# Investigation of Structural Stability for Monolithic Nano Bridges on Micro Apertures
## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Material** & \(\gamma^{p}\) **(mN m\({}^{-1}\))** & \(\gamma^{d}\) **(mN m\({}^{-1}\))** & \(\gamma\) **(= \(\gamma^{p}\) + \(\gamma^{d}\)) (mN m\({}^{-1}\))** & \(W_{\text{pdms-PUAS11}}\) **(mN m\({}^{-1}\))** \\ \hline
**DI water** & 21.8 & 51.0 & 72.8 & - \\ \hline
**Ethylene glycol** & 29.0 & 19.0 & 48.0 & - \\ \hline
**PUA 311** & 10.22 & 10.20 & 20.42 & 28.14 \\ \hline
**PDMS** & 16.86 & 21.7 & 17.58 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Surface energy of the materials. Figure 5: Measurements of the contact angle of DI water and ethylene glycol on the surface of PUA 311 and PDMS. When the nano-line length was 300 and 500 \(\upmu\)m, not only the collapse phenomenon of the nano bridges but also the fracture phenomenon at the center of the nano bridges was observed. To elucidate this phenomenon, we proposed a model of mechanical stability of the beams fixed at both ends under constant distributed loads [13]. When the beam fixed at both ends was under the uniformly distributed load, the maximum allowable stress can be derived as follows [19]:  
\[\sigma_{max}=\frac{M_{center}y_{max}}{I}=\frac{qL}{2t^{2}}\,\ \ I=\frac{Lt^{3}}{12} \tag{10}\]  
where \(M_{center}\) is the moment at the center of the beam, \(y_{max}\) is the maximum distance across the beam from the center of the beam, \(I\) is the area moment of inertia of the beam, \(q\) is the stress applied to the beam and \(h\) is the length of the beam. The allowable beam deflection at the center of the beam is assumed to not exceed the thickness of the nano bridge (\(\sim\) 400 nm), and the applied stress \(q\) is the work of adhesion required to detach the PUA 311 membrane from the PDMS mold. At the center of the nano-bridges is applied the maximum stress. It is generally known that the fracture of the structure occurs when the applied stress exceeds the ultimate tensile stress (\(\sim\)16 MPa) [20,21]. To calculate the work of adhesion, we used following harmonic mean method equations:  
\[W_{12}=4\Bigg{(}\frac{\gamma_{1}^{p}\gamma_{2}^{p}}{\gamma_{1}^{p}+\gamma_{2}^ {p}}+\frac{\gamma_{1}^{d}\gamma_{2}^{d}}{\gamma_{1}^{d}+\gamma_{2}^{d}}\Bigg{)} \tag{11}\]  
where the superscripts \(p\) and \(d\) are for the polar and dispersed components of the surface tension \(\gamma\), respectively; these values are summarized in Table 1. The calculated work of adhesion is about 28.14 mJ/m\({}^{2}\), which is a uniformly distributed load on the nano bridges when the PUA membrane was detached from the PDMS mold. Therefore, Equation (10) can be substituted with the following equation:  
\[\sigma_{max}=\frac{WL}{2t^{2}} \tag{12}\]  
From Equation (12), we can calculate the maximum applied stress depending on the length of nano bridge structures. Figure 4b shows the illustration of the work of adhesion at the interface between the PDMS mold and PUA 311 nano-layers, as well as the plotted maximum tensile stress for the cases 5 \(\upmu\)m, 10 \(\upmu\)m, 50 \(\upmu\)m, 100 \(\upmu\)m, 300 \(\upmu\)m and 500 \(\upmu\)m. As shown in the graph, the maximum applied stress of the 100 \(\upmu\)m and lower pore diameter is lower than ultimate tensile stress of PUA 311. As a result, the fracture phenomenon is not observed for nano bridges supported on micro-aperture diameters of 5 \(\upmu\)m, 10 \(\upmu\)m, 50 \(\upmu\)m and 100 \(\upmu\)m. Contrariwise, the maximum applied stress of the 300 \(\upmu\)m and 500 \(\upmu\)m pore diameters is higher than the ultimate tensile stress of PUA 311, and some fractures are observed as unstably supported on the micro-aperture. These design rules can be used as a guideline for the fabrication of flexible and free-standing membranes with nano-line structures monolithically constructed on the micro-aperture membrane.