# A Review of Passive Constant Flow Regulators for Microfluidic Applications
## 3 Mechanical Passive Constant Flow Regulators
### Devices with Flexible Membrane(s)  
#### 3.1.1 Silicon Membrane(s)  
To the best of the author's knowledge, the first passive constant flow regulator made using microelectromechanical system (MEMS) technology was conceived by [PERSON] et al. in 1988 for the treatment of hydrocephalus [29]. The device consists of a silicon wafer (membrane) between the top and bottom Pyrex covers. The rectangular Si membrane deflects, as the pressure increases, toward the bottom Pyrex cover to narrow the fluid path and therefore increase the hydraulic resistance (see Figure 3). Theoretical modeling of the membrane deflection and fluid flow was used to design the passive valve. The thin rectangular membrane was obtained by etching a 2\" Si wafer. The top (inlet port) and bottom (outlet port) Pyrex covers were anodically bonded to the middle Si wafer. Tests performed with water showed a constant flow rate of about 0.015 mL/min in the pressure range \([200;400]\) mbar for the device with a channel height of 9.5 \(\upmu\)m, a membrane thickness of 26 \(\upmu\)m, and a ratio L/W = 2, where L and W are the channel length and width, respectively. The other design configurations did not show flow saturation. The saturation flow rate and pressure requirements for hydrocephalus (about 0.35 mL/min in the pressure range \([10;35]\) mbar) were not met, but the authors used the experimental data to refine the numerical model of this passive flow regulator with a thin rectangular membrane. Spiral Microchannels  
In the device conceived by [PERSON] et al. [29], the channel cross-section varies continuously as the pressure changes since the channel is formed by the gap between the bottom of the moving membrane and the bottom cover. Considering also that the flow-induced pressure distribution along the channel leads to non-symmetrical membrane deformation, it must be stressed that the design of a constant flow regulator with such a structure is a challenging engineering task. An alternative passive flow regulator design comprising a well-defined channel cross-section was investigated by [PERSON] et al. at EPFL in 1998 [16] and further developed by [PERSON] et al. at Debiotech SA in 2001 [30]. The device structure, shown in Figure 4, comprises an inlet port, a circular Si membrane with a central hole, a spiraling microchannel etched in a Pyrex substrate, and an outlet port. As the pressure difference between the inlet and the outlet increases, the Si membrane deflects and reduces the gap  
Figure 3: Schematic cross-section of a triple-stack passive flow regulator, with a Si wafer (membrane), bonded to the inlet and outlet Pyrex covers. The flow direction is indicated by red arrows. The fluid is forced to flow through the gap (or channel) between the membrane and the bottom cover that narrows as the pressure increases. As a result, the flow–pressure relationship becomes nonlinear. between the membrane and the substrate. At low pressure, the main hydraulic resistance is due to the hole in the membrane. Once the membrane is in contact with the substrate, the fluid is forced to flow through the microchannel. The contact radius of the membrane with the substrate as a function of the pressure is obtained by finite element method (FEM) simulations since the membrane deflection is highly nonlinear. Constant microchannel cross-sections were considered in the two studies [16; 30]. Therefore, constant flow regulation was achieved by calculating a spiral channel in a way that the part of the channel covered by the membrane increases linearly with the pressure. The target water flow rates of these passive devices for implantable drug delivery systems are 0.5 mL/day [16] and 1 mL/h [30]. [PERSON] et al. used a 45 \(\upmu\)m Si membrane with a hole of 20 \(\upmu\)m in diameter obtained by laser ablation. The channel in the glass substrate was etched in a 50% hydrofluoric acid (HF) solution. Finally, the membrane and the substrate were anodically bonded together.