CHAPTER 3 : Lock-and-key interaction in nematic liquid crystals
3.2 Defect-free system as a system for colloidal assembly
Here we present an experimental system that confirms the lock-and-key mechanism for Saturn ring particles and also allows us to interrogate the behavior of the elastic dipoles. To demonstrate the lock-and-key mechanism, we designed an experimental system (i) to have a 2D-like geometry as in the original simulation, (ii) to control independently the anchoring at the wall and at the particles, (iii) to allow for clear visualization of the process of particle moving towards topographical features and finding its preferred location, (iv) to investigate particles with Saturn rings but also particles with an associated point defect, and (v) to potentially allow for the exploration of many parameters and anchoring configurations. The experimental setup, the sandwich cell, is described in Chapter 2 (Fig. 3.1a). Typically, we observe the sample from a bird’s-eye-view (Fig. 3.1b).
The mismatched anchoring condition between the colloids (homeotropic) and the top and bottom cover slips (planar) causes the colloids to be repelled away from both (Fig. 3.1a). Elastic energy Eel = Ka and potential energy Eg = 43πa3(ρp −ρ)gT2, where T denotes
thickness of the cell. Given a = 7.5µm, K = 10−11N, T = 20−28µm, g = 9.8m/s2, and ρp−ρ = 0.99×103kg/m3 for our system. Using these values to estimate the relavant
energies, Eel = 7.5×10−17J = 1.8×104kBT, and Eg = 2.14×10−16J = 5.1×104kBT.
So gravity and elastic force compete with each other to determine the equilibrium position of the particle. The equilibrium height of a homeotropic particle can be calculated by balancing the elastic repulsion and gravity. The interaction between a dipole and a flat wall is calculated by the pair interaction between the dipole and an image dipole reflected across the flat wall:
Edp= 4πK
p2
r3 (3.1)
wherep=Aa2,Ais a numerical factor A≈2.04 [65], thus the force between two dipoles is
Fdp = 3 2πKA 2 2a r 4 (3.2) In a sandwich cell [84]: ΣF =Fbottom−Ftop−Fg = 3 2πKA 2 2a T −2δ 4 −3 2πKA 2 2a T+ 2δ 4 −4 3πa 3(ρ p−ρ)g = 0 (3.3)
where Fbottom and Ftop are the repulsion between the dipole and its mirror image located
at the bottom (z = 0) and the top (z = T), where δ is the deviation from the mid-plane (z= T2) due to gravity.
For a pair of Saturn rings: Fqp = 180πK c2 r6 (3.4)
wherec=−βa3 is the quadrupole moment,β ≈0.72, and the balance in the case of Saturn ring is then: ΣF =Fbottom−Ftop−Fg = 180πKβ2 a T −2δ 6 −180πKβ2 a T + 2δ 6 −4 3πa 3(ρ p−ρ)g = 0 (3.5)
Given the same material constants, we can calculate δ = 1.4−5.1µm for quadrupole. So although the particle is not situated exactly at the mid-plane(z = T2) between the top and bottom glass slide, they are very close to it due to the elastic repulsion. The dipole equilibrate much closer to mid-plane compare to Saturn ring, due to stronger repulsion from the glass cover slips.
The wavy wall forms a series of hills and wells, with amplitude 2Ameasured from the base of the well to the highest point on a hill. Because of strong homeotropic anchoring at the wavy wall, these features impose zones of splay and bend in this domain. In particular, the valleys are sites of converging splay, the hills are sites of diverging splay, and the inflection points are sites of maximum bend. The wavelength of the structure λcan be expressed in terms of the radius of curvature R and the amplitude A: λ= 4R
q
A R(2−
A
R) (Fig. 3.1c).
Therefore, λ and R are not independent for fixed A. Different aspects of the colloid-wall interaction are best described in terms of one or the other. For example, the range of the distortion is discussed in terms ofλ, and the splay field near the well is described in terms of
R. Throughout this study, unless specified otherwise, 2A= 10µm. The gentle undulations of this wall deform the surrounding director field but do not seed defect structures into the NLC. Deeper wells, such as those discussed later in this Chapter, are also characterized by their well depth ∆, and opening Ω.
As a result of their curvature, the gentle hills and dales of the walls become sites of splay distortion in the nematic director fieldn. The distortion in the bulk decays smoothly away from the wall without introducing any topological defects. It is instructive to consider a Fourier mode of the function defining the wall shape:
h(x) =Acos(kx) (3.6)
Under the one-constant approximation and assuming small gradients in the director field. In this limit,n can be expressed simply (Fig. 3.1d):
nx=Aksin(kx)e−ky ny = 1 (3.7)
wherenx, ny are components of the director.
This director field could be realized either near walls with shallow hills and dales, or, alternatively, in regions far enough from the wall that gradients in n are weak. This form suggests hills and dales are sites with the highest splay energy density, while the inflection points are sites with highest bend energy density. The distortion created by a particle with homeotropic anchoring always induces a defect, either an elastic dipole or a Saturn ring [107].
The alignment of a colloid-free cell is examined under crossed polarizers (Fig. 3.1e, f), which shows that the bulk liquid crystal is defect-free. The much brighter texture at 45◦- 135◦(Fig. 3.1e) compared to the 0◦-90◦ (Fig. 3.1f) also shows good planar alignment along the y-direction. The defects visible in Fig. 3.1e,f are only in the thin NLC film squeezed between the top of the wavy wall and the confining glass, a region which is not accessible to the colloids.
To investigate the lock-and-key mechanism predicted in simulation [23, 97], it is essential to have particles with Saturn rings. We achieve this by utilizing large silica particles (2a= 15
Figure 3.1: Schematics and the director field. (a) Schematic of the experimental setup (N denotes rubbing direction, T denotes thickness of the cell. (b) Experimental top view. (c) Schematic of the wall shape with relevant parameters: radius of curvatureR, amplitudeA, and wavelengthλ. (d) Simulated director field in the small-slope limit. (e-f) Cross polarized images of liquid crystal near the wavy wall with the long axis either (e) at 45o angle to the polarizer or (f) perpendicular to the polarizer. The scale bars are 20 µm
µm) whose size not only matched the size of the “lock”, but also is only slightly smaller than the total thickness of the LC cell (T = 20 - 25µm). It is known that confinement from the walls of a LC cell stabilizes Saturn rings against point defects [104]. In particular, the Saturn ring becomes the most stable configuration when the ratio between the radius of the particle and the distance between the particle surface and the wall is bigger than unity. In our case, given the polydispersity of the particles and the uncertainty in the cell thickness, this ratio was between 0.9 and 2.5.