| 85 |
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proton-ordered bulk ice I$_\mathrm{h}$ in alternative orthorhombic |
| 86 |
|
cells.\cite{Hirsch04} |
| 87 |
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|
| 88 |
< |
We have using Hirsch and Ojam\"{a}e's structure 6 which is an |
| 88 |
> |
We are using Hirsch and Ojam\"{a}e's structure 6 which is an |
| 89 |
|
orthorhombic cell ($P2_12_12_1$) that produces a proton-ordered |
| 90 |
|
version of ice Ih. Table \ref{tab:equiv} contains a mapping between |
| 91 |
|
the Miller indices in the P$6_3/mmc$ crystal system and those in the |
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|
were divided into 100 artificial bins along the $z$-axis for the |
| 248 |
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VSS-RNEMD moves, which were attempted every 50 fs. The gradients were |
| 249 |
|
allowed to develop for 1 ns before data collection was began. Once |
| 250 |
< |
established, snapshots of the system were taken every 1 ps, and the |
| 250 |
> |
established, four successive 0.5 ns runs were performed for each shear rate. During these simulations, snapshots of the system were taken every 1 ps, and the |
| 251 |
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average velocities and densities of each bin were accumulated every |
| 252 |
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attempted VSS-RNEMD move. |
| 253 |
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|
| 256 |
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%For the prismatic, a gradient was not found that would give me the interfacial temperature I desired, so while imposing a thermal gradient that had the interface at 220K, I raised the temperature of the system to 230K. This resulted in a thermal gradient which gave my interface at 225K, equilibrated for ins NVT, then ins NVE while this gradient was still imposed, then I began dragging. |
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%I have run each system for 1 ns under PTgrads to allow them to develop, then ran each system for an additional 2 ns in segments of 0.5 ns in order to calculate statistics of the calculated values. |
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|
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\section{Results and discussion} |
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|
| 261 |
|
\subsection{Measuring the Width of the Interface} |
| 262 |
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Any parameter or function that varies across the interface from a bulk liquid value to a solid value can be used as a measure of the width of the interface. Here, the local order tetraherdal parameter as described by Kumar\cite{Kumar09} and |
| 263 |
< |
Errington\cite{Errington01} was used as a structural measure of the interfacial width. The orientational time-correlation function was also used to investigate the difference between the dynamic and structural width of the interface. |
| 262 |
> |
\subsubsection{Tetrahedrality Order Parameter} |
| 263 |
> |
Any parameter or function that varies across the interface from a bulk liquid value to a solid value can be used as a measure of the width of the interface. However, due to the VSS-RNEMD moves pertrurbing the momentum of the molecules, parameters such as the translational order parameter and the diffusion order parameter may be artifically skewed. A structural parameter such as the pairwise correlation function would not be influenced by the perturbations. Here, the local order tetraherdal parameter as described by Kumar\cite{Kumar09} and |
| 264 |
> |
Errington\cite{Errington01} was used as a measure of the interfacial width. |
| 265 |
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|
| 266 |
|
The local tetrahedral order parameter, $q$, is given by |
| 267 |
|
\begin{equation} |
| 271 |
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|
| 272 |
|
%If the central water molecule has a perfect tetrahedral geometry with its four nearest neighbors, the parameter goes to one, and decreases to zero as the geometry deviates from the ideal configuration. |
| 273 |
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|
| 274 |
< |
The system was divided into 100 bins of length $L$ along the $z$-axis, and a cutoff radius for the neighboring molecules was set to 3.41 \AA\. A $q_{z}$ value was then determined for each bin by averaging the $q$ values for each molecule in the bin. |
| 274 |
> |
The system was divided into 100 bins of length $L$ along the $z$-axis, and a cutoff radius for the neighboring molecules was set to 3.41 \AA\ . A $q_{z}$ value was then determined by averaging the $q$ values for each molecule in the bin. |
| 275 |
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\begin{equation} |
| 276 |
|
q_{z} \equiv \int_0^L \Bigg[1 -\frac{3}{8}\sum_{i=1}^{3} \sum_{j=i+1}^{4} \bigg[\cos\psi_{ikj}+\frac{1}{3}\bigg]^2\Bigg]\delta(z_{k}-z)\mathrm{d}z |
| 274 |
– |
\end{equation} |
| 275 |
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The $q_{z}$ values for each snapshot were then averaged to give an average tetrahedrally profile of the system about the $z$- axis. The profile was then fit with a hyperbolic tangent function given by |
| 276 |
– |
\begin{equation} |
| 277 |
– |
q_{z} \approx q_{liq}+\frac{q_{ice}-q_{liq}}{2}(\tanh(\alpha(z-I_{L,m}))-\tanh(\alpha(z-I_{R,m}))) |
| 277 |
|
\end{equation} |
| 278 |
< |
where $q_{liq}$ and $q_{ice}$ are fitting parameters for the local tetrahedral order parameter for the liquid and ice, $\alpha$ is proportional to the inverse of the width of the interface, and $I_{L,m}$ and $I_{R,m}$ are the midpoints of the left and right interface. During the simulations where a kinetic energy flux was imposed, there was found to be a thermal influence in the liquid phase region of the tetrahedrally profile due to the thermal gradient developed in the system. To maximize the fit of the interface, another term was added to the hyperbolic tangent fitting function, |
| 279 |
< |
\begin{equation} |
| 278 |
> |
The $q_{z}$ values for each snapshot were then averaged to give an average tetrahedrality profile of the system about the $z$- axis. The profile was then fit with a hyperbolic tangent function given by |
| 279 |
> |
|
| 280 |
> |
\begin{equation}\label{tet_fit} |
| 281 |
|
q_{z} \approx q_{liq}+\frac{q_{ice}-q_{liq}}{2}(\tanh(\alpha(z-I_{L,m}))-\tanh(\alpha(z-I_{R,m})))+\beta|(z-z_{mid})| |
| 282 |
|
\end{equation} |
| 283 |
– |
where $\beta$ is a fitting parameter and $z_{mid}$ is the midpoint of the z dimension of the simulation box. |
| 283 |
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|
| 284 |
+ |
where $q_{liq}$ and $q_{ice}$ are fitting parameters for the local tetrahedral order parameter for the liquid and ice, $\alpha$ is proportional to the inverse of the width of the interface, and $I_{L,m}$ and $I_{R,m}$ are the midpoints of the left and right interface. The last term in \ref{tet_fit} accounts for the influence the thermal gradient has on the tetrahedrality profile in the liquid region; here $\beta$ is a fitting parameter and $z_{mid}$ is the midpoint of the z dimension of the simulation box. |
| 285 |
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|
| 286 |
< |
\section{Results and discussion} |
| 286 |
> |
In Figures \ref{fig:bComic} and \ref{fig:pComic} we see the $z$-dimensional profiles for several parameters for the basal and prismatic systems. In panel (a) of the figures we see the tetrahedrality profile of the systems (black circles). In the liquid region of the system, the local tetrahedral order parameter is approximately 0.75 while in the solid region the parameter is approximately 0.94, indicating a more tetrahedral structure of the water molecules. The hyperbolic tangent function used to fit the tetrahedrality profiles is in red and the verticle dotted lines denote the midpoint of the interfaces. The weak thermal gradient applied to the systems in order to keep the interface at a stable temperature, 225$\pm$5K, can be seen in panel (b). Lastly, the velocity gradient across the systems can be seen in panel (c). From panel (c), we can see liquid phase water molecules 10 \AA\ to 15 \AA\ from the midpoint of the interfaces are being dragged along with the ice block, indicating that the shearing of ice water is in the stick boundary condition. |
| 287 |
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|
| 288 |
– |
%Images to include: 3-long comic strip style of <Vx>, T, q_z as a function of z for the basal and prismatic faces. q_z by z with fit for basal and prismatic. interface width as a function of deltaVx (shear rate) with basal and prismatic on the same plot, error bars in the x and y. <Vx> by flux with basal and prismatic on same graph, back out slope from xmgr and error in slope to get lambda, friction coefficient of interface. |
| 289 |
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|
| 290 |
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In Figures \ref{fig:bComic} and \ref{fig:pComic} we see the $z$-dimensional profiles for several parameters for the basal and prismatic systems respectively. In panel (a) of the figures we see the tetrahedrality profile of the systems (black circles). In the liquid region of the system, the local tetrahedral order parameter is approximately 0.75. In the solid region, the parameter is approximately 0.94, indicating a more tetrahedral structure of the water molecules. The hyperbolic tangent function used to fit the tetrahedrality profiles can be found in red and the verticle dotted lines denote the midpoint of the interfaces. The weak thermal gradient applied to the systems in order to keep the interface at a stable temperature, 225$\pm$5K, can be seen in panel (b). Lastly, the velocity gradient across the systems can be seen in panel (c). From panel (c), we can see liquid phase water molecules 5 to 12 \AA\ from the midpoint of the basal and prismatic interfaces are being dragged along with the ice block. This indicates that the shearing of ice water is in the stick boundary condition. |
| 291 |
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|
| 288 |
|
\begin{figure} |
| 289 |
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\includegraphics[width=\linewidth]{bComicStrip} |
| 290 |
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\caption{\label{fig:bComic} The basal system: (a) The local tetrahedral order parameter,$q$, (black circles) fit by a hyperbolic tangent (red line), (b) the thermal gradient imposed on the system to maintain a stable interfacial temperature, and (c) the velocity gradient imposed on the system. The verticle dotted lines indicate the midpoint of the interfaces.} |
| 291 |
|
\end{figure} |
| 292 |
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|
| 297 |
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%(a) The local tetrahedral order parameter across the z-dimension of the system (black circles) fit by a hyperbolic tangent (red line). (b) The thermal gradient imposed on the system to maintain a stable interfacial temperature. (c) The velocity gradient imposed on the system. The verticle dotted lines indicate the midpoint of the interfaces. |
| 298 |
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|
| 293 |
|
\begin{figure} |
| 294 |
|
\includegraphics[width=\linewidth]{pComicStrip} |
| 295 |
|
\caption{\label{fig:pComic} The prismatic system: (a) The local tetrahedral order parameter,$q$, (black circles) fit by a hyperbolic tangent (red line), (b) the thermal gradient imposed on the system to maintain a stable interfacial temperature, and (c) the velocity gradient imposed on the system.The verticle dotted lines indicate the midpoint of the interfaces.} |
| 296 |
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\end{figure} |
| 297 |
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|
| 298 |
+ |
From the tetrahedrality fits, we found the interfacial width for the basal and prismatic systems to be 3.2$\pm$0.4 \AA\ and 3.6$\pm$0.2 \AA\ with no applied momentum flux. Over the range of shear rates investigated, 0.6$\pm$0.3 ms\textsuperscript{-1} to 5.3$\pm$0.5 ms\textsuperscript{-1} for the basal system and 0.9$\pm$0.2 ms\textsuperscript{-1} to 4.5$\pm$0.1 ms\textsuperscript{-1}, there was no appreciable change in the interface width found. The calculated values for the interfacial width over all shear rates investigated contained the control values within their error bars. |
| 299 |
|
|
| 300 |
< |
\subsection{Interfacial Width} |
| 301 |
< |
%For the basal and prismatic systems, the ice blocks were sheared through the water at varying rates while an imposed thermal gradient kept the interface at the stable temperature range as described by Byrk and Haymet. |
| 302 |
< |
We found the interfacial width for the basal and prismatic systems to be 3.2$\pm$0.4 \AA\ and 3.6$\pm$0.2 \AA\ with no applied momentum flux. Over the range of shear rates investigated, 0.6$\pm$0.3 ms\textsuperscript{-1} to 5.3$\pm$0.5 ms\textsuperscript{-1} for the basal system and 0.9$\pm$0.2 ms\textsuperscript{-1} to 4.5$\pm$0.1 ms\textsuperscript{-1}, there was no appreciable change in the interface width found. The calculated values for the interfacial width over all shear rates investigated contained the control values within their error bars. |
| 300 |
> |
\subsubsection{Orientational Time Correlation Function} |
| 301 |
> |
The orientational time correlation function (OTCF) gives insight of the local environment of molecules. The rate at which the function decays corresponds to how hindered the motions of a molecule are. The more hindered a molecules motion is the slower the function will decay, and the function decays more rapidly for molecules with less constrained motions. |
| 302 |
> |
\begin{equation}\label{C(t)1} |
| 303 |
> |
C_{2}(t)=\langle P_{2}(\mathbf{v}_{i}(t)\mathbf{v}_{i}(t=0))\rangle |
| 304 |
> |
\end{equation} |
| 305 |
> |
In \eqref{C(t)1}, $P_{2}$ is the Legendre polynomial of the second order and $\mathbf{v}_{i}$ is the bisecting unit vector of the $i$th water molecule in the lab frame. |
| 306 |
|
|
| 307 |
< |
%Need to reword the following paragraph |
| 310 |
< |
Beaglehole and Wilson have measured the ice/water interface to have a thickness approximately 10-20 \AA\ for both the basal and prismatic face of ice by ellipticity measurements \cite{Beaglehole93}. While Haymet \emph{et al.} have reported values that agree with these measurements, our results do not. We believe this arises from the different methods used to measure the interfacial width. Haymet and co-workers use the 10-90 widths of the translational, average density, diffusion, and orientational decay times \cite{Hayward01} to measure the interface, whereas we are using the local tetrahedral order parameter. |
| 307 |
> |
Here, we are evaluating this function across the $z$-dimension of the system as another measure of the change in the local environment and behavior of water molecules from the liquid region to the slushy interfacial region. After each of the 0.5 ns simulations with an applied shear and the control simulations, the simulations were run for an additional 200 ps where the positions of every molecule in the system were recorded every 0.1 ps. The systems were then divided into 30 bins and the OTCF was evaluated for each bin. |
| 308 |
|
|
| 309 |
< |
|
| 313 |
< |
%Beaglehole and Wilson have measured the ice/water interface to have a thickness approximately 10-20 \AA\ for both the basal and prismatic face of ice by ellipticity measurements \cite{Beaglehole93}. Haymet \emph{et al.} agrees with these measurements, our results do not. We are using a parameter from the hyperbolic tangent fit of the local tetrahedrality order parameter to determine the interfacial width, whereas Haymet and co-workers use the 10-90 widths of the translational, average density, diffusion, and orientational decay times \cite{Hayward01}. |
| 314 |
< |
|
| 315 |
< |
|
| 316 |
< |
\subsection{Coefficient of Friction of the Interface} |
| 317 |
< |
As the ice is sheared through the liquid, there will be a friction between the ice and the interface. Balasubramanian has shown how to calculate the coefficient of friction for a solid-liquid interface. \cite{Balasubramanian99} |
| 309 |
> |
It has been shown that the OTCF for water can be fit by a triexponential decay\cite{Furse08}, where the three components of the decay correspond to a fast (<200 fs) reorientational piece driven by the restoring forces of existing hydrogen bonds, a middle (on the order of several ps) piece describing the large angle jumps that occur during the breaking and formation of new hydrogen bonds\cite{Laage08,Laage11}, and a slow (on the order of hundreds of ps) contribution describing the translational motion of the molecules. The OTCF data for each bin were pruned to 100 ps, and fit to the triexponential decay |
| 310 |
|
\begin{equation} |
| 311 |
< |
%<F_{x}^{w}>_{NE}(t)=-S\lambda_{wall}v_{x}(y_{wall}) |
| 320 |
< |
\langle F_{x}^{w}\rangle(t)=-S\lambda_{wall}v_{x}(y_{wall}) |
| 311 |
> |
C_{2}(t)=a_{1}e^{-t/\tau_{short}}+a_{2}e^{-t/\tau_{middle}}+a_{3}e^{-t/\tau_{long}}+a_{4} |
| 312 |
|
\end{equation} |
| 313 |
< |
In this equation, $F_{x}^{w}$ is the total force of all the atoms acting on the fluid, $S$ is the surface area the force is being applied upon, and $\lambda_{wall}$ is the coefficient of friction of the interface. Since the imposed momentum flux, $J_{z}(p_{x})$, is known in the VSS-RNEMD simulations, and the $wall$ is the ice block in our simulations, the above equation can be rewritten as |
| 323 |
< |
\begin{equation} |
| 324 |
< |
J_{z}(p_{x})=-\lambda_{ice}v_{x}(y_{ice}) |
| 325 |
< |
\end{equation} |
| 326 |
< |
and finally as |
| 327 |
< |
\begin{equation} |
| 328 |
< |
v_{x}(y_{ice})=-J_{z}(p_{x})\Bigg(\frac{1}{\lambda_{ice}}\Bigg) |
| 329 |
< |
\end{equation} |
| 313 |
> |
where $a_{1}+a_{2}+a_{3}+a_{4}=1$. An average value and standard deviation for each $\tau$ was obtained for each bin from the four runs. Lastly, the means and standard deviations were averaged about the center of the system. |
| 314 |
|
|
| 315 |
< |
In Figure \ref{fig:CoeffFric}, the average velocity of the ice is plotted against the imposed momentum flux for the basal (black circles) and prismatic (red circles) systems. From the equation above, the inverse of the slope of the linear fit of the data is $\lambda_{wall}$. The coefficient of friction of the interface for the basal face was calculated to be , and the $\lambda_{wall}$ for the prismatic face was determined to be . |
| 315 |
> |
\begin{figure} |
| 316 |
> |
\includegraphics[width=\linewidth]{basal_Tau_comic_strip} |
| 317 |
> |
\caption{\label{fig:basal_Tau_comic_strip} The orientational time correlation function for the basal system fit by the triexponential decay $a_{1}e^{-t/\tau_{short}}+a_{2}e^{-t/\tau_{middle}}+a_{3}e^{-t/\tau_{long}}+a_{4}$ where $a_{1}+a_{2}+a_{3}+a_{4}=1$. The verticle dotted line indicates the average midpoint of the interface as determined by the tetrahedrality fit. In (b) and (c) $\tau_{middle}$ and $\tau_{long}$ have a consistent value of about 5.5 ps and 50 ps in the liquid region, and increase in value approaching the interface. $\tau_{middle}$ corresponds to the breaking and making of hydrogen bonds as explained by extended jump model proposed by Laage and Hynes\cite{Laage08,Laage11} and $\tau_{long}$ relates to the translational motion of the molecules. In (a), we see that $\tau_{short}$ has a value of about 71 fs in the liquid region, and decreases in value approaching the interface. This component of the decay corresponds to the reorientational forces of existing hydrogen bonds on the inertial rotational of the molecules. Thus $\tau_{short}$ decreases approaching the interface due to the hindered range of motion of the molecules. } |
| 318 |
> |
\end{figure} |
| 319 |
|
|
| 333 |
– |
|
| 320 |
|
\begin{figure} |
| 321 |
< |
\includegraphics[width=\linewidth]{CoeffFric} |
| 322 |
< |
\caption{\label{fig:CoeffFric} The average velocity of the ice for the basal (black circles) and prismatic (red circles) systems as a function off applied momentum flux. The slope of the fit line } |
| 321 |
> |
\includegraphics[width=\linewidth]{prismatic_Tau_comic_strip} |
| 322 |
> |
\caption{\label{fig:prismatic_Tau_comic_strip} The orientational time correlation function for the prismatic system fit by the triexponential decay $a_{1}e^{-t/\tau_{short}}+a_{2}e^{-t/\tau_{middle}}+a_{3}e^{-t/\tau_{long}}+a_{4}$ where $a_{1}+a_{2}+a_{3}+a_{4}=1$. The verticle dotted line indicates the average midpoint of the interface as determined by the tetrahedrality fit. In (b) and (c) $\tau_{middle}$ and $\tau_{long}$ have a consistent value of about 3.5 ps and 30 ps in the liquid region, and increase in value approaching the interface. $\tau_{middle}$ corresponds to the breaking and making of hydrogen bonds as explained by extended jump model proposed by Laage and Hynes\cite{Laage08,Laage11} and $\tau_{long}$ relates to the translational motion of the molecules. In (a), we see that $\tau_{short}$ has a value of about 73 fs in the liquid region, and decreases in value approaching the interface. This component of the decay corresponds to the reorientational forces of existing hydrogen bonds on the inertial rotational of the molecules. Thus $\tau_{short}$ decreases approaching the interface due to the hindered range of motion of the molecules.} |
| 323 |
|
\end{figure} |
| 324 |
|
|
| 325 |
+ |
Figures \ref{fig:basal_Tau_comic_strip} and \ref{fig:prismatic_Tau_comic_strip} plots the decomposition of the OTCF at varying displacements from the center of the ice for the basal and prismatic systems. We see in (a) $\tau_{short}$, (b) $\tau_{middle}$, and (c) $\tau_{long}$ for the control system (no applied momentum flux) in black, and a system with a large shear rate in red. The verticle dotted lines at a displacement of about 17 \AA\ and 9 \AA\ denote the midpoints of the interfaces as determined by the hyperbolic tangent fit of the tetrahedrality profile. |
| 326 |
|
|
| 327 |
< |
\subsection{Orientational Time Correlation Function} |
| 341 |
< |
The orientational time correlation function (OTCF) gives insight of the local environment of molecules. The rate at which the function decays corresponds to how hindered the motions of a molecule are. The more hindered a molecules motion is the slower the function will decay, and the function decays more rapidly for molecules with less constrained motions. |
| 342 |
< |
\begin{equation} |
| 343 |
< |
C_{1}(t)=\langle P_{2}(\mathbf{v}_{i}(t)\mathbf{v}_{i}(t=0))\rangle |
| 344 |
< |
\end{equation} |
| 345 |
< |
where $P_{2}$ is the Legendre polynomial of the second order, |
| 327 |
> |
In panels (a), we see at large displacements from the center of the ice $\tau_{short}$ for the basal system has a value of about 71 fs and 72 fs for the prismatic. Decreasing in displacement from about 26 \AA\ to about 19 \AA\ in the basal system, the value of $\tau_{short}$ decreases to about 63 fs. Likewise, $\tau_{short}$ decreases to about 63 fs from roughly 20 \AA\ to 12 \AA\. This is due to the increasingly constrained motion of the water molecules as we approach the interface. In panels (b), $\tau_{middle}$ at large displacements from the ice has a value of about 5.5 ps and 3 ps for the basal and prismatic systems. We find $\tau_{middle}$ increases in value as we approach the interface in both cases. This component of the decay corresponds to the rearrangement of the hydrogen bonding network, which takes longer as the molecules motion becomes more constrained. In panels (c), $\tau_{long}$ has a value of about 50 ps for the basal and roughly 30 ps for the prismatic at large displacements from the interface. Similar to $\tau_{middle}$, $\tau_{long}$ also increases in value as we approach the interface for both systems. It is also apparent that shearing the ice water has no effect on the orientational decay time, or on any of the decomposed components. |
| 328 |
|
|
| 329 |
< |
Here, we evaluating this function across the $z$-dimension of the system as another measure of the change in the local environment and behavior of water molecules from the liquid region to the slushy interfacial region. After each of the 0.5 ns simulations with an applied shear and the control simulations, the simulations were run for an additional 200 ps where the positions of every molecule in the system were recorded every 0.1 ps. The systems were then divided into 30 bins and the OTCF was evaluated for each bin. |
| 329 |
> |
For each system, there is an apparent approximate value for $\tau_{short}$, $\tau_{middle}$, and $\tau_{long}$ at large displacements from the interface. There also appears to be a single displacement, $d_{basal}$ or $d_{prismatic}$, from the interface at which all three decay times begin to deviate from their bulk liquid values. We found $d_{basal}$ and $d_{prismatic}$ to be roughly 15 \AA\ and 8 \AA\ respectively. These two results indicate that the dynamics of the water molecules within $d_{basal}$ and $d_{prismatic}$ are being significantly perturbed by the ice and/or the interface, even though the structural width of the interface by analysis of the tetrahedrality profile indicates that bulk liquid structure of water is recovered in about 4 \AA\ from the edge of the ice. |
| 330 |
|
|
| 331 |
< |
It has been shown that the OTCF for water can be fit by a tri-exponential decay/cite{Corcelli's paper}, where the three components of the decay correspond to a fast (<200 fs) reorientational piece driven by the restoring forces of existing hydrogen bonds, a middle (on the order of several ps) piece describing the large angle jumps that occur during the breaking and formation of new hydrogen bonds\cite{JT Hynes paper}, and a slow (on the order of hundreds of ps) contribution describing something. The OTCF data for each bin were pruned to 100 ps, and fit to the following triexponential decay |
| 331 |
> |
Beaglehole and Wilson have measured the ice/water interface to have a thickness approximately 10 \AA\ for both the basal and prismatic face of ice by ellipticity measurements \cite{Beaglehole93}. Structurally, we have found the basal and prismatic interfacial width to be 3.2$\pm$0.4 \AA\ and 3.6$\pm$0.2 \AA\ . However, we have shown through decomposition of the OTCF a much larger interfacial region. |
| 332 |
|
|
| 333 |
< |
An average value and standard deviation for each bin was obtained from the four runs. Lastly, the averages and standard deviations were averaged about the center of the system, resulting in a |
| 333 |
> |
\subsection{Coefficient of Friction of the Interface} |
| 334 |
> |
As the ice is sheared through the liquid, there will be a friction between the ice and The interface. Balasubramanian has shown how to calculate the coefficient of friction for a solid-liquid interface. \cite{Balasubramanian99} |
| 335 |
> |
\begin{equation} |
| 336 |
> |
\langle F_{x}^{w}\rangle(t)=-S\lambda_{wall}v_{x}(y_{wall}) |
| 337 |
> |
\end{equation} |
| 338 |
> |
In this equation, $F_{x}^{w}$ is the total force of all the atoms acting on the fluid, $S$ is the surface area the force is being applied upon, $\lambda_{wall}$ is the coefficient of friction of the interface, and $v_{x}(y_{wall})$ is the velocity at the displacement from the interface at which the hydrodynamics breaks down. Since the total force imposed momentum flux, $J_{z}(p_{x})$, is known in the VSS-RNEMD simulations, |
| 339 |
|
|
| 340 |
|
|
| 354 |
– |
|
| 355 |
– |
\begin{figure} |
| 356 |
– |
\includegraphics[width=\linewidth]{basal_Tau_comic_strip} |
| 357 |
– |
\caption{\label{fig:basal_Tau_comic_strip} The basal system: ~~~//~~~} |
| 358 |
– |
\end{figure} |
| 359 |
– |
|
| 360 |
– |
\begin{figure} |
| 361 |
– |
\includegraphics[width=\linewidth]{prismatic_Tau_comic_strip} |
| 362 |
– |
\caption{\label{fig:prismatic_Tau_comic_strip} The prismatic system: ~~~//~~~} |
| 363 |
– |
\end{figure} |
| 364 |
– |
|
| 365 |
– |
|
| 341 |
|
\section{Conclusion} |
| 342 |
|
Here we have simulated the basal and prismatic facets of an SPC/E model of the ice Ih / water interface. Using VSS-RNEMD, the ice was sheared relative to the liquid while imposed thermal gradients kept the interface at a stable temperature. Caculation of the local tetrahedrality order parameter has shown an appearant independence of the shear rate on the interfacial width. The coefficient of friction of the interface was also calculated for each of the facets. The $\lambda_{wall}$ for the basal face was calculated to be , and for the prismatic facet. For both facets, the shearing ice water was found to be in the no-slip boundary condition. |
| 343 |
|
|
