# Modeling anisotropic sintering for fused deposition modeling of alumina

Fatima Hammoud, Jérôme Lecourt, Christelle Bilot, Charles Manière

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## To cite this version:

Fatima Hammoud, Jérôme Lecourt, Christelle Bilot, Charles Manière. Modeling anisotropic sintering for fused deposition modeling of alumina. International Journal of Advanced Manufacturing Technology, 2025, 137 (5-6), pp.2319-2334. 10.1007/s00170-025-15306-3. hal-05130081

HAL Id: hal-05130081

https://hal.science/hal-05130081v1

Submitted on 25 Jun 2025

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# Modeling Anisotropic Sintering For Fused Deposition Modeling of Alumina

Fatima Hammoud<sup>a*</sup>, Jérôme Lecourt<sup>a</sup>, Christelle Bilot<sup>a</sup> and Charles Manière<sup>a*</sup>

<sup>a</sup> Université de Caen Normandie, ENSICAEN, CNRS, CRISMAT, Normandie Univ, 14000 Caen, France

* Corresponding authors: fatima.hammoud@ensicaen.fr, charles.maniere@ensicaen.fr

## Abstract

Additive manufacturing, particularly molten filament deposition, has revolutionized the production of complex parts, offering rapid fabrication from ceramics to metals. Nanoe® uses this technology to print intricate ceramic and metal components within minutes to hours. However, challenges arise during the sintering process, including intrinsic anisotropy leading to part deformation and the risk of cracks with rapid heating, especially in porous structures. To address these challenges, our study has developed a sintering simulation framework. The primary objectives include simulating the sintering process of parts printed with Nanoe filaments, characterizing the unique behavior of each filament, and establishing a comprehensive library of material properties for accurate sintering simulation. The framework has been validated through strong agreement between experimental and simulated results, demonstrating its ability to predict anisotropic behavior in alumina parts. The analysis of shrinkage rates, densification kinetics, and mechanistic aspect has provided valuable insights into optimizing sintering processes for improved reliability and precision. In this presentation article, the sintering model identification method is presented for alumina. The method is first tested on pressed powder which is isotropic and then adapted to 3D printed sample that has an intrinsic anisotropy. Ultimately, our goal is to provide a valuable numerical tool for optimizing the additive manufacturing shapes design.

**Key words:** Additive manufacturing, ceramics, sintering process, anisotropy, simulation.

## Nomenclature

* $\theta$ Porosity

* $\dot{\theta}$ Porosity elimination rate (s⁻¹)

* $\rho$ Relative density

* $\sigma$ Stress tensor (N.m⁻²)

* $\dot{\sigma}$ Deviatoric stress tensor (N.m⁻²)

* $\dot{\dot{\sigma}}$ Strain rate tensor (s⁻¹)

* $\eta$ Material viscosity (Pa.s)

* $\eta_0$ Viscosity pre-exponential factor (Pa.s)

* Q Viscosity activation energy (J.mol⁻¹)

* R Gas constant 8.314 (J.mol⁻¹.K⁻¹)

* T Temperature (K)
* $\varphi$ Shear modulus, porosity function of the sintering model

* $\psi$ Bulk modulus, porosity function of the sintering model

* $\theta_c$ Initial critical porosity

* $P_l$ Sintering stress resulting the capillarity driving forces of sintering (N.m$^{-2}$)

* $P_{lx}, P_{ly}, P_{lz}$ Anisotropic sintering stress terms (N.m$^{-2}$)

* $\mathring{i}$ Identity tensor

* $\alpha$ Surface energy (J.m$^{-2}$)

* $r$ Particles radius (m)

* $\dot{e}$ Trace of the strain rate tensor (s$^{-1}$)

* $G$ Grain size (m)

* $\dot{G}$ grain growth rate (m.s$^{-1}$)

* $G_0$ Initial grain size (m)

* $m$ Densification model grain size exponent

* $p$ Grain growth model grain size exponent

* $K$ Temperature dependent term of grain growth model (m$^{1+p}$.s$^{-1}$)

* $k_0$ Pre-exponent term of grain growth model (m$^{1+p}$.s$^{-1}$)

* $Q_G$ Grain growth activation energy (J.mol$^{-1}$)

* $a, b, c, c', c''$ Fitting constants

* $t$ Time (s)

* $C$ Diffusion model constant

* $\Phi$ Master sintering curve integral parameter

* MSC Master Sintering Curve

# 1. Introduction

Ceramic additive manufacturing represents a significant advancement in modern fabrication technologies, enabling the creation of complex geometries and fine details that are challenging or impossible to achieve with traditional methods. This capability is particularly valuable for applications requiring multiscale and bioinspired structures, lightweight configurations, or advanced composites with superior mechanical and functional properties [1] [2] [3]. Specific techniques such as robocasting and stereolithography achieve varying resolutions, allowing for the production of green ceramic specimens through extrusion or photopolymerization of ceramic resins, respectively [4] [5].

The appeal of ceramic 3D printing lies in its ability to produce complex geometries that are often unachievable with traditional manufacturing methods. However, these complex shapes pose challenges throughout the manufacturing process, particularly during critical stages such as debinding and sintering. Thin structures, characteristic of many ceramic prints, are susceptible to gravitational deformation during high temperature sintering [6] [7] [8].

Debinding, a crucial step in removing organic binders requires precise temperature control to avoid cracking and swelling [9] [10]. The subsequent sintering process, aimed at densifying the
green ceramic parts. It can introduce additional complexities such as anisotropic shrinkage,
distortion due to thermal gradients, and interactions with support materials [11] [12] [7].

Among ceramic additive manufacturing methods, fused deposition modeling (FDM) has gained
significant attention for its ability to use ceramic-filled filaments to produce highly complex
parts. FDM-based 3D printing is particularly advantageous for creating lightweight and porous
structures, which are critical for applications such as biomedical implants and energy devices.
However, achieving high-quality prints requires addressing challenges related to filament
homogeneity, extrusion stability, and the thermal post-processing steps required to remove
binders and sinter the ceramic materials. Recent advancements include the development of
optimized ceramic filament formulations and process parameters to enhance mechanical
properties and reduce defects in the final components [13] [14] [15].

For ceramic 3D printing technologies, finite element methods (FEM) play a key role in
predicting and optimizing the shape design and adjust the sintering conditions. Beyond
ceramics, FEM has also been applied in other manufacturing sectors, such as laser cutting,
metallic coatings, and composite joints [16] [17] [18]. These studies demonstrate the versatility
of FEM in analyzing thermal effects, stress distribution, and mechanical behavior across various
materials and processes, underscoring its critical role in industrial applications. Our objective
is to be capable of predicting the sintering distortions of printed specimens during sintering.
The sintering models like Skorohod-Olevsky's model [39] are particularly useful to assess this
behavior and integrate it in a FEM code. In the literature such model is used to simulate the
behavior of printing lattices during sintering, sintering anisotropy, pressure assisted sintering
like SPS, etc [19], [20], [21]. However, the assessment of a Skorohod-Olevsky's model is
particularly challenging and long as it requires to identify the thermally activated viscosity, the
shear and bulk moduli that are porosity functions reflecting the porous skeleton and le grain
growth that disturbs the densification at the final stage[22]. The establishment of Skorohod-
Olevsky's model requires the independent identification densification (viscosity, moduli) and
grain growth behavior. This process requires extensive experimental analysis to accurately
characterize the densification mechanisms and their governing parameters. The thermal
simulation can also have a high importance for large part with internal lattice fill strategies [23].
However, this work is limited the sintering model assessment which is the most difficult part.
The densification model assessment is generally made in intermediate sintering far from the
final stage where grain growth is highly active. The identification methods generally used
dilatometry densification curves at different heating rates like the “master sintering curve”
(MSC) [24] [25], [26] or the “kinetic field”[27] [28]. We develop an approach consisting of coupling the MSC method with direct regression to assess both the temperature dependent parameter and the moduli [29]. The Grain growth is generally identified by interrupted test at different dwell time and temperature [30] [31]. We will use here a method based on final stage sintering densification curve [29] which avoid a very time consuming experimental exploration. Finally, the sintering anisotropy is one of the most important aspects for additive manufacturing as in tends to orient the grains [32] [33] and the porosity in the layer plane [34] [35]. This can be simulated by anisotropic sintering properties [36] [19] [37] [38] [23], the FEM model can then reproduce the same shape sintering shrinkage which is essential to anticipate the final shape geometry.

In this study, we focus on advancing the understanding and simulation of the sintering process for ceramic components fabricated using Nanoe®'s alumina molten filament deposition technology. Our research aims to develop a comprehensive sintering simulation framework that characterizes the unique behavior of Nanoe filaments, including their shrinkage rates, densification kinetics, and microstructural properties during the sintering process. Specifically, we employ a methodology to identify and model the anisotropic sintering behavior of alumina, initially validated on isotropic pressed pellets and subsequently adapted for 3D printed samples with inherent anisotropy.

By enhancing our ability to simulate and predict the sintering outcomes, this study seeks to optimize the additive manufacturing process monitoring at the design stage.

## 2. Theory and calculations

### 2.1. Sintering model definition

The Skorohod-Olevsky's model is used to simulate the sintering of parts in continuum approaches like FEM. The model provides a framework to predict sintering deformation and porosity evolution, critical for understanding densification mechanisms. The constitutive equation describing the linear viscous sintering behavior of a compressible medium and taking into account the sintering stress $P_l$ is defined as [39]:

$$ \underline{\sigma} = 2\eta(T) \left(\varphi(\theta)\dot{\underline{\epsilon}} + \left(\psi(\theta) - \frac{1}{3}\varphi(\theta)\right)\dot{\underline{e}}\right) + P_l\underline{\mathbf{i}} \quad (1) $$

Where $\underline{\sigma}$, $\dot{\underline{\epsilon}}$ are the stress, strain rate tensors (Pa and s⁻¹), $\eta(T)$ is the viscosity (Pa·s) as a function of temperature (T), $\varphi(\theta)$ and $\psi(\theta)$ are the shear and bulk moduli, porosity ($\theta$) functions, $\dot{\underline{e}}$ is the volumetric shrinkage rate (s⁻¹), $P_l$ is the sintering stress (Pa), and $\underline{\mathbf{i}}$ is the identity tensor. $\dot{\underline{e}}$ is the following invariant.
$$ \dot{e} = \dot{e}_x + \dot{e}_y + \dot{e}_z \qquad (2) $$

Skorohod's theory provides the following sintering stress expression [39] for the capillarity forces developed at the particle's contacts and the following shear and bulk moduli.

$$ P_l = \frac{3\alpha}{r} (1 - \theta)^2 \qquad (3) $$

With, $\alpha$ the surface energy (J/m²), $r$ is the particle radius.

$$ \varphi(\theta) = (1 - \theta)^2 \qquad (4) $$

$$ \psi(\theta) = \frac{2(1-\theta)^3}{3\theta} \qquad (5) $$

Equation (1) gives the sintering deformation behavior of a linear viscous medium. The porosity evolution rate is linked to the rate of volume change through the mass conservation equation:

$$ \frac{\dot{\theta}}{1-\theta} = \dot{e}_x + \dot{e}_y + \dot{e}_z \qquad (6) $$

## 2.2. For pressure-less isotropic sintering

For small sample sintering in conventional furnace, the stress and strain rate tensors can be reduced to [40]:

$$ \underline{\sigma} \equiv \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix} ; \quad \underline{\dot{\epsilon}} \equiv \begin{pmatrix} \dot{\epsilon}_r & 0 & 0 \\ 0 & \dot{\epsilon}_r & 0 \\ 0 & 0 & \dot{\epsilon}_r \end{pmatrix} \text{ isotropic } \dot{\epsilon} = 3\dot{\epsilon}_r \qquad (7) $$

Equation (1) becomes:

$$ 0 = 2\eta \left( \varphi \dot{\epsilon}_r + \left( \psi - \frac{1}{3}\varphi \right) 3\dot{\epsilon}_r \right) + P_l \qquad (8) $$

We obtain the reduced expression:

$$ -P_l = 2\eta 3 \dot{\epsilon}_r \psi \qquad (9) $$

Using the mass conservation (6), we obtain [39]:

$$ \psi = \frac{-P_l(1-\theta)}{2\eta\dot{\theta}} \qquad (10) $$

The analytic equation describing the rate of porosity elimination to model sintering without grain growth is the following.

$$ \dot{\theta} = \frac{-P_l(1-\theta)}{2\eta\psi} \qquad (11) $$

Equation (11) can model analytically the sintering densification. However, the final stage grain growth must be considered, as it influences the sintering stress (3) and the diffusional viscosity by extending the diffusion distances [41]. The grain growth can be modeled by the following rate equation [25]:

$$ \dot{G} = \frac{K}{G^p} = \frac{k_0}{G^p} \exp\left(\frac{-Q_G}{RT}\right) \qquad (12) $$
Where G is the grain size in (m).

The impact of grain size on linear diffusional creep mechanisms can be calculated by considering the dependence on viscosity, temperature, and grain size, [40] as given by the following expression:

$$2\eta = \left(\frac{G}{G_0}\right)^m \eta_0 T \exp\left(\frac{Q}{RT}\right) \qquad (13)$$

with $m=2$ for lattice diffusion and $m=3$ for grain boundary diffusion. The $m$ exponent corresponds to the effect of grain size on the extended diffusion distances [41].

Substituting the expression of the sintering stress (3) and the viscosity (13) in Eq. (11), and assuming particle size is close to the grain size (r~G/2), we obtain the expression of the rate of porosity evolution which takes into account the actual grain size (G) [40].

$$\dot{\theta} = \frac{-6\alpha(1-\theta)^3}{G^{m+1}\frac{\eta_0}{G_0}T\exp\left(\frac{Q}{RT}\right)\psi} \qquad (14)$$

This equation can be compared to the theoretical equations of combined stage sintering which have the general form [40]:

$$\dot{\theta} = \frac{-C f(\theta) \alpha D}{G^{w} k T} \qquad (15)$$

With $w=m+1$.

### 2.3. Combinational approach to assess the sintering parameter from MSC and regression analysis

This method described in ref [29] consist in the identification of all the sintering parameters from constant heating rate dilatometry. It combines the master sintering curves (MSC) method to identify independently the activation energy [24] and a regression approach [40] to assess the sintering moduli and the viscosity constant.

The method is divided in 4 main steps [23].

In step 1, three or more constant heating rate dilatomtries are made to identify the activation energy by the master sintering curve method [24] which can be completed for verification with the kinetic field methods [27] [28].

The second step involves using a regression equation during the intermediate stage of sintering. Equation (11) is written in the following form where the temperature dependent viscosity term is isolated.

$$Y = \ln\left(\frac{-3(1-\theta)^3}{rT\dot{\theta}\psi}\right) = \ln\left(\frac{\eta_0}{\alpha}\right) + \frac{Q}{RT} \qquad (16)$$
In this equation, Y represents the regression variable for viscosity analysis, and R is the universal gas constant (8.314 J/mol·K).

This regression gives the activation energy (Q) by the slope and the constant part of the viscosity by the origin $ln(\frac{\eta_0}{\alpha})$. However, this equation use Shorohod's bulk modulus. Has this modulus is theoretical it has an influence on the apparent activation energy obtained which differs from the MSC value. In this context the Skorohod moduli are replaced by the modified moduli reported [29] below.

$$ \psi = a \frac{(\theta_c - \theta)^b}{\theta^c} \quad \& \quad \varphi = \left(1 - \frac{\theta}{\theta_c}\right)^2 \qquad (17) $$

Here, $\theta_c$ represents the critical porosity, while a, b, and c are fitting parameters for the porosity-dependent moduli.

The $\theta_c$ term b and if needed the b term are adjusted in the regression formula up to obtain a slope similar to the value of activation energy obtained by the MSC which is a method independent from the moduli. By doing so, the experimental moduli are corrected and the term $\frac{\eta_0}{\alpha}$ can be identified.

At this stage all required densification parameters of the model are identified. The step 3 consists of simulating the MSC sintering curves by equation (11) to see if the curve reproduce the initial and intermediate stage. A minimal parameter adjustment is possible at this stage.

The final stage sintering should not fit because the grain growth is not assessed yet. The last stage consists of the grain growth and sintering mechanism identification based on the final stage influence of grain growth on the sintering kinetics. The sintering including grain growth influence is then equation (14). In the latter, the grain size term is isolated to obtain equation (18) [29].

$$ G = \left( \frac{\frac{6\alpha G_0^{w-1}}{\eta_0} \exp\left(\frac{-Q}{RT}\right)}{\frac{\partial T}{\partial T}} \frac{(1-\theta)^3}{\psi} \right)^{\frac{1}{w}} \qquad (18) $$

Where, $G_0$ is the initial grain size (m), and w is the sintering model grain size exponent, which is equal to m+1.

In this equation all parameters are already identified and the porosity, porosity rate are experimental. It is then possible to estimate two estimative grain size curves for the final stage for grain boundary diffusion (w=4) and lattice diffusion (w=3). Compared with the final microstructure grain size one of the two mechanisms will explain the final grain size.

Certain powders have more or less high agglomerates or can involve precoarseing at the initial stage with implicate the particle radius $r$ used to calculate the capillarity forces $P_l$ behave
differently from the grain size involved in grain boundary/lattice diffusion. The last equation can be written for separating the grain size diffusional sensitivity ($\frac{G}{G_0}$)$^m$ and the capillarity grain radius ($r$) this alternative equation is the following with $r_0$ the radius of agglomerates

$$G = \left( \frac{-3(1-\theta)^3 G_0^m}{\dot{\theta} r_0 \frac{\eta_0}{\alpha} T \exp\left(\frac{Q}{RT}\right) \psi} \right)^{\frac{1}{m}} \quad (19)$$

After identifying the correct grain size curve, a second linearization of normal grain growth using equation (12) with $p=2$ can be used to assess the grain growth parameters to the final stage sintering.

$$YG = \ln(\dot{G}G^2) = \ln(k_0) - \frac{Q_G}{RT} \quad (20)$$

In this equation, $YG$ represents the regression variable for grain growth, $\dot{G}$ is the grain growth rate (m/s), $k_0$ is the pre-exponential constant for grain growth, and $Q_G$ denotes the activation energy for grain growth (J/mol).

This comprehensive methodology integrates densification and grain growth behaviors, providing a robust framework for modeling sintering processes.

# 3. Experiment part

## 3.1. Samples Preparation

For the isotropic model, Nanoe® alumina powder with an average crystallite size of approximately 150 µm, doped with MgO (1000 ppm) and Na₂O (< 100 ppm), was utilized. The experimental procedure began with 100K/h heating the powder to 900°C, holding it at this temperature for 1 hour, and then cooling it to ambient temperature with 200K/h heating rate. The thermal treatment was conducted in a Pyrox muffle furnace, operating under natural air. After thermal treatment, chemical debinding was conducted using Rhodoviol. The debound powder was then compressed into pellets using hydraulic manual press under a pressure of approximately 200 MPa at ambient temperature for 30 seconds, forming pellets with dimensions of 6 mm in diameter and 3 mm in height.

For the anisotropic model, Zetamix filaments, specifically formulated for ceramic FDM printing, were utilized. These filaments consist of fine ceramic powder (< 1.0 µm) bound by a polyolefin-based thermoplastic binder system. The printing was performed using a Prusa Slicer 2.5.0 printer equipped with a 0.6 mm nozzle, used here for the Zetamix filaments. The printing parameters were set as follows: a printing temperature of 160°C, 40°C of bed temperature, a nozzle size of 0.6 mm, layer thickness of 0.2 mm, and a printing speed of 20 mm/s. The printing
process adhered to Zetamix guidelines to ensure proper filament handling and extrusion without filament grinding. After printing, the green parts underwent a two-stage debinding process to remove the thermoplastic binder. In the first stage, chemical debinding, the parts were immersed in an acetone bath at 40°C for approximately 6 hours, aiming to remove at least 5% of the part's weight. Following solvent debinding, the parts were air-dried for 2 hours at ambient conditions. In the second stage, thermal debinding, the process involved a controlled ramp-up heating rate of 8 k/h per hour up to 500°C followed by a heating rate of 50°C per hour up to 1100°C. The total thermal debinding process typically spanned over 3 days, ensuring thorough removal of the remaining binder.

## 3.2. Sintering process

Sintering of the debound ceramic parts was carried out in a high-temperature furnace. Both the isotropic and anisotropic samples underwent sintering under controlled conditions to evaluate their respective behaviors. The temperature cycle involved a ramp-up from 20°C to 1550°C at 50°C per hour, a holding time of 2 hours at 1550°C, and a cool-down from 1550°C to 20°C at 100°C per hour. The sintering process aimed to achieve densification while managing the expected shrinkage rates: x, y = 20.8% ±1%; z = 23.2% ±1%, as specified by Zetamix guidelines.

## 3.3. Characterization

Characterization of both isotropic and anisotropic samples included sintering dilatometry and microstructural analysis. Dilatometry (Setaram® TMA96 dilatometer) was utilized to measure the dimensional changes of the samples during the sintering process, with the primary objective of determining the shrinkage behavior and densification kinetics. The samples were tested at three controlled heating rates: 2 K/min, 5 K/min, and 10 K/min, to analyze their impact on the sintering behavior and densification kinetics. This technique involved heating the samples at a controlled rate while continuously measuring their length changes, allowing us to identify critical sintering stages and calculate parameters such as the coefficient of thermal expansion and sintering activation energy. Microstructural analysis was conducted using scanning electron microscopy (SEM Jeol 7200 LV) to investigate the internal structure of the sintered samples. SEM provided high-resolution images to observe grain size, shape, and distribution, as well as the presence and morphology of pores. For the isotropic samples, SEM was used to examine grain growth and porosity distribution to assess the consistency of the sintering process. For the
anisotropic samples, SEM analysis was crucial to identify any directional dependencies in grain
structure and porosity due to the layer-by-layer printing process. The mean grain size was
determined using the linear intercept method and Mendelson's [42] 1.56 scaling factor. By
integrating data from dilatometry and SEM analysis, we aimed to develop a robust
understanding of the sintering mechanisms and identify any potential issues related to
anisotropy in 3D printed ceramics, which is essential for optimizing the additive manufacturing
process and improving the quality and performance of ceramic components produced using this
technology.

4. Method

The methodology employed in this study integrates theoretical frameworks with experimental procedures to comprehensively investigate the sintering behavior of ceramic materials. Central to this approach is the utilization of dilatometry, which provides critical data on dimensional changes during sintering, facilitating the calculation of activation energies and the characterization of grain growth dynamics. Building upon these foundational elements, the study employs analytical sintering modeling using Octave-Forge software to simulate and predict densification processes under controlled thermal conditions. By bridging theory and experimentation, this methodology aims at enhancing the understanding of sintering mechanisms, optimize processing parameters, and improve the reliability of ceramic manufacturing processes.

The assessment of the sintering model includes the following stages: (i) dilatometry curve acquisition at different heating rate and on pressed specimen, (ii) the activation energy is identified by MSC method, (iii) the regression minimization study is conducted to assess the viscosity parameters, (iv) the dilatometry is made on printed samples in building and in plane direction, and the final stage dilatometry curve is exploited to assess the grain growth using equation (18), (v) the anisotropy is finally imposed using the shrinkage curves and the model is compared to the experimental curves. The main aspects of the article printing and sintering model assessment are described in Figure 1.
(A) Printing process main steps

<figure><img src="image_3.png" /></figure>

(B) Sintering model assessment main steps

<figure><img src="image_4.png" /><figcaption>Figure 1. (A) Scheme of the printing process main stages. (B) Description of the sintering model assessment stages.</figcaption></figure>

5. Results and discussions

5.1. Alumina powder analysis

The initial grain size distribution of the starting alumina powder was analyzed to provide a baseline for sintering studies. The powder, after chemical and thermal debinding, was examined using SEM, and grain sizes were determined using ImageJ software. The analysis, depicted in Figure 2, shows an average grain size of approximately 190 nm, with a distribution ranging from 50 nm to 500 nm.

This relatively broad grain size distribution can affect the sintering behavior: smaller grains, with higher surface energy, sinter at lower temperatures, while larger grains require higher temperatures and longer durations. The submicron-scale average grain size (190 nm) is advantageous for achieving high-density sintered parts with fine microstructures [29].
<figure><img src="image_5.png" /><figcaption>Figure 2. Granulometry analysis on the debinded Zetamix® alumina filament.</figcaption></figure>

Understanding this distribution is crucial for accurate sintering modeling, as the diverse grain sizes will influence the kinetics of the process. These empirical findings serve as essential inputs for the theoretical models described earlier, enhancing the accuracy and reliability of the simulations in predicting sintering behavior and guiding process optimization.

## 5.2. Sintering activation energy assessment

The relative density curves obtained from the Master Sintering Curve (MSC) study are illustrated in Figure 3A. As anticipated, the results indicate that slower heating rates lead to higher levels of densification. This trend can be attributed to the extended time the material spends at elevated temperatures, allowing for more complete sintering.

These observations align with theoretical expectations and underline the importance of optimizing heating rates to achieve desired densification levels in sintering processes. The data also validate the curves for a use by MSC.
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>Figure 3. (A) Relative density curves at constant heating rates of 2, 5 and 10 K/min and (B) Sintering activation energy identification via the master sintering curve.</figcaption></figure>

To determine the activation energy, we applied a minimization procedure according to the Master Sintering Curve (MSC) method. As shown in Figure 3B, the 1000 ppm MgO-doped Nanoe® alumina powder exhibited an activation energy of 825 kJ/mol, with each curve fitting the data points well. This activation energy aligns with values reported for alumina obtained through uniaxial dry pressing and cold isostatic pressing, typically ranging from 668 to 770 kJ/mol [26] [43]. Notably, an activation energy of 868 kJ/mol has been documented during initial/intermediate stage sintering and for alumina dry pressed with a PVA binder [44] [45]. In contrast, slip casting methods generally yield lower activation energies, between 440 and 600 kJ/mol [45] [46]. These lower values for slip casting are close to the activation energy for diffusional mechanisms of aluminum and oxygen in alumina, which range from 380 to 636 kJ/mol [47] [48].
kJ/mol [47]. The lower activation energies observed with slip casting suggest this method facilitates higher initial densification, aligning the model more closely with diffusional mechanisms in alumina. Additives like Na₂O have been shown to increase activation energy, indicating their impact on diffusion processes [43]. However, MgO additives do not significantly influence the activation energy, as noted in previous research.

### 5.3. Sintering model identification

Using the activation energy determined by the MSC study, the direct regression method was applied to identify the ratio η₀/α via equation (16). The bulk modulus may impact the outcome of equation (16) as it is located at the denominator. This indicates a potential error in the activation energy obtained by the direct method if the bulk modulus used in equation (17) is not the real one. To ensure the activation energy aligns with that determined by the MSC method, the bulk modulus must be adjusted via θ_c to a value generally higher than the initial porosity. A minimization of activation energy difference (Q-QMSC) is done to adjust θ_c if this parameter is not enough the term a, b, c can also be tuned.

Figure 4A presents the results of the direct regression method. Two key phenomena are noticeable in the linear regression: (i) at the final stage of sintering, grain growth caused a deviation from the linear trend (grain size estimation was based on this deviation) and (ii) at the initial stage, a second deviation was observed. For alumina, the onset of sintering was noted at 1200 K. In this study, the initial grain size, G₀, was determined to be 190 nm. The identified moduli are reported in equation (21). During the analytical modeling of the sintering curves (Figure 4B) using the identified ψ, Q, and η₀/α values, no significant deviations are observed for the initial and intermediate sintering. The only difference is at the final stage where expected deviations are present because the grain growth phenomenon which slower the sintering is not yet implemented. However, this issue will be addressed by incorporating grain growth into the model (later section).

$$ \psi = a \frac{(0.45-\theta)^b}{0.45} \quad \& \quad \varphi = \left(1-\frac{\theta}{0.45}\right)^2 \qquad (21) $$
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>Figure 4. (A) Modeling of the intermediate stage sintering by direct regression using MSC-derived activation energy, and in inset the bulk modulus exponent adjustment. (B) Resulting analytical modeling of the porosity curve without grain growth.</figcaption></figure>

5.4. Sintering of 3D printed powders and anisotropy

Figure 5 presents two key aspects of the sintering process: the variation of temperature and relative density over time, and the changes in axial and radial dimensions. The first part of the figure illustrates the relationship between temperature and relative density as a function of time. The results indicate a clear trend where the relative density increases with rising temperature. This behavior aligns with the expectation that higher temperatures promote densification in ceramic materials, leading to a more compact structure as sintering progresses.
The second part of Figure 5 focuses on the dimensional changes during sintering, specifically comparing axial (building direction) and radial (in plane) shrinkage. The data reveal that axial shrinkage is significantly greater than radial shrinkage. This anisotropic shrinkage behavior suggests that the material contracts more along the vertical axis than horizontally. Such differences in shrinkage can be attributed to the initial particle arrangement and the directional dependencies of the sintering process [32] [34]. These observations are crucial for understanding the overall densification and structural evolution of the material, informing future adjustments in sintering protocols to achieve more uniform shrinkage and improved material properties.

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /><figcaption>Figure 5. Sintering dilatometry of the curves. (A) Sintering dilatometry results showing temperature and relative density as functions of time. (B) Sintering dilatometry results illustrating changes in height (h0/h) and diameter (d0/d), along with relative density and temperature over time.</figcaption></figure>
To address the differences observed between the isotropic and anisotropic samples, the applied model for the pressed (isotropic) samples needs to be adjusted. The primary distinction lies in the initial density and critical porosity between the two types of samples. For the anisotropic samples, the critical porosity parameter was modified to better match the experimental observations. The results of these adjustments are presented in Figure 6, which demonstrates the improved fit of the model to the anisotropic sample data. This adjustment is essential as it accounts for the inherent differences in the microstructure and sintering behavior between pressed and printed porous skeletons, ensuring more accurate predictions and better understanding of the sintering process for both types of materials.

$$ \psi = a \frac{(0.6-\theta)^{3.465}}{0.6} \quad \& \quad \varphi = (1 - \frac{\theta}{0.6})^2 \qquad (22) $$

<figure><img src="image_12.png" /><figcaption>Figure 6. Adjusted Sintering Model for Anisotropic Samples with Modified Critical Porosity Parameter. (A) Linear regression showing the relationship between 1/RT and Y. (B) Error analysis identifying the optimal parameter b=3.465.</figcaption></figure>
5.5. 3D Printed sample sintering mechanism analysis and grain growth estimation

At this stage all densification parameters are identified. The last stage of the finale stage sintering described in the theory part is applied to assess the sintering mechanism, grain size curve and anisotropy correction. Equation (18) is used for different w mechanisms and the result is reported in figure 7a. The grain boundary diffusion (w=4) and lattice diffusion (w=3) do not explain the obtained value. As the 1000ppm of MgO doping is present in the starting powder which is known the significantly modify the diffusion involving more vacancies [48] a lower w=2 mechanism is also tested. However, all these curves are far from the final density.

The creep/capillarity separated formulation (19) is then used to see if considering the separation of capillarity forces grain radius ($r_0$) and diffusional part grain size sensitivity ($m$) is better representative. Such hypotheses affect a fixed $r_0$ corresponding to agglomerates and $G$ for the grain size. This is reported in figure 7B and give more realistic curves with an $m=1$ (equivalent to w=2) mechanism that perfectly match the final grain size. This mechanism can correspond to grain boundary sliding [49] which normally does not apply here because the sintering stress is almost isotropic. A small anisotropy result the printing but it is untenable that this small stress anisotropy could result a grain boundary sliding. Cannon *et al* [50] also introduce “interface-controlled diffusional creep” that is a low stress diffusional mechanism with a grain size sensitivity of $m=1$. We can also cite the viscous flow mechanism [51] where we assume here the MgO doping significantly increases particle deformation. However, the most probable explanation is that the 1000ppm MgO generates high vacancy concentration variations disturbing the lattice diffusion in a less nonlinear behavior. It is known the mechanistic exponent value can be modified if the system evolves during the test as it is criticized in the limit of Herring’s scaling law that assumes a same system but only scalability transformation. In our system, the experiment clearly indicates an $m=1$ behavior that is considered as an experimental law.

Based on this curve the second regression equation (20) is used to assess the grain growth behavior; this regression is reported in figure 7c. The circular like shape of the curves are due to the small oscillation of the temperature around the set temperature but a clear linear tendency is found which enable the identification of the transient stage of grain growth.
<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>Figure 7. Sintering Mechanism Analysis and Grain Growth identification: in (A) the model assuming grain size is the same as particle size is used, in (B) the equation assuming the decorrelation between the grain size and agglomerated particle size is used, (C) the identification of the grain growth law by regression is applied to the m=1 mechanism.</figcaption></figure>

Incorporating the identified sintering mechanism m=2 and the transient grain growth law it is possible the correct the final stage sintering of our model for the pressed specimens. The results are reported in figure 8a. It is clear that all the sintering stages are well reproduced using the identified grain growth behavior. The final grain sizes are not far from the experiment (figure 8b) even despite the grain growth law is identified from the printed specimens.
<figure><img src="image_16.png" /><figcaption>Figure 8. Correction of Final Stage Sintering in Pressed Specimens: (A) Sintering Model Results Incorporating Identified Sintering Mechanism and Transient Grain Growth Law, (B) Comparison of Final Grain Sizes Between Model Predictions and Experimental Data.</figcaption></figure>

5.6. Sintering anisotropy modeling

The isotropic sintering stress model is represented by Skorohod’s formula $Pl = \frac{3\alpha}{r} (1 - \theta)^2$. It was originally developed for pressed samples, which exhibit uniform densification behavior in
all directions. However, when applied to the 3D-printed anisotropic samples, this model needed to be adjusted due to the intrinsic directional dependencies of the printed material.

To address this, we modified the sintering stress equations to account for the anisotropic nature of the printed samples. Specifically, the sintering stress was adjusted for each principal direction, resulting in the following set of equations [23]:

$$ \begin{cases} Plx = Pl(1 + c\theta) \\ Ply = Pl(1 + c'\theta) \\ Plz = Pl(1 + c''\theta) \end{cases} \qquad (23) $$

In these equations, the constants c, c', and c'' were introduced to capture the different rates of shrinkage and densification in the x, y, and z directions, respectively. These constants were calibrated by fitting the model to experimental data, ensuring that the model accurately reflected the behavior of the anisotropic samples.

After adjusting the model, we observed a good agreement between the experimental data and the simulation results (see Figure 9 with an R-squared of 0.9975 for the relative density curve indicating a good fit). The modified model successfully predicted the directional variations in shrinkage and densification, as demonstrated by the close match between the experimental and simulated relative densities and dimensional changes. This agreement is illustrated in Figure 9, which shows the effectiveness of the adjusted model in capturing the anisotropic sintering behavior of the 3D printed samples.

The results highlight the importance of considering directional dependencies in the sintering stress for accurately modeling the sintering process in anisotropic materials. This adjustment allows for better prediction and control of the densification process in 3D printed materials, ultimately leading to improved material properties and performance.
<figure><img src="image_17.png" /></figure>

<figure><img src="image_18.png" /></figure>

<figure><img src="image_19.png" /><figcaption>Figure 9. Adjusted Sintering Model for Anisotropic Samples: Comparison of Experimental and Simulated Results.</figcaption></figure>
The origin of this anisotropy can arise from multiple factors, such as anisotropic porosity, grain orientation, or the architecture of the porosity distribution. It is always challenging to analyze the initial particle distribution because the specimen is fragile and cannot be polished. However, a detailed analysis of the grains suggests an orientation of alumina grains in the sintered sample. The results are presented in Figure 10. In the SEM image (Figure 10a), a certain number of grains appear to be aligned at -45° (indicated by the black arrow). To analyze this orientation, the "OrientationJ" add-on of the "ImageJ" software was used. A visual directional method was employed with a resolution of 1 and 2 pixels, identifying oriented zones (violet) corresponding to a normal direction of 45° (-45° grain orientation angle). In the inset, the color scale represents the angle normal to the preferential direction (calculated from the pixel grayscale gradient).

In Figure 10b, the orientation analysis confirms a preferential orientation angle of 45°, as indicated by the violet zones in the previous analysis. This observation suggests that the extrusion process slightly oriented the alumina grains, which could also contribute to the measured sintering anisotropy.

<figure><img src="image_20.png" /><figcaption>Figure 10. Use of OrientationJ to reveal the preferential grain orientation in a sintered SEM image: (a) SEM image analyzed using the pixel orientation method, indicating the direction of the local dominant pixel gradient (perpendicular to the grain boundary) with 1- and 2-pixel resolution; (b) normal angle distribution curve showing a preferential angle of 45°.</figcaption></figure>

**6. Conclusion**

This study aimed to advance the understanding of the sintering behavior of 3D printed alumina parts, focusing on the anisotropic nature introduced by the additive manufacturing process. The isotropic sintering model was initially applied to the pressed samples, but it was necessary to
adjust this model for the anisotropic 3D-printed samples. By introducing directional sintering
stress constants, we successfully calibrated the model to reflect the different rates of shrinkage
and densification in the x, y, and z directions. The adjusted model showed a good agreement
with experimental data, demonstrating its validity in capturing the anisotropic sintering
behavior.

The study also included an analysis of the sintering mechanism and grain growth estimation.
The onset of sintering for the alumina powder was identified at 1200°C, with an initial grain
size G0 of 190 nm. Using the activation energy determined from the Master Sintering Curve
(MSC) study, we employed a direct regression method to fine-tune the bulk modulus and other
parameters, ensuring consistency with the experimental data. This comprehensive approach
highlighted the critical role of grain growth in the final stage of sintering, emphasizing the
necessity to account for this phenomenon in the model. In particular it shows the grain size
effect on diffusional creep part and capillarity can be separated and in this alumina, the
densification sensitivity to grain growth is particularly low and unconventional. This research
lays the groundwork for further studies and potential industrial applications, where accurate
simulations are needed to adjust the part-dimensional control.

Author contribution

Fatima Hammoud, Jérôme Lecourt, Christelle Bilot and Charles Manière: Conceptualization,
Methodology, Software, Data curation, Writing – original draft, Visualization, Investigation,
Writing – review & editing.

Funding

This work has been funded by the BPI France project "ZETAFACTORY" number "259872".
We would like to thank Guillaume de Calan and Nanoe ZETAMIX's team for their help for
samples printing.

Declarations

**Competing interests** The authors declare no competing interests.

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