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\title{Solid-liquid friction at ice-I$_\mathrm{h}$ / water interfaces} |
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\begin{document} |
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\author{P. B. Louden} |
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu} |
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\affiliation[University of Notre Dame]{251 Nieuwland Science Hall\\ |
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Department of Chemistry and Biochemistry\\ University of Notre |
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Dame\\ Notre Dame, Indiana 46556} |
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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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|
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\keywords{} |
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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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\begin{document} |
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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 an ice I$_\mathrm{h}$ / water |
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interface when the solid phase is being drawn through the liquid |
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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 relative velocity of the ice |
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and liquid layers over a wide range of shear rates. Decays of |
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molecular orientational time correlation functions for gave very |
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similar estimates for the width of the interfaces, although the |
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short- and longer-time decay components of the orientational |
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correlation functions behave differently closer to the interface. |
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Although both facets of ice are in ``stick'' boundary conditions in |
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liquid water, the solid-liquid friction coefficient was found to be |
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different for the basal and prismatic facets of ice. |
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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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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, SPC/E, CF1, and TIP4P models for |
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water.\cite{Karim87,Karim88,Karim90,Hayward01,Bryk02,Hayward02,Gay02} |
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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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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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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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|
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\subsection{Stable ice I$_\mathrm{h}$ / water interfaces under shear} |
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|
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The structure of ice I$_\mathrm{h}$ is very well understood; it |
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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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the prismatic, $\{2~0~\bar{2}~1\}$, faces. Ice I$_\mathrm{h}$ is |
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normally proton disordered in bulk crystals, although the surfaces |
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probably have a preference for proton ordering along strips of |
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dangling H-atoms and Oxygen lone pairs.\cite{Buch:2008fk} |
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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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|
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For small simulated ice interfaces, it is useful to have a |
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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 useful to have |
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an orthorhombic (rectangular) box. A recent paper by Hirsch and |
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Ojam\"{a}e describes how to create proton-ordered bulk ice |
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I$_\mathrm{h}$ in alternative orthorhombic cells.\cite{Hirsch04} |
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|
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We are using Hirsch and Ojam\"{a}e's structure 6 which is an |
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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 mapping between |
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the Miller indices in the P$6_3/mmc$ crystal system and those in the |
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Hirsch and Ojam\"{a}e $P2_12_12_1$ system. |
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|
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\begin{wraptable}{r}{3.5in} |
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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 \protect\cite{Hirsch04}} |
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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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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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pryamidal & $\{2~0~\bar{2}~1\}$ & $\{2~0~1\}$ \\ \hline |
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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{wraptable} |
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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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|
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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 |
| 185 |
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version of ice I$_\mathrm{h}$. Table \ref{tab:equiv} contains a |
| 186 |
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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 $a = 4.49225$, $b = 7.78080$, $c = 7.33581$ and two water |
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molecules whose atoms reside at fractional coordinates given in table |
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\ref{tab:p212121}. To construct the basal and prismatic interfaces, |
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these crystallographic coordinates were used to construct an |
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orthorhombic unit cell which was then replicated in all three |
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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 $z$-dimension normal to the exposed face. |
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The block was then cut along two perpendicular directions in a way |
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that allowed for perfect periodic replication in the $x$ and $y$ axes, |
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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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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{wraptable}{r}{3.25in} |
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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 |
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\protect\cite{Hirsch04}} |
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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 & 0.75 & 0.1667 & 0.4375 \\ |
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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 & 0.25 & 0.6667 & 0.4375 \\ |
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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{wraptable} |
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\end{table} |
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|
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Although experimental solid/liquid coexistant temperature under normal |
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pressure are close to 273K, Haymet \emph{et al.} have done extensive |
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work on characterizing the ice/water |
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interface.\cite{Karim88,Karim90,Hayward01,Bryk02,Hayward02} They have |
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found that for the SPC/E water model,\cite{Berendsen87} which is also |
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used in this study, the ice/water interface is most stable at |
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225$\pm$5K.\cite{Bryk02} Therefore, we created our ice / water |
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interfaces, utilizing a box of liquid water that had the same |
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dimensions in $x$ and $y$ was equilibrated at 225 K and 1 atm of |
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pressure in the NPAT ensemble (with the $z$ axis allowed to fluctuate |
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to equilibrate to the correct pressure). The liquid and solid systems |
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were combined by carving out any water molecule from the liquid |
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simulation cell that was within 3 \AA\ of any atom in the ice slab. |
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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 |
| 234 |
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allowed to fluctuate to equilibrate to the correct pressure). The |
| 235 |
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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 24 \AA\ x 36 \AA\ x 99 \AA\ with 900 SPC/E molecules in the ice slab and 1846 SPC/E molecules in the liquid phase. Similarly, the prismatic system was constructed as 36 \AA\ x 36 \AA\ x 86 \AA\ with 1000 SPC/E molecules in the ice slab and 2684 SPC/E molecules in the liquid phase. |
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|
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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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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 along with the |
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momentum gradient. This can be accomplished using he velocity |
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it is necessary to apply a weak thermal gradient in combination with |
| 250 |
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the momentum gradient. This can be accomplished using the velocity |
| 251 |
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shearing and scaling (VSS) variant of reverse non-equilibrium |
| 252 |
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molecular dynamics (RNEMD), which utilizes a series of simultaneous |
| 253 |
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velocity exchanges between two regions within the simulation |
| 254 |
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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 phase |
| 255 |
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slab, while the other is centrally located in the liquid |
| 256 |
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region. VSS-RNEMD provides a set of conservation constraints for |
| 257 |
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simultaneously creating either a momentum flux or a thermal flux (or |
| 258 |
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both) between the two slabs. Satisfying the constraint equations |
| 259 |
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ensures that the new configurations are sampled from the same NVE |
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ensemble as before the VSS move. |
| 257 |
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creating either a momentum flux or a thermal flux (or both |
| 258 |
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simultaneously) between the two slabs. Satisfying the constraint |
| 259 |
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equations ensures that the new configurations are sampled from the |
| 260 |
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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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minimal, as is thermal anisotropy. This ability to generate |
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simultaneous thermal and shear fluxes has been previously utilized to |
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map out the shear viscosity of SPC/E water over a wide range of |
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temperatures (90~K) with a single 1 ns simulation.\cite{Kuang12} |
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temperatures (90~K) with a single 1~ns simulation.\cite{Kuang12} |
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|
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For this work, we are using the VSS-RNEMD method primarily to generate |
| 328 |
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a shear between the ice slab and the liquid phase, while using a weak |
| 329 |
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thermal gradient to maintaining the interface at the 225K target |
| 330 |
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value. This will insure minimal melting of the bulk ice phase and |
| 331 |
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allows us to control the exact temperature of the interface. |
| 329 |
> |
thermal gradient to maintain the interface at the 225~K target |
| 330 |
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value. This ensures minimal melting of the bulk ice phase and allows |
| 331 |
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us to control the exact temperature of the interface. |
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|
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\subsection{Computational Details} |
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All simulations were performed using OpenMD,\cite{OOPSE,openmd} with a |
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dimensions. Electrostatics were handled using the damped-shifted |
| 337 |
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force real-space electrostatic kernel.\cite{Ewald} The systems were |
| 338 |
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divided into 100 bins along the $z$-axis for the VSS-RNEMD moves, |
| 339 |
< |
which were attempted every 50 fs. |
| 339 |
> |
which were attempted every 50~fs. |
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|
| 341 |
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The interfaces were equilibrated for a total of 10 ns at equilibrium |
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conditions before being exposed to either a shear or thermal gradient. |
| 343 |
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This consisted of 5 ns under a constant temperature (NVT) integrator |
| 344 |
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set to 225K followed by 5 ns under a microcanonical integrator. Weak |
| 345 |
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thermal gradients were allowed to develop using the VSS-RNEMD (NVE) |
| 346 |
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integrator using a a small thermal flux ($-2.0\times 10^{-6}$ |
| 346 |
> |
integrator using a small thermal flux ($-2.0\times 10^{-6}$ |
| 347 |
|
kcal/mol/\AA$^2$/fs) for a duration of 5 ns to allow the gradient to |
| 348 |
< |
stabilize. The resulting temperature gradient was less than 5K over |
| 349 |
< |
the entire 1 nm box length, which was sufficient to keep the |
| 348 |
> |
stabilize. The resulting temperature gradient was $\approx$ 10K over |
| 349 |
> |
the entire 100 \AA\ box length, which was sufficient to keep the |
| 350 |
|
temperature at the interface within $\pm 1$ K of the 225K target. |
| 351 |
|
|
| 352 |
|
Velocity gradients were then imposed using the VSS-RNEMD (NVE) |
| 353 |
|
integrator with a range of momentum fluxes. These gradients were |
| 354 |
< |
allowed to stabilize for 1 ns before data collection began. Once |
| 355 |
< |
established, four successive 0.5 ns runs were performed for each shear |
| 354 |
> |
allowed to stabilize for 1~ns before data collection began. Once |
| 355 |
> |
established, four successive 0.5~ns runs were performed for each shear |
| 356 |
|
rate. During these simulations, snapshots of the system were taken |
| 357 |
< |
every 1 ps, and statistics on the structure and dynamics in each bin |
| 357 |
> |
every 1~ps, and statistics on the structure and dynamics in each bin |
| 358 |
|
were accumulated throughout the simulations. |
| 359 |
|
|
| 360 |
|
\section{Results and discussion} |
| 361 |
|
|
| 362 |
< |
\subsection{Measuring the Width of the Interface} |
| 363 |
< |
Any order parameter or correlation function that varies across the |
| 364 |
< |
interface from a bulk liquid to a solid can be used as a measure of |
| 365 |
< |
the width of the interface. However, because VSS-RNEMD imposes a |
| 366 |
< |
lateral flow, parameters that depend on translational motion of the |
| 367 |
< |
molecules (e.g. the diffusion constant) may be artifically skewed by |
| 368 |
< |
the RNEMD moves. A structural parameter like a radial distribution |
| 369 |
< |
function is not influenced by the RNEMD perturbations to the same |
| 370 |
< |
degree. Here, we have used the local tetraherdal order parameter as |
| 371 |
< |
described by Kumar\cite{Kumar09} and Errington\cite{Errington01} as a |
| 362 |
> |
\subsection{Interfacial width} |
| 363 |
> |
Any order parameter or time correlation function that changes as one |
| 364 |
> |
crosses an interface from a bulk liquid to a solid can be used to |
| 365 |
> |
measure the width of the interface. In previous work on the ice/water |
| 366 |
> |
interface, Haymet {\it et al.}\cite{Bryk02} have utilized structural |
| 367 |
> |
features (including the density) as well as dynamic properties |
| 368 |
> |
(including the diffusion constant) to estimate the width of the |
| 369 |
> |
interfaces for a number of facets of the ice crystals. Because |
| 370 |
> |
VSS-RNEMD imposes a lateral flow, parameters that depend on |
| 371 |
> |
translational motion of the molecules (e.g. diffusion) may be |
| 372 |
> |
artificially skewed by the RNEMD moves. A structural parameter is not |
| 373 |
> |
influenced by the RNEMD perturbations to the same degree. Here, we |
| 374 |
> |
have used the local tetrahedral order parameter as described by |
| 375 |
> |
Kumar\cite{Kumar09} and Errington\cite{Errington01} as our principal |
| 376 |
|
measure of the interfacial width. |
| 377 |
|
|
| 378 |
|
The local tetrahedral order parameter, $q(z)$, is given by |
| 379 |
|
\begin{equation} |
| 380 |
< |
q(z) \equiv \int_0^L \sum_{k=1}^{N} \Bigg(1 -\frac{3}{8}\sum_{i=1}^{3} |
| 381 |
< |
\sum_{j=i+1}^{4} \bigg[\cos\psi_{ikj}+\frac{1}{3}\bigg]^2\Bigg) |
| 382 |
< |
\delta(z_{k}-z)\mathrm{d}z |
| 380 |
> |
q(z) = \int_0^L \sum_{k=1}^{N} \Bigg(1 -\frac{3}{8}\sum_{i=1}^{3} |
| 381 |
> |
\sum_{j=i+1}^{4} \bigg(\cos\psi_{ikj}+\frac{1}{3}\bigg)^2\Bigg) |
| 382 |
> |
\delta(z_{k}-z)\mathrm{d}z \Bigg/ N_z |
| 383 |
|
\label{eq:qz} |
| 384 |
|
\end{equation} |
| 385 |
|
where $\psi_{ikj}$ is the angle formed between the oxygen site on |
| 386 |
< |
molecule $k$, and the oxygen sites on its two closest neighbors, |
| 387 |
< |
molecules $i$ and $j$. The local tetrahedral order parameter function |
| 388 |
< |
has a range of (0,1), where the larger the value $q$ has the more |
| 389 |
< |
tetrahedral the ordering of the local environment is. A $q$ value of |
| 390 |
< |
one describes a perfectly tetrahedral environment relative to it and |
| 391 |
< |
its four nearest neighbors, and the parameter's value decreases as the |
| 392 |
< |
local ordering becomes less tetrahedral. Equation \ref{eq:qz} |
| 393 |
< |
describes a $z$-binned tetrahedral order parameter in which the $z$ |
| 394 |
< |
coordinate of the central molecule is used to give a spatial |
| 395 |
< |
description of the local orientational ordering. |
| 386 |
> |
central molecule $k$, and the oxygen sites on two of the four closest |
| 387 |
> |
molecules, $i$ and $j$. Molecules $i$ and $j$ are further restricted |
| 388 |
> |
to lie within the first solvation shell of molecule $k$. $N_z = \int |
| 389 |
> |
\delta(z_k - z) \mathrm{d}z$ is a normalization factor to account for |
| 390 |
> |
the varying population of molecules within each finite-width bin. The |
| 391 |
> |
local tetrahedral order parameter has a range of $(0,1)$, where the |
| 392 |
> |
larger values of $q$ indicate a larger degree of tetrahedral ordering |
| 393 |
> |
of the local environment. In perfect ice I$_\mathrm{h}$ structures, |
| 394 |
> |
the parameter can approach 1 at low temperatures, while in liquid |
| 395 |
> |
water, the ordering is significantly less tetrahedral, and values of |
| 396 |
> |
$q(z) \approx 0.75$ are more common. |
| 397 |
|
|
| 398 |
< |
The system was divided into 100 bins along the $z$-dimension, and a |
| 399 |
< |
cutoff radius for the neighboring molecules was set to 3.41 \AA\ . |
| 400 |
< |
The $q_{z}$ values for each snapshot were then averaged to give a |
| 401 |
< |
tetrahedrality profile of the system about the $z$-dimension. The |
| 402 |
< |
profile was then fit with a hyperbolic tangent function given by |
| 403 |
< |
|
| 398 |
> |
To estimate the interfacial width, the system was divided into 100 |
| 399 |
> |
bins along the $z$-dimension, and a cutoff radius for the first |
| 400 |
> |
solvation shell was set to 3.41~\AA\ . The $q_{z}$ function was |
| 401 |
> |
time-averaged to give yield a tetrahedrality profile of the |
| 402 |
> |
system. The profile was then fit to a hyperbolic tangent that smoothly |
| 403 |
> |
links the liquid and solid states, |
| 404 |
|
\begin{equation}\label{tet_fit} |
| 405 |
< |
q_{z} \approx q_{liq}+\frac{q_{ice}-q_{liq}}{2}\Bigg[\tanh\bigg(\frac{z-I_{L,m}}{w}\bigg)-\tanh\bigg(\frac{z-I_{R,m}}{w}\bigg)\Bigg]+\beta|(z-z_{mid})| |
| 405 |
> |
q(z) \approx |
| 406 |
> |
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- |
| 407 |
> |
\frac{r+l}{2}\right|. |
| 408 |
|
\end{equation} |
| 409 |
+ |
Here $q_{liq}$ and $q_{ice}$ are the local tetrahedral order parameter |
| 410 |
+ |
for the bulk liquid and ice domains, respectively, $w$ is the width of |
| 411 |
+ |
the interface. $l$ and $r$ are the midpoints of the left and right |
| 412 |
+ |
interfaces, respectively. The last term in eq. \eqref{tet_fit} |
| 413 |
+ |
accounts for the influence that the weak thermal gradient has on the |
| 414 |
+ |
tetrahedrality profile in the liquid region. To estimate the |
| 415 |
+ |
10\%-90\% widths commonly used in previous studies,\cite{Bryk02} it is |
| 416 |
+ |
a simple matter to scale the widths obtained from the hyperbolic |
| 417 |
+ |
tangent fits to obtain $w_{10-90} = 2.1971 \times w$.\cite{Bryk02} |
| 418 |
|
|
| 419 |
< |
where $q_{liq}$ and $q_{ice}$ are fitting parameters for the local tetrahedral order parameter for the liquid and ice, $w$ is the width of the interface, and $I_{L,m}$ and $I_{R,m}$ are the midpoints of the left and right interface. The last term in \eqref{tet_fit} accounts for the influence the thermal gradient has on the tetrahedrality profile in the liquid region; where $\beta$ is a fitting parameter and $z_{mid}$ is the midpoint of the $z$-dimension of the simulation box. |
| 420 |
< |
|
| 421 |
< |
In Figures \ref{fig:bComic} and \ref{fig:pComic} we see the $z$-dimensional profiles for several components of the basal and prismatic systems. In panel (a) of the figures we see the tetrahedrality profile of the systems (black circles). In the liquid region of the system, the local tetrahedral order parameter is approximately 0.75 while in the solid region the parameter is approximately 0.94, indicating a more tetrahedral structure of the water molecules. The hyperbolic tangent function used to fit the tetrahedrality profiles is in red and the verticle dotted lines denote the midpoint of the interfaces. The weak thermal gradient applied to the systems in order to keep the interface at a stable temperature, 225$\pm$5K, can be seen in panel (b). Lastly, the velocity gradient across the systems can be seen in panel (c). From panel (c), we can see liquid phase water molecules 10 \AA\ to 15 \AA\ from the interfaces are being dragged along with the ice block, indicating that the shearing of ice water is in the stick boundary condition. |
| 419 |
> |
In Figures \ref{fig:bComic} and \ref{fig:pComic} we see the |
| 420 |
> |
$z$-coordinate profiles for tetrahedrality, temperature, and the |
| 421 |
> |
$x$-component of the velocity for the basal and prismatic interfaces. |
| 422 |
> |
The lower panels show the $q(z)$ (black circles) along with the |
| 423 |
> |
hyperbolic tangent fits (red lines). In the liquid region, the local |
| 424 |
> |
tetrahedral order parameter, $q(z) \approx 0.75$ while in the |
| 425 |
> |
crystalline region, $q(z) \approx 0.94$, indicating a more tetrahedral |
| 426 |
> |
environment. The vertical dotted lines denote the midpoint of the |
| 427 |
> |
interfaces ($r$ and $l$ in eq. \eqref{tet_fit}). The weak thermal |
| 428 |
> |
gradient applied to the systems in order to keep the interface at |
| 429 |
> |
225~$\pm$~5K, can be seen in middle panels. The transverse velocity |
| 430 |
> |
profile is shown in the upper panels. It is clear from the upper |
| 431 |
> |
panels that water molecules in close proximity to the surface (i.e. |
| 432 |
> |
within 10~\AA\ to 15~\AA~of the interfaces) have transverse |
| 433 |
> |
velocities quite close to the velocities within the ice block. There |
| 434 |
> |
is no velocity discontinuity at the interface, which indicates that |
| 435 |
> |
the shearing of ice/water interfaces occurs in the ``stick'' or |
| 436 |
> |
no-slip boundary conditions. |
| 437 |
|
|
| 438 |
|
\begin{figure} |
| 439 |
|
\includegraphics[width=\linewidth]{bComicStrip} |
| 440 |
< |
\caption{\label{fig:bComic} The basal system: (a) The local tetrahedral order parameter,$q$, (black circles) fit by a hyperbolic tangent (red line), (b) the thermal gradient imposed on the system to maintain a stable interfacial temperature, and (c) the velocity gradient imposed on the system. The verticle dotted lines indicate the midpoint of the interfaces.} |
| 440 |
> |
\caption{\label{fig:bComic} The basal interfaces. Lower panel: the |
| 441 |
> |
local tetrahedral order parameter, $q(z)$, (black circles) and the |
| 442 |
> |
hyperbolic tangent fit (red line). Middle panel: the imposed |
| 443 |
> |
thermal gradient required to maintain a fixed interfacial |
| 444 |
> |
temperature. Upper panel: the transverse velocity gradient that |
| 445 |
> |
develops in response to an imposed momentum flux. The vertical |
| 446 |
> |
dotted lines indicate the locations of the midpoints of the two |
| 447 |
> |
interfaces.} |
| 448 |
|
\end{figure} |
| 449 |
|
|
| 450 |
|
\begin{figure} |
| 451 |
|
\includegraphics[width=\linewidth]{pComicStrip} |
| 452 |
< |
\caption{\label{fig:pComic} The prismatic system: (a) The local tetrahedral order parameter,$q$, (black circles) fit by a hyperbolic tangent (red line), (b) the thermal gradient imposed on the system to maintain a stable interfacial temperature, and (c) the velocity gradient imposed on the system.The verticle dotted lines indicate the midpoint of the interfaces.} |
| 452 |
> |
\caption{\label{fig:pComic} The prismatic interfaces. Panel |
| 453 |
> |
descriptions match those in figure \ref{fig:bComic}} |
| 454 |
|
\end{figure} |
| 455 |
|
|
| 456 |
< |
From the tetrahedrality fits, we found the interfacial width for the basal and prismatic systems to be 3.2$\pm$0.4 \AA\ and 3.6$\pm$0.2 \AA\ with no applied momentum flux. Over the range of shear rates investigated, 0.6$\pm$0.3 ms\textsuperscript{-1} to 5.3$\pm$0.5 ms\textsuperscript{-1} for the basal system and 0.9$\pm$0.2 ms\textsuperscript{-1} to 4.5$\pm$0.1 ms\textsuperscript{-1} for the prismatic, there was no appreciable change in the interface width found. The calculated values for the interfacial width over all shear rates investigated contained the control values within their error bars. |
| 456 |
> |
From the fits using eq. \eqref{tet_fit}, we find the interfacial |
| 457 |
> |
width for the basal and prismatic systems to be 3.2~$\pm$~0.4~\AA\ and |
| 458 |
> |
3.6~$\pm$~0.2~\AA\ , respectively, with no applied momentum flux. Over |
| 459 |
> |
the range of shear rates investigated, $0.6 \pm 0.3 \mathrm{ms}^{-1} |
| 460 |
> |
\rightarrow 5.3 \pm 0.5 \mathrm{ms}^{-1}$ for the basal system and |
| 461 |
> |
$0.9 \pm 0.2 \mathrm{ms}^{-1} \rightarrow 4.5 \pm 0.1 |
| 462 |
> |
\mathrm{ms}^{-1}$ for the prismatic, we found no appreciable change in |
| 463 |
> |
the interface width. The fit values for the interfacial width ($w$) |
| 464 |
> |
over all shear rates contained the values reported above within their |
| 465 |
> |
error bars. |
| 466 |
|
|
| 467 |
< |
\subsubsection{Orientational Time Correlation Function} |
| 468 |
< |
The orientational time correlation function (OTCF) gives insight of the local environment of molecules. The rate at which the function decays corresponds to how hindered the motions of a molecule are. The more hindered a molecules motion is the slower the function will decay, and the function decays more rapidly for molecules with less constrained motions. |
| 467 |
> |
\subsubsection{Orientational Dynamics} |
| 468 |
> |
The orientational time correlation function, |
| 469 |
|
\begin{equation}\label{C(t)1} |
| 470 |
< |
C_{2}(t)=\langle P_{2}(\mathbf{v}_{i}(t)\mathbf{v}_{i}(t=0))\rangle |
| 470 |
> |
C_{2}(t)=\langle P_{2}(\mathbf{u}(0) \cdot \mathbf{u}(t)) \rangle, |
| 471 |
|
\end{equation} |
| 472 |
< |
In eq. \eqref{C(t)1}, $P_{2}$ is the Legendre polynomial of the second order and $\mathbf{v}_{i}$ is the bisecting unit vector of the $i$th water molecule in the lab frame. |
| 472 |
> |
gives insight into the local dynamic environment around the water |
| 473 |
> |
molecules. The rate at which the function decays provides information |
| 474 |
> |
about hindered motions and the timescales for relaxation. In |
| 475 |
> |
eq. \eqref{C(t)1}, $P_{2}$ is the second-order Legendre polynomial, |
| 476 |
> |
the vector $\mathbf{u}$ is often taken as HOH bisector, although |
| 477 |
> |
slightly different behavior can be observed when $\mathbf{u}$ is the |
| 478 |
> |
vector along one of the OH bonds. The angle brackets denote an |
| 479 |
> |
ensemble average over all water molecules in a given spatial region. |
| 480 |
|
|
| 481 |
< |
Here, we are evaluating this function across the $z$-dimension of the system as another measure of the change in the local environment and behavior of water molecules from the liquid region to the slushy interfacial region. After each of the 0.5 ns simulations, the systems were run for an additional 200 ps where the positions of every molecule in the system were recorded every 0.1 ps. The systems were then divided into 30 bins and the OTCF was evaluated for each bin. |
| 481 |
> |
To investigate the dynamic behavior of water at the ice interfaces, we |
| 482 |
> |
have computed $C_{2}(z,t)$ for molecules that are present within a |
| 483 |
> |
particular slab along the $z$- axis at the initial time. The change |
| 484 |
> |
in the decay behavior as a function of the $z$ coordinate is another |
| 485 |
> |
measure of the change of how the local environment changes across the |
| 486 |
> |
ice/water interface. To compute these correlation functions, each of |
| 487 |
> |
the 0.5 ns simulations was followed by a shorter 200 ps microcanonical |
| 488 |
> |
(NVE) simulation in which the positions and orientations of every |
| 489 |
> |
molecule in the system were recorded every 0.1 ps. The systems were |
| 490 |
> |
then divided into 30 bins along the $z$-axis and $C_2(t)$ was |
| 491 |
> |
evaluated for each bin. |
| 492 |
|
|
| 493 |
< |
It has been shown that the OTCF for water can be fit by a triexponential decay\cite{Furse08}, where the three components of the decay correspond to a fast (<200 fs) reorientational piece driven by the restoring forces of existing hydrogen bonds, a middle (on the order of several ps) piece describing the large angle jumps that occur during the breaking and formation of new hydrogen bonds\cite{Laage08,Laage11}, and a slow (on the order of tens of ps) contribution describing the translational motion of the molecules. The OTCF data for each bin were truncated at 100 ps, and fit to the triexponential decay |
| 493 |
> |
In simulations of water at biological interfaces, Furse {\em et al.} |
| 494 |
> |
fit $C_2(t)$ functions for water with triexponential |
| 495 |
> |
functions,\cite{Furse08} where the three components of the decay |
| 496 |
> |
correspond to a fast ($<$200 fs) reorientational piece driven by the |
| 497 |
> |
restoring forces of existing hydrogen bonds, a middle (on the order of |
| 498 |
> |
several ps) piece describing the large angle jumps that occur during |
| 499 |
> |
the breaking and formation of new hydrogen bonds,and a slow (on the |
| 500 |
> |
order of tens of ps) contribution describing the translational motion |
| 501 |
> |
of the molecules. The model for orientational decay presented |
| 502 |
> |
recently by Laage and Hynes {\em et al.}\cite{Laage08,Laage11} also |
| 503 |
> |
includes three similar decay constants, although two of the time |
| 504 |
> |
constants are linked, and the resulting decay curve has two parameters |
| 505 |
> |
governing the dynamics of decay. |
| 506 |
> |
|
| 507 |
> |
In our ice/water interfaces, we are at substantially lower |
| 508 |
> |
temperatures, and the water molecules are further perturbed by the |
| 509 |
> |
presence of the ice phase nearby. We have obtained the most |
| 510 |
> |
reasonable fits using triexponential functions with three distinct |
| 511 |
> |
time domains, as well as a constant piece to account for the water |
| 512 |
> |
stuck in the ice phase that does not experience any long-time |
| 513 |
> |
orientational decay, |
| 514 |
|
\begin{equation} |
| 515 |
< |
C_{2}(t) \approx a_{1}e^{-t/\tau_{short}}+a_{2}e^{-t/\tau_{middle}}+a_{3}e^{-t/\tau_{long}}+a_{4} |
| 515 |
> |
C_{2}(t) \approx a e^{-t/\tau_\mathrm{short}} + b e^{-t/\tau_\mathrm{middle}} + c |
| 516 |
> |
e^{-t/\tau_\mathrm{long}} + (1-a-b-c) |
| 517 |
|
\end{equation} |
| 518 |
< |
where $a_{1}+a_{2}+a_{3}+a_{4}=1$. An average value and standard deviation for each $\tau$ was obtained for each bin from the four runs. Lastly, the means and standard deviations were averaged about the center of the system. |
| 518 |
> |
Average values for the three decay constants (and error estimates) |
| 519 |
> |
were obtained for each bin. In figures \ref{fig:basal_Tau_comic_strip} |
| 520 |
> |
and \ref{fig:prismatic_Tau_comic_strip}, the three orientational decay |
| 521 |
> |
times are shown as a function of distance from the center of the ice |
| 522 |
> |
slab. |
| 523 |
|
|
| 524 |
|
\begin{figure} |
| 525 |
|
\includegraphics[width=\linewidth]{basal_Tau_comic_strip} |
| 526 |
< |
\caption{\label{fig:basal_Tau_comic_strip} The orientational time correlation function for the basal system fit by the triexponential decay $a_{1}e^{-t/\tau_{short}}+a_{2}e^{-t/\tau_{middle}}+a_{3}e^{-t/\tau_{long}}+a_{4}$ where $a_{1}+a_{2}+a_{3}+a_{4}=1$. The verticle dotted line indicates the average midpoint of the interface as determined by the tetrahedrality fit. In (b) and (c) $\tau_{middle}$ and $\tau_{long}$ have a consistent value of about 5.5 ps and 50 ps in the liquid region, and increase in value approaching the interface. $\tau_{middle}$ corresponds to the breaking and making of hydrogen bonds as explained by extended jump model proposed by Laage and Hynes\cite{Laage08,Laage11} and $\tau_{long}$ relates to the translational motion of the molecules. In (a), we see that $\tau_{short}$ has a value of about 71 fs in the liquid region, and decreases in value approaching the interface. This component of the decay corresponds to the reorientational forces of existing hydrogen bonds on the inertial rotational of the molecules. Thus $\tau_{short}$ decreases approaching the interface due to the hindered range of motion of the molecules. } |
| 526 |
> |
\caption{\label{fig:basal_Tau_comic_strip} The three decay constants |
| 527 |
> |
of the orientational time correlation function, $C_2(t)$, for water |
| 528 |
> |
as a function of distance from the center of the ice slab. The |
| 529 |
> |
dashed line indicates the location of the basal face (as determined |
| 530 |
> |
from the tetrahedrality order parameter). The moderate and long |
| 531 |
> |
time contributions slow down close to the interface which would be |
| 532 |
> |
expected under reorganizations that involve large motions of the |
| 533 |
> |
molecules (e.g. frame-reorientations and jumps). The observed |
| 534 |
> |
speed-up in the short time contribution is surprising, but appears |
| 535 |
> |
to reflect the restricted motion of librations closer to the |
| 536 |
> |
interface.} |
| 537 |
|
\end{figure} |
| 538 |
|
|
| 539 |
|
\begin{figure} |
| 540 |
|
\includegraphics[width=\linewidth]{prismatic_Tau_comic_strip} |
| 541 |
< |
\caption{\label{fig:prismatic_Tau_comic_strip} The orientational time correlation function for the prismatic system fit by the triexponential decay $a_{1}e^{-t/\tau_{short}}+a_{2}e^{-t/\tau_{middle}}+a_{3}e^{-t/\tau_{long}}+a_{4}$ where $a_{1}+a_{2}+a_{3}+a_{4}=1$. The verticle dotted line indicates the average midpoint of the interface as determined by the tetrahedrality fit. In (b) and (c) $\tau_{middle}$ and $\tau_{long}$ have a consistent value of about 3.5 ps and 30 ps in the liquid region, and increase in value approaching the interface. $\tau_{middle}$ corresponds to the breaking and making of hydrogen bonds as explained by extended jump model proposed by Laage and Hynes\cite{Laage08,Laage11} and $\tau_{long}$ relates to the translational motion of the molecules. In (a), we see that $\tau_{short}$ has a value of about 73 fs in the liquid region, and decreases in value approaching the interface. This component of the decay corresponds to the reorientational forces of existing hydrogen bonds on the inertial rotational of the molecules. Thus $\tau_{short}$ decreases approaching the interface due to the hindered range of motion of the molecules.} |
| 541 |
> |
\caption{\label{fig:prismatic_Tau_comic_strip} |
| 542 |
> |
Decay constants for $C_2(t)$ at the prismatic interface. Panel |
| 543 |
> |
descriptions match those in figure \ref{fig:basal_Tau_comic_strip}.} |
| 544 |
|
\end{figure} |
| 545 |
|
|
| 546 |
< |
Figures \ref{fig:basal_Tau_comic_strip} and \ref{fig:prismatic_Tau_comic_strip} plot the decomposition of the OTCF at varying displacements from the center of the ice for the basal and prismatic systems. We see in (a) $\tau_{short}$, (b) $\tau_{middle}$, and (c) $\tau_{long}$ for the control system (no applied momentum flux) in black, and a system with a large shear rate in red. The verticle dotted lines at a displacement of about 17 \AA\ and 9 \AA\ denote the midpoints of the interfaces as determined by the hyperbolic tangent fit of the tetrahedrality profile. |
| 546 |
> |
Figures \ref{fig:basal_Tau_comic_strip} and |
| 547 |
> |
\ref{fig:prismatic_Tau_comic_strip} show the three decay constants for |
| 548 |
> |
the orientational time correlation function for water at varying |
| 549 |
> |
displacements from the center of the ice slab for both the basal and |
| 550 |
> |
prismatic interfaces. The vertical dotted lines indicate the |
| 551 |
> |
locations of the midpoints of the interfaces as determined by the |
| 552 |
> |
tetrahedrality fits. In the liquid regions, $\tau_{middle}$ and |
| 553 |
> |
$\tau_{long}$ have consistent values around 3-4 ps and 20-40 ps, |
| 554 |
> |
respectively, and increase in value approaching the interface. |
| 555 |
> |
According to the jump model of Laage and Hynes {\em et |
| 556 |
> |
al.},\cite{Laage08,Laage11} $\tau_{middle}$ corresponds to the |
| 557 |
> |
breaking and making of hydrogen bonds and $\tau_{long}$ is explained |
| 558 |
> |
with translational motion of the molecules (i.e. frame reorientation). |
| 559 |
> |
The shortest of the three decay constants, the librational time |
| 560 |
> |
$\tau_\mathrm{short}$ has a value of about 70 fs in the liquid region, |
| 561 |
> |
and decreases in value approaching the interface. The observed |
| 562 |
> |
speed-up in the short time contribution is surprising, but appears to |
| 563 |
> |
reflect the restricted motion of librations closer to the interface. |
| 564 |
|
|
| 565 |
< |
In panels (a), we see at large displacements from the center of the ice $\tau_{short}$ for the basal system has a value of about 71 fs and 72 fs for the prismatic. Decreasing in displacement from about 26 \AA\ to about 19 \AA\ in the basal system, the value of $\tau_{short}$ decreases to about 63 fs. Likewise, $\tau_{short}$ decreases to about 63 fs from roughly 20 \AA\ to 12 \AA\ . This is due to the increasingly constrained motion of the water molecules as we approach the interface. In panels (b), $\tau_{middle}$ at large displacements from the ice has a value of about 5.5 ps and 3 ps for the basal and prismatic systems. We find $\tau_{middle}$ increases in value as we approach the interface in both cases. This component of the decay corresponds to the rearrangement of the hydrogen bonding network, which takes longer as the molecules motion becomes more constrained. In panels (c), $\tau_{long}$ has a value of about 50 ps for the basal system and roughly 30 ps for the prismatic system at large displacements from the interface. Similar to $\tau_{middle}$, $\tau_{long}$ also increases in value as we approach the interface for both systems. It is also apparent from Figures \ref{fig:basal_Tau_comic_strip} and \ref{fig:prismatic_Tau_comic_strip} that shearing the ice water has no effect on the orientational decay time, or on any of its decomposed components. |
| 565 |
> |
The control systems (with no applied momentum flux) are shown with |
| 566 |
> |
black symbols in figs. \ref{fig:basal_Tau_comic_strip} and |
| 567 |
> |
\ref{fig:prismatic_Tau_comic_strip}, while those obtained while a |
| 568 |
> |
shear was active are shown in red. |
| 569 |
|
|
| 570 |
< |
For each system, there is an apparent approximate value for $\tau_{short}$, $\tau_{middle}$, and $\tau_{long}$ at large displacements from the interface. There also appears to be a single displacement, $d_{basal}$ and $d_{prismatic}$, from the interface at which all three decay times begin to deviate from their bulk liquid values. We found $d_{basal}$ and $d_{prismatic}$ to be roughly 15 \AA\ and 8 \AA\ respectively. These two results indicate that the dynamics of the water molecules within $d_{basal}$ and $d_{prismatic}$ are being significantly perturbed by the ice and/or the interface, even though the structural width of the interface by analysis of the tetrahedrality profile indicates that bulk liquid structure of water is recovered after about 4 \AA\ from the edge of the ice. |
| 570 |
> |
Two notable features deserve clarification. First, there are |
| 571 |
> |
nearly-constant liquid-state values for $\tau_{short}$, |
| 572 |
> |
$\tau_{middle}$, and $\tau_{long}$ at large displacements from the |
| 573 |
> |
interface. Second, there appears to be a single distance, $d_{basal}$ |
| 574 |
> |
or $d_{prismatic}$, from the interface at which all three decay times |
| 575 |
> |
begin to deviate from their bulk liquid values. We find these |
| 576 |
> |
distances to be approximately 15~\AA\ and 8~\AA\, respectively, |
| 577 |
> |
although significantly finer binning of the $C_2(t)$ data would be |
| 578 |
> |
necessary to provide better estimates of a ``dynamic'' interfacial |
| 579 |
> |
thickness. |
| 580 |
|
|
| 581 |
< |
Beaglehole and Wilson have measured the ice/water interface to have a thickness approximately 10 \AA\ for both the basal and prismatic face of ice by ellipticity measurements \cite{Beaglehole93}. Structurally, we have found the basal and prismatic interfacial width to be 3.2$\pm$0.4 \AA\ and 3.6$\pm$0.2 \AA\ . However, we have shown through decomposition of the OTCF a much larger interfacial region exists in which the dynamics of the water molecules behave differently than those of the bulk liquid. |
| 581 |
> |
Beaglehole and Wilson have measured the ice/water interface using |
| 582 |
> |
ellipsometry and find a thickness of approximately 10~\AA\ for both |
| 583 |
> |
the basal and prismatic faces.\cite{Beaglehole93} Structurally, we |
| 584 |
> |
have found the basal and prismatic interfacial width to be |
| 585 |
> |
3.2~$\pm$~0.4~\AA\ and 3.6~$\pm$~0.2~\AA. However, decomposition of |
| 586 |
> |
the spatial dependence of the decay times of $C_2(t)$ indicates that a |
| 587 |
> |
somewhat thicker interfacial region exists in which the orientational |
| 588 |
> |
dynamics of the water molecules begin to resemble the trapped |
| 589 |
> |
interfacial water more than the surrounding liquid. |
| 590 |
|
|
| 591 |
+ |
Our results indicate that the dynamics of the water molecules within |
| 592 |
+ |
$d_{basal}$ and $d_{prismatic}$ are being significantly perturbed by |
| 593 |
+ |
the interface, even though the structural width of the interface via |
| 594 |
+ |
analysis of the tetrahedrality profile indicates that bulk liquid |
| 595 |
+ |
structure of water is recovered after about 4 \AA\ from the edge of |
| 596 |
+ |
the ice. |
| 597 |
+ |
|
| 598 |
|
\subsection{Coefficient of Friction of the Interface} |
| 599 |
< |
As the ice is sheared through the liquid, there will be a friction between the solid and the liquid. Pit has shown how to calculate the coefficient of friction $\lambda$ for a solid-liquid interface for a Newtonian fluid of viscosity $\eta$ and has a slip length of $\delta$. \cite{Pit99} |
| 599 |
> |
As liquid water flows over an ice interface, there is a distance from |
| 600 |
> |
the structural interface where bulk-like hydrodynamics are recovered. |
| 601 |
> |
Bocquet and Barrat constructed a theory for the hydrodynamic boundary |
| 602 |
> |
parameters, which include the slipping length |
| 603 |
> |
$\left(\delta_\mathrm{wall}\right)$ of this boundary layer and the |
| 604 |
> |
``hydrodynamic position'' of the boundary |
| 605 |
> |
$\left(z_\mathrm{wall}\right)$.\cite{PhysRevLett.70.2726,PhysRevE.49.3079} |
| 606 |
> |
This last parameter is the location (relative to a solid surface) |
| 607 |
> |
where the bulk-like behavior is recovered. Work by Mundy {\it et al.} |
| 608 |
> |
has helped to combine these parameters into a liquid-solid friction |
| 609 |
> |
coefficient, which quantifies the resistance to pulling the solid |
| 610 |
> |
interface through a liquid,\cite{Mundy1997305} |
| 611 |
> |
\begin{equation} |
| 612 |
> |
\lambda_\mathrm{wall} = \frac{\eta}{\delta_\mathrm{wall}}. |
| 613 |
> |
\end{equation} |
| 614 |
> |
This expression is nearly identical to one provided by Pit {\it et |
| 615 |
> |
al.} for the solid-liquid friction of an interface,\cite{Pit99} |
| 616 |
|
\begin{equation}\label{Pit} |
| 617 |
< |
\lambda=\eta/\delta |
| 617 |
> |
\lambda=\frac{\eta}{\delta} |
| 618 |
|
\end{equation} |
| 619 |
< |
From linear response theory, $\eta$ can be obtained from the imposed momentum flux and the slope of the velocity about the dimension of the imposed flux.\cite{Kuang12} |
| 619 |
> |
where $\delta$ is the slip length for the liquid measured at the |
| 620 |
> |
location of the interface itself. |
| 621 |
> |
|
| 622 |
> |
In both of these expressions, $\eta$ is the shear viscosity of the |
| 623 |
> |
bulk-like region of the liquid, a quantity which is easily obtained in |
| 624 |
> |
VSS-RNEMD simulations by fitting the velocity profile in the region |
| 625 |
> |
far from the surface.\cite{Kuang12} Assuming linear response in the |
| 626 |
> |
bulk-like region, |
| 627 |
|
\begin{equation}\label{Kuang} |
| 628 |
< |
j_{z}(p_{x})=-\eta\frac{\partial v_{x}}{\partial z} |
| 628 |
> |
j_{z}(p_{x})=-\eta \left(\frac{\partial v_{x}}{\partial z}\right) |
| 629 |
|
\end{equation} |
| 630 |
< |
Solving eq. \eqref{Kuang} for $\eta$ and substituting the result into eq. \eqref{Pit}, we obtain an alternate expression for the coefficient of friction. |
| 630 |
> |
Substituting this result into eq. \eqref{Pit}, we can estimate the |
| 631 |
> |
solid-liquid coefficient using the slip length, |
| 632 |
|
\begin{equation} |
| 633 |
< |
\lambda=-\frac{j_{z}(p_{x})}{\delta \frac{\partial v_{x}}{\partial z}} |
| 633 |
> |
\lambda=-\frac{j_{z}(p_{x})} {\left(\frac{\partial v_{x}}{\partial |
| 634 |
> |
z}\right) \delta} |
| 635 |
|
\end{equation} |
| 636 |
|
|
| 637 |
< |
For our simulations, we obtain $\delta$ from the difference between the structural edge of the ice block determined by the tetrahedrality profile fit, and the intersection of the linear regression of the $v_{x}$ profiles about the $z$-dimension for the ice and liquid. (See Figure \ref{fig:delta_example}) The coefficient of friction for the basal and the prismatic facets were found to be (0.07$\pm$0.01) amu fs\textsuperscript{-1} and (0.032$\pm$0.007) amu fs\textsuperscript{-1}. It is known that the basal and prismatic faces have different surface structures. The basal face is smoother than the prismatic with small alternating valleys and crests, while the prismatic surface has deep corrugating channels. We believe the reason that the prismatic face's coefficient of friction was found to be smaller than the basal's is due to the direction of the shear. The shear of the ice/water was in the same direction of the corrugating channels, allowing water molecules to pass through the channels during the shear. |
| 637 |
> |
For ice / water interfaces, the boundary conditions are markedly |
| 638 |
> |
no-slip, so projecting the bulk liquid state velocity profile yields a |
| 639 |
> |
negative slip length. This length is the difference between the |
| 640 |
> |
structural edge of the ice (determined by the tetrahedrality profile) |
| 641 |
> |
and the location where the projected velocity of the bulk liquid |
| 642 |
> |
intersects the solid phase velocity (see Figure |
| 643 |
> |
\ref{fig:delta_example}). The coefficients of friction for the basal |
| 644 |
> |
and the prismatic facets are found to be |
| 645 |
> |
0.07~$\pm$~0.01~amu~fs\textsuperscript{-1} and |
| 646 |
> |
0.032~$\pm$~0.007~amu~fs\textsuperscript{-1}, respectively. These |
| 647 |
> |
results may seem surprising as the basal face is smoother than the |
| 648 |
> |
prismatic with only small undulations of the oxygen positions, while |
| 649 |
> |
the prismatic surface has deep corrugated channels. The applied |
| 650 |
> |
momentum flux used in our simulations is parallel to these channels, |
| 651 |
> |
however, and this results in a flow of water in the same direction as |
| 652 |
> |
the corrugations, allowing water molecules to pass through the |
| 653 |
> |
channels during the shear. |
| 654 |
|
|
| 655 |
|
\begin{figure} |
| 656 |
|
\includegraphics[width=\linewidth]{delta_example} |
| 657 |
< |
\caption{\label{fig:delta_example} A schematic of determining the slip length ($\delta$). The slip length is the difference of the structural starting point of the ice and the point of intersection of the linear regressions of the liquid phase velocity profile (red) and of the solid ice velocity profile (black). The dotted line indicates the location of the ice as determined by the tetrahedrality profile.} |
| 657 |
> |
\caption{\label{fig:delta_example} Determining the (negative) slip |
| 658 |
> |
length ($\delta$) for the ice-water interfaces (which have decidedly |
| 659 |
> |
non-slip behavior). This length is the difference between the |
| 660 |
> |
structural edge of the ice (determined by the tetrahedrality |
| 661 |
> |
profile) and the location where the projected velocity of the bulk |
| 662 |
> |
liquid (dashed red line) intersects the solid phase velocity (solid |
| 663 |
> |
black line). The dotted line indicates the location of the ice as |
| 664 |
> |
determined by the tetrahedrality profile.} |
| 665 |
|
\end{figure} |
| 666 |
|
|
| 667 |
|
|
| 668 |
|
\section{Conclusion} |
| 669 |
< |
Here we have simulated the basal and prismatic facets of an SPC/E model of the ice I$_\mathrm{h}$ / water interface. Using VSS-RNEMD, the ice was sheared relative to the liquid while imposed thermal gradients kept the interface at a stable temperature. Caculation of the local tetrahedrality order parameter has shown an apparent independence of the shear rate on the interfacial width, which was found to be 3.2$\pm$0.4 \AA\ and 3.6$\pm$0.2 \AA\ for the basal and prismatic systems. The orientational time correlation function was calculated from varying displacements from the interface. Decomposition by a triexponential decay also showed an apparent independence of the shear rate. The short time decay due to the restoring forces of existing hydrogen bonds decreased at close displacements from the interface, while the middle and long time decays were found to increase. There is also an apparent displacement, $d_{basal}$ and $d_{prismatic}$, from the interface at which these deviations from bulk liquid values occurs. We found $d_{basal}$ and $d_{prismatic}$ to be approximately 15 \AA\ and 8 \AA\ . This implies that the dynamics of water molecules which are structurally equivalent to bulk phase molecules are being perturbed by the presence of the ice and/or the interface. The coefficient of friction of each of the facets was also determined. They were found to be (0.07$\pm$0.01) amu fs\textsuperscript{-1} and (0.032$\pm$0.007) amu fs\textsuperscript{-1} for the basal and prismatic facets respectively. We believe the large difference between the two friction coefficients is due to the direction of the shear and the surface structure of the crystal facets. |
| 669 |
> |
We have simulated the basal and prismatic facets of an SPC/E model of |
| 670 |
> |
the ice I$_\mathrm{h}$ / water interface. Using VSS-RNEMD, the ice |
| 671 |
> |
was sheared relative to the liquid while simultaneously being exposed |
| 672 |
> |
to a weak thermal gradient which kept the interface at a stable |
| 673 |
> |
temperature. Calculation of the local tetrahedrality order parameter |
| 674 |
> |
has shown an apparent independence of the interfacial width on the |
| 675 |
> |
shear rate. This width was found to be 3.2~$\pm$0.4~\AA\ and |
| 676 |
> |
3.6~$\pm$0.2~\AA\ for the basal and prismatic systems, respectively. |
| 677 |
|
|
| 678 |
< |
\begin{acknowledgement} |
| 678 |
> |
Orientational time correlation functions were calculated at varying |
| 679 |
> |
displacements from the interface, and were found to be similarly |
| 680 |
> |
independent of the applied momentum flux. The short decay due to the |
| 681 |
> |
restoring forces of existing hydrogen bonds decreased close to the |
| 682 |
> |
interface, while the longer-time decay constants increased in close |
| 683 |
> |
proximity to the interface. There is also an apparent dynamic |
| 684 |
> |
interface width, $d_{basal}$ and $d_{prismatic}$, at which these |
| 685 |
> |
deviations from bulk liquid values begin. We found $d_{basal}$ and |
| 686 |
> |
$d_{prismatic}$ to be approximately 15~\AA\ and 8~\AA\ . This implies |
| 687 |
> |
that the dynamics of water molecules which have similar structural |
| 688 |
> |
environments to liquid phase molecules are dynamically perturbed by |
| 689 |
> |
the presence of the ice interface. |
| 690 |
> |
|
| 691 |
> |
The coefficient of liquid-solid friction for each of the facets was |
| 692 |
> |
also determined. They were found to be |
| 693 |
> |
0.07~$\pm$~0.01~amu~fs\textsuperscript{-1} and |
| 694 |
> |
0.032~$\pm$~0.007~amu~fs\textsuperscript{-1} for the basal and |
| 695 |
> |
prismatic facets respectively. We attribute the large difference |
| 696 |
> |
between the two friction coefficients to the direction of the shear |
| 697 |
> |
and to the surface structure of the crystal facets. |
| 698 |
> |
|
| 699 |
> |
\section{Acknowledgements} |
| 700 |
|
Support for this project was provided by the National Science |
| 701 |
|
Foundation under grant CHE-0848243. Computational time was provided |
| 702 |
|
by the Center for Research Computing (CRC) at the University of |
| 703 |
|
Notre Dame. |
| 444 |
– |
\end{acknowledgement} |
| 704 |
|
|
| 705 |
|
\newpage |
| 447 |
– |
\bibstyle{achemso} |
| 706 |
|
\bibliography{iceWater} |
| 707 |
|
|
| 708 |
< |
\begin{tocentry} |
| 451 |
< |
\begin{wrapfigure}{l}{0.5\textwidth} |
| 452 |
< |
\begin{center} |
| 453 |
< |
\includegraphics[width=\linewidth]{SystemImage.png} |
| 454 |
< |
\end{center} |
| 455 |
< |
\end{wrapfigure} |
| 456 |
< |
An image of our system. |
| 457 |
< |
\end{tocentry} |
| 708 |
> |
\end{doublespace} |
| 709 |
|
|
| 710 |
+ |
% \begin{tocentry} |
| 711 |
+ |
% \begin{wrapfigure}{l}{0.5\textwidth} |
| 712 |
+ |
% \begin{center} |
| 713 |
+ |
% \includegraphics[width=\linewidth]{SystemImage.png} |
| 714 |
+ |
% \end{center} |
| 715 |
+ |
% \end{wrapfigure} |
| 716 |
+ |
% The cell used to simulate liquid-solid shear in ice I$_\mathrm{h}$ / |
| 717 |
+ |
% water interfaces. Velocity gradients were applied using the velocity |
| 718 |
+ |
% shearing and scaling variant of reverse non-equilibrium molecular |
| 719 |
+ |
% dynamics (VSS-RNEMD) with a weak thermal gradient to prevent melting. |
| 720 |
+ |
% The interface width is relatively robust in both structual and dynamic |
| 721 |
+ |
% measures as a function of the applied shear. |
| 722 |
+ |
% \end{tocentry} |
| 723 |
+ |
|
| 724 |
|
\end{document} |
| 725 |
|
|
| 726 |
|
%************************************************************** |
