This chapter provides the grand potential phase field equations to three important case studies of interest. The first case deals with the situation of a multi-component, two-phase alloy where the local grand potential of each phase is assumed to be well approximated by linear deviations in its chemical potentials from its equilibrium value. The second case deals with a multi-phase binary alloy, whose local grand potential of each phase can be approximated by a quadratic function away from its equilibrium value. The third case extends this quadratic grand potential approximation to multi-component and multi-phase alloys. The coefficients kijeff form a matrix of ratios of free energy curvatures at the local equilibrium chemical potentials. The chapter considers how one chooses the parameter relations in the phase field model specializations in order to emulate the kinetics of the classic sharp interface model of solidification that governs slow solidification processes.