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High shock release in ultrafast laser irradiated metals: Scenario for material

ejection

Article  in  Physical Review B · April 2007

DOI: 10.1103/PhysRevB.75.104105 · Source: arXiv

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High shock release in ultrafast laser irradiated metals:

Scenario for material ejection

J.P. Colombier,1, 2 P. Combis,1 R. Stoian,2 and E. Audouard2

1CEA/DAM Ile de France, Dept de Physique Theorique et Appliquee, BP 12, 91680 Bruyeres-le-Chatel, France2Laboratoire Hubert Curien, Universite Jean Monnet,

UMR CNRS 5516, 18 rue Benoıt Lauras, 42000 Saint-Etienne, France(Dated: February 6, 2008)

We present one-dimensional numerical simulations describing the behavior of solid matter exposedto subpicosecond near infrared pulsed laser radiation. We point out to the role of strong isochoricheating as a mechanism for producing highly non-equilibrium thermodynamic states. In the caseof metals, the conditions of material ejection from the surface are discussed in a hydrodynamiccontext, allowing correlation of the thermodynamic features with ablation mechanisms. A conve-nient synthetic representation of the thermodynamic processes is presented, emphasizing differentcompetitive pathways of material ejection. Based on the study of the relaxation and cooling pro-cesses which constrain the system to follow original thermodynamic paths, we establish that themetal surface can exhibit several kinds of phase evolution which can result in phase explosion orfragmentation. An estimation of the amount of material exceeding the specific energy required formelting is reported for copper and aluminum and a theoretical value of the limit-size of the recastmaterial after ultrashort laser irradiation is determined. Ablation by mechanical fragmentation isalso analysed and compared to experimental data for aluminum subjected to high tensile pressuresand ultrafast loading rates. Spallation is expected to occur at the rear surface of the aluminum foilsand a comparison with simulation results can determine a spall strength value related to high strainrates.

PACS numbers: 61.80.Az, 72.15.Cz, 79.20.Ap, 64.70.Dv

I. INTRODUCTION

The understanding of matter ablation under pulsedlaser irradiation is a long-standing topic of condensedmatter physics and materials science.1 The employmentof ultrashort laser pulse have opened the possibility toobserve laser-induced transformations on ultrafast timescales. To give a complete description of the mechanismspreceding and determining material ejection from sur-faces, several physical processes, sometimes occurring si-multaneously, have to be examined. The first processinvolves the energy deposition of the laser pulse and itstransfer to the matter constitutive elements. Then, en-ergy distribution and related effects are strongly corre-lated with the temporal evolution of the transport prop-erties. Depending on the intensity of the pulse, a lo-cal mesoscopic transformation occurs on a picosecondtime scale, primarily due to a thermodynamic phasetransformation.2 The final process is ablation, which isrelated to the macroscopic ejection of matter in the solid,liquid, or gas phase. It has been shown recently that thequantity of ejected matter can be numerically estimated.3

A more detailed study of solid to plasma transforma-tion, subsequent to numerous intermediate thermody-namic states and associated material deformations arerequired for a better understanding of the mechanismsinvolved in the ablation process. In this paper, attentionis drawn to the role of shock release on the productionof non-equilibrium thermodynamic states and its influ-ence on the ablation mechanisms in the case of selectedmetals.

The importance of accessing information on transitionstates should be viewed in a broad context. The under-standing of the ultrafast transformations of material iscentral to both applied and fundamental physics. Theo-retical studies present a technological interest in the op-timization of material micro-machining and represent ascheme for the understanding of multi-scale processes re-lated to matter transformations. A large amount of workhas been devoted to give successive snapshots of the ther-modynamic states of matter preceding or following ma-terial ejection. For example, time-resolved optical mi-croscopy experiments performed by Sokolowski-Tinten etal.4 on laser-excited materials have revealed the appear-ance of Newton rings within several tens of picosecondsafter irradiation, which have been attributed to the pres-ence of a metastable, inhomogeneous state of matter.4,5 Amanifestation of a rarefaction wave, generated by cross-ing particular states of the phase diagram, has been alsosuggested to explain these observations.6 Based on a vander Waals type equation of state (EOS), simulations per-formed at thermal equilibrium have shown a spinodaldecomposition process occurring near the liquid-vaporcritical point.7 Optical probing of the properties of theisentropes lying above the critical point has been previ-ously proposed by Celliers et al.8 to investigate transportproperties of such thermodynamic states. More recently,molecular-dynamics calculations have indicated variousmechanisms, such as fragmentation, phase explosion, andvaporization, resulting from different trajectories in thethermodynamic phase-space. Each one is specific to away of solid ablation in different fluence regimes.9

2

The theoretical problem consists mainly in achievingsimulations of the laser-interaction process with sufficientcompleteness to reproduce the observed dynamics andto predict ways of improvement. In this way, the in-vestigation approach has to contain optical, thermal, hy-drodynamical, and mechanical models of non-equilibriumto reproduce subsequent interaction features. Komashkoet al.10 have reported simulation results which correlatelaser properties and material removal efficiency, obtainedusing the one-dimensional radiation hydrodynamic codeHYADES enhanced with a wave-equation solver. Sim-ilarly, using the hydrodynamic code MULTI that in-cludes electronic heat conduction, Eidmann et al. per-formed solid to plasma simulations of the subpicosec-ond laser-interaction with aluminum for a wide range oflaser intensities.11 Considering the previous works, noneof them insist on the evolution of thermodynamic con-ditions. A more general view, which can be potentiallyprovided by all the mentioned models, is still lacking, de-termining a certain loss of details concerning the disinte-gration mechanisms of the heated metal. The aim of ourwork is to gather non-equilibrium excitation and trans-port models in a unified fluid approach and to extendthe thermodynamic features, as revealed by the Molecu-lar Dynamics (MD) approach on small scales, to largervolume simulations. In fact, MD possibilities are limitedto a superficial region within the irradiation spot and thesimulation dimension is restricted to a few hundreds ofnanometers in depth.13 Hydrocodes are more suitable forstudying shock wave formation and propagation for sys-tems of mesoscopic scales, and the two approaches arecomplementary. To investigate the capability of report-ing thermodynamic insights within the framework of non-equilibrium fluid models, specific investigations implyingstrong expansion dynamics have been performed usingthe 1D hydrocode ESTHER.3,12 Details about the mainassumption and the formalism of this code are providedin the next section. Using this approach, our intention isto shed some light on some of the thermodynamical andmechanical aspects of ultrafast laser ablation.

Due to the energy deposition of ultrashort laser pulsein a swift manner on a thin layer of matter, electrons areheated quasi-isochorically in the solid sample. Actually,the induced deformations are driven by the shock and rar-efaction waves which travel at a velocity where a lowerestimate is given by the sound velocity. At this velocity,it takes for the shock wave about 10 ps to travel 50 nm inthe metal. This time is similar to the electron-ion relax-ation time and expansion is insignificant on the timescaleof the laser pulse. The heated matter will expand quasi-isentropically, will pass through unconventional physicalstates, and enter new regions of the phase diagram. Inthis study, we demonstrate the role of electronic con-tribution responsible for generating specific paths thatlead to material ejection under ultrafast laser irradia-tion. Once the maximum degree of non-equilibrium isreached, the role of the electronic pressure during re-laxation is outlined in order to investigate the sequence

of non-conventional thermodynamic states. To extendthe analysis of the thermodynamic paths observed dur-ing and after interaction with ultrashort pulses, we pro-vide an interpretation of the thermodynamic diagramswhich allows an estimation of the amount of metal thatundergoes mesoscopic transformation. This estimationcould present a direct interest for post-process analy-sis of ablation in terms of quality and efficiency. Fi-nally, we have also considered mechanical disintegrationfollowing a discussion related to spall experiments per-formed by Tamura et al.14 Comparison of these resultswith those obtained in our simulation show a fairly goodagreement, suggesting that the low-strain rate value ofthe spall strength can be applied on short time scales.

The paper is organized as follows. In section II webriefly describe our model of electron-ion non-equilibriuminteraction and the hydrodynamic simulation technique.Concentrating on the results of the simulations, wepresent in section III independent thermodynamic pathsfollowed by different layers inside the material. The re-sults of these simulations are discussed in the context ofthe possibility of material ejection. In section IV we pro-vide a numerical quantification of the remaining liquidon the solid substrate. In section V we compare profilesof computational stress to current experimental observa-tions of spall formation. In section VI conclusions aredrawn.

II. ELECTRON-ION NON-EQUILIBRIUM

The metal response under ultrashort pulse laser ir-radiation has been simulated using various numericaltechniques that emphasize particular issues of the mat-ter evolution and include a variety of hydrodynamicalprocesses.11,15,16 To be consistent with the experimen-tally measured properties of the material under irradia-tion, suitable models are needed to describe the absorp-tion and the intermediate stages related to phase trans-formations during the solid to plasma transition. A set ofaccurate thermophysical parameters has to be inserted inthe simulation to provide a correct picture of the excitedmatter. For a laser pulse of τ = 150 fs (FWHM), thewavelength, λ = 800 nm, is smaller than the coherencelength Lτ = cτ , where c is the speed of light. Conse-quently, to determine optical solid and plasma response,Maxwell equations can be reduced to the Helmholtz equa-tion by using slowly-varying-envelope approximation forthe electric field. In addition, if the pulse duration isless than a sonic wave period, τ < λ/cS , where cS isthe sound velocity, we can assume that hydrodynamicprocesses occur on a time scale longer than the char-acteristic scale for the evolution of the optical param-eters. As a consequence, the resolution of the station-ary equations of electrodynamics is sufficient to calculateradiation absorption. The large difference between theFermi electron velocity and the sound velocity requiresdistinguishing between the electronic and ionic thermal

3

evolutions during irradiation.17,18 The electron-ion non-equilibrium produces a hydrodynamic decoupling whereelectronic pressure contributes strongly to the materialdeformation. General aspects of the code have been de-scribed in detail elsewhere,3 and the main assumptionswere given above to facilitate understanding.

The hydrodynamic code used in this work solves theone-dimensional fluid equations for the conservation ofenergy, momentum, and mass, in a Lagrangian formal-ism. The specificity of this approach is the flexible di-mension of the cells which deform together with the ma-terial and permits an accurate description of shock prop-agation. Nevertheless, there is no mass transport be-tween the cells which imposes limitation on the descrip-tion of gas-phase transformation. The target material istreated as a compressible fluid composed of electronic andionic subsystems with independent internal energy, tem-perature, and pressure. In addition, the radiation fieldcorresponding to the incident laser pulse is absorbed inthe material through the classical inverse bremsstrahlunginvolving electron-ion collisions. Even with a plane-polarized wave reflecting from a semi-infinite metal, esti-mation of the amplitude and the phase of the field is notsimple and a numerical calculation of the self-consistentresponse is required. The absorption of the incident elec-tromagnetic wave by the inhomogeneous medium is cal-culated based on the Helmholtz equation involving a dy-namic behavior of complex refractive indices. Accordingto the Drude model, the refractive index evolution de-pends on the collision time which is initially determinedat solid density based on available experimental data.19

During the laser pulse, both the temperature increase andthe density decrease affect the laser absorption and opti-cal parameters are calculated using an interpolation be-tween the standard condition data19 and values given byconductivity models of dense plasmas.20 From a micro-scopic point of view, the energy contained in an incidentphoton is absorbed by a free electron in a Coulombianfield, which remains in the conduction zone but is pro-moted in a state with higher energy. Excited electronsinteract with lattice phonons and with other electrons,transferring their energy to the surroundings. During thelaser pulse, electrons which have absorbed photons un-dergo many collisions between themselves but few withlattice phonons. The energy absorbed by an electron isdistributed between electrons on a time scale of a few tensof femtoseconds and is then transferred to the lattice inseveral picoseconds.21 Our calculations begin with the as-sumption that the light energy is instantly transformedinto heat inside the electron gas. Thus, it is consideredthat the local equilibrium condition in the electronic sys-tem is verified during the entire pulse. Consequently, itis supposed that a certain electronic temperature is de-fined at each point. In this context, regular equations forthe description of the heat flow have been used and theTwo Temperature Model (TTM) has been implementedin our fluid treatment.17 As in Refs. [5,22], we haveemployed wide-range equations of state in tabular form,

which have been the subject of extensive development,and for which a great deal of high-quality experimentaldata is available.23 An additional electronic contributionto the total pressure has been artificially inserted in theequilibrium EOS to account for the overpressure yieldedby the strong electronic heating. The electronic pressurepe depends on the electronic temperature Te accordingto the Fermi-Dirac model and is expressed as :

pe =2

3A

∫

∞

0

E3/2

1 + exp [(E − µ)/kBTe]dE , (1)

where E is the internal electronic energy and kB is theBoltzmann constant. The chemical potential, µ, is deter-mined numerically to match the value of the electronicdensity, Ne, at a fixed electronic temperature. The Aparameter in equation (1) is defined as :

A =1

2π2

(

2me

~2

)3/2

, (2)

where me is the mass of the free electron. In order tomake a more realistic calculation of the electronic pres-sure, it is more suitable to use band-structure calcula-tions. Due to the complexity of these calculations, theyare beyond the scope of this work. Notwithstanding defi-ciencies of the Fermi-Dirac model, our simplified approxi-mations are reasonable and give a qualitative view on thebehavior of the non-equilibrium system. At solid den-sity, Fermi-Dirac model predicts an electronic pressurecontribution in the range of a few hundreds GPa for atemperature of 0K. This degeneracy pressure arises fromthe Pauli exclusion principle and our model assumes thatthe electronic contribution only considers the thermal ex-citation of the free electrons. To improve the description,strong corrections due to screening and chemical bondingmust be added and a cold pressure term is often suppliedin addition to the electronic and ionic contributions.24

To replace this usual correction, the equilibrium valueof the pressure resulting from all contributions has beenkept as a reference, and we have subtracted the electronicpressure calculated at ionic temperature and added avalue which depends on the electronic temperature. Asa consequence, the total pressure used in the above cal-culations is determined as p = pe(Te) − pe(Ti) + pa(Ti)where pe and pa are the electronic and atomic pressure,respectively. Note that the atomic pressure pa is given bythe employed EOS at the equilibrium temperature andis composed of the electronic, ionic, and cold pressurecomponents : pa = pe + pi + pc.

Fig. 1 shows the time evolution of the atomic and totalpressure profiles induced by a 5 J cm−2, 150 fs laser pulsein an aluminum sample. The atomic and total pressurecurves are identical at 6 ps due to the fact that electroniccontribution vanishes when the relaxation process ends.At this time the maximum amplitude of the atomic pres-sure is reached. For short times the curves are clearlydistinct. This is due to the role played by the electronicpressure in a non-equilibrium state. At this stage, the

4

-25 0 25 50 75 100 125

0

20

40

0

20

40

60

To

tal p

ress

ure

(GP

a)

Ato

mic

pre

ssu

re(G

Pa)

Depth (nm)

(b)

(a) t=1 ps t=2 ps t=3 ps t=6 ps

FIG. 1: (Color online). Non-equilibrium spatial distributionand time behavior of pressure for an aluminum target irradi-ated by a 150 fs, 5 J cm−2 laser pulse indicating the evolutionof (a) total pressure and (b) atomic pressure. The atomiccontribution is separated from total pressure before completerelaxation.

electronic pressure spreads at a velocity depending onthe electronic conduction, which, in turn, is governed inour simulations by the TTM. One can note that the max-imum values reached by the partial and the total pres-sure do not necessarily occur at the same depth, becauseelectronic conduction offsets the highest internal pressurerelative to the ionic component. Information about thedegree of non-equilibrium can be derived from the dif-ference between electronic and atomic pressure curves.Obviously, the gap between electronic and ionic contri-bution is more important at shortest times following ab-sorption and close to the surface, where the Fermi gas wassubjected to the electromagnetic field. This gap is pro-gressively reduced during the electronic cooling and ionicheating controlled by the TTM, ensuring at the sametime energy conservation during heat diffusion and cou-pling processes. The non-equilibrium remains observableon the first hundred of nanometers, and is especially sig-nificant up to 50 nm. The thickness of material affectedby strong non-equilibrium pressure is larger than the op-tical skin depth, which is about 8 nm for aluminum at800 nm, because the electronic diffusion pushes the elec-tronic pressure towards the bulk material. According tothe present EOS, the velocity of the total pressure wave

composed by electronic and ionic terms is limited by thesound velocity in near-solid-density aluminum. No shockprocess due to electronic diffusion velocity exceeding thesound velocity is expected to occur in the simulation.From Fig. 1, decoupling the pressure distribution con-veys information about the space and time scales affectedby the non-equilibrium dynamics. It results that, at thislaser energy density, these non-equilibrium dynamics sur-vive for several picoseconds and can propagate severaltens of nanometers under the surface. We will examinein the next section the role of the pressure decoupling ofthe material properties inside the metal.

III. SHOCK RELEASED THERMODYNAMIC

STATES

In our fluid treatment, internal properties and dynam-ics of the material evolve jointly and the ablated mate-rial is not discernible from the rest of the solid. Theuse of an ablation criterion is necessary but a defini-tion based on nucleation can be questionable in a one-dimensional simulation. Such models of matter separa-tion of the fluid and solid phases may be responsible forvoid formation and this feature was not implemented inour calculations. Consequently, the distinction betweenthe ablated and non-ablated material was chosen to bediscussed by means of the phase diagram analysis for theirradiated metal, as it will be developed below. To find apicture equivalent of the transformation phenomena thatleads to ablation, some authors have previously proposedto study the temporal evolution of the thermodynamicstate of the system in search for critical states.7,9,25 Todo this, they discussed phase-space trajectories followedby infinitesimal layers of matter after irradiation in the(T,ρ) plane of the phase diagram. If several characteris-tic paths have been identified which may result in mate-rial removal, the role of the pressure and its influence onthe non-equilibrium has not been indicated, due to thechoice of coordinates.

In the following, we consider again the case of a 5J cm−2, 150 fs laser pulse irradiating metal samples andwe will try to connect the space and time observationsto pressure-related coordinates for showing the effectsof the pressure. We present the successive thermody-namic states in a pressure-density (p,ρ) phase diagram.The choice of these thermodynamical parameters is well-suited to represent the achievement of new states. In-deed, the electronic contribution is directly visible dueto the additional features of the partial pressures. Weremind that the hydrodynamical approach used here isbased on a Lagrangian formalism and we are studyingthe behavior of several cells, initially localised at a fixeddepth inside the cold homogeneous material precedingthe laser pulse arrival. Each cell follows, independently ofthe others, a thermodynamic path governed by the EOS,providing the pressure as a function of the internal energyand the metal density. Phase-diagrams were obtained by

5

tracking down the pressure time evolution and correlat-ing it with the density behavior based on the EOS. Atthe considered fluence and due to the limited energy dif-fusion, only cells located initially in the first hundreds ofnanometers show dramatic changes. However, a region of50 µm thickness was simulated to avoid spatial confine-ment of the diffused energy and waves reflection at therear side of the sample which can happen in thin foils.Simulations have been performed for an ultrashort pulseirradiating two types of metals, aluminum and copper,which differ significantly with respect to their electron-lattice coupling (strong coupling for aluminum, and, re-spectively, weak coupling for copper). The transport andthermodynamical data for these metals are quite accu-rately known for equilibrium conditions and this datahas been extended to non-equilibrium situations usingthe Two-Temperature Model framework.3,20,26 For eachmetal the set of EOS data fully assigns thermodynamicvalues over the entire diagram region of phase transitionsowing to analytical expressions including specific coeffi-cients which are based on the available experimental andtheoretical data. Asymptotic values of the thermody-namical, optical, and transport parameters were fixed tofulfill the equilibrium conditions. This choice ensures adescription of the material properties with a good accu-racy in the region where thermodynamic constants areaccessible and constrains the unknown properties to befitted with known values. The systematic use of the de-fined constants at equilibrium for non-equilibrium coeffi-cients limits the number of free parameters and ensuresthat the specificities of each metal is preserved in thesimulation, in an optimal manner with respect to the ac-cessible data.

Fig. 2 depicts the phase diagram of aluminium in thevariables pressure and density. The local variations ofthe matter states during compression and expansion arerepresented by several thermodynamic paths correspond-ing to specific locations under the surface of the sample.(p,ρ) points belonging to a curve describe different timemoments of the evolution and no temporal correlationcan be easily made between the curves. Consequently, norelation in time can be directly made between the differ-ent paths and only the correlated evolution of the pres-sure and density remains accessible. However, similardevelopments occurring in well-specified pressure-densityranges take place on similar time scales. These charac-teristic behaviors are indicating by labels [(a)-(e)] on thefigure and will be detailed step by step in the follow-ing. Consequently, one can roughly estimate the timerequired for the transformation, as well as the sequenceof transitions. The mesh calculation employed here isbased on a geometric progression to assure a more pre-cise spatial description in the front of the sample, wheregradients are more important and accurate resolution isnecessary to sample the skin depth. The thickness of thefirst cell is fixed at 5 A and the ratio of each two con-secutive cell sizes is calculated in order to have the sumof all widths equals to 10 µm. Seven curves where the

mid-points correspond to different approximative depthsin the cold solid are represented : 0.25 nm, 5 nm, 10 nm,20 nm, 50 nm, 100 nm and 200 nm. The numerical con-vergence has been verified in reducing 5 times the gridcell, leading to less than 2% change in density and pres-sure. The transformation phase lines given by the EOS,solid-liquid (melting line) and liquid-gas (binodal line),are added in grey lines on the figure. The spinodal line isalso represented to distinguish fluid metastable states tounstable ones. This limit is a theoretical one which marksthe location of the infinite compressibility states.27

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.510-5

10-4

10-3

10-2

10-1

100

101

102

(e)

(d)

(b) (c)

(a)Mel

tingBinodal

Spin

odal

20 nm10 nm

200 nm

100 nm

50 nm

5 nm

0.25 nm

Pres

sure

(GPa

)

Density (g/cm3)

FIG. 2: (Color online). Pressure-density (p-ρ) phase diagramof aluminum exposed to ultrafast laser radiation (150 fs, 5J cm−2). Thermodynamic paths of material layers are plot-ted for several depths inside the metal. The initial depth ofthe cell is indicated on the top of each curve. The criticalpoint (⋆), binodal, spinodal, and melting curves are also rep-resented in the diagram.

The metal starts from solid density (2.7 g cm−3 foraluminum) and standard pressure condition (10−4 GPa).Then, due to laser heating, the matter expands andpasses through different states. During the first pi-coseconds, electronic heating and the associated non-equilibrium effects generate a strong electronic pressurecomponent. Consequently, a vertical transition in the (p-ρ) diagram appears, marked by the label (a). The latticeremains cold and a static description of the matter canstill be preserved. However, the isochoric heating leadsto a sharp increase of the system entropy. When thelaser irradiation stops, the maximum electronic pressure,which corresponds to the peak of compression, has al-ready been reached. The pressure decrease is due to thelocal volume increase and the cooling of the electron gasby the electron-phonon relaxation (b). After a few tens ofpicoseconds, the deeper cells enter in an overdense state,in the right part of the thermodynamic diagram (c). Theenergy transferred from the electron gas to the latticereduces the non-equilibrium. Additionally, the densitydecreases due to the progressive ionic heating. At thisstage, the thermodynamic trajectories split and follow

6

two paths.

The first one passes over the binodal region becausethe strong shock release produces a sequence of thermo-dynamic states above the critical point. The cells locatedinitially at 0.25 nm, 5 nm and 10 nm inside the sample(indicated by the label (d) on Fig. 2), represent this non-conventional case. The evolution is characterised by ahigh pressure expansion with no liquid-gas transforma-tion. The expansion of the system takes a few picosec-onds during which the fluid reaches a supercritical stateand the degree of degeneracy of the electronic subsystemdecreases. The swift heating is followed by a fast ex-pansion of the supercritical fluid during this stage. Suchthermodynamic states may cause fragmentation, phaseexplosion, or spinodal decomposition9 but these effectscan not be described in the hydrodynamic frames with-out involving nucleation theory. Several picoseconds af-ter irradiation, the density drops quickly and the pres-sure follows a typical ideal gas dependence, evolving ac-cording to a ρ5/3 law. In such exotic thermodynamicstates, the matter exhibits original features and specificplasma effects.8,28,29 The second path is followed by thecells originally at 50 nm, 100 nm and 200 nm inside themetal. The associated curves pass through the meltingboundary and the liquid maintains a high density whilethe pressure decreases. Consequently the sound veloc-ity drops and the curves intersect the binodal in the re-gion labeled by (e). The matter enters in the two-phasedomain and a liquid-gas transformation occurs. At thisstage, which occurs several tens of picosecond after the ir-radiation, the electron-ion relaxation is complete and therole of electronic pressure becomes negligible. The mostimportant feature in this case is the stepwise decrease ofthe sound velocity at the point where the quasi-isentropiccurve intersects the binodal.1,4 The drastic change in thesound velocity results in a radical structural rearrange-ment of the flow.6 In this case, the system relaxes towardsthe liquid-gas regime within 100-200 ps, and nucleationis supposed to control the main ablation process.2 Theclassical theory of homogeneous nucleation determinesthe formation kinetics of a critical nucleus, dependingon the thermodynamics and the transport properties ofthe fluid. Consequently, the ablation rate for material re-moved as liquid droplets should present a threshold effect.No consideration of the nucleus size has been accountedfor in our 1D calculations and the nucleation process isconsidered as a homogeneous one.

The limitations to the model impose a series of consid-erations concerning the material removal process. Theamount of ablated material is supposed to be composedof that quantity of matter which attains a supercriticalstate or undergoes the liquid-gas transition with a posi-tive velocity, seen as limiting cases for material removal.This gives the dimension corresponding to the ablatedthickness that we have estimated in a previous study.3

The ablated layers that we have defined were composedof the expanded gas and liquid accelerated outside thesample. This definition corresponds to the matter (mixed

liquid and gas) reaching a low density, below the criti-cal one. To explore the role of electron-phonon couplingfactor, we have performed a set of simulations with avariation of the aluminum coupling constant around thetabulated value. This parameter is characteristic of theTTM and governs the temperature dependence of the en-ergy transfer rate between electron and ion subsystems.It appears that increasing the coupling factor augmentsthe thickness of the material which undergoes supercrit-ical state transition while the thickness of matter whichsuffers liquid-gas transformation remains relatively un-changed at this fluence. In the TTM framework, theelectron thermal conduction plays an opposite role to theelectron-ion coupling constant in the control of the energydistribution. In fact, the thermal diffusion parameter al-lows the propagation of the energy before the completerelaxation. Decreasing the value of the electronic conduc-tion tends to confine the energy to the metal surface andthe electronic pressure decreases in a less significant man-ner due to the cooling associated to the diffusion process.In this case, the pressure remains high and the amountof material reaching supercritical states is more impor-tant. However, in order to reach these states, the dy-namics of energy transfer is becoming equally importantas the shock release, because the unloading wave causesa strong pressure decrease in the front-surface layer. Theacceleration of the surface and the associated shock for-mation are determined by the pressure gradient which,in turn, is governed by the energy profile since in theemployed EOS, the pressure is a function of the internalenergy and the material density. Due to the hydrody-namic equations, the rapid expansion required for reach-ing the density of the critical point is driven by a steeppressure gradient. Simulations performed in the case ofa weaker thermal conductivity show that the density de-creases quickly and the pressure drops jointly. Therefore,the balance between these three competitive processes(energy transfer, thermal diffusion and shock release) de-termines the rate of achieving supercritical states by thematerial during its transformation.

To avoid discussions on the parametric values whichcan be strongly arbitrary, we have chosen to presentour results for a second material, copper, which differsstrongly from aluminium in that concerns the strengthof the electron-phonon coupling. We have preferred thischoice of materials with different properties since it canbe verified against experimental results. One can notethat the experimental and calculated amount of ablatedmatter is similar for both materials at 5 J cm−2, and thethermodynamic states reached by these materials can bediscussed in parallel for this fluence used in our simula-tion. Although these metals have different EOS, the en-ergy deposition process can be compared since it dependsmainly on their transport properties. In fact, the criticalpoint is about ρ0/4 in density for both materials and thevalue of the critical pressure does not play a significantrole. On the other hand, copper, as mentioned, is charac-terized by a lower value of the coupling constant. Conse-

7

0 1 2 3 4 5 6 7 8 9 1010-5

10-4

10-3

10-2

10-1

100

101

102(b)

(d)

(e)

(c)

(a)

Mel

ting

Binodal

Spi

noda

l

100

nm

10 nm

200 nm

20 n

m

5 nm0.25 nm

Pres

sure

(GPa

)

Density (g/cm3)

FIG. 3: (Color online). Pressure-density (p-ρ) phase diagramof copper exposed to ultrafast laser radiation (150 fs, 5 Jcm−2). Thermodynamic paths of specific material layers areplotted for several depths inside the metal. The initial depthof the cell is indicated on the top of each curve. The criti-cal point (⋆), binodal, spinodal, and melting curves are alsorepresented in the diagram.

quently, longer non-equilibrium processes take place dueto a larger thermalisation time. Fig. 3 shows a pressure-density diagram for copper which is similar to the oneobtained for aluminum. The diagram depicts thermody-namic paths of the cells at the same depths as previouslypresented. The curve at 50 nm was skipped because cellslocated deeper than 20 nm exhibit similar behaviors. Thecells start from the solid density at 8.9 g cm−3 in this caseand follow a short phase of compression at high pressurebefore the expansion phase. Qualitatively, the liquid-gastransformation resulting from the irradiation of the cop-per sample shows the same types of paths, correspondingto the expansion of a supercritical fluid layer at the sur-face of the material. As for aluminum, nucleation maybe a possible mechanism of ejection for material situatedin the deeper region, up to 200 nm depth. For identicalinput laser fluence, at 5 J cm−2, the supercritical stateaffects a thicker layer below the aluminum surface thanin the copper case due to a stronger absorption. Cellsdeeper than 5 nm do not present supercritical behaviorin the second case while a 15 nm region is becoming su-percritical in the first case. Yet, the maximum pressurereached by the outer cell at 0.25 nm is higher for copper.Note that the electronic thermal conduction parameter ofcopper does not differ significantly from aluminum. How-ever, the electron-phonon coupling is weaker, approxi-matively 3 × 1016 WK−1m−3 for copper30 compared to3 × 1017 WK−1m−3 for aluminum,31 thus the electronicenergy is slowly transfered to the crystal lattice allowingfor even more diffusion before the energy is relaxed.32

As a consequence, the electronic temperature and pres-sure are enhanced inside the material in this case. Thedensity decrease, related to the ionic expansion, depends

strongly on the energy transfer time. Consequently, thenon-equilibrium which survives for a longer time in cop-per is an impediment for reaching supercritical states be-cause the electronic energy has more time to spread inthe metal due to electronic diffusion effects. When thedensity reaches the region corresponding to the criticalpoint, the majority of the cells have a total pressure lowerthan the critical pressure and the nucleation process issupposed to take place on a volume larger than the onewhere fragmentation may occur.

To conclude, it has been shown that the supercriti-cal states are mainly observed at the beginning of thenon-equilibrium, for a strong expansion. An interplaybetween the electromagnetic absorption profile, the ab-sorption coefficient, the electron-lattice coupling, and theelectronic thermal conductivity determine the extent ofthe non-equilibrium. In addition, the shock release, gov-erned by the hydrodynamics and the EOS in our cal-culation, is responsible for the drop of the pressure ata high or a low density. Expansion of the supercriticalstates and the slower liquid-gas transformation are iden-tified to be the main regimes of ablation. We have shownthat these regimes take place on different time scales. Aneventual fragmentation process or phase explosion mayoccur at the surface several picoseconds after the irra-diation. Subsequently, nucleation process is expected tooccur within several hundreds of picoseconds. The rela-tive occurence of these ablation processes depends on themetal transport properties which determine the rate ofsupercritical states reached during the material expan-sion. The metals considered here show different featuresmainly due to the strong electron-phonon coupling dis-parity which exhibits an order of magnitude variation.Since electron-phonon coupling in aluminum is strongerthan copper value, and the thermal diffusivities are com-parable, aluminum is favored for reaching supercriticalstates.

IV. LIQUID LAYER : NUMERICAL

ESTIMATION OF THE SIZE OF RESOLIDIFIED

MATERIAL

After the quasi-isochoric heating and after several in-termediate stages including the subsequent plasma for-mation at the surface, a thin layer confined betweenthe solid-liquid and the liquid-gas interfaces is stronglycompressed, forming a hot liquid. If the outer part ofthis liquid expands quickly, the inner part expands afterseveral picoseconds with respect to the onset of expan-sion. The expanding plasma and the external liquid layerheat and compress the material for an extended period,much longer than the laser pulse. As a consequence, themelting front propagates further into the material andthe liquid thickness increases. Moreover, this compres-sion phase leads to additional material removal via nu-cleation process. The appearance of Newton fringes onthe expanding material, found experimentally using time-

8

resolved imaging of the irradiated regions on short timescales in a large variety of materials, represents clear evi-dence on the existence of two reflecting surfaces.4 Visiblelight reflected by the liquid-gas layer interfering with theretarded light reflected from the dense liquid would ex-plain why the far-field pattern exhibits the characteristicring structure. This fringes last for several ns and theevaluation of the interference pattern suggest a mixedphase characterized by a high index of refraction. Thelifetime of the observed optical interference patterns is aconsequence of the ablation of the metal following nucle-ation. Otherwise, the liquid layer at constant density isnot ablated and remains attached to the solid. Duringthis time, thermal conduction cools down the disorderedmatter which no longer conserves fluid properties char-acteristic to the liquid state and droplet ejection stops.The long-living liquid layer cannot escape from the sam-ple and resolidification occurs. Significant change in thematerial structure can occur during the solid-liquid andliquid-solid transformations with the potential of produc-ing new structures with intrinsic properties that are per-haps different from the original ones. The aim of thissection is to provide a numerical measure of the thicknessof matter concerned by the resolidification phenomenon.

Under ultrafast laser irradiation conditions, the abla-tion process is highly dependent on the material nature.In dielectric materials, damage and ablation mechanismsare attributed to optical breakdown and plasma forma-tion. For semiconductors, laser irradiation induces elec-tronic transitions from bonding to antibonding states.The presence of a large concentration of excited electronschanges the interatomic forces and the atomic motionsat low ionic temperature may be sufficient to break in-teratomic bonds.33,34 For metals, the strong electronicexcitation does not affect the band structure and thethermal motion represents the most reasonable mecha-nism for yielding a modification of the atomic configu-ration which is able to produce deviations from the or-dered phase. Large amplitude vibrations result in the lossof the long range order present in the crystalline phase.The structure evolves into a disordered liquid state whereonly short range atomic correlations are present. Time-resolved electron diffraction experiments have shown thatcomplete solid-to-liquid transition occurs within a few pi-coseconds for aluminum thin films.35 It appears that clas-sical thermal melting features are preserved and equilib-rium conditions of melting are supposed to be valid formetals in conditions characteristic to our simulations.

To determine whether a numerical layer undergoes asolid-to-liquid transformation or not, we have examinedthe specific internal energy value E. We compare thisvalue with the melting energy Em, whose values in oursimulations are the ones corresponding to the plateauonset provided by the EOS. For aluminum at StandardTemperature and Pressure conditions (STP), ESTP

m =6.74 × 105 J kg−1, and this value grows significantly asthe pressure increases. For copper, the dependence is lesssignificant than for aluminum in the considered pressure

0.1 1 10

10

100

1000

Liq

uid

laye

r th

ickn

ess

(nm

)

Fluence (J/cm2)

Aluminum sample Copper sample

FIG. 4: (Color online). The thickness of the liquid layer re-maining on the metal substrate at the crater bottom versusthe incident laser fluence for aluminum (solid line) and cop-per (dashed line) samples. Irradiation conditions: 150 fs laserpulse.

range and the energy shows a reduced scattering aroundthe standard value ESTP

m = 4.7 × 105 J kg−1. When thelocal energy is sufficiently high, a solid-to-liquid transi-tion starts and crystal modifications should occur. Inthis study, we only consider the non-expanding materialand the thickness of the liquid layer is calculated by de-termining both the positions of the outer cell which un-dergoes liquid-gas transformation and the first one fromthe bulk that has an energy E ≥ Em. The difference be-tween these two locations of the cells provides the size ofthe liquid confined between the solid substrate and theexpanding ablated matter. Fig. 4 shows the liquid layerthickness as a function of the laser fluence for aluminum(solid line) and copper (dashed line). It demonstrates thepronounced dependence of the liquid-gas transformationand heat diffusion effect on the incident laser fluence.

While liquid droplets generation requires strict kinet-ics and thermodynamic conditions, no restrictions applyconcerning the liquid layer formation mechanism. Ex-amination of Fig. 4 and previously performed simulationsreported in Ref. [3] reveal that for fluence higher than 5 Jcm−2 the thicknesses of the liquid layer and, respectively,the ablated matter are of the same order of magnitudefor both materials. For low fluences, a large part of theheated aluminum sample is ablated and a small quan-tity of liquid remains on the solid substrate. As a conse-quence, only tens of nanometers of liquid layer result fromthe irradiation just above the threshold fluence for alu-minum ablation. This is in agreement with experimentalobservations of a limited recast material at the thresh-old fluence. For copper, the calculated liquid thickness iszero at low fluences, below the melting threshold, and in-creases quickly close to the threshold fluence, FCu

Th . Thevalue of FCu

Th is not realistic and the discrepancy has beenalready discussed in Ref. [3], but the curve in Fig. 4 shows

9

a correct dynamic behavior. The discrepancy is appar-ently due to the effects of the non-equilibrium transporton the optical properties, which was not accurately de-scribed in our simulation. The slope of the curve for cop-per is more pronounced than for aluminum due to thehigher electronic heat diffusion characteristic for copperthat spreads the energy in a larger volume. Thus, onlyfluences very close to FCu

Th allow to reach a minimal sizeof recast material. Finally, even if Em is higher for alu-minum, copper is denser than aluminum and its melt-ing temperature is 1.5 times higher. The local energyrequired for melting aluminum is lower and simulationsshow a thicker liquid layer for this material.

To control the quality of the laser micro-machiningprocess, the region undergoing mesoscopic irreversibletransformation has to be reduced. Then, due to the cor-relation between the resolidification size and the laserfluence, it is important to connect macroscopically-induced effects to the laser parameters. The numerically-estimated liquid size can be directly compared to the re-cast size in the hypothesis of a one-dimensional model.These results emphasize the interest in using the thresh-old fluence to minimize recast material and improve theablation quality. Appropriate irradiation processes maybe used to create very high quality micro-machined struc-tures and simulations are able to predict the optimal in-tensity range.

V. SPALLATION REGIME OF ABLATION

To add to the description of the potential factors thatmay end up in ablation upon ultrafast laser irradiation,this section is devoted to laser-assisted spallation processtaking place in the solid phase. The formation and prop-agation of shock waves resulting from isochoric heatingcan lead to high tensile stress close to the rear surface ofthe sample if it is thin enough. Consequently, spallationfollows. The peak of the pressure wave results in a shockfront, while its rear decreasing slope induces a rarefactionwave into the shocked material. When the shock wavereaches the free surface, the compressed material expandsfreely and accelerates the surface expansion. The super-position of the two rarefaction waves, the tail of the initialpressure wave and tensile stress wave, running in oppo-site directions, yields a residual stress whose amplitudeincreases with the distance from the free surface in theearly times. The mechanism of mechanical fracture is dueto the nucleation of microvoids which grow in the regionwhere high plastic deformation occurs. As the load on themetal increases, the microvoids eventually form a contin-uous fracture. In the simulation results reported here, thecharacteristic of the stress profile induced in aluminumsamples is compared with the experimental detection ofspallation phenomena performed by Tamura et al14 whomeasured a linear relationship between foil thickness andthe depth where spallation occurred, i.e., spall thickness.

Coupled molecular-dynamics and fluid computational

models have revealed high tensile stress leading to themechanical disintegration of thin films.36 Vidal et al.37

have recently performed hydrodynamical simulations toreproduce spallation experiments. It was argued thatstandard defect growth models are irrelevant for fasthydrodynamic regime and the spall strength used inthe above-mentioned simulations is about one order ofmagnitude higher than the spall strength measured atstrain rates in the range (103-106) s−1.38 The spallstrength value used in the respective study correspondsto the crossing-point between spinodal and cold pres-sure curves given by the Quotidian-Equation-Of-State(QEOS) model.39 This EOS displays a van der Waalsbehavior which allows tensile stress calculations but doesnot take into account defects present in solids. How-ever, the mechanical properties of solids are controlledby lattice imperfections and dislocations, which weak-ens the materials. This deviation from a perfect latticemay explain the large discrepancy between experimentaland computational conditions of irradiation proposed bythese authors in their simulations and the very high in-cident fluence levels that were used. The aim of this sec-tion is to provide new mechanical insights into the spal-lation process, while preserving the experimental condi-tions usual for ablation experiments. Additionally, wewill determine and discuss the spall strength value.

The measurements of the spall thickness reported byTamura et al.14 are of interest because they were obtainedusing irradiation levels close to the mechanical ablationthreshold. From this point of view we can assume thatlower fluences and the associated tensile stress are notsufficient to produce fractures. Consequently, at the me-chanical ablation fluence threshold, neglecting dynami-cal effects, the position corresponding to the maximumamplitude of the tensile stress can be directly comparedto the spall thickness and supplies the criterion for thedetermination of the mechanical strength. We have per-formed simulations of the effects induced by a τ = 50 fs(FWHM) laser pulse at a wavelength of 775 nm to com-pare the calculated stress evolution to the experimentalmeasurement of the fracture position. The s-polarisedincident laser irradiates the target at 57.5˚with respectto the surface normal. Note that the state of laser po-larisation was not specified in the reported experimentand different polarisation cases have been numericallytested but none of them modified the results significantly.The hydrodynamic responses of three targets of differentthicknesses: 25 µm, 50 µm, and 100 µm, were studied un-der laser intensities of 0.5×1015, 0.7×1015, and 1.4×1015

W cm−2 respectively. Fig. 5 shows the time evolution ofthe stress at the location of the maximum tensile stress(a) and the stress profiles at the time of maximum tensilestress from the rear surfaces of the targets (b).

Our main concern in these simulations was to followthe position of the maximum tensile stress and to de-duce the value which generates a void in the sample. Forthe 25 µm thick foil, numerical considerations indicate amaximum tensile stress and, therefore, a probable frac-

10

25 20 15 10 5 0

0.00

0.25

0.50

0.75

1.00

0 5 10 15 20 25-1.5-1.0-0.50.00.51.0

25 mm 50 mm 100 mm

(a)S

tres

s (G

Pa)

Time (ns)

25 mm 50 mm 100 mm

(b)

Str

ess

(GP

a)

Position (µm)

FIG. 5: (Color online). (a) Stress evolutions in time at theposition of the maximum tensile stress for three different nu-merical simulations performed in aluminum target of 25 µm,50 µm, and 100 µm thicknesses. (b) Stress profiles at thetime corresponding to the maximum tensile stress. Parame-ters used in the simulations are chosen to reproduce irradia-tion conditions of Ref. 14 : 25 µm thickness, 25 J cm−2 pulse(solid line), 50 µm thickness, 35 J cm−2 pulse (dashed line),and 100 µm thickness, 70 J cm−2 pulse (dotted line).

ture at 4.4 µm depth from the rear face of the target. Forthicker samples, the spall thickness increases and is esti-mated to be 7.2 µm and 11.3 µm for 50 and 100 µm foilthickness, respectively. The numerically-determined val-ues and the experimental values of the spall thickness arereported in table [I]. Maximum tensile stress is also indi-cated and this provides an estimation of the static tensilestrength related to a high strain rate. Good agreementis obtained between the experimental and the simulatedspall thicknesses which depend mainly on the velocity ofthe rarefaction wave front. Details and reasoning aboutthe modeling approach are given below. Following theusual approach in stress-related problems, the stress isdecomposed into a volumetric and a deviatoric part. Thevolumetric part of the stress is computed using the EOS,while the deviatoric one is calculated using an additivedecomposition of the rate of deformation into elastic andplastic parts, the von Mises yield condition,40 and a flowstress model based on Steinberg-Cochran-Guinan (SCG)formalism.41 The SCG model is applicable at high strainrates and has been used to provide a temperature andpressure variation of the yield strength and the shearmodulus. To correctly describe the material responseunder shock conditions, material-strength model shouldbe based on the evolution of the micro-structural statewhich would required the dynamics of dislocations andgrain-boundaries.

It is worth noting that the mechanical model used in

Foil Thickness F S

Th Spall Thickness σS

(µm) (J cm−2) Sim. (µm) Exp. (µm) (GPa)25 25 4.4 2.9± 0.9 0.8850 35 7.2 6.3± 1 0.72100 70 11.3 11.2 ± 1.2 0.62

TABLE I: Experimental (Exp.)14 and simulated (Sim.) valuesof the spall thickness and the associated tensile strength σS

given for three different Al foil thicknesses.

this work is an elastic-perfectly plastic model involvingstress behavior which responds elastically up to the pointwhere the stress exceeds the yield stress and after whichplastic flow occurs in the material. When the solid under-goes plastic deformation at high pressure, stresses resultin the generation and subsequent propagation of latticedislocations. Whereas solid state plastic flow is causedby the rearrangement of the atoms in the crystalline ma-trix by transporting these dislocations, such an empiricalmodel does not consider changes in lattice structure. Thedensity of the generated dislocations and their propaga-tion velocity are both dependent on the rate of the ap-plied strain.42 The parameters defined in the constitutivemodels are determined from experiments involving strainrates on the order of 103 − 106 s−1. Application of themodel is speculative at higher strain rates of 107 − 109

s−1 and should be compared carefully to experimentaldata. In practice this study is difficult and the numericalrepresentation of the stress flow is obtained by estimat-ing the physical parameters involved under deformations.Such parameters are calibrated in experiments involvingstrain rates on the order of 103−106 s−1 and the applica-bility of the SCG model for higher strain rate is question-able. Nevertheless in our simulations, high strain rateson the order of 109 s−1 are reached at the front side ofthe sample and, at the same time, high volumetric stresslevels provided by the EOS are attained. In this case,the deviatoric contribution does not play a significantrole compared to the volumetric one. As a consequence,the effects of material strength may be ignored and thehydrodynamic response governs the mechanical deforma-tion of the irradiated surface. In spite of uncertainties,we assume that the accuracy provided by the SCG modelis acceptable to take into account the pressure depen-dence of the shear modulus and the yield strength in thishydrodynamic regime. On the other hand, after suchwave propagation, the stress is much lower than at thefront side of the sample and the elasto-plastic proper-ties of the material play a significant role for pressuresbelow 1 GPa. After the shock has propagated severalmicrometers, the stain rates decrease below 106 s−1 andwe suppose that the shock response of the metal can stillbe modelled by the parameters associated to the SCGmodel. In fact, this shock compression at lower strainrates belongs in the validity domain of the SCG consti-tutive model and corresponds to a regime where solidstate material strength is strongly affected by the en-

11

hancement of pressure and the temperature. Calculatedvalues of the tensile strength are lower than the standardvalue (rate-independent) around 1 GPa for lowest strainrates. However, high strain rate spallation experimentsshow that for increasing loading rates in the tensile re-gion before spallation, the spall strength increases withthe rate of mechanical loading and reach several GPafor strain rates around 107 s−1.43 Consequently, the de-duced value of the spall strength is lower than expected,which may reveal a limitation in the mechanical modelunder such strength of material deformation. Local de-fects may also affect the experimental threshold fluenceof spallation. The observed decrease of the calculatedspall strength while the foil thickness increases may bedue to the strain rate decrease with the propagation timeof the shock wave. Nevertheless, the spall strength val-ues determined by the simulation are close to the onescorresponding to the low strain rate range.

VI. CONCLUSION

We have presented a one-dimensional numerical toolthat yields information on several aspects of ultrafastlaser metal ablation. The insertion of non-equilibriumaspects into a Lagrangian hydrodynamic code describ-ing laser-matter interaction has provided a novel ultra-fast hydrodynamic perspective of matter ejection. Thedistinction between the time evolution of the atomicpressure and the total pressure has revealed a strongelectronic contribution on the first tens of nanometersfor several picoseconds after irradiation. Due to thiselectron-ion non-equilibrium, unconventional thermody-namic states can be reached and we proposed a syntheticrepresentation of the thermodynamic matter evolution insuitable diagrams for copper and aluminum in the viewof their different efficiencies in the electron-phonon cou-pling. Then, the successive thermodynamic propertieswere plotted at different depths inside the metal to showlayers trajectories in a (p-ρ) phase-plane. The fast laserabsorption at the metal surface yields isochoric heat-ing and the solid was brought in states far from equi-

librium. The relaxation process occurs through the ex-pansion phase and we have identified different paths cor-responding to different mechanisms for matter ejection.The expansion of the supercritical fluid, which may re-sult in a fragmentation process, and a potential homo-geneous nucleation mechanism have been identified aspossible ablation processes. It has been shown that thesupercritical states responsible for the fragmentation pro-cess are mainly observed in the first picoseconds of non-equilibrium, owing to a strong expansion. We emphasizethat the materials with a high electron-phonon couplingare then favored in reaching such states. Further informa-tions on the phase transition under such irradiation con-dition may be obtained by combining molecular-dynamicdescription to fluid treatment to avoid assumptions in-herent to the use of EOS.

An estimation of the thickness of the liquid layer, lo-calised between the expanding matter and the unaffectedsolid substrate, was performed. It can be related to theablation rate and structure size in the longitudinal di-rection and could be compared with post-mortem exam-ination. The control of the quantity of recast materialwould be important to improve micro-structuring qual-ity for micro-machining applications. Mechanical frac-ture of bulk aluminum was also investigated. If experi-mental spall thicknesses were correctly reproduced, ourdeduced values of the spall strength are on the same or-der of magnitude as the ones corresponding to the lowstrain rate range. These values are far below the val-ues measured in dedicated experiments and suitable voidnucleation models are needed to correctly describe solidfragmentation. However, our simulations provide reliableinformation on both thermodynamic and mechanic prop-erties under such extreme conditions.

ACKNOWLEDGEMENTS

The authors are grateful to N. Bulgakova and F. Bon-neau for fruitful discussions on several aspects of thiswork.

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