Interpretation of time-of-flight distributions for neutral particles under pulsed laserevaporation using direct Monte Carlo simulationAlexey A. Morozov

Citation: The Journal of Chemical Physics 139, 234706 (2013); doi: 10.1063/1.4848718 View online: http://dx.doi.org/10.1063/1.4848718 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/23?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Evolution of the plasma composition of a high power impulse magnetron sputtering system studied with a time-of-flight spectrometer J. Appl. Phys. 105, 093304 (2009); 10.1063/1.3125443 Modeling of Vapor Expansion under Pulsed Laser Ablation: Timeofflight Data Analysis AIP Conf. Proc. 762, 373 (2005); 10.1063/1.1941565 Optical emission spectroscopy and time-of-flight investigations of plasmas generated from AlN targets in casesof pulsed laser deposition with sub-ps and ns ultraviolet laser pulses J. Appl. Phys. 93, 2244 (2003); 10.1063/1.1539537 Time-of-flight study of the ionic and neutral particles produced by pulsed-laser ablation of frozen glycerol J. Appl. Phys. 90, 3623 (2001); 10.1063/1.1398068 Monte Carlo simulation of the laser-induced plasma plume expansion under vacuum: Comparison withexperiments J. Appl. Phys. 83, 5075 (1998); 10.1063/1.367324

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THE JOURNAL OF CHEMICAL PHYSICS 139, 234706 (2013)

Interpretation of time-of-flight distributions for neutral particlesunder pulsed laser evaporation using direct Monte Carlo simulation

Alexey A. Morozova)

Institute of Thermophysics SB RAS, 1 Lavrentyev Ave., 630090 Novosibirsk, Russia

(Received 14 October 2013; accepted 2 December 2013; published online 19 December 2013)

A theoretical study of the time-of-flight (TOF) distributions under pulsed laser evaporation in vacuumhas been performed. A database of TOF distributions has been calculated by the direct simulationMonte Carlo (DSMC) method. It is shown that describing experimental TOF signals through theuse of the calculated TOF database combined with a simple analysis of evaporation allows deter-mining the irradiated surface temperature and the rate of evaporation. Analysis of experimental TOFdistributions under laser ablation of niobium, copper, and graphite has been performed, with the eval-uated surface temperature being well agreed with results of the thermal model calculations. Generalempirical dependences are proposed, which allow indentifying the regime of the laser induced ther-mal ablation from the TOF distributions for neutral particles without invoking the DSMC-calculateddatabase. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4848718]

I. INTRODUCTION

Time-of-flight (TOF) measurements represent an impor-tant instrument for study mechanisms and dynamics of pulsedlaser ablation and desorption.1–7 The TOF distributions char-acterize the velocity distribution of the laser-produced parti-cles, allows to reveal different populations, and to estimatetheir translational energy. In these distributions, informationon the mechanism of ablation (thermal or non-thermal), onthe thermodynamic state of the irradiated surface, and on thewhole ablation dynamics is encoded, including the amountand composition of the ablated material, gas-dynamic andplasma processes, effects of ionization, and absorption oflaser irradiation within the plume. Interpretation of the TOFdistributions could considerably assist in analysis of processesaccompanying pulsed laser ablation (PLA) and hence facili-tate developing laser ablation-based techniques on thin filmdeposition and nanomaterial synthesis.1, 2 On the other hand,the TOF interpretation is vital for analysis of laser-induceddesorption of atoms and molecules from solid surfaces instudies of fundamentals such as surface electronic structuresand dynamics.8, 9 However, this problem has not been solvedin many years of research. Even a simpler question as to whatthe temperature of the irradiated surface corresponding to aparticular TOF distribution for laser-produced neutral parti-cles is has not been presented unambiguously.

The TOF signals for a density-sensitive detector for col-lisionless expansion are fitted by a Maxwell-Boltzmann dis-tribution

I (t) ∼ 1

t4exp

{− (L/t)2

2kTf ree/m

}, (1)

where L is the distance to the detector, k is the Boltzmann con-stant, m is the molecular mass, Tfree is the surface temperature.Under collisional expansion the velocity distribution function

a)E-mail: morozov@itp.nsc.ru

is modified, and the TOF signals are commonly approximatedby Eq. (1) with a superimposed flow velocity

I (t) ∼ 1

t4exp

{−

(L/t − Vf low

)2

2kTf low/m

}, (2)

where Vflow and Tflow are some velocity and temperature,which characterize the formed flow.3

Sometimes the fitting parameters Vflow and Tflow ofEq. (2) are presumed to be equal to the parameters at theKnudsen boundary layer.3, 4 However in the context of thisapproach, the gas-dynamic parameters of the flow should notchange after the Knudsen layer, which is not correct for manyexperimental conditions when the plume molecules undergomany collisions during expansion. As a result, temperaturedetermined by this approach greatly exceeds the real irradi-ated surface temperature.7 An analytical model of TOF distri-butions which takes into consideration the gas expansion afterthe Knudsen layer formation has been proposed recently.10

However, the model includes an additional free parameter,characterizing the transition from the continuum-like flow tothe collisionless one, which complicates the analysis.

Moreover, angular distributions have shown that TOFdistributions for different angles cannot be described by onecouple of the parameters. To take into consideration the angu-lar dependence, Kools et al.11 proposed the elliptic Maxwell-Boltzmann distribution with three independent parameters(velocity and radial and axial temperatures). However, theseparameters can be considered only as fitting parameters with-out clear physical understanding. To overcome these difficul-ties, Zhigilei and Garrison12 proposed a modified functionwith two physical parameters (the plume temperature and themaximum stream velocity), which however is of limited util-ity for analysis of real TOF distributions. To summarize, thevalid theory of TOF distributions is still lacking.

The TOF distribution depends on many factors, such asmechanism of ablation (thermal, photochemical, phase explo-sion, Coulomb explosion, etc.), ionization of the plume and

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234706-2 Alexey A. Morozov J. Chem. Phys. 139, 234706 (2013)

absorption of laser irradiation within the plume, gas-dynamicand plasma processes, presence of the background gas. In theframes of the present work, we consider the simple case ofthermal evaporation followed by gas-dynamic expansion ofneutral plume into vacuum. Such a problem statement is ap-propriate for analysis of any TOF measurements of neutralparticles which are vaporized, sputtered, or desorbed due apulsed heat source.3 To simulate such flows, the direct simu-lation Monte Carlo (DSMC) method13 was used. This methodis widely used for simulation of rarefied flow, including thosetypical for PLA applications.14–22 When it is applied for PLAsimulation, it is presumed that thermal evaporation takes placeand the laser-induced plume is neutral. It was shown that thismethod can be used for adequate modeling of TOF distribu-tions under PLA in vacuum.7 The DSMC method has beenused for analysis of TOF distributions under PLA into vacuumand a background gas with special emphasis on the evapora-tion heat effect20 and on the influence of the number of evap-orated monolayers.22

This work presents results of numerical study of TOF dis-tributions by the DSMC method. It is shown that using the cal-culated database of TOF distributions combined with a simpleanalysis of evaporation allows to interpret the experimentalTOF distributions and to determine both the irradiated surfacetemperature and the amount of the evaporated material.

II. MODEL

A 2D axial-symmetric problem of pulsed evaporationof molecules into vacuum is considered. The mechanism ofnormal evaporation23 is supposed. This mechanism is com-monly considered to be adequate for describing nanosecondlaser ablation and desorption for moderate laser fluences (upto 10–20 J/cm2) for quite different materials: metals,24–26

graphite,26–28 or semiconductors,26, 29, 30 with the calculatedevaporation rate being well agreed with numerous experimen-tal measurements.24–26, 29, 30 It should be noted that for theconsidered conditions the experimentally observed evapora-tion depth can be as much as 20 ÷ 300 nm,24–26, 30 which isequivalent to hundreds of evaporated monolayers.

Molecules are evaporated with energy corresponding tothe surface temperature T. The pressure of saturated vapor isdetermined by the Clausius–Clapeyron equation

pS (T ) = pb exp

{LV

k

(1

Tb

− 1

T

)}, (3)

where Tb is the boiling temperature under pressure pb, LV isthe latent heat for evaporation. During time interval τ vap, par-ticle flux �VAP is constant and equal to

�V AP (T ) = 14nS(T )uT (T ), (4)

where nS(T) = pS(T)/kT, uT(T) = √8kT /(πm). The hard

sphere model is used. A polyatomic gas with j internal de-grees of freedom is considered. To account for the internaldegrees of freedom, the Larsen-Borgnakke model is used.13

Collisions are classified as elastic for monatomic gases andinelastic for polyatomic gases. If a collision is considered asinelastic, the total energy is reallocated between the transla-tional and internal modes by sampling from the equilibrium

distributions of these modes. All backscattered molecules areassumed to recondense at the surface.

The problem is characterized by two parameters: the spotradius R and the number of evaporated monolayers

� (T ) = �V AP (T ) τV AP S = 116nS(T )uT (T )τV AP σ, (5)

where � = σ /4 is an area occupied by one molecule at thesurface, σ = πd 2, d is the hard sphere diameter. These pa-rameters allow defining the inverse relative thickness of theinitial gas cloud

b = R/(uT τV AP ). (6)

The considered problem requires large computer resources,and initial stages for some regimes (with a dense initial cloud,e.g., � > 10 for b = 10) were calculated using the “equilib-rium” modification of the DSMC method.31

Calculations have been performed for a wide range of thedetermining parameters: � = 0.001 ÷ 100, b = 0.1 ÷ 100.To perform calculations for � > 100 was not possible due tocomputer limitations. However, it should be noted that withfurther increasing � the obtained flow should approach to alimiting continuum flow (e.g., one-dimensional simulationshave demonstrated that the TOF distribution for � = 1000is very close to the continuum one22). Thus, the consideredrange of parameters is believed to be sufficient for approxi-mate analysis.

For each pair (�, b), the calculation has been performeduntil completely collisionless stage of the expansion. Duringthe calculation in different time moments the velocity distri-bution was calculated for those molecules which were locatedwithin a 2◦ cone around the normal to the surface. Duringthe initial stage of the expansion, the velocity distribution isstrongly modified, whereas after time t = 1000 ÷ 10 000 τ

it becomes invariable. Calculations have demonstrated thatdecreasing the cone angle has small effect on the obtaineddistribution. Thus, the final (invariable) velocity distributionf (u, �, b) is considered as the velocity distribution ofmolecules, which should pass through a detector located atthe normal at the infinitely large distance to the surface. Thecalculated set of velocity distributions f (u, �, b) has beenused as a reference database for analysis of real experimentalTOF distributions in order to clarify whether it is possible todetermine the actual surface temperature.

The following approach is proposed. Suppose that wewant to determine temperature of the evaporating surfacefrom an experimental TOF distribution Iexper(t), measured atthe normal to the surface. Scanning all possible pairs (�, b)is performed, and for each pair we calculate the TOF distribu-tion

I (t,�, b, T ) = 1

t2f

(L

t√

2kT /m,�, b

),

where T is the supposed surface temperature, which is variedin some range in vicinity of the collisionless temperature Tfree

from Eq. (1). Then we calculate the mean square deviation

�(�, b, T ) =√

1N

∑N

i=1(I (ti , �, b, T ) − Iexp er (ti))2.

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234706-3 Alexey A. Morozov J. Chem. Phys. 139, 234706 (2013)

Dependence �(�, b, T) has a minimum for the best-fit tem-perature Tfit(�, b) with the corresponding minimum deviationε(�, b) = �(�, b, Tfit). This temperature Tfit is the most prob-able temperature of the evaporating surface under assumptionthat for the experimental conditions there is evaporation of� monolayers and the initial plume thickness is equal to b.It is worth noting that by deriving the best-fit temperatureTfit(�, b) no other relationship between the surface temper-ature and the number of monolayers is assumed. Later, itwill be shown that analyzing the field Tfit(�, b) and usingEqs. (3)–(5) (which describe the relation between the surfacetemperature and the number of monolayers under evapora-tion) it is possible to determine the real surface temperatureTreal.

III. ANALYSIS OF EXPERIMENTAL DATA

A. Niobium

As an example, let us consider laser ablation of niobium(Nd:YAG laser, 13 ns FWHM, 6 J/cm2).7 The detector islocated at distance L = 68 mm, the surface temperature isT0 = 6757 K (this temperature was determined based on thethermal model of PLA combined with DSMC calculations).Fitting temperatures determined using formulas (1) and (2)are Tfree = 13 160 K and Tflow = 10 200 K (at Vflow = 520 m/s)accordingly.

Figure 1 shows the experimental TOF distribution incomparison with DSMC ones for � = 0.3 and 10 and the fit-ting formula (1) (� = 0). The inset shows the mean squaredeviation �(�, b, T) as a function of temperature T for� = 0.3. One can see minimum for Tfit = 9270 K. Figure 2presents corresponding fields of the best-fit temperatureTfit(�, b) and deviation ε(�, b) for the whole range of � andb. The best-fit temperature is very sensitive to the parametersof the problem and changes from 13 160 K for collisionlessexpansion (� = 0) down to 6000 K for � = 100. The de-viation ε is fairly good (ε < 0.1) for a very wide range ofparameters without any pronounced minimum, practically forthe whole zone except for the right lower region (� > 10,b < 10). Figure 1 illustrates this result, presenting TOF dis-tributions for quite different values of �. The obtained best-fit temperature and deviation are Tfit = 13 160 K, ε = 0.070(for � = 0), Tfit = 9270 K, ε = 0.068 (for � = 0.3), and

FIG. 1. The TOF distribution under Nb ablation: experiment from Ref. 7 vsDSMC calculations for b = 10 and � = 0, 0.3, 10. The inset shows the meansquare deviation �(�, b, T) as a function of temperature T for � = 0.3.

FIG. 2. The fields of best-fit temperature Tfit(�, b) (a) and minimum devia-tion ε(�, b) (b) obtained from analysis of experimental TOF distribution forNb.7 Cycles (◦) show pairs (�, b), chosen for TOF presentation in Figure 1,the cross (×) shows solution of Bykov et al.,7 the dashed line shows solutionsobtained using relation TVAP(�).

Tfit = 6960 K, ε = 0.092 (for � = 10). All three distribu-tions fairly well describe the experimental signal. Thus, anunfavourable conclusion can be made that in the general caseit is not possible to determine the surface temperature fromthe TOF distribution, since the same experimental distributioncan be theoretically obtained under quite different conditionsof ablation. Generally, it confirms conclusion of the previousstudy22 on inability to identify the regime only from the TOFdistribution shape.

It should be noted that Figure 1 clearly illustrates con-siderable plume transformation due to interparticle collisionsduring the expansion. In particular, the very close TOF distri-butions for � = 0 and � = 10 indicate that for � = 10 thereis a strong increase of the kinetic energy of particles in theTOF distribution in the comparison with the kinetic energyof the same particles during evaporation. Really the energyof particles during evaporation is E = 2kT ≈ 1.2 eV (assum-ing the surface temperature T = 6960 K for � = 10), whileenergy in the TOF distribution is nearly equal to the energyof particles during evaporation for collisionless expansionE ≈ 2.27 eV (assuming the surface temperature T = 13 160 Kfor � = 0). Thus, one can estimate the energy increase by1.07 eV, which corresponds to 89%. Accurate calculationsshow that for the monatomic gas the energy can increase bymore than 100%.22 Usually this energy increase is attributedto the Knudsen layer effect.3 However, Kelly and Dreyfus3

have shown that for the monatomic gas the Knudsen layercan give the energy increase of no more than 26%. Thus,the Knudsen layer is responsible only for the initial stage ofmodification of the velocity distribution of the laser plumemolecules, while actually it continues to alter during pro-longed time of the expansion.

For typical nanosecond PLA applications, the ratio b≥ 10.32 As may be inferred from Figure 2(b), for b >

10 the best-fit temperature Tfit weakly depends on b, butstrongly depends on �, which allows us to consider the TOF

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234706-4 Alexey A. Morozov J. Chem. Phys. 139, 234706 (2013)

FIG. 3. The best-fit temperature Tfit and deviation ε obtained from analysisof Nb TOF distribution7 as functions of the number of evaporated monolayers� for b = 10 and 100. The relation TVAP(�) obtained using Eq. (5) for Nbfor evaporation duration τVAP = 0.5; 1; 2 τ , the critical temperature Tc, andthe solution obtained using the thermal model of PLA from Ref. 7 (� = 15,T = 6757 K) are presented as well.

distribution as a function of the number of evaporated mono-layers � only.22 Figure 3 shows temperature Tfit and deviationε for b = 10 and 100 as a function of �. Such a presentationvisually demonstrates sharp decrease of the best-fit tempera-ture with increase of � from 0.01 up to 1. One can see thatactually dependences for temperature Tfit for b = 10 and 100are closely related to each other. Figure 3 clearly demonstratesthat deviation ε begins to considerably increase only for� > 10. Given the number of evaporated monolayers �, itis possible to determine the real surface temperature Treal bythe use of Figure 3.

Since Tfit is the supposed temperature of the evaporat-ing surface, from Eq. (5) one can evaluate the number ofmonolayers which should be evaporated under this temper-ature. Given niobium properties presented in Table I, such adependence has been calculated, and its inversion TVAP(�) isshown in Figure 3. The only unknown parameter in Eq. (5)is the evaporation duration τVAP. For nanosecond laser abla-tion, this duration is known to correlate with the laser pulseduration τ .7, 33, 34 We used values of duration τVAP = 0.5; 1;2 τ . The median curve (τVAP = τ ) intersects the Tfit curves at(b = 10, � = 18, Treal = 6710 K) and (b = 100, � = 24, Treal

= 6860 K), which are very close to the correct values (b = 25,� = 15, T0 = 6757 K). Taking into account smaller and largerevaporation duration (0.5 and 2 τ ), the obtained real temper-atures are Treal = 6710 ± 180 K for b = 10 and Treal = 6860± 240 K for b = 100, which corresponds to possible error of

3%. Generally, all three curves are closely related, and in thefollowing analysis we use τVAP = τ .

In Figure 3, one can see the critical temperature for nio-bium (Tc = 7610 K) as well. This temperature is well abovethe obtained real surface temperature Treal, which indicatesplausibility of the obtained results. Generally, the critical tem-perature sets the upper bound for the found temperature Treal.

If we consider intersection of TVAP(�) with the surfaceTfit(�, b), we obtain a curve �(b), which describes all possiblesolutions in the range b = 0.1 ÷ 100. As one can see fromthis curve in Figure 2, the number of monolayers varies from� = 20 for b > 10 up to � = 100 for b = 1. For b < 1, thereis no any solution.

To summarize, from general considerations on thermalcharacter of ablation and using the DSMC-based fitting wehave determined the real surface temperature and the num-ber of evaporated monolayers for Nb, which are very close tovalues calculated using the thermal model of PLA.

B. Copper

As an another example let us consider copper ablation(KrF laser, 15 ns FWHM, 4.5 J/cm2).38 The detector is locatedat distance L = 218 mm. Unlike the first example, for this casethermal analysis has not been performed and all parameters ofablation (the surface temperature, the number of monolayers�, and the ratio b) are unknown.

Figure 4 shows the experimental TOF distribution, fittedby Eqs. (1) and (2). These fitting give quite different temper-ature: Tfree = 10 280 K for (1) and Tflow = 2750 K for (2).Figure 5 shows plots of Tfit and ε, obtained using the aboveDSMC-based fitting. For this case, there is good fitting forrange � > 0.3, with the best fitting occurring for � ≈ 20.

The used copper properties are presented in Table I.To estimate the hard sphere diameter d, parameters of theLennard-Jones model σ = 2.277 Å and ε = 0.415 eV39 havebeen used. Assuming vapor temperature 5000 K, the diametercan be estimated as d = 2.8 Å.21 The calculated dependenceTVAP(�) (5) is presented in Figure 5. One can see that it inter-sects the Tfit curves at (b = 10, � = 60, Treal = 4530 K) and(b = 100, � = 80, Treal = 4710 K), with the obtained tem-perature Treal being lower than the critical one. Let us take,for example, b = 10 and � = 50 and fit the experimental sig-nal by the corresponding DSMC distribution. As can be seenfrom Figure 4, the obtained DSMC fit fairly well describes theexperiment. The fitting temperature is Treal = 4520 K, which

TABLE I. Material properties used for calculations and analysis.

Nb Cu Graphite

Atomic mass m (amu) 93 63.5 36Boiling temperature Tb (at 105 Pa) (K) 507326 281635 4084, T < 5000 K

3261, T > 5000 K36

Critical temperature Tc (K) 761026 539035 671026

Latent heat LV (kJ/mole) 722.826 30235 851, T < 5000 K357, T > 5000 K36

Diameter of hard sphere d (Å) 2.97 2.8 1.9837

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234706-5 Alexey A. Morozov J. Chem. Phys. 139, 234706 (2013)

FIG. 4. The TOF distribution under Cu ablation: experiment from Ref. 38vs fit (1) (Tfree = 10 280 K), fit (2) (Tflow = 2750 K, Vflow = 1695 m/s), andDSMC calculations for b = 10, � = 50 (Treal = 4520 K).

is significantly different from those determined using fittings(1) (Tfree = 10 280 K) and (2) (Tflow = 2750 K).

C. Graphite

As the third example, we refer to graphite ablation(Nd:YAG laser, 7 ns FWHM, 0.45 J/cm2).10 The detector islocated at distance L = 81 mm. As in the case of niobium,the thermal model of PLA combined with DSMC calculationshas been applied, which allowed to determine parameters ofablation: the surface temperature T0 = 4920 K, � = 1.8,b = 6.10 Fitting the experimental TOF distribution by Eq. (1)gives temperature Tfree = 13 290 K,10 while Eq. (2) gives un-physical values: Tflow = 18 860 K and negative velocity Vflow

= −1460 m/s, which obviously demonstrates a limitation ofusing Eq. (2).

In contrast to the first two cases (where monatomicgases have been considered), the main detected species wasmolecule C3. Since vibrational degrees of freedom are knownto demand many collisions for their deactivation,40 only ro-tational degrees of freedom are taken into consideration. C3

molecule is known to be a linear one,41 thus we proposej = 2.

Figure 6 shows plots of Tfit and ε, obtained using theabove DSMC-based fitting. For this case, there is good fit-ting for range � < 10, with the best fitting occurring for �

FIG. 5. The best-fit temperature Tfit and deviation ε obtained from analysisof Cu TOF distribution38 as functions of the number of evaporated monolay-ers � for b = 10 and 100. The relation TVAP(�) obtained using Eq. (5) forCu and the critical temperature Tc are presented as well.

FIG. 6. The best-fit temperature Tfit and deviation ε obtained from analysisof C3 TOF distribution10 as functions of the number of evaporated monolay-ers � for b = 10 and 100. The relation TVAP(�) obtained using Eq. (5) forC3, the critical temperature Tc, and the solution obtained using the thermalmodel of PLA from Ref. 10 (� = 1.8, T = 4920 K) are presented as well.

≈ 0. Given C3 properties presented in Table I, dependenceTVAP(�) has been calculated (Figure 6). One can see that itintersects the Tfit curves at (b = 10, � = 2.7, Treal = 5030 K)and (b = 100, � = 1.6, Treal = 4880 K), which are very closeto the correct values (b = 6, � = 1.8, T0 = 4920 K).

It should be noted that despite widely using TOF spec-tra for analysis of experimental data, it is rather difficult tofind sufficient experimental TOF spectra for verification of theproposed model. Usually TOF distributions are measured forionized gases or gas mixtures or under conditions of intenseabsorption of incident laser beam within the plume, whichare beyond the limits of the current model. For TOF spectraof neutral species usually there is no information about thesurface temperature. To the best of our knowledge, the onlytwo experimental works which provide the surface tempera-ture during ablation as well as the TOF spectra are the pre-sented studies of niobium and graphite ablation.7, 10 Copperis simply taken as an example of application of the proposedapproach for analysis of arbitrary experimental TOF data.

D. Influence of spot size

Often it is difficult to determine the ratio b for experi-mental conditions, especially taking into account that sizes ofirradiated and evaporating spots can be substantially differ-ent. Therefore, it seems to be actual to estimate effect of thespot size on the determined surface temperature. To performsuch an analysis, for each value b intersection between curvesTfit(�) and TVAP(�) (5) has been calculated. The obtained re-sults can be seen in Figure 7. For Nb results are presented fordifferent values of τVAP. One can see that for b > 10 (which isof practical interest) the obtained real temperature Treal variesno more than 4%. Thus, this confirms our previous conclusionabout insignificant effect of the spot size on the determinedtemperature.

IV. GENERAL MODEL

In the previous paragraph, it was demonstrated thaton the basis of DSMC-calculated TOF distributions it is

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234706-6 Alexey A. Morozov J. Chem. Phys. 139, 234706 (2013)

FIG. 7. The real surface temperature Treal determined from intersection ofcurves Tfit(�) and TVAP(�) as a function of the initial inverse plume thicknessb from analysis of experimental data on Nb (for τVAP = 0.5; 1; 2 τ ), Cu, andC3. Results from thermal model calculations for Nb (×) from Ref. 7 andC3 (◦) from Ref. 10 are presented as well.

possible to determine the parameters of laser-induced evap-oration from experimental TOF distributions. However, us-ability of this approach is rather limited, since it requiresavailability of a reference database of TOF distributions, cal-culated using the DSMC or another numerical method for awide range of parameters. Therefore, it is important to gen-eralize the obtained results in such a way that would allowto approximately estimate the surface temperature from TOFdistributions without referring to any numerical calculations.

First, let us reduce the results of Figures 3 and 5 (ob-tained for monatomic gases Nb and Cu) to a common presen-tation. For this purpose, we should normalize temperature Tfit

by temperature Tfree, determined using Eq. (1). The obtaineddependences are presented in Figure 8. One can see that de-spite quite different experimental TOF signals, the curves arevery close. It can be proposed that fitting the experimentalTOF distribution by a calculated one is good in the case whenthe peaks of the TOF distributions coincide. Then it is possi-ble to calculate an universal dependence Tpeak(�), based oncoincidence of the TOF peaks for different values of �.

Let an experimental TOF distribution is fitted by the col-lisionless TOF distribution (� = 0, which corresponds to fit

FIG. 8. The best-fit temperature Tfit obtained from analysis of Nb and CuTOF distributions and temperature Tpeak from fitting peaks of the TOF distri-butions, normalized to Tfree from Eq. (1), for j = 0 as functions of the numberof evaporated monolayers � for b = 10 and 100. Experimental data on Xelaser desorption from Ref. 42 are presented as well.

(1)) under temperature Tfree. Then for any � = �′ > 0 theTOF distribution is shifted (in comparison with � = 0) to-wards less times, and it is possible to select such a temperatureTpeak < Tfree that the peak of the TOF distribution for �′ withtemperature Tpeak coincides with the peak of the TOF distribu-tion for � = 0 with temperature Tfree. It can be straightforwardshown that

Tpeak

Tf ree

=(

tpeak,f ree

tpeak,�

)2

,

where tpeak,� and tpeak,free are time moments corresponding tothe peaks of the TOF distribution for �′ and � = 0 corre-spondingly under the same temperature Tfree.

The temperature dependence Tpeak(�), thus defined, isshown in Figure 8. It is seen that this curve is closely relatedwith curves for Nb and Cu, although some distinction occursfor � > 1. Similar good agreement has been also obtainedunder comparison of curves Tpeak(�) for j = 2 and Tfit(�)for C3.

The obtained results on TOF acceleration with increas-ing the number of monolayers can be qualitatively confirmedby experimental data on laser desorption of xenon.42 UnderXe desorption for different values of the number of desorbedmonolayers evident acceleration of the plume has been ob-served, with the TOF peak being at tpeak = 440, 402, 361,357 μs for � = 0.3, 1.7, 6.7, 10 correspondingly. Since reli-able data on the surface temperature under desorption is ab-sent, we can link these points to the calculated dependence byproposing that Tpeak = 0.7Tfree for � = 0.3 (Figure 8). Gen-erally, these Xe data confirm the numerically observed plumeacceleration with � increasing from 0.3 to 10.

Dependences of this kind have been obtained for differ-ent values of the number of internal degrees of freedom j(Figure 9). One can see that the minimum surface tempera-ture strongly depends on j: it decreases from 0.5Tfree for j = 0down to 0.15Tfree for j = 9. Also for illustration dependenceTVAP(�) for Cu is shown, normalized to temperatures 10 000K and 3000 K. Assuming that temperature Tfree (determinedusing fitting (1) of an experimental TOF distribution) is equalto 10 000 K, we obtain crossing curves TVAP(�) and Tfit(�) for

FIG. 9. Temperature Tpeak obtained from fitting peaks of the TOF distribu-tions, normalized to Tfree from Eq. (1), for j = 0, 2, 9 as a function of the num-ber of evaporated monolayers � for b = 10 and 100. The relations TVAP(�)obtained using Eq. (5) for Cu normalized by 10 000 K and 3000 K are pre-sented as well.

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234706-7 Alexey A. Morozov J. Chem. Phys. 139, 234706 (2013)

FIG. 10. Temperature Tpeak obtained from fitting peaks of the TOF distribu-tions, normalized to Tfree from Eq. (1), as a function of the number of internaldegrees of freedom j for � = 10 and b = 10.

j = 0 at � ≈ 30, that corresponds to Treal ≈ 0.45Tfree = 4500K. Assuming, for example, that Tfree = 3000 K, we have thenormalized curve much higher in comparison with the caseTfree = 10 000 K (Figure 9), and the intersection occurs at �

≈ 0.05, that corresponds to Treal ≈ 0.85Tfree = 2550 K.Figure 9 clearly demonstrates great effect of internal

degrees of freedom on the determined surface temperature.Therefore, it seems to be informative to present a figure il-lustrating influence of the number of internal degrees of free-dom on the temperature (Figure 10). One can see that withj increase from 0 up to 100 the temperature falls from 0.48down to 0.12 Tfree. It is important to keep in mind that herethe limiting case of instantaneous internal-translational relax-ation is considered, when every collision results in an equi-librium state over all degrees of freedom. For most of themolecules with vibrational degrees of freedom, the effect willbe some less due to long-continued vibrational-translationalrelaxation.40

The general dependences presented in Figure 9 can beused for self-contained analysis of experimental TOF distri-butions and for approximate determination of the evaporatingsurface temperature and amount of evaporated matter. Surelythis approach does not allow verifying the obtained resultsby fitting of the experimental TOF distribution by DSMC-calculated ones with calculation of the mean deviation. There-fore, for accurate analysis the above DSMC-based approachis required.

Some comments on limitations and propositions of themodel are required. First, the proposed model presumesthe thermal mechanism of ablation, when there is normalevaporation23 in accordance with the Clausius–Clapeyronequation (3). Otherwise, the dependence TVAP(�) will besomewhat different from the considered one. Besides, pres-ence of clusters or droplets in the plume can substantiallychange gas-dynamics and therefore modify dependencesTfit(�).

Second, we presume that the axis of the laser-inducedplume expansion coincides with the normal to the irradiatedsurface. This usually takes place when the target is irradiatedat normal incidence, whereas for oblique incidence the plumecan deflect from the normal.43

Third, it is presumed that there is neither ionization norlaser irradiation absorption within the laser-induced plume.Once these prepositions do not occur, the plume more inten-sively accelerates forward. At that the temperature Tfree is in-creased so that the normalized curve Tvap(�)/Tfree goes downand intersects with the curve Tpeak(�) for larger values of �

or do not intersect it at all. For this situation, the real temper-ature Treal found can be higher than the critical temperatureTc. Thus, for the case of anomalous acceleration (caused byionization or irradiation absorption within the plume), the ap-proach gives overestimated values of � and Treal or fails togive any solution.

V. CONCLUSION

A theoretical study of the TOF distributions under pulsedlaser evaporation in vacuum has been performed. It is shownthat describing experimental TOF distributions with the useof the DSMC-calculated TOF database combined with a sim-ple analysis of evaporation allows determining the irradiatedsurface temperature and the rate of evaporation. Analysis ofexperimental TOF distributions under laser ablation of dif-ferent substances has been performed, with the estimated sur-face temperature being well agreed with results of the thermalmodel calculations. General empirical dependences are pro-posed, which allow identifying the parameters of evaporationfrom TOF distributions for neutral particles without using theDSMC-calculated TOF distributions.

ACKNOWLEDGMENTS

The author thanks A. V. Bulgakov for fruitful discus-sions. The work was supported by grant of the Russian Foun-dation for Basic Research (No. 11-08-00100).

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