Nanoscale patterning of materials at scales of less than 20nm remains a challenging problem. Standard techniques, such as lithography, rely on electron and photon beams to shape materials, yet these methods are difficult to employ at the sub 50nm scale. We consider the possibility of employing natural interface instabilities during solidification in order to reliably produce desirable structures. Existing continuum methods for complicated materials may break down at these scales, while direct atomistic simulation is computationally infeasible. The primary objective of the thesis is to develop methods capable of efficiently resolving the atomistic computations which may eventually be solved concurrently with continuum scale approaches. In the present work we develop atomistic algorithms to furnish closures for continuum methods. We develop novel molecular techniques capable of resolving detail only about an interface with non-periodic boundaries and accelerated by GPUs. The new algorithms provide a method to determine interfacial dynamics based on local curvature, employing wedge shaped domains. These dynamics lead to continuum closures which are used to determine necessary physical parameters to enable dendritic patterning at the sub 20nm scale.