## 2 Materials and Methods
### Analytical Modeling of Leakage Rates  
In this section, an analytical model for the estimation of the leakage rates of the NO valves in closed state is presented. Leakage is the result of imperfect contact between the steel foil and the steel valve body due to surface roughness. In order to evaluate the leakage flow, we first determine the average separation between the steel diaphragm and the valve seat. This separation represents the height of the microchannel, determining the pressure-dependent flow rate. In order to solve this problem, we apply the approach of [PERSON] [30], who studied the contact problem of two solids with surface roughness squeezed against each other. This leads to the average separation \(u(p_{contact})\) between the steel diaphragm and the valve body as a function of the contact pressure \(p_{contact}\):  
\[u(p_{contact})\ =\gamma\alpha^{-1}h_{rms}\log\left(\beta eq_{0}h_{rms}\frac{E^{* }}{p_{contact}}\ \ \right) \tag{2}\]  
where the constant \(\gamma\) is set to one, which means assuming perfectly elastic deformation, \(\alpha\) and \(\beta\) are surface roughness constants, \(h_{rms}\) is the root mean square of the surface roughness, \(\epsilon\) is a constant related to the power spectrum of the surface roughness \(C(q)\) and the lower and upper cutoff wave vectors \(q_{0}\) and \(q_{1}\) of the surface, and E* the plane-strain modulus. The contact pressure \(p_{contact}\) of the two bodies is equivalent to the maximum force exerted by the actuator diaphragm. The so-called blocking pressure \(p_{block}\) of the actuator is scaled by the ratio of valve diaphragm surface to the contact area to achieve the contact pressure and can be calculated according to [31]:  
\[p_{\text{contact}}=\frac{A_{diaphrag}}{A_{\text{contact}}}\ p_{\text{block}} \tag{3}\]  
Comprehensive derivations of Equations (2) and (3) are provided in Appendix B.  
Applying Equation (2), we are now able to calculate the leakage flow rate depending on the pressure drop over the closed valve. To this end, an appropriate flow model of the leakage flow of water through microscale channels and openings must be developed. Since water can be considered as an incompressible fluid [32] and the Reynolds number is (as in almost all microfluidic applications) much smaller than 1 [33], the flow is modeled as a simple laminar Poiseuille flow driven by the pressure difference \(\Delta p\) between the inlet and the outlet of the valve. The channel geometry is determined by the design of the sealing and the average surface separation due to surface roughness. The average separation height is on the order of some micrometers, whereas the sealing dimensions are 2-3 orders of magnitude higher. Therefore, the leakage channel is assumed to be a rectangular slit, whose width \(w\) is much larger than its height \(h=u(p)\), so that the side walls of the channel can be neglected, and the geometry can be regarded as a channel between two infinite plates [34]. Applying no-slip boundary conditions \(v(z=0)=v(z=h)=0\), the velocity profile and resulting flow rate are given by:  
\[v_{x}(z)=\frac{\Delta p}{2\mu L}((h-z)z), \tag{4}\]  
\[Q=\int_{0}^{w}\!dy\!\int_{0}^{h}\!\!dz\;v_{x}=\frac{h^{3}w}{12\mu}\frac{\Delta p }{L}, \tag{5}\]  
where \(\mu\) is the viscosity of water.  
The above-described geometrical simplification leads to a worst-case scenario estimation of the leakage rate; therefore, it is expected that the analytical model overestimates the observable leakage rate and constitutes an upper limit for the evaluation of the manufacturing-induced imperfections of the valve and the hereof resulting potential issues in its reliability in the medical application.