38 |
|
\begin{doublespace} |
39 |
|
|
40 |
|
\begin{abstract} |
41 |
< |
|
41 |
> |
We present a new method for introducing stable non-equilibrium |
42 |
> |
velocity and temperature distributions in molecular dynamics |
43 |
> |
simulations of heterogeneous systems. This method extends some |
44 |
> |
earlier Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods |
45 |
> |
which use momentum exchange swapping moves that can create |
46 |
> |
non-thermal velocity distributions (and which are difficult to use |
47 |
> |
for interfacial calculations). By using non-isotropic velocity |
48 |
> |
scaling (NIVS) on the molecules in specific regions of a system, it |
49 |
> |
is possible to impose momentum or thermal flux between regions of a |
50 |
> |
simulation and stable thermal and momentum gradients can then be |
51 |
> |
established. The scaling method we have developed conserves the |
52 |
> |
total linear momentum and total energy of the system. To test the |
53 |
> |
methods, we have computed the thermal conductivity of model liquid |
54 |
> |
and solid systems as well as the interfacial thermal conductivity of |
55 |
> |
a metal-water interface. We find that the NIVS-RNEMD improves the |
56 |
> |
problematic velocity distributions that develop in other RNEMD |
57 |
> |
methods. |
58 |
|
\end{abstract} |
59 |
|
|
60 |
|
\newpage |
65 |
|
% BODY OF TEXT |
66 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
67 |
|
|
52 |
– |
|
53 |
– |
|
68 |
|
\section{Introduction} |
69 |
|
The original formulation of Reverse Non-equilibrium Molecular Dynamics |
70 |
|
(RNEMD) obtains transport coefficients (thermal conductivity and shear |
71 |
|
viscosity) in a fluid by imposing an artificial momentum flux between |
72 |
|
two thin parallel slabs of material that are spatially separated in |
73 |
|
the simulation cell.\cite{MullerPlathe:1997xw,ISI:000080382700030} The |
74 |
< |
artificial flux is typically created by periodically ``swapping'' either |
75 |
< |
the entire momentum vector $\vec{p}$ or single components of this |
76 |
< |
vector ($p_x$) between molecules in each of the two slabs. If the two |
77 |
< |
slabs are separated along the $z$ coordinate, the imposed flux is either |
78 |
< |
directional ($j_z(p_x)$) or isotropic ($J_z$), and the response of a |
79 |
< |
simulated system to the imposed momentum flux will typically be a |
80 |
< |
velocity or thermal gradient (Fig. \ref{thermalDemo}). The transport |
81 |
< |
coefficients (shear viscosity and thermal conductivity) are easily |
82 |
< |
obtained by assuming linear response of the system, |
74 |
> |
artificial flux is typically created by periodically ``swapping'' |
75 |
> |
either the entire momentum vector $\vec{p}$ or single components of |
76 |
> |
this vector ($p_x$) between molecules in each of the two slabs. If |
77 |
> |
the two slabs are separated along the $z$ coordinate, the imposed flux |
78 |
> |
is either directional ($j_z(p_x)$) or isotropic ($J_z$), and the |
79 |
> |
response of a simulated system to the imposed momentum flux will |
80 |
> |
typically be a velocity or thermal gradient (Fig. \ref{thermalDemo}). |
81 |
> |
The shear viscosity ($\eta$) and thermal conductivity ($\lambda$) are |
82 |
> |
easily obtained by assuming linear response of the system, |
83 |
|
\begin{eqnarray} |
84 |
|
j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
85 |
|
J_z & = & \lambda \frac{\partial T}{\partial z} |
87 |
|
RNEMD has been widely used to provide computational estimates of thermal |
88 |
|
conductivities and shear viscosities in a wide range of materials, |
89 |
|
from liquid copper to monatomic liquids to molecular fluids |
90 |
< |
(e.g. ionic liquids).\cite{ISI:000246190100032} |
90 |
> |
(e.g. ionic liquids).\cite{Bedrov:2000-1,Bedrov:2000,Muller-Plathe:2002,ISI:000184808400018,ISI:000231042800044,Maginn:2007,Muller-Plathe:2008,ISI:000258460400020,ISI:000258840700015,ISI:000261835100054} |
91 |
|
|
92 |
|
\begin{figure} |
93 |
|
\includegraphics[width=\linewidth]{thermalDemo} |
94 |
< |
\caption{Demostration of thermal gradient estalished by RNEMD |
95 |
< |
method. Physical thermal flow directs from high temperature region |
96 |
< |
to low temperature region. Unphysical thermal transfer counteracts |
97 |
< |
it and maintains a steady thermal gradient.} |
94 |
> |
\caption{RNEMD methods impose an unphysical transfer of momentum or |
95 |
> |
kinetic energy between a ``hot'' slab and a ``cold'' slab in the |
96 |
> |
simulation box. The molecular system responds to this imposed flux |
97 |
> |
by generating a momentum or temperature gradient. The slope of the |
98 |
> |
gradient can then be used to compute transport properties (e.g. |
99 |
> |
shear viscosity and thermal conductivity).} |
100 |
|
\label{thermalDemo} |
101 |
|
\end{figure} |
102 |
|
|
103 |
< |
RNEMD is preferable in many ways to the forward NEMD methods because |
104 |
< |
it imposes what is typically difficult to measure (a flux or stress) |
105 |
< |
and it is typically much easier to compute momentum gradients or |
106 |
< |
strains (the response). For similar reasons, RNEMD is also preferable |
107 |
< |
to slowly-converging equilibrium methods for measuring thermal |
108 |
< |
conductivity and shear viscosity (using Green-Kubo relations or the |
109 |
< |
Helfand moment approach of Viscardy {\it et |
103 |
> |
RNEMD is preferable in many ways to the forward NEMD methods |
104 |
> |
[CITATIONS NEEDED] because it imposes what is typically difficult to measure |
105 |
> |
(a flux or stress) and it is typically much easier to compute momentum |
106 |
> |
gradients or strains (the response). For similar reasons, RNEMD is |
107 |
> |
also preferable to slowly-converging equilibrium methods for measuring |
108 |
> |
thermal conductivity and shear viscosity (using Green-Kubo relations |
109 |
> |
[CITATIONS NEEDED] or the Helfand moment approach of Viscardy {\it et |
110 |
|
al.}\cite{Viscardy:2007bh,Viscardy:2007lq}) because these rely on |
111 |
|
computing difficult to measure quantities. |
112 |
|
|
130 |
|
(conservation of linear momentum and total energy, compatibility with |
131 |
|
periodic boundary conditions) while establishing true thermal |
132 |
|
distributions in each of the two slabs. In the next section, we |
133 |
< |
develop the method for determining the scaling constraints. We then |
133 |
> |
present the method for determining the scaling constraints. We then |
134 |
|
test the method on both single component, multi-component, and |
135 |
|
non-isotropic mixtures and show that it is capable of providing |
136 |
|
reasonable estimates of the thermal conductivity and shear viscosity |
137 |
|
in these cases. |
138 |
|
|
139 |
|
\section{Methodology} |
140 |
< |
We retain the basic idea of Muller-Plathe's RNEMD method; the periodic |
141 |
< |
system is partitioned into a series of thin slabs along a particular |
140 |
> |
We retain the basic idea of M\"{u}ller-Plathe's RNEMD method; the |
141 |
> |
periodic system is partitioned into a series of thin slabs along one |
142 |
|
axis ($z$). One of the slabs at the end of the periodic box is |
143 |
|
designated the ``hot'' slab, while the slab in the center of the box |
144 |
|
is designated the ``cold'' slab. The artificial momentum flux will be |
146 |
|
hot slab. |
147 |
|
|
148 |
|
Rather than using momentum swaps, we use a series of velocity scaling |
149 |
< |
moves. For molecules $\{i\}$ located within the cold slab, |
149 |
> |
moves. For molecules $\{i\}$ located within the cold slab, |
150 |
|
\begin{equation} |
151 |
|
\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
152 |
|
x & 0 & 0 \\ |
166 |
|
\end{equation} |
167 |
|
|
168 |
|
Conservation of linear momentum in each of the three directions |
169 |
< |
($\alpha = x,y,z$) ties the values of the hot and cold bin scaling |
169 |
> |
($\alpha = x,y,z$) ties the values of the hot and cold scaling |
170 |
|
parameters together: |
171 |
|
\begin{equation} |
172 |
|
P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
200 |
|
similar manner to the linear momenta (Eq. \ref{eq:momentumdef}). |
201 |
|
Substituting in the expressions for the hot scaling parameters |
202 |
|
($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), we obtain the |
203 |
< |
{\it constraint ellipsoid equation}: |
203 |
> |
{\it constraint ellipsoid}: |
204 |
|
\begin{equation} |
205 |
|
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0 |
206 |
|
\label{eq:constraintEllipsoid} |
214 |
|
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
215 |
|
\label{eq:constraintEllipsoidConsts} |
216 |
|
\end{eqnarray} |
217 |
< |
This ellipsoid equation defines the set of cold slab scaling |
218 |
< |
parameters which can be applied while preserving both linear momentum |
219 |
< |
in all three directions as well as kinetic energy. |
217 |
> |
This ellipsoid (Eq. \ref{eq:constraintEllipsoid}) defines the set of |
218 |
> |
cold slab scaling parameters which can be applied while preserving |
219 |
> |
both linear momentum in all three directions as well as total kinetic |
220 |
> |
energy. |
221 |
|
|
222 |
|
The goal of using velocity scaling variables is to transfer linear |
223 |
|
momentum or kinetic energy from the cold slab to the hot slab. If the |
232 |
|
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 |
233 |
|
\label{eq:fluxEllipsoid} |
234 |
|
\end{equation} |
235 |
< |
The spatial extent of the {\it thermal flux ellipsoid equation} is |
236 |
< |
governed both by a targetted value, $J_z$ as well as the instantaneous |
237 |
< |
values of the kinetic energy components in the cold bin. |
235 |
> |
The spatial extent of the {\it thermal flux ellipsoid} is governed |
236 |
> |
both by a targetted value, $J_z$ as well as the instantaneous values |
237 |
> |
of the kinetic energy components in the cold bin. |
238 |
|
|
239 |
|
To satisfy an energetic flux as well as the conservation constraints, |
240 |
< |
it is sufficient to determine the points ${x,y,z}$ which lie on both |
241 |
< |
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
242 |
< |
flux ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
243 |
< |
the two ellipsoids in 3-dimensional space. |
240 |
> |
we must determine the points ${x,y,z}$ which lie on both the |
241 |
> |
constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the flux |
242 |
> |
ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of the |
243 |
> |
two ellipsoids in 3-dimensional space. |
244 |
|
|
245 |
|
\begin{figure} |
246 |
|
\includegraphics[width=\linewidth]{ellipsoids} |
253 |
|
\label{ellipsoids} |
254 |
|
\end{figure} |
255 |
|
|
256 |
< |
One may also define momentum flux (say along the $x$-direction) as: |
256 |
> |
One may also define {\it momentum} flux (say along the $x$-direction) as: |
257 |
|
\begin{equation} |
258 |
|
(1-x) P_c^x = j_z(p_x)\Delta t |
259 |
|
\label{eq:fluxPlane} |
260 |
|
\end{equation} |
261 |
< |
The above {\it momentum flux equation} is essentially a plane which is |
262 |
< |
perpendicular to the $x$-axis, with its position governed both by a |
263 |
< |
target value, $j_z(p_x)$ as well as the instantaneous value of the |
247 |
< |
momentum along the $x$-direction. |
261 |
> |
The above {\it momentum flux plane} is perpendicular to the $x$-axis, |
262 |
> |
with its position governed both by a target value, $j_z(p_x)$ as well |
263 |
> |
as the instantaneous value of the momentum along the $x$-direction. |
264 |
|
|
265 |
< |
Similarly, to satisfy a momentum flux as well as the conservation |
266 |
< |
constraints, it is sufficient to determine the points ${x,y,z}$ which |
267 |
< |
lie on both the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) |
268 |
< |
and the flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of |
269 |
< |
an ellipsoid and a plane in 3-dimensional space. |
265 |
> |
In order to satisfy a momentum flux as well as the conservation |
266 |
> |
constraints, we must determine the points ${x,y,z}$ which lie on both |
267 |
> |
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
268 |
> |
flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of an |
269 |
> |
ellipsoid and a plane in 3-dimensional space. |
270 |
|
|
271 |
< |
To summarize, by solving respective equation sets, one can determine |
272 |
< |
possible sets of scaling variables for cold slab. And corresponding |
273 |
< |
sets of scaling variables for hot slab can be determine as well. |
271 |
> |
In both the momentum and energy flux scenarios, valid scaling |
272 |
> |
parameters are arrived at by solving geometric intersection problems |
273 |
> |
in $x, y, z$ space in order to obtain cold slab scaling parameters. |
274 |
> |
Once the scaling variables for the cold slab are known, the hot slab |
275 |
> |
scaling has also been determined. |
276 |
|
|
277 |
+ |
|
278 |
|
The following problem will be choosing an optimal set of scaling |
279 |
|
variables among the possible sets. Although this method is inherently |
280 |
|
non-isotropic, the goal is still to maintain the system as isotropic |
282 |
|
energies in different directions could become as close as each other |
283 |
|
after each scaling. Simultaneously, one would also like each scaling |
284 |
|
as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
285 |
< |
large perturbation to the system. Therefore, one approach to obtain the |
286 |
< |
scaling variables would be constructing an criteria function, with |
285 |
> |
large perturbation to the system. Therefore, one approach to obtain |
286 |
> |
the scaling variables would be constructing an criteria function, with |
287 |
|
constraints as above equation sets, and solving the function's minimum |
288 |
|
by method like Lagrange multipliers. |
289 |
|
|
393 |
|
\subsection{ Water / Metal Thermal Conductivity} |
394 |
|
Another series of our simulation is the calculation of interfacial |
395 |
|
thermal conductivity of a Au/H$_2$O system. Respective calculations of |
396 |
< |
liquid water (SPC/E) and crystal gold (QSC) thermal conductivity were |
397 |
< |
performed and compared with current results to ensure the validity of |
398 |
< |
NIVS-RNEMD. After that, a mixture system was simulated. |
396 |
> |
liquid water (Extended Simple Point Charge model) and crystal gold |
397 |
> |
thermal conductivity were performed and compared with current results |
398 |
> |
to ensure the validity of NIVS-RNEMD. After that, a mixture system was |
399 |
> |
simulated. |
400 |
|
|
401 |
|
For thermal conductivity calculation of bulk water, a simulation box |
402 |
|
consisting of 1000 molecules were first equilibrated under ambient |
408 |
|
process was similar to Lennard-Jones fluid system. |
409 |
|
|
410 |
|
Thermal conductivity calculation of bulk crystal gold used a similar |
411 |
< |
protocol. The face centered cubic crystal simulation box consists of |
411 |
> |
protocol. Two types of force field parameters, Embedded Atom Method |
412 |
> |
(EAM) and Quantum Sutten-Chen (QSC) force field were used |
413 |
> |
respectively. The face-centered cubic crystal simulation box consists of |
414 |
|
2880 Au atoms. The lattice was first allowed volume change to relax |
415 |
|
under ambient temperature and pressure. Equilibrations in canonical and |
416 |
|
microcanonical ensemble were followed in order. With the simulation |
485 |
|
\end{figure} |
486 |
|
|
487 |
|
During these simulations, molecule velocities were recorded in 1000 of |
488 |
< |
all the snapshots. These velocity data were used to produce histograms |
489 |
< |
of velocity and speed distribution in different slabs. From these |
490 |
< |
histograms, it is observed that with increasing unphysical kinetic |
491 |
< |
energy flux, speed and velocity distribution of molecules in slabs |
492 |
< |
where swapping occured could deviate from Maxwell-Boltzmann |
493 |
< |
distribution. Figure \ref{histSwap} indicates how these distributions |
494 |
< |
deviate from ideal condition. In high temperature slabs, probability |
495 |
< |
density in low speed is confidently smaller than ideal distribution; |
496 |
< |
in low temperature slabs, probability density in high speed is smaller |
497 |
< |
than ideal. This phenomenon is observable even in our relatively low |
498 |
< |
swapping rate simulations. And this deviation could also leads to |
499 |
< |
deviation of distribution of velocity in various dimensions. One |
500 |
< |
feature of these deviated distribution is that in high temperature |
501 |
< |
slab, the ideal Gaussian peak was changed into a relatively flat |
502 |
< |
plateau; while in low temperature slab, that peak appears sharper. |
488 |
> |
all the snapshots of one single data collection process. These |
489 |
> |
velocity data were used to produce histograms of velocity and speed |
490 |
> |
distribution in different slabs. From these histograms, it is observed |
491 |
> |
that under relatively high unphysical kinetic energy flux, speed and |
492 |
> |
velocity distribution of molecules in slabs where swapping occured |
493 |
> |
could deviate from Maxwell-Boltzmann distribution. Figure |
494 |
> |
\ref{histSwap} illustrates how these distributions deviate from an |
495 |
> |
ideal distribution. In high temperature slab, probability density in |
496 |
> |
low speed is confidently smaller than ideal curve fit; in low |
497 |
> |
temperature slab, probability density in high speed is smaller than |
498 |
> |
ideal, while larger than ideal in low speed. This phenomenon is more |
499 |
> |
obvious in our high swapping rate simulations. And this deviation |
500 |
> |
could also leads to deviation of distribution of velocity in various |
501 |
> |
dimensions. One feature of these deviated distribution is that in high |
502 |
> |
temperature slab, the ideal Gaussian peak was changed into a |
503 |
> |
relatively flat plateau; while in low temperature slab, that peak |
504 |
> |
appears sharper. This problem is rooted in the mechanism of the |
505 |
> |
swapping method. Continually depleting low (high) speed particles in |
506 |
> |
the high (low) temperature slab could not be complemented by |
507 |
> |
diffusions of low (high) speed particles from neighbor slabs, unless |
508 |
> |
in suffciently low swapping rate. Simutaneously, surplus low speed |
509 |
> |
particles in the low temperature slab do not have sufficient time to |
510 |
> |
diffuse to neighbor slabs. However, thermal exchange rate should reach |
511 |
> |
a minimum level to produce an observable thermal gradient under noise |
512 |
> |
interference. Consequently, swapping RNEMD has a relatively narrow |
513 |
> |
choice of swapping rate to satisfy these above restrictions. |
514 |
|
|
515 |
|
\begin{figure} |
516 |
|
\includegraphics[width=\linewidth]{histSwap} |
517 |
< |
\caption{Speed distribution for thermal conductivity using swapping RNEMD.} |
517 |
> |
\caption{Speed distribution for thermal conductivity using swapping |
518 |
> |
RNEMD. Shown is from the simulation with 250 fs exchange interval.} |
519 |
|
\label{histSwap} |
520 |
|
\end{figure} |
521 |
|
|
522 |
+ |
Comparatively, NIVS-RNEMD has a speed distribution closer to the ideal |
523 |
+ |
curve fit (Figure \ref{histScale}). Essentially, after scaling, a |
524 |
+ |
Gaussian distribution function would remain Gaussian. Although a |
525 |
+ |
single scaling is non-isotropic in all three dimensions, our scaling |
526 |
+ |
coefficient criteria could help maintian the scaling region as |
527 |
+ |
isotropic as possible. On the other hand, scaling coefficients are |
528 |
+ |
preferred to be as close to 1 as possible, which also helps minimize |
529 |
+ |
the difference among different dimensions. This is possible if scaling |
530 |
+ |
interval and one-time thermal transfer energy are well |
531 |
+ |
chosen. Consequently, NIVS-RNEMD is able to impose an unphysical |
532 |
+ |
thermal flux as the previous RNEMD method without large perturbation |
533 |
+ |
to the distribution of velocity and speed in the exchange regions. |
534 |
+ |
|
535 |
|
\begin{figure} |
536 |
|
\includegraphics[width=\linewidth]{histScale} |
537 |
< |
\caption{Speed distribution for thermal conductivity using scaling RNEMD.} |
537 |
> |
\caption{Speed distribution for thermal conductivity using scaling |
538 |
> |
RNEMD. Shown is from the simulation with an equilvalent thermal flux |
539 |
> |
as an 250 fs exchange interval swapping simulation.} |
540 |
|
\label{histScale} |
541 |
|
\end{figure} |
542 |
|
|
543 |
|
\subsubsection{SPC/E Water} |
544 |
|
Our results of SPC/E water thermal conductivity are comparable to |
545 |
|
Bedrov {\it et al.}\cite{ISI:000090151400044}, which employed the |
546 |
< |
previous swapping RNEMD method for their calculation. Our simulations |
547 |
< |
were able to produce a similar temperature gradient to their |
548 |
< |
system. However, the average temperature of our system is 300K, while |
549 |
< |
theirs is 318K, which would be attributed for part of the difference |
550 |
< |
between the two series of results. Both methods yields values in |
551 |
< |
agreement with experiment. And this shows the applicability of our |
552 |
< |
method to multi-atom molecular system. |
546 |
> |
previous swapping RNEMD method for their calculation. Bedrov {\it et |
547 |
> |
al.}\cite{ISI:000090151400044} argued that exchange of the molecule |
548 |
> |
center-of-mass velocities instead of single atom velocities in a |
549 |
> |
molecule conserves the total kinetic energy and linear momentum. This |
550 |
> |
principle is adopted in our simulations. Scaling is applied to the |
551 |
> |
velocities of the rigid bodies of SPC/E model water molecules, instead |
552 |
> |
of each hydrogen and oxygen atoms in relevant water molecules. As |
553 |
> |
shown in Figure \ref{spceGrad}, temperature gradients were established |
554 |
> |
similar to their system. However, the average temperature of our |
555 |
> |
system is 300K, while theirs is 318K, which would be attributed for |
556 |
> |
part of the difference between the final calculation results (Table |
557 |
> |
\ref{spceThermal}). Both methods yields values in agreement with |
558 |
> |
experiment. And this shows the applicability of our method to |
559 |
> |
multi-atom molecular system. |
560 |
|
|
561 |
|
\begin{figure} |
562 |
|
\includegraphics[width=\linewidth]{spceGrad} |
589 |
|
\end{table*} |
590 |
|
|
591 |
|
\subsubsection{Crystal Gold} |
592 |
< |
Our results of gold thermal conductivity used QSC force field are |
593 |
< |
shown in Table \ref{AuThermal}. Although our calculation is smaller |
594 |
< |
than experimental value by an order of more than 100, this difference |
595 |
< |
is mainly attributed to the lack of electron interaction |
596 |
< |
representation in our force field parameters. Richardson {\it et |
597 |
< |
al.}\cite{ISI:A1992HX37800010} used similar force field parameters |
598 |
< |
in their metal thermal conductivity calculations. The EMD method they |
599 |
< |
employed in their simulations produced comparable results to |
600 |
< |
ours. Therefore, it is confident to conclude that NIVS-RNEMD is |
601 |
< |
applicable to metal force field system. |
592 |
> |
Our results of gold thermal conductivity using two force fields are |
593 |
> |
shown separately in Table \ref{qscThermal} and \ref{eamThermal}. In |
594 |
> |
these calculations,the end and middle slabs were excluded in thermal |
595 |
> |
gradient regession and only used as heat source and drain in the |
596 |
> |
systems. Our yielded values using EAM force field are slightly larger |
597 |
> |
than those using QSC force field. However, both series are |
598 |
> |
significantly smaller than experimental value by an order of more than |
599 |
> |
100. It has been verified that this difference is mainly attributed to |
600 |
> |
the lack of electron interaction representation in these force field |
601 |
> |
parameters. Richardson {\it et al.}\cite{Clancy:1992} used EAM |
602 |
> |
force field parameters in their metal thermal conductivity |
603 |
> |
calculations. The Non-Equilibrium MD method they employed in their |
604 |
> |
simulations produced comparable results to ours. As Zhang {\it et |
605 |
> |
al.}\cite{ISI:000231042800044} stated, thermal conductivity values |
606 |
> |
are influenced mainly by force field. Therefore, it is confident to |
607 |
> |
conclude that NIVS-RNEMD is applicable to metal force field system. |
608 |
|
|
609 |
|
\begin{figure} |
610 |
|
\includegraphics[width=\linewidth]{AuGrad} |
611 |
< |
\caption{Temperature gradients for crystal gold thermal conductivity.} |
611 |
> |
\caption{Temperature gradients for thermal conductivity calculation of |
612 |
> |
crystal gold using QSC force field.} |
613 |
|
\label{AuGrad} |
614 |
|
\end{figure} |
615 |
|
|
618 |
|
\begin{center} |
619 |
|
|
620 |
|
\caption{Calculation results for thermal conductivity of crystal gold |
621 |
< |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
622 |
< |
calculations in parentheses. } |
621 |
> |
using QSC force field at ${\langle T\rangle}$ = 300K at various |
622 |
> |
thermal exchange rates. Errors of calculations in parentheses. } |
623 |
|
|
624 |
< |
\begin{tabular}{ccc} |
624 |
> |
\begin{tabular}{cc} |
625 |
|
\hline |
626 |
|
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
564 |
– |
& This work & Previous simulations\cite{ISI:A1992HX37800010} \\ |
627 |
|
\hline |
628 |
< |
1.44 & 1.10(0.01) & \\ |
629 |
< |
2.86 & 1.08(0.02) & \\ |
630 |
< |
5.14 & 1.15(0.01) & \\ |
628 |
> |
1.44 & 1.10(0.01)\\ |
629 |
> |
2.86 & 1.08(0.02)\\ |
630 |
> |
5.14 & 1.15(0.01)\\ |
631 |
|
\hline |
632 |
|
\end{tabular} |
633 |
< |
\label{AuThermal} |
633 |
> |
\label{qscThermal} |
634 |
> |
\end{center} |
635 |
> |
\end{minipage} |
636 |
> |
\end{table*} |
637 |
> |
|
638 |
> |
\begin{figure} |
639 |
> |
\includegraphics[width=\linewidth]{eamGrad} |
640 |
> |
\caption{Temperature gradients for thermal conductivity calculation of |
641 |
> |
crystal gold using EAM force field.} |
642 |
> |
\label{eamGrad} |
643 |
> |
\end{figure} |
644 |
> |
|
645 |
> |
\begin{table*} |
646 |
> |
\begin{minipage}{\linewidth} |
647 |
> |
\begin{center} |
648 |
> |
|
649 |
> |
\caption{Calculation results for thermal conductivity of crystal gold |
650 |
> |
using EAM force field at ${\langle T\rangle}$ = 300K at various |
651 |
> |
thermal exchange rates. Errors of calculations in parentheses. } |
652 |
> |
|
653 |
> |
\begin{tabular}{cc} |
654 |
> |
\hline |
655 |
> |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
656 |
> |
\hline |
657 |
> |
1.24 & 1.24(0.06)\\ |
658 |
> |
2.06 & 1.37(0.04)\\ |
659 |
> |
2.55 & 1.41(0.03)\\ |
660 |
> |
\hline |
661 |
> |
\end{tabular} |
662 |
> |
\label{eamThermal} |
663 |
|
\end{center} |
664 |
|
\end{minipage} |
665 |
|
\end{table*} |
666 |
|
|
667 |
+ |
|
668 |
|
\subsection{Interfaciel Thermal Conductivity} |
669 |
< |
After valid simulations of homogeneous water and gold systems using |
670 |
< |
NIVS-RNEMD method, calculation of gold/water interfacial thermal |
671 |
< |
conductivity was followed. It is found out that the interfacial |
672 |
< |
conductance is low due to a hydrophobic surface in our system. Figure |
673 |
< |
\ref{interfaceDensity} demonstrates this observance. Consequently, our |
674 |
< |
reported results (Table \ref{interfaceRes}) are of two orders of |
675 |
< |
magnitude smaller than our calculations on homogeneous systems. |
669 |
> |
After simulations of homogeneous water and gold systems using |
670 |
> |
NIVS-RNEMD method were proved valid, calculation of gold/water |
671 |
> |
interfacial thermal conductivity was followed. It is found out that |
672 |
> |
the low interfacial conductance is probably due to the hydrophobic |
673 |
> |
surface in our system. Figure \ref{interfaceDensity} demonstrates mass |
674 |
> |
density change along $z$-axis, which is perpendicular to the |
675 |
> |
gold/water interface. It is observed that water density significantly |
676 |
> |
decreases when approaching the surface. Under this low thermal |
677 |
> |
conductance, both gold and water phase have sufficient time to |
678 |
> |
eliminate temperature difference inside respectively (Figure |
679 |
> |
\ref{interfaceGrad}). With indistinguishable temperature difference |
680 |
> |
within respective phase, it is valid to assume that the temperature |
681 |
> |
difference between gold and water on surface would be approximately |
682 |
> |
the same as the difference between the gold and water phase. This |
683 |
> |
assumption enables convenient calculation of $G$ using |
684 |
> |
Eq. \ref{interfaceCalc} instead of measuring temperatures of thin |
685 |
> |
layer of water and gold close enough to surface, which would have |
686 |
> |
greater fluctuation and lower accuracy. Reported results (Table |
687 |
> |
\ref{interfaceRes}) are of two orders of magnitude smaller than our |
688 |
> |
calculations on homogeneous systems, and thus have larger relative |
689 |
> |
errors than our calculation results on homogeneous systems. |
690 |
|
|
691 |
|
\begin{figure} |
692 |
|
\includegraphics[width=\linewidth]{interfaceDensity} |
693 |
|
\caption{Density profile for interfacial thermal conductivity |
694 |
< |
simulation box.} |
694 |
> |
simulation box. Significant water density decrease is observed on |
695 |
> |
gold surface.} |
696 |
|
\label{interfaceDensity} |
697 |
|
\end{figure} |
698 |
|
|
699 |
|
\begin{figure} |
700 |
|
\includegraphics[width=\linewidth]{interfaceGrad} |
701 |
|
\caption{Temperature profiles for interfacial thermal conductivity |
702 |
< |
simulation box.} |
702 |
> |
simulation box. Temperatures of different slabs in the same phase |
703 |
> |
show no significant difference.} |
704 |
|
\label{interfaceGrad} |
705 |
|
\end{figure} |
706 |
|
|
798 |
|
|
799 |
|
\section{Conclusions} |
800 |
|
NIVS-RNEMD simulation method is developed and tested on various |
801 |
< |
systems. Simulation results demonstrate its validity of thermal |
802 |
< |
conductivity calculations. NIVS-RNEMD improves non-Boltzmann-Maxwell |
803 |
< |
distributions existing in previous RNEMD methods, and extends its |
804 |
< |
applicability to interfacial systems. NIVS-RNEMD has also limited |
805 |
< |
application on shear viscosity calculations, but under high momentum |
806 |
< |
flux, it could cause temperature difference among different |
807 |
< |
dimensions. Modification is necessary to extend the applicability of |
808 |
< |
NIVS-RNEMD in shear viscosity calculations. |
801 |
> |
systems. Simulation results demonstrate its validity in thermal |
802 |
> |
conductivity calculations, from Lennard-Jones fluid to multi-atom |
803 |
> |
molecule like water and metal crystals. NIVS-RNEMD improves |
804 |
> |
non-Boltzmann-Maxwell distributions, which exist in previous RNEMD |
805 |
> |
methods. Furthermore, it develops a valid means for unphysical thermal |
806 |
> |
transfer between different species of molecules, and thus extends its |
807 |
> |
applicability to interfacial systems. Our calculation of gold/water |
808 |
> |
interfacial thermal conductivity demonstrates this advantage over |
809 |
> |
previous RNEMD methods. NIVS-RNEMD has also limited application on |
810 |
> |
shear viscosity calculations, but could cause temperature difference |
811 |
> |
among different dimensions under high momentum flux. Modification is |
812 |
> |
necessary to extend the applicability of NIVS-RNEMD in shear viscosity |
813 |
> |
calculations. |
814 |
|
|
815 |
|
\section{Acknowledgments} |
816 |
|
Support for this project was provided by the National Science |
818 |
|
the Center for Research Computing (CRC) at the University of Notre |
819 |
|
Dame. \newpage |
820 |
|
|
821 |
< |
\bibliographystyle{jcp2} |
821 |
> |
\bibliographystyle{aip} |
822 |
|
\bibliography{nivsRnemd} |
823 |
+ |
|
824 |
|
\end{doublespace} |
825 |
|
\end{document} |
826 |
|
|