trunk/iceWater2/iceWater3.tex

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\begin{document}

\title{Friction at water / ice-I\textsubscript{h} interfaces: Do the
  different facets of ice have different hydrophilicities?}

\author{Patrick B. Louden\affil{1}{Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame,
IN 46556}
\and
J. Daniel Gezelter\affil{1}{}}

\contributor{Submitted to Proceedings of the National Academy of Sciences
of the United States of America}

%%%Newly updated.
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%\significancetext{RJSM and ACAC developed the concept of the study. RJSM conducted the analysis, data interpretation and drafted the manuscript. AGB contributed to the development of the statistical methods, data interpretation and drafting of the manuscript.}

\maketitle

\begin{article}
  \begin{abstract}
    In this paper we present evidence that some of the crystal facets
    of ice-I$_\mathrm{h}$ posess structural features that can halve
    the effective hydrophilicity of the ice/water interface. The
    spreading dynamics of liquid water droplets on ice facets exhibits
    long-time behavior that differs substantially for the prismatic
    $\{1~0~\bar{1}~0\}$ and secondary prism $\{1~1~\bar{2}~0\}$ facets
    when compared with the basal $\{0001\}$ and pyramidal
    $\{2~0~\bar{2}~1\}$ facets.  We also present the results of
    simulations of solid-liquid friction of the same four crystal
    facets being drawn through liquid water. Both simulation
    techniques provide evidence that the two prismatic faces have an
    effective surface area in contact with the liquid water of
    approximately half of the total surface area of the crystal.  The
    ice / water interfacial widths for all four crystal facets are
    similar (using both structural and dynamic measures), and were
    found to be independent of the shear rate.  Additionally,
    decomposition of orientational time correlation functions show
    position-dependence for the short- and longer-time decay
    components close to the interface.
\end{abstract}

\keywords{ice | water | interfaces | hydrophobicity}
\abbreviations{QLL, quasi liquid layer; MD, molecular dynamics; RNEMD,
reverse non-equilibrium molecular dynamics}

\dropcap{S}urfaces can be characterized as hydrophobic or hydrophilic
based on the strength of the interactions with water. Hydrophobic
surfaces do not have strong enough interactions with water to overcome
the internal attraction between molecules in the liquid phase, and the
degree of hydrophilicity of a surface can be described by the extent a
droplet can spread out over the surface. The contact angle formed
between the solid and the liquid depends on the free energies of the
three interfaces involved, and is given by Young's
equation.\cite{Young05}
\begin{equation}\label{young}
\cos\theta = (\gamma_{sv} - \gamma_{sl})/\gamma_{lv} .
\end{equation} 
Here $\gamma_{sv}$, $\gamma_{sl}$, and $\gamma_{lv}$ are the free
energies of the solid/vapor, solid/liquid, and liquid/vapor interfaces
respectively.  Large contact angles, $\theta > 90^{\circ}$, correspond
to hydrophobic surfaces with low wettability, while small contact
angles, $\theta < 90^{\circ}$, correspond to hydrophilic surfaces.
Experimentally, measurements of the contact angle of sessile drops is
often used to quantify the extent of wetting on surfaces with
thermally selective wetting
characteristics.\cite{Tadanaga00,Liu04,Sun04}

Nanometer-scale structural features of a solid surface can influence
the hydrophilicity to a surprising degree.  Small changes in the
heights and widths of nano-pillars can change a surface from
superhydrophobic, $\theta \ge 150^{\circ}$, to hydrophilic, $\theta
\sim 0^{\circ}$.\cite{Koishi09} This is often referred to as the
Cassie-Baxter to Wenzel transition.  Nano-pillared surfaces with
electrically tunable Cassie-Baxter and Wenzel states have also been
observed.\cite{Herbertson06,Dhindsa06,Verplanck07,Ahuja08,Manukyan11}
Luzar and coworkers have modeled these transitions on nano-patterned
surfaces\cite{Daub07,Daub10,Daub11,Ritchie12}, and have found the
change in contact angle is due to the field-induced perturbation of
hydrogen bonding at the liquid/vapor interface.\cite{Daub07}

One would expect the interfaces of ice to be highly hydrophilic (and
possibly the most hydrophilic of all solid surfaces). In this paper we
present evidence that some of the crystal facets of ice-I$_\mathrm{h}$
have structural features that can halve the effective hydrophilicity.
Our evidence for this comes from molecular dynamics (MD) simulations
of the spreading dynamics of liquid droplets on these facets, as well
as reverse non-equilibrium molecular dynamics (RNEMD) simulations of
solid-liquid friction.

Quiescent ice-I$_\mathrm{h}$/water interfaces have been studied
extensively using computer simulations. Haymet \textit{et al.}
characterized and measured the width of these interfaces for the
SPC~\cite{Karim90}, SPC/E~\cite{Gay02,Bryk02},
CF1~\cite{Hayward01,Hayward02} and TIP4P~\cite{Karim88} models, in
both neat water and with solvated
ions~\cite{Bryk04,Smith05,Wilson08,Wilson10}. Nada and Furukawa have
studied the width of basal/water and prismatic/water
interfaces~\cite{Nada95} as well as crystal restructuring at
temperatures approaching the melting point~\cite{Nada00}.

The surface of ice exhibits a premelting layer, often called a
quasi-liquid layer (QLL), at temperatures near the melting point.  MD
simulations of the facets of ice-I$_\mathrm{h}$ exposed to vacuum have
found QLL widths of approximately 10 \AA\ at 3 K below the melting
point.\cite{Conde08} Similarly, Limmer and Chandler have used the mW
water model~\cite{Molinero09} and statistical field theory to estimate
QLL widths at similar temperatures to be about 3 nm.\cite{Limmer14}

Recently, Sazaki and Furukawa have developed a technique using laser
confocal microscopy combined with differential interference contrast
microscopy that has sufficient spatial and temporal resolution to
visulaize and quantitatively analyze QLLs on ice crystals at
temperatures near melting.\cite{Sazaki10} They have found the width of
the QLLs perpindicular to the surface at -2.2$^{o}$C to be 3-4 \AA\
wide.  They have also seen the formation of two immiscible QLLs, which
displayed different dynamics on the crystal surface.\cite{Sazaki12}

There is now significant interest in the \textit{tribological}
properties of ice/ice and ice/water interfaces in the geophysics
community.  Understanding the dynamics of solid-solid shearing that is
mediated by a liquid layer\cite{Cuffey99, Bell08} will aid in
understanding the macroscopic motion of large ice
masses.\cite{Casassa91, Sukhorukov13, Pritchard12, Lishman13}

Using molecular dynamics simulations, Samadashvili has recently shown
that when two smooth ice slabs slide past one another, a stable
liquid-like layer develops between them.\cite{Samadashvili13} In a
previous study, our RNEMD simulations of ice-I$_\mathrm{h}$ shearing
through liquid water have provided quantitative estimates of the
solid-liquid kinetic friction coefficients.\cite{Louden13} These
displayed a factor of two difference between the basal and prismatic
facets.  The friction was found to be independent of shear direction
relative to the surface orientation.  We attributed facet-based
difference in liquid-solid friction to the 6.5 \AA\ corrugation of the
prismatic face which reduces the effective surface area of the ice
that is in direct contact with liquid water.

In the sections that follow, we outline the methodology used to
simulate droplet-spreading dynamics using standard MD and tribological
properties using RNEMD simulations.  These simulation methods give
complementary results that point to the prismatic and secondary prism
facets having roughly half of their surface area in direct contact
with the liquid.

\section{Methodology}
\subsection{Construction of the Ice / Water Interfaces}
To construct the four interfacial ice/water systems, a proton-ordered,
zero-dipole crystal of ice-I$_\mathrm{h}$ with exposed strips of
H-atoms was created using Structure 6 of Hirsch and Ojam\"{a}e's set
of orthorhombic representations for ice-I$_{h}$~\cite{Hirsch04}. This
crystal structure was cleaved along the four different facets.  The
exposed face was reoriented normal to the $z$-axis of the simulation
cell, and the structures were and extended to form large exposed
facets in rectangular box geometries.  Liquid water boxes were created
with identical dimensions (in $x$ and $y$) as the ice, with a $z$
dimension of three times that of the ice block, and a density
corresponding to 1 g / cm$^3$.  Each of the ice slabs and water boxes
were independently equilibrated at a pressure of 1 atm, and the
resulting systems were merged by carving out any liquid water
molecules within 3 \AA\ of any atoms in the ice slabs.  Each of the
combined ice/water systems were then equilibrated at 225K, which is
the liquid-ice coexistence temperature for SPC/E
water~\cite{Bryk02}. Ref. \citealp{Louden13} contains a more detailed
explanation of the construction of similar ice/water interfaces. The
resulting dimensions as well as the number of ice and liquid water
molecules contained in each of these systems are shown in Table
\ref{tab:method}.

The SPC/E water model~\cite{Berendsen87} has been extensively
characterized over a wide range of liquid
conditions,\cite{Arbuckle02,Kuang12} and its phase diagram has been
well studied.\cite{Baez95,Bryk04b,Sanz04b,Fennell:2005fk} With longer
cutoff radii and careful treatment of electrostatics, SPC/E mostly
avoids metastable crystalline morphologies like
ice-\textit{i}~\cite{Fennell:2005fk} and ice-B~\cite{Baez95}.  The
free energies and melting points
\cite{Baez95,Arbuckle02,Gay02,Bryk02,Bryk04b,Sanz04b,Fennell:2005fk,Fernandez06,Abascal07,Vrbka07}
of various other crystalline polymorphs have also been calculated.
Haymet \textit{et al.} have studied quiescent Ice-I$_\mathrm{h}$/water
interfaces using the SPC/E water model, and have seen structural and
dynamic measurements of the interfacial width that agree well with
more expensive water models, although the coexistence temperature for
SPC/E is still well below the experimental melting point of real
water~\cite{Bryk02}. Given the extensive data and speed of this model,
it is a reasonable choice even though the temperatures required are
somewhat lower than real ice / water interfaces.

\subsection{Droplet Simulations}
Ice interfaces with a thickness of $\sim$~20~\AA\ were created as
described above, but were not solvated in a liquid box. The crystals
were then replicated along the $x$ and $y$ axes (parallel to the
surface) until a large surface ($>$ 126 nm\textsuperscript{2}) had
been created.  The sizes and numbers of molecules in each of the
surfaces is given in Table \ref{tab:method}.  Weak translational
restraining potentials with spring constants of 1.5-4.0~$\mathrm{kcal\
  mol}^{-1}\mathrm{~\AA}^{-2}$ were applied to the centers of mass of
each molecule in order to prevent surface melting, although the
molecules were allowed to reorient freely. A water doplet containing
2048 SPC/E molecules was created separately. Droplets of this size can
produce agreement with the Young contact angle extrapolated to an
infinite drop size~\cite{Daub10}. The surfaces and droplet were
independently equilibrated to 225 K, at which time the droplet was
placed 3-5~\AA\ above the surface.  Five statistically independent
simulations were carried out for each facet, and the droplet was
placed at unique $x$ and $y$ locations for each of these simulations.
Each simulation was 5~ns in length and was conducted in the
microcanonical (NVE) ensemble.  Representative configurations for the
droplet on the prismatic facet are shown in figure \ref{fig:Droplet}.


\subsection{Shearing Simulations (Interfaces in Bulk Water)}

To perform the shearing simulations, the velocity shearing and scaling
variant of reverse non-equilibrium molecular dynamics (VSS-RNEMD) was
employed \cite{Kuang12}. This method performs a series of simultaneous
non-equilibrium exchanges of linear momentum and kinetic energy
between two physically-separated regions of the simulation cell.  The
system responds to this unphysical flux with velocity and temperature
gradients.  When VSS-RNEMD is applied to bulk liquids, transport
properties like the thermal conductivity and the shear viscosity are
easily extracted assuming a linear response between the flux and the
gradient.  At the interfaces between dissimilar materials, the same
method can be used to extract \textit{interfacial} transport
properties (e.g. the interfacial thermal conductance and the
hydrodynamic slip length).

The kinetic energy flux (producing a thermal gradient) is necessary
when performing shearing simulations at the ice-water interface in
order to prevent the frictional heating due to the shear from melting
the interface. Reference \citealp{Louden13} provides more details on
the VSS-RNEMD method as applied to ice-water interfaces.  A
representative configuration of the solvated prismatic facet being
sheared through liquid water is shown in figure \ref{fig:Shearing}.

In the results discussed below, the exchanges between the two regions
were carried out every 2 fs (e.g. every time step). This was done to
minimize the magnitude of each individual momentum exchange.  Because
individual VSS-RNEMD exchanges conserve both total energy and linear
momentum, the method can be ``bolted-on'' to simulations in any
ensemble.  The simulations of the pyramidal interface were performed
under the canonical (NVT) ensemble.  When time correlation functions
were computed (see section \ref{sec:orient}), these simulations were
done in the microcanonical (NVE) ensemble.  All simulations of the
other interfaces were done in the microcanonical ensemble.  

\section{Results}
\subsection{Ice - Water Contact Angles}
 
To determine the extent of wetting for each of the four crystal
facets, contact angles for liquid droplets on the ice surfaces were
computed using two methods.  In the first method, the droplet is
assumed to form a spherical cap, and the contact angle is estimated
from the $z$-axis location of the droplet's center of mass
($z_\mathrm{cm}$).  This procedure was first described by Hautman and
Klein~\cite{Hautman91}, and was utilized by Hirvi and Pakkanen in
their investigation of water droplets on polyethylene and poly(vinyl
chloride) surfaces~\cite{Hirvi06}. For each stored configuration, the
contact angle, $\theta$, was found by inverting the expression for the
location of the droplet center of mass,
\begin{equation}\label{contact_1}
\langle z_\mathrm{cm}\rangle = 2^{-4/3}R_{0}\bigg(\frac{1-cos\theta}{2+cos\theta}\bigg)^{1/3}\frac{3+cos\theta}{2+cos\theta} ,
\end{equation}
where $R_{0}$ is the radius of the free water droplet. 

The second method for obtaining the contact angle was described by
Ruijter, Blake, and Coninck~\cite{Ruijter99}.  This method uses a
cylindrical averaging of the droplet's density profile.  A threshold
density of 0.5 g cm\textsuperscript{-3} is used to estimate the
location of the edge of the droplet.  The $r$ and $z$-dependence of
the droplet's edge is then fit to a circle, and the contact angle is
computed from the intersection of the fit circle with the $z$-axis
location of the solid surface.  Again, for each stored configuration,
the density profile in a set of annular shells was computed. Due to
large density fluctuations close to the ice, all shells located within
2 \AA\ of the ice surface were left out of the circular fits.  The
height of the solid surface ($z_\mathrm{suface}$) along with the best
fitting central height ($z_\mathrm{center}$) and radius
($r_\mathrm{droplet}$) of the droplet can then be used to compute the
contact angle,
\begin{equation}
\theta =  90 + \frac{180}{\pi} \sin^{-1}\left(\frac{z_\mathrm{center} -
  z_\mathrm{surface}}{r_\mathrm{droplet}} \right).
\end{equation}
Both methods provided similar estimates of the dynamic contact angle,
although the first method is significantly less prone to noise, and
is the method used to report contact angles below.

Because the initial droplet was placed above the surface, the initial
value of 180$^{\circ}$ decayed over time.  See fig.  XXXX.  Each of
these profiles were fit to a biexponential decay, with a short-time
contribution that describes the initial contact with the surface, a
long time contribution that describes the spread of the droplet over
the surface, and a constant to capture the infinite-time estimate of
the equilibrium contact angle,
\begin{equation}
\theta(t) = \theta_\infty +  (180-\theta_\infty) \left[ a e^{-k_\mathrm{contact} t} +
  (1-a) e^{-k_\mathrm{spread} t}  \right]
\end{equation}

We have found that the rate of the water droplet spreading across all
four crystal facets to be $k_\mathrm{spread} \approx$ 0.7
ns$^{-1}$. However, the basal and pyramidal facets had estimated
equilibrium contact angles of $\theta_\infty \approx$ 35$^{o}$, while
prismatic and secondary prismatic had values for $\theta_\infty$ near
43$^{o}$ as seen in Table \ref{tab:kappa}. 

These results indicate that the basal and pyramidal facets are
somewhat more hydrophilic than the prismatic and secondary prism
facets, and surprisingly, that the differential hydrophilicities of
the crystal facets is not reflected in the spreading rate of the
droplet.

% This is in good agreement with our calculations of friction
% coefficients, in which the basal
% and pyramidal had a higher coefficient of kinetic friction than the
% prismatic and secondary prismatic. Due to this, we beleive that the
% differences in friction coefficients can be attributed to the varying
% hydrophilicities of the facets. 

\subsection{Coefficient of friction of the interfaces}
While investigating the kinetic coefficient of friction, there was found 
to be a dependence for $\mu_k$ 
on the temperature of the liquid water in the system. We believe this 
dependence
arrises from the sharp discontinuity of the viscosity for the SPC/E model
at temperatures approaching 200 K\cite{Kuang12}. Due to this, we propose
a weighting to the interfacial friction coefficient, $\kappa$ by the 
shear viscosity of the fluid at 225 K. The interfacial friction coefficient 
relates the shear stress with the relative velocity of the fluid normal to the 
interface:
\begin{equation}\label{Shenyu-13}
j_{z}(p_{x}) = \kappa[v_{x}(fluid)-v_{x}(solid)]
\end{equation}
where $j_{z}(p_{x})$ is the applied momentum flux (shear stress) across $z$ 
in the 
$x$-dimension, and $v_{x}$(fluid) and $v_{x}$(solid) are the velocities
directly adjacent to the interface. The shear viscosity, $\eta(T)$, of the 
fluid can be determined under a linear response of the momentum 
gradient to the applied shear stress by
\begin{equation}\label{Shenyu-11}
j_{z}(p_{x}) = \eta(T) \frac{\partial v_{x}}{\partial z}.
\end{equation}
Using eqs \eqref{Shenyu-13} and \eqref{Shenyu-11}, we can find the following
expression for $\kappa$,
\begin{equation}\label{kappa-1}
\kappa = \eta(T) \frac{\partial v_{x}}{\partial z}\frac{1}{[v_{x}(fluid)-v_{x}(solid)]}.
\end{equation}
Here is where we will introduce the weighting term of $\eta(225)/\eta(T)$ 
giving us
\begin{equation}\label{kappa-2}
\kappa = \frac{\eta(225)}{[v_{x}(fluid)-v_{x}(solid)]}\frac{\partial v_{x}}{\partial z}.
\end{equation}

To obtain the value of $\eta(225)$ for the SPC/E model, a $31.09 \times 29.38
\times 124.39$ \AA\ box with 3744 SPC/E liquid water molecules was 
equilibrated to 225K, 
and 5 unique shearing experiments were performed. Each experiment was
conducted in the NVE and were 5 ns in 
length. The VSS were attempted every timestep, which was set to 2 fs. 
For our SPC/E systems, we found $\eta(225)$  to be 0.0148 $\pm$ 0.0007 Pa s,
roughly ten times larger than the value found for 280 K SPC/E bulk water by
Kuang\cite{Kuang12}. 

The interfacial friction coefficient, $\kappa$, can equivalently be expressed 
as the ratio of the viscosity of the fluid to the slip length, $\delta$, which
is an indication of how 'slippery' the interface is.
\begin{equation}\label{kappa-3}
\kappa = \frac{\eta}{\delta}
\end{equation} 
In each of the systems, the interfacial temperature was kept fixed to 225K, 
which ensured the viscosity of the fluid at the
interace was approximately the same. Thus, any significant variation in
$\kappa$ between
the systems indicates differences in the 'slipperiness' of the interfaces.
As each of the ice systems are sheared relative to liquid water, the
'slipperiness' of the interface can be taken as an indication of how
hydrophobic or hydrophilic the interface is. The calculated $\kappa$ values
found for the four crystal facets of Ice-I$_\mathrm{h}$ investigated are shown
in Table \ref{tab:kappa}. The basal and pyramidal facets were found to have 
similar values of $\kappa \approx$ 0.0006 
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}), while values of 
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1})
were found for the prismatic and secondary prismatic systems.
These results indicate that the basal and pyramidal facets are
more hydrophilic than the prismatic and secondary prismatic facets.

\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 structurely distinguishable 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 its 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,Hayward01,Hayward02,Karim88}  

Since the VSS-RNEMD moves used to impose the thermal and velocity gradients
 perturb the momenta of the water molecules in 
the systems, parameters that depend on 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 tetrahedral order parameter (Eq 5\cite{Louden13}) to quantify the width of the interface,
 which was originally described by Kumar\cite{Kumar09} and 
Errington\cite{Errington01}, and used by Bryk and Haymet in a previous study
of ice/water interfaces.\cite{Bryk04b} 

To determine the width of the interfaces, 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:pyrComic}
and \ref{fig:spComic} (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 (Eq. 6\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
resulting thermal and velocity gradients from an imposed kinetic energy and
momentum fluxes 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:pyrComic} and \ref{fig:spComic}) 
by eq. 6\cite{Louden13},we find the interfacial width to be
 3.2 $\pm$ 0.2 and 3.2 $\pm$ 0.2 \AA\ for the control system with no applied 
momentum flux for both the pyramidal and secondary prismatic systems. 
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 prismatic, 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 the movement of water over the ice. 


\subsection{Orientational dynamics \label{sec:orient}}
%Should we include the math here?
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}
helps indicate the local environment around the water molecules. The function 
begins with an initial value of unity, and decays to zero as the water molecule
loses memory of its former orientation. Observing the rate at which this decay
occurs can provide insight to the mechanism and timescales for the relaxation.
In eq. \eqref{C(t)1}, $P_{2}$ is the second-order Legendre polynomial, and 
$\mathbf{u}$ is the bisecting HOH vector. The angle brackets indicate
an ensemble average over all the water molecules in a given spatial region.

To investigate the dynamics of the water molecules across the interface, the 
systems were divided in the $z$-dimension into bins, each $\approx$ 3 \AA\
 wide, and eq. \eqref{C(t)1} was computed for each of the bins. A water 
molecule was allocated to a particular bin if it was initially in the bin
at time zero. To compute eq. \eqref{C(t)1}, each 0.5 ns simulation was 
followed by an additional 200 ps NVE simulation during which the 
position and orientations of each molecule were recorded every 0.1 ps.
 
The data obtained for each bin was then fit to a triexponential decay
with the three decay constants
$\tau_{short}$ corresponding to the librational motion of the water  
molecules, $\tau_{middle}$ corresponding to jumps between the breaking and 
making of hydrogen bonds, and $\tau_{long}$ corresponding to the translational
motion of the water molecules. An additive constant in the fit accounts
for the water molecules trapped in the ice which do not experience any
long-time orientational decay.

In Figures \ref{fig:PyrOrient} and \ref{fig:SPorient} we see the $z$-coordinate
profiles for the three decay constants, $\tau_{short}$ (panel a), 
$\tau_{middle}$ (panel b),
and $\tau_{long}$ (panel c) for the pyramidal and secondary prismatic systems
respectively. The control experiments (no shear) are shown in black, and 
an experiment with an imposed momentum flux is shown in red. The vertical
dotted line traversing all three panels denotes the midpoint of the 
interface as determined by the local tetrahedral order parameter fitting.
In the liquid regions of both systems, we see that $\tau_{middle}$ and
$\tau_{long}$ have approximately consistent values of $3-6$ ps and $30-40$ ps,
resepctively, and increase in value as we approach the interface. Conversely,
in panel a, we see that $\tau_{short}$ decreases from the liquid value 
of $72-76$ fs as we approach the interface. We believe this speed up is due to
the constrained motion of librations closer to the interface. Both the 
approximate values for the decays and trends approaching the interface  match 
those reported previously for the basal and prismatic interfaces.

As done previously, we have attempted to quantify the distance, $d_{pyramidal}$
and $d_{secondary prismatic}$, from the 
interface that the deviations from the bulk liquid values begin. This was done
by fitting the orientational decay constant $z$-profiles by
\begin{equation}\label{tauFit}
\tau(z)\approx\tau_{liquid}+(\tau_{wall}-\tau_{liquid})e^{-(z-z_{wall})/d}
\end{equation}
where $\tau_{liquid}$ and $\tau_{wall}$ are the liquid and projected wall
values of the decay constants, $z_{wall}$ is the location of the interface,
and $d$ is the displacement from the interface at which these deviations
occur. The values for $d_{pyramidal}$ and $d_{secondary prismatic}$ were 
determined
for each of the decay constants, and then averaged for better statistics 
($\tau_{middle}$ was ommitted for secondary prismatic). For the pyramidal 
system,
$d_{pyramidal}$ was found to be 2.7 \AA\ for both the control and the sheared
system. We found $d_{secondary prismatic}$ to be slightly larger than 
$d_{pyramidal}$ for both the control and with an applied shear, with 
displacements of $4$ \AA\ for the control system and $3$ \AA\ for the 
experiment with the imposed momentum flux. These values are consistent with
those found for the basal ($d_{basal}\approx2.9$ \AA\ ) and prismatic 
($d_{prismatic}\approx3.5$ \AA\ ) systems.


\section{Conclusion}
We present the results of molecular dynamics simulations of the basal,
prismatic, pyrmaidal
and secondary prismatic facets of an SPC/E model of the 
Ice-I$_\mathrm{h}$/water interface, and show that the differential
coefficients of friction among the four facets are due to their
relative hydrophilicities by means
of water contact angle calculations. To obtain the coefficients of
friction, the ice was sheared through the liquid
water while being exposed to a thermal gradient to maintain a stable 
interface by using the minimally perturbing VSS RNEMD method. Water
contact angles are obtained by fitting the spreading of a liquid water
droplet over the crystal facets.

In agreement with our previous findings for the basal and prismatic facets, the interfacial 
width of the prismatic and secondary prismatic crystal faces were
found to be independent of shear rate as measured by the local 
tetrahedral order parameter. This width was found to be 
3.2~$\pm$ 0.2~\AA\ for both the pyramidal and the secondary prismatic systems.
These values are in good agreement with our previously calculated interfacial
widths for the basal (3.2~$\pm$ 0.4~\AA\ ) and prismatic (3.6~$\pm$ 0.2~\AA\ )
systems. 

Orientational dynamics of the Ice-I$_\mathrm{h}$/water interfaces were studied
by calculation of the orientational time correlation function at varying
displacements normal to the interface. The decays were fit
to a tri-exponential decay, where the three decay constants correspond to
the librational motion of the molecules driven by the restoring forces of 
existing hydrogen bonds ($\tau_{short}$ $\mathcal{O}$(10 fs)), jumps between 
two different hydrogen bonds ($\tau_{middle}$ $\mathcal{O}$(1 ps)), and 
translational motion of the molecules ($\tau_{long}$ $\mathcal{O}$(100 ps)).
$\tau_{short}$ was found to decrease approaching the interface due to the 
constrained motion of the molecules as the local environment becomes more
ice-like. Conversely, the two longer-time decay constants were found to 
increase at small displacements from the interface. As seen in our previous 
work on the basal and prismatic facets, there appears to be a dynamic
interface width at which deviations from the bulk liquid values occur.
We had previously found $d_{basal}$ and $d_{prismatic}$ to be approximately 
2.8~\AA\ and 3.5~\AA. We found good agreement of these values for the 
pyramidal and secondary prismatic systems with $d_{pyramidal}$ and
$d_{secondary prismatic}$ to be 2.7~\AA\ and 3~\AA. For all four of the
facets, no apparent dependence of the dynamic width on the shear rate was 
found.
  
The interfacial friction coefficient, $\kappa$, was determined for each facet
interface. We were able to reach an expression for $\kappa$ as a function of
the velocity profile of the system which is scaled by the viscosity of the liquid
at 225 K. In doing so, we have obtained an expression for $\kappa$ which is 
independent of temperature differences of the liquid water at far displacements
from the interface. We found the basal and pyramidal facets to have
similar $\kappa$ values, of $\kappa \approx$ 6 
(x$10^{-4}$amu \AA\textsuperscript{-2} fs\textsuperscript{-1}). However, the
prismatic and secondary prismatic facets were found to have $\kappa$ values of 
$\kappa \approx$ 3 (x$10^{-4}$amu \AA\textsuperscript{-2} fs\textsuperscript{-1}).
Believing this difference was due to the relative hydrophilicities of
the crystal faces, we have calculated the infinite decay of the water
contact angle, $\theta_{\infty}$, by watching the spreading of a water
droplet over the surface of the crystal facets. We have found
$\theta_{\infty}$ for the basal and pyramidal faces to be $\approx$ 34
degrees, while obtaining $\theta_{\infty}$ of $\approx$ 43 degrees for
the prismatic and secondary prismatic faces. This indicates that the
basal and pyramidal faces of ice-I$_\mathrm{h}$ are more hydrophilic
than the prismatic and secondary prismatic. These results also seem to
explain the differential friction coefficients obtained through the
shearing simulations, namely, that the coefficients of friction of the
ice-I$_\mathrm{h}$ crystal facets are governed by their inherent
hydrophilicities. 


\begin{acknowledgments}
  Support for this project was provided by the National
  Science Foundation under grant CHE-1362211. Computational time was
  provided by the Center for Research Computing (CRC) at the
  University of Notre Dame.
\end{acknowledgments}

\bibliography{iceWater}
% *****************************************
% There is significant interest in the properties of ice/ice and ice/water 
% interfaces in the geophysics community. Most commonly, the results of shearing
% two ice blocks past one 
% another\cite{Casassa91, Sukhorukov13, Pritchard12, Lishman13} or the shearing 
% of ice through water\cite{Cuffey99, Bell08}. Using molecular dynamics 
% simulations, Samadashvili has recently shown that when two smooth ice slabs 
% slide past one another, a stable liquid-like layer develops between 
% them\cite{Samadashvili13}. To fundamentally understand these processes, a 
% molecular understanding of the ice/water interfaces is needed.     

% Investigation of the ice/water interface is also crucial in understanding
% processes such as nucleation, crystal 
% growth,\cite{Han92, Granasy95, Vanfleet95} and crystal 
% melting\cite{Weber83, Han92, Sakai96, Sakai96B}. Insight gained to these
% properties can also be applied to biological systems of interest, such as 
% the behavior of the antifreeze protein found in winter 
% flounder,\cite{Wierzbicki07,Chapsky97} and certain terrestial
% arthropods.\cite{Duman:2001qy,Meister29012013} Elucidating the properties which
% give rise to these processes through experimental techniques can be expensive,
% complicated, and sometimes infeasible. However, through the use of molecular 
% dynamics simulations much of the problems of investigating these properties
% are alleviated.

% Understanding ice/water interfaces inherently begins with the isolated
% systems. There has been extensive work parameterizing models for liquid water,
% such as the SPC\cite{Berendsen81}, SPC/E\cite{Berendsen87}, 
% TIP4P\cite{Jorgensen85}, TIP4P/2005\cite{Abascal05}, 
% ($\dots$), and more recently, models for simulating
% the solid phases of water, such as the TIP4P/Ice\cite{Abascal05b} model. The 
% melting point of various crystal structures of ice have been calculated for 
% many of these models
% (SPC\cite{Karim90,Abascal07}, SPC/E\cite{Baez95,Arbuckle02,Gay02,Bryk02,Bryk04b,Sanz04b,Gernandez06,Abascal07,Vrbka07}, TIP4P\cite{Karim88,Gao00,Sanz04,Sanz04b,Koyama04,Wang05,Fernandez06,Abascal07}, TIP5P\cite{Sanz04,Koyama04,Wang05,Fernandez06,Abascal07}), 
% and the partial or complete phase diagram for the model has been determined
% (SPC/E\cite{Baez95,Bryk04b,Sanz04b}, TIP4P\cite{Sanz04,Sanz04b,Koyama04}, TIP5P\cite{Sanz04,Koyama04}). 
% Knowing the behavior and melting point for these models has enabled an initial 
% investigation of ice/water interfaces. 

% The Ice-I$_\mathrm{h}$/water quiescent interface has been extensively studied 
% over the past 30 years by theory and experiment. Haymet \emph{et al.} have 
% done significant work characterizing and quantifying the width of these 
% interfaces for the SPC,\cite{Karim90} SPC/E,\cite{Gay02,Bryk02}, 
% CF1,\cite{Hayward01,Hayward02} and TIP4P\cite{Karim88} models for water. In 
% recent years, Haymet has focused on investigating the effects cations and 
% anions have on crystal nucleaion and 
% melting.\cite{Bryk04,Smith05,Wilson08,Wilson10} Nada and Furukawa have studied
% the the basal- and prismatic-water interface width\cite{Nada95}, crystal 
% surface restructuring at temperatures approaching the melting 
% point\cite{Nada00}, and the mechanism of ice growth inhibition by antifreeze 
% proteins\cite{Nada08,Nada11,Nada12}. Nada has developed a six-site water model
% for ice/water interfaces near the melting point\cite{Nada03}, and studied the 
% dependence of crystal growth shape on applied pressure\cite{Nada11b}. Using 
% this model, Nada and Furukawa have established differential 
% growth rates for the basal, prismatic, and secondary prismatic facets of 
% Ice-I$_\mathrm{h}$ and found their origins due to a reordering of the hydrogen 
% bond network in water near the interface\cite{Nada05}. While the work 
% described so far has mainly focused on bulk water on ice, there is significant
% interest in thin films of water on ice surfaces as well.  

% It is well known that the surface of ice exhibits a premelting layer at 
% temperatures near the melting point, often called a quasi-liquid layer (QLL).
% Molecular dynamics simulations of the facets of ice-I$_\mathrm{h}$ exposed
% to vacuum performed by Conde, Vega and Patrykiejew have found QLL widths of
% approximately 10 \AA\ at 3 K below the melting point\cite{Conde08}.
% Similarly, Limmer and Chandler have used course grain simulations and 
% statistical field theory to estimated QLL widths at the same temperature to
% be about 3 nm\cite{Limmer14}.
% Recently, Sazaki and Furukawa have developed an experimental technique with 
% sufficient spatial and temporal resolution to visulaize and quantitatively
% analyze QLLs on ice crystals at temperatures near melting\cite{Sazaki10}. They 
% have found the width of the QLLs perpindicular to the surface  at -2.2$^{o}$C 
% to be $\mathcal{O}$(\AA). They have also seen the formation of two immiscible
% QLLs, which displayed different stabilities and dynamics on the crystal 
% surface\cite{Sazaki12}. Knowledge of the hydrophilicities of each
% of the crystal facets would help further our understanding of the properties 
% and dynamics of the QLLs.
  
% Presented here is the follow up to our previous paper\cite{Louden13}, in which
% the basal and prismatic facets of an ice-I$_\mathrm{h}$/water interface were
% investigated where the ice was sheared relative to the liquid. By using a 
% recently developed velocity shearing and scaling approach to reverse 
% non-equilibrium molecular dynamics (VSS-RNEMD), simultaneous temperature and 
% velocity gradients can be applied to the system, which allows for measurment
% of friction and thermal transport properties while maintaining a stable 
% interfacial temperature\cite{Kuang12}. Structural analysis and dynamic
% correlation functions were used to probe the interfacial response to a shear, 
% and the resulting solid/liquid kinetic friction coefficients were reported.
% In this paper we present the same analysis for the pyramidal and secondary 
% prismatic facets, and show that the differential interfacial friction 
% coefficients for the four facets of ice-I$_\mathrm{h}$ are determined by their 
% relative hydrophilicity by means of dynamics water contact angle
% simulations. 

% 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 sites of molecules
% $i$,$k$, and $j$, where the centeral oxygen is located within molecule $k$ and 
% molecules $i$ and $j$ are two of the closest four water molecules
% around molecule $k$. All four closest neighbors of molecule $k$ are also
% required to reside within the first peak of the pair distribution function
% for molecule $k$ (typically $<$ 3.41 \AA\ for water). 
% $N_z = \int\delta(z_k - z) \mathrm{d}z$ is a normalization factor to account 
% for the varying population of molecules within each finite-width bin.


% The hydrophobicity or hydrophilicity of a surface can be described by the
% extent a droplet of water wets the surface. The contact angle formed between
% the solid and the liquid, $\theta$, which relates the free energies of the 
% three interfaces involved, is given by Young's equation.
% \begin{equation}\label{young}
% \cos\theta = (\gamma_{sv} - \gamma_{sl})/\gamma_{lv}
% \end{equation} 
% Here $\gamma_{sv}$, $\gamma_{sl}$, and $\gamma_{lv}$ are the free energies
% of the solid/vapor, solid/liquid, and liquid/vapor interfaces respectively.
% Large contact angles ($\theta$ $\gg$ 90\textsuperscript{o}) correspond to low 
% wettability and hydrophobic surfaces, while small contact angles 
% ($\theta$ $\ll$ 90\textsuperscript{o}) correspond to high wettability and 
% hydrophilic surfaces. Experimentally, measurements of the contact angle
% of sessile drops has been used to quantify the extent of wetting on surfaces
% with thermally selective wetting charactaristics\cite{Tadanaga00,Liu04,Sun04},
% as well as nano-pillared surfaces with electrically tunable Cassie-Baxter and 
% Wenzel states\cite{Herbertson06,Dhindsa06,Verplanck07,Ahuja08,Manukyan11}. 
% Luzar and coworkers have done significant work modeling these transitions on 
% nano-patterned surfaces\cite{Daub07,Daub10,Daub11,Ritchie12}, and have found 
% the change in contact angle to be due to the external field perturbing the 
% hydrogen bonding of the liquid/vapor interface\cite{Daub07}.


\end{article}

\begin{figure}
\includegraphics[width=\linewidth]{Droplet}
\caption{\label{fig:Droplet} Computational model of a droplet of
  liquid water spreading over the prismatic $\{1~0~\bar{1}~0\}$ facet
  of ice, before (left) and 5 ns after (right) being introduced to the
  surface.  The contact angle ($\theta$) shrinks as the simulation
  proceeds, and the long-time behavior of this angle is used to
  estimate the hydrophilicity of the facet.}
\end{figure}

\begin{figure}
\includegraphics[width=\linewidth]{ }
\caption{\label{fig:Shearing} Computational model of a slab of ice
  being sheared through liquid water (above and below).  In this
  figure, the ice is presenting the prismatic $\{1~0~\bar{1}~0\}$
  facet towards the liquid phase.}
\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}. 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]{SP_comic_strip}
\caption{\label{fig:spComic} The secondary prismatic interface with a shear 
rate of 3.5 \
ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.}
\end{figure}

\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 
prismatic face. Panel descriptions match those in \ref{fig:PyrOrient}.}
\end{figure}


\begin{table}[h]
\centering
\caption{Sizes of the droplet and shearing simulations.  Cell
  dimensions are measured in \AA. \label{tab:method}}
\begin{tabular}{r|cccc|ccccc} 
\toprule
 \multirow{2}{*}{Interface} & \multicolumn{4}{c|}{Droplet} & \multicolumn{5}{c}{Shearing} \\
  & $N_\mathrm{ice}$ & $N_\mathrm{droplet}$ & $L_x$ & $L_y$ & $N_\mathrm{ice}$ & $N_\mathrm{liquid}$ & $L_x$ & $L_y$ & $L_z$  \\ 
\midrule
Basal  $\{0001\}$                    & 12960 & 2048 & 134.70 & 140.04 & 900 & 1846  & 23.87 & 35.83 & 98.64  \\
Pyramidal  $\{2~0~\bar{2}~1\}$       & 11136 & 2048 & 143.75 & 121.41 & 1216 & 2203 & 37.47 & 29.50 & 93.02  \\
Prismatic  $\{1~0~\bar{1}~0\}$       &  9900 & 2048 & 110.04 & 115.00 & 3000 & 5464 & 35.95 & 35.65 & 205.77 \\
Secondary Prism  $\{1~1~\bar{2}~0\}$ & 11520 & 2048 & 146.72 & 124.48 & 3840 & 8176 & 71.87 & 31.66 & 161.55 \\
\bottomrule
\end{tabular}
\end{table}


\begin{table}[h]
\centering
\caption{Structural and dynamic properties of the interfaces of Ice-I$_\mathrm{h}$
  with water.  Liquid-solid friction coefficients ($\kappa_x$ and $\kappa_y$) are expressed in 10\textsuperscript{-4} amu
  \AA\textsuperscript{-2} fs\textsuperscript{-1}. \label{tab:kappa}} 
\begin{tabular}{r|cc|cccc}  
\toprule
\multirow{2}{*}{Interface} & \multicolumn{2}{c|}{Droplet} & \multicolumn{4}{c}{Shearing}\\
  & $\theta_{\infty}$ ($^\circ$)  & $k_\mathrm{spread}$  (ns\textsuperscript{-1}) &
$\kappa_{x}$  & $\kappa_{y}$ & $d_{q_{z}}$ (\AA) &  $d_{\tau}$ (\AA) \\ 
\midrule
Basal  $\{0001\}$                    & $34.1 \pm 0.9$ &$0.60 \pm 0.07$ & $5.9 \pm 0.3$ & $6.5 \pm 0.8$ & $3.2 \pm 0.4$ & $2.9$  \\
Pyramidal  $\{2~0~\bar{2}~1\}$       & $35 \pm 3$ &  $0.7 \pm 0.1$ & $5.8 \pm 0.4$ & $6.1 \pm 0.5$ & $3.2 \pm 0.2$ & $2.7$ \\
Prismatic  $\{1~0~\bar{1}~0\}$       & $45 \pm 3$ & $0.75 \pm 0.09$ & $3.0 \pm 0.2$ & $3.0 \pm 0.1$ & $3.6 \pm 0.2$ & $3.5$ \\
Secondary Prism  $\{1~1~\bar{2}~0\}$ & $42 \pm 2$ & $0.69 \pm 0.03$ & $3.5 \pm 0.1$ & $3.3 \pm 0.2$ & $3.2 \pm 0.2$ & $3$ \\ 
\bottomrule
\end{tabular}
\end{table}

\end{document}
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