| 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 |
| 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 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,Muller-Plathe:1999ek} 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. The transport coefficients (shear |
| 81 |
< |
viscosity and thermal conductivity) are easily obtained by assuming |
| 82 |
< |
linear response of the system, |
| 73 |
> |
the simulation cell.\cite{MullerPlathe:1997xw,ISI:000080382700030} The |
| 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 & = & \lambda \frac{\partial T}{\partial z} |
| 84 |
> |
j_z(p_x) & = & -\eta \frac{\partial v_x}{\partial z}\\ |
| 85 |
> |
J_z & = & \lambda \frac{\partial T}{\partial z} |
| 86 |
|
\end{eqnarray} |
| 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} |
| 87 |
> |
RNEMD has been widely used to provide computational estimates of |
| 88 |
> |
thermal conductivities and shear viscosities in a wide range of |
| 89 |
> |
materials, from liquid copper to both monatomic and molecular fluids |
| 90 |
> |
(e.g. ionic |
| 91 |
> |
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} |
| 92 |
|
|
| 93 |
< |
RNEMD is preferable in many ways to the forward NEMD methods because |
| 94 |
< |
it imposes what is typically difficult to measure (a flux or stress) |
| 95 |
< |
and it is typically much easier to compute momentum gradients or |
| 96 |
< |
strains (the response). For similar reasons, RNEMD is also preferable |
| 97 |
< |
to slowly-converging equilibrium methods for measuring thermal |
| 98 |
< |
conductivity and shear viscosity (using Green-Kubo relations or the |
| 99 |
< |
Helfand moment approach of Viscardy {\it et |
| 93 |
> |
\begin{figure} |
| 94 |
> |
\includegraphics[width=\linewidth]{thermalDemo} |
| 95 |
> |
\caption{RNEMD methods impose an unphysical transfer of momentum or |
| 96 |
> |
kinetic energy between a ``hot'' slab and a ``cold'' slab in the |
| 97 |
> |
simulation box. The molecular system responds to this imposed flux |
| 98 |
> |
by generating a momentum or temperature gradient. The slope of the |
| 99 |
> |
gradient can then be used to compute transport properties (e.g. |
| 100 |
> |
shear viscosity and thermal conductivity).} |
| 101 |
> |
\label{thermalDemo} |
| 102 |
> |
\end{figure} |
| 103 |
> |
|
| 104 |
> |
RNEMD is preferable in many ways to the forward NEMD |
| 105 |
> |
methods\cite{ISI:A1988Q205300014,hess:209,Vasquez:2004fk,backer:154503,ISI:000266247600008} |
| 106 |
> |
because it imposes what is typically difficult to measure (a flux or |
| 107 |
> |
stress) and it is typically much easier to compute the response |
| 108 |
> |
(momentum gradients or strains. For similar reasons, RNEMD is also |
| 109 |
> |
preferable to slowly-converging equilibrium methods for measuring |
| 110 |
> |
thermal conductivity and shear viscosity (using Green-Kubo |
| 111 |
> |
relations\cite{daivis:541,mondello:9327} or the Helfand moment |
| 112 |
> |
approach of Viscardy {\it et |
| 113 |
|
al.}\cite{Viscardy:2007bh,Viscardy:2007lq}) because these rely on |
| 114 |
|
computing difficult to measure quantities. |
| 115 |
|
|
| 119 |
|
typically samples from the same manifold of states in the |
| 120 |
|
microcanonical ensemble. |
| 121 |
|
|
| 122 |
< |
Recently, Tenney and Maginn have discovered some problems with the |
| 123 |
< |
original RNEMD swap technique. Notably, large momentum fluxes |
| 124 |
< |
(equivalent to frequent momentum swaps between the slabs) can result |
| 125 |
< |
in "notched", "peaked" and generally non-thermal momentum |
| 126 |
< |
distributions in the two slabs, as well as non-linear thermal and |
| 127 |
< |
velocity distributions along the direction of the imposed flux ($z$). |
| 128 |
< |
Tenney and Maginn obtained reasonable limits on imposed flux and |
| 129 |
< |
self-adjusting metrics for retaining the usability of the method. |
| 122 |
> |
Recently, Tenney and Maginn\cite{Maginn:2010} have discovered |
| 123 |
> |
some problems with the original RNEMD swap technique. Notably, large |
| 124 |
> |
momentum fluxes (equivalent to frequent momentum swaps between the |
| 125 |
> |
slabs) can result in ``notched'', ``peaked'' and generally non-thermal |
| 126 |
> |
momentum distributions in the two slabs, as well as non-linear thermal |
| 127 |
> |
and velocity distributions along the direction of the imposed flux |
| 128 |
> |
($z$). Tenney and Maginn obtained reasonable limits on imposed flux |
| 129 |
> |
and self-adjusting metrics for retaining the usability of the method. |
| 130 |
|
|
| 131 |
|
In this paper, we develop and test a method for non-isotropic velocity |
| 132 |
< |
scaling (NIVS-RNEMD) which retains the desirable features of RNEMD |
| 132 |
> |
scaling (NIVS) which retains the desirable features of RNEMD |
| 133 |
|
(conservation of linear momentum and total energy, compatibility with |
| 134 |
|
periodic boundary conditions) while establishing true thermal |
| 135 |
< |
distributions in each of the two slabs. In the next section, we |
| 136 |
< |
develop the method for determining the scaling constraints. We then |
| 137 |
< |
test the method on both single component, multi-component, and |
| 138 |
< |
non-isotropic mixtures and show that it is capable of providing |
| 135 |
> |
distributions in each of the two slabs. In the next section, we |
| 136 |
> |
present the method for determining the scaling constraints. We then |
| 137 |
> |
test the method on both liquids and solids as well as a non-isotropic |
| 138 |
> |
liquid-solid interface and show that it is capable of providing |
| 139 |
|
reasonable estimates of the thermal conductivity and shear viscosity |
| 140 |
< |
in these cases. |
| 140 |
> |
in all of these cases. |
| 141 |
|
|
| 142 |
|
\section{Methodology} |
| 143 |
< |
We retain the basic idea of Muller-Plathe's RNEMD method; the periodic |
| 144 |
< |
system is partitioned into a series of thin slabs along a particular |
| 143 |
> |
We retain the basic idea of M\"{u}ller-Plathe's RNEMD method; the |
| 144 |
> |
periodic system is partitioned into a series of thin slabs along one |
| 145 |
|
axis ($z$). One of the slabs at the end of the periodic box is |
| 146 |
|
designated the ``hot'' slab, while the slab in the center of the box |
| 147 |
|
is designated the ``cold'' slab. The artificial momentum flux will be |
| 149 |
|
hot slab. |
| 150 |
|
|
| 151 |
|
Rather than using momentum swaps, we use a series of velocity scaling |
| 152 |
< |
moves. For molecules $\{i\}$ located within the cold slab, |
| 152 |
> |
moves. For molecules $\{i\}$ located within the cold slab, |
| 153 |
|
\begin{equation} |
| 154 |
< |
\vec{v}_i \leftarrow \left( \begin{array}{c} |
| 155 |
< |
x \\ |
| 156 |
< |
y \\ |
| 157 |
< |
z \\ |
| 154 |
> |
\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
| 155 |
> |
x & 0 & 0 \\ |
| 156 |
> |
0 & y & 0 \\ |
| 157 |
> |
0 & 0 & z \\ |
| 158 |
|
\end{array} \right) \cdot \vec{v}_i |
| 159 |
|
\end{equation} |
| 160 |
< |
where ${x, y, z}$ are a set of 3 scaling variables for each of the |
| 161 |
< |
three directions in the system. Likewise, the molecules $\{j\}$ |
| 162 |
< |
located in the hot slab will see a concomitant scaling of velocities, |
| 160 |
> |
where ${x, y, z}$ are a set of 3 velocity-scaling variables for each |
| 161 |
> |
of the three directions in the system. Likewise, the molecules |
| 162 |
> |
$\{j\}$ located in the hot slab will see a concomitant scaling of |
| 163 |
> |
velocities, |
| 164 |
|
\begin{equation} |
| 165 |
< |
\vec{v}_j \leftarrow \left( \begin{array}{c} |
| 166 |
< |
x^\prime \\ |
| 167 |
< |
y^\prime \\ |
| 168 |
< |
z^\prime \\ |
| 165 |
> |
\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
| 166 |
> |
x^\prime & 0 & 0 \\ |
| 167 |
> |
0 & y^\prime & 0 \\ |
| 168 |
> |
0 & 0 & z^\prime \\ |
| 169 |
|
\end{array} \right) \cdot \vec{v}_j |
| 170 |
|
\end{equation} |
| 171 |
|
|
| 172 |
|
Conservation of linear momentum in each of the three directions |
| 173 |
< |
($\alpha = x,y,z$) ties the values of the hot and cold bin scaling |
| 173 |
> |
($\alpha = x,y,z$) ties the values of the hot and cold scaling |
| 174 |
|
parameters together: |
| 175 |
|
\begin{equation} |
| 176 |
|
P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
| 177 |
|
\end{equation} |
| 178 |
|
where |
| 179 |
< |
\begin{equation} |
| 151 |
< |
\begin{array}{rcl} |
| 179 |
> |
\begin{eqnarray} |
| 180 |
|
P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
| 181 |
< |
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha \\ |
| 154 |
< |
\end{array} |
| 181 |
> |
P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha |
| 182 |
|
\label{eq:momentumdef} |
| 183 |
< |
\end{equation} |
| 183 |
> |
\end{eqnarray} |
| 184 |
|
Therefore, for each of the three directions, the hot scaling |
| 185 |
|
parameters are a simple function of the cold scaling parameters and |
| 186 |
|
the instantaneous linear momentum in each of the two slabs. |
| 197 |
|
Conservation of total energy also places constraints on the scaling: |
| 198 |
|
\begin{equation} |
| 199 |
|
\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
| 200 |
< |
\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha. |
| 200 |
> |
\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha |
| 201 |
|
\end{equation} |
| 202 |
< |
where the kinetic energies, $K_h^\alpha$ and $K_c^\alpha$, are computed |
| 203 |
< |
for each of the three directions in a similar manner to the linear momenta |
| 204 |
< |
(Eq. \ref{eq:momentumdef}). Substituting in the expressions for the |
| 205 |
< |
hot scaling parameters ($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), |
| 206 |
< |
we obtain the {\it constraint ellipsoid equation}: |
| 202 |
> |
where the translational kinetic energies, $K_h^\alpha$ and |
| 203 |
> |
$K_c^\alpha$, are computed for each of the three directions in a |
| 204 |
> |
similar manner to the linear momenta (Eq. \ref{eq:momentumdef}). |
| 205 |
> |
Substituting in the expressions for the hot scaling parameters |
| 206 |
> |
($\alpha^\prime$) from Eq. (\ref{eq:hotcoldscaling}), we obtain the |
| 207 |
> |
{\it constraint ellipsoid}: |
| 208 |
|
\begin{equation} |
| 209 |
< |
\sum_{\alpha = x,y,z} a_\alpha \alpha^2 + b_\alpha \alpha + c_\alpha = 0, |
| 209 |
> |
\sum_{\alpha = x,y,z} \left( a_\alpha \alpha^2 + b_\alpha \alpha + |
| 210 |
> |
c_\alpha \right) = 0 |
| 211 |
|
\label{eq:constraintEllipsoid} |
| 212 |
|
\end{equation} |
| 213 |
|
where the constants are obtained from the instantaneous values of the |
| 214 |
|
linear momenta and kinetic energies for the hot and cold slabs, |
| 215 |
< |
\begin{equation} |
| 187 |
< |
\begin{array}{rcl} |
| 215 |
> |
\begin{eqnarray} |
| 216 |
|
a_\alpha & = &\left(K_c^\alpha + K_h^\alpha |
| 217 |
|
\left(p_\alpha\right)^2\right) \\ |
| 218 |
|
b_\alpha & = & -2 K_h^\alpha p_\alpha \left( 1 + p_\alpha\right) \\ |
| 219 |
< |
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha \\ |
| 192 |
< |
\end{array} |
| 219 |
> |
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
| 220 |
|
\label{eq:constraintEllipsoidConsts} |
| 221 |
< |
\end{equation} |
| 222 |
< |
This ellipsoid equation defines the set of cold slab scaling |
| 223 |
< |
parameters which can be applied while preserving both linear momentum |
| 224 |
< |
in all three directions as well as translational kinetic energy. |
| 221 |
> |
\end{eqnarray} |
| 222 |
> |
This ellipsoid (Eq. \ref{eq:constraintEllipsoid}) defines the set of |
| 223 |
> |
cold slab scaling parameters which, when applied, preserve the linear |
| 224 |
> |
momentum of the system in all three directions as well as total |
| 225 |
> |
kinetic energy. |
| 226 |
|
|
| 227 |
< |
The goal of using velocity scaling variables is to transfer linear |
| 228 |
< |
momentum or kinetic energy from the cold slab to the hot slab. If the |
| 229 |
< |
hot and cold slabs are separated along the z-axis, the energy flux is |
| 230 |
< |
given simply by the decrease in kinetic energy of the cold bin: |
| 227 |
> |
The goal of using these velocity scaling variables is to transfer |
| 228 |
> |
linear momentum or kinetic energy from the cold slab to the hot slab. |
| 229 |
> |
If the hot and cold slabs are separated along the z-axis, the energy |
| 230 |
> |
flux is given simply by the decrease in kinetic energy of the cold |
| 231 |
> |
bin: |
| 232 |
|
\begin{equation} |
| 233 |
< |
(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z |
| 233 |
> |
(1-x^2) K_c^x + (1-y^2) K_c^y + (1-z^2) K_c^z = J_z\Delta t |
| 234 |
|
\end{equation} |
| 235 |
|
The expression for the energy flux can be re-written as another |
| 236 |
|
ellipsoid centered on $(x,y,z) = 0$: |
| 237 |
|
\begin{equation} |
| 238 |
< |
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) |
| 238 |
> |
\sum_{\alpha = x,y,z} K_c^\alpha \alpha^2 = \sum_{\alpha = x,y,z} |
| 239 |
> |
K_c^\alpha -J_z \Delta t |
| 240 |
|
\label{eq:fluxEllipsoid} |
| 241 |
|
\end{equation} |
| 242 |
< |
The spatial extent of the {\it flux ellipsoid} is governed both by a |
| 243 |
< |
targetted value, $J_z$ as well as the instantaneous values of the |
| 244 |
< |
kinetic energy components in the hot bin. |
| 242 |
> |
The spatial extent of the {\it thermal flux ellipsoid} is governed |
| 243 |
> |
both by the target flux, $J_z$ as well as the instantaneous values of |
| 244 |
> |
the kinetic energy components in the cold bin. |
| 245 |
|
|
| 246 |
|
To satisfy an energetic flux as well as the conservation constraints, |
| 247 |
< |
it is sufficient to determine the points ${x,y,z}$ which lie on both |
| 247 |
> |
we must determine the points ${x,y,z}$ that lie on both the constraint |
| 248 |
> |
ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the flux ellipsoid |
| 249 |
> |
(Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of the two |
| 250 |
> |
ellipsoids in 3-dimensional space. |
| 251 |
> |
|
| 252 |
> |
\begin{figure} |
| 253 |
> |
\includegraphics[width=\linewidth]{ellipsoids} |
| 254 |
> |
\caption{Velocity scaling coefficients which maintain both constant |
| 255 |
> |
energy and constant linear momentum of the system lie on the surface |
| 256 |
> |
of the {\it constraint ellipsoid} while points which generate the |
| 257 |
> |
target momentum flux lie on the surface of the {\it flux ellipsoid}. |
| 258 |
> |
The velocity distributions in the cold bin are scaled by only those |
| 259 |
> |
points which lie on both ellipsoids.} |
| 260 |
> |
\label{ellipsoids} |
| 261 |
> |
\end{figure} |
| 262 |
> |
|
| 263 |
> |
Since ellipsoids can be expressed as polynomials up to second order in |
| 264 |
> |
each of the three coordinates, finding the the intersection points of |
| 265 |
> |
two ellipsoids is isomorphic to finding the roots a polynomial of |
| 266 |
> |
degree 16. There are a number of polynomial root-finding methods in |
| 267 |
> |
the literature, [CITATIONS NEEDED] but numerically finding the roots |
| 268 |
> |
of high-degree polynomials is generally an ill-conditioned |
| 269 |
> |
problem.[CITATION NEEDED] One way around this is to try to maintain |
| 270 |
> |
velocity scalings that are {\it as isotropic as possible}. To do |
| 271 |
> |
this, we impose $x=y$, and to treat both the constraint and flux |
| 272 |
> |
ellipsoids as 2-dimensional ellipses. In reduced dimensionality, the |
| 273 |
> |
intersecting-ellipse problem reduces to finding the roots of |
| 274 |
> |
polynomials of degree 4. |
| 275 |
> |
|
| 276 |
> |
Depending on the target flux and current velocity distributions, the |
| 277 |
> |
ellipsoids can have between 0 and 4 intersection points. If there are |
| 278 |
> |
no intersection points, it is not possible to satisfy the constraints |
| 279 |
> |
while performing a non-equilibrium scaling move, and no change is made |
| 280 |
> |
to the dynamics. |
| 281 |
> |
|
| 282 |
> |
With multiple intersection points, any of the scaling points will |
| 283 |
> |
conserve the linear momentum and kinetic energy of the system and will |
| 284 |
> |
generate the correct target flux. Although this method is inherently |
| 285 |
> |
non-isotropic, the goal is still to maintain the system as close to an |
| 286 |
> |
isotropic fluid as possible. With this in mind, we would like the |
| 287 |
> |
kinetic energies in the three different directions could become as |
| 288 |
> |
close as each other as possible after each scaling. Simultaneously, |
| 289 |
> |
one would also like each scaling as gentle as possible, i.e. ${x,y,z |
| 290 |
> |
\rightarrow 1}$, in order to avoid large perturbation to the system. |
| 291 |
> |
To do this, we pick the intersection point which maintains the scaling |
| 292 |
> |
variables ${x=y, z}$ as well as the ratio of kinetic energies |
| 293 |
> |
${K_c^z/K_c^x, K_c^z/K_c^y}$ as close as possible to 1. |
| 294 |
> |
|
| 295 |
> |
After the valid scaling parameters are arrived at by solving geometric |
| 296 |
> |
intersection problems in $x, y, z$ space in order to obtain cold slab |
| 297 |
> |
scaling parameters, Eq. (\ref{eq:hotcoldscaling}) is used to |
| 298 |
> |
determine the conjugate hot slab scaling variables. |
| 299 |
> |
|
| 300 |
> |
\subsection{Introducing shear stress via velocity scaling} |
| 301 |
> |
Rather than using this method to induce a thermal flux, it is possible |
| 302 |
> |
to use the random fluctuations of the average momentum in each of the |
| 303 |
> |
bins to induce a momentum flux. Doing this repeatedly will create a |
| 304 |
> |
shear stress on the system which will respond with an easily-measured |
| 305 |
> |
strain. The momentum flux (say along the $x$-direction) may be |
| 306 |
> |
defined as: |
| 307 |
> |
\begin{equation} |
| 308 |
> |
(1-x) P_c^x = j_z(p_x)\Delta t |
| 309 |
> |
\label{eq:fluxPlane} |
| 310 |
> |
\end{equation} |
| 311 |
> |
This {\it momentum flux plane} is perpendicular to the $x$-axis, with |
| 312 |
> |
its position governed both by a target value, $j_z(p_x)$ as well as |
| 313 |
> |
the instantaneous value of the momentum along the $x$-direction. |
| 314 |
> |
|
| 315 |
> |
In order to satisfy a momentum flux as well as the conservation |
| 316 |
> |
constraints, we must determine the points ${x,y,z}$ which lie on both |
| 317 |
|
the constraint ellipsoid (Eq. \ref{eq:constraintEllipsoid}) and the |
| 318 |
< |
flux ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
| 319 |
< |
the two ellipsoids in 3-dimensional space. |
| 318 |
> |
flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of an |
| 319 |
> |
ellipsoid and a plane in 3-dimensional space. |
| 320 |
|
|
| 321 |
< |
One may also define momentum flux (say along the x-direction) as: |
| 321 |
> |
In the case of momentum flux transfer, we also impose another |
| 322 |
> |
constraint to set the kinetic energy transfer as zero. In another |
| 323 |
> |
word, we apply Eq. \ref{eq:fluxEllipsoid} and let ${J_z = 0}$. With |
| 324 |
> |
one variable fixed by Eq. \ref{eq:fluxPlane}, one now have a similar |
| 325 |
> |
set of quartic equations to the above kinetic energy transfer problem. |
| 326 |
> |
|
| 327 |
> |
\section{Computational Details} |
| 328 |
> |
|
| 329 |
> |
We have implemented this methodology in our molecular dynamics |
| 330 |
> |
code,\cite{Meineke:2005gd} by performing the NIVS scaling moves after |
| 331 |
> |
each MD step. We have tested it for a variety of different |
| 332 |
> |
situations, including homogeneous fluids (Lennard-Jones and SPC/E |
| 333 |
> |
water), crystalline solids (EAM and Sutton-Chen models for Gold), and |
| 334 |
> |
heterogeneous interfaces (QSC gold - SPC/E water). The last of these |
| 335 |
> |
systems would have been very difficult to study using previous RNEMD |
| 336 |
> |
methods, but using velocity scaling moves, we can even obtain |
| 337 |
> |
estimates of the interfacial thermal conductivity. |
| 338 |
> |
|
| 339 |
> |
\subsection{Lennard-Jones Fluid} |
| 340 |
> |
|
| 341 |
> |
2592 Lennard-Jones atoms were placed in an orthorhombic cell |
| 342 |
> |
${10.06\sigma \times 10.06\sigma \times 30.18\sigma}$ on a side. The |
| 343 |
> |
reduced density ${\rho^* = \rho\sigma^3}$ was thus 0.85, which enabled |
| 344 |
> |
direct comparison between our results and previous methods. These |
| 345 |
> |
simulations were carried out with a reduced timestep ${\tau^* = |
| 346 |
> |
4.6\times10^{-4}}$. For the shear viscosity calculation, the mean |
| 347 |
> |
temperature was ${T^* = k_B T/\varepsilon = 0.72}$. Simulations were |
| 348 |
> |
first thermalized in canonical ensemble (NVT), then equilibrated in |
| 349 |
> |
microcanonical ensemble (NVE) before introducing any non-equilibrium |
| 350 |
> |
method. |
| 351 |
> |
|
| 352 |
> |
We have compared the momentum gradients established using our method |
| 353 |
> |
to those obtained using the original M\"{u}ller-Plathe swapping |
| 354 |
> |
algorithm.\cite{ISI:000080382700030} In both cases, the simulation box |
| 355 |
> |
was divided into ${N = 20}$ slabs. In the swapping algorithm, the top |
| 356 |
> |
slab $(n = 1)$ exchanges the most negative $x$ momentum with the most |
| 357 |
> |
positive $x$ momentum in the center slab $(n = N/2 + 1)$. The rate at |
| 358 |
> |
which the swapping moves are carried out defines the momentum or |
| 359 |
> |
thermal flux between the two slabs. In their work, Tenney {\it et |
| 360 |
> |
al.}\cite{Maginn:2010} found problematic behavior with large swap |
| 361 |
> |
frequencies. |
| 362 |
> |
|
| 363 |
> |
According to each result from swapping RNEMD, scaling RNEMD |
| 364 |
> |
simulations were run with the target momentum flux set to produce a |
| 365 |
> |
similar momentum flux, and consequently shear rate. Furthermore, |
| 366 |
> |
various scaling frequencies can be tested for one single swapping |
| 367 |
> |
rate. To test the temperature homogeneity in our system of swapping |
| 368 |
> |
and scaling methods, temperatures of different dimensions in all the |
| 369 |
> |
slabs were observed. Most of the simulations include $10^5$ steps of |
| 370 |
> |
equilibration without imposing momentum flux, $10^5$ steps of |
| 371 |
> |
stablization with imposing unphysical momentum transfer, and $10^6$ |
| 372 |
> |
steps of data collection under RNEMD. For relatively high momentum |
| 373 |
> |
flux simulations, ${5\times10^5}$ step data collection is sufficient. |
| 374 |
> |
For some low momentum flux simulations, ${2\times10^6}$ steps were |
| 375 |
> |
necessary. |
| 376 |
> |
|
| 377 |
> |
After each simulation, the shear viscosity was calculated in reduced |
| 378 |
> |
unit. The momentum flux was calculated with total unphysical |
| 379 |
> |
transferred momentum ${P_x}$ and data collection time $t$: |
| 380 |
|
\begin{equation} |
| 381 |
< |
(1-x) P_c^x = j_z(p_x) |
| 381 |
> |
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
| 382 |
> |
\end{equation} |
| 383 |
> |
where $L_x$ and $L_y$ denotes $x$ and $y$ lengths of the simulation |
| 384 |
> |
box, and physical momentum transfer occurs in two ways due to our |
| 385 |
> |
periodic boundary condition settings. And the velocity gradient |
| 386 |
> |
${\langle \partial v_x /\partial z \rangle}$ can be obtained by a |
| 387 |
> |
linear regression of the velocity profile. From the shear viscosity |
| 388 |
> |
$\eta$ calculated with the above parameters, one can further convert |
| 389 |
> |
it into reduced unit ${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$. |
| 390 |
> |
|
| 391 |
> |
For thermal conductivity calculations, simulations were first run under |
| 392 |
> |
reduced temperature ${\langle T^*\rangle = 0.72}$ in NVE |
| 393 |
> |
ensemble. Muller-Plathe's algorithm was adopted in the swapping |
| 394 |
> |
method. Under identical simulation box parameters with our shear |
| 395 |
> |
viscosity calculations, in each swap, the top slab exchanges all three |
| 396 |
> |
translational momentum components of the molecule with least kinetic |
| 397 |
> |
energy with the same components of the molecule in the center slab |
| 398 |
> |
with most kinetic energy, unless this ``coldest'' molecule in the |
| 399 |
> |
``hot'' slab is still ``hotter'' than the ``hottest'' molecule in the |
| 400 |
> |
``cold'' slab. According to swapping RNEMD results, target energy flux |
| 401 |
> |
for scaling RNEMD simulations can be set. Also, various scaling |
| 402 |
> |
frequencies can be tested for one target energy flux. To compare the |
| 403 |
> |
performance between swapping and scaling method, distributions of |
| 404 |
> |
velocity and speed in different slabs were observed. |
| 405 |
> |
|
| 406 |
> |
For each swapping rate, thermal conductivity was calculated in reduced |
| 407 |
> |
unit. The energy flux was calculated similarly to the momentum flux, |
| 408 |
> |
with total unphysical transferred energy ${E_{total}}$ and data collection |
| 409 |
> |
time $t$: |
| 410 |
> |
\begin{equation} |
| 411 |
> |
J_z = \frac{E_{total}}{2 t L_x L_y} |
| 412 |
> |
\end{equation} |
| 413 |
> |
And the temperature gradient ${\langle\partial T/\partial z\rangle}$ |
| 414 |
> |
can be obtained by a linear regression of the temperature |
| 415 |
> |
profile. From the thermal conductivity $\lambda$ calculated, one can |
| 416 |
> |
further convert it into reduced unit ${\lambda^*=\lambda \sigma^2 |
| 417 |
> |
m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$. |
| 418 |
> |
|
| 419 |
> |
\subsection{ Water / Metal Thermal Conductivity} |
| 420 |
> |
Another series of our simulation is the calculation of interfacial |
| 421 |
> |
thermal conductivity of a Au/H$_2$O system. Respective calculations of |
| 422 |
> |
liquid water (Extended Simple Point Charge model) and crystal gold |
| 423 |
> |
thermal conductivity were performed and compared with current results |
| 424 |
> |
to ensure the validity of NIVS-RNEMD. After that, a mixture system was |
| 425 |
> |
simulated. |
| 426 |
> |
|
| 427 |
> |
For thermal conductivity calculation of bulk water, a simulation box |
| 428 |
> |
consisting of 1000 molecules were first equilibrated under ambient |
| 429 |
> |
pressure and temperature conditions using NPT ensemble, followed by |
| 430 |
> |
equilibration in fixed volume (NVT). The system was then equilibrated in |
| 431 |
> |
microcanonical ensemble (NVE). Also in NVE ensemble, establishing a |
| 432 |
> |
stable thermal gradient was followed. The simulation box was under |
| 433 |
> |
periodic boundary condition and devided into 10 slabs. Data collection |
| 434 |
> |
process was similar to Lennard-Jones fluid system. |
| 435 |
> |
|
| 436 |
> |
Thermal conductivity calculation of bulk crystal gold used a similar |
| 437 |
> |
protocol. Two types of force field parameters, Embedded Atom Method |
| 438 |
> |
(EAM) and Quantum Sutten-Chen (QSC) force field were used |
| 439 |
> |
respectively. The face-centered cubic crystal simulation box consists of |
| 440 |
> |
2880 Au atoms. The lattice was first allowed volume change to relax |
| 441 |
> |
under ambient temperature and pressure. Equilibrations in canonical and |
| 442 |
> |
microcanonical ensemble were followed in order. With the simulation |
| 443 |
> |
lattice devided evenly into 10 slabs, different thermal gradients were |
| 444 |
> |
established by applying a set of target thermal transfer flux. Data of |
| 445 |
> |
the series of thermal gradients was collected for calculation. |
| 446 |
> |
|
| 447 |
> |
After simulations of bulk water and crystal gold, a mixture system was |
| 448 |
> |
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
| 449 |
> |
molecules. Spohr potential was adopted in depicting the interaction |
| 450 |
> |
between metal atom and water molecule.\cite{ISI:000167766600035} A |
| 451 |
> |
similar protocol of equilibration was followed. Several thermal |
| 452 |
> |
gradients was built under different target thermal flux. It was found |
| 453 |
> |
out that compared to our previous simulation systems, the two phases |
| 454 |
> |
could have large temperature difference even under a relatively low |
| 455 |
> |
thermal flux. Therefore, under our low flux conditions, it is assumed |
| 456 |
> |
that the metal and water phases have respectively homogeneous |
| 457 |
> |
temperature, excluding the surface regions. In calculating the |
| 458 |
> |
interfacial thermal conductivity $G$, this assumptioin was applied and |
| 459 |
> |
thus our formula becomes: |
| 460 |
> |
|
| 461 |
> |
\begin{equation} |
| 462 |
> |
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
| 463 |
> |
\langle T_{water}\rangle \right)} |
| 464 |
> |
\label{interfaceCalc} |
| 465 |
|
\end{equation} |
| 466 |
+ |
where ${E_{total}}$ is the imposed unphysical kinetic energy transfer |
| 467 |
+ |
and ${\langle T_{gold}\rangle}$ and ${\langle T_{water}\rangle}$ are the |
| 468 |
+ |
average observed temperature of gold and water phases respectively. |
| 469 |
|
|
| 470 |
+ |
\section{Results And Discussions} |
| 471 |
+ |
\subsection{Thermal Conductivity} |
| 472 |
+ |
\subsubsection{Lennard-Jones Fluid} |
| 473 |
+ |
Our thermal conductivity calculations show that scaling method results |
| 474 |
+ |
agree with swapping method. Four different exchange intervals were |
| 475 |
+ |
tested (Table \ref{thermalLJRes}) using swapping method. With a fixed |
| 476 |
+ |
10fs exchange interval, target exchange kinetic energy was set to |
| 477 |
+ |
produce equivalent kinetic energy flux as in swapping method. And |
| 478 |
+ |
similar thermal gradients were observed with similar thermal flux in |
| 479 |
+ |
two simulation methods (Figure \ref{thermalGrad}). |
| 480 |
|
|
| 481 |
+ |
\begin{table*} |
| 482 |
+ |
\begin{minipage}{\linewidth} |
| 483 |
+ |
\begin{center} |
| 484 |
|
|
| 485 |
+ |
\caption{Calculation results for thermal conductivity of Lennard-Jones |
| 486 |
+ |
fluid at ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$, with |
| 487 |
+ |
swap and scale methods at various kinetic energy exchange rates. Results |
| 488 |
+ |
in reduced unit. Errors of calculations in parentheses.} |
| 489 |
|
|
| 490 |
+ |
\begin{tabular}{ccccc} |
| 491 |
+ |
Swapping method & & & NIVS-RNEMD\\ |
| 492 |
+ |
\hline |
| 493 |
+ |
Swap Interval (fs) & $\lambda^*_{swap}$ & & Equilvalent $J_z^*$ & |
| 494 |
+ |
$\lambda^*_{scale}$\\ |
| 495 |
+ |
\hline |
| 496 |
+ |
250 & 7.03(0.34) & & 0.16 & 7.30(0.10)\\ |
| 497 |
+ |
500 & 7.03(0.14) & & 0.09 & 6.95(0.09)\\ |
| 498 |
+ |
1000 & 6.91(0.42) & & 0.047 & 7.19(0.07)\\ |
| 499 |
+ |
2000 & 7.52(0.15) & & 0.024 & 7.19(0.28)\\ |
| 500 |
+ |
\hline |
| 501 |
+ |
\end{tabular} |
| 502 |
+ |
\label{thermalLJRes} |
| 503 |
+ |
\end{center} |
| 504 |
+ |
\end{minipage} |
| 505 |
+ |
\end{table*} |
| 506 |
|
|
| 507 |
+ |
\begin{figure} |
| 508 |
+ |
\includegraphics[width=\linewidth]{thermalGrad} |
| 509 |
+ |
\caption{NIVS-RNEMD method introduced similar temperature gradients |
| 510 |
+ |
compared to ``swapping'' method under various kinetic energy flux in |
| 511 |
+ |
thermal conductivity simulations.} |
| 512 |
+ |
\label{thermalGrad} |
| 513 |
+ |
\end{figure} |
| 514 |
+ |
|
| 515 |
+ |
During these simulations, molecule velocities were recorded in 1000 of |
| 516 |
+ |
all the snapshots of one single data collection process. These |
| 517 |
+ |
velocity data were used to produce histograms of velocity and speed |
| 518 |
+ |
distribution in different slabs. From these histograms, it is observed |
| 519 |
+ |
that under relatively high unphysical kinetic energy flux, speed and |
| 520 |
+ |
velocity distribution of molecules in slabs where swapping occured |
| 521 |
+ |
could deviate from Maxwell-Boltzmann distribution. Figure |
| 522 |
+ |
\ref{thermalHist} a) illustrates how these distributions deviate from an |
| 523 |
+ |
ideal distribution. In high temperature slab, probability density in |
| 524 |
+ |
low speed is confidently smaller than ideal curve fit; in low |
| 525 |
+ |
temperature slab, probability density in high speed is smaller than |
| 526 |
+ |
ideal, while larger than ideal in low speed. This phenomenon is more |
| 527 |
+ |
obvious in our high swapping rate simulations. And this deviation |
| 528 |
+ |
could also leads to deviation of distribution of velocity in various |
| 529 |
+ |
dimensions. One feature of these deviated distribution is that in high |
| 530 |
+ |
temperature slab, the ideal Gaussian peak was changed into a |
| 531 |
+ |
relatively flat plateau; while in low temperature slab, that peak |
| 532 |
+ |
appears sharper. This problem is rooted in the mechanism of the |
| 533 |
+ |
swapping method. Continually depleting low (high) speed particles in |
| 534 |
+ |
the high (low) temperature slab could not be complemented by |
| 535 |
+ |
diffusions of low (high) speed particles from neighbor slabs, unless |
| 536 |
+ |
in suffciently low swapping rate. Simutaneously, surplus low speed |
| 537 |
+ |
particles in the low temperature slab do not have sufficient time to |
| 538 |
+ |
diffuse to neighbor slabs. However, thermal exchange rate should reach |
| 539 |
+ |
a minimum level to produce an observable thermal gradient under noise |
| 540 |
+ |
interference. Consequently, swapping RNEMD has a relatively narrow |
| 541 |
+ |
choice of swapping rate to satisfy these above restrictions. |
| 542 |
+ |
|
| 543 |
+ |
Comparatively, NIVS-RNEMD has a speed distribution closer to the ideal |
| 544 |
+ |
curve fit (Figure \ref{thermalHist} b). Essentially, after scaling, a |
| 545 |
+ |
Gaussian distribution function would remain Gaussian. Although a |
| 546 |
+ |
single scaling is non-isotropic in all three dimensions, our scaling |
| 547 |
+ |
coefficient criteria could help maintian the scaling region as |
| 548 |
+ |
isotropic as possible. On the other hand, scaling coefficients are |
| 549 |
+ |
preferred to be as close to 1 as possible, which also helps minimize |
| 550 |
+ |
the difference among different dimensions. This is possible if scaling |
| 551 |
+ |
interval and one-time thermal transfer energy are well |
| 552 |
+ |
chosen. Consequently, NIVS-RNEMD is able to impose an unphysical |
| 553 |
+ |
thermal flux as the previous RNEMD method without large perturbation |
| 554 |
+ |
to the distribution of velocity and speed in the exchange regions. |
| 555 |
+ |
|
| 556 |
+ |
\begin{figure} |
| 557 |
+ |
\includegraphics[width=\linewidth]{thermalHist} |
| 558 |
+ |
\caption{Speed distribution for thermal conductivity using a) |
| 559 |
+ |
``swapping'' and b) NIVS- RNEMD methods. Shown is from the |
| 560 |
+ |
simulations with an exchange or equilvalent exchange interval of 250 |
| 561 |
+ |
fs. In circled areas, distributions from ``swapping'' RNEMD |
| 562 |
+ |
simulation have deviation from ideal Maxwell-Boltzmann distribution |
| 563 |
+ |
(curves fit for each distribution).} |
| 564 |
+ |
\label{thermalHist} |
| 565 |
+ |
\end{figure} |
| 566 |
+ |
|
| 567 |
+ |
\subsubsection{SPC/E Water} |
| 568 |
+ |
Our results of SPC/E water thermal conductivity are comparable to |
| 569 |
+ |
Bedrov {\it et al.}\cite{Bedrov:2000}, which employed the |
| 570 |
+ |
previous swapping RNEMD method for their calculation. Bedrov {\it et |
| 571 |
+ |
al.}\cite{Bedrov:2000} argued that exchange of the molecule |
| 572 |
+ |
center-of-mass velocities instead of single atom velocities in a |
| 573 |
+ |
molecule conserves the total kinetic energy and linear momentum. This |
| 574 |
+ |
principle is adopted in our simulations. Scaling is applied to the |
| 575 |
+ |
velocities of the rigid bodies of SPC/E model water molecules, instead |
| 576 |
+ |
of each hydrogen and oxygen atoms in relevant water molecules. As |
| 577 |
+ |
shown in Figure \ref{spceGrad}, temperature gradients were established |
| 578 |
+ |
similar to their system. However, the average temperature of our |
| 579 |
+ |
system is 300K, while theirs is 318K, which would be attributed for |
| 580 |
+ |
part of the difference between the final calculation results (Table |
| 581 |
+ |
\ref{spceThermal}). Both methods yields values in agreement with |
| 582 |
+ |
experiment. And this shows the applicability of our method to |
| 583 |
+ |
multi-atom molecular system. |
| 584 |
+ |
|
| 585 |
+ |
\begin{figure} |
| 586 |
+ |
\includegraphics[width=\linewidth]{spceGrad} |
| 587 |
+ |
\caption{Temperature gradients in SPC/E water thermal conductivity |
| 588 |
+ |
simulations.} |
| 589 |
+ |
\label{spceGrad} |
| 590 |
+ |
\end{figure} |
| 591 |
+ |
|
| 592 |
+ |
\begin{table*} |
| 593 |
+ |
\begin{minipage}{\linewidth} |
| 594 |
+ |
\begin{center} |
| 595 |
+ |
|
| 596 |
+ |
\caption{Calculation results for thermal conductivity of SPC/E water |
| 597 |
+ |
at ${\langle T\rangle}$ = 300K at various thermal exchange rates. Errors of |
| 598 |
+ |
calculations in parentheses. } |
| 599 |
+ |
|
| 600 |
+ |
\begin{tabular}{cccc} |
| 601 |
+ |
\hline |
| 602 |
+ |
$\langle dT/dz\rangle$(K/\AA) & & $\lambda$(W/m/K) & \\ |
| 603 |
+ |
& This work & Previous simulations\cite{Bedrov:2000} & |
| 604 |
+ |
Experiment[CiteWagnerKruseBook]\\ |
| 605 |
+ |
\hline |
| 606 |
+ |
0.38 & 0.816(0.044) & & 0.64\\ |
| 607 |
+ |
0.81 & 0.770(0.008) & 0.784\\ |
| 608 |
+ |
1.54 & 0.813(0.007) & 0.730\\ |
| 609 |
+ |
\hline |
| 610 |
+ |
\end{tabular} |
| 611 |
+ |
\label{spceThermal} |
| 612 |
+ |
\end{center} |
| 613 |
+ |
\end{minipage} |
| 614 |
+ |
\end{table*} |
| 615 |
+ |
|
| 616 |
+ |
\subsubsection{Crystal Gold} |
| 617 |
+ |
Our results of gold thermal conductivity using two force fields are |
| 618 |
+ |
shown in Table \ref{AuThermal}. In these calculations,the end and |
| 619 |
+ |
middle slabs were excluded in thermal gradient regession and only used |
| 620 |
+ |
as heat source and drain in the systems. Our yielded values using EAM |
| 621 |
+ |
force field are slightly larger than those using QSC force |
| 622 |
+ |
field. However, both series are significantly smaller than |
| 623 |
+ |
experimental value by a factor of more than 200. It has been verified |
| 624 |
+ |
that this difference is mainly attributed to the lack of electron |
| 625 |
+ |
interaction representation in these force field parameters. Richardson |
| 626 |
+ |
{\it et al.}\cite{Clancy:1992} used EAM force field parameters in |
| 627 |
+ |
their metal thermal conductivity calculations. The Non-Equilibrium MD |
| 628 |
+ |
method they employed in their simulations produced comparable results |
| 629 |
+ |
to ours. As Zhang {\it et al.}\cite{ISI:000231042800044} stated, |
| 630 |
+ |
thermal conductivity values are influenced mainly by force |
| 631 |
+ |
field. Another factor that affects the calculation results could be |
| 632 |
+ |
the density of the metal. Simulations were run with and without |
| 633 |
+ |
isobaric-isothermal (NPT) equilibration. When equilibrated under NPT |
| 634 |
+ |
conditions, our crystall simulation cell expanded by the order of 1\% |
| 635 |
+ |
(Table \ref{AuThermal}). Under larger lattice constant than default, |
| 636 |
+ |
lower thermal conductance were expected and observed. Furthermore, |
| 637 |
+ |
from our simulations, a decrease of thermal conductance at higher |
| 638 |
+ |
temperature is observed, and this trend agrees with experimental |
| 639 |
+ |
measurements[CiteAshcroftMerminBook]. Therefore, it is confident to apply NIVS-RNEMD |
| 640 |
+ |
to metal force field systems. |
| 641 |
+ |
|
| 642 |
+ |
\begin{table*} |
| 643 |
+ |
\begin{minipage}{\linewidth} |
| 644 |
+ |
\begin{center} |
| 645 |
+ |
|
| 646 |
+ |
\caption{Calculation results for thermal conductivity of crystal gold |
| 647 |
+ |
using different force fields at different temperatures and various |
| 648 |
+ |
thermal exchange rates. Errors of calculations in parentheses.} |
| 649 |
+ |
|
| 650 |
+ |
\begin{tabular}{ccccc} |
| 651 |
+ |
\hline |
| 652 |
+ |
Force Field Used & $\rho$ (g/cm$^3$) & ${\langle T\rangle}$ (K) & |
| 653 |
+ |
$\langle dT/dz\rangle$ (K/\AA) & $\lambda$ (W/m/K)\\ |
| 654 |
+ |
\hline |
| 655 |
+ |
QSC & 19.188 & 300 & 1.44 & 1.10(0.06)\\ |
| 656 |
+ |
& & & 2.86 & 1.08(0.05)\\ |
| 657 |
+ |
& & & 5.14 & 1.15(0.07)\\ |
| 658 |
+ |
\\ |
| 659 |
+ |
& 19.263 & 300 & 2.31 & 1.25(0.06)\\ |
| 660 |
+ |
& & & 3.02 & 1.26(0.05)\\ |
| 661 |
+ |
& & 575 & 3.02 & 1.02(0.07)\\ |
| 662 |
+ |
& & & 4.84 & 0.92(0.05)\\ |
| 663 |
+ |
\\ |
| 664 |
+ |
\hline |
| 665 |
+ |
EAM & 19.045 & 300 & 1.24 & 1.24(0.16)\\ |
| 666 |
+ |
& & & 2.06 & 1.37(0.04)\\ |
| 667 |
+ |
& & & 2.55 & 1.41(0.07)\\ |
| 668 |
+ |
\\ |
| 669 |
+ |
& 19.263 & 300 & 1.06 & 1.45(0.13)\\ |
| 670 |
+ |
& & & 2.04 & 1.41(0.07)\\ |
| 671 |
+ |
& & & 2.41 & 1.53(0.10)\\ |
| 672 |
+ |
& & 575 & 2.82 & 1.08(0.03)\\ |
| 673 |
+ |
& & & 4.14 & 1.08(0.05)\\ |
| 674 |
+ |
\hline |
| 675 |
+ |
\end{tabular} |
| 676 |
+ |
\label{AuThermal} |
| 677 |
+ |
\end{center} |
| 678 |
+ |
\end{minipage} |
| 679 |
+ |
\end{table*} |
| 680 |
+ |
|
| 681 |
+ |
|
| 682 |
+ |
\subsection{Interfaciel Thermal Conductivity} |
| 683 |
+ |
After simulations of homogeneous water and gold systems using |
| 684 |
+ |
NIVS-RNEMD method were proved valid, calculation of gold/water |
| 685 |
+ |
interfacial thermal conductivity was followed. It is found out that |
| 686 |
+ |
the low interfacial conductance is probably due to the hydrophobic |
| 687 |
+ |
surface in our system. Figure \ref{interface} (a) demonstrates mass |
| 688 |
+ |
density change along $z$-axis, which is perpendicular to the |
| 689 |
+ |
gold/water interface. It is observed that water density significantly |
| 690 |
+ |
decreases when approaching the surface. Under this low thermal |
| 691 |
+ |
conductance, both gold and water phase have sufficient time to |
| 692 |
+ |
eliminate temperature difference inside respectively (Figure |
| 693 |
+ |
\ref{interface} b). With indistinguishable temperature difference |
| 694 |
+ |
within respective phase, it is valid to assume that the temperature |
| 695 |
+ |
difference between gold and water on surface would be approximately |
| 696 |
+ |
the same as the difference between the gold and water phase. This |
| 697 |
+ |
assumption enables convenient calculation of $G$ using |
| 698 |
+ |
Eq. \ref{interfaceCalc} instead of measuring temperatures of thin |
| 699 |
+ |
layer of water and gold close enough to surface, which would have |
| 700 |
+ |
greater fluctuation and lower accuracy. Reported results (Table |
| 701 |
+ |
\ref{interfaceRes}) are of two orders of magnitude smaller than our |
| 702 |
+ |
calculations on homogeneous systems, and thus have larger relative |
| 703 |
+ |
errors than our calculation results on homogeneous systems. |
| 704 |
+ |
|
| 705 |
+ |
\begin{figure} |
| 706 |
+ |
\includegraphics[width=\linewidth]{interface} |
| 707 |
+ |
\caption{Simulation results for Gold/Water interfacial thermal |
| 708 |
+ |
conductivity: (a) Significant water density decrease is observed on |
| 709 |
+ |
crystalline gold surface, which indicates low surface contact and |
| 710 |
+ |
leads to low thermal conductance. (b) Temperature profiles for a |
| 711 |
+ |
series of simulations. Temperatures of different slabs in the same |
| 712 |
+ |
phase show no significant differences.} |
| 713 |
+ |
\label{interface} |
| 714 |
+ |
\end{figure} |
| 715 |
+ |
|
| 716 |
+ |
\begin{table*} |
| 717 |
+ |
\begin{minipage}{\linewidth} |
| 718 |
+ |
\begin{center} |
| 719 |
+ |
|
| 720 |
+ |
\caption{Calculation results for interfacial thermal conductivity |
| 721 |
+ |
at ${\langle T\rangle \sim}$ 300K at various thermal exchange |
| 722 |
+ |
rates. Errors of calculations in parentheses. } |
| 723 |
+ |
|
| 724 |
+ |
\begin{tabular}{cccc} |
| 725 |
+ |
\hline |
| 726 |
+ |
$J_z$ (MW/m$^2$) & $T_{gold}$ (K) & $T_{water}$ (K) & $G$ |
| 727 |
+ |
(MW/m$^2$/K)\\ |
| 728 |
+ |
\hline |
| 729 |
+ |
98.0 & 355.2 & 295.8 & 1.65(0.21) \\ |
| 730 |
+ |
78.8 & 343.8 & 298.0 & 1.72(0.32) \\ |
| 731 |
+ |
73.6 & 344.3 & 298.0 & 1.59(0.24) \\ |
| 732 |
+ |
49.2 & 330.1 & 300.4 & 1.65(0.35) \\ |
| 733 |
+ |
\hline |
| 734 |
+ |
\end{tabular} |
| 735 |
+ |
\label{interfaceRes} |
| 736 |
+ |
\end{center} |
| 737 |
+ |
\end{minipage} |
| 738 |
+ |
\end{table*} |
| 739 |
+ |
|
| 740 |
+ |
\subsection{Shear Viscosity} |
| 741 |
+ |
Our calculations (Table \ref{shearRate}) shows that scale RNEMD method |
| 742 |
+ |
produced comparable shear viscosity to swap RNEMD method. In Table |
| 743 |
+ |
\ref{shearRate}, the names of the calculated samples are devided into |
| 744 |
+ |
two parts. The first number refers to total slabs in one simulation |
| 745 |
+ |
box. The second number refers to the swapping interval in swap method, or |
| 746 |
+ |
in scale method the equilvalent swapping interval that the same |
| 747 |
+ |
momentum flux would theoretically result in swap method. All the scale |
| 748 |
+ |
method results were from simulations that had a scaling interval of 10 |
| 749 |
+ |
time steps. The average molecular momentum gradients of these samples |
| 750 |
+ |
are shown in Figure \ref{shear} (a) and (b). |
| 751 |
+ |
|
| 752 |
+ |
\begin{table*} |
| 753 |
+ |
\begin{minipage}{\linewidth} |
| 754 |
+ |
\begin{center} |
| 755 |
+ |
|
| 756 |
+ |
\caption{Calculation results for shear viscosity of Lennard-Jones |
| 757 |
+ |
fluid at ${T^* = 0.72}$ and ${\rho^* = 0.85}$, with swap and scale |
| 758 |
+ |
methods at various momentum exchange rates. Results in reduced |
| 759 |
+ |
unit. Errors of calculations in parentheses. } |
| 760 |
+ |
|
| 761 |
+ |
\begin{tabular}{ccccc} |
| 762 |
+ |
Swapping method & & & NIVS-RNEMD & \\ |
| 763 |
+ |
\hline |
| 764 |
+ |
Swap Interval (fs) & $\eta^*_{swap}$ & & Equilvalent $j_p^*(v_x)$ & |
| 765 |
+ |
$\eta^*_{scale}$\\ |
| 766 |
+ |
\hline |
| 767 |
+ |
500 & 3.64(0.05) & & 0.09 & 3.76(0.09)\\ |
| 768 |
+ |
1000 & 3.52(0.16) & & 0.046 & 3.66(0.06)\\ |
| 769 |
+ |
2000 & 3.72(0.05) & & 0.024 & 3.32(0.18)\\ |
| 770 |
+ |
2500 & 3.42(0.06) & & 0.019 & 3.43(0.08)\\ |
| 771 |
+ |
\hline |
| 772 |
+ |
\end{tabular} |
| 773 |
+ |
\label{shearRate} |
| 774 |
+ |
\end{center} |
| 775 |
+ |
\end{minipage} |
| 776 |
+ |
\end{table*} |
| 777 |
+ |
|
| 778 |
+ |
\begin{figure} |
| 779 |
+ |
\includegraphics[width=\linewidth]{shear} |
| 780 |
+ |
\caption{Average momentum gradients in shear viscosity simulations, |
| 781 |
+ |
using (a) ``swapping'' method and (b) NIVS-RNEMD method |
| 782 |
+ |
respectively. (c) Temperature difference among x and y, z dimensions |
| 783 |
+ |
observed when using NIVS-RNEMD with equivalent exchange interval of |
| 784 |
+ |
500 fs.} |
| 785 |
+ |
\label{shear} |
| 786 |
+ |
\end{figure} |
| 787 |
+ |
|
| 788 |
+ |
However, observations of temperatures along three dimensions show that |
| 789 |
+ |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
| 790 |
+ |
two slabs which were scaled. Figure \ref{shear} (c) indicate that with |
| 791 |
+ |
relatively large imposed momentum flux, the temperature difference among $x$ |
| 792 |
+ |
and the other two dimensions was significant. This would result from the |
| 793 |
+ |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
| 794 |
+ |
momentum gradient is set up, $P_c^x$ would be roughly stable |
| 795 |
+ |
($<0$). Consequently, scaling factor $x$ would most probably larger |
| 796 |
+ |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
| 797 |
+ |
keep increase after most scaling steps. And if there is not enough time |
| 798 |
+ |
for the kinetic energy to exchange among different dimensions and |
| 799 |
+ |
different slabs, the system would finally build up temperature |
| 800 |
+ |
(kinetic energy) difference among the three dimensions. Also, between |
| 801 |
+ |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
| 802 |
+ |
are closer to neighbor slabs. This is due to momentum transfer along |
| 803 |
+ |
$z$ dimension between slabs. |
| 804 |
+ |
|
| 805 |
+ |
Although results between scaling and swapping methods are comparable, |
| 806 |
+ |
the inherent temperature inhomogeneity even in relatively low imposed |
| 807 |
+ |
exchange momentum flux simulations makes scaling RNEMD method less |
| 808 |
+ |
attractive than swapping RNEMD in shear viscosity calculation. |
| 809 |
+ |
|
| 810 |
+ |
\section{Conclusions} |
| 811 |
+ |
NIVS-RNEMD simulation method is developed and tested on various |
| 812 |
+ |
systems. Simulation results demonstrate its validity in thermal |
| 813 |
+ |
conductivity calculations, from Lennard-Jones fluid to multi-atom |
| 814 |
+ |
molecule like water and metal crystals. NIVS-RNEMD improves |
| 815 |
+ |
non-Boltzmann-Maxwell distributions, which exist in previous RNEMD |
| 816 |
+ |
methods. Furthermore, it develops a valid means for unphysical thermal |
| 817 |
+ |
transfer between different species of molecules, and thus extends its |
| 818 |
+ |
applicability to interfacial systems. Our calculation of gold/water |
| 819 |
+ |
interfacial thermal conductivity demonstrates this advantage over |
| 820 |
+ |
previous RNEMD methods. NIVS-RNEMD has also limited application on |
| 821 |
+ |
shear viscosity calculations, but could cause temperature difference |
| 822 |
+ |
among different dimensions under high momentum flux. Modification is |
| 823 |
+ |
necessary to extend the applicability of NIVS-RNEMD in shear viscosity |
| 824 |
+ |
calculations. |
| 825 |
+ |
|
| 826 |
|
\section{Acknowledgments} |
| 827 |
|
Support for this project was provided by the National Science |
| 828 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
| 829 |
|
the Center for Research Computing (CRC) at the University of Notre |
| 830 |
|
Dame. \newpage |
| 831 |
|
|
| 832 |
< |
\bibliographystyle{jcp2} |
| 832 |
> |
\bibliographystyle{aip} |
| 833 |
|
\bibliography{nivsRnemd} |
| 834 |
+ |
|
| 835 |
|
\end{doublespace} |
| 836 |
|
\end{document} |
| 837 |
|
|
