trunk/5cb/5CB.tex

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\title{Nitrile vibrations as reporters of field-induced phase
  transitions in 4-cyano-4'-pentylbiphenyl (5CB)}  
\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 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 exhibit a $\sim 40 \mathrm{~cm}^{-1}$ red
  shift of a portion 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
  direct field contributions.
\end{abstract}

\newpage

\section{Introduction}

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 has been observed to have a direct impact
on the peak position within the spectrum.  The Stark shift in the
spectrum can be quantified and mapped onto a field that is impinging
upon the nitrile bond. This has been used extensively in biological
systems like proteins and enzymes.\cite{Tucker:2004qq,Webb:2008kn}

The response of nitrile groups to electric fields has now been
investigated for a number of small molecules,\cite{Andrews:2000qv} as
well as in biochemical settings, where nitrile groups can act as
minimally invasive probes of structure and
dynamics.\cite{Lindquist:2009fk,Fafarman:2010dq} The vibrational Stark
effect has also been used to study the effects of electric fields on
nitrile-containing self-assembled monolayers at metallic
interfaces.\cite{Oklejas:2002uq,Schkolnik:2012ty}

Recently 4-cyano-4'-pentylbiphenyl (5CB), a liquid crystalline
molecule with a terminal nitrile group, has seen renewed interest as
one way to impart order on the surfactant interfaces of
nanodroplets,\cite{Moreno-Razo:2012rz} or to drive surface-ordering
that can be used to promote particular kinds of
self-assembly.\cite{PhysRevLett.111.227801} The nitrile group in 5CB
is a particularly interesting case for studying electric field
effects, as 5CB exhibits an isotropic to nematic phase transition that
can be triggered by the application of an external field near room
temperature.\cite{Gray:1973ca,Hatta:1991ee} This presents the
possiblity that the field-induced changes in the local environment
could have dramatic effects on the vibrations of this particular CN
bond.  Although the infrared spectroscopy of 5CB has been
well-investigated, particularly as a measure of the kinetics of the
phase transition,\cite{Leyte:1997zl} the 5CB nitrile group has not yet
seen the detailed theoretical treatment that biologically-relevant
small molecules have
received.\cite{Lindquist:2008bh,Lindquist:2008qf,Oh:2008fk,Choi:2008cr,Morales:2009fp,Waegele:2010ve}
 
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. For example, 5CB has a solid to nematic
phase transition at 18C and a nematic to isotropic transition at
35C.\cite{Gray:1973ca}

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. The Gay-Berne potential, in particular,
has been widely used in the liquid crystal community to describe this
anisotropic phase
behavior.~\cite{Gay:1981yu,Berne:1972pb,Kushick:1976xy,Luckhurst:1990fy,Cleaver:1996rt}
However, these simple models are insufficient to describe liquid
crystal phases which exhibit more complex polymorphic nature.
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 transitions as a function of water concentration), often have
polar head groups or zwitterionic charge separated groups that result
in strong dipolar interactions,\cite{Collings:1997rz} and terminal
cyano groups (like the one in 5CB) can induce permanent longitudinal
dipoles.\cite{Levelut:1981eu} Modeling of the phase behavior of these
molecules either requires additional dipolar
interactions,\cite{Bose:2012eu} or a unified-atom approach utilizing
point charges on the sites that contribute to the dipole
moment.\cite{Zhang:2011hh}

Macroscopic electric fields applied using electrodes on opposing sides
of a sample of 5CB have demonstrated the phase change of the molecule
as a function of electric field.\cite{Lim:2006xq} These previous
studies have shown the nitrile group serves as an excellent indicator
of the molecular orientation within the applied field. Lee {\it et
  al.}~showed a 180 degree change in field direction could be probed
with the nitrile peak intensity as it changed along with molecular
alignment in the field.\cite{Lee:2006qd,Leyte:1997zl}

While these macroscopic fields work well at indicating 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} While macroscopic fields
are insufficient to cause a Stark effect without dielectric breakdown
of the material, small fields across nanometer-sized gaps may be of
sufficient strength.  For a gap of 5 nm between a lower electrode
having a nanoelectrode placed near it via an atomic force microscope,
a potential of 1 V applied across the electrodes is equivalent to a
field of 2x10\textsuperscript{8} $\frac{V}{M}$.  This field is
certainly strong enough to cause the isotropic-nematic phase change
and as well as Stark tuning of the nitrile bond.  This should be
readily visible experimentally through Raman or IR spectroscopy.

In the sections that follow, we outline a series of coarse-grained
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 bond-length correlation functions to predict spectral
shifts. These predictions made should be easily varifiable with
scanning electrochemical microscopy experiments.

\section{Computational Details}
The force field used for 5CB was a united-atom model that was
parameterized by 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 that involved atoms in
these substructures (but with connectivity to the rest of the
molecule) were still included in the potential and force calculations.

Periodic simulations cells containing 270 molecules in random
orientations were constructed 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 equilibration 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) partial field = 0.01 V/\AA\ , and (3) full
field = 0.024 V/\AA\ . 

After the systems had come to equilibrium under the applied fields,
additional simulations were carried out with a flexible (Morse)
nitrile bond,
\begin{displaymath}
V(r_\ce{CN}) = D_e \left(1 - e^{-\beta (r_\ce{CN}-r_e)}\right)^2
\label{eq:morse}
\end{displaymath}
where $r_e= 1.157437$ \AA , $D_e = 212.95 \mathrm{~kcal~} /
\mathrm{mol}^{-1}$ and $\beta = 2.67566 $\AA~$^{-1}$.  These
parameters correspond to a vibrational frequency of $2358
\mathrm{~cm}^{-1}$, somewhat higher than the experimental
frequency. The flexible nitrile moiety required simulation time steps
of 1~fs, 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{u}_{i
    \alpha} \hat{u}_{i \beta} - \delta_{\alpha \beta} \right)
\end{equation}
where $\alpha, \beta = x, y, z$, and $\hat{u}_i$ 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 much smaller values ($\sim 0-0.2$) for isotropic fluids.
Note that the nitrogen and the terminal chain atom were used to define
the vectors for each molecule, so the typical order parameters are
lower than if one defined a vector using only the rigid core of the
molecule.  In nematic phases, typical values for $S$ are close to 0.5.

The field-induced phase transition can be clearly seen over the course
of a 60 ns equilibration runs in figure \ref{fig:orderParameter}.  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 began exhibiting nematic ordering, the orientational order
parameter became stable for the remaining 150 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. Ellipsoidal renderings of the
final configurations of the runs shows that the full-field (0.024
V/\AA\ ) experienced a isotropic-nematic phase transition and has
ordered with a director axis that is parallel to the direction of the
applied field.

\begin{figure}[H]
  \includegraphics[width=\linewidth]{Figure1}
  \caption{Evolution of the orientational order parameters for the
    no-field, partial field, and full field simulations over the
    course of 60 ns. Each simulation was started from a
    statistically-independent isotropic configuration.  On the right
    are ellipsoids representing the final configurations at three
    different field strengths: zero field (bottom), partial field
    (middle), and full field (top)}
  \label{fig:orderParameter}
\end{figure}


\section{Sampling the CN bond frequency}

The vibrational frequency of the nitrile bond in 5CB depends on
features of the local solvent environment of the individual molecules
as well as the bond's orientation relative to the applied field.  The
primary quantity of interest for interpreting the condensed phase
spectrum of this vibration is the distribution of frequencies
exhibited by the 5CB nitrile bond under the different electric fields.
There have been a number of elegant techniques for obtaining
vibrational lineshapes from classical simulations, including a
perturbation theory approach,\cite{Morales:2009fp} the use of an
optimized QM/MM approach coupled with the fluctuating frequency
approximation,\cite{Lindquist:2008qf} and empirical frequency
correlation maps.\cite{Oh:2008fk} Three distinct (and somewhat
primitive) methods for mapping classical simulations onto vibrational
spectra were brought to bear on the simulations here:
\begin{enumerate}
\item Isolated 5CB molecules and their immediate surroundings were
  extracted from the simulations.  These nitrile bonds were stretched
  and single-point {\em ab initio} calculations were used to obtain
  Morse-oscillator fits for the local vibrational motion along that
  bond.
\item The empirical frequency correlation maps developed by Cho {\it
    et al.}~\cite{Oh:2008fk} for nitrile moieties in water were
  investigated.  This method involves mapping the electrostatic
  potential around the bond to the vibrational frequency, and is
  similar in approach to field-frequency maps for OH stretches that
  were pioneered by the Skinner
  group.\cite{Corcelli:2004ai,Auer:2007dp}
\item Classical bond-length autocorrelation functions were Fourier
  transformed to directly obtain the vibrational spectrum from
  molecular dynamics simulations.
\end{enumerate}

\subsection{CN frequencies from isolated clusters}
The size of the periodic condensed phase system prevented direct
computation of the complete library of nitrile bond frequencies using
{\it ab initio} methods.  In order to sample the nitrile frequencies
present in the condensed-phase, individual molecules were selected
randomly to serve as the center of a local (gas phase) cluster.  To
include steric, electrostatic, and other effects from molecules
located near the targeted nitrile group, portions of other molecules
nearest to the nitrile group were included in the quantum mechanical
calculations.  The surrounding solvent molecules were divided into
``body'' (the two phenyl rings and the nitrile bond) and ``tail'' (the
alkyl chain).  Any molecule which had a body atom within 6~\AA\ of the
midpoint of the target nitrile bond had its own molecular body (the
4-cyano-biphenyl moiety) included in the configuration.  Likewise, the
entire alkyl tail was included if any tail atom was within 4~\AA\ of
the target nitrile bond.  If tail atoms (but no body atoms) were
included within these distances, only the tail was included as a
capped propane molecule.

\begin{figure}[H]
  \includegraphics[width=\linewidth]{Figure2}
  \caption{Cluster calculations were performed on randomly sampled 5CB
    molecules (shown in red) from each of the simulations. Surrounding
    molecular bodies were included if any body atoms were within 6
    \AA\ of the target nitrile bond, and tails were included if they
    were within 4 \AA.  Included portions of these molecules are shown
    in green.  The CN bond on the target molecule was stretched and
    compressed, and the resulting single point energies were fit to
    Morse oscillators to obtain a distribution of frequencies.}
  \label{fig:cluster}
\end{figure}

Inferred hydrogen atom locations were added to the cluster geometries,
and the nitrile bond was stretched from 0.87 to 1.52~\AA\ at
increments of 0.05~\AA. This generated 13 configurations per gas phase
cluster. Single-point energies were computed using the B3LYP
functional~\cite{Becke:1993kq,Lee:1988qf} and the 6-311++G(d,p) basis
set.  For the cluster configurations that had been generated from
molecular dynamics running under applied fields, the density
functional calculations had a field of $5 \times 10^{-4}$ atomic units
($E_h / (e a_0)$) applied in the $+z$ direction in order to match the
molecular dynamics simulations.

The energies for the stretched / compressed nitrile bond in each of
the clusters were used to fit Morse potentials, and the frequencies
were obtained from the $0 \rightarrow 1$ transition for the energy
levels for this potential.\cite{Morse:1929xy} To obtain a spectrum,
each of the frequencies was convoluted with a Lorentzian lineshape
with a width of 1.5 $\mathrm{cm}^{-1}$.  Available computing resources
limited the sampling to 67 clusters for the zero-field spectrum, and
59 for the full field.  Comparisons of the quantum mechanical spectrum
to the classical are shown in figure \ref{fig:spectrum}.

\subsection{CN frequencies from potential-frequency maps}

One approach which has been used to successfully analyze the spectrum
of nitrile and thiocyanate probes in aqueous environments was
developed by Choi {\it et al.}~\cite{Choi:2008cr,Oh:2008fk} This
method involves finding a multi-parameter fit that maps between the
local electrostatic potential at selected sites surrounding the
nitrile bond and the vibrational frequency of that bond obtained from
more expensive {\it ab initio} methods. This approach is similar in
character to the field-frequency maps developed by the Skinner group
for OH stretches in liquid water.\cite{Corcelli:2004ai,Auer:2007dp} 

To use the potential-frequency maps, the local electrostatic
potential, $\phi_a$, is computed at 20 sites ($a = 1 \rightarrow 20$)
that surround the nitrile bond,
\begin{equation}
\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{j}
\frac{q_j}{\left|r_{aj}\right|}.
\end{equation}
Here $q_j$ is the partial site on atom $j$ (residing on a different
molecule) and $r_{aj}$ is the distance between site $a$ and atom $j$.
The original map was parameterized in liquid water and comprises a set
of parameters, $l_a$, that predict the shift in nitrile peak
frequency,
\begin{equation}
\delta\tilde{\nu} =\sum^{20}_{a=1} l_{a}\phi_{a}.
\end{equation}

The simulations of 5CB were carried out in the presence of
externally-applied uniform electric fields. Although uniform fields
exert forces on charge sites, they only contribute to the potential if
one defines a reference point that can serve as an origin. One simple
modification to the potential at each of the probe sites is to use the
centroid of the \ce{CN} bond as the origin for that site,
\begin{equation}
\phi_a^\prime = \phi_a + \frac{1}{4\pi\epsilon_{0}} \vec{E} \cdot
\left(\vec{r}_a - \vec{r}_\ce{CN} \right)  
\end{equation}
where $\vec{E}$ is the uniform electric field, $\left( \vec{r}_{a} -
  \vec{r}_\ce{CN} \right)$ is the displacement between the
cooridinates described by Choi {\it et
  al.}~\cite{Choi:2008cr,Oh:2008fk} and the \ce{CN} bond centroid.
$\phi_a^\prime$ then contains an effective potential contributed by
the uniform field in addition to the local potential contributions
from other molecules.

The sites $\{\vec{r}_a\}$ and weights $\left\{l_a \right\}$
developed by Choi {\it et al.}~\cite{Choi:2008cr,Oh:2008fk} are quite
symmetric around the \ce{CN} centroid, and even at large uniform field
values we observed nearly-complete cancellation of the potenial
contributions from the uniform field.  In order to utilize the
potential-frequency maps for this problem, one would therefore need
extensive reparameterization of the maps to include explicit
contributions from the external field.  This reparameterization is
outside the scope of the current work, but would make a useful
addition to the potential-frequency map approach.

\subsection{CN frequencies from bond length autocorrelation functions}

The distribution of nitrile vibrational frequencies can also be found
using classical time correlation functions.  This was done by
replacing the rigid \ce{CN} bond with a flexible Morse oscillator
described in Eq. \ref{eq:morse}.  Since the systems were perturbed by
the addition of a flexible high-frequency bond, they were allowed to
re-equilibrate in the canonical (NVT) ensemble for 100 ps with 1 fs
timesteps. After equilibration, each configuration was run in the
microcanonical (NVE) ensemble for 20 ps. Configurations sampled every
fs were then used to compute bond-length autocorrelation functions,
\begin{equation}
C(t) = \langle \delta r(t) \cdot \delta r(0) ) \rangle
\end{equation}
%
where $\delta r(t) = r(t) - r_0$ is the deviation from the equilibrium
bond distance at time $t$.  Ten statistically-independent correlation
functions were obtained by allowing the systems to run 10 ns with
rigid \ce{CN} bonds followed by 120 ps equilibration and data
collection using the flexible \ce{CN} bonds, and repeating this
process.  The total sampling time, from sample preparation to final
configurations, exceeded 150 ns for each of the field strengths
investigated.

The correlation functions were filtered using exponential apodization
functions,\cite{FILLER:1964yg} $f(t) = e^{-|t|/c}$, with a time
constant, $c =$ 6 ps, and were Fourier transformed to yield a
spectrum,
\begin{equation}
I(\omega) = \int_{-\infty}^{\infty} C(t) f(t) e^{-i \omega t} dt.
\end{equation}   
The sample-averaged classical nitrile spectrum can be seen in Figure
\ref{fig:spectra}.  Note that the Morse oscillator parameters listed
above yield a natural frequency of 2358 $\mathrm{cm}^{-1}$, somewhat
higher than the experimental peak near 2226 $\mathrm{cm}^{-1}$.  This
shift does not effect the ability to qualitatively compare peaks from
the classical and quantum mechanical approaches, so the classical
spectra are shown as a shift relative to the natural oscillation of
the Morse bond.

\begin{figure}
  \includegraphics[width=3.25in]{Convolved}
  \includegraphics[width=3.25in]{2Spectra}
  \caption{Quantum mechanical nitrile spectrum for the no-field simulation
    (black) and the full field simulation (red).  The lower panel
    shows the corresponding classical bond-length autocorrelation
    spectrum for the flexible nitrile measured relative to the natural
    frequency for the flexible bond.}
  \label{fig:spectra}
\end{figure}

Note that due to electrostatic interactions, the classical approach
implicitly couples \ce{CN} vibrations to the same vibrational mode on
other nearby molecules.  This coupling is not handled in the {\it ab
  initio} cluster approach.

\section{Discussion}

It is clear that united-atom simulations can reproduce the
field-induced nematic ordering of the 4-cyano-4'-pentylbiphenyl.
Because we are simulating a small electrode separation (5nm), a
voltage drop as low as 1.2 V was sufficient to induce the phase
change.  This potential is significantly lower than the 500V that is
known to cause dielectric breakdown in 5CB.\cite{XXX}

 
Ordering corresponds to shift of a portion of the nitrile spectrum to
the red.
At the same time, the system exhibits an increase in aligned and anti-a


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}.

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. 

The angle-dependent pair distribution functions,
\begin{align}
g(r, \cos \omega) = &  \frac{1}{\rho N} \left< \sum_{i}
\sum_{j} \delta \left(r - r_{ij}\right) \delta\left(\cos \omega_{ij} -
  \cos \omega\right) \right> \\ \nonumber \\
g(r, \cos \theta) = & \frac{1}{\rho N} \left< \sum_{i}
\sum_{j} \delta \left(r - r_{ij}\right) \delta\left(\cos \theta_{i} -
  \cos \theta \right) \right> 
\end{align}
provide information about the joint spatial and angular correlations
in the system. The angles $\omega$ and $\theta$ are defined by vectors
along the CN axis of each nitrile bond (see figure
\ref{fig:definition}).

\begin{figure}
  \includegraphics[width=\linewidth]{definition}
  \caption{Definitions of the angles between two nitrile bonds.}
  \label{fig:definition}
\end{figure}

In figure \ref{fig:gofromega} the effects of the field-induced phase
transition are clear.  The nematic ordering transfers population from
the perpendicular or unaligned region in the center of the plot to the
nitrile-alinged peak near $\cos\omega = 1$. Most other features are
undisturbed.  This increased population of aligned nitrile bonds in
the close solvation shells is also the population that contributes
most heavily to the low-frequency peaks in the vibrational spectrum.

\begin{figure}
  \includegraphics[width=\linewidth]{Figure4}
  \caption{Contours of the angle-dependent pair distribution functions
    for nitrile bonds on 5CB in the zero-field (upper panel) and full
    field (lower panel) simulations. Dark areas signify regions of
    enhanced density, while light areas signify depletion relative to
    the bulk density.}
  \label{fig:gofromega}
\end{figure}


\section{Conclusions}
Field dependent changes

\section{Acknowledgements}
The authors thank Steven Corcelli for helpful comments and
suggestions.  Support for this project was provided by the National
Science Foundation under grant CHE-0848243. Computational time was
provided by the Center for Research Computing (CRC) at the University
of Notre Dame.

\newpage

\bibliography{5CB}

\end{doublespace}
\end{document}
Revision:	4042
Committed:	Fri Feb 21 18:46:54 2014 UTC (10 years, 4 months ago) by gezelter
Content type:	application/x-tex
File size:	30525 byte(s)
Log Message:	edits