# Investigation of Structural Stability for Monolithic Nano Bridges on Micro Apertures
## 1 Introduction  
Thin membranes possessing micro-/nano-scaled apertures have been extensively developed for a wide range of applications, including deposition masks, lithography masks, etching masks, microfluidic devices and filters [1; 2; 3; 4; 5; 6; 7]. A variety of techniques to improve the stability of the nanostructured membrane have been explored using diverse materials, factoring in corresponding surface energy, elasticity, transparency and surface hydrophilicity, depending on its purpose. When it comes to the thin membrane with a precisely controlled aperture size and arrangement, nanostructured silicon nitride (SiN\({}_{\mathrm{x}}\)) membranes supported on rigid Si membranes have been used due to its superiority in mechanical rigidity as well as structural controllability [4]. Due to its rigidity, the ultra-thin layer of the SiN\({}_{\mathrm{x}}\) membrane is likely to suffer from mechanical damage (cracks, breakdown, fractures, etc.) during handling for fabrication processes. It also requires complicated processes with high fabrication costs. As one of the contenders, free-standing stencils with polydimethylsiloxane (PDMS) has been fabricated through a soft lithography process thanks to several advantages, such as a low cost, conformal contact, flexibility, ease of process, the low mechanical strength and low elastic modulus,which rules out possible physical damages [8]. However, an implementation of polymeric flexible materials has its benefits, such as inducing folding/crumpling phenomena during the fabrication process, which leads to a limit in the aperture size and thickness [9; 10]. To address these issues, a polymeric nano-aperture membrane monolithically constructed on a supporting layer with micro-apertures was reported previously [11; 12]. The one-step process based on the scavenging effect of the infiltrated oxygen gas is highly advantageous in the facile fabrication of free-standing membranes with nano/micro two-level apertures. Furthermore, the stability of the periodically arranged nano-holes supported on microscale circular apertures, considering the adhesion forces in the demolding process, has been investigated to realize the stable fabrication of ultrathin membranes with nano-scale apertures [13]. Unlike 2-dimensionally (2D) closed-hole patterns on the polymeric membrane, the construction of a 1D line-patterned membrane causes directionally sensitive structural instability during the demolding or handling process in two aspects: The one is the lateral-direction collapse of the neighboring nano bridges, and the other is the vertical-direction fracture of the nano bridges. To overcome these issues in reliable reproduction, a 3D approach should be considered for improved mechanical stability.  
Herein, we report the design rule in the formation of the 1D nano-line membrane over the micro-hole supports, which provides the geometrical restriction for the design of polymeric membranes with two-level apertures. To investigate the overall 3D mechanical stability, we conducted six sets of 1D line-shaped polymeric membranes with varying micro-pillar diameters (5 \(\upmu\)m, 10 \(\upmu\)m, 50 \(\upmu\)m, 100 \(\upmu\)m, 300 \(\upmu\)m and 500 \(\upmu\)m) in fixed conditions with a nano-line width (\(w\)) of 800 nm. From the results, we categorized the structures in three cases: (1) stable, (2) collapse and (3) fracture, and suggest a model that explains the physical mechanism of the lateral collapse with the vertical fracture during the demolding process when peeled-off from the micro holds.