modelling of libs plasma expansion

Ž .Spectrochimica Acta Part B 56 2001 567�586

Modelling of LIBS plasma expansion�

G. Colonna�, A. Casavola, M. CapitelliCentro Studi per la Chimica dei Plasmi, Cnr and Dipartimento di Chimica, Uni�ersita degli Studi di Bari, V. Orabona 4,`

70126 Bari, Italy

Received 12 October 2000; accepted 20 April 2001

Abstract

A one-dimensional time-dependent fluid dynamic model has been developed to describe the expansion of theplume produced by laser ablation. The model includes chemical reactions considered in local thermodynamicequilibrium to describe the expansion of a TiO plasma. The results are discussed in connection with LIBS plasmas.� 2001 Elsevier Science B.V. All rights reserved.

Keywords: LIBS; Modelling; Equilibrium; Thermodynamic; Fluid dynamic

1. Introduction

Laser ablation is a powerful technique to de-posit good quality thin films. The characteristicsof the deposited material depend on the expan-sion of the plume and the theoretical study of thedeposition should consider the properties of themass flow from the target to the substrate.

To understand the characteristics of a thin film

� This paper was presented at the 1st International Congresson Laser Induced Plasma Spectroscopy and Applications, Pisa,Italy, October 2000, and is published in the Special Issue ofSpectrochimica Acta Part B, dedicated to that conference.

� Corresponding author. Tel.: �39-080-544-3563; fax: �39-080-544-2024.

Ž .E-mail address: [email protected] G. Colonna .

deposited by laser ablation, a theoretical modelhas been developed. To obtain quantitative re-sults the model should describe the target evap-oration, the Knudsen-layer formation, the plumeexpansion, and the absorption of the ablated ma-

� �terial 1 . This approach is too complicated involv-ing poorly understood concepts. So a partial modelhas been developed, which only deals with plumeexpansion. The mechanisms of the laser�surfaceinteraction are not included in the model, and theevaporated material is formed either instanta-neously or continuously with a given density andtemperature. Various expansion conditions havebeen studied, including different temperatures ofthe newly formed material at various buffer pres-sures. Moreover, the role of the production rateof matter released from the target and the loss

0584-8547�01�$ - see front matter � 2001 Elsevier Science B.V. All rights reserved.Ž .PII: S 0 5 8 4 - 8 5 4 7 0 1 0 0 2 3 0 - 0

( )G. Colonna et al. � Spectrochimica Acta Part B: Atomic Spectroscopy 56 2001 567�586568

Žrate of molecules adsorbed onto the substrate incomplete absorption or complete reflection ap-

.proximation have also been investigated.� �In previous works 2,3 , evaporated material is

considered chemically inert and therefore, thecomposition of the plume is considered constant.This approximation is not correct because pres-sure and temperature in the plume can reach very

Ž .high values P�10 atm, T�10 000 K and inthese conditions the chemical reactions are veryfast. The similarity with the high enthalpy nozzle

� �expansion 4 leads to the conclusion that strongnon-equilibrium can be present, mainly in theexpansion zone. For materials commonly used inlaser ablation for thin film deposition, there areno data available for the chemical kinetics in gasphase and therefore, a complete kinetic modelcannot be considered. For this reason, we have

Žresorted to the local equilibrium model infinite.rate coefficients which requires only thermody-

namic data. This model allows calculation of theflux of each molecular species to the substrate.

In this work the plume expansion has beenstudied via fluid dynamic equations along theplume axis in the one-dimensional approximation,based on the observation that the angular disper-sion of the plume is small. To account for this

Ždispersion, a slightly different model quasi one-.dimensional can be used, which approximates

� �the plume to nozzle expansion flow 4 .

2. TiO thermodynamics

To calculate gas expansion in the equilibriumapproximation it is necessary to calculate thethermodynamic quantities, the equilibrium con-stants and therefore, the equilibrium compositionof the considered mixture. We have used dataŽ .partition functions calculated directly from thespectroscopic constants, for molecules, and from

� �energy levels for atoms 5 . Thermodynamic datafor atomic and molecular oxygen have been taken

� �from 5 . The relationship between all the ther-Žmodynamic quantities including the equilibrium

. � �constants can be found in 6,7 . In this way it ispossible to calculate the equilibrium gas composi-tion for various temperature and pressure values.

From gas composition, we can calculate the meanmolecular mass and internal energy per unit massof the considered mixture, which must be insertedin the equations describing equilibrium flow.

This approach has been applied to calculatethe thermodynamics of the plume formed when a

Ž . � �titanium monoxide TiO target is ablated 8 . Thespecies we have considered in the TiO mixtureincluded TiO, Ti, Ti�, Ti2�, O , O�, O, O�, O�

2 2and O2�, linked by the following reactions:

TiO�Ti�O

Ti�Ti��e�

Ti��Ti2��e�

O �2O2

O �e��O�2 2

O�e��O�

O�O��e�

O��O2��e�

The chemical reactions can be represented in ageneral way:

n nR P

Ž .a R � a P 1Ý Ýi i j ji�1 j�1

where R and a represent the reactants and theiri icoefficients, whilst P and a represent thej jproducts and their coefficients. The terms n andRn represent the number of reactants andPproducts in any reaction. The equilibrium con-stant at the temperature T can be calculated as:

nPajZŁ j

j�1 � a �E k Trea BŽ . Ž .k� k T e � 2n BRaiZŁ i

i�1

where Z is the partition function of each species,E is the reaction energy, �a is the differencerea

( )G. Colonna et al. � Spectrochimica Acta Part B: Atomic Spectroscopy 56 2001 567�586 569

between the coefficients of the products and thereactants, and k represents the Boltzmann con-Bstant.

Finally, the equilibrium composition of themixture has been calculated at different tempera-tures and pressures. As an example, the molarfractions of the relevant species in the mixture, ata pressure of 1 atm, have been reported in Fig. 1as a function of temperature. At low tempera-tures, the mixture only contains TiO, which startsdissociating at T�5000 K. For the same temper-

Ž .atures T�5000 K at atmospheric pressure, oxy-gen is completely dissociated and therefore, be-comes a minority species. At higher temperaturesŽ .T�8000 K , first and second ionization processesbegin, and neutral species disappear.

Thermodynamic properties of each species andaverage quantities have been calculated using themethods of statistical thermodynamics and chemi-

� �cal equilibrium 6,7 . When the composition ofthe gas mixture has been determined, one cancalculate the mean molecular mass and the inter-nal energy per unit mass, which can be enteredinto the equations of equilibrium flows. As anexample in Fig. 2, the internal energy was re-ported as a function of temperature at differentpressures. The abrupt change in the trend of

internal energy vs. temperature is due to theonset of dissociation and ionization processes.

Ž .These processes vary at low temperature withincreasing pressure according to the laws ofchemical equilibrium.

These structures, which are exalted in the trendof total specific heats, disappear in a frozen situa-

Žtion, i.e. the gas is considered non-reacting ideal.gas approximation .

Chemical compositions and thermodynamicproperties of the investigated system have beenreproduced with suitable analytical functions inthe temperature and pressure range of interest

Žfor the fluid dynamic system under study see.Appendix A . The use of analytical fits acceler-

ates the calculation of plume flow between thetarget and substrate, avoiding calculations at eachpoint of composition and thermodynamic proper-ties.

3. Fluid dynamic model

To calculate the plume dynamics, one-dimen-sional, time-dependent Euler equations have beenconsidered which consist of mass, momentum andenergy continuity equations:

Fig. 1. Relevant species molar fraction of the TiO mixture as a function of temperature at pressure of 1 atm.


Fig. 2. Internal energy profiles at various pressures.

�� u Ž .� �0 3�t �x

Ž 2 .��u � �u �P Ž .� �0 4�t �x

�� u� �u Ž .� �P �Q 5�t �x �x

complemented with the ideal gas state equation

� Ž .P� k T 6BM

where �, u and P respectively denote mass den-sity, flow speed and pressure, and M is the meanmolecular mass of the gas particles. The total

Ž .energy per mass unit � is expressed by theequation:

Ž .k T3 U T ,PB Ž .�� 72 M M

Ž .where U T ,P represents the internal energy thataccounts for the internal degrees of freedom andchemical reaction energies.

Therefore the contribution of translational en-ergy can be calculated by the relation:

2 � Ž .� Ž .k T� M��U T ,P . 8B 3

As a first approximation, evaporated materialcan be considered chemically inert and therefore,the composition of the plume can be considered

� �constant 2,3 . The characteristic times of chemi-cal reactions are longer than flowtimes so that thesystem is not able to follow temperature andpressure variations.

For an ideal gas without internal structure,Ž .pressure and total energy � are simply related

by the equation:

2 Ž .P� �� 93

For the equilibrium TiO mixture, it is assumedthat the system at each point between the targetand the substrate, is in total chemical equilib-rium. Keeping in mind that the total energy is the

Žsum of the translational energy proportional to.the gas temperature and internal energy U, we

Ž .can write Eq. 9 in the following form:

2 2 Ž . Ž .P� �� U T ,P 103 3

where U represents the internal energy per massunit.

Ž . Ž .The system of Eqs. 3 � 6 allows one to calcu-late the time evolution of the profiles of macros-


Žcopic quantities the density � , the flow speed u,the pressure P, the thermodynamic energy � per

.mass unit, and temperature T in the expansionŽ .region. The term Q in Eq. 5 represents the heat

Žflux in the gas volume due, for example, to the. Ž .laser�gas interaction . Introducing Eq. 10 in the

energy continuity equation and considering onlyŽ .the expansion phase i.e. Q�0 , we get:

�� u� �u2 2 Ž .� � �� U �0. 11Ž .3 3�t �x �x

The solution of continuity equations has beenobtained by converting them in a system of linearalgebraic equations by means of a discretization

Ž .process see Appendix B .Various ablation and absorption models can be

described by changing the boundary conditions.As an example, the reflection condition can beobtained by imposition of zero speed and nullderivatives of density and velocity on the subs-trate. On the other hand, the total absorptioncase can be simulated imposing null secondderivatives of all quantities. Two models havebeen considered for describing the ablationprocess. The first one considers an instantaneousevaporation of the target; as an initial condition,we assume a gas confined in a small region char-

acterized by high temperature and pressure. TheŽ .second model continuous evaporation considers

a constant flux of matter and energy from thetarget.

To test the performances of the numericalmodel, we have reported in Fig. 3 the trend ofdensity profile at different times for free expan-sion of a hard sphere model with total reflectionon the walls. The formation of a shock wave canbe taken as an indication of the stability of thefluid dynamic code.

This plot shows that initially the gas particlesare confined in a region with x�L�0.4, where Lis the target�substrate distance. As the time in-creases, matter moves towards the substrate untilit rebounds on it. Subsequently, the reflectedparticles collide with the particles, which mustarrive to the substrate and this forms a shockwave. The rapid increase of density at the frontalexpansion causes strong numerical oscillations,which have been reduced using a variable spatialgrid; large spatial steps in the initial region andsmall spatial steps at frontal expansion whereoscillations are present, were considered.

By introducing the thermodynamic equilibriumto each point between the target and substrate, aproblem arises in the calculation of pressure. This

ŽFig. 3. Density profiles at different times as a function of the reduced co-ordinate L is the distance between the target and the. 6substrate under the following initial conditions: T �10 000 K; and P �10 Pa.0 0


quantity, in fact, depends on internal energy,which in turn depends on temperature and pres-sure. Therefore, an iterative method has beenused. It consists of calculating new values ofpressure and temperature from their old values:

2 Ž . Ž .P �K ��U T ,P � 1�K Pnew old old old3

Ž .12

Ž .P M T ,Pnew old new Ž . Ž .T �K � 1�K T 13new old�R

where K is a variable parameter, which permitsto give a different weight to the old and newsolutions, depending on the convergence of themethod. Subsequently, new values of temperatureand pressure become the input to calculate inter-nal energy and mean mass. This method contin-ues until values of temperature and pressure cal-culated at various steps, remain unchanged.

4. Results and discussion

The laser ablation process has been studied, asŽ .already pointed out, by considering two cases: a

the free flow case, where the composition is keptŽ .frozen; and b the equilibrium case, where equi-

librium conditions in each point between the tar-get and the substrate have been assumed. In bothcircumstances, density and mach number profileshave been analyzed. It is worth noting that theequilibrium case can be considered as a strongapproximation in regions with low density, wherethe species have a very high velocity. On theother hand, in the high-density zone, there are alot of collisions between the particles, which ver-ify the hypothesis of chemical equilibrium.

It should also be noted that we do not considerthe formation of solid-state TiO at low gas tem-peratures, i.e. the solid�gas equilibrium is con-sidered frozen.

In our treatment, two models have been devel-oped. The first one considers the instantaneousevaporation of the target, while the second con-siders a constant flux of matter and energy from

Ž .the target continuous evaporation . In both mod-els a total adsorption of the plume on the subs-trate has been supposed.

In Fig. 4, density profiles at various times areshown for both free and equilibrium flows. Thefirst case consists of matter confined in a regionnear the target. As time increases, the plume

�1 Ž . Ž .Fig. 4. Density profiles at different times where P �10 Pa and T �300 K: a free flow; and b equilibrium flow.b b


expands from the target to the substrate until allthe matter is adsorbed by the substrate. Inspec-tion of Fig. 4 shows that the introduction ofequilibrium conditions to each point between thetarget and the substrate determines conversion oftranslational energy. In fact, chemical reactions inthe recombination regimes produce an enormousquantity of energy, which reduces the rate oftemperature decrease and increase of internalenergy in the system. Therefore, gaseousmolecules acquire translational energy and movemore rapidly towards the substrate. This behaviorcan be explained by more rapid shift of the frontalexpansion, when chemical equilibrium is intro-duced.

Fig. 5 shows the mach number profiles at vari-ous times for the same conditions as that in Fig.4. Similar to the behavior of density, one canappreciate the more rapid increase of mach num-ber when equilibrium conditions are introduced,due to the acceleration of frontal expansion. As aconsequence of the acceleration of the plume,there is an increase of density near the substrateand therefore, an increase in flow speed.

ŽThe effect of varying initial conditions pres-.sure P , and temperature T , on the target has0 0

been investigated and the results reported in Fig.

6. A strong discontinuity between target and sub-strate temperature exists, which determines theexpansion of the plume until the system attainsuniform temperature. On the other hand, a verypronounced variation of temperature can producenumerical problems, which is evident in the for-mation of a peak at the expansion front. With an

Ž .initial higher pressure curves a , the system ap-pears to move more rapidly towards the substrate.In fact, the region near the target shows a morerapid temperature decrease, while at the expan-sion front one can note the development of apeak as a result of the discontinuity, only in the

Ž 6case of high initial pressure on the target �10.Pa .The effect of varying the pressure of the bufferŽ .gas P has been also studied and the resultsb

reported in Figs. 7 and 8. We are particularlyinterested in describing the system with a buffergas pressure of 10 Pa, which is the experimental

� �pressure used to deposit a thin film of TiO 8 . Inparticular, Fig. 7 reports the temperature profileswhile Fig. 8 reports the corresponding mach num-bers. We note that the increase in buffer gaspressure reduces the speed of plume, thus, mak-ing the temperature decrease more rapidly at thevalue of buffer gas temperature.

Ž �1 . Ž . Ž .Fig. 5. Mach number profiles at different times P �10 Pa and T �300 K : a free flow; and b equilibrium flow.b b


Ž �1 . Ž . 6 Ž . 4Fig. 6. Temperature profiles at different times P �10 Pa, T �300 K and T �8000 K : a P �10 Pa; and b P �10 Pa.b b 0 0 0

In Fig. 9, the trend of density vs. target�sub-strate distance has been shown in the case ofcontinuous evaporation. Matter is continuouslyproduced from the target; this causes an incre-ment in the density profile at different times,although matter is absorbed onto the substrate.

The effect of varying released time t , duringRthe laser ablation process has been investigated in

Ž .the case of continuous evaporation see Fig. 10 .During the expansion of the plume, the matterproduces peaks due to the collisions between theplume and gas buffer. By comparing the behavior

Ž . Ž . �1 Ž . Ž .Fig. 7. Temperature profiles at different times T �300 K : a P �10 Pa; b P �10 Pa; and c P �100 Pa.b b b b


Ž . Ž . �1 Ž . Ž .Fig. 8. Mach number profiles at different times T �300 K : a P �10 Pa; b P �10 Pa; and c P �100 Pa.b b b b

of density profile for two different released timeswe can see that for t �5�10�7 s, the matterRseems to decelerate compared to when t �5�R10�5 s.

Let us consider now the effect of a differentŽ .density of released matter z . In Figs. 11 and 12

we report the behavior of density and mach num-ber profiles at different times, since there is

Fig. 9. Density profile at different times: P �10�1 Pa; T �300 K; P �106 Pa; T �10 000 K; t �5�10�5 s; and z�10 000b b 0 0 Rkg�m2�s.


Ž �1 6 2 . Ž .Fig. 10. Density profiles at different times P �10 Pa, T �300 K, P �10 Pa, T �10 000 K, and z�1000 kg�m �s : ab b 0 0�5 Ž . �7t �5�10 s; and b t �5�10 s.R R

greater evaporation from the target with greatergas density and flow speed. In fact, for a lowdensity of matter, the frontal expansion of theplume is significantly reduced because of colli-sions with molecules of the buffer gas. In Fig. 11we can see that at time t�6�10�6 s, frontal

expansion has reached the substrate only in thecase of high density.

Finally, Figs. 13 and 14 report the mass flow onthe substrate for different z values in the case ofcontinuous evaporation. In both figures we reportthe quantity of different species which arrive on

Ž �1 6 �5 .Fig. 11. Density profiles at different times P �10 Pa, T �300 K, P �10 Pa, T �10 000 K, and t �5�10 s : z equalsb b 0 0 RŽ . Ž . 2a 1000; and b 100 kg�m �s, respectively.


Ž �1 6 �5 .Fig. 12. Mach number profiles at different times P �10 Pa, T �300 K, P �10 Pa, T �10 000 K, and t �5�10 s : zb b 0 0 RŽ . Ž . 2equals a 1000; and b 100 kg�m �s.

the substrate as a function of time. In this way itis possible to have an approximate idea of filmcomposition. The quantity of TiO is rather low,which is essentially due to the buffer gas. In fact,the low pressure near the substrate gives rise tothe presence of Ti� ions and atoms of Ti. Experi-mentally, it is hoped not to have an excessively

high-energy laser, because this condition pro-duces a prevalence of ionization and dissociationprocesses, depositing films with an irregular struc-ture. For example, to deposit titanium oxides withan excimer laser, the laser fluence must be ap-proximately 1 J�cm2.

A preliminary comparison with estimated ex-

Fig. 13. Trend of mass flux vs. time: P �10�1 Pa; T �300 K; P �106 Pa; T �10 000 K; t �5�10�5 s; and z�100 kg�m2�s.b b 0 0 R


Fig. 14. Trend of mass flux vs. time: P �10�1 Pa; T �300 K; P �106 Pa; T �10 000 K; t �5�10�5 s; and z�1000b b 0 0 Rkg�m2�s.

perimental results is reported in Fig. 15. Experi-mental and theoretical concentrations of Ti andTi� species at a distance of 0.6 mm from thetarget as a function of time show satisfactoryagreement. This comparison must be consideredpreliminary, taking into account the various hy-pothesis of the model and the qualitative natureof the experimental results.

5. Conclusions

In this work, a theoretical model describing theplume expansion for both free and equilibriumconditions after laser interaction with a solid tar-get has been discussed. Different parameters en-tering in the model have been exploited. The

� Ž . Ž �6 2 . Ž .Fig. 15. Ti and Ti concentration vs. time: comparison between a experimental data P �10 torr, E �6 J�cm ; and bb laserŽ �1 6 �5 2 .theoretical results P �10 Pa, T �300 K, P �10 Pa, T �10 000 K, t �5�10 s and z�1000 kg�m �s .b b 0 0 R


model is able to give the time-dependent profilesof mass density, temperature and mach numberalong the distance between the target and thesubstrate. Future improvements of this modelshould eliminate many assumptions contained inthe model such as one-dimensional flow, totalabsorption on the substrate and equilibrium con-ditions. In doing so, dedicated experiments mustbe considered for a correct validation of themodel.

Acknowledgements

The present paper has been partially supportedby MURST under project 9802276194-004, ‘Inter-azione di plasma con laser al nano e picosecondo’.

Appendix A

Here we report the analytical expressions ofthermodynamic quantities and gas compositionused in the temperature range 50�20 000 K. Theseexpressions are represented mainly as the combi-nation of two base functions: the Gaussian:

Ž . ��Ž x�c.�� 2 Ž .� x ;c,� �e 14

and the sigmoid or asymmetric Gaussian,

eŽ x�c.��

Ž . Ž .� x ;c,� � . 15Ž x�c.�� Ž x�c.��e �e

So, we have represented different quantities vs.temperature as the linear combination of twofunctions, where each function is multiplied by aparameter a . In each expression, we have foundj

Ž .the best parameters a, c and � so that the erroris minimized.

A modified function �, was introduced, only, inthe fit of the specific heat:

x�cŽ . Ž .� x ;c,� � 16�x x

�� .c c

Subsequently, the analytical expression for eachparameter vs. pressure was found. Introducingthe variable y, to indicate the parameters a, cand �, the analytical form is:

2 b5 z Ž .y�b �b z�b z �b e �b � z ;b ,b1 2 3 4 6 7 8

Ž .17

where z� log P.The mean molecular mass equation is:

3Ž . Ž .M T � a � T ;c ,� �dÝ j j j

j�1

3Ž . Ž .� a � T �d 18Ý j j

j�1

The values of different terms for each of theparameters in the calculation of mean molecularmass have been reported in Table 1.

Table 1Values of different terms for each of the parameters in the calculation of mean molecular weight

b b b b b b b b1 2 3 4 5 6 7 8

d 6.40e�2 0.00 0.00 0.00 0.00 0.00 1.00 1.00a �3.84e�2 �5.46e�5 1.55e�4 0.00 0.00 0.00 1.00 1.001c 3.43e�3 1.55e�2 59.45 0.00 0.00 0.00 1.00 1.001D 4.24e�2 9.81 19.57 0.00 0.00 0.00 1.00 1.001a �4.11e�3 0.00 0.00 0.00 0.00 �3.69e�3 3.79 1.572c 4.36e�3 93.07 1.22e�2 0.00 0.00 0.00 1.00 1.002D 5.32e�2 �1.60e�2 71.65 0.00 0.00 0.00 1.00 1.002a �8.89e�3 9.42e�06 0.00 8.04e�08 1.41 0.00 1.00 1.003c 8.30e�3 3.44e�2 2.11e�2 0.00 0.00 0.00 1.00 1.003D 6.93e�2 0.00 0.00 2.63e�2 4.29e�1 0.00 1.00 1.003


Table 2Values of different terms for each of the parameters in the calculation of internal energy

b b b b b b b b1 2 3 4 5 6 7 8

d �6.17e�6 0.00 0.00 0.00 0.00 0.00 1.00 1.00a 2.38e7 2.03e5 3.7e4 0.00 0.00 0.00 1.00 1.001c 3.81e3 1.45e2 8.71e1 0.00 0.00 0.00 1.00 1.001D 7.00e2 6.04e1 5.69e1 0.00 0.00 0.00 1.00 1.001a 4.47e7 2.22e6 0.00 0.00 0.00 0.00 1.00 1.002c 8.03e3 7.73e2 1.58e2 0.00 0.00 0.00 1.00 1.002D 9.84e2 2.21e1 8.99e1 0.00 0.00 0.00 1.00 1.002

The internal energy has been fitted by theequation:

2Ž . Ž .U T � a � T ;c ,� �dÝ j j j

j�1

2Ž . Ž .� a � T �d. 19Ý j j

j�1

The values of different terms for each of the

parameters in the calculation of internal energyhave been reported in Table 2.

The molar fractions have been calculated bymeans of the equation:

3Ž . Ž .T � a � T ;c ,� �dÝ j j j

j�1

3Ž . Ž .� a � T �d. 20Ý j j

j�1

Table 3Values of different terms for each of the parameters in the calculation of TiO molar fraction

b b b b b b b b1 2 3 4 5 6 7 8

d 1.00 �1.24e�4 7.67e�5 0.00 0.00 0.00 1.00 1.00a �1.04 5.10e�3 0.00 0.00 0.00 0.00 1.00 1.001c 3.39e3 1.47e2 6.03e1 0.00 0.00 0.00 1.00 1.001D 4.07e2 �9.55e�1 2.13e1 0.00 0.00 0.00 1.00 1.001a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003

Table 4Values of different terms for each of the parameters in the calculation of Ti molar fraction

b b b b b b b b1 2 3 4 5 6 7 8

d 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00a 0.50 0.00 0.00 0.00 0.00 0.00 1.00 1.001c 3.36e3 1.60e2 5.40e1 0.00 0.00 0.00 1.00 1.001D 4.24e2 8.12 2.32e1 0.00 0.00 0.00 1.00 1.001a �0.50 0.00 0.00 1.53e�5 1.01 0.00 1.00 1.002c 4.22e3 4.38e1 1.44e2 0.00 0.00 0.00 1.00 1.002D 7.39e2 2.31e2 9.42e1 0.00 0.00 0.00 1.00 1.002a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003


Table 5�Values of different terms for each of the parameters in the calculation of Ti molar fraction

b b b b b b b b1 2 3 4 5 6 7 8

d 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00a 0.33 1.83e�3 0.00 2.72e�4 0.76 0.00 1.00 1.001c 4.15e3 1.42e1 1.35e2 0.00 0.00 0.00 1.00 1.001D 6.25e2 9.61e1 6.21e1 0.00 0.00 0.00 1.00 1.001a �0.34 0.00 0.00 2.04�4 0.99 0.00 1.00 1.002c 8.50e3 1.94e2 2.57e2 0.00 0.00 0.00 1.00 1.002D 8.03e2 1.59e2 4.29e1 0.00 0.00 0.00 1.00 1.002a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003

The values of various terms for each of theparameters to calculate TiO, Ti, Ti�, Ti2�, O,O�, O�, O2�, O , O� and e� molar fractions2 2have been reported respectively, in Tables 3�13.

The specific heat has been fitted by the equa-tion:

4 10Ž . Ž . Ž .c T � a � T ;c ,� � a � T ;c ,� �dÝ ÝP j j j j j j

j�1 j�5

4 10Ž . Ž .� a � T � a � T �dÝ Ýj j j j

j�1 j�5

Ž .21

The values of various terms for each of theparameters in the calculation of specific heathave been reported in Table 14.

Appendix B

To solve the numerical problem, continuityequations in the general expressions can be ex-pressed in the form:

Ž .�S � uS�F Ž .� �Q�0. 22�t �x

The finite volume method, which consists of theevaluation of the volume integral of the different

Ž .terms in Eq. 22 , has been used. It produces theexpression:

� Ž . Ž .S dV� uS�F � Q dV�0 23H H�t �V �Vj j

where �V is the jth integrated volume and Fjrepresents matter flow. Using S as the meanj

Table 62�Values of different terms for each of the parameters in the calculation of Ti molar fraction

b b b b b b b b1 2 3 4 5 6 7 8

d 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00a 0.20 0.00 0.00 0.00 0.00 0.00 1.00 1.001c 8.62e3 9.85e1 2.90e2 0.00 0.00 0.00 1.00 1.001D 7.81e2 1.36e2 5.59e1 0.00 0.00 0.00 1.00 1.001a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003


Table 7Values of different terms for each of the parameters in the calculation of the O molar fraction

b b b b b b b b1 2 3 4 5 6 7 8

d 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00a 0.53 �7.77e�3 8.23e�4 0.00 0.00 0.00 1.00 1.001c 3.38e3 1.67e2 5.48e1 0.00 0.00 0.00 1.00 1.001D 3.98e2 2.27e1 1.61e1 0.00 0.00 0.00 1.00 1.001a �0.21 0.00 0.00 1.98e�2 0.31 0.00 1.00 1.002c 4.31e3 �1.38e2 1.67e2 0.00 0.00 0.00 1.00 1.002D 4.10e2 5.11e1 2.02e1 0.00 0.00 0.00 1.00 1.002a �0.34 �1.02e�3 0.00 0.00 0.00 0.00 1.00 1.003c 8.35e3 2.32e2 2.27e2 0.00 0.00 0.00 1.00 1.003D 7.32e2 0.00 0.00 1.60e2 0.51 0.00 1.00 1.003

value of S in the jth interval, S and Sj�1�2 j�1�2as the values of S at the extremes of volume, thedifferential equation can be discretized as:

� ŽS �V � u S �u S �Fj j j�1�2 j�1�2 j�1�2 j�1�2 j�1�2�t

. Ž .�F ��Q �V �0 24j�1�2 j j

where � is the surface in which the flux matter isdifferent from zero. Dividing all terms by the

Ž .volume �V , Eq. 24 becomes:j

Table 8�Values of different terms for each of the parameters in the calculation of the O molar fraction

b b b b b b b b1 2 3 4 5 6 7 8

d 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00a 1.99e�1 �7.48e�5 0.00 �4.07e�6 1.59 0.00 1.00 1.001c 8.02e3 5.18e2 1.73e2 0.00 0.00 0.00 1.00 1.001D 7.06e2 1.44e2 2.98e1 0.00 0.00 0.00 1.00 1.001a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003

Table 9�Values of different terms for each of the parameters in the calculation of the O molar fraction

b b b b b b b b1 2 3 4 5 6 7 8

d 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00a 2.15e�6 �2.31e�6 0.00 3.87e�7 1.30 0.00 1.00 1.001c 3.56e3 1.70e2 6.27e1 0.00 0.00 0.00 1.00 1.001D 3.70e2 0.95 1.88e1 0.00 0.00 0.00 1.00 1.001a �2.40e�6 2.65e�6 0.00 �4.12e�7 1.23 0.00 1.00 1.002c 4.33e3 1.21e2 1.26e2 0.00 0.00 0.00 1.00 1.002D 8.10e2 �4.49e1 5.60e1 0.00 0.00 0.00 1.00 1.002a �1.09e�7 1.15e�7 0.00 �4.17e�8 1.46 0.00 1.00 1.003c 5.53e3 3.34e2 1.80e2 0.00 0.00 0.00 1.00 1.003D 1.62e3 2.23e2 3.92e1 0.00 0.00 0.00 1.00 1.003


Table 102�Values of different terms for each the parameters in the calculation of the O molar fraction

b b b b b b b b1 2 3 4 5 6 7 8

d 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.001c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.001D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.001a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.002a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003

u S �u Sj�1�2 j�1�2 j�1�2 j�1�2

�F �F� j�1�2 j�1�2S � �Q �0.j j�t � x j

Ž .25

The time derivatives have been calculated inthe backward finite differences, which leads tothe implicit Euler integration method. Using theindex i for time, the discretized expression for Sis:

S �S�S i , j i�1, j Ž .� . 26ž /�t � ti , j

The spatial derivatives have been approximated� �by finite differences in the upwind scheme 9 .

The integration in time was improved using a step� �adapting criterion as described in 10 :

u Si�1�2, j�1�2 i , j�1�2

�u S�uS i�1�2, j�1�2 i , j�1�2 Ž .� . 27ž /�x � xi , j j

The values of various quantities at the point jafter time i have been calculated from their valuein the point j at the time i�1. In the upwindscheme, if the flow speed is positive the quantity

Ž .S depends on the value of S in i, j�1�2 .i , j

Conversely, if the flow speed is negative the quan-Ž .tity S depends on the value of S in i, j�1�2 .i , j

But the problem is time-dependent, so the flowspeed calculated at the time i�1�2 has beenintroduced as an average of flow speed at times iand i�1. The flow speed u and the term F havebeen calculated as spatial and temporal means,respectively.

As a consequence of the discretization process,

Table 11Values of different terms for each of the parameters in the calculation of the O molar fraction2

b b b b b b b b1 2 3 4 5 6 7 8

d 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00a �5.43e�5 1.11e�4 0.00 9.21e�8 1.67 0.00 1.00 1.001c 3.29e3 1.03e2 6.57e1 0.00 0.00 0.00 1.00 1.001D 5.93e2 7.10e1 1.34e1 0.00 0.00 0.00 1.00 1.001a 5.13e�5 �5.68e�5 0.00 �1.16e�7 1.63 0.00 1.00 1.002c 3.55e3 2.23e2 5.83e1 0.00 0.00 0.00 1.00 1.002D 2.55e2 0.00 0.00 9.20e1 0.42 0.00 1.00 1.002a 7.77e�6 �6.40e�5 2.58e�6 0.00 0.00 0.00 1.00 1.003c 3.53e3 �1.08e2 1.18e2 0.00 0.00 0.00 1.00 1.003D 6.69e2 0.00 0.00 5.73e1 0.64 0.00 1.00 1.003


Table 12�Values of different terms for each of the parameters in the calculation of the O molar fraction2

b b b b b b b b1 2 3 4 5 6 7 8

d 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00a �2.9e�10 0.00 0.00 3.6e�13 2.47 0.00 1.00 1.001c 3.71e3 3.89e1 7.16e1 0.00 0.00 0.00 1.00 1.001D 4.54e2 �5.71e1 2.40e1 0.00 0.00 0.00 1.00 1.001a 2.4e�10 �1.5e�10 0.00 �7.5e�15 2.79 0.00 1.00 1.002c 1.83e3 0.00 0.00 1.62e3 0.25 0.00 1.00 1.002D 5.21e2 1.59e1 5.19e1 0.00 0.00 0.00 1.00 1.002a �1.23e�9 7.2e�10 0.00 �4.6e�13 2.40 0.00 1.00 1.003c 3.56e3 1.79e2 7.98e1 0.00 0.00 0.00 1.00 1.003D 4.47e2 �5.63e1 3.52e1 0.00 0.00 0.00 1.00 1.003

the continuity equations have been converted bya system of algebraic non-linear equations.

� tS �S �� u Si , j i�1, j i�1�2, j�1�2 i , j�1�2� x

� t� u Si�1�2, j�1�2 i , j�1�2� x

� t Ž .� F �F .i�1�2, j�1�2 i�1�2, j�1�2� xŽ .28

Ž .Eq. 11 , in particular, has been modified in theform:

� t� � �� ui , j i , j i�1, j i�1, j i�1�2, j�1�2� x

�� i , j�1�2 i , j�1�2

� t� u � �i�1�2, j�1�2 i , j�1�2 i , j�1�2� x

2 �u� � t � �i , j i , jž /3 �x i , jŽ .29

2 �u� � t � U .i , j i , jž /3 �x i , j

If the spatial derivative of the flow speed u is2 �u

positive, the term � t � � must bei, j i, jž /3 �x i, jadded to the principal diagonal; if the spatialderivative of the flow speed u is negative, the

2 �uterm � t � � must be subtracted toi, j i, jž /3 �x i, jthe known term. Through this artifice, the con-tributions to the principal diagonal are alwayspositive.

After the discretization, the problem consists of

Table 13�Values of different terms for each of the parameters in the calculation of the e molar fraction

b b b b b b b b1 2 3 4 5 6 7 8

d 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00a 0.33 0.00 0.00 �5.56e�4 0.74 0.00 1.00 1.001c 4.11e3 3.92e1 1.26e2 0.00 0.00 0.00 1.00 1.001D 5.88e2 �7.98e1 5.41e1 0.00 0.00 0.00 1.00 1.001a 0.27 2.31e�3 5.54e�5 0.00 0.00 0.00 1.00 1.002c 8.23e3 4.06e2 2.01e2 0.00 0.00 0.00 1.00 1.002D 6.66e2 0.00 0.00 2.25e2 0.47 0.00 1.00 1.002a 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003c 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003D 1.00 0.00 0.00 0.00 0.00 0.00 1.00 1.003


Table 14Values of different terms for each of the parameters in the calculation of specific heat

b b b b b b b b1 2 3 4 5 6 7 8

d 324 0.00 0.00 0.00 0.00 0.00 1.00 1.00a 8.23e1 0.00 0.00 0.00 0.00 0.00 1.00 1.001c 5.01e2 0.00 0.00 0.00 0.00 0.00 1.00 1.001D 1.59 0.00 0.00 0.00 0.00 0.00 1.00 1.001a 1.99e1 0.00 0.00 0.00 0.00 0.00 1.00 1.002c 3.07e2 0.00 0.00 0.00 0.00 0.00 1.00 1.002D 3.12 0.00 0.00 0.00 0.00 0.00 1.00 1.002a 7.06e2 8.33 0.00 �1.74e1 1.06 0.00 1.00 1.003c 6.08e3 7.93e2 6.12e1 0.00 0.00 0.00 1.00 1.003D 5.94 �0.48 �1.83e�2 0.00 0.00 0.00 1.00 1.003a 6.31e2 �2.75e1 0.00 3.15e2 1.06 0.00 1.00 1.004c 1.42e4 1.32e3 4.89 0.00 0.00 0.00 1.00 1.004D 6.23 �0.63 �6.63e�2 0.00 0.00 0.00 1.00 1.004a 9.63 0.00 0.00 0.00 0.00 0.00 1.00 1.005c 1.82e3 0.00 0.00 0.00 0.00 0.00 1.00 1.005D 1.06e3 0.00 0.00 0.00 0.00 0.00 1.00 1.005a 5.40e3 �1.02e3 0.89 0.00 0.00 0.00 1.00 1.006c 5.89e3 7.94e2 6.35e1 0.00 0.00 0.00 1.00 1.006D 9.87e2 2.58e2 2.66e1 0.00 0.00 0.00 1.00 1.006a 2.35e3 �1.49e3 9.93e1 0.00 0.00 0.00 1.00 1.007c 1.49e4 2.23e3 1.77e2 0.00 0.00 0.00 1.00 1.007D 2.36e3 4.94e2 3.91e1 0.00 0.00 0.00 1.00 1.007a 1.77e3 �8.59e2 1.58e2 0.00 0.00 0.00 1.00 1.008c 6.90e3 1.15e3 1.06e2 0.00 0.00 0.00 1.00 1.008D 2.35e3 6.49e2 0.00 0.00 0.00 0.00 1.00 1.008a 1.70e3 2.19e2 9.96e1 0.00 0.00 0.00 1.00 1.009c 1.61e4 1.89e3 5.85e1 0.00 0.00 0.00 1.00 1.009D 4.44e3 1.52e3 1.99e2 0.00 0.00 0.00 1.00 1.009a �9.88e1 4.18e1 0.70 0.00 0.00 0.00 1.00 1.0010c 3.07e3 1.03e2 �4.42 0.00 0.00 0.00 1.00 1.0010D 6.97e2 2.00e2 0.00 0.00 0.00 0.00 1.00 1.0010

solving the system Ax�q, where A is the coef-ficients matrix, x is the unknown vector and q thevector of known terms. The matrix is a tri-diago-nal matrix and so it is possible to use an oppor-

� �tune algorithm to solve the system 11 . Thismodel consists of the iterative calculation of den-sity, flow speed and energy, until these quantitiescontinue to be constant.

Acknowledgements

The present study has been partially supportedby MURST under project 9802276194-004, ‘Inter-azione di plasma con laser al nano e picosecondo’.

References

� �1 R.K. Singh, J. Narayan, Pulsed-laser evaporation tech-nique for deposition of thin films: physics and theoreti-

Ž .cal model, Phys Rev. B 41 1990 8843�8859.� �2 R. Kelly, Gas dynamics of the pulsed emission of a

perfect gas with applications to laser sputtering and toŽ .nozzle expansion, Phys. Rev. A 46 1992 860�874.

� �3 R. Kelly, A. Miotello, Pulsed laser sputtering of atomsand molecules. Part I: basics solutions for gas dynamic

Ž .effects, Appl. Phys B 57 1993 145�158.� �4 G. Colonna, M. Tuttafesta, M. Capitelli, D. Giordano,

NO Formation in one-dimensional nozzle air flow withstate-to-state non- equilibrium vibrational kinetics,

Ž .JTHT 13 1999 372�375.� � Ž .5 a K.G. Herzberg, Molecular Spectra and Molecular

Structure-I. Spectra of Diatomic Molecules, D. VanNostrand Company Inc, Princeton, USA, 1950.


Ž .b C.E. Moore, Atomic Energy Levels, vols. I�III,United States Department of Commerce, NationalBureau of Standards, 1949, Circular 467.

� � Ž .6 a D. Giordano, M. Capitelli, G. Colonna, C. Gorse,Survey of methods of calculating high temperature ther-modynamic properties of air species, ESA Report STR-

Ž . Ž .236 1994 ; b D. Giordano, M. Capitelli, G. Colonna,C. Gorse, Tables of internal partition functions andthermodynamic properties of high-temperature air

Ž .species, ESA Report STR-237 1994 .� �7 M. Capitelli, E. Molinari, Problems of determination of

high-temperature thermodynamic properties of raregases with application to mixtures, J. Plasma Physics 4Ž .1970 335�355.

� �8 V.A. Shakhatov, A. De Giacomo, V. D’Onghia, G. Chita,

A.M. Losacco, G. Bruno, O. De Pascale, Plasma assistedŽ .pulsed laser deposition PA-PLD of titanium dioxide

Ž .TiO . International Conference of ALT, Potenza-2Lecce, Sept. 20�24, 1999.

� � Ž .9 P. Comte, A primer to Numerical methods for com-pressible flowsin: M. Lesieur, P. Comte, J. Zinn-JustinŽ .Eds. , Computational Fluid Dynamics, Les Houches1993 Session LIX, Elsevier, 1996, pp. 165�219.

� �10 G. Colonna, Step adaptive method for vibrational kinet-ics and other initial value problems, Supplemento aiRendiconti del Circolo Matematico di Palermo, Serie II,

Ž .57 1997 159�163.� �11 Numerical Recipes. The Art of Scientific Computing,

Cambridge University Press, 1986.

modelling of libs plasma expansion

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