# Investigation of Structural Stability for Monolithic Nano Bridges on Micro Apertures
## 3 Results and Discussion
(**d**) The diameter of 100 \(\upmu\)m: State of the polymeric multi-layer membrane with some clustering of nano-bridges. (**e**) The diameter of 300 \(\upmu\)m: State of the polymeric multi-layer membrane with torn and clustered nano-bridges. (**f**) The diameter of 500 \(\upmu\)m: State of the polymeric multi-layer membrane with torn and clustered nano-bridges. Then, the strain energy (U) of the deformed structure can be given as follows:  
\[U=\int_{0}^{L}\frac{M(x)^{2}}{2 EI}dx=\frac{P^{2}L^{3}}{384 EI} \tag{2}\]  
The contact region is set to be located at the center of the bridge because it gives the minimum strain energy at a given geometry. Let \(l\) be the length of the non-contact region where the elastic strain energy is stored. The bridges will not collapse when \(L-l<0\). Thus, \(l\) is the minimum length of the bridge to be stable. We denote \(L_{\text{c}}\) as a critical length of \(l\). The strain energy of the non-contact area of the bridges, where bending deformation is dominant, is stored, while that of the contact area is not stored [17,18]. Based on the assumption, the total strain energy of the system is expressed as follows:  
\[U=\frac{P^{2}l^{3}}{384 EI}=\frac{P\delta}{2} \tag{3}\]  
where \(E\) is elastic modulus, \(\delta\) is deflection length of the bridge and \(I\) is second moment of inertia of the deflected bridge, \(I=ta^{3}/12\) (a: width of bridge; t: thickness of bridge). From the relation, \(U\) is defined as a function of the geometrical features. \[U(l)=\frac{8E\delta^{2}ta^{3}}{l^{3}} \tag{4}\]  
Regarding energy conservation, the energy balance between the adhesion energy and strain energy is expressed as follows:  
\[\frac{dU}{dl}+\frac{dE_{ad}}{dl}=0 \tag{5}\]  
\[E_{ad}(l)=-2\gamma_{s}t(L-l) \tag{6}\]  
Figure 4: (**a**) Schematic illustration of the collapses of the nano-bridges and the plot of k from the stability equation, considering of the geometry of the micro and nano structures, (**b**) Schematic illustration of the demolding process at the interface between the PDMS mold and the PUA311 membrane in the side view, and graph of the plotted maximum tensile stress–length relationship of the six diameters of the micro PDMS molds. where \(\gamma_{s}\) (or \(\gamma_{sw}\)) is the surface energy of the bridges, which is calculated based on the geometric mean method. \[\gamma_{lv}\ (1+cos\theta)=2\Big{(}\gamma_{sw}^{\ \ d_{lv}}\gamma_{lv}^{\ d} \Big{)}^{1/2}+2(\gamma_{sw}^{\ p}\gamma_{lv}^{\ p})^{1/2} \tag{7}\]  
where \(\gamma_{lv}\) is the surface tension of liquid, \(\theta\) is the contact angle of the liquid on the solid surface, superscript d refers to disperse and superscript p refers to polar. By measuring the contact angle of two liquid materials, which are DI water and ethylene glycol on the surface, the surface energies in the polar term and in the disperse term are calculated, respectively (Figure 5). The total surface energy (\(\gamma\)) of solid material is obtained by adding up the two terms of surface energy (Table 1). Then, the critical length is determined as follows:  
\[l=L_{\mathrm{C}}=\left(\frac{12E\delta^{2}a^{3}}{\gamma_{s}}\right)^{1/4} \tag{8}\]  
If we denote k as a stability indicator, then  
\[\mathrm{k}=\frac{L_{\mathrm{C}}}{L}=\frac{l}{L}=\frac{1}{L}\bigg{(}\frac{12E \delta^{2}a^{3}}{\gamma_{s}}\bigg{)}^{1/4} \tag{9}\]  
Then, the condition for lateral stability is k \(>\) 1. In this case, the structure would be stable. If k \(\leq\) 1, the structures would be laterally collapsed. As shown in graph in Figure 4a, in our experimental set, the lateral collapse would occur except for the case of 5 \(\upmu\)m and 10 \(\upmu\)m, in which the k value is over 1. The anticipated results based on the criteria are in good agreement with the experimental results. It can be used toward the design rule for the lateral collapse of nano-layers constructed on the micro-aperture structures.