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, 4 months 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	\documentclass[11pt]{article}
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14	%\usepackage{mathrsfs}
15	%\usepackage[ref]{overcite}
16	\usepackage[square, comma, sort&compress]{natbib}
17	\usepackage{url}
18	\pagestyle{plain} \pagenumbering{arabic} \oddsidemargin 0.0cm
19	\evensidemargin 0.0cm \topmargin -21pt \headsep 10pt \textheight
20	9.0in \textwidth 6.5in \brokenpenalty=10000
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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	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	$37.47 \times 29.50 \times 93.02$ \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	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
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	\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
133	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	From fitting the tetrahedrality profiles for each of the 0.5 nanosecond
159	simulations (panel c of Figures \ref{fig:spComic} and \ref{fig:pyrComic})
160	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 $0.9 \pm 0.2 \mathrm{ms}^{-1} \rightarrow 4.5 \pm 0.1
173	\mathrm{ms}^{-1}$ 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	\subsection{Orientational dynamics}
182	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
190
191
192	\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	\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}