Liquid crystal is a typical kind of soft matter that is intermediate between crystalline solids and isotropic fluids. The study of liquid crystals has made tremendous progress over the last four decades, which is of great importance on both fundamental scientific researches and widespread applications in industry. In this paper, we review the mathematical models and their connections of liquid crystals, and survey the developments of numerical methods for finding the rich configurations of liquid crystals.

Due to structural incommensurability, the emergence of a quasicrystal from a crystalline phase represents a challenge to computational physics. Here, the nucleation of quasicrystals is investigated by using an efficient computational method applied to a Landau free-energy functional. Specifically, transition pathways connecting different local minima of the Lifshitz–Petrich model are obtained by using the high-index saddle dynamics. Saddle points on these paths are identified as the critical nuclei of the 6-fold crystals and 12-fold quasicrystals. The results reveal that phase transitions between the crystalline and quasicrystalline phases could follow two possible pathways, corresponding to a one-stage phase transition and a two-stage phase transition involving a metastable lamellar quasicrystalline state, respectively.

How do we search for the entire family tree of possible intermediate states, without unwanted random guesses, starting from a stationary state on the energy landscape all the way down to energy minima? Here we introduce a general numerical method that constructs the pathway map, which guides our understanding of how a physical system moves on the energy landscape. The method identifies the transition state between energy minima and the energy barrier associated with such a state. As an example, we solve the Landau–de Gennes energy incorporating the Dirichlet boundary conditions to model a liquid crystal confined in a square box; we illustrate the basic concepts by examining the multiple stationary solutions and the connected pathway maps of the model.

We construct a tensor model for nematic phases of bent-core molecules from molecular theory. The form of free energy is determined by molecular symmetry, which includes the couplings and derivatives of a vector and two second-order tensors, with the coefficients determined by molecular parameters. We use the model to study the nematic phases resulting from the hardcore potential. Unlike most macroscopic models, we are able to obtain the phase diagram about the molecular parameters, but not merely some phenomenological coefficients. The tensor model is applicable to other molecules with the same symmetry, which we demonstrate by studying the phase diagram of star molecules.

In this paper, we investigate the structure of local minimizers for the isotropic–nematic interface based on the Landau-de Gennes energy. In the absence of the anisotropic energy, the uniaxial solution is the only local minimizer in 1-D. In 3-D, we propose a De Giorgi’s type conjecture and give an affirmative answer under a mild assumption. In the presence of the anisotropic energy with L2 > −1 and homeotropic anchoring, the uniaxial solution is also the only local minimizer in a class of diagonal form in 1-D.