Our theoretical argument is based on the Landau-de Gennes theory [14], in which the orientational order of a LC is described by a second-rank symmetric and traceless tensor [22]

Here λ_i and e_i are the i_th eigenvalue and the i_th 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 [23]

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 [24], the free energy density of a chiral LC is given by

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 f_bulk, 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 ε_αγβ 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 [24] and the rescaled free-energy density is written, in terms of a rescaled tensor order parameter , the rescaled form of Equations (4)-(6) can be written as

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 [24].

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 d_x and d_y 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 = d_x/2, we also prescribe fixed boundary conditions with uniaxial ordering. For the TN cell, the total twist at x = d_x/2 are ±π/2 respectively, while for the TCN cell, the total twist at x = d_x/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 d_x = 3d is enough to exhibit the behavior of the equilibrium configuration in the limit as d_x ® ∞; there is no visual difference as the value of d_x continues to increase.

We employ

as the relaxation equation to iterate the order parameter χ_αβ, 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 [7]. To discretize Equation (13), we replace the derivatives with the two-dimensional finite-difference method employed in our previous studies in [25]. In our numerical calculations, we have found that a discretization with time step given by 10⁻⁵ is sufficient to guarantee the stability of the numerical procedure. In addition, our equilibration runs take 10⁵, which have been confirmed sufficient for the system to reach equilibrium state.

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).

Figure 1 shows the director orientation diagram of s = 1/2 disclinations in the cross section through the defect line, i.e., x-z plane. The lines are “equiorientational” contours. Note that the position of the defect center (x_def,

z_def) 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 Figure 2 we plot the three eigenvalues of the matrix Q along the two lines across the defect center in both x and z directions, that is z = 0 (Figure 2(a)) and x = 0 (Figure 2(b)), respectively. Outside about 0.2d from the defect line, the tensor order parameter essentially takes the form of Equation (2). Inside this region, the order parameter becomes increasingly biaxial, in that the degeneracy between two of the eigenvalues is broken. On a ring around the defect center, of radius ρ ≈ 0.02d, the LC is maximally biaxial; one of the eigenvalues is now zero. At the defect center, the order parameter once again approximately takes the form of Equation (2). However, now the symmetry axis is the defect axis itself, and as compared with the bulk, the order parameter has changed its sign.

A more detailed analysis of the defect core, directly showing the degree of biaxiality, is given in Figure 3. There, we plot β² across the defect center along z = 0 (Figures 3(a) and (b)) and x = 0 (Figures 3(c) and (d)). Figures 3(b) and (d) are the partial enlarged detail. At the defect center, β² = 0. On increasing the distance ρ from the defect, the degree of biaxiality β² at first gradually increases; it reaches its maximum value β² = 1 at ρ ≈ 0.02d. Upon further increasing ρ, the uniaxial ordering with β² = 0 is asymptotically approached, and the nematic can be described by the director field.

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. Figure 4 shows that the position of the defect centre (x_def, z_def) is located at (−0.04167d, 0), rather than (0, 0), which results from the asymmetry distribution of the director at the system.

We should notice that Figure 4 gives the similar director orientation contours as that given by Scheffer based on the Frank theory [17], which confirms the correctness and rationality of our method.