Simulation of Carbon Nanotubes Growth by Laser Ablation

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Simulation of Carbon Nanotubes Growth by Laser

Ablation Ahmed. A. Moosa 1, Hussain A. Jawad

2, Ali Zuhair Ridha

3

1Department of Materials,

Technical College, 54 Al Zafaraniya Street,

Baghdad, Iraq. E-mail: [email protected] 2Institute of Laser for Postgraduate Studies,

University of Baghdad 3Institute of Technology,

54 Al Zafaraniya Street, Baghdad, Iraq

Abstract: One of the synthesis methods of the single-walled carbon nanotubes (SWCNTs) is to use by laser ablation process.

A simulation program was built to simulate the nucleation process of the carbon nanotubes. The simulation is based on the

classical nucleation theory and the free energy of carbon nanotubes nucleus. In order to accurately simulate the growth of

carbon nanotubes by laser ablation, the laser-material interaction in addition to the nucleation process must be considered.

In this study the simulation is separated into two parts: laser ablation and Nucleation. The laser ablation process includes

the heat transfer within the target leading to ablation. Then the nucleation process will be described using classical

nucleation theory. The simulation results showed the diameters distributions of carbon nanotubes are around (1-1.7) nm over

all temperatures. Comparison between the experimental and simulation results showed there is a good agreement in their

behavior.

Keywords: Carbon nanotubes; Laser ablation; simulation.

1. Introduction The first MWCNTs were discovered by Iijima in 1991when

he found layers of rolled grapheme structure in the soot of

electric arc discharge method. Carbon nanotubes are

cylindrical structure formed by rolling single layer or

multilayers of grapheme sheets into cylinder [1]. In 1993,

Bethune group at IBM [2] and Iijima group at NEC [3]

reported the synthesis of Single-Walled Carbon Nanotubes

(SWNTs). The pulsed laser-ablation process for the

production of single-wall carbon nanotubes was developed

by Guo et al. [4, 7] at Rice University. Other improvements

were made by Thess et al., [5] and and Rao et al., [6]. Laser

ablation method uses a high-power laser (YAG type) to

vaporize pure graphite targets inside a furnace at 1200 ± oC

under an Ar atmosphere to produce MWNTs [7]. SWCNTs

with purity as high as 90% can be produced by laser ablation

technique which are purer than CNTs produced by arc

discharge[8] .In general, the most commonly used catalysts

are Ni, Co, and Fe. These are used independently or mixed

with another metal. [9]

M. H. Rummeli (2010). The properties of CNTs prepared by

the pulsed laser deposition process are strongly dependent on

many parameters such as: the laser properties (energy

fluence, peak power, CW versus pulse, repetition rate and

oscillation wavelength), the composition of the target

material, the inert gas pressure, the substrate and ambient

temperature and the distance between the target and the

substrates [10]. In 1995 Guo et al., [4] were first to synthesis

of SWCNTs by Laser ablation technique. Later on used two

laser pulses 50ns to vaporize carbon target with a cobalt-

nickel catalyst, and achieved 70-90% efficiency [11]. The

atmospheric temperature controls the chemical reaction and

the laser intensity affects the process by which the particles

species ejected [12]. Laser ablation can be accomplished by

Ablation of a graphite target using either a pulsed laser [4, 5]

or a continuous laser [13].

In both cases, the nanotubes nucleate in the vapor phase,

coalesce, get carried away by the flowing argon and condense downstream on the water-cooled copper collector.

The soot contains amorphous carbon, metal particles, and

CNTs [13].In our previous work, CNTs was produced by

laser ablation using two targets technique (graphite and Ni)

instead of one target (Ni/graphite) to synthesis SWCNTs by

laser ablation process [14]. The aim of this work is to

simulate the growth of carbon nanotubes by laser ablation by

considering laser-material interaction in addition to the

nucleation process. The laser ablation process includes the

heat transfer within the target leading to ablation. The

nucleation process will be described using classical

nucleation theory. This simulation was design to simulate the

distribution of SWCNTs at different temperatures and then

compare these results with the experimental result.

2. Laser Ablation with Nanosecond

Laser Laser ablation can be divided into three main process,

bond breaking and plasma ignition, plasma expansion and

cooling, and particle ejection and condensation [15].

During the plasma ignition process, the dominant

mechanism is thermal vaporization: the temperature of the

solid surface increases, and a well-defined phase transition

occurs, from solid to liquid, liquid to vapor, and vapor to

plasma [16].

Nano-sized particles will be formed from condensation of

the vapor. Condensation starts when the vapor plumes

temperature reaches the boiling temperature and stops at

the condensation temperature of the material. Liquid

ejection of particles can occur by high pressure gradient

forces within the highly expanding vapor plume acting as

the molten surface. Solid sample exfoliation can occur

from the large thermal stress gradients of the fast heating




Page 41

process; thermal stresses can break the sample into

irregular shaped particles, ejecting them from the surface.

[15].

3. Nucleation and Growth of Carbon

Nano-tubes Carbon nanotubes formation and growth mechanism

during the laser ablation process started by the formation

of liquid nanoparticles of metal supersaturated with

carbon. These nanoparticles originate from condensation

of the metal plasma/vapor in the moderate temperature

zone of the laser ablation chamber. During the nucleation

the metal nanoparticles can incorporate large amount of

carbon depending on the temperature and its size [17].

Through cooling the solubility limit of carbon in liquid

nanoparticles is decreased and the carbon begin to

segregate[18]. The segregation of carbon occurs via

diffusion towards the particles surface. The occurrence of

such segregation is supported by the absence of carbon

inside nanoparticles after solidification. Then the carbon

crystalize at the surface of nanoparticle according to the

two competing transformation which lead to the formation

of graphite sheet and to nucleation of single walled carbon

nanotubes.[19].

4. Computational Method In order to accurately simulate the growth of carbon

nanotubes by laser ablation, the laser-material interaction

in addition to the nucleation process must be considered.

In this study the simulation is separated into two parts:

laser ablation and Nucleation. The laser ablation process

includes the heat transfer within the target leading to

ablation. The nucleation process will be described using

classical nucleation theory.

4.1 Interaction of laser light with target surface

Interaction of laser light with target surface will increase the

internal energy of the target. The mode of interaction is

considered to be thermal i.e. the time scale of the process is

much longer than the relaxing time of the excited electron

(10-15

s). The heat transfer is modeled as having a local

thermal equilibrium within the target. The state of the target

can be described by time-rate first law of thermodynamic

and by neglecting heat transfer by convection because the

interaction related to the target [19].

( ) ( )stored in out cond in out radE E E E E (1)

The radiation term in equation (1) can be separated into two

parts, the surface radiation term as a boundary condition and

the laser energy as a heat source. Re-writing the heat transfer

by conduction into1-D heat transfer by conduction as:

.( )

( )pC T

KT q St

(2)

where .q = energy generation per unit volume, S = source

term for energy required for phase change. For laser at1064

nm, the absorption coefficient is ˂105cm

-1 so

.

q is

neglected [20]. Within the target there is no phase change

and S is considered to be zero. The heat transfer within the

target is given by

( )( )

pC TKT

t

(3)

The internal boundaries are kept at the initial temperature

T(t) = Tintial at (t = 0), The radiation boundary condition is

given by:

4 4 ..( )

( )surr laser

KTT T q

n

(4)

The algorithm of the COMSOL Program is shown in Fig. 1

4.2 Nucleation and Growth of Carbon nano-tubes

To describe the nucleation of carbon nano-tubes

mathematically various mechanisms for the formation of

carbon filamentous deposit have been introduced. In the first

group Baker et al [21] showed a mechanism include a stage

of carbon dissolution that is independent of carbon source.

The second mechanism is based on the diffusion of carbon at

the surface of the metal particle rather than on the bulk

diffusion of carbon through the catalyst particle [22-23]. The

third mechanism is based on the catalyst particles take part

only in the tube initiation process such that the increase in the

thickness in the primary tube occurs without the further

participation of the metal particle [24].

The fourth group based on small metal clusters activates

SWCNTs by moving around the open tube tip and bonding

carbon atoms from the gas phase. [25]. The fifth group

proposed a model for the formation of cylindrical layers

around a hollow core using a thermo dynamic approach [26].

For the purpose of this study the model proposed by Tibbets

and modified by Vladimir et al [ 27] will be used on the

nucleation of the carbon nano-tubes on a nickel nano-particle.

4.3 Thermodynamic Treatment

The nucleation of carbon is a core step for the formation of

all types of carbon filament deposit. The critical size of

carbon nucleus determines the type of carbon deposit

because the size of the tube cannot be smaller than the

critical size. When the critical size of the carbon is

sufficiently small compared to the size of the metal particle

this allow the formation of on nucleus several nuclei. These

nucleus combines into small structure with the carbon atoms

arranged in hexagons which transform into grapheme sheet

bonded with its edges to the metal surfaces [28-29]. The

change in the free energy for the formation of two

dimensions nucleus can be written using Gibbs free energy

[30] as:

2

2( )I h

G G Inucleus nucleus gas nucleus surface surface gasv

m

I Estrain

ò

(5)




Page 42

Fig.1: COMSOl algorithm for Heat transfer within target.

Where I is a perimeter of nucleus, is geometrical factor

,vm is a molar volume, nucleusG is a change in the free

energy of 1mole of carbon precipitate from metal-carbon

solution, is a specific surface energy, ò is a specific edge

free energy and Estrain is a strain energy arising during the

bending of graphene layer on a metal surface.The change in

the free energy of nucleus nucleusG is given by [31]:

ln( ) ln( )a x

G k T k Tnucleus b ba x

(6)

where kb is a Boltzman constant, T is Temperature, a and

a are the carbon activities of the saturated and actual

solution respectively, x and x are the saturated and the

actual molar content of the dissolved carbon respectively

and Qc is a constant ( for CNT it is equal to 4.4 ev). Use the

continuum formalism of elasticity, the strain energy

(4.5

Q IcE

strain h ) arising from the bending of the

graphene layer on the metal is given by [32]. Substitute in

Eq. (5) to get the Gibbs free energy for the formation of

nucleus with perimeter I and height h. After differentiation

with respect to I , and rearrange to get I critical :

( )4.5

( )

Qc

hIcritical

hG

nucleus nucleus gas nucleus surface surface gasvm

ò

The critical number of carbon atoms which can form a stable

nucleus can be calculated by substituting the total number of

carbon atom in nucleus ( 2I h

nv

m

) in Eq. (7). If we

consider the nucleus has circular shape with (I=2πr and

γ=1/4 π) [33] then the critical size of the nucleus is given by

Eqs (8) and (9) :

2

( )4.5

( )

Qc

h hncritical v hm G

nucleus nucleus gas nucleus surface surface gasvm

ò

The critical radius in equation (9) means that the cluster that

has this critical size is energetically favorable to grow while

the smaller clusters will preferentially decay. For clusters

smaller than the critical cluster, the evaporation rate exceeds

the condensation rate and vice versa. An important

conclusion from the energy consideration is that the

evaporation rate of a cluster at a given temperature equals

the condensation rate of this cluster at the same temperature

and such a vapor pressure that the cluster is critical at it. The

condensation rate can be expressed by evaluating the

sticking cross-section. If Ni is the concentration of clusters

containing i monomers (i-clusters) the rate equation as

function of time can be written as follows:

1 1

1 1

( )( ) ( ) ( )

( ); 2,3,....

ii i i i i

i i

dN tC N t C E N t

dt

E N t i

(10)

Where Ci is the rate of monomer sticking with an i-cluster

or condensation rate, and Ei is the rate of monomer

evaporation from an i-cluster. Since the classical nucleation

theory considers only the initial stage of cluster

condensation, where the total cluster mass is small as

compared to the vapor mass, it is assumed that the monomer

concentration is constant [34], then ( )

0idN t

dt . To solve

(7)

1( ) ( ) ( )4.5

Qc hr Gcritical nucleus nucleus gas nucleus surface surface gash v

m

ò

(8)

(9)




3

Eq. (10), it is assumed that clusters larger than J-cluster are

instantly decomposed into monomers, so that

1 1

( )( ) ( ) ( )

( )0;

JJ J J J J

i

dN tC N t C E N t

dt

dN ti J

dt

(11)

To get an approximate analytical solution for steady state

case using classical nucleation theory, rewrite equation (11)

as [35]:

1

( )( ) ( )i

i i

dN tA t A t

dt (12)

Where Ai i the number of freshly formed i-cluster per unit

volume and time and is given by

:1 1( ) ( ) ( ); 2,3.......,

( ) ( )

i i i i i

J J J

A t C N t E N t i J

A t C N t

(13)

The number density of i-cluster at total equilibrium is

( 1 exp( )ii

GD N

KT

). The important feature of Di is

that for i>>1 1

1

i i

i i

D E

D C

(14)

Rewrite equation (13) using equation (12) as

follows: 1

1

( )ss i i

i i i i

A N N

C D D D

(15)

Where Ass is steady state of i-cluster. For over all I getting

1

1

1ss

i i i

NA

C D D (16)

Using Taylor expansion for terms of i the sum in equation

(16) can be approximated

by1 1

exp( )

crit

Ji

ss i

i ii i i

GA K

C D C KT

(17)

Where 14i iK R D is the attachment rate constant

approximated by Park-Privman [36] . Where Ri is the radius

of the single CNT (Ri = 1

31.2iR hi ). The solute

diffusion coefficient can be estimated by using Einstein

formula for diffusion of cluster in the Argon gas [37] is

6

Bk TD

h , where is the viscosity of Argon and it can

be neglected [38]. For singlet carbon particle aggregates

according to the difference between density of the primary

nucleus formed and the rate of its transformation.

1

2

( )j

j

dNdNt j

dt dt

(18)

Where is the formation of critical nuclei in (m-3

s-1

)

2( ) 4 exp( )criticalcritical

BT

Gt hn DC

K

(19)

The concentration of carbon particle (C (t)) is a function of

time and it decrease with formation and the growth of

critical nuclei. Assuming the process of formation is

irreversible the concentration as a function of time can then

be written as [6]:

( )critical

dcn t

dt (20)

The master equation for population of large cluster (i>3) can

be written as follows [39]:

1 1 1 2 1i

i i i

dNK N N k N N

dt (21)

To apply nucleation theory method numerically we used the

following algorithm as shown in Fig.2.

5. Heat Transfer within Target In the present work the heat transfer within target (graphite)

due to the interaction of the laser beam with graphite under

flowing of Argon gas and furnace temperature of 1000°C

was introduced upon heat transfer within target Eq. (3). The

laser source to perform laser ablation simulation was

Nd:YAG with fluence of 1.5 J/cm2, pulse duration = 10ns

and wavelength = 1064 nm. The laser target has a dimension

of (10*10*0.3) mm3with effective area (2*2) mm

2. The

numerical procedure was developed under COMSOl

Multiphysics Project ver (3.5).

The target was simplified to 2-D rectangle as shown as

shown in Fig.3. The target was treated with mesh size (1nm)

(boundary 3). Only the interface between heated and

unheated target was solved i.e. a half of the target is

irradiated by laser and the other half was not. Surface

boundary condition for irradiated area (boundary 3 in Fig.3

is described by Eq.(4) . The unheated are (boundary 4) had

the same equation as boundary 3 but without heat source

distribution. Left bottom and right side (boundary 1, 2 and 5)

are thermally insulated, adiabatic condition (

.( . ) 0n K T ). The computer program was run for

physical time (10ns) pulse duration. The maximum time step

was limited after several trial and errors to 2.5*10-13

s. The

computational domain was taken as having area of 4mm2

and depth of 3μm. The space step was taken to 10nm. . As a

result Fig. 4shows temperature field evolution of the target

for time 10ns during laser action. Table 1 shows some global

expressions and constants used in the simulation of the laser

ablation.

Fig. 3: Sketch of selected part of bulk sample for modeling

in COMSOL with numbers of boundary conditions with

material surface (dimensions in mm).




Page 44

Time step (dt ) to calculate the population 0f clusters

Calculation of first concentration

of C 3 cluster

Calculation of concentration of C 3 at

dt

Calculation of production rate of

critical particle at dt

Open loop for number of nuclei n from 1-1000

Update of time

step dt

Calculation of the population of clusters of

n=1,2,3.. nuclei

Calculation of CNTs size

distribution

Update

Open loop to Calculate CNTs

size distribution

Update of time

step ,dt

Update of time

step dt

Calculation of critical particle size

Calculation of production rate of

critical particle

End

Time step dt




Page 45

Fig. 4: Temperature field evolution for time 10ns

Table 1: Parameters required estimating critical nucleus

ϵ Specific free energy J/A=1.8*10-19

[40]

h Height of the nucleus A=3.4

vm Molar volume A3/mol=5.3*10

-24

graph

Specific surface free energy of the basal

plane of graphite J/m2 =0.077 [41]

Wad Work of adhesion of graphitic-metal

system J/m2 = 0.4-0.14 [42]

Figures 5 and 6 represent surface temperature versus time

and depth respectively. As shown from these figures the

temperature distribution within the target surface differs

from subsurface. This means that at the beginning before the

laser hits the target the temperature of the surface is equal to

furnace temperature 1000°C and after the laser hits the target

the surface temperature will rise to reach its maximum value

at the end of the laser pulse. Also once can notice from Fig. 6

that the temperature decreases gradually from the surface to

sub surface to reach at depth 2.3μm to the furnace

temperature. These results are agreed with Michael Shusser

[43].

Fig. 5: Average surface temperature versus time

Fig. 6: Average surface temperature versus depth.

6. Carbon Nanotubes Formation In the first step of nucleation process several carbon nuclei

precipitate on the surface of the same metal surface (catalyst)

and increase in diameter to form highly packed mosaic

structure. If the nucleation rate is highly enough, the size of

single wall nano-tubes tends toward minimal value that

corresponds to critical nucleus. So that the first step in

nucleation of carbon nano-tubes is estimation of critical

nuclei (nc).By using Eq. (8) the estimation of the critical

nucleus in the range of (500-1400)°K could be established

which is typical for the formation of different catalytic

carbon deposit. The numerical result for estimation of critical

nucleus of carbon nano-tubes as a function of reaction

temperature for Ni Catalyst is presented in Fig. 7. With

increasing the temperature, the number of nuclei inside a

critical cluster is decreased. Thus, with increasing the

temperature the cluster size and also the diameter of the

produced carbon nano-tubes are decreased. The probability

of formation of SWCNTs is high at high temperature. The

relation between the critical nuclei and critical radius rof

formation carbon nano-tube can be estimated using Eq. 9.

The critical nucleus as a function of critical radius is shown

in Fig. 8. It is clear from this figure the decrease of critical

nuclei will decrease the critical cluster radius with increasing

temperature.

Fig. 7: Critical nucleus of CNTs Versus the reaction

Temperature (K) for Ni Catalyst.

Using Eqs. 18,19, 20 and 21 ,the distribution of carbon

nanotubes diameter can be established. Some approximations

for simplification were done such as: Ignoring the cluster –

cluster interaction (aggregation) and the radius of the

primary nuclei is closed to radius of critical nuclei.




Page 46

Fig. 8: Critical nucleus of CNTs versus Critical radius (nm).

Fig. 9 show the simulation results of carbon nanotubes

distribution at different temperature (850-1000)°C. This

simulation was design to simulate the distribution of

SWCNTs at different temperatures and then compare these

results with the experimental result. The concentration of the

solution is related to C3 cluster which is (1024

)m-3

.

Fig. 9: Simulation results for carbon nanotubes distribution

at (a) 850, (b) 900 and (c) 1000°C.

From Fig 9 the diameters distributions of carbon nanotubes

are around (1-1.7) nm over all temperatures. The yield of the

nanotubes increased with increasing temperature. Finally the

yield of the larger diameter increased as temperature

increased. These results are reasonable if we consider that

each temperature has its intrinsic nc. Fig. 9 was then

compared with the experimental distribution of SWCNTs at

different temperatures Fig. 10 from our previous work [14]

which gives the same behavior but with different yield value

and the diameters of the nanotubes were around ~(1-2)nm.

This difference in the results between the simulation and the

experimental work is resulted from the cluster formed after

laser ablation is only C3 it contains C2 , C and metal particle.

In our previous work [14] it was found that at temperature

between 750-800 ºC the structure of the deposit carbon

nanotubes has a web-like structure. At temperature 850ºC -

1000ºC there is a transformation of the deposit carbon

nanotubes structure from web-like structure to spaghetti-like

structure [14] . The temperature dependence of the carbon

nanotubes diameter (nm) can be explained by changes in the

cooling rate of the ablated carbon species. As the carbon

vapor plume expands and cools down, the nanotubes

structure become kinetically fixed [21]. As the nuclei size

changes there is balance between the strain energy of the

curved graphene sheet is and strain energy of an open

graphene edge. A decrease in the nucleus diameter for fixed

number of carbon atoms. Probably the nanotubes do not

reach thermal equilibrium in rapidly expanding and cooling

plume of carbon cluster [27].

Fig. 10: Experimental SEM image Histogarm of carbon

nanotubes distribution at 850ºC ; 900ºC and 1000 ºC [14 ]

7. Conclusions The production of the carbon nanotubes by using laser

ablation is considered as an effective method for production

of SWCNTs. In order to get best results in the simulation of

carbon nanotube distribution is to take into account all types

of clusters formed and the growth time of the nanotubes

within these clusters.




Page 47

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Simulation of Carbon Nanotubes Growth by Laser Ablation

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