Using the Landau-Brazovskii model, a new numerical implementation is developed to investigate the phase behavior of the diblock copolymer system. Though the method is based on the Fourier expansion of order parameter, a priori symmetric information is not required, and more significantly, the period structure can be adjusted automatically during the iteration as well. The method enables us to calculate the phase diagram, discover new meta-stable phases, validate the epitaxial relation in the phase transition process, and find the inefficiency of the Landau-Brazovskii model for some situations.

We will also introduce a new numerical method to study the nucleation in ordered phases. Nucleation is the decay of a metastable state via the thermally activated formation and subsequent growth of droplets of the equilibrium phase. We will consider the nucleation in diblock copolymer melts, whose equilibrium phases are well understood. We apply a new numerical method, called the string method, to compute the minimum energy path (MEP). Then from the MEP, we find the size and shape of the critical droplet and the free-energy barrier to nucleation. This method is generally useful for other systems, such as nucleation of liquid crystal based on Landau-de Gennes theory and binary alloy system by phase field method etc.