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\documentclass[aps,jcp,preprint,showpacs,superscriptaddress,groupedaddress]{revtex4} % for double-spaced preprint |
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\usepackage{graphicx} % needed for figures |
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\usepackage{dcolumn} % needed for some tables |
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\usepackage{bm} % for math |
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\usepackage{amssymb} % for math |
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%\usepackage{booktabs} |
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\usepackage{multirow} |
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\usepackage{tablefootnote} |
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\usepackage{times} |
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\usepackage{mathptm} |
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\usepackage[version=3]{mhchem} |
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\begin{document} |
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|
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\title{Supporting Information for: \\ |
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The different facets of ice have different hydrophilicities: Friction at water / |
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ice-I$_\mathrm{h}$ interfaces} |
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|
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\author{Patrick B. Louden} |
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\author{J. Daniel Gezelter} |
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\email{gezelter@nd.edu} |
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\affiliation{Department of Chemistry and Biochemistry, University of |
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Notre Dame, Notre Dame, IN 46556} |
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|
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\date{\today} |
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|
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\begin{abstract} |
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The supporting information supplies figures that support the data |
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presented in the main text. |
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\end{abstract} |
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|
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\pacs{68.08.Bc, 68.08.De, 66.20.Cy} |
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|
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|
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\maketitle |
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|
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\section{The Advancing Contact Angle} |
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The advancing contact angles for the liquid droplets were computed |
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using inversion of Eq. (2) in the main text which requires finding the |
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real roots of a fourth order polynomial, |
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\begin{equation} |
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\label{eq:poly} |
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c_4 \cos^4 \theta + c_3 \cos^3 \theta + c_2 \cos^2 \theta + c_1 |
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\cos \theta + c_0 = 0 |
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\end{equation} |
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where the coefficients of the polynomial are expressed in terms of the |
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$z$ coordinate of the center of mass of the liquid droplet relative to |
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the solid surface, $z = z_\mathrm{cm} - z_\mathrm{surface}$, and a |
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factor that depends on the initial droplet radius, $k = 2^{-4/3} R_0$. |
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The coefficients are simple functions of these two quantities, |
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\begin{align} |
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c_4 &= z^3 + k^3 \\ |
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c_3 &= 8 z^3 + 8 k^3 \\ |
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c_2 &= 24 z^3 + 18 k^3 \\ |
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c_1 &= 32 z^3 \\ |
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c_0 &= 16 z^3 - 27 k^3 . |
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\end{align} |
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Solving for the values of the real roots of this polynomial |
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(Eq. \ref{eq:poly}) give estimates of the advancing contact angle. |
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The dynamics of this quantity for each of the four interfaces is shown |
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in figure 1 below. |
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|
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\section{Interfacial widths using structural information} |
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To determine the structural widths of the interfaces under shear, each |
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of the systems was divided into 100 bins along the $z$-dimension, and |
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the local tetrahedral order parameter (Eq. 5 in Reference |
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\citealp{Louden13}) was time-averaged in each bin for the duration of |
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the shearing simulation. The spatial dependence of this order |
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parameter, $q(z)$, is the tetrahedrality profile of the interface. |
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The lower panels in figures 2-5 show tetrahedrality profiles (in |
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circles) for each of the four interfaces. The $q(z)$ function has a |
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range of $(0,1)$, where a value of unity indicates a perfectly |
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tetrahedral environment. The $q(z)$ for the bulk liquid was found to |
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be $\approx~0.77$, while values of $\approx~0.92$ were more common in |
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the ice. The tetrahedrality profiles were fit using a hyperbolic |
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tangent function (see Eq. 6 in Reference \citealp{Louden13}) designed |
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to smoothly fit the bulk to ice transition while accounting for the |
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weak thermal gradient. In panels $b$ and $c$ of the same figures, the |
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resulting thermal and velocity gradients from an imposed kinetic |
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energy and momentum fluxes can be seen. The vertical dotted lines |
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traversing these figures indicate the midpoints of the interfaces as |
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determined by the tetrahedrality profiles. |
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|
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\section{Interfacial widths using dynamic information} |
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To determine the dynamic widths of the interfaces under shear, each of |
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the systems was divided into bins along the $z$-dimension ($\approx$ 3 |
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\AA\ wide) and $C_2(z,t)$ was computed using only those molecules that |
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were in the bin at the initial time. To compute these correlation |
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functions, each of the 0.5 ns simulations was followed by a shorter |
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200 ps microcanonical (NVE) simulation in which the positions and |
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orientations of every molecule in the system were recorded every 0.1 |
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ps. |
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|
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The time-dependence was fit to a triexponential decay, with three time |
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constants: $\tau_{short}$, measuring the librational motion of the |
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water molecules, $\tau_{middle}$, measuring the timescale for breaking |
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and making of hydrogen bonds, and $\tau_{long}$, corresponding to the |
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translational motion of the water molecules. An additional constant |
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was introduced in the fits to describe molecules in the crystal which |
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do not experience long-time orientational decay. |
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|
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In Figures 6-9, the $z$-coordinate profiles for the three decay |
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constants, $\tau_{short}$, $\tau_{middle}$, and $\tau_{long}$ for the |
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different interfaces are shown. (Figures 6 \& 7 are new results, |
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and Figures 8 \& 9 are updated plots from Ref \citealp{Louden13}.) |
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In the liquid regions of all four interfaces, we observe |
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$\tau_{middle}$ and $\tau_{long}$ to have approximately consistent |
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values of $3-6$ ps and $30-40$ ps, respectively. Both of these times |
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increase in value approaching the interface. Approaching the |
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interface, we also observe that $\tau_{short}$ decreases from its |
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liquid-state value of $72-76$ fs. The approximate values for the |
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decay constants and the trends approaching the interface match those |
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reported previously for the basal and prismatic interfaces. |
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|
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We have estimated the dynamic interfacial width $d_\mathrm{dyn}$ by |
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fitting the profiles of all the three orientational time constants |
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with an exponential decay to the bulk-liquid behavior, |
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\begin{equation}\label{tauFit} |
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\tau(z)\approx\tau_{liquid}+(\tau_{wall}-\tau_{liquid})e^{-(z-z_{wall})/d_\mathrm{dyn}} |
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\end{equation} |
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where $\tau_{liquid}$ and $\tau_{wall}$ are the liquid and projected |
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wall values of the decay constants, $z_{wall}$ is the location of the |
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interface, as measured by the structural order parameter. These |
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values are shown in table 1 in the main text. Because the bins must be |
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quite wide to obtain reasonable profiles of $C_2(z,t)$, the error |
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estimates for the dynamic widths of the interface are significantly |
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larger than for the structural widths. However, all four interfaces |
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exhibit dynamic widths that are significantly below 1~nm, and are in |
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reasonable agreement with the structural width above. |
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|
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\bibliographystyle{aip} |
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\bibliography{iceWater} |
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|
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\newpage |
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%S1: contact angle |
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\begin{figure} |
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\includegraphics[width=\linewidth]{ContactAngle} |
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\caption{\label{fig:ContactAngle} The dynamic contact angle of a |
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droplet after approaching each of the four ice facets. The decay to |
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an equilibrium contact angle displays similar dynamics. Although |
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all the surfaces are hydrophilic, the long-time behavior stabilizes |
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to significantly flatter droplets for the basal and pyramidal |
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facets. This suggests a difference in hydrophilicity for these |
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facets compared with the two prismatic facets.} |
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\end{figure} |
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|
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\newpage |
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|
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%S2-S5 are the z-rnemd profiles |
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\begin{figure} |
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\includegraphics[width=\linewidth]{Pyr_comic_strip} |
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\caption{\label{fig:pyrComic} Properties of the pyramidal interface |
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being sheared through water at 3.8 ms\textsuperscript{-1}. Lower |
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panel: the local tetrahedral order parameter, $q(z)$, (circles) and |
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the hyperbolic tangent fit (turquoise line). Middle panel: the |
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imposed thermal gradient required to maintain a fixed interfacial |
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temperature of 225 K. Upper panel: the transverse velocity gradient |
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that develops in response to an imposed momentum flux. The vertical |
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dotted lines indicate the locations of the midpoints of the two |
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interfaces.} |
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\end{figure} |
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\newpage |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{SP_comic_strip} |
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\caption{\label{fig:spComic} The secondary prism interface with a shear |
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rate of 3.5 \ |
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ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.} |
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\end{figure} |
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\newpage |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{B_comic_strip} |
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\caption{\label{fig:bComic} The basal interface with a shear |
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rate of 1.3 \ |
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ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.} |
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\end{figure} |
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\newpage |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{prismatic_comic_strip} |
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\caption{\label{fig:pComic} The prismatic interface with a shear |
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rate of 2 \ |
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ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.} |
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\end{figure} |
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\newpage |
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|
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%Figures S6-S9 are the z-orientation times |
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\begin{figure} |
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\includegraphics[width=\linewidth]{Pyr-orient} |
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\caption{\label{fig:PyrOrient} The three decay constants of the |
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orientational time correlation function, $C_2(z,t)$, for water as a |
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function of distance from the center of the ice slab. The vertical |
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dashed line indicates the edge of the pyramidal ice slab determined |
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by the local order tetrahedral parameter. The control (circles) and |
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sheared (squares) simulations were fit using shifted-exponential |
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decay (see Eq. 9 in Ref. \citealp{Louden13}).} |
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\end{figure} |
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\newpage |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{SP-orient} |
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\caption{\label{fig:SPorient} Decay constants for $C_2(z,t)$ at the secondary |
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prism face. Panel descriptions match those in \ref{fig:PyrOrient}.} |
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\end{figure} |
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|
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\newpage |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{B-orient} |
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\caption{\label{fig:Borient} Decay constants for $C_2(z,t)$ at the basal face. Panel descriptions match those in \ref{fig:PyrOrient}.} |
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\end{figure} |
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\newpage |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{prismatic-orient} |
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\caption{\label{fig:Porient} Decay constants for $C_2(z,t)$ at the |
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prismatic face. Panel descriptions match those in \ref{fig:PyrOrient}.} |
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\end{figure} |
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\end{document} |
