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}
Nitrile Stark shift repsonses to electric fields have been used
extensively in biology for the probing of local internal fields of
structures like proteins and DNA. Intigration of these probes into
different areas of interest are important for studing local structure
and fields within confined, nanoscopic
systems. 4-Cyano-4'-pentylbiphenyl (5CB) is a liquid crystal with a known
macroscopic structure reordering from the isotropic to nematic
phase with the application of an external
field and as the name suggests has an inherent nitrile group. Through
simulations of this molecule where application of
large, nanoscale external fields were applied, the nitrile was invenstigated
as a local field sensor. It was
found that while most computational methods for nitrile spectral
calculations rely on a correlation between local electric field and
the nitrile bond, 5CB did not have an easily obtained
correlation. Instead classical calculation through correlation of the
cyanide bond displacement in time use enabled to show a spectral
change in the formation of a red
shifted peak from the main peak as an external field was applied. This indicates
that local structure has a larger impact on the nitrile frequency then
does the local electric field. By better understanding how nitrile
groups respond to local and external fields it will help
nitrile groups branch out beyond their biological
applications to uses in electronics and surface sciences.
\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 external 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} Recently it has
seen new life with the application of droplets of the molecule in
water being used to study defect sites and nanoparticle
strcuturing.\cite{PhysRevLett.111.227801}

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 nanoscopic 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. Lee et. al. 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 of 5CB with external
fields applied. These simulations are then coupled with ab intio and
classical spectrum calculations to predict changes. These changes are
easily varifiable with experiments and should be able to replicated
experimentally.

\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. Each set of
frequencies were then convolved together with a lorezian lineshape
function over each value. The width value used was 1.5. For the zero
field spectrum, 67 frequencies were used and for the full field, 59
frequencies were used.

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 6000. 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 to a Morse oscillator and a short production run of 20 ps.

\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 change 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.
\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}

In the figure below, this phase change is represented nicely as
ellipsoids that are created by the vector formed between the nitrogen
of the nitrile group and the tail terminal carbon atom. These
illistrate the change from isotropic phase to nematic change. Both the
zero field and partial field images look mostly disordered. The
partial field does look somewhat orded but without any overall order
of the entire system. This is most likely a high point in the ordering
for the trajectory. The full field image on the other hand looks much
more ordered when compared to the two lower field simulations.
\begin{figure}
  \includegraphics[width=7in]{Elip_3}
  \caption{Ellipsoid reprsentation of 5CB at three different
    field strengths, Zero Field (Left), Partial Field (Middle), and Full
    Field (Right) Each image was created by taking the final
    snapshot of each 60 ns run}
  \label{fig:Cigars}
\end{figure}

This change in phase was followed by two courses of further
analysis. First was the 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 performed to investigate if the phase change caused any change
spectrum through quantum effects.

The classical nitrile spectrum can be seen in Figure 2. Most noticably
is the position of the two peaks. Obviously the experimental peak
position is near 2226 cm\textsuperscript{-1}. However, in this case
the peak position is shifted to the blue at a position of 2375
cm\textsuperscript{-1}. This shift is due solely to the choice of
oscillator strength in the Morse oscillator parameters. While this
shift makes the two spectra differ, it does not affect the ability to
qualitatively compare peak changes to possible experimental changes.
With this important fact out of the way, differences between the two
states are subtle but are very much present. The first and
most notable is the apperance for a strong band near 2300
cm\textsuperscript{-1}.
\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}

Before Gaussian silumations were carried out, it was attempt to apply
the method developed by Cho et. al.\cite{Oh:2008fk} This method involves the fitting
of multiple parameters to Gaussian calculated freuencies to find a
correlation between the potential around the bond and the
frequency. This is very similar to work done by Skinner et. al. with
water models like SPC/E. The general method is to find the shift in
the peak position through,
\begin{equation}
\delta\tilde{\nu} =\sum^{n}_{a=1} l_{a}\phi^{water}_{a}
\end{equation}
where $l_{a}$ are the fitting parameters and $\phi^{water}_{a}$ is the
potential from the surrounding water cluster. This $\phi^{water}_{a}$
takes the form,
\begin{equation}
\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{m} \sum_{j}
\frac{C^{H_{2}O}_{j \left(m \right) }}{r_{aj \left(m\right)}}
\end{equation}
where $C^{H_{2}O}_{j \left(m \right) }$ indicates the partial charge
on the $j$th site of the $m$th water molecule and $r_{aj \left(m\right)}$
is the distance between the site $a$ of the nitrile molecule and the $j$th
site of the $m$th water molecule. However, since these simulations
are done under the presence of external fields and in the
absence of water, the equations need a correction factor for the shift
caused by the external field. The equation is also reworked to use
electric field site data instead of partial charges from surrounding
atoms. So by modifing the original
$\phi^{water}_{a}$ to $\phi^{5CB}_{a}$ we get, 
\begin{equation}
\phi^{5CB}_{a} = \frac{1}{4\pi \epsilon_{0}} \left( \vec{E}\bullet
  \left(\vec{r}_{a}-\vec{r}_{CN}\right) \right) + \phi^{5CB}_{0}
\end{equation}
where $\vec{E}$ is the electric field at each atom, $\left( \vec{r}_{a} -
  \vec{r}_{CN} \right)$ is the vector between the nitrile bond and the
cooridinates described by Cho around the bond and $\phi^{5CB}_{0}$ is
the correction factor for the system of parameters. After these
changes, the correction factor was found for multiple values of an
external field being applied. However, the factor was no linear and
was overly large due to the fitting parameters being so small.

Due to this, Gaussian calculations were performed in lieu of this
method. A set of snapshots for the zero and full field simualtions,
they were first investigated for any dependence on the local, with
external field included, electric field. This was to see if a linear
or non-linear relationship between the two could be utilized for
generating spectra. This was done in part because of previous studies
showing the frequency dependence of nitrile bonds to the electric
fields generated locally between solvating water. It was seen that
little to no dependence could be directly shown. This data is not
shown.

Since no explicit dependence was observed between the calculated
frequency and the electric field, it was not a viable route for the
calculation of a nitrile spectrum. Instead, the frequencies were taken
and convolved together with a lorentzian line shape applied around the
frequency value. These spectra are seen below in Figure
4. While the spectrum without a field is lower in intensity and is
almost bimodel in distrobution, the external field spectrum is much
more unimodel. This tighter clustering has the affect of increasing the
intensity around 2226 cm\textsuperscript{-1} where the peak is
centered. The external field also has fewer frequencies of higher
energy in the spectrum. Unlike the the zero field, where some frequencies
reach as high as 2280 cm\textsuperscript{-1}. 
\begin{figure}
  \includegraphics[width=3.25in]{Convolved}
  \caption{Lorentzian convolved Gaussian frequencies of the zero field
  system (black) and the full field system (red)}
  \label{fig:Con}
\end{figure}
\section{Discussion}
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 short distance of 5 nm 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. However, this field
will not cause a breakdown of the 5CB since electrochemistry studies
have shown that it can be used in the presence of fields as high as
500 V macroscopically. This large of a field near the surface of the
elctrode would cause breakdown of 5CB if it could happen.

The absence of any electric field dependency of the freuquency with
the Gaussian simulations is strange. A large base of research has been
built upon the known tuning of the nitrile bond as the local field
changes. This difference may be due to the absence of water or a
molecule that induces a large internal field. Liquid water is known to have a very high internal field which
is much larger than the internal fields of neat 5CB. Even though the
application of Gaussian simulations followed by mapping it to
some classical parameter is easy and straight forward, this system
illistrates how that 'go to' method can break down.

While this makes the application of nitrile Stark effects in
simulations without water harder, these data show
that it is not a deal breaker. The classically calculated nitrile
spectrum shows changes in the spectra that will be easily seen through
experimental routes. It indicates a shifted peak lower in energy
should arise. This peak is a few wavenumbers from the leading edge of
the larger peak and almost 75 wavenumbers from the center. This
seperation between the two peaks means experimental results will show
an easily resolved peak.

The Gaussian derived spectra do indicate an applied field
and subsiquent phase change does cause a narrowing of freuency
distrobution. 
\section{Conclusions}
Field dependent changes
\newpage

\bibliography{5CB}

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\end{document}
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