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\begin{document} |
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|
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\title{On the temperature dependent properties of the soft sticky dipole (SSD) and related single point water models} |
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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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|
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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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\maketitle |
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\begin{abstract} |
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NVE and NPT molecular dynamics simulations were performed in order to |
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investigate the density maximum and temperature dependent transport |
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for SSD and related water models, both with and without the use of |
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reaction field. The constant pressure simulations of the melting of |
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both $I_h$ and $I_c$ ice showed a density maximum near 260 K. In most |
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cases, the calculated densities were significantly lower than the |
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densities calculated in simulations of other water models. Analysis of |
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particle diffusion showed SSD to capture the transport properties of |
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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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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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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. In order to correct the density behavior, SSD was |
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reparameterized for use both with and without a long-range interaction |
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correction, SSD/RF and SSD/E respectively. Compared to the density |
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corrected version of SSD (SSD1), these modified models were shown to |
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maintain or improve upon the structural and transport properties. |
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liquid regimes. We also present our re-parameterized 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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improve upon the experimental agreement with the structural and |
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transport properties that can be obtained with either the original SSD |
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or the density corrected version of the original model (SSD1). |
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Additionally, a novel low-density ice structure is presented |
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which appears to be the most stable ice structure for the entire SSD |
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family. |
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\end{abstract} |
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\newpage |
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\section{Introduction} |
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|
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One of the most important tasks in the simulation of biochemical |
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systems is the proper depiction of water and water solvation. In fact, |
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the bulk of the calculations performed in solvated simulations are of |
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interactions with or between solvent molecules. Thus, the outcomes of |
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these types of simulations are highly dependent on the physical |
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properties of water, both as individual molecules and in clusters or |
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bulk. Due to the fact that explicit solvent accounts for a massive |
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portion of the calculations, it necessary to simplify the solvent to |
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some extent in order to complete simulations in a reasonable amount of |
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time. In the case of simulating water in biomolecular studies, the |
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balance between accurate properties and computational efficiency is |
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especially delicate, and it has resulted in a variety of different |
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water models.\cite{Jorgensen83,Berendsen87,Jorgensen00} Many of these |
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models predict specific properties more accurately than their |
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predecessors, but often at the cost of other properties or of computer |
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time. As an example, compare TIP3P or TIP4P to TIP5P. TIP5P improves |
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upon the structural and transport properties of water relative to the |
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previous TIP models, yet this comes at a greater than 50\% increase in |
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computational cost.\cite{Jorgensen01,Jorgensen00} One recently |
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developed model that succeeds in both retaining the accuracy of system |
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properties and simplifying calculations to increase computational |
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efficiency is the Soft Sticky Dipole water model.\cite{Ichiye96} |
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systems is the proper depiction of the aqueous environment of the |
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molecules of interest. In some cases (such as in the simulation of |
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phospholipid bilayers), the majority of the calculations that are |
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performed involve interactions with or between solvent molecules. |
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Thus, the properties one may observe in biochemical simulations are |
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going to be highly dependent on the physical properties of the water |
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model that is chosen. |
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|
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The Soft Sticky Dipole (SSD)\ water model was developed by Ichiye |
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\emph{et al.} as a modified form of the hard-sphere water model |
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proposed by Bratko, Blum, and Luzar.\cite{Bratko85,Bratko95} SSD |
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consists of a single point dipole with a Lennard-Jones core and a |
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sticky potential that directs the particles to assume the proper |
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hydrogen bond orientation in the first solvation shell. Thus, the |
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interaction between two SSD water molecules \emph{i} and \emph{j} is |
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given by the potential |
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There is an especially delicate balance between computational |
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efficiency and the ability of the water model to accurately predict |
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the properties of bulk |
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water.\cite{Jorgensen83,Berendsen87,Jorgensen00} For example, the |
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TIP5P model improves on the structural and transport properties of |
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water relative to the previous TIP models, yet this comes at a greater |
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than 50\% increase in computational |
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cost.\cite{Jorgensen01,Jorgensen00} |
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|
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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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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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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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|
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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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\begin{equation} |
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u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\ + |
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({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ + |
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u_{ij}^{sp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j), |
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({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j), |
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\end{equation} |
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where the $\mathbf{r}_{ij}$ is the position vector between molecules |
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\emph{i} and \emph{j} with magnitude equal to the distance $r_{ij}$, and |
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$\boldsymbol{\Omega}_i$ and $\boldsymbol{\Omega}_j$ represent the |
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orientations of the respective molecules. The Lennard-Jones, dipole, |
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and sticky parts of the potential are giving by the following |
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equations: |
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where the ${\bf r}_{ij}$ is the position vector between molecules |
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\emph{i} and \emph{j} with magnitude $r_{ij}$, and |
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${\bf \Omega}_i$ and ${\bf \Omega}_j$ represent the orientations of |
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the two molecules. The Lennard-Jones and dipole interactions are given |
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by the following familiar forms: |
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\begin{equation} |
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u_{ij}^{LJ}(r_{ij}) = 4\epsilon \left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right], |
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u_{ij}^{LJ}(r_{ij}) = 4\epsilon |
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\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right] |
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\ , |
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\end{equation} |
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and |
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\begin{equation} |
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u_{ij}^{dp} = \frac{\boldsymbol{\mu}_i\cdot\boldsymbol{\mu}_j}{r_{ij}^3}-\frac{3(\boldsymbol{\mu}_i\cdot\mathbf{r}_{ij})(\boldsymbol{\mu}_j\cdot\mathbf{r}_{ij})}{r_{ij}^5}\ , |
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u_{ij}^{dp} = \frac{|\mu_i||\mu_j|}{4 \pi \epsilon_0 r_{ij}^3} \left( |
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\hat{\bf u}_i \cdot \hat{\bf u}_j - 3(\hat{\bf u}_i\cdot\hat{\bf |
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r}_{ij})(\hat{\bf u}_j\cdot\hat{\bf r}_{ij}) \right)\ , |
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\end{equation} |
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where $\hat{\bf u}_i$ and $\hat{\bf u}_j$ are the unit vectors along |
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the dipoles of molecules $i$ and $j$ respectively. $|\mu_i|$ and |
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$|\mu_j|$ are the strengths of the dipole moments, and $\hat{\bf |
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r}_{ij}$ is the unit vector pointing from molecule $j$ to molecule |
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$i$. |
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|
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The sticky potential is somewhat less familiar: |
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\begin{equation} |
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u_{ij}^{sp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = |
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\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
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({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = |
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\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) |
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+ s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf |
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\Omega}_j)]\ . |
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\label{stickyfunction} |
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\end{equation} |
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where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole |
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unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D, |
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$\nu_0$ scales the strength of the overall sticky potential, and $s$ |
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and $s^\prime$ are cubic switching functions. The $w$ and $w^\prime$ |
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functions take the following forms: |
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Here, $\nu_0$ is a strength parameter for the sticky potential, and |
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$s$ and $s^\prime$ are cubic switching functions which turn off the |
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sticky interaction beyond the first solvation shell. The $w$ function |
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can be thought of as an attractive potential with tetrahedral |
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geometry: |
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\begin{equation} |
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w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
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w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
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\end{equation} |
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while the $w^\prime$ function counters the normal aligned and |
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anti-aligned structures favored by point dipoles: |
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\begin{equation} |
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w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, |
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w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^\circ, |
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\end{equation} |
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where $w^0 = 0.07715$. The $w$ function is the tetrahedral attractive |
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term that promotes hydrogen bonding orientations within the first |
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solvation shell, and $w^\prime$ is a dipolar repulsion term that |
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repels unrealistic dipolar arrangements within the first solvation |
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shell. A more detailed description of the functional parts and |
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variables in this potential can be found in other |
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articles.\cite{Ichiye96,Ichiye99} |
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It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
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and $Y_3^{-2}$ spherical harmonics (a linear combination which |
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enhances the tetrahedral geometry for hydrogen bonded structures), |
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while $w^\prime$ is a purely empirical function. A more detailed |
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description of the functional parts and variables in this potential |
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can be found in the original SSD |
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articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} |
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|
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Being that this is a one-site point dipole model, the actual force |
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calculations are simplified significantly. In the original Monte Carlo |
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simulations using this model, Ichiye \emph{et al.} reported an |
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increase in calculation efficiency of up to an order of magnitude over |
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other comparable models, while maintaining the structural behavior of |
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water.\cite{Ichiye96} In the original molecular dynamics studies, it |
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was shown that SSD improves on the prediction of many of water's |
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dynamical properties over TIP3P and SPC/E.\cite{Ichiye99} This |
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attractive combination of speed and accurate depiction of solvent |
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properties makes SSD a model of interest for the simulation of large |
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scale biological systems, such as membrane phase behavior. |
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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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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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data for the structural features of liquid |
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water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties |
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exhibited by SSD agree with experiment better than those of more |
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computationally expensive models (like TIP3P and |
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SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction |
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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 of the key limitations of this water model, however, is that it |
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has been parameterized for use with the Ewald Sum technique for the |
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handling of long-ranged interactions. When studying very large |
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systems, the Ewald summation and even particle-mesh Ewald become |
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computational burdens, with their respective ideal $N^\frac{3}{2}$ and |
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$N\log N$ calculation scaling orders for $N$ particles.\cite{Darden99} |
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In applying this water model in these types of systems, it would be |
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useful to know its properties and behavior with the more |
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computationally efficient reaction field (RF) technique, and even with |
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a cutoff that lacks any form of long-range correction. This study |
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addresses these issues by looking at the structural and transport |
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behavior of SSD over a variety of temperatures with the purpose of |
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utilizing the RF correction technique. We then suggest alterations to |
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the parameters that result in more water-like behavior. It should be |
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noted that in a recent publication, some of the original investigators of |
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the SSD water model have put forth adjustments to the SSD water model |
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to address abnormal density behavior (also observed here), calling the |
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corrected model SSD1.\cite{Ichiye03} This study will make comparisons |
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with SSD1's behavior with the goal of improving upon the |
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depiction of water under conditions without the Ewald Sum. |
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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 |
191 |
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prohibitive when $N$ becomes large.\cite{Darden99} In applying this |
192 |
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water model in these types of systems, it would be useful to know its |
193 |
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properties and behavior under the more computationally efficient |
194 |
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reaction field (RF) technique, or even with a simple cutoff. This |
195 |
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study addresses these issues by looking at the structural and |
196 |
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transport behavior of SSD over a variety of temperatures with the |
197 |
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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 |
200 |
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of the original investigators of the SSD water model have suggested |
201 |
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adjustments to the SSD water model to address abnormal density |
202 |
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behavior (also observed here), calling the corrected model |
203 |
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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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|
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|
\section{Methods} |
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|
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As stated previously, the long-range dipole-dipole interactions were |
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accounted for in this study by using the reaction field method. The |
212 |
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magnitude of the reaction field acting on dipole \emph{i} is given by |
210 |
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Long-range dipole-dipole interactions were accounted for in this study |
211 |
> |
by using either the reaction field method or by resorting to a simple |
212 |
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cubic switching function at a cutoff radius. Under the first method, |
213 |
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the magnitude of the reaction field acting on dipole $i$ is |
214 |
|
\begin{equation} |
215 |
|
\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1} |
216 |
< |
\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} \boldsymbol{\mu}_{j} f(r_{ij})\ , |
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\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} f(r_{ij})\ , |
217 |
|
\label{rfequation} |
218 |
|
\end{equation} |
219 |
|
where $\mathcal{R}$ is the cavity defined by the cutoff radius |
220 |
|
($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the |
221 |
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system (80 in this case), $\boldsymbol{\mu}_{j}$ is the dipole moment |
222 |
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vector of particle \emph{j}, and $f(r_{ij})$ is a cubic switching |
221 |
> |
system (80 in the case of liquid water), ${\bf \mu}_{j}$ is the dipole |
222 |
> |
moment vector of particle $j$ and $f(r_{ij})$ is a cubic switching |
223 |
|
function.\cite{AllenTildesley} The reaction field contribution to the |
224 |
< |
total energy by particle \emph{i} is given by |
225 |
< |
$-\frac{1}{2}\boldsymbol{\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque |
226 |
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on dipole \emph{i} by |
227 |
< |
$\boldsymbol{\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use |
228 |
< |
of reaction field is known to alter the orientational dynamic |
229 |
< |
properties, such as the dielectric relaxation time, based on changes |
230 |
< |
in the length of the cutoff radius.\cite{Berendsen98} This variable |
231 |
< |
behavior makes reaction field a less attractive method than other |
232 |
< |
methods, like the Ewald summation; however, for the simulation of |
233 |
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large-scale systems, the computational cost benefit of reaction field |
234 |
< |
is dramatic. To address some of the dynamical property alterations due |
235 |
< |
to the use of reaction field, simulations were also performed without |
236 |
< |
a surrounding dielectric and suggestions are presented on how to make |
237 |
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SSD more accurate both with and without a reaction field. |
224 |
> |
total energy by particle $i$ is given by $-\frac{1}{2}{\bf |
225 |
> |
\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf |
226 |
> |
\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use of the reaction |
227 |
> |
field is known to alter the bulk orientational properties, such as the |
228 |
> |
dielectric relaxation time. There is particular sensitivity of this |
229 |
> |
property on changes in the length of the cutoff |
230 |
> |
radius.\cite{Berendsen98} This variable behavior makes reaction field |
231 |
> |
a less attractive method than the Ewald sum. However, for very large |
232 |
> |
systems, the computational benefit of reaction field is dramatic. |
233 |
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|
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We have also performed a companion set of simulations {\it without} a |
235 |
> |
surrounding dielectric (i.e. using a simple cubic switching function |
236 |
> |
at the cutoff radius) and as a result we have two reparamaterizations |
237 |
> |
of SSD which could be used either with or without the Reaction Field |
238 |
> |
turned on. |
239 |
|
|
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< |
Simulations were performed in both the isobaric-isothermal and |
241 |
< |
microcanonical ensembles. The constant pressure simulations were |
242 |
< |
implemented using an integral thermostat and barostat as outlined by |
243 |
< |
Hoover.\cite{Hoover85,Hoover86} All particles were treated as |
244 |
< |
non-linear rigid bodies. Vibrational constraints are not necessary in |
245 |
< |
simulations of SSD, because there are no explicit hydrogen atoms, and |
246 |
< |
thus no molecular vibrational modes need to be considered. |
240 |
> |
Simulations to obtain the preferred density were performed in the |
241 |
> |
isobaric-isothermal (NPT) ensemble, while all dynamical properties |
242 |
> |
were obtained from microcanonical (NVE) simulations done at densities |
243 |
> |
matching the NPT density for a particular target temperature. The |
244 |
> |
constant pressure simulations were implemented using an integral |
245 |
> |
thermostat and barostat as outlined by Hoover.\cite{Hoover85,Hoover86} |
246 |
> |
All molecules were treated as non-linear rigid bodies. Vibrational |
247 |
> |
constraints are not necessary in simulations of SSD, because there are |
248 |
> |
no explicit hydrogen atoms, and thus no molecular vibrational modes |
249 |
> |
need to be considered. |
250 |
|
|
251 |
|
Integration of the equations of motion was carried out using the |
252 |
< |
symplectic splitting method proposed by Dullweber \emph{et |
253 |
< |
al.}\cite{Dullweber1997} The reason for this integrator selection |
254 |
< |
deals with poor energy conservation of rigid body systems using |
255 |
< |
quaternions. While quaternions work well for orientational motion in |
256 |
< |
alternate ensembles, the microcanonical ensemble has a constant energy |
257 |
< |
requirement that is quite sensitive to errors in the equations of |
258 |
< |
motion. The original implementation of this code utilized quaternions |
259 |
< |
for rotational motion propagation; however, a detailed investigation |
260 |
< |
showed that they resulted in a steady drift in the total energy, |
216 |
< |
something that has been observed by others.\cite{Laird97} |
252 |
> |
symplectic splitting method proposed by Dullweber {\it et |
253 |
> |
al.}\cite{Dullweber1997} Our reason for selecting this integrator |
254 |
> |
centers on poor energy conservation of rigid body dynamics using |
255 |
> |
traditional quaternion integration.\cite{Evans77,Evans77b} While quaternions |
256 |
> |
may work well for orientational motion under NVT or NPT integrators, |
257 |
> |
our limits on energy drift in the microcanonical ensemble were quite |
258 |
> |
strict, and the drift under quaternions was substantially greater than |
259 |
> |
in the symplectic splitting method. This steady drift in the total |
260 |
> |
energy has also been observed by Kol {\it et al.}\cite{Laird97} |
261 |
|
|
262 |
|
The key difference in the integration method proposed by Dullweber |
263 |
|
\emph{et al.} is that the entire rotation matrix is propagated from |
264 |
< |
one time step to the next. In the past, this would not have been as |
265 |
< |
feasible an option, being that the rotation matrix for a single body is |
266 |
< |
nine elements long as opposed to 3 or 4 elements for Euler angles and |
267 |
< |
quaternions respectively. System memory has become much less of an |
224 |
< |
issue in recent times, and this has resulted in substantial benefits |
225 |
< |
in energy conservation. There is still the issue of 5 or 6 additional |
226 |
< |
elements for describing the orientation of each particle, which will |
227 |
< |
increase dump files substantially. Simply translating the rotation |
228 |
< |
matrix into its component Euler angles or quaternions for storage |
229 |
< |
purposes relieves this burden. |
264 |
> |
one time step to the next. The additional memory required by the |
265 |
> |
algorithm is inconsequential on modern computers, and translating the |
266 |
> |
rotation matrix into quaternions for storage purposes makes trajectory |
267 |
> |
data quite compact. |
268 |
|
|
269 |
|
The symplectic splitting method allows for Verlet style integration of |
270 |
< |
both linear and angular motion of rigid bodies. In this integration |
271 |
< |
method, the orientational propagation involves a sequence of matrix |
272 |
< |
evaluations to update the rotation matrix.\cite{Dullweber1997} These |
273 |
< |
matrix rotations are more costly computationally than the simpler |
274 |
< |
arithmetic quaternion propagation. With the same time step, a 1000 SSD |
275 |
< |
particle simulation shows an average 7\% increase in computation time |
276 |
< |
using the symplectic step method in place of quaternions. This cost is |
277 |
< |
more than justified when comparing the energy conservation of the two |
278 |
< |
methods as illustrated in figure \ref{timestep}. |
270 |
> |
both translational and orientational motion of rigid bodies. In this |
271 |
> |
integration method, the orientational propagation involves a sequence |
272 |
> |
of matrix evaluations to update the rotation |
273 |
> |
matrix.\cite{Dullweber1997} These matrix rotations are more costly |
274 |
> |
than the simpler arithmetic quaternion propagation. With the same time |
275 |
> |
step, a 1000 SSD particle simulation shows an average 7\% increase in |
276 |
> |
computation time using the symplectic step method in place of |
277 |
> |
quaternions. The additional expense per step is justified when one |
278 |
> |
considers the ability to use time steps that are nearly twice as large |
279 |
> |
under symplectic splitting than would be usable under quaternion |
280 |
> |
dynamics. The energy conservation of the two methods using a number |
281 |
> |
of different time steps is illustrated in figure |
282 |
> |
\ref{timestep}. |
283 |
|
|
284 |
|
\begin{figure} |
285 |
|
\begin{center} |
286 |
|
\epsfxsize=6in |
287 |
|
\epsfbox{timeStep.epsi} |
288 |
< |
\caption{Energy conservation using quaternion based integration versus |
288 |
> |
\caption{Energy conservation using both quaternion based integration and |
289 |
|
the symplectic step method proposed by Dullweber \emph{et al.} with |
290 |
< |
increasing time step. The larger time step plots are shifted up from |
291 |
< |
the true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.} |
290 |
> |
increasing time step. The larger time step plots are shifted from the |
291 |
> |
true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.} |
292 |
|
\label{timestep} |
293 |
|
\end{center} |
294 |
|
\end{figure} |
295 |
|
|
296 |
|
In figure \ref{timestep}, the resulting energy drift at various time |
297 |
|
steps for both the symplectic step and quaternion integration schemes |
298 |
< |
is compared. All of the 1000 SSD particle simulations started with the |
299 |
< |
same configuration, and the only difference was the method used to |
300 |
< |
handle rotational motion. At time steps of 0.1 and 0.5 fs, both |
301 |
< |
methods for propagating particle rotation conserve energy fairly well, |
302 |
< |
with the quaternion method showing a slight energy drift over time in |
303 |
< |
the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the |
304 |
< |
energy conservation benefits of the symplectic step method are clearly |
305 |
< |
demonstrated. Thus, while maintaining the same degree of energy |
306 |
< |
conservation, one can take considerably longer time steps, leading to |
307 |
< |
an overall reduction in computation time. |
298 |
> |
is compared. All of the 1000 SSD particle simulations started with |
299 |
> |
the same configuration, and the only difference was the method used to |
300 |
> |
handle orientational motion. At time steps of 0.1 and 0.5 fs, both |
301 |
> |
methods for propagating the orientational degrees of freedom conserve |
302 |
> |
energy fairly well, with the quaternion method showing a slight energy |
303 |
> |
drift over time in the 0.5 fs time step simulation. At time steps of 1 |
304 |
> |
and 2 fs, the energy conservation benefits of the symplectic step |
305 |
> |
method are clearly demonstrated. Thus, while maintaining the same |
306 |
> |
degree of energy conservation, one can take considerably longer time |
307 |
> |
steps, leading to an overall reduction in computation time. |
308 |
|
|
309 |
|
Energy drift in the symplectic step simulations was unnoticeable for |
310 |
< |
time steps up to three femtoseconds. A slight energy drift on the |
310 |
> |
time steps up to 3 fs. A slight energy drift on the |
311 |
|
order of 0.012 kcal/mol per nanosecond was observed at a time step of |
312 |
< |
four femtoseconds, and as expected, this drift increases dramatically |
313 |
< |
with increasing time step. To insure accuracy in the constant energy |
312 |
> |
4 fs, and as expected, this drift increases dramatically |
313 |
> |
with increasing time step. To insure accuracy in our microcanonical |
314 |
|
simulations, time steps were set at 2 fs and kept at this value for |
315 |
|
constant pressure simulations as well. |
316 |
|
|
317 |
< |
Ice crystals in both the $I_h$ and $I_c$ lattices were generated as |
318 |
< |
starting points for all simulations. The $I_h$ crystals were formed by |
319 |
< |
first arranging the centers of mass of the SSD particles into a |
320 |
< |
``hexagonal'' ice lattice of 1024 particles. Because of the crystal |
321 |
< |
structure of $I_h$ ice, the simulation box assumed a rectangular shape |
322 |
< |
with an edge length ratio of approximately |
317 |
> |
Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices |
318 |
> |
were generated as starting points for all simulations. The $I_h$ |
319 |
> |
crystals were formed by first arranging the centers of mass of the SSD |
320 |
> |
particles into a ``hexagonal'' ice lattice of 1024 particles. Because |
321 |
> |
of the crystal structure of $I_h$ ice, the simulation box assumed an |
322 |
> |
orthorhombic shape with an edge length ratio of approximately |
323 |
|
1.00$\times$1.06$\times$1.23. The particles were then allowed to |
324 |
|
orient freely about fixed positions with angular momenta randomized at |
325 |
|
400 K for varying times. The rotational temperature was then scaled |
339 |
|
\section{Results and discussion} |
340 |
|
|
341 |
|
Melting studies were performed on the randomized ice crystals using |
342 |
< |
constant pressure and temperature dynamics. During melting |
343 |
< |
simulations, the melting transition and the density maximum can both |
344 |
< |
be observed, provided that the density maximum occurs in the liquid |
345 |
< |
and not the supercooled regime. An ensemble average from five separate |
346 |
< |
melting simulations was acquired, each starting from different ice |
347 |
< |
crystals generated as described previously. All simulations were |
348 |
< |
equilibrated for 100 ps prior to a 200 ps data collection run at each |
349 |
< |
temperature setting. The temperature range of study spanned from 25 to |
350 |
< |
400 K, with a maximum degree increment of 25 K. For regions of |
351 |
< |
interest along this stepwise progression, the temperature increment |
352 |
< |
was decreased from 25 K to 10 and 5 K. The above equilibration and |
353 |
< |
production times were sufficient in that the system volume |
354 |
< |
fluctuations dampened out in all but the very cold simulations (below |
313 |
< |
225 K). |
342 |
> |
isobaric-isothermal (NPT) dynamics. During melting simulations, the |
343 |
> |
melting transition and the density maximum can both be observed, |
344 |
> |
provided that the density maximum occurs in the liquid and not the |
345 |
> |
supercooled regime. An ensemble average from five separate melting |
346 |
> |
simulations was acquired, each starting from different ice crystals |
347 |
> |
generated as described previously. All simulations were equilibrated |
348 |
> |
for 100 ps prior to a 200 ps data collection run at each temperature |
349 |
> |
setting. The temperature range of study spanned from 25 to 400 K, with |
350 |
> |
a maximum degree increment of 25 K. For regions of interest along this |
351 |
> |
stepwise progression, the temperature increment was decreased from 25 |
352 |
> |
K to 10 and 5 K. The above equilibration and production times were |
353 |
> |
sufficient in that fluctuations in the volume autocorrelation function |
354 |
> |
were damped out in all simulations in under 20 ps. |
355 |
|
|
356 |
|
\subsection{Density Behavior} |
316 |
– |
Initial simulations focused on the original SSD water model, and an |
317 |
– |
average density versus temperature plot is shown in figure |
318 |
– |
\ref{dense1}. Note that the density maximum when using a reaction |
319 |
– |
field appears between 255 and 265 K, where the calculated densities |
320 |
– |
within this range were nearly indistinguishable. The greater certainty |
321 |
– |
of the average value at 260 K makes a good argument for the actual |
322 |
– |
density maximum residing at this midpoint value. Figure \ref{dense1} |
323 |
– |
was constructed using ice $I_h$ crystals for the initial |
324 |
– |
configuration; though not pictured, the simulations starting from ice |
325 |
– |
$I_c$ crystal configurations showed similar results, with a |
326 |
– |
liquid-phase density maximum in this same region (between 255 and 260 |
327 |
– |
K). In addition, the $I_c$ crystals are more fragile than the $I_h$ |
328 |
– |
crystals, leading to deformation into a dense glassy state at lower |
329 |
– |
temperatures. This resulted in an overall low temperature density |
330 |
– |
maximum at 200 K, while still retaining a liquid state density maximum |
331 |
– |
in common with the $I_h$ simulations. |
357 |
|
|
358 |
+ |
Our initial simulations focused on the original SSD water model, and |
359 |
+ |
an average density versus temperature plot is shown in figure |
360 |
+ |
\ref{dense1}. Note that the density maximum when using a reaction |
361 |
+ |
field appears between 255 and 265 K. There were smaller fluctuations |
362 |
+ |
in the density at 260 K than at either 255 or 265, so we report this |
363 |
+ |
value as the location of the density maximum. Figure \ref{dense1} was |
364 |
+ |
constructed using ice $I_h$ crystals for the initial configuration; |
365 |
+ |
though not pictured, the simulations starting from ice $I_c$ crystal |
366 |
+ |
configurations showed similar results, with a liquid-phase density |
367 |
+ |
maximum in this same region (between 255 and 260 K). |
368 |
+ |
|
369 |
|
\begin{figure} |
370 |
|
\begin{center} |
371 |
|
\epsfxsize=6in |
372 |
|
\epsfbox{denseSSD.eps} |
373 |
< |
\caption{Density versus temperature for TIP4P,\cite{Jorgensen98b} |
374 |
< |
TIP3P,\cite{Jorgensen98b} SPC/E,\cite{Clancy94} SSD without Reaction |
375 |
< |
Field, SSD, and experiment.\cite{CRC80} The arrows indicate the |
376 |
< |
change in densities observed when turning off the reaction field. The |
377 |
< |
the lower than expected densities for the SSD model were what |
378 |
< |
prompted the original reparameterization.\cite{Ichiye03}} |
373 |
> |
\caption{Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}], |
374 |
> |
TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], SSD |
375 |
> |
without Reaction Field, SSD, and experiment [Ref. \citen{CRC80}]. The |
376 |
> |
arrows indicate the change in densities observed when turning off the |
377 |
> |
reaction field. The the lower than expected densities for the SSD |
378 |
> |
model were what prompted the original reparameterization of SSD1 |
379 |
> |
[Ref. \citen{Ichiye03}].} |
380 |
|
\label{dense1} |
381 |
|
\end{center} |
382 |
|
\end{figure} |
383 |
|
|
384 |
< |
The density maximum for SSD actually compares quite favorably to other |
385 |
< |
simple water models. Figure \ref{dense1} also shows calculated |
386 |
< |
densities of several other models and experiment obtained from other |
384 |
> |
The density maximum for SSD compares quite favorably to other simple |
385 |
> |
water models. Figure \ref{dense1} also shows calculated densities of |
386 |
> |
several other models and experiment obtained from other |
387 |
|
sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water |
388 |
< |
models, SSD has results closest to the experimentally observed water |
389 |
< |
density maximum. Of the listed water models, TIP4P has a density |
390 |
< |
maximum behavior most like that seen in SSD. Though not included in |
391 |
< |
this particular plot, it is useful to note that TIP5P has a water |
392 |
< |
density maximum nearly identical to experiment. |
388 |
> |
models, SSD has a temperature closest to the experimentally observed |
389 |
> |
density maximum. Of the {\it charge-based} models in |
390 |
> |
Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that |
391 |
> |
seen in SSD. Though not included in this plot, it is useful |
392 |
> |
to note that TIP5P has a density maximum nearly identical to the |
393 |
> |
experimentally measured temperature. |
394 |
|
|
395 |
< |
It has been observed that densities are dependent on the cutoff radius |
396 |
< |
used for a variety of water models in simulations both with and |
397 |
< |
without the use of reaction field.\cite{Berendsen98} In order to |
398 |
< |
address the possible affect of cutoff radius, simulations were |
399 |
< |
performed with a dipolar cutoff radius of 12.0 \AA\ to compliment the |
400 |
< |
previous SSD simulations, all performed with a cutoff of 9.0 \AA. All |
401 |
< |
of the resulting densities overlapped within error and showed no |
402 |
< |
significant trend toward lower or higher densities as a function of |
403 |
< |
cutoff radius, for simulations both with and without reaction |
404 |
< |
field. These results indicate that there is no major benefit in |
405 |
< |
choosing a longer cutoff radius in simulations using SSD. This is |
406 |
< |
advantageous in that the use of a longer cutoff radius results in |
407 |
< |
significant increases in the time required to obtain a single |
370 |
< |
trajectory. |
395 |
> |
It has been observed that liquid state densities in water are |
396 |
> |
dependent on the cutoff radius used both with and without the use of |
397 |
> |
reaction field.\cite{Berendsen98} In order to address the possible |
398 |
> |
effect of cutoff radius, simulations were performed with a dipolar |
399 |
> |
cutoff radius of 12.0 \AA\ to complement the previous SSD simulations, |
400 |
> |
all performed with a cutoff of 9.0 \AA. All of the resulting densities |
401 |
> |
overlapped within error and showed no significant trend toward lower |
402 |
> |
or higher densities as a function of cutoff radius, for simulations |
403 |
> |
both with and without reaction field. These results indicate that |
404 |
> |
there is no major benefit in choosing a longer cutoff radius in |
405 |
> |
simulations using SSD. This is advantageous in that the use of a |
406 |
> |
longer cutoff radius results in a significant increase in the time |
407 |
> |
required to obtain a single trajectory. |
408 |
|
|
409 |
|
The key feature to recognize in figure \ref{dense1} is the density |
410 |
|
scaling of SSD relative to other common models at any given |
411 |
< |
temperature. Note that the SSD model assumes a lower density than any |
412 |
< |
of the other listed models at the same pressure, behavior which is |
413 |
< |
especially apparent at temperatures greater than 300 K. Lower than |
414 |
< |
expected densities have been observed for other systems using a |
415 |
< |
reaction field for long-range electrostatic interactions, so the most |
416 |
< |
likely reason for the significantly lower densities seen in these |
417 |
< |
simulations is the presence of the reaction |
418 |
< |
field.\cite{Berendsen98,Nezbeda02} In order to test the effect of the |
419 |
< |
reaction field on the density of the systems, the simulations were |
420 |
< |
repeated without a reaction field present. The results of these |
421 |
< |
simulations are also displayed in figure \ref{dense1}. Without |
422 |
< |
reaction field, the densities increase considerably to more |
423 |
< |
experimentally reasonable values, especially around the freezing point |
424 |
< |
of liquid water. The shape of the curve is similar to the curve |
425 |
< |
produced from SSD simulations using reaction field, specifically the |
426 |
< |
rapidly decreasing densities at higher temperatures; however, a shift |
427 |
< |
in the density maximum location, down to 245 K, is observed. This is a |
428 |
< |
more accurate comparison to the other listed water models, in that no |
429 |
< |
long range corrections were applied in those |
393 |
< |
simulations.\cite{Clancy94,Jorgensen98b} However, even without a |
411 |
> |
temperature. SSD assumes a lower density than any of the other listed |
412 |
> |
models at the same pressure, behavior which is especially apparent at |
413 |
> |
temperatures greater than 300 K. Lower than expected densities have |
414 |
> |
been observed for other systems using a reaction field for long-range |
415 |
> |
electrostatic interactions, so the most likely reason for the |
416 |
> |
significantly lower densities seen in these simulations is the |
417 |
> |
presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order |
418 |
> |
to test the effect of the reaction field on the density of the |
419 |
> |
systems, the simulations were repeated without a reaction field |
420 |
> |
present. The results of these simulations are also displayed in figure |
421 |
> |
\ref{dense1}. Without the reaction field, the densities increase |
422 |
> |
to more experimentally reasonable values, especially around the |
423 |
> |
freezing point of liquid water. The shape of the curve is similar to |
424 |
> |
the curve produced from SSD simulations using reaction field, |
425 |
> |
specifically the rapidly decreasing densities at higher temperatures; |
426 |
> |
however, a shift in the density maximum location, down to 245 K, is |
427 |
> |
observed. This is a more accurate comparison to the other listed water |
428 |
> |
models, in that no long range corrections were applied in those |
429 |
> |
simulations.\cite{Clancy94,Jorgensen98b} However, even without the |
430 |
|
reaction field, the density around 300 K is still significantly lower |
431 |
|
than experiment and comparable water models. This anomalous behavior |
432 |
< |
was what lead Ichiye \emph{et al.} to recently reparameterize SSD and |
433 |
< |
make SSD1.\cite{Ichiye03} In discussing potential adjustments later in |
434 |
< |
this paper, all comparisons were performed with this new model. |
432 |
> |
was what lead Ichiye {\it et al.} to recently reparameterize |
433 |
> |
SSD.\cite{Ichiye03} Throughout the remainder of the paper our |
434 |
> |
reparamaterizations of SSD will be compared with the newer SSD1 model. |
435 |
|
|
436 |
|
\subsection{Transport Behavior} |
401 |
– |
Of importance in these types of studies are the transport properties |
402 |
– |
of the particles and their change in responce to altering |
403 |
– |
environmental conditions. In order to probe transport, constant energy |
404 |
– |
simulations were performed about the average density uncovered by the |
405 |
– |
constant pressure simulations. Simulations started with randomized |
406 |
– |
velocities and underwent 50 ps of temperature scaling and 50 ps of |
407 |
– |
constant energy equilibration before obtaining a 200 ps |
408 |
– |
trajectory. Diffusion constants were calculated via root-mean square |
409 |
– |
deviation analysis. The averaged results from five sets of NVE |
410 |
– |
simulations are displayed in figure \ref{diffuse}, alongside |
411 |
– |
experimental, SPC/E, and TIP5P |
412 |
– |
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} |
437 |
|
|
438 |
+ |
Accurate dynamical properties of a water model are particularly |
439 |
+ |
important when using the model to study permeation or transport across |
440 |
+ |
biological membranes. In order to probe transport in bulk water, |
441 |
+ |
constant energy (NVE) simulations were performed at the average |
442 |
+ |
density obtained by the NPT simulations at an identical target |
443 |
+ |
temperature. Simulations started with randomized velocities and |
444 |
+ |
underwent 50 ps of temperature scaling and 50 ps of constant energy |
445 |
+ |
equilibration before a 200 ps data collection run. Diffusion constants |
446 |
+ |
were calculated via linear fits to the long-time behavior of the |
447 |
+ |
mean-square displacement as a function of time. The averaged results |
448 |
+ |
from five sets of NVE simulations are displayed in figure |
449 |
+ |
\ref{diffuse}, alongside experimental, SPC/E, and TIP5P |
450 |
+ |
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} |
451 |
+ |
|
452 |
|
\begin{figure} |
453 |
|
\begin{center} |
454 |
|
\epsfxsize=6in |
455 |
|
\epsfbox{betterDiffuse.epsi} |
456 |
< |
\caption{Average diffusion coefficient over increasing temperature for |
457 |
< |
SSD, SPC/E,\cite{Clancy94} TIP5P,\cite{Jorgensen01} and Experimental |
458 |
< |
data.\cite{Gillen72,Mills73} Of the three water models shown, SSD has |
459 |
< |
the least deviation from the experimental values. The rapidly |
460 |
< |
increasing diffusion constants for TIP5P and SSD correspond to |
461 |
< |
significant decrease in density at the higher temperatures.} |
456 |
> |
\caption{Average self-diffusion constant as a function of temperature for |
457 |
> |
SSD, SPC/E [Ref. \citen{Clancy94}], TIP5P [Ref. \citen{Jorgensen01}], |
458 |
> |
and Experimental data [Refs. \citen{Gillen72} and \citen{Mills73}]. Of |
459 |
> |
the three water models shown, SSD has the least deviation from the |
460 |
> |
experimental values. The rapidly increasing diffusion constants for |
461 |
> |
TIP5P and SSD correspond to significant decrease in density at the |
462 |
> |
higher temperatures.} |
463 |
|
\label{diffuse} |
464 |
|
\end{center} |
465 |
|
\end{figure} |
466 |
|
|
467 |
|
The observed values for the diffusion constant point out one of the |
468 |
< |
strengths of the SSD model. Of the three experimental models shown, |
469 |
< |
the SSD model has the most accurate depiction of the diffusion trend |
470 |
< |
seen in experiment in both the supercooled and liquid temperature |
471 |
< |
regimes. SPC/E does a respectable job by producing values similar to |
472 |
< |
SSD and experiment around 290 K; however, it deviates at both higher |
473 |
< |
and lower temperatures, failing to predict the experimental |
474 |
< |
trend. TIP5P and SSD both start off low at colder temperatures and |
475 |
< |
tend to diffuse too rapidly at higher temperatures. This trend at |
476 |
< |
higher temperatures is not surprising in that the densities of both |
477 |
< |
TIP5P and SSD are lower than experimental water at these higher |
478 |
< |
temperatures. When calculating the diffusion coefficients for SSD at |
479 |
< |
experimental densities, the resulting values fall more in line with |
480 |
< |
experiment at these temperatures, albeit not at standard pressure. |
468 |
> |
strengths of the SSD model. Of the three models shown, the SSD model |
469 |
> |
has the most accurate depiction of self-diffusion in both the |
470 |
> |
supercooled and liquid regimes. SPC/E does a respectable job by |
471 |
> |
reproducing values similar to experiment around 290 K; however, it |
472 |
> |
deviates at both higher and lower temperatures, failing to predict the |
473 |
> |
correct thermal trend. TIP5P and SSD both start off low at colder |
474 |
> |
temperatures and tend to diffuse too rapidly at higher temperatures. |
475 |
> |
This behavior at higher temperatures is not particularly surprising |
476 |
> |
since the densities of both TIP5P and SSD are lower than experimental |
477 |
> |
water densities at higher temperatures. When calculating the |
478 |
> |
diffusion coefficients for SSD at experimental densities (instead of |
479 |
> |
the densities from the NPT simulations), the resulting values fall |
480 |
> |
more in line with experiment at these temperatures. |
481 |
|
|
482 |
|
\subsection{Structural Changes and Characterization} |
483 |
+ |
|
484 |
|
By starting the simulations from the crystalline state, the melting |
485 |
< |
transition and the ice structure can be studied along with the liquid |
485 |
> |
transition and the ice structure can be obtained along with the liquid |
486 |
|
phase behavior beyond the melting point. The constant pressure heat |
487 |
|
capacity (C$_\text{p}$) was monitored to locate the melting transition |
488 |
|
in each of the simulations. In the melting simulations of the 1024 |
490 |
|
at 245 K, indicating a first order phase transition for the melting of |
491 |
|
these ice crystals. When the reaction field is turned off, the melting |
492 |
|
transition occurs at 235 K. These melting transitions are |
493 |
< |
considerably lower than the experimental value, but this is not a |
454 |
< |
surprise considering the simplicity of the SSD model. |
493 |
> |
considerably lower than the experimental value. |
494 |
|
|
495 |
|
\begin{figure} |
496 |
|
\begin{center} |
497 |
|
\epsfxsize=6in |
498 |
|
\epsfbox{corrDiag.eps} |
499 |
|
\caption{Two dimensional illustration of angles involved in the |
500 |
< |
correlations observed in figure \ref{contour}.} |
500 |
> |
correlations observed in Fig. \ref{contour}.} |
501 |
|
\label{corrAngle} |
502 |
|
\end{center} |
503 |
|
\end{figure} |
509 |
|
\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at |
510 |
|
100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for |
511 |
|
clarity: dark areas signify peaks while light areas signify |
512 |
< |
depressions. White areas have g(\emph{r}) values below 0.5 and black |
512 |
> |
depressions. White areas have $g(r)$ values below 0.5 and black |
513 |
|
areas have values above 1.5.} |
514 |
|
\label{contour} |
515 |
|
\end{center} |
516 |
|
\end{figure} |
517 |
|
|
518 |
< |
Additional analysis of the melting phase-transition process was |
519 |
< |
performed by using two-dimensional structure and dipole angle |
520 |
< |
correlations. Expressions for these correlations are as follows: |
518 |
> |
Additional analysis of the melting process was performed using |
519 |
> |
two-dimensional structure and dipole angle correlations. Expressions |
520 |
> |
for these correlations are as follows: |
521 |
|
|
522 |
|
\begin{equation} |
523 |
< |
g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|\mathbf{r}_{ij}\right|)\rangle\ , |
523 |
> |
g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ , |
524 |
|
\end{equation} |
525 |
|
\begin{equation} |
526 |
|
g_{\text{AB}}(r,\cos\omega) = |
527 |
< |
\frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\omega-\cos\omega_{ij})\delta(r-\left|\mathbf{r}_{ij}\right|)\rangle\ , |
527 |
> |
\frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\omega-\cos\omega_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ , |
528 |
|
\end{equation} |
529 |
|
where $\theta$ and $\omega$ refer to the angles shown in figure |
530 |
|
\ref{corrAngle}. By binning over both distance and the cosine of the |
531 |
< |
desired angle between the two dipoles, the g(\emph{r}) can be |
532 |
< |
dissected to determine the common dipole arrangements that constitute |
533 |
< |
the peaks and troughs. Frames A and B of figure \ref{contour} show a |
534 |
< |
relatively crystalline state of an ice $I_c$ simulation. The first |
535 |
< |
peak of the g(\emph{r}) consists primarily of the preferred hydrogen |
531 |
> |
desired angle between the two dipoles, the $g(r)$ can be analyzed to |
532 |
> |
determine the common dipole arrangements that constitute the peaks and |
533 |
> |
troughs in the standard one-dimensional $g(r)$ plots. Frames A and B |
534 |
> |
of figure \ref{contour} show results from an ice $I_c$ simulation. The |
535 |
> |
first peak in the $g(r)$ consists primarily of the preferred hydrogen |
536 |
|
bonding arrangements as dictated by the tetrahedral sticky potential - |
537 |
< |
one peak for the donating and the other for the accepting hydrogen |
538 |
< |
bonds. Due to the high degree of crystallinity of the sample, the |
539 |
< |
second and third solvation shells show a repeated peak arrangement |
537 |
> |
one peak for the hydrogen bond donor and the other for the hydrogen |
538 |
> |
bond acceptor. Due to the high degree of crystallinity of the sample, |
539 |
> |
the second and third solvation shells show a repeated peak arrangement |
540 |
|
which decays at distances around the fourth solvation shell, near the |
541 |
|
imposed cutoff for the Lennard-Jones and dipole-dipole interactions. |
542 |
|
In the higher temperature simulation shown in frames C and D, these |
543 |
< |
longer-ranged repeated peak features deteriorate rapidly. The first |
544 |
< |
solvation shell still shows the strong effect of the sticky-potential, |
545 |
< |
although it covers a larger area, extending to include a fraction of |
546 |
< |
aligned dipole peaks within the first solvation shell. The latter |
547 |
< |
peaks lose definition as thermal motion and the competing dipole force |
548 |
< |
overcomes the sticky potential's tight tetrahedral structuring of the |
510 |
< |
fluid. |
543 |
> |
long-range features deteriorate rapidly. The first solvation shell |
544 |
> |
still shows the strong effect of the sticky-potential, although it |
545 |
> |
covers a larger area, extending to include a fraction of aligned |
546 |
> |
dipole peaks within the first solvation shell. The latter peaks lose |
547 |
> |
due to thermal motion and as the competing dipole force overcomes the |
548 |
> |
sticky potential's tight tetrahedral structuring of the crystal. |
549 |
|
|
550 |
|
This complex interplay between dipole and sticky interactions was |
551 |
|
remarked upon as a possible reason for the split second peak in the |
552 |
< |
oxygen-oxygen g(\emph{r}).\cite{Ichiye96} At low temperatures, the |
553 |
< |
second solvation shell peak appears to have two distinct components |
554 |
< |
that blend together to form one observable peak. At higher |
552 |
> |
oxygen-oxygen $g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, |
553 |
> |
the second solvation shell peak appears to have two distinct |
554 |
> |
components that blend together to form one observable peak. At higher |
555 |
|
temperatures, this split character alters to show the leading 4 \AA\ |
556 |
|
peak dominated by equatorial anti-parallel dipole orientations. There |
557 |
|
is also a tightly bunched group of axially arranged dipoles that most |
560 |
|
dipoles that assume hydrogen bond arrangements similar to those seen |
561 |
|
in the first solvation shell. This evidence indicates that the dipole |
562 |
|
pair interaction begins to dominate outside of the range of the |
563 |
< |
dipolar repulsion term. Primary energetically favorable dipole |
563 |
> |
dipolar repulsion term. The energetically favorable dipole |
564 |
|
arrangements populate the region immediately outside this repulsion |
565 |
< |
region (around 4 \AA), while arrangements that seek to ideally satisfy |
566 |
< |
both the sticky and dipole forces locate themselves just beyond this |
565 |
> |
region (around 4 \AA), while arrangements that seek to satisfy both |
566 |
> |
the sticky and dipole forces locate themselves just beyond this |
567 |
|
initial buildup (around 5 \AA). |
568 |
|
|
569 |
|
From these findings, the split second peak is primarily the product of |
570 |
|
the dipolar repulsion term of the sticky potential. In fact, the inner |
571 |
|
peak can be pushed out and merged with the outer split peak just by |
572 |
< |
extending the switching function cutoff ($s^\prime(r_{ij})$) from its |
573 |
< |
normal 4.0 \AA\ to values of 4.5 or even 5 \AA. This type of |
572 |
> |
extending the switching function ($s^\prime(r_{ij})$) from its normal |
573 |
> |
4.0 \AA\ cutoff to values of 4.5 or even 5 \AA. This type of |
574 |
|
correction is not recommended for improving the liquid structure, |
575 |
|
since the second solvation shell would still be shifted too far |
576 |
|
out. In addition, this would have an even more detrimental effect on |
577 |
|
the system densities, leading to a liquid with a more open structure |
578 |
< |
and a density considerably lower than the normal SSD behavior shown |
579 |
< |
previously. A better correction would be to include the |
580 |
< |
quadrupole-quadrupole interactions for the water particles outside of |
581 |
< |
the first solvation shell, but this reduces the simplicity and speed |
582 |
< |
advantage of SSD. |
578 |
> |
and a density considerably lower than the already low SSD density. A |
579 |
> |
better correction would be to include the quadrupole-quadrupole |
580 |
> |
interactions for the water particles outside of the first solvation |
581 |
> |
shell, but this would remove the simplicity and speed advantage of |
582 |
> |
SSD. |
583 |
|
|
584 |
|
\subsection{Adjusted Potentials: SSD/RF and SSD/E} |
585 |
+ |
|
586 |
|
The propensity of SSD to adopt lower than expected densities under |
587 |
|
varying conditions is troubling, especially at higher temperatures. In |
588 |
|
order to correct this model for use with a reaction field, it is |
590 |
|
intermolecular interactions. In undergoing a reparameterization, it is |
591 |
|
important not to focus on just one property and neglect the other |
592 |
|
important properties. In this case, it would be ideal to correct the |
593 |
< |
densities while maintaining the accurate transport properties. |
593 |
> |
densities while maintaining the accurate transport behavior. |
594 |
|
|
595 |
< |
The parameters available for tuning include the $\sigma$ and $\epsilon$ |
596 |
< |
Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky |
597 |
< |
attractive and dipole repulsive terms with their respective |
598 |
< |
cutoffs. To alter the attractive and repulsive terms of the sticky |
599 |
< |
potential independently, it is necessary to separate the terms as |
600 |
< |
follows: |
601 |
< |
\begin{equation} |
602 |
< |
u_{ij}^{sp} |
603 |
< |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = |
565 |
< |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)] + \frac{\nu_0^\prime}{2} [s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)], |
566 |
< |
\end{equation} |
567 |
< |
|
568 |
< |
where $\nu_0$ scales the strength of the tetrahedral attraction and |
569 |
< |
$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion |
570 |
< |
term. The separation was performed for purposes of the |
571 |
< |
reparameterization, but the final parameters were adjusted so that it |
572 |
< |
is unnecessary to separate the terms when implementing the adjusted |
573 |
< |
water potentials. The results of the reparameterizations are shown in |
574 |
< |
table \ref{params}. Note that the tetrahedral attractive and dipolar |
575 |
< |
repulsive terms do not share the same lower cutoff ($r_l$) in the |
576 |
< |
newly parameterized potentials - soft sticky dipole reaction field |
577 |
< |
(SSD/RF - for use with a reaction field) and soft sticky dipole |
578 |
< |
enhanced (SSD/E - an attempt to improve the liquid structure in |
595 |
> |
The parameters available for tuning include the $\sigma$ and |
596 |
> |
$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the |
597 |
> |
strength of the sticky potential ($\nu_0$), and the sticky attractive |
598 |
> |
and dipole repulsive cubic switching function cutoffs ($r_l$, $r_u$ |
599 |
> |
and $r_l^\prime$, $r_u^\prime$ respectively). The results of the |
600 |
> |
reparameterizations are shown in table \ref{params}. We are calling |
601 |
> |
these reparameterizations the Soft Sticky Dipole / Reaction Field |
602 |
> |
(SSD/RF - for use with a reaction field) and Soft Sticky Dipole |
603 |
> |
Extended (SSD/E - an attempt to improve the liquid structure in |
604 |
|
simulations without a long-range correction). |
605 |
|
|
606 |
|
\begin{table} |
608 |
|
\caption{Parameters for the original and adjusted models} |
609 |
|
\begin{tabular}{ l c c c c } |
610 |
|
\hline \\[-3mm] |
611 |
< |
\ \ \ Parameters\ \ \ & \ \ \ SSD\cite{Ichiye96} \ \ \ & \ SSD1\cite{Ichiye03}\ \ & \ SSD/E\ \ & \ SSD/RF \\ |
611 |
> |
\ \ \ Parameters\ \ \ & \ \ \ SSD [Ref. \citen{Ichiye96}] \ \ \ |
612 |
> |
& \ SSD1 [Ref. \citen{Ichiye03}]\ \ & \ SSD/E\ \ & \ SSD/RF \\ |
613 |
|
\hline \\[-3mm] |
614 |
|
\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
615 |
|
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
616 |
|
\ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\ |
617 |
|
\ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
618 |
+ |
\ \ \ $\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\ |
619 |
|
\ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\ |
620 |
|
\ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\ |
594 |
– |
\ \ \ $\nu_0^\prime$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
621 |
|
\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\ |
622 |
|
\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\ |
623 |
|
\end{tabular} |
629 |
|
\begin{center} |
630 |
|
\epsfxsize=5in |
631 |
|
\epsfbox{GofRCompare.epsi} |
632 |
< |
\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E |
632 |
> |
\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with SSD/E |
633 |
|
and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with |
634 |
|
reaction field turned on (bottom). The insets show the respective |
635 |
|
first peaks in detail. Note how the changes in parameters have lowered |
650 |
|
\end{center} |
651 |
|
\end{figure} |
652 |
|
|
653 |
< |
In the paper detailing the development of SSD, Liu and Ichiye placed |
654 |
< |
particular emphasis on an accurate description of the first solvation |
655 |
< |
shell. This resulted in a somewhat tall and narrow first peak in the |
656 |
< |
g(\emph{r}) that integrated to give similar coordination numbers to |
653 |
> |
In the original paper detailing the development of SSD, Liu and Ichiye |
654 |
> |
placed particular emphasis on an accurate description of the first |
655 |
> |
solvation shell. This resulted in a somewhat tall and narrow first |
656 |
> |
peak in $g(r)$ that integrated to give similar coordination numbers to |
657 |
|
the experimental data obtained by Soper and |
658 |
|
Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering |
659 |
|
data from the Head-Gordon lab indicates a slightly lower and shifted |
660 |
< |
first peak in the g$_\mathrm{OO}(r)$, so adjustments to SSD were made |
661 |
< |
while taking into consideration the new experimental |
660 |
> |
first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were |
661 |
> |
made while taking into consideration the new experimental |
662 |
|
findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the |
663 |
< |
relocation of the first peak of the oxygen-oxygen g(\emph{r}) by |
664 |
< |
comparing the revised SSD model (SSD1), SSD-E, and SSD-RF to the new |
663 |
> |
relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing |
664 |
> |
the revised SSD model (SSD1), SSD/E, and SSD/RF to the new |
665 |
|
experimental results. Both modified water models have shorter peaks |
666 |
< |
that are brought in more closely to the experimental peak (as seen in |
667 |
< |
the insets of figure \ref{grcompare}). This structural alteration was |
666 |
> |
that match more closely to the experimental peak (as seen in the |
667 |
> |
insets of figure \ref{grcompare}). This structural alteration was |
668 |
|
accomplished by the combined reduction in the Lennard-Jones $\sigma$ |
669 |
< |
variable and adjustment of the sticky potential strength and |
670 |
< |
cutoffs. As can be seen in table \ref{params}, the cutoffs for the |
671 |
< |
tetrahedral attractive and dipolar repulsive terms were nearly swapped |
672 |
< |
with each other. Isosurfaces of the original and modified sticky |
673 |
< |
potentials are shown in figure \ref{isosurface}. In these isosurfaces, |
674 |
< |
it is easy to see how altering the cutoffs changes the repulsive and |
675 |
< |
attractive character of the particles. With a reduced repulsive |
676 |
< |
surface (darker region), the particles can move closer to one another, |
677 |
< |
increasing the density for the overall system. This change in |
678 |
< |
interaction cutoff also results in a more gradual orientational motion |
679 |
< |
by allowing the particles to maintain preferred dipolar arrangements |
680 |
< |
before they begin to feel the pull of the tetrahedral |
681 |
< |
restructuring. As the particles move closer together, the dipolar |
682 |
< |
repulsion term becomes active and excludes unphysical nearest-neighbor |
683 |
< |
arrangements. This compares with how SSD and SSD1 exclude preferred |
684 |
< |
dipole alignments before the particles feel the pull of the ``hydrogen |
685 |
< |
bonds''. Aside from improving the shape of the first peak in the |
686 |
< |
g(\emph{r}), this modification improves the densities considerably by |
687 |
< |
allowing the persistence of full dipolar character below the previous |
688 |
< |
4.0 \AA\ cutoff. |
669 |
> |
variable and adjustment of the sticky potential strength and cutoffs. |
670 |
> |
As can be seen in table \ref{params}, the cutoffs for the tetrahedral |
671 |
> |
attractive and dipolar repulsive terms were nearly swapped with each |
672 |
> |
other. Isosurfaces of the original and modified sticky potentials are |
673 |
> |
shown in figure \ref{isosurface}. In these isosurfaces, it is easy to |
674 |
> |
see how altering the cutoffs changes the repulsive and attractive |
675 |
> |
character of the particles. With a reduced repulsive surface (darker |
676 |
> |
region), the particles can move closer to one another, increasing the |
677 |
> |
density for the overall system. This change in interaction cutoff also |
678 |
> |
results in a more gradual orientational motion by allowing the |
679 |
> |
particles to maintain preferred dipolar arrangements before they begin |
680 |
> |
to feel the pull of the tetrahedral restructuring. As the particles |
681 |
> |
move closer together, the dipolar repulsion term becomes active and |
682 |
> |
excludes unphysical nearest-neighbor arrangements. This compares with |
683 |
> |
how SSD and SSD1 exclude preferred dipole alignments before the |
684 |
> |
particles feel the pull of the ``hydrogen bonds''. Aside from |
685 |
> |
improving the shape of the first peak in the g(\emph{r}), this |
686 |
> |
modification improves the densities considerably by allowing the |
687 |
> |
persistence of full dipolar character below the previous 4.0 \AA\ |
688 |
> |
cutoff. |
689 |
|
|
690 |
< |
While adjusting the location and shape of the first peak of |
691 |
< |
g(\emph{r}) improves the densities, these changes alone are |
692 |
< |
insufficient to bring the system densities up to the values observed |
693 |
< |
experimentally. To further increase the densities, the dipole moments |
694 |
< |
were increased in both of the adjusted models. Since SSD is a dipole |
695 |
< |
based model, the structure and transport are very sensitive to changes |
696 |
< |
in the dipole moment. The original SSD simply used the dipole moment |
697 |
< |
calculated from the TIP3P water model, which at 2.35 D is |
698 |
< |
significantly greater than the experimental gas phase value of 1.84 |
699 |
< |
D. The larger dipole moment is a more realistic value and improves the |
700 |
< |
dielectric properties of the fluid. Both theoretical and experimental |
701 |
< |
measurements indicate a liquid phase dipole moment ranging from 2.4 D |
702 |
< |
to values as high as 3.11 D, providing a substantial range of |
703 |
< |
reasonable values for a dipole |
678 |
< |
moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
690 |
> |
While adjusting the location and shape of the first peak of $g(r)$ |
691 |
> |
improves the densities, these changes alone are insufficient to bring |
692 |
> |
the system densities up to the values observed experimentally. To |
693 |
> |
further increase the densities, the dipole moments were increased in |
694 |
> |
both of our adjusted models. Since SSD is a dipole based model, the |
695 |
> |
structure and transport are very sensitive to changes in the dipole |
696 |
> |
moment. The original SSD simply used the dipole moment calculated from |
697 |
> |
the TIP3P water model, which at 2.35 D is significantly greater than |
698 |
> |
the experimental gas phase value of 1.84 D. The larger dipole moment |
699 |
> |
is a more realistic value and improves the dielectric properties of |
700 |
> |
the fluid. Both theoretical and experimental measurements indicate a |
701 |
> |
liquid phase dipole moment ranging from 2.4 D to values as high as |
702 |
> |
3.11 D, providing a substantial range of reasonable values for a |
703 |
> |
dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
704 |
|
increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF, |
705 |
|
respectively, leads to significant changes in the density and |
706 |
|
transport of the water models. |
718 |
|
run at each temperature step, and the final configuration from the |
719 |
|
previous temperature simulation was used as a starting point. All NVE |
720 |
|
simulations had the same thermalization, equilibration, and data |
721 |
< |
collection times as stated earlier in this paper. |
721 |
> |
collection times as stated previously. |
722 |
|
|
723 |
|
\begin{figure} |
724 |
|
\begin{center} |
725 |
|
\epsfxsize=6in |
726 |
|
\epsfbox{ssdeDense.epsi} |
727 |
|
\caption{Comparison of densities calculated with SSD/E to SSD1 without a |
728 |
< |
reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00} |
729 |
< |
SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The window shows a |
730 |
< |
expansion around 300 K with error bars included to clarify this region |
731 |
< |
of interest. Note that both SSD1 and SSD/E show good agreement with |
728 |
> |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
729 |
> |
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and |
730 |
> |
experiment [Ref. \citen{CRC80}]. The window shows a expansion around |
731 |
> |
300 K with error bars included to clarify this region of |
732 |
> |
interest. Note that both SSD1 and SSD/E show good agreement with |
733 |
|
experiment when the long-range correction is neglected.} |
734 |
|
\label{ssdedense} |
735 |
|
\end{center} |
736 |
|
\end{figure} |
737 |
|
|
738 |
< |
Figure \ref{ssdedense} shows the density profile for the SSD/E model |
738 |
> |
Fig. \ref{ssdedense} shows the density profile for the SSD/E model |
739 |
|
in comparison to SSD1 without a reaction field, other common water |
740 |
|
models, and experimental results. The calculated densities for both |
741 |
|
SSD/E and SSD1 have increased significantly over the original SSD |
742 |
< |
model (see figure \ref{dense1}) and are in better agreement with the |
742 |
> |
model (see fig. \ref{dense1}) and are in better agreement with the |
743 |
|
experimental values. At 298 K, the densities of SSD/E and SSD1 without |
744 |
|
a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and |
745 |
|
0.999$\pm$0.001 g/cm$^3$ respectively. These both compare well with |
751 |
|
comes about via an increase in the liquid disorder through the |
752 |
|
weakening of the sticky potential and strengthening of the dipolar |
753 |
|
character. However, this increasing disorder in the SSD/E model has |
754 |
< |
little effect on the melting transition. By monitoring C$\text{p}$ |
754 |
> |
little effect on the melting transition. By monitoring $C_p$ |
755 |
|
throughout these simulations, the melting transition for SSD/E was |
756 |
< |
shown to occur at 235 K, the same transition temperature observed with |
757 |
< |
SSD and SSD1. |
756 |
> |
shown to occur at 235 K. The same transition temperature observed |
757 |
> |
with SSD and SSD1. |
758 |
|
|
759 |
|
\begin{figure} |
760 |
|
\begin{center} |
761 |
|
\epsfxsize=6in |
762 |
|
\epsfbox{ssdrfDense.epsi} |
763 |
|
\caption{Comparison of densities calculated with SSD/RF to SSD1 with a |
764 |
< |
reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00} |
765 |
< |
SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The inset shows the |
766 |
< |
necessity of reparameterization when utilizing a reaction field |
767 |
< |
long-ranged correction - SSD/RF provides significantly more accurate |
768 |
< |
densities than SSD1 when performing room temperature simulations.} |
764 |
> |
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
765 |
> |
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and |
766 |
> |
experiment [Ref. \citen{CRC80}]. The inset shows the necessity of |
767 |
> |
reparameterization when utilizing a reaction field long-ranged |
768 |
> |
correction - SSD/RF provides significantly more accurate densities |
769 |
> |
than SSD1 when performing room temperature simulations.} |
770 |
|
\label{ssdrfdense} |
771 |
|
\end{center} |
772 |
|
\end{figure} |
773 |
|
|
774 |
|
Including the reaction field long-range correction in the simulations |
775 |
< |
results in a more interesting comparison. A density profile including |
775 |
> |
results in a more interesting comparison. A density profile including |
776 |
|
SSD/RF and SSD1 with an active reaction field is shown in figure |
777 |
|
\ref{ssdrfdense}. As observed in the simulations without a reaction |
778 |
|
field, the densities of SSD/RF and SSD1 show a dramatic increase over |
779 |
|
normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density |
780 |
|
of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and |
781 |
< |
considerably better than the SSD value of 0.941$\pm$0.001 g/cm$^3$ and |
782 |
< |
the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results further |
783 |
< |
emphasize the importance of reparameterization in order to model the |
784 |
< |
density properly under different simulation conditions. Again, these |
785 |
< |
changes have only a minor effect on the melting point, which observed |
786 |
< |
at 245 K for SSD/RF, is identical to SSD and only 5 K lower than SSD1 |
787 |
< |
with a reaction field. Additionally, the difference in density maxima |
788 |
< |
is not as extreme, with SSD/RF showing a density maximum at 255 K, |
789 |
< |
fairly close to the density maxima of 260 K and 265 K, shown by SSD |
790 |
< |
and SSD1 respectively. |
781 |
> |
considerably better than the original SSD value of 0.941$\pm$0.001 |
782 |
> |
g/cm$^3$ and the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results |
783 |
> |
further emphasize the importance of reparameterization in order to |
784 |
> |
model the density properly under different simulation conditions. |
785 |
> |
Again, these changes have only a minor effect on the melting point, |
786 |
> |
which observed at 245 K for SSD/RF, is identical to SSD and only 5 K |
787 |
> |
lower than SSD1 with a reaction field. Additionally, the difference in |
788 |
> |
density maxima is not as extreme, with SSD/RF showing a density |
789 |
> |
maximum at 255 K, fairly close to the density maxima of 260 K and 265 |
790 |
> |
K, shown by SSD and SSD1 respectively. |
791 |
|
|
792 |
|
\begin{figure} |
793 |
|
\begin{center} |
794 |
|
\epsfxsize=6in |
795 |
|
\epsfbox{ssdeDiffuse.epsi} |
796 |
< |
\caption{Plots of the diffusion constants calculated from SSD/E and SSD1, |
797 |
< |
both without a reaction field, along with experimental |
798 |
< |
results.\cite{Gillen72,Mills73} The NVE calculations were performed |
799 |
< |
at the average densities observed in the 1 atm NPT simulations for |
800 |
< |
the respective models. SSD/E is slightly more fluid than experiment |
801 |
< |
at all of the temperatures, but it is closer than SSD1 without a |
796 |
> |
\caption{The diffusion constants calculated from SSD/E and SSD1, |
797 |
> |
both without a reaction field, along with experimental results |
798 |
> |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
799 |
> |
were performed at the average densities observed in the 1 atm NPT |
800 |
> |
simulations for the respective models. SSD/E is slightly more mobile |
801 |
> |
than experiment at all of the temperatures, but it is closer to |
802 |
> |
experiment at biologically relavent temperatures than SSD1 without a |
803 |
|
long-range correction.} |
804 |
|
\label{ssdediffuse} |
805 |
|
\end{center} |
808 |
|
The reparameterization of the SSD water model, both for use with and |
809 |
|
without an applied long-range correction, brought the densities up to |
810 |
|
what is expected for simulating liquid water. In addition to improving |
811 |
< |
the densities, it is important that particle transport be maintained |
812 |
< |
or improved. Figure \ref{ssdediffuse} compares the temperature |
813 |
< |
dependence of the diffusion constant of SSD/E to SSD1 without an |
814 |
< |
active reaction field, both at the densities calculated at 1 atm and |
815 |
< |
at the experimentally calculated densities for super-cooled and liquid |
816 |
< |
water. The diffusion constant for SSD/E is consistently a little |
817 |
< |
higher than experiment, while SSD1 remains lower than experiment until |
818 |
< |
relatively high temperatures (greater than 330 K). Both models follow |
819 |
< |
the shape of the experimental curve well below 300 K but tend to |
820 |
< |
diffuse too rapidly at higher temperatures, something that is |
821 |
< |
especially apparent with SSD1. This accelerated increasing of |
822 |
< |
diffusion is caused by the rapidly decreasing system density with |
823 |
< |
increasing temperature. Though it is difficult to see in figure |
824 |
< |
\ref{ssdedense}, the densities of SSD1 decay more rapidly with |
825 |
< |
temperature than do those of SSD/E, leading to more visible deviation |
826 |
< |
from the experimental diffusion trend. Thus, the changes made to |
811 |
> |
the densities, it is important that the excellent diffusive behavior |
812 |
> |
of SSD be maintained or improved. Figure \ref{ssdediffuse} compares |
813 |
> |
the temperature dependence of the diffusion constant of SSD/E to SSD1 |
814 |
> |
without an active reaction field at the densities calculated from the |
815 |
> |
NPT simulations at 1 atm. The diffusion constant for SSD/E is |
816 |
> |
consistently higher than experiment, while SSD1 remains lower than |
817 |
> |
experiment until relatively high temperatures (around 360 K). Both |
818 |
> |
models follow the shape of the experimental curve well below 300 K but |
819 |
> |
tend to diffuse too rapidly at higher temperatures, as seen in SSD1's |
820 |
> |
crossing above 360 K. This increasing diffusion relative to the |
821 |
> |
experimental values is caused by the rapidly decreasing system density |
822 |
> |
with increasing temperature. Both SSD1 and SSD/E show this deviation |
823 |
> |
in diffusive behavior, but this trend has different implications on |
824 |
> |
the diffusive behavior of the models. While SSD1 shows more |
825 |
> |
experimentally accurate diffusive behavior in the high temperature |
826 |
> |
regimes, SSD/E shows more accurate behavior in the supercooled and |
827 |
> |
biologically relavent temperature ranges. Thus, the changes made to |
828 |
|
improve the liquid structure may have had an adverse affect on the |
829 |
|
density maximum, but they improve the transport behavior of SSD/E |
830 |
< |
relative to SSD1. |
830 |
> |
relative to SSD1 under the most commonly simulated conditions. |
831 |
|
|
832 |
|
\begin{figure} |
833 |
|
\begin{center} |
834 |
|
\epsfxsize=6in |
835 |
|
\epsfbox{ssdrfDiffuse.epsi} |
836 |
< |
\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1, |
837 |
< |
both with an active reaction field, along with experimental |
838 |
< |
results.\cite{Gillen72,Mills73} The NVE calculations were performed |
839 |
< |
at the average densities observed in the 1 atm NPT simulations for |
840 |
< |
both of the models. Note how accurately SSD/RF simulates the |
841 |
< |
diffusion of water throughout this temperature range. The more |
842 |
< |
rapidly increasing diffusion constants at high temperatures for both |
843 |
< |
models is attributed to the significantly lower densities than |
844 |
< |
observed in experiment.} |
836 |
> |
\caption{The diffusion constants calculated from SSD/RF and SSD1, |
837 |
> |
both with an active reaction field, along with experimental results |
838 |
> |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
839 |
> |
were performed at the average densities observed in the 1 atm NPT |
840 |
> |
simulations for both of the models. Note how accurately SSD/RF |
841 |
> |
simulates the diffusion of water throughout this temperature |
842 |
> |
range. The more rapidly increasing diffusion constants at high |
843 |
> |
temperatures for both models is attributed to lower calculated |
844 |
> |
densities than those observed in experiment.} |
845 |
|
\label{ssdrfdiffuse} |
846 |
|
\end{center} |
847 |
|
\end{figure} |
848 |
|
|
849 |
|
In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are |
850 |
|
compared to SSD1 with an active reaction field. Note that SSD/RF |
851 |
< |
tracks the experimental results incredibly well, identical within |
852 |
< |
error throughout the temperature range shown and with only a slight |
853 |
< |
increasing trend at higher temperatures. SSD1 tends to diffuse more |
854 |
< |
slowly at low temperatures and deviates to diffuse too rapidly at |
855 |
< |
temperatures greater than 330 K. As stated in relation to SSD/E, this |
856 |
< |
deviation away from the ideal trend is due to a rapid decrease in |
857 |
< |
density at higher temperatures. SSD/RF does not suffer from this |
858 |
< |
problem as much as SSD1, because the calculated densities are closer |
859 |
< |
to the experimental value. These results again emphasize the |
860 |
< |
importance of careful reparameterization when using an altered |
832 |
< |
long-range correction. |
851 |
> |
tracks the experimental results quantitatively, identical within error |
852 |
> |
throughout most of the temperature range shown and exhibiting only a |
853 |
> |
slight increasing trend at higher temperatures. SSD1 tends to diffuse |
854 |
> |
more slowly at low temperatures and deviates to diffuse too rapidly at |
855 |
> |
temperatures greater than 330 K. As stated above, this deviation away |
856 |
> |
from the ideal trend is due to a rapid decrease in density at higher |
857 |
> |
temperatures. SSD/RF does not suffer from this problem as much as SSD1 |
858 |
> |
because the calculated densities are closer to the experimental |
859 |
> |
values. These results again emphasize the importance of careful |
860 |
> |
reparameterization when using an altered long-range correction. |
861 |
|
|
862 |
+ |
\begin{table} |
863 |
+ |
\begin{center} |
864 |
+ |
\caption{Calculated and experimental properties of the single point waters and liquid water at 298 K and 1 atm. (a) Ref. [\citen{Mills73}]. (b) Calculated by integrating the data in ref. \citen{Head-Gordon00_1}. (c) Calculated by integrating the data in ref. \citen{Soper86}. (d) Ref. [\citen{Eisenberg69}]. (e) Calculated for 298 K from data in ref. \citen{Krynicki66}.} |
865 |
+ |
\begin{tabular}{ l c c c c c } |
866 |
+ |
\hline \\[-3mm] |
867 |
+ |
\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \ |
868 |
+ |
\ & \ SSD/RF \ \ \ & \ Expt. \\ |
869 |
+ |
\hline \\[-3mm] |
870 |
+ |
\ \ \ $\rho$ (g/cm$^3$) & 0.999 $\pm$0.001 & 0.996 $\pm$0.001 & 0.972 $\pm$0.002 & 0.997 $\pm$0.001 & 0.997 \\ |
871 |
+ |
\ \ \ $C_p$ (cal/mol K) & 28.80 $\pm$0.11 & 25.45 $\pm$0.09 & 28.28 $\pm$0.06 & 23.83 $\pm$0.16 & 17.98 \\ |
872 |
+ |
\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 & 2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299$^\text{a}$ \\ |
873 |
+ |
\ \ \ Coordination Number & 3.9 & 4.3 & 3.8 & 4.4 & 4.7$^\text{b}$ \\ |
874 |
+ |
\ \ \ H-bonds per particle & 3.7 & 3.6 & 3.7 & 3.7 & 3.4$^\text{c}$ \\ |
875 |
+ |
\ \ \ $\tau_1^\mu$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & 7.2 $\pm$0.4 & 4.76$^\text{d}$ \\ |
876 |
+ |
\ \ \ $\tau_2^\mu$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2 $\pm$0.2 & 2.3$^\text{e}$ \\ |
877 |
+ |
\end{tabular} |
878 |
+ |
\label{liquidproperties} |
879 |
+ |
\end{center} |
880 |
+ |
\end{table} |
881 |
+ |
|
882 |
+ |
Table \ref{liquidproperties} gives a synopsis of the liquid state |
883 |
+ |
properties of the water models compared in this study along with the |
884 |
+ |
experimental values for liquid water at ambient conditions. The |
885 |
+ |
coordination number and hydrogen bonds per particle were calculated by |
886 |
+ |
integrating the following relation: |
887 |
+ |
\begin{equation} |
888 |
+ |
4\pi\rho\int_{0}^{a}r^2\text{g}(r)dr, |
889 |
+ |
\end{equation} |
890 |
+ |
where $\rho$ is the number density of pair interactions, $a$ is the |
891 |
+ |
radial location of the minima following the first solvation shell |
892 |
+ |
peak, and g$(r)$ is either g$_\text{OO}(r)$ or g$_\text{OH}(r)$ for |
893 |
+ |
calculation of the coordination number or hydrogen bonds per particle |
894 |
+ |
respectively. |
895 |
+ |
|
896 |
+ |
The time constants for the self orientational autocorrelation function |
897 |
+ |
are also displayed in Table \ref{liquidproperties}. The dipolar |
898 |
+ |
orientational time correlation function ($\Gamma_{l}$) is described |
899 |
+ |
by: |
900 |
+ |
\begin{equation} |
901 |
+ |
\Gamma_{l}(t) = \langle P_l[\mathbf{u}_j(0)\cdot\mathbf{u}_j(t)]\rangle, |
902 |
+ |
\end{equation} |
903 |
+ |
where $P_l$ is a Legendre polynomial of order $l$ and $\mathbf{u}_j$ |
904 |
+ |
is the unit vector of the particle dipole.\cite{Rahman71} From these |
905 |
+ |
correlation functions, the orientational relaxation time of the dipole |
906 |
+ |
vector can be calculated from an exponential fit in the long-time |
907 |
+ |
regime ($t > \tau_l^\mu$).\cite{Rothschild84} Calculation of these |
908 |
+ |
time constants were averaged from five detailed NVE simulations |
909 |
+ |
performed at the STP density for each of the respective models. |
910 |
+ |
|
911 |
|
\subsection{Additional Observations} |
912 |
|
|
913 |
|
\begin{figure} |
915 |
|
\epsfxsize=6in |
916 |
|
\epsfbox{povIce.ps} |
917 |
|
\caption{A water lattice built from the crystal structure assumed by |
918 |
< |
SSD/E when undergoing an extremely restricted temperature NPT |
919 |
< |
simulation. This form of ice is referred to as ice 0 to emphasize its |
920 |
< |
simulation origins. This image was taken of the (001) face of the |
921 |
< |
crystal.} |
918 |
> |
SSD/E when undergoing an extremely restricted temperature NPT |
919 |
> |
simulation. This form of ice is referred to as ice-{\it i} to |
920 |
> |
emphasize its simulation origins. This image was taken of the (001) |
921 |
> |
face of the crystal.} |
922 |
|
\label{weirdice} |
923 |
|
\end{center} |
924 |
|
\end{figure} |
925 |
|
|
926 |
< |
While performing restricted temperature melting sequences of SSD/E not |
927 |
< |
previously discussed, some interesting observations were made. After |
928 |
< |
melting at 235 K, two of five systems underwent crystallization events |
929 |
< |
near 245 K. As the heating process continued, the two systems remained |
930 |
< |
crystalline until finally melting between 320 and 330 K. The final |
931 |
< |
configurations of these two melting sequences show an expanded |
932 |
< |
zeolite-like crystal structure that does not correspond to any known |
933 |
< |
form of ice. For convenience, and to help distinguish it from the |
934 |
< |
experimentally observed forms of ice, this crystal structure will |
935 |
< |
henceforth be referred to as ice-zero (ice 0). The crystallinity was |
936 |
< |
extensive enough that a near ideal crystal structure of ice 0 could be |
937 |
< |
obtained. Figure \ref{weirdice} shows the repeating crystal structure |
938 |
< |
of a typical crystal at 5 K. Each water molecule is hydrogen bonded to |
939 |
< |
four others; however, the hydrogen bonds are flexed rather than |
940 |
< |
perfectly straight. This results in a skewed tetrahedral geometry |
941 |
< |
about the central molecule. Referring to figure \ref{isosurface}, |
942 |
< |
these flexed hydrogen bonds are allowed due to the conical shape of |
943 |
< |
the attractive regions, with the greatest attraction along the direct |
944 |
< |
hydrogen bond configuration. Though not ideal, these flexed hydrogen |
945 |
< |
bonds are favorable enough to stabilize an entire crystal generated |
946 |
< |
around them. In fact, the imperfect ice 0 crystals were so stable that |
870 |
< |
they melted at temperatures nearly 100 K greater than both ice I$_c$ |
871 |
< |
and I$_h$. |
926 |
> |
While performing a series of melting simulations on an early iteration |
927 |
> |
of SSD/E not discussed in this paper, we observed recrystallization |
928 |
> |
into a novel structure not previously known for water. After melting |
929 |
> |
at 235 K, two of five systems underwent crystallization events near |
930 |
> |
245 K. The two systems remained crystalline up to 320 and 330 K, |
931 |
> |
respectively. The crystal exhibits an expanded zeolite-like structure |
932 |
> |
that does not correspond to any known form of ice. This appears to be |
933 |
> |
an artifact of the point dipolar models, so to distinguish it from the |
934 |
> |
experimentally observed forms of ice, we have denoted the structure |
935 |
> |
Ice-$\sqrt{\smash[b]{-\text{I}}}$ (ice-{\it i}). A large enough |
936 |
> |
portion of the sample crystallized that we have been able to obtain a |
937 |
> |
near ideal crystal structure of ice-{\it i}. Figure \ref{weirdice} |
938 |
> |
shows the repeating crystal structure of a typical crystal at 5 |
939 |
> |
K. Each water molecule is hydrogen bonded to four others; however, the |
940 |
> |
hydrogen bonds are bent rather than perfectly straight. This results |
941 |
> |
in a skewed tetrahedral geometry about the central molecule. In |
942 |
> |
figure \ref{isosurface}, it is apparent that these flexed hydrogen |
943 |
> |
bonds are allowed due to the conical shape of the attractive regions, |
944 |
> |
with the greatest attraction along the direct hydrogen bond |
945 |
> |
configuration. Though not ideal, these flexed hydrogen bonds are |
946 |
> |
favorable enough to stabilize an entire crystal generated around them. |
947 |
|
|
948 |
< |
These initial simulations indicated that ice 0 is the preferred ice |
948 |
> |
Initial simulations indicated that ice-{\it i} is the preferred ice |
949 |
|
structure for at least the SSD/E model. To verify this, a comparison |
950 |
< |
was made between near ideal crystals of ice $I_h$, ice $I_c$, and ice |
951 |
< |
0 at constant pressure with SSD/E, SSD/RF, and SSD1. Near ideal |
952 |
< |
versions of the three types of crystals were cooled to 1 K, and the |
953 |
< |
potential energies of each were compared using all three water |
954 |
< |
models. With every water model, ice 0 turned out to have the lowest |
955 |
< |
potential energy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with |
956 |
< |
SSD/E, and 7.5\% lower with SSD/RF. |
950 |
> |
was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and |
951 |
> |
ice-{\it i} at constant pressure with SSD/E, SSD/RF, and |
952 |
> |
SSD1. Near-ideal versions of the three types of crystals were cooled |
953 |
> |
to 1 K, and the enthalpies of each were compared using all three water |
954 |
> |
models. With every model in the SSD family, ice-{\it i} had the lowest |
955 |
> |
calculated enthalpy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with |
956 |
> |
SSD/E, and 7.5\% lower with SSD/RF. The enthalpy data is summarized |
957 |
> |
in Table \ref{iceenthalpy}. |
958 |
|
|
959 |
< |
In addition to these low temperature comparisons, melting sequences |
960 |
< |
were performed with ice 0 as the initial configuration using SSD/E, |
959 |
> |
\begin{table} |
960 |
> |
\begin{center} |
961 |
> |
\caption{Enthalpies (in kcal / mol) of the three crystal structures (at 1 |
962 |
> |
K) exhibited by the SSD family of water models} |
963 |
> |
\begin{tabular}{ l c c c } |
964 |
> |
\hline \\[-3mm] |
965 |
> |
\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \ & \ |
966 |
> |
Ice-{\it i} \\ |
967 |
> |
\hline \\[-3mm] |
968 |
> |
\ \ \ SSD/E & -12.286 & -12.292 & -13.590 \\ |
969 |
> |
\ \ \ SSD/RF & -12.935 & -12.917 & -14.022 \\ |
970 |
> |
\ \ \ SSD1 & -12.496 & -12.411 & -13.417 \\ |
971 |
> |
\ \ \ SSD1 (RF) & -12.504 & -12.411 & -13.134 \\ |
972 |
> |
\end{tabular} |
973 |
> |
\label{iceenthalpy} |
974 |
> |
\end{center} |
975 |
> |
\end{table} |
976 |
> |
|
977 |
> |
In addition to these energetic comparisons, melting simulations were |
978 |
> |
performed with ice-{\it i} as the initial configuration using SSD/E, |
979 |
|
SSD/RF, and SSD1 both with and without a reaction field. The melting |
980 |
< |
transitions for both SSD/E and SSD1 without a reaction field occurred |
981 |
< |
at temperature in excess of 375 K. SSD/RF and SSD1 with a reaction |
982 |
< |
field showed more reasonable melting transitions near 325 K. These |
983 |
< |
melting point observations emphasize the preference for this crystal |
984 |
< |
structure over the most common types of ice when using these single |
891 |
< |
point water models. |
980 |
> |
transitions for both SSD/E and SSD1 without reaction field occurred at |
981 |
> |
temperature in excess of 375~K. SSD/RF and SSD1 with a reaction field |
982 |
> |
showed more reasonable melting transitions near 325~K. These melting |
983 |
> |
point observations clearly show that all of the SSD-derived models |
984 |
> |
prefer the ice-{\it i} structure. |
985 |
|
|
893 |
– |
Recognizing that the above tests show ice 0 to be both the most stable |
894 |
– |
and lowest density crystal structure for these single point water |
895 |
– |
models, it is interesting to speculate on the relative stability of |
896 |
– |
this crystal structure with charge based water models. As a quick |
897 |
– |
test, these 3 crystal types were converted from SSD type particles to |
898 |
– |
TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy |
899 |
– |
minimizations were performed on the crystals to compare the system |
900 |
– |
energies. Again, ice 0 was observed to have the lowest total system |
901 |
– |
energy. The total energy of ice 0 was ~2\% lower than ice $I_h$, which |
902 |
– |
was in turn ~3\% lower than ice $I_c$. Based on these initial studies, |
903 |
– |
it would not be surprising if results from the other common water |
904 |
– |
models show ice 0 to be the lowest energy crystal structure. A |
905 |
– |
continuation of this work studying ice 0 with multi-point water models |
906 |
– |
will be published in a coming article. |
907 |
– |
|
986 |
|
\section{Conclusions} |
909 |
– |
The density maximum and temperature dependent transport for the SSD |
910 |
– |
water model, both with and without the use of reaction field, were |
911 |
– |
studied via a series of NPT and NVE simulations. The constant pressure |
912 |
– |
simulations of the melting of both $I_h$ and $I_c$ ice showed a |
913 |
– |
density maximum near 260 K. In most cases, the calculated densities |
914 |
– |
were significantly lower than the densities calculated in simulations |
915 |
– |
of other water models. Analysis of particle diffusion showed SSD to |
916 |
– |
capture the transport properties of experimental water well in both |
917 |
– |
the liquid and super-cooled liquid regimes. In order to correct the |
918 |
– |
density behavior, the original SSD model was reparameterized for use |
919 |
– |
both with and without a reaction field (SSD/RF and SSD/E), and |
920 |
– |
comparison simulations were performed with SSD1, the density corrected |
921 |
– |
version of SSD. Both models improve the liquid structure, density |
922 |
– |
values, and diffusive properties under their respective conditions, |
923 |
– |
indicating the necessity of reparameterization when altering the |
924 |
– |
long-range correction specifics. When taking into account the |
925 |
– |
appropriate considerations, these simple water models are excellent |
926 |
– |
choices for representing explicit water in large scale simulations of |
927 |
– |
biochemical systems. |
987 |
|
|
988 |
+ |
The density maximum and temperature dependence of the self-diffusion |
989 |
+ |
constant were studied for the SSD water model, both with and without |
990 |
+ |
the use of reaction field, via a series of NPT and NVE |
991 |
+ |
simulations. The constant pressure simulations showed a density |
992 |
+ |
maximum near 260 K. In most cases, the calculated densities were |
993 |
+ |
significantly lower than the densities obtained from other water |
994 |
+ |
models (and experiment). Analysis of self-diffusion showed SSD to |
995 |
+ |
capture the transport properties of water well in both the liquid and |
996 |
+ |
super-cooled liquid regimes. |
997 |
+ |
|
998 |
+ |
In order to correct the density behavior, the original SSD model was |
999 |
+ |
reparameterized for use both with and without a reaction field (SSD/RF |
1000 |
+ |
and SSD/E), and comparisons were made with SSD1, Ichiye's density |
1001 |
+ |
corrected version of SSD. Both models improve the liquid structure, |
1002 |
+ |
densities, and diffusive properties under their respective simulation |
1003 |
+ |
conditions, indicating the necessity of reparameterization when |
1004 |
+ |
changing the method of calculating long-range electrostatic |
1005 |
+ |
interactions. In general, however, these simple water models are |
1006 |
+ |
excellent choices for representing explicit water in large scale |
1007 |
+ |
simulations of biochemical systems. |
1008 |
+ |
|
1009 |
+ |
The existence of a novel low-density ice structure that is preferred |
1010 |
+ |
by the SSD family of water models is somewhat troubling, since liquid |
1011 |
+ |
simulations on this family of water models at room temperature are |
1012 |
+ |
effectively simulations of super-cooled or metastable liquids. One |
1013 |
+ |
way to de-stabilize this unphysical ice structure would be to make the |
1014 |
+ |
range of angles preferred by the attractive part of the sticky |
1015 |
+ |
potential much narrower. This would require extensive |
1016 |
+ |
reparameterization to maintain the same level of agreement with the |
1017 |
+ |
experiments. |
1018 |
+ |
|
1019 |
+ |
Additionally, our initial calculations show that the ice-{\it i} |
1020 |
+ |
structure may also be a preferred crystal structure for at least one |
1021 |
+ |
other popular multi-point water model (TIP3P), and that much of the |
1022 |
+ |
simulation work being done using this popular model could also be at |
1023 |
+ |
risk for crystallization into this unphysical structure. A future |
1024 |
+ |
publication will detail the relative stability of the known ice |
1025 |
+ |
structures for a wide range of popular water models. |
1026 |
+ |
|
1027 |
|
\section{Acknowledgments} |
1028 |
|
Support for this project was provided by the National Science |
1029 |
|
Foundation under grant CHE-0134881. Computation time was provided by |
1030 |
|
the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
1031 |
< |
DMR 00 79647. |
1031 |
> |
DMR-0079647. |
1032 |
|
|
935 |
– |
|
1033 |
|
\newpage |
1034 |
|
|
1035 |
|
\bibliographystyle{jcp} |