Alejandro Montón

Francis Maury

Geoffroy Chevallier

Claude Estournès

Marc Ferrato

David Grossin

Alejandro Monton
email: alejandro.monconzarazaga@ensiacet.fr

Francis Mamy

Densification of surface-modified silicon carbide powder by spark-plasma-sintering

Keywords:

come

Introduction

Because of its excellent prope1t ies such as its high mechanical stiff ness, low density, wide bandgap, low coefficient of expansion, high them1al stability, and resistance to corrosive environments, silicon carbide (SiC) is an advanced cerantic material for many technological applications [START_REF] Harris Y | Institution of Electrical Engineers[END_REF]. Among these applications are high-power microwave devices for commercial and ntilitaiy systen1s; electt•onic devices (LED' s, MOSFET's); high ten1perature electt• onics/optics for automotive, laser processes ntiITors and well-logging; rugged MEMSs (nticroelectro-mechanical sensors) devices for hostile environments; gas and chentical sensors for internai combustion engines, fumaces and boilers; solai• -blind UV photodetectors [START_REF]Advanced Processing and Manufacturing Technologies for Nanostructured and Multifunctional Materials II: A Collection of Papers Presented at the 39th International Conference on Advanced Ceramics and Composites[END_REF], ve1y high ten1perature solar absorber [START_REF] Leray | Design and Proof of Concept of an Innovative Very High Temperature Ceramic Solar Absorber[END_REF] and aerospace applications as space telescope ntirrors [START_REF] Bougoin | A new technological step for sic mirrors preparing OTOS[END_REF] and more generally ve1y Iai•ge telescopes [START_REF] Bougoin | Full-SiC EUCLID's very large telescope[END_REF][START_REF] Castel | The 3.2m All SiC Telescope for SPICA[END_REF].

The manufacture of monoliths SiC paits is ve1y difficult because it is a covalent cerantic which decomposes above 2545 •c to fom1 liquid silicon and carbon [START_REF] Meyers | Additive Manufacturing of Technical Ceramics[END_REF]. Consequently, under normal conditions, it has no liquid phase. The main difficulty to fabricate massive and dense SiC pa1ts lie in the sintering stage. For example, without any additives, the theoretical density could only be achieved at 2500 •c by a classical hot-pressing process under a pressure of 50 MPa [START_REF]Very High Pressure Hot Pressing of Silicon Carbide[END_REF]. A number of aids ai• e then used to promote SiC densification, but ce1tain glassy-like phases formed at grain boundai•ies from heteroatoms may be dett'in1ental to the desired performance and, consequently, mechanical prope1ties can be lowered.

Spai•k Plasma Sintering (SPS) is an advanced process that was also used to consolidate silicon carbide powder without additives. The first step is to choose the most suitable SiC powder and the second one is to optintize SPS experin1ental conditions. The SPS process allows va1ious types of materials to be sintered [START_REF] Hu | A review of multi-physical fields induced phenomena and effects in spark plasma sintering: fundamentals and applications[END_REF] with contt•ol nticrosn•ucture [START_REF] Chaim | Densification and preservation of ceramic nanocrystalline character by spark plasma sintering[END_REF] and shape [START_REF] Manière | Spark plasma sintering and complex shapes: the deformed interfaces approach[END_REF]. lt relies on pulsed DC cmTent passing through an electrically conducting pressure die containing the sample. Uniaxial pressure is applied during sintering. The main benefit is that full density can be reached fairly easily, the whole experience taking only a few ntinutes. Unfortunately, like so many other covalent refracto1y cerainics, even by SPS SiC is ve1y difficult to sinter to a high density without using binder additives (that generally bring heteroatoms that ai•e left in the pait) and/or ve1y high pressures.

Thus, to reach high density sintered SiC functional additives are used. In previous studies, it was shown that these aids could lower the sintering temperature. Severa! compounds were tested as sintering https:/ /doi.org/10.1 0l 6/j.jeurceramsoc.2021.07.036 additives for SiC powder both to enhance the densification rate and to slow down grain growth kinetic. To achieve high densities through solidstate sintering and to avoid the appearance of low corrosion resistance intergranular phases, consolidation of SiC with boron and carbon as sintering additives seems to be the most efficient way [START_REF] Prochazka | Effect of boron and carbon on sintering of SIC[END_REF]. Therefore, boron carbide was found to enhance the hot-pressing behavior of SiC that could be so obtained in a fully dense state [START_REF] Stobierski | Sintering of silicon carbideI. Effect of carbon[END_REF][START_REF] Stobierski | Sintering of silicon carbide II. Effect of boron[END_REF]. These additives permit to reach high density slightly over 2000 • C by means of the reduction of the surface energy of the grains (surfactant effect due to boron) and reaction with residual silica (carbon reduction). Following these results, Maître et al. used SPS method and boron additive to produce fully dense SiC-based materials with several non-oxide secondary phases such as boron carbide (B 4 C) or free boron (B) plus free carbon (C) [START_REF] Maître | Role of boron on the spark plasma sintering of an α-SiC powder[END_REF]. However, for some applications these additives are considered as contaminants for final parts and as a consequence not acceptable.

For this reason, no additive was used and a stringent optimization of the process parameter was a preferred route. For instance, Guillard et al. applied a specific SPS procedure for synthesizing and densifying pure commercially available submicrometer-sized SiC powder. Using this procedure, a 92 % Relative Density (RD) was achieved. The aim of this previous study was to assess the evolution of SiC grain size and density versus temperature, holding time and pressure on a commercial powder [START_REF] Guillard | Densification of SiC by SPSeffects of time, temperature and pressure[END_REF]. As another example of pure SiC SPS, Hayun et al. obtained a density of SiC bodies of 3.1870 g/cm 3 (RD 99 %) in the absence of sintering additives. They have used an SPS treatment at 2050 • C for 10 min, a heating rate of 400 • C/min and a high applied pressure of 69 MPa to densify alpha-SiC powder [START_REF] Hayun | Microstructure and mechanical properties of silicon carbide processed by Spark Plasma Sintering (SPS)[END_REF]. However, even with the advantages of SPS, sintering aids are generally required to obtain a dense SiC body.

A potential possibility recently proposed by authors to obtain full-dense SiC body by SPS is the direct use of polymer-derived ceramics as single-source materials, i.e. without SiC powder [START_REF] Bernardo | Polymerderived SiC ceramics from polycarbosilane/boron mixtures densified by SPS[END_REF]. These preceramic molecular precursors present distinctive features such as low process temperatures and, above all, the possibility of applying forming techniques for conventional polymer processing. However, heating of preceramic polymers generally leads to the formation of cracks and pores in the ceramic products due to gas release and shrinkage resulting from polymerization and crosslinking. Crack free products can be manufactured in the case of fibers, microcellular foams and thin films thanks to the key role of the many interfaces. In these configurations, the material can tolerate both the release of a great amount of gases as by-products (mainly hydrogen and lightweight hydrocarbons) during the polymer-to-ceramic conversion and also the significant volumetric shrinkage up to 60 % [START_REF] Markusweinmann | Precursor-Derived Ceramics[END_REF]. Bernardo et al. analyzed the SPS processability of SiC derived from one of the commercial family of polycarbosilane (PCS) powders (Nipusi® type S, Nippon CarbonCo.,Ltd) featuring a molecular weight (Mw) of 1400 mixed with boron as additive [START_REF] Bernardo | Polymerderived SiC ceramics from polycarbosilane/boron mixtures densified by SPS[END_REF]. PCS compounds are inorganic polymer composed of a Si-C backbone leading to a fixed Si/C atomic ratio (characteristic of each PCS compound) and a small amount of hydrogen. Furthermore, they exhibit similar properties as organic polymers, i.e. they are meltable, soluble, and unstable at high temperatures. For conversion to silicon carbide, PCS must undergo several steps during heat treatment: formation of oligomers by pyrolysis, usually by dehydrogenation, then polymeric intermediaries, crosslinking, amorphous preceramics, crystallization, and crystal growth. When comparing PCS compounds and SiC powder as starting materials in the sintering process, there are notable differences. In the sintering process for a ceramic material as SiC, a crystallized powder is generally used, whereas PCS is used in an amorphous state after pyrolysis with a Si-or C-rich content depending on its initial composition. Therefore, starting with PCS, several steps must accompany sintering as the events above cited as well as nucleation, crystal growth, and grain growth [START_REF] Soraru | Structural evolutions from polycarbosilane to SiC ceramic[END_REF].

More authors explored the use of PCS for densification of SiC by SPS (Table 1). Lodhe et al. studied the phase transition of SiC during SPS, where a PCS was used after pyrolysis at 1400 • C combined with SiC whiskers (10 20% vol.) [START_REF] Lodhe | On correlation between β→α transformation and densification mechanisms in SiC during spark plasma sintering[END_REF], and Lee et al. sintered PCS (Mw 13600, ToBeM Tech. Co.) powder after pyrolysis at 1000 • C by SPS [START_REF] Lee | Phase transformation on spark plasma sintered dense polycarbosilane-derived SiC without additive[END_REF].

The aim of our study is to extend and to explore the use of original PCS compounds as a sintering aid by direct surface grafting on SiC powder. For that, we used a simple and volatile commercially available PCS precursor not yet tested as additive in an SPS process.

Consequently, by means of SPS, the original scientific approach of this work resides in the production of innovative SiC powders that enhance the sintering behavior of SiC by surface functionalization of grains consisting of grafting a molecular nanolayer that does not contain heteroatoms and does not induce the formation of secondary phases. The effects of the surface functionalization have been studied and compared to non-modified SiC powder.

Experimental

A commercial compound of the PCS family, namely poly(silaethylene) (PSE) was supplied from Starfire® System (supplier name CVD-4000). It was used for the surface functionalization of silicon carbide powder as SiC forming oligomer and known to lead to the growth of stoichiometric silicon carbide coatings (ratio Si:C 1:1) without the need for additional reactants by chemical vapor deposition [START_REF] Bernardo | Polymerderived SiC ceramics from polycarbosilane/boron mixtures densified by SPS[END_REF]. Its formula is [ SiH2 CH2 ]n, n 3-8), which reveals Si-C covalent bonds already formed and an atomic Si:C ratio 1:1 as in crystallized silicon carbide. It is a volatile liquid compound, easy to handle and readily available. It produces amorphous hydrogenated silicon carbide coatings by pyrolysis at 800 • C in a chemical vapor deposition process [START_REF] Boisselier | SiC coatings grown by liquid injection chemical vapor deposition using single source metal-organic precursors[END_REF].

The SiC samples were spark plasma sintered using a Dr. Sinter 2080 unit, SPS Syntex Inc., (Japan,) available at the Plateforme Nationale

Table 1

Summary of the process conditions and main features of PCS and PCS + SiC starting materials sintered by SPS. CNRS de Frittage Flash located at the Université Toulouse 3 Pau l Sabatier (France). The SiC powder was loaded into graphite dies with an inside diameter of 8 mm to produce pellets at least 10 mm thick. Prior this step, a graphite foil (PERMA-FOIL®Toyo Tanso) was systematically placed to cover the inside wall of the die, as well as at the interfaces between the punches/sample and spacers/electrodes. The outside wall of the die was systematically covered with a graphite felt to minimize the heat loss during the experiment. A heating rate of 300 • C/min was used from room temperature to 600 • C, where a 1-min dwell was applied. Then, a heating rate of 100 • C/min between 600 and 2200 • C was applied. An optical pyrometer (IR-AH series Chino), focused inside a little hole (1.8 mm in diameter and 3 mm in depth) located in the outside wall of the die was used to control and monitor the temperature. For one pure SiC powder and a surface-modified Silicon Carbide powder (namely smSiC) the heat treatment was stopped at 1600 • C to observe and compare the early stages of sintering. Pressure was set at 50 MPa at room temperature and maintained up to the end of the dwell. In each case, pressure was released at the end of the holding time and the cooling rate down to 1000 • C was around 400 • C/min. Some graphite paper, used for the die protection, remaining at the surface of the pellets after the SPS step was removed by soft polishing of the samples using abrasive paper. Then, density of the final specimens was determined according to Archimedes' principle and compared to SiC theoretical density [START_REF] Harris Y | Institution of Electrical Engineers[END_REF].

The structure of the samples was analyzed using Cu K-alpha radiation (λ 1.5406 Å) over the 2θ range of 20-80 • using X-ray powder diffraction (Diffractometer D8 2) and the phase composition was determined by the Rietveld method from XRD patterns with MAUD, following the strategy developed by Hongchao et al. [START_REF] Hongchao | Quantitative analysis of SiC polytype distributions by the rietveld method[END_REF]. Their microstructure was studied using Scanning Electron Microscopy (LEO435VP) The samples for the microstructural characterization were prepared using a standard metallographic procedure. The samples were polished with diamond paste and subsequently introduced in hard resin to study the cross section of the material. The cut was made with a cutting machine (Mecatome T210) using a diamond blade. FT-IR spectra of the PSE (CVD-4000), SiC powder and smSiC powder were obtained over the range of 4000 to 400 cm 1 using a Nicolet 5700 spectrometer. KBr discs were used, prepared by compressing a finely ground mixture of about 3 mg of the sample and 300 mg of KBr powder. Raman micro-spectroscopy using a Labram HR 800 Yvon Jobin Raman confocal spectrometer with a confocal hole of 100 mm and an argon laser emitting at 532 nm provided chemical and structural characterizations of the samples. We have recorded the Raman spectra by scanning the entire range of interest 550-1050 cm 1. References to standard transform acoustic phonons (TA), transverse optical phonons (TO) and longitudinal optical phonons (LO) were obtained from literature for the following basic polytypes: α-SiC (6H-SiC) and 4H-SiCboth have hexagonal symmetry and wurtzite structure, and 15R-SiC [START_REF] Dobrzhinetskaya | Moissanite (SiC) with metal-silicide and silicon inclusions from tuff of Israel: Raman spectroscopy and electron microscope studies[END_REF][START_REF] Lafon-Placette | Tribological characterization of silicon carbide and carbon materials[END_REF].

Results and discussion

PSE has been grafted onto the surface of SiC particles in the form of a conformal coverage molecular layer, which avoid the excess of oligomer that can be provided by the feed powder.

Alpha-SiC, powder with a mean grain size of 2 μm provided by Mersen Boostec were used in this project. As it was already mentioned by E. Lidén [START_REF] Lidén | Surface modification and dispersion of silicon nitride and silicon carbide powders[END_REF] et al. and experimentally proofed by Luo et al. [START_REF] Luo | Surface grafting onto SiC nanoparticles with glycidyl methacrylate in emulsion[END_REF], it is important not to use an excess of polymer that can be grafted onto the surface of the powder, since this could increase the specific surface area, change the morphology of the powder too much and the coupling agent could hardly infiltrate into the nanoparticle agglomerates of the powder and reduce the graft efficiency. Therefore, it was first necessary to calculate the amount of oligomer required for a conformal coverage of the powder grains with a molecular monolayer of PSE. The powder grains were supposed monodisperse and spherical with an average diameter of 2 μm. An approximate value of ∼ 1 wt.% of PSE was found.

Then the surface-modified silicon carbide (smSiC) powder was prepared by a wet route. A suspension of the alpha-SiC powder was stirred in toluene in a Schlenk tube, then a PSE solution (∼ 1 wt.%) in toluene as a solvent was added dropwise under argon atmosphere using a vacuum manifold to avoid premature degradation of this PCS compound. This suspension was magnetically stirred for 1 h at 60 • C, and then it was transferred into a rotative evaporator for 3 h at 60 • C to remove slowly the solvent and the PSE excess. At the end, the smSiC powder was stored in the dry atmosphere of a vessel before its use in SPS.

Indeed, this strategy does not change significantly the main features of the SiC powder as the specific surface area and the morphology. Furthermore, this coupling agent infiltrates into the particle aggregates and increases the grafting efficiency [START_REF] Lidén | Surface modification and dispersion of silicon nitride and silicon carbide powders[END_REF]. This original approach is highlighted in the Table 1 summarizing a brief state-of-the-art of SPS sintering of SiC using PCS compounds.

The liquid oligomer samples were examined as a thin film between KBr plates. Fig. 1 shows the FT-IR spectra. The bands are assigned with reference to literature on known organic compounds and polycarbosilane (PCS) [START_REF] Kessels | In situ probing of surface hydrides on hydrogenated amorphous silicon using attenuated total reflection infrared spectroscopy[END_REF][START_REF] Tsao | Studies of the solid-state conformation of polysilaethylene: an Organic/Inorganic hybrid polymer with an alternating C/Si backbone[END_REF]. In the spectra, we observe the CH stretching region (3100 2800 cm -1 ) and the strong band at 2150 cm -1 assigned to the SiH. Regarding smSiC powder IR spectra, SiH band is detected due to the oligomer grafting onto the SiC particles. Consequently, SiC powder IR spectra doesn't show this band.

Relative density

Table 2 summarizes the processing conditions applied to the investigated SiC-based ceramics, along with measured density. Interestingly, a high reproducibility was obtained using the as-received untreated SiC powder with a relative density of 91 % for several runs of only 1 min. For the functionalized SPS-smSiC samples the relative density in the same conditions reaches the narrow range 93-96 %, which also confirms a very good reproducibility. If this very low dispersion is considered significant, it can then be assumed that it would reflect some differences in the grafting efficiency of the different batches of smSiC powders.

Density of these various pellets shows differences versus the various powders used as starting materials. In general, SPS-smSiC samples prepared at 2200 • C for only 1 min exhibit the highest relative density (95 ± 1%). Their relative density is approximately 4% higher than that of SiC samples prepared using ungrafted powders (Table 2). Thus the density achieved is as high as those obtained previously by directly using a pretreated PCS compound and additives (Table 1). We retain that dense SiC bodies were obtained under our SPS conditions for both types of SiC powders used (with and without grafting).

We will further discuss the samples heat treated up to only 1600 • C for 1 min (namely SPS-SiC and SPS-smSiC) in section 4.3 (Sintering initiation). We will see that the differences in density mentioned above are confirmed and even reach 9%.

Microstructural and crystallographic characterization

SPS-SiC and SPS-smSiC pellets exhibit similar grain morphologies. Both polished SiC (Fig. 2) and polished smSiC samples (3) show a low porosity which increases at the edges of the pellet of 8 mm in diameter (see for instance at the bottom of micrographs of Figs. 2 and3). However, no specific changes are found between them. This observation is in accordance to the density measurement since we observe practically full dense SiC and smSiC bodies with RD ≥ 91 % (samples SPS-SiC-1 to SPS-smSiC-2 in Table 1). The SEM images in Figs. 2 and3 have been recorded using a concentric backscatter detector (CBS) and, usually, the contrast originates both from changes in the composition and topography. However, since the samples have been carefully polished, the different darker/brighter areas shown in the SEM image of Fig. 3 are preferentially due to different crystallographic orientation (or SiC polymorphs) since no chemical modification is detected by an EDS line-scan analysis, as confirmed by the constant value of the normalized Si intensity through a dark area of about 24 μm wide. Also, EDS analysis excludes the formation of elemental Si that would come from the decomposition of SiC. This is in agreement with the fact that such a decomposition is

Table 2

Experimental and relative density for SPS-SiC and SPS-smSiC. The relative intensity was determined compared to the theoretical value of alpha-SiC . The temperature and pressure experimental conditions were set constant at 50 MPa and 2200 • C with a rate of 100 • C/min for all the samples. The steady state temperature was maintained constant for 1 min.

expected only at 2545 • C [START_REF] Meyers | Additive Manufacturing of Technical Ceramics[END_REF].

The phase composition of the SiC powder used was investigated by XRD (Figs. 2 and3) and data are reported in Table 3. It is shown that asreceived SiC powder contains only two phases: the hexagonal-wurtzite SiC-6H (also called α-SiC) and the rhombohedral SiC-15R in the proportion 85 % and 15 % respectively. No significant difference was found for the smSiC powder, which was expected since the molecular grafting was a soft wet method that does not require high temperatures.

The XRD analysis of the SiC samples densified by SPS at 2200 • C as SPS-SiC-1 (Fig. 2) exhibits 3 different crystal phases: SiC-6H and SiC-15R like originally in the starting powder plus the hexagonal SiC-4H. The graphite signals at about 2theta 26.5 and 78 deg are also present on the XRD patterns but they are due to the contamination by the graphite dies and therefore graphite was not considered subsequently. Decomposition of silicon carbide did not occur since elemental silicon is not detected. From the Table 3, it can be seen that the proportion of the new phase SiC-4H is 25 % and it was preferentially produced to the detriment of SiC-15R and for a smaller part from SiC-6H.

The XRD pattern of the smSiC sample also prepared at 2200 • C as SPS-smSiC-2 (Fig. 3) exhibits the same three SiC phases with very close proportions as reported in Table 3. It is notable that the SiC-4H content is only 22 % instead of 25 % using as-received SiC powder. If it is assumed that these differences are significant, this can be attributed to an effect of molecular grafting which would limit the structural transformations.

The XRD patterns of SiC (SPS-SiC) and smSiC (SPS-smSiC) samples prepared at only 1600 • C are shown in Figs. 5 and6 respectively. Both patterns are similar to that of the as-received SiC powder, i.e. no evidence for the presence of SiC-4H was found. This is consistent with the fact that for these low temperature samples the SPS process was interrupted before reaching the sintering SiC temperature and, consequently, no SiC-4H transformation can occur [START_REF] Moberlychan | THE ROLES OF AMORPHOUS GRAIN BOUNDARIES AND THE b±a TRANSFORMATION IN TOUGHENING SiC[END_REF][START_REF] Tupitsyn | Controllable 6H-SiC to 4H-SiC polytype transformation during PVT growth[END_REF]. These results show that the use of PSE oligomer as SiC grafting molecular precursor does not change very significantly the final phase composition of the SPS samples and its conversion to SiC is fulfilled.

A RAMAN analysis was realized on the samples SPS-SiC-1 and SPS-smSiC-2 (Fig. 7). In this study, the focus is on the most intense FLO and FTO modes. The presence of three polytypes was identified at the initial state: SiC-4H, -6H and -15R; written in the Ramsdell notation. These polytypes differ by their crystalline lattice and stacking order. The associated Raman spectrum is a combination of these polytypes from which some peaks are characteristics of one polytype. For example, the peak at 789 cm 1 characterizes the SiC-6H, the peak at 777 cm 1, the SiC-4H and the peak at 785 cm 1, the SiC-15R [START_REF] Dobrzhinetskaya | Moissanite (SiC) with metal-silicide and silicon inclusions from tuff of Israel: Raman spectroscopy and electron microscope studies[END_REF][START_REF] Lafon-Placette | Tribological characterization of silicon carbide and carbon materials[END_REF]. This result is in accordance with the XRD pattern of the SiC and smSiC sample also prepared at 2200 • C as SPS-SiC-1 and SPS-smSiC-2 (Figs. 5 and6). They exhibit the same three SiC phases with very close proportions as reported.

Sintering initiation

Figs. 8 and9 indicate the temporal variation of temperature and displacement during the SPS process. The displacement is not corrected for the contribution of the components of the SPS column, i.e. it takes into account the graphite and SiC contribution. For these two experiments, the process was stopped at T 1600 • C. From the red curve of sample SPS-SiC (Fig. 8), we observe a slight expansion at the beginning of the SPS process that is caused by the thermal expansion of particles and die, as well as gas release [START_REF] Olevsky | Impact of thermal diffusion on densification during SPS[END_REF]. This low temperature expansion occurs in two main steps: the first one up to 100 s and the second up to approximately 300 s to reach a plateau. In this low temperature range, the behavior of the smSiC sample (SPS-smSiC) is close to that of SiC (SPS-SiC) except the plateau is reached after 200 s (Fig. 9) instead of 300 s for the SiC sample (Fig. 8). The great increase of the displacement curve at high temperature occurs faster for the SPS-smSiC sample (Fig. 9) compared to the SiC one (Fig. 8), approximately at 650 s and 750 s respectively. For both samples, the relationship between the temperature and the displacement shows that the displacement starts at 1060 • C in the smSiC sample and at 1240 • C in the SiC sample (Fig. 10). This change in the displacement curve means a dimension shrinkage of the powder [START_REF] Cha | Spark plasma sintering behavior of nanocrystalline WC Á/10Co cemented carbide powders[END_REF]. Wherewith it can be deduced that the sintering process begins with a difference between both powders of ca.180 • C, revealing that sintering is significantly facilitated using smSiC powder. Then, the temperature and pressure remained constant during the SPS process as

Table 3

Phase composition determined by Rietveld method [START_REF] Hongchao | Quantitative analysis of SiC polytype distributions by the rietveld method[END_REF] of XRD patterns of SiC and smSiC samples prepared by SPS compared to the powders used.

explained in section 3.

Regarding the microstructure, the porosity of the SPS-SiC (Fig. 11) and SPS-smSiC (Fig. 12) samples appear to be both closed and open (as suggested by some extensive ramifications of the porosity network). Anyways, the total porosity of these samples determined using the ImageJ software from SEM images as in Figs. 11 and12 gave approximate values of 39 % and 32 % for the SiC (SPS-SiC) and smSiC (SPS-smSiC) samples, respectively. This result is in very good agreement with the relative density measured by the Archimedes method since the values of 61 % and 72 % were found (Table 1). Clearly, SEM analysis and density measurements demonstrate that, compare to raw SiC powder, a better densification is obtained with the use of SiC powder previously grafted with the PSE oligomer. This is in good agreement with the sintering initiation difference that we measured in situ during the SPS process as reported in Fig. 10.

Despite the relatively large porosity of the SPS samples prepared at 1600 • C (Fig. 11 and Fig. 12), their SEM analysis did not reveal the obvious presence of dark zones unlike the samples densified at 2200 • C (Figs. 2 and3). We assumed that these bright/dark contrasts were mainly due to differences in crystallographic orientation as no variation in atomic composition was found by EDS (Fig. 4). According to this hypothesis, the dark structures could for example be attributed to the SiC-4H phase which is barely present in the starting powder but which forms during SPS runs at 2200 • C. The minor presence of this SiC-4H phase in samples prepared at 1600 • C was revealed by XRD analyzes. In addition, the absence of dark areas on the SEM micrographs of these low temperature samples supports our hypothesis to attribute the presence of dark areas to the SiC-4H phase. We then performed image treatment of the micrographs of Figs. 11 and 12 using ImageJ software and we found that the dark areas represented 27 % and 24 %, respectively for the SPS-SiC-1 and SPS-smSiC-2 samples. This is in very good agreement with the proportions of the SiC-4H phase determined by Rietveld analysis [START_REF] Hongchao | Quantitative analysis of SiC polytype distributions by the rietveld method[END_REF] of the XRD phases which were respectively 27 % and 22 % (Table 3) and the RAMAN spectra. This good overall consistency gives credibility to our hypothesis, which can be checked later, for example by TEM analyzes.

Conclusion

Dense silicon carbide specimens were fabricated by SPS process at 2200 • C and 50 MPa for only 1 min in this steady stage. The SiC samples were densified up to 91 % and the surface-modified smSiC sample up to 96 % using the same SPS conditions. The final porosity is very low since it lacks only 4% in density compared to theoretical bulk SiC. Interestingly, the densification is always found better for smSiC than for raw SiC sample. Consequently, the poly(silaethylene) (PSE) grafted onto the SiC grains is completely converted into SiC. This confirms that thanks to its atomic ratio Si:C 1:1 (as in stoichiometric SiC) and to the absence of heteroatoms, it is a suitable preceramic molecular precursor for SiC. This was previously demonstrated for CVD process for the growth of SiC coatings [START_REF] Boisselier | SiC coatings grown by liquid injection chemical vapor deposition using single source metal-organic precursors[END_REF] and this is extended here for the fabrication of 3D SiC parts by SPS technique.

Moreover, the surface-grafting of the oligomer-derived ceramic does not affect the final phases of the dense bodies since they are all composed of the three phases SiC-6H, SiC-4H and SiC-15R. This study allowed us to determine that the surface modification of the SiC powder decreases the sintering start temperature by about 180 • C and therefore a better densification is obtained, which is a good result for developing a more energy efficient SPS process.

Finally, compared with previous results [START_REF] Bernardo | Polymerderived SiC ceramics from polycarbosilane/boron mixtures densified by SPS[END_REF][START_REF] Lodhe | On correlation between β→α transformation and densification mechanisms in SiC during spark plasma sintering[END_REF][START_REF] Lee | Phase transformation on spark plasma sintered dense polycarbosilane-derived SiC without additive[END_REF], an innovative method has been used to graft a oligomer used as SiC molecular precursor in a controlled way on the surface of the SiC particles instead of using this preceramic precursor directly after pyrolysis or mixed with heteroatoms as boron additives to obtain a significantly improved final densification. Taking into account progress to understand and simulate SPS processes [START_REF] Manière | A spark plasma sintering densification modeling approach: from polymer, metals to ceramics[END_REF][START_REF] Yushin | Modeling process of spark plasma sintering of powder materials by finite element method[END_REF], this result is a new step towards the achievement of a 3D SiC part by SPS.

Fig. 1 .

 1 Fig. 1. PSE (CVD-4000), SiC powder and smSiC powder IR spectra.

Fig. 2 .

 2 Fig. 2. SEM fracture micrographs of the SPS-SiC-1 sample after polishing (final SPS temperature 2200 • C).

Fig. 3 .

 3 Fig. 3. SEM fracture micrographs of the SPS-smSiC-2 sample after polishing. (final SPS temperature 2200 • C).

Fig. 4 .

 4 Fig. 4. SEM fracture micrographs of the SPS-smSiC-2 sample after polishing prepared with a final SPS temperature in the steady state of 2200 • C for 1 min (left). EDS line-scan analysis along the red arrow across the dark area showing a constant Si content (right). (For interpretation of the references to colour in the Figure, the reader is referred to the web version of this article).

Fig. 5 .

 5 Fig. 5. XRD pattern of SPS-SiC-1, SPS-SiC and SiC powder.

Fig. 6 .

 6 Fig. 6. XRD pattern of SPS-smSiC, SPS-smSiC-2 and smSiC powder.

Fig. 7 .

 7 Fig. 7. Raman pattern of SPS-SiC-1 and SPS-smSiC-2.

Fig. 8 .

 8 Fig. 8. Temporal temperature and displacement changes during the SPS process of the sample SPS-SiC (final SPS temperature 1600 • C).

Fig. 9 .

 9 Fig. 9. Temporal temperature and displacement changes during the SPS process of the sample SPS-smSiC (final SPS temperature 1600 • C).

Fig. 10 .

 10 Fig. 10. Sintering initiation: variation of the displacement versus the temperature during the SPS process of the samples SiC (SPS-SiC; black curve) and smSiC (SPS-smSiC; red). Increasing displacement in graphs means dimension shrinkage of the powder compact according to [34]. (For interpretation of the references to colour in the Figure, the reader is referred to the web version of this article).

Fig. 11 .

 11 Fig. 11. SEM fracture micrographs of the SPS-SiC sample after polishing. The final temperature was stopped at 1600 • C to avoid a full densification.

Fig. 12 .

 12 Fig. 12. SEM fracture micrographs of the SPS-smSiC sample after polishing. The final temperature was stopped at 1600 • C to avoid a full densification.

Acknowledgements

This project, DOC-3D-Printing, has received funding from the European Union's Framework Program for Research and Innovation Horizon 2020 (2014-2020) under the Marie Skłodowska-Curie Grant Agreement No. [764935].

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Alexander I Epishin

Bernard Fedelich
email: bernard.fedelich@bam.de

Bernard Viguier

Sina Schriever

Igor L Svetlov

Nikolay V Petrushin

Romain Saillard

Arnaud Proietti

Dominique Poquillon

Anton Chyrkin

Creep of single-crystals of nickel-base γ-alloy at temperatures between 1150 °C and 1288 °C

come   L'archive ouverte pluridisciplinaire

Introduction

In order to remove the microporosity, single-crystal blades cast of nickel-base superalloys are subjected to hot isostatic pressing (HIP) in a narTow temperature range between the y'-solvus and the solidus temper atures [START_REF] Fritzemeier | The influence of high thermal gradient casting, hot isostatic pressing and alternative heat treatment on the structure and properties of a single crystal nickel base superalloy[END_REF][START_REF] Fullagar | Aero engine test experience with CMSX-4 alloy single-crystal turbine blades[END_REF][START_REF] Epishin | Pore annihilation in a single-crystal nickel-base superalloy during hot isostatic pressing: experiment and modelling[END_REF][START_REF] Epishin | Hot isostatic pressing of single-crystal nickel-base superalloys: mechanism of pore closure and effect on mechanical properties[END_REF][START_REF] Mujica Roncery | Influence of temperature, pressure, and cooling rate during hot isostatic pressing on the microstructure of an SX Ni-base superalloy[END_REF]. Since the strengthening y'-precipitates ar• e totally dissolved, the blade material is very soft at these superwlvus temper• atures. For exan1ple, the company Howmet Alcoa, USA, performs HIP for single-crystal blades of the nickel-base super• alloy CMSX-4 at 1288 •c. This te111per• ature is slightly above the y' -solvus temper• ature of this alloy but below its soli dus te111per• ature, respectively equal to 1282 • C and 1339 • C [START_REF] Shalin | Single Crystals of Nickel-Base Superalloys[END_REF]. However, HIPing the costly single-crystal blades at such higil te111perat u res can dan1age the111 by recrystallization or incipient melting. Ther• efore, the par a meter• s of con1Il1er• cial HIP (temper• ature T, pressure p, dtu• ation r) have to be car• efully optimized to eI1Stu• e full pore healing, while avoiding material dan1age a11d keeping processing costs low.

The HIP parameter• s could be optimized by using a physical mode! considering the mecha11isms of porosity a11nihilation dming HIP. One possible mechanism is pore closure via dislocation motion, as modeled in Refs. [START_REF] Epishin | Pore annihilation in a single-crystal nickel-base superalloy during hot isostatic pressing: experiment and modelling[END_REF][START_REF] Epishin | Hot isostatic pressing of single-crystal nickel-base superalloys: mechanism of pore closure and effect on mechanical properties[END_REF]. Therefore, in order to support HIP modeling, the creep behavior of superalloys at supersolvus temperatures has to be investi gated. In a previous work [START_REF] Epishin | Creep of single crystals of nickelbased superalloys at ultra-high homologous temperature[END_REF], we investigated creep of CMSX-4 at 1288 •c. However• , two experin1e11tal proble11is were encounter• ed:

• First, the supersolvus te111perature range in which CMSX-4 can be mechanically tested (1282-1339 °C) is too narrnw for a reliable measurement of the temperature depende11ce of the creep rate of this alloy as a y-solid solution. Moreover, the mechanical testing of CMSX-4 close to its solidus temper• ature is technically difficult because it bear• s the risk of incipient melting of the tested materfal. This makes it impossible to determine the cr• eep activation energy Q,,

which is an important physical parameter related to the creep mechanism. • Second, the γ ′ -phase reprecipitates during the cooling following the creep tests, which alters the dislocation structure and hinders the identification of the deformation mechanisms by Transmission Electron Microscopy (TEM).

To solve these problems, a model alloy compositionally similar to CMSX-4 but structurally free of the strengthening γ ′ -phase has been designed in this work. The alloy has been solidified as single crystals of different orientations, tested under creep conditions in the temperature range between 1150 • C and 1288 • C, and investigated by electron microscopy. The investigations aimed at understanding the creep behavior of the γ-solid solution of nickel at very high temperatures.

Experimental

The investigated model alloy is a γ-analogue of the nickel-base superalloy CMSX-4® [START_REF] Harris | Development of the rhenium containing superalloys CMSX-4 & CM 186 LC for single crystal blade and directionally solidified vane applications in advanced turbine engines[END_REF], which is widely used in the gas turbine industry and in aero-engines as blade material. In the following, we will refer to this model alloy as the "γ-alloy". It has been designed under the two following conditions:

• First, the alloy should be compositionally similar to CMSX-4 but structurally free of the strengthening γ ′ -phase in the entire temperature range. • Second, its creep resistance at temperatures above the γ ′ -solvus temperature of CMSX-4 (1282 • C) should be close to the creep resistance of CMSX-4; that is, at these temperatures the degree of its solid solution strengthening should be comparable to CMSX-4.

Therefore, as compared to CMSX-4, the content of γ ′ -formers (Al, Ti, Ta) in the γ-alloy has been significantly reduced while the content of the main solid solution strengtheners (Cr, Co, W, Re) has been maintained at the same level. Ti and Ta have been completely eliminated while the Al content has been decreased down to a level where no γ ′ -precipitation occurs. Fig. 1 shows the Al-Ni section of the phase diagram of CMSX-4 (Ti and Ta are removed) calculated with Thermo-Calc [START_REF] Andersson | Thermo-calc & dictra, computational tools for materials science[END_REF] using the thermodynamic database for Ni-based alloys TTNi7 [START_REF] Ttni | Ni-based Superalloys Database in[END_REF]. It can be seen that the alloy is γ ′ -free at Al concentrations below 1.85 wt% (4.15 at%). Accordingly, the content Al was reduced to 4 at%. For the sake of simplicity, Mo, which concentration in CMSX-4 is only 0.38 at%, has been removed. Additionally, Mo has a negative effect on oxidation resistance, which is important for high-temperature creep tests.

A small sample with the planned composition (see "γ-alloy planned" in Table 1) and a weight of about 20 g has been cast by induction levitation melting and investigated in a TEM as well as by differential scanning calorimetry (DSC). The TEM analysis has shown no γ ′ -precipitates in dark-field images (see Fig. 2a) as well as no superlattice reflections from the γ ′ -phase in electron diffraction patterns (see Fig. 2b). The shape of the measured DSC curve (Fig. 3) has confirmed the absence of γ ′ -precipitation: No characteristic γ ′ -peaks are observed during heating and subsequent cooling. The DSC analysis has given the values 1430 • C and 1475 • C for the solidus and liquidus temperatures, respectively. These values have been used to select the thermal conditions for the solidification of the γ-alloy specimens.

Single-crystals of the γ-alloy have been manufactured at the All-Russian Institute of Aviation Materials (VIAM) Moscow, Russia. The master alloy of the planned composition has been alloyed in a vacuum induction furnace at a temperature between 1550 • C and 1570 • C with a hold time of about 10 min. The composition of the cast alloy has been checked by atomic emission spectroscopy. The results are presented in Table 1, see "γ-alloy as cast". It is seen that the concentrations of the alloying elements in the cast alloy are close to the planned ones.

Single-crystals of axial orientations [001], [011], [123] and [111] (middle of a stereographic triangle) have been directionally solidified by the Bridgman method in the furnace UNVK-9A VIAM having a liquid metal cooling system (cooling in liquid aluminum). The different orientations have been obtained by seeds placed in the crucible bottom, thereby enabling a selection of the crystal growth direction. The obtained single-crystal bars were cylinders of diameter 15 mm and length 170 mm. After solidification, the bars have been subjected to a two-step homogenization heat treatment under vacuum, 2 h at 1285 • C and 18 h at 1320 • C, followed by cooling at 60 • C/min down to 1000 • C. The orientations of the cylinders' axes have been measured by X-ray diffraction (XRD) and the results are graphically presented in Fig. 4. It can be seen that the orientations of the solidified single-crystals are close to the exact orientations [001], [011], [123] and [111].

Cylindrical creep specimens of 8 mm diameter and 115 mm total length have been machined from these bars and tested in air under tensile creep conditions according to ISO 204. The tests have been carried out in constant load arm machines, equipped by resistive ATSfurnaces with SiC heating elements operating at temperatures up to 1500 • C. These furnaces have a short hot zone of about 75 mm. However, within the gage length of 25 mm, the temperature field is quite uniform. Indeed, temperature measurements have shown that the difference between the center and the ends of the gage length is less than 5 • C. The longitudinal creep strain has been measured within this 25 mm gage length with an axial MTS-extensometer protected from radiation by a water-cooled shield, which allows to use it up to 1450 • C. Note that all the measuring equipment has been calibrated.

The first creep test carried out with an uncoated specimen of γ-alloy at 1288 • C has shown strong oxidation after a short testing time. Such an oxidation behavior is very different from the CMSX-4, which demonstrated a good oxidation resistance at this temperature [START_REF] Epishin | Creep of single crystals of nickelbased superalloys at ultra-high homologous temperature[END_REF]. Obviously, the reason for this difference is the much lower Al content in the γ-alloy, which protects the material at high temperatures by forming an alumina scale. Therefore, the remaining specimens have been coated by a 50 μm thick NiCoCrAlY-Ta protective layer sputtered using the HP -HVOF (high pressurehigh velocity oxygen fuel) technique.

In order to determine the creep activation energy Q c , specimens have been tested at three temperatures, 1150 • C, 1225 • C and 1288 • C (the last one is the CMSX-4 HIP temperature) for each orientation. Most of the tests have been performed under the stress of 10 MPa, which allowed fitting Q c independently from the stress exponent n of Norton creep law. In addition, a few specimens have been tested at 1225 • C under different stress levels, i.e., 6 and 13 MPa, to enable the estimation of the stress exponent n at this temperature. Most of the tests have been interrupted at creep strains between 5 and 10 % for subsequent microstructural investigations. In order to "freeze" the dislocation structure, the specimens have been rapidly cooled down by air-jet under load. In the temperature interval 1288 • C → 400 • C, the cooling rate was about 15 • C/s.

After testing, the specimens have been investigated by different methods aiming at the identification of the deformation mechanisms at 1288 • C. In order to identify the active slip systems, the cross-section shape of one [011] specimen tested at 1288 • C has been characterized in an optical microscope, as described in Ref. [START_REF] Epishin | Creep of single crystals of nickelbased superalloys at ultra-high homologous temperature[END_REF]. Electron microscopy characterization has been performed at the Raimond Castaing Microanalysis Center (Toulouse, France). Scanning Electron Microscopy (SEM) and Electron Back Scattered Diffraction (EBSD) analysis have been carried out in order to follow the evolution of subgrain structure during high temperature creep tests. A Jeol JSM 7100F microscope equipped with an Oxford EBSD system has been used. The samples have been mechanically ground and then electrolytically polished before observations. The dislocation structures have been inspected by TEM, using a Jeol JEM 2100F apparatus. A classical electropolishing process has been conducted for thin foil preparation. Note that in order to get access to a large number of diffraction vectors, the foils have been cut along a (101) plane, i.e., inclined 45 • off the load axis. A similar creep anisotropy was observed for CMSX-4 in Ref. [START_REF] Epishin | Creep of single crystals of nickelbased superalloys at ultra-high homologous temperature[END_REF] and explained by octahedral slip 〈011〉{111}. In Fig. 5a, the creep curves of the γ-alloy (solid lines) are compared with the corresponding creep curves of CMSX-4 (dashed lines). It follows that at 1288 • C the creep resistance of the γ-alloy is in the same order of magnitude as that of CMSX-4, but the CMSX-4 is noteworthily stronger. To characterize the difference in creep resistance, the average strain rate between 0 and 5 % has been considered. The corresponding ratios for the γ-alloy to CMSX-4, i.e. εγ av (0 5%)/ε single-crystals of CMSX-4 supplied by VIAM, DPC, and Howmet (HM) Alcoa, USA. The softening behavior is emphasized by plotting the creep strain as a function of the normalized time t/t 2.5% , where t 2.5% is the time to reach the creep strain ε 2.5 %. Despite the smallness of the orientation differences (θ within 0.4-4.7 • ), the specimens exhibit a significantly different softening behavior. The most pronounced softening is shown by the VIAM specimen, followed by the DPC, and finally the HM specimen. In this regard, it should be noted that the specimens produced by VIAM have been solidified by liquid metal cooling (LMC), while the specimens received from DPC and HM by radiation cooling. In the same order, the primary dendrite arm spacing λ PDA , characterizing the density of the dendritic structure of these specimens, takes values of about 210, 280, and 360 μm, which correlates with the different solidification rates.

Results

Analysis of the creep curves

According to the results of investigations performed at the VIAM, the creep behavior of single-crystal superalloys is significantly influenced by their subgrain structure, which is more developed in the single-crystals grown under high-temperature gradient as in the case of the LMC method.

Among the available results for CMSX-4 in our group, only one [001] specimen tested at 1288 • C under 10 MPa can be directly used for comparison with the γ-alloy. Indeed, this CMSX-4 single-crystal and the γ-alloy crystals have been grown in the furnace UNVK-9A of VIAM under similar LMC solidification conditions. The comparison between this specimen and γ-alloy specimens of several orientations is presented in Fig. 6b in the same way as in Fig. 6a. First, irrespective of their orientations, the normalized creep curves ε γ hkl f(t /t 2.5 ) of the γ-specimens are almost identical and straight, pointing to the absence of softening or hardening. On the opposite, the CMSX-4 creep curve ε CMSX-4 hkl f(t /t 2.5 ) shows significant softening indicated by a pronounced creep rate acceleration.

All creep curves of the γ-alloy have been analyzed and the minimum strain rates εmin determined in order to identify the creep activation energy Q c and the stress exponent n of Norton's creep law. In this section, Q c and n have been evaluated by a simple linear regression of the creep rates separately for each orientation. A global fitting of an anisotropic Norton-type creep model with all test results requires an analytical description of the creep rate anisotropy, εmin f(θ,ρ), where θ and ρ are the Euler angles characterizing the specimen orientation. Such an approach will be presented below in the Discussion section. The value of Q c has been evaluated by linear fitting of the Arrhenius plots, assuming that εmin ∼ exp( Q c /RT), where R is the universal gas constant and T is the absolute temperature.

The Arrhenius plots in Fig. 7a C but under different applied stresses σ: 6, 10, and 13 MPa (see the corresponding creep curves in Fig. 5d). Fig. 7b shows the fitting results assuming a power-law i.e., εmin ∼ σ n . The line of best fit for [011] is slightly shifted down with respect to that for [123],

Investigation of the shape of a deformed cross-section

The previous section has shown that the creep anisotropy of the γ-alloy observed in this work is similar to that reported in Ref. [START_REF] Epishin | Creep of single crystals of nickelbased superalloys at ultra-high homologous temperature[END_REF] for CMSX-4 at 1288 • C, for which the main creep mechanism was identified as octahedral slip 〈011〉{111}. Another evidence for the operation of octahedral slip could be obtained from the analysis of the deformed cross-section of a [011] specimen after an interrupted creep test, as shown in Ref. [START_REF] Epishin | Creep of single crystals of nickelbased superalloys at ultra-high homologous temperature[END_REF]. Therefore, the same analysis has been performed for a [011] specimen of γ-alloy deformed at 1288 • C under 10 MPa. The test has been interrupted after 2.1 h at an accumulated creep strain of 7.3 %. The measured values of the cross-section radius are plotted in Fig. 8 (dots) as a function of the azimuthal angle measured counterclockwise from the [001] direction. The shape of the deformed cross-section, as calculated by the large strain formalism in Ref. [START_REF] Epishin | Creep of single crystals of nickelbased superalloys at ultra-high homologous temperature[END_REF], and assuming that the material deforms by octahedral slip only is an ellipse, with its major axis along [011] and its minor axis along [001]. The eccentricity of the ellipse is directly related to the longitudinal creep strain ε 7.3 %. The measurements agree very well with the elliptical shape predicted under the assumption of octahedral slip.

It was shown in Ref. [START_REF] Epishin | Creep of single crystals of nickelbased superalloys at ultra-high homologous temperature[END_REF] that octahedral slip is the only mechanism that leads to the observed elliptical cross-section. Other potential creep mechanisms like 〈011〉{100} cubic slip [START_REF] Sass | Orientation dependence of dislocation structures and deformation mechanisms in creep deformed CMSX-4 single crystals[END_REF][START_REF] Bettge | Cube slip" in near-[111] oriented specimens of a singlecrystal nickel-base superalloy[END_REF] or 〈011〉{011} climb [START_REF] Mesarovic | Dislocation creep: climb and glide in the lattice continuum[END_REF][START_REF] Barnes | The climb of edge dislocations in face-centred cubic crystals[END_REF][START_REF] Abu-Odeh | Insights into dislocation climb efficiency in FCC metals from atomistic simulations[END_REF] also lead to an elliptical shape of the cross-section, but rotated by 90 • ; that is, with the major axis along [100]. Creep controlled by bulk diffusion (Nabarro-Herring model [START_REF] Nabarro | Deformation of crystals by the motion of single ions[END_REF][START_REF] Herring | Diffusional viscosity of a polycrystalline solid[END_REF]) isotropically reduces the specimen cross-section without changing its shape; that is, the cross-section remains circular.

SEM & TEM investigations

A few specimens of the γ-alloy have been selected for investigations in SEM by EBSD and in TEM. Fig. 9 shows the comparison of the subgrain structure of [001] specimens before (a, b) and after (c, d) creep at 1288 • C, 10 MPa and 7 % creep strain. Low angle boundaries (LABs) are highlighted in the Kernel Average Misorientation (KAM) mapping (a, c) deduced from EBSD analysis of both specimens. Some LABs have a clearly higher degree of misorientation, others lower. It is known that two types of LABs exist in single-crystals of nickel-base superalloys: interdendritic LABs (highlighted in blue in Fig. 9b andd) between the dendritic subgrains with misorientation up to a few degrees and intradendritic LABs (highlighted in red in Fig. 9b andd) located inside the dendritic subgrains and separating the mosaic blocks, which misorientation is one order of magnitude lower [START_REF] Shalin | Single Crystals of Nickel-Base Superalloys[END_REF][START_REF] Brückner | Local X-ray diffraction analysis of the structure of dendrites in single-crystal nickel-base superalloys[END_REF].

To unambiguously distinguish these two types of LABs, the dendrites have been identified through the density contrast of scattered electron images taken in EBSD mode using a Forward Scatter Detector (FSD). They are marked by crosses in Fig. 9a,c. The visible LABs in the EBSD map of the undeformed specimen (Fig. 9a) are mainly interdendritic, while in the map of the deformed specimen (Fig. 9c) numerous intradendritic LABs can be observed in addition. Schematic drawings of these LABs have been reported in the FSD images in Fig. 9b,d. The misorientations of these LABs have been measured and it appears that they increase after creep: the mean misorientation of interdendritic LABs is 0.62 • before creep and 1.11 are below the angular resolution of the EBSD method and therefore such LABs cannot be seen in Fig. 9a. However, intradendritic LABs become visible after creep, with a mean misorientation below 0.5 • , which clearly indicates dislocation activity within the subgrains. During creep, dislocations crossing the subgrains can either be arrested by reacting with the dislocations of preexisting intradendritic LABs, leading to an increase of their misorientation, or react one with each other, thereby knitting new LAB networks.

The SEM observations have been confirmed by TEM analysis of dislocation structure. Fig. 10, which shows a LAB with its surrounding area, is an example of TEM observation for the same deformed specimen as in Fig. 9. The LAB mesh size is about 100 nm, from which the misorientation can be estimated as about 0.1-0.2 • . Thus, it is presumably an intradendritic LAB. As a striking feature of these TEM observations, although several images have been taken under different diffraction conditions, only very few dislocations could be observed inside the cells limited by such LABs. In Fig. 10a one can observe only two short individual segments (marked by red arrows), which were nearly the only ones visible in the cell interior. All in all, the dislocation landscape mainly consists of dislocation networks forming LABs and delimiting cells, the interior of which is nearly free from dislocations. What might be surprising at first glance can be understood from a theoretical estimate of the density of mobile dislocations ρ m .

The assumption of dislocation motion controlled by the diffusion of solute atoms leads to high velocities at 1288 • C, for which the density ρ m with dislocations of opposite Burgers vector by climb or react with other dislocations, thereby rearranging in new LABs that aggregate the dislocations locally in excess (often called geometrically necessary dislocations). In this regard, it should be noticed that extinction contrast analysis performed on dislocations within the LABs has indicated that the Burgers vector of all these dislocations is of the type a/2 〈011〉. A similar refinement of the subgrain structure during creep, leading to low energy dislocation configurations, has been often observed and regarded as a recovery mechanism in polycrystalline metals (see, e.g. Refs. [START_REF] Kassner | Fundamentals of Creep in Metals and Alloys[END_REF][START_REF] Blum | Dislocation mechanics of creep[END_REF]). At the high testing temperatures considered here, it is expected that these recovery mechanisms are extremely fast, hence explaining the

Table 2

Results of the identification of the Burgers vectors of the dislocations in Fig. 11 by the invisibility criterion ( g The results are presented in Table 2. All dislocations have a/2 〈011〉 type Burgers vectors, which confirms the activity of the usual fcc system of 〈011〉{111} octahedral slip. The dislocations marked by b, c, and d have a nonzero last index, which means that they experience an elastic force due to applied load. Most of these dislocations are obviously mobile, as for example the b-group in Fig. 11a

→ ⋅ b → ) = 0.

Discussion

A γ-analogue of CMSX-4 has been designed on the basis of Thermo-Calc calculations. The solidified single-crystals of this alloy have shown the same orientation dependence of the creep rate as CMSX-4 (see Fig. 5a-c) as well as the same transformation of the cross-section shape of [011] specimens (see Fig. 8), which both indicate the dominance of 〈011〉{111} octahedral slip as deformation mechanism. This result is in agreement with the widely known Weertman model for creep of solid solution alloys assuming that dislocations viscously glide under the control of solute dragging [START_REF] Zhang | High Temperature Deformation and Fracture of Materials, Chapter 5.4.1 Weertman Model[END_REF].

However, it has been already mentioned that the γ-alloy creeps at 1288 • C about two times faster than CMSX-4, see Fig. 5a. This lower strength could be due to a reduced content of the γ-stabilizing elements (Cr, Co, Mo) as well as total removing of the γ ′ -forming element Ta, see Table 1. According to Ref. [START_REF] Rehman | On the temperature dependent strengthening of nickel by transition metal solutes[END_REF], Ta can strengthen the γ-solid solution if its concentration is significant, which is the case of CMSX-4 at supersolvus temperatures. An additional cause for the higher creep resistance of CMSX-4 could be a small residual amount of the γ ′ -phase, which according to Ref. [START_REF] Epishin | Creep of single crystals of nickelbased superalloys at ultra-high homologous temperature[END_REF] still remains in the interdendritic regions of this alloy at 1288 • C. This could also be the origin of the remarkable softening of CMSX-4 during creep, which is barely distinguishable for the γ-alloy, see Fig. 6b. However, the concentration of the important elements Re and W has been kept identical to CMSX-4 and it has been verified that the creep strengths of both alloys are in the same order of magnitude. In summary, despite the mentioned differences in creep resistance, it is obvious that similar deformation mechanisms operate in both materials in the absence of the strengthening γ ′ -phase.

The values of the stress exponent n and the creep activated energy Q c have been preliminarily evaluated in section 3.1 (Analysis of the creep curves) by simple linear regression of the creep rates, separately for different orientations. In the following, Q c and n will be evaluated by a joint fitting considering all investigated orientations as well as the deviations from the ideal exact orientations measured by XRD and temperatures. This requires an analytical description of the orientation dependence of the creep rate. Such an expression was derived, e.g., in Ref. [START_REF] Epishin | Creep of single crystals of nickelbased superalloys at ultra-high homologous temperature[END_REF], for octahedral slip and a Norton power-law at the slip system level:

εmin (σ, θ, ρ) ∼ σ n Φ(θ, ρ) with Φ(θ, ρ) ∑ 12 s 1 m n+1 s (θ, ρ) (1)
where εmin (σ, θ, ρ) is the minimum creep rate of a specimen, which orientation is defined by the Euler angles θ and ρ, Φ(θ, ρ) is an orientation factor depending on θ and ρ, n is the Norton exponent and m s (θ, ϱ) εmin (σ, T, θ, ρ) A σ n Φ(θ, ρ) exp( Q c /RT), [START_REF] Fullagar | Aero engine test experience with CMSX-4 alloy single-crystal turbine blades[END_REF] where A, n and Q c are slip system independent material parameters. These parameters have been determined by maximizing the determination coefficient:

r 2 (lA, Q c , n) 1 SS res SS tot , ( 3
)
where

SS res ∑ N i 1 [ ln εi ( lA Q c RT + n ln σ + ln [ ∑ 12 j 1 m s,j (θ i , ρ i ) n+1 ] ) ] 2 , ( 4
)
with lA ln A, N is the number of tests and

SS tot ∑ N i 1 ( ln εi ln εi ) 2 with ln εi 1 N ∑ N i 1 ln εi (5)
The maximization of r 2 (lA, Q c , n) has been performed in two steps: In the first one, n is fixed, but lA and Q c are fitted to the creep data, which corresponds to a linear regression leading to the n-dependent parameters lA(n) and Q c (n). In the second step, the resulting function r 2 (n) has been maximized with respect to n.

The graph r 2 f(n), where 1 ≤ n ≤ 10, is shown in Fig. 12a. It is seen that r 2 f(n) has only one clear maximum with r 2 ≈0.97 at n 4.95. The corresponding activation energy Q c 442 kJ/mol is within the range 430-459 kJ/mol separately determined for the different orientations, see Fig. 7a. The pre-factor A in Eq. ( 2) is A

3.3 × 10 6 MPa -(n+1) s -1 .
The obtained values of n and Q c can be compared with the results reported in the literature for Ni-solid solutions, see Table 3. For the sake of comparison, only references that do not consider a back stress (or internal stress) are presented. It is seen that n slightly varies around the average value equal to 4.91 ± 0.18, which is very close to the present result. This value is also in agreement with a statement of J. Weertman and J.R. Weertman [START_REF] Weertman | Mechanical properties, strongly temperature dependent[END_REF]: "It is shown by experiments that for pure metals and many alloys stress exponent n 4-5". In addition, it is remarkable that the n values given in Table 3 do not correlate with the alloy composition, stress, and temperature. Fig. 13 now shows the n-values measured for superalloys at different temperatures. These data, with the exception of CMSX-4 (our previous work [START_REF] Epishin | Creep of single crystals of nickelbased superalloys at ultra-high homologous temperature[END_REF]) and the γ-alloy (this work), are taken from Ref. [START_REF] Wollgramm | On the role of Re in the stress and temperature dependence of creep of Ni-base single crystal superalloys[END_REF]. Vertical and horizontal bars respectively indicate the scatter of n and the investigated temperature ranges. There is a clear tendency for the exponent n to decrease with temperature, as shown by the dashed line. At lower temperatures of about 850 • C the average of n is about 10, while at 1288 • C it is equal to 6.1 for CMSX-4 and about 5 for the γ-alloy. Obviously, this decrease of n is related to a change of the deformation mechanism due to the dissolution of the strengthening γ ′ -phase. When the γ ′ -phase totally dissolves, n should approach the typical values for Ni-solid solutions, as confirmed in the case of the γ-alloy. Note that the slightly higher exponent n previously found for CMSX-4 at 1288 • C could be also due to the small residual amount of the γ ′ -phase in CMSX-4, as mentioned above.

Dislocation glide can be retarded by the large solute atoms segregating at the dislocation core, as was recently [START_REF] Wu | Unveiling the Re effect in Ni-based single crystal superalloys[END_REF] established by high-resolution TEM and atom probe tomography for Re in the nickel-base superalloy ERBO/1 under creep conditions. Accordingly, the dislocation velocity at high temperatures and low stresses is controlled by the diffusion mobility of these atoms, a phenomenon that has been analyzed, e.g., in Refs. [START_REF] Takeuchi | Glide and climb resistance to the motion of an edge dislocation due to dragging a cottrell atmosphere[END_REF][START_REF] Fuentes-Samaniego | Solute drag on moving edge dislocations[END_REF]. The calculations presented in the Appendix (see Fig. 15) also confirm that the experimental stress range lies below the maximum drag force. Therefore, it is reasonable to expect that the creep activation energy Q c of Ni-solid solutions correlates with the activation energy of diffusion Q Ni X of the solid solution strengthening (SSS) elements X in Ni. However, only a qualitative comparison is possible because of the uncertainty measurements for Q Ni X . Fig. 14, which is based on the data from Table 3, shows the correlations between Q c and Q Ni X for some elements contained in the γ-alloy. The horizontal scale gives the content of the key SSS-element in the considered Ni-solid solution (Cr, W or Re) and the vertical scale the corresponding creep activation energy. The value for pure Ni, Q c 276 kJ/mol [START_REF] Monma | High-temperature creep of nickel-chromium alloys (On the relation between high-temperature creep and diffusion in nickel base solid solutions, IV)[END_REF], is taken as a reference point. It should be noted that this value is close to the activation energy of self-diffusion in Ni, Q Ni Ni 278 ± 12 kJ/mol, calculated by averaging the literature data from Ref. [START_REF] Macewan | Self-diffusion in polycrystalline nickel[END_REF]. It is seen that small Cr additions have no influence on Q c , which agrees with the result of [START_REF] Engstrom | Assessment of diffusional mobilities in face-centered cubic Ni-Cr-Al Alloys[END_REF] according to which

Q Ni Cr ≈ Q Ni
Ni . An increase of Q c visible at larger Cr concentrations of 20-30 at% is not due the SSS-effect but to strengthening by the ordered intermetallic phase Ni 2 Cr forming in Ni-base alloys at high Cr concentrations, see e.g. Ref. [START_REF] Martinsson | Aging Influence on Nickel-Based Superalloys at Intermediate Temperatures[END_REF]. In contrast to Cr, the effect of W on Q c is achieved at significantly lower additions. For example, an addition of 4, respectively 6 at% of W yields an increase of Q c up to 311 ± 12, respectively 317 ± 8 kJ/mol (see Table 3). These values are close to the measurements of Q Ni W , 312 kJ/mol, and 310 ± 5 kJ/mol, respectively reported in Refs. [START_REF] Karunaratne | Modelling of the micro-segregation in CMSX-4 superalloy and its homogenization during heat treatment[END_REF][START_REF] Epishin | Diffusion processes in multicomponent nickel-base superalloy-nickel system[END_REF].

However, it is seen in Fig. 14 that the γ-alloy has a much higher creep activation energy Q c ≈442 kJ/mol than the Ni-Cr and Ni-W solid solutions. According to Ref. [START_REF] Fleischmann | Influence of solid solution hardening on creep properties of single-crystal nickel-based superalloys[END_REF], the creep resistance of the γ-matrix highly alloyed with Cr, Co, and W is raised by a factor of 20 by adding about 3 at% of Re. Therefore, it is logical to assume that the high Q c -value found for the γ-alloy is related to the Re addition. However, due to the large uncertainty regarding the activation energy for the diffusion of Re in Ni, Q Ni Re , it is difficult to make a direct comparison between Q c and Q Ni Re . Indeed, the values reported for Q Ni Re range between 196 kJ/mol and 412 kJ/mol, as discussed in Ref. [START_REF] Epishin | Interdiffusion in binary Ni-Re alloys[END_REF]. The cause of this uncertainty is the extremely low diffusion mobility of Re, which makes it difficult to accurately measure Q Ni Re . In order to reduce this uncertainty, the theoretical estimations of the Q Ni Re -value could be used. According to the first-principle calculations performed by Janotti et al. [START_REF] Janotti | Solute diffusion in metals: larger atoms can move faster[END_REF] the dependence of the Q Ni X -value of the transition metals of 5d series on the atomic number Z has a maximum of about 337 kJ/mol at Z 75, which corresponds to Re. This value is significantly higher than that predicted for W, which is about 287 kJ/mol.

In summary, it is likely that the high creep activation energy Q c of the γ-alloy is related to the role of Re and the high value of Q Ni Re . However, the question remains: why is Q c significantly higher than Q Ni

Re ? The answer to this question requires a deeper and broader analysis, which is beyond the scope of this paper. Therefore, only two possible reasons will be noted here. First, in the γ-alloy, diffusion of Re doesn't take place in pure Ni but in the Ni-solid solution, where every fourth atom is a foreign atom. Second, the dislocation velocity and thus the creep rate are related to Q Ni Re through different parameters that are also temperaturedependent, e.g., the elastic constants of the alloy lattice and the volumes of solute atoms (see the appendix).

Conclusions

A γ-analogue of the superalloy CMSX-4 that represents the γ-solid solution of nickel at all temperatures and does not contain the strengthening γ ′ -phase has been designed on the base of Thermo-Calc calculations. Single crystals of this alloy of different crystallographic orientations have been tested under creep conditions in the temperature range between 1150 • C and 1288 • C.

• The tests have revealed a very high creep anisotropy of this alloy as previously observed for CMSX-4 at its supersolvus temperature of is the Schmid factor for the slip system with index s in the case of uniaxial loading along the specimen axis. Assuming an Arrhenius dependence of ε ˙min on the temperature T, one can extend Eq. ( 1) as follows:

1288 • C. This anisotropy results from the operation of 〈011〉{111} octahedral slip, identified as the dominant deformation mechanism. This has been confirmed by the change of the shape of [011] specimens as well as by the orientation dependence of the creep rates. • The tests in the temperature range 1150-1288 • C yield the estimate Q c ≅ 442 kJ/mol of the creep activation energy, which correlates with the high activation energy of Re diffusion in Ni. This result supports the hypothesis that the motion of dislocations in the γ-matrix of Re-containing superalloys is controlled by the slow diffusion of Re atoms segregating at the dislocation core (Cottrell atmosphere). The Norton stress exponent n has been found to be equal to 5, which is a typical value for pure metals and single-phase metallic alloys.

Appendix. Estimates of the dislocation velocity and the density of mobile dislocations

Following e.g. Refs. [START_REF] Blum | Understanding creep-a review[END_REF][START_REF] Takeuchi | Steady-state creep of alloys due to viscous motion of dislocations[END_REF], it is assumed that creep is controlled by the viscous glide of edge dislocations dragging a cloud of solute atoms (Cottrell atmosphere). The dislocation velocity v d is then calculated under the assumption that glide is controlled by the diffusion of the large solute elements, Re and W, segregating around the dislocation core. Numerical and approximative solutions of the problem were derived for example in Refs. [START_REF] Takeuchi | Glide and climb resistance to the motion of an edge dislocation due to dragging a cottrell atmosphere[END_REF][START_REF] Hirth | Theory of Dislocations[END_REF][START_REF] Sakamoto | Diffusion equation and Cottrell atmosphere dragging of edge dislocation in high concentration solid solutions[END_REF]. A rigorous analysis of the diffusion problem around the gliding dislocation was developed in the case of an infinite region in Ref. [START_REF] Fuentes-Samaniego | Solute drag on moving edge dislocations[END_REF].

The cloud of diffusing solute atoms exerts a drag force per unit dislocation length F drag B drag v d on the gliding dislocation, where B drag is the drag coefficient. The dislocation velocity results from the equilibrium condition

F drag + F mech 0 ⇒ v d F mech B drag (6)
where the mechanical force corresponds to the glide component of the Peach-Kohler force, i.e., F mech b τ, in which b is the Burgers vector and τ the Resolved Shear Stress (RSS). The drag force was found to increase in a quasi-linear regime until a peak force is reached, corresponding to a critical dislocation velocity approximately equal to v c ≅ D k T β for one solute, where D is the corresponding interdiffusion coefficient and β is given by

β μ b 3 π 1 + ν 1 ν (Ω solute Ω solvent ), (7)
where Ω solute and Ω solvent are the atomic volumes of the solute, respectively the solvent atoms, μ is the shear modulus, and ν is Poisson's ratio. Since the γ-alloy is compositionally close to CMSX-4 it has been assumed that Ω solvent Ω CMSX4 . The diffusion problem for the atomic concentration c of the solutes around the moving edge dislocation has been solved by Finite Elements (FE) in a disk of radius R ext ≫b centered at the dislocation core and with the boundary condition c(R ext ) c 0 , where c 0 is the mean atomic concentration of the solutes. Additional details are not reproduced here since similar solutions have been already presented in the literature (see e.g. Ref. [START_REF] Sakamoto | Diffusion equation and Cottrell atmosphere dragging of edge dislocation in high concentration solid solutions[END_REF]). In the quasi-linear regime, the drag coefficient is usually written in the form

B drag c 0 β 2 ln(λ/r 0 ) Ω D k T , ( 8
)
where r 0 is an inner cut-off radius, λ is an outer cut-off radius, and Ω is the atomic volume of the alloy. The inner cut-off radius is usually taken of the size of the dislocation core, i.e., r 0 ≈ b. Regarding the quantity λ, it seems that there is no universally accepted recommendation. The rigorous analysis developed in Ref. [START_REF] Fuentes-Samaniego | Solute drag on moving edge dislocations[END_REF] for an infinite domain shows that λ is proportional to D/v for small velocities, i.e., β v 2 D ≪1. The parameters entering Eqs. ( 7) and ( 8) have been chosen as follows:

The classical treatment of solute drag by a gliding edge dislocation makes use of isotropic elasticity (see e.g. Ref. [START_REF] Blum | Understanding creep-a review[END_REF]). Hence, the question arises of an appropriate choice of the shear modulus μ iso and Poisson's ratio ν iso in Eq. ( 7). These constants have been chosen in such a way that the pre-factor )

, leading to μ iso K cubic (1 ν iso ), where the expression of K cubic as a function of the components of the elastic stiffness (c 11 , c 12 , c 44 ) is given in Ref. [START_REF] Hirth | Theory of Dislocations[END_REF]. In addition, the bulk modulus κ iso of the isotropic material is taken identical to that of the cubic material 3 κ 2 μ iso )/(6 κ + 2 μ iso ). After resolution of the corresponding system of equations, the values μ iso 42 GPa and ν iso 0.38 are obtained for 1288 • C with the elastic constants of the γ-alloy provided in Ref. [START_REF] Epishin | Investigation of elastic properties of the single-crystal nickel-base superalloy CMSX-4 in the temperature interval between Room temperature and 1300[END_REF]. These values are also close to the Hill average of the Voigt and Reuss estimates for a polycrystal yielding to μ iso 38 GPa and ν iso 0.39.

The atomic volumes at Room Temperature (RT) for Re and W have been taken as Ω Re 1.47 × 10 -29 m 3 and Ω W 1.57 × 10 -29 m 3 [START_REF] Singman | Atomic volume and allotropy of the elements[END_REF]. The lattice spacing for CMSX-4 at RT [START_REF] Epishin | Influence of small rhenium additions on the lattice spacing of nickel solid solution[END_REF] has been taken as a 3.6 × 10 -10 m, which leads to the atomic volume of the alloy Ω Ω solvent 1.16 × 10 -29 m 3 . Estimates of the atomic volumes Ω(T) at high temperature have been obtained by applying the linear thermal dilatation coefficients α(T) ΔL L0 , that is Ω(T) Ω 0 (1 + α(T)) 3 . The thermal dilatation for Re α(T)

9.7910 5 + 5.8910 6 T + 8.5710 10 T 2 , (

with T in • C has been taken from Ref. [START_REF] Conway | Properties of Some Refractory Metals. Part III: Thermal Expansion Characteristics of Tungsten, Rhenium[END_REF]. The thermal dilatation for W α(T) 1.389610 3 8.279710 7 T + 4.055710 9 T 2 1.216410 12 T 3 + 1.703410 16 T 4 , [START_REF] Ttni | Ni-based Superalloys Database in[END_REF] with T in K has been taken from [START_REF] Müller | Thermal expansion of tungsten in the range 1500-3600 K by a transient interferometric technique[END_REF]. For the γ-alloy, the thermal dilatation determined for CMSX-4 reported in Ref. [START_REF] Epishin | Investigation of elastic properties of the single-crystal nickel-base superalloy CMSX-4 in the temperature interval between Room temperature and 1300[END_REF] has been used. The interdiffusion coefficients for Re and W in the γ-alloy are not known as mentioned above. For an estimate, the coefficients for interdiffusion in pure Ni are considered. For Re, this coefficient is given in Ref. [START_REF] Epishin | Interdiffusion in binary Ni-Re alloys[END_REF] in the form

D Re D Re,0 exp ( Q Re R T ) (11)
where D Re,0 1.2 × 10 -4 m 2 /s and Q Re 317 kJ /mol. At 1288 • C, D Re 2.9 × 10 -15 m 2 /s. For W it is given in Ref. [START_REF] Epishin | Diffusion processes in multicomponent nickel-base superalloy-nickel system[END_REF] as

D W D W,0 exp ( Q W R T ) ( 12
)
where D W,0 2.5 × 10 -4 m 2 /s and Q W 310 kJ /mol. At 1288 • C, D W 1.1 × 10 -14 m 2 /s.

In the case of two species (Re and W), the total drag force is the sum of the contributions of both species, and we can write

where the atomic concentrations c W,0 and c Re,0 are indicated in Table 1 for the γ-alloy. For comparison with the applied stress levels, the total drag force F drag per unit line is divided by the Burgers vector and plotted in Fig. 15. The FE solution agrees well with that derived by Fuentes-Samaniego et al. in Ref. [START_REF] Fuentes-Samaniego | Solute drag on moving edge dislocations[END_REF]. It follows from this analysis that at the stress levels of the creep tests the dislocation motion is within the range of control by solute diffusion with the dislocation core.

It can be also seen that the quasi-linear regime applies at RSS below 10 MPa. In this range, with the value ln(λ /r 0 ) 3, the linear model matches the FE results. For example, at T 1288 • C, we obtain B drag 3.7 kN/m -2 s.

Finally, for the slip system s, the density of mobile dislocations ρ m,s can be inferred from the Orowan relation γs b ρ m,s v s b 2 ρ m,s τ s / B drag , where τ g is the resolved shear stress. In the case of a [001] specimen, the previous equation can be rewritten for the minimum creep rate ρ m,s B drag εmin 8 m 2 001 b 2 σ [START_REF] Barnes | The climb of edge dislocations in face-centred cubic crystals[END_REF] where m 001 0.41 is the Schmid factor in the direction [001]. A power-law of [fx] with an exponent n 1 is thus obtained for ρ m,s . Note that the previous Eq. doesn't account for internal stresses. A friction stress of the type τ c α μ b ρ m,s

√

following Taylor's model can be accounted for, which amounts to replace τ s m 001 σ by m 001 σ τ c in Eq. ( 14). Thereby, the numerical factor α is usually taken as α ≈ 0.

The densities ρ m,s estimated with the identified creep law (1) for both estimates ( 14) and ( 15) are plotted at 1288 • C in Fig. 16. In the considered stress range, the friction stress has only a limited influence. For 10 MPa, the density of mobile dislocations ρ m,s in each slip system of a [001] specimen is equal to about 0.1 μm -2 . Thus, the total mobile dislocation density for all 8 active slip systems is ρ m 0.1 × 8 ≈ 0.8⋅μm 2

Fig. 1 .

 1 Fig. 1. Al-Ni section of the phase diagram of CMSX-4 (Ti and Ta are removed) calculated by the software Thermo-Calc.

  The creep curves of the [001], [011], [123], and [111] specimens of the γ-alloy are presented in Fig. 5 by solid lines. Fig. 5a-c shows the creep curves obtained under the stress of 10 MPa but at different temperatures, i.e., at 1150 • C, 1225 • C, and 1288 • C. At all temperatures, a pronounced creep anisotropy is observed. The highest creep resistance is shown by the [111] orientation and the lowest by the [001] orientation. The [011] and [123] single-crystals have an intermediate creep resistance with a slightly higher strength of the [011] orientation with respect to [123]. It is also seen that the creep rate is strongly temperature-dependent, as shown by the different time scales of the creep curves in Fig. 5a-c obtained at 1150 • C, 1225 • C, 1288 • C, and 10 MPa. It is also strongly stress-dependent, as demonstrated by the creep curves in Fig. 5d obtained at 1225 • C for the stress levels 6, 10, and 13 MPa.

Fig. 2 .

 2 Fig. 2. TEM confirmation of the single-phase γ-structure of designed alloy. (a) Dark-field image. (b) Selected area electron diffraction pattern.

Fig. 3 .

 3 Fig. 3. DSC curves of γ-alloy measured in helium with heating/cooling rates ±20 • C/min.

Fig. 4 .

 4 Fig. 4. Distribution of the axial orientations of the γ-alloy single-crystals in the standard stereographic triangle.

Fig. 5 .

 5 Fig. 5. Creep curves of γ-alloy single-crystals of different orientations at 1288 • C (a), 1225 • C (b, d), and 1150 • C (c). (a-c) Different temperatures but the same stress of 10 MPa. (d) Temperature of 1225 • C but different stresses, 6, 10, and 13 MPa. (a) Comparison of γ-alloy with CMSX-4 at 1288 • C and 10 MPa.

Fig. 6 .

 6 Fig. 6. Creep strain vs. normalized time t/t 2.5% , 1288 • C, and 10 MPa. (a) [001] CMSX-4 crystals from different producers, VIAM, DPC, and HM. The deviations θ from the [001] ideal orientation are given in the graph. (b) Comparison between the γ-alloy and CMSX-4. The solid lines correspond to γ-alloy crystals of different orientations, the dashed line to a [001] CMSX-4 specimen.

  summarize the results for the [001], [011], [123] and [111] specimens tested at the temperatures 1150 • C, 1225 • C, and 1288 • C, and at the same stress of 10 MPa (see the corresponding creep curves in Fig. 5a-c). The experimental results are well approximated by straight lines. The trend lines corresponding to different orientations are vertically shifted due to the different creep resistances but have nearly the same slope. The linear regressions give close Q c values for the different orientations, respectively 442, 430, 440, and 459 kJ/mol for [001], [011], [123], and [111], with an average value of 443 ( 13/+16) kJ/mol. The stress exponent n has been evaluated with the results obtained for the [011] and [123] specimens tested at the same temperature of 1225

Fig. 7 .

 7 Fig. 7. Evaluation of the creep activation energy Q c and the stress exponent n of the γ-alloy by the separated fitting of different orientations. (a) Fitting of Q c for the creep tests at 1150 • C, 1225 • C, 1288 • C and the stress level 10 MPa. (b) Fitting of n for the creep tests at 1225 • C and stresses 6, 10, and 13 MPa.

Fig. 8 .

 8 Fig. 8. Shape of the cross-section of a [011] specimen of the γ-alloy tested at 1288 • C, 10 MPa. The test has been interrupted at the creep strain of 7.3 %. The red dots and the elliptical red curve are respectively the experimental results and the theoretical prediction assuming 〈011〉{111} octahedral slip. (For interpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

Fig. 9 .

 9 Fig. 9. Comparison of the subgrain structure of [001] specimens of γ-alloy before (a, b) and after (c, d) creep at 1288 • C, 10 MPa and 7 % creep strain. EBSD analysis with KAM mapping (a,c) and scattered electrons images (b,d). See text for details.

Fig. 10 .

 10 Fig. 10. Overview of the area near a LAB in a [001] specimen tested at 1288 • C, 10 MPa up to 7 % creep strain. Bright field two-beam TEM micrographs, the diffraction vector is indicated.

Fig. 11 .

 11 Fig. 11. Dislocations inside a subgrain in the necking area of a [001] specimen ruptured during a creep test at 1288 • C and 10 MPa. The overall creep strain is 33 % and the estimated local strain is 55 %. Bright field two-beam TEM micrographs, the diffraction vector is indicated.

Fig. 12 .

 12 Fig. 12. Joint fitting of the creep results considering the measured specimen orientations (θ, ρ). (a) Variation of the coefficient of determination r 2 with n. (b) Comparison between measured and predicted creep rates.

Fig. 13 .

 13 Fig. 13. Stress exponent n measured for superalloys at different temperatures (this work, [7,27]). Vertical and horizontal bars show respectively the scatter of n and the investigated temperature ranges. The dashed line shows the decreasing tendency of n with temperature.

  and b (shown by red arrow), which has a typical slip band configuration. However, some dislocations have a zero last index, a priori indicating the absence of elastic force acting on these dislocations. These dislocations could have formed by reaction between mobile dislocations belonging to different slip systems, e.g., by a/2 [101] + a/2 [ 011 ] a/2 [110]. Such a reaction product is visible near the left subgrain border and marked by a yellow arrow in Fig. 11b. These reactions result in a/2 [110] junctions linking the dislocations and gradually knitting new LAB networks, as seen in Fig. 9d.

Fig. 14 .

 14 Fig. 14. Correlation of the creep activation energy Q c with the content of the key solid solution strengthener (SSS) for different Ni-solid solutions.

Fig. 15 .

 15 Fig.15. Total drag force computed by FE analysis (FEA) compared to the solution[START_REF] Fuentes-Samaniego | Solute drag on moving edge dislocations[END_REF] and linear approximation for the elements Re and W.

Fig. 16 .

 16 Fig. 16. Density of mobile dislocations ρ m,s for one slip system at 1288 • C in a [001] specimen.

K

  cubic contained in the energy per line unit of the straight edge dislocation W L Kcubic b 2 4 π ln ( R r0 ) of the anisotropic material matches that of the isotropic solution, i.e., W L μ iso b 2 4 π (1-νiso) ln ( R r0

3 .drag εmin 8 m 001 b 2 (m

 32 In accordance, ρ m,s is found by solving the equation ρ m,s B 001 σ α μ b ρ m,s √ )

Table

Table 3

 3 Literature data for creep of Ni-solid solutions.

	Alloy	Reference	σ-range,	T-range, • C	n	Qc, kJ/
			MPa			mol
	Ni	[23]	13-59	650-1150	5.0	276
	Ni-10Cr				5.2	276
	Ni-20Cr				4.6	289
	Ni-25Cr				4.9	315
	Ni-30Cr				4.6	307
	Ni-20Cr-14Mo	[24]	49-176	800-900	5.2	415
	Ni-2W	[25]	21-48	852-1005	4.8 ±	298 ± 8
	Ni-4W			910-1005	0.2	311 ±
						12
	Ni-6W			854-1046		317 ± 8
	γ-alloy	This work	6-13	1150-1288	4.95	442
	Average				4.91 ±
					0.18

  • The absence of the γ ′ -reprecipitation during cooling after hightemperature creep tests facilitates microstructural investigations. It has been shown by EBSD that the creep deformation leads to an increase in the misorientations of the existing LABs. According to TEM observations, new LABs also form, which are due to reactions between the mobile a/2 〈011〉 dislocations, leading to low energy configurations after rearrangement.

	CRediT authorship contribution statement

A.I. Epishin: Conceptualization, Project administration, Writingoriginal draft. B. Fedelich: Conceptualization, Formal analysis, Writing original draft. B. Viguier: Conceptualization, Investigation, Writingoriginal draft. S. Schriever: Investigation. I.L. Svetlov: Resources. N.V. Petrushin: Resources, Investigation. R. Saillard: Investigation. A. Proietti: Investigation. D. Poquillon: Investigation. A. Chyrkin: Formal analysis.

Acknowledgements

This work was supported by the German Research Foundation DFG (projects Re 688/77-1 and FE 933/2-1), the French National Research Agency ANR (project ANR15-MERA-0003-04) and the Russian Foundation of Basic research (grant no. 18-508-12010). The authors also thank Prof. Robert Vaβen from the Forschungszentrum Jülich for help with protective coating of creep specimens.

Data availability statement

The data that support the findings of this study are available on request from the corresponding author (BF).

κ iso κ cubic κ, leading to the second equation ν iso (

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.