Read the complete paper:
Synchronous Universal Droplet Logic and Control, Nature Physics
The basic aspects that enable "droplet fluidic logic" are described in the recent (2015) Nature Physics paper . The key feature of this work is the ability implement a synchronous logic scheme instead of an asynchronous logic scheme as described in our 2007 microfluidic logic paper  and in other works . In this work, we use a multiphase system of water-based ferrofluid droplets in oil. Several schemes for single phase microfluidics have been described before  and are outside the scope of what we would like to cover here.
The fundamental principle of operation includes the generation of a magnetic landscapes in two dimensions . These 2D landscapes are determined by the geometry of permalloy bars laid out on a surface and an in-plane magnetic field that magnetizes the bars. Arrangement of these different bars and a clocked, rotating magnetic field results in a periodically varying energy landscapes that guide the magnetized ferrofluid droplets around. A similar scheme has been exploited previously to move magnetic domains in electric circuits (with a crucial distinction that magnetic bubble memory enabled magnetic domains embedded in garnet films to be flipped - and no physical material was actually transported) .
Above, you see droplets can be steered by magnetic landscapes in arbitrary "paths", analogous to electric circuit guiding current. It gets more exciting when you droplets interact with each other. These interactions allows us to make boolean logic circuits - with a special caveat: This is a "conservative logic scheme" - which means that information can not be erased completely (matter can not vanish). This is an important rule, and thus any logic gate you develop will need to have physical material be routed somewhere (e.g. trash line) since physical entities (like droplets) can not just vanish into thin air. A simple way to test this in your circuit is to check that number of droplets coming into a logic gate - is equal to the number of droplets coming out of the logic gate. If that is not true; something is not right.
Thus structuring patterns such that droplets can interact allows us to construct any arbitrary logic gate- as demonstrated in the figures from the Nature Physics paper above. This opens up a vast design space of circuits. Since we've demonstrated "Universal Digital Boolean Logic", any circuit can be constructed using the building blocks presented here.
Please read the "Basic Design Rules" and an example walk through in order to get the basic introduction of circuit design in our droplet world.
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