| 101 |
|
dipolar fluids.\cite{Kirkwood39,Onsagar36} Along with projections of |
| 102 |
|
the frequency-dependent dielectric to zero frequency, these |
| 103 |
|
fluctuation formulae are now widely used to predict the static |
| 104 |
< |
dielectric constant of simulated materials. |
| 104 |
> |
dielectric constants of simulated materials. |
| 105 |
|
|
| 106 |
|
If we consider a system of dipolar or quadrupolar molecules under the |
| 107 |
|
influence of an external field or field gradient, the net polarization |
| 185 |
|
|
| 186 |
|
Under the influence of weak external fields, the bulk polarization of |
| 187 |
|
the system is primarily a linear response to the perturbation, where |
| 188 |
< |
proportionality constant depends on the electrostatic interactions |
| 188 |
> |
the proportionality constant depends on the electrostatic interactions |
| 189 |
|
between the multipoles. The fluctuation formulae connect bulk |
| 190 |
|
properties of the fluid to equilibrium fluctuations in the system |
| 191 |
|
multipolar moments during a simulation. These fluctuations also depend |
| 259 |
|
$v_{22}$ are shifted separately. In this expression, |
| 260 |
|
$\hat{\mathbf{r}}$ is the unit vector connecting the two multipoles |
| 261 |
|
($a$ and $b$) in space, and $\mathsf{A}$ and $\mathsf{B}$ represent |
| 262 |
< |
the orientations the multipoles. Because this procedure is equivalent |
| 262 |
> |
the orientations of the multipoles. Because this procedure is equivalent |
| 263 |
|
to using the gradient of an image multipole placed at the cutoff |
| 264 |
|
sphere for shifting the force, this method is called the gradient |
| 265 |
|
shifted-force (GSF) approach. |
| 304 |
|
reorientations towards the direction of the applied field. There is an |
| 305 |
|
added complication that in the presence of external field, the |
| 306 |
|
perturbation experienced by any single molecule is not only due to |
| 307 |
< |
external field but also to the fields produced by the all other |
| 307 |
> |
the external field but also to the fields produced by the all other |
| 308 |
|
molecules in the system. |
| 309 |
|
|
| 310 |
|
\subsection{Response to External Perturbations} |
| 489 |
|
d\mathbf{r} \mathbf{T}(\mathbf{r}) |
| 490 |
|
\end{equation} |
| 491 |
|
which is the $k \rightarrow 0$ limit of |
| 492 |
< |
$\tilde{\mathbf{T}}(\mathbf{r})$. Note that the integration of the |
| 492 |
> |
$\tilde{\mathbf{T}}(\mathbf{k})$. Note that the integration of the |
| 493 |
|
dipole tensors, Eqs. (\ref{dipole-chargeTensor}) and |
| 494 |
|
(\ref{dipole-diopleTensor}), over spherical volumes yields values only |
| 495 |
|
along the diagonal. Additionally, the spherical symmetry of |
| 664 |
|
\label{propConstQuad} |
| 665 |
|
\end{equation} |
| 666 |
|
Note that as in the dipolar case, $\alpha_Q$ and $\chi_Q$ are distinct |
| 667 |
< |
quantities. $\chi_Q$ measures bulk response assuming an infinite |
| 667 |
> |
quantities. $\chi_Q$ measures the bulk response assuming an infinite |
| 668 |
|
system and exact electrostatics, while $\alpha_Q$ is relatively simple |
| 669 |
|
to compute from numerical simulations. As in the dipolar case, |
| 670 |
|
estimation of the true bulk property requires correction for |
| 753 |
|
where $T_{\gamma\delta}(\mathbf{r})$ is a dipole-dipole interaction |
| 754 |
|
tensor that depends on the level of the approximation (see |
| 755 |
|
Eq. (\ref{dipole-diopleTensor})).\cite{PaperI,PaperII} Similarly |
| 756 |
< |
$v_{21}(r)$ and $v_{22}(r)$ are the radial function for different real |
| 756 |
> |
$v_{21}(r)$ and $v_{22}(r)$ are the radial functions for different real |
| 757 |
|
space cutoff methods defined in the first paper in this |
| 758 |
|
series.\cite{PaperI} |
| 759 |
|
|
| 778 |
|
parts, $|\mathbf{r}-\mathbf{r}^\prime|\rightarrow 0 $ and |
| 779 |
|
$|\mathbf{r}-\mathbf{r}^\prime|> 0$. Since the self-contribution to |
| 780 |
|
the field gradient vanishes at the singularity (see the supplemental |
| 781 |
< |
material),\cite{supplemental} Eq. (\ref{gradMaxwell}) can be |
| 781 |
> |
material), Eq. (\ref{gradMaxwell}) can be |
| 782 |
|
written as, |
| 783 |
|
\begin{equation} |
| 784 |
|
\partial_\alpha E_\beta(\mathbf{r}) = \partial_\alpha {E}^\circ_\beta(\mathbf{r}) + |
| 1054 |
|
system, the divergence of the field will be zero, \textit{i.e.} |
| 1055 |
|
$\nabla \cdot \mathbf{E} = 0$. This condition can be satisfied |
| 1056 |
|
by using the relatively simple applied potential as described in the |
| 1057 |
< |
supplemental material.\cite{supplemental} |
| 1057 |
> |
supplemental material. |
| 1058 |
|
|
| 1059 |
|
When a constant electric field or field gradient is applied to the |
| 1060 |
|
system, the molecules align along the direction of the applied field, |
| 1072 |
|
multipolar fluids. The box multipolar moments were computed as simple |
| 1073 |
|
sums over the instantaneous molecular moments, and fluctuations in |
| 1074 |
|
these quantities were obtained from Eqs. (\ref{eq:flucDip}) and |
| 1075 |
< |
(\ref{eq:flucQuad}). The macroscopic polarizabilities of the system at |
| 1076 |
< |
a were derived using Eqs.(\ref{flucDipole}) and (\ref{flucQuad}). |
| 1075 |
> |
(\ref{eq:flucQuad}). The macroscopic polarizabilities of the system |
| 1076 |
> |
were derived using Eqs.(\ref{flucDipole}) and (\ref{flucQuad}). |
| 1077 |
|
|
| 1078 |
|
The final system consists of dipolar or quadrupolar fluids with two |
| 1079 |
|
oppositely charged ions embedded within the fluid. These ions are |
| 1113 |
|
out over a 1~ns simulation in the microcanonical (NVE) ensemble. Box |
| 1114 |
|
dipole moments were sampled every fs. For simulations with external |
| 1115 |
|
perturbations, field strengths ranging from |
| 1116 |
< |
$0 - 10 \times 10^{-4}$~V/\AA\ with increments of $ 10^{-4}$~V/\AA\ |
| 1116 |
> |
0 to 10 $10^{-3}$~V/\AA\ with increments of $ 10^{-4}$~V/\AA\ |
| 1117 |
|
were carried out for each system. For dipolar systems the interaction |
| 1118 |
|
potential between molecules $i$ and $j$, |
| 1119 |
|
\begin{equation} |
| 1240 |
|
|
| 1241 |
|
It is also notable that the TSF method again displays smaller |
| 1242 |
|
perturbations away from the correct dielectric screening behavior. We |
| 1243 |
< |
also observe that for TSF method yields high dielectric screening even |
| 1244 |
< |
for lower values of $\alpha$. |
| 1243 |
> |
also observe that for TSF, the method yields high dielectric screening |
| 1244 |
> |
even for lower values of $\alpha$. |
| 1245 |
|
|
| 1246 |
|
At short distances, the presence of the ions creates a strong local |
| 1247 |
|
field that acts to align nearby dipoles nearly perfectly in opposition |
| 1251 |
|
that the local ordering behavior is being captured by all of the |
| 1252 |
|
moderately-damped real-space methods. |
| 1253 |
|
|
| 1254 |
< |
\subsubsection{Distance-dependent Kirkwood factors} |
| 1254 |
> |
\subsubsection*{Distance-dependent Kirkwood factors} |
| 1255 |
|
One of the most sensitive measures of dipolar ordering in a liquid is |
| 1256 |
|
the disance dependent Kirkwood factor, |
| 1257 |
|
\begin{equation} |
| 1279 |
|
\end{figure} |
| 1280 |
|
Note that like the dielectric constant, $G_K(r)$ can also be corrected |
| 1281 |
|
using the expressions for $A$ in table \ref{tab:A}. This is discussed |
| 1282 |
< |
in more detail in the supplemental material.\cite{supplemental} |
| 1282 |
> |
in more detail in the supplemental material. |
| 1283 |
|
|
| 1284 |
|
\subsection{Quadrupolar fluid} |
| 1285 |
|
\begin{figure} |
| 1468 |
|
become less consequential for higher order multipoles. |
| 1469 |
|
|
| 1470 |
|
For this reason, our recommendation is that the moderately-damped |
| 1471 |
< |
($\alpha = 0.24-0.25$~\AA$^{-1}$) GSF method is a good choice for |
| 1471 |
> |
($\alpha = 0.25-0.27$~\AA$^{-1}$) GSF method is a good choice for |
| 1472 |
|
molecular dynamics simulations where point-multipole interactions are |
| 1473 |
|
being utilized to compute bulk dielectric properties of fluids. |
| 1474 |
+ |
|
| 1475 |
+ |
\section*{Supplementary Material} |
| 1476 |
+ |
See supplementary material for information on interactions with |
| 1477 |
+ |
spatially varying fields, Boltzmann averages, self-contributions from |
| 1478 |
+ |
quadrupoles, and corrections to distance-dependent Kirkwood factors. |
| 1479 |
+ |
|
| 1480 |
+ |
\begin{acknowledgments} |
| 1481 |
+ |
Support for this project was provided by the National Science Foundation |
| 1482 |
+ |
under grant CHE-1362211. Computational time was provided by the |
| 1483 |
+ |
Center for Research Computing (CRC) at the University of Notre |
| 1484 |
+ |
Dame. The authors would like to thank the reviewer for helpful |
| 1485 |
+ |
comments and suggestions. |
| 1486 |
+ |
\end{acknowledgments} |
| 1487 |
|
|
| 1488 |
|
|
| 1489 |
|
\appendix |
| 1633 |
|
&=& -\frac{8r_c^3 \kappa^5 e^{-\kappa^2 r_c^2}}{15\sqrt{\pi}}. |
| 1634 |
|
\end{eqnarray} |
| 1635 |
|
|
| 1623 |
– |
\begin{acknowledgments} |
| 1624 |
– |
Support for this project was provided by the National Science Foundation |
| 1625 |
– |
under grant CHE-1362211. Computational time was provided by the |
| 1626 |
– |
Center for Research Computing (CRC) at the University of Notre |
| 1627 |
– |
Dame. The authors would like to thank the reviewer for helpful |
| 1628 |
– |
comments and suggestions. |
| 1629 |
– |
\end{acknowledgments} |
| 1636 |
|
|
| 1637 |
|
\newpage |
| 1638 |
|
\bibliography{dielectric_new} |
