Solidifying metal alloys develop complex microstructures by the interplay between surface energy and heat and solute transport. The phase field model replaces the sharp solid/liquid interface by a diffuse one and trades a geometrically complicated free boundary problem for a stiff set of partial differential equations with a small parameter. Recent advances in the asymptotic analysis have greatly improved the accuracy and hence the potential of this model to provide quantitative information for real materials. We show how this new second-order asymptotics can be constructed for the case of asymmetric diffusivities, important for real alloy solidification, and discuss other interesting aspects of the model.