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%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. |
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The system was divided into 100 bins along the $z$-axis, and a $q$ value was determined for each snapshot of the system for each bin. The $q$ values for each bin 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 |
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The system was divided into 100 bins of length $L$ along the $z$-axis, and a $q$ value was determined for each snapshot of the system for each bin by the following equation. |
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\begin{equation} |
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q_{z}=q_{liq}+\frac{q_{ice}-q_{liq}}{2}(\tanh(\alpha(z-I_{L,m}))-\tanh(\alpha(z-I_{R,m}))) |
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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 |
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\end{equation} |
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The $q$ values for each bin 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 |
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\begin{equation} |
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q_{z} \approx q_{liq}+\frac{q_{ice}-q_{liq}}{2}(\tanh(\alpha(z-I_{L,m}))-\tanh(\alpha(z-I_{R,m}))) |
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\end{equation} |
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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, |
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\begin{equation} |
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q_{z}=q_{liq}+\frac{q_{ice}-q_{liq}}{2}(\tanh(\alpha(z-I_{L,m}))-\tanh(\alpha(z-I_{R,m})))+\beta|(z-z_{mid})| |
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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})| |
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\end{equation} |
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where $\beta$ is a fitting parameter and $z_{mid}$ is the midpoint of the z dimension of the simulation box. |
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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. |
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%Need to reword the following paragraph |
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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}. |
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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. |
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%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}. |
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\subsection{Coefficient of Friction of the Interface} |
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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} |
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\begin{equation} |
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%Ask dan about truncating versus rounding the values for lambda. |
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%The coefficient of friction of the interface for the basal face was calculated to be 11.02808 $\pm$ 0.4489844 \AA^{-2}fs^{-1}, and the $\lambda_{wall}$ for the prismatic face was determined to be 19.95948, $\pm$ 0.5370894 \AA^{-2}fs^{-1}. |
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\begin{figure} |
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\includegraphics[width=\linewidth]{CoeffFric} |
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\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 } |
