1 |
gezelter |
3524 |
\documentclass[11pt]{article} |
2 |
|
|
\usepackage{amsmath} |
3 |
|
|
\usepackage{amssymb} |
4 |
|
|
\usepackage{setspace} |
5 |
|
|
\usepackage{endfloat} |
6 |
|
|
\usepackage{caption} |
7 |
|
|
%\usepackage{tabularx} |
8 |
|
|
\usepackage{graphicx} |
9 |
|
|
%\usepackage{booktabs} |
10 |
|
|
%\usepackage{bibentry} |
11 |
|
|
%\usepackage{mathrsfs} |
12 |
|
|
\usepackage[ref]{overcite} |
13 |
|
|
\pagestyle{plain} \pagenumbering{arabic} \oddsidemargin 0.0cm |
14 |
|
|
\evensidemargin 0.0cm \topmargin -21pt \headsep 10pt \textheight |
15 |
|
|
9.0in \textwidth 6.5in \brokenpenalty=10000 |
16 |
|
|
|
17 |
|
|
% double space list of tables and figures |
18 |
|
|
\AtBeginDelayedFloats{\renewcommand{\baselinestretch}{1.66}} |
19 |
|
|
\setlength{\abovecaptionskip}{20 pt} |
20 |
|
|
\setlength{\belowcaptionskip}{30 pt} |
21 |
|
|
|
22 |
|
|
\renewcommand\citemid{\ } % no comma in optional referenc note |
23 |
|
|
|
24 |
|
|
\begin{document} |
25 |
|
|
|
26 |
|
|
\title{A gentler approach to RNEMD: Non-isotropic Velocity Scaling for computing Thermal Conductivity and Shear Viscosity} |
27 |
|
|
|
28 |
|
|
\author{Shenyu Kuang and J. Daniel |
29 |
|
|
Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
30 |
|
|
Department of Chemistry and Biochemistry,\\ |
31 |
|
|
University of Notre Dame\\ |
32 |
|
|
Notre Dame, Indiana 46556} |
33 |
|
|
|
34 |
|
|
\date{\today} |
35 |
|
|
|
36 |
|
|
\maketitle |
37 |
|
|
|
38 |
|
|
\begin{doublespace} |
39 |
|
|
|
40 |
|
|
\begin{abstract} |
41 |
gezelter |
3583 |
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 |
gezelter |
3524 |
\end{abstract} |
59 |
|
|
|
60 |
|
|
\newpage |
61 |
|
|
|
62 |
|
|
%\narrowtext |
63 |
|
|
|
64 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
65 |
|
|
% BODY OF TEXT |
66 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
67 |
|
|
|
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 |
skuang |
3534 |
the simulation cell.\cite{MullerPlathe:1997xw,ISI:000080382700030} The |
74 |
gezelter |
3583 |
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 |
gezelter |
3524 |
\begin{eqnarray} |
84 |
skuang |
3532 |
j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
85 |
skuang |
3575 |
J_z & = & \lambda \frac{\partial T}{\partial z} |
86 |
gezelter |
3524 |
\end{eqnarray} |
87 |
skuang |
3528 |
RNEMD has been widely used to provide computational estimates of thermal |
88 |
gezelter |
3524 |
conductivities and shear viscosities in a wide range of materials, |
89 |
|
|
from liquid copper to monatomic liquids to molecular fluids |
90 |
skuang |
3587 |
(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 |
gezelter |
3524 |
|
92 |
skuang |
3574 |
\begin{figure} |
93 |
|
|
\includegraphics[width=\linewidth]{thermalDemo} |
94 |
gezelter |
3583 |
\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 |
skuang |
3574 |
\label{thermalDemo} |
101 |
|
|
\end{figure} |
102 |
|
|
|
103 |
skuang |
3588 |
RNEMD is preferable in many ways to the forward NEMD methods\cite{ISI:A1988Q205300014} because it imposes what is typically difficult to measure |
104 |
gezelter |
3583 |
(a flux or stress) and it is typically much easier to compute momentum |
105 |
|
|
gradients or strains (the response). For similar reasons, RNEMD is |
106 |
|
|
also preferable to slowly-converging equilibrium methods for measuring |
107 |
|
|
thermal conductivity and shear viscosity (using Green-Kubo relations |
108 |
|
|
[CITATIONS NEEDED] or the Helfand moment approach of Viscardy {\it et |
109 |
skuang |
3527 |
al.}\cite{Viscardy:2007bh,Viscardy:2007lq}) because these rely on |
110 |
gezelter |
3524 |
computing difficult to measure quantities. |
111 |
|
|
|
112 |
|
|
Another attractive feature of RNEMD is that it conserves both total |
113 |
|
|
linear momentum and total energy during the swaps (as long as the two |
114 |
|
|
molecules have the same identity), so the swapped configurations are |
115 |
|
|
typically samples from the same manifold of states in the |
116 |
|
|
microcanonical ensemble. |
117 |
|
|
|
118 |
skuang |
3588 |
Recently, Tenney and Maginn\cite{Maginn:2010} have discovered |
119 |
skuang |
3565 |
some problems with the original RNEMD swap technique. Notably, large |
120 |
|
|
momentum fluxes (equivalent to frequent momentum swaps between the |
121 |
skuang |
3575 |
slabs) can result in ``notched'', ``peaked'' and generally non-thermal |
122 |
|
|
momentum distributions in the two slabs, as well as non-linear thermal |
123 |
|
|
and velocity distributions along the direction of the imposed flux |
124 |
|
|
($z$). Tenney and Maginn obtained reasonable limits on imposed flux |
125 |
|
|
and self-adjusting metrics for retaining the usability of the method. |
126 |
gezelter |
3524 |
|
127 |
|
|
In this paper, we develop and test a method for non-isotropic velocity |
128 |
|
|
scaling (NIVS-RNEMD) which retains the desirable features of RNEMD |
129 |
|
|
(conservation of linear momentum and total energy, compatibility with |
130 |
|
|
periodic boundary conditions) while establishing true thermal |
131 |
|
|
distributions in each of the two slabs. In the next section, we |
132 |
gezelter |
3583 |
present the method for determining the scaling constraints. We then |
133 |
gezelter |
3524 |
test the method on both single component, multi-component, and |
134 |
|
|
non-isotropic mixtures and show that it is capable of providing |
135 |
|
|
reasonable estimates of the thermal conductivity and shear viscosity |
136 |
|
|
in these cases. |
137 |
|
|
|
138 |
|
|
\section{Methodology} |
139 |
gezelter |
3583 |
We retain the basic idea of M\"{u}ller-Plathe's RNEMD method; the |
140 |
|
|
periodic system is partitioned into a series of thin slabs along one |
141 |
gezelter |
3524 |
axis ($z$). One of the slabs at the end of the periodic box is |
142 |
|
|
designated the ``hot'' slab, while the slab in the center of the box |
143 |
|
|
is designated the ``cold'' slab. The artificial momentum flux will be |
144 |
|
|
established by transferring momentum from the cold slab and into the |
145 |
|
|
hot slab. |
146 |
|
|
|
147 |
|
|
Rather than using momentum swaps, we use a series of velocity scaling |
148 |
gezelter |
3583 |
moves. For molecules $\{i\}$ located within the cold slab, |
149 |
gezelter |
3524 |
\begin{equation} |
150 |
skuang |
3565 |
\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
151 |
|
|
x & 0 & 0 \\ |
152 |
|
|
0 & y & 0 \\ |
153 |
|
|
0 & 0 & z \\ |
154 |
gezelter |
3524 |
\end{array} \right) \cdot \vec{v}_i |
155 |
|
|
\end{equation} |
156 |
|
|
where ${x, y, z}$ are a set of 3 scaling variables for each of the |
157 |
|
|
three directions in the system. Likewise, the molecules $\{j\}$ |
158 |
skuang |
3528 |
located in the hot slab will see a concomitant scaling of velocities, |
159 |
gezelter |
3524 |
\begin{equation} |
160 |
skuang |
3565 |
\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
161 |
|
|
x^\prime & 0 & 0 \\ |
162 |
|
|
0 & y^\prime & 0 \\ |
163 |
|
|
0 & 0 & z^\prime \\ |
164 |
gezelter |
3524 |
\end{array} \right) \cdot \vec{v}_j |
165 |
|
|
\end{equation} |
166 |
|
|
|
167 |
|
|
Conservation of linear momentum in each of the three directions |
168 |
gezelter |
3583 |
($\alpha = x,y,z$) ties the values of the hot and cold scaling |
169 |
gezelter |
3524 |
parameters together: |
170 |
|
|
\begin{equation} |
171 |
skuang |
3528 |
P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
172 |
gezelter |
3524 |
\end{equation} |
173 |
|
|
where |
174 |
skuang |
3565 |
\begin{eqnarray} |
175 |
skuang |
3528 |
P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
176 |
skuang |
3565 |
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha |
177 |
gezelter |
3524 |
\label{eq:momentumdef} |
178 |
skuang |
3565 |
\end{eqnarray} |
179 |
skuang |
3528 |
Therefore, for each of the three directions, the hot scaling |
180 |
|
|
parameters are a simple function of the cold scaling parameters and |
181 |
gezelter |
3524 |
the instantaneous linear momentum in each of the two slabs. |
182 |
|
|
\begin{equation} |
183 |
skuang |
3528 |
\alpha^\prime = 1 + (1 - \alpha) p_\alpha |
184 |
gezelter |
3524 |
\label{eq:hotcoldscaling} |
185 |
|
|
\end{equation} |
186 |
skuang |
3528 |
where |
187 |
|
|
\begin{equation} |
188 |
|
|
p_\alpha = \frac{P_c^\alpha}{P_h^\alpha} |
189 |
|
|
\end{equation} |
190 |
|
|
for convenience. |
191 |
gezelter |
3524 |
|
192 |
|
|
Conservation of total energy also places constraints on the scaling: |
193 |
|
|
\begin{equation} |
194 |
|
|
\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
195 |
skuang |
3565 |
\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha |
196 |
gezelter |
3524 |
\end{equation} |
197 |
skuang |
3575 |
where the translational kinetic energies, $K_h^\alpha$ and |
198 |
|
|
$K_c^\alpha$, are computed for each of the three directions in a |
199 |
|
|
similar manner to the linear momenta (Eq. \ref{eq:momentumdef}). |
200 |
|
|
Substituting in the expressions for the hot scaling parameters |
201 |
|
|
($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), we obtain the |
202 |
gezelter |
3583 |
{\it constraint ellipsoid}: |
203 |
gezelter |
3524 |
\begin{equation} |
204 |
skuang |
3565 |
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0 |
205 |
gezelter |
3524 |
\label{eq:constraintEllipsoid} |
206 |
|
|
\end{equation} |
207 |
|
|
where the constants are obtained from the instantaneous values of the |
208 |
|
|
linear momenta and kinetic energies for the hot and cold slabs, |
209 |
skuang |
3565 |
\begin{eqnarray} |
210 |
skuang |
3528 |
a_\alpha & = &\left(K_c^\alpha + K_h^\alpha |
211 |
|
|
\left(p_\alpha\right)^2\right) \\ |
212 |
|
|
b_\alpha & = & -2 K_h^\alpha p_\alpha \left( 1 + p_\alpha\right) \\ |
213 |
skuang |
3565 |
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
214 |
gezelter |
3524 |
\label{eq:constraintEllipsoidConsts} |
215 |
skuang |
3565 |
\end{eqnarray} |
216 |
gezelter |
3583 |
This ellipsoid (Eq. \ref{eq:constraintEllipsoid}) defines the set of |
217 |
|
|
cold slab scaling parameters which can be applied while preserving |
218 |
|
|
both linear momentum in all three directions as well as total kinetic |
219 |
|
|
energy. |
220 |
gezelter |
3524 |
|
221 |
|
|
The goal of using velocity scaling variables is to transfer linear |
222 |
|
|
momentum or kinetic energy from the cold slab to the hot slab. If the |
223 |
|
|
hot and cold slabs are separated along the z-axis, the energy flux is |
224 |
skuang |
3528 |
given simply by the decrease in kinetic energy of the cold bin: |
225 |
gezelter |
3524 |
\begin{equation} |
226 |
skuang |
3534 |
(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z\Delta t |
227 |
gezelter |
3524 |
\end{equation} |
228 |
|
|
The expression for the energy flux can be re-written as another |
229 |
|
|
ellipsoid centered on $(x,y,z) = 0$: |
230 |
|
|
\begin{equation} |
231 |
skuang |
3534 |
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 |
232 |
gezelter |
3524 |
\label{eq:fluxEllipsoid} |
233 |
|
|
\end{equation} |
234 |
gezelter |
3583 |
The spatial extent of the {\it thermal flux ellipsoid} is governed |
235 |
|
|
both by a targetted value, $J_z$ as well as the instantaneous values |
236 |
|
|
of the kinetic energy components in the cold bin. |
237 |
gezelter |
3524 |
|
238 |
|
|
To satisfy an energetic flux as well as the conservation constraints, |
239 |
gezelter |
3583 |
we must determine the points ${x,y,z}$ which lie on both the |
240 |
|
|
constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the flux |
241 |
|
|
ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of the |
242 |
|
|
two ellipsoids in 3-dimensional space. |
243 |
gezelter |
3524 |
|
244 |
gezelter |
3569 |
\begin{figure} |
245 |
|
|
\includegraphics[width=\linewidth]{ellipsoids} |
246 |
|
|
\caption{Scaling points which maintain both constant energy and |
247 |
|
|
constant linear momentum of the system lie on the surface of the |
248 |
|
|
{\it constraint ellipsoid} while points which generate the target |
249 |
|
|
momentum flux lie on the surface of the {\it flux ellipsoid}. The |
250 |
skuang |
3575 |
velocity distributions in the cold bin are scaled by only those |
251 |
gezelter |
3569 |
points which lie on both ellipsoids.} |
252 |
|
|
\label{ellipsoids} |
253 |
|
|
\end{figure} |
254 |
|
|
|
255 |
gezelter |
3583 |
One may also define {\it momentum} flux (say along the $x$-direction) as: |
256 |
gezelter |
3524 |
\begin{equation} |
257 |
skuang |
3565 |
(1-x) P_c^x = j_z(p_x)\Delta t |
258 |
skuang |
3531 |
\label{eq:fluxPlane} |
259 |
gezelter |
3524 |
\end{equation} |
260 |
gezelter |
3583 |
The above {\it momentum flux plane} is perpendicular to the $x$-axis, |
261 |
|
|
with its position governed both by a target value, $j_z(p_x)$ as well |
262 |
|
|
as the instantaneous value of the momentum along the $x$-direction. |
263 |
gezelter |
3524 |
|
264 |
gezelter |
3583 |
In order to satisfy a momentum flux as well as the conservation |
265 |
|
|
constraints, we must determine the points ${x,y,z}$ which lie on both |
266 |
|
|
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
267 |
|
|
flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of an |
268 |
|
|
ellipsoid and a plane in 3-dimensional space. |
269 |
gezelter |
3524 |
|
270 |
gezelter |
3583 |
In both the momentum and energy flux scenarios, valid scaling |
271 |
|
|
parameters are arrived at by solving geometric intersection problems |
272 |
|
|
in $x, y, z$ space in order to obtain cold slab scaling parameters. |
273 |
|
|
Once the scaling variables for the cold slab are known, the hot slab |
274 |
|
|
scaling has also been determined. |
275 |
gezelter |
3524 |
|
276 |
gezelter |
3583 |
|
277 |
skuang |
3531 |
The following problem will be choosing an optimal set of scaling |
278 |
|
|
variables among the possible sets. Although this method is inherently |
279 |
|
|
non-isotropic, the goal is still to maintain the system as isotropic |
280 |
|
|
as possible. Under this consideration, one would like the kinetic |
281 |
|
|
energies in different directions could become as close as each other |
282 |
|
|
after each scaling. Simultaneously, one would also like each scaling |
283 |
|
|
as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
284 |
gezelter |
3583 |
large perturbation to the system. Therefore, one approach to obtain |
285 |
|
|
the scaling variables would be constructing an criteria function, with |
286 |
skuang |
3531 |
constraints as above equation sets, and solving the function's minimum |
287 |
|
|
by method like Lagrange multipliers. |
288 |
gezelter |
3524 |
|
289 |
skuang |
3531 |
In order to save computation time, we have a different approach to a |
290 |
|
|
relatively good set of scaling variables with much less calculation |
291 |
|
|
than above. Here is the detail of our simplification of the problem. |
292 |
gezelter |
3524 |
|
293 |
skuang |
3531 |
In the case of kinetic energy transfer, we impose another constraint |
294 |
|
|
${x = y}$, into the equation sets. Consequently, there are two |
295 |
|
|
variables left. And now one only needs to solve a set of two {\it |
296 |
|
|
ellipses equations}. This problem would be transformed into solving |
297 |
|
|
one quartic equation for one of the two variables. There are known |
298 |
|
|
generic methods that solve real roots of quartic equations. Then one |
299 |
|
|
can determine the other variable and obtain sets of scaling |
300 |
|
|
variables. Among these sets, one can apply the above criteria to |
301 |
|
|
choose the best set, while much faster with only a few sets to choose. |
302 |
|
|
|
303 |
|
|
In the case of momentum flux transfer, we impose another constraint to |
304 |
|
|
set the kinetic energy transfer as zero. In another word, we apply |
305 |
|
|
Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. After that, with one |
306 |
|
|
variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar set |
307 |
|
|
of equations on the above kinetic energy transfer problem. Therefore, |
308 |
|
|
an approach similar to the above would be sufficient for this as well. |
309 |
|
|
|
310 |
|
|
\section{Computational Details} |
311 |
skuang |
3576 |
\subsection{Lennard-Jones Fluid} |
312 |
skuang |
3534 |
Our simulation consists of a series of systems. All of these |
313 |
skuang |
3565 |
simulations were run with the OpenMD simulation software |
314 |
skuang |
3576 |
package\cite{Meineke:2005gd} integrated with RNEMD codes. |
315 |
skuang |
3531 |
|
316 |
skuang |
3532 |
A Lennard-Jones fluid system was built and tested first. In order to |
317 |
|
|
compare our method with swapping RNEMD, a series of simulations were |
318 |
|
|
performed to calculate the shear viscosity and thermal conductivity of |
319 |
skuang |
3534 |
argon. 2592 atoms were in a orthorhombic cell, which was ${10.06\sigma |
320 |
|
|
\times 10.06\sigma \times 30.18\sigma}$ by size. The reduced density |
321 |
|
|
${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled direct |
322 |
|
|
comparison between our results and others. These simulations used |
323 |
skuang |
3565 |
velocity Verlet algorithm with reduced timestep ${\tau^* = |
324 |
skuang |
3534 |
4.6\times10^{-4}}$. |
325 |
skuang |
3532 |
|
326 |
|
|
For shear viscosity calculation, the reduced temperature was ${T^* = |
327 |
skuang |
3565 |
k_B T/\varepsilon = 0.72}$. Simulations were first equilibrated in canonical |
328 |
|
|
ensemble (NVT), then equilibrated in microcanonical ensemble |
329 |
|
|
(NVE). Establishing and stablizing momentum gradient were followed |
330 |
|
|
also in NVE ensemble. For the swapping part, Muller-Plathe's algorithm was |
331 |
skuang |
3532 |
adopted.\cite{ISI:000080382700030} The simulation box was under |
332 |
skuang |
3534 |
periodic boundary condition, and devided into ${N = 20}$ slabs. In each swap, |
333 |
|
|
the top slab ${(n = 1)}$ exchange the most negative $x$ momentum with the |
334 |
|
|
most positive $x$ momentum in the center slab ${(n = N/2 + 1)}$. Referred |
335 |
skuang |
3565 |
to Tenney {\it et al.}\cite{ISI:000273472300004}, a series of swapping |
336 |
skuang |
3534 |
frequency were chosen. According to each result from swapping |
337 |
skuang |
3532 |
RNEMD, scaling RNEMD simulations were run with the target momentum |
338 |
skuang |
3576 |
flux set to produce a similar momentum flux, and consequently shear |
339 |
skuang |
3534 |
rate. Furthermore, various scaling frequencies can be tested for one |
340 |
skuang |
3576 |
single swapping rate. To test the temperature homogeneity in our |
341 |
|
|
system of swapping and scaling methods, temperatures of different |
342 |
|
|
dimensions in all the slabs were observed. Most of the simulations |
343 |
|
|
include $10^5$ steps of equilibration without imposing momentum flux, |
344 |
|
|
$10^5$ steps of stablization with imposing unphysical momentum |
345 |
|
|
transfer, and $10^6$ steps of data collection under RNEMD. For |
346 |
|
|
relatively high momentum flux simulations, ${5\times10^5}$ step data |
347 |
|
|
collection is sufficient. For some low momentum flux simulations, |
348 |
|
|
${2\times10^6}$ steps were necessary. |
349 |
skuang |
3532 |
|
350 |
skuang |
3534 |
After each simulation, the shear viscosity was calculated in reduced |
351 |
|
|
unit. The momentum flux was calculated with total unphysical |
352 |
skuang |
3565 |
transferred momentum ${P_x}$ and data collection time $t$: |
353 |
skuang |
3534 |
\begin{equation} |
354 |
|
|
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
355 |
|
|
\end{equation} |
356 |
skuang |
3576 |
where $L_x$ and $L_y$ denotes $x$ and $y$ lengths of the simulation |
357 |
|
|
box, and physical momentum transfer occurs in two ways due to our |
358 |
|
|
periodic boundary condition settings. And the velocity gradient |
359 |
|
|
${\langle \partial v_x /\partial z \rangle}$ can be obtained by a |
360 |
|
|
linear regression of the velocity profile. From the shear viscosity |
361 |
|
|
$\eta$ calculated with the above parameters, one can further convert |
362 |
|
|
it into reduced unit ${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$. |
363 |
skuang |
3532 |
|
364 |
skuang |
3576 |
For thermal conductivity calculations, simulations were first run under |
365 |
|
|
reduced temperature ${\langle T^*\rangle = 0.72}$ in NVE |
366 |
|
|
ensemble. Muller-Plathe's algorithm was adopted in the swapping |
367 |
|
|
method. Under identical simulation box parameters with our shear |
368 |
|
|
viscosity calculations, in each swap, the top slab exchanges all three |
369 |
|
|
translational momentum components of the molecule with least kinetic |
370 |
|
|
energy with the same components of the molecule in the center slab |
371 |
|
|
with most kinetic energy, unless this ``coldest'' molecule in the |
372 |
|
|
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the |
373 |
|
|
``cold'' slab. According to swapping RNEMD results, target energy flux |
374 |
|
|
for scaling RNEMD simulations can be set. Also, various scaling |
375 |
skuang |
3534 |
frequencies can be tested for one target energy flux. To compare the |
376 |
|
|
performance between swapping and scaling method, distributions of |
377 |
|
|
velocity and speed in different slabs were observed. |
378 |
|
|
|
379 |
|
|
For each swapping rate, thermal conductivity was calculated in reduced |
380 |
|
|
unit. The energy flux was calculated similarly to the momentum flux, |
381 |
skuang |
3565 |
with total unphysical transferred energy ${E_{total}}$ and data collection |
382 |
skuang |
3534 |
time $t$: |
383 |
|
|
\begin{equation} |
384 |
|
|
J_z = \frac{E_{total}}{2 t L_x L_y} |
385 |
|
|
\end{equation} |
386 |
|
|
And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
387 |
|
|
can be obtained by a linear regression of the temperature |
388 |
|
|
profile. From the thermal conductivity $\lambda$ calculated, one can |
389 |
|
|
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
390 |
|
|
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
391 |
|
|
|
392 |
skuang |
3576 |
\subsection{ Water / Metal Thermal Conductivity} |
393 |
|
|
Another series of our simulation is the calculation of interfacial |
394 |
skuang |
3573 |
thermal conductivity of a Au/H$_2$O system. Respective calculations of |
395 |
skuang |
3579 |
liquid water (Extended Simple Point Charge model) and crystal gold |
396 |
skuang |
3580 |
thermal conductivity were performed and compared with current results |
397 |
|
|
to ensure the validity of NIVS-RNEMD. After that, a mixture system was |
398 |
|
|
simulated. |
399 |
skuang |
3563 |
|
400 |
skuang |
3573 |
For thermal conductivity calculation of bulk water, a simulation box |
401 |
|
|
consisting of 1000 molecules were first equilibrated under ambient |
402 |
skuang |
3576 |
pressure and temperature conditions using NPT ensemble, followed by |
403 |
|
|
equilibration in fixed volume (NVT). The system was then equilibrated in |
404 |
|
|
microcanonical ensemble (NVE). Also in NVE ensemble, establishing a |
405 |
skuang |
3573 |
stable thermal gradient was followed. The simulation box was under |
406 |
|
|
periodic boundary condition and devided into 10 slabs. Data collection |
407 |
skuang |
3576 |
process was similar to Lennard-Jones fluid system. |
408 |
skuang |
3573 |
|
409 |
skuang |
3576 |
Thermal conductivity calculation of bulk crystal gold used a similar |
410 |
skuang |
3580 |
protocol. Two types of force field parameters, Embedded Atom Method |
411 |
|
|
(EAM) and Quantum Sutten-Chen (QSC) force field were used |
412 |
|
|
respectively. The face-centered cubic crystal simulation box consists of |
413 |
skuang |
3576 |
2880 Au atoms. The lattice was first allowed volume change to relax |
414 |
|
|
under ambient temperature and pressure. Equilibrations in canonical and |
415 |
|
|
microcanonical ensemble were followed in order. With the simulation |
416 |
|
|
lattice devided evenly into 10 slabs, different thermal gradients were |
417 |
|
|
established by applying a set of target thermal transfer flux. Data of |
418 |
|
|
the series of thermal gradients was collected for calculation. |
419 |
|
|
|
420 |
skuang |
3573 |
After simulations of bulk water and crystal gold, a mixture system was |
421 |
|
|
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
422 |
|
|
molecules. Spohr potential was adopted in depicting the interaction |
423 |
|
|
between metal atom and water molecule.\cite{ISI:000167766600035} A |
424 |
skuang |
3576 |
similar protocol of equilibration was followed. Several thermal |
425 |
|
|
gradients was built under different target thermal flux. It was found |
426 |
|
|
out that compared to our previous simulation systems, the two phases |
427 |
|
|
could have large temperature difference even under a relatively low |
428 |
|
|
thermal flux. Therefore, under our low flux conditions, it is assumed |
429 |
skuang |
3573 |
that the metal and water phases have respectively homogeneous |
430 |
skuang |
3576 |
temperature, excluding the surface regions. In calculating the |
431 |
|
|
interfacial thermal conductivity $G$, this assumptioin was applied and |
432 |
|
|
thus our formula becomes: |
433 |
skuang |
3573 |
|
434 |
|
|
\begin{equation} |
435 |
|
|
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
436 |
|
|
\langle T_{water}\rangle \right)} |
437 |
|
|
\label{interfaceCalc} |
438 |
|
|
\end{equation} |
439 |
|
|
where ${E_{total}}$ is the imposed unphysical kinetic energy transfer |
440 |
|
|
and ${\langle T_{gold}\rangle}$ and ${\langle T_{water}\rangle}$ are the |
441 |
|
|
average observed temperature of gold and water phases respectively. |
442 |
|
|
|
443 |
skuang |
3577 |
\section{Results And Discussions} |
444 |
skuang |
3538 |
\subsection{Thermal Conductivity} |
445 |
skuang |
3573 |
\subsubsection{Lennard-Jones Fluid} |
446 |
skuang |
3577 |
Our thermal conductivity calculations show that scaling method results |
447 |
|
|
agree with swapping method. Four different exchange intervals were |
448 |
|
|
tested (Table \ref{thermalLJRes}) using swapping method. With a fixed |
449 |
|
|
10fs exchange interval, target exchange kinetic energy was set to |
450 |
|
|
produce equivalent kinetic energy flux as in swapping method. And |
451 |
|
|
similar thermal gradients were observed with similar thermal flux in |
452 |
|
|
two simulation methods (Figure \ref{thermalGrad}). |
453 |
skuang |
3538 |
|
454 |
skuang |
3563 |
\begin{table*} |
455 |
|
|
\begin{minipage}{\linewidth} |
456 |
|
|
\begin{center} |
457 |
skuang |
3538 |
|
458 |
skuang |
3563 |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
459 |
skuang |
3565 |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
460 |
skuang |
3563 |
swap and scale methods at various kinetic energy exchange rates. Results |
461 |
|
|
in reduced unit. Errors of calculations in parentheses.} |
462 |
|
|
|
463 |
skuang |
3565 |
\begin{tabular}{ccc} |
464 |
skuang |
3563 |
\hline |
465 |
skuang |
3577 |
(Equilvalent) Exchange Interval (fs) & $\lambda^*_{swap}$ & |
466 |
|
|
$\lambda^*_{scale}$\\ |
467 |
skuang |
3565 |
\hline |
468 |
skuang |
3577 |
250 & 7.03(0.34) & 7.30(0.10)\\ |
469 |
|
|
500 & 7.03(0.14) & 6.95(0.09)\\ |
470 |
|
|
1000 & 6.91(0.42) & 7.19(0.07)\\ |
471 |
|
|
2000 & 7.52(0.15) & 7.19(0.28)\\ |
472 |
skuang |
3566 |
\hline |
473 |
skuang |
3563 |
\end{tabular} |
474 |
skuang |
3577 |
\label{thermalLJRes} |
475 |
skuang |
3563 |
\end{center} |
476 |
|
|
\end{minipage} |
477 |
|
|
\end{table*} |
478 |
|
|
|
479 |
|
|
\begin{figure} |
480 |
skuang |
3567 |
\includegraphics[width=\linewidth]{thermalGrad} |
481 |
skuang |
3577 |
\caption{Temperature gradients under various kinetic energy flux of |
482 |
|
|
thermal conductivity simulations} |
483 |
skuang |
3567 |
\label{thermalGrad} |
484 |
skuang |
3563 |
\end{figure} |
485 |
|
|
|
486 |
|
|
During these simulations, molecule velocities were recorded in 1000 of |
487 |
skuang |
3578 |
all the snapshots of one single data collection process. These |
488 |
|
|
velocity data were used to produce histograms of velocity and speed |
489 |
|
|
distribution in different slabs. From these histograms, it is observed |
490 |
|
|
that under relatively high unphysical kinetic energy flux, speed and |
491 |
|
|
velocity distribution of molecules in slabs where swapping occured |
492 |
|
|
could deviate from Maxwell-Boltzmann distribution. Figure |
493 |
|
|
\ref{histSwap} illustrates how these distributions deviate from an |
494 |
|
|
ideal distribution. In high temperature slab, probability density in |
495 |
|
|
low speed is confidently smaller than ideal curve fit; in low |
496 |
|
|
temperature slab, probability density in high speed is smaller than |
497 |
|
|
ideal, while larger than ideal in low speed. This phenomenon is more |
498 |
|
|
obvious in our high swapping rate simulations. And this deviation |
499 |
|
|
could also leads to deviation of distribution of velocity in various |
500 |
|
|
dimensions. One feature of these deviated distribution is that in high |
501 |
|
|
temperature slab, the ideal Gaussian peak was changed into a |
502 |
|
|
relatively flat plateau; while in low temperature slab, that peak |
503 |
|
|
appears sharper. This problem is rooted in the mechanism of the |
504 |
|
|
swapping method. Continually depleting low (high) speed particles in |
505 |
|
|
the high (low) temperature slab could not be complemented by |
506 |
|
|
diffusions of low (high) speed particles from neighbor slabs, unless |
507 |
|
|
in suffciently low swapping rate. Simutaneously, surplus low speed |
508 |
|
|
particles in the low temperature slab do not have sufficient time to |
509 |
|
|
diffuse to neighbor slabs. However, thermal exchange rate should reach |
510 |
|
|
a minimum level to produce an observable thermal gradient under noise |
511 |
|
|
interference. Consequently, swapping RNEMD has a relatively narrow |
512 |
|
|
choice of swapping rate to satisfy these above restrictions. |
513 |
skuang |
3563 |
|
514 |
|
|
\begin{figure} |
515 |
skuang |
3565 |
\includegraphics[width=\linewidth]{histSwap} |
516 |
skuang |
3578 |
\caption{Speed distribution for thermal conductivity using swapping |
517 |
|
|
RNEMD. Shown is from the simulation with 250 fs exchange interval.} |
518 |
skuang |
3563 |
\label{histSwap} |
519 |
|
|
\end{figure} |
520 |
|
|
|
521 |
skuang |
3578 |
Comparatively, NIVS-RNEMD has a speed distribution closer to the ideal |
522 |
|
|
curve fit (Figure \ref{histScale}). Essentially, after scaling, a |
523 |
|
|
Gaussian distribution function would remain Gaussian. Although a |
524 |
|
|
single scaling is non-isotropic in all three dimensions, our scaling |
525 |
|
|
coefficient criteria could help maintian the scaling region as |
526 |
|
|
isotropic as possible. On the other hand, scaling coefficients are |
527 |
|
|
preferred to be as close to 1 as possible, which also helps minimize |
528 |
|
|
the difference among different dimensions. This is possible if scaling |
529 |
|
|
interval and one-time thermal transfer energy are well |
530 |
|
|
chosen. Consequently, NIVS-RNEMD is able to impose an unphysical |
531 |
|
|
thermal flux as the previous RNEMD method without large perturbation |
532 |
|
|
to the distribution of velocity and speed in the exchange regions. |
533 |
|
|
|
534 |
skuang |
3568 |
\begin{figure} |
535 |
|
|
\includegraphics[width=\linewidth]{histScale} |
536 |
skuang |
3578 |
\caption{Speed distribution for thermal conductivity using scaling |
537 |
|
|
RNEMD. Shown is from the simulation with an equilvalent thermal flux |
538 |
|
|
as an 250 fs exchange interval swapping simulation.} |
539 |
skuang |
3568 |
\label{histScale} |
540 |
|
|
\end{figure} |
541 |
|
|
|
542 |
skuang |
3573 |
\subsubsection{SPC/E Water} |
543 |
|
|
Our results of SPC/E water thermal conductivity are comparable to |
544 |
|
|
Bedrov {\it et al.}\cite{ISI:000090151400044}, which employed the |
545 |
skuang |
3579 |
previous swapping RNEMD method for their calculation. Bedrov {\it et |
546 |
|
|
al.}\cite{ISI:000090151400044} argued that exchange of the molecule |
547 |
|
|
center-of-mass velocities instead of single atom velocities in a |
548 |
|
|
molecule conserves the total kinetic energy and linear momentum. This |
549 |
|
|
principle is adopted in our simulations. Scaling is applied to the |
550 |
|
|
velocities of the rigid bodies of SPC/E model water molecules, instead |
551 |
|
|
of each hydrogen and oxygen atoms in relevant water molecules. As |
552 |
|
|
shown in Figure \ref{spceGrad}, temperature gradients were established |
553 |
|
|
similar to their system. However, the average temperature of our |
554 |
|
|
system is 300K, while theirs is 318K, which would be attributed for |
555 |
|
|
part of the difference between the final calculation results (Table |
556 |
|
|
\ref{spceThermal}). Both methods yields values in agreement with |
557 |
|
|
experiment. And this shows the applicability of our method to |
558 |
|
|
multi-atom molecular system. |
559 |
skuang |
3563 |
|
560 |
skuang |
3570 |
\begin{figure} |
561 |
|
|
\includegraphics[width=\linewidth]{spceGrad} |
562 |
|
|
\caption{Temperature gradients for SPC/E water thermal conductivity.} |
563 |
|
|
\label{spceGrad} |
564 |
|
|
\end{figure} |
565 |
|
|
|
566 |
|
|
\begin{table*} |
567 |
|
|
\begin{minipage}{\linewidth} |
568 |
|
|
\begin{center} |
569 |
|
|
|
570 |
|
|
\caption{Calculation results for thermal conductivity of SPC/E water |
571 |
|
|
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
572 |
|
|
calculations in parentheses. } |
573 |
|
|
|
574 |
|
|
\begin{tabular}{cccc} |
575 |
|
|
\hline |
576 |
|
|
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
577 |
skuang |
3573 |
& This work & Previous simulations\cite{ISI:000090151400044} & |
578 |
|
|
Experiment$^a$\\ |
579 |
skuang |
3570 |
\hline |
580 |
skuang |
3573 |
0.38 & 0.816(0.044) & & 0.64\\ |
581 |
|
|
0.81 & 0.770(0.008) & 0.784\\ |
582 |
|
|
1.54 & 0.813(0.007) & 0.730\\ |
583 |
skuang |
3570 |
\hline |
584 |
|
|
\end{tabular} |
585 |
|
|
\label{spceThermal} |
586 |
|
|
\end{center} |
587 |
|
|
\end{minipage} |
588 |
|
|
\end{table*} |
589 |
|
|
|
590 |
skuang |
3573 |
\subsubsection{Crystal Gold} |
591 |
skuang |
3580 |
Our results of gold thermal conductivity using two force fields are |
592 |
|
|
shown separately in Table \ref{qscThermal} and \ref{eamThermal}. In |
593 |
|
|
these calculations,the end and middle slabs were excluded in thermal |
594 |
|
|
gradient regession and only used as heat source and drain in the |
595 |
|
|
systems. Our yielded values using EAM force field are slightly larger |
596 |
|
|
than those using QSC force field. However, both series are |
597 |
|
|
significantly smaller than experimental value by an order of more than |
598 |
|
|
100. It has been verified that this difference is mainly attributed to |
599 |
|
|
the lack of electron interaction representation in these force field |
600 |
skuang |
3582 |
parameters. Richardson {\it et al.}\cite{Clancy:1992} used EAM |
601 |
skuang |
3580 |
force field parameters in their metal thermal conductivity |
602 |
|
|
calculations. The Non-Equilibrium MD method they employed in their |
603 |
|
|
simulations produced comparable results to ours. As Zhang {\it et |
604 |
|
|
al.}\cite{ISI:000231042800044} stated, thermal conductivity values |
605 |
|
|
are influenced mainly by force field. Therefore, it is confident to |
606 |
|
|
conclude that NIVS-RNEMD is applicable to metal force field system. |
607 |
skuang |
3570 |
|
608 |
|
|
\begin{figure} |
609 |
|
|
\includegraphics[width=\linewidth]{AuGrad} |
610 |
skuang |
3580 |
\caption{Temperature gradients for thermal conductivity calculation of |
611 |
|
|
crystal gold using QSC force field.} |
612 |
skuang |
3570 |
\label{AuGrad} |
613 |
|
|
\end{figure} |
614 |
|
|
|
615 |
|
|
\begin{table*} |
616 |
|
|
\begin{minipage}{\linewidth} |
617 |
|
|
\begin{center} |
618 |
|
|
|
619 |
|
|
\caption{Calculation results for thermal conductivity of crystal gold |
620 |
skuang |
3580 |
using QSC force field at ${\langle T\rangle}$ = 300K at various |
621 |
|
|
thermal exchange rates. Errors of calculations in parentheses. } |
622 |
skuang |
3570 |
|
623 |
skuang |
3579 |
\begin{tabular}{cc} |
624 |
skuang |
3570 |
\hline |
625 |
|
|
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
626 |
|
|
\hline |
627 |
skuang |
3579 |
1.44 & 1.10(0.01)\\ |
628 |
|
|
2.86 & 1.08(0.02)\\ |
629 |
|
|
5.14 & 1.15(0.01)\\ |
630 |
skuang |
3570 |
\hline |
631 |
|
|
\end{tabular} |
632 |
skuang |
3580 |
\label{qscThermal} |
633 |
skuang |
3570 |
\end{center} |
634 |
|
|
\end{minipage} |
635 |
|
|
\end{table*} |
636 |
|
|
|
637 |
skuang |
3580 |
\begin{figure} |
638 |
|
|
\includegraphics[width=\linewidth]{eamGrad} |
639 |
|
|
\caption{Temperature gradients for thermal conductivity calculation of |
640 |
|
|
crystal gold using EAM force field.} |
641 |
|
|
\label{eamGrad} |
642 |
|
|
\end{figure} |
643 |
|
|
|
644 |
|
|
\begin{table*} |
645 |
|
|
\begin{minipage}{\linewidth} |
646 |
|
|
\begin{center} |
647 |
|
|
|
648 |
|
|
\caption{Calculation results for thermal conductivity of crystal gold |
649 |
|
|
using EAM force field at ${\langle T\rangle}$ = 300K at various |
650 |
|
|
thermal exchange rates. Errors of calculations in parentheses. } |
651 |
|
|
|
652 |
|
|
\begin{tabular}{cc} |
653 |
|
|
\hline |
654 |
|
|
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
655 |
|
|
\hline |
656 |
|
|
1.24 & 1.24(0.06)\\ |
657 |
|
|
2.06 & 1.37(0.04)\\ |
658 |
|
|
2.55 & 1.41(0.03)\\ |
659 |
|
|
\hline |
660 |
|
|
\end{tabular} |
661 |
|
|
\label{eamThermal} |
662 |
|
|
\end{center} |
663 |
|
|
\end{minipage} |
664 |
|
|
\end{table*} |
665 |
|
|
|
666 |
|
|
|
667 |
skuang |
3573 |
\subsection{Interfaciel Thermal Conductivity} |
668 |
skuang |
3581 |
After simulations of homogeneous water and gold systems using |
669 |
|
|
NIVS-RNEMD method were proved valid, calculation of gold/water |
670 |
|
|
interfacial thermal conductivity was followed. It is found out that |
671 |
|
|
the low interfacial conductance is probably due to the hydrophobic |
672 |
|
|
surface in our system. Figure \ref{interfaceDensity} demonstrates mass |
673 |
|
|
density change along $z$-axis, which is perpendicular to the |
674 |
|
|
gold/water interface. It is observed that water density significantly |
675 |
|
|
decreases when approaching the surface. Under this low thermal |
676 |
|
|
conductance, both gold and water phase have sufficient time to |
677 |
|
|
eliminate temperature difference inside respectively (Figure |
678 |
|
|
\ref{interfaceGrad}). With indistinguishable temperature difference |
679 |
|
|
within respective phase, it is valid to assume that the temperature |
680 |
|
|
difference between gold and water on surface would be approximately |
681 |
|
|
the same as the difference between the gold and water phase. This |
682 |
|
|
assumption enables convenient calculation of $G$ using |
683 |
|
|
Eq. \ref{interfaceCalc} instead of measuring temperatures of thin |
684 |
|
|
layer of water and gold close enough to surface, which would have |
685 |
|
|
greater fluctuation and lower accuracy. Reported results (Table |
686 |
|
|
\ref{interfaceRes}) are of two orders of magnitude smaller than our |
687 |
|
|
calculations on homogeneous systems, and thus have larger relative |
688 |
|
|
errors than our calculation results on homogeneous systems. |
689 |
skuang |
3573 |
|
690 |
skuang |
3571 |
\begin{figure} |
691 |
|
|
\includegraphics[width=\linewidth]{interfaceDensity} |
692 |
|
|
\caption{Density profile for interfacial thermal conductivity |
693 |
skuang |
3581 |
simulation box. Significant water density decrease is observed on |
694 |
|
|
gold surface.} |
695 |
skuang |
3571 |
\label{interfaceDensity} |
696 |
|
|
\end{figure} |
697 |
|
|
|
698 |
skuang |
3572 |
\begin{figure} |
699 |
|
|
\includegraphics[width=\linewidth]{interfaceGrad} |
700 |
|
|
\caption{Temperature profiles for interfacial thermal conductivity |
701 |
skuang |
3581 |
simulation box. Temperatures of different slabs in the same phase |
702 |
|
|
show no significant difference.} |
703 |
skuang |
3572 |
\label{interfaceGrad} |
704 |
|
|
\end{figure} |
705 |
|
|
|
706 |
|
|
\begin{table*} |
707 |
|
|
\begin{minipage}{\linewidth} |
708 |
|
|
\begin{center} |
709 |
|
|
|
710 |
|
|
\caption{Calculation results for interfacial thermal conductivity |
711 |
|
|
at ${\langle T\rangle \sim}$ 300K at various thermal exchange |
712 |
|
|
rates. Errors of calculations in parentheses. } |
713 |
|
|
|
714 |
|
|
\begin{tabular}{cccc} |
715 |
|
|
\hline |
716 |
|
|
$J_z$(MW/m$^2$) & $T_{gold}$ & $T_{water}$ & $G$(MW/m$^2$/K)\\ |
717 |
|
|
\hline |
718 |
skuang |
3573 |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
719 |
|
|
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
720 |
|
|
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
721 |
|
|
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
722 |
skuang |
3572 |
\hline |
723 |
|
|
\end{tabular} |
724 |
skuang |
3574 |
\label{interfaceRes} |
725 |
skuang |
3572 |
\end{center} |
726 |
|
|
\end{minipage} |
727 |
|
|
\end{table*} |
728 |
|
|
|
729 |
skuang |
3576 |
\subsection{Shear Viscosity} |
730 |
|
|
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
731 |
|
|
produced comparable shear viscosity to swap RNEMD method. In Table |
732 |
|
|
\ref{shearRate}, the names of the calculated samples are devided into |
733 |
|
|
two parts. The first number refers to total slabs in one simulation |
734 |
|
|
box. The second number refers to the swapping interval in swap method, or |
735 |
|
|
in scale method the equilvalent swapping interval that the same |
736 |
|
|
momentum flux would theoretically result in swap method. All the scale |
737 |
|
|
method results were from simulations that had a scaling interval of 10 |
738 |
|
|
time steps. The average molecular momentum gradients of these samples |
739 |
|
|
are shown in Figure \ref{shearGrad}. |
740 |
|
|
|
741 |
|
|
\begin{table*} |
742 |
|
|
\begin{minipage}{\linewidth} |
743 |
|
|
\begin{center} |
744 |
|
|
|
745 |
|
|
\caption{Calculation results for shear viscosity of Lennard-Jones |
746 |
|
|
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
747 |
|
|
methods at various momentum exchange rates. Results in reduced |
748 |
|
|
unit. Errors of calculations in parentheses. } |
749 |
|
|
|
750 |
|
|
\begin{tabular}{ccc} |
751 |
|
|
\hline |
752 |
|
|
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
753 |
|
|
\hline |
754 |
|
|
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
755 |
|
|
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
756 |
|
|
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
757 |
|
|
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
758 |
|
|
\hline |
759 |
|
|
\end{tabular} |
760 |
|
|
\label{shearRate} |
761 |
|
|
\end{center} |
762 |
|
|
\end{minipage} |
763 |
|
|
\end{table*} |
764 |
|
|
|
765 |
|
|
\begin{figure} |
766 |
|
|
\includegraphics[width=\linewidth]{shearGrad} |
767 |
|
|
\caption{Average momentum gradients of shear viscosity simulations} |
768 |
|
|
\label{shearGrad} |
769 |
|
|
\end{figure} |
770 |
|
|
|
771 |
|
|
\begin{figure} |
772 |
|
|
\includegraphics[width=\linewidth]{shearTempScale} |
773 |
|
|
\caption{Temperature profile for scaling RNEMD simulation.} |
774 |
|
|
\label{shearTempScale} |
775 |
|
|
\end{figure} |
776 |
|
|
However, observations of temperatures along three dimensions show that |
777 |
|
|
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
778 |
|
|
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
779 |
|
|
relatively large imposed momentum flux, the temperature difference among $x$ |
780 |
|
|
and the other two dimensions was significant. This would result from the |
781 |
|
|
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
782 |
|
|
momentum gradient is set up, $P_c^x$ would be roughly stable |
783 |
|
|
($<0$). Consequently, scaling factor $x$ would most probably larger |
784 |
|
|
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
785 |
|
|
keep increase after most scaling steps. And if there is not enough time |
786 |
|
|
for the kinetic energy to exchange among different dimensions and |
787 |
|
|
different slabs, the system would finally build up temperature |
788 |
|
|
(kinetic energy) difference among the three dimensions. Also, between |
789 |
|
|
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
790 |
|
|
are closer to neighbor slabs. This is due to momentum transfer along |
791 |
|
|
$z$ dimension between slabs. |
792 |
|
|
|
793 |
|
|
Although results between scaling and swapping methods are comparable, |
794 |
|
|
the inherent temperature inhomogeneity even in relatively low imposed |
795 |
|
|
exchange momentum flux simulations makes scaling RNEMD method less |
796 |
|
|
attractive than swapping RNEMD in shear viscosity calculation. |
797 |
|
|
|
798 |
skuang |
3574 |
\section{Conclusions} |
799 |
|
|
NIVS-RNEMD simulation method is developed and tested on various |
800 |
skuang |
3581 |
systems. Simulation results demonstrate its validity in thermal |
801 |
|
|
conductivity calculations, from Lennard-Jones fluid to multi-atom |
802 |
|
|
molecule like water and metal crystals. NIVS-RNEMD improves |
803 |
|
|
non-Boltzmann-Maxwell distributions, which exist in previous RNEMD |
804 |
|
|
methods. Furthermore, it develops a valid means for unphysical thermal |
805 |
|
|
transfer between different species of molecules, and thus extends its |
806 |
|
|
applicability to interfacial systems. Our calculation of gold/water |
807 |
|
|
interfacial thermal conductivity demonstrates this advantage over |
808 |
|
|
previous RNEMD methods. NIVS-RNEMD has also limited application on |
809 |
|
|
shear viscosity calculations, but could cause temperature difference |
810 |
|
|
among different dimensions under high momentum flux. Modification is |
811 |
|
|
necessary to extend the applicability of NIVS-RNEMD in shear viscosity |
812 |
|
|
calculations. |
813 |
skuang |
3572 |
|
814 |
gezelter |
3524 |
\section{Acknowledgments} |
815 |
|
|
Support for this project was provided by the National Science |
816 |
|
|
Foundation under grant CHE-0848243. Computational time was provided by |
817 |
|
|
the Center for Research Computing (CRC) at the University of Notre |
818 |
|
|
Dame. \newpage |
819 |
|
|
|
820 |
gezelter |
3583 |
\bibliographystyle{aip} |
821 |
gezelter |
3524 |
\bibliography{nivsRnemd} |
822 |
gezelter |
3583 |
|
823 |
gezelter |
3524 |
\end{doublespace} |
824 |
|
|
\end{document} |
825 |
|
|
|