Phase field method has become a popular tool to investigate the microstructure evolution during the solidification. Quantitative predictions using this method is still limited, and in this dissertation, we try to study this problem from different perspectives. Most of the phase field models consider the solid-liquid interface to be in local equilibrium. Solidification during some manufacturing processes like selective laser melting, and electron beam additive manufacturing is rapid and far from equilibrium which can result in supersaturated solid solutions, segregation-free crystals, or metastable phases. Before obtaining any conclusions from the phase field simulations, we must know the answer for “which phase field model works for rapid s...