# Spark plasma sintering of UO2 nanopowders: pressure, heating rate and current effects

Alexandre Margueret, Luca Balice, Karin Popa, Michael Holzhäuser, Emanuele de Bona, Walter Bonani, Antonio Bulgheroni, Fabienne Audubert, Marco Cologna

► To cite this version:

Alexandre Margueret, Luca Balice, Karin Popa, Michael Holzhäuser, Emanuele de Bona, et al. Spark plasma sintering of UO2 nanopowders: pressure, heating rate and current effects. Journal of the European Ceramic Society, 2022, 42, pp.6056-6066. 10.1016/j.jeurceramsoc.2022.05.070. cea-05117634

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# Spark Plasma Sintering of UO₂ Nanopowders: Pressure, Heating Rate and Current Effects

Alexandre Margueret¹, Luca Balice¹,², Karin Popa³, Michael Holzhäuser³, Emanuele De Bona³, Walter Bonani³, Antonio Bulgheroni³, Fabienne Audubert¹, Marco Cologna*³

1. CEA, DES, CAD, IRESNE, DEC, F-13108, Saint-Paul-lez-Durance Cedex, France

2. Université Grenoble Alpes, 621 Avenue Centrale, 38400, Saint-Martin-d'Hères, France

3. European Commission, Joint Research Centre (JRC), Karlsruhe, Germany

## Abstract

We analysed with different methods the densification of UO₂ nanopowders in SPS under constant heating rate (CHR) and isothermal sintering conditions. The apparent activation energy of densification in SPS (75 kJ/mol with CHR method) is significantly smaller than in conventional sintering. It is shown that this is likely not an effect of the applied current.

We also observed a threshold stress at 64 MPa for the transition from pressure-insensitive sintering (stress exponent $n \approx 0$) to pressure-assisted sintering, suggesting that the contribution of the capillary stresses in such nanopowders is comparable with the typical stress applied in SPS.

## Keywords

Spark Plasma Sintering (SPS); Electric Field Assisted Sintering (EFAST); UO₂; nanopowders; sintering mechanisms; CHR; isothermal; MSC.

## 1. Introduction

Uranium dioxide (UO₂) is the main nuclear fuel used in commercial reactors, thanks to its stability under irradiation, non-volatility, excellent corrosion resistance in water and satisfactory compatibility with cladding material [1] [2] [3]. The typical microstructure of a reactor fuel pellet (95% of the theoretical density, TD) shows grains up to several micrometres size. During operation, both the pellets chemistry and microstructure change, especially in the rim zone, where the so-called high burn-up structure (HBS) can form. The HBS is characterised by a decrease of the grain size (from 10 µm to few hundreds nanometres) and an increase of total porosity [4] [5] [6].

Spark plasma sintering (SPS) is a field assisted sintering technology (FAST), where an electric current is coupled with external pressure to enhance the sintering kinetics. This process allows sintering UO₂ at a much lower temperatures and shorter times than conventional sintering methods. High heating rates (up to 1000°C/min) can be achieved thanks to the direct heating of the sample and/or graphite dies by Joule effect [7]. These features allow to limit grain growth and to densify nanostructured pellets, enabling the access to important information also for nuclear materials [8], such as the determination of mechanical properties, grain growth kinetics, oxidation behaviour or radiation damage as a function of the grain size, down to the nanodomain. [4], [9], [10], [11], [12], [13].

Other applications consist in the incorporation of second phases with higher thermal conductivity than UO₂, for the development of enhanced accident tolerant fuels (EATF) [14], [15], [16], [17], [18], [19] or simulated irradiated fuels (also called Simfuels) containing non-radioactive volatile fission products to better investigate their behaviour [20], [21], [22].

The effects of the current on the matter during sintering or on other diffusion controlled processes are still controversial [7], [23], [24], [25], [26], [27], [28], [29], [30]. The densification mechanism of UO₂ during
SPS was addressed by two recent studies; in both cases the apparent activation energy for densification in SPS (96 - 140 kJ/mol [31], [32]) is significantly smaller than the values typically reported for conventional sintering (250 - 450 kJ/mol) [33], [34], [35], [36]. The reason for a low apparent activation energy was not clearly identified; however, it was suggested that the electrical field could be partially responsible.

In this work, we studied the sintering behaviour of nanocrystalline $UO_2$ powder with an average crystallite size of 17 nm, obtained from hydrothermal decomposition of U(IV) oxalate [37], [38], [39]. In particular, we investigated the influence of pressure, temperature and heating rate on the sintering behaviour, and the role of the electrical current in SPS. The densification curves were analysed to obtain sintering parameters such as the apparent activation energy $Q_{act}$, the stress exponent $n$, and the grain size exponent $m$. In order to assess the effect of the particular experimental condition and the specific analytical model used, two types of sintering regimes (isothermal and constant heating rate) were taken into consideration, and four different analytical methods were applied. In details, densification curves were analysed either by comparing data at the same density level, or relying on explicit equations for effective stress ($\sigma_{eff}$) and effective shear modulus ($\mu_{eff}$). We performed also sintering experiments at constant heating rate with a boron nitride coating applied at the graphite punch-sample interface, with the aim of electrically insulating the sintering body from the current flow.

## 1.1. *Description of the methods to determine the densification parameters*

Two types of experiments were performed: constant heating rate (CHR) sintering and isothermal sintering. In CHR experiments, the densification is analysed during the heating phase, and different runs are performed varying either the heating rate or the applied pressure. In isothermal sintering experiments, the densification is analysed during the dwell phase at constant temperature, and the dwell temperature is varied at each run.

The characteristic parameters for densification ($Q_{act}$, $n$, $m$) were obtained by analysing the experimental data with four different methods: (i) constant heating rate analysis, (ii) master sintering curve analysis, (iii) isothermal analysis, and (iv) solution of the sintering equation with an algorithm based on the generalized reduced gradient (GRG) method. The four methods and their main features are summarised in Table 1 and will be described more in detail below.

<table><caption>**Table 1.** Overview of the methods employed to analyse the sintering curves and main features.</caption><thead><tr><th>Method</th><th>Set of experiments</th><th>Parameters determined</th><th>Advantages</th><th>Drawbacks</th></tr></thead><tbody><tr><td>CHR analysis</td><td>Varying heating rates and pressures</td><td>Qact, n</td><td>- allows extracting Qact and n for each density</td><td>- need to compute the derivatives</td></tr><tr><td>MSC analysis</td><td>Varying heating rates</td><td>Qact</td><td>- allows predictions for arbitrary time/temperatures profiles</td><td>- does not allow to compute Qact for each density (global point of view)</td></tr><tr><td>Isothermal analysis</td><td>Varying dwell temperatures</td><td>Qact, n</td><td>- few experiments needed</td><td>- stronger assumption (no grain growth)<br/>- need to compute σeff and μeff</td></tr><tr><td>GRG method</td><td>Varying heating rates or dwell temperatures</td><td>Qact, n, m</td><td>- allows to extract the three parameters</td><td>- many experiments needed (sintering trajectory)<br/>- more complex and multiple data analysis steps<br/>- does not allow to compute the parameters for each density<br/>- need to compute σeff and μeff</td></tr></tbody></table>
The methods apply the equation of steady-state creep for solid bodies [40] to SPS at intermediate sintering stage, which requires the following modifications [40], [41]:

- since the applied load is uniaxial and the lateral shrinkage is negligible, the strain rate is considered equivalent to the densification rate divided by the density.

- since creep refers to solid bodies, a correction is needed to take into account the porosity and the surface curvature, typical of porous powders systems. The stress intensification factor, $\phi$, accounts for the porosity (Eq. 1):

$$ \frac{1}{\rho} \frac{d\rho}{dt} = -\dot{\epsilon} = \frac{HD}{kTG^m}(\phi\sigma)^n \quad (1) $$

In Eq. 1, $\rho$ is the density, $t$ the time, $\epsilon$ the linear shrinkage, $H$ a constant, $D$ the diffusion coefficient, $k$ the Boltzmann constant, $T$ the temperature, $G$ the grain size, $\phi$ the stress intensification factor (function only of the density), $\sigma$ the applied stress. Finally, $m$ is the grain size exponent and $n$ the stress exponent and both depend on the densification mechanism. The diffusion coefficient is defined in Eq. 2, where $D_0$ is the pre-exponential factor, $Q_{act}$ the activation energy, $R$ the universal gas constant and $T$ the absolute temperature.

$$ D = D_0 \exp\left(-\frac{Q_{act}}{RT}\right) \quad (2) $$

### 1.1.1. Constant heating rate (CHR) analysis

This method is an adaptation of the classical Wang and Raj model [42] to pressure-assisted sintering, as presented in detail elsewhere [31]. To do so, Eq. 1 is rearranged after taking the logarithm on both sides:

$$ \ln\left(\frac{1}{\rho} \frac{d\rho}{dT} \frac{dT}{dt} T\right) = \ln\left(\frac{HD_0}{kG^m}\right) + n\ln(\phi) + n\ln(\sigma) - \frac{Q_{act}}{RT} \quad (3) $$

Here two additional assumptions are made, which should be fulfilled at least in the density interval considered for the analysis: the grain size is a function of the density only and the stress exponent $n$ is constant at the same density. If we compare data at the same density, we can group the constant terms and obtain Eq. 4.

$$ \ln\left(\frac{1}{\rho} \frac{d\rho}{dT} \frac{dT}{dt} T\right) = C_0 - \frac{Q_{act}}{RT} \quad (4) $$

Data points at the same density and different temperatures can be obtained by comparing experiments at different heating rates. The apparent activation energy was calculated by plotting the left-hand side of the equation versus 1/T. The stress exponent $n$ can be calculated in a similar way, by rearranging Eq. 1:

$$ \left[\ln\left(\frac{1}{\rho} \frac{d\rho}{dT} \frac{dT}{dt} T\right) + \frac{Q_{act}}{RT}\right] = C_1 + n\ln(\sigma) \quad (5) $$

To determine $n$, only one heating rate is used, and data at the same density are obtained by conducting experiments at different pressures.

### 1.1.2. Master sintering curve
The MSC analysis was also originally developed for pressureless sintering [43] and later applied to pressure-assisted sintering [32], [44]. Equation 1 is rearranged by separating the temperature-depending terms from the others, and then, by integrating them over time (Eq. 6).

$$ \frac{k}{H D_0 \sigma^n} \int_{\rho_0}^{\rho} \frac{G^m}{\rho \varphi^n} d\rho = \int_{0}^{t} \frac{1}{T} \exp(-\frac{Q_{\text{act}}}{RT}) dt \equiv \Theta(t, T) \quad (6) $$

The experimental points are fitted by using an appropriate function. Here we used a Boltzmann fitting (Eq. 7) [44].

$$ \rho = \frac{A_1 - A_2}{1 + e^{(\Theta(t,T) - \Theta_0)/d\Theta}} + A_2 \quad (7) $$

The activation energy, $Q_{\text{act}}$, is then calculated by minimising the error between the experimental data and the fitting.

### 1.1.3. Isothermal analysis

This method was described in detail by Bernard-Granger et al. [26] and Santanach et al. [45]. It uses Eq. 8, which is similar to Eq. 1 but is based on a stronger assumption, i.e. that the grain size is constant in the considered density interval (instead of being dependent on the density, but independent on the thermal history). It requires using an explicit form for $\sigma_{\text{eff}}$ (the instantaneous effective stress), and $\mu_{\text{eff}}$ (the effective shear modulus). In Eq. 8, b is the Burgers vector and K is a constant.

$$ \frac{1}{\mu_{\text{eff}}} \frac{1}{\rho} \frac{d\rho}{dt} = K \frac{\exp\left(\frac{-Q_{\text{act}}}{RT}\right)}{T} \left(\frac{b}{G}\right)^m \left(\frac{\sigma_{\text{eff}}}{\mu_{\text{eff}}}\right)^n \quad (8) $$

For an isothermal experiment and assuming constant grain size (as done typically for relative densities up to ~90% TD), by grouping all constant terms in K1, one can obtain Eq. 9:

$$ \ln\left(\frac{1}{\mu_{\text{eff}}} \frac{1}{\rho} \frac{d\rho}{dt}\right) = n \ln\left(\frac{\sigma_{\text{eff}}}{\mu_{\text{eff}}}\right) + K_1 \quad (9) $$

The stress exponent n can be calculated by plotting the left-hand side versus the right-hand side. Once n is known, and assuming it to be constant, the activation energy can be calculated from Eq. 10:

$$ \ln\left(\frac{T}{\mu_{\text{eff}}} \frac{1}{\rho} \frac{d\rho}{dt}\right) * \left(\frac{\mu_{\text{eff}}}{\sigma_{\text{eff}}}\right)^n = -\frac{Q_{\text{act}}}{RT} + K_2 \quad (10) $$

Different expressions of $\sigma_{\text{eff}}$ and $\mu_{\text{eff}}$ have been proposed [46], depending also on the powder morphology. Here we used Eq. 11 [45] for $\sigma_{\text{eff}}$ and Eq. 12 [45] for $\mu_{\text{eff}}$,

$$ \mu_{\text{eff}} = \frac{E_{\text{th}}}{2(1 + \nu_{\text{eff}})} * \frac{\rho - \rho_0}{1 - \rho_0} \quad (11) $$
$$ \sigma_{\text{eff}} = \frac{1 - \rho_0}{\rho * (\rho - \rho_0)} * \sigma \qquad (12) $$

where $E_{th}$ is the Young's modulus (228 GPa for $UO_2$ [47]), $v_{eff}$ the effective Poisson's ratio (which is also a function of the density, but often a constant value is assumed [26], for $UO_2$ 0.31 [47]), $\rho$ and $\rho_0$ are the density and the green density, respectively, and $\sigma$ is the applied macroscopic pressure.

### 1.1.4. Solution of the sintering equation with the GRG method

This method, which allows extracting directly all the sintering parameters, was described by Bernard-Granger and Guizard [48]. It is based on Eq. 13, which is directly obtained from Eq. 8:

$$ \ln\left(\frac{T}{\mu_{\text{eff}}} \frac{1}{\rho} \frac{d\rho}{dT} \frac{dT}{dt}\right) = -\frac{Q_{\text{act}}}{RT} - m \ln(G) + n \ln\left(\frac{\sigma_{\text{eff}}}{\mu_{\text{eff}}}\right) + K' \qquad (13) $$

It requires finding an equation for the grain size as a function of density by fitting the experimental data, i.e. the sintering path, also called sintering trajectory. The same assumptions made for the CHR and MSC are valid. The parameters $n$, $m$, and $Q_{act}$ and the constant $K'$ are found by fitting the two sides of Eq. 13 using an appropriate algorithm. Here we used the function Solver provided by Microsoft Office Excel, which is based on a generalized reduced gradient (GRG) algorithm [49]. The advantage of this method is that it allows extracting each of the three densification parameters, and it can use either curves at constant heating rates or isothermal ones. Here we used the CHR experiments at varying heating rates.

## 2. Experimental procedure

### 2.1. Powder preparation

The nanocrystalline $UO_2$ powder was prepared via hydrothermal decomposition of U(IV)-oxalate $U(C_2O_4)_2 \cdot nH_2O$, as reported originally by Walter et al. [38].

A 0.5 mol/L aqueous solution of U(IV)-oxalate was obtained by electrochemical reduction of a U(VI) solution ($UO_2(NO_3)_2 \cdot 6H_2O$) in nitric acid in the presence of an equimolar solution of hydrazinium hydroxide ($N_2H_4 \cdot H_2O$). The formed U(IV) was precipitated from solution by adding dropwise a 0.5 mol/L solution of oxalic acid in excess (20 %). After resting overnight, the precipitate was separated from the supernatant by centrifugation and repeatedly washed with milli-Q water, to remove nitrate ions. This separation step is crucial, since the nitrogen oxides (which would evolve under high temperature and high pressure conditions) could re-oxidise the U(IV) to U(VI) during the hydrothermal treatment.

Typically, 27 g of wet oxalate, 24 mL of milli-Q water and 1 mL hydrazinium hydroxide were inserted under constant argon flushing in a 75 ml high temperature – high pressure stainless steel autoclave equipped with thermocouple and pressure gauge (Parr Instruments, Moline, IL, US). The autoclave was heated with a custom-made aluminium heating mantle. A 4h heat treatment at 220°C was performed under static conditions, allowing the decomposition of U(IV)-oxalate to $UO_{2(+x)}$. The pressure in the autoclave rose continuously during heating and stabilized at about 150 bar one hour after that the maximum temperature of 220°C was reached. After the heat treatment, the residual pressure in the autoclave at room temperature was about 62 bar due to gases evolved during the decomposition reaction.

The resulting solid product (about 14 g in wet form) was washed with milli-Q water, ethanol and acetone and dried overnight at 80°C under argon flow. The dry powder was then thermally conditioned in a
quartz crucible for 5 h at 550°C (heating rate: 100°C/h, cooling rate: 200°C/h) under reducing atmosphere (Ar + 4 % H₂). Typically, 9.6 g of quasi-stoichiometric UO₂ with a crystallite size of 16 ± 4 nm were obtained with the process. The crystallographic structure was identified by X-rays diffraction (XRD, Rigaku Miniflex PXRD) and crystallite size was determined with Scherrer equation from peak broadening in the diffraction pattern. However, the conditioned powder was not stable and tended to re-oxidise over time even under low oxygen atmosphere in a nitrogen glovebox. Therefore, a second heat treatment under reducing atmosphere (Ar + 4 % H₂) was performed immediately before each set of SPS experiments (2 h at 600°C, heating rate: 100°C/h, cooling rate: 200°C/h, quartz crucible), in order to have always consistent initial conditions for the sintering tests. The second calcination did not seem to impact the crystallite size of the powder (17 ± 6 nm).

## 2.2. Densification by SPS

The as-calcined powder was sintered in vacuum in a SPS (FCT, GmbH, Rauenstein, Germany), enclosed in a glovebox [50]. No graphite separation foils were used, to allow for higher precision in the measurement of the pellets height during sintering, and a better control of the radial dimension. In order to maximise the reproducibility, a new graphite set-up was used for each run. About 400 mg of powder was inserted in a 6 mm diameter graphite die, manually pre-pressed at 18 MPa and inserted in the SPS chamber. The first step of all the SPS runs consisted in pressing at 81 MPa at 100°C, in order to ensure the same starting density for all the experiments.

The CHR experiments are summarised in Table 2. The final temperature was chosen to be 1600 °C, with a dwell time of 2 minutes, in order to achieve a final density close to the theoretical one and simplify the analysis of the sintering curves. The heating rate was varied between 25 and 200°C/min and the applied pressure between 21 and 78 MPa. The cooling rate and the applied pressure during cooling were the same as the heating phase. The densification curves were plotted after subtracting the thermal expansion and system compliance, determined by a blank run, i.e. a full cycle up to 1600 °C of an empty graphite matrix.

<table><caption>**Table 2.** Experimental conditions for the constant heating rate experiments. “H” stays for heating rate and “P” for pressure.</caption><thead><tr><th>Pressure (MPa)</th><th>Т₁ = 25 °C/min</th><th>Т₂ = 50 °C/min</th><th>Т₃ = 100 °C/min</th><th>Т₄ = 200 °C/min</th></tr></thead><tbody><tr><td>σ₁ = 21</td><td></td><td></td><td>P21</td><td></td></tr><tr><td>σ₂ = 35</td><td></td><td></td><td>P35</td><td></td></tr><tr><td>σ₃ = 50</td><td>H25</td><td>H50</td><td>H100 – P50</td><td>H200</td></tr><tr><td>σ₄ = 64</td><td></td><td></td><td>P64</td><td></td></tr><tr><td>σ₅ = 78</td><td></td><td></td><td>P78</td><td></td></tr></tbody></table>

The CHR experiments with 50 MPa and heating rate 25-200°C, were carried out also in a quasi-hot-pressing mode. This was achieved by insulating the piston surfaces in contact with the powder with a ~35 µm layer of sprayed BN powders (BN spray HENZE HeBoCoat® 21E). The scope was to insulate the current from the sample, forcing it to flow only through the graphite die. Boron nitride spray on graphite is commonly used to insulate the current in SPS [51]. The effectiveness of the insulating sprayed BN layer was checked in pre-tests at room temperature, by the application of tensions above 40 V with an external power supply.

The isothermal experiments were performed with 50 MPa external pressure and with a heating rate of 200°C/min up to the selected dwell temperature (560°C-960°C) and a dwell time of 60 min, except for the sample at 960°C, which had a dwell time of 15 min. Such samples are identified with an “I” (isothermal) and the respective temperature, e.g. I560.
Two additional samples were sintered without dwell, and rapidly cooled with the maximum cooling rate of the SPS by turning off the heating current, with the aim of obtaining intermediate densities. Such samples are identified with a “Q” (quenched): Q775 (heating rate 25 °C/min to 775°C) and Q800 (heating rate 200°C/min to 800°C).

## 2.3. Density measurements

The measurement of the density was rather challenging because of the small dimension of the samples, and the tendency to fragment, especially when the highest sintering temperature was used (1600°C). At such high temperature, as no graphite separation foils were used, some samples resulted attached to the pistons, further complicating the measurements. The best estimation was the geometrical density using the thickness measured on SEM images: the samples were placed with their fracture surface facing the electron beam, and the height was measured in five different positions with the microscope software. The geometrical density was then calculated from the average thickness, the die diameter (6 mm) and the sample weight. Where possible, the thickness was measured with a micrometre screw, which confirmed the thickness measured by SEM within ± 1% TD. The data used to build the sintering trajectory are reported in Table 3, with the geometrical density calculated from the SEM measurements.

For the CHR experiments, all samples reached a plateau in the sintering curve well before 1600°C. All the measured samples showed a relative density within 96 ± 2 %. Due to the uncertainties of the measurements, and the impossibility to determine the density for the few samples attached to the pistons, the final density was set to 96% at 1600°C for all samples sintered at that temperature. Using a different final density has only a marginal effect on the values of $Q_{act}$, n and m (within the given uncertainties).

## 2.4. Grain size measurement

The grain size was measured via SEM on fracture surfaces with the line intercept method without correction factor, which was possible as the fracture was predominantly intergranular [52]. For the samples with the lowest densities (up to 77% TD), because of the limited grain growth, the SEM resolution was not high enough to distinguish the single grains, therefore XRD measurements were performed as complementary analysis. The crystallite size was determined by XRD (Rigaku Miniflex 600 diffractometer) with the Scherrer method and the assumption that crystallites and grains have the same dimension in case of very limited grain growth was made. The results are presented in Table 3.

<table><caption>**Table 3.** Data points used to construct the sintering path.</caption><thead><tr><th>Sample ID</th><th>Density (% TD)</th><th>Grain Size (nm)</th></tr></thead><tbody><tr><td>Q800</td><td>49.7 ± 0.6</td><td>17 ± 5</td></tr><tr><td>I560</td><td>54.1 ± 0.5</td><td>22 ± 6</td></tr><tr><td>I620</td><td>66.1 ± 1.5</td><td>35 ± 8</td></tr><tr><td>Q775</td><td>76.9 ± 0.6</td><td>42 ± 8</td></tr><tr><td>I680</td><td>96.2 ± 2.3</td><td>305 ± 80</td></tr><tr><td>I740</td><td>96.8 ± 1.0</td><td>710 ± 240</td></tr><tr><td>I800</td><td>97.1 ± 1.6</td><td>944 ± 400</td></tr><tr><td>I960(15)</td><td>97.0 ± 2.5</td><td>974 ± 250</td></tr><tr><td>I920</td><td>97.4 ± 1.5</td><td>1080 ± 300</td></tr><tr><td>P78</td><td>99.0 ± 0.5</td><td>1402 ± 450</td></tr><tr><td>I880</td><td>97.2 ± 1.5</td><td>1424 ± 480</td></tr></tbody></table>
### 3. Results

The densification curves at different heating rates (with a pressure of 50 MPa) and the relative derivatives are shown in Fig. 1. The corresponding experiments with the pistons isolated by the BN layer are plotted with dash lines. When the heating rate increases, the sintering curves are shifted to higher temperatures and the maximal value of the derivative is decreased. Sintering starts between 525°C and 700°C and ends between 1000°C and 1325°C, for a heating rate of 25°C/min and 200°C/min, respectively. The maximum sintering rate is between 725°C and 1025°C, for the heating rate of 25°C/min and 200°C/min, respectively. The use of a BN coating to insulate the current shifts the sintering curves towards higher temperatures (~50°C).

<figure><img src="image_2.png" /><figcaption>**Figure 1.** Constant heating rate sintering at 50 MPa: relative density (left) and temperature derivative (right). The solid lines indicate SPS in the conventional configuration; the dashed lines refer to sintering with an insulating BN coating.</figcaption></figure>

The curves under different pressures (and fixed heating rate of 100°C/min) are given in Fig 2. Notably, the curves with an applied pressure of 21 to 68 MPa are almost identical. Only the use of 78 MPa shifts the sintering curve to lower temperatures.

<figure><img src="image_3.png" /><figcaption>**Figure 2.** Sintering at different pressures at 100°C/min heating rate: relative density (left) and temperature derivative (right).</figcaption></figure>
The isothermal sintering curves and the time derivatives at low and intermediate temperatures (560°C to 960°C) are given in Fig 3. A high final density is reached within 1h, starting from a temperature as low as 680°C. The densification curves in the lower temperature range (560°C to 680°C) show an unusual shape, with two sintering steps. At 740°C and above, the curves reach a plateau within 10 min in a single step, and higher the temperature, faster the transition to the plateau. The first 10 minutes of the densification curves and the corresponding derivatives are highlighted in Fig. 4.

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Isothermal sintering at 50 MPa: relative density as a function of dwell time at 560°C to 960°C.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>**Figure 4.** First 10 min of isothermal sintering at 50 MPa: relative density (left) and time derivative (right) as a function of dwell time at 740°C to 960°C.</figcaption></figure>

### 3.1. Activation energy and stress exponent calculation via CHR analysis

The activation energy, *Q*<sub>act</sub>, was calculated from the CHR sintering curves (Fig 1), according to Eq. 4. The value of *Q*<sub>act</sub> is the slope of the interpolation lines at a given density in Fig. 5. The activation energy is slightly increasing with the density, and the average in the 50-85% density range is 75 ± 4 kJ/mol.
<figure><img src="image_6.png" /><figcaption>**Figure 5.** CHR method: determination of the activation energy, *Q*<sub>act</sub>.</figcaption></figure>

The stress exponent, *n*, was then calculated from the densification rates at different pressures (Fig. 2), using the average value of the activation energy found. According to Eq. 5, the stress exponent is the slope of the interpolation lines in Fig. 6 at given density. Two distinct regimes can be seen: up to 64 MPa, the exponent *n* is close to 0, while above 64 MPa the stress exponent assumes positive values. An exponent of *n* = 0 indicates that the sintering is insensitive to the magnitude of the applied pressure, which is consistent with what can be observed in Fig. 2. The applied pressure begins to have an influence only above 64 MPa. The data points are too few to allow a quantitative interpolation, but they suggest a value of *n* close to 1.

<figure><img src="image_7.png" /><figcaption>**Figure 6.** CHR method: estimation of the stress exponent, *n*. Red lines are only a guide for the eye.</figcaption></figure>
## 3.2. Activation energy calculation via MSC method

The activation energy was also calculated with the MSC method, using the data of Fig 1, between 400°C and 1600°C. The activation energy was found by minimising the mean relative absolute error using the Solver function of MS Excel and then confirmed by plotting Fig. 7 (right). The value that minimises the error is $Q_{act} = 75.5$ kJ/mol (Fig. 7).

<figure><img src="image_8.png" /><figcaption>**Figure 7.** MSC method: fitting curves for all heating rates (left), sum of square residuals as function of $Q_{act}$ (right). $Q_{act} = 75.5$ kJ/mol is the best fitting value.</figcaption></figure>

The MSC method was also applied to determine the activation energy in the quasi-hot pressing configuration (Fig. 1, with BN coating, between 400°C and 1600°C) and a value of 85 kJ/mol was found, which is only slightly higher than the one found in the SPS configuration without BN.

## 3.3. Stress exponent and activation energy with isothermal analysis

The stress exponent and activation energy were also determined from the isothermal sintering curves (Fig. 4) in the 740°C-960°C range. Figure 8 shows the calculation of the stress exponent following Eq. 9. Two regimes can be identified: at lower densities $n \approx 0.5$, while at higher densities $n \approx 1$. The threshold between these two regimes is 65% TD for the two lowest temperatures (740°C and 800°C), while it is 70% TD for the highest temperatures (for 960°C the density at the start of the dwell is above 65% TD). The activation energy was thus computed with Eq. 10 by using $n = 0.5$ for 50-65% TD and $n = 1$ for 70-85% TD (Fig 8, right). The average values in the two regimes are $82 \pm 8$ kJ/mol and $105 \pm 5$ kJ/mol, respectively. It is to be noted that the choice of the exponent has in this case only a limited impact on the calculation of $Q_{act}$. For example by choosing an exponent of $n = 0$ instead of 0.5, the activation energy for the low density and high density regime, become $78 \pm 8$ kJ/mol and $104 \pm 6$ kJ/mol, respectively.
<figure><img src="image_9.png" /><figcaption>Figure 8. Isothermal analysis: determination of the stress exponent n (left), and of the activation energy Q$_{act}$.</figcaption></figure>

### 3.4. Solution of the sintering equation with the GRG algorithm using Excel Solver

The sintering trajectory was determined from the data of Table 3. The best fit (Adj. R-Square 0.9929) was obtained by plotting the density as a function of grain size (Fig. 9) and using Eq. 14.

$$ \rho = \rho_{\infty} - A \exp \left( \frac{-G(\rho)}{B} \right) \qquad (14) $$

The fitting parameters were $\rho_{\infty} = 97.21 \pm 0.56$, A = 80.99 ± 5.10 and B = 33.47 ± 2.86.

<figure><img src="image_10.png" /><figcaption>Figure 9. Density vs grain size and determination of the sintering trajectory.</figcaption></figure>

Finally, the Solver function of MS Excel was used to solve Eq. 13, using the experimental data of the CHR curves (Fig 1) in the 50%-85% TD range. This function solves the input equation by finding the lowest mean absolute error. Since the GRG algorithm works on the local minima of the function, different initial conditions were tested to ensure that the convergence was towards the global minimum. As a first iteration, different couples of n and m were imposed, and the algorithm found the values of Q$_{act}$ and K' which minimise the error between the left hand side and right hand side of Eq. 13 (first seven couples of Fig. 10, left). As second iteration
also *m* and *n* were optimised, using (*m*, *n*) = (1, 1) as starting point. The result was *n* = 0.00 and *m* = 1.98 (mean error 0.0922), which were simply rounded to the couple (*m*, *n*) = (0, 2), resulting in *Q*<sub>act</sub> = 72.9 kJ/mol and a mean error of 0.0924 (last point of Fig 10, left).

These optimised parameters were used to plot the graph in Fig. 10 (right), where it can be seen that all data points converge to a single line. The value of the activation energy is also consistent with what found with the CHR and MSC methods and, more interestingly, the value of *n* is 0, without any variation of the external pressure.

<figure><img src="image_11.png" /><figcaption>**Fig. 10.** Optimization of the fitting values (*n*, *m*, *Q*<sub>act</sub>, and *K*'). Right: plot of Eq. 13 for the four experiments and the choice of the optimised parameters.</figcaption></figure>

# 4. Discussion

## 4.1. *Comparison among the different models*

The densification parameters calculated with each method are summarised in Table 5, which allows making some simple considerations:

1) The apparent activation energy *Q*<sub>act</sub> was calculated with all models. Three models are in excellent agreement. The isothermal model gives slightly higher values.

2) The stress exponent *n* is close to zero when calculated with the CHR analysis (only up to 64 MPa) and with the iterative fitting method. Interestingly, to calculate the stress exponent, the CHR analysis relies on a change in the external applied pressure, while the iterative fitting uses only data at 50 MPa (the effective stress *σ*<sub>eff</sub> varies with the density). The isothermal model gives values of 0.5 and 1.

3) The grain size exponent *m* was calculated only with the GRG method as *m* = 2.

<table><caption>**Table 5.** Densification parameters for each method</caption><thead><tr><th>Method</th><th>Density range %</th><th>Q<sub>act</sub> (kJ/mol)</th><th>n</th><th>m</th></tr></thead><tbody><tr><td>CHR</td><td>50 - 85</td><td>75</td><td>0 (up to 64 MPa)</td><td>-</td></tr><tr><td>MSC</td><td>40 - 96</td><td>75</td><td>-</td><td>-</td></tr><tr><td rowspan="2">Isothermal</td><td>50 - 65</td><td>82</td><td>0.5</td><td>-</td></tr><tr><td>70 - 85</td><td>105</td><td>1</td><td>-</td></tr><tr><td>GRG method</td><td>50 - 85</td><td>73</td><td>0</td><td>2</td></tr></tbody></table>
For a further evaluation of this results we can recall the limitations of each model. The CHR and MSC analysis
are based on the same experimental data (Fig. 2a) and assumptions. Nevertheless, the MSC gives a global point
of view of the activation energy overlooking possible changes in the densification mechanism with density.

The GRG method analyses the same data during heating. Contrary to the CHR and MSC methods, GRG
requires to assume an explicit form for the functions σ<sub>eff</sub>(ρ) and μ<sub>eff</sub>(ρ). However it gives the same value for the
activation energy and stress exponent (n = 0), suggesting that a variation in the external pressure has no
influence on the sintering behaviour in the selected pressure range. The choice of the fitting function for the
grain versus density data can affect the values of n and m.

The isothermal method analyses data at constant temperature. Besides the need of using an explicit equation for σ<sub>eff</sub> and μ<sub>eff</sub>, it relies on the strongest assumption that G is constant during sintering up to the maximum density considered. In reality, although grain growth starts to be significant only above ~90% TD, it is not necessarily constant in the lower density range. Here a more than twofold increase in grain size was measured from relative densities of 50% to 76%. These potential sources of error can be eliminated by interpolating only data points at fixed density [41], [53] which is possible only if enough data points at the same density and different dwell temperatures are available. It requires thus a careful selection of the temperature steps and range.

4.2.1. Activation energy and effect of the current

This work confirms the remarkably low value for the apparent activation energy for densification of UO₂ in SPS (75 kJ/mol here and 96 to 140 kJ/mol [31], [32], recently measured by CHR and MSC), when compared to conventional sintering (250 - 450 kJ/mol). A low activation energy was also reported for the flash sintering of UO₂ (108 kJ/mol for flash sintering vs 380 kJ/mol for conventional sintering) [54]. The electrical fields in SPS are however much lower than what is typically applied in flash sintering [55], [56].

In the sintering of polycrystalline ceramics, the apparent activation energy should represent the one for
diffusion of the rate-limiting specie (U for UO2), along the fastest diffusion path giving densification (lattice
or grain boundary). However, the low values measured in SPS are not consistent with any of the cation
diffusion mechanism reported, which are typically in 250-500 kJ/mol range. The large scattering of the reported
values is likely due to the experimental uncertainties (typically measured by α-tracer or creep experiments)
and the dependence on the stoichiometry. [31], [57].

A lower activation energy for densification in SPS has been already reported in the literature also for other materials [58], and often attributed to an accelerating effect of the current for mass transport. However, in the case of UO₂, only a minor fraction of the current passes through the sample, which is being heated predominantly by heat transport from the graphite dies. The fraction of the current flowing through the sample increases with the electrical conductivity, and thus with the temperature, deviation from stoichiometry and density [59]. The experiments conducted here by insulating the current with BN, although not conclusive, indicate that the low activation energy in SPS is likely not an effect of the current: the Qₐct measured in the quasi-hot-pressing configuration is 85 kJ/mol, comparable to the 75 kJ/mol measured in SPS configuration and very far from the conventional sintering one. It is nevertheless interesting to note that the use of a current-insulating layer shifts the sintering curves towards higher temperatures, which is compatible with an effect on the pre-exponential term in Eq. 2.

A possible explanation for the low activation energy measured is that densification in the intermediate stage
of SPS does not occur predominantly by lattice or volume diffusion but by different concurrent mechanisms.
Volume diffusion, grain boundary diffusion and surface diffusion have typically activation energies that are in
relationship Q<sub>S</sub> < Q<sub>V</sub> < Q<sub>GB</sub>. Surface diffusion is thus dominating at lower temperatures, but it is generally
considered a non-densifying mechanism; however, it can play a role in densification if particle or grain
rearrangement and rotation take place during SPS. [44] Densification by particle rotation and repacking was reported to be particularly significant for nanocrystalline ceramics at low temperature and densities [60], [61]. It is expected to proceed rather rapidly because entire grains, rather than atoms, migrate simultaneously; the rapid heating rate of SPS limits also particle coarsening favouring particle rearrangements [62] up to higher densities. The activation energy for surface diffusion may decrease with the particle radius, when entering the nanodomain [63], and thus the lower activation energy for densification of nano-UO₂ compared to commercial UO₂ [31] may be a size-effect. Because of the extreme surface-to-volume ratio, surface controlled phenomena are also expected to be more pronounced in nanopowder.

At higher density, towards the final stage of sintering, densification proceeds by diffusional processes as grain boundary or volume diffusion [64]. The increase of the activation energy observed above 80% TD is possibly an indication of the onset of the change in the mechanism.

An alternative interpretation would imply that the activation energy of grain boundary diffusion of U is significantly lower than the one for bulk diffusion. Molecular dynamics simulations point into this direction: the activation energy of U migration at symmetrical grain boundary was indeed found to be 77 kJ/mol. [65]. If that was the case, the activation energy found in SPS experiments would be compatible with a grain boundary diffusion mechanism.

It is worth noting that the value of the activation energy alone does not allow to determine the dominant diffusion mechanism, and the exponents n and m are typically used to discern among them lattice or grain boundary diffusion, plastic flow or particle rearrangement and grain boundary sliding. [40]

### 4.2.2. Effect of the pressure

In the case of commercial UO₂ powders (primary particle size 0.2~0.3 µm) pressures of the same magnitude of the ones used here shift the sintering curves to lower temperatures, with a stress exponent n of 1.4. [31]. The response of nanopowders to the applied pressure is drastically different: densification is insensitive to the applied pressure up to 64 MPa (Fig. 1). The stress exponent n assumes thus a value close to zero up to 64 MPa and starts to be positive only above. Although our data do not allow to calculate its value from only two pressures, an estimation of its value above 64 MPa with the CHR method gives a value close to the unity. The other methods give values of n from 0 to 1. A value of 0 is not considered by the theory, while 1 is compatible with lattice diffusion, grain boundary diffusion or particle rearrangement. [40]. A transition from pressure-insensitive to pressure assisted-sintering was also observed for nanoparticles of zirconia [66]. A possible interpretation was given by Skandan et al [66]: the driving force for sintering is the sum of the intrinsic driving force (curvature dependent) and the externally applied stress. Most pressure-assisted models assume that the external pressure is significantly higher than the intrinsic sintering stress; however, in nanocrystalline particles the extreme curvature can give rise to higher stresses than the ones typically applied [66]. Several expressions for the sintering stress exist in the literature. Here we estimate it with the expression proposed by Olevsky (Eq. 14) [67],

$$ \sigma_S = \frac{6 \gamma_{SV}}{D} \rho^2 \qquad (14) $$

where D is the particle diameter and γ<sub>SV</sub> the solid-vapour surface tension (~0.7 J/m² for UO₂ at 800°C [68]). Interestingly, the intrinsic sintering stress varies from 52 to 61 MPa in the 50-77% relative density range (data from the first four rows of Table 3), and is thus very close to the observed threshold stress of 64 MPa.
### 4.3. Grain growth

Up to a density of ~80% the growth of the grains as a function of density is limited (but not negligible) and follows a quasi-linear trend. At high densities, after the microstructure reaches a closed porosity configuration, the grains starts to grow rapidly with density (Fig. 9). The calculated grain growth exponent *m* assumes a value of 2, which should correspond to lattice diffusion [40]. However, as discussed before, it is unlikely that lattice diffusion is dominating at the low temperature considered here, especially for nanopowders with very large surface and/or grain boundary area. Furthermore, the other parameters are not consistent with lattice diffusion (*n* = 0 and low Q<sub>act</sub>). The value of *m* is also sensitive to the choice of the fitting function for the sintering path (Fig. 9).

We did not find in the literature other calculations for the exponent of UO<sub>2</sub> during SPS or hot pressing, which would allow a direct comparison. Yao et al. [9] performed isothermal annealing of dense nanostructured (100~165 nm) UO<sub>2.03</sub> and UO<sub>2.11</sub> at comparably low temperatures (700-900°C) and found activation energies for grain growth in the 100~200 kJ/mol range, significantly lower than usually reported at higher temperatures, and compatible with values obtained by SPS for densification at low temperatures.

## 5. Conclusions

In CHR conditions, densification is complete at 1000°C and 1325°C, for a heating rate of 25°C/min and 200°C/min, respectively, with an activation energy of 75 kJ/mol. In isothermal sintering, densification can be achieved at 680°C within 1h dwell (final grain size ~300 nm), or within less than 10 min for temperatures of 740°C and above, with an activation energy ranging from 82 to 105 kJ/mol.

Such activation energies are significantly smaller than the values generally reported for conventional sintering. When the current flow through the punches-UO₂ interfaces is prevented by a BN coating, the activation energy in CHR experiments becomes 85 kJ/mol, and the sintering curves are retarded by ~50°C. The results indicate that an effect of the current cannot explain the low activation energy observed in SPS of UO₂.

This suggests that densification in the early and intermediate stages does not proceed by grain boundary or lattice diffusion, but is more likely dominated by concurrent mechanisms, such as particle rotation and repacking.

An alternative explanation would imply that the activation energy for grain boundary diffusion is much lower than previously reported, as suggested by molecular dynamic simulations.

A transition from pressure-insensitive sintering (from 21 to 64 MPa) to pressure-assisted sintering was also identified. The threshold stress of 64 MPa is not far from the calculated intrinsic sintering stress of 17 nm UO₂ particles. The contribution of the capillary stresses in nanopowders are thus comparable with the typical applied stress in SPS.

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