Periodic dynamics, localization metastability, and elastic interaction of colloidal particles with confining surfaces and helicoidal structure of cholesteric liquid crystals

Nematic and cholesteric liquid crystals are three-dimensional fluids that possess long-range orientational ordering and can support both topological defects and chiral superstructures. Implications of this ordering remain unexplored even for simple dynamic processes such as the ones found in so-called “fall experiments,” or motion of a spherical inclusion under the effects of gravity. Here we show that elastic and surface anchoring interactions prompt periodic dynamics of colloidal microparticles in confined cholesterics when gravity acts along the helical axis. We explore elastic interactions between colloidal microparticles and confining surfaces as well as with an aligned ground-state helical structure of cholesterics for different sizes of spheres relative to the cholesteric pitch, demonstrating unexpected departures from Stokes-like behavior at very low Reynolds numbers. We characterize metastable localization of microspheres under the effects of elastic and surface anchoring periodic potential landscapes seen by moving spheres, demonstrating the important roles played by anchoring memory, confinement, and topological defect transformation. These experimental findings are consistent with the results of numerical modeling performed through minimizing the total free energy due to colloidal inclusions at different locations along the helical axis and with respect to the confining substrates. A potential application emerging from this work is colloidal sorting based on particle shapes and sizes.

Figure 1

(a) Solenoids arranged in a Cartesian aluminum frame mounted on an inverted microscope (not shown) coupled to an optical imaging system capable of both POM and 3PEF-PM imaging. (b) Colloid center-of-mass position along the cholesteric helical axis χ (which is aligned along the $z$ axis) vs relative rotation angle β for multiple colloid species (SPMB and GaN nanowire) in various configurations. The inset is a SEM image of a SPMB showing surface roughness. (c) Predicted dynamics of SPMB in a CLC cell when only gravitational and elastic particle-substrate interactions are present. Region I denotes a so-called “wind-up and lift-off,” Region II denotes a “fall” region, and Region III is the “levitation” regime.

Figure 2

Metastable states evident in a $P=5\mu \mathrm{m}$ CLC infused in a $h\approx 30\mu \mathrm{m}$ planar aligned cell. (Region I) A SPMB is wound up to the top cell surface using magnetic manipulation and then released. The inset plot shows a histogram of the number of counts per 3° bin over the range of Region II metastable states. The blue and black solid curves (Regions I and III respectively) shows the repulsive motion of the center of mass of the colloidal particle away from the bottom and top confining surfaces, respectively. Note that the colloidal particle shown is scaled to the cholesteric pitch corresponding to 2π rotation. The right vertical axis represents the distance traveled by the colloidal particle's center of mass from its original position at the top cell wall.

Figure 3

Elastic repulsion of a SPMB from cell surfaces in a $P=5\mu \mathrm{m}$ CLC infused into a $h\approx 30\mu \mathrm{m}$ planar aligned cell. (a) Analysis of the Region I top cell wall lift-off from Fig. 2. An exponential fit yields a time constant of ≈21.3 s. The inset is the time derivative of this fit. (b) Analysis of the Region I bottom cell wall lift-off from Fig. 2. An exponential fit yields a time constant of ≈11.9 s. The inset is the time derivative of this fitting curve. The dashed line indicates the “levitation height” where the elastic repulsion between the SPMB and the cell wall balance the buoyant and gravitational forces.

Figure 4

Metastable states evident in a $P=2.5\mu \mathrm{m}$ CLC infused in a $h\approx 60\mu \mathrm{m}$ planar cell. A SPMB is wound up to the top cell surface using magnetic manipulation and released, subsequently undergoing an elastically mediated lift-off from the cell surface followed by periodic dynamic metastability with an angular periodicity of 2π rad, or one cholesteric pitch. Unlike in the $P=5\mu \mathrm{m}$ CLC, periodic dynamic metastability is apparent even close to substrates. In the inset, the elastic interaction between the colloid and the top confining surface is highlighted by locating the intercept points at each cholesteric pitch [red (gray) circles] and plotting them with respect to time. Data between $0<\Delta \beta <12\pi \mathrm{rad}$ (Region I) are fitted with an exponential function of time constant of about 2870 s. A linear fit with a slope of 0.005 rad/s ($0.002\mu \mathrm{m}/\mathrm{s}$) is used for Region II, or $\Delta \beta \ge 10\pi \mathrm{rad}$. Note that the colloidal particle is scaled to cholesteric pitch.

Figure 5

Metastable states evident in a $P=10\mu \mathrm{m}$ CLC infused in a $h\approx 60\mu \mathrm{m}$ planar aligned cell. A SPMB is wound up to the top cell surface using magnetic manipulation and released, undergoing an elasticity-mediated lift-off from the cell surface followed by “metastable states” with an angular periodicity of 2π rad, or one cholesteric pitch. The strongly pronounced elastic repulsion of the SPMB from the top surface extends within about one cholesteric pitch. Note that the colloidal particle is scaled to cholesteric pitch.

Figure 6

Particle topography enhancement of metastable states in a $P=2.5\mu \mathrm{m}$ CLC infused in a $h\approx 60\mu \mathrm{m}$ planar aligned cell. A SPMB with a spherical protrusion of diameter $\sim \frac{1}{15}$ of that of the bead enhanced the metastable states. The inset shows the region of elastic repulsion from the cell wall. Note that the colloidal particle in the inset is scaled to pitch.

Figure 7

Fall experiments with SPMB dimer. A SPMB dimer in 5-μm-pitch CLC is allowed to fall in Region II in a $h\approx 30\mu \mathrm{m}$ cell. The dimer tends to situate at an angle of 10°–15° with respect to the plane of the CLC lamella. This dependence exhibits a distinct “double period” and the corresponding metastable states.

Figure 8

Fall experiments with uncoated and coated GaN nanowires in a $P=5\mu \mathrm{m}$ CLC infused in a $h\approx 60\mu \mathrm{m}$ cell. (a) Fall data for an uncoated GaN nanowire of nominal length of 10 μm and 350 nm diameter with planar anchoring. (b) SEM images of (top) an uncoated GaN nanowire and (bottom) Al-coated GaN nanowires with nanopyramidlike surface structures. (c) Fall data for an Al-coated GaN nanowire of nominal length of 10 μm and 350 nm diameter show that the metastable states are suppressed. (d) The data for Al-coated GaN nanowire after three weeks exhibit weakly pronounced metastable states.

Figure 9

Director configurations at the particle surface and in the bulk around it obtained numerically. Colloidal particle is in the center of the cell and far from confining walls. $L=2.5P$, $h=5P$, $R=0.1\mu \mathrm{m}$. (a) $P=R$. (b) $P=2R$. (c) $P=3R$. (d) $P=4R$. The color code represents $n_x^2$, and the isosurfaces of the reduced scalar-order parameter (depicting defects) $Q=0.25$ are shown using a green color.

Figure 10

The total free energy as a function of the distance $d$ of a microsphere from a planar cell wall. (a–c) Data for the radius of the particle $R=0.1\mu \mathrm{m}$. $L=2.5P$, $h=5P$. (d–f) Data for $R=0.25\mu \mathrm{m}$. The two sets of data correspond to the cholesteric pitch $P=R$, $2R$, $4R$, respectively, as marked on the figure parts. (d) $L=5P$, $h=10P$; (e, f) $L=2.5P$, $h=5P$.

Figure 11

Particle-induced cholesteric structures obtained numerically as a function of the distance $d$ to the wall (at the right). $R=0.1\mu \mathrm{m}$, $P=R$, $L=2.5P$, $h=5P$. (a) $d=1.3P$; (b) $d=1.5P$; (c) $d=1.8P$; (d) $d=2.1P$. The color code represents $n_x^2$, and the isosurfaces of the reduced scalar-order parameter $Q=0.25$ are shown in green.

Figure 12

Particle-induced cholesteric structures obtained numerically as a function of the distance $d$ to the wall (at the right). $R=0.1\mu \mathrm{m}$, $P=3R$, $L=2.5P$, $h=5P$. (a) $d=0.48P$; (b) $d=0.91P$; (c) $d=1.2P$; (d) $d=1.92P$. The color code represents $n_x^2$, and the isosurfaces of the reduced scalar-order parameter $Q=0.25$ are shown by green color.