  A vector order parameter phase field model derived from a grand potential
functional is presented as a new approach for modeling polycrystalline
solidification of alloys. In this approach, the grand potential density is
designed to contain a discrete set of finite wells, a feature that naturally
allows for the growth and controlled interaction of multiple grains using a
single vector field. We verify that dendritic solidification in binary alloys
follows the well-established quantitative behavior in the thin interface limit.
In addition, it is shown that grain boundary energy and solute back-diffusion
are quantitatively consistent with earlier theoretical work, with grain
boundary energy being controlled through a simple solid-solid interaction
parameter. Moreover, when considering polycrystalline aggregates and their
coarsening, we show that the kinetics follow the expected parabolic growth law.
Finally, we demonstrate how this new vector order parameter model can be used
to describe nucleation in polycrystalline systems via thermal fluctuations of
the vector order parameter, a feature which has been thus far lacking from
multi-phase or multi-order parameter based phase field models. The presented
vector order parameter model serves as a practical and efficient computational
tool for simulating polycrystalline materials. We also discuss the extension of
the order parameter to higher dimensions as a simple method for modeling
multiple solid phases.
