1 |
chrisfen |
861 |
%\documentclass[prb,aps,times,twocolumn,tabularx]{revtex4} |
2 |
chrisfen |
862 |
\documentclass[11pt]{article} |
3 |
|
|
\usepackage{endfloat} |
4 |
chrisfen |
743 |
\usepackage{amsmath} |
5 |
chrisfen |
862 |
\usepackage{epsf} |
6 |
|
|
\usepackage{berkeley} |
7 |
|
|
\usepackage{setspace} |
8 |
|
|
\usepackage{tabularx} |
9 |
chrisfen |
743 |
\usepackage{graphicx} |
10 |
chrisfen |
862 |
\usepackage[ref]{overcite} |
11 |
chrisfen |
743 |
%\usepackage{berkeley} |
12 |
|
|
%\usepackage{curves} |
13 |
chrisfen |
862 |
\pagestyle{plain} |
14 |
|
|
\pagenumbering{arabic} |
15 |
|
|
\oddsidemargin 0.0cm \evensidemargin 0.0cm |
16 |
|
|
\topmargin -21pt \headsep 10pt |
17 |
|
|
\textheight 9.0in \textwidth 6.5in |
18 |
|
|
\brokenpenalty=10000 |
19 |
|
|
\renewcommand{\baselinestretch}{1.2} |
20 |
|
|
\renewcommand\citemid{\ } % no comma in optional reference note |
21 |
chrisfen |
743 |
|
22 |
|
|
\begin{document} |
23 |
|
|
|
24 |
gezelter |
921 |
\title{On the structural and transport properties of the soft sticky |
25 |
|
|
dipole (SSD) and related single point water models} |
26 |
chrisfen |
743 |
|
27 |
chrisfen |
862 |
\author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
28 |
|
|
Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
29 |
chrisfen |
743 |
Notre Dame, Indiana 46556} |
30 |
|
|
|
31 |
|
|
\date{\today} |
32 |
|
|
|
33 |
chrisfen |
862 |
\maketitle |
34 |
|
|
|
35 |
chrisfen |
743 |
\begin{abstract} |
36 |
gezelter |
921 |
The density maximum and temperature dependence of the self-diffusion |
37 |
|
|
constant were investigated for the soft sticky dipole (SSD) water |
38 |
|
|
model and two related re-parameterizations of this single-point model. |
39 |
|
|
A combination of microcanonical and isobaric-isothermal molecular |
40 |
|
|
dynamics simulations were used to calculate these properties, both |
41 |
|
|
with and without the use of reaction field to handle long-range |
42 |
|
|
electrostatics. The isobaric-isothermal (NPT) simulations of the |
43 |
|
|
melting of both ice-$I_h$ and ice-$I_c$ showed a density maximum near |
44 |
|
|
260 K. In most cases, the use of the reaction field resulted in |
45 |
|
|
calculated densities which were were significantly lower than |
46 |
|
|
experimental densities. Analysis of self-diffusion constants shows |
47 |
|
|
that the original SSD model captures the transport properties of |
48 |
chrisfen |
861 |
experimental water very well in both the normal and super-cooled |
49 |
gezelter |
921 |
liquid regimes. We also present our re-parameterized versions of SSD |
50 |
|
|
for use both with the reaction field or without any long-range |
51 |
|
|
electrostatic corrections. These are called the SSD/RF and SSD/E |
52 |
|
|
models respectively. These modified models were shown to maintain or |
53 |
|
|
improve upon the experimental agreement with the structural and |
54 |
|
|
transport properties that can be obtained with either the original SSD |
55 |
|
|
or the density corrected version of the original model (SSD1). |
56 |
|
|
Additionally, a novel low-density ice structure is presented |
57 |
|
|
which appears to be the most stable ice structure for the entire SSD |
58 |
|
|
family. |
59 |
chrisfen |
743 |
\end{abstract} |
60 |
|
|
|
61 |
chrisfen |
862 |
\newpage |
62 |
chrisfen |
743 |
|
63 |
|
|
%\narrowtext |
64 |
|
|
|
65 |
|
|
|
66 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
67 |
|
|
% BODY OF TEXT |
68 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
69 |
|
|
|
70 |
|
|
\section{Introduction} |
71 |
|
|
|
72 |
chrisfen |
862 |
One of the most important tasks in the simulation of biochemical |
73 |
gezelter |
921 |
systems is the proper depiction of the aqueous environment of the |
74 |
|
|
molecules of interest. In some cases (such as in the simulation of |
75 |
|
|
phospholipid bilayers), the majority of the calculations that are |
76 |
|
|
performed involve interactions with or between solvent molecules. |
77 |
|
|
Thus, the properties one may observe in biochemical simulations are |
78 |
|
|
going to be highly dependent on the physical properties of the water |
79 |
|
|
model that is chosen. |
80 |
chrisfen |
743 |
|
81 |
gezelter |
921 |
There is an especially delicate balance between computational |
82 |
|
|
efficiency and the ability of the water model to accurately predict |
83 |
|
|
the properties of bulk |
84 |
|
|
water.\cite{Jorgensen83,Berendsen87,Jorgensen00} For example, the |
85 |
|
|
TIP5P model improves on the structural and transport properties of |
86 |
|
|
water relative to the previous TIP models, yet this comes at a greater |
87 |
|
|
than 50\% increase in computational |
88 |
|
|
cost.\cite{Jorgensen01,Jorgensen00} |
89 |
|
|
|
90 |
|
|
One recently developed model that largely succeeds in retaining the |
91 |
|
|
accuracy of bulk properties while greatly reducing the computational |
92 |
|
|
cost is the Soft Sticky Dipole (SSD) water |
93 |
|
|
model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The SSD model was |
94 |
|
|
developed by Ichiye \emph{et al.} as a modified form of the |
95 |
|
|
hard-sphere water model proposed by Bratko, Blum, and |
96 |
|
|
Luzar.\cite{Bratko85,Bratko95} SSD is a {\it single point} model which |
97 |
|
|
has an interaction site that is both a point dipole along with a |
98 |
|
|
Lennard-Jones core. However, since the normal aligned and |
99 |
|
|
anti-aligned geometries favored by point dipoles are poor mimics of |
100 |
|
|
local structure in liquid water, a short ranged ``sticky'' potential |
101 |
|
|
is also added. The sticky potential directs the molecules to assume |
102 |
|
|
the proper hydrogen bond orientation in the first solvation |
103 |
|
|
shell. |
104 |
|
|
|
105 |
|
|
The interaction between two SSD water molecules \emph{i} and \emph{j} |
106 |
|
|
is given by the potential |
107 |
chrisfen |
743 |
\begin{equation} |
108 |
|
|
u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
109 |
gezelter |
921 |
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ + |
110 |
chrisfen |
743 |
u_{ij}^{sp} |
111 |
gezelter |
921 |
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j), |
112 |
chrisfen |
743 |
\end{equation} |
113 |
gezelter |
921 |
where the ${\bf r}_{ij}$ is the position vector between molecules |
114 |
|
|
\emph{i} and \emph{j} with magnitude $r_{ij}$, and |
115 |
|
|
${\bf \Omega}_i$ and ${\bf \Omega}_j$ represent the orientations of |
116 |
|
|
the two molecules. The Lennard-Jones and dipole interactions are given |
117 |
|
|
by the following familiar forms: |
118 |
chrisfen |
743 |
\begin{equation} |
119 |
gezelter |
921 |
u_{ij}^{LJ}(r_{ij}) = 4\epsilon |
120 |
|
|
\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right] |
121 |
|
|
\ , |
122 |
chrisfen |
743 |
\end{equation} |
123 |
gezelter |
921 |
and |
124 |
chrisfen |
743 |
\begin{equation} |
125 |
gezelter |
921 |
u_{ij}^{dp} = \frac{|\mu_i||\mu_j|}{4 \pi \epsilon_0 r_{ij}^3} \left( |
126 |
|
|
\hat{\bf u}_i \cdot \hat{\bf u}_j - 3(\hat{\bf u}_i\cdot\hat{\bf |
127 |
|
|
r}_{ij})(\hat{\bf u}_j\cdot\hat{\bf r}_{ij}) \right)\ , |
128 |
chrisfen |
743 |
\end{equation} |
129 |
gezelter |
921 |
where $\hat{\bf u}_i$ and $\hat{\bf u}_j$ are the unit vectors along |
130 |
|
|
the dipoles of molecules $i$ and $j$ respectively. $|\mu_i|$ and |
131 |
|
|
$|\mu_j|$ are the strengths of the dipole moments, and $\hat{\bf |
132 |
|
|
r}_{ij}$ is the unit vector pointing from molecule $j$ to molecule |
133 |
|
|
$i$. |
134 |
|
|
|
135 |
|
|
The sticky potential is somewhat less familiar: |
136 |
chrisfen |
743 |
\begin{equation} |
137 |
|
|
u_{ij}^{sp} |
138 |
gezelter |
921 |
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = |
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 |
chrisfen |
743 |
\end{equation} |
143 |
gezelter |
921 |
Here, $\nu_0$ is a strength parameter for the sticky potential, and |
144 |
|
|
$s$ and $s^\prime$ are cubic switching functions which turn off the |
145 |
|
|
sticky interaction beyond the first solvation shell. The $w$ function |
146 |
|
|
can be thought of as an attractive potential with tetrahedral |
147 |
|
|
geometry: |
148 |
chrisfen |
743 |
\begin{equation} |
149 |
gezelter |
921 |
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
150 |
chrisfen |
743 |
\end{equation} |
151 |
gezelter |
921 |
while the $w^\prime$ function counters the normal aligned and |
152 |
|
|
anti-aligned structures favored by point dipoles: |
153 |
chrisfen |
743 |
\begin{equation} |
154 |
gezelter |
921 |
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 |
chrisfen |
743 |
\end{equation} |
156 |
gezelter |
921 |
It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
157 |
|
|
and $Y_3^{-2}$ spherical harmonics (a linear combination which |
158 |
|
|
enhances the tetrahedral geometry for hydrogen bonded structures), |
159 |
|
|
while $w^\prime$ is a purely empirical function. A more detailed |
160 |
|
|
description of the functional parts and variables in this potential |
161 |
|
|
can be found in the original SSD |
162 |
|
|
articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} |
163 |
chrisfen |
743 |
|
164 |
gezelter |
921 |
Since SSD is a single-point {\it dipolar} model, the force |
165 |
|
|
calculations are simplified significantly relative to the standard |
166 |
|
|
{\it charged} multi-point models. In the original Monte Carlo |
167 |
|
|
simulations using this model, Ichiye {\it et al.} reported that using |
168 |
|
|
SSD decreased computer time by a factor of 6-7 compared to other |
169 |
|
|
models.\cite{Ichiye96} What is most impressive is that this savings |
170 |
|
|
did not come at the expense of accurate depiction of the liquid state |
171 |
|
|
properties. Indeed, SSD maintains reasonable agreement with the Soper |
172 |
|
|
data for the structural features of liquid |
173 |
|
|
water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties |
174 |
|
|
exhibited by SSD agree with experiment better than those of more |
175 |
|
|
computationally expensive models (like TIP3P and |
176 |
|
|
SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction |
177 |
|
|
of solvent properties makes SSD a very attractive model for the |
178 |
|
|
simulation of large scale biochemical simulations. |
179 |
chrisfen |
743 |
|
180 |
gezelter |
921 |
One feature of the SSD model is that it was parameterized for use with |
181 |
|
|
the Ewald sum to handle long-range interactions. This would normally |
182 |
|
|
be the best way of handling long-range interactions in systems that |
183 |
|
|
contain other point charges. However, our group has recently become |
184 |
|
|
interested in systems with point dipoles as mimics for neutral, but |
185 |
|
|
polarized regions on molecules (e.g. the zwitterionic head group |
186 |
|
|
regions of phospholipids). If the system of interest does not contain |
187 |
|
|
point charges, the Ewald sum and even particle-mesh Ewald become |
188 |
|
|
computational bottlenecks. Their respective ideal $N^\frac{3}{2}$ and |
189 |
|
|
$N\log N$ calculation scaling orders for $N$ particles can become |
190 |
|
|
prohibitive when $N$ becomes large.\cite{Darden99} In applying this |
191 |
|
|
water model in these types of systems, it would be useful to know its |
192 |
|
|
properties and behavior under the more computationally efficient |
193 |
|
|
reaction field (RF) technique, or even with a simple cutoff. This |
194 |
|
|
study addresses these issues by looking at the structural and |
195 |
|
|
transport behavior of SSD over a variety of temperatures with the |
196 |
|
|
purpose of utilizing the RF correction technique. We then suggest |
197 |
|
|
modifications to the parameters that result in more realistic bulk |
198 |
|
|
phase behavior. It should be noted that in a recent publication, some |
199 |
|
|
of the original investigators of the SSD water model have suggested |
200 |
|
|
adjustments to the SSD water model to address abnormal density |
201 |
|
|
behavior (also observed here), calling the corrected model |
202 |
|
|
SSD1.\cite{Ichiye03} In what follows, we compare our |
203 |
|
|
reparamaterization of SSD with both the original SSD and SSD1 models |
204 |
|
|
with the goal of improving the bulk phase behavior of an SSD-derived |
205 |
|
|
model in simulations utilizing the Reaction Field. |
206 |
chrisfen |
757 |
|
207 |
chrisfen |
743 |
\section{Methods} |
208 |
|
|
|
209 |
gezelter |
921 |
Long-range dipole-dipole interactions were accounted for in this study |
210 |
|
|
by using either the reaction field method or by resorting to a simple |
211 |
|
|
cubic switching function at a cutoff radius. Under the first method, |
212 |
|
|
the magnitude of the reaction field acting on dipole $i$ is |
213 |
chrisfen |
743 |
\begin{equation} |
214 |
|
|
\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1} |
215 |
gezelter |
921 |
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} f(r_{ij})\ , |
216 |
chrisfen |
743 |
\label{rfequation} |
217 |
|
|
\end{equation} |
218 |
|
|
where $\mathcal{R}$ is the cavity defined by the cutoff radius |
219 |
|
|
($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the |
220 |
gezelter |
921 |
system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole |
221 |
|
|
moment vector of particle $j$ and $f(r_{ij})$ is a cubic switching |
222 |
chrisfen |
743 |
function.\cite{AllenTildesley} The reaction field contribution to the |
223 |
gezelter |
921 |
total energy by particle $i$ is given by $-\frac{1}{2}{\bf |
224 |
|
|
\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf |
225 |
|
|
\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use of the reaction |
226 |
|
|
field is known to alter the bulk orientational properties, such as the |
227 |
|
|
dielectric relaxation time. There is particular sensitivity of this |
228 |
|
|
property on changes in the length of the cutoff |
229 |
|
|
radius.\cite{Berendsen98} This variable behavior makes reaction field |
230 |
|
|
a less attractive method than the Ewald sum. However, for very large |
231 |
|
|
systems, the computational benefit of reaction field is dramatic. |
232 |
|
|
|
233 |
|
|
We have also performed a companion set of simulations {\it without} a |
234 |
|
|
surrounding dielectric (i.e. using a simple cubic switching function |
235 |
|
|
at the cutoff radius) and as a result we have two reparamaterizations |
236 |
|
|
of SSD which could be used either with or without the Reaction Field |
237 |
|
|
turned on. |
238 |
chrisfen |
777 |
|
239 |
gezelter |
921 |
Simulations to obtain the preferred density were performed in the |
240 |
|
|
isobaric-isothermal (NPT) ensemble, while all dynamical properties |
241 |
|
|
were obtained from microcanonical (NVE) simulations done at densities |
242 |
|
|
matching the NPT density for a particular target temperature. The |
243 |
|
|
constant pressure simulations were implemented using an integral |
244 |
|
|
thermostat and barostat as outlined by Hoover.\cite{Hoover85,Hoover86} |
245 |
|
|
All molecules were treated as non-linear rigid bodies. Vibrational |
246 |
|
|
constraints are not necessary in simulations of SSD, because there are |
247 |
|
|
no explicit hydrogen atoms, and thus no molecular vibrational modes |
248 |
|
|
need to be considered. |
249 |
chrisfen |
743 |
|
250 |
|
|
Integration of the equations of motion was carried out using the |
251 |
gezelter |
921 |
symplectic splitting method proposed by Dullweber {\it et |
252 |
|
|
al.}\cite{Dullweber1997} Our reason for selecting this integrator |
253 |
|
|
centers on poor energy conservation of rigid body dynamics using |
254 |
|
|
traditional quaternion integration.\cite{Evans77,Evans77b} While quaternions |
255 |
|
|
may work well for orientational motion under NVT or NPT integrators, |
256 |
|
|
our limits on energy drift in the microcanonical ensemble were quite |
257 |
|
|
strict, and the drift under quaternions was substantially greater than |
258 |
|
|
in the symplectic splitting method. This steady drift in the total |
259 |
|
|
energy has also been observed by Kol {\it et al.}\cite{Laird97} |
260 |
chrisfen |
743 |
|
261 |
|
|
The key difference in the integration method proposed by Dullweber |
262 |
|
|
\emph{et al.} is that the entire rotation matrix is propagated from |
263 |
gezelter |
921 |
one time step to the next. The additional memory required by the |
264 |
|
|
algorithm is inconsequential on modern computers, and translating the |
265 |
|
|
rotation matrix into quaternions for storage purposes makes trajectory |
266 |
|
|
data quite compact. |
267 |
chrisfen |
743 |
|
268 |
|
|
The symplectic splitting method allows for Verlet style integration of |
269 |
gezelter |
921 |
both translational and orientational motion of rigid bodies. In this |
270 |
|
|
integration method, the orientational propagation involves a sequence |
271 |
|
|
of matrix evaluations to update the rotation |
272 |
|
|
matrix.\cite{Dullweber1997} These matrix rotations are more costly |
273 |
|
|
than the simpler arithmetic quaternion propagation. With the same time |
274 |
|
|
step, a 1000 SSD particle simulation shows an average 7\% increase in |
275 |
|
|
computation time using the symplectic step method in place of |
276 |
|
|
quaternions. The additional expense per step is justified when one |
277 |
|
|
considers the ability to use time steps that are nearly twice as large |
278 |
|
|
under symplectic splitting than would be usable under quaternion |
279 |
|
|
dynamics. The energy conservation of the two methods using a number |
280 |
|
|
of different time steps is illustrated in figure |
281 |
|
|
\ref{timestep}. |
282 |
chrisfen |
743 |
|
283 |
|
|
\begin{figure} |
284 |
chrisfen |
862 |
\begin{center} |
285 |
|
|
\epsfxsize=6in |
286 |
|
|
\epsfbox{timeStep.epsi} |
287 |
gezelter |
921 |
\caption{Energy conservation using both quaternion based integration and |
288 |
chrisfen |
743 |
the symplectic step method proposed by Dullweber \emph{et al.} with |
289 |
gezelter |
921 |
increasing time step. The larger time step plots are shifted from the |
290 |
|
|
true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.} |
291 |
chrisfen |
743 |
\label{timestep} |
292 |
chrisfen |
862 |
\end{center} |
293 |
chrisfen |
743 |
\end{figure} |
294 |
|
|
|
295 |
|
|
In figure \ref{timestep}, the resulting energy drift at various time |
296 |
|
|
steps for both the symplectic step and quaternion integration schemes |
297 |
gezelter |
921 |
is compared. All of the 1000 SSD particle simulations started with |
298 |
|
|
the same configuration, and the only difference was the method used to |
299 |
|
|
handle orientational motion. At time steps of 0.1 and 0.5 fs, both |
300 |
|
|
methods for propagating the orientational degrees of freedom conserve |
301 |
|
|
energy fairly well, with the quaternion method showing a slight energy |
302 |
|
|
drift over time in the 0.5 fs time step simulation. At time steps of 1 |
303 |
|
|
and 2 fs, the energy conservation benefits of the symplectic step |
304 |
|
|
method are clearly demonstrated. Thus, while maintaining the same |
305 |
|
|
degree of energy conservation, one can take considerably longer time |
306 |
|
|
steps, leading to an overall reduction in computation time. |
307 |
chrisfen |
743 |
|
308 |
chrisfen |
862 |
Energy drift in the symplectic step simulations was unnoticeable for |
309 |
gezelter |
921 |
time steps up to 3 fs. A slight energy drift on the |
310 |
chrisfen |
743 |
order of 0.012 kcal/mol per nanosecond was observed at a time step of |
311 |
gezelter |
921 |
4 fs, and as expected, this drift increases dramatically |
312 |
|
|
with increasing time step. To insure accuracy in our microcanonical |
313 |
chrisfen |
743 |
simulations, time steps were set at 2 fs and kept at this value for |
314 |
|
|
constant pressure simulations as well. |
315 |
|
|
|
316 |
gezelter |
921 |
Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices |
317 |
|
|
were generated as starting points for all simulations. The $I_h$ |
318 |
|
|
crystals were formed by first arranging the centers of mass of the SSD |
319 |
|
|
particles into a ``hexagonal'' ice lattice of 1024 particles. Because |
320 |
|
|
of the crystal structure of $I_h$ ice, the simulation box assumed an |
321 |
|
|
orthorhombic shape with an edge length ratio of approximately |
322 |
chrisfen |
743 |
1.00$\times$1.06$\times$1.23. The particles were then allowed to |
323 |
|
|
orient freely about fixed positions with angular momenta randomized at |
324 |
|
|
400 K for varying times. The rotational temperature was then scaled |
325 |
chrisfen |
862 |
down in stages to slowly cool the crystals to 25 K. The particles were |
326 |
|
|
then allowed to translate with fixed orientations at a constant |
327 |
chrisfen |
743 |
pressure of 1 atm for 50 ps at 25 K. Finally, all constraints were |
328 |
|
|
removed and the ice crystals were allowed to equilibrate for 50 ps at |
329 |
|
|
25 K and a constant pressure of 1 atm. This procedure resulted in |
330 |
|
|
structurally stable $I_h$ ice crystals that obey the Bernal-Fowler |
331 |
chrisfen |
862 |
rules.\cite{Bernal33,Rahman72} This method was also utilized in the |
332 |
chrisfen |
743 |
making of diamond lattice $I_c$ ice crystals, with each cubic |
333 |
|
|
simulation box consisting of either 512 or 1000 particles. Only |
334 |
|
|
isotropic volume fluctuations were performed under constant pressure, |
335 |
|
|
so the ratio of edge lengths remained constant throughout the |
336 |
|
|
simulations. |
337 |
|
|
|
338 |
|
|
\section{Results and discussion} |
339 |
|
|
|
340 |
|
|
Melting studies were performed on the randomized ice crystals using |
341 |
gezelter |
921 |
isobaric-isothermal (NPT) dynamics. During melting simulations, the |
342 |
|
|
melting transition and the density maximum can both be observed, |
343 |
|
|
provided that the density maximum occurs in the liquid and not the |
344 |
|
|
supercooled regime. An ensemble average from five separate melting |
345 |
|
|
simulations was acquired, each starting from different ice crystals |
346 |
|
|
generated as described previously. All simulations were equilibrated |
347 |
|
|
for 100 ps prior to a 200 ps data collection run at each temperature |
348 |
|
|
setting. The temperature range of study spanned from 25 to 400 K, with |
349 |
|
|
a maximum degree increment of 25 K. For regions of interest along this |
350 |
|
|
stepwise progression, the temperature increment was decreased from 25 |
351 |
|
|
K to 10 and 5 K. The above equilibration and production times were |
352 |
|
|
sufficient in that fluctuations in the volume autocorrelation function |
353 |
|
|
were damped out in all simulations in under 20 ps. |
354 |
chrisfen |
743 |
|
355 |
|
|
\subsection{Density Behavior} |
356 |
|
|
|
357 |
gezelter |
921 |
Our initial simulations focused on the original SSD water model, and |
358 |
|
|
an average density versus temperature plot is shown in figure |
359 |
|
|
\ref{dense1}. Note that the density maximum when using a reaction |
360 |
|
|
field appears between 255 and 265 K. There were smaller fluctuations |
361 |
|
|
in the density at 260 K than at either 255 or 265, so we report this |
362 |
|
|
value as the location of the density maximum. Figure \ref{dense1} was |
363 |
|
|
constructed using ice $I_h$ crystals for the initial configuration; |
364 |
|
|
though not pictured, the simulations starting from ice $I_c$ crystal |
365 |
|
|
configurations showed similar results, with a liquid-phase density |
366 |
|
|
maximum in this same region (between 255 and 260 K). |
367 |
|
|
|
368 |
chrisfen |
743 |
\begin{figure} |
369 |
chrisfen |
862 |
\begin{center} |
370 |
|
|
\epsfxsize=6in |
371 |
|
|
\epsfbox{denseSSD.eps} |
372 |
gezelter |
921 |
\caption{Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}], |
373 |
|
|
TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], SSD |
374 |
|
|
without Reaction Field, SSD, and experiment [Ref. \citen{CRC80}]. The |
375 |
|
|
arrows indicate the change in densities observed when turning off the |
376 |
|
|
reaction field. The the lower than expected densities for the SSD |
377 |
|
|
model were what prompted the original reparameterization of SSD1 |
378 |
|
|
[Ref. \citen{Ichiye03}].} |
379 |
chrisfen |
861 |
\label{dense1} |
380 |
chrisfen |
862 |
\end{center} |
381 |
chrisfen |
743 |
\end{figure} |
382 |
|
|
|
383 |
gezelter |
921 |
The density maximum for SSD compares quite favorably to other simple |
384 |
|
|
water models. Figure \ref{dense1} also shows calculated densities of |
385 |
|
|
several other models and experiment obtained from other |
386 |
chrisfen |
743 |
sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water |
387 |
gezelter |
921 |
models, SSD has a temperature closest to the experimentally observed |
388 |
|
|
density maximum. Of the {\it charge-based} models in |
389 |
|
|
Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that |
390 |
|
|
seen in SSD. Though not included in this plot, it is useful |
391 |
|
|
to note that TIP5P has a density maximum nearly identical to the |
392 |
|
|
experimentally measured temperature. |
393 |
chrisfen |
743 |
|
394 |
gezelter |
921 |
It has been observed that liquid state densities in water are |
395 |
|
|
dependent on the cutoff radius used both with and without the use of |
396 |
|
|
reaction field.\cite{Berendsen98} In order to address the possible |
397 |
|
|
effect of cutoff radius, simulations were performed with a dipolar |
398 |
|
|
cutoff radius of 12.0 \AA\ to complement the previous SSD simulations, |
399 |
|
|
all performed with a cutoff of 9.0 \AA. All of the resulting densities |
400 |
|
|
overlapped within error and showed no significant trend toward lower |
401 |
|
|
or higher densities as a function of cutoff radius, for simulations |
402 |
|
|
both with and without reaction field. These results indicate that |
403 |
|
|
there is no major benefit in choosing a longer cutoff radius in |
404 |
|
|
simulations using SSD. This is advantageous in that the use of a |
405 |
|
|
longer cutoff radius results in a significant increase in the time |
406 |
|
|
required to obtain a single trajectory. |
407 |
chrisfen |
743 |
|
408 |
chrisfen |
862 |
The key feature to recognize in figure \ref{dense1} is the density |
409 |
|
|
scaling of SSD relative to other common models at any given |
410 |
gezelter |
921 |
temperature. SSD assumes a lower density than any of the other listed |
411 |
|
|
models at the same pressure, behavior which is especially apparent at |
412 |
|
|
temperatures greater than 300 K. Lower than expected densities have |
413 |
|
|
been observed for other systems using a reaction field for long-range |
414 |
|
|
electrostatic interactions, so the most likely reason for the |
415 |
|
|
significantly lower densities seen in these simulations is the |
416 |
|
|
presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order |
417 |
|
|
to test the effect of the reaction field on the density of the |
418 |
|
|
systems, the simulations were repeated without a reaction field |
419 |
|
|
present. The results of these simulations are also displayed in figure |
420 |
|
|
\ref{dense1}. Without the reaction field, the densities increase |
421 |
|
|
to more experimentally reasonable values, especially around the |
422 |
|
|
freezing point of liquid water. The shape of the curve is similar to |
423 |
|
|
the curve produced from SSD simulations using reaction field, |
424 |
|
|
specifically the rapidly decreasing densities at higher temperatures; |
425 |
|
|
however, a shift in the density maximum location, down to 245 K, is |
426 |
|
|
observed. This is a more accurate comparison to the other listed water |
427 |
|
|
models, in that no long range corrections were applied in those |
428 |
|
|
simulations.\cite{Clancy94,Jorgensen98b} However, even without the |
429 |
chrisfen |
861 |
reaction field, the density around 300 K is still significantly lower |
430 |
|
|
than experiment and comparable water models. This anomalous behavior |
431 |
gezelter |
921 |
was what lead Ichiye {\it et al.} to recently reparameterize |
432 |
|
|
SSD.\cite{Ichiye03} Throughout the remainder of the paper our |
433 |
|
|
reparamaterizations of SSD will be compared with the newer SSD1 model. |
434 |
chrisfen |
861 |
|
435 |
chrisfen |
743 |
\subsection{Transport Behavior} |
436 |
|
|
|
437 |
gezelter |
921 |
Accurate dynamical properties of a water model are particularly |
438 |
|
|
important when using the model to study permeation or transport across |
439 |
|
|
biological membranes. In order to probe transport in bulk water, |
440 |
|
|
constant energy (NVE) simulations were performed at the average |
441 |
|
|
density obtained by the NPT simulations at an identical target |
442 |
|
|
temperature. Simulations started with randomized velocities and |
443 |
|
|
underwent 50 ps of temperature scaling and 50 ps of constant energy |
444 |
|
|
equilibration before a 200 ps data collection run. Diffusion constants |
445 |
|
|
were calculated via linear fits to the long-time behavior of the |
446 |
|
|
mean-square displacement as a function of time. The averaged results |
447 |
|
|
from five sets of NVE simulations are displayed in figure |
448 |
|
|
\ref{diffuse}, alongside experimental, SPC/E, and TIP5P |
449 |
|
|
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} |
450 |
|
|
|
451 |
chrisfen |
743 |
\begin{figure} |
452 |
chrisfen |
862 |
\begin{center} |
453 |
|
|
\epsfxsize=6in |
454 |
|
|
\epsfbox{betterDiffuse.epsi} |
455 |
gezelter |
921 |
\caption{Average self-diffusion constant as a function of temperature for |
456 |
|
|
SSD, SPC/E [Ref. \citen{Clancy94}], TIP5P [Ref. \citen{Jorgensen01}], |
457 |
|
|
and Experimental data [Refs. \citen{Gillen72} and \citen{Mills73}]. Of |
458 |
|
|
the three water models shown, SSD has the least deviation from the |
459 |
|
|
experimental values. The rapidly increasing diffusion constants for |
460 |
|
|
TIP5P and SSD correspond to significant decrease in density at the |
461 |
|
|
higher temperatures.} |
462 |
chrisfen |
743 |
\label{diffuse} |
463 |
chrisfen |
862 |
\end{center} |
464 |
chrisfen |
743 |
\end{figure} |
465 |
|
|
|
466 |
|
|
The observed values for the diffusion constant point out one of the |
467 |
gezelter |
921 |
strengths of the SSD model. Of the three models shown, the SSD model |
468 |
|
|
has the most accurate depiction of self-diffusion in both the |
469 |
|
|
supercooled and liquid regimes. SPC/E does a respectable job by |
470 |
|
|
reproducing values similar to experiment around 290 K; however, it |
471 |
|
|
deviates at both higher and lower temperatures, failing to predict the |
472 |
|
|
correct thermal trend. TIP5P and SSD both start off low at colder |
473 |
|
|
temperatures and tend to diffuse too rapidly at higher temperatures. |
474 |
|
|
This behavior at higher temperatures is not particularly surprising |
475 |
|
|
since the densities of both TIP5P and SSD are lower than experimental |
476 |
|
|
water densities at higher temperatures. When calculating the |
477 |
|
|
diffusion coefficients for SSD at experimental densities (instead of |
478 |
|
|
the densities from the NPT simulations), the resulting values fall |
479 |
|
|
more in line with experiment at these temperatures. |
480 |
chrisfen |
743 |
|
481 |
|
|
\subsection{Structural Changes and Characterization} |
482 |
gezelter |
921 |
|
483 |
chrisfen |
743 |
By starting the simulations from the crystalline state, the melting |
484 |
gezelter |
921 |
transition and the ice structure can be obtained along with the liquid |
485 |
chrisfen |
862 |
phase behavior beyond the melting point. The constant pressure heat |
486 |
|
|
capacity (C$_\text{p}$) was monitored to locate the melting transition |
487 |
|
|
in each of the simulations. In the melting simulations of the 1024 |
488 |
|
|
particle ice $I_h$ simulations, a large spike in C$_\text{p}$ occurs |
489 |
|
|
at 245 K, indicating a first order phase transition for the melting of |
490 |
|
|
these ice crystals. When the reaction field is turned off, the melting |
491 |
|
|
transition occurs at 235 K. These melting transitions are |
492 |
gezelter |
921 |
considerably lower than the experimental value. |
493 |
chrisfen |
743 |
|
494 |
chrisfen |
862 |
\begin{figure} |
495 |
|
|
\begin{center} |
496 |
|
|
\epsfxsize=6in |
497 |
|
|
\epsfbox{corrDiag.eps} |
498 |
|
|
\caption{Two dimensional illustration of angles involved in the |
499 |
gezelter |
921 |
correlations observed in Fig. \ref{contour}.} |
500 |
chrisfen |
862 |
\label{corrAngle} |
501 |
|
|
\end{center} |
502 |
|
|
\end{figure} |
503 |
|
|
|
504 |
|
|
\begin{figure} |
505 |
|
|
\begin{center} |
506 |
|
|
\epsfxsize=6in |
507 |
|
|
\epsfbox{fullContours.eps} |
508 |
chrisfen |
743 |
\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at |
509 |
|
|
100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for |
510 |
|
|
clarity: dark areas signify peaks while light areas signify |
511 |
gezelter |
921 |
depressions. White areas have $g(r)$ values below 0.5 and black |
512 |
chrisfen |
743 |
areas have values above 1.5.} |
513 |
|
|
\label{contour} |
514 |
chrisfen |
862 |
\end{center} |
515 |
chrisfen |
743 |
\end{figure} |
516 |
|
|
|
517 |
gezelter |
921 |
Additional analysis of the melting process was performed using |
518 |
|
|
two-dimensional structure and dipole angle correlations. Expressions |
519 |
|
|
for these correlations are as follows: |
520 |
chrisfen |
861 |
|
521 |
chrisfen |
862 |
\begin{equation} |
522 |
gezelter |
921 |
g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ , |
523 |
chrisfen |
862 |
\end{equation} |
524 |
|
|
\begin{equation} |
525 |
|
|
g_{\text{AB}}(r,\cos\omega) = |
526 |
gezelter |
921 |
\frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\omega-\cos\omega_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ , |
527 |
chrisfen |
862 |
\end{equation} |
528 |
chrisfen |
861 |
where $\theta$ and $\omega$ refer to the angles shown in figure |
529 |
|
|
\ref{corrAngle}. By binning over both distance and the cosine of the |
530 |
gezelter |
921 |
desired angle between the two dipoles, the $g(r)$ can be analyzed to |
531 |
|
|
determine the common dipole arrangements that constitute the peaks and |
532 |
|
|
troughs in the standard one-dimensional $g(r)$ plots. Frames A and B |
533 |
|
|
of figure \ref{contour} show results from an ice $I_c$ simulation. The |
534 |
|
|
first peak in the $g(r)$ consists primarily of the preferred hydrogen |
535 |
chrisfen |
861 |
bonding arrangements as dictated by the tetrahedral sticky potential - |
536 |
gezelter |
921 |
one peak for the hydrogen bond donor and the other for the hydrogen |
537 |
|
|
bond acceptor. Due to the high degree of crystallinity of the sample, |
538 |
|
|
the second and third solvation shells show a repeated peak arrangement |
539 |
chrisfen |
743 |
which decays at distances around the fourth solvation shell, near the |
540 |
|
|
imposed cutoff for the Lennard-Jones and dipole-dipole interactions. |
541 |
chrisfen |
861 |
In the higher temperature simulation shown in frames C and D, these |
542 |
gezelter |
921 |
long-range features deteriorate rapidly. The first solvation shell |
543 |
|
|
still shows the strong effect of the sticky-potential, although it |
544 |
|
|
covers a larger area, extending to include a fraction of aligned |
545 |
|
|
dipole peaks within the first solvation shell. The latter peaks lose |
546 |
|
|
due to thermal motion and as the competing dipole force overcomes the |
547 |
|
|
sticky potential's tight tetrahedral structuring of the crystal. |
548 |
chrisfen |
743 |
|
549 |
|
|
This complex interplay between dipole and sticky interactions was |
550 |
|
|
remarked upon as a possible reason for the split second peak in the |
551 |
gezelter |
921 |
oxygen-oxygen $g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, |
552 |
|
|
the second solvation shell peak appears to have two distinct |
553 |
|
|
components that blend together to form one observable peak. At higher |
554 |
chrisfen |
862 |
temperatures, this split character alters to show the leading 4 \AA\ |
555 |
|
|
peak dominated by equatorial anti-parallel dipole orientations. There |
556 |
|
|
is also a tightly bunched group of axially arranged dipoles that most |
557 |
|
|
likely consist of the smaller fraction of aligned dipole pairs. The |
558 |
|
|
trailing component of the split peak at 5 \AA\ is dominated by aligned |
559 |
|
|
dipoles that assume hydrogen bond arrangements similar to those seen |
560 |
|
|
in the first solvation shell. This evidence indicates that the dipole |
561 |
|
|
pair interaction begins to dominate outside of the range of the |
562 |
gezelter |
921 |
dipolar repulsion term. The energetically favorable dipole |
563 |
chrisfen |
862 |
arrangements populate the region immediately outside this repulsion |
564 |
gezelter |
921 |
region (around 4 \AA), while arrangements that seek to satisfy both |
565 |
|
|
the sticky and dipole forces locate themselves just beyond this |
566 |
chrisfen |
862 |
initial buildup (around 5 \AA). |
567 |
chrisfen |
743 |
|
568 |
|
|
From these findings, the split second peak is primarily the product of |
569 |
chrisfen |
861 |
the dipolar repulsion term of the sticky potential. In fact, the inner |
570 |
|
|
peak can be pushed out and merged with the outer split peak just by |
571 |
gezelter |
921 |
extending the switching function ($s^\prime(r_{ij})$) from its normal |
572 |
|
|
4.0 \AA\ cutoff to values of 4.5 or even 5 \AA. This type of |
573 |
chrisfen |
861 |
correction is not recommended for improving the liquid structure, |
574 |
chrisfen |
862 |
since the second solvation shell would still be shifted too far |
575 |
chrisfen |
861 |
out. In addition, this would have an even more detrimental effect on |
576 |
|
|
the system densities, leading to a liquid with a more open structure |
577 |
gezelter |
921 |
and a density considerably lower than the already low SSD density. A |
578 |
|
|
better correction would be to include the quadrupole-quadrupole |
579 |
|
|
interactions for the water particles outside of the first solvation |
580 |
|
|
shell, but this would remove the simplicity and speed advantage of |
581 |
|
|
SSD. |
582 |
chrisfen |
743 |
|
583 |
chrisfen |
861 |
\subsection{Adjusted Potentials: SSD/RF and SSD/E} |
584 |
gezelter |
921 |
|
585 |
chrisfen |
743 |
The propensity of SSD to adopt lower than expected densities under |
586 |
|
|
varying conditions is troubling, especially at higher temperatures. In |
587 |
chrisfen |
861 |
order to correct this model for use with a reaction field, it is |
588 |
|
|
necessary to adjust the force field parameters for the primary |
589 |
|
|
intermolecular interactions. In undergoing a reparameterization, it is |
590 |
|
|
important not to focus on just one property and neglect the other |
591 |
|
|
important properties. In this case, it would be ideal to correct the |
592 |
gezelter |
921 |
densities while maintaining the accurate transport behavior. |
593 |
chrisfen |
743 |
|
594 |
chrisfen |
862 |
The parameters available for tuning include the $\sigma$ and $\epsilon$ |
595 |
chrisfen |
743 |
Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky |
596 |
|
|
attractive and dipole repulsive terms with their respective |
597 |
|
|
cutoffs. To alter the attractive and repulsive terms of the sticky |
598 |
|
|
potential independently, it is necessary to separate the terms as |
599 |
|
|
follows: |
600 |
|
|
\begin{equation} |
601 |
|
|
u_{ij}^{sp} |
602 |
gezelter |
921 |
({\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 |
chrisfen |
743 |
\end{equation} |
605 |
|
|
where $\nu_0$ scales the strength of the tetrahedral attraction and |
606 |
gezelter |
921 |
$\nu_0^\prime$ 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 |
chrisfen |
862 |
repulsive terms do not share the same lower cutoff ($r_l$) in the |
613 |
gezelter |
921 |
newly parameterized potentials. We are calling these |
614 |
|
|
reparameterizations the Soft Sticky Dipole / Reaction Field |
615 |
|
|
(SSD/RF - for use with a reaction field) and Soft Sticky Dipole |
616 |
|
|
Enhanced (SSD/E - an attempt to improve the liquid structure in |
617 |
chrisfen |
862 |
simulations without a long-range correction). |
618 |
chrisfen |
743 |
|
619 |
|
|
\begin{table} |
620 |
chrisfen |
862 |
\begin{center} |
621 |
chrisfen |
743 |
\caption{Parameters for the original and adjusted models} |
622 |
chrisfen |
856 |
\begin{tabular}{ l c c c c } |
623 |
chrisfen |
743 |
\hline \\[-3mm] |
624 |
gezelter |
921 |
\ \ \ Parameters\ \ \ & \ \ \ SSD [Ref. \citen{Ichiye96}] \ \ \ |
625 |
|
|
& \ SSD1 [Ref. \citen{Ichiye03}]\ \ & \ SSD/E\ \ & \ SSD/RF \\ |
626 |
chrisfen |
743 |
\hline \\[-3mm] |
627 |
chrisfen |
856 |
\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
628 |
|
|
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
629 |
|
|
\ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\ |
630 |
|
|
\ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
631 |
|
|
\ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\ |
632 |
|
|
\ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\ |
633 |
|
|
\ \ \ $\nu_0^\prime$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
634 |
|
|
\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\ |
635 |
|
|
\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\ |
636 |
chrisfen |
743 |
\end{tabular} |
637 |
|
|
\label{params} |
638 |
chrisfen |
862 |
\end{center} |
639 |
chrisfen |
743 |
\end{table} |
640 |
|
|
|
641 |
chrisfen |
862 |
\begin{figure} |
642 |
|
|
\begin{center} |
643 |
|
|
\epsfxsize=5in |
644 |
|
|
\epsfbox{GofRCompare.epsi} |
645 |
gezelter |
921 |
\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with SSD/E |
646 |
chrisfen |
856 |
and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with |
647 |
chrisfen |
743 |
reaction field turned on (bottom). The insets show the respective |
648 |
chrisfen |
862 |
first peaks in detail. Note how the changes in parameters have lowered |
649 |
|
|
and broadened the first peak of SSD/E and SSD/RF.} |
650 |
chrisfen |
743 |
\label{grcompare} |
651 |
chrisfen |
862 |
\end{center} |
652 |
chrisfen |
743 |
\end{figure} |
653 |
|
|
|
654 |
chrisfen |
862 |
\begin{figure} |
655 |
|
|
\begin{center} |
656 |
|
|
\epsfxsize=6in |
657 |
|
|
\epsfbox{dualsticky.ps} |
658 |
chrisfen |
856 |
\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \& |
659 |
chrisfen |
743 |
SSD/RF (right). Light areas correspond to the tetrahedral attractive |
660 |
chrisfen |
862 |
component, and darker areas correspond to the dipolar repulsive |
661 |
|
|
component.} |
662 |
chrisfen |
743 |
\label{isosurface} |
663 |
chrisfen |
862 |
\end{center} |
664 |
chrisfen |
743 |
\end{figure} |
665 |
|
|
|
666 |
gezelter |
921 |
In the original paper detailing the development of SSD, Liu and Ichiye |
667 |
|
|
placed particular emphasis on an accurate description of the first |
668 |
|
|
solvation shell. This resulted in a somewhat tall and narrow first |
669 |
|
|
peak in $g(r)$ that integrated to give similar coordination numbers to |
670 |
chrisfen |
862 |
the experimental data obtained by Soper and |
671 |
|
|
Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering |
672 |
|
|
data from the Head-Gordon lab indicates a slightly lower and shifted |
673 |
gezelter |
921 |
first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were |
674 |
|
|
made while taking into consideration the new experimental |
675 |
chrisfen |
862 |
findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the |
676 |
gezelter |
921 |
relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing |
677 |
|
|
the revised SSD model (SSD1), SSD/E, and SSD/RF to the new |
678 |
chrisfen |
862 |
experimental results. Both modified water models have shorter peaks |
679 |
gezelter |
921 |
that match more closely to the experimental peak (as seen in the |
680 |
|
|
insets of figure \ref{grcompare}). This structural alteration was |
681 |
chrisfen |
862 |
accomplished by the combined reduction in the Lennard-Jones $\sigma$ |
682 |
gezelter |
921 |
variable and adjustment of the sticky potential strength and cutoffs. |
683 |
|
|
As can be seen in table \ref{params}, the cutoffs for the tetrahedral |
684 |
|
|
attractive and dipolar repulsive terms were nearly swapped with each |
685 |
|
|
other. Isosurfaces of the original and modified sticky potentials are |
686 |
|
|
shown in figure \ref{isosurface}. In these isosurfaces, it is easy to |
687 |
|
|
see how altering the cutoffs changes the repulsive and attractive |
688 |
|
|
character of the particles. With a reduced repulsive surface (darker |
689 |
|
|
region), the particles can move closer to one another, increasing the |
690 |
|
|
density for the overall system. This change in interaction cutoff also |
691 |
|
|
results in a more gradual orientational motion by allowing the |
692 |
|
|
particles to maintain preferred dipolar arrangements before they begin |
693 |
|
|
to feel the pull of the tetrahedral restructuring. As the particles |
694 |
|
|
move closer together, the dipolar repulsion term becomes active and |
695 |
|
|
excludes unphysical nearest-neighbor arrangements. This compares with |
696 |
|
|
how SSD and SSD1 exclude preferred dipole alignments before the |
697 |
|
|
particles feel the pull of the ``hydrogen bonds''. Aside from |
698 |
|
|
improving the shape of the first peak in the g(\emph{r}), this |
699 |
|
|
modification improves the densities considerably by allowing the |
700 |
|
|
persistence of full dipolar character below the previous 4.0 \AA\ |
701 |
|
|
cutoff. |
702 |
chrisfen |
743 |
|
703 |
gezelter |
921 |
While adjusting the location and shape of the first peak of $g(r)$ |
704 |
|
|
improves the densities, these changes alone are insufficient to bring |
705 |
|
|
the system densities up to the values observed experimentally. To |
706 |
|
|
further increase the densities, the dipole moments were increased in |
707 |
|
|
both of our adjusted models. Since SSD is a dipole based model, the |
708 |
|
|
structure and transport are very sensitive to changes in the dipole |
709 |
|
|
moment. The original SSD simply used the dipole moment calculated from |
710 |
|
|
the TIP3P water model, which at 2.35 D is significantly greater than |
711 |
|
|
the experimental gas phase value of 1.84 D. The larger dipole moment |
712 |
|
|
is a more realistic value and improves the dielectric properties of |
713 |
|
|
the fluid. Both theoretical and experimental measurements indicate a |
714 |
|
|
liquid phase dipole moment ranging from 2.4 D to values as high as |
715 |
|
|
3.11 D, providing a substantial range of reasonable values for a |
716 |
|
|
dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
717 |
chrisfen |
862 |
increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF, |
718 |
|
|
respectively, leads to significant changes in the density and |
719 |
|
|
transport of the water models. |
720 |
chrisfen |
743 |
|
721 |
chrisfen |
861 |
In order to demonstrate the benefits of these reparameterizations, a |
722 |
chrisfen |
743 |
series of NPT and NVE simulations were performed to probe the density |
723 |
|
|
and transport properties of the adapted models and compare the results |
724 |
|
|
to the original SSD model. This comparison involved full NPT melting |
725 |
|
|
sequences for both SSD/E and SSD/RF, as well as NVE transport |
726 |
chrisfen |
861 |
calculations at the calculated self-consistent densities. Again, the |
727 |
chrisfen |
862 |
results are obtained from five separate simulations of 1024 particle |
728 |
|
|
systems, and the melting sequences were started from different ice |
729 |
|
|
$I_h$ crystals constructed as described previously. Each NPT |
730 |
chrisfen |
861 |
simulation was equilibrated for 100 ps before a 200 ps data collection |
731 |
chrisfen |
862 |
run at each temperature step, and the final configuration from the |
732 |
|
|
previous temperature simulation was used as a starting point. All NVE |
733 |
|
|
simulations had the same thermalization, equilibration, and data |
734 |
gezelter |
921 |
collection times as stated previously. |
735 |
chrisfen |
743 |
|
736 |
chrisfen |
862 |
\begin{figure} |
737 |
|
|
\begin{center} |
738 |
|
|
\epsfxsize=6in |
739 |
|
|
\epsfbox{ssdeDense.epsi} |
740 |
chrisfen |
861 |
\caption{Comparison of densities calculated with SSD/E to SSD1 without a |
741 |
gezelter |
921 |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
742 |
|
|
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and |
743 |
|
|
experiment [Ref. \citen{CRC80}]. The window shows a expansion around |
744 |
|
|
300 K with error bars included to clarify this region of |
745 |
|
|
interest. Note that both SSD1 and SSD/E show good agreement with |
746 |
chrisfen |
856 |
experiment when the long-range correction is neglected.} |
747 |
chrisfen |
743 |
\label{ssdedense} |
748 |
chrisfen |
862 |
\end{center} |
749 |
chrisfen |
743 |
\end{figure} |
750 |
|
|
|
751 |
gezelter |
921 |
Fig. \ref{ssdedense} shows the density profile for the SSD/E model |
752 |
chrisfen |
862 |
in comparison to SSD1 without a reaction field, other common water |
753 |
|
|
models, and experimental results. The calculated densities for both |
754 |
|
|
SSD/E and SSD1 have increased significantly over the original SSD |
755 |
gezelter |
921 |
model (see fig. \ref{dense1}) and are in better agreement with the |
756 |
chrisfen |
862 |
experimental values. At 298 K, the densities of SSD/E and SSD1 without |
757 |
|
|
a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and |
758 |
|
|
0.999$\pm$0.001 g/cm$^3$ respectively. These both compare well with |
759 |
|
|
the experimental value of 0.997 g/cm$^3$, and they are considerably |
760 |
|
|
better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The changes to |
761 |
|
|
the dipole moment and sticky switching functions have improved the |
762 |
|
|
structuring of the liquid (as seen in figure \ref{grcompare}, but they |
763 |
|
|
have shifted the density maximum to much lower temperatures. This |
764 |
|
|
comes about via an increase in the liquid disorder through the |
765 |
|
|
weakening of the sticky potential and strengthening of the dipolar |
766 |
|
|
character. However, this increasing disorder in the SSD/E model has |
767 |
gezelter |
921 |
little effect on the melting transition. By monitoring $C_p$ |
768 |
chrisfen |
862 |
throughout these simulations, the melting transition for SSD/E was |
769 |
gezelter |
921 |
shown to occur at 235 K. The same transition temperature observed |
770 |
|
|
with SSD and SSD1. |
771 |
chrisfen |
743 |
|
772 |
chrisfen |
862 |
\begin{figure} |
773 |
|
|
\begin{center} |
774 |
|
|
\epsfxsize=6in |
775 |
|
|
\epsfbox{ssdrfDense.epsi} |
776 |
chrisfen |
861 |
\caption{Comparison of densities calculated with SSD/RF to SSD1 with a |
777 |
gezelter |
921 |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
778 |
|
|
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and |
779 |
|
|
experiment [Ref. \citen{CRC80}]. The inset shows the necessity of |
780 |
|
|
reparameterization when utilizing a reaction field long-ranged |
781 |
|
|
correction - SSD/RF provides significantly more accurate densities |
782 |
|
|
than SSD1 when performing room temperature simulations.} |
783 |
chrisfen |
743 |
\label{ssdrfdense} |
784 |
chrisfen |
862 |
\end{center} |
785 |
chrisfen |
743 |
\end{figure} |
786 |
|
|
|
787 |
chrisfen |
862 |
Including the reaction field long-range correction in the simulations |
788 |
gezelter |
921 |
results in a more interesting comparison. A density profile including |
789 |
chrisfen |
862 |
SSD/RF and SSD1 with an active reaction field is shown in figure |
790 |
|
|
\ref{ssdrfdense}. As observed in the simulations without a reaction |
791 |
|
|
field, the densities of SSD/RF and SSD1 show a dramatic increase over |
792 |
|
|
normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density |
793 |
|
|
of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and |
794 |
gezelter |
921 |
considerably better than the original SSD value of 0.941$\pm$0.001 |
795 |
|
|
g/cm$^3$ and the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results |
796 |
|
|
further emphasize the importance of reparameterization in order to |
797 |
|
|
model the density properly under different simulation conditions. |
798 |
|
|
Again, these changes have only a minor effect on the melting point, |
799 |
|
|
which observed at 245 K for SSD/RF, is identical to SSD and only 5 K |
800 |
|
|
lower than SSD1 with a reaction field. Additionally, the difference in |
801 |
|
|
density maxima is not as extreme, with SSD/RF showing a density |
802 |
|
|
maximum at 255 K, fairly close to the density maxima of 260 K and 265 |
803 |
|
|
K, shown by SSD and SSD1 respectively. |
804 |
chrisfen |
743 |
|
805 |
chrisfen |
862 |
\begin{figure} |
806 |
|
|
\begin{center} |
807 |
|
|
\epsfxsize=6in |
808 |
|
|
\epsfbox{ssdeDiffuse.epsi} |
809 |
chrisfen |
861 |
\caption{Plots of the diffusion constants calculated from SSD/E and SSD1, |
810 |
gezelter |
921 |
both without a reaction field, along with experimental results |
811 |
|
|
[Refs. \citen{Gillen72} and \citen{Mills73}]. The NVE calculations were |
812 |
|
|
performed at the average densities observed in the 1 atm NPT |
813 |
|
|
simulations for the respective models. SSD/E is slightly more fluid |
814 |
|
|
than experiment at all of the temperatures, but it is closer than SSD1 |
815 |
|
|
without a long-range correction.} |
816 |
chrisfen |
861 |
\label{ssdediffuse} |
817 |
chrisfen |
862 |
\end{center} |
818 |
chrisfen |
861 |
\end{figure} |
819 |
|
|
|
820 |
chrisfen |
743 |
The reparameterization of the SSD water model, both for use with and |
821 |
|
|
without an applied long-range correction, brought the densities up to |
822 |
|
|
what is expected for simulating liquid water. In addition to improving |
823 |
gezelter |
921 |
the densities, it is important that the excellent diffusive behavior |
824 |
|
|
of SSD be maintained or improved. Figure \ref{ssdediffuse} compares |
825 |
|
|
the temperature dependence of the diffusion constant of SSD/E to SSD1 |
826 |
|
|
without an active reaction field, both at the densities calculated at |
827 |
|
|
1 atm and at the experimentally calculated densities for super-cooled |
828 |
|
|
and liquid water. The diffusion constant for SSD/E is consistently |
829 |
chrisfen |
862 |
higher than experiment, while SSD1 remains lower than experiment until |
830 |
|
|
relatively high temperatures (greater than 330 K). Both models follow |
831 |
|
|
the shape of the experimental curve well below 300 K but tend to |
832 |
|
|
diffuse too rapidly at higher temperatures, something that is |
833 |
gezelter |
921 |
especially apparent with SSD1. This increasing diffusion relative to |
834 |
|
|
the experimental values is caused by the rapidly decreasing system |
835 |
|
|
density with increasing temperature. The densities of SSD1 decay more |
836 |
|
|
rapidly with temperature than do those of SSD/E, leading to more |
837 |
|
|
visible deviation from the experimental diffusion trend. Thus, the |
838 |
|
|
changes made to improve the liquid structure may have had an adverse |
839 |
|
|
affect on the density maximum, but they improve the transport behavior |
840 |
|
|
of SSD/E relative to SSD1. |
841 |
chrisfen |
743 |
|
842 |
chrisfen |
862 |
\begin{figure} |
843 |
|
|
\begin{center} |
844 |
|
|
\epsfxsize=6in |
845 |
|
|
\epsfbox{ssdrfDiffuse.epsi} |
846 |
chrisfen |
856 |
\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1, |
847 |
gezelter |
921 |
both with an active reaction field, along with experimental results |
848 |
|
|
[Refs. \citen{Gillen72} and \citen{Mills73}]. The NVE calculations |
849 |
|
|
were performed at the average densities observed in the 1 atm NPT |
850 |
|
|
simulations for both of the models. Note how accurately SSD/RF |
851 |
|
|
simulates the diffusion of water throughout this temperature |
852 |
|
|
range. The more rapidly increasing diffusion constants at high |
853 |
|
|
temperatures for both models is attributed to the significantly lower |
854 |
|
|
densities than observed in experiment.} |
855 |
chrisfen |
856 |
\label{ssdrfdiffuse} |
856 |
chrisfen |
862 |
\end{center} |
857 |
chrisfen |
743 |
\end{figure} |
858 |
|
|
|
859 |
|
|
In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are |
860 |
chrisfen |
862 |
compared to SSD1 with an active reaction field. Note that SSD/RF |
861 |
gezelter |
921 |
tracks the experimental results quantitatively, identical within error |
862 |
|
|
throughout the temperature range shown and with only a slight |
863 |
chrisfen |
861 |
increasing trend at higher temperatures. SSD1 tends to diffuse more |
864 |
|
|
slowly at low temperatures and deviates to diffuse too rapidly at |
865 |
gezelter |
921 |
temperatures greater than 330 K. As stated above, this deviation away |
866 |
|
|
from the ideal trend is due to a rapid decrease in density at higher |
867 |
|
|
temperatures. SSD/RF does not suffer from this problem as much as SSD1 |
868 |
|
|
because the calculated densities are closer to the experimental |
869 |
|
|
values. These results again emphasize the importance of careful |
870 |
|
|
reparameterization when using an altered long-range correction. |
871 |
chrisfen |
743 |
|
872 |
|
|
\subsection{Additional Observations} |
873 |
|
|
|
874 |
|
|
\begin{figure} |
875 |
chrisfen |
862 |
\begin{center} |
876 |
|
|
\epsfxsize=6in |
877 |
|
|
\epsfbox{povIce.ps} |
878 |
|
|
\caption{A water lattice built from the crystal structure assumed by |
879 |
gezelter |
921 |
SSD/E when undergoing an extremely restricted temperature NPT |
880 |
|
|
simulation. This form of ice is referred to as ice-{\it i} to |
881 |
|
|
emphasize its simulation origins. This image was taken of the (001) |
882 |
|
|
face of the crystal.} |
883 |
chrisfen |
743 |
\label{weirdice} |
884 |
chrisfen |
862 |
\end{center} |
885 |
chrisfen |
743 |
\end{figure} |
886 |
|
|
|
887 |
gezelter |
921 |
While performing a series of melting simulations on an early iteration |
888 |
|
|
of SSD/E not discussed in this paper, we observed recrystallization |
889 |
|
|
into a novel structure not previously known for water. After melting |
890 |
|
|
at 235 K, two of five systems underwent crystallization events near |
891 |
|
|
245 K. The two systems remained crystalline up to 320 and 330 K, |
892 |
|
|
respectively. The crystal exhibits an expanded zeolite-like structure |
893 |
|
|
that does not correspond to any known form of ice. This appears to be |
894 |
|
|
an artifact of the point dipolar models, so to distinguish it from the |
895 |
|
|
experimentally observed forms of ice, we have denoted the structure |
896 |
|
|
Ice-$\sqrt{\smash[b]{-\text{I}}}$ (ice-{\it i}). A large enough |
897 |
|
|
portion of the sample crystallized that we have been able to obtain a |
898 |
|
|
near ideal crystal structure of ice-{\it i}. Figure \ref{weirdice} |
899 |
|
|
shows the repeating crystal structure of a typical crystal at 5 |
900 |
|
|
K. Each water molecule is hydrogen bonded to four others; however, the |
901 |
|
|
hydrogen bonds are bent rather than perfectly straight. This results |
902 |
|
|
in a skewed tetrahedral geometry about the central molecule. In |
903 |
|
|
figure \ref{isosurface}, it is apparent that these flexed hydrogen |
904 |
|
|
bonds are allowed due to the conical shape of the attractive regions, |
905 |
|
|
with the greatest attraction along the direct hydrogen bond |
906 |
chrisfen |
863 |
configuration. Though not ideal, these flexed hydrogen bonds are |
907 |
gezelter |
921 |
favorable enough to stabilize an entire crystal generated around them. |
908 |
chrisfen |
743 |
|
909 |
gezelter |
921 |
Initial simulations indicated that ice-{\it i} is the preferred ice |
910 |
|
|
structure for at least the SSD/E model. To verify this, a comparison |
911 |
|
|
was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and |
912 |
|
|
ice-{\it i} at constant pressure with SSD/E, SSD/RF, and |
913 |
|
|
SSD1. Near-ideal versions of the three types of crystals were cooled |
914 |
|
|
to 1 K, and the enthalpies of each were compared using all three water |
915 |
|
|
models. With every model in the SSD family, ice-{\it i} had the lowest |
916 |
|
|
calculated enthalpy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with |
917 |
|
|
SSD/E, and 7.5\% lower with SSD/RF. The enthalpy data is summarized |
918 |
|
|
in Table \ref{iceenthalpy}. |
919 |
chrisfen |
743 |
|
920 |
gezelter |
921 |
\begin{table} |
921 |
|
|
\begin{center} |
922 |
|
|
\caption{Enthalpies (in kcal / mol) of the three crystal structures (at 1 |
923 |
|
|
K) exhibited by the SSD family of water models} |
924 |
|
|
\begin{tabular}{ l c c c } |
925 |
|
|
\hline \\[-3mm] |
926 |
|
|
\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \ & \ |
927 |
|
|
Ice-{\it i} \\ |
928 |
|
|
\hline \\[-3mm] |
929 |
|
|
\ \ \ SSD/E & -12.286 & -12.292 & -13.590 \\ |
930 |
|
|
\ \ \ SSD/RF & -12.935 & -12.917 & -14.022 \\ |
931 |
|
|
\ \ \ SSD1 & -12.496 & -12.411 & -13.417 \\ |
932 |
|
|
\ \ \ SSD1 (RF) & -12.504 & -12.411 & -13.134 \\ |
933 |
|
|
\end{tabular} |
934 |
|
|
\label{iceenthalpy} |
935 |
|
|
\end{center} |
936 |
|
|
\end{table} |
937 |
chrisfen |
743 |
|
938 |
gezelter |
921 |
In addition to these energetic comparisons, melting simulations were |
939 |
|
|
performed with ice-{\it i} as the initial configuration using SSD/E, |
940 |
|
|
SSD/RF, and SSD1 both with and without a reaction field. The melting |
941 |
|
|
transitions for both SSD/E and SSD1 without reaction field occurred at |
942 |
|
|
temperature in excess of 375~K. SSD/RF and SSD1 with a reaction field |
943 |
|
|
showed more reasonable melting transitions near 325~K. These melting |
944 |
|
|
point observations clearly show that all of the SSD-derived models |
945 |
|
|
prefer the ice-{\it i} structure. |
946 |
chrisfen |
743 |
|
947 |
|
|
\section{Conclusions} |
948 |
|
|
|
949 |
gezelter |
921 |
The density maximum and temperature dependence of the self-diffusion |
950 |
|
|
constant were studied for the SSD water model, both with and without |
951 |
|
|
the use of reaction field, via a series of NPT and NVE |
952 |
|
|
simulations. The constant pressure simulations showed a density |
953 |
|
|
maximum near 260 K. In most cases, the calculated densities were |
954 |
|
|
significantly lower than the densities obtained from other water |
955 |
|
|
models (and experiment). Analysis of self-diffusion showed SSD to |
956 |
|
|
capture the transport properties of water well in both the liquid and |
957 |
|
|
super-cooled liquid regimes. |
958 |
|
|
|
959 |
|
|
In order to correct the density behavior, the original SSD model was |
960 |
|
|
reparameterized for use both with and without a reaction field (SSD/RF |
961 |
|
|
and SSD/E), and comparisons were made with SSD1, Ichiye's density |
962 |
|
|
corrected version of SSD. Both models improve the liquid structure, |
963 |
|
|
densities, and diffusive properties under their respective simulation |
964 |
|
|
conditions, indicating the necessity of reparameterization when |
965 |
|
|
changing the method of calculating long-range electrostatic |
966 |
|
|
interactions. In general, however, these simple water models are |
967 |
|
|
excellent choices for representing explicit water in large scale |
968 |
|
|
simulations of biochemical systems. |
969 |
|
|
|
970 |
|
|
The existence of a novel low-density ice structure that is preferred |
971 |
|
|
by the SSD family of water models is somewhat troubling, since liquid |
972 |
|
|
simulations on this family of water models at room temperature are |
973 |
|
|
effectively simulations of super-cooled or metastable liquids. One |
974 |
|
|
way to de-stabilize this unphysical ice structure would be to make the |
975 |
|
|
range of angles preferred by the attractive part of the sticky |
976 |
|
|
potential much narrower. This would require extensive |
977 |
|
|
reparameterization to maintain the same level of agreement with the |
978 |
|
|
experiments. |
979 |
|
|
|
980 |
|
|
Additionally, our initial calculations show that the ice-{\it i} |
981 |
|
|
structure may also be a preferred crystal structure for at least one |
982 |
|
|
other popular multi-point water model (TIP3P), and that much of the |
983 |
|
|
simulation work being done using this popular model could also be at |
984 |
|
|
risk for crystallization into this unphysical structure. A future |
985 |
|
|
publication will detail the relative stability of the known ice |
986 |
|
|
structures for a wide range of popular water models. |
987 |
|
|
|
988 |
chrisfen |
743 |
\section{Acknowledgments} |
989 |
chrisfen |
777 |
Support for this project was provided by the National Science |
990 |
|
|
Foundation under grant CHE-0134881. Computation time was provided by |
991 |
|
|
the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
992 |
gezelter |
921 |
DMR-0079647. |
993 |
chrisfen |
743 |
|
994 |
chrisfen |
862 |
\newpage |
995 |
|
|
|
996 |
chrisfen |
743 |
\bibliographystyle{jcp} |
997 |
|
|
\bibliography{nptSSD} |
998 |
|
|
|
999 |
|
|
%\pagebreak |
1000 |
|
|
|
1001 |
|
|
\end{document} |