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\begin{document} |
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\title{Friction at water / ice-I\textsubscript{h} interfaces: Do the |
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different facets of ice have different hydrophilicities?} |
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\title{The different facets of ice have different hydrophilicities: |
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Friction at water / ice-I\textsubscript{h} interfaces} |
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\author{Patrick B. Louden\affil{1}{Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, |
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IN 46556} |
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\begin{article} |
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\begin{abstract} |
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In this paper we present evidence that some of the crystal facets |
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of ice-I$_\mathrm{h}$ posess structural features that can halve |
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the effective hydrophilicity of the ice/water interface. The |
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spreading dynamics of liquid water droplets on ice facets exhibits |
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long-time behavior that differs substantially for the prismatic |
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We present evidence that some of the crystal facets of |
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ice-I$_\mathrm{h}$ posess structural features that can reduce the |
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effective hydrophilicity of the ice/water interface. The spreading |
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dynamics of liquid water droplets on ice facets exhibits long-time |
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behavior that differs substantially for the prismatic |
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$\{1~0~\bar{1}~0\}$ and secondary prism $\{1~1~\bar{2}~0\}$ facets |
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when compared with the basal $\{0001\}$ and pyramidal |
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$\{2~0~\bar{2}~1\}$ facets. We also present the results of |
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surfaces do not have strong enough interactions with water to overcome |
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the internal attraction between molecules in the liquid phase, and the |
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degree of hydrophilicity of a surface can be described by the extent a |
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droplet can spread out over the surface. The contact angle formed |
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between the solid and the liquid depends on the free energies of the |
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three interfaces involved, and is given by Young's |
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equation.\cite{Young05} |
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droplet can spread out over the surface. The contact angle, $\theta$, |
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formed between the solid and the liquid depends on the free energies |
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of the three interfaces involved, and is given by Young's |
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equation~\cite{Young05}, |
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\begin{equation}\label{young} |
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\cos\theta = (\gamma_{sv} - \gamma_{sl})/\gamma_{lv} . |
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\end{equation} |
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Here $\gamma_{sv}$, $\gamma_{sl}$, and $\gamma_{lv}$ are the free |
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energies of the solid/vapor, solid/liquid, and liquid/vapor interfaces |
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energies of the solid/vapor, solid/liquid, and liquid/vapor interfaces, |
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respectively. Large contact angles, $\theta > 90^{\circ}$, correspond |
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to hydrophobic surfaces with low wettability, while small contact |
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angles, $\theta < 90^{\circ}$, correspond to hydrophilic surfaces. |
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Experimentally, measurements of the contact angle of sessile drops is |
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often used to quantify the extent of wetting on surfaces with |
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thermally selective wetting |
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characteristics.\cite{Tadanaga00,Liu04,Sun04} |
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characteristics~\cite{Tadanaga00,Liu04,Sun04}. |
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|
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Nanometer-scale structural features of a solid surface can influence |
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the hydrophilicity to a surprising degree. Small changes in the |
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\sim 0^{\circ}$.\cite{Koishi09} This is often referred to as the |
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Cassie-Baxter to Wenzel transition. Nano-pillared surfaces with |
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electrically tunable Cassie-Baxter and Wenzel states have also been |
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observed.\cite{Herbertson06,Dhindsa06,Verplanck07,Ahuja08,Manukyan11} |
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observed~\cite{Herbertson06,Dhindsa06,Verplanck07,Ahuja08,Manukyan11}. |
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Luzar and coworkers have modeled these transitions on nano-patterned |
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surfaces\cite{Daub07,Daub10,Daub11,Ritchie12}, and have found the |
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surfaces~\cite{Daub07,Daub10,Daub11,Ritchie12}, and have found the |
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change in contact angle is due to the field-induced perturbation of |
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hydrogen bonding at the liquid/vapor interface.\cite{Daub07} |
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hydrogen bonding at the liquid/vapor interface~\cite{Daub07}. |
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|
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One would expect the interfaces of ice to be highly hydrophilic (and |
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possibly the most hydrophilic of all solid surfaces). In this paper we |
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present evidence that some of the crystal facets of ice-I$_\mathrm{h}$ |
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have structural features that can halve the effective hydrophilicity. |
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have structural features that can reduce the effective hydrophilicity. |
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Our evidence for this comes from molecular dynamics (MD) simulations |
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of the spreading dynamics of liquid droplets on these facets, as well |
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as reverse non-equilibrium molecular dynamics (RNEMD) simulations of |
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quasi-liquid layer (QLL), at temperatures near the melting point. MD |
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simulations of the facets of ice-I$_\mathrm{h}$ exposed to vacuum have |
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found QLL widths of approximately 10 \AA\ at 3 K below the melting |
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point.\cite{Conde08} Similarly, Limmer and Chandler have used the mW |
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point~\cite{Conde08}. Similarly, Limmer and Chandler have used the mW |
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water model~\cite{Molinero09} and statistical field theory to estimate |
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QLL widths at similar temperatures to be about 3 nm.\cite{Limmer14} |
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QLL widths at similar temperatures to be about 3 nm~\cite{Limmer14}. |
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|
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Recently, Sazaki and Furukawa have developed a technique using laser |
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confocal microscopy combined with differential interference contrast |
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microscopy that has sufficient spatial and temporal resolution to |
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visulaize and quantitatively analyze QLLs on ice crystals at |
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temperatures near melting.\cite{Sazaki10} They have found the width of |
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temperatures near melting~\cite{Sazaki10}. They have found the width of |
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the QLLs perpindicular to the surface at -2.2$^{o}$C to be 3-4 \AA\ |
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wide. They have also seen the formation of two immiscible QLLs, which |
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displayed different dynamics on the crystal surface.\cite{Sazaki12} |
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displayed different dynamics on the crystal surface~\cite{Sazaki12}. |
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|
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There is now significant interest in the \textit{tribological} |
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properties of ice/ice and ice/water interfaces in the geophysics |
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community. Understanding the dynamics of solid-solid shearing that is |
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mediated by a liquid layer\cite{Cuffey99, Bell08} will aid in |
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mediated by a liquid layer~\cite{Cuffey99, Bell08} will aid in |
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understanding the macroscopic motion of large ice |
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masses.\cite{Casassa91, Sukhorukov13, Pritchard12, Lishman13} |
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masses~\cite{Casassa91, Sukhorukov13, Pritchard12, Lishman13}. |
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|
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Using molecular dynamics simulations, Samadashvili has recently shown |
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that when two smooth ice slabs slide past one another, a stable |
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liquid-like layer develops between them.\cite{Samadashvili13} In a |
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liquid-like layer develops between them~\cite{Samadashvili13}. In a |
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previous study, our RNEMD simulations of ice-I$_\mathrm{h}$ shearing |
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through liquid water have provided quantitative estimates of the |
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solid-liquid kinetic friction coefficients.\cite{Louden13} These |
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solid-liquid kinetic friction coefficients~\cite{Louden13}. These |
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displayed a factor of two difference between the basal and prismatic |
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facets. The friction was found to be independent of shear direction |
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relative to the surface orientation. We attributed facet-based |
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molecules within 3 \AA\ of any atoms in the ice slabs. Each of the |
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combined ice/water systems were then equilibrated at 225K, which is |
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the liquid-ice coexistence temperature for SPC/E |
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water~\cite{Bryk02}. Ref. \citealp{Louden13} contains a more detailed |
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explanation of the construction of similar ice/water interfaces. The |
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resulting dimensions as well as the number of ice and liquid water |
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molecules contained in each of these systems are shown in Table |
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\ref{tab:method}. |
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water~\cite{Bryk02}. Reference \citealp{Louden13} contains a more |
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detailed explanation of the construction of similar ice/water |
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interfaces. The resulting dimensions as well as the number of ice and |
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liquid water molecules contained in each of these systems are shown in |
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Table \ref{tab:method}. |
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|
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The SPC/E water model~\cite{Berendsen87} has been extensively |
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characterized over a wide range of liquid |
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conditions,\cite{Arbuckle02,Kuang12} and its phase diagram has been |
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well studied.\cite{Baez95,Bryk04b,Sanz04b,Fennell:2005fk} With longer |
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conditions~\cite{Arbuckle02,Kuang12}, and its phase diagram has been |
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well studied~\cite{Baez95,Bryk04b,Sanz04b,Fennell:2005fk}, With longer |
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cutoff radii and careful treatment of electrostatics, SPC/E mostly |
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avoids metastable crystalline morphologies like |
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ice-\textit{i}~\cite{Fennell:2005fk} and ice-B~\cite{Baez95}. The |
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free energies and melting points |
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\cite{Baez95,Arbuckle02,Gay02,Bryk02,Bryk04b,Sanz04b,Fennell:2005fk,Fernandez06,Abascal07,Vrbka07} |
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free energies and melting |
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points~\cite{Baez95,Arbuckle02,Gay02,Bryk02,Bryk04b,Sanz04b,Fennell:2005fk,Fernandez06,Abascal07,Vrbka07} |
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of various other crystalline polymorphs have also been calculated. |
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Haymet \textit{et al.} have studied quiescent Ice-I$_\mathrm{h}$/water |
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interfaces using the SPC/E water model, and have seen structural and |
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surface) until a large surface ($>$ 126 nm\textsuperscript{2}) had |
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been created. The sizes and numbers of molecules in each of the |
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surfaces is given in Table \ref{tab:method}. Weak translational |
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restraining potentials with spring constants of 1.5-4.0~$\mathrm{kcal\ |
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mol}^{-1}\mathrm{~\AA}^{-2}$ were applied to the centers of mass of |
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each molecule in order to prevent surface melting, although the |
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molecules were allowed to reorient freely. A water doplet containing |
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2048 SPC/E molecules was created separately. Droplets of this size can |
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produce agreement with the Young contact angle extrapolated to an |
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infinite drop size~\cite{Daub10}. The surfaces and droplet were |
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independently equilibrated to 225 K, at which time the droplet was |
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placed 3-5~\AA\ above the surface. Five statistically independent |
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simulations were carried out for each facet, and the droplet was |
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placed at unique $x$ and $y$ locations for each of these simulations. |
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Each simulation was 5~ns in length and was conducted in the |
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microcanonical (NVE) ensemble. Representative configurations for the |
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droplet on the prismatic facet are shown in figure \ref{fig:Droplet}. |
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restraining potentials with spring constants of 1.5~$\mathrm{kcal\ |
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mol}^{-1}\mathrm{~\AA}^{-2}$ (prismatic and pyramidal facets) or |
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4.0~$\mathrm{kcal\ mol}^{-1}\mathrm{~\AA}^{-2}$ (basal facet) were |
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applied to the centers of mass of each molecule in order to prevent |
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surface melting, although the molecules were allowed to reorient |
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freely. A water doplet containing 2048 SPC/E molecules was created |
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separately. Droplets of this size can produce agreement with the Young |
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contact angle extrapolated to an infinite drop size~\cite{Daub10}. The |
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surfaces and droplet were independently equilibrated to 225 K, at |
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which time the droplet was placed 3-5~\AA\ above the surface. Five |
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statistically independent simulations were carried out for each facet, |
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and the droplet was placed at unique $x$ and $y$ locations for each of |
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these simulations. Each simulation was 5~ns in length and was |
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conducted in the microcanonical (NVE) ensemble. Representative |
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configurations for the droplet on the prismatic facet are shown in |
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figure \ref{fig:Droplet}. |
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|
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– |
|
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\subsection{Shearing Simulations (Interfaces in Bulk Water)} |
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|
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To perform the shearing simulations, the velocity shearing and scaling |
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The kinetic energy flux (producing a thermal gradient) is necessary |
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when performing shearing simulations at the ice-water interface in |
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order to prevent the frictional heating due to the shear from melting |
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the interface. Reference \citealp{Louden13} provides more details on |
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the VSS-RNEMD method as applied to ice-water interfaces. A |
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representative configuration of the solvated prismatic facet being |
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sheared through liquid water is shown in figure \ref{fig:Shearing}. |
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the crystal. Reference \citealp{Louden13} provides more details on the |
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VSS-RNEMD method as applied to ice-water interfaces. A representative |
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configuration of the solvated prismatic facet being sheared through |
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liquid water is shown in figure \ref{fig:Shearing}. |
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|
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In the results discussed below, the exchanges between the two regions |
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were carried out every 2 fs (e.g. every time step). This was done to |
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minimize the magnitude of each individual momentum exchange. Because |
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individual VSS-RNEMD exchanges conserve both total energy and linear |
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momentum, the method can be ``bolted-on'' to simulations in any |
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ensemble. The simulations of the pyramidal interface were performed |
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under the canonical (NVT) ensemble. When time correlation functions |
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were computed (see section \ref{sec:orient}), these simulations were |
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done in the microcanonical (NVE) ensemble. All simulations of the |
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other interfaces were done in the microcanonical ensemble. |
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The exchanges between the two regions were carried out every 2 fs |
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(e.g. every time step). This was done to minimize the magnitude of |
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each individual momentum exchange. Because individual VSS-RNEMD |
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exchanges conserve both total energy and linear momentum, the method |
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can be ``bolted-on'' to simulations in any ensemble. The simulations |
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of the pyramidal interface were performed under the canonical (NVT) |
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ensemble. When time correlation functions were computed, the RNEMD |
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simulations were done in the microcanonical (NVE) ensemble. All |
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simulations of the other interfaces were carried out in the |
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microcanonical ensemble. |
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|
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\section{Results} |
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\subsection{Ice - Water Contact Angles} |
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large density fluctuations close to the ice, all shells located within |
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2 \AA\ of the ice surface were left out of the circular fits. The |
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height of the solid surface ($z_\mathrm{suface}$) along with the best |
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fitting central height ($z_\mathrm{center}$) and radius |
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fitting origin ($z_\mathrm{droplet}$) and radius |
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($r_\mathrm{droplet}$) of the droplet can then be used to compute the |
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contact angle, |
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\begin{equation} |
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\theta = 90 + \frac{180}{\pi} \sin^{-1}\left(\frac{z_\mathrm{center} - |
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\theta = 90 + \frac{180}{\pi} \sin^{-1}\left(\frac{z_\mathrm{droplet} - |
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z_\mathrm{surface}}{r_\mathrm{droplet}} \right). |
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\end{equation} |
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Both methods provided similar estimates of the dynamic contact angle, |
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prismatic and secondary prismatic had values for $\theta_\infty$ near |
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43$^{o}$ as seen in Table \ref{tab:kappa}. |
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|
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These results indicate that the basal and pyramidal facets are |
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somewhat more hydrophilic than the prismatic and secondary prism |
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facets, and surprisingly, that the differential hydrophilicities of |
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the crystal facets is not reflected in the spreading rate of the |
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droplet. |
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These results indicate that the basal and pyramidal facets are more |
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hydrophilic than the prismatic and secondary prism facets, and |
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surprisingly, that the differential hydrophilicities of the crystal |
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facets is not reflected in the spreading rate of the droplet. |
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|
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% This is in good agreement with our calculations of friction |
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% coefficients, in which the basal |
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% hydrophilicities of the facets. |
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|
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\subsection{Coefficient of friction of the interfaces} |
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While investigating the kinetic coefficient of friction, there was found |
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to be a dependence for $\mu_k$ |
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on the temperature of the liquid water in the system. We believe this |
| 357 |
< |
dependence |
| 358 |
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arrises from the sharp discontinuity of the viscosity for the SPC/E model |
| 359 |
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at temperatures approaching 200 K\cite{Kuang12}. Due to this, we propose |
| 360 |
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a weighting to the interfacial friction coefficient, $\kappa$ by the |
| 361 |
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shear viscosity of the fluid at 225 K. The interfacial friction coefficient |
| 362 |
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relates the shear stress with the relative velocity of the fluid normal to the |
| 363 |
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interface: |
| 364 |
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\begin{equation}\label{Shenyu-13} |
| 365 |
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j_{z}(p_{x}) = \kappa[v_{x}(fluid)-v_{x}(solid)] |
| 366 |
< |
\end{equation} |
| 367 |
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where $j_{z}(p_{x})$ is the applied momentum flux (shear stress) across $z$ |
| 368 |
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in the |
| 369 |
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$x$-dimension, and $v_{x}$(fluid) and $v_{x}$(solid) are the velocities |
| 370 |
< |
directly adjacent to the interface. The shear viscosity, $\eta(T)$, of the |
| 371 |
< |
fluid can be determined under a linear response of the momentum |
| 372 |
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gradient to the applied shear stress by |
| 354 |
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In a bulk fluid, the shear viscosity, $\eta$, can be determined |
| 355 |
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assuming a linear response to a shear stress, |
| 356 |
|
\begin{equation}\label{Shenyu-11} |
| 357 |
< |
j_{z}(p_{x}) = \eta(T) \frac{\partial v_{x}}{\partial z}. |
| 357 |
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j_{z}(p_{x}) = \eta \frac{\partial v_{x}}{\partial z}. |
| 358 |
|
\end{equation} |
| 359 |
< |
Using eqs \eqref{Shenyu-13} and \eqref{Shenyu-11}, we can find the following |
| 360 |
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expression for $\kappa$, |
| 361 |
< |
\begin{equation}\label{kappa-1} |
| 362 |
< |
\kappa = \eta(T) \frac{\partial v_{x}}{\partial z}\frac{1}{[v_{x}(fluid)-v_{x}(solid)]}. |
| 359 |
> |
Here $j_{z}(p_{x})$ is the flux (in $x$-momentum) that is transferred |
| 360 |
> |
in the $z$ direction (i.e. the shear stress). The RNEMD simulations |
| 361 |
> |
impose an artificial momentum flux between two regions of the |
| 362 |
> |
simulation, and the velocity gradient is the fluid's response. This |
| 363 |
> |
technique has now been applied quite widely to determine the |
| 364 |
> |
viscosities of a number of bulk fluids~\cite{}. |
| 365 |
> |
|
| 366 |
> |
At the interface between two phases (e.g. liquid / solid) the same |
| 367 |
> |
momentum flux creates a velocity difference between the two materials, |
| 368 |
> |
and this can be used to define an interfacial friction coefficient |
| 369 |
> |
($\kappa$), |
| 370 |
> |
\begin{equation}\label{Shenyu-13} |
| 371 |
> |
j_{z}(p_{x}) = \kappa \left[ v_{x}(liquid) - v_{x}(solid) \right] |
| 372 |
|
\end{equation} |
| 373 |
< |
Here is where we will introduce the weighting term of $\eta(225)/\eta(T)$ |
| 374 |
< |
giving us |
| 373 |
> |
where $v_{x}(liquid)$ and $v_{x}(solid)$ are the velocities measured |
| 374 |
> |
directly adjacent to the interface. |
| 375 |
> |
|
| 376 |
> |
The simulations described here contain significant quantities of both |
| 377 |
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liquid and solid phases, and the momentum flux must traverse a region |
| 378 |
> |
of the liquid that is simultaneously under a thermal gradient. Since |
| 379 |
> |
the liquid has a temperature-dependent shear viscosity, $\eta(T)$, |
| 380 |
> |
estimates of the solid-liquid friction coefficient can be obtained if |
| 381 |
> |
one knows the viscosity of the liquid at the interface (i.e. at the |
| 382 |
> |
melting temperature, $T_m$), |
| 383 |
|
\begin{equation}\label{kappa-2} |
| 384 |
< |
\kappa = \frac{\eta(225)}{[v_{x}(fluid)-v_{x}(solid)]}\frac{\partial v_{x}}{\partial z}. |
| 384 |
> |
\kappa = \frac{\eta(T_{m})}{\left[v_{x}(fluid)-v_{x}(solid)\right]}\left(\frac{\partial v_{x}}{\partial z}\right). |
| 385 |
|
\end{equation} |
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< |
|
| 387 |
< |
To obtain the value of $\eta(225)$ for the SPC/E model, a $31.09 \times 29.38 |
| 388 |
< |
\times 124.39$ \AA\ box with 3744 SPC/E liquid water molecules was |
| 389 |
< |
equilibrated to 225K, |
| 390 |
< |
and 5 unique shearing experiments were performed. Each experiment was |
| 391 |
< |
conducted in the NVE and were 5 ns in |
| 392 |
< |
length. The VSS were attempted every timestep, which was set to 2 fs. |
| 393 |
< |
For our SPC/E systems, we found $\eta(225)$ to be 0.0148 $\pm$ 0.0007 Pa s, |
| 394 |
< |
roughly ten times larger than the value found for 280 K SPC/E bulk water by |
| 395 |
< |
Kuang\cite{Kuang12}. |
| 386 |
> |
For SPC/E, the melting temperature of Ice-I$_\mathrm{h}$ is estimated |
| 387 |
> |
to be 225~K~\cite{Bryk02}. To obtain the value of |
| 388 |
> |
$\eta(225\mathrm{~K})$ for the SPC/E model, a $31.09 \times 29.38 |
| 389 |
> |
\times 124.39$ \AA\ box with 3744 water molecules in a disordered |
| 390 |
> |
configuration was equilibrated to 225~K, and five |
| 391 |
> |
statistically-independent shearing simulations were performed (with |
| 392 |
> |
imposed fluxes that spanned a range of XXXX-YYYY). Each simulation |
| 393 |
> |
was conducted in the microcanonical ensemble with total simulation |
| 394 |
> |
times of 5 ns. The VSS-RNEMD exchanges were carried out every 2 fs. We |
| 395 |
> |
estimate $\eta(225\mathrm{~K})$ to be 0.0148 $\pm$ 0.0007 Pa s for |
| 396 |
> |
SPC/E, roughly ten times larger than the shear viscosity previously |
| 397 |
> |
computed at 280~K~\cite{Kuang12}. |
| 398 |
|
|
| 399 |
< |
The interfacial friction coefficient, $\kappa$, can equivalently be expressed |
| 400 |
< |
as the ratio of the viscosity of the fluid to the slip length, $\delta$, which |
| 401 |
< |
is an indication of how 'slippery' the interface is. |
| 399 |
> |
The interfacial friction coefficient, $\kappa$, can equivalently be |
| 400 |
> |
expressed as the ratio of the viscosity of the fluid to the |
| 401 |
> |
hydrodynamic slip length, $\delta$, which is an indication of strength |
| 402 |
> |
of the interactions between the solid and liquid phases, |
| 403 |
|
\begin{equation}\label{kappa-3} |
| 404 |
|
\kappa = \frac{\eta}{\delta} |
| 405 |
|
\end{equation} |
| 406 |
< |
In each of the systems, the interfacial temperature was kept fixed to 225K, |
| 407 |
< |
which ensured the viscosity of the fluid at the |
| 408 |
< |
interace was approximately the same. Thus, any significant variation in |
| 409 |
< |
$\kappa$ between |
| 410 |
< |
the systems indicates differences in the 'slipperiness' of the interfaces. |
| 411 |
< |
As each of the ice systems are sheared relative to liquid water, the |
| 412 |
< |
'slipperiness' of the interface can be taken as an indication of how |
| 413 |
< |
hydrophobic or hydrophilic the interface is. The calculated $\kappa$ values |
| 414 |
< |
found for the four crystal facets of Ice-I$_\mathrm{h}$ investigated are shown |
| 415 |
< |
in Table \ref{tab:kappa}. The basal and pyramidal facets were found to have |
| 416 |
< |
similar values of $\kappa \approx$ 0.0006 |
| 417 |
< |
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}), while values of |
| 418 |
< |
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1}) |
| 416 |
< |
were found for the prismatic and secondary prismatic systems. |
| 417 |
< |
These results indicate that the basal and pyramidal facets are |
| 418 |
< |
more hydrophilic than the prismatic and secondary prismatic facets. |
| 406 |
> |
The connection between slip length and surface hydrophobicity is not |
| 407 |
> |
yet clear. In some simulations, the slip length has been found to have |
| 408 |
> |
a link to the effective surface hydrophobicity~\cite{Sendner:2009uq}, |
| 409 |
> |
although Ho \textit{et al.} have found that liquid water can also slip |
| 410 |
> |
on hydrophilic surfaces~\cite{Ho:2011zr}. Experimental evidence for a |
| 411 |
> |
direct tie between slip length and hydrophobicity is also not |
| 412 |
> |
definitive. Total-internal reflection particle image velocimetry |
| 413 |
> |
(TIR-PIV) studies have suggested that there is a link between slip |
| 414 |
> |
length and effective |
| 415 |
> |
hydrophobicity~\cite{Lasne:2008vn,Bouzigues:2008ys}. However, recent |
| 416 |
> |
surface sensitive cross-correlation spectroscopy (TIR-FCCS) |
| 417 |
> |
measurements have seen similar slip behavior for both hydrophobic and |
| 418 |
> |
hydrophilic surfaces~\cite{Schaeffel:2013kx}. |
| 419 |
|
|
| 420 |
< |
\subsection{Interfacial width} |
| 421 |
< |
In the literature there is good agreement that between the solid ice and |
| 422 |
< |
the bulk water, there exists a region of 'slush-like' water molecules. |
| 423 |
< |
In this region, the water molecules are structurely distinguishable and |
| 424 |
< |
behave differently than those of the solid ice or the bulk water. |
| 425 |
< |
The characteristics of this region have been defined by both structural |
| 426 |
< |
and dynamic properties; and its width has been measured by the change of these |
| 427 |
< |
properties from their bulk liquid values to those of the solid ice. |
| 428 |
< |
Examples of these properties include the density, the diffusion constant, and |
| 429 |
< |
the translational order profile. \cite{Bryk02,Karim90,Gay02,Hayward01,Hayward02,Karim88} |
| 420 |
> |
In each of the systems studied here, the interfacial temperature was |
| 421 |
> |
kept fixed to 225K, which ensured the viscosity of the fluid at the |
| 422 |
> |
interace was identical. Thus, any significant variation in $\kappa$ |
| 423 |
> |
between the systems is a direct indicator of the slip length and the |
| 424 |
> |
effective interaction strength between the solid and liquid phases. |
| 425 |
|
|
| 426 |
< |
Since the VSS-RNEMD moves used to impose the thermal and velocity gradients |
| 427 |
< |
perturb the momenta of the water molecules in |
| 428 |
< |
the systems, parameters that depend on translational motion may give |
| 429 |
< |
faulty results. A stuructural parameter will be less effected by the |
| 430 |
< |
VSS-RNEMD perturbations to the system. Due to this, we have used the |
| 431 |
< |
local tetrahedral order parameter (Eq 5\cite{Louden13}) to quantify the width of the interface, |
| 432 |
< |
which was originally described by Kumar\cite{Kumar09} and |
| 433 |
< |
Errington\cite{Errington01}, and used by Bryk and Haymet in a previous study |
| 434 |
< |
of ice/water interfaces.\cite{Bryk04b} |
| 435 |
< |
|
| 436 |
< |
To determine the width of the interfaces, each of the systems were |
| 437 |
< |
divided into 100 artificial bins along the |
| 438 |
< |
$z$-dimension, and the local tetrahedral order parameter, $q(z)$, was |
| 439 |
< |
time-averaged for each of the bins, resulting in a tetrahedrality profile of |
| 445 |
< |
the system. These profiles are shown across the $z$-dimension of the systems |
| 446 |
< |
in panel $a$ of Figures \ref{fig:pyrComic} |
| 447 |
< |
and \ref{fig:spComic} (black circles). The $q(z)$ function has a range of |
| 448 |
< |
(0,1), where a larger value indicates a more tetrahedral environment. |
| 449 |
< |
The $q(z)$ for the bulk liquid was found to be $\approx $ 0.77, while values of |
| 450 |
< |
$\approx $ 0.92 were more common for the ice. The tetrahedrality profiles were |
| 451 |
< |
fit using a hyperbolic tangent (Eq. 6\cite{Louden13}) designed to smoothly fit the |
| 452 |
< |
bulk to ice |
| 453 |
< |
transition, while accounting for the thermal influence on the profile by the |
| 454 |
< |
kinetic energy exchanges of the VSS-RNEMD moves. In panels $b$ and $c$, the |
| 455 |
< |
resulting thermal and velocity gradients from an imposed kinetic energy and |
| 456 |
< |
momentum fluxes can be seen. The verticle dotted |
| 457 |
< |
lines traversing all three panels indicate the midpoints of the interface |
| 458 |
< |
as determined by the hyperbolic tangent fit of the tetrahedrality profiles. |
| 459 |
< |
|
| 460 |
< |
From fitting the tetrahedrality profiles for each of the 0.5 nanosecond |
| 461 |
< |
simulations (panel c of Figures \ref{fig:pyrComic} and \ref{fig:spComic}) |
| 462 |
< |
by eq. 6\cite{Louden13},we find the interfacial width to be |
| 463 |
< |
3.2 $\pm$ 0.2 and 3.2 $\pm$ 0.2 \AA\ for the control system with no applied |
| 464 |
< |
momentum flux for both the pyramidal and secondary prismatic systems. |
| 465 |
< |
Over the range of shear rates investigated, |
| 466 |
< |
0.6 $\pm$ 0.2 $\mathrm{ms}^{-1} \rightarrow$ 5.6 $\pm$ 0.4 $\mathrm{ms}^{-1}$ |
| 467 |
< |
for the pyramidal system and 0.9 $\pm$ 0.3 $\mathrm{ms}^{-1} \rightarrow$ 5.4 |
| 468 |
< |
$\pm$ 0.1 $\mathrm{ms}^{-1}$ for the secondary prismatic, we found no |
| 469 |
< |
significant change in the interfacial width. This follows our previous |
| 470 |
< |
findings of the basal and |
| 471 |
< |
prismatic systems, in which the interfacial width was invarient of the |
| 472 |
< |
shear rate of the ice. The interfacial width of the quiescent basal and |
| 473 |
< |
prismatic systems was found to be 3.2 $\pm$ 0.4 \AA\ and 3.6 $\pm$ 0.2 \AA\ |
| 474 |
< |
respectively, over the range of shear rates investigated, 0.6 $\pm$ 0.3 |
| 475 |
< |
$\mathrm{ms}^{-1} \rightarrow$ 5.3 $\pm$ 0.5 $\mathrm{ms}^{-1}$ for the basal |
| 476 |
< |
system and 0.9 $\pm$ 0.2 $\mathrm{ms}^{-1} \rightarrow$ 4.5 $\pm$ 0.1 |
| 477 |
< |
$\mathrm{ms}^{-1}$ for the prismatic. |
| 426 |
> |
The calculated $\kappa$ values found for the four crystal facets of |
| 427 |
> |
Ice-I$_\mathrm{h}$ are shown in Table \ref{tab:kappa}. The basal and |
| 428 |
> |
pyramidal facets were found to have similar values of $\kappa \approx |
| 429 |
> |
6$ ($\times 10^{-4} \mathrm{amu~\AA}^{-2}\mathrm{fs}^{-1}$), while the |
| 430 |
> |
prismatic and secondary prism facets exhibited $\kappa \approx 3$ |
| 431 |
> |
($\times 10^{-4} \mathrm{amu~\AA}^{-2}\mathrm{fs}^{-1}$). These |
| 432 |
> |
results are also essentially independent of shearing direction |
| 433 |
> |
relative to features on the surface of the facets. The friction |
| 434 |
> |
coefficients indicate that the basal and pyramidal facets have |
| 435 |
> |
significantly stronger interactions with liquid water than either of |
| 436 |
> |
the two prismatic facets. This is in agreement with the contact angle |
| 437 |
> |
results above - both of the high-friction facets exhbited smaller |
| 438 |
> |
contact angles, suggesting that the solid-liquid friction is |
| 439 |
> |
correlated with the hydrophilicity of these facets. |
| 440 |
|
|
| 441 |
< |
These results indicate that the surface structure of the exposed ice crystal |
| 442 |
< |
has little to no effect on how far into the bulk the ice-like structural |
| 443 |
< |
ordering is. Also, it appears that the interface is not structurally effected |
| 444 |
< |
by the movement of water over the ice. |
| 441 |
> |
\subsection{Structural measures of interfacial width under shear} |
| 442 |
> |
One of the open questions about ice/water interfaces is whether the |
| 443 |
> |
thickness of the 'slush-like' quasi-liquid layer (QLL) depends on the |
| 444 |
> |
facet of ice presented to the water. In the QLL region, the water |
| 445 |
> |
molecules are ordered differently than in either the solid or liquid |
| 446 |
> |
phases, and also exhibit distinct dynamical behavior. The width of |
| 447 |
> |
this quasi-liquid layer has been estimated by finding the distance |
| 448 |
> |
over which structural order parameters or dynamic properties change |
| 449 |
> |
from their bulk liquid values to those of the solid ice. The |
| 450 |
> |
properties used to find interfacial widths have included the local |
| 451 |
> |
density, the diffusion constant, and the translational and |
| 452 |
> |
orientational order |
| 453 |
> |
parameters~\cite{Karim88,Karim90,Hayward01,Hayward02,Bryk02,Gay02,Louden13}. |
| 454 |
|
|
| 455 |
+ |
The VSS-RNEMD simulations impose thermal and velocity gradients. |
| 456 |
+ |
These gradients perturb the momenta of the water molecules, so |
| 457 |
+ |
parameters that depend on translational motion are often measuring the |
| 458 |
+ |
momentum exchange, and not physical properties of the interface. As a |
| 459 |
+ |
structural measure of the interface, we have used the local |
| 460 |
+ |
tetrahedral order parameter to estimate the width of the interface. |
| 461 |
+ |
This quantity was originally described by Errington and |
| 462 |
+ |
Debenedetti~\cite{Errington01} and has been used in bulk simulations |
| 463 |
+ |
by Kumar \textit{et al.}~\cite{Kumar09}. It has previously been used |
| 464 |
+ |
in ice/water interfaces by by Bryk and Haymet~\cite{Bryk04b}. |
| 465 |
|
|
| 466 |
< |
\subsection{Orientational dynamics \label{sec:orient}} |
| 467 |
< |
%Should we include the math here? |
| 466 |
> |
To determine the structural widths of the interfaces under shear, each |
| 467 |
> |
of the systems was divided into 100 bins along the $z$-dimension, and |
| 468 |
> |
the local tetrahedral order parameter (Eq. 5 in Reference |
| 469 |
> |
\citealp{Louden13}) was time-averaged in each bin for the duration of |
| 470 |
> |
the shearing simulation. The spatial dependence of this order |
| 471 |
> |
parameter, $q(z)$, is the tetrahedrality profile of the interface. A |
| 472 |
> |
representative profile for the pyramidal facet is shown in circles in |
| 473 |
> |
panel $a$ of figure \ref{fig:pyrComic}. The $q(z)$ function has a |
| 474 |
> |
range of $(0,1)$, where a value of unity indicates a perfectly |
| 475 |
> |
tetrahedral environment. The $q(z)$ for the bulk liquid was found to |
| 476 |
> |
be $\approx~0.77$, while values of $\approx~0.92$ were more common in |
| 477 |
> |
the ice. The tetrahedrality profiles were fit using a hyperbolic |
| 478 |
> |
tangent function (see Eq. 6 in Reference \citealp{Louden13}) designed |
| 479 |
> |
to smoothly fit the bulk to ice transition while accounting for the |
| 480 |
> |
weak thermal gradient. In panels $b$ and $c$, the resulting thermal |
| 481 |
> |
and velocity gradients from an imposed kinetic energy and momentum |
| 482 |
> |
fluxes can be seen. The vertical dotted lines traversing all three |
| 483 |
> |
panels indicate the midpoints of the interface as determined by the |
| 484 |
> |
tetrahedrality profiles. |
| 485 |
> |
|
| 486 |
> |
We find the interfacial width to be $3.2 \pm 0.2$ \AA\ (pyramidal) and |
| 487 |
> |
$3.2 \pm 0.2$ \AA\ (secondary prism) for the control systems with no |
| 488 |
> |
applied momentum flux. This is similar to our previous results for the |
| 489 |
> |
interfacial widths of the quiescent basal ($3.2 \pm 0.4$ \AA) and |
| 490 |
> |
prismatic systems ($3.6 \pm 0.2$ \AA). |
| 491 |
> |
|
| 492 |
> |
Over the range of shear rates investigated, $0.4 \rightarrow |
| 493 |
> |
6.0~\mathrm{~m~s}^{-1}$ for the pyramidal system and $0.6 \rightarrow |
| 494 |
> |
5.2~\mathrm{~m~s}^{-1}$ for the secondary prism, we found no |
| 495 |
> |
significant change in the interfacial width. The mean interfacial |
| 496 |
> |
widths are collected in table \ref{tab:kappa}. This follows our |
| 497 |
> |
previous findings of the basal and prismatic systems, in which the |
| 498 |
> |
interfacial widths of the basal and prismatic facets were also found |
| 499 |
> |
to be insensitive to the shear rate~\cite{Louden13}. |
| 500 |
> |
|
| 501 |
> |
The similarity of these interfacial width estimates indicate that the |
| 502 |
> |
particular facet of the exposed ice crystal has little to no effect on |
| 503 |
> |
how far into the bulk the ice-like structural ordering persists. Also, |
| 504 |
> |
it appears that for the shearing rates imposed in this study, the |
| 505 |
> |
interfacial width is not structurally modified by the movement of |
| 506 |
> |
water over the ice. |
| 507 |
> |
|
| 508 |
> |
\subsection{Dynamic measures of interfacial width under shear} |
| 509 |
|
The orientational time correlation function, |
| 510 |
|
\begin{equation}\label{C(t)1} |
| 511 |
|
C_{2}(t)=\langle P_{2}(\mathbf{u}(0)\cdot \mathbf{u}(t))\rangle, |
| 577 |
|
($d_{prismatic}\approx3.5$ \AA\ ) systems. |
| 578 |
|
|
| 579 |
|
|
| 558 |
– |
|
| 559 |
– |
|
| 560 |
– |
|
| 580 |
|
\section{Conclusion} |
| 581 |
|
We present the results of molecular dynamics simulations of the basal, |
| 582 |
|
prismatic, pyrmaidal |
