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%\url{www.pnas.org/cgi/doi/10.1073/pnas.0709640104} |
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\copyrightyear{2014} |
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\begin{document} |
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|
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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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|
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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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\and |
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J. Daniel Gezelter\affil{1}{}} |
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|
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\contributor{Submitted to Proceedings of the National Academy of Sciences |
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of the United States of America} |
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|
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%%%Newly updated. |
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%%% If significance statement need, then can use the below command otherwise just delete it. |
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\significancetext{Surface hydrophilicity is a measure of the |
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interaction strength between a solid surface and liquid water. Our |
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simulations show that the solid that is thought to be extremely |
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hydrophilic (ice) displays different behavior depending on which |
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crystal facet is presented to the liquid. This behavior is |
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potentially important in geophysics, in recognition of ice surfaces |
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by anti-freeze proteins, and in understanding how the friction |
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between ice and other solids may be mediated by a quasi-liquid layer |
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of water.} |
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|
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\maketitle |
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|
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\begin{article} |
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\begin{abstract} |
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We present evidence that the prismatic and secondary prism facets |
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of ice-I$_\mathrm{h}$ crystals posess structural features that can |
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reduce the effective hydrophilicity of the ice/water |
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interface. The spreading dynamics of liquid water droplets on ice |
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facets exhibits long-time behavior that differs substantially for |
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the prismatic $\{1~0~\bar{1}~0\}$ and secondary prism |
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$\{1~1~\bar{2}~0\}$ facets when compared with the basal $\{0001\}$ |
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and pyramidal $\{2~0~\bar{2}~1\}$ facets. We also present the |
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results of simulations of solid-liquid friction of the same four |
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crystal facets being drawn through liquid water. These simulation |
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techniques provide evidence that the two prismatic faces have an |
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effective surface area in contact with the liquid water of |
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approximately half of the total surface area of the crystal. The |
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ice / water interfacial widths for all four crystal facets are |
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similar (using both structural and dynamic measures), and were |
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found to be independent of the shear rate. Additionally, |
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decomposition of orientational time correlation functions show |
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position-dependence for the short- and longer-time decay |
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components close to the interface. |
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\end{abstract} |
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|
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\keywords{ice | water | interfaces | hydrophobicity} |
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\abbreviations{QLL, quasi liquid layer; MD, molecular dynamics; RNEMD, |
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reverse non-equilibrium molecular dynamics} |
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|
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\dropcap{S}urfaces can be characterized as hydrophobic or hydrophilic |
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based on the strength of the interactions with water. Hydrophobic |
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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, $\theta$, |
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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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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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|
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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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heights and widths of nano-pillars can change a surface from |
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superhydrophobic, $\theta \ge 150^{\circ}$, to hydrophilic, $\theta |
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\sim 0^{\circ}$.\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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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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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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|
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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 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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solid-liquid friction. |
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|
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Quiescent ice-I$_\mathrm{h}$/water interfaces have been studied |
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extensively using computer simulations. Haymet \textit{et al.} |
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characterized and measured the width of these interfaces for the |
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SPC~\cite{Karim90}, SPC/E~\cite{Gay02,Bryk02}, |
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CF1~\cite{Hayward01,Hayward02} and TIP4P~\cite{Karim88} models, in |
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both neat water and with solvated |
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ions~\cite{Bryk04,Smith05,Wilson08,Wilson10}. Nada and Furukawa have |
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studied the width of basal/water and prismatic/water |
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interfaces~\cite{Nada95} as well as crystal restructuring at |
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temperatures approaching the melting point~\cite{Nada00}. |
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|
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The surface of ice exhibits a premelting layer, often called a |
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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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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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|
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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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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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|
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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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% understanding the macroscopic motion of large ice |
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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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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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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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difference in liquid-solid friction to the 6.5 \AA\ corrugation of the |
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prismatic face which reduces the effective surface area of the ice |
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that is in direct contact with liquid water. |
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|
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In the sections that follow, we outline the methodology used to |
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simulate droplet-spreading dynamics using standard MD and tribological |
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properties using RNEMD simulations. These simulation methods give |
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complementary results that point to the prismatic and secondary prism |
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facets having roughly half of their surface area in direct contact |
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with the liquid. |
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|
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\section{Methodology} |
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\subsection{Construction of the Ice / Water Interfaces} |
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To construct the four interfacial ice/water systems, a proton-ordered, |
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zero-dipole crystal of ice-I$_\mathrm{h}$ with exposed strips of |
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H-atoms was created using Structure 6 of Hirsch and Ojam\"{a}e's set |
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of orthorhombic representations for ice-I$_{h}$~\cite{Hirsch04}. This |
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crystal structure was cleaved along the four different facets. The |
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exposed face was reoriented normal to the $z$-axis of the simulation |
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cell, and the structures were and extended to form large exposed |
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facets in rectangular box geometries. Liquid water boxes were created |
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with identical dimensions (in $x$ and $y$) as the ice, with a $z$ |
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dimension of three times that of the ice block, and a density |
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corresponding to 1 g / cm$^3$. Each of the ice slabs and water boxes |
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were independently equilibrated at a pressure of 1 atm, and the |
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resulting systems were merged by carving out any liquid water |
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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}. 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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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 |
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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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dynamic measurements of the interfacial width that agree well with |
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more expensive water models, although the coexistence temperature for |
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SPC/E is still well below the experimental melting point of real |
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water~\cite{Bryk02}. Given the extensive data and speed of this model, |
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it is a reasonable choice even though the temperatures required are |
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somewhat lower than real ice / water interfaces. |
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|
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\subsection{Droplet Simulations} |
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Ice interfaces with a thickness of $\sim$~20~\AA\ were created as |
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described above, but were not solvated in a liquid box. The crystals |
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were then replicated along the $x$ and $y$ axes (parallel to the |
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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~$\mathrm{kcal\ |
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mol}^{-1}\mathrm{~\AA}^{-2}$ (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 |
248 |
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 |
252 |
configurations for the droplet on the prismatic facet are shown in |
253 |
figure \ref{fig:Droplet}. |
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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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variant of reverse non-equilibrium molecular dynamics (VSS-RNEMD) was |
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employed \cite{Kuang12}. This method performs a series of simultaneous |
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non-equilibrium exchanges of linear momentum and kinetic energy |
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between two physically-separated regions of the simulation cell. The |
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system responds to this unphysical flux with velocity and temperature |
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gradients. When VSS-RNEMD is applied to bulk liquids, transport |
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properties like the thermal conductivity and the shear viscosity are |
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easily extracted assuming a linear response between the flux and the |
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gradient. At the interfaces between dissimilar materials, the same |
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method can be used to extract \textit{interfacial} transport |
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properties (e.g. the interfacial thermal conductance and the |
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hydrodynamic slip length). |
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|
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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 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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The exchanges between the two regions were carried out every 2 fs |
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(i.e. every time step). Although computationally expensive, this was |
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done to minimize the magnitude of each individual momentum exchange. |
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Because individual VSS-RNEMD exchanges conserve both total energy and |
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linear 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, the RNEMD simulations were done in the microcanonical |
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(NVE) ensemble. All simulations of the other interfaces were carried |
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out in the 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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|
293 |
To determine the extent of wetting for each of the four crystal |
294 |
facets, contact angles for liquid droplets on the ice surfaces were |
295 |
computed using two methods. In the first method, the droplet is |
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assumed to form a spherical cap, and the contact angle is estimated |
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from the $z$-axis location of the droplet's center of mass |
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($z_\mathrm{cm}$). This procedure was first described by Hautman and |
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Klein~\cite{Hautman91}, and was utilized by Hirvi and Pakkanen in |
300 |
their investigation of water droplets on polyethylene and poly(vinyl |
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chloride) surfaces~\cite{Hirvi06}. For each stored configuration, the |
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contact angle, $\theta$, was found by inverting the expression for the |
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location of the droplet center of mass, |
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\begin{equation}\label{contact_1} |
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\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} , |
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\end{equation} |
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where $R_{0}$ is the radius of the free water droplet. |
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|
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The second method for obtaining the contact angle was described by |
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Ruijter, Blake, and Coninck~\cite{Ruijter99}. This method uses a |
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cylindrical averaging of the droplet's density profile. A threshold |
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density of 0.5 g cm\textsuperscript{-3} is used to estimate the |
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location of the edge of the droplet. The $r$ and $z$-dependence of |
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the droplet's edge is then fit to a circle, and the contact angle is |
315 |
computed from the intersection of the fit circle with the $z$-axis |
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location of the solid surface. Again, for each stored configuration, |
317 |
the density profile in a set of annular shells was computed. Due to |
318 |
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 |
320 |
height of the solid surface ($z_\mathrm{suface}$) along with the best |
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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} |
325 |
\theta = 90 + \frac{180}{\pi} \sin^{-1}\left(\frac{z_\mathrm{droplet} - |
326 |
z_\mathrm{surface}}{r_\mathrm{droplet}} \right). |
327 |
\end{equation} |
328 |
Both methods provided similar estimates of the dynamic contact angle, |
329 |
although the first method is significantly less prone to noise, and |
330 |
is the method used to report contact angles below. |
331 |
|
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Because the initial droplet was placed above the surface, the initial |
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value of 180$^{\circ}$ decayed over time (See figure |
334 |
\ref{fig:ContactAngle}). Each of these profiles were fit to a |
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biexponential decay, with a short-time contribution ($\tau_c$) that |
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describes the initial contact with the surface, a long time |
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contribution ($\tau_s$) that describes the spread of the droplet over |
338 |
the surface, and a constant ($\theta_\infty$) to capture the |
339 |
infinite-time estimate of the equilibrium contact angle, |
340 |
\begin{equation} |
341 |
\theta(t) = \theta_\infty + (180-\theta_\infty) \left[ a e^{-t/\tau_c} + |
342 |
(1-a) e^{-t/\tau_s} \right] |
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\end{equation} |
344 |
We have found that the rate for water droplet spreading across all |
345 |
four crystal facets, $k_\mathrm{spread} = 1/\tau_s \approx$ 0.7 |
346 |
ns$^{-1}$. However, the basal and pyramidal facets produced estimated |
347 |
equilibrium contact angles, $\theta_\infty \approx$ 35$^{o}$, while |
348 |
prismatic and secondary prismatic had values for $\theta_\infty$ near |
349 |
43$^{o}$ as seen in Table \ref{tab:kappa}. |
350 |
|
351 |
These results indicate that the basal and pyramidal facets are more |
352 |
hydrophilic by traditional measures than the prismatic and secondary |
353 |
prism facets, and surprisingly, that the differential hydrophilicities |
354 |
of the crystal facets is not reflected in the spreading rate of the |
355 |
droplet. |
356 |
|
357 |
% This is in good agreement with our calculations of friction |
358 |
% coefficients, in which the basal |
359 |
% and pyramidal had a higher coefficient of kinetic friction than the |
360 |
% prismatic and secondary prismatic. Due to this, we beleive that the |
361 |
% differences in friction coefficients can be attributed to the varying |
362 |
% hydrophilicities of the facets. |
363 |
|
364 |
\subsection{Solid-liquid friction of the interfaces} |
365 |
In a bulk fluid, the shear viscosity, $\eta$, can be determined |
366 |
assuming a linear response to a shear stress, |
367 |
\begin{equation}\label{Shenyu-11} |
368 |
j_{z}(p_{x}) = \eta \frac{\partial v_{x}}{\partial z}. |
369 |
\end{equation} |
370 |
Here $j_{z}(p_{x})$ is the flux (in $x$-momentum) that is transferred |
371 |
in the $z$ direction (i.e. the shear stress). The RNEMD simulations |
372 |
impose an artificial momentum flux between two regions of the |
373 |
simulation, and the velocity gradient is the fluid's response. This |
374 |
technique has now been applied quite widely to determine the |
375 |
viscosities of a number of bulk fluids~\cite{}. |
376 |
|
377 |
At the interface between two phases (e.g. liquid / solid) the same |
378 |
momentum flux creates a velocity difference between the two materials, |
379 |
and this can be used to define an interfacial friction coefficient |
380 |
($\kappa$), |
381 |
\begin{equation}\label{Shenyu-13} |
382 |
j_{z}(p_{x}) = \kappa \left[ v_{x}(liquid) - v_{x}(solid) \right] |
383 |
\end{equation} |
384 |
where $v_{x}(liquid)$ and $v_{x}(solid)$ are the velocities measured |
385 |
directly adjacent to the interface. |
386 |
|
387 |
The simulations described here contain significant quantities of both |
388 |
liquid and solid phases, and the momentum flux must traverse a region |
389 |
of the liquid that is simultaneously under a thermal gradient. Since |
390 |
the liquid has a temperature-dependent shear viscosity, $\eta(T)$, |
391 |
estimates of the solid-liquid friction coefficient can be obtained if |
392 |
one knows the viscosity of the liquid at the interface (i.e. at the |
393 |
melting temperature, $T_m$), |
394 |
\begin{equation}\label{kappa-2} |
395 |
\kappa = \frac{\eta(T_{m})}{\left[v_{x}(fluid)-v_{x}(solid)\right]}\left(\frac{\partial v_{x}}{\partial z}\right). |
396 |
\end{equation} |
397 |
For SPC/E, the melting temperature of Ice-I$_\mathrm{h}$ is estimated |
398 |
to be 225~K~\cite{Bryk02}. To obtain the value of |
399 |
$\eta(225\mathrm{~K})$ for the SPC/E model, a $31.09 \times 29.38 |
400 |
\times 124.39$ \AA\ box with 3744 water molecules in a disordered |
401 |
configuration was equilibrated to 225~K, and five |
402 |
statistically-independent shearing simulations were performed (with |
403 |
imposed fluxes that spanned a range of $3 \rightarrow 13 |
404 |
\mathrm{~m~s}^{-1}$ ). Each simulation was conducted in the |
405 |
microcanonical ensemble with total simulation times of 5 ns. The |
406 |
VSS-RNEMD exchanges were carried out every 2 fs. We estimate |
407 |
$\eta(225\mathrm{~K})$ to be 0.0148 $\pm$ 0.0007 Pa s for SPC/E, |
408 |
roughly ten times larger than the shear viscosity previously computed |
409 |
at 280~K~\cite{Kuang12}. |
410 |
|
411 |
The interfacial friction coefficient can equivalently be expressed as |
412 |
the ratio of the viscosity of the fluid to the hydrodynamic slip |
413 |
length, $\kappa = \eta / \delta$. The slip length is an indication of |
414 |
strength of the interactions between the solid and liquid phases, |
415 |
although the connection between slip length and surface hydrophobicity |
416 |
is not yet clear. In some simulations, the slip length has been found |
417 |
to have a link to the effective surface |
418 |
hydrophobicity~\cite{Sendner:2009uq}, although Ho \textit{et al.} have |
419 |
found that liquid water can also slip on hydrophilic |
420 |
surfaces~\cite{Ho:2011zr}. Experimental evidence for a direct tie |
421 |
between slip length and hydrophobicity is also not |
422 |
definitive. Total-internal reflection particle image velocimetry |
423 |
(TIR-PIV) studies have suggested that there is a link between slip |
424 |
length and effective |
425 |
hydrophobicity~\cite{Lasne:2008vn,Bouzigues:2008ys}. However, recent |
426 |
surface sensitive cross-correlation spectroscopy (TIR-FCCS) |
427 |
measurements have seen similar slip behavior for both hydrophobic and |
428 |
hydrophilic surfaces~\cite{Schaeffel:2013kx}. |
429 |
|
430 |
In each of the systems studied here, the interfacial temperature was |
431 |
kept fixed to 225K, which ensured the viscosity of the fluid at the |
432 |
interace was identical. Thus, any significant variation in $\kappa$ |
433 |
between the systems is a direct indicator of the slip length and the |
434 |
effective interaction strength between the solid and liquid phases. |
435 |
|
436 |
The calculated $\kappa$ values found for the four crystal facets of |
437 |
Ice-I$_\mathrm{h}$ are shown in Table \ref{tab:kappa}. The basal and |
438 |
pyramidal facets were found to have similar values of $\kappa \approx |
439 |
6$ ($\times 10^{-4} \mathrm{amu~\AA}^{-2}\mathrm{fs}^{-1}$), while the |
440 |
prismatic and secondary prism facets exhibited $\kappa \approx 3$ |
441 |
($\times 10^{-4} \mathrm{amu~\AA}^{-2}\mathrm{fs}^{-1}$). These |
442 |
results are also essentially independent of shearing direction |
443 |
relative to features on the surface of the facets. The friction |
444 |
coefficients indicate that the basal and pyramidal facets have |
445 |
significantly stronger interactions with liquid water than either of |
446 |
the two prismatic facets. This is in agreement with the contact angle |
447 |
results above - both of the high-friction facets exhbited smaller |
448 |
contact angles, suggesting that the solid-liquid friction is |
449 |
correlated with the hydrophilicity of these facets. |
450 |
|
451 |
\subsection{Structural measures of interfacial width under shear} |
452 |
One of the open questions about ice/water interfaces is whether the |
453 |
thickness of the 'slush-like' quasi-liquid layer (QLL) depends on the |
454 |
facet of ice presented to the water. In the QLL region, the water |
455 |
molecules are ordered differently than in either the solid or liquid |
456 |
phases, and also exhibit distinct dynamical behavior. The width of |
457 |
this quasi-liquid layer has been estimated by finding the distance |
458 |
over which structural order parameters or dynamic properties change |
459 |
from their bulk liquid values to those of the solid ice. The |
460 |
properties used to find interfacial widths have included the local |
461 |
density, the diffusion constant, and the translational and |
462 |
orientational order |
463 |
parameters~\cite{Karim88,Karim90,Hayward01,Hayward02,Bryk02,Gay02,Louden13}. |
464 |
|
465 |
The VSS-RNEMD simulations impose thermal and velocity gradients. |
466 |
These gradients perturb the momenta of the water molecules, so |
467 |
parameters that depend on translational motion are often measuring the |
468 |
momentum exchange, and not physical properties of the interface. As a |
469 |
structural measure of the interface, we have used the local |
470 |
tetrahedral order parameter to estimate the width of the interface. |
471 |
This quantity was originally described by Errington and |
472 |
Debenedetti~\cite{Errington01} and has been used in bulk simulations |
473 |
by Kumar \textit{et al.}~\cite{Kumar09}. It has previously been used |
474 |
in ice/water interfaces by by Bryk and Haymet~\cite{Bryk04b}. |
475 |
|
476 |
To determine the structural widths of the interfaces under shear, each |
477 |
of the systems was divided into 100 bins along the $z$-dimension, and |
478 |
the local tetrahedral order parameter (Eq. 5 in Reference |
479 |
\citealp{Louden13}) was time-averaged in each bin for the duration of |
480 |
the shearing simulation. The spatial dependence of this order |
481 |
parameter, $q(z)$, is the tetrahedrality profile of the interface. A |
482 |
representative profile for the pyramidal facet is shown in circles in |
483 |
panel $a$ of figure \ref{fig:pyrComic}. The $q(z)$ function has a |
484 |
range of $(0,1)$, where a value of unity indicates a perfectly |
485 |
tetrahedral environment. The $q(z)$ for the bulk liquid was found to |
486 |
be $\approx~0.77$, while values of $\approx~0.92$ were more common in |
487 |
the ice. The tetrahedrality profiles were fit using a hyperbolic |
488 |
tangent function (see Eq. 6 in Reference \citealp{Louden13}) designed |
489 |
to smoothly fit the bulk to ice transition while accounting for the |
490 |
weak thermal gradient. In panels $b$ and $c$, the resulting thermal |
491 |
and velocity gradients from an imposed kinetic energy and momentum |
492 |
fluxes can be seen. The vertical dotted lines traversing all three |
493 |
panels indicate the midpoints of the interface as determined by the |
494 |
tetrahedrality profiles. |
495 |
|
496 |
We find the interfacial width to be $3.2 \pm 0.2$ \AA\ (pyramidal) and |
497 |
$3.2 \pm 0.2$ \AA\ (secondary prism) for the control systems with no |
498 |
applied momentum flux. This is similar to our previous results for the |
499 |
interfacial widths of the quiescent basal ($3.2 \pm 0.4$ \AA) and |
500 |
prismatic systems ($3.6 \pm 0.2$ \AA). |
501 |
|
502 |
Over the range of shear rates investigated, $0.4 \rightarrow |
503 |
6.0~\mathrm{~m~s}^{-1}$ for the pyramidal system and $0.6 \rightarrow |
504 |
5.2~\mathrm{~m~s}^{-1}$ for the secondary prism, we found no |
505 |
significant change in the interfacial width. The mean interfacial |
506 |
widths are collected in table \ref{tab:kappa}. This follows our |
507 |
previous findings of the basal and prismatic systems, in which the |
508 |
interfacial widths of the basal and prismatic facets were also found |
509 |
to be insensitive to the shear rate~\cite{Louden13}. |
510 |
|
511 |
The similarity of these interfacial width estimates indicate that the |
512 |
particular facet of the exposed ice crystal has little to no effect on |
513 |
how far into the bulk the ice-like structural ordering persists. Also, |
514 |
it appears that for the shearing rates imposed in this study, the |
515 |
interfacial width is not structurally modified by the movement of |
516 |
water over the ice. |
517 |
|
518 |
\subsection{Dynamic measures of interfacial width under shear} |
519 |
The spatially-resolved orientational time correlation function, |
520 |
\begin{equation}\label{C(t)1} |
521 |
C_{2}(z,t)=\langle P_{2}(\mathbf{u}_i(0)\cdot \mathbf{u}_i(t)) |
522 |
\delta(z_i(0) - z) \rangle, |
523 |
\end{equation} |
524 |
provides local information about the decorrelation of molecular |
525 |
orientations in time. Here, $P_{2}$ is the second-order Legendre |
526 |
polynomial, and $\mathbf{u}_i$ is the molecular vector that bisects |
527 |
the HOH angle of molecule $i$. The angle brackets indicate an average |
528 |
over all the water molecules, and the delta function restricts the |
529 |
average to specific regions. In the crystal, decay of $C_2(z,t)$ is |
530 |
incomplete, while liquid water correlation times are typically |
531 |
measured in ps. Observing the spatial-transition between the decay |
532 |
regimes can define a dynamic measure of the interfacial width. |
533 |
|
534 |
Each of the systems was divided into bins along the $z$-dimension |
535 |
($\approx$ 3 \AA\ wide) and $C_2(z,t)$ was computed using only those |
536 |
molecules that were in the bin at the initial time. The |
537 |
time-dependence was fit to a triexponential decay, with three time |
538 |
constants: $\tau_{short}$, measuring the librational motion of the |
539 |
water molecules, $\tau_{middle}$, measuring the timescale for breaking |
540 |
and making of hydrogen bonds, and $\tau_{long}$, corresponding to the |
541 |
translational motion of the water molecules. An additional constant |
542 |
was introduced in the fits to describe molecules in the crystal which |
543 |
do not experience long-time orientational decay. |
544 |
|
545 |
In Figures S5-S8 in the supporting information, the $z$-coordinate |
546 |
profiles for the three decay constants, $\tau_{short}$, |
547 |
$\tau_{middle}$, and $\tau_{long}$ for the different interfaces are |
548 |
shown. Figures S5 \& S6 are new results, and Figures S7 \& S8 are |
549 |
updated plots from Ref \citealp{Louden13}. In the liquid regions of |
550 |
all four interfaces, we observe $\tau_{middle}$ and $\tau_{long}$ to |
551 |
have approximately consistent values of $3-6$ ps and $30-40$ ps, |
552 |
respectively. Both of these times increase in value approaching the |
553 |
interface. Approaching the interface, we also observe that |
554 |
$\tau_{short}$ decreases from its liquid-state value of $72-76$ fs. |
555 |
The approximate values for the decay constants and the trends |
556 |
approaching the interface match those reported previously for the |
557 |
basal and prismatic interfaces. |
558 |
|
559 |
We have estimated the dynamic interfacial width $d_\mathrm{dyn}$ by |
560 |
fitting the profiles of all the three orientational time constants |
561 |
with an exponential decay to the bulk-liquid behavior, |
562 |
\begin{equation}\label{tauFit} |
563 |
\tau(z)\approx\tau_{liquid}+(\tau_{wall}-\tau_{liquid})e^{-(z-z_{wall})/d_\mathrm{dyn}} |
564 |
\end{equation} |
565 |
where $\tau_{liquid}$ and $\tau_{wall}$ are the liquid and projected |
566 |
wall values of the decay constants, $z_{wall}$ is the location of the |
567 |
interface, as measured by the structural order parameter. These |
568 |
values are shown in table \ref{tab:kappa}. Because the bins must be |
569 |
quite wide to obtain reasonable profiles of $C_2(z,t)$, the error |
570 |
estimates for the dynamic widths of the interface are significantly |
571 |
larger than for the structural widths. However, all four interfaces |
572 |
exhibit dynamic widths that are significantly below 1~nm, and are in |
573 |
reasonable agreement with the structural width above. |
574 |
|
575 |
\section{Conclusions} |
576 |
In this work, we used MD simulations to measure the advancing contact |
577 |
angles of water droplets on the basal, prismatic, pyramidal, and |
578 |
secondary prism facets of Ice-I$_\mathrm{h}$. Although there was no |
579 |
significant change in the \textit{rate} at which the droplets spread |
580 |
over the surface, the long-time behavior indicates that we should |
581 |
expect to see larger equilibrium contact angles for the two prismatic |
582 |
facets. |
583 |
|
584 |
We have also used RNEMD simulations of water interfaces with the same |
585 |
four crystal facets to compute solid-liquid friction coefficients. We |
586 |
have observed coefficients of friction that differ by a factor of two |
587 |
between the two prismatic facets and the basal and pyramidal facets. |
588 |
Because the solid-liquid friction coefficient is directly tied to the |
589 |
hydrodynamic slip length, this suggests that there are significant |
590 |
differences in the overall interaction strengths between these facets |
591 |
and the liquid layers immediately in contact with them. |
592 |
|
593 |
The agreement between these two measures have lead us to conclude that |
594 |
the two prismatic facets have a lower hydrophilicity than either the |
595 |
basal or pyramidal facets. One possible explanation of this behavior |
596 |
is that the face presented by both prismatic facets consists of deep, |
597 |
narrow channels (i.e. stripes of adjacent rows of pairs of |
598 |
hydrodgen-bound water molecules). At the surfaces of these facets, |
599 |
the channels are 6.35 \AA\ wide and the sub-surface ice layer is 2.25 |
600 |
\AA\ below (and therefore blocked from hydrogen bonding with the |
601 |
liquid). This means that only 1/2 of the surface molecules can form |
602 |
hydrogen bonds with liquid-phase molecules. |
603 |
|
604 |
In the basal plane, the surface features are narrower (4.49 \AA) and |
605 |
shallower (1.3 \AA), while the pyramidal face has much wider channels |
606 |
(8.65 \AA) which are also quite shallow (1.37 \AA). These features |
607 |
allow liquid phase molecules to form hydrogen bonds with all of the |
608 |
surface molecules in the basal and pyramidal facets. This means that |
609 |
for similar surface areas, the two prismatic facets have an effective |
610 |
hydrogen bonding surface area of half of the basal and pyramidal |
611 |
facets. The reduction in the effective surface area would explain |
612 |
much of the behavior observed in our simulations. |
613 |
|
614 |
\begin{acknowledgments} |
615 |
Support for this project was provided by the National |
616 |
Science Foundation under grant CHE-1362211. Computational time was |
617 |
provided by the Center for Research Computing (CRC) at the |
618 |
University of Notre Dame. |
619 |
\end{acknowledgments} |
620 |
|
621 |
\bibliography{iceWater} |
622 |
% ***************************************** |
623 |
% There is significant interest in the properties of ice/ice and ice/water |
624 |
% interfaces in the geophysics community. Most commonly, the results of shearing |
625 |
% two ice blocks past one |
626 |
% another\cite{Casassa91, Sukhorukov13, Pritchard12, Lishman13} or the shearing |
627 |
% of ice through water\cite{Cuffey99, Bell08}. Using molecular dynamics |
628 |
% simulations, Samadashvili has recently shown that when two smooth ice slabs |
629 |
% slide past one another, a stable liquid-like layer develops between |
630 |
% them\cite{Samadashvili13}. To fundamentally understand these processes, a |
631 |
% molecular understanding of the ice/water interfaces is needed. |
632 |
|
633 |
% Investigation of the ice/water interface is also crucial in understanding |
634 |
% processes such as nucleation, crystal |
635 |
% growth,\cite{Han92, Granasy95, Vanfleet95} and crystal |
636 |
% melting\cite{Weber83, Han92, Sakai96, Sakai96B}. Insight gained to these |
637 |
% properties can also be applied to biological systems of interest, such as |
638 |
% the behavior of the antifreeze protein found in winter |
639 |
% flounder,\cite{Wierzbicki07,Chapsky97} and certain terrestial |
640 |
% arthropods.\cite{Duman:2001qy,Meister29012013} Elucidating the properties which |
641 |
% give rise to these processes through experimental techniques can be expensive, |
642 |
% complicated, and sometimes infeasible. However, through the use of molecular |
643 |
% dynamics simulations much of the problems of investigating these properties |
644 |
% are alleviated. |
645 |
|
646 |
% Understanding ice/water interfaces inherently begins with the isolated |
647 |
% systems. There has been extensive work parameterizing models for liquid water, |
648 |
% such as the SPC\cite{Berendsen81}, SPC/E\cite{Berendsen87}, |
649 |
% TIP4P\cite{Jorgensen85}, TIP4P/2005\cite{Abascal05}, |
650 |
% ($\dots$), and more recently, models for simulating |
651 |
% the solid phases of water, such as the TIP4P/Ice\cite{Abascal05b} model. The |
652 |
% melting point of various crystal structures of ice have been calculated for |
653 |
% many of these models |
654 |
% (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}), |
655 |
% and the partial or complete phase diagram for the model has been determined |
656 |
% (SPC/E\cite{Baez95,Bryk04b,Sanz04b}, TIP4P\cite{Sanz04,Sanz04b,Koyama04}, TIP5P\cite{Sanz04,Koyama04}). |
657 |
% Knowing the behavior and melting point for these models has enabled an initial |
658 |
% investigation of ice/water interfaces. |
659 |
|
660 |
% The Ice-I$_\mathrm{h}$/water quiescent interface has been extensively studied |
661 |
% over the past 30 years by theory and experiment. Haymet \emph{et al.} have |
662 |
% done significant work characterizing and quantifying the width of these |
663 |
% interfaces for the SPC,\cite{Karim90} SPC/E,\cite{Gay02,Bryk02}, |
664 |
% CF1,\cite{Hayward01,Hayward02} and TIP4P\cite{Karim88} models for water. In |
665 |
% recent years, Haymet has focused on investigating the effects cations and |
666 |
% anions have on crystal nucleaion and |
667 |
% melting.\cite{Bryk04,Smith05,Wilson08,Wilson10} Nada and Furukawa have studied |
668 |
% the the basal- and prismatic-water interface width\cite{Nada95}, crystal |
669 |
% surface restructuring at temperatures approaching the melting |
670 |
% point\cite{Nada00}, and the mechanism of ice growth inhibition by antifreeze |
671 |
% proteins\cite{Nada08,Nada11,Nada12}. Nada has developed a six-site water model |
672 |
% for ice/water interfaces near the melting point\cite{Nada03}, and studied the |
673 |
% dependence of crystal growth shape on applied pressure\cite{Nada11b}. Using |
674 |
% this model, Nada and Furukawa have established differential |
675 |
% growth rates for the basal, prismatic, and secondary prismatic facets of |
676 |
% Ice-I$_\mathrm{h}$ and found their origins due to a reordering of the hydrogen |
677 |
% bond network in water near the interface\cite{Nada05}. While the work |
678 |
% described so far has mainly focused on bulk water on ice, there is significant |
679 |
% interest in thin films of water on ice surfaces as well. |
680 |
|
681 |
% It is well known that the surface of ice exhibits a premelting layer at |
682 |
% temperatures near the melting point, often called a quasi-liquid layer (QLL). |
683 |
% Molecular dynamics simulations of the facets of ice-I$_\mathrm{h}$ exposed |
684 |
% to vacuum performed by Conde, Vega and Patrykiejew have found QLL widths of |
685 |
% approximately 10 \AA\ at 3 K below the melting point\cite{Conde08}. |
686 |
% Similarly, Limmer and Chandler have used course grain simulations and |
687 |
% statistical field theory to estimated QLL widths at the same temperature to |
688 |
% be about 3 nm\cite{Limmer14}. |
689 |
% Recently, Sazaki and Furukawa have developed an experimental technique with |
690 |
% sufficient spatial and temporal resolution to visulaize and quantitatively |
691 |
% analyze QLLs on ice crystals at temperatures near melting\cite{Sazaki10}. They |
692 |
% have found the width of the QLLs perpindicular to the surface at -2.2$^{o}$C |
693 |
% to be $\mathcal{O}$(\AA). They have also seen the formation of two immiscible |
694 |
% QLLs, which displayed different stabilities and dynamics on the crystal |
695 |
% surface\cite{Sazaki12}. Knowledge of the hydrophilicities of each |
696 |
% of the crystal facets would help further our understanding of the properties |
697 |
% and dynamics of the QLLs. |
698 |
|
699 |
% Presented here is the follow up to our previous paper\cite{Louden13}, in which |
700 |
% the basal and prismatic facets of an ice-I$_\mathrm{h}$/water interface were |
701 |
% investigated where the ice was sheared relative to the liquid. By using a |
702 |
% recently developed velocity shearing and scaling approach to reverse |
703 |
% non-equilibrium molecular dynamics (VSS-RNEMD), simultaneous temperature and |
704 |
% velocity gradients can be applied to the system, which allows for measurment |
705 |
% of friction and thermal transport properties while maintaining a stable |
706 |
% interfacial temperature\cite{Kuang12}. Structural analysis and dynamic |
707 |
% correlation functions were used to probe the interfacial response to a shear, |
708 |
% and the resulting solid/liquid kinetic friction coefficients were reported. |
709 |
% In this paper we present the same analysis for the pyramidal and secondary |
710 |
% prismatic facets, and show that the differential interfacial friction |
711 |
% coefficients for the four facets of ice-I$_\mathrm{h}$ are determined by their |
712 |
% relative hydrophilicity by means of dynamics water contact angle |
713 |
% simulations. |
714 |
|
715 |
% The local tetrahedral order parameter, $q(z)$, is given by |
716 |
% \begin{equation} |
717 |
% q(z) = \int_0^L \sum_{k=1}^{N} \Bigg(1 -\frac{3}{8}\sum_{i=1}^{3} |
718 |
% \sum_{j=i+1}^{4} \bigg(\cos\psi_{ikj}+\frac{1}{3}\bigg)^2\Bigg) |
719 |
% \delta(z_{k}-z)\mathrm{d}z \Bigg/ N_z |
720 |
% \label{eq:qz} |
721 |
% \end{equation} |
722 |
% where $\psi_{ikj}$ is the angle formed between the oxygen sites of molecules |
723 |
% $i$,$k$, and $j$, where the centeral oxygen is located within molecule $k$ and |
724 |
% molecules $i$ and $j$ are two of the closest four water molecules |
725 |
% around molecule $k$. All four closest neighbors of molecule $k$ are also |
726 |
% required to reside within the first peak of the pair distribution function |
727 |
% for molecule $k$ (typically $<$ 3.41 \AA\ for water). |
728 |
% $N_z = \int\delta(z_k - z) \mathrm{d}z$ is a normalization factor to account |
729 |
% for the varying population of molecules within each finite-width bin. |
730 |
|
731 |
|
732 |
% The hydrophobicity or hydrophilicity of a surface can be described by the |
733 |
% extent a droplet of water wets the surface. The contact angle formed between |
734 |
% the solid and the liquid, $\theta$, which relates the free energies of the |
735 |
% three interfaces involved, is given by Young's equation. |
736 |
% \begin{equation}\label{young} |
737 |
% \cos\theta = (\gamma_{sv} - \gamma_{sl})/\gamma_{lv} |
738 |
% \end{equation} |
739 |
% Here $\gamma_{sv}$, $\gamma_{sl}$, and $\gamma_{lv}$ are the free energies |
740 |
% of the solid/vapor, solid/liquid, and liquid/vapor interfaces respectively. |
741 |
% Large contact angles ($\theta$ $\gg$ 90\textsuperscript{o}) correspond to low |
742 |
% wettability and hydrophobic surfaces, while small contact angles |
743 |
% ($\theta$ $\ll$ 90\textsuperscript{o}) correspond to high wettability and |
744 |
% hydrophilic surfaces. Experimentally, measurements of the contact angle |
745 |
% of sessile drops has been used to quantify the extent of wetting on surfaces |
746 |
% with thermally selective wetting charactaristics\cite{Tadanaga00,Liu04,Sun04}, |
747 |
% as well as nano-pillared surfaces with electrically tunable Cassie-Baxter and |
748 |
% Wenzel states\cite{Herbertson06,Dhindsa06,Verplanck07,Ahuja08,Manukyan11}. |
749 |
% Luzar and coworkers have done significant work modeling these transitions on |
750 |
% nano-patterned surfaces\cite{Daub07,Daub10,Daub11,Ritchie12}, and have found |
751 |
% the change in contact angle to be due to the external field perturbing the |
752 |
% hydrogen bonding of the liquid/vapor interface\cite{Daub07}. |
753 |
|
754 |
% SI stuff: |
755 |
|
756 |
% Correlation functions: |
757 |
% To compute eq. \eqref{C(t)1}, each 0.5 ns simulation was |
758 |
% followed by an additional 200 ps NVE simulation during which the |
759 |
% position and orientations of each molecule were recorded every 0.1 ps. |
760 |
|
761 |
|
762 |
|
763 |
|
764 |
\end{article} |
765 |
|
766 |
\begin{figure} |
767 |
\includegraphics[width=\linewidth]{Droplet} |
768 |
\caption{\label{fig:Droplet} Computational model of a droplet of |
769 |
liquid water spreading over the prismatic $\{1~0~\bar{1}~0\}$ facet |
770 |
of ice, before (left) and 2.6 ns after (right) being introduced to the |
771 |
surface. The contact angle ($\theta$) shrinks as the simulation |
772 |
proceeds, and the long-time behavior of this angle is used to |
773 |
estimate the hydrophilicity of the facet.} |
774 |
\end{figure} |
775 |
|
776 |
\begin{figure} |
777 |
\includegraphics[width=2in]{Shearing} |
778 |
\caption{\label{fig:Shearing} Computational model of a slab of ice |
779 |
being sheared through liquid water. In this figure, the ice is |
780 |
presenting two copies of the prismatic $\{1~0~\bar{1}~0\}$ facet |
781 |
towards the liquid phase. The RNEMD simulation exchanges both |
782 |
linear momentum (indicated with arrows) and kinetic energy between |
783 |
the central box and the box that spans the cell boundary. The |
784 |
system responds with weak thermal gradient and a velocity profile |
785 |
that shears the ice relative to the surrounding liquid.} |
786 |
\end{figure} |
787 |
|
788 |
\begin{figure} |
789 |
\includegraphics[width=\linewidth]{ContactAngle} |
790 |
\caption{\label{fig:ContactAngle} The dynamic contact angle of a |
791 |
droplet after approaching each of the four ice facets. The decay to |
792 |
an equilibrium contact angle displays similar dynamics. Although |
793 |
all the surfaces are hydrophilic, the long-time behavior stabilizes |
794 |
to significantly flatter droplets for the basal and pyramidal |
795 |
facets. This suggests a difference in hydrophilicity for these |
796 |
facets compared with the two prismatic facets.} |
797 |
\end{figure} |
798 |
|
799 |
% \begin{figure} |
800 |
% \includegraphics[width=\linewidth]{Pyr_comic_strip} |
801 |
% \caption{\label{fig:pyrComic} Properties of the pyramidal interface |
802 |
% being sheared through water at 3.8 ms\textsuperscript{-1}. Lower |
803 |
% panel: the local tetrahedral order parameter, $q(z)$, (circles) and |
804 |
% the hyperbolic tangent fit (turquoise line). Middle panel: the |
805 |
% imposed thermal gradient required to maintain a fixed interfacial |
806 |
% temperature of 225 K. Upper panel: the transverse velocity gradient |
807 |
% that develops in response to an imposed momentum flux. The vertical |
808 |
% dotted lines indicate the locations of the midpoints of the two |
809 |
% interfaces.} |
810 |
% \end{figure} |
811 |
|
812 |
% \begin{figure} |
813 |
% \includegraphics[width=\linewidth]{SP_comic_strip} |
814 |
% \caption{\label{fig:spComic} The secondary prismatic interface with a shear |
815 |
% rate of 3.5 \ |
816 |
% ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.} |
817 |
% \end{figure} |
818 |
|
819 |
% \begin{figure} |
820 |
% \includegraphics[width=\linewidth]{Pyr-orient} |
821 |
% \caption{\label{fig:PyrOrient} The three decay constants of the |
822 |
% orientational time correlation function, $C_2(z,t)$, for water as a |
823 |
% function of distance from the center of the ice slab. The vertical |
824 |
% dashed line indicates the edge of the pyramidal ice slab determined |
825 |
% by the local order tetrahedral parameter. The control (circles) and |
826 |
% sheared (squares) simulations were fit using shifted-exponential |
827 |
% decay (see Eq. 9 in Ref. \citealp{Louden13}).} |
828 |
% \end{figure} |
829 |
|
830 |
% \begin{figure} |
831 |
% \includegraphics[width=\linewidth]{SP-orient-less} |
832 |
% \caption{\label{fig:SPorient} Decay constants for $C_2(z,t)$ at the secondary |
833 |
% prismatic face. Panel descriptions match those in \ref{fig:PyrOrient}.} |
834 |
% \end{figure} |
835 |
|
836 |
|
837 |
\begin{table}[h] |
838 |
\centering |
839 |
\caption{Sizes of the droplet and shearing simulations. Cell |
840 |
dimensions are measured in \AA. \label{tab:method}} |
841 |
\begin{tabular}{r|cccc|ccccc} |
842 |
\toprule |
843 |
\multirow{2}{*}{Interface} & \multicolumn{4}{c|}{Droplet} & \multicolumn{5}{c}{Shearing} \\ |
844 |
& $N_\mathrm{ice}$ & $N_\mathrm{droplet}$ & $L_x$ & $L_y$ & $N_\mathrm{ice}$ & $N_\mathrm{liquid}$ & $L_x$ & $L_y$ & $L_z$ \\ |
845 |
\midrule |
846 |
Basal $\{0001\}$ & 12960 & 2048 & 134.70 & 140.04 & 900 & 1846 & 23.87 & 35.83 & 98.64 \\ |
847 |
Pyramidal $\{2~0~\bar{2}~1\}$ & 11136 & 2048 & 143.75 & 121.41 & 1216 & 2203 & 37.47 & 29.50 & 93.02 \\ |
848 |
Prismatic $\{1~0~\bar{1}~0\}$ & 9900 & 2048 & 110.04 & 115.00 & 3000 & 5464 & 35.95 & 35.65 & 205.77 \\ |
849 |
Secondary Prism $\{1~1~\bar{2}~0\}$ & 11520 & 2048 & 146.72 & 124.48 & 3840 & 8176 & 71.87 & 31.66 & 161.55 \\ |
850 |
\bottomrule |
851 |
\end{tabular} |
852 |
\end{table} |
853 |
|
854 |
|
855 |
\begin{table}[h] |
856 |
\centering |
857 |
\caption{Structural and dynamic properties of the interfaces of |
858 |
Ice-I$_\mathrm{h}$ with water.\label{tab:kappa}} |
859 |
\begin{tabular}{r|cc|cc|cccc} |
860 |
\toprule |
861 |
\multirow{2}{*}{Interface} & \multicolumn{2}{c|}{Channel Size} &\multicolumn{2}{c|}{Droplet} & \multicolumn{4}{c}{Shearing\footnotemark[1]}\\ |
862 |
& Width (\AA) & Depth (\AA) & $\theta_{\infty}$ ($^\circ$) & $k_\mathrm{spread}$ (ns\textsuperscript{-1}) & |
863 |
$\kappa_{x}$ & $\kappa_{y}$ & $d_\mathrm{struct}$ (\AA) & $d_\mathrm{dyn}$ (\AA) \\ |
864 |
\midrule |
865 |
Basal $\{0001\}$ & 4.49 & 1.30 & $34.1(9)$ &$0.60(7)$ |
866 |
& $5.9(3)$ & $6.5(8)$ & $3.2(4)$ & $2(1)$ \\ |
867 |
Pyramidal $\{2~0~\bar{2}~1\}$ & 8.65 & 1.37 & $35(3)$ & $0.7(1)$ & |
868 |
$5.8(4)$ & $6.1(5)$ & $3.2(2)$ & $2.5(3)$\\ |
869 |
Prismatic $\{1~0~\bar{1}~0\}$ & 6.35 & 2.25 & $45(3)$ & $0.75(9)$ & |
870 |
$3.0(2)$ & $3.0(1)$ & $3.6(2)$ & $4(2)$ \\ |
871 |
Secondary Prism $\{1~1~\bar{2}~0\}$ & 6.35 & 2.25 & $43(2)$ & $0.69(3)$ & |
872 |
$3.5(1)$ & $3.3(2)$ & $3.2(2)$ & $5(3)$ \\ |
873 |
\bottomrule |
874 |
\end{tabular} |
875 |
\begin{flushleft} |
876 |
\footnotemark[1]\footnotesize{Liquid-solid friction coefficients ($\kappa_x$ and |
877 |
$\kappa_y$) are expressed in 10\textsuperscript{-4} amu |
878 |
\AA\textsuperscript{-2} fs\textsuperscript{-1}.} \\ |
879 |
\footnotemark[2]\footnotesize{Uncertainties in |
880 |
the last digits are given in parentheses.} |
881 |
\end{flushleft} |
882 |
\end{table} |
883 |
|
884 |
% Basal $\{0001\}$ & 4.49 & 1.30 & $34.1 \pm 0.9$ &$0.60 \pm 0.07$ |
885 |
% & $5.9 \pm 0.3$ & $6.5 \pm 0.8$ & $3.2 \pm 0.4$ & $2 \pm 1$ \\ |
886 |
% Pyramidal $\{2~0~\bar{2}~1\}$ & 8.65 & 1.37 & $35 \pm 3$ & $0.7 \pm 0.1$ & |
887 |
% $5.8 \pm 0.4$ & $6.1 \pm 0.5$ & $3.2 \pm 0.2$ & $2.5 \pm 0.3$\\ |
888 |
% Prismatic $\{1~0~\bar{1}~0\}$ & 6.35 & 2.25 & $45 \pm 3$ & $0.75 \pm 0.09$ & |
889 |
% $3.0 \pm 0.2$ & $3.0 \pm 0.1$ & $3.6 \pm 0.2$ & $4 \pm 2$ \\ |
890 |
% Secondary Prism $\{1~1~\bar{2}~0\}$ & 6.35 & 2.25 & $43 \pm 2$ & $0.69 \pm 0.03$ & |
891 |
% $3.5 \pm 0.1$ & $3.3 \pm 0.2$ & $3.2 \pm 0.2$ & $5 \pm 3$ \\ |
892 |
|
893 |
|
894 |
\end{document} |