| 220 |
|
V(r_\ce{CN}) = D_e \left(1 - e^{-\beta (r_\ce{CN}-r_e)}\right)^2 |
| 221 |
|
\end{displaymath} |
| 222 |
|
where $r_e= 1.157437$ \AA , $D_e = 212.95 \mathrm{~kca~l} / |
| 223 |
< |
\mathrm{mol}^{-1}$ and $\beta = 2.67566 $\AA~$^{-1}$, corresponding to a |
| 224 |
< |
vibrational frequency of $2375 \mathrm{~cm}^{-1}$, a |
| 225 |
< |
bit higher than the experimental frequency. The flexible nitrile |
| 226 |
< |
moiety required simulation time steps of 1~fs, so the additional |
| 227 |
< |
flexibility was introducuced only after the rigid systems had come to |
| 228 |
< |
equilibrium under the applied fields. Whenever time correlation |
| 229 |
< |
functions were computed from the flexible simulations, |
| 223 |
> |
\mathrm{mol}^{-1}$ and $\beta = 2.67566 $\AA~$^{-1}$. These |
| 224 |
> |
parameters correspond to a vibrational frequency of $2375 |
| 225 |
> |
\mathrm{~cm}^{-1}$, a bit higher than the experimental frequency. The |
| 226 |
> |
flexible nitrile moiety required simulation time steps of 1~fs, so the |
| 227 |
> |
additional flexibility was introducuced only after the rigid systems |
| 228 |
> |
had come to equilibrium under the applied fields. Whenever time |
| 229 |
> |
correlation functions were computed from the flexible simulations, |
| 230 |
|
statistically-independent configurations were sampled from the last ns |
| 231 |
|
of the induced-field runs. These configurations were then |
| 232 |
|
equilibrated with the flexible nitrile moiety for 100 ps, and time |
| 254 |
|
defined a vector using only the rigid core of the molecule. In |
| 255 |
|
nematic phases, typical values for $S$ are close to 0.5. |
| 256 |
|
|
| 257 |
< |
In Figure \ref{fig:orderParameter}, the field-induced phase change can |
| 258 |
< |
be clearly seen over the course of a 60 ns equilibration run. All |
| 257 |
> |
The field-induced phase transition can be clearly seen over the course |
| 258 |
> |
of a 60 ns equilibration runs in figure \ref{fig:orderParameter}. All |
| 259 |
|
three of the systems started in a random (isotropic) packing, with |
| 260 |
|
order parameters near 0.2. Over the course 10 ns, the full field |
| 261 |
|
causes an alignment of the molecules (due primarily to the interaction |
| 262 |
|
of the nitrile group dipole with the electric field). Once this |
| 263 |
< |
system landed in the nematic-ordered state, it became stable for the |
| 264 |
< |
remaining 50 ns of simulation time. It is possible that the |
| 265 |
< |
partial-field simulation is meta-stable and given enough time, it |
| 266 |
< |
would eventually find a nematic-ordered phase, but the partial-field |
| 267 |
< |
simulation was stable as an isotropic phase for the full duration of a |
| 268 |
< |
60 ns simulation. |
| 263 |
> |
system started exhibiting nematic ordering, the orientational order |
| 264 |
> |
parameter became stable for the remaining 50 ns of simulation time. |
| 265 |
> |
It is possible that the partial-field simulation is meta-stable and |
| 266 |
> |
given enough time, it would eventually find a nematic-ordered phase, |
| 267 |
> |
but the partial-field simulation was stable as an isotropic phase for |
| 268 |
> |
the full duration of a 60 ns simulation. |
| 269 |
|
\begin{figure} |
| 270 |
< |
\includegraphics[trim = 5mm 10mm 3mm 10mm, clip, width=3.25in]{P2} |
| 271 |
< |
\caption{Ordering of each external field application over the course |
| 272 |
< |
of 60 ns with a sampling every 100 ps. Each trajectory was started |
| 273 |
< |
after equilibration with zero field applied.} |
| 270 |
> |
\includegraphics[trim = 5mm 10mm 3mm 10mm, clip, width=4in]{P2} |
| 271 |
> |
\caption{Evolution of the orientational order parameter 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.} |
| 275 |
|
\label{fig:orderParameter} |
| 276 |
|
\end{figure} |
| 277 |
|
|
| 278 |
< |
In figure \ref{fig:Cigars}, the field-induced isotropic-nematic |
| 279 |
< |
transition is represented using ellipsoids aligned along the long-axis |
| 280 |
< |
of each molecule. The vector between the nitrogen of the nitrile |
| 281 |
< |
group and the terminal tail atom is used to orient each |
| 282 |
< |
ellipsoid. Both the zero field and partial field simulations appear |
| 282 |
< |
isotropic, while the full field simulations has clearly been |
| 283 |
< |
orientationally ordered |
| 278 |
> |
The field-induced isotropic-nematic transition can be visualized in |
| 279 |
> |
figure \ref{fig:Cigars}, where each molecule has been represented |
| 280 |
> |
using and ellipsoids aligned along the long-axis of each molecule. |
| 281 |
> |
Both the zero field and partial field simulations appear isotropic, |
| 282 |
> |
while the full field simulations has been orientationally ordered |
| 283 |
|
\begin{figure} |
| 284 |
|
\includegraphics[width=7in]{Elip_3} |
| 285 |
< |
\caption{Ellipsoid reprsentation of 5CB at three different |
| 286 |
< |
field strengths, Zero Field (Left), Partial Field (Middle), and Full |
| 287 |
< |
Field (Right) Each image was created by taking the final |
| 288 |
< |
snapshot of each 60 ns run} |
| 285 |
> |
\caption{Ellipsoid reprsentation of 5CB at three different field |
| 286 |
> |
strengths, Zero Field (Left), Partial Field (Middle), and Full |
| 287 |
> |
Field (Right) Each image was created from the final configuration |
| 288 |
> |
of each 60 ns equilibration run.} |
| 289 |
|
\label{fig:Cigars} |
| 290 |
|
\end{figure} |
| 291 |
|
|
| 292 |
< |
\section{Analysis} |
| 292 |
> |
\section{Sampling the CN bond frequency} |
| 293 |
|
|
| 294 |
+ |
The primary quantity of interest is the distribution of vibrational |
| 295 |
+ |
frequencies exhibited by the 5CB nitrile bond under the different |
| 296 |
+ |
electric fields. Three distinct methods for mapping classical |
| 297 |
+ |
simulations onto vibrational spectra were brought to bear on these |
| 298 |
+ |
simulations: |
| 299 |
+ |
\begin{enumerate} |
| 300 |
+ |
\item Isolated 5CB molecules and their immediate surroundings were |
| 301 |
+ |
extracted from the simulations, their nitrile bonds were stretched |
| 302 |
+ |
and single-point {\em ab initio} calculations were used to obtain |
| 303 |
+ |
Morse-oscillator fits for the local vibrational motion along that |
| 304 |
+ |
bond. |
| 305 |
+ |
\item The potential - frequency maps developed by Cho {\it et |
| 306 |
+ |
al.}~\cite{Oh:2008fk} for nitrile moieties in water were |
| 307 |
+ |
investigated. This method involves mapping the electrostatic |
| 308 |
+ |
potential around the bond to the vibrational frequency, and is |
| 309 |
+ |
similar in approach to field-frequency maps that were pioneered by |
| 310 |
+ |
work done by Skinner {\it et al.}\cite{XXXX} |
| 311 |
+ |
\item Classical bond-length autocorrelation functions were Fourier |
| 312 |
+ |
transformed to directly obtain the vibrational spectrum from |
| 313 |
+ |
molecular dynamics simulations. |
| 314 |
+ |
\end{enumerate} |
| 315 |
+ |
|
| 316 |
+ |
\subsection{CN frequencies from isolated clusters} |
| 317 |
+ |
|
| 318 |
|
For quantum calculation of the nitrile bond frequency, Gaussian 09 was |
| 319 |
|
used. A single 5CB molecule was selected for the center of the |
| 320 |
|
cluster. For effects from molecules located near the chosen nitrile |
| 338 |
|
field spectrum, 67 frequencies were used and for the full field, 59 |
| 339 |
|
frequencies were used. |
| 340 |
|
|
| 341 |
+ |
\subsection{CN frequencies from potential-frequency maps} |
| 342 |
+ |
Before Gaussian silumations were carried out, it was attempt to apply |
| 343 |
+ |
the method developed by Cho {\it et al.}~\cite{Oh:2008fk} This method involves the fitting |
| 344 |
+ |
of multiple parameters to Gaussian calculated freuencies to find a |
| 345 |
+ |
correlation between the potential around the bond and the |
| 346 |
+ |
frequency. This is very similar to work done by Skinner {\it et al.}~with |
| 347 |
+ |
water models like SPC/E. The general method is to find the shift in |
| 348 |
+ |
the peak position through, |
| 349 |
+ |
\begin{equation} |
| 350 |
+ |
\delta\tilde{\nu} =\sum^{n}_{a=1} l_{a}\phi^{water}_{a} |
| 351 |
+ |
\end{equation} |
| 352 |
+ |
where $l_{a}$ are the fitting parameters and $\phi^{water}_{a}$ is the |
| 353 |
+ |
potential from the surrounding water cluster. This $\phi^{water}_{a}$ |
| 354 |
+ |
takes the form, |
| 355 |
+ |
\begin{equation} |
| 356 |
+ |
\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{m} \sum_{j} |
| 357 |
+ |
\frac{C^{H_{2}O}_{j \left(m \right) }}{r_{aj \left(m\right)}} |
| 358 |
+ |
\end{equation} |
| 359 |
+ |
where $C^{H_{2}O}_{j \left(m \right) }$ indicates the partial charge |
| 360 |
+ |
on the $j$th site of the $m$th water molecule and $r_{aj \left(m\right)}$ |
| 361 |
+ |
is the distance between the site $a$ of the nitrile molecule and the $j$th |
| 362 |
+ |
site of the $m$th water molecule. However, since these simulations |
| 363 |
+ |
are done under the presence of external fields and in the |
| 364 |
+ |
absence of water, the equations need a correction factor for the shift |
| 365 |
+ |
caused by the external field. The equation is also reworked to use |
| 366 |
+ |
electric field site data instead of partial charges from surrounding |
| 367 |
+ |
atoms. So by modifing the original |
| 368 |
+ |
$\phi^{water}_{a}$ to $\phi^{5CB}_{a}$ we get, |
| 369 |
+ |
\begin{equation} |
| 370 |
+ |
\phi^{5CB}_{a} = \frac{1}{4\pi \epsilon_{0}} \left( \vec{E}\bullet |
| 371 |
+ |
\left(\vec{r}_{a}-\vec{r}_{CN}\right) \right) + \phi^{5CB}_{0} |
| 372 |
+ |
\end{equation} |
| 373 |
+ |
where $\vec{E}$ is the electric field at each atom, $\left( \vec{r}_{a} - |
| 374 |
+ |
\vec{r}_{CN} \right)$ is the vector between the nitrile bond and the |
| 375 |
+ |
cooridinates described by Cho around the bond and $\phi^{5CB}_{0}$ is |
| 376 |
+ |
the correction factor for the system of parameters. After these |
| 377 |
+ |
changes, the correction factor was found for multiple values of an |
| 378 |
+ |
external field being applied. However, the factor was no linear and |
| 379 |
+ |
was overly large due to the fitting parameters being so small. |
| 380 |
+ |
|
| 381 |
+ |
|
| 382 |
+ |
\subsection{CN frequencies from bond length autocorrelation functions} |
| 383 |
+ |
|
| 384 |
|
Classical nitrile bond frequencies were found by replacing the rigid |
| 385 |
|
cyanide bond with a flexible Morse oscillator bond |
| 386 |
|
($r_0= 1.157437$ \AA , $D_0 = 212.95$ and |
| 403 |
|
with a rigid bond followed by an equilibrium run with the bond |
| 404 |
|
switched back to a Morse oscillator and a short production run of 20 ps. |
| 405 |
|
|
| 340 |
– |
\section{Results} |
| 406 |
|
|
| 407 |
|
This change in phase was followed by two courses of further |
| 408 |
|
analysis. First was the replacement of the static nitrile bond with a |
| 431 |
|
\label{fig:twoSpectra} |
| 432 |
|
\end{figure} |
| 433 |
|
|
| 369 |
– |
Before Gaussian silumations were carried out, it was attempt to apply |
| 370 |
– |
the method developed by Cho {\it et al.}~\cite{Oh:2008fk} This method involves the fitting |
| 371 |
– |
of multiple parameters to Gaussian calculated freuencies to find a |
| 372 |
– |
correlation between the potential around the bond and the |
| 373 |
– |
frequency. This is very similar to work done by Skinner {\it et al.}~with |
| 374 |
– |
water models like SPC/E. The general method is to find the shift in |
| 375 |
– |
the peak position through, |
| 376 |
– |
\begin{equation} |
| 377 |
– |
\delta\tilde{\nu} =\sum^{n}_{a=1} l_{a}\phi^{water}_{a} |
| 378 |
– |
\end{equation} |
| 379 |
– |
where $l_{a}$ are the fitting parameters and $\phi^{water}_{a}$ is the |
| 380 |
– |
potential from the surrounding water cluster. This $\phi^{water}_{a}$ |
| 381 |
– |
takes the form, |
| 382 |
– |
\begin{equation} |
| 383 |
– |
\phi_{a} = \frac{1}{4\pi \epsilon_{0}} \sum_{m} \sum_{j} |
| 384 |
– |
\frac{C^{H_{2}O}_{j \left(m \right) }}{r_{aj \left(m\right)}} |
| 385 |
– |
\end{equation} |
| 386 |
– |
where $C^{H_{2}O}_{j \left(m \right) }$ indicates the partial charge |
| 387 |
– |
on the $j$th site of the $m$th water molecule and $r_{aj \left(m\right)}$ |
| 388 |
– |
is the distance between the site $a$ of the nitrile molecule and the $j$th |
| 389 |
– |
site of the $m$th water molecule. However, since these simulations |
| 390 |
– |
are done under the presence of external fields and in the |
| 391 |
– |
absence of water, the equations need a correction factor for the shift |
| 392 |
– |
caused by the external field. The equation is also reworked to use |
| 393 |
– |
electric field site data instead of partial charges from surrounding |
| 394 |
– |
atoms. So by modifing the original |
| 395 |
– |
$\phi^{water}_{a}$ to $\phi^{5CB}_{a}$ we get, |
| 396 |
– |
\begin{equation} |
| 397 |
– |
\phi^{5CB}_{a} = \frac{1}{4\pi \epsilon_{0}} \left( \vec{E}\bullet |
| 398 |
– |
\left(\vec{r}_{a}-\vec{r}_{CN}\right) \right) + \phi^{5CB}_{0} |
| 399 |
– |
\end{equation} |
| 400 |
– |
where $\vec{E}$ is the electric field at each atom, $\left( \vec{r}_{a} - |
| 401 |
– |
\vec{r}_{CN} \right)$ is the vector between the nitrile bond and the |
| 402 |
– |
cooridinates described by Cho around the bond and $\phi^{5CB}_{0}$ is |
| 403 |
– |
the correction factor for the system of parameters. After these |
| 404 |
– |
changes, the correction factor was found for multiple values of an |
| 405 |
– |
external field being applied. However, the factor was no linear and |
| 406 |
– |
was overly large due to the fitting parameters being so small. |
| 434 |
|
|
| 435 |
|
Due to this, Gaussian calculations were performed in lieu of this |
| 436 |
|
method. A set of snapshots for the zero and full field simualtions, |
