Novel Colloidal Interactions in Anisotropic Fluids
Original Entry: Aaron Goldfain, AP 225, Fall 2012
Authors: Phillippe Poulin, Holger Stark, T. C. Lubensky, and D. A. Weitz.
Publication: Phillippe Poulin, Holger Stark, T. C. Lubensky, and D. A. Weitz, Novel Colloidal Interactions in Anisotropic Fluids. J. Science (1997) vol. 275 page 1770.
This article describes equilibrium positions of colloidal particles and their companion defects in a nematic liquid crystal. It presents experimental observations of colloidal particles contained within nematic drops, along with a qualitative explanation of the observed features. After that, it gives a detailed theoretical description of the system, which the authors claim can be readily extended to describe colloidal interactions within other anisotropic fluids.
The experimental system used by the authors is a double emulsion. It consists of small water droplets within nematic drops in a continuous water phase. Each nematic drop is treated as an individual system, and the water droplets are treated as the colloidal particles. The emulsion is imaged between crossed polarizers, enabling the director field of the nematic to be determined. For the particular nematic used (5CB), at the nematic-water interfaces the director is normal to the interface.
When the nematic droplets are dispersed in water, the authors observe diploar chains of colloidal particles and defects in the middle of the nematic droplets, as seen in Figure 1. They do not detect any thermal fluctuations in the particles' positions, indicating they are in deep potential wells. The authors also confine the nematic droplets between parallel plates that impose a tangential boundary condition on the director at the plates. In these systems, the colloidal particles initially form chains, but then create large complex structures in which each particle is associated with a defect.
Based on the textures observed in the nematic liquid crystal, the authors claim that each defect is a hedgehog defect, with the director field coming straight out of its center. Each hedgehog defect has a topological charge of Q = 1, but pairs of hedgehogs can either have Q = 0 or 2. Normal boundary conditions around the nematic drop impose a total charge of Q = 1 within the droplet, while parallel boundary conditions dictate Q = 0. Based on this constraint, the authors qualitatively describe the observed structures within the nematic drops.
The authors then develop a detailed theoretical model for system based on the Frank elastic energy of the nematic director. They first develop an expression for the energy of a droplet-defect dipole as a function of dipole-droplet separation distnace. Then, the authors complete their model by determining an expression for the energy of a dipole in an arbitrary director field. Their model implies that the equilibrium separation distance between a defect and a droplet is 1.17<math>a</math>, where <math>a</math> is the droplet radius, and that the energy of this interaction is on order of <math>10^5kT</math>. This separation distance is quantitatively consistent with their observed dipoles, and the energy of the interactions is consistent with them not observing thermal fluctuations in the dipoles. Additionally, their model predicts a dipole-dipole attraction and the inter-dipole energy in minimized when the dipoles align in the same direction, explaining the tendency of chains to form. Finally, their model also predicts that dipoles are at their lowest energy when the background director field has maximal splay, explaining why dipoles are observed in the center of the nematic drops.
As a further test of their model, the authors briefly describe their investigations of systems with different boundary conditions. Nematic drops with tangential boundary conditions on the director were examined, and the interior water droplets were also made to have tangential boundary conditions at their surfaces. In theses systems, the authors also found that their theory was able to explain all of the observed features they describe.
This is a very well written article that provides an essentially complete experimental and theoretical description of an interesting soft matter system. In soft matter, it is rare to see such a complete understanding of a system. It is an excellent demonstration of a calculation of elastic forces in liquid crystals. The one thing I feel this article could benefit from is more quantitative experimental results. It would be very interesting to see the details of their theory tested, for example, by measuring the interaction energy between a defect and droplet as a function of separation distance.
 P. Poulin, H. Stark, T. C. Lubensky, and D. A. Weitz, Novel Colloidal Interactions in Anisotropic Fluids. J. Science (1997) vol. 275 page 1770.