trunk/5cb/5CB.tex

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\title{Nitrile vibrations as reporters of field-induced phase
  transitions in liquid crystals}  
\author{James M. Marr} 
\author{J. Daniel Gezelter} 
\email{gezelter@nd.edu} 
\affiliation[University of Notre Dame]{251 Nieuwland Science Hall\\ 
Department of Chemistry and Biochemistry\\ 
University of Notre Dame\\ 
Notre Dame, Indiana 46556}

\date{\today}

\begin{document}

\maketitle 

\begin{doublespace}

\begin{abstract}
  The behavior of the spectral lineshape of the nitrile group in
  4-Cyano-4'-pentylbiphenyl (5CB) in response to an applied electric
  field has been simulated using both classical molecular dynamics
  simulations and {\it ab initio} quantum mechanical calculations of
  liquid-like clusters.  This nitrile group is a well-known reporter
  of local field effects in other condensed phase settings, and our
  simulations suggest that there is a significant response when 5CB
  liquids are exposed to a relatively large external field.  However,
  this response is due largely to the field-induced phase transition.
  We observe a peak shift to the red of nearly 40
  cm\textsuperscript{-1}. These results indicate that applied fields
  can play a role in the observed peak shape and position even if
  those fields are significantly weaker than the local electric fields
  that are normally felt within polar liquids.
\end{abstract}

\newpage

\section{Introduction}

The fundamental characteristic of liquid crystal mesophases is that
they maintain some degree of orientational order while translational
order is limited or absent. This orientational order produces a
complex direction-dependent response to external perturbations like
electric fields and mechanical distortions.  The anisotropy of the
macroscopic phases originates in the anisotropy of the constituent
molecules, which typically have highly non-spherical structures with a
significant degree of internal rigidity.  In nematic phases, rod-like
molecules are orientationally ordered with isotropic distributions of
molecular centers of mass, while in smectic phases, the molecules
arrange themselves into layers with their long (symmetry) axis normal
($S_{A}$) or tilted ($S_{C}$) with respect to the layer planes.

The behavior of the $S_{A}$ phase can be partially explained with
models mainly based on geometric factors and van der Waals
interactions.  However, these simple models are insufficient to
describe liquid crystal phases which exhibit more complex polymorphic
nature.  X-ray diffraction studies have shown that the ratio between
lamellar spacing ($s$) and molecular length ($l$) can take on a wide
range of values.\cite{Gray:1984hc,Leadbetter:1976vf,Hardouin:1980yq}
Typical $S_{A}$ phases have $s/l$ ratios on the order of $0.8$, while
for some compounds, e.g. the 4-alkyl-4'-cyanobiphenyls, the $s/l$
ratio is on the order of $1.4$.  Molecules which form $S_{A}$ phases
can exhibit a wide variety of subphases like monolayers ($S_{A1}$),
uniform bilayers ($S_{A2}$), partial bilayers ($S_{\tilde A}$) as well
as interdigitated bilayers ($S_{A_{d}}$), and often have a terminal
cyano or nitro group.  In particular lyotropic liquid crystals (those
exhibiting liquid crystal phase transition as a function of water
concentration) often have polar head groups or zwitterionic charge
separated groups that result in strong dipolar
interactions.\cite{Collings97} Because of their versatile polymorphic
nature, polar liquid crystalline materials have important
technological applications in addition to their immense relevance to
biological systems.\cite{Collings97}

Experimental studies by Levelut {\it et al.}~\cite{Levelut:1981eu}
revealed that terminal cyano or nitro groups usually induce permanent
longitudinal dipole moments on the molecules.

Liquid crystalline materials with dipole moments located at the ends
of the molecules have important applications in display technologies
in addition to their relevance in biological systems.\cite{LCapp}

Many of the technological applications of the lyotropic mesogens
manipulate the orientation and structuring of the liquid crystal
through application of local electric fields.\cite{?}
Macroscopically, this restructuring is visible in the interactions the
bulk phase has with scattered or transmitted light.\cite{?}  

4-Cyano-4'-pentylbiphenyl (5CB), has been a model for field-induced
phase changes due to the known electric field response of the liquid
crystal\cite{Hatta:1991ee}.  It was discovered (along with three
similar compounds) in 1973 in an effort to find a LC that had a
melting point near room temperature.\cite{Gray:1973ca} It's known to
have a crystalline to nematic phase transition at 18 C and a nematic
to isotropic transition at 35 C.\cite{Gray:1973ca}

Nitrile groups can serve as very precise electric field reporters via
their distinctive Raman and IR signatures.\cite{Boxer:2009xw} The
triple bond between the nitrogen and the carbon atom is very sensitive
to local field changes and is observed to have a direct impact on the
peak position within the spectrum.  The Stark shift in the spectrum
can be quantified and mapped into a field value that is impinging upon
the nitrile bond. This has been used extensively in biological systems
like proteins and enzymes.\cite{Tucker:2004qq,Webb:2008kn} 

To date, the nitrile electric field response of
4-Cyano-4'-n-alkylbiphenyl liquid crystals has not been investigated.
While macroscopic electric fields applied across macro electrodes show
the phase change of the molecule as a function of electric
field,\cite{Lim:2006xq} the effect of a microscopic field application
has not been probed. These previous studies have shown the nitrile
group serves as an excellent indicator of the molecular orientation
within the field applied. Blank showed the 180 degree change in field
direction could be probed with the nitrile peak intensity as it
decreased and increased with molecule alignment in the
field.\cite{Lee:2006qd,Leyte:97}

While these macroscopic fields worked well at showing the bulk
response, the atomic scale response has not been studied. With the
advent of nano-electrodes and coupling them with atomic force
microscopy, control of electric fields applied across nanometer
distances is now possible\cite{citation1}. This application of
nanometer length is interesting in the case of a nitrile group on the
molecule. While macroscopic fields are insufficient to cause a Stark
effect, small fields across nanometer-sized gaps are of sufficient
strength. If one were to assume a gap of 5 nm between a lower
electrode having a nanoelectrode placed near it via an atomic force
microscope, a field of 1 V applied across the electrodes would
translate into a field of 2x10\textsuperscript{8} $\frac{V}{M}$. This
field is theoretically strong enough to cause a phase change from
isotropic to nematic, as well as Stark tuning of the nitrile
bond. This should be readily visible experimentally through Raman or
IR spectroscopy.

Herein, we show the computational investigation of these electric field effects through atomistic simulations. These are then coupled with ab intio and classical spectrum calculations to predict changes experiments should be able to replicate.

\section{Computational Details}
The force field was mainly taken from Guo et al.\cite{Zhang:2011hh} A
deviation from this force field was made to create a rigid body from
the phenyl rings. Bond distances within the rigid body were taken from
equilibrium bond distances. While the phenyl rings were held rigid,
bonds, bends, torsions and inversion centers still included the rings.

Simulations were with boxes of 270 molecules locked at experimental
densities with periodic boundaries. The molecules were thermalized
from 0 kelvin to 300 kelvin. To equilibrate, each was first run in NVT
for 1 ns. This was followed by NVE for simulations used in the data
collection.

External electric fields were applied in a simplistic charge-field
interaction. Forces were calculated by multiplying the electric field
being applied by the charge at each atom. For the potential, the
origin of the box was used as a point of reference. This allows for a
potential value to be added to each atom as the molecule move in space
within the box. Fields values were applied in a manner representing
those that would be applable in an experimental set-up. The assumed
electrode seperation was 5 nm and the field was input as
$\frac{V}{\text{\AA}}$. The three field environments were, 1) no field
  applied, 2) 0.01 $\frac{V}{\text{\AA}}$ (0.5 V) and 3) 0.024
    $\frac{V}{\text{\AA}}$ (1.2 V). Each field was applied in the
    Z-axis of the simulation box. For the simplicity of this paper,
    each field will be called zero, partial and full, respectively.

For quantum calculation of the nitrile bond frequency, Gaussian 09 was
used. A single 5CB molecule was selected for the center of the
cluster. For effects from molecules located near the chosen nitrile
group, parts of molecules nearest to the nitrile group were
included. For the body not including the tail, any atom within 6~\AA 
of the midpoint of the nitrile group was included. For the tail
structure, the whole tail was included if a tail atom was within 4~\AA 
of the midpoint. If the tail did not include any atoms from the ring
structure, it was considered a propane molecule and included as
such. Once the clusters were generated, input files were created that
stretched the nitrile bond along its axis from 0.87 to 1.52~\AA at
increments of 0.05~\AA. This generated 13 single point energies to be
calculated. The Gaussian files were run with B3LYP/6-311++G(d,p) with
no other keywords for the zero field simulation. Simulations with
fields applied included the keyword ''Field=Z+5'' to match the
external field applied in molecular dynamic runs. Once completed, the central nitrile bond frequency
was calculated with a Morse fit. Using this fit and the solved energy
levels for a Morse oscillator, the frequency was found.

Classical nitrile bond frequencies were found by replacing the rigid
cyanide bond with a flexible Morse oscillator bond
($r_0= 1.157437$ \AA , $D_0 = 212.95$ and
$\beta = 2.67566$) . Once replaced, the
systems were allowed to re-equilibrate in NVT for 100 ps. After
re-equilibration, the system was run in NVE for 20 ps with a snapshot
spacing of 1 fs. These snapshot were then used in bond correlation
calculation to find the decay structure of the bond in time using the
average bond displacement in time,
\begin{equation}
C(t) = \langle \left(r(t) - r_0 \right) \cdot \left(r(0) - r_0 \right) \rangle
\end{equation}
%
where $r_0$ is the equilibrium bond distance and $r(t)$ is the
instantaneous bond displacement at time $t$. Once calculated,
smoothing was applied by adding an exponential decay on top of the
decay with a $\tau$ of 3000 (have to check this). Further smoothing
was applied by padding 20,000 zeros on each side of the symmetric
data. This was done five times by allowing the systems to run 1 ns
with a rigid bond followed by an equilibrium run with the bond
switched back on and the short production run.

\section{Results}

In order to characterize the orientational ordering of the system, the
primary quantity of interest is the nematic (orientational) order
parameter. This is determined using the tensor
\begin{equation}
Q_{\alpha \beta} = \frac{1}{2N} \sum_{i=1}^{N} \left(3 \hat{e}_{i
    \alpha} \hat{e}_{i \beta} - \delta_{\alpha \beta} \right)
\end{equation}
where $\alpha, \beta = x, y, z$, and $\hat{e}_i$ is the molecular
end-to-end unit vector for molecule $i$. The nematic order parameter
$S$ is the largest eigenvalue of $Q_{\alpha \beta}$, and the
corresponding eigenvector defines the director axis for the phase.
$S$ takes on values close to 1 in highly ordered phases, but falls to
zero for isotropic fluids. In the context of 5CB, this value would be
close to zero for its isotropic phase and raise closer to one as it
moved to the nematic and crystalline phases.

This value indicates phases changes at temperature boundaries but also
when a phase changes occurs due to external field applications. In
Figure 1, this phase change can be clearly seen over the course of 60
ns. Each system starts with an ordering paramter near 0.1 to 0.2,
which is an isotropic phase. Over the course 10 ns, the full external field
causes a shift in the ordering of the system to 0.5, the nematic phase
of the liquid crystal. This change is consistent over the full 50 ns
with no drop back into the isotropic phase. This change is clearly
field induced and stable over a long period of simulation time.

Interestingly, the field that is needed to switch the phase of 5CB
macroscopically is larger than 1 V. However, in this case, only a
voltage of 1.2 V was need to induce a phase change. This is impart due
to the distance the field is being applied across. At such a small
distance, the field is much larger than the macroscopic and thus
easily induces a field dependent phase change.

This change in phase was followed by two courses of further
simulation. First, was replacement of the static nitrile bond with a
morse oscillator bond. This was then simulated for a period of time
and a classical spetrum was calculated. Second, ab intio calcualtions were performe to investigate
if the phase change caused any change spectrum from quantum
effects.

In respect to the classical calculations, it was first seen if previous
studies of nitriles within water and neat simulation done by Cho
et. al. could be used for the spectrum.

After Gaussian calculations were performed on a set of snapshots, any 

\begin{figure}
  \includegraphics[trim = 5mm 10mm 3mm 10mm, clip, width=3.25in]{P2}
  \caption{Ordering of each external field application over the course
    of 60 ns with a sampling every 100 ps. Each trajectory was started
  after equilibration with zero field applied.}
  \label{fig:orderParameter}
\end{figure}

\begin{figure}
  \includegraphics[width=3.25in]{2Spectra}
  \caption{The classically calculated nitrile bond spetrum for no
    external field application (black) and full external field
    application (red)}
  \label{fig:twoSpectra}
\end{figure}

\begin{figure}
        \centering
        \begin{subfigure}[b]{0.3\textwidth}
                \includegraphics[trim = 100mm 20mm 50mm 20mm, clip, width=2in]{0x00V_Elip-1}
                \label{fig:zeroField}
        \end{subfigure}
        \begin{subfigure}[b]{0.3\textwidth}
                \includegraphics[trim = 100mm 20mm 50mm 20mm, clip, width=2in]{0x50V_Elip-1}
                \label{fig:partialField}
        \end{subfigure}
        \begin{subfigure}[b]{0.3\textwidth}
                \includegraphics[trim = 100mm 20mm 50mm 20mm, clip, width=2in]{1x20V_Elip-1}
                \label{fig:fullField}
        \end{subfigure}
        \caption{Ellipsoid reprsentation of 5CB at three different
          field strengths, Zero Field (Left), Partial Field (Middle), and Full
        Field (Right)}\label{fig:Cigars}
\end{figure}

\section{Discussion}

\section{Conclusions}
\newpage

\bibliography{5CB}

\end{doublespace}
\end{document}
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40
41
42			\title{Nitrile vibrations as reporters of field-induced phase
43			transitions in liquid crystals}
44			\author{James M. Marr}
45			\author{J. Daniel Gezelter}
46			\email{gezelter@nd.edu}
47			\affiliation[University of Notre Dame]{251 Nieuwland Science Hall\\
48			Department of Chemistry and Biochemistry\\
49			University of Notre Dame\\
50			Notre Dame, Indiana 46556}
51
52			\date{\today}
53
54			\begin{document}
55
56			\maketitle
57
58			\begin{doublespace}
59
60			\begin{abstract}
61			The behavior of the spectral lineshape of the nitrile group in
62			4-Cyano-4'-pentylbiphenyl (5CB) in response to an applied electric
63			field has been simulated using both classical molecular dynamics
64			simulations and {\it ab initio} quantum mechanical calculations of
65			liquid-like clusters. This nitrile group is a well-known reporter
66			of local field effects in other condensed phase settings, and our
67			simulations suggest that there is a significant response when 5CB
68			liquids are exposed to a relatively large external field. However,
69			this response is due largely to the field-induced phase transition.
70			We observe a peak shift to the red of nearly 40
71			cm\textsuperscript{-1}. These results indicate that applied fields
72			can play a role in the observed peak shape and position even if
73			those fields are significantly weaker than the local electric fields
74			that are normally felt within polar liquids.
75			\end{abstract}
76
77			\newpage
78
79			\section{Introduction}
80
81			The fundamental characteristic of liquid crystal mesophases is that
82			they maintain some degree of orientational order while translational
83			order is limited or absent. This orientational order produces a
84			complex direction-dependent response to external perturbations like
85			electric fields and mechanical distortions. The anisotropy of the
86			macroscopic phases originates in the anisotropy of the constituent
87			molecules, which typically have highly non-spherical structures with a
88			significant degree of internal rigidity. In nematic phases, rod-like
89			molecules are orientationally ordered with isotropic distributions of
90			molecular centers of mass, while in smectic phases, the molecules
91			arrange themselves into layers with their long (symmetry) axis normal
92			($S_{A}$) or tilted ($S_{C}$) with respect to the layer planes.
93
94			The behavior of the $S_{A}$ phase can be partially explained with
95			models mainly based on geometric factors and van der Waals
96			interactions. However, these simple models are insufficient to
97			describe liquid crystal phases which exhibit more complex polymorphic
98			nature. X-ray diffraction studies have shown that the ratio between
99			lamellar spacing ($s$) and molecular length ($l$) can take on a wide
100			range of values.\cite{Gray:1984hc,Leadbetter:1976vf,Hardouin:1980yq}
101			Typical $S_{A}$ phases have $s/l$ ratios on the order of $0.8$, while
102			for some compounds, e.g. the 4-alkyl-4'-cyanobiphenyls, the $s/l$
103			ratio is on the order of $1.4$. Molecules which form $S_{A}$ phases
104			can exhibit a wide variety of subphases like monolayers ($S_{A1}$),
105			uniform bilayers ($S_{A2}$), partial bilayers ($S_{\tilde A}$) as well
106			as interdigitated bilayers ($S_{A_{d}}$), and often have a terminal
107			cyano or nitro group. In particular lyotropic liquid crystals (those
108			exhibiting liquid crystal phase transition as a function of water
109			concentration) often have polar head groups or zwitterionic charge
110			separated groups that result in strong dipolar
111			interactions.\cite{Collings97} Because of their versatile polymorphic
112			nature, polar liquid crystalline materials have important
113			technological applications in addition to their immense relevance to
114			biological systems.\cite{Collings97}
115
116			Experimental studies by Levelut {\it et al.}~\cite{Levelut:1981eu}
117			revealed that terminal cyano or nitro groups usually induce permanent
118			longitudinal dipole moments on the molecules.
119
120			Liquid crystalline materials with dipole moments located at the ends
121			of the molecules have important applications in display technologies
122			in addition to their relevance in biological systems.\cite{LCapp}
123
124			Many of the technological applications of the lyotropic mesogens
125			manipulate the orientation and structuring of the liquid crystal
126			through application of local electric fields.\cite{?}
127			Macroscopically, this restructuring is visible in the interactions the
128			bulk phase has with scattered or transmitted light.\cite{?}
129
130			4-Cyano-4'-pentylbiphenyl (5CB), has been a model for field-induced
131			phase changes due to the known electric field response of the liquid
132			crystal\cite{Hatta:1991ee}. It was discovered (along with three
133			similar compounds) in 1973 in an effort to find a LC that had a
134			melting point near room temperature.\cite{Gray:1973ca} It's known to
135			have a crystalline to nematic phase transition at 18 C and a nematic
136			to isotropic transition at 35 C.\cite{Gray:1973ca}
137
138			Nitrile groups can serve as very precise electric field reporters via
139			their distinctive Raman and IR signatures.\cite{Boxer:2009xw} The
140			triple bond between the nitrogen and the carbon atom is very sensitive
141			to local field changes and is observed to have a direct impact on the
142			peak position within the spectrum. The Stark shift in the spectrum
143			can be quantified and mapped into a field value that is impinging upon
144			the nitrile bond. This has been used extensively in biological systems
145			like proteins and enzymes.\cite{Tucker:2004qq,Webb:2008kn}
146
147			To date, the nitrile electric field response of
148			4-Cyano-4'-n-alkylbiphenyl liquid crystals has not been investigated.
149			While macroscopic electric fields applied across macro electrodes show
150			the phase change of the molecule as a function of electric
151			field,\cite{Lim:2006xq} the effect of a microscopic field application
152			has not been probed. These previous studies have shown the nitrile
153			group serves as an excellent indicator of the molecular orientation
154			within the field applied. Blank showed the 180 degree change in field
155			direction could be probed with the nitrile peak intensity as it
156			decreased and increased with molecule alignment in the
157			field.\cite{Lee:2006qd,Leyte:97}
158
159			While these macroscopic fields worked well at showing the bulk
160			response, the atomic scale response has not been studied. With the
161			advent of nano-electrodes and coupling them with atomic force
162			microscopy, control of electric fields applied across nanometer
163			distances is now possible\cite{citation1}. This application of
164			nanometer length is interesting in the case of a nitrile group on the
165			molecule. While macroscopic fields are insufficient to cause a Stark
166			effect, small fields across nanometer-sized gaps are of sufficient
167			strength. If one were to assume a gap of 5 nm between a lower
168			electrode having a nanoelectrode placed near it via an atomic force
169			microscope, a field of 1 V applied across the electrodes would
170			translate into a field of 2x10\textsuperscript{8} $\frac{V}{M}$. This
171			field is theoretically strong enough to cause a phase change from
172			isotropic to nematic, as well as Stark tuning of the nitrile
173			bond. This should be readily visible experimentally through Raman or
174			IR spectroscopy.
175
176			Herein, we show the computational investigation of these electric field effects through atomistic simulations. These are then coupled with ab intio and classical spectrum calculations to predict changes experiments should be able to replicate.
177
178			\section{Computational Details}
179			The force field was mainly taken from Guo et al.\cite{Zhang:2011hh} A
180			deviation from this force field was made to create a rigid body from
181			the phenyl rings. Bond distances within the rigid body were taken from
182			equilibrium bond distances. While the phenyl rings were held rigid,
183			bonds, bends, torsions and inversion centers still included the rings.
184
185			Simulations were with boxes of 270 molecules locked at experimental
186			densities with periodic boundaries. The molecules were thermalized
187			from 0 kelvin to 300 kelvin. To equilibrate, each was first run in NVT
188			for 1 ns. This was followed by NVE for simulations used in the data
189			collection.
190
191			External electric fields were applied in a simplistic charge-field
192			interaction. Forces were calculated by multiplying the electric field
193			being applied by the charge at each atom. For the potential, the
194			origin of the box was used as a point of reference. This allows for a
195			potential value to be added to each atom as the molecule move in space
196	jmarr	4008	within the box. Fields values were applied in a manner representing
197			those that would be applable in an experimental set-up. The assumed
198			electrode seperation was 5 nm and the field was input as
199			$\frac{V}{\text{\AA}}$. The three field environments were, 1) no field
200			applied, 2) 0.01 $\frac{V}{\text{\AA}}$ (0.5 V) and 3) 0.024
201			$\frac{V}{\text{\AA}}$ (1.2 V). Each field was applied in the
202	jmarr	4017	Z-axis of the simulation box. For the simplicity of this paper,
203			each field will be called zero, partial and full, respectively.
204	gezelter	4007
205			For quantum calculation of the nitrile bond frequency, Gaussian 09 was
206			used. A single 5CB molecule was selected for the center of the
207			cluster. For effects from molecules located near the chosen nitrile
208			group, parts of molecules nearest to the nitrile group were
209	jmarr	4008	included. For the body not including the tail, any atom within 6~\AA
210	gezelter	4007	of the midpoint of the nitrile group was included. For the tail
211	jmarr	4008	structure, the whole tail was included if a tail atom was within 4~\AA
212	gezelter	4007	of the midpoint. If the tail did not include any atoms from the ring
213			structure, it was considered a propane molecule and included as
214			such. Once the clusters were generated, input files were created that
215	jmarr	4008	stretched the nitrile bond along its axis from 0.87 to 1.52~\AA at
216	gezelter	4007	increments of 0.05~\AA. This generated 13 single point energies to be
217			calculated. The Gaussian files were run with B3LYP/6-311++G(d,p) with
218	jmarr	4008	no other keywords for the zero field simulation. Simulations with
219			fields applied included the keyword ''Field=Z+5'' to match the
220			external field applied in molecular dynamic runs. Once completed, the central nitrile bond frequency
221	gezelter	4007	was calculated with a Morse fit. Using this fit and the solved energy
222			levels for a Morse oscillator, the frequency was found.
223
224			Classical nitrile bond frequencies were found by replacing the rigid
225	jmarr	4008	cyanide bond with a flexible Morse oscillator bond
226			($r_0= 1.157437$ \AA , $D_0 = 212.95$ and
227			$\beta = 2.67566$) . Once replaced, the
228	gezelter	4007	systems were allowed to re-equilibrate in NVT for 100 ps. After
229			re-equilibration, the system was run in NVE for 20 ps with a snapshot
230			spacing of 1 fs. These snapshot were then used in bond correlation
231			calculation to find the decay structure of the bond in time using the
232			average bond displacement in time,
233			\begin{equation}
234			C(t) = \langle \left(r(t) - r_0 \right) \cdot \left(r(0) - r_0 \right) \rangle
235			\end{equation}
236			%
237			where $r_0$ is the equilibrium bond distance and $r(t)$ is the
238			instantaneous bond displacement at time $t$. Once calculated,
239			smoothing was applied by adding an exponential decay on top of the
240			decay with a $\tau$ of 3000 (have to check this). Further smoothing
241			was applied by padding 20,000 zeros on each side of the symmetric
242			data. This was done five times by allowing the systems to run 1 ns
243			with a rigid bond followed by an equilibrium run with the bond
244			switched back on and the short production run.
245
246			\section{Results}
247
248			In order to characterize the orientational ordering of the system, the
249			primary quantity of interest is the nematic (orientational) order
250			parameter. This is determined using the tensor
251			\begin{equation}
252			Q_{\alpha \beta} = \frac{1}{2N} \sum_{i=1}^{N} \left(3 \hat{e}_{i
253			\alpha} \hat{e}_{i \beta} - \delta_{\alpha \beta} \right)
254			\end{equation}
255			where $\alpha, \beta = x, y, z$, and $\hat{e}_i$ is the molecular
256			end-to-end unit vector for molecule $i$. The nematic order parameter
257			$S$ is the largest eigenvalue of $Q_{\alpha \beta}$, and the
258			corresponding eigenvector defines the director axis for the phase.
259			$S$ takes on values close to 1 in highly ordered phases, but falls to
260	jmarr	4017	zero for isotropic fluids. In the context of 5CB, this value would be
261			close to zero for its isotropic phase and raise closer to one as it
262			moved to the nematic and crystalline phases.
263	gezelter	4007
264	jmarr	4017	This value indicates phases changes at temperature boundaries but also
265			when a phase changes occurs due to external field applications. In
266			Figure 1, this phase change can be clearly seen over the course of 60
267			ns. Each system starts with an ordering paramter near 0.1 to 0.2,
268			which is an isotropic phase. Over the course 10 ns, the full external field
269			causes a shift in the ordering of the system to 0.5, the nematic phase
270			of the liquid crystal. This change is consistent over the full 50 ns
271			with no drop back into the isotropic phase. This change is clearly
272			field induced and stable over a long period of simulation time.
273
274			Interestingly, the field that is needed to switch the phase of 5CB
275			macroscopically is larger than 1 V. However, in this case, only a
276			voltage of 1.2 V was need to induce a phase change. This is impart due
277			to the distance the field is being applied across. At such a small
278			distance, the field is much larger than the macroscopic and thus
279			easily induces a field dependent phase change.
280
281			This change in phase was followed by two courses of further
282			simulation. First, was replacement of the static nitrile bond with a
283			morse oscillator bond. This was then simulated for a period of time
284			and a classical spetrum was calculated. Second, ab intio calcualtions were performe to investigate
285			if the phase change caused any change spectrum from quantum
286			effects.
287
288			In respect to the classical calculations, it was first seen if previous
289			studies of nitriles within water and neat simulation done by Cho
290			et. al. could be used for the spectrum.
291
292			After Gaussian calculations were performed on a set of snapshots, any
293
294	jmarr	4013	\begin{figure}
295			\includegraphics[trim = 5mm 10mm 3mm 10mm, clip, width=3.25in]{P2}
296	jmarr	4017	\caption{Ordering of each external field application over the course
297			of 60 ns with a sampling every 100 ps. Each trajectory was started
298			after equilibration with zero field applied.}
299	jmarr	4013	\label{fig:orderParameter}
300			\end{figure}
301	gezelter	4007
302	jmarr	4013	\begin{figure}
303			\includegraphics[width=3.25in]{2Spectra}
304	jmarr	4017	\caption{The classically calculated nitrile bond spetrum for no
305			external field application (black) and full external field
306			application (red)}
307	jmarr	4013	\label{fig:twoSpectra}
308			\end{figure}
309	jmarr	4012
310	jmarr	4013	\begin{figure}
311			\centering
312			\begin{subfigure}[b]{0.3\textwidth}
313			\includegraphics[trim = 100mm 20mm 50mm 20mm, clip, width=2in]{0x00V_Elip-1}
314			\label{fig:zeroField}
315			\end{subfigure}
316			\begin{subfigure}[b]{0.3\textwidth}
317			\includegraphics[trim = 100mm 20mm 50mm 20mm, clip, width=2in]{0x50V_Elip-1}
318			\label{fig:partialField}
319			\end{subfigure}
320			\begin{subfigure}[b]{0.3\textwidth}
321			\includegraphics[trim = 100mm 20mm 50mm 20mm, clip, width=2in]{1x20V_Elip-1}
322			\label{fig:fullField}
323			\end{subfigure}
324			\caption{Ellipsoid reprsentation of 5CB at three different
325			field strengths, Zero Field (Left), Partial Field (Middle), and Full
326			Field (Right)}\label{fig:Cigars}
327			\end{figure}
328
329	gezelter	4007	\section{Discussion}
330
331			\section{Conclusions}
332			\newpage
333
334			\bibliography{5CB}
335
336			\end{doublespace}
337			\end{document}