domain: sdv info

text:
M.-O J Nozieres

Pierre Dulphy

J.-L Peyraud

René Baumont

Prediction of forage rumen protein degradability from an in situ degradability database

Forage protein value evaluation relies on the estimation of forage protein degradability in the rumen (deg).

Deg is affected by botanical family, plant species, growth cycle and level of nitrogen fertilization, and decreases with plant maturity stage.

 To develop equations for predicting forage deg values in order to improve the estimation of forage protein values expressed in PDI.

-Results

1-Objectives

Prediction of forage rumen protein degradability from an in situ degradability database Marie-Odile Nozières 1 , Jean-Pierre Dulphy 1 , Jean-Louis Peyraud 2 , René Baumont

-Conclusion

These equations have satisfactory accuracy and will be used to improve estimation for herbage from European temperate climate areas in the INRA PDI system.

• Fresh forages

• Conserved forages

Mean values in the data base for fresh forages • Example for natural grasslands -1 st growth deg hay deg silage


J Martin

A Martinavicius

S Bruyère

H P Van Landeghem

C Gendarme

F Danoix

R Danoix

A Redjaimia

T Grosdidier

T Czerwiec

Multiscale analysis of an ODS FeAl40 intermetallic after plasma-assisted nitriding

Keywords: Intermetallic, Fe-Al alloy, Plasma nitriding, discontinuous precipitation, Transmission electron microscopy, Atom probe tomography

The binary B2 (ordered bcc-type structure) FeAl40 Grade 3 aluminide containing 40 at.% Al was nitrided at 600 °C for 15 min with a pulsed DC plasma in a 95% N 2 / 5% H 2 gas mixture. The nitrided layer was described at the micro-, nano-and atomic scales. The nitrided layer consists of a thin outer sublayer (1 µm in thickness) containing '-Fe 4 N and having a moderate hardness (870 Hv) below which a thicker (36 µm in thickness) and harder (1400 Hv) inner sublayer is formed. While the outer sublayer is completely depleted in Al, the inner sublayer is made of a a mixture of hex.

AlN and -Fe phases regularly structured at different levels: i) At the micrometre scale, observations showed a network of -Fe "wave-like" veins (2 µm in length and 100 nm in width) located at the grain boundaries and preferentially aligned parallel to the nitrided surface. ii) At the nanometre scale, observations evidenced that such veins are embedded in a "lamellar-like" morphology matrix alternately composed of -Fe and -Fe/hex. AlN lamellae (300 nm in length and 10 nm in width). iii) At the atomic scale, analysis pointed out that each -Fe / hex. AlN lamella presents a finer structural arrangement in which hex. AlN-rich clusters (4 nm in diameter) are trapped between thin -Fe walls (2 nm in thickness). The diversity of the morphologies evidenced in the present study highlights the complexity of the nitriding mechanisms of iron-based aluminides. The observations support however a mechanism of discontinuous precipitation in the inner nitrided sublayer which seems to be strongly promoted by the high amount of Al in the FeAl40 grade 3 alloy (40 at.%).

Introduction

Binary B2 iron aluminides (Fe-Al) are attractive intermetallic compounds as lightweight and structural materials [START_REF] Baker | A review of the mechanical properties of B2 compounds[END_REF]. However, they are rather brittle at room temperature and offer poor strength and creep resistance at high temperature. Efforts have been devoted to improve these properties for applications in the ambient to 600 °C temperature range [START_REF] Liu | An environmental effect as the major cause for room-temperature embrittlement in FeAl[END_REF]. A significant progress has consisted in developing an oxide dispersed strengthened (ODS) creep resistant version of the FeAl intermetallic, the FeAl Grade 3 alloy, from a powder metallurgy route involving ball milling followed by hot extrusion at 1100 °C [START_REF] Moret | Properties and applications of B2-FeAl intermetallic alloys[END_REF]. From micro-alloying with Zr and B combined with the fine dispersion of the nanometre size Y 2 O 3 resulting in a grain size reduction, the grain boundary strength was also optimised [START_REF] Moret | Properties and applications of B2-FeAl intermetallic alloys[END_REF][START_REF] Munoz-Morris | An analysis of strengthening mechanisms in a mechanically alloyed, oxide dispersion strengthened iron aluminide intermetallic[END_REF] while the oxidation resistance was improved [START_REF] Montealegre | Influence of the yttria content on the oxidation behaviour of the intermetallic Fe40Al alloy[END_REF][START_REF] Berlanga | Oxidation behaviour of yttria dispersion strengthened Fe40Al alloy foils[END_REF]. Other processing powder metallurgy routes including consolidation by forging, rolling or swaging [START_REF] Munoz-Morris | Microstructure and room temperature strength of Fe-40Al containing nanocrystalline oxide particles[END_REF], spark plasma sintering [START_REF] Ji | Bulk FeAl nanostructured materials obtained by spray forming and spark plasma sintering[END_REF][START_REF] Skiba | Mechanical properties of spark plasma sintered Fe-Al intermetallics[END_REF], thermal spraying [START_REF] Grosdidier | Hardness, thermal stability and yttrium distribution in nanostructured deposits obtained by thermal spraying from milled -Y2O3 reinforced -or atomized FeAl powders[END_REF][START_REF] Musalek | Fatigue properties of Fe-Al intermetallic coatings prepared by plasma spraying[END_REF] and even additive manufacturing [START_REF] Rolink | Laser metal deposition and selective laser melting of Fe-28 at% Al[END_REF] have been attempted. The development of this new generation of Al-rich B2 FeAl alloys has raised some hopes for promising applications as a substitution material of steels and superalloys for the fabrication of transmission shafts, brakes or valves for the aerospace and automotive industries as well as protective coatings for heat exchangers or piping [START_REF] Moret | Properties and applications of B2-FeAl intermetallic alloys[END_REF][START_REF]Industrial Applications of FeA40Grade3[END_REF].

Although major progresses have been made in improving the mechanical properties of these B2 iron aluminides and despite some additional attempts to play with the microstructure for improving ductility [START_REF] Grosdidier | Processing dense hetero-nanostructured metallic materials by spark plasma sintering[END_REF] and/or texture for improving stifness [START_REF] Grosdidier | Primary recrystallization in an ODS FeAl alloy: an effective way to modify texture and microstructure[END_REF][START_REF] Guessasma | Static recrystallisation in an ODS FeAl alloy: The effect of particles on texture and anisotropic grain growth developments[END_REF][17], these alloys often still suffer from a major lack of ductility due to (i) a wellknown moisture induced environmental embrittlement [START_REF] Stoloff | Review: Environmental embrittlement of iron aluminides[END_REF][START_REF] Liu | Recent advances in B2 iron aluminied alloys : deformation, fracture and alloy design[END_REF], a high temperature brittle-to-ductile transition temperauture (BDTT) at high (41-45 at%) Al contents [START_REF] Risanti | Dependence of the brittle-to-ductile transition temperature (BDTT) on the Al content of Fe-Al alloys[END_REF] or a lack of cohesion at the former inter-particle boundaries in powder processed materials [START_REF] Ji | Processing conditions, microstructure and mechanical properties of hetero-nanostructured ODS FeAl alloys produced by spark plasma sintering[END_REF]. In addition, the wear resistance of iron aluminides remain poor due to the insufficient surface hardness [START_REF] Maupin | Deformation of ordered Fe-Al intermetallic alloys[END_REF][START_REF] Qiu | The effects of stoichiometry on the dry sliding wear of FeAl[END_REF]. This additional handicap could slow down their future applications in which high loads and strong surface contacts are applied to the FeAl structural material. Thus, several studies have proposed surface engineering approaches such as thermal oxidation [START_REF] Liu | Recent advances in B2 iron aluminied alloys : deformation, fracture and alloy design[END_REF][START_REF] Xia | Thermal oxidation treatment of B2 iron aluminide for improved wear resistance[END_REF][START_REF] Nowak | The effect of pre-oxidation treatment on the plasticity behavior of B2 iron aluminide[END_REF], boronizing [START_REF] Ekmekçiler | Hard boride coating on iron aluminide (FeAl)[END_REF] and nitriding treatments (gaseous atmosphere, ion implantation, plasma-assisted) [START_REF] Pedraza | Nitridation effects on the oxidation mechanisms of an ods Fe-Al intermetallic alloy[END_REF][START_REF] Spies | Nitriding behaviour of the intermetallic alloy FeAl[END_REF][START_REF] Zhang | Plasma-nitriding and characterization of FeAl40 iron aluminide[END_REF] to improve the surface hardness. At the same time, as was proved by mechanical tests on preoxidized samples of B2 iron aluminides [START_REF] Liu | Recent advances in B2 iron aluminied alloys : deformation, fracture and alloy design[END_REF][START_REF] Nowak | The effect of pre-oxidation treatment on the plasticity behavior of B2 iron aluminide[END_REF], these techniques should provide a physical protecting layer to avoid the moister-induced embrittlement originating from the interaction, at the crack tip, of the aluminium atoms with the hydrogen contained in the environment. Among all the previously mentioned surface treatment processes, it has been shown that the nitriding treatments are the most promising for hardening and that they lead to thick and compact protective layers. In order to better understand the response of the B2 FeAl alloy to nitriding, particularly those containing large amounts of Al, few pioneer studies have been conducted to gain a deeper scientific understanding of the mechanisms involved during the growth of the nitride layers [START_REF] Pedraza | Nitridation effects on the oxidation mechanisms of an ods Fe-Al intermetallic alloy[END_REF][START_REF] Spies | Nitriding behaviour of the intermetallic alloy FeAl[END_REF][START_REF] Zhang | Plasma-nitriding and characterization of FeAl40 iron aluminide[END_REF]. Owing to the complexity of the nitride microstructures in the B2 alloys, it is likely that the possible proposed mechanisms for nitriding remain still debateable mainly due to the fact that the nitrided FeAl microstructure has not yet been observed and described at very local scales. To this end, the present paper reports a detailed multi-scale description of the ODS FeAl (40 at. % Al), the FeAl40 grade 3 model alloy, nitrided at 600 °C for 15 min within a pulsed d.c. plasma nitrogen discharge. Complementary analytical techniques, such as scanning electron microscopy (SEM), electron probe micro-analysis (EPMA), grazing-incidence X-ray diffraction (GIXRD), transmission electron microscopy (TEM) and atomic probe tomography (APT), were used to depict the chemical and morphological description of the nitrided layer at various scales, from the micrometre to the atomic scale. In addition, micro-hardness measurements were carried out to ensure that nitriding was successfully applied and to confirm the duplex nature of the nitride layer that was, in fact, depicted as two successive layers through the nitride depth.

Material and methods

The investigated material is an ODS aluminide intermetallic alloy manufactured using powder metallurgy route under the reference FeAl40 Grade 3 (Fe-40 at.% Al) [START_REF] Moret | Properties and applications of B2-FeAl intermetallic alloys[END_REF].

Plasma nitriding was performed at 600 °C for 15 min using a 800 Hz pulsed d.c. discharge with a 0.8 duty cycle ratio [START_REF] Czerwiec | Low pressure, high density plasma nitriding : mechanisms, technology and results[END_REF]. The plasma was ignited in a 95% N 2 -5% H 2 gas mixture under a 180 Pa pressure.

Mechanical properties of the nitrided layer were evaluated with hardness measurements using a Duramin micro-indenter (Struers). In-depth micro-hardness profiles were performed on cross-section using a constant load (100 g). Complementary to these cross-section measurements, the topmost surface hardness of the nitrided layer was also investigated by indenting the surface with increasing loads (50, 100, 300, 500 and 1000 g) and the recorded hardness were related to an equivalent penetration depth [START_REF] Kang | Conventional Vickers and true instrumented indentation hardness determined by instrumented indentation tests[END_REF][START_REF] Samih | Microstructure modifications and associated hardness and corrosion improvements in the AISI 420 martensitic stainless steel treated by high current pulsed electron beam (HCPEB)[END_REF]. In this way, from top surface hardness measurement, it was possible to estimate, by extrapolation, the hardness of the very thin layer that formed at the top surface of the treated sample.

At the micrometre scale, the morphology of the nitrided layer was observed using a scanning electron microscope in back-scattered electron imaging mode (SEM -BSE mode) while the elemental composition was quantitatively measured using a electron probe micro-analyzer (EPMA). SEM observations were performed on a Philips XL30 S-FEG microscope operating in the backscattered electron imaging mode at an acceleration voltage of 25 kV. The EPMA measurements were carried out on a JEOL JXA 8530-F electron probe micro-analyser with an acceleration voltage at 15 kV and a probe current at 30 nA. Depth profiles were carried out to determine the elemental distribution of nitrogen (N), iron (Fe) and aluminium (Al) with depth. Profiles were performed directly on the cross-section of the sample, from the topmost surface of the nitrided layer to the substrate material with a step size of 1 µm. The analyser spectrometers were calibrated with standards before measurements: an Al 2 O 3 standard was used to quantify the amount of Al while Fe and N were quantified using a '-Fe 4 N standard. The conventional ZAF matrix correction procedure was used to calculate the weigt and atomic compositions from the measured X-ray intensities [START_REF] Pouchou | Surface film X-ray microanalysis[END_REF]. The crystalline phase identification of the nitrided layer was carried out on a INEL CPS 120 diffractometer using a grazing incidence X-ray diffraction (GIXRD) geometry. The GIXRD measurements were conducted at various incidence angles ( = 6° and 3°) using a Co-K  incident radiation ( = 0.17903 nm). GIXRD patterns were recorded in the 2θ = 30° -115° angular range, with a step size of 0.027° and 25 s per step.

At the micro-and nanometre scale, transmission electron microscopy (TEM) and atom probe tomography (APT) were conducted. Specimens for TEM and APT were prepared within the nitrided layer using a focused ion beam (FIB) microscope (Zeiss® NVision-40). For TEM, the foil was extracted from a mirror polished cross section of the nitrided specimen. It was prepared perpendicular to the specimen surface, at the midpoint between the surface and the bulk. Specimens for APT were lifted-out from the same cross-section, but parallel to the sample surface, at a depth of 3 ± 0.1 µm. These specimens were thinned to their final dimensions (< 70 nm in thickness for the TEM foil, and to a final radius of curvature of about 50 nm for the APT tip) using 30 keV Ga + ions [START_REF] Giannuzzi | Introduction to Focused Ion Beams, Instrumentation, Theory, Techniques and Practice[END_REF][START_REF] Larson | Local Electrode Atom Probe Tomography[END_REF]. A final milling step with 2 keV Ga + ions was applied to reduce the ion damage on the TEM foil. TEM observations were carried out on a JEOL ARM 200cold FEG UHR transmission electron microscope operating at 200 kV and equipped with a Cs corrector. In scanning mode (STEM), EDS analyses (EDS elemental maps) were obtained using a DRY SD30VG energy-dispersive X-ray spectrometer with a probe size equal to 7 nm and a dwell time equal to 5 ms. The APT analyses were conducted on a CAMECA-LEAP 4000 HR, with a base temperature of 80 K, a pulse repetition rate of 200 kHz and an average detection rate of 0.1%. Several peak overlaps in APT mass spectra were encountered, and treated as follows: the common isobar between 54 Cr ++ and 54 Fe ++ was assigned to Fe, because of very low Cr concentration, and the possible overlap between 27 Al + and 54 Fe ++ was neglected in favor of Fe, as the presence of 27 Al + is marginal in Fe based materials. In addition, peaks at 14, 14.5 and 15 Da were assigned to N + and N 2 ++ ions. The presence of N 2 ++ , manifested by the ( 14 N-15 N) ++ peak at 14.5 Da, signifies that the peak at 14 Da is also composed of ( 14 N-14 N) ++ in addition to 14 N + ions. From the natural abundance of N isotopes, according to the procedure described in [START_REF] Bachhav | Investigation of O-18 enriched hematite (alpha-Fe2O3) by laser assisted atom probe tomography[END_REF], 38 % of counts in the main peak at 14 Da are due to ( 14 N-14 N) ++ ions. This correction was applied for the quantitative data in table 1, but not on the illustrating concentration profiles provided in Fig. 7c. Also, the standard deviations indicated on concentration profiles are calculated from detection statistics [START_REF] Danoix | Standard deviations of composition measurements in atom probe analyses-Part II: 3D atom probe[END_REF], but do not include the uncertainty due to the presence of N 2 ++ ions.

Results and discussion

The SEM micrograph in Fig. 1a reveals two distinct sublayers throughout the nitrided layer depth: a 1-2 µm-thick outer sublayer on the topmost surface (see close-up micrograph in Fig. 1c) and a 34 -36 µm-thick inner sublayer directly in contact with the substrate. The first outer sublayer is thin and appears quite homogeneous within its thickness. The second inner sublayer is thick and presents two specific features which will be the subject of a close examinations in the present study: i) a network of light contrasted "wave-like" veins of 1 -3 µm in length that mainly follow the grain boundaries and are preferentially extended parallel to the surface (Fig. 1b) and ii) some light contrasted segregated "nodular-like" domains of 500 nm -1 µm in diameter located in the inner nitrided layer (marked in Fig. 1a and1c). It is also worth noting that the BSE imaging mode used here being strongly related to the atomic number of the probed elements, suggests that the brighter areas observed in the Fig. 1a-c (the thin outer sublayer, the "wave-like" veins and the "nodular-like" regions) are much richer in Fe than in Al.

Micro-hardness results carried out on the nitrided layer are given in Fig. 2a-b.

As depicted in the experimental part, two hardness measurement protocols were used: a first which consisted in performing indentations on the sample surface with increasing loads (50, 100, 300, 500 and 1000 g) (Fig. 2a) and a second which consisted in carring out an indentation line scan-profile on the sample cross-section at constant load (100 g) (Fig. 2b). The Fig. 2a highlights differences in the hardness between the topmost nitrided surface (about 870 Hv evaluated from the lowest indentation load of 50 g at the minimum penetration depth of 1.4 µm) and the remaining nitrided layer (about 1390 Hv evaluated with the highest indentation load of 1000 g at the maximum penetration depth of 5.2 µm). Such differences are caused by the presence of the thin outer sublayer (1-2 µm in thickness) which exhibits lower mechanical properties and surely different microstructural characteristics compared to the inner nitrided sublayer. This point will be further explored within the next section. The micro-hardness line scan-profile given in Fig. 2b evidences the strong surface hardening of the FeAl40 Grade 3 iron aluminide when subjected to the nitriding treatment. While the hardness of the base material is about 410 Hv, the hardness of the inner nitrided layer is about 1400 Hv. It is also worth noting that a good agreement is observed between the two protocols of the microhardness measurement that were applied since the hardness value measured on the top surface using the maxium load (Hv 1000g = 1390 in Fig. 2a) is very close to the one measured within the inner layer (Hv 100g =1400 in Fig. 2b).

EPMA-depth profiles recorded along the cross-section of the nitrided layer are given in Fig. 3. Concerning the outer sublayer, the EPMA analysis performed at 1 µmdepth show that the nitrided layer consists mainly in Fe and N and is totally depleted in Al. The strong presence of Fe and the absence of Al in the outer sublayer are consistent with the image contrast of the previous SEM observations carried out under the BSE imaging mode (Fig. 1). Moreover, the atomic ratio N / Fe is about 0.24 which is very close to the '-Fe 4 N iron nitride composition. This is confirmed with the XRD patterns given in Fig. 4 which were recorded in grazing-incidence geometry on the nitrided surface. Indeed, the increase in the relative intensity of the '-Fe 4 N peaks with the decrease in the incidence -angle (from 6° to 3°) of the X-ray beam evidences that '-Fe 4 N is preferentially located at the topmost surface of the nitrided material. The presence of the '-Fe 4 N iron nitride in the thin outer sublayer explains the lower value in the hardness (870 Hv) measured at the topmost surface with the minimum indentation load (50 g). Interestingly, the extrapolation of the hardness data in Fig. 2a gives a hardness range between 500 and 800 Hv for this outer sublayer which is rather consistent with the values reported in the literature for the '-Fe 4 N iron nitride [START_REF] Granito | Wear behavior on the nitride steels[END_REF][START_REF] Díaz-Guillén | Surface properties of Fe4N compounds layer on AISI 4340 steel modified by pulsed plasma nitriding[END_REF].

In their pioneer work on the nitriding of a Fe-40 at.% Al alloy, the presence of the '-Fe 4 N on the processed surface was also observed by Spies et al. [START_REF] Spies | Nitriding behaviour of the intermetallic alloy FeAl[END_REF]. Using a sufficiently high temperature to enable Fe atoms mobility (> 550 °C) and a sufficiently high nitriding potential K N (>1) to control the nitrogen supply at the surface, they propose that the diffusion of Fe atoms towards the nitrogen located at the surface is the main mechanism for the growth of such iron nitride layer at the topmost surface. The nitriding conditions used in the present study are close to those recommended by Spies et al. [START_REF] Spies | Nitriding behaviour of the intermetallic alloy FeAl[END_REF] since the nitriding temperature was set at 600 °C and the pulsed dc-plasma nitriding process ensures a high nitrogen supply at the processed surface [START_REF] Czerwiec | Low pressure, high density plasma nitriding : mechanisms, technology and results[END_REF]. More recently, the unusual formation of '-Fe 4 N at the nitrided surface was also reported by Meka et al. [START_REF] Meka | Unusual nucleation and growth of ' iron nitride upon nitriding Fe-4.75 at.% Al alloy[END_REF] surface (less than 2 µm). In addition, it is not easy to explain where the aluminium atoms which are rejected from the '-Fe 4 N superficial layer are going since no accumulation of Al is observed at the outer / inner sublayer interface. Finally, another possible explanation which could explain the absence of Al in the outer sublayer consists in considering the selective evaporation of this element due to a local higher temperature at the topmost surface of the processed sample compared to the bulk material (where the temperature probe is located). Indeed, it is quite possible that some "edge" effects take place at the extreme surface of the processed sample mainly due to the ionic bombardment of the surface and the local distribution of the electric field at the vicinity of the cylindrical processed sample. These "edge" effects are known to promote a slight increase of the temperature locally at the topmost surface. This last explanation should be the subject of further investigations.

Concerning the inner nitrided sublayer, the EPMA-depth profile given in Fig. 3 shows the presence of both the Fe, Al and N elements. A weak composition gradient is observed from the surface to the substrate (36 to 40 at.% Fe, 37 to 33 at.% N and 26 to 24 at.% Al). As discussed previously, and taking into account that '-Fe 4 N is associated to the outer sublayer, the GIXRD patterns given in Fig. 4 tend to indicate that the inner sublayer consists in both the hex. AlN aluminium nitride and the -Fe ferrite. As the atomic ratio N / Al is higher than 1, an excess in nitrogen is reported into the inner sublayer which contains more nitrogen than necessary for precipitation of all Al atoms as AlN together with the equilibrium saturation with nitrogen in the -Fe solid solution.

Although the storage of this amount of excess nitrogen is still the subject of discussion, a widespread explanation in the literature considers that the nitrogen atoms in excess could be trapped at dislocations, phase interfaces and within the strained -Fe ferrite lattice [START_REF] Meka | Unusual nucleation and growth of ' iron nitride upon nitriding Fe-4.75 at.% Al alloy[END_REF][START_REF] Van Wiggen | The nitriding behaviourof iron-chromium-carbon alloys[END_REF][START_REF] Somers | Formation and Growth of Compound Layer on Nitrocarburizing Iron: Kinetics and Microstructural Evolution Surf[END_REF][START_REF] Schwarz | Pore formation upon nitriding iron and iron-based alloys: the role of alloying elements and grain boundaries[END_REF][START_REF] Schacherl | Gaseous nitriding of ironchromium alloys[END_REF][START_REF] Schacherl | The nitriding kinetics of ironchromium alloys; the role of excess nitrogen: Experiments and modelling[END_REF][START_REF] Hosmani | Nitriding behavior of Fe4wt%V and Fe-2wt%V alloys[END_REF][START_REF] Biglari | Analysis of the nitrogen absorption isotherms of cold-rolled Fe-2 at. % Al specimens with different AlN precipitate dimensions[END_REF][START_REF] Biglari | Crystal structure and morphology of AIN precipitating on nitriding of an Fe-2at.% Al alloy Phil[END_REF][START_REF] Meka | Mittemeijer The emergence and disappearance of a high density of microcracks in nitrided Fe-4.65 at.% Al alloy[END_REF][START_REF] Meka | Interrelationships of defects, nitride modification and excess nitrogen in nitrided Fe-4.75 at.% Al alloy[END_REF][START_REF] Hosmani | The nitrogenabsorption isotherm for Fe-21.5 at.% Cr alloy: dependence of excess nitrogen uptake on precipitation morphology Phil[END_REF]. As will be discussed hereafter, it is also possible that some excess N atoms form N 2 molecules that coalesce in residual porosity.

After crossing the nitrided layer / metal substrate interface, the N concentration drops sharply to zero while the Fe and Al amounts increase to reach the base composition of the FeAl40 Grade 3. Such a sharp decrease in the nitrogen content indicates that there is no diffusion zone or chemical gradient in the base substrate.

Consequently, this suggests that the interaction of Al atoms in the B2 bcc-ordered matrix and the diffusing nitrogen is of a strong character, which is to known to give rise to domains separated with a sharp diffusion boundary [START_REF] Spies | Nitriding behaviour of the intermetallic alloy FeAl[END_REF]. The combination of the Al and the N atoms to form the hex. AlN aluminium nitrides results in the decomposition of the ordered B2 FeAl base material structure and the appearance of the -Fe ferrite structure as indexed on the GIXRD patterns provided in Fig. 4.

At a sub-micrometre scale, the TEM micrographs given in Fig. 5a show that the "wave-like" veins of about 100 nm in width are interconnected and join the segregated "nodular-like" domains. The STEM / EDS elemental maps shown in Fig. 5b and the quantitative elemental analysis reported in Table 1 evidence that the "wave-like" veins and the "nodular-like" domains are exclusively composed of Fe atoms which crystallized into the -Fe ferrite bcc-structure consistently with the previous GIXRD phase identification (Fig. 4). Without having determine the crystallographic structure and the elemental composition of such "wave-like" veins, Spies et al. [START_REF] Spies | Nitriding behaviour of the intermetallic alloy FeAl[END_REF] were the first to report their presence within the nitrided layer of a Fe-40at.% Al iron aluminide alloy.

Recently, Zhang et al. [START_REF] Zhang | Plasma-nitriding and characterization of FeAl40 iron aluminide[END_REF] have shown that these veins are rich in Fe and they proposed a mechanism for their formation. Taking into account both the diffusion of N through the original FeAl grain boundaries and the strong interaction character between Al and N to form the aluminium nitride, they suggested that the N atoms preferentially located along the grain boundaries attract the Al atoms leaving behind the centre of the nitrided grains depleted in Al and enriched in Fe. Assuming this explanation, the Fe-rich "wavelike" veins should be located intragranularly and not at the grain boundaries. This is clearly not the case since SEM observations such as the micrographs given in Fig. 1b clearly evidence that: i) these -Fe "wave-like" veins are mainly located at the grain boundaries and ii) they are preferentially aligned parallel to the nitrided surface.

Looking at the literature corresponding to other systems, the presence of similar intergranular veins was also observed in nitrided chromium-alloyed steels. Van Wiggen et al. [START_REF] Van Wiggen | The nitriding behaviourof iron-chromium-carbon alloys[END_REF] were the first to report the presence of Fe 3 C cementite at the grain boundaries of the steel matrix. With prolonged nitriding time, they showed that the percolation of these cementite domains results in veins preferentially aligned with respect to the nitrided surface. Shortly after, using quite the same type of alloyed steel and applying quite the same nitriding conditions, Barrallier et al. [START_REF] Barrallier | Morphology of intergranular cementite arrays in nitride chromium-alloyed steels[END_REF] and Traskine et al. [START_REF] Traskine | Physicochemical mechanics of structural transformations in nitride steel[END_REF] proposed a mechanism of formation of these oriented carbides. They established that the probability of the Fe 3 C cementite precipitation is strongly dependent on the orientation of the grain boundary with respect to the local stress axis generated in the nitrided layer during the nitriding process. Specifically, they showed that the nucleation of the Fe 3 C

precipitates is promoted for the grain boundaries which are oriented parallel to the nitrided surface because they correspond to the macroscopic plane of the maximum compressive stress. The TEM micrographs in Fig. 5a and the STEM / EDS elemental maps in Fig. 5b show the presence of porosities of about 200 nm in diameter (appearing in dark under the dark field imaging in Fig. 5b) which are precisely located within the Fe-rich "nodular-like" region. In the case of nitriding of iron-based metallic alloys, and to the best of our knowledges, few authors have reported and discussed the formation of pores in nitrided layers [START_REF] Van Wiggen | The nitriding behaviourof iron-chromium-carbon alloys[END_REF][START_REF] Somers | Formation and Growth of Compound Layer on Nitrocarburizing Iron: Kinetics and Microstructural Evolution Surf[END_REF][START_REF] Schwarz | Pore formation upon nitriding iron and iron-based alloys: the role of alloying elements and grain boundaries[END_REF]. They proposed that the formation of pores proceeds by the local recombination of N atoms initially dissolved in the -Fe ferrite phase, to form N 2 molecules which coalesce and produce voids containing nitrogen gas. The fact that these pores are located within the Fe-rich "nodular-like" regions, often at the intersection of several -Fe veins (Fig. 5a-b), suggest that this mechanism can also be valid for the nitriding of the investigated FeAl40 Grade3 alloy.

At the tens of nanometre scale, the TEM micrograph in Fig. 6a shows the presence of a "lamellar-like" morphology which fills the space between the -Fe "wave-like" veins. The TEM micrographs also reveal the occurrence of many differently oriented colonies of lamellae which start from the original FeAl grain boundaries. The lamellae exhibit a length of about 300 nm on average (Fig. 6a) and a width of 10 nm (Fig 6c). The STEM /EDS elemental maps given in Fig. 6b show that Fe-rich lamellae alternate periodically with Fe-lean lamellae, the latter being mainly composed of Al and N and in lower amount of Fe. This results was confirmed with the APT elemental quantification performed locally on each type of lamella and reported in Table 1. The selected area electron diffraction (SAED) pattern recorded on a colony of lamellae (Fig. 6a) reveals the presence of both the hex. AlN aluminium nitride and the -Fe ferrite while the Fourier fast transform (FFT) pattern performed locally on a single Fe-rich lamella shows the exclusive presence of the -Fe ferrite (Fig. 6c).

Consequently, it could be deduced that the observed lamellar morphology consists in -Fe lamellae which alternate periodically with AlN / -Fe lamellae. The growth of such a "lamellar-like" morphology has been widely reported in the literature dealing with the nitriding of iron-based alloys, for example in the binary Fe-Cr and Fe-V alloys and the ternary Fe-Cr-C alloys [START_REF] Schacherl | Gaseous nitriding of ironchromium alloys[END_REF][START_REF] Schacherl | The nitriding kinetics of ironchromium alloys; the role of excess nitrogen: Experiments and modelling[END_REF][START_REF] Hosmani | Nitriding behavior of Fe4wt%V and Fe-2wt%V alloys[END_REF][START_REF] Hosmani | The nitrogenabsorption isotherm for Fe-21.5 at.% Cr alloy: dependence of excess nitrogen uptake on precipitation morphology Phil[END_REF]. The similarity between the lamellar morphology observed in the nitrided Fe-1wt.% Cr alloys forming CrN / -Fe lamellae or in the nitrided Fe-2 wt.% V alloys forming VN / -Fe lamellae and the lamellar morphology observed in the present nitrided material strongly suggests that the nitriding of the FeAl40 Grade 3 alloy proceeds also through a mechanism of discontinuous precipitation. With the exception of Zhang et al. [START_REF] Zhang | Plasma-nitriding and characterization of FeAl40 iron aluminide[END_REF], who were recently the first to observe this lamellar morphology in the FeAl40 Grade 3 alloy but without mentioning the possibility of a discontinuous precipitation mechanism, the growth of this specific lamellar morphology has never been reported in the literature for the nitriding of the Fe-Al binary alloys [START_REF] Biglari | Analysis of the nitrogen absorption isotherms of cold-rolled Fe-2 at. % Al specimens with different AlN precipitate dimensions[END_REF][START_REF] Biglari | Crystal structure and morphology of AIN precipitating on nitriding of an Fe-2at.% Al alloy Phil[END_REF][START_REF] Meka | Mittemeijer The emergence and disappearance of a high density of microcracks in nitrided Fe-4.65 at.% Al alloy[END_REF][START_REF] Meka | Interrelationships of defects, nitride modification and excess nitrogen in nitrided Fe-4.75 at.% Al alloy[END_REF]. This could be explained by the fact that most of the referenced studies remains focused on the nitriding of Fe-Al alloys with low amount of Al (a maximum of 4.75 at.% Al for Ref. [START_REF] Meka | Interrelationships of defects, nitride modification and excess nitrogen in nitrided Fe-4.75 at.% Al alloy[END_REF]) for which the precipitation proceeds only according to a "continuous" mechanism. In comparison, in the case of iron-based alloys (Fe-Cr, Fe-V and Fe-Cr-C alloys), it has been established that the concentration in the alloying elements is the main factor that influences the amount of discontinuously transformed areas in the nitrided layer [START_REF] Schacherl | The nitriding kinetics of ironchromium alloys; the role of excess nitrogen: Experiments and modelling[END_REF][START_REF] Hosmani | Nitriding behavior of Fe4wt%V and Fe-2wt%V alloys[END_REF][START_REF] Hosmani | The nitrogenabsorption isotherm for Fe-21.5 at.% Cr alloy: dependence of excess nitrogen uptake on precipitation morphology Phil[END_REF]. The larger the concentration in Cr or V alloying elements in the alloy, the larger the amount of discontinuous precipitation taking place in the ferritic matrix. For example, it has been demonstrated that below 1wt.% of Cr, the nitrided Fe-Cr alloy shows only a "continuous" precipitation morphology with dispersed coherent submicroscopic CrN "platelet-like" nitrides.

Comparatively, in the range of 1-13 wt.% of Cr, the "continuous" precipitation mechanism is concomitant with a "discontinuous" precipitation of CrN / -Fe lamellae, the latter occurring only in few grains of the nitrided layer. Finally, with amounts in Cr above 13 wt.%, the nitriding process involves only a "discontinuous" precipitation for which the resulting CrN / -Fe lamellar morphology covers the whole nitrided layer.In our case, these literature results suggest that the "discontinuous" precipitation is the main mechanism involved for the nitriding of the FeAl40 Grade 3 because of its high amount of Al.

At the nanometre and atomic scales, the APT analyses presented in Fig. 7a-b and reported in Table 1 were carried out in a probed volume speading over a Fe-rich lamella and a Fe-lean lamella. Within the Fe-lean lamella, a finer structural arrangement is observed in which Fe-lean clusters are separated by Fe-rich walls. The detailed spatial distribution of Al, Fe and N species through a 3 nm thick cross-section described by the rectangle in solid lines in Fig. 7a supports the same description. This is shown in Fig. 7b where a Fe-rich region located on the left hand side corresponds actually to the -Fe lamellae. On the right hand side, within the hex. AlN / -Fe lamellae, the Al and N species are localized in Fe-lean clusters (6 nm in thickness) which are trapped between two successive thin Fe-rich walls (2 nm in thickness) (Fig. 7c). Combined with the high resolution TEM results given in Fig. 6, it is reasonable to expect that these Fe-rich walls are composed of -Fe phase while the Fe-lean clusters probably consists in a hex.

AlN / -Fe mixture structured at the nanometric scale.

Conclusion

A binary B2 bcc-ordered FeAl40 Grade 3 iron aluminide was nitrided at 600 °C for 15 min with a pulsed DC plasma in a 95% N 2 / 5% H 2 gas mixture. The microstructure of the nitrided layer was fully described at the micro-, nano-and atomic scales in order to  An inner sublayer which is thicker (36 µm in thickness) and harder (1400 Hv).

Within this inner sublayer, the strong interaction character between the Al atoms with the diffusing N atoms results in the formation of the hex. AlN aluminium nitride and the decomposition of the B2 bcc-ordered structure (fully depleted in Al) into the -Fe ferritic structure. An excess in nitrogen was revealed in this sublayer. In addition to possible N trapping at dislocations, interfaces or within the strained lattice the presence of porosity was revealed which was attributed to recombination of N atoms, initially present in the -Fe ferrite phase, to form N 2 gas molecules and porosity exclusively located at the intersection of -Fe veins.

The microstructure within the inner sublayer was rather complex and revealed a number of interesting additional features at different scales of analysis:

 The -Fe phase formed during the nitriding process followed an intergranular network of -Fe "wave-like" veins (1-3 µm in length, 100 nm in width) that were essentially oriented along the grain boundaries parallel to the nitrided surface. It is suggested that they form specifically at these boundaries because they experience the maximum compressive stress.

 When submitted to nitriding, the FeAl40 Grade 3 containing high amount of Al (40 at.% Al) is subjected to a mechanism of "discontinuous" precipitation. This mechanism leads to the growth of a "lamellar-like" microstructure which starts from the grain boundaries and fills the space between the -Fe "wave-like" veins. The lamellar morphology is composed of -Fe lamellae which alternate periodically with AlN/-Fe lamellae (300 nm in length, 10 nm in thickness). 5b. 2 From APT analysis in regions respectively marked A, B and C in Fig. 7b.

  upon nitriding a Fe-4.75 at.% Al alloy. They proposed that the growth of the '-Fe 4 N iron nitride is due to the negligible solubility of Al within '-Fe 4 N in association with the obstructed precipitation of the AlN aluminium nitride in the -Fe ferrite. This results in the nucleation and the growth of a '-Fe 4 N needles which deeply penetrates the nitrided layer (up to about 40 µm from the surface). Although this mechanism is well adapted to describe the formation of the '-Fe 4 N phase during nitriding of Fe-Al alloys with low Al content (4.75 at.% Al), it remains debatable for Al-rich Fe-Al alloys for which '-Fe 4 N is developed as a thin layer at the top most



  propose a better understanding of the nitriding mechanisms that occurs in Al-rich Fe-Al alloy (40 at.% Al) Within the nitriding conditions used, the nitrided layer consisted of two sublayers have very different charcteristics : An outer sublayer form at the top surface because of a sufficiently high nitriding temperature (600 °C) combined with the use of the pulsed-dc plasma nitriding technology which ensures a high nitrogen supply at the metal surface. This resulted in the formation of a thin outer sublayer composed of the '-Fe 4 N iron nitride (1-2 µm in thickness) which exhibits a moderate hardness (between 550 and 850 Hv).



  The formation of lamellae containing both -Fe and AlN is rather unusual. The observations conducted by high resolution transmission electron microscopy and atom probe tomography show that within an AlN / -Fe lamella, a finer structural arrangement is observed with the presence of AlN rich clusters (6 nm in thickness) separated by thin -Fe walls (2 nm thick). The diversity of the morphologies evidenced in the present study highlights the complexity of the nitriding mechanisms of iron-based aluminides, particularly those containing large amounts of Al. Further experiments and analysis should be undertaken to confirm the proposed mechanisms of formation of the multi-scale morphological features observed in the present study.

Figure captions Fig. 1 .

 captions1 Figure captions

Fig. 2 .

 2 Fig. 2. Variation of the hardness with depth measured: on the top surface of the nitrided

Fig. 3 .

 3 Fig. 3. Chemical composition (in at%. of Fe, Al and N) of the nitrided layer determined

Fig. 4 .

 4 Fig. 4. Phase composition of the nitrided layer determined from grazing-incidence XRD

Fig. 5 .

 5 Fig. 5. (a) TEM micrographs of the FeAl40 grade 3 sample nitrided at 600 °C in 95%

Fig. 6 .

 6 Fig. 6. (a) TEM micrographs of the FeAl40 grade 3 sample nitrided at 600 °C in 95%

Figure 1

 1 Figure 1

TableTable 1 .

 1 Atomic composition of the different morphologies observed at various scales in the FeAl40 grade 3 nitrided sample at 600 °C for 15 min in a 95% N 2 + 5% H 2 d.c.

	plasma.
		"Wave-like"	"Nodular-like"	-Fe	hex. AlN /-Fe
		region 1	region 1	lamellae 2	lamellae 2
					Fe-rich walls	Fe-lean clusters
	Fe	98 ± 1	100 ± 1	99.69 ± 0.02 79.82 ± 0.56	19.00 ± 0.28
	Al	1 ± 1	-	0.03 ± 0.01	11.75 ± 0.16	48.28 ± 0.36
	N	1 ± 1	-	0.28 ± 0.02	8.37 ± 0.14	32.31 ± 0.34
	Cr	-	-	-	0.12 ± 0.04	0.23 ± 0.03
	C	-	-	-	0.02 ± 0.02	0.18 ± 0.03

1

From STEM/EDS analysis in regions respectively reported in Fig.

Acknowledgments

This work and the post-doctoral stay of A. Martinavicius were supported by the French Government through the program "Investissements d'avenir" operated by the French National Research Agency (ANR) and referenced to as ANR-11-LABX-0008-01 (LabEx DAMAS). The authors would like to thank Dr.-Ing. P. Boulet (Institut Jean Lamour, CC X-Gamma) for grazing incidence XRD measurements.


Jieli Lyu

Claire Goldmann

Cyrille Hamon
email: cyrille.hamon@universite-paris-saclay.fr

Doru Constantin
email: constantin@unistra.fr

Extracting the morphology of gold bipyramids from SAXS experiments via form factor modeling

Keywords: SAXS, colloids, bipyramids, form factor, TEM

Accurate shape description is a challenge in material science. Small-angle X-ray scattering (SAXS) can provide the shape, size and polydispersity of nanoparticles by form factor modeling.

However, simple geometrical models such as the ellipsoid may not be enough to describe objects with complex shapes. In this work we show that the form factor of gold nanobipyramids is accurately described by a truncated bicone model, which is validated by comparison with transmission electron microscopy (TEM) data for nine different synthesis batches: we obtain the average shape parameters (width, height, and truncation) and the sample polydispersity. In contrast, the ellipsoid model yields worse fits of the SAXS data and exhibits systematic discrepancies with the TEM results.

I. INTRODUCTION

Recent progress in materials chemistry has allowed the synthesis of nanoparticles with very well defined shape and size [1]. In order to stay relevant, characterization techniques must also evolve to keep up with this progress. Form factors for a wide variety of shapes have therefore been implemented in many small-angle scattering software suites [2][3][START_REF] Doucet | SasView version 4.0[END_REF][START_REF] Ginsburg | [END_REF][6], but their analytical expressions can be quite complicated and the numerical evaluation very time-consuming, especially since a double integral over the orientation is usually required.

An alternative (model-free) strategy consists in describing the objects as a collection of small beads (also referred to as dummy atom models, or DAMs) [7]. The number and positions of these "atoms" are then adjusted until the scattering signal of the model approaches the experimental data. Initially developed for the study of biological macromolecules, this approach has recently been applied to inorganic nanocrystals [8][9][10]. We do not consider these models here, for two main reasons: the difficulty of converting between DAMs and geometrical shapes (which are very good descriptions for the nanoparticles we are interested in) and that of accounting for polydispersity, although some progress has been made on the latter aspect [11].

A natural question to ask in this context is: how detailed must the models be in order to extract as much information as possible about the morphology of the objects? Is it really useful to go beyond simple shapes, such as spheres or ellipsoids? The answer is a resounding "yes" in the case of cubes. Previous work [12] has shown that the difference between cubes and rhombocuboctahedra in composite Au@Ag objects is both detectable by SAXS and important in view of applications. In our group, we have followed by SAXS the morphological transition from spheres to cubes in such objects and confirmed these results by TEM [13]. A noteworthy conclusion is that both the asphericity and the polydispersity reduce the amplitude of the characteristic oscillations of the sphere form factor, but in slightly different ways: the former reduces the contrast of the fringes and preserves their number and overall profile, while the latter "smears" them in the manner of a Debye-Waller factor. Although both are isometric, cubes and spheres are easily distinguished if monodisperse enough. Moreover, these shapes are instances of a more general family, that of the superball. The position of the nanoparticles along the continuum defined by the associated shape parameter can be estimated via SAXS [14] and modulates the particle packing in supercrystals [15].

Introducing anisometry (by elongating or flattening the object of interest) renders the problem more complicated, unless the resulting object has a constant section (as in a rod or a plate). Since they are amenable to factorization, these limiting cases are easily treated analytically and have been extensively used in the literature. Let us note that factorization can also be used in the case of curved plates, considerably simplifying the calculations [16].

Here, we are specifically interested in spindles, elongated objects whose section varies along the length (so factorization does not apply): are they adequately described by their equivalent ellipsoids (with an appropriate polydispersity), or would we benefit from using more realistic models? The experimental system we have investigated consists of gold nanobipyramids (Au NBPs). The synthesis of these objects was refined over the last decade and the interest in their optical properties and subsequent applications has grown constantly [17][18][19]. Advanced modeling and simulations have shown how the optical response of the objects (e.g. the position of the surface plasmon resonance) depends on shape features such as the truncation [20][21][22]. The precise morphology of the NBPs influences their assembly in two [23,24] or three [25] dimensions, which in turn further modulates their optical properties.

Accurate characterization of the Au NBPs is therefore imperative in view of any applications. The shape information can of course be obtained by TEM, but SAXS and other scattering techniques exhibit two major advantages: they are non-intrusive (and thus can investigate synthesis, reshaping or assembly processes) and they average over a large collection of objects (obviating the statistical issues that might affect imaging techniques). On the downside, they only yield indirect and orientation-averaged information: the comparison with TEM is of course needed for validating the models.

In this paper, we describe Au NBPs as truncated bicones: the model is used to extract morphological parameters (from both SAXS and TEM data) and compared to the simpler ellipsoid model. Extensive analysis of the results obtained with nine different synthesis batches shows that the bicone model is accurate enough to capture the width, height opening angle (or, equivalently, truncation) and polydispersity of the particles, while the ellipsoid model exhibits systematic discrepancies or, in the case of the opening angle, simply does not account for this feature.

II. METHODS

A. Materials

Gold chloride trihydrate (HAuCl 4 • 3 H 2 O ≥ 99.9%), silver nitrate (AgNO 3 > 99%), hydrochloric acid (HCl 37%), sodium borohydride (NaBH 4 ≥ 96%), L-ascorbic acid (AA, ≥ 99%), trisodium citrate dihydrate (≥ 99%), cetyltrimethylammonium bromide (CTAB ≥ 99%), cetyltrimethylammonium chloride (CTAC, 25 wt % in H 2 O), and benzyldimethylhexadecylammonium chloride (BDAC 99%) were purchased from Merck. Water purified by reverse osmosis with a resistivity above 15 MΩ.cm) was used in all experiments.

B. Bipyramid Synthesis and Purification

Au NBPs were synthesized as described previously [21,26,27].

a. Seed synthesis 2.65 mL of CTAC (25wt % in water) and 33 mL of water were heated of AA (100 mM) was added, followed by a various amount of seeds: 3.5, 3.5, 3.6, 3.6 mL for samples A to D. After 4 h at 30 • C, the bipyramids were centrifuged twice and purified by depletion during one night at 30 • C in 15 mL BDAC (350 mM) [28]. The supernatant was removed, and the precipitate was redispersed in water and washed twice with 1 mM CTAC.

The Au NBPs were finally redispersed in 2 mL of 1 mM CTAC. c. Growth of particle batches E to I The protocol was in all points similar to that used for samples A to D, except the solution volumes were halved and the final CTAC concentration was 2.5 mM. The seed volumes were: 5, 2, 1, 1 and 0.1 ml for samples E to I.

C. TEM

The solutions were concentrated by slow centrifugation to a final Au 0 concentration of 0.75 mM in 0.55 mM CTAC. 10 µL of this solution was then dropped on a carbon-coated grid and dried at 70 • C. Transmission electron microscopy (TEM) images were obtained with a JEOL 1400 microscope, operating at an acceleration voltage of 120 kV.

The bicone model is defined as in Figure 1: The total width is denoted by W , the total (effective) length by L, and the total length of the bicone (without truncation) is H. The truncation t = (H -L)/2 and the full tip angle is α. These parameters are not independent, so in the following we will use the set (W, L, α) to fully describe the shape of one particle.

The ellipsoid model only has two parameters, the major and minor axes a and b, which correspond to W and L of the bicone model, as the length along the symmetry axis and the transverse diameter, respectively. For a complete description of the particle population in one sample we also need the polydispersity p (to be discussed below).

The TEM images are treated using Igor Pro 7.0 [29]: first, the particles are separated from the background using a bimodal fit: ImageThreshold operation, with the M=(2) option. The contours of each particle are then identified using the ImageAnalyzeParticles operation, with options /E/W/M=3/FILL/EBPC. Option /E computes the equivalent ellipse for each particle, defined by the five parameters (X c , Y c , a, b, θ): x and y coordinates of the particle center, major and minor semi-axes and orientation angle. They are used as a first approximation for the bicone shape (or, more precisely, for its plane projection: a truncated diamond, see Figure 1).

Both the extracted contour and the model are represented in polar coordinates (as R e (φ) and R m (φ), respectively) and the difference between them quantified as χ 2 = dφ|R e (φ) -R m (φ)| 2 . Optimizing χ 2 is not straightforward, but we obtained good results by a twostep approach: simulated annealing (which is more robust, but does not always reach the minimum) followed by line search (to further refine the parameter values). Both steps are performed using the Optimize operation, with options M = {3, 0} and M = {0, 0} (default),

respectively.

The extracted contour and the model are then presented to the user for inspection. We reject inadequately fitted contours, composite objects (where several particles are superposed and cannot be discriminated), and some round objects (possibly spheres or unreacted decahedral seeds).

D. SAXS

SAXS measurements were performed on the SWING beamline of the SOLEIL synchrotron (Saint-Aubin, France) at a beam energy of E = 16 keV. The sample-to-detector distance was 6.22 m, covering a scattering vector range 0.0014 < q < 0. (affecting all dimensions similarly) with relative standard deviation p.

III. RESULTS AND DISCUSSION

A. SAXS

A detailed fit example is shown in Figure 2 for sample A. Fits for the other samples (B-I) are shown in Figure 3.

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)@A6< B ' D ' 3 ' E ! "# ! "$ ! "% ! " ! ! ! ! % ! $ & ' ( ) ) " * !+!! % $ # , -. / 0 !+! % $ # , -. / 0 !+ % 1'(2 " * & ' ' ' 3 ' 4 ' 5

B. TEM

A fit example is shown in Figure 4 for one particle from sample H. !@%A !@%! !@ A !@ ! !@!A !@!! > +B7C +4'DE-1 +7??'>FE'(

TEM analysis)

. This parameter is shown in Figure 6. Except for sample C (where the TEM data is lacking) and for sample I (where the fit quality is low), the BC values are always within the standard deviation of the TEM distribution.

Note that the ellipsoid model cannot measure α (or, conversely, the virtual length H).

One could of course build an angle β from the aspect ratio, e.g. as tan(β) = W/L, but this is an arbitrary choice and would severely overestimate the true tip angle, because it neglects truncation: β is between 41 • and 52 • for all our samples.

All fit parameters and some other details are given in Table I. The AS curves are presented in Appendix B and representative TEM images are shown in Appendix C. Both the bicone and the ellipsoid models yield bad fits for sample I: in particular, the polydispersity is severely overestimated (see Figure 3 andTable I), because the particle shape is often irregular (see Figure 9). This is because the attempted particle size (about 150 nm long) is at the upper limit for NBP synthesis: above it, one obtains nanojavelins [21].

∆λ c Peak ratio d Unit [nm] [nm] [nm] [nm] • - -[nm] [nm] [nm] [nm] • - - - [nm] [nm] - A 77.

IV. CONCLUSION

The ellipsoid model yields reasonable values for the length and width of the objects, although they are always slightly underestimated. The polydispersity is significantly overestimated, and the tip angle cannot be inferred from this model.

On the other hand, the bicone model clearly yields much better fits to the SAXS data than the ellipsoid one and the resulting coefficients are in very good agreement with the TEM results, in particular for the tip angle α. We conclude that this model is appropriate for describing Au NBPs. Potential applications include monitoring the growth of these objects in solution, but also the evolution of composite nanoparticles obtained by the deposition of a different metal (e.g. silver) onto Au NBPs [30,31].

The SAXS data and the two models presented in this work are available as supporting information.

Appendix A: Form factor models and fit quality

Ellipsoid

The ellipsoid form factor is implemented as in the NIST SANS macros [2]: due to the symmetry of the NBPs, we only consider spheroids (with major axes a = b = c). All fits yield prolate results (with a > b).

Bicone

The form factor for a (full or truncated) cone is given in the literature, e.g. in [32], and implemented in SASFIT [3] and BornAgain [6]. For completeness, we present here its derivation for a truncated bicone, using the notation in Figure 1. Since the body has azimuthal symmetry (around the z axis), we can assume without loss of generality that the scattering vector q is contained in the (x, z) plane: q = (q r , 0, q z ), and makes an angle θ with the z axis. The transverse radius of the complete cone at any height z between -H and H is given by:

r(z) = R 1 - |z| H .
The form factor of the object is:

F (q) = R 3 dr 3 e -iqr ρ(r) = ∆ρ V dr 3 e -iqr
where ρ(r) is the scattering length density (SLD), which depends on the space position. ∆ρ is the SLD difference between the object (which is homogeneous and occupies the volume V)

and the surrounding medium. In cylindrical coordinates, the current vector is: r = (ξ, φ, z),

where the azimuthal angle φ is measured with respect to the x axis and the phase factor e -iqr = e -iξqr cos(φ) e -iqzz . We can express the form factor as:

F (q) = ∆ρ H/2-t -(H/2-t) dz e -iqzz r(z) 0 ξdξ 2π 0 dφ e -iξqr cos(φ) = 2π∆ρ H/2-t -(H/2-t) dz e -iqzz r(z) 0 ξdξJ 0 (ξq r ) = 2π∆ρ H/2-t -(H/2-t) dz r 2 (z) e -iqzz J 1 [q r r(z)] q r r(z) = 4π∆ρ H/2-t 0 dz r 2 (z) cos(q z z) J 1 [q r r(z)] q r r(z) . (A1)
For a fixed amplitude q = |q| of the scattering vector, the formula above yields the form factor for a given orientation θ. In solution, the scattering signal results from an incoherent average over all orientations, so the relevant quantity is:

P (q) = |F (q)| 2 θ = π 0
sin(θ)dθ|F ((q cos(θ), 0, q sin(θ)))| 2 .

(A2)

Polydispersity

In both the ellipsoid and the bicone models we account for the polydispersity by introducing a homothetical size distribution: all dimensions are scaled by a parameter λ with respect to their reference (λ = 1) values (W 0 , L 0 , α) or (a 0 , b 0 ) -in which case the scattered signal is I 0 (q)-and λ is distributed along a Gaussian:

g(λ) = 1 √ 2π p exp - 1 2 λ -1 p 2 (A3)
Scaling all sizes by λ or the scattering vector q by the same factor preserves the signal, up to a λ 6 prefactor (easily understood if we recall that the scattered intensity is proportional to the particle volume squared.) The polydisperse signal can then be obtained as:

I avg (q) = dλ g(λ)I λ (q) = dλ g(λ)λ 6 I 0 (λq) (A4)

Relation (A4) applies to any particle shape. In particular, we checked that the sphere model yields very close results to the analytical Schulz distribution as implemented in Igor

Pro [2].

This algorithm has the advantage of being very fast, as it requires only one calculation of the form factor (for the reference values of the dimensions), followed by the averaging step (A4). It is also very general, applying to homogeneous or composite nanoparticles of any shape. Its only disadvantage is the intrinsic limitation to homothetical polydispersity.

Fit quality

There is a significant difference in fit quality between the two models, but it is difficult to discern from the graphs in Figures 2 and3. We give the goodness-of-fit χ 2 for both models in Table II:

at 30 •

 30 C. Then, 400 µL of HAuCl 4 • 3 H 2 O (25 mM) and 4 mL of trisodium citrate (50 mM) were added and the mixture was kept at 30 • C for 30 minutes. Under fast stirring, 1 mL of NaBH 4 (25 mM) was added quickly. Stirring was continued for 1 minute and the resulting solution was put in the oven for 5 days at 40 • C prior to use. b. Growth of particle batches A to D 2 mL of AgNO 3 (10 mM), 4 mL of HAuCl 4 • 3 H 2 O (25 mM), and 4 mL of HCl (1 M) were added in 200 mL of CTAB (100 mM). Then, 1.6 mL

FIG. 1 .

 1 FIG.1. TEM image of a particle, with the morphology parameters of the bicone model.

24 Å- 1 .FIG. 2 .

 2412 FIG. 2. Fits to the scattering data for sample A (black dots) with the ellipsoid (red dashed line) and bicone (solid green line) models. The residues are shown in the top panel. A zoomed-in view of the oscillations is shown to the right: for clarity, only one data point in ten is displayed.

FIG. 3 .FIG. 4 .

 34 FIG. 3. Fits to the scattering data for samples B through I (black dots) with the ellipsoid (red dashed line) and bicone (solid green line) models.

FIG. 5 .FIG. 6 .

 56 FIG. 5. Fit parameters for the bicone and ellipsoid models compared with the TEM values. Top panel: polydispersity p (green symbols). Middle panel: total length (red). Bottom panel: total width (black and gray). The symbols are the same for all panels: solid dots for the bicone model, open dots for the ellipsoid model, squares for the TEM data.

a

  Number of particles used in the analysis. b Position of the longitudinal plasmon peak. c Width of the longitudinal plasmon peak. d Between the intensity of the longitudinal and the transverse plasmon peaks.

FIG. 7 .

 7 FIG. 7. AS spectra for all samples.

FIG. 9 .FIG. 10 .

 910 FIG. 9. TEM images for samples F to I. The image of sample H was taken prior to purification, which explains the large number of spheres still present. Most of them are no longer present in the final sample.

TABLE I .

 I Parameters obtained by the three techniques (AS, SAXS and TEM) for all samples.

TABLE II .

 II Goodness-of-fit χ 2 obtained with the bicone and ellipsoid models for all samples (data and fits in Figures2 and 3.).

	Sample χ 2 BC χ 2 Ell
	A	103.2 1112
	B	14.8 51.5
	C	7.4 20.5
	D	2.9 4.0
	E	18.0 22.7
	F	3.6 8.0
	G	19.8 18.6
	H	4.9 8.5
	I	115.8 134.2

ACKNOWLEDGMENTS

JL acknowledges financial support by the China Scholarship Council (CSC). The authors acknowledge SOLEIL for the provision of synchrotron radiation facilities (experiment 20201118) and Thomas Bizien for help using the SWING beamline. The present work has benefited from the electronic microscopy facility of Imagerie-Gif, (http://www.i2bc.parissaclay.fr), member of IBiSA (http://www.ibisa.net), supported by "France-BioImaging" (ANR10-INBS-04-01), and the Labex "Saclay Plant Science" (ANR-11-IDEX-0003-02).

Appendix B: UV-Vis-IR absorbance spectroscopy

Absorbance spectroscopy (AS) is the most common technique for characterizing plasmonic nanoparticles. In Figure 7, we present the spectra of all samples, normalized to an extinction value of 0.5 at 400 nm.

Appendix C: TEM images

We present in Figures 8 and9 representative images for all samples.

Appendix D: Distance distribution function

In Figure 10 we present the distance distribution function P (r) for some samples, computed using the denss.fit data routine in the DENSS suite [33]. For ease of comparison, the curves are normalized to 1 at the mode. We could not obtain reliable P (r) values for curves E, F and I, presumably due to small-q imperfections (for the first two) and to the large particle size for the last one.