  The finite solid-liquid interface width in phase field models results in
non-equilibrium effects, including solute trapping. Prior phase field modeling
has shown that this extra degree of freedom, when compared to sharp-interface
models, results in solute trapping that is well captured when realistic
parameters, such as interface width, are employed. However, increasing the
interface width, which is desirable for computational reasons, leads to
artificially enhanced trapping thus making it difficult to model departure from
equilibrium quantitatively. In the present work, we develop a variational phase
field model with independent kinetic equations for the solid and liquid phases.
Separate kinetic equations for the phase concentrations obviate the assumption
of point wise equality of diffusion potentials, as is done in previous works.
Non-equilibrium effects such as solute trapping, drag and interface kinetics
can be introduced in a controlled manner in the present model. In addition, the
model parameters can be tuned to obtain ``experimentally-relevant" trapping
while using significantly larger interface widths than prior efforts. A
comparison with these other phase field models suggests that interface width of
about three to twenty-five times larger than current best-in-class models can
be employed depending upon the material system at hand leading to a speed-up by
a factor of $W^{(d+2)}$, where $W$ and $d$ denote the interface width and
spatial dimension, respectively. Finally the capacity to model non-equilibrium
phenomena is demonstrated by simulating oscillatory instability leading to the
formation of solute bands.
