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%\url{www.pnas.org/cgi/doi/10.1073/pnas.0709640104} |
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\copyrightyear{2014} |
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\begin{document} |
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\title{Supporting Information for: \\ |
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The different facets of ice have different hydrophilicities: Friction at water / |
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ice-I\textsubscript{h} interfaces} |
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|
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\author{Patrick B. Louden\affil{1}{Department of Chemistry and Biochemistry, University of Notre Dame, Notre Dame, |
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IN 46556} |
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\and |
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J. Daniel Gezelter\affil{1}{}} |
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|
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\contributor{Submitted to Proceedings of the National Academy of Sciences |
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of the United States of America} |
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|
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\maketitle |
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|
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\begin{article} |
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|
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\section{Overview} |
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The supporting information contains further details about the model |
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construction, analysis methods, and supplies figures that support the |
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data presented in the main text. |
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|
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\section{Construction of the Ice / Water interfaces} |
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Ice I$_\mathrm{h}$ crystallizes in the hexagonal space group |
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P$6_3/mmc$, and common ice crystals form hexagonal plates with the |
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basal face $\{0~0~0~1\}$ forming the top and bottom of each plate, and |
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the prismatic facet $\{1~0~\bar{1}~0\}$ forming the sides. In extreme |
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temperatures or low water saturation conditions, ice crystals can |
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easily form as hollow columns, needles and dendrites. These are |
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structures that expose other crystalline facets of the ice to the |
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surroundings. Among the more common facets are the secondary prism, |
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$\{1~1~\bar{2}~0\}$, and pyramidal, $\{2~0~\bar{2}~1\}$, faces. |
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|
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We found it most useful to work with proton-ordered, zero-dipole |
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crystals that expose strips of dangling H-atoms and lone |
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pairs.\cite{Buch:2008fk} Our structures were created starting from |
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Structure 6 of Hirsch and Ojam\"{a}e's set of orthorhombic |
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representations for ice-I$_{h}$~\cite{Hirsch04}. This crystal |
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structure was cleaved along the four different facets. The exposed |
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face was reoriented normal to the $z$-axis of the simulation cell, and |
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the structures were and extended to form large exposed facets in |
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rectangular box geometries. Liquid water boxes were created with |
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identical dimensions (in $x$ and $y$) as the ice, with a $z$ dimension |
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of three times that of the ice block, and a density corresponding to 1 |
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g / cm$^3$. Each of the ice slabs and water boxes were independently |
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equilibrated at a pressure of 1 atm, and the resulting systems were |
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merged by carving out any liquid water molecules within 3 \AA\ of any |
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atoms in the ice slabs. Each of the combined ice/water systems were |
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then equilibrated at 225K, which is the liquid-ice coexistence |
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temperature for SPC/E water~\cite{Bryk02}. Reference |
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\citealp{Louden13} contains a more detailed explanation of the |
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construction of similar ice/water interfaces. The resulting dimensions |
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as well as the number of ice and liquid water molecules contained in |
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each of these systems are shown in Table S1. |
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|
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\section{A second method for computing contact angles} |
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In addition to the spherical cap method outlined in the main text, a |
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second method for obtaining the contact angle was described by |
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Ruijter, Blake, and Coninck~\cite{Ruijter99}. This method uses a |
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cylindrical averaging of the droplet's density profile. A threshold |
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density of 0.5 g cm\textsuperscript{-3} is used to estimate the |
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location of the edge of the droplet. The $r$ and $z$-dependence of |
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the droplet's edge is then fit to a circle, and the contact angle is |
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computed from the intersection of the fit circle with the $z$-axis |
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location of the solid surface. Again, for each stored configuration, |
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the density profile in a set of annular shells was computed. Due to |
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large density fluctuations close to the ice, all shells located within |
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2 \AA\ of the ice surface were left out of the circular fits. The |
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height of the solid surface ($z_\mathrm{suface}$) along with the best |
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fitting origin ($z_\mathrm{droplet}$) and radius |
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($r_\mathrm{droplet}$) of the droplet can then be used to compute the |
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contact angle, |
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\begin{equation} |
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\theta = 90 + \frac{180}{\pi} \sin^{-1}\left(\frac{z_\mathrm{droplet} - |
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z_\mathrm{surface}}{r_\mathrm{droplet}} \right). |
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\end{equation} |
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|
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\section{Determining 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 S2-S5 in the SI show tetrahedrality |
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profiles (in circles) for each of the four interfaces. The $q(z)$ |
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function has a range of $(0,1)$, where a value of unity indicates a |
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perfectly tetrahedral environment. The $q(z)$ for the bulk liquid was |
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found to be $\approx~0.77$, while values of $\approx~0.92$ were more |
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common in the ice. The tetrahedrality profiles were fit using a |
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hyperbolic tangent function (see Eq. 6 in Reference |
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\citealp{Louden13}) designed to smoothly fit the bulk to ice |
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transition while accounting for the weak thermal gradient. In panels |
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$b$ and $c$ of the same figures, the resulting thermal and velocity |
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gradients from an imposed kinetic energy and momentum fluxes can be |
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seen. The vertical dotted lines traversing these figures indicate the |
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midpoints of the interfaces as determined by the tetrahedrality |
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profiles. |
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|
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\section{Determining 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. The time-dependence was fit to a |
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triexponential decay, with three time constants: $\tau_{short}$, |
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measuring the librational motion of the water molecules, |
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$\tau_{middle}$, measuring the timescale for breaking and making of |
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hydrogen bonds, and $\tau_{long}$, corresponding to the translational |
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motion of the water molecules. An additional constant was introduced |
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in the fits to describe molecules in the crystal which do not |
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experience long-time orientational decay. |
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|
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In Figures S6-S9 in the supporting information, the $z$-coordinate |
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profiles for the three decay constants, $\tau_{short}$, |
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$\tau_{middle}$, and $\tau_{long}$ for the different interfaces are |
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shown. (Figures S6 \& S7 are new results, and Figures S8 \& S9 are |
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updated plots from Ref \citealp{Louden13}.) In the liquid regions of |
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all four interfaces, we observe $\tau_{middle}$ and $\tau_{long}$ to |
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have approximately consistent values of $3-6$ ps and $30-40$ ps, |
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respectively. Both of these times increase in value approaching the |
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interface. Approaching the interface, we also observe that |
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$\tau_{short}$ decreases from its liquid-state value of $72-76$ fs. |
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The approximate values for the decay constants and the trends |
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approaching the interface match those reported previously for the |
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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 \ref{tab:kappa}. 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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\end{article} |
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|
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\begin{table}[h] |
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\centering |
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\caption{Sizes of the droplet and shearing simulations. Cell |
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dimensions are measured in \AA. \label{tab:method}} |
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\begin{tabular}{r|cccc|ccccc} |
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\toprule |
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\multirow{2}{*}{Interface} & \multicolumn{4}{c|}{Droplet} & \multicolumn{5}{c}{Shearing} \\ |
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& $N_\mathrm{ice}$ & $N_\mathrm{droplet}$ & $L_x$ & $L_y$ & $N_\mathrm{ice}$ & $N_\mathrm{liquid}$ & $L_x$ & $L_y$ & $L_z$ \\ |
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\midrule |
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Basal $\{0001\}$ & 12960 & 2048 & 134.70 & 140.04 & 900 & 1846 & 23.87 & 35.83 & 98.64 \\ |
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Pyramidal $\{2~0~\bar{2}~1\}$ & 11136 & 2048 & 143.75 & 121.41 & 1216 & 2203 & 37.47 & 29.50 & 93.02 \\ |
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Prismatic $\{1~0~\bar{1}~0\}$ & 9900 & 2048 & 110.04 & 115.00 & 3000 & 5464 & 35.95 & 35.65 & 205.77 \\ |
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Secondary Prism $\{1~1~\bar{2}~0\}$ & 11520 & 2048 & 146.72 & 124.48 & 3840 & 8176 & 71.87 & 31.66 & 161.55 \\ |
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\bottomrule |
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\end{tabular} |
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\end{table} |
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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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|
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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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\bibliography{iceWater} |
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\end{document} |
