1 |
gezelter |
3914 |
\documentclass[journal = jpccck, manuscript = article]{achemso} |
2 |
|
|
\setkeys{acs}{usetitle = true} |
3 |
|
|
\usepackage{achemso} |
4 |
|
|
\usepackage{natbib} |
5 |
|
|
\usepackage{multirow} |
6 |
|
|
\usepackage{wrapfig} |
7 |
|
|
\usepackage{fixltx2e} |
8 |
|
|
%\mciteErrorOnUnknownfalse |
9 |
gezelter |
3897 |
|
10 |
gezelter |
3914 |
\usepackage[version=3]{mhchem} % this is a great package for formatting chemical reactions |
11 |
|
|
\usepackage{url} |
12 |
gezelter |
3897 |
|
13 |
|
|
|
14 |
gezelter |
3914 |
\title{Do the facets of ice $I_\mathrm{h}$ crystals have different |
15 |
|
|
friction coefficients? Simulating shear in ice/water interfaces} |
16 |
gezelter |
3897 |
|
17 |
|
|
\author{P. B. Louden} |
18 |
gezelter |
3914 |
\author{J. Daniel Gezelter} |
19 |
|
|
\email{gezelter@nd.edu} |
20 |
|
|
\affiliation[University of Notre Dame]{251 Nieuwland Science Hall\\ |
21 |
|
|
Department of Chemistry and Biochemistry\\ University of Notre |
22 |
|
|
Dame\\ Notre Dame, Indiana 46556} |
23 |
gezelter |
3897 |
|
24 |
gezelter |
3914 |
\keywords{} |
25 |
gezelter |
3897 |
|
26 |
gezelter |
3914 |
\begin{document} |
27 |
gezelter |
3897 |
|
28 |
gezelter |
3914 |
\begin{abstract} |
29 |
|
|
We have investigated the structural properties of the basal and |
30 |
|
|
prismatic facets of an SPC/E model of the ice Ih / water interface |
31 |
|
|
when the solid phase is being drawn through liquid water (i.e. sheared |
32 |
|
|
relative to the fluid phase). To impose the shear, we utilized a |
33 |
|
|
reverse non-equilibrium molecular dynamics (RNEMD) method that creates |
34 |
|
|
non-equilibrium conditions using velocity shearing and scaling (VSS) |
35 |
|
|
moves of the molecules in two physically separated slabs in the |
36 |
|
|
simulation cell. This method can create simultaneous temperature and |
37 |
|
|
velocity gradients and allow the measurement of friction transport |
38 |
|
|
properties at interfaces. We present calculations of the interfacial |
39 |
|
|
friction coefficients and the apparent independence of shear rate on |
40 |
|
|
interfacial width and show that water moving over a flat ice/water |
41 |
|
|
interface is close to the no-slip boundary condition. |
42 |
|
|
\end{abstract} |
43 |
gezelter |
3897 |
|
44 |
gezelter |
3914 |
\newpage |
45 |
gezelter |
3897 |
|
46 |
|
|
\section{Introduction} |
47 |
|
|
|
48 |
|
|
%Other people looking at the ice/water interface |
49 |
|
|
%Geologists are concerned with the flow of water over ice |
50 |
|
|
%Antifreeze protein in fish--Haymet's group has cited this before |
51 |
|
|
|
52 |
plouden |
3899 |
%Paragraph explaining why the ice/water interface is important |
53 |
|
|
%Paragraph on what other people have done / lead into what hasn't been done |
54 |
|
|
%Paragraph on what I'm going to do |
55 |
plouden |
3898 |
|
56 |
|
|
|
57 |
gezelter |
3897 |
|
58 |
|
|
|
59 |
plouden |
3902 |
With the recent development of velocity shearing and scaling reverse non-equilibrium molecular dynamics (VSS-RNEMD), it is now possible to calculate transport properties from heterogeneous systems.\cite{Kuang12} This method can create simultaneous temperature and velocity gradients and allow the measurement of friction and thermal transport properties at interfaces. This allows for the study of the width of the ice/water interface as the ice is sheared through the liquid, while imposing a thermal gradient to prevent frictional heating of the interface. |
60 |
plouden |
3898 |
|
61 |
plouden |
3902 |
as well as determining the friction coefficient of the interface. |
62 |
|
|
|
63 |
plouden |
3899 |
In this paper, we investigate the width and the friction coefficient of the ice/water interface as the ice is sheared through the liquid. |
64 |
|
|
|
65 |
|
|
|
66 |
|
|
|
67 |
gezelter |
3897 |
\section{Methodology} |
68 |
|
|
|
69 |
gezelter |
3914 |
\subsection{Stable ice I$_\mathrm{h}$ / water interfaces} |
70 |
|
|
|
71 |
|
|
The structure of ice I$_\mathrm{h}$ is well understood; it |
72 |
|
|
crystallizes in a hexagonal space group P$6_3/mmc$, and the hexagonal |
73 |
|
|
crystals of ice have two faces that are commonly exposed, the basal |
74 |
|
|
face $\{0~0~0~1\}$, which forms the top and bottom of each hexagonal |
75 |
|
|
plate, and the prismatic $\{1~0~\bar{1}~0\}$ face which forms the |
76 |
|
|
sides of the plate. Other less-common, but still important, faces of |
77 |
|
|
ice I$_\mathrm{h}$ are the secondary prism face, $\{1~1~\bar{2}~0\}$, |
78 |
|
|
and the prismatic face, $\{2~0~\bar{2}~1\}$. |
79 |
|
|
|
80 |
|
|
Ice I$_\mathrm{h}$ is normally proton disordered in bulk crystals, |
81 |
|
|
although the surfaces probably have a preference for proton ordering |
82 |
|
|
along strips of dangling H-atoms and Oxygen lone |
83 |
|
|
pairs.\cite{Buch:2008fk} |
84 |
|
|
|
85 |
|
|
For small simulated ice interfaces, it is useful to have a |
86 |
|
|
proton-ordered, but zero-dipole crystal that exposes these strips of |
87 |
|
|
dangling H-atoms and lone pairs. Also, if we're going to place |
88 |
|
|
another material in contact with one of the ice crystalline planes, it |
89 |
|
|
is useful to have an orthorhombic (rectangular) box to work with. A |
90 |
|
|
recent paper by Hirsch and Ojam\"{a}e describes how to create |
91 |
|
|
proton-ordered bulk ice I$_\mathrm{h}$ in alternative orthorhombic |
92 |
|
|
cells.\cite{doi:10.1021/jp048434u} |
93 |
|
|
|
94 |
|
|
We have using Hirsch and Ojam\"{a}e's structure 6 which is an |
95 |
|
|
orthorhombic cell ($P2_12_12_1$) that produces a proton-ordered |
96 |
|
|
version of ice Ih. Table \ref{tab:equiv} contains a mapping between |
97 |
|
|
the Miller indices in the P$6_3/mmc$ crystal system and those in the |
98 |
|
|
Hirsch and Ojam\"{a}e $P2_12_12_1$ system. |
99 |
|
|
|
100 |
|
|
\begin{wraptable}{r}{3.5in} |
101 |
|
|
\begin{tabular}{|ccc|} \hline |
102 |
|
|
& hexagonal & orthorhombic \\ |
103 |
|
|
& ($P6_3/mmc$) & ($P2_12_12_1$) \\ |
104 |
|
|
crystal face & Miller indices & equivalent \\ \hline |
105 |
|
|
basal & $\{0~0~0~1\}$ & $\{0~0~1\}$ \\ |
106 |
|
|
prism & $\{1~0~\bar{1}~0\}$ & $\{1~0~0\}$ \\ |
107 |
|
|
secondary prism & $\{1~1~\bar{2}~0\}$ & $\{1~3~0\}$ \\ |
108 |
|
|
pryamidal & $\{2~0~\bar{2}~1\}$ & $\{2~0~1\}$ \\ \hline |
109 |
|
|
\end{tabular} |
110 |
|
|
\end{wraptable} |
111 |
|
|
|
112 |
|
|
Structure 6 from the Hirsch and Ojam\"{a}e paper has lattice |
113 |
|
|
parameters $a = 4.49225$, $b = 7.78080$, $c = 7.33581$ and two water |
114 |
|
|
molecules whose atoms reside at the following fractional coordinates: |
115 |
|
|
|
116 |
|
|
\begin{wraptable}{r}{3.25in} |
117 |
|
|
\begin{tabular}{|ccccc|} \hline |
118 |
|
|
atom label & type & x & y & z \\ \hline |
119 |
|
|
O$_{a}$ & O & 0.75 & 0.1667 & 0.4375 \\ |
120 |
|
|
H$_{1a}$ & H & 0.5735 & 0.2202 & 0.4836 \\ |
121 |
|
|
H$_{2a}$ & H & 0.7420 & 0.0517 & 0.4836 \\ |
122 |
|
|
O$_{b}$ & O & 0.25 & 0.6667 & 0.4375 \\ |
123 |
|
|
H$_{1b}$ & H & 0.2580 & 0.6693 & 0.3071 \\ |
124 |
|
|
H$_{2b}$ & H & 0.4265 & 0.7255 & 0.4756 \\ \hline |
125 |
|
|
\end{tabular} |
126 |
|
|
\end{wraptable} |
127 |
|
|
|
128 |
|
|
To construct the basal and prismatic interfaces, the crystallographic |
129 |
|
|
coordinates above were used to construct an orthorhombic unit cell |
130 |
|
|
which was then replicated in all three dimensions yielding a |
131 |
|
|
proton-ordered block of ice I$_{h}$. To expose the desired face, the |
132 |
|
|
orthorhombic representation was then cut along the ($001$) or ($100$) |
133 |
|
|
planes for the basal and prismatic faces respectively. The resulting |
134 |
|
|
block was rotated so that the exposed faces were aligned with the $z$ |
135 |
|
|
axis normal to the exposed face. The block was then cut along two |
136 |
|
|
perpendicular directions in a way that allowed for perfect periodic |
137 |
|
|
replication in the $x$ and $y$ axes, creating a slab with either the |
138 |
|
|
basal or prismatic faces exposed along the $z$ axis. The slab was |
139 |
|
|
then replicated in the $x$ and $y$ dimensions until a desired sample |
140 |
|
|
size was obtained. |
141 |
|
|
|
142 |
|
|
Although experimental solid/liquid coexistant temperature under normal |
143 |
|
|
pressure are close to 273K, Haymet \emph{et al.} have done extensive |
144 |
|
|
work on characterizing the ice/water |
145 |
|
|
interface.\cite{Karim88,Karim90,Hayward01,Bryk02,Hayward02} They have |
146 |
|
|
found for the SPC/E water model,\cite{Berendsen87} which is also used |
147 |
|
|
in this study, the ice/water interface is most stable at |
148 |
|
|
225$\pm$5K.\cite{Bryk02} To create a ice / water interface, a box of |
149 |
|
|
liquid water that had the same dimensions in $x$ and $y$ was |
150 |
|
|
equilibrated at 225 K and 1 atm of pressure in the NPAT ensemble (with |
151 |
|
|
the $z$ axis allowed to fluctuate to equilibrate to the correct |
152 |
|
|
pressure). The liquid and solid systems were combined by carving out |
153 |
|
|
any water molecule from the liquid simulation cell that was within 3 |
154 |
|
|
\AA\ of any atom in the ice slab. |
155 |
|
|
|
156 |
|
|
Molecular translation and orientational restraints were applied in the |
157 |
|
|
early stages of equilibration to prevent melting of the ice slab. |
158 |
|
|
These restraints were removed during NVT equilibration, well before |
159 |
|
|
data collection was carried out. |
160 |
|
|
|
161 |
gezelter |
3897 |
\subsection{Computational Details} |
162 |
|
|
All simulations were performed using OpenMD with a time step of 2 fs, and periodic boundary conditions in all three dimensions. The systems were divided into 100 artificial bins along the $z$-axis for the VSS-RNEMD moves, which were attempted every 50 fs. The gradients were allowed to develop for 1 ns before data collection was began. Once established, snapshots of the system were taken every 1 ps, and the average velocities and densities of each bin were accumulated every attempted VSS-RNEMD move. |
163 |
|
|
|
164 |
|
|
|
165 |
|
|
%A paragraph on the equilibration procedure of the system? Shenyu did some amount of equilibration to the files and then I was handed them. I performed 5 ns of NVT at 225K for both systems, then 5 ns of NVE at 225K for both systems, with no gradients imposed. |
166 |
|
|
%For the basal, once the thermal gradient was found which gave me the interfacial temperature I wanted (-2.0E-6 kcal/mol/A^2/fs), I equilibrated the file for 5 ns letting this gradient stabilize. Then I continued to use this thermal gradient as I imposed momentum gradients and watched the response of the interface. |
167 |
|
|
%For the prismatic, a gradient was not found that would give me the interfacial temperature I desired, so while imposing a thermal gradient that had the interface at 220K, I raised the temperature of the system to 230K. This resulted in a thermal gradient which gave my interface at 225K, equilibrated for ins NVT, then ins NVE while this gradient was still imposed, then I began dragging. |
168 |
|
|
%I have run each system for 1 ns under PTgrads to allow them to develop, then ran each system for an additional 2 ns in segments of 0.5 ns in order to calculate statistics of the calculated values. |
169 |
|
|
|
170 |
|
|
\subsection{Measuring the Width of the Interface} |
171 |
|
|
In order to characterize the ice/water interface, the local tetrahedral order parameter as described by Kumar\cite{Kumar09} and Errinton\cite{Errington01} was used. The local tetrahedral order parameter, $q$, is given by |
172 |
|
|
\begin{equation} |
173 |
|
|
q_{k} \equiv 1 -\frac{3}{8}\sum_{i=1}^{3} \sum_{j=i+1}^{4} \Bigg[\cos\psi_{ikj}+\frac{1}{3}\Bigg]^2 |
174 |
|
|
\end{equation} |
175 |
plouden |
3909 |
where $\psi_{ikj}$ is the angle formed by the oxygen sites on molecule $k$, and the oxygen site on its two closest neighbors, molecules $i$ and $j$. The local tetrahedral order parameter function has a range of (0,1), where the larger the value $q$ has the more tetrahedral the ordering of the local environment is. A $q$ value of one describes a perfectly tetrahedral environment relative to it and its four nearest neighbors, and the parameter's value decreases as the local ordering becomes less tetrahedral. |
176 |
|
|
|
177 |
|
|
%If the central water molecule has a perfect tetrahedral geometry with its four nearest neighbors, the parameter goes to one, and decreases to zero as the geometry deviates from the ideal configuration. |
178 |
|
|
|
179 |
gezelter |
3897 |
The system was divided into 100 bins along the $z$-axis, and a $q$ value was determined for each snapshot of the system for each bin. The $q$ values for each bin were then averaged to give an average tetrahedrally profile of the system about the $z$- axis. The profile was then fit with a hyperbolic tangent function given by |
180 |
|
|
\begin{equation} |
181 |
|
|
q_{z}=q_{liq}+\frac{q_{ice}-q_{liq}}{2}(\tanh(\alpha(z-I_{L,m}))-\tanh(\alpha(z-I_{R,m}))) |
182 |
|
|
\end{equation} |
183 |
|
|
where $q_{liq}$ and $q_{ice}$ are fitting parameters for the local tetrahedral order parameter for the liquid and ice, $\alpha$ is proportional to the inverse of the width of the interface, and $I_{L,m}$ and $I_{R,m}$ are the midpoints of the left and right interface. During the simulations where a kinetic energy flux was imposed, there was found to be a thermal influence in the liquid phase region of the tetrahedrally profile due to the thermal gradient developed in the system. To maximize the fit of the interface, another term was added to the hyperbolic tangent fitting function, |
184 |
|
|
\begin{equation} |
185 |
|
|
q_{z}=q_{liq}+\frac{q_{ice}-q_{liq}}{2}(\tanh(\alpha(z-I_{L,m}))-\tanh(\alpha(z-I_{R,m})))+\beta|(z-z_{mid})| |
186 |
|
|
\end{equation} |
187 |
|
|
where $\beta$ is a fitting parameter and $z_{mid}$ is the midpoint of the z dimension of the simulation box. |
188 |
|
|
|
189 |
|
|
|
190 |
|
|
\section{Results and discussion} |
191 |
plouden |
3898 |
|
192 |
|
|
%Images to include: 3-long comic strip style of <Vx>, T, q_z as a function of z for the basal and prismatic faces. q_z by z with fit for basal and prismatic. interface width as a function of deltaVx (shear rate) with basal and prismatic on the same plot, error bars in the x and y. <Vx> by flux with basal and prismatic on same graph, back out slope from xmgr and error in slope to get lambda, friction coefficient of interface. |
193 |
|
|
|
194 |
plouden |
3909 |
In Figures \ref{fig:bComic} and \ref{fig:pComic} we see the $z$-dimensional profiles for several parameters for the basal and prismatic systems respectively. 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. 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 can be found 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$5K, can be seen in panel (b). Lastly, the velocity gradient across the systems can be seen in panel (c). From panel (c), we can see liquid phase water molecules 5 to 12 \AA\ from the midpoint of the basal and prismatic interfaces are being dragged along with the ice block. This indicates that the shearing of ice water is in the stick boundary condition. |
195 |
plouden |
3904 |
|
196 |
gezelter |
3914 |
\begin{figure} |
197 |
|
|
\includegraphics[width=\linewidth]{bComicStrip} |
198 |
|
|
\caption{\label{fig:bComic} The basal system: (a) The local tetrahedral order parameter,$q$, (black circles) fit by a hyperbolic tangent (red line), (b) the thermal gradient imposed on the system to maintain a stable interfacial temperature, and (c) the velocity gradient imposed on the system. The verticle dotted lines indicate the midpoint of the interfaces.} |
199 |
|
|
\end{figure} |
200 |
plouden |
3904 |
|
201 |
gezelter |
3914 |
%(a) The local tetrahedral order parameter across the z-dimension of the system (black circles) fit by a hyperbolic tangent (red line). (b) The thermal gradient imposed on the system to maintain a stable interfacial temperature. (c) The velocity gradient imposed on the system. The verticle dotted lines indicate the midpoint of the interfaces. |
202 |
|
|
|
203 |
|
|
\begin{figure} |
204 |
|
|
\includegraphics[width=\linewidth]{pComicStrip} |
205 |
|
|
\caption{\label{fig:pComic} The prismatic system: (a) The local tetrahedral order parameter,$q$, (black circles) fit by a hyperbolic tangent (red line), (b) the thermal gradient imposed on the system to maintain a stable interfacial temperature, and (c) the velocity gradient imposed on the system.The verticle dotted lines indicate the midpoint of the interfaces.} |
206 |
|
|
\end{figure} |
207 |
|
|
|
208 |
|
|
|
209 |
gezelter |
3897 |
\subsection{Interfacial Width} |
210 |
plouden |
3909 |
%For the basal and prismatic systems, the ice blocks were sheared through the water at varying rates while an imposed thermal gradient kept the interface at the stable temperature range as described by Byrk and Haymet. |
211 |
|
|
We found the interfacial width for the basal and prismatic systems to be 3.2$\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}, 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. |
212 |
gezelter |
3897 |
|
213 |
|
|
\subsection{Coefficient of Friction of the Interface} |
214 |
|
|
As the ice is sheared through the liquid, there will be a friction between the ice and the interface. Balasubramanian has shown how to calculate the coefficient of friction for a solid-liquid interface. \cite{Balasumramnian99} |
215 |
|
|
\begin{equation} |
216 |
|
|
%<F_{x}^{w}>_{NE}(t)=-S\lambda_{wall}v_{x}(y_{wall}) |
217 |
|
|
\langle F_{x}^{w}\rangle(t)=-S\lambda_{wall}v_{x}(y_{wall}) |
218 |
|
|
\end{equation} |
219 |
|
|
In this equation, $F_{x}^{w}$ is the total force of all the atoms acting on the fluid, $S$ is the surface area the force is being applied upon, and $\lambda_{wall}$ is the coefficient of friction of the interface. Since the imposed momentum flux, $J_{z}(p_{x})$, is known in the VSS-RNEMD simulations, and the $wall$ is the ice block in our simulations, the above equation can be rewritten as |
220 |
|
|
\begin{equation} |
221 |
|
|
J_{z}(p_{x})=-\lambda_{ice}v_{x}(y_{ice}). |
222 |
|
|
\end{equation} |
223 |
|
|
|
224 |
plouden |
3909 |
In Figure \ref{fig:CoeffFric}, the average velocity of the ice is plotted against the imposed momentum flux for the basal (black circles) and prismatic (red circles) systems. From the equation above, the slope of the linear fit of the data is $\lambda_{wall}$. The coefficient of friction of the interface for the basal face was calculated to be 11.0 $\pm$ 0.4 \AA\textsuperscript{-2}fs\textsuperscript{-1}, and the $\lambda_{wall}$ for the prismatic face was determined to be 19.9, $\pm$ 0.5 \AA\textsuperscript{-2}fs\textsuperscript{-1}. |
225 |
gezelter |
3897 |
|
226 |
plouden |
3904 |
%Ask dan about truncating versus rounding the values for lambda. |
227 |
plouden |
3907 |
%The coefficient of friction of the interface for the basal face was calculated to be 11.02808 $\pm$ 0.4489844 \AA^{-2}fs^{-1}, and the $\lambda_{wall}$ for the prismatic face was determined to be 19.95948, $\pm$ 0.5370894 \AA^{-2}fs^{-1}. |
228 |
gezelter |
3914 |
\begin{figure} |
229 |
|
|
\includegraphics[width=\linewidth]{CoeffFric} |
230 |
|
|
\caption{\label{fig:CoeffFric} The average velocity of the ice for the basal (black circles) and prismatic (red circles) systems as a function off applied momentum flux. The slope of the fit line } |
231 |
|
|
\end{figure} |
232 |
plouden |
3904 |
|
233 |
gezelter |
3897 |
\section{Conclusion} |
234 |
plouden |
3909 |
Here we have simulated the basal and prismatic facets of an SPC/E model of the ice Ih / 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 appearant independence of the shear rate on the interfacial width. The coefficient of friction of the interface was also calculated for each of the facets. The $\lambda_{wall}$ for the basal face was calculated to be 11.0 $\pm$ 0.4 \AA\textsuperscript{-2}fs\textsuperscript{-1}, and 19.9, $\pm$ 0.5 \AA\textsuperscript{-2}fs\textsuperscript{-1} for the prismatic facet. For both facets, the shearing ice water was found to be in the no-slip boundary condition. |
235 |
gezelter |
3897 |
|
236 |
plouden |
3909 |
|
237 |
gezelter |
3914 |
\begin{acknowledgement} |
238 |
|
|
Support for this project was provided by the National Science |
239 |
|
|
Foundation under grant CHE-0848243. Computational time was provided |
240 |
|
|
by the Center for Research Computing (CRC) at the University of |
241 |
|
|
Notre Dame. |
242 |
|
|
\end{acknowledgement} |
243 |
gezelter |
3897 |
|
244 |
gezelter |
3914 |
\newpage |
245 |
|
|
\bibstyle{achemso} |
246 |
plouden |
3909 |
\bibliography{iceWater} |
247 |
gezelter |
3897 |
|
248 |
gezelter |
3914 |
\begin{tocentry} |
249 |
|
|
\begin{wrapfigure}{l}{0.5\textwidth} |
250 |
|
|
\begin{center} |
251 |
|
|
\includegraphics[width=\linewidth]{SystemImage.png} |
252 |
|
|
\end{center} |
253 |
|
|
\end{wrapfigure} |
254 |
|
|
An image of our system. |
255 |
|
|
\end{tocentry} |
256 |
gezelter |
3897 |
|
257 |
gezelter |
3914 |
\end{document} |
258 |
gezelter |
3897 |
|
259 |
plouden |
3904 |
% basal: slope=11.02808, error in slope = 0.4489844 |
260 |
|
|
%prismatic: slope = 19.95948, error in slope = 0.5370894 |