  We present a detailed derivation and thin interface analysis of a phase-field
model that can accurately simulate microstructural pattern formation for
low-speed directional solidification of a dilute binary alloy. This advance
with respect to previous phase-field models is achieved by the addition of a
phenomenological "antitrapping" solute current in the mass conservation
relation [A. Karma, Phys. Rev. Lett 87, 115701 (2001)]. This antitrapping
current counterbalances the physical, albeit artificially large, solute
trapping effect generated when a mesoscopic interface thickness is used to
simulate the interface evolution on experimental length and time scales.
Furthermore, it provides additional freedom in the model to suppress other
spurious effects that scale with this thickness when the diffusivity is unequal
in solid and liquid [R. F. Almgren, SIAM J. Appl. Math 59, 2086 (1999)], which
include surface diffusion and a curvature correction to the Stefan condition.
This freedom can also be exploited to make the kinetic undercooling of the
interface arbitrarily small even for mesoscopic values of both the interface
thickness and the phase-field relaxation time, as for the solidification of
pure melts [A. Karma and W.-J. Rappel, Phys. Rev. E 53, R3017 (1996)]. The
performance of the model is demonstrated by calculating accurately for the
first time within a phase-field approach the Mullins-Sekerka stability spectrum
of a planar interface and nonlinear cellular shapes for realistic alloy
parameters and growth conditions.
