# Temperature Estimation during Pulsed Laser Sintering of Silver Nanoparticles
## 3 Numerical Calculations  
A three-dimensional numerical calculation was carried out in COMSOL Multiphysics for thermal analysis of the pulsed laser sintering process of inkjet-printed Ag NPs lines on glass substrates. The two-dimensional cross-sectional schematic of the three-dimensional calculation model is presented in Figure 2. The three-dimensional model is depicted in Figure S2 (Supplementary Materials). The following assumptions were adopted for the calculation model. The absorbed laser energy was completely converted into heat and no phase change was considered for the Ag NPs because of the high melting temperatures of Ag NPs. Constant specific heat and density were adopted for Ag NPs. The initial condition was set at 20 \({}^{\circ}\)C. The convection and radiation boundary conditions were applied on the surface of the printed conductive ink line and the glass substrate in the form of external natural convection. The natural convection coefficient is expressed in Equation (S1) (Supplementary Materials). Figure 1: Schematic diagram of the experimental setup for pulsed laser sintering of the Ag NPs conductive ink lines. The dimensions of the conductive ink line along with the pads printed on either side are depicted on the top left corner. Figure 2: Two-dimensional cross-sectional schematic of the three-dimensional model used for temperature calculation during the pulsed laser sintering of Ag NPs, where the two sides and the bottom surface of the glass substrate are thermally insulated and the top surface of the ink and the glass substrate is subjected to convection and radiation boundary conditions. Thermal insulation was applied at the two sides and bottom part of the glass substrate, where the temperature gradient across these boundaries was zero. The pulsed laser sintering of Ag NPs lines inkjet-printed on glass substrates was analyzed by a transient three-dimensional heat conduction equation as follows:  
\[\rho_{l}C_{p,i}\left(\frac{\partial T_{i}}{\partial t}+u_{i}\frac{\partial T_{i }}{\partial x}\right)=\frac{\partial}{\partial x}\left(k_{i}\frac{\partial T_{ i}}{\partial x}\right)+\frac{\partial}{\partial y}\left(k_{i}\frac{\partial T_{i }}{\partial y}\right)+\frac{\partial}{\partial z}\left(k_{i}\frac{\partial T_ {i}}{\partial z}\right)+Q_{i}(t), \tag{1}\]  
where \(\rho\), \(C_{p}\), \(k\), and \(u\) are density, specific heat capacity, thermal conductivity of the Ag NPs/glass substrate, and moving velocity of the laser beam, respectively. The subscript \(i=1\) and \(i=2\) represent Ag NPs and glass substrate, respectively (See Table 1). In addition, \(u_{1}\) is the scanning velocity of the laser beam and \(u_{2}\) is zero. The term \(Q_{1}(t)\) is the laser heat source, which is given as follows:  
\[Q_{1}(t)=\{1-R(T)\}\frac{2P}{\pi r^{2}}\exp\left\{2\frac{\left(x-x_{\text{fp} }+vt\right)^{2}+\left(y-y_{\text{fp}}\right)^{2}}{r^{2}}\right\}p(t)\exp(- \frac{z}{\gamma}),\quad Q_{2}=0, \tag{2}\]  
where \(R(T)\), \(P\), \(r\), \(p(t)\), and \(\gamma\) are reflectivity, laser power, laser beam radius at \(1/e^{2}\) of the Gaussian distribution, pulse time, and optical absorption length, respectively. The terms \(v\), \(t\), \(x_{\text{fp}}\), and \(y_{\text{fp}}\) stand for speed of laser scanning speed, time after the start, and initial laser focal \(x\), \(y\)-position at the start in terms of the peak of the Gaussian beam, respectively. The reflectivity and the absorption length at the laser wavelength were acquired experimentally. Figure S3 (Supplementary Materials) shows the plot for the normal reflectivity and absorption length of the incident laser beam against different sintering temperatures. The data used in these plots were calculated from the optical properties measured from ellipsometry [32]. To obtain the density of the printed Ag NPs lines, the porosity in the printed lines was considered and determined as 26% (the method has been described in the reference [33]). We assumed the empty space as air and the rest as Ag NPs.