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Revision 3909 by plouden, Wed Jul 10 20:00:23 2013 UTC vs.
Revision 3914 by gezelter, Fri Jul 12 18:14:40 2013 UTC

# Line 1 | Line 1
1 < \documentclass[%
2 < jcp,
3 < jmp,%
4 < amsmath,amssymb,
5 < preprint,%
6 < % reprint,%
7 < %author-year,%
8 < %author-numerical,%
9 < ]{revtex4-1}
1 > \documentclass[journal = jpccck, manuscript = article]{achemso}
2 > \setkeys{acs}{usetitle = true}
3 > \usepackage{achemso}
4 > \usepackage{natbib}
5 > \usepackage{multirow}
6 > \usepackage{wrapfig}
7 > \usepackage{fixltx2e}
8 > %\mciteErrorOnUnknownfalse
9  
10 < \usepackage{graphicx}% Include figure files
11 < \usepackage{dcolumn}% Align table columns on decimal point
13 < \usepackage{bm}% bold math
14 < %\usepackage[mathlines]{lineno}% Enable numbering of text and display math
15 < %\linenumbers\relax % Commence numbering lines
10 > \usepackage[version=3]{mhchem}  % this is a great package for formatting chemical reactions
11 > \usepackage{url}
12  
13  
14 < \begin{document}
14 > \title{Do the facets of ice $I_\mathrm{h}$ crystals have different
15 >  friction coefficients?  Simulating shear in ice/water interfaces}
16  
20 – %\preprint{AIP/123-QED}
21 –
22 – \title[VSS-RNEMD Simulation of Shearing the Basal and Prismatic Faces of Ice I_{h}]{VSS-RNEMD Simulation of Shearing the Basal and Prismatic Faces of Ice I_{h}}% Force line breaks with \\
23 – %\thanks{Footnote to title of article.}
24 –
17   \author{P. B. Louden}
18 < %\altaffiliation[Also at]{Physics Department, XYZ University.}%Lines break automatically or can be forced with \\
19 < \author{J. D. Gezelter}
20 < \homepage{To whom correspondence should be addressed. E-mail: gezelter@nd.edu}
21 < \affiliation{%
22 < Department of Chemistry and Biochemistry, The University of Notre Dame, Notre Dame, IN 46556%\\This line break forced% with \\
31 < }%
18 > \author{J. Daniel Gezelter}
19 > \email{gezelter@nd.edu}
20 > \affiliation[University of Notre Dame]{251 Nieuwland Science Hall\\
21 >  Department of Chemistry and Biochemistry\\ University of Notre
22 >  Dame\\ Notre Dame, Indiana 46556}
23  
24 < \date{\today}% It is always \today, today,
34 <             %  but any date may be explicitly specified
24 > \keywords{}
25  
26 < \maketitle
26 > \begin{document}
27  
28 < \section{Abstract}
29 < We have investigated the structural properties of the basal and prismatic facets of an SPC/E model of the ice Ih / water interface when the solid phase is being drawn through liquid water (i.e. sheared relative to the fluid phase). To impose the shear, we utilized a reverse non-equilibrium molecular dynamics (RNEMD) method that creates non-equilibrium conditions using velocity shearing and scaling (VSS) moves of the molecules in two physically separated slabs in the simulation cell. This method can create simultaneous temperature and velocity gradients and allow the measurement of friction transport properties at interfaces. We present calculations of the interfacial friction coefficients and the apparent independence of shear rate on interfacial width and show that water moving over a flat ice/water interface is close to the no-slip boundary condition.
28 > \begin{abstract}
29 > We have investigated the structural properties of the basal and
30 > prismatic facets of an SPC/E model of the ice Ih / water interface
31 > when the solid phase is being drawn through liquid water (i.e. sheared
32 > relative to the fluid phase). To impose the shear, we utilized a
33 > reverse non-equilibrium molecular dynamics (RNEMD) method that creates
34 > non-equilibrium conditions using velocity shearing and scaling (VSS)
35 > moves of the molecules in two physically separated slabs in the
36 > simulation cell. This method can create simultaneous temperature and
37 > velocity gradients and allow the measurement of friction transport
38 > properties at interfaces. We present calculations of the interfacial
39 > friction coefficients and the apparent independence of shear rate on
40 > interfacial width and show that water moving over a flat ice/water
41 > interface is close to the no-slip boundary condition.
42 > \end{abstract}
43  
44 + \newpage
45  
46   \section{Introduction}
47  
# Line 61 | Line 65 | In this paper, we investigate the width and the fricti
65  
66  
67   \section{Methodology}
64 – \subsection{System Construction}
65 – To construct the basal and prismatic systems, first the ice lattices were created. Hirsch and Ojam\"{a}e recently determined possible proton-ordered structures of ice Ih for an orthorhombic unit cell containing eight water molecules. \cite{Hirsch04} The crystallographic coordinates for structure 6 (P$2_{1}2_{1}2_{1}$) were used to construct an orthorhombic unit cell which was then replicated in all three dimensions yielding a proton-ordered block of ice Ih. To expose the desired face, the ice block was then cut along the ($0001$) plane or the ($10\overline{1}0$) plane for the basal and prismatic faces respectively. The ice block was also cut perpendicular to the initial cuts, and oriented so that the desired face is exposed to the $z$-axis. The ice block was then replicated in the $x$ and $y$ dimensions, and lastly liquid phase water molecules were added to the system. Haymet \emph{et al.} have done extensive work on studying and characterizing the ice/water interface.\cite{Karim88,Karim90,Hayward01,Bryk02,Hayward02} They have found for the SPC/E water model\cite{Berendsen87} (used here), the ice/water interface is most stable at 225$\pm$5K.\cite{Bryk02} Therefore, the temperature of the interface for each simulation was  225$\pm$5K. Molecular translation and orientation resrtaints were imposed in the early stages of equilibration to prevent melting of the ice block. These restraints were removed during NVT equilibration, long before data collection.
68  
69 + \subsection{Stable ice I$_\mathrm{h}$ / water interfaces}
70 +
71 + The structure of ice I$_\mathrm{h}$ is well understood; it
72 + crystallizes in a hexagonal space group P$6_3/mmc$, and the hexagonal
73 + crystals of ice have two faces that are commonly exposed, the basal
74 + face $\{0~0~0~1\}$, which forms the top and bottom of each hexagonal
75 + plate, and the prismatic $\{1~0~\bar{1}~0\}$ face which forms the
76 + sides of the plate. Other less-common, but still important, faces of
77 + ice I$_\mathrm{h}$ are the secondary prism face, $\{1~1~\bar{2}~0\}$,
78 + and the prismatic face, $\{2~0~\bar{2}~1\}$.
79 +
80 + Ice I$_\mathrm{h}$ is normally proton disordered in bulk crystals,
81 + although the surfaces probably have a preference for proton ordering
82 + along strips of dangling H-atoms and Oxygen lone
83 + pairs.\cite{Buch:2008fk}
84 +
85 + For small simulated ice interfaces, it is useful to have a
86 + proton-ordered, but zero-dipole crystal that exposes these strips of
87 + dangling H-atoms and lone pairs.  Also, if we're going to place
88 + another material in contact with one of the ice crystalline planes, it
89 + is useful to have an orthorhombic (rectangular) box to work with.  A
90 + recent paper by Hirsch and Ojam\"{a}e describes how to create
91 + proton-ordered bulk ice I$_\mathrm{h}$ in alternative orthorhombic
92 + cells.\cite{doi:10.1021/jp048434u}
93 +
94 + We have using Hirsch and Ojam\"{a}e's structure 6 which is an
95 + orthorhombic cell ($P2_12_12_1$) that produces a proton-ordered
96 + version of ice Ih.  Table \ref{tab:equiv} contains a mapping between
97 + the Miller indices in the P$6_3/mmc$ crystal system and those in the
98 + Hirsch and Ojam\"{a}e $P2_12_12_1$ system.
99 +
100 + \begin{wraptable}{r}{3.5in}
101 + \begin{tabular}{|ccc|} \hline
102 + & hexagonal & orthorhombic \\
103 + & ($P6_3/mmc$) & ($P2_12_12_1$) \\
104 + crystal face  & Miller indices & equivalent \\ \hline
105 + basal & $\{0~0~0~1\}$ & $\{0~0~1\}$ \\
106 + prism & $\{1~0~\bar{1}~0\}$ & $\{1~0~0\}$ \\
107 + secondary prism & $\{1~1~\bar{2}~0\}$ & $\{1~3~0\}$ \\
108 + pryamidal & $\{2~0~\bar{2}~1\}$ & $\{2~0~1\}$ \\ \hline
109 + \end{tabular}
110 + \end{wraptable}
111 +
112 + Structure 6 from the Hirsch and Ojam\"{a}e paper has lattice
113 + parameters $a = 4.49225$, $b = 7.78080$, $c = 7.33581$ and two water
114 + molecules whose atoms reside at the following fractional coordinates:
115 +
116 + \begin{wraptable}{r}{3.25in}
117 + \begin{tabular}{|ccccc|}  \hline
118 + atom label & type & x & y & z \\ \hline
119 + O$_{a}$    & O & 0.75 & 0.1667 & 0.4375 \\
120 + H$_{1a}$ & H & 0.5735 & 0.2202 & 0.4836 \\
121 + H$_{2a}$ & H & 0.7420 & 0.0517 & 0.4836 \\
122 + O$_{b}$    & O & 0.25 & 0.6667 & 0.4375 \\
123 + H$_{1b}$ & H & 0.2580 & 0.6693 & 0.3071 \\
124 + H$_{2b}$ & H & 0.4265 & 0.7255 & 0.4756 \\ \hline
125 + \end{tabular}
126 + \end{wraptable}
127 +
128 + To construct the basal and prismatic interfaces, the crystallographic
129 + coordinates above were used to construct an orthorhombic unit cell
130 + which was then replicated in all three dimensions yielding a
131 + proton-ordered block of ice I$_{h}$. To expose the desired face, the
132 + orthorhombic representation was then cut along the ($001$) or ($100$)
133 + planes for the basal and prismatic faces respectively.  The resulting
134 + block was rotated so that the exposed faces were aligned with the $z$
135 + axis normal to the exposed face.  The block was then cut along two
136 + perpendicular directions in a way that allowed for perfect periodic
137 + replication in the $x$ and $y$ axes, creating a slab with either the
138 + basal or prismatic faces exposed along the $z$ axis.  The slab was
139 + then replicated in the $x$ and $y$ dimensions until a desired sample
140 + size was obtained.  
141 +
142 + Although experimental solid/liquid coexistant temperature under normal
143 + pressure are close to 273K, Haymet \emph{et al.} have done extensive
144 + work on characterizing the ice/water
145 + interface.\cite{Karim88,Karim90,Hayward01,Bryk02,Hayward02} They have
146 + found for the SPC/E water model,\cite{Berendsen87} which is also used
147 + in this study, the ice/water interface is most stable at
148 + 225$\pm$5K.\cite{Bryk02} To create a ice / water interface, a box of
149 + liquid water that had the same dimensions in $x$ and $y$ was
150 + equilibrated at 225 K and 1 atm of pressure in the NPAT ensemble (with
151 + the $z$ axis allowed to fluctuate to equilibrate to the correct
152 + pressure).  The liquid and solid systems were combined by carving out
153 + any water molecule from the liquid simulation cell that was within 3
154 + \AA\ of any atom in the ice slab.
155 +
156 + Molecular translation and orientational restraints were applied in the
157 + early stages of equilibration to prevent melting of the ice slab.
158 + These restraints were removed during NVT equilibration, well before
159 + data collection was carried out.
160 +
161   \subsection{Computational Details}
162   All simulations were performed using OpenMD with a time step of 2 fs, and periodic boundary conditions in all three dimensions. The systems were divided into 100 artificial bins along the $z$-axis for the VSS-RNEMD moves, which were attempted every 50 fs. The gradients were allowed to develop for 1 ns before data collection was began. Once established,  snapshots of the system were taken every 1 ps, and the average velocities and densities of each bin were accumulated every attempted VSS-RNEMD move.
163  
# Line 99 | Line 193 | In Figures \ref{fig:bComic} and \ref{fig:pComic} we se
193  
194   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$5K, can be seen in panel (b).  Lastly, the velocity gradient across the systems can be seen in panel (c). From panel (c), we can see liquid phase water molecules 5 to 12 \AA\ from the midpoint of the basal and prismatic interfaces are being dragged along with the ice block. This indicates that the shearing of ice water is in the stick boundary condition.  
195  
196 + \begin{figure}
197 + \includegraphics[width=\linewidth]{bComicStrip}
198 + \caption{\label{fig:bComic} The basal system: (a) The local tetrahedral order parameter,$q$, (black circles) fit by a hyperbolic tangent (red line), (b) the thermal gradient imposed on the system to maintain a stable interfacial temperature, and (c) the velocity gradient imposed on the system. The verticle dotted lines indicate the midpoint of the interfaces.}
199 + \end{figure}
200  
201 + %(a) The local tetrahedral order parameter across the z-dimension of the system (black circles) fit by a hyperbolic tangent (red line). (b) The thermal gradient imposed on the system to maintain a stable interfacial temperature. (c) The velocity gradient imposed on the system. The verticle dotted lines indicate the midpoint of the interfaces.
202 +
203 + \begin{figure}
204 + \includegraphics[width=\linewidth]{pComicStrip}
205 + \caption{\label{fig:pComic} The prismatic system: (a) The local tetrahedral order parameter,$q$, (black circles) fit by a hyperbolic tangent (red line), (b) the thermal gradient imposed on the system to maintain a stable interfacial temperature, and (c) the velocity gradient imposed on the system.The verticle dotted lines indicate the midpoint of the interfaces.}
206 + \end{figure}
207 +
208 +
209   \subsection{Interfacial Width}
210   %For the basal and prismatic systems, the ice blocks were sheared through the water at varying rates while an imposed thermal gradient kept the interface at the stable temperature range as described by Byrk and Haymet.
211   We found the interfacial width for the basal and prismatic systems to be 3.2$\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.
# Line 119 | Line 225 | In Figure \ref{fig:CoeffFric}, the average velocity of
225  
226   %Ask dan about truncating versus rounding the values for lambda.
227   %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}.
228 + \begin{figure}
229 + \includegraphics[width=\linewidth]{CoeffFric}
230 + \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   }
231 + \end{figure}
232  
123 –
233   \section{Conclusion}
234   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 11.0 $\pm$ 0.4 \AA\textsuperscript{-2}fs\textsuperscript{-1}, and 19.9, $\pm$ 0.5 \AA\textsuperscript{-2}fs\textsuperscript{-1} for the prismatic facet. For both facets, the shearing ice water was found to be in the no-slip boundary condition.  
235  
236  
237 < \section{Acknowledgements}
237 > \begin{acknowledgement}
238 >  Support for this project was provided by the National Science
239 >  Foundation under grant CHE-0848243. Computational time was provided
240 >  by the Center for Research Computing (CRC) at the University of
241 >  Notre Dame.
242 > \end{acknowledgement}
243  
244 <
245 < \section{References}
244 > \newpage
245 > \bibstyle{achemso}
246   \bibliography{iceWater}
247  
248 < %Below is an example of how a Figure is imported and referenced in the paper.
249 < %here, FIGURENAME = fig:dip
250 < %\pagebreak
251 < %\begin{figure}
252 < %\includegraphics{dipolemoment.eps}% Here is how to import EPS art
253 < %\caption{\label{fig:dip} The condensation coefficient as a function of the dipole moment.}
254 < %\end{figure}
248 > \begin{tocentry}
249 > \begin{wrapfigure}{l}{0.5\textwidth}
250 > \begin{center}
251 > \includegraphics[width=\linewidth]{SystemImage.png}
252 > \end{center}
253 > \end{wrapfigure}
254 > An image of our system.
255 > \end{tocentry}
256  
257 < \pagebreak
143 < \begin{figure}
144 < \includegraphics{bComicStrip.eps}
145 < \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.}
146 < \end{figure}
257 > \end{document}
258  
148 – %(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.
149 –
150 –
151 – \pagebreak
152 – \begin{figure}
153 – \includegraphics{pComicStrip.eps}
154 – \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.}
155 – \end{figure}
156 –
157 – \pagebreak
158 – \begin{figure}
159 – \includegraphics{CoeffFric.eps}
160 – \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   }
161 – \end{figure}
162 –
259   % basal: slope=11.02808, error in slope = 0.4489844
260   %prismatic: slope = 19.95948, error in slope  = 0.5370894
165 –
166 – \end{document}

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