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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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Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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\author{Christopher J. Fennell and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: |
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gezelter@nd.edu} \\ Department of Chemistry and Biochemistry\\ University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle |
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\doublespacing |
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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 re-parameterizations of this single-point model. |
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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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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 re-parameterized versions of SSD |
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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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%\narrowtext |
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– |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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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 along with a |
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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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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, Ichiye {\it et al.} reported that using |
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SSD decreased computer time by a factor of 6-7 compared to other |
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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 |
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Soper data for the structural features of liquid |
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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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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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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 method or by resorting to a simple |
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cubic switching function at a cutoff radius. The reaction field |
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method was actually first used in Monte Carlo simulations of liquid |
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water.\cite{Barker73} Under this method, the magnitude of the reaction |
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field acting on dipole $i$ is |
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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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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, such as the |
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dielectric relaxation time. There is particular sensitivity of this |
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property on changes in the length of the cutoff |
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radius.\cite{Berendsen98} This variable behavior makes reaction field |
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a less attractive method than the Ewald sum. However, for very large |
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systems, the computational benefit of reaction field is dramatic. |
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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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|
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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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\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 plots |
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are shifted from the true energy baseline (that of $\Delta t$ = 0.1 |
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fs) for clarity.} |
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\caption{Energy conservation using both quaternion-based integration and the |
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{\sc dlm} method with increasing time step. The larger time step plots |
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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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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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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 |
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drift over time in the 0.5 fs time step simulation. At time steps of 1 |
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and 2 fs, the energy conservation benefits of the {\sc dlm} method are |
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drift over time in the 0.5~fs time step simulation. At time steps of 1 |
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and 2~fs, the energy conservation benefits of the {\sc dlm} method are |
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clearly demonstrated. Thus, while maintaining the same degree of |
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energy conservation, one can take considerably longer time steps, |
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leading to an overall reduction in computation time. |
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|
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Energy drift in the simulations using {\sc dlm} integration was |
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unnoticeable for time steps up to 3 fs. A slight energy drift on the |
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order of 0.012 kcal/mol per nanosecond was observed at a time step of |
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4 fs, and as expected, this drift increases dramatically with |
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unnoticeable for time steps up to 3~fs. A slight energy drift on the |
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order of 0.012~kcal/mol per nanosecond was observed at a time step of |
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4~fs, and as expected, this drift increases dramatically with |
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increasing time step. To insure accuracy in our microcanonical |
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simulations, time steps were set at 2 fs and kept at this value for |
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simulations, time steps were set at 2~fs and kept at this value for |
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constant pressure simulations as well. |
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|
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Proton-disordered ice crystals in both the $I_h$ and $I_c$ lattices |
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were generated as starting points for all simulations. The $I_h$ |
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crystals were formed by first arranging the centers of mass of the |
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SSD particles into a ``hexagonal'' ice lattice of 1024 |
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particles. Because of the crystal structure of $I_h$ ice, the |
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simulation box assumed an orthorhombic shape with an edge length ratio |
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of approximately 1.00$\times$1.06$\times$1.23. The particles were then |
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allowed to orient freely about fixed positions with angular momenta |
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randomized at 400 K for varying times. The rotational temperature was |
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then scaled down in stages to slowly cool the crystals to 25 K. The |
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particles were then allowed to translate with fixed orientations at a |
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constant pressure of 1 atm for 50 ps at 25 K. Finally, all constraints |
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were removed and the ice crystals were allowed to equilibrate for 50 |
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ps at 25 K and a constant pressure of 1 atm. This procedure resulted |
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in structurally stable $I_h$ ice crystals that obey the Bernal-Fowler |
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crystals were formed by first arranging the centers of mass of the SSD |
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particles into a ``hexagonal'' ice lattice of 1024 particles. Because |
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of the crystal structure of $I_h$ ice, the simulation box assumed an |
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orthorhombic shape with an edge length ratio of approximately |
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1.00$\times$1.06$\times$1.23. The particles were then allowed to |
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orient freely about fixed positions with angular momenta randomized at |
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400~K for varying times. The rotational temperature was then scaled |
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down in stages to slowly cool the crystals to 25~K. The particles were |
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then allowed to translate with fixed orientations at a constant |
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pressure of 1 atm for 50~ps at 25~K. Finally, all constraints were |
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removed and the ice crystals were allowed to equilibrate for 50~ps at |
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25~K and a constant pressure of 1~atm. This procedure resulted in |
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structurally stable $I_h$ ice crystals that obey the Bernal-Fowler |
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rules.\cite{Bernal33,Rahman72} This method was also utilized in the |
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making of diamond lattice $I_c$ ice crystals, with each cubic |
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simulation box consisting of either 512 or 1000 particles. Only |
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supercooled regime. An ensemble average from five separate melting |
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simulations was acquired, each starting from different ice crystals |
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generated as described previously. All simulations were equilibrated |
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for 100 ps prior to a 200 ps data collection run at each temperature |
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setting. The temperature range of study spanned from 25 to 400 K, with |
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a maximum degree increment of 25 K. For regions of interest along this |
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stepwise progression, the temperature increment was decreased from 25 |
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K to 10 and 5 K. The above equilibration and production times were |
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for 100~ps prior to a 200~ps data collection run at each temperature |
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setting. The temperature range of study spanned from 25 to 400~K, with |
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a maximum degree increment of 25~K. For regions of interest along this |
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stepwise progression, the temperature increment was decreased from |
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25~K to 10 and 5~K. The above equilibration and production times were |
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sufficient in that fluctuations in the volume autocorrelation function |
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were damped out in all simulations in under 20 ps. |
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were damped out in all simulations in under 20~ps. |
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|
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\subsection{Density Behavior} |
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Our initial simulations focused on the original SSD water model, |
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and an average density versus temperature plot is shown in figure |
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\ref{dense1}. Note that the density maximum when using a reaction |
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field appears between 255 and 265 K. There were smaller fluctuations |
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in the density at 260 K than at either 255 or 265, so we report this |
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field appears between 255 and 265~K. There were smaller fluctuations |
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in the density at 260~K than at either 255 or 265~K, so we report this |
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value as the location of the density maximum. Figure \ref{dense1} was |
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constructed using ice $I_h$ crystals for the initial configuration; |
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though not pictured, the simulations starting from ice $I_c$ crystal |
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configurations showed similar results, with a liquid-phase density |
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maximum in this same region (between 255 and 260 K). |
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maximum in this same region (between 255 and 260~K). |
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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{denseSSDnew.eps} |
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\caption{Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}], |
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\caption{ Density versus temperature for TIP4P [Ref. \citen{Jorgensen98b}], |
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TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], SSD |
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without Reaction Field, SSD, and experiment [Ref. \citen{CRC80}]. The |
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arrows indicate the change in densities observed when turning off the |
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dependent on the cutoff radius used both with and without the use of |
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reaction field.\cite{Berendsen98} In order to address the possible |
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effect of cutoff radius, simulations were performed with a dipolar |
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cutoff radius of 12.0 \AA\ to complement the previous SSD |
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simulations, all performed with a cutoff of 9.0 \AA. All of the |
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cutoff radius of 12.0~\AA\ to complement the previous SSD |
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simulations, all performed with a cutoff of 9.0~\AA. All of the |
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resulting densities overlapped within error and showed no significant |
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trend toward lower or higher densities as a function of cutoff radius, |
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for simulations both with and without reaction field. These results |
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scaling of SSD relative to other common models at any given |
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temperature. SSD assumes a lower density than any of the other |
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listed models at the same pressure, behavior which is especially |
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apparent at temperatures greater than 300 K. Lower than expected |
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apparent at temperatures greater than 300~K. Lower than expected |
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densities have been observed for other systems using a reaction field |
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for long-range electrostatic interactions, so the most likely reason |
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for the significantly lower densities seen in these simulations is the |
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freezing point of liquid water. The shape of the curve is similar to |
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the curve produced from SSD simulations using reaction field, |
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specifically the rapidly decreasing densities at higher temperatures; |
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however, a shift in the density maximum location, down to 245 K, is |
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however, a shift in the density maximum location, down to 245~K, is |
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observed. This is a more accurate comparison to the other listed water |
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models, in that no long range corrections were applied in those |
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simulations.\cite{Clancy94,Jorgensen98b} However, even without the |
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reaction field, the density around 300 K is still significantly lower |
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reaction field, the density around 300~K is still significantly lower |
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than experiment and comparable water models. This anomalous behavior |
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was what lead Tan {\it et al.} to recently reparameterize |
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SSD.\cite{Ichiye03} Throughout the remainder of the paper our |
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constant energy (NVE) simulations were performed at the average |
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density obtained by the NPT simulations at an identical target |
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temperature. Simulations started with randomized velocities and |
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underwent 50 ps of temperature scaling and 50 ps of constant energy |
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equilibration before a 200 ps data collection run. Diffusion constants |
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underwent 50~ps of temperature scaling and 50~ps of constant energy |
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equilibration before a 200~ps data collection run. Diffusion constants |
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were calculated via linear fits to the long-time behavior of the |
| 450 |
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mean-square displacement as a function of time. The averaged results |
| 451 |
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from five sets of NVE simulations are displayed in figure |
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\begin{center} |
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\epsfxsize=6in |
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\epsfbox{betterDiffuse.epsi} |
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\caption{Average self-diffusion constant as a function of temperature for |
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\caption{ Average self-diffusion constant as a function of temperature for |
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SSD, SPC/E [Ref. \citen{Clancy94}], and TIP5P |
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[Ref. \citen{Jorgensen01}] compared with experimental data |
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[Refs. \citen{Gillen72} and \citen{Holz00}]. Of the three water models |
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strengths of the SSD model. Of the three models shown, the SSD model |
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has the most accurate depiction of self-diffusion in both the |
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supercooled and liquid regimes. SPC/E does a respectable job by |
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reproducing values similar to experiment around 290 K; however, it |
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reproducing values similar to experiment around 290~K; however, it |
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deviates at both higher and lower temperatures, failing to predict the |
| 476 |
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correct thermal trend. TIP5P and SSD both start off low at colder |
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temperatures and tend to diffuse too rapidly at higher temperatures. |
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capacity (C$_\text{p}$) was monitored to locate the melting transition |
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in each of the simulations. In the melting simulations of the 1024 |
| 492 |
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particle ice $I_h$ simulations, a large spike in C$_\text{p}$ occurs |
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at 245 K, indicating a first order phase transition for the melting of |
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at 245~K, indicating a first order phase transition for the melting of |
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these ice crystals. When the reaction field is turned off, the melting |
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transition occurs at 235 K. These melting transitions are |
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transition occurs at 235~K. These melting transitions are |
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considerably lower than the experimental value. |
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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{corrDiag.eps} |
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\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.} |
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\label{corrAngle} |
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– |
\end{center} |
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– |
\end{figure} |
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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{fullContours.eps} |
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\caption{Contour plots of 2D angular pair correlation functions for |
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512 SSD molecules at 100 K (A \& B) and 300 K (C \& D). Dark areas |
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> |
\caption{ Contour plots of 2D angular pair correlation functions for |
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512 SSD molecules at 100~K (A \& B) and 300~K (C \& D). Dark areas |
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signify regions of enhanced density while light areas signify |
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depletion relative to the bulk density. White areas have pair |
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correlation values below 0.5 and black areas have values above 1.5.} |
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\end{center} |
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\end{figure} |
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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{corrDiag.eps} |
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\caption{ An illustration of angles involved in the correlations observed in Fig. \ref{contour}.} |
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\label{corrAngle} |
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\end{center} |
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\end{figure} |
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|
| 520 |
|
Additional analysis of the melting process was performed using |
| 521 |
|
two-dimensional structure and dipole angle correlations. Expressions |
| 522 |
|
for these correlations are as follows: |
| 555 |
|
$g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, the second |
| 556 |
|
solvation shell peak appears to have two distinct components that |
| 557 |
|
blend together to form one observable peak. At higher temperatures, |
| 558 |
< |
this split character alters to show the leading 4 \AA\ peak dominated |
| 558 |
> |
this split character alters to show the leading 4~\AA\ peak dominated |
| 559 |
|
by equatorial anti-parallel dipole orientations. There is also a |
| 560 |
|
tightly bunched group of axially arranged dipoles that most likely |
| 561 |
|
consist of the smaller fraction of aligned dipole pairs. The trailing |
| 562 |
< |
component of the split peak at 5 \AA\ is dominated by aligned dipoles |
| 562 |
> |
component of the split peak at 5~\AA\ is dominated by aligned dipoles |
| 563 |
|
that assume hydrogen bond arrangements similar to those seen in the |
| 564 |
|
first solvation shell. This evidence indicates that the dipole pair |
| 565 |
|
interaction begins to dominate outside of the range of the dipolar |
| 566 |
|
repulsion term. The energetically favorable dipole arrangements |
| 567 |
|
populate the region immediately outside this repulsion region (around |
| 568 |
< |
4 \AA), while arrangements that seek to satisfy both the sticky and |
| 568 |
> |
4~\AA), while arrangements that seek to satisfy both the sticky and |
| 569 |
|
dipole forces locate themselves just beyond this initial buildup |
| 570 |
< |
(around 5 \AA). |
| 570 |
> |
(around 5~\AA). |
| 571 |
|
|
| 572 |
|
From these findings, the split second peak is primarily the product of |
| 573 |
|
the dipolar repulsion term of the sticky potential. In fact, the inner |
| 574 |
|
peak can be pushed out and merged with the outer split peak just by |
| 575 |
|
extending the switching function ($s^\prime(r_{ij})$) from its normal |
| 576 |
< |
4.0 \AA\ cutoff to values of 4.5 or even 5 \AA. This type of |
| 576 |
> |
4.0~\AA\ cutoff to values of 4.5 or even 5~\AA. This type of |
| 577 |
|
correction is not recommended for improving the liquid structure, |
| 578 |
|
since the second solvation shell would still be shifted too far |
| 579 |
|
out. In addition, this would have an even more detrimental effect on |
| 608 |
|
|
| 609 |
|
\begin{table} |
| 610 |
|
\begin{center} |
| 611 |
< |
\caption{Parameters for the original and adjusted models} |
| 611 |
> |
\caption{ Parameters for the original and adjusted models} |
| 612 |
|
\begin{tabular}{ l c c c c } |
| 613 |
|
\hline \\[-3mm] |
| 614 |
|
\ \ \ Parameters\ \ \ & \ \ \ SSD [Ref. \citen{Ichiye96}] \ \ \ |
| 615 |
< |
& \ SSD1 [Ref. \citen{Ichiye03}]\ \ & \ SSD/E\ \ & \ SSD/RF \\ |
| 615 |
> |
& \ SSD1 [Ref. \citen{Ichiye03}]\ \ & \ SSD/E\ \ & \ \ SSD/RF \\ |
| 616 |
|
\hline \\[-3mm] |
| 617 |
|
\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
| 618 |
|
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
| 632 |
|
\begin{center} |
| 633 |
|
\epsfxsize=5in |
| 634 |
|
\epsfbox{GofRCompare.epsi} |
| 635 |
< |
\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with |
| 635 |
> |
\caption{ Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with |
| 636 |
|
SSD/E and SSD1 without reaction field (top), as well as |
| 637 |
|
SSD/RF and SSD1 with reaction field turned on |
| 638 |
|
(bottom). The insets show the respective first peaks in detail. Note |
| 646 |
|
\begin{center} |
| 647 |
|
\epsfxsize=6in |
| 648 |
|
\epsfbox{dualsticky_bw.eps} |
| 649 |
< |
\caption{Positive and negative isosurfaces of the sticky potential for |
| 649 |
> |
\caption{ Positive and negative isosurfaces of the sticky potential for |
| 650 |
|
SSD1 (left) and SSD/E \& SSD/RF (right). Light areas |
| 651 |
|
correspond to the tetrahedral attractive component, and darker areas |
| 652 |
|
correspond to the dipolar repulsive component.} |
| 688 |
|
particles feel the pull of the ``hydrogen bonds''. Aside from |
| 689 |
|
improving the shape of the first peak in the g(\emph{r}), this |
| 690 |
|
modification improves the densities considerably by allowing the |
| 691 |
< |
persistence of full dipolar character below the previous 4.0 \AA\ |
| 691 |
> |
persistence of full dipolar character below the previous 4.0~\AA\ |
| 692 |
|
cutoff. |
| 693 |
|
|
| 694 |
|
While adjusting the location and shape of the first peak of $g(r)$ |
| 695 |
|
improves the densities, these changes alone are insufficient to bring |
| 696 |
|
the system densities up to the values observed experimentally. To |
| 697 |
|
further increase the densities, the dipole moments were increased in |
| 698 |
< |
both of our adjusted models. Since SSD is a dipole based model, |
| 699 |
< |
the structure and transport are very sensitive to changes in the |
| 700 |
< |
dipole moment. The original SSD simply used the dipole moment |
| 701 |
< |
calculated from the TIP3P water model, which at 2.35 D is |
| 702 |
< |
significantly greater than the experimental gas phase value of 1.84 |
| 703 |
< |
D. The larger dipole moment is a more realistic value and improves the |
| 704 |
< |
dielectric properties of the fluid. Both theoretical and experimental |
| 705 |
< |
measurements indicate a liquid phase dipole moment ranging from 2.4 D |
| 706 |
< |
to values as high as 3.11 D, providing a substantial range of |
| 707 |
< |
reasonable values for a dipole |
| 708 |
< |
moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
| 709 |
< |
increasing the dipole moments to 2.42 and 2.48 D for SSD/E and |
| 710 |
< |
SSD/RF, respectively, leads to significant changes in the |
| 711 |
< |
density and transport of the water models. |
| 698 |
> |
both of our adjusted models. Since SSD is a dipole based model, the |
| 699 |
> |
structure and transport are very sensitive to changes in the dipole |
| 700 |
> |
moment. The original SSD simply used the dipole moment calculated from |
| 701 |
> |
the TIP3P water model, which at 2.35~D is significantly greater than |
| 702 |
> |
the experimental gas phase value of 1.84~D. The larger dipole moment |
| 703 |
> |
is a more realistic value and improves the dielectric properties of |
| 704 |
> |
the fluid. Both theoretical and experimental measurements indicate a |
| 705 |
> |
liquid phase dipole moment ranging from 2.4~D to values as high as |
| 706 |
> |
3.11~D, providing a substantial range of reasonable values for a |
| 707 |
> |
dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
| 708 |
> |
increasing the dipole moments to 2.42 and 2.48~D for SSD/E and SSD/RF, |
| 709 |
> |
respectively, leads to significant changes in the density and |
| 710 |
> |
transport of the water models. |
| 711 |
|
|
| 712 |
|
In order to demonstrate the benefits of these reparameterizations, a |
| 713 |
|
series of NPT and NVE simulations were performed to probe the density |
| 718 |
|
results are obtained from five separate simulations of 1024 particle |
| 719 |
|
systems, and the melting sequences were started from different ice |
| 720 |
|
$I_h$ crystals constructed as described previously. Each NPT |
| 721 |
< |
simulation was equilibrated for 100 ps before a 200 ps data collection |
| 721 |
> |
simulation was equilibrated for 100~ps before a 200~ps data collection |
| 722 |
|
run at each temperature step, and the final configuration from the |
| 723 |
|
previous temperature simulation was used as a starting point. All NVE |
| 724 |
|
simulations had the same thermalization, equilibration, and data |
| 728 |
|
\begin{center} |
| 729 |
|
\epsfxsize=6in |
| 730 |
|
\epsfbox{ssdeDense.epsi} |
| 731 |
< |
\caption{Comparison of densities calculated with SSD/E to |
| 731 |
> |
\caption{ Comparison of densities calculated with SSD/E to |
| 732 |
|
SSD1 without a reaction field, TIP3P [Ref. \citen{Jorgensen98b}], |
| 733 |
|
TIP5P [Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and |
| 734 |
|
experiment [Ref. \citen{CRC80}]. The window shows a expansion around |
| 758 |
|
strengthening of the dipolar character. However, this increasing |
| 759 |
|
disorder in the SSD/E model has little effect on the melting |
| 760 |
|
transition. By monitoring $C_p$ throughout these simulations, the |
| 761 |
< |
melting transition for SSD/E was shown to occur at 235 K. The |
| 761 |
> |
melting transition for SSD/E was shown to occur at 235~K. The |
| 762 |
|
same transition temperature observed with SSD and SSD1. |
| 763 |
|
|
| 764 |
|
\begin{figure} |
| 765 |
|
\begin{center} |
| 766 |
|
\epsfxsize=6in |
| 767 |
|
\epsfbox{ssdrfDense.epsi} |
| 768 |
< |
\caption{Comparison of densities calculated with SSD/RF to |
| 768 |
> |
\caption{ Comparison of densities calculated with SSD/RF to |
| 769 |
|
SSD1 with a reaction field, TIP3P [Ref. \citen{Jorgensen98b}], |
| 770 |
|
TIP5P [Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and |
| 771 |
|
experiment [Ref. \citen{CRC80}]. The inset shows the necessity of |
| 789 |
|
further emphasize the importance of reparameterization in order to |
| 790 |
|
model the density properly under different simulation conditions. |
| 791 |
|
Again, these changes have only a minor effect on the melting point, |
| 792 |
< |
which observed at 245 K for SSD/RF, is identical to SSD and only 5 K |
| 792 |
> |
which observed at 245~K for SSD/RF, is identical to SSD and only 5~K |
| 793 |
|
lower than SSD1 with a reaction field. Additionally, the difference in |
| 794 |
|
density maxima is not as extreme, with SSD/RF showing a density |
| 795 |
< |
maximum at 255 K, fairly close to the density maxima of 260 K and 265 |
| 796 |
< |
K, shown by SSD and SSD1 respectively. |
| 795 |
> |
maximum at 255~K, fairly close to the density maxima of 260~K and |
| 796 |
> |
265~K, shown by SSD and SSD1 respectively. |
| 797 |
|
|
| 798 |
|
\begin{figure} |
| 799 |
|
\begin{center} |
| 800 |
|
\epsfxsize=6in |
| 801 |
|
\epsfbox{ssdeDiffuse.epsi} |
| 802 |
< |
\caption{The diffusion constants calculated from SSD/E and |
| 802 |
> |
\caption{ The diffusion constants calculated from SSD/E and |
| 803 |
|
SSD1 (both without a reaction field) along with experimental results |
| 804 |
|
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations were |
| 805 |
|
performed at the average densities observed in the 1 atm NPT |
| 822 |
|
SSD/E is consistently higher than experiment, while SSD1 remains lower |
| 823 |
|
than experiment until relatively high temperatures (around 360 |
| 824 |
|
K). Both models follow the shape of the experimental curve well below |
| 825 |
< |
300 K but tend to diffuse too rapidly at higher temperatures, as seen |
| 826 |
< |
in SSD1's crossing above 360 K. This increasing diffusion relative to |
| 825 |
> |
300~K but tend to diffuse too rapidly at higher temperatures, as seen |
| 826 |
> |
in SSD1's crossing above 360~K. This increasing diffusion relative to |
| 827 |
|
the experimental values is caused by the rapidly decreasing system |
| 828 |
|
density with increasing temperature. Both SSD1 and SSD/E show this |
| 829 |
|
deviation in particle mobility, but this trend has different |
| 840 |
|
\begin{center} |
| 841 |
|
\epsfxsize=6in |
| 842 |
|
\epsfbox{ssdrfDiffuse.epsi} |
| 843 |
< |
\caption{The diffusion constants calculated from SSD/RF and |
| 843 |
> |
\caption{ The diffusion constants calculated from SSD/RF and |
| 844 |
|
SSD1 (both with an active reaction field) along with |
| 845 |
|
experimental results [Refs. \citen{Gillen72} and \citen{Holz00}]. The |
| 846 |
|
NVE calculations were performed at the average densities observed in |
| 859 |
|
throughout most of the temperature range shown and exhibiting only a |
| 860 |
|
slight increasing trend at higher temperatures. SSD1 tends to diffuse |
| 861 |
|
more slowly at low temperatures and deviates to diffuse too rapidly at |
| 862 |
< |
temperatures greater than 330 K. As stated above, this deviation away |
| 862 |
> |
temperatures greater than 330~K. As stated above, this deviation away |
| 863 |
|
from the ideal trend is due to a rapid decrease in density at higher |
| 864 |
|
temperatures. SSD/RF does not suffer from this problem as much as SSD1 |
| 865 |
|
because the calculated densities are closer to the experimental |
| 870 |
|
\begin{minipage}{\linewidth} |
| 871 |
|
\renewcommand{\thefootnote}{\thempfootnote} |
| 872 |
|
\begin{center} |
| 873 |
< |
\caption{Properties of the single-point water models compared with |
| 874 |
< |
experimental data at ambient conditions} |
| 873 |
> |
\caption{ Properties of the single-point water models compared with |
| 874 |
> |
experimental data at ambient conditions. Deviations of the of the |
| 875 |
> |
averages are given in parentheses.} |
| 876 |
|
\begin{tabular}{ l c c c c c } |
| 877 |
|
\hline \\[-3mm] |
| 878 |
< |
\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \ |
| 879 |
< |
\ & \ SSD/RF \ \ \ & \ Expt. \\ |
| 878 |
> |
\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ \ SSD/E \ \ \ & \ \ SSD1 (RF) \ \ |
| 879 |
> |
\ & \ \ SSD/RF \ \ \ & \ \ Expt. \\ |
| 880 |
|
\hline \\[-3mm] |
| 881 |
< |
\ \ \ $\rho$ (g/cm$^3$) & 0.999 $\pm$0.001 & 0.996 $\pm$0.001 & 0.972 $\pm$0.002 & 0.997 $\pm$0.001 & 0.997 \\ |
| 882 |
< |
\ \ \ $C_p$ (cal/mol K) & 28.80 $\pm$0.11 & 25.45 $\pm$0.09 & 28.28 $\pm$0.06 & 23.83 $\pm$0.16 & 17.98 \\ |
| 883 |
< |
\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 & |
| 884 |
< |
2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299\cite{Mills73} \\ |
| 885 |
< |
\ \ \ Coordination Number ($n_C$) & 3.9 & 4.3 & 3.8 & 4.4 & |
| 881 |
> |
\ \ $\rho$ (g/cm$^3$) & 0.999(0.001) & 0.996(0.001) & 0.972(0.002) & 0.997(0.001) & 0.997 \\ |
| 882 |
> |
\ \ $C_p$ (cal/mol K) & 28.80(0.11) & 25.45(0.09) & 28.28(0.06) & 23.83(0.16) & 17.98 \\ |
| 883 |
> |
\ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78(0.7) & 2.51(0.18) & 2.00(0.17) & 2.32(0.06) & 2.299\cite{Mills73} \\ |
| 884 |
> |
\ \ Coordination Number ($n_C$) & 3.9 & 4.3 & 3.8 & 4.4 & |
| 885 |
|
4.7\footnote{Calculated by integrating $g_{\text{OO}}(r)$ in |
| 886 |
|
Ref. \citen{Head-Gordon00_1}} \\ |
| 887 |
< |
\ \ \ H-bonds per particle ($n_H$) & 3.7 & 3.6 & 3.7 & 3.7 & |
| 887 |
> |
\ \ H-bonds per particle ($n_H$) & 3.7 & 3.6 & 3.7 & 3.7 & |
| 888 |
|
3.5\footnote{Calculated by integrating $g_{\text{OH}}(r)$ in |
| 889 |
|
Ref. \citen{Soper86}} \\ |
| 890 |
< |
\ \ \ $\tau_1$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & |
| 891 |
< |
7.2 $\pm$0.4 & 5.7\footnote{Calculated for 298 K from data in Ref. \citen{Eisenberg69}} \\ |
| 893 |
< |
\ \ \ $\tau_2$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2 |
| 894 |
< |
$\pm$0.2 & 2.3\footnote{Calculated for 298 K from data in |
| 890 |
> |
\ \ $\tau_1$ (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. \citen{Eisenberg69}} \\ |
| 891 |
> |
\ \ $\tau_2$ (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 |
| 892 |
|
Ref. \citen{Krynicki66}} |
| 893 |
|
\end{tabular} |
| 894 |
|
\label{liquidproperties} |
| 940 |
|
the commonly cited value of 1.9 ps for $\tau_2$ was determined from |
| 941 |
|
the NMR data in Ref. \citen{Krynicki66} at a temperature near |
| 942 |
|
34$^\circ$C.\cite{Rahman71} Because of the strong temperature |
| 943 |
< |
dependence of $\tau_2$, it is necessary to recalculate it at 298 K to |
| 943 |
> |
dependence of $\tau_2$, it is necessary to recalculate it at 298~K to |
| 944 |
|
make proper comparisons. The value shown in Table |
| 945 |
|
\ref{liquidproperties} was calculated from the same NMR data in the |
| 946 |
|
fashion described in Ref. \citen{Krynicki66}. Similarly, $\tau_1$ was |
| 947 |
< |
recomputed for 298 K from the data in Ref. \citen{Eisenberg69}. |
| 947 |
> |
recomputed for 298~K from the data in Ref. \citen{Eisenberg69}. |
| 948 |
|
Again, SSD/E and SSD/RF show improved behavior over SSD1, both with |
| 949 |
|
and without an active reaction field. Turning on the reaction field |
| 950 |
|
leads to much improved time constants for SSD1; however, these results |
| 959 |
|
\begin{center} |
| 960 |
|
\epsfxsize=6in |
| 961 |
|
\epsfbox{icei_bw.eps} |
| 962 |
< |
\caption{The most stable crystal structure assumed by the SSD family |
| 962 |
> |
\caption{ The most stable crystal structure assumed by the SSD family |
| 963 |
|
of water models. We refer to this structure as Ice-{\it i} to |
| 964 |
|
indicate its origins in computer simulation. This image was taken of |
| 965 |
|
the (001) face of the crystal.} |
| 970 |
|
While performing a series of melting simulations on an early iteration |
| 971 |
|
of SSD/E not discussed in this paper, we observed |
| 972 |
|
recrystallization into a novel structure not previously known for |
| 973 |
< |
water. After melting at 235 K, two of five systems underwent |
| 974 |
< |
crystallization events near 245 K. The two systems remained |
| 975 |
< |
crystalline up to 320 and 330 K, respectively. The crystal exhibits |
| 973 |
> |
water. After melting at 235~K, two of five systems underwent |
| 974 |
> |
crystallization events near 245~K. The two systems remained |
| 975 |
> |
crystalline up to 320 and 330~K, respectively. The crystal exhibits |
| 976 |
|
an expanded zeolite-like structure that does not correspond to any |
| 977 |
|
known form of ice. This appears to be an artifact of the point |
| 978 |
|
dipolar models, so to distinguish it from the experimentally observed |
| 1006 |
|
|
| 1007 |
|
\begin{table} |
| 1008 |
|
\begin{center} |
| 1009 |
< |
\caption{Enthalpies of Formation (in kcal / mol) of the three crystal |
| 1009 |
> |
\caption{ Enthalpies of Formation (in kcal / mol) of the three crystal |
| 1010 |
|
structures (at 1 K) exhibited by the SSD family of water models} |
| 1011 |
|
\begin{tabular}{ l c c c } |
| 1012 |
|
\hline \\[-3mm] |
| 1013 |
< |
\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \ & \ |
| 1014 |
< |
Ice-{\it i} \\ |
| 1013 |
> |
\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ \ \ Ice-$I_c$ \ \ \ & |
| 1014 |
> |
\ \ \ \ Ice-{\it i} \\ |
| 1015 |
|
\hline \\[-3mm] |
| 1016 |
|
\ \ \ SSD/E & -72.444 & -72.450 & -73.748 \\ |
| 1017 |
|
\ \ \ SSD/RF & -73.093 & -73.075 & -74.180 \\ |
| 1023 |
|
\end{table} |
| 1024 |
|
|
| 1025 |
|
In addition to these energetic comparisons, melting simulations were |
| 1026 |
< |
performed with ice-{\it i} as the initial configuration using SSD/E, |
| 1026 |
> |
performed with Ice-{\it i} as the initial configuration using SSD/E, |
| 1027 |
|
SSD/RF, and SSD1 both with and without a reaction field. The melting |
| 1028 |
|
transitions for both SSD/E and SSD1 without reaction field occurred at |
| 1029 |
|
temperature in excess of 375~K. SSD/RF and SSD1 with a reaction field |
| 1081 |
|
\newpage |
| 1082 |
|
|
| 1083 |
|
\bibliographystyle{jcp} |
| 1084 |
< |
\bibliography{nptSSD} |
| 1084 |
> |
\bibliography{nptSSD} |
| 1085 |
|
|
| 1089 |
– |
%\pagebreak |
| 1086 |
|
|
| 1087 |
|
\end{document} |
