On 3D non-Riemannian geometry of twisted nematic liquid crystals

A teleparallel geometrical description of the nematic phases of liquid crystals is proposed.In the case of the twisted geometry of nematics Cartan torsion is given by a spatial helical form which depends on the twist angle. Geodesics of a test particle around the teleparallel liquid crystal are given. PACS numbers : 0420,0450 Departamento de F́ısica Teórica Instituto de F́ısica UERJ Rua São Fco. Xavier 524, Rio de Janeiro, RJ Maracanã, CEP:20550-003 , Brasil. E-mail : [email protected] On non-Riemannian geometry of nematic liquid crystals 2 Several kinds of geometries have been used in the investigation of the elastic properties of solids.Among them Riemann-Cartan has been used by Katanaev and Volovich [1] to show that it describes the defects (disclinations and dislocations) as solutions of 3D Einstein-Cartan equations.Earlier S.Amari [2] has built a Finsler geometrical model of ferromagnetic substances.Riemannian geometry has also been used recently by myself [3] to show that a Heisenberg ferromagnet yields a conical 2D solution of vacuum Einstein equations describing a monopole (topological defect)which could be match with Katanaev-Volovich solutions.This conical metric describes also cosmic strings in the Early Universe [4].Teleparallelism has also been applied with success [5] to investigate a Bloch wall with spin density where the Cartan torsion [6] is a distributional Dirac delta forming a sort of torsion wall.More recently Mullen [7] has also considered geometrical aspects of defects in semi-conductors in Josephson arrays.F.Moraes and C.Furtado [8] have also recently used the Katanaev and Volovich ideas to investigate the graphite monolayer.In this letter we consider another application of teleparallelism [9], this time to nematic liquid crystals [10],where the elastic properties of the nematic phases are investigated.Nematic liquid crystals can transmite torques and the application of torsion geometries will prove useful in the investigation of his geometrical and physical properties.Let us built the Riemann metric in analogy to Katanaev and Volovich as the deformation of the crystals eij = n(j,i) where this deformation represents the perturbation on the metric gij given by gij = δij + eij (1) where δij is the delta Kronecker symbol and the vector ni where (i = 1, 2, 3) is the director field of nematic liquid crystal.The teleparallel geometry was used by Einstein in 1928 [10] to build the first teleparallel 4D geometrical unified theory of electromagnetism and gravitation.This theory has as one of his main carachteristics that the Riemann curvature tensor vanishes which implies that the anti-symmetric connection (torsion tensor) is given in terms of the metric by Tijk = gjk,i − gji,k (2) Here the comma denotes partial derivatives with respect to the lower index.Substitution of expression (1) into (2) yields the following torsion vector Tk = [∇ nk − δk(divn)] (3) On non-Riemannian geometry of nematic liquid crystals 3 therefore and the Weitzenböck condition for teleparallelism on the curvature Riemann tensor Rijkl(Γ) = 0 ,where Γ is the Riemann-Cartan affine connection.Later on we shall make an application of this formula to a special case of nematic crystal.In the meantime let us compute the geodesics equations for the corresponding metric of the liquid crystal.The geodesics equations ẍi + Γijkẋ j ẋ = 0 (4) and ẍi + δnk,ljẋ ẋ = 0 (5) where Γijk = 1 2 δ[glj,k + glk,j − gjk,l] and we use the Euclidean 3D metric δij to raise and lower indices.Let us now apply these ideas to the pure twist geometry of the nematic liquid crystals,where the director field is now given by the components nz = cosθ(y) (6) and nx = sinθ(y) (7) Where θ is the twist angle and the planar crystal is orthogonal to the ycoordinate direction.Substituting expressions (6) and (7) into (3) one obtains the following components of the torsion vector T i ki = Tk Ty(θ) = −∂y(divn) (8) and Tz(θ) = −∇ nz (9) Since divn = nx,x + ny,y + nz,z = 0 , there is no torsion vector component along the orthogonal direction as happens in some domain walls with torsion [11].Thus the only non-vanishing torsion component of torsion reads Tz(θ) = −[cosθ( dθ dy ) + sinθ dθ dy ] (10) Since local equilibrium conditions on the Liquid crystals yields [10] dθ dy = constant = K (11) On non-Riemannian geometry of nematic liquid crystals 4 Thus equation (11) reduces to Tz(θ) = −K cosθ (12) This last expression tell us that the twisted geometry of the crystal leads to a 3D helical torsion.Geodesics of the twisted geometry of the liquid crystal leads to the results ẍ+ nx,yy ẏ 2 = 0 (13) and ÿ = 0 (14) and finally z̈ + nz,yy ẏ 2 = 0 (15) Here the dots means derivation with respect to time coordinate.Substitution of equation (14) into expressions (13) and (15) yields yields ẍ+K1 sinθ(y) = 0 (16) and z̈ +K1 cosθ(y) = 0 (17) To simplify matters let us solve these euqtions in the approximation of small twist angles θ <<< 0 where the equations (16) and (17) reduces to ẍ+K1 θ = 0 (18) and z̈ +K1 2 = 0 (19) Therefore from expression (21) one obtains the following solution z = −K1 t+ c (20) where c is an integration constant.To solve the remaining equations we need an explicit form of the twist angle with respect to time ,this can be obtained from the integration of the expression dθ dy = 0 which yields