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\begin{document} |
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\begin{abstract} |
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We present a new method for introducing stable non-equilibrium |
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velocity and temperature distributions in molecular dynamics |
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simulations of heterogeneous systems. This method extends earlier |
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Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods which use |
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momentum exchange swapping moves that can create non-thermal |
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velocity distributions and are difficult to use for interfacial |
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calculations. By using non-isotropic velocity scaling (NIVS) on the |
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molecules in specific regions of a system, it is possible to impose |
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momentum or thermal flux between regions of a simulation and stable |
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thermal and momentum gradients can then be established. The scaling |
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method we have developed conserves the total linear momentum and |
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total energy of the system. To test the methods, we have computed |
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the thermal conductivity of model liquid and solid systems as well |
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as the interfacial thermal conductivity of a metal-water interface. |
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We find that the NIVS-RNEMD improves the problematic velocity |
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> |
velocity and temperature gradients in molecular dynamics simulations |
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of heterogeneous systems. This method extends earlier Reverse |
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Non-Equilibrium Molecular Dynamics (RNEMD) methods which use |
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momentum exchange swapping moves. The standard swapping moves can |
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create non-thermal velocity distributions and are difficult to use |
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> |
for interfacial calculations. By using non-isotropic velocity |
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scaling (NIVS) on the molecules in specific regions of a system, it |
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is possible to impose momentum or thermal flux between regions of a |
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simulation while conserving the linear momentum and total energy of |
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the system. To test the methods, we have computed the thermal |
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conductivity of model liquid and solid systems as well as the |
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interfacial thermal conductivity of a metal-water interface. We |
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find that the NIVS-RNEMD improves the problematic velocity |
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distributions that develop in other RNEMD methods. |
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\end{abstract} |
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|
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typically samples from the same manifold of states in the |
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microcanonical ensemble. |
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|
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Recently, Tenney and Maginn\cite{Maginn:2010} have discovered |
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some problems with the original RNEMD swap technique. Notably, large |
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Recently, Tenney and Maginn\cite{Maginn:2010} have discovered some |
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problems with the original RNEMD swap technique. Notably, large |
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momentum fluxes (equivalent to frequent momentum swaps between the |
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slabs) can result in ``notched'', ``peaked'' and generally non-thermal |
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momentum distributions in the two slabs, as well as non-linear thermal |
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and velocity distributions along the direction of the imposed flux |
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($z$). Tenney and Maginn obtained reasonable limits on imposed flux |
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and self-adjusting metrics for retaining the usability of the method. |
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and proposed self-adjusting metrics for retaining the usability of the |
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method. |
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|
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In this paper, we develop and test a method for non-isotropic velocity |
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scaling (NIVS) which retains the desirable features of RNEMD |
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\end{equation} |
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where |
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\begin{eqnarray} |
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P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i \left[\vec{v}_i\right]_\alpha \\ |
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P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j \left[\vec{v}_j\right]_\alpha |
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P_c^\alpha & = & \sum_{i = 1}^{N_c} m_i v_{i\alpha} \\ |
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P_h^\alpha & = & \sum_{j = 1}^{N_h} m_j v_{j\alpha} |
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\label{eq:momentumdef} |
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\end{eqnarray} |
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Therefore, for each of the three directions, the hot scaling |
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parameters are a simple function of the cold scaling parameters and |
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the instantaneous linear momentum in each of the two slabs. |
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the instantaneous linear momenta in each of the two slabs. |
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\begin{equation} |
| 192 |
|
\alpha^\prime = 1 + (1 - \alpha) p_\alpha |
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\label{eq:hotcoldscaling} |
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|
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Conservation of total energy also places constraints on the scaling: |
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\begin{equation} |
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< |
\sum_{\alpha = x,y,z} K_h^\alpha + K_c^\alpha = \sum_{\alpha = x,y,z} |
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\left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha |
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> |
\sum_{\alpha = x,y,z} \left\{ K_h^\alpha + K_c^\alpha \right\} = \sum_{\alpha = x,y,z} |
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> |
\left\{ \left(\alpha^\prime\right)^2 K_h^\alpha + \alpha^2 K_c^\alpha \right\} |
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\end{equation} |
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where the translational kinetic energies, $K_h^\alpha$ and |
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$K_c^\alpha$, are computed for each of the three directions in a |
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degree 16. There are a number of polynomial root-finding methods in |
| 270 |
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the literature,\cite{Hoffman:2001sf,384119} but numerically finding |
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the roots of high-degree polynomials is generally an ill-conditioned |
| 272 |
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problem.\cite{Hoffman:2001sf}[P157] One simplification is to maintain velocity |
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< |
scalings that are {\it as isotropic as possible}. To do this, we |
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impose $x=y$, and to treat both the constraint and flux ellipsoids as |
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2-dimensional ellipses. In reduced dimensionality, the |
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problem.\cite{Hoffman:2001sf} One simplification is to maintain |
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velocity scalings that are {\it as isotropic as possible}. To do |
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this, we impose $x=y$, and treat both the constraint and flux |
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ellipsoids as 2-dimensional ellipses. In reduced dimensionality, the |
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intersecting-ellipse problem reduces to finding the roots of |
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polynomials of degree 4. |
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|
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|
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We have implemented this methodology in our molecular dynamics code, |
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OpenMD,\cite{Meineke:2005gd,openmd} performing the NIVS scaling moves |
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< |
after an MD step with a variable frequency. We have tested the method |
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< |
in a variety of different systems, including homogeneous fluids |
| 335 |
< |
(Lennard-Jones and SPC/E water), crystalline solids ({\sc |
| 336 |
< |
eam}~\cite{PhysRevB.33.7983} and quantum Sutton-Chen ({\sc |
| 337 |
< |
q-sc})~\cite{PhysRevB.59.3527} models for Gold), and heterogeneous |
| 333 |
> |
with a variable frequency after the molecular dynamics (MD) steps. We |
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> |
have tested the method in a variety of different systems, including: |
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homogeneous fluids (Lennard-Jones and SPC/E water), crystalline |
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> |
solids, using both the embedded atom method |
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> |
(EAM)~\cite{PhysRevB.33.7983} and quantum Sutton-Chen |
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> |
(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 using |
| 341 |
< |
velocity scaling moves, we can even obtain estimates of the |
| 342 |
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interfacial thermal conductivities ($G$). |
| 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 ($G$). |
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|
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\subsection{Simulation Cells} |
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|
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|
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In order to compare our new methodology with the original |
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M\"{u}ller-Plathe swapping algorithm,\cite{ISI:000080382700030} we |
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first performed simulations using the original technique. |
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> |
first performed simulations using the original technique. At fixed |
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> |
intervals, kinetic energy or momentum exchange moves were performed |
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> |
between the hot and the cold slabs. The interval between exchange |
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> |
moves governs the effective momentum flux ($j_z(p_x)$) or energy flux |
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> |
($J_z$) between the two slabs so to vary this quantity, we performed |
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> |
simulations with a variety of delay intervals between the swapping moves. |
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|
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+ |
For thermal conductivity measurements, the particle with smallest |
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+ |
speed in the hot slab and the one with largest speed in the cold slab |
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had their entire momentum vectors swapped. In the test cases run |
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here, all particles had the same chemical identity and mass, so this |
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move preserves both total linear momentum and total energy. It is |
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also possible to exchange energy by assuming an elastic collision |
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between the two particles which are exchanging energy. |
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|
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For shear stress simulations, the particle with the most negative |
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$p_x$ in the hot slab and the one with the most positive $p_x$ in the |
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cold slab exchanged only this component of their momentum vectors. |
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+ |
|
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|
\subsection{RNEMD with NIVS scaling} |
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|
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For each simulation utilizing the swapping method, a corresponding |
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NIVS-RNEMD simulation was carried out using a target momentum flux set |
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< |
to produce a the same momentum or energy flux exhibited in the |
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swapping simulation. |
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> |
to produce the same flux experienced in the swapping simulation. |
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|
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< |
To test the temperature homogeneity (and to compute transport |
| 387 |
< |
coefficients), directional momentum and temperature distributions were |
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accumulated for molecules in each of the slabs. |
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> |
To test the temperature homogeneity, directional momentum and |
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> |
temperature distributions were accumulated for molecules in each of |
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> |
the slabs. Transport coefficients were computed using the temperature |
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> |
(and momentum) gradients across the $z$-axis as well as the total |
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> |
momentum flux the system experienced during data collection portion of |
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> |
the simulation. |
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|
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\subsection{Shear viscosities} |
| 394 |
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|
| 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 occurs in two directions due to our periodic |
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< |
boundary conditions. The velocity gradient ${\langle \partial v_x |
| 404 |
< |
/\partial z \rangle}$ was obtained using linear regression of the |
| 405 |
< |
velocity profiles in the bins. For Lennard-Jones simulations, shear |
| 406 |
< |
viscosities are reporte in reduced units (${\eta^* = \eta \sigma^2 |
| 407 |
< |
(\varepsilon m)^{-1/2}}$). |
| 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 (${\eta^* = \eta \sigma^2 (\varepsilon m)^{-1/2}}$). |
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|
| 409 |
|
\subsection{Thermal Conductivities} |
| 410 |
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|
| 411 |
< |
The energy flux was calculated similarly to the momentum flux, using |
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< |
the total non-physical energy transferred (${E_{total}}$) and the data |
| 413 |
< |
collection time $t$: |
| 411 |
> |
The energy flux was calculated in a similar manner to the momentum |
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> |
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} |
| 422 |
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|
| 423 |
|
\subsection{Interfacial Thermal Conductivities} |
| 424 |
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|
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< |
For materials with a relatively low interfacial conductance, and in |
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< |
cases where the flux between the materials is small, the bulk regions |
| 427 |
< |
on either side of an interface rapidly come to a state in which the |
| 428 |
< |
two phases have relatively homogeneous (but distinct) temperatures. |
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< |
In calculating the interfacial thermal conductivity $G$, this |
| 406 |
< |
assumption was made, and the conductance can be approximated as: |
| 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 |
|
\begin{equation} |
| 432 |
|
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_{gold}\rangle - |
| 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 |
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< |
water phases respectively. |
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> |
water phases respectively. If the interfacial conductance is {\it |
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> |
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. |
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|
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\section{Results} |
| 446 |
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|
| 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 first run under reduced |
| 455 |
> |
conductivity calculations, simulations were run under reduced |
| 456 |
|
temperature ${\langle T^*\rangle = 0.72}$ in the microcanonical |
| 457 |
< |
ensemble, but other temperatures ([XXX, YYY, and ZZZ]) were also |
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< |
sampled. The simulations included $10^5$ steps of equilibration |
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> |
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 |
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data collection under RNEMD. |
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\subsubsection*{Thermal Conductivity} |
| 463 |
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|
| 464 |
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Our thermal conductivity calculations show that the NIVS method agrees |
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< |
well with the swapping method. Four different swap intervals were |
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tested (Table \ref{LJ}). With a fixed scaling interval of 10 time steps, |
| 467 |
< |
the target exchange kinetic energy produced equivalent kinetic energy |
| 468 |
< |
flux as in the swapping method. Similar thermal gradients were |
| 469 |
< |
observed with similar thermal flux under the two different methods |
| 470 |
< |
(Figure \ref{thermalGrad}). |
| 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} |
| 489 |
|
\multicolumn{2}{|c|}{NIVS-RNEMD} \\ |
| 490 |
|
\hline |
| 491 |
|
\multirow{2}{*}{Swap Interval (fs)} & Equivalent $J_z^*$ or & |
| 464 |
– |
|
| 492 |
|
\multirow{2}{*}{$\lambda^*_{swap}$} & |
| 493 |
|
\multirow{2}{*}{$\eta^*_{swap}$} & |
| 494 |
|
\multirow{2}{*}{$\lambda^*_{scale}$} & |
| 495 |
|
\multirow{2}{*}{$\eta^*_{scale}$} \\ |
| 496 |
< |
& $j_p^*(v_x)$ (reduced units) & & & & \\ |
| 496 |
> |
& $j_z^*(p_x)$ (reduced units) & & & & \\ |
| 497 |
|
\hline |
| 498 |
< |
250 & 0.16 & 7.03(0.34) & & 7.30(0.10) & \\ |
| 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 & & 3.42(0.06) & & 3.43(0.08)\\ |
| 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} |
| 509 |
|
|
| 510 |
|
\begin{figure} |
| 511 |
|
\includegraphics[width=\linewidth]{thermalGrad} |
| 512 |
< |
\caption{NIVS-RNEMD method creates similar temperature gradients |
| 513 |
< |
compared with the swapping method under a variety of imposed kinetic |
| 514 |
< |
energy flux values.} |
| 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 was used to produce distributions of both velocity and speed in |
| 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 |
| 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 |
| 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. |
| 533 |
> |
phenomenon is more obvious at higher swapping rates. |
| 534 |
|
|
| 535 |
< |
In the velocity distributions, the ideal Gaussian peak is |
| 536 |
< |
substantially flattened in the hot slab, and is overly sharp (with |
| 537 |
< |
truncated wings) in the cold slab. This problem is rooted in the |
| 538 |
< |
mechanism of the swapping method. Continually depleting low (high) |
| 539 |
< |
speed particles in the high (low) temperature slab is not complemented |
| 540 |
< |
by diffusions of low (high) speed particles from neighboring slabs, |
| 541 |
< |
unless the swapping rate is sufficiently small. Simutaneously, surplus |
| 542 |
< |
low speed particles in the low temperature slab do not have sufficient |
| 543 |
< |
time to diffuse to neighboring slabs. Since the thermal exchange rate |
| 544 |
< |
must reach a minimum level to produce an observable thermal gradient, |
| 545 |
< |
the swapping-method RNEMD has a relatively narrow choice of exchange |
| 546 |
< |
times that can be utilized. |
| 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 |
| 557 |
|
|
| 558 |
|
\begin{figure} |
| 559 |
|
\includegraphics[width=\linewidth]{thermalHist} |
| 560 |
< |
\caption{Speed distribution for thermal conductivity using a) |
| 561 |
< |
``swapping'' and b) NIVS- RNEMD methods. Shown is from the |
| 562 |
< |
simulations with an exchange or equilvalent exchange interval of 250 |
| 563 |
< |
fs. In circled areas, distributions from ``swapping'' RNEMD |
| 564 |
< |
simulation have deviation from ideal Maxwell-Boltzmann distribution |
| 565 |
< |
(curves fit for each distribution).} |
| 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 |
| 575 |
< |
RNEMD predicted comparable shear viscosities to swap RNEMD method. All |
| 576 |
< |
the scale method results were from simulations that had a scaling |
| 577 |
< |
interval of 10 time steps. The average molecular momentum gradients of |
| 546 |
< |
these samples are shown in Figure \ref{shear} (a) and (b). |
| 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 (a) ``swapping'' method and (b) NIVS-RNEMD method |
| 583 |
< |
respectively. (c) Temperature difference among x and y, z dimensions |
| 584 |
< |
observed when using NIVS-RNEMD with equivalent exchange interval of |
| 585 |
< |
500 fs.} |
| 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 |
< |
However, observations of temperatures along three dimensions show that |
| 590 |
< |
inhomogeneity occurs in scaling RNEMD simulations, particularly in the |
| 591 |
< |
two slabs which were scaled. Figure \ref{shear} (c) indicate that with |
| 592 |
< |
relatively large imposed momentum flux, the temperature difference among $x$ |
| 593 |
< |
and the other two dimensions was significant. This would result from the |
| 594 |
< |
algorithm of scaling method. From Eq. \ref{eq:fluxPlane}, after |
| 595 |
< |
momentum gradient is set up, $P_c^x$ would be roughly stable |
| 596 |
< |
($<0$). Consequently, scaling factor $x$ would most probably larger |
| 597 |
< |
than 1. Therefore, the kinetic energy in $x$-dimension $K_c^x$ would |
| 598 |
< |
keep increase after most scaling steps. And if there is not enough time |
| 599 |
< |
for the kinetic energy to exchange among different dimensions and |
| 600 |
< |
different slabs, the system would finally build up temperature |
| 601 |
< |
(kinetic energy) difference among the three dimensions. Also, between |
| 571 |
< |
$y$ and $z$ dimensions in the scaled slabs, temperatures of $z$-axis |
| 572 |
< |
are closer to neighbor slabs. This is due to momentum transfer along |
| 573 |
< |
$z$ dimension between slabs. |
| 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 inhomogeneity even in relatively low imposed |
| 605 |
< |
exchange momentum flux simulations makes scaling RNEMD method less |
| 606 |
< |
attractive than swapping RNEMD in shear viscosity calculation. |
| 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 |
|
|
| 580 |
– |
|
| 609 |
|
\subsection{Bulk SPC/E water} |
| 610 |
|
|
| 611 |
|
We compared the thermal conductivity of SPC/E water using NIVS-RNEMD |
| 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 in our simulations. Scaling was applied to |
| 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 |
|
|
| 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 |
< |
100ps period under constant-NVE conditions without any momentum |
| 628 |
< |
flux. 100ps was allowed to stabilize the system with an imposed |
| 629 |
< |
momentum transfer to create a gradient, and 1ns was alotted for |
| 630 |
< |
data collection under RNEMD. |
| 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 |
< |
As shown in Figure \ref{spceGrad}, temperature gradients were |
| 633 |
< |
established similar to the previous work. However, the average |
| 634 |
< |
temperature of our system is 300K, while that in Bedrov {\it et al.} |
| 635 |
< |
is 318K, which would be attributed for part of the difference between |
| 636 |
< |
the final calculation results (Table \ref{spceThermal}). [WHY DIDN'T |
| 609 |
< |
WE DO 318 K?] Both methods yield values in reasonable agreement with |
| 610 |
< |
experiment [DONE AT WHAT TEMPERATURE?] |
| 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 |
|
|
| 612 |
– |
\begin{figure} |
| 613 |
– |
\includegraphics[width=\linewidth]{spceGrad} |
| 614 |
– |
\caption{Temperature gradients in SPC/E water thermal conductivity |
| 615 |
– |
simulations.} |
| 616 |
– |
\label{spceGrad} |
| 617 |
– |
\end{figure} |
| 618 |
– |
|
| 638 |
|
\begin{table*} |
| 639 |
|
\begin{minipage}{\linewidth} |
| 640 |
|
\begin{center} |
| 643 |
|
imposed thermal gradients. Uncertainties are indicated in |
| 644 |
|
parentheses.} |
| 645 |
|
|
| 646 |
< |
\begin{tabular}{|c|ccc|} |
| 646 |
> |
\begin{tabular}{|c|c|ccc|} |
| 647 |
|
\hline |
| 648 |
< |
\multirow{2}{*}{$\langle dT/dz\rangle$(K/\AA)} & \multicolumn{3}{|c|}{$\lambda |
| 649 |
< |
(\mathrm{W m}^{-1} \mathrm{K}^{-1})$} \\ |
| 650 |
< |
& This work (300K) & Previous simulations (318K)\cite{Bedrov:2000} & |
| 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 |
< |
0.38 & 0.816(0.044) & & 0.64\\ |
| 656 |
< |
0.81 & 0.770(0.008) & 0.784 & \\ |
| 657 |
< |
1.54 & 0.813(0.007) & 0.730 & \\ |
| 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} |
| 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 ({\sc eam})~\cite{PhysRevB.33.7983} and Sutton-Chen ({\sc |
| 677 |
< |
sc})~\cite{Chen90} potential have been used to study a wide range of |
| 678 |
< |
phenomena in both bulk materials and |
| 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 sc} potential has a simple form that closely |
| 684 |
< |
resembles the Lennard Jones potential, |
| 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] , |
| 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 ({\sc q-sc}) formulation matches these properties while |
| 705 |
< |
including zero-point quantum corrections for different transition |
| 706 |
< |
metals.\cite{PhysRevB.59.3527} The {\sc eam} functional forms differ |
| 707 |
< |
slightly from {\sc sc} but the overall method is very similar. |
| 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 {\sc eam} and the {\sc q-sc} |
| 710 |
< |
potentials to test the behavior of scaling RNEMD. |
| 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 |
| 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. {\sc eam} predicts |
| 726 |
< |
slightly larger thermal conductivities than {\sc q-sc}. However, both |
| 727 |
< |
values are smaller than experimental value by a factor of more than |
| 728 |
< |
200. This behavior has been observed previously by Richardson and |
| 729 |
< |
Clancy, and has been attributed to the lack of electronic effects in |
| 730 |
< |
these force fields.\cite{Clancy:1992} The non-equilibrium MD method |
| 731 |
< |
employed in their simulations gave an thermal conductance estimation |
| 732 |
< |
of [FORCE FIELD] gold as [RESULT IN REF], which is comparable to ours. It |
| 733 |
< |
should be noted that the density of the metal being simulated also |
| 734 |
< |
greatly affects the thermal conductivity. With an expanded lattice, |
| 735 |
< |
lower thermal conductance is expected (and observed). We also observed |
| 710 |
< |
a decrease in thermal conductance at higher temperatures, a trend that |
| 711 |
< |
agrees with experimental measurements [PAGE |
| 712 |
< |
NUMBERS?].\cite{AshcroftMermin} |
| 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} |
| 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. [SWAPPING COMPARISON?]} |
| 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}{*}{\sc q-sc} & \multirow{3}{*}{19.188} & \multirow{3}{*}{300} & 1.44 & 1.10(0.06)\\ |
| 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)\\ |
| 760 |
|
& & \multirow{2}{*}{575} & 3.02 & 1.02(0.07)\\ |
| 761 |
|
& & & 4.84 & 0.92(0.05)\\ |
| 762 |
|
\hline |
| 763 |
< |
\multirow{8}{*}{\sc eam} & \multirow{3}{*}{19.045} & \multirow{3}{*}{300} & 1.24 & 1.24(0.16)\\ |
| 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)\\ |
| 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 |
| 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 |
< |
After simulations of bulk water and crystal gold, a mixture system was |
| 792 |
< |
constructed, consisting of 1188 Au atoms and 1862 H$_2$O |
| 793 |
< |
molecules. Spohr potential was adopted in depicting the interaction |
| 794 |
< |
between metal atom and water molecule.\cite{ISI:000167766600035} A |
| 795 |
< |
similar protocol of equilibration was followed. Several thermal |
| 770 |
< |
gradients was built under different target thermal flux. It was found |
| 771 |
< |
out that compared to our previous simulation systems, the two phases |
| 772 |
< |
could have large temperature difference even under a relatively low |
| 773 |
< |
thermal flux. |
| 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 |
< |
After simulations of homogeneous water and gold systems using |
| 812 |
< |
NIVS-RNEMD method were proved valid, calculation of gold/water |
| 813 |
< |
interfacial thermal conductivity was followed. It is found out that |
| 814 |
< |
the low interfacial conductance is probably due to the hydrophobic |
| 815 |
< |
surface in our system. Figure \ref{interface} (a) demonstrates mass |
| 816 |
< |
density change along $z$-axis, which is perpendicular to the |
| 817 |
< |
gold/water interface. It is observed that water density significantly |
| 818 |
< |
decreases when approaching the surface. Under this low thermal |
| 819 |
< |
conductance, both gold and water phase have sufficient time to |
| 785 |
< |
eliminate temperature difference inside respectively (Figure |
| 786 |
< |
\ref{interface} b). With indistinguishable temperature difference |
| 787 |
< |
within respective phase, it is valid to assume that the temperature |
| 788 |
< |
difference between gold and water on surface would be approximately |
| 789 |
< |
the same as the difference between the gold and water phase. This |
| 790 |
< |
assumption enables convenient calculation of $G$ using |
| 791 |
< |
Eq. \ref{interfaceCalc} instead of measuring temperatures of thin |
| 792 |
< |
layer of water and gold close enough to surface, which would have |
| 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 |
| 824 |
|
|
| 825 |
|
\begin{figure} |
| 826 |
|
\includegraphics[width=\linewidth]{interface} |
| 827 |
< |
\caption{Simulation results for Gold/Water interfacial thermal |
| 828 |
< |
conductivity: (a) Significant water density decrease is observed on |
| 829 |
< |
crystalline gold surface, which indicates low surface contact and |
| 830 |
< |
leads to low thermal conductance. (b) Temperature profiles for a |
| 831 |
< |
series of simulations. Temperatures of different slabs in the same |
| 805 |
< |
phase show no significant differences.} |
| 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 |
|
|
| 841 |
|
300K using a range of energy fluxes. Uncertainties are |
| 842 |
|
indicated in parentheses. } |
| 843 |
|
|
| 844 |
< |
\begin{tabular}{|cccc|} |
| 844 |
> |
\begin{tabular}{|cccc| } |
| 845 |
|
\hline |
| 846 |
|
$J_z$ (MW/m$^2$) & $\langle T_{gold} \rangle$ (K) & $\langle |
| 847 |
|
T_{water} \rangle$ (K) & $G$ |
| 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 in previous RNEMD |
| 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 |
| 876 |
|
calculations. |
| 877 |
|
|
| 878 |
|
\section{Acknowledgments} |
| 879 |
< |
Support for this project was provided by the National Science |
| 880 |
< |
Foundation under grant CHE-0848243. Computational time was provided by |
| 881 |
< |
the Center for Research Computing (CRC) at the University of Notre |
| 882 |
< |
Dame. \newpage |
| 879 |
> |
The authors would like to thank Craig Tenney and Ed Maginn for many |
| 880 |
> |
helpful discussions. Support for this project was provided by the |
| 881 |
> |
National Science Foundation under grant CHE-0848243. Computational |
| 882 |
> |
time was provided by the Center for Research Computing (CRC) at the |
| 883 |
> |
University of Notre Dame. |
| 884 |
> |
\newpage |
| 885 |
|
|
| 858 |
– |
\bibliographystyle{aip} |
| 886 |
|
\bibliography{nivsRnemd} |
| 887 |
|
|
| 888 |
|
\end{doublespace} |
