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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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\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{RJSM and ACAC developed the concept of the study. RJSM conducted the analysis, data interpretation and drafted the manuscript. AGB contributed to the development of the statistical methods, data interpretation and drafting of the manuscript.} |
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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 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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simulations of solid-liquid friction of the same four crystal |
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facets being drawn through liquid water. Both 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 |
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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 |
243 |
figure \ref{fig:Droplet}. |
244 |
|
245 |
\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 |
249 |
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 |
254 |
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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(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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|
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To determine the extent of wetting for each of the four crystal |
284 |
facets, contact angles for liquid droplets on the ice surfaces were |
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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 |
287 |
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 |
290 |
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. |
298 |
|
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The second method for obtaining the contact angle was described by |
300 |
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 |
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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, |
307 |
the density profile in a set of annular shells was computed. Due to |
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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 |
311 |
fitting origin ($z_\mathrm{droplet}$) and radius |
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($r_\mathrm{droplet}$) of the droplet can then be used to compute the |
313 |
contact angle, |
314 |
\begin{equation} |
315 |
\theta = 90 + \frac{180}{\pi} \sin^{-1}\left(\frac{z_\mathrm{droplet} - |
316 |
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, |
319 |
although the first method is significantly less prone to noise, and |
320 |
is the method used to report contact angles below. |
321 |
|
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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 |
324 |
\ref{fig:ContactAngle}). Each of these profiles were fit to a |
325 |
biexponential decay, with a short-time contribution ($\tau_c$) that |
326 |
describes the initial contact with the surface, a long time |
327 |
contribution ($\tau_s$) that describes the spread of the droplet over |
328 |
the surface, and a constant ($\theta_\infty$) to capture the |
329 |
infinite-time estimate of the equilibrium contact angle, |
330 |
\begin{equation} |
331 |
\theta(t) = \theta_\infty + (180-\theta_\infty) \left[ a e^{-t/\tau_c} + |
332 |
(1-a) e^{-t/\tau_s} \right] |
333 |
\end{equation} |
334 |
We have found that the rate for water droplet spreading across all |
335 |
four crystal facets, $k_\mathrm{spread} = 1/\tau_s \approx$ 0.7 |
336 |
ns$^{-1}$. However, the basal and pyramidal facets produced estimated |
337 |
equilibrium contact angles, $\theta_\infty \approx$ 35$^{o}$, while |
338 |
prismatic and secondary prismatic had values for $\theta_\infty$ near |
339 |
43$^{o}$ as seen in Table \ref{tab:kappa}. |
340 |
|
341 |
These results indicate that the basal and pyramidal facets are more |
342 |
hydrophilic than the prismatic and secondary prism facets, and |
343 |
surprisingly, that the differential hydrophilicities of the crystal |
344 |
facets is not reflected in the spreading rate of the droplet. |
345 |
|
346 |
% This is in good agreement with our calculations of friction |
347 |
% coefficients, in which the basal |
348 |
% and pyramidal had a higher coefficient of kinetic friction than the |
349 |
% prismatic and secondary prismatic. Due to this, we beleive that the |
350 |
% differences in friction coefficients can be attributed to the varying |
351 |
% hydrophilicities of the facets. |
352 |
|
353 |
\subsection{Coefficient of friction of the interfaces} |
354 |
In a bulk fluid, the shear viscosity, $\eta$, can be determined |
355 |
assuming a linear response to a shear stress, |
356 |
\begin{equation}\label{Shenyu-11} |
357 |
j_{z}(p_{x}) = \eta \frac{\partial v_{x}}{\partial z}. |
358 |
\end{equation} |
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 |
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 |
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(T_{m})}{\left[v_{x}(fluid)-v_{x}(solid)\right]}\left(\frac{\partial v_{x}}{\partial z}\right). |
385 |
\end{equation} |
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 |
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 |
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 |
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 |
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 |
\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 |
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, |
512 |
\end{equation} |
513 |
helps indicate the local environment around the water molecules. The function |
514 |
begins with an initial value of unity, and decays to zero as the water molecule |
515 |
loses memory of its former orientation. Observing the rate at which this decay |
516 |
occurs can provide insight to the mechanism and timescales for the relaxation. |
517 |
In eq. \eqref{C(t)1}, $P_{2}$ is the second-order Legendre polynomial, and |
518 |
$\mathbf{u}$ is the bisecting HOH vector. The angle brackets indicate |
519 |
an ensemble average over all the water molecules in a given spatial region. |
520 |
|
521 |
To investigate the dynamics of the water molecules across the interface, the |
522 |
systems were divided in the $z$-dimension into bins, each $\approx$ 3 \AA\ |
523 |
wide, and eq. \eqref{C(t)1} was computed for each of the bins. A water |
524 |
molecule was allocated to a particular bin if it was initially in the bin |
525 |
at time zero. To compute eq. \eqref{C(t)1}, each 0.5 ns simulation was |
526 |
followed by an additional 200 ps NVE simulation during which the |
527 |
position and orientations of each molecule were recorded every 0.1 ps. |
528 |
|
529 |
The data obtained for each bin was then fit to a triexponential decay |
530 |
with the three decay constants |
531 |
$\tau_{short}$ corresponding to the librational motion of the water |
532 |
molecules, $\tau_{middle}$ corresponding to jumps between the breaking and |
533 |
making of hydrogen bonds, and $\tau_{long}$ corresponding to the translational |
534 |
motion of the water molecules. An additive constant in the fit accounts |
535 |
for the water molecules trapped in the ice which do not experience any |
536 |
long-time orientational decay. |
537 |
|
538 |
In Figure \ref{fig:PyrOrient} we see the $z$-coordinate profiles for |
539 |
the three decay constants, $\tau_{short}$ (panel a), $\tau_{middle}$ |
540 |
(panel b), and $\tau_{long}$ (panel c) for the pyramidal and secondary |
541 |
prismatic systems respectively. The control experiments (no shear) are |
542 |
shown with circles, and an experiment with an imposed momentum flux is |
543 |
shown with squares. The vertical dotted line traversing all three |
544 |
panels denotes the midpoint of the interface determined using the |
545 |
local tetrahedral order parameter. In the liquid regions of both |
546 |
systems, we see that $\tau_{middle}$ and $\tau_{long}$ have |
547 |
approximately consistent values of $3-6$ ps and $30-40$ ps, |
548 |
resepctively, and increase in value as we approach the |
549 |
interface. Conversely, in panel a, we see that $\tau_{short}$ |
550 |
decreases from the liquid value of $72-76$ fs as we approach the |
551 |
interface. We believe this speed up is due to the constrained motion |
552 |
of librations closer to the interface. Both the approximate values for |
553 |
the decays and trends approaching the interface match those reported |
554 |
previously for the basal and prismatic interfaces. |
555 |
|
556 |
As done previously, we have attempted to quantify the distance, $d_{pyramidal}$ |
557 |
and $d_{secondary prismatic}$, from the |
558 |
interface that the deviations from the bulk liquid values begin. This was done |
559 |
by fitting the orientational decay constant $z$-profiles by |
560 |
\begin{equation}\label{tauFit} |
561 |
\tau(z)\approx\tau_{liquid}+(\tau_{wall}-\tau_{liquid})e^{-(z-z_{wall})/d} |
562 |
\end{equation} |
563 |
where $\tau_{liquid}$ and $\tau_{wall}$ are the liquid and projected wall |
564 |
values of the decay constants, $z_{wall}$ is the location of the interface, |
565 |
and $d$ is the displacement from the interface at which these deviations |
566 |
occur. The values for $d_{pyramidal}$ and $d_{secondary prismatic}$ were |
567 |
determined |
568 |
for each of the decay constants, and then averaged for better statistics |
569 |
($\tau_{middle}$ was ommitted for secondary prismatic). For the pyramidal |
570 |
system, |
571 |
$d_{pyramidal}$ was found to be 2.7 \AA\ for both the control and the sheared |
572 |
system. We found $d_{secondary prismatic}$ to be slightly larger than |
573 |
$d_{pyramidal}$ for both the control and with an applied shear, with |
574 |
displacements of $4$ \AA\ for the control system and $3$ \AA\ for the |
575 |
experiment with the imposed momentum flux. These values are consistent with |
576 |
those found for the basal ($d_{basal}\approx2.9$ \AA\ ) and prismatic |
577 |
($d_{prismatic}\approx3.5$ \AA\ ) systems. |
578 |
|
579 |
|
580 |
\section{Conclusion} |
581 |
We present the results of molecular dynamics simulations of the basal, |
582 |
prismatic, pyrmaidal |
583 |
and secondary prismatic facets of an SPC/E model of the |
584 |
Ice-I$_\mathrm{h}$/water interface, and show that the differential |
585 |
coefficients of friction among the four facets are due to their |
586 |
relative hydrophilicities by means |
587 |
of water contact angle calculations. To obtain the coefficients of |
588 |
friction, the ice was sheared through the liquid |
589 |
water while being exposed to a thermal gradient to maintain a stable |
590 |
interface by using the minimally perturbing VSS RNEMD method. Water |
591 |
contact angles are obtained by fitting the spreading of a liquid water |
592 |
droplet over the crystal facets. |
593 |
|
594 |
In agreement with our previous findings for the basal and prismatic facets, the interfacial |
595 |
width of the prismatic and secondary prismatic crystal faces were |
596 |
found to be independent of shear rate as measured by the local |
597 |
tetrahedral order parameter. This width was found to be |
598 |
3.2~$\pm$ 0.2~\AA\ for both the pyramidal and the secondary prismatic systems. |
599 |
These values are in good agreement with our previously calculated interfacial |
600 |
widths for the basal (3.2~$\pm$ 0.4~\AA\ ) and prismatic (3.6~$\pm$ 0.2~\AA\ ) |
601 |
systems. |
602 |
|
603 |
Orientational dynamics of the Ice-I$_\mathrm{h}$/water interfaces were studied |
604 |
by calculation of the orientational time correlation function at varying |
605 |
displacements normal to the interface. The decays were fit |
606 |
to a tri-exponential decay, where the three decay constants correspond to |
607 |
the librational motion of the molecules driven by the restoring forces of |
608 |
existing hydrogen bonds ($\tau_{short}$ $\mathcal{O}$(10 fs)), jumps between |
609 |
two different hydrogen bonds ($\tau_{middle}$ $\mathcal{O}$(1 ps)), and |
610 |
translational motion of the molecules ($\tau_{long}$ $\mathcal{O}$(100 ps)). |
611 |
$\tau_{short}$ was found to decrease approaching the interface due to the |
612 |
constrained motion of the molecules as the local environment becomes more |
613 |
ice-like. Conversely, the two longer-time decay constants were found to |
614 |
increase at small displacements from the interface. As seen in our previous |
615 |
work on the basal and prismatic facets, there appears to be a dynamic |
616 |
interface width at which deviations from the bulk liquid values occur. |
617 |
We had previously found $d_{basal}$ and $d_{prismatic}$ to be approximately |
618 |
2.8~\AA\ and 3.5~\AA. We found good agreement of these values for the |
619 |
pyramidal and secondary prismatic systems with $d_{pyramidal}$ and |
620 |
$d_{secondary prismatic}$ to be 2.7~\AA\ and 3~\AA. For all four of the |
621 |
facets, no apparent dependence of the dynamic width on the shear rate was |
622 |
found. |
623 |
|
624 |
The interfacial friction coefficient, $\kappa$, was determined for each facet |
625 |
interface. We were able to reach an expression for $\kappa$ as a function of |
626 |
the velocity profile of the system which is scaled by the viscosity of the liquid |
627 |
at 225 K. In doing so, we have obtained an expression for $\kappa$ which is |
628 |
independent of temperature differences of the liquid water at far displacements |
629 |
from the interface. We found the basal and pyramidal facets to have |
630 |
similar $\kappa$ values, of $\kappa \approx$ 6 |
631 |
(x$10^{-4}$amu \AA\textsuperscript{-2} fs\textsuperscript{-1}). However, the |
632 |
prismatic and secondary prismatic facets were found to have $\kappa$ values of |
633 |
$\kappa \approx$ 3 (x$10^{-4}$amu \AA\textsuperscript{-2} fs\textsuperscript{-1}). |
634 |
Believing this difference was due to the relative hydrophilicities of |
635 |
the crystal faces, we have calculated the infinite decay of the water |
636 |
contact angle, $\theta_{\infty}$, by watching the spreading of a water |
637 |
droplet over the surface of the crystal facets. We have found |
638 |
$\theta_{\infty}$ for the basal and pyramidal faces to be $\approx$ 34 |
639 |
degrees, while obtaining $\theta_{\infty}$ of $\approx$ 43 degrees for |
640 |
the prismatic and secondary prismatic faces. This indicates that the |
641 |
basal and pyramidal faces of ice-I$_\mathrm{h}$ are more hydrophilic |
642 |
than the prismatic and secondary prismatic. These results also seem to |
643 |
explain the differential friction coefficients obtained through the |
644 |
shearing simulations, namely, that the coefficients of friction of the |
645 |
ice-I$_\mathrm{h}$ crystal facets are governed by their inherent |
646 |
hydrophilicities. |
647 |
|
648 |
|
649 |
\begin{acknowledgments} |
650 |
Support for this project was provided by the National |
651 |
Science Foundation under grant CHE-1362211. Computational time was |
652 |
provided by the Center for Research Computing (CRC) at the |
653 |
University of Notre Dame. |
654 |
\end{acknowledgments} |
655 |
|
656 |
\bibliography{iceWater} |
657 |
% ***************************************** |
658 |
% There is significant interest in the properties of ice/ice and ice/water |
659 |
% interfaces in the geophysics community. Most commonly, the results of shearing |
660 |
% two ice blocks past one |
661 |
% another\cite{Casassa91, Sukhorukov13, Pritchard12, Lishman13} or the shearing |
662 |
% of ice through water\cite{Cuffey99, Bell08}. Using molecular dynamics |
663 |
% simulations, Samadashvili has recently shown that when two smooth ice slabs |
664 |
% slide past one another, a stable liquid-like layer develops between |
665 |
% them\cite{Samadashvili13}. To fundamentally understand these processes, a |
666 |
% molecular understanding of the ice/water interfaces is needed. |
667 |
|
668 |
% Investigation of the ice/water interface is also crucial in understanding |
669 |
% processes such as nucleation, crystal |
670 |
% growth,\cite{Han92, Granasy95, Vanfleet95} and crystal |
671 |
% melting\cite{Weber83, Han92, Sakai96, Sakai96B}. Insight gained to these |
672 |
% properties can also be applied to biological systems of interest, such as |
673 |
% the behavior of the antifreeze protein found in winter |
674 |
% flounder,\cite{Wierzbicki07,Chapsky97} and certain terrestial |
675 |
% arthropods.\cite{Duman:2001qy,Meister29012013} Elucidating the properties which |
676 |
% give rise to these processes through experimental techniques can be expensive, |
677 |
% complicated, and sometimes infeasible. However, through the use of molecular |
678 |
% dynamics simulations much of the problems of investigating these properties |
679 |
% are alleviated. |
680 |
|
681 |
% Understanding ice/water interfaces inherently begins with the isolated |
682 |
% systems. There has been extensive work parameterizing models for liquid water, |
683 |
% such as the SPC\cite{Berendsen81}, SPC/E\cite{Berendsen87}, |
684 |
% TIP4P\cite{Jorgensen85}, TIP4P/2005\cite{Abascal05}, |
685 |
% ($\dots$), and more recently, models for simulating |
686 |
% the solid phases of water, such as the TIP4P/Ice\cite{Abascal05b} model. The |
687 |
% melting point of various crystal structures of ice have been calculated for |
688 |
% many of these models |
689 |
% (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}), |
690 |
% and the partial or complete phase diagram for the model has been determined |
691 |
% (SPC/E\cite{Baez95,Bryk04b,Sanz04b}, TIP4P\cite{Sanz04,Sanz04b,Koyama04}, TIP5P\cite{Sanz04,Koyama04}). |
692 |
% Knowing the behavior and melting point for these models has enabled an initial |
693 |
% investigation of ice/water interfaces. |
694 |
|
695 |
% The Ice-I$_\mathrm{h}$/water quiescent interface has been extensively studied |
696 |
% over the past 30 years by theory and experiment. Haymet \emph{et al.} have |
697 |
% done significant work characterizing and quantifying the width of these |
698 |
% interfaces for the SPC,\cite{Karim90} SPC/E,\cite{Gay02,Bryk02}, |
699 |
% CF1,\cite{Hayward01,Hayward02} and TIP4P\cite{Karim88} models for water. In |
700 |
% recent years, Haymet has focused on investigating the effects cations and |
701 |
% anions have on crystal nucleaion and |
702 |
% melting.\cite{Bryk04,Smith05,Wilson08,Wilson10} Nada and Furukawa have studied |
703 |
% the the basal- and prismatic-water interface width\cite{Nada95}, crystal |
704 |
% surface restructuring at temperatures approaching the melting |
705 |
% point\cite{Nada00}, and the mechanism of ice growth inhibition by antifreeze |
706 |
% proteins\cite{Nada08,Nada11,Nada12}. Nada has developed a six-site water model |
707 |
% for ice/water interfaces near the melting point\cite{Nada03}, and studied the |
708 |
% dependence of crystal growth shape on applied pressure\cite{Nada11b}. Using |
709 |
% this model, Nada and Furukawa have established differential |
710 |
% growth rates for the basal, prismatic, and secondary prismatic facets of |
711 |
% Ice-I$_\mathrm{h}$ and found their origins due to a reordering of the hydrogen |
712 |
% bond network in water near the interface\cite{Nada05}. While the work |
713 |
% described so far has mainly focused on bulk water on ice, there is significant |
714 |
% interest in thin films of water on ice surfaces as well. |
715 |
|
716 |
% It is well known that the surface of ice exhibits a premelting layer at |
717 |
% temperatures near the melting point, often called a quasi-liquid layer (QLL). |
718 |
% Molecular dynamics simulations of the facets of ice-I$_\mathrm{h}$ exposed |
719 |
% to vacuum performed by Conde, Vega and Patrykiejew have found QLL widths of |
720 |
% approximately 10 \AA\ at 3 K below the melting point\cite{Conde08}. |
721 |
% Similarly, Limmer and Chandler have used course grain simulations and |
722 |
% statistical field theory to estimated QLL widths at the same temperature to |
723 |
% be about 3 nm\cite{Limmer14}. |
724 |
% Recently, Sazaki and Furukawa have developed an experimental technique with |
725 |
% sufficient spatial and temporal resolution to visulaize and quantitatively |
726 |
% analyze QLLs on ice crystals at temperatures near melting\cite{Sazaki10}. They |
727 |
% have found the width of the QLLs perpindicular to the surface at -2.2$^{o}$C |
728 |
% to be $\mathcal{O}$(\AA). They have also seen the formation of two immiscible |
729 |
% QLLs, which displayed different stabilities and dynamics on the crystal |
730 |
% surface\cite{Sazaki12}. Knowledge of the hydrophilicities of each |
731 |
% of the crystal facets would help further our understanding of the properties |
732 |
% and dynamics of the QLLs. |
733 |
|
734 |
% Presented here is the follow up to our previous paper\cite{Louden13}, in which |
735 |
% the basal and prismatic facets of an ice-I$_\mathrm{h}$/water interface were |
736 |
% investigated where the ice was sheared relative to the liquid. By using a |
737 |
% recently developed velocity shearing and scaling approach to reverse |
738 |
% non-equilibrium molecular dynamics (VSS-RNEMD), simultaneous temperature and |
739 |
% velocity gradients can be applied to the system, which allows for measurment |
740 |
% of friction and thermal transport properties while maintaining a stable |
741 |
% interfacial temperature\cite{Kuang12}. Structural analysis and dynamic |
742 |
% correlation functions were used to probe the interfacial response to a shear, |
743 |
% and the resulting solid/liquid kinetic friction coefficients were reported. |
744 |
% In this paper we present the same analysis for the pyramidal and secondary |
745 |
% prismatic facets, and show that the differential interfacial friction |
746 |
% coefficients for the four facets of ice-I$_\mathrm{h}$ are determined by their |
747 |
% relative hydrophilicity by means of dynamics water contact angle |
748 |
% simulations. |
749 |
|
750 |
% The local tetrahedral order parameter, $q(z)$, is given by |
751 |
% \begin{equation} |
752 |
% q(z) = \int_0^L \sum_{k=1}^{N} \Bigg(1 -\frac{3}{8}\sum_{i=1}^{3} |
753 |
% \sum_{j=i+1}^{4} \bigg(\cos\psi_{ikj}+\frac{1}{3}\bigg)^2\Bigg) |
754 |
% \delta(z_{k}-z)\mathrm{d}z \Bigg/ N_z |
755 |
% \label{eq:qz} |
756 |
% \end{equation} |
757 |
% where $\psi_{ikj}$ is the angle formed between the oxygen sites of molecules |
758 |
% $i$,$k$, and $j$, where the centeral oxygen is located within molecule $k$ and |
759 |
% molecules $i$ and $j$ are two of the closest four water molecules |
760 |
% around molecule $k$. All four closest neighbors of molecule $k$ are also |
761 |
% required to reside within the first peak of the pair distribution function |
762 |
% for molecule $k$ (typically $<$ 3.41 \AA\ for water). |
763 |
% $N_z = \int\delta(z_k - z) \mathrm{d}z$ is a normalization factor to account |
764 |
% for the varying population of molecules within each finite-width bin. |
765 |
|
766 |
|
767 |
% The hydrophobicity or hydrophilicity of a surface can be described by the |
768 |
% extent a droplet of water wets the surface. The contact angle formed between |
769 |
% the solid and the liquid, $\theta$, which relates the free energies of the |
770 |
% three interfaces involved, is given by Young's equation. |
771 |
% \begin{equation}\label{young} |
772 |
% \cos\theta = (\gamma_{sv} - \gamma_{sl})/\gamma_{lv} |
773 |
% \end{equation} |
774 |
% Here $\gamma_{sv}$, $\gamma_{sl}$, and $\gamma_{lv}$ are the free energies |
775 |
% of the solid/vapor, solid/liquid, and liquid/vapor interfaces respectively. |
776 |
% Large contact angles ($\theta$ $\gg$ 90\textsuperscript{o}) correspond to low |
777 |
% wettability and hydrophobic surfaces, while small contact angles |
778 |
% ($\theta$ $\ll$ 90\textsuperscript{o}) correspond to high wettability and |
779 |
% hydrophilic surfaces. Experimentally, measurements of the contact angle |
780 |
% of sessile drops has been used to quantify the extent of wetting on surfaces |
781 |
% with thermally selective wetting charactaristics\cite{Tadanaga00,Liu04,Sun04}, |
782 |
% as well as nano-pillared surfaces with electrically tunable Cassie-Baxter and |
783 |
% Wenzel states\cite{Herbertson06,Dhindsa06,Verplanck07,Ahuja08,Manukyan11}. |
784 |
% Luzar and coworkers have done significant work modeling these transitions on |
785 |
% nano-patterned surfaces\cite{Daub07,Daub10,Daub11,Ritchie12}, and have found |
786 |
% the change in contact angle to be due to the external field perturbing the |
787 |
% hydrogen bonding of the liquid/vapor interface\cite{Daub07}. |
788 |
|
789 |
|
790 |
|
791 |
\end{article} |
792 |
|
793 |
\begin{figure} |
794 |
\includegraphics[width=\linewidth]{Droplet} |
795 |
\caption{\label{fig:Droplet} Computational model of a droplet of |
796 |
liquid water spreading over the prismatic $\{1~0~\bar{1}~0\}$ facet |
797 |
of ice, before (left) and 2.6 ns after (right) being introduced to the |
798 |
surface. The contact angle ($\theta$) shrinks as the simulation |
799 |
proceeds, and the long-time behavior of this angle is used to |
800 |
estimate the hydrophilicity of the facet.} |
801 |
\end{figure} |
802 |
|
803 |
\begin{figure} |
804 |
\includegraphics[width=2in]{Shearing} |
805 |
\caption{\label{fig:Shearing} Computational model of a slab of ice |
806 |
being sheared through liquid water. In this figure, the ice is |
807 |
presenting two copies of the prismatic $\{1~0~\bar{1}~0\}$ facet |
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towards the liquid phase. The RNEMD simulation exchanges both |
809 |
linear momentum (indicated with arrows) and kinetic energy between |
810 |
the central box and the box that spans the cell boundary. The |
811 |
system responds with weak thermal gradient and a velocity profile |
812 |
that shears the ice relative to the surrounding liquid.} |
813 |
\end{figure} |
814 |
|
815 |
\begin{figure} |
816 |
\includegraphics[width=\linewidth]{ContactAngle} |
817 |
\caption{\label{fig:ContactAngle} The dynamic contact angle of a |
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droplet after approaching each of the four ice facets. The decay to |
819 |
an equilibrium contact angle displays similar dynamics. Although |
820 |
all the surfaces are hydrophilic, the long-time behavior stabilizes |
821 |
to significantly flatter droplets for the basal and pyramidal |
822 |
facets. This suggests a difference in hydrophilicity for these |
823 |
facets compared with the two prismatic facets.} |
824 |
\end{figure} |
825 |
|
826 |
\begin{figure} |
827 |
\includegraphics[width=\linewidth]{Pyr_comic_strip} |
828 |
\caption{\label{fig:pyrComic} Properties of the pyramidal interface |
829 |
being sheared through water at 3.8 ms\textsuperscript{-1}. Lower |
830 |
panel: the local tetrahedral order parameter, $q(z)$, (circles) and |
831 |
the hyperbolic tangent fit (turquoise line). Middle panel: the |
832 |
imposed thermal gradient required to maintain a fixed interfacial |
833 |
temperature of 225 K. Upper panel: the transverse velocity gradient |
834 |
that develops in response to an imposed momentum flux. The vertical |
835 |
dotted lines indicate the locations of the midpoints of the two |
836 |
interfaces.} |
837 |
\end{figure} |
838 |
|
839 |
% \begin{figure} |
840 |
% \includegraphics[width=\linewidth]{SP_comic_strip} |
841 |
% \caption{\label{fig:spComic} The secondary prismatic interface with a shear |
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% rate of 3.5 \ |
843 |
% ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.} |
844 |
% \end{figure} |
845 |
|
846 |
\begin{figure} |
847 |
\includegraphics[width=\linewidth]{Pyr-orient} |
848 |
\caption{\label{fig:PyrOrient} The three decay constants of the |
849 |
orientational time correlation function, $C_2(t)$, for water as a |
850 |
function of distance from the center of the ice slab. The vertical |
851 |
dashed line indicates the edge of the pyramidal ice slab determined |
852 |
by the local order tetrahedral parameter. The control (circles) and |
853 |
sheared (squares) simulations were fit using shifted-exponential |
854 |
decay (see Eq. 9 in Ref. \citealp{Louden13}).} |
855 |
\end{figure} |
856 |
|
857 |
% \begin{figure} |
858 |
% \includegraphics[width=\linewidth]{SP-orient-less} |
859 |
% \caption{\label{fig:SPorient} Decay constants for $C_2(t)$ at the secondary |
860 |
% prismatic face. Panel descriptions match those in \ref{fig:PyrOrient}.} |
861 |
% \end{figure} |
862 |
|
863 |
|
864 |
\begin{table}[h] |
865 |
\centering |
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\caption{Sizes of the droplet and shearing simulations. Cell |
867 |
dimensions are measured in \AA. \label{tab:method}} |
868 |
\begin{tabular}{r|cccc|ccccc} |
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\toprule |
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\multirow{2}{*}{Interface} & \multicolumn{4}{c|}{Droplet} & \multicolumn{5}{c}{Shearing} \\ |
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& $N_\mathrm{ice}$ & $N_\mathrm{droplet}$ & $L_x$ & $L_y$ & $N_\mathrm{ice}$ & $N_\mathrm{liquid}$ & $L_x$ & $L_y$ & $L_z$ \\ |
872 |
\midrule |
873 |
Basal $\{0001\}$ & 12960 & 2048 & 134.70 & 140.04 & 900 & 1846 & 23.87 & 35.83 & 98.64 \\ |
874 |
Pyramidal $\{2~0~\bar{2}~1\}$ & 11136 & 2048 & 143.75 & 121.41 & 1216 & 2203 & 37.47 & 29.50 & 93.02 \\ |
875 |
Prismatic $\{1~0~\bar{1}~0\}$ & 9900 & 2048 & 110.04 & 115.00 & 3000 & 5464 & 35.95 & 35.65 & 205.77 \\ |
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Secondary Prism $\{1~1~\bar{2}~0\}$ & 11520 & 2048 & 146.72 & 124.48 & 3840 & 8176 & 71.87 & 31.66 & 161.55 \\ |
877 |
\bottomrule |
878 |
\end{tabular} |
879 |
\end{table} |
880 |
|
881 |
|
882 |
\begin{table}[h] |
883 |
\centering |
884 |
\caption{Structural and dynamic properties of the interfaces of Ice-I$_\mathrm{h}$ |
885 |
with water. Liquid-solid friction coefficients ($\kappa_x$ and $\kappa_y$) are expressed in 10\textsuperscript{-4} amu |
886 |
\AA\textsuperscript{-2} fs\textsuperscript{-1}. \label{tab:kappa}} |
887 |
\begin{tabular}{r|cc|cccc} |
888 |
\toprule |
889 |
\multirow{2}{*}{Interface} & \multicolumn{2}{c|}{Droplet} & \multicolumn{4}{c}{Shearing}\\ |
890 |
& $\theta_{\infty}$ ($^\circ$) & $k_\mathrm{spread}$ (ns\textsuperscript{-1}) & |
891 |
$\kappa_{x}$ & $\kappa_{y}$ & $d_{q_{z}}$ (\AA) & $d_{\tau}$ (\AA) \\ |
892 |
\midrule |
893 |
Basal $\{0001\}$ & $34.1 \pm 0.9$ &$0.60 \pm 0.07$ |
894 |
& $5.9 \pm 0.3$ & $6.5 \pm 0.8$ & $3.2 \pm 0.4$ & $2 \pm 1$ \\ |
895 |
Pyramidal $\{2~0~\bar{2}~1\}$ & $35 \pm 3$ & $0.7 \pm 0.1$ & |
896 |
$5.8 \pm 0.4$ & $6.1 \pm 0.5$ & $3.2 \pm 0.2$ & $2.5 \pm 0.3$\\ |
897 |
Prismatic $\{1~0~\bar{1}~0\}$ & $45 \pm 3$ & $0.75 \pm 0.09$ & |
898 |
$3.0 \pm 0.2$ & $3.0 \pm 0.1$ & $3.6 \pm 0.2$ & $4 \pm 2$ \\ |
899 |
Secondary Prism $\{1~1~\bar{2}~0\}$ & $43 \pm 2$ & $0.69 \pm 0.03$ & |
900 |
$3.5 \pm 0.1$ & $3.3 \pm 0.2$ & $3.2 \pm 0.2$ & $5 \pm 3$ \\ |
901 |
\bottomrule |
902 |
\end{tabular} |
903 |
\end{table} |
904 |
|
905 |
\end{document} |