Saturn rings

Fig. 1. Schematic diagrams of the director field distortions black lines) around particles in an aligned nematic liquid crystal. For a normal anchoring of the liquid crystal molecules at the surface of the particles, there are two possible configurations, a Dipole configuration with a companion point defect (indicated by an arrow) located in the immediate vicinity of the particle, b Quadrupolar Saturn-ring configuration with a disclination ring surrounding the particle at the equator...

However, the amplitude and the anisotropy of the interactions between drops exhibiting the Saturn ring configuration are completely different. The expression for the quadrupolar interaction energy is [4-7] ... 

While the field is on, the Saturn ring configuration is maintained and appears to be the most stable configuration in the presence of the field. However, when the field is turned off, the ring continuously shrinks back to the hyperbolic hedgehog defect within a time scale of a few tens of seconds on average [47]. This... 

Fig. 11. Series of pictures showing the ring defect relaxation once the electric field has been tiuned off. 1 Saturn ring 2 and 3 intermediate configurations 4 dipole. Drop diameter 35 pm...

Fig. 4.17 a Colloidal GNPs (100 rnn diameter) trapped into the Saturn ring disclination line surrounding a large spherical silica particle (1 mm diameter) suspended in a thermotropic N LC host, b The calculated stable colloidal ring superstructure for this combination of colloids (inset shows a schematic of the corresponding electrical circuit) [113]. Copyright from American Physical Society 2008... 

Figure 5. Size of the Saturn ring (in units of the particle radius) as a function of the particle size.

If we examine the system at various separations, we observe that for large s each sphere is surrounded by its own Saturn ring. However, for s < J we observe that the two disclination rings interact and form a new type of defect structure, shown in Fig. 9 two incomplete equatorial Saturn rings connected to a third ring in the plane normal to ri2. 

In Fig. 10, we compaxe the structure of the three-ring defect obtained from simulation and theory for s = 0.3ii with R = Soq. In both cases, the director field is shown superimposed to a contour plot of 5, the scalar order parameter. Both theory and simulation exhibit the third ring in addition to the usual two Saturn rings. In these plots we can observe how the strength of S decreases continuously from its bulk value to a minimum at the defects. Once more, in contrast to the field theory results, the Monte Carlo data show layers of low and high values of S close to the spheres surfaces that are correlated with modulations in density see Fig. 10). 

It is of interest to discuss what type of experiments may be able to detect the new defect structure. A first comment is that this structure is not expected to arise for large particles in the bulk [21]. Saturn rings au e not stable for large particles instead, hyperbolic hedgehogs are the stable structures. However, experimentally it is known that Saturn rings can be stabilized by confinement [12]. In this case, it would not be... 

The first detailed examination of ring dynamics was accomplished by James Clerk Maxwell in his 1857 Smith Prize essay, a study that still repays reading. He dealt with the question of the composition and stability of Saturn s rings, demonstrating that they must consist of a swarm of small particles trapped in planar orbits. The demonstration of differential rotation in the Saturn rings by the observation of Doppler shifts in a reflected solar spectrum was first performed by Keeler about 30 years later. 

---