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\begin{document} |
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\title{On the temperature dependent properties of the soft sticky dipole (SSD) and related single point water models} |
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\author{Christopher J. Fennell and J. Daniel Gezelter{\thefootnote} |
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\footnote[1]{Corresponding author. \ Electronic mail: gezelter@nd.edu}} |
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\address{Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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\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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maintain or improve upon the structural and transport properties. |
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\end{abstract} |
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\maketitle |
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\newpage |
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%\narrowtext |
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\section{Introduction} |
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|
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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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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 |
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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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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 accuracy of system |
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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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|
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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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\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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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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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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(\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, $s$ 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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$\nu_0$ 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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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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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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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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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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a cutoff that lacks any form of long-range correction. This study |
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addresses these issues by looking at the structural and transport |
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behavior of SSD over a variety of temperatures, with the purpose of |
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utilizing the RF correction technique. Toward the end, we suggest |
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alterations to the parameters that result in more water-like |
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behavior. It should be noted that in a recent publication, some the |
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original investigators of the SSD water model have put forth |
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adjustments to the SSD water model to address abnormal density |
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behavior (also observed here), calling the corrected model |
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SSD1.\cite{Ichiye03} This study will make comparisons with this new |
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model's behavior with the goal of improving upon the depiction of |
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water under conditions without the Ewald Sum. |
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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, 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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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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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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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 proposed on how to make |
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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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Simulations were performed in both the isobaric-isothermal and |
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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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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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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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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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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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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 end up being more costly computationally than the |
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simpler arithmetic quaternion propagation. With the same time step, a |
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1000 SSD particle simulation shows an average 7\% increase in |
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computation time using the symplectic step method in place of |
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quaternions. This cost is more than justified when comparing the |
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energy conservation of the two methods as illustrated in figure |
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\ref{timestep}. |
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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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\includegraphics[width=61mm, angle=-90]{timeStep.epsi} |
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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. 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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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 for |
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handling rotational motion. At time steps of 0.1 and 0.5 fs, both |
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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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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 these SSD particle simulations was unnoticeable for |
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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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constant pressure simulations as well. |
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|
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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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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 down to 25 K. The particles |
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were then allowed translate with fixed orientations at a constant |
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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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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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\section{Results and discussion} |
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|
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Melting studies were performed on the randomized ice crystals using |
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constant pressure and temperature dynamics. By performing melting |
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simulations, the melting transition can be determined by monitoring |
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the heat capacity, in addition to determining the density maximum - |
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provided that the density maximum occurs in the liquid and not the |
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supercooled regime. An ensemble average from five separate melting |
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simulations was acquired, each starting from different ice crystals |
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generated as described previously. All simulations were equilibrated |
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for 100 ps prior to a 200 ps data collection run at each temperature |
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setting. The temperature range of study spanned from 25 to 400 K, with |
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a maximum degree increment of 25 K. For regions of interest along this |
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stepwise progression, the temperature increment was decreased from 25 |
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K to 10 and 5 K. The above equilibration and production times were |
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sufficient in that the system volume fluctuations dampened out in all |
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but the very cold simulations (below 225 K). |
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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 |
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production times were sufficient in that the system volume |
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fluctuations dampened out in all but the very cold simulations (below |
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225 K). |
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|
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\subsection{Density Behavior} |
| 316 |
|
Initial simulations focused on the original SSD water model, and an |
| 321 |
|
of the average value at 260 K makes a good argument for the actual |
| 322 |
|
density maximum residing at this midpoint value. Figure \ref{dense1} |
| 323 |
|
was constructed using ice $I_h$ crystals for the initial |
| 324 |
< |
configuration; and though not pictured, the simulations starting from |
| 325 |
< |
ice $I_c$ crystal configurations showed similar results, with a |
| 324 |
> |
configuration; though not pictured, the simulations starting from ice |
| 325 |
> |
$I_c$ crystal configurations showed similar results, with a |
| 326 |
|
liquid-phase density maximum in this same region (between 255 and 260 |
| 327 |
|
K). In addition, the $I_c$ crystals are more fragile than the $I_h$ |
| 328 |
< |
crystals, leading them to deform into a dense glassy state at lower |
| 328 |
> |
crystals, leading to deformation into a dense glassy state at lower |
| 329 |
|
temperatures. This resulted in an overall low temperature density |
| 330 |
< |
maximum at 200 K, but they still retained a common liquid state |
| 331 |
< |
density maximum with the $I_h$ simulations. |
| 330 |
> |
maximum at 200 K, while still retaining a liquid state density maximum |
| 331 |
> |
in common with the $I_h$ simulations. |
| 332 |
|
|
| 333 |
|
\begin{figure} |
| 334 |
< |
\includegraphics[width=65mm,angle=-90]{dense2.eps} |
| 335 |
< |
\caption{Density versus temperature for TIP4P\cite{Jorgensen98b}, |
| 336 |
< |
TIP3P\cite{Jorgensen98b}, SPC/E\cite{Clancy94}, SSD without Reaction |
| 337 |
< |
Field, SSD, and Experiment\cite{CRC80}. The arrows indicate the |
| 334 |
> |
\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 |
|
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 |
|
\label{dense1} |
| 344 |
+ |
\end{center} |
| 345 |
|
\end{figure} |
| 346 |
|
|
| 347 |
|
The density maximum for SSD actually compares quite favorably to other |
| 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 |
< |
the resulting densities overlapped within error and showed no |
| 364 |
< |
significant trend in lower or higher densities as a function of cutoff |
| 365 |
< |
radius, both for simulations with and without reaction field. These |
| 366 |
< |
results indicate that there is no major benefit in choosing a longer |
| 367 |
< |
cutoff radius in simulations using SSD. This is comforting in that the |
| 368 |
< |
use of a longer cutoff radius results in significant increases in the |
| 369 |
< |
time required to obtain a single trajectory. |
| 363 |
> |
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 |
|
|
| 372 |
< |
The most important thing to recognize in figure \ref{dense1} is the |
| 373 |
< |
density scaling of SSD relative to other common models at any given |
| 372 |
> |
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 |
|
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 |
< |
expected densities have been observed for other systems with the use |
| 378 |
< |
of a reaction field for long-range electrostatic interactions, so the |
| 379 |
< |
most likely reason for these significantly lower densities in these |
| 377 |
> |
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 |
|
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 |
< |
reaction field, these densities increase considerably to more |
| 385 |
> |
reaction field, the densities increase considerably to more |
| 386 |
|
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 |
< |
in the density maximum location, down to 245 K, is observed. This is |
| 391 |
< |
probably a more accurate comparison to the other listed water models, |
| 392 |
< |
in that no long range corrections were applied in those |
| 390 |
> |
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 |
|
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 |
| 399 |
|
|
| 400 |
|
\subsection{Transport Behavior} |
| 401 |
|
Of importance in these types of studies are the transport properties |
| 402 |
< |
of the particles and how they change when altering the environmental |
| 403 |
< |
conditions. In order to probe transport, constant energy simulations |
| 404 |
< |
were performed about the average density uncovered by the constant |
| 405 |
< |
pressure simulations. Simulations started with randomized velocities |
| 406 |
< |
and underwent 50 ps of temperature scaling and 50 ps of constant |
| 407 |
< |
energy equilibration before obtaining a 200 ps trajectory. Diffusion |
| 408 |
< |
constants were calculated via root-mean square deviation analysis. The |
| 409 |
< |
averaged results from 5 sets of these NVE simulations is displayed in |
| 410 |
< |
figure \ref{diffuse}, alongside experimental, SPC/E, and TIP5P |
| 411 |
< |
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} |
| 402 |
> |
of the particles and their change in responce to altering |
| 403 |
> |
environmental conditions. In order to probe transport, constant energy |
| 404 |
> |
simulations were performed about the average density uncovered by the |
| 405 |
> |
constant pressure simulations. Simulations started with randomized |
| 406 |
> |
velocities and underwent 50 ps of temperature scaling and 50 ps of |
| 407 |
> |
constant energy equilibration before obtaining a 200 ps |
| 408 |
> |
trajectory. Diffusion constants were calculated via root-mean square |
| 409 |
> |
deviation analysis. The averaged results from five sets of NVE |
| 410 |
> |
simulations are displayed in figure \ref{diffuse}, alongside |
| 411 |
> |
experimental, SPC/E, and TIP5P |
| 412 |
> |
results.\cite{Gillen72,Mills73,Clancy94,Jorgensen01} |
| 413 |
|
|
| 414 |
|
\begin{figure} |
| 415 |
< |
\includegraphics[width=65mm, angle=-90]{betterDiffuse.epsi} |
| 415 |
> |
\begin{center} |
| 416 |
> |
\epsfxsize=6in |
| 417 |
> |
\epsfbox{betterDiffuse.epsi} |
| 418 |
|
\caption{Average diffusion coefficient over increasing temperature for |
| 419 |
< |
SSD, SPC/E\cite{Clancy94}, TIP5P\cite{Jorgensen01}, and Experimental |
| 420 |
< |
data from Gillen \emph{et al.}\cite{Gillen72}, and from |
| 421 |
< |
Mills\cite{Mills73}.} |
| 419 |
> |
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 |
|
\label{diffuse} |
| 425 |
+ |
\end{center} |
| 426 |
|
\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 |
< |
seen in experiment in both the supercooled and normal regimes. SPC/E |
| 432 |
< |
does a respectable job by getting similar values as SSD and experiment |
| 433 |
< |
around 290 K; however, it deviates at both higher and lower |
| 434 |
< |
temperatures, failing to predict the experimental trend. TIP5P and SSD |
| 435 |
< |
both start off low at the colder temperatures and tend to diffuse too |
| 436 |
< |
rapidly at the higher temperatures. This type of trend at the higher |
| 437 |
< |
temperatures is not surprising in that the densities of both TIP5P and |
| 438 |
< |
SSD are lower than experimental water at temperatures higher than room |
| 439 |
< |
temperature. When calculating the diffusion coefficients for SSD at |
| 431 |
> |
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 |
|
experimental densities, the resulting values fall more in line with |
| 441 |
|
experiment at these temperatures, albeit not at standard pressure. |
| 442 |
|
|
| 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 |
< |
phase behavior beyond the melting point. To locate the melting |
| 447 |
< |
transition, the constant pressure heat capacity (C$_\text{p}$) was |
| 448 |
< |
monitored in each of the simulations. In the melting simulations of |
| 449 |
< |
the 1024 particle ice $I_h$ simulations, a large spike in C$_\text{p}$ |
| 450 |
< |
occurs at 245 K, indicating a first order phase transition for the |
| 451 |
< |
melting of these ice crystals. When the reaction field is turned off, |
| 452 |
< |
the melting transition occurs at 235 K. These melting transitions are |
| 453 |
< |
considerably lower than the experimental value, but this is not |
| 454 |
< |
surprising when considering the simplicity of the SSD model. |
| 446 |
> |
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 |
|
|
| 456 |
< |
\begin{figure} |
| 457 |
< |
\includegraphics[width=85mm]{fullContours.eps} |
| 456 |
> |
\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 |
|
\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 |
+ |
\end{center} |
| 477 |
|
\end{figure} |
| 478 |
|
|
| 465 |
– |
\begin{figure} |
| 466 |
– |
\includegraphics[width=45mm]{corrDiag.eps} |
| 467 |
– |
\caption{Two dimensional illustration of the angles involved in the |
| 468 |
– |
correlations observed in figure \ref{contour}.} |
| 469 |
– |
\label{corrAngle} |
| 470 |
– |
\end{figure} |
| 471 |
– |
|
| 479 |
|
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 |
< |
\begin{multline} |
| 484 |
< |
g_{\text{AB}}(r,\cos\theta) = \\ |
| 485 |
< |
\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\ , |
| 486 |
< |
\end{multline} |
| 487 |
< |
\begin{multline} |
| 481 |
< |
g_{\text{AB}}(r,\cos\omega) = \\ |
| 483 |
> |
\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 |
|
\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 |
< |
\end{multline} |
| 489 |
> |
\end{equation} |
| 490 |
|
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 |
|
desired angle between the two dipoles, the g(\emph{r}) can be |
| 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 |
< |
second solvation shell peak appears to have two distinct parts that |
| 516 |
< |
blend together to form one observable peak. At higher temperatures, |
| 517 |
< |
this split character alters to show the leading 4 \AA\ peak dominated |
| 518 |
< |
by equatorial anti-parallel dipole orientations, and there is tightly |
| 519 |
< |
bunched group of axially arranged dipoles that most likely consist of |
| 520 |
< |
the smaller fraction aligned dipole pairs. The trailing part of the |
| 521 |
< |
split peak at 5 \AA\ is dominated by aligned dipoles that range |
| 522 |
< |
primarily within the axial to the chief hydrogen bond arrangements |
| 523 |
< |
similar to those seen in the first solvation shell. This evidence |
| 524 |
< |
indicates that the dipole pair interaction begins to dominate outside |
| 525 |
< |
of the range of the dipolar repulsion term, with the primary |
| 526 |
< |
energetically favorable dipole arrangements populating the region |
| 527 |
< |
immediately outside this repulsion region (around 4 \AA), and |
| 528 |
< |
arrangements that seek to ideally satisfy both the sticky and dipole |
| 529 |
< |
forces locate themselves just beyond this initial buildup (around 5 |
| 524 |
< |
\AA). |
| 515 |
> |
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 |
|
|
| 531 |
|
From these findings, the split second peak is primarily the product of |
| 532 |
|
the dipolar repulsion term of the sticky potential. In fact, the inner |
| 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 |
< |
because the second solvation shell will still be shifted too far |
| 537 |
> |
since the second solvation shell would still be shifted too far |
| 538 |
|
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 |
| 553 |
|
important properties. In this case, it would be ideal to correct the |
| 554 |
|
densities while maintaining the accurate transport properties. |
| 555 |
|
|
| 556 |
< |
The possible parameters for tuning include the $\sigma$ and $\epsilon$ |
| 556 |
> |
The parameters available for tuning include the $\sigma$ and $\epsilon$ |
| 557 |
|
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} |
| 558 |
– |
\begin{split} |
| 563 |
|
u_{ij}^{sp} |
| 564 |
< |
(\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)]\\ |
| 562 |
< |
& \quad \ + \frac{\nu_0^\prime}{2} |
| 563 |
< |
[s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)], |
| 564 |
< |
\end{split} |
| 564 |
> |
(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j) = |
| 565 |
> |
\frac{\nu_0}{2}[s(r_{ij})w(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)] + \frac{\nu_0^\prime}{2} [s^\prime(r_{ij})w^\prime(\mathbf{r}_{ij},\boldsymbol{\Omega}_i,\boldsymbol{\Omega}_j)], |
| 566 |
|
\end{equation} |
| 567 |
|
|
| 568 |
|
where $\nu_0$ scales the strength of the tetrahedral attraction and |
| 569 |
|
$\nu_0^\prime$ acts in an identical fashion on the dipole repulsion |
| 570 |
< |
term. For purposes of the reparameterization, the separation was |
| 571 |
< |
performed, but the final parameters were adjusted so that it is |
| 572 |
< |
unnecessary to separate the terms when implementing the adjusted water |
| 573 |
< |
potentials. The results of the reparameterizations are shown in table |
| 574 |
< |
\ref{params}. Note that both the tetrahedral attractive and dipolar |
| 575 |
< |
repulsive don't share the same lower cutoff ($r_l$) in the newly |
| 576 |
< |
parameterized potentials - soft sticky dipole reaction field (SSD/RF - |
| 577 |
< |
for use with a reaction field) and soft sticky dipole enhanced (SSD/E |
| 578 |
< |
- an attempt to improve the liquid structure in simulations without a |
| 579 |
< |
long-range correction). |
| 570 |
> |
term. The separation was performed for purposes of the |
| 571 |
> |
reparameterization, but the final parameters were adjusted so that it |
| 572 |
> |
is unnecessary to separate the terms when implementing the adjusted |
| 573 |
> |
water potentials. The results of the reparameterizations are shown in |
| 574 |
> |
table \ref{params}. Note that the tetrahedral attractive and dipolar |
| 575 |
> |
repulsive terms do not share the same lower cutoff ($r_l$) in the |
| 576 |
> |
newly parameterized potentials - soft sticky dipole reaction field |
| 577 |
> |
(SSD/RF - for use with a reaction field) and soft sticky dipole |
| 578 |
> |
enhanced (SSD/E - an attempt to improve the liquid structure in |
| 579 |
> |
simulations without a long-range correction). |
| 580 |
|
|
| 581 |
|
\begin{table} |
| 582 |
+ |
\begin{center} |
| 583 |
|
\caption{Parameters for the original and adjusted models} |
| 584 |
|
\begin{tabular}{ l c c c c } |
| 585 |
|
\hline \\[-3mm] |
| 586 |
< |
\ \ \ Parameters\ \ \ & \ \ \ SSD$^\dagger$ \ \ \ & \ SSD1$^\ddagger$\ \ & \ SSD/E\ \ & \ SSD/RF \\ |
| 586 |
> |
\ \ \ Parameters\ \ \ & \ \ \ SSD\cite{Ichiye96} \ \ \ & \ SSD1\cite{Ichiye03}\ \ & \ SSD/E\ \ & \ SSD/RF \\ |
| 587 |
|
\hline \\[-3mm] |
| 588 |
|
\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
| 589 |
|
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
| 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\\ |
| 595 |
– |
\\[-2mm]$^\dagger$ ref. \onlinecite{Ichiye96} |
| 596 |
– |
\\$^\ddagger$ ref. \onlinecite{Ichiye03} |
| 597 |
|
\end{tabular} |
| 598 |
|
\label{params} |
| 599 |
+ |
\end{center} |
| 600 |
|
\end{table} |
| 601 |
|
|
| 602 |
< |
\begin{figure} |
| 603 |
< |
\includegraphics[width=85mm]{GofRCompare.epsi} |
| 602 |
> |
\begin{figure} |
| 603 |
> |
\begin{center} |
| 604 |
> |
\epsfxsize=5in |
| 605 |
> |
\epsfbox{GofRCompare.epsi} |
| 606 |
|
\caption{Plots comparing experiment\cite{Head-Gordon00_1} with SSD/E |
| 607 |
|
and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with |
| 608 |
|
reaction field turned on (bottom). The insets show the respective |
| 609 |
< |
first peaks in detail. Solid Line - experiment, dashed line - SSD/E |
| 610 |
< |
and SSD/RF, and dotted line - SSD1 (with and without reaction field).} |
| 609 |
> |
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 |
|
\label{grcompare} |
| 612 |
+ |
\end{center} |
| 613 |
|
\end{figure} |
| 614 |
|
|
| 615 |
< |
\begin{figure} |
| 616 |
< |
\includegraphics[width=85mm]{dualsticky.ps} |
| 615 |
> |
\begin{figure} |
| 616 |
> |
\begin{center} |
| 617 |
> |
\epsfxsize=6in |
| 618 |
> |
\epsfbox{dualsticky.ps} |
| 619 |
|
\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \& |
| 620 |
|
SSD/RF (right). Light areas correspond to the tetrahedral attractive |
| 621 |
< |
part, and the darker areas correspond to the dipolar repulsive part.} |
| 621 |
> |
component, and darker areas correspond to the dipolar repulsive |
| 622 |
> |
component.} |
| 623 |
|
\label{isosurface} |
| 624 |
+ |
\end{center} |
| 625 |
|
\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 |
< |
shell. This resulted in a somewhat tall and sharp first peak that |
| 630 |
< |
integrated to give similar coordination numbers to the experimental |
| 631 |
< |
data obtained by Soper and Phillips.\cite{Ichiye96,Soper86} New |
| 632 |
< |
experimental x-ray scattering data from the Head-Gordon lab indicates |
| 633 |
< |
a slightly lower and shifted first peak in the g$_\mathrm{OO}(r)$, so |
| 634 |
< |
adjustments to SSD were made while taking into consideration the new |
| 635 |
< |
experimental findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} |
| 636 |
< |
shows the relocation of the first peak of the oxygen-oxygen |
| 637 |
< |
g(\emph{r}) by comparing the revised SSD model (SSD1), SSD-E, and |
| 638 |
< |
SSD-RF to the new experimental results. Both the modified water models |
| 639 |
< |
have shorter peaks that are brought in more closely to the |
| 640 |
< |
experimental peak (as seen in the insets of figure \ref{grcompare}). |
| 641 |
< |
This structural alteration was accomplished by the combined reduction |
| 642 |
< |
in the Lennard-Jones $\sigma$ variable and adjustment of the sticky |
| 643 |
< |
potential strength and cutoffs. As can be seen in table \ref{params}, |
| 644 |
< |
the cutoffs for the tetrahedral attractive and dipolar repulsive terms |
| 645 |
< |
were nearly swapped with each other. Isosurfaces of the original and |
| 646 |
< |
modified sticky potentials are shown in figure \cite{isosurface}. In |
| 647 |
< |
these isosurfaces, it is easy to see how altering the cutoffs changes |
| 648 |
< |
the repulsive and attractive character of the particles. With a |
| 649 |
< |
reduced repulsive surface (the darker region), the particles can move |
| 650 |
< |
closer to one another, increasing the density for the overall |
| 651 |
< |
system. This change in interaction cutoff also results in a more |
| 652 |
< |
gradual orientational motion by allowing the particles to maintain |
| 653 |
< |
preferred dipolar arrangements before they begin to feel the pull of |
| 654 |
< |
the tetrahedral restructuring. Upon moving closer together, the |
| 655 |
< |
dipolar repulsion term becomes active and excludes unphysical |
| 656 |
< |
nearest-neighbor arrangements. This compares with how SSD and SSD1 |
| 657 |
< |
exclude preferred dipole alignments before the particles feel the pull |
| 658 |
< |
of the ``hydrogen bonds''. Aside from improving the shape of the first |
| 659 |
< |
peak in the g(\emph{r}), this improves the densities considerably by |
| 629 |
> |
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 |
|
allowing the persistence of full dipolar character below the previous |
| 662 |
|
4.0 \AA\ cutoff. |
| 663 |
|
|
| 664 |
|
While adjusting the location and shape of the first peak of |
| 665 |
|
g(\emph{r}) improves the densities, these changes alone are |
| 666 |
|
insufficient to bring the system densities up to the values observed |
| 667 |
< |
experimentally. To finish bringing up the densities, the dipole |
| 668 |
< |
moments were increased in both the adjusted models. Being a dipole |
| 667 |
> |
experimentally. To further increase the densities, the dipole moments |
| 668 |
> |
were increased in both of the adjusted models. Since SSD is a dipole |
| 669 |
|
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 |
| 673 |
|
D. The larger dipole moment is a more realistic value and improves the |
| 674 |
|
dielectric properties of the fluid. Both theoretical and experimental |
| 675 |
|
measurements indicate a liquid phase dipole moment ranging from 2.4 D |
| 676 |
< |
to values as high as 3.11 D, so there is quite a range of available |
| 677 |
< |
values for a reasonable dipole |
| 678 |
< |
moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} The increasing of |
| 679 |
< |
the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF |
| 680 |
< |
respectively is moderate in this range; however, it leads to |
| 681 |
< |
significant changes in the density and transport of the water models. |
| 676 |
> |
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 |
|
|
| 683 |
|
In order to demonstrate the benefits of these reparameterizations, a |
| 684 |
|
series of NPT and NVE simulations were performed to probe the density |
| 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 |
|
calculations at the calculated self-consistent densities. Again, the |
| 689 |
< |
results come from five separate simulations of 1024 particle systems, |
| 690 |
< |
and the melting sequences were started from different ice $I_h$ |
| 691 |
< |
crystals constructed as stated earlier. Like before, each NPT |
| 689 |
> |
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 |
|
simulation was equilibrated for 100 ps before a 200 ps data collection |
| 693 |
< |
run at each temperature step, and they used the final configuration |
| 694 |
< |
from the previous temperature simulation as a starting point. All of |
| 695 |
< |
the NVE simulations had the same thermalization, equilibration, and |
| 696 |
< |
data collection times stated earlier in this paper. |
| 693 |
> |
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 |
|
|
| 698 |
< |
\begin{figure} |
| 699 |
< |
\includegraphics[width=62mm, angle=-90]{ssdeDense.epsi} |
| 698 |
> |
\begin{figure} |
| 699 |
> |
\begin{center} |
| 700 |
> |
\epsfxsize=6in |
| 701 |
> |
\epsfbox{ssdeDense.epsi} |
| 702 |
|
\caption{Comparison of densities calculated with SSD/E to SSD1 without a |
| 703 |
< |
reaction field, TIP3P\cite{Jorgensen98b}, TIP5P\cite{Jorgensen00}, |
| 704 |
< |
SPC/E\cite{Clancy94}, and Experiment\cite{CRC80}. The window shows a |
| 703 |
> |
reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00} |
| 704 |
> |
SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The window shows a |
| 705 |
|
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 |
|
\label{ssdedense} |
| 709 |
+ |
\end{center} |
| 710 |
|
\end{figure} |
| 711 |
|
|
| 712 |
|
Figure \ref{ssdedense} shows the density profile for the SSD/E model |
| 713 |
< |
in comparison to SSD1 without a reaction field, experiment, and other |
| 714 |
< |
common water models. The calculated densities for both SSD/E and SSD1 |
| 715 |
< |
have increased significantly over the original SSD model (see figure |
| 716 |
< |
\ref{dense1} and are in significantly better agreement with the |
| 717 |
< |
experimental values. At 298 K, the density of SSD/E and SSD1 without a |
| 718 |
< |
long-range correction are 0.996$\pm$0.001 g/cm$^3$ and 0.999$\pm$0.001 |
| 719 |
< |
g/cm$^3$ respectively. These both compare well with the experimental |
| 720 |
< |
value of 0.997 g/cm$^3$, and they are considerably better than the SSD |
| 721 |
< |
value of 0.967$\pm$0.003 g/cm$^3$. The changes to the dipole moment |
| 722 |
< |
and sticky switching functions have improved the structuring of the |
| 723 |
< |
liquid (as seen in figure \ref{grcompare}, but they have shifted the |
| 724 |
< |
density maximum to much lower temperatures. This comes about via an |
| 725 |
< |
increase of the liquid disorder through the weakening of the sticky |
| 726 |
< |
potential and strengthening of the dipolar character. However, this |
| 727 |
< |
increasing disorder in the SSD/E model has little affect on the |
| 728 |
< |
melting transition. By monitoring C$\text{p}$ throughout these |
| 729 |
< |
simulations, the melting transition for SSD/E occurred at 235 K, the |
| 730 |
< |
same transition temperature observed with SSD and SSD1. |
| 713 |
> |
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 |
|
|
| 733 |
< |
\begin{figure} |
| 734 |
< |
\includegraphics[width=62mm, angle=-90]{ssdrfDense.epsi} |
| 733 |
> |
\begin{figure} |
| 734 |
> |
\begin{center} |
| 735 |
> |
\epsfxsize=6in |
| 736 |
> |
\epsfbox{ssdrfDense.epsi} |
| 737 |
|
\caption{Comparison of densities calculated with SSD/RF to SSD1 with a |
| 738 |
< |
reaction field, TIP3P\cite{Jorgensen98b}, TIP5P\cite{Jorgensen00}, |
| 739 |
< |
SPC/E\cite{Clancy94}, and Experiment\cite{CRC80}. The inset shows the |
| 738 |
> |
reaction field, TIP3P,\cite{Jorgensen98b} TIP5P,\cite{Jorgensen00} |
| 739 |
> |
SPC/E,\cite{Clancy94} and experiment.\cite{CRC80} The inset shows the |
| 740 |
|
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 |
|
\label{ssdrfdense} |
| 744 |
+ |
\end{center} |
| 745 |
|
\end{figure} |
| 746 |
|
|
| 747 |
< |
Including the reaction field long-range correction results in a more |
| 748 |
< |
interesting comparison. A density profile including SSD/RF and SSD1 |
| 749 |
< |
with an active reaction field is shown in figure \ref{ssdrfdense}. As |
| 750 |
< |
observed in the simulations without a reaction field, the densities of |
| 751 |
< |
SSD/RF and SSD1 show a dramatic increase over normal SSD (see figure |
| 752 |
< |
\ref{dense1}). At 298 K, SSD/RF has a density of 0.997$\pm$0.001 |
| 753 |
< |
g/cm$^3$, right in line with experiment and considerably better than |
| 754 |
< |
the SSD value of 0.941$\pm$0.001 g/cm$^3$ and the SSD1 value of |
| 755 |
< |
0.972$\pm$0.002 g/cm$^3$. These results further emphasize the |
| 756 |
< |
importance of reparameterization in order to model the density |
| 757 |
< |
properly under different simulation conditions. Again, these changes |
| 758 |
< |
don't have that profound an effect on the melting point which is |
| 759 |
< |
observed at 245 K for SSD/RF, identical to SSD and only 5 K lower than |
| 760 |
< |
SSD1 with a reaction field. However, the difference in density maxima |
| 761 |
< |
is not quite as extreme with SSD/RF showing a density maximum at 255 |
| 762 |
< |
K, fairly close to 260 and 265 K, the density maxima for SSD and SSD1 |
| 763 |
< |
respectively. |
| 764 |
< |
|
| 765 |
< |
\begin{figure} |
| 766 |
< |
\includegraphics[width=65mm, angle=-90]{ssdeDiffuse.epsi} |
| 767 |
< |
\caption{Plots of the diffusion constants calculated from SSD/E and SSD1, |
| 768 |
< |
both without a reaction field, along with experimental results are |
| 769 |
< |
from Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The |
| 770 |
< |
NVE calculations were performed at the average densities observed in |
| 771 |
< |
the 1 atm NPT simulations for the respective models. SSD/E is |
| 772 |
< |
slightly more fluid than experiment at all of the temperatures, but |
| 773 |
< |
it is closer than SSD1 without a long-range correction.} |
| 747 |
> |
Including the reaction field long-range correction in the simulations |
| 748 |
> |
results in a more interesting comparison. A density profile including |
| 749 |
> |
SSD/RF and SSD1 with an active reaction field is shown in figure |
| 750 |
> |
\ref{ssdrfdense}. As observed in the simulations without a reaction |
| 751 |
> |
field, the densities of SSD/RF and SSD1 show a dramatic increase over |
| 752 |
> |
normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density |
| 753 |
> |
of 0.997$\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 |
> |
|
| 765 |
> |
\begin{figure} |
| 766 |
> |
\begin{center} |
| 767 |
> |
\epsfxsize=6in |
| 768 |
> |
\epsfbox{ssdeDiffuse.epsi} |
| 769 |
> |
\caption{Plots of the diffusion constants calculated from SSD/E and SSD1, |
| 770 |
> |
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 |
|
\label{ssdediffuse} |
| 777 |
+ |
\end{center} |
| 778 |
|
\end{figure} |
| 779 |
|
|
| 780 |
|
The reparameterization of the SSD water model, both for use with and |
| 785 |
|
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 |
< |
water. In the upper plot, the diffusion constant for SSD/E is |
| 789 |
< |
consistently a little faster than experiment, while SSD1 remains |
| 790 |
< |
slower than experiment until relatively high temperatures (greater |
| 791 |
< |
than 330 K). Both models follow the shape of the experimental trend |
| 792 |
< |
well below 300 K, but the trend leans toward diffusing too rapidly at |
| 793 |
< |
higher temperatures, something that is especially apparent with |
| 794 |
< |
SSD1. This accelerated increasing of diffusion is caused by the |
| 795 |
< |
rapidly decreasing system density with increasing temperature. Though |
| 796 |
< |
it is difficult to see in figure \ref{ssdedense}, the densities of SSD1 |
| 797 |
< |
decay more rapidly with temperature than do those of SSD/E, leading to |
| 798 |
< |
more visible deviation from the experimental diffusion trend. Thus, |
| 799 |
< |
the changes made to improve the liquid structure may have had an |
| 800 |
< |
adverse affect on the density maximum, but they improve the transport |
| 801 |
< |
behavior of the water model. |
| 788 |
> |
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 |
|
|
| 803 |
< |
\begin{figure} |
| 804 |
< |
\includegraphics[width=65mm, angle=-90]{ssdrfDiffuse.epsi} |
| 803 |
> |
\begin{figure} |
| 804 |
> |
\begin{center} |
| 805 |
> |
\epsfxsize=6in |
| 806 |
> |
\epsfbox{ssdrfDiffuse.epsi} |
| 807 |
|
\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1, |
| 808 |
< |
both with an active reaction field, along with experimental results |
| 809 |
< |
from Gillen \emph{et al.}\cite{Gillen72} and Mills\cite{Mills73}. The |
| 810 |
< |
NVE calculations were performed at the average densities observed in |
| 811 |
< |
the 1 atm NPT simulations for both of the models. Note how accurately |
| 812 |
< |
SSD/RF simulates the diffusion of water throughout this temperature |
| 813 |
< |
range. The more rapidly increasing diffusion constants at high |
| 814 |
< |
temperatures for both models is attributed to the significantly lower |
| 815 |
< |
densities than observed in experiment.} |
| 808 |
> |
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 |
|
\label{ssdrfdiffuse} |
| 817 |
+ |
\end{center} |
| 818 |
|
\end{figure} |
| 819 |
|
|
| 820 |
|
In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are |
| 821 |
< |
compared with SSD1 with an active reaction field. Note that SSD/RF |
| 821 |
> |
compared to SSD1 with an active reaction field. Note that SSD/RF |
| 822 |
|
tracks the experimental results incredibly well, identical within |
| 823 |
< |
error throughout the temperature range shown and only showing a slight |
| 823 |
> |
error throughout the temperature range shown and with only a slight |
| 824 |
|
increasing trend at higher temperatures. SSD1 tends to diffuse more |
| 825 |
|
slowly at low temperatures and deviates to diffuse too rapidly at |
| 826 |
< |
temperatures greater than 330 K. As was stated in the SSD/E |
| 827 |
< |
comparisons, this deviation away from the ideal trend is due to a |
| 828 |
< |
rapid decrease in density at higher temperatures. SSD/RF doesn't |
| 829 |
< |
suffer from this problem as much as SSD1, because the calculated |
| 830 |
< |
densities are more true to experiment. These results again emphasize |
| 831 |
< |
the importance of careful reparameterization when using an altered |
| 826 |
> |
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 |
|
long-range correction. |
| 833 |
|
|
| 834 |
|
\subsection{Additional Observations} |
| 835 |
|
|
| 836 |
|
\begin{figure} |
| 837 |
< |
\includegraphics[width=85mm]{povIce.ps} |
| 838 |
< |
\caption{A water lattice built from the crystal structure that SSD/E |
| 839 |
< |
assumed when undergoing an extremely restricted temperature NPT |
| 837 |
> |
\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 |
|
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 |
|
\label{weirdice} |
| 846 |
+ |
\end{center} |
| 847 |
|
\end{figure} |
| 848 |
|
|
| 849 |
|
While performing restricted temperature melting sequences of SSD/E not |
| 850 |
< |
discussed earlier in this paper, some interesting observations were |
| 851 |
< |
made. After melting at 235 K, two of five systems underwent |
| 852 |
< |
crystallization events near 245 K. As the heating process continued, |
| 853 |
< |
the two systems remained crystalline until finally melting between 320 |
| 854 |
< |
and 330 K. The final configurations of these two melting sequences |
| 855 |
< |
show an expanded zeolite-like crystal structure that does not |
| 856 |
< |
correspond to any known form of ice. For convenience and to help |
| 857 |
< |
distinguish it from the experimentally observed forms of ice, this |
| 858 |
< |
crystal structure will henceforth be referred to as ice-zero (ice |
| 859 |
< |
0). The crystallinity was extensive enough that a near ideal crystal |
| 860 |
< |
structure could be obtained. Figure \ref{weirdice} shows the repeating |
| 861 |
< |
crystal structure of a typical crystal at 5 K. Each water molecule is |
| 862 |
< |
hydrogen bonded to four others; however, the hydrogen bonds are flexed |
| 863 |
< |
rather than perfectly straight. This results in a skewed tetrahedral |
| 864 |
< |
geometry about the central molecule. Looking back at figure |
| 865 |
< |
\ref{isosurface}, it is easy to see how these flexed hydrogen bonds |
| 866 |
< |
are allowed in that the attractive regions are conical in shape, with |
| 867 |
< |
the greatest attraction in the central region. Though not ideal, these |
| 868 |
< |
flexed hydrogen bonds are favorable enough to stabilize an entire |
| 869 |
< |
crystal generated around them. In fact, the imperfect ice 0 crystals |
| 870 |
< |
were so stable that they melted at temperatures nearly 100 K greater |
| 871 |
< |
than both ice I$_c$ and I$_h$. |
| 850 |
> |
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 |
|
|
| 873 |
|
These initial simulations indicated that ice 0 is the preferred ice |
| 874 |
< |
structure for at least SSD/E. To verify this, a comparison was made |
| 875 |
< |
between near ideal crystals of ice $I_h$, ice $I_c$, and ice 0 at |
| 876 |
< |
constant pressure with SSD/E, SSD/RF, and SSD1. Near ideal versions of |
| 877 |
< |
the three types of crystals were cooled to 1 K, and the potential |
| 878 |
< |
energies of each were compared using all three water models. With |
| 879 |
< |
every water model, ice 0 turned out to have the lowest potential |
| 880 |
< |
energy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with SSD/E, and |
| 881 |
< |
7.5\% lower with SSD/RF. |
| 874 |
> |
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 |
|
|
| 883 |
|
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 |
< |
field had more reasonable melting transitions, down near 325 K. These |
| 889 |
< |
melting point observations emphasize how preferred this crystal |
| 890 |
< |
structure is over the most common types of ice when using these single |
| 888 |
> |
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 |
|
point water models. |
| 892 |
|
|
| 893 |
|
Recognizing that the above tests show ice 0 to be both the most stable |
| 896 |
|
this crystal structure with charge based water models. As a quick |
| 897 |
|
test, these 3 crystal types were converted from SSD type particles to |
| 898 |
|
TIP3P waters and read into CHARMM.\cite{Karplus83} Identical energy |
| 899 |
< |
minimizations were performed on all of these crystals to compare the |
| 900 |
< |
system energies. Again, ice 0 was observed to have the lowest total |
| 901 |
< |
system energy. The total energy of ice 0 was ~2\% lower than ice |
| 902 |
< |
$I_h$, which was in turn ~3\% lower than ice $I_c$. From these initial |
| 903 |
< |
results, we would not be surprised if results from the other common |
| 904 |
< |
water models show ice 0 to be the lowest energy crystal structure. A |
| 905 |
< |
continuation on work studying ice 0 with multi-point water models will |
| 906 |
< |
be published in a coming article. |
| 899 |
> |
minimizations were performed on the crystals to compare the system |
| 900 |
> |
energies. Again, ice 0 was observed to have the lowest total system |
| 901 |
> |
energy. The total energy of ice 0 was ~2\% lower than ice $I_h$, which |
| 902 |
> |
was in turn ~3\% lower than ice $I_c$. Based on these initial studies, |
| 903 |
> |
it would not be surprising if results from the other common water |
| 904 |
> |
models show ice 0 to be the lowest energy crystal structure. A |
| 905 |
> |
continuation of this work studying ice 0 with multi-point water models |
| 906 |
> |
will be published in a coming article. |
| 907 |
|
|
| 908 |
|
\section{Conclusions} |
| 909 |
|
The density maximum and temperature dependent transport for the SSD |
| 913 |
|
density maximum near 260 K. In most cases, the calculated densities |
| 914 |
|
were significantly lower than the densities calculated in simulations |
| 915 |
|
of other water models. Analysis of particle diffusion showed SSD to |
| 916 |
< |
capture the transport properties of experimental very well in both the |
| 917 |
< |
normal and super-cooled liquid regimes. In order to correct the |
| 916 |
> |
capture the transport properties of experimental water well in both |
| 917 |
> |
the liquid and super-cooled liquid regimes. In order to correct the |
| 918 |
|
density behavior, the original SSD model was reparameterized for use |
| 919 |
|
both with and without a reaction field (SSD/RF and SSD/E), and |
| 920 |
|
comparison simulations were performed with SSD1, the density corrected |
| 921 |
|
version of SSD. Both models improve the liquid structure, density |
| 922 |
|
values, and diffusive properties under their respective conditions, |
| 923 |
|
indicating the necessity of reparameterization when altering the |
| 924 |
< |
long-range correction specifics. When taking the appropriate |
| 925 |
< |
considerations, these simple water models are excellent choices for |
| 926 |
< |
representing explicit water in large scale simulations of biochemical |
| 927 |
< |
systems. |
| 924 |
> |
long-range correction specifics. When taking into account the |
| 925 |
> |
appropriate considerations, these simple water models are excellent |
| 926 |
> |
choices for representing explicit water in large scale simulations of |
| 927 |
> |
biochemical systems. |
| 928 |
|
|
| 929 |
|
\section{Acknowledgments} |
| 930 |
|
Support for this project was provided by the National Science |
| 932 |
|
the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
| 933 |
|
DMR 00 79647. |
| 934 |
|
|
| 910 |
– |
\bibliographystyle{jcp} |
| 935 |
|
|
| 936 |
+ |
\newpage |
| 937 |
+ |
|
| 938 |
+ |
\bibliographystyle{jcp} |
| 939 |
|
\bibliography{nptSSD} |
| 940 |
|
|
| 941 |
|
%\pagebreak |
