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\begin{document} |
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\title{Simulations of solid-liquid friction at ice-I$_\mathrm{h}$ / |
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water interfaces} |
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\author{Patrick B. Louden and J. Daniel |
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Gezelter\footnote{Corresponding author. \ Electronic mail: |
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gezelter@nd.edu} \\ |
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Department of Chemistry and Biochemistry,\\ |
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University of Notre Dame\\ |
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Notre Dame, Indiana 46556} |
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\date{\today} |
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\maketitle |
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\begin{doublespace} |
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\begin{abstract} |
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We have investigated the structural and dynamic properties of the |
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basal and prismatic facets of the ice I$_\mathrm{h}$ / water |
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interface when the solid phase is drawn through the liquid |
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(i.e. sheared relative to the fluid phase). To impose the shear, we |
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utilized a velocity-shearing and scaling (VSS) approach to reverse |
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non-equilibrium molecular dynamics (RNEMD). This method can create |
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simultaneous temperature and velocity gradients and allow the |
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measurement of transport properties at interfaces. The interfacial |
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width was found to be independent of the relative velocity of the |
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ice and liquid layers over a wide range of shear rates. Decays of |
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molecular orientational time correlation functions gave similar |
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estimates for the width of the interfaces, although the short- and |
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longer-time decay components behave differently closer to the |
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interface. Although both facets of ice are in ``stick'' boundary |
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conditions in liquid water, the solid-liquid friction coefficients |
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were found to be significantly different for the basal and prismatic |
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facets of ice. |
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\end{abstract} |
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|
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\newpage |
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\section{Introduction} |
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%-----Outline of Intro--------------- |
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% in general, ice/water interface is important b/c .... |
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% here are some people who have worked on ice/water, trying to understand the processes above .... |
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% with the recent development of VSS-RNEMD, we can now look at the shearing problem |
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% talk about what we will present in this paper |
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% -------End Intro------------------ |
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|
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%Gay02: cites many other ice/water papers, make sure to cite them. |
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|
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Understanding the ice/water interface is essential for explaining |
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complex processes such as nucleation and crystal |
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growth,\cite{Han92,Granasy95,Vanfleet95} crystal |
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melting,\cite{Weber83,Han92,Sakai96,Sakai96B} and some fascinating |
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biological processes, such as the behavior of the antifreeze proteins |
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found in winter flounder,\cite{Wierzbicki07, Chapsky97} and certain |
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terrestrial arthropods.\cite{Duman:2001qy,Meister29012013} There has |
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been significant progress on understanding the structure and dynamics |
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of quiescent ice/water interfaces utilizing both theory and |
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experiment. Haymet \emph{et al.} have done extensive work on ice I$_\mathrm{h}$, |
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including characterizing and determining the width of the ice/water |
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interface for the SPC,\cite{Karim90} SPC/E,\cite{Gay02,Bryk02} CF1,\cite{Hayward01,Hayward02} and TIP4P~\cite{Karim88} models for |
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water. |
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More recently, Haymet \emph{et al.} have investigated the effects |
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cations and anions have on crystal |
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nucleation.\cite{Bryk04,Smith05,Wilson08,Wilson10} Nada \emph{et al.} |
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have also studied ice/water |
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interfaces,\cite{Nada95,Nada00,Nada03,Nada12} and have found that the |
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differential growth rates of the facets of ice I$_\mathrm{h}$ are due to the the |
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reordering of the hydrogen bonding network.\cite{Nada05} |
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|
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The movement of liquid water over the facets of ice has been less |
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thoroughly studied than the quiescent surfaces. This process is |
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potentially important in understanding transport of large blocks of |
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ice in water (which has important implications in the earth sciences), |
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as well as the relative motion of crystal-crystal interfaces that have |
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been separated by nanometer-scale fluid domains. In addition to |
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understanding both the structure and thickness of the interfacial |
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regions, it is important to understand the molecular origin of |
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friction, drag, and other changes in dynamical properties of the |
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liquid in the regions close to the surface that are altered by the |
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presence of a shearing of the bulk fluid relative to the solid phase. |
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|
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In this work, we apply a recently-developed velocity shearing and |
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scaling approach to reverse non-equilibrium molecular dynamics |
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(VSS-RNEMD). This method makes it possible to calculate transport |
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properties like the interfacial thermal conductance across |
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heterogeneous interfaces,\cite{Kuang12} and can create simultaneous |
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temperature and velocity gradients and allow the measurement of |
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friction and thermal transport properties at interfaces. This has |
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allowed us to investigate the width of the ice/water interface as the |
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ice is sheared through the liquid, while simultaneously imposing a |
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weak thermal gradient to prevent frictional heating of the interface. |
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In the sections that follow, we discuss the methodology for creating |
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and simulating ice/water interfaces under shear and provide results |
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from both structural and dynamical correlation functions. We also |
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show that the solid-liquid interfacial friction coefficient depends |
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sensitively on the details of the surface morphology. |
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\section{Methodology} |
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\subsection{Stable ice I$_\mathrm{h}$ / water interfaces under shear} |
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The structure of ice I$_\mathrm{h}$ is well understood; it |
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crystallizes in a hexagonal space group P$6_3/mmc$, and the hexagonal |
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crystals of ice have two faces that are commonly exposed, the basal |
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face $\{0~0~0~1\}$, which forms the top and bottom of each hexagonal |
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plate, and the prismatic $\{1~0~\bar{1}~0\}$ face which forms the |
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sides of the plate. Other less-common, but still important, faces of |
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ice I$_\mathrm{h}$ are the secondary prism, $\{1~1~\bar{2}~0\}$, and |
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pyramidal, $\{2~0~\bar{2}~1\}$, faces. Ice I$_\mathrm{h}$ is normally |
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proton disordered in bulk crystals, although the surfaces probably |
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have a preference for proton ordering along strips of dangling H-atoms |
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and Oxygen lone pairs.\cite{Buch:2008fk} |
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\begin{table}[h] |
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\centering |
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\caption{Mapping between the Miller indices of four facets of ice in |
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the $P6_3/mmc$ crystal system to the orthorhombic $P2_12_12_1$ |
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system in reference \bibpunct{}{}{,}{n}{}{,} \protect\citep{Hirsch04}.} |
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\label{tab:equiv} |
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\begin{tabular}{|ccc|} \hline |
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& hexagonal & orthorhombic \\ |
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& ($P6_3/mmc$) & ($P2_12_12_1$) \\ |
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crystal face & Miller indices & equivalent \\ \hline |
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basal & $\{0~0~0~1\}$ & $\{0~0~1\}$ \\ |
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prism & $\{1~0~\bar{1}~0\}$ & $\{1~0~0\}$ \\ |
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secondary prism & $\{1~1~\bar{2}~0\}$ & $\{1~3~0\}$ \\ |
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pyramidal & $\{2~0~\bar{2}~1\}$ & $\{2~0~1\}$ \\ \hline |
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\end{tabular} |
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\end{table} |
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For small simulated ice interfaces, it is useful to work with |
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proton-ordered, but zero-dipole crystal that exposes these strips of |
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dangling H-atoms and lone pairs. When placing another material in |
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contact with one of the ice crystalline planes, it is also quite |
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useful to have an orthorhombic (rectangular) box. Recent work by |
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Hirsch and Ojam\"{a}e describes a number of alternative crystal |
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systems for proton-ordered bulk ice I$_\mathrm{h}$ using orthorhombic |
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cells.\cite{Hirsch04} |
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In this work, we are using Hirsch and Ojam\"{a}e's structure 6 which |
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is an orthorhombic cell ($P2_12_12_1$) that produces a proton-ordered |
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version of ice I$_\mathrm{h}$. Table \ref{tab:equiv} contains a |
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mapping between the Miller indices of common ice facets in the |
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P$6_3/mmc$ crystal system and those in the Hirsch and Ojam\"{a}e |
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$P2_12_12_1$ system. |
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|
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Structure 6 from the Hirsch and Ojam\"{a}e paper has lattice |
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parameters $a = 4.49225$ \AA\ , $b = 7.78080$ \AA\ , $c = 7.33581$ \AA\ |
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and two water molecules whose atoms reside at fractional coordinates |
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given in table \ref{tab:p212121}. To construct the basal and prismatic |
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interfaces, these crystallographic coordinates were used to construct |
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an orthorhombic unit cell which was then replicated in all three |
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dimensions yielding a proton-ordered block of ice I$_{h}$. To expose |
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the desired face, the orthorhombic representation was then cut along |
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the ($001$) or ($100$) planes for the basal and prismatic faces |
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respectively. The resulting block was rotated so that the exposed |
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faces were aligned with the $z$-axis normal to the exposed face. The |
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block was then cut along two perpendicular directions in a way that |
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allowed for perfect periodic replication in the $x$ and $y$ axes, |
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creating a slab with either the basal or prismatic faces exposed along |
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the $z$ axis. The slab was then replicated in the $x$ and $y$ |
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dimensions until a desired sample size was obtained. |
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|
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\begin{table}[h] |
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\centering |
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\caption{Fractional coordinates for water in the orthorhombic |
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$P2_12_12_1$ system for ice I$_\mathrm{h}$ in reference \bibpunct{}{}{,}{n}{}{,} \protect\citep{Hirsch04}.} |
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\label{tab:p212121} |
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\begin{tabular}{|cccc|} \hline |
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atom type & x & y & z \\ \hline |
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O & 0.7500 & 0.1667 & 0.4375 \\ |
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H & 0.5735 & 0.2202 & 0.4836 \\ |
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H & 0.7420 & 0.0517 & 0.4836 \\ |
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O & 0.2500 & 0.6667 & 0.4375 \\ |
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H & 0.2580 & 0.6693 & 0.3071 \\ |
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H & 0.4265 & 0.7255 & 0.4756 \\ \hline |
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\end{tabular} |
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\end{table} |
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Our ice / water interfaces were created using a box of liquid water |
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that had the same dimensions (in $x$ and $y$) as the ice block. |
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Although the experimental solid/liquid coexistence temperature under |
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atmospheric pressure is close to 273~K, Haymet \emph{et al.} have done |
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extensive work on characterizing the ice/water interface, and find |
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that the coexistence temperature for simulated water is often quite a |
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bit different.\cite{Karim88,Karim90,Hayward01,Bryk02,Hayward02} They |
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have found that for the SPC/E water model,\cite{Berendsen87} which is |
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also used in this study, the ice/water interface is most stable at |
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225~$\pm$5~K.\cite{Bryk02} This liquid box was therefore equilibrated at |
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225~K and 1~atm of pressure in the NPAT ensemble (with the $z$ axis |
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allowed to fluctuate to equilibrate to the correct pressure). The |
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liquid and solid systems were combined by carving out any water |
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molecule from the liquid simulation cell that was within 3~\AA\ of any |
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atom in the ice slab. The resulting basal system was $23.87 \times 35.83 |
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\times 98.64$ \AA\ with 900 SPC/E molecules in the ice slab, and 1846 in |
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the liquid phase. Similarly, the prismatic system was $36.12 \times 36.43 |
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\times 86.10$ \AA\ with 1000 SPC/E molecules in the ice slab and 2684 in |
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the liquid. |
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Molecular translation and orientational restraints were applied in the |
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early stages of equilibration to prevent melting of the ice slab. |
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These restraints were removed during NVT equilibration, well before |
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data collection was carried out. |
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\subsection{Shearing ice / water interfaces without bulk melting} |
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As a solid is dragged through a liquid, there is frictional heating |
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that will act to melt the interface. To study the behavior of the |
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interface under a shear stress without causing the interface to melt, |
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it is necessary to apply a weak thermal gradient in combination with |
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the momentum gradient. This can be accomplished using the velocity |
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shearing and scaling (VSS) variant of reverse non-equilibrium |
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molecular dynamics (RNEMD), which utilizes a series of simultaneous |
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velocity exchanges between two regions within the simulation |
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cell.\cite{Kuang12} One of these regions is centered within the ice |
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slab, while the other is centrally located in the liquid |
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region. VSS-RNEMD provides a set of conservation constraints for |
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creating either a momentum flux or a thermal flux (or both |
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simultaneously) between the two slabs. Satisfying the constraint |
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equations ensures that the new configurations are sampled from the |
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same NVE ensemble as before the VSS move. |
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|
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The VSS moves are applied periodically to scale and shift the particle |
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velocities ($\mathbf{v}_i$ and $\mathbf{v}_j$) in two slabs ($H$ and |
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$C$) which are separated by half of the simulation box, |
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\begin{displaymath} |
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\begin{array}{rclcl} |
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& \underline{\mathrm{shearing}} & & |
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\underline{~~~~~~~~~~~~\mathrm{scaling}~~~~~~~~~~~~} \\ |
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\mathbf{v}_i \leftarrow & |
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\mathbf{a}_c\ & + & c\cdot\left(\mathbf{v}_i - \langle\mathbf{v}_c |
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\rangle\right) + \langle\mathbf{v}_c\rangle \\ |
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\mathbf{v}_j \leftarrow & |
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\mathbf{a}_h & + & h\cdot\left(\mathbf{v}_j - \langle\mathbf{v}_h |
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\rangle\right) + \langle\mathbf{v}_h\rangle . |
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\end{array} |
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\end{displaymath} |
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Here $\langle\mathbf{v}_c\rangle$ and $\langle\mathbf{v}_h\rangle$ are |
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the center of mass velocities in the $C$ and $H$ slabs, respectively. |
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Within the two slabs, particles receive incremental changes or a |
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``shear'' to their velocities. The amount of shear is governed by the |
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imposed momentum flux, $\mathbf{j}_z(\mathbf{p})$ |
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\begin{eqnarray} |
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\mathbf{a}_c & = & - \mathbf{j}_z(\mathbf{p}) \Delta t / M_c \label{vss1}\\ |
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\mathbf{a}_h & = & + \mathbf{j}_z(\mathbf{p}) \Delta t / M_h \label{vss2} |
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\end{eqnarray} |
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where $M_{\{c,h\}}$ is the total mass of particles within each of the |
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slabs and $\Delta t$ is the interval between two separate operations. |
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To simultaneously impose a thermal flux ($J_z$) between the slabs we |
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use energy conservation constraints, |
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\begin{eqnarray} |
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K_c - J_z\Delta t & = & c^2 (K_c - \frac{1}{2}M_c \langle\mathbf{v}_c |
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\rangle^2) + \frac{1}{2}M_c (\langle \mathbf{v}_c \rangle + \mathbf{a}_c)^2 \label{vss3}\\ |
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K_h + J_z\Delta t & = & h^2 (K_h - \frac{1}{2}M_h \langle\mathbf{v}_h |
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\rangle^2) + \frac{1}{2}M_h (\langle \mathbf{v}_h \rangle + |
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|
\mathbf{a}_h)^2 \label{vss4}. |
303 |
|
|
\label{constraint} |
304 |
|
|
\end{eqnarray} |
305 |
|
|
Simultaneous solution of these quadratic formulae for the scaling |
306 |
|
|
coefficients, $c$ and $h$, will ensure that the simulation samples from |
307 |
|
|
the original microcanonical (NVE) ensemble. Here $K_{\{c,h\}}$ is the |
308 |
|
|
instantaneous translational kinetic energy of each slab. At each time |
309 |
|
|
interval, it is a simple matter to solve for $c$, $h$, $\mathbf{a}_c$, |
310 |
|
|
and $\mathbf{a}_h$, subject to the imposed momentum flux, |
311 |
|
|
$j_z(\mathbf{p})$, and thermal flux, $J_z$, values. Since the VSS |
312 |
|
|
operations do not change the kinetic energy due to orientational |
313 |
|
|
degrees of freedom or the potential energy of a system, configurations |
314 |
|
|
after the VSS move have exactly the same energy (and linear |
315 |
|
|
momentum) as before the move. |
316 |
|
|
|
317 |
|
|
As the simulation progresses, the VSS moves are performed on a regular |
318 |
|
|
basis, and the system develops a thermal and/or velocity gradient in |
319 |
gezelter |
3945 |
response to the applied flux. In a bulk material, it is quite simple |
320 |
gezelter |
3918 |
to use the slope of the temperature or velocity gradients to obtain |
321 |
gezelter |
3945 |
either the thermal conductivity or shear viscosity. |
322 |
gezelter |
3918 |
|
323 |
|
|
The VSS-RNEMD approach is versatile in that it may be used to |
324 |
|
|
implement thermal and shear transport simultaneously. Perturbations |
325 |
|
|
of velocities away from the ideal Maxwell-Boltzmann distributions are |
326 |
|
|
minimal, as is thermal anisotropy. This ability to generate |
327 |
|
|
simultaneous thermal and shear fluxes has been previously utilized to |
328 |
|
|
map out the shear viscosity of SPC/E water over a wide range of |
329 |
gezelter |
3957 |
temperatures (90~K) with a single 1~ns simulation.\cite{Kuang12} |
330 |
gezelter |
3918 |
|
331 |
gezelter |
3945 |
For this work, we are using the VSS-RNEMD method primarily to generate |
332 |
|
|
a shear between the ice slab and the liquid phase, while using a weak |
333 |
gezelter |
3957 |
thermal gradient to maintain the interface at the 225~K target |
334 |
gezelter |
3946 |
value. This ensures minimal melting of the bulk ice phase and allows |
335 |
|
|
us to control the exact temperature of the interface. |
336 |
gezelter |
3918 |
|
337 |
gezelter |
3897 |
\subsection{Computational Details} |
338 |
gezelter |
3945 |
All simulations were performed using OpenMD,\cite{OOPSE,openmd} with a |
339 |
|
|
time step of 2 fs and periodic boundary conditions in all three |
340 |
|
|
dimensions. Electrostatics were handled using the damped-shifted |
341 |
|
|
force real-space electrostatic kernel.\cite{Ewald} The systems were |
342 |
|
|
divided into 100 bins along the $z$-axis for the VSS-RNEMD moves, |
343 |
gezelter |
3957 |
which were attempted every 50~fs. |
344 |
gezelter |
3897 |
|
345 |
gezelter |
3945 |
The interfaces were equilibrated for a total of 10 ns at equilibrium |
346 |
|
|
conditions before being exposed to either a shear or thermal gradient. |
347 |
|
|
This consisted of 5 ns under a constant temperature (NVT) integrator |
348 |
|
|
set to 225K followed by 5 ns under a microcanonical integrator. Weak |
349 |
|
|
thermal gradients were allowed to develop using the VSS-RNEMD (NVE) |
350 |
plouden |
3966 |
integrator using a small thermal flux ($-2.0\times 10^{-6}$ |
351 |
gezelter |
3945 |
kcal/mol/\AA$^2$/fs) for a duration of 5 ns to allow the gradient to |
352 |
gezelter |
3946 |
stabilize. The resulting temperature gradient was $\approx$ 10K over |
353 |
|
|
the entire 100 \AA\ box length, which was sufficient to keep the |
354 |
gezelter |
3945 |
temperature at the interface within $\pm 1$ K of the 225K target. |
355 |
gezelter |
3897 |
|
356 |
gezelter |
3945 |
Velocity gradients were then imposed using the VSS-RNEMD (NVE) |
357 |
gezelter |
3970 |
integrator with a range of momentum fluxes. These gradients were |
358 |
gezelter |
3957 |
allowed to stabilize for 1~ns before data collection began. Once |
359 |
|
|
established, four successive 0.5~ns runs were performed for each shear |
360 |
gezelter |
3945 |
rate. During these simulations, snapshots of the system were taken |
361 |
gezelter |
3957 |
every 1~ps, and statistics on the structure and dynamics in each bin |
362 |
gezelter |
3967 |
were accumulated throughout the simulations. Although there was some |
363 |
|
|
small variation in the measured interfacial width between succcessive |
364 |
gezelter |
3970 |
runs, no indication of bulk melting or crystallization was observed. |
365 |
|
|
That is, no large scale changes in the positions of the left and right |
366 |
|
|
interfaces occurred during the simulations. |
367 |
gezelter |
3945 |
|
368 |
plouden |
3941 |
\section{Results and discussion} |
369 |
|
|
|
370 |
gezelter |
3946 |
\subsection{Interfacial width} |
371 |
|
|
Any order parameter or time correlation function that changes as one |
372 |
|
|
crosses an interface from a bulk liquid to a solid can be used to |
373 |
|
|
measure the width of the interface. In previous work on the ice/water |
374 |
gezelter |
3954 |
interface, Haymet {\it et al.}\cite{Bryk02} have utilized structural |
375 |
gezelter |
3946 |
features (including the density) as well as dynamic properties |
376 |
|
|
(including the diffusion constant) to estimate the width of the |
377 |
|
|
interfaces for a number of facets of the ice crystals. Because |
378 |
|
|
VSS-RNEMD imposes a lateral flow, parameters that depend on |
379 |
|
|
translational motion of the molecules (e.g. diffusion) may be |
380 |
gezelter |
3954 |
artificially skewed by the RNEMD moves. A structural parameter is not |
381 |
gezelter |
3946 |
influenced by the RNEMD perturbations to the same degree. Here, we |
382 |
gezelter |
3954 |
have used the local tetrahedral order parameter as described by |
383 |
gezelter |
3946 |
Kumar\cite{Kumar09} and Errington\cite{Errington01} as our principal |
384 |
gezelter |
3967 |
measure of the interfacial width. A previous study by Bryk and Haymet |
385 |
|
|
also used local tetrahedrality as an order parameter for ice/water |
386 |
|
|
interfaces.\cite{Bryk2004b} |
387 |
plouden |
3936 |
|
388 |
gezelter |
3945 |
The local tetrahedral order parameter, $q(z)$, is given by |
389 |
gezelter |
3897 |
\begin{equation} |
390 |
gezelter |
3946 |
q(z) = \int_0^L \sum_{k=1}^{N} \Bigg(1 -\frac{3}{8}\sum_{i=1}^{3} |
391 |
|
|
\sum_{j=i+1}^{4} \bigg(\cos\psi_{ikj}+\frac{1}{3}\bigg)^2\Bigg) |
392 |
|
|
\delta(z_{k}-z)\mathrm{d}z \Bigg/ N_z |
393 |
gezelter |
3945 |
\label{eq:qz} |
394 |
gezelter |
3897 |
\end{equation} |
395 |
gezelter |
3945 |
where $\psi_{ikj}$ is the angle formed between the oxygen site on |
396 |
gezelter |
3946 |
central molecule $k$, and the oxygen sites on two of the four closest |
397 |
|
|
molecules, $i$ and $j$. Molecules $i$ and $j$ are further restricted |
398 |
gezelter |
3967 |
to lie withing the first peak in the pair distribution function for |
399 |
|
|
molecule $k$ (typically $<$ 3.41 \AA\ for water). $N_z = \int |
400 |
gezelter |
3946 |
\delta(z_k - z) \mathrm{d}z$ is a normalization factor to account for |
401 |
|
|
the varying population of molecules within each finite-width bin. The |
402 |
|
|
local tetrahedral order parameter has a range of $(0,1)$, where the |
403 |
|
|
larger values of $q$ indicate a larger degree of tetrahedral ordering |
404 |
|
|
of the local environment. In perfect ice I$_\mathrm{h}$ structures, |
405 |
|
|
the parameter can approach 1 at low temperatures, while in liquid |
406 |
|
|
water, the ordering is significantly less tetrahedral, and values of |
407 |
|
|
$q(z) \approx 0.75$ are more common. |
408 |
plouden |
3909 |
|
409 |
gezelter |
3946 |
To estimate the interfacial width, the system was divided into 100 |
410 |
gezelter |
3967 |
bins along the $z$-dimension. The $q_{z}$ function was time-averaged |
411 |
|
|
to give yield a tetrahedrality profile of the system. The profile was |
412 |
|
|
then fit to a hyperbolic tangent that smoothly links the liquid and |
413 |
|
|
solid states, |
414 |
plouden |
3941 |
\begin{equation}\label{tet_fit} |
415 |
gezelter |
3946 |
q(z) \approx |
416 |
|
|
q_{liq}+\frac{q_{ice}-q_{liq}}{2}\left[\tanh\left(\frac{z-l}{w}\right)-\tanh\left(\frac{z-r}{w}\right)\right]+\beta\left|z- |
417 |
|
|
\frac{r+l}{2}\right|. |
418 |
gezelter |
3897 |
\end{equation} |
419 |
gezelter |
3946 |
Here $q_{liq}$ and $q_{ice}$ are the local tetrahedral order parameter |
420 |
|
|
for the bulk liquid and ice domains, respectively, $w$ is the width of |
421 |
|
|
the interface. $l$ and $r$ are the midpoints of the left and right |
422 |
gezelter |
3957 |
interfaces, respectively. The last term in eq. \eqref{tet_fit} |
423 |
gezelter |
3946 |
accounts for the influence that the weak thermal gradient has on the |
424 |
gezelter |
3967 |
tetrahedrality profile in the liquid region. |
425 |
gezelter |
3897 |
|
426 |
gezelter |
3946 |
In Figures \ref{fig:bComic} and \ref{fig:pComic} we see the |
427 |
|
|
$z$-coordinate profiles for tetrahedrality, temperature, and the |
428 |
|
|
$x$-component of the velocity for the basal and prismatic interfaces. |
429 |
|
|
The lower panels show the $q(z)$ (black circles) along with the |
430 |
|
|
hyperbolic tangent fits (red lines). In the liquid region, the local |
431 |
|
|
tetrahedral order parameter, $q(z) \approx 0.75$ while in the |
432 |
|
|
crystalline region, $q(z) \approx 0.94$, indicating a more tetrahedral |
433 |
|
|
environment. The vertical dotted lines denote the midpoint of the |
434 |
gezelter |
3957 |
interfaces ($r$ and $l$ in eq. \eqref{tet_fit}). The weak thermal |
435 |
gezelter |
3946 |
gradient applied to the systems in order to keep the interface at |
436 |
gezelter |
3957 |
225~$\pm$~5K, can be seen in middle panels. The transverse velocity |
437 |
gezelter |
3946 |
profile is shown in the upper panels. It is clear from the upper |
438 |
|
|
panels that water molecules in close proximity to the surface (i.e. |
439 |
gezelter |
3967 |
within 10~\AA\ to 15~\AA~of the interfaces) have transverse velocities |
440 |
|
|
quite close to the velocities within the ice block. There is no |
441 |
|
|
velocity discontinuity at the interface, which indicates that the |
442 |
|
|
shearing of ice/water interfaces occurs in the ``stick'' or no-slip |
443 |
|
|
boundary conditions. |
444 |
gezelter |
3897 |
|
445 |
gezelter |
3914 |
\begin{figure} |
446 |
gezelter |
3957 |
\includegraphics[width=\linewidth]{bComicStrip} |
447 |
gezelter |
3967 |
\caption{\label{fig:bComic} The basal interface with a shear rate of |
448 |
gezelter |
3970 |
1.3 ms\textsuperscript{-1}. Lower panel: the local tetrahedral |
449 |
|
|
order parameter, $q(z)$, (black circles) and the hyperbolic tangent |
450 |
|
|
fit (red line). Middle panel: the imposed thermal gradient required |
451 |
|
|
to maintain a fixed interfacial temperature. Upper panel: the |
452 |
|
|
transverse velocity gradient that develops in response to an imposed |
453 |
|
|
momentum flux. The vertical dotted lines indicate the locations of |
454 |
|
|
the midpoints of the two interfaces.} |
455 |
gezelter |
3914 |
\end{figure} |
456 |
plouden |
3904 |
|
457 |
gezelter |
3914 |
\begin{figure} |
458 |
gezelter |
3957 |
\includegraphics[width=\linewidth]{pComicStrip} |
459 |
gezelter |
3967 |
\caption{\label{fig:pComic} The prismatic interface with a shear rate |
460 |
plouden |
3968 |
of 2.0 ms\textsuperscript{-1}. Panel |
461 |
gezelter |
3946 |
descriptions match those in figure \ref{fig:bComic}} |
462 |
gezelter |
3914 |
\end{figure} |
463 |
|
|
|
464 |
gezelter |
3967 |
From the fits using eq. \eqref{tet_fit}, we find the interfacial width |
465 |
|
|
for the basal and prismatic systems to be 3.2~$\pm$~0.4~\AA\ and |
466 |
gezelter |
3957 |
3.6~$\pm$~0.2~\AA\ , respectively, with no applied momentum flux. Over |
467 |
gezelter |
3946 |
the range of shear rates investigated, $0.6 \pm 0.3 \mathrm{ms}^{-1} |
468 |
|
|
\rightarrow 5.3 \pm 0.5 \mathrm{ms}^{-1}$ for the basal system and |
469 |
|
|
$0.9 \pm 0.2 \mathrm{ms}^{-1} \rightarrow 4.5 \pm 0.1 |
470 |
|
|
\mathrm{ms}^{-1}$ for the prismatic, we found no appreciable change in |
471 |
|
|
the interface width. The fit values for the interfacial width ($w$) |
472 |
|
|
over all shear rates contained the values reported above within their |
473 |
gezelter |
3967 |
error bars. Note that the interfacial widths reported here are based |
474 |
|
|
on the hyperbolic tangent parameter $w$ in Eq. \ref{tet_fit}. This is |
475 |
|
|
related to, but not identical with, the 10\%-90\% intefacial widths |
476 |
|
|
commonly used in previous studies.\cite{Bryk02,Bryk2004b} To estimate |
477 |
|
|
the 10\%-90\% widths, it is a simple matter to scale the widths |
478 |
|
|
obtained from the hyperbolic tangent fits to obtain $w_{10-90} = |
479 |
gezelter |
3970 |
2.197 \times w$.\cite{Bryk02,Bryk2004b} This results in $w_{10-90}$ |
480 |
gezelter |
3967 |
values of 7.0~$\pm$~0.9~\AA\ for the basal face, and 7.9~$\pm$~0.4 |
481 |
|
|
\AA\ for the prismatic face. These are somewhat smaller than |
482 |
|
|
previously reported values. |
483 |
gezelter |
3914 |
|
484 |
gezelter |
3967 |
\begin{figure} |
485 |
|
|
\includegraphics[width=\linewidth]{interface_width_by_shear_rate} |
486 |
|
|
\caption{\label{fig:widthByShear} The width of the ice water |
487 |
|
|
interfaces (as measured by Eq. \ref{tet_fit}) exhibits no dependence |
488 |
|
|
on the applied shear rate between the ice and water regions.} |
489 |
|
|
\end{figure} |
490 |
|
|
|
491 |
|
|
|
492 |
|
|
|
493 |
gezelter |
3954 |
\subsubsection{Orientational Dynamics} |
494 |
gezelter |
3946 |
The orientational time correlation function, |
495 |
plouden |
3941 |
\begin{equation}\label{C(t)1} |
496 |
gezelter |
3946 |
C_{2}(t)=\langle P_{2}(\mathbf{u}(0) \cdot \mathbf{u}(t)) \rangle, |
497 |
plouden |
3941 |
\end{equation} |
498 |
gezelter |
3946 |
gives insight into the local dynamic environment around the water |
499 |
|
|
molecules. The rate at which the function decays provides information |
500 |
|
|
about hindered motions and the timescales for relaxation. In |
501 |
|
|
eq. \eqref{C(t)1}, $P_{2}$ is the second-order Legendre polynomial, |
502 |
gezelter |
3948 |
the vector $\mathbf{u}$ is often taken as HOH bisector, although |
503 |
|
|
slightly different behavior can be observed when $\mathbf{u}$ is the |
504 |
gezelter |
3946 |
vector along one of the OH bonds. The angle brackets denote an |
505 |
|
|
ensemble average over all water molecules in a given spatial region. |
506 |
gezelter |
3897 |
|
507 |
gezelter |
3946 |
To investigate the dynamic behavior of water at the ice interfaces, we |
508 |
|
|
have computed $C_{2}(z,t)$ for molecules that are present within a |
509 |
|
|
particular slab along the $z$- axis at the initial time. The change |
510 |
|
|
in the decay behavior as a function of the $z$ coordinate is another |
511 |
gezelter |
3948 |
measure of the change of how the local environment changes across the |
512 |
|
|
ice/water interface. To compute these correlation functions, each of |
513 |
|
|
the 0.5 ns simulations was followed by a shorter 200 ps microcanonical |
514 |
|
|
(NVE) simulation in which the positions and orientations of every |
515 |
|
|
molecule in the system were recorded every 0.1 ps. The systems were |
516 |
gezelter |
3954 |
then divided into 30 bins along the $z$-axis and $C_2(t)$ was |
517 |
|
|
evaluated for each bin. |
518 |
plouden |
3919 |
|
519 |
gezelter |
3948 |
In simulations of water at biological interfaces, Furse {\em et al.} |
520 |
|
|
fit $C_2(t)$ functions for water with triexponential |
521 |
|
|
functions,\cite{Furse08} where the three components of the decay |
522 |
gezelter |
3957 |
correspond to a fast ($<$200 fs) reorientational piece driven by the |
523 |
gezelter |
3948 |
restoring forces of existing hydrogen bonds, a middle (on the order of |
524 |
|
|
several ps) piece describing the large angle jumps that occur during |
525 |
|
|
the breaking and formation of new hydrogen bonds,and a slow (on the |
526 |
|
|
order of tens of ps) contribution describing the translational motion |
527 |
|
|
of the molecules. The model for orientational decay presented |
528 |
|
|
recently by Laage and Hynes {\em et al.}\cite{Laage08,Laage11} also |
529 |
|
|
includes three similar decay constants, although two of the time |
530 |
|
|
constants are linked, and the resulting decay curve has two parameters |
531 |
|
|
governing the dynamics of decay. |
532 |
|
|
|
533 |
|
|
In our ice/water interfaces, we are at substantially lower |
534 |
|
|
temperatures, and the water molecules are further perturbed by the |
535 |
|
|
presence of the ice phase nearby. We have obtained the most |
536 |
|
|
reasonable fits using triexponential functions with three distinct |
537 |
gezelter |
3954 |
time domains, as well as a constant piece to account for the water |
538 |
gezelter |
3948 |
stuck in the ice phase that does not experience any long-time |
539 |
|
|
orientational decay, |
540 |
gezelter |
3897 |
\begin{equation} |
541 |
gezelter |
3946 |
C_{2}(t) \approx a e^{-t/\tau_\mathrm{short}} + b e^{-t/\tau_\mathrm{middle}} + c |
542 |
|
|
e^{-t/\tau_\mathrm{long}} + (1-a-b-c) |
543 |
gezelter |
3897 |
\end{equation} |
544 |
gezelter |
3948 |
Average values for the three decay constants (and error estimates) |
545 |
|
|
were obtained for each bin. In figures \ref{fig:basal_Tau_comic_strip} |
546 |
|
|
and \ref{fig:prismatic_Tau_comic_strip}, the three orientational decay |
547 |
|
|
times are shown as a function of distance from the center of the ice |
548 |
|
|
slab. |
549 |
gezelter |
3897 |
|
550 |
plouden |
3941 |
\begin{figure} |
551 |
gezelter |
3957 |
\includegraphics[width=\linewidth]{basal_Tau_comic_strip} |
552 |
gezelter |
3949 |
\caption{\label{fig:basal_Tau_comic_strip} The three decay constants |
553 |
|
|
of the orientational time correlation function, $C_2(t)$, for water |
554 |
|
|
as a function of distance from the center of the ice slab. The |
555 |
|
|
dashed line indicates the location of the basal face (as determined |
556 |
plouden |
3969 |
from the tetrahedrality order parameter) and the black and red lines |
557 |
gezelter |
3970 |
are fits using Eq. \ref{tauFit}. The moderate and long time |
558 |
|
|
contributions slow down close to the interface which would be |
559 |
gezelter |
3954 |
expected under reorganizations that involve large motions of the |
560 |
gezelter |
3949 |
molecules (e.g. frame-reorientations and jumps). The observed |
561 |
|
|
speed-up in the short time contribution is surprising, but appears |
562 |
|
|
to reflect the restricted motion of librations closer to the |
563 |
|
|
interface.} |
564 |
plouden |
3941 |
\end{figure} |
565 |
gezelter |
3897 |
|
566 |
gezelter |
3914 |
\begin{figure} |
567 |
gezelter |
3957 |
\includegraphics[width=\linewidth]{prismatic_Tau_comic_strip} |
568 |
gezelter |
3949 |
\caption{\label{fig:prismatic_Tau_comic_strip} |
569 |
|
|
Decay constants for $C_2(t)$ at the prismatic interface. Panel |
570 |
|
|
descriptions match those in figure \ref{fig:basal_Tau_comic_strip}.} |
571 |
gezelter |
3914 |
\end{figure} |
572 |
plouden |
3904 |
|
573 |
gezelter |
3949 |
Figures \ref{fig:basal_Tau_comic_strip} and |
574 |
|
|
\ref{fig:prismatic_Tau_comic_strip} show the three decay constants for |
575 |
|
|
the orientational time correlation function for water at varying |
576 |
|
|
displacements from the center of the ice slab for both the basal and |
577 |
|
|
prismatic interfaces. The vertical dotted lines indicate the |
578 |
|
|
locations of the midpoints of the interfaces as determined by the |
579 |
gezelter |
3970 |
tetrahedrality fits. In the liquid regions, $\tau_\mathrm{middle}$ and |
580 |
|
|
$\tau_\mathrm{long}$ have consistent values around 3-4 ps and 20-40 ps, |
581 |
gezelter |
3949 |
respectively, and increase in value approaching the interface. |
582 |
|
|
According to the jump model of Laage and Hynes {\em et |
583 |
gezelter |
3970 |
al.},\cite{Laage08,Laage11} $\tau_\mathrm{middle}$ corresponds to the |
584 |
|
|
breaking and making of hydrogen bonds and $\tau_\mathrm{long}$ is explained |
585 |
gezelter |
3949 |
with translational motion of the molecules (i.e. frame reorientation). |
586 |
|
|
The shortest of the three decay constants, the librational time |
587 |
|
|
$\tau_\mathrm{short}$ has a value of about 70 fs in the liquid region, |
588 |
|
|
and decreases in value approaching the interface. The observed |
589 |
|
|
speed-up in the short time contribution is surprising, but appears to |
590 |
|
|
reflect the restricted motion of librations closer to the interface. |
591 |
plouden |
3937 |
|
592 |
gezelter |
3949 |
The control systems (with no applied momentum flux) are shown with |
593 |
|
|
black symbols in figs. \ref{fig:basal_Tau_comic_strip} and |
594 |
|
|
\ref{fig:prismatic_Tau_comic_strip}, while those obtained while a |
595 |
gezelter |
3970 |
shear was active are shown in red. There are two notable features |
596 |
|
|
from the spatial dependence of the orientational correlation times |
597 |
|
|
seen in figures \ref{fig:basal_Tau_comic_strip} and |
598 |
|
|
\ref{fig:prismatic_Tau_comic_strip}. First, there are nearly-constant |
599 |
|
|
liquid-state values for $\tau_\mathrm{short}$, $\tau_\mathrm{middle}$, |
600 |
|
|
and $\tau_\mathrm{long}$ at large displacements from the |
601 |
|
|
interface. Second, there appears to be a single distance, $d$ from the |
602 |
|
|
interface at which all three decay times begin to deviate from their |
603 |
|
|
bulk liquid values. To quantify this ``dynamic interface width,'' |
604 |
|
|
each of the decay constant profiles were fit to |
605 |
plouden |
3968 |
\begin{equation}\label{tauFit} |
606 |
gezelter |
3970 |
\tau_{a}(z) = \tau^{\mathrm{liquid}}_{a}+(\tau^{\mathrm{wall}}_a-\tau^{\mathrm{liquid}}_a)e^{-(z-z^{\mathrm{wall}})/d} |
607 |
plouden |
3968 |
\end{equation} |
608 |
gezelter |
3970 |
where $a = \{ \mathrm{short, middle,} ~or~ \mathrm{long} \}$. In this |
609 |
|
|
expression, $\tau^{\mathrm{liquid}}_a$ and $\tau^{\mathrm{wall}}_a$ |
610 |
|
|
are the bulk liquid and projected values of the decay constants at the |
611 |
|
|
wall, $z^{\mathrm{wall}}$ is the location of the interface, and $d$ is |
612 |
|
|
the dynamic interface width. The widths for the two facets of ice, |
613 |
|
|
$d_{basal}$ and $d_{prismatic}$, were determined by averaging $d$ |
614 |
|
|
values for each of the three decay times. For the basal system, we |
615 |
|
|
find $d_{basal}$ to be 2.9 \AA\ for the quiescent control set, and 2.8 |
616 |
|
|
\AA\ for a simulation with a shear rate of 1.3 |
617 |
|
|
ms\textsuperscript{-1}. We also find $d_{prismatic}$ to be slightly |
618 |
plouden |
3968 |
larger than $d_{basal}$ for both the control and an applied shear, |
619 |
gezelter |
3970 |
with widths of 3.6 \AA\ for the quiescent system and 3.5 \AA\ for a |
620 |
|
|
simulation with a 2 ms\textsuperscript{-1} shear rate. There appears |
621 |
|
|
to be little or no dependence of the dynamic interface width on the |
622 |
|
|
applied shear rate. |
623 |
plouden |
3937 |
|
624 |
gezelter |
3970 |
The spatial dependence of the decay times in $C_2(t)$ determines a |
625 |
|
|
dynamic interfacial width that is in good agreement with the |
626 |
|
|
structural interfacial width determined by the local tetrahedrality. |
627 |
|
|
For exponential models like the one in Eq. \ref{tauFit}, it is again a |
628 |
|
|
simple matter to estimate the 10\%-90\% interfacial width by |
629 |
|
|
multiplying the $d$ constant by $2.19723$. For the interfaces |
630 |
|
|
studied, we obtain 10\%-90\% dynamic interface width $(d_{10-90})$ |
631 |
|
|
values of $\sim$~6.3~\AA\ for the basal face and 7.8~\AA\ for the |
632 |
|
|
prismatic face. |
633 |
plouden |
3937 |
|
634 |
plouden |
3941 |
\subsection{Coefficient of Friction of the Interface} |
635 |
gezelter |
3954 |
As liquid water flows over an ice interface, there is a distance from |
636 |
|
|
the structural interface where bulk-like hydrodynamics are recovered. |
637 |
|
|
Bocquet and Barrat constructed a theory for the hydrodynamic boundary |
638 |
|
|
parameters, which include the slipping length |
639 |
gezelter |
3952 |
$\left(\delta_\mathrm{wall}\right)$ of this boundary layer and the |
640 |
|
|
``hydrodynamic position'' of the boundary |
641 |
gezelter |
3954 |
$\left(z_\mathrm{wall}\right)$.\cite{PhysRevLett.70.2726,PhysRevE.49.3079} |
642 |
|
|
This last parameter is the location (relative to a solid surface) |
643 |
gezelter |
3952 |
where the bulk-like behavior is recovered. Work by Mundy {\it et al.} |
644 |
gezelter |
3954 |
has helped to combine these parameters into a liquid-solid friction |
645 |
gezelter |
3952 |
coefficient, which quantifies the resistance to pulling the solid |
646 |
gezelter |
3954 |
interface through a liquid,\cite{Mundy1997305} |
647 |
gezelter |
3952 |
\begin{equation} |
648 |
|
|
\lambda_\mathrm{wall} = \frac{\eta}{\delta_\mathrm{wall}}. |
649 |
|
|
\end{equation} |
650 |
gezelter |
3954 |
This expression is nearly identical to one provided by Pit {\it et |
651 |
|
|
al.} for the solid-liquid friction of an interface,\cite{Pit99} |
652 |
plouden |
3942 |
\begin{equation}\label{Pit} |
653 |
gezelter |
3952 |
\lambda=\frac{\eta}{\delta} |
654 |
plouden |
3942 |
\end{equation} |
655 |
gezelter |
3952 |
where $\delta$ is the slip length for the liquid measured at the |
656 |
gezelter |
3967 |
location of the interface itself. In our simulations, the shoulder on |
657 |
|
|
the velocity profile indicating the location of the hydrodynamic |
658 |
|
|
boundary in the liquid is not always apparent. In some cases, the |
659 |
|
|
linear behavior persists nearly up to the interfacial region. For |
660 |
|
|
this reason, the hydrodynamic position of the boundary is not always |
661 |
|
|
computable, while the Pit approach (Eq. \ref{Pit}) can be used to find |
662 |
|
|
the solid-liquid friction coefficient more reliably. |
663 |
gezelter |
3952 |
|
664 |
gezelter |
3967 |
In both the Pit and hydrodynamic boundary expressions, $\eta$ is the |
665 |
|
|
shear viscosity of the bulk-like region of the liquid, a quantity |
666 |
|
|
which is easily obtained in VSS-RNEMD simulations by fitting the |
667 |
|
|
velocity profile in the region far from the surface.\cite{Kuang12} |
668 |
|
|
Assuming linear response in the bulk-like region, |
669 |
plouden |
3942 |
\begin{equation}\label{Kuang} |
670 |
gezelter |
3952 |
j_{z}(p_{x})=-\eta \left(\frac{\partial v_{x}}{\partial z}\right) |
671 |
plouden |
3942 |
\end{equation} |
672 |
gezelter |
3952 |
Substituting this result into eq. \eqref{Pit}, we can estimate the |
673 |
|
|
solid-liquid coefficient using the slip length, |
674 |
plouden |
3941 |
\begin{equation} |
675 |
gezelter |
3954 |
\lambda=-\frac{j_{z}(p_{x})} {\left(\frac{\partial v_{x}}{\partial |
676 |
|
|
z}\right) \delta} |
677 |
plouden |
3941 |
\end{equation} |
678 |
plouden |
3937 |
|
679 |
gezelter |
3967 |
For ice / water interfaces, the boundary conditions are no-slip, so |
680 |
|
|
projecting the bulk liquid state velocity profile yields a negative |
681 |
|
|
slip length. This length is the difference between the structural edge |
682 |
|
|
of the ice (determined by the tetrahedrality profile) and the location |
683 |
|
|
where the projected velocity of the bulk liquid intersects the solid |
684 |
|
|
phase velocity (see Figure \ref{fig:delta_example}). The coefficients |
685 |
|
|
of friction for the basal and the prismatic facets were determined for |
686 |
|
|
shearing along both the $x$ and $y$ axes. The values are given in |
687 |
|
|
table \ref{tab:lambda}. |
688 |
plouden |
3939 |
|
689 |
gezelter |
3967 |
Note that the measured friction coefficient for the basal face is |
690 |
|
|
twice that of the prismatic face (regardless of drag direction). |
691 |
|
|
These results may seem surprising as the basalface appears smoother |
692 |
|
|
than the prismatic with only small undulations of the oxygen |
693 |
|
|
positions, while the prismatic surface has deep corrugated channels |
694 |
|
|
along the $x$ direction in the crystal system used in this work. |
695 |
|
|
However, the corrugations are relatively thin, and the liquid phase |
696 |
|
|
water does not appear to populate the channels. The prismatic face |
697 |
|
|
therefore effectively presents stripes of solid-phase molecules |
698 |
|
|
(making up approximately half of the exposed surface area) with nearly |
699 |
|
|
empty space between them. The interfacial friction appears to be |
700 |
|
|
independent of the drag direction, so flow parallel to these channels |
701 |
|
|
does not explain the lower friction of the prismatic face. A more |
702 |
|
|
likely explanation is that the effective contact between the liquid |
703 |
|
|
phase and the prismatic face is reduced by the empty corrugations. |
704 |
|
|
|
705 |
|
|
\begin{figure} |
706 |
gezelter |
3957 |
\includegraphics[width=\linewidth]{delta_example} |
707 |
gezelter |
3952 |
\caption{\label{fig:delta_example} Determining the (negative) slip |
708 |
|
|
length ($\delta$) for the ice-water interfaces (which have decidedly |
709 |
|
|
non-slip behavior). This length is the difference between the |
710 |
|
|
structural edge of the ice (determined by the tetrahedrality |
711 |
|
|
profile) and the location where the projected velocity of the bulk |
712 |
|
|
liquid (dashed red line) intersects the solid phase velocity (solid |
713 |
|
|
black line). The dotted line indicates the location of the ice as |
714 |
gezelter |
3967 |
determined by the tetrahedrality profile. This example is taken |
715 |
gezelter |
3970 |
from the basal-face simulation with an applied shear rate of 3.0 |
716 |
|
|
ms\textsuperscript{-1}.} |
717 |
plouden |
3942 |
\end{figure} |
718 |
|
|
|
719 |
|
|
|
720 |
gezelter |
3967 |
\begin{table}[h] |
721 |
|
|
\centering |
722 |
|
|
\caption{Solid-liquid friction coefficients (measured in amu~fs\textsuperscript{-1}) } |
723 |
|
|
\label{tab:lambda} |
724 |
|
|
\begin{tabular}{|ccc|} \hline |
725 |
|
|
& \multicolumn{2}{c|}{Drag direction} \\ |
726 |
|
|
Interface & $x$ & $y$ \\ \hline |
727 |
|
|
basal & $0.08 \pm 0.02$ & $0.09 \pm 0.03$ \\ |
728 |
|
|
prismatic & $0.037 \pm 0.008$ & $0.04 \pm 0.01$ \\ \hline |
729 |
|
|
\end{tabular} |
730 |
|
|
\end{table} |
731 |
|
|
|
732 |
|
|
|
733 |
gezelter |
3897 |
\section{Conclusion} |
734 |
gezelter |
3952 |
We have simulated the basal and prismatic facets of an SPC/E model of |
735 |
|
|
the ice I$_\mathrm{h}$ / water interface. Using VSS-RNEMD, the ice |
736 |
|
|
was sheared relative to the liquid while simultaneously being exposed |
737 |
|
|
to a weak thermal gradient which kept the interface at a stable |
738 |
|
|
temperature. Calculation of the local tetrahedrality order parameter |
739 |
|
|
has shown an apparent independence of the interfacial width on the |
740 |
gezelter |
3957 |
shear rate. This width was found to be 3.2~$\pm$0.4~\AA\ and |
741 |
|
|
3.6~$\pm$0.2~\AA\ for the basal and prismatic systems, respectively. |
742 |
gezelter |
3970 |
Using the 10\%-90\% width estimates common in previous studies, these |
743 |
|
|
interfacial widths are 7.0~$\pm$~0.9~\AA\ for the basal face, and |
744 |
|
|
7.9~$\pm$~0.4 \AA\ for the prismatic face. |
745 |
gezelter |
3897 |
|
746 |
gezelter |
3952 |
Orientational time correlation functions were calculated at varying |
747 |
|
|
displacements from the interface, and were found to be similarly |
748 |
|
|
independent of the applied momentum flux. The short decay due to the |
749 |
|
|
restoring forces of existing hydrogen bonds decreased close to the |
750 |
|
|
interface, while the longer-time decay constants increased in close |
751 |
|
|
proximity to the interface. There is also an apparent dynamic |
752 |
|
|
interface width, $d_{basal}$ and $d_{prismatic}$, at which these |
753 |
gezelter |
3954 |
deviations from bulk liquid values begin. We found $d_{basal}$ and |
754 |
gezelter |
3970 |
$d_{prismatic}$ to be approximately 2.8~\AA\ and 3.5~\AA\ . Using the |
755 |
|
|
10\%-90\% estimates common in previous simulation studies, the dynamic |
756 |
|
|
interface widths are $\sim$~6.3~\AA\ for the basal face and 7.8~\AA\ |
757 |
|
|
for the prismatic face. These interfacial widths are in good |
758 |
|
|
agreement with values determined by the structural analysis of the |
759 |
|
|
interface using the local tetrahedrality order parameter. |
760 |
gezelter |
3952 |
|
761 |
gezelter |
3954 |
The coefficient of liquid-solid friction for each of the facets was |
762 |
gezelter |
3970 |
also determined and were found to be essentially independent of shear |
763 |
|
|
direction relative to the surface orientation. We attribute the |
764 |
|
|
factor of two difference between the basal and prismatic friction |
765 |
|
|
coefficients to the corrugation of the prismatic face and the |
766 |
|
|
resulting reduction of effective surface area for interaction with the |
767 |
|
|
liquid. |
768 |
gezelter |
3952 |
|
769 |
gezelter |
3957 |
\section{Acknowledgements} |
770 |
gezelter |
3970 |
The authors thank the reviewers for helpful comments and suggestions. |
771 |
|
|
Support for this project was provided by the National Science |
772 |
|
|
Foundation under grant CHE-0848243. Computational time was provided by |
773 |
|
|
the Center for Research Computing (CRC) at the University of Notre |
774 |
|
|
Dame. |
775 |
gezelter |
3897 |
|
776 |
gezelter |
3914 |
\newpage |
777 |
plouden |
3909 |
\bibliography{iceWater} |
778 |
gezelter |
3897 |
|
779 |
gezelter |
3957 |
\end{doublespace} |
780 |
gezelter |
3897 |
|
781 |
gezelter |
3957 |
% \begin{tocentry} |
782 |
|
|
% \begin{wrapfigure}{l}{0.5\textwidth} |
783 |
|
|
% \begin{center} |
784 |
|
|
% \includegraphics[width=\linewidth]{SystemImage.png} |
785 |
|
|
% \end{center} |
786 |
|
|
% \end{wrapfigure} |
787 |
|
|
% The cell used to simulate liquid-solid shear in ice I$_\mathrm{h}$ / |
788 |
|
|
% water interfaces. Velocity gradients were applied using the velocity |
789 |
|
|
% shearing and scaling variant of reverse non-equilibrium molecular |
790 |
|
|
% dynamics (VSS-RNEMD) with a weak thermal gradient to prevent melting. |
791 |
|
|
% The interface width is relatively robust in both structual and dynamic |
792 |
|
|
% measures as a function of the applied shear. |
793 |
|
|
% \end{tocentry} |
794 |
|
|
|
795 |
gezelter |
3914 |
\end{document} |
796 |
gezelter |
3897 |
|
797 |
plouden |
3924 |
%************************************************************** |
798 |
|
|
%Non-mass weighted slopes (amu*Angstroms^-2 * fs^-1) |
799 |
|
|
% basal: slope=0.090677616, error in slope = 0.003691743 |
800 |
|
|
%prismatic: slope = 0.050101506, error in slope = 0.001348181 |
801 |
|
|
%Mass weighted slopes (Angstroms^-2 * fs^-1) |
802 |
|
|
%basal slope = 4.76598E-06, error in slope = 1.94037E-07 |
803 |
|
|
%prismatic slope = 3.23131E-06, error in slope = 8.69514E-08 |
804 |
|
|
%************************************************************** |