Continuing the program initiated in [17], we analyze a model problem based on highly disparate elastic constants that we propose in order to understand corners and cusps that form on the boundary between the nematic and isotropic phases in a liquid crystal. For a bounded planar domain Ω we investigate the ε → 0 asymptotics of the variational problemwithin various parameter regimes for L ε > 0. Here u : Ω → R 2 and W is a potential vanishing on the unit circle and at the origin. When ε L ε → 0, we show that these functionals Γ−converge to a constant multiple of the perimeter of the phase boundary and the divergence penalty is not felt. However, when L ε ≡ L > 0, we find that a tangency requirement along the phase boundary for competitors in the conjectured Γ-limit becomes a mechanism for development of singularities. We establish criticality conditions for this limit and under a non-degeneracy assumption on the potential we prove compactness of energy bounded sequences in L 2 . The role played by this tangency condition on the formation of interfacial singularities is investigated through several examples: each of these examples involves analytically rigorous reasoning motivated by numerical experiments. We argue that generically, "wall" singularities between S 1 -valued states of the kind analyzed in [17] are expected near the defects along the phase boundary. *
We demonstrate that a first order isotropic-to-nematic phase transition in liquid crystals can be succesfully modeled within the generalized Landau-de Gennes theory by selecting an appropriate combination of elastic constants. The numerical simulations of the model established in this paper qualitatively reproduce the experimentally observed configurations that include interfaces and topological defects in the nematic phase.
Within the framework of the generalized 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 minimization problem is well-posed for a significantly wider choice of elastic constants. In particular, 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 Landaude Gennes theory, we establish a rigorous connection between this theory and its Oseen-Frank counterpart via a Γ-convergence argument in the limit of vanishing nematic correlation length. We also prove strong convergence of the associated minimizers. *
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.