C Salvan

T Baffie

CuCrZr alloy produced by Laser Powder Bed Fusion: microstructure, nanoscale strengthening mechanisms, electrical and mechanical properties

Keywords: Laser Powder Bed Fusion, mechanical properties, electrical properties, CuCrZr alloy, nanoscale strengthening mechanisms, heat treatments

HV1 hardness, YS=527 MPa, UTS=585MPa) while keeping a good elongation (14%) and electrical conductivity (42 MS.m -1 ). These results are due to a combination between (i) a high Cr nano-precipitates density, leading to a high precipitation hardening, and (ii) a high dislocation density associated to the presence of remaining solidification cells.

Introduction

The Copper-Chromium-Zirconium (Cr: 0.5-1.2 wt.%; Zr: 0.03-0.3 wt.%) alloy is a precipitation-hardening alloy, which offers high electrical, thermal and mechanical properties.

It is used, for example, in nuclear fusion reactor components as heat exchangers [START_REF] Ma | Selective laser melting of Cu-Cr-Zr copper alloy: Parameter optimization, microstructure and mechanical properties[END_REF][START_REF] Zinkle | Applicability of copper alloys for DEMO high heat flux components[END_REF]. Laser Powder Bed Fusion (L-PBF) is a metallic additive manufacturing process, which has multiple advantages such as the ability to create easily complex shape components. It is used to build heat exchangers with small internal channels, improving their thermal exchange properties [START_REF] Ma | Selective laser melting of Cu-Cr-Zr copper alloy: Parameter optimization, microstructure and mechanical properties[END_REF].

L-PBF process involves depositing a thin layer of metal powder that is melted by a laser according to a pattern defined by a Computer Aided Design (CAD) file. This operation is repeated to create a three-dimensional object.

Due to high optical reflectivity of copper-rich powders at the laser wavelength (1070 nm), several authors have studied the effect of coatings on Cu on CuCr powders, both for increasing absorption and forming in-situ alloys [START_REF] Lindström | Laser Powder Bed Fusion of Metal Coated Copper Powders[END_REF][START_REF] Jadhav | Highly conductive and strong CuSn0.3 alloy processed via laser powder bed fusion starting from a tin-coated copper powder[END_REF][START_REF] Jadhav | Laser powder bed fusion additive manufacturing of highly conductive parts made of optically absorptive carburized CuCr1 powder[END_REF]. Even if this approach is effective in decreasing laser reflectivity, and thus the energy density required for melting the powder, neither Sn coating [START_REF] Lindström | Laser Powder Bed Fusion of Metal Coated Copper Powders[END_REF][START_REF] Jadhav | Highly conductive and strong CuSn0.3 alloy processed via laser powder bed fusion starting from a tin-coated copper powder[END_REF], nor Ni coating [START_REF] Lindström | Laser Powder Bed Fusion of Metal Coated Copper Powders[END_REF] lead to a final alloy as mechanically resistant as CuCrZr. The formation of a carburized layer at the surface of a CuCr powder, thanks to a nanometer carbon black powder deposition followed by a 750°C heat treatment [START_REF] Jadhav | Laser powder bed fusion additive manufacturing of highly conductive parts made of optically absorptive carburized CuCr1 powder[END_REF], allowed to get relatively high tensile properties of L-PBF parts (Yield Strength YS = 347 MPa) after direct ageing, but this was only reached with a laser power over 400W. Indeed, CuCrZr gas-atomized powder can be processed efficiently in the 300-400W power range without costly additional powder modifications.

Despite the difficulties to process CuCr or CuCrZr alloy by L-PBF due to their high thermal conductivity and high optical reflectivity at 1070 nm [START_REF] Popovich | Microstructure and mechanical properties of additive manufactured copper alloy[END_REF], several recent studies [START_REF] Ma | Selective laser melting of Cu-Cr-Zr copper alloy: Parameter optimization, microstructure and mechanical properties[END_REF][START_REF] Jahns | Additive manufacturing of CuCr1Zr by development of a gas atomization and laser powder bed fusion routine[END_REF][START_REF] Buchmayr | Laser Powder Bed Fusion -Materials Issues and Optimized Processing Parameters for Tool steels, AlSiMg-and CuCrZr-Alloys[END_REF][START_REF] Wallis | Effect of heat treatments on microstructure and properties of CuCrZr produced by laser-powder bed fusion[END_REF][START_REF] Zhang | Microstructure and properties in QCr0.8 alloy produced by selective laser melting with different heat treatment[END_REF][START_REF] Zeng | Effect of temperature history on thermal properties of additively manufactured C-18150 alloy samples[END_REF] showed the ability to create dense CuCr and CuCrZr parts (relative density > 99%) in particular by using laser power over 300 W. Zeng et al. [START_REF] Zeng | Effect of temperature history on thermal properties of additively manufactured C-18150 alloy samples[END_REF] studied the evolution of the Cu lattice parameter by X-rays diffraction (XRD) after heat treatments and the thermal properties of Cu-1.5Cr-0.5Zr alloy. This short communication only addresses the thermal properties; microstructures and mechanical properties were not deeply studied. The authors indicate that as-built L-PBF CuCrZr is not stable when re-heated and must be stabilized at temperatures above 500°C to form precipitates.

CuCr and CuCrZr wrought alloy needs to be heat-treated in order to induce the precipitation of nano-phases responsible for the hardening of the alloy. The different stages of heat treatments consist in a solution annealing (SA) of the Cr and Zr elements in the Cu matrix, followed by water quenching (WQ) leading to a supersaturated matrix. Finally, an age hardening (AH) treatment is necessary to precipitate (Cr, Zr) nano-phases in the matrix [START_REF] Holzwarth | The precipitation behaviour of ITER-grade Cu-Cr-Zr alloy after simulating the thermal cycle of hot isostatic pressing[END_REF][START_REF] Chbihi | Atomic scale investigation of Cr precipitation in copper[END_REF].

As the high cooling rate involved during the L-PBF process (of the order of 10 6 K/s) has the effect of quenching the microstructure, a direct age hardening (DAH) heat treatment can be applied to the as-built samples. Indeed, several studies [START_REF] Jahns | Additive manufacturing of CuCr1Zr by development of a gas atomization and laser powder bed fusion routine[END_REF][START_REF] Wallis | Effect of heat treatments on microstructure and properties of CuCrZr produced by laser-powder bed fusion[END_REF][START_REF] Guan | Effect of selective laser melting process parameters and aging heat treatment on properties of CuCrZr alloy[END_REF] report that the L-PBF as-built CuCr or CuCrZr alloys show no evidence of Cr or Zr precipitation at micrometric scale. More interesting, such a heat treatment leads to an increase of the Vickers hardness [START_REF] Jahns | Additive manufacturing of CuCr1Zr by development of a gas atomization and laser powder bed fusion routine[END_REF][START_REF] Wallis | Effect of heat treatments on microstructure and properties of CuCrZr produced by laser-powder bed fusion[END_REF], the ultimate tensile stress (UTS) [START_REF] Jahns | Additive manufacturing of CuCr1Zr by development of a gas atomization and laser powder bed fusion routine[END_REF][START_REF] Guan | Effect of selective laser melting process parameters and aging heat treatment on properties of CuCrZr alloy[END_REF] and the electrical conductivity [START_REF] Jahns | Additive manufacturing of CuCr1Zr by development of a gas atomization and laser powder bed fusion routine[END_REF][START_REF] Guan | Effect of selective laser melting process parameters and aging heat treatment on properties of CuCrZr alloy[END_REF] with a thermal conductivity [START_REF] Wallis | Effect of heat treatments on microstructure and properties of CuCrZr produced by laser-powder bed fusion[END_REF] comparable to that of wrought samples subjected to a conventional SA+AH heat treatment.

Zhang et al. [START_REF] Zhang | Microstructure and properties in QCr0.8 alloy produced by selective laser melting with different heat treatment[END_REF] compared the effect of several heat treatments (SA between 900°C and 1020°C; SA at 1020°C + AH at 480°C; DAH at 480°C) on the lattice parameter, microstructure, mechanical properties, electrical properties and corrosion performances of Cu-0.5Cr alloy.

Nano-precipitates were identified as nano-Cr face centered cubic (FCC) or body centered cubic (BCC) phases with specific orientation relationship as determined by Transmission Electron Microscopy (TEM)/selected area electron diffraction pattern. Concentrations of alloying elements in matrix range from 0.46 wt% in the as-built state, to 0.29 wt% in the SA+AH state and 0.03 % in the DAH, showing the interest of this last heat-treatment on the precipitation.

The volume fraction of nano-precipitates varies from 0.05% in as-built state to 0.25 % in SA+AH state and up to 0.58 % in the DAH state. The experimental yield strength reaches 377 MPa after DAH, the elongation being 19% and the electrical conductivity 98 %IACS. The authors compared their YS experimental values to modelled ones using an addition of the different hardening contributions and experimental data from electron backscattered diffraction (EBSD) and TEM analyses; they found a good correlation.

Several studies dedicated to wrought CuCrZr alloy have shown that adding small amounts of zirconium to the Cu-Cr alloy plays a role in the precipitation of Cr and significantly increases the mechanical properties of the alloy [START_REF] Jinshui | Effect of different Zr contents on properties and microstructure of Cu-Cr-Zr alloys[END_REF][START_REF] Chen | Contribution of Zr to strength and grain refinement in Cu Cr Zr alloy[END_REF][START_REF] Watanabe | Mechanical properties of Cu-Cr system alloys with and without Zr and Ag[END_REF]. With a small addition of Zr in a Cu-0.5Cr alloy, increases of 10 % on the yield strength and 6 % on the Vickers hardness are observed [START_REF] Watanabe | Mechanical properties of Cu-Cr system alloys with and without Zr and Ag[END_REF]. So, in a similar way as it was done by [START_REF] Zhang | Microstructure and properties in QCr0.8 alloy produced by selective laser melting with different heat treatment[END_REF] for L-PBF Cu-Cr, there is an important interest to determine the contribution of each strengthening mechanisms in the case of a L-PBF CuCrZr alloy.

The main objective of the present work is to clarify the impact of the L-PBF process and postfabrication heat treatments on the nano-scale microstructure of L-PBF CuCrZr alloy and the different strengthening mechanisms (related to nano-precipitation, dislocation strengthening, solid solution strengthening, grain size refinement). The understanding of these mechanisms are essential to be able to tailor the mechanical and electrical properties of the alloy to the targeted application. For this purpose, a study was performed to select the process parameters leading to the highest density. Then, scanning electron microcopy (SEM) associated to electron backscattered diffraction (EBSD) and transmission electron microscopy (TEM) associated to energy-dispersive X-rays spectroscopy (X-EDS) were used to characterize the samples microstructures. Vickers hardness and tensile tests were used to compare the mechanical properties, whereas eddy current measurements allowed determining the electrical properties.

Based on EBSD and TEM analyses, mean grain size, dislocation density, mean radius and volume fraction of nano-precipitates were measured on the different heat-treated samples.

Those measurements were used to calculate the yield strengths and compare them to the experimental YS values. The results obtained on L-PBF samples (As-built, SA+AH and DAH) were compared to those of a reference wrought SA+AH material. For the first time, the main hardening mechanisms of L-PBF CuCrZr are quantified and the interest of the "L-PBF+DAH" process route to get a high amount of chromium nano-precipitates is shown.

Materials and methods

Materials

L-PBF CuCrZr specimens were produced using a spherical 10-45 µm nitrogen-atomized powder (TLS Technik GmbH & Co). Chemical compositions of the powder, the L-PBF samples and the wrought CuCrZr plate, used as a reference, were measured by Instrumental Gas Analysis (IGA) and Inductively Coupled Plasma Optical Emission Spectroscopy (ICP-OES) (Table 1). Chromium and zirconium contents in all samples are conform to the EN12420:2014 standard [START_REF] Afnor | Cuivre et alliages de cuivre -Pièces forgées[END_REF]. The iron content in the powder and the L-PBF samples is a bit higher than the minimum value expected by the standard.

L-PBF specimens

All specimens were produced on a FS271 L-PBF device (Farsoon, China). The device is equipped with a 480 W fiber laser with a wavelength of 1070 nm and a 130 µm Gaussian spot diameter. The powder was handled in a glovebox, dried before L-PBF and the oxygen level was maintained below 250 ppm during the whole fabrication process. During the process, the powder was spread using a silicon blade, and the specimens were produced on a 270 mm x 270 mm C45 steel platform. The layer thickness was set to 30 µm. A zigzag laser strategy on 10 mm wide stripes, forming a 45° angle with the cubes or pieces edge, with a 90° rotation between layers was used. The chamber was filled with argon gas. Cubes (10x10x10 mm 3 , Figure 1.b)

were built with a different set of parameters and the resulting density was measured. Their base was rotated 45° with respect to the blade path. The laser power (P) varied from 360 to 480 W, the scan speed (v) from 200 to 700 mm.s -1 and the hatch (h) from 100 to 120 µm. The optimized parameters, i.e. the parameters that lead to the highest density (99.4%), are a power of 480 W, a scan speed of 700 mm.s -1 and a hatch of 100 µm (Figure 1.a). Those parameters were then used to fabricate mechanical and electrical samples (Figure 1.c). 55x13x13 mm 3 parallelepipeds were built for mechanical samples; their small bases was rotated 45° in the YZ plane and their main axis was rotated 45° relative to the direction of the blade. 18 mm diameter and 22 mm high cylinders were built for electrical samples.

Material characterization techniques

The density of L-PBF CuCrZr was measured on polished samples (SiC P320), by Archimedes method using isopropanol thanks to the equation (1) :

1.

ρ relative density = M L-PBF sample M HIP (1)
where 𝑀 𝐿-𝑃𝐵𝐹 𝑠𝑎𝑚𝑝𝑙𝑒 is the density of L-PBF CuCrZr, 𝑀 𝐻𝐼𝑃 is the density of Hot Isostatically Pressed (HIPed) CuCrZr samples obtained by HIPping the CuCrZr powder at 1040°C for 2 hours under 1400 bar; their density was measured with helium pycnometer on an AccuPyc II 1340 Micrometrics and used as a reference. The standard deviation on the relative density is ±0.25%. The chemical composition was verified by IGA and ICP-OES on three different parts for each CuCrZr (Table 1). The SA + AH heat treatments have been applied to L-PBF and wrought samples. They consist in a solution annealing followed by water quenching (WQ), and subsequent thermal ageing under air followed by air-cooling (Table 2). The temperature and exposure time were chosen according to previous research on this material for fusion nuclear reactor applications [START_REF] Frayssines | CuCrZr alloy microstructure and mechanical properties after hot isostatic pressing bonding cycles[END_REF][START_REF] Ivanov | Effect of heat treatments on the properties of CuCrZr alloys[END_REF]. A DAH heat treatment, which consists in an age hardening under air followed by air-cooling [START_REF] Wallis | Effect of heat treatments on microstructure and properties of CuCrZr produced by laser-powder bed fusion[END_REF] (Table 2), has been applied to as-built L-PBF samples. This heat treatment corresponds to the best compromise between mechanical and thermal properties according to [START_REF] Wallis | Effect of heat treatments on microstructure and properties of CuCrZr produced by laser-powder bed fusion[END_REF]. For the SEM and TEM observations, samples were cut normal to the building direction (noted BD plan in Figure 2) or parallel to the building direction (noted ↑ BD plan in Figure 2). The plates were mechanically polished down to 100 µm thickness, punched out into 3 mm disks and electro-polished. The electro-polishing was performed at room temperature with a solution of 25 vol.% ethylene glycol, 25 vol.% ortho-phosphoric acid and 50 vol.% deionized water at 20°C, under an applied potential of 12 V, and with a 200-300 mA current. For SEM analysis, the electro-polishing was stopped quickly, to remove hundreds of nanometers of damaged surface. For TEM analysis, the electro-polishing was carried on to obtain a thickness of around 20 µm. The disks were then polished using an argon ion beam polishing (PIPS 2 GATAN 691)

to create a hole and avoid oxidation issues observed after an electro-polishing.

SEM observations were performed on a field emission gun (FEG) SEM Zeiss Merlin equipped with two Energy Dispersive Spectroscopy (EDS) detectors and an Electron Back-scattered Diffraction (EBSD) QUANTAX detector from Bruker. EBSD analyses using Esprit 2.2 software for data processing were used to assess the average grain size and the median misorientation angle. Grains boundaries are defined with a misorientation angle higher than 15°

and subgrains boundaries with a misorientation angle lower than 15°. The grain size is defined as the equivalent diameter of a circle of the same surface as the observed grains. The average grain size is weighted by the surface. A Kernel Average Misorientation map (KAM) analysis, which measures the local misorientation of a pixel between 0.5° and 5° against the closest pixels, was used to define the median misorientation angle. This angle (𝜃 [rad]) is then used to estimate the dislocation density (𝜌 0 ) thanks to equation (2) [START_REF] Zhilyaev | Grain refinement kinetics and strengthening mechanisms in Cu-0.3Cr-0.5Zr alloy subjected to intense plastic deformation[END_REF].

𝜌 0 = 2√3 3 𝜃 𝑏𝑠 (2)
With b [m], the Burgers vector and s [m] the step size used during the EBSD analysis. The Burgers vector is taken equal to 0.256 nm for a Cu matrix [START_REF] Holzwarth | The precipitation behaviour of ITER-grade Cu-Cr-Zr alloy after simulating the thermal cycle of hot isostatic pressing[END_REF].

TEM analyses were performed on a TITAN Themis (FEI), operating at 200kV and equipped with a Cs probe corrector, a Super-X system (4 Silicon Drift Detectors) for EDS analyses and a Gatan Imaging Filter (GIF Quantum) for Electron Energy Loss Spectroscopy (EELS) analysis. Images were acquired in scanning transmission electron microscopy (STEM) mode using either a HAADF (high angle annular dark field) or a BF (bright field) detector.

Mechanical and electrical analyses

Vickers hardness was measured on a Matsuzawa MMT-X7B micro-hardness tester according to EN ISO 6507-(1-4) [START_REF]1 : méthode d'essai -Partie 2 : vérification et étalonnage des machines -Partie 3 : étalonnage des blocs de référence -Partie 4 : tableaux des valeurs de dureté[END_REF] with a load of 1 kg during 15 seconds; 10 indents were made per sample. Tensile tests perpendicular to the building direction were done on 3 specimens per CuCrZr state under air, at 20°C on a MTS servo-hydraulic testing machine with a 100 kN load cell at a 2.5x10 -4 s -1 strain rate. Tensile bars were made according to NF EN ISO 6892-1 [START_REF]Matériaux métalliques -Essai de traction-Partie 1: Méthode d'essai à température ambiante[END_REF].

Electrical conductivity was measured, according to EN 2004-1 standard [START_REF]Série aérospatiale -Méthodes d'essais applicables aux produits en aluminium et alliages d'aluminium -Partie 1 : détermination de la conductivité électrique des alliages d'aluminium corroyés[END_REF], at 20°C on 3 polished specimens per CuCrZr state (SiC P4000), by Eddy current on a Fischer FS40 machine.

Experimental results

Relative density and chemical composition

𝐸 𝑣 = 𝑃 𝑣 * ℎ * 𝑙 (3)
The relative density of the alloy increases with increasing Ev input. A minimum Ev of approximatively 300 J.mm -3 is needed to achieve a relative density equal to or higher than 99%.

At a laser power of 360 W, only a few sets of parameters leading to a high Ev with small scan speeds (200 and 300 mm.s -1 ), offer a relative density higher than 99%. Whereas, 480 W laser power offers the widest range of Ev which can achieve a relative density higher than 99% even with a laser speed of 700 mm.s -1 that allows higher productivity. Consequently, a laser power of 480 W, a scan speed of 700 mm.s -1 and a hatch of 100 µm were used for building the samples of this work.

The chemical composition of the samples is comparable to that of the powder (Table 1). The oxygen content is stable, showing no evidence of oxidation during the process. In addition, no evidence of impurities, cross contamination, or Zr loss, was detected. Inside the subgrains, TEM observations revealed the presence of dislocation cells having an average size of 300 nm (Figure 3f). They probably are solidification cells resulting from the high solidification rate of the L-PBF process and are observed on other L-PBF alloys [START_REF] Krakhmalev | Microstructure, Solidification Texture, and Thermal Stability of 316 L Stainless Steel Manufactured by Laser Powder Bed Fusion[END_REF].

Microstructure of the L-PBF as-built CuCrZr

KAM analysis is used to estimate the dislocation density in the CuCrZr (equation ( 1)). The measurement was performed at two different magnifications: at low magnification, which underestimates the dislocation density but provides an average dislocation density in this nonhomogeneous CuCrZr, and at high magnification, with a step size of 0.1 µm that is smaller than the solidification cells (300 nm) for a more relevant estimation. These values are used to compare the different metallurgical states and the results are reported in Table 4. The as-built

CuCrZr presents a high dislocation density (5 ± 1 x 10 14 m -2 ), comparable to the dislocation density obtained on cold worked CuCrZr (5 x 10 14 m -2 ) [START_REF] Huang | Microstructure and strengthening mechanisms of CuCrZr alloy by two-step thermomechanical treatment[END_REF]. The dislocations appear during the L-PBF process due to high stresses induced by the thermal gradients generated during the rapid solidification. The thermal history of the CuCrZr during the process is complex and inhomogeneous, and so is the microstructure.

Effect of the heat treatment on L-PBF CuCrZr and comparison with wrought

CuCrZr

Figure 4 shows the SEM and EBSD microstructures of the wrought and the different L-PBF heat-treated CuCrZr (heat treatments are reported in Table 2). The wrought SA+AH CuCrZr whose microstructure (Figure 4.a and d) is different to the L-PBF alloy, shows large equiaxed grains with numerous twins, leading to an average grain size (including twins) of 470 ± 150 µm (Table 3). A dislocation density of 3 x 10 13 was estimated using KAM analysis (Table 5). This density is similar to the value found by [START_REF] Huang | Microstructure and strengthening mechanisms of CuCrZr alloy by two-step thermomechanical treatment[END_REF] on wrought Cu-0.8Cr-0.1Zr (wt.%) in the same metallurgical state.

For the L-PBF alloy, the SA + AH heat treatment caused an evolution of the microstructure (Figure 4.b and e) compared to the as-built state (Figure 3). The pavement shape of the grain cross-section in the BD plan and the elongated grain cross-section in the ↑ BD plan is partially preserved however in some area, recrystallization has occurred and has resulted in larger twinned grains. The average grain size is 276 ± 150 µm in the BD plan and 334 ± 140 µm in the ↑BD plan (Table 3). This partial recrystallization (60% of the surface) is confirmed by the KAM analysis ( µm was used for KAM analysis to estimate the dislocation density (Table 4). Dislocation density in the recrystallized zones is found to be close to the value of the wrought SA+AH CuCrZr (3 ± 1 x 10 13 m -2 ), while in the non-recrystallized zone it is higher (7 ± 1 x 10 13 m -2 ). On the contrary, the DAH heat treatment involved no change of grain morphology and size compared to the as-built state (Figure 4.c and f), a similar average grain sizes of 112 ± 60 µm, in the BD plan and 242 ± 100 µm in the ↑ BD plan were measured (Table 3). Moreover, the dislocation cells with similar size as the as-built state were still observed by TEM. It is however likely that the heat treatment at 490°C allows a dislocation recovery in the cell walls that can explain the slightly lower dislocation density measured by KAM analysis (Figure 4.h) than in the as-built state (4 x 10 14 m -2 versus 5 x 10 14 m -2 ) although this difference is within the measurement error bar.

Micro-and nano-precipitation

In the as-built L-PBF CuCrZr, Cr, and Zr micro-or nano-precipitates are not detected (Figure Zr nano-precipitates are observed in different locations and with different shapes after SA+AH or DAH heat treatments: at grain boundaries, next to Cr nano-precipitates (the Zr precipitate size is then between 5 and 30 nm as in Figure 6.d), inside the Cu matrix (Zr hexagonal precipitate with 10-40 nm of diameter). EDS analyses of these Zr nano-precipitates show a particularly high Cu peak suggesting these precipitates could be Cu5Zr or Cu4Zr according to [START_REF] Holzwarth | The precipitation behaviour of ITER-grade Cu-Cr-Zr alloy after simulating the thermal cycle of hot isostatic pressing[END_REF].

The Cr nano-precipitates were observed in all heat-treated CuCrZr samples; they are uniformly dispersed in the Cu matrix (Figure 6.b and c), and display an ellipsoidal shape (Figure 6.e and f). They have a face centered cubic (FCC) structure and are coherent with the matrix. In the wrought SA+AH CuCrZr, the equivalent radius (r) is uniform at 1.8 ± 1.0 nm (Table 5). In the L-PBF DAH CuCrZr, r varies depending on the nano-precipitate location. For the precipitates, which have grown on dislocations or grain boundaries, it is around 3.2 ± 1.6 nm, whereas for one located in the Cu matrix it is around 2.1 ± 1.5 nm (Table 5). This could be explained by a specific precipitation occurring at the grain boundaries in DAH CuCrZr. Note the equivalent radius measurement gives a high standard deviation. To determine the volume fraction of the precipitates, the thickness of the CuCrZr thin foils where TEM analysis was performed was roughly estimated by EELS at 30 nm. Considering this value, the volume fraction of the Cr nano-precipitates in the CuCrZr matrix of the L-PBF DAH CuCrZr was estimated (3 images were analyzed) to be 0.6 % (Table 5). This is consistent with the theoretical volume fraction of Cr in solid solution in the L-PBF as built CuCrZr which is 0.62 vol% (estimation made with the chemical composition of Table 1, and thanks to the theoretical density of Cu, Cr and Zr [START_REF] Lide | CRC handbook of chemistry and physics: a ready-reference book of chemical and physical data, 90[END_REF]). Then the volume fraction of Cr nano-precipitates is twice higher in the L-PBF DAH CuCrZr than in the wrought SA+AH CuCrZr (Table 5). As the size of the precipitates in both L-PBF DAH and wrought SA+AH CuCrZr is equivalent, the number of precipitates is lower, and thus the distance between them is greater in the L-PBF DAH CuCrZr.

Mechanical properties

HV1 Vickers hardness performed on the different materials are presented in Table 6 and, as indicated by the standard deviation, the hardness testing has good repeatability. The hardness of the as-built L-PBF CuCrZr is 101 ± 4 HV1, which is close to the values found by [START_REF] Jahns | Additive manufacturing of CuCr1Zr by development of a gas atomization and laser powder bed fusion routine[END_REF][START_REF] Wallis | Effect of heat treatments on microstructure and properties of CuCrZr produced by laser-powder bed fusion[END_REF] for similar process parameters. This obtained value is high compared to that of the CuCrZr (SA+WQ) from literature (65-77 HV1 [START_REF] Merola | Influence of the manufacturing heat cycles on the CuCrZr properties[END_REF]), which is also a microstructural state free from nano-precipitates. Hardness was found to be similar for L-PBF SA+AH and wrought SA+AH CuCrZr tested here and compared to similar CuCrZr SA+AH from the literature [START_REF] Buchmayr | Laser Powder Bed Fusion -Materials Issues and Optimized Processing Parameters for Tool steels, AlSiMg-and CuCrZr-Alloys[END_REF][START_REF] Wallis | Effect of heat treatments on microstructure and properties of CuCrZr produced by laser-powder bed fusion[END_REF][START_REF] Holzwarth | The precipitation behaviour of ITER-grade Cu-Cr-Zr alloy after simulating the thermal cycle of hot isostatic pressing[END_REF][START_REF] Ivanov | Effect of heat treatments on the properties of CuCrZr alloys[END_REF]. This is probably due to the recrystallization occurring during the SA heat treatment. By contrast, the L-PBF DAH CuCrZr hardness is much higher than the others, in agreement with literature [START_REF] Jahns | Additive manufacturing of CuCr1Zr by development of a gas atomization and laser powder bed fusion routine[END_REF][START_REF] Wallis | Effect of heat treatments on microstructure and properties of CuCrZr produced by laser-powder bed fusion[END_REF]. different mechanical behavior: an elongation half as high as wrought SA+AH CuCrZr and a YS 95% higher, even higher than the L-PBF DAH CuCrZr of [START_REF] Guan | Effect of selective laser melting process parameters and aging heat treatment on properties of CuCrZr alloy[END_REF]. A limited strain hardening is also observed on the L-PBF DAH CuCrZr. The presence of cells and extremely high dislocation density in L-PBF DAH CuCrZr is probably due to an equilibrium between creation and annihilation of dislocations, which explains the shape of the curves. Moreover, in all tested samples (Figure 7.c), dimples are visible on the fracture surface.

Electrical properties

The low electrical conductivity of L-PBF as-built CuCrZr is consistent with a supersaturated solid solution state of the Cu matrix (Table 7) and the observations made in ref [START_REF] Jahns | Additive manufacturing of CuCr1Zr by development of a gas atomization and laser powder bed fusion routine[END_REF]. After SA+AH, both the L-PBF and wrought CuCrZr samples show similar electrical conductivity, even compared to the wrought SA+AH from literature [START_REF] Ivanov | Effect of heat treatments on the properties of CuCrZr alloys[END_REF]. Finally, L-PBF DAH CuCrZr displays a lower conductivity compared to SA+AH samples, due to the still high amount of dislocation density [START_REF] Li | Study on improvement of conductivity of Cu-Cr-Zr alloys[END_REF]. This tendency is in agreement with the values of other studies [START_REF] Jahns | Additive manufacturing of CuCr1Zr by development of a gas atomization and laser powder bed fusion routine[END_REF][START_REF] Guan | Effect of selective laser melting process parameters and aging heat treatment on properties of CuCrZr alloy[END_REF].

Microstructural evolution of L-PBF CuCrZr

During L-PBF process, the heat transfer through the already densified part tends to create elongated grains in the building direction. Gorsse et al. [START_REF] Gorsse | Additive manufacturing of metals: a brief review of the characteristic microstructures and properties of steels, Ti-6Al-4V and high-entropy alloys[END_REF] found similar grain morphology, i.e. elongated grains in the ↑BD plan and pavement of the size of the laser hatch in the BD plan. Moreover, a cellular growth is observed due to high cooling rate typical of materials fabricated by L-PBF [START_REF] Gorsse | Additive manufacturing of metals: a brief review of the characteristic microstructures and properties of steels, Ti-6Al-4V and high-entropy alloys[END_REF][START_REF] Krakhmalev | Microstructure, Solidification Texture, and Thermal Stability of 316 L Stainless Steel Manufactured by Laser Powder Bed Fusion[END_REF]. This leads to a high dislocation density present in the as-built part.

They do not show any evidence of micro-or nano-precipitates (Figure 8). The solubility limit of Cr is then 0.78 wt.%, more than the theoretical solubility limit of Cr in Cu-Cr alloy (0.72 wt.% [START_REF] Que | Microstructure Evolution of the Cu-Cr Hypereutectic Alloys During Directional Solidification[END_REF]). metallurgical states in the following sections. Some parameters have been evaluated experimentally whereas the others have been taken from the literature (Table 8). The elements remaining in solid solution in the Cu matrix can play a strengthening effect caused by the induced crystal lattice distortion affecting dislocation movement (equation ( 5)) [START_REF] Zhang | Microstructure and properties in QCr0.8 alloy produced by selective laser melting with different heat treatment[END_REF]:

∆𝜎 𝑠𝑠 = 𝑀𝐺𝜀 3/2 √ 𝑥 𝑎 3 (5)
With 𝑥 𝑎 [𝑤𝑡. %] the mass concentration of the element in the solid solution. The Zr, O, Si and Fe contributions are neglected due to their low content and low solubility limits in the Cu matrix [START_REF] Caron | Copper: Alloying[END_REF].

In the as-built L-PBF CuCrZr, all Cr atoms are in solid solution (𝑥 𝑎 is then equal to 0.78 wt.%, Table 1), as no evidence of precipitation has been found during TEM observations. Thus, the contribution of solid solution strengthening to YS is 35 MPa. After SA+AH or DAH, nearly all Cr atoms have precipitated as the electrical conductivity proves. We decided to discard this contribution for these heat treatments. Indeed, even with 10 wt.% of the total content of Cr in solid solution, this contribution would represent less than 10 MPa.

Grain size

The contribution of the grain size to the YS is given by the Hall-Petch model ( 6) [START_REF] Hansen | Hall-Petch relation and boundary strengthening[END_REF][START_REF] Thompson | The effect of grain size on fatigue[END_REF].

∆𝜎 𝑔𝑠 = k √D (6)
With D [µm] the average grain size. As shown in Figure 3, the grains of L-PBF CuCrZr are columnar. Thus, an assumption has been made on the grain size. We decided to take the critical situation and thus consider the smallest grain size, i.e. in the BD plan. In addition, twins appear in the SA+AH CuCrZr. As twins also impede the dislocation motion, they have been considered as grain boundaries, lowering the values given in Table 3. The calculated gs for the different CuCrZr are reported in Table 9. The twins in SA+AH CuCrZr do not affect significantly the grain contribution to 𝑌𝑆. Whatever the metallurgical state, the grain size contribution to the YS is low and accounts for 3% to 8% of the total YS. The dislocation strengthening mechanism, due to the dislocation forest effect, can be estimated in an isotropic material thanks to equation ( 7) [START_REF] Huang | Microstructure and strengthening mechanisms of CuCrZr alloy by two-step thermomechanical treatment[END_REF]43] (the heterogeneous distribution of dislocation density is neglected).

∆𝝈 𝑑𝑠 = 𝛼𝑀𝐺𝑏√𝜌 0 (7)
With 𝜌 0 [m -2 ] the dislocation density, measured by KAM analysis (section 3.2 and Table 4). The dislocation strengthening magnitude depends on the process and the applied heat treatment (Table 9). On a wrought SA+AH CuCrZr, this contribution to the YS is low, since the heat treatment leads to a rather low dislocation density. By contrast, it is 4.5 times higher for the asbuilt L-PBF sample due to the high dislocation density induced by the high cooling rate. The SA part of the SA+AH strongly decreases the dislocation strengthening contribution to YS. The remaining non-recrystallized grains in this sample however contribute to an intermediate dislocation strengthening, which is 25% higher than the one of wrought SA+AH. With the DAH, the dislocation density slightly decreases due to a slight recovery of the dislocations located on the solidification cells walls. However, this contribution is 3.7 times higher than the one of wrought SA+AH.

Precipitation strengthening

In this part, we assumed that precipitation strengthening is driven by the Orowan-Ashby mechanism [START_REF] Holzwarth | The precipitation behaviour of ITER-grade Cu-Cr-Zr alloy after simulating the thermal cycle of hot isostatic pressing[END_REF][START_REF] Huang | Microstructure and strengthening mechanisms of CuCrZr alloy by two-step thermomechanical treatment[END_REF][START_REF] Gladman | Precipitation hardening in metals[END_REF] which strongly depends on the channel size between the nanoprecipitates, directly related to the duration and temperature of the age hardening step. The smaller this length, the harder it is for the dislocations to curve and to cross the precipitates.

This impedes the dislocation motion and hardens the alloy. The contribution of the nanoprecipitation to the YS is estimated using equation ( 8) [START_REF] Ardell | Precipitation hardening[END_REF]43,[START_REF] Aikin | The role of equiaxed particles on the yield stress of composites[END_REF].

∆𝜎 𝑝𝑠 = 0.8 * M𝐺𝑏 2𝜋√1 - • 𝑙𝑛 ( 2r r 0 ) 𝐿 𝑝 𝑤𝑖𝑡ℎ 𝐿 𝑝 = 𝑟 (√ 2𝜋 3𝑓 - 𝜋 2 ) (8)
The 0.8 factor is a statistic factor, reflecting the random repartition of the precipitates in the slip plan [START_REF] Torma | Hardening mechanisms in Al-Sc alloys[END_REF]. With r [nm] the mean radius of the nano-precipitates, r 0 [nm] the cut-off radius or radius of dislocation core, that can be taken equal to the Burgers vector [START_REF] Aikin | The role of equiaxed particles on the yield stress of composites[END_REF], 𝐿 𝑝 [nm] the channel size between the nano-precipitates and f the volume fraction of precipitates. As the quantity of Zr nano-precipitates in the Cu matrix is low, only the Cr nano-precipitation is taken into account to estimate this contribution. r and f have been measured by TEM (Table 5).

Moreover, we assume that the L-PBF and wrought CuCrZr have the same Cr nano-precipitation after the SA+AH heat treatments.

The calculated precipitation strengthening contribution (Table 9) is overestimated as the experimental YS does not exceed 300 MPa (Table 6). This overestimation can come from the imprecise determination of precipitates size (50% inaccuracy due to high standard deviation, Table 5. Some high resolution TEM analyses made on the size of precipitate (Figure 6.e-f) seem to confirm the underestimation of the radius. It is also possible that a shear hardening mechanism is acting instead of the Orowan hardening mechanism. Nevertheless, the obtained trends are consistent: this contribution is predominant in all heat-treated CuCrZr, and the biggest in the L-PBF DAH CuCrZr. Such a heat treatment provides a high nano-precipitation density in the Cu matrix reducing the average channel size, and thus increasing the precipitation strengthening contribution.

Comparison between the experimental YS and the calculated YS contributions

The objective of the previous calculations was to quantify the contribution of the different nanoscale strengthening mechanisms (nano-precipitation, dislocations, solid solution, grain size refinement) for each CuCrZr sample (L-PBF as-built, L-PBF SA+AH, L-PBF DAH and the reference wrought SA+AH). Some interesting results emerge from this analysis. Figure 9 summarizes the different contributions to the YS calculated for each heat-treated CuCrZr and compared to the experimental YS (black spots). Qualitatively, we found the right trends for every metallurgical states. In the L-PBF SA+AH and wrought SA+AH CuCrZr, the predominant strengthening mechanism is the nano-precipitation strengthening, accounting for, respectively, 80% and 85% of the total strengthening. The overestimation of 48% to 62% of the calculated YS compared to the experimental one, respectively for the L-PBF SA+AH and wrought SA+AH CuCrZr, can come from the high standard deviation on the mean radius of the Chromium nano-precipitates. For the as-built L-PBF CuCrZr, there are two main acting mechanisms: dislocation strengthening (78% of calculated YS) and solid solution strengthening (15% of calculated YS). The difference between the measured and estimated YS is less than 15%. For the L-PBF DAH CuCrZr, the precipitation strengthening and the dislocation strengthening account, respectively, for 70% and 27% of the total estimated YS. The difference between the measured and estimated YS is less than 8%. Thus, the supersaturated Cu solid solution obtained thanks to L-PBF process offers the possibility to precipitate more nano-Cr after a DAH (0.6 %vol. as shown in Figure 8) than what is possible with a SA+AH (0.3 %vol.), increasing the precipitation strengthening.

Conclusions

In this study, we have shown that:

(1) CuCrZr samples with a relative density higher than 99% have been successfully

processed by L-PBF.

(2) the L-PBF as-built CuCrZr alloy microstructure shows columnar grains, a high dislocation density due to the presence of solidification cells of 294 ± 32 nm of diameter, and it is in a supersaturated state without Cr micro-nor nano-precipitate, which is very different from the well-known wrought CuCrZr alloy. In L-PBF as-built sample, the volume fraction of Cr in solid solution is twice the one of the SA+WQ wrought alloy; this is explained by the higher cooling rate of the L-PBF process.

(3) Applying a solution annealing at 980°C for 1h followed by an ageing at 490°C for 6h on a L-PBF CuCrZr sample recrystallizes the microstructure at 60%. During the solution annealing, a part of the Cr (0.4 wt%) forms micro-precipitates while the other part remains in solid solution which will form nano-precipitates after the age hardening treatment. The mechanical properties are comparable to wrought CuCrZr in the direction perpendicular to the building one.

(4) A direct ageing at 490°C for 1 hour allows to precipitate all the Cr (0.78 wt%) at nanometric scale but does not modify the microstructure of the as built L-PBF CuCrZr (columnar grain, high dislocation density) to a huge extent. This particular heat treatment offers high mechanical properties (Vickers hardness of 184 HV1, YS=527

MPa and UTS=585 MPa) while conserving a significant elongation (14%) and good electrical properties (42 MS.m -1 ) compared to SA+AH heat treatments. This is due to the high dislocation density and the high Cr nano-precipitates density, which lead to a lower distance between precipitates and a stronger structural hardening compared to the one after SA+AH.

Figure 1 .

 1 Figure 1.b shows the cubic L-PBF specimens. Their relative density as a function of the energy density Ev [J.mm -3 ] input during the process is presented in Figure 1.a. Ev depends on the laser power energy P [W], the laser scanning velocity v [mm.s -1 ], the hatch h [mm] and the layer

Figure 3 .

 3 Figure 3.a and Figure 3.d show SEM images of the L-PBF as-built CuCrZr taken in the BD

Figure 4 .

 4 g). No solidification cells are observed in both, recrystallized or nonrecrystallized grains. The dislocation recovery should occur prior to the recrystallization during the 980°C SA heat treatment. Given the absence of solidification cells, a step size of 1.7 or 2.1

5.b and Figure 6

 6 .a), which confirms that Cr and Zr form a solid solution in the Cu matrix of this material. After heat treatment and due to the low solubility of the Cr and Zr in the copper matrix, Cr and Zr form micro or nano-precipitates.Cr micro-precipitates were observed on SEM images (Figure5). In the wrought SA+AH CuCrZr, they are intragranular precipitates (Figure5.a). In contrast, in the L-PBF SA+AH CuCrZr, they are intergranular or located along sub-grain boundaries in the non-recrystallized areas, but they are also intragranular in the recrystallized areas (Figure5.d). Contrary to SA+AH, the DAH heat treatment did not induce Cr micro-precipitation (Figure5.c). Thus, during the DAH heat treatment, a larger amount of Cr was involved in the nanoprecipitation as confirmed in the following.

Table 6 :Figure 7

 67 Figure 7 shows tensile test curves for the different alloys recorded at 20°C (as the curves were

Further

  investigations on the SA+AH heat treatment on L-PBF CuCrZr may offer the possibility to get totally rid of the anisotropic microstructure induced by the process. A further investigation on the DAH heat treatment may offer the possibility to determine precisely the hardening peak depending on the Cr concentration in the alloy. A TEM study of the Cr nanoprecipitates on CuCrZr tensile samples may confirm the structural hardening mechanism (shear or bypass mechanism) playing a role here.

Figure 1 :

 1 Figure 1: (a) Relative parts density (compared to CuCrZr HIPped samples density of 8.908 g.cm - 3 ) as a function of the energy density for various parameters sets. Varying scanning speed at a laser power of 360 W (white marks), 420 W (orange marks) and 480 W (black marks) using different hatch (100, 110 and 120 µm) and a layer thickness fixed at 30 µm. (b) L-PBF as-built cubes (10x10x10 mm 3 ) and (c) mechanical (55x13x13 mm 3 parallelepiped) and electrical samples (18 mm diameter and 22 mm high cylinder).

Figure 2 :Figure 3 :

 23 Figure 2 : Schema of the BD plan and ↑ BD plan orientations in regard to L-PBF scan path and sample edges

Figure 4 :

 4 Figure 4 : SEM-EBSD analysis of the (a, d) wrought, (b, e, g) L-PBF SA+AH and (c, f, h) L-PBF DAH CuCrZr samples. (a-c) SEM images in the BD plan; (d-f) quality pattern images with grain boundaries and twins of (a-c) respectively. (g, h) KAM analysis of respectively L-PBF SA + AH and L-PBF DAH CuCrZr in the BD plan.

Figure 5 :

 5 Figure 5: Study of the micro-precipitation: SEM images of (a) Wrought CuCrZr, (b) L-PBF as-built CuCrZr and (c) L-PBF DAH CuCrZr, both exempt of micro-precipitates, (d) L-PBF SA+AH CuCrZr. (e) TEM image focused on a Cr micro-precipitate, and (f) an EDX analysis of one of them.

Figure 6 :Figure 7 :

 67 Figure 6: HAADF/STEM and Cr and Zr EDX analysis of the (a) L-PBF as built CuCrZr showing no nano-precipitates; (b, e) wrought SA+AH CuCrZr showing Cr nano-precipitation (no Zr detected; matrix is pure Cu on these images); (c, f) L-PBF DAH CuCrZr showing Cr nano-precipitation (no Zr detected; matrix is pure Cu on these images); (d) L-PBF DAH CuCrZr showing a Zr nano-precipitate next to a Cr nano-precipitate

Figure 8 :

 8 Figure 8 : Microstructural evolution of L-PBF and wrought CuCrZr as a function of the heat treatments. The Cr content in solid solution is noted next to the pictures. The final volume fractions of Cr nano-precipitates are given at the bottom.

Figure 9 :

 9 Figure 9: Calculated hardening mechanisms contributions to the yield strengths of CuCrZr alloys processed by four different routes, compared to experimental values (black spots)

Table 1 :

 1 Chemical compositions of CuCrZr powder upon receipt, L-PBF CuCrZr and wrought

	CuCrZr samples compared to standard (values correspond to the average of three
			measurements).
	Element [wt.%]	Cr	Zr	Fe	Si	O	Others Cu
	CuCrZr -EN12420:2014 standard	0.5-1.2 0.03-0.3 < 0.008 < 0.1	< 0.2	Bal.
	Wrought CuCrZr (reference)	0.596 ± 0.014	0.071 ± 0.001	0.004 ± 0.001	<0.002	0.005 ± 0.001	-	Bal.
	Powder upon receipt	0.751 ± 0.01	0.086 ± 0.002	0.020 ± 0.001	0.005 ± 0.002	0.018 ± 0.002	0.02 Bal.
	L-PBF CuCrZr (studied material)	0.781 ± 0.012	0.088 ± 0.003	0.029 ± 0.001	0.008 ± 0.005	0.012 ± 0.001	-	Bal.

Table 2 :

 2 Heat treatments applied to the CuCrZr samples

	Heat treatment	Temperature [°C] / Atmosphere	Exposure time [min]	Cooling type	Samples concerned
	Solution Annealing +	980 / Ar +	60 +	Water quenching	L-PBF
	Age Hardening (SA+ AH)	490 / Air	360	+ Air cooling	Wrought

Table 3 :

 3 Average grain size of the different CuCrZr samples.

	CuCrZr	L-PBF DAH BD ↑BD	L-PBF SA+AH BD ↑ BD	Wrought SA+AH
	Average grain size [µm]	112 ± 60 242 ± 100 276 ± 150 334 ± 140 471 ± 150
	Number of grains counted	785	368	338	406	44

Table 4 :

 4 KAM median misorientation angle θ and average dislocation density ρ0 as a function of the step size used s, estimated for L-PBF as built, L-PBF SA+AH and L-PBF DAH, and wrought SA+AH CuCrZr alloys.

	CuCrZr	Analysed surface	Cells size [nm]	s [µm]	θ [°]	ρ0 [m -2 ]
	L-PBF as-built	1 cm² 4.5 mm²	300	2.0 0.1	1.9 8 ± 1 x 10 13 0.7 5 ± 1 x 10 14
		1 cm²	No cell	1.7	0.9 4 ± 1 x 10 13

Table 5 :

 5 TEM analysis of the equivalent radius and volume fraction of Cr nano-precipitates on heat treated CuCrZr.

	CuCrZr	Equivalent radius [nm]	Number of precipitates measured	Volume fraction of precipitates [%vol]
	L-PBF DAH	2.1 ± 1.5	233	0.6
	Wrought SA+AH	1.8 ± 1.0	54	0.3

Table 7 :

 7 Electrical conductivity measured by Eddy current

	CuCrZr	Electrical Conductivity

Table 8 :

 8 Value of the parameters for pure Copper used in the calculations

		Parameters	Value	Unit	Reference
	𝜎 0 Peierls-Nabarro stress	20	MPa	[16]
	G	Shear modulus of the matrix	45.5	GPa	[34,36-37]
	k	Hall Petch slope	180	MPa.µm 1/2	[16,34]
	M	Taylor factor	3.06	-	[10,28,36]
	b	Burgers vector	0.256	nm	[12,36]
	α	-	0.23	-	[10]
	ε	Misfit	0.015	-	[14,38]
	ν	Poisson's ratio	0.34	-	[34,39]
	4.2.1. Solid solution strengthening

Table 9 :

 9 Grain size contribution, dislocation density measured thanks to KAM analysis, dislocation strengthening contribution in the different CuCrZr, channel size between the nanoprecipitates and strengthening contribution for the different CuCrZr.

	CuCrZr	L-PBF As-built	L-PBF DAH	L-PBF SA+AH	Wrought SA + AH
	Average equivalent grain diameter [µm]	115 ± 60	112 ± 55	80 ± 40	186 ± 108
	Number of grains	902	785	1095	514
	∆𝜎 𝑔𝑠 [MPa]	17	17	20	13
	𝜌 0 [m -2 ]	5 ± 1 x10 14	4 ± 1 x10 14	4 ± 1 x10 13	3 ± 1 x10 13
	∆𝜎 𝑑𝑠 [MPa]	184	153	52	41
	𝐿 𝑝 [nm]	Ø	36	Taken equal to wrought	45
	∆𝜎 𝑝𝑠 [MPa]	Ø	402	SA+AH values	304
	4.2.3. Dislocation strengthening

Acknowledgments

The authors gratefully acknowledge CEA-LITEN and the CARNOT "Energies du Futur" program for their financial support. The TEM and SEM measurements were performed at the

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

After SA+AH on L-PBF CuCrZr, recrystallization is observed on 60% of the surface. During SA, at 980°C, partial recrystallization is possible, as shown by [START_REF] Liu | Effect of the ITER FW Manufacturing Process on the Microstructure and Properties of a CuCrZr Alloy[END_REF] on cold rolled wrought CuCrZr. In the recrystallized zones, the microstructure is similar to the wrought CuCrZr, with large twined grains showing no particular orientation. After WQ of the SA treatment, the Cr and the Zr partially precipitate at a micrometric scale and are partially maintained in solid solution (0.4 wt.% maximum [START_REF] Que | Microstructure Evolution of the Cu-Cr Hypereutectic Alloys During Directional Solidification[END_REF]) (Figure 8). The AH treatment allows the Cr and Zr in solid solution to precipitate at a nanometric scale (Figure 8).

After DAH, the specific microstructure of L-PBF as-built CuCrZr is globally preserved. A slight dislocation recovery is observed, but the solidification cells remain. The age hardening temperature (490°C) is not high enough to trigger recovery. All the Cr, which was in a supersaturated solid solution state, precipitates at a nanometric scale, leading to a volume fraction of nano-precipitates twice as high as after SA+AH (Figure 8).

Hardening mechanisms

In the following, the different hardening mechanisms acting on the mechanical behavior of our four samples will be estimated, to discuss the effect of the processes and heat treatments. The yield strength (𝑌𝑆) of such materials depends on several contributions according to equation (4) [START_REF] Zhang | Microstructure and properties in QCr0.8 alloy produced by selective laser melting with different heat treatment[END_REF][START_REF] Liu | The microstructure evolution and properties of a Cu-Cr-Ag alloy during thermal-mechanical treatment[END_REF][START_REF] Starink | A model for the yield strength of overaged Al-Zn-Mg-Cu alloys[END_REF]. Riccetti, M. Pellat and P.E. Frayssines for their help in this work.


B Morel

R Giust

K Ardaneh

F Courvoisier

Two-fluid plasma model for ultrashort laser-induced electron-hole nanoplasmas

We present a two-fluid plasma model to describe localized interaction, at sub-micrometer scale, of highly focused ultra-short laser pulses within dielectrics and semiconductors. This model includes transport of the electron-hole plasma, electron-hole scattering, photoionization, impact ionization and interactions between charge carriers and phonons. We start with a plasma kinetic description and we consistently derive the two-fluid plasma model equations. Our numerical results highlight that transport of the electron-hole plasma can strongly affect the plasma density profile in the case of extremely localized interaction.

I. INTRODUCTION

Ultrashort laser pulses are a powerful tool to induce modifications of materials [1][2][3][4] . Particularly in transparent dielectrics, ultrashort laser pulses can be used to locally modify, inside the material's bulk, the chemical structure, the index of refraction, the density of color centers, photopolymerize, generate nano-gratings, surface nanostructures or internal voids. A large number of application fields benefit from fundamental advances: surgery and biomedical applications, photonics, microfluidics, highspeed laser manufacturing 2,[5][6][7] .

Pushing forward these applications to nanometric structuring requires the support of numerical modelling 8 . Under laser-induced strong-field, bound electrons transit from the valence band to the conduction band 1,9,10 , leaving a hole in the valence band. Particles of the electronhole plasma are accelerated in the laser field, which results in the multiplication of the free carrier density via impact ionization, and potentially in the creation of a dense electron-hole plasma. Finally, at timescales much larger than several picoseconds, thermal and structural events takes place inside the material 1 . Our model is focused on the plasma density build-up, at timescales up to some picoseconds.

A large number of different models have been developed to study the propagation of ultra-short laser pulses (∼100 fs) in the high-intensity regime (∼ 10 14 W/cm 2 ) inside dielectrics and subsequent ionization. These models can be classified into two groups. In the first, several models were developed to study the filamentation regime inside transparent solids with propagation scales on the order of several tens of micrometers 11,12 . The pulse propagation model, such as the Unidirectional Pulse Propagation Equation 13, is derived from Maxwell's equations and coupled with a rate equation for free electron generation 14 . These models are well suited to describe the spatio-temporal evolution of the pulse under the different linear and nonlinear phenomena: Kerr and Raman effects, group velocity dispersion, diffraction, nonlinear ionization, plasma absorption and recombinations. They can be refined to describe non-paraxial propagation 15,16 . However, the derivation of the pulse propagation equation requires the assumptions of unidirectionality, local neutrality and transverse fields. These assumptions are not well suited to the case of nanoplasmas (< 1 µm) induced by tightly focused beams 3 .

In the second group, the models are intended to describe energy deposition in dielectrics at micrometric scale and are computationally very expensive because the full set of Maxwell equations is solved. Several descriptions of the plasma have been used: in references [17][18][19] , plasma formation and heating is described by rate equations. A more refined model is the single fluid hydrodynamic approach 20,21 . An approach based on a twotemperature model (TTM) was used in Ref. 22 .

In these previous works, the transport phenomena is conventionally neglected since valence holes are conventionally heavy compared to the effective electron mass (m * h = 5 -10m e ) 21,23,24 .

Because of Coulomb attraction, transport of electrons and holes remain quasi-identical. The fluid of the heaviest particle (electron or hole) limit the transport of lightest one. If electron and hole have comparable light masses, then actual transport can be significant. Neglecting transport phenomena can be therefore inappropriate for materials with light holes such as sapphire 25 , magnesium oxide 26 , silicon 27 , or zinc oxide 28 . Ultrafast transport at high-speed (1.5×10 6 m/s) was recently observed experimentally in the latter material with pump-probe imaging-ellipsometry 28 . A densitydependent two temperature model (nTTM) was recently developed 29 and includes the electron-holes transport in semi-conductors. This model describes the evolution of free carrier density, carriers temperature, lattice temperature, density and energy transport, but hypothesizes plasma neutrality and same temperature for conduction electrons and valence holes.

Here, we propose a more general theoretical description of laser-induced excitation of dielectrics and semiconductors, that takes into account the transport phenomena and hence is valid for any effective mass ratio. We will particularly emphasize here the case where mass ratio is around 1, since it is the case that differs at most from the state of the art described above. In this case, we observed a significant transport, but also important consequences resulting from the holes dynamics (modification of the cut-off frequency, additional impact ionization, warm holes). Our fluid model includes 5-moments transport equations for each fluid, photoionization, impact ionization, electron-hole scattering and interactions between carriers and phonons. The approach is only valid when the band structure of the material remains, and is intended to describe the transient regime before the occurrence of nonthermal melting 30 or of the transformation of the plasma into warm dense matter [31][32][33] .

The paper is organized as follows: in section II, we recall the two-fluid plasma model equations for an electronhole plasma inside a bandgap material with parabolic energy bands, and we briefly review the different physical phenomena included in the model. In section III, we derive the source terms describing respectively electron-hole scattering, interactions between carriers and phonons, photoionization and impact ionization. Finally, in section IV, our numerical results of Bessel beam interaction with sapphire highlight that transport can significantly affect the plasma profile, throughout extremely localized interactions inducing nano-plasmas.

II. THE TWO-FLUID PLASMA MODEL

The fluid model developed in this article is intended for materials that can be described by an isotropic parabolic band structure: the effective masses of conduction electrons and valence holes are independent of direction and position in Brillouin zone. The fluid model can be derived consistently by calculating moments of the kinetic equation with respect to velocity 34 and by assuming a local thermodynamic equilibrium 35 for electrons and holes. This implies Maxwellian distributions for both particle species. The fluid description allows saving computational effort in comparison with the kinetic approach 36 . In our case, we consider that the plasma is sufficiently collisional to keep the particles distribution close to a Maxwellian distribution. The equations of two-fluid plasma model under balance laws form (i.e. a conservative form on the left side of fluid equations and a source term on the right one) are given by 37,38 :

∂ ∂t        n e n e u e ǫ e n h n h u h ǫ h        + ∇ •         n e u e n e u e ⊗ u e + pe m * e I (ǫ e + p e )u e n h u h n h u h ⊗ u h + p h m * h I (ǫ h + p h )u h         =         0 neqe m * e (E + u e ×B) n e q e u e • E 0 n h q h m * h (E + u h ×B) n h q h u h • E         + S Coll e S Coll h (1)
The system of equations (1) corresponds to three conservation's laws for each fluids (electrons and holes): conservation of particles number, momentum and energy. The term with the divergence operator is called the transport term whereas the terms on the right member is called the source terms. For a specie of particles a (a = e for conduction electrons and a = h for valence holes), q a is the electric charge, m * a the effective mass, n a the density, u a the mean velocity, p a the scalar pressure, ǫ a the fluid energy density, E the electric field and B the magnetic field. I is the identity matrix and ⊗ is tensor product. To obtain the system of equations (1), we additionally made the assumption that electrons and holes fluids are non-viscous. The effects of viscosity are usually unimportant in ultra-fast dynamics 39 . It allows us to consider scalar pressures instead of pressure tensors.

The system of equations for electrons and hole fluids is closed by assuming local thermodynamic equilibrium (Maxwellian distribution) for both species:

ǫ a ≡ p a γ -1 + 1 2 n a m * a u a 2 (2)
where γ is the adiabatic index. The local thermodynamic equilibrium allows linking pressure and temper-ature through the ideal gas law 35 : p a = n a k B T a , where T a is the temperature and k B the Boltzmann constant.

The terms S Coll e and S Coll h are the collisional terms for electron and hole fluids. We use the same general method as in Ref. 40,41 in order to include different physical phenomena, i.e. we split the collisional terms of Eq. ( 1

+ S h-p e S h-p h ≡S h-p (3)
where the superscript e-h is related to electrons-holes scattering, e-p to interactions between electrons and phonons, h-p to interactions between holes and phonons, Ph to photoionization, Imp-e to electron-initiated impact ionization and Imp-h to hole-initiated impact ionization.

The electric and magnetic fields in Eq. ( 1) are determined by Maxwell equations:

∇ • ε r E = 1 ε 0 [q e n e + q h n h ] (4) ∇ • B = 0 (5) ∇×E = - ∂B ∂t (6) ∇×B = µ 0 J + 1 c 2 ∂ε r E ∂t (7)
where ε 0 and µ 0 are respectively the vacuum permittivity and permeability. c = (ε 0 µ 0 ) -1/2 is the speed of light. ε r is the relative permittivity of the background dielectric. The total current density J is composed of the contribution from conduction electrons and of the one from valence holes:

J = q e n e u e + q h n h u h (8)
In our description, we consider an electron-hole plasma localized in background dielectric of relative permittivity ε r . The effect of bound charges is also taken into account through relative permittivity ε r whereas the effect of charge carriers is included through the current density J of Eq. ( 8). We finally remark that, here, the medium response to an electromagnetic excitation is taken into account through a fluid approach.

This approach is valid only when the spatial scales are larger than the Debye length 42 . The typical densities involved in numerical results of section IV are around 10 21 cm -3 and the temperatures are in a range of 1-100 eV, leading to a Debye length between 0.2 nm and 2 nm.

III. COLLISIONAL SOURCE TERMS

In this section, we specify the source terms S Coll e and S Coll h of Eq. ( 3). We include the different physical phenomena one by one in our model.

A. Electron-hole scattering

We start with the elastic scattering between conduction electrons and valence holes. We have chosen to include this phenomenon via the Bhatnagar-Gross-Krook (BGK) approach 43 . This approach assumes that the elastic scattering restores a local equilibrium with an exponential decay in time. This description usually gives a first approximation to the problem under consideration 35 and is relatively easy to link to the Drude model of plasmas 44 . The inclusion of the BGK term in the twofluid plasma equations is performed via the following source term 45 :

S e-h = S e-h e S e-h h =          0 -n e ν eh µ m * e (u e -u h ) -κν eh ne γ-1 (T e -T h ) + κν eh ne 2 m * e u e 2 -m * h u h 2 + (m * h -m * e )u e • u h 0 -n h ν he µ m * h (u h -u e ) -κν he n h γ-1 (T h -T e ) + κν he n h 2 m * h u h 2 -m * e u e 2 + (m * e -m * h )u h • u e          (9)
where µ =

m * e m * h m * e +m * h is the reduced mass, κ = 2µ
m * e +m * h and ν eh is the scattering frequency for momentum transfer from electrons to holes whereas ν he is from holes to elec-trons. Eq. ( 9) is symmetrical to the permutation e ↔ h, and conservation of momentum and energy imposes the following condition 45 :

n e ν eh = n h ν he (10)

B. Interactions between charge carriers and phonons

As in Ref. 41 , we consider two scattering phenomena for the interactions between charge carriers and phonons. The first one is the charge carrier -phonon elastic scattering, where charge carriers undergo a momentum change due to phonon emission/absorption. The second phenomenon is the carrier-phonon-photon interaction. In these two phenomena, the loss of energy to the phonon bath is negligible at sub-picosecond timescale 9,33 . The impact of these scattering events on the conduction electron fluid and on the valence hole fluid is a modification of the distribution of the particle, i.e., a reduction of the mean velocity and an increase of the thermal energy. We include these phenomena in a classical way (i.e., in a non-quantum treatment) via frictions terms where ν epp is the electron-phonon-photon interaction rate and ν ep is the electron-phonon scattering rate. Similarly, we have for the hole fluid:

S h-p = S h-p e S h-p h ≡        0 0 0 0 -n h (ν hpp + ν hp )u h 0        (12)
where ν hpp is the hole-phonon-photon interaction rate and ν hp is the hole-phonon scattering rate.

In this description with friction terms, the total energies ǫ a of the fluids do not change (no term in components 3 and 6 of Eq. ( 11) and Eq. ( 12)). The scattering induces a conversion of directed energy 1 2 n a m * a u a 2 into thermal energy pa γ-1 . It also induces a weakening of the current density J, thus absorbing energy from the electromagnetic fields.

We note that here, we discard the energy transfer from the carriers to the phonon bath since it is effective at timescales of several picoseconds while our model is intended to model typically sub-picosecond dynamics 9,33 .

C. Electron-electron and hole-hole scattering

Electron-electron and hole-hole elastic scattering do not affect fluid equations (1) 29,35 . The reason is that in the fluid approach, the shape of the distribution is fixed to a Maxwellian one, before and after the scattering process. In addition, during a scattering event, the change of momentum (resp. energy) of a particle is equal to the change of momentum (resp. energy) of the other particle. This implies that the total momentum (resp. energy) of the fluid remain constant during elastic scattering between particles of same specie 35 .

D. Photoionization

Photoionization, impact ionization and recombination can be considered as inelastic collisions. 35 . However, it is conventionally difficult to rigorously describe inelastic collisions in the framework of the fluid description 46 . In this section, we insert photoionization (promotion of electrons in the conduction band by absorption of field energy) in the two-fluid plasma model. Impact ionization will be modelled in the following section. We start by defining n ev as the density of electron available in the valence band: this number is a function of time and space. We set n 0 as the initial density of electron available in the valence band, and α Ph as the photoionization rate in m -3 s -1 .

To include photoionization, we make the following assumptions: (i) the number of created fluid particles (electrons and holes) through photoionization is given by rate equation used in Ref. 17 ( ∂ ∂t n e = nev n0 α Ph ). We use the Keldysh ionization rate in solid 47 (appendix A), but it could be replaced by more sophisticated models 48 without changing the overall architecture of our model. (ii) as in Ref. 49 , we consider that during photoionization, valence electrons are promoted to the bottom of the conduction band. This is equivalent to neglecting the residual energy transfer to charge carriers after photoionization: fluid particles are generated with zero energy (p a = 0 and u a = 0).

The source term for photoionization of Eq. ( 3) is therefore given by:

S Ph ≡ S Ph e S Ph h ≡         nev n0 α Ph 0 0 nev n0 α Ph 0 0         (13)
For each electron-hole pair generated, one valence electron must be removed. This leads to the following rate equation for the density of valence electrons:

∂ ∂t n ev = - n ev n 0 α Ph (14)