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\begin{document} |
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\title{On the structural and transport properties of the soft sticky |
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dipole (SSD) and related single point water models} |
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\author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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\author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu}} |
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\affiliation{Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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|
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– |
\maketitle |
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|
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\begin{abstract} |
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The density maximum and temperature dependence of the self-diffusion |
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constant were investigated for the soft sticky dipole (SSD) water |
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model and two related re-parameterizations of this single-point model. |
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model and two related reparameterizations of this single-point model. |
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A combination of microcanonical and isobaric-isothermal molecular |
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dynamics simulations were used to calculate these properties, both |
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with and without the use of reaction field to handle long-range |
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electrostatics. The isobaric-isothermal (NPT) simulations of the |
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melting of both ice-$I_h$ and ice-$I_c$ showed a density maximum near |
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260 K. In most cases, the use of the reaction field resulted in |
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260~K. In most cases, the use of the reaction field resulted in |
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calculated densities which were were significantly lower than |
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experimental densities. Analysis of self-diffusion constants shows |
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that the original SSD model captures the transport properties of |
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experimental water very well in both the normal and super-cooled |
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liquid regimes. We also present our re-parameterized versions of SSD |
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liquid regimes. We also present our reparameterized versions of SSD |
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for use both with the reaction field or without any long-range |
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electrostatic corrections. These are called the SSD/RF and SSD/E |
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models respectively. These modified models were shown to maintain or |
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family. |
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\end{abstract} |
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|
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\maketitle |
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\newpage |
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%\narrowtext |
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One recently developed model that largely succeeds in retaining the |
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accuracy of bulk properties while greatly reducing the computational |
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cost is the Soft Sticky Dipole (SSD) water |
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model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The SSD model was |
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developed by Ichiye \emph{et al.} as a modified form of the |
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model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The SSD model |
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was developed by Ichiye \emph{et al.} as a modified form of the |
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hard-sphere water model proposed by Bratko, Blum, and |
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Luzar.\cite{Bratko85,Bratko95} SSD is a {\it single point} model which |
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has an interaction site that is both a point dipole along with a |
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Luzar.\cite{Bratko85,Bratko95} SSD is a {\it single point} model |
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which has an interaction site that is both a point dipole and a |
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Lennard-Jones core. However, since the normal aligned and |
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anti-aligned geometries favored by point dipoles are poor mimics of |
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local structure in liquid water, a short ranged ``sticky'' potential |
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is also added. The sticky potential directs the molecules to assume |
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the proper hydrogen bond orientation in the first solvation |
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shell. |
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the proper hydrogen bond orientation in the first solvation shell. |
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|
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The interaction between two SSD water molecules \emph{i} and \emph{j} |
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is given by the potential |
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Since SSD is a single-point {\it dipolar} model, the force |
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calculations are simplified significantly relative to the standard |
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{\it charged} multi-point models. In the original Monte Carlo |
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simulations using this model, Ichiye {\it et al.} reported that using |
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SSD decreased computer time by a factor of 6-7 compared to other |
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simulations using this model, Liu and Ichiye reported that using SSD |
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decreased computer time by a factor of 6-7 compared to other |
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models.\cite{Ichiye96} What is most impressive is that this savings |
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did not come at the expense of accurate depiction of the liquid state |
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properties. Indeed, SSD maintains reasonable agreement with the Soper |
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of solvent properties makes SSD a very attractive model for the |
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simulation of large scale biochemical simulations. |
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|
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One feature of the SSD model is that it was parameterized for use with |
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the Ewald sum to handle long-range interactions. This would normally |
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be the best way of handling long-range interactions in systems that |
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contain other point charges. However, our group has recently become |
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interested in systems with point dipoles as mimics for neutral, but |
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polarized regions on molecules (e.g. the zwitterionic head group |
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regions of phospholipids). If the system of interest does not contain |
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point charges, the Ewald sum and even particle-mesh Ewald become |
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computational bottlenecks. Their respective ideal $N^\frac{3}{2}$ and |
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$N\log N$ calculation scaling orders for $N$ particles can become |
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prohibitive when $N$ becomes large.\cite{Darden99} In applying this |
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water model in these types of systems, it would be useful to know its |
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properties and behavior under the more computationally efficient |
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reaction field (RF) technique, or even with a simple cutoff. This |
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study addresses these issues by looking at the structural and |
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transport behavior of SSD over a variety of temperatures with the |
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purpose of utilizing the RF correction technique. We then suggest |
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modifications to the parameters that result in more realistic bulk |
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phase behavior. It should be noted that in a recent publication, some |
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of the original investigators of the SSD water model have suggested |
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adjustments to the SSD water model to address abnormal density |
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behavior (also observed here), calling the corrected model |
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SSD1.\cite{Ichiye03} In what follows, we compare our |
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reparamaterization of SSD with both the original SSD and SSD1 models |
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with the goal of improving the bulk phase behavior of an SSD-derived |
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model in simulations utilizing the Reaction Field. |
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One feature of the SSD model is that it was parameterized for |
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use with the Ewald sum to handle long-range interactions. This would |
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> |
normally be the best way of handling long-range interactions in |
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> |
systems that contain other point charges. However, our group has |
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recently become interested in systems with point dipoles as mimics for |
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neutral, but polarized regions on molecules (e.g. the zwitterionic |
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> |
head group regions of phospholipids). If the system of interest does |
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not contain point charges, the Ewald sum and even particle-mesh Ewald |
| 192 |
> |
become computational bottlenecks. Their respective ideal |
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$N^\frac{3}{2}$ and $N\log N$ calculation scaling orders for $N$ |
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particles can become prohibitive when $N$ becomes |
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large.\cite{Darden99} In applying this water model in these types of |
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> |
systems, it would be useful to know its properties and behavior under |
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> |
the more computationally efficient reaction field (RF) technique, or |
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> |
even with a simple cutoff. This study addresses these issues by |
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> |
looking at the structural and transport behavior of SSD over a |
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> |
variety of temperatures with the purpose of utilizing the RF |
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correction technique. We then suggest modifications to the parameters |
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> |
that result in more realistic bulk phase behavior. It should be noted |
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that in a recent publication, some of the original investigators of |
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the SSD water model have suggested adjustments to the SSD |
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water model to address abnormal density behavior (also observed here), |
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calling the corrected model SSD1.\cite{Ichiye03} In what |
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follows, we compare our reparamaterization of SSD with both the |
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original SSD and SSD1 models with the goal of improving |
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the bulk phase behavior of an SSD-derived model in simulations |
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utilizing the reaction field. |
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|
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\section{Methods} |
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|
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Long-range dipole-dipole interactions were accounted for in this study |
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by using either the reaction field method or by resorting to a simple |
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cubic switching function at a cutoff radius. The reaction field |
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method was actually first used in Monte Carlo simulations of liquid |
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water.\cite{Barker73} Under this method, the magnitude of the reaction |
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field acting on dipole $i$ is |
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by using either the reaction field technique or by resorting to a |
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simple cubic switching function at a cutoff radius. One of the early |
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applications of a reaction field was actually in Monte Carlo |
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simulations of liquid water.\cite{Barker73} Under this method, the |
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magnitude of the reaction field acting on dipole $i$ is |
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\begin{equation} |
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\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1} |
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\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} f(r_{ij})\ , |
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\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} s(r_{ij}), |
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\label{rfequation} |
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\end{equation} |
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where $\mathcal{R}$ is the cavity defined by the cutoff radius |
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($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the |
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system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole |
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moment vector of particle $j$ and $f(r_{ij})$ is a cubic switching |
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moment vector of particle $j$, and $s(r_{ij})$ is a cubic switching |
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function.\cite{AllenTildesley} The reaction field contribution to the |
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total energy by particle $i$ is given by $-\frac{1}{2}{\bf |
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\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf |
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\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use of the reaction |
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field is known to alter the bulk orientational properties, such as the |
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dielectric relaxation time. There is particular sensitivity of this |
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property on changes in the length of the cutoff |
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radius.\cite{Berendsen98} This variable behavior makes reaction field |
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a less attractive method than the Ewald sum. However, for very large |
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systems, the computational benefit of reaction field is dramatic. |
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field is known to alter the bulk orientational properties of simulated |
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water, and there is particular sensitivity of these properties on |
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changes in the length of the cutoff radius.\cite{Berendsen98} This |
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variable behavior makes reaction field a less attractive method than |
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the Ewald sum. However, for very large systems, the computational |
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benefit of reaction field is dramatic. |
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|
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We have also performed a companion set of simulations {\it without} a |
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surrounding dielectric (i.e. using a simple cubic switching function |
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at the cutoff radius) and as a result we have two reparamaterizations |
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of SSD which could be used either with or without the Reaction Field |
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turned on. |
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at the cutoff radius), and as a result we have two reparamaterizations |
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of SSD which could be used either with or without the reaction |
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field turned on. |
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|
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Simulations to obtain the preferred density were performed in the |
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isobaric-isothermal (NPT) ensemble, while all dynamical properties |
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were obtained from microcanonical (NVE) simulations done at densities |
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matching the NPT density for a particular target temperature. The |
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constant pressure simulations were implemented using an integral |
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thermostat and barostat as outlined by Hoover.\cite{Hoover85,Hoover86} |
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All molecules were treated as non-linear rigid bodies. Vibrational |
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constraints are not necessary in simulations of SSD, because there are |
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no explicit hydrogen atoms, and thus no molecular vibrational modes |
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need to be considered. |
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Simulations to obtain the preferred densities of the models were |
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performed in the isobaric-isothermal (NPT) ensemble, while all |
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dynamical properties were obtained from microcanonical (NVE) |
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simulations done at densities matching the NPT density for a |
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particular target temperature. The constant pressure simulations were |
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implemented using an integral thermostat and barostat as outlined by |
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Hoover.\cite{Hoover85,Hoover86} All molecules were treated as |
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non-linear rigid bodies. Vibrational constraints are not necessary in |
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simulations of SSD, because there are no explicit hydrogen |
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atoms, and thus no molecular vibrational modes need to be considered. |
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|
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Integration of the equations of motion was carried out using the |
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symplectic splitting method proposed by Dullweber {\it et |
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al.}\cite{Dullweber1997} Our reason for selecting this integrator |
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centers on poor energy conservation of rigid body dynamics using |
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traditional quaternion integration.\cite{Evans77,Evans77b} While quaternions |
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may work well for orientational motion under NVT or NPT integrators, |
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our limits on energy drift in the microcanonical ensemble were quite |
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strict, and the drift under quaternions was substantially greater than |
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in the symplectic splitting method. This steady drift in the total |
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energy has also been observed by Kol {\it et al.}\cite{Laird97} |
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symplectic splitting method proposed by Dullweber, Leimkuhler, and |
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McLachlan ({\sc dlm}).\cite{Dullweber1997} Our reason for selecting |
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this integrator centers on poor energy conservation of rigid body |
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dynamics using traditional quaternion |
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integration.\cite{Evans77,Evans77b} In typical microcanonical ensemble |
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simulations, the energy drift when using quaternions was substantially |
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greater than when using the {\sc dlm} method (fig. \ref{timestep}). |
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This steady drift in the total energy has also been observed by Kol |
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{\it et al.}\cite{Laird97} |
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|
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The key difference in the integration method proposed by Dullweber |
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\emph{et al.} is that the entire rotation matrix is propagated from |
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rotation matrix into quaternions for storage purposes makes trajectory |
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data quite compact. |
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|
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The symplectic splitting method allows for Verlet style integration of |
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both translational and orientational motion of rigid bodies. In this |
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The {\sc dlm} method allows for Verlet style integration of both |
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translational and orientational motion of rigid bodies. In this |
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integration method, the orientational propagation involves a sequence |
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of matrix evaluations to update the rotation |
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matrix.\cite{Dullweber1997} These matrix rotations are more costly |
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than the simpler arithmetic quaternion propagation. With the same time |
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step, a 1000 SSD particle simulation shows an average 7\% increase in |
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computation time using the symplectic step method in place of |
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step, a 1000 SSD particle simulation shows an average 7\% |
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increase in computation time using the {\sc dlm} method in place of |
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quaternions. The additional expense per step is justified when one |
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considers the ability to use time steps that are nearly twice as large |
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under symplectic splitting than would be usable under quaternion |
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dynamics. The energy conservation of the two methods using a number |
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of different time steps is illustrated in figure |
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under {\sc dlm} than would be usable under quaternion dynamics. The |
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energy conservation of the two methods using a number of different |
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time steps is illustrated in figure |
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\ref{timestep}. |
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|
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\begin{figure} |
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\begin{center} |
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\epsfxsize=6in |
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\epsfbox{timeStep.epsi} |
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\caption{Energy conservation using both quaternion based integration and |
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the symplectic step method proposed by Dullweber \emph{et al.} with |
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increasing time step. The larger time step plots are shifted from the |
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true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.} |
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\label{timestep} |
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\end{center} |
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\end{figure} |
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%\begin{figure} |
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%\begin{center} |
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%\epsfxsize=6in |
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%\epsfbox{timeStep.epsi} |
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%\caption{Energy conservation using both quaternion-based integration and |
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%the {\sc dlm} method with increasing time step. The larger time step |
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%plots are shifted from the true energy baseline (that of $\Delta t$ = |
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%0.1~fs) for clarity.} |
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%\label{timestep} |
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%\end{center} |
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%\end{figure} |
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|
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In figure \ref{timestep}, the resulting energy drift at various time |
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steps for both the symplectic step and quaternion integration schemes |
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is compared. All of the 1000 SSD particle simulations started with |
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steps for both the {\sc dlm} and quaternion integration schemes is |
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compared. All of the 1000 SSD particle simulations started with |
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the same configuration, and the only difference was the method used to |
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handle orientational motion. At time steps of 0.1 and 0.5 fs, both |
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handle orientational motion. At time steps of 0.1 and 0.5~fs, both |
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methods for propagating the orientational degrees of freedom conserve |
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energy fairly well, with the quaternion method showing a slight energy |
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drift over time in the 0.5 fs time step simulation. At time steps of 1 |
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and 2 fs, the energy conservation benefits of the symplectic step |
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method are clearly demonstrated. Thus, while maintaining the same |
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degree of energy conservation, one can take considerably longer time |
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steps, leading to an overall reduction in computation time. |
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drift over time in the 0.5~fs time step simulation. At time steps of 1 |
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and 2~fs, the energy conservation benefits of the {\sc dlm} method are |
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clearly demonstrated. Thus, while maintaining the same degree of |
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energy conservation, one can take considerably longer time steps, |
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leading to an overall reduction in computation time. |
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|
|
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Energy drift in the symplectic step simulations was unnoticeable for |
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time steps up to 3 fs. A slight energy drift on the |
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order of 0.012 kcal/mol per nanosecond was observed at a time step of |
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4 fs, and as expected, this drift increases dramatically |
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with increasing time step. To insure accuracy in our microcanonical |
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simulations, time steps were set at 2 fs and kept at this value for |
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Energy drift in the simulations using {\sc dlm} integration was |
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unnoticeable for time steps up to 3~fs. A slight energy drift on the |
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order of 0.012~kcal/mol per nanosecond was observed at a time step of |
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> |
4~fs, and as expected, this drift increases dramatically with |
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> |
increasing time step. To insure accuracy in our microcanonical |
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simulations, time steps were set at 2~fs and kept at this value for |
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constant pressure simulations as well. |
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|
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Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices |
| 328 |
|
orthorhombic shape with an edge length ratio of approximately |
| 329 |
|
1.00$\times$1.06$\times$1.23. The particles were then allowed to |
| 330 |
|
orient freely about fixed positions with angular momenta randomized at |
| 331 |
< |
400 K for varying times. The rotational temperature was then scaled |
| 332 |
< |
down in stages to slowly cool the crystals to 25 K. The particles were |
| 331 |
> |
400~K for varying times. The rotational temperature was then scaled |
| 332 |
> |
down in stages to slowly cool the crystals to 25~K. The particles were |
| 333 |
|
then allowed to translate with fixed orientations at a constant |
| 334 |
< |
pressure of 1 atm for 50 ps at 25 K. Finally, all constraints were |
| 335 |
< |
removed and the ice crystals were allowed to equilibrate for 50 ps at |
| 336 |
< |
25 K and a constant pressure of 1 atm. This procedure resulted in |
| 334 |
> |
pressure of 1 atm for 50~ps at 25~K. Finally, all constraints were |
| 335 |
> |
removed and the ice crystals were allowed to equilibrate for 50~ps at |
| 336 |
> |
25~K and a constant pressure of 1~atm. This procedure resulted in |
| 337 |
|
structurally stable $I_h$ ice crystals that obey the Bernal-Fowler |
| 338 |
|
rules.\cite{Bernal33,Rahman72} This method was also utilized in the |
| 339 |
|
making of diamond lattice $I_c$ ice crystals, with each cubic |
| 351 |
|
supercooled regime. An ensemble average from five separate melting |
| 352 |
|
simulations was acquired, each starting from different ice crystals |
| 353 |
|
generated as described previously. All simulations were equilibrated |
| 354 |
< |
for 100 ps prior to a 200 ps data collection run at each temperature |
| 355 |
< |
setting. The temperature range of study spanned from 25 to 400 K, with |
| 356 |
< |
a maximum degree increment of 25 K. For regions of interest along this |
| 357 |
< |
stepwise progression, the temperature increment was decreased from 25 |
| 358 |
< |
K to 10 and 5 K. The above equilibration and production times were |
| 354 |
> |
for 100~ps prior to a 200~ps data collection run at each temperature |
| 355 |
> |
setting. The temperature range of study spanned from 25 to 400~K, with |
| 356 |
> |
a maximum degree increment of 25~K. For regions of interest along this |
| 357 |
> |
stepwise progression, the temperature increment was decreased from |
| 358 |
> |
25~K to 10 and 5~K. The above equilibration and production times were |
| 359 |
|
sufficient in that fluctuations in the volume autocorrelation function |
| 360 |
< |
were damped out in all simulations in under 20 ps. |
| 360 |
> |
were damped out in all simulations in under 20~ps. |
| 361 |
|
|
| 362 |
|
\subsection{Density Behavior} |
| 363 |
|
|
| 364 |
< |
Our initial simulations focused on the original SSD water model, and |
| 365 |
< |
an average density versus temperature plot is shown in figure |
| 364 |
> |
Our initial simulations focused on the original SSD water model, |
| 365 |
> |
and an average density versus temperature plot is shown in figure |
| 366 |
|
\ref{dense1}. Note that the density maximum when using a reaction |
| 367 |
< |
field appears between 255 and 265 K. There were smaller fluctuations |
| 368 |
< |
in the density at 260 K than at either 255 or 265, so we report this |
| 367 |
> |
field appears between 255 and 265~K. There were smaller fluctuations |
| 368 |
> |
in the density at 260~K than at either 255 or 265~K, so we report this |
| 369 |
|
value as the location of the density maximum. Figure \ref{dense1} was |
| 370 |
|
constructed using ice $I_h$ crystals for the initial configuration; |
| 371 |
|
though not pictured, the simulations starting from ice $I_c$ crystal |
| 372 |
|
configurations showed similar results, with a liquid-phase density |
| 373 |
< |
maximum in this same region (between 255 and 260 K). |
| 373 |
> |
maximum in this same region (between 255 and 260~K). |
| 374 |
|
|
| 375 |
< |
\begin{figure} |
| 376 |
< |
\begin{center} |
| 377 |
< |
\epsfxsize=6in |
| 378 |
< |
\epsfbox{denseSSD.eps} |
| 379 |
< |
\caption{Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}], |
| 380 |
< |
TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], SSD |
| 381 |
< |
without Reaction Field, SSD, and experiment [Ref. \citen{CRC80}]. The |
| 382 |
< |
arrows indicate the change in densities observed when turning off the |
| 383 |
< |
reaction field. The the lower than expected densities for the SSD |
| 384 |
< |
model were what prompted the original reparameterization of SSD1 |
| 385 |
< |
[Ref. \citen{Ichiye03}].} |
| 386 |
< |
\label{dense1} |
| 387 |
< |
\end{center} |
| 388 |
< |
\end{figure} |
| 375 |
> |
%\begin{figure} |
| 376 |
> |
%\begin{center} |
| 377 |
> |
%\epsfxsize=6in |
| 378 |
> |
%\epsfbox{denseSSDnew.eps} |
| 379 |
> |
%\caption{Density versus temperature for TIP4P [Ref. \onlinecite{Jorgensen98b}], |
| 380 |
> |
% TIP3P [Ref. \onlinecite{Jorgensen98b}], SPC/E [Ref. \onlinecite{Clancy94}], SSD |
| 381 |
> |
% without Reaction Field, SSD, and experiment [Ref. \onlinecite{CRC80}]. The |
| 382 |
> |
% arrows indicate the change in densities observed when turning off the |
| 383 |
> |
% reaction field. The the lower than expected densities for the SSD |
| 384 |
> |
% model were what prompted the original reparameterization of SSD1 |
| 385 |
> |
% [Ref. \onlinecite{Ichiye03}].} |
| 386 |
> |
%\label{dense1} |
| 387 |
> |
%\end{center} |
| 388 |
> |
%\end{figure} |
| 389 |
|
|
| 390 |
< |
The density maximum for SSD compares quite favorably to other simple |
| 391 |
< |
water models. Figure \ref{dense1} also shows calculated densities of |
| 392 |
< |
several other models and experiment obtained from other |
| 390 |
> |
The density maximum for SSD compares quite favorably to other |
| 391 |
> |
simple water models. Figure \ref{dense1} also shows calculated |
| 392 |
> |
densities of several other models and experiment obtained from other |
| 393 |
|
sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water |
| 394 |
< |
models, SSD has a temperature closest to the experimentally observed |
| 395 |
< |
density maximum. Of the {\it charge-based} models in |
| 394 |
> |
models, SSD has a temperature closest to the experimentally |
| 395 |
> |
observed density maximum. Of the {\it charge-based} models in |
| 396 |
|
Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that |
| 397 |
< |
seen in SSD. Though not included in this plot, it is useful |
| 398 |
< |
to note that TIP5P has a density maximum nearly identical to the |
| 397 |
> |
seen in SSD. Though not included in this plot, it is useful to |
| 398 |
> |
note that TIP5P has a density maximum nearly identical to the |
| 399 |
|
experimentally measured temperature. |
| 400 |
|
|
| 401 |
|
It has been observed that liquid state densities in water are |
| 402 |
|
dependent on the cutoff radius used both with and without the use of |
| 403 |
|
reaction field.\cite{Berendsen98} In order to address the possible |
| 404 |
|
effect of cutoff radius, simulations were performed with a dipolar |
| 405 |
< |
cutoff radius of 12.0 \AA\ to complement the previous SSD simulations, |
| 406 |
< |
all performed with a cutoff of 9.0 \AA. All of the resulting densities |
| 407 |
< |
overlapped within error and showed no significant trend toward lower |
| 408 |
< |
or higher densities as a function of cutoff radius, for simulations |
| 409 |
< |
both with and without reaction field. These results indicate that |
| 410 |
< |
there is no major benefit in choosing a longer cutoff radius in |
| 411 |
< |
simulations using SSD. This is advantageous in that the use of a |
| 412 |
< |
longer cutoff radius results in a significant increase in the time |
| 413 |
< |
required to obtain a single trajectory. |
| 405 |
> |
cutoff radius of 12.0~\AA\ to complement the previous SSD |
| 406 |
> |
simulations, all performed with a cutoff of 9.0~\AA. All of the |
| 407 |
> |
resulting densities overlapped within error and showed no significant |
| 408 |
> |
trend toward lower or higher densities as a function of cutoff radius, |
| 409 |
> |
for simulations both with and without reaction field. These results |
| 410 |
> |
indicate that there is no major benefit in choosing a longer cutoff |
| 411 |
> |
radius in simulations using SSD. This is advantageous in that |
| 412 |
> |
the use of a longer cutoff radius results in a significant increase in |
| 413 |
> |
the time required to obtain a single trajectory. |
| 414 |
|
|
| 415 |
|
The key feature to recognize in figure \ref{dense1} is the density |
| 416 |
|
scaling of SSD relative to other common models at any given |
| 417 |
< |
temperature. SSD assumes a lower density than any of the other listed |
| 418 |
< |
models at the same pressure, behavior which is especially apparent at |
| 419 |
< |
temperatures greater than 300 K. Lower than expected densities have |
| 420 |
< |
been observed for other systems using a reaction field for long-range |
| 421 |
< |
electrostatic interactions, so the most likely reason for the |
| 422 |
< |
significantly lower densities seen in these simulations is the |
| 417 |
> |
temperature. SSD assumes a lower density than any of the other |
| 418 |
> |
listed models at the same pressure, behavior which is especially |
| 419 |
> |
apparent at temperatures greater than 300~K. Lower than expected |
| 420 |
> |
densities have been observed for other systems using a reaction field |
| 421 |
> |
for long-range electrostatic interactions, so the most likely reason |
| 422 |
> |
for the significantly lower densities seen in these simulations is the |
| 423 |
|
presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order |
| 424 |
|
to test the effect of the reaction field on the density of the |
| 425 |
|
systems, the simulations were repeated without a reaction field |
| 429 |
|
freezing point of liquid water. The shape of the curve is similar to |
| 430 |
|
the curve produced from SSD simulations using reaction field, |
| 431 |
|
specifically the rapidly decreasing densities at higher temperatures; |
| 432 |
< |
however, a shift in the density maximum location, down to 245 K, is |
| 432 |
> |
however, a shift in the density maximum location, down to 245~K, is |
| 433 |
|
observed. This is a more accurate comparison to the other listed water |
| 434 |
|
models, in that no long range corrections were applied in those |
| 435 |
|
simulations.\cite{Clancy94,Jorgensen98b} However, even without the |
| 436 |
< |
reaction field, the density around 300 K is still significantly lower |
| 436 |
> |
reaction field, the density around 300~K is still significantly lower |
| 437 |
|
than experiment and comparable water models. This anomalous behavior |
| 438 |
< |
was what lead Ichiye {\it et al.} to recently reparameterize |
| 438 |
> |
was what lead Tan {\it et al.} to recently reparameterize |
| 439 |
|
SSD.\cite{Ichiye03} Throughout the remainder of the paper our |
| 440 |
< |
reparamaterizations of SSD will be compared with the newer SSD1 model. |
| 440 |
> |
reparamaterizations of SSD will be compared with their newer SSD1 |
| 441 |
> |
model. |
| 442 |
|
|
| 443 |
|
\subsection{Transport Behavior} |
| 444 |
|
|
| 448 |
|
constant energy (NVE) simulations were performed at the average |
| 449 |
|
density obtained by the NPT simulations at an identical target |
| 450 |
|
temperature. Simulations started with randomized velocities and |
| 451 |
< |
underwent 50 ps of temperature scaling and 50 ps of constant energy |
| 452 |
< |
equilibration before a 200 ps data collection run. Diffusion constants |
| 451 |
> |
underwent 50~ps of temperature scaling and 50~ps of constant energy |
| 452 |
> |
equilibration before a 200~ps data collection run. Diffusion constants |
| 453 |
|
were calculated via linear fits to the long-time behavior of the |
| 454 |
|
mean-square displacement as a function of time. The averaged results |
| 455 |
|
from five sets of NVE simulations are displayed in figure |
| 456 |
|
\ref{diffuse}, alongside experimental, SPC/E, and TIP5P |
| 457 |
< |
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} |
| 457 |
> |
results.\cite{Gillen72,Holz00,Clancy94,Jorgensen01} |
| 458 |
|
|
| 459 |
< |
\begin{figure} |
| 460 |
< |
\begin{center} |
| 461 |
< |
\epsfxsize=6in |
| 462 |
< |
\epsfbox{betterDiffuse.epsi} |
| 463 |
< |
\caption{Average self-diffusion constant as a function of temperature for |
| 464 |
< |
SSD, SPC/E [Ref. \citen{Clancy94}], TIP5P [Ref. \citen{Jorgensen01}], |
| 465 |
< |
and Experimental data [Refs. \citen{Gillen72} and \citen{Mills73}]. Of |
| 466 |
< |
the three water models shown, SSD has the least deviation from the |
| 467 |
< |
experimental values. The rapidly increasing diffusion constants for |
| 468 |
< |
TIP5P and SSD correspond to significant decrease in density at the |
| 469 |
< |
higher temperatures.} |
| 470 |
< |
\label{diffuse} |
| 471 |
< |
\end{center} |
| 472 |
< |
\end{figure} |
| 459 |
> |
%\begin{figure} |
| 460 |
> |
%\begin{center} |
| 461 |
> |
%\epsfxsize=6in |
| 462 |
> |
%\epsfbox{betterDiffuse.epsi} |
| 463 |
> |
%\caption{Average self-diffusion constant as a function of temperature for |
| 464 |
> |
%SSD, SPC/E [Ref. \onlinecite{Clancy94}], and TIP5P |
| 465 |
> |
%[Ref. \onlinecite{Jorgensen01}] compared with experimental data |
| 466 |
> |
%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. Of the three water models |
| 467 |
> |
%shown, SSD has the least deviation from the experimental values. The |
| 468 |
> |
%rapidly increasing diffusion constants for TIP5P and SSD correspond to |
| 469 |
> |
%significant decreases in density at the higher temperatures.} |
| 470 |
> |
%\label{diffuse} |
| 471 |
> |
%\end{center} |
| 472 |
> |
%\end{figure} |
| 473 |
|
|
| 474 |
|
The observed values for the diffusion constant point out one of the |
| 475 |
|
strengths of the SSD model. Of the three models shown, the SSD model |
| 476 |
|
has the most accurate depiction of self-diffusion in both the |
| 477 |
|
supercooled and liquid regimes. SPC/E does a respectable job by |
| 478 |
< |
reproducing values similar to experiment around 290 K; however, it |
| 478 |
> |
reproducing values similar to experiment around 290~K; however, it |
| 479 |
|
deviates at both higher and lower temperatures, failing to predict the |
| 480 |
|
correct thermal trend. TIP5P and SSD both start off low at colder |
| 481 |
|
temperatures and tend to diffuse too rapidly at higher temperatures. |
| 482 |
|
This behavior at higher temperatures is not particularly surprising |
| 483 |
|
since the densities of both TIP5P and SSD are lower than experimental |
| 484 |
|
water densities at higher temperatures. When calculating the |
| 485 |
< |
diffusion coefficients for SSD at experimental densities (instead of |
| 486 |
< |
the densities from the NPT simulations), the resulting values fall |
| 487 |
< |
more in line with experiment at these temperatures. |
| 485 |
> |
diffusion coefficients for SSD at experimental densities |
| 486 |
> |
(instead of the densities from the NPT simulations), the resulting |
| 487 |
> |
values fall more in line with experiment at these temperatures. |
| 488 |
|
|
| 489 |
|
\subsection{Structural Changes and Characterization} |
| 490 |
|
|
| 494 |
|
capacity (C$_\text{p}$) was monitored to locate the melting transition |
| 495 |
|
in each of the simulations. In the melting simulations of the 1024 |
| 496 |
|
particle ice $I_h$ simulations, a large spike in C$_\text{p}$ occurs |
| 497 |
< |
at 245 K, indicating a first order phase transition for the melting of |
| 497 |
> |
at 245~K, indicating a first order phase transition for the melting of |
| 498 |
|
these ice crystals. When the reaction field is turned off, the melting |
| 499 |
< |
transition occurs at 235 K. These melting transitions are |
| 499 |
> |
transition occurs at 235~K. These melting transitions are |
| 500 |
|
considerably lower than the experimental value. |
| 501 |
|
|
| 502 |
< |
\begin{figure} |
| 503 |
< |
\begin{center} |
| 504 |
< |
\epsfxsize=6in |
| 505 |
< |
\epsfbox{corrDiag.eps} |
| 506 |
< |
\caption{Two dimensional illustration of angles involved in the |
| 507 |
< |
correlations observed in Fig. \ref{contour}.} |
| 508 |
< |
\label{corrAngle} |
| 509 |
< |
\end{center} |
| 505 |
< |
\end{figure} |
| 502 |
> |
%\begin{figure} |
| 503 |
> |
%\begin{center} |
| 504 |
> |
%\epsfxsize=6in |
| 505 |
> |
%\epsfbox{corrDiag.eps} |
| 506 |
> |
%\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.} |
| 507 |
> |
%\label{corrAngle} |
| 508 |
> |
%\end{center} |
| 509 |
> |
%\end{figure} |
| 510 |
|
|
| 511 |
< |
\begin{figure} |
| 512 |
< |
\begin{center} |
| 513 |
< |
\epsfxsize=6in |
| 514 |
< |
\epsfbox{fullContours.eps} |
| 515 |
< |
\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at |
| 516 |
< |
100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for |
| 517 |
< |
clarity: dark areas signify peaks while light areas signify |
| 518 |
< |
depressions. White areas have $g(r)$ values below 0.5 and black |
| 519 |
< |
areas have values above 1.5.} |
| 520 |
< |
\label{contour} |
| 521 |
< |
\end{center} |
| 522 |
< |
\end{figure} |
| 511 |
> |
%\begin{figure} |
| 512 |
> |
%\begin{center} |
| 513 |
> |
%\epsfxsize=6in |
| 514 |
> |
%\epsfbox{fullContours.eps} |
| 515 |
> |
%\caption{Contour plots of 2D angular pair correlation functions for |
| 516 |
> |
%512 SSD molecules at 100~K (A \& B) and 300~K (C \& D). Dark areas |
| 517 |
> |
%signify regions of enhanced density while light areas signify |
| 518 |
> |
%depletion relative to the bulk density. White areas have pair |
| 519 |
> |
%correlation values below 0.5 and black areas have values above 1.5.} |
| 520 |
> |
%\label{contour} |
| 521 |
> |
%\end{center} |
| 522 |
> |
%\end{figure} |
| 523 |
|
|
| 524 |
|
Additional analysis of the melting process was performed using |
| 525 |
|
two-dimensional structure and dipole angle correlations. Expressions |
| 555 |
|
|
| 556 |
|
This complex interplay between dipole and sticky interactions was |
| 557 |
|
remarked upon as a possible reason for the split second peak in the |
| 558 |
< |
oxygen-oxygen $g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, |
| 559 |
< |
the second solvation shell peak appears to have two distinct |
| 560 |
< |
components that blend together to form one observable peak. At higher |
| 561 |
< |
temperatures, this split character alters to show the leading 4 \AA\ |
| 562 |
< |
peak dominated by equatorial anti-parallel dipole orientations. There |
| 563 |
< |
is also a tightly bunched group of axially arranged dipoles that most |
| 564 |
< |
likely consist of the smaller fraction of aligned dipole pairs. The |
| 565 |
< |
trailing component of the split peak at 5 \AA\ is dominated by aligned |
| 566 |
< |
dipoles that assume hydrogen bond arrangements similar to those seen |
| 567 |
< |
in the first solvation shell. This evidence indicates that the dipole |
| 568 |
< |
pair interaction begins to dominate outside of the range of the |
| 569 |
< |
dipolar repulsion term. The energetically favorable dipole |
| 570 |
< |
arrangements populate the region immediately outside this repulsion |
| 571 |
< |
region (around 4 \AA), while arrangements that seek to satisfy both |
| 572 |
< |
the sticky and dipole forces locate themselves just beyond this |
| 573 |
< |
initial buildup (around 5 \AA). |
| 558 |
> |
oxygen-oxygen pair correlation function, |
| 559 |
> |
$g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, the second |
| 560 |
> |
solvation shell peak appears to have two distinct components that |
| 561 |
> |
blend together to form one observable peak. At higher temperatures, |
| 562 |
> |
this split character alters to show the leading 4~\AA\ peak dominated |
| 563 |
> |
by equatorial anti-parallel dipole orientations. There is also a |
| 564 |
> |
tightly bunched group of axially arranged dipoles that most likely |
| 565 |
> |
consist of the smaller fraction of aligned dipole pairs. The trailing |
| 566 |
> |
component of the split peak at 5~\AA\ is dominated by aligned dipoles |
| 567 |
> |
that assume hydrogen bond arrangements similar to those seen in the |
| 568 |
> |
first solvation shell. This evidence indicates that the dipole pair |
| 569 |
> |
interaction begins to dominate outside of the range of the dipolar |
| 570 |
> |
repulsion term. The energetically favorable dipole arrangements |
| 571 |
> |
populate the region immediately outside this repulsion region (around |
| 572 |
> |
4~\AA), while arrangements that seek to satisfy both the sticky and |
| 573 |
> |
dipole forces locate themselves just beyond this initial buildup |
| 574 |
> |
(around 5~\AA). |
| 575 |
|
|
| 576 |
|
From these findings, the split second peak is primarily the product of |
| 577 |
|
the dipolar repulsion term of the sticky potential. In fact, the inner |
| 578 |
|
peak can be pushed out and merged with the outer split peak just by |
| 579 |
|
extending the switching function ($s^\prime(r_{ij})$) from its normal |
| 580 |
< |
4.0 \AA\ cutoff to values of 4.5 or even 5 \AA. This type of |
| 580 |
> |
4.0~\AA\ cutoff to values of 4.5 or even 5~\AA. This type of |
| 581 |
|
correction is not recommended for improving the liquid structure, |
| 582 |
|
since the second solvation shell would still be shifted too far |
| 583 |
|
out. In addition, this would have an even more detrimental effect on |
| 584 |
|
the system densities, leading to a liquid with a more open structure |
| 585 |
< |
and a density considerably lower than the already low SSD density. A |
| 586 |
< |
better correction would be to include the quadrupole-quadrupole |
| 587 |
< |
interactions for the water particles outside of the first solvation |
| 588 |
< |
shell, but this would remove the simplicity and speed advantage of |
| 589 |
< |
SSD. |
| 585 |
> |
and a density considerably lower than the already low SSD |
| 586 |
> |
density. A better correction would be to include the |
| 587 |
> |
quadrupole-quadrupole interactions for the water particles outside of |
| 588 |
> |
the first solvation shell, but this would remove the simplicity and |
| 589 |
> |
speed advantage of SSD. |
| 590 |
|
|
| 591 |
|
\subsection{Adjusted Potentials: SSD/RF and SSD/E} |
| 592 |
|
|
| 601 |
|
|
| 602 |
|
The parameters available for tuning include the $\sigma$ and |
| 603 |
|
$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the |
| 604 |
< |
strength of the sticky potential ($\nu_0$), and the sticky attractive |
| 605 |
< |
and dipole repulsive cubic switching function cutoffs ($r_l$, $r_u$ |
| 606 |
< |
and $r_l^\prime$, $r_u^\prime$ respectively). The results of the |
| 607 |
< |
reparameterizations are shown in table \ref{params}. We are calling |
| 608 |
< |
these reparameterizations the Soft Sticky Dipole / Reaction Field |
| 609 |
< |
(SSD/RF - for use with a reaction field) and Soft Sticky Dipole |
| 610 |
< |
Extended (SSD/E - an attempt to improve the liquid structure in |
| 611 |
< |
simulations without a long-range correction). |
| 604 |
> |
strength of the sticky potential ($\nu_0$), and the cutoff distances |
| 605 |
> |
for the sticky attractive and dipole repulsive cubic switching |
| 606 |
> |
function cutoffs ($r_l$, $r_u$ and $r_l^\prime$, $r_u^\prime$ |
| 607 |
> |
respectively). The results of the reparameterizations are shown in |
| 608 |
> |
table \ref{params}. We are calling these reparameterizations the Soft |
| 609 |
> |
Sticky Dipole / Reaction Field (SSD/RF - for use with a reaction |
| 610 |
> |
field) and Soft Sticky Dipole Extended (SSD/E - an attempt to improve |
| 611 |
> |
the liquid structure in simulations without a long-range correction). |
| 612 |
|
|
| 613 |
|
\begin{table} |
| 614 |
|
\begin{center} |
| 615 |
|
\caption{Parameters for the original and adjusted models} |
| 616 |
|
\begin{tabular}{ l c c c c } |
| 617 |
|
\hline \\[-3mm] |
| 618 |
< |
\ \ \ Parameters\ \ \ & \ \ \ SSD [Ref. \citen{Ichiye96}] \ \ \ |
| 619 |
< |
& \ SSD1 [Ref. \citen{Ichiye03}]\ \ & \ SSD/E\ \ & \ SSD/RF \\ |
| 618 |
> |
\ \ \ Parameters\ \ \ & \ \ \ SSD [Ref. \onlinecite{Ichiye96}] \ \ \ |
| 619 |
> |
& \ SSD1 [Ref. \onlinecite{Ichiye03}]\ \ & \ SSD/E\ \ & \ \ SSD/RF \\ |
| 620 |
|
\hline \\[-3mm] |
| 621 |
|
\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
| 622 |
|
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
| 632 |
|
\end{center} |
| 633 |
|
\end{table} |
| 634 |
|
|
| 635 |
< |
\begin{figure} |
| 636 |
< |
\begin{center} |
| 637 |
< |
\epsfxsize=5in |
| 638 |
< |
\epsfbox{GofRCompare.epsi} |
| 639 |
< |
\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with SSD/E |
| 640 |
< |
and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with |
| 641 |
< |
reaction field turned on (bottom). The insets show the respective |
| 642 |
< |
first peaks in detail. Note how the changes in parameters have lowered |
| 643 |
< |
and broadened the first peak of SSD/E and SSD/RF.} |
| 644 |
< |
\label{grcompare} |
| 645 |
< |
\end{center} |
| 646 |
< |
\end{figure} |
| 635 |
> |
%\begin{figure} |
| 636 |
> |
%\begin{center} |
| 637 |
> |
%\epsfxsize=5in |
| 638 |
> |
%\epsfbox{GofRCompare.epsi} |
| 639 |
> |
%\caption{Plots comparing experiment [Ref. \onlinecite{Head-Gordon00_1}] with |
| 640 |
> |
%SSD/E and SSD1 without reaction field (top), as well as |
| 641 |
> |
%SSD/RF and SSD1 with reaction field turned on |
| 642 |
> |
%(bottom). The insets show the respective first peaks in detail. Note |
| 643 |
> |
%how the changes in parameters have lowered and broadened the first |
| 644 |
> |
%peak of SSD/E and SSD/RF.} |
| 645 |
> |
%\label{grcompare} |
| 646 |
> |
%\end{center} |
| 647 |
> |
%\end{figure} |
| 648 |
|
|
| 649 |
< |
\begin{figure} |
| 650 |
< |
\begin{center} |
| 651 |
< |
\epsfxsize=6in |
| 652 |
< |
\epsfbox{dualsticky.ps} |
| 653 |
< |
\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \& |
| 654 |
< |
SSD/RF (right). Light areas correspond to the tetrahedral attractive |
| 655 |
< |
component, and darker areas correspond to the dipolar repulsive |
| 656 |
< |
component.} |
| 657 |
< |
\label{isosurface} |
| 658 |
< |
\end{center} |
| 659 |
< |
\end{figure} |
| 649 |
> |
%\begin{figure} |
| 650 |
> |
%\begin{center} |
| 651 |
> |
%\epsfxsize=6in |
| 652 |
> |
%\epsfbox{dualsticky_bw.eps} |
| 653 |
> |
%\caption{Positive and negative isosurfaces of the sticky potential for |
| 654 |
> |
%SSD1 (left) and SSD/E \& SSD/RF (right). Light areas |
| 655 |
> |
%correspond to the tetrahedral attractive component, and darker areas |
| 656 |
> |
%correspond to the dipolar repulsive component.} |
| 657 |
> |
%\label{isosurface} |
| 658 |
> |
%\end{center} |
| 659 |
> |
%\end{figure} |
| 660 |
|
|
| 661 |
|
In the original paper detailing the development of SSD, Liu and Ichiye |
| 662 |
|
placed particular emphasis on an accurate description of the first |
| 666 |
|
Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering |
| 667 |
|
data from the Head-Gordon lab indicates a slightly lower and shifted |
| 668 |
|
first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were |
| 669 |
< |
made while taking into consideration the new experimental |
| 669 |
> |
made after taking into consideration the new experimental |
| 670 |
|
findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the |
| 671 |
|
relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing |
| 672 |
|
the revised SSD model (SSD1), SSD/E, and SSD/RF to the new |
| 682 |
|
see how altering the cutoffs changes the repulsive and attractive |
| 683 |
|
character of the particles. With a reduced repulsive surface (darker |
| 684 |
|
region), the particles can move closer to one another, increasing the |
| 685 |
< |
density for the overall system. This change in interaction cutoff also |
| 686 |
< |
results in a more gradual orientational motion by allowing the |
| 685 |
> |
density for the overall system. This change in interaction cutoff |
| 686 |
> |
also results in a more gradual orientational motion by allowing the |
| 687 |
|
particles to maintain preferred dipolar arrangements before they begin |
| 688 |
|
to feel the pull of the tetrahedral restructuring. As the particles |
| 689 |
|
move closer together, the dipolar repulsion term becomes active and |
| 692 |
|
particles feel the pull of the ``hydrogen bonds''. Aside from |
| 693 |
|
improving the shape of the first peak in the g(\emph{r}), this |
| 694 |
|
modification improves the densities considerably by allowing the |
| 695 |
< |
persistence of full dipolar character below the previous 4.0 \AA\ |
| 695 |
> |
persistence of full dipolar character below the previous 4.0~\AA\ |
| 696 |
|
cutoff. |
| 697 |
|
|
| 698 |
|
While adjusting the location and shape of the first peak of $g(r)$ |
| 702 |
|
both of our adjusted models. Since SSD is a dipole based model, the |
| 703 |
|
structure and transport are very sensitive to changes in the dipole |
| 704 |
|
moment. The original SSD simply used the dipole moment calculated from |
| 705 |
< |
the TIP3P water model, which at 2.35 D is significantly greater than |
| 706 |
< |
the experimental gas phase value of 1.84 D. The larger dipole moment |
| 705 |
> |
the TIP3P water model, which at 2.35~D is significantly greater than |
| 706 |
> |
the experimental gas phase value of 1.84~D. The larger dipole moment |
| 707 |
|
is a more realistic value and improves the dielectric properties of |
| 708 |
|
the fluid. Both theoretical and experimental measurements indicate a |
| 709 |
< |
liquid phase dipole moment ranging from 2.4 D to values as high as |
| 710 |
< |
3.11 D, providing a substantial range of reasonable values for a |
| 709 |
> |
liquid phase dipole moment ranging from 2.4~D to values as high as |
| 710 |
> |
3.11~D, providing a substantial range of reasonable values for a |
| 711 |
|
dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
| 712 |
< |
increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF, |
| 712 |
> |
increasing the dipole moments to 2.42 and 2.48~D for SSD/E and SSD/RF, |
| 713 |
|
respectively, leads to significant changes in the density and |
| 714 |
|
transport of the water models. |
| 715 |
|
|
| 722 |
|
results are obtained from five separate simulations of 1024 particle |
| 723 |
|
systems, and the melting sequences were started from different ice |
| 724 |
|
$I_h$ crystals constructed as described previously. Each NPT |
| 725 |
< |
simulation was equilibrated for 100 ps before a 200 ps data collection |
| 725 |
> |
simulation was equilibrated for 100~ps before a 200~ps data collection |
| 726 |
|
run at each temperature step, and the final configuration from the |
| 727 |
|
previous temperature simulation was used as a starting point. All NVE |
| 728 |
|
simulations had the same thermalization, equilibration, and data |
| 729 |
|
collection times as stated previously. |
| 730 |
|
|
| 731 |
< |
\begin{figure} |
| 732 |
< |
\begin{center} |
| 733 |
< |
\epsfxsize=6in |
| 734 |
< |
\epsfbox{ssdeDense.epsi} |
| 735 |
< |
\caption{Comparison of densities calculated with SSD/E to SSD1 without a |
| 736 |
< |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
| 737 |
< |
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and |
| 738 |
< |
experiment [Ref. \citen{CRC80}]. The window shows a expansion around |
| 739 |
< |
300 K with error bars included to clarify this region of |
| 740 |
< |
interest. Note that both SSD1 and SSD/E show good agreement with |
| 741 |
< |
experiment when the long-range correction is neglected.} |
| 742 |
< |
\label{ssdedense} |
| 743 |
< |
\end{center} |
| 744 |
< |
\end{figure} |
| 731 |
> |
%\begin{figure} |
| 732 |
> |
%\begin{center} |
| 733 |
> |
%\epsfxsize=6in |
| 734 |
> |
%\epsfbox{ssdeDense.epsi} |
| 735 |
> |
%\caption{Comparison of densities calculated with SSD/E to |
| 736 |
> |
%SSD1 without a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
| 737 |
> |
%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}] and |
| 738 |
> |
%experiment [Ref. \onlinecite{CRC80}]. The window shows a expansion around |
| 739 |
> |
%300 K with error bars included to clarify this region of |
| 740 |
> |
%interest. Note that both SSD1 and SSD/E show good agreement with |
| 741 |
> |
%experiment when the long-range correction is neglected.} |
| 742 |
> |
%\label{ssdedense} |
| 743 |
> |
%\end{center} |
| 744 |
> |
%\end{figure} |
| 745 |
|
|
| 746 |
< |
Fig. \ref{ssdedense} shows the density profile for the SSD/E model |
| 747 |
< |
in comparison to SSD1 without a reaction field, other common water |
| 748 |
< |
models, and experimental results. The calculated densities for both |
| 749 |
< |
SSD/E and SSD1 have increased significantly over the original SSD |
| 750 |
< |
model (see fig. \ref{dense1}) and are in better agreement with the |
| 751 |
< |
experimental values. At 298 K, the densities of SSD/E and SSD1 without |
| 746 |
> |
Fig. \ref{ssdedense} shows the density profile for the SSD/E |
| 747 |
> |
model in comparison to SSD1 without a reaction field, other |
| 748 |
> |
common water models, and experimental results. The calculated |
| 749 |
> |
densities for both SSD/E and SSD1 have increased |
| 750 |
> |
significantly over the original SSD model (see |
| 751 |
> |
fig. \ref{dense1}) and are in better agreement with the experimental |
| 752 |
> |
values. At 298 K, the densities of SSD/E and SSD1 without |
| 753 |
|
a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and |
| 754 |
|
0.999$\pm$0.001 g/cm$^3$ respectively. These both compare well with |
| 755 |
|
the experimental value of 0.997 g/cm$^3$, and they are considerably |
| 756 |
< |
better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The changes to |
| 757 |
< |
the dipole moment and sticky switching functions have improved the |
| 758 |
< |
structuring of the liquid (as seen in figure \ref{grcompare}, but they |
| 759 |
< |
have shifted the density maximum to much lower temperatures. This |
| 760 |
< |
comes about via an increase in the liquid disorder through the |
| 761 |
< |
weakening of the sticky potential and strengthening of the dipolar |
| 762 |
< |
character. However, this increasing disorder in the SSD/E model has |
| 763 |
< |
little effect on the melting transition. By monitoring $C_p$ |
| 764 |
< |
throughout these simulations, the melting transition for SSD/E was |
| 765 |
< |
shown to occur at 235 K. The same transition temperature observed |
| 766 |
< |
with SSD and SSD1. |
| 756 |
> |
better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The |
| 757 |
> |
changes to the dipole moment and sticky switching functions have |
| 758 |
> |
improved the structuring of the liquid (as seen in figure |
| 759 |
> |
\ref{grcompare}, but they have shifted the density maximum to much |
| 760 |
> |
lower temperatures. This comes about via an increase in the liquid |
| 761 |
> |
disorder through the weakening of the sticky potential and |
| 762 |
> |
strengthening of the dipolar character. However, this increasing |
| 763 |
> |
disorder in the SSD/E model has little effect on the melting |
| 764 |
> |
transition. By monitoring $C_p$ throughout these simulations, the |
| 765 |
> |
melting transition for SSD/E was shown to occur at 235~K. The |
| 766 |
> |
same transition temperature observed with SSD and SSD1. |
| 767 |
|
|
| 768 |
< |
\begin{figure} |
| 769 |
< |
\begin{center} |
| 770 |
< |
\epsfxsize=6in |
| 771 |
< |
\epsfbox{ssdrfDense.epsi} |
| 772 |
< |
\caption{Comparison of densities calculated with SSD/RF to SSD1 with a |
| 773 |
< |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
| 774 |
< |
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and |
| 775 |
< |
experiment [Ref. \citen{CRC80}]. The inset shows the necessity of |
| 776 |
< |
reparameterization when utilizing a reaction field long-ranged |
| 777 |
< |
correction - SSD/RF provides significantly more accurate densities |
| 778 |
< |
than SSD1 when performing room temperature simulations.} |
| 779 |
< |
\label{ssdrfdense} |
| 780 |
< |
\end{center} |
| 781 |
< |
\end{figure} |
| 768 |
> |
%\begin{figure} |
| 769 |
> |
%\begin{center} |
| 770 |
> |
%\epsfxsize=6in |
| 771 |
> |
%\epsfbox{ssdrfDense.epsi} |
| 772 |
> |
%\caption{Comparison of densities calculated with SSD/RF to |
| 773 |
> |
%SSD1 with a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
| 774 |
> |
%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}], and |
| 775 |
> |
%experiment [Ref. \onlinecite{CRC80}]. The inset shows the necessity of |
| 776 |
> |
%reparameterization when utilizing a reaction field long-ranged |
| 777 |
> |
%correction - SSD/RF provides significantly more accurate |
| 778 |
> |
%densities than SSD1 when performing room temperature |
| 779 |
> |
%simulations.} |
| 780 |
> |
%\label{ssdrfdense} |
| 781 |
> |
%\end{center} |
| 782 |
> |
%\end{figure} |
| 783 |
|
|
| 784 |
|
Including the reaction field long-range correction in the simulations |
| 785 |
|
results in a more interesting comparison. A density profile including |
| 793 |
|
further emphasize the importance of reparameterization in order to |
| 794 |
|
model the density properly under different simulation conditions. |
| 795 |
|
Again, these changes have only a minor effect on the melting point, |
| 796 |
< |
which observed at 245 K for SSD/RF, is identical to SSD and only 5 K |
| 796 |
> |
which observed at 245~K for SSD/RF, is identical to SSD and only 5~K |
| 797 |
|
lower than SSD1 with a reaction field. Additionally, the difference in |
| 798 |
|
density maxima is not as extreme, with SSD/RF showing a density |
| 799 |
< |
maximum at 255 K, fairly close to the density maxima of 260 K and 265 |
| 800 |
< |
K, shown by SSD and SSD1 respectively. |
| 799 |
> |
maximum at 255~K, fairly close to the density maxima of 260~K and |
| 800 |
> |
265~K, shown by SSD and SSD1 respectively. |
| 801 |
|
|
| 802 |
< |
\begin{figure} |
| 803 |
< |
\begin{center} |
| 804 |
< |
\epsfxsize=6in |
| 805 |
< |
\epsfbox{ssdeDiffuse.epsi} |
| 806 |
< |
\caption{The diffusion constants calculated from SSD/E and SSD1, |
| 807 |
< |
both without a reaction field, along with experimental results |
| 808 |
< |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
| 809 |
< |
were performed at the average densities observed in the 1 atm NPT |
| 810 |
< |
simulations for the respective models. SSD/E is slightly more mobile |
| 811 |
< |
than experiment at all of the temperatures, but it is closer to |
| 812 |
< |
experiment at biologically relavent temperatures than SSD1 without a |
| 813 |
< |
long-range correction.} |
| 814 |
< |
\label{ssdediffuse} |
| 815 |
< |
\end{center} |
| 816 |
< |
\end{figure} |
| 802 |
> |
%\begin{figure} |
| 803 |
> |
%\begin{center} |
| 804 |
> |
%\epsfxsize=6in |
| 805 |
> |
%\epsfbox{ssdeDiffuse.epsi} |
| 806 |
> |
%\caption{The diffusion constants calculated from SSD/E and |
| 807 |
> |
%SSD1 (both without a reaction field) along with experimental results |
| 808 |
> |
%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The NVE calculations were |
| 809 |
> |
%performed at the average densities observed in the 1 atm NPT |
| 810 |
> |
%simulations for the respective models. SSD/E is slightly more mobile |
| 811 |
> |
%than experiment at all of the temperatures, but it is closer to |
| 812 |
> |
%experiment at biologically relevant temperatures than SSD1 without a |
| 813 |
> |
%long-range correction.} |
| 814 |
> |
%\label{ssdediffuse} |
| 815 |
> |
%\end{center} |
| 816 |
> |
%\end{figure} |
| 817 |
|
|
| 818 |
|
The reparameterization of the SSD water model, both for use with and |
| 819 |
|
without an applied long-range correction, brought the densities up to |
| 820 |
|
what is expected for simulating liquid water. In addition to improving |
| 821 |
< |
the densities, it is important that the excellent diffusive behavior |
| 822 |
< |
of SSD be maintained or improved. Figure \ref{ssdediffuse} compares |
| 823 |
< |
the temperature dependence of the diffusion constant of SSD/E to SSD1 |
| 824 |
< |
without an active reaction field at the densities calculated from the |
| 825 |
< |
NPT simulations at 1 atm. The diffusion constant for SSD/E is |
| 826 |
< |
consistently higher than experiment, while SSD1 remains lower than |
| 827 |
< |
experiment until relatively high temperatures (around 360 K). Both |
| 828 |
< |
models follow the shape of the experimental curve well below 300 K but |
| 829 |
< |
tend to diffuse too rapidly at higher temperatures, as seen in SSD1's |
| 830 |
< |
crossing above 360 K. This increasing diffusion relative to the |
| 831 |
< |
experimental values is caused by the rapidly decreasing system density |
| 832 |
< |
with increasing temperature. Both SSD1 and SSD/E show this deviation |
| 833 |
< |
in diffusive behavior, but this trend has different implications on |
| 834 |
< |
the diffusive behavior of the models. While SSD1 shows more |
| 835 |
< |
experimentally accurate diffusive behavior in the high temperature |
| 836 |
< |
regimes, SSD/E shows more accurate behavior in the supercooled and |
| 837 |
< |
biologically relavent temperature ranges. Thus, the changes made to |
| 838 |
< |
improve the liquid structure may have had an adverse affect on the |
| 839 |
< |
density maximum, but they improve the transport behavior of SSD/E |
| 840 |
< |
relative to SSD1 under the most commonly simulated conditions. |
| 821 |
> |
the densities, it is important that the diffusive behavior of SSD be |
| 822 |
> |
maintained or improved. Figure \ref{ssdediffuse} compares the |
| 823 |
> |
temperature dependence of the diffusion constant of SSD/E to SSD1 |
| 824 |
> |
without an active reaction field at the densities calculated from |
| 825 |
> |
their respective NPT simulations at 1 atm. The diffusion constant for |
| 826 |
> |
SSD/E is consistently higher than experiment, while SSD1 remains lower |
| 827 |
> |
than experiment until relatively high temperatures (around 360 |
| 828 |
> |
K). Both models follow the shape of the experimental curve well below |
| 829 |
> |
300~K but tend to diffuse too rapidly at higher temperatures, as seen |
| 830 |
> |
in SSD1's crossing above 360~K. This increasing diffusion relative to |
| 831 |
> |
the experimental values is caused by the rapidly decreasing system |
| 832 |
> |
density with increasing temperature. Both SSD1 and SSD/E show this |
| 833 |
> |
deviation in particle mobility, but this trend has different |
| 834 |
> |
implications on the diffusive behavior of the models. While SSD1 |
| 835 |
> |
shows more experimentally accurate diffusive behavior in the high |
| 836 |
> |
temperature regimes, SSD/E shows more accurate behavior in the |
| 837 |
> |
supercooled and biologically relevant temperature ranges. Thus, the |
| 838 |
> |
changes made to improve the liquid structure may have had an adverse |
| 839 |
> |
affect on the density maximum, but they improve the transport behavior |
| 840 |
> |
of SSD/E relative to SSD1 under the most commonly simulated |
| 841 |
> |
conditions. |
| 842 |
|
|
| 843 |
< |
\begin{figure} |
| 844 |
< |
\begin{center} |
| 845 |
< |
\epsfxsize=6in |
| 846 |
< |
\epsfbox{ssdrfDiffuse.epsi} |
| 847 |
< |
\caption{The diffusion constants calculated from SSD/RF and SSD1, |
| 848 |
< |
both with an active reaction field, along with experimental results |
| 849 |
< |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
| 850 |
< |
were performed at the average densities observed in the 1 atm NPT |
| 851 |
< |
simulations for both of the models. Note how accurately SSD/RF |
| 852 |
< |
simulates the diffusion of water throughout this temperature |
| 853 |
< |
range. The more rapidly increasing diffusion constants at high |
| 854 |
< |
temperatures for both models is attributed to lower calculated |
| 855 |
< |
densities than those observed in experiment.} |
| 856 |
< |
\label{ssdrfdiffuse} |
| 857 |
< |
\end{center} |
| 858 |
< |
\end{figure} |
| 843 |
> |
%\begin{figure} |
| 844 |
> |
%\begin{center} |
| 845 |
> |
%\epsfxsize=6in |
| 846 |
> |
%\epsfbox{ssdrfDiffuse.epsi} |
| 847 |
> |
%\caption{The diffusion constants calculated from SSD/RF and |
| 848 |
> |
%SSD1 (both with an active reaction field) along with |
| 849 |
> |
%experimental results [Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The |
| 850 |
> |
%NVE calculations were performed at the average densities observed in |
| 851 |
> |
%the 1 atm NPT simulations for both of the models. SSD/RF |
| 852 |
> |
%simulates the diffusion of water throughout this temperature range |
| 853 |
> |
%very well. The rapidly increasing diffusion constants at high |
| 854 |
> |
%temperatures for both models can be attributed to lower calculated |
| 855 |
> |
%densities than those observed in experiment.} |
| 856 |
> |
%\label{ssdrfdiffuse} |
| 857 |
> |
%\end{center} |
| 858 |
> |
%\end{figure} |
| 859 |
|
|
| 860 |
|
In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are |
| 861 |
|
compared to SSD1 with an active reaction field. Note that SSD/RF |
| 863 |
|
throughout most of the temperature range shown and exhibiting only a |
| 864 |
|
slight increasing trend at higher temperatures. SSD1 tends to diffuse |
| 865 |
|
more slowly at low temperatures and deviates to diffuse too rapidly at |
| 866 |
< |
temperatures greater than 330 K. As stated above, this deviation away |
| 866 |
> |
temperatures greater than 330~K. As stated above, this deviation away |
| 867 |
|
from the ideal trend is due to a rapid decrease in density at higher |
| 868 |
|
temperatures. SSD/RF does not suffer from this problem as much as SSD1 |
| 869 |
|
because the calculated densities are closer to the experimental |
| 871 |
|
reparameterization when using an altered long-range correction. |
| 872 |
|
|
| 873 |
|
\begin{table} |
| 874 |
+ |
\begin{minipage}{\linewidth} |
| 875 |
+ |
\renewcommand{\thefootnote}{\thempfootnote} |
| 876 |
|
\begin{center} |
| 877 |
< |
\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}.} |
| 877 |
> |
\caption{Properties of the single-point water models compared with |
| 878 |
> |
experimental data at ambient conditions. Deviations of the of the |
| 879 |
> |
averages are given in parentheses.} |
| 880 |
|
\begin{tabular}{ l c c c c c } |
| 881 |
|
\hline \\[-3mm] |
| 882 |
< |
\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \ |
| 883 |
< |
\ & \ SSD/RF \ \ \ & \ Expt. \\ |
| 882 |
> |
\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ \ SSD/E \ \ \ & \ \ SSD1 (RF) \ \ |
| 883 |
> |
\ & \ \ SSD/RF \ \ \ & \ \ Expt. \\ |
| 884 |
|
\hline \\[-3mm] |
| 885 |
< |
\ \ \ $\rho$ (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 \\ |
| 886 |
< |
\ \ \ $C_p$ (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 \\ |
| 887 |
< |
\ \ \ $D$ ($10^{-5}$ 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}$ \\ |
| 888 |
< |
\ \ \ Coordination Number & 3.9 & 4.3 & 3.8 & 4.4 & 4.7$^\text{b}$ \\ |
| 889 |
< |
\ \ \ H-bonds per particle & 3.7 & 3.6 & 3.7 & 3.7 & 3.4$^\text{c}$ \\ |
| 890 |
< |
\ \ \ $\tau_1^\mu$ (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}$ \\ |
| 891 |
< |
\ \ \ $\tau_2^\mu$ (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}$ \\ |
| 885 |
> |
\ \ $\rho$ (g/cm$^3$) & 0.999(0.001) & 0.996(0.001) & 0.972(0.002) & 0.997(0.001) & 0.997 \\ |
| 886 |
> |
\ \ $C_p$ (cal/mol K) & 28.80(0.11) & 25.45(0.09) & 28.28(0.06) & 23.83(0.16) & 17.98 \\ |
| 887 |
> |
\ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78(0.7) & 2.51(0.18) & 2.00(0.17) & 2.32(0.06) & 2.299\cite{Mills73} \\ |
| 888 |
> |
\ \ Coordination Number ($n_C$) & 3.9 & 4.3 & 3.8 & 4.4 & |
| 889 |
> |
4.7\footnote{Calculated by integrating $g_{\text{OO}}(r)$ in |
| 890 |
> |
Ref. \onlinecite{Head-Gordon00_1}} \\ |
| 891 |
> |
\ \ H-bonds per particle ($n_H$) & 3.7 & 3.6 & 3.7 & 3.7 & |
| 892 |
> |
3.5\footnote{Calculated by integrating $g_{\text{OH}}(r)$ in |
| 893 |
> |
Ref. \onlinecite{Soper86}} \\ |
| 894 |
> |
\ \ $\tau_1$ (ps) & 10.9(0.6) & 7.3(0.4) & 7.5(0.7) & 7.2(0.4) & 5.7\footnote{Calculated for 298 K from data in Ref. \onlinecite{Eisenberg69}} \\ |
| 895 |
> |
\ \ $\tau_2$ (ps) & 4.7(0.4) & 3.1(0.2) & 3.5(0.3) & 3.2(0.2) & 2.3\footnote{Calculated for 298 K from data in |
| 896 |
> |
Ref. \onlinecite{Krynicki66}} |
| 897 |
|
\end{tabular} |
| 898 |
|
\label{liquidproperties} |
| 899 |
|
\end{center} |
| 900 |
+ |
\end{minipage} |
| 901 |
|
\end{table} |
| 902 |
|
|
| 903 |
|
Table \ref{liquidproperties} gives a synopsis of the liquid state |
| 904 |
|
properties of the water models compared in this study along with the |
| 905 |
|
experimental values for liquid water at ambient conditions. The |
| 906 |
< |
coordination number and hydrogen bonds per particle were calculated by |
| 907 |
< |
integrating the following relation: |
| 906 |
> |
coordination number ($n_C$) and number of hydrogen bonds per particle |
| 907 |
> |
($n_H$) were calculated by integrating the following relations: |
| 908 |
|
\begin{equation} |
| 909 |
< |
4\pi\rho\int_{0}^{a}r^2\text{g}(r)dr, |
| 909 |
> |
n_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2\text{g}_{\text{OO}}(r)dr, |
| 910 |
|
\end{equation} |
| 911 |
< |
where $\rho$ is the number density of pair interactions, $a$ is the |
| 912 |
< |
radial location of the minima following the first solvation shell |
| 913 |
< |
peak, and g$(r)$ is either g$_\text{OO}(r)$ or g$_\text{OH}(r)$ for |
| 914 |
< |
calculation of the coordination number or hydrogen bonds per particle |
| 915 |
< |
respectively. The number of hydrogen bonds stays relatively constant |
| 916 |
< |
across all of the models, but the coordination numbers of SSD/E and |
| 917 |
< |
SSD/RF show an improvement over SSD1. This improvement is primarily |
| 918 |
< |
due to the widening of the first solvation shell peak, allowing the |
| 919 |
< |
first minima to push outward. Comparing the coordination number with |
| 920 |
< |
the number of hydrogen bonds can lead to more insight into the |
| 921 |
< |
structural character of the liquid. Because of the near identical |
| 922 |
< |
values for SSD1, it appears to be a little too exclusive, in that all |
| 923 |
< |
molecules in the first solvation shell are involved in forming ideal |
| 924 |
< |
hydrogen bonds. The differing numbers for the newly parameterized |
| 925 |
< |
models indicate the allowance of more fluid configurations in addition |
| 926 |
< |
to the formation of an acceptable number of ideal hydrogen bonds. |
| 911 |
> |
\begin{equation} |
| 912 |
> |
n_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2\text{g}_{\text{OH}}(r)dr, |
| 913 |
> |
\end{equation} |
| 914 |
> |
where $\rho$ is the number density of the specified pair interactions, |
| 915 |
> |
$a$ and $b$ are the radial locations of the minima following the first |
| 916 |
> |
peak in g$_\text{OO}(r)$ or g$_\text{OH}(r)$ respectively. The number |
| 917 |
> |
of hydrogen bonds stays relatively constant across all of the models, |
| 918 |
> |
but the coordination numbers of SSD/E and SSD/RF show an |
| 919 |
> |
improvement over SSD1. This improvement is primarily due to |
| 920 |
> |
extension of the first solvation shell in the new parameter sets. |
| 921 |
> |
Because $n_H$ and $n_C$ are nearly identical in SSD1, it appears |
| 922 |
> |
that all molecules in the first solvation shell are involved in |
| 923 |
> |
hydrogen bonds. Since $n_H$ and $n_C$ differ in the newly |
| 924 |
> |
parameterized models, the orientations in the first solvation shell |
| 925 |
> |
are a bit more ``fluid''. Therefore SSD1 overstructures the |
| 926 |
> |
first solvation shell and our reparameterizations have returned this |
| 927 |
> |
shell to more realistic liquid-like behavior. |
| 928 |
|
|
| 929 |
< |
The time constants for the self orientational autocorrelation function |
| 929 |
> |
The time constants for the orientational autocorrelation functions |
| 930 |
|
are also displayed in Table \ref{liquidproperties}. The dipolar |
| 931 |
< |
orientational time correlation function ($\Gamma_{l}$) is described |
| 931 |
> |
orientational time correlation functions ($C_{l}$) are described |
| 932 |
|
by: |
| 933 |
|
\begin{equation} |
| 934 |
< |
\Gamma_{l}(t) = \langle P_l[\mathbf{u}_j(0)\cdot\mathbf{u}_j(t)]\rangle, |
| 934 |
> |
C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{\mathbf{u}}_j(t)]\rangle, |
| 935 |
|
\end{equation} |
| 936 |
< |
where $P_l$ is a Legendre polynomial of order $l$ and $\mathbf{u}_j$ |
| 937 |
< |
is the unit vector of the particle dipole.\cite{Rahman71} From these |
| 938 |
< |
correlation functions, the orientational relaxation time of the dipole |
| 939 |
< |
vector can be calculated from an exponential fit in the long-time |
| 940 |
< |
regime ($t > \tau_l^\mu$).\cite{Rothschild84} Calculation of these |
| 941 |
< |
time constants were averaged from five detailed NVE simulations |
| 942 |
< |
performed at the STP density for each of the respective models. Again, |
| 943 |
< |
SSD/E and SSD/RF show improved behavior over SSD1 both with and |
| 944 |
< |
without an active reaction field. Numbers published from the original |
| 945 |
< |
SSD dynamics studies appear closer to the experimental values, and we |
| 946 |
< |
attribute this discrepancy to the implimentation of an Ewald sum |
| 947 |
< |
versus a reaction field. |
| 936 |
> |
where $P_l$ are Legendre polynomials of order $l$ and |
| 937 |
> |
$\hat{\mathbf{u}}_j$ is the unit vector pointing along the molecular |
| 938 |
> |
dipole.\cite{Rahman71} From these correlation functions, the |
| 939 |
> |
orientational relaxation time of the dipole vector can be calculated |
| 940 |
> |
from an exponential fit in the long-time regime ($t > |
| 941 |
> |
\tau_l$).\cite{Rothschild84} Calculation of these time constants were |
| 942 |
> |
averaged over five detailed NVE simulations performed at the ambient |
| 943 |
> |
conditions for each of the respective models. It should be noted that |
| 944 |
> |
the commonly cited value of 1.9 ps for $\tau_2$ was determined from |
| 945 |
> |
the NMR data in Ref. \onlinecite{Krynicki66} at a temperature near |
| 946 |
> |
34$^\circ$C.\cite{Rahman71} Because of the strong temperature |
| 947 |
> |
dependence of $\tau_2$, it is necessary to recalculate it at 298~K to |
| 948 |
> |
make proper comparisons. The value shown in Table |
| 949 |
> |
\ref{liquidproperties} was calculated from the same NMR data in the |
| 950 |
> |
fashion described in Ref. \onlinecite{Krynicki66}. Similarly, $\tau_1$ was |
| 951 |
> |
recomputed for 298~K from the data in Ref. \onlinecite{Eisenberg69}. |
| 952 |
> |
Again, SSD/E and SSD/RF show improved behavior over SSD1, both with |
| 953 |
> |
and without an active reaction field. Turning on the reaction field |
| 954 |
> |
leads to much improved time constants for SSD1; however, these results |
| 955 |
> |
also include a corresponding decrease in system density. |
| 956 |
> |
Orientational relaxation times published in the original SSD dynamics |
| 957 |
> |
papers are smaller than the values observed here, and this difference |
| 958 |
> |
can be attributed to the use of the Ewald sum.\cite{Ichiye99} |
| 959 |
|
|
| 960 |
|
\subsection{Additional Observations} |
| 961 |
|
|
| 962 |
< |
\begin{figure} |
| 963 |
< |
\begin{center} |
| 964 |
< |
\epsfxsize=6in |
| 965 |
< |
\epsfbox{povIce.ps} |
| 966 |
< |
\caption{A water lattice built from the crystal structure assumed by |
| 967 |
< |
SSD/E when undergoing an extremely restricted temperature NPT |
| 968 |
< |
simulation. This form of ice is referred to as ice-{\it i} to |
| 969 |
< |
emphasize its simulation origins. This image was taken of the (001) |
| 970 |
< |
face of the crystal.} |
| 971 |
< |
\label{weirdice} |
| 972 |
< |
\end{center} |
| 942 |
< |
\end{figure} |
| 962 |
> |
%\begin{figure} |
| 963 |
> |
%\begin{center} |
| 964 |
> |
%\epsfxsize=6in |
| 965 |
> |
%\epsfbox{icei_bw.eps} |
| 966 |
> |
%\caption{The most stable crystal structure assumed by the SSD family |
| 967 |
> |
%of water models. We refer to this structure as Ice-{\it i} to |
| 968 |
> |
%indicate its origins in computer simulation. This image was taken of |
| 969 |
> |
%the (001) face of the crystal.} |
| 970 |
> |
%\label{weirdice} |
| 971 |
> |
%\end{center} |
| 972 |
> |
%\end{figure} |
| 973 |
|
|
| 974 |
|
While performing a series of melting simulations on an early iteration |
| 975 |
< |
of SSD/E not discussed in this paper, we observed recrystallization |
| 976 |
< |
into a novel structure not previously known for water. After melting |
| 977 |
< |
at 235 K, two of five systems underwent crystallization events near |
| 978 |
< |
245 K. The two systems remained crystalline up to 320 and 330 K, |
| 979 |
< |
respectively. The crystal exhibits an expanded zeolite-like structure |
| 980 |
< |
that does not correspond to any known form of ice. This appears to be |
| 981 |
< |
an artifact of the point dipolar models, so to distinguish it from the |
| 982 |
< |
experimentally observed forms of ice, we have denoted the structure |
| 983 |
< |
Ice-$\sqrt{\smash[b]{-\text{I}}}$ (ice-{\it i}). A large enough |
| 975 |
> |
of SSD/E not discussed in this paper, we observed |
| 976 |
> |
recrystallization into a novel structure not previously known for |
| 977 |
> |
water. After melting at 235~K, two of five systems underwent |
| 978 |
> |
crystallization events near 245~K. The two systems remained |
| 979 |
> |
crystalline up to 320 and 330~K, respectively. The crystal exhibits |
| 980 |
> |
an expanded zeolite-like structure that does not correspond to any |
| 981 |
> |
known form of ice. This appears to be an artifact of the point |
| 982 |
> |
dipolar models, so to distinguish it from the experimentally observed |
| 983 |
> |
forms of ice, we have denoted the structure |
| 984 |
> |
Ice-$\sqrt{\smash[b]{-\text{I}}}$ (Ice-{\it i}). A large enough |
| 985 |
|
portion of the sample crystallized that we have been able to obtain a |
| 986 |
< |
near ideal crystal structure of ice-{\it i}. Figure \ref{weirdice} |
| 986 |
> |
near ideal crystal structure of Ice-{\it i}. Figure \ref{weirdice} |
| 987 |
|
shows the repeating crystal structure of a typical crystal at 5 |
| 988 |
|
K. Each water molecule is hydrogen bonded to four others; however, the |
| 989 |
|
hydrogen bonds are bent rather than perfectly straight. This results |
| 994 |
|
configuration. Though not ideal, these flexed hydrogen bonds are |
| 995 |
|
favorable enough to stabilize an entire crystal generated around them. |
| 996 |
|
|
| 997 |
< |
Initial simulations indicated that ice-{\it i} is the preferred ice |
| 997 |
> |
Initial simulations indicated that Ice-{\it i} is the preferred ice |
| 998 |
|
structure for at least the SSD/E model. To verify this, a comparison |
| 999 |
|
was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and |
| 1000 |
< |
ice-{\it i} at constant pressure with SSD/E, SSD/RF, and |
| 1000 |
> |
Ice-{\it i} at constant pressure with SSD/E, SSD/RF, and |
| 1001 |
|
SSD1. Near-ideal versions of the three types of crystals were cooled |
| 1002 |
< |
to 1 K, and the enthalpies of each were compared using all three water |
| 1003 |
< |
models. With every model in the SSD family, ice-{\it i} had the lowest |
| 1004 |
< |
calculated enthalpy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with |
| 1005 |
< |
SSD/E, and 7.5\% lower with SSD/RF. The enthalpy data is summarized |
| 1006 |
< |
in Table \ref{iceenthalpy}. |
| 1002 |
> |
to 1 K, and enthalpies of formation of each were compared using all |
| 1003 |
> |
three water models. Enthalpies were estimated from the |
| 1004 |
> |
isobaric-isothermal simulations using $H=U+P_{\text ext}V$ where |
| 1005 |
> |
$P_{\text ext}$ is the applied pressure. A constant value of -60.158 |
| 1006 |
> |
kcal / mol has been added to place our zero for the enthalpies of |
| 1007 |
> |
formation for these systems at the traditional state (elemental forms |
| 1008 |
> |
at standard temperature and pressure). With every model in the SSD |
| 1009 |
> |
family, Ice-{\it i} had the lowest calculated enthalpy of formation. |
| 1010 |
|
|
| 1011 |
|
\begin{table} |
| 1012 |
|
\begin{center} |
| 1013 |
< |
\caption{Enthalpies (in kcal / mol) of the three crystal structures (at 1 |
| 1014 |
< |
K) exhibited by the SSD family of water models} |
| 1013 |
> |
\caption{Enthalpies of Formation (in kcal / mol) of the three crystal |
| 1014 |
> |
structures (at 1 K) exhibited by the SSD family of water models} |
| 1015 |
|
\begin{tabular}{ l c c c } |
| 1016 |
|
\hline \\[-3mm] |
| 1017 |
< |
\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \ & \ |
| 1018 |
< |
Ice-{\it i} \\ |
| 1017 |
> |
\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ \ \ Ice-$I_c$ \ \ \ & |
| 1018 |
> |
\ \ \ \ Ice-{\it i} \\ |
| 1019 |
|
\hline \\[-3mm] |
| 1020 |
< |
\ \ \ SSD/E & -12.286 & -12.292 & -13.590 \\ |
| 1021 |
< |
\ \ \ SSD/RF & -12.935 & -12.917 & -14.022 \\ |
| 1022 |
< |
\ \ \ SSD1 & -12.496 & -12.411 & -13.417 \\ |
| 1023 |
< |
\ \ \ SSD1 (RF) & -12.504 & -12.411 & -13.134 \\ |
| 1020 |
> |
\ \ \ SSD/E & -72.444 & -72.450 & -73.748 \\ |
| 1021 |
> |
\ \ \ SSD/RF & -73.093 & -73.075 & -74.180 \\ |
| 1022 |
> |
\ \ \ SSD1 & -72.654 & -72.569 & -73.575 \\ |
| 1023 |
> |
\ \ \ SSD1 (RF) & -72.662 & -72.569 & -73.292 \\ |
| 1024 |
|
\end{tabular} |
| 1025 |
|
\label{iceenthalpy} |
| 1026 |
|
\end{center} |
| 1027 |
|
\end{table} |
| 1028 |
|
|
| 1029 |
|
In addition to these energetic comparisons, melting simulations were |
| 1030 |
< |
performed with ice-{\it i} as the initial configuration using SSD/E, |
| 1030 |
> |
performed with Ice-{\it i} as the initial configuration using SSD/E, |
| 1031 |
|
SSD/RF, and SSD1 both with and without a reaction field. The melting |
| 1032 |
|
transitions for both SSD/E and SSD1 without reaction field occurred at |
| 1033 |
|
temperature in excess of 375~K. SSD/RF and SSD1 with a reaction field |
| 1038 |
|
\section{Conclusions} |
| 1039 |
|
|
| 1040 |
|
The density maximum and temperature dependence of the self-diffusion |
| 1041 |
< |
constant were studied for the SSD water model, both with and without |
| 1042 |
< |
the use of reaction field, via a series of NPT and NVE |
| 1041 |
> |
constant were studied for the SSD water model, both with and |
| 1042 |
> |
without the use of reaction field, via a series of NPT and NVE |
| 1043 |
|
simulations. The constant pressure simulations showed a density |
| 1044 |
|
maximum near 260 K. In most cases, the calculated densities were |
| 1045 |
|
significantly lower than the densities obtained from other water |
| 1046 |
< |
models (and experiment). Analysis of self-diffusion showed SSD to |
| 1047 |
< |
capture the transport properties of water well in both the liquid and |
| 1048 |
< |
super-cooled liquid regimes. |
| 1046 |
> |
models (and experiment). Analysis of self-diffusion showed SSD |
| 1047 |
> |
to capture the transport properties of water well in both the liquid |
| 1048 |
> |
and supercooled liquid regimes. |
| 1049 |
|
|
| 1050 |
|
In order to correct the density behavior, the original SSD model was |
| 1051 |
|
reparameterized for use both with and without a reaction field (SSD/RF |
| 1059 |
|
simulations of biochemical systems. |
| 1060 |
|
|
| 1061 |
|
The existence of a novel low-density ice structure that is preferred |
| 1062 |
< |
by the SSD family of water models is somewhat troubling, since liquid |
| 1063 |
< |
simulations on this family of water models at room temperature are |
| 1064 |
< |
effectively simulations of super-cooled or metastable liquids. One |
| 1065 |
< |
way to de-stabilize this unphysical ice structure would be to make the |
| 1062 |
> |
by the SSD family of water models is somewhat troubling, since |
| 1063 |
> |
liquid simulations on this family of water models at room temperature |
| 1064 |
> |
are effectively simulations of supercooled or metastable liquids. One |
| 1065 |
> |
way to destabilize this unphysical ice structure would be to make the |
| 1066 |
|
range of angles preferred by the attractive part of the sticky |
| 1067 |
|
potential much narrower. This would require extensive |
| 1068 |
|
reparameterization to maintain the same level of agreement with the |
| 1069 |
|
experiments. |
| 1070 |
|
|
| 1071 |
< |
Additionally, our initial calculations show that the ice-{\it i} |
| 1071 |
> |
Additionally, our initial calculations show that the Ice-{\it i} |
| 1072 |
|
structure may also be a preferred crystal structure for at least one |
| 1073 |
|
other popular multi-point water model (TIP3P), and that much of the |
| 1074 |
|
simulation work being done using this popular model could also be at |
| 1085 |
|
\newpage |
| 1086 |
|
|
| 1087 |
|
\bibliographystyle{jcp} |
| 1088 |
< |
\bibliography{nptSSD} |
| 1088 |
> |
\bibliography{nptSSD} |
| 1089 |
|
|
| 1090 |
< |
%\pagebreak |
| 1090 |
> |
\newpage |
| 1091 |
|
|
| 1092 |
+ |
\begin{list} |
| 1093 |
+ |
{Figure \arabic{captions}: }{\usecounter{captions} |
| 1094 |
+ |
\setlength{\rightmargin}{\leftmargin}} |
| 1095 |
+ |
|
| 1096 |
+ |
\item Energy conservation using both quaternion-based integration and |
| 1097 |
+ |
the {\sc dlm} method with increasing time step. The larger time step |
| 1098 |
+ |
plots are shifted from the true energy baseline (that of $\Delta t$ = |
| 1099 |
+ |
0.1~fs) for clarity. |
| 1100 |
+ |
|
| 1101 |
+ |
\item Density versus temperature for TIP4P [Ref. \onlinecite{Jorgensen98b}], |
| 1102 |
+ |
TIP3P [Ref. \onlinecite{Jorgensen98b}], SPC/E |
| 1103 |
+ |
[Ref. \onlinecite{Clancy94}], SSD without Reaction Field, SSD, and |
| 1104 |
+ |
experiment [Ref. \onlinecite{CRC80}]. The arrows indicate the change |
| 1105 |
+ |
in densities observed when turning off the reaction field. The the |
| 1106 |
+ |
lower than expected densities for the SSD model were what prompted the |
| 1107 |
+ |
original reparameterization of SSD1 [Ref. \onlinecite{Ichiye03}]. |
| 1108 |
+ |
|
| 1109 |
+ |
\item Average self-diffusion constant as a function of temperature for |
| 1110 |
+ |
SSD, SPC/E [Ref. \onlinecite{Clancy94}], and TIP5P |
| 1111 |
+ |
[Ref. \onlinecite{Jorgensen01}] compared with experimental data |
| 1112 |
+ |
[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. Of the three |
| 1113 |
+ |
water models shown, SSD has the least deviation from the experimental |
| 1114 |
+ |
values. The rapidly increasing diffusion constants for TIP5P and SSD |
| 1115 |
+ |
correspond to significant decreases in density at the higher |
| 1116 |
+ |
temperatures. |
| 1117 |
+ |
|
| 1118 |
+ |
\item An illustration of angles involved in the correlations observed in |
| 1119 |
+ |
Fig. \ref{contour}. |
| 1120 |
+ |
|
| 1121 |
+ |
\item Contour plots of 2D angular pair correlation functions for |
| 1122 |
+ |
512 SSD molecules at 100~K (A \& B) and 300~K (C \& D). Dark areas |
| 1123 |
+ |
signify regions of enhanced density while light areas signify |
| 1124 |
+ |
depletion relative to the bulk density. White areas have pair |
| 1125 |
+ |
correlation values below 0.5 and black areas have values above 1.5. |
| 1126 |
+ |
|
| 1127 |
+ |
\item Plots comparing experiment [Ref. \onlinecite{Head-Gordon00_1}] with |
| 1128 |
+ |
SSD/E and SSD1 without reaction field (top), as well as SSD/RF and |
| 1129 |
+ |
SSD1 with reaction field turned on (bottom). The insets show the |
| 1130 |
+ |
respective first peaks in detail. Note how the changes in parameters |
| 1131 |
+ |
have lowered and broadened the first peak of SSD/E and SSD/RF. |
| 1132 |
+ |
|
| 1133 |
+ |
\item Positive and negative isosurfaces of the sticky potential for |
| 1134 |
+ |
SSD1 (left) and SSD/E \& SSD/RF (right). Light areas |
| 1135 |
+ |
correspond to the tetrahedral attractive component, and darker areas |
| 1136 |
+ |
correspond to the dipolar repulsive component. |
| 1137 |
+ |
|
| 1138 |
+ |
\item Comparison of densities calculated with SSD/E to |
| 1139 |
+ |
SSD1 without a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
| 1140 |
+ |
TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}] and |
| 1141 |
+ |
experiment [Ref. \onlinecite{CRC80}]. The window shows a expansion around |
| 1142 |
+ |
300 K with error bars included to clarify this region of |
| 1143 |
+ |
interest. Note that both SSD1 and SSD/E show good agreement with |
| 1144 |
+ |
experiment when the long-range correction is neglected. |
| 1145 |
+ |
|
| 1146 |
+ |
\item Comparison of densities calculated with SSD/RF to |
| 1147 |
+ |
SSD1 with a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
| 1148 |
+ |
TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}], and |
| 1149 |
+ |
experiment [Ref. \onlinecite{CRC80}]. The inset shows the necessity of |
| 1150 |
+ |
reparameterization when utilizing a reaction field long-ranged |
| 1151 |
+ |
correction - SSD/RF provides significantly more accurate |
| 1152 |
+ |
densities than SSD1 when performing room temperature |
| 1153 |
+ |
simulations. |
| 1154 |
+ |
|
| 1155 |
+ |
\item The diffusion constants calculated from SSD/E and |
| 1156 |
+ |
SSD1 (both without a reaction field) along with experimental results |
| 1157 |
+ |
[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The NVE calculations were |
| 1158 |
+ |
performed at the average densities observed in the 1 atm NPT |
| 1159 |
+ |
simulations for the respective models. SSD/E is slightly more mobile |
| 1160 |
+ |
than experiment at all of the temperatures, but it is closer to |
| 1161 |
+ |
experiment at biologically relevant temperatures than SSD1 without a |
| 1162 |
+ |
long-range correction. |
| 1163 |
+ |
|
| 1164 |
+ |
\item The diffusion constants calculated from SSD/RF and |
| 1165 |
+ |
SSD1 (both with an active reaction field) along with |
| 1166 |
+ |
experimental results [Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The |
| 1167 |
+ |
NVE calculations were performed at the average densities observed in |
| 1168 |
+ |
the 1 atm NPT simulations for both of the models. SSD/RF |
| 1169 |
+ |
simulates the diffusion of water throughout this temperature range |
| 1170 |
+ |
very well. The rapidly increasing diffusion constants at high |
| 1171 |
+ |
temperatures for both models can be attributed to lower calculated |
| 1172 |
+ |
densities than those observed in experiment. |
| 1173 |
+ |
|
| 1174 |
+ |
\item The most stable crystal structure assumed by the SSD family |
| 1175 |
+ |
of water models. We refer to this structure as Ice-{\it i} to |
| 1176 |
+ |
indicate its origins in computer simulation. This image was taken of |
| 1177 |
+ |
the (001) face of the crystal. |
| 1178 |
+ |
\end{list} |
| 1179 |
+ |
|
| 1180 |
+ |
\newpage |
| 1181 |
+ |
|
| 1182 |
+ |
\begin{figure} |
| 1183 |
+ |
\begin{center} |
| 1184 |
+ |
\epsfxsize=6in |
| 1185 |
+ |
\epsfbox{timeStep.epsi} |
| 1186 |
+ |
%\caption{Energy conservation using both quaternion-based integration and |
| 1187 |
+ |
%the {\sc dlm} method with increasing time step. The larger time step |
| 1188 |
+ |
%plots are shifted from the true energy baseline (that of $\Delta t$ = |
| 1189 |
+ |
%0.1~fs) for clarity.} |
| 1190 |
+ |
\label{timestep} |
| 1191 |
+ |
\end{center} |
| 1192 |
+ |
\end{figure} |
| 1193 |
+ |
|
| 1194 |
+ |
\newpage |
| 1195 |
+ |
|
| 1196 |
+ |
\begin{figure} |
| 1197 |
+ |
\begin{center} |
| 1198 |
+ |
\epsfxsize=6in |
| 1199 |
+ |
\epsfbox{denseSSDnew.eps} |
| 1200 |
+ |
%\caption{Density versus temperature for TIP4P [Ref. \onlinecite{Jorgensen98b}], |
| 1201 |
+ |
% TIP3P [Ref. \onlinecite{Jorgensen98b}], SPC/E [Ref. \onlinecite{Clancy94}], SSD |
| 1202 |
+ |
% without Reaction Field, SSD, and experiment [Ref. \onlinecite{CRC80}]. The |
| 1203 |
+ |
% arrows indicate the change in densities observed when turning off the |
| 1204 |
+ |
% reaction field. The the lower than expected densities for the SSD |
| 1205 |
+ |
% model were what prompted the original reparameterization of SSD1 |
| 1206 |
+ |
% [Ref. \onlinecite{Ichiye03}].} |
| 1207 |
+ |
\label{dense1} |
| 1208 |
+ |
\end{center} |
| 1209 |
+ |
\end{figure} |
| 1210 |
+ |
|
| 1211 |
+ |
\newpage |
| 1212 |
+ |
|
| 1213 |
+ |
\begin{figure} |
| 1214 |
+ |
\begin{center} |
| 1215 |
+ |
\epsfxsize=6in |
| 1216 |
+ |
\epsfbox{betterDiffuse.epsi} |
| 1217 |
+ |
%\caption{Average self-diffusion constant as a function of temperature for |
| 1218 |
+ |
%SSD, SPC/E [Ref. \onlinecite{Clancy94}], and TIP5P |
| 1219 |
+ |
%[Ref. \onlinecite{Jorgensen01}] compared with experimental data |
| 1220 |
+ |
%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. Of the three water models |
| 1221 |
+ |
%shown, SSD has the least deviation from the experimental values. The |
| 1222 |
+ |
%rapidly increasing diffusion constants for TIP5P and SSD correspond to |
| 1223 |
+ |
%significant decreases in density at the higher temperatures.} |
| 1224 |
+ |
\label{diffuse} |
| 1225 |
+ |
\end{center} |
| 1226 |
+ |
\end{figure} |
| 1227 |
+ |
|
| 1228 |
+ |
\newpage |
| 1229 |
+ |
|
| 1230 |
+ |
\begin{figure} |
| 1231 |
+ |
\begin{center} |
| 1232 |
+ |
\epsfxsize=6in |
| 1233 |
+ |
\epsfbox{corrDiag.eps} |
| 1234 |
+ |
%\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.} |
| 1235 |
+ |
\label{corrAngle} |
| 1236 |
+ |
\end{center} |
| 1237 |
+ |
\end{figure} |
| 1238 |
+ |
|
| 1239 |
+ |
\newpage |
| 1240 |
+ |
|
| 1241 |
+ |
\begin{figure} |
| 1242 |
+ |
\begin{center} |
| 1243 |
+ |
\epsfxsize=6in |
| 1244 |
+ |
\epsfbox{fullContours.eps} |
| 1245 |
+ |
%\caption{Contour plots of 2D angular pair correlation functions for |
| 1246 |
+ |
%512 SSD molecules at 100~K (A \& B) and 300~K (C \& D). Dark areas |
| 1247 |
+ |
%signify regions of enhanced density while light areas signify |
| 1248 |
+ |
%depletion relative to the bulk density. White areas have pair |
| 1249 |
+ |
%correlation values below 0.5 and black areas have values above 1.5.} |
| 1250 |
+ |
\label{contour} |
| 1251 |
+ |
\end{center} |
| 1252 |
+ |
\end{figure} |
| 1253 |
+ |
|
| 1254 |
+ |
\newpage |
| 1255 |
+ |
|
| 1256 |
+ |
\begin{figure} |
| 1257 |
+ |
\begin{center} |
| 1258 |
+ |
\epsfxsize=6in |
| 1259 |
+ |
\epsfbox{GofRCompare.epsi} |
| 1260 |
+ |
%\caption{Plots comparing experiment [Ref. \onlinecite{Head-Gordon00_1}] with |
| 1261 |
+ |
%SSD/E and SSD1 without reaction field (top), as well as |
| 1262 |
+ |
%SSD/RF and SSD1 with reaction field turned on |
| 1263 |
+ |
%(bottom). The insets show the respective first peaks in detail. Note |
| 1264 |
+ |
%how the changes in parameters have lowered and broadened the first |
| 1265 |
+ |
%peak of SSD/E and SSD/RF.} |
| 1266 |
+ |
\label{grcompare} |
| 1267 |
+ |
\end{center} |
| 1268 |
+ |
\end{figure} |
| 1269 |
+ |
|
| 1270 |
+ |
\newpage |
| 1271 |
+ |
|
| 1272 |
+ |
\begin{figure} |
| 1273 |
+ |
\begin{center} |
| 1274 |
+ |
\epsfxsize=7in |
| 1275 |
+ |
\epsfbox{dualsticky_bw.eps} |
| 1276 |
+ |
%\caption{Positive and negative isosurfaces of the sticky potential for |
| 1277 |
+ |
%SSD1 (left) and SSD/E \& SSD/RF (right). Light areas |
| 1278 |
+ |
%correspond to the tetrahedral attractive component, and darker areas |
| 1279 |
+ |
%correspond to the dipolar repulsive component.} |
| 1280 |
+ |
\label{isosurface} |
| 1281 |
+ |
\end{center} |
| 1282 |
+ |
\end{figure} |
| 1283 |
+ |
|
| 1284 |
+ |
\newpage |
| 1285 |
+ |
|
| 1286 |
+ |
\begin{figure} |
| 1287 |
+ |
\begin{center} |
| 1288 |
+ |
\epsfxsize=6in |
| 1289 |
+ |
\epsfbox{ssdeDense.epsi} |
| 1290 |
+ |
%\caption{Comparison of densities calculated with SSD/E to |
| 1291 |
+ |
%SSD1 without a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
| 1292 |
+ |
%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}] and |
| 1293 |
+ |
%experiment [Ref. \onlinecite{CRC80}]. The window shows a expansion around |
| 1294 |
+ |
%300 K with error bars included to clarify this region of |
| 1295 |
+ |
%interest. Note that both SSD1 and SSD/E show good agreement with |
| 1296 |
+ |
%experiment when the long-range correction is neglected.} |
| 1297 |
+ |
\label{ssdedense} |
| 1298 |
+ |
\end{center} |
| 1299 |
+ |
\end{figure} |
| 1300 |
+ |
|
| 1301 |
+ |
\newpage |
| 1302 |
+ |
|
| 1303 |
+ |
\begin{figure} |
| 1304 |
+ |
\begin{center} |
| 1305 |
+ |
\epsfxsize=6in |
| 1306 |
+ |
\epsfbox{ssdrfDense.epsi} |
| 1307 |
+ |
%\caption{Comparison of densities calculated with SSD/RF to |
| 1308 |
+ |
%SSD1 with a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
| 1309 |
+ |
%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}], and |
| 1310 |
+ |
%experiment [Ref. \onlinecite{CRC80}]. The inset shows the necessity of |
| 1311 |
+ |
%reparameterization when utilizing a reaction field long-ranged |
| 1312 |
+ |
%correction - SSD/RF provides significantly more accurate |
| 1313 |
+ |
%densities than SSD1 when performing room temperature |
| 1314 |
+ |
%simulations.} |
| 1315 |
+ |
\label{ssdrfdense} |
| 1316 |
+ |
\end{center} |
| 1317 |
+ |
\end{figure} |
| 1318 |
+ |
|
| 1319 |
+ |
\newpage |
| 1320 |
+ |
|
| 1321 |
+ |
\begin{figure} |
| 1322 |
+ |
\begin{center} |
| 1323 |
+ |
\epsfxsize=6in |
| 1324 |
+ |
\epsfbox{ssdeDiffuse.epsi} |
| 1325 |
+ |
%\caption{The diffusion constants calculated from SSD/E and |
| 1326 |
+ |
%SSD1 (both without a reaction field) along with experimental results |
| 1327 |
+ |
%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The NVE calculations were |
| 1328 |
+ |
%performed at the average densities observed in the 1 atm NPT |
| 1329 |
+ |
%simulations for the respective models. SSD/E is slightly more mobile |
| 1330 |
+ |
%than experiment at all of the temperatures, but it is closer to |
| 1331 |
+ |
%experiment at biologically relevant temperatures than SSD1 without a |
| 1332 |
+ |
%long-range correction.} |
| 1333 |
+ |
\label{ssdediffuse} |
| 1334 |
+ |
\end{center} |
| 1335 |
+ |
\end{figure} |
| 1336 |
+ |
|
| 1337 |
+ |
\newpage |
| 1338 |
+ |
|
| 1339 |
+ |
\begin{figure} |
| 1340 |
+ |
\begin{center} |
| 1341 |
+ |
\epsfxsize=6in |
| 1342 |
+ |
\epsfbox{ssdrfDiffuse.epsi} |
| 1343 |
+ |
%\caption{The diffusion constants calculated from SSD/RF and |
| 1344 |
+ |
%SSD1 (both with an active reaction field) along with |
| 1345 |
+ |
%experimental results [Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The |
| 1346 |
+ |
%NVE calculations were performed at the average densities observed in |
| 1347 |
+ |
%the 1 atm NPT simulations for both of the models. SSD/RF |
| 1348 |
+ |
%simulates the diffusion of water throughout this temperature range |
| 1349 |
+ |
%very well. The rapidly increasing diffusion constants at high |
| 1350 |
+ |
%temperatures for both models can be attributed to lower calculated |
| 1351 |
+ |
%densities than those observed in experiment.} |
| 1352 |
+ |
\label{ssdrfdiffuse} |
| 1353 |
+ |
\end{center} |
| 1354 |
+ |
\end{figure} |
| 1355 |
+ |
|
| 1356 |
+ |
\newpage |
| 1357 |
+ |
|
| 1358 |
+ |
\begin{figure} |
| 1359 |
+ |
\begin{center} |
| 1360 |
+ |
\epsfxsize=6in |
| 1361 |
+ |
\epsfbox{icei_bw.eps} |
| 1362 |
+ |
%\caption{The most stable crystal structure assumed by the SSD family |
| 1363 |
+ |
%of water models. We refer to this structure as Ice-{\it i} to |
| 1364 |
+ |
%indicate its origins in computer simulation. This image was taken of |
| 1365 |
+ |
%the (001) face of the crystal.} |
| 1366 |
+ |
\label{weirdice} |
| 1367 |
+ |
\end{center} |
| 1368 |
+ |
\end{figure} |
| 1369 |
+ |
|
| 1370 |
|
\end{document} |
