Dynamical field theories for biaxial liquid crystals

Phase field crystal (PFC) models constitute central tools for a microscopic understanding of the dynamics of complex systems in soft-matter physics. They have found widespread application in the modeling of the uniaxial orientational ordering of liquid crystals. However, only very limited progress has been made in applying them to the more complex cases of biaxial phases and biaxial particles. Here we discuss the microscopic derivation of PFC models for biaxial liquid crystals. We illustrate it by presenting two models, one involving four scalar orientational order parameters relevant for the dynamics of biaxial particles and one involving two scalar order parameters and all three directors to describe biaxial phases in a three-dimensional molecularly uniaxial nematic liquid crystal. These models allow for an efficient simulation of spatially inhomogeneous biaxial orientational ordering dynamics. We also combine our microscopic with a macroscopic approach to extract model coefficients for a full biaxial model from the microscopic derivation for a simple special case. This universal method also enables to perform derivations for other low-symmetry particles where, due to the complexity of the more general cases, this has not been previously attempted.