# Modeling and Design Enhancement of Electothermal Actuators for Microgripping Applications
## 2 Materials and Methods
### Electrothermal Actuator Design Enhancement  
In order to improve the performance of micro-electrothermal actuators, a single row of square-shaped perforations with a width of 5 um are added on the top surface of the hot arm of both actuator 1 and actuator 2. The position of these perforations on the surface of the hot arm are depicted in Figure 7a. Additionally, Figure 7 shows the considered configurations for the placement of the perforations considered in this study. Indeed, Figure 7b-d shows the geometry and position of perforations for different scenarios on the surface of the non-extended hot arm. The radius of circular perforations is selected to be 2.82 um so that the volume reductions caused by square and circular perforation are the same. We plot, in Figure 8, the variations of the tip displacement and temperature with the applied voltage for perforated and non-perforated actuators 1 and 2 when considering a single/double row of square/circular-shaped perforations on the top surface of the hot arm for both actuators, as illustrated in Figure 7a. The presence of perforations on the hot arm of the actuator enhances the response of the thermal actuator, as manifested by the increase in the displacement and the decrease in the operating temperature. Indeed, the displacement magnitude for actuator 1 without perforations at 8 V is found to be equal to 1.793 um, whereas for the single-row square-perforated actuator 1, the displacement at the same voltage reaches 1.921 um. The tip displacement is further increased to 2.061 um  
Figure 6: Variations of the tip displacement with the applied voltage and comparison against previous experimental results and analytical solution: (**a**) actuator 1 and (**b**) actuator 2. Figure 7: Schematic diagrams of the top surface of the hot arm of the electrothermal actuator illustrating the placement of the perforations. The length of the non-extended hot arm is 250 μm and the thickness is 25 μm in all cases. when switching to a double-row square perforation. The same observations are made for actuator 2 in which the tip displacement magnitude at 8 V increases from 3.090 \(\upmu\)m to 3.7286 \(\upmu\)m when adding a double row of square perforations. This behavior may be attributed to the decrease in the overall stiffness of the system and hence an increase in their displacements. As per Equation (2), the perforated arms have higher electrical resistance given the associated reduction in the surface area, and this leads to higher power generation induced by Joule heating, as shown in Equation (1). This would explain the capability to achieve larger deformation at the same electric actuation level. Furthermore, using the analytical approach based on beam theory, Equation (11), we notice that the incorporation of perforation reduces the effective width and thickness of the arm, and this results in a larger tip displacement for the same electric actuation. The temperatures at the tip are reduced when perforations are added on the top of the hot arm and there is larger tip displacement as the temperatures are reduced; these are two of the most important effects of perforations on the U-shaped electrothermal micro-actuators. We report, in Table 2, the relative changes in tip displacement and temperature for both actuators 1 and 2 after adding perforations when operating at 8 V.  
Figure 8: Comparison between perforated and non-perforated electrothermal actuators: (**a**) tip displacement of actuator 1, (**b**) tip displacement of actuator 2, (**c**) tip temperature of actuator 1, and (**d**) tip temperature of actuator 2. Contour plots displaying the temperature and displacement for both single-row square-perforated and non-perforated actuators when applying 5 V are shown in Figure 9. Clearly, the incorporation of perforations results in lower temperatures and higher displacement for both actuators. Displacement contour plots show maximum displacements occurring at the tip, since the other side of actuator is connected to the anchor and the boundary has a fixed constraint condition.