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 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}

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,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,Berne72,Kushick:1976xy,Luckhurst90,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 transition 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}

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 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 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 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
\end{displaymath}
where $r_e= 1.157437$ \AA , $D_e = 212.95 \mathrm{~kca~l} /
\mathrm{mol}^{-1}$ and $\beta = 2.67566 $\AA~$^{-1}$.  These
parameters correspond to a vibrational frequency of $2375
\mathrm{~cm}^{-1}$, a bit 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{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.  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 started exhibiting nematic ordering, the orientational order
parameter 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. 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.
Three distinct methods for mapping classical simulations onto
vibrational spectra were brought to bear on these simulations:
\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 potential - frequency 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 that were pioneered by
  Skinner {\it et al.}\cite{XXXX}
\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.  For the alkyl
tail, the entire 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 frequency distributions.}
  \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 oscillators, 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 Skinner {\it et
  al.} for OH stretches in liquid water.\cite{XXXX}

To use the potential-frequency maps, the local electrostatic
potential, $\phi$, 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 $j$, 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 in the presence of external
electric fields, out without water present, so it is not clear if they
can be applied to this situation without extensive
reparameterization.  We do, however, suggest a small modification that
would help 

, so the equations need to be corrected
for the frequency shift caused by the electric field. We attempted to
make small modifications 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.


\subsection{CN frequencies from bond length autocorrelation functions}

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 the canonical (NVT) ensemble
for 100 ps. After re-equilibration, the system was run in the
microcanonical (NVE) ensemble for 20 ps.  Configurations sampled every
fs were then used to compute bond-length autocorrelation functions 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.


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}


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}
\end{document}
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