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\title{Nitrile vibrations as reporters of field-induced phase |
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transitions in 4-cyano-4'-pentylbiphenyl (5CB)} |
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\author{James M. Marr} |
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu} |
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\affiliation[University of Notre Dame]{251 Nieuwland Science Hall\\ |
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Department of Chemistry and Biochemistry\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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|
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\date{\today} |
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|
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\begin{document} |
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|
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\maketitle |
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|
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\begin{doublespace} |
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|
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\begin{abstract} |
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4-cyano-4'-pentylbiphenyl (5CB) is a liquid-crystal-forming compound |
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with a terminal nitrile group aligned with the long axis of the |
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molecule. Simulations of condensed-phase 5CB were carried out both |
64 |
with and without applied electric fields to provide an understanding |
65 |
of the Stark shift of the terminal nitrile group. A field-induced |
66 |
isotropic-nematic phase transition was observed in the simulations, |
67 |
and the effects of this transition on the distribution of nitrile |
68 |
frequencies were computed. Classical bond displacement correlation |
69 |
functions exhibit a $\sim 40 \mathrm{~cm}^{-1}$ red shift of a |
70 |
portion of the main nitrile peak, and this shift was observed only |
71 |
when the fields were large enough to induce orientational ordering |
72 |
of the bulk phase. Joint spatial-angular distribution functions |
73 |
indicate that phase-induced anti-caging of the nitrile bond is |
74 |
contributing to the change in the nitrile spectrum. |
75 |
\end{abstract} |
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|
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\newpage |
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|
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\section{Introduction} |
80 |
|
81 |
Nitrile groups can serve as very precise electric field reporters via |
82 |
their distinctive Raman and IR signatures.\cite{Boxer:2009xw} The |
83 |
triple bond between the nitrogen and the carbon atom is very sensitive |
84 |
to local field changes and has been observed to have a direct impact |
85 |
on the peak position within the spectrum. The Stark shift in the |
86 |
spectrum can be quantified and mapped onto a field that is impinging |
87 |
upon the nitrile bond. The response of nitrile groups to electric |
88 |
fields has now been investigated for a number of small |
89 |
molecules,\cite{Andrews:2000qv} as well as in biochemical settings, |
90 |
where nitrile groups can act as minimally invasive probes of structure |
91 |
and |
92 |
dynamics.\cite{Tucker:2004qq,Webb:2008kn,Lindquist:2009fk,Fafarman:2010dq} |
93 |
The vibrational Stark effect has also been used to study the effects |
94 |
of electric fields on nitrile-containing self-assembled monolayers at |
95 |
metallic interfaces.\cite{Oklejas:2002uq,Schkolnik:2012ty} |
96 |
|
97 |
|
98 |
Recently 4-cyano-4'-pentylbiphenyl (5CB), a liquid crystalline |
99 |
molecule with a terminal nitrile group, has seen renewed interest as |
100 |
one way to impart order on the surfactant interfaces of |
101 |
nanodroplets,\cite{Moreno-Razo:2012rz} or to drive surface-ordering |
102 |
that can be used to promote particular kinds of |
103 |
self-assembly.\cite{PhysRevLett.111.227801} The nitrile group in 5CB |
104 |
is a particularly interesting case for studying electric field |
105 |
effects, as 5CB exhibits an isotropic to nematic phase transition that |
106 |
can be triggered by the application of an external field near room |
107 |
temperature.\cite{Gray:1973ca,Hatta:1991ee} This presents the |
108 |
possiblity that the field-induced changes in the local environment |
109 |
could have dramatic effects on the vibrations of this particular CN |
110 |
bond. Although the infrared spectroscopy of 5CB has been |
111 |
well-investigated, particularly as a measure of the kinetics of the |
112 |
phase transition,\cite{Leyte:1997zl} the 5CB nitrile group has not yet |
113 |
seen the detailed theoretical treatment that biologically-relevant |
114 |
small molecules have |
115 |
received.\cite{Lindquist:2008bh,Lindquist:2008qf,Oh:2008fk,Choi:2008cr,Morales:2009fp,Waegele:2010ve} |
116 |
|
117 |
The fundamental characteristic of liquid crystal mesophases is that |
118 |
they maintain some degree of orientational order while translational |
119 |
order is limited or absent. This orientational order produces a |
120 |
complex direction-dependent response to external perturbations like |
121 |
electric fields and mechanical distortions. The anisotropy of the |
122 |
macroscopic phases originates in the anisotropy of the constituent |
123 |
molecules, which typically have highly non-spherical structures with a |
124 |
significant degree of internal rigidity. In nematic phases, rod-like |
125 |
molecules are orientationally ordered with isotropic distributions of |
126 |
molecular centers of mass. For example, 5CB has a solid to nematic |
127 |
phase transition at 18C and a nematic to isotropic transition at |
128 |
35C.\cite{Gray:1973ca} |
129 |
|
130 |
In smectic phases, the molecules arrange themselves into layers with |
131 |
their long (symmetry) axis normal ($S_{A}$) or tilted ($S_{C}$) with |
132 |
respect to the layer planes. The behavior of the $S_{A}$ phase can be |
133 |
partially explained with models mainly based on geometric factors and |
134 |
van der Waals interactions. The Gay-Berne potential, in particular, |
135 |
has been widely used in the liquid crystal community to describe this |
136 |
anisotropic phase |
137 |
behavior.~\cite{Gay:1981yu,Berne:1972pb,Kushick:1976xy,Luckhurst:1990fy,Cleaver:1996rt} |
138 |
However, these simple models are insufficient to describe liquid |
139 |
crystal phases which exhibit more complex polymorphic nature. |
140 |
Molecules which form $S_{A}$ phases can exhibit a wide variety of |
141 |
subphases like monolayers ($S_{A1}$), uniform bilayers ($S_{A2}$), |
142 |
partial bilayers ($S_{\tilde A}$) as well as interdigitated bilayers |
143 |
($S_{A_{d}}$), and often have a terminal cyano or nitro group. In |
144 |
particular, lyotropic liquid crystals (those exhibiting liquid crystal |
145 |
phase transitions as a function of water concentration), often have |
146 |
polar head groups or zwitterionic charge separated groups that result |
147 |
in strong dipolar interactions,\cite{Collings:1997rz} and terminal |
148 |
cyano groups (like the one in 5CB) can induce permanent longitudinal |
149 |
dipoles.\cite{Levelut:1981eu} Modeling of the phase behavior of these |
150 |
molecules either requires additional dipolar |
151 |
interactions,\cite{Bose:2012eu} or a unified-atom approach utilizing |
152 |
point charges on the sites that contribute to the dipole |
153 |
moment.\cite{Zhang:2011hh} |
154 |
|
155 |
Macroscopic electric fields applied using electrodes on opposing sides |
156 |
of a sample of 5CB have demonstrated the phase change of the molecule |
157 |
as a function of electric field.\cite{Lim:2006xq} These previous |
158 |
studies have shown the nitrile group serves as an excellent indicator |
159 |
of the molecular orientation within the applied field. Lee {\it et |
160 |
al.}~showed a 180 degree change in field direction could be probed |
161 |
with the nitrile peak intensity as it changed along with molecular |
162 |
alignment in the field.\cite{Lee:2006qd,Leyte:1997zl} |
163 |
|
164 |
While these macroscopic fields work well at indicating the bulk |
165 |
response, the response at a molecular scale has not been studied. With |
166 |
the advent of nano-electrodes and the ability to couple these |
167 |
electrodes to atomic force microscopy, control of electric fields |
168 |
applied across nanometer distances is now possible.\cite{citation1} In |
169 |
special cases where the macroscopic fields are insufficient to cause |
170 |
an observable Stark effect without dielectric breakdown of the |
171 |
material, small potentials across nanometer-sized gaps may be of |
172 |
sufficient strength. For a gap of 5 nm between a lower electrode |
173 |
having a nanoelectrode placed near it via an atomic force microscope, |
174 |
a potential of 1 V applied across the electrodes is equivalent to a |
175 |
field of 2x10\textsuperscript{8} $\frac{V}{M}$. This field is |
176 |
certainly strong enough to cause the isotropic-nematic phase change |
177 |
and as well as a visible Stark tuning of the nitrile bond. We expect |
178 |
that this would be readily visible experimentally through Raman or IR |
179 |
spectroscopy. |
180 |
|
181 |
In the sections that follow, we outline a series of coarse-grained |
182 |
classical molecular dynamics simulations of 5CB that were done in the |
183 |
presence of static electric fields. These simulations were then |
184 |
coupled with both {\it ab intio} calculations of CN-deformations and |
185 |
classical bond-length correlation functions to predict spectral |
186 |
shifts. These predictions made should be easily varifiable with |
187 |
scanning electrochemical microscopy experiments. |
188 |
|
189 |
\section{Computational Details} |
190 |
The force field used for 5CB was a united-atom model that was |
191 |
parameterized by Guo {\it et al.}\cite{Zhang:2011hh} However, for most |
192 |
of the simulations, each of the phenyl rings was treated as a rigid |
193 |
body to allow for larger time steps and very long simulation times. |
194 |
The geometries of the rigid bodies were taken from equilibrium bond |
195 |
distances and angles. Although the individual phenyl rings were held |
196 |
rigid, bonds, bends, torsions and inversion centers that involved |
197 |
atoms in these substructures (but with connectivity to the rest of the |
198 |
molecule) were still included in the potential and force calculations. |
199 |
|
200 |
Periodic simulations cells containing 270 molecules in random |
201 |
orientations were constructed and were locked at experimental |
202 |
densities. Electrostatic interactions were computed using damped |
203 |
shifted force (DSF) electrostatics.\cite{Fennell:2006zl} The molecules |
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were equilibrated for 1~ns at a temperature of 300K. Simulations with |
205 |
applied fields were carried out in the microcanonical (NVE) ensemble |
206 |
with an energy corresponding to the average energy from the canonical |
207 |
(NVT) equilibration runs. Typical applied-field equilibration runs |
208 |
were more than 60ns in length. |
209 |
|
210 |
Static electric fields with magnitudes similar to what would be |
211 |
available in an experimental setup were applied to the different |
212 |
simulations. With an assumed electrode seperation of 5 nm and an |
213 |
electrostatic potential that is limited by the voltage required to |
214 |
split water (1.23V), the maximum realistic field that could be applied |
215 |
is $\sim 0.024$ V/\AA. Three field environments were investigated: |
216 |
(1) no field applied, (2) partial field = 0.01 V/\AA\ , and (3) full |
217 |
field = 0.024 V/\AA\ . |
218 |
|
219 |
After the systems had come to equilibrium under the applied fields, |
220 |
additional simulations were carried out with a flexible (Morse) |
221 |
nitrile bond, |
222 |
\begin{displaymath} |
223 |
V(r_\ce{CN}) = D_e \left(1 - e^{-\beta (r_\ce{CN}-r_e)}\right)^2 |
224 |
\label{eq:morse} |
225 |
\end{displaymath} |
226 |
where $r_e= 1.157437$ \AA , $D_e = 212.95 \mathrm{~kcal~} / |
227 |
\mathrm{mol}^{-1}$ and $\beta = 2.67566 $\AA~$^{-1}$. These |
228 |
parameters correspond to a vibrational frequency of $2358 |
229 |
\mathrm{~cm}^{-1}$, somewhat higher than the experimental |
230 |
frequency. The flexible nitrile moiety required simulation time steps |
231 |
of 1~fs, so the additional flexibility was introducuced only after the |
232 |
rigid systems had come to equilibrium under the applied fields. |
233 |
Whenever time correlation functions were computed from the flexible |
234 |
simulations, statistically-independent configurations were sampled |
235 |
from the last ns of the induced-field runs. These configurations were |
236 |
then equilibrated with the flexible nitrile moiety for 100 ps, and |
237 |
time correlation functions were computed using data sampled from an |
238 |
additional 200 ps of run time carried out in the microcanonical |
239 |
ensemble. |
240 |
|
241 |
\section{Field-induced Nematic Ordering} |
242 |
|
243 |
In order to characterize the orientational ordering of the system, the |
244 |
primary quantity of interest is the nematic (orientational) order |
245 |
parameter. This was determined using the tensor |
246 |
\begin{equation} |
247 |
Q_{\alpha \beta} = \frac{1}{2N} \sum_{i=1}^{N} \left(3 \hat{u}_{i |
248 |
\alpha} \hat{u}_{i \beta} - \delta_{\alpha \beta} \right) |
249 |
\end{equation} |
250 |
where $\alpha, \beta = x, y, z$, and $\hat{u}_i$ is the molecular |
251 |
end-to-end unit vector for molecule $i$. The nematic order parameter |
252 |
$S$ is the largest eigenvalue of $Q_{\alpha \beta}$, and the |
253 |
corresponding eigenvector defines the director axis for the phase. |
254 |
$S$ takes on values close to 1 in highly ordered (smectic A) phases, |
255 |
but falls to much smaller values ($0 \rightarrow 0.3$) for isotropic |
256 |
fluids. Note that the nitrogen and the terminal chain atom were used |
257 |
to define the vectors for each molecule, so the typical order |
258 |
parameters are lower than if one defined a vector using only the rigid |
259 |
core of the molecule. In nematic phases, typical values for $S$ are |
260 |
close to 0.5. |
261 |
|
262 |
The field-induced phase transition can be clearly seen over the course |
263 |
of a 60 ns equilibration runs in figure \ref{fig:orderParameter}. All |
264 |
three of the systems started in a random (isotropic) packing, with |
265 |
order parameters near 0.2. Over the course 10 ns, the full field |
266 |
causes an alignment of the molecules (due primarily to the interaction |
267 |
of the nitrile group dipole with the electric field). Once this |
268 |
system began exhibiting nematic ordering, the orientational order |
269 |
parameter became stable for the remaining 150 ns of simulation time. |
270 |
It is possible that the partial-field simulation is meta-stable and |
271 |
given enough time, it would eventually find a nematic-ordered phase, |
272 |
but the partial-field simulation was stable as an isotropic phase for |
273 |
the full duration of a 60 ns simulation. Ellipsoidal renderings of the |
274 |
final configurations of the runs shows that the full-field (0.024 |
275 |
V/\AA\ ) experienced a isotropic-nematic phase transition and has |
276 |
ordered with a director axis that is parallel to the direction of the |
277 |
applied field. |
278 |
|
279 |
\begin{figure}[H] |
280 |
\includegraphics[width=\linewidth]{Figure1} |
281 |
\caption{Evolution of the orientational order parameters for the |
282 |
no-field, partial field, and full field simulations over the |
283 |
course of 60 ns. Each simulation was started from a |
284 |
statistically-independent isotropic configuration. On the right |
285 |
are ellipsoids representing the final configurations at three |
286 |
different field strengths: zero field (bottom), partial field |
287 |
(middle), and full field (top)} |
288 |
\label{fig:orderParameter} |
289 |
\end{figure} |
290 |
|
291 |
|
292 |
\section{Sampling the CN bond frequency} |
293 |
|
294 |
The vibrational frequency of the nitrile bond in 5CB depends on |
295 |
features of the local solvent environment of the individual molecules |
296 |
as well as the bond's orientation relative to the applied field. The |
297 |
primary quantity of interest for interpreting the condensed phase |
298 |
spectrum of this vibration is the distribution of frequencies |
299 |
exhibited by the 5CB nitrile bond under the different electric fields. |
300 |
There have been a number of elegant techniques for obtaining |
301 |
vibrational lineshapes from classical simulations, including a |
302 |
perturbation theory approach,\cite{Morales:2009fp} the use of an |
303 |
optimized QM/MM approach coupled with the fluctuating frequency |
304 |
approximation,\cite{Lindquist:2008qf} and empirical frequency |
305 |
correlation maps.\cite{Oh:2008fk} Three distinct (and comparatively |
306 |
primitive) methods for mapping classical simulations onto vibrational |
307 |
spectra were brought to bear on the simulations in this work: |
308 |
\begin{enumerate} |
309 |
\item Isolated 5CB molecules and their immediate surroundings were |
310 |
extracted from the simulations. These nitrile bonds were stretched |
311 |
and single-point {\em ab initio} calculations were used to obtain |
312 |
Morse-oscillator fits for the local vibrational motion along that |
313 |
bond. |
314 |
\item A static-field extension of the empirical frequency correlation |
315 |
maps developed by Cho {\it et al.}~\cite{Oh:2008fk} for nitrile |
316 |
moieties in water was attempted. |
317 |
\item Classical bond-length autocorrelation functions were Fourier |
318 |
transformed to directly obtain the vibrational spectrum from |
319 |
molecular dynamics simulations. |
320 |
\end{enumerate} |
321 |
|
322 |
\subsection{CN frequencies from isolated clusters} |
323 |
The size of the periodic condensed phase system prevented direct |
324 |
computation of the complete library of nitrile bond frequencies using |
325 |
{\it ab initio} methods. In order to sample the nitrile frequencies |
326 |
present in the condensed-phase, individual molecules were selected |
327 |
randomly to serve as the center of a local (gas phase) cluster. To |
328 |
include steric, electrostatic, and other effects from molecules |
329 |
located near the targeted nitrile group, portions of other molecules |
330 |
nearest to the nitrile group were included in the quantum mechanical |
331 |
calculations. The surrounding solvent molecules were divided into |
332 |
``body'' (the two phenyl rings and the nitrile bond) and ``tail'' (the |
333 |
alkyl chain). Any molecule which had a body atom within 6~\AA\ of the |
334 |
midpoint of the target nitrile bond had its own molecular body (the |
335 |
4-cyano-biphenyl moiety) included in the configuration. Likewise, the |
336 |
entire alkyl tail was included if any tail atom was within 4~\AA\ of |
337 |
the target nitrile bond. If tail atoms (but no body atoms) were |
338 |
included within these distances, only the tail was included as a |
339 |
capped propane molecule. |
340 |
|
341 |
\begin{figure}[H] |
342 |
\includegraphics[width=\linewidth]{Figure2} |
343 |
\caption{Cluster calculations were performed on randomly sampled 5CB |
344 |
molecules (shown in red) from each of the simulations. Surrounding |
345 |
molecular bodies were included if any body atoms were within 6 |
346 |
\AA\ of the target nitrile bond, and tails were included if they |
347 |
were within 4 \AA. Included portions of these molecules are shown |
348 |
in green. The CN bond on the target molecule was stretched and |
349 |
compressed, and the resulting single point energies were fit to |
350 |
Morse oscillators to obtain a distribution of frequencies.} |
351 |
\label{fig:cluster} |
352 |
\end{figure} |
353 |
|
354 |
Inferred hydrogen atom locations were added to the cluster geometries, |
355 |
and the nitrile bond was stretched from 0.87 to 1.52~\AA\ at |
356 |
increments of 0.05~\AA. This generated 13 configurations per gas phase |
357 |
cluster. Single-point energies were computed using the B3LYP |
358 |
functional~\cite{Becke:1993kq,Lee:1988qf} and the 6-311++G(d,p) basis |
359 |
set. For the cluster configurations that had been generated from |
360 |
molecular dynamics running under applied fields, the density |
361 |
functional calculations had a field of $5 \times 10^{-4}$ atomic units |
362 |
($E_h / (e a_0)$) applied in the $+z$ direction in order to match the |
363 |
molecular dynamics simulations. |
364 |
|
365 |
The energies for the stretched / compressed nitrile bond in each of |
366 |
the clusters were used to fit Morse potentials, and the frequencies |
367 |
were obtained from the $0 \rightarrow 1$ transition for the energy |
368 |
levels for this potential.\cite{Morse:1929xy} To obtain a spectrum, |
369 |
each of the frequencies was convoluted with a Lorentzian lineshape |
370 |
with a width of 1.5 $\mathrm{cm}^{-1}$. Available computing resources |
371 |
limited the sampling to 67 clusters for the zero-field spectrum, and |
372 |
59 for the full field. Comparisons of the quantum mechanical spectrum |
373 |
to the classical are shown in figure \ref{fig:spectrum}. |
374 |
|
375 |
\subsection{CN frequencies from potential-frequency maps} |
376 |
|
377 |
One approach which has been used to successfully analyze the spectrum |
378 |
of nitrile and thiocyanate probes in aqueous environments was |
379 |
developed by Choi {\it et al.}~\cite{Choi:2008cr,Oh:2008fk} This |
380 |
method involves finding a multi-parameter fit that maps between the |
381 |
local electrostatic potential at selected sites surrounding the |
382 |
nitrile bond and the vibrational frequency of that bond obtained from |
383 |
more expensive {\it ab initio} methods. This approach is similar in |
384 |
character to the field-frequency maps developed by the Skinner group |
385 |
for OH stretches in liquid water.\cite{Corcelli:2004ai,Auer:2007dp} |
386 |
|
387 |
To use the potential-frequency maps, the local electrostatic |
388 |
potential, $\phi_a$, is computed at 20 sites ($a = 1 \rightarrow 20$) |
389 |
that surround the nitrile bond, |
390 |
\begin{equation} |
391 |
\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{j} |
392 |
\frac{q_j}{\left|r_{aj}\right|}. |
393 |
\end{equation} |
394 |
Here $q_j$ is the partial site on atom $j$ (residing on a different |
395 |
molecule) and $r_{aj}$ is the distance between site $a$ and atom $j$. |
396 |
The original map was parameterized in liquid water and comprises a set |
397 |
of parameters, $l_a$, that predict the shift in nitrile peak |
398 |
frequency, |
399 |
\begin{equation} |
400 |
\delta\tilde{\nu} =\sum^{20}_{a=1} l_{a}\phi_{a}. |
401 |
\end{equation} |
402 |
|
403 |
The simulations of 5CB were carried out in the presence of |
404 |
externally-applied uniform electric fields. Although uniform fields |
405 |
exert forces on charge sites, they only contribute to the potential if |
406 |
one defines a reference point that can serve as an origin. One simple |
407 |
modification to the potential at each of the probe sites is to use the |
408 |
centroid of the \ce{CN} bond as the origin for that site, |
409 |
\begin{equation} |
410 |
\phi_a^\prime = \phi_a + \frac{1}{4\pi\epsilon_{0}} \vec{E} \cdot |
411 |
\left(\vec{r}_a - \vec{r}_\ce{CN} \right) |
412 |
\end{equation} |
413 |
where $\vec{E}$ is the uniform electric field, $\left( \vec{r}_{a} - |
414 |
\vec{r}_\ce{CN} \right)$ is the displacement between the |
415 |
cooridinates described by Choi {\it et |
416 |
al.}~\cite{Choi:2008cr,Oh:2008fk} and the \ce{CN} bond centroid. |
417 |
$\phi_a^\prime$ then contains an effective potential contributed by |
418 |
the uniform field in addition to the local potential contributions |
419 |
from other molecules. |
420 |
|
421 |
The sites $\{\vec{r}_a\}$ and weights $\left\{l_a \right\}$ |
422 |
developed by Choi {\it et al.}~\cite{Choi:2008cr,Oh:2008fk} are quite |
423 |
symmetric around the \ce{CN} centroid, and even at large uniform field |
424 |
values we observed nearly-complete cancellation of the potenial |
425 |
contributions from the uniform field. In order to utilize the |
426 |
potential-frequency maps for this problem, one would therefore need |
427 |
extensive reparameterization of the maps to include explicit |
428 |
contributions from the external field. This reparameterization is |
429 |
outside the scope of the current work, but would make a useful |
430 |
addition to the potential-frequency map approach. |
431 |
|
432 |
\subsection{CN frequencies from bond length autocorrelation functions} |
433 |
|
434 |
The distribution of nitrile vibrational frequencies can also be found |
435 |
using classical time correlation functions. This was done by |
436 |
replacing the rigid \ce{CN} bond with a flexible Morse oscillator |
437 |
described in Eq. \ref{eq:morse}. Since the systems were perturbed by |
438 |
the addition of a flexible high-frequency bond, they were allowed to |
439 |
re-equilibrate in the canonical (NVT) ensemble for 100 ps with 1 fs |
440 |
timesteps. After equilibration, each configuration was run in the |
441 |
microcanonical (NVE) ensemble for 20 ps. Configurations sampled every |
442 |
fs were then used to compute bond-length autocorrelation functions, |
443 |
\begin{equation} |
444 |
C(t) = \langle \delta r(t) \cdot \delta r(0) ) \rangle |
445 |
\end{equation} |
446 |
% |
447 |
where $\delta r(t) = r(t) - r_0$ is the deviation from the equilibrium |
448 |
bond distance at time $t$. Because the other atomic sites have very |
449 |
small partial charges, this correlation function is an approximation |
450 |
to the dipole autocorrelation function for the molecule, which would |
451 |
be particularly relevant to computing the IR spectrum. Ten |
452 |
statistically-independent correlation functions were obtained by |
453 |
allowing the systems to run 10 ns with rigid \ce{CN} bonds followed by |
454 |
120 ps equilibration and data collection using the flexible \ce{CN} |
455 |
bonds. This process was repeated 10 times, and the total sampling |
456 |
time, from sample preparation to final configurations, exceeded 150 ns |
457 |
for each of the field strengths investigated. |
458 |
|
459 |
The correlation functions were filtered using exponential apodization |
460 |
functions,\cite{FILLER:1964yg} $f(t) = e^{-|t|/c}$, with a time |
461 |
constant, $c =$ 3.5 ps, and were Fourier transformed to yield a |
462 |
spectrum, |
463 |
\begin{equation} |
464 |
I(\omega) = \int_{-\infty}^{\infty} C(t) f(t) e^{-i \omega t} dt. |
465 |
\end{equation} |
466 |
The sample-averaged classical nitrile spectrum can be seen in Figure |
467 |
\ref{fig:spectra}. Note that the Morse oscillator parameters listed |
468 |
above yield a natural frequency of 2358 $\mathrm{cm}^{-1}$, somewhat |
469 |
higher than the experimental peak near 2226 $\mathrm{cm}^{-1}$. This |
470 |
shift does not effect the ability to qualitatively compare peaks from |
471 |
the classical and quantum mechanical approaches, so the classical |
472 |
spectra are shown as a shift relative to the natural oscillation of |
473 |
the Morse bond. |
474 |
|
475 |
\begin{figure} |
476 |
\includegraphics[width=3.25in]{Convolved} |
477 |
\includegraphics[width=3.25in]{2Spectra} |
478 |
\caption{Quantum mechanical nitrile spectrum for the no-field simulation |
479 |
(black) and the full field simulation (red). The lower panel |
480 |
shows the corresponding classical bond-length autocorrelation |
481 |
spectrum for the flexible nitrile measured relative to the natural |
482 |
frequency for the flexible bond.} |
483 |
\label{fig:spectra} |
484 |
\end{figure} |
485 |
|
486 |
Note that due to electrostatic interactions, the classical approach |
487 |
implicitly couples \ce{CN} vibrations to the same vibrational mode on |
488 |
other nearby molecules. This coupling is not handled in the {\it ab |
489 |
initio} cluster approach. |
490 |
|
491 |
\section{Discussion} |
492 |
|
493 |
Our simulations show that the united-atom model can reproduce the |
494 |
field-induced nematic ordering of the 4-cyano-4'-pentylbiphenyl. |
495 |
Because we are simulating a very small electrode separation (5~nm), a |
496 |
voltage drop as low as 1.2~V was sufficient to induce the phase |
497 |
change. This potential is significantly smaller than the 500~V that is |
498 |
known to cause dielectric breakdown in 5CB,\cite{XXX} and suggests |
499 |
that by using close electrode separation, it would be relatively |
500 |
straightforward to observe the nitrile Stark shift in 5CB. |
501 |
|
502 |
Both the classical correlation function and the isolated cluster |
503 |
approaches to estimating the IR spectrum show that a small population |
504 |
of nitrile stretches shift by $\sim 40 \mathrm{cm}^{-1}$ to the red of |
505 |
the unperturbed vibrational line. To understand the origin of this |
506 |
shift, a more complete picture of the spatial ordering around the |
507 |
nitrile bonds is required. We have computed the angle-dependent pair |
508 |
distribution functions, |
509 |
\begin{align} |
510 |
g(r, \cos \omega) = & \frac{1}{\rho N} \left< \sum_{i} |
511 |
\sum_{j} \delta \left(r - r_{ij}\right) \delta\left(\cos \omega_{ij} - |
512 |
\cos \omega\right) \right> \\ \nonumber \\ |
513 |
g(r, \cos \theta) = & \frac{1}{\rho N} \left< \sum_{i} |
514 |
\sum_{j} \delta \left(r - r_{ij}\right) \delta\left(\cos \theta_{i} - |
515 |
\cos \theta \right) \right> |
516 |
\end{align} |
517 |
which provide information about the joint spatial and angular |
518 |
correlations present in the system. The angles $\omega$ and $\theta$ |
519 |
are defined by vectors along the CN axis of each nitrile bond (see |
520 |
figure \ref{fig:definition}). |
521 |
\begin{figure} |
522 |
\includegraphics[width=4in]{definition} |
523 |
\caption{Definitions of the angles between two nitrile bonds.} |
524 |
\label{fig:definition} |
525 |
\end{figure} |
526 |
|
527 |
The primary structural effect of the field-induced phase transition is |
528 |
apparent in figure \ref{fig:gofromega}. The nematic ordering transfers |
529 |
population from the perpendicular ($\cos\omega\approx 0$) and |
530 |
anti-aligned ($\cos\omega\approx -1$) to the nitrile-alinged peak |
531 |
near $\cos\omega\approx 1$, leaving most other features undisturbed. This |
532 |
change is visible in the simulations as an increased population of |
533 |
aligned nitrile bonds in the first solvation shell. |
534 |
\begin{figure} |
535 |
\includegraphics[width=\linewidth]{Figure4} |
536 |
\caption{Contours of the angle-dependent pair distribution functions |
537 |
for nitrile bonds on 5CB in the no field (upper panel) and full |
538 |
field (lower panel) simulations. Dark areas signify regions of |
539 |
enhanced density, while light areas signify depletion relative to |
540 |
the bulk density.} |
541 |
\label{fig:gofromega} |
542 |
\end{figure} |
543 |
Although it is certainly possible that the coupling between |
544 |
closely-spaced nitrile pairs is responsible for some of the red-shift, |
545 |
that is not the only structural change that is taking place. The |
546 |
second two-dimensional pair distribution function, $g(r,\cos\theta)$, |
547 |
shows that nematic ordering also transfers population that is directly |
548 |
in line with the nitrile bond (see figure \ref{fig:gofrtheta}) to the |
549 |
sides of the molecule, thereby freeing steric blockage can directly |
550 |
influence the nitrile vibration. We are suggesting here that the |
551 |
nematic ordering provides an anti-caging of the nitrile vibration, and |
552 |
given that the oscillator is fairly anharmonic, this provides a |
553 |
fraction of the nitrile bonds with a significant red-shift. |
554 |
\begin{figure} |
555 |
\includegraphics[width=\linewidth]{Figure6} |
556 |
\caption{Contours of the angle-dependent pair distribution function, |
557 |
$g(r,\cos \theta)$, for finding any other atom at a distance and |
558 |
angular deviation from the center of a nitrile bond. The top edge |
559 |
of each contour plot corresponds to local density along the |
560 |
direction of the nitrogen in the CN bond, while the bottom is in |
561 |
the direction of the carbon atom. Bottom panel: $g(z)$ data taken |
562 |
by following the \ce{C -> N} vector for each nitrile bond shows |
563 |
that the field-induced phase transition reduces the population of |
564 |
atoms that are directly in line with the nitrogen motion.} |
565 |
\label{fig:gofrtheta} |
566 |
\end{figure} |
567 |
|
568 |
The cause of this shift does not appear to be related to the alignment |
569 |
of those nitrile bonds with the field, but rather to the change in |
570 |
local environment that is brought about by the isotropic-nematic |
571 |
transition. We have compared configurations for many of the cluster |
572 |
calculations that exhibited the frequencies between (2190 and 2215 |
573 |
$\mathrm{cm}^{-1}$) , and have observed some similar features. The |
574 |
lowest frequencies appear to come from configurations which have |
575 |
nearly-empty pockets directly opposite the nitrogen atom from the |
576 |
nitrile carbon. Because we have so few clusters, this is certainly not |
577 |
quantitative confirmation of this effect. |
578 |
|
579 |
|
580 |
While this makes the application of nitrile Stark effects in |
581 |
simulations without water harder, these data show |
582 |
that it is not a deal breaker. The classically calculated nitrile |
583 |
spectrum shows changes in the spectra that will be easily seen through |
584 |
experimental routes. It indicates a shifted peak lower in energy |
585 |
should arise. This peak is a few wavenumbers from the leading edge of |
586 |
the larger peak and almost 75 wavenumbers from the center. This |
587 |
seperation between the two peaks means experimental results will show |
588 |
an easily resolved peak. |
589 |
|
590 |
The Gaussian derived spectra do indicate an applied field |
591 |
and subsiquent phase change does cause a narrowing of freuency |
592 |
distrobution. With narrowing, it would indicate an increased |
593 |
homogeneous distrobution of the local field near the nitrile. |
594 |
|
595 |
|
596 |
|
597 |
\section{Conclusions} |
598 |
Field dependent changes |
599 |
|
600 |
\section{Acknowledgements} |
601 |
The authors thank Steven Corcelli for helpful comments and |
602 |
suggestions. Support for this project was provided by the National |
603 |
Science Foundation under grant CHE-0848243. Computational time was |
604 |
provided by the Center for Research Computing (CRC) at the University |
605 |
of Notre Dame. |
606 |
|
607 |
\newpage |
608 |
|
609 |
\bibliography{5CB} |
610 |
|
611 |
\end{doublespace} |
612 |
\end{document} |