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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\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle |
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\begin{abstract} |
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NVE and NPT molecular dynamics simulations were performed in order to |
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investigate the density maximum and temperature dependent transport |
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for SSD and related water models, both with and without the use of |
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reaction field. The constant pressure simulations of the melting of |
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both $I_h$ and $I_c$ ice showed a density maximum near 260 K. In most |
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cases, the calculated densities were significantly lower than the |
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densities calculated in simulations of other water models. Analysis of |
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particle diffusion showed SSD to capture the transport properties of |
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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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\newpage |
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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 the simulation of biochemical |
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systems is the proper depiction of water and water solvation. In fact, |
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the bulk of the calculations performed in solvated simulations are of |
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interactions with or between solvent molecules. Thus, the outcomes of |
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these types of simulations are highly dependent on the physical |
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properties of water, both as individual molecules and in clusters or |
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bulk. Due to the fact that explicit solvent accounts for a massive |
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portion of the calculations, it necessary to simplify the solvent to |
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some extent in order to complete simulations in a reasonable amount of |
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time. In the case of simulating water in biomolecular studies, the |
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balance between accurate properties and computational efficiency is |
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especially delicate, and it has resulted in a variety of different |
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water models.\cite{Jorgensen83,Berendsen87,Jorgensen00} Many of these |
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models predict specific properties more accurately than their |
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predecessors, but often at the cost of other properties or of computer |
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time. As an example, compare TIP3P or TIP4P to TIP5P. TIP5P improves |
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upon the structural and transport properties of water relative to the |
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previous TIP models, yet this comes at a greater than 50\% increase in |
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computational cost.\cite{Jorgensen01,Jorgensen00} One recently |
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developed model that succeeds in both retaining the accuracy of system |
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properties and simplifying calculations to increase computational |
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efficiency is the Soft Sticky Dipole water model.\cite{Ichiye96} |
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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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u_{ij}^{sp} |
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(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = |
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\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) + s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)]\ , |
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\end{equation} |
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where $\boldsymbol{\mu}_i$ and $\boldsymbol{\mu}_j$ are the dipole |
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unit vectors of particles \emph{i} and \emph{j} with magnitude 2.35 D, |
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$\nu_0$ scales the strength of the overall sticky potential, and $s$ |
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and $s^\prime$ are cubic switching functions. The $w$ and $w^\prime$ |
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functions take the following forms: |
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\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 an |
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increase in calculation efficiency of up to an order of magnitude over |
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other comparable models, while maintaining the structural behavior of |
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water.\cite{Ichiye96} In the original molecular dynamics studies, it |
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was shown that SSD improves on the prediction of many of water's |
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dynamical properties over TIP3P and SPC/E.\cite{Ichiye99} This |
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attractive combination of speed and accurate depiction of solvent |
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properties makes SSD a model of interest for the simulation of large |
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scale biological systems, such as membrane phase behavior. |
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One of the key limitations of this water model, however, is that it |
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has been parameterized for use with the Ewald Sum technique for the |
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handling of long-ranged interactions. When studying very large |
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systems, the Ewald summation and even particle-mesh Ewald become |
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computational burdens, with their respective ideal $N^\frac{3}{2}$ and |
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$N\log N$ calculation scaling orders for $N$ particles.\cite{Darden99} |
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In applying this water model in these types of systems, it would be |
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useful to know its properties and behavior with the more |
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computationally efficient reaction field (RF) technique, and even with |
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a cutoff that lacks any form of long-range correction. This study |
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addresses these issues by looking at the structural and transport |
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behavior of SSD over a variety of temperatures with the purpose of |
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utilizing the RF correction technique. We then suggest alterations to |
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the parameters that result in more water-like behavior. It should be |
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noted that in a recent publication, some of the original investigators of |
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the SSD water model have put forth adjustments to the SSD water model |
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to address abnormal density behavior (also observed here), calling the |
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corrected model SSD1.\cite{Ichiye03} This study will make comparisons |
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with SSD1's behavior with the goal of improving upon the |
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depiction of water under conditions without the Ewald Sum. |
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\section{Methods} |
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As stated previously, the long-range dipole-dipole interactions were |
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accounted for in this study by using the reaction field method. The |
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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 systems, 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 presented on how to make |
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SSD more accurate both with and without a reaction field. |
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|
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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 an 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 this 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 are more costly computationally than the simpler |
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arithmetic quaternion propagation. With the same time step, a 1000 SSD |
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particle simulation shows an average 7\% increase in computation time |
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using the symplectic step method in place of quaternions. This cost is |
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more than justified when comparing the energy conservation of the two |
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methods as illustrated in figure \ref{timestep}. |
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\begin{figure} |
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\begin{center} |
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\epsfxsize=6in |
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\epsfbox{timeStep.epsi} |
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\caption{Energy conservation using 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. The larger time step plots are shifted up from |
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the true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.} |
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\label{timestep} |
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\end{center} |
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\end{figure} |
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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 used to |
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handle 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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|
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Energy drift in the symplectic step 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 simulations. The $I_h$ crystals were formed by |
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first arranging the centers of mass of the SSD particles into a |
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``hexagonal'' ice lattice of 1024 particles. Because of the crystal |
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structure of $I_h$ ice, the simulation box assumed a rectangular shape |
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with an 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 to 25 K. The particles were |
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then allowed to 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. During melting |
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simulations, the melting transition and the density maximum can both |
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be observed, provided that the density maximum occurs in the liquid |
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and not the supercooled regime. An ensemble average from five separate |
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melting simulations was acquired, each starting from different ice |
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crystals generated as described previously. All simulations were |
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equilibrated for 100 ps prior to a 200 ps data collection run at each |
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temperature setting. The temperature range of study spanned from 25 to |
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400 K, with a maximum degree increment of 25 K. For regions of |
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interest along this stepwise progression, the temperature increment |
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|
was decreased from 25 K to 10 and 5 K. The above equilibration and |
311 |
|
|
production times were sufficient in that the system volume |
312 |
|
|
fluctuations dampened out in all but the very cold simulations (below |
313 |
|
|
225 K). |
314 |
chrisfen |
743 |
|
315 |
|
|
\subsection{Density Behavior} |
316 |
chrisfen |
861 |
Initial simulations focused on the original SSD water model, and an |
317 |
|
|
average density versus temperature plot is shown in figure |
318 |
|
|
\ref{dense1}. Note that the density maximum when using a reaction |
319 |
|
|
field appears between 255 and 265 K, where the calculated densities |
320 |
|
|
within this range were nearly indistinguishable. The greater certainty |
321 |
|
|
of the average value at 260 K makes a good argument for the actual |
322 |
|
|
density maximum residing at this midpoint value. Figure \ref{dense1} |
323 |
|
|
was constructed using ice $I_h$ crystals for the initial |
324 |
chrisfen |
862 |
configuration; though not pictured, the simulations starting from ice |
325 |
|
|
$I_c$ crystal configurations showed similar results, with a |
326 |
chrisfen |
861 |
liquid-phase density maximum in this same region (between 255 and 260 |
327 |
|
|
K). In addition, the $I_c$ crystals are more fragile than the $I_h$ |
328 |
chrisfen |
862 |
crystals, leading to deformation into a dense glassy state at lower |
329 |
chrisfen |
861 |
temperatures. This resulted in an overall low temperature density |
330 |
chrisfen |
862 |
maximum at 200 K, while still retaining a liquid state density maximum |
331 |
|
|
in common with the $I_h$ simulations. |
332 |
chrisfen |
743 |
|
333 |
|
|
\begin{figure} |
334 |
chrisfen |
862 |
\begin{center} |
335 |
|
|
\epsfxsize=6in |
336 |
|
|
\epsfbox{denseSSD.eps} |
337 |
|
|
\caption{Density versus temperature for TIP4P,\cite{Jorgensen98b} |
338 |
|
|
TIP3P,\cite{Jorgensen98b} SPC/E,\cite{Clancy94} SSD without Reaction |
339 |
|
|
Field, SSD, and experiment.\cite{CRC80} The arrows indicate the |
340 |
chrisfen |
856 |
change in densities observed when turning off the reaction field. The |
341 |
|
|
the lower than expected densities for the SSD model were what |
342 |
|
|
prompted the original reparameterization.\cite{Ichiye03}} |
343 |
chrisfen |
861 |
\label{dense1} |
344 |
chrisfen |
862 |
\end{center} |
345 |
chrisfen |
743 |
\end{figure} |
346 |
|
|
|
347 |
|
|
The density maximum for SSD actually compares quite favorably to other |
348 |
chrisfen |
861 |
simple water models. Figure \ref{dense1} also shows calculated |
349 |
|
|
densities of several other models and experiment obtained from other |
350 |
chrisfen |
743 |
sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water |
351 |
|
|
models, SSD has results closest to the experimentally observed water |
352 |
|
|
density maximum. Of the listed water models, TIP4P has a density |
353 |
chrisfen |
861 |
maximum behavior most like that seen in SSD. Though not included in |
354 |
|
|
this particular plot, it is useful to note that TIP5P has a water |
355 |
|
|
density maximum nearly identical to experiment. |
356 |
chrisfen |
743 |
|
357 |
|
|
It has been observed that densities are dependent on the cutoff radius |
358 |
|
|
used for a variety of water models in simulations both with and |
359 |
|
|
without the use of reaction field.\cite{Berendsen98} In order to |
360 |
|
|
address the possible affect of cutoff radius, simulations were |
361 |
|
|
performed with a dipolar cutoff radius of 12.0 \AA\ to compliment the |
362 |
|
|
previous SSD simulations, all performed with a cutoff of 9.0 \AA. All |
363 |
chrisfen |
862 |
of the resulting densities overlapped within error and showed no |
364 |
|
|
significant trend toward lower or higher densities as a function of |
365 |
|
|
cutoff radius, for simulations both with and without reaction |
366 |
|
|
field. These results indicate that there is no major benefit in |
367 |
|
|
choosing a longer cutoff radius in simulations using SSD. This is |
368 |
|
|
advantageous in that the use of a longer cutoff radius results in |
369 |
|
|
significant increases in the time required to obtain a single |
370 |
|
|
trajectory. |
371 |
chrisfen |
743 |
|
372 |
chrisfen |
862 |
The key feature to recognize in figure \ref{dense1} is the density |
373 |
|
|
scaling of SSD relative to other common models at any given |
374 |
chrisfen |
861 |
temperature. Note that the SSD model assumes a lower density than any |
375 |
|
|
of the other listed models at the same pressure, behavior which is |
376 |
|
|
especially apparent at temperatures greater than 300 K. Lower than |
377 |
chrisfen |
862 |
expected densities have been observed for other systems using a |
378 |
|
|
reaction field for long-range electrostatic interactions, so the most |
379 |
|
|
likely reason for the significantly lower densities seen in these |
380 |
chrisfen |
861 |
simulations is the presence of the reaction |
381 |
|
|
field.\cite{Berendsen98,Nezbeda02} In order to test the effect of the |
382 |
|
|
reaction field on the density of the systems, the simulations were |
383 |
|
|
repeated without a reaction field present. The results of these |
384 |
|
|
simulations are also displayed in figure \ref{dense1}. Without |
385 |
chrisfen |
862 |
reaction field, the densities increase considerably to more |
386 |
chrisfen |
861 |
experimentally reasonable values, especially around the freezing point |
387 |
|
|
of liquid water. The shape of the curve is similar to the curve |
388 |
|
|
produced from SSD simulations using reaction field, specifically the |
389 |
|
|
rapidly decreasing densities at higher temperatures; however, a shift |
390 |
chrisfen |
862 |
in the density maximum location, down to 245 K, is observed. This is a |
391 |
|
|
more accurate comparison to the other listed water models, in that no |
392 |
|
|
long range corrections were applied in those |
393 |
chrisfen |
861 |
simulations.\cite{Clancy94,Jorgensen98b} However, even without a |
394 |
|
|
reaction field, the density around 300 K is still significantly lower |
395 |
|
|
than experiment and comparable water models. This anomalous behavior |
396 |
|
|
was what lead Ichiye \emph{et al.} to recently reparameterize SSD and |
397 |
|
|
make SSD1.\cite{Ichiye03} In discussing potential adjustments later in |
398 |
|
|
this paper, all comparisons were performed with this new model. |
399 |
|
|
|
400 |
chrisfen |
743 |
\subsection{Transport Behavior} |
401 |
|
|
Of importance in these types of studies are the transport properties |
402 |
chrisfen |
862 |
of the particles and their change in responce to altering |
403 |
|
|
environmental conditions. In order to probe transport, constant energy |
404 |
|
|
simulations were performed about the average density uncovered by the |
405 |
|
|
constant pressure simulations. Simulations started with randomized |
406 |
|
|
velocities and underwent 50 ps of temperature scaling and 50 ps of |
407 |
|
|
constant energy equilibration before obtaining a 200 ps |
408 |
|
|
trajectory. Diffusion constants were calculated via root-mean square |
409 |
|
|
deviation analysis. The averaged results from five sets of NVE |
410 |
|
|
simulations are displayed in figure \ref{diffuse}, alongside |
411 |
|
|
experimental, SPC/E, and TIP5P |
412 |
|
|
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} |
413 |
chrisfen |
743 |
|
414 |
|
|
\begin{figure} |
415 |
chrisfen |
862 |
\begin{center} |
416 |
|
|
\epsfxsize=6in |
417 |
|
|
\epsfbox{betterDiffuse.epsi} |
418 |
chrisfen |
743 |
\caption{Average diffusion coefficient over increasing temperature for |
419 |
chrisfen |
862 |
SSD, SPC/E,\cite{Clancy94} TIP5P,\cite{Jorgensen01} and Experimental |
420 |
|
|
data.\cite{Gillen72,Mills73} Of the three water models shown, SSD has |
421 |
|
|
the least deviation from the experimental values. The rapidly |
422 |
|
|
increasing diffusion constants for TIP5P and SSD correspond to |
423 |
|
|
significant decrease in density at the higher temperatures.} |
424 |
chrisfen |
743 |
\label{diffuse} |
425 |
chrisfen |
862 |
\end{center} |
426 |
chrisfen |
743 |
\end{figure} |
427 |
|
|
|
428 |
|
|
The observed values for the diffusion constant point out one of the |
429 |
|
|
strengths of the SSD model. Of the three experimental models shown, |
430 |
|
|
the SSD model has the most accurate depiction of the diffusion trend |
431 |
chrisfen |
862 |
seen in experiment in both the supercooled and liquid temperature |
432 |
|
|
regimes. SPC/E does a respectable job by producing values similar to |
433 |
|
|
SSD and experiment around 290 K; however, it deviates at both higher |
434 |
|
|
and lower temperatures, failing to predict the experimental |
435 |
|
|
trend. TIP5P and SSD both start off low at colder temperatures and |
436 |
|
|
tend to diffuse too rapidly at higher temperatures. This trend at |
437 |
|
|
higher temperatures is not surprising in that the densities of both |
438 |
|
|
TIP5P and SSD are lower than experimental water at these higher |
439 |
|
|
temperatures. When calculating the diffusion coefficients for SSD at |
440 |
chrisfen |
743 |
experimental densities, the resulting values fall more in line with |
441 |
chrisfen |
861 |
experiment at these temperatures, albeit not at standard pressure. |
442 |
chrisfen |
743 |
|
443 |
|
|
\subsection{Structural Changes and Characterization} |
444 |
|
|
By starting the simulations from the crystalline state, the melting |
445 |
|
|
transition and the ice structure can be studied along with the liquid |
446 |
chrisfen |
862 |
phase behavior beyond the melting point. The constant pressure heat |
447 |
|
|
capacity (C$_\text{p}$) was monitored to locate the melting transition |
448 |
|
|
in each of the simulations. In the melting simulations of the 1024 |
449 |
|
|
particle ice $I_h$ simulations, a large spike in C$_\text{p}$ occurs |
450 |
|
|
at 245 K, indicating a first order phase transition for the melting of |
451 |
|
|
these ice crystals. When the reaction field is turned off, the melting |
452 |
|
|
transition occurs at 235 K. These melting transitions are |
453 |
|
|
considerably lower than the experimental value, but this is not a |
454 |
|
|
surprise considering the simplicity of the SSD model. |
455 |
chrisfen |
743 |
|
456 |
chrisfen |
862 |
\begin{figure} |
457 |
|
|
\begin{center} |
458 |
|
|
\epsfxsize=6in |
459 |
|
|
\epsfbox{corrDiag.eps} |
460 |
|
|
\caption{Two dimensional illustration of angles involved in the |
461 |
|
|
correlations observed in figure \ref{contour}.} |
462 |
|
|
\label{corrAngle} |
463 |
|
|
\end{center} |
464 |
|
|
\end{figure} |
465 |
|
|
|
466 |
|
|
\begin{figure} |
467 |
|
|
\begin{center} |
468 |
|
|
\epsfxsize=6in |
469 |
|
|
\epsfbox{fullContours.eps} |
470 |
chrisfen |
743 |
\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at |
471 |
|
|
100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for |
472 |
|
|
clarity: dark areas signify peaks while light areas signify |
473 |
|
|
depressions. White areas have g(\emph{r}) values below 0.5 and black |
474 |
|
|
areas have values above 1.5.} |
475 |
|
|
\label{contour} |
476 |
chrisfen |
862 |
\end{center} |
477 |
chrisfen |
743 |
\end{figure} |
478 |
|
|
|
479 |
chrisfen |
861 |
Additional analysis of the melting phase-transition process was |
480 |
|
|
performed by using two-dimensional structure and dipole angle |
481 |
|
|
correlations. Expressions for these correlations are as follows: |
482 |
|
|
|
483 |
chrisfen |
862 |
\begin{equation} |
484 |
|
|
g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|\mathbf{r}_{ij}\right|)\rangle\ , |
485 |
|
|
\end{equation} |
486 |
|
|
\begin{equation} |
487 |
|
|
g_{\text{AB}}(r,\cos\omega) = |
488 |
chrisfen |
743 |
\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\ , |
489 |
chrisfen |
862 |
\end{equation} |
490 |
chrisfen |
861 |
where $\theta$ and $\omega$ refer to the angles shown in figure |
491 |
|
|
\ref{corrAngle}. By binning over both distance and the cosine of the |
492 |
chrisfen |
743 |
desired angle between the two dipoles, the g(\emph{r}) can be |
493 |
|
|
dissected to determine the common dipole arrangements that constitute |
494 |
|
|
the peaks and troughs. Frames A and B of figure \ref{contour} show a |
495 |
|
|
relatively crystalline state of an ice $I_c$ simulation. The first |
496 |
chrisfen |
861 |
peak of the g(\emph{r}) consists primarily of the preferred hydrogen |
497 |
|
|
bonding arrangements as dictated by the tetrahedral sticky potential - |
498 |
chrisfen |
743 |
one peak for the donating and the other for the accepting hydrogen |
499 |
|
|
bonds. Due to the high degree of crystallinity of the sample, the |
500 |
|
|
second and third solvation shells show a repeated peak arrangement |
501 |
|
|
which decays at distances around the fourth solvation shell, near the |
502 |
|
|
imposed cutoff for the Lennard-Jones and dipole-dipole interactions. |
503 |
chrisfen |
861 |
In the higher temperature simulation shown in frames C and D, these |
504 |
|
|
longer-ranged repeated peak features deteriorate rapidly. The first |
505 |
|
|
solvation shell still shows the strong effect of the sticky-potential, |
506 |
|
|
although it covers a larger area, extending to include a fraction of |
507 |
|
|
aligned dipole peaks within the first solvation shell. The latter |
508 |
|
|
peaks lose definition as thermal motion and the competing dipole force |
509 |
|
|
overcomes the sticky potential's tight tetrahedral structuring of the |
510 |
|
|
fluid. |
511 |
chrisfen |
743 |
|
512 |
|
|
This complex interplay between dipole and sticky interactions was |
513 |
|
|
remarked upon as a possible reason for the split second peak in the |
514 |
|
|
oxygen-oxygen g(\emph{r}).\cite{Ichiye96} At low temperatures, the |
515 |
chrisfen |
862 |
second solvation shell peak appears to have two distinct components |
516 |
|
|
that blend together to form one observable peak. At higher |
517 |
|
|
temperatures, this split character alters to show the leading 4 \AA\ |
518 |
|
|
peak dominated by equatorial anti-parallel dipole orientations. There |
519 |
|
|
is also a tightly bunched group of axially arranged dipoles that most |
520 |
|
|
likely consist of the smaller fraction of aligned dipole pairs. The |
521 |
|
|
trailing component of the split peak at 5 \AA\ is dominated by aligned |
522 |
|
|
dipoles that assume hydrogen bond arrangements similar to those seen |
523 |
|
|
in the first solvation shell. This evidence indicates that the dipole |
524 |
|
|
pair interaction begins to dominate outside of the range of the |
525 |
|
|
dipolar repulsion term. Primary energetically favorable dipole |
526 |
|
|
arrangements populate the region immediately outside this repulsion |
527 |
|
|
region (around 4 \AA), while arrangements that seek to ideally satisfy |
528 |
|
|
both the sticky and dipole forces locate themselves just beyond this |
529 |
|
|
initial buildup (around 5 \AA). |
530 |
chrisfen |
743 |
|
531 |
|
|
From these findings, the split second peak is primarily the product of |
532 |
chrisfen |
861 |
the dipolar repulsion term of the sticky potential. In fact, the inner |
533 |
|
|
peak can be pushed out and merged with the outer split peak just by |
534 |
|
|
extending the switching function cutoff ($s^\prime(r_{ij})$) from its |
535 |
|
|
normal 4.0 \AA\ to values of 4.5 or even 5 \AA. This type of |
536 |
|
|
correction is not recommended for improving the liquid structure, |
537 |
chrisfen |
862 |
since the second solvation shell would still be shifted too far |
538 |
chrisfen |
861 |
out. In addition, this would have an even more detrimental effect on |
539 |
|
|
the system densities, leading to a liquid with a more open structure |
540 |
|
|
and a density considerably lower than the normal SSD behavior shown |
541 |
|
|
previously. A better correction would be to include the |
542 |
|
|
quadrupole-quadrupole interactions for the water particles outside of |
543 |
|
|
the first solvation shell, but this reduces the simplicity and speed |
544 |
|
|
advantage of SSD. |
545 |
chrisfen |
743 |
|
546 |
chrisfen |
861 |
\subsection{Adjusted Potentials: SSD/RF and SSD/E} |
547 |
chrisfen |
743 |
The propensity of SSD to adopt lower than expected densities under |
548 |
|
|
varying conditions is troubling, especially at higher temperatures. In |
549 |
chrisfen |
861 |
order to correct this model for use with a reaction field, it is |
550 |
|
|
necessary to adjust the force field parameters for the primary |
551 |
|
|
intermolecular interactions. In undergoing a reparameterization, it is |
552 |
|
|
important not to focus on just one property and neglect the other |
553 |
|
|
important properties. In this case, it would be ideal to correct the |
554 |
|
|
densities while maintaining the accurate transport properties. |
555 |
chrisfen |
743 |
|
556 |
chrisfen |
862 |
The parameters available for tuning include the $\sigma$ and $\epsilon$ |
557 |
chrisfen |
743 |
Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky |
558 |
|
|
attractive and dipole repulsive terms with their respective |
559 |
|
|
cutoffs. To alter the attractive and repulsive terms of the sticky |
560 |
|
|
potential independently, it is necessary to separate the terms as |
561 |
|
|
follows: |
562 |
|
|
\begin{equation} |
563 |
|
|
u_{ij}^{sp} |
564 |
chrisfen |
862 |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = |
565 |
|
|
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)] + \frac{\nu_0^\prime}{2} [s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)], |
566 |
chrisfen |
743 |
\end{equation} |
567 |
|
|
|
568 |
|
|
where $\nu_0$ scales the strength of the tetrahedral attraction and |
569 |
|
|
$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion |
570 |
chrisfen |
862 |
term. The separation was performed for purposes of the |
571 |
|
|
reparameterization, but the final parameters were adjusted so that it |
572 |
|
|
is unnecessary to separate the terms when implementing the adjusted |
573 |
|
|
water potentials. The results of the reparameterizations are shown in |
574 |
|
|
table \ref{params}. Note that the tetrahedral attractive and dipolar |
575 |
|
|
repulsive terms do not share the same lower cutoff ($r_l$) in the |
576 |
|
|
newly parameterized potentials - soft sticky dipole reaction field |
577 |
|
|
(SSD/RF - for use with a reaction field) and soft sticky dipole |
578 |
|
|
enhanced (SSD/E - an attempt to improve the liquid structure in |
579 |
|
|
simulations without a long-range correction). |
580 |
chrisfen |
743 |
|
581 |
|
|
\begin{table} |
582 |
chrisfen |
862 |
\begin{center} |
583 |
chrisfen |
743 |
\caption{Parameters for the original and adjusted models} |
584 |
chrisfen |
856 |
\begin{tabular}{ l c c c c } |
585 |
chrisfen |
743 |
\hline \\[-3mm] |
586 |
chrisfen |
862 |
\ \ \ Parameters\ \ \ & \ \ \ SSD\cite{Ichiye96} \ \ \ & \ SSD1\cite{Ichiye03}\ \ & \ SSD/E\ \ & \ SSD/RF \\ |
587 |
chrisfen |
743 |
\hline \\[-3mm] |
588 |
chrisfen |
856 |
\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
589 |
|
|
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
590 |
|
|
\ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\ |
591 |
|
|
\ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
592 |
|
|
\ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\ |
593 |
|
|
\ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\ |
594 |
|
|
\ \ \ $\nu_0^\prime$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
595 |
|
|
\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\ |
596 |
|
|
\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\ |
597 |
chrisfen |
743 |
\end{tabular} |
598 |
|
|
\label{params} |
599 |
chrisfen |
862 |
\end{center} |
600 |
chrisfen |
743 |
\end{table} |
601 |
|
|
|
602 |
chrisfen |
862 |
\begin{figure} |
603 |
|
|
\begin{center} |
604 |
|
|
\epsfxsize=5in |
605 |
|
|
\epsfbox{GofRCompare.epsi} |
606 |
chrisfen |
743 |
\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E |
607 |
chrisfen |
856 |
and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with |
608 |
chrisfen |
743 |
reaction field turned on (bottom). The insets show the respective |
609 |
chrisfen |
862 |
first peaks in detail. Note how the changes in parameters have lowered |
610 |
|
|
and broadened the first peak of SSD/E and SSD/RF.} |
611 |
chrisfen |
743 |
\label{grcompare} |
612 |
chrisfen |
862 |
\end{center} |
613 |
chrisfen |
743 |
\end{figure} |
614 |
|
|
|
615 |
chrisfen |
862 |
\begin{figure} |
616 |
|
|
\begin{center} |
617 |
|
|
\epsfxsize=6in |
618 |
|
|
\epsfbox{dualsticky.ps} |
619 |
chrisfen |
856 |
\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \& |
620 |
chrisfen |
743 |
SSD/RF (right). Light areas correspond to the tetrahedral attractive |
621 |
chrisfen |
862 |
component, and darker areas correspond to the dipolar repulsive |
622 |
|
|
component.} |
623 |
chrisfen |
743 |
\label{isosurface} |
624 |
chrisfen |
862 |
\end{center} |
625 |
chrisfen |
743 |
\end{figure} |
626 |
|
|
|
627 |
|
|
In the paper detailing the development of SSD, Liu and Ichiye placed |
628 |
|
|
particular emphasis on an accurate description of the first solvation |
629 |
chrisfen |
862 |
shell. This resulted in a somewhat tall and narrow first peak in the |
630 |
|
|
g(\emph{r}) that integrated to give similar coordination numbers to |
631 |
|
|
the experimental data obtained by Soper and |
632 |
|
|
Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering |
633 |
|
|
data from the Head-Gordon lab indicates a slightly lower and shifted |
634 |
|
|
first peak in the g$_\mathrm{OO}(r)$, so adjustments to SSD were made |
635 |
|
|
while taking into consideration the new experimental |
636 |
|
|
findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the |
637 |
|
|
relocation of the first peak of the oxygen-oxygen g(\emph{r}) by |
638 |
|
|
comparing the revised SSD model (SSD1), SSD-E, and SSD-RF to the new |
639 |
|
|
experimental results. Both modified water models have shorter peaks |
640 |
|
|
that are brought in more closely to the experimental peak (as seen in |
641 |
|
|
the insets of figure \ref{grcompare}). This structural alteration was |
642 |
|
|
accomplished by the combined reduction in the Lennard-Jones $\sigma$ |
643 |
|
|
variable and adjustment of the sticky potential strength and |
644 |
|
|
cutoffs. As can be seen in table \ref{params}, the cutoffs for the |
645 |
|
|
tetrahedral attractive and dipolar repulsive terms were nearly swapped |
646 |
|
|
with each other. Isosurfaces of the original and modified sticky |
647 |
|
|
potentials are shown in figure \ref{isosurface}. In these isosurfaces, |
648 |
|
|
it is easy to see how altering the cutoffs changes the repulsive and |
649 |
|
|
attractive character of the particles. With a reduced repulsive |
650 |
|
|
surface (darker region), the particles can move closer to one another, |
651 |
|
|
increasing the density for the overall system. This change in |
652 |
|
|
interaction cutoff also results in a more gradual orientational motion |
653 |
|
|
by allowing the particles to maintain preferred dipolar arrangements |
654 |
|
|
before they begin to feel the pull of the tetrahedral |
655 |
|
|
restructuring. As the particles move closer together, the dipolar |
656 |
|
|
repulsion term becomes active and excludes unphysical nearest-neighbor |
657 |
|
|
arrangements. This compares with how SSD and SSD1 exclude preferred |
658 |
|
|
dipole alignments before the particles feel the pull of the ``hydrogen |
659 |
|
|
bonds''. Aside from improving the shape of the first peak in the |
660 |
|
|
g(\emph{r}), this modification improves the densities considerably by |
661 |
chrisfen |
861 |
allowing the persistence of full dipolar character below the previous |
662 |
|
|
4.0 \AA\ cutoff. |
663 |
chrisfen |
743 |
|
664 |
|
|
While adjusting the location and shape of the first peak of |
665 |
chrisfen |
861 |
g(\emph{r}) improves the densities, these changes alone are |
666 |
|
|
insufficient to bring the system densities up to the values observed |
667 |
chrisfen |
862 |
experimentally. To further increase the densities, the dipole moments |
668 |
|
|
were increased in both of the adjusted models. Since SSD is a dipole |
669 |
chrisfen |
861 |
based model, the structure and transport are very sensitive to changes |
670 |
|
|
in the dipole moment. The original SSD simply used the dipole moment |
671 |
|
|
calculated from the TIP3P water model, which at 2.35 D is |
672 |
chrisfen |
743 |
significantly greater than the experimental gas phase value of 1.84 |
673 |
chrisfen |
861 |
D. The larger dipole moment is a more realistic value and improves the |
674 |
chrisfen |
743 |
dielectric properties of the fluid. Both theoretical and experimental |
675 |
|
|
measurements indicate a liquid phase dipole moment ranging from 2.4 D |
676 |
chrisfen |
862 |
to values as high as 3.11 D, providing a substantial range of |
677 |
|
|
reasonable values for a dipole |
678 |
|
|
moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
679 |
|
|
increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF, |
680 |
|
|
respectively, leads to significant changes in the density and |
681 |
|
|
transport of the water models. |
682 |
chrisfen |
743 |
|
683 |
chrisfen |
861 |
In order to demonstrate the benefits of these reparameterizations, a |
684 |
chrisfen |
743 |
series of NPT and NVE simulations were performed to probe the density |
685 |
|
|
and transport properties of the adapted models and compare the results |
686 |
|
|
to the original SSD model. This comparison involved full NPT melting |
687 |
|
|
sequences for both SSD/E and SSD/RF, as well as NVE transport |
688 |
chrisfen |
861 |
calculations at the calculated self-consistent densities. Again, the |
689 |
chrisfen |
862 |
results are obtained from five separate simulations of 1024 particle |
690 |
|
|
systems, and the melting sequences were started from different ice |
691 |
|
|
$I_h$ crystals constructed as described previously. Each NPT |
692 |
chrisfen |
861 |
simulation was equilibrated for 100 ps before a 200 ps data collection |
693 |
chrisfen |
862 |
run at each temperature step, and the final configuration from the |
694 |
|
|
previous temperature simulation was used as a starting point. All NVE |
695 |
|
|
simulations had the same thermalization, equilibration, and data |
696 |
|
|
collection times as stated earlier in this paper. |
697 |
chrisfen |
743 |
|
698 |
chrisfen |
862 |
\begin{figure} |
699 |
|
|
\begin{center} |
700 |
|
|
\epsfxsize=6in |
701 |
|
|
\epsfbox{ssdeDense.epsi} |
702 |
chrisfen |
861 |
\caption{Comparison of densities calculated with SSD/E to SSD1 without a |
703 |
chrisfen |
862 |
reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00} |
704 |
|
|
SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The window shows a |
705 |
chrisfen |
856 |
expansion around 300 K with error bars included to clarify this region |
706 |
|
|
of interest. Note that both SSD1 and SSD/E show good agreement with |
707 |
|
|
experiment when the long-range correction is neglected.} |
708 |
chrisfen |
743 |
\label{ssdedense} |
709 |
chrisfen |
862 |
\end{center} |
710 |
chrisfen |
743 |
\end{figure} |
711 |
|
|
|
712 |
chrisfen |
861 |
Figure \ref{ssdedense} shows the density profile for the SSD/E model |
713 |
chrisfen |
862 |
in comparison to SSD1 without a reaction field, other common water |
714 |
|
|
models, and experimental results. The calculated densities for both |
715 |
|
|
SSD/E and SSD1 have increased significantly over the original SSD |
716 |
|
|
model (see figure \ref{dense1}) and are in better agreement with the |
717 |
|
|
experimental values. At 298 K, the densities of SSD/E and SSD1 without |
718 |
|
|
a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and |
719 |
|
|
0.999$\pm$0.001 g/cm$^3$ respectively. These both compare well with |
720 |
|
|
the experimental value of 0.997 g/cm$^3$, and they are considerably |
721 |
|
|
better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The changes to |
722 |
|
|
the dipole moment and sticky switching functions have improved the |
723 |
|
|
structuring of the liquid (as seen in figure \ref{grcompare}, but they |
724 |
|
|
have shifted the density maximum to much lower temperatures. This |
725 |
|
|
comes about via an increase in the liquid disorder through the |
726 |
|
|
weakening of the sticky potential and strengthening of the dipolar |
727 |
|
|
character. However, this increasing disorder in the SSD/E model has |
728 |
|
|
little effect on the melting transition. By monitoring C$\text{p}$ |
729 |
|
|
throughout these simulations, the melting transition for SSD/E was |
730 |
|
|
shown to occur at 235 K, the same transition temperature observed with |
731 |
|
|
SSD and SSD1. |
732 |
chrisfen |
743 |
|
733 |
chrisfen |
862 |
\begin{figure} |
734 |
|
|
\begin{center} |
735 |
|
|
\epsfxsize=6in |
736 |
|
|
\epsfbox{ssdrfDense.epsi} |
737 |
chrisfen |
861 |
\caption{Comparison of densities calculated with SSD/RF to SSD1 with a |
738 |
chrisfen |
862 |
reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00} |
739 |
|
|
SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The inset shows the |
740 |
chrisfen |
856 |
necessity of reparameterization when utilizing a reaction field |
741 |
|
|
long-ranged correction - SSD/RF provides significantly more accurate |
742 |
|
|
densities than SSD1 when performing room temperature simulations.} |
743 |
chrisfen |
743 |
\label{ssdrfdense} |
744 |
chrisfen |
862 |
\end{center} |
745 |
chrisfen |
743 |
\end{figure} |
746 |
|
|
|
747 |
chrisfen |
862 |
Including the reaction field long-range correction in the simulations |
748 |
|
|
results in a more interesting comparison. A density profile including |
749 |
|
|
SSD/RF and SSD1 with an active reaction field is shown in figure |
750 |
|
|
\ref{ssdrfdense}. As observed in the simulations without a reaction |
751 |
|
|
field, the densities of SSD/RF and SSD1 show a dramatic increase over |
752 |
|
|
normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density |
753 |
|
|
of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and |
754 |
|
|
considerably better than the SSD value of 0.941$\pm$0.001 g/cm$^3$ and |
755 |
|
|
the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results further |
756 |
|
|
emphasize the importance of reparameterization in order to model the |
757 |
|
|
density properly under different simulation conditions. Again, these |
758 |
|
|
changes have only a minor effect on the melting point, which observed |
759 |
|
|
at 245 K for SSD/RF, is identical to SSD and only 5 K lower than SSD1 |
760 |
|
|
with a reaction field. Additionally, the difference in density maxima |
761 |
|
|
is not as extreme, with SSD/RF showing a density maximum at 255 K, |
762 |
|
|
fairly close to the density maxima of 260 K and 265 K, shown by SSD |
763 |
|
|
and SSD1 respectively. |
764 |
chrisfen |
743 |
|
765 |
chrisfen |
862 |
\begin{figure} |
766 |
|
|
\begin{center} |
767 |
|
|
\epsfxsize=6in |
768 |
|
|
\epsfbox{ssdeDiffuse.epsi} |
769 |
chrisfen |
861 |
\caption{Plots of the diffusion constants calculated from SSD/E and SSD1, |
770 |
chrisfen |
862 |
both without a reaction field, along with experimental |
771 |
|
|
results.\cite{Gillen72,Mills73} The NVE calculations were performed |
772 |
|
|
at the average densities observed in the 1 atm NPT simulations for |
773 |
|
|
the respective models. SSD/E is slightly more fluid than experiment |
774 |
|
|
at all of the temperatures, but it is closer than SSD1 without a |
775 |
|
|
long-range correction.} |
776 |
chrisfen |
861 |
\label{ssdediffuse} |
777 |
chrisfen |
862 |
\end{center} |
778 |
chrisfen |
861 |
\end{figure} |
779 |
|
|
|
780 |
chrisfen |
743 |
The reparameterization of the SSD water model, both for use with and |
781 |
|
|
without an applied long-range correction, brought the densities up to |
782 |
|
|
what is expected for simulating liquid water. In addition to improving |
783 |
|
|
the densities, it is important that particle transport be maintained |
784 |
|
|
or improved. Figure \ref{ssdediffuse} compares the temperature |
785 |
chrisfen |
861 |
dependence of the diffusion constant of SSD/E to SSD1 without an |
786 |
|
|
active reaction field, both at the densities calculated at 1 atm and |
787 |
|
|
at the experimentally calculated densities for super-cooled and liquid |
788 |
chrisfen |
862 |
water. The diffusion constant for SSD/E is consistently a little |
789 |
|
|
higher than experiment, while SSD1 remains lower than experiment until |
790 |
|
|
relatively high temperatures (greater than 330 K). Both models follow |
791 |
|
|
the shape of the experimental curve well below 300 K but tend to |
792 |
|
|
diffuse too rapidly at higher temperatures, something that is |
793 |
|
|
especially apparent with SSD1. This accelerated increasing of |
794 |
|
|
diffusion is caused by the rapidly decreasing system density with |
795 |
|
|
increasing temperature. Though it is difficult to see in figure |
796 |
|
|
\ref{ssdedense}, the densities of SSD1 decay more rapidly with |
797 |
|
|
temperature than do those of SSD/E, leading to more visible deviation |
798 |
|
|
from the experimental diffusion trend. Thus, the changes made to |
799 |
|
|
improve the liquid structure may have had an adverse affect on the |
800 |
|
|
density maximum, but they improve the transport behavior of SSD/E |
801 |
|
|
relative to SSD1. |
802 |
chrisfen |
743 |
|
803 |
chrisfen |
862 |
\begin{figure} |
804 |
|
|
\begin{center} |
805 |
|
|
\epsfxsize=6in |
806 |
|
|
\epsfbox{ssdrfDiffuse.epsi} |
807 |
chrisfen |
856 |
\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1, |
808 |
chrisfen |
862 |
both with an active reaction field, along with experimental |
809 |
|
|
results.\cite{Gillen72,Mills73} The NVE calculations were performed |
810 |
|
|
at the average densities observed in the 1 atm NPT simulations for |
811 |
|
|
both of the models. Note how accurately SSD/RF simulates the |
812 |
|
|
diffusion of water throughout this temperature range. The more |
813 |
|
|
rapidly increasing diffusion constants at high temperatures for both |
814 |
|
|
models is attributed to the significantly lower densities than |
815 |
|
|
observed in experiment.} |
816 |
chrisfen |
856 |
\label{ssdrfdiffuse} |
817 |
chrisfen |
862 |
\end{center} |
818 |
chrisfen |
743 |
\end{figure} |
819 |
|
|
|
820 |
|
|
In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are |
821 |
chrisfen |
862 |
compared to SSD1 with an active reaction field. Note that SSD/RF |
822 |
chrisfen |
861 |
tracks the experimental results incredibly well, identical within |
823 |
chrisfen |
862 |
error throughout the temperature range shown and with only a slight |
824 |
chrisfen |
861 |
increasing trend at higher temperatures. SSD1 tends to diffuse more |
825 |
|
|
slowly at low temperatures and deviates to diffuse too rapidly at |
826 |
chrisfen |
862 |
temperatures greater than 330 K. As stated in relation to SSD/E, this |
827 |
|
|
deviation away from the ideal trend is due to a rapid decrease in |
828 |
|
|
density at higher temperatures. SSD/RF does not suffer from this |
829 |
|
|
problem as much as SSD1, because the calculated densities are closer |
830 |
|
|
to the experimental value. These results again emphasize the |
831 |
|
|
importance of careful reparameterization when using an altered |
832 |
chrisfen |
861 |
long-range correction. |
833 |
chrisfen |
743 |
|
834 |
|
|
\subsection{Additional Observations} |
835 |
|
|
|
836 |
|
|
\begin{figure} |
837 |
chrisfen |
862 |
\begin{center} |
838 |
|
|
\epsfxsize=6in |
839 |
|
|
\epsfbox{povIce.ps} |
840 |
|
|
\caption{A water lattice built from the crystal structure assumed by |
841 |
|
|
SSD/E when undergoing an extremely restricted temperature NPT |
842 |
chrisfen |
861 |
simulation. This form of ice is referred to as ice 0 to emphasize its |
843 |
|
|
simulation origins. This image was taken of the (001) face of the |
844 |
|
|
crystal.} |
845 |
chrisfen |
743 |
\label{weirdice} |
846 |
chrisfen |
862 |
\end{center} |
847 |
chrisfen |
743 |
\end{figure} |
848 |
|
|
|
849 |
chrisfen |
861 |
While performing restricted temperature melting sequences of SSD/E not |
850 |
chrisfen |
862 |
previously discussed, some interesting observations were made. After |
851 |
|
|
melting at 235 K, two of five systems underwent crystallization events |
852 |
|
|
near 245 K. As the heating process continued, the two systems remained |
853 |
|
|
crystalline until finally melting between 320 and 330 K. The final |
854 |
|
|
configurations of these two melting sequences show an expanded |
855 |
|
|
zeolite-like crystal structure that does not correspond to any known |
856 |
|
|
form of ice. For convenience, and to help distinguish it from the |
857 |
|
|
experimentally observed forms of ice, this crystal structure will |
858 |
|
|
henceforth be referred to as ice-zero (ice 0). The crystallinity was |
859 |
|
|
extensive enough that a near ideal crystal structure of ice 0 could be |
860 |
|
|
obtained. Figure \ref{weirdice} shows the repeating crystal structure |
861 |
|
|
of a typical crystal at 5 K. Each water molecule is hydrogen bonded to |
862 |
|
|
four others; however, the hydrogen bonds are flexed rather than |
863 |
|
|
perfectly straight. This results in a skewed tetrahedral geometry |
864 |
|
|
about the central molecule. Referring to figure \ref{isosurface}, |
865 |
|
|
these flexed hydrogen bonds are allowed due to the conical shape of |
866 |
|
|
the attractive regions, with the greatest attraction along the direct |
867 |
|
|
hydrogen bond configuration. Though not ideal, these flexed hydrogen |
868 |
|
|
bonds are favorable enough to stabilize an entire crystal generated |
869 |
|
|
around them. In fact, the imperfect ice 0 crystals were so stable that |
870 |
|
|
they melted at temperatures nearly 100 K greater than both ice I$_c$ |
871 |
|
|
and I$_h$. |
872 |
chrisfen |
743 |
|
873 |
chrisfen |
861 |
These initial simulations indicated that ice 0 is the preferred ice |
874 |
chrisfen |
862 |
structure for at least the SSD/E model. To verify this, a comparison |
875 |
|
|
was made between near ideal crystals of ice $I_h$, ice $I_c$, and ice |
876 |
|
|
0 at constant pressure with SSD/E, SSD/RF, and SSD1. Near ideal |
877 |
|
|
versions of the three types of crystals were cooled to 1 K, and the |
878 |
|
|
potential energies of each were compared using all three water |
879 |
|
|
models. With every water model, ice 0 turned out to have the lowest |
880 |
|
|
potential energy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with |
881 |
|
|
SSD/E, and 7.5\% lower with SSD/RF. |
882 |
chrisfen |
743 |
|
883 |
chrisfen |
861 |
In addition to these low temperature comparisons, melting sequences |
884 |
|
|
were performed with ice 0 as the initial configuration using SSD/E, |
885 |
|
|
SSD/RF, and SSD1 both with and without a reaction field. The melting |
886 |
|
|
transitions for both SSD/E and SSD1 without a reaction field occurred |
887 |
|
|
at temperature in excess of 375 K. SSD/RF and SSD1 with a reaction |
888 |
chrisfen |
862 |
field showed more reasonable melting transitions near 325 K. These |
889 |
|
|
melting point observations emphasize the preference for this crystal |
890 |
|
|
structure over the most common types of ice when using these single |
891 |
chrisfen |
743 |
point water models. |
892 |
|
|
|
893 |
|
|
Recognizing that the above tests show ice 0 to be both the most stable |
894 |
|
|
and lowest density crystal structure for these single point water |
895 |
chrisfen |
861 |
models, it is interesting to speculate on the relative stability of |
896 |
|
|
this crystal structure with charge based water models. As a quick |
897 |
chrisfen |
743 |
test, these 3 crystal types were converted from SSD type particles to |
898 |
|
|
TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy |
899 |
chrisfen |
862 |
minimizations were performed on the crystals to compare the system |
900 |
|
|
energies. Again, ice 0 was observed to have the lowest total system |
901 |
|
|
energy. The total energy of ice 0 was ~2\% lower than ice $I_h$, which |
902 |
|
|
was in turn ~3\% lower than ice $I_c$. Based on these initial studies, |
903 |
|
|
it would not be surprising if results from the other common water |
904 |
|
|
models show ice 0 to be the lowest energy crystal structure. A |
905 |
|
|
continuation of this work studying ice 0 with multi-point water models |
906 |
|
|
will be published in a coming article. |
907 |
chrisfen |
743 |
|
908 |
|
|
\section{Conclusions} |
909 |
|
|
The density maximum and temperature dependent transport for the SSD |
910 |
|
|
water model, both with and without the use of reaction field, were |
911 |
|
|
studied via a series of NPT and NVE simulations. The constant pressure |
912 |
|
|
simulations of the melting of both $I_h$ and $I_c$ ice showed a |
913 |
|
|
density maximum near 260 K. In most cases, the calculated densities |
914 |
|
|
were significantly lower than the densities calculated in simulations |
915 |
|
|
of other water models. Analysis of particle diffusion showed SSD to |
916 |
chrisfen |
862 |
capture the transport properties of experimental water well in both |
917 |
|
|
the liquid and super-cooled liquid regimes. In order to correct the |
918 |
chrisfen |
861 |
density behavior, the original SSD model was reparameterized for use |
919 |
|
|
both with and without a reaction field (SSD/RF and SSD/E), and |
920 |
|
|
comparison simulations were performed with SSD1, the density corrected |
921 |
|
|
version of SSD. Both models improve the liquid structure, density |
922 |
|
|
values, and diffusive properties under their respective conditions, |
923 |
|
|
indicating the necessity of reparameterization when altering the |
924 |
chrisfen |
862 |
long-range correction specifics. When taking into account the |
925 |
|
|
appropriate considerations, these simple water models are excellent |
926 |
|
|
choices for representing explicit water in large scale simulations of |
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biochemical systems. |
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chrisfen |
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\section{Acknowledgments} |
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chrisfen |
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Support for this project was provided by the National Science |
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Foundation under grant CHE-0134881. Computation time was provided by |
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the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
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DMR 00 79647. |
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\newpage |
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\bibliographystyle{jcp} |
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\bibliography{nptSSD} |
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%\pagebreak |
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\end{document} |