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\section{Introduction} |
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|
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Non-equilibrium Molecular Dynamics (NEMD) methods impose a temperature |
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or velocity {\it gradient} on a |
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system,\cite{ASHURST:1975tg,Evans:1982zk,ERPENBECK:1984sp,MAGINN:1993hc,Berthier:2002ij,Evans:2002ai,Schelling:2002dp,PhysRevA.34.1449,JiangHao_jp802942v} |
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and use linear response theory to connect the resulting thermal or |
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momentum flux to transport coefficients of bulk materials. However, |
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for heterogeneous systems, such as phase boundaries or interfaces, it |
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is often unclear what shape of gradient should be imposed at the |
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boundary between materials. |
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|
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\begin{figure} |
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\includegraphics[width=\linewidth]{VSS2} |
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\caption{Schematics of periodic (left) and non-periodic (right) |
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Velocity Shearing and Scaling RNEMD. A kinetic energy or momentum |
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flux is applied from region B to region A. Thermal gradients are |
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depicted by a color gradient. Linear or angular velocity gradients |
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are shown as arrows.} |
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\label{fig:VSS} |
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\end{figure} |
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|
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Reverse Non-Equilibrium Molecular Dynamics (RNEMD) methods impose an |
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unphysical {\it flux} between different regions or ``slabs'' of the |
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simulation |
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box.\cite{MullerPlathe:1997xw,ISI:000080382700030,Kuang:2010uq} The |
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system responds by developing a temperature or velocity {\it gradient} |
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between the two regions. The gradients which develop in response to |
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the applied flux are then related (via linear response theory) to the |
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transport coefficient of interest. Since the amount of the applied |
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flux is known exactly, and measurement of a gradient is generally less |
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complicated, imposed-flux methods typically take shorter simulation |
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times to obtain converged results. At interfaces, the observed |
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gradients often exhibit near-discontinuities at the boundaries between |
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dissimilar materials. RNEMD methods do not need many trajectories to |
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provide information about transport properties, and they have become |
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widely used to compute thermal and mechanical transport in both |
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homogeneous liquids and |
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solids~\cite{MullerPlathe:1997xw,ISI:000080382700030,Maginn:2010} as |
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well as heterogeneous |
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interfaces.\cite{garde:nl2005,garde:PhysRevLett2009,kuang:AuThl} |
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|
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|
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% **METHODOLOGY** |
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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< |
\section{Methodology} |
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> |
\section{Velocity Shearing and Scaling (VSS) for non-periodic systems} |
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> |
The VSS-RNEMD approach uses a series of simultaneous velocity shearing |
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> |
and scaling exchanges between the two slabs.\cite{2012MolPh.110..691K} |
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> |
This method imposes energy and momentum conservation constraints while |
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simultaneously creating a desired flux between the two slabs. These |
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constraints ensure that all configurations are sampled from the same |
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> |
microcanonical (NVE) ensemble. |
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|
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We have extended the VSS method for use in {\it non-periodic} |
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simulations, in which the ``slabs'' have been generalized to two |
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separated regions of space. These regions could be defined as |
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concentric spheres (as in figure \ref{fig:VSS}), or one of the regions |
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can be defined in terms of a dynamically changing ``hull'' comprising |
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the surface atoms of the cluster. This latter definition is identical |
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to the hull used in the Langevin Hull algorithm. |
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|
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We present here a new set of constraints that are more general than |
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the VSS constraints. For the non-periodic variant, the constraints |
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fix both the total energy and total {\it angular} momentum of the |
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system while simultaneously imposing a thermal and angular momentum |
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flux between the two regions. |
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|
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After each $\Delta t$ time interval, the particle velocities |
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($\mathbf{v}_i$ and $\mathbf{v}_j$) in the two shells ($A$ and $B$) |
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are modified by a velocity scaling coefficient ($a$ and $b$) and by a |
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rotational shearing term ($\mathbf{c}_a$ and $\mathbf{c}_b$). |
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\begin{displaymath} |
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\begin{array}{rclcl} |
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& \underline{~~~~~~~~\mathrm{scaling}~~~~~~~~} & & |
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\underline{\mathrm{rotational~shearing}} \\ \\ |
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\mathbf{v}_i $~~~$\leftarrow & |
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a \left(\mathbf{v}_i - \langle \omega_a |
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+ |
\rangle \times \mathbf{r}_i\right) & + & \mathbf{c}_a \times \mathbf{r}_i \\ |
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+ |
\mathbf{v}_j $~~~$\leftarrow & |
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b \left(\mathbf{v}_j - \langle \omega_b |
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+ |
\rangle \times \mathbf{r}_j\right) & + & \mathbf{c}_b \times \mathbf{r}_j |
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+ |
\end{array} |
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+ |
\end{displaymath} |
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Here $\langle\mathbf{\omega}_a\rangle$ and |
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+ |
$\langle\mathbf{\omega}_b\rangle$ are the instantaneous angular |
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velocities of each shell, and $\mathbf{r}_i$ is the position of |
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+ |
particle $i$ relative to a fixed point in space (usually the center of |
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+ |
mass of the cluster). Particles in the shells also receive an |
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additive ``angular shear'' to their velocities. The amount of shear |
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+ |
is governed by the imposed angular momentum flux, |
| 170 |
+ |
$\mathbf{j}_r(\mathbf{L})$, |
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+ |
\begin{eqnarray} |
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\mathbf{c}_a & = & - \mathbf{j}_r(\mathbf{L}) \cdot |
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\overleftrightarrow{I_a}^{-1} \Delta t + \langle \omega_a \rangle \label{eq:bc}\\ |
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+ |
\mathbf{c}_b & = & + \mathbf{j}_r(\mathbf{L}) \cdot |
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+ |
\overleftrightarrow{I_b}^{-1} \Delta t + \langle \omega_b \rangle \label{eq:bh} |
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+ |
\end{eqnarray} |
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where $\overleftrightarrow{I}_{\{a,b\}}$ is the moment of inertia tensor for |
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each of the two shells. |
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+ |
|
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To simultaneously impose a thermal flux ($J_r$) between the shells we |
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use energy conservation constraints, |
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+ |
\begin{eqnarray} |
| 183 |
+ |
K_a - J_r \Delta t & = & a^2 \left(K_a - \frac{1}{2}\langle |
| 184 |
+ |
\omega_a \rangle \cdot \overleftrightarrow{I_a} \cdot \langle \omega_a |
| 185 |
+ |
\rangle \right) + \frac{1}{2} \mathbf{c}_a \cdot \overleftrightarrow{I_a} |
| 186 |
+ |
\cdot \mathbf{c}_a \label{eq:Kc}\\ |
| 187 |
+ |
K_b + J_r \Delta t & = & b^2 \left(K_b - \frac{1}{2}\langle |
| 188 |
+ |
\omega_b \rangle \cdot \overleftrightarrow{I_b} \cdot \langle \omega_b |
| 189 |
+ |
\rangle \right) + \frac{1}{2} \mathbf{c}_b \cdot \overleftrightarrow{I_b} \cdot \mathbf{c}_b \label{eq:Kh} |
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+ |
\end{eqnarray} |
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+ |
Simultaneous solution of these quadratic formulae for the scaling |
| 192 |
+ |
coefficients, $a$ and $b$, will ensure that the simulation samples |
| 193 |
+ |
from the original microcanonical (NVE) ensemble. Here $K_{\{a,b\}}$ |
| 194 |
+ |
is the instantaneous translational kinetic energy of each shell. At |
| 195 |
+ |
each time interval, we solve for $a$, $b$, $\mathbf{c}_a$, and |
| 196 |
+ |
$\mathbf{c}_b$, subject to the imposed angular momentum flux, |
| 197 |
+ |
$j_r(\mathbf{L})$, and thermal flux, $J_r$ values. The new particle |
| 198 |
+ |
velocities are computed, and the simulation continues. System |
| 199 |
+ |
configurations after the transformations have exactly the same energy |
| 200 |
+ |
({\it and} angular momentum) as before the moves. |
| 201 |
+ |
|
| 202 |
+ |
As the simulation progresses, the velocity transformations can be |
| 203 |
+ |
performed on a regular basis, and the system will develop a |
| 204 |
+ |
temperature and/or angular velocity gradient in response to the |
| 205 |
+ |
applied flux. Using the slope of the radial temperature or velocity |
| 206 |
+ |
gradients, it is quite simple to obtain both the thermal conductivity |
| 207 |
+ |
($\lambda$) and shear viscosity ($\eta$), |
| 208 |
+ |
\begin{equation} |
| 209 |
+ |
J_r = -\lambda \frac{\partial T}{\partial |
| 210 |
+ |
r} \hspace{2in} j_r(\mathbf{L}_z) = -\eta \frac{\partial |
| 211 |
+ |
\omega_z}{\partial r} |
| 212 |
+ |
\end{equation} |
| 213 |
+ |
of a liquid cluster. |
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+ |
|
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|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
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% NON-PERIODIC DYNAMICS |
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|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
