## 2 Materials and Methods
### Experimental Characterization  
The mechanical and fluidic characteristics of the NO valve are extensively tested in order to assess its performance and to estimate its reliability in the field. Additionally, the measurements are used to extract parameters necessary for model calibration, theoretical design studies, and analytical calculations of the leakage rates.  
The operation of the diaphragm microvalve is fundamentally defined by the piezo-electrically driven diaphragm movement. Free actuator movement is allowed in open state of the microvalve, whereas further actuator movement is restricted when the actuator sits down on the valve seat in closed state. The mechanical stroke of the actuator diaphragm is measured optically using a white light profilometer (Fries Research and Technology; sensor range of 3 mm, maximal resolution of 30 nm) with a quasi-static voltage actuation (amplifier SVR 500-3, piezosystem jena GmbH) ranging from \(-0.4\) kV/mm to \(2.0\) kV/mm to detect field-dependent open and closed state. Repetitive measurements using this setup reveal a measurement accuracy of the total actuator stroke of 2 um. Additionally, the profilometer is used for surface inspection of the microvalve steel diaphragms in order to determine the fractal dimension as well as the maximum and minimum wave vectors \(q_{0}\) and \(q_{1}\) of the surface, respectively, serving as input parameters for analytical leakage modeling (see Section 2.3).  
Subsequently, all manufactured valves are characterized with a view to the fluidic performance using deionized water at room temperature. The instruments used for these investigations are Coriflow sensors of different ranges (Bronkhorst MINI CORI-FLOW M14: range 0.5 mL/min to 167 mL/min, accuracy: \(\pm 0.2\%\) and ML120V00: range 0.8 mL/min to 500 \(\mu\)L/min, accuracy: \(\pm 0.2\%\)) as well as a pressure controller (Mensor CPC3000; range \(-50\) kPa to 200 kPa, accuracy: \(\pm 50\) Pa).  
Applying this measurement setup, the assembled microvalves undergo different fluidic tests. As an initial characterization, we measure their pressure-dependent NO flow rate, their quasi-static opening and closing flow characteristics, and the closed state leakage rate. NO flow rates are evaluated without electrical actuation of the piezoceramic and increasing water pressure at the inlet of the microvalve up to 100 kPa. Quasi-static opening and closing characterization of the microvalve at an inlet pressure of 20 kPa is achieved by sinusoidal actuation of the piezoceramic at a frequency of f = 0.01 Hz and electric fields between \(-0.4\) kV/mm and 2.0 kV/mm. For evaluations of the pressure-dependent leakage rate, the valve's inlet pressure is increased from 0 kPa to 20 kPa, while the piezoceramic is subjected to a constant electric field of 2.0 kV/mm.  
Finally, a fatigue test of the microvalves is performed with 5 microvalves of each design variant. The valves are operated at a constant fluidic pressure load of 20 kPa, a sinusoidal electric signal of 100 Hz, and an electric field of \(-0.4\) kV/mm to 2.0 kV/mm for \(1\times 10^{6}\) actuation cycles. After this long-term excitation, the above-described fluidic characterization methods are repeated in order to evaluate the fatigue of the microvalves, e.g., in terms of increased leakage rates or piezoceramic material failure.