| 56 |
|
(RNEMD) obtains transport coefficients (thermal conductivity and shear |
| 57 |
|
viscosity) in a fluid by imposing an artificial momentum flux between |
| 58 |
|
two thin parallel slabs of material that are spatially separated in |
| 59 |
< |
the simulation cell.\cite{MullerPlathe:1997xw,Muller-Plathe:1999ek} The |
| 60 |
< |
artificial flux is typically created by periodically "swapping" either |
| 59 |
> |
the simulation cell.\cite{MullerPlathe:1997xw,ISI:000080382700030} The |
| 60 |
> |
artificial flux is typically created by periodically ``swapping'' either |
| 61 |
|
the entire momentum vector $\vec{p}$ or single components of this |
| 62 |
|
vector ($p_x$) between molecules in each of the two slabs. If the two |
| 63 |
|
slabs are separated along the z coordinate, the imposed flux is either |
| 64 |
< |
directional ($J_z(p_x)$) or isotropic ($J_z$), and the response of a |
| 64 |
> |
directional ($j_z(p_x)$) or isotropic ($J_z$), and the response of a |
| 65 |
|
simulated system to the imposed momentum flux will typically be a |
| 66 |
|
velocity or thermal gradient. The transport coefficients (shear |
| 67 |
|
viscosity and thermal conductivity) are easily obtained by assuming |
| 68 |
|
linear response of the system, |
| 69 |
|
\begin{eqnarray} |
| 70 |
< |
J_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
| 70 |
> |
j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
| 71 |
|
J & = & \lambda \frac{\partial T}{\partial z} |
| 72 |
|
\end{eqnarray} |
| 73 |
< |
RNEMD been widely used to provide computational estimates of thermal |
| 73 |
> |
RNEMD has been widely used to provide computational estimates of thermal |
| 74 |
|
conductivities and shear viscosities in a wide range of materials, |
| 75 |
|
from liquid copper to monatomic liquids to molecular fluids |
| 76 |
< |
(e.g. ionic liquids). |
| 76 |
> |
(e.g. ionic liquids).\cite{ISI:000246190100032} |
| 77 |
|
|
| 78 |
|
RNEMD is preferable in many ways to the forward NEMD methods because |
| 79 |
|
it imposes what is typically difficult to measure (a flux or stress) |
| 91 |
|
typically samples from the same manifold of states in the |
| 92 |
|
microcanonical ensemble. |
| 93 |
|
|
| 94 |
< |
Recently, Tenney and Maginn have discovered some problems with the |
| 95 |
< |
original RNEMD swap technique. Notably, large momentum fluxes |
| 96 |
< |
(equivalent to frequent momentum swaps between the slabs) can result |
| 97 |
< |
in "notched", "peaked" and generally non-thermal momentum |
| 94 |
> |
Recently, Tenney and Maginn\cite{ISI:000273472300004} have discovered |
| 95 |
> |
some problems with the original RNEMD swap technique. Notably, large |
| 96 |
> |
momentum fluxes (equivalent to frequent momentum swaps between the |
| 97 |
> |
slabs) can result in ``notched'', ``peaked'' and generally non-thermal momentum |
| 98 |
|
distributions in the two slabs, as well as non-linear thermal and |
| 99 |
|
velocity distributions along the direction of the imposed flux ($z$). |
| 100 |
|
Tenney and Maginn obtained reasonable limits on imposed flux and |
| 121 |
|
hot slab. |
| 122 |
|
|
| 123 |
|
Rather than using momentum swaps, we use a series of velocity scaling |
| 124 |
< |
moves. For molecules $\{i\}$ located within the hot slab, |
| 124 |
> |
moves. For molecules $\{i\}$ located within the cold slab, |
| 125 |
|
\begin{equation} |
| 126 |
< |
\vec{v}_i \leftarrow \left( \begin{array}{c} |
| 127 |
< |
x \\ |
| 128 |
< |
y \\ |
| 129 |
< |
z \\ |
| 126 |
> |
\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
| 127 |
> |
x & 0 & 0 \\ |
| 128 |
> |
0 & y & 0 \\ |
| 129 |
> |
0 & 0 & z \\ |
| 130 |
|
\end{array} \right) \cdot \vec{v}_i |
| 131 |
|
\end{equation} |
| 132 |
|
where ${x, y, z}$ are a set of 3 scaling variables for each of the |
| 133 |
|
three directions in the system. Likewise, the molecules $\{j\}$ |
| 134 |
< |
located in the cold bin will see a concomitant scaling of velocities, |
| 134 |
> |
located in the hot slab will see a concomitant scaling of velocities, |
| 135 |
|
\begin{equation} |
| 136 |
< |
\vec{v}_j \leftarrow \left( \begin{array}{c} |
| 137 |
< |
x^\prime \\ |
| 138 |
< |
y^\prime \\ |
| 139 |
< |
z^\prime \\ |
| 136 |
> |
\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
| 137 |
> |
x^\prime & 0 & 0 \\ |
| 138 |
> |
0 & y^\prime & 0 \\ |
| 139 |
> |
0 & 0 & z^\prime \\ |
| 140 |
|
\end{array} \right) \cdot \vec{v}_j |
| 141 |
|
\end{equation} |
| 142 |
|
|
| 144 |
|
($\alpha = x,y,z$) ties the values of the hot and cold bin scaling |
| 145 |
|
parameters together: |
| 146 |
|
\begin{equation} |
| 147 |
< |
P_h^\alpha + P_c^\alpha = \alpha P_h^\alpha + \alpha^\prime P_c^\alpha |
| 147 |
> |
P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
| 148 |
|
\end{equation} |
| 149 |
|
where |
| 150 |
< |
\begin{equation} |
| 151 |
< |
\begin{array}{rcl} |
| 152 |
< |
P_h^\alpha & = & \sum_{i = 1}^{N_h} m_i \left[\vec{v}_i\right]_\alpha \\ |
| 153 |
< |
P_c^\alpha & = & \sum_{j = 1}^{N_c} m_j \left[\vec{v}_j\right]_\alpha |
| 154 |
< |
\\ |
| 155 |
< |
\end{array} |
| 150 |
> |
\begin{eqnarray} |
| 151 |
> |
P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
| 152 |
> |
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha |
| 153 |
|
\label{eq:momentumdef} |
| 154 |
< |
\end{equation} |
| 155 |
< |
Therefore, for each of the three directions, the cold scaling |
| 156 |
< |
parameters are a simple function of the hot scaling parameters and |
| 154 |
> |
\end{eqnarray} |
| 155 |
> |
Therefore, for each of the three directions, the hot scaling |
| 156 |
> |
parameters are a simple function of the cold scaling parameters and |
| 157 |
|
the instantaneous linear momentum in each of the two slabs. |
| 158 |
|
\begin{equation} |
| 159 |
< |
\alpha^\prime = 1 + (1 - \alpha) \frac{P_h^\alpha}{P_c^\alpha}. |
| 159 |
> |
\alpha^\prime = 1 + (1 - \alpha) p_\alpha |
| 160 |
|
\label{eq:hotcoldscaling} |
| 161 |
|
\end{equation} |
| 162 |
+ |
where |
| 163 |
+ |
\begin{equation} |
| 164 |
+ |
p_\alpha = \frac{P_c^\alpha}{P_h^\alpha} |
| 165 |
+ |
\end{equation} |
| 166 |
+ |
for convenience. |
| 167 |
|
|
| 168 |
|
Conservation of total energy also places constraints on the scaling: |
| 169 |
|
\begin{equation} |
| 170 |
|
\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
| 171 |
< |
\alpha^2 K_h^\alpha + \left(\alpha^\prime\right)^2 K_c^\alpha. |
| 171 |
> |
\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha |
| 172 |
|
\end{equation} |
| 173 |
|
where the kinetic energies, $K_h^\alpha$ and $K_c^\alpha$, are computed |
| 174 |
|
for each of the three directions in a similar manner to the linear momenta |
| 175 |
|
(Eq. \ref{eq:momentumdef}). Substituting in the expressions for the |
| 176 |
< |
cold scaling parameters ($\alpha^\prime$) from |
| 177 |
< |
Eq. (\ref{eq:hotcoldscaling}), we obtain the {\it constraint ellipsoid}: |
| 176 |
> |
hot scaling parameters ($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), |
| 177 |
> |
we obtain the {\it constraint ellipsoid equation}: |
| 178 |
|
\begin{equation} |
| 179 |
< |
C_x (1+x)^2 + C_y (1+y)^2 + C_z (1+z)^2 = 0, |
| 179 |
> |
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0 |
| 180 |
|
\label{eq:constraintEllipsoid} |
| 181 |
|
\end{equation} |
| 182 |
|
where the constants are obtained from the instantaneous values of the |
| 183 |
|
linear momenta and kinetic energies for the hot and cold slabs, |
| 184 |
< |
\begin{equation} |
| 185 |
< |
C_\alpha = \left( K_h^\alpha + K_c^\alpha |
| 186 |
< |
\left(\frac{P_h^\alpha}{P_c^\alpha}\right)^2\right) |
| 184 |
> |
\begin{eqnarray} |
| 185 |
> |
a_\alpha & = &\left(K_c^\alpha + K_h^\alpha |
| 186 |
> |
\left(p_\alpha\right)^2\right) \\ |
| 187 |
> |
b_\alpha & = & -2 K_h^\alpha p_\alpha \left( 1 + p_\alpha\right) \\ |
| 188 |
> |
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
| 189 |
|
\label{eq:constraintEllipsoidConsts} |
| 190 |
< |
\end{equation} |
| 191 |
< |
This ellipsoid defines the set of hot slab scaling parameters which can be |
| 192 |
< |
applied while preserving both linear momentum in all three directions |
| 193 |
< |
as well as total energy. |
| 190 |
> |
\end{eqnarray} |
| 191 |
> |
This ellipsoid equation defines the set of cold slab scaling |
| 192 |
> |
parameters which can be applied while preserving both linear momentum |
| 193 |
> |
in all three directions as well as kinetic energy. |
| 194 |
|
|
| 195 |
|
The goal of using velocity scaling variables is to transfer linear |
| 196 |
|
momentum or kinetic energy from the cold slab to the hot slab. If the |
| 197 |
|
hot and cold slabs are separated along the z-axis, the energy flux is |
| 198 |
< |
given simply by the increase in kinetic energy of the hot bin: |
| 198 |
> |
given simply by the decrease in kinetic energy of the cold bin: |
| 199 |
|
\begin{equation} |
| 200 |
< |
(x^2-1) K_h^x + (y^2-1) K_h^y + (z^2-1) K_h^z = J_z |
| 200 |
> |
(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z\Delta t |
| 201 |
|
\end{equation} |
| 202 |
|
The expression for the energy flux can be re-written as another |
| 203 |
|
ellipsoid centered on $(x,y,z) = 0$: |
| 204 |
|
\begin{equation} |
| 205 |
< |
x^2 K_h^x + y^2 K_h^y + z^2 K_h^z = (J_z + K_h^x + K_h^y + K_h^z) |
| 205 |
> |
x^2 K_c^x + y^2 K_c^y + z^2 K_c^z = K_c^x + K_c^y + K_c^z - J_z\Delta t |
| 206 |
|
\label{eq:fluxEllipsoid} |
| 207 |
|
\end{equation} |
| 208 |
< |
The spatial extent of the {\it flux ellipsoid} is governed both by a |
| 209 |
< |
targetted value, $J_z$ as well as the instantaneous values of the |
| 210 |
< |
kinetic energy components in the hot bin. |
| 208 |
> |
The spatial extent of the {\it flux ellipsoid equation} is governed |
| 209 |
> |
both by a targetted value, $J_z$ as well as the instantaneous values of the |
| 210 |
> |
kinetic energy components in the cold bin. |
| 211 |
|
|
| 212 |
|
To satisfy an energetic flux as well as the conservation constraints, |
| 213 |
|
it is sufficient to determine the points ${x,y,z}$ which lie on both |
| 214 |
|
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
| 215 |
|
flux ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
| 216 |
< |
the two ellipsoids in 3-space. |
| 216 |
> |
the two ellipsoids in 3-dimensional space. |
| 217 |
|
|
| 218 |
+ |
\begin{figure} |
| 219 |
+ |
\includegraphics[width=\linewidth]{ellipsoids} |
| 220 |
+ |
\caption{Scaling points which maintain both constant energy and |
| 221 |
+ |
constant linear momentum of the system lie on the surface of the |
| 222 |
+ |
{\it constraint ellipsoid} while points which generate the target |
| 223 |
+ |
momentum flux lie on the surface of the {\it flux ellipsoid}. The |
| 224 |
+ |
velocity distributions in the hot bin are scaled by only those |
| 225 |
+ |
points which lie on both ellipsoids.} |
| 226 |
+ |
\label{ellipsoids} |
| 227 |
+ |
\end{figure} |
| 228 |
|
|
| 229 |
|
One may also define momentum flux (say along the x-direction) as: |
| 230 |
|
\begin{equation} |
| 231 |
< |
(x-1) P_h^x = j_z(p_x) |
| 231 |
> |
(1-x) P_c^x = j_z(p_x)\Delta t |
| 232 |
> |
\label{eq:fluxPlane} |
| 233 |
> |
\end{equation} |
| 234 |
> |
The above {\it flux equation} is essentially a plane which is |
| 235 |
> |
perpendicular to the x-axis, with its position governed both by a |
| 236 |
> |
targetted value, $j_z(p_x)$ as well as the instantaneous value of the |
| 237 |
> |
momentum along the x-direction. |
| 238 |
> |
|
| 239 |
> |
Similarly, to satisfy a momentum flux as well as the conservation |
| 240 |
> |
constraints, it is sufficient to determine the points ${x,y,z}$ which |
| 241 |
> |
lie on both the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) |
| 242 |
> |
and the flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of |
| 243 |
> |
an ellipsoid and a plane in 3-dimensional space. |
| 244 |
> |
|
| 245 |
> |
To summarize, by solving respective equation sets, one can determine |
| 246 |
> |
possible sets of scaling variables for cold slab. And corresponding |
| 247 |
> |
sets of scaling variables for hot slab can be determine as well. |
| 248 |
> |
|
| 249 |
> |
The following problem will be choosing an optimal set of scaling |
| 250 |
> |
variables among the possible sets. Although this method is inherently |
| 251 |
> |
non-isotropic, the goal is still to maintain the system as isotropic |
| 252 |
> |
as possible. Under this consideration, one would like the kinetic |
| 253 |
> |
energies in different directions could become as close as each other |
| 254 |
> |
after each scaling. Simultaneously, one would also like each scaling |
| 255 |
> |
as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
| 256 |
> |
large perturbation to the system. Therefore, one approach to obtain the |
| 257 |
> |
scaling variables would be constructing an criteria function, with |
| 258 |
> |
constraints as above equation sets, and solving the function's minimum |
| 259 |
> |
by method like Lagrange multipliers. |
| 260 |
> |
|
| 261 |
> |
In order to save computation time, we have a different approach to a |
| 262 |
> |
relatively good set of scaling variables with much less calculation |
| 263 |
> |
than above. Here is the detail of our simplification of the problem. |
| 264 |
> |
|
| 265 |
> |
In the case of kinetic energy transfer, we impose another constraint |
| 266 |
> |
${x = y}$, into the equation sets. Consequently, there are two |
| 267 |
> |
variables left. And now one only needs to solve a set of two {\it |
| 268 |
> |
ellipses equations}. This problem would be transformed into solving |
| 269 |
> |
one quartic equation for one of the two variables. There are known |
| 270 |
> |
generic methods that solve real roots of quartic equations. Then one |
| 271 |
> |
can determine the other variable and obtain sets of scaling |
| 272 |
> |
variables. Among these sets, one can apply the above criteria to |
| 273 |
> |
choose the best set, while much faster with only a few sets to choose. |
| 274 |
> |
|
| 275 |
> |
In the case of momentum flux transfer, we impose another constraint to |
| 276 |
> |
set the kinetic energy transfer as zero. In another word, we apply |
| 277 |
> |
Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. After that, with one |
| 278 |
> |
variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar set |
| 279 |
> |
of equations on the above kinetic energy transfer problem. Therefore, |
| 280 |
> |
an approach similar to the above would be sufficient for this as well. |
| 281 |
> |
|
| 282 |
> |
\section{Computational Details} |
| 283 |
> |
Our simulation consists of a series of systems. All of these |
| 284 |
> |
simulations were run with the OpenMD simulation software |
| 285 |
> |
package\cite{Meineke:2005gd} integrated with RNEMD methods. |
| 286 |
> |
|
| 287 |
> |
A Lennard-Jones fluid system was built and tested first. In order to |
| 288 |
> |
compare our method with swapping RNEMD, a series of simulations were |
| 289 |
> |
performed to calculate the shear viscosity and thermal conductivity of |
| 290 |
> |
argon. 2592 atoms were in a orthorhombic cell, which was ${10.06\sigma |
| 291 |
> |
\times 10.06\sigma \times 30.18\sigma}$ by size. The reduced density |
| 292 |
> |
${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled direct |
| 293 |
> |
comparison between our results and others. These simulations used |
| 294 |
> |
velocity Verlet algorithm with reduced timestep ${\tau^* = |
| 295 |
> |
4.6\times10^{-4}}$. |
| 296 |
> |
|
| 297 |
> |
For shear viscosity calculation, the reduced temperature was ${T^* = |
| 298 |
> |
k_B T/\varepsilon = 0.72}$. Simulations were first equilibrated in canonical |
| 299 |
> |
ensemble (NVT), then equilibrated in microcanonical ensemble |
| 300 |
> |
(NVE). Establishing and stablizing momentum gradient were followed |
| 301 |
> |
also in NVE ensemble. For the swapping part, Muller-Plathe's algorithm was |
| 302 |
> |
adopted.\cite{ISI:000080382700030} The simulation box was under |
| 303 |
> |
periodic boundary condition, and devided into ${N = 20}$ slabs. In each swap, |
| 304 |
> |
the top slab ${(n = 1)}$ exchange the most negative $x$ momentum with the |
| 305 |
> |
most positive $x$ momentum in the center slab ${(n = N/2 + 1)}$. Referred |
| 306 |
> |
to Tenney {\it et al.}\cite{ISI:000273472300004}, a series of swapping |
| 307 |
> |
frequency were chosen. According to each result from swapping |
| 308 |
> |
RNEMD, scaling RNEMD simulations were run with the target momentum |
| 309 |
> |
flux set to produce a similar momentum flux and shear |
| 310 |
> |
rate. Furthermore, various scaling frequencies can be tested for one |
| 311 |
> |
single swapping rate. To compare the performance between swapping and |
| 312 |
> |
scaling method, temperatures of different dimensions in all the slabs |
| 313 |
> |
were observed. Most of the simulations include $10^5$ steps of |
| 314 |
> |
equilibration without imposing momentum flux, $10^5$ steps of |
| 315 |
> |
stablization with imposing momentum transfer, and $10^6$ steps of data |
| 316 |
> |
collection under RNEMD. For relatively high momentum flux simulations, |
| 317 |
> |
${5\times10^5}$ step data collection is sufficient. For some low momentum |
| 318 |
> |
flux simulations, ${2\times10^6}$ steps were necessary. |
| 319 |
> |
|
| 320 |
> |
After each simulation, the shear viscosity was calculated in reduced |
| 321 |
> |
unit. The momentum flux was calculated with total unphysical |
| 322 |
> |
transferred momentum ${P_x}$ and data collection time $t$: |
| 323 |
> |
\begin{equation} |
| 324 |
> |
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
| 325 |
> |
\end{equation} |
| 326 |
> |
And the velocity gradient ${\langle \partial v_x /\partial z \rangle}$ |
| 327 |
> |
can be obtained by a linear regression of the velocity profile. From |
| 328 |
> |
the shear viscosity $\eta$ calculated with the above parameters, one |
| 329 |
> |
can further convert it into reduced unit ${\eta^* = \eta \sigma^2 |
| 330 |
> |
(\varepsilon m)^{-1/2}}$. |
| 331 |
> |
|
| 332 |
> |
For thermal conductivity calculation, simulations were first run under |
| 333 |
> |
reduced temperature ${T^* = 0.72}$ in NVE ensemble. Muller-Plathe's |
| 334 |
> |
algorithm was adopted in the swapping method. Under identical |
| 335 |
> |
simulation box parameters, in each swap, the top slab exchange the |
| 336 |
> |
molecule with least kinetic energy with the molecule in the center |
| 337 |
> |
slab with most kinetic energy, unless this ``coldest'' molecule in the |
| 338 |
> |
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the ``cold'' |
| 339 |
> |
slab. According to swapping RNEMD results, target energy flux for |
| 340 |
> |
scaling RNEMD simulations can be set. Also, various scaling |
| 341 |
> |
frequencies can be tested for one target energy flux. To compare the |
| 342 |
> |
performance between swapping and scaling method, distributions of |
| 343 |
> |
velocity and speed in different slabs were observed. |
| 344 |
> |
|
| 345 |
> |
For each swapping rate, thermal conductivity was calculated in reduced |
| 346 |
> |
unit. The energy flux was calculated similarly to the momentum flux, |
| 347 |
> |
with total unphysical transferred energy ${E_{total}}$ and data collection |
| 348 |
> |
time $t$: |
| 349 |
> |
\begin{equation} |
| 350 |
> |
J_z = \frac{E_{total}}{2 t L_x L_y} |
| 351 |
|
\end{equation} |
| 352 |
+ |
And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
| 353 |
+ |
can be obtained by a linear regression of the temperature |
| 354 |
+ |
profile. From the thermal conductivity $\lambda$ calculated, one can |
| 355 |
+ |
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
| 356 |
+ |
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
| 357 |
|
|
| 358 |
+ |
Another series of our simulation is to calculate the interfacial |
| 359 |
+ |
thermal conductivity of a Au/H${_2}$O system. Respective calculations of |
| 360 |
+ |
water (SPC/E) and gold (QSC) thermal conductivity were performed and |
| 361 |
+ |
compared with current results to ensure the validity of |
| 362 |
+ |
NIVS-RNEMD. After that, the mixture system was simulated. |
| 363 |
|
|
| 364 |
+ |
\section{Results And Discussion} |
| 365 |
+ |
\subsection{Shear Viscosity} |
| 366 |
+ |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
| 367 |
+ |
produced comparable shear viscosity to swap RNEMD method. In Table |
| 368 |
+ |
\ref{shearRate}, the names of the calculated samples are devided into |
| 369 |
+ |
two parts. The first number refers to total slabs in one simulation |
| 370 |
+ |
box. The second number refers to the swapping interval in swap method, or |
| 371 |
+ |
in scale method the equilvalent swapping interval that the same |
| 372 |
+ |
momentum flux would theoretically result in swap method. All the scale |
| 373 |
+ |
method results were from simulations that had a scaling interval of 10 |
| 374 |
+ |
time steps. The average molecular momentum gradients of these samples |
| 375 |
+ |
are shown in Figure \ref{shearGrad}. |
| 376 |
|
|
| 377 |
+ |
\begin{table*} |
| 378 |
+ |
\begin{minipage}{\linewidth} |
| 379 |
+ |
\begin{center} |
| 380 |
|
|
| 381 |
+ |
\caption{Calculation results for shear viscosity of Lennard-Jones |
| 382 |
+ |
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
| 383 |
+ |
methods at various momentum exchange rates. Results in reduced |
| 384 |
+ |
unit. Errors of calculations in parentheses. } |
| 385 |
|
|
| 386 |
+ |
\begin{tabular}{ccc} |
| 387 |
+ |
\hline |
| 388 |
+ |
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
| 389 |
+ |
\hline |
| 390 |
+ |
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
| 391 |
+ |
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
| 392 |
+ |
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
| 393 |
+ |
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
| 394 |
+ |
\hline |
| 395 |
+ |
\end{tabular} |
| 396 |
+ |
\label{shearRate} |
| 397 |
+ |
\end{center} |
| 398 |
+ |
\end{minipage} |
| 399 |
+ |
\end{table*} |
| 400 |
+ |
|
| 401 |
+ |
\begin{figure} |
| 402 |
+ |
\includegraphics[width=\linewidth]{shearGrad} |
| 403 |
+ |
\caption{Average momentum gradients of shear viscosity simulations} |
| 404 |
+ |
\label{shearGrad} |
| 405 |
+ |
\end{figure} |
| 406 |
+ |
|
| 407 |
+ |
\begin{figure} |
| 408 |
+ |
\includegraphics[width=\linewidth]{shearTempScale} |
| 409 |
+ |
\caption{Temperature profile for scaling RNEMD simulation.} |
| 410 |
+ |
\label{shearTempScale} |
| 411 |
+ |
\end{figure} |
| 412 |
+ |
However, observations of temperatures along three dimensions show that |
| 413 |
+ |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
| 414 |
+ |
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
| 415 |
+ |
relatively large imposed momentum flux, the temperature difference among $x$ |
| 416 |
+ |
and the other two dimensions was significant. This would result from the |
| 417 |
+ |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
| 418 |
+ |
momentum gradient is set up, $P_c^x$ would be roughly stable |
| 419 |
+ |
($<0$). Consequently, scaling factor $x$ would most probably larger |
| 420 |
+ |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
| 421 |
+ |
keep increase after most scaling steps. And if there is not enough time |
| 422 |
+ |
for the kinetic energy to exchange among different dimensions and |
| 423 |
+ |
different slabs, the system would finally build up temperature |
| 424 |
+ |
(kinetic energy) difference among the three dimensions. Also, between |
| 425 |
+ |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
| 426 |
+ |
are closer to neighbor slabs. This is due to momentum transfer along |
| 427 |
+ |
$z$ dimension between slabs. |
| 428 |
+ |
|
| 429 |
+ |
Although results between scaling and swapping methods are comparable, |
| 430 |
+ |
the inherent temperature inhomogeneity even in relatively low imposed |
| 431 |
+ |
exchange momentum flux simulations makes scaling RNEMD method less |
| 432 |
+ |
attractive than swapping RNEMD in shear viscosity calculation. |
| 433 |
+ |
|
| 434 |
+ |
\subsection{Thermal Conductivity} |
| 435 |
+ |
|
| 436 |
+ |
Our thermal conductivity calculations also show that scaling method results |
| 437 |
+ |
agree with swapping method. Table \ref{thermal} lists our simulation |
| 438 |
+ |
results with similar manner we used in shear viscosity |
| 439 |
+ |
calculation. All the data reported from scaling method were obtained |
| 440 |
+ |
by simulations of 10-step exchange frequency, and the target exchange |
| 441 |
+ |
kinetic energy were set to produce equivalent kinetic energy flux as |
| 442 |
+ |
in swapping method. Figure \ref{thermalGrad} exhibits similar thermal |
| 443 |
+ |
gradients of respective similar kinetic energy flux. |
| 444 |
+ |
|
| 445 |
+ |
\begin{table*} |
| 446 |
+ |
\begin{minipage}{\linewidth} |
| 447 |
+ |
\begin{center} |
| 448 |
+ |
|
| 449 |
+ |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
| 450 |
+ |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
| 451 |
+ |
swap and scale methods at various kinetic energy exchange rates. Results |
| 452 |
+ |
in reduced unit. Errors of calculations in parentheses.} |
| 453 |
+ |
|
| 454 |
+ |
\begin{tabular}{ccc} |
| 455 |
+ |
\hline |
| 456 |
+ |
Series & $\lambda^*_{swap}$ & $\lambda^*_{scale}$\\ |
| 457 |
+ |
\hline |
| 458 |
+ |
20-250 & 7.03(0.34) & 7.30(0.10)\\ |
| 459 |
+ |
20-500 & 7.03(0.14) & 6.95(0.09)\\ |
| 460 |
+ |
20-1000 & 6.91(0.42) & 7.19(0.07)\\ |
| 461 |
+ |
20-2000 & 7.52(0.15) & 7.19(0.28)\\ |
| 462 |
+ |
\hline |
| 463 |
+ |
\end{tabular} |
| 464 |
+ |
\label{thermal} |
| 465 |
+ |
\end{center} |
| 466 |
+ |
\end{minipage} |
| 467 |
+ |
\end{table*} |
| 468 |
+ |
|
| 469 |
+ |
\begin{figure} |
| 470 |
+ |
\includegraphics[width=\linewidth]{thermalGrad} |
| 471 |
+ |
\caption{Temperature gradients of thermal conductivity simulations} |
| 472 |
+ |
\label{thermalGrad} |
| 473 |
+ |
\end{figure} |
| 474 |
+ |
|
| 475 |
+ |
During these simulations, molecule velocities were recorded in 1000 of |
| 476 |
+ |
all the snapshots. These velocity data were used to produce histograms |
| 477 |
+ |
of velocity and speed distribution in different slabs. From these |
| 478 |
+ |
histograms, it is observed that with increasing unphysical kinetic |
| 479 |
+ |
energy flux, speed and velocity distribution of molecules in slabs |
| 480 |
+ |
where swapping occured could deviate from Maxwell-Boltzmann |
| 481 |
+ |
distribution. Figure \ref{histSwap} indicates how these distributions |
| 482 |
+ |
deviate from ideal condition. In high temperature slabs, probability |
| 483 |
+ |
density in low speed is confidently smaller than ideal distribution; |
| 484 |
+ |
in low temperature slabs, probability density in high speed is smaller |
| 485 |
+ |
than ideal. This phenomenon is observable even in our relatively low |
| 486 |
+ |
swapping rate simulations. And this deviation could also leads to |
| 487 |
+ |
deviation of distribution of velocity in various dimensions. One |
| 488 |
+ |
feature of these deviated distribution is that in high temperature |
| 489 |
+ |
slab, the ideal Gaussian peak was changed into a relatively flat |
| 490 |
+ |
plateau; while in low temperature slab, that peak appears sharper. |
| 491 |
+ |
|
| 492 |
+ |
\begin{figure} |
| 493 |
+ |
\includegraphics[width=\linewidth]{histSwap} |
| 494 |
+ |
\caption{Speed distribution for thermal conductivity using swapping RNEMD.} |
| 495 |
+ |
\label{histSwap} |
| 496 |
+ |
\end{figure} |
| 497 |
+ |
|
| 498 |
+ |
\begin{figure} |
| 499 |
+ |
\includegraphics[width=\linewidth]{histScale} |
| 500 |
+ |
\caption{Speed distribution for thermal conductivity using scaling RNEMD.} |
| 501 |
+ |
\label{histScale} |
| 502 |
+ |
\end{figure} |
| 503 |
+ |
|
| 504 |
+ |
\subsection{Interfaciel Thermal Conductivity} |
| 505 |
+ |
|
| 506 |
+ |
\begin{figure} |
| 507 |
+ |
\includegraphics[width=\linewidth]{spceGrad} |
| 508 |
+ |
\caption{Temperature gradients for SPC/E water thermal conductivity.} |
| 509 |
+ |
\label{spceGrad} |
| 510 |
+ |
\end{figure} |
| 511 |
+ |
|
| 512 |
+ |
\begin{table*} |
| 513 |
+ |
\begin{minipage}{\linewidth} |
| 514 |
+ |
\begin{center} |
| 515 |
+ |
|
| 516 |
+ |
\caption{Calculation results for thermal conductivity of SPC/E water |
| 517 |
+ |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
| 518 |
+ |
calculations in parentheses. } |
| 519 |
+ |
|
| 520 |
+ |
\begin{tabular}{cccc} |
| 521 |
+ |
\hline |
| 522 |
+ |
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
| 523 |
+ |
& This work & Previous simulations$^a$ & Experiment$^b$\\ |
| 524 |
+ |
\hline |
| 525 |
+ |
0.3 & 0.82() & 0.784 & 0.64\\ |
| 526 |
+ |
0.8 & 0.770() & 0.730\\ |
| 527 |
+ |
1.5 & 0.813() & \\ |
| 528 |
+ |
\hline |
| 529 |
+ |
\end{tabular} |
| 530 |
+ |
\label{spceThermal} |
| 531 |
+ |
\end{center} |
| 532 |
+ |
\end{minipage} |
| 533 |
+ |
\end{table*} |
| 534 |
+ |
|
| 535 |
+ |
|
| 536 |
+ |
\begin{figure} |
| 537 |
+ |
\includegraphics[width=\linewidth]{AuGrad} |
| 538 |
+ |
\caption{Temperature gradients for crystal gold thermal conductivity.} |
| 539 |
+ |
\label{AuGrad} |
| 540 |
+ |
\end{figure} |
| 541 |
+ |
|
| 542 |
+ |
\begin{table*} |
| 543 |
+ |
\begin{minipage}{\linewidth} |
| 544 |
+ |
\begin{center} |
| 545 |
+ |
|
| 546 |
+ |
\caption{Calculation results for thermal conductivity of crystal gold |
| 547 |
+ |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
| 548 |
+ |
calculations in parentheses. } |
| 549 |
+ |
|
| 550 |
+ |
\begin{tabular}{ccc} |
| 551 |
+ |
\hline |
| 552 |
+ |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
| 553 |
+ |
& This work & Previous simulations$^a$ \\ |
| 554 |
+ |
\hline |
| 555 |
+ |
1.4 & 1.10() & \\ |
| 556 |
+ |
2.8 & 1.08() & \\ |
| 557 |
+ |
5.1 & 1.15() & \\ |
| 558 |
+ |
\hline |
| 559 |
+ |
\end{tabular} |
| 560 |
+ |
\label{AuThermal} |
| 561 |
+ |
\end{center} |
| 562 |
+ |
\end{minipage} |
| 563 |
+ |
\end{table*} |
| 564 |
+ |
|
| 565 |
+ |
|
| 566 |
|
\section{Acknowledgments} |
| 567 |
|
Support for this project was provided by the National Science |
| 568 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
