Question 5.2.7: Determine the force acting on a straight disclination with t......
Determine the force acting on a straight disclination with the Frank index m, if it is located at a distance l from the boundary plane of the nematic with the fixed orientation of directors (see Figure 5.11).
This problem can be solved by exploring the above mentioned electrostatic analogy. As seen from Figure 5.11, the equilibrium distribution of directors, which satisfies the required boundary condition, can be formed by two disclinations: the original one, and the “reflected” one with the same Frank index.
Indeed, the angles θ_{1} and θ_{2} are equal to each other; therefore, while moving along the boundary surface, the one discinlation provides the clockwise rotation of the director there, while the other one makes it counterclockwise with the same pace. As a result, the orientation of directors at the boundary does not change. Then, it follows from the previous problem, that the disclination under consideration is repelled from the boundary with the force per unit length equal to πKm^{2}/2l.