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In the light of φ-mapping method and topological current thcory the effect of disclination lines on thefree energy density of nematic liquid crystals is studied. It is pointed out that the total Frank free energy density canbe divided into two parts. One is the distorted energy density of director field around the disclination lines. The otheris the saddle-splay energy density which is shown to be centralized at the disclination lines and to be topologicallyquantized in tfe unit of kπ/2 when the Jacobian determinant of the director field does not vanish at the singularitiesof the director field. The topological quantum numbers are determined by the Hopf indices and Brouwer degrees of thedirector field at the disclination lines, i.e., the disclination strengthes. When the Jacobian determinant vanishes, thegeneration, annihilation, intersection, splitting and merging processes of the saddle-splay energy density are detailed inthe neghborhoods of the limit points and bifurcation points, respectively. It is shown that the disclination Iine withhigh topologcal quantum number is unstable and will evolve to the low topological quantum number states through thesplitting process.
The is of pointed out that the total Frank free energy density canbe divided into two parts. One is the distorted energy density of director field around the disclination lines. The otheris the saddle-splay energy density which is shown to be centralized at the disclination lines and to be topologicallyquantized in tfe 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 thedirector field at the disclination lines, ie, the disclination strengthes. When the Jacobian determinant vanishes, the generation, annihilation, intersection, splitting and merging processes of the saddle-splay energy density are detailed inthe neghborhoods of the limit points and bifurcation points, respectively. It is shown that the disclination Iine with high topologcal quantum number is unstable and will evolve to the low topological quantum number states through thesplitting process.