Surface ordering in nano-drops containing nematic liquid crystals (original) (raw)

The behavior and configuration of nematic liquid crystals within nanodroplets was simulated using a molecular theory and an un-symmetric radial basis function collocation approach. liquid crystals | asymmetric radial basis functions | Landau-de-Gennes Abbreviations: RBF, Radial Basis Function; TPS, thin-plate spline I n this article we describe the free energy functional minimization of nematic liquid crystals nanodroplets. History and Motivation (LC general, LC-substrate, LC-confinement, LC-drops) Theoretical model: Liquid Crystal Liquid crystals (LCs) appear as phases which posses properties intermediate to those of crystalline solids and amorphous liquids. These phases have a certain degree of long-range order of anisotropic, crystalline solids, but deform continuously under the application of stresses, as do fluids. Systems that form these anisotropic fluids are composed molecules that present themselves a degree of structural anisotropy, such as rod-like or disk-like molecules [1, 2, 3]. In general, highly symmetric molecular systems, composed of neutral and spherical molecules, exhibit a direct transition from a highly ordered crystalline state to a disordered-isotropic liquid state [4, 5]. In contrast, LC materials can undergo several mesophase transitions according to concentration and/or temperature and external fields like electrostatics, hydrodynamics, magnetic or confinement [3, 6, 7, 8]. There are mesophases that appear as intermediate stages between the isotropic liquid and the crystalline solid. They possesses long-range orientation, yet deforms continuously under the application of stresses. These phases are characterized by a director field n. They are known to occur either by heating a solid crystal up to a critical temperature, or by varying the concentration of the molecules. The first are called thermotropic, while the second are lyotropic LCs. The liquid crystalline mesophases are classified into three types: nematic, cholesteric and smectic. The nematic LC has a long-range orientational order of the molecules along the direction of the director n, but it does not present positional order. Cholesteric liquid crystals characterize a phase similar to the nematics, except that in addition to the nematic order at small distances, it exhibits a long-range molecular structure that corresponds to the helical rotation of the direction along a neutral direction. This phase is only formed by chiral molecules, i.e. without mirror symmetry. The smectics corresponds to a mesophase which exhibits positional as well as orientation order. There are several categories each one with different positional and orientational ordering of the molecules [2, 6]. The thermodynamics of phase transitions in liquid crystals requires the introduction of an additional internal structural parameter in order to characterize the degree of alignment. The average directional cosine between a particular molecule direction u and the director n is not an appropriate option because this average will vanish for most situations (relying that n and −n are equivalent). Thus, a higher moment of the molecular orientation is used; the lower moment that gives a non-trivial answer is defined by the second moment as follows [6, 3] Q (x, t) = MII (x, t) − δ 3 , [ 1 ] where δ is the 3 × 3 identity tensor and the second moment MII is given by MII (x, t) = nnψ (n, x, t) dn, [ 2 ] where ψ (n, x, t) is the configuration distribution function of orien-tations. According to this definitions tr(Q) = 0 while tr(MII) = 1. The tensor order parameter Q, in an appropriate coordinate system, can be diagonalized in terms of its eigenvalues, i.e. Q =   2S 3 0 0 0 η−S 3 0 0 0