Defects on spherical drops and shells have been already investigated and we are now interested in studying topology of nematic phase on non-spherical spaces. The main reason for studying different geometries is that the number and type of defects in a drop of Liquid Crystal is subject to the geometrical constraint that the total topological charge on the surface, i.e. the sum of the charge of the single defects, must equal the Euler characteristic χ of the surface itself. The Euler characteristic can be expressed in terms of the genus g of the surface that equals the number of “handles” of the closed surface. χ = 2(1 - g). That means that for a sphere we will have χ = 2, on a torus χ = 0, on a figure eight χ = -2. Though on a torus the geometrical constraint allows the Liquid Crystal to arrange in a way that doesn’t present any defect, it is possible that it will be energetically favorable for the nematic phase to have defects: in that case we expect these defects’ charge to cancel out. The negative characteristic of the figure eight ensures that we will need to have at least two defect of negative -2 charge.

Non-spherical drops are produced using a rotating continuous phase that exerts a viscous force on the inner phase while it is injected into the first one using a needle. The system is stabilized using a viscoelastic continuous phase that allows the formation of the drop while rotating, but that “freezes” the shape once the rotation is stopped.

References:

[1] D. R. Nelson, Defects and Geometry in Condensed Matter Physics (Cambridge University Press)

[2] V. Vitelli and D. R. Nelson, Phys. Rev. E 74, 021711 (2006)

[3] E. Pairam A. Fernandez-Nieves, Phys. Rev. Lett. 102, 234501 (2009)