Gabriel Kerbart

Charles Manière

Christelle Harnois

Sylvain Marinel

Predicting final stage sintering grain growth affected by porosity

Keywords: Grain growth, Sintering, Microstructure development, MgAl 2 O 4, Mechanisms, Modeling Nomenclature

Grain growth has a definitive impact on the quality of transparent sintered materials in areas such as ballistics, biomaterials, jewelry, etc. Controlling the sintering trajectory at the precise moment of final stage sintering is one of the main sintering challenges for obtaining highperformance, fully-dense nano-ceramics. However, the final stage of sintering involves a very complex coupling between the rate of porosity elimination/grain growth and transition mechanisms. This complexity makes predicting the sintering trajectory very difficult, and most transparent material production escapes this problem by using expensive high-pressure methods such as hot isostatic pressing (HIP). In the quest for a pressureless transparent material process, this paper addresses the challenge of predicting grain growth in the transition domain from the grain growth onset (in a high porosity region) to full density for MgAl 2 O 4 spinel. We present a comprehensive modeling approach linking theoretical models such as Zhao & Harmer's and Olevsky's equations to accurately predict the complex grain growth transition region of final stage sintering. This modeling approach opens up the possibility for numerical exploration of microstructure development via underlying kinetics experimental identification.

I. Introduction

The scientific community and industry takes much interest in the development of transparent polycrystalline ceramic technology due to their numerous applications for jewelry, laser hosts, spacecrafts and IR windows for military applications [START_REF] Wang | Transparent ceramics: Processing, materials and applications[END_REF]. Ceramics' main attractive features are their thermo-mechanical properties, fracture toughness, and high hardness from room temperature to high temperatures (>1000°C). For some of them, their intrinsic transparency in the visible-IR range and their low cost of raw materials [START_REF] Petit | Drying step optimization to obtain large-size transparent magnesiumaluminate spinel samples[END_REF] also enhance their attractiveness. Several ceramic compositions present advantageous mechanical properties along with transparency, including alumina Al 2 O 3 , aluminum oxynitride spinel Al 23 O 27 N 5 and magnesium aluminate spinel MgAl 2 O 4 [START_REF] Wang | Transparent ceramics: Processing, materials and applications[END_REF]. Compared to alumina, which is known for its transparency and better mechanical properties, MgAl 2 O 4 does not exhibit birefringence properties. This birefringence has an impact on transparency and implies a minimal grain size (<400 nm for alumina) in order to avoid the degradation of optical properties. Moreover, transparency can only be achieved for highly pure and fully dense materials. In the case of magnesium aluminate spinel (MgAl 2 O 4 ), hereafter termed spinel, these applications also require minimal grain growth to optimize mechanical properties [START_REF] Salem | Transparent Armor Ceramics as Spacecraft Windows[END_REF][START_REF] Rubat Du Merac | Fifty Years of Research and Development Coming to Fruition; Unraveling the Complex Interactions during Processing of Transparent Magnesium Aluminate (MgAl 2 O 4 ) Spinel[END_REF]. The impact of grain size on the mechanical properties of spinel is well documented. It's known that for this type of ceramic material, smaller grain size improves mechanical properties due to its Hall-Petch tendency [START_REF] Sokol | An inverse Hall-Petch relation in nanocrystalline MgAl 2 O 4 spinel consolidated by high pressure spark plasma sintering (HPSPS)[END_REF].

The latest methods for sintering MgAl 2 O 4 include Hot Isostatic Pressure (HIP) and Spark Plasma Sintering (SPS) [START_REF] Krell | Advanced spinel and sub-μm Al2O3 for transparent armour applications[END_REF][START_REF] Sutorik | The Production of Transparent MgAl2O4 Ceramic Using Calcined Powder Mixtures of Mg(OH)2 and γ-Al2O3 or AlOOH[END_REF][START_REF] Goldstein | Transparent polycrystalline MgAl2O4 spinel with submicron grains, by low temperature sintering[END_REF][START_REF] Bonnefont | Fine-grained transparent MgAl2O4 spinel obtained by spark plasma sintering of commercially available nanopowders[END_REF][START_REF] Sokol | Nano-structured MgAl 2 O 4 spinel consolidated by high pressure spark plasma sintering (HPSPS)[END_REF]. These sintering methods limit grain growth through the application of mechanical pressure (gas pressure for HIP and uniaxial mechanical pressure for SPS). The best result for HIP sintering of spinel has been obtained with pretreatment, such as conventional sintering [START_REF] Krell | Advanced spinel and sub-μm Al2O3 for transparent armour applications[END_REF][START_REF] Sutorik | The Production of Transparent MgAl2O4 Ceramic Using Calcined Powder Mixtures of Mg(OH)2 and γ-Al2O3 or AlOOH[END_REF]. A pre-sintering stage followed by a HIP treatment at 1400°C leads to a transparent spinel with a grain size of 400-600 nm [START_REF] Krell | Advanced spinel and sub-μm Al2O3 for transparent armour applications[END_REF]. Similarly, Goldstein et al [START_REF] Goldstein | Transparent polycrystalline MgAl2O4 spinel with submicron grains, by low temperature sintering[END_REF] obtained equivalent grain size with cold-isostatic pressing, conventional pre-sintering and HIP sintering at 1320°C-170MPa. Transparent spinel has also been obtained using SPS sintering by Bonnefont et al [START_REF] Bonnefont | Fine-grained transparent MgAl2O4 spinel obtained by spark plasma sintering of commercially available nanopowders[END_REF] who reported a grain size of 275 nm, at 1300°C-72 MPa. Sokol et al [START_REF] Sokol | Nano-structured MgAl 2 O 4 spinel consolidated by high pressure spark plasma sintering (HPSPS)[END_REF] obtained nano-sized grains of transparent spinel with a mean grain size of 50 nm by HPSPS (High Pressure Spark Plasma Sintering) at 1000MPa and 1000°C. However, those methods involve pressure and specific atmospheric conditions during sintering (argon pressure or vacuum with graphite contact, for respectively HIP and SPS). For SPS, graphite pollution has been widely reported to be a limiting transparency phenomenon [START_REF] Morita | Assessment of carbon contamination in MgAl2O4 spinel during spark-plasma-sintering (SPS) processing[END_REF][START_REF] Hammoud | Mechanism of Carbon Contamination in Transparent MgAl2O4 and Y3Al5O12 Ceramics Sintered by Spark Plasma Sintering[END_REF]. Dopants were also studied for the sintering of spinel, notably LiF [START_REF] Villalobos | Degradation of Magnesium Aluminum Spinel by Lithium Fluoride Sintering Aid[END_REF][START_REF] Rozenburg | Chemical Interaction Between LiF and MgAl 2 O 4 Spinel During Sintering[END_REF][START_REF] Rozenburg | Sintering Kinetics of a MgAl 2 O 4 Spinel Doped with LiF[END_REF]. However, this sintering aid seems to initiate exaggerated grain growth for sintering temperature around 1620-1650°C [START_REF] Gilde | Evaluation of Hot Pressing and Hot Isostastic Pressing Parameters on the Optical Properties of Spinel[END_REF].

The study's objective is to model the spinel grain growth during sintering. This modeling will be useful for predicting the sintering trajectory and therefore for finding optimized thermal treatment cycles [START_REF] Chen | Sintering dense nanocrystalline ceramics without final-stage grain growth[END_REF]. Sintering trajectory models have proven to be a powerful tool for obtaining nano-grain, fully-dense "Zpex" grade Tosoh zirconia [START_REF] Manière | Modeling zirconia sintering trajectory for obtaining translucent submicronic ceramics for dental implant applications[END_REF]. In this approach, the sintering response is modeled via a combination of densification and the grain growth model. Indeed, in addition to the detrimental effect on the ceramic 's mechanical properties, grain growth slows down the sintering kinetic by extending the diffusion path [START_REF] Rahaman | Sintering of Ceramics[END_REF][START_REF] German | Sintering Theory and Practice[END_REF][START_REF] Bordia | Current understanding and future research directions at the onset of the next century of sintering science and technology[END_REF]. This effect is very active in the final stage of sintering and was used to correct the sintering model densification curve at the end of sintering [START_REF] Manière | Spark-plasma-sintering and finite element method: From the identification of the sintering parameters of a submicronic α -alumina powder to the development of complex shapes[END_REF][START_REF] Manière | A predictive model to reflect the final stage of spark plasma sintering of submicronic α-alumina[END_REF]. In the wellknown equation (1) below, which originated from the analytical form of various solid-state pressureless sintering mechanisms [START_REF] Hansen | Combined-Stage Sintering Model[END_REF], the final stage effect takes his origin in the term G w which divides the porosity elimination rate [START_REF] Rahaman | Sintering of Ceramics[END_REF].

𝜃 ̇= -𝐶 𝑓(𝜃)𝛼𝐷 𝐺 𝑤 𝑘𝑇 (1)
In this formula, the grain size exponent w is 3 for lattice diffusion and 4 for grain boundary diffusion [START_REF] Rahaman | Sintering of Ceramics[END_REF]. Equation ( 1) is the base of numerous sintering methods for studying sintering mechanisms, such as master sintering curves [START_REF] Su | Master Sintering Curve: A Practical Approach to Sintering[END_REF], kinetic fields and the Wang and Raj approach [START_REF] Wang | Estimate of the Activation Energies for Boundary Diffusion from Rate-Controlled Sintering of Pure Alumina, and Alumina Doped with Zirconia or Titania[END_REF], etc.

The same approach is applied to a spinel powder for controlling the sintering trajectory. We concentrate our efforts on the temperature onset region of the final sintering stage where the grain growth takes place. This transition region, where both densification and grain growth are active, represents the best opportunity for sintering optimization [START_REF] Cameron | Grain-Growth Transition During Sintering of Colloidally Prepared Alumina Powder Compacts[END_REF][START_REF] Marder | Grain growth stagnation in fully dense nanocrystalline Y2O3 by spark plasma sintering[END_REF]. The difficulty and the interesting aspect of identifying grain growth behavior in this region lies in the influence of porosity on grain growth. At low temperatures, the grain growth kinetic is slow due to the pinning effect of the porosity located on the grain boundaries. After the separation of pores from the grain boundaries at higher temperatures, the grain growth kinetic accelerates [START_REF] Manière | Modeling zirconia sintering trajectory for obtaining translucent submicronic ceramics for dental implant applications[END_REF]. This separation has been studied by numerous authors [START_REF] Hsueh | Microstructure development during final/intermediate stage sintering-I. Pore/grain boundary separation[END_REF][START_REF] Spears | Microstructure development during final/ intermediate stage sintering-II. Grain and pore coarsening[END_REF][START_REF] Brook | Pore-Grain Boundary Interactions and Grain Growth[END_REF] and is known to be a key aspect of sintering, particularly with respect to obtaining transparent ceramics. In order to model the grain growth in this interesting transition region, we will consider the two grain growth models from Zhao and Olevsky which take into account the porosity contribution. Zhao and Harmer published a three-part study [START_REF] Zhao | Effect of Pore Distribution on Microstructure Development: I, Matrix Pores[END_REF][START_REF] Zhao | Effect of Pore Distribution on Microstructure Development: II, First-and Second-Generation Pores[END_REF][START_REF] Zhao | Effect of Pore Distribution on Microstructure Development: III, Model Experiments[END_REF] where the grain growth is modeled theoretically via a model which takes into account porosity and pore size distribution. In the Olevsky et al model [START_REF] Olevsky | Fundamental Aspects of Spark Plasma Sintering: II. Finite Element Analysis of Scalability[END_REF], porosity is implemented via a function which includes a critical porosity and easily models the transition between the region of porosity influence and the traditional form of grain growth for fully-dense materials at high temperatures.

In this work, we will compare these two models and identify the grain growth of spinel at different temperatures in the transition region where grain growth is active. Special attention will be paid to the description of the evolution of the grain growth mechanisms in the transition region.

II. Theory and calculation

II.1. Theoretical model of grain growth with porosity

The traditional grain growth model based on grain boundary mobility (with or without pores) obeys the following relation of the grain growth rate [START_REF] Rahaman | Sintering of Ceramics[END_REF][START_REF] Park | Master Sintering Curve Formulated from Constitutive Models[END_REF][START_REF] Besson | Grain growth enhancement in alumina during hot isostatic pressing[END_REF]:

Ġ = 𝐾(𝑇) 𝐺 𝑝 (2)
with K(T) defined by various parameters that can be expressed by an Arrhenius relation:

𝐾(𝑇) = 𝐾 0 𝑒𝑥𝑝 ( -𝑄 𝑅𝑇 ) (3)
For sintering, the equation ( 2) is often known by its integration form 𝐺 𝑝+1 = 𝐺 0 𝑝+1 + 𝐾′𝑡 for fully dense or porous specimens and for isothermal conditions (𝐾′ constant) [START_REF] Kingery | Grain Growth in Porous Compacts[END_REF][START_REF] Nichols | Theory of Grain Growth in Porous Compacts[END_REF]. For anisothermal conditions, the term 𝐾′𝑡 of the integration form is replaced (mathematically) by 𝐴 ∫ 𝑒𝑥𝑝 (

-𝑄 𝑅𝑇 ) 𝑡 𝑡0
𝑑𝑡 in order to take into account the thermal history of 𝐾(𝑇) as in the equivalent rate formulation (2) [START_REF] Rahaman | Sintering of Ceramics[END_REF].

The exponent of the integrated form (p+1 between 2 and 4) can be related to various grain growth mechanisms such as: 4 for pore control by surface diffusion, 3 for pore control by lattice diffusion or 2 for boundary control of pure system or evaporation/condensation pore drag [START_REF] Brook | Controlled Grain Growth[END_REF]. There are numerous grain growth mechanisms and exponents but, as highlighted by Rahaman, the experimental exponent value during sintering is often close to 3 and may also evolve with temperature and additional factors such as the impurity segregation [START_REF] Rahaman | Sintering of Ceramics[END_REF]. Similarly, a transition of the grain growth kinetic is often observed with temperatures corresponding to a change in the grain growth regime [START_REF] Manière | Modeling zirconia sintering trajectory for obtaining translucent submicronic ceramics for dental implant applications[END_REF][START_REF] Chaim | Activation energy and grain growth in nanocrystalline Y-TZP ceramics[END_REF]. This change has supposedly been linked to a change in the pore drag regime on grain growth. In the present approach, we will investigate the impact of both the porosity and the change of the grain growth mechanism during the sintering of spinel.

The impact of the porosity on the grain growth kinetic is mainly investigated by various identified mechanisms [START_REF] Brook | Controlled Grain Growth[END_REF] using equation [START_REF] Petit | Drying step optimization to obtain large-size transparent magnesiumaluminate spinel samples[END_REF]. However, the evolution of the porosity elimination may directly impact the grain growth kinetic regardless of the exponent. Very few theoretical studies have been published in this regard [START_REF] Bordia | Current understanding and future research directions at the onset of the next century of sintering science and technology[END_REF], but one can cite the works of Zhao and Harmer [START_REF] Zhao | Effect of Pore Distribution on Microstructure Development: I, Matrix Pores[END_REF][START_REF] Zhao | Effect of Pore Distribution on Microstructure Development: II, First-and Second-Generation Pores[END_REF][START_REF] Zhao | Effect of Pore Distribution on Microstructure Development: III, Model Experiments[END_REF]. One of their main achievements is the grain growth equation depending on the porosity 𝜃 and the mean number of pores per grains N g :

Ġ = 𝐾(𝑇) 𝐺 𝑝 ( 𝑁 𝑔 𝑚 𝜃 𝑛 ) (4)
With p, n and m, exponents depend on the control mechanisms (Surface diffusion: p=3, n=4/3 and m=1/3, lattice diffusion: p=2, n=1 and m=0). Riedel and Svoboda propose various grain growth models with a similar form [START_REF] Riedel | A theoretical study of grain growth in porous solids during sintering[END_REF]. Olevsky's grain growth model corrected the traditional theory by modeling the effect of porosity on grain growth via a critical porosity function:

Ġ = 𝐾(𝑇) 𝐺 𝑝 ( 𝜃 𝑐 𝜃 + 𝜃 𝑐 ) 𝑛 (5)
These equations modulate grain growth by the introduction of a porosity influence function which reduces the grain growth kinetics for various porosity levels. However, when samples approach maximum density, Zhao's function of porosity tends toward infinity while Olevsky's empirical function tends toward 1, which reduces the model to its conventional form for normal grain growth in fully-dense materials.

Zhao and Harmer use equation ( 4) in combination with a densification model to model the sintering trajectory. However the consequence of the grain growth function singularity for full density makes the grain growth rate tend toward infinity close to full densification (1/𝜃 𝑛  ∞). Furthermore, the determination of the mean number of pores per grain (Ng) is very difficult for classical microstructures with a high variety of pores. For this reason, Zhao and Harmer used the pore former to ease the experimental identification of Ng. Therefore, Olevsky's model appears to be more suitable for routine experimental use. However, it's important to understand if this empirical function describes the grain growth mechanism in the same way as the theoretical equations (like those of Zhao and Harmer) do.

In order to compare Olevsky and Zhao's and Harmer's grain growth law porosity function, the experimental results of Zhao and Harmer's function (𝑁 𝑔 𝑚 /𝜃 𝑛 ) were plotted in Fig. 1 for a p exponent of 2 and 3, corresponding respectively to surface and lattice diffusion pore control mechanisms [START_REF] Zhao | Effect of Pore Distribution on Microstructure Development: III, Model Experiments[END_REF].

Olevsky's function [START_REF] Olevsky | Fundamental Aspects of Spark Plasma Sintering: II. Finite Element Analysis of Scalability[END_REF] in the form 𝑎(

𝜃 𝑐 𝜃+𝜃 𝑐 ) 𝑛
was also plotted in Fig. 1, with a fitting parameter "a" that can be assumed to be taken from the term K(T) in equation ( 5). It's clearly demonstrated in Fig. 1 that Olevsky's empirical critical porosity function has the same tendency of Zhao and Harmer's grain growth model. Nevertheless, compared to the latter, Olevsky's function depends only on the porosity and does not suffer from the singularity for full densification case (due to the critical porosity term). Based on these results, the following study will consider Olevsky's function to estimate the porosity influence on the grain growth of spinel. It will be demonstrated that this porosity function is inevitable for describing grain growth behavior in the transition region mentioned in the introduction.

Figure 1 Zhao and Harmer's theoretical grain growth porosity dependent term fitted by Olevsky's empirical function

II.2. Identification methodology

In order to identify the parameters of the grain growth model, four main steps are required and are defined below. This methodology is based on interrupted isothermal sintering tests at various temperatures and a double regression for determining the grain growth kinetics (see method scheme in Fig. 2). Figure 2 The four steps of the grain growth model identification method

Step 1 consists of the experimental collection of porosity and grain size data for obtaining isothermal grain growth curves at three different temperatures. For each temperature, more points are taken at the beginning of the dwell temperature where the grain growth is faster.

After data collection, step 2 consists of a mathematical fitting of each isothermal grain size point for determining the grain growth rate curve Ġ. The fitting mathematical equation is below:

𝐺 𝑓𝑖𝑡 = 𝑎 + 𝑏 𝑙𝑛(𝑡 + 𝑐) (6)
At this stage, G and Ġ are known. It's then possible to determine the grain growth model constants K and p for each of the three holding temperatures (step 3). The linear regression equation appears below:

ln(Ġ) = ln(𝐾(𝑇)) -𝑝 ln(𝐺) (7)
Knowing K(T) for the three different temperatures, the final step (step 4) is the identification of the pre-exponential constant K 0 and activation energy Q via the second linear regression below originating from equation ( This traditional identification method [START_REF] Manière | A predictive model to reflect the final stage of spark plasma sintering of submicronic α-alumina[END_REF][START_REF] Besson | Grain growth enhancement in alumina during hot isostatic pressing[END_REF] is used to determine the grain growth parameters. However, in this study, we chose a temperature range where the porosity has a high impact on grain growth. Consequently, the traditional method is not accurate enough to describe the results correctly (for the present study). The addition of the effect of porosity is required for a better description of the microstructure evolution. In our methodology, the porosity function is introduced during step 3 (Fig. 2c) to correct the regression curves which are not completely linear due to the porosity effect on isothermal grain growth (see later). We will also consider the grain growth exponent p change with the sintering temperature.

III. Experiment

Samples have been produced by cold isostatic pressing at 300 MPa using a commercial powder S30CR from Baikowski. This powder has a purity of more than 99%, a specific surface area between 25 and 28 m²/g (BET) and a d50 between 0.15 and 0.3 µm. The samples were sintered with dwelling times of 0min, 30min, 1h and 4h at three different temperatures with the same heating ramp (2 K/min) (see Fig. 3.a). These experiments do not readily correspond to interruptions in order to avoid the deterioration of the heating element of the furnace (cooled by inertia). Additionally, a finite element analysis of the furnace thermal inertia was conducted to estimate the error generated by the heating and cooling latency on grain growth (see the dedicated part in the results section). The mean grain size was measured using the intercept method. The measurement has been done with at least 5 lines intercepting 10 grains on different SEM images for each sample. As established in Mendelson's publication [START_REF] Mendelson | Average Grain Size in Polycrystalline Ceramics[END_REF], a corrective factor of 1.56 has been used. The microstructures were obtained using a Jeol 7200 LV scanning electron microscope (Fig. 3.b). The micrograph acquisition was conducted under low tension of the beam (< 3keV) to avoid a charging effect on the samples' surfaces. With this method, no thermal etching was required. The analysis was conducted with BSE electrons to highlight the microstructure and grains. The sample surfaces were polished down to 1 µm with a diamond suspension and 0.25 µm with colloidal silica. Density was measured using Archimedes's method.

IV. Results and discussion

IV.1. Finite element study of furnace inertia

The furnace temperature during the sintering tests was simulated to accurately obtain the thermal history of the samples. The main purpose was to determine the potential disturbance of the furnace thermal inertia on the grain growth curves, particularly for cooling. A finite element simulation was conducted using COMSOL Multiphysics software (Fig. 4) using the same conditions of ref [START_REF] Manière | Fluid dynamics thermo-mechanical simulation of sintering: Uniformity of temperature and density distributions[END_REF]. In order to calibrate the furnace thermal inertia of the simulation, two temperature measurements (Fig. 4 We can see on Fig. 4. a) the existence of a small temperature offset during the heating ramp. This shift disappears at high temperature where sintering takes place, and therefore has no impact on the results. Furthermore, furnace cooling takes place very quickly, meaning that it may not have a significant impact on grain growth. The simulation Fig. 4. b) shows the temperature distribution inside the chamber with the convection fluxes indicated by gray arrows. To test the impact of cooling on grain growth, we modeled on Fig. 4. c) the case of the onset of grain growth 1673 K -0 h, and the opposite case of high temperature and high density, 1873 K -4h.

The result shown on Fig. 4. d. is obtained using the grain growth model described later in this paper. We simulated the grain growth for the sintering cycles taking into account the furnace inertia (simulated curve Fig. 4. c). We show in this figure that there is essentially no grain growth during the cooling of the 1673 K -0 h experiment. For the 1873 K -4h experiment, we find 0.8% grain growth (blue arrow Fig. 4. d) during cooling. These experiments, under two different conditions, show that the error on the grain growth value due to the thermal inertia is less than 1% . Therefore, this phenomenon can be ignored.

IV.2. Step 1 dilatometry and choice of the exploration temperature domain

Figure 5 Interest temperature determined by a dilatometry experiment

A dilatometry experiment was conducted in order to identify the experimental temperatures of interest for the sintering and grain growth study (Fig. 5). On the dilatometry curve, we see a significant slowdown in densification from an onset temperature (indicated on Fig. 5), which is an indication of the grain growth onset [START_REF] Manière | Modeling zirconia sintering trajectory for obtaining translucent submicronic ceramics for dental implant applications[END_REF][START_REF] Manière | Spark-plasma-sintering and finite element method: From the identification of the sintering parameters of a submicronic α -alumina powder to the development of complex shapes[END_REF]. Consequently, the three temperature holdings were taken in this region of interest, between 1673 K and 1873 K. Since porosity plays a major role in grain growth, long dwell times will be conducted in order to gain valuable insights into this phenomenon. Therefore, we choose to sinter the samples with dwelling times varying from 0 min to 4h.

Once determination of the region of interest is conducted, determination of grain growth parameters can be subsequently conducted. The influence of porosity on grain growth will then be deduced. Fig. 6. a) shows the evolution of grain growth with dwell times. Equation ( 6) was used to determine the evolution of the grain size as a function of time for each sintering temperature. After that, the grain growth rate (Ġ) can be calculated as shown in Fig. 6. b). As expected, we notice a lower level of grain growth for lower dwell temperatures (1673 K and 1773 K) and significant grain growth at 1873 K.

From the dilatometric curve, the density evolution with the heating ramp can be obtained. With the values of the sintered samples' density, we can estimate the evolution of the porosity during the sintering process (Fig. 6. c). The analysis of the evolution of the porosity with the sintering time, correlated to the grain growth rate, seems to indicate a strong influence of porosity on the growth rate for low porosity levels.

Figure 6 a) Grain size fitted as a function of the dwell time b) grain growth rate as a function of the dwell time c) evolution of porosity during sintering

With knowledge of the grain size and grain growth rate, we can determine the grain growth exponent p and the pre-exponential factor K(T) as stated in the equation [START_REF] Sutorik | The Production of Transparent MgAl2O4 Ceramic Using Calcined Powder Mixtures of Mg(OH)2 and γ-Al2O3 or AlOOH[END_REF]. Fig. 7. a) presents the regression of the logarithm of grain size as a function of the logarithm of the grain growth rate. From this point onward, the identification method is split between the traditional model (the model without the porosity effect) and Olevsky's model (the model with the porosity effect). The grain growth exponent p changes from 3 to 2 with the sintering temperature. This phenomenon is apparent in both models, and can be explained by the evolution of the grain growth mechanism changing from surface diffusion for high porosity (p=3) to lattice diffusion for low porosity (p=2) [START_REF] Rahaman | Sintering of Ceramics[END_REF]. Porosity diminishes with the increasing sintering temperature, which explains the transition from high porosity surface control mechanisms to lattice control. ). The blue zone symbolizes the region of porosity we have explored in this paper. This clearly shows the presence of a 0.3 coefficient between samples with low porosity vs high porosity. The effect of this correction is significant for a dwelling at 1873 K, with the first points misaligned. We will see later that the porosity function also plays an important role in the modeling of the grain growth behavior versus the dwell temperatures.

With the determination of the grain growth exponent and the pre-exponential factor, the next step is to determine the activation energy of grain growth.

IV.3.3 Step 4, determination of K0 and Q

As stated by the equation ( 8), the activation energy of the grain growth mechanism can be obtained by the regression of the logarithm of K(T) as a function of 1/RT. Both regressions are presented in Fig. 8. The traditional model (Fig. 8. a) fits the experimental data with low accuracy, despite taking into account the change of the p exponent. The model parameters are K 0 =1.430E24 m 1+p .s -1 , Q=1673 kJ/mol, and the high value of the activation energy is due to the change of mechanism (p). In comparison, Olevsky's model (Fig. 8. b) has a much more accurate alignment of the data points and a parameters value of K 0 = 3.433E+18 m 1+p .s -1 , Q=1515 kJ/mol. The impact of porosity on grain growth is the only factor that can explain these differences in point alignment since it is the only change that exists between the two models. Determining the activation energy was the final step before modeling the grain growth during sintering. Two different models were devised and computed with Octave software for the traditional model (equation ( 2)) as well as for Olevsky's (equation ( 5)) model.

IV.4 Analytical grain growth modeling without the porosity effect

In Fig. 9., we show the results of the simulation of grain growth using the traditional model. It seems that, although the grain growth exponent change has been considered, it is not sufficient to predict the experimental data. There are two noticeable inaccuracies.

On one hand, a high level of discrepancy is noticed in the beginning of the dwell which generates a high error while the grain growth rate (slope of the curve) looks better at the end of each dwell.

On the other hand, the modeled differences between the three dwell temperatures are not well described, particularly for 1773 K. These errors are attributed to the misalignment of the data in Fig. 8. a) and to the absence of the porosity function, which has the ability to modulate grain growth with the porosity difference during each dwell and the average porosity variation between each dwell (see Fig.

Figure 10 Olevsky's grain growth model with the porosity effect

In Fig. 10, we show the results of the simulation using Olevsky's model. A high correlation can be observed between the experimental data and the modeling results, especially in the early stage of grain growth. This simulation used the porosity function reported in Fig. 7. b) and the values K 0 = 6.014E+18 m 1+p .s -1 , Q=1515 kJ/mol. Compared to the values from Fig. 8. b), K 0 was slightly adjusted to improve the overall grain growth model accuracy. In comparing Fig. 9. and Fig. 10., the role of the porosity function is clearly demonstrated. This function corrects the excessive grain growth rate at the beginning of holding and very efficiently modulates the difference between the curves at various temperatures.

The corrective porosity function is then necessary to explain grain growth in the domain where a transition mechanism as well as a high porosity variation takes place. It's also important to note that the transition region, from the surface diffusion to the lattice diffusion mechanism, has a combined effect of p vs θ, which makes the model highly sensitive to variations compared to the traditional approach where the porosity is generally lower and is dominated by a single mechanism.

V. Conclusion

In this paper, we explored various models of grain growth allowing for the prediction of the final stage sintering grain growth transition with porosity. The theoretical reference model for this transition region comes from Zhao and Harmer, and we can easily apply this theoretical principle to the real ceramic materials in which Olevsky's model was employed. We have shown the correlation between these two models and the limitations of the theoretical model, in particular, for the transition to full density, which is better described by Olevsky's model. In this paper, we outlined a comprehensive method based on interrupted isothermal tests to determine, step by step, the constitutive parameters of the grain growth model. Furthermore, we used a finite element simulation to estimate the influence of furnace inertia (during heating and cooling) on grain growth data collection. This simulation showed that furnace inertia may generate only a negligible deviation of the results (<1%).

We have demonstrated two main effects of porosity variation on grain growth for spinel samples:

-High porosity at low temperature favors grain growth mechanisms by diffusion at the pores' surfaces while at high temperature and lower porosity, lattice diffusion grain growth mechanisms are dominant.

-The grain growth porosity function has an important role in modulating the grain growth mechanism transition from low to high temperatures and the grain growth rate inhibition at the beginning dwell.

Olevsky's model is a relevant tool for explaining spinel grain growth in a complex but interesting range of temperatures for sintering optimization purposes. In this grain growth model, the porosity evolution was taken from the experimental curves data. However, the main endeavor of this work will be to combine this grain growth model with a sintering model in order to predict the sintering trajectory of spinel, thus avoiding a very time-consuming experimental design.

Figures captions

  3): ln(𝐾(𝑇)) = ln (𝐾 0 ) -

Figure 3 a

 3 Figure 3 a) Experimental sintering cycles. b) Microstructures of spinel polished surfaces used to determine grain size

  . a) were made in two locations (Tfurnace & Tsample, see Fig. 4. b). Tfurnace is the temperature inside the furnace used for the PID regulation. Another thermocouple was placed close to the sample (Tsample) to detect the inertia of this zone where an alumina box was present.

Figure 4 a

 4 Figure 4 a) Simulated thermal cycle compared to experimental data. b) Furnace simulation with thermocouple location. c) Difference between simulated thermal cycle and furnace programming. d) modeled grain growth for two experiments 1673 K -0 h and 1873 K -4 h

1

 1 Step 2, collections of experimental data and mathematical fittings.

IV. 3 . 2

 32 Step 3, determination of the grain growth mechanism with porosity correction.

Figure 7 a

 7 Figure 7 a) Linear regression of the logarithm of the grain growth rate, as a function of the logarithm of the grain size. b) Olevsky's porosity function

  model implies a porosity function, and is shown on Fig. 7. b

Figure 8 a

 8 Figure 8 a) Determining the activation energy of grain growth using traditional models (without the porosity effect), b) Determining the activation energy of grain growth using Olevsky's model

  6. c).

Figure 9 IV. 5

 95 Figure 9 Traditional grain growth model curves

Figure 2 Figure 5 Figure 9

 259 Figure 1 Zhao and Harmer's theoretical grain growth porosity dependent term fitted by Olevsky's empirical function Figure 2 The four steps of the grain growth model identification method Figure 3 a) Experimental sintering cycles. b) Microstructures of spinel polished surfaces used to determine grain size Figure 4 a) Simulated thermal cycle compared to experimental data. b) Furnace simulation with thermocouple location. c) Difference between simulated thermal cycle and furnace programming. d) modeled grain growth for two experiments 1673 K -0 h and 1873 K -4 h Figure 5 Interest temperature determined by a dilatometry experiment Figure 6 a) Grain size fitted as a function of the dwell time b) grain growth rate as a function of the dwell time c) evolution of porosity during sintering Figure 7 a) Linear regression of the logarithm of the grain growth rate, as a function of the logarithm of the grain size. b) Olevsky's porosity function Figure 8 a) Determining the activation energy of grain growth using traditional models (without the porosity effect), b) Determining the activation energy of grain growth using Olevsky's model Figure 9 Traditional grain growth model curves Figure 10 Olevsky's grain growth model with the porosity effect

Acknowledgements

The author thanks Baikowski for providing us free samples of spinel powder to use in this study. The author also thanks Jérome Lecourt and Christelle Bilot for their help in the progression of this study. We would like to thank the French Ministry of Research for the PhD subvention of Gabriel Kerbart.

Data availability

The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.

Credit authorship contribution statement


Charles Manière
email: charles.maniere@ensicaen.fr

Guillaume Riquet

Sylvain Marinel

DIELECTRIC PROPERTIES OF FLASH SPARK PLASMA SINTERED BaTiO 3 AND CaCu 3 Ti 4 O 12

Keywords: Flash Spark Plasma Sintering, Barium Titanate, CaCu 3 Ti 4 O 12, Dielectric Constant, Phase Decomposition

Flash sintering is an approach allowing reducing the sintering time to mere seconds. To improve the microstructures and the properties of flash sintered specimens, this process has been successfully adapted to pressure assisted sintering such as the spark plasma sintering. This work is the exploration of the potential of this ultra-rapid sintering process for the improvement of the dielectric properties of well-known materials such as BaTiO 3 and CaCu 3 Ti 4 O 12 . In particular, we focus on the potential improvement of the dielectric loss, colossal permittivity and microstructures of these materials after ~20 sec sintering and annealing.

Dielectric materials such as BaTiO 3 and CaCu 3 Ti 4 O 12 are of high interest for the improvement of capacitors, resonators, supercapacitors, etc [1][START_REF] More | AIP Conf. Proc[END_REF][START_REF] Jia | [END_REF][4][5]. In this field, the research is oriented in the miniaturization of the electronic components by keeping microstructures at nanoscale while maintaining high performances [6][7][8]. Sintering is a key step for electroceramic materials and has a high impact on the microstructure [9,10], the crystal defects, the stoichiometry, and the dielectric properties [11,12]. In particular, Spark Plasma Sintering (SPS) has demonstrated a strong potential to generate colossal dielectric properties [11,13]. These properties are often associated with the compound reduction phenomenon [14] possibly combined with an internal barrier layer capacitor behavior (IBLC) [15], electrode effects and/or hopping conduction mechanism [11,16,17]. The IBLC ideal model consists of polycrystalline materials with thin electrically insulating grain boundaries and semiconducting large grains. It is generally assumed that the SPS highly reductive atmosphere induces oxygen vacancies playing a key role in the semiconducting behavior in both BaTiO 3 [16] and CaCu 3 Ti 4 O 12 [14] oxides. The hopping is another mechanism which may occur in mixed valence semiconductor materials by charge hopping from one site to another, therefore impacting [11,16] the dielectric properties.

However, the compound reduction occurring in SPS has a very detrimental impact on the dielectric losses, which are generally very high for colossal permittivity materials. It is often observed that long SPS cycle at high temperatures generates colossal permittivity with high dielectric loss [13]. A subsequent annealing in air is usually required to decrease the dielectric losses; however, it is generally accompanied by a significant reduction of the permittivity [11].

Controlling the sintering process is therefore of primary importance to attain the objective of high permittivity and low dielectric losses. For this goal, different strategies can be implemented. For instance, some groups [ [18][19][20] have designed core-shell composite structure, where semiconducting grains are surrounded by silica, prior sintering. This composite structure was shown to be efficient for decreasing the dielectric losses.

In this work, our strategy is to reduce the sintering time down to a few seconds to partially decrease the compound reduction phenomenon while using the abrupt and high thermal profile for promoting "flash grain growth" [21]. In this way, we can expect to gather the optimal conditions of the IBLC theory [22], i.e., large semiconducting grains and fine electrically insulated grain boundaries. We also expect to obtain a balanced semiconducting behavior by limiting the reduction phenomena which generally ends-up to high dielectric losses. In sum, an ultra-rapid process is explored to find a window between excessive reduction of SPS (high permittivity, high loss) and conventional ways (moderate permittivity, low loss).

Since the discovery of flash sintering [23][START_REF] Raj | Methods of Flash Sintering[END_REF][25], this process has been applied to various approaches such as microwave sintering [26,27] or SPS [28,29]. The last approach is very promising as it allows combining high electrical current and high pressures [30]. By an electric current concentration [31], it has been shown that this "flashSPS" approach can be extended to nearly all materials [32] (from dielectric to metals). In this work, a similar flashSPS configuration is used for the ultra-rapid sintering of BaTiO 3 and CaCu 3 Ti 4 O 12 . All sintering experiments were performed on SPS device model "FCT HPD25" using a punch and die set for the fabrication of pellets of 15 mm diameter and 2 mm height. The BaTiO 3 were purchased from Chempur (99.6 % pure, 85-128 nm) and CaCu 3 Ti 4 O 12 powder were conventionally synthesized [33] using the following powders, CaCO 3 (99,95% Alfa Aesar), CuO (99,7% Alfa Aesar) and anatase-TiO 2 (99,9% Materion). A comparison of microstructure and dielectric properties is investigated using a 100 K/min usual heating rate and in flash condition (~ 1000 K/min). The flashSPS configurations are the same as for the ref [32]: the electrical current is concentrated around the powder bed by electrically insulating the die (via a boron nitride coating on the inner die surface). Compared to ref [32], the flash process is PID controlled by imposing a heating rate of 1000 K/min on the punch temperature.

The punch temperature is different from the powder temperature and this difference may increase for high heating rates due to the electric and thermal contact resistance at the interfaces [34][35][36]. The measured temperature is used for the regulation of the process, but it is not relevant for indicating the sintering conditions of the powder. The sintering advancement is controlled by the displacement curves and for all experiments (SPS and flashSPS), the heating ramp is stopped when the displacement curve indicates the ending stage plateau. For CaCu 3 Ti 4 O 12 in classical SPS configuration, a dwell at 950 °C is added to prevent the phenomenon of phase decomposition (see further). For flashSPS, the sintering time is about 20 sec. The dielectric properties were measured from 5 to 1000 kHz, with FLUKE PM6306 RCL meter. The X-ray diffraction diagrams were recorded on XRD Theta-Theta PANalytical'pert Pro diffractometer with CuKα λ = 1.54059 Å. The Scanning Electron Microscopy and EDX map were collected on "SEM Zeiss EVO MA 15".

For both BaTiO 3 and CaCu 3 Ti 4 O 12 , all the sintered pellets were black. For removing the carbon contamination and re-oxidizing the compounds, an annealing in air (150 K/h heating up to 800 °C, holding of 4 hours and cooling at 150 K/h to room temperature) was performed for both materials. After this annealing, the CaCu 3 Ti 4 O 12 pellets are still black and BaTiO 3 pellets become gray for conventional SPS and white for flashSPS (see figure 1a). The black color was expected for CaCu 3 Ti 4 O 12 [33], since this oxide is naturally black. For BaTiO 3 SPS sintered, the long sintering duration implies a quite significant compound reduction [11]. As a result, a thermal annealing at 800 °C for 4 hours is not long enough to fully re-oxidize the material (it remains gray). On the contrary, the flashSPS sample (sintered in 20 sec) becomes white after annealing, clearly showing that the reduction mechanism was limited. The X-ray diffraction of BaTiO 3 is reported in figure 2b, for SPS, flashSPS specimens before and after 800 °C annealing, BaTiO 3 SPS sintered, XRD pattern peaks can be indexed to BaTiO 3 phase.

It is also remarkably observed the broadening of the (200) and other related peaks (220), (103) etc, which can be interpreted by the presence of both a cubic and a tetragonal perovskite phase. This phenomenon has been already observed and discussed in ref [16]. On the contrary, for the 'flashSPS' sintered materials, the corresponding XRD peaks are sharps with the splitting of the ( 002 The polished microstructures of all samples are reported in figure 2. For classical SPS BaTiO 3 sintered sample, the grain size is submicronic and the amount of porosity is about 5 %.

FlashSPS sintered material has larger grains (~10 µm) with a higher porosity level. In a previous study [32], it was shown by the finite element simulation that a significant temperature gradient develops through the flashSPS sintered sample (for dielectric material), with the highest temperatures on the edges. Experimentally, it is observed that the grain size is significantly larger in the surface than in the bulk of the sample, which is consistent with the modeling result (temperature gradient). Concerning CaCu 3 Ti 4 O 12 , the grains and porosity are similar between classical SPS and flashSPS configurations. The main difference is the CaCu 3 Ti 4 O 12 phase decomposition (for flash sintered material), as revealed by the XRD pattern. In figure 2 (for flashSPS), a white phase appears in the edge of the specimen and 'dark phases' are also observed in some localized areas. The EDX map of this black phase reveals a titanium rich zone (possibly TiO 2 observed in XRD) and the white phase is copper rich (possibly CuO observed in XRD). This suggests an overheating on the edges of the specimen as it was previously shown by modeling on dielectric flashSPS [32].

The dielectric measurements (5-1000 kHz) are reported in figure 3. In the following, for simplicity, the real part of the permittivity will be called "permittivity". CaCu 3 Ti 4 O 12 has a permittivity between 10 3 -10 4 and a loss tangent between 0.1-1. After the 800 °C annealing, the permittivity is almost twice and the loss tangent corresponding to flashSPS is interestingly reduced to a stable value close to 0.1. Concerning BaTiO 3 and before annealing, a clear conductive/semiconductive behavior is observed with a very high loss tangent (tanδ > 1-10 2 ) and a high permittivity (for flashSPS). After annealing, the "SPS" experiment has a high permittivity and a loss tangent closed to 1 (which is still highly dissipative). In comparison, the flashSPS experiment succeeds to preserve the classical values obtained for BaTiO 3 by conventional sintering with a permittivity closed to 2000 and a loss tangent below 0.1. Among all experiments, the optimal property is obtained for flashSPS of CaCu 3 Ti 4 O 12 after annealing with ~ 4000 of permittivity and a loss tangent of 0.1.

To conclude, we have investigated the impact of flashSPS on the properties of classical dielectric materials such as BaTiO 3 and CaCu 3 Ti 4 O 12 . In general, the SPS reductive atmosphere tends to generate colossal permittivity but the dielectric nature of the materials is annihilated in favor of a conductive/semiconductive behavior. This phenomenon is highly active for BaTiO 3 with long sintering time. The significant reduction of the sintering times of flashSPS (20 sec) has a positive impact on this phenomenon while allowing preserving the tetragonal phase of BaTiO 3 . After annealing it is even possible to reproduce the classical dielectric behavior of BaTiO 3 by "flash" approach. For CaCu 3 Ti 4 O 12 , the reduction phenomenon is less intense than BaTiO 3 and a phenomenon of phase decomposition is also observed. The annealing (800°C, 4 hours) helps decrease the flashSPS dielectric losses while increase the permittivity to 4000. The potential role of the TiO 2 , CuO phases in situ generated by the "flash" process should be investigated by a modeling approach. To go further, the impact of flashSPS/annealing, time, temperatures should be finely explored in order to decrease the dielectric loss while more significantly increasing the permittivity via IBLC, local hopping mechanisms, etc. FlashSPS is an original opening to escape the classical trend in SPS where colossal dielectric properties are accompanied by high dielectric losses. In particular, the capacity of flashSPS to preserve the powder phase like for BaTiO 3 represents an interesting prospect for composites and core-shell structures.

Figure captions

Fig. 1: a) Sintered pellets after annealing at 800 °C (all pellets were black before annealing), XRD of the flash and conventional SPS before and after annealing for b) BaTiO 3 and c) CCTO; the other phases creating during the sintering process are indexed using colored symbols (see lower legend).

  )/(200) peaks and other related peaks, showing only the presence of the tetragonal BaTiO 3 phase. This is consistent with the fact that flashSPS process is beneficial for limiting the material reduction. Concerning the XRD diagrams of CaCu 3 Ti 4 O 12 (figure 1c), the difference between the classical SPS and flashSPS suggests a higher temperature for the sample obtained by flashSPS. The XRD pattern of the classical SPS sintered samples only has the peaks corresponding to CaCu 3 Ti 4 O 12 . For the flashSPS sample, four additional phases appear, namely TiO 2 , CaTiO 3 and Cu 2 O (which is fully converted to CuO after annealing at 800 °C). Because of the high temperatures of flashSPS, CaCu 3 Ti 4 O 12 phase partially decomposes during this flash thermal treatment.

Fig. 2 :

 2 Fig. 2:Polished SEM images of the microstructures in the center and the edge of

Fig. 3 :

 3 Fig. 3: Dielectric properties of BaTiO 3 and CCTO obtained by conventional SPS

FiguresFig. 1

 1 Figures

  Fig.1:a) Sintered pellets after annealing at 800 °C (all pellets were black before

Fig. 2 :Fig. 3 :

 23 Fig.2:Polished SEM images of the microstructures in the center and the edge of

Acknowledgements

The help and support of Christelle Bilot and Jérôme Lecourt is gratefully acknowledged.

The authors thank the French Ministry of Research for the financial support of Guillaume Riquet's PhD.


Charles Manière
email: charles.maniere@ensicaen.fr

Flora Borie

Sylvain Marinel

Impact of convection and radiation on direct/hybrid heating stability of field assisted sintering

Keywords: Thermal Instability, Multiphysics Modeling, Hybrid Heating, Sintering, Ceramic

Field assisted sintering (FAST) processes allow a direct transmission of the heating energy to the specimen (through the electric, magnetic fields or the electrical current). FAST allows higher heating rates, faster sintering response and a better control of the sintered microstructures. However, FAST suffers high heating instability in direct heating configurations which generally takes the form of a hot spot. The origin of these hot spots is well known and is correlated to the convective/radiative cooling at the specimen surfaces and the thermal dissipation in the specimen. Nevertheless, the impact of these cooling fluxes evolves with the sample dimensions, thermal insulation, heating rate and hybrid heating conditions and there is not clear quantification of the relative importance of these fluxes in regards to the previous cited heating conditions. In this work we develop a finite element (FE) tool which can easily explore the heating stability of an "Equivalent Thermal Cavity" (ETC).

We illustrate the ETC concept by the case study of the microwave sintering of zirconia. We show that the dominant heat transfer is radiative, but the convective fluxes have a high importance for the temperatures homogenization, in particular in the case of a hybrid heating configuration.

Introduction

Field assisted sintering (FAST) approaches such as microwave sintering, inductive sintering, or spark plasma sintering obey a complex Multiphysics problem including [START_REF] Bordia | Current understanding and future research directions at the onset of the next century of sintering science and technology[END_REF][START_REF] Olevsky | Field-Assisted Sintering[END_REF][START_REF] Orrù | Consolidation/synthesis of materials by electric current activated/assisted sintering[END_REF][START_REF] Bouvard | Finite Element Modelling of Microwave Sintering[END_REF]: electromagnetic fields, electrical current, heat transfer and sintering. The complex coupling between all these Multiphysics parameters is a cause of instabilities like field concentration, thermal runaway, thermal gradients [START_REF] Anselmi-Tamburini | Fundamental investigations on the spark plasma sintering/synthesis process[END_REF][START_REF] Vanmeensel | Modelling of the temperature distribution during field assisted sintering[END_REF][START_REF] Todd | Electrical characteristics of flash sintering: thermal runaway of Joule heating[END_REF][START_REF] Dong | Thermal Runaway in Mold-Assisted Flash Sintering[END_REF][START_REF] Giuntini | Localized Overheating Phenomena and Optimization of Spark-Plasma Sintering Tooling Design[END_REF]. However, the most detrimental phenomenon is the formation of hot spots [START_REF] Demirskyi | Hot-spots generation, exaggerated grain growth and mechanical performance of silicon carbide bulks consolidated by flash spark plasma sintering[END_REF][START_REF] Marinel | Sintering of CuO and ZnO in a Single Mode Microwave Cavity with Shrinkage Control[END_REF][START_REF] Dong | On the Hotspot Problem in Flash Sintering[END_REF]. Materials having a dissipative parameter which increases with temperature (like for negative temperature coefficient resistivity NTC materials) tends to experience hot spots in direct heating [START_REF] Manière | Inherent heating instability of direct microwave sintering process: Sample analysis for porous 3Y-ZrO2[END_REF]. This is due to the surfaces cooling implying higher center specimen temperature. In addition, for NTC materials like zirconia, the higher the temperature rises, the greater is the heat dissipation [START_REF] Yakovlev | Computational Study of Microwave Processing of thermal runaway in microwave processing Zirconia[END_REF]. Consequently, the thermal instabilities originated from the surfaces cooling are amplified by the NTC behavior of some materials, ending with hot spot formation and thermal runaway [START_REF] Manière | Fully coupled electromagnetic-thermal-mechanical comparative simulation of direct vs hybrid microwave sintering of 3Y-ZrO2[END_REF]. This problem is particularly present in flash sintering. In addition to previous cited thermal inhomogeneity problems, this new field assisted and ultra-rapid sintering approach experiments abrupt sintering and thermal profiles which end up with inherent instabilities [START_REF] Todd | Electrical characteristics of flash sintering: thermal runaway of Joule heating[END_REF][START_REF] Cologna | Flash Sintering of Nanograin Zirconia in <5 s at 850°C[END_REF][START_REF] Yu | Review of flash sintering: materials, mechanisms and modelling[END_REF][START_REF] Biesuz | Flash sintering of ceramics[END_REF][START_REF] Zhang | Thermal runaway, flash sintering and asymmetrical microstructural development of ZnO and ZnO-Bi2O3 under direct currents[END_REF]. In flash sintering, the homogeneity problem is a key issue necessary for the scalability of this phenomenon [START_REF] Manière | All-Materials-Inclusive Flash Spark Plasma Sintering[END_REF]. In order to stabilize FAST processes, hybrid heating approaches are generally investigated. In microwave sintering, a susceptor (a material which strongly couples with microwaves) is generally employed to initiate the specimens heating and homogenize the temperature field by reducing the specimen natural cooling from its surface [START_REF] Oghbaei | Microwave versus conventional sintering: A review of fundamentals, advantages and applications[END_REF][START_REF] Rybakov | Microwave Sintering: Fundamentals and Modeling[END_REF].

Simulation works have been done to evaluate the direct/hybrid heating and the homogeneity of the temperature field, but these studies are generally limited to the heating and does not take into account the air convection [START_REF] Bouvard | Finite Element Modelling of Microwave Sintering[END_REF][START_REF] Manière | Fully coupled electromagnetic-thermal-mechanical comparative simulation of direct vs hybrid microwave sintering of 3Y-ZrO2[END_REF][START_REF] Thuault | Microwave sintering of large size pieces with complex shape[END_REF][START_REF] Heuguet | Effects of the Susceptor Dielectric Properties on the Microwave Sintering of Alumina[END_REF][START_REF] Santos | Electromagnetic and thermal history during microwave heating[END_REF][START_REF] Santos | Microwave processing of porcelain tableware using a multiple generator configuration[END_REF][START_REF] Manière | Microwave sintering of complex shapes: From multiphysics simulation to improvements of process scalability[END_REF].

All these FAST processes need to be stabilized through a better understanding of their cooling mechanisms and the help of a predictive simulation tool. The finite element simulation is generally employed to detect the dominant heat transfer parameter and predict optimization configurations. However, the high Multiphysics nature of FAST process makes this task difficult. For instance, a comprehensive microwave sintering model requires implementing an electromagnetic part, for the microwave distribution in the cavity, thermal/fluid dynamic parts for the heat transfer by conduction/radiation/convection [START_REF] Manière | Fluid dynamics thermo-mechanical simulation of sintering: Uniformity of temperature and density distributions[END_REF] and a mechanical part, for the sintering and a numerical PID to regulate the system [START_REF] Manière | Fully coupled electromagnetic-thermal-mechanical comparative simulation of direct vs hybrid microwave sintering of 3Y-ZrO2[END_REF]. All these physics have parameters coupled to each other and represent a challenge of computation stability and time.

In this context, we develop a modeling approach called "ETC", standing for "Equivalent Thermal Cavity", consisting of a cavity where it is simple to investigate the impact of the cooling fluxes and different parameters (dimensions, heating rates, susceptors) on the heating homogeneity. This work focuses on the hot spot formation during the direct/hybrid heating of a regular shape specimen and the study of the contribution of the radiative/convective cooling fluxes. In the following, we will describe first the finite element ETC model and how the different heat fluxes are calculated. Then, the results of the ETC model for the different parameters will be presented and discussed.

Theory and calculations

The ETC model governing equations is presented first, and is followed by the description of the main boundary conditions.

Heat transfer in the solid and gas

Heat transfer is different whether if considered in fluid or solid. In solid parts, no convection phenomenon is possible, then heat is classically described by [START_REF] Anselmi-Tamburini | Fundamental investigations on the spark plasma sintering/synthesis process[END_REF]:

𝜌𝐶 𝑝 𝜕𝑇 𝜕𝑡 + ∇. (-𝜅∇T) = 𝑄 (1)
.

In the fluid part like the gas area in a furnace, two additional heat transfers should be taken into account, the surface to surface thermal radiation which is here considered by boundary conditions (see next section) and the natural convection of the air which requires additional fluid dynamic equations. The convection is a key point for heat transfer problems like a furnace and heating system because the convective fluxes can be a source of intense cooling [START_REF] Egorov | Role of convective heat removal and electromagnetic field structure in the microwave heating of materials[END_REF] or of homogenization when heating elements (or susceptors) are used [START_REF] Manière | Fluid dynamics thermo-mechanical simulation of sintering: Uniformity of temperature and density distributions[END_REF]. The Navier-Stokes convective problem is employed and is defined by the following three main governing equations [START_REF] Bernardi | Couplage des équations de Navier-Stokes et de la chaleur : le modèle et son approximation par éléments finis[END_REF]:

the mass conservation equation:

.

Boundary conditions

A scheme of the "Equivalent Thermal Cavity" (ETC) is reported in figure 1a. The ETC model is an insulation box cavity where a cylindrical zirconia specimen and eventual susceptor are placed. We consider the 2D axisymmetric approximation to reduce the computation time.

The external insulation box boundaries are subjected to convective and surface to ambient cooling fluxes described by the equations:

𝜑 𝑟𝑠𝑎 = 𝜎 𝑠 ϵ(𝑇 𝑎𝑖𝑟 4 -𝑇 4 ) (5)
𝜑 𝑐𝑠𝑎 = ℎ 𝑖𝑎 (𝑇 𝑎𝑖𝑟 -𝑇) (6)
. The inner solid/air interfaces are subjected to the fluid dynamics part where "wall" no penetration conditions are used and to the mutual surface-to-surface radiation which considers each material emissivity. The surface to surface radiation can be defined by the total outgoing radiative flux (called radiosity) J which includes the irradiated power 𝑒 𝑏 (𝑇) and the incoming irradiation G:

𝐽 = 𝜌 𝑟 𝐺 + ϵ𝑒 𝑏 (𝑇) = 𝜌 𝑟 𝐺 + ϵ𝑛 2 𝜎 𝑠 𝑇 4 (7)
by the ideal gray body simplification, we have:

𝜖 = 1 -𝜌 𝑟 (8)
the expression of the net inward radiative heat flux 𝜑 𝑟𝑠𝑠 is:

𝜑 𝑟𝑠𝑠 = ϵ(𝐺 -𝑒 𝑏 (𝑇)) (9)
.

For this study, we use a triangular mesh with refinement at the interfaces (figure 1b).

Classically, the temperature is measured by a pyrometer on the top of the specimen through the upper hole of the insulation box. In the model, the temperature upper specimen temperature is regulated through a proportional integral derivative (PID) numerical regulator

Experiment and method

The "Equivalent Thermal Cavity" (ETC) is used to simulate the impact of the main heat transfer mechanism in the formation of hot spot and heating instabilities for the FAST sintering approaches. In our method, we do not model the fields or current distribution in the material which are assimilated to a uniform dissipation term which is PID regulated (the term 𝑄 𝑃𝐼𝐷 ). Instead, we focus on the different radiative/convective heat fluxes in the cavity that we suspect to be mainly responsible for the hot spot formation and the heating heterogeneity. The ETC configuration (figure 1a) has the potential to approximate the heating homogeneity of most of the FAST approach by adjusting the cavity dimensions to the equivalent specimen, insulation, heating element and air volumes. In this work, we will test the microwave heating response of a 50 % dense zirconia specimen placed in an insulation box cavity. The reference thermal cycle of the simulation work is a 10 K/min heating ramp from room temperature to 1000 °C. Indeed, the heating rate value of 10 K/min is considered as fast for conventional sintering and lower range for field assisted sintering. For simplicity, the simulation study does not consider sintering starting at 1000 °C, the thermal simulations are then stopped at this temperature. For direct microwave heating, the microwave thermal dissipation in the zirconia sample is modeled by 𝑄 𝑃𝐼𝐷 . For hybrid heating, a silicon carbide (SiC) susceptor is generally added next to the specimen. The susceptor couples more with microwaves than the zirconia sample; in our case it will be modeled by a dissipation term 𝑄 𝑃𝐼𝐷 . 𝐹𝑐 , where Fc is a multiplicative factor which increase or reduce the dissipation in the susceptor. With the help of the ETC model, we will test the impact of the sample dimension, heating rate and hybrid heating on the heating stability of the zirconia specimen. The temperature dependence properties of zirconia, silicon carbide and the insulation material can be found in our previous work in ref [START_REF] Manière | Fully coupled electromagnetic-thermal-mechanical comparative simulation of direct vs hybrid microwave sintering of 3Y-ZrO2[END_REF][START_REF] Manière | Microwave sintering of complex shapes: From multiphysics simulation to improvements of process scalability[END_REF].

We report in figure 2 an initial case of 10 K/min direct heating where it can be seen the temperature distribution (figure 1a) and the convective velocity field (figure 1b). In order to estimate the relative magnitude of the cooling fluxes of the zirconia sample, we calculate the integral of the radiative and convective fluxes in the solid/air interfaces and the integral of the zirconia/insulation interface conductive flux (see figure 1c). This gives the curves reported in figure 1d in watts. Using these curves, it is possible to determine the ratio of each of these three fluxes contribution versus temperature, giving the graph reported in figure 1e. ) to obtain 50% dense 12 mm green pellets after die pressing (with a hydraulic press "Specac GS1501", under an applied pressure of 1 ton).

Results and discussions

In this section is presented, the outcome of the ETC simulation in different direct and hybrid heating configurations.

Direct heating cooling fluxes

The initial direct heating model presented in figure 2 This phenomenon partially stabilizes the heating and explains why the convection contribution is only 30%. This mechanism also explains why convective cooling fluxes can be high when the specimen is placed in a large cavity [START_REF] Egorov | Role of convective heat removal and electromagnetic field structure in the microwave heating of materials[END_REF]. To check the impact of the surface to surface radiation on the hot spot formation, we simulate the direct heating case, reported in figure 1, without the convection in the gas. The ETC cavity is assumed to be in vacuum with pure radiation in the inner surfaces, the results are reported in figure 3. The temperature field shows that the thermal radiation from the specimen raises the temperature of the adjacent insulation box inner surfaces. The proportion of the specimen radiative cooling flux is 90% at the end of the cycle. A hot spot is still observed in the specimen at the end of the cycle, with a temperature difference Tmax-Tmin = 1008 K which is closed to the previous case with convection. In direct heating, the thermal inhomogeneity is high and originates from an important cooling at the sample surfaces by radiation and/or convection. In the following, we will try two approaches to stabilize the direct heating inhomogeneity. One by reducing the air volume in the cavity (reduction of convective cooling flux) and another by delaying the hot spot formation using high heating rates.

Impact of dimension and cavity filling in the heating stabilization

Increasing the dimensions of the specimen in the reference configuration figure 2a decreases the air volume in the cavity. We expect that this parameter reduces the cooling fluxes by convection and also by radiation as the insulator surface becomes very close to the specimen.

The ETC results are reported in figure 4 where the cavity filling ratio is increased from the reference configuration (11%) to 87%. In figure 5 are reported, the specimen temperature curves corresponding to the reference configuration (such as in figure 2) and the graph of the specimen temperature difference and average cavity material temperatures. First of all, we can see from the simulated temperature field (figure 4a) and in (figure 5b) that the specimen temperature difference (Tmax-Tmin) varies from 1000 K to almost 400 K when the cavity filling is increased to more than 80%. In figure 4a specimen and the insulation box leads to an increase of both the insulation box inner surface temperature and air temperature (figure 4a and figure 5b). These two temperatures tend to be very close to the sample temperature when increasing the cavity filling ratio. The radiative fluxes are then reduced for high cavity filling ratio as the insulation box inner surface irradiates a significant heat back to the specimen.

Increasing the cavity filling ratio contribute to a significant reduction of the convective/radiative cooling fluxes which improves the specimen homogeneity of 60% (in terms of temperature difference, Tmax-Tmin) compared to the reference direct heating configuration.

Heating rate stabilization effect

With higher heating rates, an improved temperature stabilization is expected by delaying the hot-spot formation. As hot spots originate from the cooling of the surfaces of the specimen, it is easy to figure out that the reduction of the heating time will end up with a lower total heat loss energy in the sample surfaces, and then, in the homogenization of the temperatures. We test here different heating rates from the reference direct heating case (10K/min figure 2) to 200 K/min. The results are reported in figure 6 and figure 7. The simulated temperature field in figure 6a and the curves in figure 7 shows that an increase of the heating rate decreases the air and insulation material temperature and homogenizes the specimen temperature where Tmax-Tmin goes from 1000 K for 10 K/min to 800 K for 200 K/min. This homogenization effect is weaker than the one previously discussed (cavity filling effect); however, the effect of the heating rate is not negligible.

The homogenization effect is not explained by the relative heat flux diagrams figure 6b but by the heat loss energy on the sample surfaces (see figure 6c). Compared to the previous case (cooling flux comparison), the present case on heating rates should take into account the cooling time. We then choose to compare the cooling flux energy (in W.h) which represents the consumed energy through the specimen surfaces. These curves clearly indicate a high reduction of the total energy loss from the surface for high heating rates. The thermal heterogeneity remains high because we consider a low cavity filling configuration (11%). In this configuration, the cooling flux increases because the air and insulation temperatures remain at low temperatures for high heating rate (figure 6a and figure 7a). The weak homogenization for high heating rates, results from the delay of the hot spot formation (a time factor). As showed in figure 7b, it is possible to couple the high cavity filling case (87 %) with high heating rate (200 K/min). This coupling reduces the convective/radiative heat loss while delay the gradient formation which homogenize the system. The temperature difference in the sample is decreased of 63 % which is slightly better than the pure effect of high cavity filling ratio (figure 5b).

For direct heating, the cavity filling parameter can improve the sample homogeneity of 60%, high heating rate of 20% and both parameters of 63%. The results figure 7b can be compared to Biesuz et al [START_REF] Biesuz | Thermally-insulated flash sintering[END_REF] study on "thermally insulated flash sintering", a configuration which combines high filling ratio (thermal insulation) and very high heating rates (>1000 K/min).

They showed this configuration is more energy efficient and enhance densification while having a relatively homogeneous microstructure. Indeed, attaining stable high heating rate systems (like in the configuration figure 7b) is the main targeted challenge to favor interesting materials properties via methods like: two steps sintering [START_REF] Chen | Sintering dense nanocrystalline ceramics without final-stage grain growth[END_REF], microwave sintering [START_REF] Rybakov | Microwave Sintering: Fundamentals and Modeling[END_REF], flash sintering [START_REF] Biesuz | Flash sintering of ceramics[END_REF], etc. Nevertheless, temperature differences (in figure 7a) of a few 100 K are still present due to the cooling through the insulation box and may disturb the sintering and generates distortions [START_REF] Manière | Fully coupled electromagnetic-thermal-mechanical comparative simulation of direct vs hybrid microwave sintering of 3Y-ZrO2[END_REF]. To efficiently homogenize the temperatures, a hybrid heating configuration is generally employed; this configuration is studied in the next section.

Hybrid heating

As shown in figure 1a, the susceptor is a ring of silicon carbide as it is usually employed in microwave sintering [START_REF] Heuguet | Effects of the Susceptor Dielectric Properties on the Microwave Sintering of Alumina[END_REF]. The heating of the zirconia specimen is imposed by a 𝑄 𝑃𝐼𝐷 heat source, which regulates the upper specimen temperature. Hybrid heating considers an additional heat source term 𝑄 𝑃𝐼𝐷 . 𝐹𝑐 in the susceptor. We will vary the multiplicative factor Fc from 1 to 10. The higher Fc is, the higher is the heat dissipation in the susceptor and lower is the relative dissipation in the zirconia specimen. For high Fc, the specimen is essentially heated by the susceptor, which tends to conventional heating. The results of simulated hybrid temperature field, heat fluxes and temperature curves are reported in figure 8, figure 9 and figure 10, respectively.

The simulation in figure 8 shows the circular convective flux motion is inverted compared to direct heating (figure 2) and corresponds to a convection motion that brings the energy from the susceptor ring to the zirconia specimen. The average temperature in figure 10 indicates that an increase of Fc raises the tooling and air temperatures. Therefore, the temperature distribution in the zirconia specimen is homogenized. At the end of the cycle, it must be noted for Fc=1X that a hot spot was developed (figure 8). This is due to the low external heat coming from the susceptor (not enough to homogenize the hot spot). The specimen temperature is regulated by the upper surface (see scheme in figure 5). However, if we compare the sample temperature distribution, in the beginning (600 s) and for high Fc, T PID overestimates the specimen temperatures (external heating) while in the end (6000 s) and for low Fc, T PID underestimates the specimen temperatures (hot spot). The temperature difference in the sample is significantly reduced for Fc higher than 4 (figure 10). Compared to the direct case, the homogenization effect of hybrid heating is of 90% which is far higher than the impact of the cavity filling and heating rates in direct heating. The analysis of the specimen surfaces fluxes in figure 9a reveals, the low Fc is essentially governed by cooling fluxes (negative). In the transition (Fc=4X) and for high Fc, the radiative and convective fluxes stat to become positive meaning the specimen gains heat from the susceptor (external heating).

The hybrid case at Fc=4X has a well-balanced heat flux between the heat produced in the specimen and the heat coming from the external part (susceptor). The transition of heating/cooling fluxes for Fc=4X generates the radiative/convective distribution obtained in figure 9b. In this case, at the beginning of the heating (600 s in figure 8), the heating is external but not excessively heterogeneous, which is interesting to allow an easy debinding of the polymeric phase usually present in green specimens. If the temperature is high in the center of the specimen, like in direct configuration (figure 2a), the debinding could make the specimen swells generating cracks. In hybrid configuration, at the end of the cycle (still for Fc=4X at 6000 s), the heating is homogeneous and the temperature differences do not exceed 100 K, which is required for an homogeneous sintering. The 'Tmax-Tmin' curve (figure 10) reveals that this temperature difference slightly increases for Fc value higher than 4X. This is due to the conductive cooling through the support which is not compensated by the specimen volume dissipation like in the Fc=4X case.

This study shows that a pure external heating can be less homogeneous than hybrid heating, even if conventional heating is anyway more homogeneous than pure direct heating. This characteristic is interesting as it justifies the interest of the Microwaves Assisted Technologies (MAT) which inject microwaves to assist the conventional sintering in furnaces [START_REF] Oghbaei | Microwave versus conventional sintering: A review of fundamentals, advantages and applications[END_REF][START_REF] Wroe | Evidence for a non-thermal microwave effect in the sintering of partially stabilized zirconia[END_REF]. This simulation reveals the ideal hybrid heating case is a situation where the susceptor heating behavior follows the specimen heating with a slight advance. To approach this ideal case experimentally, it is required to adjust the thermal dissipation in the susceptor by composites/multimaterials approaches or by the addition of an external heating element. The ETC model developed here will be of great help to conduct this adjustment study. In this section, we test experimentally the different hybrid and direct heating configurations.

The aim is to observe the impact of the heating mode on the sintering of a zirconia sample, especially its homogeneity. Three microwaves sintering configurations were tested to study direct and hybrid heating stability. One is similar to the reference configuration (figure 2)

where the zirconia specimen is directly heated in a large air volume (where convection cooling can easily develop). As suggested by section 4.2, the heating can be significantly reduced for high cavity filling values (corresponding to a configuration close to thermal insulation). We test this "insulated direct" configuration by surrounding the sample with insulation material powder. Finally, we try the hybrid heating configuration using two silicon carbide plates. We choose a heating cycle of 40 K/min to 800 °C and 10 K/min for the sintering zone up to 1300 °C. The results are reported in figure 11. The "direct" sintering experiment which undergoes ~ 1000 K of temperature difference in the simulation (figure 2a) leads to the rapid destruction of the specimen (experimentally). As suggested in figure 4a, the "insulated direct" configuration is much more stable even if a small concave shape can be seen on the specimen surface indicating a higher temperature in the specimen center. As expected the "hybrid" configuration shows the most homogeneous results.

Conclusion

Heating modes have a key impact on the homogeneity of the field assisted sintering (FAST)

processes. We focus our approach on the simulation of direct and hybrid heating of a zirconia green specimen in an insulation box cavity, a classical case in microwave sintering. Our simulation work aims to study the impact of the surface to surface thermal radiation and the natural convection in the cavity on the homogeneity of the specimen heating. These parameters are classically approximated by analytical relations that highly idealize the complexity of the cavity heat exchanges. In the present simulations, the radiative fluxes take into account the radiative contribution coming from all surrounding surfaces in the inner cavity. The convective flux is also very important as it homogenizes the heating in all the cavity areas filled with air. We determine by our analysis that the specimen heating is influenced by conduction loss flux on the support which represents about 10% and convection/radiation exchanges flux in the inner cavity which represents about 30% and 60% respectively. In direct heating configuration, the thermal inhomogeneity generated by theses cooling fluxes induces the formation of a hot spot in the specimen center with temperature differences as high as 1000 K (from the center to the corner).

The parametric simulation analyses reveal that these heterogeneities can be reduced by three approaches. Lowering the air volume in the cavity is an efficient way to reduce the convective motion while raising the adjacent tooling temperatures which decreases 60% of the initial heterogeneity. High heating rates as an interesting potential to delay the hot spot formation and homogenize the temperatures, but this decreases only by 20% the initial heterogeneity.

Hybrid heating is the most efficient way to homogenize up to 90% of the temperatures initial heterogeneity. The success of this approach lies in the heating balance between the heat generated in the sample volume and the external heat provided by the susceptor through radiative/convective fluxes. The microwave sintering experiments performed in similar direct/hybrid configurations confirm the presence of an important hot spot in direct heating and homogeneous sintering in insulated direct and hybrid configurations.

The optimum heating configurations have been determined to be (i) a coupled conditions of high thermal insulation and high heating rate, for direct heating, and (ii) a hybrid configuration where the susceptor is adjusted to follow the specimen heating. The "Equivalent Thermal Cavity" (ETC) simulation developed in this study in an efficient way to study the heating stability of the highly varied FAST sintering approaches, where thermal gradient problems are often limiting the advanced material properties advantages. Figure 11 Direct, thermally insulated and hybrid microwave heating of 12 mm green zirconia specimen; the heating cycle imposed to all specimen surfaces is reported on the left.

Figure captions

3 𝜇

 3 where appears the pressure, viscous and body forces terms: ∇𝒖) = -∇𝑝 + ∇. (𝜇(∇𝐮 + (∇𝐮) 𝑇 ) -2 (∇. 𝐮)𝕚) + 𝑭 𝒈 (3) and the energy balance equation which is the equivalent of the heat equation in fluid and takes into account the convective term and the additional pressure 𝑄 𝑝 and viscous dissipation 𝑄 𝑣𝑑 heat source term which are often very low and neglected for our application: . ∇)𝑇) + ∇. (-𝜅∇T) = 𝑄 + 𝑄 𝑝 + 𝑄 𝑣𝑑

Figure 1 a

 1 Figure 1 a) Scheme of the equivalent thermal cavity with the values of the boundary parameters, b) mesh.

  value of the specimen heat source term 𝑄 𝑃𝐼𝐷 to follow a constant heating rate of 10 K/min. The PID expression is as follows.𝑄 𝑃𝐼𝐷 = 𝐾 𝑝 𝑒(𝑡) + 𝐾 𝐼 ∫ 𝑒(

Figure 2

 2 Figure 2 Direct 10 K/min heating of a zirconia specimen; a) temperature and b) convection velocity field at the end of the cycle; c) scheme of the specimen interfaces considered in the specimen cooling fluxes analysis, d) integral of the specimen cooling fluxes through the specimen surfaces, e) cooling fluxes ratio diagram.

  leads to the formation of a hot spot into the zirconia sample. This result is consistent with temperature differences (Tmax-Tmin) as high as ~ 1000 K (figure2a). It is clear from figures 2a and figures 2e that the solid/air interface represents the highest cooling fluxes with a contribution of about 60% of radiation and 30% of convection. The conductive flux from the sample to the support is low, explaining why the hot spot is near this interface. The cooling convection fluxes homogenize the heat in the cavity through gas convection velocities between 0.1-0.3 m/s. The convection fluxes follow a circular motion where a fraction of the heat loss from the sample is coming back.

Figure 3

 3 Figure 3 Direct 10 k/min heating configuration in vacuum (without convection), a) simulated temperature field at the end of the cycle and b) the simulated specimen cooling fluxes diagram.

Figure 4

 4 Figure 4 Simulation of the ETC configuration for different cavity filling ratio, a) simulated temperatures at the end of the cycle, b) simulated specimen cooling fluxes diagram and c) the cooling fluxes curves.

Figure 5 a

 5 Figure 5 a) Simulated maximum, minimum and PID regulated specimen temperatures for the reference direct heating configuration (11% cavity filling ratio), b) simulated specimen maximum temperature difference and average specimen, air, insulation temperatures.

  , the convection velocity arrows almost disappear for high filling ratio. The 20% convective flux in figure 4b is essentially due to air conduction. Nevertheless, we should keep in mind that the diagrams in figure 4b just indicate the relative proportion of the specimen cooling fluxes. In order to explain the stabilization effect of the cavity filling ratio, we should compare directly the cooling fluxes curves reported in figure 4c. These fluxes curves indicate a clear decrease of the radiative and convective fluxes when increasing the cavity filling ratio. The convection fluxes are decreased because the air cavity space becomes very thin for high cavity filling ratio and the circular convection fluxes cannot be developed. On the other hand, the thin air distance between the

Figure 6

 6 Figure 6 Simulation of the ETC configuration for different heating rate, a) simulated temperatures at the end of the cycle, b) simulated specimen cooling fluxes diagram and c) boundary heat transfer energy curves.

Figure 7 a

 7 Figure 7 a) Simulated specimen maximum temperature difference and average specimen, air, insulation temperatures for different heating rates, b) simulated temperature field corresponding to high heating rate (200 K/min) and high cavity filling ratio (87 %).

Figure 8 Figure 9 a

 89 Figure 8 Simulated temperature field for hybrid heating ETC configurations using a volumetric heat dissipation in the susceptor ring multiplied by a factor of 1×, 4× and 10× the dissipation of the specimen.

Figure 10 20 4. 5 .

 10205 Figure 10 Simulated specimen maximum temperature difference and average specimen, air, insulation temperatures for different susceptor dissipation factor.

Figure 11

 11 Figure 11 Direct, thermally insulated and hybrid microwave heating of 12 mm green zirconia specimen; the heating cycle imposed to all specimen surfaces is reported on the left.

Figure 1 a

 1 Figure 1 a) Scheme of the equivalent thermal cavity with the values of the boundary

Figure 2

 2 Figure 2 Direct 10 K/min heating of a zirconia specimen; a) temperature and b) convection

Figure 3

 3 Figure 3 Direct 10 k/min heating configuration in vacuum (without convection), a) simulated

Figure 4

 4 Figure 4 Simulation of the ETC configuration for different cavity filling ratio, a) simulated

Figure 5 a

 5 Figure 5 a) Simulated maximum, minimum and PID regulated specimen temperatures for the

Figure 6

 6 Figure 6 Simulation of the ETC configuration for different heating rate, a) simulated

Figure 7 a

 7 Figure 7 a) Simulated specimen maximum temperature difference and average specimen, air,

Figure 8

 8 Figure 8 Simulated temperature field for hybrid heating ETC configurations using a

Figure 9 a

 9 Figure 9 a) Boundary cooling fluxes curves and b) fluxes diagrams, for 1×, 4× and 10× hybrid

Figure 10

 10 Figure 10 Simulated specimen maximum temperature difference and average specimen, air,

Acknowledgements

The help and support of Christelle Bilot and Anthony Bilot is gratefully acknowledged.