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 |
|
|
|
42 |
|
|
\end{abstract} |
43 |
|
|
|
44 |
|
|
\newpage |
45 |
|
|
|
46 |
|
|
%\narrowtext |
47 |
|
|
|
48 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
49 |
|
|
% BODY OF TEXT |
50 |
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
51 |
|
|
|
52 |
|
|
|
53 |
|
|
|
54 |
|
|
\section{Introduction} |
55 |
|
|
The original formulation of Reverse Non-equilibrium Molecular Dynamics |
56 |
|
|
(RNEMD) obtains transport coefficients (thermal conductivity and shear |
57 |
|
|
viscosity) in a fluid by imposing an artificial momentum flux between |
58 |
|
|
two thin parallel slabs of material that are spatially separated in |
59 |
skuang |
3534 |
the simulation cell.\cite{MullerPlathe:1997xw,ISI:000080382700030} The |
60 |
skuang |
3531 |
artificial flux is typically created by periodically ``swapping'' either |
61 |
gezelter |
3524 |
the entire momentum vector $\vec{p}$ or single components of this |
62 |
|
|
vector ($p_x$) between molecules in each of the two slabs. If the two |
63 |
|
|
slabs are separated along the z coordinate, the imposed flux is either |
64 |
skuang |
3532 |
directional ($j_z(p_x)$) or isotropic ($J_z$), and the response of a |
65 |
gezelter |
3524 |
simulated system to the imposed momentum flux will typically be a |
66 |
|
|
velocity or thermal gradient. The transport coefficients (shear |
67 |
|
|
viscosity and thermal conductivity) are easily obtained by assuming |
68 |
|
|
linear response of the system, |
69 |
|
|
\begin{eqnarray} |
70 |
skuang |
3532 |
j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
71 |
gezelter |
3524 |
J & = & \lambda \frac{\partial T}{\partial z} |
72 |
|
|
\end{eqnarray} |
73 |
skuang |
3528 |
RNEMD has been widely used to provide computational estimates of thermal |
74 |
gezelter |
3524 |
conductivities and shear viscosities in a wide range of materials, |
75 |
|
|
from liquid copper to monatomic liquids to molecular fluids |
76 |
skuang |
3528 |
(e.g. ionic liquids).\cite{ISI:000246190100032} |
77 |
gezelter |
3524 |
|
78 |
|
|
RNEMD is preferable in many ways to the forward NEMD methods because |
79 |
|
|
it imposes what is typically difficult to measure (a flux or stress) |
80 |
|
|
and it is typically much easier to compute momentum gradients or |
81 |
|
|
strains (the response). For similar reasons, RNEMD is also preferable |
82 |
|
|
to slowly-converging equilibrium methods for measuring thermal |
83 |
|
|
conductivity and shear viscosity (using Green-Kubo relations or the |
84 |
|
|
Helfand moment approach of Viscardy {\it et |
85 |
skuang |
3527 |
al.}\cite{Viscardy:2007bh,Viscardy:2007lq}) because these rely on |
86 |
gezelter |
3524 |
computing difficult to measure quantities. |
87 |
|
|
|
88 |
|
|
Another attractive feature of RNEMD is that it conserves both total |
89 |
|
|
linear momentum and total energy during the swaps (as long as the two |
90 |
|
|
molecules have the same identity), so the swapped configurations are |
91 |
|
|
typically samples from the same manifold of states in the |
92 |
|
|
microcanonical ensemble. |
93 |
|
|
|
94 |
skuang |
3565 |
Recently, Tenney and Maginn\cite{ISI:000273472300004} have discovered |
95 |
|
|
some problems with the original RNEMD swap technique. Notably, large |
96 |
|
|
momentum fluxes (equivalent to frequent momentum swaps between the |
97 |
|
|
slabs) can result in ``notched'', ``peaked'' and generally non-thermal momentum |
98 |
gezelter |
3524 |
distributions in the two slabs, as well as non-linear thermal and |
99 |
|
|
velocity distributions along the direction of the imposed flux ($z$). |
100 |
|
|
Tenney and Maginn obtained reasonable limits on imposed flux and |
101 |
|
|
self-adjusting metrics for retaining the usability of the method. |
102 |
|
|
|
103 |
|
|
In this paper, we develop and test a method for non-isotropic velocity |
104 |
|
|
scaling (NIVS-RNEMD) which retains the desirable features of RNEMD |
105 |
|
|
(conservation of linear momentum and total energy, compatibility with |
106 |
|
|
periodic boundary conditions) while establishing true thermal |
107 |
|
|
distributions in each of the two slabs. In the next section, we |
108 |
|
|
develop the method for determining the scaling constraints. We then |
109 |
|
|
test the method on both single component, multi-component, and |
110 |
|
|
non-isotropic mixtures and show that it is capable of providing |
111 |
|
|
reasonable estimates of the thermal conductivity and shear viscosity |
112 |
|
|
in these cases. |
113 |
|
|
|
114 |
|
|
\section{Methodology} |
115 |
|
|
We retain the basic idea of Muller-Plathe's RNEMD method; the periodic |
116 |
|
|
system is partitioned into a series of thin slabs along a particular |
117 |
|
|
axis ($z$). One of the slabs at the end of the periodic box is |
118 |
|
|
designated the ``hot'' slab, while the slab in the center of the box |
119 |
|
|
is designated the ``cold'' slab. The artificial momentum flux will be |
120 |
|
|
established by transferring momentum from the cold slab and into the |
121 |
|
|
hot slab. |
122 |
|
|
|
123 |
|
|
Rather than using momentum swaps, we use a series of velocity scaling |
124 |
skuang |
3528 |
moves. For molecules $\{i\}$ located within the cold slab, |
125 |
gezelter |
3524 |
\begin{equation} |
126 |
skuang |
3565 |
\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
127 |
|
|
x & 0 & 0 \\ |
128 |
|
|
0 & y & 0 \\ |
129 |
|
|
0 & 0 & z \\ |
130 |
gezelter |
3524 |
\end{array} \right) \cdot \vec{v}_i |
131 |
|
|
\end{equation} |
132 |
|
|
where ${x, y, z}$ are a set of 3 scaling variables for each of the |
133 |
|
|
three directions in the system. Likewise, the molecules $\{j\}$ |
134 |
skuang |
3528 |
located in the hot slab will see a concomitant scaling of velocities, |
135 |
gezelter |
3524 |
\begin{equation} |
136 |
skuang |
3565 |
\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
137 |
|
|
x^\prime & 0 & 0 \\ |
138 |
|
|
0 & y^\prime & 0 \\ |
139 |
|
|
0 & 0 & z^\prime \\ |
140 |
gezelter |
3524 |
\end{array} \right) \cdot \vec{v}_j |
141 |
|
|
\end{equation} |
142 |
|
|
|
143 |
|
|
Conservation of linear momentum in each of the three directions |
144 |
|
|
($\alpha = x,y,z$) ties the values of the hot and cold bin scaling |
145 |
|
|
parameters together: |
146 |
|
|
\begin{equation} |
147 |
skuang |
3528 |
P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
148 |
gezelter |
3524 |
\end{equation} |
149 |
|
|
where |
150 |
skuang |
3565 |
\begin{eqnarray} |
151 |
skuang |
3528 |
P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
152 |
skuang |
3565 |
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha |
153 |
gezelter |
3524 |
\label{eq:momentumdef} |
154 |
skuang |
3565 |
\end{eqnarray} |
155 |
skuang |
3528 |
Therefore, for each of the three directions, the hot scaling |
156 |
|
|
parameters are a simple function of the cold scaling parameters and |
157 |
gezelter |
3524 |
the instantaneous linear momentum in each of the two slabs. |
158 |
|
|
\begin{equation} |
159 |
skuang |
3528 |
\alpha^\prime = 1 + (1 - \alpha) p_\alpha |
160 |
gezelter |
3524 |
\label{eq:hotcoldscaling} |
161 |
|
|
\end{equation} |
162 |
skuang |
3528 |
where |
163 |
|
|
\begin{equation} |
164 |
|
|
p_\alpha = \frac{P_c^\alpha}{P_h^\alpha} |
165 |
|
|
\end{equation} |
166 |
|
|
for convenience. |
167 |
gezelter |
3524 |
|
168 |
|
|
Conservation of total energy also places constraints on the scaling: |
169 |
|
|
\begin{equation} |
170 |
|
|
\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
171 |
skuang |
3565 |
\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha |
172 |
gezelter |
3524 |
\end{equation} |
173 |
|
|
where the kinetic energies, $K_h^\alpha$ and $K_c^\alpha$, are computed |
174 |
|
|
for each of the three directions in a similar manner to the linear momenta |
175 |
|
|
(Eq. \ref{eq:momentumdef}). Substituting in the expressions for the |
176 |
skuang |
3528 |
hot scaling parameters ($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), |
177 |
|
|
we obtain the {\it constraint ellipsoid equation}: |
178 |
gezelter |
3524 |
\begin{equation} |
179 |
skuang |
3565 |
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0 |
180 |
gezelter |
3524 |
\label{eq:constraintEllipsoid} |
181 |
|
|
\end{equation} |
182 |
|
|
where the constants are obtained from the instantaneous values of the |
183 |
|
|
linear momenta and kinetic energies for the hot and cold slabs, |
184 |
skuang |
3565 |
\begin{eqnarray} |
185 |
skuang |
3528 |
a_\alpha & = &\left(K_c^\alpha + K_h^\alpha |
186 |
|
|
\left(p_\alpha\right)^2\right) \\ |
187 |
|
|
b_\alpha & = & -2 K_h^\alpha p_\alpha \left( 1 + p_\alpha\right) \\ |
188 |
skuang |
3565 |
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
189 |
gezelter |
3524 |
\label{eq:constraintEllipsoidConsts} |
190 |
skuang |
3565 |
\end{eqnarray} |
191 |
skuang |
3528 |
This ellipsoid equation defines the set of cold slab scaling |
192 |
|
|
parameters which can be applied while preserving both linear momentum |
193 |
skuang |
3530 |
in all three directions as well as kinetic energy. |
194 |
gezelter |
3524 |
|
195 |
|
|
The goal of using velocity scaling variables is to transfer linear |
196 |
|
|
momentum or kinetic energy from the cold slab to the hot slab. If the |
197 |
|
|
hot and cold slabs are separated along the z-axis, the energy flux is |
198 |
skuang |
3528 |
given simply by the decrease in kinetic energy of the cold bin: |
199 |
gezelter |
3524 |
\begin{equation} |
200 |
skuang |
3534 |
(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z\Delta t |
201 |
gezelter |
3524 |
\end{equation} |
202 |
|
|
The expression for the energy flux can be re-written as another |
203 |
|
|
ellipsoid centered on $(x,y,z) = 0$: |
204 |
|
|
\begin{equation} |
205 |
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 |
206 |
gezelter |
3524 |
\label{eq:fluxEllipsoid} |
207 |
|
|
\end{equation} |
208 |
skuang |
3529 |
The spatial extent of the {\it flux ellipsoid equation} is governed |
209 |
|
|
both by a targetted value, $J_z$ as well as the instantaneous values of the |
210 |
skuang |
3530 |
kinetic energy components in the cold bin. |
211 |
gezelter |
3524 |
|
212 |
|
|
To satisfy an energetic flux as well as the conservation constraints, |
213 |
|
|
it is sufficient to determine the points ${x,y,z}$ which lie on both |
214 |
|
|
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
215 |
|
|
flux ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
216 |
skuang |
3528 |
the two ellipsoids in 3-dimensional space. |
217 |
gezelter |
3524 |
|
218 |
gezelter |
3569 |
\begin{figure} |
219 |
|
|
\includegraphics[width=\linewidth]{ellipsoids} |
220 |
|
|
\caption{Scaling points which maintain both constant energy and |
221 |
|
|
constant linear momentum of the system lie on the surface of the |
222 |
|
|
{\it constraint ellipsoid} while points which generate the target |
223 |
|
|
momentum flux lie on the surface of the {\it flux ellipsoid}. The |
224 |
|
|
velocity distributions in the hot bin are scaled by only those |
225 |
|
|
points which lie on both ellipsoids.} |
226 |
|
|
\label{ellipsoids} |
227 |
|
|
\end{figure} |
228 |
|
|
|
229 |
gezelter |
3524 |
One may also define momentum flux (say along the x-direction) as: |
230 |
|
|
\begin{equation} |
231 |
skuang |
3565 |
(1-x) P_c^x = j_z(p_x)\Delta t |
232 |
skuang |
3531 |
\label{eq:fluxPlane} |
233 |
gezelter |
3524 |
\end{equation} |
234 |
skuang |
3531 |
The above {\it flux equation} is essentially a plane which is |
235 |
|
|
perpendicular to the x-axis, with its position governed both by a |
236 |
|
|
targetted value, $j_z(p_x)$ as well as the instantaneous value of the |
237 |
|
|
momentum along the x-direction. |
238 |
gezelter |
3524 |
|
239 |
skuang |
3531 |
Similarly, to satisfy a momentum flux as well as the conservation |
240 |
|
|
constraints, it is sufficient to determine the points ${x,y,z}$ which |
241 |
|
|
lie on both the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) |
242 |
|
|
and the flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of |
243 |
|
|
an ellipsoid and a plane in 3-dimensional space. |
244 |
gezelter |
3524 |
|
245 |
skuang |
3531 |
To summarize, by solving respective equation sets, one can determine |
246 |
|
|
possible sets of scaling variables for cold slab. And corresponding |
247 |
|
|
sets of scaling variables for hot slab can be determine as well. |
248 |
gezelter |
3524 |
|
249 |
skuang |
3531 |
The following problem will be choosing an optimal set of scaling |
250 |
|
|
variables among the possible sets. Although this method is inherently |
251 |
|
|
non-isotropic, the goal is still to maintain the system as isotropic |
252 |
|
|
as possible. Under this consideration, one would like the kinetic |
253 |
|
|
energies in different directions could become as close as each other |
254 |
|
|
after each scaling. Simultaneously, one would also like each scaling |
255 |
|
|
as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
256 |
|
|
large perturbation to the system. Therefore, one approach to obtain the |
257 |
|
|
scaling variables would be constructing an criteria function, with |
258 |
|
|
constraints as above equation sets, and solving the function's minimum |
259 |
|
|
by method like Lagrange multipliers. |
260 |
gezelter |
3524 |
|
261 |
skuang |
3531 |
In order to save computation time, we have a different approach to a |
262 |
|
|
relatively good set of scaling variables with much less calculation |
263 |
|
|
than above. Here is the detail of our simplification of the problem. |
264 |
gezelter |
3524 |
|
265 |
skuang |
3531 |
In the case of kinetic energy transfer, we impose another constraint |
266 |
|
|
${x = y}$, into the equation sets. Consequently, there are two |
267 |
|
|
variables left. And now one only needs to solve a set of two {\it |
268 |
|
|
ellipses equations}. This problem would be transformed into solving |
269 |
|
|
one quartic equation for one of the two variables. There are known |
270 |
|
|
generic methods that solve real roots of quartic equations. Then one |
271 |
|
|
can determine the other variable and obtain sets of scaling |
272 |
|
|
variables. Among these sets, one can apply the above criteria to |
273 |
|
|
choose the best set, while much faster with only a few sets to choose. |
274 |
|
|
|
275 |
|
|
In the case of momentum flux transfer, we impose another constraint to |
276 |
|
|
set the kinetic energy transfer as zero. In another word, we apply |
277 |
|
|
Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. After that, with one |
278 |
|
|
variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar set |
279 |
|
|
of equations on the above kinetic energy transfer problem. Therefore, |
280 |
|
|
an approach similar to the above would be sufficient for this as well. |
281 |
|
|
|
282 |
|
|
\section{Computational Details} |
283 |
skuang |
3534 |
Our simulation consists of a series of systems. All of these |
284 |
skuang |
3565 |
simulations were run with the OpenMD simulation software |
285 |
skuang |
3534 |
package\cite{Meineke:2005gd} integrated with RNEMD methods. |
286 |
skuang |
3531 |
|
287 |
skuang |
3532 |
A Lennard-Jones fluid system was built and tested first. In order to |
288 |
|
|
compare our method with swapping RNEMD, a series of simulations were |
289 |
|
|
performed to calculate the shear viscosity and thermal conductivity of |
290 |
skuang |
3534 |
argon. 2592 atoms were in a orthorhombic cell, which was ${10.06\sigma |
291 |
|
|
\times 10.06\sigma \times 30.18\sigma}$ by size. The reduced density |
292 |
|
|
${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled direct |
293 |
|
|
comparison between our results and others. These simulations used |
294 |
skuang |
3565 |
velocity Verlet algorithm with reduced timestep ${\tau^* = |
295 |
skuang |
3534 |
4.6\times10^{-4}}$. |
296 |
skuang |
3532 |
|
297 |
|
|
For shear viscosity calculation, the reduced temperature was ${T^* = |
298 |
skuang |
3565 |
k_B T/\varepsilon = 0.72}$. Simulations were first equilibrated in canonical |
299 |
|
|
ensemble (NVT), then equilibrated in microcanonical ensemble |
300 |
|
|
(NVE). Establishing and stablizing momentum gradient were followed |
301 |
|
|
also in NVE ensemble. For the swapping part, Muller-Plathe's algorithm was |
302 |
skuang |
3532 |
adopted.\cite{ISI:000080382700030} The simulation box was under |
303 |
skuang |
3534 |
periodic boundary condition, and devided into ${N = 20}$ slabs. In each swap, |
304 |
|
|
the top slab ${(n = 1)}$ exchange the most negative $x$ momentum with the |
305 |
|
|
most positive $x$ momentum in the center slab ${(n = N/2 + 1)}$. Referred |
306 |
skuang |
3565 |
to Tenney {\it et al.}\cite{ISI:000273472300004}, a series of swapping |
307 |
skuang |
3534 |
frequency were chosen. According to each result from swapping |
308 |
skuang |
3532 |
RNEMD, scaling RNEMD simulations were run with the target momentum |
309 |
skuang |
3534 |
flux set to produce a similar momentum flux and shear |
310 |
|
|
rate. Furthermore, various scaling frequencies can be tested for one |
311 |
|
|
single swapping rate. To compare the performance between swapping and |
312 |
|
|
scaling method, temperatures of different dimensions in all the slabs |
313 |
skuang |
3538 |
were observed. Most of the simulations include $10^5$ steps of |
314 |
|
|
equilibration without imposing momentum flux, $10^5$ steps of |
315 |
|
|
stablization with imposing momentum transfer, and $10^6$ steps of data |
316 |
|
|
collection under RNEMD. For relatively high momentum flux simulations, |
317 |
|
|
${5\times10^5}$ step data collection is sufficient. For some low momentum |
318 |
|
|
flux simulations, ${2\times10^6}$ steps were necessary. |
319 |
skuang |
3532 |
|
320 |
skuang |
3534 |
After each simulation, the shear viscosity was calculated in reduced |
321 |
|
|
unit. The momentum flux was calculated with total unphysical |
322 |
skuang |
3565 |
transferred momentum ${P_x}$ and data collection time $t$: |
323 |
skuang |
3534 |
\begin{equation} |
324 |
|
|
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
325 |
|
|
\end{equation} |
326 |
|
|
And the velocity gradient ${\langle \partial v_x /\partial z \rangle}$ |
327 |
|
|
can be obtained by a linear regression of the velocity profile. From |
328 |
|
|
the shear viscosity $\eta$ calculated with the above parameters, one |
329 |
|
|
can further convert it into reduced unit ${\eta^* = \eta \sigma^2 |
330 |
|
|
(\varepsilon m)^{-1/2}}$. |
331 |
skuang |
3532 |
|
332 |
skuang |
3534 |
For thermal conductivity calculation, simulations were first run under |
333 |
|
|
reduced temperature ${T^* = 0.72}$ in NVE ensemble. Muller-Plathe's |
334 |
|
|
algorithm was adopted in the swapping method. Under identical |
335 |
skuang |
3536 |
simulation box parameters, in each swap, the top slab exchange the |
336 |
|
|
molecule with least kinetic energy with the molecule in the center |
337 |
|
|
slab with most kinetic energy, unless this ``coldest'' molecule in the |
338 |
|
|
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the ``cold'' |
339 |
skuang |
3534 |
slab. According to swapping RNEMD results, target energy flux for |
340 |
|
|
scaling RNEMD simulations can be set. Also, various scaling |
341 |
|
|
frequencies can be tested for one target energy flux. To compare the |
342 |
|
|
performance between swapping and scaling method, distributions of |
343 |
|
|
velocity and speed in different slabs were observed. |
344 |
|
|
|
345 |
|
|
For each swapping rate, thermal conductivity was calculated in reduced |
346 |
|
|
unit. The energy flux was calculated similarly to the momentum flux, |
347 |
skuang |
3565 |
with total unphysical transferred energy ${E_{total}}$ and data collection |
348 |
skuang |
3534 |
time $t$: |
349 |
|
|
\begin{equation} |
350 |
|
|
J_z = \frac{E_{total}}{2 t L_x L_y} |
351 |
|
|
\end{equation} |
352 |
|
|
And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
353 |
|
|
can be obtained by a linear regression of the temperature |
354 |
|
|
profile. From the thermal conductivity $\lambda$ calculated, one can |
355 |
|
|
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
356 |
|
|
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
357 |
|
|
|
358 |
skuang |
3565 |
Another series of our simulation is to calculate the interfacial |
359 |
skuang |
3563 |
thermal conductivity of a Au/H${_2}$O system. Respective calculations of |
360 |
skuang |
3565 |
water (SPC/E) and gold (QSC) thermal conductivity were performed and |
361 |
skuang |
3563 |
compared with current results to ensure the validity of |
362 |
|
|
NIVS-RNEMD. After that, the mixture system was simulated. |
363 |
|
|
|
364 |
skuang |
3534 |
\section{Results And Discussion} |
365 |
|
|
\subsection{Shear Viscosity} |
366 |
skuang |
3538 |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
367 |
|
|
produced comparable shear viscosity to swap RNEMD method. In Table |
368 |
|
|
\ref{shearRate}, the names of the calculated samples are devided into |
369 |
|
|
two parts. The first number refers to total slabs in one simulation |
370 |
|
|
box. The second number refers to the swapping interval in swap method, or |
371 |
|
|
in scale method the equilvalent swapping interval that the same |
372 |
|
|
momentum flux would theoretically result in swap method. All the scale |
373 |
skuang |
3563 |
method results were from simulations that had a scaling interval of 10 |
374 |
|
|
time steps. The average molecular momentum gradients of these samples |
375 |
skuang |
3565 |
are shown in Figure \ref{shearGrad}. |
376 |
skuang |
3534 |
|
377 |
skuang |
3538 |
\begin{table*} |
378 |
|
|
\begin{minipage}{\linewidth} |
379 |
|
|
\begin{center} |
380 |
|
|
|
381 |
|
|
\caption{Calculation results for shear viscosity of Lennard-Jones |
382 |
|
|
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
383 |
|
|
methods at various momentum exchange rates. Results in reduced |
384 |
|
|
unit. Errors of calculations in parentheses. } |
385 |
|
|
|
386 |
skuang |
3565 |
\begin{tabular}{ccc} |
387 |
skuang |
3538 |
\hline |
388 |
skuang |
3565 |
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
389 |
skuang |
3538 |
\hline |
390 |
|
|
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
391 |
skuang |
3539 |
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
392 |
|
|
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
393 |
skuang |
3566 |
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
394 |
|
|
\hline |
395 |
skuang |
3538 |
\end{tabular} |
396 |
|
|
\label{shearRate} |
397 |
|
|
\end{center} |
398 |
|
|
\end{minipage} |
399 |
|
|
\end{table*} |
400 |
|
|
|
401 |
|
|
\begin{figure} |
402 |
skuang |
3565 |
\includegraphics[width=\linewidth]{shearGrad} |
403 |
|
|
\caption{Average momentum gradients of shear viscosity simulations} |
404 |
|
|
\label{shearGrad} |
405 |
skuang |
3538 |
\end{figure} |
406 |
|
|
|
407 |
|
|
\begin{figure} |
408 |
skuang |
3565 |
\includegraphics[width=\linewidth]{shearTempScale} |
409 |
skuang |
3538 |
\caption{Temperature profile for scaling RNEMD simulation.} |
410 |
|
|
\label{shearTempScale} |
411 |
|
|
\end{figure} |
412 |
|
|
However, observations of temperatures along three dimensions show that |
413 |
|
|
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
414 |
|
|
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
415 |
skuang |
3563 |
relatively large imposed momentum flux, the temperature difference among $x$ |
416 |
|
|
and the other two dimensions was significant. This would result from the |
417 |
|
|
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
418 |
|
|
momentum gradient is set up, $P_c^x$ would be roughly stable |
419 |
|
|
($<0$). Consequently, scaling factor $x$ would most probably larger |
420 |
|
|
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
421 |
|
|
keep increase after most scaling steps. And if there is not enough time |
422 |
|
|
for the kinetic energy to exchange among different dimensions and |
423 |
|
|
different slabs, the system would finally build up temperature |
424 |
|
|
(kinetic energy) difference among the three dimensions. Also, between |
425 |
|
|
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
426 |
|
|
are closer to neighbor slabs. This is due to momentum transfer along |
427 |
|
|
$z$ dimension between slabs. |
428 |
skuang |
3538 |
|
429 |
|
|
Although results between scaling and swapping methods are comparable, |
430 |
skuang |
3563 |
the inherent temperature inhomogeneity even in relatively low imposed |
431 |
|
|
exchange momentum flux simulations makes scaling RNEMD method less |
432 |
skuang |
3538 |
attractive than swapping RNEMD in shear viscosity calculation. |
433 |
|
|
|
434 |
|
|
\subsection{Thermal Conductivity} |
435 |
|
|
|
436 |
skuang |
3567 |
Our thermal conductivity calculations also show that scaling method results |
437 |
|
|
agree with swapping method. Table \ref{thermal} lists our simulation |
438 |
skuang |
3563 |
results with similar manner we used in shear viscosity |
439 |
|
|
calculation. All the data reported from scaling method were obtained |
440 |
|
|
by simulations of 10-step exchange frequency, and the target exchange |
441 |
|
|
kinetic energy were set to produce equivalent kinetic energy flux as |
442 |
skuang |
3567 |
in swapping method. Figure \ref{thermalGrad} exhibits similar thermal |
443 |
|
|
gradients of respective similar kinetic energy flux. |
444 |
skuang |
3538 |
|
445 |
skuang |
3563 |
\begin{table*} |
446 |
|
|
\begin{minipage}{\linewidth} |
447 |
|
|
\begin{center} |
448 |
skuang |
3538 |
|
449 |
skuang |
3563 |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
450 |
skuang |
3565 |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
451 |
skuang |
3563 |
swap and scale methods at various kinetic energy exchange rates. Results |
452 |
|
|
in reduced unit. Errors of calculations in parentheses.} |
453 |
|
|
|
454 |
skuang |
3565 |
\begin{tabular}{ccc} |
455 |
skuang |
3563 |
\hline |
456 |
skuang |
3567 |
Series & $\lambda^*_{swap}$ & $\lambda^*_{scale}$\\ |
457 |
skuang |
3565 |
\hline |
458 |
skuang |
3564 |
20-250 & 7.03(0.34) & 7.30(0.10)\\ |
459 |
|
|
20-500 & 7.03(0.14) & 6.95(0.09)\\ |
460 |
skuang |
3563 |
20-1000 & 6.91(0.42) & 7.19(0.07)\\ |
461 |
skuang |
3566 |
20-2000 & 7.52(0.15) & 7.19(0.28)\\ |
462 |
|
|
\hline |
463 |
skuang |
3563 |
\end{tabular} |
464 |
|
|
\label{thermal} |
465 |
|
|
\end{center} |
466 |
|
|
\end{minipage} |
467 |
|
|
\end{table*} |
468 |
|
|
|
469 |
|
|
\begin{figure} |
470 |
skuang |
3567 |
\includegraphics[width=\linewidth]{thermalGrad} |
471 |
|
|
\caption{Temperature gradients of thermal conductivity simulations} |
472 |
|
|
\label{thermalGrad} |
473 |
skuang |
3563 |
\end{figure} |
474 |
|
|
|
475 |
|
|
During these simulations, molecule velocities were recorded in 1000 of |
476 |
|
|
all the snapshots. These velocity data were used to produce histograms |
477 |
|
|
of velocity and speed distribution in different slabs. From these |
478 |
|
|
histograms, it is observed that with increasing unphysical kinetic |
479 |
|
|
energy flux, speed and velocity distribution of molecules in slabs |
480 |
|
|
where swapping occured could deviate from Maxwell-Boltzmann |
481 |
|
|
distribution. Figure \ref{histSwap} indicates how these distributions |
482 |
|
|
deviate from ideal condition. In high temperature slabs, probability |
483 |
|
|
density in low speed is confidently smaller than ideal distribution; |
484 |
|
|
in low temperature slabs, probability density in high speed is smaller |
485 |
|
|
than ideal. This phenomenon is observable even in our relatively low |
486 |
skuang |
3568 |
swapping rate simulations. And this deviation could also leads to |
487 |
skuang |
3563 |
deviation of distribution of velocity in various dimensions. One |
488 |
|
|
feature of these deviated distribution is that in high temperature |
489 |
|
|
slab, the ideal Gaussian peak was changed into a relatively flat |
490 |
|
|
plateau; while in low temperature slab, that peak appears sharper. |
491 |
|
|
|
492 |
|
|
\begin{figure} |
493 |
skuang |
3565 |
\includegraphics[width=\linewidth]{histSwap} |
494 |
skuang |
3563 |
\caption{Speed distribution for thermal conductivity using swapping RNEMD.} |
495 |
|
|
\label{histSwap} |
496 |
|
|
\end{figure} |
497 |
|
|
|
498 |
skuang |
3568 |
\begin{figure} |
499 |
|
|
\includegraphics[width=\linewidth]{histScale} |
500 |
|
|
\caption{Speed distribution for thermal conductivity using scaling RNEMD.} |
501 |
|
|
\label{histScale} |
502 |
|
|
\end{figure} |
503 |
|
|
|
504 |
skuang |
3563 |
\subsection{Interfaciel Thermal Conductivity} |
505 |
|
|
|
506 |
skuang |
3570 |
\begin{figure} |
507 |
|
|
\includegraphics[width=\linewidth]{spceGrad} |
508 |
|
|
\caption{Temperature gradients for SPC/E water thermal conductivity.} |
509 |
|
|
\label{spceGrad} |
510 |
|
|
\end{figure} |
511 |
|
|
|
512 |
|
|
\begin{table*} |
513 |
|
|
\begin{minipage}{\linewidth} |
514 |
|
|
\begin{center} |
515 |
|
|
|
516 |
|
|
\caption{Calculation results for thermal conductivity of SPC/E water |
517 |
|
|
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
518 |
|
|
calculations in parentheses. } |
519 |
|
|
|
520 |
|
|
\begin{tabular}{cccc} |
521 |
|
|
\hline |
522 |
|
|
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
523 |
|
|
& This work & Previous simulations$^a$ & Experiment$^b$\\ |
524 |
|
|
\hline |
525 |
skuang |
3571 |
0.38 & 0.816(0.044) & 0.784 & 0.64\\ |
526 |
|
|
0.81 & 0.770(0.008) & 0.730\\ |
527 |
|
|
1.54 & 0.813(0.007) & \\ |
528 |
skuang |
3570 |
\hline |
529 |
|
|
\end{tabular} |
530 |
|
|
\label{spceThermal} |
531 |
|
|
\end{center} |
532 |
|
|
\end{minipage} |
533 |
|
|
\end{table*} |
534 |
|
|
|
535 |
|
|
|
536 |
|
|
\begin{figure} |
537 |
|
|
\includegraphics[width=\linewidth]{AuGrad} |
538 |
|
|
\caption{Temperature gradients for crystal gold thermal conductivity.} |
539 |
|
|
\label{AuGrad} |
540 |
|
|
\end{figure} |
541 |
|
|
|
542 |
|
|
\begin{table*} |
543 |
|
|
\begin{minipage}{\linewidth} |
544 |
|
|
\begin{center} |
545 |
|
|
|
546 |
|
|
\caption{Calculation results for thermal conductivity of crystal gold |
547 |
|
|
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
548 |
|
|
calculations in parentheses. } |
549 |
|
|
|
550 |
|
|
\begin{tabular}{ccc} |
551 |
|
|
\hline |
552 |
|
|
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
553 |
|
|
& This work & Previous simulations$^a$ \\ |
554 |
|
|
\hline |
555 |
skuang |
3571 |
1.44 & 1.10(0.01) & \\ |
556 |
|
|
2.86 & 1.08(0.02) & \\ |
557 |
|
|
5.14 & 1.15(0.01) & \\ |
558 |
skuang |
3570 |
\hline |
559 |
|
|
\end{tabular} |
560 |
|
|
\label{AuThermal} |
561 |
|
|
\end{center} |
562 |
|
|
\end{minipage} |
563 |
|
|
\end{table*} |
564 |
|
|
|
565 |
|
|
|
566 |
skuang |
3571 |
\begin{figure} |
567 |
|
|
\includegraphics[width=\linewidth]{interfaceDensity} |
568 |
|
|
\caption{Density profile for interfacial thermal conductivity |
569 |
|
|
simulation box.} |
570 |
|
|
\label{interfaceDensity} |
571 |
|
|
\end{figure} |
572 |
|
|
|
573 |
|
|
|
574 |
gezelter |
3524 |
\section{Acknowledgments} |
575 |
|
|
Support for this project was provided by the National Science |
576 |
|
|
Foundation under grant CHE-0848243. Computational time was provided by |
577 |
|
|
the Center for Research Computing (CRC) at the University of Notre |
578 |
|
|
Dame. \newpage |
579 |
|
|
|
580 |
|
|
\bibliographystyle{jcp2} |
581 |
|
|
\bibliography{nivsRnemd} |
582 |
|
|
\end{doublespace} |
583 |
|
|
\end{document} |
584 |
|
|
|