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\begin{document} |
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\title{On the structural and transport properties of the soft sticky |
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dipole (SSD) and related single point water models} |
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\author{Christopher J. Fennell and J. Daniel Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu}} |
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\affiliation{Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\begin{abstract} |
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The density maximum and temperature dependence of the self-diffusion |
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constant were investigated for the soft sticky dipole (SSD) water |
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model and two related reparameterizations of this single-point model. |
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A combination of microcanonical and isobaric-isothermal molecular |
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dynamics simulations were used to calculate these properties, both |
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with and without the use of reaction field to handle long-range |
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electrostatics. The isobaric-isothermal (NPT) simulations of the |
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melting of both ice-$I_h$ and ice-$I_c$ showed a density maximum near |
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260~K. In most cases, the use of the reaction field resulted in |
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calculated densities which were were significantly lower than |
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experimental densities. Analysis of self-diffusion constants shows |
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that the original SSD model captures the transport properties of |
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experimental water very well in both the normal and super-cooled |
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liquid regimes. We also present our reparameterized versions of SSD |
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for use both with the reaction field or without any long-range |
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electrostatic corrections. These are called the SSD/RF and SSD/E |
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models respectively. These modified models were shown to maintain or |
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improve upon the experimental agreement with the structural and |
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transport properties that can be obtained with either the original SSD |
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or the density corrected version of the original model (SSD1). |
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Additionally, a novel low-density ice structure is presented |
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which appears to be the most stable ice structure for the entire SSD |
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family. |
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\end{abstract} |
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\maketitle |
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\newpage |
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%\narrowtext |
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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 the aqueous environment of the |
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molecules of interest. In some cases (such as in the simulation of |
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phospholipid bilayers), the majority of the calculations that are |
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performed involve interactions with or between solvent molecules. |
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Thus, the properties one may observe in biochemical simulations are |
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going to be highly dependent on the physical properties of the water |
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model that is chosen. |
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|
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There is an especially delicate balance between computational |
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efficiency and the ability of the water model to accurately predict |
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the properties of bulk |
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water.\cite{Jorgensen83,Berendsen87,Jorgensen00} For example, the |
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TIP5P model improves on the structural and transport properties of |
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water relative to the previous TIP models, yet this comes at a greater |
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than 50\% increase in computational |
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cost.\cite{Jorgensen01,Jorgensen00} |
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One recently developed model that largely succeeds in retaining the |
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accuracy of bulk properties while greatly reducing the computational |
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cost is the Soft Sticky Dipole (SSD) water |
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model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The SSD model |
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was developed by Ichiye \emph{et al.} as a modified form of the |
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hard-sphere water model proposed by Bratko, Blum, and |
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Luzar.\cite{Bratko85,Bratko95} SSD is a {\it single point} model |
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which has an interaction site that is both a point dipole and a |
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Lennard-Jones core. However, since the normal aligned and |
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anti-aligned geometries favored by point dipoles are poor mimics of |
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local structure in liquid water, a short ranged ``sticky'' potential |
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is also added. The sticky potential directs the molecules to assume |
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the proper hydrogen bond orientation in the first solvation shell. |
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|
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The interaction between two SSD water molecules \emph{i} and \emph{j} |
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is given by the potential |
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\begin{equation} |
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u_{ij} = u_{ij}^{LJ} (r_{ij})\ + u_{ij}^{dp} |
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({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)\ + |
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u_{ij}^{sp} |
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({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j), |
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\end{equation} |
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where the ${\bf r}_{ij}$ is the position vector between molecules |
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\emph{i} and \emph{j} with magnitude $r_{ij}$, and |
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${\bf \Omega}_i$ and ${\bf \Omega}_j$ represent the orientations of |
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the two molecules. The Lennard-Jones and dipole interactions are given |
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by the following familiar forms: |
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\begin{equation} |
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u_{ij}^{LJ}(r_{ij}) = 4\epsilon |
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\left[\left(\frac{\sigma}{r_{ij}}\right)^{12}-\left(\frac{\sigma}{r_{ij}}\right)^{6}\right] |
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\ , |
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\end{equation} |
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and |
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\begin{equation} |
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u_{ij}^{dp} = \frac{|\mu_i||\mu_j|}{4 \pi \epsilon_0 r_{ij}^3} \left( |
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\hat{\bf u}_i \cdot \hat{\bf u}_j - 3(\hat{\bf u}_i\cdot\hat{\bf |
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r}_{ij})(\hat{\bf u}_j\cdot\hat{\bf r}_{ij}) \right)\ , |
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\end{equation} |
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where $\hat{\bf u}_i$ and $\hat{\bf u}_j$ are the unit vectors along |
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the dipoles of molecules $i$ and $j$ respectively. $|\mu_i|$ and |
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$|\mu_j|$ are the strengths of the dipole moments, and $\hat{\bf |
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r}_{ij}$ is the unit vector pointing from molecule $j$ to molecule |
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$i$. |
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The sticky potential is somewhat less familiar: |
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\begin{equation} |
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u_{ij}^{sp} |
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({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = |
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\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) |
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+ s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf |
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\Omega}_j)]\ . |
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\label{stickyfunction} |
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\end{equation} |
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Here, $\nu_0$ is a strength parameter for the sticky potential, and |
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$s$ and $s^\prime$ are cubic switching functions which turn off the |
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sticky interaction beyond the first solvation shell. The $w$ function |
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can be thought of as an attractive potential with tetrahedral |
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geometry: |
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\begin{equation} |
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w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)=\sin\theta_{ij}\sin2\theta_{ij}\cos2\phi_{ij}, |
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\end{equation} |
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while the $w^\prime$ function counters the normal aligned and |
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anti-aligned structures favored by point dipoles: |
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\begin{equation} |
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w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^\circ, |
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\end{equation} |
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It should be noted that $w$ is proportional to the sum of the $Y_3^2$ |
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and $Y_3^{-2}$ spherical harmonics (a linear combination which |
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enhances the tetrahedral geometry for hydrogen bonded structures), |
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while $w^\prime$ is a purely empirical function. A more detailed |
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description of the functional parts and variables in this potential |
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can be found in the original SSD |
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articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} |
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|
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Since SSD is a single-point {\it dipolar} model, the force |
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calculations are simplified significantly relative to the standard |
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{\it charged} multi-point models. In the original Monte Carlo |
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simulations using this model, Liu and Ichiye reported that using SSD |
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decreased computer time by a factor of 6-7 compared to other |
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models.\cite{Ichiye96} What is most impressive is that this savings |
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did not come at the expense of accurate depiction of the liquid state |
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properties. Indeed, SSD maintains reasonable agreement with the Soper |
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data for the structural features of liquid |
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water.\cite{Soper86,Ichiye96} Additionally, the dynamical properties |
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exhibited by SSD agree with experiment better than those of more |
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computationally expensive models (like TIP3P and |
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SPC/E).\cite{Ichiye99} The combination of speed and accurate depiction |
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of solvent properties makes SSD a very attractive model for the |
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simulation of large scale biochemical simulations. |
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One feature of the SSD model is that it was parameterized for |
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use with the Ewald sum to handle long-range interactions. This would |
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normally be the best way of handling long-range interactions in |
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systems that contain other point charges. However, our group has |
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recently become interested in systems with point dipoles as mimics for |
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neutral, but polarized regions on molecules (e.g. the zwitterionic |
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head group regions of phospholipids). If the system of interest does |
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not contain point charges, the Ewald sum and even particle-mesh Ewald |
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become computational bottlenecks. Their respective ideal |
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$N^\frac{3}{2}$ and $N\log N$ calculation scaling orders for $N$ |
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particles can become prohibitive when $N$ becomes |
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large.\cite{Darden99} In applying this water model in these types of |
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systems, it would be useful to know its properties and behavior under |
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the more computationally efficient reaction field (RF) technique, or |
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even with a simple cutoff. This study addresses these issues by |
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looking at the structural and transport behavior of SSD over a |
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variety of temperatures with the purpose of utilizing the RF |
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correction technique. We then suggest modifications to the parameters |
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that result in more realistic bulk phase behavior. It should be noted |
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that in a recent publication, some of the original investigators of |
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the SSD water model have suggested adjustments to the SSD |
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water model to address abnormal density behavior (also observed here), |
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calling the corrected model SSD1.\cite{Ichiye03} In what |
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follows, we compare our reparamaterization of SSD with both the |
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original SSD and SSD1 models with the goal of improving |
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the bulk phase behavior of an SSD-derived model in simulations |
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utilizing the reaction field. |
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\section{Methods} |
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Long-range dipole-dipole interactions were accounted for in this study |
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by using either the reaction field technique or by resorting to a |
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simple cubic switching function at a cutoff radius. One of the early |
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applications of a reaction field was actually in Monte Carlo |
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simulations of liquid water.\cite{Barker73} Under this method, the |
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magnitude of the reaction field acting on dipole $i$ is |
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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}}} {\bf \mu}_{j} s(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 the case of liquid water), ${\bf \mu}_{j}$ is the dipole |
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moment vector of particle $j$, and $s(r_{ij})$ 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 $i$ is given by $-\frac{1}{2}{\bf |
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\mu}_{i}\cdot\mathcal{E}_{i}$ and the torque on dipole $i$ by ${\bf |
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\mu}_{i}\times\mathcal{E}_{i}$.\cite{AllenTildesley} Use of the reaction |
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field is known to alter the bulk orientational properties of simulated |
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water, and there is particular sensitivity of these properties on |
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changes in the length of the cutoff radius.\cite{Berendsen98} This |
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variable behavior makes reaction field a less attractive method than |
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the Ewald sum. However, for very large systems, the computational |
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benefit of reaction field is dramatic. |
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We have also performed a companion set of simulations {\it without} a |
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surrounding dielectric (i.e. using a simple cubic switching function |
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at the cutoff radius), and as a result we have two reparamaterizations |
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of SSD which could be used either with or without the reaction |
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field turned on. |
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|
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Simulations to obtain the preferred densities of the models were |
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performed in the isobaric-isothermal (NPT) ensemble, while all |
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dynamical properties were obtained from microcanonical (NVE) |
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simulations done at densities matching the NPT density for a |
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particular target temperature. 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 molecules 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 |
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atoms, and 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, Leimkuhler, and |
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McLachlan ({\sc dlm}).\cite{Dullweber1997} Our reason for selecting |
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this integrator centers on poor energy conservation of rigid body |
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dynamics using traditional quaternion |
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integration.\cite{Evans77,Evans77b} In typical microcanonical ensemble |
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simulations, the energy drift when using quaternions was substantially |
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greater than when using the {\sc dlm} method (fig. \ref{timestep}). |
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This steady drift in the total energy has also been observed by Kol |
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{\it et al.}\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. The additional memory required by the |
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algorithm is inconsequential on modern computers, and translating the |
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rotation matrix into quaternions for storage purposes makes trajectory |
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data quite compact. |
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|
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The {\sc dlm} method allows for Verlet style integration of both |
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translational and orientational motion of rigid bodies. In this |
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integration method, the orientational propagation involves a sequence |
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of matrix evaluations to update the rotation |
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matrix.\cite{Dullweber1997} These matrix rotations are more costly |
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than the simpler arithmetic quaternion propagation. With the same time |
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step, a 1000 SSD particle simulation shows an average 7\% |
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increase in computation time using the {\sc dlm} method in place of |
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quaternions. The additional expense per step is justified when one |
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considers the ability to use time steps that are nearly twice as large |
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under {\sc dlm} than would be usable under quaternion dynamics. The |
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energy conservation of the two methods using a number of different |
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time steps is illustrated in figure |
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\ref{timestep}. |
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|
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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 both quaternion-based integration and |
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%the {\sc dlm} method with increasing time step. The larger time step |
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%plots are shifted from the true energy baseline (that of $\Delta t$ = |
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%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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|
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In figure \ref{timestep}, the resulting energy drift at various time |
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steps for both the {\sc dlm} and quaternion integration schemes is |
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compared. All of the 1000 SSD particle simulations started with |
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the same configuration, and the only difference was the method used to |
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handle orientational motion. At time steps of 0.1 and 0.5~fs, both |
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methods for propagating the orientational degrees of freedom conserve |
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|
energy fairly well, with the quaternion method showing a slight energy |
309 |
chrisfen |
1033 |
drift over time in the 0.5~fs time step simulation. At time steps of 1 |
310 |
|
|
and 2~fs, the energy conservation benefits of the {\sc dlm} method are |
311 |
chrisfen |
1030 |
clearly demonstrated. Thus, while maintaining the same degree of |
312 |
|
|
energy conservation, one can take considerably longer time steps, |
313 |
|
|
leading to an overall reduction in computation time. |
314 |
chrisfen |
743 |
|
315 |
chrisfen |
1030 |
Energy drift in the simulations using {\sc dlm} integration was |
316 |
chrisfen |
1033 |
unnoticeable for time steps up to 3~fs. A slight energy drift on the |
317 |
|
|
order of 0.012~kcal/mol per nanosecond was observed at a time step of |
318 |
|
|
4~fs, and as expected, this drift increases dramatically with |
319 |
chrisfen |
1030 |
increasing time step. To insure accuracy in our microcanonical |
320 |
chrisfen |
1033 |
simulations, time steps were set at 2~fs and kept at this value for |
321 |
chrisfen |
1030 |
constant pressure simulations as well. |
322 |
chrisfen |
743 |
|
323 |
gezelter |
921 |
Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices |
324 |
|
|
were generated as starting points for all simulations. The $I_h$ |
325 |
chrisfen |
1033 |
crystals were formed by first arranging the centers of mass of the SSD |
326 |
|
|
particles into a ``hexagonal'' ice lattice of 1024 particles. Because |
327 |
|
|
of the crystal structure of $I_h$ ice, the simulation box assumed an |
328 |
|
|
orthorhombic shape with an edge length ratio of approximately |
329 |
|
|
1.00$\times$1.06$\times$1.23. The particles were then allowed to |
330 |
|
|
orient freely about fixed positions with angular momenta randomized at |
331 |
|
|
400~K for varying times. The rotational temperature was then scaled |
332 |
|
|
down in stages to slowly cool the crystals to 25~K. The particles were |
333 |
|
|
then allowed to translate with fixed orientations at a constant |
334 |
|
|
pressure of 1 atm for 50~ps at 25~K. Finally, all constraints were |
335 |
|
|
removed and the ice crystals were allowed to equilibrate for 50~ps at |
336 |
|
|
25~K and a constant pressure of 1~atm. This procedure resulted in |
337 |
|
|
structurally stable $I_h$ ice crystals that obey the Bernal-Fowler |
338 |
chrisfen |
862 |
rules.\cite{Bernal33,Rahman72} This method was also utilized in the |
339 |
chrisfen |
743 |
making of diamond lattice $I_c$ ice crystals, with each cubic |
340 |
|
|
simulation box consisting of either 512 or 1000 particles. Only |
341 |
|
|
isotropic volume fluctuations were performed under constant pressure, |
342 |
|
|
so the ratio of edge lengths remained constant throughout the |
343 |
|
|
simulations. |
344 |
|
|
|
345 |
|
|
\section{Results and discussion} |
346 |
|
|
|
347 |
|
|
Melting studies were performed on the randomized ice crystals using |
348 |
gezelter |
921 |
isobaric-isothermal (NPT) dynamics. During melting simulations, the |
349 |
|
|
melting transition and the density maximum can both be observed, |
350 |
|
|
provided that the density maximum occurs in the liquid and not the |
351 |
|
|
supercooled regime. An ensemble average from five separate melting |
352 |
|
|
simulations was acquired, each starting from different ice crystals |
353 |
|
|
generated as described previously. All simulations were equilibrated |
354 |
chrisfen |
1033 |
for 100~ps prior to a 200~ps data collection run at each temperature |
355 |
|
|
setting. The temperature range of study spanned from 25 to 400~K, with |
356 |
|
|
a maximum degree increment of 25~K. For regions of interest along this |
357 |
|
|
stepwise progression, the temperature increment was decreased from |
358 |
|
|
25~K to 10 and 5~K. The above equilibration and production times were |
359 |
gezelter |
921 |
sufficient in that fluctuations in the volume autocorrelation function |
360 |
chrisfen |
1033 |
were damped out in all simulations in under 20~ps. |
361 |
chrisfen |
743 |
|
362 |
|
|
\subsection{Density Behavior} |
363 |
|
|
|
364 |
chrisfen |
1030 |
Our initial simulations focused on the original SSD water model, |
365 |
|
|
and an average density versus temperature plot is shown in figure |
366 |
gezelter |
921 |
\ref{dense1}. Note that the density maximum when using a reaction |
367 |
chrisfen |
1033 |
field appears between 255 and 265~K. There were smaller fluctuations |
368 |
|
|
in the density at 260~K than at either 255 or 265~K, so we report this |
369 |
gezelter |
921 |
value as the location of the density maximum. Figure \ref{dense1} was |
370 |
|
|
constructed using ice $I_h$ crystals for the initial configuration; |
371 |
|
|
though not pictured, the simulations starting from ice $I_c$ crystal |
372 |
|
|
configurations showed similar results, with a liquid-phase density |
373 |
chrisfen |
1033 |
maximum in this same region (between 255 and 260~K). |
374 |
gezelter |
921 |
|
375 |
chrisfen |
1033 |
%\begin{figure} |
376 |
|
|
%\begin{center} |
377 |
|
|
%\epsfxsize=6in |
378 |
|
|
%\epsfbox{denseSSDnew.eps} |
379 |
|
|
%\caption{Density versus temperature for TIP4P [Ref. \onlinecite{Jorgensen98b}], |
380 |
|
|
% TIP3P [Ref. \onlinecite{Jorgensen98b}], SPC/E [Ref. \onlinecite{Clancy94}], SSD |
381 |
|
|
% without Reaction Field, SSD, and experiment [Ref. \onlinecite{CRC80}]. The |
382 |
|
|
% arrows indicate the change in densities observed when turning off the |
383 |
|
|
% reaction field. The the lower than expected densities for the SSD |
384 |
|
|
% model were what prompted the original reparameterization of SSD1 |
385 |
|
|
% [Ref. \onlinecite{Ichiye03}].} |
386 |
|
|
%\label{dense1} |
387 |
|
|
%\end{center} |
388 |
|
|
%\end{figure} |
389 |
chrisfen |
743 |
|
390 |
chrisfen |
1030 |
The density maximum for SSD compares quite favorably to other |
391 |
|
|
simple water models. Figure \ref{dense1} also shows calculated |
392 |
|
|
densities of several other models and experiment obtained from other |
393 |
chrisfen |
743 |
sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water |
394 |
chrisfen |
1030 |
models, SSD has a temperature closest to the experimentally |
395 |
|
|
observed density maximum. Of the {\it charge-based} models in |
396 |
gezelter |
921 |
Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that |
397 |
chrisfen |
1030 |
seen in SSD. Though not included in this plot, it is useful to |
398 |
|
|
note that TIP5P has a density maximum nearly identical to the |
399 |
gezelter |
921 |
experimentally measured temperature. |
400 |
chrisfen |
743 |
|
401 |
gezelter |
921 |
It has been observed that liquid state densities in water are |
402 |
|
|
dependent on the cutoff radius used both with and without the use of |
403 |
|
|
reaction field.\cite{Berendsen98} In order to address the possible |
404 |
|
|
effect of cutoff radius, simulations were performed with a dipolar |
405 |
chrisfen |
1033 |
cutoff radius of 12.0~\AA\ to complement the previous SSD |
406 |
|
|
simulations, all performed with a cutoff of 9.0~\AA. All of the |
407 |
chrisfen |
1030 |
resulting densities overlapped within error and showed no significant |
408 |
|
|
trend toward lower or higher densities as a function of cutoff radius, |
409 |
|
|
for simulations both with and without reaction field. These results |
410 |
|
|
indicate that there is no major benefit in choosing a longer cutoff |
411 |
|
|
radius in simulations using SSD. This is advantageous in that |
412 |
|
|
the use of a longer cutoff radius results in a significant increase in |
413 |
|
|
the time required to obtain a single trajectory. |
414 |
chrisfen |
743 |
|
415 |
chrisfen |
862 |
The key feature to recognize in figure \ref{dense1} is the density |
416 |
chrisfen |
1030 |
scaling of SSD relative to other common models at any given |
417 |
|
|
temperature. SSD assumes a lower density than any of the other |
418 |
|
|
listed models at the same pressure, behavior which is especially |
419 |
chrisfen |
1033 |
apparent at temperatures greater than 300~K. Lower than expected |
420 |
chrisfen |
1030 |
densities have been observed for other systems using a reaction field |
421 |
|
|
for long-range electrostatic interactions, so the most likely reason |
422 |
|
|
for the significantly lower densities seen in these simulations is the |
423 |
gezelter |
921 |
presence of the reaction field.\cite{Berendsen98,Nezbeda02} In order |
424 |
|
|
to test the effect of the reaction field on the density of the |
425 |
|
|
systems, the simulations were repeated without a reaction field |
426 |
|
|
present. The results of these simulations are also displayed in figure |
427 |
|
|
\ref{dense1}. Without the reaction field, the densities increase |
428 |
|
|
to more experimentally reasonable values, especially around the |
429 |
|
|
freezing point of liquid water. The shape of the curve is similar to |
430 |
chrisfen |
1030 |
the curve produced from SSD simulations using reaction field, |
431 |
gezelter |
921 |
specifically the rapidly decreasing densities at higher temperatures; |
432 |
chrisfen |
1033 |
however, a shift in the density maximum location, down to 245~K, is |
433 |
gezelter |
921 |
observed. This is a more accurate comparison to the other listed water |
434 |
|
|
models, in that no long range corrections were applied in those |
435 |
|
|
simulations.\cite{Clancy94,Jorgensen98b} However, even without the |
436 |
chrisfen |
1033 |
reaction field, the density around 300~K is still significantly lower |
437 |
chrisfen |
861 |
than experiment and comparable water models. This anomalous behavior |
438 |
chrisfen |
1027 |
was what lead Tan {\it et al.} to recently reparameterize |
439 |
chrisfen |
1030 |
SSD.\cite{Ichiye03} Throughout the remainder of the paper our |
440 |
|
|
reparamaterizations of SSD will be compared with their newer SSD1 |
441 |
gezelter |
1029 |
model. |
442 |
chrisfen |
861 |
|
443 |
chrisfen |
743 |
\subsection{Transport Behavior} |
444 |
|
|
|
445 |
gezelter |
921 |
Accurate dynamical properties of a water model are particularly |
446 |
|
|
important when using the model to study permeation or transport across |
447 |
|
|
biological membranes. In order to probe transport in bulk water, |
448 |
|
|
constant energy (NVE) simulations were performed at the average |
449 |
|
|
density obtained by the NPT simulations at an identical target |
450 |
|
|
temperature. Simulations started with randomized velocities and |
451 |
chrisfen |
1033 |
underwent 50~ps of temperature scaling and 50~ps of constant energy |
452 |
|
|
equilibration before a 200~ps data collection run. Diffusion constants |
453 |
gezelter |
921 |
were calculated via linear fits to the long-time behavior of the |
454 |
|
|
mean-square displacement as a function of time. The averaged results |
455 |
|
|
from five sets of NVE simulations are displayed in figure |
456 |
|
|
\ref{diffuse}, alongside experimental, SPC/E, and TIP5P |
457 |
chrisfen |
1022 |
results.\cite{Gillen72,Holz00,Clancy94,Jorgensen01} |
458 |
gezelter |
921 |
|
459 |
chrisfen |
1033 |
%\begin{figure} |
460 |
|
|
%\begin{center} |
461 |
|
|
%\epsfxsize=6in |
462 |
|
|
%\epsfbox{betterDiffuse.epsi} |
463 |
|
|
%\caption{Average self-diffusion constant as a function of temperature for |
464 |
|
|
%SSD, SPC/E [Ref. \onlinecite{Clancy94}], and TIP5P |
465 |
|
|
%[Ref. \onlinecite{Jorgensen01}] compared with experimental data |
466 |
|
|
%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. Of the three water models |
467 |
|
|
%shown, SSD has the least deviation from the experimental values. The |
468 |
|
|
%rapidly increasing diffusion constants for TIP5P and SSD correspond to |
469 |
|
|
%significant decreases in density at the higher temperatures.} |
470 |
|
|
%\label{diffuse} |
471 |
|
|
%\end{center} |
472 |
|
|
%\end{figure} |
473 |
chrisfen |
743 |
|
474 |
|
|
The observed values for the diffusion constant point out one of the |
475 |
chrisfen |
1030 |
strengths of the SSD model. Of the three models shown, the SSD model |
476 |
gezelter |
921 |
has the most accurate depiction of self-diffusion in both the |
477 |
|
|
supercooled and liquid regimes. SPC/E does a respectable job by |
478 |
chrisfen |
1033 |
reproducing values similar to experiment around 290~K; however, it |
479 |
gezelter |
921 |
deviates at both higher and lower temperatures, failing to predict the |
480 |
chrisfen |
1030 |
correct thermal trend. TIP5P and SSD both start off low at colder |
481 |
gezelter |
921 |
temperatures and tend to diffuse too rapidly at higher temperatures. |
482 |
|
|
This behavior at higher temperatures is not particularly surprising |
483 |
chrisfen |
1030 |
since the densities of both TIP5P and SSD are lower than experimental |
484 |
gezelter |
921 |
water densities at higher temperatures. When calculating the |
485 |
chrisfen |
1030 |
diffusion coefficients for SSD at experimental densities |
486 |
|
|
(instead of the densities from the NPT simulations), the resulting |
487 |
|
|
values fall more in line with experiment at these temperatures. |
488 |
chrisfen |
743 |
|
489 |
|
|
\subsection{Structural Changes and Characterization} |
490 |
gezelter |
921 |
|
491 |
chrisfen |
743 |
By starting the simulations from the crystalline state, the melting |
492 |
gezelter |
921 |
transition and the ice structure can be obtained along with the liquid |
493 |
chrisfen |
862 |
phase behavior beyond the melting point. The constant pressure heat |
494 |
|
|
capacity (C$_\text{p}$) was monitored to locate the melting transition |
495 |
|
|
in each of the simulations. In the melting simulations of the 1024 |
496 |
|
|
particle ice $I_h$ simulations, a large spike in C$_\text{p}$ occurs |
497 |
chrisfen |
1033 |
at 245~K, indicating a first order phase transition for the melting of |
498 |
chrisfen |
862 |
these ice crystals. When the reaction field is turned off, the melting |
499 |
chrisfen |
1033 |
transition occurs at 235~K. These melting transitions are |
500 |
gezelter |
921 |
considerably lower than the experimental value. |
501 |
chrisfen |
743 |
|
502 |
chrisfen |
1033 |
%\begin{figure} |
503 |
|
|
%\begin{center} |
504 |
|
|
%\epsfxsize=6in |
505 |
|
|
%\epsfbox{corrDiag.eps} |
506 |
|
|
%\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.} |
507 |
|
|
%\label{corrAngle} |
508 |
|
|
%\end{center} |
509 |
|
|
%\end{figure} |
510 |
chrisfen |
862 |
|
511 |
chrisfen |
1033 |
%\begin{figure} |
512 |
|
|
%\begin{center} |
513 |
|
|
%\epsfxsize=6in |
514 |
|
|
%\epsfbox{fullContours.eps} |
515 |
|
|
%\caption{Contour plots of 2D angular pair correlation functions for |
516 |
|
|
%512 SSD molecules at 100~K (A \& B) and 300~K (C \& D). Dark areas |
517 |
|
|
%signify regions of enhanced density while light areas signify |
518 |
|
|
%depletion relative to the bulk density. White areas have pair |
519 |
|
|
%correlation values below 0.5 and black areas have values above 1.5.} |
520 |
|
|
%\label{contour} |
521 |
|
|
%\end{center} |
522 |
|
|
%\end{figure} |
523 |
chrisfen |
743 |
|
524 |
gezelter |
921 |
Additional analysis of the melting process was performed using |
525 |
|
|
two-dimensional structure and dipole angle correlations. Expressions |
526 |
|
|
for these correlations are as follows: |
527 |
chrisfen |
861 |
|
528 |
chrisfen |
862 |
\begin{equation} |
529 |
gezelter |
921 |
g_{\text{AB}}(r,\cos\theta) = \frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\theta-\cos\theta_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ , |
530 |
chrisfen |
862 |
\end{equation} |
531 |
|
|
\begin{equation} |
532 |
|
|
g_{\text{AB}}(r,\cos\omega) = |
533 |
gezelter |
921 |
\frac{V}{N_\text{A}N_\text{B}}\langle\sum_{i\in\text{A}}\sum_{j\in\text{B}}\delta(\cos\omega-\cos\omega_{ij})\delta(r-\left|{\bf r}_{ij}\right|)\rangle\ , |
534 |
chrisfen |
862 |
\end{equation} |
535 |
chrisfen |
861 |
where $\theta$ and $\omega$ refer to the angles shown in figure |
536 |
|
|
\ref{corrAngle}. By binning over both distance and the cosine of the |
537 |
gezelter |
921 |
desired angle between the two dipoles, the $g(r)$ can be analyzed to |
538 |
|
|
determine the common dipole arrangements that constitute the peaks and |
539 |
|
|
troughs in the standard one-dimensional $g(r)$ plots. Frames A and B |
540 |
|
|
of figure \ref{contour} show results from an ice $I_c$ simulation. The |
541 |
|
|
first peak in the $g(r)$ consists primarily of the preferred hydrogen |
542 |
chrisfen |
861 |
bonding arrangements as dictated by the tetrahedral sticky potential - |
543 |
gezelter |
921 |
one peak for the hydrogen bond donor and the other for the hydrogen |
544 |
|
|
bond acceptor. Due to the high degree of crystallinity of the sample, |
545 |
|
|
the second and third solvation shells show a repeated peak arrangement |
546 |
chrisfen |
743 |
which decays at distances around the fourth solvation shell, near the |
547 |
|
|
imposed cutoff for the Lennard-Jones and dipole-dipole interactions. |
548 |
chrisfen |
861 |
In the higher temperature simulation shown in frames C and D, these |
549 |
gezelter |
921 |
long-range features deteriorate rapidly. The first solvation shell |
550 |
|
|
still shows the strong effect of the sticky-potential, although it |
551 |
|
|
covers a larger area, extending to include a fraction of aligned |
552 |
|
|
dipole peaks within the first solvation shell. The latter peaks lose |
553 |
|
|
due to thermal motion and as the competing dipole force overcomes the |
554 |
|
|
sticky potential's tight tetrahedral structuring of the crystal. |
555 |
chrisfen |
743 |
|
556 |
|
|
This complex interplay between dipole and sticky interactions was |
557 |
|
|
remarked upon as a possible reason for the split second peak in the |
558 |
gezelter |
1029 |
oxygen-oxygen pair correlation function, |
559 |
|
|
$g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, the second |
560 |
|
|
solvation shell peak appears to have two distinct components that |
561 |
|
|
blend together to form one observable peak. At higher temperatures, |
562 |
chrisfen |
1033 |
this split character alters to show the leading 4~\AA\ peak dominated |
563 |
gezelter |
1029 |
by equatorial anti-parallel dipole orientations. There is also a |
564 |
|
|
tightly bunched group of axially arranged dipoles that most likely |
565 |
|
|
consist of the smaller fraction of aligned dipole pairs. The trailing |
566 |
chrisfen |
1033 |
component of the split peak at 5~\AA\ is dominated by aligned dipoles |
567 |
gezelter |
1029 |
that assume hydrogen bond arrangements similar to those seen in the |
568 |
|
|
first solvation shell. This evidence indicates that the dipole pair |
569 |
|
|
interaction begins to dominate outside of the range of the dipolar |
570 |
|
|
repulsion term. The energetically favorable dipole arrangements |
571 |
|
|
populate the region immediately outside this repulsion region (around |
572 |
chrisfen |
1033 |
4~\AA), while arrangements that seek to satisfy both the sticky and |
573 |
gezelter |
1029 |
dipole forces locate themselves just beyond this initial buildup |
574 |
chrisfen |
1033 |
(around 5~\AA). |
575 |
chrisfen |
743 |
|
576 |
|
|
From these findings, the split second peak is primarily the product of |
577 |
chrisfen |
861 |
the dipolar repulsion term of the sticky potential. In fact, the inner |
578 |
|
|
peak can be pushed out and merged with the outer split peak just by |
579 |
gezelter |
921 |
extending the switching function ($s^\prime(r_{ij})$) from its normal |
580 |
chrisfen |
1033 |
4.0~\AA\ cutoff to values of 4.5 or even 5~\AA. This type of |
581 |
chrisfen |
861 |
correction is not recommended for improving the liquid structure, |
582 |
chrisfen |
862 |
since the second solvation shell would still be shifted too far |
583 |
chrisfen |
861 |
out. In addition, this would have an even more detrimental effect on |
584 |
|
|
the system densities, leading to a liquid with a more open structure |
585 |
chrisfen |
1030 |
and a density considerably lower than the already low SSD |
586 |
|
|
density. A better correction would be to include the |
587 |
|
|
quadrupole-quadrupole interactions for the water particles outside of |
588 |
|
|
the first solvation shell, but this would remove the simplicity and |
589 |
|
|
speed advantage of SSD. |
590 |
chrisfen |
743 |
|
591 |
chrisfen |
1030 |
\subsection{Adjusted Potentials: SSD/RF and SSD/E} |
592 |
gezelter |
921 |
|
593 |
chrisfen |
1030 |
The propensity of SSD to adopt lower than expected densities under |
594 |
chrisfen |
743 |
varying conditions is troubling, especially at higher temperatures. In |
595 |
chrisfen |
861 |
order to correct this model for use with a reaction field, it is |
596 |
|
|
necessary to adjust the force field parameters for the primary |
597 |
|
|
intermolecular interactions. In undergoing a reparameterization, it is |
598 |
|
|
important not to focus on just one property and neglect the other |
599 |
|
|
important properties. In this case, it would be ideal to correct the |
600 |
gezelter |
921 |
densities while maintaining the accurate transport behavior. |
601 |
chrisfen |
743 |
|
602 |
chrisfen |
1017 |
The parameters available for tuning include the $\sigma$ and |
603 |
|
|
$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the |
604 |
gezelter |
1029 |
strength of the sticky potential ($\nu_0$), and the cutoff distances |
605 |
|
|
for the sticky attractive and dipole repulsive cubic switching |
606 |
|
|
function cutoffs ($r_l$, $r_u$ and $r_l^\prime$, $r_u^\prime$ |
607 |
|
|
respectively). The results of the reparameterizations are shown in |
608 |
|
|
table \ref{params}. We are calling these reparameterizations the Soft |
609 |
chrisfen |
1030 |
Sticky Dipole / Reaction Field (SSD/RF - for use with a reaction |
610 |
|
|
field) and Soft Sticky Dipole Extended (SSD/E - an attempt to improve |
611 |
gezelter |
1029 |
the liquid structure in simulations without a long-range correction). |
612 |
chrisfen |
743 |
|
613 |
|
|
\begin{table} |
614 |
chrisfen |
862 |
\begin{center} |
615 |
chrisfen |
743 |
\caption{Parameters for the original and adjusted models} |
616 |
chrisfen |
856 |
\begin{tabular}{ l c c c c } |
617 |
chrisfen |
743 |
\hline \\[-3mm] |
618 |
chrisfen |
1033 |
\ \ \ Parameters\ \ \ & \ \ \ SSD [Ref. \onlinecite{Ichiye96}] \ \ \ |
619 |
|
|
& \ SSD1 [Ref. \onlinecite{Ichiye03}]\ \ & \ SSD/E\ \ & \ \ SSD/RF \\ |
620 |
chrisfen |
743 |
\hline \\[-3mm] |
621 |
chrisfen |
856 |
\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
622 |
|
|
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
623 |
|
|
\ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\ |
624 |
|
|
\ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ |
625 |
chrisfen |
1017 |
\ \ \ $\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\ |
626 |
chrisfen |
856 |
\ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\ |
627 |
|
|
\ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\ |
628 |
|
|
\ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\ |
629 |
|
|
\ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\ |
630 |
chrisfen |
743 |
\end{tabular} |
631 |
|
|
\label{params} |
632 |
chrisfen |
862 |
\end{center} |
633 |
chrisfen |
743 |
\end{table} |
634 |
|
|
|
635 |
chrisfen |
1033 |
%\begin{figure} |
636 |
|
|
%\begin{center} |
637 |
|
|
%\epsfxsize=5in |
638 |
|
|
%\epsfbox{GofRCompare.epsi} |
639 |
|
|
%\caption{Plots comparing experiment [Ref. \onlinecite{Head-Gordon00_1}] with |
640 |
|
|
%SSD/E and SSD1 without reaction field (top), as well as |
641 |
|
|
%SSD/RF and SSD1 with reaction field turned on |
642 |
|
|
%(bottom). The insets show the respective first peaks in detail. Note |
643 |
|
|
%how the changes in parameters have lowered and broadened the first |
644 |
|
|
%peak of SSD/E and SSD/RF.} |
645 |
|
|
%\label{grcompare} |
646 |
|
|
%\end{center} |
647 |
|
|
%\end{figure} |
648 |
chrisfen |
743 |
|
649 |
chrisfen |
1033 |
%\begin{figure} |
650 |
|
|
%\begin{center} |
651 |
|
|
%\epsfxsize=6in |
652 |
|
|
%\epsfbox{dualsticky_bw.eps} |
653 |
|
|
%\caption{Positive and negative isosurfaces of the sticky potential for |
654 |
|
|
%SSD1 (left) and SSD/E \& SSD/RF (right). Light areas |
655 |
|
|
%correspond to the tetrahedral attractive component, and darker areas |
656 |
|
|
%correspond to the dipolar repulsive component.} |
657 |
|
|
%\label{isosurface} |
658 |
|
|
%\end{center} |
659 |
|
|
%\end{figure} |
660 |
chrisfen |
743 |
|
661 |
chrisfen |
1030 |
In the original paper detailing the development of SSD, Liu and Ichiye |
662 |
gezelter |
921 |
placed particular emphasis on an accurate description of the first |
663 |
|
|
solvation shell. This resulted in a somewhat tall and narrow first |
664 |
|
|
peak in $g(r)$ that integrated to give similar coordination numbers to |
665 |
chrisfen |
862 |
the experimental data obtained by Soper and |
666 |
|
|
Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering |
667 |
|
|
data from the Head-Gordon lab indicates a slightly lower and shifted |
668 |
chrisfen |
1030 |
first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were |
669 |
gezelter |
1029 |
made after taking into consideration the new experimental |
670 |
chrisfen |
862 |
findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the |
671 |
gezelter |
921 |
relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing |
672 |
chrisfen |
1030 |
the revised SSD model (SSD1), SSD/E, and SSD/RF to the new |
673 |
chrisfen |
862 |
experimental results. Both modified water models have shorter peaks |
674 |
gezelter |
921 |
that match more closely to the experimental peak (as seen in the |
675 |
|
|
insets of figure \ref{grcompare}). This structural alteration was |
676 |
chrisfen |
862 |
accomplished by the combined reduction in the Lennard-Jones $\sigma$ |
677 |
gezelter |
921 |
variable and adjustment of the sticky potential strength and cutoffs. |
678 |
|
|
As can be seen in table \ref{params}, the cutoffs for the tetrahedral |
679 |
|
|
attractive and dipolar repulsive terms were nearly swapped with each |
680 |
|
|
other. Isosurfaces of the original and modified sticky potentials are |
681 |
|
|
shown in figure \ref{isosurface}. In these isosurfaces, it is easy to |
682 |
|
|
see how altering the cutoffs changes the repulsive and attractive |
683 |
|
|
character of the particles. With a reduced repulsive surface (darker |
684 |
|
|
region), the particles can move closer to one another, increasing the |
685 |
chrisfen |
1030 |
density for the overall system. This change in interaction cutoff |
686 |
|
|
also results in a more gradual orientational motion by allowing the |
687 |
gezelter |
921 |
particles to maintain preferred dipolar arrangements before they begin |
688 |
|
|
to feel the pull of the tetrahedral restructuring. As the particles |
689 |
|
|
move closer together, the dipolar repulsion term becomes active and |
690 |
|
|
excludes unphysical nearest-neighbor arrangements. This compares with |
691 |
chrisfen |
1030 |
how SSD and SSD1 exclude preferred dipole alignments before the |
692 |
gezelter |
921 |
particles feel the pull of the ``hydrogen bonds''. Aside from |
693 |
|
|
improving the shape of the first peak in the g(\emph{r}), this |
694 |
|
|
modification improves the densities considerably by allowing the |
695 |
chrisfen |
1033 |
persistence of full dipolar character below the previous 4.0~\AA\ |
696 |
gezelter |
921 |
cutoff. |
697 |
chrisfen |
743 |
|
698 |
gezelter |
921 |
While adjusting the location and shape of the first peak of $g(r)$ |
699 |
|
|
improves the densities, these changes alone are insufficient to bring |
700 |
|
|
the system densities up to the values observed experimentally. To |
701 |
|
|
further increase the densities, the dipole moments were increased in |
702 |
chrisfen |
1033 |
both of our adjusted models. Since SSD is a dipole based model, the |
703 |
|
|
structure and transport are very sensitive to changes in the dipole |
704 |
|
|
moment. The original SSD simply used the dipole moment calculated from |
705 |
|
|
the TIP3P water model, which at 2.35~D is significantly greater than |
706 |
|
|
the experimental gas phase value of 1.84~D. The larger dipole moment |
707 |
|
|
is a more realistic value and improves the dielectric properties of |
708 |
|
|
the fluid. Both theoretical and experimental measurements indicate a |
709 |
|
|
liquid phase dipole moment ranging from 2.4~D to values as high as |
710 |
|
|
3.11~D, providing a substantial range of reasonable values for a |
711 |
|
|
dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
712 |
|
|
increasing the dipole moments to 2.42 and 2.48~D for SSD/E and SSD/RF, |
713 |
|
|
respectively, leads to significant changes in the density and |
714 |
|
|
transport of the water models. |
715 |
chrisfen |
743 |
|
716 |
chrisfen |
861 |
In order to demonstrate the benefits of these reparameterizations, a |
717 |
chrisfen |
743 |
series of NPT and NVE simulations were performed to probe the density |
718 |
|
|
and transport properties of the adapted models and compare the results |
719 |
chrisfen |
1030 |
to the original SSD model. This comparison involved full NPT melting |
720 |
|
|
sequences for both SSD/E and SSD/RF, as well as NVE transport |
721 |
chrisfen |
861 |
calculations at the calculated self-consistent densities. Again, the |
722 |
chrisfen |
862 |
results are obtained from five separate simulations of 1024 particle |
723 |
|
|
systems, and the melting sequences were started from different ice |
724 |
|
|
$I_h$ crystals constructed as described previously. Each NPT |
725 |
chrisfen |
1033 |
simulation was equilibrated for 100~ps before a 200~ps data collection |
726 |
chrisfen |
862 |
run at each temperature step, and the final configuration from the |
727 |
|
|
previous temperature simulation was used as a starting point. All NVE |
728 |
|
|
simulations had the same thermalization, equilibration, and data |
729 |
gezelter |
921 |
collection times as stated previously. |
730 |
chrisfen |
743 |
|
731 |
chrisfen |
1033 |
%\begin{figure} |
732 |
|
|
%\begin{center} |
733 |
|
|
%\epsfxsize=6in |
734 |
|
|
%\epsfbox{ssdeDense.epsi} |
735 |
|
|
%\caption{Comparison of densities calculated with SSD/E to |
736 |
|
|
%SSD1 without a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
737 |
|
|
%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}] and |
738 |
|
|
%experiment [Ref. \onlinecite{CRC80}]. The window shows a expansion around |
739 |
|
|
%300 K with error bars included to clarify this region of |
740 |
|
|
%interest. Note that both SSD1 and SSD/E show good agreement with |
741 |
|
|
%experiment when the long-range correction is neglected.} |
742 |
|
|
%\label{ssdedense} |
743 |
|
|
%\end{center} |
744 |
|
|
%\end{figure} |
745 |
chrisfen |
743 |
|
746 |
chrisfen |
1030 |
Fig. \ref{ssdedense} shows the density profile for the SSD/E |
747 |
|
|
model in comparison to SSD1 without a reaction field, other |
748 |
|
|
common water models, and experimental results. The calculated |
749 |
|
|
densities for both SSD/E and SSD1 have increased |
750 |
|
|
significantly over the original SSD model (see |
751 |
|
|
fig. \ref{dense1}) and are in better agreement with the experimental |
752 |
|
|
values. At 298 K, the densities of SSD/E and SSD1 without |
753 |
chrisfen |
862 |
a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and |
754 |
|
|
0.999$\pm$0.001 g/cm$^3$ respectively. These both compare well with |
755 |
|
|
the experimental value of 0.997 g/cm$^3$, and they are considerably |
756 |
chrisfen |
1030 |
better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The |
757 |
|
|
changes to the dipole moment and sticky switching functions have |
758 |
|
|
improved the structuring of the liquid (as seen in figure |
759 |
|
|
\ref{grcompare}, but they have shifted the density maximum to much |
760 |
|
|
lower temperatures. This comes about via an increase in the liquid |
761 |
|
|
disorder through the weakening of the sticky potential and |
762 |
|
|
strengthening of the dipolar character. However, this increasing |
763 |
|
|
disorder in the SSD/E model has little effect on the melting |
764 |
|
|
transition. By monitoring $C_p$ throughout these simulations, the |
765 |
chrisfen |
1033 |
melting transition for SSD/E was shown to occur at 235~K. The |
766 |
chrisfen |
1030 |
same transition temperature observed with SSD and SSD1. |
767 |
chrisfen |
743 |
|
768 |
chrisfen |
1033 |
%\begin{figure} |
769 |
|
|
%\begin{center} |
770 |
|
|
%\epsfxsize=6in |
771 |
|
|
%\epsfbox{ssdrfDense.epsi} |
772 |
|
|
%\caption{Comparison of densities calculated with SSD/RF to |
773 |
|
|
%SSD1 with a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
774 |
|
|
%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}], and |
775 |
|
|
%experiment [Ref. \onlinecite{CRC80}]. The inset shows the necessity of |
776 |
|
|
%reparameterization when utilizing a reaction field long-ranged |
777 |
|
|
%correction - SSD/RF provides significantly more accurate |
778 |
|
|
%densities than SSD1 when performing room temperature |
779 |
|
|
%simulations.} |
780 |
|
|
%\label{ssdrfdense} |
781 |
|
|
%\end{center} |
782 |
|
|
%\end{figure} |
783 |
chrisfen |
743 |
|
784 |
chrisfen |
862 |
Including the reaction field long-range correction in the simulations |
785 |
gezelter |
921 |
results in a more interesting comparison. A density profile including |
786 |
chrisfen |
1030 |
SSD/RF and SSD1 with an active reaction field is shown in figure |
787 |
chrisfen |
862 |
\ref{ssdrfdense}. As observed in the simulations without a reaction |
788 |
chrisfen |
1030 |
field, the densities of SSD/RF and SSD1 show a dramatic increase over |
789 |
|
|
normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density |
790 |
chrisfen |
862 |
of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and |
791 |
chrisfen |
1030 |
considerably better than the original SSD value of 0.941$\pm$0.001 |
792 |
|
|
g/cm$^3$ and the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results |
793 |
gezelter |
921 |
further emphasize the importance of reparameterization in order to |
794 |
|
|
model the density properly under different simulation conditions. |
795 |
|
|
Again, these changes have only a minor effect on the melting point, |
796 |
chrisfen |
1033 |
which observed at 245~K for SSD/RF, is identical to SSD and only 5~K |
797 |
chrisfen |
1030 |
lower than SSD1 with a reaction field. Additionally, the difference in |
798 |
|
|
density maxima is not as extreme, with SSD/RF showing a density |
799 |
chrisfen |
1033 |
maximum at 255~K, fairly close to the density maxima of 260~K and |
800 |
|
|
265~K, shown by SSD and SSD1 respectively. |
801 |
chrisfen |
743 |
|
802 |
chrisfen |
1033 |
%\begin{figure} |
803 |
|
|
%\begin{center} |
804 |
|
|
%\epsfxsize=6in |
805 |
|
|
%\epsfbox{ssdeDiffuse.epsi} |
806 |
|
|
%\caption{The diffusion constants calculated from SSD/E and |
807 |
|
|
%SSD1 (both without a reaction field) along with experimental results |
808 |
|
|
%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The NVE calculations were |
809 |
|
|
%performed at the average densities observed in the 1 atm NPT |
810 |
|
|
%simulations for the respective models. SSD/E is slightly more mobile |
811 |
|
|
%than experiment at all of the temperatures, but it is closer to |
812 |
|
|
%experiment at biologically relevant temperatures than SSD1 without a |
813 |
|
|
%long-range correction.} |
814 |
|
|
%\label{ssdediffuse} |
815 |
|
|
%\end{center} |
816 |
|
|
%\end{figure} |
817 |
chrisfen |
861 |
|
818 |
chrisfen |
1030 |
The reparameterization of the SSD water model, both for use with and |
819 |
chrisfen |
743 |
without an applied long-range correction, brought the densities up to |
820 |
|
|
what is expected for simulating liquid water. In addition to improving |
821 |
chrisfen |
1030 |
the densities, it is important that the diffusive behavior of SSD be |
822 |
gezelter |
1029 |
maintained or improved. Figure \ref{ssdediffuse} compares the |
823 |
chrisfen |
1030 |
temperature dependence of the diffusion constant of SSD/E to SSD1 |
824 |
chrisfen |
1027 |
without an active reaction field at the densities calculated from |
825 |
|
|
their respective NPT simulations at 1 atm. The diffusion constant for |
826 |
chrisfen |
1030 |
SSD/E is consistently higher than experiment, while SSD1 remains lower |
827 |
chrisfen |
1027 |
than experiment until relatively high temperatures (around 360 |
828 |
|
|
K). Both models follow the shape of the experimental curve well below |
829 |
chrisfen |
1033 |
300~K but tend to diffuse too rapidly at higher temperatures, as seen |
830 |
|
|
in SSD1's crossing above 360~K. This increasing diffusion relative to |
831 |
chrisfen |
1027 |
the experimental values is caused by the rapidly decreasing system |
832 |
chrisfen |
1030 |
density with increasing temperature. Both SSD1 and SSD/E show this |
833 |
chrisfen |
1027 |
deviation in particle mobility, but this trend has different |
834 |
chrisfen |
1030 |
implications on the diffusive behavior of the models. While SSD1 |
835 |
chrisfen |
1027 |
shows more experimentally accurate diffusive behavior in the high |
836 |
chrisfen |
1030 |
temperature regimes, SSD/E shows more accurate behavior in the |
837 |
chrisfen |
1027 |
supercooled and biologically relevant temperature ranges. Thus, the |
838 |
|
|
changes made to improve the liquid structure may have had an adverse |
839 |
|
|
affect on the density maximum, but they improve the transport behavior |
840 |
chrisfen |
1030 |
of SSD/E relative to SSD1 under the most commonly simulated |
841 |
chrisfen |
1027 |
conditions. |
842 |
chrisfen |
743 |
|
843 |
chrisfen |
1033 |
%\begin{figure} |
844 |
|
|
%\begin{center} |
845 |
|
|
%\epsfxsize=6in |
846 |
|
|
%\epsfbox{ssdrfDiffuse.epsi} |
847 |
|
|
%\caption{The diffusion constants calculated from SSD/RF and |
848 |
|
|
%SSD1 (both with an active reaction field) along with |
849 |
|
|
%experimental results [Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The |
850 |
|
|
%NVE calculations were performed at the average densities observed in |
851 |
|
|
%the 1 atm NPT simulations for both of the models. SSD/RF |
852 |
|
|
%simulates the diffusion of water throughout this temperature range |
853 |
|
|
%very well. The rapidly increasing diffusion constants at high |
854 |
|
|
%temperatures for both models can be attributed to lower calculated |
855 |
|
|
%densities than those observed in experiment.} |
856 |
|
|
%\label{ssdrfdiffuse} |
857 |
|
|
%\end{center} |
858 |
|
|
%\end{figure} |
859 |
chrisfen |
743 |
|
860 |
chrisfen |
1030 |
In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are |
861 |
|
|
compared to SSD1 with an active reaction field. Note that SSD/RF |
862 |
gezelter |
921 |
tracks the experimental results quantitatively, identical within error |
863 |
chrisfen |
1017 |
throughout most of the temperature range shown and exhibiting only a |
864 |
chrisfen |
1030 |
slight increasing trend at higher temperatures. SSD1 tends to diffuse |
865 |
chrisfen |
1017 |
more slowly at low temperatures and deviates to diffuse too rapidly at |
866 |
chrisfen |
1033 |
temperatures greater than 330~K. As stated above, this deviation away |
867 |
gezelter |
921 |
from the ideal trend is due to a rapid decrease in density at higher |
868 |
chrisfen |
1030 |
temperatures. SSD/RF does not suffer from this problem as much as SSD1 |
869 |
gezelter |
921 |
because the calculated densities are closer to the experimental |
870 |
|
|
values. These results again emphasize the importance of careful |
871 |
|
|
reparameterization when using an altered long-range correction. |
872 |
chrisfen |
743 |
|
873 |
chrisfen |
1017 |
\begin{table} |
874 |
gezelter |
1029 |
\begin{minipage}{\linewidth} |
875 |
|
|
\renewcommand{\thefootnote}{\thempfootnote} |
876 |
chrisfen |
1017 |
\begin{center} |
877 |
gezelter |
1029 |
\caption{Properties of the single-point water models compared with |
878 |
chrisfen |
1033 |
experimental data at ambient conditions. Deviations of the of the |
879 |
|
|
averages are given in parentheses.} |
880 |
chrisfen |
1017 |
\begin{tabular}{ l c c c c c } |
881 |
|
|
\hline \\[-3mm] |
882 |
chrisfen |
1033 |
\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ \ SSD/E \ \ \ & \ \ SSD1 (RF) \ \ |
883 |
|
|
\ & \ \ SSD/RF \ \ \ & \ \ Expt. \\ |
884 |
chrisfen |
1017 |
\hline \\[-3mm] |
885 |
chrisfen |
1033 |
\ \ $\rho$ (g/cm$^3$) & 0.999(0.001) & 0.996(0.001) & 0.972(0.002) & 0.997(0.001) & 0.997 \\ |
886 |
|
|
\ \ $C_p$ (cal/mol K) & 28.80(0.11) & 25.45(0.09) & 28.28(0.06) & 23.83(0.16) & 17.98 \\ |
887 |
|
|
\ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78(0.7) & 2.51(0.18) & 2.00(0.17) & 2.32(0.06) & 2.299\cite{Mills73} \\ |
888 |
|
|
\ \ Coordination Number ($n_C$) & 3.9 & 4.3 & 3.8 & 4.4 & |
889 |
gezelter |
1029 |
4.7\footnote{Calculated by integrating $g_{\text{OO}}(r)$ in |
890 |
chrisfen |
1033 |
Ref. \onlinecite{Head-Gordon00_1}} \\ |
891 |
|
|
\ \ H-bonds per particle ($n_H$) & 3.7 & 3.6 & 3.7 & 3.7 & |
892 |
gezelter |
1029 |
3.5\footnote{Calculated by integrating $g_{\text{OH}}(r)$ in |
893 |
chrisfen |
1033 |
Ref. \onlinecite{Soper86}} \\ |
894 |
|
|
\ \ $\tau_1$ (ps) & 10.9(0.6) & 7.3(0.4) & 7.5(0.7) & 7.2(0.4) & 5.7\footnote{Calculated for 298 K from data in Ref. \onlinecite{Eisenberg69}} \\ |
895 |
|
|
\ \ $\tau_2$ (ps) & 4.7(0.4) & 3.1(0.2) & 3.5(0.3) & 3.2(0.2) & 2.3\footnote{Calculated for 298 K from data in |
896 |
|
|
Ref. \onlinecite{Krynicki66}} |
897 |
chrisfen |
1017 |
\end{tabular} |
898 |
|
|
\label{liquidproperties} |
899 |
|
|
\end{center} |
900 |
gezelter |
1029 |
\end{minipage} |
901 |
chrisfen |
1017 |
\end{table} |
902 |
|
|
|
903 |
|
|
Table \ref{liquidproperties} gives a synopsis of the liquid state |
904 |
|
|
properties of the water models compared in this study along with the |
905 |
|
|
experimental values for liquid water at ambient conditions. The |
906 |
gezelter |
1029 |
coordination number ($n_C$) and number of hydrogen bonds per particle |
907 |
|
|
($n_H$) were calculated by integrating the following relations: |
908 |
chrisfen |
1017 |
\begin{equation} |
909 |
gezelter |
1029 |
n_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2\text{g}_{\text{OO}}(r)dr, |
910 |
chrisfen |
1017 |
\end{equation} |
911 |
chrisfen |
1027 |
\begin{equation} |
912 |
gezelter |
1029 |
n_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2\text{g}_{\text{OH}}(r)dr, |
913 |
chrisfen |
1027 |
\end{equation} |
914 |
|
|
where $\rho$ is the number density of the specified pair interactions, |
915 |
|
|
$a$ and $b$ are the radial locations of the minima following the first |
916 |
gezelter |
1029 |
peak in g$_\text{OO}(r)$ or g$_\text{OH}(r)$ respectively. The number |
917 |
|
|
of hydrogen bonds stays relatively constant across all of the models, |
918 |
chrisfen |
1030 |
but the coordination numbers of SSD/E and SSD/RF show an |
919 |
|
|
improvement over SSD1. This improvement is primarily due to |
920 |
|
|
extension of the first solvation shell in the new parameter sets. |
921 |
|
|
Because $n_H$ and $n_C$ are nearly identical in SSD1, it appears |
922 |
|
|
that all molecules in the first solvation shell are involved in |
923 |
|
|
hydrogen bonds. Since $n_H$ and $n_C$ differ in the newly |
924 |
|
|
parameterized models, the orientations in the first solvation shell |
925 |
|
|
are a bit more ``fluid''. Therefore SSD1 overstructures the |
926 |
|
|
first solvation shell and our reparameterizations have returned this |
927 |
|
|
shell to more realistic liquid-like behavior. |
928 |
chrisfen |
1017 |
|
929 |
gezelter |
1029 |
The time constants for the orientational autocorrelation functions |
930 |
chrisfen |
1017 |
are also displayed in Table \ref{liquidproperties}. The dipolar |
931 |
gezelter |
1029 |
orientational time correlation functions ($C_{l}$) are described |
932 |
chrisfen |
1017 |
by: |
933 |
|
|
\begin{equation} |
934 |
gezelter |
1029 |
C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{\mathbf{u}}_j(t)]\rangle, |
935 |
chrisfen |
1017 |
\end{equation} |
936 |
gezelter |
1029 |
where $P_l$ are Legendre polynomials of order $l$ and |
937 |
|
|
$\hat{\mathbf{u}}_j$ is the unit vector pointing along the molecular |
938 |
|
|
dipole.\cite{Rahman71} From these correlation functions, the |
939 |
|
|
orientational relaxation time of the dipole vector can be calculated |
940 |
|
|
from an exponential fit in the long-time regime ($t > |
941 |
|
|
\tau_l$).\cite{Rothschild84} Calculation of these time constants were |
942 |
|
|
averaged over five detailed NVE simulations performed at the ambient |
943 |
|
|
conditions for each of the respective models. It should be noted that |
944 |
|
|
the commonly cited value of 1.9 ps for $\tau_2$ was determined from |
945 |
chrisfen |
1033 |
the NMR data in Ref. \onlinecite{Krynicki66} at a temperature near |
946 |
gezelter |
1029 |
34$^\circ$C.\cite{Rahman71} Because of the strong temperature |
947 |
chrisfen |
1033 |
dependence of $\tau_2$, it is necessary to recalculate it at 298~K to |
948 |
gezelter |
1029 |
make proper comparisons. The value shown in Table |
949 |
chrisfen |
1022 |
\ref{liquidproperties} was calculated from the same NMR data in the |
950 |
chrisfen |
1033 |
fashion described in Ref. \onlinecite{Krynicki66}. Similarly, $\tau_1$ was |
951 |
|
|
recomputed for 298~K from the data in Ref. \onlinecite{Eisenberg69}. |
952 |
chrisfen |
1030 |
Again, SSD/E and SSD/RF show improved behavior over SSD1, both with |
953 |
chrisfen |
1027 |
and without an active reaction field. Turning on the reaction field |
954 |
chrisfen |
1030 |
leads to much improved time constants for SSD1; however, these results |
955 |
gezelter |
1029 |
also include a corresponding decrease in system density. |
956 |
chrisfen |
1030 |
Orientational relaxation times published in the original SSD dynamics |
957 |
gezelter |
1029 |
papers are smaller than the values observed here, and this difference |
958 |
|
|
can be attributed to the use of the Ewald sum.\cite{Ichiye99} |
959 |
chrisfen |
1017 |
|
960 |
chrisfen |
743 |
\subsection{Additional Observations} |
961 |
|
|
|
962 |
chrisfen |
1033 |
%\begin{figure} |
963 |
|
|
%\begin{center} |
964 |
|
|
%\epsfxsize=6in |
965 |
|
|
%\epsfbox{icei_bw.eps} |
966 |
|
|
%\caption{The most stable crystal structure assumed by the SSD family |
967 |
|
|
%of water models. We refer to this structure as Ice-{\it i} to |
968 |
|
|
%indicate its origins in computer simulation. This image was taken of |
969 |
|
|
%the (001) face of the crystal.} |
970 |
|
|
%\label{weirdice} |
971 |
|
|
%\end{center} |
972 |
|
|
%\end{figure} |
973 |
chrisfen |
743 |
|
974 |
gezelter |
921 |
While performing a series of melting simulations on an early iteration |
975 |
chrisfen |
1030 |
of SSD/E not discussed in this paper, we observed |
976 |
|
|
recrystallization into a novel structure not previously known for |
977 |
chrisfen |
1033 |
water. After melting at 235~K, two of five systems underwent |
978 |
|
|
crystallization events near 245~K. The two systems remained |
979 |
|
|
crystalline up to 320 and 330~K, respectively. The crystal exhibits |
980 |
chrisfen |
1030 |
an expanded zeolite-like structure that does not correspond to any |
981 |
|
|
known form of ice. This appears to be an artifact of the point |
982 |
|
|
dipolar models, so to distinguish it from the experimentally observed |
983 |
|
|
forms of ice, we have denoted the structure |
984 |
gezelter |
1029 |
Ice-$\sqrt{\smash[b]{-\text{I}}}$ (Ice-{\it i}). A large enough |
985 |
gezelter |
921 |
portion of the sample crystallized that we have been able to obtain a |
986 |
gezelter |
1029 |
near ideal crystal structure of Ice-{\it i}. Figure \ref{weirdice} |
987 |
gezelter |
921 |
shows the repeating crystal structure of a typical crystal at 5 |
988 |
|
|
K. Each water molecule is hydrogen bonded to four others; however, the |
989 |
|
|
hydrogen bonds are bent rather than perfectly straight. This results |
990 |
|
|
in a skewed tetrahedral geometry about the central molecule. In |
991 |
|
|
figure \ref{isosurface}, it is apparent that these flexed hydrogen |
992 |
|
|
bonds are allowed due to the conical shape of the attractive regions, |
993 |
|
|
with the greatest attraction along the direct hydrogen bond |
994 |
chrisfen |
863 |
configuration. Though not ideal, these flexed hydrogen bonds are |
995 |
gezelter |
921 |
favorable enough to stabilize an entire crystal generated around them. |
996 |
chrisfen |
743 |
|
997 |
gezelter |
1029 |
Initial simulations indicated that Ice-{\it i} is the preferred ice |
998 |
chrisfen |
1030 |
structure for at least the SSD/E model. To verify this, a comparison |
999 |
|
|
was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and |
1000 |
|
|
Ice-{\it i} at constant pressure with SSD/E, SSD/RF, and |
1001 |
|
|
SSD1. Near-ideal versions of the three types of crystals were cooled |
1002 |
|
|
to 1 K, and enthalpies of formation of each were compared using all |
1003 |
|
|
three water models. Enthalpies were estimated from the |
1004 |
|
|
isobaric-isothermal simulations using $H=U+P_{\text ext}V$ where |
1005 |
|
|
$P_{\text ext}$ is the applied pressure. A constant value of -60.158 |
1006 |
|
|
kcal / mol has been added to place our zero for the enthalpies of |
1007 |
|
|
formation for these systems at the traditional state (elemental forms |
1008 |
|
|
at standard temperature and pressure). With every model in the SSD |
1009 |
|
|
family, Ice-{\it i} had the lowest calculated enthalpy of formation. |
1010 |
chrisfen |
743 |
|
1011 |
gezelter |
921 |
\begin{table} |
1012 |
|
|
\begin{center} |
1013 |
gezelter |
1029 |
\caption{Enthalpies of Formation (in kcal / mol) of the three crystal |
1014 |
chrisfen |
1030 |
structures (at 1 K) exhibited by the SSD family of water models} |
1015 |
gezelter |
921 |
\begin{tabular}{ l c c c } |
1016 |
|
|
\hline \\[-3mm] |
1017 |
chrisfen |
1033 |
\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ \ \ Ice-$I_c$ \ \ \ & |
1018 |
|
|
\ \ \ \ Ice-{\it i} \\ |
1019 |
gezelter |
921 |
\hline \\[-3mm] |
1020 |
chrisfen |
1030 |
\ \ \ SSD/E & -72.444 & -72.450 & -73.748 \\ |
1021 |
|
|
\ \ \ SSD/RF & -73.093 & -73.075 & -74.180 \\ |
1022 |
|
|
\ \ \ SSD1 & -72.654 & -72.569 & -73.575 \\ |
1023 |
|
|
\ \ \ SSD1 (RF) & -72.662 & -72.569 & -73.292 \\ |
1024 |
gezelter |
921 |
\end{tabular} |
1025 |
|
|
\label{iceenthalpy} |
1026 |
|
|
\end{center} |
1027 |
|
|
\end{table} |
1028 |
chrisfen |
743 |
|
1029 |
gezelter |
921 |
In addition to these energetic comparisons, melting simulations were |
1030 |
chrisfen |
1033 |
performed with Ice-{\it i} as the initial configuration using SSD/E, |
1031 |
chrisfen |
1030 |
SSD/RF, and SSD1 both with and without a reaction field. The melting |
1032 |
|
|
transitions for both SSD/E and SSD1 without reaction field occurred at |
1033 |
|
|
temperature in excess of 375~K. SSD/RF and SSD1 with a reaction field |
1034 |
gezelter |
921 |
showed more reasonable melting transitions near 325~K. These melting |
1035 |
chrisfen |
1030 |
point observations clearly show that all of the SSD-derived models |
1036 |
gezelter |
921 |
prefer the ice-{\it i} structure. |
1037 |
chrisfen |
743 |
|
1038 |
|
|
\section{Conclusions} |
1039 |
|
|
|
1040 |
gezelter |
921 |
The density maximum and temperature dependence of the self-diffusion |
1041 |
chrisfen |
1030 |
constant were studied for the SSD water model, both with and |
1042 |
|
|
without the use of reaction field, via a series of NPT and NVE |
1043 |
gezelter |
921 |
simulations. The constant pressure simulations showed a density |
1044 |
|
|
maximum near 260 K. In most cases, the calculated densities were |
1045 |
|
|
significantly lower than the densities obtained from other water |
1046 |
chrisfen |
1030 |
models (and experiment). Analysis of self-diffusion showed SSD |
1047 |
|
|
to capture the transport properties of water well in both the liquid |
1048 |
|
|
and supercooled liquid regimes. |
1049 |
gezelter |
921 |
|
1050 |
chrisfen |
1030 |
In order to correct the density behavior, the original SSD model was |
1051 |
|
|
reparameterized for use both with and without a reaction field (SSD/RF |
1052 |
|
|
and SSD/E), and comparisons were made with SSD1, Ichiye's density |
1053 |
|
|
corrected version of SSD. Both models improve the liquid structure, |
1054 |
gezelter |
921 |
densities, and diffusive properties under their respective simulation |
1055 |
|
|
conditions, indicating the necessity of reparameterization when |
1056 |
|
|
changing the method of calculating long-range electrostatic |
1057 |
|
|
interactions. In general, however, these simple water models are |
1058 |
|
|
excellent choices for representing explicit water in large scale |
1059 |
|
|
simulations of biochemical systems. |
1060 |
|
|
|
1061 |
|
|
The existence of a novel low-density ice structure that is preferred |
1062 |
chrisfen |
1030 |
by the SSD family of water models is somewhat troubling, since |
1063 |
|
|
liquid simulations on this family of water models at room temperature |
1064 |
|
|
are effectively simulations of supercooled or metastable liquids. One |
1065 |
chrisfen |
1027 |
way to destabilize this unphysical ice structure would be to make the |
1066 |
gezelter |
921 |
range of angles preferred by the attractive part of the sticky |
1067 |
|
|
potential much narrower. This would require extensive |
1068 |
|
|
reparameterization to maintain the same level of agreement with the |
1069 |
|
|
experiments. |
1070 |
|
|
|
1071 |
gezelter |
1029 |
Additionally, our initial calculations show that the Ice-{\it i} |
1072 |
gezelter |
921 |
structure may also be a preferred crystal structure for at least one |
1073 |
|
|
other popular multi-point water model (TIP3P), and that much of the |
1074 |
|
|
simulation work being done using this popular model could also be at |
1075 |
|
|
risk for crystallization into this unphysical structure. A future |
1076 |
|
|
publication will detail the relative stability of the known ice |
1077 |
|
|
structures for a wide range of popular water models. |
1078 |
|
|
|
1079 |
chrisfen |
743 |
\section{Acknowledgments} |
1080 |
chrisfen |
777 |
Support for this project was provided by the National Science |
1081 |
|
|
Foundation under grant CHE-0134881. Computation time was provided by |
1082 |
|
|
the Notre Dame Bunch-of-Boxes (B.o.B) computer cluster under NSF grant |
1083 |
gezelter |
921 |
DMR-0079647. |
1084 |
chrisfen |
743 |
|
1085 |
chrisfen |
862 |
\newpage |
1086 |
|
|
|
1087 |
chrisfen |
743 |
\bibliographystyle{jcp} |
1088 |
chrisfen |
1033 |
\bibliography{nptSSD} |
1089 |
chrisfen |
743 |
|
1090 |
chrisfen |
1033 |
\newpage |
1091 |
chrisfen |
743 |
|
1092 |
chrisfen |
1033 |
\begin{list} |
1093 |
|
|
{Figure \arabic{captions}: }{\usecounter{captions} |
1094 |
|
|
\setlength{\rightmargin}{\leftmargin}} |
1095 |
|
|
|
1096 |
|
|
\item Energy conservation using both quaternion-based integration and |
1097 |
|
|
the {\sc dlm} method with increasing time step. The larger time step |
1098 |
|
|
plots are shifted from the true energy baseline (that of $\Delta t$ = |
1099 |
|
|
0.1~fs) for clarity. |
1100 |
|
|
|
1101 |
|
|
\item Density versus temperature for TIP4P [Ref. \onlinecite{Jorgensen98b}], |
1102 |
|
|
TIP3P [Ref. \onlinecite{Jorgensen98b}], SPC/E |
1103 |
|
|
[Ref. \onlinecite{Clancy94}], SSD without Reaction Field, SSD, and |
1104 |
|
|
experiment [Ref. \onlinecite{CRC80}]. The arrows indicate the change |
1105 |
|
|
in densities observed when turning off the reaction field. The the |
1106 |
|
|
lower than expected densities for the SSD model were what prompted the |
1107 |
|
|
original reparameterization of SSD1 [Ref. \onlinecite{Ichiye03}]. |
1108 |
|
|
|
1109 |
|
|
\item Average self-diffusion constant as a function of temperature for |
1110 |
|
|
SSD, SPC/E [Ref. \onlinecite{Clancy94}], and TIP5P |
1111 |
|
|
[Ref. \onlinecite{Jorgensen01}] compared with experimental data |
1112 |
|
|
[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. Of the three |
1113 |
|
|
water models shown, SSD has the least deviation from the experimental |
1114 |
|
|
values. The rapidly increasing diffusion constants for TIP5P and SSD |
1115 |
|
|
correspond to significant decreases in density at the higher |
1116 |
|
|
temperatures. |
1117 |
|
|
|
1118 |
|
|
\item An illustration of angles involved in the correlations observed in |
1119 |
|
|
Fig. \ref{contour}. |
1120 |
|
|
|
1121 |
|
|
\item Contour plots of 2D angular pair correlation functions for |
1122 |
|
|
512 SSD molecules at 100~K (A \& B) and 300~K (C \& D). Dark areas |
1123 |
|
|
signify regions of enhanced density while light areas signify |
1124 |
|
|
depletion relative to the bulk density. White areas have pair |
1125 |
|
|
correlation values below 0.5 and black areas have values above 1.5. |
1126 |
|
|
|
1127 |
|
|
\item Plots comparing experiment [Ref. \onlinecite{Head-Gordon00_1}] with |
1128 |
|
|
SSD/E and SSD1 without reaction field (top), as well as SSD/RF and |
1129 |
|
|
SSD1 with reaction field turned on (bottom). The insets show the |
1130 |
|
|
respective first peaks in detail. Note how the changes in parameters |
1131 |
|
|
have lowered and broadened the first peak of SSD/E and SSD/RF. |
1132 |
|
|
|
1133 |
|
|
\item Positive and negative isosurfaces of the sticky potential for |
1134 |
|
|
SSD1 (left) and SSD/E \& SSD/RF (right). Light areas |
1135 |
|
|
correspond to the tetrahedral attractive component, and darker areas |
1136 |
|
|
correspond to the dipolar repulsive component. |
1137 |
|
|
|
1138 |
|
|
\item Comparison of densities calculated with SSD/E to |
1139 |
|
|
SSD1 without a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
1140 |
|
|
TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}] and |
1141 |
|
|
experiment [Ref. \onlinecite{CRC80}]. The window shows a expansion around |
1142 |
|
|
300 K with error bars included to clarify this region of |
1143 |
|
|
interest. Note that both SSD1 and SSD/E show good agreement with |
1144 |
|
|
experiment when the long-range correction is neglected. |
1145 |
|
|
|
1146 |
|
|
\item Comparison of densities calculated with SSD/RF to |
1147 |
|
|
SSD1 with a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
1148 |
|
|
TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}], and |
1149 |
|
|
experiment [Ref. \onlinecite{CRC80}]. The inset shows the necessity of |
1150 |
|
|
reparameterization when utilizing a reaction field long-ranged |
1151 |
|
|
correction - SSD/RF provides significantly more accurate |
1152 |
|
|
densities than SSD1 when performing room temperature |
1153 |
|
|
simulations. |
1154 |
|
|
|
1155 |
|
|
\item The diffusion constants calculated from SSD/E and |
1156 |
|
|
SSD1 (both without a reaction field) along with experimental results |
1157 |
|
|
[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The NVE calculations were |
1158 |
|
|
performed at the average densities observed in the 1 atm NPT |
1159 |
|
|
simulations for the respective models. SSD/E is slightly more mobile |
1160 |
|
|
than experiment at all of the temperatures, but it is closer to |
1161 |
|
|
experiment at biologically relevant temperatures than SSD1 without a |
1162 |
|
|
long-range correction. |
1163 |
|
|
|
1164 |
|
|
\item The diffusion constants calculated from SSD/RF and |
1165 |
|
|
SSD1 (both with an active reaction field) along with |
1166 |
|
|
experimental results [Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The |
1167 |
|
|
NVE calculations were performed at the average densities observed in |
1168 |
|
|
the 1 atm NPT simulations for both of the models. SSD/RF |
1169 |
|
|
simulates the diffusion of water throughout this temperature range |
1170 |
|
|
very well. The rapidly increasing diffusion constants at high |
1171 |
|
|
temperatures for both models can be attributed to lower calculated |
1172 |
|
|
densities than those observed in experiment. |
1173 |
|
|
|
1174 |
|
|
\item The most stable crystal structure assumed by the SSD family |
1175 |
|
|
of water models. We refer to this structure as Ice-{\it i} to |
1176 |
|
|
indicate its origins in computer simulation. This image was taken of |
1177 |
|
|
the (001) face of the crystal. |
1178 |
|
|
\end{list} |
1179 |
|
|
|
1180 |
|
|
\newpage |
1181 |
|
|
|
1182 |
|
|
\begin{figure} |
1183 |
|
|
\begin{center} |
1184 |
|
|
\epsfxsize=6in |
1185 |
|
|
\epsfbox{timeStep.epsi} |
1186 |
|
|
%\caption{Energy conservation using both quaternion-based integration and |
1187 |
|
|
%the {\sc dlm} method with increasing time step. The larger time step |
1188 |
|
|
%plots are shifted from the true energy baseline (that of $\Delta t$ = |
1189 |
|
|
%0.1~fs) for clarity.} |
1190 |
|
|
\label{timestep} |
1191 |
|
|
\end{center} |
1192 |
|
|
\end{figure} |
1193 |
|
|
|
1194 |
|
|
\newpage |
1195 |
|
|
|
1196 |
|
|
\begin{figure} |
1197 |
|
|
\begin{center} |
1198 |
|
|
\epsfxsize=6in |
1199 |
|
|
\epsfbox{denseSSDnew.eps} |
1200 |
|
|
%\caption{Density versus temperature for TIP4P [Ref. \onlinecite{Jorgensen98b}], |
1201 |
|
|
% TIP3P [Ref. \onlinecite{Jorgensen98b}], SPC/E [Ref. \onlinecite{Clancy94}], SSD |
1202 |
|
|
% without Reaction Field, SSD, and experiment [Ref. \onlinecite{CRC80}]. The |
1203 |
|
|
% arrows indicate the change in densities observed when turning off the |
1204 |
|
|
% reaction field. The the lower than expected densities for the SSD |
1205 |
|
|
% model were what prompted the original reparameterization of SSD1 |
1206 |
|
|
% [Ref. \onlinecite{Ichiye03}].} |
1207 |
|
|
\label{dense1} |
1208 |
|
|
\end{center} |
1209 |
|
|
\end{figure} |
1210 |
|
|
|
1211 |
|
|
\newpage |
1212 |
|
|
|
1213 |
|
|
\begin{figure} |
1214 |
|
|
\begin{center} |
1215 |
|
|
\epsfxsize=6in |
1216 |
|
|
\epsfbox{betterDiffuse.epsi} |
1217 |
|
|
%\caption{Average self-diffusion constant as a function of temperature for |
1218 |
|
|
%SSD, SPC/E [Ref. \onlinecite{Clancy94}], and TIP5P |
1219 |
|
|
%[Ref. \onlinecite{Jorgensen01}] compared with experimental data |
1220 |
|
|
%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. Of the three water models |
1221 |
|
|
%shown, SSD has the least deviation from the experimental values. The |
1222 |
|
|
%rapidly increasing diffusion constants for TIP5P and SSD correspond to |
1223 |
|
|
%significant decreases in density at the higher temperatures.} |
1224 |
|
|
\label{diffuse} |
1225 |
|
|
\end{center} |
1226 |
|
|
\end{figure} |
1227 |
|
|
|
1228 |
|
|
\newpage |
1229 |
|
|
|
1230 |
|
|
\begin{figure} |
1231 |
|
|
\begin{center} |
1232 |
|
|
\epsfxsize=6in |
1233 |
|
|
\epsfbox{corrDiag.eps} |
1234 |
|
|
%\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.} |
1235 |
|
|
\label{corrAngle} |
1236 |
|
|
\end{center} |
1237 |
|
|
\end{figure} |
1238 |
|
|
|
1239 |
|
|
\newpage |
1240 |
|
|
|
1241 |
|
|
\begin{figure} |
1242 |
|
|
\begin{center} |
1243 |
|
|
\epsfxsize=6in |
1244 |
|
|
\epsfbox{fullContours.eps} |
1245 |
|
|
%\caption{Contour plots of 2D angular pair correlation functions for |
1246 |
|
|
%512 SSD molecules at 100~K (A \& B) and 300~K (C \& D). Dark areas |
1247 |
|
|
%signify regions of enhanced density while light areas signify |
1248 |
|
|
%depletion relative to the bulk density. White areas have pair |
1249 |
|
|
%correlation values below 0.5 and black areas have values above 1.5.} |
1250 |
|
|
\label{contour} |
1251 |
|
|
\end{center} |
1252 |
|
|
\end{figure} |
1253 |
|
|
|
1254 |
|
|
\newpage |
1255 |
|
|
|
1256 |
|
|
\begin{figure} |
1257 |
|
|
\begin{center} |
1258 |
|
|
\epsfxsize=6in |
1259 |
|
|
\epsfbox{GofRCompare.epsi} |
1260 |
|
|
%\caption{Plots comparing experiment [Ref. \onlinecite{Head-Gordon00_1}] with |
1261 |
|
|
%SSD/E and SSD1 without reaction field (top), as well as |
1262 |
|
|
%SSD/RF and SSD1 with reaction field turned on |
1263 |
|
|
%(bottom). The insets show the respective first peaks in detail. Note |
1264 |
|
|
%how the changes in parameters have lowered and broadened the first |
1265 |
|
|
%peak of SSD/E and SSD/RF.} |
1266 |
|
|
\label{grcompare} |
1267 |
|
|
\end{center} |
1268 |
|
|
\end{figure} |
1269 |
|
|
|
1270 |
|
|
\newpage |
1271 |
|
|
|
1272 |
|
|
\begin{figure} |
1273 |
|
|
\begin{center} |
1274 |
|
|
\epsfxsize=7in |
1275 |
|
|
\epsfbox{dualsticky_bw.eps} |
1276 |
|
|
%\caption{Positive and negative isosurfaces of the sticky potential for |
1277 |
|
|
%SSD1 (left) and SSD/E \& SSD/RF (right). Light areas |
1278 |
|
|
%correspond to the tetrahedral attractive component, and darker areas |
1279 |
|
|
%correspond to the dipolar repulsive component.} |
1280 |
|
|
\label{isosurface} |
1281 |
|
|
\end{center} |
1282 |
|
|
\end{figure} |
1283 |
|
|
|
1284 |
|
|
\newpage |
1285 |
|
|
|
1286 |
|
|
\begin{figure} |
1287 |
|
|
\begin{center} |
1288 |
|
|
\epsfxsize=6in |
1289 |
|
|
\epsfbox{ssdeDense.epsi} |
1290 |
|
|
%\caption{Comparison of densities calculated with SSD/E to |
1291 |
|
|
%SSD1 without a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
1292 |
|
|
%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}] and |
1293 |
|
|
%experiment [Ref. \onlinecite{CRC80}]. The window shows a expansion around |
1294 |
|
|
%300 K with error bars included to clarify this region of |
1295 |
|
|
%interest. Note that both SSD1 and SSD/E show good agreement with |
1296 |
|
|
%experiment when the long-range correction is neglected.} |
1297 |
|
|
\label{ssdedense} |
1298 |
|
|
\end{center} |
1299 |
|
|
\end{figure} |
1300 |
|
|
|
1301 |
|
|
\newpage |
1302 |
|
|
|
1303 |
|
|
\begin{figure} |
1304 |
|
|
\begin{center} |
1305 |
|
|
\epsfxsize=6in |
1306 |
|
|
\epsfbox{ssdrfDense.epsi} |
1307 |
|
|
%\caption{Comparison of densities calculated with SSD/RF to |
1308 |
|
|
%SSD1 with a reaction field, TIP3P [Ref. \onlinecite{Jorgensen98b}], |
1309 |
|
|
%TIP5P [Ref. \onlinecite{Jorgensen00}], SPC/E [Ref. \onlinecite{Clancy94}], and |
1310 |
|
|
%experiment [Ref. \onlinecite{CRC80}]. The inset shows the necessity of |
1311 |
|
|
%reparameterization when utilizing a reaction field long-ranged |
1312 |
|
|
%correction - SSD/RF provides significantly more accurate |
1313 |
|
|
%densities than SSD1 when performing room temperature |
1314 |
|
|
%simulations.} |
1315 |
|
|
\label{ssdrfdense} |
1316 |
|
|
\end{center} |
1317 |
|
|
\end{figure} |
1318 |
|
|
|
1319 |
|
|
\newpage |
1320 |
|
|
|
1321 |
|
|
\begin{figure} |
1322 |
|
|
\begin{center} |
1323 |
|
|
\epsfxsize=6in |
1324 |
|
|
\epsfbox{ssdeDiffuse.epsi} |
1325 |
|
|
%\caption{The diffusion constants calculated from SSD/E and |
1326 |
|
|
%SSD1 (both without a reaction field) along with experimental results |
1327 |
|
|
%[Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The NVE calculations were |
1328 |
|
|
%performed at the average densities observed in the 1 atm NPT |
1329 |
|
|
%simulations for the respective models. SSD/E is slightly more mobile |
1330 |
|
|
%than experiment at all of the temperatures, but it is closer to |
1331 |
|
|
%experiment at biologically relevant temperatures than SSD1 without a |
1332 |
|
|
%long-range correction.} |
1333 |
|
|
\label{ssdediffuse} |
1334 |
|
|
\end{center} |
1335 |
|
|
\end{figure} |
1336 |
|
|
|
1337 |
|
|
\newpage |
1338 |
|
|
|
1339 |
|
|
\begin{figure} |
1340 |
|
|
\begin{center} |
1341 |
|
|
\epsfxsize=6in |
1342 |
|
|
\epsfbox{ssdrfDiffuse.epsi} |
1343 |
|
|
%\caption{The diffusion constants calculated from SSD/RF and |
1344 |
|
|
%SSD1 (both with an active reaction field) along with |
1345 |
|
|
%experimental results [Refs. \onlinecite{Gillen72} and \onlinecite{Holz00}]. The |
1346 |
|
|
%NVE calculations were performed at the average densities observed in |
1347 |
|
|
%the 1 atm NPT simulations for both of the models. SSD/RF |
1348 |
|
|
%simulates the diffusion of water throughout this temperature range |
1349 |
|
|
%very well. The rapidly increasing diffusion constants at high |
1350 |
|
|
%temperatures for both models can be attributed to lower calculated |
1351 |
|
|
%densities than those observed in experiment.} |
1352 |
|
|
\label{ssdrfdiffuse} |
1353 |
|
|
\end{center} |
1354 |
|
|
\end{figure} |
1355 |
|
|
|
1356 |
|
|
\newpage |
1357 |
|
|
|
1358 |
|
|
\begin{figure} |
1359 |
|
|
\begin{center} |
1360 |
|
|
\epsfxsize=6in |
1361 |
|
|
\epsfbox{icei_bw.eps} |
1362 |
|
|
%\caption{The most stable crystal structure assumed by the SSD family |
1363 |
|
|
%of water models. We refer to this structure as Ice-{\it i} to |
1364 |
|
|
%indicate its origins in computer simulation. This image was taken of |
1365 |
|
|
%the (001) face of the crystal.} |
1366 |
|
|
\label{weirdice} |
1367 |
|
|
\end{center} |
1368 |
|
|
\end{figure} |
1369 |
|
|
|
1370 |
chrisfen |
743 |
\end{document} |