| 301 |
|
rate. Furthermore, various scaling frequencies can be tested for one |
| 302 |
|
single swapping rate. To compare the performance between swapping and |
| 303 |
|
scaling method, temperatures of different dimensions in all the slabs |
| 304 |
< |
were observed. |
| 304 |
> |
were observed. Most of the simulations include $10^5$ steps of |
| 305 |
> |
equilibration without imposing momentum flux, $10^5$ steps of |
| 306 |
> |
stablization with imposing momentum transfer, and $10^6$ steps of data |
| 307 |
> |
collection under RNEMD. For relatively high momentum flux simulations, |
| 308 |
> |
${5\times10^5}$ step data collection is sufficient. For some low momentum |
| 309 |
> |
flux simulations, ${2\times10^6}$ steps were necessary. |
| 310 |
|
|
| 311 |
|
After each simulation, the shear viscosity was calculated in reduced |
| 312 |
|
unit. The momentum flux was calculated with total unphysical |
| 348 |
|
|
| 349 |
|
\section{Results And Discussion} |
| 350 |
|
\subsection{Shear Viscosity} |
| 351 |
+ |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
| 352 |
+ |
produced comparable shear viscosity to swap RNEMD method. In Table |
| 353 |
+ |
\ref{shearRate}, the names of the calculated samples are devided into |
| 354 |
+ |
two parts. The first number refers to total slabs in one simulation |
| 355 |
+ |
box. The second number refers to the swapping interval in swap method, or |
| 356 |
+ |
in scale method the equilvalent swapping interval that the same |
| 357 |
+ |
momentum flux would theoretically result in swap method. All the scale |
| 358 |
+ |
method results were from simulations that had 10 time steps of scaling |
| 359 |
+ |
interval. The average molecular momentum gradients of these samples |
| 360 |
+ |
are shown in Figures \ref{shearGradSwap} and \ref{shearGradScale} |
| 361 |
+ |
respectively. |
| 362 |
+ |
|
| 363 |
+ |
\begin{table*} |
| 364 |
+ |
\begin{minipage}{\linewidth} |
| 365 |
+ |
\begin{center} |
| 366 |
+ |
|
| 367 |
+ |
\caption{Calculation results for shear viscosity of Lennard-Jones |
| 368 |
+ |
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
| 369 |
+ |
methods at various momentum exchange rates. Results in reduced |
| 370 |
+ |
unit. Errors of calculations in parentheses. } |
| 371 |
+ |
|
| 372 |
+ |
\begin{tabular} |
| 373 |
+ |
\hline |
| 374 |
+ |
Name & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
| 375 |
+ |
\hline |
| 376 |
+ |
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
| 377 |
+ |
20-1000 & 3.52(0.16) & -\\ |
| 378 |
+ |
20-2000 & - & 3.32(0.18)\\ |
| 379 |
+ |
20-2500 & - & 3.43(0.08)\\ |
| 380 |
+ |
\end{tabular} |
| 381 |
+ |
\label{shearRate} |
| 382 |
+ |
\end{center} |
| 383 |
+ |
\end{minipage} |
| 384 |
+ |
\end{table*} |
| 385 |
+ |
|
| 386 |
+ |
\begin{figure} |
| 387 |
+ |
\includegraphics[width=\linewidth]{shearGradSwap.eps} |
| 388 |
+ |
\caption{Average momentum gradients of simulations using swap method.} |
| 389 |
+ |
\label{shearGradSwap} |
| 390 |
+ |
\end{figure} |
| 391 |
|
|
| 392 |
+ |
\begin{figure} |
| 393 |
+ |
\includegraphics[width=\linewidth]{shearGradScale.eps} |
| 394 |
+ |
\caption{Average momentum gradients of simulations using scale |
| 395 |
+ |
method.} |
| 396 |
+ |
\label{shearGradScale} |
| 397 |
+ |
\end{figure} |
| 398 |
+ |
|
| 399 |
+ |
\begin{figure} |
| 400 |
+ |
\includegraphics[width=\linewidth]{shearTempScale.eps} |
| 401 |
+ |
\caption{Temperature profile for scaling RNEMD simulation.} |
| 402 |
+ |
\label{shearTempScale} |
| 403 |
+ |
\end{figure} |
| 404 |
+ |
However, observations of temperatures along three dimensions show that |
| 405 |
+ |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
| 406 |
+ |
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
| 407 |
+ |
increased imposed momentum flux, the temperature difference among $x$ |
| 408 |
+ |
and the other two dimensions were larger. This would result from the |
| 409 |
+ |
scaling method. From Eq. \ref{eq:fluxPlane}, after momentum gradient |
| 410 |
+ |
is set up, $P_c^x$ would be roughly stable ($<0$). Consequently, scaling |
| 411 |
+ |
factor $x$ would most probably larger than 1. Therefore, the kinetic |
| 412 |
+ |
energy in $x$-dimension $K_c^x$ would keep increase after most scaling |
| 413 |
+ |
step. And if there is not enough time for the kinetic energy to |
| 414 |
+ |
exchange among different dimensions and different slabs, the system would finally build up temperature (kinetic energy) difference among the three dimensions. |
| 415 |
+ |
Also, between $y$ and $z$ dimensions in the scaled slabs, temperatures of |
| 416 |
+ |
$z$-axis are closer to neighbor slabs. This is due to momentum |
| 417 |
+ |
transfer along $z$ dimension between slabs. |
| 418 |
+ |
|
| 419 |
+ |
Although results between scaling and swapping methods are comparable, |
| 420 |
+ |
the inherent temperature inhomogeneity makes scaling RNEMD method less |
| 421 |
+ |
attractive than swapping RNEMD in shear viscosity calculation. |
| 422 |
+ |
|
| 423 |
+ |
\subsection{Thermal Conductivity} |
| 424 |
+ |
|
| 425 |
+ |
|
| 426 |
+ |
|
| 427 |
|
\section{Acknowledgments} |
| 428 |
|
Support for this project was provided by the National Science |
| 429 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
