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# Constrained viscous sintering of melting grains

Domenica Braile, Colin Hare, Marco Ramaioli, Chuan-Yu Wu

► To cite this version:

Domenica Braile, Colin Hare, Marco Ramaioli, Chuan-Yu Wu. Constrained viscous sintering of melting grains. Powder Technology, 2024, 433, pp.119200. 10.1016/j.powtec.2023.119200. hal-05301583

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# Constrained viscous sintering of melting grains

Domenica Braile<sup>a</sup>, Colin Hare<sup>a, b</sup>, Marco Ramaioli<sup>c</sup>, Chuan-Yu Wu<sup>a, *</sup>

## HIGHLIGHTS

* Heat transfer and grain size effects on sintering dynamics of melting chocolate.

* Bridge growth and surface temperature captured with IR and optical cameras.

* Sintering is faster for small spheres, at high air flow temperature and velocity.

* The coalescence shows a lag time corresponding to a critical viscosity.

* Pokluda model was modified and successfully used to predict the bridge growth.

## GRAPHICAL ABSTRACT

<figure><img src="image_4.png" /></figure>

## ARTICLE INFO

### Keywords:

Viscous sintering

Heat transfer

Dark chocolate

Bridge growth

## ABSTRACT

Viscous sintering of food grains can occur spontaneously and cause undesired caking or be induced to drive a change in food product structure. The dynamics of this phenomenon are still poorly understood, particularly when the sintering is coupled with heat or mass transfer. The aim of this work is to investigate experimentally the effects of heat transfer and grain size on the sintering dynamics of partially melting fat-based food products, i.e., dark chocolate. Two spherical particles were heated through forced convection using an air stream at different temperatures and velocities. Optical and IR thermal cameras were employed to capture the bridge growth and surface temperature evolution. Subsequently, the optical images were analysed to extract the evolution of the size of the bridge between the grains and a first order kinetics model was used to describe the sintering dynamics. The experimental observation showed that the viscous sintering is faster for smaller spheres, and at higher air flow temperature and velocity. When the spheres are heated from ambient conditions, a lag time is observed, which was found to be related to the time required by the material to reach a temperature at which its viscosity decreases to about 54 Pa·s allowing flow. Therefore, the lag time is temperature-dependent and decreases with decreasing particle size and increasing air temperature and velocity. Moreover, a semiempirical model involving a temperature dependent viscosity was proposed to describe the viscous coalescence. It is shown that considering some correction parameters accounting for the experimental conditions leads to an accurate prediction of the bridge growth. The model could be incorporated into the discrete element method to simulate the viscous sintering of a bulk of chocolate grains.

<footnote><sup>a</sup> School of Chemistry and Chemical Engineering, University of Surrey, Guildford GU2 7XH, United Kingdom</footnote>

<footnote><sup>b</sup> School of Chemical Engineering, Newcastle University, Newcastle upon Tyne NE1 7RU, United Kingdom</footnote>

<footnote><sup>c</sup> Université Paris Saclay, INRAE, AgroParisTech, UMR SayFood, 91120 Palaiseau, France</footnote>

<footnote>* Corresponding author.
E-mail address: c.y.wu@surrey.ac.uk (C.-Y. Wu).</footnote>
# 1. Introduction

Viscous sintering occurs when two grains, upon contact, coalesce to reduce their total surface area. The driving force is the reduction of surface energy, while viscous and inertial forces hinder the bridge growth. In amorphous materials, an increase of temperature leads to a higher molecular mobility, a decrease in viscosity, and faster flow under internal or external forces. The large curvature around the contact point between two particles leads to a high surface energy, which the system minimises by mass flow from inside the particles to the contact point with the formation of a bridge or neck [1]. Unless external forces hinder the growth, the bridge grows until the surface energy is minimum and the full coalescence of the particles is achieved.

Viscous sintering of granular materials is common in many industries such as polymers, fertilisers, bio-industries, food and pharmaceuticals [1]. In the food industry, viscous sintering is often undesirable as it causes caking of food powders [1], yet desired in other applications, such as additive manufacturing [2]. One of the materials commonly used in additive manufacturing processes is chocolate. This material can be defined as a mixture of amorphous and crystalline components. Its amorphous characteristics are given mainly by sugars, while its crystallinity is attributed to the cocoa butter which is a mixture of triglycerides. Depending on its composition and manufacturing process, the cocoa butter can crystallise into six polymorphic forms. Crystals in the form (V) are the preferred ones because these are the most palatable form, hence, chocolate used in the food industry is usually subjected to a tempering process, which increases the polymorphic form (V) and makes the material more suitable to produce confectionery products [2].

Few studies were performed on viscous sintering of food grains [2]–[4], Rando et al. investigated, using the finite element method (FEM), the sintering coupled with moisture transfer for maltodextrins [3], as an amorphous material, and coupled with heat transfer for chocolate [5], as a semi-crystalline material. In the latter study, they proposed a generic FEM model which couples the heat transfer and sintering processes and applied it to investigate the effect of heat transfer on the sintering phenomenon between two unconstrained 3D printed filaments of chocolate. They found that for materials with a viscosity strongly dependent on the temperature, the bridge growth prediction cannot ignore the viscosity gradient caused by thermal treatments. Furthermore, they proposed and experimentally validated a modified hyperbolic model to evaluate the viscosity of chocolate as a function of the temperature. However, despite providing useful insights, their research was mainly numerical. Experimental studies on sintering of food grains are scarce.

Nevertheless, viscous sintering of polymeric materials was widely investigated experimentally [6,7] and numerically [8]. Le Boterff et al. [7] experimentally investigated the viscous coalescence of two molten polymer beads for various temperatures and employed a theoretical model reported by Pokluda [9] to predict the grain coalescence considering surface tension and viscosity as temperature-dependent properties. Their results showed that using these two temperature-dependent parameters allows to predict accurately the viscous coalescence of polymers grains. Additionally, Alvarez et al. [8] developed a novel discrete element method (DEM) for simulating the visco-elastic sintering of polymer powders. In their model, the viscous sintering process is modelled as a three-stage process, each stage is characterised by a different driving force, i.e., adhesive contact force, adhesive inter-surface force and surface tension. Their results showed that their model can predict the coalescence of polymers powders with greater accuracy compared to traditional theoretical sintering models such as Frenkel's.

Using DEM to simulate bulk of grains which experience sintering is possible only if the dynamics of the coalescence at the particle scale is well understood and described. This understanding was achieved through theoretical models for metals and ceramics [10,11] and hence, many authors were able to employ the DEM to simulate the sintering of such materials [12,13]. As previously discussed, the viscous sintering of

polymer grain pairs was experimentally investigated [7,14] and models to predict the coalescence of polymers at the particle scale are available [8]. However, due to lack of experimental studies, the knowledge of sintering of food grain is still scarce and to the authors' knowledge, no experimental investigation exists on the sintering of partially melting fat-based food products, such as chocolate grains. Therefore, this project developed a novel experimental setup to investigate the bridge growth of dark chocolate spheres, heated through forced convection at different temperatures and velocities. Additionally, the Pokluda model [9] was modified to predict the coalesce angle during viscous sintering of chocolate particles.

# 2. Methods

In this section, the material, sample preparation and experimental setup are first presented. The image analysis and IR thermography is then introduced together with a discussion on the material properties for various temperature. The Pokluda model [9] and its modification for the coalescence prediction is introduced in Section 2.6.

## 2.1. Material and sample preparation

In this study, spheres of dark chocolate BC-811 (Barry Callebaut, UK) consisting of 37.8% fat and 54.5% cocoa solids were used. Chocolate was first melted and tempered using a Mini Rev. Tempering Machine (Revolution 2B, ChocoVision, United Kingdom). In this process, the temperature of the material is first increased from room temperature to 45 °C to melt the cocoa butter crystals and subsequently cooled down to desired temperature (32 °C). The melted material was then injected into a mould and allowed to cool down and solidify to obtain spheres of 0.8, 1.4, 2 and 3 mm in diameter ($D_0$). The mould was created by employing Ecoflex™ and using glass beads having a specific size. As previously mentioned, the tempering of chocolate increases the polymorphic form (V) of cocoa butter.

## 2.2. Experimental setup

A 2D representation of the experimental setup is provided in Fig. 1. One chocolate sphere (1) is placed on top of a metal plate (not heated), while the other one (2) is fixed on a needle which can be moved using a laboratory elevator represented in black. The latter allows manual vertical and horizontal adjustment of the needle. Such a system permits positioning a sphere (2) on top of a second sphere (1), ensuring a near perfect alignment. However, during the viscous coalescence the needle is fixed and cannot move in any direction. The system is heated through forced convection with air at a specific temperature ($T_∞$) and flow velocity ($v_∞$). For each experiment, air flow temperature (Tair) and velocity (vair) are measured in the vicinity of the spheres (red cross in Fig. 1) with a thermometer and an anemometer, respectively.

Two types of experiments were performed, i.e., isothermal and non-isothermal. For the former, the spheres are first heated at around 35 °C (surface temperature detected with the IR camera) then placed in contact, hence, in such case the temperature of the spheres surface can be considered constant and the bridge formation occurs as soon as the spheres are set in contact. For the latter, the two spheres are first aligned and placed in contact at ambient temperature then the forced convection is initiated. Hence, in this case, the sphere surface temperature is in non-isothermal condition and the bridge formation occurs only after the temperature reaches a certain value at which the viscosity is low enough for the material to flow.

An optical and an infrared camera were used to capture the bridge size growth and the temperature on the bridge surface. In the non-isothermal experiments, the two cameras were actioned at the same time of the forced convection. During isothermal experiments, the recordings were initiated as soon as the temperature on the spheres surface, detected by the IR camera, was about 35 °C.
All the experiments were repeated in triplicate and ImageJ was used to extract the bridge size (a), while the IR thermal camera data were used to extract the time evolution of the temperature on the bridge surface (T). In all the graphs presented in this paper, the solid lines indicate the average of the repetitions, while shades represent the standard deviation.

## 2.3. Image analysis

An optical camera Basler Ace 2 (Basler AG, Germany) with lens f = 50 mm 1:2.4 (Computar, China) was employed to record image sequences of the viscous sintering. A Java code algorithm was developed and implemented in ImageJ [15] to process the images as follows. Fig. 2 shows a representation of the stages performed in the imaging processing. The first step (Fig. 2a) is performed manually and involves converting the image in 8-bit greyscale and cutting the image in the area of interest (contact point between the spheres). The algorithm is then used to set a black and white threshold (Fig. 2b) followed by the automatic bridge detection (c) on each image of the sequence. The black and white threshold is performed by using the tool “threshold” available in ImageJ which allows to set lower- and upper-pixels threshold values automatically. The bridge is then detected by scanning the pixels in the z and y direction of the image. For each z-pixel, the software detects the number of black pixels on the y-direction, hence finding a certain segment (e.g., a₁ and a₂ dashed white lines in Fig. 2c, corresponding to the pixels y₁ and y₂). After detecting two consecutive segments, the algorithm compares their size and registers the one which has the minimum length. This minimum length is then compared to the next detected segment and so on, until the minimum black horizontal length of the entire image is detected, that is considered as the bridge size (a), which is represented with a solid red line in Fig. 2c. The number of pixels corresponding to the bridge is then converted in mm based on the ratio pixel/mm evaluated with the calibration tool available in the software (note that the dimension of the spheres is known).

## 2.4. Infrared thermal camera calibration and data extraction

An infrared camera Testo 885 (S.A. Testo N-V, Belgium) was employed to detect the temperature on the bridge surface during the viscous coalescence of the spheres. IR thermography is a non-invasive technique which does not affect the viscous phenomenon; however, calibrating the thermal camera is an important step to obtain accurate and reliable temperature readings. In this work the calibration of the thermal camera was carried out by changing the emissivity parameter while comparing the value measured by the camera and the value measured by two thermometers. These two were placed in two different locations within the melted chocolate, but as close as possible to the surface and the IR camera readings were extracted in proximity to the

thermometer locations. Setting the emissivity to 0.95 allowed to obtain temperature readings with the thermal camera, which are in good agreement with the measurements using the thermometers. Fig. 3 shows a typical image captured by the IR thermal camera. In the next sections the graphs representing the time evolution of temperature, during the experiments, refer to the temperature detected on the bridge surface, i. e., black cross in Fig. 3.

## 2.5. Effect of temperature on the properties of dark chocolate

The temperature variation of the material leads to a change in its thermal properties. Density (ρ), surface tension (γ), and conductivity (k) were considered constant with temperature, while the impact of the heat transfer on its viscosity (μ) could not be neglected [5]. Rando et al. [5] proposed and validated a model to predict the viscosity of chocolate BC-811 with temperature:

$$ \begin{cases} \mu = \mu_{max}, & \text{for } T \le T^* \\ \mu = \mu_0 + \frac{k_1}{T - T_s}, & \text{for } T > T^* \end{cases} \quad (1) $$

In which

* $\mu_0$ and $k_1$ are model parameters,

$$ T^* = T_s + 0.25^\circ C, $$

* $T_s$ is the solidification temperature,

* $\mu_{max}$ is a high-viscosity cutoff.

In this work, the model above is used to evaluate the viscosity of the material based on the temperature detected by the IR thermal camera on the bridge surface.

If not otherwise specified, all these parameters are chosen as suggested by Rando et al. [5] and are listed in Table 1.

The effect of the temperature on the viscosity of chocolate BC-811, evaluated with Eq. (1), considering a temperature range from 18 to 80 °C are shown in Fig. A1 of the Appendix.

## 2.6. Coalescence model

A simple theoretical model, available in literature, for predicting the viscous coalescence evolution for isothermal spherical particles was proposed by Pokluda [9]:

$$ \frac{d\theta}{dt} = \frac{2^{-5/3} \cos\theta \sin\theta [2 - \cos\theta]^{1/3}}{t_{\text{visc}} [1 - \cos\theta] [1 + \cos\theta]} \quad (2) $$

where θ is the angle of coalescence as shown in the representation in

<figure><img src="image_5.png" /><figcaption>Fig. 1. 2D representation of the experimental methodology.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 2. Stages of the image processing to extract the bridge size.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 3. Typical thermal image captured by the IR camera. The black cross indicates the location where the surface temperature data are extracted.</figcaption></figure>

<table><caption>**Table 1**

List of parameters employed to estimate the viscosity [5].</caption><thead><tr><td>Parameter</td><td>Value</td></tr></thead><tbody><tr><td>μ<sub>0</sub></td><td>37.08 Pa s</td></tr><tr><td>k<sub>1</sub></td><td>173.60 Pa s °C</td></tr><tr><td>T<sub>s</sub></td><td>21.50 °C</td></tr><tr><td>μ<sub>max</sub></td><td>789.80 Pa s</td></tr></tbody></table>

<caption>Fig. 4 and $t_{visc}$ is defined based on the initial diameter of the spheres ($D_0$) (as)</caption>

$$t_{visc} = \frac{D_0\mu}{2\gamma} \qquad (3)$$

This and all the theoretical models available in literature on the sintering of two spherical particles assume free movement of the grains, which under the action of the driving force are free to move towards each other along their axis and reach the full coalescence with a final diameter equal to 1.41 $D_0$ [5,9] or a final coalescence angle ($\theta_f$) equal to 1.57 rad (90°). However, this is not valid for our experimental setup in which there are two constraints, one between the top sphere and the needle, and the other one between the bottom sphere and the metal plate. Hence, in our experiment, the spheres are not free to move during sintering and the full coalescence is unlikely to be achieved. Previous experimental studies on polymers have used an oil based liquid to reduce the interaction between the polymer grains and the surface [7], however, due to the fat content of chocolate, oils interact with the chocolate grains and influence the bridge growth; hence, the use of the same methodology was unsuccessful with our material. A representation of the various sintering stages as observed in our experiments is provided in Fig. 5.

The first two stages (Fig. 5), which consists of contact and bridge formation between the two grains, are similar to the ones described by the theoretical models in literature [5,9]. On the other hand, the bridge growth (stage 3 in Fig. 5) is affected by the presence of constraints which slows down the coalescence and, therefore, reduces the bridge growth kinetic. And in the final state (stage 4 in Fig. 5) the full coalescence is not achieved, rather the two-grains system reaches an equilibrium state with a final bridge size equal to $a_{fin}$. Additionally, as shown in stage 5 of Fig. 5

<figure><img src="image_8.png" /><figcaption>Fig. 4. Representation of the coalescence angle $\theta$.</figcaption></figure>
the surface curvature around the metal-sphere contact point leads to another bridge formation which competes with the spheres coalescence. Yet, because of the heat conducted through the cold metal plate, the temperature rise at the sphere-metal contact point is slower compared to temperature rise at the sphere-sphere contact. Hence, the metal-sphere bridge forms later than the sphere-sphere bridge and the impact of the sphere-metal contact can be regarded negligible only prior to the formation of the sphere-metal bridge. Hence, our experiment is concluded in stage 4.

Therefore, in presence of constraints, the bridge growth kinetic and a<sub>fin</sub> significantly differs from the case of two spheres which are free to move, and therefore the theoretical models must be adapted to predict the constraint viscous coalescence. In this work, we propose a semi-empirical model (Eq. 4 and 5) in which the Pokluda model [9] is modified with two parameters λ and ε, which describe the sintering dynamic and maximum coalescence during constrained viscous sintering, respectively.

$$ \frac{d\theta}{dt} = \frac{2^{-5/3} \cos(\varepsilon\theta) \sin(\varepsilon\theta) [2 - \cos(\varepsilon\theta)]^{1/3}}{t_{\text{visc}}[1 - \cos(\varepsilon\theta)][1 + \cos(\varepsilon\theta)]} \quad (4) $$

$$ t_{\text{visc}} = \frac{\lambda D_0 \mu}{2\gamma} \quad (5) $$

The parameter ε is equal to 1 in absence of constraints, when the full coalescence can be achieved, while when constraints are present it is impossible to determine the value of ε theoretically. Hence, to obtain an accurate fit of the experimental observation, the parameter ε needs to be evaluated based on the final angle θ observed experimentally (θ<sub>f,ex</sub>) and theoretically (θ<sub>f,th</sub>):

$$ \varepsilon = \frac{\theta_{f,th}}{\theta_{f,ex}} = \frac{\frac{\pi}{2}}{\theta_{f,ex}} \quad (6) $$

As for ε, if constraints are present, it is impossible to determine a theoretical value of the parameter λ which, hence, needs experimental calibration. In our case, an optimisation algorithm was implemented to find the λ value which minimises the error of the model predictions.

### 3. Dimensional analysis

In this section a dimensional analysis is carried out to evaluate gravitational, viscous, inertial and surface tensions forces. In particular, the Eötvös number [16] compares the gravitational forces with the surface tensions forces as follows:

$$ E_0 = \frac{\rho g D^2}{\gamma} \quad (7) $$

where g is the gravitational acceleration. Systems where the surface

tension dominates the gravitational stress are characterised by low values of the Eötvös number (E<sub>0</sub> << 1), on the opposite, a high Eötvös number indicates that a system is unaffected by the effect of the surface tension.

The Ohnesorge number compares the viscous forces with the inertial and surface tension forces and is defined as [16]:

$$ Oh = \frac{\mu}{\sqrt{\rho\gamma D}} \quad (8) $$

where μ is the viscosity of the material. High values of the Ohnesorge number (Oh > 1) indicate that the viscous forces dominate the inertial forces resulting from the motion driven by the surface tension. Hence the inertial effects can be neglected [16].

The Eotvos number does not depend on the temperature because all the parameters considered in Eq. 7 are considered constant. Hence, for both the experiments (isothermal and non-isothermal conditions) the Eotvos number is equivalent and is reported in Table 2 and evaluated considering ρ = 1235 kg/m³ and γ = 2.6 10⁻² N/m as suggested by Ref. [5]. The results show that E<sub>0</sub> < 1 for particle diameters smaller than 2 mm, while for larger diameters E<sub>0</sub> is on the threshold. Hence, for the latter, the gravitational forces and surface tension are concurrent, while for smaller spheres, the gravitational effect can be neglected, and the surface tension effect can be considered dominant.

The Ohnesorge number evaluated for the isothermal experiment is also reported in Table 2 and its values are significantly higher than 1 for all particle diameters considered. This dimensionless number is temperature-dependent because it is evaluated based on the viscosity (Eq. 8) and, hence, in non-isothermal conditions its value is not constant. However, in the non-isothermal experiments, the viscosity is the highest at the start (ambient T) and decreases to the lowest values, when the temperature reaches the steady state. Therefore, since the viscosity appears in the numerator in Eq. 8, it can be concluded that the Oh number in the non-isothermal condition is higher or equal to the one evaluated for the isothermal case. Hence in all cases Oh >> 1, which implies that for all experiments during mass flow the viscous forces dominate over the inertial forces.

<caption>**Table 2**</caption>

The Eotvos and Ohnesorge (for the isothermal condition) numbers evaluated based on the chocolate grain size

<table><thead><tr><th>D<sub>0</sub> (mm)</th><th>E<sub>0</sub> (-)</th><th>Oh (-)</th></tr></thead><tbody><tr><td>0.8</td><td>0.30</td><td>311.96</td></tr><tr><td>1.4</td><td>0.91</td><td>235.82</td></tr><tr><td>2.0</td><td>1.86</td><td>197.30</td></tr><tr><td>3.0</td><td>4.19</td><td>161.10</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>**Fig. 5.** Representation of the constrained viscous sintering of two particles heated through forced convection</figcaption></figure>
## 4. Results and discussion

The evolution of the bridge growth from stage 1 to stage 4 (in Fig. 5) as captured in our experiment is shown in Fig. 6. Note that these optical images refer to a test performed in non-isothermal conditions, however the same bridge growth evolution was observed for the isothermal case. As visible in Fig. 6, for experiments reported in this study, the variation of sphere diameter during the coalescence is very small, hence, the grain size is assumed constant, i.e., D(t) = D₀. Moreover, it is important to mention that the initial value of the bridge is also not considered in our analysis because its value could be influenced by geometrical factors, such as grain and optical camera position. Hence, for the reasons highlighted above, this study focuses primarily on the effect of particle size and heating condition on the sintering dynamics, which will be discussed in the following sections. The last image in Fig. 6 shows the typical final state observed in our experiments which is characterised by a non-complete coalescence of the two spheres. As discussed in the methodology, the non-full coalescence was expected since the experimental setup presents constraint to the grain movements, however there is another possibility related to the non-Newtonian nature of the material. Pawar et al. [17] showed that the coalescence of viscoelastic droplets with internal microstructure can be arrested when the interfacial energy and elastic energy balance one another. The sintering reduces the interfacial energy, but the coalescence causes a compression of the internal structure leading to an increase of the elastic energy. Hence, since our material is non-Newtonian, its elasticity could play a key role in the arrested coalescence.

### 4.1. Assuming a constant surface temperature

The temperature detected on the bridge surface with time (in Fig. 7) is not constant but its variation is small and can be considered as isothermal. From Fig. 7 the temperature is slightly higher for larger particles, which is believed to be caused by the distance between the sphere and the source of heat. The distance between the needle and the heat source is fixed, hence larger particles have their surface closer to the heat source. The time evolution of the dimensionless bridge size ($a/D_0$) for various particle sizes is shown in Fig. 8. As soon as the contact between the two spheres takes place (time = 0 s), the bridge grows instantaneously with a certain dynamic until the final bridge size is reached. Note that the experiments on D₀ = 1.4 mm present a general low repeatability because the mould was created using glass beads having a particle size distribution (1.2 to 1.6 mm) rather than a single size. On the other hand, the other glass beads (0.8, 2, 3 mm) were bought with the specific size from the manufacturer.

The abscissas axis of the time series data in Fig. 8 can be normalised dividing by the viscous time (Eq. 3) as proposed in literature [5,19]:

<figure><img src="image_10.png" /><figcaption>Fig. 7. Time evolution of the temperature detected on the bridge surface for various particle sizes during the isothermal experiments (T<sub>air</sub> = 38 °C - v<sub>air</sub> = 0.4 m/s)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>Fig. 8. Time evolution of the dimensionless bridge size (a/D₀) for various particle sizes during the isothermal experiments (T<sub>air</sub> = 38 °C - v<sub>air</sub> = 0.4 m/s)</figcaption></figure>

$$t^* = \frac{2t\gamma}{D_0\mu} \qquad (9)$$

<figure><img src="image_12.png" /><figcaption>Fig. 6. Images of the viscous coalescence (T<sub>air</sub> = 38 °C - v<sub>air</sub> = 1.15 m/s - D₀ = 2 mm) captured with the optical camera showing the typical bridge size evolution observed in our experiments.</figcaption></figure>
where the surface tension is considered equal to 2.6 10⁻² N/m [5], while the viscosity is evaluated with (Eq. 1) based on the temperature detected on the bridge surface (Fig. 7). Moreover, a dimensionless bridge cross section parameter (A') is introduced as:

$$A' = \left( \frac{a - a_0}{a_f - a_0} \right)^2 \qquad (10)$$

where the bridge size (a) is normalised with the initial (a₀) and final (a_f) bridge values. Fig. 9 shows how the dimensionless bridge cross section (A') evolves with t*, and compared to Fig. 8, the time normalisation causes the small spheres data points to expand to the right of the abscissas axis, while the points referring to large grains shift to the left direction of the axis. Note that Fig. 9 also shows a comparison with data extracted from Rando et al. [5] represented with the green cross symbol. They employed the finite element method (FEM) to investigate the viscous coalescence of the cross section of 3D printed unconstrained cylindrical layers of the same material employed in this work. The trend extracted from their paper and represented in Fig. 9 refer to a cross section diameter equal to 0.8 mm. However, despite using a normalised time and bridge size, the trends are different implying that the results from this work do not compare well with the FEM observation. However, as previously mentioned in this work the experimental setup presents some constraints which makes our results not comparable with simulations employing theoretical models. Indeed, it is shown that at the initial stage of the sintering (A' < 0.3) the trends almost overlap. This suggests that the constraints posed in the current experiments lead to a different sintering behaviour when A > 0.3.

To quantitatively analyse the sintering dynamics for various particle sizes, a first order kinetics model (Eq. 11) is then used to fit the experimental values as shown by the solid lines in Fig. 9.

$$A' = 1 - \exp(-k \cdot t) \qquad (11)$$

The kinetic parameter (k) is used to quantify the dynamics of the sintering phenomenon as impacted by the particle size. As shown in Table 3, the value of k decreases with the particle size, meaning that the sintering rate reduces as the particle size is increased. This can be explained by the greater surface area to volume ratio for smaller particles, which in terms of heat transfer means supplying more heat per unit mass.

Polychronopoulos et al. [18] have demonstrated that when two

<figure><img src="image_13.png" /><figcaption>Fig. 9. Dimensionless bridge cross section (A') as a function of the dimensionless time t* for different particle size (D₀) in isothermal conditions (Tair = 38 °C - vair = 0.4 m/s). Note that Rando et al. [5] data refer to FEM simulations of the unconstrained viscous sintering of cylinders. Solid lines represent the fitting with a first order kinetics equation (Eq. 11).</figcaption></figure>

equal-sized spheres undergo sintering with minimal variation in their diameters (D=D₀), Pokluda's model (Eq. 2) can be simplified as follows:

$$\left(\frac{a}{D_0}\right)^2 = 1 - \exp\left(-\frac{1}{t_{visc}} \cdot t\right) \qquad (12)$$

In our specific scenario, A' (Eq. 10) is proportional to Eq. 12:

$$A' = \left( \frac{a - a_0}{a_f - a_0} \right)^2 = 1 - \exp(-k \cdot t) \propto \left( \frac{a}{D_0} \right)^2 = 1 - \exp\left(-\frac{1}{t_{visc}} \cdot t\right) \qquad (13)$$

This implies that the kinetic parameter (k) in Eq. 11 is proportional to 1/tvisc, where tvisc (Eq. 3) represents the viscous time. This demonstrates that k has a tangible physical interpretation. Nevertheless, using Eq. 12 to predict the bridge growth in our experiments proved unsuccessful because the presence of constraints hindered the complete coalescence of the grains, in contrast to what was observed in the referenced paper.

The Pokluda model (Eq. 2) and the modified model (Eq. 4) were employed to predict the sintering of spheres (D₀ = 0.8 mm) in isothermal conditions (35 °C), surface tension (γ) equal to 2.6 10⁻² N/m and constant viscosity (μ) evaluated at 35 °C with Eq.1 as 50.8 Pa s.

Note that the coalescence angle (θ) is normalised based on the initial (θ₀) and final value (θ_f) as:

$$\theta = \frac{\theta - \theta_0}{\theta_f - \theta_0} \qquad (14)$$

The evolution of the normalised coalescence angle (θ') as predicted by the Pokluda model (dashed line) and the modified Pokluda model (solid line) is compared to the experimental results (symbols) in Fig. 10. It is shown that the Pokluda model slightly overpredicts the kinetic of the observation. However, employing a modified viscous time (Eq. 5) with λ equal to 3.1, allows to obtain a better prediction of the coalescence (R² = 0.85). The calibration procedure for λ was applied to all the experiments in isothermal conditions (Fig. 9) and Fig. 11 shows the model prediction (lines) using the optimum λ in comparison with the experiments (symbols). The model does not show a high accuracy for all the cases, indeed, the coefficient of determination R² ranges from 0.60 to 0.99, but it is able to express the general viscous coalescence trend for all the D₀ with greater accuracy compared to the original Pokluda model. In particular, the model predictions for D₀ = 1.4 mm present a low accuracy (R² = 0.60) and λ equal to 8 significantly different from the other cases. Such differences are due to the low repeatability of the experiment for D₀ = 1.4 mm as discussed above.

## 4.2. Non-isothermal sintering

In this experiment, initially at the ambient temperature, the viscosity is too high for the material to flow and the time evolution of the bridge growth shows an initial constant value, e.g., Fig. 12. When the temperature increases, the viscosity decreases to values which allow mass flow, the bridge starts growing at a certain rate which is affected by particle size and heating conditions. The time taken to pass from the no-flow to flow condition is defined as the lag time.

### 4.2.1. Influence of particle size on the coalescence dynamics

The impact of particle size on the viscous sintering was investigated by setting the air temperature and flow at 38 °C and 0.4 m/s. The time

<table><caption>**Table 3**
Kinetic parameter (k) and accuracy obtained by fitting the experiments with Eq. 11 at different D₀ (Isothermal conditions - Tair = 38 °C - vair = 0.4 m/s)</caption><thead><tr><th>D₀ (mm)</th><th>k (s⁻¹)</th><th>R²</th><th>RMSE</th></tr></thead><tbody><tr><td>0.8</td><td>0.163</td><td>0.98</td><td>0.031</td></tr><tr><td>1.4</td><td>0.065</td><td>0.92</td><td>0.059</td></tr><tr><td>2</td><td>0.061</td><td>0.98</td><td>0.029</td></tr><tr><td>3</td><td>0.055</td><td>0.98</td><td>0.041</td></tr></tbody></table>
<figure><img src="image_14.png" /><figcaption>**Fig. 10.** Prediction of the normalised viscous coalescence angle (θ') with Pokluda model (Eq. 2 - dashed line) and modified Pokluda (Eq. 4 - solid line) in comparison with the experimental observations (symbols) in isothermal conditions (T<sub>air</sub> = 38 °C - v<sub>air</sub> = 0.4 m/s – D<sub>0</sub> = 0.8 mm)</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 11.** Prediction of the normalised viscous coalescence angle (θ') with the modified Pokluda model (Eq. 4 - lines) in comparison with the experimental observations (symbols) in isothermal conditions (T<sub>air</sub> = 38 °C - v<sub>air</sub> = 0.4 m/s)</figcaption></figure>

evolution of the dimensionless bridge size ($\bar{D}_0$) for various particle sizes is shown in Fig. 12, while the temperature detected on the bridge surface as a function of time is shown in Fig. 13. At time equal to zero, the initial temperature is not affected by the particle size and the small differences are due to environmental conditions. When the forced convection is initiated, the temperature increases, and it can be noticed that small spheres have a generally higher temperature after a given time, which could be related to the ratio between sphere surface area and volume. For a smaller sphere this ratio between surface area and volume is higher, leading to a greater heat energy absorbed per unit mass. Additionally, the rate of heat dissipation by conduction from the surface to the centre of the chocolate grain is directly proportional to the diameter of the sphere. Hence, the smaller the particle, the lower the dissipation and the higher the temperature on the surface. On the contrary, for the steady state experiment previously discussed, as shown in Fig. 7, the surface temperature is higher for larger particles; in such case it is believed that the phenomenon of heat dissipation ends before placing the spheres in contact.

<figure><img src="image_16.png" /><figcaption>**Fig. 12.** Time evolution of the dimensionless bridge size (a/D₀) for various particle sizes during the non-isothermal experiments (T<sub>air</sub> = 38 °C - v<sub>air</sub> = 0.4 m/s)</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 13.** Time evolution of the temperature detected on the bridge surface for various particle sizes during the non-isothermal condition experiments (T<sub>air</sub> = 38 °C - v<sub>air</sub> = 0.4 m/s)</figcaption></figure>

The lag time ($t_{lag}$) and the temperature corresponding to the lag time ($T_{lag}$) are plotted against the particle size in Fig. 14, from which it is visible that for each size the bridge starts growing when its surface temperature reaches values above 32 °C. Based on Eq. (1) at this temperature the material has a viscosity equal to 53.6 Pa s, which can be considered as the viscosity threshold, above which the material is not able to flow. Because the heat must be conducted from the surface to the centre of the sphere, it is expected that the time required to reach the minimum temperature and viscosity threshold increases with size. Indeed, as shown in Fig. 14, the lag time increases linearly with the particle size and the experimental data can be fitted with a linear equation, as displayed in Fig. 14.

The dimensionless bridge cross section $A'$ is plotted against the dimensionless time (Eq. 9) in Fig. 15. Interestingly, the normalised lag time almost overlaps, which means that the variables in Eq. 9 strongly affect the time required by the material to pass from the no-flow to the flow regime. Since $\gamma$ is considered constant, the only variables in Eq. 9 that can affect the lag time are $D_0$ and $\mu$. Even if the viscosity is affected by the heat transfer, its effect can be considered minor compared to $D_0$.
<figure><img src="image_18.png" /><figcaption>**Fig. 14.** Plot of the lag time as a function of the initial particle size ($D_0$) in blue ($T_{air} = 38$ °C - $v_{air} = 0.4$ m/s). The secondary axis in red shows the temperature detected on the bridge surface at the lag time. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

Indeed, changing the size of the spheres affects the surface area to volume ratio which determines the heat transferred per unit mass. The larger the spheres, the lower is the heat supplied. Hence, the results show that the lag time scales perfectly with $D_0$. First order kinetics (Eq. 11) was also employed to fit the experimental data (symbols) without the lag time, as shown by the solid lines superimposed in Fig. 15. The kinetic parameter $k$ and statistical parameters ($R^2$ and RMSE) are listed in Table 4 and it is shown that even in non-isothermal conditions, the kinetic parameter overall decreases with the particle size. An exception is observed for $D_0$ equal to 2 mm for which the parameter $k$ (0.070 $s^{-1}$) is slightly lower than the one obtained for $D_0 = 3$ mm (0.077 $s^{-1}$). It is believed that the experimental errors (see error bars for 2 mm in Fig. 12) could have caused this inconsistency.

The modified Pokluda model (Eq. 4) can be also applied in non-isothermal conditions. However, having a non-constant temperature implies that the viscosity must be evaluated with the hyperbolic model (Eq. 1) using the surface temperature measured with the IR camera as an approximated temperature. As shown in Fig. 16, despite the optimisation of $\lambda$ in the model, using the parameters in Table 1 led to a poor prediction of the coalescence, which was found to be related to the values of maximum viscosity ($\mu_{max}$) and solidification temperature ($T_s$).

<figure><img src="image_19.png" /><figcaption>**Fig. 15.** Dimensionless bridge cross section (A') as a function of the dimensionless time $t^*$ for different particle size ($D_0$) in non-isothermal conditions ($T_{air} = 38$ °C - $v_{air} = 0.4$ m/s). Solid lines represent the fitting with a first order kinetics (FOK) equation.</figcaption></figure>

<caption>**Table 4**</caption>

Kinetic parameter ($k$) and accuracy obtained by fitting the experiments with Eq. 11 at different $D_0$ (non-isothermal conditions - $T_{air} = 38$ °C - $v_{air} = 0.4$ m/s)

<table><thead><tr><th>D<sub>0</sub> (mm)</th><th>k (s<sup>-1</sup>)</th><th>R<sup>2</sup></th><th>RMSE</th></tr></thead><tbody><tr><td>0.8</td><td>0.409</td><td>0.98</td><td>0.044</td></tr><tr><td>1.4</td><td>0.093</td><td>0.90</td><td>0.069</td></tr><tr><td>2</td><td>0.070</td><td>0.98</td><td>0.041</td></tr><tr><td>3</td><td>0.077</td><td>0.98</td><td>0.042</td></tr></tbody></table>

Choosing a suitable value of $T_s$ allows the correct prediction of the lag time, while $\mu_{max}$ is responsible for the estimation of the constant neck size prior to the lag time. Both these two parameters are intrinsic properties of the material and are not related to the heating conditions ($v_{air}$, $T_{air}$) and particle diameter ($D_0$), which means that a unique value of $\mu_{max}$ and $T_s$ must be determined and used for all the experimental conditions. Hence, $T_s$ is increased from 21.5 °C (in Table 1) to 32 °C corresponding to the average surface temperature at the lag time (Fig. 14). Note that in our case the need of modifying $T_s$ might be related to evaluating the viscosity with the surface temperature, which is certainly higher than the temperature within the chocolate grain. On the other hand, the parameter $\mu_{max}$ is the viscosity of chocolate at the solid state and since solids are not able to flow, $\mu_{max}$ is set to be infinite. Note that the effect of the temperature on the viscosity, evaluated with Eq. (1), considering the modification on of $\mu_{max}$ and $T_s$ are shown in Fig. A1 of the Appendix. Therefore, the modified Pokluda model (Eq. 4) was applied to several non-isothermal experimental cases through calibration of the parameters $\varepsilon$ (Eq. 6) and $\lambda$ (optimisation algorithm). The results are shown in Fig. 17a, b and c for $T_{air}$ equal to 38, 63 and 82 °C, respectively. The optimal values of $\lambda$ and the accuracy ($R^2$) of the predictions are listed in Table 5. It is shown that the proposed model can predict the viscous coalescence with good accuracy, however it is worth noting that the accuracy decreases with increasing air temperature. The reason behind this observation is believed to be the temperature employed to evaluate the viscosity. As previously mentioned, the surface temperature is not representative of the temperature gradient within the grain, and theoretically a greater heating temperature leads to a greater temperature difference between the surface and the centre of the particle. Hence, using the surface temperature to evaluate the viscosity is a good approximation only when $T_{air}$ is low, while at high $T_{air}$ the heat transfer within the grain must be solved to obtain more accurate predictions. Moreover, looking at Fig. 17, the prediction curves are not

<figure><img src="image_20.png" /><figcaption>**Fig. 16.** Prediction of the viscous coalescence angle (0) with modified Pokluda model (solid line) using a thermo-dependent viscosity (properties in Table 1). The prediction is plotted in comparison with the experimental observations represented by symbols (non-isothermal conditions - $T_{air} = 38$ °C - $v_{air} = 0.4$ m/s)</figcaption></figure>
<figure><img src="image_21.png" /></figure>

<figure><img src="image_22.png" /></figure>

<figure><img src="image_23.png" /><figcaption>**Fig. 17.** Prediction of the viscous coalescence angle (θ) with the modified Pokluda model (solid lines) in comparison with the experimental observations (Symbols) in non-isothermal conditions (v<sub>air</sub> = 0.4 m/s) with T<sub>air</sub> equal to (a)38 °C, (b) 63 °C and (c) 82 °C</figcaption></figure>

<caption>**Table 5**</caption>

Optimal λ based on D<sub>0</sub> and T<sub>air</sub> and its accuracy in predicting the spheres coalescence (in non-isothermal conditions).

<table><thead><tr><th rowspan="2">D<sub>0</sub> (mm)</th><th colspan="2">T<sub>air</sub> = 38 °C</th><th colspan="2">T<sub>air</sub> = 63 °C</th><th colspan="2">T<sub>air</sub> = 82 °C</th></tr><tr><th>λ</th><th>R<sup>2</sup></th><th>λ</th><th>R<sup>2</sup></th><th>λ</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>0.8</td><td>0.5</td><td>0.99</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1.4</td><td>2.7</td><td>0.98</td><td>0.9</td><td>0.98</td><td>3.9</td><td>0.83</td></tr><tr><td>2.0</td><td>1.7</td><td>0.99</td><td>3.6</td><td>0.94</td><td>2.1</td><td>0.90</td></tr><tr><td>3.0</td><td>1.8</td><td>0.99</td><td>3.9</td><td>0.97</td><td>1.8</td><td>0.93</td></tr></tbody></table>

smooth and show some abrupt changes which are associated with the experimental measurement of surface temperature, i.e., the time interval between two readings is too high to capture a thorough temperature variation. Indeed, the abrupt changes are even more visible at high Tair (Fig. 17 (c)), at which the temperature on the grain surface increases faster, while the measurement intervals of the IR thermal camera remain the same, i.e., the lowest limit of the IR camera.

### 4.2.2. Influence of air flow temperature on the coalesce dynamics

The investigation of the effect of the air temperature (T<sub>air</sub>) on the viscous coalescence was carried out in non-isothermal conditions, keeping a constant particle diameter (D<sub>0</sub> = 2 mm) and changing the T<sub>air</sub>

<figure><img src="image_24.png" /><figcaption>**Fig. 18.** Time evolution of the temperature detected on the bridge surface for various T<sub>air</sub> (non-isothermal conditions- D<sub>0</sub> = 2 mm - v<sub>air</sub> = 0.4 m/s)</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>**Fig. 19.** Time evolution of the dimensionless bridge size (a/D<sub>0</sub>) for various T<sub>air</sub> (non-isothermal conditions- D<sub>0</sub> = 2 mm - v<sub>air</sub> = 0.4 m/s)</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>**Fig. 20.** Time evolution of the dimensionless bridge cross section (A') as a function of the dimensionless time t* at different Tair (non-isothermal conditions- D₀ = 2 mm - vair = 0.4 m/s). Solid lines represent the fitting with a first order kinetics (FOK) equation.</figcaption></figure>

<caption>**Table 6**</caption>

Kinetic parameter (k) and accuracy obtained by fitting with a first order kinetics model the experiments at different T<sub>air</sub> (non-isothermal conditions- D<sub>0</sub> = 2 mm - v<sub>air</sub> = 0.4 m/s)

<table><thead><tr><th>T<sub>air</sub> (°C)</th><th>k (s<sup>-1</sup>)</th><th>R2</th><th>RMSE</th></tr></thead><tbody><tr><td>38</td><td>0.070</td><td>0.96</td><td>0.048</td></tr><tr><td>40</td><td>0.097</td><td>0.97</td><td>0.043</td></tr><tr><td>63</td><td>0.214</td><td>0.97</td><td>0.039</td></tr><tr><td>81</td><td>0.403</td><td>0.98</td><td>0.042</td></tr></tbody></table>

from 38 to 82 °C. The temperature on the bridge surface as recorded by the IR camera at different T<sub>air</sub> is reported in Fig. 18, while Fig. 19 shows the corresponding time evolution of the dimensionless bridge size (a/ D₀). When T<sub>air</sub> is equal to 38 and 40 °C, the temperature on the bridge surface is almost identical as expected, hence the corresponding dimensionless bridge size (Fig. 19) shows a negligible difference for the two cases. When the temperature increases from ambient temperature to

<figure><img src="image_27.png" /><figcaption>**Fig. 21.** Evolution of the first order kinetic parameter (k) with T<sub>air</sub> (non-isothermal conditions - D<sub>0</sub> = 2 mm - vair = 0.4 m/s). The data points in red were fitted with an exponential equation (k(s<sup>-1</sup>) = 0.0186e<sup>0.0383 T</sup>(°C), R<sup>2</sup> = 0.996) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

<figure><img src="image_28.png" /></figure>

<figure><img src="image_29.png" /><figcaption>**Fig. 22.** Time evolution of the dimensionless bridge cross section (A') for various v<sub>air</sub> (non-isothermal conditions- D<sub>0</sub> = 2 mm - T<sub>air</sub> = 40 °C) for spheres having a diameter equal to (a) 2 mm and (b) 3 mm</figcaption></figure>

around 32 °C the curves in Fig. 18 overlap implying that the dynamics with which the temperature increases is similar for all T<sub>air</sub>, while after 32 °C the higher is T<sub>air</sub> the faster the bridge growth rate. This observation appears to suggest that there is a different heat transfer regime before and after the lag time, which as detected in Fig. 19 for T<sub>air</sub> equal to 63 and 82 °C is around 19 s. The latter corresponds also to the regime transition observed in Fig. 18. Hence the reason behind the heat transfer regime transition is related to the phase transition from solid to liquid. As expected, as T<sub>air</sub> increases the material reaches the threshold temperature of 32 °C faster, implying a lower the lag time.

The bridge cross section was normalised based on its initial and final values, as shown in Eq. 10 and the results are presented in Fig. 20. Note that expressing the time in dimensionless terms (Eq. 9) in Fig. 20 does not produce any significant difference on the trends (in terms of scaling of lag time with T<sub>air</sub>) compared to those in Fig. 19. Having used the same particle diameter in all the tests, the only variable in Eq. 9 is the
<caption>**Table 7**</caption>

Kinetic parameter (k) and accuracy obtained by fitting with a first order kinetic model the experiments at different $v_{\text{air}}$ and $D_0$ (non-isothermal conditions - $T_{\text{air}} = 40$ °C)

<table><thead><tr><th rowspan="2"></th><th colspan="3">D<sub>0</sub> = 2 mm</th><th colspan="3">D<sub>0</sub> = 3 mm</th></tr><tr><th>k (s<sup>-1</sup>)</th><th>R2</th><th>RMSE</th><th>k (s<sup>-1</sup>)</th><th>R2</th><th>RMSE</th></tr></thead><tbody><tr><td>v<sub>air</sub> = 0.4 m/s</td><td>0.070</td><td>0.96</td><td>0.06</td><td>0.048</td><td>0.93</td><td>0.06</td></tr><tr><td>v<sub>air</sub> = 1.2 m/s</td><td>0.124</td><td>0.95</td><td>0.08</td><td>0.092</td><td>0.99</td><td>0.03</td></tr></tbody></table>

viscosity, and the fact that the plots do not experience any modification confirms our previous hypothesis of negligible effect of viscosity in Eq. 9. In terms of bridge growth rate, the superimposed solid lines in Fig. 20 represent the fitting with the first order kinetics equation (Eq. 11) of the experimental observation without the lag time, while Table 6 shows the kinetic parameter (k) and accuracy of the model. The first order kinetics equation can predict with good accuracy ($R^2$ above 0.96 in Table 6) the experimental bridge growth and demonstrates that the viscous sintering dynamics is faster at higher air temperature. Fig. 21 shows that the parameter k rises exponentially with increasing $T_{\text{air}}$.

### 4.2.3. Influence of air flow velocity on the coalescence dynamics

The impact of air flow velocity ($v_{\text{air}}$) on the viscous sintering dynamics was carried out in non-isothermal conditions employing two values of $v_{\text{air}}$, i.e. 0.4 m/s and 1.2 m/s. Fig. 22 shows how the normalised size of the bridge (Eq. 9) evolves with time based on the air flow velocity, for spheres having a diameter equal to 2 mm (Fig. 22a) and 3 mm (Fig. 22b). At a higher velocity the convective heat from the heat source to the sphere is increased, hence the temperature increase on the grain surface is faster. Therefore, at a higher velocity the lag time is expected to be lower which is confirmed for grains with $D_0$ equal to 2.0 mm (Fig. 22a), i.e., the blue curve ($v_{\text{air}} = 1.2$ m/s) has a lower lag time (20 s) compared to the red curve (35 s). On the contrary, the opposite is observed for spheres having a $D_0$ of 3.0 mm (Fig. 22b) for which the lag time is lower at $v_{\text{air}} = 0.4$ m/s; however, the difference between the two $t_{\text{lag}}$ is small (lower than 5 s). The faster increase of surface temperature at higher $v_{\text{air}}$ is confirmed by the viscous coalescence dynamics, which was quantified by fitting with a first order kinetic model (Eq. 11). As reported in Table 7, the kinetic parameter (k) is higher with increasing air flow velocity.

## 5. Conclusions

In this work, the effect of particle size and heating conditions on the constrained viscous sintering of chocolate spheres was investigated. The sintering dynamics was quantitatively analysed employing a first order kinetics equation. Results showed that the value of the kinetic parameter (k) decreases with increasing particle size, indicating a slowing of the sintering process, which is associated with the heat conduction rate within the spheres. Larger spheres have a lower surface area to volume ratio, therefore less heat is supplied per unit mass. Increasing the temperature and velocity of the forced convection causes a more rapid

increase of the temperature of the grains, which leads to the increase in the bridge growth rate. Specifically, the kinetic parameter increases exponentially with the air temperature. Nevertheless, experiments in non-isothermal conditions present a temperature dependent lag time which is found to be the time required by the material to reach a temperature of about 32 °C. Indeed, it is shown that the lag time decreases with decreasing particle size and increasing air temperature and velocity, as the heat transfer in such cases is faster. Additionally, a modified Pokluda model [9] was introduced to evaluate the constrained viscous coalescence. It was demonstrated that this model can successfully predict the viscous coalescence in both isothermal and non-isothermal conditions, provided the temperature evolution is known. Considering a temperature dependent viscosity is crucial to accurately predict the lag time in the non-isothermal case. The parameters introduced in this modified model require experimental calibration. Nevertheless, this modified model could be implemented in the discrete element method to simulate the viscous coalescence at the bulk scale and to simulate undesired caking under different chocolate storage scenarios.

### CRediT authorship contribution statement

**Domenica Braile:** Methodology, Software, Validation, Formal analysis, Investigation, Data curation, Writing – original draft, Writing – review & editing, Visualization, Project administration. **Colin Hare:** Methodology, Validation, Writing – review & editing, Supervision, Project administration. **Marco Ramaioli:** Conceptualization, Methodology, Validation, Resources, Writing – review & editing, Supervision, Project administration. **Chuan-Yu Wu:** Methodology, Validation, Writing – review & editing, Supervision, Project administration, Funding acquisition.

### Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Data availability

Data will be made available on request.

### Acknowledgments

This project is funded through Marie SKŁODOWSKA-CURIE Innovative Training Network MATHEGRAM, the People Programme (Marie SKŁODOWSKA-CURIE Actions) of the European Union's Horizon 2020 Programme H2020 under REA grant agreement No. 813202.

The authors would like to thank INRAE and AgroParisTech (Massy, France) for hosting this research work and all the employees in UMR SayFood for the fruitful research discussions and for the technical help.
Appendix A

<figure><img src="image_30.png" /><figcaption>Fig. A1. Temperature evolution of the viscosity of chocolate BC-811 evaluated with Eq. (1).</figcaption></figure>

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