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\begin{document} |
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\title{On the temperature dependent structural and transport 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 the SSD water model, both with and without the use of reaction |
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field. The constant pressure simulations of the melting of both $I_h$ |
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and $I_c$ ice showed a density maximum near 260 K. In most cases, the |
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calculated densities were significantly lower than the densities |
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calculated in simulations of other water models. Analysis of particle |
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diffusion showed SSD to capture the transport properties of |
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experimental very well in both the normal and super-cooled liquid |
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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. In addition to correcting |
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the abnormally low densities, these new versions were show to maintain |
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or improve upon the transport and structural features of the original |
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water model. |
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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 of |
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SSD, it was shown that it actually improves upon the prediction of |
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water's dynamical properties 3 and 4-point models.\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, a specific |
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interest within our group. |
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Up to this point, a detailed look at the model's structure and ion |
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solvation abilities has been performed.\cite{Ichiye96} In addition, a |
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thorough investigation of the dynamic properties of SSD was performed |
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by Chandra and Ichiye focusing on translational and orientational |
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properties at 298 K.\cite{Ichiye99} This study focuses on determining |
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the density maximum for SSD utilizing both microcanonical and |
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isobaric-isothermal ensemble molecular dynamics, while using the |
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reaction field method for handling long-ranged dipolar interactions. A |
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reaction field method has been previously implemented in Monte Carlo |
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simulations by Liu and Ichiye in order to study the static dielectric |
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constant for the model.\cite{Ichiye96b} This paper will expand the |
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scope of these original simulations to look on how the reaction field |
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affects the physical and dynamic properties of SSD systems. |
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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 compatible with 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} For the constant pressure |
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simulations, the \emph{Q} parameter for the was set to 5.0 amu |
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\(\cdot\)\AA\(^{2}\), and the relaxation time (\(\tau\))\ was set at |
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100 ps. |
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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 actually quite sensitive to errors in the |
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equations of motion. The original implementation of this code utilized |
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quaternions for rotational motion propagation; however, a detailed |
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investigation showed that they resulted in a steady drift in the total |
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energy, 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 an additional 5 or |
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6 additional elements for describing the orientation of each particle, |
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which will increase dump files substantially. Simply translating the |
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rotation matrix into its component Euler angles or quaternions for |
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storage 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. On average, a 1000 SSD |
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particle simulation shows a 7\% increase in simulation time using the |
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symplectic step method in place of quaternions. This cost is more than |
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justified when comparing the energy conservation of the two methods as |
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illustrated in figure \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. |
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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. This involved an initial |
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randomization of velocities about the starting temperature of 25 K for |
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varying amounts of time. The systems were all equilibrated for 100 ps |
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prior to a 200 ps data collection run at each temperature setting, |
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ranging from 25 to 400 K, with a maximum degree increment of 25 K. For |
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regions of interest along this stepwise progression, the temperature |
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increment was decreased from 25 K to 10 and then 5 K. The above |
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equilibration and production times were sufficient in that the system |
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volume fluctuations dampened out in all but the very cold simulations |
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(below 225 K). In order to further improve statistics, five separate |
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simulation progressions were performed, and the averaged results from |
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the $I_h$ melting simulations are shown in figure \ref{dense1}. |
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\begin{figure} |
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\includegraphics[width=65mm, angle=-90]{1hdense.epsi} |
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\caption{Average density of SSD water at increasing temperatures |
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starting from ice $I_h$ lattice.} |
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\label{dense1} |
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\end{figure} |
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\subsection{Density Behavior} |
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In the initial average density versus temperature plot, the density |
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maximum clearly appears between 255 and 265 K. The calculated |
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densities within this range were nearly indistinguishable, as can be |
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seen in the zoom of this region of interest, shown in figure |
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\ref{dense1}. The greater certainty of the average value at 260 K makes |
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a good argument for the actual density maximum residing at this |
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midpoint value. Figure \ref{dense1} was constructed using ice $I_h$ |
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|
crystals for the initial configuration; and though not pictured, the |
329 |
|
|
simulations starting from ice $I_c$ crystal configurations showed |
330 |
|
|
similar results, with a liquid-phase density maximum in this same |
331 |
|
|
region (between 255 and 260 K). In addition, the $I_c$ crystals are |
332 |
|
|
more fragile than the $I_h$ crystals, leading them to deform into a |
333 |
|
|
dense glassy state at lower temperatures. This resulted in an overall |
334 |
|
|
low temperature density maximum at 200 K, but they still retained a |
335 |
|
|
common liquid state density maximum with the $I_h$ simulations. |
336 |
|
|
|
337 |
|
|
\begin{figure} |
338 |
|
|
\includegraphics[width=65mm,angle=-90]{dense2.eps} |
339 |
|
|
\caption{Density versus temperature for TIP4P\cite{Jorgensen98b}, |
340 |
|
|
TIP3P\cite{Jorgensen98b}, SPC/E\cite{Clancy94}, SSD without Reaction |
341 |
|
|
Field, SSD, and Experiment\cite{CRC80}. } |
342 |
|
|
\label{dense2} |
343 |
|
|
\end{figure} |
344 |
|
|
|
345 |
|
|
The density maximum for SSD actually compares quite favorably to other |
346 |
|
|
simple water models. Figure \ref{dense2} shows a plot of these |
347 |
|
|
findings with the density progression of several other models and |
348 |
|
|
experiment obtained from other |
349 |
|
|
sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water |
350 |
|
|
models, SSD has results closest to the experimentally observed water |
351 |
|
|
density maximum. Of the listed water models, TIP4P has a density |
352 |
|
|
maximum behavior most like that seen in SSD. Though not shown, it is |
353 |
|
|
useful to note that TIP5P has a water density maximum nearly identical |
354 |
|
|
to experiment. |
355 |
|
|
|
356 |
|
|
Possibly of more importance is the density scaling of SSD relative to |
357 |
|
|
other common models at any given temperature (Fig. \ref{dense2}). Note |
358 |
|
|
that the SSD model assumes a lower density than any of the other |
359 |
|
|
listed models at the same pressure, behavior which is especially |
360 |
|
|
apparent at temperatures greater than 300 K. Lower than expected |
361 |
|
|
densities have been observed for other systems with the use of a |
362 |
|
|
reaction field for long-range electrostatic interactions, so the most |
363 |
|
|
likely reason for these significantly lower densities in these |
364 |
|
|
simulations is the presence of the reaction field.\cite{Berendsen98} |
365 |
|
|
In order to test the effect of the reaction field on the density of |
366 |
|
|
the systems, the simulations were repeated for the temperature region |
367 |
|
|
of interest without a reaction field present. The results of these |
368 |
|
|
simulations are also displayed in figure \ref{dense2}. Without |
369 |
|
|
reaction field, these densities increase considerably to more |
370 |
|
|
experimentally reasonable values, especially around the freezing point |
371 |
|
|
of liquid water. The shape of the curve is similar to the curve |
372 |
|
|
produced from SSD simulations using reaction field, specifically the |
373 |
|
|
rapidly decreasing densities at higher temperatures; however, a slight |
374 |
|
|
shift in the density maximum location, down to 245 K, is |
375 |
|
|
observed. This is probably a more accurate comparison to the other |
376 |
|
|
listed water models in that no long range corrections were applied in |
377 |
|
|
those simulations.\cite{Clancy94,Jorgensen98b} |
378 |
|
|
|
379 |
|
|
It has been observed that densities are dependent on the cutoff radius |
380 |
|
|
used for a variety of water models in simulations both with and |
381 |
|
|
without the use of reaction field.\cite{Berendsen98} In order to |
382 |
|
|
address the possible affect of cutoff radius, simulations were |
383 |
|
|
performed with a dipolar cutoff radius of 12.0 \AA\ to compliment the |
384 |
|
|
previous SSD simulations, all performed with a cutoff of 9.0 \AA. All |
385 |
|
|
the resulting densities overlapped within error and showed no |
386 |
|
|
significant trend in lower or higher densities as a function of cutoff |
387 |
|
|
radius, both for simulations with and without reaction field. These |
388 |
|
|
results indicate that there is no major benefit in choosing a longer |
389 |
|
|
cutoff radius in simulations using SSD. This is comforting in that the |
390 |
|
|
use of a longer cutoff radius results in a near doubling of the time |
391 |
|
|
required to compute a single trajectory. |
392 |
|
|
|
393 |
|
|
\subsection{Transport Behavior} |
394 |
|
|
Of importance in these types of studies are the transport properties |
395 |
|
|
of the particles and how they change when altering the environmental |
396 |
|
|
conditions. In order to probe transport, constant energy simulations |
397 |
|
|
were performed about the average density uncovered by the constant |
398 |
|
|
pressure simulations. Simulations started with randomized velocities |
399 |
|
|
and underwent 50 ps of temperature scaling and 50 ps of constant |
400 |
|
|
energy equilibration before obtaining a 200 ps trajectory. Diffusion |
401 |
|
|
constants were calculated via root-mean square deviation analysis. The |
402 |
|
|
averaged results from 5 sets of these NVE simulations is displayed in |
403 |
|
|
figure \ref{diffuse}, alongside experimental, SPC/E, and TIP5P |
404 |
|
|
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} |
405 |
|
|
|
406 |
|
|
\begin{figure} |
407 |
|
|
\includegraphics[width=65mm, angle=-90]{betterDiffuse.epsi} |
408 |
|
|
\caption{Average diffusion coefficient over increasing temperature for |
409 |
|
|
SSD, SPC/E\cite{Clancy94}, TIP5P\cite{Jorgensen01}, and Experimental |
410 |
|
|
data from Gillen \emph{et al.}\cite{Gillen72}, and from |
411 |
|
|
Mills\cite{Mills73}.} |
412 |
|
|
\label{diffuse} |
413 |
|
|
\end{figure} |
414 |
|
|
|
415 |
|
|
The observed values for the diffusion constant point out one of the |
416 |
|
|
strengths of the SSD model. Of the three experimental models shown, |
417 |
|
|
the SSD model has the most accurate depiction of the diffusion trend |
418 |
|
|
seen in experiment in both the supercooled and normal regimes. SPC/E |
419 |
|
|
does a respectable job by getting similar values as SSD and experiment |
420 |
|
|
around 290 K; however, it deviates at both higher and lower |
421 |
|
|
temperatures, failing to predict the experimental trend. TIP5P and SSD |
422 |
|
|
both start off low at the colder temperatures and tend to diffuse too |
423 |
|
|
rapidly at the higher temperatures. This type of trend at the higher |
424 |
|
|
temperatures is not surprising in that the densities of both TIP5P and |
425 |
|
|
SSD are lower than experimental water at temperatures higher than room |
426 |
|
|
temperature. When calculating the diffusion coefficients for SSD at |
427 |
|
|
experimental densities, the resulting values fall more in line with |
428 |
|
|
experiment at these temperatures, albeit not at standard |
429 |
|
|
pressure. Results under these conditions can be found later in this |
430 |
|
|
paper. |
431 |
|
|
|
432 |
|
|
\subsection{Structural Changes and Characterization} |
433 |
|
|
By starting the simulations from the crystalline state, the melting |
434 |
|
|
transition and the ice structure can be studied along with the liquid |
435 |
|
|
phase behavior beyond the melting point. To locate the melting |
436 |
|
|
transition, the constant pressure heat capacity (C$_\text{p}$) was |
437 |
|
|
monitored in each of the simulations. In the melting simulations of |
438 |
|
|
the 1024 particle ice $I_h$ simulations, a large spike in C$_\text{p}$ |
439 |
|
|
occurs at 245 K, indicating a first order phase transition for the |
440 |
|
|
melting of these ice crystals. When the reaction field is turned off, |
441 |
|
|
the melting transition occurs at 235 K. These melting transitions are |
442 |
|
|
considerably lower than the experimental value, but this is not |
443 |
|
|
surprising in that SSD is a simple rigid body model with a fixed |
444 |
|
|
dipole. |
445 |
|
|
|
446 |
|
|
\begin{figure} |
447 |
|
|
\includegraphics[width=85mm]{fullContours.eps} |
448 |
|
|
\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at |
449 |
|
|
100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for |
450 |
|
|
clarity: dark areas signify peaks while light areas signify |
451 |
|
|
depressions. White areas have g(\emph{r}) values below 0.5 and black |
452 |
|
|
areas have values above 1.5.} |
453 |
|
|
\label{contour} |
454 |
|
|
\end{figure} |
455 |
|
|
|
456 |
|
|
Additional analyses for understanding the melting phase-transition |
457 |
|
|
process were performed via two-dimensional structure and dipole angle |
458 |
|
|
correlations. Expressions for the correlations are as follows: |
459 |
|
|
|
460 |
|
|
\begin{figure} |
461 |
|
|
\includegraphics[width=45mm]{corrDiag.eps} |
462 |
|
|
\caption{Two dimensional illustration of the angles involved in the |
463 |
|
|
correlations observed in figure \ref{contour}.} |
464 |
|
|
\label{corrAngle} |
465 |
|
|
\end{figure} |
466 |
|
|
|
467 |
|
|
\begin{multline} |
468 |
|
|
g_{\text{AB}}(r,\cos\theta) = \\ |
469 |
|
|
\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\ , |
470 |
|
|
\end{multline} |
471 |
|
|
\begin{multline} |
472 |
|
|
g_{\text{AB}}(r,\cos\omega) = \\ |
473 |
|
|
\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\ , |
474 |
|
|
\end{multline} |
475 |
|
|
where $\theta$ and $\omega$ refer to the angles shown in the above |
476 |
|
|
illustration. By binning over both distance and the cosine of the |
477 |
|
|
desired angle between the two dipoles, the g(\emph{r}) can be |
478 |
|
|
dissected to determine the common dipole arrangements that constitute |
479 |
|
|
the peaks and troughs. Frames A and B of figure \ref{contour} show a |
480 |
|
|
relatively crystalline state of an ice $I_c$ simulation. The first |
481 |
|
|
peak of the g(\emph{r}) primarily consists of the preferred hydrogen |
482 |
|
|
bonding arrangements as dictated by the tetrahedral sticky potential, |
483 |
|
|
one peak for the donating and the other for the accepting hydrogen |
484 |
|
|
bonds. Due to the high degree of crystallinity of the sample, the |
485 |
|
|
second and third solvation shells show a repeated peak arrangement |
486 |
|
|
which decays at distances around the fourth solvation shell, near the |
487 |
|
|
imposed cutoff for the Lennard-Jones and dipole-dipole interactions. |
488 |
|
|
In the higher temperature simulation shown in frames C and D, the |
489 |
|
|
repeated peak features are significantly blurred. The first solvation |
490 |
|
|
shell still shows the strong effect of the sticky-potential, although |
491 |
|
|
it covers a larger area, extending to include a fraction of aligned |
492 |
|
|
dipole peaks within the first solvation shell. The latter peaks lose |
493 |
|
|
definition as thermal motion and the competing dipole force overcomes |
494 |
|
|
the sticky potential's tight tetrahedral structuring of the fluid. |
495 |
|
|
|
496 |
|
|
This complex interplay between dipole and sticky interactions was |
497 |
|
|
remarked upon as a possible reason for the split second peak in the |
498 |
|
|
oxygen-oxygen g(\emph{r}).\cite{Ichiye96} At low temperatures, the |
499 |
|
|
second solvation shell peak appears to have two distinct parts that |
500 |
|
|
blend together to form one observable peak. At higher temperatures, |
501 |
|
|
this split character alters to show the leading 4 \AA\ peak dominated |
502 |
|
|
by equatorial anti-parallel dipole orientations, and there is tightly |
503 |
|
|
bunched group of axially arranged dipoles that most likely consist of |
504 |
|
|
the smaller fraction aligned dipole pairs. The trailing part of the |
505 |
|
|
split peak at 5 \AA\ is dominated by aligned dipoles that range |
506 |
|
|
primarily within the axial to the chief hydrogen bond arrangements |
507 |
|
|
similar to those seen in the first solvation shell. This evidence |
508 |
|
|
indicates that the dipole pair interaction begins to dominate outside |
509 |
|
|
of the range of the dipolar repulsion term, with the primary |
510 |
|
|
energetically favorable dipole arrangements populating the region |
511 |
|
|
immediately outside of it's range (around 4 \AA), and arrangements |
512 |
|
|
that seek to ideally satisfy both the sticky and dipole forces locate |
513 |
|
|
themselves just beyond this region (around 5 \AA). |
514 |
|
|
|
515 |
|
|
From these findings, the split second peak is primarily the product of |
516 |
|
|
the dipolar repulsion term of the sticky potential. In fact, the |
517 |
|
|
leading of the two peaks can be pushed out and merged with the outer |
518 |
|
|
split peak just by extending the switching function cutoff |
519 |
|
|
($s^\prime(r_{ij})$) from its normal 4.0 \AA\ to values of 4.5 or even |
520 |
|
|
5 \AA. This type of correction is not recommended for improving the |
521 |
|
|
liquid structure, because the second solvation shell will still be |
522 |
|
|
shifted too far out. In addition, this would have an even more |
523 |
|
|
detrimental effect on the system densities, leading to a liquid with a |
524 |
|
|
more open structure and a density considerably lower than the normal |
525 |
|
|
SSD behavior shown previously. A better correction would be to include |
526 |
|
|
the quadrupole-quadrupole interactions for the water particles outside |
527 |
|
|
of the first solvation shell, but this reduces the simplicity and |
528 |
|
|
speed advantage of SSD, so it is not the most desirable path to take. |
529 |
|
|
|
530 |
|
|
\subsection{Adjusted Potentials: SSD/E and SSD/RF} |
531 |
|
|
The propensity of SSD to adopt lower than expected densities under |
532 |
|
|
varying conditions is troubling, especially at higher temperatures. In |
533 |
|
|
order to correct this behavior, it's necessary to adjust the force |
534 |
|
|
field parameters for the primary intermolecular interactions. In |
535 |
|
|
undergoing a reparameterization, it is important not to focus on just |
536 |
|
|
one property and neglect the other important properties. In this case, |
537 |
|
|
it would be ideal to correct the densities while maintaining the |
538 |
|
|
accurate transport properties. |
539 |
|
|
|
540 |
|
|
The possible parameters for tuning include the $\sigma$ and $\epsilon$ |
541 |
|
|
Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky |
542 |
|
|
attractive and dipole repulsive terms with their respective |
543 |
|
|
cutoffs. To alter the attractive and repulsive terms of the sticky |
544 |
|
|
potential independently, it is necessary to separate the terms as |
545 |
|
|
follows: |
546 |
|
|
\begin{equation} |
547 |
|
|
\begin{split} |
548 |
|
|
u_{ij}^{sp} |
549 |
|
|
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) &= |
550 |
|
|
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\\ |
551 |
|
|
& \quad \ + \frac{\nu_0^\prime}{2} |
552 |
|
|
[s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)], |
553 |
|
|
\end{split} |
554 |
|
|
\end{equation} |
555 |
|
|
|
556 |
|
|
where $\nu_0$ scales the strength of the tetrahedral attraction and |
557 |
|
|
$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion |
558 |
|
|
term. For purposes of the reparameterization, the separation was |
559 |
|
|
performed, but the final parameters were adjusted so that it is |
560 |
|
|
unnecessary to separate the terms when implementing the adjusted water |
561 |
|
|
potentials. The results of the reparameterizations are shown in table |
562 |
|
|
\ref{params}. Note that both the tetrahedral attractive and dipolar |
563 |
|
|
repulsive don't share the same lower cutoff ($r_l$) in the newly |
564 |
|
|
parameterized potentials - soft sticky dipole enhanced (SSD/E) and |
565 |
|
|
soft sticky dipole reaction field (SSD/RF). |
566 |
|
|
|
567 |
|
|
\begin{table} |
568 |
|
|
\caption{Parameters for the original and adjusted models} |
569 |
|
|
\begin{tabular}{ l c c c } |
570 |
|
|
\hline \\[-3mm] |
571 |
|
|
\ Parameters & \ \ \ SSD$^\dagger$\ \ \ \ & \ SSD/E\ \ & \ SSD/RF\ \ \\ |
572 |
|
|
\hline \\[-3mm] |
573 |
|
|
\ \ \ $\sigma$ (\AA) & 3.051 & 3.035 & 3.019\\ |
574 |
|
|
\ \ \ $\epsilon$ (kcal/mol)\ \ & 0.152 & 0.152 & 0.152\\ |
575 |
|
|
\ \ \ $\mu$ (D) & 2.35 & 2.418 & 2.480\\ |
576 |
|
|
\ \ \ $\nu_0$ (kcal/mol)\ \ & 3.7284 & 3.90 & 3.90\\ |
577 |
|
|
\ \ \ $r_l$ (\AA) & 2.75 & 2.40 & 2.40\\ |
578 |
|
|
\ \ \ $r_u$ (\AA) & 3.35 & 3.80 & 3.80\\ |
579 |
|
|
\ \ \ $\nu_0^\prime$ (kcal/mol)\ \ & 3.7284 & 3.90 & 3.90\\ |
580 |
|
|
\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75\\ |
581 |
|
|
\ \ \ $r_u^\prime$ (\AA) & 4.00 & 3.35 & 3.35\\ |
582 |
|
|
\\[-2mm]$^\dagger$ ref. \onlinecite{Ichiye96} |
583 |
|
|
\end{tabular} |
584 |
|
|
\label{params} |
585 |
|
|
\end{table} |
586 |
|
|
|
587 |
|
|
\begin{figure} |
588 |
|
|
\includegraphics[width=85mm]{gofrCompare.epsi} |
589 |
|
|
\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E |
590 |
|
|
and SSD without reaction field (top), as well as SSD/RF and SSD with |
591 |
|
|
reaction field turned on (bottom). The insets show the respective |
592 |
|
|
first peaks in detail. Solid Line - experiment, dashed line - SSD/E |
593 |
|
|
and SSD/RF, and dotted line - SSD (with and without reaction field).} |
594 |
|
|
\label{grcompare} |
595 |
|
|
\end{figure} |
596 |
|
|
|
597 |
|
|
\begin{figure} |
598 |
|
|
\includegraphics[width=85mm]{dualsticky.ps} |
599 |
|
|
\caption{Isosurfaces of the sticky potential for SSD (left) and SSD/E \& |
600 |
|
|
SSD/RF (right). Light areas correspond to the tetrahedral attractive |
601 |
|
|
part, and the darker areas correspond to the dipolar repulsive part.} |
602 |
|
|
\label{isosurface} |
603 |
|
|
\end{figure} |
604 |
|
|
|
605 |
|
|
In the paper detailing the development of SSD, Liu and Ichiye placed |
606 |
|
|
particular emphasis on an accurate description of the first solvation |
607 |
|
|
shell. This resulted in a somewhat tall and sharp first peak that |
608 |
|
|
integrated to give similar coordination numbers to the experimental |
609 |
|
|
data obtained by Soper and Phillips.\cite{Ichiye96,Soper86} New |
610 |
|
|
experimental x-ray scattering data from the Head-Gordon lab indicates |
611 |
|
|
a slightly lower and shifted first peak in the g$_\mathrm{OO}(r)$, so |
612 |
|
|
adjustments to SSD were made while taking into consideration the new |
613 |
|
|
experimental findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} |
614 |
|
|
shows the relocation of the first peak of the oxygen-oxygen |
615 |
|
|
g(\emph{r}) by comparing the original SSD (with and without reaction |
616 |
|
|
field), SSD-E, and SSD-RF to the new experimental results. Both the |
617 |
|
|
modified water models have shorter peaks that are brought in more |
618 |
|
|
closely to the experimental peak (as seen in the insets of figure |
619 |
|
|
\ref{grcompare}). This structural alteration was accomplished by a |
620 |
|
|
reduction in the Lennard-Jones $\sigma$ variable as well as adjustment |
621 |
|
|
of the sticky potential strength and cutoffs. The cutoffs for the |
622 |
|
|
tetrahedral attractive and dipolar repulsive terms were nearly swapped |
623 |
|
|
with each other. Isosurfaces of the original and modified sticky |
624 |
|
|
potentials are shown in figure \cite{isosurface}. In these |
625 |
|
|
isosurfaces, it is easy to see how altering the cutoffs changes the |
626 |
|
|
repulsive and attractive character of the particles. With a reduced |
627 |
|
|
repulsive surface (the darker region), the particles can move closer |
628 |
|
|
to one another, increasing the density for the overall system. This |
629 |
|
|
change in interaction cutoff also results in a more gradual |
630 |
|
|
orientational motion by allowing the particles to maintain preferred |
631 |
|
|
dipolar arrangements before they begin to feel the pull of the |
632 |
|
|
tetrahedral restructuring. Upon moving closer together, the dipolar |
633 |
|
|
repulsion term becomes active and excludes the unphysical |
634 |
|
|
arrangements. This compares with the original SSD's excluding dipolar |
635 |
|
|
before the particles feel the pull of the ``hydrogen bonds''. Aside |
636 |
|
|
from improving the shape of the first peak in the g(\emph{r}), this |
637 |
|
|
improves the densities considerably by allowing the persistence of |
638 |
|
|
full dipolar character below the previous 4.0 \AA\ cutoff. |
639 |
|
|
|
640 |
|
|
While adjusting the location and shape of the first peak of |
641 |
|
|
g(\emph{r}) improves the densities to some degree, these changes alone |
642 |
|
|
are insufficient to bring the system densities up to the values |
643 |
|
|
observed experimentally. To finish bringing up the densities, the |
644 |
|
|
dipole moments were increased in both the adjusted models. Being a |
645 |
|
|
dipole based model, the structure and transport are very sensitive to |
646 |
|
|
changes in the dipole moment. The original SSD simply used the dipole |
647 |
|
|
moment calculated from the TIP3P water model, which at 2.35 D is |
648 |
|
|
significantly greater than the experimental gas phase value of 1.84 |
649 |
|
|
D. The larger dipole moment is a more realistic value and improve the |
650 |
|
|
dielectric properties of the fluid. Both theoretical and experimental |
651 |
|
|
measurements indicate a liquid phase dipole moment ranging from 2.4 D |
652 |
|
|
to values as high as 3.11 D, so there is quite a range available for |
653 |
|
|
adjusting the dipole |
654 |
|
|
moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} The increasing of |
655 |
|
|
the dipole moments to 2.418 and 2.48 D for SSD/E and SSD/RF |
656 |
|
|
respectively is moderate in the range of the experimental values; |
657 |
|
|
however, it leads to significant changes in the density and transport |
658 |
|
|
of the water models. |
659 |
|
|
|
660 |
|
|
In order to demonstrate the benefits of this reparameterization, a |
661 |
|
|
series of NPT and NVE simulations were performed to probe the density |
662 |
|
|
and transport properties of the adapted models and compare the results |
663 |
|
|
to the original SSD model. This comparison involved full NPT melting |
664 |
|
|
sequences for both SSD/E and SSD/RF, as well as NVE transport |
665 |
|
|
calculations at both self-consistent and experimental |
666 |
|
|
densities. Again, the results come from five separate simulations of |
667 |
|
|
1024 particle systems, and the melting sequences were started from |
668 |
|
|
different ice $I_h$ crystals constructed as stated previously. Like |
669 |
|
|
before, all of the NPT simulations were equilibrated for 100 ps before |
670 |
|
|
a 200 ps data collection run, and they used the previous temperature's |
671 |
|
|
final configuration as a starting point. All of the NVE simulations |
672 |
|
|
had the same thermalization, equilibration, and data collection times |
673 |
|
|
stated earlier in this paper. |
674 |
|
|
|
675 |
|
|
\begin{figure} |
676 |
|
|
\includegraphics[width=85mm]{ssdecompare.epsi} |
677 |
|
|
\caption{Comparison of densities calculated with SSD/E to SSD without a |
678 |
|
|
reaction field, TIP4P\cite{Jorgensen98b}, TIP3P\cite{Jorgensen98b}, |
679 |
|
|
SPC/E\cite{Clancy94}, and Experiment\cite{CRC80}. The upper plot |
680 |
|
|
includes error bars, and the calculated results from the other |
681 |
|
|
references were removed for clarity.} |
682 |
|
|
\label{ssdedense} |
683 |
|
|
\end{figure} |
684 |
|
|
|
685 |
|
|
Figure \ref{ssdedense} shows the density profile for the SSD/E water |
686 |
|
|
model in comparison to the original SSD without a reaction field, |
687 |
|
|
experiment, and the other common water models considered |
688 |
|
|
previously. The calculated densities have increased significantly over |
689 |
|
|
the original SSD model and match the experimental value just below 298 |
690 |
|
|
K. At 298 K, the density of SSD/E is 0.995$\pm$0.001 g/cm$^3$, which |
691 |
|
|
compares well with the experimental value of 0.997 g/cm$^3$ and is |
692 |
|
|
considerably better than the SSD value of 0.967$\pm$0.003 |
693 |
|
|
g/cm$^3$. The increased dipole moment in SSD/E has helped to flatten |
694 |
|
|
out the curve at higher temperatures, only the improvement is marginal |
695 |
|
|
at best. This steep drop in densities is due to the dipolar rather |
696 |
|
|
than charge based interactions which decay more rapidly at longer |
697 |
|
|
distances. |
698 |
|
|
|
699 |
|
|
By monitoring C$\text{p}$ throughout these simulations, the melting |
700 |
|
|
transition for SSD/E was observed at 230 K, about 5 degrees lower than |
701 |
|
|
SSD. The resulting density maximum is located at 240 K, again about 5 |
702 |
|
|
degrees lower than the SSD value of 245 K. Though there is a decrease |
703 |
|
|
in both of these values, the corrected densities near room temperature |
704 |
|
|
justify the modifications taken. |
705 |
|
|
|
706 |
|
|
\begin{figure} |
707 |
|
|
\includegraphics[width=85mm]{ssdrfcompare.epsi} |
708 |
|
|
\caption{Comparison of densities calculated with SSD/RF to SSD with a |
709 |
|
|
reaction field, TIP4P\cite{Jorgensen98b}, TIP3P\cite{Jorgensen98b}, |
710 |
|
|
SPC/E\cite{Clancy94}, and Experiment\cite{CRC80}. The upper plot |
711 |
|
|
includes error bars, and the calculated results from the other |
712 |
|
|
references were removed for clarity.} |
713 |
|
|
\label{ssdrfdense} |
714 |
|
|
\end{figure} |
715 |
|
|
|
716 |
|
|
Figure \ref{ssdrfdense} shows a density comparison between SSD/RF and |
717 |
|
|
SSD with an active reaction field. Like in the simulations of SSD/E, |
718 |
|
|
the densities show a dramatic increase over normal SSD. At 298 K, |
719 |
|
|
SSD/RF has a density of 0.997$\pm$0.001 g/cm$^3$, right in line with |
720 |
|
|
experiment and considerably better than the SSD value of |
721 |
|
|
0.941$\pm$0.001 g/cm$^3$. The melting point is observed at 240 K, |
722 |
|
|
which is 5 degrees lower than SSD with a reaction field, and the |
723 |
|
|
density maximum at 255 K, again 5 degrees lower than SSD. The density |
724 |
|
|
at higher temperature still drops off more rapidly than the charge |
725 |
|
|
based models but is in better agreement than SSD/E. |
726 |
|
|
|
727 |
|
|
The reparameterization of the SSD water model, both for use with and |
728 |
|
|
without an applied long-range correction, brought the densities up to |
729 |
|
|
what is expected for simulating liquid water. In addition to improving |
730 |
|
|
the densities, it is important that particle transport be maintained |
731 |
|
|
or improved. Figure \ref{ssdediffuse} compares the temperature |
732 |
|
|
dependence of the diffusion constant of SSD/E to SSD without an active |
733 |
|
|
reaction field, both at the densities calculated at 1 atm and at the |
734 |
|
|
experimentally calculated densities for super-cooled and liquid |
735 |
|
|
water. In the upper plot, the diffusion constant for SSD/E is |
736 |
|
|
consistently a little faster than experiment, while SSD starts off |
737 |
|
|
slower than experiment and crosses to merge with SSD/E at high |
738 |
|
|
temperatures. Both models follow the experimental trend well, but |
739 |
|
|
diffuse too rapidly at higher temperatures. This abnormally fast |
740 |
|
|
diffusion is caused by the decreased system density. Since the |
741 |
|
|
densities of SSD/E don't deviate as much from experiment as those of |
742 |
|
|
SSD, it follows the experimental trend more closely. This observation |
743 |
|
|
is backed up by looking at the lower plot. The diffusion constants for |
744 |
|
|
SSD/E track with the experimental values while SSD deviates on the low |
745 |
|
|
side of the trend with increasing temperature. This is again a product |
746 |
|
|
of SSD/E having densities closer to experiment, and not deviating to |
747 |
|
|
lower densities with increasing temperature as rapidly. |
748 |
|
|
|
749 |
|
|
\begin{figure} |
750 |
|
|
\includegraphics[width=85mm]{ssdediffuse.epsi} |
751 |
|
|
\caption{Plots of the diffusion constants calculated from SSD/E and SSD, |
752 |
|
|
both without a reaction field along with experimental results from |
753 |
|
|
Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The |
754 |
|
|
upper plot is at densities calculated from the NPT simulations at a |
755 |
|
|
pressure of 1 atm, while the lower plot is at the experimentally |
756 |
|
|
calculated densities.} |
757 |
|
|
\label{ssdediffuse} |
758 |
|
|
\end{figure} |
759 |
|
|
|
760 |
|
|
\begin{figure} |
761 |
|
|
\includegraphics[width=85mm]{ssdrfdiffuse.epsi} |
762 |
|
|
\caption{Plots of the diffusion constants calculated from SSD/RF and SSD, |
763 |
|
|
both with an active reaction field along with experimental results |
764 |
|
|
from Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The |
765 |
|
|
upper plot is at densities calculated from the NPT simulations at a |
766 |
|
|
pressure of 1 atm, while the lower plot is at the experimentally |
767 |
|
|
calculated densities.} |
768 |
|
|
\label{ssdrfdiffuse} |
769 |
|
|
\end{figure} |
770 |
|
|
|
771 |
|
|
In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are |
772 |
|
|
compared with SSD with an active reaction field. In the upper plot, |
773 |
|
|
SSD/RF tracks with the experimental results incredibly well, identical |
774 |
|
|
within error throughout the temperature range and only showing a |
775 |
|
|
slight increasing trend at higher temperatures. SSD also tracks |
776 |
|
|
experiment well, only it tends to diffuse a little more slowly at low |
777 |
|
|
temperatures and deviates to diffuse too rapidly at high |
778 |
|
|
temperatures. As was stated in the SSD/E comparisons, this deviation |
779 |
|
|
away from the ideal trend is due to a rapid decrease in density at |
780 |
|
|
higher temperatures. SSD/RF doesn't suffer from this problem as much |
781 |
|
|
as SSD, because the calculated densities are more true to |
782 |
|
|
experiment. This is again emphasized in the lower plot, where SSD/RF |
783 |
|
|
tracks the experimental diffusion exactly while SSD's diffusion |
784 |
|
|
constants are slightly too low due to its need for a lower density at |
785 |
|
|
the specified temperature. |
786 |
|
|
|
787 |
|
|
\subsection{Additional Observations} |
788 |
|
|
|
789 |
|
|
While performing the melting sequences of SSD/E, some interesting |
790 |
|
|
observations were made. After melting at 230 K, two of the systems |
791 |
|
|
underwent crystallization events near 245 K. As the heating process |
792 |
|
|
continued, the two systems remained crystalline until finally melting |
793 |
|
|
between 320 and 330 K. These simulations were excluded from the data |
794 |
|
|
set shown in figure \ref{ssdedense} and replaced with two additional |
795 |
|
|
melting sequences that did not undergo this anomalous phase |
796 |
|
|
transition, while this crystallization event was investigated |
797 |
|
|
separately. |
798 |
|
|
|
799 |
|
|
\begin{figure} |
800 |
|
|
\includegraphics[width=85mm]{povIce.ps} |
801 |
|
|
\caption{Crystal structure of an ice 0 lattice shown from the (001) face.} |
802 |
|
|
\label{weirdice} |
803 |
|
|
\end{figure} |
804 |
|
|
|
805 |
|
|
The final configurations of these two melting sequences shows an |
806 |
|
|
expanded zeolite-like crystal structure that does not correspond to |
807 |
|
|
any known form of ice. For convenience and to help distinguish it from |
808 |
|
|
the experimentally observed forms of ice, this crystal structure will |
809 |
|
|
henceforth be referred to as ice-zero (ice 0). The crystallinity was |
810 |
|
|
extensive enough than a near ideal crystal structure could be |
811 |
|
|
obtained. Figure \ref{weirdice} shows the repeating crystal structure |
812 |
|
|
of a typical crystal at 5 K. The unit cell contains eight molecules, |
813 |
|
|
and figure \ref{unitcell} shows a unit cell built from the water |
814 |
|
|
particle center of masses that can be used to construct a repeating |
815 |
|
|
lattice, similar to figure \ref{weirdice}. Each molecule is hydrogen |
816 |
|
|
bonded to four other water molecules; however, the hydrogen bonds are |
817 |
|
|
flexed rather than perfectly straight. This results in a skewed |
818 |
|
|
tetrahedral geometry about the central molecule. Looking back at |
819 |
|
|
figure \ref{isosurface}, it is easy to see how these flexed hydrogen |
820 |
|
|
bonds are allowed in that the attractive regions are conical in shape, |
821 |
|
|
with the greatest attraction in the central region. Though not ideal, |
822 |
|
|
these flexed hydrogen bonds are favorable enough to stabilize an |
823 |
|
|
entire crystal generated around them. In fact, the imperfect ice 0 |
824 |
|
|
crystals were so stable that they melted at greater than room |
825 |
|
|
temperature. |
826 |
|
|
|
827 |
|
|
\begin{figure} |
828 |
|
|
\includegraphics[width=65mm]{ice0cell.eps} |
829 |
|
|
\caption{Simple unit cell for constructing ice 0. In this cell, $c$ is |
830 |
|
|
equal to $0.4714\times a$, and a typical value for $a$ is 8.25 \AA.} |
831 |
|
|
\label{unitcell} |
832 |
|
|
\end{figure} |
833 |
|
|
|
834 |
|
|
The initial simulations indicated that ice 0 is the preferred ice |
835 |
|
|
structure for at least SSD/E. To verify this, a comparison was made |
836 |
|
|
between near ideal crystals of ice $I_h$, ice $I_c$, and ice 0 at |
837 |
|
|
constant pressure with SSD/E, SSD/RF, and SSD. Near ideal versions of |
838 |
|
|
the three types of crystals were cooled to ~1 K, and the potential |
839 |
|
|
energies of each were compared using all three water models. With |
840 |
|
|
every water model, ice 0 turned out to have the lowest potential |
841 |
|
|
energy: 5\% lower than $I_h$ with SSD, 6.5\% lower with SSD/E, and |
842 |
|
|
7.5\% lower with SSD/RF. In all three of these water models, ice $I_c$ |
843 |
|
|
was observed to be ~2\% less stable than ice $I_h$. In addition to |
844 |
|
|
having the lowest potential energy, ice 0 was the most expanded of the |
845 |
|
|
three ice crystals, ~5\% less dense than ice $I_h$ with all of the |
846 |
|
|
water models. In all three water models, ice $I_c$ was observed to be |
847 |
|
|
~2\% more dense than ice $I_h$. |
848 |
|
|
|
849 |
|
|
In addition to the low temperature comparisons, melting sequences were |
850 |
|
|
performed with ice 0 as the initial configuration using SSD/E, SSD/RF, |
851 |
|
|
and SSD both with and without a reaction field. The melting |
852 |
|
|
transitions for both SSD/E and SSD without a reaction field occurred |
853 |
|
|
at temperature in excess of 375 K. SSD/RF and SSD with a reaction |
854 |
|
|
field had more reasonable melting transitions, down near 325 K. These |
855 |
|
|
melting point observations emphasize how preferred this crystal |
856 |
|
|
structure is over the most common types of ice when using these single |
857 |
|
|
point water models. |
858 |
|
|
|
859 |
|
|
Recognizing that the above tests show ice 0 to be both the most stable |
860 |
|
|
and lowest density crystal structure for these single point water |
861 |
|
|
models, it is interesting to speculate on the favorability of this |
862 |
|
|
crystal structure with the different charge based models. As a quick |
863 |
|
|
test, these 3 crystal types were converted from SSD type particles to |
864 |
|
|
TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy |
865 |
|
|
minimizations were performed on all of these crystals to compare the |
866 |
|
|
system energies. Again, ice 0 was observed to have the lowest total |
867 |
|
|
system energy. The total energy of ice 0 was ~2\% lower than ice |
868 |
|
|
$I_h$, which was in turn ~3\% lower than ice $I_c$. From these initial |
869 |
|
|
results, we would not be surprised if results from the other common |
870 |
|
|
water models show ice 0 to be the lowest energy crystal structure. A |
871 |
|
|
continuation on work studing ice 0 with multipoint water models will |
872 |
|
|
be published in a coming article. |
873 |
|
|
|
874 |
|
|
\section{Conclusions} |
875 |
|
|
The density maximum and temperature dependent transport for the SSD |
876 |
|
|
water model, both with and without the use of reaction field, were |
877 |
|
|
studied via a series of NPT and NVE simulations. The constant pressure |
878 |
|
|
simulations of the melting of both $I_h$ and $I_c$ ice showed a |
879 |
|
|
density maximum near 260 K. In most cases, the calculated densities |
880 |
|
|
were significantly lower than the densities calculated in simulations |
881 |
|
|
of other water models. Analysis of particle diffusion showed SSD to |
882 |
|
|
capture the transport properties of experimental very well in both the |
883 |
|
|
normal and super-cooled liquid regimes. In order to correct the |
884 |
|
|
density behavior, SSD was reparameterized for use both with and |
885 |
|
|
without a long-range interaction correction, SSD/RF and SSD/E |
886 |
|
|
respectively. In addition to correcting the abnormally low densities, |
887 |
|
|
these new versions were show to maintain or improve upon the transport |
888 |
|
|
and structural features of the original water model, all while |
889 |
|
|
maintaining the fast performance of the SSD water model. This work |
890 |
|
|
shows these simple water models, and in particular SSD/E and SSD/RF, |
891 |
|
|
to be excellent choices to represent explicit water in future |
892 |
|
|
simulations of biochemical systems. |
893 |
|
|
|
894 |
|
|
\section{Acknowledgments} |
895 |
|
|
The authors would like to thank the National Science Foundation for |
896 |
|
|
funding under grant CHE-0134881. Computation time was provided by the |
897 |
|
|
Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant DMR |
898 |
|
|
00 79647. |
899 |
|
|
|
900 |
|
|
\bibliographystyle{jcp} |
901 |
|
|
|
902 |
|
|
\bibliography{nptSSD} |
903 |
|
|
|
904 |
|
|
%\pagebreak |
905 |
|
|
|
906 |
|
|
\end{document} |