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Ab Initio Nonequilibrium Thermodynamic and
Transport Properties of Ultrafast Laser Irradiated 316L
Stainless Steel
Emile Bevillon, Jean-Philippe Colombier, Biswanath Dutta, Razvan Stoian
To cite this version:
Emile Bevillon, Jean-Philippe Colombier, Biswanath Dutta, Razvan Stoian. Ab Ini-tio Nonequilibrium Thermodynamic and Transport Properties of Ultrafast Laser Irradi-ated 316L Stainless Steel. Journal of Physical Chemistry C, American Chemical Society,2015, 119 (21), pp.11438-11446. <http://pubs.acs.org/doi/abs/10.1021/acs.jpcc.5b02085>.<10.1021/acs.jpcc.5b02085>. <ujm-01159667>
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Ab Initio Nonequilibrium Thermodynamic andTransport Properties of Ultrafast Laser Irradiated
316L Stainless Steel
E. Bevillon,† J.P. Colombier,∗,† B. Dutta,‡ and R. Stoian†
†Laboratoire Hubert Curien, UMR CNRS 5516, Universite de Lyon,Universite Jean-Monnet 42000, Saint-Etienne, France
‡Max-Planck-Institut fur Eisenforschung GmbH, D-40237, Dusseldorf, Germany
E-mail: [email protected]: +334 77 91 58 82
Abstract
We present calculations of transient behaviorof thermodynamic and transport coefficientson the timescale of electron-phonon relaxationupon ultrashort laser excitation of ferrous al-loys. Their role defining energy depositionand primary microscopic material response tothe laser irradiation is outlined. Nonequilib-rium thermodynamic properties of 316L stain-less steel are determined from first-principlescalculations. Taking into account the complex-ity of multi-metallic materials, the density func-tional theory is first applied to describe theelectronic density of states of an alloy stainlesssteel matrix as a function of electronic heat-ing. An increase of the localization degree ofthe charge density was found to be responsi-ble for the modification of the electronic struc-ture upon electronic heating, with consequenceson chemical potential, electronic capacity andpressure. It is shown that the electronic tem-perature dependence of stainless steel thermo-dynamic properties are consistent with the be-havior observed for pure γ-Fe, outlining the roleof the main constituent in the same atomic ar-rangement. Assuming that similar behaviorsextend to the transport properties, the tran-sient electron-phonon coupling, optical proper-ties and thermal conductivities of γ-Fe are de-rived based on density functional perturbation
theory and ab initio molecular dynamics andextrapolated for steel. The insertion of accuratetransport coefficients allows to improve currentmodels and to achieve more realistic descriptionof femtosecond pulse laser processing. Effects offast temperature variation driving phase tran-sitions and strong thermal stresses induced bythe laser pulse are finally presented by combin-ing first principle results to a nonequilibriumhydrodynamic approach.
Introduction
The development of material processing islargely driven by off-stoichiometric disorderedmaterials, whereas their complex microscopicbehavior under ultrafast laser irradiation re-mains mostly inaccessible with current theoret-ical approaches. Particularly, stainless steelsconstitute a specific kind of alloys containinga certain amount of chromium that via sur-face oxidation produces a dense passive layerconferring corrosion resistance to the material.This crucial property is responsible for a large-scale use of stainless steels spreading into manyindustrial domains where both the tailorableproperties of steels and corrosion resistance arerequired. Unsurprisingly, these materials arealso of interest for technological applications re-lated to laser manufacturing, especially as they
1
have been found to respond favourably to ul-trashort laser irradiation.1 For instance, theyare used in laser process development and re-cently in nanostructuring with applications intribology,2 wettability3 and optical functional-ization4 fields. Requiring accurate design andcontrol, these applications put forward the chal-lenge of an accurate description of laser-matterinteraction from optical, thermal and mechan-ical point of view. This remains difficult dueto the lack of nonequilibrium thermodynamicproperties evolving with the electronic excita-tion induced by ultrashort laser exposure.
A general laser-matter interaction scenariogoes on the following lines. By focusing anintense energy beam into a small and con-fined volume of matter in solid state, ultra-short laser irradiations produce a large excita-tion of electrons within the band structure.5
Via electron-electron collisions, these excitedelectrons redistribute momentum and energyand reach a Fermi-Dirac distribution withintens of femtoseconds timescale.6 The electronicsubsystem is thus rapidly thermalized and canbe defined by an electronic temperature Tereaching thousands of Kelvin, while the lat-tice system remains mostly unaffected and char-acterized by a low temperature Ti. The in-crease of Te with respect to Ti refers to theso-called electron-phonon nonequilibrium state.At larger timescales, up to few picoseconds,7
the energy stored in the electronic subsystemis transferred to the lattice by electron-phononcoupling. This progressively leads to the returnof the equilibrium (Te = Ti) while the energy isfurther dissipated through the thermal diffusiv-ity inside the material.
These temperature evolutions are modelledthrough the two-temperature model rely-ing on numerous thermodynamic parameters,among them the electronic heat capacities, theelectron-phonon coupling and the thermal con-ductivity,8,9 all of them subject of approxima-tion or sources of uncertainties. The electronicpressure is also of importance as it plays a sig-nificantly role in hydrodynamic models.10 Thismodel is based on the assumption that bothionic and electronic systems can be efficientlydescribed by two separated and effective tem-
peratures which is found to be consistent withexperimental observations.11,12 The thermody-namic conditions experienced by the materialunder irradiations strongly depends on thesenonequilibrium properties. However, experi-mental evaluation of these properties is diffi-cult as excited solids in steady states cannotbe created.13–15 This is the main advantagesof first-principles methods, where a finite elec-tronic temperature can be applied and proper-ties derived accordingly.16,17
In the present study, we investigate the evolu-tion of 316L stainless steel properties under ul-trafast laser-induced electron-phonon nonequi-librium. From electronic structure calculations,the temperature dependent density of electronicstates are obtained. They give access to elec-tronic thermodynamic quantities, as chemicalpotentials, heat capacities, pressures and freeelectron numbers. This is followed by the eval-uation of transient transport properties, withthe calculation of the electron-phonon coupling,the optical indices and thermal conductivities.Finally, the thermodynamic and kinetic condi-tions leading to the transient phase transforma-tion in a surface region of a stainless steel sam-ple irradiated by an ultrashort laser pulse arediscussed in light of the results of a nonequilib-rium hydrodynamic simulation.
Ab initio calculations details
The calculations are carried out in theframework of the density functional theory(DFT)18,19 extended to non-zero tempera-tures.20 The finite electronic temperature istaken into account directly and indirectlythrough the Fermi-Dirac distribution of elec-trons and by minimizing the free energy withrespect to the electronic entropy term.21 Theexchange and correlation part is treated bygeneralized gradient approximations (GGA)in the form parameterized by Perdew, Burkeand Ernzerhof.22 The Abinit package is used,23
which is based on a plane-waves description ofthe electronic wave functions. In order to takeinto account nuclei and core electrons, projec-tor augmented-waves (PAW) atomic data24–26
2
are used for the atoms constituting the 316Lstainless steel. For γ-Fe a Troullier-Martins(TM) pseudo-potential27 is considered to sim-plify linear response function for the calculationof the electron phonon coupling constant. TheMonkhorst-Pack mesh of the reciprocal space,28
as well as the cutoff energy of plane-waves havebeen designed depending on the size of the celland the kind of calculations.
Off-stoichiometric disorder refers to materialshaving multiple constituents in solid solutionon the material lattice or sub-lattices. In ourcalculations, we have used the concept of spe-cial quasirandom structures (SQS), which hasemerged in recent times as a powerful approachfor the calculation of total energy and relatedproperties in disordered phase.29,30 Proposed byZunger et al.,31 this method is specially de-signed to order supercells with a limited num-ber of atoms constructed in such a way thatthe most relevant pair and multi-site correlationfunctions closely mimic the ensemble-averagedcorrelation functions of the random substitu-tional alloy. Since the atomic distributions onlattice sites in an SQS are determined on the ba-sis of matching correlation functions, it can ac-curately capture the local environments in ran-dom alloys. Austenitic steels crystallize withina face centered cubic structure (FCC), the solidsolution is modelled within a 108 atoms super-cell generated by the ATAT package.32 It corre-sponds to a 3 × 3 × 3 conventional cell of FCC.316L stainless steel has a constituent mass com-position of : 65.9% Fe, 16-18% Cr, 10.5-13% Ni,2-2.5% Mo, 2% Mn, 1% Si, 0.04% P, 0.03%Sand 0.02% C. We first approximated this com-position to the most abundant constituents, i.e.Fe, Cr and Ni, leading to the Fe73Cr21Ni14 su-percell and corresponding to a mass composi-tion of 68.1% Fe, 18.2% Cr, 13.7% Ni. A pic-ture of the corresponding supercell is providedin Figure 1. Beforehand, we conducted a ge-ometry optimization, including cell volume andatomic positions. The cubic shape of the cellwas conserved to maintain the FCC arrange-ment. To reduce computational costs whilekeeping accuracy, we performed a set of teststhat led to this minimalistic set of convergenceparameters: a k-point grid of 3 × 3 × 3 with
Figure 1: The 108 atoms supercell of 316L stain-less steel generated with the SQS method (left),and a representation of the electronic density atTe = 4× 104 K (right).
a cutoff energy of 15 Ha and a smearing pa-rameter of 0.001 Ha. Once forces were below5× 10−4 eV/A the convergence was consideredto be reached, leading to a supercell parameterof 10.47 A, close to the experimental value of10.77 A. Further electronic temperature depen-dent calculations were performed with a 10 ×10 × 10 k-point grid and with a similar cutoffenergy.
To ensure the accuracy of PAW atomic dataand TM pseudo-potentials, we first focused onthe determination of key properties of the puremetals formed by the main constituents of thesteel. Accordingly, we first considered Fe, Crand Ni phases at ambient conditions and ad-ditionally the high temperature phase of Fe asit adopts similar crystal structure as austeniticsteels. The electronic configurations consideredare 3d64s2, 3s23p63d54s1 and 3d84s2 for Fe, Crand Ni respectively. In standard conditions,Fe crystallizes in a body cubic centered struc-ture (BCC) that corresponds to the so-calledα phase, it shows a ferromagnetic (FM) or-der. Ni adopts a FCC structure with a FMorder, and Cr crystallizes in a BCC structure,exhibiting a complex antiferromagnetic (AFM)spin-density-wave35 as a ground state. Havinga dominant composition of Fe atoms, austeniticstainless steel results are compared to those ob-tained on Fe with the FCC structure. It corre-sponds to a high temperature allotrope knownas the γ-Fe phase characterized by a complexspin spiral magnetism leading to local FM or-ders and global AFM properties.36 As a precise
3
Table 1: Theoretical and experimental values of lattice parameters (A), bulk moduli(GPa) and magnetic moments per atom for the pure metals, as main constituents ofthe steel.
Metal Chem. Struc. ath aref Bth Bref µthB µref
B
α-Fe BCC 2.84 2.87a 175 170a 2.22 2.22a
γ-Fe FCC 3.51 [3.39-3.45]b 275 [294-346]b - -Cr BCC 2.87 2.91a 171 160a 0.27 -Ni FCC 3.52 3.52a 188 180a 0.62 0.61a
a Experimental data from Ref. 33; b Theoretical data from Ref. 34.
determination of these magnetic properties isnot the scope of this paper, the magnetic ordersare simplified to FM orders for α-Fe and Ni, aclassic AFM order for Cr and a non-magneticorder for γ-Fe. Calculations are performed witha 40 × 40 × 40 or equivalent mesh of the re-ciprocal space and a cutoff energy of 40 Ha.Lattice parameters, bulk moduli and magneticmoments of these metal phases are provided inTable 1. Theoretical results show a good agree-ment with experimental data, confirming thereliability of the potentials used for the mod-elling of the 316L stainless steel.
Te dependent Density of
States and nonequilibrium
thermodynamic properties
Laser irradiation effects are first taken intoaccount through the electronic temperatures,which constitute, to a first approximation, pri-mary excitation effect. They are applied on thefixed and cold lattice previously optimized. Thedensities of electronic states (DOS) are com-puted at 7 different electronic temperatures,from 10−2 to 4 × 104 K. The bottom of thevalence band of the electronic structure com-puted at 10−2 K is arbitrarily shifted in orderto start at 0 eV, then all Te dependent DOSare shifted from the same value. The Te de-pendent electronic structures of 316L austeniticstainless steel and γ-Fe are provided in the Fig-ure 2. The valence DOS of these metals areconstituted of two distinct zones. A continuousbackground consisting of low density of states -having a shape of a roughly square root shape
of the energy - corresponds to the energy dis-persion of spatially delocalized bands as s or pbands. Between 4 and 12 eV a part consistingof high density of electronic states correspondsto bands having a higher spatial localizationdegree, i.e. 3d bands that we refer to as thed block. The difference in behaviour between3d band and 4sp bands mainly lies in the mainquantum number, whose value controls the spa-tial extension of the orbital. Accordingly, 3d or-bitals are less spatially extended than 4sp ones,leading to weaker overlaps and interactions, andfinally to a lower energy dispersion from whichemerges the d block.
Considering 316L stainless steel and γ-Fe atTe = 10−2 K, the densities of states are quitesimilar (Figure 2a and b). Stainless steel showsa larger and smoother d block than γ-Fe, dueto multiple constituents that modify the nucleifield and induce a symmetry lowering that al-lows atomic displacements from the positionsof high symmetry. The Fermi energy is locatedwithin the d block in both cases, indicating anelectronic filling around 2/3. The d block partis the most affected by the increase of Te. Inboth materials it tends to lower the density ofthe d blocks in conjunction with an increaseof their width. A shift of the d block towardhigher energies also occurs with the increase ofthe electronic temperature. This is attributedto an increase of the electronic screening withTe that originates in an electron transfer fromspatially delocalized sp bands toward localizedd bands.17 This induces a stronger localizationof the electronic density around nucleus and re-duces the effective electron-ion potential. Thisphenomenon occurs when the electronic chemi-cal potential µ(Te) is located within a partially
4
0.0
0.5
1.0
1.5
2.0
2.5
0 5 10 15 20 25 300.0
0.8
1.6
2.4
3.2
4.0
b) -Fe
a) Stainless steel
DOS [states/eV/atom]
0 K 10000 K 40000 K
...
Energy [eV]
Figure 2: Electronic structures of 316L austeniticstainless steel (a) and γ-Fe (b) with the dependenceon the electronic temperature. Blue, green and or-ange colors respectively correspond to Te = 10−2,104 and 4 × 104 K. In dotted curves, the FermiDirac distributions of electrons are provided withtheir characteristic electronic chemical potential indashed vertical lines.
occupied d block, as already noticed for Ti andW in Ref. 17.
Figure 3 shows thermodynamic properties ofboth 316L stainless steel and γ-Fe. We fo-cus on the state parameters µe, Ce, Pe and Ne
with a dependency on the electronic tempera-ture, as an approximate approach used to de-scribe the conditions experienced by the ma-terial during the irradiation process. First theelectronic chemical potential is provided in Fig-ure 3a. This parameter is a signature of theelectronic filling of the electronic structures. Itis increasing with Te, showing strong similari-ties between steel and γ-Fe. This increase withTe is first due to the balance of electronic statesaround the Fermi level with higher densities offilled electronic states at its left side and lowerdensities of empty electronic states at its right.
Depopulated electrons are thus accommodatedby a displacement of the electronic potential to-ward higher energies. To a lower extent, µ(Te)is also affected by the shift of the d block towardhigher energy with Te. Evolution of the elec-tronic chemical potential toward high energiesresults from two superimposed effects, asymme-try of occupied and unoccupied electronic stateson one side, and shift of the d block on the otherside.
Being an indicator of the Te rise as a functionof the deposited energy, the electronic heat ca-pacity Ce is obtained from the variation of theinternal energy E with respect to the electronictemperature as Ce = ∂E/∂Te. In magnitude,this thermodynamic quantity is similar for steeland Fe, as observable in Figure 3b. However,stainless steel rapidly shows an asymptotic be-havior starting from 104 K while a more com-plex change is observable for Fe at higher tem-peratures. The comparison of Ce(Te) values tothe ones obtained for other pure metals37 con-firms a quite linear relation of the electronicheat capacities with the number of valence elec-trons, as they respond collectively to the heat-ing of the electronic subsystem.
With the increase of the electronic tempera-ture, irradiated samples undergo an importantincrease of the electronic pressure Pe.
17 Bothparameters may have important consequencesin term of phase stability.38 The pressure iscomputed from the variation of the free en-ergy F with respect to the volume V as Pe =−∂F/∂V = −∂E/∂V +Te∂S/∂V , with S beingthe entropy of the system. The change of stain-less steel electronic pressure upon excitation isprovided in Figure 3c. It linearly increases upto 150 GPa at 4 × 104 K, a behavior which isagain very similar to the one of iron within itsFCC phase. Such important modifications ofelectronic conditions (Te and Pe) might inducephases instabilities, especially since martensitictransformations are frequent for steels,39 butlaser-induced diffusionless transformations arebeyond the scope of this study.
While it has various meaning and definitiondepending on the applied field,17,37 the free elec-tron number is a key parameter used in con-tinuous nanoscopic optical and thermodynamic
5
8.5
9.0
9.5
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11.0
0
10
20
30
40
50
0.0
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1.2
1.5
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
1.3
1.5
1.6
1.8
1.9
2.1
a)
b)
...
Ce [10
5 Jm
-3K
-1]
Pe [10
2 GPa]
[eV]
Ne [per atom
]
Te [10
4 K]
c)
Stainless Steel -Fe
d)
Figure 3: Evolution of a) electronic chemical po-tentials, b) electronic heat capacities, c) electronicpressures and d) number of free electrons as a func-tion of the electronic temperature. Solid curvesstand for stainless steel and dashed curves for γ-Fe.
models. Classically, it is evaluated according tothe electronic configuration of isolated atoms40
and thus it disregards electronic transfers in-duced by solid state band structure as well aselectronic temperature effects. According tothis method, the free electrons are those belong-ing the most extended orbital of the electronicconfiguration i.e. the one having the highestmain quantum number. Thus, free electronnumbers from the classical approach are 2 forFe, 1 for Cr and 2 for Ni. For our modelled steelFe73Cr21Ni14, it leads to a number of free elec-trons per atom around of 1.8. To overcome re-strictions induced by the classical approach, wedeveloped an original and simple methodologybased on density of electronic states. The mainidea is that free electrons belong to spatiallydelocalized states that strongly interact, lead-ing to a strong energy dispersion and to a DOShaving a roughly square root shape of low den-
sity (as Al). Accordingly, the d block, that cor-responds to spatially localized electronic statesis manually removed and replaced by a squareroot function. The procedure is repeated for allTe dependent density of states and the resultingDOS is integrated with the unchanged Fermi-Dirac function, leading to a number of freeelectrons evolving with the electronic temper-ature.17 These numbers are provided for 316Lstainless steel and γ-Fe in Figure 3d. At lowelectronic temperature, Ne(Te) = 1.3 electronsfor both stainless steel and Fe. These values aresignificantly lower than expected from the clas-sical approach, which originates in the capabil-ity of the band structure to allow the transferof electrons between bands. With the rise ofTe, Ne(Te) increase for both materials, reachingabout 2 free electrons at the highest tempera-ture considered here.
Transient transport proper-
ties
Electron-phonon coupling
The effective electron-phonon coupling controlsthe energy transfer from the electronic subsys-tem to the lattice.9 It is a crucial parametergiving access to relaxation times of the nonequi-librium state and describes the effectiveness ofexcitation conversion to heat. However, thederivation of this property requires extensive re-sponse linear calculations and is generally ap-proximated. More specifically, the Te depen-dent effective electron phonon coupling G(Te)can be obtained from the knowledge of λ〈ω2〉,associated to the density of electronic statesg:41
G(Te) =π~kBλ〈ω2〉g(εF , Te)
∫ ∞−∞−g2(ε, Te)
(∂f
∂ε
)dε, (1)
where λ is the electron phonon coupling con-stant, 〈ω2〉 is the second moment of the phononspectrum. The latter is also related to the De-bye temperature 〈ω2〉 ≈ θ2D/2. ~ and kB respec-tively correspond to the Planck and Boltzmannconstants and εF is the Fermi energy. The
6
unknown parameters here, λ and 〈ω2〉 can beobtained either from experimental approachesas thermoreflectivity measurements,42 or fromelectron-phonon coupling calculations. With acell of 108 atoms, the theoretical modelling ofelectron-phonon interaction is out of calculationcapabilities. Consequently, we used the electronresistivity experimental approach of Nath andMajumbar43 to evaluate λ〈ω2〉:
λ =β~g(εF )〈υF 〉2e2
6πkBand θD =
hcs2kB
3
√6N
πV. (2)
Here, β is the stainless electronic resistivityof Fe66Cr20Ni14 the closest composition to ourmodelled stainless steel. υF is the Fermi veloc-ity, cs is the effective speed of sound in the ma-terial, N and V respectively stand for the num-ber of atoms and the volume of the cell. Theseequations give λ = 0.58 and θD = 492 K, andare used to estimate G(Te) which is providedin Figure 4. From Te = 10−2 to 8 × 103 K,G(Te) exhibits high values, this is followed by a50% decrease at intermediate temperatures, be-fore it reaches a levelled behavior at high tem-peratures. This decrease is attributed to a Te-induced Fermi broadening, bringing a signifi-cant amount of electrons into an energy rangeof lower density of states above 10-12 eV (seeFigure 2a). This induces a decrease of the avail-able transition space able to accommodate theenergy transfers from electrons to phonons. Toa lower extent, this behavior has already beenobserved for thermalized electrons for Ni.6,41
Despite these significant evolutions with the in-crease of Te, these values are very high compareto the most of pure metals,41 making the stain-less steel a material that releases rapidly theenergy of its electronic subsystem to the lat-tice. An other method provided by Petrov andAnisimov44 is available to assess the electron-phonon coupling. Based on the electronic ther-mal conductivity and the low temperature elec-tronic structure, one gets the value of 0.43 forλ (θD unchanged). This is 25% lower than inthe previous evaluation, leading to an effectiveelectron-phonon coupling 25% lower.
In order to compare these results with γ-Fe,
0 1 2 3 4
20
30
40
50
60
.
.
G [10
17 Wm
-3K
-1] Stainless Steel
-Fe
Te [10
4 K]
Figure 4: Te dependent effective electron-phononcoupling for stainless steel (solid line) and γ-Fe(dashed line).
7
we computed the electron-phonon coupling con-stant from density functional perturbation the-ory.45 λ〈ω2〉 is defined as the second moment ofthe Eliashberg function α2F (Ω):11
λ〈ω2〉 = 2
∫ ∞0
ωα2F (Ω)dω. (3)
A q-point grid of 4 × 4 × 4 was used to obtaina converged Eliashberg spectral function, lead-ing to an isotropic constant λ equals to 0.62and an associated Debye temperature of 458 K.These values are very close to the experimentalones obtained for the steel. Accordingly, differ-ences between stainless steel and γ-Fe G(Te) areweak, especially at low temperature. At highTe, differences emerge and are related to theelectronic structure evolution with Te. For γ-Fe, the electronic chemical potential leaving thed block at high temperature leads to a signif-icant decrease of the electron-phonon couplingand is not yet asymptotic despite the high Teconsidered here. This is a different behaviorcompared to stainless steel, where the d blockwidth increases with Te, maintaining the elec-tronic chemical potential in an energy region ofhigh density of electronic states.
This different behavior originates from thevarious constituents of the stainless steel. Sim-ple Cr, Fe and Ni metals have d block statesof various density, with different energy loca-tion and electronic filling. Once added into astainless steel, these initially proper character-istics are mixed and contributes to a larger dblock for steel than for γ-Fe (6.7 versus 5.8 eVof width). Moreover, with the rise of Te, thed block width increases significantly more forsteel than for γ-Fe, a phenomenon likely relatedto a stronger increase of the localization of thecharge density for the steel. This is attributedto a lower electronic occupation of the d block,since the composition percentage of Cr (6 va-lence electrons per atom) is greater than theone of Ni (10 valence electrons per atom) lead-ing to a 13% deficit of electrons with respectto γ-Fe (8 valence electrons per atom). Addi-tionally, the partial occupation of the d block islower for steel than for Fe, decreasing the pos-
sibility for the electronic chemical potential toescape from the d block. All these phenomenacontribute to maintain µ(Te) in a region of highdensity of states at high temperature leading tothe difference between steel and Fe at high Te.
Optical properties and thermalconductivity
In laser irradiation conditions, optical proper-ties are crucial parameters as they characterizethe material reflectivity and thus the amountand distribution of absorbed light energy. Inaddition, the optical properties determine thepossibility of generating evanescent penetrationof light below the surface. Optical coupling de-fines therefore the interaction with the possibil-ity of large excursion of optical properties uponelectronic heating. The thermal conductivitycontrols the diffusion of the thermal energy overlengths larger than the penetration depth, andthus is responsible for thermal energy confine-ment or spreading, with consequences on thecooling and matter state of the irradiated mate-rial. Both quantities are dependent on tempera-ture and pressure conditions that are taken intoaccount in a first-principles approach, proceed-ing in two steps. First an ab initio moleculardynamical simulation is performed in the de-sired conditions of ionic temperature and pres-sure. Once a pseudo-equilibrium is reached,ionic configurations are extracted and stand asbeing representative of the state of the matterin those conditions. Then, accurate DFT calcu-lations are performed giving access to the corre-sponding condition-dependent electronic struc-tures, that are used for the derivation of theproperties of interest. Here, assuming that thesimilar behavior of γ-Fe and 316L stainless steelextends to their optical properties and ther-mal conductivities, and in the aim of decreasingcomputational costs, the optical properties areonly computed for Fe, in its FCC phase.
The main point is to evaluate how opticalproperties evolve for Fe FCC at standard con-ditions, while its electronic temperature in-creases. It corresponds to the first step of theulrashort irradiation prior any significant en-ergy transfer from the electronic subsystem to
8
the lattice. Accordingly, the molecular dynamicsimulation is performed at Ti = Te = 300 K,in the framework of the isokinetic ensemble tokeep the ionic temperature constant. A super-cell of 16 iron atoms in a FCC lattice is used,with a reciprocal space meshed by a k-pointgrid equivalent to 3 × 3 × 3. The modellinglasted 2 ps, from which an ionic configurationis extracted at the end of the run. Then, accu-rate electronic structure calculations are carriedout involving a k-point grid equivalent to 10 ×10 × 10 and a number of bands significantlyincreased to take into account the electronictemperatures. These calculations are coupledto the Kubo-Greenwood formalism for the de-termination of the optical properties,46 and tothe Onsager coefficients giving access to thethermal conductivity.47 Here, optical proper-ties and thermal conductivities are computedat Ti = 300 K and Te equals to 300, 1.5 × 104
and 3× 104 K.The corresponding Te dependent complex op-
tical indices n = n + ik are provided in Fig-ure 5. The smooth evolution of the curveswith the photon energy indicates a good conver-gence with respect to the mesh of the reciprocalspace. Being a high temperature phase, opticalproperties of γ-Fe (FCC phase) are not avail-able, consequently, we provided experimentaldata related to α-Fe, the BCC ambient condi-tion phase. Both electronic structures of thesephases are comparable, with d blocks havingsimilar width and electronic occupation. Themain difference lies in the presence of a pseudo-band gap in the d block for BCC while a morecontinuous density of states characterizes theone of FCC phase. This difference producesa dumping of optical properties for α-Fe com-pared to γ-Fe, explaining the difference of op-tical properties between theoretical and exper-imental data. If we focus on theoretical data,the intraband component of n and k dominatesin the photon energy range up to 1 eV. Sinceat Te = 0 K the Fermi energy is located withinthe d band, an interband contribution is alsoeffective within the intraband part. The inter-band component extends up to 3 and 4 eV forn and k respectively, corresponding to the en-ergy width of the occupied d block, an energy
0 1 2 3 4 5 6 7 8 9 10
1
3
5
7
1
3
5
7
0.01.53.00.0
0.2
0.4
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1.0
1.2
1.4
1.6
Exp. [-Fe]
c)
b)
a)
k
300 K 15000 K [-Fe] 30000 K
Photon energy [eV]
.
Kth [10
3 Wm
-1K
-1]
Te [10
4 K]
n
Figure 5: Te dependent optical indices of γ-Fe,real part in a) and imaginary part in b). The ex-perimental data of α-Fe is also provided in blackdots.48,49 c) the thermal conductivity (Kth), ex-hibiting a strong dependency on the electronic tem-perature.
domains that concentrates interband electronictransitions. The increase of Te produces an im-portant redistribution of the electrons withinthe band structure as can be seen in Figure 2.The corresponding Fermi broadening activatesan increasing number of electronic transitions,whose effect is to dilute the interband signal,leading to a dumping of the interband compo-nent. Consequently, optical indices decrease inthe photon energy range corresponding to theinterband component as shown in Figure 5a andb. The Te dependent thermal conductivity isplotted in Figure 5c. A substantial increase ofthis quantity occurs with Te, in agreement withthe strong augmentation determined by Petrovet al.50 At low temperature, the computed valueis 13 Wm−1K−1, close to the experimental valueKexp
th = 15 Wm−1K−1,51 it then increases to 800and 1600 Wm−1K−1 at 1.5×104 and 3×104 K,
9
respectively. The significant modification of op-tical indices and the strong increase of the ther-mal conductivity with Te have important con-sequences on the energy absorption and trans-fer, and has to be taken into account for laser-matter interaction modelling as described in thenext section.
Two-temperature hydrody-
namic simulation
The hydrodynamic response of a stainless steelsurface to ultrashort laser pulse heating hasbeen investigated using the two-temperaturehydrodynamics code Esther. Details of thiscode can be found elsewhere.10,52 Followinglaser excitation, the one-dimensional in-depthcalculation of laser-matter interaction allows in-sight into the successive thermodynamic statesfor both electrons and ionic material. Thermo-mechanical excitation and subsequent relax-ation stages define the energy distribution in-side the material and are described accuratelyfor laser fluence ranging from sub-ablationregime to warm dense plasma generation. Toexploit the full range of electron properties cal-culated for temperatures up to 4 eV, the flu-ence has been set at 0.6 Jcm−2, correspondingto 5 times the ablation threshold for a pulseduration of 100 fs (FWHM). We note therethat the calculated threshold (0.12 Jcm−2) isin good agreement with experimental values.53
The spatio-temporal dynamics of the irradiatedsurface is represented in Figure 6, where theevolution of electron and matter temperatures,density and total pressure in the expanding lay-ers are given. Thermal excitation of the elec-trons constitutes the initial stage observed dur-ing laser absorption in the skin depth of thematerial as shown in Figure 6a.
Electronic properties determined by DFTmethod have been implemented to calculate thetransient electronic heat capacity and pressuredefining the thermodynamic state of the elec-tron subsystem, as done in Refs.21,54,55 Thethermodynamic and kinetic response of themetal is conditioned by the sharp electrontemperature gradient but also by the strong
electron-phonon coupling which leads to a fastenergy transfer from the hot electrons to thelattice. This is reflected by the ionic tempera-ture increase reaching a maximal value 1 ps af-ter laser excitation as shown in Figure 6b. Thisshort relaxation time is less than the hydrody-namic time scale τh = lT/cs = 10 ps, wherelT ' 45 nm is the thickness of the heated sur-face region while cs is the sound velocity. Thecharacteristic size of the heated layer is mainlyestablished by the competition of electron heatdiffusion and electron-phonon relaxation andthe dependence of the electron thermal conduc-tivity with temperature as discussed in section3 is crucial to determine the effective heat pen-etration depth. One can note that lT is partic-ularly low for a metal, resulting to reduce theregion of stress confinement. This relaxes bygenerating a high hydrodynamic momentum atthe surface which can locally modify the sur-face topography. In that context, materials ex-hibiting a strong confinement of the energy havebeen found to foster the growth of laser-inducedperiodic surface structures.56 High contrast ofthese periodic pattern structures has been ex-perimentally observed on stainless steel, whichis likely related to the low extension of theheated layer.
At this fluence regime, the peak ionic tem-perature does not exceed the critical point,and material decomposition is supposed to takeplace through phase explosion.57–59 The dy-namics of the ablated layers is clearly visible inFigure 6c depicting the kinetics of phase trans-formation through spatio-temporal density evo-lution. The solid-liquid interface is also shownby a dashed line showing that a thin liquid layerof 10 nm thickness was not expelled from thesurface. Solidification process takes place 150ps after irradiation and the ablation layer rep-resents around 30 nm of material at 0.6 Jcm−2.This calculated ablation depth is in excellentagreement with the experimental ones in simi-lar ultrashort irradiation conditions.60,61 Turn-ing to the investigation of the pressure evolu-tion shown in Figure 6d, we predominantly ob-serve the electronic contribution in the first pi-cosecond which has not time to propagate be-fore electron-ion equilibrium. A high pressure
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Figure 6: Spatio-temporal evolution of a) electronic temperature, b) matter temperature, c) materialdensity and d) total pressure in a hydrodynamic simulation of a 316L stainless steel target irradiated witha 100 fs laser pulse at an incident fluence of 0.6 Jcm−2. Simulation parameters integrate nonequilibriumelectronic data presented in previous sections.
layer is formed near the surface, with a maximalpressure reaching 150 GPa at Te ' 4 × 104 K.Following the stage of energy transfer below thesurface, an ionic wave propagates in-depth andtoward the surface. The component reachingthe free surface is reflected and transforms intoa rarefaction wave resulting in tensile stresses(in white color in Figure 6d in the solid or liq-uid material, following the compression wave.54
These observations suggest that upon femtosec-ond irradiation relatively far from the melt-ing threshold, extensive nonequilibrium DFTdata are required to capture the complex phasetransformation of the irradiated material.
Conclusion
In this study, the first instants of an ultrashortlaser excited 316L stainless steel are modelledin the framework of the density functional the-ory. The thermal nonequilibrium propertieshave been calculated assuming a thermalizationof the electronic subsystem and disregardingionic temperature effects. The structure of thestainless steel has been rounded to its main con-stituents and generated according to the special
quasirandom structure method in order to get arepresentative solid solution. Results have beencompared to γ-Fe, as it is the main constituentof 316L stainless steel, that has an austenitestructure, i.e. a FCC atomic arrangement.
Within the electronic density of states, the dblock is found to enlarge and to shift towardhigher energies due to the excitation process.This is in agreement with an increase of theelectronic screening corresponding to a changeof the electronic distribution from spatially de-localized states to more localized ones, as al-ready observed for the pure metals W and Ti.According to the unbalance of occupied elec-tronic states and unoccupied ones from bothside of the Fermi energy, the electronic chemi-cal potential is found to shift toward higher en-ergies to accommodate the excess of electronsfor both metals.
The electronic heat capacities, electronicpressure, free electron numbers, effectiveelectron-phonon coupling, and optical and ther-mal conductivities have been computed witha dependence on the electronic temperature.Electronic heat capacities are in agreement withthe number of valence electrons per atom show-
11
ing a clear asymptotic behavior for stainlesssteel. Electronic pressures increase significantlywith Te, as expected for intense laser irradiationconditions. However, at the difference of mostof pure metals, the electron-phonon couplingis very high at low temperature, decreasingby a factor two at intermediate temperatures,and reaching an asymptotic behavior at hightemperature. The interband signal for opticalconductivities is weakened while the thermalconductivity significantly increases with theelectronic temperature.
These nonequilibrium thermodynamic ortransport properties are quite similar between316L stainless steel and γ-Fe, with some de-viations at high temperature, especially con-cerning the number of free electrons and theelectron-phonon coupling. This similar behav-ior tends to show that these results concerningboth 316L stainless steels and γ-Fe can beextended to a set of similar austenitic steels.Accordingly, the dynamics of femtosecond laserablation of a stainless steel target was studiedbased on a hydrodynamic simulation, relyingon nonequilibrium DFT calculated data forelectron temperature range commonly reachedin materials processing.
Acknowledgement This work was sup-ported by the ANR project DYLIPSS(ANR-12-IS04-0002-01) and by the LABEXMANUTECH-SISE (ANR-10-LABX-0075) ofthe Universite de Lyon, within the program“Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Re-search Agency (ANR). Part of the numericalcalculations has been performed using resourcesfrom GENCI, project gen7041.
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