6 |
|
\usepackage{caption} |
7 |
|
%\usepackage{tabularx} |
8 |
|
\usepackage{graphicx} |
9 |
+ |
\usepackage{multirow} |
10 |
|
%\usepackage{booktabs} |
11 |
|
%\usepackage{bibentry} |
12 |
|
%\usepackage{mathrsfs} |
13 |
< |
\usepackage[ref]{overcite} |
13 |
> |
%\usepackage[ref]{overcite} |
14 |
> |
\usepackage[square, comma, sort&compress]{natbib} |
15 |
> |
\usepackage{url} |
16 |
|
\pagestyle{plain} \pagenumbering{arabic} \oddsidemargin 0.0cm |
17 |
|
\evensidemargin 0.0cm \topmargin -21pt \headsep 10pt \textheight |
18 |
|
9.0in \textwidth 6.5in \brokenpenalty=10000 |
22 |
|
\setlength{\abovecaptionskip}{20 pt} |
23 |
|
\setlength{\belowcaptionskip}{30 pt} |
24 |
|
|
25 |
< |
\renewcommand\citemid{\ } % no comma in optional referenc note |
25 |
> |
%\renewcommand\citemid{\ } % no comma in optional referenc note |
26 |
> |
\bibpunct{[}{]}{,}{s}{}{;} |
27 |
> |
\bibliographystyle{aip} |
28 |
|
|
29 |
|
\begin{document} |
30 |
|
|
44 |
|
|
45 |
|
\begin{abstract} |
46 |
|
We present a new method for introducing stable non-equilibrium |
47 |
< |
velocity and temperature distributions in molecular dynamics |
48 |
< |
simulations of heterogeneous systems. This method extends some |
49 |
< |
earlier Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods |
50 |
< |
which use momentum exchange swapping moves that can create |
51 |
< |
non-thermal velocity distributions (and which are difficult to use |
52 |
< |
for interfacial calculations). By using non-isotropic velocity |
47 |
> |
velocity and temperature gradients in molecular dynamics simulations |
48 |
> |
of heterogeneous systems. This method extends earlier Reverse |
49 |
> |
Non-Equilibrium Molecular Dynamics (RNEMD) methods which use |
50 |
> |
momentum exchange swapping moves. The standard swapping moves can |
51 |
> |
create non-thermal velocity distributions and are difficult to use |
52 |
> |
for interfacial calculations. By using non-isotropic velocity |
53 |
|
scaling (NIVS) on the molecules in specific regions of a system, it |
54 |
|
is possible to impose momentum or thermal flux between regions of a |
55 |
< |
simulation and stable thermal and momentum gradients can then be |
56 |
< |
established. The scaling method we have developed conserves the |
57 |
< |
total linear momentum and total energy of the system. To test the |
58 |
< |
methods, we have computed the thermal conductivity of model liquid |
59 |
< |
and solid systems as well as the interfacial thermal conductivity of |
60 |
< |
a metal-water interface. We find that the NIVS-RNEMD improves the |
56 |
< |
problematic velocity distributions that develop in other RNEMD |
57 |
< |
methods. |
55 |
> |
simulation while conserving the linear momentum and total energy of |
56 |
> |
the system. To test the methods, we have computed the thermal |
57 |
> |
conductivity of model liquid and solid systems as well as the |
58 |
> |
interfacial thermal conductivity of a metal-water interface. We |
59 |
> |
find that the NIVS-RNEMD improves the problematic velocity |
60 |
> |
distributions that develop in other RNEMD methods. |
61 |
|
\end{abstract} |
62 |
|
|
63 |
|
\newpage |
68 |
|
% BODY OF TEXT |
69 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
70 |
|
|
68 |
– |
|
69 |
– |
|
71 |
|
\section{Introduction} |
72 |
|
The original formulation of Reverse Non-equilibrium Molecular Dynamics |
73 |
|
(RNEMD) obtains transport coefficients (thermal conductivity and shear |
87 |
|
j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
88 |
|
J_z & = & \lambda \frac{\partial T}{\partial z} |
89 |
|
\end{eqnarray} |
90 |
< |
RNEMD has been widely used to provide computational estimates of thermal |
91 |
< |
conductivities and shear viscosities in a wide range of materials, |
92 |
< |
from liquid copper to monatomic liquids to molecular fluids |
93 |
< |
(e.g. ionic liquids).\cite{ISI:000246190100032} [MORE CITATIONS HERE] |
90 |
> |
RNEMD has been widely used to provide computational estimates of |
91 |
> |
thermal conductivities and shear viscosities in a wide range of |
92 |
> |
materials, from liquid copper to both monatomic and molecular fluids |
93 |
> |
(e.g. ionic |
94 |
> |
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} |
95 |
|
|
96 |
|
\begin{figure} |
97 |
|
\includegraphics[width=\linewidth]{thermalDemo} |
104 |
|
\label{thermalDemo} |
105 |
|
\end{figure} |
106 |
|
|
107 |
< |
RNEMD is preferable in many ways to the forward NEMD methods |
108 |
< |
[CITATIONS NEEDED] because it imposes what is typically difficult to measure |
109 |
< |
(a flux or stress) and it is typically much easier to compute momentum |
110 |
< |
gradients or strains (the response). For similar reasons, RNEMD is |
111 |
< |
also preferable to slowly-converging equilibrium methods for measuring |
112 |
< |
thermal conductivity and shear viscosity (using Green-Kubo relations |
113 |
< |
[CITATIONS NEEDED] or the Helfand moment approach of Viscardy {\it et |
107 |
> |
RNEMD is preferable in many ways to the forward NEMD |
108 |
> |
methods\cite{ISI:A1988Q205300014,hess:209,Vasquez:2004fk,backer:154503,ISI:000266247600008} |
109 |
> |
because it imposes what is typically difficult to measure (a flux or |
110 |
> |
stress) and it is typically much easier to compute the response |
111 |
> |
(momentum gradients or strains). For similar reasons, RNEMD is also |
112 |
> |
preferable to slowly-converging equilibrium methods for measuring |
113 |
> |
thermal conductivity and shear viscosity (using Green-Kubo |
114 |
> |
relations\cite{daivis:541,mondello:9327} or the Helfand moment |
115 |
> |
approach of Viscardy {\it et |
116 |
|
al.}\cite{Viscardy:2007bh,Viscardy:2007lq}) because these rely on |
117 |
|
computing difficult to measure quantities. |
118 |
|
|
122 |
|
typically samples from the same manifold of states in the |
123 |
|
microcanonical ensemble. |
124 |
|
|
125 |
< |
Recently, Tenney and Maginn\cite{ISI:000273472300004} have discovered |
126 |
< |
some problems with the original RNEMD swap technique. Notably, large |
125 |
> |
Recently, Tenney and Maginn\cite{Maginn:2010} have discovered some |
126 |
> |
problems with the original RNEMD swap technique. Notably, large |
127 |
|
momentum fluxes (equivalent to frequent momentum swaps between the |
128 |
|
slabs) can result in ``notched'', ``peaked'' and generally non-thermal |
129 |
|
momentum distributions in the two slabs, as well as non-linear thermal |
130 |
|
and velocity distributions along the direction of the imposed flux |
131 |
|
($z$). Tenney and Maginn obtained reasonable limits on imposed flux |
132 |
< |
and self-adjusting metrics for retaining the usability of the method. |
132 |
> |
and proposed self-adjusting metrics for retaining the usability of the |
133 |
> |
method. |
134 |
|
|
135 |
|
In this paper, we develop and test a method for non-isotropic velocity |
136 |
< |
scaling (NIVS-RNEMD) which retains the desirable features of RNEMD |
136 |
> |
scaling (NIVS) which retains the desirable features of RNEMD |
137 |
|
(conservation of linear momentum and total energy, compatibility with |
138 |
|
periodic boundary conditions) while establishing true thermal |
139 |
< |
distributions in each of the two slabs. In the next section, we |
139 |
> |
distributions in each of the two slabs. In the next section, we |
140 |
|
present the method for determining the scaling constraints. We then |
141 |
< |
test the method on both single component, multi-component, and |
142 |
< |
non-isotropic mixtures and show that it is capable of providing |
141 |
> |
test the method on both liquids and solids as well as a non-isotropic |
142 |
> |
liquid-solid interface and show that it is capable of providing |
143 |
|
reasonable estimates of the thermal conductivity and shear viscosity |
144 |
< |
in these cases. |
144 |
> |
in all of these cases. |
145 |
|
|
146 |
|
\section{Methodology} |
147 |
|
We retain the basic idea of M\"{u}ller-Plathe's RNEMD method; the |
161 |
|
0 & 0 & z \\ |
162 |
|
\end{array} \right) \cdot \vec{v}_i |
163 |
|
\end{equation} |
164 |
< |
where ${x, y, z}$ are a set of 3 scaling variables for each of the |
165 |
< |
three directions in the system. Likewise, the molecules $\{j\}$ |
166 |
< |
located in the hot slab will see a concomitant scaling of velocities, |
164 |
> |
where ${x, y, z}$ are a set of 3 velocity-scaling variables for each |
165 |
> |
of the three directions in the system. Likewise, the molecules |
166 |
> |
$\{j\}$ located in the hot slab will see a concomitant scaling of |
167 |
> |
velocities, |
168 |
|
\begin{equation} |
169 |
|
\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
170 |
|
x^\prime & 0 & 0 \\ |
181 |
|
\end{equation} |
182 |
|
where |
183 |
|
\begin{eqnarray} |
184 |
< |
P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
185 |
< |
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha |
184 |
> |
P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i v_{i\alpha} \\ |
185 |
> |
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j v_{j\alpha} |
186 |
|
\label{eq:momentumdef} |
187 |
|
\end{eqnarray} |
188 |
|
Therefore, for each of the three directions, the hot scaling |
189 |
|
parameters are a simple function of the cold scaling parameters and |
190 |
< |
the instantaneous linear momentum in each of the two slabs. |
190 |
> |
the instantaneous linear momenta in each of the two slabs. |
191 |
|
\begin{equation} |
192 |
|
\alpha^\prime = 1 + (1 - \alpha) p_\alpha |
193 |
|
\label{eq:hotcoldscaling} |
200 |
|
|
201 |
|
Conservation of total energy also places constraints on the scaling: |
202 |
|
\begin{equation} |
203 |
< |
\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
204 |
< |
\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha |
203 |
> |
\sum_{\alpha = x,y,z} \left\{ K_h^\alpha + K_c^\alpha \right\} = \sum_{\alpha = x,y,z} |
204 |
> |
\left\{ \left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha \right\} |
205 |
|
\end{equation} |
206 |
|
where the translational kinetic energies, $K_h^\alpha$ and |
207 |
|
$K_c^\alpha$, are computed for each of the three directions in a |
210 |
|
($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), we obtain the |
211 |
|
{\it constraint ellipsoid}: |
212 |
|
\begin{equation} |
213 |
< |
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0 |
213 |
> |
\sum_{\alpha = x,y,z} \left( a_\alpha \alpha^2 + b_\alpha \alpha + |
214 |
> |
c_\alpha \right) = 0 |
215 |
|
\label{eq:constraintEllipsoid} |
216 |
|
\end{equation} |
217 |
|
where the constants are obtained from the instantaneous values of the |
224 |
|
\label{eq:constraintEllipsoidConsts} |
225 |
|
\end{eqnarray} |
226 |
|
This ellipsoid (Eq. \ref{eq:constraintEllipsoid}) defines the set of |
227 |
< |
cold slab scaling parameters which can be applied while preserving |
228 |
< |
both linear momentum in all three directions as well as total kinetic |
229 |
< |
energy. |
227 |
> |
cold slab scaling parameters which, when applied, preserve the linear |
228 |
> |
momentum of the system in all three directions as well as total |
229 |
> |
kinetic energy. |
230 |
|
|
231 |
< |
The goal of using velocity scaling variables is to transfer linear |
232 |
< |
momentum or kinetic energy from the cold slab to the hot slab. If the |
233 |
< |
hot and cold slabs are separated along the z-axis, the energy flux is |
234 |
< |
given simply by the decrease in kinetic energy of the cold bin: |
231 |
> |
The goal of using these velocity scaling variables is to transfer |
232 |
> |
kinetic energy from the cold slab to the hot slab. If the hot and |
233 |
> |
cold slabs are separated along the z-axis, the energy flux is given |
234 |
> |
simply by the decrease in kinetic energy of the cold bin: |
235 |
|
\begin{equation} |
236 |
|
(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z\Delta t |
237 |
|
\end{equation} |
238 |
|
The expression for the energy flux can be re-written as another |
239 |
|
ellipsoid centered on $(x,y,z) = 0$: |
240 |
|
\begin{equation} |
241 |
< |
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 |
241 |
> |
\sum_{\alpha = x,y,z} K_c^\alpha \alpha^2 = \sum_{\alpha = x,y,z} |
242 |
> |
K_c^\alpha -J_z \Delta t |
243 |
|
\label{eq:fluxEllipsoid} |
244 |
|
\end{equation} |
245 |
|
The spatial extent of the {\it thermal flux ellipsoid} is governed |
246 |
< |
both by a targetted value, $J_z$ as well as the instantaneous values |
247 |
< |
of the kinetic energy components in the cold bin. |
246 |
> |
both by the target flux, $J_z$ as well as the instantaneous values of |
247 |
> |
the kinetic energy components in the cold bin. |
248 |
|
|
249 |
|
To satisfy an energetic flux as well as the conservation constraints, |
250 |
< |
we must determine the points ${x,y,z}$ which lie on both the |
251 |
< |
constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the flux |
252 |
< |
ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of the |
253 |
< |
two ellipsoids in 3-dimensional space. |
250 |
> |
we must determine the points ${x,y,z}$ that lie on both the constraint |
251 |
> |
ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the flux ellipsoid |
252 |
> |
(Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of the two |
253 |
> |
ellipsoids in 3-dimensional space. |
254 |
|
|
255 |
|
\begin{figure} |
256 |
|
\includegraphics[width=\linewidth]{ellipsoids} |
257 |
< |
\caption{Scaling points which maintain both constant energy and |
258 |
< |
constant linear momentum of the system lie on the surface of the |
259 |
< |
{\it constraint ellipsoid} while points which generate the target |
260 |
< |
momentum flux lie on the surface of the {\it flux ellipsoid}. The |
261 |
< |
velocity distributions in the cold bin are scaled by only those |
257 |
> |
\caption{Velocity scaling coefficients which maintain both constant |
258 |
> |
energy and constant linear momentum of the system lie on the surface |
259 |
> |
of the {\it constraint ellipsoid} while points which generate the |
260 |
> |
target momentum flux lie on the surface of the {\it flux ellipsoid}. |
261 |
> |
The velocity distributions in the cold bin are scaled by only those |
262 |
|
points which lie on both ellipsoids.} |
263 |
|
\label{ellipsoids} |
264 |
|
\end{figure} |
265 |
|
|
266 |
< |
One may also define {\it momentum} flux (say along the $x$-direction) as: |
266 |
> |
Since ellipsoids can be expressed as polynomials up to second order in |
267 |
> |
each of the three coordinates, finding the the intersection points of |
268 |
> |
two ellipsoids is isomorphic to finding the roots a polynomial of |
269 |
> |
degree 16. There are a number of polynomial root-finding methods in |
270 |
> |
the literature,\cite{Hoffman:2001sf,384119} but numerically finding |
271 |
> |
the roots of high-degree polynomials is generally an ill-conditioned |
272 |
> |
problem.\cite{Hoffman:2001sf} One simplification is to maintain |
273 |
> |
velocity scalings that are {\it as isotropic as possible}. To do |
274 |
> |
this, we impose $x=y$, and treat both the constraint and flux |
275 |
> |
ellipsoids as 2-dimensional ellipses. In reduced dimensionality, the |
276 |
> |
intersecting-ellipse problem reduces to finding the roots of |
277 |
> |
polynomials of degree 4. |
278 |
> |
|
279 |
> |
Depending on the target flux and current velocity distributions, the |
280 |
> |
ellipsoids can have between 0 and 4 intersection points. If there are |
281 |
> |
no intersection points, it is not possible to satisfy the constraints |
282 |
> |
while performing a non-equilibrium scaling move, and no change is made |
283 |
> |
to the dynamics. |
284 |
> |
|
285 |
> |
With multiple intersection points, any of the scaling points will |
286 |
> |
conserve the linear momentum and kinetic energy of the system and will |
287 |
> |
generate the correct target flux. Although this method is inherently |
288 |
> |
non-isotropic, the goal is still to maintain the system as close to an |
289 |
> |
isotropic fluid as possible. With this in mind, we would like the |
290 |
> |
kinetic energies in the three different directions could become as |
291 |
> |
close as each other as possible after each scaling. Simultaneously, |
292 |
> |
one would also like each scaling as gentle as possible, i.e. ${x,y,z |
293 |
> |
\rightarrow 1}$, in order to avoid large perturbation to the system. |
294 |
> |
To do this, we pick the intersection point which maintains the three |
295 |
> |
scaling variables ${x, y, z}$ as well as the ratio of kinetic energies |
296 |
> |
${K_c^z/K_c^x, K_c^z/K_c^y}$ as close as possible to 1. |
297 |
> |
|
298 |
> |
After the valid scaling parameters are arrived at by solving geometric |
299 |
> |
intersection problems in $x, y, z$ space in order to obtain cold slab |
300 |
> |
scaling parameters, Eq. (\ref{eq:hotcoldscaling}) is used to |
301 |
> |
determine the conjugate hot slab scaling variables. |
302 |
> |
|
303 |
> |
\subsection{Introducing shear stress via velocity scaling} |
304 |
> |
It is also possible to use this method to magnify the random |
305 |
> |
fluctuations of the average momentum in each of the bins to induce a |
306 |
> |
momentum flux. Doing this repeatedly will create a shear stress on |
307 |
> |
the system which will respond with an easily-measured strain. The |
308 |
> |
momentum flux (say along the $x$-direction) may be defined as: |
309 |
|
\begin{equation} |
310 |
|
(1-x) P_c^x = j_z(p_x)\Delta t |
311 |
|
\label{eq:fluxPlane} |
312 |
|
\end{equation} |
313 |
< |
The above {\it momentum flux plane} is perpendicular to the $x$-axis, |
314 |
< |
with its position governed both by a target value, $j_z(p_x)$ as well |
315 |
< |
as the instantaneous value of the momentum along the $x$-direction. |
313 |
> |
This {\it momentum flux plane} is perpendicular to the $x$-axis, with |
314 |
> |
its position governed both by a target value, $j_z(p_x)$ as well as |
315 |
> |
the instantaneous value of the momentum along the $x$-direction. |
316 |
|
|
317 |
|
In order to satisfy a momentum flux as well as the conservation |
318 |
|
constraints, we must determine the points ${x,y,z}$ which lie on both |
319 |
|
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
320 |
|
flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of an |
321 |
< |
ellipsoid and a plane in 3-dimensional space. |
321 |
> |
ellipsoid and a plane in 3-dimensional space. |
322 |
|
|
323 |
< |
In both the momentum and energy flux scenarios, valid scaling |
324 |
< |
parameters are arrived at by solving geometric intersection problems |
325 |
< |
in $x, y, z$ space in order to obtain cold slab scaling parameters. |
326 |
< |
Once the scaling variables for the cold slab are known, the hot slab |
327 |
< |
scaling has also been determined. |
323 |
> |
In the case of momentum flux transfer, we also impose another |
324 |
> |
constraint to set the kinetic energy transfer as zero. In other |
325 |
> |
words, we apply Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. With |
326 |
> |
one variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar |
327 |
> |
set of quartic equations to the above kinetic energy transfer problem. |
328 |
|
|
329 |
+ |
\section{Computational Details} |
330 |
|
|
331 |
< |
The following problem will be choosing an optimal set of scaling |
332 |
< |
variables among the possible sets. Although this method is inherently |
333 |
< |
non-isotropic, the goal is still to maintain the system as isotropic |
334 |
< |
as possible. Under this consideration, one would like the kinetic |
335 |
< |
energies in different directions could become as close as each other |
336 |
< |
after each scaling. Simultaneously, one would also like each scaling |
337 |
< |
as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
338 |
< |
large perturbation to the system. Therefore, one approach to obtain |
339 |
< |
the scaling variables would be constructing an criteria function, with |
340 |
< |
constraints as above equation sets, and solving the function's minimum |
341 |
< |
by method like Lagrange multipliers. |
331 |
> |
We have implemented this methodology in our molecular dynamics code, |
332 |
> |
OpenMD,\cite{Meineke:2005gd,openmd} performing the NIVS scaling moves |
333 |
> |
with a variable frequency after the molecular dynamics (MD) steps. We |
334 |
> |
have tested the method in a variety of different systems, including: |
335 |
> |
homogeneous fluids (Lennard-Jones and SPC/E water), crystalline |
336 |
> |
solids, using both the embedded atom method |
337 |
> |
(EAM)~\cite{PhysRevB.33.7983} and quantum Sutton-Chen |
338 |
> |
(QSC)~\cite{PhysRevB.59.3527} models for Gold, and heterogeneous |
339 |
> |
interfaces (QSC gold - SPC/E water). The last of these systems would |
340 |
> |
have been difficult to study using previous RNEMD methods, but the |
341 |
> |
current method can easily provide estimates of the interfacial thermal |
342 |
> |
conductivity ($G$). |
343 |
|
|
344 |
< |
In order to save computation time, we have a different approach to a |
293 |
< |
relatively good set of scaling variables with much less calculation |
294 |
< |
than above. Here is the detail of our simplification of the problem. |
344 |
> |
\subsection{Simulation Cells} |
345 |
|
|
346 |
< |
In the case of kinetic energy transfer, we impose another constraint |
347 |
< |
${x = y}$, into the equation sets. Consequently, there are two |
348 |
< |
variables left. And now one only needs to solve a set of two {\it |
349 |
< |
ellipses equations}. This problem would be transformed into solving |
350 |
< |
one quartic equation for one of the two variables. There are known |
351 |
< |
generic methods that solve real roots of quartic equations. Then one |
352 |
< |
can determine the other variable and obtain sets of scaling |
353 |
< |
variables. Among these sets, one can apply the above criteria to |
354 |
< |
choose the best set, while much faster with only a few sets to choose. |
346 |
> |
In each of the systems studied, the dynamics was carried out in a |
347 |
> |
rectangular simulation cell using periodic boundary conditions in all |
348 |
> |
three dimensions. The cells were longer along the $z$ axis and the |
349 |
> |
space was divided into $N$ slabs along this axis (typically $N=20$). |
350 |
> |
The top slab ($n=1$) was designated the ``hot'' slab, while the |
351 |
> |
central slab ($n= N/2 + 1$) was designated the ``cold'' slab. In all |
352 |
> |
cases, simulations were first thermalized in canonical ensemble (NVT) |
353 |
> |
using a Nos\'{e}-Hoover thermostat.\cite{Hoover85} then equilibrated in |
354 |
> |
microcanonical ensemble (NVE) before introducing any non-equilibrium |
355 |
> |
method. |
356 |
|
|
357 |
< |
In the case of momentum flux transfer, we impose another constraint to |
307 |
< |
set the kinetic energy transfer as zero. In another word, we apply |
308 |
< |
Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. After that, with one |
309 |
< |
variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar set |
310 |
< |
of equations on the above kinetic energy transfer problem. Therefore, |
311 |
< |
an approach similar to the above would be sufficient for this as well. |
357 |
> |
\subsection{RNEMD with M\"{u}ller-Plathe swaps} |
358 |
|
|
359 |
< |
\section{Computational Details} |
360 |
< |
\subsection{Lennard-Jones Fluid} |
361 |
< |
Our simulation consists of a series of systems. All of these |
362 |
< |
simulations were run with the OpenMD simulation software |
363 |
< |
package\cite{Meineke:2005gd} integrated with RNEMD codes. |
359 |
> |
In order to compare our new methodology with the original |
360 |
> |
M\"{u}ller-Plathe swapping algorithm,\cite{ISI:000080382700030} we |
361 |
> |
first performed simulations using the original technique. At fixed |
362 |
> |
intervals, kinetic energy or momentum exchange moves were performed |
363 |
> |
between the hot and the cold slabs. The interval between exchange |
364 |
> |
moves governs the effective momentum flux ($j_z(p_x)$) or energy flux |
365 |
> |
($J_z$) between the two slabs so to vary this quantity, we performed |
366 |
> |
simulations with a variety of delay intervals between the swapping moves. |
367 |
|
|
368 |
< |
A Lennard-Jones fluid system was built and tested first. In order to |
369 |
< |
compare our method with swapping RNEMD, a series of simulations were |
370 |
< |
performed to calculate the shear viscosity and thermal conductivity of |
371 |
< |
argon. 2592 atoms were in a orthorhombic cell, which was ${10.06\sigma |
372 |
< |
\times 10.06\sigma \times 30.18\sigma}$ by size. The reduced density |
373 |
< |
${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled direct |
374 |
< |
comparison between our results and others. These simulations used |
326 |
< |
velocity Verlet algorithm with reduced timestep ${\tau^* = |
327 |
< |
4.6\times10^{-4}}$. |
368 |
> |
For thermal conductivity measurements, the particle with smallest |
369 |
> |
speed in the hot slab and the one with largest speed in the cold slab |
370 |
> |
had their entire momentum vectors swapped. In the test cases run |
371 |
> |
here, all particles had the same chemical identity and mass, so this |
372 |
> |
move preserves both total linear momentum and total energy. It is |
373 |
> |
also possible to exchange energy by assuming an elastic collision |
374 |
> |
between the two particles which are exchanging energy. |
375 |
|
|
376 |
< |
For shear viscosity calculation, the reduced temperature was ${T^* = |
377 |
< |
k_B T/\varepsilon = 0.72}$. Simulations were first equilibrated in canonical |
378 |
< |
ensemble (NVT), then equilibrated in microcanonical ensemble |
332 |
< |
(NVE). Establishing and stablizing momentum gradient were followed |
333 |
< |
also in NVE ensemble. For the swapping part, Muller-Plathe's algorithm was |
334 |
< |
adopted.\cite{ISI:000080382700030} The simulation box was under |
335 |
< |
periodic boundary condition, and devided into ${N = 20}$ slabs. In each swap, |
336 |
< |
the top slab ${(n = 1)}$ exchange the most negative $x$ momentum with the |
337 |
< |
most positive $x$ momentum in the center slab ${(n = N/2 + 1)}$. Referred |
338 |
< |
to Tenney {\it et al.}\cite{ISI:000273472300004}, a series of swapping |
339 |
< |
frequency were chosen. According to each result from swapping |
340 |
< |
RNEMD, scaling RNEMD simulations were run with the target momentum |
341 |
< |
flux set to produce a similar momentum flux, and consequently shear |
342 |
< |
rate. Furthermore, various scaling frequencies can be tested for one |
343 |
< |
single swapping rate. To test the temperature homogeneity in our |
344 |
< |
system of swapping and scaling methods, temperatures of different |
345 |
< |
dimensions in all the slabs were observed. Most of the simulations |
346 |
< |
include $10^5$ steps of equilibration without imposing momentum flux, |
347 |
< |
$10^5$ steps of stablization with imposing unphysical momentum |
348 |
< |
transfer, and $10^6$ steps of data collection under RNEMD. For |
349 |
< |
relatively high momentum flux simulations, ${5\times10^5}$ step data |
350 |
< |
collection is sufficient. For some low momentum flux simulations, |
351 |
< |
${2\times10^6}$ steps were necessary. |
376 |
> |
For shear stress simulations, the particle with the most negative |
377 |
> |
$p_x$ in the hot slab and the one with the most positive $p_x$ in the |
378 |
> |
cold slab exchanged only this component of their momentum vectors. |
379 |
|
|
380 |
< |
After each simulation, the shear viscosity was calculated in reduced |
381 |
< |
unit. The momentum flux was calculated with total unphysical |
382 |
< |
transferred momentum ${P_x}$ and data collection time $t$: |
380 |
> |
\subsection{RNEMD with NIVS scaling} |
381 |
> |
|
382 |
> |
For each simulation utilizing the swapping method, a corresponding |
383 |
> |
NIVS-RNEMD simulation was carried out using a target momentum flux set |
384 |
> |
to produce the same flux experienced in the swapping simulation. |
385 |
> |
|
386 |
> |
To test the temperature homogeneity, directional momentum and |
387 |
> |
temperature distributions were accumulated for molecules in each of |
388 |
> |
the slabs. Transport coefficients were computed using the temperature |
389 |
> |
(and momentum) gradients across the $z$-axis as well as the total |
390 |
> |
momentum flux the system experienced during data collection portion of |
391 |
> |
the simulation. |
392 |
> |
|
393 |
> |
\subsection{Shear viscosities} |
394 |
> |
|
395 |
> |
The momentum flux was calculated using the total non-physical momentum |
396 |
> |
transferred (${P_x}$) and the data collection time ($t$): |
397 |
|
\begin{equation} |
398 |
|
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
399 |
|
\end{equation} |
400 |
< |
where $L_x$ and $L_y$ denotes $x$ and $y$ lengths of the simulation |
401 |
< |
box, and physical momentum transfer occurs in two ways due to our |
402 |
< |
periodic boundary condition settings. And the velocity gradient |
403 |
< |
${\langle \partial v_x /\partial z \rangle}$ can be obtained by a |
404 |
< |
linear regression of the velocity profile. From the shear viscosity |
405 |
< |
$\eta$ calculated with the above parameters, one can further convert |
406 |
< |
it into reduced unit ${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$. |
400 |
> |
where $L_x$ and $L_y$ denote the $x$ and $y$ lengths of the simulation |
401 |
> |
box. The factor of two in the denominator is present because physical |
402 |
> |
momentum transfer between the slabs occurs in two directions ($+z$ and |
403 |
> |
$-z$). The velocity gradient ${\langle \partial v_x /\partial z |
404 |
> |
\rangle}$ was obtained using linear regression of the mean $x$ |
405 |
> |
component of the velocity, $\langle v_x \rangle$, in each of the bins. |
406 |
> |
For Lennard-Jones simulations, shear viscosities are reported in |
407 |
> |
reduced units (${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$). |
408 |
|
|
409 |
< |
For thermal conductivity calculations, simulations were first run under |
368 |
< |
reduced temperature ${\langle T^*\rangle = 0.72}$ in NVE |
369 |
< |
ensemble. Muller-Plathe's algorithm was adopted in the swapping |
370 |
< |
method. Under identical simulation box parameters with our shear |
371 |
< |
viscosity calculations, in each swap, the top slab exchanges all three |
372 |
< |
translational momentum components of the molecule with least kinetic |
373 |
< |
energy with the same components of the molecule in the center slab |
374 |
< |
with most kinetic energy, unless this ``coldest'' molecule in the |
375 |
< |
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the |
376 |
< |
``cold'' slab. According to swapping RNEMD results, target energy flux |
377 |
< |
for scaling RNEMD simulations can be set. Also, various scaling |
378 |
< |
frequencies can be tested for one target energy flux. To compare the |
379 |
< |
performance between swapping and scaling method, distributions of |
380 |
< |
velocity and speed in different slabs were observed. |
409 |
> |
\subsection{Thermal Conductivities} |
410 |
|
|
411 |
< |
For each swapping rate, thermal conductivity was calculated in reduced |
412 |
< |
unit. The energy flux was calculated similarly to the momentum flux, |
413 |
< |
with total unphysical transferred energy ${E_{total}}$ and data collection |
385 |
< |
time $t$: |
411 |
> |
The energy flux was calculated in a similar manner to the momentum |
412 |
> |
flux, using the total non-physical energy transferred (${E_{total}}$) |
413 |
> |
and the data collection time $t$: |
414 |
|
\begin{equation} |
415 |
|
J_z = \frac{E_{total}}{2 t L_x L_y} |
416 |
|
\end{equation} |
417 |
< |
And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
418 |
< |
can be obtained by a linear regression of the temperature |
419 |
< |
profile. From the thermal conductivity $\lambda$ calculated, one can |
420 |
< |
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
421 |
< |
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
417 |
> |
The temperature gradient ${\langle\partial T/\partial z\rangle}$ was |
418 |
> |
obtained by a linear regression of the temperature profile. For |
419 |
> |
Lennard-Jones simulations, thermal conductivities are reported in |
420 |
> |
reduced units (${\lambda^*=\lambda \sigma^2 m^{1/2} |
421 |
> |
k_B^{-1}\varepsilon^{-1/2}}$). |
422 |
|
|
423 |
< |
\subsection{ Water / Metal Thermal Conductivity} |
396 |
< |
Another series of our simulation is the calculation of interfacial |
397 |
< |
thermal conductivity of a Au/H$_2$O system. Respective calculations of |
398 |
< |
liquid water (Extended Simple Point Charge model) and crystal gold |
399 |
< |
thermal conductivity were performed and compared with current results |
400 |
< |
to ensure the validity of NIVS-RNEMD. After that, a mixture system was |
401 |
< |
simulated. |
423 |
> |
\subsection{Interfacial Thermal Conductivities} |
424 |
|
|
425 |
< |
For thermal conductivity calculation of bulk water, a simulation box |
426 |
< |
consisting of 1000 molecules were first equilibrated under ambient |
427 |
< |
pressure and temperature conditions using NPT ensemble, followed by |
428 |
< |
equilibration in fixed volume (NVT). The system was then equilibrated in |
429 |
< |
microcanonical ensemble (NVE). Also in NVE ensemble, establishing a |
408 |
< |
stable thermal gradient was followed. The simulation box was under |
409 |
< |
periodic boundary condition and devided into 10 slabs. Data collection |
410 |
< |
process was similar to Lennard-Jones fluid system. |
425 |
> |
For interfaces with a relatively low interfacial conductance, the bulk |
426 |
> |
regions on either side of an interface rapidly come to a state in |
427 |
> |
which the two phases have relatively homogeneous (but distinct) |
428 |
> |
temperatures. The interfacial thermal conductivity $G$ can therefore |
429 |
> |
be approximated as: |
430 |
|
|
412 |
– |
Thermal conductivity calculation of bulk crystal gold used a similar |
413 |
– |
protocol. Two types of force field parameters, Embedded Atom Method |
414 |
– |
(EAM) and Quantum Sutten-Chen (QSC) force field were used |
415 |
– |
respectively. The face-centered cubic crystal simulation box consists of |
416 |
– |
2880 Au atoms. The lattice was first allowed volume change to relax |
417 |
– |
under ambient temperature and pressure. Equilibrations in canonical and |
418 |
– |
microcanonical ensemble were followed in order. With the simulation |
419 |
– |
lattice devided evenly into 10 slabs, different thermal gradients were |
420 |
– |
established by applying a set of target thermal transfer flux. Data of |
421 |
– |
the series of thermal gradients was collected for calculation. |
422 |
– |
|
423 |
– |
After simulations of bulk water and crystal gold, a mixture system was |
424 |
– |
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
425 |
– |
molecules. Spohr potential was adopted in depicting the interaction |
426 |
– |
between metal atom and water molecule.\cite{ISI:000167766600035} A |
427 |
– |
similar protocol of equilibration was followed. Several thermal |
428 |
– |
gradients was built under different target thermal flux. It was found |
429 |
– |
out that compared to our previous simulation systems, the two phases |
430 |
– |
could have large temperature difference even under a relatively low |
431 |
– |
thermal flux. Therefore, under our low flux conditions, it is assumed |
432 |
– |
that the metal and water phases have respectively homogeneous |
433 |
– |
temperature, excluding the surface regions. In calculating the |
434 |
– |
interfacial thermal conductivity $G$, this assumptioin was applied and |
435 |
– |
thus our formula becomes: |
436 |
– |
|
431 |
|
\begin{equation} |
432 |
|
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
433 |
|
\langle T_{water}\rangle \right)} |
434 |
|
\label{interfaceCalc} |
435 |
|
\end{equation} |
436 |
< |
where ${E_{total}}$ is the imposed unphysical kinetic energy transfer |
437 |
< |
and ${\langle T_{gold}\rangle}$ and ${\langle T_{water}\rangle}$ are the |
438 |
< |
average observed temperature of gold and water phases respectively. |
436 |
> |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
437 |
> |
transfer and ${\langle T_{gold}\rangle}$ and ${\langle |
438 |
> |
T_{water}\rangle}$ are the average observed temperature of gold and |
439 |
> |
water phases respectively. If the interfacial conductance is {\it |
440 |
> |
not} small, it is also be possible to compute the interfacial |
441 |
> |
thermal conductivity using this method utilizing the change in the |
442 |
> |
slope of the thermal gradient ($\partial^2 \langle T \rangle / \partial |
443 |
> |
z^2$) at the interface. |
444 |
|
|
445 |
< |
\section{Results And Discussions} |
447 |
< |
\subsection{Thermal Conductivity} |
448 |
< |
\subsubsection{Lennard-Jones Fluid} |
449 |
< |
Our thermal conductivity calculations show that scaling method results |
450 |
< |
agree with swapping method. Four different exchange intervals were |
451 |
< |
tested (Table \ref{thermalLJRes}) using swapping method. With a fixed |
452 |
< |
10fs exchange interval, target exchange kinetic energy was set to |
453 |
< |
produce equivalent kinetic energy flux as in swapping method. And |
454 |
< |
similar thermal gradients were observed with similar thermal flux in |
455 |
< |
two simulation methods (Figure \ref{thermalGrad}). |
445 |
> |
\section{Results} |
446 |
|
|
447 |
< |
\begin{table*} |
448 |
< |
\begin{minipage}{\linewidth} |
449 |
< |
\begin{center} |
447 |
> |
\subsection{Lennard-Jones Fluid} |
448 |
> |
2592 Lennard-Jones atoms were placed in an orthorhombic cell |
449 |
> |
${10.06\sigma \times 10.06\sigma \times 30.18\sigma}$ on a side. The |
450 |
> |
reduced density ${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled |
451 |
> |
direct comparison between our results and previous methods. These |
452 |
> |
simulations were carried out with a reduced timestep ${\tau^* = |
453 |
> |
4.6\times10^{-4}}$. For the shear viscosity calculations, the mean |
454 |
> |
temperature was ${T^* = k_B T/\varepsilon = 0.72}$. For thermal |
455 |
> |
conductivity calculations, simulations were run under reduced |
456 |
> |
temperature ${\langle T^*\rangle = 0.72}$ in the microcanonical |
457 |
> |
ensemble. The simulations included $10^5$ steps of equilibration |
458 |
> |
without any momentum flux, $10^5$ steps of stablization with an |
459 |
> |
imposed momentum transfer to create a gradient, and $10^6$ steps of |
460 |
> |
data collection under RNEMD. |
461 |
|
|
462 |
< |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
462 |
< |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
463 |
< |
swap and scale methods at various kinetic energy exchange rates. Results |
464 |
< |
in reduced unit. Errors of calculations in parentheses.} |
462 |
> |
\subsubsection*{Thermal Conductivity} |
463 |
|
|
464 |
< |
\begin{tabular}{ccc} |
465 |
< |
\hline |
466 |
< |
(Equilvalent) Exchange Interval (fs) & $\lambda^*_{swap}$ & |
467 |
< |
$\lambda^*_{scale}$\\ |
468 |
< |
\hline |
469 |
< |
250 & 7.03(0.34) & 7.30(0.10)\\ |
470 |
< |
500 & 7.03(0.14) & 6.95(0.09)\\ |
471 |
< |
1000 & 6.91(0.42) & 7.19(0.07)\\ |
472 |
< |
2000 & 7.52(0.15) & 7.19(0.28)\\ |
473 |
< |
\hline |
474 |
< |
\end{tabular} |
475 |
< |
\label{thermalLJRes} |
476 |
< |
\end{center} |
477 |
< |
\end{minipage} |
464 |
> |
Our thermal conductivity calculations show that the NIVS method agrees |
465 |
> |
well with the swapping method. Five different swap intervals were |
466 |
> |
tested (Table \ref{LJ}). Similar thermal gradients were observed with |
467 |
> |
similar thermal flux under the two different methods (Figure |
468 |
> |
\ref{thermalGrad}). Furthermore, the 1-d temperature profiles showed |
469 |
> |
no observable differences between the $x$, $y$ and $z$ axes (Figure |
470 |
> |
\ref{thermalGrad} c), so even though we are using a non-isotropic |
471 |
> |
scaling method, none of the three directions are experience |
472 |
> |
disproportionate heating due to the velocity scaling. |
473 |
> |
|
474 |
> |
\begin{table*} |
475 |
> |
\begin{minipage}{\linewidth} |
476 |
> |
\begin{center} |
477 |
> |
|
478 |
> |
\caption{Thermal conductivity ($\lambda^*$) and shear viscosity |
479 |
> |
($\eta^*$) (in reduced units) of a Lennard-Jones fluid at |
480 |
> |
${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$ computed |
481 |
> |
at various momentum fluxes. The original swapping method and |
482 |
> |
the velocity scaling method give similar results. |
483 |
> |
Uncertainties are indicated in parentheses.} |
484 |
> |
|
485 |
> |
\begin{tabular}{|cc|cc|cc|} |
486 |
> |
\hline |
487 |
> |
\multicolumn{2}{|c}{Momentum Exchange} & |
488 |
> |
\multicolumn{2}{|c}{Swapping RNEMD} & |
489 |
> |
\multicolumn{2}{|c|}{NIVS-RNEMD} \\ |
490 |
> |
\hline |
491 |
> |
\multirow{2}{*}{Swap Interval (fs)} & Equivalent $J_z^*$ or & |
492 |
> |
\multirow{2}{*}{$\lambda^*_{swap}$} & |
493 |
> |
\multirow{2}{*}{$\eta^*_{swap}$} & |
494 |
> |
\multirow{2}{*}{$\lambda^*_{scale}$} & |
495 |
> |
\multirow{2}{*}{$\eta^*_{scale}$} \\ |
496 |
> |
& $j_z^*(p_x)$ (reduced units) & & & & \\ |
497 |
> |
\hline |
498 |
> |
250 & 0.16 & 7.03(0.34) & 3.57(0.06) & 7.30(0.10) & 3.54(0.04)\\ |
499 |
> |
500 & 0.09 & 7.03(0.14) & 3.64(0.05) & 6.95(0.09) & 3.76(0.09)\\ |
500 |
> |
1000 & 0.047 & 6.91(0.42) & 3.52(0.16) & 7.19(0.07) & 3.66(0.06)\\ |
501 |
> |
2000 & 0.024 & 7.52(0.15) & 3.72(0.05) & 7.19(0.28) & 3.32(0.18)\\ |
502 |
> |
2500 & 0.019 & 7.41(0.29) & 3.42(0.06) & 7.98(0.33) & 3.43(0.08)\\ |
503 |
> |
\hline |
504 |
> |
\end{tabular} |
505 |
> |
\label{LJ} |
506 |
> |
\end{center} |
507 |
> |
\end{minipage} |
508 |
|
\end{table*} |
509 |
|
|
510 |
|
\begin{figure} |
511 |
< |
\includegraphics[width=\linewidth]{thermalGrad} |
512 |
< |
\caption{Temperature gradients under various kinetic energy flux of |
513 |
< |
thermal conductivity simulations} |
514 |
< |
\label{thermalGrad} |
511 |
> |
\includegraphics[width=\linewidth]{thermalGrad} |
512 |
> |
\caption{The NIVS-RNEMD method creates similar temperature gradients |
513 |
> |
compared with the swapping method under a variety of imposed |
514 |
> |
kinetic energy flux values. Furthermore, the implementation of |
515 |
> |
Non-Isotropic Velocity Scaling does not cause temperature |
516 |
> |
anisotropy to develop in thermal conductivity calculations.} |
517 |
> |
\label{thermalGrad} |
518 |
|
\end{figure} |
519 |
|
|
520 |
< |
During these simulations, molecule velocities were recorded in 1000 of |
490 |
< |
all the snapshots of one single data collection process. These |
491 |
< |
velocity data were used to produce histograms of velocity and speed |
492 |
< |
distribution in different slabs. From these histograms, it is observed |
493 |
< |
that under relatively high unphysical kinetic energy flux, speed and |
494 |
< |
velocity distribution of molecules in slabs where swapping occured |
495 |
< |
could deviate from Maxwell-Boltzmann distribution. Figure |
496 |
< |
\ref{histSwap} illustrates how these distributions deviate from an |
497 |
< |
ideal distribution. In high temperature slab, probability density in |
498 |
< |
low speed is confidently smaller than ideal curve fit; in low |
499 |
< |
temperature slab, probability density in high speed is smaller than |
500 |
< |
ideal, while larger than ideal in low speed. This phenomenon is more |
501 |
< |
obvious in our high swapping rate simulations. And this deviation |
502 |
< |
could also leads to deviation of distribution of velocity in various |
503 |
< |
dimensions. One feature of these deviated distribution is that in high |
504 |
< |
temperature slab, the ideal Gaussian peak was changed into a |
505 |
< |
relatively flat plateau; while in low temperature slab, that peak |
506 |
< |
appears sharper. This problem is rooted in the mechanism of the |
507 |
< |
swapping method. Continually depleting low (high) speed particles in |
508 |
< |
the high (low) temperature slab could not be complemented by |
509 |
< |
diffusions of low (high) speed particles from neighbor slabs, unless |
510 |
< |
in suffciently low swapping rate. Simutaneously, surplus low speed |
511 |
< |
particles in the low temperature slab do not have sufficient time to |
512 |
< |
diffuse to neighbor slabs. However, thermal exchange rate should reach |
513 |
< |
a minimum level to produce an observable thermal gradient under noise |
514 |
< |
interference. Consequently, swapping RNEMD has a relatively narrow |
515 |
< |
choice of swapping rate to satisfy these above restrictions. |
520 |
> |
\subsubsection*{Velocity Distributions} |
521 |
|
|
522 |
< |
\begin{figure} |
523 |
< |
\includegraphics[width=\linewidth]{histSwap} |
524 |
< |
\caption{Speed distribution for thermal conductivity using swapping |
525 |
< |
RNEMD. Shown is from the simulation with 250 fs exchange interval.} |
526 |
< |
\label{histSwap} |
527 |
< |
\end{figure} |
522 |
> |
During these simulations, velocities were recorded every 1000 steps |
523 |
> |
and were used to produce distributions of both velocity and speed in |
524 |
> |
each of the slabs. From these distributions, we observed that under |
525 |
> |
relatively high non-physical kinetic energy flux, the speed of |
526 |
> |
molecules in slabs where swapping occured could deviate from the |
527 |
> |
Maxwell-Boltzmann distribution. This behavior was also noted by Tenney |
528 |
> |
and Maginn.\cite{Maginn:2010} Figure \ref{thermalHist} shows how these |
529 |
> |
distributions deviate from an ideal distribution. In the ``hot'' slab, |
530 |
> |
the probability density is notched at low speeds and has a substantial |
531 |
> |
shoulder at higher speeds relative to the ideal MB distribution. In |
532 |
> |
the cold slab, the opposite notching and shouldering occurs. This |
533 |
> |
phenomenon is more obvious at higher swapping rates. |
534 |
|
|
535 |
< |
Comparatively, NIVS-RNEMD has a speed distribution closer to the ideal |
536 |
< |
curve fit (Figure \ref{histScale}). Essentially, after scaling, a |
537 |
< |
Gaussian distribution function would remain Gaussian. Although a |
538 |
< |
single scaling is non-isotropic in all three dimensions, our scaling |
539 |
< |
coefficient criteria could help maintian the scaling region as |
540 |
< |
isotropic as possible. On the other hand, scaling coefficients are |
541 |
< |
preferred to be as close to 1 as possible, which also helps minimize |
542 |
< |
the difference among different dimensions. This is possible if scaling |
543 |
< |
interval and one-time thermal transfer energy are well |
544 |
< |
chosen. Consequently, NIVS-RNEMD is able to impose an unphysical |
545 |
< |
thermal flux as the previous RNEMD method without large perturbation |
546 |
< |
to the distribution of velocity and speed in the exchange regions. |
535 |
> |
The peak of the velocity distribution is substantially flattened in |
536 |
> |
the hot slab, and is overly sharp (with truncated wings) in the cold |
537 |
> |
slab. This problem is rooted in the mechanism of the swapping method. |
538 |
> |
Continually depleting low (high) speed particles in the high (low) |
539 |
> |
temperature slab is not complemented by diffusions of low (high) speed |
540 |
> |
particles from neighboring slabs, unless the swapping rate is |
541 |
> |
sufficiently small. Simutaneously, surplus low speed particles in the |
542 |
> |
low temperature slab do not have sufficient time to diffuse to |
543 |
> |
neighboring slabs. Since the thermal exchange rate must reach a |
544 |
> |
minimum level to produce an observable thermal gradient, the |
545 |
> |
swapping-method RNEMD has a relatively narrow choice of exchange times |
546 |
> |
that can be utilized. |
547 |
|
|
548 |
+ |
For comparison, NIVS-RNEMD produces a speed distribution closer to the |
549 |
+ |
Maxwell-Boltzmann curve (Figure \ref{thermalHist}). The reason for |
550 |
+ |
this is simple; upon velocity scaling, a Gaussian distribution remains |
551 |
+ |
Gaussian. Although a single scaling move is non-isotropic in three |
552 |
+ |
dimensions, our criteria for choosing a set of scaling coefficients |
553 |
+ |
helps maintain the distributions as close to isotropic as possible. |
554 |
+ |
Consequently, NIVS-RNEMD is able to impose an unphysical thermal flux |
555 |
+ |
as the previous RNEMD methods but without large perturbations to the |
556 |
+ |
velocity distributions in the two slabs. |
557 |
+ |
|
558 |
|
\begin{figure} |
559 |
< |
\includegraphics[width=\linewidth]{histScale} |
560 |
< |
\caption{Speed distribution for thermal conductivity using scaling |
561 |
< |
RNEMD. Shown is from the simulation with an equilvalent thermal flux |
562 |
< |
as an 250 fs exchange interval swapping simulation.} |
563 |
< |
\label{histScale} |
559 |
> |
\includegraphics[width=\linewidth]{thermalHist} |
560 |
> |
\caption{Velocity and speed distributions that develop under the |
561 |
> |
swapping and NIVS-RNEMD methods at high flux. The distributions for |
562 |
> |
the hot bins (upper panels) and cold bins (lower panels) were |
563 |
> |
obtained from Lennard-Jones simulations with $\langle T^* \rangle = |
564 |
> |
4.5$ with a flux of $J_z^* \sim 5$ (equivalent to a swapping interval |
565 |
> |
of 10 time steps). This is a relatively large flux which shows the |
566 |
> |
non-thermal distributions that develop under the swapping method. |
567 |
> |
NIVS does a better job of producing near-thermal distributions in |
568 |
> |
the bins.} |
569 |
> |
\label{thermalHist} |
570 |
|
\end{figure} |
571 |
|
|
545 |
– |
\subsubsection{SPC/E Water} |
546 |
– |
Our results of SPC/E water thermal conductivity are comparable to |
547 |
– |
Bedrov {\it et al.}\cite{ISI:000090151400044}, which employed the |
548 |
– |
previous swapping RNEMD method for their calculation. Bedrov {\it et |
549 |
– |
al.}\cite{ISI:000090151400044} argued that exchange of the molecule |
550 |
– |
center-of-mass velocities instead of single atom velocities in a |
551 |
– |
molecule conserves the total kinetic energy and linear momentum. This |
552 |
– |
principle is adopted in our simulations. Scaling is applied to the |
553 |
– |
velocities of the rigid bodies of SPC/E model water molecules, instead |
554 |
– |
of each hydrogen and oxygen atoms in relevant water molecules. As |
555 |
– |
shown in Figure \ref{spceGrad}, temperature gradients were established |
556 |
– |
similar to their system. However, the average temperature of our |
557 |
– |
system is 300K, while theirs is 318K, which would be attributed for |
558 |
– |
part of the difference between the final calculation results (Table |
559 |
– |
\ref{spceThermal}). Both methods yields values in agreement with |
560 |
– |
experiment. And this shows the applicability of our method to |
561 |
– |
multi-atom molecular system. |
572 |
|
|
573 |
+ |
\subsubsection*{Shear Viscosity} |
574 |
+ |
Our calculations (Table \ref{LJ}) show that velocity-scaling RNEMD |
575 |
+ |
predicted comparable shear viscosities to swap RNEMD method. The |
576 |
+ |
average molecular momentum gradients of these samples are shown in |
577 |
+ |
Figure \ref{shear} (a) and (b). |
578 |
+ |
|
579 |
|
\begin{figure} |
580 |
< |
\includegraphics[width=\linewidth]{spceGrad} |
581 |
< |
\caption{Temperature gradients for SPC/E water thermal conductivity.} |
582 |
< |
\label{spceGrad} |
580 |
> |
\includegraphics[width=\linewidth]{shear} |
581 |
> |
\caption{Average momentum gradients in shear viscosity simulations, |
582 |
> |
using ``swapping'' method (top panel) and NIVS-RNEMD method |
583 |
> |
(middle panel). NIVS-RNEMD produces a thermal anisotropy artifact |
584 |
> |
in the hot and cold bins when used for shear viscosity. This |
585 |
> |
artifact does not appear in thermal conductivity calculations.} |
586 |
> |
\label{shear} |
587 |
|
\end{figure} |
588 |
|
|
589 |
< |
\begin{table*} |
590 |
< |
\begin{minipage}{\linewidth} |
591 |
< |
\begin{center} |
589 |
> |
Observations of the three one-dimensional temperatures in each of the |
590 |
> |
slabs shows that NIVS-RNEMD does produce some thermal anisotropy, |
591 |
> |
particularly in the hot and cold slabs. Figure \ref{shear} (c) |
592 |
> |
indicates that with a relatively large imposed momentum flux, |
593 |
> |
$j_z(p_x)$, the $x$ direction approaches a different temperature from |
594 |
> |
the $y$ and $z$ directions in both the hot and cold bins. This is an |
595 |
> |
artifact of the scaling constraints. After the momentum gradient has |
596 |
> |
been established, $P_c^x < 0$. Consequently, the scaling factor $x$ |
597 |
> |
is nearly always greater than one in order to satisfy the constraints. |
598 |
> |
This will continually increase the kinetic energy in $x$-dimension, |
599 |
> |
$K_c^x$. If there is not enough time for the kinetic energy to |
600 |
> |
exchange among different directions and different slabs, the system |
601 |
> |
will exhibit the observed thermal anisotropy in the hot and cold bins. |
602 |
|
|
603 |
< |
\caption{Calculation results for thermal conductivity of SPC/E water |
604 |
< |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
605 |
< |
calculations in parentheses. } |
603 |
> |
Although results between scaling and swapping methods are comparable, |
604 |
> |
the inherent temperature anisotropy does make NIVS-RNEMD method less |
605 |
> |
attractive than swapping RNEMD for shear viscosity calculations. We |
606 |
> |
note that this problem appears only when momentum flux is applied, and |
607 |
> |
does not appear in thermal flux calculations. |
608 |
|
|
609 |
< |
\begin{tabular}{cccc} |
578 |
< |
\hline |
579 |
< |
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
580 |
< |
& This work & Previous simulations\cite{ISI:000090151400044} & |
581 |
< |
Experiment$^a$\\ |
582 |
< |
\hline |
583 |
< |
0.38 & 0.816(0.044) & & 0.64\\ |
584 |
< |
0.81 & 0.770(0.008) & 0.784\\ |
585 |
< |
1.54 & 0.813(0.007) & 0.730\\ |
586 |
< |
\hline |
587 |
< |
\end{tabular} |
588 |
< |
\label{spceThermal} |
589 |
< |
\end{center} |
590 |
< |
\end{minipage} |
591 |
< |
\end{table*} |
609 |
> |
\subsection{Bulk SPC/E water} |
610 |
|
|
611 |
< |
\subsubsection{Crystal Gold} |
612 |
< |
Our results of gold thermal conductivity using two force fields are |
613 |
< |
shown separately in Table \ref{qscThermal} and \ref{eamThermal}. In |
614 |
< |
these calculations,the end and middle slabs were excluded in thermal |
615 |
< |
gradient regession and only used as heat source and drain in the |
616 |
< |
systems. Our yielded values using EAM force field are slightly larger |
617 |
< |
than those using QSC force field. However, both series are |
618 |
< |
significantly smaller than experimental value by an order of more than |
619 |
< |
100. It has been verified that this difference is mainly attributed to |
602 |
< |
the lack of electron interaction representation in these force field |
603 |
< |
parameters. Richardson {\it et al.}\cite{Clancy:1992} used EAM |
604 |
< |
force field parameters in their metal thermal conductivity |
605 |
< |
calculations. The Non-Equilibrium MD method they employed in their |
606 |
< |
simulations produced comparable results to ours. As Zhang {\it et |
607 |
< |
al.}\cite{ISI:000231042800044} stated, thermal conductivity values |
608 |
< |
are influenced mainly by force field. Therefore, it is confident to |
609 |
< |
conclude that NIVS-RNEMD is applicable to metal force field system. |
611 |
> |
We compared the thermal conductivity of SPC/E water using NIVS-RNEMD |
612 |
> |
to the work of Bedrov {\it et al.}\cite{Bedrov:2000}, which employed |
613 |
> |
the original swapping RNEMD method. Bedrov {\it et |
614 |
> |
al.}\cite{Bedrov:2000} argued that exchange of the molecule |
615 |
> |
center-of-mass velocities instead of single atom velocities in a |
616 |
> |
molecule conserves the total kinetic energy and linear momentum. This |
617 |
> |
principle is also adopted Fin our simulations. Scaling was applied to |
618 |
> |
the center-of-mass velocities of the rigid bodies of SPC/E model water |
619 |
> |
molecules. |
620 |
|
|
621 |
< |
\begin{figure} |
622 |
< |
\includegraphics[width=\linewidth]{AuGrad} |
623 |
< |
\caption{Temperature gradients for thermal conductivity calculation of |
624 |
< |
crystal gold using QSC force field.} |
625 |
< |
\label{AuGrad} |
626 |
< |
\end{figure} |
621 |
> |
To construct the simulations, a simulation box consisting of 1000 |
622 |
> |
molecules were first equilibrated under ambient pressure and |
623 |
> |
temperature conditions using the isobaric-isothermal (NPT) |
624 |
> |
ensemble.\cite{melchionna93} A fixed volume was chosen to match the |
625 |
> |
average volume observed in the NPT simulations, and this was followed |
626 |
> |
by equilibration, first in the canonical (NVT) ensemble, followed by a |
627 |
> |
100~ps period under constant-NVE conditions without any momentum flux. |
628 |
> |
Another 100~ps was allowed to stabilize the system with an imposed |
629 |
> |
momentum transfer to create a gradient, and 1~ns was allotted for data |
630 |
> |
collection under RNEMD. |
631 |
|
|
632 |
< |
\begin{table*} |
633 |
< |
\begin{minipage}{\linewidth} |
634 |
< |
\begin{center} |
632 |
> |
In our simulations, the established temperature gradients were similar |
633 |
> |
to the previous work. Our simulation results at 318K are in good |
634 |
> |
agreement with those from Bedrov {\it et al.} (Table |
635 |
> |
\ref{spceThermal}). And both methods yield values in reasonable |
636 |
> |
agreement with experimental values. |
637 |
|
|
638 |
< |
\caption{Calculation results for thermal conductivity of crystal gold |
639 |
< |
using QSC force field at ${\langle T\rangle}$ = 300K at various |
640 |
< |
thermal exchange rates. Errors of calculations in parentheses. } |
641 |
< |
|
642 |
< |
\begin{tabular}{cc} |
643 |
< |
\hline |
644 |
< |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
645 |
< |
\hline |
646 |
< |
1.44 & 1.10(0.01)\\ |
647 |
< |
2.86 & 1.08(0.02)\\ |
648 |
< |
5.14 & 1.15(0.01)\\ |
649 |
< |
\hline |
650 |
< |
\end{tabular} |
651 |
< |
\label{qscThermal} |
652 |
< |
\end{center} |
653 |
< |
\end{minipage} |
638 |
> |
\begin{table*} |
639 |
> |
\begin{minipage}{\linewidth} |
640 |
> |
\begin{center} |
641 |
> |
|
642 |
> |
\caption{Thermal conductivity of SPC/E water under various |
643 |
> |
imposed thermal gradients. Uncertainties are indicated in |
644 |
> |
parentheses.} |
645 |
> |
|
646 |
> |
\begin{tabular}{|c|c|ccc|} |
647 |
> |
\hline |
648 |
> |
\multirow{2}{*}{$\langle T\rangle$(K)} & |
649 |
> |
\multirow{2}{*}{$\langle dT/dz\rangle$(K/\AA)} & |
650 |
> |
\multicolumn{3}{|c|}{$\lambda (\mathrm{W m}^{-1} |
651 |
> |
\mathrm{K}^{-1})$} \\ |
652 |
> |
& & This work & Previous simulations\cite{Bedrov:2000} & |
653 |
> |
Experiment\cite{WagnerKruse}\\ |
654 |
> |
\hline |
655 |
> |
\multirow{3}{*}{300} & 0.38 & 0.816(0.044) & & |
656 |
> |
\multirow{3}{*}{0.61}\\ |
657 |
> |
& 0.81 & 0.770(0.008) & & \\ |
658 |
> |
& 1.54 & 0.813(0.007) & & \\ |
659 |
> |
\hline |
660 |
> |
\multirow{2}{*}{318} & 0.75 & 0.750(0.032) & 0.784 & |
661 |
> |
\multirow{2}{*}{0.64}\\ |
662 |
> |
& 1.59 & 0.778(0.019) & 0.730 & \\ |
663 |
> |
\hline |
664 |
> |
\end{tabular} |
665 |
> |
\label{spceThermal} |
666 |
> |
\end{center} |
667 |
> |
\end{minipage} |
668 |
|
\end{table*} |
669 |
|
|
670 |
< |
\begin{figure} |
641 |
< |
\includegraphics[width=\linewidth]{eamGrad} |
642 |
< |
\caption{Temperature gradients for thermal conductivity calculation of |
643 |
< |
crystal gold using EAM force field.} |
644 |
< |
\label{eamGrad} |
645 |
< |
\end{figure} |
670 |
> |
\subsection{Crystalline Gold} |
671 |
|
|
672 |
< |
\begin{table*} |
673 |
< |
\begin{minipage}{\linewidth} |
674 |
< |
\begin{center} |
672 |
> |
To see how the method performed in a solid, we calculated thermal |
673 |
> |
conductivities using two atomistic models for gold. Several different |
674 |
> |
potential models have been developed that reasonably describe |
675 |
> |
interactions in transition metals. In particular, the Embedded Atom |
676 |
> |
Model (EAM)~\cite{PhysRevB.33.7983} and Sutton-Chen (SC)~\cite{Chen90} |
677 |
> |
potential have been used to study a wide range of phenomena in both |
678 |
> |
bulk materials and |
679 |
> |
nanoparticles.\cite{ISI:000207079300006,Clancy:1992,Vardeman:2008fk,Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} |
680 |
> |
Both potentials are based on a model of a metal which treats the |
681 |
> |
nuclei and core electrons as pseudo-atoms embedded in the electron |
682 |
> |
density due to the valence electrons on all of the other atoms in the |
683 |
> |
system. The SC potential has a simple form that closely resembles the |
684 |
> |
Lennard Jones potential, |
685 |
> |
\begin{equation} |
686 |
> |
\label{eq:SCP1} |
687 |
> |
U_{tot}=\sum _{i}\left[ \frac{1}{2}\sum _{j\neq i}D_{ij}V^{pair}_{ij}(r_{ij})-c_{i}D_{ii}\sqrt{\rho_{i}}\right] , |
688 |
> |
\end{equation} |
689 |
> |
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
690 |
> |
\begin{equation} |
691 |
> |
\label{eq:SCP2} |
692 |
> |
V^{pair}_{ij}(r)=\left( \frac{\alpha_{ij}}{r_{ij}}\right)^{n_{ij}}, \rho_{i}=\sum_{j\neq i}\left( \frac{\alpha_{ij}}{r_{ij}}\right) ^{m_{ij}}. |
693 |
> |
\end{equation} |
694 |
> |
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
695 |
> |
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
696 |
> |
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
697 |
> |
the interactions between the valence electrons and the cores of the |
698 |
> |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
699 |
> |
scale, $c_i$ scales the attractive portion of the potential relative |
700 |
> |
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
701 |
> |
that assures a dimensionless form for $\rho$. These parameters are |
702 |
> |
tuned to various experimental properties such as the density, cohesive |
703 |
> |
energy, and elastic moduli for FCC transition metals. The quantum |
704 |
> |
Sutton-Chen (QSC) formulation matches these properties while including |
705 |
> |
zero-point quantum corrections for different transition |
706 |
> |
metals.\cite{PhysRevB.59.3527} The EAM functional forms differ |
707 |
> |
slightly from SC but the overall method is very similar. |
708 |
|
|
709 |
< |
\caption{Calculation results for thermal conductivity of crystal gold |
710 |
< |
using EAM force field at ${\langle T\rangle}$ = 300K at various |
653 |
< |
thermal exchange rates. Errors of calculations in parentheses. } |
709 |
> |
In this work, we have utilized both the EAM and the QSC potentials to |
710 |
> |
test the behavior of scaling RNEMD. |
711 |
|
|
712 |
< |
\begin{tabular}{cc} |
713 |
< |
\hline |
714 |
< |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
715 |
< |
\hline |
716 |
< |
1.24 & 1.24(0.06)\\ |
717 |
< |
2.06 & 1.37(0.04)\\ |
718 |
< |
2.55 & 1.41(0.03)\\ |
719 |
< |
\hline |
720 |
< |
\end{tabular} |
721 |
< |
\label{eamThermal} |
722 |
< |
\end{center} |
723 |
< |
\end{minipage} |
712 |
> |
A face-centered-cubic (FCC) lattice was prepared containing 2880 Au |
713 |
> |
atoms (i.e. ${6\times 6\times 20}$ unit cells). Simulations were run |
714 |
> |
both with and without isobaric-isothermal (NPT)~\cite{melchionna93} |
715 |
> |
pre-equilibration at a target pressure of 1 atm. When equilibrated |
716 |
> |
under NPT conditions, our simulation box expanded by approximately 1\% |
717 |
> |
in volume. Following adjustment of the box volume, equilibrations in |
718 |
> |
both the canonical and microcanonical ensembles were carried out. With |
719 |
> |
the simulation cell divided evenly into 10 slabs, different thermal |
720 |
> |
gradients were established by applying a set of target thermal |
721 |
> |
transfer fluxes. |
722 |
> |
|
723 |
> |
The results for the thermal conductivity of gold are shown in Table |
724 |
> |
\ref{AuThermal}. In these calculations, the end and middle slabs were |
725 |
> |
excluded in thermal gradient linear regession. EAM predicts slightly |
726 |
> |
larger thermal conductivities than QSC. However, both values are |
727 |
> |
smaller than experimental value by a factor of more than 200. This |
728 |
> |
behavior has been observed previously by Richardson and Clancy, and |
729 |
> |
has been attributed to the lack of electronic contribution in these |
730 |
> |
force fields.\cite{Clancy:1992} It should be noted that the density of |
731 |
> |
the metal being simulated has an effect on thermal conductance. With |
732 |
> |
an expanded lattice, lower thermal conductance is expected (and |
733 |
> |
observed). We also observed a decrease in thermal conductance at |
734 |
> |
higher temperatures, a trend that agrees with experimental |
735 |
> |
measurements.\cite{AshcroftMermin} |
736 |
> |
|
737 |
> |
\begin{table*} |
738 |
> |
\begin{minipage}{\linewidth} |
739 |
> |
\begin{center} |
740 |
> |
|
741 |
> |
\caption{Calculated thermal conductivity of crystalline gold |
742 |
> |
using two related force fields. Calculations were done at both |
743 |
> |
experimental and equilibrated densities and at a range of |
744 |
> |
temperatures and thermal flux rates. Uncertainties are |
745 |
> |
indicated in parentheses. Richardson {\it et |
746 |
> |
al.}\cite{Clancy:1992} give an estimate of 1.74 $\mathrm{W |
747 |
> |
m}^{-1}\mathrm{K}^{-1}$ for EAM gold |
748 |
> |
at a density of 19.263 g / cm$^3$.} |
749 |
> |
|
750 |
> |
\begin{tabular}{|c|c|c|cc|} |
751 |
> |
\hline |
752 |
> |
Force Field & $\rho$ (g/cm$^3$) & ${\langle T\rangle}$ (K) & |
753 |
> |
$\langle dT/dz\rangle$ (K/\AA) & $\lambda (\mathrm{W m}^{-1} \mathrm{K}^{-1})$\\ |
754 |
> |
\hline |
755 |
> |
\multirow{7}{*}{QSC} & \multirow{3}{*}{19.188} & \multirow{3}{*}{300} & 1.44 & 1.10(0.06)\\ |
756 |
> |
& & & 2.86 & 1.08(0.05)\\ |
757 |
> |
& & & 5.14 & 1.15(0.07)\\\cline{2-5} |
758 |
> |
& \multirow{4}{*}{19.263} & \multirow{2}{*}{300} & 2.31 & 1.25(0.06)\\ |
759 |
> |
& & & 3.02 & 1.26(0.05)\\\cline{3-5} |
760 |
> |
& & \multirow{2}{*}{575} & 3.02 & 1.02(0.07)\\ |
761 |
> |
& & & 4.84 & 0.92(0.05)\\ |
762 |
> |
\hline |
763 |
> |
\multirow{8}{*}{EAM} & \multirow{3}{*}{19.045} & \multirow{3}{*}{300} & 1.24 & 1.24(0.16)\\ |
764 |
> |
& & & 2.06 & 1.37(0.04)\\ |
765 |
> |
& & & 2.55 & 1.41(0.07)\\\cline{2-5} |
766 |
> |
& \multirow{5}{*}{19.263} & \multirow{3}{*}{300} & 1.06 & 1.45(0.13)\\ |
767 |
> |
& & & 2.04 & 1.41(0.07)\\ |
768 |
> |
& & & 2.41 & 1.53(0.10)\\\cline{3-5} |
769 |
> |
& & \multirow{2}{*}{575} & 2.82 & 1.08(0.03)\\ |
770 |
> |
& & & 4.14 & 1.08(0.05)\\ |
771 |
> |
\hline |
772 |
> |
\end{tabular} |
773 |
> |
\label{AuThermal} |
774 |
> |
\end{center} |
775 |
> |
\end{minipage} |
776 |
|
\end{table*} |
777 |
|
|
778 |
+ |
\subsection{Thermal Conductance at the Au/H$_2$O interface} |
779 |
+ |
The most attractive aspect of the scaling approach for RNEMD is the |
780 |
+ |
ability to use the method in non-homogeneous systems, where molecules |
781 |
+ |
of different identities are segregated in different slabs. To test |
782 |
+ |
this application, we simulated a Gold (111) / water interface. To |
783 |
+ |
construct the interface, a box containing a lattice of 1188 Au atoms |
784 |
+ |
(with the 111 surface in the $+z$ and $-z$ directions) was allowed to |
785 |
+ |
relax under ambient temperature and pressure. A separate (but |
786 |
+ |
identically sized) box of SPC/E water was also equilibrated at ambient |
787 |
+ |
conditions. The two boxes were combined by removing all water |
788 |
+ |
molecules within 3 \AA radius of any gold atom. The final |
789 |
+ |
configuration contained 1862 SPC/E water molecules. |
790 |
|
|
791 |
< |
\subsection{Interfaciel Thermal Conductivity} |
792 |
< |
After simulations of homogeneous water and gold systems using |
793 |
< |
NIVS-RNEMD method were proved valid, calculation of gold/water |
794 |
< |
interfacial thermal conductivity was followed. It is found out that |
795 |
< |
the low interfacial conductance is probably due to the hydrophobic |
796 |
< |
surface in our system. Figure \ref{interfaceDensity} demonstrates mass |
797 |
< |
density change along $z$-axis, which is perpendicular to the |
798 |
< |
gold/water interface. It is observed that water density significantly |
799 |
< |
decreases when approaching the surface. Under this low thermal |
800 |
< |
conductance, both gold and water phase have sufficient time to |
801 |
< |
eliminate temperature difference inside respectively (Figure |
802 |
< |
\ref{interfaceGrad}). With indistinguishable temperature difference |
803 |
< |
within respective phase, it is valid to assume that the temperature |
804 |
< |
difference between gold and water on surface would be approximately |
805 |
< |
the same as the difference between the gold and water phase. This |
806 |
< |
assumption enables convenient calculation of $G$ using |
807 |
< |
Eq. \ref{interfaceCalc} instead of measuring temperatures of thin |
808 |
< |
layer of water and gold close enough to surface, which would have |
791 |
> |
The Spohr potential was adopted in depicting the interaction between |
792 |
> |
metal atoms and water molecules.\cite{ISI:000167766600035} A similar |
793 |
> |
protocol of equilibration to our water simulations was followed. We |
794 |
> |
observed that the two phases developed large temperature differences |
795 |
> |
even under a relatively low thermal flux. |
796 |
> |
|
797 |
> |
The low interfacial conductance is probably due to an acoustic |
798 |
> |
impedance mismatch between the solid and the liquid |
799 |
> |
phase.\cite{Cahill:793,RevModPhys.61.605} Experiments on the thermal |
800 |
> |
conductivity of gold nanoparticles and nanorods in solvent and in |
801 |
> |
glass cages have predicted values for $G$ between 100 and 350 |
802 |
> |
(MW/m$^2$/K). The experiments typically have multiple gold surfaces |
803 |
> |
that have been protected by a capping agent (citrate or CTAB) or which |
804 |
> |
are in direct contact with various glassy solids. In these cases, the |
805 |
> |
acoustic impedance mismatch would be substantially reduced, leading to |
806 |
> |
much higher interfacial conductances. It is also possible, however, |
807 |
> |
that the lack of electronic effects that gives rise to the low thermal |
808 |
> |
conductivity of EAM gold is also causing a low reading for this |
809 |
> |
particular interface. |
810 |
> |
|
811 |
> |
Under this low thermal conductance, both gold and water phase have |
812 |
> |
sufficient time to eliminate temperature difference inside |
813 |
> |
respectively (Figure \ref{interface} b). With indistinguishable |
814 |
> |
temperature difference within respective phase, it is valid to assume |
815 |
> |
that the temperature difference between gold and water on surface |
816 |
> |
would be approximately the same as the difference between the gold and |
817 |
> |
water phase. This assumption enables convenient calculation of $G$ |
818 |
> |
using Eq. \ref{interfaceCalc} instead of measuring temperatures of |
819 |
> |
thin layer of water and gold close enough to surface, which would have |
820 |
|
greater fluctuation and lower accuracy. Reported results (Table |
821 |
|
\ref{interfaceRes}) are of two orders of magnitude smaller than our |
822 |
|
calculations on homogeneous systems, and thus have larger relative |
823 |
|
errors than our calculation results on homogeneous systems. |
824 |
|
|
825 |
|
\begin{figure} |
826 |
< |
\includegraphics[width=\linewidth]{interfaceDensity} |
827 |
< |
\caption{Density profile for interfacial thermal conductivity |
828 |
< |
simulation box. Significant water density decrease is observed on |
829 |
< |
gold surface.} |
830 |
< |
\label{interfaceDensity} |
826 |
> |
\includegraphics[width=\linewidth]{interface} |
827 |
> |
\caption{Temperature profiles of the Gold / Water interface at four |
828 |
> |
different values for the thermal flux. Temperatures for slabs |
829 |
> |
either in the gold or in the water show no significant differences, |
830 |
> |
although there is a large discontinuity between the materials |
831 |
> |
because of the relatively low interfacial thermal conductivity.} |
832 |
> |
\label{interface} |
833 |
|
\end{figure} |
834 |
|
|
701 |
– |
\begin{figure} |
702 |
– |
\includegraphics[width=\linewidth]{interfaceGrad} |
703 |
– |
\caption{Temperature profiles for interfacial thermal conductivity |
704 |
– |
simulation box. Temperatures of different slabs in the same phase |
705 |
– |
show no significant difference.} |
706 |
– |
\label{interfaceGrad} |
707 |
– |
\end{figure} |
708 |
– |
|
835 |
|
\begin{table*} |
836 |
< |
\begin{minipage}{\linewidth} |
837 |
< |
\begin{center} |
838 |
< |
|
839 |
< |
\caption{Calculation results for interfacial thermal conductivity |
840 |
< |
at ${\langle T\rangle \sim}$ 300K at various thermal exchange |
841 |
< |
rates. Errors of calculations in parentheses. } |
842 |
< |
|
843 |
< |
\begin{tabular}{cccc} |
844 |
< |
\hline |
845 |
< |
$J_z$(MW/m$^2$) & $T_{gold}$ & $T_{water}$ & $G$(MW/m$^2$/K)\\ |
846 |
< |
\hline |
847 |
< |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
848 |
< |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
849 |
< |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
850 |
< |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
851 |
< |
\hline |
852 |
< |
\end{tabular} |
853 |
< |
\label{interfaceRes} |
854 |
< |
\end{center} |
855 |
< |
\end{minipage} |
836 |
> |
\begin{minipage}{\linewidth} |
837 |
> |
\begin{center} |
838 |
> |
|
839 |
> |
\caption{Computed interfacial thermal conductivity ($G$) values |
840 |
> |
for the Au(111) / water interface at ${\langle T\rangle \sim}$ |
841 |
> |
300K using a range of energy fluxes. Uncertainties are |
842 |
> |
indicated in parentheses. } |
843 |
> |
|
844 |
> |
\begin{tabular}{|cccc| } |
845 |
> |
\hline |
846 |
> |
$J_z$ (MW/m$^2$) & $\langle T_{gold} \rangle$ (K) & $\langle |
847 |
> |
T_{water} \rangle$ (K) & $G$ |
848 |
> |
(MW/m$^2$/K)\\ |
849 |
> |
\hline |
850 |
> |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
851 |
> |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
852 |
> |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
853 |
> |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
854 |
> |
\hline |
855 |
> |
\end{tabular} |
856 |
> |
\label{interfaceRes} |
857 |
> |
\end{center} |
858 |
> |
\end{minipage} |
859 |
|
\end{table*} |
860 |
|
|
732 |
– |
\subsection{Shear Viscosity} |
733 |
– |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
734 |
– |
produced comparable shear viscosity to swap RNEMD method. In Table |
735 |
– |
\ref{shearRate}, the names of the calculated samples are devided into |
736 |
– |
two parts. The first number refers to total slabs in one simulation |
737 |
– |
box. The second number refers to the swapping interval in swap method, or |
738 |
– |
in scale method the equilvalent swapping interval that the same |
739 |
– |
momentum flux would theoretically result in swap method. All the scale |
740 |
– |
method results were from simulations that had a scaling interval of 10 |
741 |
– |
time steps. The average molecular momentum gradients of these samples |
742 |
– |
are shown in Figure \ref{shearGrad}. |
861 |
|
|
744 |
– |
\begin{table*} |
745 |
– |
\begin{minipage}{\linewidth} |
746 |
– |
\begin{center} |
747 |
– |
|
748 |
– |
\caption{Calculation results for shear viscosity of Lennard-Jones |
749 |
– |
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
750 |
– |
methods at various momentum exchange rates. Results in reduced |
751 |
– |
unit. Errors of calculations in parentheses. } |
752 |
– |
|
753 |
– |
\begin{tabular}{ccc} |
754 |
– |
\hline |
755 |
– |
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
756 |
– |
\hline |
757 |
– |
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
758 |
– |
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
759 |
– |
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
760 |
– |
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
761 |
– |
\hline |
762 |
– |
\end{tabular} |
763 |
– |
\label{shearRate} |
764 |
– |
\end{center} |
765 |
– |
\end{minipage} |
766 |
– |
\end{table*} |
767 |
– |
|
768 |
– |
\begin{figure} |
769 |
– |
\includegraphics[width=\linewidth]{shearGrad} |
770 |
– |
\caption{Average momentum gradients of shear viscosity simulations} |
771 |
– |
\label{shearGrad} |
772 |
– |
\end{figure} |
773 |
– |
|
774 |
– |
\begin{figure} |
775 |
– |
\includegraphics[width=\linewidth]{shearTempScale} |
776 |
– |
\caption{Temperature profile for scaling RNEMD simulation.} |
777 |
– |
\label{shearTempScale} |
778 |
– |
\end{figure} |
779 |
– |
However, observations of temperatures along three dimensions show that |
780 |
– |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
781 |
– |
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
782 |
– |
relatively large imposed momentum flux, the temperature difference among $x$ |
783 |
– |
and the other two dimensions was significant. This would result from the |
784 |
– |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
785 |
– |
momentum gradient is set up, $P_c^x$ would be roughly stable |
786 |
– |
($<0$). Consequently, scaling factor $x$ would most probably larger |
787 |
– |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
788 |
– |
keep increase after most scaling steps. And if there is not enough time |
789 |
– |
for the kinetic energy to exchange among different dimensions and |
790 |
– |
different slabs, the system would finally build up temperature |
791 |
– |
(kinetic energy) difference among the three dimensions. Also, between |
792 |
– |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
793 |
– |
are closer to neighbor slabs. This is due to momentum transfer along |
794 |
– |
$z$ dimension between slabs. |
795 |
– |
|
796 |
– |
Although results between scaling and swapping methods are comparable, |
797 |
– |
the inherent temperature inhomogeneity even in relatively low imposed |
798 |
– |
exchange momentum flux simulations makes scaling RNEMD method less |
799 |
– |
attractive than swapping RNEMD in shear viscosity calculation. |
800 |
– |
|
862 |
|
\section{Conclusions} |
863 |
|
NIVS-RNEMD simulation method is developed and tested on various |
864 |
|
systems. Simulation results demonstrate its validity in thermal |
865 |
|
conductivity calculations, from Lennard-Jones fluid to multi-atom |
866 |
|
molecule like water and metal crystals. NIVS-RNEMD improves |
867 |
< |
non-Boltzmann-Maxwell distributions, which exist in previous RNEMD |
867 |
> |
non-Boltzmann-Maxwell distributions, which exist inb previous RNEMD |
868 |
|
methods. Furthermore, it develops a valid means for unphysical thermal |
869 |
|
transfer between different species of molecules, and thus extends its |
870 |
|
applicability to interfacial systems. Our calculation of gold/water |
876 |
|
calculations. |
877 |
|
|
878 |
|
\section{Acknowledgments} |
879 |
< |
Support for this project was provided by the National Science |
880 |
< |
Foundation under grant CHE-0848243. Computational time was provided by |
881 |
< |
the Center for Research Computing (CRC) at the University of Notre |
882 |
< |
Dame. \newpage |
879 |
> |
The authors would like to thank Craig Tenney and Ed Maginn for many |
880 |
> |
helpful discussions. Support for this project was provided by the |
881 |
> |
National Science Foundation under grant CHE-0848243. Computational |
882 |
> |
time was provided by the Center for Research Computing (CRC) at the |
883 |
> |
University of Notre Dame. |
884 |
> |
\newpage |
885 |
|
|
823 |
– |
\bibliographystyle{aip} |
886 |
|
\bibliography{nivsRnemd} |
887 |
|
|
888 |
|
\end{doublespace} |