  We introduce a new phase-field formulation of rapid alloy solidification that
quantitatively incorporates nonequilibrium effects at the solid-liquid
interface over a very wide range of interface velocities. Simulations identify
a new dynamical instability of dendrite tip growth driven by solute trapping at
velocities approaching the absolute stability limit. They also reproduce the
formation of the widely observed banded microstructures, revealing how this
instability triggers transitions between dendritic and microsegregation-free
solidification. Predicted band spacings agree quantitatively with observations
in rapidly solidified Al-Cu thin films.
