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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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dipole ({\sc ssd}) and related single point water models} |
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|
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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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\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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constant were investigated for the soft sticky dipole ({\sc ssd}) water |
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model and two related re-parameterizations 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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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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that the original {\sc 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 re-parameterized versions of {\sc 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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electrostatic corrections. These are called the {\sc ssd/rf} and {\sc 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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transport properties that can be obtained with either the original {\sc ssd} |
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or the density corrected version of the original model ({\sc 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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which appears to be the most stable ice structure for the entire {\sc ssd} |
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family. |
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\end{abstract} |
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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 was |
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cost is the Soft Sticky Dipole ({\sc ssd}) water |
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model.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} The {\sc ssd} model was |
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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 which |
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Luzar.\cite{Bratko85,Bratko95} {\sc ssd} is a {\it single point} model which |
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has an interaction site that is both a point dipole along with 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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the proper hydrogen bond orientation in the first solvation |
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shell. |
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The interaction between two SSD water molecules \emph{i} and \emph{j} |
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The interaction between two {\sc 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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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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can be found in the original {\sc ssd} |
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articles.\cite{Ichiye96,Ichiye96b,Ichiye99,Ichiye03} |
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Since SSD is a single-point {\it dipolar} model, the force |
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Since {\sc 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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{\sc ssd} 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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properties. Indeed, {\sc 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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exhibited by {\sc 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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of solvent properties makes {\sc 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 use with |
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One feature of the {\sc ssd} model is that it was parameterized for use with |
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the Ewald sum to handle long-range interactions. This would normally |
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be the best way of handling long-range interactions in systems that |
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contain other point charges. However, our group has recently become |
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properties and behavior under the more computationally efficient |
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reaction field (RF) technique, or even with a simple cutoff. This |
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study addresses these issues by looking at the structural and |
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transport behavior of SSD over a variety of temperatures with the |
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transport behavior of {\sc ssd} over a variety of temperatures with the |
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purpose of utilizing the RF correction technique. We then suggest |
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modifications to the parameters that result in more realistic bulk |
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phase behavior. It should be noted that in a recent publication, some |
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of the original investigators of the SSD water model have suggested |
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adjustments to the SSD water model to address abnormal density |
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of the original investigators of the {\sc ssd} water model have suggested |
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adjustments to the {\sc ssd} water model to address abnormal density |
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behavior (also observed here), calling the corrected model |
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SSD1.\cite{Ichiye03} In what follows, we compare our |
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reparamaterization of SSD with both the original SSD and SSD1 models |
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with the goal of improving the bulk phase behavior of an SSD-derived |
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{\sc ssd1}.\cite{Ichiye03} In what follows, we compare our |
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reparamaterization of {\sc ssd} with both the original {\sc ssd} and {\sc ssd1} models |
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with the goal of improving the bulk phase behavior of an {\sc ssd}-derived |
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model in simulations utilizing the Reaction Field. |
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|
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\section{Methods} |
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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} f(r_{ij}), |
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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 $f(r_{ij})$ is a cubic switching |
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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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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 field |
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of {\sc ssd} which could be used either with or without the reaction field |
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turned on. |
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Simulations to obtain the preferred density were performed in the |
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isobaric-isothermal (NPT) ensemble, while all dynamical properties |
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were obtained from microcanonical (NVE) simulations done at densities |
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matching the NPT density for a particular target temperature. The |
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constant pressure simulations were implemented using an integral |
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thermostat and barostat as outlined by Hoover.\cite{Hoover85,Hoover86} |
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All molecules were treated as non-linear rigid bodies. Vibrational |
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constraints are not necessary in simulations of SSD, because there are |
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no explicit hydrogen atoms, and thus no molecular vibrational modes |
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need to be considered. |
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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 {\sc ssd}, because there are no explicit hydrogen atoms, and |
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thus no molecular vibrational modes need to be considered. |
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|
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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 (DLM).\cite{Dullweber1997} Our reason for selecting this |
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McLachlan ({\sc dlm}).\cite{Dullweber1997} Our reason for selecting this |
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integrator centers on poor energy conservation of rigid body dynamics |
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using traditional quaternion integration.\cite{Evans77,Evans77b} In |
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typical microcanonical ensemble simulations, the energy drift when |
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using quaternions was substantially greater than when using the DLM |
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using quaternions was substantially greater than when using the {\sc dlm} |
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method (fig. \ref{timestep}). This steady drift in the total energy |
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has also been observed by Kol {\it et al.}\cite{Laird97} |
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|
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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 DML method allows for Verlet style integration of both |
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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\% increase in |
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computation time using the DML method in place of quaternions. The |
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step, a 1000 {\sc ssd} particle simulation shows an average 7\% increase in |
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computation time using the {\sc dlm} method in place of quaternions. The |
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additional expense per step is justified when one considers the |
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ability to use time steps that are nearly twice as large under DML |
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ability to use time steps that are nearly twice as large under {\sc dlm} |
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than would be usable under quaternion dynamics. The energy |
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conservation of the two methods using a number of different time steps |
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is illustrated in 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 symplectic splitting method proposed by Dullweber \emph{et al.} |
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with increasing time step. The larger time step plots are shifted from |
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the true energy baseline (that of $\Delta t$ = 0.1 fs) for clarity.} |
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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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\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 DML and quaternion integration schemes is compared. |
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All of the 1000 SSD particle simulations started with the same |
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steps for both the {\sc dlm} and quaternion integration schemes is compared. |
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All of the 1000 {\sc ssd} particle simulations started with the same |
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configuration, and the only difference was the method used to handle |
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orientational motion. At time steps of 0.1 and 0.5 fs, both methods |
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for propagating the orientational degrees of freedom conserve energy |
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fairly well, with the quaternion method showing a slight energy drift |
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over time in the 0.5 fs time step simulation. At time steps of 1 and 2 |
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fs, the energy conservation benefits of the DML method are clearly |
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fs, the energy conservation benefits of the {\sc dlm} method are clearly |
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demonstrated. Thus, while maintaining the same degree of energy |
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conservation, one can take considerably longer time steps, leading to |
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an overall reduction in computation time. |
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|
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Energy drift in the simulations using DML integration was unnoticeable |
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Energy drift in the simulations using {\sc dlm} integration was unnoticeable |
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for time steps up to 3 fs. A slight energy drift on the order of 0.012 |
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kcal/mol per nanosecond was observed at a time step of 4 fs, and as |
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expected, this drift increases dramatically with increasing time |
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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 SSD |
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crystals were formed by first arranging the centers of mass of the {\sc 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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|
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\subsection{Density Behavior} |
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|
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Our initial simulations focused on the original SSD water model, and |
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Our initial simulations focused on the original {\sc ssd} water model, and |
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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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\epsfxsize=6in |
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\epsfbox{denseSSD.eps} |
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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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TIP3P [Ref. \citen{Jorgensen98b}], SPC/E [Ref. \citen{Clancy94}], {\sc ssd} |
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without Reaction Field, {\sc 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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reaction field. The the lower than expected densities for the SSD |
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model were what prompted the original reparameterization of SSD1 |
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reaction field. The the lower than expected densities for the {\sc ssd} |
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model were what prompted the original reparameterization of {\sc ssd1} |
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[Ref. \citen{Ichiye03}].} |
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\label{dense1} |
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\end{center} |
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\end{figure} |
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|
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The density maximum for SSD compares quite favorably to other simple |
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The density maximum for {\sc ssd} compares quite favorably to other simple |
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water models. Figure \ref{dense1} also shows calculated densities of |
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several other models and experiment obtained from other |
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sources.\cite{Jorgensen98b,Clancy94,CRC80} Of the listed simple water |
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models, SSD has a temperature closest to the experimentally observed |
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models, {\sc ssd} has a temperature closest to the experimentally observed |
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density maximum. Of the {\it charge-based} models in |
| 391 |
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Fig. \ref{dense1}, TIP4P has a density maximum behavior most like that |
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seen in SSD. Though not included in this plot, it is useful |
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seen in {\sc ssd}. Though not included in this plot, it is useful |
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to note that TIP5P has a density maximum nearly identical to the |
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experimentally measured temperature. |
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|
| 397 |
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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 |
| 399 |
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effect of cutoff radius, simulations were performed with a dipolar |
| 400 |
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cutoff radius of 12.0 \AA\ to complement the previous SSD simulations, |
| 400 |
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cutoff radius of 12.0 \AA\ to complement the previous {\sc ssd} simulations, |
| 401 |
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all performed with a cutoff of 9.0 \AA. All of the resulting densities |
| 402 |
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overlapped within error and showed no significant trend toward lower |
| 403 |
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or higher densities as a function of cutoff radius, for simulations |
| 404 |
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both with and without reaction field. These results indicate that |
| 405 |
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there is no major benefit in choosing a longer cutoff radius in |
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simulations using SSD. This is advantageous in that the use of a |
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simulations using {\sc ssd}. This is advantageous in that the use of a |
| 407 |
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longer cutoff radius results in a significant increase in the time |
| 408 |
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required to obtain a single trajectory. |
| 409 |
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|
| 410 |
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The key feature to recognize in figure \ref{dense1} is the density |
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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 listed |
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scaling of {\sc ssd} relative to other common models at any given |
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temperature. {\sc ssd} assumes a lower density than any of the other listed |
| 413 |
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models at the same pressure, behavior which is especially apparent at |
| 414 |
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temperatures greater than 300 K. Lower than expected densities have |
| 415 |
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been observed for other systems using a reaction field for long-range |
| 422 |
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\ref{dense1}. Without the reaction field, the densities increase |
| 423 |
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to more experimentally reasonable values, especially around the |
| 424 |
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freezing point of liquid water. The shape of the curve is similar to |
| 425 |
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the curve produced from SSD simulations using reaction field, |
| 425 |
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the curve produced from {\sc ssd} simulations using reaction field, |
| 426 |
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specifically the rapidly decreasing densities at higher temperatures; |
| 427 |
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however, a shift in the density maximum location, down to 245 K, is |
| 428 |
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observed. This is a more accurate comparison to the other listed water |
| 431 |
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reaction field, the density around 300 K is still significantly lower |
| 432 |
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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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reparamaterizations of SSD will be compared with the newer SSD1 model. |
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{\sc ssd}.\cite{Ichiye03} Throughout the remainder of the paper our |
| 435 |
> |
reparamaterizations of {\sc ssd} will be compared with their newer {\sc ssd1} |
| 436 |
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model. |
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|
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\subsection{Transport Behavior} |
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|
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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 |
| 459 |
< |
SSD, SPC/E [Ref. \citen{Clancy94}], TIP5P [Ref. \citen{Jorgensen01}], |
| 460 |
< |
and Experimental data [Refs. \citen{Gillen72} and \citen{Holz00}]. Of |
| 461 |
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the three water models shown, SSD has the least deviation from the |
| 462 |
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experimental values. The rapidly increasing diffusion constants for |
| 463 |
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TIP5P and SSD correspond to significant decrease in density at the |
| 464 |
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higher temperatures.} |
| 459 |
> |
{\sc ssd}, SPC/E [Ref. \citen{Clancy94}], and TIP5P |
| 460 |
> |
[Ref. \citen{Jorgensen01}] compared with experimental data |
| 461 |
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[Refs. \citen{Gillen72} and \citen{Holz00}]. Of the three water models |
| 462 |
> |
shown, {\sc ssd} has the least deviation from the experimental values. The |
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rapidly increasing diffusion constants for TIP5P and {\sc ssd} correspond to |
| 464 |
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significant decreases in density at the higher temperatures.} |
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\label{diffuse} |
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\end{center} |
| 467 |
|
\end{figure} |
| 468 |
|
|
| 469 |
|
The observed values for the diffusion constant point out one of the |
| 470 |
< |
strengths of the SSD model. Of the three models shown, the SSD model |
| 470 |
> |
strengths of the {\sc ssd} model. Of the three models shown, the {\sc ssd} model |
| 471 |
|
has the most accurate depiction of self-diffusion in both the |
| 472 |
|
supercooled and liquid regimes. SPC/E does a respectable job by |
| 473 |
|
reproducing values similar to experiment around 290 K; however, it |
| 474 |
|
deviates at both higher and lower temperatures, failing to predict the |
| 475 |
< |
correct thermal trend. TIP5P and SSD both start off low at colder |
| 475 |
> |
correct thermal trend. TIP5P and {\sc ssd} both start off low at colder |
| 476 |
|
temperatures and tend to diffuse too rapidly at higher temperatures. |
| 477 |
|
This behavior at higher temperatures is not particularly surprising |
| 478 |
< |
since the densities of both TIP5P and SSD are lower than experimental |
| 478 |
> |
since the densities of both TIP5P and {\sc ssd} are lower than experimental |
| 479 |
|
water densities at higher temperatures. When calculating the |
| 480 |
< |
diffusion coefficients for SSD at experimental densities (instead of |
| 480 |
> |
diffusion coefficients for {\sc ssd} at experimental densities (instead of |
| 481 |
|
the densities from the NPT simulations), the resulting values fall |
| 482 |
|
more in line with experiment at these temperatures. |
| 483 |
|
|
| 498 |
|
\begin{center} |
| 499 |
|
\epsfxsize=6in |
| 500 |
|
\epsfbox{corrDiag.eps} |
| 501 |
< |
\caption{Two dimensional illustration of angles involved in the |
| 501 |
< |
correlations observed in Fig. \ref{contour}.} |
| 501 |
> |
\caption{An illustration of angles involved in the correlations observed in Fig. \ref{contour}.} |
| 502 |
|
\label{corrAngle} |
| 503 |
|
\end{center} |
| 504 |
|
\end{figure} |
| 507 |
|
\begin{center} |
| 508 |
|
\epsfxsize=6in |
| 509 |
|
\epsfbox{fullContours.eps} |
| 510 |
< |
\caption{Contour plots of 2D angular g($r$)'s for 512 SSD systems at |
| 511 |
< |
100 K (A \& B) and 300 K (C \& D). Contour colors are inverted for |
| 512 |
< |
clarity: dark areas signify peaks while light areas signify |
| 513 |
< |
depressions. White areas have $g(r)$ values below 0.5 and black |
| 514 |
< |
areas have values above 1.5.} |
| 510 |
> |
\caption{Contour plots of 2D angular pair correlation functions for |
| 511 |
> |
512 {\sc ssd} molecules at 100 K (A \& B) and 300 K (C \& D). Dark areas |
| 512 |
> |
signify regions of enhanced density while light areas signify |
| 513 |
> |
depletion relative to the bulk density. White areas have pair |
| 514 |
> |
correlation values below 0.5 and black areas have values above 1.5.} |
| 515 |
|
\label{contour} |
| 516 |
|
\end{center} |
| 517 |
|
\end{figure} |
| 550 |
|
|
| 551 |
|
This complex interplay between dipole and sticky interactions was |
| 552 |
|
remarked upon as a possible reason for the split second peak in the |
| 553 |
< |
oxygen-oxygen $g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, |
| 554 |
< |
the second solvation shell peak appears to have two distinct |
| 555 |
< |
components that blend together to form one observable peak. At higher |
| 556 |
< |
temperatures, this split character alters to show the leading 4 \AA\ |
| 557 |
< |
peak dominated by equatorial anti-parallel dipole orientations. There |
| 558 |
< |
is also a tightly bunched group of axially arranged dipoles that most |
| 559 |
< |
likely consist of the smaller fraction of aligned dipole pairs. The |
| 560 |
< |
trailing component of the split peak at 5 \AA\ is dominated by aligned |
| 561 |
< |
dipoles that assume hydrogen bond arrangements similar to those seen |
| 562 |
< |
in the first solvation shell. This evidence indicates that the dipole |
| 563 |
< |
pair interaction begins to dominate outside of the range of the |
| 564 |
< |
dipolar repulsion term. The energetically favorable dipole |
| 565 |
< |
arrangements populate the region immediately outside this repulsion |
| 566 |
< |
region (around 4 \AA), while arrangements that seek to satisfy both |
| 567 |
< |
the sticky and dipole forces locate themselves just beyond this |
| 568 |
< |
initial buildup (around 5 \AA). |
| 553 |
> |
oxygen-oxygen pair correlation function, |
| 554 |
> |
$g_\mathrm{OO}(r)$.\cite{Ichiye96} At low temperatures, the second |
| 555 |
> |
solvation shell peak appears to have two distinct components that |
| 556 |
> |
blend together to form one observable peak. At higher temperatures, |
| 557 |
> |
this split character alters to show the leading 4 \AA\ peak dominated |
| 558 |
> |
by equatorial anti-parallel dipole orientations. There is also a |
| 559 |
> |
tightly bunched group of axially arranged dipoles that most likely |
| 560 |
> |
consist of the smaller fraction of aligned dipole pairs. The trailing |
| 561 |
> |
component of the split peak at 5 \AA\ is dominated by aligned dipoles |
| 562 |
> |
that assume hydrogen bond arrangements similar to those seen in the |
| 563 |
> |
first solvation shell. This evidence indicates that the dipole pair |
| 564 |
> |
interaction begins to dominate outside of the range of the dipolar |
| 565 |
> |
repulsion term. The energetically favorable dipole arrangements |
| 566 |
> |
populate the region immediately outside this repulsion region (around |
| 567 |
> |
4 \AA), while arrangements that seek to satisfy both the sticky and |
| 568 |
> |
dipole forces locate themselves just beyond this initial buildup |
| 569 |
> |
(around 5 \AA). |
| 570 |
|
|
| 571 |
|
From these findings, the split second peak is primarily the product of |
| 572 |
|
the dipolar repulsion term of the sticky potential. In fact, the inner |
| 577 |
|
since the second solvation shell would still be shifted too far |
| 578 |
|
out. In addition, this would have an even more detrimental effect on |
| 579 |
|
the system densities, leading to a liquid with a more open structure |
| 580 |
< |
and a density considerably lower than the already low SSD density. A |
| 580 |
> |
and a density considerably lower than the already low {\sc ssd} density. A |
| 581 |
|
better correction would be to include the quadrupole-quadrupole |
| 582 |
|
interactions for the water particles outside of the first solvation |
| 583 |
|
shell, but this would remove the simplicity and speed advantage of |
| 584 |
< |
SSD. |
| 584 |
> |
{\sc ssd}. |
| 585 |
|
|
| 586 |
< |
\subsection{Adjusted Potentials: SSD/RF and SSD/E} |
| 586 |
> |
\subsection{Adjusted Potentials: {\sc ssd/rf} and {\sc ssd/e}} |
| 587 |
|
|
| 588 |
< |
The propensity of SSD to adopt lower than expected densities under |
| 588 |
> |
The propensity of {\sc ssd} to adopt lower than expected densities under |
| 589 |
|
varying conditions is troubling, especially at higher temperatures. In |
| 590 |
|
order to correct this model for use with a reaction field, it is |
| 591 |
|
necessary to adjust the force field parameters for the primary |
| 596 |
|
|
| 597 |
|
The parameters available for tuning include the $\sigma$ and |
| 598 |
|
$\epsilon$ Lennard-Jones parameters, the dipole strength ($\mu$), the |
| 599 |
< |
strength of the sticky potential ($\nu_0$), and the sticky attractive |
| 600 |
< |
and dipole repulsive cubic switching function cutoffs ($r_l$, $r_u$ |
| 601 |
< |
and $r_l^\prime$, $r_u^\prime$ respectively). The results of the |
| 602 |
< |
reparameterizations are shown in table \ref{params}. We are calling |
| 603 |
< |
these reparameterizations the Soft Sticky Dipole / Reaction Field |
| 604 |
< |
(SSD/RF - for use with a reaction field) and Soft Sticky Dipole |
| 605 |
< |
Extended (SSD/E - an attempt to improve the liquid structure in |
| 606 |
< |
simulations without a long-range correction). |
| 599 |
> |
strength of the sticky potential ($\nu_0$), and the cutoff distances |
| 600 |
> |
for the sticky attractive and dipole repulsive cubic switching |
| 601 |
> |
function cutoffs ($r_l$, $r_u$ and $r_l^\prime$, $r_u^\prime$ |
| 602 |
> |
respectively). The results of the reparameterizations are shown in |
| 603 |
> |
table \ref{params}. We are calling these reparameterizations the Soft |
| 604 |
> |
Sticky Dipole / Reaction Field ({\sc ssd/rf} - for use with a reaction |
| 605 |
> |
field) and Soft Sticky Dipole Extended ({\sc ssd/e} - an attempt to improve |
| 606 |
> |
the liquid structure in simulations without a long-range correction). |
| 607 |
|
|
| 608 |
|
\begin{table} |
| 609 |
|
\begin{center} |
| 610 |
|
\caption{Parameters for the original and adjusted models} |
| 611 |
|
\begin{tabular}{ l c c c c } |
| 612 |
|
\hline \\[-3mm] |
| 613 |
< |
\ \ \ Parameters\ \ \ & \ \ \ SSD [Ref. \citen{Ichiye96}] \ \ \ |
| 614 |
< |
& \ SSD1 [Ref. \citen{Ichiye03}]\ \ & \ SSD/E\ \ & \ SSD/RF \\ |
| 613 |
> |
\ \ \ Parameters\ \ \ & \ \ \ {\sc ssd} [Ref. \citen{Ichiye96}] \ \ \ |
| 614 |
> |
& \ {\sc ssd1} [Ref. \citen{Ichiye03}]\ \ & \ {\sc ssd/e}\ \ & \ {\sc ssd/rf} \\ |
| 615 |
|
\hline \\[-3mm] |
| 616 |
|
\ \ \ $\sigma$ (\AA) & 3.051 & 3.016 & 3.035 & 3.019\\ |
| 617 |
|
\ \ \ $\epsilon$ (kcal/mol) & 0.152 & 0.152 & 0.152 & 0.152\\ |
| 631 |
|
\begin{center} |
| 632 |
|
\epsfxsize=5in |
| 633 |
|
\epsfbox{GofRCompare.epsi} |
| 634 |
< |
\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with SSD/E |
| 635 |
< |
and SSD1 without reaction field (top), as well as SSD/RF and SSD1 with |
| 634 |
> |
\caption{Plots comparing experiment [Ref. \citen{Head-Gordon00_1}] with {\sc ssd/e} |
| 635 |
> |
and {\sc ssd1} without reaction field (top), as well as {\sc ssd/rf} and {\sc ssd1} with |
| 636 |
|
reaction field turned on (bottom). The insets show the respective |
| 637 |
|
first peaks in detail. Note how the changes in parameters have lowered |
| 638 |
< |
and broadened the first peak of SSD/E and SSD/RF.} |
| 638 |
> |
and broadened the first peak of {\sc ssd/e} and {\sc ssd/rf}.} |
| 639 |
|
\label{grcompare} |
| 640 |
|
\end{center} |
| 641 |
|
\end{figure} |
| 644 |
|
\begin{center} |
| 645 |
|
\epsfxsize=6in |
| 646 |
|
\epsfbox{dualsticky_bw.eps} |
| 647 |
< |
\caption{Isosurfaces of the sticky potential for SSD1 (left) and SSD/E \& |
| 648 |
< |
SSD/RF (right). Light areas correspond to the tetrahedral attractive |
| 649 |
< |
component, and darker areas correspond to the dipolar repulsive |
| 650 |
< |
component.} |
| 647 |
> |
\caption{Positive and negative isosurfaces of the sticky potential for |
| 648 |
> |
{\sc ssd1} (left) and {\sc ssd/e} \& {\sc ssd/rf} (right). Light areas correspond to the |
| 649 |
> |
tetrahedral attractive component, and darker areas correspond to the |
| 650 |
> |
dipolar repulsive component.} |
| 651 |
|
\label{isosurface} |
| 652 |
|
\end{center} |
| 653 |
|
\end{figure} |
| 654 |
|
|
| 655 |
< |
In the original paper detailing the development of SSD, Liu and Ichiye |
| 655 |
> |
In the original paper detailing the development of {\sc ssd}, Liu and Ichiye |
| 656 |
|
placed particular emphasis on an accurate description of the first |
| 657 |
|
solvation shell. This resulted in a somewhat tall and narrow first |
| 658 |
|
peak in $g(r)$ that integrated to give similar coordination numbers to |
| 659 |
|
the experimental data obtained by Soper and |
| 660 |
|
Phillips.\cite{Ichiye96,Soper86} New experimental x-ray scattering |
| 661 |
|
data from the Head-Gordon lab indicates a slightly lower and shifted |
| 662 |
< |
first peak in the g$_\mathrm{OO}(r)$, so our adjustments to SSD were |
| 663 |
< |
made while taking into consideration the new experimental |
| 662 |
> |
first peak in the g$_\mathrm{OO}(r)$, so our adjustments to {\sc ssd} were |
| 663 |
> |
made after taking into consideration the new experimental |
| 664 |
|
findings.\cite{Head-Gordon00_1} Figure \ref{grcompare} shows the |
| 665 |
|
relocation of the first peak of the oxygen-oxygen $g(r)$ by comparing |
| 666 |
< |
the revised SSD model (SSD1), SSD/E, and SSD/RF to the new |
| 666 |
> |
the revised {\sc ssd} model ({\sc ssd1}), {\sc ssd/e}, and {\sc ssd/rf} to the new |
| 667 |
|
experimental results. Both modified water models have shorter peaks |
| 668 |
|
that match more closely to the experimental peak (as seen in the |
| 669 |
|
insets of figure \ref{grcompare}). This structural alteration was |
| 682 |
|
to feel the pull of the tetrahedral restructuring. As the particles |
| 683 |
|
move closer together, the dipolar repulsion term becomes active and |
| 684 |
|
excludes unphysical nearest-neighbor arrangements. This compares with |
| 685 |
< |
how SSD and SSD1 exclude preferred dipole alignments before the |
| 685 |
> |
how {\sc ssd} and {\sc ssd1} exclude preferred dipole alignments before the |
| 686 |
|
particles feel the pull of the ``hydrogen bonds''. Aside from |
| 687 |
|
improving the shape of the first peak in the g(\emph{r}), this |
| 688 |
|
modification improves the densities considerably by allowing the |
| 693 |
|
improves the densities, these changes alone are insufficient to bring |
| 694 |
|
the system densities up to the values observed experimentally. To |
| 695 |
|
further increase the densities, the dipole moments were increased in |
| 696 |
< |
both of our adjusted models. Since SSD is a dipole based model, the |
| 696 |
> |
both of our adjusted models. Since {\sc ssd} is a dipole based model, the |
| 697 |
|
structure and transport are very sensitive to changes in the dipole |
| 698 |
< |
moment. The original SSD simply used the dipole moment calculated from |
| 698 |
> |
moment. The original {\sc ssd} simply used the dipole moment calculated from |
| 699 |
|
the TIP3P water model, which at 2.35 D is significantly greater than |
| 700 |
|
the experimental gas phase value of 1.84 D. The larger dipole moment |
| 701 |
|
is a more realistic value and improves the dielectric properties of |
| 703 |
|
liquid phase dipole moment ranging from 2.4 D to values as high as |
| 704 |
|
3.11 D, providing a substantial range of reasonable values for a |
| 705 |
|
dipole moment.\cite{Sprik91,Kusalik02,Badyal00,Barriol64} Moderately |
| 706 |
< |
increasing the dipole moments to 2.42 and 2.48 D for SSD/E and SSD/RF, |
| 706 |
> |
increasing the dipole moments to 2.42 and 2.48 D for {\sc ssd/e} and {\sc ssd/rf}, |
| 707 |
|
respectively, leads to significant changes in the density and |
| 708 |
|
transport of the water models. |
| 709 |
|
|
| 710 |
|
In order to demonstrate the benefits of these reparameterizations, a |
| 711 |
|
series of NPT and NVE simulations were performed to probe the density |
| 712 |
|
and transport properties of the adapted models and compare the results |
| 713 |
< |
to the original SSD model. This comparison involved full NPT melting |
| 714 |
< |
sequences for both SSD/E and SSD/RF, as well as NVE transport |
| 713 |
> |
to the original {\sc ssd} model. This comparison involved full NPT melting |
| 714 |
> |
sequences for both {\sc ssd/e} and {\sc ssd/rf}, as well as NVE transport |
| 715 |
|
calculations at the calculated self-consistent densities. Again, the |
| 716 |
|
results are obtained from five separate simulations of 1024 particle |
| 717 |
|
systems, and the melting sequences were started from different ice |
| 726 |
|
\begin{center} |
| 727 |
|
\epsfxsize=6in |
| 728 |
|
\epsfbox{ssdeDense.epsi} |
| 729 |
< |
\caption{Comparison of densities calculated with SSD/E to SSD1 without a |
| 729 |
> |
\caption{Comparison of densities calculated with {\sc ssd/e} to {\sc ssd1} without a |
| 730 |
|
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
| 731 |
|
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}] and |
| 732 |
|
experiment [Ref. \citen{CRC80}]. The window shows a expansion around |
| 733 |
|
300 K with error bars included to clarify this region of |
| 734 |
< |
interest. Note that both SSD1 and SSD/E show good agreement with |
| 734 |
> |
interest. Note that both {\sc ssd1} and {\sc ssd/e} show good agreement with |
| 735 |
|
experiment when the long-range correction is neglected.} |
| 736 |
|
\label{ssdedense} |
| 737 |
|
\end{center} |
| 738 |
|
\end{figure} |
| 739 |
|
|
| 740 |
< |
Fig. \ref{ssdedense} shows the density profile for the SSD/E model |
| 741 |
< |
in comparison to SSD1 without a reaction field, other common water |
| 740 |
> |
Fig. \ref{ssdedense} shows the density profile for the {\sc ssd/e} model |
| 741 |
> |
in comparison to {\sc ssd1} without a reaction field, other common water |
| 742 |
|
models, and experimental results. The calculated densities for both |
| 743 |
< |
SSD/E and SSD1 have increased significantly over the original SSD |
| 743 |
> |
{\sc ssd/e} and {\sc ssd1} have increased significantly over the original {\sc ssd} |
| 744 |
|
model (see fig. \ref{dense1}) and are in better agreement with the |
| 745 |
< |
experimental values. At 298 K, the densities of SSD/E and SSD1 without |
| 745 |
> |
experimental values. At 298 K, the densities of {\sc ssd/e} and {\sc ssd1} without |
| 746 |
|
a long-range correction are 0.996$\pm$0.001 g/cm$^3$ and |
| 747 |
|
0.999$\pm$0.001 g/cm$^3$ respectively. These both compare well with |
| 748 |
|
the experimental value of 0.997 g/cm$^3$, and they are considerably |
| 749 |
< |
better than the SSD value of 0.967$\pm$0.003 g/cm$^3$. The changes to |
| 749 |
> |
better than the {\sc ssd} value of 0.967$\pm$0.003 g/cm$^3$. The changes to |
| 750 |
|
the dipole moment and sticky switching functions have improved the |
| 751 |
|
structuring of the liquid (as seen in figure \ref{grcompare}, but they |
| 752 |
|
have shifted the density maximum to much lower temperatures. This |
| 753 |
|
comes about via an increase in the liquid disorder through the |
| 754 |
|
weakening of the sticky potential and strengthening of the dipolar |
| 755 |
< |
character. However, this increasing disorder in the SSD/E model has |
| 755 |
> |
character. However, this increasing disorder in the {\sc ssd/e} model has |
| 756 |
|
little effect on the melting transition. By monitoring $C_p$ |
| 757 |
< |
throughout these simulations, the melting transition for SSD/E was |
| 757 |
> |
throughout these simulations, the melting transition for {\sc ssd/e} was |
| 758 |
|
shown to occur at 235 K. The same transition temperature observed |
| 759 |
< |
with SSD and SSD1. |
| 759 |
> |
with {\sc ssd} and {\sc ssd1}. |
| 760 |
|
|
| 761 |
|
\begin{figure} |
| 762 |
|
\begin{center} |
| 763 |
|
\epsfxsize=6in |
| 764 |
|
\epsfbox{ssdrfDense.epsi} |
| 765 |
< |
\caption{Comparison of densities calculated with SSD/RF to SSD1 with a |
| 765 |
> |
\caption{Comparison of densities calculated with {\sc ssd/rf} to {\sc ssd1} with a |
| 766 |
|
reaction field, TIP3P [Ref. \citen{Jorgensen98b}], TIP5P |
| 767 |
|
[Ref. \citen{Jorgensen00}], SPC/E [Ref. \citen{Clancy94}], and |
| 768 |
|
experiment [Ref. \citen{CRC80}]. The inset shows the necessity of |
| 769 |
|
reparameterization when utilizing a reaction field long-ranged |
| 770 |
< |
correction - SSD/RF provides significantly more accurate densities |
| 771 |
< |
than SSD1 when performing room temperature simulations.} |
| 770 |
> |
correction - {\sc ssd/rf} provides significantly more accurate densities |
| 771 |
> |
than {\sc ssd1} when performing room temperature simulations.} |
| 772 |
|
\label{ssdrfdense} |
| 773 |
|
\end{center} |
| 774 |
|
\end{figure} |
| 775 |
|
|
| 776 |
|
Including the reaction field long-range correction in the simulations |
| 777 |
|
results in a more interesting comparison. A density profile including |
| 778 |
< |
SSD/RF and SSD1 with an active reaction field is shown in figure |
| 778 |
> |
{\sc ssd/rf} and {\sc ssd1} with an active reaction field is shown in figure |
| 779 |
|
\ref{ssdrfdense}. As observed in the simulations without a reaction |
| 780 |
< |
field, the densities of SSD/RF and SSD1 show a dramatic increase over |
| 781 |
< |
normal SSD (see figure \ref{dense1}). At 298 K, SSD/RF has a density |
| 780 |
> |
field, the densities of {\sc ssd/rf} and {\sc ssd1} show a dramatic increase over |
| 781 |
> |
normal {\sc ssd} (see figure \ref{dense1}). At 298 K, {\sc ssd/rf} has a density |
| 782 |
|
of 0.997$\pm$0.001 g/cm$^3$, directly in line with experiment and |
| 783 |
< |
considerably better than the original SSD value of 0.941$\pm$0.001 |
| 784 |
< |
g/cm$^3$ and the SSD1 value of 0.972$\pm$0.002 g/cm$^3$. These results |
| 783 |
> |
considerably better than the original {\sc ssd} value of 0.941$\pm$0.001 |
| 784 |
> |
g/cm$^3$ and the {\sc ssd1} value of 0.972$\pm$0.002 g/cm$^3$. These results |
| 785 |
|
further emphasize the importance of reparameterization in order to |
| 786 |
|
model the density properly under different simulation conditions. |
| 787 |
|
Again, these changes have only a minor effect on the melting point, |
| 788 |
< |
which observed at 245 K for SSD/RF, is identical to SSD and only 5 K |
| 789 |
< |
lower than SSD1 with a reaction field. Additionally, the difference in |
| 790 |
< |
density maxima is not as extreme, with SSD/RF showing a density |
| 788 |
> |
which observed at 245 K for {\sc ssd/rf}, is identical to {\sc ssd} and only 5 K |
| 789 |
> |
lower than {\sc ssd1} with a reaction field. Additionally, the difference in |
| 790 |
> |
density maxima is not as extreme, with {\sc ssd/rf} showing a density |
| 791 |
|
maximum at 255 K, fairly close to the density maxima of 260 K and 265 |
| 792 |
< |
K, shown by SSD and SSD1 respectively. |
| 792 |
> |
K, shown by {\sc ssd} and {\sc ssd1} respectively. |
| 793 |
|
|
| 794 |
|
\begin{figure} |
| 795 |
|
\begin{center} |
| 796 |
|
\epsfxsize=6in |
| 797 |
|
\epsfbox{ssdeDiffuse.epsi} |
| 798 |
< |
\caption{The diffusion constants calculated from SSD/E and SSD1, |
| 799 |
< |
both without a reaction field, along with experimental results |
| 800 |
< |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
| 801 |
< |
were performed at the average densities observed in the 1 atm NPT |
| 802 |
< |
simulations for the respective models. SSD/E is slightly more mobile |
| 803 |
< |
than experiment at all of the temperatures, but it is closer to |
| 804 |
< |
experiment at biologically relevant temperatures than SSD1 without a |
| 805 |
< |
long-range correction.} |
| 798 |
> |
\caption{The diffusion constants calculated from {\sc ssd/e} and {\sc ssd1} (both |
| 799 |
> |
without a reaction field) along with experimental results |
| 800 |
> |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations were |
| 801 |
> |
performed at the average densities observed in the 1 atm NPT |
| 802 |
> |
simulations for the respective models. {\sc ssd/e} is slightly more mobile |
| 803 |
> |
than experiment at all of the temperatures, but it is closer to |
| 804 |
> |
experiment at biologically relevant temperatures than {\sc ssd1} without a |
| 805 |
> |
long-range correction.} |
| 806 |
|
\label{ssdediffuse} |
| 807 |
|
\end{center} |
| 808 |
|
\end{figure} |
| 809 |
|
|
| 810 |
< |
The reparameterization of the SSD water model, both for use with and |
| 810 |
> |
The reparameterization of the {\sc ssd} water model, both for use with and |
| 811 |
|
without an applied long-range correction, brought the densities up to |
| 812 |
|
what is expected for simulating liquid water. In addition to improving |
| 813 |
< |
the densities, it is important that the excellent diffusive behavior |
| 814 |
< |
of SSD be maintained or improved. Figure \ref{ssdediffuse} compares |
| 815 |
< |
the temperature dependence of the diffusion constant of SSD/E to SSD1 |
| 813 |
> |
the densities, it is important that the diffusive behavior of {\sc ssd} be |
| 814 |
> |
maintained or improved. Figure \ref{ssdediffuse} compares the |
| 815 |
> |
temperature dependence of the diffusion constant of {\sc ssd/e} to {\sc ssd1} |
| 816 |
|
without an active reaction field at the densities calculated from |
| 817 |
|
their respective NPT simulations at 1 atm. The diffusion constant for |
| 818 |
< |
SSD/E is consistently higher than experiment, while SSD1 remains lower |
| 818 |
> |
{\sc ssd/e} is consistently higher than experiment, while {\sc ssd1} remains lower |
| 819 |
|
than experiment until relatively high temperatures (around 360 |
| 820 |
|
K). Both models follow the shape of the experimental curve well below |
| 821 |
|
300 K but tend to diffuse too rapidly at higher temperatures, as seen |
| 822 |
< |
in SSD1's crossing above 360 K. This increasing diffusion relative to |
| 822 |
> |
in {\sc ssd1}'s crossing above 360 K. This increasing diffusion relative to |
| 823 |
|
the experimental values is caused by the rapidly decreasing system |
| 824 |
< |
density with increasing temperature. Both SSD1 and SSD/E show this |
| 824 |
> |
density with increasing temperature. Both {\sc ssd1} and {\sc ssd/e} show this |
| 825 |
|
deviation in particle mobility, but this trend has different |
| 826 |
< |
implications on the diffusive behavior of the models. While SSD1 |
| 826 |
> |
implications on the diffusive behavior of the models. While {\sc ssd1} |
| 827 |
|
shows more experimentally accurate diffusive behavior in the high |
| 828 |
< |
temperature regimes, SSD/E shows more accurate behavior in the |
| 828 |
> |
temperature regimes, {\sc ssd/e} shows more accurate behavior in the |
| 829 |
|
supercooled and biologically relevant temperature ranges. Thus, the |
| 830 |
|
changes made to improve the liquid structure may have had an adverse |
| 831 |
|
affect on the density maximum, but they improve the transport behavior |
| 832 |
< |
of SSD/E relative to SSD1 under the most commonly simulated |
| 832 |
> |
of {\sc ssd/e} relative to {\sc ssd1} under the most commonly simulated |
| 833 |
|
conditions. |
| 834 |
|
|
| 835 |
|
\begin{figure} |
| 836 |
|
\begin{center} |
| 837 |
|
\epsfxsize=6in |
| 838 |
|
\epsfbox{ssdrfDiffuse.epsi} |
| 839 |
< |
\caption{The diffusion constants calculated from SSD/RF and SSD1, |
| 840 |
< |
both with an active reaction field, along with experimental results |
| 841 |
< |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations |
| 842 |
< |
were performed at the average densities observed in the 1 atm NPT |
| 843 |
< |
simulations for both of the models. Note how accurately SSD/RF |
| 844 |
< |
simulates the diffusion of water throughout this temperature |
| 845 |
< |
range. The more rapidly increasing diffusion constants at high |
| 846 |
< |
temperatures for both models is attributed to lower calculated |
| 847 |
< |
densities than those observed in experiment.} |
| 839 |
> |
\caption{The diffusion constants calculated from {\sc ssd/rf} and {\sc ssd1} (both |
| 840 |
> |
with an active reaction field) along with experimental results |
| 841 |
> |
[Refs. \citen{Gillen72} and \citen{Holz00}]. The NVE calculations were |
| 842 |
> |
performed at the average densities observed in the 1 atm NPT |
| 843 |
> |
simulations for both of the models. {\sc ssd/rf} simulates the diffusion of |
| 844 |
> |
water throughout this temperature range very well. The rapidly |
| 845 |
> |
increasing diffusion constants at high temperatures for both models |
| 846 |
> |
can be attributed to lower calculated densities than those observed in |
| 847 |
> |
experiment.} |
| 848 |
|
\label{ssdrfdiffuse} |
| 849 |
|
\end{center} |
| 850 |
|
\end{figure} |
| 851 |
|
|
| 852 |
< |
In figure \ref{ssdrfdiffuse}, the diffusion constants for SSD/RF are |
| 853 |
< |
compared to SSD1 with an active reaction field. Note that SSD/RF |
| 852 |
> |
In figure \ref{ssdrfdiffuse}, the diffusion constants for {\sc ssd/rf} are |
| 853 |
> |
compared to {\sc ssd1} with an active reaction field. Note that {\sc ssd/rf} |
| 854 |
|
tracks the experimental results quantitatively, identical within error |
| 855 |
|
throughout most of the temperature range shown and exhibiting only a |
| 856 |
< |
slight increasing trend at higher temperatures. SSD1 tends to diffuse |
| 856 |
> |
slight increasing trend at higher temperatures. {\sc ssd1} tends to diffuse |
| 857 |
|
more slowly at low temperatures and deviates to diffuse too rapidly at |
| 858 |
|
temperatures greater than 330 K. As stated above, this deviation away |
| 859 |
|
from the ideal trend is due to a rapid decrease in density at higher |
| 860 |
< |
temperatures. SSD/RF does not suffer from this problem as much as SSD1 |
| 860 |
> |
temperatures. {\sc ssd/rf} does not suffer from this problem as much as {\sc ssd1} |
| 861 |
|
because the calculated densities are closer to the experimental |
| 862 |
|
values. These results again emphasize the importance of careful |
| 863 |
|
reparameterization when using an altered long-range correction. |
| 864 |
|
|
| 865 |
|
\begin{table} |
| 866 |
+ |
\begin{minipage}{\linewidth} |
| 867 |
+ |
\renewcommand{\thefootnote}{\thempfootnote} |
| 868 |
|
\begin{center} |
| 869 |
< |
\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) Calculated for 298 K from data in ref. [\citen{Eisenberg69}]. (e) Calculated for 298 K from data in ref. \citen{Krynicki66}.} |
| 869 |
> |
\caption{Properties of the single-point water models compared with |
| 870 |
> |
experimental data at ambient conditions} |
| 871 |
|
\begin{tabular}{ l c c c c c } |
| 872 |
|
\hline \\[-3mm] |
| 873 |
< |
\ \ \ \ \ \ & \ \ \ SSD1 \ \ \ & \ SSD/E \ \ \ & \ SSD1 (RF) \ \ |
| 874 |
< |
\ & \ SSD/RF \ \ \ & \ Expt. \\ |
| 873 |
> |
\ \ \ \ \ \ & \ \ \ {\sc ssd1} \ \ \ & \ {\sc ssd/e} \ \ \ & \ {\sc ssd1} (RF) \ \ |
| 874 |
> |
\ & \ {\sc ssd/rf} \ \ \ & \ Expt. \\ |
| 875 |
|
\hline \\[-3mm] |
| 876 |
|
\ \ \ $\rho$ (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 \\ |
| 877 |
|
\ \ \ $C_p$ (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 \\ |
| 878 |
< |
\ \ \ $D$ ($10^{-5}$ 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}$ \\ |
| 879 |
< |
\ \ \ Coordination Number & 3.9 & 4.3 & 3.8 & 4.4 & 4.7$^\text{b}$ \\ |
| 880 |
< |
\ \ \ H-bonds per particle & 3.7 & 3.6 & 3.7 & 3.7 & 3.5$^\text{c}$ \\ |
| 881 |
< |
\ \ \ $\tau_1$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & 7.2 $\pm$0.4 & 5.7$^\text{d}$ \\ |
| 882 |
< |
\ \ \ $\tau_2$ (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}$ \\ |
| 878 |
> |
\ \ \ $D$ ($10^{-5}$ cm$^2$/s) & 1.78 $\pm$0.07 & 2.51 $\pm$0.18 & |
| 879 |
> |
2.00 $\pm$0.17 & 2.32 $\pm$0.06 & 2.299\cite{Mills73} \\ |
| 880 |
> |
\ \ \ Coordination Number ($n_C$) & 3.9 & 4.3 & 3.8 & 4.4 & |
| 881 |
> |
4.7\footnote{Calculated by integrating $g_{\text{OO}}(r)$ in |
| 882 |
> |
Ref. \citen{Head-Gordon00_1}} \\ |
| 883 |
> |
\ \ \ H-bonds per particle ($n_H$) & 3.7 & 3.6 & 3.7 & 3.7 & |
| 884 |
> |
3.5\footnote{Calculated by integrating $g_{\text{OH}}(r)$ in |
| 885 |
> |
Ref. \citen{Soper86}} \\ |
| 886 |
> |
\ \ \ $\tau_1$ (ps) & 10.9 $\pm$0.6 & 7.3 $\pm$0.4 & 7.5 $\pm$0.7 & |
| 887 |
> |
7.2 $\pm$0.4 & 5.7\footnote{Calculated for 298 K from data in Ref. \citen{Eisenberg69}} \\ |
| 888 |
> |
\ \ \ $\tau_2$ (ps) & 4.7 $\pm$0.4 & 3.1 $\pm$0.2 & 3.5 $\pm$0.3 & 3.2 |
| 889 |
> |
$\pm$0.2 & 2.3\footnote{Calculated for 298 K from data in |
| 890 |
> |
Ref. \citen{Krynicki66}} |
| 891 |
|
\end{tabular} |
| 892 |
|
\label{liquidproperties} |
| 893 |
|
\end{center} |
| 894 |
+ |
\end{minipage} |
| 895 |
|
\end{table} |
| 896 |
|
|
| 897 |
|
Table \ref{liquidproperties} gives a synopsis of the liquid state |
| 898 |
|
properties of the water models compared in this study along with the |
| 899 |
|
experimental values for liquid water at ambient conditions. The |
| 900 |
< |
coordination number ($N_C$) and hydrogen bonds per particle ($N_H$) |
| 901 |
< |
were calculated by integrating the following relations: |
| 900 |
> |
coordination number ($n_C$) and number of hydrogen bonds per particle |
| 901 |
> |
($n_H$) were calculated by integrating the following relations: |
| 902 |
|
\begin{equation} |
| 903 |
< |
N_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2\text{g}_{\text{OO}}(r)dr, |
| 903 |
> |
n_C = 4\pi\rho_{\text{OO}}\int_{0}^{a}r^2\text{g}_{\text{OO}}(r)dr, |
| 904 |
|
\end{equation} |
| 905 |
|
\begin{equation} |
| 906 |
< |
N_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2\text{g}_{\text{OH}}(r)dr, |
| 906 |
> |
n_H = 4\pi\rho_{\text{OH}}\int_{0}^{b}r^2\text{g}_{\text{OH}}(r)dr, |
| 907 |
|
\end{equation} |
| 908 |
|
where $\rho$ is the number density of the specified pair interactions, |
| 909 |
|
$a$ and $b$ are the radial locations of the minima following the first |
| 910 |
< |
solvation shell peak in g$_\text{OO}(r)$ or g$_\text{OH}(r)$ |
| 911 |
< |
respectively. The number of hydrogen bonds stays relatively constant |
| 912 |
< |
across all of the models, but the coordination numbers of SSD/E and |
| 913 |
< |
SSD/RF show an improvement over SSD1. This improvement is primarily |
| 914 |
< |
due to the widening of the first solvation shell peak, allowing the |
| 915 |
< |
first minima to push outward. Comparing the coordination number with |
| 916 |
< |
the number of hydrogen bonds can lead to more insight into the |
| 917 |
< |
structural character of the liquid. Because of the near identical |
| 918 |
< |
values for SSD1, it appears to be a little too exclusive, in that all |
| 919 |
< |
molecules in the first solvation shell are involved in forming ideal |
| 920 |
< |
hydrogen bonds. The differing numbers for the newly parameterized |
| 921 |
< |
models indicate the allowance of more fluid configurations in addition |
| 922 |
< |
to the formation of an acceptable number of ideal hydrogen bonds. |
| 910 |
< |
|
| 911 |
< |
The time constants for the self orientational autocorrelation function |
| 910 |
> |
peak in g$_\text{OO}(r)$ or g$_\text{OH}(r)$ respectively. The number |
| 911 |
> |
of hydrogen bonds stays relatively constant across all of the models, |
| 912 |
> |
but the coordination numbers of {\sc ssd/e} and {\sc ssd/rf} show an improvement |
| 913 |
> |
over {\sc ssd1}. This improvement is primarily due to extension of the |
| 914 |
> |
first solvation shell in the new parameter sets. Because $n_H$ and |
| 915 |
> |
$n_C$ are nearly identical in {\sc ssd1}, it appears that all molecules in |
| 916 |
> |
the first solvation shell are involved in hydrogen bonds. Since $n_H$ |
| 917 |
> |
and $n_C$ differ in the newly parameterized models, the orientations |
| 918 |
> |
in the first solvation shell are a bit more ``fluid''. Therefore {\sc ssd1} |
| 919 |
> |
overstructures the first solvation shell and our reparameterizations |
| 920 |
> |
have returned this shell to more realistic liquid-like behavior. |
| 921 |
> |
|
| 922 |
> |
The time constants for the orientational autocorrelation functions |
| 923 |
|
are also displayed in Table \ref{liquidproperties}. The dipolar |
| 924 |
< |
orientational time correlation function ($\Gamma_{l}$) is described |
| 924 |
> |
orientational time correlation functions ($C_{l}$) are described |
| 925 |
|
by: |
| 926 |
|
\begin{equation} |
| 927 |
< |
\Gamma_{l}(t) = \langle P_l[\mathbf{u}_j(0)\cdot\mathbf{u}_j(t)]\rangle, |
| 927 |
> |
C_{l}(t) = \langle P_l[\hat{\mathbf{u}}_j(0)\cdot\hat{\mathbf{u}}_j(t)]\rangle, |
| 928 |
|
\end{equation} |
| 929 |
< |
where $P_l$ is a Legendre polynomial of order $l$ and $\mathbf{u}_j$ |
| 930 |
< |
is the unit vector of the particle dipole.\cite{Rahman71} From these |
| 931 |
< |
correlation functions, the orientational relaxation time of the dipole |
| 932 |
< |
vector can be calculated from an exponential fit in the long-time |
| 933 |
< |
regime ($t > \tau_l$).\cite{Rothschild84} Calculation of these |
| 934 |
< |
time constants were averaged from five detailed NVE simulations |
| 935 |
< |
performed at the STP density for each of the respective models. It |
| 936 |
< |
should be noted that the commonly cited value for $\tau_2$ of 1.9 ps |
| 937 |
< |
was determined from the NMR data in reference \citen{Krynicki66} at a |
| 938 |
< |
temperature near 34$^\circ$C.\cite{Rahman71} Because of the strong |
| 939 |
< |
temperature dependence of $\tau_2$, it is necessary to recalculate it |
| 940 |
< |
at 298 K to make proper comparisons. The value shown in Table |
| 929 |
> |
where $P_l$ are Legendre polynomials of order $l$ and |
| 930 |
> |
$\hat{\mathbf{u}}_j$ is the unit vector pointing along the molecular |
| 931 |
> |
dipole.\cite{Rahman71} From these correlation functions, the |
| 932 |
> |
orientational relaxation time of the dipole vector can be calculated |
| 933 |
> |
from an exponential fit in the long-time regime ($t > |
| 934 |
> |
\tau_l$).\cite{Rothschild84} Calculation of these time constants were |
| 935 |
> |
averaged over five detailed NVE simulations performed at the ambient |
| 936 |
> |
conditions for each of the respective models. It should be noted that |
| 937 |
> |
the commonly cited value of 1.9 ps for $\tau_2$ was determined from |
| 938 |
> |
the NMR data in Ref. \citen{Krynicki66} at a temperature near |
| 939 |
> |
34$^\circ$C.\cite{Rahman71} Because of the strong temperature |
| 940 |
> |
dependence of $\tau_2$, it is necessary to recalculate it at 298 K to |
| 941 |
> |
make proper comparisons. The value shown in Table |
| 942 |
|
\ref{liquidproperties} was calculated from the same NMR data in the |
| 943 |
< |
fashion described in reference \citen{Krynicki66}. Similarly, $\tau_1$ |
| 944 |
< |
was recomputed for 298 K from the data in ref \citen{Eisenberg69}. |
| 945 |
< |
Again, SSD/E and SSD/RF show improved behavior over SSD1, both with |
| 943 |
> |
fashion described in Ref. \citen{Krynicki66}. Similarly, $\tau_1$ was |
| 944 |
> |
recomputed for 298 K from the data in Ref. \citen{Eisenberg69}. |
| 945 |
> |
Again, {\sc ssd/e} and {\sc ssd/rf} show improved behavior over {\sc ssd1}, both with |
| 946 |
|
and without an active reaction field. Turning on the reaction field |
| 947 |
< |
leads to much improved time constants for SSD1; however, these results |
| 948 |
< |
also include a corresponding decrease in system density. Numbers |
| 949 |
< |
published from the original SSD dynamics studies are shorter than the |
| 950 |
< |
values observed here, and this difference can be attributed to the use |
| 951 |
< |
of the Ewald sum technique versus a reaction field.\cite{Ichiye99} |
| 947 |
> |
leads to much improved time constants for {\sc ssd1}; however, these results |
| 948 |
> |
also include a corresponding decrease in system density. |
| 949 |
> |
Orientational relaxation times published in the original {\sc ssd} dynamics |
| 950 |
> |
papers are smaller than the values observed here, and this difference |
| 951 |
> |
can be attributed to the use of the Ewald sum.\cite{Ichiye99} |
| 952 |
|
|
| 953 |
|
\subsection{Additional Observations} |
| 954 |
|
|
| 956 |
|
\begin{center} |
| 957 |
|
\epsfxsize=6in |
| 958 |
|
\epsfbox{icei_bw.eps} |
| 959 |
< |
\caption{A water lattice built from the crystal structure assumed by |
| 960 |
< |
SSD/E when undergoing an extremely restricted temperature NPT |
| 961 |
< |
simulation. This form of ice is referred to as ice-{\it i} to |
| 962 |
< |
emphasize its simulation origins. This image was taken of the (001) |
| 951 |
< |
face of the crystal.} |
| 959 |
> |
\caption{The most stable crystal structure assumed by the {\sc ssd} family |
| 960 |
> |
of water models. We refer to this structure as Ice-{\it i} to |
| 961 |
> |
indicate its origins in computer simulation. This image was taken of |
| 962 |
> |
the (001) face of the crystal.} |
| 963 |
|
\label{weirdice} |
| 964 |
|
\end{center} |
| 965 |
|
\end{figure} |
| 966 |
|
|
| 967 |
|
While performing a series of melting simulations on an early iteration |
| 968 |
< |
of SSD/E not discussed in this paper, we observed recrystallization |
| 968 |
> |
of {\sc ssd/e} not discussed in this paper, we observed recrystallization |
| 969 |
|
into a novel structure not previously known for water. After melting |
| 970 |
|
at 235 K, two of five systems underwent crystallization events near |
| 971 |
|
245 K. The two systems remained crystalline up to 320 and 330 K, |
| 973 |
|
that does not correspond to any known form of ice. This appears to be |
| 974 |
|
an artifact of the point dipolar models, so to distinguish it from the |
| 975 |
|
experimentally observed forms of ice, we have denoted the structure |
| 976 |
< |
Ice-$\sqrt{\smash[b]{-\text{I}}}$ (ice-{\it i}). A large enough |
| 976 |
> |
Ice-$\sqrt{\smash[b]{-\text{I}}}$ (Ice-{\it i}). A large enough |
| 977 |
|
portion of the sample crystallized that we have been able to obtain a |
| 978 |
< |
near ideal crystal structure of ice-{\it i}. Figure \ref{weirdice} |
| 978 |
> |
near ideal crystal structure of Ice-{\it i}. Figure \ref{weirdice} |
| 979 |
|
shows the repeating crystal structure of a typical crystal at 5 |
| 980 |
|
K. Each water molecule is hydrogen bonded to four others; however, the |
| 981 |
|
hydrogen bonds are bent rather than perfectly straight. This results |
| 986 |
|
configuration. Though not ideal, these flexed hydrogen bonds are |
| 987 |
|
favorable enough to stabilize an entire crystal generated around them. |
| 988 |
|
|
| 989 |
< |
Initial simulations indicated that ice-{\it i} is the preferred ice |
| 990 |
< |
structure for at least the SSD/E model. To verify this, a comparison |
| 991 |
< |
was made between near ideal crystals of ice~$I_h$, ice~$I_c$, and |
| 992 |
< |
ice-{\it i} at constant pressure with SSD/E, SSD/RF, and |
| 993 |
< |
SSD1. Near-ideal versions of the three types of crystals were cooled |
| 994 |
< |
to 1 K, and the enthalpies of each were compared using all three water |
| 995 |
< |
models. With every model in the SSD family, ice-{\it i} had the lowest |
| 996 |
< |
calculated enthalpy: 5\% lower than $I_h$ with SSD1, 6.5\% lower with |
| 997 |
< |
SSD/E, and 7.5\% lower with SSD/RF. The enthalpy data is summarized |
| 998 |
< |
in Table \ref{iceenthalpy}. |
| 989 |
> |
Initial simulations indicated that Ice-{\it i} is the preferred ice |
| 990 |
> |
structure for at least the {\sc ssd/e} model. To verify this, a |
| 991 |
> |
comparison was made between near ideal crystals of ice~$I_h$, |
| 992 |
> |
ice~$I_c$, and Ice-{\it i} at constant pressure with {\sc ssd/e}, {\sc |
| 993 |
> |
ssd/rf}, and {\sc ssd1}. Near-ideal versions of the three types of |
| 994 |
> |
crystals were cooled to 1 K, and enthalpies of formation of each were |
| 995 |
> |
compared using all three water models. Enthalpies were estimated from |
| 996 |
> |
the isobaric-isothermal simulations using $H=U+P_{\text ext}V$ where |
| 997 |
> |
$P_{\text ext}$ is the applied pressure. A constant value of |
| 998 |
> |
-60.158 kcal / mol has been added to place our zero for the |
| 999 |
> |
enthalpies of formation for these systems at the traditional state |
| 1000 |
> |
(elemental forms at standard temperature and pressure). With every |
| 1001 |
> |
model in the {\sc ssd} family, Ice-{\it i} had the lowest calculated |
| 1002 |
> |
enthalpy of formation. |
| 1003 |
|
|
| 1004 |
|
\begin{table} |
| 1005 |
|
\begin{center} |
| 1006 |
< |
\caption{Enthalpies (in kcal / mol) of the three crystal structures (at 1 |
| 1007 |
< |
K) exhibited by the SSD family of water models} |
| 1006 |
> |
\caption{Enthalpies of Formation (in kcal / mol) of the three crystal |
| 1007 |
> |
structures (at 1 K) exhibited by the {\sc ssd} family of water models} |
| 1008 |
|
\begin{tabular}{ l c c c } |
| 1009 |
|
\hline \\[-3mm] |
| 1010 |
|
\ \ \ Water Model \ \ \ & \ \ \ Ice-$I_h$ \ \ \ & \ Ice-$I_c$\ \ & \ |
| 1011 |
|
Ice-{\it i} \\ |
| 1012 |
|
\hline \\[-3mm] |
| 1013 |
< |
\ \ \ SSD/E & -12.286 & -12.292 & -13.590 \\ |
| 1014 |
< |
\ \ \ SSD/RF & -12.935 & -12.917 & -14.022 \\ |
| 1015 |
< |
\ \ \ SSD1 & -12.496 & -12.411 & -13.417 \\ |
| 1016 |
< |
\ \ \ SSD1 (RF) & -12.504 & -12.411 & -13.134 \\ |
| 1013 |
> |
\ \ \ {\sc ssd/e} & -12.286 & -12.292 & -13.590 \\ |
| 1014 |
> |
\ \ \ {\sc ssd/rf} & -12.935 & -12.917 & -14.022 \\ |
| 1015 |
> |
\ \ \ {\sc ssd1} & -12.496 & -12.411 & -13.417 \\ |
| 1016 |
> |
\ \ \ {\sc ssd1} (RF) & -12.504 & -12.411 & -13.134 \\ |
| 1017 |
|
\end{tabular} |
| 1018 |
|
\label{iceenthalpy} |
| 1019 |
|
\end{center} |
| 1020 |
|
\end{table} |
| 1021 |
|
|
| 1022 |
|
In addition to these energetic comparisons, melting simulations were |
| 1023 |
< |
performed with ice-{\it i} as the initial configuration using SSD/E, |
| 1024 |
< |
SSD/RF, and SSD1 both with and without a reaction field. The melting |
| 1025 |
< |
transitions for both SSD/E and SSD1 without reaction field occurred at |
| 1026 |
< |
temperature in excess of 375~K. SSD/RF and SSD1 with a reaction field |
| 1023 |
> |
performed with ice-{\it i} as the initial configuration using {\sc ssd/e}, |
| 1024 |
> |
{\sc ssd/rf}, and {\sc ssd1} both with and without a reaction field. The melting |
| 1025 |
> |
transitions for both {\sc ssd/e} and {\sc ssd1} without reaction field occurred at |
| 1026 |
> |
temperature in excess of 375~K. {\sc ssd/rf} and {\sc ssd1} with a reaction field |
| 1027 |
|
showed more reasonable melting transitions near 325~K. These melting |
| 1028 |
< |
point observations clearly show that all of the SSD-derived models |
| 1028 |
> |
point observations clearly show that all of the {\sc ssd}-derived models |
| 1029 |
|
prefer the ice-{\it i} structure. |
| 1030 |
|
|
| 1031 |
|
\section{Conclusions} |
| 1032 |
|
|
| 1033 |
|
The density maximum and temperature dependence of the self-diffusion |
| 1034 |
< |
constant were studied for the SSD water model, both with and without |
| 1034 |
> |
constant were studied for the {\sc ssd} water model, both with and without |
| 1035 |
|
the use of reaction field, via a series of NPT and NVE |
| 1036 |
|
simulations. The constant pressure simulations showed a density |
| 1037 |
|
maximum near 260 K. In most cases, the calculated densities were |
| 1038 |
|
significantly lower than the densities obtained from other water |
| 1039 |
< |
models (and experiment). Analysis of self-diffusion showed SSD to |
| 1039 |
> |
models (and experiment). Analysis of self-diffusion showed {\sc ssd} to |
| 1040 |
|
capture the transport properties of water well in both the liquid and |
| 1041 |
|
supercooled liquid regimes. |
| 1042 |
|
|
| 1043 |
< |
In order to correct the density behavior, the original SSD model was |
| 1044 |
< |
reparameterized for use both with and without a reaction field (SSD/RF |
| 1045 |
< |
and SSD/E), and comparisons were made with SSD1, Ichiye's density |
| 1046 |
< |
corrected version of SSD. Both models improve the liquid structure, |
| 1043 |
> |
In order to correct the density behavior, the original {\sc ssd} model was |
| 1044 |
> |
reparameterized for use both with and without a reaction field ({\sc ssd/rf} |
| 1045 |
> |
and {\sc ssd/e}), and comparisons were made with {\sc ssd1}, Ichiye's density |
| 1046 |
> |
corrected version of {\sc ssd}. Both models improve the liquid structure, |
| 1047 |
|
densities, and diffusive properties under their respective simulation |
| 1048 |
|
conditions, indicating the necessity of reparameterization when |
| 1049 |
|
changing the method of calculating long-range electrostatic |
| 1052 |
|
simulations of biochemical systems. |
| 1053 |
|
|
| 1054 |
|
The existence of a novel low-density ice structure that is preferred |
| 1055 |
< |
by the SSD family of water models is somewhat troubling, since liquid |
| 1055 |
> |
by the {\sc ssd} family of water models is somewhat troubling, since liquid |
| 1056 |
|
simulations on this family of water models at room temperature are |
| 1057 |
|
effectively simulations of supercooled or metastable liquids. One |
| 1058 |
|
way to destabilize this unphysical ice structure would be to make the |
| 1061 |
|
reparameterization to maintain the same level of agreement with the |
| 1062 |
|
experiments. |
| 1063 |
|
|
| 1064 |
< |
Additionally, our initial calculations show that the ice-{\it i} |
| 1064 |
> |
Additionally, our initial calculations show that the Ice-{\it i} |
| 1065 |
|
structure may also be a preferred crystal structure for at least one |
| 1066 |
|
other popular multi-point water model (TIP3P), and that much of the |
| 1067 |
|
simulation work being done using this popular model could also be at |
