Furnace geometry effects on plume dynamics in laser ablation for nanotube synthesis

Mathematics and Computers in Simulation 65 (2004) 365–383

Furnace geometry effects on plume dynamicsin laser ablation for nanotube synthesis

Diomar Cesar Lobãoa,∗, Alex Povitskyb

a CERCA, Centre de recherche en calcul appliqué (CERCA), 5600 boul. Décarie, bureau 400,Montréal, Que., Canada H3X 2H9

b Department of Mechanical Engineering, University of Akron, Akron, OH, USA

Abstract

Laser ablation (LA) has become a popular method for production of carbon nanotubes where formation of gaseouscarbon plume and liberation of catalyst particles are caused by the laser pulse. The plume dynamics in laser ablationis somewhat similar to point explosion where a large amount of energy is liberated in a small volume. The aim ofthis study is to explore thermal dynamics of catalyst particles that is crucial for the formation of carbon nanotubes.The challenge of this case is to describe numerically the strong shock wave propagation and reflections in theworking space of laser furnace and its effect on plume dynamics and temperature regime of catalyst particles. Theproposed model includes a multi-species formulation for concentration of chemical components combined with thecompressible Euler equations. An axisymmetric unsteady computational gas dynamic model of plume expansioninto ambient gas has been developed. In the present work, the system of governing PDEs is solved numerically usingthe relaxing TVD scheme in generalized curvilinear coordinates. To obtain the thermal behavior of catalyst particlesthe Eulerian solver has been combined with Lagrangian tracking of catalyst particles. The developed software hasbeen implemented and tested using the set of test cases described in the paper. Reflected shock waves interact withthe plume and catalyst particles, heat them, and cause temperature oscillations. The contradictory requirements ofsustained high temperature of catalyst particles and minimum of temperature oscillations are discussed using theseries of representative working space shapes.© 2003 Published by Elsevier B.V. on behalf of IMACS.

Keywords:Laser ablation; Euler equations; Shock wave-plume interaction; Explosion wave; Laser furnace; Relaxing TVDscheme

1. Introduction

The known processes of carbon nanotubes production include laser ablation (LA), chemical vapordeposition (CVD), and decomposition of high-pressure carbon oxide (HiPco). All these processes are

∗ Corresponding author.E-mail address:[email protected] (D.C. Lobão).

0378-4754/$30.00 © 2003 Published by Elsevier B.V. on behalf of IMACS.doi:10.1016/j.matcom.2004.01.008

366 D.C. Lobão, A. Povitsky / Mathematics and Computers in Simulation 65 (2004) 365–383

controlled by metal catalyst particles that initiate synthesis of carbon nanotubes from feed-stock gas orejected plume. In the laser ablation process, the solid carbon substrate is gasified by the energy of laserbeam and emerges into the ambient gas of laser furnace. In the laser vaporization process, the feed-stockplume loaded with catalyst particles expands explosively into the background gas. The thermal behavior ofcatalyst particles is critical for nanotubes synthesis as it has been recently shown for the HiPco process[1].

The distinctive feature of laser ablation is that the process is based on wave propagation and reflectionin the working space of the laser furnace. Ambient gas pressure varies in computational and physicalexperiments from near-vacuum to atmospheric pressure[2,3]. Since the gas pressure in the plume exceeds100 atm, shock waves are formed in the ambient gas in the case of the explosive laser ablation. The highpressure initial conditions are similar to that of explosions. In turn, the propagation of incident and reflectedshock waves affects the plume behind it. The plume mixing is caused by the instability at the plume togas interface while the reflected shock wave interacts with the plume[4]. Reflected shock waves interactwith the plume and catalyst particles, heat them, and, on the other hand, cause temperature oscillations.The contradictory requirements of sustained high temperature of catalyst particles and minimum oftemperature oscillations are discussed using the series of representative working space shapes.

While the plume development is a relatively long-range phenomena (of order of 1 ms), the laser ablationtakes 10–20 ns. In the current study, the results of laser ablation are taken as boundary conditions forplume propagation. The carbon injection velocity is kept on during the plume injection process. Afterthe end of plume injection period, the injection velocity is turned off and replaced by either the chamberfreestream conditions or adiabatic target conditions. The numerical data (temperature of ablated gaseouscarbon, its pressure, and injection speed) are taken from experiments[5]. Nanotubes grow in the plumeand the furnace shape does not affect the laser ablation so the experimental parameters of injected plumecan be utilized. Similar approach was adopted by Greendyke et al.[6]. In another study by Lobao andPovitsky[1]the influence of injection velocity of ablated gas on the plume dynamics was investigated forbroader prospective.

Similar to the HiPco modeling[4], the LA CFD model is based on the combined Eulerian solverand Lagrangian postprocessing approach applied to the tracking of the trajectories of catalyst particles.However, the explosive-type flow in laser ablation is completely different from steady turbulent jets inHiPco process.

The Lagrangian approach to track temperature of the catalyst particles in the plume is implemented toevaluate the efficiency of the process of synthesis of nanotubes. Catalyst particles follow the streaklinesof the unsteady flow and their temperature is equal to the local gas temperature. Using the temperatureregime of catalyst particles, chemistry of formation of nanotubes growing on catalyst particles shouldbe considered separately of the thermo-fluid problem as it was done in the study[1] for HiPco process.The concentration of catalyst particles and synthesized nanotubes is small and does not affect the flow;therefore, catalyst particles are not included in the multi-species Euler equations.

In the present study, the implemented numerical model has to be accurate and stable in a broad range ofgas pressure ranking from very low ambient pressure (≈10-2 atm) to the very high pressure (≈100 atm).For the assessment of the proposed mathematical model and numerical method, numerical results arecompared with numerical and analytical solutions available for shock wave propagation caused by initialdiscontinuities.

Unfortunately, no experimental data are available about temperature of individual catalyst particles.On the other hand, analytical results relevant to the plume dynamics of laser ablation are limited to thepoint explosion and to spherical expansion into vacuum[7].

D.C. Lobão, A. Povitsky / Mathematics and Computers in Simulation 65 (2004) 365–383 367

The goal of this paper is to obtain the temperature of catalyst particles as a function of time for asingle plume emerging as a result of laser ablation. As it has been shown for simple cylindrical shapein our previous study[8], the temperature and trajectories of catalyst particles are greatly affected bythe working pressure and the periodicity of plume emerging. In this paper, we are focused on the effectof geometry of the working space on the temperature and trajectories of catalyst particles. This paper isaiming at to enhance heating of catalyst particles or avoid thermal oscillation of particles by control ofreflected shock waves by means of design of curvilinear walls of laser furnace. Similar ideas are usedfor design of supersonic combustion engines, nozzles, and intakes in aeronautics[9]. The contradictoryrequirements of sustained high temperature of catalyst particles and minimum of temperature oscillationsare discussed in the current paper using the series of representative working space shapes. Working spaceof different shapes can be manufactured relatively easy and experimental laser ablation tests are feasible.

This paper is composed as follows. In Section 2, the mathematical model of plume dynamics is intro-duced. In Section 3, the relaxing TVD numerical scheme is discussed, implemented, and tested for thebenchmark 1D and 2D cases. In Section 4, results of numerical simulation for different furnace geometriesand carbon target boundary conditions are discussed.

2. Governing equations

The model is based on compressible Euler equations written in general curvilinear coordinatesξ, η

for unsteady flow. The short duration of investigated phenomena allows to neglect viscous effects. Theeffective viscous length-scale is defined aslvisc = √

ντ, whereν is effective kinematic viscosity andτis the time length of modeled phenomena. This method to evaluate the influence of viscous forces hasbeen used by Quirk and Karni[10] and by Picone and Boris[11] for shock wave interaction with densityinhomogeneities. Under computational conditions of ambient gas Argon (Section 3.3), the kinematicviscosity of Argon at 1500 K is 2.04 × 10−4 cm2/s [12,13]. For 200 ms of modeled time period, thelength-scale is 0.2 mm. This is one order of magnitude smaller than the initial plume radius and two ordersof magnitude smaller than the radius of chamber. The plume occupies most of the furnace cross-sectionsoon after its ablation. Thus, the viscous effects are negligible for the time interval simulated and thespatial scale of the plume. If the turbulence is fully developed in the surrounding gas, the effectiveviscosity would be much larger. However, the flow of surrounding gas is slow and corresponds to laminarRenumber in the available experiments.

Diffusion of chemical species and momentum transfer by viscosity are related by Pr number. Sinceviscous effects are negligible (see above), the diffusion effects are assumed to be minor compare toconvective mixing.

Although cylindrical geometry of the experimental set-up is used in the available experiments, gener-alized coordinates are needed in the current study, to model how shock waves reflected by curvilinearfurnace walls interact with the plume. For generality, all variables are non–dimensionalized as follows:

ρ = ρ

ρ∞, p = p

ρ∞ c2∞, E = E

ρ∞ c2∞, h = hρ∞

p∞, t = c∞ t

lr, u = u

c∞,

v = v

c∞, c = c

c∞(1)


The freestream conditions are denoted by the subscript∞ andlr denotes the reference length which istaken as the maximum geometric length dimension.

Substituting the relations given byEq. (1)to theEuler equationsyields∂Q

∂t+ ∂F

∂ξ+ ∂G

∂η= S (2)

where the conserved variable vector and fluxes in non-dimensional coordinates are given by:

Q = J

ρ

ρu

ρv

E

ρC1

ρC2

, F = J

ρU

ρuU + ξxp

ρvU + ξyp

(E + p)U

ρC1U

ρC2U

, G = J

ρV

ρuV+ ηxp

ρvV + ηyp

(E + p)V

ρC1V

ρC2V

S = J

0

0

p

0

0

0

(3)

Hereρ is the density,�m = (ρU, ρV, ρC1, ρC2)T is the vector of mass fluxes,E is the total energy per

unit volume,p is pressure, andC1, C2 are mass fractions of two species. More detailed chemistry ofplume dynamics including recombination of ionized atoms will be considered in our future research. Theimplemented numerical method allows for easy account of several species in such a way that amount ofcomputational work is proportional to number of species. Such a refinement will help to determine theplume dynamics more accurately.

U = ξxu+ ξyv

V = ηxu+ ηyv

HereU,V are the contra-variant velocities[14] in directions normal to constantξ andη surfaces, respec-tively, J = 1/[xξyη − xηyξ] is the Jacobian of coordinate transformation. The quantitiesξx andηx are themetrics of the transformation(x, y) → (ξ, η):

ξx = Jyη, ξy = −Jxη

ηx = −Jyξ, ηy = Jxξ

In the numerical scheme, the metrics are approximated by second-order finite differences. Second-orderthree-point backward finite difference are used for the boundary points. The total energy is defined by:

E = −p+ ρ (u2 + v2)

2+ ρh (4)

whereh = h(hi) is the enthalpy per unit mass for each species. The equation of state for multi-speciesflow can be written as:

p =2∑i=1

pi =2∑i=1

ρCiRiT = ρ

(2∑i=1

CiRi

)T = ρRT, (5)

whereT is the temperature andR is defined as,R = ∑2i=1CiRi. In the present model calorically perfect

gas is adopted.


3. Numerical scheme

To discretize the Euler equations (Eq. (2)), the second-order upwind scheme for conservation laws(MUSCL) is used in the current study. Relaxation methods make use of the characteristic variables of thesystem and finite speed of propagation and do not need Riemann solvers (seeSection 3.1for details). Therelaxing TVD scheme is similar to central schemes described in[15,16]. In 2D case, the system(2) canbe split using Strang technique[17] into two separate 1D equations which are solved in sequential way.The relaxing TVD scheme offers an easier implementation than conventional TVD since there is no needfor calculating the flux Jacobian eigenvectors matrices for Euler equations, which makes the numericalalgorithm economical and efficient for multi-species and/or reactive chemical systems.

3.1. Relaxing TVD scheme

First a 1D system of conservation laws is described using the relaxing TVD scheme. Its generalizationto multi-dimensional cases is straightforward.

∂q

∂t+ ∂F(q)

∂x= 0 (6)

This system is replaced by the relaxing system as:

∂q

∂t+ ∂cv

∂x= 0,

∂v

∂t+ ∂cq

∂x= 0 (7)

wherec(x, t) is a free parameter calledfreezing speed[18]. This system is also called relaxation systemwith v = F(q)/c and it is used to calculate fluxes, while an explicit time integration is implemented usingsecond-order Runge–Kutta method. In[18] is shown that this scheme is TVD under the constraint thatc

is greater than the characteristic speedu given by∂F/∂q. One way to satisfy this condition is to definecas:c = |u| + a, wherea is the local speed of sound. To solve the relaxed system, it is decoupled usingvariablesw1 = (q + v)/2 andw2 = (q − v)/2. The linear system of equations obtained with respect tow1 andw2 is discretized in space using MUSCL scheme as described in[18]. The conserved averagedquantitiesw are defined at integer grid cellsxi. Is need to define the fluxes at the cell boundaries,F = cw,whereF = F(w1)−F(w2) atxi+1/2. Then,∂xcw = F(xi+1/2)−F(xi−1/2). The remaining step is to definethe fluxF at the half cells. The first order upwind definition is simplyF(xi+1/2) = cw(xi), assumingthat the flow is to the right. There are two second-order choices: (1)(cw(xi) + cw(xi+1/2))/2 and (2)(3/2)cw(xi−(1/2)cw(xi−1)). Generalizing these choices is possible by writingF(xi+1/2) = cw(xi)+%w,where

%w+ = cw(xi+1)− cw(xi)

2, %w− = cw(xi)− cw(xi−1)

2(8)

Define the limiter minmod as minmod(a, b) = (sign(a)+ sign(b))Min(|a|, |b|)/2. The%w = minmod(%w+,%w−) limiter is the simplest TVD MUSCL choice. At the extrema,minmod= 0, and thesecond-order scheme reverts to a first order upwind scheme. The mathematical justification for thiscan be found in[19]. The time step is determined by satisfying the CFL condition,(cmax%t)/(%x) ≤ 1.Note that a new freezing speed is computed for each partial time step in the Runge–Kutta scheme. Thecmax has to be larger than max(cti) and max(ct+%t/2i ).


Tube Length

Vel

,Den

s,P

res

0 0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Vel Rtvd

Analytical

Den Rtvd

Analytical

Pres Rtvd

Analytical

Tube Length

Vel

,Den

s,P

res

0.25 0.5 0.75 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Vel YHtvdAnalyticalDen YHtvdAnalyticalPres YHtvdAnalytical

(a) (b)

Fig. 1. 1D Sod shock tube test: (a) relaxing TVD and (b) Yee–Harten TVD.

3.2. Shock tube test

As a validation of numerical discretization, Sod’s shock tube 1D test case was carried out usingthe relaxing TVD and the upwind TVD[20,21] type scheme. The numerical results and the analyticalsolutions are shown inFig. 1a and b, respectively. The solution consists of a left rare-fraction wave, acontact and right shock wave. The initial conditions for(ρ, u, p) are(1.0, 0, 2.5)T if 0 ≤ x < 0.5 and(0.125, 0, 0.25)T if 0.5 ≤ x < 1.0. The solution consists of a left-moving rare-fraction wave, a contactand right-moving shock wave. In numerical simulations, CFL is equal to 0.75, number of numerical gridpoints is 500 and the results are shown att = 0.1644.

As can be seen inFig. 1a and bthe shock and contact discontinuity are captured in their expectedpositions by both schemes. The shock is captured over two numerical grid cells for the upwind TVDscheme and over one cell for the relaxing TVD scheme. For the upwind TVD solution, the discontinuityis smeared as compared to the relaxing TVD solution. These results demonstrate the better resolutionquality of the relaxing TVD scheme.

3.3. 2D shock wave about bump

For more than one spatial dimension Strang[17] splitting sweep technique is applied on the above twoequations given by system(7). The sweep has to be symmetrical, first the forward sweep is carried out as,qt+%t = LξLηq

t and them the reverse sweep is carried out as,qt+2%t = LηLξqt+%t. The operatorsLξ(·)

andLη(·) are defined as:Lξ(·) = (1−(F(·)i+1/2,j−F(·)i−1/2,j)/(%ξ)) and similarly forη direction. Notethat the same time step%t has to be used in both sweeps in order to keep the second-order accuracy intime. Using dimensional splitting, it is a straightforward task to implement the code in vector processingenvironment.

A validation test for blast wave 2D problem using the Relaxing TVD scheme is carried out. The initialconditions are prescribed in a 2D rectangular domain [3.8 × 1.5] to simulate the movement of a blastwave passing over a bump given byyb = 1/12.0(1.0 − cos(2.0πxi)). The initial conditions are set up(3.45,2.18,0,11.12)T for (ρ, u, v, p), respectively. The 300× 100 numerical mesh is used and CFL is


Length

Hig

h

-1 0 10

0.5

1

1.5

2

2.5

Length

Hig

h

-1 0 10

0.5

1

1.5

2

2.5

(a) (b)

Fig. 2. Pressure isolines for the 2D blast wave test: (a) 300 time steps and (b) 550 time steps.

taken equal to 0.50. In Fig. 2a and bthe pressure contours are depicted using 15 uniformly distributedlevels after 300 and 550 time steps, respectively. Computational results show the blast wave movingdownstream as at passing over the bump, when the refraction wave is already visible moving upstream,and after it has passed over the bump.

3.4. Point explosion

The Sedov–Taylor 2D blast shock wave problem was used as a test case of apoint explosionwherea large amount of energy is liberated in a small volume. Detailed discussion on this test case can befound in [22]. Since the surrounding pressure is negligible small, the shock wave is very strong. Theexplosion-type flow in this case is similar to the plume dynamics caused by the laser pulse beam. Thechallenge of this case is to describe the shock wave propagation in the gas emerging from the point wherethe energy is released.

Initially, air with constant densityρ = 1, very small pressure of 10−20, andγ = 1.4 fills the 2D domain.The pulse is introduced by high pressurep = 1 in the vicinity of origin (two numerical grid nodes in bothspatial directions). Therefore, certain amount of energyE0 is released at the center of the domain at timet = 0. The numerical value of Eo is not needed as an input datum since the energy recovers automaticallyby Eq. (4). The 100× 100 numerical grid is chosen and a CFL number of 0.25 is taken. The propagationof the wave front in spherical coordinates is described by a power function of time[23] as

R = ζ

(E

ρ1

)1/5

t2/5 (9)

whereζ is a non-dimensional parameter dependent on the energy conservation condition and forγ = 1.4give ζ = 1.033, as in[23] and ρ1 is the gas density in the domain, In cylindrical coordinates, theSedov–Taylor 2D blast wave front coordinate is given by[24]:

R =[

4(γ − 1)(γ + 1)2

π(3γ − 1)

]1/4(E

ρ1

)1/4

t1/2 (10)


X coord

Y c

oo

rd

1.5 2 2.5 3

1.25

1.5

1.75

2

2.25

2.5

2.75

3

3.68E-033.43E-033.19E-032.94E-032.70E-032.45E-032.21E-031.96E-031.72E-031.47E-031.23E-039.81E-047.35E-044.90E-042.45E-04

Time

R D

ista

nce

1 20.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

ST-2/5ST-1/2RelaxingTVD

(a) (b)

Fig. 3. Pressure isolines for the 2D Sedov–Taylor point blast wave at t = 2.53: (a) pressure contours and (b) shock coordinate,respectively.

Numerical simulations were conducted in 2D planar domain using generalized coordinates. Pressurecontours are shown in Fig. 3a at t = 2.53 where the outgoing shock wave is clearly seen. Numericalsimulation show that R as a function of time agrees well with the 2/5 power law of S-T analyticalexpression (9) (see Fig. 3b).

3.5. Lagrangian approach to model dynamics of catalyst particles

In the current study, Lagrangian approach is used to post-processing only. The 10 uniformly-distributedinitial positions for the streaklinesare chosen at the laser spot. Since catalyst particles are of sub-micronsize, they follow the streaklines and do not have their own inertia. A Runge–Kutta second-order explicitscheme is used to integrate in time the equations for streaklines. The Lagrangian approach has an advantageover the Euler approach since it gives the temperature of each catalyst particle as a function of time.Streaklines were selected in such a way that their starting points are uniformly distributed across theablated target. In some cases, all streaklines show similar behavior while in other cases there are groupsof streamlines with different behavior (see Fig. 10a).

For example, in Sedov–Taylor explosion in infinite space all streamline behave the same way andsingle streamline is sufficient to describe the thermal regime of particles. The idea of classification ofstreaklines to groups and related optimization of reactors has been developed by Povitsky and Salas forHiPco process [4]. Definition of necessary and sufficient number of streamlines is an important issue thatwill be addressed in our future research. Having the temperature of catalyst particles as a function of time,it will be possible to model kinetics of carbon nanotubes formation as it was done for HiPco process [1].

3.6. Computational model setup

Radiation of a laser beam is focused on Carbon C3 solid target. The diameter of the focusing spot is5 mm, while the target is placed inside a chamber containing Argon at pressure of 1 atm and temperature


of 1500 K [6,5]. The duration of plume emerging is 20 ns. Under these conditions, the laser delivers100 atm pressure at 5000 K for the gaseous plume. Initial conditions at the grid points correspondingto the 5 mm spot was kept on during the plume emerging (20.0 ns). At the end of this time, the plumefraction boundary conditions are turned off and replaced by the Argon flowfield. The outflow is setas adjustable sub/supersonic boundary. The solid boundary is treated imposing the tangential velocitysatisfying �n · �V = 0, �n is the normal surface unit vector. On the axisymmetrical boundary, simpleextrapolation boundary condition is adopted.

The numerical code solves the discretized Euler equations in time. In simulations, a sequence of CFLnumbers was used in order to integrate the equations with the maximum time step and to avoid the blowup of the solution due to the large pressure gradients, which may cause non-linear instability. For thefirst 180 steps CFL of 0.0001 is used; therefore, the time step is of the order of 10−11 s. The series ofincreasing CFL numbers is used 0.001, 0.01, 0.1, 0.3 and 0.5 with an increase of CFL after each 180steps. The maximum CFL used is equal to 0.5. The time step%t is computed at the beginning of each stepand is given by the maximum flowfield characteristic velocity λξ,η. The CFL number and the time stepare spatially uniform at each time moment. Since the time step is uniform in space, the variable-in-timeCFL number is applicable to time-dependent flows.

Carbon injection velocity is equal to the rate of carbon mass ablation as it is injected into the flowfield bythe laser firing process [6]. To find carbon injection velocity, one should model melting and vaporizationprocesses, that is out of scope of this paper. The features of temperature dynamics of catalyst particlesas a function of injection velocity are discussed in previous work [8]. Gaseous carbon is released with avelocity rate that is non-dimensionalized by the freestream sound speed.

4. Numerical results

In most of experiments available in literature [5,6], the chamber pressure is close to standard atmosphericpressure. The numerical simulations carried out shown here are for proposed generic furnace geometriesincluding the typical cylinder geometry, the bulb geometry, target at the middle of the bulb geometry (i.e.,semi-bulb geometry), and the constricted cylinder.

4.1. Cylinder geometry

This case corresponds to experimental conditions described in Section 3.3. The axisymmetrical cylinderchamber has diameter d = 5.0 cm and the length of 25 cm. The physical domain is discretized in acomputational domain using a uniform grid, having 200 × 50 nodes in the axial and radial directions,respectively (see Fig. 4a).

Gaseous carbon is released with a velocity rate that is non-dimensionalized by the freestream soundspeed. The carbon injection velocity is kept on during the plume injection process. After the end of plumeinjection period, the injection velocity is turned off and replaced by the chamber freestream conditions(see Section 3.6).

For injection velocity equal to 15c∞, the carbon C3 plume concentration is depicted in Fig. 5a, thetemperature along streaklines is shown in Fig. 5b, and the peak temperature profile as a function of timeis shown in Fig. 5c. To show that temperature oscillations of catalyst particles are caused by crossing theshock waves, the pressure along the streaklines is shown in Fig. 5d. To show the influence of stagnation


Symmetrical axis

Rad

ial a

xis

0 0.1 0.20

0.1

0.2

Symmetrical axis

Rad

ial a

xis

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0.3

Symmetrical axis

Rad

ial a

xis

0.2 0.3 0.40 0.5

0

0.1

0.2

Symmetrical axis

Rad

ial a

xis

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

(c) (d)

(b)(a)

Fig. 4. The 200 × 50 numerical grid for four representative furnace geometry: (a) cylinder; (b) bulb; (c) semi-bulb; and (d)constricted cylinder.

points in the flowfield on trajectories of particles and on temperature oscillations the trajectories of catalystparticles are shown in Fig. 5e at t = 243 �s.

To explain the complex vortical motion of the particles observed in Fig. 5e, the flowfield is shown at thetime moment when the boundary slip-line is formed (see Fig. 5f). This dynamically adjusted recirculationzone drives the particles back toward the laser target and forms the mushroom-type shape of the carbonplume.

4.2. Bulb cylinder geometry

The axisymmetrical bulb cylinder chamber has diameter of d = 5.0 cm and the length of 25 cm. Thebulb geometry is described by the following equation:


Symmetrical axis

Rad

ial a

xis

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4 0.690.640.600.550.510.460.410.370.320.280.230.180.140.090.05

Symmetrical axis

Tem

per

atu

re,T

/300

K

0 0.0001 0.0002

17

18

19

20

21

22

23

24

25

26

Time, microsecond

Tem

per

atu

re, T

/300

K

0 0.0001 0.0002

20

25

30

35

40

Time, microsecond

Pre

ssu

re

0.00E+00 1.00E-04 2.00E-04

10

20

30

40

50

60

70

Symmetrical axis

Rad

ial a

xis

0 0.1 0.2

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Symmetrical axis

Rad

ial a

xis

0.15 0.2 0.25 0.3 0.35

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.880.820.770.710.650.590.530.470.410.350.290.240.180.120.06

(a) (b)

(d)(c)

(e) (f)

Fig. 5. Cylindrical furnace, snapshot at t = 243 �s: (a) carbon mass concentration; (b) temperature along streaklines; (c) peaktemperature as a function of time; (d) pressure along streaklines; (e) trajectories of catalyst particles (streaklines); and (f) velocityvector field at t = 60 �s.


ywall = d

2.0+ α(1.0 − cos(2.0πxi)) (11)

where α = 1.2 in the current paper, positive α accounts for the bulb expansion while negative α accountsfor constriction. For α = 0 the cylinder geometry is recovered. The physical domain is discretized in acomputational domain using a grid with 200 × 50 nodes in the axial and radial directions, respectively(see Fig. 4b). An exponential stretching function is used to concentrate lines close to the symmetricalaxis and is defined as:

%yj = εd/2.0

((1.0 + ε)ny − 1)(1.0 + ε)j, (12)

where ny is the total number of grid points in the y direction and j is the number of grid line. For thepresent grid ε = 0.01.

The numerical simulations are conducted for the same injection velocity as for the basic cylinder ge-ometry 15×c∞. In Fig. 6a–c the carbon C3 mass concentration is depicted for t = 30.0, 60.0 and 161 �s,respectively. In Fig. 6d and e the pressure contours and pressure streaklines are depicted for t = 30.0and 161 �s, respectively. In Fig. 6f, the streaklines of catalyst particles are shown at t = 161 �s. Thecarbon C3 plume evolves to the mushroom-type shape and the bulb wall reflects a strong shock waveat 40 �s (its formation can be seen in Fig. 6d). Two others weaker reflected shocks hit the plume at50.0 and 60.0 �s respectively as can be seen in Fig. 6e and f. At later time (≈100.0 �s), the tempera-ture of catalyst particles drops being affected by other reflected shocks of minor intensity as shown inFig. 6f.

4.3. Semi-bulb geometry

The bulb geometry is the same as that described in the previous section; however, only half a bulbis included as part of the furnace. The carbon target is located at the middle of the bulb section (seeFig. 4c). In Fig. 7a–c the carbon C3 mass concentration is shown for t = 30.0, 60.0 and 96 �s,respectively. In Fig. 7d and e the pressure contours at t = 30.0 �s and pressure along streaklinesare shown for t ≤ 96 �s. Fig. 7f depicts the trajectories of catalyst particles (streaklines)at t = 96 �s.

The temperature along streaklines is shown in Fig. 8a for t ≤ 96.0 �s. Pressure contours at later timemoments are depicted in Fig. 8b and cfor t = 60.0 and 96.0 �s, respectively. The carbon C3 plume issqueezed by the upstream pressure that is built up by the shock wave (see Fig. 7d). The carbon C3 plumeis affected by the upstream pressure which restrain its propagation further from the median bulb section.The streaklines in Fig. 7f confirm that the catalyst particles are confined close to the centerline of thebulb section. This furnace geometry reflects shock waves in a way that the temperature profiles becomesmoother than those in the previous cases (see Fig. 8a).

4.4. Constricted cylinder geometry

The axisymmetrical constricted cylinder chamber has diameter of d = 5.0 cm and the length of 25 cm.The constricted geometry is described by Eq. (11) with α = −1.2. The physical domain is discretized in acomputational domain using a grid, having 200 × 50 nodes in the axial and radial directions, respectively(see Fig. 4d). For the present grid ε = 0.02.


Symmetrical axis

Rad

ial a

xis

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0.3

0.970.910.840.780.710.650.580.520.450.390.320.260.190.130.06

Symmetrical axis

Rad

ial a

xis

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4

0.5

0.830.780.720.670.610.550.500.440.390.330.280.220.170.110.06

Symmetrical axis

Rad

ial a

xis

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0.3

0.40.740.690.640.590.540.490.440.390.340.300.250.200.150.100.05

Symmetrical axis

Rad

ial a

xis

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0.3

28.6026.7024.8122.9221.0319.1417.2515.3613.4711.58

9.697.795.904.012.12

Time, microsecond

Pre

ssu

re

0 2.5E-05 5E-05 7.5E-05

10

20

30

40

50

60

70

Time,microsecond

Tem

per

atu

re, T

/300

K

0.00E+00 5.00E-05 1.00E-04 1.50E-0416

17

18

19

20

21

22

23

24

(a)

(c)

(e) (f)

(b)

(d)

Fig. 6. Bulb furnace: (a–c) carbon C3 mass concentration at t = 30.0, 60.0, and 161.0 �s, respectively (d) pressure isolines att = 30.0 �s; (e) pressure along the streaklines for t ≤ 161.0 �s; and (f) streaklines at t = 161.0 �s.


Symmetrical axis

Rad

ial a

xis

0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.950.890.820.760.700.630.570.510.440.380.320.250.190.130.06

Symmetrical axis

Rad

ial a

xis

0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.86670.80900.75120.69340.63560.57780.52000.46230.40450.34670.28890.23110.17330.11560.0578

Symmetrical axis

Rad

ial a

xis

0.2 0.3 0.40

0.1

0.2

0.640.600.550.510.470.430.380.340.300.260.210.170.130.090.04

Symmetrical axis

Rad

ial a

xis

0.20 0.3 0.40 0.050

0.1

0.2

0.3

10.219.558.898.227.566.896.235.574.904.243.572.912.251.580.92

Time, microsecond

Pre

ssu

re

0 2.5E-05 5E-05 7.5E-05

10

20

30

40

50

60

70

Symmetrical axis

Rad

ial a

xis

0.2 0.25 0.3 0.350.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

0.055

0.06

(a)

(c) (d)

(f)(e)

(b)

Fig. 7. Semi-bulb furnace: (a–c) carbon C3 mass concentration at t = 30.0, 60.0, and 96.0 �s, respectively; (d) pressure isolinesat t = 30.0 �s; (e) pressure along the streaklines; and (f) streaklines at t = 96.0 �s.


Time, microsecond

Tem

per

atu

re, T

/300

K

0 2.5E-05 5E-05 7.5E-0516

17

18

19

20

21

22

23

24

25

26

27

Symmetrical axis

Rad

ial a

xis

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7 4.23983.98673.73373.48063.22752.97452.72142.46832.21531.96221.70921.45611.20300.95000.6969

Symmetrical axis

Rad

ial a

xis

0.2 0.4 0.6 0.8 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

5.955.575.194.814.434.063.683.302.922.542.161.791.411.030.65

(a) (b)

(c)

Fig. 8. Semi-bulb furnace: (a) temperature along streaklines at t = 96.0 �s; (b) and (c) pressure isolines at t = 60.0 and 96.0 �s,respectively.

In Fig. 9a–c the carbon C3 mass concentration is depicted for t = 30.0, 90.0 and 162 �s, respectively.In Fig. 9d and e the pressure contours and pressure along the streaklines are depicted for t = 30.0 and90 �s, respectively. Fig. 9f depict the trajectories streaklines at t = 162 �s.

The temperature streaklines are depicted in Fig. 10a for t = 162.0 �s, pressure contours are depictedin Fig. 10b and c for t = 90.0 and 162.0 �s, respectively. The throat section of this geometry and thenarrow region in the convergent section cause the carbon C3 plume to be compressed into this region.The plume expands and eventually it occupies the entire convergent section. Part of it flows through thethroat section and start to expand into the divergent section after 30.0 �s. The first strong reflected shockwave hits the plume at 15.0 �s, and two reflected shock waves of small intensity hit the plume at 20.0and 30 �s. The temperature along streaklines confirm the presence of these reflected shock waves as thetemperature rise distinctly at those time intervals (see Fig. 10a).


Symmetrical axis

Symmetrical axis

Symmetrical axis

Symmetrical axis

Rad

ial a

xis

Rad

ial a

xis

Rad

ial a

xis

Rad

ial a

xis

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.930.870.810.750.680.620.560.500.430.370.310.250.190.120.06

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

0.4

0.820.770.710.660.600.550.490.440.380.330.270.220.160.110.05

0 0.1 0.2 0.3 0.4

0

0.1

0.2

0.3

0.4

0.690.650.600.550.510.460.420.370.320.280.230.180.140.090.05

0 0.1 0.2 0.3 0.40

0.1

0.2

0.3

8.257.727.196.666.135.615.084.554.023.492.962.431.901.370.85

Time, microsecond

Pre

ssu

re

0 5E-05 0.0001 0.00015

10

20

30

40

50

60

70

X coord

Y c

oo

rd

0.05 0.1 0.15 0.2

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

(e) (f)

(d)(c)

(a) (b)

Fig. 9. Constricted cylinder furnace: (a–c) carbon C3 mass concentration at t = 30.0, 90.0, and 162.0 �s, respectively; (d)pressure isolines at t = 30.0 �s; (e) pressure along the streaklines for t ≤ 162.0 �s; and (f) streaklines at t = 162.0 �s.


Time, microsecond

Tem

per

atu

re, T

/300

K

0.00E+00 5.00E-05 1.00E-04 1.50E-04

10

15

20

25

30

Symmetrical axis

Rad

ial a

xis

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

6.135.735.344.954.554.163.773.372.982.592.191.801.411.010.62

Symmetrical axis

Rad

ial a

xis

0 0.25 0.5 0.75 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

4.624.334.043.753.463.162.872.582.292.001.711.421.130.840.55

(a) (b)

(c)

Fig. 10. Constricted cylinder surface: (a) temperature along streaklines at t = 162.0 �s; (b) and (c) pressure isolines at t = 90.0and 162.0 �s, respectively.

4.5. Influence of the graphite target boundary

The effect of the inlet cross-section on the plume dynamics is discussed in this section. It is shownthat the type of boundary condition applied on the graphite target plays an important role in the plumedynamics. In experiments, the graphite target may be pressed into the solid disk of larger diameter;therefore, solid wall boundary condition has to be applied after the plume emergence has been completed.The following results show the influence of the chosen type of boundary conditions as related to the plumedynamics behavior. For the solid wall boundary condition, the solid wall is assumed to occupy 50% ofthe chamber cross-section. For the rest of the entrance cross-section, the outflow boundary is assumedwith free-stream conditions for Argon. For the remaining solid target, after the end of plasma injection


Symmetrical axis

Rad

ial a

xis

0 0.1 0.2 0.3 0.4 0.50

0.1

0.2

0.3

0.4 0.720.670.620.570.530.480.430.380.340.290.240.190.140.100.05

Time, microsecond

Tem

per

atu

re, T

/300

K

0.00E+00 1.00E-04 2.00E-04

12

13

14

15

16

17

18

19

20

(a) (b)

Fig. 11. Cylindrical furnace, solid target boundary: (a) concentration of carbon plume at t = 243.0 �s; and (b) temperature alongstreaklines.

the adiabatic condition is used, dE/dn = 0, where n is the direction perpendicular to the surface. Forthe removed target, the free-stream boundary conditions are used for all variables after the end of plumeemergence. The effect of the set-up of solid wall boundary conditions at the entrance is two-fold. First, theplume velocity propagation appears to be 43.5% larger than that for ambient conditions at the entrancesee Fig. 11a and Fig. 5a for comparison. Second, the temperature peak reveal a decrease of 26.0% of itsmaximum peak value (compare Figs. 11b and 5b).

5. Summary and conclusions

The presented mathematical model of plume dynamics emerging from laser firing in various furnacegeometry is based on inviscid multi-species equations of gas dynamics. Relaxing TVD scheme appearsto have important numerical features suitable for modeling of plume dynamics in laser ablation. TheLagrangian approach reveals important features of heating of catalyst particles due to shock interactionwith the plume that are not explicitly observed in frame of the Eulerian approach. The proposed math-ematical model gives a physical insight into the plume dynamics and dynamics of individual particlesin laser ablation as the furnace geometry varies. The simulation software has been designed and testedthrough a series of test cases and simulation results were compared with available analytical or numericalsolutions.

The reasons for non-monotonic behavior of temperature of catalyst particles are interaction of par-ticles with reflected shock waves from the furnace wall and circular motion of the particles caused byformation of vortical zones, stagnation points, and slip-lines. The position of the front of the plumedepends on several factors including the geometry of the furnace wall and the carbon target boundarycondition. The location of the carbon target in the middle section of the bulb (semi-bulb geometry)produces relatively smooth temperature profile of catalyst particles required for synthesis of carbonnanotubes.


Acknowledgements

Authors were supported by the NSERC grant of the second author. The authors wish to thank theCentre de recherche en calcul appliqué—CERCA (Montreal) for the logistic and computational facilitiesavailable during this research. The first author thanks Dr. Ian Stewart, University of Bristol, UK, for hiscomments. The second author wishes to acknowledge the University of Akron start-up funds that helphim to work on the final part of this study.

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Furnace geometry effects on plume dynamics in laser ablation for nanotube synthesis