| 139 |
|
\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) |
| 140 |
|
+ s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf |
| 141 |
|
\Omega}_j)]\ . |
| 142 |
+ |
\label{stickyfunction} |
| 143 |
|
\end{equation} |
| 144 |
|
Here, $\nu_0$ is a strength parameter for the sticky potential, and |
| 145 |
|
$s$ and $s^\prime$ are cubic switching functions which turn off the |
| 152 |
|
while the $w^\prime$ function counters the normal aligned and |
| 153 |
|
anti-aligned structures favored by point dipoles: |
| 154 |
|
\begin{equation} |
| 155 |
< |
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
| 155 |
> |
w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^\circ, |
| 156 |
|
\end{equation} |
| 157 |
|
It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
| 158 |
|
and $Y_3^{-2}$ spherical harmonics (a linear combination which |
| 209 |
|
|
| 210 |
|
Long-range dipole-dipole interactions were accounted for in this study |
| 211 |
|
by using either the reaction field method or by resorting to a simple |
| 212 |
< |
cubic switching function at a cutoff radius. Under the first method, |
| 213 |
< |
the magnitude of the reaction field acting on dipole $i$ is |
| 212 |
> |
cubic switching function at a cutoff radius. The reaction field |
| 213 |
> |
method was actually first used in Monte Carlo simulations of liquid |
| 214 |
> |
water.\cite{Barker73} Under this method, the magnitude of the reaction |
| 215 |
> |
field acting on dipole $i$ is |
| 216 |
|
\begin{equation} |
| 217 |
|
\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1} |
| 218 |
|
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} f(r_{ij})\ , |
| 235 |
|
|
| 236 |
|
We have also performed a companion set of simulations {\it without} a |
| 237 |
|
surrounding dielectric (i.e. using a simple cubic switching function |
| 238 |
< |
at the cutoff radius) and as a result we have two reparamaterizations |
| 239 |
< |
of SSD which could be used either with or without the Reaction Field |
| 238 |
> |
at the cutoff radius), and as a result we have two reparamaterizations |
| 239 |
> |
of SSD which could be used either with or without the reaction field |
| 240 |
|
turned on. |
| 241 |
|
|
| 242 |
|
Simulations to obtain the preferred density were performed in the |
| 254 |
|
symplectic splitting method proposed by Dullweber {\it et |
| 255 |
|
al.}\cite{Dullweber1997} Our reason for selecting this integrator |
| 256 |
|
centers on poor energy conservation of rigid body dynamics using |
| 257 |
< |
traditional quaternion integration.\cite{Evans77,Evans77b} While quaternions |
| 258 |
< |
may work well for orientational motion under NVT or NPT integrators, |
| 259 |
< |
our limits on energy drift in the microcanonical ensemble were quite |
| 260 |
< |
strict, and the drift under quaternions was substantially greater than |
| 261 |
< |
in the symplectic splitting method. This steady drift in the total |
| 259 |
< |
energy has also been observed by Kol {\it et al.}\cite{Laird97} |
| 257 |
> |
traditional quaternion integration.\cite{Evans77,Evans77b} In typical |
| 258 |
> |
microcanonical ensemble simulations, the energy drift when using |
| 259 |
> |
quaternions was substantially greater than when using the symplectic |
| 260 |
> |
splitting method (fig. \ref{timestep}). This steady drift in the |
| 261 |
> |
total energy has also been observed by Kol {\it et al.}\cite{Laird97} |
| 262 |
|
|
| 263 |
|
The key difference in the integration method proposed by Dullweber |
| 264 |
|
\emph{et al.} is that the entire rotation matrix is propagated from |
| 448 |
|
mean-square displacement as a function of time. The averaged results |
| 449 |
|
from five sets of NVE simulations are displayed in figure |
| 450 |
|
\ref{diffuse}, alongside experimental, SPC/E, and TIP5P |
| 451 |
< |
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} |
| 451 |
> |
results.\cite{Gillen72,Holz00,Clancy94,Jorgensen01} |
| 452 |
|
|
| 453 |
|
\begin{figure} |
| 454 |
|
\begin{center} |
| 456 |
|
\epsfbox{betterDiffuse.epsi} |
| 457 |
|
\caption{Average self-diffusion constant as a function of temperature for |
| 458 |
|
SSD, SPC/E [Ref. \citen{Clancy94}], TIP5P [Ref. \citen{Jorgensen01}], |
| 459 |
< |
and Experimental data [Refs. \citen{Gillen72} and \citen{Mills73}]. Of |
| 459 |
> |
and Experimental data [Refs. \citen{Gillen72} and \citen{Holz00}]. Of |
| 460 |
|
the three water models shown, SSD has the least deviation from the |
| 461 |
|
experimental values. The rapidly increasing diffusion constants for |
| 462 |
|
TIP5P and SSD correspond to significant decrease in density at the |
| 593 |
|
important properties. In this case, it would be ideal to correct the |
| 594 |
|
densities while maintaining the accurate transport behavior. |
| 595 |
|
|
| 596 |
< |
The parameters available for tuning include the $\sigma$ and $\epsilon$ |
| 597 |
< |
Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky |
| 598 |
< |
attractive and dipole repulsive terms with their respective |
| 599 |
< |
cutoffs. To alter the attractive and repulsive terms of the sticky |
| 600 |
< |
potential independently, it is necessary to separate the terms as |
| 601 |
< |
follows: |
| 602 |
< |
\begin{equation} |
| 601 |
< |
u_{ij}^{sp} |
| 602 |
< |
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = |
| 603 |
< |
\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)] + \frac{\nu_0^\prime}{2} [s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)], |
| 604 |
< |
\end{equation} |
| 605 |
< |
where $\nu_0$ scales the strength of the tetrahedral attraction and |
| 606 |
< |
$\nu_0^\prime$ scales the dipole repulsion term independently. The |
| 607 |
< |
separation was performed for purposes of the reparameterization, but |
| 608 |
< |
the final parameters were adjusted so that it is not necessary to |
| 609 |
< |
separate the terms when implementing the adjusted water |
| 610 |
< |
potentials. The results of the reparameterizations are shown in table |
| 611 |
< |
\ref{params}. Note that the tetrahedral attractive and dipolar |
| 612 |
< |
repulsive terms do not share the same lower cutoff ($r_l$) in the |
| 613 |
< |
newly parameterized potentials. We are calling these |
| 614 |
< |
reparameterizations the Soft Sticky Dipole / Reaction Field |
| 596 |
> |
The parameters available for tuning include the $\sigma$ and |
| 597 |
> |
$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the |
| 598 |
> |
strength of the sticky potential ($\nu_0$), and the sticky attractive |
| 599 |
> |
and dipole repulsive cubic switching function cutoffs ($r_l$, $r_u$ |
| 600 |
> |
and $r_l^\prime$, $r_u^\prime$ respectively). The results of the |
| 601 |
> |
reparameterizations are shown in table \ref{params}. We are calling |
| 602 |
> |
these reparameterizations the Soft Sticky Dipole / Reaction Field |
| 603 |
|
(SSD/RF - for use with a reaction field) and Soft Sticky Dipole |
| 604 |
< |
Enhanced (SSD/E - an attempt to improve the liquid structure in |
| 604 |
> |
Extended (SSD/E - an attempt to improve the liquid structure in |
| 605 |
|
simulations without a long-range correction). |
| 606 |
|
|
| 607 |
|
\begin{table} |
| 616 |
|
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
| 617 |
|
\ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\ |
| 618 |
|
\ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
| 619 |
+ |
\ \ \ $\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\ |
| 620 |
|
\ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\ |
| 621 |
|
\ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\ |
| 633 |
– |
\ \ \ $\nu_0^\prime$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
| 622 |
|
\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\ |
| 623 |
|
\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\ |
| 624 |
|
\end{tabular} |
| 794 |
|
\begin{center} |
| 795 |
|
\epsfxsize=6in |
| 796 |
|
\epsfbox{ssdeDiffuse.epsi} |
| 797 |
< |
\caption{Plots of the diffusion constants calculated from SSD/E and SSD1, |
| 798 |
< |
both without a reaction field, along with experimental results |
| 799 |
< |
[Refs. \citen{Gillen72} and \citen{Mills73}]. The NVE calculations were |
| 800 |
< |
performed at the average densities observed in the 1 atm NPT |
| 801 |
< |
simulations for the respective models. SSD/E is slightly more fluid |
| 802 |
< |
than experiment at all of the temperatures, but it is closer than SSD1 |
| 803 |
< |
without a long-range correction.} |
| 797 |
> |
\caption{The diffusion constants calculated from SSD/E and SSD1, |
| 798 |
> |
both without a reaction field, along with experimental results |
| 799 |
> |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
| 800 |
> |
were performed at the average densities observed in the 1 atm NPT |
| 801 |
> |
simulations for the respective models. SSD/E is slightly more mobile |
| 802 |
> |
than experiment at all of the temperatures, but it is closer to |
| 803 |
> |
experiment at biologically relavent temperatures than SSD1 without a |
| 804 |
> |
long-range correction.} |
| 805 |
|
\label{ssdediffuse} |
| 806 |
|
\end{center} |
| 807 |
|
\end{figure} |
| 812 |
|
the densities, it is important that the excellent diffusive behavior |
| 813 |
|
of SSD be maintained or improved. Figure \ref{ssdediffuse} compares |
| 814 |
|
the temperature dependence of the diffusion constant of SSD/E to SSD1 |
| 815 |
< |
without an active reaction field, both at the densities calculated at |
| 816 |
< |
1 atm and at the experimentally calculated densities for super-cooled |
| 817 |
< |
and liquid water. The diffusion constant for SSD/E is consistently |
| 818 |
< |
higher than experiment, while SSD1 remains lower than experiment until |
| 819 |
< |
relatively high temperatures (greater than 330 K). Both models follow |
| 820 |
< |
the shape of the experimental curve well below 300 K but tend to |
| 821 |
< |
diffuse too rapidly at higher temperatures, something that is |
| 822 |
< |
especially apparent with SSD1. This increasing diffusion relative to |
| 823 |
< |
the experimental values is caused by the rapidly decreasing system |
| 824 |
< |
density with increasing temperature. The densities of SSD1 decay more |
| 825 |
< |
rapidly with temperature than do those of SSD/E, leading to more |
| 826 |
< |
visible deviation from the experimental diffusion trend. Thus, the |
| 827 |
< |
changes made to improve the liquid structure may have had an adverse |
| 828 |
< |
affect on the density maximum, but they improve the transport behavior |
| 829 |
< |
of SSD/E relative to SSD1. |
| 815 |
> |
without an active reaction field at the densities calculated from the |
| 816 |
> |
NPT simulations at 1 atm. The diffusion constant for SSD/E is |
| 817 |
> |
consistently higher than experiment, while SSD1 remains lower than |
| 818 |
> |
experiment until relatively high temperatures (around 360 K). Both |
| 819 |
> |
models follow the shape of the experimental curve well below 300 K but |
| 820 |
> |
tend to diffuse too rapidly at higher temperatures, as seen in SSD1's |
| 821 |
> |
crossing above 360 K. This increasing diffusion relative to the |
| 822 |
> |
experimental values is caused by the rapidly decreasing system density |
| 823 |
> |
with increasing temperature. Both SSD1 and SSD/E show this deviation |
| 824 |
> |
in diffusive behavior, but this trend has different implications on |
| 825 |
> |
the diffusive behavior of the models. While SSD1 shows more |
| 826 |
> |
experimentally accurate diffusive behavior in the high temperature |
| 827 |
> |
regimes, SSD/E shows more accurate behavior in the supercooled and |
| 828 |
> |
biologically relavent temperature ranges. Thus, the changes made to |
| 829 |
> |
improve the liquid structure may have had an adverse affect on the |
| 830 |
> |
density maximum, but they improve the transport behavior of SSD/E |
| 831 |
> |
relative to SSD1 under the most commonly simulated conditions. |
| 832 |
|
|
| 833 |
|
\begin{figure} |
| 834 |
|
\begin{center} |
| 835 |
|
\epsfxsize=6in |
| 836 |
|
\epsfbox{ssdrfDiffuse.epsi} |
| 837 |
< |
\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1, |
| 837 |
> |
\caption{The diffusion constants calculated from SSD/RF and SSD1, |
| 838 |
|
both with an active reaction field, along with experimental results |
| 839 |
< |
[Refs. \citen{Gillen72} and \citen{Mills73}]. The NVE calculations |
| 839 |
> |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
| 840 |
|
were performed at the average densities observed in the 1 atm NPT |
| 841 |
|
simulations for both of the models. Note how accurately SSD/RF |
| 842 |
|
simulates the diffusion of water throughout this temperature |
| 843 |
|
range. The more rapidly increasing diffusion constants at high |
| 844 |
< |
temperatures for both models is attributed to the significantly lower |
| 845 |
< |
densities than observed in experiment.} |
| 844 |
> |
temperatures for both models is attributed to lower calculated |
| 845 |
> |
densities than those observed in experiment.} |
| 846 |
|
\label{ssdrfdiffuse} |
| 847 |
|
\end{center} |
| 848 |
|
\end{figure} |
| 850 |
|
In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are |
| 851 |
|
compared to SSD1 with an active reaction field. Note that SSD/RF |
| 852 |
|
tracks the experimental results quantitatively, identical within error |
| 853 |
< |
throughout the temperature range shown and with only a slight |
| 854 |
< |
increasing trend at higher temperatures. SSD1 tends to diffuse more |
| 855 |
< |
slowly at low temperatures and deviates to diffuse too rapidly at |
| 853 |
> |
throughout most of the temperature range shown and exhibiting only a |
| 854 |
> |
slight increasing trend at higher temperatures. SSD1 tends to diffuse |
| 855 |
> |
more slowly at low temperatures and deviates to diffuse too rapidly at |
| 856 |
|
temperatures greater than 330 K. As stated above, this deviation away |
| 857 |
|
from the ideal trend is due to a rapid decrease in density at higher |
| 858 |
|
temperatures. SSD/RF does not suffer from this problem as much as SSD1 |
| 860 |
|
values. These results again emphasize the importance of careful |
| 861 |
|
reparameterization when using an altered long-range correction. |
| 862 |
|
|
| 863 |
+ |
\begin{table} |
| 864 |
+ |
\begin{center} |
| 865 |
+ |
\caption{Calculated and experimental properties of the single point waters and liquid water at 298 K and 1 atm. (a) Ref. [\citen{Mills73}]. (b) Calculated by integrating the data in ref. \citen{Head-Gordon00_1}. (c) Calculated by integrating the data in ref. \citen{Soper86}. (d) Ref. [\citen{Eisenberg69}]. (e) Calculated for 298 K from data in ref. \citen{Krynicki66}.} |
| 866 |
+ |
\begin{tabular}{ l c c c c c } |
| 867 |
+ |
\hline \\[-3mm] |
| 868 |
+ |
\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \ |
| 869 |
+ |
\ & \ SSD/RF \ \ \ & \ Expt. \\ |
| 870 |
+ |
\hline \\[-3mm] |
| 871 |
+ |
\ \ \ $\rho$ (g/cm$^3$) & 0.999 $\pm$0.001 & 0.996 $\pm$0.001 & 0.972 $\pm$0.002 & 0.997 $\pm$0.001 & 0.997 \\ |
| 872 |
+ |
\ \ \ $C_p$ (cal/mol K) & 28.80 $\pm$0.11 & 25.45 $\pm$0.09 & 28.28 $\pm$0.06 & 23.83 $\pm$0.16 & 17.98 \\ |
| 873 |
+ |
\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 & 2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299$^\text{a}$ \\ |
| 874 |
+ |
\ \ \ Coordination Number & 3.9 & 4.3 & 3.8 & 4.4 & 4.7$^\text{b}$ \\ |
| 875 |
+ |
\ \ \ H-bonds per particle & 3.7 & 3.6 & 3.7 & 3.7 & 3.4$^\text{c}$ \\ |
| 876 |
+ |
\ \ \ $\tau_1^\mu$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & 7.2 $\pm$0.4 & 4.76$^\text{d}$ \\ |
| 877 |
+ |
\ \ \ $\tau_2^\mu$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2 $\pm$0.2 & 2.3$^\text{e}$ \\ |
| 878 |
+ |
\end{tabular} |
| 879 |
+ |
\label{liquidproperties} |
| 880 |
+ |
\end{center} |
| 881 |
+ |
\end{table} |
| 882 |
+ |
|
| 883 |
+ |
Table \ref{liquidproperties} gives a synopsis of the liquid state |
| 884 |
+ |
properties of the water models compared in this study along with the |
| 885 |
+ |
experimental values for liquid water at ambient conditions. The |
| 886 |
+ |
coordination number and hydrogen bonds per particle were calculated by |
| 887 |
+ |
integrating the following relation: |
| 888 |
+ |
\begin{equation} |
| 889 |
+ |
4\pi\rho\int_{0}^{a}r^2\text{g}(r)dr, |
| 890 |
+ |
\end{equation} |
| 891 |
+ |
where $\rho$ is the number density of pair interactions, $a$ is the |
| 892 |
+ |
radial location of the minima following the first solvation shell |
| 893 |
+ |
peak, and g$(r)$ is either g$_\text{OO}(r)$ or g$_\text{OH}(r)$ for |
| 894 |
+ |
calculation of the coordination number or hydrogen bonds per particle |
| 895 |
+ |
respectively. The number of hydrogen bonds stays relatively constant |
| 896 |
+ |
across all of the models, but the coordination numbers of SSD/E and |
| 897 |
+ |
SSD/RF show an improvement over SSD1. This improvement is primarily |
| 898 |
+ |
due to the widening of the first solvation shell peak, allowing the |
| 899 |
+ |
first minima to push outward. Comparing the coordination number with |
| 900 |
+ |
the number of hydrogen bonds can lead to more insight into the |
| 901 |
+ |
structural character of the liquid. Because of the near identical |
| 902 |
+ |
values for SSD1, it appears to be a little too exclusive, in that all |
| 903 |
+ |
molecules in the first solvation shell are involved in forming ideal |
| 904 |
+ |
hydrogen bonds. The differing numbers for the newly parameterized |
| 905 |
+ |
models indicate the allowance of more fluid configurations in addition |
| 906 |
+ |
to the formation of an acceptable number of ideal hydrogen bonds. |
| 907 |
+ |
|
| 908 |
+ |
The time constants for the self orientational autocorrelation function |
| 909 |
+ |
are also displayed in Table \ref{liquidproperties}. The dipolar |
| 910 |
+ |
orientational time correlation function ($\Gamma_{l}$) is described |
| 911 |
+ |
by: |
| 912 |
+ |
\begin{equation} |
| 913 |
+ |
\Gamma_{l}(t) = \langle P_l[\mathbf{u}_j(0)\cdot\mathbf{u}_j(t)]\rangle, |
| 914 |
+ |
\end{equation} |
| 915 |
+ |
where $P_l$ is a Legendre polynomial of order $l$ and $\mathbf{u}_j$ |
| 916 |
+ |
is the unit vector of the particle dipole.\cite{Rahman71} From these |
| 917 |
+ |
correlation functions, the orientational relaxation time of the dipole |
| 918 |
+ |
vector can be calculated from an exponential fit in the long-time |
| 919 |
+ |
regime ($t > \tau_l^\mu$).\cite{Rothschild84} Calculation of these |
| 920 |
+ |
time constants were averaged from five detailed NVE simulations |
| 921 |
+ |
performed at the STP density for each of the respective models. It |
| 922 |
+ |
should be noted that the commonly cited value for $\tau_2$ of 1.9 ps |
| 923 |
+ |
was determined from the NMR data in reference \citen{Krynicki66} at a |
| 924 |
+ |
temperature near 34$^\circ$C.\cite{Rahman73} Because of the strong |
| 925 |
+ |
temperature dependence of $\tau_2$, it is necessary to recalculate it |
| 926 |
+ |
at 298 K to make proper comparisons. The value shown in Table |
| 927 |
+ |
\ref{liquidproperties} was calculated from the same NMR data in the |
| 928 |
+ |
fashion described in reference \citen{Krynicki66}. Again, SSD/E and |
| 929 |
+ |
SSD/RF show improved behavior over SSD1, both with and without an |
| 930 |
+ |
active reaction field. Turning on the reaction field leads to much |
| 931 |
+ |
improved time constants for SSD1; however, these results also include |
| 932 |
+ |
a corresponding decrease in system density. Numbers published from the |
| 933 |
+ |
original SSD dynamics studies appear closer to the experimental |
| 934 |
+ |
values, and this difference can be attributed to the use of the Ewald |
| 935 |
+ |
sum technique versus a reaction field.\cite{Ichiye99} |
| 936 |
+ |
|
| 937 |
|
\subsection{Additional Observations} |
| 938 |
|
|
| 939 |
|
\begin{figure} |
