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The use of monolithic normally-closed (NC) valves within microfluidic systems is currently limited by the poor scalability of mitigation strategies developed to prevent the formation of a permanent bond between the valve 'seat' and elastomeric membrane during fabrication. Herein we report the highly-scalable design and characterization of a slightly-open doormat (SOD) valve that exhibits properties traditionally associated with NC valves including operability at low actuation pressures, and the capacity to produce a complete seal. In addition, these valves are comparable in their scalability and in responsiveness, and exhibit a capacity to eliminate resting channel resistance that is absent in their NC counterparts. We demonstrate the utility of this novel valve design through the design and operation of a gas-on-gas multiplexer containing 32 device valves controlled by a multiplexer containing 160 SOD valves using only 10 external solenoid valves and a single common pressure source. [ABSTRACT FROM AUTHOR]