| 75 |
|
from liquid copper to monatomic liquids to molecular fluids |
| 76 |
|
(e.g. ionic liquids).\cite{ISI:000246190100032} |
| 77 |
|
|
| 78 |
+ |
\begin{figure} |
| 79 |
+ |
\includegraphics[width=\linewidth]{thermalDemo} |
| 80 |
+ |
\caption{Demostration of thermal gradient estalished by RNEMD method.} |
| 81 |
+ |
\label{thermalDemo} |
| 82 |
+ |
\end{figure} |
| 83 |
+ |
|
| 84 |
|
RNEMD is preferable in many ways to the forward NEMD methods because |
| 85 |
|
it imposes what is typically difficult to measure (a flux or stress) |
| 86 |
|
and it is typically much easier to compute momentum gradients or |
| 97 |
|
typically samples from the same manifold of states in the |
| 98 |
|
microcanonical ensemble. |
| 99 |
|
|
| 100 |
< |
Recently, Tenney and Maginn have discovered some problems with the |
| 101 |
< |
original RNEMD swap technique. Notably, large momentum fluxes |
| 102 |
< |
(equivalent to frequent momentum swaps between the slabs) can result |
| 103 |
< |
in "notched", "peaked" and generally non-thermal momentum |
| 100 |
> |
Recently, Tenney and Maginn\cite{ISI:000273472300004} have discovered |
| 101 |
> |
some problems with the original RNEMD swap technique. Notably, large |
| 102 |
> |
momentum fluxes (equivalent to frequent momentum swaps between the |
| 103 |
> |
slabs) can result in ``notched'', ``peaked'' and generally non-thermal momentum |
| 104 |
|
distributions in the two slabs, as well as non-linear thermal and |
| 105 |
|
velocity distributions along the direction of the imposed flux ($z$). |
| 106 |
|
Tenney and Maginn obtained reasonable limits on imposed flux and |
| 129 |
|
Rather than using momentum swaps, we use a series of velocity scaling |
| 130 |
|
moves. For molecules $\{i\}$ located within the cold slab, |
| 131 |
|
\begin{equation} |
| 132 |
< |
\vec{v}_i \leftarrow \left( \begin{array}{c} |
| 133 |
< |
x \\ |
| 134 |
< |
y \\ |
| 135 |
< |
z \\ |
| 132 |
> |
\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
| 133 |
> |
x & 0 & 0 \\ |
| 134 |
> |
0 & y & 0 \\ |
| 135 |
> |
0 & 0 & z \\ |
| 136 |
|
\end{array} \right) \cdot \vec{v}_i |
| 137 |
|
\end{equation} |
| 138 |
|
where ${x, y, z}$ are a set of 3 scaling variables for each of the |
| 139 |
|
three directions in the system. Likewise, the molecules $\{j\}$ |
| 140 |
|
located in the hot slab will see a concomitant scaling of velocities, |
| 141 |
|
\begin{equation} |
| 142 |
< |
\vec{v}_j \leftarrow \left( \begin{array}{c} |
| 143 |
< |
x^\prime \\ |
| 144 |
< |
y^\prime \\ |
| 145 |
< |
z^\prime \\ |
| 142 |
> |
\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
| 143 |
> |
x^\prime & 0 & 0 \\ |
| 144 |
> |
0 & y^\prime & 0 \\ |
| 145 |
> |
0 & 0 & z^\prime \\ |
| 146 |
|
\end{array} \right) \cdot \vec{v}_j |
| 147 |
|
\end{equation} |
| 148 |
|
|
| 153 |
|
P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
| 154 |
|
\end{equation} |
| 155 |
|
where |
| 156 |
< |
\begin{equation} |
| 151 |
< |
\begin{array}{rcl} |
| 156 |
> |
\begin{eqnarray} |
| 157 |
|
P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
| 158 |
< |
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha \\ |
| 154 |
< |
\end{array} |
| 158 |
> |
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha |
| 159 |
|
\label{eq:momentumdef} |
| 160 |
< |
\end{equation} |
| 160 |
> |
\end{eqnarray} |
| 161 |
|
Therefore, for each of the three directions, the hot scaling |
| 162 |
|
parameters are a simple function of the cold scaling parameters and |
| 163 |
|
the instantaneous linear momentum in each of the two slabs. |
| 174 |
|
Conservation of total energy also places constraints on the scaling: |
| 175 |
|
\begin{equation} |
| 176 |
|
\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
| 177 |
< |
\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha. |
| 177 |
> |
\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha |
| 178 |
|
\end{equation} |
| 179 |
|
where the kinetic energies, $K_h^\alpha$ and $K_c^\alpha$, are computed |
| 180 |
|
for each of the three directions in a similar manner to the linear momenta |
| 182 |
|
hot scaling parameters ($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), |
| 183 |
|
we obtain the {\it constraint ellipsoid equation}: |
| 184 |
|
\begin{equation} |
| 185 |
< |
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0, |
| 185 |
> |
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0 |
| 186 |
|
\label{eq:constraintEllipsoid} |
| 187 |
|
\end{equation} |
| 188 |
|
where the constants are obtained from the instantaneous values of the |
| 189 |
|
linear momenta and kinetic energies for the hot and cold slabs, |
| 190 |
< |
\begin{equation} |
| 187 |
< |
\begin{array}{rcl} |
| 190 |
> |
\begin{eqnarray} |
| 191 |
|
a_\alpha & = &\left(K_c^\alpha + K_h^\alpha |
| 192 |
|
\left(p_\alpha\right)^2\right) \\ |
| 193 |
|
b_\alpha & = & -2 K_h^\alpha p_\alpha \left( 1 + p_\alpha\right) \\ |
| 194 |
< |
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha \\ |
| 192 |
< |
\end{array} |
| 194 |
> |
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
| 195 |
|
\label{eq:constraintEllipsoidConsts} |
| 196 |
< |
\end{equation} |
| 196 |
> |
\end{eqnarray} |
| 197 |
|
This ellipsoid equation defines the set of cold slab scaling |
| 198 |
|
parameters which can be applied while preserving both linear momentum |
| 199 |
|
in all three directions as well as kinetic energy. |
| 220 |
|
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
| 221 |
|
flux ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
| 222 |
|
the two ellipsoids in 3-dimensional space. |
| 223 |
+ |
|
| 224 |
+ |
\begin{figure} |
| 225 |
+ |
\includegraphics[width=\linewidth]{ellipsoids} |
| 226 |
+ |
\caption{Scaling points which maintain both constant energy and |
| 227 |
+ |
constant linear momentum of the system lie on the surface of the |
| 228 |
+ |
{\it constraint ellipsoid} while points which generate the target |
| 229 |
+ |
momentum flux lie on the surface of the {\it flux ellipsoid}. The |
| 230 |
+ |
velocity distributions in the hot bin are scaled by only those |
| 231 |
+ |
points which lie on both ellipsoids.} |
| 232 |
+ |
\label{ellipsoids} |
| 233 |
+ |
\end{figure} |
| 234 |
|
|
| 235 |
|
One may also define momentum flux (say along the x-direction) as: |
| 236 |
|
\begin{equation} |
| 237 |
< |
(1-x) P_c^x = j_z(p_x)\Delta t |
| 237 |
> |
(1-x) P_c^x = j_z(p_x)\Delta t |
| 238 |
|
\label{eq:fluxPlane} |
| 239 |
|
\end{equation} |
| 240 |
|
The above {\it flux equation} is essentially a plane which is |
| 287 |
|
|
| 288 |
|
\section{Computational Details} |
| 289 |
|
Our simulation consists of a series of systems. All of these |
| 290 |
< |
simulations were run with the OOPSE simulation software |
| 290 |
> |
simulations were run with the OpenMD simulation software |
| 291 |
|
package\cite{Meineke:2005gd} integrated with RNEMD methods. |
| 292 |
|
|
| 293 |
|
A Lennard-Jones fluid system was built and tested first. In order to |
| 297 |
|
\times 10.06\sigma \times 30.18\sigma}$ by size. The reduced density |
| 298 |
|
${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled direct |
| 299 |
|
comparison between our results and others. These simulations used |
| 300 |
< |
Verlet time-stepping algorithm with reduced timestep ${\tau^* = |
| 300 |
> |
velocity Verlet algorithm with reduced timestep ${\tau^* = |
| 301 |
|
4.6\times10^{-4}}$. |
| 302 |
|
|
| 303 |
|
For shear viscosity calculation, the reduced temperature was ${T^* = |
| 304 |
< |
k_B T/\varepsilon = 0.72}$. Simulations were run in microcanonical |
| 305 |
< |
ensemble (NVE). For the swapping part, Muller-Plathe's algorithm was |
| 304 |
> |
k_B T/\varepsilon = 0.72}$. Simulations were first equilibrated in canonical |
| 305 |
> |
ensemble (NVT), then equilibrated in microcanonical ensemble |
| 306 |
> |
(NVE). Establishing and stablizing momentum gradient were followed |
| 307 |
> |
also in NVE ensemble. For the swapping part, Muller-Plathe's algorithm was |
| 308 |
|
adopted.\cite{ISI:000080382700030} The simulation box was under |
| 309 |
|
periodic boundary condition, and devided into ${N = 20}$ slabs. In each swap, |
| 310 |
|
the top slab ${(n = 1)}$ exchange the most negative $x$ momentum with the |
| 311 |
|
most positive $x$ momentum in the center slab ${(n = N/2 + 1)}$. Referred |
| 312 |
< |
to Tenney {\it et al.}\cite{tenneyANDmaginn}, a series of swapping |
| 312 |
> |
to Tenney {\it et al.}\cite{ISI:000273472300004}, a series of swapping |
| 313 |
|
frequency were chosen. According to each result from swapping |
| 314 |
|
RNEMD, scaling RNEMD simulations were run with the target momentum |
| 315 |
|
flux set to produce a similar momentum flux and shear |
| 316 |
|
rate. Furthermore, various scaling frequencies can be tested for one |
| 317 |
|
single swapping rate. To compare the performance between swapping and |
| 318 |
|
scaling method, temperatures of different dimensions in all the slabs |
| 319 |
< |
were observed. |
| 319 |
> |
were observed. Most of the simulations include $10^5$ steps of |
| 320 |
> |
equilibration without imposing momentum flux, $10^5$ steps of |
| 321 |
> |
stablization with imposing momentum transfer, and $10^6$ steps of data |
| 322 |
> |
collection under RNEMD. For relatively high momentum flux simulations, |
| 323 |
> |
${5\times10^5}$ step data collection is sufficient. For some low momentum |
| 324 |
> |
flux simulations, ${2\times10^6}$ steps were necessary. |
| 325 |
|
|
| 326 |
|
After each simulation, the shear viscosity was calculated in reduced |
| 327 |
|
unit. The momentum flux was calculated with total unphysical |
| 328 |
< |
transferred momentum ${P_x}$ and simulation time $t$: |
| 328 |
> |
transferred momentum ${P_x}$ and data collection time $t$: |
| 329 |
|
\begin{equation} |
| 330 |
|
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
| 331 |
|
\end{equation} |
| 338 |
|
For thermal conductivity calculation, simulations were first run under |
| 339 |
|
reduced temperature ${T^* = 0.72}$ in NVE ensemble. Muller-Plathe's |
| 340 |
|
algorithm was adopted in the swapping method. Under identical |
| 341 |
< |
simulation box, in each swap, the top slab exchange the molecule with |
| 342 |
< |
least kinetic energy with the molecule in the center slab with most |
| 343 |
< |
kinetic energy, unless this ``coldest'' molecule in the ``hot'' slab |
| 344 |
< |
is still ``hotter'' than the ``hottest'' molecule in the ``cold'' |
| 341 |
> |
simulation box parameters, in each swap, the top slab exchange the |
| 342 |
> |
molecule with least kinetic energy with the molecule in the center |
| 343 |
> |
slab with most kinetic energy, unless this ``coldest'' molecule in the |
| 344 |
> |
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the ``cold'' |
| 345 |
|
slab. According to swapping RNEMD results, target energy flux for |
| 346 |
|
scaling RNEMD simulations can be set. Also, various scaling |
| 347 |
|
frequencies can be tested for one target energy flux. To compare the |
| 350 |
|
|
| 351 |
|
For each swapping rate, thermal conductivity was calculated in reduced |
| 352 |
|
unit. The energy flux was calculated similarly to the momentum flux, |
| 353 |
< |
with total unphysical transferred energy ${E_{total}}$ and simulation |
| 353 |
> |
with total unphysical transferred energy ${E_{total}}$ and data collection |
| 354 |
|
time $t$: |
| 355 |
|
\begin{equation} |
| 356 |
|
J_z = \frac{E_{total}}{2 t L_x L_y} |
| 360 |
|
profile. From the thermal conductivity $\lambda$ calculated, one can |
| 361 |
|
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
| 362 |
|
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
| 363 |
+ |
|
| 364 |
+ |
Another series of our simulation is to calculate the interfacial |
| 365 |
+ |
thermal conductivity of a Au/H$_2$O system. Respective calculations of |
| 366 |
+ |
water (SPC/E) and gold (QSC) thermal conductivity were performed and |
| 367 |
+ |
compared with current results to ensure the validity of |
| 368 |
+ |
NIVS-RNEMD. After that, a mixture system was simulated. |
| 369 |
+ |
|
| 370 |
+ |
For thermal conductivity calculation of bulk water, a simulation box |
| 371 |
+ |
consisting of 1000 molecules were first equilibrated under ambient |
| 372 |
+ |
pressure and temperature conditions (NPT), followed by equilibration |
| 373 |
+ |
in fixed volume (NVT). The system was then equilibrated in |
| 374 |
+ |
microcanonical ensemble (NVE). Also in NVE ensemble, establishing |
| 375 |
+ |
stable thermal gradient was followed. The simulation box was under |
| 376 |
+ |
periodic boundary condition and devided into 10 slabs. Data collection |
| 377 |
+ |
process was similar to Lennard-Jones fluid system. Thermal |
| 378 |
+ |
conductivity calculation of bulk crystal gold used a similar |
| 379 |
+ |
protocol. And the face centered cubic crystal simulation box consists |
| 380 |
+ |
of 2880 Au atoms. |
| 381 |
|
|
| 382 |
+ |
After simulations of bulk water and crystal gold, a mixture system was |
| 383 |
+ |
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
| 384 |
+ |
molecules. Spohr potential was adopted in depicting the interaction |
| 385 |
+ |
between metal atom and water molecule.\cite{ISI:000167766600035} A |
| 386 |
+ |
similar protocol of equilibration was followed. A thermal gradient was |
| 387 |
+ |
built. It was found out that compared to homogeneous systems, the two |
| 388 |
+ |
phases could have large temperature difference under a relatively low |
| 389 |
+ |
thermal flux. Therefore, under our low flux condition, it is assumed |
| 390 |
+ |
that the metal and water phases have respectively homogeneous |
| 391 |
+ |
temperature. In calculating the interfacial thermal conductivity $G$, |
| 392 |
+ |
this assumptioin was applied and thus our formula becomes: |
| 393 |
+ |
|
| 394 |
+ |
\begin{equation} |
| 395 |
+ |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
| 396 |
+ |
\langle T_{water}\rangle \right)} |
| 397 |
+ |
\label{interfaceCalc} |
| 398 |
+ |
\end{equation} |
| 399 |
+ |
where ${E_{total}}$ is the imposed unphysical kinetic energy transfer |
| 400 |
+ |
and ${\langle T_{gold}\rangle}$ and ${\langle T_{water}\rangle}$ are the |
| 401 |
+ |
average observed temperature of gold and water phases respectively. |
| 402 |
+ |
|
| 403 |
|
\section{Results And Discussion} |
| 404 |
|
\subsection{Shear Viscosity} |
| 405 |
+ |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
| 406 |
+ |
produced comparable shear viscosity to swap RNEMD method. In Table |
| 407 |
+ |
\ref{shearRate}, the names of the calculated samples are devided into |
| 408 |
+ |
two parts. The first number refers to total slabs in one simulation |
| 409 |
+ |
box. The second number refers to the swapping interval in swap method, or |
| 410 |
+ |
in scale method the equilvalent swapping interval that the same |
| 411 |
+ |
momentum flux would theoretically result in swap method. All the scale |
| 412 |
+ |
method results were from simulations that had a scaling interval of 10 |
| 413 |
+ |
time steps. The average molecular momentum gradients of these samples |
| 414 |
+ |
are shown in Figure \ref{shearGrad}. |
| 415 |
|
|
| 416 |
+ |
\begin{table*} |
| 417 |
+ |
\begin{minipage}{\linewidth} |
| 418 |
+ |
\begin{center} |
| 419 |
+ |
|
| 420 |
+ |
\caption{Calculation results for shear viscosity of Lennard-Jones |
| 421 |
+ |
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
| 422 |
+ |
methods at various momentum exchange rates. Results in reduced |
| 423 |
+ |
unit. Errors of calculations in parentheses. } |
| 424 |
+ |
|
| 425 |
+ |
\begin{tabular}{ccc} |
| 426 |
+ |
\hline |
| 427 |
+ |
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
| 428 |
+ |
\hline |
| 429 |
+ |
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
| 430 |
+ |
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
| 431 |
+ |
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
| 432 |
+ |
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
| 433 |
+ |
\hline |
| 434 |
+ |
\end{tabular} |
| 435 |
+ |
\label{shearRate} |
| 436 |
+ |
\end{center} |
| 437 |
+ |
\end{minipage} |
| 438 |
+ |
\end{table*} |
| 439 |
+ |
|
| 440 |
+ |
\begin{figure} |
| 441 |
+ |
\includegraphics[width=\linewidth]{shearGrad} |
| 442 |
+ |
\caption{Average momentum gradients of shear viscosity simulations} |
| 443 |
+ |
\label{shearGrad} |
| 444 |
+ |
\end{figure} |
| 445 |
+ |
|
| 446 |
+ |
\begin{figure} |
| 447 |
+ |
\includegraphics[width=\linewidth]{shearTempScale} |
| 448 |
+ |
\caption{Temperature profile for scaling RNEMD simulation.} |
| 449 |
+ |
\label{shearTempScale} |
| 450 |
+ |
\end{figure} |
| 451 |
+ |
However, observations of temperatures along three dimensions show that |
| 452 |
+ |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
| 453 |
+ |
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
| 454 |
+ |
relatively large imposed momentum flux, the temperature difference among $x$ |
| 455 |
+ |
and the other two dimensions was significant. This would result from the |
| 456 |
+ |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
| 457 |
+ |
momentum gradient is set up, $P_c^x$ would be roughly stable |
| 458 |
+ |
($<0$). Consequently, scaling factor $x$ would most probably larger |
| 459 |
+ |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
| 460 |
+ |
keep increase after most scaling steps. And if there is not enough time |
| 461 |
+ |
for the kinetic energy to exchange among different dimensions and |
| 462 |
+ |
different slabs, the system would finally build up temperature |
| 463 |
+ |
(kinetic energy) difference among the three dimensions. Also, between |
| 464 |
+ |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
| 465 |
+ |
are closer to neighbor slabs. This is due to momentum transfer along |
| 466 |
+ |
$z$ dimension between slabs. |
| 467 |
+ |
|
| 468 |
+ |
Although results between scaling and swapping methods are comparable, |
| 469 |
+ |
the inherent temperature inhomogeneity even in relatively low imposed |
| 470 |
+ |
exchange momentum flux simulations makes scaling RNEMD method less |
| 471 |
+ |
attractive than swapping RNEMD in shear viscosity calculation. |
| 472 |
+ |
|
| 473 |
+ |
\subsection{Thermal Conductivity} |
| 474 |
+ |
\subsubsection{Lennard-Jones Fluid} |
| 475 |
+ |
Our thermal conductivity calculations also show that scaling method results |
| 476 |
+ |
agree with swapping method. Table \ref{thermal} lists our simulation |
| 477 |
+ |
results with similar manner we used in shear viscosity |
| 478 |
+ |
calculation. All the data reported from scaling method were obtained |
| 479 |
+ |
by simulations of 10-step exchange frequency, and the target exchange |
| 480 |
+ |
kinetic energy were set to produce equivalent kinetic energy flux as |
| 481 |
+ |
in swapping method. Figure \ref{thermalGrad} exhibits similar thermal |
| 482 |
+ |
gradients of respective similar kinetic energy flux. |
| 483 |
+ |
|
| 484 |
+ |
\begin{table*} |
| 485 |
+ |
\begin{minipage}{\linewidth} |
| 486 |
+ |
\begin{center} |
| 487 |
+ |
|
| 488 |
+ |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
| 489 |
+ |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
| 490 |
+ |
swap and scale methods at various kinetic energy exchange rates. Results |
| 491 |
+ |
in reduced unit. Errors of calculations in parentheses.} |
| 492 |
+ |
|
| 493 |
+ |
\begin{tabular}{ccc} |
| 494 |
+ |
\hline |
| 495 |
+ |
Series & $\lambda^*_{swap}$ & $\lambda^*_{scale}$\\ |
| 496 |
+ |
\hline |
| 497 |
+ |
20-250 & 7.03(0.34) & 7.30(0.10)\\ |
| 498 |
+ |
20-500 & 7.03(0.14) & 6.95(0.09)\\ |
| 499 |
+ |
20-1000 & 6.91(0.42) & 7.19(0.07)\\ |
| 500 |
+ |
20-2000 & 7.52(0.15) & 7.19(0.28)\\ |
| 501 |
+ |
\hline |
| 502 |
+ |
\end{tabular} |
| 503 |
+ |
\label{thermal} |
| 504 |
+ |
\end{center} |
| 505 |
+ |
\end{minipage} |
| 506 |
+ |
\end{table*} |
| 507 |
+ |
|
| 508 |
+ |
\begin{figure} |
| 509 |
+ |
\includegraphics[width=\linewidth]{thermalGrad} |
| 510 |
+ |
\caption{Temperature gradients of thermal conductivity simulations} |
| 511 |
+ |
\label{thermalGrad} |
| 512 |
+ |
\end{figure} |
| 513 |
+ |
|
| 514 |
+ |
During these simulations, molecule velocities were recorded in 1000 of |
| 515 |
+ |
all the snapshots. These velocity data were used to produce histograms |
| 516 |
+ |
of velocity and speed distribution in different slabs. From these |
| 517 |
+ |
histograms, it is observed that with increasing unphysical kinetic |
| 518 |
+ |
energy flux, speed and velocity distribution of molecules in slabs |
| 519 |
+ |
where swapping occured could deviate from Maxwell-Boltzmann |
| 520 |
+ |
distribution. Figure \ref{histSwap} indicates how these distributions |
| 521 |
+ |
deviate from ideal condition. In high temperature slabs, probability |
| 522 |
+ |
density in low speed is confidently smaller than ideal distribution; |
| 523 |
+ |
in low temperature slabs, probability density in high speed is smaller |
| 524 |
+ |
than ideal. This phenomenon is observable even in our relatively low |
| 525 |
+ |
swapping rate simulations. And this deviation could also leads to |
| 526 |
+ |
deviation of distribution of velocity in various dimensions. One |
| 527 |
+ |
feature of these deviated distribution is that in high temperature |
| 528 |
+ |
slab, the ideal Gaussian peak was changed into a relatively flat |
| 529 |
+ |
plateau; while in low temperature slab, that peak appears sharper. |
| 530 |
+ |
|
| 531 |
+ |
\begin{figure} |
| 532 |
+ |
\includegraphics[width=\linewidth]{histSwap} |
| 533 |
+ |
\caption{Speed distribution for thermal conductivity using swapping RNEMD.} |
| 534 |
+ |
\label{histSwap} |
| 535 |
+ |
\end{figure} |
| 536 |
+ |
|
| 537 |
+ |
\begin{figure} |
| 538 |
+ |
\includegraphics[width=\linewidth]{histScale} |
| 539 |
+ |
\caption{Speed distribution for thermal conductivity using scaling RNEMD.} |
| 540 |
+ |
\label{histScale} |
| 541 |
+ |
\end{figure} |
| 542 |
+ |
|
| 543 |
+ |
\subsubsection{SPC/E Water} |
| 544 |
+ |
Our results of SPC/E water thermal conductivity are comparable to |
| 545 |
+ |
Bedrov {\it et al.}\cite{ISI:000090151400044}, which employed the |
| 546 |
+ |
previous swapping RNEMD method for their calculation. Our simulations |
| 547 |
+ |
were able to produce a similar temperature gradient to their |
| 548 |
+ |
system. However, the average temperature of our system is 300K, while |
| 549 |
+ |
theirs is 318K, which would be attributed for part of the difference |
| 550 |
+ |
between the two series of results. Both methods yields values in |
| 551 |
+ |
agreement with experiment. And this shows the applicability of our |
| 552 |
+ |
method to multi-atom molecular system. |
| 553 |
+ |
|
| 554 |
+ |
\begin{figure} |
| 555 |
+ |
\includegraphics[width=\linewidth]{spceGrad} |
| 556 |
+ |
\caption{Temperature gradients for SPC/E water thermal conductivity.} |
| 557 |
+ |
\label{spceGrad} |
| 558 |
+ |
\end{figure} |
| 559 |
+ |
|
| 560 |
+ |
\begin{table*} |
| 561 |
+ |
\begin{minipage}{\linewidth} |
| 562 |
+ |
\begin{center} |
| 563 |
+ |
|
| 564 |
+ |
\caption{Calculation results for thermal conductivity of SPC/E water |
| 565 |
+ |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
| 566 |
+ |
calculations in parentheses. } |
| 567 |
+ |
|
| 568 |
+ |
\begin{tabular}{cccc} |
| 569 |
+ |
\hline |
| 570 |
+ |
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
| 571 |
+ |
& This work & Previous simulations\cite{ISI:000090151400044} & |
| 572 |
+ |
Experiment$^a$\\ |
| 573 |
+ |
\hline |
| 574 |
+ |
0.38 & 0.816(0.044) & & 0.64\\ |
| 575 |
+ |
0.81 & 0.770(0.008) & 0.784\\ |
| 576 |
+ |
1.54 & 0.813(0.007) & 0.730\\ |
| 577 |
+ |
\hline |
| 578 |
+ |
\end{tabular} |
| 579 |
+ |
\label{spceThermal} |
| 580 |
+ |
\end{center} |
| 581 |
+ |
\end{minipage} |
| 582 |
+ |
\end{table*} |
| 583 |
+ |
|
| 584 |
+ |
\subsubsection{Crystal Gold} |
| 585 |
+ |
Our results of gold thermal conductivity used QSC force field are |
| 586 |
+ |
shown in Table \ref{AuThermal}. Although our calculation is smaller |
| 587 |
+ |
than experimental value by an order of more than 100, this difference |
| 588 |
+ |
is mainly attributed to the lack of electron interaction |
| 589 |
+ |
representation in our force field parameters. Richardson {\it et |
| 590 |
+ |
al.}\cite{ISI:A1992HX37800010} used similar force field parameters |
| 591 |
+ |
in their metal thermal conductivity calculations. The EMD method they |
| 592 |
+ |
employed in their simulations produced comparable results to |
| 593 |
+ |
ours. Therefore, it is confident to conclude that NIVS-RNEMD is |
| 594 |
+ |
applicable to metal force field system. |
| 595 |
+ |
|
| 596 |
+ |
\begin{figure} |
| 597 |
+ |
\includegraphics[width=\linewidth]{AuGrad} |
| 598 |
+ |
\caption{Temperature gradients for crystal gold thermal conductivity.} |
| 599 |
+ |
\label{AuGrad} |
| 600 |
+ |
\end{figure} |
| 601 |
+ |
|
| 602 |
+ |
\begin{table*} |
| 603 |
+ |
\begin{minipage}{\linewidth} |
| 604 |
+ |
\begin{center} |
| 605 |
+ |
|
| 606 |
+ |
\caption{Calculation results for thermal conductivity of crystal gold |
| 607 |
+ |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
| 608 |
+ |
calculations in parentheses. } |
| 609 |
+ |
|
| 610 |
+ |
\begin{tabular}{ccc} |
| 611 |
+ |
\hline |
| 612 |
+ |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
| 613 |
+ |
& This work & Previous simulations\cite{ISI:A1992HX37800010} \\ |
| 614 |
+ |
\hline |
| 615 |
+ |
1.44 & 1.10(0.01) & \\ |
| 616 |
+ |
2.86 & 1.08(0.02) & \\ |
| 617 |
+ |
5.14 & 1.15(0.01) & \\ |
| 618 |
+ |
\hline |
| 619 |
+ |
\end{tabular} |
| 620 |
+ |
\label{AuThermal} |
| 621 |
+ |
\end{center} |
| 622 |
+ |
\end{minipage} |
| 623 |
+ |
\end{table*} |
| 624 |
+ |
|
| 625 |
+ |
\subsection{Interfaciel Thermal Conductivity} |
| 626 |
+ |
After valid simulations of homogeneous water and gold systems using |
| 627 |
+ |
NIVS-RNEMD method, calculation of gold/water interfacial thermal |
| 628 |
+ |
conductivity was followed. It is found out that the interfacial |
| 629 |
+ |
conductance is low due to a hydrophobic surface in our system. Figure |
| 630 |
+ |
\ref{interfaceDensity} demonstrates this observance. Consequently, our |
| 631 |
+ |
reported results (Table \ref{interfaceRes}) are of two orders of |
| 632 |
+ |
magnitude smaller than our calculations on homogeneous systems. |
| 633 |
+ |
|
| 634 |
+ |
\begin{figure} |
| 635 |
+ |
\includegraphics[width=\linewidth]{interfaceDensity} |
| 636 |
+ |
\caption{Density profile for interfacial thermal conductivity |
| 637 |
+ |
simulation box.} |
| 638 |
+ |
\label{interfaceDensity} |
| 639 |
+ |
\end{figure} |
| 640 |
+ |
|
| 641 |
+ |
\begin{figure} |
| 642 |
+ |
\includegraphics[width=\linewidth]{interfaceGrad} |
| 643 |
+ |
\caption{Temperature profiles for interfacial thermal conductivity |
| 644 |
+ |
simulation box.} |
| 645 |
+ |
\label{interfaceGrad} |
| 646 |
+ |
\end{figure} |
| 647 |
+ |
|
| 648 |
+ |
\begin{table*} |
| 649 |
+ |
\begin{minipage}{\linewidth} |
| 650 |
+ |
\begin{center} |
| 651 |
+ |
|
| 652 |
+ |
\caption{Calculation results for interfacial thermal conductivity |
| 653 |
+ |
at ${\langle T\rangle \sim}$ 300K at various thermal exchange |
| 654 |
+ |
rates. Errors of calculations in parentheses. } |
| 655 |
+ |
|
| 656 |
+ |
\begin{tabular}{cccc} |
| 657 |
+ |
\hline |
| 658 |
+ |
$J_z$(MW/m$^2$) & $T_{gold}$ & $T_{water}$ & $G$(MW/m$^2$/K)\\ |
| 659 |
+ |
\hline |
| 660 |
+ |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
| 661 |
+ |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
| 662 |
+ |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
| 663 |
+ |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
| 664 |
+ |
\hline |
| 665 |
+ |
\end{tabular} |
| 666 |
+ |
\label{interfaceRes} |
| 667 |
+ |
\end{center} |
| 668 |
+ |
\end{minipage} |
| 669 |
+ |
\end{table*} |
| 670 |
+ |
|
| 671 |
+ |
\section{Conclusions} |
| 672 |
+ |
NIVS-RNEMD simulation method is developed and tested on various |
| 673 |
+ |
systems. Simulation results demonstrate its validity of thermal |
| 674 |
+ |
conductivity calculations. NIVS-RNEMD improves non-Boltzmann-Maxwell |
| 675 |
+ |
distributions existing in previous RNEMD methods, and extends its |
| 676 |
+ |
applicability to interfacial systems. NIVS-RNEMD has also limited |
| 677 |
+ |
application on shear viscosity calculations, but under high momentum |
| 678 |
+ |
flux, it could cause temperature difference among different |
| 679 |
+ |
dimensions. Modification is necessary to extend the applicability of |
| 680 |
+ |
NIVS-RNEMD in shear viscosity calculations. |
| 681 |
+ |
|
| 682 |
|
\section{Acknowledgments} |
| 683 |
|
Support for this project was provided by the National Science |
| 684 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
