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\begin{document} |
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\title{Friction at Water / Ice-I$_\mathrm{h}$ interfaces: Do the |
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Facets of Ice Have Different Hydrophilicity?} |
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|
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\author{Patrick B. Louden} |
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|
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu.} |
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\affiliation{Department of Chemistry and Biochemistry, University |
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of Notre Dame, Notre Dame, IN 46556} |
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\date{\today} |
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\begin{abstract} |
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In this follow up paper of the basal and prismatic facets of the |
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Ice-I$_\mathrm{h}$/water interface, we present the |
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pyramidal and secondary prismatic |
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interfaces for both the quiescent and sheared systems. The structural and |
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dynamic interfacial widths for all four crystal facets were found to be in good |
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agreement, and were found to be independent of the shear rate over the shear |
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rates investigated. |
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Decomposition of the molecular orientational time correlation function showed |
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different behavior for the short- and longer-time decay components approaching |
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normal to the interface. Lastly we show through calculation of the interfacial |
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friction coefficient that the basal and pyramidal facets are more |
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hydrophilic than the prismatic and secondary prismatic facets. |
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|
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\end{abstract} |
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|
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\maketitle |
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|
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\section{Introduction} |
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|
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% Why are ice and ice/water interfaces important? |
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There is significant interest in the properties of ice/ice and ice/water |
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interfaces in the geophysics community. Most commonly, the results of shearing |
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two ice blocks past one |
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another\cite{Casassa91, Sukhorukov13, Pritchard12, Lishman13} or the shearing |
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of ice through water\cite{Cuffey99, Bell08}. Using molecular dynamics |
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simulations, Samadashvili has recently shown that when two smooth ice slabs |
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slide past one another, a stable liquid-like layer develops between |
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them\cite{Samadashvili13}. To fundamentally understand these processes, a |
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molecular understanding of the ice/water interfaces is needed. |
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|
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Investigation of the ice/water interface is also crucial in understanding |
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processes such as nucleation, crystal |
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growth,\cite{Han92, Granasy95, Vanfleet95} and crystal |
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melting\cite{Weber83, Han92, Sakai96, Sakai96B}. Insight gained to these |
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properties can also be applied to biological systems of interest, such as |
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the behavior of the antifreeze protein found in winter |
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flounder,\cite{Wierzbicki07,Chapsky97} and certain terrestial |
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arthropods.\cite{Duman:2001qy,Meister29012013} Elucidating the properties which |
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give rise to these processes through experimental techniques can be expensive, |
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complicated, and sometimes infeasible. However, through the use of molecular |
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dynamics simulations much of the problems of investigating these properties |
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are alleviated. |
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|
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Understanding ice/water interfaces inherently begins with the isolated |
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systems. There has been extensive work parameterizing models for liquid water, |
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such as the SPC\cite{Berendsen81}, SPC/E\cite{Berendsen87}, |
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TIP4P\cite{Jorgensen85}, TIP4P/2005\cite{Abascal05}, |
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($\dots$), and more recently, models for simulating |
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the solid phases of water, such as the TIP4P/Ice\cite{Abascal05b} model. The |
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melting point of various crystal structures of ice have been calculated for |
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many of these models |
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(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}), |
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and the partial or complete phase diagram for the model has been determined |
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(SPC/E\cite{Baez95,Bryk04b,Sanz04b}, TIP4P\cite{Sanz04,Sanz04b,Koyama04}, TIP5P\cite{Sanz04,Koyama04}). |
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Knowing the behavior and melting point for these models has enabled an initial |
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investigation of ice/water interfaces. |
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|
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The Ice-I$_\mathrm{h}$/water quiescent interface has been extensively studied |
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over the past 30 years by theory and experiment. Haymet \emph{et al.} have |
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done significant work characterizing and quantifying the width of these |
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interfaces for the SPC,\cite{Karim90} SPC/E,\cite{Gay02,Bryk02}, |
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CF1,\cite{Hayward01,Hayward02} and TIP4P\cite{Karim88} models for water. In |
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recent years, Haymet has focused on investigating the effects cations and |
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anions have on crystal nucleaion and |
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melting.\cite{Bryk04,Smith05,Wilson08,Wilson10} Nada and Furukawa have studied |
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the the basal- and prismatic-water interface width\cite{Nada95}, crystal |
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surface restructuring at temperatures approaching the melting |
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point\cite{Nada00}, and the mechanism of ice growth inhibition by antifreeze |
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proteins\cite{Nada08,Nada11,Nada12}. Nada has developed a six-site water model |
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for ice/water interfaces near the melting point\cite{Nada03}, and studied the |
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dependence of crystal growth shape on applied pressure\cite{Nada11b}. Using |
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this model, Nada and Furukawa have established differential |
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growth rates for the basal, prismatic, and secondary prismatic facets of |
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Ice-I$_\mathrm{h}$ and found their origins due to a reordering of the hydrogen |
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bond network in water near the interface\cite{Nada05}. While the work |
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described so far has mainly focused on bulk water on ice, there is significant |
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interest in thin films of water on ice surfaces as well. |
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|
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It is well known that the surface of ice exhibits a premelting layer at |
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temperatures near the melting point, often called a quasi-liquid layer (QLL). |
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Molecular dynamics simulations of the facets of ice-I$_\mathrm{h}$ exposed |
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to vacuum performed by Conde, Vega and Patrykiejew have found QLL widths of |
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approximately 10 \AA\ at 3 K below the melting point\cite{Conde08}. |
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Similarly, Limmer and Chandler have used course grain simulations and |
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statistical field theory to estimated QLL widths at the same temperature to |
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be about 3 nm\cite{Limmer14}. |
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Recently, Sazaki and Furukawa have developed an experimental technique with |
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sufficient spatial and temporal resolution to visulaize and quantitatively |
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analyze QLLs on ice crystals at temperatures near melting\cite{Sazaki10}. They |
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have found the width of the QLLs perpindicular to the surface at -2.2$^{o}$C |
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to be $\mathcal{O}$(\AA). They have also seen the formation of two immiscible |
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QLLs, which displayed different stabilities and dynamics on the crystal |
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surface\cite{Sazaki12}. Knowledge of the hydrophilicities of each |
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of the crystal facets would help further our understanding of the properties |
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and dynamics of the QLLs. |
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|
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Presented here is the follow up to our previous paper\cite{Louden13}, in which |
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the basal and prismatic facets of an ice-I$_\mathrm{h}$/water interface were |
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investigated where the ice was sheared relative to the liquid. By using a |
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recently developed velocity shearing and scaling approach to reverse |
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non-equilibrium molecular dynamics (VSS-RNEMD), simultaneous temperature and |
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velocity gradients can be applied to the system, which allows for measurment |
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of friction and thermal transport properties while maintaining a stable |
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interfacial temperature\cite{Kuang12}. Structural analysis and dynamic |
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correlation functions were used to probe the interfacial response to a shear, |
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and the resulting solid/liquid kinetic friction coefficients were reported. |
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In this paper we present the same analysis for the pyramidal and secondary |
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prismatic facets, and show that the differential interfacial friction |
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coefficients for the four facets of ice-I$_\mathrm{h}$ are determined by their |
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relative hydrophilicity by means of dynamics water contact angle simulations. |
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|
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\section{Methodology} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{Pyr_comic_strip} |
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\caption{\label{fig:pyrComic} The pyramidal interface with a shear |
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rate of 3.8 ms\textsuperscript{-1}. Lower panel: the local tetrahedral order |
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parameter, $q(z)$, (black circles) and the hyperbolic tangent fit (red line). |
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Middle panel: the imposed thermal gradient required to maintain a fixed |
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interfacial temperature. Upper panel: the transverse velocity gradient that |
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develops in response to an imposed momentum flux. The vertical dotted lines |
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indicate the locations of the midpoints of the two interfaces.} |
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\end{figure} |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{SP_comic_strip} |
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\caption{\label{fig:spComic} The secondary prismatic interface with a shear |
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rate of 3.5 \ |
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ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.} |
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\end{figure} |
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|
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\subsection{Pyramidal and secondary prismatic system construction} |
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|
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The construction of the pyramidal and secondary prismatic systems follows that |
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of |
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the basal and prismatic systems presented elsewhere\cite{Louden13}, however |
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the ice crystals and water boxes were equilibrated and combined at 50K |
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instead of 225K. The ice / water systems generated were then equilibrated |
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to 225K. The resulting pyramidal system was |
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$37.47 \times 29.50 \times 93.02$ \AA\ with 1216 |
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SPC/E\cite{Berendsen87} molecules in the ice slab, and 2203 in the liquid |
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phase. The secondary |
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prismatic system generated was $71.87 \times 31.66 \times 161.55$ \AA\ with |
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3840 |
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SPC/E molecules in the ice slab and 8176 molecules in the liquid phase. |
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|
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\subsection{Computational details} |
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% Do we need to justify the sims at 225K? |
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% No crystal growth or shrinkage over 2 successive 1 ns NVT simulations for |
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% either the pyramidal or sec. prismatic ice/water systems. |
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|
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The computational details performed here were equivalent to those reported |
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in our previous publication\cite{Louden13}. The only changes made to the |
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previously reported procedure were the following. VSS-RNEMD moves were |
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attempted every 2 fs instead of every 50 fs. This was done to minimize |
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the magnitude of each individual VSS-RNEMD perturbation to the system. |
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|
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All pyramidal simulations were performed under the canonical (NVT) ensamble |
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except those |
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during which statistics were accumulated for the orientational correlation |
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function, which were performed under the microcanonical (NVE) ensamble. All |
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secondary prismatic |
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simulations were performed under the NVE ensamble. |
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|
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\section{Results and discussion} |
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\subsection{Interfacial width} |
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In the literature there is good agreement that between the solid ice and |
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the bulk water, there exists a region of 'slush-like' water molecules. |
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In this region, the water molecules are structurely distinguishable and |
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behave differently than those of the solid ice or the bulk water. |
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The characteristics of this region have been defined by both structural |
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and dynamic properties; and its width has been measured by the change of these |
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properties from their bulk liquid values to those of the solid ice. |
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Examples of these properties include the density, the diffusion constant, and |
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the translational order profile. \cite{Bryk02,Karim90,Gay02,Hayward01,Hayward02,Karim88} |
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|
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Since the VSS-RNEMD moves used to impose the thermal and velocity gradients |
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perturb the momenta of the water molecules in |
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the systems, parameters that depend on translational motion may give |
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faulty results. A stuructural parameter will be less effected by the |
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VSS-RNEMD perturbations to the system. Due to this, we have used the |
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local tetrahedral order parameter to quantify the width of the interface, |
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which was originally described by Kumar\cite{Kumar09} and |
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Errington\cite{Errington01}, and used by Bryk and Haymet in a previous study |
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of ice/water interfaces.\cite{Bryk04b} |
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|
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The local tetrahedral order parameter, $q(z)$, is given by |
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\begin{equation} |
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q(z) = \int_0^L \sum_{k=1}^{N} \Bigg(1 -\frac{3}{8}\sum_{i=1}^{3} |
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\sum_{j=i+1}^{4} \bigg(\cos\psi_{ikj}+\frac{1}{3}\bigg)^2\Bigg) |
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\delta(z_{k}-z)\mathrm{d}z \Bigg/ N_z |
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\label{eq:qz} |
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\end{equation} |
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where $\psi_{ikj}$ is the angle formed between the oxygen sites of molecules |
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$i$,$k$, and $j$, where the centeral oxygen is located within molecule $k$ and |
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molecules $i$ and $j$ are two of the closest four water molecules |
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around molecule $k$. All four closest neighbors of molecule $k$ are also |
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required to reside within the first peak of the pair distribution function |
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for molecule $k$ (typically $<$ 3.41 \AA\ for water). |
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$N_z = \int\delta(z_k - z) \mathrm{d}z$ is a normalization factor to account |
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for the varying population of molecules within each finite-width bin. |
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|
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To determine the width of the interfaces, each of the systems were |
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divided into 100 artificial bins along the |
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$z$-dimension, and the local tetrahedral order parameter, $q(z)$, was |
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time-averaged for each of the bins, resulting in a tetrahedrality profile of |
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the system. These profiles are shown across the $z$-dimension of the systems |
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in panel $a$ of Figures \ref{fig:pyrComic} |
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and \ref{fig:spComic} (black circles). The $q(z)$ function has a range of |
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(0,1), where a larger value indicates a more tetrahedral environment. |
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The $q(z)$ for the bulk liquid was found to be $\approx $ 0.77, while values of |
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$\approx $ 0.92 were more common for the ice. The tetrahedrality profiles were |
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fit using a hyperbolic tangent\cite{Louden13} designed to smoothly fit the |
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bulk to ice |
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transition, while accounting for the thermal influence on the profile by the |
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kinetic energy exchanges of the VSS-RNEMD moves. In panels $b$ and $c$, the |
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resulting thermal and velocity gradients from an imposed kinetic energy and |
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momentum fluxes can be seen. The verticle dotted |
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lines traversing all three panels indicate the midpoints of the interface |
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as determined by the hyperbolic tangent fit of the tetrahedrality profiles. |
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|
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From fitting the tetrahedrality profiles for each of the 0.5 nanosecond |
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simulations (panel c of Figures \ref{fig:pyrComic} and \ref{fig:spComic}) |
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by eq. 6\cite{Louden13},we find the interfacial width to be |
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3.2 $\pm$ 0.2 and 3.2 $\pm$ 0.2 \AA\ for the control system with no applied |
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momentum flux for both the pyramidal and secondary prismatic systems. |
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Over the range of shear rates investigated, |
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0.6 $\pm$ 0.2 $\mathrm{ms}^{-1} \rightarrow$ 5.6 $\pm$ 0.4 $\mathrm{ms}^{-1}$ |
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for the pyramidal system and 0.9 $\pm$ 0.3 $\mathrm{ms}^{-1} \rightarrow$ 5.4 |
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$\pm$ 0.1 $\mathrm{ms}^{-1}$ for the secondary prismatic, we found no |
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significant change in the interfacial width. This follows our previous |
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findings of the basal and |
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prismatic systems, in which the interfacial width was invarient of the |
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shear rate of the ice. The interfacial width of the quiescent basal and |
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prismatic systems was found to be 3.2 $\pm$ 0.4 \AA\ and 3.6 $\pm$ 0.2 \AA\ |
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respectively, over the range of shear rates investigated, 0.6 $\pm$ 0.3 |
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$\mathrm{ms}^{-1} \rightarrow$ 5.3 $\pm$ 0.5 $\mathrm{ms}^{-1}$ for the basal |
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system and 0.9 $\pm$ 0.2 $\mathrm{ms}^{-1} \rightarrow$ 4.5 $\pm$ 0.1 |
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$\mathrm{ms}^{-1}$ for the prismatic. |
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|
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These results indicate that the surface structure of the exposed ice crystal |
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has little to no effect on how far into the bulk the ice-like structural |
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ordering is. Also, it appears that the interface is not structurally effected |
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by shearing the ice through water. |
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|
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|
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\subsection{Orientational dynamics} |
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%Should we include the math here? |
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The orientational time correlation function, |
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\begin{equation}\label{C(t)1} |
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C_{2}(t)=\langle P_{2}(\mathbf{u}(0)\cdot \mathbf{u}(t))\rangle, |
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\end{equation} |
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helps indicate the local environment around the water molecules. The function |
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begins with an initial value of unity, and decays to zero as the water molecule |
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loses memory of its former orientation. Observing the rate at which this decay |
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occurs can provide insight to the mechanism and timescales for the relaxation. |
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In eq. \eqref{C(t)1}, $P_{2}$ is the second-order Legendre polynomial, and |
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$\mathbf{u}$ is the bisecting HOH vector. The angle brackets indicate |
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an ensemble average over all the water molecules in a given spatial region. |
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|
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To investigate the dynamics of the water molecules across the interface, the |
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systems were divided in the $z$-dimension into bins, each $\approx$ 3 \AA\ |
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wide, and eq. \eqref{C(t)1} was computed for each of the bins. A water |
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molecule was allocated to a particular bin if it was initially in the bin |
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at time zero. To compute eq. \eqref{C(t)1}, each 0.5 ns simulation was |
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followed by an additional 200 ps NVE simulation during which the |
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position and orientations of each molecule were recorded every 0.1 ps. |
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|
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The data obtained for each bin was then fit to a triexponential decay given by |
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\begin{equation}\label{C(t)_fit} |
| 323 |
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C_{2}(t) \approx a e^{-t/\tau_\mathrm{short}} + b e^{-t/\tau_\mathrm{middle}} +\ |
| 324 |
< |
c |
| 325 |
< |
e^{-t/\tau_\mathrm{long}} + (1-a-b-c) |
| 326 |
< |
\end{equation} |
| 327 |
< |
where $\tau_{short}$ corresponds to the librational motion of the water |
| 328 |
< |
molecules, $\tau_{middle}$ corresponds to jumps between the breaking and |
| 329 |
< |
making of hydrogen bonds, and $\tau_{long}$ corresponds to the translational |
| 330 |
< |
motion of the water molecules. The last term in \eqref{C(t)_fit} accounts |
| 331 |
< |
for the water molecules trapped in the ice which do not experience any |
| 332 |
< |
long-time orientational decay. |
| 333 |
< |
|
| 334 |
< |
In Figures \ref{fig:PyrOrient} and \ref{fig:SPorient} we see the $z$-coordinate |
| 335 |
< |
profiles for the three decay constants, $\tau_{short}$ (panel a), |
| 336 |
< |
$\tau_{middle}$ (panel b), |
| 337 |
< |
and $\tau_{long}$ (panel c) for the pyramidal and secondary prismatic systems |
| 338 |
< |
respectively. The control experiments (no shear) are shown in black, and |
| 339 |
< |
an experiment with an imposed momentum flux is shown in red. The vertical |
| 340 |
< |
dotted line traversing all three panels denotes the midpoint of the |
| 341 |
< |
interface as determined by the local tetrahedral order parameter fitting. |
| 342 |
< |
In the liquid regions of both systems, we see that $\tau_{middle}$ and |
| 343 |
< |
$\tau_{long}$ have approximately consistent values of $3-6$ ps and $30-40$ ps, |
| 344 |
< |
resepctively, and increase in value as we approach the interface. Conversely, |
| 345 |
< |
in panel a, we see that $\tau_{short}$ decreases from the liquid value |
| 346 |
< |
of $72-76$ fs as we approach the interface. We believe this speed up is due to |
| 347 |
< |
the constrained motion of librations closer to the interface. Both the |
| 348 |
< |
approximate values for the decays and trends approaching the interface match |
| 349 |
< |
those reported previously for the basal and prismatic interfaces. |
| 350 |
< |
|
| 351 |
< |
As done previously, we have attempted to quantify the distance, $d_{pyramidal}$ |
| 352 |
< |
and $d_{secondary prismatic}$, from the |
| 353 |
< |
interface that the deviations from the bulk liquid values begin. This was done |
| 354 |
< |
by fitting the orientational decay constant $z$-profiles by |
| 355 |
< |
\begin{equation}\label{tauFit} |
| 356 |
< |
\tau(z)\approx\tau_{liquid}+(\tau_{wall}-\tau_{liquid})e^{-(z-z_{wall})/d} |
| 357 |
< |
\end{equation} |
| 358 |
< |
where $\tau_{liquid}$ and $\tau_{wall}$ are the liquid and projected wall |
| 359 |
< |
values of the decay constants, $z_{wall}$ is the location of the interface, |
| 360 |
< |
and $d$ is the displacement from the interface at which these deviations |
| 361 |
< |
occur. The values for $d_{pyramidal}$ and $d_{secondary prismatic}$ were |
| 362 |
< |
determined |
| 363 |
< |
for each of the decay constants, and then averaged for better statistics |
| 364 |
< |
($\tau_{middle}$ was ommitted for secondary prismatic). For the pyramidal |
| 365 |
< |
system, |
| 366 |
< |
$d_{pyramidal}$ was found to be 2.7 \AA\ for both the control and the sheared |
| 367 |
< |
system. We found $d_{secondary prismatic}$ to be slightly larger than |
| 368 |
< |
$d_{pyramidal}$ for both the control and with an applied shear, with |
| 369 |
< |
displacements of $4$ \AA\ for the control system and $3$ \AA\ for the |
| 370 |
< |
experiment with the imposed momentum flux. These values are consistent with |
| 371 |
< |
those found for the basal ($d_{basal}\approx2.9$ \AA\ ) and prismatic |
| 372 |
< |
($d_{prismatic}\approx3.5$ \AA\ ) systems. |
| 373 |
< |
|
| 374 |
< |
\subsection{Coefficient of friction of the interfaces} |
| 375 |
< |
While investigating the kinetic coefficient of friction, there was found |
| 376 |
< |
to be a dependence for $\mu_k$ |
| 377 |
< |
on the temperature of the liquid water in the system. We believe this |
| 378 |
< |
dependence |
| 379 |
< |
arrises from the sharp discontinuity of the viscosity for the SPC/E model |
| 380 |
< |
at temperatures approaching 200 K\cite{Kuang12}. Due to this, we propose |
| 381 |
< |
a weighting to the interfacial friction coefficient, $\kappa$ by the |
| 382 |
< |
shear viscosity of the fluid at 225 K. The interfacial friction coefficient |
| 383 |
< |
relates the shear stress with the relative velocity of the fluid normal to the |
| 384 |
< |
interface: |
| 385 |
< |
\begin{equation}\label{Shenyu-13} |
| 386 |
< |
j_{z}(p_{x}) = \kappa[v_{x}(fluid)-v_{x}(solid)] |
| 387 |
< |
\end{equation} |
| 388 |
< |
where $j_{z}(p_{x})$ is the applied momentum flux (shear stress) across $z$ |
| 389 |
< |
in the |
| 390 |
< |
$x$-dimension, and $v_{x}$(fluid) and $v_{x}$(solid) are the velocities |
| 391 |
< |
directly adjacent to the interface. The shear viscosity, $\eta(T)$, of the |
| 392 |
< |
fluid can be determined under a linear response of the momentum |
| 393 |
< |
gradient to the applied shear stress by |
| 394 |
< |
\begin{equation}\label{Shenyu-11} |
| 395 |
< |
j_{z}(p_{x}) = \eta(T) \frac{\partial v_{x}}{\partial z}. |
| 396 |
< |
\end{equation} |
| 397 |
< |
Using eqs \eqref{Shenyu-13} and \eqref{Shenyu-11}, we can find the following |
| 398 |
< |
expression for $\kappa$, |
| 399 |
< |
\begin{equation}\label{kappa-1} |
| 400 |
< |
\kappa = \eta(T) \frac{\partial v_{x}}{\partial z}\frac{1}{[v_{x}(fluid)-v_{x}(solid)]}. |
| 401 |
< |
\end{equation} |
| 402 |
< |
Here is where we will introduce the weighting term of $\eta(225)/\eta(T)$ |
| 403 |
< |
giving us |
| 404 |
< |
\begin{equation}\label{kappa-2} |
| 405 |
< |
\kappa = \frac{\eta(225)}{[v_{x}(fluid)-v_{x}(solid)]}\frac{\partial v_{x}}{\partial z}. |
| 406 |
< |
\end{equation} |
| 407 |
< |
|
| 408 |
< |
To obtain the value of $\eta(225)$ for the SPC/E model, a $31.09 \times 29.38 |
| 409 |
< |
\times 124.39$ \AA\ box with 3744 SPC/E liquid water molecules was |
| 410 |
< |
equilibrated to 225K, |
| 411 |
< |
and 5 unique shearing experiments were performed. Each experiment was |
| 412 |
< |
conducted in the NVE and were 5 ns in |
| 413 |
< |
length. The VSS were attempted every timestep, which was set to 2 fs. |
| 414 |
< |
For our SPC/E systems, we found $\eta(225)$ to be 0.0148 $\pm$ 0.0007 Pa s, |
| 415 |
< |
roughly ten times larger than the value found for 280 K SPC/E bulk water by |
| 416 |
< |
Kuang\cite{Kuang12}. |
| 417 |
< |
|
| 418 |
< |
The interfacial friction coefficient, $\kappa$, can equivalently be expressed |
| 419 |
< |
as the ratio of the viscosity of the fluid to the slip length, $\delta$, which |
| 420 |
< |
is an indication of how 'slippery' the interface is. |
| 421 |
< |
\begin{equation}\label{kappa-3} |
| 422 |
< |
\kappa = \frac{\eta}{\delta} |
| 423 |
< |
\end{equation} |
| 424 |
< |
In each of the systems, the interfacial temperature was kept fixed to 225K, |
| 425 |
< |
which ensured the viscosity of the fluid at the |
| 426 |
< |
interace was approximately the same. Thus, any significant variation in |
| 427 |
< |
$\kappa$ between |
| 428 |
< |
the systems indicates differences in the 'slipperiness' of the interfaces. |
| 429 |
< |
As each of the ice systems are sheared relative to liquid water, the |
| 430 |
< |
'slipperiness' of the interface can be taken as an indication of how |
| 431 |
< |
hydrophobic or hydrophilic the interface is. The calculated $\kappa$ values |
| 432 |
< |
found for the four crystal facets of Ice-I$_\mathrm{h}$ investigated are shown |
| 433 |
< |
in Table \ref{tab:kappa}. The basal and pyramidal facets were found to have |
| 434 |
< |
similar values of $\kappa \approx$ 0.0006 |
| 435 |
< |
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}), while values of |
| 436 |
< |
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1}) |
| 437 |
< |
were found for the prismatic and secondary prismatic systems. |
| 438 |
< |
These results indicate that the basal and pyramidal facets are |
| 439 |
< |
more hydrophilic than the prismatic and secondary prismatic facets. |
| 440 |
< |
|
| 441 |
< |
\subsection{Dynamic water contact angle} |
| 442 |
< |
The hydrophobicity or hydrophilicity of a surface can be described by the |
| 443 |
< |
extent a droplet of water wets the surface. The contact angle formed between |
| 444 |
< |
the solid and the liquid, $\theta$, which relates the free energies of the |
| 445 |
< |
three interfaces involved, is given by Young's equation. |
| 446 |
< |
\begin{equation}\label{young} |
| 447 |
< |
\cos\theta = (\gamma_{sv} - \gamma_{sl})/\gamma_{lv} |
| 448 |
< |
\end{equation} |
| 449 |
< |
Here $\gamma_{sv}$, $\gamma_{sl}$, and $\gamma_{lv}$ are the free energies |
| 450 |
< |
of the solid/vapor, solid/liquid, and liquid/vapor interfaces respectively. |
| 451 |
< |
Large contact angles ($\theta$ $\gg$ 90\textsuperscript{o}) correspond to low |
| 452 |
< |
wettability and hydrophobic surfaces, while small contact angles |
| 453 |
< |
($\theta$ $\ll$ 90\textsuperscript{o}) correspond to high wettability and |
| 454 |
< |
hydrophilic surfaces. Experimentally, measurements of the contact angle |
| 455 |
< |
of sessile drops has been used to quantify the extent of wetting on surfaces |
| 456 |
< |
with thermally selective wetting charactaristics\cite{Tadanaga00,Liu04,Sun04}, |
| 457 |
< |
as well as nano-pillared surfaces with electrically tunable Cassie-Baxter and |
| 458 |
< |
Wenzel states\cite{Herbertson06,Dhindsa06,Verplanck07,Ahuja08,Manukyan11}. |
| 459 |
< |
Luzar and coworkers have done significant work modeling these transitions on |
| 460 |
< |
nano-patterned surfaces\cite{Daub07,Daub10,Daub11,Ritchie12}, and have found |
| 461 |
< |
the change in contact angle to be due to the external field perturbing the |
| 462 |
< |
hydrogen bonding of the liquid/vapor interface\cite{Daub07}. |
| 463 |
< |
|
| 464 |
< |
Here, we will calculate the contact angle of a water droplet as it spreads |
| 465 |
< |
across each of the four ice I$_\mathrm{h}$ crystal facets in order to |
| 466 |
< |
determine the surface's relative hydrophilicites. The ice surfaces were |
| 467 |
< |
oriented so that the desired facet was exposed to the positive z dimension. |
| 468 |
< |
The sizes and number of molecules in each of the surfaces is given in Table |
| 469 |
< |
\ref{tab:ice_sheets}. Molecular restraints were applied to the center of mass |
| 470 |
< |
of the rigid bodies to prevent surface melting, however the molecules were |
| 471 |
< |
allowed to reorient themselves freely. The water doplet to be placed on the |
| 472 |
< |
surface contained 2048 SPC/E molecules, which has been found to produce |
| 473 |
< |
agreement for the Young contact angle extrapolated to an infinite drop |
| 474 |
< |
size\cite{Daub10}. The surfaces and droplet were equilibrated to 225 K, at |
| 475 |
< |
which time the droplet was placed 3-5 \AA\ above the surface at 5 unique |
| 476 |
< |
locations. Each simulation was 5 ns in length and conducted in the NVE |
| 477 |
< |
ensemble. |
| 478 |
< |
|
| 479 |
< |
|
| 480 |
< |
|
| 481 |
< |
|
| 482 |
< |
\begin{table}[h] |
| 483 |
< |
\centering |
| 484 |
< |
\caption{Phyiscal properties of the basal, prismatic, pyramidal, and secondary prismatic facets of Ice-I$_\mathrm{h}$} |
| 485 |
< |
\label{tab:kappa} |
| 486 |
< |
\begin{tabular}{|ccccc|} \hline |
| 487 |
< |
& \multicolumn{2}{c}{$\kappa_{Drag direction}$ (amu \AA\textsuperscript{-2} fs\textsuperscript{-1})} & & \\ |
| 488 |
< |
Interface & $\kappa_{x}$ & $\kappa_{y}$ & $\theta_{\infty}$ & K_{spread} (ns^{-1}) \\ \hline |
| 489 |
< |
basal & $0.00059 \pm 0.00003$ & $0.00065 \pm 0.00008$ & $34.1 \pm 0.9$ & $0.60 \pm 0.07$ \\ |
| 490 |
< |
pyramidal & $0.00058 \pm 0.00004$ & $0.00061 \pm 0.00005$ & $35 \pm 3$ & $0.7 \pm 0.1$ \\ |
| 491 |
< |
prismatic & $0.00030 \pm 0.00002$ & $0.00030 \pm 0.00001$ & $46 \pm 4$ & $0.75 \pm 0.09$ \\ |
| 492 |
< |
secondary prismatic & $0.00035 \pm 0.00001$ & $0.00033 \pm 0.00002$ & $42 \pm 2$ & $0.69 \pm 0.03$ \\ \hline |
| 493 |
< |
\end{tabular} |
| 494 |
< |
\end{table} |
| 495 |
< |
|
| 496 |
< |
|
| 497 |
< |
|
| 498 |
< |
\begin{figure} |
| 499 |
< |
\includegraphics[width=\linewidth]{Pyr-orient} |
| 500 |
< |
\caption{\label{fig:PyrOrient} The three decay constants of the |
| 501 |
< |
orientational time correlation function, $C_2(t)$, for water as a function |
| 502 |
< |
of distance from the center of the ice slab. The vertical dashed line |
| 503 |
< |
indicates the edge of the pyramidal ice slab determined by the local order |
| 504 |
< |
tetrahedral parameter. The control (black circles) and sheared (red squares) |
| 505 |
< |
experiments were fit by a shifted exponential decay (Eq. 9\cite{Louden13}) |
| 506 |
< |
shown by the black and red lines respectively. The upper two panels show that |
| 507 |
< |
translational and hydrogen bond making and breaking events slow down |
| 508 |
< |
through the interface while approaching the ice slab. The bottom most panel |
| 509 |
< |
shows the librational motion of the water molecules speeding up approaching |
| 510 |
< |
the ice block due to the confined region of space allowed for the molecules |
| 511 |
< |
to move in.} |
| 512 |
< |
\end{figure} |
| 513 |
< |
|
| 514 |
< |
\begin{figure} |
| 515 |
< |
\includegraphics[width=\linewidth]{SP-orient-less} |
| 516 |
< |
\caption{\label{fig:SPorient} Decay constants for $C_2(t)$ at the secondary |
| 517 |
< |
prismatic face. Panel descriptions match those in \ref{fig:PyrOrient}.} |
| 518 |
< |
\end{figure} |
| 519 |
< |
|
| 520 |
< |
|
| 521 |
< |
|
| 522 |
< |
\section{Conclusion} |
| 523 |
< |
We present the results of molecular dynamics simulations of the pyrmaidal |
| 524 |
< |
and secondary prismatic facets of an SPC/E model of the |
| 525 |
< |
Ice-I$_\mathrm{h}$/water interface. The ice was sheared through the liquid |
| 526 |
< |
water while being exposed to a thermal gradient to maintain a stable |
| 527 |
< |
interface by using the minimally perturbing VSS RNEMD method. In agreement |
| 528 |
< |
with our previous findings for the basal and prismatic facets, the interfacial |
| 529 |
< |
width was found to be independent of shear rate as measured by the local |
| 530 |
< |
tetrahedral order parameter. This width was found to be |
| 531 |
< |
3.2~$\pm$ 0.2~\AA\ for both the pyramidal and the secondary prismatic systems. |
| 532 |
< |
These values are in good agreement with our previously calculated interfacial |
| 533 |
< |
widths for the basal (3.2~$\pm$ 0.4~\AA\ ) and prismatic (3.6~$\pm$ 0.2~\AA\ ) |
| 534 |
< |
systems. |
| 535 |
< |
|
| 536 |
< |
Orientational dynamics of the Ice-I$_\mathrm{h}$/water interfaces were studied |
| 537 |
< |
by calculation of the orientational time correlation function at varying |
| 538 |
< |
displacements normal to the interface. The decays were fit |
| 539 |
< |
to a tri-exponential decay, where the three decay constants correspond to |
| 540 |
< |
the librational motion of the molecules driven by the restoring forces of |
| 541 |
< |
existing hydrogen bonds ($\tau_{short}$ $\mathcal{O}$(10 fs)), jumps between |
| 542 |
< |
two different hydrogen bonds ($\tau_{middle}$ $\mathcal{O}$(1 ps)), and |
| 543 |
< |
translational motion of the molecules ($\tau_{long}$ $\mathcal{O}$(100 ps)). |
| 544 |
< |
$\tau_{short}$ was found to decrease approaching the interface due to the |
| 545 |
< |
constrained motion of the molecules as the local environment becomes more |
| 546 |
< |
ice-like. Conversely, the two longer-time decay constants were found to |
| 547 |
< |
increase at small displacements from the interface. As seen in our previous |
| 548 |
< |
work on the basal and prismatic facets, there appears to be a dynamic |
| 549 |
< |
interface width at which deviations from the bulk liquid values occur. |
| 550 |
< |
We had previously found $d_{basal}$ and $d_{prismatic}$ to be approximately |
| 551 |
< |
2.8~\AA\ and 3.5~\AA. We found good agreement of these values for the |
| 552 |
< |
pyramidal and secondary prismatic systems with $d_{pyramidal}$ and |
| 553 |
< |
$d_{secondary prismatic}$ to be 2.7~\AA\ and 3~\AA. For all four of the |
| 554 |
< |
facets, no apparent dependence of the dynamic width on the shear rate was |
| 555 |
< |
found. |
| 556 |
< |
|
| 557 |
< |
%Paragraph summarizing the Kappa values |
| 558 |
< |
The interfacial friction coefficient, $\kappa$, was determined for each facet |
| 559 |
< |
interface. We were able to reach an expression for $\kappa$ as a function of |
| 560 |
< |
the velocity profile of the system which is scaled by the viscosity of the liquid |
| 561 |
< |
at 225 K. In doing so, we have obtained an expression for $\kappa$ which is |
| 562 |
< |
independent of temperature differences of the liquid water at far displacements |
| 563 |
< |
from the interface. We found the basal and pyramidal facets to have |
| 564 |
< |
similar $\kappa$ values, of $\kappa \approx$ 0.0006 |
| 565 |
< |
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}). However, the |
| 566 |
< |
prismatic and secondary prismatic facets were found to have $\kappa$ values of |
| 567 |
< |
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1}). |
| 568 |
< |
As these ice facets are being sheared relative to liquid water, with the |
| 569 |
< |
structural and dynamic width of all four |
| 570 |
< |
interfaces being approximately the same, the difference in the coefficient of |
| 571 |
< |
friction indicates the hydrophilicity of the crystal facets are not |
| 572 |
< |
equivalent. Namely, that the basal and pyramidal facets of Ice-I$_\mathrm{h}$ |
| 573 |
< |
are more hydrophilic than the prismatic and secondary prismatic facets. |
| 574 |
< |
|
| 575 |
< |
|
| 576 |
< |
\begin{acknowledgments} |
| 577 |
< |
Support for this project was provided by the National |
| 578 |
< |
Science Foundation under grant CHE-1362211. Computational time was |
| 579 |
< |
provided by the Center for Research Computing (CRC) at the |
| 580 |
< |
University of Notre Dame. |
| 581 |
< |
\end{acknowledgments} |
| 582 |
< |
|
| 583 |
< |
\newpage |
| 584 |
< |
|
| 585 |
< |
\bibliography{iceWater} |
| 586 |
< |
|
| 587 |
< |
There is significant interest in the properties of ice/ice and ice/water |
| 588 |
< |
interfaces in the geophysics community. Most commonly, the results of shearing |
| 589 |
< |
two ice blocks past one |
| 590 |
< |
another\cite{Casassa91, Sukhorukov13, Pritchard12, Lishman13} or the shearing |
| 591 |
< |
of ice through water\cite{Cuffey99, Bell08}. Using molecular dynamics |
| 592 |
< |
simulations, Samadashvili has recently shown that when two smooth ice slabs |
| 593 |
< |
slide past one another, a stable liquid-like layer develops between |
| 594 |
< |
them\cite{Samadashvili13}. To fundamentally understand these processes, a |
| 595 |
< |
molecular understanding of the ice/water interfaces is needed. |
| 596 |
< |
|
| 597 |
< |
Investigation of the ice/water interface is also crucial in understanding |
| 598 |
< |
processes such as nucleation, crystal |
| 599 |
< |
growth,\cite{Han92, Granasy95, Vanfleet95} and crystal |
| 600 |
< |
melting\cite{Weber83, Han92, Sakai96, Sakai96B}. Insight gained to these |
| 601 |
< |
properties can also be applied to biological systems of interest, such as |
| 602 |
< |
the behavior of the antifreeze protein found in winter |
| 603 |
< |
flounder,\cite{Wierzbicki07,Chapsky97} and certain terrestial |
| 604 |
< |
arthropods.\cite{Duman:2001qy,Meister29012013} Elucidating the properties which |
| 605 |
< |
give rise to these processes through experimental techniques can be expensive, |
| 606 |
< |
complicated, and sometimes infeasible. However, through the use of molecular |
| 607 |
< |
dynamics simulations much of the problems of investigating these properties |
| 608 |
< |
are alleviated. |
| 609 |
< |
|
| 610 |
< |
Understanding ice/water interfaces inherently begins with the isolated |
| 611 |
< |
systems. There has been extensive work parameterizing models for liquid water, |
| 612 |
< |
such as the SPC\cite{Berendsen81}, SPC/E\cite{Berendsen87}, |
| 613 |
< |
TIP4P\cite{Jorgensen85}, TIP4P/2005\cite{Abascal05}, |
| 614 |
< |
($\dots$), and more recently, models for simulating |
| 615 |
< |
the solid phases of water, such as the TIP4P/Ice\cite{Abascal05b} model. The |
| 616 |
< |
melting point of various crystal structures of ice have been calculated for |
| 617 |
< |
many of these models |
| 618 |
< |
(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}), |
| 619 |
< |
and the partial or complete phase diagram for the model has been determined |
| 620 |
< |
(SPC/E\cite{Baez95,Bryk04b,Sanz04b}, TIP4P\cite{Sanz04,Sanz04b,Koyama04}, TIP5P\cite{Sanz04,Koyama04}). |
| 621 |
< |
Knowing the behavior and melting point for these models has enabled an initial |
| 622 |
< |
investigation of ice/water interfaces. |
| 623 |
< |
|
| 624 |
< |
The Ice-I$_\mathrm{h}$/water quiescent interface has been extensively studied |
| 625 |
< |
over the past 30 years by theory and experiment. Haymet \emph{et al.} have |
| 626 |
< |
done significant work characterizing and quantifying the width of these |
| 627 |
< |
interfaces for the SPC,\cite{Karim90} SPC/E,\cite{Gay02,Bryk02}, |
| 628 |
< |
CF1,\cite{Hayward01,Hayward02} and TIP4P\cite{Karim88} models for water. In |
| 629 |
< |
recent years, Haymet has focused on investigating the effects cations and |
| 630 |
< |
anions have on crystal nucleaion and |
| 631 |
< |
melting.\cite{Bryk04,Smith05,Wilson08,Wilson10} Nada and Furukawa have studied |
| 632 |
< |
the the basal- and prismatic-water interface width\cite{Nada95}, crystal |
| 633 |
< |
surface restructuring at temperatures approaching the melting |
| 634 |
< |
point\cite{Nada00}, and the mechanism of ice growth inhibition by antifreeze |
| 635 |
< |
proteins\cite{Nada08,Nada11,Nada12}. Nada has developed a six-site water model |
| 636 |
< |
for ice/water interfaces near the melting point\cite{Nada03}, and studied the |
| 637 |
< |
dependence of crystal growth shape on applied pressure\cite{Nada11b}. Using |
| 638 |
< |
this model, Nada and Furukawa have established differential |
| 639 |
< |
growth rates for the basal, prismatic, and secondary prismatic facets of |
| 640 |
< |
Ice-I$_\mathrm{h}$ and found their origins due to a reordering of the hydrogen |
| 641 |
< |
bond network in water near the interface\cite{Nada05}. While the work |
| 642 |
< |
described so far has mainly focused on bulk water on ice, there is significant |
| 643 |
< |
interest in thin films of water on ice surfaces as well. |
| 644 |
< |
|
| 645 |
< |
It is well known that the surface of ice exhibits a premelting layer at |
| 646 |
< |
temperatures near the melting point, often called a quasi-liquid layer (QLL). |
| 647 |
< |
Molecular dynamics simulations of the facets of ice-I$_\mathrm{h}$ exposed |
| 648 |
< |
to vacuum performed by Conde, Vega and Patrykiejew have found QLL widths of |
| 649 |
< |
approximately 10 \AA\ at 3 K below the melting point\cite{Conde08}. |
| 650 |
< |
Similarly, Limmer and Chandler have used course grain simulations and |
| 651 |
< |
statistical field theory to estimated QLL widths at the same temperature to |
| 652 |
< |
be about 3 nm\cite{Limmer14}. |
| 653 |
< |
Recently, Sazaki and Furukawa have developed an experimental technique with |
| 654 |
< |
sufficient spatial and temporal resolution to visulaize and quantitatively |
| 655 |
< |
analyze QLLs on ice crystals at temperatures near melting\cite{Sazaki10}. They |
| 656 |
< |
have found the width of the QLLs perpindicular to the surface at -2.2$^{o}$C |
| 657 |
< |
to be $\mathcal{O}$(\AA). They have also seen the formation of two immiscible |
| 658 |
< |
QLLs, which displayed different stabilities and dynamics on the crystal |
| 659 |
< |
surface\cite{Sazaki12}. Knowledge of the hydrophilicities of each |
| 660 |
< |
of the crystal facets would help further our understanding of the properties |
| 661 |
< |
and dynamics of the QLLs. |
| 662 |
< |
|
| 663 |
< |
Presented here is the follow up to our previous paper\cite{Louden13}, in which |
| 664 |
< |
the basal and prismatic facets of an ice-I$_\mathrm{h}$/water interface were |
| 665 |
< |
investigated where the ice was sheared relative to the liquid. By using a |
| 666 |
< |
recently developed velocity shearing and scaling approach to reverse |
| 667 |
< |
non-equilibrium molecular dynamics (VSS-RNEMD), simultaneous temperature and |
| 668 |
< |
velocity gradients can be applied to the system, which allows for measurment |
| 669 |
< |
of friction and thermal transport properties while maintaining a stable |
| 670 |
< |
interfacial temperature\cite{Kuang12}. Structural analysis and dynamic |
| 671 |
< |
correlation functions were used to probe the interfacial response to a shear, |
| 672 |
< |
and the resulting solid/liquid kinetic friction coefficients were reported. |
| 673 |
< |
In this paper we present the same analysis for the pyramidal and secondary |
| 674 |
< |
prismatic facets, and show that the differential interfacial friction |
| 675 |
< |
coefficients for the four facets of ice-I$_\mathrm{h}$ are determined by their |
| 676 |
< |
relative hydrophilicity by means of dynamics water contact angle simulations. |
| 677 |
< |
|
| 678 |
< |
\end{document} |
| 679 |
< |
|
| 680 |
< |
|
| 1 |
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%% PNAStwoS.tex |
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%% For PNAS Only: |
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\url{www.pnas.org/cgi/doi/10.1073/pnas.0709640104} |
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\copyrightyear{2008} |
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\issuedate{Issue Date} |
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\volume{Volume} |
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\issuenumber{Issue Number} |
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%\setcounter{page}{2687} %Set page number here if desired |
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%%%%%%%%%%%% |
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\begin{document} |
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|
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\title{Friction at Water / Ice-I$_\mathrm{h}$ interfaces: Do the |
| 31 |
> |
Different Facets of Ice Have Different Hydrophilicity?} |
| 32 |
> |
|
| 33 |
> |
\author{Patrick B. Louden |
| 34 |
> |
\and |
| 35 |
> |
J. Daniel Gezelter\affil{1}{Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, |
| 36 |
> |
IN 46556}} |
| 37 |
> |
|
| 38 |
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\contributor{Submitted to Proceedings of the National Academy of Sciences |
| 39 |
> |
of the United States of America} |
| 40 |
> |
|
| 41 |
> |
%%%Newly updated. |
| 42 |
> |
%%% If significance statement need, then can use the below command otherwise just delete it. |
| 43 |
> |
\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.} |
| 44 |
> |
|
| 45 |
> |
\maketitle |
| 46 |
> |
|
| 47 |
> |
\begin{article} |
| 48 |
> |
\begin{abstract} |
| 49 |
> |
{In this follow up paper of the basal and prismatic facets of the |
| 50 |
> |
Ice-I$_\mathrm{h}$/water interface, we present the |
| 51 |
> |
pyramidal and secondary prismatic |
| 52 |
> |
interfaces for both the quiescent and sheared systems. The structural and |
| 53 |
> |
dynamic interfacial widths for all four crystal facets were found to be in good |
| 54 |
> |
agreement, and were found to be independent of the shear rate over the shear |
| 55 |
> |
rates investigated. |
| 56 |
> |
Decomposition of the molecular orientational time correlation function showed |
| 57 |
> |
different behavior for the short- and longer-time decay components approaching |
| 58 |
> |
normal to the interface. Lastly we show through calculation of the interfacial |
| 59 |
> |
friction coefficient that the basal and pyramidal facets are more |
| 60 |
> |
hydrophilic than the prismatic and secondary prismatic facets.} |
| 61 |
> |
\end{abstract} |
| 62 |
> |
|
| 63 |
> |
\keywords{ice|water|interface|contact angle|molecular dynamics} |
| 64 |
> |
|
| 65 |
> |
\abbreviations{QLL, quasi liquid layer; MD, molecular dynamics} |
| 66 |
> |
|
| 67 |
> |
%\dropcap{I}n this article we study the evolution of ``almost-sharp'' fronts |
| 68 |
> |
%for the surface quasi-geostrophic equation. This 2-D active scalar |
| 69 |
> |
%equation reads for the surface quasi-geostrophic equation. |
| 70 |
> |
%\begin{equation} |
| 71 |
> |
%\mfrac{D \theta}{Dt}=\mfrac{\pr \theta}{\pr t} + u\cdot \nabla |
| 72 |
> |
%\theta=0 \label{qg1} |
| 73 |
> |
%\end{equation} |
| 74 |
> |
|
| 75 |
> |
The Ice-I$_\mathrm{h}$/water quiescent interface has been extensively studied |
| 76 |
> |
over the past 30 years by theory and experiment. Haymet \emph{et al.} have |
| 77 |
> |
done significant work characterizing and quantifying the width of these |
| 78 |
> |
interfaces for the SPC,\cite{Karim90} SPC/E,\cite{Gay02,Bryk02}, |
| 79 |
> |
CF1,\cite{Hayward01,Hayward02} and TIP4P\cite{Karim88} models for water.\cite{Bryk04,Smith05,Wilson08,Wilson10} Nada and Furukawa have studied |
| 80 |
> |
the the basal- and prismatic-water interface width\cite{Nada95} and crystal |
| 81 |
> |
surface restructuring at temperatures approaching the melting |
| 82 |
> |
point\cite{Nada00}. |
| 83 |
> |
|
| 84 |
> |
It is well known that the surface of ice exhibits a premelting layer at |
| 85 |
> |
temperatures near the melting point, often called a quasi-liquid layer (QLL). |
| 86 |
> |
Molecular dynamics simulations of the facets of ice-I$_\mathrm{h}$ exposed |
| 87 |
> |
to vacuum performed by Conde, Vega and Patrykiejew have found QLL widths of |
| 88 |
> |
approximately 10 \AA\ at 3 K below the melting point\cite{Conde08}. |
| 89 |
> |
Similarly, Limmer and Chandler have used course grain simulations and |
| 90 |
> |
statistical field theory to estimated QLL widths at the same temperature to |
| 91 |
> |
be about 3 nm\cite{Limmer14}. |
| 92 |
> |
Recently, Sazaki and Furukawa have developed an experimental technique with |
| 93 |
> |
sufficient spatial and temporal resolution to visulaize and quantitatively |
| 94 |
> |
analyze QLLs on ice crystals at temperatures near melting\cite{Sazaki10}. They |
| 95 |
> |
have found the width of the QLLs perpindicular to the surface at -2.2$^{o}$C |
| 96 |
> |
to be 3-4 \AA\ wide. They have also seen the formation of two immiscible |
| 97 |
> |
QLLs, which displayed different stabilities and dynamics on the crystal |
| 98 |
> |
surface\cite{Sazaki12}. |
| 99 |
> |
|
| 100 |
> |
There is significant interest in the properties of ice/ice and ice/water |
| 101 |
> |
interfaces in the geophysics community. Most commonly, the results of shearing |
| 102 |
> |
two ice blocks past one |
| 103 |
> |
another\cite{Casassa91, Sukhorukov13, Pritchard12, Lishman13} or the shearing |
| 104 |
> |
of ice through water\cite{Cuffey99, Bell08}. Using molecular dynamics |
| 105 |
> |
simulations, Samadashvili has recently shown that when two smooth ice slabs |
| 106 |
> |
slide past one another, a stable liquid-like layer develops between |
| 107 |
> |
them\cite{Samadashvili13}. To fundamentally understand these processes, a |
| 108 |
> |
molecular understanding of the ice/water interfaces is needed. |
| 109 |
> |
Investigation of the ice/water interface is also crucial in understanding |
| 110 |
> |
processes such as nucleation, crystal |
| 111 |
> |
growth,\cite{Han92, Granasy95, Vanfleet95} and crystal |
| 112 |
> |
melting\cite{Weber83, Han92, Sakai96, Sakai96B}. |
| 113 |
> |
|
| 114 |
> |
In a previous study\cite{Louden13}, we investigated |
| 115 |
> |
the basal and prismatic facets of an ice-I$_\mathrm{h}$/water |
| 116 |
> |
interface where the ice was sheared relative to the liquid. Using |
| 117 |
> |
velocity shearing and scaling approach to reverse |
| 118 |
> |
non-equilibrium molecular dynamics (VSS-RNEMD), simultaneous temperature and |
| 119 |
> |
velocity gradients were applied to the system, allowing for measurment |
| 120 |
> |
of friction and thermal transport properties while maintaining a stable |
| 121 |
> |
interfacial temperature\cite{Kuang12}. |
| 122 |
> |
|
| 123 |
> |
Paragraph here about hydrophobicity and hydrophilicity, maybe move up |
| 124 |
> |
more in the paper as well. Talk about physically what it means for a |
| 125 |
> |
surface to by hydrophobic or hydrophilic, and then we move into |
| 126 |
> |
how do we define it (mathematically) and then measure the degree |
| 127 |
> |
of wetting experimentally and theoretically. |
| 128 |
> |
|
| 129 |
> |
The hydrophobicity or hydrophilicity of a surface can be described by the |
| 130 |
> |
extent a droplet of water wets the surface. The contact angle formed between |
| 131 |
> |
the solid and the liquid, $\theta$, which relates the free energies of the |
| 132 |
> |
three interfaces involved, is given by Young's equation. |
| 133 |
> |
\begin{equation}\label{young} |
| 134 |
> |
\cos\theta = (\gamma_{sv} - \gamma_{sl})/\gamma_{lv} |
| 135 |
> |
\end{equation} |
| 136 |
> |
Here $\gamma_{sv}$, $\gamma_{sl}$, and $\gamma_{lv}$ are the free energies |
| 137 |
> |
of the solid/vapor, solid/liquid, and liquid/vapor interfaces respectively. |
| 138 |
> |
Large contact angles ($\theta$ $\gg$ 90\textsuperscript{o}) correspond to low |
| 139 |
> |
wettability and hydrophobic surfaces, while small contact angles |
| 140 |
> |
($\theta$ $\ll$ 90\textsuperscript{o}) correspond to high wettability and |
| 141 |
> |
hydrophilic surfaces. Experimentally, measurements of the contact angle |
| 142 |
> |
of sessile drops has been used to quantify the extent of wetting on surfaces |
| 143 |
> |
with thermally selective wetting charactaristics\cite{Tadanaga00,Liu04,Sun04}, |
| 144 |
> |
as well as nano-pillared surfaces with electrically tunable Cassie-Baxter and |
| 145 |
> |
Wenzel states\cite{Herbertson06,Dhindsa06,Verplanck07,Ahuja08,Manukyan11}. |
| 146 |
> |
Luzar and coworkers have done significant work modeling these transitions on |
| 147 |
> |
nano-patterned surfaces\cite{Daub07,Daub10,Daub11,Ritchie12}, and have found |
| 148 |
> |
the change in contact angle to be due to the external field perturbing the |
| 149 |
> |
hydrogen bonding of the liquid/vapor interface\cite{Daub07}. |
| 150 |
> |
|
| 151 |
> |
The resulting solid/liquid kinetic friction coefficients were |
| 152 |
> |
reported, and displayed a factor of two difference between the |
| 153 |
> |
basal and prismatic facets. |
| 154 |
> |
In this paper we present the same analysis for the pyramidal and secondary |
| 155 |
> |
prismatic facets, and show that the differential interfacial friction |
| 156 |
> |
coefficients for the four facets of ice-I$_\mathrm{h}$ are determined by their |
| 157 |
> |
relative hydrophilicity by means of dynamics water contact angle simulations. |
| 158 |
> |
|
| 159 |
> |
\section{Methodology} |
| 160 |
> |
|
| 161 |
> |
\subsection{Water Model} |
| 162 |
> |
Understanding ice/water interfaces inherently begins with the isolated |
| 163 |
> |
systems. There has been extensive work parameterizing models for liquid water, |
| 164 |
> |
such as the SPC\cite{Berendsen81}, SPC/E\cite{Berendsen87}, |
| 165 |
> |
TIP4P\cite{Jorgensen85}, TIP4P/2005\cite{Abascal05}, |
| 166 |
> |
($\dots$), and more recently, models for simulating |
| 167 |
> |
the solid phases of water, such as the TIP4P/Ice\cite{Abascal05b} model. The |
| 168 |
> |
melting point of various crystal structures of ice have been calculated for |
| 169 |
> |
many of these models |
| 170 |
> |
(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}), |
| 171 |
> |
and the partial or complete phase diagram for the model has been determined |
| 172 |
> |
(SPC/E\cite{Baez95,Bryk04b,Sanz04b}, TIP4P\cite{Sanz04,Sanz04b,Koyama04}, TIP5P\cite{Sanz04,Koyama04}). |
| 173 |
> |
|
| 174 |
> |
Haymet et al. have studied the quiescent Ice-I$_\mathrm{h}$/water interface |
| 175 |
> |
using the rigid SPC, SPC/E, TIP4P, and the flexible CF1 water models, and has seen good |
| 176 |
> |
agreement for structural and dynamic measurements of the interfacial |
| 177 |
> |
width. Given the expansive size of our systems of interest, and to |
| 178 |
> |
compare with our previous work, we have chosen to use rigid SPC/E |
| 179 |
> |
water model in this study. |
| 180 |
> |
|
| 181 |
> |
\subsection{Pyramidal and secondary prismatic system construction} |
| 182 |
> |
|
| 183 |
> |
The construction of the pyramidal and secondary prismatic systems follows that |
| 184 |
> |
of |
| 185 |
> |
the basal and prismatic systems presented elsewhere\cite{Louden13}, however |
| 186 |
> |
the ice crystals and water boxes were equilibrated and combined at 50K |
| 187 |
> |
instead of 225K. The ice / water systems generated were then equilibrated |
| 188 |
> |
to 225K. The resulting pyramidal system was |
| 189 |
> |
$37.47 \times 29.50 \times 93.02$ \AA\ with 1216 |
| 190 |
> |
SPC/E\cite{Berendsen87} molecules in the ice slab, and 2203 in the liquid |
| 191 |
> |
phase. The secondary |
| 192 |
> |
prismatic system generated was $71.87 \times 31.66 \times 161.55$ \AA\ with |
| 193 |
> |
3840 |
| 194 |
> |
SPC/E molecules in the ice slab and 8176 molecules in the liquid phase. |
| 195 |
> |
|
| 196 |
> |
\subsection{Shearing simulations} |
| 197 |
> |
% Do we need to justify the sims at 225K? |
| 198 |
> |
% No crystal growth or shrinkage over 2 successive 1 ns NVT simulations for |
| 199 |
> |
% either the pyramidal or sec. prismatic ice/water systems. |
| 200 |
> |
|
| 201 |
> |
The computational details performed here were equivalent to those reported |
| 202 |
> |
in our previous publication\cite{Louden13}. The only changes made to the |
| 203 |
> |
previously reported procedure were the following. VSS-RNEMD moves were |
| 204 |
> |
attempted every 2 fs instead of every 50 fs. This was done to minimize |
| 205 |
> |
the magnitude of each individual VSS-RNEMD perturbation to the system. |
| 206 |
> |
|
| 207 |
> |
All pyramidal simulations were performed under the canonical (NVT) ensamble |
| 208 |
> |
except those |
| 209 |
> |
during which statistics were accumulated for the orientational correlation |
| 210 |
> |
function, which were performed under the microcanonical (NVE) ensamble. All |
| 211 |
> |
secondary prismatic |
| 212 |
> |
simulations were performed under the NVE ensamble. |
| 213 |
> |
|
| 214 |
> |
\subsection{Droplet simulations} |
| 215 |
> |
Here, we will calculate the contact angle of a water droplet as it spreads |
| 216 |
> |
across each of the four ice I$_\mathrm{h}$ crystal facets in order to |
| 217 |
> |
determine the surface's relative hydrophilicites. The ice surfaces were |
| 218 |
> |
oriented so that the desired facet was exposed to the positive z dimension. |
| 219 |
> |
The sizes and number of molecules in each of the surfaces is given in Table |
| 220 |
> |
\ref{tab:ice_sheets}. Molecular restraints were applied to the center of mass |
| 221 |
> |
of the rigid bodies to prevent surface melting, however the molecules were |
| 222 |
> |
allowed to reorient themselves freely. The water doplet to be placed on the |
| 223 |
> |
surface contained 2048 SPC/E molecules, which has been found to produce |
| 224 |
> |
agreement for the Young contact angle extrapolated to an infinite drop |
| 225 |
> |
size\cite{Daub10}. The surfaces and droplet were equilibrated to 225 K, at |
| 226 |
> |
which time the droplet was placed 3-5 \AA\ above the surface at 5 unique |
| 227 |
> |
locations. Each simulation was 5 ns in length and conducted in the NVE |
| 228 |
> |
ensemble. |
| 229 |
> |
|
| 230 |
> |
|
| 231 |
> |
\section{Results and discussion} |
| 232 |
> |
\subsection{Interfacial width} |
| 233 |
> |
In the literature there is good agreement that between the solid ice and |
| 234 |
> |
the bulk water, there exists a region of 'slush-like' water molecules. |
| 235 |
> |
In this region, the water molecules are structurely distinguishable and |
| 236 |
> |
behave differently than those of the solid ice or the bulk water. |
| 237 |
> |
The characteristics of this region have been defined by both structural |
| 238 |
> |
and dynamic properties; and its width has been measured by the change of these |
| 239 |
> |
properties from their bulk liquid values to those of the solid ice. |
| 240 |
> |
Examples of these properties include the density, the diffusion constant, and |
| 241 |
> |
the translational order profile. \cite{Bryk02,Karim90,Gay02,Hayward01,Hayward02,Karim88} |
| 242 |
> |
|
| 243 |
> |
Since the VSS-RNEMD moves used to impose the thermal and velocity gradients |
| 244 |
> |
perturb the momenta of the water molecules in |
| 245 |
> |
the systems, parameters that depend on translational motion may give |
| 246 |
> |
faulty results. A stuructural parameter will be less effected by the |
| 247 |
> |
VSS-RNEMD perturbations to the system. Due to this, we have used the |
| 248 |
> |
local tetrahedral order parameter (Eq 5\cite{Louden13} to quantify the width of the interface, |
| 249 |
> |
which was originally described by Kumar\cite{Kumar09} and |
| 250 |
> |
Errington\cite{Errington01}, and used by Bryk and Haymet in a previous study |
| 251 |
> |
of ice/water interfaces.\cite{Bryk04b} |
| 252 |
> |
|
| 253 |
> |
To determine the width of the interfaces, each of the systems were |
| 254 |
> |
divided into 100 artificial bins along the |
| 255 |
> |
$z$-dimension, and the local tetrahedral order parameter, $q(z)$, was |
| 256 |
> |
time-averaged for each of the bins, resulting in a tetrahedrality profile of |
| 257 |
> |
the system. These profiles are shown across the $z$-dimension of the systems |
| 258 |
> |
in panel $a$ of Figures \ref{fig:pyrComic} |
| 259 |
> |
and \ref{fig:spComic} (black circles). The $q(z)$ function has a range of |
| 260 |
> |
(0,1), where a larger value indicates a more tetrahedral environment. |
| 261 |
> |
The $q(z)$ for the bulk liquid was found to be $\approx $ 0.77, while values of |
| 262 |
> |
$\approx $ 0.92 were more common for the ice. The tetrahedrality profiles were |
| 263 |
> |
fit using a hyperbolic tangent\cite{Louden13} designed to smoothly fit the |
| 264 |
> |
bulk to ice |
| 265 |
> |
transition, while accounting for the thermal influence on the profile by the |
| 266 |
> |
kinetic energy exchanges of the VSS-RNEMD moves. In panels $b$ and $c$, the |
| 267 |
> |
resulting thermal and velocity gradients from an imposed kinetic energy and |
| 268 |
> |
momentum fluxes can be seen. The verticle dotted |
| 269 |
> |
lines traversing all three panels indicate the midpoints of the interface |
| 270 |
> |
as determined by the hyperbolic tangent fit of the tetrahedrality profiles. |
| 271 |
> |
|
| 272 |
> |
From fitting the tetrahedrality profiles for each of the 0.5 nanosecond |
| 273 |
> |
simulations (panel c of Figures \ref{fig:pyrComic} and \ref{fig:spComic}) |
| 274 |
> |
by eq. 6\cite{Louden13},we find the interfacial width to be |
| 275 |
> |
3.2 $\pm$ 0.2 and 3.2 $\pm$ 0.2 \AA\ for the control system with no applied |
| 276 |
> |
momentum flux for both the pyramidal and secondary prismatic systems. |
| 277 |
> |
Over the range of shear rates investigated, |
| 278 |
> |
0.6 $\pm$ 0.2 $\mathrm{ms}^{-1} \rightarrow$ 5.6 $\pm$ 0.4 $\mathrm{ms}^{-1}$ |
| 279 |
> |
for the pyramidal system and 0.9 $\pm$ 0.3 $\mathrm{ms}^{-1} \rightarrow$ 5.4 |
| 280 |
> |
$\pm$ 0.1 $\mathrm{ms}^{-1}$ for the secondary prismatic, we found no |
| 281 |
> |
significant change in the interfacial width. This follows our previous |
| 282 |
> |
findings of the basal and |
| 283 |
> |
prismatic systems, in which the interfacial width was invarient of the |
| 284 |
> |
shear rate of the ice. The interfacial width of the quiescent basal and |
| 285 |
> |
prismatic systems was found to be 3.2 $\pm$ 0.4 \AA\ and 3.6 $\pm$ 0.2 \AA\ |
| 286 |
> |
respectively, over the range of shear rates investigated, 0.6 $\pm$ 0.3 |
| 287 |
> |
$\mathrm{ms}^{-1} \rightarrow$ 5.3 $\pm$ 0.5 $\mathrm{ms}^{-1}$ for the basal |
| 288 |
> |
system and 0.9 $\pm$ 0.2 $\mathrm{ms}^{-1} \rightarrow$ 4.5 $\pm$ 0.1 |
| 289 |
> |
$\mathrm{ms}^{-1}$ for the prismatic. |
| 290 |
> |
|
| 291 |
> |
These results indicate that the surface structure of the exposed ice crystal |
| 292 |
> |
has little to no effect on how far into the bulk the ice-like structural |
| 293 |
> |
ordering is. Also, it appears that the interface is not structurally effected |
| 294 |
> |
by shearing the ice through water. |
| 295 |
> |
|
| 296 |
> |
|
| 297 |
> |
\subsection{Orientational dynamics} |
| 298 |
> |
%Should we include the math here? |
| 299 |
> |
The orientational time correlation function, |
| 300 |
> |
\begin{equation}\label{C(t)1} |
| 301 |
> |
C_{2}(t)=\langle P_{2}(\mathbf{u}(0)\cdot \mathbf{u}(t))\rangle, |
| 302 |
> |
\end{equation} |
| 303 |
> |
helps indicate the local environment around the water molecules. The function |
| 304 |
> |
begins with an initial value of unity, and decays to zero as the water molecule |
| 305 |
> |
loses memory of its former orientation. Observing the rate at which this decay |
| 306 |
> |
occurs can provide insight to the mechanism and timescales for the relaxation. |
| 307 |
> |
In eq. \eqref{C(t)1}, $P_{2}$ is the second-order Legendre polynomial, and |
| 308 |
> |
$\mathbf{u}$ is the bisecting HOH vector. The angle brackets indicate |
| 309 |
> |
an ensemble average over all the water molecules in a given spatial region. |
| 310 |
> |
|
| 311 |
> |
To investigate the dynamics of the water molecules across the interface, the |
| 312 |
> |
systems were divided in the $z$-dimension into bins, each $\approx$ 3 \AA\ |
| 313 |
> |
wide, and eq. \eqref{C(t)1} was computed for each of the bins. A water |
| 314 |
> |
molecule was allocated to a particular bin if it was initially in the bin |
| 315 |
> |
at time zero. To compute eq. \eqref{C(t)1}, each 0.5 ns simulation was |
| 316 |
> |
followed by an additional 200 ps NVE simulation during which the |
| 317 |
> |
position and orientations of each molecule were recorded every 0.1 ps. |
| 318 |
> |
|
| 319 |
> |
The data obtained for each bin was then fit to a triexponential decay |
| 320 |
> |
with the three decay constants |
| 321 |
> |
$\tau_{short}$ corresponding to the librational motion of the water |
| 322 |
> |
molecules, $\tau_{middle}$ corresponding to jumps between the breaking and |
| 323 |
> |
making of hydrogen bonds, and $\tau_{long}$ corresponding to the translational |
| 324 |
> |
motion of the water molecules. An additive constant in the fit accounts |
| 325 |
> |
for the water molecules trapped in the ice which do not experience any |
| 326 |
> |
long-time orientational decay. |
| 327 |
> |
|
| 328 |
> |
In Figures \ref{fig:PyrOrient} and \ref{fig:SPorient} we see the $z$-coordinate |
| 329 |
> |
profiles for the three decay constants, $\tau_{short}$ (panel a), |
| 330 |
> |
$\tau_{middle}$ (panel b), |
| 331 |
> |
and $\tau_{long}$ (panel c) for the pyramidal and secondary prismatic systems |
| 332 |
> |
respectively. The control experiments (no shear) are shown in black, and |
| 333 |
> |
an experiment with an imposed momentum flux is shown in red. The vertical |
| 334 |
> |
dotted line traversing all three panels denotes the midpoint of the |
| 335 |
> |
interface as determined by the local tetrahedral order parameter fitting. |
| 336 |
> |
In the liquid regions of both systems, we see that $\tau_{middle}$ and |
| 337 |
> |
$\tau_{long}$ have approximately consistent values of $3-6$ ps and $30-40$ ps, |
| 338 |
> |
resepctively, and increase in value as we approach the interface. Conversely, |
| 339 |
> |
in panel a, we see that $\tau_{short}$ decreases from the liquid value |
| 340 |
> |
of $72-76$ fs as we approach the interface. We believe this speed up is due to |
| 341 |
> |
the constrained motion of librations closer to the interface. Both the |
| 342 |
> |
approximate values for the decays and trends approaching the interface match |
| 343 |
> |
those reported previously for the basal and prismatic interfaces. |
| 344 |
> |
|
| 345 |
> |
As done previously, we have attempted to quantify the distance, $d_{pyramidal}$ |
| 346 |
> |
and $d_{secondary prismatic}$, from the |
| 347 |
> |
interface that the deviations from the bulk liquid values begin. This was done |
| 348 |
> |
by fitting the orientational decay constant $z$-profiles by |
| 349 |
> |
\begin{equation}\label{tauFit} |
| 350 |
> |
\tau(z)\approx\tau_{liquid}+(\tau_{wall}-\tau_{liquid})e^{-(z-z_{wall})/d} |
| 351 |
> |
\end{equation} |
| 352 |
> |
where $\tau_{liquid}$ and $\tau_{wall}$ are the liquid and projected wall |
| 353 |
> |
values of the decay constants, $z_{wall}$ is the location of the interface, |
| 354 |
> |
and $d$ is the displacement from the interface at which these deviations |
| 355 |
> |
occur. The values for $d_{pyramidal}$ and $d_{secondary prismatic}$ were |
| 356 |
> |
determined |
| 357 |
> |
for each of the decay constants, and then averaged for better statistics |
| 358 |
> |
($\tau_{middle}$ was ommitted for secondary prismatic). For the pyramidal |
| 359 |
> |
system, |
| 360 |
> |
$d_{pyramidal}$ was found to be 2.7 \AA\ for both the control and the sheared |
| 361 |
> |
system. We found $d_{secondary prismatic}$ to be slightly larger than |
| 362 |
> |
$d_{pyramidal}$ for both the control and with an applied shear, with |
| 363 |
> |
displacements of $4$ \AA\ for the control system and $3$ \AA\ for the |
| 364 |
> |
experiment with the imposed momentum flux. These values are consistent with |
| 365 |
> |
those found for the basal ($d_{basal}\approx2.9$ \AA\ ) and prismatic |
| 366 |
> |
($d_{prismatic}\approx3.5$ \AA\ ) systems. |
| 367 |
> |
|
| 368 |
> |
\subsection{Coefficient of friction of the interfaces} |
| 369 |
> |
While investigating the kinetic coefficient of friction, there was found |
| 370 |
> |
to be a dependence for $\mu_k$ |
| 371 |
> |
on the temperature of the liquid water in the system. We believe this |
| 372 |
> |
dependence |
| 373 |
> |
arrises from the sharp discontinuity of the viscosity for the SPC/E model |
| 374 |
> |
at temperatures approaching 200 K\cite{Kuang12}. Due to this, we propose |
| 375 |
> |
a weighting to the interfacial friction coefficient, $\kappa$ by the |
| 376 |
> |
shear viscosity of the fluid at 225 K. The interfacial friction coefficient |
| 377 |
> |
relates the shear stress with the relative velocity of the fluid normal to the |
| 378 |
> |
interface: |
| 379 |
> |
\begin{equation}\label{Shenyu-13} |
| 380 |
> |
j_{z}(p_{x}) = \kappa[v_{x}(fluid)-v_{x}(solid)] |
| 381 |
> |
\end{equation} |
| 382 |
> |
where $j_{z}(p_{x})$ is the applied momentum flux (shear stress) across $z$ |
| 383 |
> |
in the |
| 384 |
> |
$x$-dimension, and $v_{x}$(fluid) and $v_{x}$(solid) are the velocities |
| 385 |
> |
directly adjacent to the interface. The shear viscosity, $\eta(T)$, of the |
| 386 |
> |
fluid can be determined under a linear response of the momentum |
| 387 |
> |
gradient to the applied shear stress by |
| 388 |
> |
\begin{equation}\label{Shenyu-11} |
| 389 |
> |
j_{z}(p_{x}) = \eta(T) \frac{\partial v_{x}}{\partial z}. |
| 390 |
> |
\end{equation} |
| 391 |
> |
Using eqs \eqref{Shenyu-13} and \eqref{Shenyu-11}, we can find the following |
| 392 |
> |
expression for $\kappa$, |
| 393 |
> |
\begin{equation}\label{kappa-1} |
| 394 |
> |
\kappa = \eta(T) \frac{\partial v_{x}}{\partial z}\frac{1}{[v_{x}(fluid)-v_{x}(solid)]}. |
| 395 |
> |
\end{equation} |
| 396 |
> |
Here is where we will introduce the weighting term of $\eta(225)/\eta(T)$ |
| 397 |
> |
giving us |
| 398 |
> |
\begin{equation}\label{kappa-2} |
| 399 |
> |
\kappa = \frac{\eta(225)}{[v_{x}(fluid)-v_{x}(solid)]}\frac{\partial v_{x}}{\partial z}. |
| 400 |
> |
\end{equation} |
| 401 |
> |
|
| 402 |
> |
To obtain the value of $\eta(225)$ for the SPC/E model, a $31.09 \times 29.38 |
| 403 |
> |
\times 124.39$ \AA\ box with 3744 SPC/E liquid water molecules was |
| 404 |
> |
equilibrated to 225K, |
| 405 |
> |
and 5 unique shearing experiments were performed. Each experiment was |
| 406 |
> |
conducted in the NVE and were 5 ns in |
| 407 |
> |
length. The VSS were attempted every timestep, which was set to 2 fs. |
| 408 |
> |
For our SPC/E systems, we found $\eta(225)$ to be 0.0148 $\pm$ 0.0007 Pa s, |
| 409 |
> |
roughly ten times larger than the value found for 280 K SPC/E bulk water by |
| 410 |
> |
Kuang\cite{Kuang12}. |
| 411 |
> |
|
| 412 |
> |
The interfacial friction coefficient, $\kappa$, can equivalently be expressed |
| 413 |
> |
as the ratio of the viscosity of the fluid to the slip length, $\delta$, which |
| 414 |
> |
is an indication of how 'slippery' the interface is. |
| 415 |
> |
\begin{equation}\label{kappa-3} |
| 416 |
> |
\kappa = \frac{\eta}{\delta} |
| 417 |
> |
\end{equation} |
| 418 |
> |
In each of the systems, the interfacial temperature was kept fixed to 225K, |
| 419 |
> |
which ensured the viscosity of the fluid at the |
| 420 |
> |
interace was approximately the same. Thus, any significant variation in |
| 421 |
> |
$\kappa$ between |
| 422 |
> |
the systems indicates differences in the 'slipperiness' of the interfaces. |
| 423 |
> |
As each of the ice systems are sheared relative to liquid water, the |
| 424 |
> |
'slipperiness' of the interface can be taken as an indication of how |
| 425 |
> |
hydrophobic or hydrophilic the interface is. The calculated $\kappa$ values |
| 426 |
> |
found for the four crystal facets of Ice-I$_\mathrm{h}$ investigated are shown |
| 427 |
> |
in Table \ref{tab:kappa}. The basal and pyramidal facets were found to have |
| 428 |
> |
similar values of $\kappa \approx$ 0.0006 |
| 429 |
> |
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}), while values of |
| 430 |
> |
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1}) |
| 431 |
> |
were found for the prismatic and secondary prismatic systems. |
| 432 |
> |
These results indicate that the basal and pyramidal facets are |
| 433 |
> |
more hydrophilic than the prismatic and secondary prismatic facets. |
| 434 |
> |
|
| 435 |
> |
\subsection{Dynamic water contact angle} |
| 436 |
> |
|
| 437 |
> |
|
| 438 |
> |
|
| 439 |
> |
|
| 440 |
> |
\section{Conclusion} |
| 441 |
> |
We present the results of molecular dynamics simulations of the pyrmaidal |
| 442 |
> |
and secondary prismatic facets of an SPC/E model of the |
| 443 |
> |
Ice-I$_\mathrm{h}$/water interface. The ice was sheared through the liquid |
| 444 |
> |
water while being exposed to a thermal gradient to maintain a stable |
| 445 |
> |
interface by using the minimally perturbing VSS RNEMD method. In agreement |
| 446 |
> |
with our previous findings for the basal and prismatic facets, the interfacial |
| 447 |
> |
width was found to be independent of shear rate as measured by the local |
| 448 |
> |
tetrahedral order parameter. This width was found to be |
| 449 |
> |
3.2~$\pm$ 0.2~\AA\ for both the pyramidal and the secondary prismatic systems. |
| 450 |
> |
These values are in good agreement with our previously calculated interfacial |
| 451 |
> |
widths for the basal (3.2~$\pm$ 0.4~\AA\ ) and prismatic (3.6~$\pm$ 0.2~\AA\ ) |
| 452 |
> |
systems. |
| 453 |
> |
|
| 454 |
> |
Orientational dynamics of the Ice-I$_\mathrm{h}$/water interfaces were studied |
| 455 |
> |
by calculation of the orientational time correlation function at varying |
| 456 |
> |
displacements normal to the interface. The decays were fit |
| 457 |
> |
to a tri-exponential decay, where the three decay constants correspond to |
| 458 |
> |
the librational motion of the molecules driven by the restoring forces of |
| 459 |
> |
existing hydrogen bonds ($\tau_{short}$ $\mathcal{O}$(10 fs)), jumps between |
| 460 |
> |
two different hydrogen bonds ($\tau_{middle}$ $\mathcal{O}$(1 ps)), and |
| 461 |
> |
translational motion of the molecules ($\tau_{long}$ $\mathcal{O}$(100 ps)). |
| 462 |
> |
$\tau_{short}$ was found to decrease approaching the interface due to the |
| 463 |
> |
constrained motion of the molecules as the local environment becomes more |
| 464 |
> |
ice-like. Conversely, the two longer-time decay constants were found to |
| 465 |
> |
increase at small displacements from the interface. As seen in our previous |
| 466 |
> |
work on the basal and prismatic facets, there appears to be a dynamic |
| 467 |
> |
interface width at which deviations from the bulk liquid values occur. |
| 468 |
> |
We had previously found $d_{basal}$ and $d_{prismatic}$ to be approximately |
| 469 |
> |
2.8~\AA\ and 3.5~\AA. We found good agreement of these values for the |
| 470 |
> |
pyramidal and secondary prismatic systems with $d_{pyramidal}$ and |
| 471 |
> |
$d_{secondary prismatic}$ to be 2.7~\AA\ and 3~\AA. For all four of the |
| 472 |
> |
facets, no apparent dependence of the dynamic width on the shear rate was |
| 473 |
> |
found. |
| 474 |
> |
|
| 475 |
> |
%Paragraph summarizing the Kappa values |
| 476 |
> |
The interfacial friction coefficient, $\kappa$, was determined for each facet |
| 477 |
> |
interface. We were able to reach an expression for $\kappa$ as a function of |
| 478 |
> |
the velocity profile of the system which is scaled by the viscosity of the liquid |
| 479 |
> |
at 225 K. In doing so, we have obtained an expression for $\kappa$ which is |
| 480 |
> |
independent of temperature differences of the liquid water at far displacements |
| 481 |
> |
from the interface. We found the basal and pyramidal facets to have |
| 482 |
> |
similar $\kappa$ values, of $\kappa \approx$ 0.0006 |
| 483 |
> |
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}). However, the |
| 484 |
> |
prismatic and secondary prismatic facets were found to have $\kappa$ values of |
| 485 |
> |
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1}). |
| 486 |
> |
As these ice facets are being sheared relative to liquid water, with the |
| 487 |
> |
structural and dynamic width of all four |
| 488 |
> |
interfaces being approximately the same, the difference in the coefficient of |
| 489 |
> |
friction indicates the hydrophilicity of the crystal facets are not |
| 490 |
> |
equivalent. Namely, that the basal and pyramidal facets of Ice-I$_\mathrm{h}$ |
| 491 |
> |
are more hydrophilic than the prismatic and secondary prismatic facets. |
| 492 |
> |
|
| 493 |
> |
|
| 494 |
> |
\begin{acknowledgments} |
| 495 |
> |
Support for this project was provided by the National |
| 496 |
> |
Science Foundation under grant CHE-1362211. Computational time was |
| 497 |
> |
provided by the Center for Research Computing (CRC) at the |
| 498 |
> |
University of Notre Dame. |
| 499 |
> |
\end{acknowledgments} |
| 500 |
> |
|
| 501 |
> |
\newpage |
| 502 |
> |
|
| 503 |
> |
\bibliography{iceWater.bib} |
| 504 |
> |
|
| 505 |
> |
There is significant interest in the properties of ice/ice and ice/water |
| 506 |
> |
interfaces in the geophysics community. Most commonly, the results of shearing |
| 507 |
> |
two ice blocks past one |
| 508 |
> |
another\cite{Casassa91, Sukhorukov13, Pritchard12, Lishman13} or the shearing |
| 509 |
> |
of ice through water\cite{Cuffey99, Bell08}. Using molecular dynamics |
| 510 |
> |
simulations, Samadashvili has recently shown that when two smooth ice slabs |
| 511 |
> |
slide past one another, a stable liquid-like layer develops between |
| 512 |
> |
them\cite{Samadashvili13}. To fundamentally understand these processes, a |
| 513 |
> |
molecular understanding of the ice/water interfaces is needed. |
| 514 |
> |
|
| 515 |
> |
Investigation of the ice/water interface is also crucial in understanding |
| 516 |
> |
processes such as nucleation, crystal |
| 517 |
> |
growth,\cite{Han92, Granasy95, Vanfleet95} and crystal |
| 518 |
> |
melting\cite{Weber83, Han92, Sakai96, Sakai96B}. Insight gained to these |
| 519 |
> |
properties can also be applied to biological systems of interest, such as |
| 520 |
> |
the behavior of the antifreeze protein found in winter |
| 521 |
> |
flounder,\cite{Wierzbicki07,Chapsky97} and certain terrestial |
| 522 |
> |
arthropods.\cite{Duman:2001qy,Meister29012013} Elucidating the properties which |
| 523 |
> |
give rise to these processes through experimental techniques can be expensive, |
| 524 |
> |
complicated, and sometimes infeasible. However, through the use of molecular |
| 525 |
> |
dynamics simulations much of the problems of investigating these properties |
| 526 |
> |
are alleviated. |
| 527 |
> |
|
| 528 |
> |
Understanding ice/water interfaces inherently begins with the isolated |
| 529 |
> |
systems. There has been extensive work parameterizing models for liquid water, |
| 530 |
> |
such as the SPC\cite{Berendsen81}, SPC/E\cite{Berendsen87}, |
| 531 |
> |
TIP4P\cite{Jorgensen85}, TIP4P/2005\cite{Abascal05}, |
| 532 |
> |
($\dots$), and more recently, models for simulating |
| 533 |
> |
the solid phases of water, such as the TIP4P/Ice\cite{Abascal05b} model. The |
| 534 |
> |
melting point of various crystal structures of ice have been calculated for |
| 535 |
> |
many of these models |
| 536 |
> |
(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}), |
| 537 |
> |
and the partial or complete phase diagram for the model has been determined |
| 538 |
> |
(SPC/E\cite{Baez95,Bryk04b,Sanz04b}, TIP4P\cite{Sanz04,Sanz04b,Koyama04}, TIP5P\cite{Sanz04,Koyama04}). |
| 539 |
> |
Knowing the behavior and melting point for these models has enabled an initial |
| 540 |
> |
investigation of ice/water interfaces. |
| 541 |
> |
|
| 542 |
> |
The Ice-I$_\mathrm{h}$/water quiescent interface has been extensively studied |
| 543 |
> |
over the past 30 years by theory and experiment. Haymet \emph{et al.} have |
| 544 |
> |
done significant work characterizing and quantifying the width of these |
| 545 |
> |
interfaces for the SPC,\cite{Karim90} SPC/E,\cite{Gay02,Bryk02}, |
| 546 |
> |
CF1,\cite{Hayward01,Hayward02} and TIP4P\cite{Karim88} models for water. In |
| 547 |
> |
recent years, Haymet has focused on investigating the effects cations and |
| 548 |
> |
anions have on crystal nucleaion and |
| 549 |
> |
melting.\cite{Bryk04,Smith05,Wilson08,Wilson10} Nada and Furukawa have studied |
| 550 |
> |
the the basal- and prismatic-water interface width\cite{Nada95}, crystal |
| 551 |
> |
surface restructuring at temperatures approaching the melting |
| 552 |
> |
point\cite{Nada00}, and the mechanism of ice growth inhibition by antifreeze |
| 553 |
> |
proteins\cite{Nada08,Nada11,Nada12}. Nada has developed a six-site water model |
| 554 |
> |
for ice/water interfaces near the melting point\cite{Nada03}, and studied the |
| 555 |
> |
dependence of crystal growth shape on applied pressure\cite{Nada11b}. Using |
| 556 |
> |
this model, Nada and Furukawa have established differential |
| 557 |
> |
growth rates for the basal, prismatic, and secondary prismatic facets of |
| 558 |
> |
Ice-I$_\mathrm{h}$ and found their origins due to a reordering of the hydrogen |
| 559 |
> |
bond network in water near the interface\cite{Nada05}. While the work |
| 560 |
> |
described so far has mainly focused on bulk water on ice, there is significant |
| 561 |
> |
interest in thin films of water on ice surfaces as well. |
| 562 |
> |
|
| 563 |
> |
It is well known that the surface of ice exhibits a premelting layer at |
| 564 |
> |
temperatures near the melting point, often called a quasi-liquid layer (QLL). |
| 565 |
> |
Molecular dynamics simulations of the facets of ice-I$_\mathrm{h}$ exposed |
| 566 |
> |
to vacuum performed by Conde, Vega and Patrykiejew have found QLL widths of |
| 567 |
> |
approximately 10 \AA\ at 3 K below the melting point\cite{Conde08}. |
| 568 |
> |
Similarly, Limmer and Chandler have used course grain simulations and |
| 569 |
> |
statistical field theory to estimated QLL widths at the same temperature to |
| 570 |
> |
be about 3 nm\cite{Limmer14}. |
| 571 |
> |
Recently, Sazaki and Furukawa have developed an experimental technique with |
| 572 |
> |
sufficient spatial and temporal resolution to visulaize and quantitatively |
| 573 |
> |
analyze QLLs on ice crystals at temperatures near melting\cite{Sazaki10}. They |
| 574 |
> |
have found the width of the QLLs perpindicular to the surface at -2.2$^{o}$C |
| 575 |
> |
to be $\mathcal{O}$(\AA). They have also seen the formation of two immiscible |
| 576 |
> |
QLLs, which displayed different stabilities and dynamics on the crystal |
| 577 |
> |
surface\cite{Sazaki12}. Knowledge of the hydrophilicities of each |
| 578 |
> |
of the crystal facets would help further our understanding of the properties |
| 579 |
> |
and dynamics of the QLLs. |
| 580 |
> |
|
| 581 |
> |
Presented here is the follow up to our previous paper\cite{Louden13}, in which |
| 582 |
> |
the basal and prismatic facets of an ice-I$_\mathrm{h}$/water interface were |
| 583 |
> |
investigated where the ice was sheared relative to the liquid. By using a |
| 584 |
> |
recently developed velocity shearing and scaling approach to reverse |
| 585 |
> |
non-equilibrium molecular dynamics (VSS-RNEMD), simultaneous temperature and |
| 586 |
> |
velocity gradients can be applied to the system, which allows for measurment |
| 587 |
> |
of friction and thermal transport properties while maintaining a stable |
| 588 |
> |
interfacial temperature\cite{Kuang12}. Structural analysis and dynamic |
| 589 |
> |
correlation functions were used to probe the interfacial response to a shear, |
| 590 |
> |
and the resulting solid/liquid kinetic friction coefficients were reported. |
| 591 |
> |
In this paper we present the same analysis for the pyramidal and secondary |
| 592 |
> |
prismatic facets, and show that the differential interfacial friction |
| 593 |
> |
coefficients for the four facets of ice-I$_\mathrm{h}$ are determined by their |
| 594 |
> |
relative hydrophilicity by means of dynamics water contact angle |
| 595 |
> |
simulations. |
| 596 |
> |
|
| 597 |
> |
The local tetrahedral order parameter, $q(z)$, is given by |
| 598 |
> |
\begin{equation} |
| 599 |
> |
q(z) = \int_0^L \sum_{k=1}^{N} \Bigg(1 -\frac{3}{8}\sum_{i=1}^{3} |
| 600 |
> |
\sum_{j=i+1}^{4} \bigg(\cos\psi_{ikj}+\frac{1}{3}\bigg)^2\Bigg) |
| 601 |
> |
\delta(z_{k}-z)\mathrm{d}z \Bigg/ N_z |
| 602 |
> |
\label{eq:qz} |
| 603 |
> |
\end{equation} |
| 604 |
> |
where $\psi_{ikj}$ is the angle formed between the oxygen sites of molecules |
| 605 |
> |
$i$,$k$, and $j$, where the centeral oxygen is located within molecule $k$ and |
| 606 |
> |
molecules $i$ and $j$ are two of the closest four water molecules |
| 607 |
> |
around molecule $k$. All four closest neighbors of molecule $k$ are also |
| 608 |
> |
required to reside within the first peak of the pair distribution function |
| 609 |
> |
for molecule $k$ (typically $<$ 3.41 \AA\ for water). |
| 610 |
> |
$N_z = \int\delta(z_k - z) \mathrm{d}z$ is a normalization factor to account |
| 611 |
> |
for the varying population of molecules within each finite-width bin. |
| 612 |
> |
|
| 613 |
> |
|
| 614 |
> |
The hydrophobicity or hydrophilicity of a surface can be described by the |
| 615 |
> |
extent a droplet of water wets the surface. The contact angle formed between |
| 616 |
> |
the solid and the liquid, $\theta$, which relates the free energies of the |
| 617 |
> |
three interfaces involved, is given by Young's equation. |
| 618 |
> |
\begin{equation}\label{young} |
| 619 |
> |
\cos\theta = (\gamma_{sv} - \gamma_{sl})/\gamma_{lv} |
| 620 |
> |
\end{equation} |
| 621 |
> |
Here $\gamma_{sv}$, $\gamma_{sl}$, and $\gamma_{lv}$ are the free energies |
| 622 |
> |
of the solid/vapor, solid/liquid, and liquid/vapor interfaces respectively. |
| 623 |
> |
Large contact angles ($\theta$ $\gg$ 90\textsuperscript{o}) correspond to low |
| 624 |
> |
wettability and hydrophobic surfaces, while small contact angles |
| 625 |
> |
($\theta$ $\ll$ 90\textsuperscript{o}) correspond to high wettability and |
| 626 |
> |
hydrophilic surfaces. Experimentally, measurements of the contact angle |
| 627 |
> |
of sessile drops has been used to quantify the extent of wetting on surfaces |
| 628 |
> |
with thermally selective wetting charactaristics\cite{Tadanaga00,Liu04,Sun04}, |
| 629 |
> |
as well as nano-pillared surfaces with electrically tunable Cassie-Baxter and |
| 630 |
> |
Wenzel states\cite{Herbertson06,Dhindsa06,Verplanck07,Ahuja08,Manukyan11}. |
| 631 |
> |
Luzar and coworkers have done significant work modeling these transitions on |
| 632 |
> |
nano-patterned surfaces\cite{Daub07,Daub10,Daub11,Ritchie12}, and have found |
| 633 |
> |
the change in contact angle to be due to the external field perturbing the |
| 634 |
> |
hydrogen bonding of the liquid/vapor interface\cite{Daub07}. |
| 635 |
> |
|
| 636 |
> |
|
| 637 |
> |
|
| 638 |
> |
\end{article} |
| 639 |
> |
|
| 640 |
> |
\begin{figure} |
| 641 |
> |
\includegraphics[width=\linewidth]{Pyr_comic_strip} |
| 642 |
> |
\caption{\label{fig:pyrComic} The pyramidal interface with a shear |
| 643 |
> |
rate of 3.8 ms\textsuperscript{-1}. Lower panel: the local tetrahedral order |
| 644 |
> |
parameter, $q(z)$, (black circles) and the hyperbolic tangent fit (red line). |
| 645 |
> |
Middle panel: the imposed thermal gradient required to maintain a fixed |
| 646 |
> |
interfacial temperature. Upper panel: the transverse velocity gradient that |
| 647 |
> |
develops in response to an imposed momentum flux. The vertical dotted lines |
| 648 |
> |
indicate the locations of the midpoints of the two interfaces.} |
| 649 |
> |
\end{figure} |
| 650 |
> |
|
| 651 |
> |
\begin{figure} |
| 652 |
> |
\includegraphics[width=\linewidth]{SP_comic_strip} |
| 653 |
> |
\caption{\label{fig:spComic} The secondary prismatic interface with a shear |
| 654 |
> |
rate of 3.5 \ |
| 655 |
> |
ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.} |
| 656 |
> |
\end{figure} |
| 657 |
> |
|
| 658 |
> |
\begin{figure} |
| 659 |
> |
\includegraphics[width=\linewidth]{Pyr-orient} |
| 660 |
> |
\caption{\label{fig:PyrOrient} The three decay constants of the |
| 661 |
> |
orientational time correlation function, $C_2(t)$, for water as a function |
| 662 |
> |
of distance from the center of the ice slab. The vertical dashed line |
| 663 |
> |
indicates the edge of the pyramidal ice slab determined by the local order |
| 664 |
> |
tetrahedral parameter. The control (black circles) and sheared (red squares) |
| 665 |
> |
experiments were fit by a shifted exponential decay (Eq. 9\cite{Louden13}) |
| 666 |
> |
shown by the black and red lines respectively. The upper two panels show that |
| 667 |
> |
translational and hydrogen bond making and breaking events slow down |
| 668 |
> |
through the interface while approaching the ice slab. The bottom most panel |
| 669 |
> |
shows the librational motion of the water molecules speeding up approaching |
| 670 |
> |
the ice block due to the confined region of space allowed for the molecules |
| 671 |
> |
to move in.} |
| 672 |
> |
\end{figure} |
| 673 |
> |
|
| 674 |
> |
\begin{figure} |
| 675 |
> |
\includegraphics[width=\linewidth]{SP-orient-less} |
| 676 |
> |
\caption{\label{fig:SPorient} Decay constants for $C_2(t)$ at the secondary |
| 677 |
> |
prismatic face. Panel descriptions match those in \ref{fig:PyrOrient}.} |
| 678 |
> |
\end{figure} |
| 679 |
> |
|
| 680 |
> |
|
| 681 |
> |
\begin{table}[h] |
| 682 |
> |
\centering |
| 683 |
> |
\caption{Phyiscal properties of the basal, prismatic, pyramidal, and secondary prismatic facets of Ice-I$_\mathrm{h}$} |
| 684 |
> |
\label{tab:kappa} |
| 685 |
> |
\begin{tabular}{|ccccc|} \hline |
| 686 |
> |
& \multicolumn{2}{c}{$\kappa_{Drag direction}$ |
| 687 |
> |
(x10\textsuperscript{-4} amu \AA\textsuperscript{-2} fs\textsuperscript{-1})} & & \\ |
| 688 |
> |
Interface & $\kappa_{x}$ & $\kappa_{y}$ & $\theta_{\infty}$ & $K_{spread} (ns^{-1})$ \\ \hline |
| 689 |
> |
basal & $5.9 \pm 0.3$ & $6.5 \pm 0.8$ & $34.1 \pm 0.9$ & $0.60 \pm 0.07$ \\ |
| 690 |
> |
pyramidal & $5.8 \pm 0.4$ & $6.1 \pm 0.5$ & $35 \pm 3$ & $0.7 \pm 0.1$ \\ |
| 691 |
> |
prismatic & $3.0 \pm 0.2$ & $3.0 \pm 0.1$ & $45 \pm 3$ & $0.75 \pm 0.09$ \\ |
| 692 |
> |
secondary prismatic & $3.5 \pm 0.1$ & $3.3 \pm 0.2$ & $42 \pm 2$ & $0.69 \pm 0.03$ \\ \hline |
| 693 |
> |
\end{tabular} |
| 694 |
> |
\end{table} |
| 695 |
> |
|
| 696 |
> |
|
| 697 |
> |
\begin{table}[h] |
| 698 |
> |
\centering |
| 699 |
> |
\caption{Shearing and Droplet simulation parameters} |
| 700 |
> |
\label{tab:method} |
| 701 |
> |
\begin{tabular}{|cccc|ccc|} \hline |
| 702 |
> |
& \multicolumn{3}{c}{Shearing} & \multicolumn{3}{c}{Droplet}\\ |
| 703 |
> |
Interface & $N_{ice}$ & $N_{liquid}$ & Lx, Ly, Lz (\AA) & |
| 704 |
> |
$N_{ice}$ & $N_{droplet}$ & Lx, Ly (\AA) \\ \hline |
| 705 |
> |
Basal & 900 & 1846 & (23.87, 35.83, 98.64) & 12960 & 2048 & (134.70, 140.04)\\ |
| 706 |
> |
Prismatic & 3000 & 5464 & (35.95, 35.65, 205.77) & 9900 & 2048 & |
| 707 |
> |
(110.04, 115.00)\\ |
| 708 |
> |
Pyramidal & 1216 & 2203& (37.47, 29.50, 93.02) & 11136 & 2048 & |
| 709 |
> |
(143.75, 121.41)\\ |
| 710 |
> |
Secondary Prismatic & 3840 & 8176 & (71.87, 31.66, 161.55) & 11520 & |
| 711 |
> |
2048 & (146.72, 124.48)\\ |
| 712 |
> |
\hline |
| 713 |
> |
\end{tabular} |
| 714 |
> |
\end{table} |
| 715 |
> |
|
| 716 |
> |
\end{document} |
