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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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\keywords{} |
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|
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\begin{document} |
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|
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|
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|
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\begin{tocentry} |
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%\includegraphics[width=9cm]{Elip_3} |
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\includegraphics[width=9cm]{cluster.pdf} |
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\end{tocentry} |
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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 |
52 |
with a terminal nitrile group aligned with the long axis of the |
53 |
molecule. Simulations of condensed-phase 5CB were carried out both |
54 |
with and without applied electric fields to provide an understanding |
55 |
of the Stark shift of the terminal nitrile group. A field-induced |
56 |
isotropic-nematic phase transition was observed in the simulations, |
57 |
and the effects of this transition on the distribution of nitrile |
58 |
frequencies were computed. Classical bond displacement correlation |
59 |
functions exhibit a $\sim~3~\mathrm{cm}^{-1}$ red shift of a portion |
60 |
of the main nitrile peak, and this shift was observed only when the |
61 |
fields were large enough to induce orientational ordering of the |
62 |
bulk phase. Distributions of frequencies obtained via cluster-based |
63 |
fits to quantum mechanical energies of nitrile bond deformations |
64 |
exhibit a similar $\sim~2.7~\mathrm{cm}^{-1}$ red shift. Joint |
65 |
spatial-angular distribution functions indicate that phase-induced |
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anti-caging of the nitrile bond is contributing to the change in the |
67 |
nitrile spectrum. |
68 |
\end{abstract} |
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|
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\newpage |
71 |
|
72 |
\section{Introduction} |
73 |
|
74 |
Because the triple bond between nitrogen and carbon is sensitive to |
75 |
local electric fields, nitrile groups can report on field strengths |
76 |
via their distinctive Raman and IR signatures.\cite{Boxer:2009xw} The |
77 |
response of nitrile groups to electric fields has now been |
78 |
investigated for a number of small molecules,\cite{Andrews:2000qv} as |
79 |
well as in biochemical settings, where nitrile groups can act as |
80 |
minimally invasive probes of structure and |
81 |
dynamics.\cite{Tucker:2004qq,Webb:2008kn,Lindquist:2009fk,Fafarman:2010dq} |
82 |
The vibrational Stark effect has also been used to study the effects |
83 |
of electric fields on nitrile-containing self-assembled monolayers at |
84 |
metallic interfaces.\cite{Oklejas:2002uq,Schkolnik:2012ty} |
85 |
|
86 |
Recently 4-cyano-4'-pentylbiphenyl (5CB), a liquid crystalline |
87 |
molecule with a terminal nitrile group, has seen renewed interest as |
88 |
one way to impart order on the surfactant interfaces of |
89 |
nanodroplets,\cite{Moreno-Razo:2012rz} or to drive surface-ordering |
90 |
that can be used to promote particular kinds of |
91 |
self-assembly.\cite{PhysRevLett.111.227801} The nitrile group in 5CB |
92 |
is a particularly interesting case for studying electric field |
93 |
effects, as 5CB exhibits an isotropic to nematic phase transition that |
94 |
can be triggered by the application of an external field near room |
95 |
temperature.\cite{Gray:1973ca,Hatta:1991ee} This presents the |
96 |
possibility that the field-induced changes in the local environment |
97 |
could have dramatic effects on the vibrations of this particular nitrile |
98 |
bond. Although the infrared spectroscopy of 5CB has been |
99 |
well-investigated, particularly as a measure of the kinetics of the |
100 |
phase transition,\cite{Leyte:1997zl} the 5CB nitrile group has not yet |
101 |
seen the detailed theoretical treatment that biologically-relevant |
102 |
small molecules have |
103 |
received.\cite{Lindquist:2008bh,Lindquist:2008qf,Oh:2008fk,Choi:2008cr,Morales:2009fp,Waegele:2010ve} |
104 |
|
105 |
The fundamental characteristic of liquid crystal mesophases is that |
106 |
they maintain some degree of orientational order while translational |
107 |
order is limited or absent. This orientational order produces a |
108 |
complex direction-dependent response to external perturbations like |
109 |
electric fields and mechanical distortions. The anisotropy of the |
110 |
macroscopic phases originates in the anisotropy of the constituent |
111 |
molecules, which typically have highly non-spherical structures with a |
112 |
significant degree of internal rigidity. In nematic phases, rod-like |
113 |
molecules are orientationally ordered with isotropic distributions of |
114 |
molecular centers of mass. For example, 5CB has a solid to nematic |
115 |
phase transition at 18C and a nematic to isotropic transition at |
116 |
35C.\cite{Gray:1973ca} |
117 |
|
118 |
In smectic phases, the molecules arrange themselves into layers with |
119 |
their long (symmetry) axis normal ($S_{A}$) or tilted ($S_{C}$) with |
120 |
respect to the layer planes. The behavior of the $S_{A}$ phase can be |
121 |
explained with models based solely on geometric factors and van der |
122 |
Waals interactions. The Gay-Berne potential, in particular, has been |
123 |
widely used in the liquid crystal community to describe this |
124 |
anisotropic phase |
125 |
behavior.~\cite{Gay:1981yu,Berne:1972pb,Kushick:1976xy,Luckhurst:1990fy,Cleaver:1996rt} |
126 |
However, these simple models are insufficient to describe liquid |
127 |
crystal phases which exhibit more complex polymorphic nature. |
128 |
Molecules which form $S_{A}$ phases can exhibit a wide variety of |
129 |
sub-phases like monolayers ($S_{A1}$), uniform bilayers ($S_{A2}$), |
130 |
partial bilayers ($S_{\tilde A}$) as well as interdigitated bilayers |
131 |
($S_{A_{d}}$), and often have a terminal cyano or nitro group. In |
132 |
particular, lyotropic liquid crystals (those exhibiting liquid crystal |
133 |
phase transitions as a function of water concentration), often have |
134 |
polar head groups or zwitterionic charge separated groups that result |
135 |
in strong dipolar interactions,\cite{Collings:1997rz} and terminal |
136 |
cyano groups (like the one in 5CB) can induce permanent longitudinal |
137 |
dipoles.\cite{Levelut:1981eu} Modeling of the phase behavior of these |
138 |
molecules either requires additional dipolar |
139 |
interactions,\cite{Bose:2012eu} or a unified-atom approach utilizing |
140 |
point charges on the sites that contribute to the dipole |
141 |
moment.\cite{Zhang:2011hh} |
142 |
|
143 |
Macroscopic electric fields applied using electrodes on opposing sides |
144 |
of a sample of 5CB have demonstrated the phase change of the molecule |
145 |
as a function of electric field.\cite{Lim:2006xq} These previous |
146 |
studies have shown the nitrile group serves as an excellent indicator |
147 |
of the molecular orientation within the applied field. Lee {\it et |
148 |
al.}~showed a 180 degree change in field direction could be probed |
149 |
with the nitrile peak intensity as it changed along with molecular |
150 |
alignment in the field.\cite{Lee:2006qd,Leyte:1997zl} |
151 |
|
152 |
While these macroscopic fields work well at indicating the bulk |
153 |
response, the response at a molecular scale has not been studied. With |
154 |
the advent of nano-electrodes and the ability to couple these |
155 |
electrodes to atomic force microscopy, control of electric fields |
156 |
applied across nanometer distances is now possible.\cite{C3AN01651J} |
157 |
In special cases where the macroscopic fields are insufficient to |
158 |
cause an observable Stark effect without dielectric breakdown of the |
159 |
material, small potentials across nanometer-sized gaps may have |
160 |
sufficient strength. For a gap of 5 nm between a lower electrode |
161 |
having a nanoelectrode placed near it via an atomic force microscope, |
162 |
a potential of 1 V applied across the electrodes is equivalent to a |
163 |
field of $2 \times 10^8~\mathrm{V/m}$. This field is certainly strong |
164 |
enough to cause the isotropic-nematic phase change and an observable |
165 |
Stark tuning of the nitrile bond. We expect that this would be readily |
166 |
visible experimentally through Raman or IR spectroscopy. |
167 |
|
168 |
In the sections that follow, we outline a series of coarse-grained |
169 |
(united atom) classical molecular dynamics simulations of 5CB that |
170 |
were done in the presence of static electric fields. These simulations |
171 |
were then coupled with both {\it ab intio} calculations of |
172 |
CN-deformations and classical bond-length correlation functions to |
173 |
predict spectral shifts. These predictions should be verifiable via |
174 |
scanning electrochemical microscopy. |
175 |
|
176 |
\section{Computational Details} |
177 |
The force-field used to model 5CB was a united-atom model that was |
178 |
parameterized by Guo {\it et al.}\cite{Zhang:2011hh} However, for most |
179 |
of the simulations, both of the phenyl rings and the nitrile bond were |
180 |
treated as rigid bodies to allow for larger time steps and longer |
181 |
simulation times. The geometries of the rigid bodies were taken from |
182 |
equilibrium bond distances and angles. Although the individual phenyl |
183 |
rings were held rigid, bonds, bends, torsions and inversion centers |
184 |
that involved atoms in these substructures (but with connectivity to |
185 |
the rest of the molecule) were still included in the potential and |
186 |
force calculations. |
187 |
|
188 |
Periodic simulations cells containing 270 molecules in random |
189 |
orientations were constructed and were locked at experimental |
190 |
densities. Electrostatic interactions were computed using damped |
191 |
shifted force (DSF) electrostatics.\cite{Fennell:2006zl} The molecules |
192 |
were equilibrated for 1~ns at a temperature of 300K. Simulations with |
193 |
applied fields were carried out in the microcanonical (NVE) ensemble |
194 |
with an energy corresponding to the average energy from the canonical |
195 |
(NVT) equilibration runs. Typical applied-field equilibration runs |
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were more than 60~ns in length. |
197 |
|
198 |
Static electric fields with magnitudes similar to what would be |
199 |
available in an experimental setup were applied to the different |
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simulations. With an assumed electrode separation of 5 nm and an |
201 |
electrostatic potential that is limited by the voltage required to |
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split water (1.23V), the maximum realistic field that could be applied |
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is $\sim 0.024$ V/\AA. Three field environments were investigated: |
204 |
(1) no field applied, (2) partial field = 0.01 V/\AA\ , and (3) full |
205 |
field = 0.024 V/\AA\ . |
206 |
|
207 |
After the systems had come to equilibrium under the applied fields, |
208 |
additional simulations were carried out with a flexible (Morse) |
209 |
nitrile bond, |
210 |
\begin{equation} |
211 |
V(r_\ce{CN}) = D_e \left(1 - e^{-\beta (r_\ce{CN}-r_e)}\right)^2 |
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\label{eq:morse} |
213 |
\end{equation} |
214 |
where $r_e= 1.157$ \AA (the fixed CN bond length from the force field |
215 |
of Guo {\it et al.}\cite{Zhang:2011hh}), $D_e = 212.95 \mathrm{~kcal~} |
216 |
/ \mathrm{mol}^{-1}$ (the average bond energy for CN triple bonds) and |
217 |
$\beta = 2.526 $\AA~$^{-1}$. These parameters correspond to a |
218 |
vibrational frequency of $\approx 2226 \mathrm{~cm}^{-1}$, which is |
219 |
very close to the frequency of the nitrile peak in the vibrational |
220 |
spectrum of neat 5CB. The flexible nitrile moiety required simulation |
221 |
time steps of 1~fs, so the additional flexibility was introduced only |
222 |
after the rigid systems had come to equilibrium under the applied |
223 |
fields. Whenever time correlation functions were computed from the |
224 |
flexible simulations, statistically-independent configurations |
225 |
(separated in time by 10 ns) were sampled from the last 110 ns of the |
226 |
induced-field runs. These configurations were then equilibrated with |
227 |
the flexible nitrile moiety for 100 ps, and time correlation functions |
228 |
were computed using data sampled from an additional 20 ps of run time |
229 |
carried out in the microcanonical ensemble. |
230 |
|
231 |
\section{Field-induced Nematic Ordering} |
232 |
|
233 |
In order to characterize the orientational ordering of the system, the |
234 |
primary quantity of interest is the nematic (orientational) order |
235 |
parameter. This was determined using the tensor |
236 |
\begin{equation} |
237 |
Q_{\alpha \beta} = \frac{1}{2N} \sum_{i=1}^{N} \left(3 \hat{u}_{i |
238 |
\alpha} \hat{u}_{i \beta} - \delta_{\alpha \beta} \right) |
239 |
\end{equation} |
240 |
where $\alpha, \beta = x, y, z$, and $\hat{u}_i$ is the molecular |
241 |
end-to-end unit vector for molecule $i$. The nematic order parameter |
242 |
$S$ is the largest eigenvalue of $Q_{\alpha \beta}$, and the |
243 |
corresponding eigenvector defines the director axis for the phase. |
244 |
$S$ takes on values close to 1 in highly ordered (smectic A) phases, |
245 |
but falls to much smaller values ($0 \rightarrow 0.3$) for isotropic |
246 |
fluids. Note that the nitrogen and the terminal chain atom were used |
247 |
to define the vector for each molecule, so the typical order |
248 |
parameters are lower than if one defined a vector using only the rigid |
249 |
core of the molecule. In nematic phases, typical values for $S$ are |
250 |
close to 0.5. |
251 |
|
252 |
The field-induced phase transition can be clearly seen over the course |
253 |
of a 60 ns equilibration runs in figure \ref{fig:orderParameter}. All |
254 |
three of the systems started in a random (isotropic) packing, with |
255 |
order parameters near 0.2. Over the course 10 ns, the full field |
256 |
causes an alignment of the molecules (due primarily to the interaction |
257 |
of the nitrile group dipole with the electric field). Once this |
258 |
system began exhibiting nematic ordering, the orientational order |
259 |
parameter became stable for the remaining 150 ns of simulation time. |
260 |
It is possible that the partial-field simulation is meta-stable and |
261 |
given enough time, it would eventually find a nematic-ordered phase, |
262 |
but the partial-field simulation was stable as an isotropic phase for |
263 |
the full duration of the 60 ns simulation. Ellipsoidal renderings of |
264 |
the final configurations of the runs show that the full-field (0.024 |
265 |
V/\AA\ ) experienced a isotropic-nematic phase transition and has |
266 |
ordered with a director axis that is parallel to the direction of the |
267 |
applied field. |
268 |
|
269 |
\begin{figure}[H] |
270 |
\includegraphics[width=\linewidth]{orderParameter.pdf} |
271 |
\caption{Evolution of the orientational order parameters for the |
272 |
no-field, partial field, and full field simulations over the |
273 |
course of 60 ns. Each simulation was started from a |
274 |
statistically-independent isotropic configuration. On the right |
275 |
are ellipsoids representing the final configurations at three |
276 |
different field strengths: zero field (bottom), partial field |
277 |
(middle), and full field (top)} |
278 |
\label{fig:orderParameter} |
279 |
\end{figure} |
280 |
|
281 |
|
282 |
\section{Sampling the CN bond frequency} |
283 |
|
284 |
The vibrational frequency of the nitrile bond in 5CB depends on |
285 |
features of the local solvent environment of the individual molecules |
286 |
as well as the bond's orientation relative to the applied field. The |
287 |
primary quantity of interest for interpreting the condensed phase |
288 |
spectrum of this vibration is the distribution of frequencies |
289 |
exhibited by the 5CB nitrile bond under the different electric fields. |
290 |
There have been a number of elegant techniques for obtaining |
291 |
vibrational line shapes from classical simulations, including a |
292 |
perturbation theory approach,\cite{Morales:2009fp} the use of an |
293 |
optimized QM/MM approach coupled with the fluctuating frequency |
294 |
approximation,\cite{Lindquist:2008qf} and empirical frequency |
295 |
correlation maps.\cite{Choi:2008cr,Oh:2008fk} Three distinct (and |
296 |
comparatively primitive) methods for mapping classical simulations |
297 |
onto vibrational spectra were brought to bear on the simulations in |
298 |
this work: |
299 |
\begin{enumerate} |
300 |
\item Isolated 5CB molecules and their immediate surroundings were |
301 |
extracted from the simulations. These nitrile bonds were stretched |
302 |
by displacing the nitrogen along the CN bond vector with the carbon |
303 |
atom remaining stationary. Single-point {\em ab initio} calculations |
304 |
were used to obtain Morse-oscillator fits for the local vibrational |
305 |
motion along that bond. |
306 |
\item The empirical frequency correlation maps developed by Choi {\it |
307 |
et al.}~\cite{Choi:2008cr,Oh:2008fk} for nitrile moieties in water |
308 |
were utilized by adding an electric field contribution to the local |
309 |
electrostatic potential. |
310 |
\item Classical bond-length autocorrelation functions were Fourier |
311 |
transformed to directly obtain the vibrational spectrum from |
312 |
molecular dynamics simulations. |
313 |
\end{enumerate} |
314 |
|
315 |
\subsection{CN frequencies from isolated clusters} |
316 |
The size of the condensed phase liquid crystal system prevented direct |
317 |
computation of the complete library of nitrile bond frequencies using |
318 |
{\it ab initio} methods. In order to sample the nitrile frequencies |
319 |
present in the condensed-phase, individual molecules were selected |
320 |
randomly to serve as the center of a local (gas phase) cluster. To |
321 |
include steric, electrostatic, and other effects from molecules |
322 |
located near the targeted nitrile group, portions of other molecules |
323 |
nearest to the nitrile group were included in the quantum mechanical |
324 |
calculations. Steric interactions are generally shorter ranged than |
325 |
electrostatic interactions, so portions of surrounding molecules that |
326 |
cause electrostatic perturbations to the central nitrile (e.g. the |
327 |
biphenyl core and nitrile moieties) must be included if they fall |
328 |
anywhere near the CN bond. Portions of these molecules that interact |
329 |
primarily via dispersion and steric repulsion (e.g. the alkyl tails) |
330 |
can be truncated at a shorter distance. |
331 |
|
332 |
The surrounding solvent molecules were therefore divided into ``body'' |
333 |
(the two phenyl rings and the nitrile bond) and ``tail'' (the alkyl |
334 |
chain). Any molecule which had a body atom within 6~\AA\ of the |
335 |
midpoint of the target nitrile bond had its own molecular body (the |
336 |
4-cyano-biphenyl moiety) included in the configuration. Likewise, the |
337 |
entire alkyl tail was included if any tail atom was within 4~\AA\ of |
338 |
the target nitrile bond. If tail atoms (but no body atoms) were |
339 |
included within these distances, only the tail was included as a |
340 |
capped propane molecule. |
341 |
|
342 |
\begin{figure}[H] |
343 |
\includegraphics[width=\linewidth]{cluster.pdf} |
344 |
\caption{Cluster calculations were performed on randomly sampled 5CB |
345 |
molecules (shown in red) from the full-field and no-field |
346 |
simulations. Surrounding molecular bodies were included if any |
347 |
body atoms were within 6 \AA\ of the target nitrile bond, and |
348 |
tails were included if they were within 4 \AA. Included portions |
349 |
of these molecules are shown in green. The CN bond on the target |
350 |
molecule was stretched and compressed, and the resulting single |
351 |
point energies were fit to Morse oscillators to obtain a |
352 |
distribution of frequencies.} |
353 |
\label{fig:cluster} |
354 |
\end{figure} |
355 |
|
356 |
Inferred hydrogen atom locations were added to the cluster geometries, |
357 |
and the nitrile bond was stretched from 0.87 to 1.52~\AA\ at |
358 |
increments of 0.05~\AA. This generated 13 configurations per gas phase |
359 |
cluster. Single-point energies were computed using the B3LYP |
360 |
functional~\cite{Becke:1993kq,Lee:1988qf} and the 6-311++G(d,p) basis |
361 |
set. For the cluster configurations that had been generated from |
362 |
molecular dynamics running under applied fields, the density |
363 |
functional calculations had a field of $5 \times 10^{-4}$ atomic units |
364 |
($E_h / (e a_0)$) applied in the $+z$ direction in order to match the |
365 |
molecular dynamics simulations. |
366 |
|
367 |
The energies for the stretched / compressed nitrile bond in each of |
368 |
the clusters were used to fit Morse potentials, and the frequencies |
369 |
were obtained from the $0 \rightarrow 1$ transition for the energy |
370 |
levels for this potential.\cite{Morse:1929xy} To obtain a spectrum, |
371 |
each of the frequencies was convoluted with a Lorentzian line shape |
372 |
with a width of 1.5 $\mathrm{cm}^{-1}$. This linewidth corresponds to |
373 |
a vibrational lifetime of $\sim 3.5$ ps, which is within the reported |
374 |
ranges ($\sim 1 - 5$ ps) for CN stretching vibrational lifetimes in |
375 |
other molecules.\cite{Ghosh:2009qf,Ha:2009xy,Waegele:2010ve}. |
376 |
Available computing resources limited the sampling to 100 clusters for |
377 |
both the no-field and full-field spectra. Comparisons of the quantum |
378 |
mechanical spectrum to the classical are shown in figure |
379 |
\ref{fig:spectra}. The mean frequencies obtained from the |
380 |
distributions give a field-induced red shift of |
381 |
$2.68~\mathrm{cm}^{-1}$. |
382 |
|
383 |
\subsection{CN frequencies from potential-frequency maps} |
384 |
|
385 |
One approach which has been used to successfully analyze the spectrum |
386 |
of nitrile and thiocyanate probes in aqueous environments was |
387 |
developed by Choi {\it et al.}~\cite{Choi:2008cr,Oh:2008fk} This |
388 |
method involves finding a multi-parameter fit that maps between the |
389 |
local electrostatic potential at selected sites surrounding the |
390 |
nitrile bond and the vibrational frequency of that bond obtained from |
391 |
more expensive {\it ab initio} methods. This approach is similar in |
392 |
character to the field-frequency maps developed by the Skinner group |
393 |
for OH stretches in liquid water.\cite{Corcelli:2004ai,Auer:2007dp} |
394 |
|
395 |
To use the potential-frequency maps, the local electrostatic |
396 |
potential, $\phi_a$, is computed at 20 sites ($a = 1 \rightarrow 20$) |
397 |
that surround the nitrile bond, |
398 |
\begin{equation} |
399 |
\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{j} |
400 |
\frac{q_j}{\left|r_{aj}\right|}. |
401 |
\end{equation} |
402 |
Here $q_j$ is the partial charge on atom $j$ (residing on a different |
403 |
molecule) and $r_{aj}$ is the distance between site $a$ and atom $j$. |
404 |
The original map was parameterized in liquid water and comprises a set |
405 |
of parameters, $l_a$, that predict the shift in nitrile peak |
406 |
frequency, |
407 |
\begin{equation} |
408 |
\delta\tilde{\nu} =\sum^{20}_{a=1} l_{a}\phi_{a}. |
409 |
\end{equation} |
410 |
|
411 |
The simulations of 5CB were carried out in the presence of |
412 |
externally-applied uniform electric fields. Although uniform fields |
413 |
exert forces on charge sites, they only contribute to the potential if |
414 |
one defines a reference point that can serve as an origin. One simple |
415 |
modification to the potential at each of the probe sites is to use the |
416 |
centroid of the \ce{CN} bond as the origin for that site, |
417 |
\begin{equation} |
418 |
\phi_a^\prime = \phi_a + \frac{1}{4\pi\epsilon_{0}} \vec{E} \cdot |
419 |
\left(\vec{r}_a - \vec{r}_\ce{CN} \right) |
420 |
\end{equation} |
421 |
where $\vec{E}$ is the uniform electric field, $\left( \vec{r}_{a} - |
422 |
\vec{r}_\ce{CN} \right)$ is the displacement between the |
423 |
coordinates described by Choi {\it et |
424 |
al.}~\cite{Choi:2008cr,Oh:2008fk} and the \ce{CN} bond centroid. |
425 |
$\phi_a^\prime$ then contains an effective potential contributed by |
426 |
the uniform field in addition to the local potential contributions |
427 |
from other molecules. |
428 |
|
429 |
The sites $\{\vec{r}_a\}$ and weights $\left\{l_a \right\}$ |
430 |
developed by Choi {\it et al.}~\cite{Choi:2008cr,Oh:2008fk} are quite |
431 |
symmetric around the \ce{CN} centroid, and even at large uniform field |
432 |
values we observed nearly-complete cancellation of the potential |
433 |
contributions from the uniform field. |
434 |
|
435 |
The frequency shifts were computed for 4000 configurations sampled |
436 |
every 1 ps after the systems had equilibrated. The potential |
437 |
frequency map produces a small blue shift of 0.34 cm$^{-1}$, and the |
438 |
frequency shifts are quite narrowly distributed. However, the |
439 |
parameters for the potential frequency maps were derived for nitrile |
440 |
bonds in aqueous solutions, where the magnitudes of the local fields |
441 |
and electrostatic potentials are much larger than they would be in |
442 |
neat 5CB. |
443 |
|
444 |
We note that in 5CB there does not appear to be a particularly strong |
445 |
correlation between the electric field strengths observed at the |
446 |
nitrile centroid and the calculated vibrational frequencies. In |
447 |
Fig. \ref{fig:fieldMap} we show the calculated frequencies plotted |
448 |
against the field magnitude as well as the parallel and perpendicular |
449 |
components of that field. |
450 |
|
451 |
\begin{figure} |
452 |
\includegraphics[width=\linewidth]{fieldMap.pdf} |
453 |
\caption{The observed cluster frequencies have no apparent |
454 |
correlation with the electric field felt at the centroid of the |
455 |
nitrile bond. Upper panel: vibrational frequencies plotted |
456 |
against the component of the field parallel to the CN bond. |
457 |
Middle panel: plotted against the magnitude of the field |
458 |
components perpendicular to the CN bond. Lower panel: plotted |
459 |
against the total field magnitude.} |
460 |
\label{fig:fieldMap} |
461 |
\end{figure} |
462 |
|
463 |
|
464 |
\subsection{CN frequencies from bond length autocorrelation functions} |
465 |
|
466 |
The distribution of nitrile vibrational frequencies can also be found |
467 |
using classical time correlation functions. This was done by |
468 |
replacing the rigid \ce{CN} bond with a flexible Morse oscillator |
469 |
described in Eq. \ref{eq:morse}. Since the systems were perturbed by |
470 |
the addition of a flexible high-frequency bond, they were allowed to |
471 |
re-equilibrate in the canonical (NVT) ensemble for 100 ps with 1 fs |
472 |
time steps. After equilibration, each configuration was run in the |
473 |
microcanonical (NVE) ensemble for 20 ps. Configurations sampled every |
474 |
fs were then used to compute bond-length autocorrelation functions, |
475 |
\begin{equation} |
476 |
C(t) = \langle \delta r(t) \cdot \delta r(0) ) \rangle |
477 |
\end{equation} |
478 |
% |
479 |
where $\delta r(t) = r(t) - r_0$ is the deviation from the equilibrium |
480 |
bond distance at time $t$. Because the other atomic sites have very |
481 |
small partial charges, this correlation function is an approximation |
482 |
to the dipole autocorrelation function for the molecule, which would |
483 |
be particularly relevant to computing the IR spectrum. Eleven |
484 |
statistically-independent correlation functions were obtained by |
485 |
allowing the systems to run 10 ns with rigid \ce{CN} bonds followed by |
486 |
120 ps equilibration and data collection using the flexible \ce{CN} |
487 |
bonds. This process was repeated 11 times, and the total sampling |
488 |
time, from sample preparation to final configurations, exceeded 160 ns |
489 |
for each of the field strengths investigated. |
490 |
|
491 |
The correlation functions were filtered using exponential apodization |
492 |
functions,\cite{FILLER:1964yg} $f(t) = e^{-|t|/c}$, with a time |
493 |
constant, $c =$ 3.5 ps, and were Fourier transformed to yield a |
494 |
spectrum, |
495 |
\begin{equation} |
496 |
I(\omega) = \int_{-\infty}^{\infty} C(t) f(t) e^{-i \omega t} dt. |
497 |
\end{equation} |
498 |
This time constant was chosen to match the Lorentzian linewidth that |
499 |
was used for computing the quantum mechanical spectra, and falls |
500 |
within the range of reported lifetimes for CN vibrations in other |
501 |
nitrile-containing molecules. The sample-averaged classical nitrile |
502 |
spectrum can be seen in Figure \ref{fig:spectra}. The Morse oscillator |
503 |
parameters listed above yield a natural frequency of 2226 |
504 |
$\mathrm{cm}^{-1}$ (close to the experimental value). To compare peaks |
505 |
from the classical and quantum mechanical approaches, both are |
506 |
displayed on an axis centered on the experimental nitrile frequency. |
507 |
|
508 |
\begin{figure} |
509 |
\includegraphics[width=\linewidth]{spectra.pdf} |
510 |
\caption{Spectrum of nitrile frequency shifts for the no-field |
511 |
(black) and the full-field (red) simulations. Upper panel: |
512 |
frequency shifts obtained from {\it ab initio} cluster |
513 |
calculations. Lower panel: classical bond-length autocorrelation |
514 |
spectrum for the flexible nitrile measured relative to the natural |
515 |
frequency for the flexible bond. The dashed lines indicate the |
516 |
mean frequencies for each of the distributions. The cluster |
517 |
calculations exhibit a $2.68~\mathrm{cm}^{-1}$ field-induced red |
518 |
shift, while the classical correlation functions predict a red |
519 |
shift of $2.29~\mathrm{cm}^{-1}$.} |
520 |
\label{fig:spectra} |
521 |
\end{figure} |
522 |
|
523 |
The classical approach includes both intramolecular and electrostatic |
524 |
interactions, and so it implicitly couples \ce{CN} vibrations to other |
525 |
vibrations within the molecule as well as to nitrile vibrations on |
526 |
other nearby molecules. The classical frequency spectrum is |
527 |
significantly broader because of this coupling. The {\it ab initio} |
528 |
cluster approach exercises only the targeted nitrile bond, with no |
529 |
additional coupling to other degrees of freedom. As a result the |
530 |
quantum calculations are quite narrowly peaked around the experimental |
531 |
nitrile frequency. Although the spectra are quite noisy, the main |
532 |
effect seen in both distributions is a moderate shift to the red |
533 |
($2.29~\mathrm{cm}^{-1}$ classical and $2.68~\mathrm{cm}^{-1}$ |
534 |
quantum) after the electrostatic field had induced the nematic phase |
535 |
transition. |
536 |
|
537 |
\section{Discussion} |
538 |
Our simulations show that the united-atom model can reproduce the |
539 |
field-induced nematic ordering of the 4-cyano-4'-pentylbiphenyl. |
540 |
Because we are simulating a very small electrode separation (5~nm), a |
541 |
voltage drop as low as 1.2~V was sufficient to induce the phase |
542 |
change. This potential is significantly smaller than 100~V that was |
543 |
used with a 5~$\mu$m gap to study the electrochemiluminescence of |
544 |
rubrene in neat 5CB,\cite{Kojima19881789} and suggests that by using |
545 |
electrodes separated by a nanometer-scale gap, it will be relatively |
546 |
straightforward to observe the nitrile Stark shift in 5CB. |
547 |
|
548 |
Both the classical correlation function and the isolated cluster |
549 |
approaches to estimating the IR spectrum show that a population of |
550 |
nitrile stretches shift by $\sim~3~\mathrm{cm}^{-1}$ to the red of |
551 |
the unperturbed vibrational line. To understand the origin of this |
552 |
shift, a more complete picture of the spatial ordering around the |
553 |
nitrile bonds is required. We have computed the angle-dependent pair |
554 |
distribution functions, |
555 |
\begin{align} |
556 |
g(r, \cos \omega) = & \frac{1}{\rho N} \left< \sum_{i} \sum_{j} |
557 |
\delta \left(r - r_{ij}\right) \delta\left(\cos \omega_{ij} - |
558 |
\cos \omega\right) \right> \\ \nonumber \\ |
559 |
g(r, \cos \theta) = & \frac{1}{\rho N} \left< \sum_{i} |
560 |
\sum_{j} \delta \left(r - r_{ij}\right) \delta\left(\cos \theta_{i} - |
561 |
\cos \theta \right) \right> |
562 |
\end{align} |
563 |
which provide information about the joint spatial and angular |
564 |
correlations present in the system. The angles $\omega$ and $\theta$ |
565 |
are defined by vectors along the CN axis of each nitrile bond (see |
566 |
figure \ref{fig:definition}). |
567 |
\begin{figure} |
568 |
\includegraphics[width=4in]{definition.pdf} |
569 |
\caption{Definitions of the angles between two nitrile bonds.} |
570 |
\label{fig:definition} |
571 |
\end{figure} |
572 |
|
573 |
The primary structural effect of the field-induced phase transition is |
574 |
apparent in figure \ref{fig:gofromega}. The nematic ordering transfers |
575 |
population from the perpendicular ($\cos\omega\approx 0$) and |
576 |
anti-aligned ($\cos\omega\approx -1$) to the nitrile-aligned peak |
577 |
near $\cos\omega\approx 1$, leaving most other features undisturbed. This |
578 |
change is visible in the simulations as an increased population of |
579 |
aligned nitrile bonds in the first solvation shell. |
580 |
|
581 |
\begin{figure} |
582 |
\includegraphics[width=\linewidth]{gofrOmega.pdf} |
583 |
\caption{Contours of the angle-dependent pair distribution functions |
584 |
for nitrile bonds on 5CB in the no field (upper panel) and full |
585 |
field (lower panel) simulations. Dark areas signify regions of |
586 |
enhanced density, while light areas signify depletion relative to |
587 |
the bulk density.} |
588 |
\label{fig:gofromega} |
589 |
\end{figure} |
590 |
|
591 |
Although it is certainly possible that the coupling between |
592 |
closely-spaced nitrile pairs is responsible for some of the red-shift, |
593 |
that is not the only structural change that is taking place. The |
594 |
second two-dimensional pair distribution function, $g(r,\cos\theta)$, |
595 |
shows that nematic ordering also transfers population that is directly |
596 |
in line with the nitrile bond (see figure \ref{fig:gofrtheta}) to the |
597 |
sides of the molecule, thereby freeing steric blockage which can |
598 |
directly influence the nitrile vibration. This is confirmed by |
599 |
observing the one-dimensional $g(z)$ obtained by following the \ce{C |
600 |
-> N} vector for each nitrile bond and observing the local density |
601 |
($\rho(z)/\rho$) of other atoms at a distance $z$ along this |
602 |
direction. The full-field simulation shows a significant drop in the |
603 |
first peak of $g(z)$, indicating that the nematic ordering has moved |
604 |
density away from the region that is directly in line with the |
605 |
nitrogen side of the CN bond. |
606 |
|
607 |
\begin{figure} |
608 |
\includegraphics[width=\linewidth]{gofrTheta.pdf} |
609 |
\caption{Contours of the angle-dependent pair distribution function, |
610 |
$g(r,\cos \theta)$, for finding any other atom at a distance and |
611 |
angular deviation from the center of a nitrile bond. The top edge |
612 |
of each contour plot corresponds to local density along the |
613 |
direction of the nitrogen in the CN bond, while the bottom is in |
614 |
the direction of the carbon atom. Bottom panel: $g(z)$ data taken |
615 |
by following the \ce{C -> N} vector for each nitrile bond shows |
616 |
that the field-induced phase transition reduces the population of |
617 |
atoms that are directly in line with the nitrogen motion.} |
618 |
\label{fig:gofrtheta} |
619 |
\end{figure} |
620 |
|
621 |
We are suggesting an anti-caging mechanism here -- the nematic |
622 |
ordering provides additional space directly inline with the nitrile |
623 |
vibration, and since the oscillator is fairly anharmonic, this freedom |
624 |
provides a fraction of the nitrile bonds with a significant red-shift. |
625 |
|
626 |
The cause of this shift does not appear to be related to the alignment |
627 |
of those nitrile bonds with the field, but rather to the change in |
628 |
local steric environment that is brought about by the |
629 |
isotropic-nematic transition. We have compared configurations for many |
630 |
of the cluster that exhibited the lowest frequencies (between 2190 and |
631 |
2215 $\mathrm{cm}^{-1}$) and have observed some similar structural |
632 |
features. The lowest frequencies appear to come from configurations |
633 |
which have nearly-empty pockets directly opposite the nitrogen atom |
634 |
from the nitrile carbon. However, because we do not have a |
635 |
particularly large cluster population to interrogate, this is |
636 |
certainly not quantitative confirmation of this effect. |
637 |
|
638 |
The prediction of a small red-shift of the nitrile peak in 5CB in |
639 |
response to a field-induced nematic ordering is the primary result of |
640 |
this work, and although the proposed anti-caging mechanism is somewhat |
641 |
speculative, this work provides some impetus for further theory and |
642 |
experiments. |
643 |
|
644 |
\section{Acknowledgements} |
645 |
The authors thank Steven Corcelli and Zac Schultz for helpful comments |
646 |
and suggestions. Support for this project was provided by the National |
647 |
Science Foundation under grant CHE-0848243. Computational time was |
648 |
provided by the Center for Research Computing (CRC) at the University |
649 |
of Notre Dame. |
650 |
|
651 |
\newpage |
652 |
|
653 |
\bibliography{5CB} |
654 |
|
655 |
\end{document} |