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# User Rev Content
1 gezelter 4268 \documentclass[aps,jcp,preprint,showpacs,superscriptaddress,groupedaddress]{revtex4} % for double-spaced preprint
2     \usepackage{graphicx} % needed for figures
3     \usepackage{dcolumn} % needed for some tables
4     \usepackage{bm} % for math
5     \usepackage{amssymb} % for math
6     %\usepackage{booktabs}
7     \usepackage{multirow}
8     \usepackage{tablefootnote}
9     \usepackage{times}
10     \usepackage{mathptm}
11     \usepackage[version=3]{mhchem}
12    
13     \begin{document}
14    
15     \title{Supporting Information for: \\
16     The different facets of ice have different hydrophilicities: Friction at water /
17     ice-I$_\mathrm{h}$ interfaces}
18    
19     \author{Patrick B. Louden}
20     \author{J. Daniel Gezelter}
21     \email{gezelter@nd.edu}
22     \affiliation{Department of Chemistry and Biochemistry, University of
23     Notre Dame, Notre Dame, IN 46556}
24    
25     \date{\today}
26    
27     \begin{abstract}
28     The supporting information supplies figures that support the data
29     presented in the main text.
30     \end{abstract}
31    
32     \pacs{68.08.Bc, 68.08.De, 66.20.Cy}
33    
34    
35     \maketitle
36    
37     \section{The Advancing Contact Angle}
38     The advancing contact angles for the liquid droplets were computed
39     using inversion of Eq. (2) in the main text which requires finding the
40     real roots of a fourth order polynomial,
41     \begin{equation}
42     \label{eq:poly}
43     c_4 \cos^4 \theta + c_3 \cos^3 \theta + c_2 \cos^2 \theta + c_1
44     \cos \theta + c_0 = 0
45     \end{equation}
46     where the coefficients of the polynomial are expressed in terms of the
47     $z$ coordinate of the center of mass of the liquid droplet relative to
48     the solid surface, $z = z_\mathrm{cm} - z_\mathrm{surface}$, and a
49     factor that depends on the initial droplet radius, $k = 2^{-4/3} R_0$.
50     The coefficients are simple functions of these two quantities,
51     \begin{align}
52     c_4 &= z^3 + k^3 \\
53     c_3 &= 8 z^3 + 8 k^3 \\
54     c_2 &= 24 z^3 + 18 k^3 \\
55     c_1 &= 32 z^3 \\
56     c_0 &= 16 z^3 - 27 k^3 .
57     \end{align}
58     Solving for the values of the real roots of this polynomial
59     (Eq. \ref{eq:poly}) give estimates of the advancing contact angle.
60     The dynamics of this quantity for each of the four interfaces is shown
61     in figure 1 below.
62    
63     \section{Interfacial widths using structural information}
64     To determine the structural widths of the interfaces under shear, each
65     of the systems was divided into 100 bins along the $z$-dimension, and
66     the local tetrahedral order parameter (Eq. 5 in Reference
67     \citealp{Louden13}) was time-averaged in each bin for the duration of
68     the shearing simulation. The spatial dependence of this order
69     parameter, $q(z)$, is the tetrahedrality profile of the interface.
70     The lower panels in figures 2-5 show tetrahedrality profiles (in
71     circles) for each of the four interfaces. The $q(z)$ function has a
72     range of $(0,1)$, where a value of unity indicates a perfectly
73     tetrahedral environment. The $q(z)$ for the bulk liquid was found to
74     be $\approx~0.77$, while values of $\approx~0.92$ were more common in
75     the ice. The tetrahedrality profiles were fit using a hyperbolic
76     tangent function (see Eq. 6 in Reference \citealp{Louden13}) designed
77     to smoothly fit the bulk to ice transition while accounting for the
78     weak thermal gradient. In panels $b$ and $c$ of the same figures, the
79     resulting thermal and velocity gradients from an imposed kinetic
80     energy and momentum fluxes can be seen. The vertical dotted lines
81     traversing these figures indicate the midpoints of the interfaces as
82     determined by the tetrahedrality profiles.
83    
84     \section{Interfacial widths using dynamic information}
85     To determine the dynamic widths of the interfaces under shear, each of
86     the systems was divided into bins along the $z$-dimension ($\approx$ 3
87     \AA\ wide) and $C_2(z,t)$ was computed using only those molecules that
88     were in the bin at the initial time. To compute these correlation
89     functions, each of the 0.5 ns simulations was followed by a shorter
90     200 ps microcanonical (NVE) simulation in which the positions and
91     orientations of every molecule in the system were recorded every 0.1
92     ps.
93    
94     The time-dependence was fit to a triexponential decay, with three time
95     constants: $\tau_{short}$, measuring the librational motion of the
96     water molecules, $\tau_{middle}$, measuring the timescale for breaking
97     and making of hydrogen bonds, and $\tau_{long}$, corresponding to the
98     translational motion of the water molecules. An additional constant
99     was introduced in the fits to describe molecules in the crystal which
100     do not experience long-time orientational decay.
101    
102     In Figures 6-9, the $z$-coordinate profiles for the three decay
103     constants, $\tau_{short}$, $\tau_{middle}$, and $\tau_{long}$ for the
104     different interfaces are shown. (Figures 6 \& 7 are new results,
105     and Figures 8 \& 9 are updated plots from Ref \citealp{Louden13}.)
106     In the liquid regions of all four interfaces, we observe
107     $\tau_{middle}$ and $\tau_{long}$ to have approximately consistent
108     values of $3-6$ ps and $30-40$ ps, respectively. Both of these times
109     increase in value approaching the interface. Approaching the
110     interface, we also observe that $\tau_{short}$ decreases from its
111     liquid-state value of $72-76$ fs. The approximate values for the
112     decay constants and the trends approaching the interface match those
113     reported previously for the basal and prismatic interfaces.
114    
115     We have estimated the dynamic interfacial width $d_\mathrm{dyn}$ by
116     fitting the profiles of all the three orientational time constants
117     with an exponential decay to the bulk-liquid behavior,
118     \begin{equation}\label{tauFit}
119     \tau(z)\approx\tau_{liquid}+(\tau_{wall}-\tau_{liquid})e^{-(z-z_{wall})/d_\mathrm{dyn}}
120     \end{equation}
121     where $\tau_{liquid}$ and $\tau_{wall}$ are the liquid and projected
122     wall values of the decay constants, $z_{wall}$ is the location of the
123     interface, as measured by the structural order parameter. These
124     values are shown in table 1 in the main text. Because the bins must be
125     quite wide to obtain reasonable profiles of $C_2(z,t)$, the error
126     estimates for the dynamic widths of the interface are significantly
127     larger than for the structural widths. However, all four interfaces
128     exhibit dynamic widths that are significantly below 1~nm, and are in
129     reasonable agreement with the structural width above.
130    
131     \bibliographystyle{aip}
132     \bibliography{iceWater}
133    
134     \newpage
135     %S1: contact angle
136     \begin{figure}
137     \includegraphics[width=\linewidth]{ContactAngle}
138     \caption{\label{fig:ContactAngle} The dynamic contact angle of a
139     droplet after approaching each of the four ice facets. The decay to
140     an equilibrium contact angle displays similar dynamics. Although
141     all the surfaces are hydrophilic, the long-time behavior stabilizes
142     to significantly flatter droplets for the basal and pyramidal
143     facets. This suggests a difference in hydrophilicity for these
144     facets compared with the two prismatic facets.}
145     \end{figure}
146    
147     \newpage
148    
149     %S2-S5 are the z-rnemd profiles
150     \begin{figure}
151     \includegraphics[width=\linewidth]{Pyr_comic_strip}
152     \caption{\label{fig:pyrComic} Properties of the pyramidal interface
153     being sheared through water at 3.8 ms\textsuperscript{-1}. Lower
154     panel: the local tetrahedral order parameter, $q(z)$, (circles) and
155     the hyperbolic tangent fit (turquoise line). Middle panel: the
156     imposed thermal gradient required to maintain a fixed interfacial
157     temperature of 225 K. Upper panel: the transverse velocity gradient
158     that develops in response to an imposed momentum flux. The vertical
159     dotted lines indicate the locations of the midpoints of the two
160     interfaces.}
161     \end{figure}
162     \newpage
163    
164     \begin{figure}
165     \includegraphics[width=\linewidth]{SP_comic_strip}
166     \caption{\label{fig:spComic} The secondary prism interface with a shear
167     rate of 3.5 \
168     ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.}
169     \end{figure}
170     \newpage
171    
172     \begin{figure}
173     \includegraphics[width=\linewidth]{B_comic_strip}
174     \caption{\label{fig:bComic} The basal interface with a shear
175     rate of 1.3 \
176     ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.}
177     \end{figure}
178     \newpage
179    
180     \begin{figure}
181     \includegraphics[width=\linewidth]{prismatic_comic_strip}
182     \caption{\label{fig:pComic} The prismatic interface with a shear
183     rate of 2 \
184     ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.}
185     \end{figure}
186     \newpage
187    
188     %Figures S6-S9 are the z-orientation times
189     \begin{figure}
190     \includegraphics[width=\linewidth]{Pyr-orient}
191     \caption{\label{fig:PyrOrient} The three decay constants of the
192     orientational time correlation function, $C_2(z,t)$, for water as a
193     function of distance from the center of the ice slab. The vertical
194     dashed line indicates the edge of the pyramidal ice slab determined
195     by the local order tetrahedral parameter. The control (circles) and
196     sheared (squares) simulations were fit using shifted-exponential
197     decay (see Eq. 9 in Ref. \citealp{Louden13}).}
198     \end{figure}
199     \newpage
200    
201     \begin{figure}
202     \includegraphics[width=\linewidth]{SP-orient}
203     \caption{\label{fig:SPorient} Decay constants for $C_2(z,t)$ at the secondary
204     prism face. Panel descriptions match those in \ref{fig:PyrOrient}.}
205     \end{figure}
206    
207     \newpage
208    
209     \begin{figure}
210     \includegraphics[width=\linewidth]{B-orient}
211     \caption{\label{fig:Borient} Decay constants for $C_2(z,t)$ at the basal face. Panel descriptions match those in \ref{fig:PyrOrient}.}
212     \end{figure}
213     \newpage
214    
215     \begin{figure}
216     \includegraphics[width=\linewidth]{prismatic-orient}
217     \caption{\label{fig:Porient} Decay constants for $C_2(z,t)$ at the
218     prismatic face. Panel descriptions match those in \ref{fig:PyrOrient}.}
219     \end{figure}
220    
221    
222     \end{document}