Properties of surface Landau–de GennesQ-tensor models

Abstract

Uniaxial nematic liquid crystals whose molecular orientation is subjected to a tangential anchoring on a curved surface offer a non trivial interplay between the geometry and the topology of the surface and the orientational degree of freedom. We consider a general thin film limit of a Landau-de Gennes Q-tensor model which retains the characteristics of the 3D model. From this, previously proposed surface models follow as special cases. We compare fundamental properties, such as alignment of the orientational degrees of freedom with principle curvature lines, order parameter symmetry and phase transition type for these models, and suggest experiments to identify proper model assumptions.1 simulations and experimental observations for thin films and 2D systems is provided. The situation on curved surfaces should somehow reflect these properties. However, in some of the proposed surface models first order phase transitions are not possible. Another aspect highlighting the differences is discussed in [35] by comparing mean field theories in 2D [57] and 3D [56]. The definitions yield different eigenvalue spectra in the order tensor, which corresponds to either a symmetry under in plane rotations by 90 • in 2D or a rotational symmetry (w.r.t. the average particle direction) in 3D. How a nematic liquid crystal on a curved surface fits into this picture is open. Some of the proposed surface models yield an eigenvalue spectrum as in 2D, others as in 3D. The third aspect considers the mean orientation. If not induced by external fields or boundary conditions the mean orientation in nematic liquid crystals is arbitrary in 2D and 3D. This changes on curved surfaces, where it should be preferential to align with the principle curvature lines. This aspect has been discussed previously and is identified with the influence of extrinsic curvature terms [41,40] in surface models, which again are considered in some but not all of the proposed models.We will review the proposed surface Landau-de Gennes Q-tensor models under the aspect of these fundamental properties. Furthermore we propose a version of a surface Landau-de Gennes Q-tensor model which effectively describes surface liquid crystals retaining the 3D phase transition type and eigenvalue spectra. The previously proposed models follow as special cases and we discuss under which assumptions fundamental properties get lost. The paper is structured as follows. In Section 2 we briefly review the 3D Landau-de Gennes Q-tensor model [10] and propose its thin film limit under generic anchoring conditions. In Section 3 this model together with its special cases is discussed in a flat 2D scenario with respect to their fundamental properties. We demonstrate the general model to retain the 3D properties. With this established, we apply the models to curved surfaces and discuss the effects of curvature on the ordering of the liquid crystal in Section 4. We summarize our findings and discuss them in a general framework in Section 5. As all these results do not depend on specific materia...

Active nematodynamics on curved surfaces -- the influence of geometric forces on motion patterns of topological defects

“…Turning to the spatial distribution of the directed distortion energy density, see Figure 4[B], shows low magnitudes covering the bulk of prolate geometries, while on oblate geometries the energy density reaches high magnitudes on the rim with almost zero magnitude on quasi planar top and bottom part. As detailed in [25] this distribution can be attributed to the alignment of nematic texture with lines of minimal curvature. The stationary defect configurations for pure isotropic distortion (β = 0) and including directed distortion energy (β = −S * /3) match for prolate geometries and the sphere.…”

“…The first is present already in the passive case, α = 0, and does not required hydrodynamic interactions. The additional directed geometric forces in the system with β = −S * /3 have an impact on the nematic texture [25]. Reviewing the contributions of isotropic and directed distortion energy in the steady state configurations, see Figure 4[A], we observe a strong directed contribution providing a strong energy sink for geometries with broken symmetry.…”

“…Given the discussion of [25] we highlight the impact of the choice of β. With β = 0 we yield a surface model for uniaxial nematics [19,16], which essentially has the same properties as a two-dimensional model, while β = 0 results in thin film models of nematics [24,14,15,35,28,31] with several properties shared with three-dimensional models.…”

“…However, these models are based on a simplified surface Landau-de Gennes energy neglecting various curvature contributions [19]. For more detailed surface Landau-de Gennes models which also take extrinsic curvature contributions into account, see [15,28,25]. Another critical issue is the considered numerical approach for the Navier-Stokes-like equations.…”

Topological fine structure of smectic grain boundaries and tetratic disclination lines within three-dimensional smectic liquid crystals

“…The most prominent type of ordering, which is typically found in liquid crystals, is orientational (nematic) ordering, where the characteristically shaped subunits, i.e, molecules or colloidal particles in close proximity, show a tendency to align. If this preferred order gets frustrated, e.g., by confinement to a finite container [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49], constraining on a surface [50][51][52][53][54][55][56][57] or insertion of obstacles [58][59][60][61][62][63][64][65][66][67], topological defects emerge, which are discontinuities in the ordered structures that can display particlelike properties themselves [6,16,[68][69][70]. * Rene.Wittmann@hhu.de FIG.…”