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In this paper, the stability of disclination ring in nematic liquid crystals is studied. In the presence of saddle-splay elasticity (characterized by k24) the disclination ring has a universal equilibrium radius. Depending on the values of the saddle-splay constant k24, the universal equilibrium radius is altered. When k24 > 0.92k (m = 1/2) and k24 > 0.88k (m = −1/2), the disclination will be a point rather than a ring, where k is the Frank elastic constant in the one-constant approximation.

In the light of -mapping method and topological current theory, the stability of disclinations around a spherical particle in nematic liquid crystals is studied. We consider two different defect structures around a spherical particle: disclination ring and point defect at the north or south pole of the particle. We calculate the free energy of these different defects in the elastic theory. It is pointed out that the total Frank free energy density can be divided into two parts. One is the distorted energy density of director field around the disclinations. The other is the free energy density of disclinations themselves, which is shown to be concentrated at the defect and to be topologically quantized in the unit of (k−k24)π/2. It is shown that in the presence of saddle-splay elasticity a dipole (radial and hyperbolic hedgehog) configuration that accompanies a particle with strong homeotropic anchoring takes the structure of a small disclination ring, not a point defect.

In the light of -mapping method and topological current theory, the effect of disclination lines on the free energy density of nematic liquid crystals is studied. It is pointed out that the total Frank free energy density can be divided into two parts. One is the distorted energy density of director field around the disclination lines. The other is the saddle-splay energy density, which is shown to be centralized at the disclination lines and to be topologically quantized in the unit of kπ/2 when the Jacobian determinant of the director field does not vanish at the singularities of the director field. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director field at the disclination lines, i.e., the disclination strengthes. When the Jacobian determinant vanishes, the generation, annihilation, intersection, splitting and merging processes of the saddle-splay energy density are detailed in the neighborhoods of the limit points and bifurcation points, respectively. It is shown that the disclination line with high topological quantum number is unstable and will evolve to the low topological quantum number states through the splitting process.

In the light of -mapping method and topological current theory, the contribution of disclination lines to free energy density of liquid crystals is studied in the single-elastic constant approximation. It is pointed out that the total free energy density can be divided into two parts. One is the usual distorted energy density of director field around the disclination lines. The other is the free energy density of disclination lines themselves, which is shown to be centralized at the disclination lines and to be topologically quantized in the unit of . The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of the director field at the disclination lines, i.e. the disclination strengths. From the Lagrange's method of multipliers, the equilibrium equation and the molecular field of liquid crystals are also obtained. The physical meaning of the Lagrangian multiplier is just the distorted energy density.