Guillaume Bernard-Granger

Nassira Benameur

Ahmed Addad

Mats Nygren

Christian Guizard

Sylvain Deville

Phenomenological analysis of densification mechanism during spark plasma sintering of MgAl 2 O 4

Spark plasma sintering (SPS) of MgAl2O4 powder was investigated at temperatures between 1200 and 1300°C. A significant grain growth was observed during densification. The densification rate always exhibits at least one strong minimum, and resumes after an incubation period. Transmission electron microscopy investigations performed on sintered samples never revealed extensive dislocation activity in the elemental grains. The densification mechanism involved during SPS was determined by anisothermal (investigation of the heating stage of a SPS run) and isothermal methods (investigation at given soak temperatures). Grain-boundary sliding, accommodated by an in-series {interface-reaction/lattice diffusion of the O 2-anions} mechanism controlled by the interface-reaction step, governs densification. The zero-densification-rate period, detected for all soak temperatures, arise from the difficulty of annealing vacancies, necessary for the densification to proceed. The detection of atomic ledges at grain boundaries and the modification of the stoichiometry of spinel during SPS could be related to the difficulty to anneal vacancies at temperature soaks.

I. Introduction

Fully dense polycrystalline alumina-magnesia spinel (referred to as spinel hereafter), MgAl2O4, is an attractive material for its excellent optical properties (in-line transmittance) in the visible to mid-infrared ranges. [START_REF] Harris | History of development of polycrystalline optical spinel in the U[END_REF][START_REF] Mroz | Recent advances in spinel optical ceramics[END_REF] It is currently considered as a cost-effective alternative to monocrystalline sapphire for the manufacturing of infrared-domes, intended to be mounted on the new generation of high speed air-to-air/ground-to-air missiles coming onto the market. However, most of the polycrystalline materials developed up to now exhibit a grain size in the 10-150 µm range, explaining the disappointing mechanical/thermomechanical properties reported. [START_REF] Roy | Hot-pressed MgAl2O4 for ultraviolet (UV), visible, and infrared (IR) optical requirements[END_REF][START_REF] Baudin | High-temperature mechanical behavior of stoichiometric magnesium spinel[END_REF] A different approach has recently proved possible to obtain fully dense polycrystalline spinel with a grain size well below the micrometer, [START_REF] Krell | Transparent polycrystalline sintered ceramic of cubic crystal structure[END_REF][START_REF] Krell | Ballistic strength of opaque and transparent armors[END_REF] using a sinter/ hot-isostatic pressing (HIP) strategy. a) Address all correspondence to this author.

To simultaneously limit grain growth and obtain nearly maximum densification, spark plasma sintering (SPS) has been successfully applied to other materials, such as TiN, [START_REF] Groza | Field-assisted sintering of nanocrystalline titanium nitride[END_REF] Al2O3, [START_REF] Shen | Spark plasma sintering of alumina[END_REF][START_REF] Kim | Spark plasma sintering of transparent alumina[END_REF] Si3N4, [START_REF] Suganuma | Pulsed electric current sintering of silicon nitride[END_REF] 3, and 8 mol% yttria-stabilized ZrO2 [START_REF] Bernard-Granger | Spark plasma sintering of a commercially available granulated zirconia powder: I. Sintering path and hypotheses about the mechanism(s) controlling densification[END_REF][START_REF] Bangchao | Spark-plasma sintering the 8-mol% yttriastabilized zirconia electrolyte[END_REF] and b-SiC. [START_REF] Yamamoto | Consolidation of nanostructured b-SiC by spark plasma sintering[END_REF] Dense polycrystalline spinel with acceptable optical properties (residual pores are still present in the material) can be processed using SPS. [START_REF] Frage | Spark plasma sintering (SPS) of transparent magnesium-aluminate spinel[END_REF] The grain size obtained is nevertheless still in the order of tens of micrometers and should be defined as coarse. [START_REF] Frage | Spark plasma sintering (SPS) of transparent magnesium-aluminate spinel[END_REF] We therefore decided to investigate SPS as a densification method for spinel, aiming at combining high densification and grain size below the micrometer.

The sintering behavior of a commercially available spinel powder was investigated for a soak temperature in the range 1200-1300°C, a soak time of 15 min, a heating rate of 100 °C/min, and an applied macroscopic compaction pressure of 25 MPa. The relative density of the sintered samples has been correlated with their average grain size. The mechanism controlling densification during the SPS experiments are investigated and discussed. TABLE I. Impurities, humidity level, and specific surface area of the raw powder. Note: ppm in weight.

II. Raw powder and green samples ready for SPS experiments

The commercially available S30CR raw powder (Baikowski Chimie, La Balme de Sillingy, France) was selected as the starting material. The main impurities determined by inductively coupled plasma spectrometry (ICP; Varian Vista Pro, Varian Inc., Palo Alto, CA) are listed in Table I. Because the raw powder is crystallized from an alumbased process, the residual sulfur concentration appears relatively high (around 400 ppm in weight).

Using Brunauer-Emmett-Teller analysis (BET) measurements (Nova 2000, Quantachrome Instruments, Boynton Beach, FL) the specific surface area (SSA) of the raw powder was found in the range 30-31 m 2 /g. Scanning electron microscope (SEM; JSM-6301F, JEOL Ltd., Tokyo, Japan) examinations showed aggregation of the raw powder. The elemental crystallites constituting the aggregates have a spherical shape and an average diameter in the range 55-70 nm, in good agreement with the specific surface area.

X-ray diffraction (XRD; Bruker D5000, Bruker AXS Gmbh, Karlsruhe, Germany) confirmed that only the cubic MgAl2O4 species (space group 227, 𝐹𝑑3𝑚, lattice parameter of 8.0831 Å) is present in the raw powder.

High solids loading (53 wt%) water-based slurries were prepared from the S30CR raw powder (ammonium polyacrylate was incorporated as a dispersant). After deagglomeration, optimal blending, and degassing, samples were slip-casted in porous plaster molds.

Once setting was completed, green samples were left in a drying oven for a few hours. After a debinding step in air (480°C/3 h), samples were ready for the SPS experiments. The diameter of the samples was typically 19.7 mm and their thickness 6 mm (the diameter/thickness ratio was always above 2.5, which strongly minimized the axial density gradient during the SPS experiments). For all samples, the relative green density was around 42% (the theoretical density for spinel has been calculated to be 3.579 g/cc from the elemental lattice). The fracture surface in Fig. 1 shows the typical microstructure observed in green samples.

III. Experimental Procedure

All the tests were conducted in vacuum, with the slipcasted debinded samples, on equipment (SPS-2080, SPS Syntex Inc., Kanagawa, Japan) located at the Arrhenius Laboratory (Stockholm University, Sweden). FIG. 1. Typical aspect of the green microstructure obtained after slip casting-fracture surface.

For each test, a graphite die (internal diameter of 20 mm, thickness of 15 mm) was filled with one green sample and mounted on the SPS equipment (graphite punches). A heating rate of 100°C/min, a macroscopic compaction pressure of 25 MPa (applied at room temperature), and the standard 12:2 pulse sequence for the direct current (dc) [START_REF] Shen | Spark plasma sintering of alumina[END_REF] were chosen.

The temperature was obtained from an optical pyrometer focused on the outer surface of the graphite die. This temperature was therefore not the one actually seen by the powder. Modeling of the temperature distribution during field-assisted sintering, performed on a TZ3Y raw powder, has been recently conducted. [START_REF] Vanmeensel | Modelling of the temperature distribution during field assisted sintering[END_REF] For graphite die geometry similar to that used here, the temperature difference between the specimen center (4.25 mm thick) and the external pyrometer focused on the outer die wall surface was calculated to be around 80°C. [START_REF] Vanmeensel | Modelling of the temperature distribution during field assisted sintering[END_REF] The thermal conductivity of spinel is much higher than that of TZ3Y (respectively 14.6 and 2.5 W/m/K). The maximum temperature difference between the outer surface of the die and the powder compact should therefore be around 80°C (probably less) during the SPS experiments performed on spinel samples. During all tests (heating, soak, and cooling), the height variation of the powder bed (DL = L-L0 < 0, L is the instantaneous height and L0 the initial height of the powder bed when the macroscopic pressure is applied at room temperature) was precisely measured. Each test was corrected to account for the dimensional variations of the SPS equipment (a blank test with a fully dense polycrystalline spinel sample positioned in the die was performed and then subtracted to the test result). The instantaneous sample height variation and the relative density D are linked by the following relationship:

𝐷 = 𝐷 (eq. 1)
where Lf is the final height, L the instantaneous height, and Df the final relative density.

The apparent density of the sintered samples was measured using the Archimedes method with deionized water (three measurements were made for each sample). The final relative density, Df, was obtained using a theoretical density of 3.579 g/cc for stoichiometric spinel.

Thin foils were prepared from the central zone of as sintered samples by slicing and mechanical polishing, followed by ion milling. The foils were covered with a thin layer of graphite and observed using a Philips CM30 microscope (Philips Research Laboratories, Eindhoven, The Netherlands, acceleration voltage of 300 kV, point-to-point resolution of 0.19 nm) equipped with an energy dispersive spectroscopy (EDS) microanalysis system (Thermo Electron Corporation, Waltham, MA, Noran system equipped with an ultrathin window). The general microstructure was observed in bright field mode. Local EDS analyses were performed on one sample using the scanning transmission electron microscopy (STEM) mode in the center of the elemental grains (10 measurements), at grain boundaries (10 measurements) and at triple points (9 measurements) using a probe size of 5.6 nm. Quantitative analyses were carried out using the Doukhan-Van Cappellen method, based on electroneutrality of the specimen to access the local thin foil thickness. [START_REF] Van Cappellen | Quantitative transmission x-ray microanalysis of ionic compounds[END_REF] Additional investigations were also performed using the high-resolution transmission electron microscopy (HRTEM) mode at grain boundaries.

TEM was also used to evaluate the grain size for each sintered sample. A line-intercept method taking into account at least 150 grains (with a three-dimensional correction factor determined to be 1.2, approximating the grains to spheres [START_REF] Bernard-Granger | Influence of codoping on the sintering path and on the optical properties of a submicronic alumina material[END_REF] ) was used.

The densification rate variation (relative to temperature), 1/D dD/dT, is shown as a function of temperature up to 1400°C in Fig. 2. Densification starts around 750°C and three regimes are observed. (i) From 750 to 1150°C the densification rate (relative to temperature) increases continuously, (ii) between 1150 and 1315°C the densification rate (relative to temperature) has a constant value around 2x10 -3 /°C, and (iii) above 1315°C the densification rate (relative to temperature) increases abruptly.

FIG. 2. Densification rate (relative to temperature) curve obtained on a slip casted sample. The heating rate is fixed to 100°C/min, the applied macroscopic compaction pressure to 25 MPa (the compaction pressure is applied at room temperature). The rectangle determines the temperature range where subsequent SPS experiments are performed.

For this study, the 1200 to 1300°C temperature range, where the densification rate (relative to temperature) is always around 2x10 -3 /°C, was selected. Even if the real temperature in the compact is not precisely known [finite element modeling (FEM) calculations are required] we assume that the temperature difference between the matrix, where the pyrometer is focalized, and the powder bed is at maximum 80°C and constant in the range 1200-1300°C. The following SPS conditions in vacuum were then selected: (i) soak temperature = 1200-1225-1250-1275-1300°C; (ii) soak time = 15 min; (iii) heating rate (HR) = 100°C/min; (iv) macroscopic compaction stress = 25 MPa (applied at room temperature); and (v) 12:2 pulse configuration.

In the past, mass transport during sintering, with or without an external load, has been considered close to that occurring in high temperature creep. [START_REF] Brook | Sintering-Theory and Practice[END_REF][START_REF] Coble | Diffusion models for hot pressing with surface energy and pressure effects as driving forces[END_REF] Assuming an approach similar to Mukherjee for the creep of dense metals, [START_REF] Mukherjee | Experimental correlations for hightemperature creep[END_REF] it has been proposed that the SPS kinetic equation can be written as [START_REF] Bernard-Granger | Spark plasma sintering of a commercially available granulated zirconia powder: I. Sintering path and hypotheses about the mechanism(s) controlling densification[END_REF][START_REF] Bernard-Granger | Densification mechanism involved during spark plasma sintering of a co-doped a-alumina material. I: Formal sintering analysis[END_REF] :

1 1 = 𝐾 (eq. 2)
where D is the instantaneous relative density of the compact, t the time, µeff the instantaneous shear modulus of the compact, K a constant, R the gas constant, T the absolute temperature, Qd the apparent activation energy of the mechanism controlling densification, b the Burgers vector (close to the lattice parameter), G the grain size, and seff the instantaneous effective stress acting on the compact. It was also proposed that µeff, and seff can be written as [START_REF] Bernard-Granger | Spark plasma sintering of a commercially available granulated zirconia powder: I. Sintering path and hypotheses about the mechanism(s) controlling densification[END_REF][START_REF] Bernard-Granger | Densification mechanism involved during spark plasma sintering of a co-doped a-alumina material. I: Formal sintering analysis[END_REF] :

𝜇 = ℎ 2(1+ ) -0 1-0 (eq. 3) 𝜎 = 1-0 2 (1-0 ) 𝜎 (eq. 4)
where Eth is the Young's modulus of the theoretically dense MgAl2O4 material (its variation as a function of temperature can be found in Ref. 22), neff the effective Poisson's ratio (a value of 0.26 is chosen for all experimental conditions), D0 the starting green density of the powder compact (42%), and smac the macroscopic compaction pressure (25 MPa).

Equation ( 4), proposed for the effective stress, approximates the individual grains to spheres, independently of the relative density value. To be perfectly correct, the real effective stress should also incorporate an additional term related to a pressureless kind of driving force, which becomes prominent when the relative density approaches 1. An intuitive expression could be 2g/r, the same as for pressureless sintering, where g is the surface energy and r the pore radius. It is also possible that this pressureless kind of driving force is counterbalanced by another force related to gas pressure that develops within closing pores during the SPS runs. More investigations (theoretical and experimental) are needed to develop a more precise expression for the effective stress. In the meantime, the macroscopic compaction pressure alone is assumed to be responsible for the driving force operating during the SPS experiments we completed.

From Eq. ( 2), following the procedure suggested by Brook, [START_REF] Brook | Sintering-Theory and Practice[END_REF] Qd, p, and n can be determined. These are the key parameters enabling the identification of the mechanisms controlling densification of the powder bed during the SPS experiments.

IV. Results

Figure 3 shows the densification curves at soak. The higher the temperature, the higher the relative density is after sintering. The shape of the curves is nevertheless uncommon and different to that reported for TZ3Y. [START_REF] Bernard-Granger | Spark plasma sintering of a commercially available granulated zirconia powder: I. Sintering path and hypotheses about the mechanism(s) controlling densification[END_REF] The densification curves for all temperatures, but 1200°C, adopt a wavy shape (the temperature regulation of the SPS equipment is good enough to ensure it has no influence on the curves shape). Some "plateaus," where the relative density remains almost constant during a certain period of time, are observed before densification resumes. This behavior is very similar to that observed during hot pressing of spinel powders. [START_REF] Ting | Hot pressing of magnesium aluminate spinel: I. Kinetics and densification mechanism[END_REF] It is also close to that observed during creep experiments on fully dense polycrystalline spinel. [START_REF] Panda | Superplastic deformation in fine-grained MgO2Al2O3 spinel[END_REF][START_REF] Morita | Strain softening and hardening during superplastic-like flow in a fine-grained MgAl2O4 spinel polycrystal[END_REF] The variations of 1/D dD/dt as a function of D for all the test temperatures are shown in Fig. 4. When SPS was performed between 1225 and 1300°C, strong minima where densification of the compact is strongly reduced are detected. This corresponds to the "plateau" periods observed in Fig. 3. After the minima, 1/D dD/dt increased again and finally collapsed at the end of the test. When the SPS temperature is 1200°C, the same behavior is observed, though clearly less pronounced. It is also interesting to note that the minima move to higher relative densities with an increase in the test temperature. To understand this behavior, it could be interesting to perform interrupted SPS experiments (with a high cooling rate to "freeze" the structure) and observe the resulting microstructure by TEM.

The grain size versus relative density trajectory, referred to as sintering path, and obtained with results from the different samples sintered by SPS is shown in Fig. 5. It is compared to the sintering path obtained for the same material (slip-casted samples) densified using pressureless sintering (PS) in air (the heating rate is only 10°C/min, in comparison to 100°C/min for the SPS experiments) at 1500 °C. In the green samples, the elemental crystallite size is around 55 to 70 nm (Fig. 1). A significant grain growth is observed during both the SPS and PS runs. The sintering path of the SPS material is below the one of the PS material. For a given relative density, the final grain size obtained using SPS will be smaller. A good approximation for the grain size/relative density trajectory obtained by SPS is given by the dashed line visible in Fig. 5. Subsequently, for the rest of the work, the following expression linking G to D will be used:

𝐺 = 𝛼𝑒 (eq. 5)
with a = 0.0018 and b = 5.3586. Even if the residual porosity level is different, the typical microstructures for samples sintered at 1200°C (76.58% relative density, 110-nm grain size), 1250°C (85.08% relative density, 177-nm grain size), and 1275°C (93.99% relative density, 260-nm grain size) for 15 min are qualitatively similar. As an example, the microstructure observed on the sample sintered at 1250°C is shown in Fig. 6. For all samples (i) the residual porosity is located at grain boundaries (intergranular porosity), and is homogeneous in size (no aggregation to form large pores). Its spatial distribution is also homogeneous; (ii) the spinel grains constituting the matrix exhibit an equiaxed shape and a narrow grain size distribution; and (iii) no dislocation activity was detected in the elemental grains or at grain boundaries (multi two-axis tilting of the thin foils).

The typical microstructure developed in the sample sintered at 1300°C for 15 min (98.82% relative density, 379-nm grain size) is shown in Figs. 7(a) and 7(b). Most of the porosity has disappeared, only a few residual pores are observed at triple points and homogenously distributed throughout the thin foil. Some intragranular pores are detected [see white arrow in Fig. 7(a)], though their total content is low. The grains still exhibit an equiaxed shape but it seems that the average grain size from one elemental crystal to another one is more variable, as compared to previous samples. Some grains also exhibit an intragranular dislocation activity [Fig. 7(b)]. The slip systems activated were not investigated (use of the weak-beam method, determination of the g vector). Such dislocations are nevertheless clearly homogeneously detected in the sample sintered at 1300°C and not in the other ones sintered at a lower temperature. However, despite the clear presence of such events, the general dislocation activity is quantitatively considered as low. At that time, it is not clear why a dislocation activity is only observed for a SPS temperature of 1300°C. It could be interesting in the future For all sintering temperatures, densification during SPS of the spinel samples investigated is not controlled by a mechanism involving dislocations (gliding or climbing) contribution. In fact, the microstructures observed are in good agreement with a densification mechanism based on a grain-boundary sliding/diffusion accommodated process.

Figure 8(a) shows the typical aspect of grain boundaries observed using HRTEM. Grain boundaries appear depleted of any amorphous thin film. Using higher magnification, with a nonoptimal focus, most of the grain boundaries do not appear perfectly flat and exhibit atomic ledges, as shown in Fig. 8(b).

EDS nanoanalyses (Fig. 9) have been performed for the sample sintered at 1300°C for 15 min (98.82% relative density, 379-nm grain size). For all locations (center of grain, grain boundary, triple point), the atomic Al/Mg ratio is clearly above 2 (standard deviation is in all cases +/-0.03). For comparison, the O/Al ratio is constant, with an average value of 1.96+/-0.01 for the grain centers versus 1.93+/-0.01 for grain boundaries and triple points. The following chemical compositions have been determined: (i) Mg0.922Al2O3.922 for grain centers; (ii) Mg0.855Al2O3.855 for grain boundaries; and (iii) Mg0.862Al2O3.862 for triple points.

Using similar experimental conditions, EDS nanoanalyses have been conducted on the elemental crystallites constituting the raw powder. Results are also shown in Fig. 9. Therefore, the average composition Mg0.939Al2O3.939 for crystallite centers has been determined. The O/Al ratio is constant for all crystallites with an average value of 1.98 0.02.

Clearly, there is a change of stoichiometry during SPS of spinel and an impoverishment in MgO is observed, especially at grain boundaries and at triple points. However, at that time, the modification of stoichiometry was not understood (critical temperature, critical relative density, pressure effect, etc). This change of stoichiometry might also have an influence on the space charge at grain boundaries and, consequently, on the diffusion process that controls densification and grain growth. We will come back later on to that point in Sec. V. Finally, it is also possible that the slight brown color of the sintered samples does matter with the astoichiometry finally obtained. But carbon contamination, from a CO containing residual atmosphere during the SPS experiments, cannot be excluded (graphite heating die; a rotary pump is generating vacuum in which the residual gases can give rise to an atmosphere containing CO2 possibly transforming to CO at high pressure within the shrinking pores). If such a contamination occurs during the SPS tests, the expression proposed for the densification rate should be modified accordingly (in that case the driving force is not the effective pressure alone anymore, a contribution from gas pressure developing within closed pores has to be subtracted). Complementary investigations are needed to clarify this point.

V. Discussion

To discriminate the mechanisms controlling the densification of the spinel powder during SPS, it is necessary to determine the values of the Qd, p, and n parameters in Eq. ( 2). Rearranging Eq. ( 2) yields:

𝐿𝑛 1 = - -𝑝𝐿𝑛 𝐺 + 𝑛𝐿𝑛 + 𝐾′ (eq. 6)
where G is given by relation ( 5) and dT/dt is the heating rate during the SPS experiment.

Phenomenological models have been developed to describe high-temperature creep behavior for ceramic polycrystals. [START_REF] Ashby | Diffusion-accommodated flow and superplasticity[END_REF][START_REF] Burton | The characteristic equation for superplastic flow[END_REF] Such models can be adapted to an SPS problematic, where the densification mechanism is not based on dislocations activity, as it is the case for the runs performed on spinel (see TEM observations reported in Sec. IV).

If the grain boundaries are perfect sources/sinks of vacancies, the n and p parameters in relations ( 2) and ( 6) can have the following values 26 : (i) n = 1, p = 2: the densification mechanism is grain-boundary sliding accommodated by volume diffusion and the apparent activation energy has a bulk character; and (ii) n = 1, p = 3: the densification mechanism is grain-boundary sliding accommodated by grain-boundary diffusion and the apparent activation energy has a grain-boundary character.

If the grain boundaries are not perfect sources/sinks of vacancies, the n and p parameters in relations ( 2) and ( 6) have the following values27: (i) n = 2, p = 1: the densification mechanism is grain-boundary sliding accommodated by an in-series {interfacereaction/lattice diffusion} mechanism controlled by the interface-reaction step and the apparent activation energy has a bulk character; and (ii) n = 2, p = 2: the densification mechanism is grain boundary sliding accommodated by an in-series {interface reaction/grain-boundary diffusion} mechanism controlled by the interface-reaction step and the apparent activation energy has a grain-boundary character.

The heating part of an SPS run, for temperatures between 940 and 1300°C, can also be exploited. Imposing a given value for the activation energy Q, it is possible, using the Excel Solver function (Microsoft Excel, Microsoft France, Courtaboeuf, France), to calculate the corresponding p, n, and K0 parameters that enable the left side of relation ( 6) to be equal to its right side by minimization of the residual sum of squares (RSS).

FIG. 10. Determination of the densification mechanism using an anisothermal method.

The heating rate is fixed to 100°C/min and the applied macroscopic compaction pressure to 25 MPa. The heating portion of a SPS experiment is used, activation energy values are imposed (Q) and the corresponding p, n, and K0 parameters involved in relation ( 6) are calculated using Excel Solver function. Best result is for the lowest RSS (residual sum of squares).

The p, n, K0, and RSS values, obtained for each value of Q imposed, are summarized in Fig. 10. Clearly, the minimum RSS value is obtained when n and p have a value of 2.0 ±0.1 and 1.2±0.0, respectively. In that case, K0 is 51.1±2.1 and Qd has a value of 500±20 kJ/ mol. Because of the obtained values for n (around 2) and p (close to 1), the apparent activation energy has a bulk character. In the past, it has been shown that the activation energy for oxygen self-diffusion in monocrystalline stoichiometric spinel is in the range 415-500 kJ/mol, [START_REF] Reddy | Oxygen diffusion in magnesium aluminate spinel[END_REF][START_REF] Ando | Effect of ratio of surface area to volume on self-diffusion coefficients determined for crushed MgO-Al2O3 spinels[END_REF][START_REF] Ting | Defect reactions and the controlling mechanism in the sintering of magnesium aluminate spinel[END_REF] comparable to the value of 500 kJ/mol obtained there. For comparison, the activation energies for the selfdiffusion of Mg 2+ cations, in the same kind of spinel monocrystal, has been determined to be around 200 kJ/ mol. [START_REF] Liermann | Diffusion kinetics of Fe 2+ and Mg in aluminous spinel. Experimental determination and applications[END_REF] No value was found in the literature for the selfdiffusion of the Al 3+ cations, although it can be indirectly estimated. Martinelli et al. [START_REF] Martinelli | Mobility of cations in magnesium aluminate spinel[END_REF] concluded from conductivity experiments that, at 1000°C, magnesium is the more mobile cation. Independently, a Mg 2+ «Al 3+ interdiffusion activation energy of 235 kJ/mol was determined in by Watson. [START_REF] Watson | Kinetics of the reaction MgO + Al2O3 ® MgAl2O4 and Mg-Al interdiffusion in spinel at 1200 to 2000°C and 1.0 to 4.0 GPa[END_REF] Assuming that migration of the aluminum cation is the rate limiting step in this process, as suggested by Martinelli, then Murphy et al. [START_REF] Murphy | Cation diffusion in magnesium aluminate spinel[END_REF] concluded that 235 kJ/mol could be a good evaluation for the activation energy for Al 3+ self-diffusion in spinel. Both values reported for Mg 2+ and Al 3+ cations are much lower than the apparent activation energy for densification obtained from these SPS experiments on spinel.

We therefore propose that grain-boundary sliding, accommodated by an in-series {interface-reaction/lattice diffusion of the O2 anions} mechanism controlled by the interface reaction step, governs densification of our spinel samples during the heating portion of the SPS experiments we performed (at least between 940 and 1300°C).

Regarding the densification curves obtained for the different soak temperatures, combining relations (2) and ( 5) yields:

1 1 = 𝐾 0 - 𝑒 -𝛽 (eq. 7)
where K0 is a constant.

Assuming only one constant value of Qd (coherency with the results obtained using the anisothermal method, no reason for a change in densification mechanism between the heating portion and the early stages of the soaks), the slope of the straight line obtained when plotting Using a fixed value of 1 for p, Fig. 11 shows the variations of

𝐿𝑛 1 𝜇 1 𝐷 𝑑𝐷 𝑑𝑡 + 𝛽𝑝𝐷
as a function of Ln(seff/µeff) for the different soak temperatures selected. It seems that n exhibits an average value of 2 when densification progresses at the beginning of the soaks, for temperatures between 1200 and 1275°C, in good agreement with the ideal combination (n = 2, p = 1) determined from the heating portion of an SPS experiment (see above). But in all cases the

𝐿𝑛 1 𝜇 1 𝐷 𝑑𝐷 𝑑𝑡 + 𝛽𝑝𝐷 = 𝑓 𝐿𝑛 𝜎 𝜇
trajectories differ from a straight line after some period of time at soak. When the soak temperature is fixed to 1300°C, the apparent value of n is lower than 2. as a function of 1/T is plotted (Fig. 12). According to relation (10), a value of Qd around 530±30 kJ/mol is calculated (XLSTAT solver, Addinsoft France, Paris, France).

This value is similar to what has been obtained using the heating portion of an SPS experiment (500±20 kJ/mol). Therefore, it is proposed that grain-boundary sliding, accommodated by an in-series {interface-reaction/lattice diffusion of the O2-anions} mechanism controlled by the interface-reaction step, is still governing densification of our spinel samples at the beginning of the soak, for SPS temperatures in the range 1200-1275°C. For soak temperatures of 1200, 1225, and 1250°C, the period where n is close to 2 is followed by an abrupt densification hardening regime (the instantaneous value of n increases continuously), corresponding to a strong decrease of the instantaneous relative densification rate (Fig. 4). A similar trend is observed for a soak temperature of 1300°C, where n has a value between 1 and 2. When the soak temperature is 1275°C, the densification hardening regime is preceded by a period where n is close to 0. After the hardening period, densification resumes, for all soak temperatures. This corresponds to a densification softening period.

Such densification or strain hardening/densification or strain softening behavior has already been observed during hot pressing of spinel powder [START_REF] Panda | Superplastic deformation in fine-grained MgO2Al2O3 spinel[END_REF] and high temperature creep experiments on polycrystalline spinel. [START_REF] Morita | Strain softening and hardening during superplastic-like flow in a fine-grained MgAl2O4 spinel polycrystal[END_REF] The softening and hardening were related to a change in the internal stresses, depending on a decrease and increase in the density of the intragranular dislocations, respectively, whose motions contribute to the relaxation of stress concentrations exerted through the predominant mechanism of grainboundary sliding. It was proposed that densification/deformation was controlled by the continuous recovery of the dislocations, limited by lattice diffusion of the oxygen ions.

Clearly, this hypothesis is not in agreement with the typical microstructures observed for the different samples obtained here by SPS. For all temperatures, no extensive dislocation activity has been reported in the elemental grains constituting the sintered samples (see Sec. IV). It is proposed here that the densification hardening, corresponding to a zero-densification-rate period, is originating from the difficulty to anneal vacancies, which is the driving force for the densification to proceed. At each soak temperature, after a certain period of time, vacancies are accumulating because the annealing step stops. Then densification also stops. An incubation time is then necessary to anneal enough vacancies to resume densification. Atomic ledges have been detected at grain boundaries using HRTEM [Fig. 8(b)]. The EDS nanoanalyses have also shown that the stoichiometry of spinel is changing during SPS (Fig. 9). Both events, at the nanoscale, could be related to the difficulty to anneal vacancies at the soak temperatures.

The astoichiometry is amplified at grain boundaries and at triple points, in comparison to the center of the grains. The EDS analyses suggest an excess of Mg and O vacancies in these areas. If the concentrations for each kind of vacancy are not the same, the consequence will be an excess of negative positive charge at grain boundaries and triple points, depending on which concentration is the highest. Therefore, the residual electrical charge will be compensated by an opposite electrical charge cloud in the surrounding grains, at the close vicinity of the grain boundaries/triple points, called space charge. [START_REF] Kliever | Space charge in ionic crystals: I. General approach with application to NaCl[END_REF] The space-charge region creates an electric potential and therefore modifies the conditions of the charged defect formation and diffusion that can explain the perturbation of the interface-reaction (zero densification-rate period), claimed to control densification during our SPS experiments. Interesting works have been published on the stoichiometry variation of polycrystalline spinel at grain boundaries. [START_REF] Chiang | Grain boundary migration in nonstoichiometric solid solutions of magnesium aluminate spinel: II. Effects of grain boundary nonstoichiometry[END_REF][START_REF] Nuns | Space charge characterization by EDS microanalysis in spinel MgAl2O4[END_REF] No successes have been reported for the characterization of the space charge region alone in such materials. In fact, such investigations are difficult, since the space-charge dimension could be lower than the lowest spot size available on the best TEM/STEM equipments. The fact that the SPS technology also involves the submission of the compact to an electric field has perhaps also an influence on the properties of a possible spacecharge region and consequently on the possible perturbation of the interface-reaction.

To investigate such a last effect, the same kind of spinel samples will be densified using standard hot pressing in the future.

Another comment may be added, regarding the determination of n, p, and Qd. The method investigating the heating part of a SPS run is fairly straightforward. Inversely, more questionable are the results obtained when investigating the densification curves at soak. In such case, it may be difficult to determine a precise value for n because the hardening phenomenon appears rapidly at soak (Fig. 11, 1225 and 1250°C). At least an approximated value may be extrapolated using few points at the beginning of the soak.

VI. Conclusions

SPS of a stoichiometric alumina-magnesia spinel powder, shaped by slip casting, has been investigated in vacuum, in the 1200 to 1300°C temperature range. The other experimental parameters were a heating rate of 100°C/min, an applied macroscopic compaction pressure of 25 MPa, a soak time of 15 min, and the use of the standard 12:2 pulse configuration.

For all soak temperatures, grain growth is significant during densification. Accounting for this phenomenon, the mechanism controlling the densification of spinel powder during the SPS experiments has been identified. It is proposed that grain-boundary sliding, accommodated by an in-series {interface-reaction/lattice diffusion of the O 2 -anions} mechanism controlled by the interface reaction step, governs densification. This hypothesis is in good agreement with microstructural observations, performed on the SPS samples, using TEM. For each soak temperature, a zero-densification-rate period is observed. In our case, the lack of dislocation activity in the elemental grains after SPS implies that the zero-densification-rate period is related to the stop of the interface reaction that controls the annealing of vacancies. More investigations are now required to identify precisely the mechanisms responsible for such a zero-densification-rate period.

FIG. 3 .FIG. 4 .FIG. 6 .

 346 FIG. 3. Densification curves obtained for different SPS temperatures as a function of soak time. The applied macroscopic compaction pressure is fixed to 25 MPa.

FIG. 8 .

 8 FIG. 7. Typical aspect of the microstructure of the sample obtained by SPS at 1300°C during 15 min. The heating rate is fixed to 100 °C/ min and the applied macroscopic compaction pressure to 25 MPa. (a) No residual intergranular pores are detected but few entrapped intragranular pores are observed (arrow), (b) some grains contain an intragranular dislocation activity.

  n value. Knowing the n value, the slope of the straight line obtained when plotting

FIG. 11 .

 11 FIG. 11. Initial effective stress exponent calculated with relation (11) for the different soak temperatures. The heating rate is fixed to 100°C/min and the applied macroscopic compaction pressure to 25 MPa. The stress exponent values are the slopes of the different straight lines.

Figure 4

 4 Figure 4 presents the formalism used to calculate Qd with a stress exponent of 2. A fixed value of 1/D dD/dt (8x10 -4 /s) has been chosen to fall within the regime where n has a value of 2, for all the soak temperatures investigated. The corresponding relative density values are then accessible (dashed lines, Fig. 4), which allow us to calculate the corresponding values of seff and µeff. Finally the variation of


Sylvain Deville
email: sylvain.deville@saint-gobain.com

Eric Maire

Guillaume Bernard-Granger

Audrey Lasalle

Agnès Bogner

Catherine Gauthier

Jérôme Leloup

Christian Guizard

Metastable and unstable cellular solidification of colloidal suspensions

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.

Investigations of a solid/liquid interface stability have direct implications and practical applications in fields as diverse as metallurgy and materials science [START_REF] Zhang | Aligned two-and three-dimensional structures by directional freezing of polymers and nanoparticles[END_REF][START_REF] Binks | Phase inversion of particle-stabilized materials from foams to dry water[END_REF] , crystal growth [START_REF] Granasy | Growth of 'dizzy dendrites' in a random field of foreign particles[END_REF] , cryopreservation [START_REF] Karlsson | Cryopreservation: Freezing and vitrification[END_REF] , microfluidics [START_REF] Abécassis | Boosting migration of large particles by solute contrasts[END_REF] , chromatography [START_REF] Guiochon | Monolithic columns in high-performance liquid chromatography[END_REF] , food engineering [START_REF] Mezzenga | Understanding foods as soft materials[END_REF] , earth science 8 or biology 9 . In materials science, the cellular solidification of colloidal suspensions is becoming an attractive processing route for porous ceramic [START_REF] Deville | Freezing as a path to build complex composites[END_REF][START_REF] Deville | Freeze-Casting of Porous Ceramics: A Review of Current Achievements and Issues[END_REF] , polymeric [START_REF] Zhang | Aligned two-and three-dimensional structures by directional freezing of polymers and nanoparticles[END_REF] and metallic materials [START_REF] Yook | Porous titanium (Ti) scaffolds by freezing TiH2/camphene slurries[END_REF] or composites [START_REF] Munch | Tough, Bio-Inspired Hybrid Materials[END_REF] , with applications from tissue engineering to gas sensors [START_REF] Colard | Conducting Nanocomposite Polymer Foams from Ice-Crystal-Templated Assembly of Mixtures of Colloids[END_REF] and catalysis [START_REF] Colombo | Materials science: In praise of pores[END_REF] and where materials integrity and homogeneity is a requirement. Understanding the critical parameters controlling the stability of solidification interfaces in colloidal systems is a necessary step, its absence currently preventing a rationale control of the materials processing conditions. Already published investigations [START_REF] Peppin | Solidification of colloidal suspensions[END_REF][START_REF] Peppin | Experimental verification of morphological instability in freezing aqueous colloidal suspensions[END_REF][START_REF] Peppin | Morphological instability in freezing colloidal suspensions[END_REF] of the solidification of colloidal suspensions are limited to the case of planar interface moving at low velocities (0.1-1 µm/s) by optical observations, where enough time is given to the system to reach equilibrium by Brownian diffusion. Typical velocities used when processing materials through solidification of colloidal suspensions are greater by at least one order of magnitude (10-100 µm/s). The system in these conditions is likely to be out of equilibrium.

What we understand so far of the solidification of colloidal suspensions is derived primarily from the analogies with dilute alloys systems, or the investigated behaviour of single particles in front of a moving interface and is still a subject of intense work. A more realistic, multi-particles model should account for the particles movement, the various possible interactions between the particles and the multiple interactions between the particles and the solid/liquid cellular interface. We can tackle the problem using the "colloids as big atoms" approach, applying Brownian diffusion mechanisms and Fick's law. Several specificities of colloidal systems must nevertheless be accounted for. Unlike alloys, colloidal suspensions exhibit various non-linear physical properties dependence with the concentration of particles, such as the freezing temperature curve, a variation of the segregation coefficient with the interface velocity and a dependence of particle diffusivity upon concentration [START_REF] Peppin | Experimental verification of morphological instability in freezing aqueous colloidal suspensions[END_REF] . The equilibrium situation with a planar interface at low velocity has recently been greatly clarified, both experimentally and theoretically [START_REF] Peppin | Solidification of colloidal suspensions[END_REF][START_REF] Peppin | Experimental verification of morphological instability in freezing aqueous colloidal suspensions[END_REF][START_REF] Peppin | Morphological instability in freezing colloidal suspensions[END_REF] . Attempts towards modelling the system out of equilibrium with a nonplanar interface are extremely complex. Direct experimental observations on which further theoretical analyses could be built and validated are still lacking, preventing further progress in this field.

In order to bring new experimental observations, we chose to investigate the stability of a cellular interface during directional solidification of colloidal suspensions by using X-ray radiography and tomography, under conditions typically used in processing routes. X-ray radiography and tomography experiments provide several advantages in this context [START_REF] Maire | On the application of X-ray microtomography in the field of materials science[END_REF] : transparent materials, like those used in reference [START_REF] Sekhar | Solidification microstructure evolution in the presence of inert particles[END_REF] , are no longer required since opaque materials can be imaged using the X-rays absorption contrast; a high spatial resolution (submicronic) can be obtained on synchrotron instruments, so that in some cases individual particles can be resolved, and a three-dimensional reconstruction of the particle's arrangement and the solvent crystals -ice crystals in this case-after complete solidification is possible.

In the cellular regime we investigate here, ice crystals are nucleating and growing into the suspension, eventually reaching a regime where they exhibit a cellular morphology. Particles are repelled by the moving cellular interface, and concentrate laterally between the growing crystals. The particle concentration in these intercrystals regions (referred to as the freezing zone in the paper) is increasing until the osmotic pressure exceeds the capillary pressure; the solid/liquid interface invades the interparticle space at a breakthrough concentration Cmax -typically 0.55-0.58-usually close to the maximum particle random packing [START_REF] Shanti | Particle Redistribution During Dendritic Solidification of Particle Suspensions[END_REF] . The solidified body (or frozen zone) can therefore be divided into two domains: particle-free regions, corresponding to the cellular ice crystals free of particles, and particle-rich regions, corresponding to the concentrated particles entrapped in ice. This templating effect by the ice crystals is the structuring mechanism used to control the morphology of materials processed by directional solidification [START_REF] Deville | Freeze-Casting of Porous Ceramics: A Review of Current Achievements and Issues[END_REF] , yielding materials with unprecedented properties, as we have shown in the past [START_REF] Deville | Freezing as a path to build complex composites[END_REF][START_REF] Munch | Tough, Bio-Inspired Hybrid Materials[END_REF] . We can differentiate these two types of domains using X-ray tomography images, the absorption of the beam in such regions being very different. The continuous cellular morphology along the solidification direction results in the observation of vertical absorption contrast bands in X-ray radiography, as seen in figure 1a. Since the absorption signal is averaged through the whole thickness of the sample in radiography, the pictures cannot provide a direct view of the crystal structure. The propagation of the interface can nevertheless be clearly observed, as well as the region close to the crystals tips, as shown later. We can use the absorption contrast obtained by tomography after a proper image treatment and analysis to quantify the relative particle-free and particle-rich phase fraction and to obtain three-dimensional representation of the crystals morphology in the frozen body.

Our measurements show kinetic and structural evidences of global instabilities of the displacement of the interface during solidification experiments at moderate velocities (figure 1, movie S1). The series of radiographs such as the one shown in figure 1a can be used to measure the evolution of the position of the interface with time. This, in turn, can be used to calculate the displacement velocity of the interface. The macroscopic contraction of the cold finger supporting the mold and the suspension, resulting from the temperature change during solidification, is negligible. These global instabilities are first revealed by sudden jumps of the interface (movie S1) corresponding to a local increase of the interface velocity (figure 1c), and resulting in bright bands parallel to the interface on the radiograph observations during solidification (figure 1a) associated with a local decrease of the particle-rich phase fraction (figure 1b).

The whole interface is affected at the same time by the global instability: bright bands are spanning the entire cross-section at a given location along the solidification direction in figure 1a. Limited edge effects, due to the difference of thermal conductivities between the suspension and the mold, induce a slight macroscopic curvature of the interface; for clarity, this is better illustrated in the magnified image of figure S1. We can describe the impact of the global instabilities on the crystals structure more precisely through the tomography reconstruction of the frozen body. Three-dimensional reconstruction after the tomography scans reveals that the ice crystals exhibit the same morphology before and after the instability with a continuity of the ice crystals along the solidification direction (figure 1d). The instability can be understood as inducing a growth of the ice crystals in the plane perpendicular to the solidification direction, which in turn induces a local decrease of the entrapped particles fraction, due to particles being rejected below and above the crystals growing transversally. Such transverse crystals can clearly be observed in figure 1d (black arrows). When we solidify the same suspension at higher velocities or using larger particles, instabilities are no longer observed, the obtained frozen structure is homogeneous and continuous along the solidification direction (figure 1e). These global instabilities are different from the local and already observed side branch instabilities leading to tip splitting or neighbouring primary dendrites (figure 1f). Localized instabilities, appearing only between the crystals, can also be observed at the same time, resulting in transverse crystals (white arrows in figure 1d), and inducing the creation of a large number of defects in the materials after sublimation of the ice crystals (figure S2). Such localized instabilities cannot be observed with the radiography observations, due to their local nature and averaging of the signal through the sample thickness. Repeating the solidification experiments with different interfacial velocities and particle sizes, we can assess the conditions under which the global instabilities appear. We can build a stability diagram with two domains: a metastability domain (green) where no instabilities are observed and therefore homogeneous crystals are obtained, corresponding to high interface velocity and/or large particles, and an instability domain (red) where global instabilities occur, for small particle size and/or low interface velocities (figure 2a). The reason why we refer to this situation as metastable and not as stable is discussed later in the paper.

We propose that these instabilities are due to partial diffusion of the particles ahead of the solidifying interface, leading to constitutional supercooling situations. The freezing point depression dependency on particles content has been reported previously [START_REF] Peppin | Experimental verification of morphological instability in freezing aqueous colloidal suspensions[END_REF] and is, unlike in the case of metallic alloys, non-linear. For convenience, these data have been replotted in figure S3. Although the plot shown here in figure S3 has been calculated and measured for bentonite particles, we can assume that the trend is similar for the alumina particles used in this study, the calculations being based on morphological parameters such as the particles hydrodynamic radius. The configuration of the system, with the definition of the various interfaces: the frozen zone/freezing zone interface and the freezing zone/suspension interface -although the latter one does not correspond to a physical interface-is shown in figure 3a. The corresponding experimental observation for metastable solidification is given in figure 3b. Under metastable conditions, where no instabilities occur, it is clear that the concentration of particles is restricted to the freezing zone. The diffusivity of the particles can be estimated using the Stokes-Einstein diffusivity of an isolated colloidal particle, corrected by the dimensionless diffusivity which accounts for the particle-particle interactions effects on the diffusivity [START_REF] Peppin | Morphological instability in freezing colloidal suspensions[END_REF] . The diffusivity can thus be expressed by:

𝐷 𝑅 = 𝐷 0 (𝑅)𝐷 (equation 1)
where the dimensionless diffusivity 𝐷 = 0.502 (see reference 18 for the details), and

𝐷 𝑅 = 6𝜋 𝜇 (equation 2)
is the Stokes-Einstein diffusivity of an isolated colloidal particle (R is the particle radius). The dimensionless diffusivity D ˆcan be calculated using the following parameters: f=0.28 (nominal fraction of particles in suspension), f p =0.64 (monodispersed spheres, random packing), dynamic viscosity of water µ=1.787.10 -3 Ns/m², T=273K. The actual dimensionless diffusivity is likely to be slightly lower, although the concentration dependence for particles of 0.1 µm and larger is low for a particle fraction below 0.55, see for example figure 3 of reference [START_REF] Peppin | Morphological instability in freezing colloidal suspensions[END_REF] .

We can construct the expected particle concentration profile in the liquid phase along the position of the interface when the interface velocity is high (figure 4a), which corresponds to the experimental observations of figure 3b. The particle concentration in the particle-rich region of the solidified body is the breakthrough concentration Cmax defined previously. The initial concentration Cini is given by the formulation of the suspension (it has been verified that no segregation of the particles occurs within the time frame of the experiments). If the lateral growth velocity of the crystals is independent of the distance to the crystal tip, the shape of the growing crystals should be triangular, as sketched in figure 3a. Then, assuming that no Brownian diffusion can take place at high interface velocity, the particle concentration in the freezing zone should vary linearly from Cini up to a concentration where the interparticle interactions make further concentration more difficult; the particle concentration then progressively reaches Cmax, deviating from the linear dependency. Using the freezing point depression of fig S3, we can construct the actual freezing point profile for this situation, along with the temperature profile. The discontinuity at the fr/fz interface is due to the latent heat (fig. 4b). The relative position of the actual temperature profile and the freezing point cannot be a priori determined. We make the hypothesis here that the temperature gradient is large enough that the actual temperature is lower than the freezing temperature.

From the construction of figure 4b, the freezing zone appears to be in a constitutionally supercooled state, with the actual temperature below the solidification temperature. The zone ahead of the crystals tip is in a stable state. For high interface velocity, nucleation and growth cannot occur, so that the solidification occurs regularly with no instabilities, yielding homogeneous materials. This situation is therefore referred to as metastable, and corresponds to the metastable state zone defined in figure 2a.

The unstable situation occurs when enough time is given to the system for diffusion to occur, redistributing particles away from the concentrated freezing zone. The modified concentration and freezing point profile are given in figure 4c-d: the supercooled zone now extends beyond the crystals tips. We can verify qualitatively the concentration profile in this configuration experimentally, by plotting the grey level, directly dependent on the particle concentration, across the interface and within the freezing zone (figure 5). Note that the location of the line profile was selected to be located away from a pure ice crystal so that it is directly linked to the concentration of particles in the liquid phase. The observed particle concentration profile is qualitatively in excellent agreement with that expected from the model proposed here, providing a very good validation of the theory:

the linear section of the particle concentration profile in between the crystals tips indicates that the concentration of the particles is induced by the lateral growth of the ice crystals exhibiting a triangular shape, -the deviation from the linear trend when the particle concentration is high indicates that the particles interactions become non negligible and make further concentration difficult when approaching the fr/fz limit, -the concentration profile with particle concentration greater than Cini, extending beyond the crystals tips, indicates that the diffusion is partly occurring, redistributing particles away from the concentrated zone between the crystals tips. One intriguing result is the dependency on particle size of the velocity leading to a transition between the stability domains (dashed area in figure 2a). We can expect the metastable to unstable transition to depend -at least-on two mechanisms: the development of a constitutionally supercooled zone ahead of the crystals tips, which provides the necessary conditions for the instabilities to appear, and the nucleation of the crystals in the supercooled zone. The first mechanism is directly related to the particle diffusivity (see equations 1 and 2), the dependency on particle size is therefore straightforward. The mechanisms controlling nucleation in such a complex system are more difficult to assess. Particles can probably act as preferential nucleation sites in the constitutionally supercooled zones, so that some influence of the particle size might also be expected: smaller particles mean more nucleation sites for an equivalent particle content. Assuming that the nucleation sites are related to the particle surface available at the crystal's tips for nucleation, a power-law behaviour can also be expected, although with an exponent 2 as the surface available for nucleation increases with the square of the particle size for a constant particle content. This is nevertheless still speculation only. An additional effect should be accounted for: the diffusion of the particles is somehow linked to the interface velocity, since the enrichment of particles in the freezing zone is due to the lateral growth of ice crystals. Faster growth kinetics will lead to faster enrichment of particles and therefore favour the diffusion away from the freezing zone. The particle size dependency of the transition between the stability regimes is still mysterious, but we can certainly expect the particle diffusivity to play a crucial role. Which mechanism is dominant and rate limiting is still to be assessed and deserves further exploration in the future.

We can now explain the development of the localized and global instabilities (figure 6). When the constitutionally supercooled zone extends beyond the tips of the crystals, new crystals are able to nucleate homogeneously in the supercooled zones. The strong anisotropy of growth kinetics of ice leads to a rapid growth of the favourably oriented crystals, leading to large crystals in the constitutionally supercooled zone, and similar to that observed for low interface velocity (see Fig. 1c of reference [START_REF] Peppin | Morphological instability in freezing colloidal suspensions[END_REF] ). In the meantime, the cellular crystals entering the constitutionally supercooled zone should exhibit a much faster growth in the region zfz/diff-zdiff/s until the equilibrium is reached again (the actual temperature matches the solidification temperature, zfz/diff = zdiff/s) when the particle concentration at the crystals tips corresponds to the nominal particle concentration in the suspension. This explains both the continuity of the cellular crystals in the solidified body (figure 1d) and the acceleration of the interface observed experimentally (figure 1c). Since the acceleration of the interface velocity corresponds to an unstable situation, the actual growth velocity is probably much greater than experimentally measured; the measure is currently limited by the radiography acquisition frequency used in these experiments (3 Hz). Furthermore, this mechanism accounts for the experimental observation that localized instabilities occur more often than global instabilities, comparing for instance figure 1a andS2, the periodicity of the global and localized instabilities being respectively a few hundreds of micrometers and a few micrometers. The nucleation and growth time available is more important in the freezing zone than ahead of the crystals tip. This should imply that a limited zone with only localized instabilities should exist in the stability diagram shown in figure 2a; this zone is represented by the dashed area between the metastable and unstable regions and its existence should be verified experimentally.

We can now plot a general stability diagram for the solidification of colloidal suspensions as a function of particle size and interface velocity (figure 2b), putting these results in a larger context. Instabilities of both type, affecting the integrity and homogeneity of the materials, are a major drawback of these processing routes, which must certainly be avoided. The proposed stability diagram is expected to be of great practical importance for the people processing materials through directional solidification. Using powders in the nanometre range, which can be required for materials with high surface area such as for catalysis applications, obtaining homogeneous and defectfree structures will be extremely difficult. The stability of the system being related to the diffusivity of the particles, workaround solutions might nevertheless be found: a modification of the suspension characteristics such as the viscosity can be used to lower the particles diffusivity. Since the transition between the stability domains is likely to be -at least partly, as discussed previously-determined by the particle diffusivity, this implies that a different stability diagram will be obtained when the formulation of the colloidal suspension is modified. Additives used to modify the crystals growth morphology, for example [START_REF] Munch | Architectural control of freeze-cast ceramics through additives and templating[END_REF] , will certainly affect the stability domain boundaries. To be of practical importance, the stability diagram will have to be assessed for each formulation.

The recent identification of a stable to unstable morphology transition [START_REF] Peppin | Experimental verification of morphological instability in freezing aqueous colloidal suspensions[END_REF] of the planar solidification interface in colloidal suspensions provided an accessible steady state to investigate the growth of morphological instabilities, which have numerous implications beyond materials processing. With the identified metastability region and corresponding metastable to unstable transition revealed here, it is not only possible to control the structure and the integrity of the resulting materials, but also to investigate the growth of morphological instabilities originating from a metastable state in a cellular regime, which could have implications not only in materials synthesis but also in various areas such as the design of microstructures in cast metal-matrix composites, organic electronics, microfluidics, geology, food engineering, biomaterials or cryobiology.

Methods

Slurries were prepared by mixing distilled water with a small amount (1 wt% of the powder) of ammonium polymethacrylate anionic dispersant (Darvan C, R. T. Vanderbilt Co., Norwalk, CT) and the alumina powder (see figure S4 and table S1). Slurries were ball-milled for 20 hrs with alumina balls and de-aired by stirring in a vacuum desiccator, until complete removal of air bubbles (typically 30 min).

Freezing experiments were performed by pouring the suspensions into a polypropylene mold (invisible to X-rays) placed in a cooling cell cooled using liquid nitrogen. The freezing kinetics were controlled by a heater placed at the bottom of the copper rod and a thermocouple also placed at the bottom. The experimental setup allows only linear ramp of temperature during cooling. Various cooling rates were used in the experiments, to investigate a wide range of interface velocities.

The experimental part of this study focuses on the use of a three-dimensional non-destructive technique, i.e. the high-resolution X-ray absorption radiography and tomography, to directly image nucleation and growth of the crystals in the suspension and the corresponding particles redistribution. Radiography is used to provide a dynamic visualisation of the solidification of the suspension. Tomography combines information from a large number of X-ray radiographs taken with different viewing angles of the sample, which should not move during the scan. The technique includes a computed step, i.e. a recalculation step during which a 3D map of the local absorption coefficients in each elementary volume of the sample is retrieved from the set of absorption radiographs. In this study, the HST program available at the ESRF was used [START_REF]HST Program[END_REF] . The reconstructed map gives an indirect image of the microstructure. In the present study, the samples were scanned using a high-resolution X-ray tomograph located at the ESRF (beam line ID 19) in Grenoble (France). X-ray tomography was performed at a voxel size of (1.4 μm) [START_REF] Granasy | Growth of 'dizzy dendrites' in a random field of foreign particles[END_REF] . The energy was set to 20.5 keV. The distance between the sample and the detector was 20 mm. Because of the extremely high coherence of the X-ray beam on this beam line, absorption is not the only source of contrast in the obtained radiographs and phase contrast is also present, but in a small amount. A set of 1200 projections was taken within 180°. The detector was a CCD camera with 2048 × 2048 sensitive elements coupled with an X-ray-sensitive laser screen. Dynamics of solidification was followed by X-ray radiography, with an acquisition frequency of 3 Hz, which allows to precisely follow the interface evolution in two dimensions. The frozen structures after complete solidification were characterized in three dimensions afterwards using a low speed high-resolution tomography acquisition, and the results were then combined with the data obtained by radiography. To improve the acquisition frequency, the radiographs were acquired in a binning configuration (i.e. with a 1024x1024 pixels resolution, half that of tomography acquisitions).

Frozen samples were freeze-dried (Freeze dryer 8, Labconco, Kansas City, MI) for 24 hours. The green bodies thus produced were sintered in air for 2 hours at 1500°C, with heating and cooling rates of 5°C/min (1216BL, CM Furnaces Inc., Bloomfield, NJ). The microstructure of the samples was analyzed by scanning electron microscopy (ESEM, S-4300SE/N, Hitachi, Pleasanton, CA). The dashed data points correspond to experiments where no instabilities were observed, although the measurement time span might be too short to be significant considering the chaotic frequency of instabilities. The dashed area is an estimation of the domain in which the metastable to unstable regime transition can occur; further experiments are necessary to situate more precisely the transition. b, General stability and structure diagram for the solidification of colloidal suspensions. The dashed area indicates the conditions investigated in this paper. At low velocities, the solid/liquid interface is planar and all particles are rejected. The interface becomes unstable when the velocity increases, due to constitutional supercooling situations created by particles Brownian diffusion ahead of the solidification front [START_REF] Peppin | Solidification of colloidal suspensions[END_REF] . Above the critical velocities, all particles are entrapped [START_REF] Uhlman | Interaction between particles and a solid/liquid interface[END_REF] . The green areas indicate the metastable conditions suitable for processing homogeneous and defects-free cellular materials.

Figure 1 :

 1 Figure 1: Experimental observations of the development of global interface instabilities and their impact on the crystals structure, 0.2 µm particles. a, radiography evidences of global instabilities, appearing as bright bands (less absorption of the beam). Interface displacement direction: bottom to top. Scale bar: 500 µm. b, particle-rich phase fraction in the frozen body. The particle-rich phase fraction is decreasing at locations corresponding to the development of the instabilities, c, instantaneous velocity of the interface. The sequence of radiographs acquired, as the interface is moving, can be used to track the interface position and provides a local velocity measurement. Spikes in the instantaneous velocity of the interface are observed, at locations corresponding to just before the bright bands and the decrease of particles fraction. The bright bands observed in a, corresponding to the instabilities of the interface velocities identified in c, also corresponds to zones of the sample with less absorption; the particles content is therefore lower. d-e 3D local observation by X-Ray tomography of the ice crystals, revealing lateral growth of ice crystals resulting from localized (white arrows) and global (black arrows) instabilities for 0.2 µm particles suspensions under unstable conditions (d), and their absence for 1.3 µm particles suspensions under metastable conditions (e). Blocks dimensions: 360x360x360 µm 3 . (f) neighbouring primary dendrites (arrows) resulting from side branch instabilities, front view (left) and side view (right), block dimensions: 30x70x140 µm 3 . The absence of a bright band in the radiograph 1a at point G is due to the dynamics of instabilities

Figure 2 :

 2 Figure 2: Experimental conditions for the development of global instabilities: ice crystals stability and structure diagrams. a, Metastability diagram as function of particle size and interface velocity, for a given particles fraction, based on various interface velocities and particle size conditions investigated in this paper.The dashed data points correspond to experiments where no instabilities were observed, although the measurement time span might be too short to be significant considering the chaotic frequency of instabilities. The dashed area is an estimation of the domain in which the metastable to unstable regime transition can occur; further experiments are necessary to situate more precisely the transition. b, General stability and structure diagram for the solidification of colloidal suspensions. The dashed area indicates the conditions investigated in this paper. At low velocities, the solid/liquid interface is planar and all particles are rejected. The interface becomes unstable when the velocity increases, due to constitutional supercooling situations created by particles Brownian diffusion ahead of the solidification front[START_REF] Peppin | Solidification of colloidal suspensions[END_REF] . Above the critical velocities, all particles are entrapped[START_REF] Uhlman | Interaction between particles and a solid/liquid interface[END_REF] . The green areas indicate the metastable conditions suitable for processing homogeneous and defects-free cellular materials.

Figure 3 :

 3 Figure 3: Particles and crystals arrangement and definition of the interfaces location close to the crystals tips. a, schematic representation. Acronyms for the position definition: fr: frozen zone, fz: freezing zone, diff: diffusion layer. b, high resolution X-ray radiography of the particle and crystals arrangement near crystals tips in metastable solidification conditions (interface displacement direction: left to right). The morphology of the crystals tip, exhibiting triangular shapes, is in good agreement with the model shown in figure3a, supporting the hypothesis of a constant lateral growth kinetic (perpendicular to the solidification direction). The concentration of particles is restricted to the freezing zone, no diffuse layer ahead of the crystals tips can be observed. Scale bar: 50 µm.

Figure 4 :

 4 Figure 4: Proposed mechanism for the development of instabilities: particle concentration, temperature and freezing point profile, for metastable and unstable situations. a, Particle concentration profiles for high velocity (metastable situation). b, corresponding temperature and freezing point profiles. The dashed area indicates the zone of constitutional supercooling; global instabilities cannot appear ahead of the ice crystals tips. The development of localized instabilities in the freezing zone is limited by the available time for nucleation and growth of crystals in the constitutionally supercooled zone. c, particle concentration profiles for moderate velocity (unstable situation), where diffusion can occur. An increase of particle concentration DC is observed at the crystals tip position d, corresponding temperature and freezing point profile. The dashed area indicates the zone of constitutional supercooling, which now extends beyond the dendrites tips, allowing for global instabilities.

Figure 5 :

 5 Figure 5: Experimental observations of particle concentration profile under unstable conditions (0.4 µm particles, interface displacement direction: left to right): a, Radiograph of the sample, b, detail from the panel a, with the background removed and grey level scale inverted. c, corresponding concentration profile (grey level) across the interface (white line in panel b). The crystals tip position defining the freezing zone/diffusion layer interface can be estimated, although the radiograph is the average signal from the whole sample thickness. The concentration profile is in good agreement with that expected from the proposed mechanism (figure 4c). Scale bars: a 500 µm, b 100 µm. The periodic ghost pattern (vertical stripes) visible in panels a and b is an acquisition artefact of the detector formed when subtracting the background, and is inducing the grey level periodic fluctuations observed in the freezing zone. The dashed line corresponds to the experimental data corrected to account for this ghost pattern.

Figure 6 :

 6 Figure 6: Schematic sequence of the development of the instabilities, with nucleation and growth of crystals in the constitutionally supercooled zones A (freezing zone) and B (diffusion layer ahead of the crystals tip). Favourably oriented nuclei (blue) grow faster than the unfavourably oriented ones (red). The rapid growth kinetics of the crystals in the constitutionally supercooled zone leads to crystals spanning transversely the entire freezing zone, as observed in figure 1d (black arrows).

Acknowledgements

We acknowledge the European Synchrotron Radiation Facility for provision of synchrotron radiation beam time and we would like to thank Elodie Boller and Jean-Paul Valade for their irreplaceable assistance in using beamline ID19. Financial support was provided by the National Research Agency (ANR), project NACRE in the non-thematic BLANC programme, reference BLAN07-2_192446.

Authors contributions

Additional information

Supplementary information accompanies this paper on www.nature.com/naturematerials.


Fabien Robaux
email: fabien.robaux@irphe.univ-mrs.fr

Michel Benoit
email: michel.benoit@irphe.univ-mrs.fr

Modeling nonlinear wave-body interaction with the Harmonic Polynomial Cell method combined with the Immersed Boundary Method on a fixed grid

Introduction

To model the propagation of large water waves and associated loads applied to offshore structures, scientists and engineers have a need of fast and accurate models. A wide range of models have been developped in order to predict wave-fields and hydrodynamic loads at small scale, from the linear potential boundary element method to complete CFD codes, based on the Navier-Stokes equations. Although the latters are well adapted to solve the wave-structure interaction at small scale, their use is limited due to the computational cost of such models and numerical diffusion. Alternative approaches, capturing the nonlinear effects, are thus needed. Shao and Faltinsen [START_REF] Shao | A harmonic polynomial cell (HPC) method for 3D Laplace equation with application in marine hydrodynamics[END_REF] proposed an innovative technique, called "harmonic polynomial cell" (HPC) method to tackle this problem. This approach is implemented and tested in 2 dimensions (x, z), first on a standing wave problem and then to evaluate the nonlinear forces acting on a fixed submerged cylinder.

Overview of the harmonic polynomial cell method (HPC)

This method is based on the potential hypothesis, which assumes the irrotationality of the flow. Viscous effects are also neglected. The complete description of the velocity field can thus be reduced to the knowledge of the potential scalar field v = ∇(φ), where φ satisfies the Laplace equation:

∇ 2 φ = 0, -h(x) ≤ z ≤ η(x, t) (1)
To solve the potential problem, we use the HPC method introduced by Shao and Faltinsen [START_REF] Shao | A harmonic polynomial cell (HPC) method for 3D Laplace equation with application in marine hydrodynamics[END_REF]. The fluid volume is discretized into overlapping cells. In each cell, composed of 9 points, the potential is approximated as a weighted sum of the first harmonic polynomials, each of them being solution of (Eq. 1). The method can be extended to 3D cases considering cubic-like cells with 27 nodes.

In a given cell the potential is thus locally approximated as a linear combination of the 8 first harmonic polynomials (f j ), which are explicitely known (1, x, z, xz, x 2 -z 2 ...).

φ(x = (x, z)) = 8 j=1 b j f j (x) (2)
Thus, applying this equation to each node of the cell, except the center, gives This local matrix (size 8x8 with C ij = f j (x i )) can be inverted, such that the b j coefficients are obtained for the given local cell as b j = C -1 ji φ i . Then injecting this result into the interpolation equation (Eq. 2), a relation is obtained providing an approximation for the potential inside the cell using the potentials of the eight surrounding nodes:

φ i = φ(x i ) = b j f j (x i ) for i = 1..8.
φ(x) = 8 i=1 8 j=1 C -1 ji f j (x) φ i (3)
Still, the potential at the center of the cell (node 9) has not been used to obtain (Eq. 3). Thus, applying (Eq. 3) at this node where x = (0, 0), a relation is found between the nine potentials of each cell. For each cell in the fluid domain, this equation is set in a global matrix. Thus this matrix contains at most 9 non-zero values in each row. For the boundary conditions, either a Dirichlet condition can be imposed or a Neumann condition is found by deriving the local expression (Eq. 3). Following Zakharov [START_REF] Zakharov | Stability of periodic waves of finite amplitude on the surface of a deep fluid[END_REF], the kinematic and dynamic free surface nonlinear boundary conditions are formulated as:

η t = -∇η • ∇ φ + w(1 + ∇η • ∇η) (4) φt = -gη - 1 2 ∇ φ • ∇ φ + 1 2 w2 (1 + ∇η • ∇η), (5)
where φ(x, t) = φ(x, η(x, t), t), and w(x, t) = ∂φ ∂z z=η is the vertical velocity at the free surface. The gradient of η is computed by finite difference of arbitrary order. Others spatial derivatives can be computed locally by deriving the local expression for φ (Eq. 3). Moreover the Runge-Kutta method at order 4 with constant time-step is used to integrate (Eq. 4-5) in time.

Immersed boundary method (IBM) on a fixed grid

In a first step of this study, the HPC method was implemented using deforming boundary fitted grids, but a lack of stability was denoted. The results were found to be highly dependent on the mesh deformation method used, due to the difficulty to invert the local geometry matrix at some time-steps and for particular lay-outs of the cell. Recently, Ma et al. [START_REF] Ma | Local and global properties of the harmonic polynomial cell method: In-depth analysis in two dimensions[END_REF] pointed out that the method HPC is much more efficient when using fixed perfectly-squared cell. In order to work with regular fixed grids, an IBM strategy is implemented, following Hanssen et al. [START_REF] Hanssen | Free-surface tracking in 2D with the harmonic polynomial cell method: Two alternative strategies[END_REF]. In this method, the free surface is discretized with markers, evenly spaced and positionned at each vertical intersection with the background fixed grid (See schematic representation Fig. 1). In order to close the system, each point above the free surface -denoted ghost-points -must have a dedicated equation in the global matrix. Those equations are the local expressions (Eq. 3) applied at the marker position in a given cell (given C ij ). A choice must be made when two ghost-points appear to be linked to the same maker in the same cell. The second closest cell whose center is inside the fluid domain is thus chosen to prevent singularity, see arrow in Fig. 1.

Ghost Points Centers of cells used for ghost points

Markers on the free surface Interpolation points from other mesh

Inactive points

Neumann BC points To allow the background mesh to be composed of square cells, and still have the possibility to include fixed or moving objects, a boundary fitted overlapping grid is also added following Ma et al. [START_REF] Ma | Local and global properties of the harmonic polynomial cell method: In-depth analysis in two dimensions[END_REF]. In this method an immersed boundary-fitted mesh is constructed. Local expressions are also solved on this complementary mesh and added to the global matrix to solve both problems (on the background and immersed mesh) simultaneously. Communication is set and solved in the global matrix through interpolations using (Eq. 3) at the extreme points of both the background mesh and the immersed mesh, ensuring the continuity of the solution and the boundary condition on the body.

Results on two selected test cases

Standing wave freely evolving in a closed square domain This first test case is compared to the results from the method of Tsai and Jeng [START_REF] Tsai | Numerical Fourier solutions of standing waves in finite water depth[END_REF]. It consists in a standing wave of wavelength λ = 64m, steepness H/λ = 0.1 in a square domain with water depth h = λ. The initial conditions for η are set according to the results of [START_REF] Tsai | Numerical Fourier solutions of standing waves in finite water depth[END_REF] with a phase such that the velocity field is null. The wave is freely evolving for 9 periods, and the error on η is computed at each time step with an L 2 norm, and normalized with the wave height:

err = √ p (ηp-η th ) 2 H
. This normalized L 2 -error is plotted at four different times in Fig. 2 as a function of the number of nodes per wavelength (N x ) and the Courant Friedrich Levy (CFL) number, defined as the radio of the number of time-steps per period divided by N x . Results are promising, with errors down to 10 -6 . Convergence properties with mesh and time-step refinement are also investigated and both are found to be with an order close to 4 as expected (Fig. 3).

Fixed horizontal immersed cylinder

The second test case is based on the work of Chaplin [START_REF] Chaplin | Nonlinear forces on a horizontal cylinder beneath waves[END_REF],which consists in a fixed horizontal cylinder, slightly immersed, in regular waves of period T = 1s. The depth of the cylinder center is d c = 0.101m, for a radius r = 1/2d c , and a total depth of d = 0.85m. This problem is numerically difficult to solve as the cylinder is close to the free surface, involving a small water gap to be meshed, of height ≈ λ/30. The incident waves are generated using the stream function theory. The mean value and the amplitudes of the first 3 harmonics of the vertical load on the body are compared to numerical results from Guerber [START_REF] Guerber | Modélisation numérique des interactions non-linéaires entre vagues et structures immergées, appliquée à la simulation de systèmes houlomoteurs[END_REF], computed with a non-linear boundary element method for different Keulegan-Carpenter numbers (K c ). As presented in Fig. 4, results are in quite good agreement with Guerber [START_REF] Guerber | Modélisation numérique des interactions non-linéaires entre vagues et structures immergées, appliquée à la simulation de systèmes houlomoteurs[END_REF], thought some differences are observed. For smaller wave heights (ie, smaller K c ), the results found with a more classical HPC boundary fitted mesh approach were in good agreement with the litterature due to the small deformation of the mesh. Results from the two approaches will be compared and discussed during the workshop.

Figure 1 :

 1 Figure 1: Schematic representation of the immersed free surface and immersed body

Figure 2 :

 2 Figure 2: Normalized L 2 error on η -Standing wave

Figure 3 :

 3 Figure 3: Convergence in time and space

Figure 4 :

 4 Figure 4: Different harmonics of the vertical load on the cylinder. Current results (dots) compared to numerical simulations from [2] (lines with crosses).


M Czapski

T Stora

C Tardivat

Sylvain Deville

R Santos

J Leloup

F Bouville

R Fernandes

R Santos Augusto

Porous silicon carbide and aluminum oxide with unidirectional open porosity as model target materials for radioisotope beam production

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

At ISOL (Isotope Separation On-Line) facilities, a variety of radioactive ion beams (RIB) is produced by bombarding a thick solid or liquid target with highly energetic particles to initiate nuclear reactions (i.e. fission, fragmentation, spallation) in the material. One of such facilities is ISOLDE located at CERN [START_REF] Lindroos | HIE-ISOLDE: the technical options[END_REF], which has a long tradition of using different target materials, which are subjected to pulsed beams of 1.4 GeV protons from the Proton Synchrotron Booster (PSB). A search for more intense isotope beams with shorter half lives requires the use of materials characterized by faster release i.e. having better diffusion and effusion properties.

Among the different ISOLDE targets, there are molten metals, powders and foils, refractory carbides and oxides. The latter materials are particularly of interest due to their stability at high temperature (at which targets operate) in comparison to pure metals [START_REF] Koster | ISOLDE target and ion source chemistry[END_REF]. Two compounds here proposed can serve as model target materials among these groups i.e. silicon carbide (SiC) and aluminum oxide (Al2O3). Recent studies [START_REF] Fernandes | Submicro-and nano-structured porous materials for production of high-intensity exotic radioactive ion beams[END_REF] showed that further modifications of their microstructure by decreasing the grain size from micron region to tens of nanometers and by controlling the porosity fraction, can alter significantly the release properties of these materials. Following these results, nowadays slip-casted SiC of 63% porosity are a part of ISOLDE regular target materials. On the other hand such changes can also influence the material stability, which can lead to a significant drop in target's lifetime. Therefore further investigations of evolution of their microstructure in irradiative environment should be done. In this study, a synthesis of different controlled sub-microstructures of the above selected model materials with the ice-templating method is presented. Furthermore, mechanical behavior is estimated using energy deposition spectra deduced from FLUKA codes to assess mechanical integrity under irradiation. This analysis will be used to verify further experimental results of irradiation with two proton beams of 1.4 GeV and 440 GeV.

Experimental

Sample preparation

The samples were prepared using the ice-templating technique [START_REF] Deville | Freeze-casting of porous ceramics: a review of current achievements and issues[END_REF][START_REF] Deville | Ice-templated porous alumina structures[END_REF][START_REF] Deville | Ice shaping properties, similar to that of antifreeze proteins, of a zirconium acetate complex[END_REF], which allows one to produce materials with open directional porosity. In our case, initial slurries were made from SiC (Hexoloy® and Crystar®) and Al2O3(Ceralox®) and were frozen on the cold support with two different ice-front velocities. Temperature gradient within the slurry (trapped inside a Teflon mold) froze the water inside in direction perpendicular to the support's surface (Fig. 1). After the freezing process, the samples are put into a freeze-drier where, under lowered pressure, water crystals sublimate from the green body. In this way, the introduced porosity is of an open type. Two different freezing rates produce samples with two different pore sizes, without changing the porosity fraction. The later property can be controlled by changing the amount of powder in the initial slurry.

We were also able to pore shape with selected additives (NaOH, zirconium acetate (ZRA) complex). Table 1 summarizes the parameters and the properties they modify. Fig. 2 shows SEM images of different pore formations with a summary of structure modification. The so-prepared green bodies are ready for sintering. Porosity content achieved by this technique reaches from 25-90% [START_REF] Deville | Freeze-casting of porous ceramics: a review of current achievements and issues[END_REF]. Our prototypes have porosity content from 40-70% with pore sizes ranging between 1.1-26 μm (Al2O3), 11-27 μm (SiC Hexoloy) and 62-86 μm (SiC Crystar).

Mechanical calculations

In order to estimate the mechanical integrity upon irradiation, which allows the study of the impact of an irradiative environment on aging of the preselected prototypes, a simple mechanical analysis of the system was made. The irradiations took place at ISOLDE and HiRadMat [START_REF] Efthymiopoulos | HiRadMat: a new irradiation facility for material testing at CERN[END_REF] using CERN pulsed beams at 1.4 GeV (PSB) and at 440 GeV (Super Proton Synchrotron, SPS), respectively. The total number of protons used in both cases is equal to 1016 protons, due to HiRadMat facility limitations. Other parameters such as gaussian σ beam radii (3.5 mm and 2.0 mm for ISOLDE and HiRadMat, respectively) and repetition rates (1.2 s and 40 s, respectively) are predetermined by the accelerator operation (PSB and SPS, respectively).

FLUKA codes [START_REF] Ferrari | [END_REF] were used to simulate irradiation of cylindrical samples of 1 cm in diameter and 4 cm long (which is an approximation of tightly stacked pellets of the real experiment) with ice-templated channels parallel to cylinder's long axis which is also the axis of the proton beam. Fig. 3 shows the energy deposition spectra in both materials simulating the conditions of ISOLDE and HiRadMat. For both irradiation types, energy is deposited radially in a similar manner with the maximum value located in the core of the sample. Assuming that the biggest thermal shock will occur in this region, the point of maximum deposited energy per volume per primary particles (ppp) (Emaxfluka) was used for further calculation. The energy deposited in the material per volume (Edep) can be calculated from the formula:

E dep =E maxfluka •N ppp (1)
where Nppp is number of protons pulse (3•10 13 and 8•10 12 for ISOLDE and HiRadMat respectively). Under this assumption and taking different repetition rates (t), we can evaluate the power deposited (P) in the material volume (V) as:

P=Edept•V. ( 2
)
The temperature increase per pulse (ΔT) can be then calculated from the formula:

∆𝑇 = * (3)
where Cp is the specific heat and ρ * is the reduced density calculated as:

ρ * =ρ•(1-porosity), ( 4
)
where ρ is the nominal density of the material. In this study we show calculations for the porosity fraction of 50% in comparison to unmodified material. By knowing the latter, thermal stresses σ can be evaluated from:

𝜎 = ∆𝑇 . 𝐸 * . 𝛼 (5)
where E * young is the reduced Young modulus and α is the thermal expansion coefficient. It is assumed that the cylinder is subjected to deformation along its long axis. In this scenario the walls undergo only axial extension and compression [START_REF] Gibson | Cellular Solids: Structure and Properties[END_REF]. The reduced Young modulus can be calculated as:

𝐸 = * 𝐸 * (6)
where Eyoung refers to the Young modulus of the nominal density material. Generated stresses should be lower than the compression strength of the material, therefore the material should remain intact. If we assume that the properties of ice-templated ceramics are similar to ceramic foams we can calculate the (reduced) strength in compression (σ * s ) which is related to the cell wall strength (σ s ) [START_REF] Lu | The thermal shock resistance of solids[END_REF]:

𝜎 * = 0.2 𝜎 * 3 2 (7)
Table 2 summarizes all the thermo-mechanical properties of materials used in this study. The thermo-mechanical analysis results are presented in Table 3. The data reported in the last two columns show that in average the thermal shock is much below the reduced strength of the ice-templated structure, which suggests that its mechanical integrity should be guaranteed.

Conclusions

It was shown that the ice-templating technique allows the design of tailor-made microstructures of pre-defined open porosity in a relatively easy and fast way. Three parameters of sample preparation can be easily tuned to independently modify different structure properties, which opens a door for testing of a large array of target material prototypes. For the first time, ice-templated microstructures are candidates for RIB production. Those prototypes have been irradiated with 10 16 protons of 1.4 GeV and 440 GeV to verify if they can sustain mechanical integrity and serve as future model spallation targets. So far mechanical calculations proved that the temperature increase inside tested materials should generate thermal shock much below material's strength, therefore their microstructure should be preserved. The irradiation results will be published in a forthcoming paper.

Fig. 1 .

 1 Fig. 1. Schematic of the experimental setup used to directionally freeze the slurries.

Fig. 2 .

 2 Fig. 2. SEM images of ice-templates bodies of SiC (Hexoloy (A-C) and Crystar (D)) subjected to different parameter modifications listed in Table 1: (A) fast-freezing; (B) slow-freezing; (C) addition of ZRA-honeycomb-like structure; (D) over-loaded slurry.

Fig. 3 .

 3 Fig. 3. Energy depostion spectra generated by FLUKA codes for (A) HiRadMat beam and (B) ISOLDE beam. In this simulation the proton beam is passing from the left to the right through the cylindrical sample partially filled with Al2O3 and SiC. The maximum of deposited energy is 2.65•10 -2 GeV/cm 3 /ppp (A) and 5.3•10 -3 GeV/cm 3 /ppp (B).

Table 1 .

 1 List of parameters which directly control the final pore formation in a slurry.

	Experimental parameter	Green compact property
	Powder fraction in slurry	Porosity fraction
	Freezing speed	Pore size
	Additives	Pore structure

Table 2 .

 2 Mechanical properties of SiC and Al2O3.

	Material	ρ[g/cm3]	Eyoung [GPa] Cp [J/kg.K] α[1/K] • 10 -6	σ s [GPa]
	SiC Crystar 2.7		240	750	4.8		0.85
	SiC Hexoloy 3.0		350	750	4.0		3.90
	Al2O3	3.8		370	880	8.2		2.60
	Table 3. Thermo-mechanical analysis of the systems irradiated with HiRadMat and
	ISOLDE beam.
	Beam Material	Edep [J/cm3] Edep•V[J]	P [W]	ΔT [K] σ [MPa] σ *	s [MPa]
	HiRadMat
	SiC Hexoloy 34	107		2.7	30	21	276
	SiC Crystar 34	107		2.7	33	19	60
	Al2O3		34	107		2.7	20	31	184
	ISOLDE
	SiC Hexoloy 25	80		67	23	16	276
	SiC Crystar 25	80		67	25	15	60
	Al2O3		25	80		67	15	23	184

Acknowledgement

This research project has been supported by a Marie Curie Early Initial Training Network Fellowship of the European Community's FP7 Program under contract number (PITN-GA-2010-264330-CATHI).