| 82 |
|
droplet can spread out over the surface. The contact angle formed |
| 83 |
|
between the solid and the liquid depends on the free energies of the |
| 84 |
|
three interfaces involved, and is given by Young's |
| 85 |
< |
equation.\cite{Young} |
| 85 |
> |
equation.\cite{Young05} |
| 86 |
|
\begin{equation}\label{young} |
| 87 |
|
\cos\theta = (\gamma_{sv} - \gamma_{sl})/\gamma_{lv} . |
| 88 |
|
\end{equation} |
| 100 |
|
the hydrophilicity to a surprising degree. Small changes in the |
| 101 |
|
heights and widths of nano-pillars can change a surface from |
| 102 |
|
superhydrophobic, $\theta \ge 150^{\circ}$, to hydrophilic, $\theta |
| 103 |
< |
\sim 0^{\circ}$.\cite{CBW} This is often referred to as the |
| 103 |
> |
\sim 0^{\circ}$.\cite{Koishi09} This is often referred to as the |
| 104 |
|
Cassie-Baxter to Wenzel transition. Nano-pillared surfaces with |
| 105 |
|
electrically tunable Cassie-Baxter and Wenzel states have also been |
| 106 |
|
observed.\cite{Herbertson06,Dhindsa06,Verplanck07,Ahuja08,Manukyan11} |
| 191 |
|
carving out any liquid water molecules within 3 \AA\ of any atoms in |
| 192 |
|
the ice slabs. Each of the combined ice/water systems were then |
| 193 |
|
equilibrated at 225K, which is the liquid-ice coexistence temperature |
| 194 |
< |
for SPC/E water.\cite{} Ref. \citealp{Louden13} contains a more |
| 194 |
> |
for SPC/E water.\cite{Bryk02} Ref. \citealp{Louden13} contains a more |
| 195 |
|
detailed explanation of the construction of ice/water interfaces. The |
| 196 |
|
resulting dimensions, number of ice, and liquid water molecules |
| 197 |
|
contained in each of these systems can be seen in Table |
| 198 |
|
\ref{tab:method}. |
| 199 |
|
|
| 200 |
< |
We used SPC/E Why? Extensively characterized over a wide range of |
| 201 |
< |
liquid conditions. Well-studied phase diagram. Reasonably accurate |
| 202 |
< |
crystalline free energies. Mostly avoids spurious crystalline |
| 203 |
< |
morphologies like ice-i and ice-B. Most importantly, the use of SPC/E |
| 204 |
< |
has been well characterized in previous ice/water interfacial studies. |
| 200 |
> |
Mostly avoids spurious crystalline morphologies like ice-i and ice-B. |
| 201 |
|
|
| 202 |
< |
|
| 203 |
< |
|
| 204 |
< |
There has been extensive work parameterizing good models for liquid |
| 205 |
< |
water over a wide range of conditions. The melting points of various |
| 206 |
< |
crystal structures of ice have been calculated for many of these |
| 207 |
< |
models (SPC\cite{Karim90,Abascal07}, |
| 212 |
< |
SPC/E\cite{Baez95,Arbuckle02,Gay02,Bryk02,Bryk04b,Sanz04b,Fernandez06,Abascal07,Vrbka07}, |
| 213 |
< |
TIP4P\cite{Karim88,Gao00,Sanz04,Sanz04b,Koyama04,Wang05,Fernandez06,Abascal07}, |
| 214 |
< |
TIP5P\cite{Sanz04,Koyama04,Wang05,Fernandez06,Abascal07}), and the |
| 215 |
< |
partial or complete phase diagram for the model has been determined |
| 216 |
< |
(SPC/E\cite{Baez95,Bryk04b,Sanz04b}, |
| 217 |
< |
TIP4P\cite{Sanz04,Sanz04b,Koyama04}, TIP5P\cite{Sanz04,Koyama04}). |
| 218 |
< |
|
| 219 |
< |
|
| 220 |
< |
such as the SPC\cite{Berendsen81}, SPC/E\cite{Berendsen87}, |
| 221 |
< |
TIP4P\cite{Jorgensen85}, TIP4P/2005\cite{Abascal05}, ($\dots$), and |
| 222 |
< |
more recently, models for simulating the solid phases of water, such |
| 223 |
< |
as the TIP4P/Ice\cite{Abascal05b} model. |
| 224 |
< |
|
| 202 |
> |
The SPC/E water model\cite{Berendsen87} has been extensively |
| 203 |
> |
characterized over a wide range of liquid |
| 204 |
> |
conditions,\cite{Arbuckle02, Kuang12} and its phase diagram has been well studied.\cite{Baez95,Bryk04b,Sanz04b} |
| 205 |
> |
The free energies \cite{Baez95} and melting points |
| 206 |
> |
\cite{Baez95,Arbuckle02,Gay02,Bryk02,Bryk04b,Sanz04b,Fernandez06,Abascal07,Vrbka07} |
| 207 |
> |
of various crystal structures have also been calculated. |
| 208 |
|
Haymet et al. have studied the quiescent Ice-I$_\mathrm{h}$/water |
| 209 |
< |
interface using the rigid SPC, SPC/E, TIP4P, and the flexible CF1 |
| 210 |
< |
water models, and has seen good agreement for structural and dynamic |
| 211 |
< |
measurements of the interfacial width. Given the expansive size of our |
| 212 |
< |
systems of interest, and the apparent independence of water model on |
| 230 |
< |
interfacial width, we have chosen to use the rigid SPC/E water model |
| 231 |
< |
in this study. |
| 209 |
> |
interface using the SPC/E water model, and has seen good agreement for |
| 210 |
> |
structural and dynamic measurements of the interfacial width when compared with more |
| 211 |
> |
expensive water models. For these reasons, the SPC/E water |
| 212 |
> |
model was used in this study. |
| 213 |
|
|
| 214 |
|
\subsection{Shearing simulations (interfaces in bulk water)} |
| 215 |
|
% Should we mention number of runs, sim times, etc. ? |
| 233 |
|
secondary prismatic simulations were performed under the NVE ensamble. |
| 234 |
|
|
| 235 |
|
\subsection{Droplet simulations} |
| 236 |
< |
Ice interfaces with a thickness of $\sim 30 \AA$ were created as |
| 236 |
> |
Ice interfaces with a thickness of $\sim 20 \AA$ were created as |
| 237 |
|
described above, but were not solvated in a liquid box. The crystals |
| 238 |
|
were then replicated along the $x$ and $y$ axes (parallel to the |
| 239 |
|
surface) until a large surface had been created. The sizes and |
| 240 |
|
numbers of molecules in each of the surfaces is given in Table |
| 241 |
|
\ref{tab:ice_sheets}. Weak translational restraining potentials with |
| 242 |
< |
spring constants of XXXX were applied to the center of mass of each |
| 242 |
> |
spring constants of 1.5 to 4.0 UNITS were applied to the center of mass of each |
| 243 |
|
molecule in order to prevent surface melting, although the molecules |
| 244 |
|
were allowed to reorient freely. A water doplet containing 2048 SPC/E |
| 245 |
|
molecules was created separately. Droplets of this size can produce |
| 252 |
|
5 ns in length and conducted in the microcanonical (NVE) ensemble. |
| 253 |
|
|
| 254 |
|
\section{Results and discussion} |
| 255 |
+ |
\subsection{Dynamic water contact angle} |
| 256 |
+ |
|
| 257 |
+ |
To determine the extent of wetting for each of the four crystal |
| 258 |
+ |
facets, water contact angle simuations were performed. Contact angles |
| 259 |
+ |
were obtained from these simulations by two methods. In the first |
| 260 |
+ |
method, the contact angle was obtained from the $z$-center of mass |
| 261 |
+ |
($z_{c.m.}$) of the water droplet as described by Hautman and Klein\cite{Hautman91} |
| 262 |
+ |
and utilized by Hirvi and Pakkanen in their investigation of water |
| 263 |
+ |
droplets on polyethylene and poly(vinyl chloride) |
| 264 |
+ |
surface\cite{Hirvi06}. At each snapshot of the simulation, the contact |
| 265 |
+ |
angle, $\theta$, was found by |
| 266 |
+ |
\begin{equation}\label{contact_1} |
| 267 |
+ |
\langle z_{c.m.}\rangle = 2^{-4/3}R_{0}\bigg(\frac{1-cos\theta}{2+cos\theta}\bigg)^{1/3}\frac{3+cos\theta}{2+cos\theta} , |
| 268 |
+ |
\end{equation} |
| 269 |
+ |
where $R_{0}$ is the radius of the free water droplet. In the second |
| 270 |
+ |
method, the contact angle was obtained from fitting the droplet's |
| 271 |
+ |
$z$-profile after radial averaging to a |
| 272 |
+ |
circle as described by Ruijter, Blake, and |
| 273 |
+ |
Coninck\cite{Ruijter99}. At each snapshot of the simulation the water droplet was |
| 274 |
+ |
broken into bins, and the location of bin containing half-bulk density was |
| 275 |
+ |
stored. Due to fluctuations close to the ice, all bins located within |
| 276 |
+ |
2.0 \AA\ of the ice were discarded. The remaining stored bins were |
| 277 |
+ |
then fit by a circle, whose tangential intersection with the ice plane could |
| 278 |
+ |
be used to calculate the water |
| 279 |
+ |
contact angle. These results proved noisey and unreliable when |
| 280 |
+ |
compared with the first method, for these purposes we omit the data |
| 281 |
+ |
from the second method. |
| 282 |
+ |
|
| 283 |
+ |
The resulting water contact angle profiles generated by the first method |
| 284 |
+ |
had an initial value of 180$^{o}$, and decayed over time. Each of |
| 285 |
+ |
these profiles were fit to a biexponential decay, with a short time |
| 286 |
+ |
piece to account for the water droplet initially adhering to the |
| 287 |
+ |
surface, a long time piece describing the spreading of the droplet |
| 288 |
+ |
over the surface, and an added constant to capture the infinite |
| 289 |
+ |
decay of the contact angle. We have found that the rate of the water |
| 290 |
+ |
droplet spreading across all four crystal facets to be $\approx$ 0.7 |
| 291 |
+ |
ns$^{-1}$. However, the basal and pyramidal facets |
| 292 |
+ |
had an infinite decay value for $\theta$ of $\approx$ 35$^{o}$, while |
| 293 |
+ |
prismatic and secondary prismatic had values for $\theta$ near |
| 294 |
+ |
43$^{o}$ as seen in Table \ref{tab:kappa}. These results indicate that the |
| 295 |
+ |
basal and pyramidal facets are more hydrophilic than the prismatic and |
| 296 |
+ |
secondary prismatic, and surprisingly, that the differential hydrophilicities of |
| 297 |
+ |
the crystal facets is not reflected in the spreading rate of the droplet. |
| 298 |
+ |
% This is in good agreement with our calculations of friction |
| 299 |
+ |
% coefficients, in which the basal |
| 300 |
+ |
% and pyramidal had a higher coefficient of kinetic friction than the |
| 301 |
+ |
% prismatic and secondary prismatic. Due to this, we beleive that the |
| 302 |
+ |
% differences in friction coefficients can be attributed to the varying |
| 303 |
+ |
% hydrophilicities of the facets. |
| 304 |
+ |
|
| 305 |
+ |
\subsection{Coefficient of friction of the interfaces} |
| 306 |
+ |
While investigating the kinetic coefficient of friction, there was found |
| 307 |
+ |
to be a dependence for $\mu_k$ |
| 308 |
+ |
on the temperature of the liquid water in the system. We believe this |
| 309 |
+ |
dependence |
| 310 |
+ |
arrises from the sharp discontinuity of the viscosity for the SPC/E model |
| 311 |
+ |
at temperatures approaching 200 K\cite{Kuang12}. Due to this, we propose |
| 312 |
+ |
a weighting to the interfacial friction coefficient, $\kappa$ by the |
| 313 |
+ |
shear viscosity of the fluid at 225 K. The interfacial friction coefficient |
| 314 |
+ |
relates the shear stress with the relative velocity of the fluid normal to the |
| 315 |
+ |
interface: |
| 316 |
+ |
\begin{equation}\label{Shenyu-13} |
| 317 |
+ |
j_{z}(p_{x}) = \kappa[v_{x}(fluid)-v_{x}(solid)] |
| 318 |
+ |
\end{equation} |
| 319 |
+ |
where $j_{z}(p_{x})$ is the applied momentum flux (shear stress) across $z$ |
| 320 |
+ |
in the |
| 321 |
+ |
$x$-dimension, and $v_{x}$(fluid) and $v_{x}$(solid) are the velocities |
| 322 |
+ |
directly adjacent to the interface. The shear viscosity, $\eta(T)$, of the |
| 323 |
+ |
fluid can be determined under a linear response of the momentum |
| 324 |
+ |
gradient to the applied shear stress by |
| 325 |
+ |
\begin{equation}\label{Shenyu-11} |
| 326 |
+ |
j_{z}(p_{x}) = \eta(T) \frac{\partial v_{x}}{\partial z}. |
| 327 |
+ |
\end{equation} |
| 328 |
+ |
Using eqs \eqref{Shenyu-13} and \eqref{Shenyu-11}, we can find the following |
| 329 |
+ |
expression for $\kappa$, |
| 330 |
+ |
\begin{equation}\label{kappa-1} |
| 331 |
+ |
\kappa = \eta(T) \frac{\partial v_{x}}{\partial z}\frac{1}{[v_{x}(fluid)-v_{x}(solid)]}. |
| 332 |
+ |
\end{equation} |
| 333 |
+ |
Here is where we will introduce the weighting term of $\eta(225)/\eta(T)$ |
| 334 |
+ |
giving us |
| 335 |
+ |
\begin{equation}\label{kappa-2} |
| 336 |
+ |
\kappa = \frac{\eta(225)}{[v_{x}(fluid)-v_{x}(solid)]}\frac{\partial v_{x}}{\partial z}. |
| 337 |
+ |
\end{equation} |
| 338 |
+ |
|
| 339 |
+ |
To obtain the value of $\eta(225)$ for the SPC/E model, a $31.09 \times 29.38 |
| 340 |
+ |
\times 124.39$ \AA\ box with 3744 SPC/E liquid water molecules was |
| 341 |
+ |
equilibrated to 225K, |
| 342 |
+ |
and 5 unique shearing experiments were performed. Each experiment was |
| 343 |
+ |
conducted in the NVE and were 5 ns in |
| 344 |
+ |
length. The VSS were attempted every timestep, which was set to 2 fs. |
| 345 |
+ |
For our SPC/E systems, we found $\eta(225)$ to be 0.0148 $\pm$ 0.0007 Pa s, |
| 346 |
+ |
roughly ten times larger than the value found for 280 K SPC/E bulk water by |
| 347 |
+ |
Kuang\cite{Kuang12}. |
| 348 |
+ |
|
| 349 |
+ |
The interfacial friction coefficient, $\kappa$, can equivalently be expressed |
| 350 |
+ |
as the ratio of the viscosity of the fluid to the slip length, $\delta$, which |
| 351 |
+ |
is an indication of how 'slippery' the interface is. |
| 352 |
+ |
\begin{equation}\label{kappa-3} |
| 353 |
+ |
\kappa = \frac{\eta}{\delta} |
| 354 |
+ |
\end{equation} |
| 355 |
+ |
In each of the systems, the interfacial temperature was kept fixed to 225K, |
| 356 |
+ |
which ensured the viscosity of the fluid at the |
| 357 |
+ |
interace was approximately the same. Thus, any significant variation in |
| 358 |
+ |
$\kappa$ between |
| 359 |
+ |
the systems indicates differences in the 'slipperiness' of the interfaces. |
| 360 |
+ |
As each of the ice systems are sheared relative to liquid water, the |
| 361 |
+ |
'slipperiness' of the interface can be taken as an indication of how |
| 362 |
+ |
hydrophobic or hydrophilic the interface is. The calculated $\kappa$ values |
| 363 |
+ |
found for the four crystal facets of Ice-I$_\mathrm{h}$ investigated are shown |
| 364 |
+ |
in Table \ref{tab:kappa}. The basal and pyramidal facets were found to have |
| 365 |
+ |
similar values of $\kappa \approx$ 0.0006 |
| 366 |
+ |
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}), while values of |
| 367 |
+ |
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1}) |
| 368 |
+ |
were found for the prismatic and secondary prismatic systems. |
| 369 |
+ |
These results indicate that the basal and pyramidal facets are |
| 370 |
+ |
more hydrophilic than the prismatic and secondary prismatic facets. |
| 371 |
+ |
|
| 372 |
|
\subsection{Interfacial width} |
| 373 |
|
In the literature there is good agreement that between the solid ice and |
| 374 |
|
the bulk water, there exists a region of 'slush-like' water molecules. |
| 385 |
|
the systems, parameters that depend on translational motion may give |
| 386 |
|
faulty results. A stuructural parameter will be less effected by the |
| 387 |
|
VSS-RNEMD perturbations to the system. Due to this, we have used the |
| 388 |
< |
local tetrahedral order parameter (Eq 5\cite{Louden13} to quantify the width of the interface, |
| 388 |
> |
local tetrahedral order parameter (Eq 5\cite{Louden13}) to quantify the width of the interface, |
| 389 |
|
which was originally described by Kumar\cite{Kumar09} and |
| 390 |
|
Errington\cite{Errington01}, and used by Bryk and Haymet in a previous study |
| 391 |
|
of ice/water interfaces.\cite{Bryk04b} |
| 400 |
|
(0,1), where a larger value indicates a more tetrahedral environment. |
| 401 |
|
The $q(z)$ for the bulk liquid was found to be $\approx $ 0.77, while values of |
| 402 |
|
$\approx $ 0.92 were more common for the ice. The tetrahedrality profiles were |
| 403 |
< |
fit using a hyperbolic tangent\cite{Louden13} designed to smoothly fit the |
| 403 |
> |
fit using a hyperbolic tangent (Eq. 6\cite{Louden13}) designed to smoothly fit the |
| 404 |
|
bulk to ice |
| 405 |
|
transition, while accounting for the thermal influence on the profile by the |
| 406 |
|
kinetic energy exchanges of the VSS-RNEMD moves. In panels $b$ and $c$, the |
| 505 |
|
those found for the basal ($d_{basal}\approx2.9$ \AA\ ) and prismatic |
| 506 |
|
($d_{prismatic}\approx3.5$ \AA\ ) systems. |
| 507 |
|
|
| 410 |
– |
\subsection{Coefficient of friction of the interfaces} |
| 411 |
– |
While investigating the kinetic coefficient of friction, there was found |
| 412 |
– |
to be a dependence for $\mu_k$ |
| 413 |
– |
on the temperature of the liquid water in the system. We believe this |
| 414 |
– |
dependence |
| 415 |
– |
arrises from the sharp discontinuity of the viscosity for the SPC/E model |
| 416 |
– |
at temperatures approaching 200 K\cite{Kuang12}. Due to this, we propose |
| 417 |
– |
a weighting to the interfacial friction coefficient, $\kappa$ by the |
| 418 |
– |
shear viscosity of the fluid at 225 K. The interfacial friction coefficient |
| 419 |
– |
relates the shear stress with the relative velocity of the fluid normal to the |
| 420 |
– |
interface: |
| 421 |
– |
\begin{equation}\label{Shenyu-13} |
| 422 |
– |
j_{z}(p_{x}) = \kappa[v_{x}(fluid)-v_{x}(solid)] |
| 423 |
– |
\end{equation} |
| 424 |
– |
where $j_{z}(p_{x})$ is the applied momentum flux (shear stress) across $z$ |
| 425 |
– |
in the |
| 426 |
– |
$x$-dimension, and $v_{x}$(fluid) and $v_{x}$(solid) are the velocities |
| 427 |
– |
directly adjacent to the interface. The shear viscosity, $\eta(T)$, of the |
| 428 |
– |
fluid can be determined under a linear response of the momentum |
| 429 |
– |
gradient to the applied shear stress by |
| 430 |
– |
\begin{equation}\label{Shenyu-11} |
| 431 |
– |
j_{z}(p_{x}) = \eta(T) \frac{\partial v_{x}}{\partial z}. |
| 432 |
– |
\end{equation} |
| 433 |
– |
Using eqs \eqref{Shenyu-13} and \eqref{Shenyu-11}, we can find the following |
| 434 |
– |
expression for $\kappa$, |
| 435 |
– |
\begin{equation}\label{kappa-1} |
| 436 |
– |
\kappa = \eta(T) \frac{\partial v_{x}}{\partial z}\frac{1}{[v_{x}(fluid)-v_{x}(solid)]}. |
| 437 |
– |
\end{equation} |
| 438 |
– |
Here is where we will introduce the weighting term of $\eta(225)/\eta(T)$ |
| 439 |
– |
giving us |
| 440 |
– |
\begin{equation}\label{kappa-2} |
| 441 |
– |
\kappa = \frac{\eta(225)}{[v_{x}(fluid)-v_{x}(solid)]}\frac{\partial v_{x}}{\partial z}. |
| 442 |
– |
\end{equation} |
| 508 |
|
|
| 444 |
– |
To obtain the value of $\eta(225)$ for the SPC/E model, a $31.09 \times 29.38 |
| 445 |
– |
\times 124.39$ \AA\ box with 3744 SPC/E liquid water molecules was |
| 446 |
– |
equilibrated to 225K, |
| 447 |
– |
and 5 unique shearing experiments were performed. Each experiment was |
| 448 |
– |
conducted in the NVE and were 5 ns in |
| 449 |
– |
length. The VSS were attempted every timestep, which was set to 2 fs. |
| 450 |
– |
For our SPC/E systems, we found $\eta(225)$ to be 0.0148 $\pm$ 0.0007 Pa s, |
| 451 |
– |
roughly ten times larger than the value found for 280 K SPC/E bulk water by |
| 452 |
– |
Kuang\cite{Kuang12}. |
| 509 |
|
|
| 454 |
– |
The interfacial friction coefficient, $\kappa$, can equivalently be expressed |
| 455 |
– |
as the ratio of the viscosity of the fluid to the slip length, $\delta$, which |
| 456 |
– |
is an indication of how 'slippery' the interface is. |
| 457 |
– |
\begin{equation}\label{kappa-3} |
| 458 |
– |
\kappa = \frac{\eta}{\delta} |
| 459 |
– |
\end{equation} |
| 460 |
– |
In each of the systems, the interfacial temperature was kept fixed to 225K, |
| 461 |
– |
which ensured the viscosity of the fluid at the |
| 462 |
– |
interace was approximately the same. Thus, any significant variation in |
| 463 |
– |
$\kappa$ between |
| 464 |
– |
the systems indicates differences in the 'slipperiness' of the interfaces. |
| 465 |
– |
As each of the ice systems are sheared relative to liquid water, the |
| 466 |
– |
'slipperiness' of the interface can be taken as an indication of how |
| 467 |
– |
hydrophobic or hydrophilic the interface is. The calculated $\kappa$ values |
| 468 |
– |
found for the four crystal facets of Ice-I$_\mathrm{h}$ investigated are shown |
| 469 |
– |
in Table \ref{tab:kappa}. The basal and pyramidal facets were found to have |
| 470 |
– |
similar values of $\kappa \approx$ 0.0006 |
| 471 |
– |
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}), while values of |
| 472 |
– |
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1}) |
| 473 |
– |
were found for the prismatic and secondary prismatic systems. |
| 474 |
– |
These results indicate that the basal and pyramidal facets are |
| 475 |
– |
more hydrophilic than the prismatic and secondary prismatic facets. |
| 510 |
|
|
| 477 |
– |
\subsection{Dynamic water contact angle} |
| 478 |
– |
|
| 479 |
– |
To determine the extent of wetting for each of the four crystal |
| 480 |
– |
facets, water contact angle simuations were performed. Contact angles |
| 481 |
– |
were obtained from these simulations by two methods. In the first |
| 482 |
– |
method, the contact angle was obtained from the $z$-center of mass |
| 483 |
– |
($z_{c.m.}$) of the water droplet as described by Hautman and Klein\cite{Hautman91} |
| 484 |
– |
and utilized by Hirvi and Pakkanen in their investigation of water |
| 485 |
– |
droplets on polyethylene and poly(vinyl chloride) |
| 486 |
– |
surface\cite{Hirvi06}. At each snapshot of the simulation, the contact |
| 487 |
– |
angle, $\theta$, was found by |
| 488 |
– |
\begin{equation}\label{contact_1} |
| 489 |
– |
\langle z_{c.m.}\rangle = 2^{-4/3}R_{0}\bigg(\frac{1-cos\theta}{2+cos\theta}\bigg)^{1/3}\frac{3+cos\theta}{2+cos\theta} , |
| 490 |
– |
\end{equation} |
| 491 |
– |
where $R_{0}$ is the radius of the free water droplet. In the second |
| 492 |
– |
method, the contact angle was obtained from fitting the droplet's |
| 493 |
– |
$z$-profile after radial averaging to a |
| 494 |
– |
circle as described by Ruijter, Blake, and |
| 495 |
– |
Coninck\cite{Ruijter99}. At each snapshot of the simulation the water droplet was |
| 496 |
– |
broken into bins, and the location of bin containing half-bulk density was |
| 497 |
– |
stored. Due to fluctuations close to the ice, all bins located within |
| 498 |
– |
2.0 \AA\ of the ice were discarded. The remaining stored bins were |
| 499 |
– |
then fit by a circle, whose tangential intersection with the ice plane could |
| 500 |
– |
be used to calculate the water |
| 501 |
– |
contact angle. These results proved noisey and unreliable when |
| 502 |
– |
compared with the first method, for these purposes we omit the data |
| 503 |
– |
from the second method. |
| 511 |
|
|
| 505 |
– |
The resulting water contact angle profiles generated by the first method |
| 506 |
– |
had an initial value of 180$^{o}$, and decayed over time. Each of |
| 507 |
– |
these profiles were fit to a biexponential decay, with a short time |
| 508 |
– |
piece to account for the water droplet initially adhering to the |
| 509 |
– |
surface, a long time piece describing the spreading of the droplet |
| 510 |
– |
over the surface, and an additive constant to capture the infinite |
| 511 |
– |
decay of the contact angle. We have found that the rate of the water |
| 512 |
– |
droplet spreading across all four crystal facets to be $\approx$ 0.7 |
| 513 |
– |
ns$^{-1}$. However, the basal and pyramidal facets |
| 514 |
– |
had an infinite decay value for $\theta$ of $\approx$ 35$^{o}$, while |
| 515 |
– |
prismatic and secondary prismatic had values for $\theta$ near |
| 516 |
– |
43$^{o}$ as seen in Table \ref{tab:kappa}. This indicates that the |
| 517 |
– |
basal and pyramidal facets are more hydrophilic than the prismatic and |
| 518 |
– |
secondary prismatic. This is in good agreement |
| 519 |
– |
with our calculations of friction coefficients, in which the basal |
| 520 |
– |
and pyramidal had a higher coefficient of kinetic friction than the |
| 521 |
– |
prismatic and secondary prismatic. Due to this, we beleive that the |
| 522 |
– |
differences in friction coefficients can be attributed to the varying |
| 523 |
– |
hydrophilicities of the facets. |
| 524 |
– |
|
| 512 |
|
\section{Conclusion} |
| 513 |
|
We present the results of molecular dynamics simulations of the basal, |
| 514 |
|
prismatic, pyrmaidal |
| 765 |
|
|
| 766 |
|
\begin{table}[h] |
| 767 |
|
\centering |
| 768 |
< |
\caption{Phyiscal properties of the basal, prismatic, pyramidal, and secondary prismatic facets of Ice-I$_\mathrm{h}$} |
| 769 |
< |
\label{tab:kappa} |
| 770 |
< |
\begin{tabular}{|ccccc|} \hline |
| 771 |
< |
& \multicolumn{2}{c}{$\kappa_{Drag direction}$ |
| 772 |
< |
(x10\textsuperscript{-4} amu \AA\textsuperscript{-2} fs\textsuperscript{-1})} & & \\ |
| 773 |
< |
Interface & $\kappa_{x}$ & $\kappa_{y}$ & $\theta_{\infty}$ & $K_{spread} (ns^{-1})$ \\ \hline |
| 774 |
< |
basal & $5.9 \pm 0.3$ & $6.5 \pm 0.8$ & $34.1 \pm 0.9$ & $0.60 \pm 0.07$ \\ |
| 775 |
< |
pyramidal & $5.8 \pm 0.4$ & $6.1 \pm 0.5$ & $35 \pm 3$ & $0.7 \pm 0.1$ \\ |
| 776 |
< |
prismatic & $3.0 \pm 0.2$ & $3.0 \pm 0.1$ & $45 \pm 3$ & $0.75 \pm 0.09$ \\ |
| 777 |
< |
secondary prismatic & $3.5 \pm 0.1$ & $3.3 \pm 0.2$ & $42 \pm 2$ & $0.69 \pm 0.03$ \\ \hline |
| 768 |
> |
\caption{Droplet and Shearing simulation parameters} |
| 769 |
> |
\label{tab:method} |
| 770 |
> |
\begin{tabular}{|cccc|ccc|} \hline |
| 771 |
> |
& \multicolumn{3}{c}{Droplet} & \multicolumn{3}{c|}{Shearing}\\ |
| 772 |
> |
Interface & $N_{ice}$ & $N_{droplet}$ & Lx, Ly (\AA) & |
| 773 |
> |
$N_{ice}$ & $N_{liquid}$ & Lx, Ly, Lz (\AA) \\ \hline |
| 774 |
> |
Basal & 12960& 2048 & (134.70, 140.04) & 900 & 1846 & (23.87, 35.83, 98.64)\\ |
| 775 |
> |
Prismatic & 9900& 2048 & (110.04, 115.00) & 3000 & 5464 & |
| 776 |
> |
(35.95, 35.65, 205.77)\\ |
| 777 |
> |
Pyramidal & 11136 & 2048& (143.75, 121.41) & 1216 & 2203 & |
| 778 |
> |
(143.75, 121.41)\\ |
| 779 |
> |
Secondary Prismatic & 11520 & 2048 & (146.72, 124.48) & 3840 & |
| 780 |
> |
8176 & (71.87, 31.66, 161.55)\\ |
| 781 |
> |
\hline |
| 782 |
|
\end{tabular} |
| 783 |
|
\end{table} |
| 784 |
|
|
| 785 |
|
|
| 786 |
|
\begin{table}[h] |
| 787 |
|
\centering |
| 788 |
< |
\caption{Shearing and Droplet simulation parameters} |
| 789 |
< |
\label{tab:method} |
| 790 |
< |
\begin{tabular}{|cccc|ccc|} \hline |
| 791 |
< |
& \multicolumn{3}{c}{Shearing} & \multicolumn{3}{c}{Droplet}\\ |
| 792 |
< |
Interface & $N_{ice}$ & $N_{liquid}$ & Lx, Ly, Lz (\AA) & |
| 793 |
< |
$N_{ice}$ & $N_{droplet}$ & Lx, Ly (\AA) \\ \hline |
| 794 |
< |
Basal & 900 & 1846 & (23.87, 35.83, 98.64) & 12960 & 2048 & (134.70, 140.04)\\ |
| 795 |
< |
Prismatic & 3000 & 5464 & (35.95, 35.65, 205.77) & 9900 & 2048 & |
| 796 |
< |
(110.04, 115.00)\\ |
| 797 |
< |
Pyramidal & 1216 & 2203& (37.47, 29.50, 93.02) & 11136 & 2048 & |
| 798 |
< |
(143.75, 121.41)\\ |
| 799 |
< |
Secondary Prismatic & 3840 & 8176 & (71.87, 31.66, 161.55) & 11520 & |
| 800 |
< |
2048 & (146.72, 124.48)\\ |
| 801 |
< |
\hline |
| 788 |
> |
\caption{Phyiscal properties of the basal, prismatic, pyramidal, and secondary prismatic facets of Ice-I$_\mathrm{h}$} |
| 789 |
> |
\label{tab:kappa} |
| 790 |
> |
\begin{tabular}{|ccc|cccc|} \hline |
| 791 |
> |
& \multicolumn{2}{c}{Droplet} & \multicolumn{4}{c|}{Shearing}\\ |
| 792 |
> |
Interface & $\theta^{\circ}_{\infty}$ & $K_{spread} (ns^{-1})$ & |
| 793 |
> |
$\kappa_{x}$ & $\kappa_{y}$ & d$_{q_{z}}$ (\AA) & d$_{\tau}$ (\AA) \\ \hline |
| 794 |
> |
Basal & $34.1 \pm 0.9$ &$0.60 \pm 0.07$ & $5.9 \pm 0.3$&$6.5 \pm 0.8$ |
| 795 |
> |
& $3.2 \pm 0.4$ & $2.9$ \\ |
| 796 |
> |
Pyramidal & $35 \pm 3$ & $0.7 \pm 0.1$ & $5.8 \pm 0.4$ & $6.1 \pm |
| 797 |
> |
0.5$ & $3.2 \pm 0.2$ & $2.7$ \\ |
| 798 |
> |
Prismatic & $45 \pm 3$ & $0.75 \pm 0.09$ & $3.0 \pm 0.2$ & $3.0 \pm |
| 799 |
> |
0.1$ & $3.6 \pm 0.2$ & $3.5$ \\ |
| 800 |
> |
Secondary Prismatic & $42 \pm 2$ & $0.69 \pm 0.03$ & $3.5 \pm 0.1$ & |
| 801 |
> |
$3.3 \pm 0.2$ & $3.2 \pm 0.2$ & $3$ \\ \hline |
| 802 |
|
\end{tabular} |
| 803 |
|
\end{table} |
| 804 |
|
|
