# Temperature Estimation during Pulsed Laser Sintering of Silver Nanoparticles
## 4 Results and Discussion
At the start of the silver line, maximum value of about 44 W/(m\(\cdot\)K) is recorded against the maximum temperature of 201 \({}^{\circ}\)C. Furthermore, as we progress, the conductivity of 46 W/(m\(\cdot\)K) value is recorded against 203 \({}^{\circ}\)C at the middle of the line, and, finally, the conductivity of 49 W/(m\(\cdot\)K) value against 204.4 \({}^{\circ}\)C.  
Figure 4: Maximum temperature profile after a full pulsed laser scan over the printed Ag NP line. The maximum recorded temperature was approximately 204.4 \({}^{\circ}\)C. The full laser scan across the entire 7000 \(\upmu\)m length of the printed Ag NPs line takes around 52 s at a scanning speed of 135 \(\upmu\)m/s. Figure 5: 2D surface plots show the temperature distribution at different time steps. The first 5 ms after the peak temperature shows high cooling rate while the cooling rates reduces as the time steps progresses. The reason behind the slightly increasing temperature trend in Figure 4 can be elaborated by considering the \"residual temperature\". It is defined as an initial temperature just before the start of next laser pulse [20]. Figure 8 presents the residual temperatures after a complete laser scan of the entire 7000 \(\upmu\)m-long printed conductive line, which are extracted from transient temperature variations in Figure 4. The region away from the domain hit by a laser pulse does not cool to room temperature before the next pulse is hit. To understand this, we calculated the thermal diffusion length induced by the pulsed laser operation  
Figure 6: The transient behavior of \(T\) (temperature) and \(T_{\text{max}}\) (stored maximum temperature) plotted at a point where the maximum temperature is recorded at 27.05 s.  
Figure 7: Spatial thermal conductivity distributions are presented at the start, middle, and end throughout the sintered NP line. The magnified areas are highlighted the thermal conductivity distribution at different points. to show how far the heat diffuses after a laser shot. For the current pulsed laser process configuration, the thermal diffusion length is roughly estimated by the following relation. \[d_{L}=2\sqrt{\alpha\tau}, \tag{3}\]  
where \(d_{L}\), \(\alpha\), and \(\tau\), stand for thermal diffusion length, thermal diffusivity, and pulse duration, respectively. The thermal diffusivity is obtained from the Ag NPs thermal conductivity of 49 W/m\(\cdot\)K, density 7762.6 kg/m\({}^{3}\), and heat capacity 245 J/kg\(\cdot\)K values. With a pulse duration of 100 ms, the calculated heat diffusion length is approximately 3.2 mm, which is much larger than the beam diameter of 270.9 \(\upmu\)m. Therefore, the region ahead of the laser irradiation area is thermally affected before the irradiation of the scanning laser beam. The residual temperature increases with the progressing pulse train. The pulse repetition rate provides a shorter thermal relaxation period than the time needed for the Ag NPs to cool off completely to room temperature (\(T_{i}\) = 20 \({}^{\circ}\)C). This behavior indicates that a residual temperature acts as an initial condition for the following laser pulse. Consequently, the attained higher peak temperature cools to slightly higher residual temperature with each progressing laser pulse. High residual temperatures can be detrimental to the underlying substrate because it increases the overall process temperature and heating duration. Furthermore, Equation (3) tells us that a longer pulse duration will cause a wider thermal diffusion length. The reason is that a longer pulse duration supplies prolonged thermal energy to the underlying NPs providing enough time for the wider propagation of heat as reported in a previous study [26]. The vertically deeper heat diffusion length is known to cause complete sintering which tends to increase the possibility of the adhesion strength between the NPs layer and the underlying substrate [26; 40; 49]. Figure 9 presents the residual temperature plots under the varied experimental parameters.