  Eutectic two-phase cells, also known as eutectic colonies, are commonly
observed during the solidification of ternary alloys when the composition is
close to a binary eutectic valley. In analogy with the solidification cells
formed in dilute binary alloys, colony formation is triggered by a
morphological instability of a macroscopically planar eutectic solidification
front due to the rejection by both solid phases of a ternary impurity that
diffuses in the liquid. Here we develop a phase-field model of a binary
eutectic with a dilute ternary impurity and we investigate by dynamical
simulations both the initial linear regime of this instability, and the
subsequent highly nonlinear evolution of the interface that leads to fully
developed two-phase cells with a spacing much larger than the lamellar spacing.
We find a good overall agreement with our recent linear stability analysis [M.
Plapp and A. Karma, Phys. Rev. E 60, 6865 (1999)], which predicts a
destabilization of the front by long-wavelength modes that may be stationary or
oscillatory. A fine comparison, however, reveals that the assumption commonly
attributed to Cahn that lamella grow perpendicular to the envelope of the
solidification front is weakly violated in the phase-field simulations. We show
that, even though weak, this violation has an important quantitative effect on
the stability properties of the eutectic front. We also investigate the
dynamics of fully developed colonies and find that the large-scale envelope of
the composite eutectic front does not converge to a steady state, but exhibits
cell elimination and tip-splitting events up to the largest times simulated.
