Xavier Sauvage
email: xavier.sauvage@univ-rouen.fr

Fabien Cuvilly

Alan Russell

Kaveh Edalati

Understanding the role of Ca segregation on thermal stability, electrical resistivity and mechanical strength of nanostructured aluminum

Keywords: aluminum, nanostructure, grain boundary segregation, electrical resistivity, severe plastic deformation

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Understanding the role of Ca segregation on thermal stability, electrical resistivity and mechanical strength of nanostructured aluminum

1-Introduction

The design of low-weight electrical conductors with a high specific mechanical strength combined with a high electrical conductivity is of interest not only for automotive and aircraft industries but also for power transmission lines. Aluminum, with a relatively low production cost, an electrical conductivity of about 62% IACS (International Annealed Copper Standard) and a density of about 2.7 against 8.9 g.cm -3 for Cu, is the most interesting candidate [START_REF]Overhead Power Lines: Planning, Design[END_REF][START_REF]Prospects for novel deformation processed Al/Ca composite conductors for overhead high voltage direct current (HVDC) power transmission composite[END_REF]. However, like copper, the low mechanical strength of pure Al requires advanced microstructural design to reach high mechanical strength without substantially degrading the electrical properties. Moreover, due to its higher electrical resistivity (leading to more energy dissipation by Joule effect) and relatively low recrystallization temperature, special attention should be given to optimizing the microstructure's thermal stability.

Classical precipitation hardening is sometimes used for power transmission lines using Al-Mg-Si alloys [START_REF]Overhead Power Lines: Planning, Design[END_REF], but fine precipitate particles scatter electrons, causing significant decreases in electrical conductivity. It has been demonstrated however that a similar or even a higher mechanical strength could be achieved in ultrafine grain (UFG) or nanoscaled Al [START_REF] Xu | Strain-induced microstructure refinement in pure Al below 100 nm in size[END_REF][START_REF]High-pressure torsion of aluminum with ultrahigh purity (99.9999%) and occurrence of inverse Hall-Petch relationship, Yuki Ito[END_REF][START_REF] Haque | Mechanical behavior of 30-50 nm thick aluminum films under uniaxial tension[END_REF][START_REF] Xu | Tensile nanomechanics and the Hall-Petch effect in nanocrystalline aluminium[END_REF][START_REF] Choi | Tensile behavior of bulk nanocrystalline aluminum synthesized by hot extrusion of ball-milled powders[END_REF]. This is the well-known grain boundary (GB) strengthening as described by the Hall and Petch Law [START_REF] Xu | Tensile nanomechanics and the Hall-Petch effect in nanocrystalline aluminium[END_REF][START_REF] Choi | Tensile behavior of bulk nanocrystalline aluminum synthesized by hot extrusion of ball-milled powders[END_REF][START_REF] Hansen | Hall-Petch relation and boundary strengthening[END_REF]. For example, nanocrystalline sputtered Al films exhibit a yield stress higher than 400 MPa [START_REF] Haque | Mechanical behavior of 30-50 nm thick aluminum films under uniaxial tension[END_REF]. In such materials with grain size below 100 nm, the uniform elongation is often very much reduced, and some specific deformation mechanisms such as twinning [START_REF] Aluminum | Deformation Twinning[END_REF][START_REF] Yamakov | Deformation twinning in nanocrystalline Al by molecular dynamics simulation[END_REF][START_REF] Zhu | Nucleation and growth of deformation twins in nanocrystalline aluminum[END_REF] and GB sliding [START_REF] Edalati | Transition from poor ductility to room-temperature superplasticity in a nanostructured aluminum alloy[END_REF] or GB migration [START_REF] Gianola | Stress-assisted discontinuous grain growth and its effect on the deformation behavior of nanocrystalline aluminum thin films[END_REF] occur.

UFG structures can be achieved by Severe Plastic Deformation (SPD) [START_REF] Zhilyaev | Microstructural evolution in commercial purity aluminum during high-pressure torsion[END_REF] with the final grain size being controlled by dynamic recovery and recrystallisation processes. Solute elements are needed to achieve grain sizes in the submicrometer range [START_REF] Edalati | Influence of dislocation-solute atom interactions and stacking fault energy on grain size of singlephase alloys after severe plastic deformation using high-pressure torsion[END_REF]. Using such a strategy, it has been demonstrated that the combination of GB and precipitation hardening may give rise to optimized combinations of mechanical and electrical properties in 6000-series alloys [START_REF] Sauvage | Optimization of electrical conductivity and strength combination by nanoscale structure design in an Al-Mg-Si alloy[END_REF] and to a yield stress up to 1 GPa in other Al alloys [START_REF] Chen | Microstructural evolution, strengthening and thermal stability of an ultrafine-grained AlCuMg alloy[END_REF][START_REF] Peter | Nanostructural hierarchy increases the strength of aluminium alloys[END_REF]. However, residual alloying elements in solid solution reduce the electrical conductivity, and the overall thermal stability is relatively low. The thermal stability of UFG or nanostructured alloys might be improved by nanoscaled particles (Zener pinning) [START_REF] Duchaussoy | Structure and mechanical behavior of ultrafine-grained aluminum iron alloy stabilized by nanoscaled intermetallic particles[END_REF] or segregations to reduce the GB energy (and thus the driving force for coarsening) or directly the boundary mobility [START_REF] Murdoch | Estimation of grain boundary segregation enthalpy and its role in stable nanocrystalline alloy design[END_REF][START_REF] Wagih | Spectrum of grain boundary segregation energies in a polycrystal[END_REF][START_REF] Devaraj | Grain boundary segregation and intermetallic precipitation in coarsening resistant nanocrystalline aluminum alloys[END_REF].

Thus, the ideal UFG structure for a combination of low weight, high strength, high electrical conductivity and high thermal stability requires a small amount of an immiscible alloying element to achieve pure Al nanoscaled grains pinned by GB segregations or nanoscaled particles. The best strategy might depend on the considered system, but there is little knowledge to optimize the nanoscaled structure regarding the envisioned multifunctional properties. It is well known that the electrical resistivity could be significantly increased by crystalline defects, especially dislocations or GBs [START_REF] Peter | Nanostructural hierarchy increases the strength of aluminium alloys[END_REF][START_REF] Yu | Nanostructured Al and Cu alloys with superior strength and electrical conductivity[END_REF][START_REF] Orlova | Influence of grain boundary state on electrical resistivity of ultrafine grained aluminium[END_REF][START_REF] Karolik | Calculation of electrical resistivity produced by dislocations and grain boundaries in metals, S[END_REF].

Twins, such as those grown during electrodeposition of nanocrystalline Cu, are a kind of exception [START_REF] Lu | Ultrahigh Strength and High Electrical Conductivity in Copper[END_REF].

However, the influence of segregations, both on electrical resistivity and mechanical strength of nanostructured Al alloys is not yet completely understood. It has already been demonstrated that grain refinement and intermetallic fragmentation can be successfully achieved by SPD processing of as-cast Al alloys containing immiscible elements (Rare Earth [START_REF] Murashkin | Mechanical and electrical properties of an ultrafine grained Al-8.5 wt. % RE (RE=5.4wt.% Ce, 3.1wt.% La) alloy processed by severe plastic deformation[END_REF], Fe [START_REF] Duchaussoy | Structure and mechanical behavior of ultrafine-grained aluminum iron alloy stabilized by nanoscaled intermetallic particles[END_REF][START_REF] Senkov | Microstructure of Aluminum-Iron Alloys Subjected to Severe Plastic Deformation[END_REF] or Ca [START_REF] Rogachev | Structure and mechanical properties of Al-Ca alloys processed by severe plastic deformation[END_REF]). However, starting from as-cast structures with intermetallic compounds requires huge plastic strains, and only Zener pinning can be achieved. The aim of the present work was to investigate both the influence of segregations and of intermetallic phases in a nanostructured Al-Ca alloy. To reach this goal, it was assumed that the co-deformation of fcc Al and fcc Ca phases could lead to mechanical mixing, promote the nanoscaled structure formation and segregation of the minor element (namely Ca) along grain boundaries, as observed in various immiscible systems [START_REF]Deformation-Induced Supersaturation in Immiscible Material Systems during High-Pressure Torsion, Karoline Sophie Kormout, Reinhard Pippan and Andrea Bachmaier[END_REF][START_REF] Ashkenazy | Shear induced chemical mixing in heterogeneous systems[END_REF][START_REF] Cordero | Phase strength effects on chemical mixing in extensively deformed alloys[END_REF][START_REF] Yu | Bulk nanocrystalline ferrite stabilized through grain boundary carbon segregation[END_REF][START_REF] Quelennec | Homogeneous Cu-Fe supersaturated solid solution prepared by SPD, influence of processing parameters and physical mechanisms[END_REF][START_REF] Sauvage | Strain induced segregations in severely deformed materials[END_REF]. Thus, a bi-metallic composite material made of fcc Al and fcc Ca phases [START_REF] Liang Tian | A deformation-processed Al-matrix/Ca-nanofilamentary composite with low density, high strength, and high conductivity[END_REF] has been processed by High-Pressure Torsion (HPT) at various strains. In a second step, intermetallic particles were nucleated in a controlled post-processing manner to preserve the overall nanoscale structure. Microstructures have been characterized down to the nanoscale to correlate with mechanical and electrical properties with a special emphasis on the critical role played by the segregation of Ca atoms at GBs.

2-Experimental

The material investigated in the present study is an Al-Ca metal matrix composite produced from highpurity Al and Ca powders that have been mixed, compacted and extruded to achieve a rod of 10mm in diameter with a volume fraction of 12% Ca (Details about the processing route are given in [START_REF] Liang Tian | A deformation-processed Al-matrix/Ca-nanofilamentary composite with low density, high strength, and high conductivity[END_REF]). During the extrusion process, equiaxed Ca particles are transformed into sub-micrometer co-axial filaments.

Then, the Al-Ca rod was sliced to prepare discs (diameter 10 mm, thickness 0.8 mm) for further deformation by (HPT) under a pressure of 6 GPa with a speed of one revolution per minute. Samples were deformed by 1, 10, 100 or 1000 revolutions at room temperature. The resulting shear deformation  was estimated as:  = 2 r N / t, where r is the distance from the disc center, N the number of revolutions and t the disc thickness. Thanks to the large dimension of anvils as compared to the sample volume, the heat generated during the process was efficiently dissipated. The temperature measured with a pyrometer on the anvils near the sample always stayed below 50 °C, even after 1000 revolutions.

The mechanical strength of the severely deformed Al-Ca composite was evaluated by Vickers microhardness (HV) measurements carried out across HPT disc diameters with a load of 300 g. The electrical resistivity was evaluated at room temperature using an eddy-current electric conductivity meter calibrated with pure Al. Due to the relatively large zone probed with this technique (about 5 mm in diameter), measurements were performed in the center and near the edge of HPT discs, providing only two average values for each condition. Crystallographic phases were identified by X-ray diffraction (XRD) using Brucker D8 diffractometer with a Co anticathode (Co-K radiation,  = 0.17909 nm). Data were recorded with the torsion axis as the out-of-plane component.

Microstructures were characterized by Transmission Electron Microscopy (TEM) on a JEOL ARM 200F

microscope. Images were also recorded in Scanning TEM (STEM) mode with a bright field (BF) detector (collection angles 0-45 mrad) and a High-Angle Annular Dark-Field (HAADF) detector (collection angles 68-280 mrad). X-ray Energy Dispersive Spectroscopy (EDS) was carried out with a JEOL Centurio detector (collection angle: 1 sr). TEM samples were prepared in the cross section of HPT discs by Focused Ion Beam (FIB) using a Zeiss N-vision40 dual beam Scanning Electron Microscope (SEM). The thermal stability at 200 °C was checked using an in-situ TEM double-tilt heating holder (Gatan 652 MA).

Heating from room temperature to 200 °C was done in a one-minute interval. Based on these observations, the static annealing conditions have been established, and some HPT samples were annealed for 30 min at 200 °C.

More detailed information about the Ca distribution has been obtained by Atom Probe Tomography (APT) analyses. They were carried out with a CAMECA LEAP-4000HR apparatus in ultra-high vacuum (10 -11 mbar), with a pulse fraction of 20%, a pulse repetition rate of 200 kHz and a sample temperature of 50 K. Three-dimensional (3D) reconstructions were performed using IVAS Software and further data processing was done with the Gpm3dSoft Software. APT tips were prepared by FIB so that the shear plane was perpendicular to the analysis direction.

3-Results

To evaluate hardness changes due to SPD of the Al-Ca composite, profiles were measured across HPT discs (Fig. 1). After one revolution, the hardness remains below 50 HV, and there is only a small gradient from 25 HV in the center up to about 35 HV in the outer part. Increasing the number of revolutions (i.e., the shear strain) by a factor of ten leads to significant changes, and while the hardness remains below 50 HV in the center part, it rises to about 250 HV in the outer part. After 100 revolutions, the hardness is more homogeneous across the HPT disc, it seems to saturate between 280 and 300 HV at the periphery while the 1 mm softer region remains in the center. At higher shear strain (up to 1000 revolutions), the hardness is relatively homogeneous across the HPT disc, but a significant softening (about 50 HV) is clearly observed as compared to the sample processed up to 100 revolutions. To evaluate the thermal stability, annealing during 30 min at 200 °C was carried out on the sample processed by 100 revolutions, and Fig. 1 shows that this annealing causes a hardness drop of about 100 HV. The electrical resistivity and the hardness of the Al-Ca composite processed by HPT are plotted as a function of the shear strain on Fig. 2. At lower strains, up to  = 100, they both increase slightly without exceeding 400 10 -4 m and 60 HV respectively and thus remain relatively close to the properties of the initial Al-Ca composite material [START_REF] Liang Tian | A deformation-processed Al-matrix/Ca-nanofilamentary composite with low density, high strength, and high conductivity[END_REF]. However, at a shear strain of about  = 200, there is a very sharp increase of both the electrical resistivity and the microhardness that reach a plateau for  > 1000, respectively close to 2100±100 10 -4 m and 290±10 HV. The short time annealing (30 min at 200°C) leads to a significant drop, down to about 1200±100 10 -4 m and 200±10 HV, but still much larger than original properties of the Al-Ca composite. Circle markers are for the as-deformed material and squares for the material aged 30 min at 200 °C.

Measurements were carried out on samples processed by 1, 10 and 100 revolutions. The initial electrical resistivity is from [START_REF] Liang Tian | A deformation-processed Al-matrix/Ca-nanofilamentary composite with low density, high strength, and high conductivity[END_REF].

The evolution of phases during SPD has been characterized by XRD. In samples processed by one and 10 revolutions (Fig. 3a), peaks corresponding to fcc Al, fcc Ca, and SiC were detected. The SiC was detected only in the one-and 10-revolution samples, since these samples were very soft (Fig. 1) and ductile. After 100 revolutions, the material became much harder (Fig. 1), and no SiC particles are incorporated during polishing. In the 100-revolutions material, only the fcc Al phase remains (along with significant peak broadening), and no peaks attributable to fcc Ca were detected. This seems to indicate that some mechanical mixing led to the dissolution of the Ca phase during HPT processing. At a larger shear strain, up to 1000 revolutions (Fig. 3b), the spectrum looks similar, but additional peaks corresponding to the Al2Ca phase are clearly exhibited. After annealing for 30 min at 200 °C, the sample processed by 100 revolutions displayed significant fcc Al peak-width reduction, indicating some defect recovery (Fig. 3b). There are also numerous peaks attributed to the Al4Ca phase showing that this intermetallic phase has nucleated. The microstructure resulting from SPD of the Al-Ca composite has been investigated by TEM in the sample processed by 100 revolutions where the maximum hardness was achieved (Fig. 1) and where the fcc Ca phase could not be detected by XRD (Fig. 3a). As shown on the bright-field image of Fig. 4a, the grain size has a mean value between 20 and 30 nm. The corresponding Selected Area Electron Diffraction (SAED) pattern exhibits Debye-Scherrer rings typical of nanostructured fcc Al (Fig. 4b), fully consistent with XRD data (Fig. 3a). However, a few spots that could be indexed with reflections corresponding to the interplanar distances of the Al4Ca phase also appear. Thus, a few nanoscale intermetallic particles have also nucleated during the SPD process. To visualize the distribution of particles within the microstructure, HAADF-STEM imaging was carried out (Fig. 5a). Ca-rich regions are brightly imaged since the atomic number of Ca is much higher than that of Al. The contrast is, however, relatively poor, which is partly due to overlapping nanograins in the TEM sample. The low contrast notwithstanding, it is evident that Ca atoms are not homogeneously distributed. This feature was confirmed using EDS line profiling (Fig. 5b), and it indicates that even if the fcc Ca phase disappeared during SPD, a fully homogeneous super-saturated solid solution was not achieved.

Figure 5: STEM dark-field image showing the ultrafine grain structure of the Al-Ca composite processed by 100 revolutions by HPT (a). The slight contrast variations on the HAADF image are attributed to local fluctuations of the Ca content which are confirmed by the EDS line profile analysis (b).

To determine more accurately the distribution of Ca atoms within the nanoscaled structure, the material was analyzed by Atom Probe Tomography (APT). Two representative volumes are displayed in Fig. 6a and6b. These data confirm that Ca is not homogeneously distributed: there are segregations along linear or planar defects and Ca-rich particles. The composition profile computed across such a nanoparticle (diameter of about 5 nm) shows that it contains about 15 at.% Ca (Fig. 6c). Numerous particles were probed and for all of them, the Ca concentration stands between 10 and 20 at.% Ca.

There are also Ca-depleted regions in reconstructed volumes, which correspond to nanoscale Al grains.

The amount of Ca in solid solution was carefully measured, taking care not to include either Ca-rich particles or segregations. The average amount of Ca measured in solid solution was 0.23 at.% (± 0.05), which is much higher than the equilibrium solubility limit at room temperature [START_REF] Islam | Thermodynamic Modeling of Mg-Ca and Al-Ca Binary Systems[END_REF][START_REF]ASM Alloy Phase Diagram Database[END_REF]. The thermal stability of this nanoscale structure achieved by SPD was investigated by in-situ TEM annealing. On Fig. 7, the same region of the sample that was tracked during annealing at 200 °C is shown in as-SPD condition and after 30 min at 200 °C in STEM-BF images (Fig. 7a and7c). Comparing these images clearly shows significant grain growth, and STEM-HAADF images indicate nucleation of Ca-rich particles (top left in Fig. 7d) and the coarsening of Ca-rich regions (bottom right in Fig. 7d). Thus, a significant redistribution of Ca atoms occurred during annealing. The evolution of the grain size was estimated from bright-field TEM images (Figs. 4a and8a) and corresponding distributions are plotted on Fig. 8c. The mean grain size increases from about 25±5 nm to 35±5 nm during 30 min at 200 °C. In addition, the SAED pattern recorded in the annealed state (Fig. 8b) clearly exhibits many spots attributed to the Al4Ca phase, confirming the nucleation of this Ca-rich phase in agreement with XRD data (Fig. 3). The Ca redistribution within the nanoscaled structure was studied more accurately using APT, and a typical volume analyzed in the annealed sample is displayed on Fig. 9a. There are two kinds of Ca-rich particles, some are faceted with a size close to the measured grain size (20-40 nm) and others are much smaller with a diameter in a range of 5 to 10 nm. Composition profiles (Fig. 9b and9c) show that they all contain about 20 at.% Ca, as expected for the Al4Ca phase. The average amount of Ca left in solid solution was also measured from these APT data. It is much lower than the as-SPD state, down to only 0.045 at.% (± 0.01), which is closer to the solubility limit [START_REF] Islam | Thermodynamic Modeling of Mg-Ca and Al-Ca Binary Systems[END_REF][START_REF]ASM Alloy Phase Diagram Database[END_REF].

4-Discussion

Influence of calcium on the deformation induced nanoscaled structure

In the Al-Ca composite before SPD, there are two phases: the fcc Al matrix and elongated submicrometer fcc Ca filaments [START_REF] Liang Tian | A deformation-processed Al-matrix/Ca-nanofilamentary composite with low density, high strength, and high conductivity[END_REF]. These two phases are ductile and easily co-deform during the HPT process, which is the main difference with as-cast structures containing brittle intermetallic particles [START_REF] Duchaussoy | Structure and mechanical behavior of ultrafine-grained aluminum iron alloy stabilized by nanoscaled intermetallic particles[END_REF][START_REF] Murashkin | Mechanical and electrical properties of an ultrafine grained Al-8.5 wt. % RE (RE=5.4wt.% Ce, 3.1wt.% La) alloy processed by severe plastic deformation[END_REF][START_REF] Senkov | Microstructure of Aluminum-Iron Alloys Subjected to Severe Plastic Deformation[END_REF][START_REF] Rogachev | Structure and mechanical properties of Al-Ca alloys processed by severe plastic deformation[END_REF]. During the early stage of deformation (one revolution by HPT), only a smooth hardness gradient appears (Fig. 1), although the shear strain gradient is huge across the HPT disc diameter. This is obviously the result of the relatively easy co-deformation between the two phases. At a shear strain of about  = 200, a dramatic change both in electrical resistivity and microhardness was observed (Fig. 2). At this stage, the original sub-micrometer Ca filaments (dCa < 1m) are greatly elongated with a thickness dCa*  dCa /  of only 5 nm or less. Thus, at larger levels of deformation, Ca filaments are so much elongated that they start to dissolve in the fcc Al matrix as observed in other immiscible systems [START_REF]Deformation-Induced Supersaturation in Immiscible Material Systems during High-Pressure Torsion, Karoline Sophie Kormout, Reinhard Pippan and Andrea Bachmaier[END_REF][START_REF] Ashkenazy | Shear induced chemical mixing in heterogeneous systems[END_REF][START_REF] Cordero | Phase strength effects on chemical mixing in extensively deformed alloys[END_REF][START_REF] Yu | Bulk nanocrystalline ferrite stabilized through grain boundary carbon segregation[END_REF][START_REF] Quelennec | Homogeneous Cu-Fe supersaturated solid solution prepared by SPD, influence of processing parameters and physical mechanisms[END_REF][START_REF] Sauvage | Strain induced segregations in severely deformed materials[END_REF]. It starts at the HPT disc periphery where the shear strain is larger, and it progressively reaches the whole sample, leading to a progressive decrease and vanishing of Ca peaks on XRD spectra (Fig. 3).

Then, even if the distribution of Ca is not homogeneous within the nanoscaled structure, a kind of saturation regime seems to be reached. As shown by our TEM and APT data (Figs. 4, 5 and 6), a nanoscale structure with an extremely small grain size, much smaller than that achieved by SPD in commercially pure Al, is achieved [START_REF] Peter | Nanostructural hierarchy increases the strength of aluminium alloys[END_REF][START_REF] Zhilyaev | Microstructural evolution in commercial purity aluminum during high-pressure torsion[END_REF][START_REF] Edalati | Influence of dislocation-solute atom interactions and stacking fault energy on grain size of singlephase alloys after severe plastic deformation using high-pressure torsion[END_REF]. Ca solute atoms obviously play a critical role thanks to their strong interaction with crystalline defects, as in the Fe-C system [START_REF] Yu | Bulk nanocrystalline ferrite stabilized through grain boundary carbon segregation[END_REF]. The distribution of Ca is heterogeneous at different length scales: submillimeter, sub-micrometer, and nanoscale. At the larger scales, the non-uniform Ca distribution results mainly from the original distribution of Ca filaments in the composite [START_REF] Liang Tian | A deformation-processed Al-matrix/Ca-nanofilamentary composite with low density, high strength, and high conductivity[END_REF], and it is illustrated by the deviation between the nominal Ca concentration and the measurements done by TEM-EDS and APT (see Table 1). Then at the nanometer scale, due to the extremely low equilibrium solubility in Al [START_REF] Islam | Thermodynamic Modeling of Mg-Ca and Al-Ca Binary Systems[END_REF][START_REF]ASM Alloy Phase Diagram Database[END_REF], only a very small fraction of Ca atoms is driven into solid solution (less than 1 Ca atom out of 10), while others are segregated along crystalline defects such as GBs or located in Ca-rich particles (Fig. 6). The fraction of available sites at GBs f writes as:

𝑓~3 𝑎 2 𝑑 (1)
Where a is the lattice parameter of fcc Al (a = 0.405 nm) and d the mean grain size (25 nm in the asdeformed material). It yields to f  2% of lattice sites located at GBs, i.e. only twice less than the nominal atomic fraction of Ca and relatively close to the local concentration measured in APT volumes (table 1). Then crystalline defects, and especially GBs become easy nucleation sites for Al4Ca precipitates and considering the highly negative enthalpy of formation of this intermetallic phase (-17.2 kJ/mol [START_REF] Islam | Thermodynamic Modeling of Mg-Ca and Al-Ca Binary Systems[END_REF]), it is not surprising that few particles directly nucleate along these defects during SPD, as shown on Fig. 6 and confirmed by the SAED pattern (Fig. 4b). It is however interesting to note that their composition estimated from APT data is significantly lower than the expected stoichiometry (in a range of 10 to 20 at.% Ca, Fig. 6c). Nevertheless, they are only a few nanometers in size, and they exhibit an evaporation field slightly lower than the Al matrix (estimated from changes of the local density in reconstructed volumes), thus ion trajectory overlaps may lead to an under-estimation of the Ca content. However, the Al4Ca stoichiometry was measured in nanoscale particles after annealing (Fig. 9c), thus it could be that in as-SPD samples, particles are highly defected, resulting in a nonstoichiometric composition. Similar features have been reported for Fe3C carbides in severely deformed steels [START_REF] Sauvage | The role of carbon segregation on nanocrystallisation of pearlitic steels processed by severe plastic deformation[END_REF].

When the deformation is continued in the saturation regime (  1000), the Al2Ca phase nucleates (Fig. 3b). This is not expected from the phase diagram in an alloy containing only 4 at.% Ca [START_REF] Islam | Thermodynamic Modeling of Mg-Ca and Al-Ca Binary Systems[END_REF][START_REF]ASM Alloy Phase Diagram Database[END_REF], but locally the amount of Ca segregated on crystalline defects might be much higher and the nucleation of this phase could be favored due to a significantly lower enthalpy of formation as compared to Al4Ca (-29.3 kJ/mol for Al2Ca against -17.2 kJ/mol for Al4Ca [START_REF] Islam | Thermodynamic Modeling of Mg-Ca and Al-Ca Binary Systems[END_REF]). For such extreme deformations, the formation of metastable phases has been reported in other systems [START_REF] Edalati | Ultra-severe plastic deformation: Evolution of microstructure and phase transformations in immiscible systems[END_REF]. Both, the high pressure applied during HPT (about 6 GPa) and also possible local temperature increase [START_REF] Edalati | Effect of temperature rise on microstructural evolution during high-pressure torsion[END_REF] may also affect the thermodynamic equilibrium. In the Al-Ca composite deformed up to (  1000, it leads to a significant decrease in hardness, probably because this phase being relatively rich in Ca (33 at.%), the amount of Ca available to interact with crystalline defects is then significantly reduced.

Nanoscale structures are intrinsically thermally unstable due to the extremely high fraction of interfaces. In the present case, the low mutual solubility of Ca and Al is another driving force for the transformation of the system as soon as enough atomic mobility is provided to evolve towards a situation with a lower enthalpy. Crystalline defects, such as dislocations and GBs, are fast diffusion paths for the solute, thus in only 30 min at 200 °C, the nanoscale structure has completely changed.

Grain growth (Fig. 7 and8c) and precipitation of the Al4Ca phase occur concomitantly. If it is assumed that all Ca atoms are located in Al4Ca particles, then the volume fraction of this phase estimated from the nominal composition is about 24 vol.%. APT data revealed (Fig. 9) that precipitation occurred through two different mechanisms: i) some nanoscale (size below 10 nm) and relatively spherical Al4Ca particles nucleated homogeneously, inside Al grains, probably directly from the super-saturated solid solution, and eventually fed by pipe diffusion along dislocations; ii) others, larger (20-40 nm) and faceted like typical grains, nucleated at GBs and then grew at a faster rate. This second mechanism is often dominant in the decomposition of UFG super-saturated solid solutions where it is usually impossible to achieve homogeneous precipitation within nanoscale or submicrometer grains [START_REF] Sauvage | Optimization of electrical conductivity and strength combination by nanoscale structure design in an Al-Mg-Si alloy[END_REF][START_REF] Rodrigues | Microstructure, mechanical properties and aging behavior of nanocrystalline copper -berylium alloy, Ivan Lomakin, Miguel Castillo[END_REF][START_REF] Fillon | Influence of severe plastic deformation on the precipitation hardening of a FeSiTi steel[END_REF][START_REF] Sha | Influence of equal-channel angular pressing on precipitation in an Al-Zn-Mg-Cu alloy[END_REF].

Influence of the calcium distribution on the mechanical strength

At low levels of deformation, the easy co-deformation of Al and Ca does not lead to a significant increase in hardness; it remains below 40 HV (Fig. 1). At higher plastic strain, the Ca fcc phase dissolves, and the resulting nanoscale structure (mean grain size of 25±5 nm) gives rise to a huge increase in hardness to about 280-300 HV. The works of H.J. Choi [START_REF] Choi | Tensile behavior of bulk nanocrystalline aluminum synthesized by hot extrusion of ball-milled powders[END_REF] and S.I. Ahmed [START_REF] Ahmed | The activation of deformation mechanisms for improved tensile properties in nanocrystalline aluminum[END_REF] on nanocrystalline aluminum demonstrate that Tabor's law (y = HV (MPa) / 3 ; [START_REF] Tabor | The hardness of solids[END_REF]) provides a good estimate of the yield stress y in such materials. Then, following this approach, the corresponding yield stress of the Al-Ca composite processed by SPD was compared to other nanoscale Al alloys on the Hall Petch plot on Fig. 10. [START_REF] Xu | Tensile nanomechanics and the Hall-Petch effect in nanocrystalline aluminium[END_REF]. Data from [Liddicoat 2010] and [START_REF] Mohamed | Strengthening of A2024 alloy by high-pressure torsion and subsequent aging, Intan Fadhlina[END_REF]] were obtained respectively on 7075 and 2024 Al alloys processed by HPT [START_REF] Peter | Nanostructural hierarchy increases the strength of aluminium alloys[END_REF][START_REF] Mohamed | Strengthening of A2024 alloy by high-pressure torsion and subsequent aging, Intan Fadhlina[END_REF]. Data from [START_REF] Ma | Mechanical behavior and strengthening mechanisms in ultrafine grain precipitation-strengthened aluminum alloy[END_REF]] were obtained on a 7075 Al alloy processed by cryo-milling [START_REF] Ma | Mechanical behavior and strengthening mechanisms in ultrafine grain precipitation-strengthened aluminum alloy[END_REF]. Data from [START_REF] Choi | Tensile behavior of bulk nanocrystalline aluminum synthesized by hot extrusion of ball-milled powders[END_REF]] and [START_REF] Ahmed | The activation of deformation mechanisms for improved tensile properties in nanocrystalline aluminum[END_REF]] were obtained on consolidated ball milled pure Al [START_REF] Choi | Tensile behavior of bulk nanocrystalline aluminum synthesized by hot extrusion of ball-milled powders[END_REF][START_REF] Ahmed | The activation of deformation mechanisms for improved tensile properties in nanocrystalline aluminum[END_REF]. The grey dashed line was plotted using the HP coefficient reported by [START_REF] Hansen | Hall-Petch relation and boundary strengthening[END_REF]] in [START_REF] Hansen | Hall-Petch relation and boundary strengthening[END_REF].

For most of the data on this plot, a significant deviation to the classical Hall Petch law (y = 0 + k d -1/2 where 0 and k are constants) is exhibited. The experimental work of H.J. Choi [START_REF] Choi | Tensile behavior of bulk nanocrystalline aluminum synthesized by hot extrusion of ball-milled powders[END_REF] seems to indicate that such a deviation occurs for a grain size below 80 nm in nanocrystalline Al. At such small grain size, dislocation pile-ups are quite unlikely, and other deformation mechanisms are involved [9-11, 50, 51].

Besides, it is also well accepted that grain boundary mediated plasticity may play a critical role at such length scale [START_REF] Edalati | Transition from poor ductility to room-temperature superplasticity in a nanostructured aluminum alloy[END_REF][START_REF] Borodin | A simple mechanical model for grain boundary sliding in nanocrystalline metals[END_REF][START_REF] Hu | Grain boundary stability governs hardening and softening in extremely fine nanograined metals[END_REF], especially at relatively low strain rates. Then, it is not surprising that molecular dynamics simulation results [START_REF] Xu | Tensile nanomechanics and the Hall-Petch effect in nanocrystalline aluminium[END_REF] with a strain rate much higher than that of experimental data (10 10 s -1 against 10 -4 or 10 -3 s -1 ), exhibits the higher yield stress for a grain size of 25 nm ( 1.1 GPa [START_REF] Xu | Tensile nanomechanics and the Hall-Petch effect in nanocrystalline aluminium[END_REF], against  0.3 GPa predicted by the Hall-Petch law [START_REF] Hansen | Hall-Petch relation and boundary strengthening[END_REF]). It is important to note that all experimental data on this plot with a yield stress larger than 600 MPa were obtained on alloys where solute elements, clusters and precipitates are thought to provide additional strengthening contributions. The nanocrystalline Al-Ca alloy (as processed by HPT), with an estimated yield stress of 915-980 MPa is close to the highest reported value of 1 GPa for an ultrafine 7000-series aluminum alloy [START_REF] Peter | Nanostructural hierarchy increases the strength of aluminium alloys[END_REF]. There is however only a small amount of Ca in solid solution (about 0.2 at.%, see table1) and a low number density of nanoscaled Ca rich particles mostly located at GBs, thus most of the hardening obviously results from calcium segregations along boundaries. At such small grain sizes, dislocation pile-ups are quite unlikely and grain boundary mediated plasticity probably plays a critical role [START_REF] Edalati | Transition from poor ductility to room-temperature superplasticity in a nanostructured aluminum alloy[END_REF][START_REF] Borodin | A simple mechanical model for grain boundary sliding in nanocrystalline metals[END_REF][START_REF] Hu | Grain boundary stability governs hardening and softening in extremely fine nanograined metals[END_REF]. Ca atoms located at GBs probably affect the mobility of intergranular dislocations strongly and thus limit grain boundary plasticity which gives rise to the exceptionally high microhardness. Similar effects have been reported in other nanoscaled structures where it was demonstrated that composition modulations or segregations influence the thermal stability but also significantly increase the mechanical strength [START_REF] Hu | Grain boundary stability governs hardening and softening in extremely fine nanograined metals[END_REF][START_REF] Abramova | Grain boundary segregation induced strengthening of an ultrafine-grained 316 stainless steel[END_REF].

After annealing at 200 °C for 30 min, the hardness decreases by about 30%, down to about 200 HV (Fig. 2) as a result of microstructural changes. The mean grain size increased from d = 25±5 nm to 35±5 nm (Fig. 8) and intensive precipitation of the intermetallic Al4Ca phase occurred with a volume fraction of about 24% (Fig. 9). Such a nanoscaled structure is relatively similar to that obtained by Rogachev et al [START_REF] Rogachev | Structure and mechanical properties of Al-Ca alloys processed by severe plastic deformation[END_REF] who processed by SPD an as-cast alloy containing intermetallic particles, leading to a similar microhardness. The hardness of Al4Ca being higher than that of Al [START_REF]Phase transformation of the intermetallic compound AI4Ca, H. Zogg and P. Schwellinger[END_REF] and the classical Hall-Petch dependence of the yield stress being a function of d -1/2 , then the expected hardness drop during annealing of the nanostructured Al-Ca material of the present study should not exceed 15%. The significant difference with our experimental observations (30%) clearly indicates that Ca segregations probably inhibit partly grain boundary mediated plasticity leading to the exceptional yield stress close to 1 GPa.

Influence of the calcium distribution on electrical properties

Once the level of deformation becomes moderate (below  = 200), the electrical resistivity remains relatively stable (Fig. 2) with only a minor increase as compared to the initial value of about 300 10 - 4 m [START_REF] Liang Tian | A deformation-processed Al-matrix/Ca-nanofilamentary composite with low density, high strength, and high conductivity[END_REF]. Then, at higher shear strains, when the fcc Ca phase starts dissolving, the electrical resistivity changes dramatically and becomes larger than 2000 10 -4 m (Fig. 2). The electrical resistivity of Al is affected by crystalline defects (vacancies, dislocations and GBs) and Ca in solid solution. For the as-processed nanoscaled Al-Ca material, the influence of nanoscaled Al4Ca particles will be at first neglected since the volume fraction is relatively small, and the contribution of Al/Al4Ca interfaces will be discussed later on. Thus, the electrical resistivity Al can be simply written using the so-called Mattissen-Flemming rule [START_REF]Aluminum-1. a review of resistive mechanisms in aluminum, F R Fickett[END_REF]:

𝜌 𝐴𝑙 = 𝜌 𝐴𝑙 0 + 𝐶 𝑣 ∆𝜌 𝐴𝑙 𝑣 + 𝐿 𝑑 ∆𝜌 𝐴𝑙 𝑑 + 𝑆 𝐺𝐵 ∆𝜌 𝐴𝑙 𝐺𝐵 + 𝐶 𝐶𝑎 ∆𝜌 𝐴𝑙 𝐶𝑎 (2)
Where Al 0 is the electrical resistivity of pure Al (265 10 -4 m), Cv the vacancy concentration, Ld the dislocation density (m -2 ), SGB the surface of GBs per unit volume (m -1 ), CCa the fraction of Ca in solid solution, and Al v , Al d , Al GB and Al Ca the corresponding constants reflecting each contribution (respective values are listed in Table 2).

As shown by Y. Miyajima [START_REF] Peter | Nanostructural hierarchy increases the strength of aluminium alloys[END_REF], even if the vacancy concentration could be up to 10 -3 % in severely deformed alloys, their contribution to the total electrical resistivity remains several orders of magnitude lower than Al 0 . The contribution of dislocations could be significantly higher than vacancies and rises up to about 10% of Al 0 ( 27 10 -4 m) for a dislocation density of 10 16 m -2 (corresponding to a mean distance between dislocations of about 10 nm) [START_REF] Peter | Nanostructural hierarchy increases the strength of aluminium alloys[END_REF][START_REF] Schrank | Effect of high deformation of electrical resistivity in pure aluminium[END_REF]. Considering the amount of Ca in solid solution measured by APT (<0.5 at.%, see Table 1), then the corresponding increase in electrical resistivity is less than 15 10 -4 m. Lastly, grain boundaries, assuming spherical grains with a mean diameter of 25 nm, gives a maximum electrical contribution to the overall resistivity of 300 10 -4 m.

These estimates are listed in Table 2. They lead to an upper theoretical limit of the electrical resistivity of about 610 10 -4 m, which is significantly lower than the experimental value of 2100±100 10 -4 m (Fig. 2).

It should be noted however that the above approach completely neglects the contribution of Ca atoms segregated along GBs. It has been shown that the grain boundary scattering writes as [START_REF] Liang Tian | Modeling the electrical resistivity of deformation processed metal-metal composites[END_REF][START_REF] Mayadas | Electrical resistivity model for polycristalline films: the case of specular reflection at external surfaces[END_REF]:

𝜌 𝐺𝐵 = 𝜌 𝐴𝑙 0 3( 1 3 - 𝛼 2 +𝛼 2 -𝛼 3 𝑙𝑛(1+ 1 𝛼 )) (3)
With

𝛼 = 𝜆 𝑅 𝑑 (1-𝑅) (4)
Where 𝝀 is the mean free path of electrons inside Al grains (𝝀 = 18.9 nm [START_REF] Gall | Electron mean free path in elemental metals[END_REF]), and R the reflection coefficient of grain boundaries (R = 0 means that GBs do not scatter any electron and thus have no influence on the electrical resistivity). H. Schwarz and R. Lück have shown that in metals the value of R is often close to 0.5 [START_REF] Schwarz | The influence of sample surface quality and grain boundaries on the electrical resistivity of metals[END_REF]. On Fig. 11, the grain boundary contribution to resistivity estimated from eq.

(3) was plotted as a function of the grain size for various values of R. The comparison with the experimental dependence as reported by Andrews [START_REF] Andrews | The effect of grain boundaries on the electrical resistivity of polycrystalline copper and aluminium[END_REF] and Kasen [START_REF] Kasen | Grain boundary resistivity of aluminium[END_REF] indicates that 30% < R < 50% in pure Al. To fit our experimental measurements of electrical resistivity in the nanostructured AlCa, a GB contribution of about 1500 10 -4 m is required (Table 3). Then, the plot on Fig. 11 also reveals that in this material, the fraction of electrons reflected or trapped GBs is much higher than that in pure Al, up to about 85%, and this large difference can only be attributed to Ca atoms segregated at GBs. 3)). These theoretical data are compared to experimental data of intrinsic GB resistivity of pure Al reported by Andrews [START_REF] Andrews | The effect of grain boundaries on the electrical resistivity of polycrystalline copper and aluminium[END_REF] and Kasen [START_REF] Kasen | Grain boundary resistivity of aluminium[END_REF]. The experimentally estimated boundary contributions to electrical resistivity are also indicated for Al GBs in the as-HPT nanostructured alloy and for Al/Al4Ca interfaces in the nanostructured alloy after annealing during 30min at 200°C.

Annealing at 200 °C during 30 min leads to a strong decrease of the amount of Ca in solid solution in Al (down to 0.045 at.%, see Table 1) and to the precipitation of Al4Ca particles (Fig. 9). Assuming that the nanostructured material is a mixture of pure Al and Al4Ca, then the electrical resistivity can be estimated with a simple rule of mixture [START_REF] Liang Tian | A deformation-processed Al-matrix/Ca-nanofilamentary composite with low density, high strength, and high conductivity[END_REF] :

𝜌 𝐴𝑙-𝐶𝑎 = 𝜌 𝐴𝑙 0 𝑓 𝑣 𝐴𝑙 + 𝜌 𝐴𝑙4𝐶𝑎 0 𝑓 𝑣 𝐴𝑙4𝐶𝑎 (4)
Where Al4Ca 0 is the intrinsic electrical resistivity of the Al4Ca phase (517 10 -4 m [START_REF] Liang Tian | A deformation-processed Al-matrix/Ca-nanofilamentary composite with low density, high strength, and high conductivity[END_REF]), fv Al and fv Al4Ca

the volume fraction of Al and Al4Ca phases (about 75% and 25% respectively). It leads to an estimated total resistivity after annealing of about 330 10 -4 m. This is much lower than our experimental measurements (about 1200 10 -4 m, Table 3), confirming that at such a small grain size (d = 35±5 nm), electron scattering or trapping by boundaries cannot be neglected. If it is assumed that Al4Ca grains are homogeneously distributed, then only two kinds of interfaces prevail in the system, namely Al/Al and Al/Al4Ca boundaries. Thus, neglecting the contribution of other defects, the electrical resistivity of the annealed nanostructured Al-Ca annealed materials may be written from a combination of eq. ( 2) and ( 4):

𝜌 𝐴𝑙-𝐶𝑎 𝑎𝑛𝑛𝑒𝑎𝑙𝑒𝑑 = 𝜌 𝐴𝑙 0 𝑓 𝑣 𝐴𝑙 + 𝜌 𝐴𝑙 𝐺𝐵 𝐹 𝐺𝐵 𝐴𝑙/𝐴𝑙 + 𝜌 𝐴𝑙4𝐶𝑎 0 𝑓 𝑣 𝐴𝑙4𝐶𝑎 + 𝜌 𝐴𝑙4𝐶𝑎 𝐺𝐵 𝐹 𝐺𝐵 𝐴𝑙/𝐴𝑙4𝐶𝑎 (5)
Where Al GB is the Al/Al GB contribution to the electrical resistivity (as estimated by eq. ( 3), with R = 0.5 and d = 35 nm, Al GB = 200 10 -4 m). FGB Al/Al and FGB Al/Al4Ca are the fractions of Al/Al and Al/Al4Ca

boundaries respectively. If it is assumed that Al4Ca and Al grains have similar sizes (35 nm), then 25% volume fraction of Al4Ca leads to FGB Al/Al = FGB Al/Al4Ca = 50%. Then, as summarized in Table 3, eq. ( 5) gives Al4Ca GB  1500 10 -4 m. This contribution is plotted on Fig. 11 and it shows that it corresponds to a fraction of electrons reflected or trapped by Al/Al4Ca boundaries higher than 85%.

5-Conclusions

i) An Al-Ca composite was severely deformed by HPT at various strain levels. The micro-hardness increased sharply for a shear strain in a range of 100 to 200, and then it saturated at 290±10 HV for larger strain levels. XRD and TEM data showed that the hardness evolution is connected with both the dissolution of the fcc Ca phase and the formation of a nanoscaled structure with a mean grain size of only 25 nm.

ii)

The electrical resistivity followed exactly the same trend, reaching a saturation value of 2100±100 10 -4 m. This is considerably higher than the initial resistivity of about 300 10 -4 m, and it is obviously connected with the mechanical mixing of Ca and Al. APT analyses revealed, however, that Ca is not homogeneously distributed within the Al matrix. Only a small fraction is in solid solution (0.23±0.05 at.%), the rest being segregated along crystalline defects.

iii)

The yield stress of this nanoscaled structure, estimated from the Tabor's Law, is in a range of 915 to 980 MPa, much higher than predictions based on the Hall-Petch Law. Due to the relatively low calcium super saturated solution and the low number density of Ca-rich particles, it is mainly attributed to calcium segregations along grain boundaries. The influence of such segregation on plasticity mechanisms will be investigated in the near future. v)

The analysis of the electrical resistivity data indicated that the proportion of electrons reflected or trapped by GBs is in range of only 30 to 50% in pure Al, but it rose to about 85% in Al/Ca due to Ca segregations. This is the main contribution to the electrical resistivity measured in the as-deformed state. After annealing, there was no more segregation, but Al/Al4Ca boundaries significantly affect the properties with a ratio slightly higher, in a range of 85 to 90%. Table2: Summary of the different contributions to the electrical resistivity of the as-processed Al-Ca material (as-HPT) for a shear strain  > 200. The total estimated electrical resistivity was obtained from eq. ( 2) and data are compared with experimental values (see Fig. 2). The symbol (*) refers to 10 -4 m unit.

Figure 1 :

 1 Figure 1: Micro-hardness profiles measured across the discs processed by HPT (1, 10, 100 and 1000 revolutions) and after post-HPT annealing (100 revolutions followed by 30 min at 200 °C).

Figure 2 :

 2 Figure 2: Evolution of the microhardness (HV, black markers, left vertical scale) and electrical resistivity (10 -4 m, red markers, right vertical scale) as a function of the shear deformation induced by HPT.

Figure 3 :

 3 Figure 3: (a) XRD patterns showing the progressive dissolution of Ca in fcc Al during deformation up to 100 revolutions by HPT, (b) XRD patterns showing the nucleation of Al2Ca phase during HPT up to 1000 turns and the precipitation of the Al4Ca phase in the material deformed up to 100 revolutions and annealed 30 min at 200 °C.

Figure 4 :

 4 Figure 4: a) Bright field TEM image of the Al-Ca composite processed by 100 revolutions by HPT showing a nanoscale grain size, b) corresponding SAED pattern exhibiting Debye-Scherrer rings corresponding only to fcc Al. A few additional spots corresponding to the Al4Ca phase are also exhibited.

Figure 6 :

 6 Figure 6: (a) and (b) 3D reconstructions of volumes analyzed by APT in the Al-Ca composite processed by 100 revolutions by HPT (Al in blue, Ca in orange). (c) Concentration profile plotted across a Ca-rich nanoscale particle (sampling box thickness 1 nm).

Figure 7 :

 7 Figure 7: STEM BF (a, c) and HAADF (b, d) images of the nanoscale structure of the Al-Ca composite processed by 100 revolutions by HPT showing the microstructure evolution during in-situ annealing. The same regions are displayed in the initial state (a, b) and after 30 min at 200 °C (c, d). Grain growth and precipitation of Ca-rich clusters are highlighted by the red circles.

Figure 8 :

 8 Figure 8: (a) TEM BF image showing the nanoscaled structure after in-situ annealing 30 min at 200 °C and (b) corresponding SAED pattern showing diffraction spots attributed to fcc Al and Al4Ca phase. (c) Grain size distribution measured after 30 min at 200 °C and compared to the initial state.

Figure 9 :

 9 Figure 9: (a) 3D reconstruction of a volume analyzed by APT in the Al-Ca composite processed by 100 revolutions by HPT and annealed during 30 min at 200 °C (Al in blue, Ca in orange). The arrows indicate the location and the direction of the Ca concentration profiles plotted in (b) and (c) (sampling box thickness 1 nm).

Figure 10 :

 10 Figure 10: Hall-Petch plot (Yield stress as a function of d -1/2 ) comparing the yield stress of the nanostructured Al-Ca composites of the present study (estimated from Tabor's law) with data from the literature. These are all experimental data except for [XU 2018], which refers to molecular dynamic simulations at high strain rate [6]. Data from [Liddicoat 2010] and [Mohamed 2017] were obtained

Figure 11 :

 11 Figure 11: Contribution of GBs to electrical resistivity of pure Al as a function of the fraction of electrons reflected or trapped by GBs (plotted from eq. (3)). These theoretical data are compared to experimental

  iv)Annealing for 30 min at 200 °C led to a slight grain growth (mean size 35±5 nm) and intensive nucleation of the Al4Ca phase. Both homogeneous nucleation of nanoscaled particles (few nanometers in diameter) and heterogenous precipitation (size similar to the Al grains) occurred. APT analyses revealed that the amount of Ca in solid solution decreased to 0.045±0.01 at.%. The annealed nanoscaled structure exhibited a lower microhardness and electrical resistivity than the as-processed material (200±10 HV and 1200±100 10 -4 m).

Table 1 :

 1 Ca concentration measured experimentally in the HPT-processed Al-Ca composite (compared to the nominal composition). "HPT+aged" refers to a post HPT aging treatment of 30 min at 200 °C. A systematically lower value was found experimentally for all states and techniques, which is attributed to the heterogeneous distribution of Ca filaments in the starting composite material. The error range given for APT data refers to the scatter from one analysis to another. The last two columns give the concentration of Ca measured with sampling boxes set inside Al grains without overlapping with GBs or Ca rich particles to estimate the amount of Ca in super-saturated solid solution.

	Tables
	Nominal	STEM-EDS	APT	APT	APT		APT
	(12 vol.%	(as HPT)	(as HPT)	(HPT+aged)	(as HPT)		(HPT+aged)
		Ca)		Full data	Full data	Solid solution	Solid solution
						in Al		in Al
	at.%Ca	4	2.3 ± 0.5	1.2 ± 0.4	2.27 ± 0.5	0.23 ± 0.05	0.045 ± 0.01
								Total	Measured
	Fcc Al		Vacancies	Dislocations	Ca in sol. sol.	GBs	estimated	electrical
			[23, 26]	[23, 26, 56]	[55]	[26, 57, 58]	electrical	resistivity
							resistivity
	Intrinsic
	elec. res.

Table 3 :

 3 Summary of the different contributions to the electrical resistivity of the nanostructured Al-Ca material after HPT ( > 200) annealed 30min at 200°C. See text for details.

Data availability

The raw and processed data required to reproduce these findings are available on request to the corresponding author.

Acknowledgements:

G. Zaher and I. Lomakin are gratefully acknowledged for the electrical resistivity measurements. Z.

Grou is acknowledged for the help in


Bin Yuan

Xiaodong Yue

Ying Lv

Thierry Denoeux

Evidential Deep Neural Networks for Uncertain Data Classification

Keywords: Uncertain data classification, Evidence theory, Deep neural networks

Uncertain data classification makes it possible to reduce the decision risk through abstaining from classifying uncertain cases. Incorporating this idea into the process of computer aided diagnosis can greatly reduce the risk of misdiagnosis. However, for deep neural networks, most existing models lack a strategy to handle uncertain data and thus suffer the costs of serious classification errors. To tackle this problem, we utilize Dempster-Shafer evidence theory to measure the uncertainty of the prediction output by deep neural networks and thereby propose an uncertain data classification method with evidential deep neural networks (EviNet-UC). The proposed method can effectively improve the recall rate of the risky class through involving the evidence adjustment in the learning objective. Experiments on medical images show that the proposed method is effective to identify uncertain data instances and reduce the decision risk.

Introduction

In data classification tasks, the data instances that are uncertain to be classified form the main cause of prediction error [START_REF] Chow | On optimum recognition error and reject tradeoff[END_REF][START_REF] Fumera | Support vector machines with embedded reject option[END_REF][START_REF] Yue | Tri-partition neighborhood covering reduction for robust classification[END_REF]. Certain classification methods strictly assign a class label to each instance, which may produce farfetched classification results for uncertain instances. Uncertain classification methods aim to measure the uncertainty of data instances and accordingly reject uncertain cases [START_REF] Hellman | The nearest neighbor classification rule with a reject option[END_REF][START_REF] Cortes | Boosting with abstention[END_REF][START_REF] Fumera | Reject option with multiple thresholds[END_REF]. The methodology of uncertain classification is helpful to reduce the decision risk and involve domain knowledge in classification process [START_REF] Yao | Three-way decisions with probabilistic rough sets[END_REF][START_REF] Yue | Fuzzy neighborhood covering for three-way classification[END_REF][START_REF] Yue | Shadowed neighborhoods based on fuzzy rough transformation for three-way classification[END_REF]. For instance, in decision support for cancer diagnosis, filtering out uncertain cases for further cautious identification, may allow us to avoid serious misdiagnosis [START_REF] Chen | Three-way decision support for diagnosis on focal liver lesions[END_REF].

Due to their very good performance, deep neural networks have been widely ⋆ Corresponding author. Email: yswantfly@shu.edu.cn used in the classification of complex data [START_REF] Lecun | Deep learning[END_REF][START_REF] He | Deep residual learning for image recognition[END_REF], such as various kinds of medical images. However, most existing deep neural networks lack a strategy to handle uncertain data and may produce serious classification mistakes. For example, classifying CT images using convolutional neural networks without considering uncertainty may lead to overconfident decisions.

Trying to implement uncertain data classification based on deep neural networks, Geifman and El-Yaniv propose a selective classification method with deep neural networks, in which a selection function is constructed to quantify the reliability of predictions [START_REF] El-Yaniv | On the foundations of noise-free selective classification[END_REF][START_REF] Geifman | Selective classification for deep neural networks[END_REF]. The method relies on the quality of the selection function. If the quantification of reliability is not accurate, the identification of uncertain data cannot be guaranteed. Dempster-Shafer (D-S) evidence theory [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF] is also used to measure the uncertainty in machine learning models [START_REF] Denoeux | Logistic regression, neural networks and dempster-shafer theory: A new perspective[END_REF][START_REF] Denoeux | Maximum likelihood estimation from uncertain data in the belief function framework[END_REF][START_REF] Tong | Convnet and dempster-shafer theory for object recognition[END_REF]. Sensoy, Kaplan and Kandemir formulate the uncertainty in deep neural networks from the view of evidence theory [START_REF] Sensoy | Evidential deep learning to quantify classification uncertainty[END_REF]. Moreover, evidential neural networks have been constructed and applied for the uncertain classification of medical images [START_REF] Ghesu | Quantifying and leveraging classification uncertainty for chest radiograph assessment[END_REF][START_REF] Tardy | Uncertainty measurements for the reliable classification of mammograms[END_REF]. However, if the decision costs of different classes are imbalanced, evidential neural networks are not effective to classify the uncertain data instances of the risky class.

To address these problems, we construct a novel evidential deep neural network model and propose an uncertain data classification method. We formalize the uncertainty of the prediction output with evidence theory. A strategy to adjust the uncertainty in classification is also designed to improve the identification of certain and uncertain data instances in the risky class. The contributions of this paper are summarized as follows:

• Propose a novel evidential deep neural networks with the loss objective of both prediction error and evidence adjustment;

• An uncertain data classification method based on evidential deep neural networks (EviNet-UC) is proposed and applied to medical image diagnosis.

The rest of this paper is organized as follows. Section 2 presents the uncertain data classification method with evidential deep neural networks, which includes the model description and the strategy for uncertain data identification. In Section 3, we apply the proposed uncertain classification method to medical image data sets and show that the proposed method is effective to reduce the decision costs. Conclusions are given in Section 4.

Uncertain Data Classification with Evidential Deep Neural Networks

Given a dataset D = {x i , y i } N i=1 of N labeled data instances where y i is the class label of the instance x i , the loss of data classification with the evidential deep neural networks consists of the prediction error term L p i and the evidence adjustment term L e i as

L = 1 N N i=1 (L p i + λ * L e i ) , (1)
where λ = min(1.0, t/10) is the annealing coefficient to balance the two terms, t is the index of the current training epoch. At the beginning of model training, λ < 1 makes the network focus on reducing the prediction error. When t ≥ 10 the two terms play equal roles in the loss.

Prediction Error

For the binary classification of x i , we define the model output e + i , e - i as the evidence collected by the deep neural network for the positive and negative classes. The sum of the total evidence is E = e + i + e - i + 2. According to the evidence, we define the belief values of x i belonging to positive and negative classes as b

+ i = e + i /E, b - i = e - i /E, the uncertainty of classification is defined as u i = 1 -b + i -b - i .
Similar to the model proposed in [START_REF] Ghesu | Quantifying and leveraging classification uncertainty for chest radiograph assessment[END_REF], we adopt Beta distribution to formulate the distribution of the prediction with the evidences e + i , e - i . Suppose p i is the prediction of the instance x i belonging to the positive class, the probability density function of the prediction is

f (p i ; α i , β i ) = Γ (α i + β i ) Γ (α i ) Γ (β i ) p αi-1 i (1 -p i ) βi-1 dp i , (2)
where the parameters of Beta distribution are α i = e + i + 1, β i = e - i + 1 and Γ (•) is the gamma function. The prediction of the positive class can be obtained by p i = α i /E and 1 -p i = β i /E denotes the prediction of negative class.

Based on the probability density of the prediction, we construct the prediction error term for each data instance x i as the following expectation of squared error,

L p i = p i -y i 2 f (p i ; α i , β i ) dp i (3) = p i -y i 2 Γ (α i + β i ) Γ (α i ) Γ (β i ) p αi-1 i (1 -p i ) βi-1 dp i . (4)
Referring to the properties of the expectation and variance of Beta distribution, the formula (4) can be derived as

L p i = p i -y i 2 Γ (α i + β i ) Γ (α i ) Γ (β i ) p αi-1 i (1 -p i ) βi-1 dp i (5) = E p i -y i 2 (6) = E (p i ) 2 -2y i E (p i ) + y 2 i + var (p i ) (7) = (E (p i ) -y i ) 2 + var (p i ) (8) = y i - α i α i + β i 2 + 1 -y i - β i α i + β i 2 + α i β i (α i + β i ) 2 (α i + β i + 1) . (9)

Evidence Adjustment

Besides the prediction error, the uncertain cases in the classification should also be considered in real application scenarios. Identifying uncertain data instances for abstaining from classification is helpful to reduce the decision risk. In [START_REF] Ghesu | Quantifying and leveraging classification uncertainty for chest radiograph assessment[END_REF], a regularization term is integrated into the objective of neural network to reduce the evidences of uncertain instances. But this strategy ignores the difference of the risks of uncertain instances from different classes. To find out the uncertain instances of risky class effectively, we expect to rescale the data uncertainty u through adjusting the evidence and add an evidence adjustment term into the loss objective. The evidence adjustment term is constructed by the Kullback-Leibler divergence between the distributions of prediction with original and adjusted evidences. We also adopt the Beta distribution for the prediction with adjusted evidences and define λ > 1 as the evidence adjustment factor. The evidence adjustment term is expressed as

L e i = KL f p i ; αi , βi |f p i ; 1, λ , (10)
where (1, λ) = (1,

y i λ + (1 -y i )), αi , βi = ((1 -y i ) α i + y i , y i β i + (1 -y i ))
are the parameters of the Beta distributions of the prediction p i with adjusted and original evidences.

Let '1' denote the positive class and '0' denote the negative class. When the instance x i belongs to positive class,

y i = 1, (1, λ) = (1, λ) and αi , βi = (1, β i ).
If x i belongs to negative class, y i = 0, (1, λ) = (1, 1) and αi , βi = (α i , 1). For a negative-class instance, the adjustment term guides the parameter α i to 1 and thereby reduce the evidence of positive class to 0. For a positive-class instance, the adjustment term guides the parameter β i to λ. This will force the neural networks to promote the positive-class evidence for certain positive instances to reduce the prediction error.

According to the definition of KL divergence, the evidence adjustment term can be further simplified as

L e i = f p i ; αi , βi log f p i ; αi , βi f p i ; 1, λ dp i (11) = f p i ; αi , βi log f p i ; αi , βi dp i -f p i ; αi , βi log f p i ; 1, λ dp i (12) = E log f p i ; αi , βi -E B( αi βi) log f p i ; 1, λ . (13)
Referring to the properties of Beta distribution, the expectations in ( 13) can be further derived for computation as

E log f p i ; αi , βi (14)
=E

  log Γ αi + βi Γ (α i ) Γ βi p αi-1 i (1 -p i ) βi-1   ( 15
) =E   log Γ αi + βi Γ (α i ) Γ βi + ( αi -1) log p i + βi -1 log (1 -p i )   (16)
= log

Γ αi + βi Γ (α i ) Γ βi + (α i -1) E (log p i ) + βi -1 E (log (1 -p i )) (17)
= log

Γ αi + βi Γ (α i ) Γ βi + 2 -αi -βi ψ αi + βi + (α i -1) ψ (α i ) + βi -1 ψ βi , (18)
and

E B( αi, βi) log f p i ; 1, λ (19)
=E B( αi, βi) log Γ (1 + λ) Γ (1)Γ ( λ) p 1-1 i (1 -p i ) λ-1 (20) =E B( αi, βi) log Γ (1 + λ) Γ (1)Γ ( λ) + ( λ -1) log (1 -p i ) (21) = log Γ (1 + λ) Γ (1)Γ ( λ) + ( λ -1)E B(ãi, βi) (log (1 -p i )) (22)
= log

Γ (1 + λ) Γ (1)Γ ( λ) + ( λ -1) ψ βi -ψ αi + βi , (23)
in which ψ(•) denotes the digamma function. (a) rejection rate=0%, (b) rejection rate=10%, (c) rejection rate=20%, (d) rejection rate=30%.

Classification of uncertain data

As explained above, based on the belief values of x i belonging to positive and negative classes b + i = e + i /E, b - i = e - i /E, we can measure the classification uncertainty of x i by

u i = 1 -b + i -b - i .
With this uncertainty measure, applying the evidential neural networks to classify data, we can not only assign class labels to instances but also identify the uncertain ones. Through sorting the classified instances according to their uncertainty in ascending order, we select the top k uncertain instances for classification rejection to reduce the prediction risk.

Applying the proposed evidential neural network to the Breast IDC dataset (see the Section 3), Fig. 1(a) shows the evidence distribution of all the instances of positive class for multiple values of λ. We can see that the factor λ adjusts the evidences of instances and the evidences of certain positive instances are promoted. The data instances with low-level evidences for both classes have high uncertainty in classification. Thus the instances located in the bottom-left corner indicate uncertain cases. Fig. 1(b-d) display the evidence distribution of data instances after filtering out 10%, 20%, 30% uncertain instances, respectively. Based on the uncertain data identification strategy, we implement the uncertain data classification method with an evidential neural network (EviNet-UC). The effectiveness of the proposed method will be demonstrated in the following section.

Experimental results

To show that the uncertain classification method with evidential neural network is effective to reduce decision costs, we tested the proposed method on the medical datasets Breast IDC [START_REF] Cruz-Roa | Automatic detection of invasive ductal carcinoma in whole slide images with convolutional neural networks[END_REF] and Chest Xray [START_REF] Kermany | Identifying medical diagnoses and treatable diseases by image-based deep learning[END_REF]. The Breast IDC dataset consists of the pathological images of patients with infiltrating ductal carcinoma of the breast. The training set has 155314 images and the test set has 36904 images. We set the cancer case as positive class and the normal case as negative class. The Chest Xray dataset has 2838 chest radiographs, in which 427 images are chosen as the test data and the rest are set as the training data. The pneumonia and normal cases are set as positive class and negative class, respectively. For the algorithm implementation, we constructed the evidential deep neural networks based on the resnet18 architecture [START_REF] He | Deep residual learning for image recognition[END_REF] and we modified the activation function of the output layer to the ReLU function.

To achieve the overall evaluation of the classification methods, we adopt the measures of accuracy, F1 score, precision, recall rate and decision cost. Suppose the number of the instances of negative class is N and the number of positiveclass instances is P, TP and FP denote the numbers of true positive and false positive instances, TN and FN denote the true negative and false negative instances respectively. The measures are defined as accuracy = (T P + T N )/(P + N ), F1 score = (2 * T P )/(2 * T P + F N + F P ), precision = T P/(T P + F P ), recall = T P/(T P + F N ).

Assuming correct classification to have zero cost, cost N P , cos t P N denote the costs of false-positive classification and false-negative classification, respectively. The average decision cost of classification can be calculated as

decision cost = cost N P • F P P + N + cos t P N • F N P + N .
Based on the measures above, we carried out two experiments to evaluate the performance of the proposed uncertain classification method with evidential neural network (EviNet-UC). The first experiment aims to verify the superiority of the classification of the proposed method. Specifically, we compared the EviNet-UC method with other four uncertain classification methods based on deep neural networks: EvidentialNet [START_REF] Ghesu | Quantifying and leveraging classification uncertainty for chest radiograph assessment[END_REF], SelectiveNet [START_REF] Geifman | Selectivenet: A deep neural network with an integrated reject option[END_REF], Resnet-pd and Resnet-md [START_REF] Tardy | Uncertainty measurements for the reliable classification of mammograms[END_REF]. For fair comparison, we implemented all the methods above based on the resnet18 architecture. We set rejection rate = 0 (no rejection), cost P N = 5, cos t N P =1 and applied all the classification methods to the Breast IDC dataset. The comparative experimental results are presented in Fig. 2 and Table 1. We can find that the proposed EviNet-UC method achieves the highest recall rate and the lowest decision cost among all the comparative methods. This means that the proposed method is effective to reduce the misclassification of the risky class (cancer case). Moreover, we changed the rejection rate from 0 to 0.5 to further compare the classification methods. Fig. 3 presents the recall rates and the decision costs of different methods with different rejection rates. We can see that the EviNet-UC method achieves the best performance for all rejection rates. Compared to other methods, the proposed method is more effective to reduce the classification risk.

The second experiment aims to show that the proposed method is effective to identify uncertain data. Varying the rejection rate in [0, 1] and applying EviNet-UC on the Chest Xray dataset, we obtained the classification results for different numbers of rejected uncertain radiographs. Fig. 4 illustrates the evaluation of the classification based on EviNet-UC with varying rejection rates. It can be seen The uncertain radiographs will be rejected for cautious examination to further reduce the cost of misclassification.

Conclusions

Certain classification methods with deep neural networks strictly assign a class label to each data instance, which may produce overconfident classification results for uncertain cases. In this paper, we propose an uncertain classification method with evidential neural networks which measures the uncertainty of the data instances with evidence theory. Experiments on medical images validate the effectiveness of the proposed method for uncertain data identification and decision cost reduction. Our method currently focuses on only binary classifica-tion problem and the relationship between the decision cost and the evidence adjustment factor also requires theoretical analysis. Exploring the evidence adjustment factor in multi-class classification problems and constructing the precise uncertainty measurement for reducing decision risk will be future works.

Fig. 1 .

 1 Fig. 1. Evidence distribution of positive-class instances with different rejection rates. (a) rejection rate=0%, (b) rejection rate=10%, (c) rejection rate=20%, (d) rejection rate=30%.

Fig. 2 .

 2 Fig. 2. Comparative experimental results on Breast IDC dataset.

Fig. 3 .

 3 Fig. 3. (a) Recall rates of different classification methods with varying rejection rates, (b) decision costs with rejection rates.

Fig. 4 .

 4 Fig. 4. Classification evaluation of EviNet-UC with varying rejection rates.

Fig. 5 .

 5 Fig. 5. (a) certain negative-class instance (normal case), (b) certain positive-class instance (pneumonia), (c) uncertain normal case, (d) uncertain pneumonia case.

Fig. 5 (

 5 Fig.5 (c) displays an uncertain normal case. In general, the lung area is clear but there exists a dense area of nodule in the right part (marked in red circle). EviNet-UC produces high uncertainty u = 0.44 to indicate the judgement is not confident. Fig.5 (d)shows another uncertain case of pneumonia. In the radiograph, there exists a shadow area in the right lung but the symptom is not prominent, which leads to the uncertainty u = 0.35 for pneumonia identification. The uncertain radiographs will be rejected for cautious examination to further reduce the cost of misclassification.

Table 1 .

 1 Comparative experimental results on Breast IDC dataset.

	Methods	Accuracy F1-score Precision Recall Decision cost
	EvidentialNet 0.8874	0.8477	0.8566	0.8389	0.3532
	SelectiveNet	0.8874	0.8493	0.8488	0.8498	0.3370
	ResNet-PN	0.8912	0.8512	0.8702	0.8330	0.3583
	ResNet-MD	0.8911	0.8511	0.8699	0.8330	0.3584
	EviNet-UC	0.8915	0.8572	0.8432 0.8716	0.3004

Acknowledgment

This work was supported by National Natural Science Foundation of China (Nos. 61976134, 61573235, 61991410, 61991415) and Open Project Foundation of Intelligent Information Processing Key Laboratory of Shanxi Province (No. CICIP2018001).