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\begin{document} |
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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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\author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote} |
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\footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}} |
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\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\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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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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\end{abstract} |
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\maketitle |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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\section{Introduction} |
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One of the most important tasks in simulations of biochemical systems |
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is the proper depiction of water and water solvation. In fact, the |
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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 |
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groups/bulk. Due to the fact that explicit solvent accounts for a |
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massive portion of the calculations, it necessary to simplify the |
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solvent to some extent in order to complete simulations in a |
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reasonable amount of time. In the case of simulating water in |
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bio-molecular studies, the balance between accurate properties and |
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computational efficiency is especially delicate, and it has resulted |
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in a variety of different water |
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models.\cite{Jorgensen83,Berendsen87,Jorgensen00} Many of these models |
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get specific properties correct or better than their predecessors, but |
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this is often at a cost of some other properties or of computer |
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time. As an example, compare TIP3P or TIP4P to TIP5P. TIP5P succeeds |
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in improving the structural and transport properties over its |
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predecessors, 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 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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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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\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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u_{ij}^{sp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\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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\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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\end{equation} |
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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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\end{equation} |
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\begin{equation} |
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\begin{split} |
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u_{ij}^{sp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) |
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&= |
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\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)\\ |
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& \quad \ + |
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s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
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\end{split} |
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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, $s$ and |
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$s^\prime$ are cubic switching functions. The $w$ and $w^\prime$ |
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functions take the following forms, |
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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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\end{equation} |
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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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\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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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 a |
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calculation speed up of up to an order of magnitude over other |
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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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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. Toward the end, we suggest |
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alterations to the parameters that result in more water-like |
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behavior. It should be noted that in a recent publication, some the |
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original investigators of the SSD water model have put forth |
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adjustments to the SSD water model to address abnormal density |
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behavior (also observed here), calling the corrected model |
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SSD1.\cite{Ichiye03} This study will make comparisons with this new |
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model's behavior with the goal of improving upon the depiction of |
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water under conditions without the Ewald Sum. |
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\section{Methods} |
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As stated previously, in this study the long-range dipole-dipole |
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interactions were accounted for using the reaction field method. The |
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magnitude of the reaction field acting on dipole \emph{i} is given by |
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\begin{equation} |
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\mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1} |
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\frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} \boldsymbol{\mu}_{j} f(r_{ij})\ , |
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\label{rfequation} |
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\end{equation} |
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where $\mathcal{R}$ is the cavity defined by the cutoff radius |
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($r_{c}$), $\varepsilon_{s}$ is the dielectric constant imposed on the |
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system (80 in this case), $\boldsymbol{\mu}_{j}$ is the dipole moment |
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vector of particle \emph{j}, and $f(r_{ij})$ is a cubic switching |
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function.\cite{AllenTildesley} The reaction field contribution to the |
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total energy by particle \emph{i} is given by |
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$-\frac{1}{2}\boldsymbol{\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque |
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on dipole \emph{i} by |
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$\boldsymbol{\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use |
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of reaction field is known to alter the orientational dynamic |
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properties, such as the dielectric relaxation time, based on changes |
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in the length of the cutoff radius.\cite{Berendsen98} This variable |
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behavior makes reaction field a less attractive method than other |
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methods, like the Ewald summation; however, for the simulation of |
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large-scale system, the computational cost benefit of reaction field |
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is dramatic. To address some of the dynamical property alterations due |
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to the use of reaction field, simulations were also performed without |
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a surrounding dielectric and suggestions are proposed on how to make |
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SSD more accurate both with and without a reaction field. |
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Simulations were performed in both the isobaric-isothermal and |
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microcanonical ensembles. The constant pressure simulations were |
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implemented using an integral thermostat and barostat as outlined by |
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Hoover.\cite{Hoover85,Hoover86} All particles were treated as |
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non-linear rigid bodies. Vibrational constraints are not necessary in |
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simulations of SSD, because there are no explicit hydrogen atoms, and |
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thus no molecular vibrational modes need to be considered. |
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Integration of the equations of motion was carried out using the |
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symplectic splitting method proposed by Dullweber \emph{et |
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al.}.\cite{Dullweber1997} The reason for this integrator selection |
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deals with poor energy conservation of rigid body systems using |
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quaternions. While quaternions work well for orientational motion in |
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alternate ensembles, the microcanonical ensemble has a constant energy |
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requirement that is quite sensitive to errors in the equations of |
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motion. The original implementation of this code utilized quaternions |
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for rotational motion propagation; however, a detailed investigation |
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showed that they resulted in a steady drift in the total energy, |
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something that has been observed by others.\cite{Laird97} |
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The key difference in the integration method proposed by Dullweber |
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\emph{et al.} is that the entire rotation matrix is propagated from |
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one time step to the next. In the past, this would not have been as |
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feasible a option, being that the rotation matrix for a single body is |
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nine elements long as opposed to 3 or 4 elements for Euler angles and |
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quaternions respectively. System memory has become much less of an |
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issue in recent times, and this has resulted in substantial benefits |
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in energy conservation. There is still the issue of 5 or 6 additional |
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elements for describing the orientation of each particle, which will |
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increase dump files substantially. Simply translating the rotation |
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matrix into its component Euler angles or quaternions for storage |
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purposes relieves this burden. |
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The symplectic splitting method allows for Verlet style integration of |
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both linear and angular motion of rigid bodies. In the integration |
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method, the orientational propagation involves a sequence of matrix |
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evaluations to update the rotation matrix.\cite{Dullweber1997} These |
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matrix rotations end up being more costly computationally than the |
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simpler arithmetic quaternion propagation. With the same time step, a |
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1000 SSD particle simulation shows an average 7\% increase in |
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computation time using the symplectic step method in place of |
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quaternions. This cost is more than justified when comparing the |
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energy conservation of the two methods as illustrated in figure |
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\ref{timestep}. |
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\begin{figure} |
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\includegraphics[width=61mm, angle=-90]{timeStep.epsi} |
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\caption{Energy conservation using quaternion based integration versus |
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the symplectic step method proposed by Dullweber \emph{et al.} with |
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increasing time step. For each time step, the dotted line is total |
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energy using the symplectic step integrator, and the solid line comes |
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from the quaternion integrator. The larger time step plots are shifted |
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up from the true energy baseline for clarity.} |
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\label{timestep} |
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\end{figure} |
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In figure \ref{timestep}, the resulting energy drift at various time |
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steps for both the symplectic step and quaternion integration schemes |
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is compared. All of the 1000 SSD particle simulations started with the |
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same configuration, and the only difference was the method for |
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handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
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methods for propagating particle rotation conserve energy fairly well, |
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with the quaternion method showing a slight energy drift over time in |
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the 0.5 fs time step simulation. At time steps of 1 and 2 fs, the |
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energy conservation benefits of the symplectic step method are clearly |
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demonstrated. Thus, while maintaining the same degree of energy |
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conservation, one can take considerably longer time steps, leading to |
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an overall reduction in computation time. |
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Energy drift in these SSD particle simulations was unnoticeable for |
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time steps up to three femtoseconds. A slight energy drift on the |
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order of 0.012 kcal/mol per nanosecond was observed at a time step of |
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four femtoseconds, and as expected, this drift increases dramatically |
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with increasing time step. To insure accuracy in the constant energy |
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simulations, time steps were set at 2 fs and kept at this value for |
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constant pressure simulations as well. |
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Ice crystals in both the $I_h$ and $I_c$ lattices were generated as |
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starting points for all the simulations. The $I_h$ crystals were |
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formed by first arranging the center of masses of the SSD particles |
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into a ``hexagonal'' ice lattice of 1024 particles. Because of the |
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crystal structure of $I_h$ ice, the simulation box assumed a |
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rectangular shape with a edge length ratio of approximately |
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1.00$\times$1.06$\times$1.23. The particles were then allowed to |
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orient freely about fixed positions with angular momenta randomized at |
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400 K for varying times. The rotational temperature was then scaled |
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down in stages to slowly cool the crystals down to 25 K. The particles |
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were then allowed translate with fixed orientations at a constant |
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pressure of 1 atm for 50 ps at 25 K. Finally, all constraints were |
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removed and the ice crystals were allowed to equilibrate for 50 ps at |
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25 K and a constant pressure of 1 atm. This procedure resulted in |
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structurally stable $I_h$ ice crystals that obey the Bernal-Fowler |
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rules\cite{Bernal33,Rahman72}. This method was also utilized in the |
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making of diamond lattice $I_c$ ice crystals, with each cubic |
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simulation box consisting of either 512 or 1000 particles. Only |
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isotropic volume fluctuations were performed under constant pressure, |
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so the ratio of edge lengths remained constant throughout the |
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simulations. |
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\section{Results and discussion} |
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Melting studies were performed on the randomized ice crystals using |
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constant pressure and temperature dynamics. By performing melting |
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simulations, the melting transition can be determined by monitoring |
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the heat capacity, in addition to determining the density maximum - |
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provided that the density maximum occurs in the liquid and not the |
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supercooled regime. An ensemble average from five separate melting |
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simulations was acquired, each starting from different ice crystals |
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generated as described previously. All simulations were equilibrated |
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for 100 ps prior to a 200 ps data collection run at each temperature |
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setting. The temperature range of study spanned from 25 to 400 K, with |
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a maximum degree increment of 25 K. For regions of interest along this |
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stepwise progression, the temperature increment was decreased from 25 |
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K to 10 and 5 K. The above equilibration and production times were |
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sufficient in that the system volume fluctuations dampened out in all |
322 |
|
|
but the very cold simulations (below 225 K). |
323 |
chrisfen |
743 |
|
324 |
|
|
\subsection{Density Behavior} |
325 |
chrisfen |
861 |
Initial simulations focused on the original SSD water model, and an |
326 |
|
|
average density versus temperature plot is shown in figure |
327 |
|
|
\ref{dense1}. Note that the density maximum when using a reaction |
328 |
|
|
field appears between 255 and 265 K, where the calculated densities |
329 |
|
|
within this range were nearly indistinguishable. The greater certainty |
330 |
|
|
of the average value at 260 K makes a good argument for the actual |
331 |
|
|
density maximum residing at this midpoint value. Figure \ref{dense1} |
332 |
|
|
was constructed using ice $I_h$ crystals for the initial |
333 |
|
|
configuration; and though not pictured, the simulations starting from |
334 |
|
|
ice $I_c$ crystal configurations showed similar results, with a |
335 |
|
|
liquid-phase density maximum in this same region (between 255 and 260 |
336 |
|
|
K). In addition, the $I_c$ crystals are more fragile than the $I_h$ |
337 |
|
|
crystals, leading them to deform into a dense glassy state at lower |
338 |
|
|
temperatures. This resulted in an overall low temperature density |
339 |
|
|
maximum at 200 K, but they still retained a common liquid state |
340 |
|
|
density maximum with the $I_h$ simulations. |
341 |
chrisfen |
743 |
|
342 |
|
|
\begin{figure} |
343 |
|
|
\includegraphics[width=65mm,angle=-90]{dense2.eps} |
344 |
|
|
\caption{Density versus temperature for TIP4P\cite{Jorgensen98b}, |
345 |
chrisfen |
856 |
TIP3P\cite{Jorgensen98b}, SPC/E\cite{Clancy94}, SSD without Reaction |
346 |
|
|
Field, SSD, and Experiment\cite{CRC80}. The arrows indicate the |
347 |
|
|
change in densities observed when turning off the reaction field. The |
348 |
|
|
the lower than expected densities for the SSD model were what |
349 |
|
|
prompted the original reparameterization.\cite{Ichiye03}} |
350 |
chrisfen |
861 |
\label{dense1} |
351 |
chrisfen |
743 |
\end{figure} |
352 |
|
|
|
353 |
|
|
The density maximum for SSD actually compares quite favorably to other |
354 |
chrisfen |
861 |
simple water models. Figure \ref{dense1} also shows calculated |
355 |
|
|
densities of several other models and experiment obtained from other |
356 |
chrisfen |
743 |
sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water |
357 |
|
|
models, SSD has results closest to the experimentally observed water |
358 |
|
|
density maximum. Of the listed water models, TIP4P has a density |
359 |
chrisfen |
861 |
maximum behavior most like that seen in SSD. Though not included in |
360 |
|
|
this particular plot, it is useful to note that TIP5P has a water |
361 |
|
|
density maximum nearly identical to experiment. |
362 |
chrisfen |
743 |
|
363 |
|
|
It has been observed that densities are dependent on the cutoff radius |
364 |
|
|
used for a variety of water models in simulations both with and |
365 |
|
|
without the use of reaction field.\cite{Berendsen98} In order to |
366 |
|
|
address the possible affect of cutoff radius, simulations were |
367 |
|
|
performed with a dipolar cutoff radius of 12.0 \AA\ to compliment the |
368 |
|
|
previous SSD simulations, all performed with a cutoff of 9.0 \AA. All |
369 |
|
|
the resulting densities overlapped within error and showed no |
370 |
|
|
significant trend in lower or higher densities as a function of cutoff |
371 |
|
|
radius, both for simulations with and without reaction field. These |
372 |
|
|
results indicate that there is no major benefit in choosing a longer |
373 |
|
|
cutoff radius in simulations using SSD. This is comforting in that the |
374 |
chrisfen |
861 |
use of a longer cutoff radius results in significant increases in the |
375 |
|
|
time required to obtain a single trajectory. |
376 |
chrisfen |
743 |
|
377 |
chrisfen |
861 |
The most important thing to recognize in figure \ref{dense1} is the |
378 |
|
|
density scaling of SSD relative to other common models at any given |
379 |
|
|
temperature. Note that the SSD model assumes a lower density than any |
380 |
|
|
of the other listed models at the same pressure, behavior which is |
381 |
|
|
especially apparent at temperatures greater than 300 K. Lower than |
382 |
|
|
expected densities have been observed for other systems with the use |
383 |
|
|
of a reaction field for long-range electrostatic interactions, so the |
384 |
|
|
most likely reason for these significantly lower densities in these |
385 |
|
|
simulations is the presence of the reaction |
386 |
|
|
field.\cite{Berendsen98,Nezbeda02} In order to test the effect of the |
387 |
|
|
reaction field on the density of the systems, the simulations were |
388 |
|
|
repeated without a reaction field present. The results of these |
389 |
|
|
simulations are also displayed in figure \ref{dense1}. Without |
390 |
|
|
reaction field, these densities increase considerably to more |
391 |
|
|
experimentally reasonable values, especially around the freezing point |
392 |
|
|
of liquid water. The shape of the curve is similar to the curve |
393 |
|
|
produced from SSD simulations using reaction field, specifically the |
394 |
|
|
rapidly decreasing densities at higher temperatures; however, a shift |
395 |
|
|
in the density maximum location, down to 245 K, is observed. This is |
396 |
|
|
probably a more accurate comparison to the other listed water models, |
397 |
|
|
in that no long range corrections were applied in those |
398 |
|
|
simulations.\cite{Clancy94,Jorgensen98b} However, even without a |
399 |
|
|
reaction field, the density around 300 K is still significantly lower |
400 |
|
|
than experiment and comparable water models. This anomalous behavior |
401 |
|
|
was what lead Ichiye \emph{et al.} to recently reparameterize SSD and |
402 |
|
|
make SSD1.\cite{Ichiye03} In discussing potential adjustments later in |
403 |
|
|
this paper, all comparisons were performed with this new model. |
404 |
|
|
|
405 |
chrisfen |
743 |
\subsection{Transport Behavior} |
406 |
|
|
Of importance in these types of studies are the transport properties |
407 |
|
|
of the particles and how they change when altering the environmental |
408 |
|
|
conditions. In order to probe transport, constant energy simulations |
409 |
|
|
were performed about the average density uncovered by the constant |
410 |
|
|
pressure simulations. Simulations started with randomized velocities |
411 |
|
|
and underwent 50 ps of temperature scaling and 50 ps of constant |
412 |
|
|
energy equilibration before obtaining a 200 ps trajectory. Diffusion |
413 |
|
|
constants were calculated via root-mean square deviation analysis. The |
414 |
|
|
averaged results from 5 sets of these NVE simulations is displayed in |
415 |
|
|
figure \ref{diffuse}, alongside experimental, SPC/E, and TIP5P |
416 |
|
|
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} |
417 |
|
|
|
418 |
|
|
\begin{figure} |
419 |
|
|
\includegraphics[width=65mm, angle=-90]{betterDiffuse.epsi} |
420 |
|
|
\caption{Average diffusion coefficient over increasing temperature for |
421 |
|
|
SSD, SPC/E\cite{Clancy94}, TIP5P\cite{Jorgensen01}, and Experimental |
422 |
|
|
data from Gillen \emph{et al.}\cite{Gillen72}, and from |
423 |
|
|
Mills\cite{Mills73}.} |
424 |
|
|
\label{diffuse} |
425 |
|
|
\end{figure} |
426 |
|
|
|
427 |
|
|
The observed values for the diffusion constant point out one of the |
428 |
|
|
strengths of the SSD model. Of the three experimental models shown, |
429 |
|
|
the SSD model has the most accurate depiction of the diffusion trend |
430 |
|
|
seen in experiment in both the supercooled and normal regimes. SPC/E |
431 |
|
|
does a respectable job by getting similar values as SSD and experiment |
432 |
|
|
around 290 K; however, it deviates at both higher and lower |
433 |
|
|
temperatures, failing to predict the experimental trend. TIP5P and SSD |
434 |
|
|
both start off low at the colder temperatures and tend to diffuse too |
435 |
|
|
rapidly at the higher temperatures. This type of trend at the higher |
436 |
|
|
temperatures is not surprising in that the densities of both TIP5P and |
437 |
|
|
SSD are lower than experimental water at temperatures higher than room |
438 |
|
|
temperature. When calculating the diffusion coefficients for SSD at |
439 |
|
|
experimental densities, the resulting values fall more in line with |
440 |
chrisfen |
861 |
experiment at these temperatures, albeit not at standard pressure. |
441 |
chrisfen |
743 |
|
442 |
|
|
\subsection{Structural Changes and Characterization} |
443 |
|
|
By starting the simulations from the crystalline state, the melting |
444 |
|
|
transition and the ice structure can be studied along with the liquid |
445 |
|
|
phase behavior beyond the melting point. To locate the melting |
446 |
|
|
transition, the constant pressure heat capacity (C$_\text{p}$) was |
447 |
|
|
monitored in each of the simulations. In the melting simulations of |
448 |
|
|
the 1024 particle ice $I_h$ simulations, a large spike in C$_\text{p}$ |
449 |
|
|
occurs at 245 K, indicating a first order phase transition for the |
450 |
|
|
melting of these ice crystals. When the reaction field is turned off, |
451 |
|
|
the melting transition occurs at 235 K. These melting transitions are |
452 |
|
|
considerably lower than the experimental value, but this is not |
453 |
chrisfen |
861 |
surprising when considering the simplicity of the SSD model. |
454 |
chrisfen |
743 |
|
455 |
|
|
\begin{figure} |
456 |
|
|
\includegraphics[width=85mm]{fullContours.eps} |
457 |
|
|
\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at |
458 |
|
|
100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for |
459 |
|
|
clarity: dark areas signify peaks while light areas signify |
460 |
|
|
depressions. White areas have g(\emph{r}) values below 0.5 and black |
461 |
|
|
areas have values above 1.5.} |
462 |
|
|
\label{contour} |
463 |
|
|
\end{figure} |
464 |
|
|
|
465 |
|
|
\begin{figure} |
466 |
|
|
\includegraphics[width=45mm]{corrDiag.eps} |
467 |
|
|
\caption{Two dimensional illustration of the angles involved in the |
468 |
|
|
correlations observed in figure \ref{contour}.} |
469 |
|
|
\label{corrAngle} |
470 |
|
|
\end{figure} |
471 |
|
|
|
472 |
chrisfen |
861 |
Additional analysis of the melting phase-transition process was |
473 |
|
|
performed by using two-dimensional structure and dipole angle |
474 |
|
|
correlations. Expressions for these correlations are as follows: |
475 |
|
|
|
476 |
chrisfen |
743 |
\begin{multline} |
477 |
|
|
g_{\text{AB}}(r,\cos\theta) = \\ |
478 |
|
|
\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\ , |
479 |
|
|
\end{multline} |
480 |
|
|
\begin{multline} |
481 |
|
|
g_{\text{AB}}(r,\cos\omega) = \\ |
482 |
|
|
\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\ , |
483 |
|
|
\end{multline} |
484 |
chrisfen |
861 |
where $\theta$ and $\omega$ refer to the angles shown in figure |
485 |
|
|
\ref{corrAngle}. By binning over both distance and the cosine of the |
486 |
chrisfen |
743 |
desired angle between the two dipoles, the g(\emph{r}) can be |
487 |
|
|
dissected to determine the common dipole arrangements that constitute |
488 |
|
|
the peaks and troughs. Frames A and B of figure \ref{contour} show a |
489 |
|
|
relatively crystalline state of an ice $I_c$ simulation. The first |
490 |
chrisfen |
861 |
peak of the g(\emph{r}) consists primarily of the preferred hydrogen |
491 |
|
|
bonding arrangements as dictated by the tetrahedral sticky potential - |
492 |
chrisfen |
743 |
one peak for the donating and the other for the accepting hydrogen |
493 |
|
|
bonds. Due to the high degree of crystallinity of the sample, the |
494 |
|
|
second and third solvation shells show a repeated peak arrangement |
495 |
|
|
which decays at distances around the fourth solvation shell, near the |
496 |
|
|
imposed cutoff for the Lennard-Jones and dipole-dipole interactions. |
497 |
chrisfen |
861 |
In the higher temperature simulation shown in frames C and D, these |
498 |
|
|
longer-ranged repeated peak features deteriorate rapidly. The first |
499 |
|
|
solvation shell still shows the strong effect of the sticky-potential, |
500 |
|
|
although it covers a larger area, extending to include a fraction of |
501 |
|
|
aligned dipole peaks within the first solvation shell. The latter |
502 |
|
|
peaks lose definition as thermal motion and the competing dipole force |
503 |
|
|
overcomes the sticky potential's tight tetrahedral structuring of the |
504 |
|
|
fluid. |
505 |
chrisfen |
743 |
|
506 |
|
|
This complex interplay between dipole and sticky interactions was |
507 |
|
|
remarked upon as a possible reason for the split second peak in the |
508 |
|
|
oxygen-oxygen g(\emph{r}).\cite{Ichiye96} At low temperatures, the |
509 |
|
|
second solvation shell peak appears to have two distinct parts that |
510 |
|
|
blend together to form one observable peak. At higher temperatures, |
511 |
|
|
this split character alters to show the leading 4 \AA\ peak dominated |
512 |
|
|
by equatorial anti-parallel dipole orientations, and there is tightly |
513 |
|
|
bunched group of axially arranged dipoles that most likely consist of |
514 |
|
|
the smaller fraction aligned dipole pairs. The trailing part of the |
515 |
|
|
split peak at 5 \AA\ is dominated by aligned dipoles that range |
516 |
|
|
primarily within the axial to the chief hydrogen bond arrangements |
517 |
|
|
similar to those seen in the first solvation shell. This evidence |
518 |
|
|
indicates that the dipole pair interaction begins to dominate outside |
519 |
|
|
of the range of the dipolar repulsion term, with the primary |
520 |
|
|
energetically favorable dipole arrangements populating the region |
521 |
chrisfen |
861 |
immediately outside this repulsion region (around 4 \AA), and |
522 |
|
|
arrangements that seek to ideally satisfy both the sticky and dipole |
523 |
|
|
forces locate themselves just beyond this initial buildup (around 5 |
524 |
|
|
\AA). |
525 |
chrisfen |
743 |
|
526 |
|
|
From these findings, the split second peak is primarily the product of |
527 |
chrisfen |
861 |
the dipolar repulsion term of the sticky potential. In fact, the inner |
528 |
|
|
peak can be pushed out and merged with the outer split peak just by |
529 |
|
|
extending the switching function cutoff ($s^\prime(r_{ij})$) from its |
530 |
|
|
normal 4.0 \AA\ to values of 4.5 or even 5 \AA. This type of |
531 |
|
|
correction is not recommended for improving the liquid structure, |
532 |
|
|
because the second solvation shell will still be shifted too far |
533 |
|
|
out. In addition, this would have an even more detrimental effect on |
534 |
|
|
the system densities, leading to a liquid with a more open structure |
535 |
|
|
and a density considerably lower than the normal SSD behavior shown |
536 |
|
|
previously. A better correction would be to include the |
537 |
|
|
quadrupole-quadrupole interactions for the water particles outside of |
538 |
|
|
the first solvation shell, but this reduces the simplicity and speed |
539 |
|
|
advantage of SSD. |
540 |
chrisfen |
743 |
|
541 |
chrisfen |
861 |
\subsection{Adjusted Potentials: SSD/RF and SSD/E} |
542 |
chrisfen |
743 |
The propensity of SSD to adopt lower than expected densities under |
543 |
|
|
varying conditions is troubling, especially at higher temperatures. In |
544 |
chrisfen |
861 |
order to correct this model for use with a reaction field, it is |
545 |
|
|
necessary to adjust the force field parameters for the primary |
546 |
|
|
intermolecular interactions. In undergoing a reparameterization, it is |
547 |
|
|
important not to focus on just one property and neglect the other |
548 |
|
|
important properties. In this case, it would be ideal to correct the |
549 |
|
|
densities while maintaining the accurate transport properties. |
550 |
chrisfen |
743 |
|
551 |
|
|
The possible parameters for tuning include the $\sigma$ and $\epsilon$ |
552 |
|
|
Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky |
553 |
|
|
attractive and dipole repulsive terms with their respective |
554 |
|
|
cutoffs. To alter the attractive and repulsive terms of the sticky |
555 |
|
|
potential independently, it is necessary to separate the terms as |
556 |
|
|
follows: |
557 |
|
|
\begin{equation} |
558 |
|
|
\begin{split} |
559 |
|
|
u_{ij}^{sp} |
560 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) &= |
561 |
|
|
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\\ |
562 |
|
|
& \quad \ + \frac{\nu_0^\prime}{2} |
563 |
|
|
[s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)], |
564 |
|
|
\end{split} |
565 |
|
|
\end{equation} |
566 |
|
|
|
567 |
|
|
where $\nu_0$ scales the strength of the tetrahedral attraction and |
568 |
|
|
$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion |
569 |
|
|
term. For purposes of the reparameterization, the separation was |
570 |
|
|
performed, but the final parameters were adjusted so that it is |
571 |
|
|
unnecessary to separate the terms when implementing the adjusted water |
572 |
|
|
potentials. The results of the reparameterizations are shown in table |
573 |
|
|
\ref{params}. Note that both the tetrahedral attractive and dipolar |
574 |
|
|
repulsive don't share the same lower cutoff ($r_l$) in the newly |
575 |
chrisfen |
861 |
parameterized potentials - soft sticky dipole reaction field (SSD/RF - |
576 |
|
|
for use with a reaction field) and soft sticky dipole enhanced (SSD/E |
577 |
|
|
- an attempt to improve the liquid structure in simulations without a |
578 |
|
|
long-range correction). |
579 |
chrisfen |
743 |
|
580 |
|
|
\begin{table} |
581 |
|
|
\caption{Parameters for the original and adjusted models} |
582 |
chrisfen |
856 |
\begin{tabular}{ l c c c c } |
583 |
chrisfen |
743 |
\hline \\[-3mm] |
584 |
chrisfen |
856 |
\ \ \ Parameters\ \ \ & \ \ \ SSD$^\dagger$ \ \ \ & \ SSD1$^\ddagger$\ \ & \ SSD/E\ \ & \ SSD/RF \\ |
585 |
chrisfen |
743 |
\hline \\[-3mm] |
586 |
chrisfen |
856 |
\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
587 |
|
|
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
588 |
|
|
\ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\ |
589 |
|
|
\ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
590 |
|
|
\ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\ |
591 |
|
|
\ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\ |
592 |
|
|
\ \ \ $\nu_0^\prime$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
593 |
|
|
\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\ |
594 |
|
|
\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\ |
595 |
chrisfen |
743 |
\\[-2mm]$^\dagger$ ref. \onlinecite{Ichiye96} |
596 |
chrisfen |
856 |
\\$^\ddagger$ ref. \onlinecite{Ichiye03} |
597 |
chrisfen |
743 |
\end{tabular} |
598 |
|
|
\label{params} |
599 |
|
|
\end{table} |
600 |
|
|
|
601 |
|
|
\begin{figure} |
602 |
chrisfen |
856 |
\includegraphics[width=85mm]{GofRCompare.epsi} |
603 |
chrisfen |
743 |
\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E |
604 |
chrisfen |
856 |
and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with |
605 |
chrisfen |
743 |
reaction field turned on (bottom). The insets show the respective |
606 |
|
|
first peaks in detail. Solid Line - experiment, dashed line - SSD/E |
607 |
chrisfen |
856 |
and SSD/RF, and dotted line - SSD1 (with and without reaction field).} |
608 |
chrisfen |
743 |
\label{grcompare} |
609 |
|
|
\end{figure} |
610 |
|
|
|
611 |
|
|
\begin{figure} |
612 |
|
|
\includegraphics[width=85mm]{dualsticky.ps} |
613 |
chrisfen |
856 |
\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \& |
614 |
chrisfen |
743 |
SSD/RF (right). Light areas correspond to the tetrahedral attractive |
615 |
|
|
part, and the darker areas correspond to the dipolar repulsive part.} |
616 |
|
|
\label{isosurface} |
617 |
|
|
\end{figure} |
618 |
|
|
|
619 |
|
|
In the paper detailing the development of SSD, Liu and Ichiye placed |
620 |
|
|
particular emphasis on an accurate description of the first solvation |
621 |
|
|
shell. This resulted in a somewhat tall and sharp first peak that |
622 |
|
|
integrated to give similar coordination numbers to the experimental |
623 |
|
|
data obtained by Soper and Phillips.\cite{Ichiye96,Soper86} New |
624 |
|
|
experimental x-ray scattering data from the Head-Gordon lab indicates |
625 |
|
|
a slightly lower and shifted first peak in the g$_\mathrm{OO}(r)$, so |
626 |
|
|
adjustments to SSD were made while taking into consideration the new |
627 |
|
|
experimental findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} |
628 |
|
|
shows the relocation of the first peak of the oxygen-oxygen |
629 |
chrisfen |
861 |
g(\emph{r}) by comparing the revised SSD model (SSD1), SSD-E, and |
630 |
|
|
SSD-RF to the new experimental results. Both the modified water models |
631 |
|
|
have shorter peaks that are brought in more closely to the |
632 |
|
|
experimental peak (as seen in the insets of figure \ref{grcompare}). |
633 |
|
|
This structural alteration was accomplished by the combined reduction |
634 |
|
|
in the Lennard-Jones $\sigma$ variable and adjustment of the sticky |
635 |
|
|
potential strength and cutoffs. As can be seen in table \ref{params}, |
636 |
|
|
the cutoffs for the tetrahedral attractive and dipolar repulsive terms |
637 |
|
|
were nearly swapped with each other. Isosurfaces of the original and |
638 |
|
|
modified sticky potentials are shown in figure \cite{isosurface}. In |
639 |
|
|
these isosurfaces, it is easy to see how altering the cutoffs changes |
640 |
|
|
the repulsive and attractive character of the particles. With a |
641 |
|
|
reduced repulsive surface (the darker region), the particles can move |
642 |
|
|
closer to one another, increasing the density for the overall |
643 |
|
|
system. This change in interaction cutoff also results in a more |
644 |
|
|
gradual orientational motion by allowing the particles to maintain |
645 |
|
|
preferred dipolar arrangements before they begin to feel the pull of |
646 |
|
|
the tetrahedral restructuring. Upon moving closer together, the |
647 |
|
|
dipolar repulsion term becomes active and excludes unphysical |
648 |
|
|
nearest-neighbor arrangements. This compares with how SSD and SSD1 |
649 |
|
|
exclude preferred dipole alignments before the particles feel the pull |
650 |
|
|
of the ``hydrogen bonds''. Aside from improving the shape of the first |
651 |
|
|
peak in the g(\emph{r}), this improves the densities considerably by |
652 |
|
|
allowing the persistence of full dipolar character below the previous |
653 |
|
|
4.0 \AA\ cutoff. |
654 |
chrisfen |
743 |
|
655 |
|
|
While adjusting the location and shape of the first peak of |
656 |
chrisfen |
861 |
g(\emph{r}) improves the densities, these changes alone are |
657 |
|
|
insufficient to bring the system densities up to the values observed |
658 |
|
|
experimentally. To finish bringing up the densities, the dipole |
659 |
|
|
moments were increased in both the adjusted models. Being a dipole |
660 |
|
|
based model, the structure and transport are very sensitive to changes |
661 |
|
|
in the dipole moment. The original SSD simply used the dipole moment |
662 |
|
|
calculated from the TIP3P water model, which at 2.35 D is |
663 |
chrisfen |
743 |
significantly greater than the experimental gas phase value of 1.84 |
664 |
chrisfen |
861 |
D. The larger dipole moment is a more realistic value and improves the |
665 |
chrisfen |
743 |
dielectric properties of the fluid. Both theoretical and experimental |
666 |
|
|
measurements indicate a liquid phase dipole moment ranging from 2.4 D |
667 |
chrisfen |
861 |
to values as high as 3.11 D, so there is quite a range of available |
668 |
|
|
values for a reasonable dipole |
669 |
chrisfen |
743 |
moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} The increasing of |
670 |
chrisfen |
861 |
the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF |
671 |
|
|
respectively is moderate in this range; however, it leads to |
672 |
|
|
significant changes in the density and transport of the water models. |
673 |
chrisfen |
743 |
|
674 |
chrisfen |
861 |
In order to demonstrate the benefits of these reparameterizations, a |
675 |
chrisfen |
743 |
series of NPT and NVE simulations were performed to probe the density |
676 |
|
|
and transport properties of the adapted models and compare the results |
677 |
|
|
to the original SSD model. This comparison involved full NPT melting |
678 |
|
|
sequences for both SSD/E and SSD/RF, as well as NVE transport |
679 |
chrisfen |
861 |
calculations at the calculated self-consistent densities. Again, the |
680 |
|
|
results come from five separate simulations of 1024 particle systems, |
681 |
|
|
and the melting sequences were started from different ice $I_h$ |
682 |
|
|
crystals constructed as stated earlier. Like before, each NPT |
683 |
|
|
simulation was equilibrated for 100 ps before a 200 ps data collection |
684 |
|
|
run at each temperature step, and they used the final configuration |
685 |
|
|
from the previous temperature simulation as a starting point. All of |
686 |
|
|
the NVE simulations had the same thermalization, equilibration, and |
687 |
|
|
data collection times stated earlier in this paper. |
688 |
chrisfen |
743 |
|
689 |
|
|
\begin{figure} |
690 |
chrisfen |
856 |
\includegraphics[width=62mm, angle=-90]{ssdeDense.epsi} |
691 |
chrisfen |
861 |
\caption{Comparison of densities calculated with SSD/E to SSD1 without a |
692 |
chrisfen |
856 |
reaction field, TIP3P\cite{Jorgensen98b}, TIP5P\cite{Jorgensen00}, |
693 |
|
|
SPC/E\cite{Clancy94}, and Experiment\cite{CRC80}. The window shows a |
694 |
|
|
expansion around 300 K with error bars included to clarify this region |
695 |
|
|
of interest. Note that both SSD1 and SSD/E show good agreement with |
696 |
|
|
experiment when the long-range correction is neglected.} |
697 |
chrisfen |
743 |
\label{ssdedense} |
698 |
|
|
\end{figure} |
699 |
|
|
|
700 |
chrisfen |
861 |
Figure \ref{ssdedense} shows the density profile for the SSD/E model |
701 |
|
|
in comparison to SSD1 without a reaction field, experiment, and other |
702 |
|
|
common water models. The calculated densities for both SSD/E and SSD1 |
703 |
|
|
have increased significantly over the original SSD model (see figure |
704 |
|
|
\ref{dense1} and are in significantly better agreement with the |
705 |
|
|
experimental values. At 298 K, the density of SSD/E and SSD1 without a |
706 |
|
|
long-range correction are 0.996$\pm$0.001 g/cm$^3$ and 0.999$\pm$0.001 |
707 |
|
|
g/cm$^3$ respectively. These both compare well with the experimental |
708 |
|
|
value of 0.997 g/cm$^3$, and they are considerably better than the SSD |
709 |
|
|
value of 0.967$\pm$0.003 g/cm$^3$. The changes to the dipole moment |
710 |
|
|
and sticky switching functions have improved the structuring of the |
711 |
|
|
liquid (as seen in figure \ref{grcompare}, but they have shifted the |
712 |
|
|
density maximum to much lower temperatures. This comes about via an |
713 |
|
|
increase of the liquid disorder through the weakening of the sticky |
714 |
|
|
potential and strengthening of the dipolar character. However, this |
715 |
|
|
increasing disorder in the SSD/E model has little affect on the |
716 |
|
|
melting transition. By monitoring C$\text{p}$ throughout these |
717 |
|
|
simulations, the melting transition for SSD/E occurred at 235 K, the |
718 |
|
|
same transition temperature observed with SSD and SSD1. |
719 |
chrisfen |
743 |
|
720 |
|
|
\begin{figure} |
721 |
chrisfen |
856 |
\includegraphics[width=62mm, angle=-90]{ssdrfDense.epsi} |
722 |
chrisfen |
861 |
\caption{Comparison of densities calculated with SSD/RF to SSD1 with a |
723 |
chrisfen |
856 |
reaction field, TIP3P\cite{Jorgensen98b}, TIP5P\cite{Jorgensen00}, |
724 |
|
|
SPC/E\cite{Clancy94}, and Experiment\cite{CRC80}. The inset shows the |
725 |
|
|
necessity of reparameterization when utilizing a reaction field |
726 |
|
|
long-ranged correction - SSD/RF provides significantly more accurate |
727 |
|
|
densities than SSD1 when performing room temperature simulations.} |
728 |
chrisfen |
743 |
\label{ssdrfdense} |
729 |
|
|
\end{figure} |
730 |
|
|
|
731 |
chrisfen |
861 |
Including the reaction field long-range correction results in a more |
732 |
|
|
interesting comparison. A density profile including SSD/RF and SSD1 |
733 |
|
|
with an active reaction field is shown in figure \ref{ssdrfdense}. As |
734 |
|
|
observed in the simulations without a reaction field, the densities of |
735 |
|
|
SSD/RF and SSD1 show a dramatic increase over normal SSD (see figure |
736 |
|
|
\ref{dense1}). At 298 K, SSD/RF has a density of 0.997$\pm$0.001 |
737 |
|
|
g/cm$^3$, right in line with experiment and considerably better than |
738 |
|
|
the SSD value of 0.941$\pm$0.001 g/cm$^3$ and the SSD1 value of |
739 |
|
|
0.972$\pm$0.002 g/cm$^3$. These results further emphasize the |
740 |
|
|
importance of reparameterization in order to model the density |
741 |
|
|
properly under different simulation conditions. Again, these changes |
742 |
|
|
don't have that profound an effect on the melting point which is |
743 |
|
|
observed at 245 K for SSD/RF, identical to SSD and only 5 K lower than |
744 |
|
|
SSD1 with a reaction field. However, the difference in density maxima |
745 |
|
|
is not quite as extreme with SSD/RF showing a density maximum at 255 |
746 |
|
|
K, fairly close to 260 and 265 K, the density maxima for SSD and SSD1 |
747 |
|
|
respectively. |
748 |
chrisfen |
743 |
|
749 |
chrisfen |
861 |
\begin{figure} |
750 |
|
|
\includegraphics[width=65mm, angle=-90]{ssdeDiffuse.epsi} |
751 |
|
|
\caption{Plots of the diffusion constants calculated from SSD/E and SSD1, |
752 |
|
|
both without a reaction field, along with experimental results are |
753 |
|
|
from Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The |
754 |
|
|
NVE calculations were performed at the average densities observed in |
755 |
|
|
the 1 atm NPT simulations for the respective models. SSD/E is |
756 |
|
|
slightly more fluid than experiment at all of the temperatures, but |
757 |
|
|
it is closer than SSD1 without a long-range correction.} |
758 |
|
|
\label{ssdediffuse} |
759 |
|
|
\end{figure} |
760 |
|
|
|
761 |
chrisfen |
743 |
The reparameterization of the SSD water model, both for use with and |
762 |
|
|
without an applied long-range correction, brought the densities up to |
763 |
|
|
what is expected for simulating liquid water. In addition to improving |
764 |
|
|
the densities, it is important that particle transport be maintained |
765 |
|
|
or improved. Figure \ref{ssdediffuse} compares the temperature |
766 |
chrisfen |
861 |
dependence of the diffusion constant of SSD/E to SSD1 without an |
767 |
|
|
active reaction field, both at the densities calculated at 1 atm and |
768 |
|
|
at the experimentally calculated densities for super-cooled and liquid |
769 |
chrisfen |
743 |
water. In the upper plot, the diffusion constant for SSD/E is |
770 |
chrisfen |
861 |
consistently a little faster than experiment, while SSD1 remains |
771 |
|
|
slower than experiment until relatively high temperatures (greater |
772 |
|
|
than 330 K). Both models follow the shape of the experimental trend |
773 |
|
|
well below 300 K, but the trend leans toward diffusing too rapidly at |
774 |
|
|
higher temperatures, something that is especially apparent with |
775 |
|
|
SSD1. This accelerated increasing of diffusion is caused by the |
776 |
|
|
rapidly decreasing system density with increasing temperature. Though |
777 |
|
|
it is difficult to see in figure \ref{ssdedense}, the densities of SSD1 |
778 |
|
|
decay more rapidly with temperature than do those of SSD/E, leading to |
779 |
|
|
more visible deviation from the experimental diffusion trend. Thus, |
780 |
|
|
the changes made to improve the liquid structure may have had an |
781 |
|
|
adverse affect on the density maximum, but they improve the transport |
782 |
|
|
behavior of the water model. |
783 |
chrisfen |
743 |
|
784 |
|
|
\begin{figure} |
785 |
chrisfen |
856 |
\includegraphics[width=65mm, angle=-90]{ssdrfDiffuse.epsi} |
786 |
|
|
\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1, |
787 |
|
|
both with an active reaction field, along with experimental results |
788 |
|
|
from Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The |
789 |
|
|
NVE calculations were performed at the average densities observed in |
790 |
|
|
the 1 atm NPT simulations for both of the models. Note how accurately |
791 |
|
|
SSD/RF simulates the diffusion of water throughout this temperature |
792 |
|
|
range. The more rapidly increasing diffusion constants at high |
793 |
|
|
temperatures for both models is attributed to the significantly lower |
794 |
|
|
densities than observed in experiment.} |
795 |
|
|
\label{ssdrfdiffuse} |
796 |
chrisfen |
743 |
\end{figure} |
797 |
|
|
|
798 |
|
|
In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are |
799 |
chrisfen |
861 |
compared with SSD1 with an active reaction field. Note that SSD/RF |
800 |
|
|
tracks the experimental results incredibly well, identical within |
801 |
|
|
error throughout the temperature range shown and only showing a slight |
802 |
|
|
increasing trend at higher temperatures. SSD1 tends to diffuse more |
803 |
|
|
slowly at low temperatures and deviates to diffuse too rapidly at |
804 |
|
|
temperatures greater than 330 K. As was stated in the SSD/E |
805 |
|
|
comparisons, this deviation away from the ideal trend is due to a |
806 |
|
|
rapid decrease in density at higher temperatures. SSD/RF doesn't |
807 |
|
|
suffer from this problem as much as SSD1, because the calculated |
808 |
|
|
densities are more true to experiment. These results again emphasize |
809 |
|
|
the importance of careful reparameterization when using an altered |
810 |
|
|
long-range correction. |
811 |
chrisfen |
743 |
|
812 |
|
|
\subsection{Additional Observations} |
813 |
|
|
|
814 |
|
|
\begin{figure} |
815 |
|
|
\includegraphics[width=85mm]{povIce.ps} |
816 |
chrisfen |
861 |
\caption{A water lattice built from the crystal structure that SSD/E |
817 |
|
|
assumed when undergoing an extremely restricted temperature NPT |
818 |
|
|
simulation. This form of ice is referred to as ice 0 to emphasize its |
819 |
|
|
simulation origins. This image was taken of the (001) face of the |
820 |
|
|
crystal.} |
821 |
chrisfen |
743 |
\label{weirdice} |
822 |
|
|
\end{figure} |
823 |
|
|
|
824 |
chrisfen |
861 |
While performing restricted temperature melting sequences of SSD/E not |
825 |
|
|
discussed earlier in this paper, some interesting observations were |
826 |
|
|
made. After melting at 235 K, two of five systems underwent |
827 |
|
|
crystallization events near 245 K. As the heating process continued, |
828 |
|
|
the two systems remained crystalline until finally melting between 320 |
829 |
|
|
and 330 K. The final configurations of these two melting sequences |
830 |
|
|
show an expanded zeolite-like crystal structure that does not |
831 |
|
|
correspond to any known form of ice. For convenience and to help |
832 |
|
|
distinguish it from the experimentally observed forms of ice, this |
833 |
|
|
crystal structure will henceforth be referred to as ice-zero (ice |
834 |
|
|
0). The crystallinity was extensive enough that a near ideal crystal |
835 |
|
|
structure could be obtained. Figure \ref{weirdice} shows the repeating |
836 |
|
|
crystal structure of a typical crystal at 5 K. Each water molecule is |
837 |
|
|
hydrogen bonded to four others; however, the hydrogen bonds are flexed |
838 |
|
|
rather than perfectly straight. This results in a skewed tetrahedral |
839 |
|
|
geometry about the central molecule. Looking back at figure |
840 |
|
|
\ref{isosurface}, it is easy to see how these flexed hydrogen bonds |
841 |
|
|
are allowed in that the attractive regions are conical in shape, with |
842 |
|
|
the greatest attraction in the central region. Though not ideal, these |
843 |
|
|
flexed hydrogen bonds are favorable enough to stabilize an entire |
844 |
|
|
crystal generated around them. In fact, the imperfect ice 0 crystals |
845 |
|
|
were so stable that they melted at temperatures nearly 100 K greater |
846 |
|
|
than both ice I$_c$ and I$_h$. |
847 |
chrisfen |
743 |
|
848 |
chrisfen |
861 |
These initial simulations indicated that ice 0 is the preferred ice |
849 |
chrisfen |
743 |
structure for at least SSD/E. To verify this, a comparison was made |
850 |
|
|
between near ideal crystals of ice $I_h$, ice $I_c$, and ice 0 at |
851 |
chrisfen |
861 |
constant pressure with SSD/E, SSD/RF, and SSD1. Near ideal versions of |
852 |
|
|
the three types of crystals were cooled to 1 K, and the potential |
853 |
chrisfen |
743 |
energies of each were compared using all three water models. With |
854 |
|
|
every water model, ice 0 turned out to have the lowest potential |
855 |
chrisfen |
861 |
energy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with SSD/E, and |
856 |
|
|
7.5\% lower with SSD/RF. |
857 |
chrisfen |
743 |
|
858 |
chrisfen |
861 |
In addition to these low temperature comparisons, melting sequences |
859 |
|
|
were performed with ice 0 as the initial configuration using SSD/E, |
860 |
|
|
SSD/RF, and SSD1 both with and without a reaction field. The melting |
861 |
|
|
transitions for both SSD/E and SSD1 without a reaction field occurred |
862 |
|
|
at temperature in excess of 375 K. SSD/RF and SSD1 with a reaction |
863 |
chrisfen |
743 |
field had more reasonable melting transitions, down near 325 K. These |
864 |
|
|
melting point observations emphasize how preferred this crystal |
865 |
|
|
structure is over the most common types of ice when using these single |
866 |
|
|
point water models. |
867 |
|
|
|
868 |
|
|
Recognizing that the above tests show ice 0 to be both the most stable |
869 |
|
|
and lowest density crystal structure for these single point water |
870 |
chrisfen |
861 |
models, it is interesting to speculate on the relative stability of |
871 |
|
|
this crystal structure with charge based water models. As a quick |
872 |
chrisfen |
743 |
test, these 3 crystal types were converted from SSD type particles to |
873 |
|
|
TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy |
874 |
|
|
minimizations were performed on all of these crystals to compare the |
875 |
|
|
system energies. Again, ice 0 was observed to have the lowest total |
876 |
|
|
system energy. The total energy of ice 0 was ~2\% lower than ice |
877 |
|
|
$I_h$, which was in turn ~3\% lower than ice $I_c$. From these initial |
878 |
|
|
results, we would not be surprised if results from the other common |
879 |
|
|
water models show ice 0 to be the lowest energy crystal structure. A |
880 |
chrisfen |
861 |
continuation on work studying ice 0 with multi-point water models will |
881 |
chrisfen |
743 |
be published in a coming article. |
882 |
|
|
|
883 |
|
|
\section{Conclusions} |
884 |
|
|
The density maximum and temperature dependent transport for the SSD |
885 |
|
|
water model, both with and without the use of reaction field, were |
886 |
|
|
studied via a series of NPT and NVE simulations. The constant pressure |
887 |
|
|
simulations of the melting of both $I_h$ and $I_c$ ice showed a |
888 |
|
|
density maximum near 260 K. In most cases, the calculated densities |
889 |
|
|
were significantly lower than the densities calculated in simulations |
890 |
|
|
of other water models. Analysis of particle diffusion showed SSD to |
891 |
|
|
capture the transport properties of experimental very well in both the |
892 |
|
|
normal and super-cooled liquid regimes. In order to correct the |
893 |
chrisfen |
861 |
density behavior, the original SSD model was reparameterized for use |
894 |
|
|
both with and without a reaction field (SSD/RF and SSD/E), and |
895 |
|
|
comparison simulations were performed with SSD1, the density corrected |
896 |
|
|
version of SSD. Both models improve the liquid structure, density |
897 |
|
|
values, and diffusive properties under their respective conditions, |
898 |
|
|
indicating the necessity of reparameterization when altering the |
899 |
|
|
long-range correction specifics. When taking the appropriate |
900 |
|
|
considerations, these simple water models are excellent choices for |
901 |
|
|
representing explicit water in large scale simulations of biochemical |
902 |
|
|
systems. |
903 |
chrisfen |
743 |
|
904 |
|
|
\section{Acknowledgments} |
905 |
chrisfen |
777 |
Support for this project was provided by the National Science |
906 |
|
|
Foundation under grant CHE-0134881. Computation time was provided by |
907 |
|
|
the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
908 |
|
|
DMR 00 79647. |
909 |
chrisfen |
743 |
|
910 |
|
|
\bibliographystyle{jcp} |
911 |
|
|
|
912 |
|
|
\bibliography{nptSSD} |
913 |
|
|
|
914 |
|
|
%\pagebreak |
915 |
|
|
|
916 |
|
|
\end{document} |