# A Review of Passive Constant Flow Regulators for Microfluidic Applications
## 3 Mechanical Passive Constant Flow Regulators
### Pillar Valves in Parallel  
The passive flow regulators with a flexible Si membrane and a spiral microchannel required extended numerical simulations and are by design sensitive to particles, since the contact surface between the membrane and the substrate is large. Moreover, for square or semicircular microchannel cross-sections, flow rate varies as a power 4 with microchannel dimensions and the microchannel machining is a Critical-To-Quality (CTQ) process step due to the absence of etch stop. An alternative design made of a drilled Si membrane and a substrate with pillars, as shown in Figure 5, has been examined to overcome these abovementioned drawbacks [31]. A membrane hole and its facing pillar form a valve that closes progressively as the pressure increases. The resulting reduction of the gap between the top surface of the pillar and the bottom of the Si membrane leads to an increase in the hydrodynamic resistance of this radial valve. The valve modeling requires the determination, using 3D FEM simulations, of the membrane deflection and the fluid flow as a function of the pressure gradient. Simplified analytical models supported by experimental data were also used to simplify the design phase [32]. The total flow rate \(Q\) is the sum of the flows \(Q_{i}\) through each valve \(i\  
\(0.13\) mL/min \(\pm\) 5% in the pressure range [7; 21] bar. Two similar device structures were tested by [PERSON] et al., who reported constant flow rates of respectively \(0.95\) mL/min \(\pm\) 5% in the pressure range [8; 19] bar and \(0.21\) mL/min \(\pm\) 5% in the pressure range [6; 19] bar [34]. Numerical simulations showed that flow rate variability due to the temperature dependence of the fluid viscosity can be mitigated by the use of a polyethylene block glued at room temperature beneath the pillar substrate made flexible by an additional backside etch [35]. This polyethylene (PE) block, which exhibits a large coefficient of thermal expansion of about \(200\cdot 10^{-6}\)K\({}^{-1}\), is used as a passive thermal actuator that generates a large upward or downward force onto the substrate that respectively reduces or increases the gap between the membrane and the substrate as the temperature varies. A flow regulator delivering a constant water flow rate of \(20\) mL/h at \(20\)\({}^{\circ}\)C has been considered for the simulations. It was shown that the automatic adjustment of the gap due to the presence of the PE block keeps the flow constant at \(20\) mL/h \(\pm\) 10% in the temperature range \([+10;\ +30]\)\({}^{\circ}\)C and the pressure range [50; 150] mbar. Such a feature would improve the overall flow rate variability of a passive constant flow regulator used in an ambulatory infusion system. The simulation tools specifically developed for this regulator design showed that for a target constant flow rate lower than \(10\) mL/h at \(1\) cP, the machining tolerances of the membrane hole diameters significantly narrow and become an important yield detector in production. Infusion error larger than \(\pm\)10% at \(3\sigma\) could be expected [9]. To achieve a constant flow rate as low as \(1\) mL/day or less, [PERSON] et al. designed a triple-stack structure made of an SOI wafer to form a flexible membrane with no holes, a middle Si wafer with pillars, through holes and microchannels, and a bottom borosilicate wafer with inlet and outlet ports (see Figure 6) [9]. Flow regulation is again achieved by the deflection of a silicon membrane which gradually obstructs the fluid path as the pressure increases. The device has several valves in parallel, each valve consisting of an inlet port, a microchannel, a through hole in a pillar, and a radial diffuser (fluidic restriction between the top of the pillar and the bottom membrane). By contrast to the other elements of the valve, the hydraulic resistance of the radial diffusers varies with the pressure. The number of valves and their positions and dimensions are adjusted to meet the targeted fluidic characteristic of the passive constant flow regulator.