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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 |
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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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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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,Muller-Plathe:1999ek} 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. The transport coefficients (shear |
| 84 |
< |
viscosity and thermal conductivity) are easily obtained by assuming |
| 85 |
< |
linear response of the system, |
| 76 |
> |
the simulation cell.\cite{MullerPlathe:1997xw,ISI:000080382700030} The |
| 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 & = & \lambda \frac{\partial T}{\partial z} |
| 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 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). |
| 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 |
|
|
| 96 |
< |
RNEMD is preferable in many ways to the forward NEMD methods because |
| 97 |
< |
it imposes what is typically difficult to measure (a flux or stress) |
| 98 |
< |
and it is typically much easier to compute momentum gradients or |
| 99 |
< |
strains (the response). For similar reasons, RNEMD is also preferable |
| 100 |
< |
to slowly-converging equilibrium methods for measuring thermal |
| 101 |
< |
conductivity and shear viscosity (using Green-Kubo relations or the |
| 102 |
< |
Helfand moment approach of Viscardy {\it et |
| 96 |
> |
\begin{figure} |
| 97 |
> |
\includegraphics[width=\linewidth]{thermalDemo} |
| 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 |
| 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 have discovered some problems with the |
| 126 |
< |
original RNEMD swap technique. Notably, large momentum fluxes |
| 127 |
< |
(equivalent to frequent momentum swaps between the slabs) can result |
| 128 |
< |
in "notched", "peaked" and generally non-thermal momentum |
| 129 |
< |
distributions in the two slabs, as well as non-linear thermal and |
| 130 |
< |
velocity distributions along the direction of the imposed flux ($z$). |
| 131 |
< |
Tenney and Maginn obtained reasonable limits on imposed flux and |
| 132 |
< |
self-adjusting metrics for retaining the usability of the method. |
| 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 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 hot slab, |
| 156 |
> |
moves. For molecules $\{i\}$ located within the cold slab, |
| 157 |
|
\begin{equation} |
| 158 |
< |
\vec{v}_i \leftarrow \left( \begin{array}{c} |
| 159 |
< |
x \\ |
| 160 |
< |
y \\ |
| 161 |
< |
z \\ |
| 158 |
> |
\vec{v}_i \leftarrow \left( \begin{array}{ccc} |
| 159 |
> |
x & 0 & 0 \\ |
| 160 |
> |
0 & y & 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 cold bin 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}{c} |
| 170 |
< |
x^\prime \\ |
| 171 |
< |
y^\prime \\ |
| 172 |
< |
z^\prime \\ |
| 169 |
> |
\vec{v}_j \leftarrow \left( \begin{array}{ccc} |
| 170 |
> |
x^\prime & 0 & 0 \\ |
| 171 |
> |
0 & y^\prime & 0 \\ |
| 172 |
> |
0 & 0 & z^\prime \\ |
| 173 |
|
\end{array} \right) \cdot \vec{v}_j |
| 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 P_h^\alpha + \alpha^\prime P_c^\alpha |
| 180 |
> |
P_h^\alpha + P_c^\alpha = \alpha^\prime P_h^\alpha + \alpha P_c^\alpha |
| 181 |
|
\end{equation} |
| 182 |
|
where |
| 183 |
< |
\begin{equation} |
| 184 |
< |
\begin{array}{rcl} |
| 185 |
< |
P_h^\alpha & = & \sum_{i = 1}^{N_h} m_i \left[\vec{v}_i\right]_\alpha \\ |
| 153 |
< |
P_c^\alpha & = & \sum_{j = 1}^{N_c} m_j \left[\vec{v}_j\right]_\alpha |
| 154 |
< |
\\ |
| 155 |
< |
\end{array} |
| 183 |
> |
\begin{eqnarray} |
| 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{equation} |
| 188 |
< |
Therefore, for each of the three directions, the cold scaling |
| 189 |
< |
parameters are a simple function of the hot scaling parameters and |
| 190 |
< |
the instantaneous linear momentum in each of the two slabs. |
| 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 momenta in each of the two slabs. |
| 191 |
|
\begin{equation} |
| 192 |
< |
\alpha^\prime = 1 + (1 - \alpha) \frac{P_h^\alpha}{P_c^\alpha}. |
| 192 |
> |
\alpha^\prime = 1 + (1 - \alpha) p_\alpha |
| 193 |
|
\label{eq:hotcoldscaling} |
| 194 |
|
\end{equation} |
| 195 |
+ |
where |
| 196 |
+ |
\begin{equation} |
| 197 |
+ |
p_\alpha = \frac{P_c^\alpha}{P_h^\alpha} |
| 198 |
+ |
\end{equation} |
| 199 |
+ |
for convenience. |
| 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 |
< |
\alpha^2 K_h^\alpha + \left(\alpha^\prime\right)^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 kinetic energies, $K_h^\alpha$ and $K_c^\alpha$, are computed |
| 207 |
< |
for each of the three directions in a similar manner to the linear momenta |
| 208 |
< |
(Eq. \ref{eq:momentumdef}). Substituting in the expressions for the |
| 209 |
< |
cold scaling parameters ($\alpha^\prime$) from |
| 210 |
< |
Eq. (\ref{eq:hotcoldscaling}), we obtain the {\it constraint ellipsoid}: |
| 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}: |
| 212 |
|
\begin{equation} |
| 213 |
< |
C_x (1+x)^2 + C_y (1+y)^2 + C_z (1+z)^2 = 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 |
| 218 |
|
linear momenta and kinetic energies for the hot and cold slabs, |
| 219 |
< |
\begin{equation} |
| 220 |
< |
C_\alpha = \left( K_h^\alpha + K_c^\alpha |
| 221 |
< |
\left(\frac{P_h^\alpha}{P_c^\alpha}\right)^2\right) |
| 219 |
> |
\begin{eqnarray} |
| 220 |
> |
a_\alpha & = &\left(K_c^\alpha + K_h^\alpha |
| 221 |
> |
\left(p_\alpha\right)^2\right) \\ |
| 222 |
> |
b_\alpha & = & -2 K_h^\alpha p_\alpha \left( 1 + p_\alpha\right) \\ |
| 223 |
> |
c_\alpha & = & K_h^\alpha p_\alpha^2 + 2 K_h^\alpha p_\alpha - K_c^\alpha |
| 224 |
|
\label{eq:constraintEllipsoidConsts} |
| 225 |
< |
\end{equation} |
| 226 |
< |
This ellipsoid defines the set of hot slab scaling parameters which can be |
| 227 |
< |
applied while preserving both linear momentum in all three directions |
| 228 |
< |
as well as total energy. |
| 225 |
> |
\end{eqnarray} |
| 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 increase in kinetic energy of the hot 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 |
< |
(x^2-1) K_h^x + (y^2-1) K_h^y + (z^2-1) K_h^z = J_z |
| 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_h^x + y^2 K_h^y + z^2 K_h^z = (J_z + K_h^x + K_h^y + K_h^z) |
| 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 flux ellipsoid} is governed both by a |
| 246 |
< |
targetted value, $J_z$ as well as the instantaneous values of the |
| 247 |
< |
kinetic energy components in the hot 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 |
| 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{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 |
> |
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 |
> |
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 |
> |
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 ellipsoid (Eq. \ref{eq:fluxEllipsoid}), i.e. the intersection of |
| 321 |
< |
the two ellipsoids in 3-space. |
| 320 |
> |
flux plane (Eq. \ref{eq:fluxPlane}), i.e. the intersection of an |
| 321 |
> |
ellipsoid and a plane in 3-dimensional space. |
| 322 |
|
|
| 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 |
< |
One may also define momentum flux (say along the x-direction) as: |
| 329 |
> |
\section{Computational Details} |
| 330 |
> |
|
| 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 |
> |
\subsection{Simulation Cells} |
| 345 |
> |
|
| 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 |
> |
\subsection{RNEMD with M\"{u}ller-Plathe swaps} |
| 358 |
> |
|
| 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 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 |
> |
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 |
< |
(x-1) P_h^x = j_z(p_x) |
| 398 |
> |
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
| 399 |
|
\end{equation} |
| 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 |
+ |
\subsection{Thermal Conductivities} |
| 410 |
|
|
| 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 |
+ |
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{Interfacial Thermal Conductivities} |
| 424 |
|
|
| 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 |
|
|
| 431 |
< |
\section{Acknowledgments} |
| 432 |
< |
Support for this project was provided by the National Science |
| 433 |
< |
Foundation under grant CHE-0848243. Computational time was provided by |
| 434 |
< |
the Center for Research Computing (CRC) at the University of Notre |
| 435 |
< |
Dame. \newpage |
| 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 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 |
< |
\bibliographystyle{jcp2} |
| 445 |
> |
\section{Results} |
| 446 |
> |
|
| 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 |
> |
\subsubsection*{Thermal Conductivity} |
| 463 |
> |
|
| 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{table*} |
| 475 |
> |
\begin{minipage}{\linewidth} |
| 476 |
> |
\begin{center} |
| 477 |
> |
|
| 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]{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*{Velocity Distributions} |
| 521 |
> |
|
| 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 |
> |
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 |
> |
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 |
> |
|
| 558 |
> |
\begin{figure} |
| 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 |
> |
|
| 572 |
> |
|
| 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 |
> |
|
| 579 |
> |
\begin{figure} |
| 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 |
> |
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 |
> |
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 |
> |
\subsection{Bulk SPC/E water} |
| 610 |
> |
|
| 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 |
> |
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 |
> |
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 |
> |
\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 |
> |
\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]{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} |
| 834 |
> |
|
| 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 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 |
> |
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 |
> |
|
| 886 |
|
\bibliography{nivsRnemd} |
| 887 |
+ |
|
| 888 |
|
\end{doublespace} |
| 889 |
|
\end{document} |
| 890 |
|
|
