Though melting is a common phase transition in nature, the small scale details of the crystal breakdown in two dimensions are not yet fully understood. We present a numerical investigation of melting and freezing in two dimensions through the phase field crystal model. We start out by discussing phase transitions by means of disorder-order transitions through the phase field method. The method is suitable of describing phase transitions in heterogeneous and isotropic systems and is an important tool for modeling micro structural evolution. We continue by deriving the phase field crystal method. That is a method in which the isotropic approximation is relaxed and the free energy functional is constructed to ensure a periodic structure in the equilibrium state. By this approach, important features such as crystal orientation, defects and deformations are naturally incorporated in the model. Hence, this method provides a suitable method for describing the process of crystal growth as well as the breakdown of the crystalline structure, that is the processes of freezing and melting respectively. Additionally, a set of amplitude equations are presented for the study of the dynamical evolution in the crystalline phase.
Thereafter, we discuss the process of melting and the role of dimensionality and dislocations in the phase transition. We limit our work to two dimensional systems and investigate the evolution of melting numerically with a particular focus on the role of defects in the phase transition. The simulated melting is induced by two different protocols, that is uniform heating and applied shear stress, and we evaluate our numerical results relative to the theory of dislocation mediated melting. As a closure we summarize our results, elaborate upon the challenges in the search for the melting mechanism and propose ideas for future work which can be carried through with the phase field crystal method.