1 |
< |
\documentclass[prb,aps,times,twocolumn,tabularx]{revtex4} |
2 |
< |
%\documentclass[prb,aps,times,tabularx,preprint]{revtex4} |
1 |
> |
%\documentclass[prb,aps,times,twocolumn,tabularx]{revtex4} |
2 |
> |
\documentclass[11pt]{article} |
3 |
> |
\usepackage{endfloat} |
4 |
|
\usepackage{amsmath} |
5 |
+ |
\usepackage{epsf} |
6 |
+ |
\usepackage{berkeley} |
7 |
+ |
\usepackage{setspace} |
8 |
+ |
\usepackage{tabularx} |
9 |
|
\usepackage{graphicx} |
10 |
< |
|
6 |
< |
%\usepackage{endfloat} |
10 |
> |
\usepackage[ref]{overcite} |
11 |
|
%\usepackage{berkeley} |
8 |
– |
%\usepackage{epsf} |
9 |
– |
%\usepackage[ref]{overcite} |
10 |
– |
%\usepackage{setspace} |
11 |
– |
%\usepackage{tabularx} |
12 |
– |
%\usepackage{graphicx} |
12 |
|
%\usepackage{curves} |
13 |
< |
%\usepackage{amsmath} |
14 |
< |
%\pagestyle{plain} |
15 |
< |
%\pagenumbering{arabic} |
16 |
< |
%\oddsidemargin 0.0cm \evensidemargin 0.0cm |
17 |
< |
%\topmargin -21pt \headsep 10pt |
18 |
< |
%\textheight 9.0in \textwidth 6.5in |
19 |
< |
%\brokenpenalty=10000 |
20 |
< |
%\renewcommand{\baselinestretch}{1.2} |
22 |
< |
%\renewcommand\citemid{\ } % no comma in optional reference note |
13 |
> |
\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 |
|
|
22 |
|
\begin{document} |
23 |
|
|
24 |
< |
\title{On the temperature dependent structural and transport properties of the soft sticky dipole (SSD) and related single point water models} |
24 |
> |
\title{On the structural and transport properties of the soft sticky |
25 |
> |
dipole (SSD) and related single point water models} |
26 |
|
|
27 |
< |
\author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote} |
28 |
< |
\footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}} |
30 |
< |
|
31 |
< |
\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
27 |
> |
\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 |
|
Notre Dame, Indiana 46556} |
30 |
|
|
31 |
|
\date{\today} |
32 |
|
|
33 |
+ |
\maketitle |
34 |
+ |
|
35 |
|
\begin{abstract} |
36 |
< |
NVE and NPT molecular dynamics simulations were performed in order to |
37 |
< |
investigate the density maximum and temperature dependent transport |
38 |
< |
for the SSD water model, both with and without the use of reaction |
39 |
< |
field. The constant pressure simulations of the melting of both $I_h$ |
40 |
< |
and $I_c$ ice showed a density maximum near 260 K. In most cases, the |
41 |
< |
calculated densities were significantly lower than the densities |
42 |
< |
calculated in simulations of other water models. Analysis of particle |
43 |
< |
diffusion showed SSD to capture the transport properties of |
44 |
< |
experimental very well in both the normal and super-cooled liquid |
45 |
< |
regimes. In order to correct the density behavior, SSD was |
46 |
< |
reparameterized for use both with and without a long-range interaction |
47 |
< |
correction, SSD/RF and SSD/E respectively. In addition to correcting |
48 |
< |
the abnormally low densities, these new versions were show to maintain |
49 |
< |
or improve upon the transport and structural features of the original |
50 |
< |
water model. |
36 |
> |
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 |
> |
experimental water very well in both the normal and super-cooled |
49 |
> |
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 |
|
\end{abstract} |
60 |
|
|
61 |
< |
\maketitle |
61 |
> |
\newpage |
62 |
|
|
63 |
|
%\narrowtext |
64 |
|
|
69 |
|
|
70 |
|
\section{Introduction} |
71 |
|
|
72 |
< |
One of the most important tasks in simulations of biochemical systems |
73 |
< |
is the proper depiction of water and water solvation. In fact, the |
74 |
< |
bulk of the calculations performed in solvated simulations are of |
75 |
< |
interactions with or between solvent molecules. Thus, the outcomes of |
76 |
< |
these types of simulations are highly dependent on the physical |
77 |
< |
properties of water, both as individual molecules and in |
78 |
< |
groups/bulk. Due to the fact that explicit solvent accounts for a |
79 |
< |
massive portion of the calculations, it necessary to simplify the |
73 |
< |
solvent to some extent in order to complete simulations in a |
74 |
< |
reasonable amount of time. In the case of simulating water in |
75 |
< |
bio-molecular studies, the balance between accurate properties and |
76 |
< |
computational efficiency is especially delicate, and it has resulted |
77 |
< |
in a variety of different water |
78 |
< |
models.\cite{Jorgensen83,Berendsen87,Jorgensen00} Many of these models |
79 |
< |
get specific properties correct or better than their predecessors, but |
80 |
< |
this is often at a cost of some other properties or of computer |
81 |
< |
time. As an example, compare TIP3P or TIP4P to TIP5P. TIP5P succeeds |
82 |
< |
in improving the structural and transport properties over its |
83 |
< |
predecessors, yet this comes at a greater than 50\% increase in |
84 |
< |
computational cost.\cite{Jorgensen01,Jorgensen00} One recently |
85 |
< |
developed model that succeeds in both retaining accuracy of system |
86 |
< |
properties and simplifying calculations to increase computational |
87 |
< |
efficiency is the Soft Sticky Dipole water model.\cite{Ichiye96} |
72 |
> |
One of the most important tasks in the simulation of biochemical |
73 |
> |
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 |
|
|
81 |
< |
The Soft Sticky Dipole (SSD)\ water model was developed by Ichiye |
82 |
< |
\emph{et al.} as a modified form of the hard-sphere water model |
83 |
< |
proposed by Bratko, Blum, and Luzar.\cite{Bratko85,Bratko95} SSD |
84 |
< |
consists of a single point dipole with a Lennard-Jones core and a |
85 |
< |
sticky potential that directs the particles to assume the proper |
86 |
< |
hydrogen bond orientation in the first solvation shell. Thus, the |
87 |
< |
interaction between two SSD water molecules \emph{i} and \emph{j} is |
88 |
< |
given by the potential |
81 |
> |
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 |
|
\begin{equation} |
108 |
|
u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
109 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
109 |
> |
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ + |
110 |
|
u_{ij}^{sp} |
111 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
111 |
> |
({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j), |
112 |
|
\end{equation} |
113 |
< |
where the $\mathbf{r}_{ij}$ is the position vector between molecules |
114 |
< |
\emph{i} and \emph{j} with magnitude equal to the distance $r_ij$, and |
115 |
< |
$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
116 |
< |
orientations of the respective molecules. The Lennard-Jones, dipole, |
117 |
< |
and sticky parts of the potential are giving by the following |
108 |
< |
equations, |
113 |
> |
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 |
|
\begin{equation} |
119 |
< |
u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right], |
119 |
> |
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 |
|
\end{equation} |
123 |
+ |
and |
124 |
|
\begin{equation} |
125 |
< |
u_{ij}^{dp} = \frac{\boldsymbol{\mu}_i\cdot\boldsymbol{\mu}_j}{r_{ij}^3}-\frac{3(\boldsymbol{\mu}_i\cdot\mathbf{r}_{ij})(\boldsymbol{\mu}_j\cdot\mathbf{r}_{ij})}{r_{ij}^5}\ , |
125 |
> |
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 |
|
\end{equation} |
129 |
+ |
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 |
|
\begin{equation} |
116 |
– |
\begin{split} |
137 |
|
u_{ij}^{sp} |
138 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) |
139 |
< |
&= |
140 |
< |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\\ |
141 |
< |
& \quad \ + |
142 |
< |
s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
123 |
< |
\end{split} |
138 |
> |
({\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 |
> |
\label{stickyfunction} |
143 |
|
\end{equation} |
144 |
< |
where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole |
145 |
< |
unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D, |
146 |
< |
$\nu_0$ scales the strength of the overall sticky potential, $s$ and |
147 |
< |
$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$ |
148 |
< |
functions take the following forms, |
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 |
146 |
> |
sticky interaction beyond the first solvation shell. The $w$ function |
147 |
> |
can be thought of as an attractive potential with tetrahedral |
148 |
> |
geometry: |
149 |
|
\begin{equation} |
150 |
< |
w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
150 |
> |
w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
151 |
|
\end{equation} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\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 |
< |
where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive |
158 |
< |
term that promotes hydrogen bonding orientations within the first |
159 |
< |
solvation shell, and $w^\prime$ is a dipolar repulsion term that |
160 |
< |
repels unrealistic dipolar arrangements within the first solvation |
161 |
< |
shell. A more detailed description of the functional parts and |
162 |
< |
variables in this potential can be found in other |
163 |
< |
articles.\cite{Ichiye96,Ichiye99} |
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 |
159 |
> |
enhances the tetrahedral geometry for hydrogen bonded structures), |
160 |
> |
while $w^\prime$ is a purely empirical function. A more detailed |
161 |
> |
description of the functional parts and variables in this potential |
162 |
> |
can be found in the original SSD |
163 |
> |
articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} |
164 |
|
|
165 |
< |
Being that this is a one-site point dipole model, the actual force |
166 |
< |
calculations are simplified significantly. In the original Monte Carlo |
167 |
< |
simulations using this model, Ichiye \emph{et al.} reported a |
168 |
< |
calculation speed up of up to an order of magnitude over other |
169 |
< |
comparable models while maintaining the structural behavior of |
170 |
< |
water.\cite{Ichiye96} In the original molecular dynamics studies of |
171 |
< |
SSD, it was shown that it actually improves upon the prediction of |
172 |
< |
water's dynamical properties 3 and 4-point models.\cite{Ichiye99} This |
173 |
< |
attractive combination of speed and accurate depiction of solvent |
174 |
< |
properties makes SSD a model of interest for the simulation of large |
175 |
< |
scale biological systems, such as membrane phase behavior, a specific |
176 |
< |
interest within our group. |
165 |
> |
Since SSD is a single-point {\it dipolar} model, the force |
166 |
> |
calculations are simplified significantly relative to the standard |
167 |
> |
{\it charged} multi-point models. In the original Monte Carlo |
168 |
> |
simulations using this model, Ichiye {\it et al.} reported that using |
169 |
> |
SSD decreased computer time by a factor of 6-7 compared to other |
170 |
> |
models.\cite{Ichiye96} What is most impressive is that this savings |
171 |
> |
did not come at the expense of accurate depiction of the liquid state |
172 |
> |
properties. Indeed, SSD maintains reasonable agreement with the Soper |
173 |
> |
data for the structural features of liquid |
174 |
> |
water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties |
175 |
> |
exhibited by SSD agree with experiment better than those of more |
176 |
> |
computationally expensive models (like TIP3P and |
177 |
> |
SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction |
178 |
> |
of solvent properties makes SSD a very attractive model for the |
179 |
> |
simulation of large scale biochemical simulations. |
180 |
|
|
181 |
< |
One of the key limitations of this water model, however, is that it |
182 |
< |
has been parameterized for use with the Ewald Sum technique for the |
183 |
< |
handling of long-ranged interactions. When studying very large |
184 |
< |
systems, the Ewald summation and even particle-mesh Ewald become |
185 |
< |
computational burdens with their respective ideal $N^\frac{3}{2}$ and |
186 |
< |
$N\log N$ calculation scaling orders for $N$ particles.\cite{Darden99} |
181 |
> |
One feature of the SSD model is that it was parameterized for use with |
182 |
> |
the Ewald sum to handle long-range interactions. This would normally |
183 |
> |
be the best way of handling long-range interactions in systems that |
184 |
> |
contain other point charges. However, our group has recently become |
185 |
> |
interested in systems with point dipoles as mimics for neutral, but |
186 |
> |
polarized regions on molecules (e.g. the zwitterionic head group |
187 |
> |
regions of phospholipids). If the system of interest does not contain |
188 |
> |
point charges, the Ewald sum and even particle-mesh Ewald become |
189 |
> |
computational bottlenecks. Their respective ideal $N^\frac{3}{2}$ and |
190 |
> |
$N\log N$ calculation scaling orders for $N$ particles can become |
191 |
> |
prohibitive when $N$ becomes large.\cite{Darden99} In applying this |
192 |
> |
water model in these types of systems, it would be useful to know its |
193 |
> |
properties and behavior under the more computationally efficient |
194 |
> |
reaction field (RF) technique, or even with a simple cutoff. This |
195 |
> |
study addresses these issues by looking at the structural and |
196 |
> |
transport behavior of SSD over a variety of temperatures with the |
197 |
> |
purpose of utilizing the RF correction technique. We then suggest |
198 |
> |
modifications to the parameters that result in more realistic bulk |
199 |
> |
phase behavior. It should be noted that in a recent publication, some |
200 |
> |
of the original investigators of the SSD water model have suggested |
201 |
> |
adjustments to the SSD water model to address abnormal density |
202 |
> |
behavior (also observed here), calling the corrected model |
203 |
> |
SSD1.\cite{Ichiye03} In what follows, we compare our |
204 |
> |
reparamaterization of SSD with both the original SSD and SSD1 models |
205 |
> |
with the goal of improving the bulk phase behavior of an SSD-derived |
206 |
> |
model in simulations utilizing the Reaction Field. |
207 |
|
|
164 |
– |
Up to this point, a detailed look at the model's structure and ion |
165 |
– |
solvation abilities has been performed.\cite{Ichiye96} In addition, a |
166 |
– |
thorough investigation of the dynamic properties of SSD was performed |
167 |
– |
by Chandra and Ichiye focusing on translational and orientational |
168 |
– |
properties at 298 K.\cite{Ichiye99} This study focuses on determining |
169 |
– |
the density maximum for SSD utilizing both microcanonical and |
170 |
– |
isobaric-isothermal ensemble molecular dynamics, while using the |
171 |
– |
reaction field method for handling long-ranged dipolar interactions. A |
172 |
– |
reaction field method has been previously implemented in Monte Carlo |
173 |
– |
simulations by Liu and Ichiye in order to study the static dielectric |
174 |
– |
constant for the model.\cite{Ichiye96b} This paper will expand the |
175 |
– |
scope of these original simulations to look on how the reaction field |
176 |
– |
affects the physical and dynamic properties of SSD systems. |
177 |
– |
|
208 |
|
\section{Methods} |
209 |
|
|
210 |
< |
As stated previously, in this study the long-range dipole-dipole |
211 |
< |
interactions were accounted for using the reaction field method. The |
212 |
< |
magnitude of the reaction field acting on dipole \emph{i} is given by |
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 |
214 |
|
\begin{equation} |
215 |
|
\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1} |
216 |
< |
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} \boldsymbol{\mu}_{j} f(r_{ij})\ , |
216 |
> |
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} f(r_{ij})\ , |
217 |
|
\label{rfequation} |
218 |
|
\end{equation} |
219 |
|
where $\mathcal{R}$ is the cavity defined by the cutoff radius |
220 |
|
($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the |
221 |
< |
system (80 in this case), $\boldsymbol{\mu}_{j}$ is the dipole moment |
222 |
< |
vector of particle \emph{j}, and $f(r_{ij})$ is a cubic switching |
221 |
> |
system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole |
222 |
> |
moment vector of particle $j$ and $f(r_{ij})$ is a cubic switching |
223 |
|
function.\cite{AllenTildesley} The reaction field contribution to the |
224 |
< |
total energy by particle \emph{i} is given by |
225 |
< |
$-\frac{1}{2}\boldsymbol{\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque |
226 |
< |
on dipole \emph{i} by |
227 |
< |
$\boldsymbol{\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use |
228 |
< |
of reaction field is known to alter the orientational dynamic |
229 |
< |
properties, such as the dielectric relaxation time, based on changes |
230 |
< |
in the length of the cutoff radius.\cite{Berendsen98} This variable |
231 |
< |
behavior makes reaction field a less attractive method than other |
232 |
< |
methods, like the Ewald summation; however, for the simulation of |
202 |
< |
large-scale system, the computational cost benefit of reaction field |
203 |
< |
is dramatic. To address some of the dynamical property alterations due |
204 |
< |
to the use of reaction field, simulations were also performed without |
205 |
< |
a surrounding dielectric and suggestions are proposed on how to make |
206 |
< |
SSD more compatible with a reaction field. |
224 |
> |
total energy by particle $i$ is given by $-\frac{1}{2}{\bf |
225 |
> |
\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf |
226 |
> |
\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use of the reaction |
227 |
> |
field is known to alter the bulk orientational properties, such as the |
228 |
> |
dielectric relaxation time. There is particular sensitivity of this |
229 |
> |
property on changes in the length of the cutoff |
230 |
> |
radius.\cite{Berendsen98} This variable behavior makes reaction field |
231 |
> |
a less attractive method than the Ewald sum. However, for very large |
232 |
> |
systems, the computational benefit of reaction field is dramatic. |
233 |
|
|
234 |
< |
Simulations were performed in both the isobaric-isothermal and |
235 |
< |
microcanonical ensembles. The constant pressure simulations were |
236 |
< |
implemented using an integral thermostat and barostat as outlined by |
237 |
< |
Hoover.\cite{Hoover85,Hoover86} For the constant pressure |
238 |
< |
simulations, the \emph{Q} parameter for the was set to 5.0 amu |
213 |
< |
\(\cdot\)\AA\(^{2}\), and the relaxation time (\(\tau\))\ was set at |
214 |
< |
100 ps. |
234 |
> |
We have also performed a companion set of simulations {\it without} a |
235 |
> |
surrounding dielectric (i.e. using a simple cubic switching function |
236 |
> |
at the cutoff radius) and as a result we have two reparamaterizations |
237 |
> |
of SSD which could be used either with or without the Reaction Field |
238 |
> |
turned on. |
239 |
|
|
240 |
+ |
Simulations to obtain the preferred density were performed in the |
241 |
+ |
isobaric-isothermal (NPT) ensemble, while all dynamical properties |
242 |
+ |
were obtained from microcanonical (NVE) simulations done at densities |
243 |
+ |
matching the NPT density for a particular target temperature. The |
244 |
+ |
constant pressure simulations were implemented using an integral |
245 |
+ |
thermostat and barostat as outlined by Hoover.\cite{Hoover85,Hoover86} |
246 |
+ |
All molecules were treated as non-linear rigid bodies. Vibrational |
247 |
+ |
constraints are not necessary in simulations of SSD, because there are |
248 |
+ |
no explicit hydrogen atoms, and thus no molecular vibrational modes |
249 |
+ |
need to be considered. |
250 |
+ |
|
251 |
|
Integration of the equations of motion was carried out using the |
252 |
< |
symplectic splitting method proposed by Dullweber \emph{et |
253 |
< |
al.}.\cite{Dullweber1997} The reason for this integrator selection |
254 |
< |
deals with poor energy conservation of rigid body systems using |
255 |
< |
quaternions. While quaternions work well for orientational motion in |
256 |
< |
alternate ensembles, the microcanonical ensemble has a constant energy |
257 |
< |
requirement that is actually quite sensitive to errors in the |
258 |
< |
equations of motion. The original implementation of this code utilized |
259 |
< |
quaternions for rotational motion propagation; however, a detailed |
260 |
< |
investigation showed that they resulted in a steady drift in the total |
226 |
< |
energy, something that has been observed by others.\cite{Laird97} |
252 |
> |
symplectic splitting method proposed by Dullweber {\it et |
253 |
> |
al.}\cite{Dullweber1997} Our reason for selecting this integrator |
254 |
> |
centers on poor energy conservation of rigid body dynamics using |
255 |
> |
traditional quaternion integration.\cite{Evans77,Evans77b} While quaternions |
256 |
> |
may work well for orientational motion under NVT or NPT integrators, |
257 |
> |
our limits on energy drift in the microcanonical ensemble were quite |
258 |
> |
strict, and the drift under quaternions was substantially greater than |
259 |
> |
in the symplectic splitting method. This steady drift in the total |
260 |
> |
energy has also been observed by Kol {\it et al.}\cite{Laird97} |
261 |
|
|
262 |
|
The key difference in the integration method proposed by Dullweber |
263 |
|
\emph{et al.} is that the entire rotation matrix is propagated from |
264 |
< |
one time step to the next. In the past, this would not have been as |
265 |
< |
feasible a option, being that the rotation matrix for a single body is |
266 |
< |
nine elements long as opposed to 3 or 4 elements for Euler angles and |
267 |
< |
quaternions respectively. System memory has become much less of an |
234 |
< |
issue in recent times, and this has resulted in substantial benefits |
235 |
< |
in energy conservation. There is still the issue of an additional 5 or |
236 |
< |
6 additional elements for describing the orientation of each particle, |
237 |
< |
which will increase dump files substantially. Simply translating the |
238 |
< |
rotation matrix into its component Euler angles or quaternions for |
239 |
< |
storage purposes relieves this burden. |
264 |
> |
one time step to the next. The additional memory required by the |
265 |
> |
algorithm is inconsequential on modern computers, and translating the |
266 |
> |
rotation matrix into quaternions for storage purposes makes trajectory |
267 |
> |
data quite compact. |
268 |
|
|
269 |
|
The symplectic splitting method allows for Verlet style integration of |
270 |
< |
both linear and angular motion of rigid bodies. In the integration |
271 |
< |
method, the orientational propagation involves a sequence of matrix |
272 |
< |
evaluations to update the rotation matrix.\cite{Dullweber1997} These |
273 |
< |
matrix rotations end up being more costly computationally than the |
274 |
< |
simpler arithmetic quaternion propagation. On average, a 1000 SSD |
275 |
< |
particle simulation shows a 7\% increase in simulation time using the |
276 |
< |
symplectic step method in place of quaternions. This cost is more than |
277 |
< |
justified when comparing the energy conservation of the two methods as |
278 |
< |
illustrated in figure \ref{timestep}. |
270 |
> |
both translational and orientational motion of rigid bodies. In this |
271 |
> |
integration method, the orientational propagation involves a sequence |
272 |
> |
of matrix evaluations to update the rotation |
273 |
> |
matrix.\cite{Dullweber1997} These matrix rotations are more costly |
274 |
> |
than the simpler arithmetic quaternion propagation. With the same time |
275 |
> |
step, a 1000 SSD particle simulation shows an average 7\% increase in |
276 |
> |
computation time using the symplectic step method in place of |
277 |
> |
quaternions. The additional expense per step is justified when one |
278 |
> |
considers the ability to use time steps that are nearly twice as large |
279 |
> |
under symplectic splitting than would be usable under quaternion |
280 |
> |
dynamics. The energy conservation of the two methods using a number |
281 |
> |
of different time steps is illustrated in figure |
282 |
> |
\ref{timestep}. |
283 |
|
|
284 |
|
\begin{figure} |
285 |
< |
\includegraphics[width=61mm, angle=-90]{timeStep.epsi} |
286 |
< |
\caption{Energy conservation using quaternion based integration versus |
287 |
< |
the symplectic step method proposed by Dullweber \emph{et al.} with |
288 |
< |
increasing time step. For each time step, the dotted line is total |
289 |
< |
energy using the symplectic step integrator, and the solid line comes |
290 |
< |
from the quaternion integrator. The larger time step plots are shifted |
291 |
< |
up from the true energy baseline for clarity.} |
285 |
> |
\begin{center} |
286 |
> |
\epsfxsize=6in |
287 |
> |
\epsfbox{timeStep.epsi} |
288 |
> |
\caption{Energy conservation using both quaternion based integration and |
289 |
> |
the symplectic step method proposed by Dullweber \emph{et al.} with |
290 |
> |
increasing time step. The larger time step plots are shifted from the |
291 |
> |
true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.} |
292 |
|
\label{timestep} |
293 |
+ |
\end{center} |
294 |
|
\end{figure} |
295 |
|
|
296 |
|
In figure \ref{timestep}, the resulting energy drift at various time |
297 |
|
steps for both the symplectic step and quaternion integration schemes |
298 |
< |
is compared. All of the 1000 SSD particle simulations started with the |
299 |
< |
same configuration, and the only difference was the method for |
300 |
< |
handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
301 |
< |
methods for propagating particle rotation conserve energy fairly well, |
302 |
< |
with the quaternion method showing a slight energy drift over time in |
303 |
< |
the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the |
304 |
< |
energy conservation benefits of the symplectic step method are clearly |
305 |
< |
demonstrated. |
298 |
> |
is compared. All of the 1000 SSD particle simulations started with |
299 |
> |
the same configuration, and the only difference was the method used to |
300 |
> |
handle orientational motion. At time steps of 0.1 and 0.5 fs, both |
301 |
> |
methods for propagating the orientational degrees of freedom conserve |
302 |
> |
energy fairly well, with the quaternion method showing a slight energy |
303 |
> |
drift over time in the 0.5 fs time step simulation. At time steps of 1 |
304 |
> |
and 2 fs, the energy conservation benefits of the symplectic step |
305 |
> |
method are clearly demonstrated. Thus, while maintaining the same |
306 |
> |
degree of energy conservation, one can take considerably longer time |
307 |
> |
steps, leading to an overall reduction in computation time. |
308 |
|
|
309 |
< |
Energy drift in these SSD particle simulations was unnoticeable for |
310 |
< |
time steps up to three femtoseconds. A slight energy drift on the |
309 |
> |
Energy drift in the symplectic step simulations was unnoticeable for |
310 |
> |
time steps up to 3 fs. A slight energy drift on the |
311 |
|
order of 0.012 kcal/mol per nanosecond was observed at a time step of |
312 |
< |
four femtoseconds, and as expected, this drift increases dramatically |
313 |
< |
with increasing time step. To insure accuracy in the constant energy |
312 |
> |
4 fs, and as expected, this drift increases dramatically |
313 |
> |
with increasing time step. To insure accuracy in our microcanonical |
314 |
|
simulations, time steps were set at 2 fs and kept at this value for |
315 |
|
constant pressure simulations as well. |
316 |
|
|
317 |
< |
Ice crystals in both the $I_h$ and $I_c$ lattices were generated as |
318 |
< |
starting points for all the simulations. The $I_h$ crystals were |
319 |
< |
formed by first arranging the center of masses of the SSD particles |
320 |
< |
into a ``hexagonal'' ice lattice of 1024 particles. Because of the |
321 |
< |
crystal structure of $I_h$ ice, the simulation box assumed a |
322 |
< |
rectangular shape with a edge length ratio of approximately |
317 |
> |
Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices |
318 |
> |
were generated as starting points for all simulations. The $I_h$ |
319 |
> |
crystals were formed by first arranging the centers of mass of the SSD |
320 |
> |
particles into a ``hexagonal'' ice lattice of 1024 particles. Because |
321 |
> |
of the crystal structure of $I_h$ ice, the simulation box assumed an |
322 |
> |
orthorhombic shape with an edge length ratio of approximately |
323 |
|
1.00$\times$1.06$\times$1.23. The particles were then allowed to |
324 |
|
orient freely about fixed positions with angular momenta randomized at |
325 |
|
400 K for varying times. The rotational temperature was then scaled |
326 |
< |
down in stages to slowly cool the crystals down to 25 K. The particles |
327 |
< |
were then allowed translate with fixed orientations at a constant |
326 |
> |
down in stages to slowly cool the crystals to 25 K. The particles were |
327 |
> |
then allowed to translate with fixed orientations at a constant |
328 |
|
pressure of 1 atm for 50 ps at 25 K. Finally, all constraints were |
329 |
|
removed and the ice crystals were allowed to equilibrate for 50 ps at |
330 |
|
25 K and a constant pressure of 1 atm. This procedure resulted in |
331 |
|
structurally stable $I_h$ ice crystals that obey the Bernal-Fowler |
332 |
< |
rules\cite{Bernal33,Rahman72}. This method was also utilized in the |
332 |
> |
rules.\cite{Bernal33,Rahman72} This method was also utilized in the |
333 |
|
making of diamond lattice $I_c$ ice crystals, with each cubic |
334 |
|
simulation box consisting of either 512 or 1000 particles. Only |
335 |
|
isotropic volume fluctuations were performed under constant pressure, |
339 |
|
\section{Results and discussion} |
340 |
|
|
341 |
|
Melting studies were performed on the randomized ice crystals using |
342 |
< |
constant pressure and temperature dynamics. This involved an initial |
343 |
< |
randomization of velocities about the starting temperature of 25 K for |
344 |
< |
varying amounts of time. The systems were all equilibrated for 100 ps |
345 |
< |
prior to a 200 ps data collection run at each temperature setting, |
346 |
< |
ranging from 25 to 400 K, with a maximum degree increment of 25 K. For |
347 |
< |
regions of interest along this stepwise progression, the temperature |
348 |
< |
increment was decreased from 25 K to 10 and then 5 K. The above |
349 |
< |
equilibration and production times were sufficient in that the system |
350 |
< |
volume fluctuations dampened out in all but the very cold simulations |
351 |
< |
(below 225 K). In order to further improve statistics, five separate |
352 |
< |
simulation progressions were performed, and the averaged results from |
353 |
< |
the $I_h$ melting simulations are shown in figure \ref{dense1}. |
342 |
> |
isobaric-isothermal (NPT) dynamics. During melting simulations, the |
343 |
> |
melting transition and the density maximum can both be observed, |
344 |
> |
provided that the density maximum occurs in the liquid and not the |
345 |
> |
supercooled regime. An ensemble average from five separate melting |
346 |
> |
simulations was acquired, each starting from different ice crystals |
347 |
> |
generated as described previously. All simulations were equilibrated |
348 |
> |
for 100 ps prior to a 200 ps data collection run at each temperature |
349 |
> |
setting. The temperature range of study spanned from 25 to 400 K, with |
350 |
> |
a maximum degree increment of 25 K. For regions of interest along this |
351 |
> |
stepwise progression, the temperature increment was decreased from 25 |
352 |
> |
K to 10 and 5 K. The above equilibration and production times were |
353 |
> |
sufficient in that fluctuations in the volume autocorrelation function |
354 |
> |
were damped out in all simulations in under 20 ps. |
355 |
|
|
320 |
– |
\begin{figure} |
321 |
– |
\includegraphics[width=65mm, angle=-90]{1hdense.epsi} |
322 |
– |
\caption{Average density of SSD water at increasing temperatures |
323 |
– |
starting from ice $I_h$ lattice.} |
324 |
– |
\label{dense1} |
325 |
– |
\end{figure} |
326 |
– |
|
356 |
|
\subsection{Density Behavior} |
328 |
– |
In the initial average density versus temperature plot, the density |
329 |
– |
maximum clearly appears between 255 and 265 K. The calculated |
330 |
– |
densities within this range were nearly indistinguishable, as can be |
331 |
– |
seen in the zoom of this region of interest, shown in figure |
332 |
– |
\ref{dense1}. The greater certainty of the average value at 260 K makes |
333 |
– |
a good argument for the actual density maximum residing at this |
334 |
– |
midpoint value. Figure \ref{dense1} was constructed using ice $I_h$ |
335 |
– |
crystals for the initial configuration; and though not pictured, the |
336 |
– |
simulations starting from ice $I_c$ crystal configurations showed |
337 |
– |
similar results, with a liquid-phase density maximum in this same |
338 |
– |
region (between 255 and 260 K). In addition, the $I_c$ crystals are |
339 |
– |
more fragile than the $I_h$ crystals, leading them to deform into a |
340 |
– |
dense glassy state at lower temperatures. This resulted in an overall |
341 |
– |
low temperature density maximum at 200 K, but they still retained a |
342 |
– |
common liquid state density maximum with the $I_h$ simulations. |
357 |
|
|
358 |
+ |
Our initial simulations focused on the original SSD water model, and |
359 |
+ |
an average density versus temperature plot is shown in figure |
360 |
+ |
\ref{dense1}. Note that the density maximum when using a reaction |
361 |
+ |
field appears between 255 and 265 K. There were smaller fluctuations |
362 |
+ |
in the density at 260 K than at either 255 or 265, so we report this |
363 |
+ |
value as the location of the density maximum. Figure \ref{dense1} was |
364 |
+ |
constructed using ice $I_h$ crystals for the initial configuration; |
365 |
+ |
though not pictured, the simulations starting from ice $I_c$ crystal |
366 |
+ |
configurations showed similar results, with a liquid-phase density |
367 |
+ |
maximum in this same region (between 255 and 260 K). |
368 |
+ |
|
369 |
|
\begin{figure} |
370 |
< |
\includegraphics[width=65mm,angle=-90]{dense2.eps} |
371 |
< |
\caption{Density versus temperature for TIP4P\cite{Jorgensen98b}, |
372 |
< |
TIP3P\cite{Jorgensen98b}, SPC/E\cite{Clancy94}, SSD without Reaction |
373 |
< |
Field, SSD, and Experiment\cite{CRC80}. } |
374 |
< |
\label{dense2} |
370 |
> |
\begin{center} |
371 |
> |
\epsfxsize=6in |
372 |
> |
\epsfbox{denseSSD.eps} |
373 |
> |
\caption{Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}], |
374 |
> |
TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], SSD |
375 |
> |
without Reaction Field, SSD, and experiment [Ref. \citen{CRC80}]. The |
376 |
> |
arrows indicate the change in densities observed when turning off the |
377 |
> |
reaction field. The the lower than expected densities for the SSD |
378 |
> |
model were what prompted the original reparameterization of SSD1 |
379 |
> |
[Ref. \citen{Ichiye03}].} |
380 |
> |
\label{dense1} |
381 |
> |
\end{center} |
382 |
|
\end{figure} |
383 |
|
|
384 |
< |
The density maximum for SSD actually compares quite favorably to other |
385 |
< |
simple water models. Figure \ref{dense2} shows a plot of these |
386 |
< |
findings with the density progression of several other models and |
355 |
< |
experiment obtained from other |
384 |
> |
The density maximum for SSD compares quite favorably to other simple |
385 |
> |
water models. Figure \ref{dense1} also shows calculated densities of |
386 |
> |
several other models and experiment obtained from other |
387 |
|
sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water |
388 |
< |
models, SSD has results closest to the experimentally observed water |
389 |
< |
density maximum. Of the listed water models, TIP4P has a density |
390 |
< |
maximum behavior most like that seen in SSD. Though not shown, it is |
391 |
< |
useful to note that TIP5P has a water density maximum nearly identical |
392 |
< |
to experiment. |
388 |
> |
models, SSD has a temperature closest to the experimentally observed |
389 |
> |
density maximum. Of the {\it charge-based} models in |
390 |
> |
Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that |
391 |
> |
seen in SSD. Though not included in this plot, it is useful |
392 |
> |
to note that TIP5P has a density maximum nearly identical to the |
393 |
> |
experimentally measured temperature. |
394 |
|
|
395 |
< |
Possibly of more importance is the density scaling of SSD relative to |
396 |
< |
other common models at any given temperature (Fig. \ref{dense2}). Note |
397 |
< |
that the SSD model assumes a lower density than any of the other |
398 |
< |
listed models at the same pressure, behavior which is especially |
399 |
< |
apparent at temperatures greater than 300 K. Lower than expected |
400 |
< |
densities have been observed for other systems with the use of a |
401 |
< |
reaction field for long-range electrostatic interactions, so the most |
402 |
< |
likely reason for these significantly lower densities in these |
403 |
< |
simulations is the presence of the reaction field.\cite{Berendsen98} |
404 |
< |
In order to test the effect of the reaction field on the density of |
405 |
< |
the systems, the simulations were repeated for the temperature region |
406 |
< |
of interest without a reaction field present. The results of these |
407 |
< |
simulations are also displayed in figure \ref{dense2}. Without |
376 |
< |
reaction field, these densities increase considerably to more |
377 |
< |
experimentally reasonable values, especially around the freezing point |
378 |
< |
of liquid water. The shape of the curve is similar to the curve |
379 |
< |
produced from SSD simulations using reaction field, specifically the |
380 |
< |
rapidly decreasing densities at higher temperatures; however, a slight |
381 |
< |
shift in the density maximum location, down to 245 K, is |
382 |
< |
observed. This is probably a more accurate comparison to the other |
383 |
< |
listed water models in that no long range corrections were applied in |
384 |
< |
those simulations.\cite{Clancy94,Jorgensen98b} |
395 |
> |
It has been observed that liquid state densities in water are |
396 |
> |
dependent on the cutoff radius used both with and without the use of |
397 |
> |
reaction field.\cite{Berendsen98} In order to address the possible |
398 |
> |
effect of cutoff radius, simulations were performed with a dipolar |
399 |
> |
cutoff radius of 12.0 \AA\ to complement the previous SSD simulations, |
400 |
> |
all performed with a cutoff of 9.0 \AA. All of the resulting densities |
401 |
> |
overlapped within error and showed no significant trend toward lower |
402 |
> |
or higher densities as a function of cutoff radius, for simulations |
403 |
> |
both with and without reaction field. These results indicate that |
404 |
> |
there is no major benefit in choosing a longer cutoff radius in |
405 |
> |
simulations using SSD. This is advantageous in that the use of a |
406 |
> |
longer cutoff radius results in a significant increase in the time |
407 |
> |
required to obtain a single trajectory. |
408 |
|
|
409 |
< |
It has been observed that densities are dependent on the cutoff radius |
410 |
< |
used for a variety of water models in simulations both with and |
411 |
< |
without the use of reaction field.\cite{Berendsen98} In order to |
412 |
< |
address the possible affect of cutoff radius, simulations were |
413 |
< |
performed with a dipolar cutoff radius of 12.0 \AA\ to compliment the |
414 |
< |
previous SSD simulations, all performed with a cutoff of 9.0 \AA. All |
415 |
< |
the resulting densities overlapped within error and showed no |
416 |
< |
significant trend in lower or higher densities as a function of cutoff |
417 |
< |
radius, both for simulations with and without reaction field. These |
418 |
< |
results indicate that there is no major benefit in choosing a longer |
419 |
< |
cutoff radius in simulations using SSD. This is comforting in that the |
420 |
< |
use of a longer cutoff radius results in a near doubling of the time |
421 |
< |
required to compute a single trajectory. |
409 |
> |
The key feature to recognize in figure \ref{dense1} is the density |
410 |
> |
scaling of SSD relative to other common models at any given |
411 |
> |
temperature. SSD assumes a lower density than any of the other listed |
412 |
> |
models at the same pressure, behavior which is especially apparent at |
413 |
> |
temperatures greater than 300 K. Lower than expected densities have |
414 |
> |
been observed for other systems using a reaction field for long-range |
415 |
> |
electrostatic interactions, so the most likely reason for the |
416 |
> |
significantly lower densities seen in these simulations is the |
417 |
> |
presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order |
418 |
> |
to test the effect of the reaction field on the density of the |
419 |
> |
systems, the simulations were repeated without a reaction field |
420 |
> |
present. The results of these simulations are also displayed in figure |
421 |
> |
\ref{dense1}. Without the reaction field, the densities increase |
422 |
> |
to more experimentally reasonable values, especially around the |
423 |
> |
freezing point of liquid water. The shape of the curve is similar to |
424 |
> |
the curve produced from SSD simulations using reaction field, |
425 |
> |
specifically the rapidly decreasing densities at higher temperatures; |
426 |
> |
however, a shift in the density maximum location, down to 245 K, is |
427 |
> |
observed. This is a more accurate comparison to the other listed water |
428 |
> |
models, in that no long range corrections were applied in those |
429 |
> |
simulations.\cite{Clancy94,Jorgensen98b} However, even without the |
430 |
> |
reaction field, the density around 300 K is still significantly lower |
431 |
> |
than experiment and comparable water models. This anomalous behavior |
432 |
> |
was what lead Ichiye {\it et al.} to recently reparameterize |
433 |
> |
SSD.\cite{Ichiye03} Throughout the remainder of the paper our |
434 |
> |
reparamaterizations of SSD will be compared with the newer SSD1 model. |
435 |
|
|
436 |
|
\subsection{Transport Behavior} |
437 |
< |
Of importance in these types of studies are the transport properties |
438 |
< |
of the particles and how they change when altering the environmental |
439 |
< |
conditions. In order to probe transport, constant energy simulations |
440 |
< |
were performed about the average density uncovered by the constant |
441 |
< |
pressure simulations. Simulations started with randomized velocities |
442 |
< |
and underwent 50 ps of temperature scaling and 50 ps of constant |
443 |
< |
energy equilibration before obtaining a 200 ps trajectory. Diffusion |
444 |
< |
constants were calculated via root-mean square deviation analysis. The |
445 |
< |
averaged results from 5 sets of these NVE simulations is displayed in |
446 |
< |
figure \ref{diffuse}, alongside experimental, SPC/E, and TIP5P |
437 |
> |
|
438 |
> |
Accurate dynamical properties of a water model are particularly |
439 |
> |
important when using the model to study permeation or transport across |
440 |
> |
biological membranes. In order to probe transport in bulk water, |
441 |
> |
constant energy (NVE) simulations were performed at the average |
442 |
> |
density obtained by the NPT simulations at an identical target |
443 |
> |
temperature. Simulations started with randomized velocities and |
444 |
> |
underwent 50 ps of temperature scaling and 50 ps of constant energy |
445 |
> |
equilibration before a 200 ps data collection run. Diffusion constants |
446 |
> |
were calculated via linear fits to the long-time behavior of the |
447 |
> |
mean-square displacement as a function of time. The averaged results |
448 |
> |
from five sets of NVE simulations are displayed in figure |
449 |
> |
\ref{diffuse}, alongside experimental, SPC/E, and TIP5P |
450 |
|
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} |
451 |
|
|
452 |
|
\begin{figure} |
453 |
< |
\includegraphics[width=65mm, angle=-90]{betterDiffuse.epsi} |
454 |
< |
\caption{Average diffusion coefficient over increasing temperature for |
455 |
< |
SSD, SPC/E\cite{Clancy94}, TIP5P\cite{Jorgensen01}, and Experimental |
456 |
< |
data from Gillen \emph{et al.}\cite{Gillen72}, and from |
457 |
< |
Mills\cite{Mills73}.} |
453 |
> |
\begin{center} |
454 |
> |
\epsfxsize=6in |
455 |
> |
\epsfbox{betterDiffuse.epsi} |
456 |
> |
\caption{Average self-diffusion constant as a function of temperature for |
457 |
> |
SSD, SPC/E [Ref. \citen{Clancy94}], TIP5P [Ref. \citen{Jorgensen01}], |
458 |
> |
and Experimental data [Refs. \citen{Gillen72} and \citen{Mills73}]. Of |
459 |
> |
the three water models shown, SSD has the least deviation from the |
460 |
> |
experimental values. The rapidly increasing diffusion constants for |
461 |
> |
TIP5P and SSD correspond to significant decrease in density at the |
462 |
> |
higher temperatures.} |
463 |
|
\label{diffuse} |
464 |
+ |
\end{center} |
465 |
|
\end{figure} |
466 |
|
|
467 |
|
The observed values for the diffusion constant point out one of the |
468 |
< |
strengths of the SSD model. Of the three experimental models shown, |
469 |
< |
the SSD model has the most accurate depiction of the diffusion trend |
470 |
< |
seen in experiment in both the supercooled and normal regimes. SPC/E |
471 |
< |
does a respectable job by getting similar values as SSD and experiment |
472 |
< |
around 290 K; however, it deviates at both higher and lower |
473 |
< |
temperatures, failing to predict the experimental trend. TIP5P and SSD |
474 |
< |
both start off low at the colder temperatures and tend to diffuse too |
475 |
< |
rapidly at the higher temperatures. This type of trend at the higher |
476 |
< |
temperatures is not surprising in that the densities of both TIP5P and |
477 |
< |
SSD are lower than experimental water at temperatures higher than room |
478 |
< |
temperature. When calculating the diffusion coefficients for SSD at |
479 |
< |
experimental densities, the resulting values fall more in line with |
480 |
< |
experiment at these temperatures, albeit not at standard |
436 |
< |
pressure. Results under these conditions can be found later in this |
437 |
< |
paper. |
468 |
> |
strengths of the SSD model. Of the three models shown, the SSD model |
469 |
> |
has the most accurate depiction of self-diffusion in both the |
470 |
> |
supercooled and liquid regimes. SPC/E does a respectable job by |
471 |
> |
reproducing values similar to experiment around 290 K; however, it |
472 |
> |
deviates at both higher and lower temperatures, failing to predict the |
473 |
> |
correct thermal trend. TIP5P and SSD both start off low at colder |
474 |
> |
temperatures and tend to diffuse too rapidly at higher temperatures. |
475 |
> |
This behavior at higher temperatures is not particularly surprising |
476 |
> |
since the densities of both TIP5P and SSD are lower than experimental |
477 |
> |
water densities at higher temperatures. When calculating the |
478 |
> |
diffusion coefficients for SSD at experimental densities (instead of |
479 |
> |
the densities from the NPT simulations), the resulting values fall |
480 |
> |
more in line with experiment at these temperatures. |
481 |
|
|
482 |
|
\subsection{Structural Changes and Characterization} |
483 |
+ |
|
484 |
|
By starting the simulations from the crystalline state, the melting |
485 |
< |
transition and the ice structure can be studied along with the liquid |
486 |
< |
phase behavior beyond the melting point. To locate the melting |
487 |
< |
transition, the constant pressure heat capacity (C$_\text{p}$) was |
488 |
< |
monitored in each of the simulations. In the melting simulations of |
489 |
< |
the 1024 particle ice $I_h$ simulations, a large spike in C$_\text{p}$ |
490 |
< |
occurs at 245 K, indicating a first order phase transition for the |
491 |
< |
melting of these ice crystals. When the reaction field is turned off, |
492 |
< |
the melting transition occurs at 235 K. These melting transitions are |
493 |
< |
considerably lower than the experimental value, but this is not |
450 |
< |
surprising in that SSD is a simple rigid body model with a fixed |
451 |
< |
dipole. |
485 |
> |
transition and the ice structure can be obtained along with the liquid |
486 |
> |
phase behavior beyond the melting point. The constant pressure heat |
487 |
> |
capacity (C$_\text{p}$) was monitored to locate the melting transition |
488 |
> |
in each of the simulations. In the melting simulations of the 1024 |
489 |
> |
particle ice $I_h$ simulations, a large spike in C$_\text{p}$ occurs |
490 |
> |
at 245 K, indicating a first order phase transition for the melting of |
491 |
> |
these ice crystals. When the reaction field is turned off, the melting |
492 |
> |
transition occurs at 235 K. These melting transitions are |
493 |
> |
considerably lower than the experimental value. |
494 |
|
|
495 |
< |
\begin{figure} |
496 |
< |
\includegraphics[width=85mm]{fullContours.eps} |
495 |
> |
\begin{figure} |
496 |
> |
\begin{center} |
497 |
> |
\epsfxsize=6in |
498 |
> |
\epsfbox{corrDiag.eps} |
499 |
> |
\caption{Two dimensional illustration of angles involved in the |
500 |
> |
correlations observed in Fig. \ref{contour}.} |
501 |
> |
\label{corrAngle} |
502 |
> |
\end{center} |
503 |
> |
\end{figure} |
504 |
> |
|
505 |
> |
\begin{figure} |
506 |
> |
\begin{center} |
507 |
> |
\epsfxsize=6in |
508 |
> |
\epsfbox{fullContours.eps} |
509 |
|
\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at |
510 |
|
100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for |
511 |
|
clarity: dark areas signify peaks while light areas signify |
512 |
< |
depressions. White areas have g(\emph{r}) values below 0.5 and black |
512 |
> |
depressions. White areas have $g(r)$ values below 0.5 and black |
513 |
|
areas have values above 1.5.} |
514 |
|
\label{contour} |
515 |
+ |
\end{center} |
516 |
|
\end{figure} |
517 |
|
|
518 |
< |
Additional analyses for understanding the melting phase-transition |
519 |
< |
process were performed via two-dimensional structure and dipole angle |
520 |
< |
correlations. Expressions for the correlations are as follows: |
518 |
> |
Additional analysis of the melting process was performed using |
519 |
> |
two-dimensional structure and dipole angle correlations. Expressions |
520 |
> |
for these correlations are as follows: |
521 |
|
|
522 |
< |
\begin{figure} |
523 |
< |
\includegraphics[width=45mm]{corrDiag.eps} |
524 |
< |
\caption{Two dimensional illustration of the angles involved in the |
525 |
< |
correlations observed in figure \ref{contour}.} |
526 |
< |
\label{corrAngle} |
527 |
< |
\end{figure} |
528 |
< |
|
529 |
< |
\begin{multline} |
530 |
< |
g_{\text{AB}}(r,\cos\theta) = \\ |
531 |
< |
\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|\mathbf{r}_{ij}\right|)\rangle\ , |
532 |
< |
\end{multline} |
533 |
< |
\begin{multline} |
534 |
< |
g_{\text{AB}}(r,\cos\omega) = \\ |
535 |
< |
\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|\mathbf{r}_{ij}\right|)\rangle\ , |
536 |
< |
\end{multline} |
537 |
< |
where $\theta$ and $\omega$ refer to the angles shown in the above |
538 |
< |
illustration. By binning over both distance and the cosine of the |
539 |
< |
desired angle between the two dipoles, the g(\emph{r}) can be |
485 |
< |
dissected to determine the common dipole arrangements that constitute |
486 |
< |
the peaks and troughs. Frames A and B of figure \ref{contour} show a |
487 |
< |
relatively crystalline state of an ice $I_c$ simulation. The first |
488 |
< |
peak of the g(\emph{r}) primarily consists of the preferred hydrogen |
489 |
< |
bonding arrangements as dictated by the tetrahedral sticky potential, |
490 |
< |
one peak for the donating and the other for the accepting hydrogen |
491 |
< |
bonds. Due to the high degree of crystallinity of the sample, the |
492 |
< |
second and third solvation shells show a repeated peak arrangement |
522 |
> |
\begin{equation} |
523 |
> |
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\ , |
524 |
> |
\end{equation} |
525 |
> |
\begin{equation} |
526 |
> |
g_{\text{AB}}(r,\cos\omega) = |
527 |
> |
\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\ , |
528 |
> |
\end{equation} |
529 |
> |
where $\theta$ and $\omega$ refer to the angles shown in figure |
530 |
> |
\ref{corrAngle}. By binning over both distance and the cosine of the |
531 |
> |
desired angle between the two dipoles, the $g(r)$ can be analyzed to |
532 |
> |
determine the common dipole arrangements that constitute the peaks and |
533 |
> |
troughs in the standard one-dimensional $g(r)$ plots. Frames A and B |
534 |
> |
of figure \ref{contour} show results from an ice $I_c$ simulation. The |
535 |
> |
first peak in the $g(r)$ consists primarily of the preferred hydrogen |
536 |
> |
bonding arrangements as dictated by the tetrahedral sticky potential - |
537 |
> |
one peak for the hydrogen bond donor and the other for the hydrogen |
538 |
> |
bond acceptor. Due to the high degree of crystallinity of the sample, |
539 |
> |
the second and third solvation shells show a repeated peak arrangement |
540 |
|
which decays at distances around the fourth solvation shell, near the |
541 |
|
imposed cutoff for the Lennard-Jones and dipole-dipole interactions. |
542 |
< |
In the higher temperature simulation shown in frames C and D, the |
543 |
< |
repeated peak features are significantly blurred. The first solvation |
544 |
< |
shell still shows the strong effect of the sticky-potential, although |
545 |
< |
it covers a larger area, extending to include a fraction of aligned |
542 |
> |
In the higher temperature simulation shown in frames C and D, these |
543 |
> |
long-range features deteriorate rapidly. The first solvation shell |
544 |
> |
still shows the strong effect of the sticky-potential, although it |
545 |
> |
covers a larger area, extending to include a fraction of aligned |
546 |
|
dipole peaks within the first solvation shell. The latter peaks lose |
547 |
< |
definition as thermal motion and the competing dipole force overcomes |
548 |
< |
the sticky potential's tight tetrahedral structuring of the fluid. |
547 |
> |
due to thermal motion and as the competing dipole force overcomes the |
548 |
> |
sticky potential's tight tetrahedral structuring of the crystal. |
549 |
|
|
550 |
|
This complex interplay between dipole and sticky interactions was |
551 |
|
remarked upon as a possible reason for the split second peak in the |
552 |
< |
oxygen-oxygen g(\emph{r}).\cite{Ichiye96} At low temperatures, the |
553 |
< |
second solvation shell peak appears to have two distinct parts that |
554 |
< |
blend together to form one observable peak. At higher temperatures, |
555 |
< |
this split character alters to show the leading 4 \AA\ peak dominated |
556 |
< |
by equatorial anti-parallel dipole orientations, and there is tightly |
557 |
< |
bunched group of axially arranged dipoles that most likely consist of |
558 |
< |
the smaller fraction aligned dipole pairs. The trailing part of the |
559 |
< |
split peak at 5 \AA\ is dominated by aligned dipoles that range |
560 |
< |
primarily within the axial to the chief hydrogen bond arrangements |
561 |
< |
similar to those seen in the first solvation shell. This evidence |
562 |
< |
indicates that the dipole pair interaction begins to dominate outside |
563 |
< |
of the range of the dipolar repulsion term, with the primary |
564 |
< |
energetically favorable dipole arrangements populating the region |
565 |
< |
immediately outside of it's range (around 4 \AA), and arrangements |
566 |
< |
that seek to ideally satisfy both the sticky and dipole forces locate |
567 |
< |
themselves just beyond this region (around 5 \AA). |
552 |
> |
oxygen-oxygen $g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, |
553 |
> |
the second solvation shell peak appears to have two distinct |
554 |
> |
components that blend together to form one observable peak. At higher |
555 |
> |
temperatures, this split character alters to show the leading 4 \AA\ |
556 |
> |
peak dominated by equatorial anti-parallel dipole orientations. There |
557 |
> |
is also a tightly bunched group of axially arranged dipoles that most |
558 |
> |
likely consist of the smaller fraction of aligned dipole pairs. The |
559 |
> |
trailing component of the split peak at 5 \AA\ is dominated by aligned |
560 |
> |
dipoles that assume hydrogen bond arrangements similar to those seen |
561 |
> |
in the first solvation shell. This evidence indicates that the dipole |
562 |
> |
pair interaction begins to dominate outside of the range of the |
563 |
> |
dipolar repulsion term. The energetically favorable dipole |
564 |
> |
arrangements populate the region immediately outside this repulsion |
565 |
> |
region (around 4 \AA), while arrangements that seek to satisfy both |
566 |
> |
the sticky and dipole forces locate themselves just beyond this |
567 |
> |
initial buildup (around 5 \AA). |
568 |
|
|
569 |
|
From these findings, the split second peak is primarily the product of |
570 |
< |
the dipolar repulsion term of the sticky potential. In fact, the |
571 |
< |
leading of the two peaks can be pushed out and merged with the outer |
572 |
< |
split peak just by extending the switching function cutoff |
573 |
< |
($s^\prime(r_{ij})$) from its normal 4.0 \AA\ to values of 4.5 or even |
574 |
< |
5 \AA. This type of correction is not recommended for improving the |
575 |
< |
liquid structure, because the second solvation shell will still be |
576 |
< |
shifted too far out. In addition, this would have an even more |
577 |
< |
detrimental effect on the system densities, leading to a liquid with a |
578 |
< |
more open structure and a density considerably lower than the normal |
579 |
< |
SSD behavior shown previously. A better correction would be to include |
580 |
< |
the quadrupole-quadrupole interactions for the water particles outside |
581 |
< |
of the first solvation shell, but this reduces the simplicity and |
582 |
< |
speed advantage of SSD, so it is not the most desirable path to take. |
570 |
> |
the dipolar repulsion term of the sticky potential. In fact, the inner |
571 |
> |
peak can be pushed out and merged with the outer split peak just by |
572 |
> |
extending the switching function ($s^\prime(r_{ij})$) from its normal |
573 |
> |
4.0 \AA\ cutoff to values of 4.5 or even 5 \AA. This type of |
574 |
> |
correction is not recommended for improving the liquid structure, |
575 |
> |
since the second solvation shell would still be shifted too far |
576 |
> |
out. In addition, this would have an even more detrimental effect on |
577 |
> |
the system densities, leading to a liquid with a more open structure |
578 |
> |
and a density considerably lower than the already low SSD density. A |
579 |
> |
better correction would be to include the quadrupole-quadrupole |
580 |
> |
interactions for the water particles outside of the first solvation |
581 |
> |
shell, but this would remove the simplicity and speed advantage of |
582 |
> |
SSD. |
583 |
|
|
584 |
< |
\subsection{Adjusted Potentials: SSD/E and SSD/RF} |
584 |
> |
\subsection{Adjusted Potentials: SSD/RF and SSD/E} |
585 |
> |
|
586 |
|
The propensity of SSD to adopt lower than expected densities under |
587 |
|
varying conditions is troubling, especially at higher temperatures. In |
588 |
< |
order to correct this behavior, it's necessary to adjust the force |
589 |
< |
field parameters for the primary intermolecular interactions. In |
590 |
< |
undergoing a reparameterization, it is important not to focus on just |
591 |
< |
one property and neglect the other important properties. In this case, |
592 |
< |
it would be ideal to correct the densities while maintaining the |
593 |
< |
accurate transport properties. |
588 |
> |
order to correct this model for use with a reaction field, it is |
589 |
> |
necessary to adjust the force field parameters for the primary |
590 |
> |
intermolecular interactions. In undergoing a reparameterization, it is |
591 |
> |
important not to focus on just one property and neglect the other |
592 |
> |
important properties. In this case, it would be ideal to correct the |
593 |
> |
densities while maintaining the accurate transport behavior. |
594 |
|
|
595 |
< |
The possible parameters for tuning include the $\sigma$ and $\epsilon$ |
596 |
< |
Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky |
597 |
< |
attractive and dipole repulsive terms with their respective |
598 |
< |
cutoffs. To alter the attractive and repulsive terms of the sticky |
599 |
< |
potential independently, it is necessary to separate the terms as |
600 |
< |
follows: |
601 |
< |
\begin{equation} |
602 |
< |
\begin{split} |
603 |
< |
u_{ij}^{sp} |
604 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) &= |
557 |
< |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\\ |
558 |
< |
& \quad \ + \frac{\nu_0^\prime}{2} |
559 |
< |
[s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)], |
560 |
< |
\end{split} |
561 |
< |
\end{equation} |
595 |
> |
The parameters available for tuning include the $\sigma$ and |
596 |
> |
$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the |
597 |
> |
strength of the sticky potential ($\nu_0$), and the sticky attractive |
598 |
> |
and dipole repulsive cubic switching function cutoffs ($r_l$, $r_u$ |
599 |
> |
and $r_l^\prime$, $r_u^\prime$ respectively). The results of the |
600 |
> |
reparameterizations are shown in table \ref{params}. We are calling |
601 |
> |
these reparameterizations the Soft Sticky Dipole / Reaction Field |
602 |
> |
(SSD/RF - for use with a reaction field) and Soft Sticky Dipole |
603 |
> |
Extended (SSD/E - an attempt to improve the liquid structure in |
604 |
> |
simulations without a long-range correction). |
605 |
|
|
563 |
– |
where $\nu_0$ scales the strength of the tetrahedral attraction and |
564 |
– |
$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion |
565 |
– |
term. For purposes of the reparameterization, the separation was |
566 |
– |
performed, but the final parameters were adjusted so that it is |
567 |
– |
unnecessary to separate the terms when implementing the adjusted water |
568 |
– |
potentials. The results of the reparameterizations are shown in table |
569 |
– |
\ref{params}. Note that both the tetrahedral attractive and dipolar |
570 |
– |
repulsive don't share the same lower cutoff ($r_l$) in the newly |
571 |
– |
parameterized potentials - soft sticky dipole enhanced (SSD/E) and |
572 |
– |
soft sticky dipole reaction field (SSD/RF). |
573 |
– |
|
606 |
|
\begin{table} |
607 |
+ |
\begin{center} |
608 |
|
\caption{Parameters for the original and adjusted models} |
609 |
< |
\begin{tabular}{ l c c c } |
609 |
> |
\begin{tabular}{ l c c c c } |
610 |
|
\hline \\[-3mm] |
611 |
< |
\ Parameters & \ \ \ SSD$^\dagger$\ \ \ \ & \ SSD/E\ \ & \ SSD/RF\ \ \\ |
611 |
> |
\ \ \ Parameters\ \ \ & \ \ \ SSD [Ref. \citen{Ichiye96}] \ \ \ |
612 |
> |
& \ SSD1 [Ref. \citen{Ichiye03}]\ \ & \ SSD/E\ \ & \ SSD/RF \\ |
613 |
|
\hline \\[-3mm] |
614 |
< |
\ \ \ $\sigma$ (\AA) & 3.051 & 3.035 & 3.019\\ |
615 |
< |
\ \ \ $\epsilon$ (kcal/mol)\ \ & 0.152 & 0.152 & 0.152\\ |
616 |
< |
\ \ \ $\mu$ (D) & 2.35 & 2.418 & 2.480\\ |
617 |
< |
\ \ \ $\nu_0$ (kcal/mol)\ \ & 3.7284 & 3.90 & 3.90\\ |
618 |
< |
\ \ \ $r_l$ (\AA) & 2.75 & 2.40 & 2.40\\ |
619 |
< |
\ \ \ $r_u$ (\AA) & 3.35 & 3.80 & 3.80\\ |
620 |
< |
\ \ \ $\nu_0^\prime$ (kcal/mol)\ \ & 3.7284 & 3.90 & 3.90\\ |
621 |
< |
\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75\\ |
622 |
< |
\ \ \ $r_u^\prime$ (\AA) & 4.00 & 3.35 & 3.35\\ |
589 |
< |
\\[-2mm]$^\dagger$ ref. \onlinecite{Ichiye96} |
614 |
> |
\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
615 |
> |
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
616 |
> |
\ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\ |
617 |
> |
\ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
618 |
> |
\ \ \ $\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\ |
619 |
> |
\ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\ |
620 |
> |
\ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\ |
621 |
> |
\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\ |
622 |
> |
\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\ |
623 |
|
\end{tabular} |
624 |
|
\label{params} |
625 |
+ |
\end{center} |
626 |
|
\end{table} |
627 |
|
|
628 |
< |
\begin{figure} |
629 |
< |
\includegraphics[width=85mm]{gofrCompare.epsi} |
630 |
< |
\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E |
631 |
< |
and SSD without reaction field (top), as well as SSD/RF and SSD with |
628 |
> |
\begin{figure} |
629 |
> |
\begin{center} |
630 |
> |
\epsfxsize=5in |
631 |
> |
\epsfbox{GofRCompare.epsi} |
632 |
> |
\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with SSD/E |
633 |
> |
and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with |
634 |
|
reaction field turned on (bottom). The insets show the respective |
635 |
< |
first peaks in detail. Solid Line - experiment, dashed line - SSD/E |
636 |
< |
and SSD/RF, and dotted line - SSD (with and without reaction field).} |
635 |
> |
first peaks in detail. Note how the changes in parameters have lowered |
636 |
> |
and broadened the first peak of SSD/E and SSD/RF.} |
637 |
|
\label{grcompare} |
638 |
+ |
\end{center} |
639 |
|
\end{figure} |
640 |
|
|
641 |
< |
\begin{figure} |
642 |
< |
\includegraphics[width=85mm]{dualsticky.ps} |
643 |
< |
\caption{Isosurfaces of the sticky potential for SSD (left) and SSD/E \& |
641 |
> |
\begin{figure} |
642 |
> |
\begin{center} |
643 |
> |
\epsfxsize=6in |
644 |
> |
\epsfbox{dualsticky.ps} |
645 |
> |
\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \& |
646 |
|
SSD/RF (right). Light areas correspond to the tetrahedral attractive |
647 |
< |
part, and the darker areas correspond to the dipolar repulsive part.} |
647 |
> |
component, and darker areas correspond to the dipolar repulsive |
648 |
> |
component.} |
649 |
|
\label{isosurface} |
650 |
+ |
\end{center} |
651 |
|
\end{figure} |
652 |
|
|
653 |
< |
In the paper detailing the development of SSD, Liu and Ichiye placed |
654 |
< |
particular emphasis on an accurate description of the first solvation |
655 |
< |
shell. This resulted in a somewhat tall and sharp first peak that |
656 |
< |
integrated to give similar coordination numbers to the experimental |
657 |
< |
data obtained by Soper and Phillips.\cite{Ichiye96,Soper86} New |
658 |
< |
experimental x-ray scattering data from the Head-Gordon lab indicates |
659 |
< |
a slightly lower and shifted first peak in the g$_\mathrm{OO}(r)$, so |
660 |
< |
adjustments to SSD were made while taking into consideration the new |
661 |
< |
experimental findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} |
662 |
< |
shows the relocation of the first peak of the oxygen-oxygen |
663 |
< |
g(\emph{r}) by comparing the original SSD (with and without reaction |
664 |
< |
field), SSD-E, and SSD-RF to the new experimental results. Both the |
665 |
< |
modified water models have shorter peaks that are brought in more |
666 |
< |
closely to the experimental peak (as seen in the insets of figure |
667 |
< |
\ref{grcompare}). This structural alteration was accomplished by a |
668 |
< |
reduction in the Lennard-Jones $\sigma$ variable as well as adjustment |
669 |
< |
of the sticky potential strength and cutoffs. The cutoffs for the |
670 |
< |
tetrahedral attractive and dipolar repulsive terms were nearly swapped |
671 |
< |
with each other. Isosurfaces of the original and modified sticky |
672 |
< |
potentials are shown in figure \cite{isosurface}. In these |
673 |
< |
isosurfaces, it is easy to see how altering the cutoffs changes the |
674 |
< |
repulsive and attractive character of the particles. With a reduced |
675 |
< |
repulsive surface (the darker region), the particles can move closer |
676 |
< |
to one another, increasing the density for the overall system. This |
677 |
< |
change in interaction cutoff also results in a more gradual |
678 |
< |
orientational motion by allowing the particles to maintain preferred |
679 |
< |
dipolar arrangements before they begin to feel the pull of the |
680 |
< |
tetrahedral restructuring. Upon moving closer together, the dipolar |
681 |
< |
repulsion term becomes active and excludes the unphysical |
682 |
< |
arrangements. This compares with the original SSD's excluding dipolar |
683 |
< |
before the particles feel the pull of the ``hydrogen bonds''. Aside |
684 |
< |
from improving the shape of the first peak in the g(\emph{r}), this |
685 |
< |
improves the densities considerably by allowing the persistence of |
686 |
< |
full dipolar character below the previous 4.0 \AA\ cutoff. |
653 |
> |
In the original paper detailing the development of SSD, Liu and Ichiye |
654 |
> |
placed particular emphasis on an accurate description of the first |
655 |
> |
solvation shell. This resulted in a somewhat tall and narrow first |
656 |
> |
peak in $g(r)$ that integrated to give similar coordination numbers to |
657 |
> |
the experimental data obtained by Soper and |
658 |
> |
Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering |
659 |
> |
data from the Head-Gordon lab indicates a slightly lower and shifted |
660 |
> |
first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were |
661 |
> |
made while taking into consideration the new experimental |
662 |
> |
findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the |
663 |
> |
relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing |
664 |
> |
the revised SSD model (SSD1), SSD/E, and SSD/RF to the new |
665 |
> |
experimental results. Both modified water models have shorter peaks |
666 |
> |
that match more closely to the experimental peak (as seen in the |
667 |
> |
insets of figure \ref{grcompare}). This structural alteration was |
668 |
> |
accomplished by the combined reduction in the Lennard-Jones $\sigma$ |
669 |
> |
variable and adjustment of the sticky potential strength and cutoffs. |
670 |
> |
As can be seen in table \ref{params}, the cutoffs for the tetrahedral |
671 |
> |
attractive and dipolar repulsive terms were nearly swapped with each |
672 |
> |
other. Isosurfaces of the original and modified sticky potentials are |
673 |
> |
shown in figure \ref{isosurface}. In these isosurfaces, it is easy to |
674 |
> |
see how altering the cutoffs changes the repulsive and attractive |
675 |
> |
character of the particles. With a reduced repulsive surface (darker |
676 |
> |
region), the particles can move closer to one another, increasing the |
677 |
> |
density for the overall system. This change in interaction cutoff also |
678 |
> |
results in a more gradual orientational motion by allowing the |
679 |
> |
particles to maintain preferred dipolar arrangements before they begin |
680 |
> |
to feel the pull of the tetrahedral restructuring. As the particles |
681 |
> |
move closer together, the dipolar repulsion term becomes active and |
682 |
> |
excludes unphysical nearest-neighbor arrangements. This compares with |
683 |
> |
how SSD and SSD1 exclude preferred dipole alignments before the |
684 |
> |
particles feel the pull of the ``hydrogen bonds''. Aside from |
685 |
> |
improving the shape of the first peak in the g(\emph{r}), this |
686 |
> |
modification improves the densities considerably by allowing the |
687 |
> |
persistence of full dipolar character below the previous 4.0 \AA\ |
688 |
> |
cutoff. |
689 |
|
|
690 |
< |
While adjusting the location and shape of the first peak of |
691 |
< |
g(\emph{r}) improves the densities to some degree, these changes alone |
692 |
< |
are insufficient to bring the system densities up to the values |
693 |
< |
observed experimentally. To finish bringing up the densities, the |
694 |
< |
dipole moments were increased in both the adjusted models. Being a |
695 |
< |
dipole based model, the structure and transport are very sensitive to |
696 |
< |
changes in the dipole moment. The original SSD simply used the dipole |
697 |
< |
moment calculated from the TIP3P water model, which at 2.35 D is |
698 |
< |
significantly greater than the experimental gas phase value of 1.84 |
699 |
< |
D. The larger dipole moment is a more realistic value and improve the |
700 |
< |
dielectric properties of the fluid. Both theoretical and experimental |
701 |
< |
measurements indicate a liquid phase dipole moment ranging from 2.4 D |
702 |
< |
to values as high as 3.11 D, so there is quite a range available for |
703 |
< |
adjusting the dipole |
704 |
< |
moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} The increasing of |
705 |
< |
the dipole moments to 2.418 and 2.48 D for SSD/E and SSD/RF |
706 |
< |
respectively is moderate in the range of the experimental values; |
664 |
< |
however, it leads to significant changes in the density and transport |
665 |
< |
of the water models. |
690 |
> |
While adjusting the location and shape of the first peak of $g(r)$ |
691 |
> |
improves the densities, these changes alone are insufficient to bring |
692 |
> |
the system densities up to the values observed experimentally. To |
693 |
> |
further increase the densities, the dipole moments were increased in |
694 |
> |
both of our adjusted models. Since SSD is a dipole based model, the |
695 |
> |
structure and transport are very sensitive to changes in the dipole |
696 |
> |
moment. The original SSD simply used the dipole moment calculated from |
697 |
> |
the TIP3P water model, which at 2.35 D is significantly greater than |
698 |
> |
the experimental gas phase value of 1.84 D. The larger dipole moment |
699 |
> |
is a more realistic value and improves the dielectric properties of |
700 |
> |
the fluid. Both theoretical and experimental measurements indicate a |
701 |
> |
liquid phase dipole moment ranging from 2.4 D to values as high as |
702 |
> |
3.11 D, providing a substantial range of reasonable values for a |
703 |
> |
dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
704 |
> |
increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF, |
705 |
> |
respectively, leads to significant changes in the density and |
706 |
> |
transport of the water models. |
707 |
|
|
708 |
< |
In order to demonstrate the benefits of this reparameterization, a |
708 |
> |
In order to demonstrate the benefits of these reparameterizations, a |
709 |
|
series of NPT and NVE simulations were performed to probe the density |
710 |
|
and transport properties of the adapted models and compare the results |
711 |
|
to the original SSD model. This comparison involved full NPT melting |
712 |
|
sequences for both SSD/E and SSD/RF, as well as NVE transport |
713 |
< |
calculations at both self-consistent and experimental |
714 |
< |
densities. Again, the results come from five separate simulations of |
715 |
< |
1024 particle systems, and the melting sequences were started from |
716 |
< |
different ice $I_h$ crystals constructed as stated previously. Like |
717 |
< |
before, all of the NPT simulations were equilibrated for 100 ps before |
718 |
< |
a 200 ps data collection run, and they used the previous temperature's |
719 |
< |
final configuration as a starting point. All of the NVE simulations |
720 |
< |
had the same thermalization, equilibration, and data collection times |
721 |
< |
stated earlier in this paper. |
713 |
> |
calculations at the calculated self-consistent densities. Again, the |
714 |
> |
results are obtained from five separate simulations of 1024 particle |
715 |
> |
systems, and the melting sequences were started from different ice |
716 |
> |
$I_h$ crystals constructed as described previously. Each NPT |
717 |
> |
simulation was equilibrated for 100 ps before a 200 ps data collection |
718 |
> |
run at each temperature step, and the final configuration from the |
719 |
> |
previous temperature simulation was used as a starting point. All NVE |
720 |
> |
simulations had the same thermalization, equilibration, and data |
721 |
> |
collection times as stated previously. |
722 |
|
|
723 |
< |
\begin{figure} |
724 |
< |
\includegraphics[width=85mm]{ssdecompare.epsi} |
725 |
< |
\caption{Comparison of densities calculated with SSD/E to SSD without a |
726 |
< |
reaction field, TIP4P\cite{Jorgensen98b}, TIP3P\cite{Jorgensen98b}, |
727 |
< |
SPC/E\cite{Clancy94}, and Experiment\cite{CRC80}. The upper plot |
728 |
< |
includes error bars, and the calculated results from the other |
729 |
< |
references were removed for clarity.} |
723 |
> |
\begin{figure} |
724 |
> |
\begin{center} |
725 |
> |
\epsfxsize=6in |
726 |
> |
\epsfbox{ssdeDense.epsi} |
727 |
> |
\caption{Comparison of densities calculated with SSD/E to SSD1 without a |
728 |
> |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
729 |
> |
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and |
730 |
> |
experiment [Ref. \citen{CRC80}]. The window shows a expansion around |
731 |
> |
300 K with error bars included to clarify this region of |
732 |
> |
interest. Note that both SSD1 and SSD/E show good agreement with |
733 |
> |
experiment when the long-range correction is neglected.} |
734 |
|
\label{ssdedense} |
735 |
+ |
\end{center} |
736 |
|
\end{figure} |
737 |
|
|
738 |
< |
Figure \ref{ssdedense} shows the density profile for the SSD/E water |
739 |
< |
model in comparison to the original SSD without a reaction field, |
740 |
< |
experiment, and the other common water models considered |
741 |
< |
previously. The calculated densities have increased significantly over |
742 |
< |
the original SSD model and match the experimental value just below 298 |
743 |
< |
K. At 298 K, the density of SSD/E is 0.995$\pm$0.001 g/cm$^3$, which |
744 |
< |
compares well with the experimental value of 0.997 g/cm$^3$ and is |
745 |
< |
considerably better than the SSD value of 0.967$\pm$0.003 |
746 |
< |
g/cm$^3$. The increased dipole moment in SSD/E has helped to flatten |
747 |
< |
out the curve at higher temperatures, only the improvement is marginal |
748 |
< |
at best. This steep drop in densities is due to the dipolar rather |
749 |
< |
than charge based interactions which decay more rapidly at longer |
750 |
< |
distances. |
751 |
< |
|
752 |
< |
By monitoring C$\text{p}$ throughout these simulations, the melting |
753 |
< |
transition for SSD/E was observed at 230 K, about 5 degrees lower than |
754 |
< |
SSD. The resulting density maximum is located at 240 K, again about 5 |
755 |
< |
degrees lower than the SSD value of 245 K. Though there is a decrease |
756 |
< |
in both of these values, the corrected densities near room temperature |
757 |
< |
justify the modifications taken. |
738 |
> |
Fig. \ref{ssdedense} shows the density profile for the SSD/E model |
739 |
> |
in comparison to SSD1 without a reaction field, other common water |
740 |
> |
models, and experimental results. The calculated densities for both |
741 |
> |
SSD/E and SSD1 have increased significantly over the original SSD |
742 |
> |
model (see fig. \ref{dense1}) and are in better agreement with the |
743 |
> |
experimental values. At 298 K, the densities of SSD/E and SSD1 without |
744 |
> |
a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and |
745 |
> |
0.999$\pm$0.001 g/cm$^3$ respectively. These both compare well with |
746 |
> |
the experimental value of 0.997 g/cm$^3$, and they are considerably |
747 |
> |
better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The changes to |
748 |
> |
the dipole moment and sticky switching functions have improved the |
749 |
> |
structuring of the liquid (as seen in figure \ref{grcompare}, but they |
750 |
> |
have shifted the density maximum to much lower temperatures. This |
751 |
> |
comes about via an increase in the liquid disorder through the |
752 |
> |
weakening of the sticky potential and strengthening of the dipolar |
753 |
> |
character. However, this increasing disorder in the SSD/E model has |
754 |
> |
little effect on the melting transition. By monitoring $C_p$ |
755 |
> |
throughout these simulations, the melting transition for SSD/E was |
756 |
> |
shown to occur at 235 K. The same transition temperature observed |
757 |
> |
with SSD and SSD1. |
758 |
|
|
759 |
< |
\begin{figure} |
760 |
< |
\includegraphics[width=85mm]{ssdrfcompare.epsi} |
761 |
< |
\caption{Comparison of densities calculated with SSD/RF to SSD with a |
762 |
< |
reaction field, TIP4P\cite{Jorgensen98b}, TIP3P\cite{Jorgensen98b}, |
763 |
< |
SPC/E\cite{Clancy94}, and Experiment\cite{CRC80}. The upper plot |
764 |
< |
includes error bars, and the calculated results from the other |
765 |
< |
references were removed for clarity.} |
759 |
> |
\begin{figure} |
760 |
> |
\begin{center} |
761 |
> |
\epsfxsize=6in |
762 |
> |
\epsfbox{ssdrfDense.epsi} |
763 |
> |
\caption{Comparison of densities calculated with SSD/RF to SSD1 with a |
764 |
> |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
765 |
> |
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and |
766 |
> |
experiment [Ref. \citen{CRC80}]. The inset shows the necessity of |
767 |
> |
reparameterization when utilizing a reaction field long-ranged |
768 |
> |
correction - SSD/RF provides significantly more accurate densities |
769 |
> |
than SSD1 when performing room temperature simulations.} |
770 |
|
\label{ssdrfdense} |
771 |
+ |
\end{center} |
772 |
|
\end{figure} |
773 |
|
|
774 |
< |
Figure \ref{ssdrfdense} shows a density comparison between SSD/RF and |
775 |
< |
SSD with an active reaction field. Like in the simulations of SSD/E, |
776 |
< |
the densities show a dramatic increase over normal SSD. At 298 K, |
777 |
< |
SSD/RF has a density of 0.997$\pm$0.001 g/cm$^3$, right in line with |
778 |
< |
experiment and considerably better than the SSD value of |
779 |
< |
0.941$\pm$0.001 g/cm$^3$. The melting point is observed at 240 K, |
780 |
< |
which is 5 degrees lower than SSD with a reaction field, and the |
781 |
< |
density maximum at 255 K, again 5 degrees lower than SSD. The density |
782 |
< |
at higher temperature still drops off more rapidly than the charge |
783 |
< |
based models but is in better agreement than SSD/E. |
774 |
> |
Including the reaction field long-range correction in the simulations |
775 |
> |
results in a more interesting comparison. A density profile including |
776 |
> |
SSD/RF and SSD1 with an active reaction field is shown in figure |
777 |
> |
\ref{ssdrfdense}. As observed in the simulations without a reaction |
778 |
> |
field, the densities of SSD/RF and SSD1 show a dramatic increase over |
779 |
> |
normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density |
780 |
> |
of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and |
781 |
> |
considerably better than the original SSD value of 0.941$\pm$0.001 |
782 |
> |
g/cm$^3$ and the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results |
783 |
> |
further emphasize the importance of reparameterization in order to |
784 |
> |
model the density properly under different simulation conditions. |
785 |
> |
Again, these changes have only a minor effect on the melting point, |
786 |
> |
which observed at 245 K for SSD/RF, is identical to SSD and only 5 K |
787 |
> |
lower than SSD1 with a reaction field. Additionally, the difference in |
788 |
> |
density maxima is not as extreme, with SSD/RF showing a density |
789 |
> |
maximum at 255 K, fairly close to the density maxima of 260 K and 265 |
790 |
> |
K, shown by SSD and SSD1 respectively. |
791 |
|
|
792 |
+ |
\begin{figure} |
793 |
+ |
\begin{center} |
794 |
+ |
\epsfxsize=6in |
795 |
+ |
\epsfbox{ssdeDiffuse.epsi} |
796 |
+ |
\caption{The diffusion constants calculated from SSD/E and SSD1, |
797 |
+ |
both without a reaction field, along with experimental results |
798 |
+ |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
799 |
+ |
were performed at the average densities observed in the 1 atm NPT |
800 |
+ |
simulations for the respective models. SSD/E is slightly more mobile |
801 |
+ |
than experiment at all of the temperatures, but it is closer to |
802 |
+ |
experiment at biologically relavent temperatures than SSD1 without a |
803 |
+ |
long-range correction.} |
804 |
+ |
\label{ssdediffuse} |
805 |
+ |
\end{center} |
806 |
+ |
\end{figure} |
807 |
+ |
|
808 |
|
The reparameterization of the SSD water model, both for use with and |
809 |
|
without an applied long-range correction, brought the densities up to |
810 |
|
what is expected for simulating liquid water. In addition to improving |
811 |
< |
the densities, it is important that particle transport be maintained |
812 |
< |
or improved. Figure \ref{ssdediffuse} compares the temperature |
813 |
< |
dependence of the diffusion constant of SSD/E to SSD without an active |
814 |
< |
reaction field, both at the densities calculated at 1 atm and at the |
815 |
< |
experimentally calculated densities for super-cooled and liquid |
816 |
< |
water. In the upper plot, the diffusion constant for SSD/E is |
817 |
< |
consistently a little faster than experiment, while SSD starts off |
818 |
< |
slower than experiment and crosses to merge with SSD/E at high |
819 |
< |
temperatures. Both models follow the experimental trend well, but |
820 |
< |
diffuse too rapidly at higher temperatures. This abnormally fast |
821 |
< |
diffusion is caused by the decreased system density. Since the |
822 |
< |
densities of SSD/E don't deviate as much from experiment as those of |
823 |
< |
SSD, it follows the experimental trend more closely. This observation |
824 |
< |
is backed up by looking at the lower plot. The diffusion constants for |
825 |
< |
SSD/E track with the experimental values while SSD deviates on the low |
826 |
< |
side of the trend with increasing temperature. This is again a product |
827 |
< |
of SSD/E having densities closer to experiment, and not deviating to |
828 |
< |
lower densities with increasing temperature as rapidly. |
811 |
> |
the densities, it is important that the excellent diffusive behavior |
812 |
> |
of SSD be maintained or improved. Figure \ref{ssdediffuse} compares |
813 |
> |
the temperature dependence of the diffusion constant of SSD/E to SSD1 |
814 |
> |
without an active reaction field at the densities calculated from the |
815 |
> |
NPT simulations at 1 atm. The diffusion constant for SSD/E is |
816 |
> |
consistently higher than experiment, while SSD1 remains lower than |
817 |
> |
experiment until relatively high temperatures (around 360 K). Both |
818 |
> |
models follow the shape of the experimental curve well below 300 K but |
819 |
> |
tend to diffuse too rapidly at higher temperatures, as seen in SSD1's |
820 |
> |
crossing above 360 K. This increasing diffusion relative to the |
821 |
> |
experimental values is caused by the rapidly decreasing system density |
822 |
> |
with increasing temperature. Both SSD1 and SSD/E show this deviation |
823 |
> |
in diffusive behavior, but this trend has different implications on |
824 |
> |
the diffusive behavior of the models. While SSD1 shows more |
825 |
> |
experimentally accurate diffusive behavior in the high temperature |
826 |
> |
regimes, SSD/E shows more accurate behavior in the supercooled and |
827 |
> |
biologically relavent temperature ranges. Thus, the changes made to |
828 |
> |
improve the liquid structure may have had an adverse affect on the |
829 |
> |
density maximum, but they improve the transport behavior of SSD/E |
830 |
> |
relative to SSD1 under the most commonly simulated conditions. |
831 |
|
|
832 |
< |
\begin{figure} |
833 |
< |
\includegraphics[width=85mm]{ssdediffuse.epsi} |
834 |
< |
\caption{Plots of the diffusion constants calculated from SSD/E and SSD, |
835 |
< |
both without a reaction field along with experimental results from |
836 |
< |
Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The |
837 |
< |
upper plot is at densities calculated from the NPT simulations at a |
838 |
< |
pressure of 1 atm, while the lower plot is at the experimentally |
839 |
< |
calculated densities.} |
840 |
< |
\label{ssdediffuse} |
841 |
< |
\end{figure} |
842 |
< |
|
843 |
< |
\begin{figure} |
844 |
< |
\includegraphics[width=85mm]{ssdrfdiffuse.epsi} |
769 |
< |
\caption{Plots of the diffusion constants calculated from SSD/RF and SSD, |
770 |
< |
both with an active reaction field along with experimental results |
771 |
< |
from Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The |
772 |
< |
upper plot is at densities calculated from the NPT simulations at a |
773 |
< |
pressure of 1 atm, while the lower plot is at the experimentally |
774 |
< |
calculated densities.} |
832 |
> |
\begin{figure} |
833 |
> |
\begin{center} |
834 |
> |
\epsfxsize=6in |
835 |
> |
\epsfbox{ssdrfDiffuse.epsi} |
836 |
> |
\caption{The diffusion constants calculated from SSD/RF and SSD1, |
837 |
> |
both with an active reaction field, along with experimental results |
838 |
> |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
839 |
> |
were performed at the average densities observed in the 1 atm NPT |
840 |
> |
simulations for both of the models. Note how accurately SSD/RF |
841 |
> |
simulates the diffusion of water throughout this temperature |
842 |
> |
range. The more rapidly increasing diffusion constants at high |
843 |
> |
temperatures for both models is attributed to lower calculated |
844 |
> |
densities than those observed in experiment.} |
845 |
|
\label{ssdrfdiffuse} |
846 |
+ |
\end{center} |
847 |
|
\end{figure} |
848 |
|
|
849 |
|
In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are |
850 |
< |
compared with SSD with an active reaction field. In the upper plot, |
851 |
< |
SSD/RF tracks with the experimental results incredibly well, identical |
852 |
< |
within error throughout the temperature range and only showing a |
853 |
< |
slight increasing trend at higher temperatures. SSD also tracks |
854 |
< |
experiment well, only it tends to diffuse a little more slowly at low |
855 |
< |
temperatures and deviates to diffuse too rapidly at high |
856 |
< |
temperatures. As was stated in the SSD/E comparisons, this deviation |
857 |
< |
away from the ideal trend is due to a rapid decrease in density at |
858 |
< |
higher temperatures. SSD/RF doesn't suffer from this problem as much |
859 |
< |
as SSD, because the calculated densities are more true to |
860 |
< |
experiment. This is again emphasized in the lower plot, where SSD/RF |
790 |
< |
tracks the experimental diffusion exactly while SSD's diffusion |
791 |
< |
constants are slightly too low due to its need for a lower density at |
792 |
< |
the specified temperature. |
850 |
> |
compared to SSD1 with an active reaction field. Note that SSD/RF |
851 |
> |
tracks the experimental results quantitatively, identical within error |
852 |
> |
throughout most of the temperature range shown and exhibiting only a |
853 |
> |
slight increasing trend at higher temperatures. SSD1 tends to diffuse |
854 |
> |
more slowly at low temperatures and deviates to diffuse too rapidly at |
855 |
> |
temperatures greater than 330 K. As stated above, this deviation away |
856 |
> |
from the ideal trend is due to a rapid decrease in density at higher |
857 |
> |
temperatures. SSD/RF does not suffer from this problem as much as SSD1 |
858 |
> |
because the calculated densities are closer to the experimental |
859 |
> |
values. These results again emphasize the importance of careful |
860 |
> |
reparameterization when using an altered long-range correction. |
861 |
|
|
862 |
< |
\subsection{Additional Observations} |
862 |
> |
\begin{table} |
863 |
> |
\begin{center} |
864 |
> |
\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}.} |
865 |
> |
\begin{tabular}{ l c c c c c } |
866 |
> |
\hline \\[-3mm] |
867 |
> |
\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \ |
868 |
> |
\ & \ SSD/RF \ \ \ & \ Expt. \\ |
869 |
> |
\hline \\[-3mm] |
870 |
> |
\ \ \ $\rho$ (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 \\ |
871 |
> |
\ \ \ $C_p$ (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 \\ |
872 |
> |
\ \ \ $D$ ($10^{-5}$ 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}$ \\ |
873 |
> |
\ \ \ Coordination Number & 3.9 & 4.3 & 3.8 & 4.4 & 4.7$^\text{b}$ \\ |
874 |
> |
\ \ \ H-bonds per particle & 3.7 & 3.6 & 3.7 & 3.7 & 3.4$^\text{c}$ \\ |
875 |
> |
\ \ \ $\tau_1^\mu$ (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}$ \\ |
876 |
> |
\ \ \ $\tau_2^\mu$ (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}$ \\ |
877 |
> |
\end{tabular} |
878 |
> |
\label{liquidproperties} |
879 |
> |
\end{center} |
880 |
> |
\end{table} |
881 |
|
|
882 |
< |
While performing the melting sequences of SSD/E, some interesting |
883 |
< |
observations were made. After melting at 230 K, two of the systems |
884 |
< |
underwent crystallization events near 245 K. As the heating process |
885 |
< |
continued, the two systems remained crystalline until finally melting |
886 |
< |
between 320 and 330 K. These simulations were excluded from the data |
887 |
< |
set shown in figure \ref{ssdedense} and replaced with two additional |
888 |
< |
melting sequences that did not undergo this anomalous phase |
889 |
< |
transition, while this crystallization event was investigated |
890 |
< |
separately. |
882 |
> |
Table \ref{liquidproperties} gives a synopsis of the liquid state |
883 |
> |
properties of the water models compared in this study along with the |
884 |
> |
experimental values for liquid water at ambient conditions. The |
885 |
> |
coordination number and hydrogen bonds per particle were calculated by |
886 |
> |
integrating the following relation: |
887 |
> |
\begin{equation} |
888 |
> |
4\pi\rho\int_{0}^{a}r^2\text{g}(r)dr, |
889 |
> |
\end{equation} |
890 |
> |
where $\rho$ is the number density of pair interactions, $a$ is the |
891 |
> |
radial location of the minima following the first solvation shell |
892 |
> |
peak, and g$(r)$ is either g$_\text{OO}(r)$ or g$_\text{OH}(r)$ for |
893 |
> |
calculation of the coordination number or hydrogen bonds per particle |
894 |
> |
respectively. |
895 |
|
|
896 |
< |
\begin{figure} |
897 |
< |
\includegraphics[width=85mm]{povIce.ps} |
898 |
< |
\caption{Crystal structure of an ice 0 lattice shown from the (001) face.} |
899 |
< |
\label{weirdice} |
900 |
< |
\end{figure} |
896 |
> |
The time constants for the self orientational autocorrelation function |
897 |
> |
are also displayed in Table \ref{liquidproperties}. The dipolar |
898 |
> |
orientational time correlation function ($\Gamma_{l}$) is described |
899 |
> |
by: |
900 |
> |
\begin{equation} |
901 |
> |
\Gamma_{l}(t) = \langle P_l[\mathbf{u}_j(0)\cdot\mathbf{u}_j(t)]\rangle, |
902 |
> |
\end{equation} |
903 |
> |
where $P_l$ is a Legendre polynomial of order $l$ and $\mathbf{u}_j$ |
904 |
> |
is the unit vector of the particle dipole.\cite{Rahman71} From these |
905 |
> |
correlation functions, the orientational relaxation time of the dipole |
906 |
> |
vector can be calculated from an exponential fit in the long-time |
907 |
> |
regime ($t > \tau_l^\mu$).\cite{Rothschild84} Calculation of these |
908 |
> |
time constants were averaged from five detailed NVE simulations |
909 |
> |
performed at the STP density for each of the respective models. |
910 |
|
|
911 |
< |
The final configurations of these two melting sequences shows an |
813 |
< |
expanded zeolite-like crystal structure that does not correspond to |
814 |
< |
any known form of ice. For convenience and to help distinguish it from |
815 |
< |
the experimentally observed forms of ice, this crystal structure will |
816 |
< |
henceforth be referred to as ice-zero (ice 0). The crystallinity was |
817 |
< |
extensive enough than a near ideal crystal structure could be |
818 |
< |
obtained. Figure \ref{weirdice} shows the repeating crystal structure |
819 |
< |
of a typical crystal at 5 K. The unit cell contains eight molecules, |
820 |
< |
and figure \ref{unitcell} shows a unit cell built from the water |
821 |
< |
particle center of masses that can be used to construct a repeating |
822 |
< |
lattice, similar to figure \ref{weirdice}. Each molecule is hydrogen |
823 |
< |
bonded to four other water molecules; however, the hydrogen bonds are |
824 |
< |
flexed rather than perfectly straight. This results in a skewed |
825 |
< |
tetrahedral geometry about the central molecule. Looking back at |
826 |
< |
figure \ref{isosurface}, it is easy to see how these flexed hydrogen |
827 |
< |
bonds are allowed in that the attractive regions are conical in shape, |
828 |
< |
with the greatest attraction in the central region. Though not ideal, |
829 |
< |
these flexed hydrogen bonds are favorable enough to stabilize an |
830 |
< |
entire crystal generated around them. In fact, the imperfect ice 0 |
831 |
< |
crystals were so stable that they melted at greater than room |
832 |
< |
temperature. |
911 |
> |
\subsection{Additional Observations} |
912 |
|
|
913 |
|
\begin{figure} |
914 |
< |
\includegraphics[width=65mm]{ice0cell.eps} |
915 |
< |
\caption{Simple unit cell for constructing ice 0. In this cell, $c$ is |
916 |
< |
equal to $0.4714\times a$, and a typical value for $a$ is 8.25 \AA.} |
917 |
< |
\label{unitcell} |
914 |
> |
\begin{center} |
915 |
> |
\epsfxsize=6in |
916 |
> |
\epsfbox{povIce.ps} |
917 |
> |
\caption{A water lattice built from the crystal structure assumed by |
918 |
> |
SSD/E when undergoing an extremely restricted temperature NPT |
919 |
> |
simulation. This form of ice is referred to as ice-{\it i} to |
920 |
> |
emphasize its simulation origins. This image was taken of the (001) |
921 |
> |
face of the crystal.} |
922 |
> |
\label{weirdice} |
923 |
> |
\end{center} |
924 |
|
\end{figure} |
925 |
|
|
926 |
< |
The initial simulations indicated that ice 0 is the preferred ice |
927 |
< |
structure for at least SSD/E. To verify this, a comparison was made |
928 |
< |
between near ideal crystals of ice $I_h$, ice $I_c$, and ice 0 at |
929 |
< |
constant pressure with SSD/E, SSD/RF, and SSD. Near ideal versions of |
930 |
< |
the three types of crystals were cooled to ~1 K, and the potential |
931 |
< |
energies of each were compared using all three water models. With |
932 |
< |
every water model, ice 0 turned out to have the lowest potential |
933 |
< |
energy: 5\% lower than $I_h$ with SSD, 6.5\% lower with SSD/E, and |
934 |
< |
7.5\% lower with SSD/RF. In all three of these water models, ice $I_c$ |
935 |
< |
was observed to be ~2\% less stable than ice $I_h$. In addition to |
936 |
< |
having the lowest potential energy, ice 0 was the most expanded of the |
937 |
< |
three ice crystals, ~5\% less dense than ice $I_h$ with all of the |
938 |
< |
water models. In all three water models, ice $I_c$ was observed to be |
939 |
< |
~2\% more dense than ice $I_h$. |
926 |
> |
While performing a series of melting simulations on an early iteration |
927 |
> |
of SSD/E not discussed in this paper, we observed recrystallization |
928 |
> |
into a novel structure not previously known for water. After melting |
929 |
> |
at 235 K, two of five systems underwent crystallization events near |
930 |
> |
245 K. The two systems remained crystalline up to 320 and 330 K, |
931 |
> |
respectively. The crystal exhibits an expanded zeolite-like structure |
932 |
> |
that does not correspond to any known form of ice. This appears to be |
933 |
> |
an artifact of the point dipolar models, so to distinguish it from the |
934 |
> |
experimentally observed forms of ice, we have denoted the structure |
935 |
> |
Ice-$\sqrt{\smash[b]{-\text{I}}}$ (ice-{\it i}). A large enough |
936 |
> |
portion of the sample crystallized that we have been able to obtain a |
937 |
> |
near ideal crystal structure of ice-{\it i}. Figure \ref{weirdice} |
938 |
> |
shows the repeating crystal structure of a typical crystal at 5 |
939 |
> |
K. Each water molecule is hydrogen bonded to four others; however, the |
940 |
> |
hydrogen bonds are bent rather than perfectly straight. This results |
941 |
> |
in a skewed tetrahedral geometry about the central molecule. In |
942 |
> |
figure \ref{isosurface}, it is apparent that these flexed hydrogen |
943 |
> |
bonds are allowed due to the conical shape of the attractive regions, |
944 |
> |
with the greatest attraction along the direct hydrogen bond |
945 |
> |
configuration. Though not ideal, these flexed hydrogen bonds are |
946 |
> |
favorable enough to stabilize an entire crystal generated around them. |
947 |
|
|
948 |
< |
In addition to the low temperature comparisons, melting sequences were |
949 |
< |
performed with ice 0 as the initial configuration using SSD/E, SSD/RF, |
950 |
< |
and SSD both with and without a reaction field. The melting |
951 |
< |
transitions for both SSD/E and SSD without a reaction field occurred |
952 |
< |
at temperature in excess of 375 K. SSD/RF and SSD with a reaction |
953 |
< |
field had more reasonable melting transitions, down near 325 K. These |
954 |
< |
melting point observations emphasize how preferred this crystal |
955 |
< |
structure is over the most common types of ice when using these single |
956 |
< |
point water models. |
948 |
> |
Initial simulations indicated that ice-{\it i} is the preferred ice |
949 |
> |
structure for at least the SSD/E model. To verify this, a comparison |
950 |
> |
was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and |
951 |
> |
ice-{\it i} at constant pressure with SSD/E, SSD/RF, and |
952 |
> |
SSD1. Near-ideal versions of the three types of crystals were cooled |
953 |
> |
to 1 K, and the enthalpies of each were compared using all three water |
954 |
> |
models. With every model in the SSD family, ice-{\it i} had the lowest |
955 |
> |
calculated enthalpy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with |
956 |
> |
SSD/E, and 7.5\% lower with SSD/RF. The enthalpy data is summarized |
957 |
> |
in Table \ref{iceenthalpy}. |
958 |
|
|
959 |
< |
Recognizing that the above tests show ice 0 to be both the most stable |
960 |
< |
and lowest density crystal structure for these single point water |
961 |
< |
models, it is interesting to speculate on the favorability of this |
962 |
< |
crystal structure with the different charge based models. As a quick |
963 |
< |
test, these 3 crystal types were converted from SSD type particles to |
964 |
< |
TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy |
965 |
< |
minimizations were performed on all of these crystals to compare the |
966 |
< |
system energies. Again, ice 0 was observed to have the lowest total |
967 |
< |
system energy. The total energy of ice 0 was ~2\% lower than ice |
968 |
< |
$I_h$, which was in turn ~3\% lower than ice $I_c$. From these initial |
969 |
< |
results, we would not be surprised if results from the other common |
970 |
< |
water models show ice 0 to be the lowest energy crystal structure. A |
971 |
< |
continuation on work studing ice 0 with multipoint water models will |
972 |
< |
be published in a coming article. |
959 |
> |
\begin{table} |
960 |
> |
\begin{center} |
961 |
> |
\caption{Enthalpies (in kcal / mol) of the three crystal structures (at 1 |
962 |
> |
K) exhibited by the SSD family of water models} |
963 |
> |
\begin{tabular}{ l c c c } |
964 |
> |
\hline \\[-3mm] |
965 |
> |
\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \ & \ |
966 |
> |
Ice-{\it i} \\ |
967 |
> |
\hline \\[-3mm] |
968 |
> |
\ \ \ SSD/E & -12.286 & -12.292 & -13.590 \\ |
969 |
> |
\ \ \ SSD/RF & -12.935 & -12.917 & -14.022 \\ |
970 |
> |
\ \ \ SSD1 & -12.496 & -12.411 & -13.417 \\ |
971 |
> |
\ \ \ SSD1 (RF) & -12.504 & -12.411 & -13.134 \\ |
972 |
> |
\end{tabular} |
973 |
> |
\label{iceenthalpy} |
974 |
> |
\end{center} |
975 |
> |
\end{table} |
976 |
|
|
977 |
+ |
In addition to these energetic comparisons, melting simulations were |
978 |
+ |
performed with ice-{\it i} as the initial configuration using SSD/E, |
979 |
+ |
SSD/RF, and SSD1 both with and without a reaction field. The melting |
980 |
+ |
transitions for both SSD/E and SSD1 without reaction field occurred at |
981 |
+ |
temperature in excess of 375~K. SSD/RF and SSD1 with a reaction field |
982 |
+ |
showed more reasonable melting transitions near 325~K. These melting |
983 |
+ |
point observations clearly show that all of the SSD-derived models |
984 |
+ |
prefer the ice-{\it i} structure. |
985 |
+ |
|
986 |
|
\section{Conclusions} |
987 |
< |
The density maximum and temperature dependent transport for the SSD |
988 |
< |
water model, both with and without the use of reaction field, were |
989 |
< |
studied via a series of NPT and NVE simulations. The constant pressure |
990 |
< |
simulations of the melting of both $I_h$ and $I_c$ ice showed a |
991 |
< |
density maximum near 260 K. In most cases, the calculated densities |
992 |
< |
were significantly lower than the densities calculated in simulations |
993 |
< |
of other water models. Analysis of particle diffusion showed SSD to |
994 |
< |
capture the transport properties of experimental very well in both the |
995 |
< |
normal and super-cooled liquid regimes. In order to correct the |
996 |
< |
density behavior, SSD was reparameterized for use both with and |
997 |
< |
without a long-range interaction correction, SSD/RF and SSD/E |
998 |
< |
respectively. In addition to correcting the abnormally low densities, |
999 |
< |
these new versions were show to maintain or improve upon the transport |
1000 |
< |
and structural features of the original water model, all while |
1001 |
< |
maintaining the fast performance of the SSD water model. This work |
1002 |
< |
shows these simple water models, and in particular SSD/E and SSD/RF, |
1003 |
< |
to be excellent choices to represent explicit water in future |
987 |
> |
|
988 |
> |
The density maximum and temperature dependence of the self-diffusion |
989 |
> |
constant were studied for the SSD water model, both with and without |
990 |
> |
the use of reaction field, via a series of NPT and NVE |
991 |
> |
simulations. The constant pressure simulations showed a density |
992 |
> |
maximum near 260 K. In most cases, the calculated densities were |
993 |
> |
significantly lower than the densities obtained from other water |
994 |
> |
models (and experiment). Analysis of self-diffusion showed SSD to |
995 |
> |
capture the transport properties of water well in both the liquid and |
996 |
> |
super-cooled liquid regimes. |
997 |
> |
|
998 |
> |
In order to correct the density behavior, the original SSD model was |
999 |
> |
reparameterized for use both with and without a reaction field (SSD/RF |
1000 |
> |
and SSD/E), and comparisons were made with SSD1, Ichiye's density |
1001 |
> |
corrected version of SSD. Both models improve the liquid structure, |
1002 |
> |
densities, and diffusive properties under their respective simulation |
1003 |
> |
conditions, indicating the necessity of reparameterization when |
1004 |
> |
changing the method of calculating long-range electrostatic |
1005 |
> |
interactions. In general, however, these simple water models are |
1006 |
> |
excellent choices for representing explicit water in large scale |
1007 |
|
simulations of biochemical systems. |
1008 |
|
|
1009 |
+ |
The existence of a novel low-density ice structure that is preferred |
1010 |
+ |
by the SSD family of water models is somewhat troubling, since liquid |
1011 |
+ |
simulations on this family of water models at room temperature are |
1012 |
+ |
effectively simulations of super-cooled or metastable liquids. One |
1013 |
+ |
way to de-stabilize this unphysical ice structure would be to make the |
1014 |
+ |
range of angles preferred by the attractive part of the sticky |
1015 |
+ |
potential much narrower. This would require extensive |
1016 |
+ |
reparameterization to maintain the same level of agreement with the |
1017 |
+ |
experiments. |
1018 |
+ |
|
1019 |
+ |
Additionally, our initial calculations show that the ice-{\it i} |
1020 |
+ |
structure may also be a preferred crystal structure for at least one |
1021 |
+ |
other popular multi-point water model (TIP3P), and that much of the |
1022 |
+ |
simulation work being done using this popular model could also be at |
1023 |
+ |
risk for crystallization into this unphysical structure. A future |
1024 |
+ |
publication will detail the relative stability of the known ice |
1025 |
+ |
structures for a wide range of popular water models. |
1026 |
+ |
|
1027 |
|
\section{Acknowledgments} |
1028 |
< |
The authors would like to thank the National Science Foundation for |
1029 |
< |
funding under grant CHE-0134881. Computation time was provided by the |
1030 |
< |
Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant DMR |
1031 |
< |
00 79647. |
1028 |
> |
Support for this project was provided by the National Science |
1029 |
> |
Foundation under grant CHE-0134881. Computation time was provided by |
1030 |
> |
the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
1031 |
> |
DMR-0079647. |
1032 |
|
|
1033 |
< |
\bibliographystyle{jcp} |
1033 |
> |
\newpage |
1034 |
|
|
1035 |
+ |
\bibliographystyle{jcp} |
1036 |
|
\bibliography{nptSSD} |
1037 |
|
|
1038 |
|
%\pagebreak |