This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

At the wall in a hybrid nematic cell with strong anchoring, the nematic director is parallel to one wall and perpendicular to the other. Within the Landau-de Gennes theory, we have investigated the dynamics of _{c}* ≈ 9ξ (where ξ is the characteristic length for order-parameter changes), the exchange solution is stable, while the defect core solution becomes metastable. Comparing to the case with no initial disclination, the value at which the exchange solution becomes stable increases relatively. At a critical separation of _{c} ≈ 6ξ, the system undergoes a structural transition, and the defect core merges into a biaxial layer with large biaxiality. For weak anchoring boundary conditions, a similar structural transition takes place at a relative lower critical value. Because of the weakened frustration, the asymmetric boundary conditions repel the defect to the weak anchoring boundary and have a relatively lower critical value of _{a}, where the shape of the defect deforms. Further, the response time between two very close cell gaps is about tens of microseconds, and the response becomes slower as the defect explodes.

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 [

Eigenvalue exchange (also called order reconstruction) was first shown by Schopohl and Sluckin within the core of

Recently, the atomic force microscope (AFM) has been used as a unique tool to measure the separation dependence of surface forces with unprecedented accuracy and flexibility [

Recently, many theoretical investigations have focused on the dynamical behavior of structure transition, using the

Our theoretical argument is based on the Landau-de Gennes theory [

Here, λ_{i} and _{i} are the _{th} eigenvalue and the _{th} eigenvector of

where

Finally, when all eigenvalues of ^{2}, defined as [

which is a convenient parameter for illustrating spatial inhomogeneities of ^{2} = 0, while states with maximal biaxiality correspond to β^{2} = 1. Since ^{3}) = 3det^{2} = 1 are precisely those where det

Following the notation in [_{bulk}{_{αβ}} + _{e}{_{αβ}, ▽}, in which

is the bulk energy that describes a homogeneous phase. In _{bulk},

The free-energy, _{e}, which penalizes gradients in the tensor order parameter field, is given in the form

The coefficients, _{i}, are related to the splay, twist and bend elastic constants. In this work, we use the one-elastic-constant approximation just for simplicity, where the splay, twist and bend elastic constants have a common value that depends quadratically on the scalar order parameter ^{2}_{1}, and

The boundary conditions at the LC interfaces are taken into account by the surface term, _{s}, describing the interaction between the nematic molecules close to the substrate and the substrate itself, which is given by

where _{s} = _{s} is the value of the tensor order parameter preferred by the surface [

where _{k} is the _{th} component of the outward vector normal to the substrate. This expression allows us to impose on the surface substrate uniaxial, as well as biaxial conditions. For rigid anchoring, this is equivalent to imposing the Dirichlet condition on the surface,

After an appropriate rescaling of the variables, we can simplify the calculating process. Here, we follow the rescaling of Schopohl and Sluckin [^{4} / (9^{3}],

where the rescaled parameter,
_{0} = −_{0}

In the scaling employed above, the rescaled uniaxial ordering has the form

where

We choose a HAN cell with strong anchoring boundary conditions; the nematic director is parallel to one wall and perpendicular to the other (see _{x} and _{y}, of the cell along the _{x}_{y}

We study the structure of

At the two plates _{−}_{d}_{/2}_{+}_{d}_{/2} = π/2, that is

On the lateral walls at _{x}_{x}_{x}

We compute the evolution of the LC with a dynamic theory for its tensor order-parameter field,

The coefficient, ^{2})]^{2}, where

The numerical calculations are to be intended with respect to the scaled variables. When the functional derivatives in

with the boundary conditions,

where Γ̃ =Γ × (−_{0}) and

We use a two-dimensional finite-difference iterative method employed in our previous studies in [

In our numerical calculations, we have found that a discretization with a time step given by 4 × 10^{−10} is sufficient to guarantee the stability of the numerical procedure. In addition, our equilibration runs take 10^{6}, which has been confirmed as sufficient for the system to reach equilibrium state.

We let the system relax from an initial condition under the boundary conditions given in Section 2.2. We have calculated the tensor ^{2}, given by

In this section, we present our numerical results. According to the parameters given in [^{−6} J/m^{3}, ^{3}, ^{3}, _{1} = 2.25 × 10^{−12} J/m. In our simulations, we set the scaled temperature at
_{x} to identify the behavior of the equilibrium configuration in the limit of _{x}

^{2} (^{2} = 0), except for a small region around the defect core (_{c} ≈ 6ξ. In a word, the system has transited from the eigenvector rotation configuration with a topological defect into the eigenvalue exchange configuration by developing a thin transitional biaxial nematic layer.

A more detailed analysis of the defect core, directly showing the process of biaxial transition, is given in ^{2} across the defect center along ^{2} reaches its maximum value β^{2} = 1 (

To quantify the influence of the cell gap on the defect structure, we plot in _{x}_{z}_{x}_{z}^{2} = 1. It is shown that _{z}_{x}_{x}^{2} become increasingly prolate in the

It seems that the transition to the biaxial wall originates from the biaxiality seed concentrated at the −1/2 defect. The biaxial wall would also be created in the absence of a defect [_{c0}, the rotation solution is stable, while the exchange solution is unstable. When _{c0}, only the exchange solution exists as a stable configuration; no rotation solution exist. In other words, the rotation solutions merge continuously into the exchange solution at a critical value, _{c0}.

In order to make the mechanism of the structure transition clear, we make a comparison between the two cases (with and without initial disclination).

_{c}* ≈ 9ξ, the exchange solution is stable, while the defect core solution is metastable. Comparing to the case with no initial disclination, we find that for the configuration with a topological defect, the value at which the exchange solution becomes stable increases relatively. That is because the configuration with a topological defect has larger energy, relative to the pure HAN configuration with no defect. With further decreasing of the cell gap until a critical value, _{c}, a transition of the defect core solution into the exchange solution happens. Below _{c}, only the exchange solution exists, but no defect core solution. According to the structure given by _{c} ≈ 6ξ.

_{c}_{c}_{c} = _{c0}.

In order to make it clear how the configuration changes between two states, we make dynamical calculations of the defect core structure between different cell gaps (10ξ → 9ξ, 9ξ → 8ξ, 8ξ → 7ξ, 7ξ → 6ξ). ^{2} across the defect center along _{c}, an initial structure with a defect core cannot change to the eigenvalue exchange structure, even at the value where the exchange structure is stable. The response time is about tens of microseconds, and the response becomes slower as the defect explodes (see

To analyze the influence of the defect core on the dynamic process, we make dynamical calculations of the exchange configurations between different cell gaps. To aid the comparisons, we give the dynamics as the cell gap decreases. ^{2}, along

_{c}, there are only defect-defect and exchange-exchange transitions, but no defect-exchange or exchange-defect transitions existing. However, the response of an initial defect configuration with _{c} to _{c} (e.g., 7ξ → 6ξ) will undergo a defect-exchange transition (See _{c} ≈ 6ξ. While the cell gap,

In order to get the influence of the surface anchoring at the boundary plates on the behavior of topological defects, we focus on the Neumann conditions in this section. First, we study the condition of asymmetric boundaries. We impose strong homeotropic anchoring on the top plate and weak planar anchoring on the bottom plate. _{S}_{S}

Next, we fix the anchoring strength at the bottom plate to _{S}_{x} and height _{z} of the defect core inside the cell for different values of cell gap

_{c}(_{as}) ≈ 5ξ, the defect is transformed into a surface layer. The surface plane defect is then expelled from the cell when the cell gap reaches _{c}**(_{as}) ≈ 2.5ξ. For _{c}**(_{as}) < _{c}(_{as}), the surface biaxial layer bridges two uniaxial stares, one homeotropic in the bulk and the other planer on the boundary. The results are consistent with the results in [

_{z} is slight, while _{x} increases when the cell gap reaches a critical value, _{a}, where the defect core begins to lose its circular cross-section and stretches along the _{a}. That is because the moving of the defect as the cell gap decreases weakens the frustration on the confining surfaces. _{c}(_{as}), where the defect is transformed into a surface layer, is smaller than that of strong anchoring conditions.

Further, we study the condition of symmetric boundaries. We impose the same value of weak anchoring on both the two boundary plates and studied the effect of anchoring strength on the critical value, _{c}. In our simulation, the anchoring strength is set equal to _{S}_{c}(_{s}) = 3ξ. Compared with the strong anchoring conditions (_{c}. That is because the weak anchoring boundary weakens the frustration on the confining surfaces.

Within the Landau-de Gennes theory, we carried out a numerical study on the structure of _{c}, of a HAN layer with a topological defect, an eigenvalue exchange transition occurs, and the defect is squashed flat into a biaxial layer. Because of the presence of the defect core, the value at which the exchange solution becomes stable increases relative to the pure HAN. The dynamic simulations show that before the critical value of _{c}, an initial structure with a defect core cannot change to the eigenvalue exchange structure, even at the value where the exchange structure is stable. The response time between two very close cell gaps is about tens of microseconds, and the response becomes slower as the defect explodes. For weak anchoring conditions, plates with strong anchoring repel the defects, while plates with weak anchoring allow the defect to escape through the boundary. Because of the weakened frustration, the weak anchoring boundary condition has relatively lower critical values of _{a} and _{c}, corresponding to the value where the defect core begins to lose its circular cross-section and where the defect is transformed into a surface layer, respectively.

According to the parameters given, we can easily get the characteristic length ξ ~ 3.96 nm, and the critical separation is ~20 nm, which is consistent with the experimental result in [

Further, the parameters also give that the unit of anchoring strength is _{0} ≈ 5.7 × 10^{−4} J/m^{2}, which is reasonable. According to _{s}^{−7} m, which is larger than the cell gap of ^{−8} m, _{s}

It can be predicted that the critical separation and the response time should be affected by the temperature; the detailed results, as well as the force curve with the

This research was supported by the National Natural Science Foundation of China under Grant No. 11374087 and the Key Subject Construction Project of Hebei Province University.

The authors declare no conflicts of interest.

The geometry of the problem.

The director field profile at equilibrium state in a thin nematic layer with different thicknesses. The simulations are based on the initial conditions given in Section 2.2. (

Biaxiality β^{2} for different thicknesses in a thin hybrid alignment nematic (HAN) layer with an initial line defect. (

Biaxiality β^{2} across the defect center along

Variation of the defect core size with different thicknesses. The lengths of _{x}_{z}

The free energy as a function of

The dynamics of the biaxiality, β^{2}, across the defect center along

The dynamics of the biaxiality, β^{2}, along

The director field profile at equilibrium state in a thin nematic layer with different anchoring strength on the bottom plate. The cell gap of the simulated nematic layer is _{s}_{s}_{s}_{s}

The director field profile at equilibrium state in a thin nematic layer with different thicknesses. The anchoring strength is _{S}

Variation of the defect core size with different thicknesses for the asymmetric boundaries. The lengths of _{x}_{z}

The director field profile at equilibrium state in a thin nematic layer with different thicknesses. The anchoring strength is _{S}