Equilibrium Configurations and Energetics of Point Defects in Two-Dimensional Colloidal Crystals

The two dimensional motion of a generally non-circular non-uniform cylinder on a flat horizontal surface is investigated. Assuming that the cylinder does not slip, energy conservation is used to study the motion in general. Points of returns, and small oscillations around equilibrium configuration are studied. As examples, cylinders are studied for which the cross section is an ellipse, with the center of mass at the center of the ellipse or at a focal point, and the frequencies of small oscillations around their equilibrium configurations are found. The conditions for losing contact or sliding are also investigated. Finally, the motion is studied in more detail for the case of a nearly circular cylinder.

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Fabrication of Large-Area Two-Dimensional Colloidal Crystals

Angewandte Chemie International Edition ◽

10.1002/anie.201105439 ◽

2012 ◽

Vol 51 (25) ◽

pp. 6117-6120 ◽

Cited By ~ 121

Author(s):

Jian-Tao Zhang ◽

Luling Wang ◽

Daniel N. Lamont ◽

Sachin S. Velankar ◽

Sanford A. Asher

Keyword(s):

Colloidal Crystals ◽

Two Dimensional ◽

Large Area

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Surface, size and topological effects for some nematic equilibria on rectangular domains

Mathematics and Mechanics of Solids ◽

10.1177/1081286520902507 ◽

2020 ◽

Vol 25 (5) ◽

pp. 1101-1123 ◽

Cited By ~ 1

Author(s):

Lidong Fang ◽

Apala Majumdar ◽

Lei Zhang

Keyword(s):

Aspect Ratio ◽

Point Defects ◽

Three Dimensional ◽

Two Dimensional ◽

Bifurcation Diagrams ◽

Variable Formula ◽

Limiting Profile ◽

Switching Dynamics ◽

Line Defects ◽

Carry Over

We study nematic equilibria on rectangular domains, in a reduced two-dimensional Landau–de Gennes framework. These reduced equilibria carry over to the three-dimensional framework at a special temperature. There is one essential model variable, [Formula: see text], which is a geometry-dependent and material-dependent variable. We compute the limiting profiles exactly in two distinguished limits: the [Formula: see text] 0 limit relevant for macroscopic domains and the [Formula: see text] limit relevant for nanoscale domains. The limiting profile has line defects near the shorter edges in the [Formula: see text] limit, whereas we observe fractional point defects in the [Formula: see text] 0 limit. The analytical studies are complemented by some bifurcation diagrams for these reduced equilibria as a function of [Formula: see text] and the rectangular aspect ratio. We also introduce the concept of ‘non-trivial’ topologies and study the relaxation of non-trivial topologies to trivial topologies mediated via point and line defects, with potential consequences for non-equilibrium phenomena and switching dynamics.

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