Liquid crystals and geodesics

Overview & Synopsis

Since a unit vector n is a point on the unit sphere, any one-dimensional liquid crystal configuratior n(z), a ≤ z ≤ b, generates a path on the unit sphere. In order to see what configurations are stable, we map the unit sphere onto a surface G having the property that equilibrium configurations of a nematic liquid crystal map into geodesic lines on the surface (geodesics). The shape of the surface depends on the elastic constant ratios k1/k3 and k2/k3, and is a sphere when all three constants are equal. A picture or model of the surface for a particular material is helpful in visualizing the equilibrium configurations that correspond to prescribed boundary conditions n(a) and n(b), and in studying their energy and stability. For example, a configuration n(z) is in stable equilibrium if and only if the path on G is a curve of least length among nearby curves having the same end points, and arc length ds on G is proportional to dz. The physical interpretation of the proportionality of ds to dz is that the elastic free energy density of the liquid crystal is a constant, independent of z.