The ordered patterns formed bymicrophase-separated block copolymer systems demonstrate periodic symmetry,and all periodic structures belong to one of 230 space groups.Based on this fact,a strategy of estimating the initial values of self-consistent field theory to discover ordered patterns of block copolymers is developed.In particular,the initial period of the computational box is estimated by the Landau-Brazovskii model as well.By planting the strategy into the whole-space discrete method,several new metastable patterns are discovered in diblock copolymers.
Rod-like molecules confined on a spherical surface can organize themselves into nematic liquid crystal phases. This can give rise to novel textures displayed on the surface, which has been observed in experiments. An important theoretical question is how to find and predict these textures. Mathematically, a stable configuration of the nematic fluid corresponds to a local minimum in the free energy landscape. By applying Taylor expansion and Bingham approximation to a general molecular model, we obtain a closed-form tensor model, which gives a free energy form that is different from the classic Landau-de Gennes model. Based on the tensor model, we implement an efficient numerical algorithm to locate the local minimum of the free energy. Our model successfully predicts the splay, tennis-ball and rectangle textures. Among them, the tennis-ball configuration has the lowest free energy.
