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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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% 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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in our 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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VSS-RNEMD perturbations to the system. Due to this, we have used the |
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local order tetrahedral parameter to quantify the width of the interface, |
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which was originally described by Kumar\cite{Kumar09} and |
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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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Errington\cite{Errington01}, and used by Bryk and Haymet in a previous study |
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of ice/water interfaces.\cite{Bryk2004b} |
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|
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Paragraph and eq. for tetrahedrality here. |
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The local tetrahedral order parameter, $q(z)$, is given by |
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\begin{equation} |
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q(z) = \int_0^L \sum_{k=1}^{N} \Bigg(1 -\frac{3}{8}\sum_{i=1}^{3} |
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\sum_{j=i+1}^{4} \bigg(\cos\psi_{ikj}+\frac{1}{3}\bigg)^2\Bigg) |
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\delta(z_{k}-z)\mathrm{d}z \Bigg/ N_z |
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\label{eq:qz} |
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\end{equation} |
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where $\psi_{ikj}$ is the angle formed between the oxygen sites of molecules |
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$i$,$k$, and $j$, where the centeral oxygen is located within molecule $k$ and |
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molecules $i$ and $j$ are two of the closest four water molecules |
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around molecule $k$. All four closest neighbors of molecule $k$ are also |
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required to reside within the first peak of the pair distribution function |
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for molecule $k$ (typically $<$ 3.41 \AA\ for water). |
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$N_z = \int\delta(z_k - z) \mathrm{d}z$ is a normalization factor to account |
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for the varying population of molecules within each finite-width bin. |
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|
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To determine the width of the interfaces, each of the systems were |
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divided into 100 artificial bins along the |
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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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|
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To investigate the dynamics of the water molecules across the interface, the |
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($d_{prismatic}\approx3.5$ \AA\ ) systems. |
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|
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\subsection{Coefficient of friction of the interfaces} |
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While investigating the kinetic coefficient of friction for the larger |
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prismatic system, there was found to be a dependence for $\mu_k$ |
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While investigating the kinetic coefficient of friction, there was found |
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to be a dependence for $\mu_k$ |
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on the temperature of the liquid water in the system. We believe this |
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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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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 $\eta$ 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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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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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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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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|
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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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and 5 unique shearing experiments were performed. Each experiment was |
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conducted in the microcanonical ensemble (NVE) and were 5 ns in |
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length. The VSS were attempted every timestep, which was set to 2 fs. |
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For our SPC/E systems, we found $\eta(225)$ to be $0.0148 \pm 0.0007$ Pa s, |
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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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|
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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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|
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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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\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|}{$\kappa_{Drag direction}$} \\ |
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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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\end{table} |
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|
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|
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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 water molecules was equilibrated to 225K, |
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and 5 unique shearing experiments were performed. Each experiment was |
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conducted in the microcanonical ensemble (NVE) and were 5 ns in |
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– |
length. The VSS were attempted every timestep, which was set to 2 fs. |
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For our SPC/E systems, we found $\eta(225)$ to be $0.0148 \pm 0.0007 Pa s$, |
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roughly ten times larger than the value found for 280 K SPC/E water by |
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Kuang\cite{kuang12}. |
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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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\textsuperscript{a} See ref. \onlinecite{Louden13}. } |
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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 (T = 225)\textsuperscript{a} & $0.037 \pm 0.008$ & $0.04 \pm 0.01$ \\ |
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prismatic (T = 230) & $0.10 \pm 0.01$ & $0.070 \pm 0.006$\\ |
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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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\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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%{-1}). \\ |
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%\textsuperscript{a} See ref. \onlinecite{Louden13}. } |
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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 (T = 225)\textsuperscript{a} & $0.037 \pm 0.008$ & $0.04 \pm 0.01$ \\ |
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% prismatic (T = 230) & $0.10 \pm 0.01$ & $0.070 \pm 0.006$\\ |
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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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%\end{table} |
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|
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|
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\begin{figure} |
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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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We present the results of molecular dynamics simulations of the pyrmaidal |
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and secondary prismatic facets of an SPC/E model of the |
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Ice-I$_\mathrm{h}$/water interface. The ice was sheared through the liquid |
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water while being exposed to a thermal gradient to maintain a stable |
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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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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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|
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|
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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 |
