## 2 Material and Methods
### 3D Printing, Simulation and Characterization  
The significance of the maximum spreading diameter in the 3D printing process mechanism in this study must be introduced first. The spreading behavior of droplets determines the final printability of the binder/powder system (shown in Figure 4a) in 3D printing. The droplets spreading interference (a captured diagram of millimeter-scale droplets shown in Figure 4e) caused by the over-narrow space of the droplets position leads to the imbalance of binder penetration. As a result, part of the region of excessive penetration occurs as shown in Figure 4b; meanwhile, too much spacing between droplets leads to a binder blank area, and the desired effective adhesion cannot be formed, as shown in Figure 4d. The optimal droplet spacing distance makes the droplets spread without interference with each other, and the desired region of the powder is effectively achieved after the droplet penetration, as shown in Figure 4c. Regarding the printing accuracy, the consistency between the area of precision printed and the area of design guarantees the accuracy. On the other hand, the final penetration area minus the effective precision area is the printing error. Figure 4f,g are the schematic diagram of the spreading of two droplets, where the \(t_{max}\) is the maximum spreading time, \(T_{d}\) is the droplets' spreading interval, \(D_{max}\) is the largest spreading diameter, and \(\Delta T\) is the time difference between the two droplets.  
Figure 3: Comparison of observation experiment and simulation. (**a**) is the schematic diagram of test equipment, (**b**) is the real pictures of experimental equipment (A. CCD high-speed camera, B. computer, C. syringe with a needle, D. water droplet, E. multiple spherical substrates, F. light source) (**c**) is the CAD design model of the substrate, (**d**) is the solid substrate. (**e**) are the curves of spreading ratio with time observation experiment and simulation. (**f**) is the droplet spreading deformation process, where the red region was the droplet, and the blue region was air.  
Under the condition of no interference between droplets during spreading, the maximum spreading performance of droplets determines the minimum droplet spacing distance. That is to say, the droplets deposition distance between two droplets is greater than or at least equal to twice the maximum spreading radius (as shown in Figure 4g); the mathematical expression can be explained as follows,  
\[\Delta d\geq\ \epsilon_{max}\cdot D_{0} \tag{14}\]  
where \(\epsilon_{max}\) is the maximum spreading ratio, \(D_{0}\) is the initial diameter of the droplet. The droplet deposition distance \(\Delta d\) is defined as the distance between the centers of two adjacent droplets on the \(XY\) plane.  
In theory, according to the definition of dots per inch (\(DPI\)), the relationship between deposition distance \(\Delta d\) and printing \(DPI\) is as follows,  
\[DPI=\frac{1\textit{inch}}{\Delta d} \tag{15}\]  
Based on Equations (14) and (15), the relationship between 3D printing resolution \(DPI\) and the maximum spreading ratio, \(\epsilon_{max}\) can be expressed as,  
\[DPI\leq\frac{1\textit{inch}}{\epsilon_{max}\cdot D_{0}} \tag{16}\]  
So, the optimal \(DPI\) can be deduced by the study of the maximum spreading ratio.