## 3 Results and Discussion
### Extrusion Prediction Formula  
As shown in Figure 7a below, the paste was primarily affected by the feed pressure P (provided by the high-pressure air source for the entire extrusion process), the friction \(F_{f}\) between the paste inside the nozzle and the nozzle wall, the surface tension of the paste (\(\gamma\)) in the internal pressure of the nozzle, and the effect of the paste self-weight (\(G\)). When dealing with dense materials, such as conductive pastes, the gravity number, \(G_{p}\), in pipe flow is often used to relate the effect of gravity to the shear yield stress, \(\tau_{y}\)[27; 28], as shown in Equation (2) below:  
\[G_{p}=\frac{\rho gd_{i}}{4\tau_{y}} \tag{2}\]  
where \(\rho\) is the density of the paste, \(d_{i}\) is the inner diameter of the glass nozzle, and \(\tau_{y}\) is the yield stress of the paste. After completing the calculation, we established that \(G_{p}=0.04\ll 1\) for all the paste studied in this paper, and so the influence of gravity on the nozzle flow could be ignored. In the actual printing process, the feed pressure is used as the main driving force, which is much larger than other forces such as the material's own weight. Therefore, in order to simplify the model, these less influential forces can be ignored. The flow rate of the paste extruded through the nozzle can be expressed as  
\[q_{v}=\frac{dV}{dt} \tag{3}\]  
where \(V\) is the volume of the extruded paste and \(t\) is the duration. Figure 5: Experimental measurement data for the single-factor analysis. Figure 6: (**a**) Normal situation and actual print failure conditions of (**b**) stacking and (**c**) broken lines. According to the measurements of the scanning electron microscope (SEM), shown in Figure 7b, the shape of the extruded paste could be approximately regarded as being half oval, and then the flow rate in the Equation (3) could be approximately expressed as  
\[q_{v}=\frac{\pi ch}{4}\frac{dL}{dt}=\frac{\pi chv}{4}=\frac{\pi vc^{2}k_{ch}}{4} \tag{4}\]  
where \(c\) is the line width, h is the vertical distance from the highest point to the substrate after the paste is fixed, \(L\) is the total length of the paste deposited on the substrate over duration \(t\), \(v\) is the substrate moving speed of the platform that holds the substrate on the system, and \(k_{ch}\) is the aspect ratio, \(k_{ch}=h/c\). We assumed that the paste extrusion was a continuous viscous non-Newtonian fluid, and the conductive paste was incompressible. The inlet effect and the slight loss caused by the attachment could be ignored. As shown in Figure 7b, the force analysis of the micro-element of the conductive paste at the nozzle, the shear force acting on the paste at the wall, and the pressure drop in this section were in the mechanical equilibrium required to obtain Equation (6). Equation (6) was simplified and combined with the H-B fluid expression to obtain the nozzle outlet flow in Equation (7) [29]. The simplification from Equation (7) to Equation (8) is presented in Appendix C.  
\[\tan\theta=\frac{r_{1}-r_{2}}{l} \tag{5}\]  
\[\pi r^{2}\Delta P=2\pi r\tau dl\sec\theta\cos\theta \tag{6}\]  
\[\tau=\frac{\Delta PR}{2L}=\tau_{y}+K\dot{\gamma}^{n}=\tau_{y}+K\left(\frac{3n+ 1}{4n}\frac{4Q}{\pi R^{3}}\right)^{n} \tag{7}\]  
where \(\tau_{y}\) is the yield stress of the conductive paste, \(K\) is the consistency coefficient, and \(dl=-\frac{dr}{\tan\theta}\). Then,  
\[q_{v}=\frac{\pi n}{3n+1}\left(\frac{3n(r_{1}-r_{2})}{2 Kl\left(r_{2}^{-3n}-r_{1 }^{-3n}\right)}\left(\Delta P-\frac{2\tau_{y}l}{r_{1}-r_{2}}\ln\left(\frac{r_{ 1}}{r_{2}}\right)\right)\right)^{\frac{1}{n}} \tag{8}\]  
where \(l\) is the total length of the conical part where the front end of the glass nozzle is deformed and \(\Delta P/l\) is the pressure gradient applied by the compressed air. Figure 7: (**a**) The print force analysis diagram of the model. (**b**) A simplified model of the print head and the morphology change of the conductive paste.