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\title{Nitrile vibrations as reporters of field-induced phase |
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transitions in liquid crystals} |
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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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Nitrile Stark shift repsonses to electric fields have been used |
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extensively in biology for the probing of local internal fields of |
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structures like proteins and DNA. Intigration of these probes into |
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different areas of interest are important for studing local structure |
65 |
and fields within confined, nanoscopic |
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systems. 4-Cyano-4'-pentylbiphenyl (5CB) is a liquid crystal with a known |
67 |
macroscopic structure reordering from the isotropic to nematic |
68 |
phase with the application of an external |
69 |
field and as the name suggests has an inherent nitrile group. Through |
70 |
simulations of this molecule where application of |
71 |
large, nanoscale external fields were applied, the nitrile was invenstigated |
72 |
as a local field sensor. It was |
73 |
found that while most computational methods for nitrile spectral |
74 |
calculations rely on a correlation between local electric field and |
75 |
the nitrile bond, 5CB did not have an easily obtained |
76 |
correlation. Instead classical calculation through correlation of the |
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cyanide bond displacement in time use enabled to show a spectral |
78 |
change in the formation of a red |
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shifted peak from the main peak as an external field was applied. This indicates |
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that local structure has a larger impact on the nitrile frequency then |
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does the local electric field. By better understanding how nitrile |
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groups respond to local and external fields it will help |
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nitrile groups branch out beyond their biological |
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applications to uses in electronics and surface sciences. |
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\end{abstract} |
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|
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\newpage |
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|
89 |
\section{Introduction} |
90 |
|
91 |
The fundamental characteristic of liquid crystal mesophases is that |
92 |
they maintain some degree of orientational order while translational |
93 |
order is limited or absent. This orientational order produces a |
94 |
complex direction-dependent response to external perturbations like |
95 |
electric fields and mechanical distortions. The anisotropy of the |
96 |
macroscopic phases originates in the anisotropy of the constituent |
97 |
molecules, which typically have highly non-spherical structures with a |
98 |
significant degree of internal rigidity. In nematic phases, rod-like |
99 |
molecules are orientationally ordered with isotropic distributions of |
100 |
molecular centers of mass, while in smectic phases, the molecules |
101 |
arrange themselves into layers with their long (symmetry) axis normal |
102 |
($S_{A}$) or tilted ($S_{C}$) with respect to the layer planes. |
103 |
|
104 |
The behavior of the $S_{A}$ phase can be partially explained with |
105 |
models mainly based on geometric factors and van der Waals |
106 |
interactions. However, these simple models are insufficient to |
107 |
describe liquid crystal phases which exhibit more complex polymorphic |
108 |
nature. X-ray diffraction studies have shown that the ratio between |
109 |
lamellar spacing ($s$) and molecular length ($l$) can take on a wide |
110 |
range of values.\cite{Gray:1984hc,Leadbetter:1976vf,Hardouin:1980yq} |
111 |
Typical $S_{A}$ phases have $s/l$ ratios on the order of $0.8$, while |
112 |
for some compounds, e.g. the 4-alkyl-4'-cyanobiphenyls, the $s/l$ |
113 |
ratio is on the order of $1.4$. Molecules which form $S_{A}$ phases |
114 |
can exhibit a wide variety of subphases like monolayers ($S_{A1}$), |
115 |
uniform bilayers ($S_{A2}$), partial bilayers ($S_{\tilde A}$) as well |
116 |
as interdigitated bilayers ($S_{A_{d}}$), and often have a terminal |
117 |
cyano or nitro group. In particular lyotropic liquid crystals (those |
118 |
exhibiting liquid crystal phase transition as a function of water |
119 |
concentration) often have polar head groups or zwitterionic charge |
120 |
separated groups that result in strong dipolar |
121 |
interactions.\cite{Collings97} Because of their versatile polymorphic |
122 |
nature, polar liquid crystalline materials have important |
123 |
technological applications in addition to their immense relevance to |
124 |
biological systems.\cite{Collings97} |
125 |
|
126 |
Experimental studies by Levelut {\it et al.}~\cite{Levelut:1981eu} |
127 |
revealed that terminal cyano or nitro groups usually induce permanent |
128 |
longitudinal dipole moments on the molecules. |
129 |
|
130 |
Liquid crystalline materials with dipole moments located at the ends |
131 |
of the molecules have important applications in display technologies |
132 |
in addition to their relevance in biological systems.\cite{LCapp} |
133 |
|
134 |
Many of the technological applications of the lyotropic mesogens |
135 |
manipulate the orientation and structuring of the liquid crystal |
136 |
through application of external electric fields.\cite{?} |
137 |
Macroscopically, this restructuring is visible in the interactions the |
138 |
bulk phase has with scattered or transmitted light.\cite{?} |
139 |
|
140 |
4-Cyano-4'-pentylbiphenyl (5CB), has been a model for field-induced |
141 |
phase changes due to the known electric field response of the liquid |
142 |
crystal\cite{Hatta:1991ee}. It was discovered (along with three |
143 |
similar compounds) in 1973 in an effort to find a LC that had a |
144 |
melting point near room temperature.\cite{Gray:1973ca} It's known to |
145 |
have a crystalline to nematic phase transition at 18 C and a nematic |
146 |
to isotropic transition at 35 C.\cite{Gray:1973ca} Recently it has |
147 |
seen new life with the application of droplets of the molecule in |
148 |
water being used to study defect sites and nanoparticle |
149 |
strcuturing.\cite{PhysRevLett.111.227801} |
150 |
|
151 |
Nitrile groups can serve as very precise electric field reporters via |
152 |
their distinctive Raman and IR signatures.\cite{Boxer:2009xw} The |
153 |
triple bond between the nitrogen and the carbon atom is very sensitive |
154 |
to local field changes and is observed to have a direct impact on the |
155 |
peak position within the spectrum. The Stark shift in the spectrum |
156 |
can be quantified and mapped into a field value that is impinging upon |
157 |
the nitrile bond. This has been used extensively in biological systems |
158 |
like proteins and enzymes.\cite{Tucker:2004qq,Webb:2008kn} |
159 |
|
160 |
To date, the nitrile electric field response of |
161 |
4-Cyano-4'-n-alkylbiphenyl liquid crystals has not been investigated. |
162 |
While macroscopic electric fields applied across macro electrodes show |
163 |
the phase change of the molecule as a function of electric |
164 |
field,\cite{Lim:2006xq} the effect of a nanoscopic field application |
165 |
has not been probed. These previous studies have shown the nitrile |
166 |
group serves as an excellent indicator of the molecular orientation |
167 |
within the field applied. Lee {\it et al.}~showed the 180 degree change in field |
168 |
direction could be probed with the nitrile peak intensity as it |
169 |
decreased and increased with molecule alignment in the |
170 |
field.\cite{Lee:2006qd,Leyte:97} |
171 |
|
172 |
While these macroscopic fields worked well at showing the bulk |
173 |
response, the atomic scale response has not been studied. With the |
174 |
advent of nano-electrodes and coupling them with atomic force |
175 |
microscopy, control of electric fields applied across nanometer |
176 |
distances is now possible\cite{citation1}. This application of |
177 |
nanometer length is interesting in the case of a nitrile group on the |
178 |
molecule. While macroscopic fields are insufficient to cause a Stark |
179 |
effect, small fields across nanometer-sized gaps are of sufficient |
180 |
strength. If one were to assume a gap of 5 nm between a lower |
181 |
electrode having a nanoelectrode placed near it via an atomic force |
182 |
microscope, a field of 1 V applied across the electrodes would |
183 |
translate into a field of 2x10\textsuperscript{8} $\frac{V}{M}$. This |
184 |
field is theoretically strong enough to cause a phase change from |
185 |
isotropic to nematic, as well as Stark tuning of the nitrile |
186 |
bond. This should be readily visible experimentally through Raman or |
187 |
IR spectroscopy. |
188 |
|
189 |
Herein, we show the computational investigation of these electric |
190 |
field effects through atomistic simulations of 5CB with external |
191 |
fields applied. These simulations are then coupled with ab intio and |
192 |
classical spectrum calculations to predict changes. These changes are |
193 |
easily varifiable with experiments and should be able to replicated |
194 |
experimentally. |
195 |
|
196 |
\section{Computational Details} |
197 |
The force field was mainly taken from Guo et al.\cite{Zhang:2011hh} A |
198 |
deviation from this force field was made to create a rigid body from |
199 |
the phenyl rings. Bond distances within the rigid body were taken from |
200 |
equilibrium bond distances. While the phenyl rings were held rigid, |
201 |
bonds, bends, torsions and inversion centers still included the rings. |
202 |
|
203 |
Simulations were with boxes of 270 molecules locked at experimental |
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densities with periodic boundaries. The molecules were thermalized |
205 |
from 0 kelvin to 300 kelvin. To equilibrate, each was first run in NVT |
206 |
for 1 ns. This was followed by NVE for simulations used in the data |
207 |
collection. |
208 |
|
209 |
External electric fields were applied in a simplistic charge-field |
210 |
interaction. Forces were calculated by multiplying the electric field |
211 |
being applied by the charge at each atom. For the potential, the |
212 |
origin of the box was used as a point of reference. This allows for a |
213 |
potential value to be added to each atom as the molecule move in space |
214 |
within the box. Fields values were applied in a manner representing |
215 |
those that would be applable in an experimental set-up. The assumed |
216 |
electrode seperation was 5 nm and the field was input as |
217 |
$\frac{V}{\text{\AA}}$. The three field environments were, 1) no field |
218 |
applied, 2) 0.01 $\frac{V}{\text{\AA}}$ (0.5 V) and 3) 0.024 |
219 |
$\frac{V}{\text{\AA}}$ (1.2 V). Each field was applied in the |
220 |
Z-axis of the simulation box. For the simplicity of this paper, |
221 |
each field will be called zero, partial and full, respectively. |
222 |
|
223 |
For quantum calculation of the nitrile bond frequency, Gaussian 09 was |
224 |
used. A single 5CB molecule was selected for the center of the |
225 |
cluster. For effects from molecules located near the chosen nitrile |
226 |
group, parts of molecules nearest to the nitrile group were |
227 |
included. For the body not including the tail, any atom within 6~\AA |
228 |
of the midpoint of the nitrile group was included. For the tail |
229 |
structure, the whole tail was included if a tail atom was within 4~\AA |
230 |
of the midpoint. If the tail did not include any atoms from the ring |
231 |
structure, it was considered a propane molecule and included as |
232 |
such. Once the clusters were generated, input files were created that |
233 |
stretched the nitrile bond along its axis from 0.87 to 1.52~\AA at |
234 |
increments of 0.05~\AA. This generated 13 single point energies to be |
235 |
calculated. The Gaussian files were run with B3LYP/6-311++G(d,p) with |
236 |
no other keywords for the zero field simulation. Simulations with |
237 |
fields applied included the keyword ''Field=Z+5'' to match the |
238 |
external field applied in molecular dynamic runs. Once completed, the central nitrile bond frequency |
239 |
was calculated with a Morse fit. Using this fit and the solved energy |
240 |
levels for a Morse oscillator, the frequency was found. Each set of |
241 |
frequencies were then convolved together with a lorezian lineshape |
242 |
function over each value. The width value used was 1.5. For the zero |
243 |
field spectrum, 67 frequencies were used and for the full field, 59 |
244 |
frequencies were used. |
245 |
|
246 |
Classical nitrile bond frequencies were found by replacing the rigid |
247 |
cyanide bond with a flexible Morse oscillator bond |
248 |
($r_0= 1.157437$ \AA , $D_0 = 212.95$ and |
249 |
$\beta = 2.67566$) . Once replaced, the |
250 |
systems were allowed to re-equilibrate in NVT for 100 ps. After |
251 |
re-equilibration, the system was run in NVE for 20 ps with a snapshot |
252 |
spacing of 1 fs. These snapshot were then used in bond correlation |
253 |
calculation to find the decay structure of the bond in time using the |
254 |
average bond displacement in time, |
255 |
\begin{equation} |
256 |
C(t) = \langle \left(r(t) - r_0 \right) \cdot \left(r(0) - r_0 \right) \rangle |
257 |
\end{equation} |
258 |
% |
259 |
where $r_0$ is the equilibrium bond distance and $r(t)$ is the |
260 |
instantaneous bond displacement at time $t$. Once calculated, |
261 |
smoothing was applied by adding an exponential decay on top of the |
262 |
decay with a $\tau$ of 6000. Further smoothing |
263 |
was applied by padding 20,000 zeros on each side of the symmetric |
264 |
data. This was done five times by allowing the systems to run 1 ns |
265 |
with a rigid bond followed by an equilibrium run with the bond |
266 |
switched back to a Morse oscillator and a short production run of 20 ps. |
267 |
|
268 |
\section{Results} |
269 |
|
270 |
In order to characterize the orientational ordering of the system, the |
271 |
primary quantity of interest is the nematic (orientational) order |
272 |
parameter. This is determined using the tensor |
273 |
\begin{equation} |
274 |
Q_{\alpha \beta} = \frac{1}{2N} \sum_{i=1}^{N} \left(3 \hat{e}_{i |
275 |
\alpha} \hat{e}_{i \beta} - \delta_{\alpha \beta} \right) |
276 |
\end{equation} |
277 |
where $\alpha, \beta = x, y, z$, and $\hat{e}_i$ is the molecular |
278 |
end-to-end unit vector for molecule $i$. The nematic order parameter |
279 |
$S$ is the largest eigenvalue of $Q_{\alpha \beta}$, and the |
280 |
corresponding eigenvector defines the director axis for the phase. |
281 |
$S$ takes on values close to 1 in highly ordered phases, but falls to |
282 |
zero for isotropic fluids. In the context of 5CB, this value would be |
283 |
close to zero for its isotropic phase and raise closer to one as it |
284 |
moved to the nematic and crystalline phases. |
285 |
|
286 |
This value indicates phases changes at temperature boundaries but also |
287 |
when a phase change occurs due to external field applications. In |
288 |
Figure 1, this phase change can be clearly seen over the course of 60 |
289 |
ns. Each system starts with an ordering paramter near 0.1 to 0.2, |
290 |
which is an isotropic phase. Over the course 10 ns, the full external field |
291 |
causes a shift in the ordering of the system to 0.5, the nematic phase |
292 |
of the liquid crystal. This change is consistent over the full 50 ns |
293 |
with no drop back into the isotropic phase. This change is clearly |
294 |
field induced and stable over a long period of simulation time. |
295 |
\begin{figure} |
296 |
\includegraphics[trim = 5mm 10mm 3mm 10mm, clip, width=3.25in]{P2} |
297 |
\caption{Ordering of each external field application over the course |
298 |
of 60 ns with a sampling every 100 ps. Each trajectory was started |
299 |
after equilibration with zero field applied.} |
300 |
\label{fig:orderParameter} |
301 |
\end{figure} |
302 |
|
303 |
In the figure below, this phase change is represented nicely as |
304 |
ellipsoids that are created by the vector formed between the nitrogen |
305 |
of the nitrile group and the tail terminal carbon atom. These |
306 |
illistrate the change from isotropic phase to nematic change. Both the |
307 |
zero field and partial field images look mostly disordered. The |
308 |
partial field does look somewhat orded but without any overall order |
309 |
of the entire system. This is most likely a high point in the ordering |
310 |
for the trajectory. The full field image on the other hand looks much |
311 |
more ordered when compared to the two lower field simulations. |
312 |
\begin{figure} |
313 |
\includegraphics[width=7in]{Elip_3} |
314 |
\caption{Ellipsoid reprsentation of 5CB at three different |
315 |
field strengths, Zero Field (Left), Partial Field (Middle), and Full |
316 |
Field (Right) Each image was created by taking the final |
317 |
snapshot of each 60 ns run} |
318 |
\label{fig:Cigars} |
319 |
\end{figure} |
320 |
|
321 |
This change in phase was followed by two courses of further |
322 |
analysis. First was the replacement of the static nitrile bond with a |
323 |
morse oscillator bond. This was then simulated for a period of time |
324 |
and a classical spetrum was calculated. Second, ab intio calcualtions |
325 |
were performed to investigate if the phase change caused any change |
326 |
spectrum through quantum effects. |
327 |
|
328 |
The classical nitrile spectrum can be seen in Figure 2. Most noticably |
329 |
is the position of the two peaks. Obviously the experimental peak |
330 |
position is near 2226 cm\textsuperscript{-1}. However, in this case |
331 |
the peak position is shifted to the blue at a position of 2375 |
332 |
cm\textsuperscript{-1}. This shift is due solely to the choice of |
333 |
oscillator strength in the Morse oscillator parameters. While this |
334 |
shift makes the two spectra differ, it does not affect the ability to |
335 |
qualitatively compare peak changes to possible experimental changes. |
336 |
With this important fact out of the way, differences between the two |
337 |
states are subtle but are very much present. The first and |
338 |
most notable is the apperance for a strong band near 2300 |
339 |
cm\textsuperscript{-1}. |
340 |
\begin{figure} |
341 |
\includegraphics[width=3.25in]{2Spectra} |
342 |
\caption{The classically calculated nitrile bond spetrum for no |
343 |
external field application (black) and full external field |
344 |
application (red)} |
345 |
\label{fig:twoSpectra} |
346 |
\end{figure} |
347 |
|
348 |
Before Gaussian silumations were carried out, it was attempt to apply |
349 |
the method developed by Cho {\it et al.}~\cite{Oh:2008fk} This method involves the fitting |
350 |
of multiple parameters to Gaussian calculated freuencies to find a |
351 |
correlation between the potential around the bond and the |
352 |
frequency. This is very similar to work done by Skinner {\it et al.}~with |
353 |
water models like SPC/E. The general method is to find the shift in |
354 |
the peak position through, |
355 |
\begin{equation} |
356 |
\delta\tilde{\nu} =\sum^{n}_{a=1} l_{a}\phi^{water}_{a} |
357 |
\end{equation} |
358 |
where $l_{a}$ are the fitting parameters and $\phi^{water}_{a}$ is the |
359 |
potential from the surrounding water cluster. This $\phi^{water}_{a}$ |
360 |
takes the form, |
361 |
\begin{equation} |
362 |
\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{m} \sum_{j} |
363 |
\frac{C^{H_{2}O}_{j \left(m \right) }}{r_{aj \left(m\right)}} |
364 |
\end{equation} |
365 |
where $C^{H_{2}O}_{j \left(m \right) }$ indicates the partial charge |
366 |
on the $j$th site of the $m$th water molecule and $r_{aj \left(m\right)}$ |
367 |
is the distance between the site $a$ of the nitrile molecule and the $j$th |
368 |
site of the $m$th water molecule. However, since these simulations |
369 |
are done under the presence of external fields and in the |
370 |
absence of water, the equations need a correction factor for the shift |
371 |
caused by the external field. The equation is also reworked to use |
372 |
electric field site data instead of partial charges from surrounding |
373 |
atoms. So by modifing the original |
374 |
$\phi^{water}_{a}$ to $\phi^{5CB}_{a}$ we get, |
375 |
\begin{equation} |
376 |
\phi^{5CB}_{a} = \frac{1}{4\pi \epsilon_{0}} \left( \vec{E}\bullet |
377 |
\left(\vec{r}_{a}-\vec{r}_{CN}\right) \right) + \phi^{5CB}_{0} |
378 |
\end{equation} |
379 |
where $\vec{E}$ is the electric field at each atom, $\left( \vec{r}_{a} - |
380 |
\vec{r}_{CN} \right)$ is the vector between the nitrile bond and the |
381 |
cooridinates described by Cho around the bond and $\phi^{5CB}_{0}$ is |
382 |
the correction factor for the system of parameters. After these |
383 |
changes, the correction factor was found for multiple values of an |
384 |
external field being applied. However, the factor was no linear and |
385 |
was overly large due to the fitting parameters being so small. |
386 |
|
387 |
Due to this, Gaussian calculations were performed in lieu of this |
388 |
method. A set of snapshots for the zero and full field simualtions, |
389 |
they were first investigated for any dependence on the local, with |
390 |
external field included, electric field. This was to see if a linear |
391 |
or non-linear relationship between the two could be utilized for |
392 |
generating spectra. This was done in part because of previous studies |
393 |
showing the frequency dependence of nitrile bonds to the electric |
394 |
fields generated locally between solvating water. It was seen that |
395 |
little to no dependence could be directly shown. This data is not |
396 |
shown. |
397 |
|
398 |
Since no explicit dependence was observed between the calculated |
399 |
frequency and the electric field, it was not a viable route for the |
400 |
calculation of a nitrile spectrum. Instead, the frequencies were taken |
401 |
and convolved together with a lorentzian line shape applied around the |
402 |
frequency value. These spectra are seen below in Figure |
403 |
4. While the spectrum without a field is lower in intensity and is |
404 |
almost bimodel in distrobution, the external field spectrum is much |
405 |
more unimodel. This tighter clustering has the affect of increasing the |
406 |
intensity around 2226 cm\textsuperscript{-1} where the peak is |
407 |
centered. The external field also has fewer frequencies of higher |
408 |
energy in the spectrum. Unlike the the zero field, where some frequencies |
409 |
reach as high as 2280 cm\textsuperscript{-1}. |
410 |
\begin{figure} |
411 |
\includegraphics[width=3.25in]{Convolved} |
412 |
\caption{Lorentzian convolved Gaussian frequencies of the zero field |
413 |
system (black) and the full field system (red)} |
414 |
\label{fig:Con} |
415 |
\end{figure} |
416 |
\section{Discussion} |
417 |
Interestingly, the field that is needed to switch the phase of 5CB |
418 |
macroscopically is larger than 1 V. However, in this case, only a |
419 |
voltage of 1.2 V was need to induce a phase change. This is impart due |
420 |
to the short distance of 5 nm the field is being applied across. At such a small |
421 |
distance, the field is much larger than the macroscopic and thus |
422 |
easily induces a field dependent phase change. However, this field |
423 |
will not cause a breakdown of the 5CB since electrochemistry studies |
424 |
have shown that it can be used in the presence of fields as high as |
425 |
500 V macroscopically. This large of a field near the surface of the |
426 |
elctrode would cause breakdown of 5CB if it could happen. |
427 |
|
428 |
The absence of any electric field dependency of the freuquency with |
429 |
the Gaussian simulations is interesting. A large base of research has been |
430 |
built upon the known tuning of the nitrile bond as the local field |
431 |
changes. This difference may be due to the absence of water or a |
432 |
molecule that induces a large internal field. Liquid water is known to have a very high internal field which |
433 |
is much larger than the internal fields of neat 5CB. Even though the |
434 |
application of Gaussian simulations followed by mapping it to |
435 |
some classical parameter is easy and straight forward, this system |
436 |
illistrates how that 'go to' method can break down. |
437 |
|
438 |
While this makes the application of nitrile Stark effects in |
439 |
simulations without water harder, these data show |
440 |
that it is not a deal breaker. The classically calculated nitrile |
441 |
spectrum shows changes in the spectra that will be easily seen through |
442 |
experimental routes. It indicates a shifted peak lower in energy |
443 |
should arise. This peak is a few wavenumbers from the leading edge of |
444 |
the larger peak and almost 75 wavenumbers from the center. This |
445 |
seperation between the two peaks means experimental results will show |
446 |
an easily resolved peak. |
447 |
|
448 |
The Gaussian derived spectra do indicate an applied field |
449 |
and subsiquent phase change does cause a narrowing of freuency |
450 |
distrobution. With narrowing, it would indicate an increased |
451 |
homogeneous distrobution of the local field near the nitrile. |
452 |
\section{Conclusions} |
453 |
Field dependent changes |
454 |
\newpage |
455 |
|
456 |
\bibliography{5CB} |
457 |
|
458 |
\end{doublespace} |
459 |
\end{document} |