Ion Multivalence and Like-Charge Polyelectrolyte Attraction

It is known empirically that multivalent ions generate attractions between like-charged polyelectrolytes, with different valence requirements for different systems. How multivalent must an ion be before it can condense a given polyelectrolyte? Using charge-tunable $M13$ virus rods and a family of artificial homologous “dumbbell” divalent ions of different sizes, we have constructed a multivalent ion-polyelectrolyte phase diagram, and find an experimentally motivated general criterion for like-charged attraction based on the ion valence, ion size, and the Gouy-Chapman length.