LC : Nemantic, Smectic and Chiral Phase

I. NEMANTIC PHASE

"One of the most common Liquid Crystal (LC) phases is the nematic. The word nematic comes from the Greek νήμα (nema), which means "thread". This term originates from the thread-like topological defects observed in nematics, which are formally called 'disclinations'. Nematics also exhibit so-called "hedgehog" topological defects. In a nematic phase, the calamitic or rod-shaped organic molecules have no positional order, but they self-align to have long-range directional order with their long axes roughly parallel. Thus, the molecules are free to flow and their center of mass positions are randomly distributed as in a liquid, but still maintain their long-range directional order."
http://en.wikipedia.org/wiki/Nematic#Nematic_phase

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Mutually tangled colloidal knots and induced defect loops in nematic fields

"Colloidal dispersions in liquid crystals can serve as asoft-matter toolkit for the self-assembly of composite materials with pre-engineered properties and structures that are highly dependent on particle-induced topological defects. Here, we demonstrate that bulk and surface defects in nematic fluids can be patterned by tuning the topology of colloidal particles dispersed in them. In particular, by taking advantage of two-photon photopolymerization techniques to make knot-shaped microparticles, we show that the interplay of the topologies of the knotted particles, the nematic field and the induced defects leads to knotted, linked and other topologically non-trivial field configurations."
http://www.nature.com/nmat/journal/vaop/ncurrent/full/nmat3840.html

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Phase Separation of mixtures of isotropic liquids and liquid crystals

"Elasticity of LC-rich nematic phase affects the phase separated domain pattern significantly. For examples, we often observed the triangle and tear shaped droplets of isotropic phase. We demonstrated the two types of ordering processes, which we called isotropic and nematic spinoidal decompositions."

"When the mixture is quenched to the point, where is metastable for phase separation and unstable for nematic ordering. nematic transition occurs quickly than phase separation. In this case, the phase separation proceeds via nucleation and growth process."
http://statphys.scphys.kyoto-u.ac.jp/~araki/e_sim_pdlc.html

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Möbius strip ties liquid crystal in knots

• Liquid crystal are composed of long, thin, rod-like molecules which align themselves so they all point in the same direction. By controlling the alignment of these molecules, scientists can literally tie them in a knot.
• To do this, they simulated adding a micron sized silica particle - or colloid - to the liquid crystal. This disrupts the orientation of the liquid crystal molecules.
• For example, a colloid in the shape of a sphere will cause the liquid crystal molecules to align perpendicular to the surface of the sphere, a bit like a hedgehog’s spikes.
• Using a theoretical model, the University of Warwick scientists have taken this principle and extended it to colloids which have a knotted shape in the form a Möbius strip. http://www2.warwick.ac.uk/newsandevents/pressreleases/m246bius_strip_ties

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Liquid-crystal-mediated self-assembly at nanodroplet interfaces

Technological applications of liquid crystals have generally relied on control of molecular orientation at a surface or an interface. Such control has been achieved through topography, chemistry and the adsorption of monolayers or surfactants. The role of the substrate or interface has been to impart order over visible length scales and to confine the liquid crystal in a device. Here, we report results from a computational study of a liquid-crystal-based system in which the opposite is true: the liquid crystal is used to impart order on the interfacial arrangement of a surfactant.

http://www.nature.com/nature/journal/v485/n7396/full/nature11084.html

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II. SMECTIC PHASE

"Schematic of alignment in the smectic phases. The smectic A phase (left) has molecules organized into layers. In the smectic C phase (right), the molecules are tilted inside the layers. The smectic phases, which are found at lower temperatures than the nematic, form well-defined layers that can slide over one another in a manner similar to that of soap. The word "smetic" originates from the Latin word "smecticus", meaning cleaning, or having soap like properties.[21] The smectics are thus positionally ordered along one direction. In the Smectic A phase, the molecules are oriented along the layer normal, while in the Smectic C phase they are tilted away from the layer normal. These phases are liquid-like within the layers. There are many different smectic phases, all characterized by different types and degrees of positional and orientational order."
http://en.wikipedia.org/wiki/Nematic#Smectic_phases

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The Growth and Buckling of a Smectic Liquid Crystal Filament

Growth by permeation and drag-induced buckling instabilities have been observed in the dynamics of thin filaments in an isotropic-Smectic A ($I-S_A$) phase transition of liquid crystal fluid, and in lipid bilayer tubes evolving in a fluid medium. With motivation from the experiments with liquid crystal, we have been studying the dynamics of a growing elastic filament immersed in a Stokes fluid. By combining results from slender body theory, Green's function methods, and elasticity theory, we express the self-induced velocity of the filament as the nonlocal consequence of forces the filament exerts upon the incompressible fluid by its elastic response and growth.
http://math.nyu.edu/faculty/shelley/Fluids/Smectic/smectic.html

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III. CHIRAL PHASE

"The chiral nematic phase exhibits chirality (handedness). This phase is often called the cholesteric phase because it was first observed for cholesterol derivatives. Only chiral molecules (i.e., those that have no internal planes of symmetry) can give rise to such a phase. This phase exhibits a twisting of the molecules perpendicular to the director, with the molecular axis parallel to the director. The finite twist angle between adjacent molecules is due to their asymmetric packing"

http://en.wikipedia.org/wiki/Nematic#Chiral_phases

Test Results

A. One cw-flow colliding against one ccw-flow -> forming a central Stem-like flow:

B. Ellipsoidal vs. Spherical particles:

C. Spherical: Two circular flows (cw - ccw) quickly smoothing out into one circular flow:

D. Ellipsoidal: Out of the tessellated starting setting, flows start to emerge after a few seconds:

Spin Around & Aligning with the Hertz Model

By changing the 'Material model' in the Woo-DEM from 'Linear' to 'Hertz' it is possible to make the particles go round, be it Clock-Wise or Counter Clock-Wise, the direction is incidental.

“The Hertz-Mindlin model begins by assuming that contacting solids are isotropic and elastic, and that the representative dimensions of the contact area are very small compared to the various radii of curvature of the undeformed bodies. Another assumption of the Hertz-Mindlin model is that the two solids are perfectly smooth. Only the normal pressures that arise during contact are considered (the extensions of Hertz theory for the tangential component of traction will be discussed later). The Hertz-Mindlin contact-force-displacement law is nonlinear elastic, with path dependence and dissipation due to slip, and omits relative roll and torsion between the two spheres. Strictly speaking, the simplified contact force-displacement law is thermodynamically inconsistent (i.e., unphysical), since it permits energy generation at no cost.”
http://www.cflhd.gov/programs/techDevelopment/geotech/velocity/documents/05_chapter_3_numerical_modeling.pdf

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If you want you could check out the action yourself in Woo-DEM,  the program is free (http://www.launchpad.net/woo), and it’s just a 5 to 10 minute download & installation, copy these lines into your Ubuntu Terminal and you're good:

sudo add-apt-repository ppa:eudoxos/woo-daily
sudo apt-get update
sudo apt-get install python-woo

or use the singlecore-version:

sudo add-apt-repository ppa:eudoxos/woo-daily
sudo apt-get update
sudo apt-get install python-woo-singlecore

To start the program simply type in ‘woo’ and next you can press the F10-key to launch the control panels. Chose at the Preprocess-tab the preset plugin ‘EllGroup (woo.pre.ell2d)’, select Hertz (highlighted in yellow) to make ‘m go round and press the arrow in the lower right corner to process these  settings.

Press Play (>) at ‘Simulation’ to start all the action, you can switch ‘Trace particles’ on/off at the Trace-tab; for coloring the particles look at the Display-tab where you set ‘colorBy’ to ‘angVel’ and select ‘Z’ at ‘vecAxis’ or chose some other settings, have fun!

btw notice that the 'Restitution coefficient' (COR) is set to 0.7 if you boost it up to 0.99 then there's less disipation, and the particles will move more dynamic and for a longer time, but by doing so the subtle aligning interaction is gone and they become again chaotic, acting similar to the ‘Linear’ method ... and that *special* circular motion is gone. Hopefully this can be fixed.

For more info on Contact Models and Coefficient Of Restitution in Woo:
- http://woodem.eu/doc/theory/contact/index.html
- http://woodem.eu/doc/theory/contact/hertzian.html#coefficient-of-restitution

Packings of Ellipsoids

Unusually Dense Crystal Packings of Ellipsoids:

 http://cherrypit.princeton.edu/papers/paper-220.pdf (549 kb)

It matches with /// and ads a spiraling profile.

"We just stretched a sphere and suddenly things changed dramatically," said Torquato. "To me, it's remarkable that you can take this simple system with common candies and probe one of the deepest problems in condensed matter physics.”

http://edition.cnn.com/2004/TECH/science/02/16/science.candy.reut

Medium of Frictionless Elastic Ellipsoids

 Simulated by Václav Šmilauer with his program WOO-DEM:

http://woodem.eu - https://launchpad.net/woo

Ref.: Neighbor list collision-driven molecular dynamics simulation for nonspherical hard particles. - II. Applications to ellipses and ellipsoids. by Aleksandar Donev, Salvatore Torquato, Frank H. Stillinger 

http://cherrypit.princeton.edu/papers/paper-229.pdf

http://woodem.eu/doc/woo.dem.html#woo.dem.Cg2_Ellipsoid_Ellipsoid_L6Geom

Original movie: http://flux.doxos.eu/temp/ell2d-equal.avi (80 Mb)

Box-simulation: http://flux.doxos.eu/temp/ell2d-grp-mesh.zip 

Unpack and open with Paraview, import the whole group-package, press apply, and play …  

you can color them by velocity, angular velocity, use filters to show traces and whatnot. 

The ellipsoids of different sizes in this case are rendered as icosahedra.

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Before this group-simulation some individual 0ne on 0ne simulations were done to test the slipperiness (frictionless) of the 0vals and to see how they deflect. 

0val 'A' is set and 'B' comes flying in. The results are shown in the graphs below.

Left graph: the angle of 'B' changes - Right graph: 'B' has only one value.

'B' is coming along (+X) with differing (d); for every (d), there are three different (V2): 0, 0.1 and 0.2 radians - the lines coming from the right split in 3 parts depending on (V2).

Hitting a slippery floor; left graph, with (V1) varying; the two right tables (close-up): with density varying (changing density changes both mass and inertia); they both depend linearly on density, therefore have the same proportion and there is no difference in angle, but the angle is stretched out from (V) towards (\__/) and thus adding more linearity.

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Next up is doing some testing with all the possible settings and see if steady vortex-structures-patterns would show in this sort of medium, by using different ellipsoidal shapes, velocities and densities, as mentioned in Spiraling Figure 8.

Whirling Keratocytes

Fish cells swim circles around physics lab:

"... above this critical density the keratocytes moved in coherent groups. When the cell density was increased even further, the effect of collisions with the walls of the square-shaped incubator caused the cells to move in a whirl-like structure."

"The researchers were able to explain this phase transition by creating a simple model of the interaction between two keratocytes that is based on three forces acting at three different distances. At very short separations a repulsive force causes the keratocytes to move apart. At intermediate cell separations an attractive force causes the keratocytes to move together. At distances greater than about one cell diameter the force was set to zero. When used to simulate the behaviour of moving cells, this combination of simple forces caused the onset of collective motion at a critical density. This is unlike previous models, which assumed that the cells could respond to the motion of their neighbours."

http://physicsworld.com/cws/article/news/2006/nov/30/fish-cells-swim-circles-around-physics-lab

Vibrational projections forming a figure 8 pattern

The patterns generated by the kaleidophone (above) were described in more detail by the mathematician and physicist Jules Lissajous, later in the nineteenth century. He studied the curves created by the combination of two perpendicular vibrations. In order to visualize these curves he designed an instrument consisting of two little mirrors mounted on tuning forks. If the tuning forks sound in a ‘pure’ interval, the resulting image is static and symmetric. If the interval is not ‘pure’, the result is a chaos of lines. Smaller interferences between the two frequencies translate into subtle movements of the generated patterns. (https://en.wikipedia.org/wiki/Lissajous_curve)

(http://www.uh.edu/engines/musicforawhile/musictext.htm)

(http://www.luisaph.com/s/index.php/the-harmonics-series)

Cathode Ray Tube (CRT): If we allow one AC signal to deflect the beam up and down (connect that AC voltage source to the "vertical" deflection plates) and another AC signal to deflect the beam left and right (using the other pair of deflection plates), patterns will be produced on the screen of the CRT indicative of the ratio of these two AC frequencies. These Lissajous figures are a common means of comparative frequency measurement in electronics. (http://www.vias.org/feee/acmeter_03.html)

Flexible Spiral Structure

Here's a link to an interesting TED-talk by biologist Diane Kelly on the anatomy and function of vertebrate penises:

http://www.ted.com/talks/diane_kelly_what_we_didn_t_know_about_penis_anatomy.html

At about 7 min into the talk she explains how a spiral structure is very flexible as it can be Extended and Bend versus a cross-layered structure which is more stiff and less flexible. This adds to the idea that the inner-structure should follow a spiralling flow ...

Z