1 |
% ****** Start of file aipsamp.tex ****** |
2 |
% |
3 |
% This file is part of the AIP files in the AIP distribution for REVTeX 4. |
4 |
% Version 4.1 of REVTeX, October 2009 |
5 |
% |
6 |
% Copyright (c) 2009 American Institute of Physics. |
7 |
% |
8 |
% See the AIP README file for restrictions and more information. |
9 |
% |
10 |
% TeX'ing this file requires that you have AMS-LaTeX 2.0 installed |
11 |
% as well as the rest of the prerequisites for REVTeX 4.1 |
12 |
% |
13 |
% It also requires running BibTeX. The commands are as follows: |
14 |
% |
15 |
% 1) latex aipsamp |
16 |
% 2) bibtex aipsamp |
17 |
% 3) latex aipsamp |
18 |
% 4) latex aipsamp |
19 |
% |
20 |
% Use this file as a source of example code for your aip document. |
21 |
% Use the file aiptemplate.tex as a template for your document. |
22 |
\documentclass[% |
23 |
aip,jcp, |
24 |
amsmath,amssymb, |
25 |
preprint,% |
26 |
% reprint,% |
27 |
%author-year,% |
28 |
%author-numerical,% |
29 |
jcp]{revtex4-1} |
30 |
|
31 |
\usepackage{graphicx}% Include figure files |
32 |
\usepackage{dcolumn}% Align table columns on decimal point |
33 |
%\usepackage{bm}% bold math |
34 |
\usepackage{times} |
35 |
\usepackage[version=3]{mhchem} % this is a great package for formatting chemical reactions |
36 |
\usepackage{url} |
37 |
|
38 |
\begin{document} |
39 |
|
40 |
\title{Friction at Water / Ice-I$_\mathrm{h}$ interfaces: Do the |
41 |
Facets of Ice Have Different Hydrophilicity?} |
42 |
|
43 |
\author{Patrick B. Louden} |
44 |
|
45 |
\author{J. Daniel Gezelter} |
46 |
\email{gezelter@nd.edu.} |
47 |
\affiliation{Department of Chemistry and Biochemistry, University |
48 |
of Notre Dame, Notre Dame, IN 46556} |
49 |
|
50 |
\date{\today} |
51 |
|
52 |
\begin{abstract} |
53 |
In this follow up paper of the basal and prismatic facets of the |
54 |
Ice-I$_\mathrm{h}$/water interface, we present the |
55 |
pyramidal and secondary prismatic |
56 |
interfaces for both the quiescent and sheared systems. The structural and |
57 |
dynamic interfacial widths for all four crystal facets were found to be in good |
58 |
agreement, and were found to be independent of the shear rate over the shear |
59 |
rates investigated. |
60 |
Decomposition of the molecular orientational time correlation function showed |
61 |
different behavior for the short- and longer-time decay components approaching |
62 |
normal to the interface. Lastly we show through calculation of the interfacial |
63 |
friction coefficient that the basal and pyramidal facets are more |
64 |
hydrophilic than the prismatic and secondary prismatic facets. |
65 |
|
66 |
\end{abstract} |
67 |
|
68 |
\maketitle |
69 |
|
70 |
\section{Introduction} |
71 |
Explain a little bit about ice Ih, point group stuff. |
72 |
|
73 |
Mention previous work done / on going work by other people. Haymet and Rick |
74 |
seem to be investigating how the interfaces is perturbed by the presence of |
75 |
ions. This is the conlcusion of a recent publication of the basal and |
76 |
prismatic facets of ice Ih, now presenting the pyramidal and secondary |
77 |
prismatic facets under shear. |
78 |
|
79 |
Investigation of the ice/water interface is crucial in understanding |
80 |
the fundamental processes of nucleation,\cite{} crystal |
81 |
growth,\cite{Han92, Granasy95, Vanfleet95} and crystal |
82 |
melting,\cite{Weber83, Han92, Sakai96, Sakai96B}. Insight gained to these |
83 |
properties can also be applied to biological systems of interest, such as |
84 |
the behavior of the antifreeze protein found in winter |
85 |
flounder,\cite{Wierzbicki07, Chapsky97} and certain terrestial |
86 |
arthropods.\cite{Duman:2001qy,Meister29012013}%add more! |
87 |
|
88 |
The Ice-I$_\mathrm{h}$/water quiescent interface has been extensively studied |
89 |
over the past 30 years. Haymet \emph{et al.} have done significant work |
90 |
characterizing and quantifying the width of these interfaces for the |
91 |
SPC,\cite{Karim90} SPC/E,\cite{Gay02,Bryk02}, CF1,\cite{Hayward01,Hayward02} |
92 |
and TIP4P\cite{Karim88} models for water. In recent years, Haymet has focused |
93 |
on investigating the effects cations and anions have on crystal |
94 |
nucleaion and melting.\cite{Bryk04,Smith05,Wilson08,Wilson10} |
95 |
|
96 |
|
97 |
\section{Methodology} |
98 |
|
99 |
\begin{figure} |
100 |
\includegraphics[width=\linewidth]{Pyr_comic_strip} |
101 |
\caption{\label{fig:pyrComic} The pyramidal interface with a shear |
102 |
rate of 3.8 ms\textsuperscript{-1}. Lower panel: the local tetrahedral order |
103 |
parameter, $q(z)$, (black circles) and the hyperbolic tangent fit (red line). |
104 |
Middle panel: the imposed thermal gradient required to maintain a fixed |
105 |
interfacial temperature. Upper panel: the transverse velocity gradient that |
106 |
develops in response to an imposed momentum flux. The vertical dotted lines |
107 |
indicate the locations of the midpoints of the two interfaces.} |
108 |
\end{figure} |
109 |
|
110 |
\begin{figure} |
111 |
\includegraphics[width=\linewidth]{SP_comic_strip} |
112 |
\caption{\label{fig:spComic} The secondary prismatic interface with a shear |
113 |
rate of 3.5 \ |
114 |
ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.} |
115 |
\end{figure} |
116 |
|
117 |
\subsection{Pyramidal and secondary prismatic system construction} |
118 |
|
119 |
The construction of the pyramidal and secondary prismatic systems follows that |
120 |
of |
121 |
the basal and prismatic systems presented elsewhere\cite{Louden13}, however |
122 |
the ice crystals and water boxes were equilibrated and combined at 50K |
123 |
instead of 225K. The ice / water systems generated were then equilibrated |
124 |
to 225K. The resulting pyramidal system was |
125 |
$37.47 \times 29.50 \times 93.02$ \AA\ with 1216 |
126 |
SPC/E\cite{Berendsen97} molecules in the ice slab, and 2203 in the liquid |
127 |
phase. The secondary |
128 |
prismatic system generated was $71.87 \times 31.66 \times 161.55$ \AA\ with |
129 |
3840 |
130 |
SPC/E molecules in the ice slab and 8176 molecules in the liquid phase. |
131 |
|
132 |
\subsection{Computational details} |
133 |
% Do we need to justify the sims at 225K? |
134 |
% No crystal growth or shrinkage over 2 successive 1 ns NVT simulations for |
135 |
% either the pyramidal or sec. prismatic ice/water systems. |
136 |
|
137 |
The computational details performed here were equivalent to those reported |
138 |
in our previous publication\cite{Louden13}. The only changes made to the |
139 |
previously reported procedure were the following. VSS-RNEMD moves were |
140 |
attempted every 2 fs instead of every 50 fs. This was done to minimize |
141 |
the magnitude of each individual VSS-RNEMD perturbation to the system. |
142 |
|
143 |
All pyramidal simulations were performed under the canonical (NVT) ensamble |
144 |
except those |
145 |
during which statistics were accumulated for the orientational correlation |
146 |
function, which were performed under the microcanonical (NVE) ensamble. All |
147 |
secondary prismatic |
148 |
simulations were performed under the NVE ensamble. |
149 |
|
150 |
\section{Results and discussion} |
151 |
\subsection{Interfacial width} |
152 |
In the literature there is good agreement that between the solid ice and |
153 |
the bulk water, there exists a region of 'slush-like' water molecules. |
154 |
In this region, the water molecules are structurely distinguishable and |
155 |
behave differently than those of the solid ice or the bulk water. |
156 |
The characteristics of this region have been defined by both structural |
157 |
and dynamic properties; and its width has been measured by the change of these |
158 |
properties from their bulk liquid values to those of the solid ice. |
159 |
Examples of these properties include the density, the diffusion constant, and |
160 |
the translational order profile. \cite{Bryk02,Karim90,Gay02,Hayward01,Hayward02,Karim88} |
161 |
|
162 |
Since the VSS-RNEMD moves used to impose the thermal and velocity gradients |
163 |
perturb the momenta of the water molecules in |
164 |
the systems, parameters that depend on translational motion may give |
165 |
faulty results. A stuructural parameter will be less effected by the |
166 |
VSS-RNEMD perturbations to the system. Due to this, we have used the |
167 |
local tetrahedral order parameter to quantify the width of the interface, |
168 |
which was originally described by Kumar\cite{Kumar09} and |
169 |
Errington\cite{Errington01}, and used by Bryk and Haymet in a previous study |
170 |
of ice/water interfaces.\cite{Bryk2004b} |
171 |
|
172 |
The local tetrahedral order parameter, $q(z)$, is given by |
173 |
\begin{equation} |
174 |
q(z) = \int_0^L \sum_{k=1}^{N} \Bigg(1 -\frac{3}{8}\sum_{i=1}^{3} |
175 |
\sum_{j=i+1}^{4} \bigg(\cos\psi_{ikj}+\frac{1}{3}\bigg)^2\Bigg) |
176 |
\delta(z_{k}-z)\mathrm{d}z \Bigg/ N_z |
177 |
\label{eq:qz} |
178 |
\end{equation} |
179 |
where $\psi_{ikj}$ is the angle formed between the oxygen sites of molecules |
180 |
$i$,$k$, and $j$, where the centeral oxygen is located within molecule $k$ and |
181 |
molecules $i$ and $j$ are two of the closest four water molecules |
182 |
around molecule $k$. All four closest neighbors of molecule $k$ are also |
183 |
required to reside within the first peak of the pair distribution function |
184 |
for molecule $k$ (typically $<$ 3.41 \AA\ for water). |
185 |
$N_z = \int\delta(z_k - z) \mathrm{d}z$ is a normalization factor to account |
186 |
for the varying population of molecules within each finite-width bin. |
187 |
|
188 |
To determine the width of the interfaces, each of the systems were |
189 |
divided into 100 artificial bins along the |
190 |
$z$-dimension, and the local tetrahedral order parameter, $q(z)$, was |
191 |
time-averaged for each of the bins, resulting in a tetrahedrality profile of |
192 |
the system. These profiles are shown across the $z$-dimension of the systems |
193 |
in panel $a$ of Figures \ref{fig:pyrComic} |
194 |
and \ref{fig:spComic} (black circles). The $q(z)$ function has a range of |
195 |
(0,1), where a larger value indicates a more tetrahedral environment. |
196 |
The $q(z)$ for the bulk liquid was found to be $\approx $ 0.77, while values of |
197 |
$\approx $ 0.92 were more common for the ice. The tetrahedrality profiles were |
198 |
fit using a hyperbolic tangent\cite{Louden13} designed to smoothly fit the |
199 |
bulk to ice |
200 |
transition, while accounting for the thermal influence on the profile by the |
201 |
kinetic energy exchanges of the VSS-RNEMD moves. In panels $b$ and $c$, the |
202 |
resulting thermal and velocity gradients from the imposed kinetic energy and |
203 |
momentum fluxes can be seen. The verticle dotted |
204 |
lines traversing all three panels indicate the midpoints of the interface |
205 |
as determined by the hyperbolic tangent fit of the tetrahedrality profiles. |
206 |
|
207 |
From fitting the tetrahedrality profiles for each of the 0.5 nanosecond |
208 |
simulations (panel c of Figures \ref{fig:spComic} and \ref{fig:pyrComic}) |
209 |
by Eq. 6\cite{Louden13},we find the interfacial width to be |
210 |
3.2 $\pm$ 0.2 and 3.2 $\pm$ 0.2 \AA\ for the control system with no applied |
211 |
momentum flux for both the pyramidal and secondary prismatic systems. |
212 |
Over the range of shear rates investigated, |
213 |
0.6 $\pm$ 0.2 $\mathrm{ms}^{-1} \rightarrow$ 5.6 $\pm$ 0.4 $\mathrm{ms}^{-1}$ |
214 |
for the pyramidal system and 0.9 $\pm$ 0.3 $\mathrm{ms}^{-1} \rightarrow$ 5.4 |
215 |
$\pm$ 0.1 $\mathrm{ms}^{-1}$ for the secondary prismatic, we found no |
216 |
significant change in the interfacial width. This follows our previous |
217 |
findings of the basal and |
218 |
prismatic systems, in which the interfacial width was invarient of the |
219 |
shear rate of the ice. The interfacial width of the quiescent basal and |
220 |
prismatic systems was found to be 3.2 $\pm$ 0.4 \AA\ and 3.6 $\pm$ 0.2 \AA\ |
221 |
respectively, over the range of shear rates investigated, 0.6 $\pm$ 0.3 |
222 |
$\mathrm{ms}^{-1} \rightarrow$ 5.3 $\pm$ 0.5 $\mathrm{ms}^{-1}$ for the basal |
223 |
system and 0.9 $\pm$ 0.2 $\mathrm{ms}^{-1} \rightarrow$ 4.5 $\pm$ 0.1 |
224 |
$\mathrm{ms}^{-1}$ for the prismatic. |
225 |
|
226 |
These results indicate that the surface structure of the exposed ice crystal |
227 |
has little to no effect on how far into the bulk the ice-like structural |
228 |
ordering is. Also, it appears that the interface is not structurally effected |
229 |
by shearing the ice through water. |
230 |
|
231 |
|
232 |
\subsection{Orientational dynamics} |
233 |
%Should we include the math here? |
234 |
The orientational time correlation function, |
235 |
\begin{equation}\label{C(t)1} |
236 |
C_{2}(t)=\langle P_{2}(\mathbf{u}(0)\cdot \mathbf{u}(t))\rangle, |
237 |
\end{equation} |
238 |
helps indicate the local environment around the water molecules. The function |
239 |
begins with an initial value of unity, and decays to zero as the water molecule |
240 |
loses memory of its former orientation. Observing the rate at which this decay |
241 |
occurs can provide insight to the mechanism and timescales for the relaxation. |
242 |
In eq. \eqref{C(t)1}, $P_{2}$ is the second-order Legendre polynomial, and |
243 |
$\mathbf{u}$ is the bisecting HOH vector. The angle brackets indicate |
244 |
an ensemble average over all the water molecules in a given spatial region. |
245 |
|
246 |
To investigate the dynamics of the water molecules across the interface, the |
247 |
systems were divided in the $z$-dimension into bins, each $\approx$ 3 \AA\ |
248 |
wide, and \eqref{C(t)1} was computed for each of the bins. A water |
249 |
molecule was allocated to a particular bin if it was initially in the bin |
250 |
at time zero. To compute \eqref{C(t)1}, each 0.5 ns simulation was followed |
251 |
by an additional 200 ps NVE simulation during which the |
252 |
position and orientations of each molecule were recorded every 0.1 ps. |
253 |
|
254 |
The data obtained for each bin was then fit to a triexponential decay given by |
255 |
\begin{equation}\label{C(t)_fit} |
256 |
C_{2}(t) \approx a e^{-t/\tau_\mathrm{short}} + b e^{-t/\tau_\mathrm{middle}} +\ |
257 |
c |
258 |
e^{-t/\tau_\mathrm{long}} + (1-a-b-c) |
259 |
\end{equation} |
260 |
where $\tau_{short}$ corresponds to the librational motion of the water |
261 |
molecules, $\tau_{middle}$ corresponds to jumps between the breaking and |
262 |
making of hydrogen bonds, and $\tau_{long}$ corresponds to the translational |
263 |
motion of the water molecules. The last term in \eqref{C(t)_fit} accounts |
264 |
for the water molecules trapped in the ice which do not experience any |
265 |
long-time orientational decay. |
266 |
|
267 |
In Figures \ref{fig:PyrOrient} and \ref{fig:SPorient} we see the $z$-coordinate |
268 |
profiles for the three decay constants, $\tau_{short}$ (panel a), |
269 |
$\tau_{middle}$ (panel b), |
270 |
and $\tau_{long}$ (panel c) for the pyramidal and secondary prismatic systems |
271 |
respectively. The control experiments (no shear) are shown in black, and |
272 |
an experiment with an imposed momentum flux is shown in red. The vertical |
273 |
dotted line traversing all three panels denotes the midpoint of the |
274 |
interface as determined by the local tetrahedral order parameter fitting. |
275 |
In the liquid regions of both systems, we see that $\tau_{middle}$ and |
276 |
$\tau_{long}$ have approximately consistent values of $3-6$ ps and $30-40$ ps, |
277 |
resepctively, and increase in value as we approach the interface. Conversely, |
278 |
in panel a, we see that $\tau_{short}$ decreases from the liquid value |
279 |
of $72-76$ fs as we approach the interface. We believe this speed up is due to |
280 |
the constrained motion of librations closer to the interface. Both the |
281 |
approximate values for the decays and trends approaching the interface match |
282 |
those reported previously for the basal and prismatic interfaces. |
283 |
|
284 |
As done previously, we have attempted to quantify the distance, $d_{pyramidal}$ |
285 |
and $d_{secondary prismatic}$, from the |
286 |
interface that the deviations from the bulk liquid values begin. This was done |
287 |
by fitting the orientational decay constant $z$-profiles by |
288 |
\begin{equation}\label{tauFit} |
289 |
\tau(z)\approx\tau_{liquid}+(\tau_{wall}-\tau_{liquid})e^{-(z-z_{wall})/d} |
290 |
\end{equation} |
291 |
where $\tau_{liquid}$ and $\tau_{wall}$ are the liquid and projected wall |
292 |
values of the decay constants, $z_{wall}$ is the location of the interface, |
293 |
and $d$ is the displacement from the interface at which these deviations |
294 |
occur. The values for $d_{pyramidal}$ and $d_{secondary prismatic}$ were |
295 |
determined |
296 |
for each of the decay constants, and then averaged for better statistics |
297 |
($\tau_{middle}$ was ommitted for secondary prismatic). For the pyramidal |
298 |
system, |
299 |
$d_{pyramidal}$ was found to be 2.7 \AA\ for both the control and the sheared |
300 |
system. We found $d_{secondary prismatic}$ to be slightly larger than |
301 |
$d_{pyramidal}$ for both the control and with an applied shear, with |
302 |
displacements of $4$ \AA\ for the control system and $3$ \AA\ for the |
303 |
experiment with the imposed momentum flux. These values are consistent with |
304 |
those found for the basal ($d_{basal}\approx2.9$ \AA\ ) and prismatic |
305 |
($d_{prismatic}\approx3.5$ \AA\ ) systems. |
306 |
|
307 |
\subsection{Coefficient of friction of the interfaces} |
308 |
While investigating the kinetic coefficient of friction, there was found |
309 |
to be a dependence for $\mu_k$ |
310 |
on the temperature of the liquid water in the system. We believe this |
311 |
dependence |
312 |
arrises from the sharp discontinuity of the viscosity for the SPC/E model |
313 |
at temperatures approaching 200 K\cite{kuang12}. Due to this, we propose |
314 |
a weighting to the interfacial friction coefficient, $\kappa$ by the |
315 |
shear viscosity of the fluid at 225 K. The interfacial friction coefficient |
316 |
relates the shear stress with the relative velocity of the fluid normal to the |
317 |
interface: |
318 |
\begin{equation}\label{Shenyu-13} |
319 |
j_{z}(p_{x}) = \kappa[v_{x}(fluid)-v_{x}(solid)] |
320 |
\end{equation} |
321 |
where $j_{z}(p_{x})$ is the applied momentum flux (shear stress) across $z$ |
322 |
in the |
323 |
$x$-dimension, and $v_{x}$(fluid) and $v_{x}$(solid) are the velocities |
324 |
directly adjacent to the interface. The shear viscosity, $\eta(T)$, of the |
325 |
fluid can be determined under a linear response of the momentum |
326 |
gradient to the applied shear stress by |
327 |
\begin{equation}\label{Shenyu-11} |
328 |
j_{z}(p_{x}) = \eta(T) \frac{\partial v_{x}}{\partial z}. |
329 |
\end{equation} |
330 |
Using eqs \eqref{Shenyu-13} and \eqref{Shenyu-11}, we can find the following |
331 |
expression for $\kappa$, |
332 |
\begin{equation}\label{kappa-1} |
333 |
\kappa = \eta(T) \frac{\partial v_{x}}{\partial z}\frac{1}{[v_{x}(fluid)-v_{x}(solid)]}. |
334 |
\end{equation} |
335 |
Here is where we will introduce the weighting term of $\eta(225)/\eta(T)$ |
336 |
giving us |
337 |
\begin{equation}\label{kappa-2} |
338 |
\kappa = \frac{\eta(225)}{[v_{x}(fluid)-v_{x}(solid)]}\frac{\partial v_{x}}{\partial z}. |
339 |
\end{equation} |
340 |
|
341 |
To obtain the value of $\eta(225)$ for the SPC/E model, a $31.09 \times 29.38 |
342 |
\times 124.39$ \AA\ box with 3744 SPC/E liquid water molecules was |
343 |
equilibrated to 225K, |
344 |
and 5 unique shearing experiments were performed. Each experiment was |
345 |
conducted in the NVE and were 5 ns in |
346 |
length. The VSS were attempted every timestep, which was set to 2 fs. |
347 |
For our SPC/E systems, we found $\eta(225)$ to be 0.0148 $\pm$ 0.0007 Pa s, |
348 |
roughly ten times larger than the value found for 280 K SPC/E bulk water by |
349 |
Kuang\cite{kuang12}. |
350 |
|
351 |
The interfacial friction coefficient, $\kappa$, can equivalently be expressed |
352 |
as the ratio of the viscosity of the fluid to the slip length, $\delta$, which |
353 |
is an indication of how 'slippery' the interface is. |
354 |
\begin{equation}\label{kappa-3} |
355 |
\kappa = \frac{\eta}{\delta} |
356 |
\end{equation} |
357 |
In each of the systems, the interfacial temperature was kept fixed to 225K, |
358 |
which ensured the viscosity of the fluid at the |
359 |
interace was approximately the same. Thus, any significant variation in |
360 |
$\kappa$ between |
361 |
the systems indicates differences in the 'slipperiness' of the interfaces. |
362 |
As each of the ice systems are sheared relative to liquid water, the |
363 |
'slipperiness' of the interface can be taken as an indication of how |
364 |
hydrophobic or hydrophilic the interface is. The calculated $\kappa$ values |
365 |
found for the four crystal facets of Ice-I$_\mathrm{h}$ investigated are shown |
366 |
in Table \ref{tab:kapa}. The basal and pyramidal facets were found to have |
367 |
similar values of $\kappa \approx$ 0.0006 |
368 |
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}), while values of |
369 |
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1}) |
370 |
were found for the prismatic and secondary prismatic systems. |
371 |
These results indicate that the basal and pyramidal facets are |
372 |
more hydrophilic than the prismatic and secondary prismatic facets. |
373 |
%This indicates something about the similarity between the two facets that |
374 |
%share similar values... |
375 |
%Maybe find values for kappa for other materials to compare against? |
376 |
|
377 |
\begin{table}[h] |
378 |
\centering |
379 |
\caption{$\kappa$ values for the basal, prismatic, pyramidal, and secondary \ |
380 |
prismatic facets of Ice-I$_\mathrm{h}$} |
381 |
\label{tab:kappa} |
382 |
\begin{tabular}{|ccc|} \hline |
383 |
& \multicolumn{2}{c|}{$\kappa_{Drag direction}$ (amu \AA\textsuperscript{-2} fs\textsuperscript{-1})} \\ |
384 |
Interface & $\kappa_{x}$ & $\kappa_{y}$ \\ \hline |
385 |
basal & $0.00059 \pm 0.00003$ & $0.00065 \pm 0.00008$ \\ |
386 |
prismatic & $0.00030 \pm 0.00002$ & $0.00030 \pm 0.00001$ \\ |
387 |
pyramidal & $0.00058 \pm 0.00004$ & $0.00061 \pm 0.00005$ \\ |
388 |
secondary prismatic & $0.00035 \pm 0.00001$ & $0.00033 \pm 0.00002$ \\ \hline |
389 |
\end{tabular} |
390 |
\end{table} |
391 |
|
392 |
|
393 |
|
394 |
|
395 |
%\begin{table}[h] |
396 |
%\centering |
397 |
%\caption{Solid-liquid friction coefficients (measured in amu~fs\textsuperscript\ |
398 |
%{-1}). \\ |
399 |
%\textsuperscript{a} See ref. \onlinecite{Louden13}. } |
400 |
%\label{tab:lambda} |
401 |
%\begin{tabular}{|ccc|} \hline |
402 |
% & \multicolumn{2}{c|}{Drag direction} \\ |
403 |
% Interface & $x$ & $y$ \\ \hline |
404 |
% basal\textsuperscript{a} & $0.08 \pm 0.02$ & $0.09 \pm 0.03$ \\ |
405 |
% prismatic (T = 225)\textsuperscript{a} & $0.037 \pm 0.008$ & $0.04 \pm 0.01$ \\ |
406 |
% prismatic (T = 230) & $0.10 \pm 0.01$ & $0.070 \pm 0.006$\\ |
407 |
% pyramidal & $0.13 \pm 0.03$ & $0.14 \pm 0.03$ \\ |
408 |
% secondary prismatic & $0.13 \pm 0.02$ & $0.12 \pm 0.03$ \\ \hline |
409 |
%\end{tabular} |
410 |
%\end{table} |
411 |
|
412 |
|
413 |
\begin{figure} |
414 |
\includegraphics[width=\linewidth]{Pyr-orient} |
415 |
\caption{\label{fig:PyrOrient} The three decay constants of the |
416 |
orientational time correlation function, $C_2(t)$, for water as a function |
417 |
of distance from the center of the ice slab. The vertical dashed line |
418 |
indicates the edge of the pyramidal ice slab determined by the local order |
419 |
tetrahedral parameter. The control (black circles) and sheared (red squares) |
420 |
experiments were fit by a shifted exponential decay (Eq. 9\cite{Louden13}) |
421 |
shown by the black and red lines respectively. The upper two panels show that |
422 |
translational and hydrogen bond making and breaking events slow down |
423 |
through the interface while approaching the ice slab. The bottom most panel |
424 |
shows the librational motion of the water molecules speeding up approaching |
425 |
the ice block due to the confined region of space allowed for the molecules |
426 |
to move in.} |
427 |
\end{figure} |
428 |
|
429 |
\begin{figure} |
430 |
\includegraphics[width=\linewidth]{SP-orient-less} |
431 |
\caption{\label{fig:SPorient} Decay constants for $C_2(t)$ at the secondary |
432 |
prismatic face. Panel descriptions match those in \ref{fig:PyrOrient}.} |
433 |
\end{figure} |
434 |
|
435 |
|
436 |
|
437 |
\section{Conclusion} |
438 |
We present the results of molecular dynamics simulations of the pyrmaidal |
439 |
and secondary prismatic facets of an SPC/E model of the |
440 |
Ice-I$_\mathrm{h}$/water interface. The ice was sheared through the liquid |
441 |
water while being exposed to a thermal gradient to maintain a stable |
442 |
interface by using the minimally perturbing VSS RNEMD method. In agreement |
443 |
with our previous findings for the basal and prismatic facets, the interfacial |
444 |
width was found to be independent of shear rate as measured by the local |
445 |
tetrahedral order parameter. This width was found to be |
446 |
3.2~$\pm$ 0.2~\AA\ for both the pyramidal and the secondary prismatic systems. |
447 |
These values are in good agreement with our previously calculated interfacial |
448 |
widths for the basal (3.2~$\pm$ 0.4~\AA\ ) and prismatic (3.6~$\pm$ 0.2~\AA\ ) |
449 |
systems. |
450 |
|
451 |
Orientational dynamics of the Ice-I$_\mathrm{h}$/water interfaces were studied |
452 |
by calculation of the orientational time correlation function at varying |
453 |
displacements normal to the interface. The decays were fit |
454 |
to a tri-exponential decay, where the three decay constants correspond to |
455 |
the librational motion of the molecules driven by the restoring forces of |
456 |
existing hydrogen bonds ($\tau_{short}$ $\mathcal{O}$(10 fs)), jumps between |
457 |
two different hydrogen bonds ($\tau_{middle}$ $\mathcal{O}$(1 ps)), and |
458 |
translational motion of the molecules ($\tau_{long}$ $\mathcal{O}$(100 ps)). |
459 |
$\tau_{short}$ was found to decrease approaching the interface due to the |
460 |
constrained motion of the molecules as the local environment becomes more |
461 |
ice-like. Conversely, the two longer-time decay constants were found to |
462 |
increase at small displacements from the interface. As seen in our previous |
463 |
work on the basal and prismatic facets, there appears to be a dynamic |
464 |
interface width at which deviations from the bulk liquid values occur. |
465 |
We had previously found $d_{basal}$ and $d_{prismatic}$ to be approximately |
466 |
2.8~\AA\ and 3.5~\AA. We found good agreement of these values for the |
467 |
pyramidal and secondary prismatic systems with $d_{pyramidal}$ and |
468 |
$d_{secondary prismatic}$ to be 2.7~\AA\ and 3~\AA. For all four of the |
469 |
facets, no apparent dependence of the dynamic width on the shear rate was |
470 |
found. |
471 |
|
472 |
%Paragraph summarizing the Kappa values |
473 |
The interfacial friction coefficient, $\kappa$, was determined for each facet |
474 |
interface. We were able to reach an expression for $\kappa$ as a function of |
475 |
the velocity profile of the system which is scaled by the viscosity of the liquid |
476 |
at 225 K. In doing so, we have obtained an expression for $\kappa$ which is |
477 |
independent of temperature differences of the liquid water at far displacements |
478 |
from the interface. We found the basal and pyramidal facets to have |
479 |
similar $\kappa$ values, of $\kappa \approx$ 0.0006 |
480 |
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}). However, the |
481 |
prismatic and secondary prismatic facets were found to have $\kappa$ values of |
482 |
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1}). |
483 |
As these ice facets are being sheared relative to liquid water, with the |
484 |
structural and dynamic width of all four |
485 |
interfaces being approximately the same, the difference in the coefficient of |
486 |
friction indicates the hydrophilicity of the crystal facets are not |
487 |
equivalent. Namely, that the basal and pyramidal facets of Ice-I$_\mathrm{h}$ |
488 |
are more hydrophilic than the prismatic and secondary prismatic facets. |
489 |
|
490 |
|
491 |
\begin{acknowledgments} |
492 |
Support for this project was provided by the National |
493 |
Science Foundation under grant CHE-1362211. Computational time was |
494 |
provided by the Center for Research Computing (CRC) at the |
495 |
University of Notre Dame. |
496 |
\end{acknowledgments} |
497 |
|
498 |
\newpage |
499 |
|
500 |
\bibliography{iceWater} |
501 |
|
502 |
\end{document} |