Defects are ubiquitous in nature and are important in particle physics, cosmology, and condensed matter physics [1] . This is explained by the importance of the role of defects in the course of different processes (phase transitions, plastic deformations, electronic processes, etc.) [2] . Defects in liquid crystals (LCs) affect the manifestation of a number of optical, field, hydrodynamic, and other effects [2] . They have posed important problems in optoelectric applications [3] as well as in fundamental physics. They can trap nanoparticles [4] [5] , mediate characteristic interaction between colloidal particles [5] - [7] and provide ordered templates for colloidal microassembly [8] - [11] . Therefore, location control of topological defects is important for defect-mediated colloidal assembly, and guided polymerization and crystallization in the self-organized nematic order [5] .

Defects in LCs are topological defects [2] , also called disclinations [12] . They appear spontaneously at the isotropic-nematic transition when the symmetry of the isotropic phase is broken to the symmetry of the nematic phase [13] [14] . They are moving during the equilibrium process. Therefore, the dynamics of topological defects in the ordering process has attracted the interest of many researchers and has been largely studied theoretically, numerically and experimentally over the last decades [13] - [26] . All of these studies focus on the relationship between and during the annihilation process, where is the relative distance between the defect pair and is the time for the annihilation. Most previous studies have been confined to investigating the annihilation of an isolated defect pair [15] - [26] . However, in real systems, disclinations are never isolated, but subjected to the anchoring forces coming from the substrates of the cell containing the liquid crystal. As a consequence, in a confined geometry, the substrate anchoring is expected to strongly influence the interactions between defect lines. A similar anchoring effect has been invoked to explain the annihilation dynamics of nematic point defects confined in capillary tubes [27] and in hybrid cells [13] [14] . However, the behavior of defect lines in a confined geometry remains totally unexplored.

In this paper, based on our previous study of the relaxation dynamics of a dipole of disclination lines with in a thin hybrid aligned nematic (HAN) cell [28] , we continue to study the relaxation dynamics of the dipole in a confined geometry.

In order to analyze the influence of on the behavior of the defect pair, the time dependence of the distance between the defect pair for different with and moving velocity of the defects as a function of time are shown in Figure 2.

The squares, circles and triangles represent numerical results with different cell gap, respectively, and the full lines are the corresponding fitting curves with function of [13] [21] in Figure 2(a), where is the coalescence time. decreases and the fitting error increases with the decrease of, respectively. decreases from to and the relative error increases from 3.9% to 4.3% when decreases from to. Further decrease to, decreases to, the relative error is up to 13.4%, which means that the time dependence of has deviated from the exponential law, as shown in Figure 2(a).

The distance between two isolated oppositely charged defects is described as [13] [22] . In a confined geometry, the substrate anchoring forces influences the interactions between the defects. The thinner the cell gap, the greater the impact of the anchoring forces. Consequently, decreases and the fitting error increases with the decrease of d. When decreases to, the anchoring forces are so strong that they make the time dependence of deviate from the exponential law.

The moving velocity of the defects increases rapidly with time because the elastic force between the two defects is [12] , but decreases with the decrease of. It decreases slightly when decreases from to, but rapidly when the cell gap decreases to and, as shown in Figure 2(b). This results in the increase of the coalescence time with the decrease of. The coalescence time slightly increases from to when d decreases from to, but soars to and when d de- creases to and, respectively.

It indicates that the anchoring forces coming from the substrates produce a negligible effect on the defects when. But, when, the anchoring forces strongly affect the behavior of the dipole: inhibiting the moving movement of the defects. The thinner the cell gap, the greater the impact of the anchoring forces. Consequently, the moving velocity of the defects decreases with the decrease of the cell gap.

In order to analyze the influence of on the behavior of the defect pair, the time dependence of the distance between the defect pair and moving velocity of the defects as a function of time for different with are shown in Figure 3.

decreases with the increase of r₀. It decreases from to when increases from to. The time dependence of follows the exponential law when. When increases to, decreases to 0.300 ± 0.046, hence the relative error is up to 15%, which indicates the time dependence of has deviated from the exponential law. The direct elastic interaction between the defects gives an elastic force [12] , resulting in the rapid decrease of the moving velocity with the increase of r₀, as shown in Figure 3(b). Therefore, the coalescence time soars from to when r₀ increases from to.

Increasing the initial distance has the same effect as reducing the cell gap: inhibiting the relative movement of the defects. The behavior of the pair depends on the competition between the attractive force, determined by the distance between the defect pair, and the anchoring forces, determined by the cell gap d. The attractive force aggravates the annihilation process, while the anchoring forces inhibit the annihilation process. If the role of one kind of force is weakened, then the impact of the other will be relatively enhanced. Therefore, the impact of the anchoring forces are relatively enhanced with the increase of, which results in the decrease of with the increase of and the time dependence of deviating from the exponential law when increases to.

The anchoring forces and the attractive force must be balanced and the defects can be stable existence under certain conditions. In order to find the conditions, the dynamic behavior of the dipole is studied in detail by varying the cell gap and the initial distance. The results are shown in Figure 4 and Figure 5.

When, there are two equilibrium states, as shown in Figure 4(a), Figure 4(b) and Figure 5. The critical thicknesses are and, respectively. When, the two defects annihilate because the attractive force between them plays a major role and a uniform uniaxial state is formed when the system is equilibrium. When, the defects can be stable existence because the anchoring forces are so strong that they can inhibit the relative movement of the defects. The thinner the cell gap, the greater the impact of the anchoring forces. As a consequence, the relative moving distance decreases with the decrease of.

When, there are three equilibrium states, divided by two critical thicknesses: and, as shown in Figures 4(c)-(e) and Figure 5. The two critical thicknesses both change linearly with the initial distance. The corresponding fitting equations are and, respectively. When, the two defects annihilate. The annihilation process has delicate structures. When, the two defects annihilate and a uniform uniaxial state is formed when the system is equilibrium. When, the two defects annihilate. However, the system is not uniform uniaxial, but biaxial in the region where the defects collide. Since the two defects are oppositely charged and the total charge is conserved, the defects disappear when they collide. But, the director in the region where the defects collide distorts inhomogeneously, which creates the biaxial layers. The range of the cell gap corresponding to this state increases with the decrease of. When, the dipole can be stable existence as the system is equilibrium because the anchoring forces play a major role under this condition.

The behavior of the dipole in a confined geometry depends on the competition between two kinds of forces: the attractive force and the anchoring force. The former aggravates the annihilation process, while the latter inhibits it. The system has three equilibrium states, divided by two critical thicknesses and. When,

the pair annihilates, and the dynamics follows the function of as is sufficiently large. de-

creases with the decrease of or increase of. The annihilation process has delicate structures: when and or and, the system is uniaxial at equilibrium state; while, when and, the system is biaxial in the region where the defects collide. When, the dipole can be stable existence.

As far as we know, it is the first time to discover the stable existence of the oppositely charged defects under certain conditions. This research plays a major role in the formation and control of topological defects, and has significant academic value for mediation of defects on colloidal particles in nematic liquid crystals.

We thank the Editor and the referee for their comments. Research of Z. Zhang is funded by National Natural

Science Foundation of China grant No. 11374087 and Key Subject Construction Project of Hebei Province University. This support is greatly appreciated.

The free-energy density of a NLC without external field can be expressed as

They are, respectively, the elastic and the bulk free-energy densities. The former is induced by the inhomogeneous order in LCs. It can be given the form

where and are elastic constants. The latter is a potential that depends on. It is conventionally described by an expansion in up to the fourth order

Usually it is assumed that, where is temperature and is the supercooling temperature., and are coefficients. The traceless and symmetry of are taken into account by the introduction of the Lagrange parameter tensor [32] :. The full free-energy functional is

The calculation can be simplified by dimensionless variables. Here, we follow the rescaling of Schopohl and Sluckin [32] by defining the following dimensionless quantities

,

where is the order parameter at the isotropic-nematic phase transition point and

is the characteristic length for order-parameter changes. Hence, the dynamics

of can be described as