| 40 |
|
|
| 41 |
|
|
| 42 |
|
\title{Nitrile vibrations as reporters of field-induced phase |
| 43 |
< |
transitions in liquid crystals} |
| 43 |
> |
transitions in 4-cyano-4'-pentylbiphenyl} |
| 44 |
|
\author{James M. Marr} |
| 45 |
|
\author{J. Daniel Gezelter} |
| 46 |
|
\email{gezelter@nd.edu} |
| 58 |
|
\begin{doublespace} |
| 59 |
|
|
| 60 |
|
\begin{abstract} |
| 61 |
< |
Nitrile Stark shift repsonses to electric fields have been used |
| 62 |
< |
extensively in biology for the probing of local internal fields of |
| 63 |
< |
structures like proteins and DNA. Intigration of these probes into |
| 64 |
< |
different areas of interest are important for studing local structure |
| 65 |
< |
and fields within confined, nanoscopic |
| 66 |
< |
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 |
| 77 |
< |
cyanide bond displacement in time use enabled to show a spectral |
| 78 |
< |
change in the formation of a red |
| 79 |
< |
shifted peak from the main peak as an external field was applied. This indicates |
| 80 |
< |
that local structure has a larger impact on the nitrile frequency then |
| 81 |
< |
does the local electric field. By better understanding how nitrile |
| 82 |
< |
groups respond to local and external fields it will help |
| 83 |
< |
nitrile groups branch out beyond their biological |
| 84 |
< |
applications to uses in electronics and surface sciences. |
| 61 |
> |
4-cyano-4'-pentylbiphenyl (5CB) is a liquid-crystal-forming compound |
| 62 |
> |
with a terminal nitrile group aligned with the long axis of the |
| 63 |
> |
molecule. Simulations of condensed-phase 5CB were carried out both |
| 64 |
> |
with and without applied electric fields to provide an understanding |
| 65 |
> |
of the the Stark shift of the terminal nitrile group. A |
| 66 |
> |
field-induced isotropic-nematic phase transition was observed in the |
| 67 |
> |
simulations, and the effects of this transition on the distribution |
| 68 |
> |
of nitrile frequencies were computed. Classical bond displacement |
| 69 |
> |
correlation functions exhibit a $\sim 40 \mathrm{~cm}^{-1}$ red |
| 70 |
> |
shift of a portion of the main nitrile peak, and this shift was |
| 71 |
> |
observed only when the fields were large enough to induce |
| 72 |
> |
orientational ordering of the bulk phase. Our simulations appear to |
| 73 |
> |
indicate that phase-induced changes to the local surroundings are a |
| 74 |
> |
larger contribution to the change in the nitrile spectrum than |
| 75 |
> |
direct field contributions. |
| 76 |
|
\end{abstract} |
| 77 |
|
|
| 78 |
|
\newpage |
| 79 |
|
|
| 80 |
|
\section{Introduction} |
| 81 |
|
|
| 82 |
+ |
Nitrile groups can serve as very precise electric field reporters via |
| 83 |
+ |
their distinctive Raman and IR signatures.\cite{Boxer:2009xw} The |
| 84 |
+ |
triple bond between the nitrogen and the carbon atom is very sensitive |
| 85 |
+ |
to local field changes and has been observed to have a direct impact |
| 86 |
+ |
on the peak position within the spectrum. The Stark shift in the |
| 87 |
+ |
spectrum can be quantified and mapped into a field value that is |
| 88 |
+ |
impinging upon the nitrile bond. This has been used extensively in |
| 89 |
+ |
biological systems like proteins and |
| 90 |
+ |
enzymes.\cite{Tucker:2004qq,Webb:2008kn} |
| 91 |
+ |
|
| 92 |
+ |
The response of nitrile groups to electric fields has now been |
| 93 |
+ |
investigated for a number of small molecules,\cite{Andrews:2000qv} as |
| 94 |
+ |
well as in biochemical settings, where nitrile groups can act as |
| 95 |
+ |
minimally invasive probes of structure and |
| 96 |
+ |
dynamics.\cite{Lindquist:2009fk,Fafarman:2010dq} The vibrational Stark |
| 97 |
+ |
effect has also been used to study the effects of electric fields on |
| 98 |
+ |
nitrile-containing self-assembled monolayers at metallic |
| 99 |
+ |
interfaces.\cite{Oklejas:2002uq,Schkolnik:2012ty} |
| 100 |
+ |
|
| 101 |
+ |
Recently 4-cyano-4'-pentylbiphenyl (5CB), a liquid crystalline |
| 102 |
+ |
molecule with a terminal nitrile group, has seen renewed interest as |
| 103 |
+ |
one way to impart order on the surfactant interfaces of |
| 104 |
+ |
nanodroplets,\cite{Moreno-Razo:2012rz} or to drive surface-ordering |
| 105 |
+ |
that can be used to promote particular kinds of |
| 106 |
+ |
self-assembly.\cite{PhysRevLett.111.227801} The nitrile group in 5CB |
| 107 |
+ |
is a particularly interesting case for studying electric field |
| 108 |
+ |
effects, as 5CB exhibits an isotropic to nematic phase transition that |
| 109 |
+ |
can be triggered by the application of an external field near room |
| 110 |
+ |
temperature.\cite{Gray:1973ca,Hatta:1991ee} This presents the |
| 111 |
+ |
possiblity that the field-induced changes in the local environment |
| 112 |
+ |
could have dramatic effects on the vibrations of this particular CN |
| 113 |
+ |
bond. Although the infrared spectroscopy of 5CB has been |
| 114 |
+ |
well-investigated, particularly as a measure of the kinetics of the |
| 115 |
+ |
phase transition,\cite{Leyte:1997zl} the 5CB nitrile group has not yet |
| 116 |
+ |
seen the detailed theoretical treatment that biologically-relevant |
| 117 |
+ |
small molecules have |
| 118 |
+ |
received.\cite{Lindquist:2008bh,Lindquist:2008qf,Oh:2008fk,Choi:2008cr,Waegele:2010ve} |
| 119 |
+ |
|
| 120 |
|
The fundamental characteristic of liquid crystal mesophases is that |
| 121 |
|
they maintain some degree of orientational order while translational |
| 122 |
|
order is limited or absent. This orientational order produces a |
| 123 |
|
complex direction-dependent response to external perturbations like |
| 124 |
< |
electric fields and mechanical distortions. The anisotropy of the |
| 124 |
> |
electric fields and mechanical distortions. The anisotropy of the |
| 125 |
|
macroscopic phases originates in the anisotropy of the constituent |
| 126 |
|
molecules, which typically have highly non-spherical structures with a |
| 127 |
< |
significant degree of internal rigidity. In nematic phases, rod-like |
| 127 |
> |
significant degree of internal rigidity. In nematic phases, rod-like |
| 128 |
|
molecules are orientationally ordered with isotropic distributions of |
| 129 |
< |
molecular centers of mass, while in smectic phases, the molecules |
| 130 |
< |
arrange themselves into layers with their long (symmetry) axis normal |
| 131 |
< |
($S_{A}$) or tilted ($S_{C}$) with respect to the layer planes. |
| 129 |
> |
molecular centers of mass. For example, 5CB has a solid to nematic |
| 130 |
> |
phase transition at 18C and a nematic to isotropic transition at |
| 131 |
> |
35C.\cite{Gray:1973ca} |
| 132 |
|
|
| 133 |
< |
The behavior of the $S_{A}$ phase can be partially explained with |
| 134 |
< |
models mainly based on geometric factors and van der Waals |
| 135 |
< |
interactions. However, these simple models are insufficient to |
| 136 |
< |
describe liquid crystal phases which exhibit more complex polymorphic |
| 137 |
< |
nature. X-ray diffraction studies have shown that the ratio between |
| 138 |
< |
lamellar spacing ($s$) and molecular length ($l$) can take on a wide |
| 139 |
< |
range of values.\cite{Gray:1984hc,Leadbetter:1976vf,Hardouin:1980yq} |
| 140 |
< |
Typical $S_{A}$ phases have $s/l$ ratios on the order of $0.8$, while |
| 141 |
< |
for some compounds, e.g. the 4-alkyl-4'-cyanobiphenyls, the $s/l$ |
| 142 |
< |
ratio is on the order of $1.4$. Molecules which form $S_{A}$ phases |
| 143 |
< |
can exhibit a wide variety of subphases like monolayers ($S_{A1}$), |
| 144 |
< |
uniform bilayers ($S_{A2}$), partial bilayers ($S_{\tilde A}$) as well |
| 145 |
< |
as interdigitated bilayers ($S_{A_{d}}$), and often have a terminal |
| 146 |
< |
cyano or nitro group. In particular lyotropic liquid crystals (those |
| 147 |
< |
exhibiting liquid crystal phase transition as a function of water |
| 148 |
< |
concentration) often have polar head groups or zwitterionic charge |
| 149 |
< |
separated groups that result in strong dipolar |
| 150 |
< |
interactions.\cite{Collings97} Because of their versatile polymorphic |
| 151 |
< |
nature, polar liquid crystalline materials have important |
| 152 |
< |
technological applications in addition to their immense relevance to |
| 124 |
< |
biological systems.\cite{Collings97} |
| 133 |
> |
In smectic phases, the molecules arrange themselves into layers with |
| 134 |
> |
their long (symmetry) axis normal ($S_{A}$) or tilted ($S_{C}$) with |
| 135 |
> |
respect to the layer planes. The behavior of the $S_{A}$ phase can be |
| 136 |
> |
partially explained with models mainly based on geometric factors and |
| 137 |
> |
van der Waals interactions. The Gay-Berne potential, in particular, |
| 138 |
> |
has been widely used in the liquid crystal community to describe this |
| 139 |
> |
anisotropic phase |
| 140 |
> |
behavior.~\cite{Gay:1981yu,Berne72,Kushick:1976xy,Luckhurst90,Cleaver:1996rt} |
| 141 |
> |
However, these simple models are insufficient to describe liquid |
| 142 |
> |
crystal phases which exhibit more complex polymorphic nature. |
| 143 |
> |
Molecules which form $S_{A}$ phases can exhibit a wide variety of |
| 144 |
> |
subphases like monolayers ($S_{A1}$), uniform bilayers ($S_{A2}$), |
| 145 |
> |
partial bilayers ($S_{\tilde A}$) as well as interdigitated bilayers |
| 146 |
> |
($S_{A_{d}}$), and often have a terminal cyano or nitro group. In |
| 147 |
> |
particular, lyotropic liquid crystals (those exhibiting liquid crystal |
| 148 |
> |
phase transition as a function of water concentration), often have |
| 149 |
> |
polar head groups or zwitterionic charge separated groups that result |
| 150 |
> |
in strong dipolar interactions,\cite{Collings:1997rz} and terminal cyano |
| 151 |
> |
groups (like the one in 5CB) can induce permanent longitudinal |
| 152 |
> |
dipoles.\cite{Levelut:1981eu} |
| 153 |
|
|
| 154 |
< |
Experimental studies by Levelut {\it et al.}~\cite{Levelut:1981eu} |
| 155 |
< |
revealed that terminal cyano or nitro groups usually induce permanent |
| 156 |
< |
longitudinal dipole moments on the molecules. |
| 154 |
> |
Macroscopic electric fields applied using electrodes on opposing sides |
| 155 |
> |
of a sample of 5CB have demonstrated the phase change of the molecule |
| 156 |
> |
as a function of electric field.\cite{Lim:2006xq} These previous |
| 157 |
> |
studies have shown the nitrile group serves as an excellent indicator |
| 158 |
> |
of the molecular orientation within the applied field. Lee {\it et |
| 159 |
> |
al.}~showed a 180 degree change in field direction could be probed |
| 160 |
> |
with the nitrile peak intensity as it changed along with molecular |
| 161 |
> |
alignment in the field.\cite{Lee:2006qd,Leyte:1997zl} |
| 162 |
|
|
| 163 |
< |
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 |
| 163 |
> |
While these macroscopic fields work well at indicating the bulk |
| 164 |
|
response, the atomic scale response has not been studied. With the |
| 165 |
|
advent of nano-electrodes and coupling them with atomic force |
| 166 |
|
microscopy, control of electric fields applied across nanometer |
| 167 |
< |
distances is now possible\cite{citation1}. This application of |
| 168 |
< |
nanometer length is interesting in the case of a nitrile group on the |
| 169 |
< |
molecule. While macroscopic fields are insufficient to cause a Stark |
| 170 |
< |
effect, small fields across nanometer-sized gaps are of sufficient |
| 171 |
< |
strength. If one were to assume a gap of 5 nm between a lower |
| 172 |
< |
electrode having a nanoelectrode placed near it via an atomic force |
| 173 |
< |
microscope, a field of 1 V applied across the electrodes would |
| 174 |
< |
translate into a field of 2x10\textsuperscript{8} $\frac{V}{M}$. This |
| 175 |
< |
field is theoretically strong enough to cause a phase change from |
| 176 |
< |
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. |
| 167 |
> |
distances is now possible.\cite{citation1} While macroscopic fields |
| 168 |
> |
are insufficient to cause a Stark effect without dielectric breakdown |
| 169 |
> |
of the material, small fields across nanometer-sized gaps may be of |
| 170 |
> |
sufficient strength. For a gap of 5 nm between a lower electrode |
| 171 |
> |
having a nanoelectrode placed near it via an atomic force microscope, |
| 172 |
> |
a potential of 1 V applied across the electrodes is equivalent to a |
| 173 |
> |
field of 2x10\textsuperscript{8} $\frac{V}{M}$. This field is |
| 174 |
> |
certainly strong enough to cause the isotropic-nematic phase change |
| 175 |
> |
and as well as Stark tuning of the nitrile bond. This should be |
| 176 |
> |
readily visible experimentally through Raman or IR spectroscopy. |
| 177 |
|
|
| 178 |
< |
Herein, we show the computational investigation of these electric |
| 179 |
< |
field effects through atomistic simulations of 5CB with external |
| 180 |
< |
fields applied. These simulations are then coupled with ab intio and |
| 181 |
< |
classical spectrum calculations to predict changes. These changes are |
| 182 |
< |
easily varifiable with experiments and should be able to replicated |
| 183 |
< |
experimentally. |
| 178 |
> |
In the sections that follow, we outline a series of coarse-grained |
| 179 |
> |
classical molecular dynamics simulations of 5CB that were done in the |
| 180 |
> |
presence of static electric fields. These simulations were then |
| 181 |
> |
coupled with both {\it ab intio} calculations of CN-deformations and |
| 182 |
> |
classical bond-length correlation functions to predict spectral |
| 183 |
> |
shifts. These predictions made should be easily varifiable with |
| 184 |
> |
scanning electrochemical microscopy experiments. |
| 185 |
|
|
| 186 |
|
\section{Computational Details} |
| 187 |
< |
The force field was mainly taken from Guo et al.\cite{Zhang:2011hh} A |
| 188 |
< |
deviation from this force field was made to create a rigid body from |
| 189 |
< |
the phenyl rings. Bond distances within the rigid body were taken from |
| 190 |
< |
equilibrium bond distances. While the phenyl rings were held rigid, |
| 191 |
< |
bonds, bends, torsions and inversion centers still included the rings. |
| 187 |
> |
The force field used for 5CB was taken from Guo {\it et |
| 188 |
> |
al.}\cite{Zhang:2011hh} However, for most of the simulations, each |
| 189 |
> |
of the phenyl rings was treated as a rigid body to allow for larger |
| 190 |
> |
time steps and very long simulation times. The geometries of the |
| 191 |
> |
rigid bodies were taken from equilibrium bond distances and angles. |
| 192 |
> |
Although the phenyl rings were held rigid, bonds, bends, torsions and |
| 193 |
> |
inversion centers that involved atoms in these substructures (but with |
| 194 |
> |
connectivity to the rest of the molecule) were still included in the |
| 195 |
> |
potential and force calculations. |
| 196 |
|
|
| 197 |
< |
Simulations were with boxes of 270 molecules locked at experimental |
| 198 |
< |
densities with periodic boundaries. The molecules were thermalized |
| 199 |
< |
from 0 kelvin to 300 kelvin. To equilibrate, each was first run in NVT |
| 200 |
< |
for 1 ns. This was followed by NVE for simulations used in the data |
| 201 |
< |
collection. |
| 197 |
> |
Periodic simulations cells containing 270 molecules in random |
| 198 |
> |
orientations were constructed and were locked at experimental |
| 199 |
> |
densities. Electrostatic interactions were computed using damped |
| 200 |
> |
shifted force (DSF) electrostatics.\cite{Fennell:2006zl} The molecules |
| 201 |
> |
were equilibrated for 1~ns at a temperature of 300K. Simulations with |
| 202 |
> |
applied fields were carried out in the microcanonical (NVE) ensemble |
| 203 |
> |
with an energy corresponding to the average energy from the canonical |
| 204 |
> |
(NVT) equilibration runs. Typical applied-field runs were more than |
| 205 |
> |
60ns in length. |
| 206 |
|
|
| 207 |
< |
External electric fields were applied in a simplistic charge-field |
| 208 |
< |
interaction. Forces were calculated by multiplying the electric field |
| 209 |
< |
being applied by the charge at each atom. For the potential, the |
| 210 |
< |
origin of the box was used as a point of reference. This allows for a |
| 211 |
< |
potential value to be added to each atom as the molecule move in space |
| 212 |
< |
within the box. Fields values were applied in a manner representing |
| 213 |
< |
those that would be applable in an experimental set-up. The assumed |
| 214 |
< |
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. |
| 207 |
> |
Static electric fields with magnitudes similar to what would be |
| 208 |
> |
available in an experimental setup were applied to the different |
| 209 |
> |
simulations. With an assumed electrode seperation of 5 nm and an |
| 210 |
> |
electrostatic potential that is limited by the voltage required to |
| 211 |
> |
split water (1.23V), the maximum realistic field that could be applied |
| 212 |
> |
is $\sim 0.024$ V/\AA. Three field environments were investigated: |
| 213 |
> |
(1) no field applied, (2) partial field = 0.01 V/\AA\ , and (3) full |
| 214 |
> |
field = 0.024 V/\AA\ . |
| 215 |
|
|
| 216 |
+ |
After the systems had come to equilibrium under the applied fields, |
| 217 |
+ |
additional simulations were carried out with a flexible (Morse) |
| 218 |
+ |
nitrile bond, |
| 219 |
+ |
\begin{displaymath} |
| 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}$. 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 |
| 233 |
+ |
correlation functions were computed using data sampled from an |
| 234 |
+ |
additional 200 ps of run time carried out in the microcanonical |
| 235 |
+ |
ensemble. |
| 236 |
+ |
|
| 237 |
+ |
\section{Field-induced Nematic Ordering} |
| 238 |
+ |
|
| 239 |
+ |
In order to characterize the orientational ordering of the system, the |
| 240 |
+ |
primary quantity of interest is the nematic (orientational) order |
| 241 |
+ |
parameter. This was determined using the tensor |
| 242 |
+ |
\begin{equation} |
| 243 |
+ |
Q_{\alpha \beta} = \frac{1}{2N} \sum_{i=1}^{N} \left(3 \hat{e}_{i |
| 244 |
+ |
\alpha} \hat{e}_{i \beta} - \delta_{\alpha \beta} \right) |
| 245 |
+ |
\end{equation} |
| 246 |
+ |
where $\alpha, \beta = x, y, z$, and $\hat{e}_i$ is the molecular |
| 247 |
+ |
end-to-end unit vector for molecule $i$. The nematic order parameter |
| 248 |
+ |
$S$ is the largest eigenvalue of $Q_{\alpha \beta}$, and the |
| 249 |
+ |
corresponding eigenvector defines the director axis for the phase. |
| 250 |
+ |
$S$ takes on values close to 1 in highly ordered (smectic A) phases, |
| 251 |
+ |
but falls to zero for isotropic fluids. Note that the nitrogen and |
| 252 |
+ |
the terminal chain atom were used to define the vectors for each |
| 253 |
+ |
molecule, so the typical order parameters are lower than if one |
| 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 |
+ |
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 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=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 |
+ |
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 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{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 |
|
|
| 268 |
– |
\section{Results} |
| 406 |
|
|
| 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 |
– |
|
| 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 |
| 409 |
|
morse oscillator bond. This was then simulated for a period of time |
| 431 |
|
\label{fig:twoSpectra} |
| 432 |
|
\end{figure} |
| 433 |
|
|
| 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. |
| 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, |
