Kinematics of Topological Defects in Nematic Toroidal Droplets

Hung Le, Karthik Nayani, and Alberto Fernandez-Nieves

In this project, we will be exploring the world of topology through the lens of nematic liquid crystal. The Poincaré-Hopf theorem establishes that the total topological charge on a bounding surface must match the Euler characteristic for that geometry [1]. The Euler characteristic is known to decrease by 2 for each hole the geometry has [2], and indeed we see 2, 0, and -2 for nematic sphere, torus, and double torus respectively in previous works[3-6, 7]. The defects can be observed under a microscope equipped with a cross-polarizer.

Fig. 1: The experiment set-up

Fig. 2: Images of nematic torus (a) and double torus (b) in bright-field

Fig. 3: The same objects under cross-polarizer

In another previous work, it was demonstrated that toroidal droplets can break into a crescent shape or shrink to a sphere, and it is by controlling the aspect ratio of the torus that we can tell whether a droplet will be stable, break, or shrink [8]. This work is a combination of these two projects: we will generate unstable nematic toroidal droplets and observe changes to the defects, if any. Particularly, we want to generate double tori where one handle is unstable and the resulting gemoetry is topologically the same as a torus, and unstable torus that evolves to topologically a sphere. Below are illustrations of scenarios we are trying to create.

Fig. 4: Double torus, one handle breaks

Fig. 5: Double torus, one handle shrinks

Fig. 6: Single torus, handle breaks

Fig. 7: Single torus, handle shrinks

It is easy to generate and control the size of a nematic torus, but it is much harder to do so with a double torus. Double tori are generated by side by side then joined together--a process we call microsurgery. The process consists of sucking out the fluid between the two tori by using a needle connected to a syringe. The needle is set in one place, we change its location my moving the stage. The microscope is first focused on the edge of the tori, then the needle is lowered into focus. The suction pressure is controlled by carefully pumping and stopping the syringe manually--too much suction can mean sucking out the tori completely. Thus, the process requires extreme dexterity and patience. Also, since unstable handles are of interest, it is crucial to generate tori at an aspect ratio that will stay stable long enough during microsurgery but then undergo breaking or shrinking afterward. We give special thanks to Ekapop Pairam, who perfected the microsurgery technique and taught us how to perform it ourselves.

References:
[1] R. D. Kamien, Rev. Mod. Phys. 74, 953 (2002).
[2] Math Illuminated. Surfaces and Manifolds. http://www.learner.org/courses/mathilluminated/units/4/textbook/03.php.
[3] M. A. Bates, Nematic ordering and defects on the surface of a sphere: A Monte Carlo simulation study, J. Chem. Phys., 128(10):104707 (2008).
[4] P. K. Chan and A. D. Rey, Simulation of reorientation dynamics in bipolar nematic droplets, Liquid Crystals, 23:677-688 (1997).
[6] A. Fernández-Nieves et al, Novel defect structures in nematic liquid crystal shells, Phys. Rev. Lett., 99(15):157801 (2007).
[7] E. Pairam et al, Stable nematic droplets with handles. PNAS 1221380110 (2013).
[8] H. Le et al, Breaking of Toroidal Droplets Inside Viscoelastic Materials. http://fernandezlab.gatech.edu/research/Hung/Hung.html.

Soft Condensed Matter Laboratory, School of Physics, Georgia Institute of Technology
770 State Street NW, Atlanta, GA, 30332-0430, USA
Phone: 404-385-3667 Fax: 404-894-9958
alberto.fernandez [at] physics.gatech.edu