Monte Carlo simulations have been performed for a discotic liquid crystal composed of Gay–Berne particles. On the basis of these simulations for the nematic phase, a subset of the spherical harmonic expansion coefficients of the direct pair correlation function (DPCF) were determined from the pair distribution function (PDF) by solving the Ornstein–Zernike (OZ) equation. This was achieved by generalizing the Wiener–Hopf factorization scheme for the numerical solution of the OZ equation. Only the expansion coefficients gl1,l2,l(r) (lα⩽4) of the PDF in the laboratory frame were used when solving the OZ equation; this means that the DPCF so obtained is equivalent to that for a nematic in which the director is randomly distributed. From the DPCF, the scaled Oseen–Zöcher–Frank elastic constants K11*, K22*, and K33*, as well as the surface constant K13*, have been calculated from the subset of expansion coefficients. Generally, we find that K33*<K11*<K22*, in agreement with what is expected and found for discotic nematics. These results are quantitatively but not qualitatively different from those calculated with the help of analytical approximations for the same spherical harmonic expansion coefficients of the direct pair correlation function. For example, the values of the bulk elastic constants determined via the OZ equation are about three times larger than the bulk elasticity obtained with the low density approximation.
A density functional theory for bulk and surface elastic constants of biaxial nematic liquid crystals is developed. It is based on a functional Taylor expansion of the free energy of a distorted biaxial nematic with respect to the one-particle distribution function. Detailed microscopic expressions for the biaxial elastic constants of bulk and surface deformations are derived by expanding further the distribution functions into symmetry-adapted Wigner matrices. The final expressions depend on generalized orientational order parameters characterizing the biaxial nematic and on expansion coefficients of the direct pair correlation function. The case where the expansions are truncated at the lowest nontrivial order with respect to the momentum index of the Wigner matrices is analyzed in detail. It gives only six distinct, nonzero bulk elastic constants. The mixed elastic constants, which measure distortions of more than one director, vanish within this approximation. As in the uniaxial case, a splay-bend degeneracy for all directors is apparent. The theory is next applied to the biaxial nematic phase recently studied by Biscarini et al. [Phys. Rev. Lett. 75, 1803 (1995)] providing numerical estimates of biaxial elastic constants for the case of thermodynamically stable biaxial ordering. It is shown that the values of the elastic constants connected with secondary directors are much lower than those associated with the primary one.
The ordering of a nematic liquid crystal in the presence of a smooth surface is analyzed in detail. In particular, the force constants for homeotropic anchoring are estimated by a local density functional method with data from molecular dynamics simulations. The system studied is a model Gay-Berne nematic liquid crystal. For the molecule-surface interaction both an anisotropic and an isotropic one-particle potential are taken. In both cases a surface-induced smectic A phase is being observed even though the phase is unstable in the bulk. ͓S1063-651X͑97͒02206-X͔ PACS number͑s͒: 61.30. Cz, 64.70.Md, 05.70.Ce, 62.20.Dc Surfaces of nematic liquid crystals show a rich variety of ordered states ͓1͔. With respect to positional order smectic or even crystalline layers of the molecules may form. Concerning orientational order the long molecular axes may be anchored along one or along several discrete directions, or along a continuous set of directions.With the help of anchoring conditions one can tailor the director field in the bulk of a nematic liquid crystal such that the polarization plane of light is guided according to wish for display purposes.Anchoring is achieved by a suitable preparation of the cover glasses for the nematic cell, and although a myriad of glasses is being prepared by empirical recipes annually, the underlying microscopic mechanisms of anchoring are not yet fully understood.Phenomenologically, anchoring is described by a surface potential or anchoring free energy, which depends on the surface director n . For an anchoring configuration cylindrically symmetric about a preferred direction n p , the potential most frequently used is of the Rapini and Papoular form ͓2͔,where and p are the polar angles of n and n p , respectively. Alternatively one can expand the surface potential into a series of Legendre polynomials F anch ϭW 2 P 2 ͑ n •n p ͒ϩW 4 P 4 ͑ n •n p ͒ϩ•••. ͑2͒The coefficients W 2 ,W 4 , and c ϭϪ3 W 2 Ϫ10 W 4 of the expansions ͑1͒ and ͑2͒, denoted anchoring strengths, must depend both on the properties of the equilibrium bulk nematic liquid crystal and on the way the surface interacts with the bulk. They have been determined experimentally ͓3,4͔.Theories about the surface-bulk interaction with the introduction of a phenomenological anchoring free energy were developed by Sluckin and Poniewierski ͓5͔, by Sen and Sullivan ͓6͔, and by Osipov and Hess ͓7͔. Theoretical estimates of the anchoring strengths were performed by Tjipto-Margo and Sullivan ͓8͔, yielding rough limits for the ratio W 2 /W 4 . This paper is devoted to a microscopic study of surfaceinduced ordering in nematic liquid crystals due to homeotropic anchoring. The strength of the nematic ordering on the surface is estimated by the Rapini-Papoular constants. These are found by combining molecular dynamics simulations at constant density ͑pressure͒ and temperature with a local density functional approach. The pair interaction of the rodlike molecules was modeled by the anisotropic Gay-Berne potential ͓9͔. For the interaction of a...
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