--- trunk/ssdePaper/nptSSD.tex 2004/01/12 16:20:53 921 +++ trunk/ssdePaper/nptSSD.tex 2004/02/04 20:19:21 1019 @@ -139,6 +139,7 @@ + s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i \frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) + s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)]\ . +\label{stickyfunction} \end{equation} Here, $\nu_0$ is a strength parameter for the sticky potential, and $s$ and $s^\prime$ are cubic switching functions which turn off the @@ -151,7 +152,7 @@ w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = while the $w^\prime$ function counters the normal aligned and anti-aligned structures favored by point dipoles: \begin{equation} -w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = (\cos\theta_{ij}-0.6)^2(\cos\theta_{ij}+0.8)^2-w^0, +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, \end{equation} It should be noted that $w$ is proportional to the sum of the $Y_3^2$ and $Y_3^{-2}$ spherical harmonics (a linear combination which @@ -208,8 +209,10 @@ cubic switching function at a cutoff radius. Under th Long-range dipole-dipole interactions were accounted for in this study by using either the reaction field method or by resorting to a simple -cubic switching function at a cutoff radius. Under the first method, -the magnitude of the reaction field acting on dipole $i$ is +cubic switching function at a cutoff radius. The reaction field +method was actually first used in Monte Carlo simulations of liquid +water.\cite{Barker73} Under this method, the magnitude of the reaction +field acting on dipole $i$ is \begin{equation} \mathcal{E}_{i} = \frac{2(\varepsilon_{s} - 1)}{2\varepsilon_{s} + 1} \frac{1}{r_{c}^{3}} \sum_{j\in{\mathcal{R}}} {\bf \mu}_{j} f(r_{ij})\ , @@ -591,29 +594,15 @@ The parameters available for tuning include the $\sigm important properties. In this case, it would be ideal to correct the densities while maintaining the accurate transport behavior. -The parameters available for tuning include the $\sigma$ and $\epsilon$ -Lennard-Jones parameters, the dipole strength ($\mu$), and the sticky -attractive and dipole repulsive terms with their respective -cutoffs. To alter the attractive and repulsive terms of the sticky -potential independently, it is necessary to separate the terms as -follows: -\begin{equation} -u_{ij}^{sp} -({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j) = -\frac{\nu_0}{2}[s(r_{ij})w({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)] + \frac{\nu_0^\prime}{2} [s^\prime(r_{ij})w^\prime({\bf r}_{ij},{\bf \Omega}_i,{\bf \Omega}_j)], -\end{equation} -where $\nu_0$ scales the strength of the tetrahedral attraction and -$\nu_0^\prime$ scales the dipole repulsion term independently. The -separation was performed for purposes of the reparameterization, but -the final parameters were adjusted so that it is not necessary to -separate the terms when implementing the adjusted water -potentials. The results of the reparameterizations are shown in table -\ref{params}. Note that the tetrahedral attractive and dipolar -repulsive terms do not share the same lower cutoff ($r_l$) in the -newly parameterized potentials. We are calling these -reparameterizations the Soft Sticky Dipole / Reaction Field +The parameters available for tuning include the $\sigma$ and +$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the +strength of the sticky potential ($\nu_0$), and the sticky attractive +and dipole repulsive cubic switching function cutoffs ($r_l$, $r_u$ +and $r_l^\prime$, $r_u^\prime$ respectively). The results of the +reparameterizations are shown in table \ref{params}. We are calling +these reparameterizations the Soft Sticky Dipole / Reaction Field (SSD/RF - for use with a reaction field) and Soft Sticky Dipole -Enhanced (SSD/E - an attempt to improve the liquid structure in +Extended (SSD/E - an attempt to improve the liquid structure in simulations without a long-range correction). \begin{table} @@ -628,9 +617,9 @@ simulations without a long-range correction). \ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ \ \ \ $\mu$ (D) & 2.35 & 2.35 & 2.42 & 2.48\\ \ \ \ $\nu_0$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ +\ \ \ $\omega^\circ$ & 0.07715 & 0.07715 & 0.07715 & 0.07715\\ \ \ \ $r_l$ (\AA) & 2.75 & 2.75 & 2.40 & 2.40\\ \ \ \ $r_u$ (\AA) & 3.35 & 3.35 & 3.80 & 3.80\\ -\ \ \ $\nu_0^\prime$ (kcal/mol) & 3.7284 & 3.6613 & 3.90 & 3.90\\ \ \ \ $r_l^\prime$ (\AA) & 2.75 & 2.75 & 2.75 & 2.75\\ \ \ \ $r_u^\prime$ (\AA) & 4.00 & 4.00 & 3.35 & 3.35\\ \end{tabular} @@ -806,13 +795,14 @@ K, shown by SSD and SSD1 respectively. \begin{center} \epsfxsize=6in \epsfbox{ssdeDiffuse.epsi} -\caption{Plots of the diffusion constants calculated from SSD/E and SSD1, -both without a reaction field, along with experimental results -[Refs. \citen{Gillen72} and \citen{Mills73}]. The NVE calculations were -performed at the average densities observed in the 1 atm NPT -simulations for the respective models. SSD/E is slightly more fluid -than experiment at all of the temperatures, but it is closer than SSD1 -without a long-range correction.} +\caption{The diffusion constants calculated from SSD/E and SSD1, + both without a reaction field, along with experimental results + [Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations + were performed at the average densities observed in the 1 atm NPT + simulations for the respective models. SSD/E is slightly more mobile + than experiment at all of the temperatures, but it is closer to + experiment at biologically relavent temperatures than SSD1 without a + long-range correction.} \label{ssdediffuse} \end{center} \end{figure} @@ -823,35 +813,37 @@ without an active reaction field, both at the densitie the densities, it is important that the excellent diffusive behavior of SSD be maintained or improved. Figure \ref{ssdediffuse} compares the temperature dependence of the diffusion constant of SSD/E to SSD1 -without an active reaction field, both at the densities calculated at -1 atm and at the experimentally calculated densities for super-cooled -and liquid water. The diffusion constant for SSD/E is consistently -higher than experiment, while SSD1 remains lower than experiment until -relatively high temperatures (greater than 330 K). Both models follow -the shape of the experimental curve well below 300 K but tend to -diffuse too rapidly at higher temperatures, something that is -especially apparent with SSD1. This increasing diffusion relative to -the experimental values is caused by the rapidly decreasing system -density with increasing temperature. The densities of SSD1 decay more -rapidly with temperature than do those of SSD/E, leading to more -visible deviation from the experimental diffusion trend. Thus, the -changes made to improve the liquid structure may have had an adverse -affect on the density maximum, but they improve the transport behavior -of SSD/E relative to SSD1. +without an active reaction field at the densities calculated from the +NPT simulations at 1 atm. The diffusion constant for SSD/E is +consistently higher than experiment, while SSD1 remains lower than +experiment until relatively high temperatures (around 360 K). Both +models follow the shape of the experimental curve well below 300 K but +tend to diffuse too rapidly at higher temperatures, as seen in SSD1's +crossing above 360 K. This increasing diffusion relative to the +experimental values is caused by the rapidly decreasing system density +with increasing temperature. Both SSD1 and SSD/E show this deviation +in diffusive behavior, but this trend has different implications on +the diffusive behavior of the models. While SSD1 shows more +experimentally accurate diffusive behavior in the high temperature +regimes, SSD/E shows more accurate behavior in the supercooled and +biologically relavent temperature ranges. Thus, the changes made to +improve the liquid structure may have had an adverse affect on the +density maximum, but they improve the transport behavior of SSD/E +relative to SSD1 under the most commonly simulated conditions. \begin{figure} \begin{center} \epsfxsize=6in \epsfbox{ssdrfDiffuse.epsi} -\caption{Plots of the diffusion constants calculated from SSD/RF and SSD1, +\caption{The diffusion constants calculated from SSD/RF and SSD1, both with an active reaction field, along with experimental results - [Refs. \citen{Gillen72} and \citen{Mills73}]. The NVE calculations + [Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations were performed at the average densities observed in the 1 atm NPT simulations for both of the models. Note how accurately SSD/RF simulates the diffusion of water throughout this temperature range. The more rapidly increasing diffusion constants at high - temperatures for both models is attributed to the significantly lower - densities than observed in experiment.} + temperatures for both models is attributed to lower calculated + densities than those observed in experiment.} \label{ssdrfdiffuse} \end{center} \end{figure} @@ -859,9 +851,9 @@ throughout the temperature range shown and with only a In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are compared to SSD1 with an active reaction field. Note that SSD/RF tracks the experimental results quantitatively, identical within error -throughout the temperature range shown and with only a slight -increasing trend at higher temperatures. SSD1 tends to diffuse more -slowly at low temperatures and deviates to diffuse too rapidly at +throughout most of the temperature range shown and exhibiting only a +slight increasing trend at higher temperatures. SSD1 tends to diffuse +more slowly at low temperatures and deviates to diffuse too rapidly at temperatures greater than 330 K. As stated above, this deviation away from the ideal trend is due to a rapid decrease in density at higher temperatures. SSD/RF does not suffer from this problem as much as SSD1 @@ -869,6 +861,71 @@ reparameterization when using an altered long-range co values. These results again emphasize the importance of careful reparameterization when using an altered long-range correction. +\begin{table} +\begin{center} +\caption{Calculated and experimental properties of the single point waters and liquid water at 298 K and 1 atm. (a) Ref. [\citen{Mills73}]. (b) Calculated by integrating the data in ref. \citen{Head-Gordon00_1}. (c) Calculated by integrating the data in ref. \citen{Soper86}. (d) Ref. [\citen{Eisenberg69}]. (e) Calculated for 298 K from data in ref. \citen{Krynicki66}.} +\begin{tabular}{ l c c c c c } +\hline \\[-3mm] +\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \ +\ & \ SSD/RF \ \ \ & \ Expt. \\ + \hline \\[-3mm] +\ \ \ $\rho$ (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 \\ +\ \ \ $C_p$ (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 \\ +\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 & 2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299$^\text{a}$ \\ +\ \ \ Coordination Number & 3.9 & 4.3 & 3.8 & 4.4 & 4.7$^\text{b}$ \\ +\ \ \ H-bonds per particle & 3.7 & 3.6 & 3.7 & 3.7 & 3.4$^\text{c}$ \\ +\ \ \ $\tau_1^\mu$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & 7.2 $\pm$0.4 & 4.76$^\text{d}$ \\ +\ \ \ $\tau_2^\mu$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2 $\pm$0.2 & 2.3$^\text{e}$ \\ +\end{tabular} +\label{liquidproperties} +\end{center} +\end{table} + +Table \ref{liquidproperties} gives a synopsis of the liquid state +properties of the water models compared in this study along with the +experimental values for liquid water at ambient conditions. The +coordination number and hydrogen bonds per particle were calculated by +integrating the following relation: +\begin{equation} +4\pi\rho\int_{0}^{a}r^2\text{g}(r)dr, +\end{equation} +where $\rho$ is the number density of pair interactions, $a$ is the +radial location of the minima following the first solvation shell +peak, and g$(r)$ is either g$_\text{OO}(r)$ or g$_\text{OH}(r)$ for +calculation of the coordination number or hydrogen bonds per particle +respectively. The number of hydrogen bonds stays relatively constant +across all of the models, but the coordination numbers of SSD/E and +SSD/RF show an improvement over SSD1. This improvement is primarily +due to the widening of the first solvation shell peak, allowing the +first minima to push outward. Comparing the coordination number with +the number of hydrogen bonds can lead to more insight into the +structural character of the liquid. Because of the near identical +values for SSD1, it appears to be a little too exclusive, in that all +molecules in the first solvation shell are involved in forming ideal +hydrogen bonds. The differing numbers for the newly parameterized +models indicate the allowance of more fluid configurations in addition +to the formation of an acceptable number of ideal hydrogen bonds. + +The time constants for the self orientational autocorrelation function +are also displayed in Table \ref{liquidproperties}. The dipolar +orientational time correlation function ($\Gamma_{l}$) is described +by: +\begin{equation} +\Gamma_{l}(t) = \langle P_l[\mathbf{u}_j(0)\cdot\mathbf{u}_j(t)]\rangle, +\end{equation} +where $P_l$ is a Legendre polynomial of order $l$ and $\mathbf{u}_j$ +is the unit vector of the particle dipole.\cite{Rahman71} From these +correlation functions, the orientational relaxation time of the dipole +vector can be calculated from an exponential fit in the long-time +regime ($t > \tau_l^\mu$).\cite{Rothschild84} Calculation of these +time constants were averaged from five detailed NVE simulations +performed at the STP density for each of the respective models. Again, +SSD/E and SSD/RF show improved behavior over SSD1 both with and +without an active reaction field. Numbers published from the original +SSD dynamics studies appear closer to the experimental values, and we +attribute this discrepancy to the implimentation of an Ewald sum +versus a reaction field. + \subsection{Additional Observations} \begin{figure}