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\begin{document} |
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\begin{doublespace} |
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|
| 45 |
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\begin{abstract} |
| 46 |
< |
|
| 46 |
> |
We present a new method for introducing stable non-equilibrium |
| 47 |
> |
velocity and temperature gradients in molecular dynamics simulations |
| 48 |
> |
of heterogeneous systems. This method extends earlier Reverse |
| 49 |
> |
Non-Equilibrium Molecular Dynamics (RNEMD) methods which use |
| 50 |
> |
momentum exchange swapping moves. The standard swapping moves can |
| 51 |
> |
create non-thermal velocity distributions and are difficult to use |
| 52 |
> |
for interfacial calculations. By using non-isotropic velocity |
| 53 |
> |
scaling (NIVS) on the molecules in specific regions of a system, it |
| 54 |
> |
is possible to impose momentum or thermal flux between regions of a |
| 55 |
> |
simulation while conserving the linear momentum and total energy of |
| 56 |
> |
the system. To test the methods, we have computed the thermal |
| 57 |
> |
conductivity of model liquid and solid systems as well as the |
| 58 |
> |
interfacial thermal conductivity of a metal-water interface. We |
| 59 |
> |
find that the NIVS-RNEMD improves the problematic velocity |
| 60 |
> |
distributions that develop in other RNEMD methods. |
| 61 |
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\end{abstract} |
| 62 |
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|
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\newpage |
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% BODY OF TEXT |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 70 |
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|
| 52 |
– |
|
| 53 |
– |
|
| 71 |
|
\section{Introduction} |
| 72 |
|
The original formulation of Reverse Non-equilibrium Molecular Dynamics |
| 73 |
|
(RNEMD) obtains transport coefficients (thermal conductivity and shear |
| 74 |
|
viscosity) in a fluid by imposing an artificial momentum flux between |
| 75 |
|
two thin parallel slabs of material that are spatially separated in |
| 76 |
|
the simulation cell.\cite{MullerPlathe:1997xw,ISI:000080382700030} The |
| 77 |
< |
artificial flux is typically created by periodically ``swapping'' either |
| 78 |
< |
the entire momentum vector $\vec{p}$ or single components of this |
| 79 |
< |
vector ($p_x$) between molecules in each of the two slabs. If the two |
| 80 |
< |
slabs are separated along the $z$ coordinate, the imposed flux is either |
| 81 |
< |
directional ($j_z(p_x)$) or isotropic ($J_z$), and the response of a |
| 82 |
< |
simulated system to the imposed momentum flux will typically be a |
| 83 |
< |
velocity or thermal gradient (Fig. \ref{thermalDemo}). The transport |
| 84 |
< |
coefficients (shear viscosity and thermal conductivity) are easily |
| 85 |
< |
obtained by assuming linear response of the system, |
| 77 |
> |
artificial flux is typically created by periodically ``swapping'' |
| 78 |
> |
either the entire momentum vector $\vec{p}$ or single components of |
| 79 |
> |
this vector ($p_x$) between molecules in each of the two slabs. If |
| 80 |
> |
the two slabs are separated along the $z$ coordinate, the imposed flux |
| 81 |
> |
is either directional ($j_z(p_x)$) or isotropic ($J_z$), and the |
| 82 |
> |
response of a simulated system to the imposed momentum flux will |
| 83 |
> |
typically be a velocity or thermal gradient (Fig. \ref{thermalDemo}). |
| 84 |
> |
The shear viscosity ($\eta$) and thermal conductivity ($\lambda$) are |
| 85 |
> |
easily obtained by assuming linear response of the system, |
| 86 |
|
\begin{eqnarray} |
| 87 |
|
j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
| 88 |
|
J_z & = & \lambda \frac{\partial T}{\partial z} |
| 89 |
|
\end{eqnarray} |
| 90 |
< |
RNEMD has been widely used to provide computational estimates of thermal |
| 91 |
< |
conductivities and shear viscosities in a wide range of materials, |
| 92 |
< |
from liquid copper to monatomic liquids to molecular fluids |
| 93 |
< |
(e.g. ionic liquids).\cite{ISI:000246190100032} |
| 90 |
> |
RNEMD has been widely used to provide computational estimates of |
| 91 |
> |
thermal conductivities and shear viscosities in a wide range of |
| 92 |
> |
materials, from liquid copper to both monatomic and molecular fluids |
| 93 |
> |
(e.g. ionic |
| 94 |
> |
liquids).\cite{Bedrov:2000-1,Bedrov:2000,Muller-Plathe:2002,ISI:000184808400018,ISI:000231042800044,Maginn:2007,Muller-Plathe:2008,ISI:000258460400020,ISI:000258840700015,ISI:000261835100054} |
| 95 |
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|
| 96 |
|
\begin{figure} |
| 97 |
|
\includegraphics[width=\linewidth]{thermalDemo} |
| 98 |
< |
\caption{Demostration of thermal gradient estalished by RNEMD |
| 99 |
< |
method. Physical thermal flow directs from high temperature region |
| 100 |
< |
to low temperature region. Unphysical thermal transfer counteracts |
| 101 |
< |
it and maintains a steady thermal gradient.} |
| 98 |
> |
\caption{RNEMD methods impose an unphysical transfer of momentum or |
| 99 |
> |
kinetic energy between a ``hot'' slab and a ``cold'' slab in the |
| 100 |
> |
simulation box. The molecular system responds to this imposed flux |
| 101 |
> |
by generating a momentum or temperature gradient. The slope of the |
| 102 |
> |
gradient can then be used to compute transport properties (e.g. |
| 103 |
> |
shear viscosity and thermal conductivity).} |
| 104 |
|
\label{thermalDemo} |
| 105 |
|
\end{figure} |
| 106 |
|
|
| 107 |
< |
RNEMD is preferable in many ways to the forward NEMD methods because |
| 108 |
< |
it imposes what is typically difficult to measure (a flux or stress) |
| 109 |
< |
and it is typically much easier to compute momentum gradients or |
| 110 |
< |
strains (the response). For similar reasons, RNEMD is also preferable |
| 111 |
< |
to slowly-converging equilibrium methods for measuring thermal |
| 112 |
< |
conductivity and shear viscosity (using Green-Kubo relations or the |
| 113 |
< |
Helfand moment approach of Viscardy {\it et |
| 107 |
> |
RNEMD is preferable in many ways to the forward NEMD |
| 108 |
> |
methods\cite{ISI:A1988Q205300014,hess:209,Vasquez:2004fk,backer:154503,ISI:000266247600008} |
| 109 |
> |
because it imposes what is typically difficult to measure (a flux or |
| 110 |
> |
stress) and it is typically much easier to compute the response |
| 111 |
> |
(momentum gradients or strains). For similar reasons, RNEMD is also |
| 112 |
> |
preferable to slowly-converging equilibrium methods for measuring |
| 113 |
> |
thermal conductivity and shear viscosity (using Green-Kubo |
| 114 |
> |
relations\cite{daivis:541,mondello:9327} or the Helfand moment |
| 115 |
> |
approach of Viscardy {\it et |
| 116 |
|
al.}\cite{Viscardy:2007bh,Viscardy:2007lq}) because these rely on |
| 117 |
|
computing difficult to measure quantities. |
| 118 |
|
|
| 122 |
|
typically samples from the same manifold of states in the |
| 123 |
|
microcanonical ensemble. |
| 124 |
|
|
| 125 |
< |
Recently, Tenney and Maginn\cite{ISI:000273472300004} have discovered |
| 126 |
< |
some problems with the original RNEMD swap technique. Notably, large |
| 125 |
> |
Recently, Tenney and Maginn\cite{Maginn:2010} have discovered some |
| 126 |
> |
problems with the original RNEMD swap technique. Notably, large |
| 127 |
|
momentum fluxes (equivalent to frequent momentum swaps between the |
| 128 |
|
slabs) can result in ``notched'', ``peaked'' and generally non-thermal |
| 129 |
|
momentum distributions in the two slabs, as well as non-linear thermal |
| 130 |
|
and velocity distributions along the direction of the imposed flux |
| 131 |
|
($z$). Tenney and Maginn obtained reasonable limits on imposed flux |
| 132 |
< |
and self-adjusting metrics for retaining the usability of the method. |
| 132 |
> |
and proposed self-adjusting metrics for retaining the usability of the |
| 133 |
> |
method. |
| 134 |
|
|
| 135 |
|
In this paper, we develop and test a method for non-isotropic velocity |
| 136 |
< |
scaling (NIVS-RNEMD) which retains the desirable features of RNEMD |
| 136 |
> |
scaling (NIVS) which retains the desirable features of RNEMD |
| 137 |
|
(conservation of linear momentum and total energy, compatibility with |
| 138 |
|
periodic boundary conditions) while establishing true thermal |
| 139 |
< |
distributions in each of the two slabs. In the next section, we |
| 140 |
< |
develop the method for determining the scaling constraints. We then |
| 141 |
< |
test the method on both single component, multi-component, and |
| 142 |
< |
non-isotropic mixtures and show that it is capable of providing |
| 139 |
> |
distributions in each of the two slabs. In the next section, we |
| 140 |
> |
present the method for determining the scaling constraints. We then |
| 141 |
> |
test the method on both liquids and solids as well as a non-isotropic |
| 142 |
> |
liquid-solid interface and show that it is capable of providing |
| 143 |
|
reasonable estimates of the thermal conductivity and shear viscosity |
| 144 |
< |
in these cases. |
| 144 |
> |
in all of these cases. |
| 145 |
|
|
| 146 |
|
\section{Methodology} |
| 147 |
< |
We retain the basic idea of Muller-Plathe's RNEMD method; the periodic |
| 148 |
< |
system is partitioned into a series of thin slabs along a particular |
| 147 |
> |
We retain the basic idea of M\"{u}ller-Plathe's RNEMD method; the |
| 148 |
> |
periodic system is partitioned into a series of thin slabs along one |
| 149 |
|
axis ($z$). One of the slabs at the end of the periodic box is |
| 150 |
|
designated the ``hot'' slab, while the slab in the center of the box |
| 151 |
|
is designated the ``cold'' slab. The artificial momentum flux will be |
| 153 |
|
hot slab. |
| 154 |
|
|
| 155 |
|
Rather than using momentum swaps, we use a series of velocity scaling |
| 156 |
< |
moves. For molecules $\{i\}$ located within the cold slab, |
| 156 |
> |
moves. For molecules $\{i\}$ located within the cold slab, |
| 157 |
|
\begin{equation} |
| 158 |
|
\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
| 159 |
|
x & 0 & 0 \\ |
| 161 |
|
0 & 0 & z \\ |
| 162 |
|
\end{array} \right) \cdot \vec{v}_i |
| 163 |
|
\end{equation} |
| 164 |
< |
where ${x, y, z}$ are a set of 3 scaling variables for each of the |
| 165 |
< |
three directions in the system. Likewise, the molecules $\{j\}$ |
| 166 |
< |
located in the hot slab will see a concomitant scaling of velocities, |
| 164 |
> |
where ${x, y, z}$ are a set of 3 velocity-scaling variables for each |
| 165 |
> |
of the three directions in the system. Likewise, the molecules |
| 166 |
> |
$\{j\}$ located in the hot slab will see a concomitant scaling of |
| 167 |
> |
velocities, |
| 168 |
|
\begin{equation} |
| 169 |
|
\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
| 170 |
|
x^\prime & 0 & 0 \\ |
| 174 |
|
\end{equation} |
| 175 |
|
|
| 176 |
|
Conservation of linear momentum in each of the three directions |
| 177 |
< |
($\alpha = x,y,z$) ties the values of the hot and cold bin scaling |
| 177 |
> |
($\alpha = x,y,z$) ties the values of the hot and cold scaling |
| 178 |
|
parameters together: |
| 179 |
|
\begin{equation} |
| 180 |
|
P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
| 181 |
|
\end{equation} |
| 182 |
|
where |
| 183 |
|
\begin{eqnarray} |
| 184 |
< |
P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
| 185 |
< |
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha |
| 184 |
> |
P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i v_{i\alpha} \\ |
| 185 |
> |
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j v_{j\alpha} |
| 186 |
|
\label{eq:momentumdef} |
| 187 |
|
\end{eqnarray} |
| 188 |
|
Therefore, for each of the three directions, the hot scaling |
| 189 |
|
parameters are a simple function of the cold scaling parameters and |
| 190 |
< |
the instantaneous linear momentum in each of the two slabs. |
| 190 |
> |
the instantaneous linear momenta in each of the two slabs. |
| 191 |
|
\begin{equation} |
| 192 |
|
\alpha^\prime = 1 + (1 - \alpha) p_\alpha |
| 193 |
|
\label{eq:hotcoldscaling} |
| 200 |
|
|
| 201 |
|
Conservation of total energy also places constraints on the scaling: |
| 202 |
|
\begin{equation} |
| 203 |
< |
\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
| 204 |
< |
\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha |
| 203 |
> |
\sum_{\alpha = x,y,z} \left\{ K_h^\alpha + K_c^\alpha \right\} = \sum_{\alpha = x,y,z} |
| 204 |
> |
\left\{ \left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha \right\} |
| 205 |
|
\end{equation} |
| 206 |
|
where the translational kinetic energies, $K_h^\alpha$ and |
| 207 |
|
$K_c^\alpha$, are computed for each of the three directions in a |
| 208 |
|
similar manner to the linear momenta (Eq. \ref{eq:momentumdef}). |
| 209 |
|
Substituting in the expressions for the hot scaling parameters |
| 210 |
|
($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), we obtain the |
| 211 |
< |
{\it constraint ellipsoid equation}: |
| 211 |
> |
{\it constraint ellipsoid}: |
| 212 |
|
\begin{equation} |
| 213 |
< |
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0 |
| 213 |
> |
\sum_{\alpha = x,y,z} \left( a_\alpha \alpha^2 + b_\alpha \alpha + |
| 214 |
> |
c_\alpha \right) = 0 |
| 215 |
|
\label{eq:constraintEllipsoid} |
| 216 |
|
\end{equation} |
| 217 |
|
where the constants are obtained from the instantaneous values of the |
| 223 |
|
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
| 224 |
|
\label{eq:constraintEllipsoidConsts} |
| 225 |
|
\end{eqnarray} |
| 226 |
< |
This ellipsoid equation defines the set of cold slab scaling |
| 227 |
< |
parameters which can be applied while preserving both linear momentum |
| 228 |
< |
in all three directions as well as kinetic energy. |
| 226 |
> |
This ellipsoid (Eq. \ref{eq:constraintEllipsoid}) defines the set of |
| 227 |
> |
cold slab scaling parameters which, when applied, preserve the linear |
| 228 |
> |
momentum of the system in all three directions as well as total |
| 229 |
> |
kinetic energy. |
| 230 |
|
|
| 231 |
< |
The goal of using velocity scaling variables is to transfer linear |
| 232 |
< |
momentum or kinetic energy from the cold slab to the hot slab. If the |
| 233 |
< |
hot and cold slabs are separated along the z-axis, the energy flux is |
| 234 |
< |
given simply by the decrease in kinetic energy of the cold bin: |
| 231 |
> |
The goal of using these velocity scaling variables is to transfer |
| 232 |
> |
kinetic energy from the cold slab to the hot slab. If the hot and |
| 233 |
> |
cold slabs are separated along the z-axis, the energy flux is given |
| 234 |
> |
simply by the decrease in kinetic energy of the cold bin: |
| 235 |
|
\begin{equation} |
| 236 |
|
(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z\Delta t |
| 237 |
|
\end{equation} |
| 238 |
|
The expression for the energy flux can be re-written as another |
| 239 |
|
ellipsoid centered on $(x,y,z) = 0$: |
| 240 |
|
\begin{equation} |
| 241 |
< |
x^2 K_c^x + y^2 K_c^y + z^2 K_c^z = K_c^x + K_c^y + K_c^z - J_z\Delta t |
| 241 |
> |
\sum_{\alpha = x,y,z} K_c^\alpha \alpha^2 = \sum_{\alpha = x,y,z} |
| 242 |
> |
K_c^\alpha -J_z \Delta t |
| 243 |
|
\label{eq:fluxEllipsoid} |
| 244 |
|
\end{equation} |
| 245 |
< |
The spatial extent of the {\it thermal flux ellipsoid equation} is |
| 246 |
< |
governed both by a targetted value, $J_z$ as well as the instantaneous |
| 247 |
< |
values of the kinetic energy components in the cold bin. |
| 245 |
> |
The spatial extent of the {\it thermal flux ellipsoid} is governed |
| 246 |
> |
both by the target flux, $J_z$ as well as the instantaneous values of |
| 247 |
> |
the kinetic energy components in the cold bin. |
| 248 |
|
|
| 249 |
|
To satisfy an energetic flux as well as the conservation constraints, |
| 250 |
< |
it is sufficient to determine the points ${x,y,z}$ which lie on both |
| 251 |
< |
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
| 252 |
< |
flux ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
| 253 |
< |
the two ellipsoids in 3-dimensional space. |
| 250 |
> |
we must determine the points ${x,y,z}$ that lie on both the constraint |
| 251 |
> |
ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the flux ellipsoid |
| 252 |
> |
(Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of the two |
| 253 |
> |
ellipsoids in 3-dimensional space. |
| 254 |
|
|
| 255 |
|
\begin{figure} |
| 256 |
|
\includegraphics[width=\linewidth]{ellipsoids} |
| 257 |
< |
\caption{Scaling points which maintain both constant energy and |
| 258 |
< |
constant linear momentum of the system lie on the surface of the |
| 259 |
< |
{\it constraint ellipsoid} while points which generate the target |
| 260 |
< |
momentum flux lie on the surface of the {\it flux ellipsoid}. The |
| 261 |
< |
velocity distributions in the cold bin are scaled by only those |
| 257 |
> |
\caption{Velocity scaling coefficients which maintain both constant |
| 258 |
> |
energy and constant linear momentum of the system lie on the surface |
| 259 |
> |
of the {\it constraint ellipsoid} while points which generate the |
| 260 |
> |
target momentum flux lie on the surface of the {\it flux ellipsoid}. |
| 261 |
> |
The velocity distributions in the cold bin are scaled by only those |
| 262 |
|
points which lie on both ellipsoids.} |
| 263 |
|
\label{ellipsoids} |
| 264 |
|
\end{figure} |
| 265 |
|
|
| 266 |
< |
One may also define momentum flux (say along the $x$-direction) as: |
| 266 |
> |
Since ellipsoids can be expressed as polynomials up to second order in |
| 267 |
> |
each of the three coordinates, finding the the intersection points of |
| 268 |
> |
two ellipsoids is isomorphic to finding the roots a polynomial of |
| 269 |
> |
degree 16. There are a number of polynomial root-finding methods in |
| 270 |
> |
the literature,\cite{Hoffman:2001sf,384119} but numerically finding |
| 271 |
> |
the roots of high-degree polynomials is generally an ill-conditioned |
| 272 |
> |
problem.\cite{Hoffman:2001sf} One simplification is to maintain |
| 273 |
> |
velocity scalings that are {\it as isotropic as possible}. To do |
| 274 |
> |
this, we impose $x=y$, and treat both the constraint and flux |
| 275 |
> |
ellipsoids as 2-dimensional ellipses. In reduced dimensionality, the |
| 276 |
> |
intersecting-ellipse problem reduces to finding the roots of |
| 277 |
> |
polynomials of degree 4. |
| 278 |
> |
|
| 279 |
> |
Depending on the target flux and current velocity distributions, the |
| 280 |
> |
ellipsoids can have between 0 and 4 intersection points. If there are |
| 281 |
> |
no intersection points, it is not possible to satisfy the constraints |
| 282 |
> |
while performing a non-equilibrium scaling move, and no change is made |
| 283 |
> |
to the dynamics. |
| 284 |
> |
|
| 285 |
> |
With multiple intersection points, any of the scaling points will |
| 286 |
> |
conserve the linear momentum and kinetic energy of the system and will |
| 287 |
> |
generate the correct target flux. Although this method is inherently |
| 288 |
> |
non-isotropic, the goal is still to maintain the system as close to an |
| 289 |
> |
isotropic fluid as possible. With this in mind, we would like the |
| 290 |
> |
kinetic energies in the three different directions could become as |
| 291 |
> |
close as each other as possible after each scaling. Simultaneously, |
| 292 |
> |
one would also like each scaling as gentle as possible, i.e. ${x,y,z |
| 293 |
> |
\rightarrow 1}$, in order to avoid large perturbation to the system. |
| 294 |
> |
To do this, we pick the intersection point which maintains the three |
| 295 |
> |
scaling variables ${x, y, z}$ as well as the ratio of kinetic energies |
| 296 |
> |
${K_c^z/K_c^x, K_c^z/K_c^y}$ as close as possible to 1. |
| 297 |
> |
|
| 298 |
> |
After the valid scaling parameters are arrived at by solving geometric |
| 299 |
> |
intersection problems in $x, y, z$ space in order to obtain cold slab |
| 300 |
> |
scaling parameters, Eq. (\ref{eq:hotcoldscaling}) is used to |
| 301 |
> |
determine the conjugate hot slab scaling variables. |
| 302 |
> |
|
| 303 |
> |
\subsection{Introducing shear stress via velocity scaling} |
| 304 |
> |
It is also possible to use this method to magnify the random |
| 305 |
> |
fluctuations of the average momentum in each of the bins to induce a |
| 306 |
> |
momentum flux. Doing this repeatedly will create a shear stress on |
| 307 |
> |
the system which will respond with an easily-measured strain. The |
| 308 |
> |
momentum flux (say along the $x$-direction) may be defined as: |
| 309 |
|
\begin{equation} |
| 310 |
|
(1-x) P_c^x = j_z(p_x)\Delta t |
| 311 |
|
\label{eq:fluxPlane} |
| 312 |
|
\end{equation} |
| 313 |
< |
The above {\it momentum flux equation} is essentially a plane which is |
| 314 |
< |
perpendicular to the $x$-axis, with its position governed both by a |
| 315 |
< |
target value, $j_z(p_x)$ as well as the instantaneous value of the |
| 247 |
< |
momentum along the $x$-direction. |
| 313 |
> |
This {\it momentum flux plane} is perpendicular to the $x$-axis, with |
| 314 |
> |
its position governed both by a target value, $j_z(p_x)$ as well as |
| 315 |
> |
the instantaneous value of the momentum along the $x$-direction. |
| 316 |
|
|
| 317 |
< |
Similarly, to satisfy a momentum flux as well as the conservation |
| 318 |
< |
constraints, it is sufficient to determine the points ${x,y,z}$ which |
| 319 |
< |
lie on both the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) |
| 320 |
< |
and the flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of |
| 321 |
< |
an ellipsoid and a plane in 3-dimensional space. |
| 317 |
> |
In order to satisfy a momentum flux as well as the conservation |
| 318 |
> |
constraints, we must determine the points ${x,y,z}$ which lie on both |
| 319 |
> |
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
| 320 |
> |
flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of an |
| 321 |
> |
ellipsoid and a plane in 3-dimensional space. |
| 322 |
|
|
| 323 |
< |
To summarize, by solving respective equation sets, one can determine |
| 324 |
< |
possible sets of scaling variables for cold slab. And corresponding |
| 325 |
< |
sets of scaling variables for hot slab can be determine as well. |
| 323 |
> |
In the case of momentum flux transfer, we also impose another |
| 324 |
> |
constraint to set the kinetic energy transfer as zero. In other |
| 325 |
> |
words, we apply Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. With |
| 326 |
> |
one variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar |
| 327 |
> |
set of quartic equations to the above kinetic energy transfer problem. |
| 328 |
|
|
| 329 |
< |
The following problem will be choosing an optimal set of scaling |
| 260 |
< |
variables among the possible sets. Although this method is inherently |
| 261 |
< |
non-isotropic, the goal is still to maintain the system as isotropic |
| 262 |
< |
as possible. Under this consideration, one would like the kinetic |
| 263 |
< |
energies in different directions could become as close as each other |
| 264 |
< |
after each scaling. Simultaneously, one would also like each scaling |
| 265 |
< |
as gentle as possible, i.e. ${x,y,z \rightarrow 1}$, in order to avoid |
| 266 |
< |
large perturbation to the system. Therefore, one approach to obtain the |
| 267 |
< |
scaling variables would be constructing an criteria function, with |
| 268 |
< |
constraints as above equation sets, and solving the function's minimum |
| 269 |
< |
by method like Lagrange multipliers. |
| 329 |
> |
\section{Computational Details} |
| 330 |
|
|
| 331 |
< |
In order to save computation time, we have a different approach to a |
| 332 |
< |
relatively good set of scaling variables with much less calculation |
| 333 |
< |
than above. Here is the detail of our simplification of the problem. |
| 331 |
> |
We have implemented this methodology in our molecular dynamics code, |
| 332 |
> |
OpenMD,\cite{Meineke:2005gd,openmd} performing the NIVS scaling moves |
| 333 |
> |
with a variable frequency after the molecular dynamics (MD) steps. We |
| 334 |
> |
have tested the method in a variety of different systems, including: |
| 335 |
> |
homogeneous fluids (Lennard-Jones and SPC/E water), crystalline |
| 336 |
> |
solids, using both the embedded atom method |
| 337 |
> |
(EAM)~\cite{PhysRevB.33.7983} and quantum Sutton-Chen |
| 338 |
> |
(QSC)~\cite{PhysRevB.59.3527} models for Gold, and heterogeneous |
| 339 |
> |
interfaces (QSC gold - SPC/E water). The last of these systems would |
| 340 |
> |
have been difficult to study using previous RNEMD methods, but the |
| 341 |
> |
current method can easily provide estimates of the interfacial thermal |
| 342 |
> |
conductivity ($G$). |
| 343 |
|
|
| 344 |
< |
In the case of kinetic energy transfer, we impose another constraint |
| 276 |
< |
${x = y}$, into the equation sets. Consequently, there are two |
| 277 |
< |
variables left. And now one only needs to solve a set of two {\it |
| 278 |
< |
ellipses equations}. This problem would be transformed into solving |
| 279 |
< |
one quartic equation for one of the two variables. There are known |
| 280 |
< |
generic methods that solve real roots of quartic equations. Then one |
| 281 |
< |
can determine the other variable and obtain sets of scaling |
| 282 |
< |
variables. Among these sets, one can apply the above criteria to |
| 283 |
< |
choose the best set, while much faster with only a few sets to choose. |
| 344 |
> |
\subsection{Simulation Cells} |
| 345 |
|
|
| 346 |
< |
In the case of momentum flux transfer, we impose another constraint to |
| 347 |
< |
set the kinetic energy transfer as zero. In another word, we apply |
| 348 |
< |
Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. After that, with one |
| 349 |
< |
variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar set |
| 350 |
< |
of equations on the above kinetic energy transfer problem. Therefore, |
| 351 |
< |
an approach similar to the above would be sufficient for this as well. |
| 346 |
> |
In each of the systems studied, the dynamics was carried out in a |
| 347 |
> |
rectangular simulation cell using periodic boundary conditions in all |
| 348 |
> |
three dimensions. The cells were longer along the $z$ axis and the |
| 349 |
> |
space was divided into $N$ slabs along this axis (typically $N=20$). |
| 350 |
> |
The top slab ($n=1$) was designated the ``hot'' slab, while the |
| 351 |
> |
central slab ($n= N/2 + 1$) was designated the ``cold'' slab. In all |
| 352 |
> |
cases, simulations were first thermalized in canonical ensemble (NVT) |
| 353 |
> |
using a Nos\'{e}-Hoover thermostat.\cite{Hoover85} then equilibrated in |
| 354 |
> |
microcanonical ensemble (NVE) before introducing any non-equilibrium |
| 355 |
> |
method. |
| 356 |
|
|
| 357 |
< |
\section{Computational Details} |
| 293 |
< |
\subsection{Lennard-Jones Fluid} |
| 294 |
< |
Our simulation consists of a series of systems. All of these |
| 295 |
< |
simulations were run with the OpenMD simulation software |
| 296 |
< |
package\cite{Meineke:2005gd} integrated with RNEMD codes. |
| 357 |
> |
\subsection{RNEMD with M\"{u}ller-Plathe swaps} |
| 358 |
|
|
| 359 |
< |
A Lennard-Jones fluid system was built and tested first. In order to |
| 360 |
< |
compare our method with swapping RNEMD, a series of simulations were |
| 361 |
< |
performed to calculate the shear viscosity and thermal conductivity of |
| 362 |
< |
argon. 2592 atoms were in a orthorhombic cell, which was ${10.06\sigma |
| 363 |
< |
\times 10.06\sigma \times 30.18\sigma}$ by size. The reduced density |
| 364 |
< |
${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled direct |
| 365 |
< |
comparison between our results and others. These simulations used |
| 366 |
< |
velocity Verlet algorithm with reduced timestep ${\tau^* = |
| 306 |
< |
4.6\times10^{-4}}$. |
| 359 |
> |
In order to compare our new methodology with the original |
| 360 |
> |
M\"{u}ller-Plathe swapping algorithm,\cite{ISI:000080382700030} we |
| 361 |
> |
first performed simulations using the original technique. At fixed |
| 362 |
> |
intervals, kinetic energy or momentum exchange moves were performed |
| 363 |
> |
between the hot and the cold slabs. The interval between exchange |
| 364 |
> |
moves governs the effective momentum flux ($j_z(p_x)$) or energy flux |
| 365 |
> |
($J_z$) between the two slabs so to vary this quantity, we performed |
| 366 |
> |
simulations with a variety of delay intervals between the swapping moves. |
| 367 |
|
|
| 368 |
< |
For shear viscosity calculation, the reduced temperature was ${T^* = |
| 369 |
< |
k_B T/\varepsilon = 0.72}$. Simulations were first equilibrated in canonical |
| 370 |
< |
ensemble (NVT), then equilibrated in microcanonical ensemble |
| 371 |
< |
(NVE). Establishing and stablizing momentum gradient were followed |
| 372 |
< |
also in NVE ensemble. For the swapping part, Muller-Plathe's algorithm was |
| 373 |
< |
adopted.\cite{ISI:000080382700030} The simulation box was under |
| 374 |
< |
periodic boundary condition, and devided into ${N = 20}$ slabs. In each swap, |
| 315 |
< |
the top slab ${(n = 1)}$ exchange the most negative $x$ momentum with the |
| 316 |
< |
most positive $x$ momentum in the center slab ${(n = N/2 + 1)}$. Referred |
| 317 |
< |
to Tenney {\it et al.}\cite{ISI:000273472300004}, a series of swapping |
| 318 |
< |
frequency were chosen. According to each result from swapping |
| 319 |
< |
RNEMD, scaling RNEMD simulations were run with the target momentum |
| 320 |
< |
flux set to produce a similar momentum flux, and consequently shear |
| 321 |
< |
rate. Furthermore, various scaling frequencies can be tested for one |
| 322 |
< |
single swapping rate. To test the temperature homogeneity in our |
| 323 |
< |
system of swapping and scaling methods, temperatures of different |
| 324 |
< |
dimensions in all the slabs were observed. Most of the simulations |
| 325 |
< |
include $10^5$ steps of equilibration without imposing momentum flux, |
| 326 |
< |
$10^5$ steps of stablization with imposing unphysical momentum |
| 327 |
< |
transfer, and $10^6$ steps of data collection under RNEMD. For |
| 328 |
< |
relatively high momentum flux simulations, ${5\times10^5}$ step data |
| 329 |
< |
collection is sufficient. For some low momentum flux simulations, |
| 330 |
< |
${2\times10^6}$ steps were necessary. |
| 368 |
> |
For thermal conductivity measurements, the particle with smallest |
| 369 |
> |
speed in the hot slab and the one with largest speed in the cold slab |
| 370 |
> |
had their entire momentum vectors swapped. In the test cases run |
| 371 |
> |
here, all particles had the same chemical identity and mass, so this |
| 372 |
> |
move preserves both total linear momentum and total energy. It is |
| 373 |
> |
also possible to exchange energy by assuming an elastic collision |
| 374 |
> |
between the two particles which are exchanging energy. |
| 375 |
|
|
| 376 |
< |
After each simulation, the shear viscosity was calculated in reduced |
| 377 |
< |
unit. The momentum flux was calculated with total unphysical |
| 378 |
< |
transferred momentum ${P_x}$ and data collection time $t$: |
| 376 |
> |
For shear stress simulations, the particle with the most negative |
| 377 |
> |
$p_x$ in the hot slab and the one with the most positive $p_x$ in the |
| 378 |
> |
cold slab exchanged only this component of their momentum vectors. |
| 379 |
> |
|
| 380 |
> |
\subsection{RNEMD with NIVS scaling} |
| 381 |
> |
|
| 382 |
> |
For each simulation utilizing the swapping method, a corresponding |
| 383 |
> |
NIVS-RNEMD simulation was carried out using a target momentum flux set |
| 384 |
> |
to produce the same flux experienced in the swapping simulation. |
| 385 |
> |
|
| 386 |
> |
To test the temperature homogeneity, directional momentum and |
| 387 |
> |
temperature distributions were accumulated for molecules in each of |
| 388 |
> |
the slabs. Transport coefficients were computed using the temperature |
| 389 |
> |
(and momentum) gradients across the $z$-axis as well as the total |
| 390 |
> |
momentum flux the system experienced during data collection portion of |
| 391 |
> |
the simulation. |
| 392 |
> |
|
| 393 |
> |
\subsection{Shear viscosities} |
| 394 |
> |
|
| 395 |
> |
The momentum flux was calculated using the total non-physical momentum |
| 396 |
> |
transferred (${P_x}$) and the data collection time ($t$): |
| 397 |
|
\begin{equation} |
| 398 |
|
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
| 399 |
|
\end{equation} |
| 400 |
< |
where $L_x$ and $L_y$ denotes $x$ and $y$ lengths of the simulation |
| 401 |
< |
box, and physical momentum transfer occurs in two ways due to our |
| 402 |
< |
periodic boundary condition settings. And the velocity gradient |
| 403 |
< |
${\langle \partial v_x /\partial z \rangle}$ can be obtained by a |
| 404 |
< |
linear regression of the velocity profile. From the shear viscosity |
| 405 |
< |
$\eta$ calculated with the above parameters, one can further convert |
| 406 |
< |
it into reduced unit ${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$. |
| 400 |
> |
where $L_x$ and $L_y$ denote the $x$ and $y$ lengths of the simulation |
| 401 |
> |
box. The factor of two in the denominator is present because physical |
| 402 |
> |
momentum transfer between the slabs occurs in two directions ($+z$ and |
| 403 |
> |
$-z$). The velocity gradient ${\langle \partial v_x /\partial z |
| 404 |
> |
\rangle}$ was obtained using linear regression of the mean $x$ |
| 405 |
> |
component of the velocity, $\langle v_x \rangle$, in each of the bins. |
| 406 |
> |
For Lennard-Jones simulations, shear viscosities are reported in |
| 407 |
> |
reduced units (${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$). |
| 408 |
|
|
| 409 |
< |
For thermal conductivity calculations, simulations were first run under |
| 347 |
< |
reduced temperature ${\langle T^*\rangle = 0.72}$ in NVE |
| 348 |
< |
ensemble. Muller-Plathe's algorithm was adopted in the swapping |
| 349 |
< |
method. Under identical simulation box parameters with our shear |
| 350 |
< |
viscosity calculations, in each swap, the top slab exchanges all three |
| 351 |
< |
translational momentum components of the molecule with least kinetic |
| 352 |
< |
energy with the same components of the molecule in the center slab |
| 353 |
< |
with most kinetic energy, unless this ``coldest'' molecule in the |
| 354 |
< |
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the |
| 355 |
< |
``cold'' slab. According to swapping RNEMD results, target energy flux |
| 356 |
< |
for scaling RNEMD simulations can be set. Also, various scaling |
| 357 |
< |
frequencies can be tested for one target energy flux. To compare the |
| 358 |
< |
performance between swapping and scaling method, distributions of |
| 359 |
< |
velocity and speed in different slabs were observed. |
| 409 |
> |
\subsection{Thermal Conductivities} |
| 410 |
|
|
| 411 |
< |
For each swapping rate, thermal conductivity was calculated in reduced |
| 412 |
< |
unit. The energy flux was calculated similarly to the momentum flux, |
| 413 |
< |
with total unphysical transferred energy ${E_{total}}$ and data collection |
| 364 |
< |
time $t$: |
| 411 |
> |
The energy flux was calculated in a similar manner to the momentum |
| 412 |
> |
flux, using the total non-physical energy transferred (${E_{total}}$) |
| 413 |
> |
and the data collection time $t$: |
| 414 |
|
\begin{equation} |
| 415 |
|
J_z = \frac{E_{total}}{2 t L_x L_y} |
| 416 |
|
\end{equation} |
| 417 |
< |
And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
| 418 |
< |
can be obtained by a linear regression of the temperature |
| 419 |
< |
profile. From the thermal conductivity $\lambda$ calculated, one can |
| 420 |
< |
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
| 421 |
< |
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
| 417 |
> |
The temperature gradient ${\langle\partial T/\partial z\rangle}$ was |
| 418 |
> |
obtained by a linear regression of the temperature profile. For |
| 419 |
> |
Lennard-Jones simulations, thermal conductivities are reported in |
| 420 |
> |
reduced units (${\lambda^*=\lambda \sigma^2 m^{1/2} |
| 421 |
> |
k_B^{-1}\varepsilon^{-1/2}}$). |
| 422 |
|
|
| 423 |
< |
\subsection{ Water / Metal Thermal Conductivity} |
| 375 |
< |
Another series of our simulation is the calculation of interfacial |
| 376 |
< |
thermal conductivity of a Au/H$_2$O system. Respective calculations of |
| 377 |
< |
liquid water (SPC/E) and crystal gold (QSC) thermal conductivity were |
| 378 |
< |
performed and compared with current results to ensure the validity of |
| 379 |
< |
NIVS-RNEMD. After that, a mixture system was simulated. |
| 423 |
> |
\subsection{Interfacial Thermal Conductivities} |
| 424 |
|
|
| 425 |
< |
For thermal conductivity calculation of bulk water, a simulation box |
| 426 |
< |
consisting of 1000 molecules were first equilibrated under ambient |
| 427 |
< |
pressure and temperature conditions using NPT ensemble, followed by |
| 428 |
< |
equilibration in fixed volume (NVT). The system was then equilibrated in |
| 429 |
< |
microcanonical ensemble (NVE). Also in NVE ensemble, establishing a |
| 386 |
< |
stable thermal gradient was followed. The simulation box was under |
| 387 |
< |
periodic boundary condition and devided into 10 slabs. Data collection |
| 388 |
< |
process was similar to Lennard-Jones fluid system. |
| 425 |
> |
For interfaces with a relatively low interfacial conductance, the bulk |
| 426 |
> |
regions on either side of an interface rapidly come to a state in |
| 427 |
> |
which the two phases have relatively homogeneous (but distinct) |
| 428 |
> |
temperatures. The interfacial thermal conductivity $G$ can therefore |
| 429 |
> |
be approximated as: |
| 430 |
|
|
| 390 |
– |
Thermal conductivity calculation of bulk crystal gold used a similar |
| 391 |
– |
protocol. The face centered cubic crystal simulation box consists of |
| 392 |
– |
2880 Au atoms. The lattice was first allowed volume change to relax |
| 393 |
– |
under ambient temperature and pressure. Equilibrations in canonical and |
| 394 |
– |
microcanonical ensemble were followed in order. With the simulation |
| 395 |
– |
lattice devided evenly into 10 slabs, different thermal gradients were |
| 396 |
– |
established by applying a set of target thermal transfer flux. Data of |
| 397 |
– |
the series of thermal gradients was collected for calculation. |
| 398 |
– |
|
| 399 |
– |
After simulations of bulk water and crystal gold, a mixture system was |
| 400 |
– |
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
| 401 |
– |
molecules. Spohr potential was adopted in depicting the interaction |
| 402 |
– |
between metal atom and water molecule.\cite{ISI:000167766600035} A |
| 403 |
– |
similar protocol of equilibration was followed. Several thermal |
| 404 |
– |
gradients was built under different target thermal flux. It was found |
| 405 |
– |
out that compared to our previous simulation systems, the two phases |
| 406 |
– |
could have large temperature difference even under a relatively low |
| 407 |
– |
thermal flux. Therefore, under our low flux conditions, it is assumed |
| 408 |
– |
that the metal and water phases have respectively homogeneous |
| 409 |
– |
temperature, excluding the surface regions. In calculating the |
| 410 |
– |
interfacial thermal conductivity $G$, this assumptioin was applied and |
| 411 |
– |
thus our formula becomes: |
| 412 |
– |
|
| 431 |
|
\begin{equation} |
| 432 |
|
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
| 433 |
|
\langle T_{water}\rangle \right)} |
| 434 |
|
\label{interfaceCalc} |
| 435 |
|
\end{equation} |
| 436 |
< |
where ${E_{total}}$ is the imposed unphysical kinetic energy transfer |
| 437 |
< |
and ${\langle T_{gold}\rangle}$ and ${\langle T_{water}\rangle}$ are the |
| 438 |
< |
average observed temperature of gold and water phases respectively. |
| 436 |
> |
where ${E_{total}}$ is the imposed non-physical kinetic energy |
| 437 |
> |
transfer and ${\langle T_{gold}\rangle}$ and ${\langle |
| 438 |
> |
T_{water}\rangle}$ are the average observed temperature of gold and |
| 439 |
> |
water phases respectively. If the interfacial conductance is {\it |
| 440 |
> |
not} small, it is also be possible to compute the interfacial |
| 441 |
> |
thermal conductivity using this method utilizing the change in the |
| 442 |
> |
slope of the thermal gradient ($\partial^2 \langle T \rangle / \partial |
| 443 |
> |
z^2$) at the interface. |
| 444 |
|
|
| 445 |
< |
\section{Results And Discussion} |
| 423 |
< |
\subsection{Thermal Conductivity} |
| 424 |
< |
\subsubsection{Lennard-Jones Fluid} |
| 425 |
< |
Our thermal conductivity calculations also show that scaling method results |
| 426 |
< |
agree with swapping method. Table \ref{thermal} lists our simulation |
| 427 |
< |
results with similar manner we used in shear viscosity |
| 428 |
< |
calculation. All the data reported from scaling method were obtained |
| 429 |
< |
by simulations of 10-step exchange frequency, and the target exchange |
| 430 |
< |
kinetic energy were set to produce equivalent kinetic energy flux as |
| 431 |
< |
in swapping method. Figure \ref{thermalGrad} exhibits similar thermal |
| 432 |
< |
gradients of respective similar kinetic energy flux. |
| 445 |
> |
\section{Results} |
| 446 |
|
|
| 447 |
< |
\begin{table*} |
| 448 |
< |
\begin{minipage}{\linewidth} |
| 449 |
< |
\begin{center} |
| 447 |
> |
\subsection{Lennard-Jones Fluid} |
| 448 |
> |
2592 Lennard-Jones atoms were placed in an orthorhombic cell |
| 449 |
> |
${10.06\sigma \times 10.06\sigma \times 30.18\sigma}$ on a side. The |
| 450 |
> |
reduced density ${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled |
| 451 |
> |
direct comparison between our results and previous methods. These |
| 452 |
> |
simulations were carried out with a reduced timestep ${\tau^* = |
| 453 |
> |
4.6\times10^{-4}}$. For the shear viscosity calculations, the mean |
| 454 |
> |
temperature was ${T^* = k_B T/\varepsilon = 0.72}$. For thermal |
| 455 |
> |
conductivity calculations, simulations were run under reduced |
| 456 |
> |
temperature ${\langle T^*\rangle = 0.72}$ in the microcanonical |
| 457 |
> |
ensemble. The simulations included $10^5$ steps of equilibration |
| 458 |
> |
without any momentum flux, $10^5$ steps of stablization with an |
| 459 |
> |
imposed momentum transfer to create a gradient, and $10^6$ steps of |
| 460 |
> |
data collection under RNEMD. |
| 461 |
|
|
| 462 |
< |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
| 439 |
< |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
| 440 |
< |
swap and scale methods at various kinetic energy exchange rates. Results |
| 441 |
< |
in reduced unit. Errors of calculations in parentheses.} |
| 462 |
> |
\subsubsection*{Thermal Conductivity} |
| 463 |
|
|
| 464 |
< |
\begin{tabular}{ccc} |
| 465 |
< |
\hline |
| 466 |
< |
Series & $\lambda^*_{swap}$ & $\lambda^*_{scale}$\\ |
| 467 |
< |
\hline |
| 468 |
< |
20-250 & 7.03(0.34) & 7.30(0.10)\\ |
| 469 |
< |
20-500 & 7.03(0.14) & 6.95(0.09)\\ |
| 470 |
< |
20-1000 & 6.91(0.42) & 7.19(0.07)\\ |
| 471 |
< |
20-2000 & 7.52(0.15) & 7.19(0.28)\\ |
| 472 |
< |
\hline |
| 452 |
< |
\end{tabular} |
| 453 |
< |
\label{thermal} |
| 454 |
< |
\end{center} |
| 455 |
< |
\end{minipage} |
| 456 |
< |
\end{table*} |
| 464 |
> |
Our thermal conductivity calculations show that the NIVS method agrees |
| 465 |
> |
well with the swapping method. Five different swap intervals were |
| 466 |
> |
tested (Table \ref{LJ}). Similar thermal gradients were observed with |
| 467 |
> |
similar thermal flux under the two different methods (Figure |
| 468 |
> |
\ref{thermalGrad}). Furthermore, the 1-d temperature profiles showed |
| 469 |
> |
no observable differences between the $x$, $y$ and $z$ axes (Figure |
| 470 |
> |
\ref{thermalGrad} c), so even though we are using a non-isotropic |
| 471 |
> |
scaling method, none of the three directions are experience |
| 472 |
> |
disproportionate heating due to the velocity scaling. |
| 473 |
|
|
| 474 |
< |
\begin{figure} |
| 475 |
< |
\includegraphics[width=\linewidth]{thermalGrad} |
| 476 |
< |
\caption{Temperature gradients of thermal conductivity simulations} |
| 461 |
< |
\label{thermalGrad} |
| 462 |
< |
\end{figure} |
| 474 |
> |
\begin{table*} |
| 475 |
> |
\begin{minipage}{\linewidth} |
| 476 |
> |
\begin{center} |
| 477 |
|
|
| 478 |
< |
During these simulations, molecule velocities were recorded in 1000 of |
| 479 |
< |
all the snapshots. These velocity data were used to produce histograms |
| 480 |
< |
of velocity and speed distribution in different slabs. From these |
| 481 |
< |
histograms, it is observed that with increasing unphysical kinetic |
| 482 |
< |
energy flux, speed and velocity distribution of molecules in slabs |
| 483 |
< |
where swapping occured could deviate from Maxwell-Boltzmann |
| 484 |
< |
distribution. Figure \ref{histSwap} indicates how these distributions |
| 485 |
< |
deviate from ideal condition. In high temperature slabs, probability |
| 486 |
< |
density in low speed is confidently smaller than ideal distribution; |
| 487 |
< |
in low temperature slabs, probability density in high speed is smaller |
| 488 |
< |
than ideal. This phenomenon is observable even in our relatively low |
| 489 |
< |
swapping rate simulations. And this deviation could also leads to |
| 490 |
< |
deviation of distribution of velocity in various dimensions. One |
| 491 |
< |
feature of these deviated distribution is that in high temperature |
| 492 |
< |
slab, the ideal Gaussian peak was changed into a relatively flat |
| 493 |
< |
plateau; while in low temperature slab, that peak appears sharper. |
| 494 |
< |
|
| 495 |
< |
\begin{figure} |
| 496 |
< |
\includegraphics[width=\linewidth]{histSwap} |
| 497 |
< |
\caption{Speed distribution for thermal conductivity using swapping RNEMD.} |
| 498 |
< |
\label{histSwap} |
| 499 |
< |
\end{figure} |
| 478 |
> |
\caption{Thermal conductivity ($\lambda^*$) and shear viscosity |
| 479 |
> |
($\eta^*$) (in reduced units) of a Lennard-Jones fluid at |
| 480 |
> |
${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$ computed |
| 481 |
> |
at various momentum fluxes. The original swapping method and |
| 482 |
> |
the velocity scaling method give similar results. |
| 483 |
> |
Uncertainties are indicated in parentheses.} |
| 484 |
> |
|
| 485 |
> |
\begin{tabular}{|cc|cc|cc|} |
| 486 |
> |
\hline |
| 487 |
> |
\multicolumn{2}{|c}{Momentum Exchange} & |
| 488 |
> |
\multicolumn{2}{|c}{Swapping RNEMD} & |
| 489 |
> |
\multicolumn{2}{|c|}{NIVS-RNEMD} \\ |
| 490 |
> |
\hline |
| 491 |
> |
\multirow{2}{*}{Swap Interval (fs)} & Equivalent $J_z^*$ or & |
| 492 |
> |
\multirow{2}{*}{$\lambda^*_{swap}$} & |
| 493 |
> |
\multirow{2}{*}{$\eta^*_{swap}$} & |
| 494 |
> |
\multirow{2}{*}{$\lambda^*_{scale}$} & |
| 495 |
> |
\multirow{2}{*}{$\eta^*_{scale}$} \\ |
| 496 |
> |
& $j_z^*(p_x)$ (reduced units) & & & & \\ |
| 497 |
> |
\hline |
| 498 |
> |
250 & 0.16 & 7.03(0.34) & 3.57(0.06) & 7.30(0.10) & 3.54(0.04)\\ |
| 499 |
> |
500 & 0.09 & 7.03(0.14) & 3.64(0.05) & 6.95(0.09) & 3.76(0.09)\\ |
| 500 |
> |
1000 & 0.047 & 6.91(0.42) & 3.52(0.16) & 7.19(0.07) & 3.66(0.06)\\ |
| 501 |
> |
2000 & 0.024 & 7.52(0.15) & 3.72(0.05) & 7.19(0.28) & 3.32(0.18)\\ |
| 502 |
> |
2500 & 0.019 & 7.41(0.29) & 3.42(0.06) & 7.98(0.33) & 3.43(0.08)\\ |
| 503 |
> |
\hline |
| 504 |
> |
\end{tabular} |
| 505 |
> |
\label{LJ} |
| 506 |
> |
\end{center} |
| 507 |
> |
\end{minipage} |
| 508 |
> |
\end{table*} |
| 509 |
|
|
| 510 |
|
\begin{figure} |
| 511 |
< |
\includegraphics[width=\linewidth]{histScale} |
| 512 |
< |
\caption{Speed distribution for thermal conductivity using scaling RNEMD.} |
| 513 |
< |
\label{histScale} |
| 511 |
> |
\includegraphics[width=\linewidth]{thermalGrad} |
| 512 |
> |
\caption{The NIVS-RNEMD method creates similar temperature gradients |
| 513 |
> |
compared with the swapping method under a variety of imposed |
| 514 |
> |
kinetic energy flux values. Furthermore, the implementation of |
| 515 |
> |
Non-Isotropic Velocity Scaling does not cause temperature |
| 516 |
> |
anisotropy to develop in thermal conductivity calculations.} |
| 517 |
> |
\label{thermalGrad} |
| 518 |
|
\end{figure} |
| 519 |
|
|
| 520 |
< |
\subsubsection{SPC/E Water} |
| 494 |
< |
Our results of SPC/E water thermal conductivity are comparable to |
| 495 |
< |
Bedrov {\it et al.}\cite{ISI:000090151400044}, which employed the |
| 496 |
< |
previous swapping RNEMD method for their calculation. Our simulations |
| 497 |
< |
were able to produce a similar temperature gradient to their |
| 498 |
< |
system. However, the average temperature of our system is 300K, while |
| 499 |
< |
theirs is 318K, which would be attributed for part of the difference |
| 500 |
< |
between the two series of results. Both methods yields values in |
| 501 |
< |
agreement with experiment. And this shows the applicability of our |
| 502 |
< |
method to multi-atom molecular system. |
| 520 |
> |
\subsubsection*{Velocity Distributions} |
| 521 |
|
|
| 522 |
< |
\begin{figure} |
| 523 |
< |
\includegraphics[width=\linewidth]{spceGrad} |
| 524 |
< |
\caption{Temperature gradients for SPC/E water thermal conductivity.} |
| 525 |
< |
\label{spceGrad} |
| 526 |
< |
\end{figure} |
| 522 |
> |
During these simulations, velocities were recorded every 1000 steps |
| 523 |
> |
and were used to produce distributions of both velocity and speed in |
| 524 |
> |
each of the slabs. From these distributions, we observed that under |
| 525 |
> |
relatively high non-physical kinetic energy flux, the speed of |
| 526 |
> |
molecules in slabs where swapping occured could deviate from the |
| 527 |
> |
Maxwell-Boltzmann distribution. This behavior was also noted by Tenney |
| 528 |
> |
and Maginn.\cite{Maginn:2010} Figure \ref{thermalHist} shows how these |
| 529 |
> |
distributions deviate from an ideal distribution. In the ``hot'' slab, |
| 530 |
> |
the probability density is notched at low speeds and has a substantial |
| 531 |
> |
shoulder at higher speeds relative to the ideal MB distribution. In |
| 532 |
> |
the cold slab, the opposite notching and shouldering occurs. This |
| 533 |
> |
phenomenon is more obvious at higher swapping rates. |
| 534 |
|
|
| 535 |
< |
\begin{table*} |
| 536 |
< |
\begin{minipage}{\linewidth} |
| 537 |
< |
\begin{center} |
| 535 |
> |
The peak of the velocity distribution is substantially flattened in |
| 536 |
> |
the hot slab, and is overly sharp (with truncated wings) in the cold |
| 537 |
> |
slab. This problem is rooted in the mechanism of the swapping method. |
| 538 |
> |
Continually depleting low (high) speed particles in the high (low) |
| 539 |
> |
temperature slab is not complemented by diffusions of low (high) speed |
| 540 |
> |
particles from neighboring slabs, unless the swapping rate is |
| 541 |
> |
sufficiently small. Simutaneously, surplus low speed particles in the |
| 542 |
> |
low temperature slab do not have sufficient time to diffuse to |
| 543 |
> |
neighboring slabs. Since the thermal exchange rate must reach a |
| 544 |
> |
minimum level to produce an observable thermal gradient, the |
| 545 |
> |
swapping-method RNEMD has a relatively narrow choice of exchange times |
| 546 |
> |
that can be utilized. |
| 547 |
|
|
| 548 |
< |
\caption{Calculation results for thermal conductivity of SPC/E water |
| 549 |
< |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
| 550 |
< |
calculations in parentheses. } |
| 548 |
> |
For comparison, NIVS-RNEMD produces a speed distribution closer to the |
| 549 |
> |
Maxwell-Boltzmann curve (Figure \ref{thermalHist}). The reason for |
| 550 |
> |
this is simple; upon velocity scaling, a Gaussian distribution remains |
| 551 |
> |
Gaussian. Although a single scaling move is non-isotropic in three |
| 552 |
> |
dimensions, our criteria for choosing a set of scaling coefficients |
| 553 |
> |
helps maintain the distributions as close to isotropic as possible. |
| 554 |
> |
Consequently, NIVS-RNEMD is able to impose an unphysical thermal flux |
| 555 |
> |
as the previous RNEMD methods but without large perturbations to the |
| 556 |
> |
velocity distributions in the two slabs. |
| 557 |
|
|
| 518 |
– |
\begin{tabular}{cccc} |
| 519 |
– |
\hline |
| 520 |
– |
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
| 521 |
– |
& This work & Previous simulations\cite{ISI:000090151400044} & |
| 522 |
– |
Experiment$^a$\\ |
| 523 |
– |
\hline |
| 524 |
– |
0.38 & 0.816(0.044) & & 0.64\\ |
| 525 |
– |
0.81 & 0.770(0.008) & 0.784\\ |
| 526 |
– |
1.54 & 0.813(0.007) & 0.730\\ |
| 527 |
– |
\hline |
| 528 |
– |
\end{tabular} |
| 529 |
– |
\label{spceThermal} |
| 530 |
– |
\end{center} |
| 531 |
– |
\end{minipage} |
| 532 |
– |
\end{table*} |
| 533 |
– |
|
| 534 |
– |
\subsubsection{Crystal Gold} |
| 535 |
– |
Our results of gold thermal conductivity used QSC force field are |
| 536 |
– |
shown in Table \ref{AuThermal}. Although our calculation is smaller |
| 537 |
– |
than experimental value by an order of more than 100, this difference |
| 538 |
– |
is mainly attributed to the lack of electron interaction |
| 539 |
– |
representation in our force field parameters. Richardson {\it et |
| 540 |
– |
al.}\cite{ISI:A1992HX37800010} used similar force field parameters |
| 541 |
– |
in their metal thermal conductivity calculations. The EMD method they |
| 542 |
– |
employed in their simulations produced comparable results to |
| 543 |
– |
ours. Therefore, it is confident to conclude that NIVS-RNEMD is |
| 544 |
– |
applicable to metal force field system. |
| 545 |
– |
|
| 558 |
|
\begin{figure} |
| 559 |
< |
\includegraphics[width=\linewidth]{AuGrad} |
| 560 |
< |
\caption{Temperature gradients for crystal gold thermal conductivity.} |
| 561 |
< |
\label{AuGrad} |
| 559 |
> |
\includegraphics[width=\linewidth]{thermalHist} |
| 560 |
> |
\caption{Velocity and speed distributions that develop under the |
| 561 |
> |
swapping and NIVS-RNEMD methods at high flux. The distributions for |
| 562 |
> |
the hot bins (upper panels) and cold bins (lower panels) were |
| 563 |
> |
obtained from Lennard-Jones simulations with $\langle T^* \rangle = |
| 564 |
> |
4.5$ with a flux of $J_z^* \sim 5$ (equivalent to a swapping interval |
| 565 |
> |
of 10 time steps). This is a relatively large flux which shows the |
| 566 |
> |
non-thermal distributions that develop under the swapping method. |
| 567 |
> |
NIVS does a better job of producing near-thermal distributions in |
| 568 |
> |
the bins.} |
| 569 |
> |
\label{thermalHist} |
| 570 |
|
\end{figure} |
| 571 |
|
|
| 552 |
– |
\begin{table*} |
| 553 |
– |
\begin{minipage}{\linewidth} |
| 554 |
– |
\begin{center} |
| 572 |
|
|
| 573 |
< |
\caption{Calculation results for thermal conductivity of crystal gold |
| 574 |
< |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
| 575 |
< |
calculations in parentheses. } |
| 573 |
> |
\subsubsection*{Shear Viscosity} |
| 574 |
> |
Our calculations (Table \ref{LJ}) show that velocity-scaling RNEMD |
| 575 |
> |
predicted comparable shear viscosities to swap RNEMD method. The |
| 576 |
> |
average molecular momentum gradients of these samples are shown in |
| 577 |
> |
Figure \ref{shear} (a) and (b). |
| 578 |
|
|
| 560 |
– |
\begin{tabular}{ccc} |
| 561 |
– |
\hline |
| 562 |
– |
$\langle dT/dz\rangle$(K/\AA) & $\lambda$(W/m/K)\\ |
| 563 |
– |
& This work & Previous simulations\cite{ISI:A1992HX37800010} \\ |
| 564 |
– |
\hline |
| 565 |
– |
1.44 & 1.10(0.01) & \\ |
| 566 |
– |
2.86 & 1.08(0.02) & \\ |
| 567 |
– |
5.14 & 1.15(0.01) & \\ |
| 568 |
– |
\hline |
| 569 |
– |
\end{tabular} |
| 570 |
– |
\label{AuThermal} |
| 571 |
– |
\end{center} |
| 572 |
– |
\end{minipage} |
| 573 |
– |
\end{table*} |
| 574 |
– |
|
| 575 |
– |
\subsection{Interfaciel Thermal Conductivity} |
| 576 |
– |
After valid simulations of homogeneous water and gold systems using |
| 577 |
– |
NIVS-RNEMD method, calculation of gold/water interfacial thermal |
| 578 |
– |
conductivity was followed. It is found out that the interfacial |
| 579 |
– |
conductance is low due to a hydrophobic surface in our system. Figure |
| 580 |
– |
\ref{interfaceDensity} demonstrates this observance. Consequently, our |
| 581 |
– |
reported results (Table \ref{interfaceRes}) are of two orders of |
| 582 |
– |
magnitude smaller than our calculations on homogeneous systems. |
| 583 |
– |
|
| 579 |
|
\begin{figure} |
| 580 |
< |
\includegraphics[width=\linewidth]{interfaceDensity} |
| 581 |
< |
\caption{Density profile for interfacial thermal conductivity |
| 582 |
< |
simulation box.} |
| 583 |
< |
\label{interfaceDensity} |
| 580 |
> |
\includegraphics[width=\linewidth]{shear} |
| 581 |
> |
\caption{Average momentum gradients in shear viscosity simulations, |
| 582 |
> |
using ``swapping'' method (top panel) and NIVS-RNEMD method |
| 583 |
> |
(middle panel). NIVS-RNEMD produces a thermal anisotropy artifact |
| 584 |
> |
in the hot and cold bins when used for shear viscosity. This |
| 585 |
> |
artifact does not appear in thermal conductivity calculations.} |
| 586 |
> |
\label{shear} |
| 587 |
|
\end{figure} |
| 588 |
|
|
| 589 |
< |
\begin{figure} |
| 590 |
< |
\includegraphics[width=\linewidth]{interfaceGrad} |
| 591 |
< |
\caption{Temperature profiles for interfacial thermal conductivity |
| 592 |
< |
simulation box.} |
| 593 |
< |
\label{interfaceGrad} |
| 594 |
< |
\end{figure} |
| 589 |
> |
Observations of the three one-dimensional temperatures in each of the |
| 590 |
> |
slabs shows that NIVS-RNEMD does produce some thermal anisotropy, |
| 591 |
> |
particularly in the hot and cold slabs. Figure \ref{shear} (c) |
| 592 |
> |
indicates that with a relatively large imposed momentum flux, |
| 593 |
> |
$j_z(p_x)$, the $x$ direction approaches a different temperature from |
| 594 |
> |
the $y$ and $z$ directions in both the hot and cold bins. This is an |
| 595 |
> |
artifact of the scaling constraints. After the momentum gradient has |
| 596 |
> |
been established, $P_c^x < 0$. Consequently, the scaling factor $x$ |
| 597 |
> |
is nearly always greater than one in order to satisfy the constraints. |
| 598 |
> |
This will continually increase the kinetic energy in $x$-dimension, |
| 599 |
> |
$K_c^x$. If there is not enough time for the kinetic energy to |
| 600 |
> |
exchange among different directions and different slabs, the system |
| 601 |
> |
will exhibit the observed thermal anisotropy in the hot and cold bins. |
| 602 |
|
|
| 603 |
< |
\begin{table*} |
| 604 |
< |
\begin{minipage}{\linewidth} |
| 605 |
< |
\begin{center} |
| 603 |
> |
Although results between scaling and swapping methods are comparable, |
| 604 |
> |
the inherent temperature anisotropy does make NIVS-RNEMD method less |
| 605 |
> |
attractive than swapping RNEMD for shear viscosity calculations. We |
| 606 |
> |
note that this problem appears only when momentum flux is applied, and |
| 607 |
> |
does not appear in thermal flux calculations. |
| 608 |
|
|
| 609 |
< |
\caption{Calculation results for interfacial thermal conductivity |
| 603 |
< |
at ${\langle T\rangle \sim}$ 300K at various thermal exchange |
| 604 |
< |
rates. Errors of calculations in parentheses. } |
| 609 |
> |
\subsection{Bulk SPC/E water} |
| 610 |
|
|
| 611 |
< |
\begin{tabular}{cccc} |
| 612 |
< |
\hline |
| 613 |
< |
$J_z$(MW/m$^2$) & $T_{gold}$ & $T_{water}$ & $G$(MW/m$^2$/K)\\ |
| 614 |
< |
\hline |
| 615 |
< |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
| 616 |
< |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
| 617 |
< |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
| 618 |
< |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
| 619 |
< |
\hline |
| 615 |
< |
\end{tabular} |
| 616 |
< |
\label{interfaceRes} |
| 617 |
< |
\end{center} |
| 618 |
< |
\end{minipage} |
| 619 |
< |
\end{table*} |
| 611 |
> |
We compared the thermal conductivity of SPC/E water using NIVS-RNEMD |
| 612 |
> |
to the work of Bedrov {\it et al.}\cite{Bedrov:2000}, which employed |
| 613 |
> |
the original swapping RNEMD method. Bedrov {\it et |
| 614 |
> |
al.}\cite{Bedrov:2000} argued that exchange of the molecule |
| 615 |
> |
center-of-mass velocities instead of single atom velocities in a |
| 616 |
> |
molecule conserves the total kinetic energy and linear momentum. This |
| 617 |
> |
principle is also adopted Fin our simulations. Scaling was applied to |
| 618 |
> |
the center-of-mass velocities of the rigid bodies of SPC/E model water |
| 619 |
> |
molecules. |
| 620 |
|
|
| 621 |
< |
\subsection{Shear Viscosity} |
| 622 |
< |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
| 623 |
< |
produced comparable shear viscosity to swap RNEMD method. In Table |
| 624 |
< |
\ref{shearRate}, the names of the calculated samples are devided into |
| 625 |
< |
two parts. The first number refers to total slabs in one simulation |
| 626 |
< |
box. The second number refers to the swapping interval in swap method, or |
| 627 |
< |
in scale method the equilvalent swapping interval that the same |
| 628 |
< |
momentum flux would theoretically result in swap method. All the scale |
| 629 |
< |
method results were from simulations that had a scaling interval of 10 |
| 630 |
< |
time steps. The average molecular momentum gradients of these samples |
| 631 |
< |
are shown in Figure \ref{shearGrad}. |
| 621 |
> |
To construct the simulations, a simulation box consisting of 1000 |
| 622 |
> |
molecules were first equilibrated under ambient pressure and |
| 623 |
> |
temperature conditions using the isobaric-isothermal (NPT) |
| 624 |
> |
ensemble.\cite{melchionna93} A fixed volume was chosen to match the |
| 625 |
> |
average volume observed in the NPT simulations, and this was followed |
| 626 |
> |
by equilibration, first in the canonical (NVT) ensemble, followed by a |
| 627 |
> |
100~ps period under constant-NVE conditions without any momentum flux. |
| 628 |
> |
Another 100~ps was allowed to stabilize the system with an imposed |
| 629 |
> |
momentum transfer to create a gradient, and 1~ns was allotted for data |
| 630 |
> |
collection under RNEMD. |
| 631 |
|
|
| 632 |
< |
\begin{table*} |
| 633 |
< |
\begin{minipage}{\linewidth} |
| 634 |
< |
\begin{center} |
| 632 |
> |
In our simulations, the established temperature gradients were similar |
| 633 |
> |
to the previous work. Our simulation results at 318K are in good |
| 634 |
> |
agreement with those from Bedrov {\it et al.} (Table |
| 635 |
> |
\ref{spceThermal}). And both methods yield values in reasonable |
| 636 |
> |
agreement with experimental values. |
| 637 |
|
|
| 638 |
< |
\caption{Calculation results for shear viscosity of Lennard-Jones |
| 639 |
< |
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
| 640 |
< |
methods at various momentum exchange rates. Results in reduced |
| 641 |
< |
unit. Errors of calculations in parentheses. } |
| 642 |
< |
|
| 643 |
< |
\begin{tabular}{ccc} |
| 644 |
< |
\hline |
| 645 |
< |
Series & $\eta^*_{swap}$ & $\eta^*_{scale}$\\ |
| 646 |
< |
\hline |
| 647 |
< |
20-500 & 3.64(0.05) & 3.76(0.09)\\ |
| 648 |
< |
20-1000 & 3.52(0.16) & 3.66(0.06)\\ |
| 649 |
< |
20-2000 & 3.72(0.05) & 3.32(0.18)\\ |
| 650 |
< |
20-2500 & 3.42(0.06) & 3.43(0.08)\\ |
| 651 |
< |
\hline |
| 652 |
< |
\end{tabular} |
| 653 |
< |
\label{shearRate} |
| 654 |
< |
\end{center} |
| 655 |
< |
\end{minipage} |
| 638 |
> |
\begin{table*} |
| 639 |
> |
\begin{minipage}{\linewidth} |
| 640 |
> |
\begin{center} |
| 641 |
> |
|
| 642 |
> |
\caption{Thermal conductivity of SPC/E water under various |
| 643 |
> |
imposed thermal gradients. Uncertainties are indicated in |
| 644 |
> |
parentheses.} |
| 645 |
> |
|
| 646 |
> |
\begin{tabular}{|c|c|ccc|} |
| 647 |
> |
\hline |
| 648 |
> |
\multirow{2}{*}{$\langle T\rangle$(K)} & |
| 649 |
> |
\multirow{2}{*}{$\langle dT/dz\rangle$(K/\AA)} & |
| 650 |
> |
\multicolumn{3}{|c|}{$\lambda (\mathrm{W m}^{-1} |
| 651 |
> |
\mathrm{K}^{-1})$} \\ |
| 652 |
> |
& & This work & Previous simulations\cite{Bedrov:2000} & |
| 653 |
> |
Experiment\cite{WagnerKruse}\\ |
| 654 |
> |
\hline |
| 655 |
> |
\multirow{3}{*}{300} & 0.38 & 0.816(0.044) & & |
| 656 |
> |
\multirow{3}{*}{0.61}\\ |
| 657 |
> |
& 0.81 & 0.770(0.008) & & \\ |
| 658 |
> |
& 1.54 & 0.813(0.007) & & \\ |
| 659 |
> |
\hline |
| 660 |
> |
\multirow{2}{*}{318} & 0.75 & 0.750(0.032) & 0.784 & |
| 661 |
> |
\multirow{2}{*}{0.64}\\ |
| 662 |
> |
& 1.59 & 0.778(0.019) & 0.730 & \\ |
| 663 |
> |
\hline |
| 664 |
> |
\end{tabular} |
| 665 |
> |
\label{spceThermal} |
| 666 |
> |
\end{center} |
| 667 |
> |
\end{minipage} |
| 668 |
|
\end{table*} |
| 669 |
|
|
| 670 |
< |
\begin{figure} |
| 658 |
< |
\includegraphics[width=\linewidth]{shearGrad} |
| 659 |
< |
\caption{Average momentum gradients of shear viscosity simulations} |
| 660 |
< |
\label{shearGrad} |
| 661 |
< |
\end{figure} |
| 670 |
> |
\subsection{Crystalline Gold} |
| 671 |
|
|
| 672 |
+ |
To see how the method performed in a solid, we calculated thermal |
| 673 |
+ |
conductivities using two atomistic models for gold. Several different |
| 674 |
+ |
potential models have been developed that reasonably describe |
| 675 |
+ |
interactions in transition metals. In particular, the Embedded Atom |
| 676 |
+ |
Model (EAM)~\cite{PhysRevB.33.7983} and Sutton-Chen (SC)~\cite{Chen90} |
| 677 |
+ |
potential have been used to study a wide range of phenomena in both |
| 678 |
+ |
bulk materials and |
| 679 |
+ |
nanoparticles.\cite{ISI:000207079300006,Clancy:1992,Vardeman:2008fk,Vardeman-II:2001jn,ShibataT._ja026764r,Sankaranarayanan:2005lr,Chui:2003fk,Wang:2005qy,Medasani:2007uq} |
| 680 |
+ |
Both potentials are based on a model of a metal which treats the |
| 681 |
+ |
nuclei and core electrons as pseudo-atoms embedded in the electron |
| 682 |
+ |
density due to the valence electrons on all of the other atoms in the |
| 683 |
+ |
system. The SC potential has a simple form that closely resembles the |
| 684 |
+ |
Lennard Jones potential, |
| 685 |
+ |
\begin{equation} |
| 686 |
+ |
\label{eq:SCP1} |
| 687 |
+ |
U_{tot}=\sum _{i}\left[ \frac{1}{2}\sum _{j\neq i}D_{ij}V^{pair}_{ij}(r_{ij})-c_{i}D_{ii}\sqrt{\rho_{i}}\right] , |
| 688 |
+ |
\end{equation} |
| 689 |
+ |
where $V^{pair}_{ij}$ and $\rho_{i}$ are given by |
| 690 |
+ |
\begin{equation} |
| 691 |
+ |
\label{eq:SCP2} |
| 692 |
+ |
V^{pair}_{ij}(r)=\left( \frac{\alpha_{ij}}{r_{ij}}\right)^{n_{ij}}, \rho_{i}=\sum_{j\neq i}\left( \frac{\alpha_{ij}}{r_{ij}}\right) ^{m_{ij}}. |
| 693 |
+ |
\end{equation} |
| 694 |
+ |
$V^{pair}_{ij}$ is a repulsive pairwise potential that accounts for |
| 695 |
+ |
interactions between the pseudoatom cores. The $\sqrt{\rho_i}$ term in |
| 696 |
+ |
Eq. (\ref{eq:SCP1}) is an attractive many-body potential that models |
| 697 |
+ |
the interactions between the valence electrons and the cores of the |
| 698 |
+ |
pseudo-atoms. $D_{ij}$, $D_{ii}$ set the appropriate overall energy |
| 699 |
+ |
scale, $c_i$ scales the attractive portion of the potential relative |
| 700 |
+ |
to the repulsive interaction and $\alpha_{ij}$ is a length parameter |
| 701 |
+ |
that assures a dimensionless form for $\rho$. These parameters are |
| 702 |
+ |
tuned to various experimental properties such as the density, cohesive |
| 703 |
+ |
energy, and elastic moduli for FCC transition metals. The quantum |
| 704 |
+ |
Sutton-Chen (QSC) formulation matches these properties while including |
| 705 |
+ |
zero-point quantum corrections for different transition |
| 706 |
+ |
metals.\cite{PhysRevB.59.3527} The EAM functional forms differ |
| 707 |
+ |
slightly from SC but the overall method is very similar. |
| 708 |
+ |
|
| 709 |
+ |
In this work, we have utilized both the EAM and the QSC potentials to |
| 710 |
+ |
test the behavior of scaling RNEMD. |
| 711 |
+ |
|
| 712 |
+ |
A face-centered-cubic (FCC) lattice was prepared containing 2880 Au |
| 713 |
+ |
atoms (i.e. ${6\times 6\times 20}$ unit cells). Simulations were run |
| 714 |
+ |
both with and without isobaric-isothermal (NPT)~\cite{melchionna93} |
| 715 |
+ |
pre-equilibration at a target pressure of 1 atm. When equilibrated |
| 716 |
+ |
under NPT conditions, our simulation box expanded by approximately 1\% |
| 717 |
+ |
in volume. Following adjustment of the box volume, equilibrations in |
| 718 |
+ |
both the canonical and microcanonical ensembles were carried out. With |
| 719 |
+ |
the simulation cell divided evenly into 10 slabs, different thermal |
| 720 |
+ |
gradients were established by applying a set of target thermal |
| 721 |
+ |
transfer fluxes. |
| 722 |
+ |
|
| 723 |
+ |
The results for the thermal conductivity of gold are shown in Table |
| 724 |
+ |
\ref{AuThermal}. In these calculations, the end and middle slabs were |
| 725 |
+ |
excluded in thermal gradient linear regession. EAM predicts slightly |
| 726 |
+ |
larger thermal conductivities than QSC. However, both values are |
| 727 |
+ |
smaller than experimental value by a factor of more than 200. This |
| 728 |
+ |
behavior has been observed previously by Richardson and Clancy, and |
| 729 |
+ |
has been attributed to the lack of electronic contribution in these |
| 730 |
+ |
force fields.\cite{Clancy:1992} It should be noted that the density of |
| 731 |
+ |
the metal being simulated has an effect on thermal conductance. With |
| 732 |
+ |
an expanded lattice, lower thermal conductance is expected (and |
| 733 |
+ |
observed). We also observed a decrease in thermal conductance at |
| 734 |
+ |
higher temperatures, a trend that agrees with experimental |
| 735 |
+ |
measurements.\cite{AshcroftMermin} |
| 736 |
+ |
|
| 737 |
+ |
\begin{table*} |
| 738 |
+ |
\begin{minipage}{\linewidth} |
| 739 |
+ |
\begin{center} |
| 740 |
+ |
|
| 741 |
+ |
\caption{Calculated thermal conductivity of crystalline gold |
| 742 |
+ |
using two related force fields. Calculations were done at both |
| 743 |
+ |
experimental and equilibrated densities and at a range of |
| 744 |
+ |
temperatures and thermal flux rates. Uncertainties are |
| 745 |
+ |
indicated in parentheses. Richardson {\it et |
| 746 |
+ |
al.}\cite{Clancy:1992} give an estimate of 1.74 $\mathrm{W |
| 747 |
+ |
m}^{-1}\mathrm{K}^{-1}$ for EAM gold |
| 748 |
+ |
at a density of 19.263 g / cm$^3$.} |
| 749 |
+ |
|
| 750 |
+ |
\begin{tabular}{|c|c|c|cc|} |
| 751 |
+ |
\hline |
| 752 |
+ |
Force Field & $\rho$ (g/cm$^3$) & ${\langle T\rangle}$ (K) & |
| 753 |
+ |
$\langle dT/dz\rangle$ (K/\AA) & $\lambda (\mathrm{W m}^{-1} \mathrm{K}^{-1})$\\ |
| 754 |
+ |
\hline |
| 755 |
+ |
\multirow{7}{*}{QSC} & \multirow{3}{*}{19.188} & \multirow{3}{*}{300} & 1.44 & 1.10(0.06)\\ |
| 756 |
+ |
& & & 2.86 & 1.08(0.05)\\ |
| 757 |
+ |
& & & 5.14 & 1.15(0.07)\\\cline{2-5} |
| 758 |
+ |
& \multirow{4}{*}{19.263} & \multirow{2}{*}{300} & 2.31 & 1.25(0.06)\\ |
| 759 |
+ |
& & & 3.02 & 1.26(0.05)\\\cline{3-5} |
| 760 |
+ |
& & \multirow{2}{*}{575} & 3.02 & 1.02(0.07)\\ |
| 761 |
+ |
& & & 4.84 & 0.92(0.05)\\ |
| 762 |
+ |
\hline |
| 763 |
+ |
\multirow{8}{*}{EAM} & \multirow{3}{*}{19.045} & \multirow{3}{*}{300} & 1.24 & 1.24(0.16)\\ |
| 764 |
+ |
& & & 2.06 & 1.37(0.04)\\ |
| 765 |
+ |
& & & 2.55 & 1.41(0.07)\\\cline{2-5} |
| 766 |
+ |
& \multirow{5}{*}{19.263} & \multirow{3}{*}{300} & 1.06 & 1.45(0.13)\\ |
| 767 |
+ |
& & & 2.04 & 1.41(0.07)\\ |
| 768 |
+ |
& & & 2.41 & 1.53(0.10)\\\cline{3-5} |
| 769 |
+ |
& & \multirow{2}{*}{575} & 2.82 & 1.08(0.03)\\ |
| 770 |
+ |
& & & 4.14 & 1.08(0.05)\\ |
| 771 |
+ |
\hline |
| 772 |
+ |
\end{tabular} |
| 773 |
+ |
\label{AuThermal} |
| 774 |
+ |
\end{center} |
| 775 |
+ |
\end{minipage} |
| 776 |
+ |
\end{table*} |
| 777 |
+ |
|
| 778 |
+ |
\subsection{Thermal Conductance at the Au/H$_2$O interface} |
| 779 |
+ |
The most attractive aspect of the scaling approach for RNEMD is the |
| 780 |
+ |
ability to use the method in non-homogeneous systems, where molecules |
| 781 |
+ |
of different identities are segregated in different slabs. To test |
| 782 |
+ |
this application, we simulated a Gold (111) / water interface. To |
| 783 |
+ |
construct the interface, a box containing a lattice of 1188 Au atoms |
| 784 |
+ |
(with the 111 surface in the $+z$ and $-z$ directions) was allowed to |
| 785 |
+ |
relax under ambient temperature and pressure. A separate (but |
| 786 |
+ |
identically sized) box of SPC/E water was also equilibrated at ambient |
| 787 |
+ |
conditions. The two boxes were combined by removing all water |
| 788 |
+ |
molecules within 3 \AA radius of any gold atom. The final |
| 789 |
+ |
configuration contained 1862 SPC/E water molecules. |
| 790 |
+ |
|
| 791 |
+ |
The Spohr potential was adopted in depicting the interaction between |
| 792 |
+ |
metal atoms and water molecules.\cite{ISI:000167766600035} A similar |
| 793 |
+ |
protocol of equilibration to our water simulations was followed. We |
| 794 |
+ |
observed that the two phases developed large temperature differences |
| 795 |
+ |
even under a relatively low thermal flux. |
| 796 |
+ |
|
| 797 |
+ |
The low interfacial conductance is probably due to an acoustic |
| 798 |
+ |
impedance mismatch between the solid and the liquid |
| 799 |
+ |
phase.\cite{Cahill:793,RevModPhys.61.605} Experiments on the thermal |
| 800 |
+ |
conductivity of gold nanoparticles and nanorods in solvent and in |
| 801 |
+ |
glass cages have predicted values for $G$ between 100 and 350 |
| 802 |
+ |
(MW/m$^2$/K). The experiments typically have multiple gold surfaces |
| 803 |
+ |
that have been protected by a capping agent (citrate or CTAB) or which |
| 804 |
+ |
are in direct contact with various glassy solids. In these cases, the |
| 805 |
+ |
acoustic impedance mismatch would be substantially reduced, leading to |
| 806 |
+ |
much higher interfacial conductances. It is also possible, however, |
| 807 |
+ |
that the lack of electronic effects that gives rise to the low thermal |
| 808 |
+ |
conductivity of EAM gold is also causing a low reading for this |
| 809 |
+ |
particular interface. |
| 810 |
+ |
|
| 811 |
+ |
Under this low thermal conductance, both gold and water phase have |
| 812 |
+ |
sufficient time to eliminate temperature difference inside |
| 813 |
+ |
respectively (Figure \ref{interface} b). With indistinguishable |
| 814 |
+ |
temperature difference within respective phase, it is valid to assume |
| 815 |
+ |
that the temperature difference between gold and water on surface |
| 816 |
+ |
would be approximately the same as the difference between the gold and |
| 817 |
+ |
water phase. This assumption enables convenient calculation of $G$ |
| 818 |
+ |
using Eq. \ref{interfaceCalc} instead of measuring temperatures of |
| 819 |
+ |
thin layer of water and gold close enough to surface, which would have |
| 820 |
+ |
greater fluctuation and lower accuracy. Reported results (Table |
| 821 |
+ |
\ref{interfaceRes}) are of two orders of magnitude smaller than our |
| 822 |
+ |
calculations on homogeneous systems, and thus have larger relative |
| 823 |
+ |
errors than our calculation results on homogeneous systems. |
| 824 |
+ |
|
| 825 |
|
\begin{figure} |
| 826 |
< |
\includegraphics[width=\linewidth]{shearTempScale} |
| 827 |
< |
\caption{Temperature profile for scaling RNEMD simulation.} |
| 828 |
< |
\label{shearTempScale} |
| 826 |
> |
\includegraphics[width=\linewidth]{interface} |
| 827 |
> |
\caption{Temperature profiles of the Gold / Water interface at four |
| 828 |
> |
different values for the thermal flux. Temperatures for slabs |
| 829 |
> |
either in the gold or in the water show no significant differences, |
| 830 |
> |
although there is a large discontinuity between the materials |
| 831 |
> |
because of the relatively low interfacial thermal conductivity.} |
| 832 |
> |
\label{interface} |
| 833 |
|
\end{figure} |
| 668 |
– |
However, observations of temperatures along three dimensions show that |
| 669 |
– |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
| 670 |
– |
two slabs which were scaled. Figure \ref{shearTempScale} indicate that with |
| 671 |
– |
relatively large imposed momentum flux, the temperature difference among $x$ |
| 672 |
– |
and the other two dimensions was significant. This would result from the |
| 673 |
– |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
| 674 |
– |
momentum gradient is set up, $P_c^x$ would be roughly stable |
| 675 |
– |
($<0$). Consequently, scaling factor $x$ would most probably larger |
| 676 |
– |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
| 677 |
– |
keep increase after most scaling steps. And if there is not enough time |
| 678 |
– |
for the kinetic energy to exchange among different dimensions and |
| 679 |
– |
different slabs, the system would finally build up temperature |
| 680 |
– |
(kinetic energy) difference among the three dimensions. Also, between |
| 681 |
– |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
| 682 |
– |
are closer to neighbor slabs. This is due to momentum transfer along |
| 683 |
– |
$z$ dimension between slabs. |
| 834 |
|
|
| 835 |
< |
Although results between scaling and swapping methods are comparable, |
| 836 |
< |
the inherent temperature inhomogeneity even in relatively low imposed |
| 837 |
< |
exchange momentum flux simulations makes scaling RNEMD method less |
| 838 |
< |
attractive than swapping RNEMD in shear viscosity calculation. |
| 835 |
> |
\begin{table*} |
| 836 |
> |
\begin{minipage}{\linewidth} |
| 837 |
> |
\begin{center} |
| 838 |
> |
|
| 839 |
> |
\caption{Computed interfacial thermal conductivity ($G$) values |
| 840 |
> |
for the Au(111) / water interface at ${\langle T\rangle \sim}$ |
| 841 |
> |
300K using a range of energy fluxes. Uncertainties are |
| 842 |
> |
indicated in parentheses. } |
| 843 |
> |
|
| 844 |
> |
\begin{tabular}{|cccc| } |
| 845 |
> |
\hline |
| 846 |
> |
$J_z$ (MW/m$^2$) & $\langle T_{gold} \rangle$ (K) & $\langle |
| 847 |
> |
T_{water} \rangle$ (K) & $G$ |
| 848 |
> |
(MW/m$^2$/K)\\ |
| 849 |
> |
\hline |
| 850 |
> |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
| 851 |
> |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
| 852 |
> |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
| 853 |
> |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
| 854 |
> |
\hline |
| 855 |
> |
\end{tabular} |
| 856 |
> |
\label{interfaceRes} |
| 857 |
> |
\end{center} |
| 858 |
> |
\end{minipage} |
| 859 |
> |
\end{table*} |
| 860 |
|
|
| 861 |
+ |
|
| 862 |
|
\section{Conclusions} |
| 863 |
|
NIVS-RNEMD simulation method is developed and tested on various |
| 864 |
< |
systems. Simulation results demonstrate its validity of thermal |
| 865 |
< |
conductivity calculations. NIVS-RNEMD improves non-Boltzmann-Maxwell |
| 866 |
< |
distributions existing in previous RNEMD methods, and extends its |
| 867 |
< |
applicability to interfacial systems. NIVS-RNEMD has also limited |
| 868 |
< |
application on shear viscosity calculations, but under high momentum |
| 869 |
< |
flux, it could cause temperature difference among different |
| 870 |
< |
dimensions. Modification is necessary to extend the applicability of |
| 871 |
< |
NIVS-RNEMD in shear viscosity calculations. |
| 864 |
> |
systems. Simulation results demonstrate its validity in thermal |
| 865 |
> |
conductivity calculations, from Lennard-Jones fluid to multi-atom |
| 866 |
> |
molecule like water and metal crystals. NIVS-RNEMD improves |
| 867 |
> |
non-Boltzmann-Maxwell distributions, which exist inb previous RNEMD |
| 868 |
> |
methods. Furthermore, it develops a valid means for unphysical thermal |
| 869 |
> |
transfer between different species of molecules, and thus extends its |
| 870 |
> |
applicability to interfacial systems. Our calculation of gold/water |
| 871 |
> |
interfacial thermal conductivity demonstrates this advantage over |
| 872 |
> |
previous RNEMD methods. NIVS-RNEMD has also limited application on |
| 873 |
> |
shear viscosity calculations, but could cause temperature difference |
| 874 |
> |
among different dimensions under high momentum flux. Modification is |
| 875 |
> |
necessary to extend the applicability of NIVS-RNEMD in shear viscosity |
| 876 |
> |
calculations. |
| 877 |
|
|
| 878 |
|
\section{Acknowledgments} |
| 879 |
< |
Support for this project was provided by the National Science |
| 880 |
< |
Foundation under grant CHE-0848243. Computational time was provided by |
| 881 |
< |
the Center for Research Computing (CRC) at the University of Notre |
| 882 |
< |
Dame. \newpage |
| 879 |
> |
The authors would like to thank Craig Tenney and Ed Maginn for many |
| 880 |
> |
helpful discussions. Support for this project was provided by the |
| 881 |
> |
National Science Foundation under grant CHE-0848243. Computational |
| 882 |
> |
time was provided by the Center for Research Computing (CRC) at the |
| 883 |
> |
University of Notre Dame. |
| 884 |
> |
\newpage |
| 885 |
|
|
| 707 |
– |
\bibliographystyle{jcp2} |
| 886 |
|
\bibliography{nivsRnemd} |
| 887 |
+ |
|
| 888 |
|
\end{doublespace} |
| 889 |
|
\end{document} |
| 890 |
|
|
