# Transient Numerical Modelling of the Pin-in-Paste Technology
## 3 Results and Discussion  
The interaction between a set of through-holes (7 pcs.) was analyzed for two cases: (a) the through-holes were arranged in parallel with the travelling direction of the printing squeegee; (b) the through-holes were arranged perpendicular to it. The diameter and the \(z\)-dimension of the holes were 1.1 and 1.55 mm, respectively. These holes were distributed evenly, at distances of 2.54 mm (Figure 7), similarly to a lead arrangement of a through-hole serial connector. As expected, there was no interaction between the holes, and the hole-filling was the same for all through-holes when they were arranged perpendicular to the travelling direction (in parallel with the line) of the printing squeegee. However, an interaction between the holes was found when they were arranged in parallel with the direction of the squeegee movement. A clear and gradual reduction in the hole-filling was observed, which became more intensive at higher printing speeds. The differences in hole-filling between the first and the seventh hole were 2.4% and 9.2% for the printing speeds of 20 and 120 mm/s, respectively (Figure 8). The hole-filling was analyzed further to find a possible explanation for the differences in the filling. The rate of shear during the printing within the solder paste was analyzed by comparing the cases of having one single hole or seven consecutive holes (Figure 9--the printing squeegee travelled till the first through-hole). It was found that placing the holes after one another disturbs the flow-field, and the rate of shear is lower (by up to 5-6%) on the stencil surface compared to the case utilizing a single hole (Figure 10). Figure 6: Empirical and calculated hole-filling by the solder paste; length of the holes in \(z\)-direction was 1.55 mm, and diameters were 0.8, 1.0, 1.1, 1.4 mm; average calculation error is 6%, and maximum calculation error is 12.8%. Figure 7: Numerical model, where the through-holes are arranged in parallel with the travelling direction of the squeegee. The seven consecutive through-holes are distributed evenly at distances of 2.54 mm. Since solder pastes are shear-thinning fluid, a decrease in shear rate yields an increase in the viscosity (Figure 10(a)). This increase in viscosity means a larger reluctance of the material to flow into the through-holes. In addition, the disturbance in the flow-field resulted in a decrease in pressure in proximity to the through-holes (Figure 10(b)). Figure 8: Hole-filling by the solder paste in the seven consecutive holes at different printing speeds. Figure 10: Rate of shear on the surface of the stencil at the proximity of the through-holes for the printing speed of 120 mm/s. Figure 9: Analyzing the interaction between through-holes and the hole-filling (printing speed was 120 mm/s): (**a**) single-hole model; (**b**) model including seven consecutive through-holes. The hole-filling during stencil printing can be approximated by the Hagen-Poiseuille Equation (4). This states that the volumetric flow rate within a channel depends on the dynamic viscosity of the fluid, on the pressure gradient in the channel, and on the radius of the channel [24]:  
\[Q=-\frac{\pi}{8\eta}\left(\frac{\partial p}{\partial z}\right)\cdot R^{4} \tag{4}\]  
where \(Q\) is the volumetric flow rate, \(R\) is the radius of the hole, and \(\partial p/\partial z\) is the pressure gradient. From this expression, one can deduct that an increase in the pressure (and thus in the pressure gradient) and/or a decrease in the dynamic viscosity yields a lower volumetric flow rate in a channel, implying lower hole-filling by the solder paste in pin-in-paste technology. This decrease in hole-filling can be considered either in the design phase of the product or the manufacturing. Through-hole connectors should be oriented perpendicular to the travelling direction of the printing squeegee to minimize the interaction between the holes and the variation in the hole-filling by the solder paste.