# Simplified Model Predicts Binder Behavior in Sand Mold Printing
## 4 Discussion  
Figure 5 shows all samples and analysis results in a statistical chart of \(\mathrm{P_{inf}}\) and \(\mathrm{D_{eff}}\) data for six different binder volumes in order of increasing porosity. The red line segments and sample points represent the simulation results, and the black line segments and sample points represent the experimental results. When porosity is low, the \(\mathrm{P_{inf}}\) presents a quadratic curve distribution and generally ranges between 500-1500 \(\upmu\)m. However, as porosity increases, the overall diffusion depth increases significantly, and the trend line tends to close to a straight line. The result is that the binder cannot be retained between the particles. It infiltrates downwards, driven by inertia and gravity [37; 38]. If porosity is low, the surface force between the particles increases, which increases the resistance of the particles to infiltration, resulting in a decrease in \(\mathrm{P_{inf}}\) (but with little effect on \(\mathrm{D_{eff}}\)), thereby increasing the precision of the mold after molding. The results obtained by the simplified model proposed in this experiment are the same as the quadratic curve trend resulting from the experiments. If binder volume and porosity are used as the main variables, the \(\mathrm{P_{inf}}\) and \(\mathrm{D_{eff}}\) results are almost identical at lower porosities. However, when the porosity exceeds 35%, the \(\mathrm{P_{inf}}\) prediction results of the simplified model produce significant errors. The reason for this is that there are still many uncertainties in permeability processes under conditions of high porosity [40]. As binder jetting technology improves, studies and equipment are moving toward smaller particle sizes, which means that low porosity is becoming a trend in high-precision, high-strength binder jetting. Therefore, the simplified model proposed in this study still has a high reference value. Regarding penetration depth, the curves of 2.0 and 2.5 \(\upmu\)L demonstrate a gentle slope because of the extremely high initial absorption force in the dry sand state, which results in rapid penetration. However, as the dry sand is transformed into wet sand, the contact angle differs in the dry and wet states, as depicted in Table 5. Consequently, the capillary force and surface tension are influenced, leading to a slower penetration rate, and the binder and hardening agent begin to react, increasing the binder's viscosity. As a result, the penetration depths at 2.0 and 2.5 \(\upmu\)L did not differ significantly. By observing the samples under the microscope, we found that while the penetration depth was not significantly different, more binder remained between the powders at 2.5 \(\upmu\)L than at 2.0 \(\upmu\)L [38]. The penetration rate of the binder in dry and wet sand is influenced by the contact angle. A larger contact angle results in stronger adhesion and faster binder penetration into the powder. Conversely, a smaller contact angle leads to slower penetration. Therefore, we can understand that when the sand is in a dry state, at the moment we introduce the binder, the powder bed begins to transition from dry to wet, causing the contact angle to decrease. Additionally, the internal voids between particles are gradually filled with the binder, resulting in minimal differences between the results of 2.0 and 2.5 \(\upmu\)L binder volumes. Furthermore, this phenomenon can be discussed using the Washburn equation for capillary infiltration [43]. The infiltration velocity decreases exponentially as the state of the sand changes from dry to wet, implying that the penetration rate of the binder into the powder slows down. Subsequent binders cannot push out or push down the initially introduced binder, resulting in differences in the final penetration depth and diffusion diameter. However, our simplified model does not consider the variation in contact angle (transition between dry and wet sand) and assumes the powder bed remains in a dry state throughout the process.