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ATIF JAVED: Hey, 3016.

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Welcome to my final video
project for the class.

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My presentation will be
on Modeling and Energy

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Analysis of Liquid
Crystals using Mathematica.

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So liquid crystals
pervade many areas

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of technology in
our world today,

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including the screen you
may be watching this on.

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They're used in LCD displays,
thermometers, surfactants,

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polymers, detergents,
and even Kevlar.

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So the versatility of
this phase of matter

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is what definitely
piqued my interest

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and made me especially
enthusiastic about this project

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as an opportunity
also to explore

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this subject as a
scientist would,

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imposing my own
questions and then

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finding ways to answer them.

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So let's move on to
the subject matter.

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Liquid crystals are
the state of matter

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which retains properties from
the conventional crystalline

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solid and the
isotropic liquid state.

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So here you see an example
of the crystalline solid.

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LC molecules are
anisotropic, meaning

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that they exhibit properties
with different effects

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when oriented in
varied directions.

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And that orientation implies
a certain free energy.

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And that allows for a structure
with interesting properties,

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such as let's say birefringence.

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So here we see the
entire disorder

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of an isotropic liquid.

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And as we move right, we're
removing degrees of freedom

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to eventually form nematic LCs
that have orientational order,

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and eventually
smectic LCs, which

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have both orientational
and translational layering

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order, which, by the way, is
growing as a tool in research,

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especially in nanotechnology
and high-performance materials,

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which I thought was pretty cool.

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Everything ultimately
references back to structure.

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And here we can see the
increasing degrees of freedom

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and the characteristic
decreasing order parameter

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value as we transition
to a liquid.

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So the transitional
degree of freedom

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but restricted
rotational freedom

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is what will guide
our assumptions

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in developing a strong model.

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All three subdivisions of
the liquid crystal phase

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can exhibit phase transitions,
because of temperature

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fluctuations.

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So this will
motivate my analysis

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of the energy and entropy,
which are directly

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based on the order
of the system,

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and whose order is different
based on the temperature.

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So I will be using calamitic,
or rod-shaped, liquid crystals,

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and finish my analysis
with a statistical modeling

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of the energy and entropy
over 100,000 trials,

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and examine the
graphical representation.

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So since the molecules
are calamitic,

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I can think of them
basically as straight rods.

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Thus, the orientation is
only based on the beginning

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and the end of the rod.

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So because we'll be
looking at trends,

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we can simplify the problem
by first approaching it

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in two dimensions.

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And after a ton of
trial and error,

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I was able to put together
a graphic of these rods

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to shape the order of our
liquid crystal system,

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and demonstrate the type
of phase that we're in.

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And it's defined by
the average theta

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angle from the order director.

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And this is the random
generator for our LC

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system, where we can change the
theta value for our director.

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And from this, we can work on
extrapolating energies in them.

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So here, we show three
primary energy scenarios

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for non-polar crystals.

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And what can be considered
their relative energy

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is given by this
equation at the top.

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So by setting these conditions,
when molecules are parallel

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that gives a relative
interaction energy

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of negative 1; and
when perpendicular,

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a net energy of
0l and when angled

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45 degrees to the
director, yields an energy

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of about negative 0.5.

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So we can keep the
negative sign as we

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take the dot product
of the two directions

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to represent the fact that a
lower free energy corresponds

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to greater stability in either
translation or orientation.

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But we'll keep things positive
for our graphics in the future.

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So we can consider the molecules
to be on average equidistant

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from one another.

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And so we can grid our liquid
crystals in 2D space, like so.

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Then we can visualize how the
nearest neighbors, particularly

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the ones in directly
adjacent cells,

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affect the energy of the system.

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The energy effects
of molecules farther

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than these adjacent
ones can be considered

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negligible in comparison
to these closer ones.

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So we further our
model by taking

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the summation of energies
in the necessary locations,

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ultimately assigning a total
energy to each crystal that

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encompasses the energy
of that crystal resulting

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from orientation.

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Or other energy contributions
could exist, let's say,

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such as magnetic.

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But we'll focus on the energy
of order and the results

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from bonding and
lowering of entropy.

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Now subsequently, we
can create a collection

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of data angles
produced from 100,000

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trials of our random
liquid crystal generator,

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and obtain their energies from
the relationships described

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earlier.

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So from this, we can
develop a histogram,

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as shown here in the top
right-- in the top left.

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And we clearly see it is
relatively uniform Gaussian

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distribution, as expected.

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So moving forward,
we want to look

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to the third law
of thermodynamics

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to relate what we
are seeing to entropy

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using a fundamental law.

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And so we'll graph our
energy against entropy

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using the natural level omega
multiplied by the Boltzmann

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constant.

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Omega is the number
of microstates, i.e.,

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the number of
configurations, basically,

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that are still macroscopically
a liquid crystal.

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So by taking the log
of our first graph,

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we can obtain the second one
plotting energy versus entropy.

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And we see the curve
flattens by taking

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the natural log of the states.

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And so after normalizing,
we get the same graph,

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except slightly
wider and flatter.

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And we can see this contrast
in the lowest picture.

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But now for the most
interesting part of the problem,

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we know from the Gibbs
free energy equation

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and the first law
of thermodynamics

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that for an isolated
system such as this,

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that dU equals TdS, where
dU is the internal energy.

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And so rewriting this,
we see the relationship

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between temperature and
changing energy and entropy.

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And since we have
developed this information

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from our previous
graph, all we have to do

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is rotate the axes
of S against E,

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making the slope of the curve
equal to our temperature

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for whatever generated
liquid crystal.

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Although I was unable to create
the manipulate command that

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would demonstrate the
slope of dE versus dS

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as the temperature was
decreased, we would expect

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a narrower distribution of
beta angles, in other words

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an increase of the
order of the system

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by being more aligned
with the director,

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to compress the histogram and
result in a lower slope for T.

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And this is pretty cool.

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In my mind, it's a pretty
cool thermodynamic proof,

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that the narrower
distribution of rods

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equates to a lower
temperature, and conversely,

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a wider distribution
of rods around energies

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represents greater randomness,
and thus higher temperature.

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And so with
confidence, we can say

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that the order and temperature
are mutually dependent.

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And so I think the
beauty of this problem

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really stems from the
simplicity derived

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from taking appropriate
steps and realizing where

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the need for comparison
and trend study

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supersedes an unnecessarily
complex setup.

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I'm certain that that
kind of technique

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I'll be applying this in
the future in DMSC courses.

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And hopefully, I can
continue that in my track,

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in studying
nanomaterials as well.

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So to me, that's pretty awesome.

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And my thanks go out to
all of you, 3016 Fall 2012,

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for making the class like
quite an experience, granted

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in retrospect.

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But nonetheless, a
really cool endeavor.

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And my greatest appreciations
go out to Ray, Esther,

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and of course Professor Carter
for their tireless efforts.

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So thank you, guys.