Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory

Abstract

We study equilibrium liquid crystal configurations in three-dimensional domains, within the continuum Landau-De Gennes theory. We obtain explicit bounds for the equilibrium scalar order parameters in terms of the temperature and material-dependent constants. We explicitly quantify the temperature regimes where the Landau-De Gennes predictions match and the temperature regimes where the Landau-De Gennes predictions don't match the probabilistic second-moment definition of the Q-tensor order parameter. The regime of agreement may be interpreted as the regime of validity of the Landau-De Gennes theory since the Landau-De Gennes theory predicts large values of the equilibrium scalar order parameters -larger than unity, in the low-temperature regime. We discuss a modified Landau-De Gennes energy functional which yields physically realistic values of the equilibrium scalar order parameters in all temperature regimes.

Transitions through critical temperatures in nematic liquid crystals

“…The LdG theory describes the state of a nematic LC by a physical state variable: the LdG Q-tensor defined in terms of anisotropic macroscopic quantities [4,12,15]. In the uniaxial phase, the Q-tensor can be written as…”

“…This assumption is justified for samples with untreated boundaries wherein there are no external effects on the director field. We work with a simple form of the LdG energy functional on a threedimensional (3D) domain, Ω ⊂ R 3 , given by [11,12] …”

“…In what follows, we refer to A as temperature even though it is the re-scaled temperature. It can be explicitly shown that the stationary points of the quartic f B in (3) are either uniaxial or isotropic [4,12,15,19] . For a uniaxial Q (of the form (1)), f B (Q) reduces to a quartic polynomial in S,…”

“…The LdG theory is a variational theory and equilibrium states correspond to minimizers of an appropriately defined LdG energy functional [4,12,19]. The isotropic-nematic phase transition is driven by the LdG thermotropic energy that contains a temperature-dependent parameter, denoted by A throughout this paper; more details are given in Section 2.…”

“…The isotropic-nematic phase transition is driven by the LdG thermotropic energy that contains a temperature-dependent parameter, denoted by A throughout this paper; more details are given in Section 2. The equilibrium order parameters at a given temperature, or equivalently for a given value of A, are given by the thermotropic energy minimizers [12]. There are two static critical values of A, associated with a bifurcation in the (S, A)-phase plane, in the LdG framework.…”