A novel Landau-de Gennes model with quartic elastic terms

Within the framework of the generalised Landau-de Gennes theory, we identify a Q-tensor-based energy that reduces to the four-constant Oseen-Frank energy when it is considered over orientable uniaxial nematic states. Although the commonly considered version of the Landau-de Gennes theory has an elastic contribution that is at most cubic in components of the Q-tensor and their derivatives, the alternative offered here is quartic in these variables. One clear advantage of our approach over the cubic theory is that the associated minimisation problem is well-posed for a significantly wider choice of elastic constants. In particular, this quartic energy can be used to model nematic-to-isotropic phase transitions for highly disparate elastic constants. In addition to proving well-posedness of the proposed version of the Landau-de Gennes theory, we establish a rigorous connection between this theory and its Oseen-Frank counterpart via a Gamma-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimisers.

Automated summary

This study introduces a quartic energy based on the Q-tensor within the framework of the Landau-de Gennes theory, offering a wider range of elastic constant choices compared to the commonly considered cubic theory for modeling nematic-to-isotropic phase transitions. A rigorous connection is established between this theory and its Oseen-Frank counterpart using Gamma-convergence arguments, with strong convergence of the associated minimisers also proven.