^{1}

^{2}

^{*}

This paper investigated the dynamics of a dipole of
±
1/2 parallel wedge disclination lines in a confined geometry, based on Landau-de Gennes theory. The behavior of the pair depends on the competition between two kinds of forces: the attractive force between the two defects, aggravating the annihilation process, and the anchoring forces coming from the substrates, inhibiting the annihilation process. There are three states when the system is equilibrium, divided by two critical thicknesses d_{c1} and d_{c2} (existing when r_{0}
≤
15
ξ
, r_{0} is the initial distance between the two defects), both changing linearly with r_{0}. When the cell gap d
＞
d_{c1}
, the two defects coalesce and annihilate. The dynamics follows the function of r
∝(
t
_{0}
-t
)
^{α}
during the annihilation step when d is sufficiently large, relative to r_{0}, where r is the relative distance between the pair and t_{0} is the coalescence time.
α
decreases with the decrease of d or the increase of r_{0}. The annihilation process has delicate structures: when r_{0}
≤
15
ξ
and d
＞
d_{c2}
or r_{0}
＞
15
ξ
and d
＞
d_{c1}
, the two defects annihilate and the system is uniaxial at equilibrium state; when r_{0}
≤
15
ξ
and d_{c2}
＞
d
＞
d_{c1}
, the two defects coalesce and annihilate, and the system is not uniaxial, but biaxial in the region where the defects collide. When d
≤
d_{c1}
, the defects can be stable existence.

Defects are ubiquitous in nature and are important in particle physics, cosmology, and condensed matter physics [

Defects in LCs are topological defects [

In this paper, based on our previous study of the relaxation dynamics of a dipole of disclination lines with

The theoretical argument is based on Landau-de Gennes theory, in which the orientational order is described by a second-rank traceless and symmetric tensor [

where

where

In the reduced space defined by Schopohl and Sluckin [

Time evolution of

The reduced space is discretized into grids with the same interval of^{3}, ^{3}, ^{3},

Consider the pair positioned at

where

of coordinate and the observation point,

In order to analyze the influence of

The squares, circles and triangles represent numerical results with different cell gap

The distance between two isolated oppositely charged defects is described as

The moving velocity of the defects increases rapidly with time because the elastic force between the two defects is

It indicates that the anchoring forces coming from the substrates produce a negligible effect on the defects when

In order to analyze the influence of

_{0}. It decreases from _{0}, as shown in _{0} increases from

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

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

When

When

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

the pair annihilates, and the dynamics follows the function of

creases with the decrease of

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

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

Usually it is assumed that

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

where

of