trunk/iceWater2/iceWater2.tex

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\begin{document}

\title{Simulations of solid-liquid friction at Secondary Prism and Pyramidal ice-I$_\mathrm{h}$ / water interfaces}

\author{Patrick B. Louden and J. Daniel
Gezelter\footnote{Corresponding author. \ Electronic mail:
  gezelter@nd.edu} \\
Department of Chemistry and Biochemistry,\\
University of Notre Dame\\
Notre Dame, Indiana 46556}

\date{\today}
\maketitle
\begin{doublespace}

\begin{abstract}
Abstract abstract abstract...
\end{abstract}

\newpage

\section{Introduction}
Explain a little bit about ice Ih, point group stuff. 

Mention previous work done / on going work by other people. Haymet and Rick
seem to be investigating how the interfaces is perturbed by the presence of 
ions. This is the conlcusion of a recent publication of the basal and 
prismatic facets of ice Ih, now presenting the pyramidal and secondary 
prism facets under shear.

\section{Methodology}

\begin{figure}
\includegraphics[width=\linewidth]{SP_comic_strip}
\caption{\label{fig:spComic} The secondary prism interface with a shear
 rate of 3.5 ms\textsuperscript{-1}. Lower panel: the local tetrahedral order
parameter, $q(z)$, (black circles) and the hyperbolic tangent fit (red line).
Middle panel: the imposed thermal gradient required to maintain a fixed
interfacial temperature. Upper panel: the transverse velocity gradient that
develops in response to an imposed momentum flux. The vertical dotted lines
indicate the locations of the midpoints of the two interfaces.}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{Pyr_comic_strip}
\caption{\label{fig:pyrComic} The pyramidal interface with a shear rate of 3.8 \
ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:spComic}.}
\end{figure}

\subsection{Pyramidal and secondary prism system construction}

The construction of the pyramidal and secondary prism systems follows that of 
the basal and prismatic systems presented elsewhere\cite{Louden13}, however
the ice crystals and water boxes were equilibrated and combined at 50K 
instead of 225K. The ice / water systems generated were then equilibrated 
to 225K. The resulting pyramidal system was 
$37.47 \times 29.50 \times 93.02$ \AA\ with 1216
SPC/E molecules in the ice slab, and 2203 in the liquid phase. The secondary 
prism system generated was $71.87 \times 31.66 \times 161.55$ \AA\ with 3840 
SPC/E molecules in the ice slab and 8176 molecules in the liquid phase.

\subsection{Computational details}
% Do we need to justify the sims at 225K? 
% No crystal growth or shrinkage over 2 successive 1 ns NVT simulations for
%    either the pyramidal or sec. prism ice/water systems.

The computational details performed here were equivalent to those reported
in the previous publication\cite{Louden13}. The only changes made to the 
previously reported procedure were the following. VSS-RNEMD moves were 
attempted every 2 fs instead of every 50 fs. This was done to minimize
the magnitude of each individual VSS-RNEMD perturbation to the system.

All pyramidal simulations were performed under the NVT ensamble except those
during which statistics were accumulated for the orientational correlation
function, which were performed under the NVE ensamble. All secondary prism 
simulations were performed under the NVE ensamble. 

\section{Results and discussion}
\subsection{Interfacial width}
In the literature there is good agreement that between the solid ice and 
the bulk water, there exists a region of 'slush-like' water molecules. 
In this region, the water molecules are structured differently and 
behave differently than those of the solid ice or the bulk water.
The characteristics of this region have been defined by both structural
and dynamic properties; and width has been measured by the change of these 
properties from their bulk liquid values to those of the solid ice. 
Examples of these properties include the density, the diffusion constant, and 
the translational order profile. \cite{Bryk02,Karim90,Gay02,Hayword01,Hayword02,Karim88}  

Since the VSS-RNEMD moves perturb the velocities of the water molecules in 
the systems, parameters that depend on the translational motion may give
faulty results. A stuructural parameter will be less effected by the 
VSS-RNEMD perturbations to the system. Due to this we have used the 
local order tetrahedral parameter, which was originally described by 
Kumar\cite{Kumar09} and Errington\cite{Errington01} and explained in our
previous publication\cite{Louden13} in relation to an ice/water system. 

Each of the systems were divided into 100 artificial bins along the 
$z$-dimension, and the local tetrahedral order parameter, $q(z)$, was 
time-averaged for each of the bins, resulting in a tetrahedrality profile of 
the system. These profiles are shown across the $z$-dimension of the systems
in panel $a$ of Figures \ref{fig:spComic}
and \ref{fig:pyrComic} (black circles). The $q(z)$ function has a range of 
(0,1), where a larger value indicates a more tetrahedral environment.
The $q(z)$ for the bulk liquid was found to be $\approx $0.77, while values of
$\approx $0.92 were more common for the ice. The tetrahedrality profiles were
fit using a hyperbolic tangent\cite{Louden13} designed to smoothly fit the 
bulk to ice 
transition, while accounting for the thermal influence on the profile by the
kinetic energy exchanges of the VSS-RNEMD moves. In panels $b$ and $c$, the
imposed thermal and velocity gradients can be seen. The verticle dotted
lines traversing all three panels indicate the midpoints of the interface
as determined by the hyperbolic tangent fit of the tetrahedrality profiles.
 
From fitting the tetrahedrality profiles for each of the 0.5 nanosecond 
simulations (panel c of Figures \ref{fig:spComic} and \ref{fig:pyrComic}) 
by Eq. 6\cite{Louden13},we find the interfacial width for the pyramidal and 
secondary prism to be $3.2 \pm 0.2$ and $3.2 \pm 0.2$ \AA\ , respectively, 
with no applied momentum flux. Over the range of shear rates investigated, 
$0.6 \pm 0.2 \mathrm{ms}^{-1} \rightarrow 5.6 \pm 0.4 \mathrm{ms}^{-1}$ for 
the pyramidal system and $0.9 \pm 0.3 \mathrm{ms}^{-1} \rightarrow 5.4 \pm 0.1
\mathrm{ms}^{-1}$ for the secondary prism, we found no significant change in
the interfacial width. This follows our previous findings of the basal and
prismatic systems, in which the interfacial width was invarient of the
shear rate of the ice. The interfacial width of the quiescent basal and 
prismatic systems was found to be $3.2 \pm 0.4$ \AA\ and $3.6 \pm 0.2$ \AA\ 
respectively. Over the range of shear rates investigated, $0.6 \pm 0.3 
\mathrm{ms}^{-1} \rightarrow 5.3 \pm 0.5 \mathrm{ms}^{-1}$ for the basal 
system and $0.9 \pm 0.2 \mathrm{ms}^{-1} \rightarrow 4.5 \pm 0.1 
\mathrm{ms}^{-1}$ for the prismatic.

These results indicate that the surface structure of the exposed ice crystal
has little to no effect on how far into the bulk the ice-like structural 
ordering is. Also, it appears that the interface is not structurally effected
by shearing the ice through water. 


\subsection{Orientational dynamics}
To investigate the dynamics of the water molecules across the interface, the 
systems were divided into $n$ bins, each $\approx$ 3 \AA\ wide in $z$, and 
the orientational time 
correlation function was computed for each of the $n$ bins. This was done by 
averaging the second order Legendre polynomial of the bisecting HOH vector 
dotted with itself at an initial time and some time later, over all molecules 
in the bin.   


\subsection{Coefficient of friction of the interfaces}


\begin{table}[h]
\centering
\caption{Solid-liquid friction coefficients (measured in amu~fs\textsuperscript\
{-1}) }
\label{tab:lambda}
\begin{tabular}{|ccc|}  \hline
           & \multicolumn{2}{c|}{Drag direction} \\
 Interface & $x$               & $y$  \\ \hline
     basal\textsuperscript{a} &  $0.08 \pm 0.02$  & $0.09 \pm 0.03$ \\
 prismatic\textsuperscript{a} & $0.037 \pm 0.008$ & $0.04 \pm 0.01$ \\
 pyramidal & $0.13 \pm 0.03$   & $0.14 \pm 0.03$ \\
 secondary prism & $0.13 \pm 0.02$ & $0.12 \pm 0.03$ \\ \hline
\end{tabular}
\caption{\textsuperscript{a}Reference \cite{Louden13}}
\end{table}
 

\begin{figure}
\includegraphics[width=\linewidth]{Pyr-orient}
\caption{\label{fig:PyrOrient} The three decay constants of the 
orientational time correlation function, $C_2(t)$, for water as a function
of distance from the center of the ice slab. The vertical dashed line
indicates the edge of the pyramidal ice slab determined by the local order
tetrahedral parameter. The control (black circles) and sheared (red squares) 
experiments were fit by a shifted exponential decay (Eq. 9\cite{Louden13})
shown by the black and red lines respectively. The upper two panels show that 
translational and hydrogen bond making and breaking events slow down 
through the interface while approaching the ice slab. The bottom most panel
shows the librational motion of the water molecules speeding up approaching
the ice block due to the confined region of space allowed for the molecules
 to move in.}
\end{figure}  

\begin{figure}
\includegraphics[width=\linewidth]{SP-orient-less}
\caption{\label{fig:SPorient} Decay constants for $C_2(t)$ at the secondary 
prism face. Panel descriptions match those in \ref{fig:PyrOrient}.}
\end{figure}


\section{Conclusion}
Conclude conclude conclude...

\section{Acknowledgements}
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.


\newpage
\bibliography{iceWater}

\end{doublespace}

\end{document}
Revision:	4194
Committed:	Mon Jun 30 20:56:56 2014 UTC (11 years, 1 month ago) by plouden
Content type:	application/x-tex
File size:	11741 byte(s)
Log Message:	wrote the interfacial width results section
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1	plouden	4192	\documentclass[11pt]{article}
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16			\usepackage[square, comma, sort&compress]{natbib}
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30
31
32			% \documentclass[journal = jpccck, manuscript = article]{achemso}
33			% \setkeys{acs}{usetitle = true}
34			% \usepackage{achemso}
35			% \usepackage{natbib}
36			% \usepackage{multirow}
37			% \usepackage{wrapfig}
38			% \usepackage{fixltx2e}
39
40			\usepackage[version=3]{mhchem} % this is a great package for formatting chemical reactions
41			\usepackage{url}
42
43
44			\begin{document}
45
46			\title{Simulations of solid-liquid friction at Secondary Prism and Pyramidal ice-I$_\mathrm{h}$ / water interfaces}
47
48			\author{Patrick B. Louden and J. Daniel
49			Gezelter\footnote{Corresponding author. \ Electronic mail:
50			gezelter@nd.edu} \\
51			Department of Chemistry and Biochemistry,\\
52			University of Notre Dame\\
53			Notre Dame, Indiana 46556}
54
55			\date{\today}
56			\maketitle
57			\begin{doublespace}
58
59			\begin{abstract}
60			Abstract abstract abstract...
61			\end{abstract}
62
63			\newpage
64
65			\section{Introduction}
66			Explain a little bit about ice Ih, point group stuff.
67
68			Mention previous work done / on going work by other people. Haymet and Rick
69			seem to be investigating how the interfaces is perturbed by the presence of
70			ions. This is the conlcusion of a recent publication of the basal and
71			prismatic facets of ice Ih, now presenting the pyramidal and secondary
72			prism facets under shear.
73
74			\section{Methodology}
75
76			\begin{figure}
77			\includegraphics[width=\linewidth]{SP_comic_strip}
78			\caption{\label{fig:spComic} The secondary prism interface with a shear
79			rate of 3.5 ms\textsuperscript{-1}. Lower panel: the local tetrahedral order
80			parameter, $q(z)$, (black circles) and the hyperbolic tangent fit (red line).
81			Middle panel: the imposed thermal gradient required to maintain a fixed
82			interfacial temperature. Upper panel: the transverse velocity gradient that
83			develops in response to an imposed momentum flux. The vertical dotted lines
84			indicate the locations of the midpoints of the two interfaces.}
85			\end{figure}
86
87			\begin{figure}
88			\includegraphics[width=\linewidth]{Pyr_comic_strip}
89			\caption{\label{fig:pyrComic} The pyramidal interface with a shear rate of 3.8 \
90			ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:spComic}.}
91			\end{figure}
92
93			\subsection{Pyramidal and secondary prism system construction}
94
95			The construction of the pyramidal and secondary prism systems follows that of
96			the basal and prismatic systems presented elsewhere\cite{Louden13}, however
97	plouden	4194	the ice crystals and water boxes were equilibrated and combined at 50K
98			instead of 225K. The ice / water systems generated were then equilibrated
99			to 225K. The resulting pyramidal system was
100	plouden	4192	$37.47 \times 29.50 \times 93.02$ \AA\ with 1216
101			SPC/E molecules in the ice slab, and 2203 in the liquid phase. The secondary
102			prism system generated was $71.87 \times 31.66 \times 161.55$ \AA\ with 3840
103			SPC/E molecules in the ice slab and 8176 molecules in the liquid phase.
104
105			\subsection{Computational details}
106			% Do we need to justify the sims at 225K?
107			% No crystal growth or shrinkage over 2 successive 1 ns NVT simulations for
108			% either the pyramidal or sec. prism ice/water systems.
109
110			The computational details performed here were equivalent to those reported
111			in the previous publication\cite{Louden13}. The only changes made to the
112			previously reported procedure were the following. VSS-RNEMD moves were
113	plouden	4194	attempted every 2 fs instead of every 50 fs. This was done to minimize
114			the magnitude of each individual VSS-RNEMD perturbation to the system.
115	plouden	4192
116			All pyramidal simulations were performed under the NVT ensamble except those
117			during which statistics were accumulated for the orientational correlation
118			function, which were performed under the NVE ensamble. All secondary prism
119			simulations were performed under the NVE ensamble.
120
121			\section{Results and discussion}
122	plouden	4194	\subsection{Interfacial width}
123			In the literature there is good agreement that between the solid ice and
124			the bulk water, there exists a region of 'slush-like' water molecules.
125			In this region, the water molecules are structured differently and
126			behave differently than those of the solid ice or the bulk water.
127			The characteristics of this region have been defined by both structural
128			and dynamic properties; and width has been measured by the change of these
129			properties from their bulk liquid values to those of the solid ice.
130			Examples of these properties include the density, the diffusion constant, and
131			the translational order profile. \cite{Bryk02,Karim90,Gay02,Hayword01,Hayword02,Karim88}
132	plouden	4192
133	plouden	4194	Since the VSS-RNEMD moves perturb the velocities of the water molecules in
134			the systems, parameters that depend on the translational motion may give
135			faulty results. A stuructural parameter will be less effected by the
136			VSS-RNEMD perturbations to the system. Due to this we have used the
137			local order tetrahedral parameter, which was originally described by
138			Kumar\cite{Kumar09} and Errington\cite{Errington01} and explained in our
139			previous publication\cite{Louden13} in relation to an ice/water system.
140
141			Each of the systems were divided into 100 artificial bins along the
142			$z$-dimension, and the local tetrahedral order parameter, $q(z)$, was
143			time-averaged for each of the bins, resulting in a tetrahedrality profile of
144			the system. These profiles are shown across the $z$-dimension of the systems
145			in panel $a$ of Figures \ref{fig:spComic}
146			and \ref{fig:pyrComic} (black circles). The $q(z)$ function has a range of
147			(0,1), where a larger value indicates a more tetrahedral environment.
148			The $q(z)$ for the bulk liquid was found to be $\approx $0.77, while values of
149			$\approx $0.92 were more common for the ice. The tetrahedrality profiles were
150			fit using a hyperbolic tangent\cite{Louden13} designed to smoothly fit the
151			bulk to ice
152			transition, while accounting for the thermal influence on the profile by the
153			kinetic energy exchanges of the VSS-RNEMD moves. In panels $b$ and $c$, the
154			imposed thermal and velocity gradients can be seen. The verticle dotted
155			lines traversing all three panels indicate the midpoints of the interface
156			as determined by the hyperbolic tangent fit of the tetrahedrality profiles.
157
158	plouden	4192	From fitting the tetrahedrality profiles for each of the 0.5 nanosecond
159	plouden	4194	simulations (panel c of Figures \ref{fig:spComic} and \ref{fig:pyrComic})
160	plouden	4192	by Eq. 6\cite{Louden13},we find the interfacial width for the pyramidal and
161			secondary prism to be $3.2 \pm 0.2$ and $3.2 \pm 0.2$ \AA\ , respectively,
162			with no applied momentum flux. Over the range of shear rates investigated,
163			$0.6 \pm 0.2 \mathrm{ms}^{-1} \rightarrow 5.6 \pm 0.4 \mathrm{ms}^{-1}$ for
164			the pyramidal system and $0.9 \pm 0.3 \mathrm{ms}^{-1} \rightarrow 5.4 \pm 0.1
165			\mathrm{ms}^{-1}$ for the secondary prism, we found no significant change in
166			the interfacial width. This follows our previous findings of the basal and
167			prismatic systems, in which the interfacial width was invarient of the
168			shear rate of the ice. The interfacial width of the quiescent basal and
169			prismatic systems was found to be $3.2 \pm 0.4$ \AA\ and $3.6 \pm 0.2$ \AA\
170			respectively. Over the range of shear rates investigated, $0.6 \pm 0.3
171			\mathrm{ms}^{-1} \rightarrow 5.3 \pm 0.5 \mathrm{ms}^{-1}$ for the basal
172			system and $0.9 \pm 0.2 \mathrm{ms}^{-1} \rightarrow 4.5 \pm 0.1
173	plouden	4194	\mathrm{ms}^{-1}$ for the prismatic.
174
175			These results indicate that the surface structure of the exposed ice crystal
176			has little to no effect on how far into the bulk the ice-like structural
177			ordering is. Also, it appears that the interface is not structurally effected
178			by shearing the ice through water.
179
180
181	plouden	4192	\subsection{Orientational dynamics}
182	plouden	4194	To investigate the dynamics of the water molecules across the interface, the
183			systems were divided into $n$ bins, each $\approx$ 3 \AA\ wide in $z$, and
184			the orientational time
185			correlation function was computed for each of the $n$ bins. This was done by
186			averaging the second order Legendre polynomial of the bisecting HOH vector
187			dotted with itself at an initial time and some time later, over all molecules
188			in the bin.
189	plouden	4192
190
191
192	plouden	4194	\subsection{Coefficient of friction of the interfaces}
193
194
195			\begin{table}[h]
196			\centering
197			\caption{Solid-liquid friction coefficients (measured in amu~fs\textsuperscript\
198			{-1}) }
199			\label{tab:lambda}
200			\begin{tabular}{\|ccc\|} \hline
201			& \multicolumn{2}{c\|}{Drag direction} \\
202			Interface & $x$ & $y$ \\ \hline
203			basal\textsuperscript{a} & $0.08 \pm 0.02$ & $0.09 \pm 0.03$ \\
204			prismatic\textsuperscript{a} & $0.037 \pm 0.008$ & $0.04 \pm 0.01$ \\
205			pyramidal & $0.13 \pm 0.03$ & $0.14 \pm 0.03$ \\
206			secondary prism & $0.13 \pm 0.02$ & $0.12 \pm 0.03$ \\ \hline
207			\end{tabular}
208			\caption{\textsuperscript{a}Reference \cite{Louden13}}
209			\end{table}
210
211
212	plouden	4192	\begin{figure}
213			\includegraphics[width=\linewidth]{Pyr-orient}
214			\caption{\label{fig:PyrOrient} The three decay constants of the
215			orientational time correlation function, $C_2(t)$, for water as a function
216			of distance from the center of the ice slab. The vertical dashed line
217			indicates the edge of the pyramidal ice slab determined by the local order
218			tetrahedral parameter. The control (black circles) and sheared (red squares)
219			experiments were fit by a shifted exponential decay (Eq. 9\cite{Louden13})
220			shown by the black and red lines respectively. The upper two panels show that
221			translational and hydrogen bond making and breaking events slow down
222			through the interface while approaching the ice slab. The bottom most panel
223			shows the librational motion of the water molecules speeding up approaching
224			the ice block due to the confined region of space allowed for the molecules
225			to move in.}
226			\end{figure}
227
228			\begin{figure}
229			\includegraphics[width=\linewidth]{SP-orient-less}
230			\caption{\label{fig:SPorient} Decay constants for $C_2(t)$ at the secondary
231			prism face. Panel descriptions match those in \ref{fig:PyrOrient}.}
232			\end{figure}
233
234
235
236			\section{Conclusion}
237			Conclude conclude conclude...
238
239			\section{Acknowledgements}
240			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.
241
242
243			\newpage
244			\bibliography{iceWater}
245
246			\end{doublespace}
247
248			\end{document}