## 3 Results  
### Analytical Modelling  
Figure 4 illustrates the typical steps and parameters of printing a layer using self-leveling ink through DIW technology. During the printing process, the nozzle moves laterally at a speed of \(v_{n}\) at a height \(h_{n}\) from the substrate, while the ink is extruded from the nozzle by air pressure to print an ink line with a width \(l\) on the substrate. Surface tension causes a contact angle \(\theta\) to form between the ink line and the substrate. Multiple ink lines with a spacing of \(l_{s}\) (\(l_{s}<l\)) can be printed side by side to create an ink layer. The crucial parameter controlling the layer thickness is the line width \(l\), which depends mainly on process parameters such as the nozzle inner diameter \(d_{n}\), nozzle moving speed \(v_{n}\), and ink extrusion speed \(v_{e}\). Previous work has shown that printing thinner layers is critical for improving the interface strength of multi-layer sensor composite structures [3]. Among the related printing parameters, the line spacing l s is the crucial parameter controlling the flatness of the ink layers. When the line spacing is too small, the overlap between adjacent ink lines increases the overall layer thickness (higher than the height of a single line), making it difficult to obtain the desired thin layer. Increasing the line spacing leads to the formation of ripples on the surface of the layer during the process of forming lines into a surface due to surface tension, which affects the flatness of the layer. In addition, if the spacing between adjacent ink lines exceeds the maximum gap that can be covered by self-leveling fusion, it is unable to print a complete flat surface. To achieve controllable printing of layer thickness and surface ripple, a large number of expensive process optimization experiments are necessary, and usually only empirical rules can be summarized, making it difficult to obtain a universal theoretical model. To address the aforementioned issues, we first observed and summarized the microstructural evolution of the ink during the forming process of lines-to-layer using DIW printing, and then identified the key governing parameters. Subsequently, a processing model was established to control the surface morphology with high precision. Based on this model, high-precision printing of layers with controllable thickness and high flatness can be achieved. The work presented in this paper offers a low-cost and rational solution for achieving high-precision, controllable direct writing printing. A widely used silver ink (CD-03, Conduction) is used to print multiple ink lines, with the spacing between lines controlled to be less than the line width, so that multiple lines can merge to form a plane. However, the uncured layer with planar surface (Figure 5a-c) becomes rippled after curing (Figure 5d-f). Figure 5a shows the optical image of the ink surface before solidification, where the color at the intersection of the ink lines is darker than at other locations. Figure 5b shows the white light interference image of the ink  
Figure 3: Contact angle \(\theta\) as a function of contact diameter \(d_{c}\). The dotted line is the average value of the measured contact angles. surface before curing, indicating high flatness. Figure 5c shows the cross-sectional profile corresponding to Figure 5b. Figure 5d,e show the optical and white light interference images of the ink surface after curing, respectively, revealing that the wave-like ripples on the layer surface are formed. Figure 5f shows the cross-sectional profile corresponding to Figure 5e. It is indicated that this is caused by differences in density at the intersection of the ink lines compared to other locations, which will be discussed later. Figure 6a shows the general process of lines to layer. The extruded ink lines merge to form a layer with relatively high flatness due to the self-leveling of the ink.