| 74 |
|
seem to be investigating how the interfaces is perturbed by the presence of |
| 75 |
|
ions. This is the conlcusion of a recent publication of the basal and |
| 76 |
|
prismatic facets of ice Ih, now presenting the pyramidal and secondary |
| 77 |
< |
prism facets under shear. |
| 77 |
> |
prismatic facets under shear. |
| 78 |
|
|
| 79 |
|
\section{Methodology} |
| 80 |
|
|
| 81 |
|
\begin{figure} |
| 82 |
< |
\includegraphics[width=\linewidth]{SP_comic_strip} |
| 83 |
< |
\caption{\label{fig:spComic} The secondary prism interface with a shear |
| 84 |
< |
rate of 3.5 ms\textsuperscript{-1}. Lower panel: the local tetrahedral order |
| 82 |
> |
\includegraphics[width=\linewidth]{Pyr_comic_strip} |
| 83 |
> |
\caption{\label{fig:pyrComic} The pyramidal interface with a shear |
| 84 |
> |
rate of 3.8 ms\textsuperscript{-1}. Lower panel: the local tetrahedral order |
| 85 |
|
parameter, $q(z)$, (black circles) and the hyperbolic tangent fit (red line). |
| 86 |
|
Middle panel: the imposed thermal gradient required to maintain a fixed |
| 87 |
|
interfacial temperature. Upper panel: the transverse velocity gradient that |
| 90 |
|
\end{figure} |
| 91 |
|
|
| 92 |
|
\begin{figure} |
| 93 |
< |
\includegraphics[width=\linewidth]{Pyr_comic_strip} |
| 94 |
< |
\caption{\label{fig:pyrComic} The pyramidal interface with a shear rate of 3.8 \ |
| 95 |
< |
ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:spComic}.} |
| 93 |
> |
\includegraphics[width=\linewidth]{SP_comic_strip} |
| 94 |
> |
\caption{\label{fig:spComic} The secondary prismatic interface with a shear |
| 95 |
> |
rate of 3.5 \ |
| 96 |
> |
ms\textsuperscript{-1}. Panel descriptions match those in figure \ref{fig:pyrComic}.} |
| 97 |
|
\end{figure} |
| 98 |
|
|
| 99 |
< |
\subsection{Pyramidal and secondary prism system construction} |
| 99 |
> |
\subsection{Pyramidal and secondary prismatic system construction} |
| 100 |
|
|
| 101 |
< |
The construction of the pyramidal and secondary prism systems follows that of |
| 101 |
> |
The construction of the pyramidal and secondary prismatic systems follows that |
| 102 |
> |
of |
| 103 |
|
the basal and prismatic systems presented elsewhere\cite{Louden13}, however |
| 104 |
|
the ice crystals and water boxes were equilibrated and combined at 50K |
| 105 |
|
instead of 225K. The ice / water systems generated were then equilibrated |
| 106 |
|
to 225K. The resulting pyramidal system was |
| 107 |
|
$37.47 \times 29.50 \times 93.02$ \AA\ with 1216 |
| 108 |
|
SPC/E molecules in the ice slab, and 2203 in the liquid phase. The secondary |
| 109 |
< |
prism system generated was $71.87 \times 31.66 \times 161.55$ \AA\ with 3840 |
| 109 |
> |
prismatic system generated was $71.87 \times 31.66 \times 161.55$ \AA\ with |
| 110 |
> |
3840 |
| 111 |
|
SPC/E molecules in the ice slab and 8176 molecules in the liquid phase. |
| 112 |
|
|
| 113 |
|
\subsection{Computational details} |
| 114 |
|
% Do we need to justify the sims at 225K? |
| 115 |
|
% No crystal growth or shrinkage over 2 successive 1 ns NVT simulations for |
| 116 |
< |
% either the pyramidal or sec. prism ice/water systems. |
| 116 |
> |
% either the pyramidal or sec. prismatic ice/water systems. |
| 117 |
|
|
| 118 |
|
The computational details performed here were equivalent to those reported |
| 119 |
|
in our previous publication\cite{Louden13}. The only changes made to the |
| 121 |
|
attempted every 2 fs instead of every 50 fs. This was done to minimize |
| 122 |
|
the magnitude of each individual VSS-RNEMD perturbation to the system. |
| 123 |
|
|
| 124 |
< |
All pyramidal simulations were performed under the NVT ensamble except those |
| 124 |
> |
All pyramidal simulations were performed under the canonical (NVT) ensamble |
| 125 |
> |
except those |
| 126 |
|
during which statistics were accumulated for the orientational correlation |
| 127 |
< |
function, which were performed under the NVE ensamble. All secondary prism |
| 127 |
> |
function, which were performed under the microcanonical (NVE) ensamble. All |
| 128 |
> |
secondary prismatic |
| 129 |
|
simulations were performed under the NVE ensamble. |
| 130 |
|
|
| 131 |
|
\section{Results and discussion} |
| 145 |
|
the systems, parameters that depend on translational motion may give |
| 146 |
|
faulty results. A stuructural parameter will be less effected by the |
| 147 |
|
VSS-RNEMD perturbations to the system. Due to this, we have used the |
| 148 |
< |
local order tetrahedral parameter to quantify the width of the interface, |
| 148 |
> |
local tetrahedral order parameter to quantify the width of the interface, |
| 149 |
|
which was originally described by Kumar\cite{Kumar09} and |
| 150 |
|
Errington\cite{Errington01}, and used by Bryk and Haymet in a previous study |
| 151 |
|
of ice/water interfaces.\cite{Bryk2004b} |
| 171 |
|
$z$-dimension, and the local tetrahedral order parameter, $q(z)$, was |
| 172 |
|
time-averaged for each of the bins, resulting in a tetrahedrality profile of |
| 173 |
|
the system. These profiles are shown across the $z$-dimension of the systems |
| 174 |
< |
in panel $a$ of Figures \ref{fig:spComic} |
| 175 |
< |
and \ref{fig:pyrComic} (black circles). The $q(z)$ function has a range of |
| 174 |
> |
in panel $a$ of Figures \ref{fig:pyrComic} |
| 175 |
> |
and \ref{fig:spComic} (black circles). The $q(z)$ function has a range of |
| 176 |
|
(0,1), where a larger value indicates a more tetrahedral environment. |
| 177 |
|
The $q(z)$ for the bulk liquid was found to be $\approx $ 0.77, while values of |
| 178 |
< |
$\approx $0.92 were more common for the ice. The tetrahedrality profiles were |
| 178 |
> |
$\approx $ 0.92 were more common for the ice. The tetrahedrality profiles were |
| 179 |
|
fit using a hyperbolic tangent\cite{Louden13} designed to smoothly fit the |
| 180 |
|
bulk to ice |
| 181 |
|
transition, while accounting for the thermal influence on the profile by the |
| 182 |
|
kinetic energy exchanges of the VSS-RNEMD moves. In panels $b$ and $c$, the |
| 183 |
< |
imposed thermal and velocity gradients can be seen. The verticle dotted |
| 183 |
> |
resulting thermal and velocity gradients from the imposed kinetic energy and |
| 184 |
> |
momentum fluxes can be seen. The verticle dotted |
| 185 |
|
lines traversing all three panels indicate the midpoints of the interface |
| 186 |
|
as determined by the hyperbolic tangent fit of the tetrahedrality profiles. |
| 187 |
|
|
| 188 |
|
From fitting the tetrahedrality profiles for each of the 0.5 nanosecond |
| 189 |
|
simulations (panel c of Figures \ref{fig:spComic} and \ref{fig:pyrComic}) |
| 190 |
|
by Eq. 6\cite{Louden13},we find the interfacial width to be |
| 191 |
< |
$3.2 \pm 0.2$ and $3.2 \pm 0.2$ \AA\ for the control system with no applied |
| 192 |
< |
momentum flux for both the pyramidal and secondary prism systems. |
| 191 |
> |
3.2 $\pm$ 0.2 and 3.2 $\pm$ 0.2 \AA\ for the control system with no applied |
| 192 |
> |
momentum flux for both the pyramidal and secondary prismatic systems. |
| 193 |
|
Over the range of shear rates investigated, |
| 194 |
< |
$0.6 \pm 0.2 \mathrm{ms}^{-1} \rightarrow 5.6 \pm 0.4 \mathrm{ms}^{-1}$ for |
| 195 |
< |
the pyramidal system and $0.9 \pm 0.3 \mathrm{ms}^{-1} \rightarrow 5.4 \pm 0.1 |
| 196 |
< |
\mathrm{ms}^{-1}$ for the secondary prism, we found no significant change in |
| 197 |
< |
the interfacial width. This follows our previous findings of the basal and |
| 194 |
> |
0.6 $\pm$ 0.2 $\mathrm{ms}^{-1} \rightarrow$ 5.6 $\pm$ 0.4 $\mathrm{ms}^{-1}$ |
| 195 |
> |
for the pyramidal system and 0.9 $\pm$ 0.3 $\mathrm{ms}^{-1} \rightarrow$ 5.4 |
| 196 |
> |
$\pm$ 0.1 $\mathrm{ms}^{-1}$ for the secondary prismatic, we found no |
| 197 |
> |
significant change in the interfacial width. This follows our previous |
| 198 |
> |
findings of the basal and |
| 199 |
|
prismatic systems, in which the interfacial width was invarient of the |
| 200 |
|
shear rate of the ice. The interfacial width of the quiescent basal and |
| 201 |
< |
prismatic systems was found to be $3.2 \pm 0.4$ \AA\ and $3.6 \pm 0.2$ \AA\ |
| 202 |
< |
respectively. Over the range of shear rates investigated, $0.6 \pm 0.3 |
| 203 |
< |
\mathrm{ms}^{-1} \rightarrow 5.3 \pm 0.5 \mathrm{ms}^{-1}$ for the basal |
| 204 |
< |
system and $0.9 \pm 0.2 \mathrm{ms}^{-1} \rightarrow 4.5 \pm 0.1 |
| 205 |
< |
\mathrm{ms}^{-1}$ for the prismatic. |
| 201 |
> |
prismatic systems was found to be 3.2 $\pm$ 0.4 \AA\ and 3.6 $\pm$ 0.2 \AA\ |
| 202 |
> |
respectively, over the range of shear rates investigated, 0.6 $\pm$ 0.3 |
| 203 |
> |
$\mathrm{ms}^{-1} \rightarrow$ 5.3 $\pm$ 0.5 $\mathrm{ms}^{-1}$ for the basal |
| 204 |
> |
system and 0.9 $\pm$ 0.2 $\mathrm{ms}^{-1} \rightarrow$ 4.5 $\pm$ 0.1 |
| 205 |
> |
$\mathrm{ms}^{-1}$ for the prismatic. |
| 206 |
|
|
| 207 |
|
These results indicate that the surface structure of the exposed ice crystal |
| 208 |
|
has little to no effect on how far into the bulk the ice-like structural |
| 229 |
|
wide, and \eqref{C(t)1} was computed for each of the bins. A water |
| 230 |
|
molecule was allocated to a particular bin if it was initially in the bin |
| 231 |
|
at time zero. To compute \eqref{C(t)1}, each 0.5 ns simulation was followed |
| 232 |
< |
by an additional 200 ps microcanonical (NVE) simulation during which the |
| 232 |
> |
by an additional 200 ps NVE simulation during which the |
| 233 |
|
position and orientations of each molecule were recorded every 0.1 ps. |
| 234 |
|
|
| 235 |
|
The data obtained for each bin was then fit to a triexponential decay given by |
| 259 |
|
in panel a, we see that $\tau_{short}$ decreases from the liquid value |
| 260 |
|
of $72-76$ fs as we approach the interface. We believe this speed up is due to |
| 261 |
|
the constrained motion of librations closer to the interface. Both the |
| 262 |
< |
approximate values for the decays and relative trends match those reported |
| 263 |
< |
previously for the basal and prismatic interfaces. |
| 262 |
> |
approximate values for the decays and trends approaching the interface match |
| 263 |
> |
those reported previously for the basal and prismatic interfaces. |
| 264 |
|
|
| 265 |
|
As done previously, we have attempted to quantify the distance, $d_{pyramidal}$ |
| 266 |
< |
and $d_{secondary prism}$, from the |
| 266 |
> |
and $d_{secondary prismatic}$, from the |
| 267 |
|
interface that the deviations from the bulk liquid values begin. This was done |
| 268 |
|
by fitting the orientational decay constant $z$-profiles by |
| 269 |
|
\begin{equation}\label{tauFit} |
| 270 |
< |
\tau(z)\approx\tau_{liquid}+(\tau_{solid}-\tau_{liquid})e^{-(z-z_{wall})/d} |
| 270 |
> |
\tau(z)\approx\tau_{liquid}+(\tau_{wall}-\tau_{liquid})e^{-(z-z_{wall})/d} |
| 271 |
|
\end{equation} |
| 272 |
< |
where $\tau_{liquid}$ and $\tau_{solid}$ are the liquid and projected solid |
| 272 |
> |
where $\tau_{liquid}$ and $\tau_{wall}$ are the liquid and projected wall |
| 273 |
|
values of the decay constants, $z_{wall}$ is the location of the interface, |
| 274 |
|
and $d$ is the displacement from the interface at which these deviations |
| 275 |
|
occur. The values for $d_{pyramidal}$ and $d_{secondary prismatic}$ were |
| 276 |
|
determined |
| 277 |
|
for each of the decay constants, and then averaged for better statistics |
| 278 |
< |
($\tau_{middle}$ was ommitted for secondary prism). For the pyramidal system, |
| 278 |
> |
($\tau_{middle}$ was ommitted for secondary prismatic). For the pyramidal |
| 279 |
> |
system, |
| 280 |
|
$d_{pyramidal}$ was found to be 2.7 \AA\ for both the control and the sheared |
| 281 |
|
system. We found $d_{secondary prismatic}$ to be slightly larger than |
| 282 |
|
$d_{pyramidal}$ for both the control and with an applied shear, with |
| 293 |
|
arrises from the sharp discontinuity of the viscosity for the SPC/E model |
| 294 |
|
at temperatures approaching 200 K\cite{kuang12}. Due to this, we propose |
| 295 |
|
a weighting to the interfacial friction coefficient, $\kappa$ by the |
| 296 |
< |
shear viscosity at 225 K. The interfacial friction coefficient relates |
| 297 |
< |
the shear stress with the relative velocity of the fluid normal to the |
| 296 |
> |
shear viscosity of the fluid at 225 K. The interfacial friction coefficient |
| 297 |
> |
relates the shear stress with the relative velocity of the fluid normal to the |
| 298 |
|
interface: |
| 299 |
|
\begin{equation}\label{Shenyu-13} |
| 300 |
|
j_{z}(p_{x}) = \kappa[v_{x}(fluid)-v_{x}(solid)] |
| 303 |
|
in the |
| 304 |
|
$x$-dimension, and $v_{x}$(fluid) and $v_{x}$(solid) are the velocities |
| 305 |
|
directly adjacent to the interface. The shear viscosity, $\eta(T)$, of the |
| 306 |
< |
fluid can be determined if we assume a linear response of the momentum |
| 306 |
> |
fluid can be determined under a linear response of the momentum |
| 307 |
|
gradient to the applied shear stress by |
| 308 |
|
\begin{equation}\label{Shenyu-11} |
| 309 |
|
j_{z}(p_{x}) = \eta(T) \frac{\partial v_{x}}{\partial z}. |
| 320 |
|
\end{equation} |
| 321 |
|
|
| 322 |
|
To obtain the value of $\eta(225)$ for the SPC/E model, a $31.09 \times 29.38 |
| 323 |
< |
\times 124.39$ \AA\ box with 3744 SPC/E water molecules was equilibrated to |
| 324 |
< |
225K, |
| 323 |
> |
\times 124.39$ \AA\ box with 3744 SPC/E liquid water molecules was |
| 324 |
> |
equilibrated to 225K, |
| 325 |
|
and 5 unique shearing experiments were performed. Each experiment was |
| 326 |
< |
conducted in the microcanonical ensemble (NVE) and were 5 ns in |
| 326 |
> |
conducted in the NVE and were 5 ns in |
| 327 |
|
length. The VSS were attempted every timestep, which was set to 2 fs. |
| 328 |
< |
For our SPC/E systems, we found $\eta(225)$ to be $0.0148 \pm 0.0007$ Pa s, |
| 328 |
> |
For our SPC/E systems, we found $\eta(225)$ to be 0.0148 $\pm$ 0.0007 Pa s, |
| 329 |
|
roughly ten times larger than the value found for 280 K SPC/E bulk water by |
| 330 |
|
Kuang\cite{kuang12}. |
| 331 |
|
|
| 346 |
|
found for the four crystal facets of Ice-I$_\mathrm{h}$ investigated are shown |
| 347 |
|
in Table \ref{tab:kapa}. The basal and pyramidal facets were found to have |
| 348 |
|
similar values of $\kappa \approx$ 0.0006 |
| 349 |
< |
(amu \AA\ \textsuperscript{-2} fs\textsuperscript{-1}), while $\kappa \approx$ |
| 350 |
< |
0.0003 (amu \AA\ \textsuperscript{-2} fs\textsuperscript{-1}) were found for |
| 351 |
< |
the prismatic and secondary prismatic systems. |
| 352 |
< |
These results indicate that the prismatic and secondary prismatic facets are |
| 353 |
< |
more hydrophobic than the basal and pyramidal facets. |
| 349 |
> |
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}), while values of |
| 350 |
> |
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1}) |
| 351 |
> |
were found for the prismatic and secondary prismatic systems. |
| 352 |
> |
These results indicate that the basal and pyramidal facets are |
| 353 |
> |
more hydrophilic than the prismatic and secondary prismatic facets. |
| 354 |
|
%This indicates something about the similarity between the two facets that |
| 355 |
|
%share similar values... |
| 356 |
|
%Maybe find values for kappa for other materials to compare against? |
| 361 |
|
prismatic facets of Ice-I$_\mathrm{h}$} |
| 362 |
|
\label{tab:kappa} |
| 363 |
|
\begin{tabular}{|ccc|} \hline |
| 364 |
< |
& \multicolumn{2}{c|}{$\kappa_{Drag direction}$ (amu \AA\ \textsuperscript{-2} fs\textsuperscript{-1})} \\ |
| 364 |
> |
& \multicolumn{2}{c|}{$\kappa_{Drag direction}$ (amu \AA\textsuperscript{-2} fs\textsuperscript{-1})} \\ |
| 365 |
|
Interface & $\kappa_{x}$ & $\kappa_{y}$ \\ \hline |
| 366 |
|
basal & $0.00059 \pm 0.00003$ & $0.00065 \pm 0.00008$ \\ |
| 367 |
|
prismatic & $0.00030 \pm 0.00002$ & $0.00030 \pm 0.00001$ \\ |
| 368 |
|
pyramidal & $0.00058 \pm 0.00004$ & $0.00061 \pm 0.00005$ \\ |
| 369 |
< |
secondary prism & $0.00035 \pm 0.00001$ & $0.00033 \pm 0.00002$ \\ \hline |
| 369 |
> |
secondary prismatic & $0.00035 \pm 0.00001$ & $0.00033 \pm 0.00002$ \\ \hline |
| 370 |
|
\end{tabular} |
| 371 |
|
\end{table} |
| 372 |
|
|
| 386 |
|
% prismatic (T = 225)\textsuperscript{a} & $0.037 \pm 0.008$ & $0.04 \pm 0.01$ \\ |
| 387 |
|
% prismatic (T = 230) & $0.10 \pm 0.01$ & $0.070 \pm 0.006$\\ |
| 388 |
|
% pyramidal & $0.13 \pm 0.03$ & $0.14 \pm 0.03$ \\ |
| 389 |
< |
% secondary prism & $0.13 \pm 0.02$ & $0.12 \pm 0.03$ \\ \hline |
| 389 |
> |
% secondary prismatic & $0.13 \pm 0.02$ & $0.12 \pm 0.03$ \\ \hline |
| 390 |
|
%\end{tabular} |
| 391 |
|
%\end{table} |
| 392 |
|
|
| 410 |
|
\begin{figure} |
| 411 |
|
\includegraphics[width=\linewidth]{SP-orient-less} |
| 412 |
|
\caption{\label{fig:SPorient} Decay constants for $C_2(t)$ at the secondary |
| 413 |
< |
prism face. Panel descriptions match those in \ref{fig:PyrOrient}.} |
| 413 |
> |
prismatic face. Panel descriptions match those in \ref{fig:PyrOrient}.} |
| 414 |
|
\end{figure} |
| 415 |
|
|
| 416 |
|
|
| 420 |
|
and secondary prismatic facets of an SPC/E model of the |
| 421 |
|
Ice-I$_\mathrm{h}$/water interface. The ice was sheared through the liquid |
| 422 |
|
water while being exposed to a thermal gradient to maintain a stable |
| 423 |
< |
interface by using the minimal perturbing VSS RNEMD method. In agreement with |
| 424 |
< |
our previous findings for the basal and prismatic facets, the interfacial |
| 423 |
> |
interface by using the minimally perturbing VSS RNEMD method. In agreement |
| 424 |
> |
with our previous findings for the basal and prismatic facets, the interfacial |
| 425 |
|
width was found to be independent of shear rate as measured by the local |
| 426 |
|
tetrahedral order parameter. This width was found to be |
| 427 |
< |
3.2~$\pm$0.2~\AA\ for both the pyramidal and the secondary prismatic systems. |
| 427 |
> |
3.2~$\pm$ 0.2~\AA\ for both the pyramidal and the secondary prismatic systems. |
| 428 |
|
These values are in good agreement with our previously calculated interfacial |
| 429 |
< |
widths for the basal (3.2~$\pm$0.4~\AA\ ) and prismatic (3.6~$\pm$0.2~\AA\ ) |
| 429 |
> |
widths for the basal (3.2~$\pm$ 0.4~\AA\ ) and prismatic (3.6~$\pm$ 0.2~\AA\ ) |
| 430 |
|
systems. |
| 431 |
|
|
| 432 |
|
Orientational dynamics of the Ice-I$_\mathrm{h}$/water interfaces were studied |
| 444 |
|
work on the basal and prismatic facets, there appears to be a dynamic |
| 445 |
|
interface width at which deviations from the bulk liquid values occur. |
| 446 |
|
We had previously found $d_{basal}$ and $d_{prismatic}$ to be approximately |
| 447 |
< |
2.8~\AA\ and 3.5~\AA\~. We found good agreement of these values for the |
| 447 |
> |
2.8~\AA\ and 3.5~\AA. We found good agreement of these values for the |
| 448 |
|
pyramidal and secondary prismatic systems with $d_{pyramidal}$ and |
| 449 |
< |
$d_{secondary prism}$ to be 2.7~\AA\ and 3~\AA\~. For all of the facets, there |
| 450 |
< |
was found to be no apparent dependence of the dynamic width on the shear rate. |
| 449 |
> |
$d_{secondary prismatic}$ to be 2.7~\AA\ and 3~\AA. For all four of the |
| 450 |
> |
facets, no apparent dependence of the dynamic width on the shear rate was |
| 451 |
> |
found. |
| 452 |
|
|
| 453 |
|
%Paragraph summarizing the Kappa values |
| 454 |
< |
The interfacial friction coefficient, $\kappa$, was determined for each of the |
| 455 |
< |
interfaces. We were able to reach an expression for $\kappa$ as a function of |
| 456 |
< |
the velocity profile of the system and is scaled by the viscosity of the liquid |
| 454 |
> |
The interfacial friction coefficient, $\kappa$, was determined for each facet |
| 455 |
> |
interface. We were able to reach an expression for $\kappa$ as a function of |
| 456 |
> |
the velocity profile of the system which is scaled by the viscosity of the liquid |
| 457 |
|
at 225 K. In doing so, we have obtained an expression for $\kappa$ which is |
| 458 |
|
independent of temperature differences of the liquid water at far displacements |
| 459 |
|
from the interface. We found the basal and pyramidal facets to have |
| 460 |
|
similar $\kappa$ values, of $\kappa \approx$ 0.0006 |
| 461 |
< |
(amu \AA\ \textsuperscript{-2} fs\textsuperscript{-1}). However, the |
| 461 |
> |
(amu \AA\textsuperscript{-2} fs\textsuperscript{-1}). However, the |
| 462 |
|
prismatic and secondary prismatic facets were found to have $\kappa$ values of |
| 463 |
< |
$\kappa \approx$ 0.0003 (amu \AA\ \textsuperscript{-2} fs\textsuperscript{-1}). |
| 463 |
> |
$\kappa \approx$ 0.0003 (amu \AA\textsuperscript{-2} fs\textsuperscript{-1}). |
| 464 |
|
As these ice facets are being sheared relative to liquid water, with the |
| 465 |
|
structural and dynamic width of all four |
| 466 |
|
interfaces being approximately the same, the difference in the coefficient of |
