| 28 |
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|
| 29 |
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\begin{document} |
| 30 |
|
|
| 31 |
< |
\title{A minimal perturbation approach to RNEMD able to simultaneously |
| 32 |
< |
impose thermal and momentum gradients} |
| 31 |
> |
\title{Velocity Shearing and Scaling RNEMD: a minimally perturbing |
| 32 |
> |
method for producing temperature and momentum gradients} |
| 33 |
|
|
| 34 |
|
\author{Shenyu Kuang and J. Daniel |
| 35 |
|
Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
| 46 |
|
\begin{abstract} |
| 47 |
|
We present a new method for introducing stable nonequilibrium |
| 48 |
|
velocity and temperature gradients in molecular dynamics simulations |
| 49 |
< |
of heterogeneous systems. This method conserves the linear momentum |
| 50 |
< |
and total energy of the system and improves previous Reverse |
| 51 |
< |
Non-Equilibrium Molecular Dynamics (RNEMD) methods while maintaining |
| 52 |
< |
thermal velocity distributions. It also avoid thermal anisotropy |
| 53 |
< |
occured in previous NIVS simulations by using isotropic velocity |
| 54 |
< |
scaling on the molecules in specific regions of a system. To test |
| 55 |
< |
the method, we have computed the thermal conductivity and shear |
| 49 |
> |
of heterogeneous systems. This method conserves both the linear |
| 50 |
> |
momentum and total energy of the system and improves previous |
| 51 |
> |
reverse non-equilibrium molecular dynamics (RNEMD) methods while |
| 52 |
> |
retaining equilibrium thermal velocity distributions in each region |
| 53 |
> |
of the system. The new method avoids the thermal anisotropy |
| 54 |
> |
produced by previous methods by using isotropic velocity scaling and |
| 55 |
> |
shearing on all of the molecules in specific regions. To test the |
| 56 |
> |
method, we have computed the thermal conductivity and shear |
| 57 |
|
viscosity of model liquid systems as well as the interfacial |
| 58 |
< |
frictions of a series of metal/liquid interfaces. Its ability to |
| 59 |
< |
combine the thermal and momentum gradients allows us to obtain shear |
| 60 |
< |
viscosity data for a range of temperatures in only one trajectory. |
| 61 |
< |
|
| 58 |
> |
friction coeefficients of a series of solid / liquid interfaces. The |
| 59 |
> |
method's ability to combine the thermal and momentum gradients |
| 60 |
> |
allows us to obtain shear viscosity data for a range of temperatures |
| 61 |
> |
from a single trajectory. |
| 62 |
|
\end{abstract} |
| 63 |
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|
| 64 |
|
\newpage |
| 70 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 71 |
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|
| 72 |
|
\section{Introduction} |
| 73 |
< |
Imposed-flux methods in Molecular Dynamics (MD) |
| 74 |
< |
simulations\cite{MullerPlathe:1997xw,ISI:000080382700030,kuang:164101} |
| 75 |
< |
can establish steady state systems with an applied flux set vs a |
| 76 |
< |
corresponding gradient that can be measured. These methods does not |
| 77 |
< |
need many trajectories to provide information of transport properties |
| 78 |
< |
of a given system. Thus, they are utilized in computing thermal and |
| 79 |
< |
mechanical transfer of homogeneous bulk systems as well as |
| 80 |
< |
heterogeneous systems such as solid-liquid |
| 73 |
> |
One of the standard ways to compute transport coefficients such as the |
| 74 |
> |
viscosities and thermal conductivities of liquids is to use |
| 75 |
> |
imposed-flux non-equilibrium molecular dynamics |
| 76 |
> |
methods.\cite{MullerPlathe:1997xw,ISI:000080382700030,kuang:164101} |
| 77 |
> |
These methods establish stable non-equilibrium conditions in a |
| 78 |
> |
simulation box using an applied momentum or thermal flux between |
| 79 |
> |
different regions of the box. The corresponding temperature or |
| 80 |
> |
velocity gradients which develop in response to the applied flux is |
| 81 |
> |
related (via linear response theory) to the transport coefficient of |
| 82 |
> |
interest. These methods are quite efficient, in that they do not need |
| 83 |
> |
many trajectories to provide information about transport properties. To |
| 84 |
> |
date, they have been utilized in computing thermal and mechanical |
| 85 |
> |
transfer of both homogeneous liquids as well as heterogeneous systems |
| 86 |
> |
such as solid-liquid |
| 87 |
|
interfaces.\cite{garde:nl2005,garde:PhysRevLett2009,kuang:AuThl} |
| 88 |
|
|
| 89 |
< |
The Reverse Non-Equilibrium MD (RNEMD) methods adopt constraints that |
| 90 |
< |
satisfy linear momentum and total energy conservation of a system when |
| 91 |
< |
imposing fluxes in a simulation. Thus they are compatible with various |
| 92 |
< |
ensembles, including the micro-canonical (NVE) ensemble, without the |
| 93 |
< |
need of an external thermostat. The original approaches proposed by |
| 94 |
< |
M\"{u}ller-Plathe {\it et |
| 89 |
> |
The reverse non-equilibrium molecular dynamics (RNEMD) methods utilize |
| 90 |
> |
additional constraints that ensure conservation of linear momentum and |
| 91 |
> |
total energy of the system while imposing the desired flux. The RNEMD |
| 92 |
> |
methods are therefore capable of sampling various thermodynamically |
| 93 |
> |
relevent ensembles, including the micro-canonical (NVE) ensemble, |
| 94 |
> |
without resorting to an external thermostat. The original approaches |
| 95 |
> |
proposed by M\"{u}ller-Plathe {\it et |
| 96 |
|
al.}\cite{MullerPlathe:1997xw,ISI:000080382700030} utilize simple |
| 97 |
< |
momentum swapping for generating energy/momentum fluxes, which can |
| 98 |
< |
also be compatible with particles of different identities. Although |
| 99 |
< |
simple to implement in a simulation, this approach can create |
| 100 |
< |
nonthermal velocity distributions, as discovered by Tenney and |
| 101 |
< |
Maginn\cite{Maginn:2010}. Furthermore, this approach is less efficient |
| 102 |
< |
for kinetic energy transfer between particles of different identities, |
| 103 |
< |
especially when the mass difference between the particles becomes |
| 104 |
< |
significant. This also limits its applications on heterogeneous |
| 105 |
< |
interfacial systems. |
| 97 |
> |
momentum swapping moves for generating energy/momentum fluxes. The |
| 98 |
> |
swapping moves can also be made compatible with particles of different |
| 99 |
> |
identities. Although the swapping moves are simple to implement in |
| 100 |
> |
molecular simulations, Tenney and Maginn have shown that they create |
| 101 |
> |
nonthermal velocity distributions.\cite{Maginn:2010} Furthermore, this |
| 102 |
> |
approach is not particularly efficient for kinetic energy transfer |
| 103 |
> |
between particles of different identities, particularly when the mass |
| 104 |
> |
difference between the particles becomes significant. This problem |
| 105 |
> |
makes applying swapping-move RNEMD methods on heterogeneous |
| 106 |
> |
interfacial systems somewhat difficult. |
| 107 |
|
|
| 108 |
< |
Recently, we developed a different approach, using Non-Isotropic |
| 109 |
< |
Velocity Scaling (NIVS) \cite{kuang:164101} algorithm to impose |
| 110 |
< |
fluxes. Compared to the momentum swapping move, it scales the velocity |
| 111 |
< |
vectors in two separate regions of a simulated system with respective |
| 112 |
< |
diagonal scaling matrices. These matrices are determined by solving a |
| 113 |
< |
set of equations including linear momentum and kinetic energy |
| 114 |
< |
conservation constraints and target flux satisfaction. This method is |
| 115 |
< |
able to effectively impose a wide range of kinetic energy fluxes |
| 116 |
< |
without obvious perturbation to the velocity distributions of the |
| 117 |
< |
simulated systems, regardless of the presence of heterogeneous |
| 118 |
< |
interfaces. We have successfully applied this approach in studying the |
| 119 |
< |
interfacial thermal conductance at metal-solvent |
| 108 |
> |
Recently, we developed a somewhat different approach to applying |
| 109 |
> |
thermal fluxes in RNEMD simulation using a Non-Isotropic Velocity |
| 110 |
> |
Scaling (NIVS) algorithm.\cite{kuang:164101} This algorithm scales all |
| 111 |
> |
atomic velocity vectors in two separate regions of a simulated system |
| 112 |
> |
using two diagonal scaling matrices. The scaling matrices are |
| 113 |
> |
determined by solving single quartic equation which includes linear |
| 114 |
> |
momentum and kinetic energy conservation constraints as well as the |
| 115 |
> |
target thermal flux between the regions. The NIVS method is able to |
| 116 |
> |
effectively impose a wide range of kinetic energy fluxes without |
| 117 |
> |
significant perturbation to the velocity distributions away from ideal |
| 118 |
> |
Maxwell-Boltzmann distributions, even in the presence of heterogeneous |
| 119 |
> |
interfaces. We successfully applied this approach in studying the |
| 120 |
> |
interfacial thermal conductance at chemically-capped metal-solvent |
| 121 |
|
interfaces.\cite{kuang:AuThl} |
| 122 |
|
|
| 123 |
< |
However, the NIVS approach has limited applications in imposing |
| 124 |
< |
momentum fluxes. Temperature anisotropy could happen under high |
| 125 |
< |
momentum fluxes due to the implementation of this algorithm. Thus, |
| 126 |
< |
combining thermal and momentum flux is also difficult to obtain with |
| 127 |
< |
this approach. However, such combination may provide a means to |
| 128 |
< |
simulate thermal/momentum gradient coupled processes such as Soret |
| 129 |
< |
effect in liquid flows. Therefore, developing improved approaches to |
| 130 |
< |
extend the applications of the imposed-flux method is desirable. |
| 123 |
> |
The NIVS approach works very well for preparing stable {\it thermal} |
| 124 |
> |
gradients. However, as we pointed out in the original paper, it has |
| 125 |
> |
limited application in imposing {\it linear} momentum fluxes (which |
| 126 |
> |
are required for measuring shear viscosities). The reason for this is |
| 127 |
> |
that linear momentum flux was being imposed by scaling random |
| 128 |
> |
fluctuations of the center of the velocity distributions. Repeated |
| 129 |
> |
application of the original NIVS approach resulted in temperature |
| 130 |
> |
anisotropy, i.e. the width of the velocity distributions depended on |
| 131 |
> |
coordinates perpendicular to the desired gradient direction. For this |
| 132 |
> |
reason, combining thermal and momentum fluxes was difficult with the |
| 133 |
> |
original NIVS algorithm. However, combinations of thermal and |
| 134 |
> |
velocity gradients would provide a means to simulate thermal-linear |
| 135 |
> |
coupled processes such as Soret effect in liquid flows. Therefore, |
| 136 |
> |
developing improved approaches to the scaling imposed-flux methods |
| 137 |
> |
would be useful. |
| 138 |
|
|
| 139 |
< |
In this paper, we improve the RNEMD methods by proposing a novel |
| 140 |
< |
approach to impose fluxes. This approach separate the means of applying |
| 141 |
< |
momentum and thermal flux with operations in one time step and thus is |
| 142 |
< |
able to simutaneously impose thermal and momentum flux. Furthermore, |
| 143 |
< |
the approach retains desirable features of previous RNEMD approaches |
| 144 |
< |
and is simpler to implement compared to the NIVS method. In what |
| 145 |
< |
follows, we first present the method and its implementation in a |
| 146 |
< |
simulation. Then we compare the method on bulk fluids to previous |
| 147 |
< |
methods. Also, interfacial frictions are computed for a series of |
| 148 |
< |
interfaces. |
| 139 |
> |
In this paper, we improve the RNEMD methods by introducing a novel |
| 140 |
> |
approach to creating imposed fluxes. This approach separates the |
| 141 |
> |
shearing and scaling of the velocity distributions in different |
| 142 |
> |
spatial regions and can apply both transformations within a single |
| 143 |
> |
time step. The approach retains desirable features of previous RNEMD |
| 144 |
> |
approaches and is simpler to implement compared to the earlier NIVS |
| 145 |
> |
method. In what follows, we first present the Shearing-and-Scaling |
| 146 |
> |
(SS) RNEMD method and its implementation in a simulation. Then we |
| 147 |
> |
compare the SS-RNEMD method in bulk fluids to previous methods. We |
| 148 |
> |
also compute interfacial frictions are computed for a series of |
| 149 |
> |
heterogeneous interfaces. |
| 150 |
|
|
| 151 |
|
\section{Methodology} |
| 152 |
< |
Similar to the NIVS method,\cite{kuang:164101} we consider a |
| 153 |
< |
periodic system divided into a series of slabs along a certain axis |
| 154 |
< |
(e.g. $z$). The unphysical thermal and/or momentum flux is designated |
| 155 |
< |
from the center slab to one of the end slabs, and thus the thermal |
| 156 |
< |
flux results in a lower temperature of the center slab than the end |
| 157 |
< |
slab, and the momentum flux results in negative center slab momentum |
| 158 |
< |
with positive end slab momentum (unless these fluxes are set |
| 159 |
< |
negative). Therefore, the center slab is denoted as ``$c$'', while the |
| 160 |
< |
end slab as ``$h$''. |
| 152 |
> |
In an approach similar to the earlier NIVS method,\cite{kuang:164101} |
| 153 |
> |
we consider a periodic system which has been divided into a series of |
| 154 |
> |
slabs along a single axis (e.g. $z$). The unphysical thermal and |
| 155 |
> |
momentum fluxes are applied from one of the end slabs to the center |
| 156 |
> |
slab, and thus the thermal flux produces a higher temperature in the |
| 157 |
> |
center slab than in the end slab, and the momentum flux results in a |
| 158 |
> |
center slab moving along the positive $y$ axis (see Fig. \ref{cartoon}). |
| 159 |
> |
The applied fluxes can be set to negative values if the reverse |
| 160 |
> |
gradients are desired. For convenience the center slab is denoted as |
| 161 |
> |
the {\it hot} or {\it ``H''} slab, while the end slab is denoted {\it |
| 162 |
> |
``C''} (or {\it cold}). |
| 163 |
|
|
| 164 |
< |
To impose these fluxes, we periodically apply different set of |
| 165 |
< |
operations on velocities of particles {$i$} within the center slab and |
| 166 |
< |
those of particles {$j$} within the end slab: |
| 164 |
> |
\begin{figure} |
| 165 |
> |
\includegraphics[width=\linewidth]{cartoon} |
| 166 |
> |
\caption{The SS-RNEMD approach we are introducing imposes unphysical |
| 167 |
> |
transfer of both momentum and kinetic energy between a ``hot'' slab |
| 168 |
> |
and a ``cold'' slab in the simulation box. The molecular system |
| 169 |
> |
responds to this imposed flux by generating both momentum and |
| 170 |
> |
temperature gradients. The slope of the gradients can then be used |
| 171 |
> |
to compute transport properties (e.g. shear viscosity and thermal |
| 172 |
> |
conductivity).} |
| 173 |
> |
\label{cartoon} |
| 174 |
> |
\end{figure} |
| 175 |
> |
|
| 176 |
> |
To impose these fluxes, we periodically apply a set of operations on |
| 177 |
> |
the velocities of particles {$i$} within the cold slab and a separate |
| 178 |
> |
operation on particles {$j$} within the hot slab. |
| 179 |
|
\begin{eqnarray} |
| 180 |
|
\vec{v}_i & \leftarrow & c\cdot\left(\vec{v}_i - \langle\vec{v}_c |
| 181 |
|
\rangle\right) + \left(\langle\vec{v}_c\rangle + \vec{a}_c\right) \\ |
| 182 |
|
\vec{v}_j & \leftarrow & h\cdot\left(\vec{v}_j - \langle\vec{v}_h |
| 183 |
|
\rangle\right) + \left(\langle\vec{v}_h\rangle + \vec{a}_h\right) |
| 184 |
|
\end{eqnarray} |
| 185 |
< |
where $\langle\vec{v}_c\rangle$ and $\langle\vec{v}_h\rangle$ denotes |
| 186 |
< |
the instantaneous bulk velocity of slabs $c$ and $h$ respectively |
| 187 |
< |
before an operation is applied. When a momentum flux $\vec{j}_z(\vec{p})$ |
| 188 |
< |
presents, these bulk velocities would have a corresponding change |
| 189 |
< |
($\vec{a}_c$ and $\vec{a}_h$ respectively) according to Newton's |
| 190 |
< |
second law: |
| 185 |
> |
where $\langle\vec{v}_c\rangle$ and $\langle\vec{v}_h\rangle$ denote |
| 186 |
> |
the instantaneous average velocity of the molecules within slabs $C$ |
| 187 |
> |
and $H$ respectively. When a momentum flux $\vec{j}_z(\vec{p})$ is |
| 188 |
> |
present, these slab-averaged velocities also get corresponding |
| 189 |
> |
incremental changes ($\vec{a}_c$ and $\vec{a}_h$ respectively) that |
| 190 |
> |
are applied to all particles within each slab. The incremental |
| 191 |
> |
changes are obtained using Newton's second law: |
| 192 |
|
\begin{eqnarray} |
| 193 |
< |
M_c \vec{a}_c & = & -\vec{j}_z(\vec{p}) \Delta t \\ |
| 193 |
> |
M_c \vec{a}_c & = & -\vec{j}_z(\vec{p}) \Delta t \label{eq:newton1} \\ |
| 194 |
|
M_h \vec{a}_h & = & \vec{j}_z(\vec{p}) \Delta t |
| 195 |
+ |
\label{eq:newton2} |
| 196 |
|
\end{eqnarray} |
| 197 |
|
where $M$ denotes total mass of particles within a slab: |
| 198 |
|
\begin{eqnarray} |
| 201 |
|
\end{eqnarray} |
| 202 |
|
and $\Delta t$ is the interval between two separate operations. |
| 203 |
|
|
| 204 |
< |
The above operations already conserve the linear momentum of a |
| 205 |
< |
periodic system. To further satisfy total energy conservation as well |
| 206 |
< |
as to impose the thermal flux $J_z$, the following equations are |
| 207 |
< |
included as well: |
| 208 |
< |
[MAY PUT EXTRA MATH IN SUPPORT INFO OR APPENDIX] |
| 204 |
> |
The operations in Eqs. \ref{eq:newton1} and \ref{eq:newton2} already |
| 205 |
> |
conserve the linear momentum of a periodic system. To further satisfy |
| 206 |
> |
total energy conservation as well as to impose the thermal flux $J_z$, |
| 207 |
> |
the following constraint equations must be solved for the two scaling |
| 208 |
> |
variables $c$ and $h$: |
| 209 |
|
\begin{eqnarray} |
| 210 |
|
K_c - J_z\Delta t & = & c^2 (K_c - \frac{1}{2}M_c \langle\vec{v}_c |
| 211 |
|
\rangle^2) + \frac{1}{2}M_c (\langle \vec{v}_c \rangle + \vec{a}_c)^2 \\ |
| 212 |
|
K_h + J_z\Delta t & = & h^2 (K_h - \frac{1}{2}M_h \langle\vec{v}_h |
| 213 |
< |
\rangle^2) + \frac{1}{2}M_h (\langle \vec{v}_h \rangle + \vec{a}_h)^2 |
| 213 |
> |
\rangle^2) + \frac{1}{2}M_h (\langle \vec{v}_h \rangle + \vec{a}_h)^2. |
| 214 |
> |
\label{constraint} |
| 215 |
|
\end{eqnarray} |
| 216 |
< |
where $K_c$ and $K_h$ denotes translational kinetic energy of slabs |
| 217 |
< |
$c$ and $h$ respectively before an operation is applied. These |
| 218 |
< |
translational kinetic energy conservation equations are sufficient to |
| 219 |
< |
ensure total energy conservation, as the operations applied in our |
| 220 |
< |
method do not change the kinetic energy related to other degrees of |
| 221 |
< |
freedom or the potential energy of a system, given that its potential |
| 187 |
< |
energy does not depend on particle velocity. |
| 216 |
> |
Here $K_c$ and $K_h$ denote the translational kinetic energy of slabs |
| 217 |
> |
$C$ and $H$ respectively. These conservation equations are sufficient |
| 218 |
> |
to ensure total energy conservation, as the operations applied in our |
| 219 |
> |
method do not change the kinetic energy related to orientational |
| 220 |
> |
degrees of freedom or the potential energy of a system (as long as the |
| 221 |
> |
potential energy is independent of particle velocity). |
| 222 |
|
|
| 223 |
< |
The above sets of equations are sufficient to determine the velocity |
| 224 |
< |
scaling coefficients ($c$ and $h$) as well as $\vec{a}_c$ and |
| 225 |
< |
$\vec{a}_h$. Note that there are two roots respectively for $c$ and |
| 226 |
< |
$h$. However, the positive roots (which are closer to 1) are chosen so |
| 227 |
< |
that the perturbations to a system can be reduced to a minimum. Figure |
| 228 |
< |
\ref{method} illustrates the implementation sketch of this algorithm |
| 229 |
< |
in an individual step. |
| 223 |
> |
Equations \ref{eq:newton1}-\ref{constraint} are sufficient to |
| 224 |
> |
determine the velocity scaling coefficients ($c$ and $h$) as well as |
| 225 |
> |
$\vec{a}_c$ and $\vec{a}_h$. Note that there are usually two roots |
| 226 |
> |
respectively for $c$ and $h$. However, the positive roots (which are |
| 227 |
> |
closer to 1) are chosen so that the perturbations to the system are |
| 228 |
> |
minimal. Figure \ref{method} illustrates the implementation of this |
| 229 |
> |
algorithm in an individual step. |
| 230 |
|
|
| 231 |
|
\begin{figure} |
| 232 |
< |
\includegraphics[width=\linewidth]{method} |
| 233 |
< |
\caption{Illustration of the implementation of the algorithm in a |
| 234 |
< |
single step. Starting from an ideal velocity distribution, the |
| 235 |
< |
transformation is used to apply the effect of both a thermal and a |
| 236 |
< |
momentum flux from the ``c'' slab to the ``h'' slab. As the figure |
| 237 |
< |
shows, thermal distributions can preserve after this operation.} |
| 232 |
> |
\centering |
| 233 |
> |
\includegraphics[width=5in]{method} |
| 234 |
> |
\caption{Illustration of a single step implementation of the |
| 235 |
> |
algorithm. Starting with the velocity distributions for the two |
| 236 |
> |
slabs in a shearing fluid, the transformation is used to apply the |
| 237 |
> |
effect of both a thermal and a momentum flux from the ``c'' slab to |
| 238 |
> |
the ``h'' slab. As the figure shows, gaussian distributions are |
| 239 |
> |
preserved by both the scaling and shearing operations.} |
| 240 |
|
\label{method} |
| 241 |
|
\end{figure} |
| 242 |
|
|
| 243 |
< |
By implementing these operations at a certain frequency, a steady |
| 244 |
< |
thermal and/or momentum flux can be applied and the corresponding |
| 245 |
< |
temperature and/or momentum gradients can be established. |
| 243 |
> |
By implementing these operations at a fixed frequency, stable thermal |
| 244 |
> |
and momentum fluxes can both be applied and the corresponding |
| 245 |
> |
temperature and momentum gradients can be established. |
| 246 |
|
|
| 247 |
< |
Compared to the previous NIVS method, this approach is computationally |
| 248 |
< |
more efficient in that only quadratic equations are involved to |
| 249 |
< |
determine a set of scaling coefficients, while the NIVS method needs |
| 250 |
< |
to solve quartic equations. Furthermore, this method implements |
| 251 |
< |
isotropic scaling of velocities in respective slabs, unlike the NIVS, |
| 252 |
< |
where an extra criteria function is necessary to choose a set of |
| 253 |
< |
coefficients that performs a scaling as isotropic as possible. More |
| 254 |
< |
importantly, separating the means of momentum flux imposing from |
| 255 |
< |
velocity scaling avoids the underlying cause to thermal anisotropy in |
| 256 |
< |
NIVS when applying a momentum flux. And later sections will |
| 257 |
< |
demonstrate that this can improve the performance in shear viscosity |
| 222 |
< |
simulations. |
| 247 |
> |
Compared to the previous NIVS method, the SS-RNEMD approach is |
| 248 |
> |
computationally simpler in that only quadratic equations are involved |
| 249 |
> |
to determine a set of scaling coefficients, while the NIVS method |
| 250 |
> |
required solution of quartic equations. Furthermore, this method |
| 251 |
> |
implements {\it isotropic} scaling of velocities in respective slabs, |
| 252 |
> |
unlike NIVS, which required extra flexibility in the choice of scaling |
| 253 |
> |
coefficients to allow for the energy and momentum constraints. Most |
| 254 |
> |
importantly, separating the linear momentum flux from velocity scaling |
| 255 |
> |
avoids the underlying cause of the thermal anisotropy in NIVS. In |
| 256 |
> |
later sections, we demonstrate that this can improve the calculated |
| 257 |
> |
shear viscosities from RNEMD simulations. |
| 258 |
|
|
| 259 |
< |
This approach is advantageous over the original momentum swapping in |
| 260 |
< |
many aspects. In one swapping, the velocity vectors involved are |
| 261 |
< |
usually very different (or the generated flux is trivial to obtain |
| 262 |
< |
gradients), thus the swapping tends to incur perturbations to the |
| 263 |
< |
neighbors of the particles involved. Comparatively, our approach |
| 264 |
< |
disperse the flux to every selected particle in a slab so that |
| 265 |
< |
perturbations in the flux generating region could be |
| 266 |
< |
minimized. Additionally, because the momentum swapping steps tend to |
| 267 |
< |
result in a nonthermal distribution, when an imposed flux is |
| 268 |
< |
relatively large and diffusions from the neighboring slabs could no |
| 269 |
< |
longer remedy this effect, problematic distributions would be |
| 270 |
< |
observed. In comparison, the operations of our approach has the nature |
| 271 |
< |
of preserving the equilibrium velocity distributions (commonly |
| 272 |
< |
Maxwell-Boltzmann), and results in later section will illustrate that |
| 273 |
< |
this is helpful to retain thermal distributions in a simulation. |
| 259 |
> |
The SS-RNEMD approach has advantages over the original momentum |
| 260 |
> |
swapping in many respects. In the swapping method, the velocity |
| 261 |
> |
vectors involved are usually very different (or the generated flux |
| 262 |
> |
would be quite small), thus the swapping tends to create strong |
| 263 |
> |
perturbations in the neighborhood of the particles involved. In |
| 264 |
> |
comparison, the SS approach distributes the flux widely to all |
| 265 |
> |
particles in a slab so that perturbations in the flux generating |
| 266 |
> |
region are minimized. Additionally, momentum swapping steps tend to |
| 267 |
> |
result in nonthermal velocity distributions when the imposed flux is |
| 268 |
> |
large and diffusion from the neighboring slabs cannot carry momentum |
| 269 |
> |
away quickly enough. In comparison, the scaling and shearing moves |
| 270 |
> |
made in the SS approach preserve the shapes of the equilibrium |
| 271 |
> |
velocity disributions (e.g. Maxwell-Boltzmann). The results presented |
| 272 |
> |
in later sections will illustrate that this is quite helpful in |
| 273 |
> |
retaining reasonable thermal distributions in a simulation. |
| 274 |
|
|
| 275 |
|
\section{Computational Details} |
| 276 |
|
The algorithm has been implemented in our MD simulation code, |
| 277 |
< |
OpenMD\cite{Meineke:2005gd,openmd}. We compare the method with |
| 278 |
< |
previous RNEMD methods or equilibrium MD (EMD) methods in homogeneous fluids |
| 279 |
< |
(Lennard-Jones and SPC/E water). And taking advantage of the method, |
| 280 |
< |
we simulate the interfacial friction of different heterogeneous |
| 281 |
< |
interfaces (gold-organic solvent and gold-SPC/E water and ice-liquid |
| 282 |
< |
water). |
| 277 |
> |
OpenMD.\cite{Meineke:2005gd,openmd} We will first compare the method |
| 278 |
> |
with previous RNEMD methods and equilibrium MD in homogeneous |
| 279 |
> |
fluids (Lennard-Jones and SPC/E water). We have also used the new |
| 280 |
> |
method to simulate the interfacial friction of different heterogeneous |
| 281 |
> |
interfaces (Au (111) with organic solvents, Au(111) with SPC/E water, |
| 282 |
> |
and the Ice Ih - liquid water interface). |
| 283 |
|
|
| 284 |
|
\subsection{Simulation Protocols} |
| 285 |
< |
The systems to be investigated are set up in orthorhombic simulation |
| 285 |
> |
The systems we investigated were set up in orthorhombic simulation |
| 286 |
|
cells with periodic boundary conditions in all three dimensions. The |
| 287 |
< |
$z$ axis of these cells were longer and set as the temperature and/or |
| 288 |
< |
momentum gradient axis. And the cells were evenly divided into $N$ |
| 289 |
< |
slabs along this axis, with various $N$ depending on individual |
| 290 |
< |
system. The $x$ and $y$ axis were of the same length in homogeneous |
| 291 |
< |
systems or had length scale close to each other where heterogeneous |
| 292 |
< |
interfaces presents. In all cases, before introducing a nonequilibrium |
| 293 |
< |
method to establish steady thermal and/or momentum gradients for |
| 294 |
< |
further measurements and calculations, canonical ensemble with a |
| 295 |
< |
Nos\'e-Hoover thermostat\cite{hoover85} and microcanonical ensemble |
| 296 |
< |
equilibrations were used before data collections. For SPC/E water |
| 297 |
< |
simulations, isobaric-isothermal equilibrations\cite{melchionna93} are |
| 298 |
< |
performed before the above to reach normal pressure (1 bar); for |
| 299 |
< |
interfacial systems, similar equilibrations are used to relax the |
| 300 |
< |
surface tensions of the $xy$ plane. |
| 287 |
> |
$z$-axes of these cells were typically quite long and served as the |
| 288 |
> |
temperature and/or momentum gradient axes. The cells were evenly |
| 289 |
> |
divided into $N$ slabs along this axis, with $N$ varying for the |
| 290 |
> |
individual system. The $x$ and $y$ axes were of similar lengths in all |
| 291 |
> |
simulations. In all cases, before introducing a nonequilibrium method |
| 292 |
> |
to establish steady thermal and/or momentum gradients, equilibration |
| 293 |
> |
simulations were run under the canonical ensemble with a Nos\'e-Hoover |
| 294 |
> |
thermostat\cite{hoover85} followed by further equilibration using |
| 295 |
> |
standard constant energy (NVE) conditions. For SPC/E water |
| 296 |
> |
simulations, isobaric-isothermal equilibrations\cite{melchionna93} |
| 297 |
> |
were performed before equilibration to reach standard densities at |
| 298 |
> |
atmospheric pressure (1 bar); for interfacial systems, similar |
| 299 |
> |
equilibrations with anisotropic box relaxations are used to relax the |
| 300 |
> |
surface tension in the $xy$ plane. |
| 301 |
|
|
| 302 |
< |
While homogeneous fluid systems can be set up with rather random |
| 303 |
< |
configurations, our interfacial systems needs a series of steps to |
| 304 |
< |
ensure the interfaces be established properly for computations. The |
| 305 |
< |
preparation and equilibration of butanethiol covered gold (111) |
| 306 |
< |
surface and further solvation and equilibration process is described |
| 307 |
< |
in details as in reference \cite{kuang:AuThl}. |
| 302 |
> |
While homogeneous fluid systems can be set up with random |
| 303 |
> |
configurations, interfacial systems are typically prepared with a |
| 304 |
> |
single crystal face presented perpendicular to the $z$-axis. This |
| 305 |
> |
crystal face is aligned in the x-y plane of the periodic cell, and the |
| 306 |
> |
solvent occupies the region directly above and below a crystalline |
| 307 |
> |
slab. The preparation and equilibration of butanethiol covered |
| 308 |
> |
Au(111) surfaces, as well as the solvation and equilibration processes |
| 309 |
> |
used for these interfaces are described in detail in reference |
| 310 |
> |
\cite{kuang:AuThl}. |
| 311 |
|
|
| 312 |
< |
As for the ice/liquid water interfaces, the basal surface of ice |
| 313 |
< |
lattice was first constructed. Hirsch {\it et |
| 314 |
< |
al.}\cite{doi:10.1021/jp048434u} explored the energetics of ice |
| 315 |
< |
lattices with all possible proton order configurations. We refer to |
| 316 |
< |
their results and choose the configuration of the lowest energy after |
| 317 |
< |
geometry optimization as the unit cell for our ice lattices. Although |
| 312 |
> |
For the ice / liquid water interfaces, the basal surface of ice |
| 313 |
> |
lattice was first constructed. Hirsch and Ojam\"{a}e |
| 314 |
> |
\cite{doi:10.1021/jp048434u} explored the energetics of ice lattices |
| 315 |
> |
with all possible proton ordered configurations. We utilized Hirsch |
| 316 |
> |
and Ojam\"{a}e's structure 6 ($P2_12_12_1$) which is an orthorhombic |
| 317 |
> |
cell giving a proton-ordered version of Ice Ih. The basal face of ice |
| 318 |
> |
in this unit cell geometry is the $\{0~0~1\}$ face. Although |
| 319 |
|
experimental solid/liquid coexistant temperature under normal pressure |
| 320 |
< |
should be close to 273K, Bryk and Haymet's simulations of ice/liquid |
| 321 |
< |
water interfaces with different models suggest that for SPC/E, the |
| 322 |
< |
most stable interface is observed at 225$\pm$5K.\cite{bryk:10258} |
| 323 |
< |
Therefore, our ice/liquid water simulations were carried out at |
| 324 |
< |
225K. To have extra protection of the ice lattice during initial |
| 325 |
< |
equilibration (when the randomly generated liquid phase configuration |
| 326 |
< |
could release large amount of energy in relaxation), restraints were |
| 327 |
< |
applied to the ice lattice to avoid inadvertent melting by the heat |
| 289 |
< |
dissipated from the high enery configurations. |
| 290 |
< |
[MAY ADD A SNAPSHOT FOR BASAL PLANE] |
| 320 |
> |
are close to 273K, Bryk and Haymet's simulations of ice/liquid water |
| 321 |
> |
interfaces with different models suggest that for SPC/E, the most |
| 322 |
> |
stable interface is observed at 225$\pm$5K.\cite{bryk:10258} |
| 323 |
> |
Therefore, our ice/liquid water simulations were carried out at an |
| 324 |
> |
average temperature of 225K. Molecular translation and orientational |
| 325 |
> |
restraints were applied in the early stages of equilibration to |
| 326 |
> |
prevent melting of the ice slab. These restraints were removed during |
| 327 |
> |
NVT equilibration, well before data collection was carried out. |
| 328 |
|
|
| 329 |
|
\subsection{Force Field Parameters} |
| 330 |
< |
For comparison of our new method with previous work, we retain our |
| 330 |
> |
For comparing the SS-RNEMD method with previous work, we retained |
| 331 |
|
force field parameters consistent with previous simulations. Argon is |
| 332 |
|
the Lennard-Jones fluid used here, and its results are reported in |
| 333 |
< |
reduced unit for direct comparison purpose. |
| 333 |
> |
reduced units for purposes of direct comparison with previous |
| 334 |
> |
calculations. |
| 335 |
|
|
| 336 |
< |
As for our water simulations, SPC/E model is used throughout this work |
| 337 |
< |
for consistency. Previous work for transport properties of SPC/E water |
| 338 |
< |
model is available\cite{Bedrov:2000,10.1063/1.3330544,Medina2011} so |
| 339 |
< |
that unnecessary repetition of previous methods can be avoided. |
| 336 |
> |
For our water simulations, we utilized the SPC/E model throughout this |
| 337 |
> |
work. Previous work for transport properties of SPC/E water model is |
| 338 |
> |
available\cite{Bedrov:2000,10.1063/1.3330544,Medina2011} so direct |
| 339 |
> |
comparison with previous calculation methods is possible. |
| 340 |
|
|
| 341 |
|
The Au-Au interaction parameters in all simulations are described by |
| 342 |
|
the quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The |
| 348 |
|
|
| 349 |
|
For our gold/organic liquid interfaces, the small organic molecules |
| 350 |
|
included in our simulations are the Au surface capping agent |
| 351 |
< |
butanethiol and liquid hexane and toluene. The United-Atom |
| 351 |
> |
butanethiol as well as liquid hexane and liquid toluene. The |
| 352 |
> |
United-Atom |
| 353 |
|
models\cite{TraPPE-UA.thiols,TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} |
| 354 |
< |
for these components were used in this work for better computational |
| 354 |
> |
for these components were used in this work for computational |
| 355 |
|
efficiency, while maintaining good accuracy. We refer readers to our |
| 356 |
|
previous work\cite{kuang:AuThl} for further details of these models, |
| 357 |
|
as well as the interactions between Au and the above organic molecule |
| 358 |
|
components. |
| 359 |
|
|
| 360 |
|
\subsection{Thermal conductivities} |
| 361 |
< |
When $\vec{j}_z(\vec{p})$ is set to zero and a target $J_z$ is set to |
| 362 |
< |
impose kinetic energy transfer, the method can be used for thermal |
| 363 |
< |
conductivity computations. Similar to previous RNEMD methods, we |
| 364 |
< |
assume linear response of the temperature gradient with respect to the |
| 365 |
< |
thermal flux in general case. And the thermal conductivity ($\lambda$) |
| 366 |
< |
can be obtained with the imposed kinetic energy flux and the measured |
| 361 |
> |
When the linear momentum flux $\vec{j}_z(\vec{p})$ is set to zero and |
| 362 |
> |
the target $J_z$ is non-zero, SS-RNEMD imposes kinetic energy transfer |
| 363 |
> |
between the slabs, which can be used for computation of thermal |
| 364 |
> |
conductivities. Similar to previous RNEMD methods, we assume that we |
| 365 |
> |
are in the linear response regime of the temperature gradient with |
| 366 |
> |
respect to the thermal flux. The thermal conductivity ($\lambda$) can |
| 367 |
> |
be calculated using the imposed kinetic energy flux and the measured |
| 368 |
|
thermal gradient: |
| 369 |
|
\begin{equation} |
| 370 |
|
J_z = -\lambda \frac{\partial T}{\partial z} |
| 371 |
|
\end{equation} |
| 372 |
|
Like other imposed-flux methods, the energy flux was calculated using |
| 373 |
|
the total non-physical energy transferred (${E_{total}}$) from slab |
| 374 |
< |
``c'' to slab ``h'', which is recorded throughout a simulation, and |
| 375 |
< |
the time for data collection $t$: |
| 374 |
> |
``c'' to slab ``h'', which was recorded throughout the simulation, and |
| 375 |
> |
the total data collection time $t$: |
| 376 |
|
\begin{equation} |
| 377 |
< |
J_z = \frac{E_{total}}{2 t L_x L_y} |
| 377 |
> |
J_z = \frac{E_{total}}{2 t L_x L_y}. |
| 378 |
|
\end{equation} |
| 379 |
< |
where $L_x$ and $L_y$ denotes the dimensions of the plane in a |
| 379 |
> |
Here, $L_x$ and $L_y$ denote the dimensions of the plane in a |
| 380 |
|
simulation cell perpendicular to the thermal gradient, and a factor of |
| 381 |
< |
two in the denominator is necessary for the heat transport occurs in |
| 382 |
< |
both $+z$ and $-z$ directions. The average temperature gradient |
| 381 |
> |
two in the denominator is necessary as the heat transport occurs in |
| 382 |
> |
both the $+z$ and $-z$ directions. The average temperature gradient |
| 383 |
|
${\langle\partial T/\partial z\rangle}$ can be obtained by a linear |
| 384 |
|
regression of the temperature profile, which is recorded during a |
| 385 |
|
simulation for each slab in a cell. For Lennard-Jones simulations, |
| 387 |
|
(${\lambda^*=\lambda \sigma^2 m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$). |
| 388 |
|
|
| 389 |
|
\subsection{Shear viscosities} |
| 390 |
< |
Alternatively, the method can carry out shear viscosity calculations |
| 391 |
< |
by specify a momentum flux. In our algorithm, one can specify the |
| 392 |
< |
three components of the flux vector $\vec{j}_z(\vec{p})$ |
| 393 |
< |
respectively. For shear viscosity simulations, $j_z(p_z)$ is usually |
| 394 |
< |
set to zero. For isotropic systems, the direction of |
| 395 |
< |
$\vec{j}_z(\vec{p})$ on the $xy$ plane should not matter, but the |
| 356 |
< |
ability of arbitarily specifying the vector direction in our method |
| 357 |
< |
could provide convenience in anisotropic simulations. |
| 390 |
> |
Alternatively, when the linear momentum flux $\vec{j}_z(\vec{p})$ is |
| 391 |
> |
non-zero in either the $x$ or $y$ directions, the method can be used |
| 392 |
> |
to compute the shear viscosity. For isotropic systems, the direction |
| 393 |
> |
of $\vec{j}_z(\vec{p})$ on the $xy$ plane should not matter, but the |
| 394 |
> |
ability to arbitarily specify the vector direction in our method could |
| 395 |
> |
provide convenience when working with anisotropic interfaces. |
| 396 |
|
|
| 397 |
< |
Similar to thermal conductivity computations, for a homogeneous |
| 398 |
< |
system, linear response of the momentum gradient with respect to the |
| 399 |
< |
shear stress is assumed, and the shear viscosity ($\eta$) can be |
| 400 |
< |
obtained with the imposed momentum flux (e.g. in $x$ direction) and |
| 401 |
< |
the measured gradient: |
| 397 |
> |
In a manner similar to the thermal conductivity calculations, a linear |
| 398 |
> |
response of the momentum gradient with respect to the shear stress is |
| 399 |
> |
assumed, and the shear viscosity ($\eta$) can be obtained with the |
| 400 |
> |
imposed momentum flux (e.g. in $x$ direction) and the measured |
| 401 |
> |
velocity gradient: |
| 402 |
|
\begin{equation} |
| 403 |
|
j_z(p_x) = -\eta \frac{\partial v_x}{\partial z} |
| 404 |
|
\end{equation} |
| 414 |
|
viscosities are also reported in reduced units |
| 415 |
|
(${\eta^*=\eta\sigma^2(\varepsilon m)^{-1/2}}$). |
| 416 |
|
|
| 417 |
< |
Although $J_z$ may be switched off for shear viscosity simulations at |
| 418 |
< |
a certain temperature, our method's ability to impose both a thermal |
| 419 |
< |
and a momentum flux in one simulation allows the combination of a |
| 420 |
< |
temperature and a velocity gradient. In this case, since viscosity is |
| 421 |
< |
generally a function of temperature, the local viscosity also depends |
| 422 |
< |
on the local temperature. Therefore, in one such simulation, viscosity |
| 423 |
< |
at $z$ (corresponding to a certain $T$) can be computed with the |
| 424 |
< |
applied shear flux and the local velocity gradient (which can be |
| 425 |
< |
obtained by finite difference approximation). As a whole, the |
| 426 |
< |
viscosity can be mapped out as the function of temperature in one |
| 427 |
< |
single trajectory of simulation. Results for shear viscosity |
| 428 |
< |
computations of SPC/E water will demonstrate its effectiveness in |
| 429 |
< |
detail. |
| 417 |
> |
Although $J_z$ may be switched off for shear viscosity simulations, |
| 418 |
> |
the SS-RNEMD method allows the user the ability to simultaneously |
| 419 |
> |
impose both a thermal and a momentum flux during a single |
| 420 |
> |
simulation. This can create system with coincident temperature and a |
| 421 |
> |
velocity gradients. Since the viscosity is generally a function of |
| 422 |
> |
temperature, the local viscosity depends on the local temperature in |
| 423 |
> |
the fluid. Therefore, in a single simulation, viscosity at $z$ |
| 424 |
> |
(corresponding to a certain $T$) can be computed with the applied |
| 425 |
> |
shear flux and the local velocity gradient (which can be obtained by |
| 426 |
> |
finite difference approximation). This means that the temperature |
| 427 |
> |
dependence of the viscosity can be mapped out in only one |
| 428 |
> |
trajectory. Results for shear viscosity computations of SPC/E water |
| 429 |
> |
will demonstrate SS-RNEMD's efficiency in this respect. |
| 430 |
|
|
| 431 |
< |
\subsection{Interfacial friction and Slip length} |
| 432 |
< |
While the shear stress results in a velocity gradient within bulk |
| 433 |
< |
fluid phase, its effect at a solid-liquid interface could vary due to |
| 434 |
< |
the interaction strength between the two phases. The interfacial |
| 435 |
< |
friction coefficient $\kappa$ is defined to relate the shear stress |
| 436 |
< |
(e.g. along $x$-axis) with the relative fluid velocity tangent to the |
| 399 |
< |
interface: |
| 431 |
> |
\subsection{Interfacial friction and slip length} |
| 432 |
> |
Shear stress creates a velocity gradient within bulk fluid phases, but |
| 433 |
> |
at solid-liquid interfaces, the effects of a shear stress depend on |
| 434 |
> |
the molecular details of the interface. The interfacial friction |
| 435 |
> |
coefficient, $\kappa$, relates the shear stress (e.g. along the |
| 436 |
> |
$x$-axis) with the relative fluid velocity tangent to the interface: |
| 437 |
|
\begin{equation} |
| 438 |
< |
j_z(p_x)|_{interface} = \kappa\Delta v_x|_{interface} |
| 438 |
> |
j_z(p_x) = \kappa \left[v_x(fluid) - v_x(solid)\right] |
| 439 |
|
\end{equation} |
| 440 |
< |
Under ``stick'' boundary condition, $\Delta v_x|_{interface} |
| 441 |
< |
\rightarrow 0$, which leads to $\kappa\rightarrow\infty$. However, for |
| 442 |
< |
``slip'' boundary conditions at the solid-liquid interfaces, $\kappa$ |
| 443 |
< |
becomes finite. To characterize the interfacial boundary conditions, |
| 444 |
< |
slip length ($\delta$) is defined using $\kappa$ and the shear |
| 445 |
< |
viscocity of liquid phase ($\eta$): |
| 440 |
> |
where $v_x(fluid)$ and $v_x(solid)$ are the velocities measured |
| 441 |
> |
directly adjacent to the interface. Under ``stick'' boundary |
| 442 |
> |
condition, $\Delta v_x|_\mathrm{interface} \rightarrow 0$, which leads |
| 443 |
> |
to $\kappa\rightarrow\infty$. However, for ``slip'' boundary |
| 444 |
> |
conditions at a solid-liquid interface, $\kappa$ becomes finite. To |
| 445 |
> |
characterize the interfacial boundary conditions, the slip length, |
| 446 |
> |
$\delta$, is defined by the ratio of the fluid-phase viscosity to the |
| 447 |
> |
friction coefficient of the interface: |
| 448 |
|
\begin{equation} |
| 449 |
|
\delta = \frac{\eta}{\kappa} |
| 450 |
|
\end{equation} |
| 451 |
< |
so that $\delta\rightarrow 0$ in the ``no-slip'' boundary condition, |
| 452 |
< |
and depicts how ``slippery'' an interface is. Figure \ref{slipLength} |
| 453 |
< |
illustrates how this quantity is defined and computed for a |
| 454 |
< |
solid-liquid interface. [MAY INCLUDE SNAPSHOT IN FIGURE] |
| 451 |
> |
In ``no-slip'' or ``stick'' boundary conditions, $\delta\rightarrow |
| 452 |
> |
0$, and $\delta$ is a measure of how ``slippery'' an interface is. |
| 453 |
> |
Figure \ref{slipLength} illustrates how this quantity is defined and |
| 454 |
> |
computed for a solid-liquid interface. |
| 455 |
|
|
| 456 |
|
\begin{figure} |
| 457 |
|
\includegraphics[width=\linewidth]{defDelta} |
| 458 |
|
\caption{The slip length $\delta$ can be obtained from a velocity |
| 459 |
|
profile of a solid-liquid interface simulation, when a momentum flux |
| 460 |
< |
is applied. An example of Au/hexane interfaces is shown, and the |
| 461 |
< |
calculation for the left side is illustrated. The calculation for |
| 462 |
< |
the right side is similar to the left.} |
| 460 |
> |
is applied. The data shown is for a simulated Au/hexane interface. |
| 461 |
> |
The Au crystalline region is moving as a block (lower dots), while |
| 462 |
> |
the measured velocity gradient in the hexane phase is discontinuous |
| 463 |
> |
a the interface.} |
| 464 |
|
\label{slipLength} |
| 465 |
|
\end{figure} |
| 466 |
|
|
| 467 |
< |
In our method, a shear stress can be applied similar to shear |
| 468 |
< |
viscosity computations by applying an unphysical momentum flux |
| 469 |
< |
(e.g. $j_z(p_x)$). A corresponding velocity profile can be obtained as |
| 470 |
< |
shown in Figure \ref{slipLength}, in which the velocity gradients |
| 471 |
< |
within liquid phase and velocity difference at the liquid-solid |
| 472 |
< |
interface can be measured respectively. Further calculations and |
| 473 |
< |
characterizations of the interface can be carried out using these |
| 474 |
< |
data. |
| 467 |
> |
Since the method can be applied for interfaces as well as for bulk |
| 468 |
> |
materials, the shear stress is applied in an identical manner to the |
| 469 |
> |
shear viscosity computations, e.g. by applying an unphysical momentum |
| 470 |
> |
flux, $j_z(\vec{p})$. With the correct choice of $\vec{p}$ in the |
| 471 |
> |
$x-y$ plane, one can compute friction coefficients and slip lengths |
| 472 |
> |
for a number of different dragging vectors on a given slab. The |
| 473 |
> |
corresponding velocity profiles can be obtained as shown in Figure |
| 474 |
> |
\ref{slipLength}, in which the velocity gradients within the liquid |
| 475 |
> |
phase and the velocity difference at the liquid-solid interface can be |
| 476 |
> |
easily measured from saved simulation data. |
| 477 |
|
|
| 478 |
|
\section{Results and Discussions} |
| 479 |
|
\subsection{Lennard-Jones fluid} |
| 480 |
< |
Our orthorhombic simulation cell of Lennard-Jones fluid has identical |
| 481 |
< |
parameters to our previous work\cite{kuang:164101} to facilitate |
| 482 |
< |
comparison. Thermal conductivitis and shear viscosities were computed |
| 483 |
< |
with the algorithm applied to the simulations. The results of thermal |
| 484 |
< |
conductivity are compared with our previous NIVS algorithm. However, |
| 485 |
< |
since the NIVS algorithm could produce temperature anisotropy for |
| 486 |
< |
shear viscocity computations, these results are instead compared to |
| 487 |
< |
the momentum swapping approaches. Table \ref{LJ} lists these |
| 488 |
< |
calculations with various fluxes in reduced units. |
| 480 |
> |
Our orthorhombic simulation cell for the Lennard-Jones fluid has |
| 481 |
> |
identical parameters to our previous work\cite{kuang:164101} to |
| 482 |
> |
facilitate comparison. Thermal conductivities and shear viscosities |
| 483 |
> |
were computed with the new algorithm applied to the simulations. The |
| 484 |
> |
results of thermal conductivity are compared with our previous NIVS |
| 485 |
> |
algorithm. However, since the NIVS algorithm produced temperature |
| 486 |
> |
anisotropy for shear viscocity computations, these results are instead |
| 487 |
> |
compared to the previous momentum swapping approaches. Table \ref{LJ} |
| 488 |
> |
lists these values with various fluxes in reduced units. |
| 489 |
|
|
| 490 |
|
\begin{table*} |
| 491 |
|
\begin{minipage}{\linewidth} |
| 504 |
|
\multicolumn{2}{c}{$\lambda^*$} & |
| 505 |
|
\multicolumn{2}{c}{$\eta^*$} \\ |
| 506 |
|
\hline |
| 507 |
< |
Swap Interval $t^*$ & Equivalent $J_z^*$ or $j_z^*(p_x)$ & |
| 507 |
> |
Swap Interval & Equivalent $J_z^*$ or $j_z^*(p_x)$ & |
| 508 |
|
NIVS\cite{kuang:164101} & This work & Swapping & This work \\ |
| 509 |
|
\hline |
| 510 |
|
0.116 & 0.16 & 7.30(0.10) & 6.25(0.23) & 3.57(0.06) & 3.53(0.16)\\ |
| 540 |
|
or NIVS (total slab momentum conponemts $P^\alpha$ scaled to $\alpha |
| 541 |
|
P^\alpha$) would not achieve this effect unless thermal flux vanishes |
| 542 |
|
(i.e. $\vec{p}_i=\vec{p}_j$ or $\alpha=1$ which do not contribute to |
| 543 |
< |
applying a thermal flux). In this sense, this method aids to achieve |
| 543 |
> |
applying a thermal flux). In this sense, this method aids in achieving |
| 544 |
|
minimal perturbation to a simulation while imposing a thermal flux. |
| 545 |
|
|
| 546 |
|
\subsubsection{Shear viscosity} |
