# Droplet impact mapping in binder jetting of OPC concrete: Numerical modelling on a DEM powder bed

Asif Ur Rehman, Alexandre Pierre, Gregory Gibbons, Viktor Mechtcherine

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Asif Ur Rehman, Alexandre Pierre, Gregory Gibbons, Viktor Mechtcherine. Droplet impact mapping in binder jetting of OPC concrete: Numerical modelling on a DEM powder bed. Powder Technology, 2025, 465, pp.121328. 10.1016/j.powtec.2025.121328. hal-05245013

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# Droplet impact mapping in binder jetting of OPC concrete: Numerical modelling on a DEM powder bed

Asif Ur Rehman, Research Fellow<sup>a,b</sup>, Alexandre Pierre<sup>a,c,*</sup>, Gregory Gibbons<sup>d</sup>,
Viktor Mechtcherine<sup>e</sup>

## HIGHLIGHTS

* The study focuses on the Binder Jetting Technology process for cement powders, emphasizing binder-powder interaction.

* Computational fluid dynamics and an enhanced discrete element method allow to model droplet impact, penetration, and flow dynamics in the powder bed.

* Investigates the influence of binder droplet size on spreading, penetration depth, splashing, and flow patterns.

* Identifies capillary forces, viscosity, gravity, and inertia as critical factors governing binder interaction with the powder bed.

* Validation of the model through high-speed hard X-ray imaging from literature

## GRAPHICAL ABSTRACT

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

## ARTICLE INFO

**Keywords:**

Additive manufacturing
3D printing
Droplet size
Binder jetting
Powder bed

## ABSTRACT

Additive manufacturing (AM) has the potential to revolutionize the construction industry in a way where material, geometric complexity, building sustainability and design freedom can be interwoven [1]. Cementitious materials AM can reduce the structural environmental impact by one third (Wangler et al., 2016 [2]) along with the tailored material and mechanical properties. Binder jetting (BJT) is a powder-based AM method that enables the production of complex components for a wide range of applications. The pre densification interaction of binder and powder is vital among various parameters that can affect the BJT performance. In this study, the BJT process is studied for the binder interaction with the powder bed of Ordinary Portland Cement (OPC). The effects of capillary forces or action, viscosity, gravity and inertia are described via computational fluid dynamics (CFD) modelling of the system. The effect of the droplet-size is thoroughly analyzed. Two different droplet sizes are considered 0.3 mm and 0.5 mm. Through the application of particle flowlines, the uniformity of binder distribution over the powder bed is scrutinized. This research introduces an enhanced and more accurate discrete

<footnote><sup>a</sup> L2MGC, CY Cergy-Paris University, France</footnote>

<footnote><sup>b</sup> Barlett School of Sustainable Construction, University College London (UCL), UK</footnote>

<footnote><sup>c</sup> Univ Rennes, Laboratoire de Génie Civil et Génie Mécanique, 3 rue du Clos Courtel, Rennes 35704, France</footnote>

<footnote><sup>d</sup> WMG, University of Warwick, UK</footnote>

<footnote><sup>e</sup> TU Dresden, Institute of Construction Materials, Dresden, Germany</footnote>

<footnote>* Corresponding author at: L2MGC, CY Cergy-Paris University, France.
E-mail addresses: Asyf.rehman@gmail.com, asif.rehman@ucl.ac.uk (A. Ur Rehman), alexandre.pierre@cyu.fr (A. Pierre).</footnote>
element method (DEM) representation of the powder bed coupled with CFD analysis, providing a comprehensive understanding of binder droplet spreading and penetration dynamics within the powder bed. The significance of splashing, influenced by the size of the binder droplet, is identified as a critical parameter. The findings contribute to advancing the understanding of binder jetting processes in construction material applications, offering valuable insights for optimizing the printing parameters to achieve superior part quality.

# 1. Introduction

Additive Manufacturing (AM) has revolutionized various industries [3], facilitating the production of intricate designs with minimal preparatory time. Applications spanning aerospace [4], biomedicine [5,6], and architecture [7,8] have embraced AM processes for their flexibility and innovation. Among the AM techniques, Binder jetting (BJT), recognized by ASTM/ISO [9] as a powder-based AM process, stands out for its potential in construction applications [9].

In the construction industry, production costs pose a formidable challenge, driven by limitations such as the non-reusability of concrete formwork. While reusable formwork is an expensive but sustainable alternative, researchers explore methods like long series scaffolding to achieve cost-effectiveness [10]. However, the environmental impact, particularly greenhouse gas emissions during the scaffold life cycle, renders these methods less sustainable. Conventional practices, like using façade elements for structural elements, fall short of addressing the industry's demands for sustainability and cost-effectiveness [11,12]. This creates a compelling need for a production process that can resolve these challenges, and AM emerges as a sustainable solution in the construction industry, offering intricate geometry at competitive production costs [13]. By directly translating CAD models into structural parts, AM in construction can substantially reduce costs, reaching as low as 35-70% of the original cost by eliminating the need for formwork [14].

The BJT process, a key player in construction-oriented AM, involves spreading powder on a build plate, followed by precise deposition of binder droplets in a pattern [15]. This method, wherein layers of powder and binder are successively added, enables the creation of complex structures layer by layer. However, challenges such as distortions during the manufacturing process can impact the overall accuracy of the produced parts [16].

While experimental techniques, including X-ray visualization of the BJT process, exist [17], they often prove expensive and may not adequately predict distortions. Computational methods have emerged as a solution to these challenges, offering a cost-effective and predictive approach to analyze and optimize BJT processes [16].

CFD applications in BJT range from optimizing droplet characteristics to simulating powder compaction and understanding droplet impact dynamics [18] [19]. Previous studies, such as Charles-Williams et al. [20] and Marston et al. [21], have explored droplet impact and spreading on powders, revealing insights into capillary effects and droplet behaviour. However, the influence of varying binder droplet sizes on the powder bed, particularly in the context of optimizing binder impact conditions, remains unexplored.

This investigation aims to fill this gap by exploring the impact of different binder droplet sizes on the powder bed during BJT. Our goal is to optimize conditions for binder impact by analysing key driving forces, including inertial and capillary effects. Through the use of particle flowlines, we unveil the flow patterns of the binder over the powder bed, offering insights into the binder distribution. Leveraging a discrete element method (DEM) representation of the powder bed, we simulate variable particle sizes, closely resembling real-world powder conditions. A comparative analysis with our computational fluid dynamics (CFD) model enhances our understanding of the system dynamics. Importantly, the validity of our model is rigorously confirmed through comprehensive validation against experimental findings.

# 2. Materials and methods

## 2.1. Computational fluid dynamics (CFD) modelling

A CFD framework was developed and implemented using particular subprocesses of the FLOW-3D CFD program (Flow Science, USA), using Discrete element modelling (DEM) and Flow-3D modules in this CFD modelling. For simplicity, the research estimates several variables and generalizations that are with Newtonian flow; the viscosity change brought on by the flow shearing is disregarded. Expansion of Newtonian flow to replicate non-Newtonian fluid is relatively trivial [22]. The incompressible, variable-density Navier-Stokes (NS) equations are used to model and solve such two-phase flow:

$$ \nabla \cdot \mathbf{V} = 0 \qquad (1) $$

$$ \rho \frac{dV}{dt} + \rho(\mathbf{V} \cdot \nabla)\mathbf{V} = -\nabla P + \nabla \cdot [\mu(\nabla \mathbf{V}) + (\nabla \mathbf{V})^T] + \sigma \delta mn \qquad (2) $$

The last term represents the surface tension force and $\delta$ is Dirac delta function, $m$ and $n$ are curvature and normal vectors of the interface [22].

The interaction between the binder and powder involves changes in flow characteristics of the binder, from the impact location to penetrating inside the powder bed. To understand the flow characteristics of the binder there are a number of important parameters to consider. Here we have used the properties of the binder at room temperature and determined non-dimensional parameters from Eqs. (3) to (8) [23]. A detailed discussion about the behaviour of binder interaction with powder is presented in the next section. The corresponding impact velocities at different sizes is calculated by using the kinetic energy of the droplet [24].

$$ \text{Reynolds} \left( \frac{\rho V D}{\mu} \right) \qquad (3) $$

$$ \text{Weber} \left( \frac{W e}{\sigma} \right) \qquad (4) $$

$$ \text{Froude} \left( \frac{F r}{g D} \right) \qquad (5) $$

$$ \text{Bond} \left( B o = \frac{\rho g D^2}{\sigma} \right) \qquad (6) $$

$$ \text{Ohnesorge} \left( \frac{O h}{\sqrt{\rho \sigma D}} \right) \qquad (7) $$

$$ K d = \sqrt{W e^* R e^{0.25}} \qquad (8) $$

## 2.2. Powder modelling

The Discrete Element Method (DEM) is widely used for simulating the motion and interactions of numerous small particles such as powder. Originally developed by Cundall et al. in 1979, DEM is now synonymous with the method itself. It involves defining particles as rigid spheres with radius and density, handling multi-body contact, and calculating particle motion based on their interactions.

FLOW-3D DEM is commercial software that integrates DEM into the CFD software FLOW-3D. It accurately simulates particle motion and interactions, as well as coupled calculations of particle aggregates and
fluids.

DEM assumes particles as rigid bodies and can be categorized into hard and soft sphere models. The hard sphere model considers instantaneous collision without deformation, while the soft sphere model allows for deformation during collision, enabling handling of many-body contact. FLOW-3D DEM utilizes the soft sphere model, where particle positions are determined by Newton's second law, considering forces acting on each particle. The contact procedure is divided into normal and tangential components to handle interactions between particles sequentially.

### 2.2.1. Normal contact model

When modelling particle contact in the normal direction, assumptions include rigid particles allowing overlap, treating overlap as relative displacement, and using a linear spring model for contact force. When the overlap (relative displacement: $\delta_n$) is 0 or more, the particles are in contact, so a virtual spring appears between the particles and a contact force acts. When the particles are not in contact with each other, there is no overlap, so there is no virtual spring.

$$ \delta_n = (r_1 + r_2) - L \geq 0 \qquad (9) $$

Where $r_1$ and $r_2$ is the radius of two particles and L is the length between the centre of the two particles.

The contact force between particles $f_n$ is expressed using the spring constant $k_n$ and overlap $\delta$ according to Hooke's law.

$$ f_n = k_n \delta_n \qquad (10) $$

### 2.2.2. Tangential contact model

Similar to the normal direction, contact in the tangential direction involves translational motion only, with a linear spring model for contact force. The contact force $f_t$ between particles is expressed using the spring constant $k_t$ and overlap $\delta$ according to Hooke's law.

$$ f_t = k_t \delta_t \qquad (11) $$

### 2.2.3. Fundamental equations

DEM allows overlap between particles and relates relative displacement to force using a contact force model. This model encompasses characteristics such as elasticity, damping, and friction. The equations of motion for particles consider forces between particles and walls, as well as fluid forces, to calculate particle motion and particle-fluid coupling. Vector variables used in equations are shown in bold. The formula that expresses the normal component of the contact force $F_n$ is given below. The relative displacement vector in the normal direction is Eq. (15), $\mathbf{u}_{ij}$ is the relative velocity of the particle i and particle. The first term on the right side of Eq. (14) represents the elastic characteristics of the spring model, and the second term represents the viscous damping characteristics of the dashpot.

$$ F_{ij} = F_n + F_t \qquad (12) $$

$$ F_{ij} = -F_{ji} \qquad (13) $$

$$ F_n = -k\delta - \eta_n (\mathbf{u}_{ij} \cdot \mathbf{n}_{ij}) \mathbf{n}_{ij} \qquad (14) $$

$$ \delta = [L_{ij} - (r_i + r_j)] \mathbf{n}_{ij} \qquad (15) $$

$$ \mathbf{u}_{ij} = \mathbf{u}_i - \mathbf{u}_j \qquad (16) $$

The equation that expresses the contact force in the tangential direction $F_t$, where, $\eta_t$ is the viscosity coefficient of the dashpot

$$ F_t = -\eta_t (\mathbf{u}_{ij} - (\mathbf{u}_{ij} \cdot \mathbf{n}_{ij}) \mathbf{n}_{ij}) \qquad (17) $$

### 2.2.4. Particle equation of motion

The particle equation of motion, derived from standard FLOW-3D, incorporates an additional term representing inter-particle forces. The below Eq. (18) calculates particle velocity and position, influenced by fluid flow velocity, pressure, gravity, and drag coefficient. When particles move in voids, external forces and inter-particle forces govern their motion, while in fluid, pressure and drag forces become significant, necessitating coupled calculation with the fluid.

$$ \frac{d\mathbf{u}_i}{dt} = -\frac{1}{\rho_i}\nabla P + \mathbf{g} + \beta(\mathbf{u} - \mathbf{u}_i)|\mathbf{u} - \mathbf{u}_i|\frac{\rho_i}{\rho} + \frac{F_i}{m_i} \qquad (18) $$

$$ C_D = \frac{4}{3}d_i\beta = \left(\frac{24}{R_e} + \frac{6}{1 + \sqrt{R_e}} + 0.4\right) \qquad (19) $$

$$ R_e = \frac{d_i \rho |\mathbf{u} - \mathbf{u}_i|}{\mu} \qquad (20) $$

Where, $\mathbf{u}$ is fluid flow velocity, $P$ is fluid pressure, $\mathbf{g}$ is gravity, $\rho$ is fluid density, $\beta$ is a factor related to $C_D$, $C_D$ is drag coefficient, $Re$ is Reynolds number, $\mu$ is fluid viscosity coefficient. The subscript i means that it is a variable for the particle i. Furthermore, $u_i$, $m_i$ and $d_i$ represent the velocity, mass, and diameter of the particles i respectively.

### 2.2.5. Hydrodynamic governing equation

The fundamental equation for particle-containing fluid in standard FLOW-3D includes a term for particle drag ($F_d \cdot (\mathbf{u} - \bar{\mathbf{u}}_p)$) where $\bar{\mathbf{u}}_p$ represents the average particle velocity. Increased drag due to particle coupling with above equation induces pressure loss in the flow field, while pressure gradient effects also influence particles. This enables consideration of fluid-particle interactions via flow velocity and pressure mediation.

$$ \frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} = -\frac{1}{\rho} \nabla P + \nu \nabla^2 \mathbf{u} + \mathbf{g} - F_d (\mathbf{u} - \bar{\mathbf{u}}_p) \qquad (21) $$

$$ \nabla \cdot \mathbf{u} = \frac{R_{\text{source}}}{\rho} \qquad (22) $$

### 2.2.6. Volume fraction evaluation

The process of dividing a sphere and assessing particle volume fraction is outlined, focusing on the axis direction division method. The position and diameter of particles within the cell are dimensionless. Eq. (23) and Eq. (24) are utilized based on the cell size to perform this evaluation.

$$ d^* = \frac{d}{\Delta x} \qquad (23) $$

$$ x^* = \frac{x_p - x_c}{\Delta x} \qquad (24) $$

where, $d^*$ is dimensionless particle diameter, $d$ is particle diameter, $\Delta x$ is
cell width in the x axis direction, $x^*$ is x coordinate of dimensionless particle position, $x_p$ is x coordinate of particle position, and $x_c$ is x co-ordinate of cell centre position.

$$ \left\{ \begin{array}{l} f p_x = \frac{(d^* - 2|x^*| + 1)^2 (2|x^*| + 2d^* - 1)}{4d^{*3}} - \frac{1}{2} \le x^* < \frac{-1 + d^*}{2}, \frac{1 - d^*}{2} < x^* \le \frac{1}{2} \\ f p_x = 1 - \frac{1 + d^*}{2} \le x^* \le \frac{1 - d^*}{2} \end{array} \right. \qquad (25) $$

Volume fraction processing occurs similarly in other axial directions. After computing the volume fraction in each direction, it's converted to the particle volume fraction within the cell. Simultaneously, the volume fraction of particles distributed to adjacent cells is assessed. This particle volume fraction is expressed by Eq. (30).

$$ \left\{ \begin{array}{l} Vfp = fpx \cdot fpy \cdot fpz \frac{\Delta V_p}{\Delta V_{cell}} \\ Vfp = (1 - fpx) \cdot (1 - fpy) \cdot (1 - fpz) \frac{\Delta V_p}{\Delta V_{cell}} \end{array} \right. \qquad (26) $$

where, particle volume fraction, $f_{px}$ is x component of particle volume fraction, $f_{py}$ is y component of particle volume fraction, $f_{pz}$ is z component of particle volume fraction, $\Delta V_p$ is particle volume distributed in the cell, and $\Delta V_{cell}$ is cell volume.

The formation of OPC concrete powder was studied using the theoretical model mentioned above. The layer of powder particles was 2 mm thick depending on the experimental printing set-up. The deposition procedure is schematically represented in Fig. 1c. During the creation of the powder bed, Newton's Second Law can be used to find forces and movements that govern the DEM model. The normal and tangential forces that interact with spherical powder particles may be separated. The elastic normal contact force in the normal direction is calculated using a contact model using the non-linear elastic Hertz-Mindlin technique [25].

<table><caption>**Table 1**
Binder Properties and non-dimensional parameters.</caption><thead><tr><th>Property</th><th>Value</th><th>Units</th></tr></thead><tbody><tr><td>Surface tension (σ)</td><td>0.073</td><td>Nm<sup>-1</sup></td></tr><tr><td>Droplet rad. 1 (R1)</td><td>0.0003</td><td>m</td></tr><tr><td>Droplet rad. 2 (R2)</td><td>0.0005</td><td>m</td></tr><tr><td>Density (ρ)</td><td>997</td><td>kgm<sup>-3</sup></td></tr><tr><td>Initial velocity (v<sub>i</sub>)</td><td>4</td><td>ms<sup>-1</sup></td></tr><tr><td>Final velocity (v<sub>f</sub>)</td><td>4.140071</td><td>ms<sup>-1</sup></td></tr><tr><td>Viscosity (η)</td><td>0.001</td><td>Pa·s</td></tr></tbody></table>

This study examines the interaction of an ideal spherical droplet with the powder bed. Table 1 enlists the properties of the water used in numerical modelling and some corresponding non-dimensional parameters. In powder-based AM technologies, the size of powder particles generally lies in the micro-scale region. Herein, we assume that the powder bed consists of particles as described in the previous section and represented in Fig. 1.

Table 1 described fluid properties of the binder droplet.

Table 2 displays the nominal range of the particle sizes that form the powder used in the DEM, where 9 difference species have been simulated (10 % of each size except for 0.0250 that is at 20 %) [8,26]. The particle sizes have been averaged out to depict the OPC and Silica Sand (1:3) mixture shown in Fig. 1a (OPC particles) and Fig. 1b (Silica sand particles). The powder particles simulated by DEM are depicted in Fig. 1c, and the particles were roughly interpreted as spherical.

During the course of this study, the layer-by-layer deposition of OPC concrete powder has been modelled using the FLOW-3D DEM module with discrete microparticles used to deposit a layer of powder, with the composition in Table 2. The 2 mm powder layer produced by the model is depicted in Fig. 1c. The real part printed through AM can be seen in Fig. 1d.

<figure><img src="image_5.png" /><figcaption>**Fig. 1.** (a) OPC powder particle SEM (b) Silica sand SEM [27], (c) DEM simulation of powder bed layer and d) Printed part specimen.</figcaption></figure>
<table><caption>**Table 2**
DEM particle diameter size (cm).</caption><tr><td>1</td><td>2</td><td>3</td><td>4</td><td>5</td><td>6</td><td>7</td><td>8</td><td>9</td><td>10</td></tr><tr><td>0.0025</td><td>0.0050</td><td>0.0075</td><td>0.010</td><td>0.020</td><td>0.0250</td><td>0.0250</td><td>0.030</td><td>0.0350</td><td>0.0010</td></tr></table>

<figure><img src="image_6.png" /><figcaption>Fig. 2. (a) OPC concrete powder distribution along the x and y plane (b) the Size of binder droplet along the z-axis.</figcaption></figure>

<table><caption>**Table 3**
Computed values of dimensionless numbers.</caption><thead><tr><th>D1</th><th>0.3 mm</th><th>D2</th><th>0.5 mm</th></tr></thead><tbody><tr><td>Fr</td><td>5824</td><td>Fr</td><td>3494</td></tr><tr><td>Bo</td><td>0.012</td><td>Bo</td><td>0.033</td></tr><tr><td>Oh</td><td>0.00058</td><td>Oh</td><td>0.00045</td></tr><tr><td>We</td><td>71</td><td>We</td><td>119</td></tr><tr><td>Re</td><td>1237</td><td>Re</td><td>2062</td></tr><tr><td>Kd</td><td>50</td><td>Kd</td><td>73</td></tr></tbody></table>

## 2.3. DEM modelling set-up

Initially, the droplet is located along the x and the middle of the y plane and droplet sizes of 0.3 and 0.5 mm are used. The droplet moves along the z-direction with the initial velocity of -4 m/s ($V_z$) and in y direction with 1 m/s ($V_y$) [8]. Its velocity in the z direction increases due to gravitational acceleration before striking the powder. It also has a velocity in the y direction to depict the BJT machine movement. The schematic representation of the droplet and the powder bed is shown in Fig. 2 a and b.

The corresponding Weber number is 71 and 118 for 0.3 mm and 0.5 mm droplets, respectively. The properties given in Table 3 are derived from the formula from Eqs. (1) to (7).

<figure><img src="image_7.png" /><figcaption>Fig. 3. a). Printer set up schematic [27] b) Schematic representation of experimental X-ray setup [28].</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Fig. 4. Time-dependent analysis of the small binder drop at (a) 0.30 (b) 0.45 (c) 0.60 and (d) 0.89 ms.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 5. Time-dependent analysis of the large binder drop at (a) 0.30 (b) 0.45 (c) 0.60 and (d) 0.89 ms.</figcaption></figure>
## 2.4. Experimental setup of binder jet / binder dropping on powder bed experimental x-ray analysis

S. Barui et al. [28] investigated the printing of binder on to a powder bed in BJT using the experimental setup shown in Fig. 3, the results of which are used to validate our developed simulation model.

### 2.4.1. Powder bed preparation

Rectangular cuvettes, designed using 3D modelling software, with internal dimensions of 2 mm (W) × 5 mm (D) × 10 mm (H) and uniform wall thickness of 1 mm, were 3D Printed using a Polyjet printer (Stratasys, Eden Prairie, MN) [28]. Powders with a mean diameter of ~15 µm (ANTS Ceramics, Mumbai, India) were loaded into these cuvettes without pressure to simulate the powder bed in printing. The 15 µm mean diameter aligns well with the 16 µm average diameter used in the DEM study in this work.

### 2.4.2. In situ synchrotron imaging

The infiltration dynamics during printing were studied using a stationary single-jet piezoelectric inkjet drop generator installed on the beamline [28]. X-ray imaging provided insights into liquid front propagation through the powder bed [28].

### 2.4.3. Data analysis

Raw projection image data were analyzed using Avizo Fire 9 software [28]. ImageJ software was employed to analyze time-dependent changes in the powder bed due to infiltration, including vertical infiltration rate calculations and dynamic interface visualization [28].

## 3. Results

### 3.1. Spreading and wetting onto the powder-bed

Fig. 4a-d, presents a time-series depiction of the dynamic behaviour of the binder droplet of 0.3 mm deposited onto the OPC concrete powder providing valuable insights into its motion and absorption process. VF1 represents the volume of fluid in the simulation domain, with blue representing fully liquid and white representing fully empty. The analysis spans four sequential sub-figs. (5a to 5d), each capturing distinct stages of the binder drop event. In Fig. 4a, the binder is depicted at its initial position, signifying the starting point of the experiment. As time progresses, the binder undergoes a rapid descent, reaching Fig. 4b at 0.45 ms. During this phase, the droplet undergoes its initial descent under the influence of gravity, coupled with its initial velocity of 4 m/s, until it reaches a final velocity of 4.1 m/s (calculated via $V^2 = U^2 + 2aS$) just before impacting the powder bed. At this juncture, the binder comes into contact with the surface, commencing the absorption process. The large Froude number value indicates strong dominance of inertial forces, suggesting significant splashing and bouncing upon impact. The small Bond number value suggests weak surface tension forces, contributing to easier spreading and deformation of the droplet. Finally, the Ohnesorge value implies negligible viscous effects, allowing the droplet to deform readily. The We value is still relatively high, suggesting potential splashing but likely less intense.

Fig. 4c showcases a progressive stage in the experiment, occurring at 0.60 ms. The visual representation provides information about the dynamics of impact, such as deformation and energy dissipation. Continuing the temporal evolution, Fig. 4d captures the next stage of the binder drop at 0.89 ms. The binder, having already made contact in the previous frame, continues to be absorbed. The sustained absorption observed in this stage suggests a gradual dissipation of kinetic energy and ongoing interaction with the absorbing medium.

Fig. 5 presents a detailed temporal analysis of the impact dynamics of a large binder droplet with a diameter of 0.5 mm onto the powder bed. This aims to elucidate the intricate behaviour during the interaction, emphasizing the differences compared to the previously discussed

smaller droplet. The investigation spans four sequential sub-figs. (8a to 8d), each capturing distinct stages of the large binder droplet impact event.

In Fig. 5a, the large binder droplet is depicted at its initial position, marking the movement in the air. Where VF1 represents the volume of fluid in the simulation domain with blue representing fully liquid and white representing fully empty. As time progresses, the droplet undergoes a rapid descent, reaching Fig. 5b at 0.45 ms. This phase corresponds to the initial descent influenced by gravitational acceleration. Unlike the smaller droplet, the larger droplet experiences increased instability during this phase, that will lead (in the next stage) to the formation of splashes and an extensive spread in the y direction of its velocity.

Fig. 5c illustrates a next stage in the experiment, occurring at 0.60 ms. The visual representation provides critical insights into the dynamic aspects of impact, revealing pronounced deformation and starting of splashing due to the instability towards the y direction of the velocity. This behaviour grows significantly from the patterns observed in the smaller droplet study, emphasizing the impact size dependency.

Continuing the temporal evolution, Fig. 5d captures the further stage of the large binder droplet impact at 0.90 ms. The droplet, having already made contact in the preceding frame, exhibits splashing with sustained penetration and absorption. The observed phenomena suggest a complex interplay between the large droplet's kinetic energy dissipation and its ongoing interaction with the absorbing medium. Unlike the smaller droplet, the larger droplet's impact creates a more extensive wetting and spreading pattern, contributing to a broader coverage on the powder bed.

The intricate dynamics of this impact involve a balance between surface tension and inertial forces, with the larger droplet's increased inertia leading to enhanced instability and extensive spreading during the impact event.

### 3.2. Penetration into the powder-bed

Fig. 6 presents a time-series cross-sectional analysis illustrating the dynamic behaviour of the 0.3 mm binder droplet as it penetrates the

<figure><img src="image_10.png" /><figcaption>Fig. 6. Time-resolved cross-sectional analysis of the smaller binder droplet penetration in the powder bed at (a) 0.30 (b) 0.45 (c) 0.60 and (d) 0.89 ms.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>**Fig. 7.** Time-resolved cross-sectional analysis of the larger binder droplet penetration in the powder bed at (a) 0.30 (b) 0.45 (c) 0.60 and (d) 0.89 ms.</figcaption></figure>

powder bed. The legend colour scheme emphasizes “F” for “Fraction” of the fluid phase being tracked. If $F = 1$: the cell is fully filled with the fluid; if $F = 0$: the cell contains no fluid. When $0 < F < 1$: the cell is partially filled, representing an interface between two phases.

In Fig. 6a, the cross-section displays the initial position similar to Fig. 5. As time advances, the binder rapidly descends, reaching Fig. 6b at 0.45 ms, signifying the impact. During this phase, the binder encounters resistance from the powder bed, as depicted by the white arrows. Additionally, the droplet penetrates into the powder bed when the droplet is in proximity to the powder bed, where the penetration dominates in the y direction due to the velocity.

Fig. 6c captures a subsequent stage at 0.60 ms, revealing further penetration of the binder droplet where the penetration further increases in the y direction due to the velocity y coordinate as shown with the green arrow, as compared to the opposite direction of the $V_y$ represented by the blue arrow.

In Fig. 6d, the further progression at 0.89 ms highlights sustained absorption of the binder droplet. This phase implies a gradual dissipation of kinetic energy and continued interaction with the powder bed. The absorption trend continues further, however, it settles down with a some uniformity due to the saturation in the powder bed towards the less saturated regions towards the direction opposite to $V_y$, depicted by the green and blue arrows, and can be compared with Fig. 6c.

Fig. 7 provides an insightful cross-sectional analysis, detailing the dynamic penetration of a large binder droplet with a diameter of 0.5 mm into the powder bed. The legend colour scheme emphasizes “F” for “Fraction” of the fluid phase being tracked as in Fig. 6.

In Fig. 7a, the cross-section illustrates the initial position of the large binder droplet, marking the initiation of the experiment. As time progresses, the droplet undergoes a rapid descent, reaching Fig. 7b at 0.45 ms, indicating the impact phase. In contrast to smaller droplets, the larger droplet exhibits increased wetting as shown with arrows, extensive spreading in the direction of its velocity. It shows increased instability as depicted by the red arrows. However, similar to the smaller droplet it shows increased penetration towards the y coordinate of the velocity as shown with the green and white arrows when compared with the opposite side as depicted by the blue arrows.

Fig. 7c captures a subsequent stage at 0.60 ms. The larger droplet’s increased size and inertia result in a more pronounced spreading pattern, influencing the impact force distribution. The spreading trends continue further as described in previous figure. The increased penetrations towards the y coordinate of the velocity as shown with the green arrows when compared with the opposite side as depicted by the blue arrows. The instability transforms into splashes as highlighted by red arrows.

In Fig. 7d, the next stage at 0.90 ms highlight sustained absorption of the large binder droplet. This phase suggests a gradual dissipation of kinetic energy and continued interaction with the powder bed, with the impact distribution indicating the persistence of forces during the absorption process. Notably, the impact distribution in larger droplets shows a broader and more intense pattern compared to smaller droplets, reflecting the increased impact energy and spreading dynamics. The spreading continues further and homogenizes it towards the less saturated zones. The splashes continue their movement towards the y coordinate of the velocity and either are still in the air or fall on the powder surface.

Furthermore, the cross-sectional analysis in Fig. 7c and d reveal that more binder spreads as well as penetrates near the impact area due to the larger droplet’s lower inertial force and higher instability at the start of the impact with the powder.

## 4. Discussion

### 4.1. Penetration dynamic: Articles flow-lines and velocity vectors as a function of initial diameter size of droplets

Fig. 8 provides an intricate analysis of the impact dynamics of a binder droplet on the powder bed, incorporating velocity vectors at each point of the liquid at different locations to understand the physics. The colour legend represents impact forces, similar to the previous Fig. 6. The legend colour scheme emphasizes “F” for “Fraction” of the fluid phase being tracked as in Fig. 6.

In Fig. 8a, the cross-section of the binder with velocity vectors portrays movement towards the powder bed, during the initial stage of the

<figure><img src="image_12.png" /><figcaption>**Fig. 8.** Comprehensive analysis of the smaller binder droplet impact on the powder bed with velocity vectors at (a) 0.30 (b) 0.45 (c) 0.60 and (d) 0.89.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Fig. 9. In-depth analysis of the smaller binder droplet penetration in the powder bed through particle flowlines.</figcaption></figure>

experiment. As time progresses, the binder experiences a rapid descent, reaching Fig. 8b at 0.45 ms, signifying the initial phase when the binder strikes the powder bed. At this stage, a recoil pressure is experienced by the droplet represented by the opposite direction of the arrows in Fig. 8b. The binder makes contact with the surface, initiating the absorption process as well. Early resistance from the powder bed is observed, leading to increased instability. Additionally, an instability forms when the droplet is close to the powder bed, dominating over inertial forces represented by movement on the right side highlighted by the red circles.

Fig. 8c represents a progressive stage at 0.60 ms where the binder absorption is slightly tilted towards the velocity direction of the binder droplet. A very small droplet is still above the powder bed facing the recoil pressure highlighted by the red circle.

Continuing the temporal evolution, Fig. 8d captures the next stage at 0.89 ms, with the binder droplet still being absorbed after making contact in the previous frame. This sustained absorption indicates a gradual dissipation of kinetic energy and ongoing interaction with the absorbing medium. However, the splashes caused by the binder droplet instability (highlighted in Fig. 8b) are slowly being absorbed on the right of the cross-section highlighted by the red circle.

Fig. 9 presents a comprehensive analysis of binder droplet penetration in the powder bed, utilizing particle flowlines to track the flow path after devisualizing the powder.

In Fig. 9a, the particle flowlines depict the initial position of the binder droplet as it is dropped from the top, highlighting the dropping path of the experiment. As time progresses, the binder undergoes a rapid descent, striking the powder bed in Fig. 9b at 0.45 ms. This stage corresponds to the initial descent, characterized by gravitational acceleration and the initial release velocity of the binder from the nozzle. Upon impact, the binder makes contact with the surface, initiating the absorption process. Early resistance from the underlying powder, slightly

<figure><img src="image_14.png" /><figcaption>Fig. 10. Comprehensive analysis of the larger binder droplet impact on the powder bed with velocity vectors at (a) 0.30 (b) 0.45 (c) 0.60 and (d) 0.89.</figcaption></figure>

contributes to the instability of the droplet towards the y component of the binder velocity.

Fig. 9c represents a progressive stage at 0.60 ms, akin to changes
described by Fig. 9b. the penetration is higher towards the right side. Continuing the temporal evolution, Fig. 9d captures the next stage at 0.89 ms, with the binder droplet still being absorbed after making contact in the previous frame. This sustained absorption suggests a gradual dissipation of kinetic energy and ongoing interaction with the absorbing medium. The penetration homogenizes itself because of the saturation towards right side and unsaturation on the opposite side. The application of particle flowlines offers a clear visualization of the flow path, aiding in the interpretation of complex fluid dynamics.

Fig. 10 delves into a detailed analysis of the impact dynamics of a binder droplet (initial diameter of 0.5 mm) on a powder bed, with an added layer of complexity through the incorporation of velocity vectors at each point of the liquid. The colour legend maintains consistency with Fig. 7.

In Fig. 10a, the cross-section of the binder, coupled with velocity vectors, illustrates the initial stage of the experiment, showcasing the movement of the binder droplet towards the powder bed. As time progresses, the binder undergoes a rapid descent, reaching Fig. 10b at 0.45 ms, signifying the initial impact phase. At this stage, although the recoil pressure is experienced by the droplet, the larger droplet has significant increase in wetting as depicted by the blue and green arrows when compared with the small droplet. The instability is formed in the direction of the y coordinate of the velocity and almost half of the droplet exerts force towards it shown by the velocity vector which has been highlighted by the red circle. The penetration is also tilted towards the y coordinate of the velocity as depicted by the green arrows. The penetration inside the powder is also tilted towards it, highlighted by the white arrow. The instability of the binder droplet is visually highlighted by red circles in Fig. 10b. The contact with the surface initiates the absorption process, encountering early resistance from the powder bed leading to increased instability.

Fig. 10c represents a progressive stage at 0.60 ms. In Fig. 10c, inside the powder bed the binder absorption is slightly tilted towards the y

coordinate of the velocity direction of the droplet, as indicated by the green arrows, and has less absorption towards the opposite side as shown with the blue arrows.

Continuing the temporal evolution, Fig. 10d captures the next stage at 0.90 ms, with the binder droplet still being absorbed after making contact in the previous frame. The sustained absorption indicates a gradual dissipation of kinetic energy and ongoing interaction with the absorbing medium. Notably, the splashes caused by the instability of the binder droplet (highlighted in Fig. 10b) continue their movement in the air as highlighted by the red arbitrary region, or fall to the surface of the powder bed as highlighted by the red circle. The binder being absorbed on the right side of the cross-section, as indicated by the blue arbitrary shape experiences the opposite force due to the saturation of the binder liquid and less saturation in the opposite direction. This homogenizes the binder saturation towards the opposite side of +y.

Fig. 1 and Fig. 12 presents an in-depth exploration of the penetration dynamics of a large binder droplet with a diameter of 0.5 mm into a powder bed, utilizing particle flowlines to trace the flow path after removing the powder in Fig. 1 and the same without the particle flowlines in the top view in Fig. 12.

In Fig. 1a, the particle flowlines vividly illustrate the initial position of the large binder droplet, showcasing its dropping path from the top. As time progresses, the droplet undergoes a rapid descent, striking the powder bed in Fig. 1b at 0.45 ms. This phase corresponds to the initial descent, influenced by gravitational acceleration and the initial release velocity of the binder from the nozzle. Unlike smaller droplets, the larger droplet exhibits increased instability during impact, resulting in the creation of splashes and a more extensive spread towards the velocity direction of the droplet. Upon impact, the large binder droplet makes contact with the surface, initiating the absorption process. The early resistance attributed to underlying powder, leads to heightened instability forces. Furthermore, a instability forms when the droplet is in proximity to the powder bed. The particle flowlines in Fig. 1b highlights

<figure><img src="image_15.png" /><figcaption>Fig. 11. In-depth analysis of the larger binder droplet penetration in the powder bed through particle flowlines.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Fig. 12. (a) Position of the droplet before impact at large size (b) consecutive droplets with formation (c) and (d) spreading of binder over the powder bed.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 13. (a) and (b) x-ray analysis of binder with powder interaction, the penetration depth [28] (c) CFD analysis of binder with powder, the penetration depth presented along the plane.</figcaption></figure>
the dynamic flow pattern during this initial impact phase.

Fig. 1c represents a subsequent stage at 0.60 ms, providing crucial insights into impact dynamics, deformation, and energy dissipation. Particle flowlines in this frame further elaborate on the altered flow paths, capturing the evolving behaviour of the large binder droplet as it continues to be absorbed. The binder penetrates into the powder bed influenced by the y coordinate of the velocity of the binder droplet.

Continuing the temporal evolution, Fig. 1d captures the next stage at 0.90 ms, with the binder droplet still being absorbed after making contact in the previous frame. This sustained absorption indicates a gradual dissipation of kinetic energy and ongoing interaction with the absorbing medium. The binder absorption in the powder bed slowly homogenizes itself. The particle flowlines in Fig. 1d provide a detailed representation of the flow patterns during this prolonged absorption phase, emphasizing the intricate dynamics involved in the penetration of large binder droplets into the powder bed. Instability translated into the splashes progressing further in the air or starting to fall onto the powder bed as highlighted by the red circles. The application of particle flowlines offers a clear and comprehensive visualization of the flow path, contributing to a deeper understanding of the impact dynamics.

The above discussion shows that penetration depth and binder spread over the powder bed are essential considerations for the binder jetting process. The droplet size increases the penetration depth, and the binder spreads over the powder. The thickness of powder bed layers in binder jetting is small, and if the penetration depth is not significant, it may result in insufficient hemming of layers. It can also reduce the strength of the end part in the direction of joining layers.

However, it should also be considered while selecting the size that the powder may undergo distortions with an increase in size, and powder particles may segregate into each other, leading to dislocations or defects inside the final part. In addition, a term called splashing of the binder over the powder bed is widely used. The splashing parameter $k_d = We^{0.5}Re^{0.25}$ is defined as the parameter describing the spreading of a single drop into several tiny droplets upon impact [24]. The boundary between splashing and non-splashing was determined using $k_d$ and this threshold value was found to be 57.7 for impact of liquid drops on solid dry substrates. Droplets having a splashing parameter greater than 57.7 would splash [24]. No significant splashing occurs for the smaller size droplet used in our study as the splashing parameter is lower than 57.7 (0.3 mm). However, in comparison, increasing the size increases the $k_d$ to 74, ultimately increasing the splashing.

## 4.2. Comparison with synchrotron measurements

Our simulation model aims to reproduce experimental observations, ensuring that our computational predictions align with real-world phenomena. In Fig. 13, we then reference the work of Barui et al. [28], where high-speed X-ray imaging was employed to observe the impact of the binder droplet on the powder bed along the cross-section. The depth exhibited in this experiment serves as a valuable metric, indicating changes in the powder bed induced by powder particle movement, and importantly, showcasing minimal splashing. Provides a detailed comparison between experimental and simulated results.

Fig. 13a and Fig. 13b shows the synchrotron results for the penetration depth of the binder droplet penetration inside the powder bed along the plane. Due to the less visibility of the powder under synchrotron the penetration can be better visualized for multiple droplets (as many as 500) and it cannot be visualized with a single droplet. It may not really give the behaviour of single droplet but can give the overall behaviour of binder powder interaction. From Fig. 13a and Fig. 13b, it can be observed that the penetration depth of the binder is more near the area of impact and increases with an increase in time. The synchrotron results have been presented after every 200 milliseconds depending on the synchrotron capability. However, the single droplet can penetrate within a millisecond in the first frame of the synchrotron result.

The overall behaviour is similar to the numerical modelling results

developed in this study and illustrated in Fig. 13c. Where the interaction of the binder droplet with the powder bed as well as its penetration depth has been presented. While in Fig. 13c, the simulation results have been presented for a single droplet as the depth of binder penetration increases with the passage of time. The depth here is the region which the binder covers after impact on the powder bed. So as time increases the penetration of binder along the cross-section and the powder bed axis increases.

Crucially, our study aligns with the experimental observation that no significant splashing occurs. This is a significant validation aspect, as splashing can lead to disruptions in the powder bed, affecting the overall printing quality. The experimental findings, as well as our simulation results, reveal that minimal momentum is transferred from the binder impact to the powder bed, corroborating the accurate representation of the impact dynamics in our model.

## 4.3. Comparison with Washburn equation

In the BJT process, a liquid binder is deposited onto the powder bed, where it binds the powder particles together. The Washburn model can be used to predict how the binder will infiltrate the powder bed. However, and as reported in [30], the scaling with Washburn law is not always applied or observed in imbibition experiments due to several factors like rheological properties of the fluid, swelling of the porous media, inertial effects of the droplet injection, etc.). Penetration depth could also scale as a linear function of the time as reported by Queré [31] [32]. Several regimes happen during spontaneous imbibition. The aim herein is to distinguish which regimes apply in our numerical modelling (ie. Regime I is the inertial regime, Regime II is the Lucas-Washburn regime $h(t) \sim t^{1/2}$), Regime III is the late period).

Fig. 14a and b represent the volume of fluid (Vof) calculated through the simulation study. and the Washburn model [29] for the 0.3 mm and 0.5 mm droplets in the powder bed. The y-axis of the graph represents the volume of fluid, and the x-axis represents time. The two curves on the graph show the volume of fluid as a function of time for the simulation results and the Washburn model.

It appears that the volume of fluid in the powder bed is less than the volume of fluid predicted by the Washburn model. This could be due to a number of factors, including the porosity of the powder bed, the viscosity of the liquid binder, and the surface tension of the liquid binder. A more porous powder bed will allow more liquid to infiltrate. A more viscous liquid will flow more slowly through the powder bed. A higher surface tension will resist the infiltration of the liquid into the powder bed.

## 5. Conclusion

The study specifically delves into the Binder Jetting (BJT) process, where powder deposition and binder droplet impact dynamics play a crucial role. Computational fluid dynamics (CFD), and discrete element method (DEM) simulations, have been employed to understand the effect of binder droplet size and understand the BJT process in depth. Notably, CFD simulations have proven valuable in understanding binder impact by considering factors such as volume, forms, velocities, and defect formation of droplets.

The investigation into the impact dynamics of binder droplets, varying in size, provides valuable insights. Small droplets exhibit distinct stages, from initial descent and contact to sustained absorption, highlighting the interplay of instability and inertial forces. Time-dependent and cross-sectional analyses offer a comprehensive understanding of the behaviour of both small and large binder droplets during impact and absorption. Additionally, particle flowlines and velocity vector analyses contribute to visualizing the intricate flow patterns.

The study emphasizes the importance of considering droplet size in optimizing BJT processes. While larger droplets result in increased
<figure><img src="image_18.png" /><figcaption>Fig. 14. Volume of fluid droplet in the powder bed and the washburn model of a) 0.3 mm and b) 0.5 mm diameter droplet.</figcaption></figure>

penetration depth and broader coverage on the powder bed, it also increases splashing.

Validation of the simulation model against experimental results, particularly through high-speed hard X-ray imaging from the literature, enhances confidence in the accuracy of predictions. The observation between simulated and experimental outcomes regarding penetration depth and minimal splashing (for smaller droplet 0.3 mm) attests to the predictive capabilities of the model. The simulation has also been validated by the experimental results of splashing parameter $K_d = 57.7$ for droplet dropping onto the dry surface, where the larger simulated droplet shows splashing because the splashing parameter is well above the splashing threshold.

In summary, this research sheds light on the complex dynamics involved in BJT processes for construction AM. By understanding the impact behaviour of binder droplets, optimizing droplet size, and validating simulations with experimental data, the study contributes to advancing sustainable and cost-effective AM solutions in the construction industry.

### CRediT authorship contribution statement

**Asif Ur Rehman:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Data curation, Conceptualization. **Alexandre Pierre:** Writing – review & editing, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization. **Gregory Gibbons:** Writing – review & editing, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis. **Viktor Mechtcherine:** Writing – review & editing, Supervision, Project administration, Methodology, Investigation, Funding acquisition, Formal analysis.

### Declaration of competing interest

The authors declare no conflict of interest related to this manuscript. The research was conducted independently, and no financial, personal, or professional relationships with any organizations or entities influenced the results or interpretation of the findings.

### Acknowledgements

The project was funded by the EUTOPIA Science and Innovation Fellowship Programme (European Union's Horizon 2020 research and innovation programme under the Marie Skłodowska Curie grant agreement number 945380).

### Data availability

Data will be made available on request.

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