| 84 |
|
j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
| 85 |
|
J_z & = & \lambda \frac{\partial T}{\partial z} |
| 86 |
|
\end{eqnarray} |
| 87 |
< |
RNEMD has been widely used to provide computational estimates of thermal |
| 88 |
< |
conductivities and shear viscosities in a wide range of materials, |
| 89 |
< |
from liquid copper to monatomic liquids to molecular fluids |
| 90 |
< |
(e.g. ionic liquids).\cite{Bedrov:2000-1,Bedrov:2000,Muller-Plathe:2002,ISI:000184808400018,ISI:000231042800044,Maginn:2007,Muller-Plathe:2008,ISI:000258460400020,ISI:000258840700015,ISI:000261835100054} |
| 87 |
> |
RNEMD has been widely used to provide computational estimates of |
| 88 |
> |
thermal conductivities and shear viscosities in a wide range of |
| 89 |
> |
materials, from liquid copper to both monatomic and molecular fluids |
| 90 |
> |
(e.g. ionic |
| 91 |
> |
liquids).\cite{Bedrov:2000-1,Bedrov:2000,Muller-Plathe:2002,ISI:000184808400018,ISI:000231042800044,Maginn:2007,Muller-Plathe:2008,ISI:000258460400020,ISI:000258840700015,ISI:000261835100054} |
| 92 |
|
|
| 93 |
|
\begin{figure} |
| 94 |
|
\includegraphics[width=\linewidth]{thermalDemo} |
| 104 |
|
RNEMD is preferable in many ways to the forward NEMD |
| 105 |
|
methods\cite{ISI:A1988Q205300014,hess:209,Vasquez:2004fk,backer:154503,ISI:000266247600008} |
| 106 |
|
because it imposes what is typically difficult to measure (a flux or |
| 107 |
< |
stress) and it is typically much easier to compute momentum gradients |
| 108 |
< |
or strains (the response). For similar reasons, RNEMD is also |
| 107 |
> |
stress) and it is typically much easier to compute the response |
| 108 |
> |
(momentum gradients or strains. For similar reasons, RNEMD is also |
| 109 |
|
preferable to slowly-converging equilibrium methods for measuring |
| 110 |
|
thermal conductivity and shear viscosity (using Green-Kubo |
| 111 |
|
relations\cite{daivis:541,mondello:9327} or the Helfand moment |
| 129 |
|
and self-adjusting metrics for retaining the usability of the method. |
| 130 |
|
|
| 131 |
|
In this paper, we develop and test a method for non-isotropic velocity |
| 132 |
< |
scaling (NIVS-RNEMD) which retains the desirable features of RNEMD |
| 132 |
> |
scaling (NIVS) which retains the desirable features of RNEMD |
| 133 |
|
(conservation of linear momentum and total energy, compatibility with |
| 134 |
|
periodic boundary conditions) while establishing true thermal |
| 135 |
< |
distributions in each of the two slabs. In the next section, we |
| 135 |
> |
distributions in each of the two slabs. In the next section, we |
| 136 |
|
present the method for determining the scaling constraints. We then |
| 137 |
< |
test the method on both single component, multi-component, and |
| 138 |
< |
non-isotropic mixtures and show that it is capable of providing |
| 137 |
> |
test the method on both liquids and solids as well as a non-isotropic |
| 138 |
> |
liquid-solid interface and show that it is capable of providing |
| 139 |
|
reasonable estimates of the thermal conductivity and shear viscosity |
| 140 |
< |
in these cases. |
| 140 |
> |
in all of these cases. |
| 141 |
|
|
| 142 |
|
\section{Methodology} |
| 143 |
|
We retain the basic idea of M\"{u}ller-Plathe's RNEMD method; the |
| 157 |
|
0 & 0 & z \\ |
| 158 |
|
\end{array} \right) \cdot \vec{v}_i |
| 159 |
|
\end{equation} |
| 160 |
< |
where ${x, y, z}$ are a set of 3 scaling variables for each of the |
| 161 |
< |
three directions in the system. Likewise, the molecules $\{j\}$ |
| 162 |
< |
located in the hot slab will see a concomitant scaling of velocities, |
| 160 |
> |
where ${x, y, z}$ are a set of 3 velocity-scaling variables for each |
| 161 |
> |
of the three directions in the system. Likewise, the molecules |
| 162 |
> |
$\{j\}$ located in the hot slab will see a concomitant scaling of |
| 163 |
> |
velocities, |
| 164 |
|
\begin{equation} |
| 165 |
|
\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
| 166 |
|
x^\prime & 0 & 0 \\ |
| 206 |
|
($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), we obtain the |
| 207 |
|
{\it constraint ellipsoid}: |
| 208 |
|
\begin{equation} |
| 209 |
< |
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0 |
| 209 |
> |
\sum_{\alpha = x,y,z} \left( a_\alpha \alpha^2 + b_\alpha \alpha + |
| 210 |
> |
c_\alpha \right) = 0 |
| 211 |
|
\label{eq:constraintEllipsoid} |
| 212 |
|
\end{equation} |
| 213 |
|
where the constants are obtained from the instantaneous values of the |
| 220 |
|
\label{eq:constraintEllipsoidConsts} |
| 221 |
|
\end{eqnarray} |
| 222 |
|
This ellipsoid (Eq. \ref{eq:constraintEllipsoid}) defines the set of |
| 223 |
< |
cold slab scaling parameters which can be applied while preserving |
| 224 |
< |
both linear momentum in all three directions as well as total kinetic |
| 225 |
< |
energy. |
| 223 |
> |
cold slab scaling parameters which, when applied, preserve the linear |
| 224 |
> |
momentum of the system in all three directions as well as total |
| 225 |
> |
kinetic energy. |
| 226 |
|
|
| 227 |
< |
The goal of using velocity scaling variables is to transfer linear |
| 228 |
< |
momentum or kinetic energy from the cold slab to the hot slab. If the |
| 229 |
< |
hot and cold slabs are separated along the z-axis, the energy flux is |
| 230 |
< |
given simply by the decrease in kinetic energy of the cold bin: |
| 227 |
> |
The goal of using these velocity scaling variables is to transfer |
| 228 |
> |
linear momentum or kinetic energy from the cold slab to the hot slab. |
| 229 |
> |
If the hot and cold slabs are separated along the z-axis, the energy |
| 230 |
> |
flux is given simply by the decrease in kinetic energy of the cold |
| 231 |
> |
bin: |
| 232 |
|
\begin{equation} |
| 233 |
|
(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z\Delta t |
| 234 |
|
\end{equation} |
| 235 |
|
The expression for the energy flux can be re-written as another |
| 236 |
|
ellipsoid centered on $(x,y,z) = 0$: |
| 237 |
|
\begin{equation} |
| 238 |
< |
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 |
| 238 |
> |
\sum_{\alpha = x,y,z} K_c^\alpha \alpha^2 = -J_z \Delta t + |
| 239 |
> |
\sum_{\alpha = x,y,z} K_c^\alpha |
| 240 |
|
\label{eq:fluxEllipsoid} |
| 241 |
|
\end{equation} |
| 242 |
|
The spatial extent of the {\it thermal flux ellipsoid} is governed |
| 243 |
< |
both by a targetted value, $J_z$ as well as the instantaneous values |
| 244 |
< |
of the kinetic energy components in the cold bin. |
| 243 |
> |
both by the target flux, $J_z$ as well as the instantaneous values of |
| 244 |
> |
the kinetic energy components in the cold bin. |
| 245 |
|
|
| 246 |
|
To satisfy an energetic flux as well as the conservation constraints, |
| 247 |
< |
we must determine the points ${x,y,z}$ which lie on both the |
| 248 |
< |
constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the flux |
| 249 |
< |
ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of the |
| 250 |
< |
two ellipsoids in 3-dimensional space. |
| 247 |
> |
we must determine the points ${x,y,z}$ that lie on both the constraint |
| 248 |
> |
ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the flux ellipsoid |
| 249 |
> |
(Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of the two |
| 250 |
> |
ellipsoids in 3-dimensional space. |
| 251 |
|
|
| 252 |
|
\begin{figure} |
| 253 |
|
\includegraphics[width=\linewidth]{ellipsoids} |
| 254 |
< |
\caption{Illustration from the perspective of a space having cold |
| 255 |
< |
slab scaling coefficients as its coordinates. Scaling points which |
| 256 |
< |
maintain both constant energy and constant linear momentum of the |
| 257 |
< |
system lie on the surface of the {\it constraint ellipsoid} while |
| 258 |
< |
points which generate the target momentum flux lie on the surface of |
| 259 |
< |
the {\it flux ellipsoid}. The velocity distributions in the cold bin |
| 255 |
< |
are scaled by only those points which lie on both ellipsoids.} |
| 254 |
> |
\caption{Velocity scaling coefficients which maintain both constant |
| 255 |
> |
energy and constant linear momentum of the system lie on the surface |
| 256 |
> |
of the {\it constraint ellipsoid} while points which generate the |
| 257 |
> |
target momentum flux lie on the surface of the {\it flux ellipsoid}. |
| 258 |
> |
The velocity distributions in the cold bin are scaled by only those |
| 259 |
> |
points which lie on both ellipsoids.} |
| 260 |
|
\label{ellipsoids} |
| 261 |
|
\end{figure} |
| 262 |
|
|
| 263 |
< |
One may also define {\it momentum} flux (say along the $x$-direction) as: |
| 263 |
> |
Since ellipsoids can be expressed as polynomials up to second order in |
| 264 |
> |
each of the three coordinates, finding the the intersection points of |
| 265 |
> |
two ellipsoids is isomorphic to finding the roots a polynomial of |
| 266 |
> |
degree 16. There are a number of polynomial root-finding methods in |
| 267 |
> |
the literature, [CITATIONS NEEDED] but numerically finding the roots |
| 268 |
> |
of high-degree polynomials is generally an ill-conditioned |
| 269 |
> |
problem.[CITATION NEEDED] One way around this is to try to maintain |
| 270 |
> |
velocity scalings that are {\it as isotropic as possible}. To do |
| 271 |
> |
this, we impose $x=y$, and to treat both the constraint and flux |
| 272 |
> |
ellipsoids as 2-dimensional ellipses. In reduced dimensionality, the |
| 273 |
> |
intersecting-ellipse problem reduces to finding the roots of |
| 274 |
> |
polynomials of degree 4. |
| 275 |
> |
|
| 276 |
> |
Depending on the target flux and current velocity distributions, the |
| 277 |
> |
ellipsoids can have between 0 and 4 intersection points. If there are |
| 278 |
> |
no intersection points, it is not possible to satisfy the constraints |
| 279 |
> |
while performing a non-equilibrium scaling move, and no change is made |
| 280 |
> |
to the dynamics. |
| 281 |
> |
|
| 282 |
> |
With multiple intersection points, any of the scaling points will |
| 283 |
> |
conserve the linear momentum and kinetic energy of the system and will |
| 284 |
> |
generate the correct target flux. Although this method is inherently |
| 285 |
> |
non-isotropic, the goal is still to maintain the system as close to an |
| 286 |
> |
isotropic fluid as possible. With this in mind, we would like the |
| 287 |
> |
kinetic energies in the three different directions could become as |
| 288 |
> |
close as each other as possible after each scaling. Simultaneously, |
| 289 |
> |
one would also like each scaling as gentle as possible, i.e. ${x,y,z |
| 290 |
> |
\rightarrow 1}$, in order to avoid large perturbation to the system. |
| 291 |
> |
To do this, we pick the intersection point which maintains the scaling |
| 292 |
> |
variables ${x=y, z}$ as well as the ratio of kinetic energies |
| 293 |
> |
${K_c^z/K_c^x, K_c^z/K_c^y}$ as close as possible to 1. |
| 294 |
> |
|
| 295 |
> |
After the valid scaling parameters are arrived at by solving geometric |
| 296 |
> |
intersection problems in $x, y, z$ space in order to obtain cold slab |
| 297 |
> |
scaling parameters, Eq. (\ref{eq:hotcoldscaling}) is used to |
| 298 |
> |
determine the conjugate hot slab scaling variables. |
| 299 |
> |
|
| 300 |
> |
\subsection{Introducing shear stress via velocity scaling} |
| 301 |
> |
Rather than using this method to induce a thermal flux, it is possible |
| 302 |
> |
to use the random fluctuations of the average momentum in each of the |
| 303 |
> |
bins to induce a momentum flux. Doing this repeatedly will create a |
| 304 |
> |
shear stress on the system which will respond with an easily-measured |
| 305 |
> |
strain. The momentum flux (say along the $x$-direction) may be |
| 306 |
> |
defined as: |
| 307 |
|
\begin{equation} |
| 308 |
|
(1-x) P_c^x = j_z(p_x)\Delta t |
| 309 |
|
\label{eq:fluxPlane} |
| 310 |
|
\end{equation} |
| 311 |
< |
The above {\it momentum flux plane} is perpendicular to the $x$-axis, |
| 312 |
< |
with its position governed both by a target value, $j_z(p_x)$ as well |
| 313 |
< |
as the instantaneous value of the momentum along the $x$-direction. |
| 311 |
> |
This {\it momentum flux plane} is perpendicular to the $x$-axis, with |
| 312 |
> |
its position governed both by a target value, $j_z(p_x)$ as well as |
| 313 |
> |
the instantaneous value of the momentum along the $x$-direction. |
| 314 |
|
|
| 315 |
|
In order to satisfy a momentum flux as well as the conservation |
| 316 |
|
constraints, we must determine the points ${x,y,z}$ which lie on both |
| 317 |
|
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
| 318 |
|
flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of an |
| 319 |
< |
ellipsoid and a plane in 3-dimensional space. |
| 319 |
> |
ellipsoid and a plane in 3-dimensional space. |
| 320 |
|
|
| 321 |
< |
In both the momentum and energy flux scenarios, valid scaling |
| 322 |
< |
parameters are arrived at by solving geometric intersection problems |
| 323 |
< |
in $x, y, z$ space in order to obtain cold slab scaling parameters. |
| 324 |
< |
Once the scaling variables for the cold slab are known, the hot slab |
| 325 |
< |
scaling has also been determined. |
| 321 |
> |
In the case of momentum flux transfer, we also impose another |
| 322 |
> |
constraint to set the kinetic energy transfer as zero. In another |
| 323 |
> |
word, we apply Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. With |
| 324 |
> |
one variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar |
| 325 |
> |
set of quartic equations to the above kinetic energy transfer problem. |
| 326 |
|
|
| 327 |
+ |
\section{Computational Details} |
| 328 |
|
|
| 329 |
< |
The following problem will be choosing an optimal set of scaling |
| 330 |
< |
variables among the possible sets. Although this method is inherently |
| 331 |
< |
non-isotropic, the goal is still to maintain the system as isotropic |
| 332 |
< |
as possible. Under this consideration, one would like the kinetic |
| 333 |
< |
energies in different directions could become as close as each other |
| 334 |
< |
after each scaling. Simultaneously, one would also like each scaling |
| 335 |
< |
as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
| 336 |
< |
large perturbation to the system. Therefore, one approach to obtain |
| 337 |
< |
the scaling variables would be constructing an criteria function, with |
| 290 |
< |
constraints as above equation sets, and solving the function's minimum |
| 291 |
< |
by method like Lagrange multipliers. |
| 329 |
> |
We have implemented this methodology in our molecular dynamics |
| 330 |
> |
code,\cite{Meineke:2005gd} by performing the NIVS scaling moves after |
| 331 |
> |
each MD step. We have tested it for a variety of different |
| 332 |
> |
situations, including homogeneous fluids (Lennard-Jones and SPC/E |
| 333 |
> |
water), crystalline solids (EAM and Sutton-Chen models for Gold), and |
| 334 |
> |
heterogeneous interfaces (EAM gold - SPC/E water). The last of these |
| 335 |
> |
systems would have been very difficult to study using previous RNEMD |
| 336 |
> |
methods, but using velocity scaling moves, we can even obtain |
| 337 |
> |
estimates of the interfacial thermal conductivity. |
| 338 |
|
|
| 293 |
– |
In order to save computation time, we have a different approach to a |
| 294 |
– |
relatively good set of scaling variables with much less calculation |
| 295 |
– |
than above. Here is the detail of our simplification of the problem. |
| 296 |
– |
|
| 297 |
– |
In the case of kinetic energy transfer, we impose another constraint |
| 298 |
– |
${x = y}$, into the equation sets. Consequently, there are two |
| 299 |
– |
variables left. And now one only needs to solve a set of two {\it |
| 300 |
– |
ellipses equations}. This problem would be transformed into solving |
| 301 |
– |
one quartic equation for one of the two variables. There are known |
| 302 |
– |
generic methods that solve real roots of quartic equations. Then one |
| 303 |
– |
can determine the other variable and obtain sets of scaling |
| 304 |
– |
variables. Among these sets, one can apply the above criteria to |
| 305 |
– |
choose the best set, while much faster with only a few sets to choose. |
| 306 |
– |
|
| 307 |
– |
In the case of momentum flux transfer, we impose another constraint to |
| 308 |
– |
set the kinetic energy transfer as zero. In another word, we apply |
| 309 |
– |
Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. After that, with one |
| 310 |
– |
variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar set |
| 311 |
– |
of equations on the above kinetic energy transfer problem. Therefore, |
| 312 |
– |
an approach similar to the above would be sufficient for this as well. |
| 313 |
– |
|
| 314 |
– |
\section{Computational Details} |
| 339 |
|
\subsection{Lennard-Jones Fluid} |
| 316 |
– |
Our simulation consists of a series of systems. All of these |
| 317 |
– |
simulations were run with the OpenMD simulation software |
| 318 |
– |
package\cite{Meineke:2005gd} integrated with RNEMD codes. |
| 340 |
|
|
| 341 |
< |
A Lennard-Jones fluid system was built and tested first. In order to |
| 342 |
< |
compare our method with swapping RNEMD, a series of simulations were |
| 343 |
< |
performed to calculate the shear viscosity and thermal conductivity of |
| 344 |
< |
argon. 2592 atoms were in a orthorhombic cell, which was ${10.06\sigma |
| 345 |
< |
\times 10.06\sigma \times 30.18\sigma}$ by size. The reduced density |
| 346 |
< |
${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled direct |
| 347 |
< |
comparison between our results and others. These simulations used |
| 348 |
< |
velocity Verlet algorithm with reduced timestep ${\tau^* = |
| 349 |
< |
4.6\times10^{-4}}$. |
| 341 |
> |
2592 Lennard-Jones atoms were placed in an orthorhombic cell |
| 342 |
> |
${10.06\sigma \times 10.06\sigma \times 30.18\sigma}$ on a side. The |
| 343 |
> |
reduced density ${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled |
| 344 |
> |
direct comparison between our results and previous methods. These |
| 345 |
> |
simulations were carried out with a reduced timestep ${\tau^* = |
| 346 |
> |
4.6\times10^{-4}}$. For the shear viscosity calculation, the mean |
| 347 |
> |
temperature was ${T^* = k_B T/\varepsilon = 0.72}$. Simulations were |
| 348 |
> |
first thermalized in canonical ensemble (NVT), then equilibrated in |
| 349 |
> |
microcanonical ensemble (NVE) before introducing any non-equilibrium |
| 350 |
> |
method. |
| 351 |
|
|
| 352 |
< |
For shear viscosity calculation, the reduced temperature was ${T^* = |
| 353 |
< |
k_B T/\varepsilon = 0.72}$. Simulations were first equilibrated in canonical |
| 354 |
< |
ensemble (NVT), then equilibrated in microcanonical ensemble |
| 355 |
< |
(NVE). Establishing and stablizing momentum gradient were followed |
| 356 |
< |
also in NVE ensemble. For the swapping part, Muller-Plathe's algorithm was |
| 357 |
< |
adopted.\cite{ISI:000080382700030} The simulation box was under |
| 358 |
< |
periodic boundary condition, and devided into ${N = 20}$ slabs. In each swap, |
| 359 |
< |
the top slab ${(n = 1)}$ exchange the most negative $x$ momentum with the |
| 360 |
< |
most positive $x$ momentum in the center slab ${(n = N/2 + 1)}$. Referred |
| 361 |
< |
to Tenney {\it et al.}\cite{Maginn:2010}, a series of swapping |
| 362 |
< |
frequency were chosen. According to each result from swapping |
| 363 |
< |
RNEMD, scaling RNEMD simulations were run with the target momentum |
| 364 |
< |
flux set to produce a similar momentum flux, and consequently shear |
| 365 |
< |
rate. Furthermore, various scaling frequencies can be tested for one |
| 366 |
< |
single swapping rate. To test the temperature homogeneity in our |
| 367 |
< |
system of swapping and scaling methods, temperatures of different |
| 368 |
< |
dimensions in all the slabs were observed. Most of the simulations |
| 369 |
< |
include $10^5$ steps of equilibration without imposing momentum flux, |
| 370 |
< |
$10^5$ steps of stablization with imposing unphysical momentum |
| 371 |
< |
transfer, and $10^6$ steps of data collection under RNEMD. For |
| 372 |
< |
relatively high momentum flux simulations, ${5\times10^5}$ step data |
| 373 |
< |
collection is sufficient. For some low momentum flux simulations, |
| 374 |
< |
${2\times10^6}$ steps were necessary. |
| 352 |
> |
We have compared the momentum gradients established using our method |
| 353 |
> |
to those obtained using the original M\"{u}ller-Plathe swapping |
| 354 |
> |
algorithm.\cite{ISI:000080382700030} In both cases, the simulation box |
| 355 |
> |
was divided into ${N = 20}$ slabs. In the swapping algorithm, the top |
| 356 |
> |
slab $(n = 1)$ exchanges the most negative $x$ momentum with the most |
| 357 |
> |
positive $x$ momentum in the center slab $(n = N/2 + 1)$. The rate at |
| 358 |
> |
which the swapping moves are carried out defines the momentum or |
| 359 |
> |
thermal flux between the two slabs. In their work, Tenney {\it et |
| 360 |
> |
al.}\cite{Maginn:2010} found problematic behavior with large swap |
| 361 |
> |
frequencies. |
| 362 |
> |
|
| 363 |
> |
According to each result from swapping RNEMD, scaling RNEMD |
| 364 |
> |
simulations were run with the target momentum flux set to produce a |
| 365 |
> |
similar momentum flux, and consequently shear rate. Furthermore, |
| 366 |
> |
various scaling frequencies can be tested for one single swapping |
| 367 |
> |
rate. To test the temperature homogeneity in our system of swapping |
| 368 |
> |
and scaling methods, temperatures of different dimensions in all the |
| 369 |
> |
slabs were observed. Most of the simulations include $10^5$ steps of |
| 370 |
> |
equilibration without imposing momentum flux, $10^5$ steps of |
| 371 |
> |
stablization with imposing unphysical momentum transfer, and $10^6$ |
| 372 |
> |
steps of data collection under RNEMD. For relatively high momentum |
| 373 |
> |
flux simulations, ${5\times10^5}$ step data collection is sufficient. |
| 374 |
> |
For some low momentum flux simulations, ${2\times10^6}$ steps were |
| 375 |
> |
necessary. |
| 376 |
|
|
| 377 |
|
After each simulation, the shear viscosity was calculated in reduced |
| 378 |
|
unit. The momentum flux was calculated with total unphysical |
| 614 |
|
|
| 615 |
|
\subsubsection{Crystal Gold} |
| 616 |
|
Our results of gold thermal conductivity using two force fields are |
| 617 |
< |
shown separately in Table \ref{qscThermal} and \ref{eamThermal}. In |
| 618 |
< |
these calculations,the end and middle slabs were excluded in thermal |
| 619 |
< |
gradient regession and only used as heat source and drain in the |
| 620 |
< |
systems. Our yielded values using EAM force field are slightly larger |
| 621 |
< |
than those using QSC force field. However, both series are |
| 622 |
< |
significantly smaller than experimental value by an order of more than |
| 623 |
< |
100. It has been verified that this difference is mainly attributed to |
| 624 |
< |
the lack of electron interaction representation in these force field |
| 625 |
< |
parameters. Richardson {\it et al.}\cite{Clancy:1992} used EAM |
| 626 |
< |
force field parameters in their metal thermal conductivity |
| 627 |
< |
calculations. The Non-Equilibrium MD method they employed in their |
| 628 |
< |
simulations produced comparable results to ours. As Zhang {\it et |
| 629 |
< |
al.}\cite{ISI:000231042800044} stated, thermal conductivity values |
| 630 |
< |
are influenced mainly by force field. Therefore, it is confident to |
| 631 |
< |
conclude that NIVS-RNEMD is applicable to metal force field system. |
| 632 |
< |
|
| 633 |
< |
\begin{figure} |
| 634 |
< |
\includegraphics[width=\linewidth]{AuGrad} |
| 635 |
< |
\caption{Temperature gradients for thermal conductivity calculation of |
| 636 |
< |
crystal gold using QSC force field.} |
| 637 |
< |
\label{AuGrad} |
| 638 |
< |
\end{figure} |
| 616 |
< |
|
| 617 |
< |
\begin{table*} |
| 618 |
< |
\begin{minipage}{\linewidth} |
| 619 |
< |
\begin{center} |
| 620 |
< |
|
| 621 |
< |
\caption{Calculation results for thermal conductivity of crystal gold |
| 622 |
< |
using QSC force field at ${\langle T\rangle}$ = 300K at various |
| 623 |
< |
thermal exchange rates. Errors of calculations in parentheses. } |
| 624 |
< |
|
| 625 |
< |
\begin{tabular}{cc} |
| 626 |
< |
\hline |
| 627 |
< |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
| 628 |
< |
\hline |
| 629 |
< |
1.44 & 1.10(0.01)\\ |
| 630 |
< |
2.86 & 1.08(0.02)\\ |
| 631 |
< |
5.14 & 1.15(0.01)\\ |
| 632 |
< |
\hline |
| 633 |
< |
\end{tabular} |
| 634 |
< |
\label{qscThermal} |
| 635 |
< |
\end{center} |
| 636 |
< |
\end{minipage} |
| 637 |
< |
\end{table*} |
| 617 |
> |
shown in Table \ref{AuThermal}. In these calculations,the end and |
| 618 |
> |
middle slabs were excluded in thermal gradient regession and only used |
| 619 |
> |
as heat source and drain in the systems. Our yielded values using EAM |
| 620 |
> |
force field are slightly larger than those using QSC force |
| 621 |
> |
field. However, both series are significantly smaller than |
| 622 |
> |
experimental value by a factor of more than 200. It has been verified |
| 623 |
> |
that this difference is mainly attributed to the lack of electron |
| 624 |
> |
interaction representation in these force field parameters. Richardson |
| 625 |
> |
{\it et al.}\cite{Clancy:1992} used EAM force field parameters in |
| 626 |
> |
their metal thermal conductivity calculations. The Non-Equilibrium MD |
| 627 |
> |
method they employed in their simulations produced comparable results |
| 628 |
> |
to ours. As Zhang {\it et al.}\cite{ISI:000231042800044} stated, |
| 629 |
> |
thermal conductivity values are influenced mainly by force |
| 630 |
> |
field. Another factor that affects the calculation results could be |
| 631 |
> |
the density of the metal. After equilibration under |
| 632 |
> |
isobaric-isothermal conditions, our crystall simulation cell expanded |
| 633 |
> |
by the order of 1\%. Under longer lattice constant than default, |
| 634 |
> |
lower thermal conductance would be expected. Furthermore, the result |
| 635 |
> |
of Richardson {\it et al.} were obtained between 300K and 850K, which |
| 636 |
> |
are significantly higher than in our simulations. Therefore, it is |
| 637 |
> |
still confident to conclude that NIVS-RNEMD is applicable to metal |
| 638 |
> |
force field system. |
| 639 |
|
|
| 639 |
– |
\begin{figure} |
| 640 |
– |
\includegraphics[width=\linewidth]{eamGrad} |
| 641 |
– |
\caption{Temperature gradients for thermal conductivity calculation of |
| 642 |
– |
crystal gold using EAM force field.} |
| 643 |
– |
\label{eamGrad} |
| 644 |
– |
\end{figure} |
| 645 |
– |
|
| 640 |
|
\begin{table*} |
| 641 |
|
\begin{minipage}{\linewidth} |
| 642 |
|
\begin{center} |
| 643 |
|
|
| 644 |
|
\caption{Calculation results for thermal conductivity of crystal gold |
| 645 |
< |
using EAM force field at ${\langle T\rangle}$ = 300K at various |
| 646 |
< |
thermal exchange rates. Errors of calculations in parentheses. } |
| 645 |
> |
using different force fields at ${\langle T\rangle}$ = 300K at |
| 646 |
> |
various thermal exchange rates. Errors of calculations in parentheses.} |
| 647 |
|
|
| 648 |
< |
\begin{tabular}{cc} |
| 648 |
> |
\begin{tabular}{ccc} |
| 649 |
|
\hline |
| 650 |
< |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
| 650 |
> |
Force Field Used & $\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
| 651 |
|
\hline |
| 652 |
< |
1.24 & 1.24(0.06)\\ |
| 653 |
< |
2.06 & 1.37(0.04)\\ |
| 654 |
< |
2.55 & 1.41(0.03)\\ |
| 652 |
> |
& 1.44 & 1.10(0.01)\\ |
| 653 |
> |
QSC & 2.86 & 1.08(0.02)\\ |
| 654 |
> |
& 5.14 & 1.15(0.01)\\ |
| 655 |
|
\hline |
| 656 |
+ |
& 1.24 & 1.24(0.06)\\ |
| 657 |
+ |
EAM & 2.06 & 1.37(0.04)\\ |
| 658 |
+ |
& 2.55 & 1.41(0.03)\\ |
| 659 |
+ |
\hline |
| 660 |
|
\end{tabular} |
| 661 |
< |
\label{eamThermal} |
| 661 |
> |
\label{AuThermal} |
| 662 |
|
\end{center} |
| 663 |
|
\end{minipage} |
| 664 |
|
\end{table*} |
| 669 |
|
NIVS-RNEMD method were proved valid, calculation of gold/water |
| 670 |
|
interfacial thermal conductivity was followed. It is found out that |
| 671 |
|
the low interfacial conductance is probably due to the hydrophobic |
| 672 |
< |
surface in our system. Figure \ref{interfaceDensity} demonstrates mass |
| 672 |
> |
surface in our system. Figure \ref{interface} (a) demonstrates mass |
| 673 |
|
density change along $z$-axis, which is perpendicular to the |
| 674 |
|
gold/water interface. It is observed that water density significantly |
| 675 |
|
decreases when approaching the surface. Under this low thermal |
| 676 |
|
conductance, both gold and water phase have sufficient time to |
| 677 |
|
eliminate temperature difference inside respectively (Figure |
| 678 |
< |
\ref{interfaceGrad}). With indistinguishable temperature difference |
| 678 |
> |
\ref{interface} b). With indistinguishable temperature difference |
| 679 |
|
within respective phase, it is valid to assume that the temperature |
| 680 |
|
difference between gold and water on surface would be approximately |
| 681 |
|
the same as the difference between the gold and water phase. This |
| 688 |
|
errors than our calculation results on homogeneous systems. |
| 689 |
|
|
| 690 |
|
\begin{figure} |
| 691 |
< |
\includegraphics[width=\linewidth]{interfaceDensity} |
| 692 |
< |
\caption{Density profile for interfacial thermal conductivity |
| 693 |
< |
simulation box. Significant water density decrease is observed on |
| 694 |
< |
gold surface.} |
| 695 |
< |
\label{interfaceDensity} |
| 691 |
> |
\includegraphics[width=\linewidth]{interface} |
| 692 |
> |
\caption{Simulation results for Gold/Water interfacial thermal |
| 693 |
> |
conductivity: (a) Significant water density decrease is observed on |
| 694 |
> |
crystalline gold surface, which indicates low surface contact and |
| 695 |
> |
leads to low thermal conductance. (b) Temperature profiles for a |
| 696 |
> |
series of simulations. Temperatures of different slabs in the same |
| 697 |
> |
phase show no significant differences.} |
| 698 |
> |
\label{interface} |
| 699 |
|
\end{figure} |
| 700 |
|
|
| 700 |
– |
\begin{figure} |
| 701 |
– |
\includegraphics[width=\linewidth]{interfaceGrad} |
| 702 |
– |
\caption{Temperature profiles for interfacial thermal conductivity |
| 703 |
– |
simulation box. Temperatures of different slabs in the same phase |
| 704 |
– |
show no significant difference.} |
| 705 |
– |
\label{interfaceGrad} |
| 706 |
– |
\end{figure} |
| 707 |
– |
|
| 701 |
|
\begin{table*} |
| 702 |
|
\begin{minipage}{\linewidth} |
| 703 |
|
\begin{center} |
| 744 |
|
|
| 745 |
|
\begin{tabular}{ccc} |
| 746 |
|
\hline |
| 747 |
< |
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
| 747 |
> |
(Equilvalent) Exchange Interval (fs) & $\eta^*_{swap}$ & |
| 748 |
> |
$\eta^*_{scale}$\\ |
| 749 |
|
\hline |
| 750 |
< |
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
| 751 |
< |
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
| 752 |
< |
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
| 753 |
< |
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
| 750 |
> |
500 & 3.64(0.05) & 3.76(0.09)\\ |
| 751 |
> |
1000 & 3.52(0.16) & 3.66(0.06)\\ |
| 752 |
> |
2000 & 3.72(0.05) & 3.32(0.18)\\ |
| 753 |
> |
2500 & 3.42(0.06) & 3.43(0.08)\\ |
| 754 |
|
\hline |
| 755 |
|
\end{tabular} |
| 756 |
|
\label{shearRate} |
