# Transient Numerical Modelling of the Pin-in-Paste Technology
## 2 Materials and Methods  
### The Numerical Model  
The numerical model for predicting the hole-filling by the solder paste was created based on the finite volume method. The solder paste can be considered as an incompressible fluid (\(\rho\) is constant). Therefore, the equation of mass continuity reduces to the volume continuity Equation (1) [20]:  
\[\
abla\cdot\mathbf{u}=0 \tag{1}\]  
where \(\mathbf{u}\) is the flow velocity. In the range of the generally used printing speeds (20-200 mm/s) and by the high viscosity of solder pastes (over 30-100 Pa-s), the Reynolds number is much lower than 1. This indicates that the Stokes flow Equation (2) can describe the system in a steady state [21]. Additionally, the flow is considered laminar due to the low Reynolds number.  
\[\
abla p=\mu\
abla^{2}\mathbf{u}+\mathbf{f} \tag{2}\]  
where \(p\) is the pressure [Pa], \(\rho\) is the density of the fluid [kg/m\({}^{3}\)], \(\mu\) is the dynamic viscosity [Pa-s], and \(\mathbf{f}\) is an applied body force [N], which is 0 in our case.  
The geometry of the model includes the printing squeegee, the solder paste (fluid), the stencil, and the through-hole(s) of the printed circuit board, as illustrated in Figure 2.  
The transient, two-phase model was built using ANSYS Fluent. In the model, the air and the solder paste were considered incompressible. The two-phase model was based on the Volume of Fluid (VOF) method. The movement of the squeegee was addressed by the dynamic meshing method, using the bottom layering technique. The following boundary conditions were applied in the transient model: No-slip wall boundary conditions were applied at the stencil, the squeegee, and the sidewalls of the through-holes. The other walls possessed pressure outlet condition (Figure 3).  
Since the heat generated by the moving solder paste is negligible [22], the model was isothermal. Gravitational acceleration was considered as 9.8 m/s\({}^{2}\). The dynamic meshing parameters were as follows: the split factor = 0.4, collapse factor = 0.2, cell height = exact value of meshing element size, and the implicit update was activated. The mesh size was optimized by grid analyses and validated by an empirical approach (Section 2.2). The final mesh element size for the fluid area and through-hole was 900 and 200 um, respectively.  
The rheological properties of the solder paste were described by the Al-Ma'aiteh model (3) [19], with parameters valid for a Type-4 solder paste (particle size range is 20-38 um).  
\[\eta_{a}=\beta(\eta_{\infty}+\frac{\eta_{0,1}-\eta_{\infty,1}}{\left[1+ \left(\lambda_{1}\dot{\gamma}\right)^{\beta}\right]^{\frac{1-\eta_{1}}{a}}}) +(1-\beta)(\eta_{\infty}+\frac{\eta_{0,2}-\eta_{\infty,2}}{1+\left(\lambda_{ 2}\dot{\gamma}\right)^{\eta_{2}}})\quad\text{where}\quad\beta(\dot{\gamma})= \left\{\begin{array}{ll}1&\text{if}\quad\dot{\gamma}\leq\dot{\gamma}_{t}\\ 0&\text{if}\quad\dot{\gamma}>\dot{\gamma}_{t}\end{array}\right\} \tag{3}\]  
where \(\eta_{a}\) is the apparent viscosity; \(\eta_{0,1}\) and \(\eta_{0,2}\) are the asymptotic viscosity values belonging to the zero rate of shear (10,560 and 31,500 Pa-s); \(\dot{\gamma}\) is the rate of shear; \(\eta_{\infty,1}\) and \(\eta_{\infty,2}\) are the asymptotic viscosity values belonging to the infinite rate of shear (24 and 30 Pa-s);  
Figure 3: Boundary conditions of the numerical model.  
Figure 2: Schematic of the numerical model for analyzing the hole-filling by the solder paste.  
\(\lambda_{1}\) and \(\lambda_{2}\) are time constants (110 and 340 s); \(n_{1}\), \(n_{2}\) and \(a\) are dimensionless exponents (0.36, 0.69 and 7); and \(\beta\) is a weighting parameter depending on \(\hat{\gamma}_{t}\) (0.0125 s\({}^{-1}\)).