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\begin{document} |
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\title{Simulations of solid-liquid friction at Secondary Prism and Pyramidal ice-I$_\mathrm{h}$ / water interfaces} |
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|
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\author{Patrick B. Louden and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: |
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gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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|
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\date{\today} |
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\maketitle |
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\begin{doublespace} |
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\begin{abstract} |
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Abstract abstract abstract... |
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\end{abstract} |
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|
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\newpage |
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|
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\section{Introduction} |
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Explain a little bit about ice Ih, point group stuff. |
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|
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Mention previous work done / on going work by other people. Haymet and Rick |
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seem to be investigating how the interfaces is perturbed by the presence of |
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ions. This is the conlcusion of a recent publication of the basal and |
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prismatic facets of ice Ih, now presenting the pyramidal and secondary |
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prism facets under shear. |
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|
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\section{Methodology} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{SP_comic_strip} |
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\caption{\label{fig:spComic} The secondary prism interface with a shear |
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rate of 3.5 ms\textsuperscript{-1}. Lower panel: the local tetrahedral order |
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parameter, $q(z)$, (black circles) and the hyperbolic tangent fit (red line). |
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Middle panel: the imposed thermal gradient required to maintain a fixed |
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interfacial temperature. Upper panel: the transverse velocity gradient that |
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develops in response to an imposed momentum flux. The vertical dotted lines |
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indicate the locations of the midpoints of the two interfaces.} |
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\end{figure} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{Pyr_comic_strip} |
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\caption{\label{fig:pyrComic} The pyramidal interface with a shear rate of 3.8 \ |
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ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:spComic}.} |
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\end{figure} |
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|
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\subsection{Pyramidal and secondary prism system construction} |
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|
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The construction of the pyramidal and secondary prism systems follows that of |
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the basal and prismatic systems presented elsewhere\cite{Louden13}, however |
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the ice crystals and water boxes were equilibrated and combined at 50K |
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instead of 225K. The ice / water systems generated were then equilibrated |
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to 225K. The resulting pyramidal system was |
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$37.47 \times 29.50 \times 93.02$ \AA\ with 1216 |
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SPC/E molecules in the ice slab, and 2203 in the liquid phase. The secondary |
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prism system generated was $71.87 \times 31.66 \times 161.55$ \AA\ with 3840 |
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SPC/E molecules in the ice slab and 8176 molecules in the liquid phase. |
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|
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\subsection{Computational details} |
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% Do we need to justify the sims at 225K? |
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% No crystal growth or shrinkage over 2 successive 1 ns NVT simulations for |
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% either the pyramidal or sec. prism ice/water systems. |
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|
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The computational details performed here were equivalent to those reported |
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in the previous publication\cite{Louden13}. The only changes made to the |
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previously reported procedure were the following. VSS-RNEMD moves were |
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attempted every 2 fs instead of every 50 fs. This was done to minimize |
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the magnitude of each individual VSS-RNEMD perturbation to the system. |
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|
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All pyramidal simulations were performed under the NVT ensamble except those |
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during which statistics were accumulated for the orientational correlation |
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function, which were performed under the NVE ensamble. All secondary prism |
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simulations were performed under the NVE ensamble. |
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|
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\section{Results and discussion} |
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\subsection{Interfacial width} |
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In the literature there is good agreement that between the solid ice and |
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the bulk water, there exists a region of 'slush-like' water molecules. |
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In this region, the water molecules are structured differently and |
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behave differently than those of the solid ice or the bulk water. |
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The characteristics of this region have been defined by both structural |
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and dynamic properties; and width has been measured by the change of these |
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properties from their bulk liquid values to those of the solid ice. |
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Examples of these properties include the density, the diffusion constant, and |
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the translational order profile. \cite{Bryk02,Karim90,Gay02,Hayword01,Hayword02,Karim88} |
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|
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Since the VSS-RNEMD moves perturb the velocities of the water molecules in |
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the systems, parameters that depend on the translational motion may give |
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faulty results. A stuructural parameter will be less effected by the |
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VSS-RNEMD perturbations to the system. Due to this we have used the |
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local order tetrahedral parameter, which was originally described by |
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Kumar\cite{Kumar09} and Errington\cite{Errington01} and explained in our |
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previous publication\cite{Louden13} in relation to an ice/water system. |
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|
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Each of the systems were divided into 100 artificial bins along the |
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$z$-dimension, and the local tetrahedral order parameter, $q(z)$, was |
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time-averaged for each of the bins, resulting in a tetrahedrality profile of |
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the system. These profiles are shown across the $z$-dimension of the systems |
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in panel $a$ of Figures \ref{fig:spComic} |
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and \ref{fig:pyrComic} (black circles). The $q(z)$ function has a range of |
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(0,1), where a larger value indicates a more tetrahedral environment. |
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The $q(z)$ for the bulk liquid was found to be $\approx $0.77, while values of |
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$\approx $0.92 were more common for the ice. The tetrahedrality profiles were |
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fit using a hyperbolic tangent\cite{Louden13} designed to smoothly fit the |
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bulk to ice |
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transition, while accounting for the thermal influence on the profile by the |
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kinetic energy exchanges of the VSS-RNEMD moves. In panels $b$ and $c$, the |
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imposed thermal and velocity gradients can be seen. The verticle dotted |
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lines traversing all three panels indicate the midpoints of the interface |
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as determined by the hyperbolic tangent fit of the tetrahedrality profiles. |
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|
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From fitting the tetrahedrality profiles for each of the 0.5 nanosecond |
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simulations (panel c of Figures \ref{fig:spComic} and \ref{fig:pyrComic}) |
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by Eq. 6\cite{Louden13},we find the interfacial width for the pyramidal and |
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secondary prism to be $3.2 \pm 0.2$ and $3.2 \pm 0.2$ \AA\ , respectively, |
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with no applied momentum flux. Over the range of shear rates investigated, |
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$0.6 \pm 0.2 \mathrm{ms}^{-1} \rightarrow 5.6 \pm 0.4 \mathrm{ms}^{-1}$ for |
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the pyramidal system and $0.9 \pm 0.3 \mathrm{ms}^{-1} \rightarrow 5.4 \pm 0.1 |
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\mathrm{ms}^{-1}$ for the secondary prism, we found no significant change in |
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the interfacial width. This follows our previous findings of the basal and |
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prismatic systems, in which the interfacial width was invarient of the |
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shear rate of the ice. The interfacial width of the quiescent basal and |
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prismatic systems was found to be $3.2 \pm 0.4$ \AA\ and $3.6 \pm 0.2$ \AA\ |
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respectively. Over the range of shear rates investigated, $0.6 \pm 0.3 |
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\mathrm{ms}^{-1} \rightarrow 5.3 \pm 0.5 \mathrm{ms}^{-1}$ for the basal |
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system and $0.9 \pm 0.2 \mathrm{ms}^{-1} \rightarrow 4.5 \pm 0.1 |
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\mathrm{ms}^{-1}$ for the prismatic. |
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|
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These results indicate that the surface structure of the exposed ice crystal |
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has little to no effect on how far into the bulk the ice-like structural |
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ordering is. Also, it appears that the interface is not structurally effected |
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by shearing the ice through water. |
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|
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|
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\subsection{Orientational dynamics} |
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To investigate the dynamics of the water molecules across the interface, the |
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systems were divided into $n$ bins, each $\approx$ 3 \AA\ wide in $z$, and |
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the orientational time |
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correlation function was computed for each of the $n$ bins. This was done by |
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averaging the second order Legendre polynomial of the bisecting HOH vector |
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dotted with itself at an initial time and some time later, over all molecules |
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in the bin. |
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|
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|
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|
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\subsection{Coefficient of friction of the interfaces} |
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|
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|
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\begin{table}[h] |
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\centering |
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\caption{Solid-liquid friction coefficients (measured in amu~fs\textsuperscript\ |
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{-1}) } |
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\label{tab:lambda} |
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\begin{tabular}{|ccc|} \hline |
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& \multicolumn{2}{c|}{Drag direction} \\ |
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Interface & $x$ & $y$ \\ \hline |
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basal\textsuperscript{a} & $0.08 \pm 0.02$ & $0.09 \pm 0.03$ \\ |
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prismatic\textsuperscript{a} & $0.037 \pm 0.008$ & $0.04 \pm 0.01$ \\ |
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pyramidal & $0.13 \pm 0.03$ & $0.14 \pm 0.03$ \\ |
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secondary prism & $0.13 \pm 0.02$ & $0.12 \pm 0.03$ \\ \hline |
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\end{tabular} |
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\caption{\textsuperscript{a}Reference \cite{Louden13}} |
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\end{table} |
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|
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{Pyr-orient} |
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\caption{\label{fig:PyrOrient} The three decay constants of the |
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orientational time correlation function, $C_2(t)$, for water as a function |
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of distance from the center of the ice slab. The vertical dashed line |
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indicates the edge of the pyramidal ice slab determined by the local order |
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tetrahedral parameter. The control (black circles) and sheared (red squares) |
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experiments were fit by a shifted exponential decay (Eq. 9\cite{Louden13}) |
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shown by the black and red lines respectively. The upper two panels show that |
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translational and hydrogen bond making and breaking events slow down |
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through the interface while approaching the ice slab. The bottom most panel |
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shows the librational motion of the water molecules speeding up approaching |
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the ice block due to the confined region of space allowed for the molecules |
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to move in.} |
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\end{figure} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{SP-orient-less} |
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\caption{\label{fig:SPorient} Decay constants for $C_2(t)$ at the secondary |
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prism face. Panel descriptions match those in \ref{fig:PyrOrient}.} |
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\end{figure} |
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|
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|
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|
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\section{Conclusion} |
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Conclude conclude conclude... |
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|
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\section{Acknowledgements} |
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Support for this progect was provided by the National Science Foundation under grant CHE-0848243. Computational time was provided by the Center for Research Computing (CRC) at the University of Notre Dame. |
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\newpage |
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\bibliography{iceWater} |
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\end{doublespace} |
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\end{document} |