Juan Macchi
email: juan.macchi@univ-lorraine.fr

Steve Gaudez

Guillaume Géandier

Julien Teixeira

Sabine Denis

Frédéric Bonnet

Sebastien Y P Allain

Macchi Juan

Gaudez Steve

Geandier Guillaume

Teixeira Julien

Denis Sabine

Allain Sébastien

Campus Artem

Allée André Guinier

Dislocation densities in a low-carbon steel during martensite transformation determined by in situ high energy X-Ray diffraction

Keywords: Steel, Martensite, Synchrotron, X-ray analysis, Dislocation, Mechanical behavior

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Although lath martensites are one of the key microstructure constituents of conventional steels, of hot-stamped boron-bearing steels and also of several 3 rd generation multi-phased steels for automotive construction [START_REF] Fonstein | ADVANCED HIGH STRENGTH SHEET STEELS : physical metallurgy, design, processing, and properties[END_REF], the correlation between their mechanical behavior and their specific hierarchical microstructure is not completely established yet. A major reason is that most of the studies have considered martensite as an homogeneous structure. Recent works in this field have tried to highlight that martensite microstructures are in fact dispersed in terms of size, residual stresses, defects and composition and behave like a multiphase aggregate [START_REF] Allain | Toward a New Interpretation of the Mechanical Behaviour of As-quenched Low Alloyed Martensitic Steels[END_REF][START_REF] Ungár | Composite Behavior of Lath Martensite Steels Induced by Plastic Strain, a New Paradigm for the Elastic-Plastic Response of Martensitic Steels[END_REF][START_REF] Hutchinson | Plastic yielding in lath martensites -An alternative viewpoint[END_REF]. These results sustain thus the micromechanical approach considering martensite as a continuum composite material (CCA) [START_REF] Allain | Toward a New Interpretation of the Mechanical Behaviour of As-quenched Low Alloyed Martensitic Steels[END_REF][START_REF] Ungár | Composite Behavior of Lath Martensite Steels Induced by Plastic Strain, a New Paradigm for the Elastic-Plastic Response of Martensitic Steels[END_REF] proposed by the present authors already in 2012 and assuming a distribution of local yield strengths at the microstructure scale. Since then, this interpretation has been supported either by local mechanical measurements or by direct microstructure observations. Some authors have found a wide dispersion of the mechanical properties between different martensite laths thanks to nanohardness investigations. Li et al. [START_REF] Li | Microstructure effect on nanohardness distribution for medium-carbon martensitic steel[END_REF] showed that the nanohardness within the martensite studied can vary between 5 GPa and 11 GPa. It has been observed that the smaller blocks (aggregate of laths with low angle misorientation) have a higher hardness in [START_REF] He | Revealing the Intrinsic Nanohardness of Lath Martensite in Low Carbon Steel[END_REF] with an average nanohardness of about 5.5 GPa for the large blocks and 6.9 GPa in the small blocks with a width estimated by Electron Backscattered Diffraction (EBSD) equal to 6 and 3.5 μm respectively. He et al. [START_REF] He | On the nanoindentation behaviour of complex ferritic phases[END_REF] studied ferritic microstructures (ferrite, lower and upper bainite, and martensite) highlighting a significantly wider scattering in the martensite results. In addition a high heterogeneity in the deformation of this phase was reported in [START_REF] Morsdorf | Multiple mechanisms of lath martensite plasticity[END_REF] by digital image correlation based on successive EBSD cartographies taken during the test. The heterogeneity was reported to increase with deformation indicating strain localization and indicating a huge dispersion in the local yield strengths of the laths.

These local dispersions and the resulting composite behavior are related to the martensite transformation itself, which is essentially displacive and athermal. Very few studies have been dedicated to the formation of microstructure heterogeneities in martensite. Starting from post mortem observations, Morsdorf et al. [START_REF] Morsdorf | 3D structural and atomic-scale analysis of lath martensite: Effect of the transformation sequence[END_REF] have analyzed how the martensite microstructure appears progressively. The study shows the presence of a wide dispersion of thickness, with thin laths (from ≈50 to ≈500 nm) and coarse lath, up to ≈3.5μm. A quantitative analysis by atom probe tomography in thin and coarse laths reveals that carbon segregation is more intense in the coarse laths than in the thin ones. Nanohardness characterization presented in the same work shows that the thinner laths were ≈10% harder than the coarse ones. It suggests that the first martensite blocks/lath to appear will be the largest, the most segregated and the softest. On the contrary, the last structures to form will be the hardest domains of the microstructure [START_REF] Morsdorf | 3D structural and atomic-scale analysis of lath martensite: Effect of the transformation sequence[END_REF]. Similar conclusions were drawn by Badinier et al. [START_REF] Badinier | Effect of Carbon Segregation and Carbide Precipitation on the Mechanical Response of Martensite[END_REF][START_REF] Badinier | Microstructural heterogeneity and its relationship to the strength of martensite[END_REF].

To the authors' best knowledge, Christien et al. [START_REF] Christien | Neutron diffraction in situ monitoring of the dislocation density during martensitic transformation in a stainless steel[END_REF] were the only group to study in situ the evolution of the mean dislocation density during the martensite transformation in both martensite and austenite. The study was conducted by neutron diffraction and the studied steel was a carbon-free stainless steel permitting long acquisition time (1 pattern every 2 mins).

Results showed that the higher the martensite phase fraction, the higher the mean dislocation densities in both phases.

The present work consists also in the detailed characterization of the evolution of mean dislocation density in both martensite and austenite phases in situ during the transformation.

We have addressed the challenge to follow these evolutions in a low carbon steel in which martensitic transformation is faster and can be accompanied by self-tempering mechanisms. A set-up with an excellent time resolution suitable for these particular steels and transformation kinetics (1 pattern every 0.1s) was used. Moreover, an analysis to determine in the final microstructure the associated spatial dispersion of local hardening due to these dislocations was developed. This new insight permits to go farther in the understanding of the mechanical properties of this heterogeneous phase.

Material and methods

An industrial low-carbon alloyed steel, provided by ArcelorMittal Maizières Research Center, France, with a composition Fe-0.215C-0.25Si-1.82Mn-0.18Cr wt.% was investigated. The steel was received after the cold-rolling stage (1.5 mm thickness) and samples were machined by 10 mm x 4 mm x 1.5 mm plates. No surface preparation was performed.

High Energy X-ray diffraction (HEXRD) experiments were performed in situ at the Deutsche-Elektronen-Synchrotron (DESY, PETRA-III) P07 beam line with a monochromatic beam with an energy of 100 keV (λ=0.012398 nm) allowing to work in transmission mode. The association with a fast 2D Perkin-Elmer detector enabled high acquisition rates (10 Hz) needed to study fast processes on bulk samples (400 x 400 µm beam size by 1.5 mm thickness) with a sufficient time resolution, as it is the case of the martensitic transformation in low carbon steels. The detector was positioned about 1.5 m behind the sample, giving access to full Debye-Scherrer rings with a maximum 2θ angle of 8° (5 Debye Scherrer rings per phase).

Martensitic quench treatment was performed with a modified Bähr DIL805 A/D dilatometer available on the beam line. The heat treatment used to investigate the martensite transformation consists of a heating at 20 °C/s up to 880 °C, followed by an isothermal holding during 180 s and finally of a fast cooling by argon gas blowing (aiming at a cooling rate of 50°C/s) down to the room temperature. The austenitization heat treatment leads to a mean prior austenite grain size of 6.5 µm. This value has been determined by Prior Austenite Grain reconstruction using Decrypt® software on the basis of EBSD cartographies (results not shown here). For the studied cooling, 500 diffraction patterns have been acquired between the Ms temperature (394°C) of the alloy and the room temperature.

The 2D diffraction patterns produced during the experiments were integrated circularly using pyFAI python library [START_REF] Ashiotis | The fast azimuthal integration Python library: PyFAI[END_REF]. The deduced 1D diffractograms (Intensity vs 2θ) were analyzed with a Rietveld refinement procedure using pseudo-Voigt functions to reproduce diffraction peaks. The Rietveld method here has served to obtain the phase fractions with an uncertainty of ±1% (absolute value) during the martensitic transformation. According to 1D diagrams, two phases were considered during the analysis: martensite with a body centered tetragonal structure (I/4 mmm) and austenite with a cubic face centered structure (Fm-3m). No carbide (transition iron carbides and/or cementite) resulting from martensite self-tempering was detected during the quench by HEXRD, even if such configurations is known to enable their detection if present in low amount [START_REF] Allain | In Situ Investigation of the Iron Carbide Precipitation Process in a Fe-C-Mn-Si Q&amp[END_REF]. Three experiments have been conducted in order to ensure the repeatability.

For simplicity, and to avoid redundancy, only one experiment is presented and discussed in the present paper. The other experiments show strong similarity with the one presented here, and the same conclusions would be extracted from each investigation. Nevertheless, the results of each individual experiments are available in the supplementary data.

Since the pioneer works of Williamson & Hall [START_REF] Williamson | X-ray line broadening from filed aluminium and wolfram[END_REF], it is well known that dislocations present in polycrystals will affect the width of their diffraction peaks. The higher the dislocation density (resp. the lower the crystallite size), the larger the peak width [START_REF] Sallez | In situ characterization of microstructural instabilities: Recovery, recrystallization and abnormal growth in nanoreinforced steel powder[END_REF]. The Rietveld method does not permit to determine properly the Full Width Half Maximum (FWHM(2θ)) of each diffraction peaks as it is only a mean description of a full line profile. In fact, the FWHM in Fullprof is based on the Caglioti et al.'s theory [START_REF] -Iuc | The Rietveld Method[END_REF] which required the refinement of three parameters per phase to describe the mean FWHM for each of its peak. This description does not permit to account in particular for the respective crystallographic anisotropy of the phases as suggested by Ungar et al. [START_REF] Ungár | The effect of dislocation contrast on x-ray line broadening: A new approach to line profile analysis[END_REF].

Besides the Rietveld refinement of the full diffractograms, a separate refinement has thus been conducted peak by peak using Pearson VII functions in order to obtain the individual FWHM and angular position of the phase diffraction peaks. The instrumental contribution to the peaks broadening was measured with a silicon powder and subtracted from the measured FWHM by considering a square additive law [START_REF] Sallez | In situ characterization of microstructural instabilities: Recovery, recrystallization and abnormal growth in nanoreinforced steel powder[END_REF]. The instrumental contribution was considered constant during the proposed experiments accounting for the stability of the beam during each experiment and considering that the only parameter that changes is the temperature of the sample, no movement has been made even on the sample or the detector.

While a tetragonal cell was used for the Rietveld refinement, it was not possible to model each peak individually considering tetragonality with one Pearson VII function per diffraction plane family. The diffraction peaks were then fitted with a single symmetric Pearson VII function. This introduces an over estimation in the dislocation density as the peaks will be slightly larger than expected due to the tetragonal lattice. At the end of the martensitic transformation, one diffractogram was analyzed both by a single function per diffraction peak and by one Pearson VII function per diffraction plane families considering tetragonality imposing the difference between positions due to the c/a determined by the Rietveld refinement (1.006). For clarity, in the case of the peak (200) the diffraction plane families are (hk0) and (00l) and the effect of the tetragonality (1.006) in the experimental set-up produces a difference of 0.03° 2θ between both families. The dislocation density found in the second case was only 12% lower than only considering one function by peak. The error introduced considering a single peak instead of two for the fitting procedure is low in regard to the experimental uncertainties for the low carbon steel studied and would not modify the tendencies found and conclusions extracted.

From the instrumently-corrected FWHMs, the dislocation densities in martensite and in austenite for each studied diffraction patterns were estimated by the modified Williamson-Hall method (mWH), introduced by Ungár et al. [START_REF] Ungár | The effect of dislocation contrast on x-ray line broadening: A new approach to line profile analysis[END_REF]. This method imposes a contrast factor for each refracting plane family considering the anisotropy of the material [START_REF] Ungár | The contrast factors of dislocations in cubic crystals: the dislocation model of strain anisotropy in practice[END_REF]. Under the assumption that the only feature introducing broadening in the materials reflections are the dislocations and crystallite sizes, the peak broadening can be described by the following equation: ΔK = ƺ/D + (((πMb 2 )/2) 1/2 * ρ̅ 1/2 )*K*C̅ hkl 1/2 + OK 2 C̅ hkl [START_REF] Fonstein | ADVANCED HIGH STRENGTH SHEET STEELS : physical metallurgy, design, processing, and properties[END_REF] where ΔK=FWHM(θ)*2cos(θ))/λ , K is the norm of the scattering vector defined by K=2sin(θ)/λ, D is the crystallite size, M is a parameter depending on the dislocation density, b is the Burgers vector, ρ̅ is the mean dislocation density and C̅ hkl is the average contrast factor of dislocations for the specific reflection. OK 2 C̅ hkl is a higher order term with no meaning established [START_REF] Ungár | The Effect of Dislocation Contrast on X-Ray Line Profiles in Untextured Polycrystals[END_REF], where O is much smaller than the coefficient before KC̅ hkl , and it is not considered here [START_REF] Takaki | Proposal of simplified modified Williamson-Hall equation[END_REF]. For the present work ƺ was set equal to 0.9 as done in [START_REF] Williamson | X-ray line broadening from filed aluminium and wolfram[END_REF] and b to 2.5 10 -10 m (ca. 2.49 10 -10 m and 2.517 10 -10 m respectively for the ferrite and the austenite) for both phases analyzed. M is a dimensionless parameter linked to the outer cut-off radius of the dislocations and the dislocation arrangement inside the phase, this parameter can be evaluated with the Warren-Averbach method. In the present study this parameter was considered constant during the thermal treatment. While it can vary [START_REF] Sallez | In situ characterization of microstructural instabilities: Recovery, recrystallization and abnormal growth in nanoreinforced steel powder[END_REF], it is set at 2.2 considering the average value from [START_REF] Shi | Study of tempering behavior of lath martensite using in situ neutron diffraction[END_REF].

The dislocation contrast factor for each reflection (C̅ hkl ) is taken from ANIZC software [START_REF] Borbély | Computer program ANIZC for the calculation of diffraction contrast factors of dislocations in elastically anisotropic cubic, hexagonal and trigonal crystals[END_REF] which gives the theoretical contrast factor for screw and for edge type dislocations. It accounts from the type of lattice (considered as cubic here), the second order elastic constants and reflections [START_REF] Borbély | Computer program ANIZC for the calculation of diffraction contrast factors of dislocations in elastically anisotropic cubic, hexagonal and trigonal crystals[END_REF]. The slip systems considered for edge dislocation are <111> {110} and <110> {111} respectively for the body-centered and face-centered-cubic phases. The dislocation lines considered for screw are <111> and <110> respectively for the bodycentered and face-centered-cubic. As the elastic constants (C 11 , C 12 and C 44 ) vary with the temperature [START_REF] Leese | Elastic stiffness coefficients of single-crystal iron from room temperature to 500°C[END_REF][START_REF] Dever | Temperature dependence of the elastic constants in α-iron single crystals: Relationship to spin order and diffusion anomalies[END_REF][START_REF] Isaak | Elastic and viscoelastic properties of α iron at high temperatures[END_REF], the dislocation contrast factors vary as well. The dislocation contrast factors for screw and edge type dislocations are considered as temperature dependent and have been calculated using the values in the literature for ferrite [START_REF] Leese | Elastic stiffness coefficients of single-crystal iron from room temperature to 500°C[END_REF][START_REF] Dever | Temperature dependence of the elastic constants in α-iron single crystals: Relationship to spin order and diffusion anomalies[END_REF][START_REF] Isaak | Elastic and viscoelastic properties of α iron at high temperatures[END_REF]. For austenite, they have been considered constant due to the lack of available data [START_REF] Zhao | Stress distribution correlated with damage in duplex stainless steel studied by synchrotron diffraction during plastic necking[END_REF]. Many models propose a description of isotropic elastic constants but the data considering the elastic anisotropy of the lattice are rare. However, as the phases are considered separately in this second posttreatment, the error made in austenite has not effect on the results for martensite Table 1 shows the dislocation contrast factor calculated at room temperature for both martensite and austenite considering a ratio of 50% edge and 50% screw type dislocations as [START_REF] Sallez | In situ characterization of microstructural instabilities: Recovery, recrystallization and abnormal growth in nanoreinforced steel powder[END_REF]. The contrast factor evolutions with temperature for martensite were calculated up to 700°C and fitted with a second order polynomial function (aT 2 + bT + c with T in K). For the {200} reflection fitted parameters are a=1.73 10 -7 , b=4.854 10 -5 and c=0.277 while for {211}

{220}{321} they are a=2.266 10 -8 , b=7.042 10 -7 and c=0.1408 for the considered proportion of dislocation types.

An estimation of the error introduces by neglecting the elastic constants thermal dependence was performed on the martensite phase at 350 °C. The dislocation density was found 7% lower than accounting for the thermal dependence. The influence of this sensitivity in thus limited in the studied temperature range. Based on this result, we could expect similar results

for austenite in the investigated temperature range (between Ms down to RT). In order to estimate an error introduced by the possible texture of the steel and the experimental scattering, one 2D image (Debye Scherrer rings) from the end of the quench was integrated circularly each degree yielding to 360 1D diagrams. Each diagram was analyzed with the mWH procedure. The calculations (results not shown here) show that 95% of the dislocations densities calculated, for both martensite and austenite phases, where in between the mean value and ±22.5% (i.e. the dispersion from the mean value for two standard deviations (95%) was lower than ±22.5%).

Results:

The cooling curve of the most representative in situ experiment is presented in Figure 1(a)

with the corresponding dilatometric signal. The first evidence of martensitic transformation is observed on the dilatometric curve due to the transformation strain from austenite to martensite. In addition, as martensitic transformation is by nature an exothermic reaction, it affects the thermal evolution. The apparent martensite start temperature (Ms) was found equal to 394±4°C by the analysis of 3 dilatometric curves and applying the offset method proposed by Yang and Bhadeshia [START_REF] Yang | Uncertainties in dilatometric determination of martensite start temperature[END_REF]. This temperature corresponds also to a deviation in the temperature evolution (sudden change in slope). Nevertheless, in the studied condition, the transformation appears to be sluggish at the beginning as only 5% of martensite is formed at 360°C. These points will be clarified and discussed later at the light of the XRD data. Some selected 1D diffractograms of the same experiment are presented in Figure 1(b) They correspond to the conditions highlighted by dots in Figure 1(a). The same color code has been followed. The respective reflections of austenite and martensite are indicated (α standing for ferrite and martensite and γ for austenite). Along the cooling sequence, the alloy is first fully austenitic at 880°C. At the apparent Ms temperature (394°C), the diffraction pattern shows already the presence of small fraction of ferritic phase (<5wt.%). This fraction is attributed to a ferritic or a bainitic transformation prior to the martensitic transformation. At 200°C and at RT, the alloy is mainly martensitic. The martensite phase fraction measured at room temperature is 94 wt.% and is similar between the experiments, meaning that 6 wt.% of austenite is retained at RT. initially not following a kinetics as it could be predicted by Koistinen and Marburger and then a rapid transformation is observed. The sudden increase in the transformation kinetics by the Rietveld analysis occurs around 365°C which is 32°C lower than the apparent Ms reported above. This second value agrees well with the theoretical Ms temperature of the investigated steel (369°C) accounting for the nominal steel composition and the prior austenite grain size (PAGS) [START_REF] Van Bohemen | The kinetics of bainite and martensite formation in steels during cooling[END_REF][START_REF] Van Bohemen | Predicting the Ms temperature of steels with a thermodynamic based model including the effect of the prior austenite grain size[END_REF]. This two steps transformation is common in industrial steels and is often explained by microsegregations or decarburized surface layer which affect the local transformation start temperature [START_REF] Van Bohemen | Bainite and martensite start temperature calculated with exponential carbon dependence[END_REF][START_REF] Bhadeshia | Thermodynamic extrapolation and martensite-start temperature of substitutionally alloyed steels[END_REF]. As a consequence, there is a spatial and time distribution of the transformation progression. The transformation starts earlier in microstructure regions where the Ms is high (low C, low Mn, high PAGS) and later where Ms is lower. In the following, 365°C will be considered as the true Ms temperature of the alloy considering it represents better the value of the alloy mean composition. Different evolutions of the peak broadening with the temperature are experimentally observed for the austenite and for the martensite. A more gradual increase is presented in the former whereas martensite seems to saturate at an early stage of the transformation. While the authors cannot give at the moment a full explanation of the presented different behaviors some elements can be numbered:

• The saturation of the FWHM of the martensite could be partially related to the increase of the martensite fraction, i.e. a newly fraction of martensite with a high FWHM will produce a bigger increase on the average FWHM when the total martensite fraction is low than when martensite fraction is high. A similar reasoning can be postulated for the austenite; as the percentage of the phase decreases the increase of the FWHM in a fraction of the residual phase produces a more important effect on the average FWHM.

• The continuous increase of the FWHM of the austenite might be also a consequence of the martensitic transformation of the austenite with lower dislocation density.

Although the high deformed austenite is less stable from a thermodynamic consideration, the crystal defects can mechanically stabilize the mother phase [START_REF] Sipos | Influence of Austenite Predeformation on Mechanical Properties and Strain-Induced Martensitic Transformations of a High Manganese Steel[END_REF].

The transformation of the austenite with a lower dislocation density produces an increase of the average dislocation density observed.

• The increase of the peak broadening of the austenite at the final states of the transformation might be also related to the presence of distributions of internal stresses as reported in [START_REF] Nakada | Self-stabilization of untransformed austenite by hydrostatic pressure via martensitic transformation[END_REF]. The progressive confinement of austenitic regions makes relaxation mechanisms more and more difficult, increasing the residual stresses.

Let us mention that during phase transformation the interpretation of the FWHM is complex since it is affected by several contributions: crystal defects, crystallites size (size of diffraction domains), thermal, chemical and internal stresses heterogeneities within the phases/grains, free surface effects. Due to the set-up used to investigate martensite transformation (transmission mode through the 1,5 mm thickness of the sample), the sample surface effect is low and assumed negligible. In addition investigations showed that the mean crystallites size of both phases are mostly constant. Presently, as the transformation is displacive, no chemical composition distribution is considered (we do not take into account possible self tempering during the martensite transformation). However, there are internal stresses in the phases resulting from the martensitic transformation during cooling due to the phase transformation deformations and their precise knowledge at the different scales is complex and is still an open question. At the scale of the phases, the mean internal stresses generated during martensitic transformation have been clearly evidenced experimentally by HEXDR [START_REF] Wang | Real time observation of martensite transformation for a 0.4C low alloyed steel by neutron diffraction[END_REF][START_REF] Allain | Internal stresses and carbon enrichment in austenite of Quenching and Partitioning steels from high energy X-ray diffraction experiments[END_REF][START_REF] Dehmas | Apport de la diffraction synchrotron à l'étude de la transformation martensitique dans les aciers[END_REF]. They are hydrostatic and lead to a shift of the diffraction peaks of the phases.

Internal stress heterogeneities in the martensitic microstructure have been fewly addressed in literature. Thus, on one hand, Archie et al. [START_REF] Archie | Anisotropic distribution of the micro residual stresses in lath martensite revealed by FIB ring-core milling technique[END_REF] and Fukui et al. [START_REF] Fukui | Internal residual stress originated from Bain strain and its effect on hardness in Fe-Ni martensite[END_REF] recently reported an anisotropic strain distribution by SEM-FIB based ring-core method at the scale of the martensite variants, while Nakada et al. have reported an anisotropic distribution of the micro residual strains in the austenite [START_REF] Nakada | Self-stabilization of untransformed austenite by hydrostatic pressure via martensitic transformation[END_REF]. It is possible to assume that the effect of the internal stresses on the FWHM might become more important with the progress of the transformation as the size effect is more important.

In our case, the experimental method used is not able to deconvolute internal stress heterogeneities from the dislocation density contribution within the phases. Nevertheless, the high contribution of the dislocation density to the FWHM in the martensite is supported by the good correlation of the values of dislocation densities measured by TEM and HEXRD in the literature [START_REF] Morito | Dislocation Density within Lath Martensite in Fe-C and Fe-Ni Alloys[END_REF] and by our own investigations (TEM: 8x10 14 1/m 2 , HEXRD: 2 x10 15 1/m 2 ).

In our approach we will link the evolutions of FWHM to the evolutions of dislocation densities in both phases.

The Figure 3(b) shows the mean dislocation densities (ρ ), deduced from the FWHMs shown ̅ in Figure 3(a) with the modified Williamson-Hall method, in martensite (α) and austenite (γ) as a function of the martensite fraction during the quench. The data have been plotted as a function of the martensite fraction instead of the temperature. As the martensite transformation proceeds, the dislocation density increases both in martensite and in austenite as reported in [START_REF] Christien | Neutron diffraction in situ monitoring of the dislocation density during martensitic transformation in a stainless steel[END_REF]. This is obviously related to the displacive nature of the martensitic transformation and the accommodation of the phase transformation strain which affects both the transformation product and the austenitic matrix in which the transformation occurs.

(Please note that the densities have been plotted in a log scale). At the beginning of the martensite transformation, preexisting bcc phase shows already a high density of dislocations (2 10 14 m/m 3 ). This value lets us suspect that the observed ferritic phase observed prior the martensitic transformation (above 394°C) corresponds in fact to bainitic ferrite appearing just before the martensitic transformation. During the first 20% of the martensitic transformation (mainly occurring below 365°C), the dislocation density increases drastically to reach a value close to 9 10 14 m/m 3 . During the rest of the transformation, down to room temperature, the density of dislocations continues to increase but at an apparent lower rate, to reach a final value of 2 10 15 m/m 3 . This value is in good agreement with the values generally reported at RT in low carbon martensitic steels [START_REF] Krauss | Martensite in steel: strength and structure[END_REF][START_REF] Saha | Effects of tempering mode on the structural changes of martensite[END_REF].

By TEM observations in a Fe-0.18C steel, Morito et al. reported for instance mean densities around 1.1 10 15 m/m 3 [START_REF] Morito | Dislocation Density within Lath Martensite in Fe-C and Fe-Ni Alloys[END_REF]. The mean dislocation density in martensite phase is thus 10 times higher at the end of the transformation compared to the very first 5 wt.% transformed. It should be emphasized that the dispersion of values along the transformation is lower than in the study of Christien et al. [START_REF] Christien | Neutron diffraction in situ monitoring of the dislocation density during martensitic transformation in a stainless steel[END_REF]. This has been permitted by our faster acquisition rate ( 10Hz). At the fastest transformation rates (around 350°C), diffractograms and thus dislocation densities are measured every 2.5% of transformed martensite. The evolution of the dislocation density of the martensite does not present a linear tendency as reported by Christien et al.

[12], thus, a square root function was chosen to fit the experimental data by its simplicity and general acceptable fit as can be observed in Figure 4.

The evolution of the dislocation density in the austenite does not saturate before 80 wt.% of martensite is formed. The evolutions in austenite and in martensite seem correlated. Before the martensitic transformation, the density is 1.7 10 13 m/m 3 as expected from an austenitic phase annealed at high temperature. The density increases during the first 15% of transformation to reach 8 10 13 m/m 3 . After this apparent first fast regime, the density continues to increase very rapidly but at almost the same rate. It should be noted that the data scattering is higher in austenite than in martensite at the beginning of the transformation. This is related to the low density values reported in that case. However, the measured values do not evolve much during the cooling before the martensitic transformation, as shown in Figure 3(a). This permits to ascertain the determination procedure described above, as no plastic events are expected in austenite.

Discussion

The discussion will be segmented in three parts. The first will focus on the insight brought by the dislocation density measurements in the martensite phase during cooling.

The succeeding section will analyze the effect of the dislocation density on the austenite strength and the last one is dedicated to analyze the distribution of local and spatial yield strength distribution in martensite due to the distribution of dislocation densities.

Instantaneous dislocation density in martensite:

The studied steel shows at room temperature a very conventional lath martensitic microstructure as observed by [START_REF] Morsdorf | 3D structural and atomic-scale analysis of lath martensite: Effect of the transformation sequence[END_REF] on similar steels. It is now well admitted that this typical microstructure appears in a displacive way by the nucleation of new laths; gathered into blocks of similar variants and packets of coplanar variants inside prior austenite grains [START_REF] Morsdorf | 3D structural and atomic-scale analysis of lath martensite: Effect of the transformation sequence[END_REF]. As said in the introduction, the size, the dislocation structure, the segregations, the residual stresses and thus the local strength of these laths at room temperature should strongly depend on their respective transformation temperatures.

As a consequence, it is likely that the dislocations are not homogeneously distributed in the final martensitic microstructure.

The increase in the mean dislocation density in martensite during the transformation can be interpreted either by the fact that already formed laths undergo plastic deformation or by the fact that new laths present a higher dislocations density than the previous ones arising the mean value. The plastic deformation in already formed lath structures is unlikely for at least two reasons. First of all, plastic accommodations due to the transformation strain will occur to a large extent in the softest phase, i.e. here in the austenite. This is the reason why its mean dislocation density increases drastically all along the transformation contrary to martensite.

Secondly, higher degree of plastic interactions is expected after the percolation of the martensitic structure (about typically 20%-30% of transformation) [START_REF] Allain | Internal stresses and carbon enrichment in austenite of Quenching and Partitioning steels from high energy X-ray diffraction experiments[END_REF][START_REF] Nambu | In situ observations and crystallographic analysis of martensitic transformation in steel[END_REF]. Figure 3(b) shows on the contrary, a decrease in the rate of dislocation creation. As a consequence, we assume that the increase in the mean dislocation density in martensite is due the progressive nucleation of new laths containing more dislocations than the previous ones. Since the data are acquired all along the martensite phase transformation, it is possible to determine the dislocation density in these new laths, considering that the forming martensite lath is not affected by the earlier ones. It is also assumed that recovery process that could occur during martensite tempering is surely limited due to the applied cooling rate. However possible selftempering (carbon segregation, carbide precipitation) cannot be ruled out even if no carbide can be observed and identified on the diffractograms.

In the following, we call the instantaneous dislocation density ρ inst. the dislocation density in the laths forming when the martensite fraction is F. At the beginning of the transformation, the instantaneous density is equal to the mean density of course. With what is said before, this function progressively increases and contributes to increase the mean density of dislocations (ρ) in previously formed martensite F. By construction, it comes:

̅ = 1/F ∫ ρ̅ 0 F ρ inst (f) df (2)
By deriving Equation ( 2) with respect to F, the instantaneous density of dislocations at F is given by:

d(F (F)) = ρ ρ̅ inst (F)dF (3)
Using a step by step integration, the instantaneous dislocation density for martensite is plotted in Figure 4. As the transformation moves forward, the new formed martensite laths contain a higher dislocation density than the earlier ones. This increase can be explained by the fact that the first martensite fraction is formed in a soft austenitic matrix. The following martensite laths would form in a harder matrix and with higher shearing modulus (temperature decrease) [START_REF] Christien | Neutron diffraction in situ monitoring of the dislocation density during martensitic transformation in a stainless steel[END_REF][START_REF] Isaak | Elastic and viscoelastic properties of α iron at high temperatures[END_REF]. The inheritance of the austenite dislocation structure into martensite may also contribute to the total dislocation density in martensite. In fact, the final dislocation density in austenite at room temperature represents about 30% of the dislocation density expected in the very last formed laths of martensite.

Dislocations and strength of austenite

The dislocation density in the austenite before the martensitic transformation is of the order of 10 13 m/m 3 and increases with a rate higher than in the martensite during the transformation.

For the austenite phase, ρ inst is not presented as the increase in the mean dislocation density cannot reasonably be explained by a martensite transformation occurring solely in the dislocation poor austenitic grains. The increase results more probably from a progressive plasticization due to the accommodation of the transformation strains. However, it is likely that dislocations are not well and homogeneously distributed. This could explain in some extent the dispersions observed in the values at the beginning of the transformation.

The final dislocation density in austenite is 1 10 15 m/m 3 at room temperature. This density is even higher than the ones found in the very first laths of martensite. Dislocations contribute to hardening, and this contribution σ disl , can be calculated with the Taylor equation [START_REF] Taylor | Plastic Strain in Metals[END_REF]:

σ disl = αC T μbρ ̅ ½ ( 4
)
where α is a geometrical constant equal to 0.4, C T the Taylor factor equal to 3, μ is the shearing modulus (84 GPa), b and ρ are respectively the Burgers vectors (2.5 10 -10 m) of perfect dislocations in martensite and the mean dislocation density in austenite. The shear modulus was calculated at room temperature taking into account the steel composition [START_REF] Ghosh | The isotropic shear modulus of multicomponent Fe-base solid solutions[END_REF].

The contribution to hardening of dislocations due to the martensitic transformation is thus equal to ca. 795 MPa in austenite. Retained austenite in martensite is thus far from being a soft phase, especially since the previous contribution does not account for solid solution hardening and size effect. In fact, retained austenite is often located as thin films in between martensite laths.

As a consequence, austenite cannot contribute to the plastic behavior of martensitic microstructures, at least not as much as supposed by certain models [START_REF] Maresca | Deformation behaviour of lath martensite in multi-phase steels[END_REF]. Retained austenite is in fact already intensively plasticized at room temperature due to the transformation strain and thus is also a hard phase of the microstructure.

Local and spatial yield strength distribution in martensite:

The instantaneous dislocation density shown in Figure 4 The dislocation contribution to the yield strength is two times higher in the last formed martensite lath than in the first one, respectively ca. 1250 and 500 MPa.

The density of probability of yielding at a given stress caused by the dislocations density distribution (called stress spectrum f(σ)) was obtained by evaluating the dislocation hardening related to each newly formed martensite between the limits of 50 MPa bins from 0 to 4500 MPa. Once the bin in which the dislocation hardening belonged was identified (evaluated with the empirical root-law calculated instantaneous dislocation density, red dotted curve in Figure 4), the newly formed martensite fraction was summed to the fraction related to that interval.

tempering (carbide precipitation and carbon segregations) are also expected in martensite [START_REF] Hutchinson | Microstructures and hardness of as-quenched martensites (0.1-0.5%C)[END_REF] but they cannot explain the behavior of martensite by themselves as shown by Hutchinson et al. [START_REF] Hutchinson | Plastic yielding in lath martensites -An alternative viewpoint[END_REF].

Hutchinson et al. introduces calculated residual stress heterogeneities in the martensite (at the scale of the grains) resulting from the phase transformation during cooling to predict the tensile behavior of martensite microstructure [START_REF] Hutchinson | Plastic yielding in lath martensites -An alternative viewpoint[END_REF]. The main assumption of the authors was to consider that the stress gradient contribution on the martensite FWHM is alone able to describe the martensite FWHM decrease during the tensile test as observed experimentally.

However, it is well known that other contributions are present as tetragonality even for low carbon steels [START_REF] Maruyama | Excess Solute Carbon and Tetragonality in As-Quenched Fe-1Mn-C (C:0.07 to 0.8 Mass Pct) Martensite[END_REF], crystallite size, dislocation density.

We consider that the distribution of flow stress (which could describe the stress-strain curve) in the martensite is originated by the microstructure principally (laths sizes, dislocations densities); internal stresses contribute to the phenomenon without being the only explanation as reported by Wang et al. [START_REF] Wang | Strain hardening behaviour of as-quenched and tempered martensite[END_REF]. Indeed, if the FWHMs are only due to the internal stresses and decrease with the further deformation (as reported by Hutchinson et al. [START_REF] Hutchinson | Plastic yielding in lath martensites -An alternative viewpoint[END_REF]), the Bauschinger effect would decrease as well which is the opposite to the experimental observation [START_REF] Allain | Toward a New Interpretation of the Mechanical Behaviour of As-quenched Low Alloyed Martensitic Steels[END_REF][START_REF] Badinier | Effect of Carbon Segregation and Carbide Precipitation on the Mechanical Response of Martensite[END_REF][START_REF] Wang | Strain hardening behaviour of as-quenched and tempered martensite[END_REF].

Hence, it appears from this work that the local yield strength distribution in martensite is the result of a complex convolution of different mechanisms, having their own spatial distributions. However, it seems that the distribution of dislocation densities is one of the most important sources of spread (nearly one half of the distribution width), contrary to lath size distribution.

The functional form for the local yield stress spectrum in the CCA model (an Avrami type law [START_REF] Allain | Toward a New Interpretation of the Mechanical Behaviour of As-quenched Low Alloyed Martensitic Steels[END_REF]) can be selected differently based on the type of distribution found experimentally.

However, for obtaining the real local yield stress spectrum not only the distribution of dislocation densities has to be considered, but also the other previously mentioned spatial distributions. The convolution of the mentioned distributions might produce a different type of functional form than the one that might be inferred only from the distribution of the dislocation densities. As consequence, the authors do not find convenient at the moment change the functional form, while the mentioned modification might be address subsequently while analyzing the other sources of distributions.

Conclusion

The evolution of the dislocation densities in martensite and in austenite during the quench of a low-carbon (0.215wt.%C) steel has been investigated in situ by the mean of X-ray diffraction experiments on a synchrotron beamline. These measurements were conducted with the mWH methodology developed by Ungar et al. and a more conventional Rietveld refinement. The line configuration offers an excellent time resolution adapted to the kinetics of the studied steels.

The mean dislocations density in martensite was shown to increase progressively as the martensitic transformation proceeds confirming that dislocations are not homogeneously distributed between the laths, in agreement with recent post mortem observations. The resulting spatial distribution has been estimated introducing the concept of instantaneous dislocation density in martensite. The associated spread in the local yield strengths of laths has been compared to the one calculated on a micromechanical basis (CCA approach), i.e.

based on an inverse analysis from the macroscopic mechanical behavior. It was shown that microstructural heterogeneities (dislocation densities, lath sizes, carbon segregations) as well as internal stresses resulting from the transformation deformation must be taken into account to explain the unique behavior of these steels [START_REF] Wang | Strain hardening behaviour of as-quenched and tempered martensite[END_REF]. The density of dislocations appears to be a major contribution explaining almost one half of the stress distribution.

The evolution of the dislocation density in austenite has also been analyzed in details and discussed at the light of the martensite transformation strains. Low carbon martensitic steels contain significant fraction of retained austenite but this latter phase cannot be considered as a soft phase as it contains almost the same density of dislocations as the martensite. The strength of retained austenite is necessarily higher than the softest martensite lath as it contains 10 times higher density of dislocations. Hence, retained austenite cannot be considered as the interphase medium permitting to explain the plastic deformation of martensitic microstructures (if excluding TRIP effect).

of N. Schell and A. Stark was much appreciated and widely contributed to the success of the study.

The Laboratory of Excellence on Design of Alloy Metals for low-mAss Structures (Labex DAMAS) from the Université de Lorraine (France) is also fully acknowledged for its support.

Figure 1 :

 1 Figure 1: a) Cooling and dilatometric curves as function of time and b) 1D diffractograms recorded during the HEXRD in situ experiment corresponding to the temperatures highlighted on the cooling curve, respectively 880, 394 (Ms), 200 and 30 °C. The deformation and the time were set at 0 at the end of the austenitization stage.

Figure 2

 2 Figure 2 presents the progress of the martensitic transformation, i.e. the martensite weight fraction evolution during the quench as a function of the temperature by XRD. The transformation kinetics obtained by dilatometry have been plotted from 0 (before the start of the transformation) to 100 (end of detected transformation) in the same figure. The values measured by XRD encompass all ferritic phases (possible ferrite transformation before Ms and martensitic transformation). Both methods show that the transformation kinetics is

Figure 2 :

 2 Figure 2: body-centred phase fraction determined by XRD and martensite fraction measured by dilatometry as function of the temperature during the quench. The fractions obtained by dilatometry were calculated from the transformation kinetics and considering the start and end fractions equal to the XRD measurements.

Figure 3 (

 3 Figure 3(a) shows the relative change in the FWHM of austenite/retained austenite and martensite peaks during the quenching treatment as a function of the temperature. The values have been normalized by the position of the studied peaks. During the cooling between the austenitization temperature down to the Ms temperature, the FWHM of austenite is almost constant as expected, meaning no change in crystallite size and defect density in the absence of phase transformation and plastic events. The observations are consistent with the literature [12,34]. The cooling rate is too high to enable recovery process. During the first stage of martensite transformation (not following the Koistinen and Marburger's athermal equation), between 394 and 365°C, no notable changes are observed while below 365°C the austenitic FWHMs of all diffraction peaks increase drastically. As the martensite transformation progresses, the austenite FWHMs continue to increase down to the room temperature. For martensite, the observed behavior is rather different. Due to very low peak intensities at the beginning of the transformation, up to 5 wt.% of body-centered phase, the uncertainties are high. The measured FWHMs are thus not relevant down to 365°C. Below this true Ms

Figure 3 :

 3 Figure 3: a) FWHMs of austenite (dotted lines) and martensite (continuous lines) diffraction peaks as function of the temperature during the whole studied cooling sequence (after austenitization down to room temperature) and b) deduced mean dislocation densities in both martensite (circle) and austenite (triangles) as a function of the transformed martensite phase fraction.

Figure 4 :

 4 Figure 4: Mean dislocation density (filled circles) and instantaneous dislocation density calculated with ca. 2 wt.% steps (hollow circles) in martensite as a function of the martensite phase fraction during the quenching. The black discontinuous line corresponds to an empirical square-root law calibrated on the experimental results for a better readability of the results, while the red dotted continuous line corresponds to the analytical solution related to the empirical square-root law calibrated.

  introduced a dislocation density distribution inside the martensite laths. The first formed at high temperatures (lower martensite fraction) show a lower dislocation density than those formed at lower temperature (higher martensite fraction). As a consequence, the contribution of dislocations to their respective hardening are surely different. Applying again the Taylor equation to the instantaneous density of dislocations instead of the average dislocation density permits to estimate the related distribution of strengths in the microstructure. The parameters used are α=0.4, C T =3, b=2.5 10 -10 m, and µ the shearing modulus for a martensite with the initial steel composition (76 GPa). The method presented in the present work allows to estimate the dislocation hardening contribution to the yield strength in the forming martensite during the whole cooling, as shown in the Erreur : source de la référence non trouvée.Erreur : source de la référence non trouvée(a) shows the dislocation contribution to the yield strength of the instantaneous dislocation density, for increasing F values. The Taylor equation was applied considering each instantaneous dislocation density while the newly formed phase fraction was summed to the total transformed martensite, allowing to show the increase of the dislocation strengthening with the martensite transformation. As expected, a similar trend with the instantaneous dislocation density in martensite phase is observed.

Figure 5 :

 5 Figure 5: a) The dislocation contribution to the yield stress for each newly formed martensite fraction b) The density probability to find a lath with a given local yield stress considering only the measured dislocation. These experimental values are compared to the expected stress distribution necessary to explain the mechanical behavior of the studied steel according to [2].

Figure 7 :

 7 Figure 7: Mean dislocation density (filled circles) and instantaneous dislocation density (hollow circles) in martensite as a function of the martensite phase fraction during the quenching. The black discontinuous line corresponds to an empirical square-root law calibrated on the experimental results for a better readability of the results, while the red dotted continuous line corresponds to the analytical solution related to the empirical square-root law calibrated.

Figure 8 :

 8 Figure 8: a) The dislocation contribution to the yield stress for each newly formed martensite fraction b) The density probability to find a lath with a given local yield stress considering only the measured dislocation. These experimental values are compared to the expected stress distribution necessary to explain the mechanical behavior of the studied steel according to [2].

Figure 10 :

 10 Figure 10: Mean dislocation density (filled circles) and instantaneous dislocation density (hollow circles) in martensite as a function of the martensite phase fraction during the quenching. The black discontinuous line corresponds to an empirical square-root law calibrated on the experimental results for a better readability of the results, while the red dotted continuous line corresponds to the analytical solution related to the empirical square-root law calibrated.

Figure 11 :

 11 Figure 11: a) The dislocation contribution to the yield stress for each newly formed martensite fraction b) The density probability to find a lath with a given local yield stress considering only the measured dislocation. These experimental values are compared to the expected stress distribution necessary to explain the mechanical behavior of the studied steel according to [2].

Table 1 :

 1 Dislocation contrast factors at room temperature for both ferrite and austenite considering a ratio of 50% edge and 50% screw type dislocations.

	Ferrite	{200}	{211}	{220}	{321}
	C̅ hkl	0.278	0.141	0.141	0.141
	Austenite	{200}	{220}	{311}	{420}
	C̅ hkl	0.299	0.148	0.204	0.202

Acknowledgements

This work was funded by the i-SITE Lorraine Université d'Excellence program (LUE) and supported by ArcelorMittal Maizières les Metz (Product Research Centre). The HEXRD experiments were conducted at DESY (PETRAIII-P07 beamline) in Hamburg under the I-20180973 EC grant. A special thanks is dedicated to the team of the P-07 line. The expertise

Data availability

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

By this method, at the end of the transformation, it was possible to obtain the total martensite fraction for each bin of 50 MPa. The following step was to normalize the integral of the fractions versus the local yield strength in order to obtain a distribution of probability which integral would be equal to 1 (the whole martensite transformed).

The mentioned density of probability has been plotted and presented Erreur : source de la référence non trouvée(b). The first spike in the obtained distribution is due to the fact that in the first calculated point the amount of fraction already transformed is significant.

The obtained distribution has been compared to the stress spectrum calculated by the Continuous Composite Approach (CCA) [START_REF] Allain | Toward a New Interpretation of the Mechanical Behaviour of As-quenched Low Alloyed Martensitic Steels[END_REF] to explain the behavior of the studied steels. The stress spectrum, f(σ), in the CCA is obtained by reverse resolution of the following equation:

where Σ is the macroscopic stress, σ min is the minimum value in the local yield stress spectrum, σ L is the stress state of phases remaining elastic.

The CCA distribution presented in this work is obtained by the calculated tensile behavior of the studied alloy (0.215wt.% C) with [START_REF] Allain | Toward a New Interpretation of the Mechanical Behaviour of As-quenched Low Alloyed Martensitic Steels[END_REF]. In the CCA, all the contributions affect the distribution as solid solution strengthening due to carbon and substitutional alloying elements, friction of pure iron, internal stresses, dislocation densities. It explains the initial shift at higher strength of the distribution expected from the CCA compared to the distribution obtained by the dislocation density distribution alone. It also clearly appears that the ratio between the harder and the softer fraction formed (ca. 2.5 considering the dislocation distribution only) is lower than the actual ratio expected by the CCA model (ca. 6). The higher values of the density of probability of spectrum due to the dislocation hardening are due to that both curves are normalized and the lower ratio of this spectrum.

The lower ratio of the spectrum due to the dislocation density distribution means that the measured distributions of dislocations in martensite is not sufficient alone to explain the micromechanical behavior of the steel. Dislocation strengthening is not the sole contribution of the local yield of martensitic laths. As shown by Morsdorf et al. [START_REF] Morsdorf | 3D structural and atomic-scale analysis of lath martensite: Effect of the transformation sequence[END_REF], the lath size effect must evidently be considered, but according to Badinier [START_REF] Badinier | Effect of Carbon Segregation and Carbide Precipitation on the Mechanical Response of Martensite[END_REF], the obtained spread of local yield is far from being sufficient to explain the stress spectrum. Different degrees of self-

Conflicts of Interest

The authors declare no conflict of interest. The funders had no role in the design of the study;

in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

Supplementary data

Three experiments were conducted to assess the reproducibility. All the experiments were investigated as describe in the present paper. The FWHM evolutions and mean dislocation density are shown for both martensite and austenite phases during the cooling treatment. In addition the mean dislocation and instantaneous dislocation densities were also show. As one experiment were detailed in the paper, here only the two other experiments results are displayed.

Experience 1


Uli Castanet

Cédric Feral-Martin

Alain Demourgues

Rachel L Neale

Dean C Sayle

Francesco Caddeo

Joseph M Flitcroft

Robert Caygill

Ben J Pointon

Marco Molinari

Jerome Majimel

Controlling the {111}/{110} surface Ratio of Cuboidal Ceria Nanoparticles

Keywords: ceria nanocube, faceting, liquid-like catalysis, molecular modelling, ceria catalysis, ceria nanoparticle

The ability to control size and morphology is crucial in optimizing nanoceria catalytic activity as this is governed by the atomistic arrangement of species and structural features at the surfaces.

Here, we show that cuboidal cerium oxide nanoparticles can be obtained via microwave-assisted hydrothermal synthesis in highly alkaline media. HRTEM revealed that the cube edges were truncated by CeO2{110} surfaces and the cube corners by CeO2{111} surfaces. When adjusting synthesis conditions by increasing NaOH concentration, the average particle size increased.

Although this was accompanied by an increase of the cube faces, CeO2{100}, the cube edges, CeO2{110}, and cube corners, CeO2{111} remained of constant size. Molecular Dynamics (MD) was used to rationalise this behaviour and revealed that energetically, the corners and edges cannot be atomically sharp, rather they are truncated by {111} and {110} surfaces respectively to stabilise the nanocube; both experiment and simulation agreed a minimum size of ~1.6 nm associated with this truncation. Moreover, HRTEM and MD revealed {111}/{110} faceting of the {110} edges, which balances the surface energy associated with the exposed surfaces, which follows {111}>{110}>{100}, although only the {110} surface facets because of the ease of extracting oxygen from its surface, which follows {111}>{100}>{110}. Finally, MD revealed that the {100} surfaces are 'liquid-like' with a surface oxygen mobility 5 orders of magnitude higher than that on the {111} surfaces; this arises from the flexibility of the surface species network that can access many different surface arrangements due to very small energy differences. This finding has implications for understanding the surface chemistry of nanoceria and provides avenues to rationalize the design of catalytically active materials at the nanoscale.

Introduction

Since the first nanoparticle syntheses, research has focused on producing various nano-catalysts of different compositions and applications [START_REF] Sayle | Environment-mediated structure, surface redox activity and reactivity of ceria nanoparticles[END_REF][START_REF] Morgan | Protecting Ceria Nanocatalysts -The Role of Sacrificial Barriers[END_REF][START_REF] Rodriguez | Ceria-based model catalysts: fundamental studies on the importance of the metal-ceria interface in CO oxidation, the water-gas shift, CO2 hydrogenation, and methane and alcohol reforming[END_REF] . Over the years, the importance of their shape, hence their exposed facets, grew in interest [START_REF] Bhatta | Morphology and Surface Analysis of Pure and Doped Cuboidal Ceria Nanoparticles[END_REF][START_REF] Li | Morphology-dependent nanocatalysts: Rod-shaped oxides[END_REF][START_REF] Trovarelli | Ceria Catalysts at Nanoscale: How Do Crystal Shapes Shape Catalysis?[END_REF] . The differences between facets, may sometimes remain unclear and controversial, but more and more effects in terms of reactivity and/or selectivity are shown in recent works [START_REF] Li | Morphology-dependent nanocatalysts: Rod-shaped oxides[END_REF][START_REF] Lee | Utilization of shape-controlled nanoparticles as catalysts with enhanced activity and selectivity[END_REF][START_REF] Sayle | Strain and Architecture-Tuned Reactivity in Ceria Nanostructures[END_REF][START_REF] Sayle | Mechanical properties of ceria nanorods and nanochains; the effect of dislocations, grain-boundaries and oriented attachment[END_REF][START_REF] John Chandler | Studies in Environmental Science[END_REF][START_REF] Karakoti | Redox-active radical scavenging nanomaterials[END_REF][START_REF] Sreeremya | Shape-Selective Oriented Cerium Oxide Nanocrystals Permit Assessment of the Effect of the Exposed Facets on Catalytic Activity and Oxygen Storage Capacity[END_REF] . Furthermore different morphologies are active towards oxidative catalysis at different temperatures, with nanocubes exposing the {100} surfaces being active at temperatures as low as 150°C [START_REF] Zhang | Extra-low-temperature oxygen storage capacity of CeO2 nanocrystals with cubic facets[END_REF] .

CeO2 is now well known for its catalytic redox properties [START_REF] Trovarelli | Catalysis by Ceria and Related Materials[END_REF] , associated or not with other materials (Au on a reconstructed ceria surface [START_REF] Tinoco | Critical Influence of Nanofaceting on the Preparation and Perormance of Supported Gold Catalysts[END_REF] and other noble metals, TiO2) especially towards CO oxidation, even at low temperature. After years of research, numerous shapes of ceria nanoparticles became available, as described in several papers [START_REF] Trovarelli | Ceria Catalysts at Nanoscale: How Do Crystal Shapes Shape Catalysis?[END_REF][START_REF] Tinoco | Critical Influence of Nanofaceting on the Preparation and Perormance of Supported Gold Catalysts[END_REF][START_REF] Florea | Three-Dimensional Tomographic Analyses of CeO2 Nanoparticles[END_REF][START_REF] Si | Shape and crystal-plane effects of nanoscale ceria on the activity of Au-CeO2 catalysts for the water-gas shift reaction[END_REF][START_REF] Mai | Shape-Selective Synthesis and Oxygen Storage Behavior of Ceria Nanopolyhedra, Nanorods, and Nanocubes[END_REF][START_REF] Asahina | Exploitation of Surface-Sensitive Electrons in Scanning Electron Microscopy Reveals the Formation Mechanism of New Cubic and Truncated Octahedral CeO2 Nanoparticles[END_REF][START_REF] Yang | Surface Faceting and Reconstruction of Ceria Nanoparticles[END_REF][START_REF] Crozier | In situ environmental TEM studies of dynamic changes in cerium-based oxides nanoparticles during redox processes[END_REF][START_REF] Lin | Imaging the Atomic Surface Structure of CeO2 Nanoparticles[END_REF] like rods, octahedrons, cubes, etc.

Among those particle shapes, the rods seem to be the most reactive nanoscale morphology, mainly because of the inherent amount of defects and the exposed facets, which are extremely reactive [START_REF] Mai | Shape-Selective Synthesis and Oxygen Storage Behavior of Ceria Nanopolyhedra, Nanorods, and Nanocubes[END_REF][START_REF] Li | Morphology-dependent nanocatalysts: Rod-shaped oxides[END_REF] . The exposed {110} surfaces of nanorods have been shown to undergo reconstruction, with {111}-type faceting, providing an explanation for the higher surface reactivity observed [START_REF] Yang | Surface Faceting and Reconstruction of Ceria Nanoparticles[END_REF] .

However, these morphologies are so reactive that they tend to evolve under catalytic conditions, lowering or even inhibiting reactivity, by encapsulation of deposited precious metals and cocatalysts [START_REF] Asahina | Exploitation of Surface-Sensitive Electrons in Scanning Electron Microscopy Reveals the Formation Mechanism of New Cubic and Truncated Octahedral CeO2 Nanoparticles[END_REF][START_REF] Yang | Surface Faceting and Reconstruction of Ceria Nanoparticles[END_REF][START_REF] Crozier | In situ environmental TEM studies of dynamic changes in cerium-based oxides nanoparticles during redox processes[END_REF][START_REF] Lin | Imaging the Atomic Surface Structure of CeO2 Nanoparticles[END_REF][START_REF] Li | Morphology-dependent nanocatalysts: Rod-shaped oxides[END_REF] . For example shape-dependent activity of nanoceria has been demonstrated in in soot oxidation [START_REF] Aneggi | Chape-Dependant Activity of Ceria in Soot Combustion[END_REF] . Cubes remain more stable than rods and exhibit mostly {001} facets, where the surface mobility of Ce ions can be controlled by the chemical atmosphere [START_REF] Bhatta | Morphology and Surface Analysis of Pure and Doped Cuboidal Ceria Nanoparticles[END_REF][START_REF] Bugnet | Direct Visualization and Control of Atomic Mobility at {100} Surfaces of Ceria in the Environmental Transmission Electron Microscope[END_REF] . Cubes are more reactive for CO oxidation than the major {111} facets encountered on octahedrons [START_REF] Mai | Shape-Selective Synthesis and Oxygen Storage Behavior of Ceria Nanopolyhedra, Nanorods, and Nanocubes[END_REF][START_REF] Wu | On the structure dependence of CO oxidation over CeO2 nanocrystals with well-defined surface planes[END_REF] . Hence, cubes seem to be a good compromise between reactivity, stability and a promising morphology for further development.

As shown by earlier work, cuboidal cerium oxide nanoparticles can be obtained via microwaveassisted hydrothermal synthesis working on highly alkaline medium made of a mixture of aqueous sodium and ammonium hydroxide [START_REF] Florea | Three-Dimensional Tomographic Analyses of CeO2 Nanoparticles[END_REF] . By using the ability of electron tomography to solve the third dimension of the object under study, main microstructural features of these nanocubes were evidenced: existence of a cubic shape with {002} facets, {011} (faceted) edges and {111} corners.

A fitting procedure applied to the external surface of the particles allowed a precise determination of their aspect ratio, also revealing that they should be considered as parallelepipeds rather than isotropic cubes, as other groups have also shown [START_REF] Bhatta | Morphology and Surface Analysis of Pure and Doped Cuboidal Ceria Nanoparticles[END_REF][START_REF] Xu | Reconstruction of 3D morphology of polyhedral nanoparticles[END_REF] . Here we chose to combine simulations and experimental data in order to provide further insights on those cuboidal nanoparticle morphological features.

Experimental

CeO2 cuboidal particles microwave heating-hydrothermal synthesis assisted

Syntheses were performed using MARS-5 and Masterwave BTR microwave digestion systems (CEM and Anton-Paar Corp. respectively). Temperature was regulated by percent increments of the microwave power (300 W, 2.45 GHz frequency) and controlled by an optical fibre.

All reagents were analytical grade and use without further purification. To obtain pure cerium oxide sample, 1.74 ml of Ce(NO3)3 (496 g.L -1 ) were dissolved in 50 mL of distilled water. This solution was added drop by drop at room temperature and under vigorous agitation in the basic medium (pH=14) composed by a mixture of NaOH (carlo erba reagenti 35%) and NH4OH (B.T. baker 30%) and placed in a closed Teflon container (100 mL) for microwave hydrothermal synthesis (XP-1500 plus model). The system was heated at 180°C for different reaction times ranging from 5 to 45 min and then cooled down at room temperature. The precipitate was collected by centrifugation at 5000 rpm for 10 min, washed three times with distilled water and dried overnight at 100°C. Table 1 summarizes all the information about synthesis parameters used to obtain the desired morphologies. The different compounds were characterized by X-Ray powder diffraction (XRD) using a

Philips PW 1050 diffractometer in a Bragg-Brentano geometry with Cu KR radiations of 1.54059 and 1.54441 Å (KR1 and KR2 respectively). The intensity data were collected at room temperature aver a 2θ range of 8-80° with 0.02 steps and an integration time of 70 sec. Pattern profile matching was performed with the FULLPROF program 33 .

Molecular Modelling

Molecular dynamics calculations use the potential model of Minervini et al. [START_REF] Minervini | Defect cluster formation in M2O3-doped CeO2[END_REF] and are carried out using the DL_POLY code [START_REF] Smith | 0: A general-purpose parallel molecular dynamics simulation package[END_REF] . Model structures are made using the METADISE code [START_REF] Watson | Atomistic simulation of dislocations, surfaces and interfaces in MgO[END_REF] .

The minimized bulk structure was cut along Miller indices to obtain the {111}, {011}, and {100}

surfaces. 3D boundary conditions were used throughout, and hence, the surfaces were modelled using the slab method [START_REF] Oliver | Computer simulation of the morphology of NiO[END_REF] in which a finite number of crystal layers is used to generate two identical surfaces via the introduction of a vacuum gap (~40 Å) perpendicular to the surface to minimize the interaction between images. The {100} and {110} slabs with a p(4 × 4) and p(4 × 3) expansion of the surface unit cell included 10 and 12 repeat units respectively, while the {111} slab with a p(3 × 4) expansion included 8 repeat units. The 3 outermost layers of each slabs are allowed to relax whereas the middle of the slab is kept fixed. All slabs were equilibrated in the temperature range of 1500 K -3600 K at 300 K intervals for 0.5 ns with a time step of 1fs in an NVT ensemble using the Nose -Hoover thermostat. All equilibrated configurations were then run for extra 5 ns allowing for data collection.

The generation of a model of a ceria cuboidal nanoparticle with molecular dynamics simulations has been achieved by simulating amorphisation and re-crystallization (A&R) [START_REF] Sayle | Environment-mediated structure, surface redox activity and reactivity of ceria nanoparticles[END_REF][START_REF] Bhatta | Morphology and Surface Analysis of Pure and Doped Cuboidal Ceria Nanoparticles[END_REF][START_REF] Sayle | Shape of CeO2 nanoparticles using simulated amorphisation and recrystallisation[END_REF] . However this has not been trivial, as the ceria nanoparticles naturally crystallize exposing the most stable {111} surface, rendering a polyhedral shape. In order to avoid this, a crystalline seed of fluorite structure with cubic shape has been used in order to direct the crystallization. A cube of CeO2 comprising of 20736 atoms (6912 Ce atoms, 13824 O atoms) (Fig. 1a) was melted holding fixed a cubic seed comprising of 12000 atoms (4000 Ce, 8000 O) in the centre of the nanoparticle performing constant volume MD at 8000 K for 50 ps (Fig. 1b). Crystallization was then performed applying constant volume MD simulation at 1500 K for 300 ps, allowing the atoms of the seed to move (Fig. 1c). The nanocube has then been annealed performing constant volume MD at 3750 K for 28 ns. During this time the eventual transition to a polyhedral shape has not been observed, supporting the relative stability of the cuboidal shape observed experimentally. The nanoparticle was then cooled down performing constant volume MD at 1K for 20 ps (Fig. 1d). All A&R simulations used a time step of 4 fs in an NVT ensemble using the Nose -Hoover thermostat. previously reported HRTEM images for cuboidal ceria nanoparticles found in the literature [START_REF] Trovarelli | Ceria Catalysts at Nanoscale: How Do Crystal Shapes Shape Catalysis?[END_REF][START_REF] Aneggi | Chape-Dependant Activity of Ceria in Soot Combustion[END_REF][START_REF] Bugnet | Direct Visualization and Control of Atomic Mobility at {100} Surfaces of Ceria in the Environmental Transmission Electron Microscope[END_REF] .

Figure 3 shows the size distributions of ceria particles. They were obtained by counting more than a hundred particles from TEM images for each sample and the measured particle size corresponds to an average of both projected sides for each cuboid. Mean particle sizes range from 10.4 to 23.5 nm with increasing NaOH/NH4OH bases ratio while standard deviation enlarges (Fig.

3) from 13.5% to 16.6%. One has to note the obvious effect of stirring, as the solvent becomes viscous with increasing NaOH proportion. Particles prepared via a synthesis route using a more efficient stirring (Anton-Paar Masterwave BTR system), due to differences in reactor volume and shape, show a decrease in median and average size (see orange and purple histograms on Fig. 3), compared to the same synthesis conditions, but with a less effective stirring. This indicates that a better stirring favours nucleation at the expense of growth of the particles. This feature has to be taken in account for future microwave-assisted syntheses, when one is targeting a given particle size. We show the surface structure of ceria nanocubes characterized by HRTEM (Fig. 5a). The faceting of the lateral facet of a cube (Fig. 5b) show a characteristic irregular {111}/{110} subfaceting in the {110} mean surface plane. The faceting observed here is consistent with other studies of ceria nanoparticles in the literature, which observed faceting of both nanorods enclosed by {110} surfaces [START_REF] Yang | Surface Faceting and Reconstruction of Ceria Nanoparticles[END_REF] , nanopowders [START_REF] Crozier | In situ environmental TEM studies of dynamic changes in cerium-based oxides nanoparticles during redox processes[END_REF] and nanocubes [START_REF] Tinoco | Critical Influence of Nanofaceting on the Preparation and Perormance of Supported Gold Catalysts[END_REF][START_REF] Lin | Imaging the Atomic Surface Structure of CeO2 Nanoparticles[END_REF][START_REF] Bugnet | Direct Visualization and Control of Atomic Mobility at {100} Surfaces of Ceria in the Environmental Transmission Electron Microscope[END_REF] .

In Fig. 5c the surface morphology of the simulated ceria cuboidal nanoparticle is shown. The cuboidal nanoparticle exposes mostly the {100} surface and it is truncated at the edge and corners exposing {110} and {111} surfaces, respectively. However, the simulations reveal that the edges are not wholly flat, but faceted [START_REF] Bezkrovnyi | Thermally induced reconstruction of ceria nanocubes into zigzag {111}-nanofacetted structures and its influence on catalytic activity in CO oxidation[END_REF][START_REF] Cao | Getting Insights into the Influence of Crystal Plane Effect of Shaped Ceria on Their Catalytic Performances[END_REF] : the {110} surface in this case can be described as a faceted {111}/{110} (Fig. 5d). In Fig. 5e and 5f two projection along the <111> and <100> directions of the crystal respectively are shown; corrugation (i.e. faceting) at the edges is visible for the projection along the <111> direction. The lengths of all projected corners of the model have been measured, being between 1.5 and 1.7 nm, in agreement with experiments. This minimum length of projected edges and corners stabilizes the cuboidal morphology; it is five times the inter-planar spacing of the {111} surface (0.32 nm 42 ) as pointed out by both experiments and simulations. As this minimal length is a necessary requirement to stabilize the cube, it is not proportional/dependent to the size of the cube. As a consequence all corners must have at least three sides that are no smaller than 1.6 nm (Fig. 5c-f). To confirm the existence of {111}/{110} faceting on the {110}, we have performed MD calculations on the three surface, i.e. {100}, {110} and {111} separately. The {111} surface shows no faceting or reconstruction at any temperature, whereas the {100} shows a very mobile liquidlike surface layer with both the Ce and O ions being highly mobile [START_REF] Sayle | Environment-mediated structure, surface redox activity and reactivity of ceria nanoparticles[END_REF][START_REF] Mullins | The surface chemistry of cerium oxide[END_REF] .

The {100} surface is the most unstable of the three low Miller Index surfaces of ceria, with the order of stability being {111} > {110} > {100} [START_REF] Molinari | Water Adsorption and Its Effect on the Stability of Low Index Stoichiometric and Reduced Surfaces of Ceria[END_REF] . According to the Tasker notation [START_REF] Tasker | The stability of ionic crystal surfaces[END_REF] , the {110} is a type 1 surface, the {111} is a type 2 surface and the {100} is a type 3 surface. The type 1 {110}

surface is neutral with double O species per cation on each plane, which also correspond to the surface repeat unit. The type 2 {111} surfaces has charged planes, although the surface repeat unit is charge neutral, thus there is no dipole moment perpendicular to the surface. Both these surfaces show relatively low surface energy, with the {111} surface displaying the lowest as it contains 7fold coordinated surface Ce cations, compared to the 6-fold coordinated surface Ce cations on the {110} surface. The type 3 {100} surface is charged and has a dipole moment in the repeat surface unit perpendicular to the surface. Type 3 surfaces are stabilized by reconstruction to quench the dipole. This reconstruction occurs in our MD simulations. The liquid-like behavior of the {100} surface arises from the flexibility of the surface oxygen network, and so of the surface Ce ions, which can easily access many different surface oxygen arrangements due to very small energy differences between these arrangements [START_REF] Mullins | The surface chemistry of cerium oxide[END_REF] . The {110} and the {111} surfaces do not display this behavior and as such they do not show a liquid-like diffusion behavior.

The origin of the faceting is a balance between the surface energy of the ceria surface which follows the {100} > {110} > {111} and the heat of reduction of the surface which follows the {111} > {100} > {110} [START_REF] Molinari | Water Adsorption and Its Effect on the Stability of Low Index Stoichiometric and Reduced Surfaces of Ceria[END_REF] . As the {111} surface is the most stable any surface that undergoes faceting will favour its appearance followed by the {110}. However it is only the {110} surface The diffusion of surface oxygen species affects surface structure and availability of oxygen species to catalytic activity [START_REF] Lawrence | Defect Engineering in Cubic Cerium Oxide Nanostructures for Catalytic Oxidation[END_REF] . To note is the liquid-like behaviour of the {100} surface. Indeed oxygen diffuses faster on the intrinsically defective {100} surface compared to the {110} and {111} surfaces [START_REF] Sayle | Environment-mediated structure, surface redox activity and reactivity of ceria nanoparticles[END_REF][START_REF] Mullins | The surface chemistry of cerium oxide[END_REF] . At 2400 K oxygen diffusion at the {100} surface is 1 and 5 order of magnitude higher than on the {110} and {111} surfaces; these remain at 1 order of magnitude higher for the {110} and reduce to 2 orders of magnitude higher for the {111}, at 3600 K, compared to the {100}. HRTEM and molecular dynamics calculations provide evidence that there is a minimum length of the {110} edges, which is 5 times (~1.6nm) the inter-planar spacing of the {111} surface, independently on particle size. Furthermore the {110} surfaces are faceted surfaces expressing {111} facets. This increases the nanoparticle surface areas proportionally to the inter-planar spacing of the {111} surface.

These findings allow for a better understanding of the surface chemistry of nanoceria, and gives ways to a rationalization of the design and engineering of active surfaces for materials design, particularly in views of face-selective catalysis [START_REF] Trovarelli | Ceria Catalysts at Nanoscale: How Do Crystal Shapes Shape Catalysis?[END_REF] .

Future work should focus on testing the catalytic activity of these morphologies as well as on the effect of a chemical atmosphere on the structural changes of nanocubes synthesized via simulated amorphisation and re-crystallization and via hydrothermal microwave-assisted synthesis.

Figure 1 . 3 .

 13 Figure 1. Amorphisation and crystallization of a ceria cubic nanoparticle; a) a pristine cube of

Figure 2 .Figure 3 .

 23 Figure 2. HRTEM micrograph of pure CeO2 cuboidal nanoparticles (sample Ce1) and its

Figure 4 .

 4 Figure 4. (Color online) Typical 3D cube models showing {001} faces (blue), {011} edges (green)

Figure 5 .

 5 Figure 5. (a) HRTEM micrograph of a lateral facet of a cube (Ce5 sample) along a <110> zone

Figure 6 . 3 . 3 Figure 7 .

 6337 Figure 6. Schematic of different surface facets. a)

Figure 8 .

 8 Figure 8. Section of the nanocube along one of the edges showing the dynamic mobility of cerium

Table 1

 1

	. Microwave-assisted synthesis conditions (45 min / 180 °C) and corresponding
	microstructural characterizations of pure cuboidal CeO2 samples

Ref. sample %NaOH/ %NH4OH Median cube side length (nm) Standard deviation (nm)

	Ce1	5/95	10.4	1.4
	Ce2	25/75	13.3	2
	Ce3	50/50	16.5	2.5
	Ce4	75/25	18.5	2.5
	Ce5	95/5	23.5	3.9
	2.2 Characterization and measurement
	Samples for transmission electron microscopy (TEM) were prepared by suspending the powders
	in alcohol by ultrasonication and depositing a drop of the suspension on a copper grid covered
	with a carbon film. The grid was finally air-dried for 15 min. TEM, high-resolution TEM

(HRTEM) experiments were performed on a JEOL 2200 FS equipped with a field emission gun operating at 200 kV with a point-to-point resolution of 0.23 nm. High-resolution micrographs and videos were acquired with a Gatan Ultrascan CCD 2kx2k camera and digital diffractograms and calculated with the Gatan Digital Micrograph software. Moreover, in order to be representative and statistically meaningful, many images from several regions of various samples were recorded and the most characteristic results are presented here.

Acknowledgements

The authors would like to thank the PLACAMAT structure (University of Bordeaux-CNRS) for its electron microscopy facilities and the Anton-Paar company for its efficient scientific support. EPSRC (EP/R010366/1); Collaborative Computational Project 5 (CCP5) funded via EPSRC (EP/J010480/1); Orion Computing facility at the University of Huddersfield; ARCHER UK National Supercomputing Service (http://www.archer.ac.uk) via the HEC Materials Chemistry Consortium funded via EPSRC (EP/L000202, EP/R029431); THOMAS, UK Materials and Molecular Modelling Hub (MMM Hub) for computational resources (EPSRC -EP/P020194/1).

Notes and references 20

Corresponding authors: * M. Molinari -M.Molinari@hud.ac.uk J. Majimel -majimel@icmcb-bordeaux.cnrs.fr

cuboids on Fig. 2). If we consider particles as perfect cubes with a as the cube length and c the truncation width, facets, edges and corners surfaces can be written as follows:

Calculated contributions of those three surfaces to the overall particle area are plotted on Fig. 4.

By keeping constant the size of the truncation, {001}area/{011}area ratio can be modulated from approximately 2.1 to 9.1 for smaller and larger cuboids respectively. The low energy {111} planes corresponding to cubes corners represent 5.1% of the cube surface for 5 nm-sized particles and can be lowered to 0.2% when the size is reaching 30 nm. It is worth noting that these values seem to adopt an asymptotic behaviour and can no longer be significantly adjusted for cuboid sizes beyond 30 nm. This suggests that a particle size of 30 nm may be a cut off for exploitation of enhanced catalytic activity. This "critical size" sees nanoparticles that are large enough to have a significant portion of the {100} surface, but they are still small enough to have a sufficient total surface area, i.e. high surface to volume ratio, ideal for catalysis [START_REF] Tinoco | Critical Influence of Nanofaceting on the Preparation and Perormance of Supported Gold Catalysts[END_REF][START_REF] Aneggi | Chape-Dependant Activity of Ceria in Soot Combustion[END_REF][START_REF] Vile | Opposite Facve Sensitivity in CeO2 in Hydrogenation and Oxidation Catalysis[END_REF][START_REF] Bezkrovnyi | Thermally induced reconstruction of ceria nanocubes into zigzag {111}-nanofacetted structures and its influence on catalytic activity in CO oxidation[END_REF] .

that undergoes faceting due to the ease of removing oxygen from its outermost layer. Visual inspections shows that oxygen species pop out the pristine flat surface and build up small ridges that then tend to crystallize into the faced {111}/{110} surface as Ce species are following the move. We noticed that faceting only occurs for the {110} surface at temperature above 2400 K.

We can use the inter-planar spacing values for the {111}, {110} and {100} surfaces to define the surface unit areas of the pristine flat {110} surface and the faceted {110}/{111} surface:

, {ℎ } is the number of Ce ions (must be > 1) present at the {hkl} surface and |ℎ | represents the inter-planar spacing of the {hkl} surface (i.e. 0.32 nm, 0.19 nm and 0.27 nm for the {111}, {110} and {100} surfaces respectively [START_REF] Jeyaranjan | Morphology and Crystal Planes Effects on Supercapacitance of CeO2 Nanostructures: Electrochemical and Molecular Dynamics Studies[END_REF] ). For the faceted surface the total area is a summation

of all the {hkl} surface facets (i.e. the {111} and {110} surfaces that make up the faceting. The difference between the surface unit area of the pristine flat {110} surface and the faceted {110}/{111} surface gives the excess surface unit area gained by the nanoparticle:

Conflict of Interest

There are no conflicts to declare.

Author