The defect structures of s = ±1/2 twist disclinations in twisted nematic and twisted chiral liquid crystals have been investigated within the Landau-de Gennes theory numerically. Our results show that there exists eigenvalue exchange across the defect core of both the two models. The defect core is essentially biaxial and never isotropic. The defect centre is uniaxial and is surrounded by a strong biaxial region.
Topological defects arise as a result of broken continuous symmetry and are ubiquitous in nature, from microscopic condensed matter systems governed by quantum mechanics to a universe in which gravity plays a decisive role [1-3]. Defects in liquid crystals (LCs) have been the subject of much interest, still offering unsolved problems. Commonly observed defects in the uniaxial nematic phase are typically point defects with topological charge s = 1 and line defects with topological charge s = ±1/2 [
The structures of s = ±1/2 twist disclination have been observed in a twisted nematic (TN) cell with planar orientations on both plates with the easy axes perpendicular to each other (called 90˚ twisted structure). These defects essentially involve two symmetric twist distortions, respectively left-handed and right-handed with degenerate energy, corresponding to ±π/2 twists over the cell thickness [
The helical structure of twisted chiral nematic (TCN) cells with planar anchoring have been studied [
Dislocations also appear in chiral LCs as a result of temperature quenching from the isotropic phase, applied fields, and surface boundary conditions [
In this study, we will investigate the defect structure of a TN cell and a TCN cell of thickness p0/2, both of which have 90˚ twisted structure. Our study is based on the Landau-de Gennes theory describing the orientational order of a LC in terms of a second-rank tensor Q, which encompasses both uniaxial and biaxial state.
Our theoretical argument is based on the Landau-de Gennes theory [
Here λi and ei are the ith eigenvalue and the ith eigenvector of Q, respectively. In isotropic phase, Q vanishes. In the uniaxial ordering, Q has two degenerate eigenvalues and can be represented by
where S is the uniaxial scalar parameter, and the unit vector n is the nematic director pointing along the local uniaxial ordering direction. In Equation (2), S can have either sign: when it is positive the ensemble of molecules represented by Q tends to be aligned along n, whereas when S is negative it tends to lie in the plane orthogonal to n.
Finally, when all eigenvalues of Q are distinct, the LC is in a biaxial state. The degree of biaxiality is measured by the biaxiality parameter β2, defined as [
which ranges in the interval [0, 1]. In all uniaxial states with two degenerate eigenvalues, β2 = 0, while states with maximal biaxiality correspond to β2 = 1. Since , the states with maximal biaxiality are precisely those where detQ = 0, which further implies that at least one eigenvalue of Q vanishes in the biaxial states.
Following the notation in Reference [
in which
is the local free energy in the Landau-de Gennes expansion,
is the free energy due to the inhomogeneity of LC order. In fbulk, a and b are positive constants and c is assumed to vary with temperature. In Equation (5), summations over repeated indices are implied, , and with εαγβ being the Levi-Civita symbol. The variables K1 and K0 are the elastic constants, and characterizes the strength and sign of the chirality (hereafter, we consider the case with q0 > 0). We also note that following the spirit of the Landau expansion, we assume that the temperature dependence appears only in the parameter c and neglect the temperature dependence of the other material parameters. Moreover, the bulk equilibrium value of the uniaxial scalar order parameter in Equation (2):
, also depends on the temperature.
When q0 = 0, the condition reduces to the ordinary nematic limit with infinite pitch, and Equation (5) reduces to
After an appropriate rescaling of the variables, we can reduce the number of relevant parameters. Here we follow the rescaling of Wright and Mermin [
where the rescaled parameters
denote temperature, strength of chirality, and the anisotropy of elasticity, respectively, where is the coherence length, the rescaled spatial derivative is in Equation (8) and in Equation (9). We notice that the isotropic-nematic transition takes place at, while the isotropic-cholesteric transition takes place at
which depends on the chirality [
In the scaling employed above, the rescaled uniaxial ordering has the form
where is the rescaled uniaxial scalar parameter at equilibrium.
We choose a TN cell of thickness d = 15ξ and a TCN cell of thickness d = p0/2 with 90˚ twisted structure. Both of the two models have two stable degenerate configurations with a twist difference of π. The plates are placed at z = ±d/2 of a Cartesian coordinate system. The lengths dx and dy of the cell along the x and the y axes are much larger than.
We study the structure of s = ±1/2 twist disclinations. The disclination line is parallel to the y axis of the simulation cells, and we seek solutions independent of y. At the two plates z = ±d/2, we enforce the fixed uniaxial anchoring represented by and
On the lateral walls at x = dx/2, we also prescribe fixed boundary conditions with uniaxial ordering. For the TN cell, the total twist at x = dx/2 are ±π/2 respectively, while for the TCN cell, the total twist at x = dx/2 are π/2 and 3π/2 respectively. These boundary conditions are compatible with the generation of line defects with topological charge s = 1/2, with which we shall deal also apply to line defects with topological charge s = −1/2, with the conditions at the lateral walls exchanged.
In the rescaled space, the length is rescaled so that the cell thickness d = 15ξ is rescaled to d = 15 for the TN cell and d = p0/2 is rescaled to d = 2π for the TCN cell, respectively. The numerical calculations are to be intended with respect to the scaled variables.
We let the system relax from an initial condition under the fixed boundary conditions given above. In our simulation, we have identified that the value of dx = 3d is enough to exhibit the behavior of the equilibrium configuration in the limit as dx ® ∞; there is no visual difference as the value of dx continues to increase.
We employ
as the relaxation equation to iterate the order parameter χαβ, where f is the total rescaled free energy of the system,
and λ0 is the Lagrange multiplier ensuring Trχ ≡ 0 which can be eliminated in the numerical procedure [
We have calculated the tensor Q after the system to reach equilibrium state, and then the current tensor is diagonalized. Accordingly, the director is identified by the eigenvector possessing the largest eigenvalue, and the eigenvalues are scaled to the largest eigenvalue at the boundary. Then the degree of biaxiality is measured by the biaxiality parameter β2 given by Equation (3).
In this section, we present our numerical results. In our simulations, we set η = 1 (one-constant approximation) just for simplicity, and the scaled temperature is set equal to τ = −1, corresponding to.
zdef) is located at the center of x-z plane, i.e., (0, 0), and the major disturbances of the defect structure in x-z space take place within a nearly-circular region around the defect center.
In
A more detailed analysis of the defect core, directly showing the degree of biaxiality, is given in
All of the above analysis indicates that there exists eigenvalue exchange across the defect core, where two uniaxial states with orthogonal directors are changed into each other through a transformation that does not involve any director rotation, but instead implies a wealth of biaxial configurations bridging the uniaxial limits.
In the simulation of the s = 1/2 disclination in TCN, we choose κ = 0.4 that amount to the cholesteric pitch p0 ≌ 280 nm, corresponding to relatively strong chirality. According to [26,27], and the isotropic-cholesteric transition temperature given above, the most stable phases at τ = −1 is a nematic helical phase.
Figures 4-6 give the corresponding diagrams of director orientation “equiorientational” contours, eigenvalues of Q and biaxiality β2, respectively, which show the similar features as in TN cell. However, there are also subtle differences.
We should notice that
We carried out a numerical study on the structure of s =
±1/2 twist disclinations in TN and TCN cells with 90˚ twisted structure, based on the Landau-de Gennes approach, in which the orientational order of the LC is taken into account by introducing a second-rank tensor order parameter. In numerical calculations, we chose a relatively low temterature τ = −1, where the nematic LCs are in deep nematic phase and respond to distortions by entering biaxial states, rather than melting [
We have confirmed that two symmetric twist distortions with equal elastic energy in the TN cell may give rise to steady twist disclinations with topological charge s = ±1/2; in the chiral LC model we proposed, the frustrations between the helical structure and the confining surfaces induce s = ±1/2 disclinations, which also appears in the Grandjean-Cano wedges [18-21].
The detailed analysis of the defect structures indicates that the defect core is essentially biaxial and never isotropic. In the very central core region, the nematic is uniaxial and constrained to lie in the direction of the disclination line; however, away from it, the texture is almost uniaxial and can approximately be described by the director field. Moreover, we can conclude that there exists eigenvalue exchange across the defect core of both TN and TCN cells, which confirms that eigenvalue
exchange is a generally existent phenomenon in defects.
A natural question is whether our simulation describes a realistic scenario within a nematic cell. When the LC is cooled down from the isotropic to the nematic phase within the cell, by symmetry breaking, line defects is formed and dominates the Scene [9,28]. Individual defects that are not too close to one another tend to straighten up to equalize the elastic distortions in their vicinity. By topological and energetic reasons, the most probable scenario is that a sequence of line defects in a given direction with alternating charges s = ±1/2 form disclination loops or terminate on the surface of the sample because of the prohibitive energy cost of the free line end [29,30]. However, the details of loop are well beyond the scope of our study. Our simulation describes reasonably well such an isolated and stabilized defect, where the y axis is set along the average local line defect (on a scale comparable to the cell thickness).
The defects we studied can be used to the operation of bistable LC devices (so-called bistable twisted nematics) that have attracted considerable attention over past few decades [31-34], and can be used for a better understanding of the defect structure in Grandjean-Cano wedges
[18-21]. We should note that most of previous study on defects in Grandjean-Cano wedges have focused on the case in which the confining surfaces imposes parallel anchoring of the same direction, while our study provided the similar defect structures in the particular 90˚ twist cell.
It can be predicted that for the two models we proposed, the core size should be affected by the cell thickness of the TN cell, the chirality of chiral LC, as well as the temperature, the detailed results is the task for the future.
This research was supported by Natural Science Foundation of Hebei Province under Grant No. A2010000004 and Key Subject Construction Project of Hebei Province University.
The Frank elastic energy density of Chiral LCs can be described in terms of n as
where Ks is the surface-like elastic constant. The director of chiral LCs is written as
with by. Equation (A.2) gives
Substituting Equations (A.3-A.6) into (A.1) leads to
The Euler-Lagrange equation is
The general solution of Equation (A.8) is
with boundary conditions
Through simple calculation we can get
When, the Frank elastic energy density is
Apparently, m = 1 and m = 0 have the same and the lowest energy, , i.e., for the condition we considered, the two helical structures and have the same and lowest energy.