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}

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\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 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 $\sim~3~\mathrm{cm}^{-1}$ 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.  Joint spatial-angular distribution functions indicate
  that phase-induced anti-caging of the nitrile bond is contributing
  to the change in the nitrile spectrum.  Distributions of frequencies
  obtained via cluster-based fits to quantum mechanical energies of
  nitrile bond deformations exhibit a similar
  $\sim~2.7~\mathrm{cm}^{-1}$ red shift.
\end{abstract}

\newpage

\section{Introduction}

Because the triple bond between nitrogen and carbon is sensitive to
local fields, nitrile groups can report on field strengths via their
distinctive Raman and IR signatures.\cite{Boxer:2009xw} 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{Tucker:2004qq,Webb:2008kn,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 response at a molecular scale has not been studied. With
the advent of nano-electrodes and the ability to couple these
electrodes to atomic force microscopy, control of electric fields
applied across nanometer distances is now possible.\cite{C3AN01651J}
In special cases where the macroscopic fields are insufficient to
cause an observable Stark effect without dielectric breakdown of the
material, small potentials 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 $2 \times 10^8~\mathrm{V/m}$. This field is
certainly strong enough to cause the isotropic-nematic phase change
and as well as a visible Stark tuning of the nitrile bond. We expect
that this would 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 verifiable with
scanning electrochemical microscopy experiments.

\section{Computational Details}
The force-field used to model 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 longer simulation times. The
geometries of the rigid bodies were taken from equilibrium bond
distances and angles. Although the individual 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 60~ns 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 ($0 \rightarrow 0.3$) 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]{orderParameter/orderParameter.pdf}
  \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 comparatively
primitive) methods for mapping classical simulations onto vibrational
spectra were brought to bear on the simulations in this work:
\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 A static-field extension of the empirical frequency correlation
  maps developed by Cho {\it et al.}~\cite{Oh:2008fk} for nitrile
  moieties in water was attempted.
\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]{cluster/cluster.pdf}
  \caption{Cluster calculations were performed on randomly sampled 5CB
    molecules (shown in red) from the full-field and no-field
    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 100 clusters for both the zero-field and full
field spectra.  Comparisons of the quantum mechanical spectrum to the
classical are shown in figure \ref{fig:spectra}.  The mean frequencies
obtained from the distributions give a field-induced red shift of
$2.68~\mathrm{cm}^{-1}$.

\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.

We note that in 5CB there does not appear to be a particularly strong
correlation between the electric field observed at the nitrile
centroid and the calculated vibrational frequency.  In
Fig. \ref{fig:fieldMap} we show the calculated frequencies plotted
against the field magnitude and the parallel and perpendicular
components of the field.

\begin{figure}
  \includegraphics[width=\linewidth]{fieldMap/fieldMap.pdf}
  \caption{The observed cluster frequencies have no apparent
    correlation with the electric field felt at the centroid of the
    nitrile bond.  Upper panel: vibrational frequencies plotted
    against the component of the field parallel to the CN bond.
    Middle panel: mapped to the magnitude of the field components
    perpendicular to the CN bond.  Lower panel: mapped to the total
    field magnitude.}
  \label{fig:fieldMap}
\end{figure}


\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 $t$.  Because the other atomic sites have very
small partial charges, this correlation function is an approximation
to the dipole autocorrelation function for the molecule, which would
be particularly relevant to computing the IR spectrum. 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.  This process was repeated 10 times, and 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, $c =$ 3.5 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=\linewidth]{spectra/spectra.pdf}
  \caption{Spectrum of nitrile frequency shifts for the no-field
    (black) and the full-field (red) simulations. Upper panel:
    frequency shifts obtained from {\it ab initio} cluster
    calculations. Lower panel: classical bond-length autocorrelation
    spectrum for the flexible nitrile measured relative to the natural
    frequency for the flexible bond.  The dashed lines indicate the
    mean frequencies for each of the distributions.  The cluster
    calculations exhibit a $2.68~\mathrm{cm}^{-1}$ field-induced red
    shift, while the classical correlation functions predict a red
    shift of $3.05~\mathrm{cm}^{-1}$.}
  \label{fig:spectra}
\end{figure}

The classical approach includes both intramolecular and electrostatic
interactions, and so it implicitly couples \ce{CN} vibrations to other
vibrations within the molecule as well as to nitrile vibrations on
other nearby molecules. The classical frequency spectrum is
significantly broader because of this coupling. The {\it ab initio}
cluster approach exercises only the targeted nitrile bond, with no
additional coupling to other degrees of freedom. As a result the
quantum calculations are quite narrowly peaked around the experimental
nitrile frequency. Although the spectra are quite noisy, the main
effect seen in both distributions is a moderate shift to the red
($3.05~\mathrm{cm}^{-1}$ classical and $2.68~\mathrm{cm}^{-1}$
quantum) when the full electrostatic field had induced the nematic
phase transition.

\section{Discussion}
Our simulations show that the united-atom model can reproduce the
field-induced nematic ordering of the 4-cyano-4'-pentylbiphenyl.
Because we are simulating a very small electrode separation (5~nm), a
voltage drop as low as 1.2~V was sufficient to induce the phase
change. This potential is significantly smaller than 100~V that was
used with a 5~$\mu$m gap to study the electrochemiluminescence of
rubrene in neat 5CB,\cite{Kojima19881789} and suggests that by using
electrodes separated by a nanometer-scale gap, it will be relatively
straightforward to observe the nitrile Stark shift in 5CB.

Both the classical correlation function and the isolated cluster
approaches to estimating the IR spectrum show that a population of
nitrile stretches shift by $\sim~3~\mathrm{cm}^{-1}$ to the red of
the unperturbed vibrational line. To understand the origin of this
shift, a more complete picture of the spatial ordering around the
nitrile bonds is required. We have computed 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}
which provide information about the joint spatial and angular
correlations present 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=4in]{definition/definition.pdf}
  \caption{Definitions of the angles between two nitrile bonds.}
  \label{fig:definition}
\end{figure}

The primary structural effect of the field-induced phase transition is
apparent in figure \ref{fig:gofromega}. The nematic ordering transfers
population from the perpendicular ($\cos\omega\approx 0$) and
anti-aligned ($\cos\omega\approx -1$) to the nitrile-alinged peak
near $\cos\omega\approx 1$, leaving most other features undisturbed.  This
change is visible in the simulations as an increased population of
aligned nitrile bonds in the first solvation shell.

\begin{figure}
  \includegraphics[width=\linewidth]{gofrOmega/gofrOmega.pdf}
  \caption{Contours of the angle-dependent pair distribution functions
    for nitrile bonds on 5CB in the no 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} 

Although it is certainly possible that the coupling between
closely-spaced nitrile pairs is responsible for some of the red-shift,
that is not the only structural change that is taking place. The
second two-dimensional pair distribution function, $g(r,\cos\theta)$,
shows that nematic ordering also transfers population that is directly
in line with the nitrile bond (see figure \ref{fig:gofrtheta}) to the
sides of the molecule, thereby freeing steric blockage can directly
influence the nitrile vibration. This is confirmed by observing the
one-dimensional $g(z)$ obtained by following the \ce{C -> N} vector
for each nitrile bond and observing the local density ($\rho(z)/\rho$)
of other atoms at a distance $z$ along this direction. The full-field
simulation shows a significant drop in the first peak of $g(z)$,
indicating that the nematic ordering has moved density away from the
region that is directly in line with the nitrogen side of the CN bond.

\begin{figure}
  \includegraphics[width=\linewidth]{gofrTheta/gofrTheta.pdf}
  \caption{Contours of the angle-dependent pair distribution function,
    $g(r,\cos \theta)$, for finding any other atom at a distance and
    angular deviation from the center of a nitrile bond.  The top edge
    of each contour plot corresponds to local density along the
    direction of the nitrogen in the CN bond, while the bottom is in
    the direction of the carbon atom.  Bottom panel: $g(z)$ data taken
    by following the \ce{C -> N} vector for each nitrile bond shows
    that the field-induced phase transition reduces the population of
    atoms that are directly in line with the nitrogen motion.}
  \label{fig:gofrtheta}
\end{figure}

We are suggesting an anti-caging mechanism here -- the nematic
ordering provides additional space directly inline with the nitrile
vibration, and since the oscillator is fairly anharmonic, this freedom
provides a fraction of the nitrile bonds with a significant red-shift.

The cause of this shift does not appear to be related to the alignment
of those nitrile bonds with the field, but rather to the change in
local steric environment that is brought about by the
isotropic-nematic transition. We have compared configurations for many
of the cluster that exhibited the lowest frequencies (between 2190 and
2215 $\mathrm{cm}^{-1}$) and have observed some similar structural
features. The lowest frequencies appear to come from configurations
which have nearly-empty pockets directly opposite the nitrogen atom
from the nitrile carbon. Because we do not have a particularly large
cluster population to interrogate, this is certainly not quantitative
confirmation of this effect.

The prediction of a small red-shift of the nitrile peak in 5CB in
response to a field-induced nematic ordering is the primary result of
this work, and although the proposed anti-caging mechanism is somewhat
speculative, this work provides some impetus for further theory and
experiments.

\section{Acknowledgements}
The authors thank Steven Corcelli and Zac Schultz 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{document}
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