# Investigation of Structural Stability for Monolithic Nano Bridges on Micro Apertures
## 3 Results and Discussion
With the prepared PDMS molds and PUA pre-polymer resin, the two-level multiscale polymeric membranes have been successfully constructed through the fabrication process, and the six types of nano bridges monolithically constructed on the micro-aperture membrane were observed through SEM, as shown in Figure 3. The results show that the nano bridges are stably sustained on the supporting layer of the micro-aperture structures with a diameter of 5 \(\upmu\)m and 10 \(\upmu\)m. In other cases, structural failures of nano bridges occurred, meaning that the nano bridges were deformed, fractured or unsupported by the layer. The failures are classified by two cases, which are the collapse of thin nano bridges in the lateral direction and fracture of the bridges in the direction perpendicular to the line direction. In the cases of the 50 \(\upmu\)m and 100 \(\upmu\)m sizes of the supporting layer, which are relatively smaller than that of the 300 \(\upmu\)m and 500 \(\upmu\)m sizes, the centers of the nano bridges were collapsed or clustered together. In turn, in the case of the 300 \(\upmu\)m and 500 \(\upmu\)m micro aperture sizes of the supporting  
Figure 2: Fabrication process of the free-standing dot-line multiscale polymeric membranes. (**a**) Schematic illustration of the assembled PDMS micro-pillars mold, PUA polymers and PDMS nano-lines mold. (**b**) Peeling off the PDMS nano-lines mold. (**c**) Peeling off the nano-line and micro-apertures multiscale membrane that is made of PUA; the thin nano-line membrane, which was supported and combined with the micro-apertures membrane. (**d**) Digital camera image of the free-standing and flexible multiscale polymeric membrane with a large area. layer, fracture and collapse of the nano bridges occurred simultaneously during the detaching process. As the micro aperture size of the supporting layers increases, the thin nano bridge structures get unstable, as expected. From the observation, the finally constructed two-level multiscale polymeric membrane can be categorized in mainly three cases: (1) stable, (2) collapse and (3) fracture and collapse. To analyze these phenomena and set the criteria for each stability, it is necessary to separate the observational view of the structures. Since the collapse, which is determined by the geometrical features of line width, the gap width of the lines and the micro aperture size should be considered in the top view of the structure. In turn, the fracture, which is determined by the geometrical features of line thickness, and the micro aperture size should be considered in the side view of the structures. In order to propose simple theoretical models to set the stability criteria for the collapse phenomenon in the lateral direction and fracture phenomenon in the vertical direction, a model where the beam structure and nano bridge are fixed at both ends is assumed. First, to establish a criterion for lateral stability of the nano bridges, we model the collapsing of the bridges as depicted in the inset of Figure 4a, in which two neighboring line bridges adhere to each other. The adhesion between the bridges induces a distributed bending moment \(M(x)\) with an adhesion force, \(P\). \[M(x)=\left\{ \begin{array}{cc}\frac{P}{2}x-\frac{P_{L}}{8},&x=\left[0,\frac{L }{2}\right]\\ -\frac{P}{2}x+\frac{3P_{L}}{8},&x=\left[\frac{L}{2},\,L\right]\end{array} \right. \tag{1}\]  
where L is length of the bridge, which is same with the diameter of the micro aperture. Figure 3: The stability criteria of the nano-line membrane monolithically constructed on the micro-aperture membrane. (**a–f**) SEM images and schematic illustration of the two-level polymeric membrane that consists of an 800 nm line layer and diverse supporting micro-pore-sized membranes. (**a**) The diameter of 5 \(\upmu\)m: State of the polymeric multi-layer membrane is stable. (**b**) The diameter of 10 \(\upmu\)m: State of the polymeric multi-layer membrane is stable. (**c**) The diameter of 50 \(\upmu\)m: State of the polymeric multi-layer membrane with some clustering of nano-bridges.