## 2 Material and Methods
### Low Impulse Droplet Impact Model  
Many effective methods have been proposed and applied during the 3D printing process to avoid the excessive impact of droplets on the powder bed. The previous studies [8; 12; 13; 16; 41; 42] showed that an appropriate packing density could avoid the excessive impact of droplets on the powder bed and achieve better printing quality. In the studies [43; 44], moisture was applied to the powder bed to improve the powder bed stability.  
In a previously conducted study [30], a diameter of 2.4 mm and a 50% concentration of glycerol drop impacting the glass bed were studied. With an impact momentum of \(9.15\times 10^{-6}\) kg\(\cdot\)m/s and the Weber number of 47.56, the powder bed was obviously not damaged (Figure 2b).  
In a similar study [12], a digital syringe was used to generate droplets impacting the CaP:CaSO\({}_{4}\) powder bed to simulate the 3D printing process. In order to protect the powder bed from excessive droplet impact, the height of droplet volume of 30 \(\upmu\)L to the powder bed was 8 mm. The impact momentum can be calculated according to the experimental parameters:  
\[\overset{\rightarrow}{P}=m\cdot\overset{\rightarrow}{v}=\rho\cdot V\cdot \overset{\rightarrow}{v} \tag{1}\]  
Figure 2: The modeling of binder droplets impact in 3D printing. (**a**) is the schematic illustration of binder droplets impact on powder bed in the 3D printing. (**b**) is the sequence of events for the typical low impulse droplet impact. (**c**) is the two-dimensional numerical simulation model.  
where \(\overset{\rightarrow}{P}\) is the impact momentum, \(m\) is the mass, \(\overset{\rightarrow}{v}\) is the impact velocity, \(\rho\) represents density (the value is 1.02 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\) here), \(V\) is the volume of droplet.  
For the free-fall velocity \(v\),  
\[v=g\cdot t=\sqrt[2]{2g\hbar} \tag{2}\]  
where \(g\) is gravity, \(h\) is the height of the droplet (8 mm), \(t\) is the time during free fall. The momentum was evaluated to be 1.2 \(\times\) 10\({}^{-5}\) kg\(\cdot\)m/s.  
The Weber number \(W_{e}\) can be given as follows:  
\[W_{e}=\frac{\rho v^{2}d}{\sigma} \tag{3}\]  
where \(\rho\) is the droplet density, \(d\) is the droplet diameter, \(\sigma\) is surface tension (48 mN/m).  
In this study, the print head of our homemade 3D printer was designed so that the binder droplet impact velocity was 2 m/s, the droplet diameter was 0.1 mm, and the binder density was 1.03 \(\times\) 10\({}^{3}\) kg/m\({}^{3}\), the surface tension coefficient was evaluated to be 48 mN/m. The impact momentum and Weber number were compared in Table 1.  
Table 1 reveals that the impact momentum and Weber number in currently employed 3D printing is lower than that mentioned in the previously conducted typical studies that examined the droplet impact without damaging the surface of the powder bed.  
Furthermore, according to [PERSON] et al. [45; 46], when the Weber number is \(W_{e}<\) 300, the droplets will spread on the surface of the powder without damaging the surface of the powder bed. Based on the comparison of the data given in Table 1, the low impulse droplet impact theory is applicable to the 3D printing scene that the droplet impacts and spreads without destroying the powder bed. The problem of micro binder droplet impact in the typical process of 3D printing can be simplified to the model of a droplet impacting on multi-particles/spheres bed/substrate, whereas in the simplified model, the powder particles were well deposited and compacted, and there is no deformation and no slip between the small particles during droplet spreading. The idealized model of binder droplets impacting the powder bed under low impact momentum is shown in Figure 2a.