  We investigate lamellar three-phase patterns that form during the directional
solidification of ternary eutectic alloys in thin samples. A distinctive
feature of this system is that many different geometric arrangements of the
three phases are possible, contrary to the widely studied two-phase patterns in
binary eutectics. Here, we first analyze the case of stable lamellar coupled
growth of a symmetric model ternary eutectic alloy, using a Jackson-Hunt type
calculation in thin film morphology, for arbitrary configurations, and derive
expressions for the front undercooling as a function of velocity and spacing.
Next, we carry out phase-field simulations to test our analytic predictions and
to study the instabilities of the simplest periodic lamellar arrays. For large
spacings, we observe different oscillatory modes that are similar to those
found previously for binary eutectics and that can be classified using the
symmetry elements of the steady-state pattern. For small spacings, we observe a
new instability that leads to a change in the sequence of the phases. Its onset
can be well predicted by our analytic calculations. Finally, some preliminary
phase-field simulations of three-dimensional growth structures are also
presented.
