## 2 Materials and Methods
### Theoretical Model  
In this study, both PDMS and photoresist are assumed as incompressible Newton fluids [18,19]. Additionally, the thermal reflow process is described by Navier-Stokes and continuity equations. At the given dimension scale of microstructures, the gravitational effect can be ignored when compared with interfacial tension [18]. The momentum equation is shown as follows:  
\[\frac{\partial}{\partial t}\left(\rho\overset{\rightarrow}{v}\right)+\
abla \cdot\left(\rho\overset{\rightarrow}{v}\overset{\rightarrow}{v}\right)=- \
abla p+\
abla\cdot\left[\mu\left(\
abla\overset{\rightarrow}{v}+\
abla \overset{\rightarrow}{v}^{T}\right)\right]+\overset{\rightarrow}{F} \tag{1}\]  
Here, \(p,\overset{\rightarrow}{v},\rho\) and \(\mu\) are static pressure, velocity, density and viscosity, respectively. The source term \(\overset{\rightarrow}{F}\) in our case is related with interfacial tension. The reflow process involves interaction between PDMS and photoresist. Herein, we introduce the multiphase flow system with the volume of the fluid model to calculate the interface evolution of PDMS and photoresist. In this way, only a single momentum equation is required to be solved throughout the domain to obtain the resulting velocity field shared among the phases. Therefore, the mixture velocity is given as follows:  
\[\overset{\rightarrow}{v}=\frac{\alpha_{r}\rho_{r}\overset{\rightarrow}{v}_{ r}+\alpha_{p}\rho_{p}\overset{\rightarrow}{v}_{p}}{\alpha_{r}\rho_{r}+\alpha_{p} \rho_{p}} \tag{2}\]  
where the subscripts \(r\) and \(p\) represent photoresist and PDMS, respectively. Additionally, the volume fraction of photoresist \(\alpha_{r}\) and the volume fraction of PDMS \(\alpha_{p}\) satisfy \(\alpha_{r}+\alpha_{p}=1\). Similarly, the mixture density and mixture viscosity are decided by volume fraction and basic material properties and presented by \(\alpha_{r},\rho_{r}+\alpha_{p}\)\(\rho_{p}\) and \(\alpha_{r}\mu_{r}+\alpha_{p}\)\(\mu_{p}\), respectively. Interfacial tension is the dominated force in the reflow process. It is critical to introduce it into the momentum equation. Here, the continuum surface force model [26] is utilized to express the interfacial tension, where the interface curvature is computed from the local gradient in the interface normal at the interface. In our case, interface normal is defined as the gradient of the \(\alpha_{p}\), and the curvature k is defined in terms of the divergence of the unit interface normal:  
\[k=\
abla\cdot\frac{\
abla\alpha_{p}}{\left|\
abla\alpha_{p}\right|}. \tag{3}\]  
Then, the interfacial tension expressed as volume force \(F_{vol}\), which is decided by the interfacial tension coefficient of PDMS and photoresist \(\sigma_{rp}\) and the curvature of the interface, has following form:  
\[F_{vol}=\sigma_{rp}\frac{\left(\alpha_{r}\rho_{r}+\alpha_{p}\rho_{p}\right)k_{ p}\
abla\alpha_{p}}{\frac{1}{2}\left(\rho_{r}+\rho_{p}\right)}. \tag{4}\]  
Previous models usually ignore dynamic wetting along substrate [16,17], while it is a significant factor for exploring mechanism of PDMS assisted thermal reflow. The dynamic boundary condition associated with the contact angle \(\theta_{w}\) (about 43\({}^{\circ}\) in our case) is introduced in this paper. It is implemented by replacing the unit normal in Equation (3) with the normal near the wall shown in the following:  
\[\hat{n}=\hat{n}_{w}\cos\theta_{w}+\hat{t}_{w}\sin\theta_{w} \tag{5}\]  
where \(\hat{n}_{w}\) and \(\hat{t}_{w}\) are the unit vectors normal and tangential to the wall, respectively. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Unit** & **PDMS** & **Photoresist** \\ \hline Specific heat & J\(\cdot\)kg\({}^{-1}\cdot\)K\({}^{-1}\) & 1540 & 1470 \\ Thermal conductivity & W\(\cdot\)m\({}^{-1}\cdot\)K\({}^{-1}\) & 0.18 & 0.19 \\ Density & kg\(\cdot\)m\({}^{-3}\) & 1030 & 1085 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Specific heat, thermal conductivity and density of PDMS and photoresist.