B De Canditiis

G Duchêne

Simulations using the PSCS technique on an S-type AGATA segmented HPGe gamma-ray detector

Keywords: HPGe detectors, Full volume characterization, Pulse Shape Analysis (PSA), AGATA, Scanning

Monte Carlo simulations are used to test the Pulse Shape Comparison Scanning (PSCS) technique implemented at the IPHC scanning table . The tests, performed on a high purity germanium (HPGe) detector unit of the AGATA array, aim to quantify the accuracy of the system and validate it. The simulations use a combination of tools such as Geant4, SIMION and the Agata Detector Library (ADL). Quality parameters are extracted at various gamma-ray energies and the impact of the input statistics on the parameters is also analyzed.

Introduction

The Advanced GAmma Tracking Array (AGATA) [START_REF] Akkoyun | AGATA -Advanced GAmma Tracking Array[END_REF] is an array of high purity germanium (HPGe) segmented detectors designed and built in the framework of an international collaboration which involves 11 European countries. With its 180 detectors, 41 of which are already in operation in GANIL, the array will cover a solid angle of ∼ 80% of 4π. Since 2010, the incomplete array is used in experiments to study the structure of atomic nuclei as a function of angular momentum, isospin, and temperature.

The 36-fold segmentation of the AGATA detector units makes them position sensitive, as for a given interaction position an uniquely-shaped pulse is generated by the detector. This characteristic leads to the main feature of AGATA, namely its ability to reconstruct the history of the interactions of a gamma ray propagating in the array, which is possible thanks to dedicated tracking algorithms [START_REF] Venturelli | Adaptive grid search as pulse shape analysis algorithm for gammatracking and results[END_REF][START_REF] Lopez-Martens | Gamma-ray tracking algorithms: a comparison[END_REF][START_REF] Suliman | Fuzzy clustering algorithm for gamma ray tracking in segmented detectors[END_REF][START_REF] Didierjean | The Deterministic Annealing Filter: A new clustering method for gammaray tracking algorithms[END_REF][START_REF] Tashenov | Tango -new tracking algorithm for gamma-rays[END_REF]. This makes obsolete the use of Compton shields, as in the previous generation HPGe detector arrays such as EUROBALL [START_REF] Beck | EUROBALL: Large gamma ray spectrometers through european collaborations[END_REF][START_REF] Simpson | The EUROBALL spectrometer[END_REF] and GAM-MASPHERE [START_REF] Beausang | Measurements on prototype Ge and BGO detectors for the EUROGAM array[END_REF], increasing the geometrical efficiency of the tracking array. Moreover, the reconstruction of the gamma-ray trajectories leads to a substantial improvement of the Doppler correction and therefore of the energy resolution. It also allows the measurement of the linear polarization of gamma rays, which is useful for the determination of the parity of the nuclear states of interest.

In order to reconstruct an event, the tracking algorithm needs as input the positions of each interaction of the gamma ray. These are determined with pulse shape analysis (PSA) algorithms [START_REF] Bruyneel | Pulse shape analysis and position determination in segmented HPGe detectors: The AGATA detector library[END_REF][START_REF] Mihailescu | Principles and Methods for gamma ray tracking with large volume Germanium Detectors[END_REF][START_REF] Mihailescu | The influence of anisotropic electron drift velocity on the signal shapes of closedend HPGe detectors[END_REF][START_REF] Medina | A simple method for the characterization of HPGe detectors[END_REF][START_REF] Schlarb | Pulse shape analysis for gamma-ray tracking (Part I): Pulse shape simulation with JASS[END_REF][START_REF] Mateu | Simulation of the charge collection and signal response of a HPGe double sided strip detector using MGS[END_REF]. For a given interaction, the algorithm compares the experimental pulse shape with a database of calculated pulse shapes associated with spatial coordinates within the volume of the detector. After PSA and gamma-ray tracking, the spatial resolution obtained is ∼ 5mm FWHM for 1 M eV interactions [START_REF] Bruyneel | Pulse shape analysis and position determination in segmented HPGe detectors: The AGATA detector library[END_REF][START_REF] Recchia | Position resolution of the prototype AGATA triple-cluster detector from an inbeam experiment[END_REF][START_REF] Soederstroem | Interaction position resolution simulations and in-beam measurements of the AGATA HPGe detectors[END_REF].

Presently, databases of pulse shapes are calculated with the AGATA Detector Library (ADL) software [START_REF] Bruyneel | Pulse shape analysis and position determination in segmented HPGe detectors: The AGATA detector library[END_REF] leading to good tracking performances, although some characteristics of the detector such as the real impurity distribution and the thickness of the dead layers are not always correctly known and their implementation is subject to uncertainties. A novel approach to database construction is the implementation of full volume characterization of the detectors via three-dimensional scans. These measurements are realized with dedicated tools called scanning tables [START_REF] Dimmock | Validation of Pulse Shape Simulations for an AGATA prototype detector[END_REF][START_REF] Ha | New setup for the characterisation of the AGATA detectors[END_REF][START_REF] Goel | Spatial calibration via imaging techniques of a novel scanning system for the pulse shape characterisation of position sensitive HPGe detectors[END_REF][START_REF] Hernandez-Prieto | Study of accuracy in the position determination with SALSA, a gammascanning system for the characterization of segmented HPGe detectors[END_REF].

One of such scanning tables is based at the IPHC Strasbourg [START_REF] Sigward | Pulse-Shape Comparison Scan of germanium detectors using the IPHC scanning table[END_REF] and is realized within the framework of the AGATA collaboration. This scanning table is designed to perform three-dimensional scans with a good spatial resolution, a large amount of scanned points and short running times. The table implements the Pulse Shape Comparison Scan (PSCS) technique [START_REF] Crespi | A novel technique for the characterization of a HPGe detector response based on pulse shape comparison[END_REF] that allows the reconstruction of a database by comparing two datasets of pulses. The two datasets are obtained by performing, with a collimated 137 Cs source, two twodimensional scans, in which the detector is oriented in two different directions perpendicular to each other (eg. vertical and horizontal orientations).

The aim of the work presented here is to quantify the characteristics of the PSCS technique implemented by the Strasbourg scanning table [START_REF] Canditiis | 3D characterization of multisegmented HPGe detectors[END_REF]. The whole scanning procedure is recreated with Monte Carlo simulations of an S-type AGATA detector unit and the data obtained are analyzed with the same algorithms used for the real measurements.

In the first part of this paper, the AGATA detector unit is briefly described along with the signal generation. Subsequently the PSCS technique and its implementation by the IPHC scanning table is presented. The tools of the simulation and the data treatment are then shown. Finally the results of the simulation are described and interpreted. Conclusions and perspectives are given in the last section.

HPGe segmented AGATA detector unit

The AGATA detector units (figure 1) [START_REF] Akkoyun | AGATA -Advanced GAmma Tracking Array[END_REF] are segmented, closed-end, coaxial, n-type HPGe crystals that have a tapered hexagonal geometry with an asymmetric shape. Three slightly different tapered shapes exist labeled as A, B and C. Moreover a fourth totally symmetrically shaped unit, the S-type (in figure 2), exists. Only thee detectors with such shape were built as prototypes, only used in tests, and they are not actually part of the array [START_REF] Simpson | AGATA[END_REF]. The length of each crystal is 90 mm and the coaxial diameter is 80 mm. The tapering angle of the S-type unit is 10 • . The central hole, drilled from the back, has a diameter of 10 mm and extends up to 13 mm from the front face of the detector. The 6-fold sectorwise segmentation goes through the middle of each flat hexagonal side. The 6-fold longitudinal segmentation forms segment rings of 8, 13, 15, 18, 18 and 18 mm in thickness starting from the front face.

Thus 37 signals are collected from each crystal: 36 segment signals plus 1 total energy signal collected by the central contact (core). Each crystal has an impurity concentration between 0.4 and 1.8 × 10 10 cm -3 . In particular, the S-type detector, which is used as model for the simulations in this work (S001) has an impurity concentration of 1.5 × 10 10 cm -3 at the front and 0.43 × 10 10 cm -3 at the back, as indicated by the manufacturer. The crystals are encapsulated in hermetically sealed aluminum canisters [START_REF] Eberth | Encapsulated Ge detectors: Development and first tests[END_REF] with a 0.8 mm wall thickness. The spacing between the crystal and the canister wall varies from 0.4 mm to 0.7 mm.

Details about the embeded cold and warm preamplifier stages are given in [START_REF] Akkoyun | AGATA -Advanced GAmma Tracking Array[END_REF][START_REF] Pullia | An advanced preamplifier for highly segmented germanium detectors[END_REF][START_REF] Zocca | A time-over-threshold technique for wide dynamic range gamma-ray spectroscopy with the AGATA detector[END_REF][START_REF] Pascovici | Low noise, dual gain preamplifier with built in spectroscopic pulser for highly segmented high-purity germanium detectors[END_REF]. Finally the signals generated by the segments and the core for each crystal are sampled with 14 bit ADCs at a 100 M Hz rate. This is the case for the AGATA data acquisition [START_REF] Akkoyun | AGATA -Advanced GAmma Tracking Array[END_REF] as well as for the Strasbourg scanning table experimental setup [START_REF] Sigward | Pulse-Shape Comparison Scan of germanium detectors using the IPHC scanning table[END_REF][START_REF] Ginsz | Characterization of high-purity, multi-segmented germanium detectors[END_REF] and will also be used for the simulations.

Signal generation

The shape of the signals generated by a detector can be calculated with the Shockley-Ramo theorem [START_REF] Shockley | Currents to conductors induced by a moving point charge[END_REF][START_REF] Ramo | Currents induced by electron motion[END_REF][START_REF] He | Review of the Shockley-Ramo theorem and its application in semiconductor gammaray detectors[END_REF].

The theorem states that the charge Q and the current i on an electrode induced by a moving point-like charge q are given by:

Q = -qφ 0 (x) (1) i = qv • E 0 (x) ( 2
)
where v is the instantaneous velocity of the charge q.

The quantities φ 0 (x) and E 0 (x) are called weighting potential and weighting field, respectively. They correspond to the electric potential and field that would exist at the instantaneous position x of the charge q if the selected electrode is considered at 1 V , all the other electrodes being at 0 V , and the charge q removed. As supplementary assumptions, it is considered that the magnetic effects are negligible and that the electric field propagates instantaneously (quasi-static approximation).

The electric field can be calculated analytically or numerically (depending on the complexity of the geometry of the detector) by solving the Poisson equation:

φ = - ρ (3)
Typically, simple symmetric detector geometries, such as planar or true coaxial configurations, have analytical solutions, while the solution for more complex geometries is generally obtained numerically.

The weighting potentials can be obtained by solving the Laplace equation:

∇ 2 φ i (x) = 0 ( 4
)
where:

φ i |S j = δ i,j (5)
with S j the surface of the electrode j. From equations 1 and 2 one can deduce that for a charge q, located at position x, and traveling towards the electrode S i , this electrode will collect an induced charge Q proportional to the weighting potential φ i (x). At the same time, the other electrodes S j =i will collect an induced charge Q proportional to the value of their respective weighting potential φ j (x). However, from equation 4, when q reaches the electrode S i the net charge induced on it will be Q = -q, while the net charge collected on the other electrodes will be Q = 0. The charge collected on the electrodes S j =i for a fixed radius along an arc. As the radius is fixed, the rise-time of the main pulse remains constant (there are slight changes due to the fact that charge velocities depend on the germanium crystal orientation). On the other hand the amplitude of the transient signals changes with the interaction position: as the interaction approaches a neighboring segment, the corresponding transient signal (left or right) grows in amplitude.

The Pulse Shape Coincidence Scanning method

The pulse shape coincidence scan (PSCS) [START_REF] Crespi | A novel technique for the characterization of a HPGe detector response based on pulse shape comparison[END_REF] method allows full volume characterization of a given position sensitive detector starting from two sets of data obtained by irradiating the detector with a collimated gamma-ray source. The detector is put at first in vertical configuration (see figure 4), then the collimated source is moved by regular steps in a plane perpendicular to the detector axis. For each position of the collimator all the pulses generated by the detector along the gamma-ray beam are collected and a vertical-beam dataset V is obtained. The detector is then moved to horizontal configuration by rotating it by 90 • and a horizontal-beam dataset H is obtained by repeating the same scanning procedure. The positions of the beams of the V and H datasets are chosen so that they cross in a three-dimensional reference frame, forming a threedimensional scanning grid.

For a position sensitive detector (such as AGATA) two identical gamma rays that interact in the same point will produce the same signals. If V and H datasets are compared, their pulses will differ except where the beams intersect. The similarity between two pulses can be evaluated with the following χ 2 -like formula

χ 2 = N i=0 (v i -h i ) 2 σ • N (6)
where v i and h i are the samples of two pulses v and h coming from the V and H datasets, respectively, N is the total number of samples of each signal (this implies that v and h are sampled at the same rate) and σ is the noise level of the signals. The lower the χ 2 value, the most similar are the two compared pulses. An absolute threshold for the χ 2 value below which two pulses can be considered identical does not exist. This value, in fact, depends also on the experimental conditions, so that only relative evaluations can be done. In this regard, the main disadvantage of the PSCS technique lies in the fact that multiple interaction events could give rise to small χ 2 and appear as fake single interaction events originating at the crossing point. These kind of events may have a negative impact on the final database quality and should be excluded if possible.

Scanning Table

The IPHC scanning table [START_REF] Sigward | Pulse-Shape Comparison Scan of germanium detectors using the IPHC scanning table[END_REF][START_REF] Ginsz | Characterization of high-purity, multi-segmented germanium detectors[END_REF] was designed with the aim of performing scans in short scan durations, with good spatial resolution and with variable scanning energies. The table allows to perform a full-volume scan of an AGATA detector in times of the order of 15 days, leading to the creation of a database of ∼ 45000 points for a 2 mm scan pitch. The table setup is shown in figure 3.2 of ref [START_REF] Ginsz | Characterization of high-purity, multi-segmented germanium detectors[END_REF]. A metal collimator sits on two motorized perpendicular axes that allow its planar movement in a range of 300 mm in both X T and Y T directions with a precision of about 10 µm. Above the collimator, two fixed plates allow the placement of the detector in vertical and horizontal positions. The detector is actually placed in an adjustment frame, which is fixed on the plates by two centering studs. The adjustment frame allows the adjustment between the detector and the table axes, by tilting it using micrometric screws and by freehand rotating it. Finally a laser alignment system is set on the mechanical support of the collimator and is used to keep the relative alignment of the detector when going from the vertical to the horizontal position.

The collimator, shown in figure 3.4 of [START_REF] Ginsz | Characterization of high-purity, multi-segmented germanium detectors[END_REF], consists of a metallic cylindrical block, 189 mm high with a diameter of 220 mm, composed of iron, lead and tungsten, materials that absorb the gamma rays which are not traveling through the central hole. When scanning an AGATA detector, the distance of the front face of the collimator from the center of the detector in both vertical and horizontal configurations is 54 mm and 95 mm, respectively. The base of the collimator holds a cylindrical capsule containing the gamma-ray source. The capsule and the overall collimator are coaxial and pierced, along their central axes up to the top of the collimator. Different gamma-ray sources are used for the scans, each one housed in a different capsule. The collimator can be opened by sliding the bottom part and allowing the insertion of the needed source capsule. The central section of the collimator can be mechanically swapped as well and it is possible to choose between three central collimating parts which have diameters of 1.0 mm, 0.5 mm and 0.2 mm.

Three sources are used for scanning. Each emits gamma rays of different energies that can be used accordingly to their penetration power. An 241 Am source, which emits monochromatic gamma rays with an energy of 59.5 keV , is used for surface and two-dimensional (2D) scans. A 137 Cs source, which emits monochromatic gamma rays with an energy of 661.7 keV , is used for deep scanning and for the full-volume characterization (3D) scans. Using the 1 mm diameter collimator, the 137 Cs beam spot diameter at a distance corresponding to the center of an AGATA detector was measured, with a finely pixelated CdTe sensor, to be 1.47 ± 0.03 mm and 1.81 ± 0.03 mm in vertical and horizontal configuration, respectively. These characteristics are reproduced in the simulations.

Finally a 152 Eu source, which emits gamma rays of several energies in the range of 122 keV to 1408 keV , is used to perform several advanced tests on detectors.

Tools of the simulation

The main tool used for the simulation is the Agata Detector Library (ADL) [START_REF] Bruyneel | Pulse shape analysis and position determination in segmented HPGe detectors: The AGATA detector library[END_REF], a software library developed at IKP, Universität zu Köln. The library is used by the AGATA collaboration to calculate the signals generated from the AGATA detectors and build the relative pulse-shapes databases to be used for PSA. ADL uses the theorem of Shockley and Ramo to numerically calculate the pulse shapes.

A simple block scheme of the architecture of ADL is shown in figure 5. Different inputs and parameters should be set in the code before running. At first the Fig. 5 ADL I/O scheme (adapted from [START_REF] Bruyneel | Pulse shape analysis and position determination in segmented HPGe detectors: The AGATA detector library[END_REF]). In blue and red are represented the parameters or inputs to be set in the code while in green are represented the different routines that form the code. In the scheme the crosstalk component is struck off as it is not implemented in the present simulations.

electric field and the weighting potentials for each electrode of the detector are to be defined, as well as the electric field generated by the space charges. These quantities are calculated through a Poisson solver. The ADL software package provides already the weighting potentials, the electric fields and the space charge fields for the 4 different types of AGATA detector (type A, B, C and S). These are calculated with SIMION [START_REF]SIMION[END_REF], a software which can be used for discrete calculation of electric fields with complex geometries. The second group of parameters to be set are the mobility parameters for the electrons and holes in the crystal and its axis orientation. For the AGATA detectors these values were determined experimentally in [START_REF] Bruyneel | Pulse shape analysis and position determination in segmented HPGe detectors: The AGATA detector library[END_REF].

Once the fields and the mobility parameters are set, it is possible to calculate the pulse shapes generated in an event by giving the positions and the deposited energies as input. These were calculated using the Monte Carlo software Geant4 [35]. Both the detector and collimator were modelized with their exact geometry and materials in order to have an accurate reproduction of the scanning system.

For this work, the pulses calculated by ADL are 1 µs long, sampled with a rate of 100 M Hz (100 samples). In order to have a realistic output, the resulting signals are convoluted with the response function of the preamplifiers. The response function used was taken from the AGAPRO code [36], a software developed by the AGATA collaboration to analyze experimental and simulated data. Realistic noise was added, as well, to the calculated pulses. The characteristics of the noise were extracted from a real S-type detector. Finally, ADL allows the user to convolute the output signal with a crosstalk function, although for this work the crosstalk was not considered.

An example of signals calculated with ADL can be seen in figure 3. In the example, different signals generated by single interactions in a S-type segmented detector are shown. Figure 6, on the other hand, shows a signal extracted from the database of detector S001, measured with the IPHC scanning table, compared to the corresponding signal calculated with ADL.

Data treatment

The simulation work-flow is shown in figure 7. The beam propagation in the detector is simulated with Geant4 and ADL is then used to calculate the pulse shapes generated by the interactions of each gammaray (event). Only the events in which the corresponding gamma ray is fully absorbed in a single segment of the detector are accepted. For each position of the collimator an independent beam file is stored which contains 6000 events. For the full volume scan presented in this work, a total of 2975 beam files are used, 1257 for the vertical configuration and 1718 for the horizontal configuration.

The beam files are processed with a code that implements the pulse-shape comparison algorithm described in section 3. The overall code design follows the one de- scribed in [START_REF] Ginsz | Characterization of high-purity, multi-segmented germanium detectors[END_REF]. The comparison between two pulses is performed by calculating the χ 2 value with equation

χ 2 = seg∈M i f i=i0 (v seg,i -h seg,i ) 2 M • (i f -i 0 ) ( 7
)
where seg is the segment index, M is the M dimensioned ensemble of segments formed by the Core segment, the hit segment and the neighboring segments of the hit segment (4 segments if the hit segment is in the middle of the detector or 3 segments if the hit segment is at the top or bottom of the detector) and v i and h i are the samples of the vertical and horizontal signals, respectively. Finally, the comparison is performed on a subset of samples in the window [i 0 , i f ] containing the leading edge of the pulse. Only the signals produced by events happening in the same segment are compared and this reduces the total amount of couples to compare from 3.6 • 10 7 to a few 10 6 . Each pair of events compared is inserted in a χ 2 value ordered list and at the end of the procedure only the 400 couples with the lowest χ 2 values are selected. Among these, some events may have been selected more than once. The duplicates are discarded from the list before the mean pulse-shape refinement.

The refinement procedure, schematized in figure 8, rejects the most diverging pulses giving a more coherent final selection of events, since the χ 2 procedure doesn't insure that the selected pairs are similar within each others. At the beginning of the procedure, the mean pulse of all the events of the final list is calculated. Each pulse is then compared with this mean and the corresponding χ 2 is calculated with equation 7. After that, 5% of the events associated to the higher χ 2 value are rejected and the procedure is repeated for 10 it- erations. The remaining signals are averaged and the resulting signal will be part of the final database.

The computational times for the selection procedure are of the order of ∼ 7 s per grid point (on a 2.60 GHz processor machine) leading to a total processing time of ∼ 4.7 days for a typical simulated scan of an AGATA detector of ∼ 60000 points which produces (considering the tapering of the detector) a database of ∼ 45000 points for a S-type detector and ∼ 48000 for asymmetric crystals.

7 Three-dimensional scan results with the 137 Cs source

The PSCS technique was tested on a S-type detector by simulating various scans, a list of which is given in table 1. For all the scans the 1 mm diameter collimator was used. In figure 9 the reference frame in which the results of the analysis will be presented is shown.

F1

Fig. 12 Example of a supertrace extracted from the database. The supertrace is presented in two panels to fit the text column. The signal is normalized with respect to the main signal amplitude (Ampl.). The graph insert shows a detail of the residual trace for segment F1 (hit segment). Note that the red line is superimposed on the black which is therefore not visible.

At first a full volume 137 Cs scan is performed. The first parameter analyzed is the percentage of pulses originating from single-interacting gamma rays (or singleinteraction events) selected by the pulse shape comparison (PSC) algorithm. This value is quite homogeneous across each transversal, 2 mm thick (with respect to the Z C direction) slice of the database, as shown in figure 10. However, figure 11 shows that the average value per slice, at 662 keV (blue markers), varies along the depth of the detector, decreasing as Z C increments. The maximum average value of single-interaction events is of about 56% at the front of the detector while the minimum average value is of around 44% at the back. The reasons for this trend can be found in two factors. The first one has to do with the geometrical properties of the segments: the active volume of the segments of the S-type detector increases from the front to the back of the crystal and this increases the fraction of selected multi-interaction events among the considered events that release all their energy in a single sector. The second factor concerns the absorption of gamma rays in the germanium: as the beams dips deep in the detector, the available number of events for the χ 2 selection is reduced and this leads to a less efficient selection of single-interaction events (the impact of the number of events used for the χ 2 selection will be described later in section 9).

A good single-interaction selection is preferred for the building of a database, since multiple interaction events can, in some way, spoil the database. Ideally, the PSC should only select pulses generated by single interactions near the considered database point. This is the case for the Liverpool scanning table [START_REF] Dimmock | Validation of Pulse Shape Simulations for an AGATA prototype detector[END_REF], which exploits a coincidence scanning technique and reaches a selection of single interactions of ∼ 90%, a simulated value much larger than the ones shown in figures 10 and 11.

For each database point simulated, however, the PSCS technique selects single-interaction and multipleinteraction events which are quite similar. Let's define a supetrace as a compact, continuous representation of all the signals leading edges coming from the 36 segments and the core of the detector (see figure 12). The average supetraces of single-interaction events and multiple-interaction events, for each database point, are calculated as well as their difference called residual supertrace. The maximum absolute value of the residual supertrace (or maximum residual) divided by the amplitude of the main signal, given in percentage, is taken as comparison parameter. The variation of its average value per database slice along the depth of the detector is shown in figure 13. The average values are quite homogeneous in the volume of the detector (< 5%) with values in the bulk of the segments of 1 -1.5%. Some anomalous high values can be registered for points that are towards the segmentation lines due to border effects. In fact, the signals at the borders, especially transient signals, vary very rapidly meaning that a small difference in the interaction position can lead to large differences in the shape of the transient signals.

The similarity between the pulses generated by singleinteraction events and multiple-interaction events can be understood by considering the average distance of the PSCS selected events from the position of their corresponding database point. The values for different transversal database slices at different depths of the detector (see figure 14) show that the average distance is homogeneous across each individual 2 mm slice. The graph in figure 15 shows a slightly growing trend along Z C for the average values of each slice. At 662 keV (blue markers), the average distance, in fact, has a value of ∼ 2 mm in the front of the detector and then rises slightly until it reaches ∼ 3 mm at the back. These values, nevertheless, are comparable to the PSA performances (see table 4 in [START_REF] Akkoyun | AGATA -Advanced GAmma Tracking Array[END_REF]) and the nominal position resolution of an AGATA detector.

Finally, a second reason that explains the similarity between the selected single-interaction events and multiple-interaction events pulses is found by considering, for multiple-interaction events, the energy released by the furthest interaction from the first interaction point as a function of the distance between these two points. The examples in figure 16 show that generally the furthest interaction releases the least energy and thus contributes less to the signal formation. In other words, the bulk of the signal is formed near the position of the first interaction.

In the next two sections the PSCS technique will be further tested with scans performed with different gamma-ray energies and different number of pulses used as input of the χ 2 selection.

Impact of the energy on the PSCS technique performance

As a further test of the PSCS technique, the impact of the gamma-ray energy used for the scan was evalu- ated. Four simulated partial scans (only one sector of the detector was scanned) were performed with energies of 122 keV , 344 keV , 779 keV and 1408 keV which correspond to four of the various gamma-ray energies emitted by a 152 Eu source. The parameters considered for the present analysis are:

the percentage of selected single-interaction events; the average distance of the selected events from the position of their corresponding database point.

Figure 11 shows the percentage of selected single-interaction events at different depths of the detector for the different scanning energies. The graphs show that data series follow the same trend as already discussed for 662 keV in 7. However, the gamma-ray energy has an impact on the PSCS results. For low energies (122 keV and 344 keV ), the fraction of selected single-interaction events is sensibly higher with respect to the energies above 500 keV . This result is due to the fact that gamma rays of lower energies rays are more likely to undergo photo-absorption. This means that for each pool of events for a given scanned position, the fraction of single-interaction events is already larger before the χ 2 selection than for higher energies. The average distance of the selected events from the position of their corresponding database point exhibits a tiny energy dependence, the higher energy gammarays having the largest average distance values (figure 15). However, the overall values are quite comparable and range between 1 mm and 3 mm. It is noteworthy that at positions around Z C = 14 mm, 22 mm, 36 mm, 56 mm, and 74 mm the average distance peaks at ∼ 4 mm due to the presence of the segment boundaries (see figure 17). Fig. 16 For multiple-interaction events, energy released by the furthest interaction from the first interaction point as a function of the distance between these two points. Each graph gathers the interactions of all the points for a slice of the database with a thickness of 1 grid unit (2 mm) for Z C = 6 mm (top) and Z C = 80 mm (bottom).

Impact of the statistics on the PSCS technique performance

The impact on the PSCS technique of the number of pulses used as input for the χ 2 selection is evaluated. A scan of six points along the Z C direction was performed, as shown in figure 17, so that each point is roughly in the center of each segment of a sector (sector D). The χ 2 selection was performed by comparing 1000 × 1000 (low statistics), 6000 × 6000 (regular simulation statistics) and 60000 × 60000 (high statistics) pulses for each point of the scan. Two gamma-ray energies were used, 779 keV and 1408 keV and, as in the previous Fig. 18 Values of the two considered parameters along the Z C direction for the 779 keV scan. In the top panel: percentage of selected single-interaction events. In the bottom panel: average distance of the selected events from the position of their corresponding database point.

Fig. 19 Values of the two considered parameters along the Z C direction for the 1408 keV scan. n the top panel: percentage of selected single-interaction events. In the bottom panel: average distance of the selected events from the position of their corresponding database point.

paragraph, the two main parameters considered for this analysis are:

the percentage of selected single-interaction events; the average distance of the selected events from the position of their corresponding database point.

The results, plotted in figure 18 [START_REF] Ha | New setup for the characterisation of the AGATA detectors[END_REF] for the 779 keV (1408 keV ) gamma-ray energy scans show that statistics plays an important role in the PSCS selection. In fact, although the trends of the data series of the two parameters previously listed is identical, their values are noticeably affected. The percentage of selected single-interaction events increases substantially and the average distance of the selected events from the position of their corresponding database point decreases with the number of pulses used for the χ 2 selection. This last phenomenon is foreseen in the bottom panel of figure 14 where a general, slight increase of the average distance from the database point is visible in correspondence of sectors E and F (Y C < 0). In fact, to reach this area the beam traverses the largest thickness of germanium in both the vertical and horizontal (see the red arrow in the top panel) scans, getting absorbed in the process. As a result, the statistics for the χ 2 selection is affected, leading to the effects described in this section. In conclusion, it is clear that for these two parameters the PSCS technique performs better when a higher number of pulses is used for the χ 2 selection. As a further analysis it would be worth to extend the range of the test to even lower statistics (few hundreds of events) and higher statistics (orders of magnitude > 10 4 ).

Conclusions and Perspectives

A study of the PSCS technique implemented at IPHC to scan germanium detectors in 2D and 3D was performed via simulations and the results were discussed. This work was aimed to characterize the accuracy and the critical points of the scanning procedure in order to estimate whether the PSCS technique is able to provide full crystal volume, experimental, pulse-shape databases. A simulation of a full volume scan of a S-type AGATA detector was performed with a gamma-ray beam energy of 662 keV ( 137 Cs source) using the GEANT4 and ADL softwares. The pulse shapes generated were convoluted with realistic noise and the response functions of the preamplifiers.

The first parameter extracted from the analysis was the percentage of the selected pulses generated by a single interaction. The values obtained indicated a percentage of single interactions that ranges, in average between 55% at the front of the detector and 45% at its back, which is lower than expected.

The influence of multiple interaction events on the average pulse shape at each point of the scan grid was investigated. The analysis showed that the many interaction points of multiple interaction events occur relatively close to each other and that the average value of their distance from the corresponding scan grid point is around 2 mm. Moreover it was observed that the multiple interaction pulses selected by the PSCS technique are similar to those of single interactions. In addition, the simulations showed that for a multiple scattering event, the energy deposition of the interaction point furthest from the first interaction decreases as the distance between these two points increases. Typically, the largest energy release occurs in the vicinity of the first interaction where most of the signal is generated. All these elements explained the similarity between the pulses generated by single and multiple interaction events which lead us to conclude that the multiple interactions pulses selected by the PSCS technique do not contaminate the database.

The impact of the gamma-ray energy on the extracted pulse shapes was examined at 122 keV , 344 keV , 779 keV and 1408 keV (europium source energies). The analysis showed that the energy of the gamma-ray beam impacted significantly only the percentage of selected single interaction events, enhancing this value for low energies gamma rays. The average distance of the selected events from their corresponding database point, instead, showed only a minimal energy dependence with values that are overall comparable within the data series and with the results obtained for the 662 keV scan.

The impact of the input statistics was also analyzed using different numbers of input events to be compared with the χ 2 selection algorithm for both the vertical and the horizontal datasets. The input statistics proved to have a larger impact than the beam energy, the PSCS technique performing better with higher input statistics. The overall results indicate that the PSCS technique applied to an AGATA unit detector is reliable.

This work shows that further studies can be performed. Different values of the power in equation 6 can be used as in [START_REF] Lewandowski | Pulse-Shape Analysis and position resolution in highly segmented HPGe AGATA detectors[END_REF]. The χ-like formula would then become χ α where α is a value that can be optimized in order to maximize the percentage of selected single interactions and minimize the average distance of the interaction points from the respective database position. Another way to improve the χ 2 selection can be the use of weights for the transient signals to give them more or less impact on the database construction. This option was preliminarily tested in [START_REF] Ginsz | Characterization of high-purity, multi-segmented germanium detectors[END_REF] and a more systematic study with extensive simulations can be done. Scans with a finer scanning pitch and smaller diameter of the collimator can be simulated. These scans can lead to improvements in the PSCS selection especially near the segmentation surfaces where it is critical to have a finer resolution for the database since in these areas the transient signals change very rapidly for small changes in the position.

Finally, real scans with an 152 Eu source have been performed recently for the first time and the results will be presented in an upcoming publication [START_REF] Canditiis | 3D characterization of multisegmented HPGe detectors[END_REF]38]. These measurements will likely give hints to further improve the PSCS technique.

Fig. 1

 1 Fig. 1 Exploded rendering of an AGATA detector unit. The six slices of the detector are shown with their respective labeling. The hexagonal shaped slices are towards the front of the detector. The segment labeling corresponds to the detector seen from its back side. Adapted from [1].

Fig. 2

 2 Fig. 2 Dimensions (in mm) of the S-type detector. Adapted from [1].

  is called transient charge (or image charge) and the signals obtained from it are called transient signals (or image signals). Since the weighting potentials strongly depend on the spatial coordinates, the signals generated on the electrodes (especially for the transient signals) can be used to deduce the position of the gamma-ray interaction that generated the charge, making segmented HPGe detectors position sensitive.

Fig. 3

 3 Fig. 3 Pulse shapes calculated with ADL (see section 5) for interactions at different radii (top panels) and at a fixed radius and different lateral positions (bottom panel). For each panel the hit segment (central plot) and the top/bottom and left/right segments are represented. See text for details.

Fig. 4

 4 Fig.4Simple scheme representing the operations needed to perform the PSCS technique. On the left-hand side the detector is in vertical configuration while on the right-hand side the detector is in horizontal configuration. The red arrows show the direction of the beam during the ongoing data acquisition positions while the green arrows show the direction of the beam at the positions for which data is already acquired.

Fig. 6

 6 Fig. 6 Comparison between a signal extracted from the measured database of detector S001 (in red) and the corresponding signal calculated with ADL (in black). The first two panels on top show the core and hit segment (B5) signals. A5 and C5 are the right and left neighbor segments, respectively. B6 and B4 are the top and bottom neighbor segments, respectively.

Fig. 7

 7 Fig. 7 Work-flow scheme.

Fig. 8

 8 Fig. 8 Refinement procedure scheme.

Fig. 9

 9 Fig. 9 Reference frame used to present the results of the PSCS technique analysis.

Fig. 10

 10 Fig.10Percentage of single-interaction events selected by the PSCS technique over the total number of events. The graphs are relative to slices with a thickness of 1 grid unit (2 mm) for Z C = 6 mm (top) and Z C = 80 mm (bottom). The red arrow in the first panel shows the direction of the beam during the horizontal scan.

Fig. 11

 11 Fig.11Percentage (pct.) of selected single-interaction events along the Z C direction for different gamma-ray energies. The bars show the standard deviation for slices at the front, center and back of the detector. Since low energy gamma rays are more likely to be absorbed in the first few centimeters of germanium, no data is available after Z C = 34 mm for the 122 keV scan.

Fig. 13

 13 Fig.[START_REF] Medina | A simple method for the characterization of HPGe detectors[END_REF] Maximum residual value, in percent (pct.) with respect to the maximum amplitude of the main signal, between the averaged signals of single-interaction events and the averaged signals of multiple-interaction events along the Z C direction. The bars show the standard deviation for slices at the front, center and back of the detector.

Fig. 14

 14 Fig.[START_REF] Schlarb | Pulse shape analysis for gamma-ray tracking (Part I): Pulse shape simulation with JASS[END_REF] Average distance (in cm) of the interactions of the PSCS selected events from the position of their database point. The graphs are relative to slices with a thickness of 1 grid unit (2 mm) for Z C = 6 mm (top) and Z C = 80 mm (bottom). The red arrow in the first panel shows the direction of the beam during the horizontal scan.

Fig. 15

 15 Fig.[START_REF] Mateu | Simulation of the charge collection and signal response of a HPGe double sided strip detector using MGS[END_REF] Average distance of interactions of the PSCS selected events from the position of their database point along the Z C direction for different gamma-ray energies. The bars show the standard deviation for slices at the front, center and back of the detector.

Fig. 17

 17 Fig. 17 Separation surface of the segments due to the electric field. In the picture the central section of the detector is shown. The Z C positions of the six scanned points at X C = -20 mm and Y C = 0 mm analyzed in section 9 are also shown.

Table 1

 1 Summary table of the simulated scans. All the scans have a pitch of 2 mm along both axes except the ones in the last two lines, where just 6 points (one per slice) were scanned.

	Description	Energy	Inputs events
	Full volume scan 662 keV	6000 2
	Sector scan	122 keV	6000 2
	Sector scan	344 keV	6000 2
	Sector scan	779 keV	6000 2
	Sector scan	1408 keV	6000 2
	6-points scan	779 keV	1000 2 , 6000 2 , 60000 2
	6-points scan	1408 keV	1000 2 , 6000 2 , 60000 2

Acknowledgements This work was partially supported by the MIRION Technologies Canberra Company under the contract No 152251.


Martin Giraud

Stéphane Vaudez

Cendrine Gatumel

Jeremy Nos

Thierry Gervais

Guillaume Bernard-Granger

Henri Berthiaux
email: henri.berthiaux@mines-albi.fr

Predicting the flowability of powder mixtures from their single components properties through the multi-component population-dependent granular Bond number; extension to ground powder mixtures

Keywords:

The granular Bond number, defined as the ratio between interparticle attractive forces and particle's weight, can be computed to predict the flow behavior of powders. Previous studies used this dimensionless number to predict the flowability of various pharmaceutical or ceramic powders, exhibiting polydispersed particle size distributions. In this paper, we employ a multi-component population-dependent granular Bond number in order to apply this model to powder mixtures. Some binary and ternary mixtures are prepared using two different techniques: a Turbula® mixer and a ball mill. The flowability predictions appear to be in very good agreement with the empirical measurements, carried out with a powder shear tester. However, the model parameters seem to be slightly different between milled and raw powders. The model discussed in this paper allows a prediction of the flowability of powder mixtures according to their composition and serve as a guide for product formulation and equipment design.

Introduction

Particulate systems are commonly handled in most industries involving food, pigments, pharmaceutical and chemical compounds or minerals for example. Yet, the behavior of bulk powders is known to be highly unpredictable and may lead to production slowdown or equipment damaging if not handled with care. However, unlike fluid mechanics, the study of flow behavior of particulate solids has only been developed in recent years and remains not fully understood [START_REF] Seville | Interparticle forces in fluidisation: a review[END_REF].

The poor flowing ability of some powders results from the force balance at particle scales. Indeed, under a certain size, generally considered around 100 µm, some attractive interparticle forces, such as the Van der Waals forces, start to overcome the weight contribution of the particles, resulting in a cohesive behavior of the powder at macroscopic scale [START_REF] Aulton | Pharmaceutics : the science of dosage form design[END_REF]. Thus, some authors have attempted to establish a link between the bulk behavior of powders and the interparticle forces that depends on the particles properties. Li [START_REF] Li | Interparticle van der Waals force in powder flowability and compactibility[END_REF] has successfully compared the pull off force measured by atomic force microscopy (AFM) to the adhesion forces computed using the Rumpf's theory and commonly used Van der Waals force model taking into account particle properties such as shape, elasticity or plasticity. Then, he attempted to derive a model allowing a prediction of the flowability and the compactability of pharmaceutical powders from AFM measurements. Tomasetta [START_REF] Tomasetta | Correlation of powder flow properties to interparticle interactions at ambient and high temperatures[END_REF] also built a model based on Rumpf's theory assuming elastic or plastic deformations. The model links the particle properties to the flow behavior of the powder, measured with a powder shear tester under different temperatures. Moreover, Liu [START_REF] Liu | Effect of particle properties on the flowability of ibuprofen powders[END_REF] developed a similar model predicting the flow behavior of powders from the force balance at the individual particles level. The model provides a good prediction of the flowability measured with a powder shear tester although it has only been tested for one single powder. Finally, Capece [START_REF] Capece | On the relationship of inter-particle cohesiveness and bulk powder behavior: Flowability of pharmaceutical powders[END_REF] established a correlation (see equation [START_REF] Seville | Interparticle forces in fluidisation: a review[END_REF]) between the flow function coefficient

, measured with shear tests, and a dimensionless population dependent granular Bond number (PGB) that depends on the particle properties such as the particles' surface energy, surface roughness, true density and particle size distribution. This model has been established and tested experimentally for powders commonly used in the pharmaceutical industry.

(

) 1
where and are adjustable model parameters.

One of the main advantages of this correlation is that it allows a prediction of the macroscopic flowability of a given powder from simple particle properties that are easily measurable or available in the literature. Moreover, the PGB takes into account the whole particle size distribution of polydispersed powders while most of the previous models used mean or median diameters to represent the particle size. The correlation represented by equation (1) has also been observed experimentally using ceramic powders by Bernard-Granger who attempted to take into account the capillary forces in the PGB calculations [START_REF] Bernard-Granger | Rheological properties of alumina powder mixtures investigated using shear tests[END_REF]. Finally, Giraud also extended the model to other powders and showed that the empirical equation ( 1) was actually consistent with Rumpf's theory [START_REF] Giraud | Investigation of a granular Bond number based rheological model for polydispersed particulate systems[END_REF]. In particular, the exponent parameter seems to be material independent and comes from the Rumpf's theory, while the proportionality factor may vary with the consolidation state of the powder bed investigated.

According to this last study, equation [START_REF] Aulton | Pharmaceutics : the science of dosage form design[END_REF] describes the flow behavior of aluminum, zirconium and yttrium oxide powders with a total squared error of between the experimental data and the model.

(

) 2
The PGB, , is defined by equation (3),Error! Reference source not found. where the size classes of the particle size distribution are indexed from 1 to M, is the center diameter of the classes and , and represents the surface distribution function derived from the volume distribution, assuming that the particles are spherical. Finally, is the individual granular Bond number (GB), corresponding to the ratio between the interparticle attractive forces and the weight of particles of diameters and , described later in this work. Then, a high PGB value reflects strong cohesive phenomenon within the powder bed, which are likely to reduce its flowability.

(∑ ∑ ( ) ( ) ) (3)
However, in most industries handling powders, the granular materials processed are actually multicomponent mixtures made of various elementary powders, such as excipients and active ingredients in pharmaceutic compounds. These elementary powders may present various flow behaviors. The amount of each component is then critical for the flowability of the resulting blend. Indeed, it was shown that the flowability of a powder mixture varies significantly depending on its formulation.

Actually, it might even exhibit a flow behavior beyond the limits of the flowability of elementary powders [START_REF] Vasilenko | Shear and flow behavior of pharmaceutical blends -Method comparison study[END_REF], [START_REF] Vlachos | Investigation of flow properties of metal powders from narrow particle size distribution to polydisperse mixtures through an improved Hall-flowmeter[END_REF]. Some studies attempted to predict the flowability of powder mixtures in order to guide industrial R&D labs during formulation development. Most models presented in the scientific literature are black boxes models linking the flowability of powder mixtures to the flowability [START_REF] Wang | Predicting flow behavior of pharmaceutical blends using shear cell methodology: A quality by design approach[END_REF] or the particle characteristics [START_REF] Yu | Prediction of Bulk Powder Flow Performance Using Comprehensive Particle Size and Particle Shape Distributions[END_REF] of its elementary powders. These models generally consist in cubic equations whose parameters are fitted by a least square method. Although such an approach gives valuable information on the relative influence of each parameter on the flowability, it only applies to the powders investigated or to powders that share the same physical and chemical properties.

In order to predict the flowability of powder mixtures as a function of their composition, Capece extended the concept of GB and PGB to multi-component mixtures. Then, the multi-component population dependent granular Bond number (MPGB) is defined similarly to the PGB by equation ( 4),

where the number of constituents of the powder mixture are indexed from 1 to N and the size classes of the particle size distributions are indexed from 1 to M [START_REF] Capece | On the relationship of inter-particle cohesiveness and bulk powder behavior: Flowability of pharmaceutical powders[END_REF].

(∑ ∑ ∑ ∑ ) (4)
In the above equation, (resp.

) is the ponderation coefficient attributed to the particles of diameter (resp. ) belonging to powder (resp. powder ). can be computed according to equation (5) from the surface particle size distribution, , of powder and the surface fraction of powder within the blend, denoted . This means that the model considers a perfect powder mixture, assuming that the contact distribution of a given particle is directly related to the composition of the blend. For example, in a binary mixture containing 50% of particles A and 50% of particles B, we assume that a given particle is in contact with other particles A or B evenly, whatever the size and shape of these particles.

( ) (5)
where is deduced from the mass fraction of powder (resp. ) within the blend, (resp. ), using equation [START_REF] Capece | On the relationship of inter-particle cohesiveness and bulk powder behavior: Flowability of pharmaceutical powders[END_REF], which involves the true density of the particles (resp.

) and the mean diameter of the surface particle size distribution, (resp.

).

* ∑ ( ) + (6)
The term, of equation ( 4) is given by equation ( 7) and corresponds to the individual GB between a particle of powder of diameter (denoted A on Figure 1) and a particle of powder of diameter (denoted B on Figure 1).

where is the geometric mean of the weight of particles A and B, given by equation [START_REF] Giraud | Investigation of a granular Bond number based rheological model for polydispersed particulate systems[END_REF], and is the attractive interparticle forces between particles A and B. If the capillary and electrostatic forces are neglected, the Van der Waals force becomes the only relevant interparticle contribution and equation ( 7) can be modified with . In, this paper, the modified Rumpf equation ( 9) was used for Van der Waals force calculations because of its simplicity and accuracy as compared to experimental results [13].

√ (8)
where is the gravity constant and (resp.

) is the true density of the powder (resp. ).

√ ( ̂ ( ̂ ) ̂ ̂ ̂ ̂ ) (9)
In the above equation, is the Hamaker constant of the particles constituting the powders and , is the interparticle distance [START_REF] Israelachvili | Chapter 13 -Van der Waals Forces between Particles and Surfaces[END_REF], ̂ is the harmonic mean of the diameters and and ̂ is the harmonic mean of the size of the asperities present at the surface of the particles constituting the powders and .

This multi-component dimensionless Bond number was shown to correlate well with the flow function coefficient measurements, according to equation ( 1), for mixtures composed of pharmaceutical powders [START_REF] Capece | On the relationship of inter-particle cohesiveness and bulk powder behavior: Flowability of pharmaceutical powders[END_REF] and ceramic oxide powders [START_REF] Bernard-Granger | Rheological properties of alumina powder mixtures investigated using shear tests[END_REF] even if the and empirical parameters of the correlation varies among the authors. However, these differences could be explained by the density of the powders investigated and also by the flow function coefficient measurement protocol [START_REF] Giraud | Investigation of a granular Bond number based rheological model for polydispersed particulate systems[END_REF]. However, despite its efficiency for predicting the flowability of pharmaceutical powder mixtures according to their composition [START_REF] Capece | On the relationship of inter-particle cohesiveness and bulk powder behavior: Flowability of pharmaceutical powders[END_REF], the model still needs to be confirmed with other powder mixtures. Moreover, considering its efficiency, the model could be further extended to other process unit operations handling granular material, such as the grinding process.

In this study, we will extend and confirm the model equation (1) using binary and ternary mixtures of oxide powders exhibiting true densities in a range of 4 to 6 g.cm -3 , quite far from the true densities of pharmaceutical powders tested in previous studies. Various alumina, zirconia and yttrium oxide powders were tested, thus, the results can be compared directly through equation ( 2) which was found with the same powders [START_REF] Giraud | Investigation of a granular Bond number based rheological model for polydispersed particulate systems[END_REF]. The powder mixtures were prepared thanks to different mixing techniques:

a Turbula® mixer (WAB, Sausheim, France) and a ball mill. The Turbula® mixer is a drum rotating through a complex motion that is supposed to ensure a good and quick homogenization of the powder blend [START_REF] Mayer-Laigle | Mixing dynamics for easy flowing powders in a lab scale Turbula® mixer[END_REF]. The ball mill consists in a horizontal rotating drum in which pebbles are the grinding media. In such an equipment, a reduction of the particle size takes place simultaneously with the mixing operation. This allows homogenization at a much finer scale [START_REF] Chaudhuri | The effect of mixing aids on the kinetics of mixing in a rotating drum[END_REF]. Indeed, not only smaller particle size increases the number of particles, leading to finer potential homogeneity [START_REF] Lacey | Developments in the theory of particle mixing[END_REF], but the grinding pebbles also acts as mixing aids, acting positively on the kinetics of the diffusion mixing mechanism [START_REF] Shoji | Further studies of ball and powder filling effects in ball milling[END_REF]. The aim of this paper is to predict the flowability of oxide powder mixtures prepared in a Turbula® mixer and in a ball mill according to their composition and from the particle properties of the elementary bulk powders the blends are made of.

Materials and Methods

Powders

Five different ceramic oxide powders have been studied. They all exhibit different flow behaviors, from free flowing to very poor flowability, and different particle properties, such as particle size distributions, particle surface roughness, surface energy and true density. The powders investigated are listed below:

 Two different grades of alumina powders: GE15 and CR6 (Baikowski®, Poisy, France).

 Two different grades of zirconia powders: GY3Z-R60 and CY3Z-RA (Saint Gobain®, Courbevoie, France).

 An yttrium oxide powder: grade C (H.C. Starck®, Newton, USA).

In this paper, alumina GE15, alumina CR6, zirconia GY3R-R60, zirconia CR3Z-RA and yttria grade

C are referred to as Al 2 O 3 G, Al 2 O 3 C, ZrO 2 G, ZrO 2 C and Y 2 O 3 C respectively.

Characterization methods

All the powders investigated were characterized in terms of flowability, particle size distribution, true density and surface energy. The PGB of single powders were computed from these characterization results for all the powders, according to the methodology described previously for N = 1.

Powder flowability

Before any rheological measurement, all the samples were dried in a stove at 110 °C for 24 h in order to remove the residual moisture content of the powders that may induce capillary forces between the particles under ambient conditions (23°C and 40% relative humidity). Shear testing were performed with a FT4® powder rheometer (Freeman, Tewkesbury, UK) in a 10 mL cylindrical cell. After conditioning and pre-consolidation steps of the powder bed under a normal stress of 9 kPa, shear tests were performed successively under 3, 4, 5, 6 and 7 kPa normal stresses following the Jenike standard procedure [START_REF] Efce | Standard shear testing technique for particulate solids using the jenike shear cell[END_REF]. According to Mohr-Coulomb's theory, the yield locus points (see figure 2), obtained by shearing the powder under different normal stresses, are expected to be aligned in the Mohr's space, thus forming the following yield locus line : [START_REF] Vlachos | Investigation of flow properties of metal powders from narrow particle size distribution to polydisperse mixtures through an improved Hall-flowmeter[END_REF] where and represent the shear and normal stresses respectively, is the cohesion of the powder, and is the friction coefficient which is related to the angle of internal friction by equation ( 11).

(

) 11
The cohesion of the powder represents its natural resistance to flowability under no normal stress application; it originates from the interparticle forces. On the other hand, the friction coefficient comes from the mechanical friction that occurs between the surfaces of the particles in contact.

The Mohr circles of Figure 2 are constructed geometrically from the yield locus line: the small Mohr circle is tangent to the yield locus and passes through the origin while the large Mohr circle is also tangent to the yield locus but passes through the pre-consolidation point, denoted by C on Figure 2.

The cohesion stress and the major consolidation stress correspond to the shear stress values of the highest intersection between the x-axis and the small and large Mohr circles, respectively. Finally, the flow function coefficient corresponds to the ratio between the major consolidation stress and the cohesion stress, as shown in equation ( 12).

(

) 12
The classification given in Table 1 is often used in order to assess and rank the flow behavior of a

given powder [START_REF] Leturia | Characterization of flow properties of cohesive powders: A comparative study of traditional and new testing methods[END_REF]. The flow function coefficient is expected to vary with the pre-consolidation stress, meaning that the flowability may vary according to the mechanical stress applied to the powder bed.

The flow function coefficient

, the cohesion and the angle of internal friction of the powders investigated are reported in Table 2. Each measurement was carried out twice. The values in the table correspond to the average of both measurement and the incertitude corresponds to the standard deviation. Concerning the ZrO 2 G powder, the cohesion measured was slightly below zero, meaning that this powder exhibits a free flowing behavior under that level of consolidation. Thus, its cohesion was kept to 0 kPa and the flow function coefficient was not computed. According to

True density

The true density of the particles constituting the powders investigated were measured with a helium pycnometer AccuPyc II 1340® (Micromeritics, Mérignac, France) in a 10 mL cell. Five samples were considered for each powder, the average value being retained. The uncertainty intervals were calculated through the standard deviation. For each sample, between three and six cycles of measurements were performed at 23 °C and 19.6 PSI (135 kPa). Each cycle includes 25 purges followed by 25 measurements. The true densities of the powders are given in Table 3.

Particle size distribution

The particle size distributions of the powders were measured by LASER diffraction with a Mastersizer 3000® (Malvern, Malvern, UK) which gives the particle size distribution through size classes. The liquid dispersion unit (Hydro MV®) was employed in order to avoid re-agglomeration that may occur during dry dispersion. The Mie theory was used for the particle diameters calculations.

Ten samples were measured per powder and each sample was measured ten times after a defined immersion time inside the liquid dispersion unit. The average particle size distribution was kept for each powder and the incertitude intervals are given by the standard deviations. The particle size distributions in volume fraction are given on Figure 3. They may be converted in surface fraction by assuming that particles are spherical. It appears that the powders containing the smallest particles are those exhibiting the worst flowability (see Table 2). The Sauter mean diameters corresponding to each powder is also given in Table 3.

Particle surface energy

The Hamaker constants of the powders were calculated from the Frenkel equation ( 13) [START_REF] Israelachvili | Chapter 13 -Van der Waals Forces between Particles and Surfaces[END_REF], where is a cut-off distance and is the dispersive surface energy of the particles.

(

) 13
The dispersive surface energy is measured by dynamic vapor sorption DVS (SMS, London, UK) from the sorption isotherm of heptane at 25 °C, using the Fowkes model [START_REF] Levoguer | Measurement of the surface energies of pharmaceutical powders using a novel vapor adsorption method[END_REF]. This method was shown to give results that are accurate and similar to those obtained by inverse gas chromatography for alumina powders [START_REF] Tisserand | Comparison of two techniques for the surface analysis of alumina (Al2O3): Inverse Gas Chromatography at Finite Concentration (IGC-FC) and Dynamic Vapor Sorption (DVS)[END_REF]. The Hamaker constant values are given in Table 3.

Population dependent granular Bond number of the bulk powders

In this study, the Van der Waals forces are the only interparticle forces considered. Indeed, electrostatic forces were neglected after measuring the electrostatic charge developed by the powders using a tribo-electrification technique. Capillary forces are also neglected since all the samples are dried in a stove before flowability measurements. The PGB is then computed from the particles properties using equation [START_REF] Li | Interparticle van der Waals force in powder flowability and compactibility[END_REF]. The Rumpf's modified equation ( 7) is used for the Van der Waals forces calculation, in which the asperity diameter is assumed to be for each powder. This assumption is consistent with AFM measurements carried out in contact mode with a confocal Raman microscope alpha300R (WITec®, Ulm, Germany). More details concerning the PGB calculation for the raw powders are available in [START_REF] Giraud | Investigation of a granular Bond number based rheological model for polydispersed particulate systems[END_REF]. The corresponding PGB values are given in Table 3.

Preparation of the powder mixtures

Preparations in a Turbula® mixer

The powder mixtures were prepared in a Turbula® T2F mixer (WAB, Sausheim, France) for 10 minutes, at a rotational speed of 32 rpm, which are standard conditions to ensure a good homogeneity of the mixtures [START_REF] Mayer-Laigle | Mixing dynamics for easy flowing powders in a lab scale Turbula® mixer[END_REF]. Furthermore, a complementary study, focusing on the homogeneity of such mixtures has been carried out by analyzing the composition of various 2 mL samples, randomly taken from the blends. The composition of the samples could be measured from helium pycnometry tests, knowing that zirconia and alumina have significantly different true densities (see section 2.2.2). The results showed that the variation of composition over a large amount of samples was not significant (variation coefficient below 3% after ten samples), allowing to conclude that the blends must be homogeneous at the scale of FT4® measurements (10 mL cells). The blends were prepared in a 300 mL polyethylene vessel, filled at a filling ratio of 40% in volume and fixed inside the 2 L Turbula® vessel.

Preparations using a ball mill

In order to compare different mixing techniques and to investigate the effect of particle size reduction on the flowability, some binary mixtures containing ZrO 2 C and Al 2 O 3 G powders have been prepared

in a ball mill. The mill is made of a 1 L cylindrical INOX vessel filled with 500 cylindrical 8x8 mm steel pebbles, which are used as grinding media and mixing aids. The amount of pebbles and powder incorporated is defined in order to get a filling ratio and a powder level of and respectively. and are defined by equations ( 14) and ( 15) respectively. Such filling conditions correspond to optimal operating conditions of a ball mill [START_REF] Shoji | Further studies of ball and powder filling effects in ball milling[END_REF], [START_REF] Petrakis | Identification of Optimal Mill Operating Parameters during Grinding of Quartz with the Use of Population Balance Modeling[END_REF]. [START_REF] Israelachvili | Chapter 13 -Van der Waals Forces between Particles and Surfaces[END_REF] and, [START_REF] Mayer-Laigle | Mixing dynamics for easy flowing powders in a lab scale Turbula® mixer[END_REF] where is the porosity of the pebbles bed, which was observed to be around by measuring the height of the pebble bed filled in a transparent vessel of approximately the same size and shape than the milling vessel.

The vessel, filled with the powders and the pebbles, is rotating horizontally at 25 rpm around its longitudinal axis. The rotational speed is chosen in order to get a Froude number Fr, defined in equation ( 16), equal to , so as to ensure a cascade motion regime of the pebbles inside the vessel [START_REF] Mellmann | The transverse motion of solids in rotating cylinders-forms of motion and transition behavior[END_REF].

(

) 16
where is the angular rotational speed and corresponds to the vessel diameter.

The cascade motion regime, represented on Figure 4, is obtained when the rotational speed is sufficient to eject some pebbles from the pebble bed, but not too high in order to ensure that the impact point of the dropping pebbles is located on the pebble's bed, and not on the walls of the vessel [START_REF] Mellmann | The transverse motion of solids in rotating cylinders-forms of motion and transition behavior[END_REF], [START_REF] Hans | Particle technology[END_REF]. This motion regime is suitable for both mixing, combining convection and diffusion mixing mechanisms, and size reduction of the particles, combining impact, compression and shear fragmentation mechanisms [START_REF] Arai | Chemistry of Powder Production[END_REF].

Experimental results and discussion

Validity of the power law parameters

Before comparing the experimental results to the predictive power law represented by equation (2), we must make sure of its validity. Indeed, equation ( 2) not only comes from the Rumpf theory, which actually gives a link between the granular Bond number and the cohesion, but also depends on the relation linking the flow function coefficient to the cohesion, both being measured during the shear tests. Actually, Giraud previously found a power law relation given by equation ( 17) between the flow function coefficient and the cohesion, leading to the exponent in equation ( 2) [START_REF] Giraud | Investigation of a granular Bond number based rheological model for polydispersed particulate systems[END_REF].

Thus, such correlation has been made up using all the powder samples analyzed in this study; this includes the raw powders, the powder mixtures and the milled and co-milled powders. The relation [START_REF] Shoji | Further studies of ball and powder filling effects in ball milling[END_REF] has been found, whose coefficients are close enough to equation [START_REF] Lacey | Developments in the theory of particle mixing[END_REF]. This is not surprising since the powder used in this study are the same than those tested to obtain equation ( 17).

Therefore, the exponent can be considered in the model equation ( 1) to make predictions for both the mixed and ground powders prepared in this study.

Predicting the flowability of the powder mixtures

Various binary and ternary mixtures have been prepared using the powders described in section 2.1 for different compositions. The mixing conditions are detailed in section 2.4.1. The blends are summarized in Table 4, some of them being carried out twice in order to check the repeatability of the mixing operation. The flowability of each blend was then characterized with the FT4® rheometer using exactly the same conditions than for the raw powders, described in section 2.2.1. This includes drying the blends in a stove at 110 °C for 24 h before the measurements. At least two samples were characterized per blend, the average being kept and the standard deviation being used as an uncertainty interval.

The flow function coefficient and the cohesion of each blend are given in Table 4. Then, the evolution of the flow function coefficient along with the composition of some binary mixtures is plotted on between two spherical particles is mainly fixed by the size of the smallest, and thus, worst flowing particle [START_REF] Hamaker | The London-van der Waals attraction between spherical particles[END_REF].

Since all five elementary powders under investigation have been fully characterized (Table 3 and

Figure 3), the MPGB related to all the corresponding mixtures were computed using the methodology presented in section 1. Indeed, the MPGB can be computed for any mixture knowing its composition and the properties of its raw constituting powders. Thus, the flow function coefficient of the mixtures is plotted on Figure 6 as a function of the MPGB. The black dots on Figure 6 corresponds to the raw powders and the model established for these single powders, described by equation (2) ( and ), is represented by the dotted line. All the other points of this graph correspond to various binary and ternary mixtures of Table 4 prepared and characterized for this study. Finally, the continuous line corresponds to the correlation given by equation ( 1) where the exponent comes from the Rumpf's theory, and the proportionality factor is adjusted with a least square method in order to fit the experimental data, with a total squared error of . This error has to be compared with the total squared error obtained for the model ( 2) established for single powders:

. Although the error corresponding to the model established with powder mixtures is higher, we can conclude that it is still relevant, considering the fact that much more experimental data have been recorded. It should also be noted that the total squared error computed is strongly influenced by the presence of the mixture containing the Al 2 O 3 G and the ZrO 2 C powders, which seems to behave differently, when compared to the other powder mixtures according to Figure 6.

Indeed, the total squared error falls to when the contribution of the Al 2 O 3 G / ZrO 2 C mixture is removed. An explanation for the behavior of this particular mixture is provided later. The correlation obtained for mixtures of powders is then given by equation [START_REF] Efce | Standard shear testing technique for particulate solids using the jenike shear cell[END_REF].

(

) 19
We can notice on Figure 6 that the correlation obtained for powder mixtures is very close to that found for the raw powders only. Indeed, the proportionality factor found for the raw powder ( in equation ( 2)) is close to that obtained when the powder mixtures are added for completing the fit ( in equation ( 19)). Thus, we can conclude that the correlation also applies to powder mixtures, as suggested previously by Capece [START_REF] Capece | On the relationship of inter-particle cohesiveness and bulk powder behavior: Flowability of pharmaceutical powders[END_REF]. Accordingly, it can be employed as a predictive model to assess the flow function coefficient of powder mixtures according to their formulation and raw powders characteristics. Moreover, we can notice that the ternary mixtures, represented by the triangles on Figure 6, seem to behave according to the same power law (equation ( 19)) than the binary mixtures. This suggests that the model applies to multi-component mixtures. However, this should be completed by more experimental data in future studies.

The value of the MPGB was computed for all the mixtures investigated using the raw powders properties. Then, equation ( 19) is employed to predict the flow function of all the blends. The evolution of the flow function coefficient predicted by the model is represented by the dotted lines on Figure 5, for a confidence interval of 90% along with the experimental data represented by the crosses.

For all mixtures, the model predictions are in good agreement with the empirical results. The blend for which the predictions are the less accurate is the mixture containing ZrO 2 C and Al 2 O 3 G powders, both raw powders exhibiting very similar flowability. This may be explained by the fact that those powders share a similar particle size as suggested by their Sauter mean diameters provided in Table 3.

Thus, we can assume that the flowability of such a mixture is mostly governed by secondary factors, which are not taken into account in this model such as the particle shape or the packing structure.

However, the predictions are very accurate for mixtures involving powders exhibiting various flow behaviors and can be used to guide industrial manufacturers for the formulation of their powder mixtures. In particular, the model seems to predict accurately the fact that the flowability of the whole mixture is driven by the flowability of the worst flowing raw powder. Indeed, a sensitivity analysis of the PGB carried out in a previous study [START_REF] Giraud | Investigation of a granular Bond number based rheological model for polydispersed particulate systems[END_REF] showed that the amount of fine particles among the particle size distribution is a first order parameter, having a great influence on the value of the granular Bond number. Then, adding a small amount of fine particles to a powder mixture will have a significant detrimental impact on the value of the Bond number, and then, on the predicted flow function coefficient.

As an example of application, the model predicts a flow function coefficient of 6 (easy flowing in corresponding to an easy flow behavior.

Application of the model for milled and co-milled powders

In order to investigate the kinetics of size reduction during the ball mill operation, three different milling conditions were tested for the following milling times: 1, 4 and 8 minutes. These correspond respectively to 25, 100 and 200 full revolutions of the drum. These operating conditions are summarized in Table 5 and referred to as B01, B02 and B03 respectively. Otherwise, all millings were performed at the same rotational speed and with the same powder/pebbles filling ratios, as described in section 2.4.2. Milling was first performed with the raw Al 2 O 3 G and ZrO 2 C powders only, and then for binary mixtures of these two powders containing 30% mass fraction of Al 2 O 3 G. Another blend containing 50% of each powder has also been milled but only in the B02 condition. In each case, the powders have been introduced as layers in the drum, starting with the worst flowing powder, without any pre-mixing step. As for mixture preparations, a complementary study focusing on the homogeneity of these co-millings has been carried out following the same procedure used for Turbula® preparations, described in section 2.4.1. The results were roughly the same, showing that the mixtures obtained by co-milling in these conditions were homogeneous at a much finer scale than the FT4® 10 mL cell.

After milling, the flowability of the collected powders were characterized using the FT4® powder shear tester with the same analyzing conditions than those described in section 2.2.1 for the raw powders. The results are given in Table 5. As expected, the flowability of the powders decreases with the milling time. It can be noted that the flowability of the ZrO 2 C powder becomes worse than that of the Al 2 O 3 G powder after milling under the same conditions. This could be due to the fact that ZrO 2 C particles seem quite spherical as compared to other powders and become more angular after milling, while Al 2 O 3 G particles are already non spherical and irregular before milling, as shown on Figure 7.

Then, one can expect the flowability of ZrO 2 C powder to be more sensible to milling. Regarding the co-milled powders, the value of their flow function coefficient is always between the values of the flow function coefficients of the raw powders milled under the same conditions. In addition, for the co-milled powders using the B02 condition listed in Table 5, the mixture containing 70% of ZrO 2 C powder, which is the worst flowing compound after milling, exhibits a lower flowability than the mixture containing 50% of ZrO 2 C.

The particle size distribution of the powders after milling and co-milling have also been measured by LASER diffraction. The measurements conditions are the same than those described in section 2.2.3

for the raw powders. The particle size distribution of the ZrO 2 C and Al 2 O 3 G powders after milling under conditions B01, B02 and B03 are represented on Figure 8 and Figure 9 respectively. As expected, the amount of fine particles increases with the milling time. We can also notice that the ZrO 2 C powder seems to be more sensitive to milling than the Al 2 O 3 G powder. That could explain why the flowability of the ZrO 2 C powder becomes worse than that of the Al 2 O 3 G powder after milling under the same conditions. DVS and helium pycnometry measurements have also been carried out for some milled powders. It resulted in the fact that the Hamaker constant and the true densities of the powders remained identical, whatever the milling conditions. We also assumed that the particle asperities size was still approximately in all cases. Then, the PGB of the milled Al 2 O 3 G and ZrO 2 C powders have been computed from their particle sizes, using the Hamaker constants, true densities and surface asperity diameters of the raw powders. The MPGB of the co-milled powders have also been computed from the properties of the elementary powders milled under the same conditions. For example, the MPGB of the co-milled powder containing 70% of ZrO 2 C and 30% of Al 2 O 3 G using the B03 condition is computed from the properties of the ZrO 2 C and Al 2 O 3 G raw powders milled under the same condition. Thereby, we assume that the flowability of the co-milled powder is the same as the flowability of the mixture containing both pre-milled powders.

The flow function coefficients of the milled and co-milled powders are then plotted as a function of the PGB, for milled powders, and of the MPGB, for co-milled powders (see Figure 10). Black dots correspond to the bulk powders and the correlation [START_REF] Efce | Standard shear testing technique for particulate solids using the jenike shear cell[END_REF] obtained in section 3.2 for powder mixtures is represented by the continuous line. The experimental data corresponding to milled powders (triangles for milled Al 2 O 3 G powder and squares for milled ZrO 2 C powder) and co-milled powders (crosses) are situated on the same power law, as stated in equation [START_REF] Seville | Interparticle forces in fluidisation: a review[END_REF]. In this case, the coefficients of the power law are , given by the Rumpf theory, and fitted in order to match the experimental data using a least squares method with a squared error of . This value may be compared with the total squared error obtained for the model represented by equation ( 2) established for single raw powders: . The correlation represented by equation ( 20), obtained with milled and co-milled powders is not exactly the same than that obtained for raw powders although they are quite close.

(

) 20
The difference can be explained by the fact that the powder bed structure changes after milling the powders. Indeed, it was shown that the proportionality factor may vary with the powder bed porosity and coordination number. Moreover, the model parameters might also be affected by the value of the asperity size diameter, which has been kept constant at while this might be no longer true after milling the powders.

The evolution of the flow function coefficient of the co-milled ZrO 2 C and Al 2 O 3 G powders is plotted as a function of the alumina mass fraction on Figure 11 for the different milling conditions investigated. The crosses correspond to the experimental measurements presented in Table 5 and the dotted lines correspond to the model predictions based on the MPGB calculations and equation [START_REF] Leturia | Characterization of flow properties of cohesive powders: A comparative study of traditional and new testing methods[END_REF] for a confidence interval of 90%. Unlike for powder mixtures, presented on Figure 5, the flow function coefficient increases when the mass fraction of alumina powder increases. This is because the ZrO 2 C powder becomes the worst flowing compound after milling. According to this Figure 11, the model predictions seem to be in good agreement with the experimental data for each co-milled powders formulations. However, they seem to be a bit less accurate for a powder containing very small particles such as the ZrO 2 C powder milled in B03 conditions. Indeed, this powder media exhibits a large amount of fine particles, as shown on Figure 8, giving a very high MPGB that in turn gives rise to a predicted flow function coefficient of 0.57 according the model equation [START_REF] Leturia | Characterization of flow properties of cohesive powders: A comparative study of traditional and new testing methods[END_REF]. For comparison, the flow function coefficient measured is actually 1.29. One possible explanation is that such small particles may re-agglomerate easily together to form bigger particles, making the measured particle size distribution not representative of the size of the particles during flow measurements. Then, some care should be taken when exploiting such results because the calculated MPGB value is very sensitive to the amount of fine particles. Moreover, the shear testing method is not the most accurate when dealing with very poor flowing powders for which the flow function coefficient is under 1.5. In this case, other flow measurement techniques, allowing smaller pre-consolidation stresses, such as the Schultz ring tester [START_REF] Vasilenko | Shear and flow behavior of pharmaceutical blends -Method comparison study[END_REF], might provide more accurate and relevant results.

Conclusion and perspectives

As a conclusion, the model described by equation ( 1) gives a link between the microscopic properties of the particles and the macroscopic flow behavior of powders, as suggested in the literature. In this paper, we have shown that this model can also be employed to predict the evolution of the flowability of powder mixtures according to their composition, given the particle properties of the raw powders.

The multi-component population dependent granular Bond number has been used successfully to represent the force balance at the particle level within the mixture. It also takes into account the polydispersity of each powder constituting the blend. Thus, the model predictions represented by equation [START_REF] Efce | Standard shear testing technique for particulate solids using the jenike shear cell[END_REF] are in good agreement with the experimental results obtained by preparing binary and ternary mixtures in a Turbula® mixer. This allows us to anticipate the flow function coefficient of a powder mixture for a given composition. Therefore, this model may be used to quickly estimate the influence of a variation of the composition of a given powder mixture on its flowability. For example, it is possible to assess which range of product formulations are compatible with the conveying and storage equipment implemented in a given process.

Moreover, the macro-micro link represented by equation ( 1) was also shown to be accurate for powders ground in a ball mill for different milling conditions. However, the proportionality factor obtained in equation ( 20) was slightly different from that obtained in equation ( 19) for raw and mixed powders. This may be due to the fact that this proportionality factor depends on the powder bed packing structure, which is likely to change after milling. Nonetheless, the evolution of the flowability of the powders according to their size reduction during ball milling was shown to follow accurately the model predictions. Then, the evolution of the flowability of powders according to the milling time can be predicted through equation [START_REF] Leturia | Characterization of flow properties of cohesive powders: A comparative study of traditional and new testing methods[END_REF] if the evolution of the size reduction according to the milling time is known, through a population balance approach for example [START_REF] Matijašić | Batch Grinding in Laboratory Ball Mills: Selection Function[END_REF]. This can help to define the optimal residence time in a continuous or discontinuous mill by taking into account the evolution of the flow behavior of the milled powder. Moreover, the model also may be used in reverse, to get an idea of the particle size distribution, or at least its population-dependent granular Bond number, after grinding thanks to a simple shear test. Such reverse correlation, allowing estimating the particle size distribution from flowability tests, should be further explored in future works.

Furthermore, it was shown that equation ( 20) was also consistent with the properties of co-milled powders for various blend compositions and different milling conditions. The flowability of a comilled powder can then be predicted through this model by using the multi-component population dependent granular Bond number, computed from the particle properties of the raw powders milled in the same conditions. This suggests that a co-milled powder behaves like a powder mixture of the same powders milled in the same conditions.

Despite its strong ability to predict the flowability of powder mixtures, milled and co-milled powders, this model can still be improved by a better knowledge of the elementary particles properties. In particular, the powder bed packing structure [START_REF] Legoix | Characterizing powders in order to determine their flow behavior in a mixer: From small scale observations to macroscopic in-mixer rheology for powders of various flowabilities[END_REF], especially for powder mixtures, the asperity diameter at the surface of an individual particle, and the particle shape [START_REF] Thomas | Modeling the mean interaction forces between powder particles. Application to silica gel-magnesium stearate mixtures[END_REF] should be further investigated and taken into account in the Bond number calculations.

List of figures:

List of tables:

Table 1: Classification of the flowability of powders according to their flow function coefficient

Highlights:

 A predictive model for the flowability of powder mixtures is built  The mixtures are prepared either in a Turbula® mixer or in a ball mill  Interparticle forces are investigated through the granular Bond number  The Bond number accounts for particle size polydispersity of real powders  Model predicts the flow behavior of the mixtures according to their compositions

Figure 5 .

 5 Figure 5. The mass fraction of the worst flowing powder is represented on the x-axis. The crosses

Figure 1 :Figure 2 :Figure 3 :Figure 4 :Figure 5 :Figure 6 :

 123456 Figure 1: Forces applied between two spherical particles A, of nature i and size d k , and B, of nature j and size d l , in close contact

Figure 7 :Figure 8 :Figure 9 :Figure 10 :Figure 11 :

 7891011 Figure 7: SEM pictures (x800 magnitude) of the Al 2 O 3 G (up) and ZrO 2 C (down) powders, before (left) and after milling in B02 conditions (right)

Figure 4

 4 Figure 4

Figure 5

 5 Figure 5

Figure 6

 6 Figure 6

Figure 7

 7 Figure 7

Figure 8

 8 Figure 8

Figure 9

 9 Figure 9

Figure 10

 10 Figure 10

Figure 11

 11 Figure 11

Table 1 ,

 1 the Al 2 O 2 C and the Y 2 O 3 C powders seem to exhibit very poor flowability, while the Al 2 O 3 G powder flows easily and the ZrO 2 C and ZrO 2 G powders are free flowing. However, it should be noted that the flowability of the ZrO 2 C powder is smaller than that of the ZrO 2 G powder despite the fact that they belong to the same flowability grade according to Table1.

Table 1 )

 1 for powder mixtures containing ZrO 2 C and Al 2 O 3 C powders for a mass content of alumina of 10%. The model also predicts that changing the alumina content of this mixture from 10% to 20% would decrease the flow function coefficient under 4, leading then to a poor flowability. However, it is still possible to get an easy flowing zirconia and alumina powder mixture containing 20% of alumina by replacing the ZrO 2 C powder by ZrO 2 G in the mixture. Indeed, the predicted flow function coefficient of the mixture containing 80% of ZrO 2 G and 20% of Al 2 O 3 C is approximately 5,

Table 2 :

 2 Flowability of the bulk powders investigated, as measured by a FT4® powder rheometer in a 10 mL cylindrical cell under a 9 kPa pre-consolidation stress

Table 3 :

 3 Bulk powder properties and population dependent granular Bond number calculated from these properties

Table 4 :

 4 Flow function coefficient and cohesion of the binary and ternary powder mixtures prepared in a Turbula® mixer

Table 5 :

 5 Flow function coefficient and cohesion of the milled and co-milled powders obtained in a ball mill

Table 1 .

 1

	Flow function coefficient	Flowability
		Not flowing
		Very poor
		Poor
		Easy
		Free flowing

Table 2 .

 2

	Powder	Flow function coefficient	(-	Cohesion (kPa)	Friction coefficient (-)
		)
	ZrO 2 G	---		0	0.47 ± 0.004
	ZrO 2 C	12.5 ± 1.43		0.34 ± 0.04	0.48 ± 0.011
	Al 2 O 2 G	8.42 ± 0.11		0.54 ± 0.01	0.68 ± 0.006
	Y 2 O 3 C	1.58 ± 0.05		3.32 ± 0.14	0.65 ± 0.007
	Al 2 O 3 C	1.19 ± 0.04		4.57 ± 0.31	0.72 ± 0.003

Powder

True density (g.cm