trunk/iceWater/iceWater.tex

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\title{Solid-liquid friction at ice-I$_\mathrm{h}$ / water interfaces}

\author{P. B. Louden}
\author{J. Daniel Gezelter}
\email{gezelter@nd.edu}
\affiliation[University of Notre Dame]{251 Nieuwland Science Hall\\
  Department of Chemistry and Biochemistry\\ University of Notre
  Dame\\ Notre Dame, Indiana 46556}

\keywords{}

\begin{document}

\begin{abstract}
  We have investigated the structural and dynamic properties of the
  basal and prismatic facets of an ice I$_\mathrm{h}$ / water
  interface when the solid phase is being drawn through the liquid
  (i.e. sheared relative to the fluid phase). To impose the shear, we
  utilized a velocity-shearing and scaling (VSS) approach to reverse
  non-equilibrium molecular dynamics (RNEMD).  This method can create
  simultaneous temperature and velocity gradients and allow the
  measurement of transport properties at interfaces.  The interfacial
  width was found to be independent of relative velocity of the ice
  and liquid layers over a wide range of shear rates.  Decays of
  molecular orientational time correlation functions for gave very
  similar estimates for the width of the interfaces, although the
  short- and longer-time decay components of the orientational
  correlation functions behave differently closer to the interface.
  Although both facets of ice are in ``stick'' boundary conditions in
  liquid water, the solid-liquid friction coefficient was found to be
  different for the basal and prismatic facets of ice.
\end{abstract}

\newpage

\section{Introduction}
%-----Outline of Intro---------------
% in general, ice/water interface is important b/c ....
% here are some people who have worked on ice/water, trying to understand the processes above ....
% with the recent development of VSS-RNEMD, we can now look at the shearing problem
% talk about what we will present in this paper
% -------End Intro------------------

%Gay02: cites many other ice/water papers, make sure to cite them.

Understanding the ice/water interface is essential for explaining
complex processes such as nucleation and crystal
growth,\cite{Han92,Granasy95,Vanfleet95} crystal
melting,\cite{Weber83,Han92,Sakai96,Sakai96B} and some fascinating
biological processes, such as the behavior of the antifreeze proteins
found in winter flounder,\cite{Wierzbicki07, Chapsky97} and certain
terrestrial arthropods.\cite{Duman:2001qy,Meister29012013} There has
been significant progress on understanding the structure and dynamics
of quiescent ice/water interfaces utilizing both theory and
experiment.  Haymet \emph{et al.} have done extensive work on ice I$_\mathrm{h}$,
including characterizing and determining the width of the ice/water
interface for the SPC, SPC/E, CF1, and TIP4P models for
water.\cite{Karim87,Karim88,Karim90,Hayward01,Bryk02,Hayward02,Gay02}
More recently, Haymet \emph{et al.} have investigated the effects
cations and anions have on crystal
nucleation\cite{Bryk04,Smith05,Wilson08,Wilson10}. Nada \emph{et al.}
have also studied ice/water
interfaces,\cite{Nada95,Nada00,Nada03,Nada12} and have found that the
differential growth rates of the facets of ice I$_\mathrm{h}$ are due to the the
reordering of the hydrogen bonding network\cite{Nada05}.

The movement of liquid water over the facets of ice has been less
thoroughly studied than the quiescent surfaces. This process is
potentially important in understanding transport of large blocks of
ice in water (which has important implications in the earth sciences),
as well as the relative motion of crystal-crystal interfaces that have
been separated by nanometer-scale fluid domains.  In addition to
understanding both the structure and thickness of the interfacial
regions, it is important to understand the molecular origin of
friction, drag, and other changes in dynamical properties of the
liquid in the regions close to the surface that are altered by the
presence of a shearing of the bulk fluid relative to the solid phase.

In this work, we apply a recently-developed velocity shearing and
scaling approach to reverse non-equilibrium molecular dynamics
(VSS-RNEMD). This method makes it possible to calculate transport
properties like the interfacial thermal conductance across
heterogeneous interfaces,\cite{Kuang12} and can create simultaneous
temperature and velocity gradients and allow the measurement of
friction and thermal transport properties at interfaces.  This has
allowed us to investigate the width of the ice/water interface as the
ice is sheared through the liquid, while simultaneously imposing a
weak thermal gradient to prevent frictional heating of the interface.
In the sections that follow, we discuss the methodology for creating
and simulating ice/water interfaces under shear and provide results
from both structural and dynamical correlation functions.  We also
show that the solid-liquid interfacial friction coefficient depends
sensitively on the details of the surface morphology.

\section{Methodology}

\subsection{Stable ice I$_\mathrm{h}$ / water interfaces under shear}

The structure of ice I$_\mathrm{h}$ is well understood; it
crystallizes in a hexagonal space group P$6_3/mmc$, and the hexagonal
crystals of ice have two faces that are commonly exposed, the basal
face $\{0~0~0~1\}$, which forms the top and bottom of each hexagonal
plate, and the prismatic $\{1~0~\bar{1}~0\}$ face which forms the
sides of the plate. Other less-common, but still important, faces of
ice I$_\mathrm{h}$ are the secondary prism, $\{1~1~\bar{2}~0\}$, and
pyramidal, $\{2~0~\bar{2}~1\}$, faces.  Ice I$_\mathrm{h}$ is normally
proton disordered in bulk crystals, although the surfaces probably
have a preference for proton ordering along strips of dangling H-atoms
and Oxygen lone pairs.\cite{Buch:2008fk}

\begin{wraptable}{r}{3.5in}
  \caption{Mapping between the Miller indices of four facets of ice in
    the $P6_3/mmc$ crystal system to the orthorhombic $P2_12_12_1$
    system in reference \protect\cite{Hirsch04}}
\label{tab:equiv}
\begin{tabular}{|ccc|} \hline
 & hexagonal & orthorhombic \\
 & ($P6_3/mmc$) & ($P2_12_12_1$) \\
 crystal face  & Miller indices & equivalent \\ \hline
basal & $\{0~0~0~1\}$ & $\{0~0~1\}$ \\
prism & $\{1~0~\bar{1}~0\}$ & $\{1~0~0\}$ \\
secondary prism & $\{1~1~\bar{2}~0\}$ & $\{1~3~0\}$ \\
pyramidal & $\{2~0~\bar{2}~1\}$ & $\{2~0~1\}$ \\ \hline
\end{tabular}
\end{wraptable}

For small simulated ice interfaces, it is useful to work with
proton-ordered, but zero-dipole crystal that exposes these strips of
dangling H-atoms and lone pairs.  When placing another material in
contact with one of the ice crystalline planes, it is also quite
useful to have an orthorhombic (rectangular) box. Recent work by
Hirsch and Ojam\"{a}e describes a number of alternative crystal
systems for proton-ordered bulk ice I$_\mathrm{h}$ using orthorhombic
cells.\cite{Hirsch04}

In this work, we are using Hirsch and Ojam\"{a}e's structure 6 which
is an orthorhombic cell ($P2_12_12_1$) that produces a proton-ordered
version of ice I$_\mathrm{h}$.  Table \ref{tab:equiv} contains a
mapping between the Miller indices of common ice facets in the
P$6_3/mmc$ crystal system and those in the Hirsch and Ojam\"{a}e
$P2_12_12_1$ system.

Structure 6 from the Hirsch and Ojam\"{a}e paper has lattice
parameters $a = 4.49225$, $b = 7.78080$, $c = 7.33581$ and two water
molecules whose atoms reside at fractional coordinates given in table
\ref{tab:p212121}.  To construct the basal and prismatic interfaces,
these crystallographic coordinates were used to construct an
orthorhombic unit cell which was then replicated in all three
dimensions yielding a proton-ordered block of ice I$_{h}$. To expose
the desired face, the orthorhombic representation was then cut along
the ($001$) or ($100$) planes for the basal and prismatic faces
respectively.  The resulting block was rotated so that the exposed
faces were aligned with the $z$-axis normal to the exposed face.  The
block was then cut along two perpendicular directions in a way that
allowed for perfect periodic replication in the $x$ and $y$ axes,
creating a slab with either the basal or prismatic faces exposed along
the $z$ axis.  The slab was then replicated in the $x$ and $y$
dimensions until a desired sample size was obtained.

\begin{wraptable}{r}{2.85in}
  \caption{Fractional coordinates for water in the orthorhombic
    $P2_12_12_1$ system for ice I$_\mathrm{h}$ in reference
    \protect\cite{Hirsch04}}
\label{tab:p212121}
\begin{tabular}{|cccc|}  \hline
atom type & x & y & z \\ \hline
 O & 0.75 & 0.1667 & 0.4375 \\
 H & 0.5735 & 0.2202 & 0.4836 \\
 H & 0.7420 & 0.0517 & 0.4836 \\
 O & 0.25 & 0.6667 & 0.4375 \\
 H & 0.2580 & 0.6693 & 0.3071 \\
 H & 0.4265 & 0.7255 & 0.4756 \\ \hline
\end{tabular}
\end{wraptable}

Our ice / water interfaces were created using a box of liquid water
that had the same dimensions (in $x$ and $y$) as the ice block.
Although the experimental solid/liquid coexistence temperature under
atmospheric pressure is close to 273K, Haymet \emph{et al.} have done
extensive work on characterizing the ice/water interface, and find
that the coexistence temperature for simulated water is often quite a
bit different.\cite{Karim88,Karim90,Hayward01,Bryk02,Hayward02} They
have found that for the SPC/E water model,\cite{Berendsen87} which is
also used in this study, the ice/water interface is most stable at
225$\pm$5K.\cite{Bryk02} This liquid box was therefore equilibrated at
225 K and 1 atm of pressure in the NPAT ensemble (with the $z$ axis
allowed to fluctuate to equilibrate to the correct pressure).  The
liquid and solid systems were combined by carving out any water
molecule from the liquid simulation cell that was within 3 \AA\ of any
atom in the ice slab.

Molecular translation and orientational restraints were applied in the
early stages of equilibration to prevent melting of the ice slab.
These restraints were removed during NVT equilibration, well before
data collection was carried out.  

\subsection{Shearing ice / water interfaces without bulk melting}

As a solid is dragged through a liquid, there is frictional heating
that will act to melt the interface.  To study the behavior of the
interface under a shear stress without causing the interface to melt,
it is necessary to apply a weak thermal gradient in combination with
the momentum gradient.  This can be accomplished using the velocity
shearing and scaling (VSS) variant of reverse non-equilibrium
molecular dynamics (RNEMD), which utilizes a series of simultaneous
velocity exchanges between two regions within the simulation
cell.\cite{Kuang12} One of these regions is centered within the ice
slab, while the other is centrally located in the liquid
region. VSS-RNEMD provides a set of conservation constraints for
creating either a momentum flux or a thermal flux (or both
simultaneously) between the two slabs.  Satisfying the constraint
equations ensures that the new configurations are sampled from the
same NVE ensemble as before the VSS move.

The VSS moves are applied periodically to scale and shift the particle
velocities ($\mathbf{v}_i$ and $\mathbf{v}_j$) in two slabs ($H$ and
$C$) which are separated by half of the simulation box,
\begin{displaymath}
\begin{array}{rclcl}

 & \underline{\mathrm{shearing}} & &
 \underline{~~~~~~~~~~~~\mathrm{scaling}~~~~~~~~~~~~} \\
\mathbf{v}_i \leftarrow & 
  \mathbf{a}_c\ & + & c\cdot\left(\mathbf{v}_i - \langle\mathbf{v}_c
  \rangle\right)  +  \langle\mathbf{v}_c\rangle \\
\mathbf{v}_j \leftarrow & 
  \mathbf{a}_h & + & h\cdot\left(\mathbf{v}_j - \langle\mathbf{v}_h
    \rangle\right) + \langle\mathbf{v}_h\rangle .

\end{array}
\end{displaymath}
Here $\langle\mathbf{v}_c\rangle$ and $\langle\mathbf{v}_h\rangle$ are
the center of mass velocities in the $C$ and $H$ slabs, respectively.
Within the two slabs, particles receive incremental changes or a
``shear'' to their velocities.  The amount of shear is governed by the
imposed momentum flux, $\mathbf{j}_z(\mathbf{p})$
\begin{eqnarray}
\mathbf{a}_c & = & - \mathbf{j}_z(\mathbf{p}) \Delta t / M_c \label{vss1}\\
\mathbf{a}_h & = & + \mathbf{j}_z(\mathbf{p}) \Delta t / M_h \label{vss2}
\end{eqnarray}
where $M_{\{c,h\}}$ is the total mass of particles within each of the
slabs and $\Delta t$ is the interval between two separate operations.

To simultaneously impose a thermal flux ($J_z$) between the slabs we
use energy conservation constraints,
\begin{eqnarray}
K_c - J_z\Delta t & = & c^2 (K_c - \frac{1}{2}M_c \langle\mathbf{v}_c
\rangle^2) + \frac{1}{2}M_c (\langle \mathbf{v}_c \rangle + \mathbf{a}_c)^2 \label{vss3}\\
K_h + J_z\Delta t & = & h^2 (K_h - \frac{1}{2}M_h \langle\mathbf{v}_h
\rangle^2) + \frac{1}{2}M_h (\langle \mathbf{v}_h \rangle +
\mathbf{a}_h)^2 \label{vss4}.
\label{constraint}
\end{eqnarray}
Simultaneous solution of these quadratic formulae for the scaling
coefficients, $c$ and $h$, will ensure that the simulation samples from
the original microcanonical (NVE) ensemble.  Here $K_{\{c,h\}}$ is the
instantaneous translational kinetic energy of each slab.  At each time
interval, it is a simple matter to solve for $c$, $h$, $\mathbf{a}_c$,
and $\mathbf{a}_h$, subject to the imposed momentum flux,
$j_z(\mathbf{p})$, and thermal flux, $J_z$, values.  Since the VSS
operations do not change the kinetic energy due to orientational
degrees of freedom or the potential energy of a system, configurations
after the VSS move have exactly the same energy (and linear
momentum) as before the move.

As the simulation progresses, the VSS moves are performed on a regular
basis, and the system develops a thermal and/or velocity gradient in
response to the applied flux.  In a bulk material, it is quite simple
to use the slope of the temperature or velocity gradients to obtain
either the thermal conductivity or shear viscosity.

The VSS-RNEMD approach is versatile in that it may be used to
implement thermal and shear transport simultaneously.  Perturbations
of velocities away from the ideal Maxwell-Boltzmann distributions are
minimal, as is thermal anisotropy.  This ability to generate
simultaneous thermal and shear fluxes has been previously utilized to
map out the shear viscosity of SPC/E water over a wide range of
temperatures (90~K) with a single 1 ns simulation.\cite{Kuang12}

For this work, we are using the VSS-RNEMD method primarily to generate
a shear between the ice slab and the liquid phase, while using a weak
thermal gradient to maintain the interface at the 225K target
value. This ensures minimal melting of the bulk ice phase and allows
us to control the exact temperature of the interface.

\subsection{Computational Details}
All simulations were performed using OpenMD,\cite{OOPSE,openmd} with a
time step of 2 fs and periodic boundary conditions in all three
dimensions.  Electrostatics were handled using the damped-shifted
force real-space electrostatic kernel.\cite{Ewald} The systems were
divided into 100 bins along the $z$-axis for the VSS-RNEMD moves,
which were attempted every 50 fs. 

The interfaces were equilibrated for a total of 10 ns at equilibrium
conditions before being exposed to either a shear or thermal gradient.
This consisted of 5 ns under a constant temperature (NVT) integrator
set to 225K followed by 5 ns under a microcanonical integrator.  Weak
thermal gradients were allowed to develop using the VSS-RNEMD (NVE)
integrator using a a small thermal flux ($-2.0\times 10^{-6}$
kcal/mol/\AA$^2$/fs) for a duration of 5 ns to allow the gradient to
stabilize.  The resulting temperature gradient was $\approx$ 10K over
the entire 100 \AA\  box length, which was sufficient to keep the
temperature at the interface within $\pm 1$ K of the 225K target.

Velocity gradients were then imposed using the VSS-RNEMD (NVE)
integrator with a range of momentum fluxes.  These gradients were
allowed to stabilize for 1 ns before data collection began. Once
established, four successive 0.5 ns runs were performed for each shear
rate.  During these simulations, snapshots of the system were taken
every 1 ps, and statistics on the structure and dynamics in each bin
were accumulated throughout the simulations.

\section{Results and discussion}

\subsection{Interfacial width}
Any order parameter or time correlation function that changes as one
crosses an interface from a bulk liquid to a solid can be used to
measure the width of the interface.  In previous work on the ice/water
interface, Haymet {\it et al.}\cite{} have utilized structural
features (including the density) as well as dynamic properties
(including the diffusion constant) to estimate the width of the
interfaces for a number of facets of the ice crystals.  Because
VSS-RNEMD imposes a lateral flow, parameters that depend on
translational motion of the molecules (e.g. diffusion) may be
artifically skewed by the RNEMD moves.  A structural parameter is not
influenced by the RNEMD perturbations to the same degree. Here, we
have used the local tetraherdal order parameter as described by
Kumar\cite{Kumar09} and Errington\cite{Errington01} as our principal
measure of the interfacial width.

The local tetrahedral order parameter, $q(z)$, is given by
\begin{equation}
q(z) = \int_0^L \sum_{k=1}^{N} \Bigg(1 -\frac{3}{8}\sum_{i=1}^{3}
\sum_{j=i+1}^{4} \bigg(\cos\psi_{ikj}+\frac{1}{3}\bigg)^2\Bigg)
\delta(z_{k}-z)\mathrm{d}z \Bigg/ N_z
\label{eq:qz}
\end{equation}
where $\psi_{ikj}$ is the angle formed between the oxygen site on
central molecule $k$, and the oxygen sites on two of the four closest
molecules, $i$ and $j$.  Molecules $i$ and $j$ are further restricted
to lie within the first solvation shell of molecule $k$.  $N_z = \int
\delta(z_k - z) \mathrm{d}z$ is a normalization factor to account for
the varying population of molecules within each finite-width bin.  The
local tetrahedral order parameter has a range of $(0,1)$, where the
larger values of $q$ indicate a larger degree of tetrahedral ordering
of the local environment.  In perfect ice I$_\mathrm{h}$ structures,
the parameter can approach 1 at low temperatures, while in liquid
water, the ordering is significantly less tetrahedral, and values of
$q(z) \approx 0.75$ are more common.

To estimate the interfacial width, the system was divided into 100
bins along the $z$-dimension, and a cutoff radius for the first
solvation shell was set to 3.41 \AA\ .  The $q_{z}$ function was
time-averaged to give yield a tetrahedrality profile of the
system. The profile was then fit to a hyperbolic tangent that smoothly
links the liquid and solid states,
\begin{equation}\label{tet_fit}
q(z) \approx
q_{liq}+\frac{q_{ice}-q_{liq}}{2}\left[\tanh\left(\frac{z-l}{w}\right)-\tanh\left(\frac{z-r}{w}\right)\right]+\beta\left|z-
\frac{r+l}{2}\right|.
\end{equation}
Here $q_{liq}$ and $q_{ice}$ are the local tetrahedral order parameter
for the bulk liquid and ice domains, respectively, $w$ is the width of
the interface.  $l$ and $r$ are the midpoints of the left and right
interfaces, respectively.  The last term in equation \eqref{tet_fit}
accounts for the influence that the weak thermal gradient has on the
tetrahedrality profile in the liquid region.  To estimate the
10\%-90\% widths commonly used in previous studies,\cite{} it is a
simple matter to scale the widths obtained from the hyberbolic tangent
fits to obtain $w_{10-90} = 2.9 w$.\cite{}

In Figures \ref{fig:bComic} and \ref{fig:pComic} we see the
$z$-coordinate profiles for tetrahedrality, temperature, and the
$x$-component of the velocity for the basal and prismatic interfaces.
The lower panels show the $q(z)$ (black circles) along with the
hyperbolic tangent fits (red lines). In the liquid region, the local
tetrahedral order parameter, $q(z) \approx 0.75$ while in the
crystalline region, $q(z) \approx 0.94$, indicating a more tetrahedral
environment.  The vertical dotted lines denote the midpoint of the
interfaces ($r$ and $l$ in equation \eqref{tet_fit}). The weak thermal
gradient applied to the systems in order to keep the interface at
225$\pm$5K, can be seen in middle panels.  The tranverse velocity
profile is shown in the upper panels.  It is clear from the upper
panels that water molecules in close proximity to the surface (i.e.
within 10 \AA\ to 15 \AA\ of the interfaces) have transverse
velocities quite close to the velocities within the ice block.  There
is no velocity discontinuity at the interface, which indicates that
the shearing of ice/water interfaces occurs in the ``stick'' or
no-slip boundary conditions.

\begin{figure}
\includegraphics[width=\linewidth]{bComicStrip.pdf}
\caption{\label{fig:bComic} The basal interfaces.  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]{pComicStrip.pdf}
\caption{\label{fig:pComic} The prismatic interfaces.  Panel
  descriptions match those in figure \ref{fig:bComic}}
\end{figure}

From the fits using equation \eqref{tet_fit}, we find the interfacial
width for the basal and prismatic systems to be 3.2$\pm$0.4 \AA\ and
3.6$\pm$0.2 \AA\ , respectively, with no applied momentum flux. 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, we found no appreciable change in
the interface width. The fit values for the interfacial width ($w$)
over all shear rates contained the values reported above within their
error bars.

\subsubsection{Orientational Time Correlation Function}
The orientational time correlation function,
\begin{equation}\label{C(t)1}
  C_{2}(t)=\langle P_{2}(\mathbf{u}(0) \cdot \mathbf{u}(t)) \rangle,
\end{equation}
gives insight into the local dynamic environment around the water
molecules.  The rate at which the function decays provides information
about hindered motions and the timescales for relaxation.  In
eq. \eqref{C(t)1}, $P_{2}$ is the second-order Legendre polynomial,
the vector $\mathbf{u}$ is often taken as HOH bisector, although
slightly different behavior can be observed when $\mathbf{u}$ is the
vector along one of the OH bonds.  The angle brackets denote an
ensemble average over all water molecules in a given spatial region.

To investigate the dynamic behavior of water at the ice interfaces, we
have computed $C_{2}(z,t)$ for molecules that are present within a
particular slab along the $z$- axis at the initial time.  The change
in the decay behavior as a function of the $z$ coordinate is another
measure of the change of how the local environment changes across the
ice/water interface.  To compute these correlation functions, each of
the 0.5 ns simulations was followed by a shorter 200 ps microcanonical
(NVE) simulation in which the positions and orientations of every
molecule in the system were recorded every 0.1 ps. The systems were
then divided into 30 bins and $C_2(t)$ was evaluated for each bin.

In simulations of water at biological interfaces, Furse {\em et al.}
fit $C_2(t)$ functions for water with triexponential
functions,\cite{Furse08} where the three components of the decay
correspond to a fast (<200 fs) reorientational piece driven by the
restoring forces of existing hydrogen bonds, a middle (on the order of
several ps) piece describing the large angle jumps that occur during
the breaking and formation of new hydrogen bonds,and a slow (on the
order of tens of ps) contribution describing the translational motion
of the molecules.  The model for orientational decay presented
recently by Laage and Hynes {\em et al.}\cite{Laage08,Laage11} also
includes three similar decay constants, although two of the time
constants are linked, and the resulting decay curve has two parameters
governing the dynamics of decay. 

In our ice/water interfaces, we are at substantially lower
temperatures, and the water molecules are further perturbed by the
presence of the ice phase nearby.  We have obtained the most
reasonable fits using triexponential functions with three distinct
time domains, as well as a constant piece that accounts for the water
stuck in the ice phase that does not experience any long-time
orientational decay,
\begin{equation}
C_{2}(t) \approx a e^{-t/\tau_\mathrm{short}} + b e^{-t/\tau_\mathrm{middle}} + c
e^{-t/\tau_\mathrm{long}} + (1-a-b-c)
\end{equation}
Average values for the three decay constants (and error estimates)
were obtained for each bin. In figures \ref{fig:basal_Tau_comic_strip}
and \ref{fig:prismatic_Tau_comic_strip}, the three orientational decay
times are shown as a function of distance from the center of the ice
slab.

\begin{figure}
\includegraphics[width=\linewidth]{basal_Tau_comic_strip.pdf}
\caption{\label{fig:basal_Tau_comic_strip} The orientational time correlation function for the basal system fit by the triexponential decay $a_{1}e^{-t/\tau_{short}}+a_{2}e^{-t/\tau_{middle}}+a_{3}e^{-t/\tau_{long}}+a_{4}$ where $a_{1}+a_{2}+a_{3}+a_{4}=1$. The verticle dotted line indicates the average midpoint of the interface as determined by the tetrahedrality fit. In (b) and (c) $\tau_{middle}$ and $\tau_{long}$ have a consistent value of about 5.5 ps and 50 ps in the liquid region, and increase in value approaching the interface. $\tau_{middle}$ corresponds to the breaking and making of hydrogen bonds as explained by extended jump model proposed by Laage and Hynes\cite{Laage08,Laage11} and $\tau_{long}$ relates to the translational motion of the molecules. In (a), we see that $\tau_{short}$ has a value of about 71 fs in the liquid region, and decreases in value approaching the interface. This component of the decay corresponds to the reorientational forces of existing hydrogen bonds on the inertial rotational of the molecules. Thus $\tau_{short}$ decreases approaching the interface due to the hindered range of motion of the molecules. }
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{prismatic_Tau_comic_strip.pdf}
\caption{\label{fig:prismatic_Tau_comic_strip} The orientational time correlation function for the prismatic system fit by the triexponential decay $a_{1}e^{-t/\tau_{short}}+a_{2}e^{-t/\tau_{middle}}+a_{3}e^{-t/\tau_{long}}+a_{4}$ where $a_{1}+a_{2}+a_{3}+a_{4}=1$. The verticle dotted line indicates the average midpoint of the interface as determined by the tetrahedrality fit. In (b) and (c) $\tau_{middle}$ and $\tau_{long}$ have a consistent value of about 3.5 ps and 30 ps in the liquid region, and increase in value approaching the interface. $\tau_{middle}$ corresponds to the breaking and making of hydrogen bonds as explained by extended jump model proposed by Laage and Hynes\cite{Laage08,Laage11} and $\tau_{long}$ relates to the translational motion of the molecules. In (a), we see that $\tau_{short}$ has a value of about 73 fs in the liquid region, and decreases in value approaching the interface. This component of the decay corresponds to the reorientational forces of existing hydrogen bonds on the inertial rotational of the molecules. Thus $\tau_{short}$ decreases approaching the interface due to the hindered range of motion of the molecules.}
\end{figure}

Figures \ref{fig:basal_Tau_comic_strip} and \ref{fig:prismatic_Tau_comic_strip} plot the decomposition of the OTCF at varying displacements from the center of the ice for the basal and prismatic systems. We see in (a) $\tau_{short}$, (b) $\tau_{middle}$, and (c) $\tau_{long}$ for the control system (no applied momentum flux)  in black, and a system with a large shear rate in red. The verticle dotted lines at a displacement of about 17 \AA\ and 9 \AA\ denote the midpoints of the interfaces as determined by the hyperbolic tangent fit of the tetrahedrality profile. 

In panels (a), we see at large displacements from the center of the ice $\tau_{short}$ for the basal system has a value of about 71 fs and 72 fs for the prismatic. Decreasing in displacement from about 26 \AA\ to about 19 \AA\ in the basal system, the value of $\tau_{short}$ decreases to about 63 fs. Likewise, $\tau_{short}$ decreases to about 63 fs from roughly 20 \AA\ to 12 \AA\ . This is due to the increasingly constrained motion of the water molecules as we approach the interface. In panels (b), $\tau_{middle}$ at large displacements from the ice has a value of about 5.5 ps and 3 ps for the basal and prismatic systems. We find $\tau_{middle}$ increases in value as we approach the interface in both cases. This component of the decay corresponds to the rearrangement of the hydrogen bonding network, which takes longer as the molecules motion becomes more constrained. In panels (c), $\tau_{long}$ has a value of about 50 ps for the basal system and roughly 30 ps for the prismatic system at large displacements from the interface. Similar to $\tau_{middle}$, $\tau_{long}$ also increases in value as we approach the interface for both systems. It is also apparent from Figures \ref{fig:basal_Tau_comic_strip} and \ref{fig:prismatic_Tau_comic_strip} that shearing the ice water has no effect on the orientational decay time, or on any of its decomposed components.

For each system, there is an apparent approximate value for $\tau_{short}$, $\tau_{middle}$, and $\tau_{long}$ at large displacements from the interface. There also appears to be a single displacement, $d_{basal}$ and $d_{prismatic}$, from the interface at which all three decay times begin to deviate from their bulk liquid values. We found $d_{basal}$ and $d_{prismatic}$ to be roughly 15 \AA\ and 8 \AA\ respectively. These two results indicate that the dynamics of the water molecules within  $d_{basal}$ and $d_{prismatic}$ are being significantly perturbed by the ice and/or the interface, even though the structural width of the interface by analysis of the tetrahedrality profile indicates that bulk liquid structure of water is recovered after about 4 \AA\ from the edge of the ice. 

Beaglehole and Wilson have measured the ice/water interface to have a thickness approximately 10 \AA\ for both the basal and prismatic face of ice by ellipticity measurements \cite{Beaglehole93}. Structurally, we have found the basal and prismatic  interfacial width to be 3.2$\pm$0.4 \AA\ and 3.6$\pm$0.2 \AA\ . However, we have shown through decomposition of the OTCF a much larger interfacial region exists in which the dynamics of the water molecules behave differently than those of the bulk liquid.

\subsection{Coefficient of Friction of the Interface}
As the ice is sheared through the liquid, there will be a friction between the solid and the liquid. Pit has shown how to calculate the coefficient of friction $\lambda$ for a solid-liquid interface for a Newtonian fluid of viscosity $\eta$ and has a slip length of $\delta$. \cite{Pit99}
\begin{equation}\label{Pit}
\lambda=\eta/\delta
\end{equation}
From linear response theory, $\eta$ can be obtained from the imposed momentum flux  and the slope of the velocity about the dimension of the imposed flux.\cite{Kuang12}
\begin{equation}\label{Kuang}
j_{z}(p_{x})=-\eta\frac{\partial v_{x}}{\partial z}
\end{equation}
Solving eq. \eqref{Kuang} for $\eta$ and substituting the result into eq. \eqref{Pit}, we obtain an alternate expression for the coefficient of friction.
\begin{equation}
\lambda=-\frac{j_{z}(p_{x})}{\delta \frac{\partial v_{x}}{\partial z}}
\end{equation}

For our simulations, we obtain $\delta$ from the difference between the structural edge of the ice block determined by the tetrahedrality profile fit, and the intersection of the linear regression of the $v_{x}$ profiles about the $z$-dimension for the ice and liquid. (See Figure \ref{fig:delta_example}) The coefficient of friction for the basal and the prismatic facets were found to be (0.07$\pm$0.01) amu fs\textsuperscript{-1} and (0.032$\pm$0.007) amu fs\textsuperscript{-1}. It is known that the basal and prismatic faces have different surface structures. The basal face is smoother than the prismatic with small alternating valleys and crests, while the prismatic surface has deep corrugating channels. We believe the reason that the prismatic face's coefficient of friction was found to be smaller than the basal's is due to the direction of the shear. The shear of the ice/water was in the same direction of the corrugating channels, allowing water molecules to pass through the channels during the shear. 

 \begin{figure}
\includegraphics[width=\linewidth]{delta_example.pdf}
\caption{\label{fig:delta_example} A schematic of determining the slip length ($\delta$). The slip length is the difference of the structural starting point of the ice and the point of intersection of the linear regressions of the liquid phase velocity profile (red) and of the solid ice velocity profile (black). The dotted line indicates the location of the ice as determined by the tetrahedrality profile.}
\end{figure}


\section{Conclusion}
Here we have simulated the basal and prismatic facets of an SPC/E model of the ice I$_\mathrm{h}$ / 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 apparent independence of the shear rate on the interfacial width, which was found to be 3.2$\pm$0.4 \AA\ and 3.6$\pm$0.2 \AA\ for the basal and prismatic systems. The orientational time correlation function was calculated from varying displacements from the interface. Decomposition by a triexponential decay also showed an apparent independence of the shear rate. The short time decay due to the restoring forces of existing hydrogen bonds decreased at close displacements from the interface, while the middle and long time decays were found to increase. There is also an apparent displacement, $d_{basal}$ and $d_{prismatic}$, from the interface at which these deviations from bulk liquid values occurs. We found $d_{basal}$ and $d_{prismatic}$ to be approximately 15 \AA\ and 8 \AA\ . This implies that the dynamics of water molecules which are structurally equivalent to bulk phase molecules are being perturbed by the presence of the ice and/or the interface. The coefficient of friction of each of the facets was also determined. They were found to be (0.07$\pm$0.01) amu fs\textsuperscript{-1} and (0.032$\pm$0.007) amu fs\textsuperscript{-1} for the basal and prismatic facets respectively. We believe the large difference between the two friction coefficients is due to the direction of the shear and the surface structure of the crystal facets. 

\begin{acknowledgement}
  Support for this project 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.
\end{acknowledgement}

\newpage
\bibstyle{achemso}
\bibliography{iceWater}

\begin{tocentry}
\begin{wrapfigure}{l}{0.5\textwidth}
\begin{center}
\includegraphics[width=\linewidth]{SystemImage.png}
\end{center}
\end{wrapfigure}
An image of our system.
\end{tocentry}

\end{document}

%**************************************************************
%Non-mass weighted slopes (amu*Angstroms^-2 * fs^-1)
% basal: slope=0.090677616, error in slope = 0.003691743
%prismatic: slope = 0.050101506, error in slope  = 0.001348181
%Mass weighted slopes (Angstroms^-2 * fs^-1)
%basal slope = 4.76598E-06, error in slope = 1.94037E-07
%prismatic slope = 3.23131E-06, error in slope = 8.69514E-08
%**************************************************************
Revision:	3948
Committed:	Fri Sep 6 20:29:20 2013 UTC (11 years, 10 months ago) by gezelter
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