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Spectrochimica Acta Part B 59 (2004) 271289

0584-8547/04/$ - see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.sab.2003.12.017

Review

Laser-induced plasma expansion: theoretical and experimental aspects

M. Capitelli *, A. Casavola , G. Colonna , A. De Giacomoa,b, a b a

Chemistry Department, University of Bari, 70126 Bari, Italya

CNR-IMIP (National Research Council-Institute of Inorganic Methodologies and of Plasmas), Chemistry Department, University of Bari,b

70126 Bari, Italy

Received 28 July 2003; accepted 19 December 2003

Abstract

Theoretical and experimental efforts to describe the expansion of laser-induced plasma plume are reviewed. Particular attentionhas been devoted to a comparison between experimental and theoretical time of flight (TOF) profiles. The theoretical resultsobtained by inserting plasma kinetics in one-dimensional fluid-dynamic code show that chemical processes strongly influence thefluid dynamic expansion. Moreover fluid-dynamics codes based on both NavierStokes and DSMC models, which in additioninclude the possibility of different temperatures for electrons and heavy particles, are discussed. 2004 Elsevier B.V. All rights reserved.

Keywords: Laser-induced plasma; Fluid-dynamics; Time of flight; LIBS

1. Introduction

Laser induced plasma, LIP, is a topic of growinginterest in different fields such as material processing,diagnostic techniques and space applications. Pulsedlaser deposition (PLD) has been successfully employedfor the deposition of thin films of classical and novelmaterials w1,2x. The possibility of producing species inLIP with electronic states far from chemical equilibriumenlarges the potential of making novel materials thatwould be unattainable under thermal conditions. Anexcellent example is the production of single carbon

nanotubes (SCN) by laser-oven method proposed bySmalley w3x. However, a great effort has been done indeveloping handable sensors based on laser inducedplasma spectroscopy (LIPS) for elemental chemicalanalysis in a wide range of applications: heavy metaldetection in soils w4x, bio-molecules detection w5x, on-line control over different laser processes w6x etc. Finally,the development of compact laser source is leading theLIP based techniques in space applications as chemical

*Corresponding author.E-mail address:[email protected] (M. Capitelli).

sensor for Mars exploration w7x or micro-propulsiontechnique for the exact positioning of satellites w8x.

LIP technological applications require a parallel inves-

tigation of the formed plasma from a more fundamentalpoint of view. Indeed, while the generation of LIP

requires an easy experimental set-up, the LIP phenomena

need a lot of theoretical and experimental work to beunderstood.

By the interaction of high power laser with matter,

the vaporization of surface layers leads to the formationof an expanding atomic plasma. During nanosecond

laser ablation, this high-density plasma (10 1017 19

cm ) is heated to high temperatures (6000 20 000 K)y3

and is ionized by the inverse Bremsstrahlung and the

photoionization processes, expanding rapidly (approx.10 cmys) perpendicularly to the surface. During the6

expansion, the main mechanism of transition of bound

electrons from the lower level to the upper level andvice versa is driven by inelastic collisions of electrons

with heavy particles, while the concentration of charged

particles is controlled by the electron impact ionizationand the three-body recombination of electrons with ions.Radiative processes as reabsorption, spontaneous andstimulated emission are also important in determining


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272 M. Capitelli et al. / Spectrochimica Acta Part B 59 (2004) 271289

the concentration of emitting levels w9x. The evolutionof the hierarchy of the basic processes occurring duringthe expansion makes the analysis of the plume reallycomplex and theoretical studies are needed for theinterpretation of the experimental data.

The traditional set of methods for measuring the

parameters of the LIP includes the optical emissionspectroscopy(OES) w10x,the laser-induced fluorescence(LIF) w11x, laser absorption, probing w12x, mass spec-trometry w13x and beam deflection w14x.Charged particledetection and laser excitation techniques give the bestresults in what concerns the investigation of LIP farfrom the targets whereas the OES is the simplest wayto investigate the brightness zone of LIP (010 mm).OES is based on the intrinsic light emission of the LIPand does not need any other excitation source orintrusive systems. This peculiarity makes the OES exper-imental set-up very simple and adaptable to automation

and remote sensing w7,15,16x. The classical approach ofthis technique is based on some assumptions, mostimportantly those of the existence of local thermody-namic equilibrium (LTE) conditions and of opticallythin plasma w17x. Unfortunately, as a consequence ofthe fast expansion, the plasma parameters can change intimes shorter than those necessary for the establishmentof elementary process balances w18x, thus leading tonon-equilibrium excitation and chemical processes. Theknowledge of the deviations from the LTE is reallyimportant to understand the limits of theory to be takeninto account for practical applications w18,19x.

The growing interest on laser-induced plasma has

generated numerous theoretical models in order toimprove the basic knowledge and determine the bestexperimental conditions of LIP processes. The maindifficulty in modeling LIP is that both the dynamics ofplume and plasma chemical reactions occurring in itshould be investigated. To reproduce the temporal andspatial evolution of the plasma a lot of theoreticalinvestigations have been devoted to the simulation ofthe plume expansion w20x. In this frame differentapproaches have been proposed: an analytical approachw21,22x, a fluid dynamic model w2325x and MonteCarlo methods w26,27x. Moreover the role of plasma

chemical processes and the possibility of deviationsfrom equilibrium conditions have been taken intoaccount.

The aim of this review is to report recent theoreticaland experimental efforts made to describe the expansionof plasma plume. In this sense, this paper complementsthe recent theoretical work w18x on analyzing micro-scopic aspects of LIP.

A complete discussion on laser-induced plasma is atoo vast argument to be reported in a single paper sothat we will focus our attention on the luminous plasmaproduced by nanosecond laser ablation. The laser solidinteraction as well as the dependence of ablation pro-

cesses on laser parameters has been considered beyondthe aim of the present work.

2. Laser-induced plasma spectroscopy

2.1. Optical emission spectra

The experimental set-up for laser induced plasmaspectroscopy (LIPS) basically consists of a laser sourcefor the vaporization of the target and for plasma for-mation, a high-resolution monochromator and a timeresolved detector. In this review paper, the attention isfocused on fluid dynamic aspects of the plume andplume monitoring by spectroscopic diagnostics.

To show the typical behavior of a laser-inducedplasma we report the results of a simple experiment ofKrF laser ablation on metallic titanium(wavelength 248nm, laser fluence 5 J cm , pulse duration 30 ns).y2

The threshold energy to produce a luminous plumedepends on the target material, its morphology as wellas the laser pulse wavelength and duration w28x. Thetemporization of the spectroscopic system is extremelyimportant because of the rapid expansion of the plume.The main trigger is obtained by the signal of a fastphotodiode detecting a portion of the laser beam. Thepulse generator, then, controls the detector acquisitiontimes including the delay between the laser pulse andthe gate width. The delay time and the gate width shouldbe optimized with respect to the experiments to obtainclear and readable spectra.

The spatial resolution is obtained by focusing the

plume image in a plane where the optical fiber entranceis placed. The magnitude of the plume image and thesize of the optical fiber aperture determine the spatialresolution generally within a few hundreds of microns.

The dynamic aspect of the LIP is a complex questionfor diagnostics since plasma parameters change quicklyin time and in space. It means that it is necessary tofind the experimental condition to approximate the LIPto an uniform plasma. Hermann et al. studying the lineshape of Ti 348.38 nm found out that the plasma isq

rather symmetric during the early stage (delay timelower than 150 ns) and close to the target surface. The

authors proposed two successive steps in LIP evolutionwith different expansion modes. Initially, for a timewhere the distance of the LIP front from the targetsurface is smaller than the laser spot dimension, thepropagation of plasma can be considered one-dimen-sional and strictly directed perpendicularly to the targetsurface. At later times a non-uniform distribution withcylindrical symmetry takes place and Abel inversion isrequired for the LIP diagnostics w9x.

A typical temporal evolution of a fragment of Ti-LIPemission spectrum at a fixed distance (0.5 mm, gatewidth 40 ns) is shown inFig. 1.Inspection of the figureshows the initial spectral continuum contribution, essen-


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273M. Capitelli et al. / Spectrochimica Acta Part B 59 (2004) 271289

Fig. 1. Typical temporal distribution of a fragment of LIP spectra atthe distance of 0.5 mm in vacuum. For figure clarity are reported onlythe spectra with an increasing temporal step of 50 ns. Experimentalconditions: metallic titanium target, laser fluence 5 J cm , pulsey2

duration 30 ns.

Fig. 2. Electron number density temporal evolution (N) at differentebackground pressures. Experimental conditions: metallic titanium tar-get, laser fluence 5 J cm , pulse duration 30 ns.y2

tially due to mechanisms involving free electrons(inverse Bremsstrahlung, radiative recombination)w29,30x. At this stage it is not possible to retrieveinformation about atoms and ions. Some tens of nano-seconds after the laser pulse is over, atomic and espe-cially ionic lines rise up from the spectral continuum.The emission lines become progressively narrower dur-ing the expansion as a consequence of the electronnumber density evolution (Fig. 2). As has been exten-sively reported in literature the fast decrease of theelectron number density is caused by plasma expansion

and three body recombination w9,10x. As a consequencethe excitation temperatures must decrease during thetime evolution. The maximum intensity of the spectrallines is reached after a characteristic time, depending onthe observation distance, and it represents the mostpopulated section of the LIP. Because of the highionization degree, for the most part of spectra the ioniclines are more intense than those of atoms, while on thetails of the temporal distribution of the LIP line inten-sities, which correspond to the cold part of the plasma,one can observe the disappearance of ionic lines.

Under the assumption of an optically thin plasma, i.e.all the emitted photons escape from the plasma w31x,

the spectral line intensity corresponding to a transitionfrom an upper excited level u to a lower excited level lis given by:

uI sG N hn A P n n (1) .ul ul ul ul ul

where N is the population density of the upper levelu,uA is the transition probability from level u to l, n isul ulthe frequency, h is the Planck constant and G is anulexperimental factor which depends on the analyzedplasma volume, the solid angle of observation and thespectral response of the optical system. The functionP(n ,n) is the normalized line profile of the spectralulline.

If the plasma is optically thick, it is necessary to addto the Eq. (1) the terms taking into account self-absorption and-induced emission. Generally in LIP emis-sion spectra the contribution on spectral lines due to the

induced emission can be neglected as exp(hvykT)4

1w31,32x. Many methods have been developed to takeinto account self-absorption in LIP spectral lines (seefor example Refs. w9,3136x). At high backgroundpressure this effect can play an important role as aconsequence of the confinement of LIP in smallervolume.

The spectral line intensity, including the self-absorp-tion and considering the plasma as an homogeneousmedium in LTE (local thermodynamic equilibrium), isexpressed as w31,37x:

thinIulI s 1yexp yt (2) ..ul ltl

where is the intensity as described in Eq. (1) andthinIult is the dimensionless optical thickness at wavelengthlldefined by t sa R with l, the absorption coefficientl lin cm and R path length in centimeters.y1


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Fig. 3. Experimental Boltzmann plots after 150 ns from the laser pulse, at distance from the target of 1 mm and in vacuum (a) for Ti and (b)for Ti . Experimental conditions: metallic titanium target, laser fluence 5 J cm , pulse duration 30 ns.q y2

Optical thick plasma inhomogeneity is reflected onthe spatial gradient of the temperature and of the atom,

ion and electron concentrations. This leads to the for-mation of strongly asymmetric self reversed emissionlines within asymmetric absorption lines having a valleyin the middle and line wings asymmetric respect to thefrequency at which the intensity of self reversal isminimal. In this case, only by theoretical modeling it ispossible to retrieve information from the experimentalspectra w38,39x.

Due to the difficulty in determining the optical thick-ness without assuming strong approximations (plasmahomogeneity, LTE etc.), when it is possible, transitionsinvolving high energy levels should be selected for the

spectral analysis in order to minimize the self-absorptionproblem and considering the LIP as a thin medium.

The gate width of the detector should be adapted tothe spontaneous emission time of the selected spectrallines so that high transition probability allows highertime resolution of the detection system.

The spectral line profile is mainly a convolution ofthe Lorentz function due to the Stark effect. In addition,a Gauss function due to the Doppler effect (generallynegligible, -10% w33,34x) and the apparatus functionof the entire spectroscopic system also contribute to theprofile. However, because other effects are less pro-nounced, the study of line broadening allows in deter-mining the electron number density by Stark effectwithout assuming LTE condition (seeFig. 2) w31x.

2.2. Investigation of atomic and ionic state distribution

function by LIPS

The relative line intensities of atomic and ionic linescan be used to determine the characteristic temperatureof each species when the mechanisms of excitation andde-excitation of atoms and ions involve mainly electronimpact. In this case, the excited state population of theconsidered species follows the Boltzmann distribution,

and the following equation for homogeneous plasma(GsG ) can be employed:ul

B EIulC Fln sm E qq (3)A u AD Gg n Au ul ul

B E1 NAC Fm sy ; q sln G (4)A AD GkT Z T .A A A

where k is the Boltzmanns constant, E and g are,u urespectively, the energy and the statistical weight of theupper excited energy level, Z is the partition functionAof the chemical species A calculated at the temperatureT , N is the concentration of the chemical speciesA inA Athe ground state. It is important to note that if LTEconditions are established and if the target evaporationis assumed congruent, the Boltzmann plot analysiswould lead to the free-calibration method proposed inRef. w40x. In this paper, the authors determine thetemperature by the Boltzmann distribution of differentexcited electronic state of the same element byEq. (1)and then, in order to remove the unknown experimentalfactor G from the intercept q , they use the normaliza-Ation relation SNs1. The method, indeed, it is noti ialways reliable because the accuracy is strictly depend-ent on the assumption of LTE condition. In any case,

care should be taken when calculating the internalpartition function especially at high temperature w18x.As a typical example Fig. 3 shows the Boltzmann

plot of atoms and ions during the laser ablation of themetallic titanium target at 10 Torr.Table 1reports they5

temperature at different delay times recovered by theslope of the Boltmann plot.

To improve the accuracy of the temperature deter-mined by Boltzmann plot analysis Yalcin et al. applythe Saha equation to enlarge the energy axis by usingthe transitions from different ionization stages w17x inthe same plot. Unfortunately, this methodology, as pre-viously established by Van der Mullen w41x, works only


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Table 1Results of Boltzmann plot analysis at different delay times from thelaser pulse and pressures (distance from the target 1 mm): internaltemperature T (K) for Ti and Ti species, intercept of Boltzmannq

plots, q(N ) and .q N q .Ti Ti

Delay(ns) Ti Tiq

T"800 (K) q(N )"0.5Ti T"800 (K) "0.5q N q .Ti

Vacuum(10 Torr)y5

50 11 800 y30.2 11 200 y25.4100 8700 y28.6 8850 y23.3150 8800 y28.6 8600 y23.2200 9600 y29.0 8950 y23.9

3.4=10 Torry2

50 10 900 y29.3 10 700 y24.6100 10 100 y28.9 9800 y23.5150 8900 y28.2 8700 y23.2200 8600 y28.1 8300 y23.7

Fig. 4. Ionization degree as function of time during laser ablation ofmetallic titanium target at 10 Torr at 2 and 3 mm from the target.y5

Experimental conditions: metallic titanium target, laser fluence 5 Jcm , pulse duration 30 ns.y2

in LTE conditions and so cannot be proposed as ageneral method in LIP diagnostics.If the high ionization degree suggests a Maxwell

distribution for the electron energy distribution functionand Boltzmann distribution for the excited states, thenatural conclusion of the LTE conditions should betaken with caution. The system indeed may becomemuch more ionized than that obtained by applying theSaha equation at the temperature obtained by the Boltz-mann plot and at the electron number density as recov-ered by the Stark effect as has been observed in Ref.w42x.

2.3. Temporal evolution of LIP

If the ground state of all atomic and ionic species isin the Boltzmann equilibrium with the excited statepopulation of the species, then Eq. (1), neglecting theshape function and assuming the homogeneity of thesampled portion of the plasma, can be written as:

I Nul AI f s exp yE ykT (5) .ul u exc

Ghn A g Z T .ul ul u A exc

where T is the (internal) temperature of electronicexclevels of the species A, which can be different from T .A

In this hypothesis, the ratio of the ground statedensities of two different species (e.g. A and B) can beexpressed as a function of the ratio of excited statedensities of those two species, which is the ratio of aline intensity ofA to a line intensity ofB,

BI Z T B E .u9l9 B excN E yEB u9 uC Fs exp y (6)A

D GN I Z T kT .A ul A exc exc

where u9l9 indicate the upper and lower level of speciesB.

This equation allows determining the ratio betweenthe ion and the atom concentrations (generally in therange of 10 10 cm ) from the selected spectral15 18 y3

lines of ionic and atomic species. In fact, the assumptionof the Boltzmann equilibrium between ground andexcited states densities may lead to underestimation of

the ground state population if the spontaneous emissiondecay process dominates over collisional de-excitationmechanisms, as suggested in Ref. w43x.

Fig. 4shows the ratio between ion and atom concen-trations of Ti-LIP obtained by the method describedabove at 2 and 3 mm from the target as function oftime. In the figure, the ratio decreases by two orders ofmagnitude in few hundreds of nanoseconds, as a con-sequence of the recombination processes and of thephysical expansion.

The evolution of LIP species by temporally andspatially resolved OES at different distances from the

target can be used to determine the expansion rate closeto the target w40,44x. The determination of the meanexpansion velocity can be achieved by measuring theionic emission temporal profiles, suitably corrected bytabulated spectroscopic data (the so called time of flight(TOF) profiles). Actually this procedure for the deter-mination of plasma velocity should be used with caredue to the superposition of both expansion and forwardmovement of the plasma plume. Moreover the proceduredoes not take into account the different populations andthe different origins of the same species. Ionic lineshave been preferred with respect to the atomic onesbecause, as will be explained later, the Ti TOF can

present an additional delay due to the three-body recom-bination mechanism.Fig. 5a,b show normalized TOF ofTi species in vacuum (10 Torr) and at 3.4=10q y5 y2

Torr. From the time shift of the TOF peaks for eachdistance it is possible to obtain the mean velocities of


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Fig. 5. TOF profiles of Ti species at 0.5, 1 and 2 mm from the target(a) in vacuum, (b) at 3.4=10 Torr. Experimental conditions: metallicq y2

titanium target, laser fluence 5 J cm , pulse duration 30 ns.y2

Table 2Peaks velocity (cm s ) at different distances from the target aty1

3.4=10 and 1=10 Torry2 y5

Distance"0.1 (mm) Peaks velocity"5=10 (cm s )4 y1

Ps3.4=10 Torry2 Ps1=10 Torry5

0.50 1.25=106 8.33=105

1.00 1.11=106 1.25=106

2.00 9.09=105 1.54=106

Fig. 6. Set of spectral fragments corresponding to the maximum ofTi TOF signal at different distances at 3.4=10 Torr. Experimentalq y2

conditions: metallic titanium target, laser fluence 5 J cm , pulsey2

duration 30 ns.

LIP along the propagation axis. The velocities can be

determined, as suggested in Ref. w

10x, by the ratio dyt(peak), where d represents the difference in the obser-

vation distance and t (peak) the corresponding timedifference calculated at the peak of TOF for eachselected distance. Calculated velocities have been report-ed inTable 2and are in good agreement with the valuespresent in literature (see for example Refs.w10,11,13,19x). Inspection of Table 2 shows that thebackground pressure modifies the expansion dynamics.In the range of distances investigated in this experimentunder vacuum condition the velocity of LIP increasesand probably will reach a maximum not so far from thelast observed distance (2 mm). This trend of the LIP

velocity is due to the initial acceleration of the ablatedparticles from the initial velocity V s0, before the laser0pulse energy reaches the ablation threshold, to a maxi-mum of velocity,V . Then the velocity decreases sincemaxpart of the energy is expended in intra-plume collisions,in heating and moving the ambient gas. At 3.4=10y2

Torr, the velocity immediately decreases with the dis-tance, denoting that the deceleration of LIP due to thecollisions of the background gas particles with the fronthead of LIP occurs at shorter distance.

As already underlined, the LIP is a dynamic system.To investigate the same section of the plasma, the spectra

corresponding to the maximum of the temporal evolutionof the Ti TOF collected at different distance has beenq

analyzed and plotted in Fig. 6. In this way for each

distance there is a time delay when the temporal evo-

lution of Ti line emission reaches its maximum. As

q

can be seen fromFig. 6,the characteristics of the spectraare similar to those described previously for the temporalevolution (Fig. 1) at a fixed distance. The electronnumber density, N of the same section of plasmaecorresponding to the maximum of Ti TOF curves,q

obtained by Stark effect, has been reported in Fig. 7.The decrease of N is mainly governed by the recom-ebination between electrons and ions, plasma expansionplaying a minor role. The electron number densityevolution reported in Fig. 7 can be expressed by the


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particles first form a Knudsen layer (KL), then passinto UAE.

An analytical approach to solve the problem concern-ing the UAE plus the KL has been widely discussed byKools et al. and by Kelly w13,48x. They solve thehydrodynamic equations for an unsteady adiabatic

expansion with the correct boundary conditions. Thisapproach is justified in all cases where the gas densityis so large that the number of collisions between particlesis sufficient for thermodynamics or hydrodynamicsdescription. Since the necessary number of collisions isvery low w27x, it is appropriate to follow this approachfor cases when more than one monolayer desorbs on atime scale of tens of nanoseconds. Moreover the use offlow equations, which assume local thermodynamicequilibrium, for an expansion, which takes place intovacuum, could appear questionable. The usual way toovercome this contradiction is to assume that the flow

equations remain valid until the density falls into acritical value and the particles go abruptly into freeflight w48x.

Detailed analytical descriptions of the adiabatic plas-ma expansion have been discussed by several authorsw21,22x. For example, Anisimov et al. w21x have calcu-lated the profile of a film produced by pulsed-laserdeposition from a solid target. A simple analyticalsolution has been obtained for the range of parameterstypically employed in PLD, by solving the gas-dynam-ical equations under the assumption of an adiabaticexpansion of the plume into vacuum.

Let us consider now numerical fluid dynamic models,

which allow calculating spatial and temporal evolutionof several macroscopic quantities, such as density, tem-perature and velocity, which can be directly comparedwith experimental results.

The most rigorous fluid-dynamic approach to describethe plasma expansion is the solution of the three-dimensional axisymmetric compressible NavierStokesequations for a multispecies gas, including the effectsof mass and forced diffusion and thermal conductionand viscosity w24x:

w zri x |qdiv r nqn sv (8) .y ~i di it

rnqdiv rnnyPIqt s0 (9) .

t

w zEa x |qdiv Eny yP Iqt nqq sS (10) .y ~a a at

w zEe x |qdiv E qP nqn sS qF (11) . .e e die e ey ~t

In these equations r , and v are the density, theni di i

forced diffusion due to the presence of an electric field,

and mass source term of species i, respectively. is thenvelocity, r is the density of the mixture, P and P area ethe atomion and electron pressure, respectively, P isthe overall pressure and is the viscous shear stressttensor.E and E are the atomion and electron energies,a eand S and S are the atom ion and electron energya esource terms.

Le et al. also assume an ideal-gas equation of statefor each species (heavy particle, electron) w24x. Theoverall pressure is given by:

RT RTePsP qP s r qr (12)a e i e8 M Mi ei/e

where is the molecular weight of species i, is M Mi ethe molecular weight of the electron and R denotes the

universal gas constant. The plasma is assumed to bequasi-neutral and two-temperature modeling (T, T) iseused to describe the ionization non-equilibrium condi-tion. The electron temperature T deviates from theeheavy-particle temperature T because of the slow rateof energy transfer between electrons and heavy particlescaused by the large difference in the mass of electronsand heavy particles.

At the beginning of the plasma expansion and closeto the surface (initial conditions), the temperatures ofelectrons and atoms are assumed to be equal as aconsequence of local thermodynamic equilibrium duringthe laser pulse. However, a two-temperature modeling

is used due to the non-equilibrium ionization hypothesis.In Fig. 9a,b the temporal evolution of the electron andheavy particles temperatures is plotted at different dis-tances z from the target. It can be observed that at adistance of 1 mm from the target the electron tempera-ture is lower than that of the heavy particles in theshock front. Between 3 and 5 mm the electron andheavy particle temperature become almost equal. At adistance of 7 mm from the target, T is globally higherethan T. This is probably due to the slower coolingmechanism of electrons, with respect to that of heavyparticles w24x.

Le et al. w24x have also considered the ambipolardiffusion due to the presence of an electric field.Electrons have higher mobility and diffuse faster thanions, leaving behind them an excess of positive charge.The electric field produced by this initial charge sepa-ration tends toward a slowing down of the electrons andan acceleration of the ions. Thus the electric field affectsthe diffusion process of electrons and ions. The rigoroustreatment of the problem of forced diffusion by anelectric field requires a complex system of equations, asshown by Lee w60x. A simplified method has been usedby the same authors, who consider this effect by intro-ducing the ambipolar diffusion concept, occurring when


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Fig. 9. Results of two-dimensional NavierStokes gas dynamic cal-culation with a laser fluence of 1.5 Jycm and 400 mTorr of Ar. Time2

evolution of electron and heavy-particle temperatures at different zpositions, 1 and 3 mm (a) and 5 and 7 mm (b). (Taken from Ref.

w24x with permission from the authors and the American PhysicalSociety).

Fig. 10. Comparison of two-dimensional results from Euler andNavierStokes gas dynamics calculations, with a laser fluence of 1.5Jycm and 400 mTorr of Ar. Profiles of Si density as a function ofZ2

position and at times ts0.12 ms (a) and ts1.1 ms (b). (Taken fromRef. w24x with permission from the authors and the American PhysicalSociety).

an electron pressure gradient exists in an ionized gas, inthe presence of small charge separation.

The same authors have shown that a more rigorousapproach w24x, i.e. the solution of NavierStokes equa-tions, is necessary only for expansions in a backgroundgas at not negligible pressures ()0.4 mTorr) or forexpansion to longer times (1 ms). To understand thispoint, density profiles of Si neutral atoms calculated byLe et al. by using NavierStokes and Euler equations

have been reported in Fig. 10ab at different times. Itcan be observed that in the early stage of expansion(ts0.12 ms) the density profiles provided by the twomodels are similar, while at longer times (ts1.1 ms)and at shorter distances (z-0.4 cm) the Si density ishigher in the NavierStokes model w24x. This behavior

is strictly linked to inserting(NavierStokes)or neglect-ing(Euler)the diffusion processes behind the expandingfront operated by the two models.

Euler equations can be used to simulate the processin the early stage of the evolution. The simple Eulerapproach allows inserting detailed kinetic schemes.


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Euler equations have been extensively used in Refs.w23,25,4954x. For instance, Aden et al. have describedthe plume expansion through the compressible and non-dissipative conservation equations for mass-, momen-tum- and energy density. Since, in the gas-dynamicaldescription, the time-averaged electric field of the laser

radiation is zero, electrons and heavy particles movewith the same velocity w50x. Moreover, Aden introducesthe energy equation for electrons, because the electroncomponent of the plasmayplasma gains energy from thelaser field, so that the electron temperature may bedifferent from the atomic one.

A simplification of the three-dimensional Euler equa-tion could be obtained supposing that after the pulsetermination the cloud starts expanding spherically w53x.For moderate background pressures, the hypothesis of aspherical expansion appears to be reasonable for thedescription of near-axis plume behavior w50x,but in high

vacuum the plume is essentially forward directed and,therefore a one-dimensional code can describe the sys-tem w23,25,44,54,61,62x.

Taking into account all these information, we haveinvestigated the expansion of a laser-ablation plume bymeans of a one-dimensional fluid dynamic modelw23,25,44,54x. This model considers only the stage ofexpansion during and after the laser pulse withoutdetailing the evaporation process. At the end of the laserpulse, a plume is assumed to be formed in the directionperpendicular to the target, in the same position of theirradiated spot center. The laser energy is spent onmelting, vaporization and heating of the target material,

and on heating and ionization of the vaporized particles.Before the expansion, the plasma can be considered inthermal equilibrium. After pulse termination, the cloudbegins to expand perpendicularly to the target into anambient gas at pressure P w53x. The plasma has beenbtreated as a single fluid characterized by one velocity uand one temperature T. This fluid dynamic approachdisregards the effects of space charge separation, as thetime-averaged electric field of the laser radiation is zero.The validity of the one-dimensional approximation islargely demonstrated in literature. In fact, numerousstudies have shown that the angular distribution of the

laser-generated flux is often much more strongly forwardpeaked than the flux obtainable from small-area effusivesources operating under collision-less conditions. Thisforward peaking phenomenon for deposition in vacuumis now generally accepted as arising from collisions ofthe plume species among themselves. In the presence ofa background gas, additional scattering effects must beconsidered w28x, as well as the possibility of plumeconfinement w28x.

Quite different approaches to simulate the laser-induced plasma expansion have been proposed by Woodet al. w6368x and Nemirovsky w69x.The Woods modelis based on a combination of multiple scattering and

hydrodynamic approaches. The plume is allowed to bebroken into components, or scattering orders, whoseparticles can undergo many collisions with the back-ground. Particles can only be transferred from one orderto the next higher order by collisions. The densities inthe individual orders propagate according to the usual

conservation equations to give the overall plume expan-sion. Nemirovsky et al. w69x, starting from the Boltz-mann equation, have derived the hydrodynamicequations of motion for partially ionized plasma whenthe charged component and the neutral component havedifferent flow velocities and different temperatures. Theyhave developed a general approach for the hydrodynam-ics of a gas in a binary mixture, when the interactionbetween particles of the same species is much strongerthan that between particles of different species. Thefollowing processes of interactions were studied: self-collisions, interspecies collisions, ionization, recombi-

nation and charge exchange.All gas-dynamical models, however, which have beenso far discussed, are invalid when the velocity distribu-tion of the ablated particles deviates from the Maxwel-lian distribution. As a result, they cannot correctly beapplied for the regions of strong density gradients.Moreover diffusive processes are not well described bythe continuous gas-dynamics and the mixing of theablated species with the background gas cannot berigorously studied with these models. This mixing occursat the later stage of the plume expansion, so that it canbe neglected if only the early stage of plasma evolutionis considered. It is necessary to investigate the effective

mixing between the ablated and the buffer particles inthe reactive laser ablation, because it provides thecondition required for the formation of the reactiveproducts.

To overcome the limits of the gas-dynamic approach,many authors have investigated the plume expansionand the effects of collisions amongst particles desorbedfrom solid surfaces by means of Direct Monte Carlosimulation w26,27,70,71x.

Urbassek et al. in Ref. w70x obtained preliminaryinformation about sputtering yield by developing asimple model, based on the assumption of a collision-

free molecular flow. Then Urbassek et al. w71x developeda Monte Carlo simulation to investigate the plasmaexpansion, also taking into account collisions betweenparticles, assuming that particles desorb during a timeinterval t from a circular area (radius r) of a plane0surface into vacuum. The desorption mechanism issupposed thermal, so that all species can be describedby a MaxwellBoltzmann distribution at the same tem-perature. Above the surface, particles may collide witheach other. The code is based on Birds direct simulationMonte Carlo algorithm w72x. In Fig. 11, the timeevolution of the density of the desorbed gas cloud isshown. At the end of the desorption period t the gas


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Fig. 11. Contour plot of the density of the axisymmetric gas cloud atthree different times after desorption. z denotes distance from surface;rthe radial distance from the center of the laser spot. Black shadingindicates particle density. Upper part: light species; lower part: heavyspecies. (Taken from Ref. w70x with permission from the authors andthe American Physical Society).

Fig. 12. Mean energyE(x)of light(a)and heavy(b)particles depos-ited at a plane substrate as a function of radial distance from the centerof the substrate. The distribution were normalized by E s2kT, where0 0T is the temperature of the surface.(Taken from Ref. w26x with per-0mission from the authors and the American Institute of Physics.)

cloud is very oblate, corresponding to the large desorp-tion area chosen for this case study. It becomes moreand more spherical at 2t, assuming a prolate form at4t. In fact, as a result of the steep pressure gradientinitially present in the direction normal to the target, theexpansion occurs preferentially in this direction. More-over Fig. 11 displays an interesting spatial segregation

effect of light and heavy particles; in particular, due tothe different velocity of desorbed heavy and light par-ticles, in the back part of the cloud both species appearto be well mixed while in the front part of the cloudconsists mainly of light species w71x.

More recently, a three-dimensional Monte Carlo sim-ulation of the laser-induced plasma plume expansionunder a non-reactive atmosphere has been developedw26x. Itina et al. used a three-dimensional algorithm thatcombines both the direct Monte Carlo (DSMC) tech-nique w27,71x, to simulate the desorption and the initialcloud expansion, and the Monte Carlo simulation of the

random trajectories (MCRT) of the ablated particles inthe ambient gas w55x. This kind of approach allowsconsidering both the interactions between ablated parti-cles and ablated particle-background gas atom collisions.The latter were found to decrease the kinetic energy ofparticles. Moreover at higher background pressures, theparticles are thermalized and randomly scattered fromtheir initial trajectories, producing a more uniform flow.This thermalization occurs at different pressures for lightand heavy particles. This behavior can be observed inFig. 12, where the distribution of mean kinetic energyE(x) of particles deposited at the plane substrate as afunction of the radial distance from the center of the

substrate is plotted. At low pressure (Ps0.01 mTorr)heavy particles are more energetic than light particlesand their distribution of mean energy is more focusedtoward the center. As the background pressure increases,the mean energies of both species diminish and thedistributions become less focused toward the center.

This effect is more evident for light particles than forheavy particles. The decrease of the kinetic energy ofthe plume particles is due to the collisions with thebackground gas, which thermalize the particles. Thedecline of energy is more pronounced for light species,because they lose energy more efficiently in a collision.Therefore a smaller background pressure is required forthermalization of light components. At high pressures(Ps100 mTorr) most of the particles are thermalizedand energy distributions become broadened along thenormal of the surface w26x.

Therefore the difficulties of the gas-dynamic descrip-

tion of the diffusive processes are avoided in direct


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Fig. 13. Contour maps of the calculated spatial distribution of the plume number density as a function of time obtained in calculation with oxygenpressure. (Taken from Ref. w56x with permission from the authors and the American Physical Society.) (1a): P s0.7 Pa, ts0.6 ms, (1b): P sb b0.7 Pa, ts1.5 ms, (1c): P s0.7 Pa, ts2.7ms; (2a): P s13 Pa, ts0.6 ms, (2b): P s13 Pa, ts1.5ms, (2c): P s13 Pa, ts2.7ms; (3a): P sb b b b b70 Pa, ts0.6 ms, (3b): P s70 Pa, ts1.5 ms, (3c): P s70 Pa, ts2.7 ms, (3d): P s70 Pa, ts6.3 ms.b b b

simulation Monte Carlo (DSMC). However, other limi-tations arise in this kind of method. When the DSMC

is used to simulate the plume expansion process, mostof the computer resources are used to calculate only theinitial stage of the expansion. In the presence of abackground gas, additional collisions between the plumeand the buffer molecules should be considered. As aresult, the DSMC becomes computationally expensivein terms of computer memory and calculation time, inthe case of a high rate laser ablation and in the presenceof a background gas with pressure more than ;10 Paw56x.

To solve the limits of both the continuous and micro-scopic approach, recently Itina et al. have proposed a

new hybrid method w57x.The three-dimensional modelwith axial symmetry combines the continuum gas-dynamic approach for the first stage of the plumeexpansion with the direct Monte Carlo simulation of theablated flux into a background gas at the later stage. Inparticular, at the first stage the plasma plume is consid-ered as a non-viscous and non-heat conductive plasmacontaining atoms, ions and electrons. The electron tem-perature deviates from the atom and ion temperature,due to the slow rate of energy transfer between electronsand heavy particles. Based on these considerations, aone-fluid two-temperature gas dynamics is used todescribe the plume movement. The continuum gas-

dynamic calculations are performed until a certain timet ; then the motion of the ablated and background0particles is followed by using a microscopic MonteCarlo method. The switch timet should be long enough0for the plume density to diminish by several orders ofmagnitude with respect to the initial values. Moreover,t should be short enough so that the mass diffusion and0the heat exchange between the plume and the back-ground gas are insignificant during t . This approach0allows determining the background pressure needed forthe beginning of the snowplough effect, from the plumeand gas density profiles obtained at different pressures.

In particular, the combined method has been used tostudy the laser ablation of aluminium in the presence of

an oxygen background gas.Fig. 13shows the dynamicsof the plume expansion at three different pressures,respectively, Ps0.7 Pa, 13 Pa and 70 Pa. At lowpressure (Fig. 13-1a1c) a strong expansion of theplume is observed and no compressed layer is formed,neither in the plume, nor in the background gas. Theleading edge of the plume moves away with a practicallyconstant velocity, while the center has a velocity oneorder of magnitude smaller (see Fig. 14). At interme-diate pressure (Fig. 13-2a2c) the plume expansion issimilar to that in the first case at the beginning and thenthe plume starts being decelerated. The plume edgedecelerates and stops at approximately t;2.5 ms (see


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Fig. 14. (a)(c) Time evolution of the z-position of the plume center(h), of the plume edge(s), of the gas edge (m)and of the gas peak(%). (a) P s0.7 Pa, (b) P s13 Pa, (c) P s70 Pa. (d) Time evo-b b blution of the gas peak number density at two different oxygen pres-sures. (Taken from Ref. w56x with permission from the authors andthe American Physical Society.)

Fig. 14b). The gas density peak rises at the first stagesof the expansion and then it decreases (see Fig. 14d)When pressure is further increased (Fig. 13-3a3d) theplumebackground interaction leads to the so-called

snow-plough effect: the ablated material pushes theambient gas away from the target as long as the plumepressure overcomes that one of the background gas. Asa result, both the ablated material and the ambient gasare compressed. The plume compression and decelera-tion can be observed also at ts0.6ms. At t;2.5ms the

plume pressure becomes equal to the background pres-sure, the plume stops and its energy is dissipated by theinteraction of the compressed plume zone with thecompressed background gas. As a result, the plume goesbackward and equilibrates. The plume center goes back,thus preventing the gas background filling up the regioninitially occupied by the plume. Only after the equili-bration and if there is still enough energy the plumewill move outward w57x.

The limit of this work is that, to reduce the compu-tational time, it did not include any chemical reactions,although in previous works, with a simpler fluid dynam-

ic approach, also chemical non-equilibrium were takeninto account w58x.

3.2. Chemical models: a macroscopic and a microscopic

approach

The large effort dedicated to the plume expansiondynamics is generally not accompanied by a similareffort in the understanding of the plume chemicalreactivity. Generally, infinitely slow chemical processrates are supposed, which generate an expansion withfrozen plasma chemical reactivity w48x. To simulate thereal system under study, it is necessary to include

chemical models into the fluid dynamic code.As a first attempt to simulate a reactive fluid dynamic,

a TiO plasma in local thermodynamic equilibrium (LTE)was studied w25x. In the LTE approximation, the plasmathermodynamics is completely defined by two independ-ent parameters, such as pressure and temperature, orenthalpy and pressure. A strong non-linear couplingoccurs between the chemical kinetics and the fluiddynamics. Implicitly in the LTE approach one supposesthat the rate coefficients of chemical processes areinfinite.

This approximation, assumed by different authors

w25,73x for the interpretation of experiments as well asfor the development of theoretical codes, is still an openproblem w18,74x. Therefore the development of a newmodel, which allows us to include the effect of non-equilibrium chemical processes during the plume expan-sion is essential.

Le et al. w24x observe a pronounced differencebetween heavy-particle and electron temperatures: thisclearly demonstrates a persistent lack of equilibriumbetween the electron and the heavy particle in the plasmaplume expansion. To take into account this effect, Le etal. couples the fluid dynamic method with a kineticapproach. The source terms in Eqs. (8)(11) depend


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on ionization and recombination rate constants, whichare given by Zeldovich and Raizer w75x. The processesincluded in the Le et al.s model are the ionization ofthe ground state and the three-body recombination intothe ground state and the photo-recombination into theground state.

In addition, other authors w50x take into account theabsorption of laser radiation in the plasma and thedynamics of its ionization state. To investigate the effectsof finite rate coefficients on the plume expansion, afluid dynamic model including the following processesfor a titanium plasma was developed w23,44,54x:

a ionization by electron impact(direct process)b three-body recombination (inverse process)

y q yTi qe mTi q2e (13) .k

c radiative recombination

Aekq yTi qe Ti qhn (14) . k

The ionization cross sections s ()were calculatedion,kusing the classical Grizinskii approach from each inter-nal level and for a wide range of the electron energyw76x. The ionization rate coefficients K from a givenion,klevel khave been calculated by integrating the relevantcross sections s () w76x over a Maxwellian electronion,kenergy distribution function f w77x:

`

K s f v s d (15) . . .ion,k ion,k|0The total ionization rate coefficients K have beenion

calculated by summing the state selected rate coeffi-cients over a Boltzmann distribution n at the samektemperature as the free electrons.

K s K n (16)ion ion,k k8k

The three-body recombination process, which can beseen as the inverse of the electron impact ionization,involves two electrons and one ion. One of the electronsrecombines with the ion, forming an atom in state k.The excess energy released in this process is transferredto the other electron. The global three-body recombina-tion rate is the sum over all the levels k. The ionizationand recombination rates are related through the principleof detailed balance.

The radiative recombination cross sections s .RR,khave been calculated in the hydrogen-like approxima-tion, by means of Kramers formulas w23,44,54,78x:

4 232Z Ry2s spa (17) .RR,k 0

3 3y3 3 137 F n . .k

where a is the Bohr radius, Z is the ion charge, Ry is0the Rydberg constant,n is the principal quantum number,F is the photon energy and is the electron energy.k

The radiative recombination rate on level k is calcu-lated by integrating the cross section (Eq. (17)) overthe electron energy distribution f():

`

K s f v s d (18) . . .RR,k RR,k|0

where f() is the electron energy distribution function,approximated with a Maxwellian distribution (eV ),y1

v()is the electron velocity. To obtain the total radiativerecombination rate we have summed over all the atomiclevels k:

K s K (19)RR RR,K8k

The radiative recombination converts electron trans-lational energy in electromagnetic energy (continuousemission light). The energy emission rate for ionsrecombining on atomic level k is given by:

`

R s f v s F d (20) . . . .RR,k RR,k k|0

and the total irradiated energy per time unit and volumeunit (J m s ) is given by:y3 y1

nlevels

Q sN N R (21)y qe e Ti RR,k8ks1

where and are the electron and titanium ionN Ny qe Tinumber densities. To take into account the loss of energydue to the radiative recombination, the energy continuityequation has been modified as:

rE ruE uq qP syQ (22)e

t x x

where r is the density, E is the energy per mass unit, uis the flow speed and P is the pressure. In this modelthe inverse Bremsstrahlung absorption w12x, extremelyeffective during the laser pulse, has been not consideredas we focus on the temporal evolution of the LIP speciesafter the laser pulse.

The role of plasma chemical processes in affectingthe macroscopic physical quantities, such as temperature,density, pressure, and composition profiles was investi-gated w25,44x. It was found that the kinetic processesstrongly affect the time of flight (TOF) plots, whichcan be simply obtained from the spatial profiles of eachquantity.


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Fig. 15. TOF plot of Ti and Ti molar fractions atds0.5 mm fromq

the target. T s30000 K, v s7000 mys, z s10 Kg m s , as17 y3 y10 0 d(ionization degree).

Fig. 16. TOF plot of normalized concentrations at ds2 mm from thetarget for T s50 000 K. v s7000 mys, z s10 Kg m s , as1.7 y3 y10 0 d

Fig. 17. (a) Comparison between theoretical ( i) and experimental ( ii) temporal profile of Ti (continuous line and circle) molar fractions at xs0.5 mm from the target. The estimated experimental error is ;10%.(b)Comparison between theoretical (i)and experimental (ii)temporal profileof Ti (dashed line and triangles) molar fractions at xs0.5 mm from the target. The estimated experimental error is ;10%.q

The plasma initially produced by the laser matterinteraction was supposed to be completely ionized. Inthe free flow case, when no plasma chemical reactionsare included, the time of flight plots of the titanium

molar fractions show a null value everywhere. When

kinetics is included, the TOF plots of Ti become

significant, due to recombination processes.

As an example, the time evolution of the atom and

ion molar fractions at a distance of 0.5 mm from the

target is represented in Fig. 15. At the first instant, the

plume arriving in that point is essentially composed ofions, as the kinetic effect is still negligible. As time

increases, recombination takes place, and consequently,

the Ti molar fraction grows up, while the ion molar

fraction decreases and becomes less than 0.5.

The developed kinetic model shows the fundamental

role of kinetics during the plume expansion. TOF plot

analysis highlights an apparent separation between atom

and ion concentrations (see Fig. 16) as has been

observed experimentally w44x, when kinetics is intro-

duced in the numerical modeling.

Model results strongly depend on the initial condi-

tions, which in the case of a continuous evaporation

approximation with a fluence of approximately 5 Jy

cm are temperature, rate of evaporation, initial velocity2

and plume composition. The difficulties to find initial

adjustable parameters for our model from experiments

have been overcome by using the Simplex method,

i.e. by minimizing the differences between theoretical

and experimental molar fractions at a distance of 0.5

mm from the target w23x. The initial parameter values,

such as temperature, rate production of matter, velocity

and ionization degree are the following: T s25410.4 K,0

v s8012.34 my

s, z s3.06493=

10 Kg m s , as

6 y3 y1

0 d

0.972948 (x s0.0270521, s s0.486474).x xq yTi Ti eThese are reasonable if we consider that the rate of

expansion, as experimentally determined at the closestdistance (0.5 mm) from the target, is approximately10 mys and the temperature, as obtained by the exper-3

imental Boltzmann plot, is 11 000 K (at 0.5 mm after100 ns from the laser pulse) w42x.


16/19


Fig. 18. (a) Comparison between theoretical (i) and experimental (ii) temporal profile of Ti (continuous line and circle) molar fractions at xs1mm from the target. The estimated experimental error is ;10%. (b) Comparison between theoretical (i) and experimental (ii) temporal profileof Ti (dashed line and triangles) molar fractions at xs1 mm from the target. The estimated experimental error is ;10%.q

Fig. 19. (a) Comparison between theoretical (i) and experimental (ii) temporal profile of Ti (continuous line and circle) molar fractions at xs2mm from the target. The estimated experimental error is ;10%. (b) Comparison between theoretical (i) and experimental (ii) temporal profileof Ti (dashed line and triangles) molar fractions at xs2 mm from the target. The estimated experimental error is ;10%.q

To validate the calculated initial parameters, we cancompare experimental and theoretical molar fractions atdifferent distances from the target, where no optimiza-tion has been performed. In particular, inFig. 17a,b, theTi and Ti molar fractions are reported as a function ofq

time, at the distance of 0.5 mm from the target, wherethe optimization between theoretical and experimentalresults has been performed. Then inFig. 18a,bFig. 19a,ba comparison between calculated and measured molarfractions is reported, respectively, at 1 mm and 2 mmfrom the target for both the atom and the ion molarfractions.

From all these figures, we can note that the agreement

between theoretical and experimental results is good,taking into account all the approximations that havebeen used in the theoretical approach. In particular, wecan observe that our theoretical approach seems tooverestimate the atom molar fraction with respect to theexperimental values. Probably, this is due to the factthat we have neglected the re-absorption effect of laser

pulse and that we did not include the photo-ionizationprocess.From a more microscopic point of view, non-equilib-

rium conditions can also arise during the plume expan-sion in the electron energy distribution functions and inthe atomic level distributions.

In previous works w74x we investigated conditionstypically met under laser induced breakdown spectros-copy (LIBS) and characterized by large ionizationdegrees. To verify the existence of LTE conditions, oneshould solve a collisional-radiative model for the popu-lation densities of excited states coupled to an adequateBoltzmann equation for the electron energy distribution

function (eedf). Both kinetics should be then insertedin a fluid-dynamic code describing the plasma expan-sion. In Ref. w74x, we focus attention on a single pointof this procedure, i.e. on the possibility of deviation ofthe eedf from Maxwell distribution. These deviationscan be caused by an unbalance between inelastic andsuperelastic electron atom collisions caused by a differ-


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Fig. 20. Evolution of electron energy distribution function.T relaxesefrom 5000 to 10 000 K.

ence between the temperature of electrons and that ofexcited atoms. Since the first stage of the LIPS plasmaevolution is the lowering in plasma density and temper-ature, while the ionization degree and excited state

population distribution are changed slower, the plasmaduring its evolution can be considered in an over-coolednon-equilibrium state. The main source of deviationfrom the Maxwellian form in the overcooled LIBSplasma is superelastic collisions of electrons with excitedatoms. Inelastic collisions as well as electronelectronand electronion collisions tend to Maxwellize eedf.

As an example, in Fig. 20 it is reported the casereferring to the heating of electron temperature from5000 to 10 000 K, with an ionization degree and apopulation density of excited states obeying Saha andBoltzmann equilibrium at 10 000 K. As it is seen, the

eedf deviates from its Maxwellian form at practicallyall the times during its evolution. In particular, theformation of a hump at 11 eV is the result of superelasticprocesses. Electronelectron Coulomb collisions areable at times larger than 10 s to establish a Maxwelly8

distribution function.It should be noted that in a real simulation the

relaxation of eedf through the inelastic and superelasticcollisions is accompanied by the alteration in the ioni-zation degree and excited atom population distribution.Both these groups of processes promote the relaxation.Thus, for a more accurate conclusion the task related toevolution of eedf should be solved simultaneously with

the collisional-radiative state-to-state kinetics of relaxa-tion of the ionization degree and excited atoms popula-tion distribution w79x.

4. Conclusions

In the present study we have reported recent effortsboth experimental and theoretical for characterizing theplume expansion in laser-induced plasmas, restrictingour analysis to nanosecond regime.

From the experimental point of view, spectroscopicmethods used to investigate the time evolution of theplume have been discussed, with particular attention tothe characterization of TOF profiles. The fundamentalassumptions for OES experiments have been consideredin order to point out the limits and the constraints ofexperimental techniques and to underline the necessityof theoretical modeling.

Several fluid-dynamic models including NavierStokes and DSMC to characterize the plume expansionhave been discussed. Three-dimensional axisymmetriccompressible NavierStokes equations for a multispe-cies gas w24x, including two-temperature modeling (T,T) and the ambipolar diffusion are presented. Theedeviation of velocity distribution of the ablated particlesfrom the Maxwellian one and the difficulty of investi-gating diffusive processes have been considered in theframe of direct Monte Carlo simulation w26,27,70,71x

and of a new hybrid method w57x.An Euler code including non-equilibrium plasma

kinetics has been used to understand the role of chem-

istry in affecting the fluid-dynamics w23,25,44,54x.Moreover, the same approach has been validated bycomparing theoretical and experimental TOF profiles.

Finally, the development of microscopic models whichtake into account collisional-radiative models as well asnon-Maxwell velocity distribution functions of free elec-trons seems to be a promising tool to completely shedlight on the complex LIP phenomenology.

Acknowledgments

This work has been partially supported by MIUR(

PON TECSIS and contract 2001031223-009)

.

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