to be

remarkable, and this is due to structural integrity, high thermal

conductivity, chemical stability and geometry of the CNTs.

However, the situation becomes highly complex for cathodes

comprising an ensemble of CNTs, where the individual CNTs

are not always aligned normalto thesubstrate surface. Figure 1

shows an SEM image in which the CNT tips are oriented

in a random manner.This is the most common situation,

which can evolve from an initially ordered state of uniformly

distributed and vertically oriented CNTs. Such an evolution

process must be analyzed accurately from the viewpoint of

long-term performance of the device. The authors’ interests

towards such an analysis and design studies stem from the

problem of precision biomedical x-ray generation.

In this paper, we focus on a diode configuration, where

the cathode contains a CNT thin film grown/deposited on a

0957-4484/08/025701+12$30.00

© 2008 IOP Publishing LtdPrinted in the UK

1

Page 2

Nanotechnology 19 (2008) 025701N Sinha et al

Figure 1. SEM image showing randomly oriented tips of CNTs in a

thin film.

metallic substrate and the anode is a magnesium–aluminum

alloy plate acting as emission current collector. Here, the most

important requirement is to have a stable field emission current

without compromising on the lifetime of the device. With

the behavior of the CNT films remaining less predictable, the

mechanisms responsible for CNT failures are also not clearly

understood so far. Failure could be due to electromechanical

deformation and stress, residual gases, ballistic transport

induced impulse and partial discharge (PD) between cathode

and anode.Several studies have reported experimental

observations confirming the degradation and failure of CNT

cathodes [8–11]. The degradation of a CNT thin film emitter is

always gradual [11] but with several small-scale fluctuations

and spikes.Such an overall gradual degradation occurs

either during initial current–voltage measurement or during

measurements under constant applied voltage over a long time

duration.

To date, from mathematical modeling and design points

of view, the detailed models and methods of system

characterization are available only for vertically aligned CNTs

grown on a patterned surface [12, 13]. In a CNT thin film, the

array of CNTs may ideally be aligned vertically. However, in

such a film, it is desired that the individual CNTs be evenly

separated, so that their spacing is greater than their height

to minimize the screening effect.

screening effect is minimized in this manner, the emission

properties as well as the lifetimes of the cathodes are adversely

affected due to a significant reduction in the density of CNTs.

Onthe otherhand, for a cathode withrandomlyorientedCNTs,

the field emission current is produced by two types of source:

(i) CNTs that point towards the current collector (anode)

and (ii) oriented CNTs subjected to electromechanical forces

causing reorientation. Therefore, a possible advantage of a

cathode with randomly oriented CNTs is that a large number

of CNTs always take part in the field emission over a longer

time duration [7]. For this reason, a thin film with randomly

oriented multi-walled carbon nanotubes (MWNTs) has been

Unfortunately, when the

considered while formulating the mathematical model. From

the modeling aspect, this becomes a general case, but is much

more challenging compared to the case of a thin film with

uniformly aligned CNTs. Although some preliminary works

have been reported (see e.g., [14–16]), a much more detailed

characterization needs to be done, especially in situations

where the array of CNTs may undergo complicated dynamics

during the process of charge transport.

accountfortherandomlyorientedCNTsandinteractionamong

them, it is necessary to consider the space charge and the

electromechanical forces responsible for their realignment.

Some of the related factors are also important from device

design considerations.

In this paper, the process of evolution, reorientation (self-

assembly of CNTs) and electrodynamic interaction among

the CNTs in the film, and the influence of these processes

on the device output current are analyzed, based on a

multiphysicsmodeldeveloped recently bythe authors(see [16]

for details). Here, first a homogeneous nucleation rate model

is employed by considering a representative volume element

(cell) to idealize the degradation and fragmentation of CNTs,

phenomenologically. The film volume is discretized by several

such cells across the planer dimensions of the film.

cell contains a specified number of CNTs with prescribed

parametric shape distribution and a certain amount of carbon

clusters as the initial description of the state. At a given time,

the evolved concentration of carbon clusters due to the process

of degradation and CNT fragmentation is obtained from the

homogeneous nucleation rate model. This information is then

used in a time-incremental manner to describe the evolved

state of the CNTs in the cells.

induced current density at the anode (cell-wise) is calculated

by using the CNT tip orientation angles and the effective

electric field in the Fowler–Nordheim equation. The diode

current is then computed by integrating the cell-wise current

density over the anode area. In this paper, while carrying out

numerical simulationsbased on the above scheme, we focus on

visualizing various details, e.g., the reorientation of the CNTs

in the ensemble. From a system perspective, such a detailed

study proves to be helpful in understanding the reason behind

the experimentally observed fluctuation in the device current.

To this end, we show experimental results and a quantitative

comparison with simulations, confirming the reorientation and

degradation of the CNTs.

Also, in order to

A

Finally, the field emission

2. Model formulation

2.1. Background

The physics of field emission from metallic surfaces is fairly

well understood. The current density (J) due to field emission

from a metallic surface is usually obtained by using the

Fowler–Nordheim equation [17], which can be expressed as

?

where E is the electric field, ? is the work function for the

cathode material, and B and C are constants. In the CNT

J =BE2

?

exp

−C?3/2

E

?

,

(1)

2

Page 3

Nanotechnology 19 (2008) 025701 N Sinha et al

thin film problem, under the influence of sufficiently high

voltage at ultra-high vacuum, the electrons emitted from the

CNTs (mainly from the CNT tip region and emitted parallel

to the axis of the tubes) reach the anode. Unlike metallic

emitters, here, the surface of the cathode is not smooth. The

cathode consists of CNTs (often in curved shapes) with certain

spacings.In addition, a certain amount of impurities and

carbon clusters may be present within the otherwise empty

spaces in the film. Moreover, the CNTs undergo reorientation

due to electromechanical interactions with the neighboring

CNTs during field emission. This makes the determination of

current density more difficult.

2.2. Mathematical model

We employ a homogeneous nucleation model to include the

effect of degradation and CNT fragmentation, phenomenolog-

ically. This nucleation model is then coupled with a model of

electromechanical interaction within the cell-wise ensemble of

CNTs via the carbon cluster concentration. These two steps of

our modeling approach are discussed in the following subsec-

tions.

2.2.1. Nucleation coupled model for degradation of CNTs.

Let NTbe the total number of carbon atoms (in the CNTs and

in the cluster form) in a representative volume element (Vcell)

ofthe thinfilm, N be thenumber ofCNTsinthe cell, and NCNT

be the total number of carbon atoms present in a CNT. Hence,

NT= NNCNT+ Ncluster,

(2)

where Nclusteris the total number of carbon atoms in the cluster

form in a cell at time t, and it is given by

Ncluster= Vcell

?t

0

dn1(t),

(3)

where n1is the concentration of carbon atoms in the cluster

form in the cell.Therefore, by combining equations (2)

and (3), one can obtain the number of CNTs in the cell as

?

In order to determine n1(t), we introduce a homogeneous

nucleation model [18, 19], which is to describe the evolution

in n1(t). Here, we modify the original model (it was proposed

in the context of an aerosol and a chemical growth process)

by assuming the degradation as a reverse process of growth

and model the phenomena of CNT degradation (the nucleation

theory has been used for the growth of CNTs [20] and other

nanoparticles [21, 22] also). Based on this modified model, the

evolution equations are expressed as

N =

1

NCNT

NT− Vcell

?t

0

dn1(t)

?

.

(4)

dNkin

dt

= Jkin,

(5)

dS

dt

dM1

dt

= −JkinSg∗

n1

− (S − 1)B1An

2v1

,

(6)

= Jkind∗

p+ (S − 1)B1Nkin,

(7)

dAn

dt

=JkinSg∗2/3s1

n1

+2πB1S(S − 1)M1

n1

,

(8)

where Nkin is the kinetic normalization constant, Jkin is the

kinetic nucleation rate, S is the cluster saturation ratio, g∗

is the critical cluster size, Anis the total surface area of the

cluster, v1is the monomer volume, M1is the moment of the

cluster size distribution, d∗

the critical cluster, and s1is the surface area of a monomer.

In equations (5)–(8), the various other quantities involved are

given by

pis the diameter of the particle of

S =n1

ns,

M1=

?dmax

p

d∗

p

(n(dp,t)dp)d(dp),

(9)

Nkin=n1

Sexp(?),

B1= 2nsv1

?

?

kT

2πm1,

(10)

Jkin=βijn2

1

12S

?2

?

?

2πexp

?3

? −

4?3

27(ln S)2

?

,

(11)

g∗=

3

?

ln S

,

d∗

p=

4σv1

kT ln S,

(12)

where nsis the equilibrium saturation monomer concentration,

dpis the cluster diameter, dmax

p

is the maximum diameter of the

clusters, n(dp,t) is the particle size distribution function, ? is

the dimensionless surface tension, k is Boltzmann’s constant,

T is the temperature, m1is the mass of monomer, βij is the

collision frequency function for collisions between i-mers and

j-mers, and σ is the surface tension. The collision frequency

function (βij) is given by

?1/6?

βij=

?3v1

4π

6kT

ρp

?1

i+1

j

?

(i1/3+ j1/3)2.

(13)

The dimensionless surface tension (?) is expressed as [18]

? =σs1

kT,

(14)

where ρpis the particle mass density. In this paper, we have

considered i = 1 and j = 1 for numerical simulations; that is,

onlymonomertypeclustersare considered. Our mainintention

in this context is to find n1 using equation (5).

during the growth of clusters, the change in the particle

size (distribution) with time is governed by the saturation

ratio (ratio of atomic concentration to the concentration at

saturation). Therefore, equation (6) is required. In growth

related kinetics, the growth is measured by the moment of

cluster size distribution [23]. M1in equations (7) and (8) tells

us about the distribution of the cluster size over time. That is,

at different locations in the cell, different sizes of cluster can

form, and that also affects how the equilibrium occurs. During

growth of the clusters, a critical size nucleus is a cluster of

size such that its rate of growth is equal to its rate of decay.

The quantity Anindicates the area of the stable carbon cluster.

To know the area of the cluster, An is required. So, based

on the established kinetic theory, we need to include all the

four equations. For instance, in equation (8) for dAn/dt, the

However,

3

Page 4

Nanotechnology 19 (2008) 025701 N Sinha et al

first term on the right-hand side of the equation describes an

increase/decrease in the surface area due to a newly formed

stable cluster and the second term means an increase in area

of the existing stable cluster. Equations (5)–(8) form a set

of four nonlinear coupled ordinary differential equations in

n1(t), S(t), M1(t) and An(t). This system of equations is

solved with the help of a finite difference scheme. Finally, the

average numberofCNTsinacell, andhence theaverage height

distributionwithaknownnumberofCNTsintheensemble, are

obtained with the help of equation (4).

2.2.2. EffectsofCNT geometryand orientation.

in section 1, the geometry and the orientation of CNTs play

important roles in the field emission performance of the film

and hence must be considered in the model. In section 2.2.1,

a model of degradation and fragmentation of CNTs has been

formulated. Following this model, if ?h is the decrease in the

length of the CNT over a time interval ?t due to degradation

and if dtis the diameter of the CNT, then the surface area of

the CNT decreases by πdt?h. By using the geometry of the

CNT, the decreased surface area can be expressed as

As discussed

πdt?h = Vcelln1(t)[s(s − a1)(s − a2)(s − a3)]1/2,

where a1, a2, a3are lattice constants, and s =1

The chiral vector for the CNT is expressed as

(15)

2(a1+a2+a3).

− →

Ch= n? a1+ m? a2,

(16)

where n and m are integers (n ? |m| ? 0) and the pair (n,m)

defines the chirality of the CNT. By using the fact that ? a1·? a1=

|a1|2, ? a2· ? a2= |a2|2, and 2? a1· ? a2= |a1|2+ |a2|2− |a3|2, the

circumference and the diameter of the CNT can be expressed

as [24], respectively,

?

dt=|− →

π

|− →

Ch| =

n2a2

1+ m2a2

2+ nm(a2

Ch|

1+ a2

2− a2

3),

.

(17)

By defining the rate of degradation of CNT as vburn

lim?t→0?h/?t, and by dividing both sides of equation (15)

by ?t, one has

=

πdtvburn= Vcelldn1(t)

dt

[s(s − a1)(s − a2)(s − a3)]1/2. (18)

By combining equations (17) and (18), vburncan be simplified

as

?

vburn= Vcelldn1(t)

dt

s(s − a1)(s − a2)(s − a3)

n2a2

1+ m2a2

2+ nm(a2

1+ a2

2− a2

3)

?1/2

(19)

.

Next, we calculate the effective electric field at a given time

step. The electric field at the deflected tip (see figure 2) is

approximated as

?

Ez? =

1 −x2+ y2

R2

(Present height)

(Present gap)

E0,

(20)

Figure 2. CNT array configuration.

where x and y are the deflections of the tip with respect to

its original location, 2R is the spacing between two adjacent

CNTs at the cathode substrate, and E0= V/d with V as the

applied DC voltage. As the CNTs degrade, their height also

decreases with time. On the other hand, the distance between

the tip and the anode increases with time due to reduction

in the height of the CNTs. With the above assumption, the

present height of the CNT can be written as h0− vburnt, and

hence the present distance between the tip and the anode can

be expressed as d − h0+ vburnt. Here, h0is the initial average

height of the CNTs and d is the distance between the cathode

substrate and the anode (see in figure 2). Equation (20) can

now be rewritten as

?

Subsequently, the effective electric field, which is required for

field emission calculation, can be expressed as

Ez? =

1 −x2+ y2

R2

(h0− vburnt)E0

(d − h0+ vburnt).

(21)

Ez= Ez? cosθ(t),

(22)

where θ(t) is the tip orientation angle that a CNT makes

with the Z-axis, as shown in figure 2. The current density

is calculated by substituting the value of Ez in the Fowler–

Nordheim equation. The above formulation takes into account

the effect of CNT tip orientations, and one can perform

a statistical analysis of the device current for randomly

distributed and randomly oriented CNTs. However, due to the

deformation of the CNTs under electromechanical forces, the

evolution process requires a much more detailed treatment. In

order to account for the changing orientations, we estimate the

effects of electromechanical forces as discussed next.

2.2.3.

orientation angle θ is dependent on the electromechanical

forces. Based on the studies reported in the literature, it is

reasonable to expect that a major contribution is by the Lorentz

force due to the flow of electron gas along the CNT and the

ponderomotive force due to electrons in an oscillatory electric

field. In addition, the electrostatic force and the van der Waals

force are also important.

The components of the Lorentz force along the Z- and X-

directions (see figure 2) are approximated as

Electromechanical forces.

For each CNT, the

flz= πdteˆ nE1z,

flx= πdteˆ nE1x≈ 0,

(23)

4

Page 5

Nanotechnology 19 (2008) 025701N Sinha et al

where e is the electronic charge (positive), ˆ n is the surface

electron density and E1z is the fluctuating electric field due

to electron flow in the CNTs. We now decompose the surface

electrondensityintoasteady(unstrained)partandafluctuating

part; that is,

ˆ n = ˆ n0+ ˆ n1,

where the steady part ˆ n0corresponds to the Fermi level energy

in an unstrained CNT, and it can be approximated as [25]

(24)

ˆ n0=

kT

πb2?,

(25)

whereb istheinteratomicdistanceand?istheoverlapintegral

(≈2 eV for carbon). In equation (24), the fluctuating part ˆ n1is

inhomogeneous along the length of the CNTs. Actually, ˆ n1

should be coupled nonlinearly with the deformation and the

electromagnetic field [26]. However, in a simplified form, ˆ n1is

primarily governed by the quantum-hydrodynamic continuity

equation:

˙ˆ n1+ ˆ n0∂ ˙ uz?

with uz? as the longitudinal displacement, and we employ this

simplified equation to compute ˆ n1 at a given time step. In

equation (23), the fluctuating electric field E1 = (E1x, E1z)

is computed using a Green’s function approach as discussed

in [27].

The ponderomotive force, which acts on free charges on

the surface of CNTs under oscillatory high field, tends to

straighten the bent CNTs in the Z-direction. Furthermore,

the ponderomotive forces induced by the applied electric field

stretch every CNT [28]. In order to estimate the components

of the ponderomotive force ( fpx, fpz) acting on the tip region

(see [29]), the following approximations are used:

∂z?= 0,

(26)

fpz≈

q2

2meω2E1z∂E1z

∂z

,

fpx≈ 0,

(27)

where q = (πdteˆ n)ds is the total charge on an elemental

segmentds of a CNT,meisthemass of an electron, ω = 2π/τ,

τ istherelaxationfrequency, and fpzisthe Z-componentofthe

ponderomotive force. The X-component of the ponderomotive

force fpxis assumed to be negligible.

In order to calculate the electrostatic force, interactions

among the neighboring CNTs are considered. Let us assume

two small segments of two neighboring CNTs of lengths ds1

and ds2, respectively. The charges on these two segments are

given by, respectively,

q1= eˆ n(πd(1)

where d(1)

t

(1) and (2), respectively. The electrostatic force on the segment

ds1by the segment ds2is given by

t )ds1,

q2= eˆ n(πd(2)

t )ds2,

(28)

and d(2)

t

are the diameters of two neighboring CNTs

1

4π??0

q1q2

r2

12

,

where ??0denotes the effective permittivity of the aggregate

of CNTs and carbon clusters and r12is the effective distance

between the centroids of the segments ds1and ds2. The force

on the segment ds1of one CNT due to the entire segment (s2)

of the neighboring CNT can be expressed as

?s2

Therefore, the force per unit length on s1due to s2is

1

4π??0

0

1

r2

12

(eˆ nπd(1)

t

ds1eˆ nπd(2)

t )ds2.

fc=

1

4π??0

?s2

0

(πeˆ n)2d(1)

t d(2)

t

r2

12

ds2.

(29)

The differential of the force d fc acts along the line joining

the centroids of the segments ds1 and ds2. Therefore, the

components of the force fc in the X-direction and the Z-

direction are, respectively,

?

1

4π??0

j=1

?

1

4π??0

j=1

Here, φ is the angle that the force vector d?fcmakes with the X-

axis, j is the node number, and ?s2is the length of discretized

segments(assumeduniform inthepresentstudy). Theeffective

distance (r12) between the centroids of the segments ds1and

ds2is obtained as

fcx=

d fccosφ =

1

4π??0

(πeˆ n)2d(1)

?s2

t d(2)

0

(πeˆ n)2d(1)

t d(2)

t

r2

12

cosφ ds2

≡

h0/?s2

?

t

r2

12

cosφ?s2,

(30)

fcz=

d fcsinφ =

1

4π??0

(πeˆ n)2d(1)

?s2

t d(2)

0

(πeˆ n)2d(1)

t d(2)

t

r2

12

sinφ ds2

≡

h0/?s2

?

t

r2

12

sinφ?s2.

(31)

r12= [(d1−lx2+lx1)2+ (lz1−lz2)2]1/2,

where d1is the spacing between the CNTs while in contact

with the surface of the cathode substrate, and lx1and lx2are

the deflections of the segments of two neighboring CNTs

(relative deflection considering the two end nodes of each of

the segments), respectively, which are parallel to the X-axis.

Similarly, lz1and lz2are the deflections of the two segments

which are parallel to the Z-axis.

Next, we consider the van der Waals effect. The van der

Waals force plays important roles, not only in the interaction of

the CNTswiththe substrate, but alsoin the interaction between

the walls of MWNTs and CNT bundles.

type deformation, the cylindrical symmetry of the CNTs is

no longer preserved, leading to axial–radial coupling [30].

For simplicity, let us restrict ourselves to a two-dimensional

(2D) situation described with respect to the (X, Z) coordinate

system shown in figure 2, and let us assume that the lateral

and longitudinal displacements of a CNT are ux? and uz?,

respectively. Due to their large aspect ratio, it is reasonable

to idealize the CNTs as one-dimensional elastic members (as

in an Euler–Bernoulli beam [31]). Therefore, the kinematics

can be expressed as

(32)

Under bending

u(m)

z?

= u(m)

z?0− r(m)∂u(m)

x?

∂z?,

(33)

5