Order parameters in the Landau–de Gennes theory: the static and dynamic scenarios

We obtain quantitative estimates for the scalar order parameters of liquid crystal configurations in three-dimensional geometries, within the Landau–de Gennes framework. We consider both static equilibria and non-equilibrium dynamics and we include external fields and surface anchoring energies in our formulation. Using maximum principle-type arguments, we obtain explicit bounds for the corresponding scalar order parameters in both static and dynamic situations; these bounds are given in terms of the material-dependent thermotropic coefficients, electric field strength and surface anchoring coefficients. These bounds provide estimates for the degree of orientational ordering, quantify the competing effects of the different energetic contributions and can be used to test the accuracy of numerical simulations.