| 38 |
|
\begin{doublespace} |
| 39 |
|
|
| 40 |
|
\begin{abstract} |
| 41 |
< |
|
| 41 |
> |
We present a new method for introducing stable non-equilibrium |
| 42 |
> |
velocity and temperature distributions in molecular dynamics |
| 43 |
> |
simulations of heterogeneous systems. This method extends some |
| 44 |
> |
earlier Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods |
| 45 |
> |
which use momentum exchange swapping moves that can create |
| 46 |
> |
non-thermal velocity distributions (and which are difficult to use |
| 47 |
> |
for interfacial calculations). By using non-isotropic velocity |
| 48 |
> |
scaling (NIVS) on the molecules in specific regions of a system, it |
| 49 |
> |
is possible to impose momentum or thermal flux between regions of a |
| 50 |
> |
simulation and stable thermal and momentum gradients can then be |
| 51 |
> |
established. The scaling method we have developed conserves the |
| 52 |
> |
total linear momentum and total energy of the system. To test the |
| 53 |
> |
methods, we have computed the thermal conductivity of model liquid |
| 54 |
> |
and solid systems as well as the interfacial thermal conductivity of |
| 55 |
> |
a metal-water interface. We find that the NIVS-RNEMD improves the |
| 56 |
> |
problematic velocity distributions that develop in other RNEMD |
| 57 |
> |
methods. |
| 58 |
|
\end{abstract} |
| 59 |
|
|
| 60 |
|
\newpage |
| 65 |
|
% BODY OF TEXT |
| 66 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 67 |
|
|
| 52 |
– |
|
| 53 |
– |
|
| 68 |
|
\section{Introduction} |
| 69 |
|
The original formulation of Reverse Non-equilibrium Molecular Dynamics |
| 70 |
|
(RNEMD) obtains transport coefficients (thermal conductivity and shear |
| 71 |
|
viscosity) in a fluid by imposing an artificial momentum flux between |
| 72 |
|
two thin parallel slabs of material that are spatially separated in |
| 73 |
|
the simulation cell.\cite{MullerPlathe:1997xw,ISI:000080382700030} The |
| 74 |
< |
artificial flux is typically created by periodically ``swapping'' either |
| 75 |
< |
the entire momentum vector $\vec{p}$ or single components of this |
| 76 |
< |
vector ($p_x$) between molecules in each of the two slabs. If the two |
| 77 |
< |
slabs are separated along the z coordinate, the imposed flux is either |
| 78 |
< |
directional ($j_z(p_x)$) or isotropic ($J_z$), and the response of a |
| 79 |
< |
simulated system to the imposed momentum flux will typically be a |
| 80 |
< |
velocity or thermal gradient. The transport coefficients (shear |
| 81 |
< |
viscosity and thermal conductivity) are easily obtained by assuming |
| 82 |
< |
linear response of the system, |
| 74 |
> |
artificial flux is typically created by periodically ``swapping'' |
| 75 |
> |
either the entire momentum vector $\vec{p}$ or single components of |
| 76 |
> |
this vector ($p_x$) between molecules in each of the two slabs. If |
| 77 |
> |
the two slabs are separated along the $z$ coordinate, the imposed flux |
| 78 |
> |
is either directional ($j_z(p_x)$) or isotropic ($J_z$), and the |
| 79 |
> |
response of a simulated system to the imposed momentum flux will |
| 80 |
> |
typically be a velocity or thermal gradient (Fig. \ref{thermalDemo}). |
| 81 |
> |
The shear viscosity ($\eta$) and thermal conductivity ($\lambda$) are |
| 82 |
> |
easily obtained by assuming linear response of the system, |
| 83 |
|
\begin{eqnarray} |
| 84 |
|
j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
| 85 |
< |
J & = & \lambda \frac{\partial T}{\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{ISI:000246190100032} |
| 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} |
| 91 |
|
|
| 92 |
|
\begin{figure} |
| 93 |
|
\includegraphics[width=\linewidth]{thermalDemo} |
| 94 |
< |
\caption{Demostration of thermal gradient estalished by RNEMD method.} |
| 94 |
> |
\caption{RNEMD methods impose an unphysical transfer of momentum or |
| 95 |
> |
kinetic energy between a ``hot'' slab and a ``cold'' slab in the |
| 96 |
> |
simulation box. The molecular system responds to this imposed flux |
| 97 |
> |
by generating a momentum or temperature gradient. The slope of the |
| 98 |
> |
gradient can then be used to compute transport properties (e.g. |
| 99 |
> |
shear viscosity and thermal conductivity).} |
| 100 |
|
\label{thermalDemo} |
| 101 |
|
\end{figure} |
| 102 |
|
|
| 103 |
< |
RNEMD is preferable in many ways to the forward NEMD methods because |
| 104 |
< |
it imposes what is typically difficult to measure (a flux or stress) |
| 105 |
< |
and it is typically much easier to compute momentum gradients or |
| 106 |
< |
strains (the response). For similar reasons, RNEMD is also preferable |
| 107 |
< |
to slowly-converging equilibrium methods for measuring thermal |
| 108 |
< |
conductivity and shear viscosity (using Green-Kubo relations or the |
| 109 |
< |
Helfand moment approach of Viscardy {\it et |
| 103 |
> |
RNEMD is preferable in many ways to the forward NEMD |
| 104 |
> |
methods\cite{ISI:A1988Q205300014,hess:209,Vasquez:2004fk,backer:154503,ISI:000266247600008} |
| 105 |
> |
because it imposes what is typically difficult to measure (a flux or |
| 106 |
> |
stress) and it is typically much easier to compute momentum gradients |
| 107 |
> |
or strains (the response). For similar reasons, RNEMD is also |
| 108 |
> |
preferable to slowly-converging equilibrium methods for measuring |
| 109 |
> |
thermal conductivity and shear viscosity (using Green-Kubo |
| 110 |
> |
relations\cite{daivis:541,mondello:9327} or the Helfand moment |
| 111 |
> |
approach of Viscardy {\it et |
| 112 |
|
al.}\cite{Viscardy:2007bh,Viscardy:2007lq}) because these rely on |
| 113 |
|
computing difficult to measure quantities. |
| 114 |
|
|
| 118 |
|
typically samples from the same manifold of states in the |
| 119 |
|
microcanonical ensemble. |
| 120 |
|
|
| 121 |
< |
Recently, Tenney and Maginn\cite{ISI:000273472300004} have discovered |
| 121 |
> |
Recently, Tenney and Maginn\cite{Maginn:2010} have discovered |
| 122 |
|
some problems with the original RNEMD swap technique. Notably, large |
| 123 |
|
momentum fluxes (equivalent to frequent momentum swaps between the |
| 124 |
< |
slabs) can result in ``notched'', ``peaked'' and generally non-thermal momentum |
| 125 |
< |
distributions in the two slabs, as well as non-linear thermal and |
| 126 |
< |
velocity distributions along the direction of the imposed flux ($z$). |
| 127 |
< |
Tenney and Maginn obtained reasonable limits on imposed flux and |
| 128 |
< |
self-adjusting metrics for retaining the usability of the method. |
| 124 |
> |
slabs) can result in ``notched'', ``peaked'' and generally non-thermal |
| 125 |
> |
momentum distributions in the two slabs, as well as non-linear thermal |
| 126 |
> |
and velocity distributions along the direction of the imposed flux |
| 127 |
> |
($z$). Tenney and Maginn obtained reasonable limits on imposed flux |
| 128 |
> |
and self-adjusting metrics for retaining the usability of the method. |
| 129 |
|
|
| 130 |
|
In this paper, we develop and test a method for non-isotropic velocity |
| 131 |
|
scaling (NIVS-RNEMD) which retains the desirable features of RNEMD |
| 132 |
|
(conservation of linear momentum and total energy, compatibility with |
| 133 |
|
periodic boundary conditions) while establishing true thermal |
| 134 |
|
distributions in each of the two slabs. In the next section, we |
| 135 |
< |
develop the method for determining the scaling constraints. We then |
| 135 |
> |
present the method for determining the scaling constraints. We then |
| 136 |
|
test the method on both single component, multi-component, and |
| 137 |
|
non-isotropic mixtures and show that it is capable of providing |
| 138 |
|
reasonable estimates of the thermal conductivity and shear viscosity |
| 139 |
|
in these cases. |
| 140 |
|
|
| 141 |
|
\section{Methodology} |
| 142 |
< |
We retain the basic idea of Muller-Plathe's RNEMD method; the periodic |
| 143 |
< |
system is partitioned into a series of thin slabs along a particular |
| 142 |
> |
We retain the basic idea of M\"{u}ller-Plathe's RNEMD method; the |
| 143 |
> |
periodic system is partitioned into a series of thin slabs along one |
| 144 |
|
axis ($z$). One of the slabs at the end of the periodic box is |
| 145 |
|
designated the ``hot'' slab, while the slab in the center of the box |
| 146 |
|
is designated the ``cold'' slab. The artificial momentum flux will be |
| 148 |
|
hot slab. |
| 149 |
|
|
| 150 |
|
Rather than using momentum swaps, we use a series of velocity scaling |
| 151 |
< |
moves. For molecules $\{i\}$ located within the cold slab, |
| 151 |
> |
moves. For molecules $\{i\}$ located within the cold slab, |
| 152 |
|
\begin{equation} |
| 153 |
|
\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
| 154 |
|
x & 0 & 0 \\ |
| 168 |
|
\end{equation} |
| 169 |
|
|
| 170 |
|
Conservation of linear momentum in each of the three directions |
| 171 |
< |
($\alpha = x,y,z$) ties the values of the hot and cold bin scaling |
| 171 |
> |
($\alpha = x,y,z$) ties the values of the hot and cold scaling |
| 172 |
|
parameters together: |
| 173 |
|
\begin{equation} |
| 174 |
|
P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
| 197 |
|
\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
| 198 |
|
\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha |
| 199 |
|
\end{equation} |
| 200 |
< |
where the kinetic energies, $K_h^\alpha$ and $K_c^\alpha$, are computed |
| 201 |
< |
for each of the three directions in a similar manner to the linear momenta |
| 202 |
< |
(Eq. \ref{eq:momentumdef}). Substituting in the expressions for the |
| 203 |
< |
hot scaling parameters ($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), |
| 204 |
< |
we obtain the {\it constraint ellipsoid equation}: |
| 200 |
> |
where the translational kinetic energies, $K_h^\alpha$ and |
| 201 |
> |
$K_c^\alpha$, are computed for each of the three directions in a |
| 202 |
> |
similar manner to the linear momenta (Eq. \ref{eq:momentumdef}). |
| 203 |
> |
Substituting in the expressions for the hot scaling parameters |
| 204 |
> |
($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), we obtain the |
| 205 |
> |
{\it constraint ellipsoid}: |
| 206 |
|
\begin{equation} |
| 207 |
|
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0 |
| 208 |
|
\label{eq:constraintEllipsoid} |
| 216 |
|
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
| 217 |
|
\label{eq:constraintEllipsoidConsts} |
| 218 |
|
\end{eqnarray} |
| 219 |
< |
This ellipsoid equation defines the set of cold slab scaling |
| 220 |
< |
parameters which can be applied while preserving both linear momentum |
| 221 |
< |
in all three directions as well as kinetic energy. |
| 219 |
> |
This ellipsoid (Eq. \ref{eq:constraintEllipsoid}) defines the set of |
| 220 |
> |
cold slab scaling parameters which can be applied while preserving |
| 221 |
> |
both linear momentum in all three directions as well as total kinetic |
| 222 |
> |
energy. |
| 223 |
|
|
| 224 |
|
The goal of using velocity scaling variables is to transfer linear |
| 225 |
|
momentum or kinetic energy from the cold slab to the hot slab. If the |
| 234 |
|
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 |
| 235 |
|
\label{eq:fluxEllipsoid} |
| 236 |
|
\end{equation} |
| 237 |
< |
The spatial extent of the {\it flux ellipsoid equation} is governed |
| 238 |
< |
both by a targetted value, $J_z$ as well as the instantaneous values of the |
| 239 |
< |
kinetic energy components in the cold bin. |
| 237 |
> |
The spatial extent of the {\it thermal flux ellipsoid} is governed |
| 238 |
> |
both by a targetted value, $J_z$ as well as the instantaneous values |
| 239 |
> |
of the kinetic energy components in the cold bin. |
| 240 |
|
|
| 241 |
|
To satisfy an energetic flux as well as the conservation constraints, |
| 242 |
< |
it is sufficient to determine the points ${x,y,z}$ which lie on both |
| 243 |
< |
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
| 244 |
< |
flux ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
| 245 |
< |
the two ellipsoids in 3-dimensional space. |
| 242 |
> |
we must determine the points ${x,y,z}$ which lie on both the |
| 243 |
> |
constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the flux |
| 244 |
> |
ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of the |
| 245 |
> |
two ellipsoids in 3-dimensional space. |
| 246 |
|
|
| 247 |
|
\begin{figure} |
| 248 |
|
\includegraphics[width=\linewidth]{ellipsoids} |
| 249 |
< |
\caption{Scaling points which maintain both constant energy and |
| 250 |
< |
constant linear momentum of the system lie on the surface of the |
| 251 |
< |
{\it constraint ellipsoid} while points which generate the target |
| 252 |
< |
momentum flux lie on the surface of the {\it flux ellipsoid}. The |
| 253 |
< |
velocity distributions in the hot bin are scaled by only those |
| 254 |
< |
points which lie on both ellipsoids.} |
| 249 |
> |
\caption{Illustration from the perspective of a space having cold |
| 250 |
> |
slab scaling coefficients as its coordinates. Scaling points which |
| 251 |
> |
maintain both constant energy and constant linear momentum of the |
| 252 |
> |
system lie on the surface of the {\it constraint ellipsoid} while |
| 253 |
> |
points which generate the target momentum flux lie on the surface of |
| 254 |
> |
the {\it flux ellipsoid}. The velocity distributions in the cold bin |
| 255 |
> |
are scaled by only those points which lie on both ellipsoids.} |
| 256 |
|
\label{ellipsoids} |
| 257 |
|
\end{figure} |
| 258 |
|
|
| 259 |
< |
One may also define momentum flux (say along the x-direction) as: |
| 259 |
> |
One may also define {\it momentum} flux (say along the $x$-direction) as: |
| 260 |
|
\begin{equation} |
| 261 |
|
(1-x) P_c^x = j_z(p_x)\Delta t |
| 262 |
|
\label{eq:fluxPlane} |
| 263 |
|
\end{equation} |
| 264 |
< |
The above {\it flux equation} is essentially a plane which is |
| 265 |
< |
perpendicular to the x-axis, with its position governed both by a |
| 266 |
< |
targetted value, $j_z(p_x)$ as well as the instantaneous value of the |
| 243 |
< |
momentum along the x-direction. |
| 264 |
> |
The above {\it momentum flux plane} is perpendicular to the $x$-axis, |
| 265 |
> |
with its position governed both by a target value, $j_z(p_x)$ as well |
| 266 |
> |
as the instantaneous value of the momentum along the $x$-direction. |
| 267 |
|
|
| 268 |
< |
Similarly, to satisfy a momentum flux as well as the conservation |
| 269 |
< |
constraints, it is sufficient to determine the points ${x,y,z}$ which |
| 270 |
< |
lie on both the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) |
| 271 |
< |
and the flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of |
| 272 |
< |
an ellipsoid and a plane in 3-dimensional space. |
| 268 |
> |
In order to satisfy a momentum flux as well as the conservation |
| 269 |
> |
constraints, we must determine the points ${x,y,z}$ which lie on both |
| 270 |
> |
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
| 271 |
> |
flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of an |
| 272 |
> |
ellipsoid and a plane in 3-dimensional space. |
| 273 |
|
|
| 274 |
< |
To summarize, by solving respective equation sets, one can determine |
| 275 |
< |
possible sets of scaling variables for cold slab. And corresponding |
| 276 |
< |
sets of scaling variables for hot slab can be determine as well. |
| 274 |
> |
In both the momentum and energy flux scenarios, valid scaling |
| 275 |
> |
parameters are arrived at by solving geometric intersection problems |
| 276 |
> |
in $x, y, z$ space in order to obtain cold slab scaling parameters. |
| 277 |
> |
Once the scaling variables for the cold slab are known, the hot slab |
| 278 |
> |
scaling has also been determined. |
| 279 |
|
|
| 280 |
+ |
|
| 281 |
|
The following problem will be choosing an optimal set of scaling |
| 282 |
|
variables among the possible sets. Although this method is inherently |
| 283 |
|
non-isotropic, the goal is still to maintain the system as isotropic |
| 285 |
|
energies in different directions could become as close as each other |
| 286 |
|
after each scaling. Simultaneously, one would also like each scaling |
| 287 |
|
as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
| 288 |
< |
large perturbation to the system. Therefore, one approach to obtain the |
| 289 |
< |
scaling variables would be constructing an criteria function, with |
| 288 |
> |
large perturbation to the system. Therefore, one approach to obtain |
| 289 |
> |
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. |
| 292 |
|
|
| 312 |
|
an approach similar to the above would be sufficient for this as well. |
| 313 |
|
|
| 314 |
|
\section{Computational Details} |
| 315 |
+ |
\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 methods. |
| 318 |
> |
package\cite{Meineke:2005gd} integrated with RNEMD codes. |
| 319 |
|
|
| 320 |
|
A Lennard-Jones fluid system was built and tested first. In order to |
| 321 |
|
compare our method with swapping RNEMD, a series of simulations were |
| 336 |
|
periodic boundary condition, and devided into ${N = 20}$ slabs. In each swap, |
| 337 |
|
the top slab ${(n = 1)}$ exchange the most negative $x$ momentum with the |
| 338 |
|
most positive $x$ momentum in the center slab ${(n = N/2 + 1)}$. Referred |
| 339 |
< |
to Tenney {\it et al.}\cite{ISI:000273472300004}, a series of swapping |
| 339 |
> |
to Tenney {\it et al.}\cite{Maginn:2010}, a series of swapping |
| 340 |
|
frequency were chosen. According to each result from swapping |
| 341 |
|
RNEMD, scaling RNEMD simulations were run with the target momentum |
| 342 |
< |
flux set to produce a similar momentum flux and shear |
| 342 |
> |
flux set to produce a similar momentum flux, and consequently shear |
| 343 |
|
rate. Furthermore, various scaling frequencies can be tested for one |
| 344 |
< |
single swapping rate. To compare the performance between swapping and |
| 345 |
< |
scaling method, temperatures of different dimensions in all the slabs |
| 346 |
< |
were observed. Most of the simulations include $10^5$ steps of |
| 347 |
< |
equilibration without imposing momentum flux, $10^5$ steps of |
| 348 |
< |
stablization with imposing momentum transfer, and $10^6$ steps of data |
| 349 |
< |
collection under RNEMD. For relatively high momentum flux simulations, |
| 350 |
< |
${5\times10^5}$ step data collection is sufficient. For some low momentum |
| 351 |
< |
flux simulations, ${2\times10^6}$ steps were necessary. |
| 344 |
> |
single swapping rate. To test the temperature homogeneity in our |
| 345 |
> |
system of swapping and scaling methods, temperatures of different |
| 346 |
> |
dimensions in all the slabs were observed. Most of the simulations |
| 347 |
> |
include $10^5$ steps of equilibration without imposing momentum flux, |
| 348 |
> |
$10^5$ steps of stablization with imposing unphysical momentum |
| 349 |
> |
transfer, and $10^6$ steps of data collection under RNEMD. For |
| 350 |
> |
relatively high momentum flux simulations, ${5\times10^5}$ step data |
| 351 |
> |
collection is sufficient. For some low momentum flux simulations, |
| 352 |
> |
${2\times10^6}$ steps were necessary. |
| 353 |
|
|
| 354 |
|
After each simulation, the shear viscosity was calculated in reduced |
| 355 |
|
unit. The momentum flux was calculated with total unphysical |
| 357 |
|
\begin{equation} |
| 358 |
|
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
| 359 |
|
\end{equation} |
| 360 |
< |
And the velocity gradient ${\langle \partial v_x /\partial z \rangle}$ |
| 361 |
< |
can be obtained by a linear regression of the velocity profile. From |
| 362 |
< |
the shear viscosity $\eta$ calculated with the above parameters, one |
| 363 |
< |
can further convert it into reduced unit ${\eta^* = \eta \sigma^2 |
| 364 |
< |
(\varepsilon m)^{-1/2}}$. |
| 360 |
> |
where $L_x$ and $L_y$ denotes $x$ and $y$ lengths of the simulation |
| 361 |
> |
box, and physical momentum transfer occurs in two ways due to our |
| 362 |
> |
periodic boundary condition settings. And the velocity gradient |
| 363 |
> |
${\langle \partial v_x /\partial z \rangle}$ can be obtained by a |
| 364 |
> |
linear regression of the velocity profile. From the shear viscosity |
| 365 |
> |
$\eta$ calculated with the above parameters, one can further convert |
| 366 |
> |
it into reduced unit ${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$. |
| 367 |
|
|
| 368 |
< |
For thermal conductivity calculation, simulations were first run under |
| 369 |
< |
reduced temperature ${T^* = 0.72}$ in NVE ensemble. Muller-Plathe's |
| 370 |
< |
algorithm was adopted in the swapping method. Under identical |
| 371 |
< |
simulation box parameters, in each swap, the top slab exchange the |
| 372 |
< |
molecule with least kinetic energy with the molecule in the center |
| 373 |
< |
slab with most kinetic energy, unless this ``coldest'' molecule in the |
| 374 |
< |
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the ``cold'' |
| 375 |
< |
slab. According to swapping RNEMD results, target energy flux for |
| 376 |
< |
scaling RNEMD simulations can be set. Also, various scaling |
| 368 |
> |
For thermal conductivity calculations, simulations were first run under |
| 369 |
> |
reduced temperature ${\langle T^*\rangle = 0.72}$ in NVE |
| 370 |
> |
ensemble. Muller-Plathe's algorithm was adopted in the swapping |
| 371 |
> |
method. Under identical simulation box parameters with our shear |
| 372 |
> |
viscosity calculations, in each swap, the top slab exchanges all three |
| 373 |
> |
translational momentum components of the molecule with least kinetic |
| 374 |
> |
energy with the same components of the molecule in the center slab |
| 375 |
> |
with most kinetic energy, unless this ``coldest'' molecule in the |
| 376 |
> |
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the |
| 377 |
> |
``cold'' slab. According to swapping RNEMD results, target energy flux |
| 378 |
> |
for scaling RNEMD simulations can be set. Also, various scaling |
| 379 |
|
frequencies can be tested for one target energy flux. To compare the |
| 380 |
|
performance between swapping and scaling method, distributions of |
| 381 |
|
velocity and speed in different slabs were observed. |
| 393 |
|
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
| 394 |
|
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
| 395 |
|
|
| 396 |
< |
Another series of our simulation is to calculate the interfacial |
| 396 |
> |
\subsection{ Water / Metal Thermal Conductivity} |
| 397 |
> |
Another series of our simulation is the calculation of interfacial |
| 398 |
|
thermal conductivity of a Au/H$_2$O system. Respective calculations of |
| 399 |
< |
water (SPC/E) and gold (QSC) thermal conductivity were performed and |
| 400 |
< |
compared with current results to ensure the validity of |
| 401 |
< |
NIVS-RNEMD. After that, a mixture system was simulated. |
| 399 |
> |
liquid water (Extended Simple Point Charge model) and crystal gold |
| 400 |
> |
thermal conductivity were performed and compared with current results |
| 401 |
> |
to ensure the validity of NIVS-RNEMD. After that, a mixture system was |
| 402 |
> |
simulated. |
| 403 |
|
|
| 404 |
|
For thermal conductivity calculation of bulk water, a simulation box |
| 405 |
|
consisting of 1000 molecules were first equilibrated under ambient |
| 406 |
< |
pressure and temperature conditions (NPT), followed by equilibration |
| 407 |
< |
in fixed volume (NVT). The system was then equilibrated in |
| 408 |
< |
microcanonical ensemble (NVE). Also in NVE ensemble, establishing |
| 406 |
> |
pressure and temperature conditions using NPT ensemble, followed by |
| 407 |
> |
equilibration in fixed volume (NVT). The system was then equilibrated in |
| 408 |
> |
microcanonical ensemble (NVE). Also in NVE ensemble, establishing a |
| 409 |
|
stable thermal gradient was followed. The simulation box was under |
| 410 |
|
periodic boundary condition and devided into 10 slabs. Data collection |
| 411 |
< |
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. |
| 411 |
> |
process was similar to Lennard-Jones fluid system. |
| 412 |
|
|
| 413 |
+ |
Thermal conductivity calculation of bulk crystal gold used a similar |
| 414 |
+ |
protocol. Two types of force field parameters, Embedded Atom Method |
| 415 |
+ |
(EAM) and Quantum Sutten-Chen (QSC) force field were used |
| 416 |
+ |
respectively. The face-centered cubic crystal simulation box consists of |
| 417 |
+ |
2880 Au atoms. The lattice was first allowed volume change to relax |
| 418 |
+ |
under ambient temperature and pressure. Equilibrations in canonical and |
| 419 |
+ |
microcanonical ensemble were followed in order. With the simulation |
| 420 |
+ |
lattice devided evenly into 10 slabs, different thermal gradients were |
| 421 |
+ |
established by applying a set of target thermal transfer flux. Data of |
| 422 |
+ |
the series of thermal gradients was collected for calculation. |
| 423 |
+ |
|
| 424 |
|
After simulations of bulk water and crystal gold, a mixture system was |
| 425 |
|
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
| 426 |
|
molecules. Spohr potential was adopted in depicting the interaction |
| 427 |
|
between metal atom and water molecule.\cite{ISI:000167766600035} A |
| 428 |
< |
similar protocol of equilibration was followed. A thermal gradient was |
| 429 |
< |
built. It was found out that compared to homogeneous systems, the two |
| 430 |
< |
phases could have large temperature difference under a relatively low |
| 431 |
< |
thermal flux. Therefore, under our low flux condition, it is assumed |
| 428 |
> |
similar protocol of equilibration was followed. Several thermal |
| 429 |
> |
gradients was built under different target thermal flux. It was found |
| 430 |
> |
out that compared to our previous simulation systems, the two phases |
| 431 |
> |
could have large temperature difference even under a relatively low |
| 432 |
> |
thermal flux. Therefore, under our low flux conditions, it is assumed |
| 433 |
|
that the metal and water phases have respectively homogeneous |
| 434 |
< |
temperature. In calculating the interfacial thermal conductivity $G$, |
| 435 |
< |
this assumptioin was applied and thus our formula becomes: |
| 434 |
> |
temperature, excluding the surface regions. In calculating the |
| 435 |
> |
interfacial thermal conductivity $G$, this assumptioin was applied and |
| 436 |
> |
thus our formula becomes: |
| 437 |
|
|
| 438 |
|
\begin{equation} |
| 439 |
|
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
| 444 |
|
and ${\langle T_{gold}\rangle}$ and ${\langle T_{water}\rangle}$ are the |
| 445 |
|
average observed temperature of gold and water phases respectively. |
| 446 |
|
|
| 447 |
< |
\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 |
< |
|
| 447 |
> |
\section{Results And Discussions} |
| 448 |
|
\subsection{Thermal Conductivity} |
| 449 |
|
\subsubsection{Lennard-Jones Fluid} |
| 450 |
< |
Our thermal conductivity calculations also show that scaling method results |
| 451 |
< |
agree with swapping method. Table \ref{thermal} lists our simulation |
| 452 |
< |
results with similar manner we used in shear viscosity |
| 453 |
< |
calculation. All the data reported from scaling method were obtained |
| 454 |
< |
by simulations of 10-step exchange frequency, and the target exchange |
| 455 |
< |
kinetic energy were set to produce equivalent kinetic energy flux as |
| 456 |
< |
in swapping method. Figure \ref{thermalGrad} exhibits similar thermal |
| 482 |
< |
gradients of respective similar kinetic energy flux. |
| 450 |
> |
Our thermal conductivity calculations show that scaling method results |
| 451 |
> |
agree with swapping method. Four different exchange intervals were |
| 452 |
> |
tested (Table \ref{thermalLJRes}) using swapping method. With a fixed |
| 453 |
> |
10fs exchange interval, target exchange kinetic energy was set to |
| 454 |
> |
produce equivalent kinetic energy flux as in swapping method. And |
| 455 |
> |
similar thermal gradients were observed with similar thermal flux in |
| 456 |
> |
two simulation methods (Figure \ref{thermalGrad}). |
| 457 |
|
|
| 458 |
|
\begin{table*} |
| 459 |
|
\begin{minipage}{\linewidth} |
| 466 |
|
|
| 467 |
|
\begin{tabular}{ccc} |
| 468 |
|
\hline |
| 469 |
< |
Series & $\lambda^*_{swap}$ & $\lambda^*_{scale}$\\ |
| 469 |
> |
(Equilvalent) Exchange Interval (fs) & $\lambda^*_{swap}$ & |
| 470 |
> |
$\lambda^*_{scale}$\\ |
| 471 |
|
\hline |
| 472 |
< |
20-250 & 7.03(0.34) & 7.30(0.10)\\ |
| 473 |
< |
20-500 & 7.03(0.14) & 6.95(0.09)\\ |
| 474 |
< |
20-1000 & 6.91(0.42) & 7.19(0.07)\\ |
| 475 |
< |
20-2000 & 7.52(0.15) & 7.19(0.28)\\ |
| 472 |
> |
250 & 7.03(0.34) & 7.30(0.10)\\ |
| 473 |
> |
500 & 7.03(0.14) & 6.95(0.09)\\ |
| 474 |
> |
1000 & 6.91(0.42) & 7.19(0.07)\\ |
| 475 |
> |
2000 & 7.52(0.15) & 7.19(0.28)\\ |
| 476 |
|
\hline |
| 477 |
|
\end{tabular} |
| 478 |
< |
\label{thermal} |
| 478 |
> |
\label{thermalLJRes} |
| 479 |
|
\end{center} |
| 480 |
|
\end{minipage} |
| 481 |
|
\end{table*} |
| 482 |
|
|
| 483 |
|
\begin{figure} |
| 484 |
|
\includegraphics[width=\linewidth]{thermalGrad} |
| 485 |
< |
\caption{Temperature gradients of thermal conductivity simulations} |
| 485 |
> |
\caption{NIVS-RNEMD method introduced similar temperature gradients |
| 486 |
> |
compared to ``swapping'' method under various kinetic energy flux in |
| 487 |
> |
thermal conductivity simulations.} |
| 488 |
|
\label{thermalGrad} |
| 489 |
|
\end{figure} |
| 490 |
|
|
| 491 |
|
During these simulations, molecule velocities were recorded in 1000 of |
| 492 |
< |
all the snapshots. These velocity data were used to produce histograms |
| 493 |
< |
of velocity and speed distribution in different slabs. From these |
| 494 |
< |
histograms, it is observed that with increasing unphysical kinetic |
| 495 |
< |
energy flux, speed and velocity distribution of molecules in slabs |
| 496 |
< |
where swapping occured could deviate from Maxwell-Boltzmann |
| 497 |
< |
distribution. Figure \ref{histSwap} indicates how these distributions |
| 498 |
< |
deviate from ideal condition. In high temperature slabs, probability |
| 499 |
< |
density in low speed is confidently smaller than ideal distribution; |
| 500 |
< |
in low temperature slabs, probability density in high speed is smaller |
| 501 |
< |
than ideal. This phenomenon is observable even in our relatively low |
| 502 |
< |
swapping rate simulations. And this deviation could also leads to |
| 503 |
< |
deviation of distribution of velocity in various dimensions. One |
| 504 |
< |
feature of these deviated distribution is that in high temperature |
| 505 |
< |
slab, the ideal Gaussian peak was changed into a relatively flat |
| 506 |
< |
plateau; while in low temperature slab, that peak appears sharper. |
| 492 |
> |
all the snapshots of one single data collection process. These |
| 493 |
> |
velocity data were used to produce histograms of velocity and speed |
| 494 |
> |
distribution in different slabs. From these histograms, it is observed |
| 495 |
> |
that under relatively high unphysical kinetic energy flux, speed and |
| 496 |
> |
velocity distribution of molecules in slabs where swapping occured |
| 497 |
> |
could deviate from Maxwell-Boltzmann distribution. Figure |
| 498 |
> |
\ref{thermalHist} a) illustrates how these distributions deviate from an |
| 499 |
> |
ideal distribution. In high temperature slab, probability density in |
| 500 |
> |
low speed is confidently smaller than ideal curve fit; in low |
| 501 |
> |
temperature slab, probability density in high speed is smaller than |
| 502 |
> |
ideal, while larger than ideal in low speed. This phenomenon is more |
| 503 |
> |
obvious in our high swapping rate simulations. And this deviation |
| 504 |
> |
could also leads to deviation of distribution of velocity in various |
| 505 |
> |
dimensions. One feature of these deviated distribution is that in high |
| 506 |
> |
temperature slab, the ideal Gaussian peak was changed into a |
| 507 |
> |
relatively flat plateau; while in low temperature slab, that peak |
| 508 |
> |
appears sharper. This problem is rooted in the mechanism of the |
| 509 |
> |
swapping method. Continually depleting low (high) speed particles in |
| 510 |
> |
the high (low) temperature slab could not be complemented by |
| 511 |
> |
diffusions of low (high) speed particles from neighbor slabs, unless |
| 512 |
> |
in suffciently low swapping rate. Simutaneously, surplus low speed |
| 513 |
> |
particles in the low temperature slab do not have sufficient time to |
| 514 |
> |
diffuse to neighbor slabs. However, thermal exchange rate should reach |
| 515 |
> |
a minimum level to produce an observable thermal gradient under noise |
| 516 |
> |
interference. Consequently, swapping RNEMD has a relatively narrow |
| 517 |
> |
choice of swapping rate to satisfy these above restrictions. |
| 518 |
|
|
| 519 |
< |
\begin{figure} |
| 520 |
< |
\includegraphics[width=\linewidth]{histSwap} |
| 521 |
< |
\caption{Speed distribution for thermal conductivity using swapping RNEMD.} |
| 522 |
< |
\label{histSwap} |
| 523 |
< |
\end{figure} |
| 519 |
> |
Comparatively, NIVS-RNEMD has a speed distribution closer to the ideal |
| 520 |
> |
curve fit (Figure \ref{thermalHist} b). Essentially, after scaling, a |
| 521 |
> |
Gaussian distribution function would remain Gaussian. Although a |
| 522 |
> |
single scaling is non-isotropic in all three dimensions, our scaling |
| 523 |
> |
coefficient criteria could help maintian the scaling region as |
| 524 |
> |
isotropic as possible. On the other hand, scaling coefficients are |
| 525 |
> |
preferred to be as close to 1 as possible, which also helps minimize |
| 526 |
> |
the difference among different dimensions. This is possible if scaling |
| 527 |
> |
interval and one-time thermal transfer energy are well |
| 528 |
> |
chosen. Consequently, NIVS-RNEMD is able to impose an unphysical |
| 529 |
> |
thermal flux as the previous RNEMD method without large perturbation |
| 530 |
> |
to the distribution of velocity and speed in the exchange regions. |
| 531 |
|
|
| 532 |
|
\begin{figure} |
| 533 |
< |
\includegraphics[width=\linewidth]{histScale} |
| 534 |
< |
\caption{Speed distribution for thermal conductivity using scaling RNEMD.} |
| 535 |
< |
\label{histScale} |
| 533 |
> |
\includegraphics[width=\linewidth]{thermalHist} |
| 534 |
> |
\caption{Speed distribution for thermal conductivity using a) |
| 535 |
> |
``swapping'' and b) NIVS- RNEMD methods. Shown is from the |
| 536 |
> |
simulations with an exchange or equilvalent exchange interval of 250 |
| 537 |
> |
fs. In circled areas, distributions from ``swapping'' RNEMD |
| 538 |
> |
simulation have deviation from ideal Maxwell-Boltzmann distribution |
| 539 |
> |
(curves fit for each distribution).} |
| 540 |
> |
\label{thermalHist} |
| 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. |
| 545 |
> |
Bedrov {\it et al.}\cite{Bedrov:2000}, which employed the |
| 546 |
> |
previous swapping RNEMD method for their calculation. Bedrov {\it et |
| 547 |
> |
al.}\cite{Bedrov:2000} argued that exchange of the molecule |
| 548 |
> |
center-of-mass velocities instead of single atom velocities in a |
| 549 |
> |
molecule conserves the total kinetic energy and linear momentum. This |
| 550 |
> |
principle is adopted in our simulations. Scaling is applied to the |
| 551 |
> |
velocities of the rigid bodies of SPC/E model water molecules, instead |
| 552 |
> |
of each hydrogen and oxygen atoms in relevant water molecules. As |
| 553 |
> |
shown in Figure \ref{spceGrad}, temperature gradients were established |
| 554 |
> |
similar to their system. However, the average temperature of our |
| 555 |
> |
system is 300K, while theirs is 318K, which would be attributed for |
| 556 |
> |
part of the difference between the final calculation results (Table |
| 557 |
> |
\ref{spceThermal}). Both methods yields values in agreement with |
| 558 |
> |
experiment. And this shows the applicability of our method to |
| 559 |
> |
multi-atom molecular system. |
| 560 |
|
|
| 561 |
|
\begin{figure} |
| 562 |
|
\includegraphics[width=\linewidth]{spceGrad} |
| 563 |
< |
\caption{Temperature gradients for SPC/E water thermal conductivity.} |
| 563 |
> |
\caption{Temperature gradients in SPC/E water thermal conductivity |
| 564 |
> |
simulations.} |
| 565 |
|
\label{spceGrad} |
| 566 |
|
\end{figure} |
| 567 |
|
|
| 576 |
|
\begin{tabular}{cccc} |
| 577 |
|
\hline |
| 578 |
|
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
| 579 |
< |
& This work & Previous simulations\cite{ISI:000090151400044} & |
| 579 |
> |
& This work & Previous simulations\cite{Bedrov:2000} & |
| 580 |
|
Experiment$^a$\\ |
| 581 |
|
\hline |
| 582 |
|
0.38 & 0.816(0.044) & & 0.64\\ |
| 590 |
|
\end{table*} |
| 591 |
|
|
| 592 |
|
\subsubsection{Crystal Gold} |
| 593 |
< |
Our results of gold thermal conductivity used QSC force field are |
| 594 |
< |
shown in Table \ref{AuThermal}. Although our calculation is smaller |
| 595 |
< |
than experimental value by an order of more than 100, this difference |
| 596 |
< |
is mainly attributed to the lack of electron interaction |
| 597 |
< |
representation in our force field parameters. Richardson {\it et |
| 598 |
< |
al.}\cite{ISI:A1992HX37800010} used similar force field parameters |
| 599 |
< |
in their metal thermal conductivity calculations. The EMD method they |
| 600 |
< |
employed in their simulations produced comparable results to |
| 601 |
< |
ours. Therefore, it is confident to conclude that NIVS-RNEMD is |
| 602 |
< |
applicable to metal force field system. |
| 593 |
> |
Our results of gold thermal conductivity using two force fields are |
| 594 |
> |
shown separately in Table \ref{qscThermal} and \ref{eamThermal}. In |
| 595 |
> |
these calculations,the end and middle slabs were excluded in thermal |
| 596 |
> |
gradient regession and only used as heat source and drain in the |
| 597 |
> |
systems. Our yielded values using EAM force field are slightly larger |
| 598 |
> |
than those using QSC force field. However, both series are |
| 599 |
> |
significantly smaller than experimental value by an order of more than |
| 600 |
> |
100. It has been verified that this difference is mainly attributed to |
| 601 |
> |
the lack of electron interaction representation in these force field |
| 602 |
> |
parameters. Richardson {\it et al.}\cite{Clancy:1992} used EAM |
| 603 |
> |
force field parameters in their metal thermal conductivity |
| 604 |
> |
calculations. The Non-Equilibrium MD method they employed in their |
| 605 |
> |
simulations produced comparable results to ours. As Zhang {\it et |
| 606 |
> |
al.}\cite{ISI:000231042800044} stated, thermal conductivity values |
| 607 |
> |
are influenced mainly by force field. Therefore, it is confident to |
| 608 |
> |
conclude that NIVS-RNEMD is applicable to metal force field system. |
| 609 |
|
|
| 610 |
|
\begin{figure} |
| 611 |
|
\includegraphics[width=\linewidth]{AuGrad} |
| 612 |
< |
\caption{Temperature gradients for crystal gold thermal conductivity.} |
| 612 |
> |
\caption{Temperature gradients for thermal conductivity calculation of |
| 613 |
> |
crystal gold using QSC force field.} |
| 614 |
|
\label{AuGrad} |
| 615 |
|
\end{figure} |
| 616 |
|
|
| 619 |
|
\begin{center} |
| 620 |
|
|
| 621 |
|
\caption{Calculation results for thermal conductivity of crystal gold |
| 622 |
< |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
| 623 |
< |
calculations in parentheses. } |
| 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}{ccc} |
| 625 |
> |
\begin{tabular}{cc} |
| 626 |
|
\hline |
| 627 |
|
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
| 613 |
– |
& This work & Previous simulations\cite{ISI:A1992HX37800010} \\ |
| 628 |
|
\hline |
| 629 |
< |
1.44 & 1.10(0.01) & \\ |
| 630 |
< |
2.86 & 1.08(0.02) & \\ |
| 631 |
< |
5.14 & 1.15(0.01) & \\ |
| 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{AuThermal} |
| 634 |
> |
\label{qscThermal} |
| 635 |
|
\end{center} |
| 636 |
|
\end{minipage} |
| 637 |
|
\end{table*} |
| 638 |
|
|
| 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 |
+ |
|
| 646 |
+ |
\begin{table*} |
| 647 |
+ |
\begin{minipage}{\linewidth} |
| 648 |
+ |
\begin{center} |
| 649 |
+ |
|
| 650 |
+ |
\caption{Calculation results for thermal conductivity of crystal gold |
| 651 |
+ |
using EAM force field at ${\langle T\rangle}$ = 300K at various |
| 652 |
+ |
thermal exchange rates. Errors of calculations in parentheses. } |
| 653 |
+ |
|
| 654 |
+ |
\begin{tabular}{cc} |
| 655 |
+ |
\hline |
| 656 |
+ |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
| 657 |
+ |
\hline |
| 658 |
+ |
1.24 & 1.24(0.06)\\ |
| 659 |
+ |
2.06 & 1.37(0.04)\\ |
| 660 |
+ |
2.55 & 1.41(0.03)\\ |
| 661 |
+ |
\hline |
| 662 |
+ |
\end{tabular} |
| 663 |
+ |
\label{eamThermal} |
| 664 |
+ |
\end{center} |
| 665 |
+ |
\end{minipage} |
| 666 |
+ |
\end{table*} |
| 667 |
+ |
|
| 668 |
+ |
|
| 669 |
|
\subsection{Interfaciel Thermal Conductivity} |
| 670 |
< |
After valid simulations of homogeneous water and gold systems using |
| 671 |
< |
NIVS-RNEMD method, calculation of gold/water interfacial thermal |
| 672 |
< |
conductivity was followed. It is found out that the interfacial |
| 673 |
< |
conductance is low due to a hydrophobic surface in our system. Figure |
| 674 |
< |
\ref{interfaceDensity} demonstrates this observance. Consequently, our |
| 675 |
< |
reported results (Table \ref{interfaceRes}) are of two orders of |
| 676 |
< |
magnitude smaller than our calculations on homogeneous systems. |
| 670 |
> |
After simulations of homogeneous water and gold systems using |
| 671 |
> |
NIVS-RNEMD method were proved valid, calculation of gold/water |
| 672 |
> |
interfacial thermal conductivity was followed. It is found out that |
| 673 |
> |
the low interfacial conductance is probably due to the hydrophobic |
| 674 |
> |
surface in our system. Figure \ref{interfaceDensity} demonstrates mass |
| 675 |
> |
density change along $z$-axis, which is perpendicular to the |
| 676 |
> |
gold/water interface. It is observed that water density significantly |
| 677 |
> |
decreases when approaching the surface. Under this low thermal |
| 678 |
> |
conductance, both gold and water phase have sufficient time to |
| 679 |
> |
eliminate temperature difference inside respectively (Figure |
| 680 |
> |
\ref{interfaceGrad}). With indistinguishable temperature difference |
| 681 |
> |
within respective phase, it is valid to assume that the temperature |
| 682 |
> |
difference between gold and water on surface would be approximately |
| 683 |
> |
the same as the difference between the gold and water phase. This |
| 684 |
> |
assumption enables convenient calculation of $G$ using |
| 685 |
> |
Eq. \ref{interfaceCalc} instead of measuring temperatures of thin |
| 686 |
> |
layer of water and gold close enough to surface, which would have |
| 687 |
> |
greater fluctuation and lower accuracy. Reported results (Table |
| 688 |
> |
\ref{interfaceRes}) are of two orders of magnitude smaller than our |
| 689 |
> |
calculations on homogeneous systems, and thus have larger relative |
| 690 |
> |
errors than our calculation results on homogeneous systems. |
| 691 |
|
|
| 692 |
|
\begin{figure} |
| 693 |
|
\includegraphics[width=\linewidth]{interfaceDensity} |
| 694 |
|
\caption{Density profile for interfacial thermal conductivity |
| 695 |
< |
simulation box.} |
| 695 |
> |
simulation box. Significant water density decrease is observed on |
| 696 |
> |
gold surface.} |
| 697 |
|
\label{interfaceDensity} |
| 698 |
|
\end{figure} |
| 699 |
|
|
| 700 |
|
\begin{figure} |
| 701 |
|
\includegraphics[width=\linewidth]{interfaceGrad} |
| 702 |
|
\caption{Temperature profiles for interfacial thermal conductivity |
| 703 |
< |
simulation box.} |
| 703 |
> |
simulation box. Temperatures of different slabs in the same phase |
| 704 |
> |
show no significant difference.} |
| 705 |
|
\label{interfaceGrad} |
| 706 |
|
\end{figure} |
| 707 |
|
|
| 728 |
|
\end{minipage} |
| 729 |
|
\end{table*} |
| 730 |
|
|
| 731 |
+ |
\subsection{Shear Viscosity} |
| 732 |
+ |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
| 733 |
+ |
produced comparable shear viscosity to swap RNEMD method. In Table |
| 734 |
+ |
\ref{shearRate}, the names of the calculated samples are devided into |
| 735 |
+ |
two parts. The first number refers to total slabs in one simulation |
| 736 |
+ |
box. The second number refers to the swapping interval in swap method, or |
| 737 |
+ |
in scale method the equilvalent swapping interval that the same |
| 738 |
+ |
momentum flux would theoretically result in swap method. All the scale |
| 739 |
+ |
method results were from simulations that had a scaling interval of 10 |
| 740 |
+ |
time steps. The average molecular momentum gradients of these samples |
| 741 |
+ |
are shown in Figure \ref{shear} (a) and (b). |
| 742 |
+ |
|
| 743 |
+ |
\begin{table*} |
| 744 |
+ |
\begin{minipage}{\linewidth} |
| 745 |
+ |
\begin{center} |
| 746 |
+ |
|
| 747 |
+ |
\caption{Calculation results for shear viscosity of Lennard-Jones |
| 748 |
+ |
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
| 749 |
+ |
methods at various momentum exchange rates. Results in reduced |
| 750 |
+ |
unit. Errors of calculations in parentheses. } |
| 751 |
+ |
|
| 752 |
+ |
\begin{tabular}{ccc} |
| 753 |
+ |
\hline |
| 754 |
+ |
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
| 755 |
+ |
\hline |
| 756 |
+ |
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
| 757 |
+ |
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
| 758 |
+ |
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
| 759 |
+ |
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
| 760 |
+ |
\hline |
| 761 |
+ |
\end{tabular} |
| 762 |
+ |
\label{shearRate} |
| 763 |
+ |
\end{center} |
| 764 |
+ |
\end{minipage} |
| 765 |
+ |
\end{table*} |
| 766 |
+ |
|
| 767 |
+ |
\begin{figure} |
| 768 |
+ |
\includegraphics[width=\linewidth]{shear} |
| 769 |
+ |
\caption{Average momentum gradients in shear viscosity simulations, |
| 770 |
+ |
using (a) ``swapping'' method and (b) NIVS-RNEMD method |
| 771 |
+ |
respectively. (c) Temperature difference among x and y, z dimensions |
| 772 |
+ |
observed when using NIVS-RNEMD with equivalent exchange interval of |
| 773 |
+ |
500 fs.} |
| 774 |
+ |
\label{shear} |
| 775 |
+ |
\end{figure} |
| 776 |
+ |
|
| 777 |
+ |
However, observations of temperatures along three dimensions show that |
| 778 |
+ |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
| 779 |
+ |
two slabs which were scaled. Figure \ref{shear} (c) indicate that with |
| 780 |
+ |
relatively large imposed momentum flux, the temperature difference among $x$ |
| 781 |
+ |
and the other two dimensions was significant. This would result from the |
| 782 |
+ |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
| 783 |
+ |
momentum gradient is set up, $P_c^x$ would be roughly stable |
| 784 |
+ |
($<0$). Consequently, scaling factor $x$ would most probably larger |
| 785 |
+ |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
| 786 |
+ |
keep increase after most scaling steps. And if there is not enough time |
| 787 |
+ |
for the kinetic energy to exchange among different dimensions and |
| 788 |
+ |
different slabs, the system would finally build up temperature |
| 789 |
+ |
(kinetic energy) difference among the three dimensions. Also, between |
| 790 |
+ |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
| 791 |
+ |
are closer to neighbor slabs. This is due to momentum transfer along |
| 792 |
+ |
$z$ dimension between slabs. |
| 793 |
+ |
|
| 794 |
+ |
Although results between scaling and swapping methods are comparable, |
| 795 |
+ |
the inherent temperature inhomogeneity even in relatively low imposed |
| 796 |
+ |
exchange momentum flux simulations makes scaling RNEMD method less |
| 797 |
+ |
attractive than swapping RNEMD in shear viscosity calculation. |
| 798 |
+ |
|
| 799 |
|
\section{Conclusions} |
| 800 |
|
NIVS-RNEMD simulation method is developed and tested on various |
| 801 |
< |
systems. Simulation results demonstrate its validity of thermal |
| 802 |
< |
conductivity calculations. NIVS-RNEMD improves non-Boltzmann-Maxwell |
| 803 |
< |
distributions existing in previous RNEMD methods, and extends its |
| 804 |
< |
applicability to interfacial systems. NIVS-RNEMD has also limited |
| 805 |
< |
application on shear viscosity calculations, but under high momentum |
| 806 |
< |
flux, it could cause temperature difference among different |
| 807 |
< |
dimensions. Modification is necessary to extend the applicability of |
| 808 |
< |
NIVS-RNEMD in shear viscosity calculations. |
| 801 |
> |
systems. Simulation results demonstrate its validity in thermal |
| 802 |
> |
conductivity calculations, from Lennard-Jones fluid to multi-atom |
| 803 |
> |
molecule like water and metal crystals. NIVS-RNEMD improves |
| 804 |
> |
non-Boltzmann-Maxwell distributions, which exist in previous RNEMD |
| 805 |
> |
methods. Furthermore, it develops a valid means for unphysical thermal |
| 806 |
> |
transfer between different species of molecules, and thus extends its |
| 807 |
> |
applicability to interfacial systems. Our calculation of gold/water |
| 808 |
> |
interfacial thermal conductivity demonstrates this advantage over |
| 809 |
> |
previous RNEMD methods. NIVS-RNEMD has also limited application on |
| 810 |
> |
shear viscosity calculations, but could cause temperature difference |
| 811 |
> |
among different dimensions under high momentum flux. Modification is |
| 812 |
> |
necessary to extend the applicability of NIVS-RNEMD in shear viscosity |
| 813 |
> |
calculations. |
| 814 |
|
|
| 815 |
|
\section{Acknowledgments} |
| 816 |
|
Support for this project was provided by the National Science |
| 818 |
|
the Center for Research Computing (CRC) at the University of Notre |
| 819 |
|
Dame. \newpage |
| 820 |
|
|
| 821 |
< |
\bibliographystyle{jcp2} |
| 821 |
> |
\bibliographystyle{aip} |
| 822 |
|
\bibliography{nivsRnemd} |
| 823 |
+ |
|
| 824 |
|
\end{doublespace} |
| 825 |
|
\end{document} |
| 826 |
|
|
