# Modeling the sintering trajectory of ZnO by cold sintering process

Nicolas Albar, Charles Manière, Thomas Hérisson de Beauvoir, Emeric

Sanchez, Geoffroy Chevallier, Alicia Weibel, Claude Estournès

► To cite this version:

Nicolas Albar, Charles Manière, Thomas Hérisson de Beauvoir, Emeric Sanchez, Geoffroy Chevallier, et al.. Modeling the sintering trajectory of ZnO by cold sintering process. Acta Materialia, 2025, 290, pp.120974. 10.1016/J.ACTAMAT.2025.120974. hal-05015110

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Distributed under a Creative Commons Attribution 4.0 International License
Modeling the sintering trajectory of ZnO by Cold Sintering Process

Nicolas Albar¹, Charles Manière², Thomas Hérisson de Beauvoir¹, Emeric Sanchez¹,

Geoffroy Chevallier¹, Alicia Weibel¹, Claude Estournès¹

**Abstract:** It was recently demonstrated that the apparent activation energy for densifying ZnO by Cold Sintering Process in the presence of acetic acid was reduced by a factor of four compared to Spark Plasma Sintering of the same dry raw powder. Building on this study, the sintering curves of zinc oxide sintered by Cold Sintering Process are modeled for the first time. By applying load sintering models to Cold Sintering Process, it is possible to predict microstructures (densities and grain sizes) through simulating the experimental sintering curves. Key sintering parameters (activation energy of densification mechanisms and grain growth, creep law stress exponent, creep law grain size sensitivity exponent) are determined, enabling a discussion of the mechanisms involved during the Cold Sintering Process of ZnO in acetic acid.

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<footnote>1: CIRIMAT, Université Toulouse 3 Paul Sabatier, CNRS-INP-UT3, Université de Toulouse, Toulouse 31062, France</footnote>

<footnote>2: Normandie Univ, ENSICAEN, UNICAEN, CNRS, CRISMAT, 14000 Caen, France</footnote>
## Nomenclature

<table><tr><td>Θ Sintering work</td><td>R Gas constant (8.314 J.mol<sup>-1</sup>.K<sup>-1</sup>)</td></tr><tr><td>θ Porosity</td><td>T Temperature (K)</td></tr><tr><td>θ̇ Porosity elimination rate (s<sup>-1</sup>)</td><td>φ Shear modulus</td></tr><tr><td>ρ Relative density</td><td>ψ Bulk modulus</td></tr><tr><td>σ<sub>z</sub> Applied uniaxial stress</td><td>P<sub>1</sub> Sintering stress (Pa)</td></tr><tr><td>n Creep law stress exponent</td><td>Ġ Grain growth rate (s<sup>-1</sup>)</td></tr><tr><td>A Creep law deformability term (s<sup>-1</sup>.Pa<sup>-n</sup>)</td><td>G Grain size (m)</td></tr><tr><td>A<sub>0</sub> Creep pre-exponential factor (K.s<sup>-1</sup>.Pa<sup>-n</sup>)</td><td>G<sub>0</sub> Initial grain size (m)</td></tr><tr><td>m Creep law grain size sensitivity exponent</td><td>K<sub>0</sub> Grain growth pre-exponential factor</td></tr><tr><td>Q<sub>d</sub> Densification activation energy (J.mol<sup>-1</sup>)</td><td>Q<sub>g</sub> Grain growth activation energy (J.mol<sup>-1</sup>)</td></tr></table>

## 1. Introduction

The Cold Sintering Process (CSP) has been under development for a few years [1], demonstrating significant advantages, and offering new possibilities in the elaboration of ceramic and composite materials [2,3]. Despite major advances in sintering applications, the development of CSP shed light on the complexity of the involved mechanisms [4,5]. This led to some limitations in its application, particularly for materials with poor and/or incongruent dissolution [6]. These drawbacks have motivated the development of alternative routes to obtain dense, pure materials, such as the use of ionic liquids [7,8], or reactive sintering [9]. Although these approaches showed interesting opportunities, the necessity for a better understanding of the underlying mechanisms was demonstrated.

Different modeling studies were carried on CSP to determine both the underlying chemical reactions leading to densification and the macroscopic strain evolution during the sintering process. Local simulations, using Reactive Force Field approach, have demonstrated the possibility to identify chemical reactions at solid-liquid interfaces, and applied to model materials such as ZnO [10,11] or functional materials like BaTiO<sub>3</sub> [7]. These studies revealed the high sensitivity of the hydration state of the solid, as well as its surface in promoting or inhibiting dissolution processes. They also control the nature of precipitation, that can both lead to epitaxial growth on existing grains or nucleation of new ones, with various crystallinity. Other approaches, focused on the kinetics of sintering, were also developed and adapted to CSP. Some researchers have examined the kinetics of early sintering stage to determine the limiting
step (dissolution – transport – precipitation) [12], while others have focused on the intermediate sintering stage to quantify deformation during the sintering plateau [4,13] and grain growth during the last stages. Liang et al. showed that ZnO sintered with acetic acid with high temperature ramps allows to reduce by 4 the activation energy of granular growth [14] determined by the Kingery model. Although these works have confirmed some differences in the thermal behavior, especially the activation energies associated with the densification process [15,16], they have shown the possibility to adapt the simulation parameters to mathematically describe the materials deformation and densification. In particular, kinetics modeling has shown the importance of chemical reactions that can lead to both densifying and/or non-densifying mechanisms.

Recently, a kinetics study has shown that CSP can be analyzed using classical approaches, such as the Master Sintering Curve (MSC) method. To further investigate the kinetic behavior, the present work aims at investigating the use of an analytic densification/grain growth model to predict the sintering trajectory of a ZnO powder.

## 2. Theory and calculations

### 2.1. Determination of Qd

MSC model introduced by Su and Jonhson [17] allows for the determination of activation energy over a range of density of the densification mechanism using three different heating rates. It is possible to determine a Θ parameter (Equation 1) that provides a unique curve (master sintering curve) when plotted with the relative density regardless of the sintering profile.

$$ \Theta = \int_{0}^{t} \frac{1}{T} \exp \left( \frac{-Q_d}{RT} \right) dt \quad (1) $$

Θ only depends on Q and temperature profile. To determine Qd, a mean residual method is used to minimize the difference between ρ(Θ) at different temperature ramps in the desired density range. This method was originally developed for conventional sintering, but as demonstrated by Guillon and Langer [18], the sintering model integration is also compatible with pressure-assisted sintering at constant pressure.
## 2.2. Determination of n

It is possible to determine the value of n by carrying out two complications at the same temperature and different pressures. The value of n is determined by Equation 2 [19].

$$n = \frac{\ln(\frac{\dot{\theta}_1}{\theta_2})}{\ln(\frac{\sigma_{z,1}-P_l}{\sigma_{z,2}-P_l})} \quad (2)$$

When the capillarity forces are negligible, Equation 3 is used. For CSP, the high pressures involved (typically in the range 0.1-1 GPa) and the low temperatures allow to neglect the sintering stress $P_l$ which is the Laplace stress originated from particle capillarity forces [19–21].

$$n = \frac{\ln(\frac{\dot{\theta}_1}{\theta_2})}{\ln(\frac{\sigma_{z,1}}{\sigma_{z,2}})} \quad (3)$$

## 2.3. Skorohod-Olevsky SPS equation

The Skorohod-Olevsky SPS analytic model (Equation 4) [19,22] allows to describe the evolution of the relative density as a function of time and temperature.

$$-\frac{1}{\rho} \frac{d\rho}{dt} = A(T, G) \left(\psi + \frac{2}{3}\varphi\right)^{\frac{-n-1}{2}} (1-\theta)^{\frac{1-n}{2}} (\sigma_z - P_l)^n \quad (4)$$

With $A(T, G) = \frac{A_0}{T} (\frac{G_0}{G})^m \exp(\frac{-Q}{RT})$ for the thermal dependence.

Regarding the $\varphi$ and $\psi$ moduli, which are porosity functions reflecting the mechanical behavior of the porous material, the theoretical Skorohod moduli can be used [23]. However, it has been shown that these moduli often differ significantly from the experimental values [19], and therefore need to be determined using the following modified Skorohod functions [24], $\Psi = a(\frac{\theta_{cr}-\theta}{\theta^c})$ and $\varphi = (1-\frac{\theta}{\theta_{cr}})^2$ with the parameters $\theta_{cr}$, a, b and c to be determined using the regression equation (Equation 5) derived from the Skorohod-Olevsky equation [22]. By isolating the parameters Q and $\ln(A_0)$ we define the parameter Y which depends on T, $\theta$, n, $\sigma_z$, $\varphi$ and $\psi$.

$$Y = \ln(T|\dot{\theta}||\sigma_z - P_l|^{-n} (\psi + \frac{2}{3}\varphi)^{\frac{n+1}{2}} (1-\theta)^{\frac{n-3}{2}}) = \ln(A_0) - \frac{Q}{RT} \quad (5)$$

By adjusting the parameters $\theta_{cr}$, a, b and c, the Skorohod theoretical modulus can be corrected to impose a slope that corresponds to the activation energy determined using the MSC method.
This adjustment must be made to match the slope of the activation energy determined by MSC for each apparent densification regime. It is important not to consider the high-temperature region, which reflects a slowdown of densification due to grain growth impact. This phenomenon is marked by a decrease in the parameter Y in the linear regression. The value of A₀ is obtained from the intercept of each slope for each mechanism. If the final stage of sintering is strongly correlated with grain growth, the grain size curve in the high-temperature region can be estimated using Equation 6 [22].

$$G = \left(\frac{T}{A_0 G_0^m}\right) \exp\left(\frac{Q}{RT}\right) |\dot{\theta}| |\sigma_z - P_l|^{-n} \left(\psi + \frac{2}{3}\varphi\right)^{\frac{n+1}{2}} (1 - \theta)^{\frac{n-3}{2}} \frac{1}{m} \quad (6)$$

Three different values of m are used to plot three estimated grain size curves, which correspond to (m = 3) grain boundary liquid phase matter transport [25], (m = 2) solid-state diffusion and grain boundary sliding, and (m = 1) dislocation motion adjacent to the grain boundary. These mechanisms primarily applied to hot pressing sintering models [26]. Interrupted experiments can be conducted to determine which value of m corresponds to the experimentally measured grain sizes.

When m is identified, the others grain growth parameters Qg and K₀ can be determined using the following linear regression (Equation 7):

$$\ln(\dot{G}G^2) = \ln(K_0) - \frac{Q_g}{RT} \quad (7)$$

The slope of the curve ln(˙GG²) = f(1/RT) corresponds to Qg and the y-intercept is ln(K₀). The determination of parameters n, Qd, A₀ for each densification mechanism and parameters m, Qg and K₀ for grain growth allows for the plotting of simulated sintering curves and the grain growth thus predicting the evolution of the material microstructure with sintering conditions grain sizes.

### 3. Experimental

The starting powder used is a commercially available ZnO powder from Alfa Aesar, NanoArc (> 99%) with a particle size in the range 40-100 nm. A 1 M acetic acid solution in distilled water was prepared from Sigma Aldrich glacial acetic acid (> 99%). For the sintering experiments, 1 g of ZnO powder was mixed with 0.1 mL of 1 M acetic acid solution and manually mixed in a mortar. After 5 min of manual mixing, the powder was placed in a 8 mm
diameter WC-Co die. To prevent direct contact between the die, the punches and the powder, a
0.2 mm graphite foil (PERMAFOIL®Toyo Tanso) was placed between the powder and each
punch, but also vertically between the inner wall of the die and the powder.

Sintering experiments were performed in a Spark Plasma Sintering (SPS) apparatus (FUJI
632Lx from Fuji Electronic Industrial CO., Japan) using DC pulsed current, located at the
*Plateforme Nationale CNRS de Frittage Flash* at the Université Toulouse 3 Paul Sabatier.
Temperature was monitored using a K type thermocouple placed in a hole halfway up the
external surface of the die. Typical CSP setup does not allow the application of high temperature
ramps (> 20-30 °C/min) and therefore the use of SPS was chosen to overcome this technical
limitation. Therefore, the use of the term CSP-SPS will be used to clarify this point throughout
the manuscript. In previous studies on *in-operando* impedance characterization of the same
powder/liquid preparation it has been shown that the mixture exhibits a resistivity in the range
of 10² Ω·cm [27,28]. This high resistivity likely forces the current to pass through the
conductive WC-Co die rather than through the sample itself. As a result, the effect of current
flow through the sample can be considered negligible in the case of CSP-SPS of ZnO with
acetic acid. Pressure application is controlled by the SPS equipment, along with dilatometric
measurements that are performed during the whole sintering experiment. A uniaxial load
(equivalent to 100 MPa on the sample) was applied under vacuum (≈ 10 Pa) or in air at room
temperature and left to stabilize the pressure related displacement for 1 min at room temperature
and reach the green density (ρ₀). Then, in the same atmosphere, a temperature ramp (20 °C/min,
50 °C/min and 100 °C/min) was applied to reach 300 °C with a 5 min dwell time of applied
once this targeted temperature attained.

A blank measurement was made on the whole setup after sintering to remove contribution from
the thermal expansion of the column (sample + punches + spacers). This blank analysis allowed
the correction of the dilatometric data to only consider the shrinkage of the powder compact.
Density measurements were further made on sintered specimens by geometrical shape
measurements and weighting. The obtained data were compared to the theoretical ZnO density
of 5.606 g/cm³ to determine the relative density.

SEM observations were done with a TESCAN Vega 3 microscope using 20 kV acceleration
voltage on fractured samples. The intercept method is used for the determination of grain size,
it is realized on an average of hundred grains per sample.
# 4. Results and discussion

## Determination of activation energy of densification (Qd) using MSC approach

**Figure 1a, b** shows the plot of the relative density for ZnO sintered at three different heating rates (20, 50 and 100 °C/min), the pressure in the SPS chamber of the experiment realized at 50 °C/min and the mean residual calculation between 70 and 90% of relative density for several activation energy values. In this range of density, Qd is equal to 79 kJ/mol. **Figure 1c** shows 2 different regimes of densification marked by a break in the slope of the relative density as a function of ln(Θ) described by **Equation 1**. The break seems to appear at 115 °C for the three experiments. To determine the activation energy of the low temperature regime (regime 1, T < 115 °C) and the high temperature regime (regime 2, T > 115 °C), the mean residual method is applied only on the relative density range 70 -75% and 80-85% for Qd,1 and Qd,2 respectively. Between 75% and 80% of relative density, the transition of densification mechanisms takes place. This density range is therefore not taken into account for the measurement of activation energies by the MSC method. Qd,1 and Qd,2 are equal to 90 and 80 kJ/mol, respectively (**Fig. 1d**). The activation energy measured over the 70-90% range is underestimated compare to those found for the two mechanisms separately.
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Figure 1: a) Sintering curves plotted at different heating rates and pressure in the SPS chamber for the experiment at 50 °C/min adapted from [16], b) Mean residual calculated between 70 and 90% of relative density for various values of activation energy, c) Density vs ln(Θ) for Qd = 79 kJ/mol and d) Mean residual calculated between 70 and 75% of relative density for regime 1 and between 80 and 85% of relative density for regime 2 for various activation energy values.

Determination of n exponent

**Figure 2a, c** represents the isothermal evolution of porosity as a function of time at 50 and 100 MPa, with a 30 min dwell at 60 °C (**Figure 2a**) and 125 °C (**Figure 2c**). The two temperatures were chosen to represent regime 1 at 60 °C and regime 2 at 125 °C, in order to determine the n value for each mechanism. **Figure 2b, d** show the plots of n values as a function of porosity determined using **Equation 2**. The n values are 1.5 and 1 for regime 1 and 2, respectively. For hot-pressing, n = 1 is associated to grain boundary diffusion or lattice diffusion [26]. In the case of ZnO with a small amount of acetic acid, the diffusion could occur on the surface of grains due to the solubility of ZnO in acetic acid. The mechanism of dissolution and reprecipitation by grain boundary diffusion thanks to a liquid phase [29] could thus justify an n value of 1. For regime 1, there are a lot of perturbations for the determination of the n value, which can be
explained by the presence of two concomitant diffusion mechanisms, grain boundary diffusion (n = 1) and grain boundary sliding (n = 2) [26]. The elimination rate being slower at this temperature, the dilotometry measurement tends to oscillate slightly, justifying the larger standard deviation than for the measurement at 125°C where the porosity elimination rate is much higher. For regime 1, an average n value of 1.5 was selected for further use in the sintering model.

<figure><img src="image_7.png" /><figcaption>Figure 2: a) Porosity elimination during isothermal sintering at 60 °C during 30 min under 50 MPa and 100 MPa, b) n value for different porosity before 75% of relative density, c) Porosity elimination during isothermal sintering at 125 °C during 30 min under 50 MPa and 100 MPa and d) n value for different porosity after 80% of relative density.</figcaption></figure>

## Determination of ψ

The plot of the linear regression of Equation 5 (Figure 3) allows to adjust shear and bulk moduli to fit the slope with the activation energy determined by MSC for each regime. The first parameter to adapt is the θcr which is included in ψ and φ parameters. θcr corresponds to the porosity at which the effective porous body loses its strength, and the stress intensification tends to infinity due to contact points between particles. This idealized state of the material closely resembles the initial state of the powder, which is characterized by low strength. However, it is
not possible to choose θcr = θinitial, as this would cause the stress intensification factor to start
at an infinite (+∞) value, leading to errors in the calculation. Furthermore, the initial state of
compacted powder possesses some strength due to particle friction and agglomeration.
Therefore, a slightly higher value must be selected. In general, a margin of 10% is often chosen,
as this accounts for possible variations in finite element modeling (FEM). When using FEM
simulations, higher porosities may be encountered due to complex shapes, edge artifacts, or
inverted elements. If a single point exceeds the critical porosity θcr, the simulation may
terminate prematurely. For this reason, a 10% margin is commonly adopted to ensure robust
simulations [30]. θcr is set at 0.375. Using a single ψ does not allow to fit the two regions
before and after 115 °C with the same linear regression (Fig.3a). The first method to fit the
activation energies to linear regression is to define a bulk modulus for each regime. Figure 3b
shows the plot of the linear regression with two ψ parameters adapted to be closer to the
activation energy of each mechanism determined using the MSC method. For the two regimes,
the fitted parameters a and b are the same, 2/3 and 2 respectively, and the parameter c is equal to
3.9 for the mechanisms that occurs at T < 115 °C and 1.4 for the one that occurs at T > 115 °C.
The bulk modulus is related to the stress applied locally at the contact between two grains.
Determining two different ψ means that the surface of the grains is brutally deformed at 115
°C. This can be caused by intense degassing at this temperature as evidenced by the increase of
the pressure in the SPS chamber (Fig.1a) that can cause cracks leading to a modification of the
contact surface between the grains and therefore a sudden drop in the psi parameter at the time
of degassing. Figure 3c shows an example of the evolution of the bulk modulus for the sample
at 20 °C/min. Considering ψ₁ = 2/3 (θcr-θ/θ³.₉)² and ψ₂ = 2/3 (θcr-θ/θ¹.₄)², it is possible to determine Qd₁
and Qd₂ that are equal to 88 kJ/mol and 80 kJ/mol, respectively and ln(A₀₁) and ln(A₀₂) that are
equal to 1.71 and 4.35, respectively. At the end of regime 2, the linear regression deviates from
the experimental curve, indicating the beginning of significant grain growth.
<figure><img src="image_8.png" /><figcaption>Figure 3: a) Regression analysis for correction of the modulus and the identification of A₀ for a single modulus for both regimes, b) Regression analysis and determination of ψ₁ and ψ₂ for regime 1 and 2 in accordance to the activation energy calculated using the MSC method, c) Evolution of the modulus during the sintering for the sample realized at 20 °C/min.</figcaption></figure>

<table><caption>Table 1: Values of coefficients ψ, ψ₁ & ψ₂, along with activation energies Qₐ and ln(A₀).</caption><thead><tr><th></th><th>ψ</th><th>ψ₁</th><th>ψ₂</th></tr></thead><tbody><tr><td>a</td><td>2/3</td><td>2/3</td><td>2/3</td></tr><tr><td>b</td><td>2</td><td>2</td><td>2</td></tr><tr><td>c</td><td>3.9</td><td>3.9</td><td>1.4</td></tr><tr><td>Qₐ (kJ/mol)</td><td>-</td><td>88</td><td>80</td></tr><tr><td>ln(A₀)</td><td>-</td><td>1.71</td><td>4.35</td></tr></tbody></table>

For the second method considered, a single value of ψ is taken into account during the whole sintering step. We can consider that the liquid phase that was added at the start will, due to the effect of the temperature, occupy a larger volume in the porosities. The pressure exerted by the vaporization of this liquid phase is called swelling pressure (Ps). This pressure is opposite to the applied pressure σz which has the effect of slowing down the sintering during the first stage of the sintering process [31]. This pressure increases with the increase of temperature until the degassing of the gas phase at 115 °C (Fig.1a). Ps is determined using Equation 8:
$$Ps = \sigma_z - e^{-\frac{\left(\ln(A_{01}) - \frac{Q_1}{RT} - \ln(T|\theta|(\psi + \frac{2}{3}\varphi))^{\frac{n_1+1}{2}}(1-\theta)^{\frac{n_1-3}{2}}\right)}{n_1}} \quad (8)$$

The determination of ln(A₀₁) is possible by considering that the Y value for regime 1 is the same as the Y value for regime 2 at 115 °C, as evidenced in Equation 9. The activation energy used for regime 1 is 90 kJ/mol calculated by the MSC method. Knowing Qd₁ (90 kJ/mol), Qd₂ (80 kJ/mol) and ln(A₀₂) (4.35), ln(A₀₁) is determined to be 7.46.

$$-\frac{Q_{d1}}{RT} + \ln(A_{01}) = -\frac{Q_{d2}}{RT} + \ln(A_{02}) \quad (9)$$

**Figure 4a** shows the Ps plot calculated for heating rates of 20 °C/min, 50 °C/min and 100 °C/min, the Ps value is calculated to compensate the difference between the slope corresponding to Qd₁ determined by MSC and the value calculated by linear regression (Equation 5). The samples sintered at 20 and 50 °C/min seem to have the same Ps values. A discrepancy is observed for the sample sintered at 100 °C/min, which can be explained by differences in sample preparation and/or mold tightness. For the continuation of this work, one fit of Ps is considered for 20 and 50 °C/min, and another one. This results in a fully linear curve for regime 1, as shown in Figure 4b. In this way, a continuous ψ value across both densification regimes is observed in Figure 4c.
<figure><img src="image_9.png" /><figcaption>Figure 4: a) Calculation of Ps using dilatometric data for different heating rates and determination of the temperature dependent function Ps = f(T) during regime 1, b) Regression analysis for correction of the bulk modulus and the identification of A₀ for a single bulk modulus for both regimes using Ps values and c) Evolution of the bulk modulus during sintering.</figcaption></figure>

To verify the validity of the model, simulated curves of ρ = f(T) are plotted along with experimental data for the two methods (method 1 considering two ψ method and method 2 considering a single ψ with Ps) in **Figure 5**. Both methods generate simulated densification curves in really good agreement with the experimental point for ρ < 0.9. However, the last stage of sintering (ρ > 0.9) is not simulated correctly since at this stage the grain growth impact is not taken into consideration. To implement the simulation, grain growth parameters should be determined. For that, the method 2 will be used with a single bulk modulus and the swelling pressure Ps.
<figure><img src="image_10.png" /><figcaption>Figure 5: Simulated $\rho = f(T)$ curves obtained using the two methods for $\psi$ determination and experimental ones at a) 20 °C/min, b) 50 °C/min and c) 100 °C/min.</figcaption></figure>

**Figure 6** shows the results of grain sizes calculated using Equation 6 for the 3 heating rates assuming 3 probable mechanisms (grain boundary sliding (m = 1), lattice diffusion (m = 2) and grain boundary diffusion (m = 3)). The comparison of the nine curves with the grain size measured from interrupted tests at 275 °C for each heating rate shows good agreement for m = 3. Grain boundary diffusion is thus the relevant grain growth mechanism for the CSP of ZnO with acetic acid. The CSP grain growth mechanism seems to be similar to the one of liquid phase sintering under load for which the creep law grain size sensitivity exponent is also equal to 3 [25,26].
<figure><img src="image_11.png" /><figcaption>Figure 6: Estimated grain size as a function of temperature when a mechanism of grain boundary sliding (m = 1), lattice diffusion (m = 2) or grain-boundary diffusion (m = 3) is assumed.</figcaption></figure>

It is now possible to calculate grain size with Equation 6 using m = 3. The determination of grain growth parameters is possible by applying the grain size calculated values in equation 7.
**Figure 7a** shows the linear regression that allows to determine grain growth activation energy Qg and ln(K0). On **Figure 7b**, the simulation of the grain size is compared to the experimental data. The slope of the linearization is set to a value of Qg = 89 kJ/mol, the value of ln(K0) has to be adapted to fit correctly the experimental measured grain sizes and is ln(K0) = -28.92.

<figure><img src="image_12.png" /><figcaption>Figure 7: a) Identification of grain growth model based on grain boundary diffusion (m = 3) and b) Simulation of grain growth using the average of Qg and ln(K0) for each heating rate and using adapted ln(K0) to fit with experimental data.</figcaption></figure>

All the parameters (Qd1, Qd2, Qg, ln(A01), ln(A02) and ln(K0)) have been determined and allow to plot the simulated curves in **Figure 8**. The simulated curves fit well the experimental data on whole sintering stages. As expected, the introduction of grain growth parameters slows down densification in the last stage of sintering.
<figure><img src="image_13.png" /><figcaption>Figure 8: Simulated $\rho = f(T)$ curves with and without grain growth and experimental ones at a) 20 °C/min, b) 50 °C/min and c) 100 °C/min.</figcaption></figure>

To confirm the viability of the model, the measured grain size after a 5 min dwell at 300 °C for each heating rate are compared with the simulation (Figure 9). There is a good agreement between the simulation and the experimental data, which validates the model. The simulation of the sintering is thus applicable to CSP and shows a good predicting capacity on the densification and grain size. It is possible, using simulations, to optimize the sintering parameters and find ideal cycle to limit grain growth while maximizing densification.
<figure><img src="image_14.png" /><figcaption>Figure 9: Simulation and experimental measurement of grain growth during the temperature ramp and at 300 °C after 5 min of dwell time with different heating rates.</figcaption></figure>

**5. Conclusion**

This work demonstrates, for the first time, that a simulation model of sintering under pressure is applicable to the Cold Sintering Process (CSP). ZnO, in the presence of 1 mol/L acetic acid, exhibits two distinct densification regimes: one before and one after 115 °C, the temperature at which part of the liquid phase is eliminated, slowing its effect on sintering. The stress exponents of the creep law, n, determined as 1.5 for regime 1 (T < 115 °C) and 1 for regime 2 (T > 115 °C), indicate the presence of two concomitant diffusion mechanisms (grain boundary diffusion and grain boundary sliding) for regime 1, and only grain boundary diffusion for regime 2. A mechanism of dissolution and reprecipitation, facilitating grain boundary diffusion in the presence of a liquid phase, can explain the CSP of ZnO with a small amount of acetic acid. A value of m = 3 for the grain size sensitivity exponent in the creep law suggests that grain boundary diffusion is the relevant grain growth mechanism in the CSP of ZnO with acetic acid. The values of n = 1–1.5 and m = 3 confirm that CSP of ZnO predominantly occurs by grain boundary diffusion.
The presence of the liquid phase in the first stage of densification is responsible for the slowdown of the densification process, as evidenced by the decrease in densification activation energies between the two regimes ($Q_{d1} \sim 90$ kJ/mol and $Q_{d2} \sim 80$ kJ/mol). The grain growth activation energy ($Q_g \sim 90$ kJ/mol) is similar to that of densification, making it difficult to optimize the sintering cycle to achieve densification while limiting grain growth.

Two methods were investigated for simulating the densification part of the sintering curves. The first method involves adapting the bulk modulus between the two densification regimes, marking a change in the value of $\psi$ during densification. The second method integrates the swelling pressure (Ps) that opposes densification. All parameters were determined for each stage of densification to accurately fit experimental curves. Grain growth parameters were also calculated, allowing for the determination of grain size at any time during sintering.

Future work can focus on specifically studying the impact of acetic acid concentration on grain growth. It is also possible to limit Ps by decreasing the quantity of acetic acid while increasing its concentration to maintain an equivalent number of moles.

## 6. Acknowledgements

N.A. acknowledge the *Groupe Français de la céramique* (GFC) for the mobility scholarship to CRISMAT in Caen.

## 7. References

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