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\date{\today} |
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\begin{abstract} |
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Abstract abstract abstract... |
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\end{abstract} |
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\maketitle |
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loses memory of its former orientation. Observing the rate at which this decay |
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occurs can provide insight to the mechanism and timescales for the relaxation. |
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In eq. \eqref{C(t)1}, $P_{2}$ is the second-order Legendre polynomial, and |
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$\mathbf{u}$ is the the bisecting HOH vector. The angle brackets indicate |
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$\mathbf{u}$ is the bisecting HOH vector. The angle brackets indicate |
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an ensemble average over all the water molecules in a given spatial region. |
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To investigate the dynamics of the water molecules across the interface, the |
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dependence |
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arrises from the sharp discontinuity of the viscosity for the SPC/E model |
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at temperatures approaching 200 K\cite{kuang12}. Due to this, we propose |
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a weighting to the structural interfacial parameter, $\kappa$ by the |
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viscosity at $225$ K, the temperature of the interface. $\kappa$ is |
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traditionally defined as |
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\begin{equation}\label{kappa} |
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\kappa = \eta/\delta |
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\end{equation} |
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where $\eta$ is the viscosity and $\delta$ is the slip length. |
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In our ice/water shearing simulations, the system has reached a steady state |
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when the applied force, |
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|
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\begin{equation} |
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f_{applied} = \mathbf{j}_z(\mathbf{p})L_x L_y |
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\end{equation} |
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is equal to the frictional force resisting the motion of the ice block |
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\begin{equation} |
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f_{friction} = \frac{\eta \mathbf{v} L_x L_y}{\delta} |
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a weighting to the interfacial friction coefficient, $\kappa$ by the |
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shear viscosity at 225 K. The interfacial friction coefficient relates |
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the shear stress with the relative velocity of the fluid normal to the |
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interface: |
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\begin{equation}\label{Shenyu-13} |
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j_{z}(p_{x}) = \kappa[v_{x}(fluid)-v_{x}(solid)] |
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\end{equation} |
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where $\mathbf{v}$ is the relative velocity of the liquid from the ice. |
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When this condition is met, we are able to solve the resulting expression to |
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obtain, |
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\begin{equation}\label{force_equality} |
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\frac{\eta}{\delta} = \frac{\mathbf{j}_z(\mathbf{p})}{\mathbf{v}} |
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where $j_{z}(p_{x})$ is the applied momentum flux (shear stress) across $z$ |
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in the |
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$x$-dimension, and $v_{x}$(fluid) and $v_{x}$(solid) are the velocities |
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directly adjacent to the interface. The shear viscosity, $\eta(T)$, of the |
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fluid can be determined if we assume a linear response of the momentum |
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gradient to the applied shear stress by |
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\begin{equation}\label{Shenyu-11} |
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j_{z}(p_{x}) = \eta(T) \frac{\partial v_{x}}{\partial z}. |
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\end{equation} |
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Using eqs \eqref{Shenyu-13} and \eqref{Shenyu-11}, we can find the following |
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expression for $\kappa$, |
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\begin{equation}\label{kappa-1} |
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\kappa = \eta(T) \frac{\partial v_{x}}{\partial z}\frac{1}{[v_{x}(fluid)-v_{x}(solid)]}. |
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\end{equation} |
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From \eqref{kappa}, \eqref{force_equality} becomes |
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\begin{equation} |
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\kappa = \frac{\mathbf{j}_z(\mathbf{p})}{\mathbf{v}} |
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Here is where we will introduce the weighting term of $\eta(225)/\eta(T)$ |
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giving us |
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\begin{equation}\label{kappa-2} |
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\kappa = \frac{\eta(225)}{[v_{x}(fluid)-v_{x}(solid)]}\frac{\partial v_{x}}{\partial z}. |
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\end{equation} |
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which we will multiply by a viscosity weighting term to reach |
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\begin{equation} \label{kappa2} |
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\kappa = \frac{\mathbf{j}_z(\mathbf{p})}{\mathbf{v}} \frac{\eta(225)}{\eta(T)} |
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\end{equation} |
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Assuming linear response theory is valid, an expression for ($\eta$) can |
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be found from the imposed momentum flux and the measured velocity gradient. |
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\begin{equation}\label{eta_eq} |
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\eta = \frac{\mathbf{j}_z(\mathbf{p})}{\frac{\partial v_x}{\partial z}} |
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\end{equation} |
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Substituting eq \eqref{eta_eq} into eq \eqref{kappa2} we arrive at |
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\begin{equation} |
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\kappa = \frac{\frac{\partial v_x}{\partial z}}{\mathbf{v}}\eta(225) |
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\end{equation} |
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|
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\begin{table}[h] |
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\centering |
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\caption{$\kappa$ values for the basal, prismatic, pyramidal, and secondary prismatic facets of Ice-I$_\mathrm{h}$} |
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\label{tab:kappa} |
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\begin{tabular}{|ccc|} \hline |
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& \multicolumn{2}{c|}{$\kappa_{Drag direction}$} \\ |
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Interface & $\kappa_{x}$ & $\kappa_{y}$ \\ \hline |
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basal & $0.00059 \pm 0.00003$ & $0.00065 \pm 0.00008$ \\ |
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prismatic & $0.00030 \pm 0.00002$ & $0.00030 \pm 0.00001$ \\ |
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pyramidal & $0.00058 \pm 0.00004$ & $0.00061 \pm 0.00005$ \\ |
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secondary prism & $0.00035 \pm 0.00001$ & $0.00033 \pm 0.00002$ \\ \hline |
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\end{tabular} |
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\end{table} |
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To obtain the value of $\eta(225)$ for the SPC/E model, a $31.09 \times 29.38 |
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\times 124.39$ \AA\ box with 3744 SPC/E water molecules was equilibrated to |
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225K, |
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roughly ten times larger than the value found for 280 K SPC/E bulk water by |
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Kuang\cite{kuang12}. |
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The resulting $\kappa$ values found for the four crystal |
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facets of Ice-I$_\mathrm{h}$ investigated are shown in Table \ref{tab:kappa}. |
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The basal and pyramidal facets were found to have similar values of |
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$\kappa \approx$ 0.0006, while $\kappa \approx$ 0.0003 were found for the |
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prismatic and secondary prismatic facets. |
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The interfacial friction coefficient, $\kappa$, can equivalently be expressed |
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as the ratio of the viscosity of the fluid to the slip length, $\delta$, which |
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is an indication of how 'slippery' the interface is. |
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\begin{equation}\label{kappa-3} |
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\kappa = \frac{\eta}{\delta} |
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\end{equation} |
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In each of the systems, the interfacial temperature was kept fixed to 225K, |
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which ensured the viscosity of the fluid at the |
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interace was approximately the same. Thus, any significant variation in |
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$\kappa$ between |
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the systems indicates differences in the 'slipperiness' of the interfaces. |
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As each of the ice systems are sheared relative to liquid water, the |
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'slipperiness' of the interface can be taken as an indication of how |
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hydrophobic or hydrophilic the interface is. The calculated $\kappa$ values |
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found for the four crystal facets of Ice-I$_\mathrm{h}$ investigated are shown |
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in Table \ref{tab:kapa}. The basal and pyramidal facets were found to have |
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similar values of $\kappa \approx$ 0.0006 (units), while $\kappa \approx$ |
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0.0003 (units) were found for the prismatic and secondary prismatic systems. |
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These results indicate that the prismatic and secondary prismatic facets are |
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more hydrophobic than the basal and pyramidal facets. |
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%This indicates something about the similarity between the two facets that |
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%share similar values... |
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%Maybe find values for kappa for other materials to compare against? |
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\begin{table}[h] |
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\centering |
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\caption{$\kappa$ values for the basal, prismatic, pyramidal, and secondary \ |
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prismatic facets of Ice-I$_\mathrm{h}$} |
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\label{tab:kappa} |
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\begin{tabular}{|ccc|} \hline |
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& \multicolumn{2}{c|}{$\kappa_{Drag direction}$ (units)} \\ |
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Interface & $\kappa_{x}$ & $\kappa_{y}$ \\ \hline |
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basal & $0.00059 \pm 0.00003$ & $0.00065 \pm 0.00008$ \\ |
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prismatic & $0.00030 \pm 0.00002$ & $0.00030 \pm 0.00001$ \\ |
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pyramidal & $0.00058 \pm 0.00004$ & $0.00061 \pm 0.00005$ \\ |
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secondary prism & $0.00035 \pm 0.00001$ & $0.00033 \pm 0.00002$ \\ \hline |
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\end{tabular} |
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\end{table} |
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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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interface by using the minimal perturbing VSS RNEMD method. In agreement with |
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our previous findings for the basal and prismatic facets, the interfacial |
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width was found to be independent of shear rate as measured by the local |
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order tetrahedral ordering parameter. This width was found to be |
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3.2~$\pm$0.2~\AA\ for both the pyramidal and the secondary prismatic systems. |
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tetrahedral order parameter. This width was found to be |
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3.2~$\pm$0.2~\AA\ for both the pyramidal and the secondary prismatic systems. |
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These values are in good agreement with our previously calculated interfacial |
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widths for the basal (3.2~$\pm$0.4~\AA\ ) and prismatic (3.6~$\pm$0.2~\AA\ ) |
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systems. |
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|
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Orientational dynamics of the Ice-I$_\mathrm{h}$/water interfaces were studied |
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by calculation of the orientational time correlation function at varying |
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displacements normal to the interface. The decays were fit |
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to a tri-exponential decay, where the three decay constants correspond to |
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the librational motion of the molecules driven by the restoring forces of |
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existing hydrogen bonds ($\tau_{short}$ $\mathcal{O}$(10 fs)), jumps between |
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two different hydrogen bonds ($\tau_{middle}$ $\mathcal{O}$(1 ps)), and |
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translational motion of the molecules ($\tau_{long}$ $\mathcal{O}$(100 ps)). |
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$\tau_{short}$ was found to decrease approaching the interface due to the |
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constrained motion of the molecules as the local environment becomes more |
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ice-like. Conversely, the two longer-time decay constants were found to |
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increase at small displacements from the interface. As seen in our previous |
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work on the basal and prismatic facets, there appears to be a dynamic |
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interface width at which deviations from the bulk liquid values occur. |
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We had previously found $d_{basal}$ and $d_{prismatic}$ to be approximately |
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2.8~\AA\ and 3.5~\AA\~. We found good agreement of these values for the |
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pyramidal and secondary prismatic systems with $d_{pyramidal}$ and |
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$d_{secondary prism}$ to be 2.7~\AA\ and 3~\AA\~. For all of the facets, there |
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was found to be no apparent dependence of the dynamic width on the shear rate. |
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|
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%Paragraph summarizing the Kappa values |
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The interfacial friction coefficient, $\kappa$, was determined for each of the |
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interfaces. We were able to reach an expression for $\kappa$ as a function of |
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the velocity profile of the system and is scaled by the viscosity of the liquid |
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at 225 K. In doing so, we have obtained an expression for $\kappa$ which is |
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independent of temperature differences of the liquid water at far displacements |
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from the interface. We found the basal and pyramidal facets to have |
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similar $\kappa$ values, of $\kappa \approx$ 0.0006 (units). However, the |
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prismatic and secondary prismatic facets were found to have $\kappa$ values of |
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$\kappa \approx$ 0.0003 (units). As these ice facets are being sheared |
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relative to liquid water, with the structural and dynamic width of all four |
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interfaces being approximately the same, the difference in the coefficient of |
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friction indicates the hydrophilicity of the crystal facets are not |
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equivalent. Namely, that the basal and pyramidal facets of Ice-I$_\mathrm{h}$ |
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are more hydrophilic than the prismatic and secondary prismatic facets. |
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\begin{acknowledgments} |
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Support for this project was provided by the National |
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Science Foundation under grant CHE-1362211. Computational time was |
