Many microfluidic applications typically require the integration of a number of valves, pumps, and other active components, most of which require auxiliary off-chip control. Although recent advances in pneumatic microvalves [1-2] have enabled large- scale integration of microfluidic components to perform hundreds of operations in parallel by multiplexing control of embedded operational valves [1-4], the need for a large number of dedicated external control lines limits the practical use of integrated microfluidic systems. The complex valve control can be simplified by integrating microfluidic devices using on-chip fluidic logic networks [5]. Much like the development of electronic logic gates simplified the construction and operation of complex electronic devices, pneumatic logic gates could considerably reduce the number of required off- chip controllers. Various microfluidic logic gates operated on Boolean rules have been proposed to direct internal flows in complex networks and perform simple on-chip calculations [6-10]. Although conceptually powerful, most of the developed microfluidic logic gates rely on different input/output types and, therefore, the output cannot be used as an input to directly actuate subsequent logic gates. This non-cascadable nature inhibits further scaling and feedback routing for more complex circuits. Other approaches utilizing droplet-based input/output for logic gates exhibit excellent cascadability but construction of control platforms from these systems is difficult since the trajectories or existences of droplets cannot be used for actuation of active control components [9-10]. Most importantly, no parallel operations have been achieved from programmable serial input signals.
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