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}
  4-Cyano-4'-pentylbiphenyl (5CB) is a liquid-crystal-forming compound
  with a terminal nitrile group aligned with the long axis of the
  molecule.  Simulations of condensed-phase 5CB were carried out both
  with and without applied electric fields to provide an understanding
  of the various contributions to the Stark shift of the terminal
  nitrile group.  A field-induced isotropic-nematic phase transition
  was observed in the simulations, and the effects of this transition
  on the distribution of nitrile frequencies were computed. Classical
  bond displacement correlation functions exhibited a ($\sim 40
  \mathrm{cm}^{-1}$ red shift of a fraction of the main nitrile peak,
  and this shift was observed only when the fields were large enough
  to induce orientational ordering of the bulk phase.  Our simulations
  appear to indicate that phase-induced changes to the local
  surroundings are a larger contribution to the change in the nitrile
  spectrum than the direct field contribution.
\end{abstract}

\newpage

\section{Introduction}
The Stark shift of nitrile groups in response to applied electric
fields have been used extensively in biology for probing the internal
fields of structures like proteins and DNA.  Integration of these
probes into different materials is also important for studying local
structure in confined fluids. This work centers on the vibrational
response of the terminal nitrile group in 4-Cyano-4'-pentylbiphenyl
(5CB), a liquid crystalline molecule with an isotropic to nematic
phase transition that can be triggered by the application of an
external field.

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 well-studied electric field
response,\cite{Hatta:1991ee} and the fact that it has a set of phase
transitions near room temperature.\cite{Gray:1973ca} The have a solid
to nematic phase transition at 18 C and a nematic to isotropic
transition at 35 C.\cite{Gray:1973ca} Recently there has been renewed
interest in 5CB in nanodroplets to understand defect sites and
nanoparticle structuring.\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 {\it 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.

In the rest of this paper, we outline a series of classical molecular
dynamics simulations of 5CB that were done in the presence of static
electric fields. These simulations were then coupled with both {\it ab
  intio} calculations of CN-deformations and classical correlation
functions to predict spectral shifts. These predictions should be
easily varifiable with scanning electrochemical microscopy
experiments.

\section{Computational Details}
The force field used for 5CB was taken from Guo {\it et
  al.}\cite{Zhang:2011hh} However, for most of the simulations, each
of the phenyl rings was treated as a rigid body to allow for larger
time steps and very long simulation times.  The geometries of the
rigid bodies were taken from equilibrium bond distances and angles.
Although the phenyl rings were held rigid, bonds, bends, torsions and
inversion centers included in these bodies (but with connectivity to
the rest of the molecule) were still included in the potential and
force calculations.  

Periodic simulations cells contained 270 molecules and were locked at
experimental densities.  Electrostatic interactions were computed
using damped shifted force (DSF) electrostatics.\cite{Fennell:2006zl}
The molecules were equilibrated for 1~ns at a temperature of 300K.
Simulations with applied fields were carried out in the microcanonical
(NVE) ensemble with an energy corresponding to the average energy from
the canonical (NVT) equilibration runs. Typical applied-field runs
were more than 60ns in length.

Static electric fields with magnitudes similar to what would be
available in an experimental setup were applied to the different
simulations.  With an assumed electrode seperation of 5 nm and an
electrostatic potential that is limited by the voltage required to
split water (1.23V), the maximum realistic field that could be applied
is $\sim 0.024 V / \AA$.   Three field environments were investigated:
(1) no field applied, (2) $0.01 V / \AA$ (0.5 V), and (3) $0.024 V /
\AA$ (1.2 V).  Each field was applied along the $z$-axis of the
simulation cell.   For simplicity, these field strengths will be
referred to as zero, partial, and full field.  

After the systems had come to equilibrium under the applied fields,
additional simulations were carried out with a flexible (harmonic)
nitrile bond with an equilibrium bond distance of XXX \AA and a force
constant of XXX kcal / mol $\AA^2$, corresponding to a vibrational
frequency of YYYY $\mathrm{cm}^{-1}$.  The flexible nitrile moiety
required simualtion time steps of 1fs, so the additional flexibility
was introducuced only after the rigid systems had come to equilibrium
under the applied fields.  Whenever time correlation functions were
computed from the flexible simulations, statistically-independent
configurations were sampled from the last ns of the induced-field
runs.  These configurations were then equilibrated with the flexible
nitrile moiety for 100 ps, and time correlation functions were
computed using data sampled from an additional 200 ps of run time
carried out in the microcanonical ensemble.

\section{Field-induced Nematic Ordering}

In order to characterize the orientational ordering of the system, the
primary quantity of interest is the nematic (orientational) order
parameter. This was 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 (smectic A) phases,
but falls to zero for isotropic fluids.   In nematic phases, typical
values are close to 0.5.

In Figure \ref{fig:orderParameter}, the field-induced phase change can
be clearly seen over the course of a 60 ns equilibration run. All
three of the systems started in a random (isotropic) packing, with
order parameters near 0.2. Over the course 10 ns, the full field
causes an alignment of the molecules (due primarily to the interaction
of the nitrile group dipole with the electric field).  Once this
system landed in the nematic-ordered state, it became stable for the
remaining 50 ns of simulation time.  It is possible that the
partial-field simulation is meta-stable and given enough time, it
would eventually find a nematic-ordered phase, but the partial-field
simulation was stable as an isotropic phase for the full duration of a
60 ns simulation.
\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 figure \ref{fig:Cigars}, the field-induced isotropic-nematic
transition is represented using ellipsoids aligned along the long-axis
of each molecule.  The vector between the nitrogen of the nitrile
group and the terminal tail atom is used to orient each
ellipsoid.  Both the zero field and partial field simulations appear
isotropic, while the full field simulations has clearly been
orientationally ordered
\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}

\section{Analysis}

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}


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  {\it 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  {\it 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 interesting. 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. With narrowing, it would indicate an increased
homogeneous distrobution of the local field near the nitrile. 
\section{Conclusions}
Field dependent changes
\newpage

\bibliography{5CB}

\end{doublespace}
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