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 |
3527 |
the simulation cell.\cite{MullerPlathe:1997xw,Muller-Plathe:1999ek} 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 |
|
|
Recently, Tenney and Maginn have discovered some problems with the |
95 |
|
|
original RNEMD swap technique. Notably, large momentum fluxes |
96 |
|
|
(equivalent to frequent momentum swaps between the slabs) can result |
97 |
|
|
in "notched", "peaked" and generally non-thermal momentum |
98 |
|
|
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 |
|
|
\vec{v}_i \leftarrow \left( \begin{array}{c} |
127 |
|
|
x \\ |
128 |
|
|
y \\ |
129 |
|
|
z \\ |
130 |
|
|
\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 |
|
|
\vec{v}_j \leftarrow \left( \begin{array}{c} |
137 |
|
|
x^\prime \\ |
138 |
|
|
y^\prime \\ |
139 |
|
|
z^\prime \\ |
140 |
|
|
\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 |
|
|
\begin{equation} |
151 |
|
|
\begin{array}{rcl} |
152 |
skuang |
3528 |
P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
153 |
|
|
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha \\ |
154 |
gezelter |
3524 |
\end{array} |
155 |
|
|
\label{eq:momentumdef} |
156 |
|
|
\end{equation} |
157 |
skuang |
3528 |
Therefore, for each of the three directions, the hot scaling |
158 |
|
|
parameters are a simple function of the cold scaling parameters and |
159 |
gezelter |
3524 |
the instantaneous linear momentum in each of the two slabs. |
160 |
|
|
\begin{equation} |
161 |
skuang |
3528 |
\alpha^\prime = 1 + (1 - \alpha) p_\alpha |
162 |
gezelter |
3524 |
\label{eq:hotcoldscaling} |
163 |
|
|
\end{equation} |
164 |
skuang |
3528 |
where |
165 |
|
|
\begin{equation} |
166 |
|
|
p_\alpha = \frac{P_c^\alpha}{P_h^\alpha} |
167 |
|
|
\end{equation} |
168 |
|
|
for convenience. |
169 |
gezelter |
3524 |
|
170 |
|
|
Conservation of total energy also places constraints on the scaling: |
171 |
|
|
\begin{equation} |
172 |
|
|
\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
173 |
skuang |
3528 |
\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha. |
174 |
gezelter |
3524 |
\end{equation} |
175 |
|
|
where the kinetic energies, $K_h^\alpha$ and $K_c^\alpha$, are computed |
176 |
|
|
for each of the three directions in a similar manner to the linear momenta |
177 |
|
|
(Eq. \ref{eq:momentumdef}). Substituting in the expressions for the |
178 |
skuang |
3528 |
hot scaling parameters ($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), |
179 |
|
|
we obtain the {\it constraint ellipsoid equation}: |
180 |
gezelter |
3524 |
\begin{equation} |
181 |
skuang |
3528 |
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0, |
182 |
gezelter |
3524 |
\label{eq:constraintEllipsoid} |
183 |
|
|
\end{equation} |
184 |
|
|
where the constants are obtained from the instantaneous values of the |
185 |
|
|
linear momenta and kinetic energies for the hot and cold slabs, |
186 |
|
|
\begin{equation} |
187 |
skuang |
3528 |
\begin{array}{rcl} |
188 |
|
|
a_\alpha & = &\left(K_c^\alpha + K_h^\alpha |
189 |
|
|
\left(p_\alpha\right)^2\right) \\ |
190 |
|
|
b_\alpha & = & -2 K_h^\alpha p_\alpha \left( 1 + p_\alpha\right) \\ |
191 |
|
|
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha \\ |
192 |
|
|
\end{array} |
193 |
gezelter |
3524 |
\label{eq:constraintEllipsoidConsts} |
194 |
|
|
\end{equation} |
195 |
skuang |
3528 |
This ellipsoid equation defines the set of cold slab scaling |
196 |
|
|
parameters which can be applied while preserving both linear momentum |
197 |
skuang |
3530 |
in all three directions as well as kinetic energy. |
198 |
gezelter |
3524 |
|
199 |
|
|
The goal of using velocity scaling variables is to transfer linear |
200 |
|
|
momentum or kinetic energy from the cold slab to the hot slab. If the |
201 |
|
|
hot and cold slabs are separated along the z-axis, the energy flux is |
202 |
skuang |
3528 |
given simply by the decrease in kinetic energy of the cold bin: |
203 |
gezelter |
3524 |
\begin{equation} |
204 |
skuang |
3528 |
(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z |
205 |
gezelter |
3524 |
\end{equation} |
206 |
|
|
The expression for the energy flux can be re-written as another |
207 |
|
|
ellipsoid centered on $(x,y,z) = 0$: |
208 |
|
|
\begin{equation} |
209 |
skuang |
3529 |
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) |
210 |
gezelter |
3524 |
\label{eq:fluxEllipsoid} |
211 |
|
|
\end{equation} |
212 |
skuang |
3529 |
The spatial extent of the {\it flux ellipsoid equation} is governed |
213 |
|
|
both by a targetted value, $J_z$ as well as the instantaneous values of the |
214 |
skuang |
3530 |
kinetic energy components in the cold bin. |
215 |
gezelter |
3524 |
|
216 |
|
|
To satisfy an energetic flux as well as the conservation constraints, |
217 |
|
|
it is sufficient to determine the points ${x,y,z}$ which lie on both |
218 |
|
|
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
219 |
|
|
flux ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
220 |
skuang |
3528 |
the two ellipsoids in 3-dimensional space. |
221 |
gezelter |
3524 |
|
222 |
|
|
One may also define momentum flux (say along the x-direction) as: |
223 |
|
|
\begin{equation} |
224 |
skuang |
3528 |
(1-x) P_c^x = j_z(p_x) |
225 |
skuang |
3531 |
\label{eq:fluxPlane} |
226 |
gezelter |
3524 |
\end{equation} |
227 |
skuang |
3531 |
The above {\it flux equation} is essentially a plane which is |
228 |
|
|
perpendicular to the x-axis, with its position governed both by a |
229 |
|
|
targetted value, $j_z(p_x)$ as well as the instantaneous value of the |
230 |
|
|
momentum along the x-direction. |
231 |
gezelter |
3524 |
|
232 |
skuang |
3531 |
Similarly, to satisfy a momentum flux as well as the conservation |
233 |
|
|
constraints, it is sufficient to determine the points ${x,y,z}$ which |
234 |
|
|
lie on both the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) |
235 |
|
|
and the flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of |
236 |
|
|
an ellipsoid and a plane in 3-dimensional space. |
237 |
gezelter |
3524 |
|
238 |
skuang |
3531 |
To summarize, by solving respective equation sets, one can determine |
239 |
|
|
possible sets of scaling variables for cold slab. And corresponding |
240 |
|
|
sets of scaling variables for hot slab can be determine as well. |
241 |
gezelter |
3524 |
|
242 |
skuang |
3531 |
The following problem will be choosing an optimal set of scaling |
243 |
|
|
variables among the possible sets. Although this method is inherently |
244 |
|
|
non-isotropic, the goal is still to maintain the system as isotropic |
245 |
|
|
as possible. Under this consideration, one would like the kinetic |
246 |
|
|
energies in different directions could become as close as each other |
247 |
|
|
after each scaling. Simultaneously, one would also like each scaling |
248 |
|
|
as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
249 |
|
|
large perturbation to the system. Therefore, one approach to obtain the |
250 |
|
|
scaling variables would be constructing an criteria function, with |
251 |
|
|
constraints as above equation sets, and solving the function's minimum |
252 |
|
|
by method like Lagrange multipliers. |
253 |
gezelter |
3524 |
|
254 |
skuang |
3531 |
In order to save computation time, we have a different approach to a |
255 |
|
|
relatively good set of scaling variables with much less calculation |
256 |
|
|
than above. Here is the detail of our simplification of the problem. |
257 |
gezelter |
3524 |
|
258 |
skuang |
3531 |
In the case of kinetic energy transfer, we impose another constraint |
259 |
|
|
${x = y}$, into the equation sets. Consequently, there are two |
260 |
|
|
variables left. And now one only needs to solve a set of two {\it |
261 |
|
|
ellipses equations}. This problem would be transformed into solving |
262 |
|
|
one quartic equation for one of the two variables. There are known |
263 |
|
|
generic methods that solve real roots of quartic equations. Then one |
264 |
|
|
can determine the other variable and obtain sets of scaling |
265 |
|
|
variables. Among these sets, one can apply the above criteria to |
266 |
|
|
choose the best set, while much faster with only a few sets to choose. |
267 |
|
|
|
268 |
|
|
In the case of momentum flux transfer, we impose another constraint to |
269 |
|
|
set the kinetic energy transfer as zero. In another word, we apply |
270 |
|
|
Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. After that, with one |
271 |
|
|
variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar set |
272 |
|
|
of equations on the above kinetic energy transfer problem. Therefore, |
273 |
|
|
an approach similar to the above would be sufficient for this as well. |
274 |
|
|
|
275 |
|
|
\section{Computational Details} |
276 |
skuang |
3532 |
Our simulation consists of a series of systems. |
277 |
skuang |
3531 |
|
278 |
skuang |
3532 |
A Lennard-Jones fluid system was built and tested first. In order to |
279 |
|
|
compare our method with swapping RNEMD, a series of simulations were |
280 |
|
|
performed to calculate the shear viscosity and thermal conductivity of |
281 |
|
|
argon. 2592 atoms were in a orthorhombic cell, which was ${10.06 \sigma |
282 |
|
|
\times 10.06 \sigma \times 30.18 \sigma}$ by size. The reduced density |
283 |
|
|
${\rho^* = \rho\sigma^3}$ was thus 0.849, which enabled direct |
284 |
|
|
comparison between our results and others. |
285 |
|
|
|
286 |
|
|
For shear viscosity calculation, the reduced temperature was ${T^* = |
287 |
|
|
k_B T / \epsilon = 0.72}$. Simulations were run in microcanonical |
288 |
|
|
ensemble (NVE). For the swapping part, Muller-Plathe's algorithm was |
289 |
|
|
adopted.\cite{ISI:000080382700030} The simulation box was under |
290 |
|
|
periodic boundary condition, and devided into 20 slabs. In each swap, |
291 |
|
|
the top slab ${(n = 0)}$ exchange the most negative $x$ momentum with the |
292 |
|
|
most positive $x$ momentum in the center slab ${(n = N/2)}$. Referring |
293 |
|
|
to Tenney {\it et al.}\cite{tenneyANDmaginn}, a series of swapping |
294 |
|
|
frequency were chosen. Corresponding to each result from swapping |
295 |
|
|
RNEMD, scaling RNEMD simulations were run with the target momentum |
296 |
|
|
flux parameter set to produce a similar momentum flux and shear |
297 |
|
|
rate. Furthermore, various scaling frequencies and corresponding flux |
298 |
|
|
can be tested for one swapping rate. |
299 |
|
|
|
300 |
|
|
After each simulation, the shear viscosities were calculated in |
301 |
|
|
reduced unit. |
302 |
|
|
|
303 |
gezelter |
3524 |
\section{Acknowledgments} |
304 |
|
|
Support for this project was provided by the National Science |
305 |
|
|
Foundation under grant CHE-0848243. Computational time was provided by |
306 |
|
|
the Center for Research Computing (CRC) at the University of Notre |
307 |
|
|
Dame. \newpage |
308 |
|
|
|
309 |
|
|
\bibliographystyle{jcp2} |
310 |
|
|
\bibliography{nivsRnemd} |
311 |
|
|
\end{doublespace} |
312 |
|
|
\end{document} |
313 |
|
|
|