This thesis summarizes work done in the simulation of cellular and dendritic growth in directional solidification of dilute binary alloys using a phase-field model. This model is solved on a dynamic adaptive grid using a linear isoparametric formulation of the finite element method. The spacing of the primary dendritic branches are examined for a wide range of thermal gradients and alloy compositions using a power spectral analysis technique. This spacing is found to undergo a maximum as a function of increasing velocity, in agreement with experimental observations. Our simulations are compared to directional solidification experiments of PVA-ETH, SCN-ACE and SCN-SAL alloys and we demonstrate that the spacing selection is described by a crossover scaling function from the emergence of cellular growth into the dendritic growth regime. This scaling function is dependent upon only dimensionless groupings of the material dependent length scales and the process parameters at which the system is cooled. We validate our results by showing that both the simulated materials and published experimental data collapse onto this single universal curve. / Thesis / Master of Applied Science (MASc)