# Investigation of the Influence of Sublayer Thickness on Pairing of Metallic MEMS Shutter Blades
## 1 Introduction
A further study shows the measurements of the Casimir forces between the two metal plates (Figure 2c) using a cantilever that freely oscillates at its clamping point and has the dimensions of 2 cm \(\times\) 5 mm \(\times\) 50:100 \(\upmu\)m. The separation gap between the two metal plates varied from 2 to 12 \(\upmu\)m considering that multiple material variances exist in that geometry, including various coatings on hybrid heterostructures. A further theory was published on the possibility of attractive or repulsive Casimir forces between two symmetrical bodies, such as hemispheres (Figure 2d). The theory states that when one is considering two identical dielectric objects, the Casimir force is attractive. Another investigation on repulsive and attractive Casimir forces demonstrated them as having a zipper-like geometry (Figure 2e). Figure 1: Schematic of the attracting net force arising between two identical uncharged conducting plates. An article was published on the dependence of the Casimir force on the geometry, and the researchers considered two identical rectangular gratings of silicon with dimensions as follows: a thickness of 2.58 \(\upmu\)m, a periodicity of 2 \(\upmu\)m, a width of 908 nm. and a height of 1.5 \(\upmu\)m, revealing a MEMS comb actuator geometry (Figure 2f) and a minimum separation distance between the gratings of 92 nm. The Casimir force shows a strong nonlinearity regarding the distance \(d\) between the two plates, which ranges between \(d^{-3}\) and \(d^{-4}\). In this paper, we use an effect called the \"stiction dilemma\" in the MEMS field, which refers to the undesired stiction of MEMS parts. The Casimir forces are used to self-assemble the shutter blades into arrays of paired ones to obtain a unique structure and further study the Casimir forces between the shutter blades. We define our structure as paired shutters that are in a Yin-Yang shape. The geometry was studied using the research in the fields of origami, 3D self-assembly, and Casimir forces. We measured the separation distance between the attached shutter blades to be 10 nm using SEM images, with a calculated threshold distance of 21 nm to keep the shutter blades tightly attached, according to our previous study [16], which is superior to other geometries. Yet, performing the measurements and calculation of the Casimir force remains challenging for us. This paper is part of a series of investigations on self-assembled structures, i.e., paired shutter blades. It focuses on identifying (i) the usable processing window when we are manipulating the stress within one metal layer via monitoring the changes in the whole comprising system and (ii) the possibilities of geometrical changes while we keep the shutter blades as a pair. In addition to self-assembly, in general, many applications could be achieved in the future: (a) Particle transport in magnetic stray field tubes [28] for drug delivery, lab on a chip systems, mass transport, and sensorics. The Yin-and-Yang shape is superior to the circular diameter used in [28], and it introduces additional attractive research fields using these new structures. (b) Shock absorbers for mobile systems. (c) Lightweight construction for material resource efficiency and weight reduction in construction is another possible application of macro- and micro-sized systems. Attractive applications could be enrolled in the field of the lightweight construction of  
Figure 2: Examples of different geometries demonstrating the existence of Casimir forces: (**a**) a lens and a metal plate [21], (**b**) two crossed cylinders of mica [27], (**c**) two metal plates [23] (**d**) two hemispheres [24], (**e**) a zipper-like geometry [25], and (**f**) a MEMS comb actuator via a silicon grating geometry [26]. All subfigures have been redrawn from the respective references. microsystem technology and mobile systems (planes, cars, and drones, etc.). This paper is organised as follows: Section 2 describes the arrangement and the design of the unpaired, curled blades and the paired shutter blades.