| 28 |
|
|
| 29 |
|
\begin{document} |
| 30 |
|
|
| 31 |
< |
\title{ENTER TITLE HERE} |
| 31 |
> |
\title{A minimal perturbation approach to RNEMD able to simultaneously |
| 32 |
> |
impose thermal and momentum gradients} |
| 33 |
|
|
| 34 |
|
\author{Shenyu Kuang and J. Daniel |
| 35 |
|
Gezelter\footnote{Corresponding author. \ Electronic mail: gezelter@nd.edu} \\ |
| 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 and maintains |
| 51 |
> |
Non-Equilibrium Molecular Dynamics (RNEMD) methods while maintaining |
| 52 |
|
thermal velocity distributions. It also avoid thermal anisotropy |
| 53 |
< |
occured in NIVS simulations by using isotropic velocity scaling on |
| 54 |
< |
the molecules in specific regions of a system. To test the method, |
| 55 |
< |
we have computed the thermal conductivity and shear viscosity of |
| 56 |
< |
model liquid systems as well as the interfacial frictions of a |
| 57 |
< |
series of metal/liquid interfaces. |
| 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 |
| 56 |
> |
viscosity of model liquid systems as well as the interfacial |
| 57 |
> |
frictions of a series of metal/liquid interfaces. Its ability to |
| 58 |
> |
combine the thermal and momentum gradients allows us to obtain shear |
| 59 |
> |
viscosity data for a range of temperatures in only one trajectory. |
| 60 |
|
|
| 61 |
|
\end{abstract} |
| 62 |
|
|
| 69 |
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% |
| 70 |
|
|
| 71 |
|
\section{Introduction} |
| 69 |
– |
[REFINE LATER, ADD MORE REF.S] |
| 72 |
|
Imposed-flux methods in Molecular Dynamics (MD) |
| 73 |
< |
simulations\cite{MullerPlathe:1997xw} can establish steady state |
| 74 |
< |
systems with a set applied flux vs a corresponding gradient that can |
| 75 |
< |
be measured. These methods does not need many trajectories to provide |
| 76 |
< |
information of transport properties of a given system. Thus, they are |
| 77 |
< |
utilized in computing thermal and mechanical transfer of homogeneous |
| 78 |
< |
or bulk systems as well as heterogeneous systems such as liquid-solid |
| 79 |
< |
interfaces.\cite{kuang:AuThl} |
| 73 |
> |
simulations\cite{MullerPlathe:1997xw,ISI:000080382700030,kuang:164101} |
| 74 |
> |
can establish steady state systems with an applied flux set vs a |
| 75 |
> |
corresponding gradient that can be measured. These methods does not |
| 76 |
> |
need many trajectories to provide information of transport properties |
| 77 |
> |
of a given system. Thus, they are utilized in computing thermal and |
| 78 |
> |
mechanical transfer of homogeneous bulk systems as well as |
| 79 |
> |
heterogeneous systems such as solid-liquid |
| 80 |
> |
interfaces.\cite{garde:nl2005,garde:PhysRevLett2009,kuang:AuThl} |
| 81 |
|
|
| 82 |
|
The Reverse Non-Equilibrium MD (RNEMD) methods adopt constraints that |
| 83 |
|
satisfy linear momentum and total energy conservation of a system when |
| 84 |
|
imposing fluxes in a simulation. Thus they are compatible with various |
| 85 |
|
ensembles, including the micro-canonical (NVE) ensemble, without the |
| 86 |
< |
need of an external thermostat. The original approaches by |
| 86 |
> |
need of an external thermostat. The original approaches proposed by |
| 87 |
|
M\"{u}ller-Plathe {\it et |
| 88 |
|
al.}\cite{MullerPlathe:1997xw,ISI:000080382700030} utilize simple |
| 89 |
< |
momentum swapping for generating energy/momentum fluxes, which is also |
| 90 |
< |
compatible with particles of different identities. Although simple to |
| 91 |
< |
implement in a simulation, this approach can create nonthermal |
| 92 |
< |
velocity distributions, as discovered by Tenney and |
| 93 |
< |
Maginn\cite{Maginn:2010}. Furthermore, this approach to kinetic energy |
| 94 |
< |
transfer between particles of different identities is less efficient |
| 95 |
< |
when the mass difference between the particles becomes significant, |
| 96 |
< |
which also limits its application on heterogeneous interfacial |
| 97 |
< |
systems. |
| 89 |
> |
momentum swapping for generating energy/momentum fluxes, which can |
| 90 |
> |
also be compatible with particles of different identities. Although |
| 91 |
> |
simple to implement in a simulation, this approach can create |
| 92 |
> |
nonthermal velocity distributions, as discovered by Tenney and |
| 93 |
> |
Maginn\cite{Maginn:2010}. Furthermore, this approach is less efficient |
| 94 |
> |
for kinetic energy transfer between particles of different identities, |
| 95 |
> |
especially when the mass difference between the particles becomes |
| 96 |
> |
significant. This also limits its applications on heterogeneous |
| 97 |
> |
interfacial systems. |
| 98 |
|
|
| 99 |
|
Recently, we developed a different approach, using Non-Isotropic |
| 100 |
|
Velocity Scaling (NIVS) \cite{kuang:164101} algorithm to impose |
| 110 |
|
interfacial thermal conductance at metal-solvent |
| 111 |
|
interfaces.\cite{kuang:AuThl} |
| 112 |
|
|
| 113 |
< |
However, the NIVS approach limits its application in imposing momentum |
| 114 |
< |
fluxes. Temperature anisotropy can happen under high momentum fluxes, |
| 115 |
< |
due to the nature of the algorithm. Thus, combining thermal and |
| 116 |
< |
momentum flux is also difficult to implement with this |
| 117 |
< |
approach. However, such combination may provide a means to simulate |
| 118 |
< |
thermal/momentum gradient coupled processes such as freeze |
| 119 |
< |
desalination. Therefore, developing novel approaches to extend the |
| 120 |
< |
application of imposed-flux method is desired. |
| 113 |
> |
However, the NIVS approach has limited applications in imposing |
| 114 |
> |
momentum fluxes. Temperature anisotropy could happen under high |
| 115 |
> |
momentum fluxes due to the implementation of this algorithm. Thus, |
| 116 |
> |
combining thermal and momentum flux is also difficult to obtain with |
| 117 |
> |
this approach. However, such combination may provide a means to |
| 118 |
> |
simulate thermal/momentum gradient coupled processes such as Soret |
| 119 |
> |
effect in liquid flows. Therefore, developing improved approaches to |
| 120 |
> |
extend the applications of the imposed-flux method is desirable. |
| 121 |
|
|
| 122 |
< |
In this paper, we improve the NIVS method and propose a novel approach |
| 123 |
< |
to impose fluxes. This approach separate the means of applying |
| 122 |
> |
In this paper, we improve the RNEMD methods by proposing a novel |
| 123 |
> |
approach to impose fluxes. This approach separate the means of applying |
| 124 |
|
momentum and thermal flux with operations in one time step and thus is |
| 125 |
|
able to simutaneously impose thermal and momentum flux. Furthermore, |
| 126 |
|
the approach retains desirable features of previous RNEMD approaches |
| 127 |
|
and is simpler to implement compared to the NIVS method. In what |
| 128 |
< |
follows, we first present the method to implement the method in a |
| 128 |
> |
follows, we first present the method and its implementation in a |
| 129 |
|
simulation. Then we compare the method on bulk fluids to previous |
| 130 |
|
methods. Also, interfacial frictions are computed for a series of |
| 131 |
|
interfaces. |
| 132 |
|
|
| 133 |
|
\section{Methodology} |
| 134 |
< |
Similar to the NIVS methodology,\cite{kuang:164101} we consider a |
| 134 |
> |
Similar to the NIVS method,\cite{kuang:164101} we consider a |
| 135 |
|
periodic system divided into a series of slabs along a certain axis |
| 136 |
|
(e.g. $z$). The unphysical thermal and/or momentum flux is designated |
| 137 |
< |
from the center slab to one of the end slabs, and thus the center slab |
| 138 |
< |
would have a lower temperature than the end slab (unless the thermal |
| 139 |
< |
flux is negative). Therefore, the center slab is denoted as ``$c$'' |
| 140 |
< |
while the end slab as ``$h$''. |
| 137 |
> |
from the center slab to one of the end slabs, and thus the thermal |
| 138 |
> |
flux results in a lower temperature of the center slab than the end |
| 139 |
> |
slab, and the momentum flux results in negative center slab momentum |
| 140 |
> |
with positive end slab momentum (unless these fluxes are set |
| 141 |
> |
negative). Therefore, the center slab is denoted as ``$c$'', while the |
| 142 |
> |
end slab as ``$h$''. |
| 143 |
|
|
| 144 |
< |
To impose these fluxes, we periodically apply separate operations to |
| 145 |
< |
velocities of particles {$i$} within the center slab and of particles |
| 146 |
< |
{$j$} within the end slab: |
| 144 |
> |
To impose these fluxes, we periodically apply different set of |
| 145 |
> |
operations on velocities of particles {$i$} within the center slab and |
| 146 |
> |
those of particles {$j$} within the end slab: |
| 147 |
|
\begin{eqnarray} |
| 148 |
|
\vec{v}_i & \leftarrow & c\cdot\left(\vec{v}_i - \langle\vec{v}_c |
| 149 |
|
\rangle\right) + \left(\langle\vec{v}_c\rangle + \vec{a}_c\right) \\ |
| 152 |
|
\end{eqnarray} |
| 153 |
|
where $\langle\vec{v}_c\rangle$ and $\langle\vec{v}_h\rangle$ denotes |
| 154 |
|
the instantaneous bulk velocity of slabs $c$ and $h$ respectively |
| 155 |
< |
before an operation occurs. When a momentum flux $\vec{j}_z(\vec{p})$ |
| 155 |
> |
before an operation is applied. When a momentum flux $\vec{j}_z(\vec{p})$ |
| 156 |
|
presents, these bulk velocities would have a corresponding change |
| 157 |
|
($\vec{a}_c$ and $\vec{a}_h$ respectively) according to Newton's |
| 158 |
|
second law: |
| 160 |
|
M_c \vec{a}_c & = & -\vec{j}_z(\vec{p}) \Delta t \\ |
| 161 |
|
M_h \vec{a}_h & = & \vec{j}_z(\vec{p}) \Delta t |
| 162 |
|
\end{eqnarray} |
| 163 |
< |
where |
| 163 |
> |
where $M$ denotes total mass of particles within a slab: |
| 164 |
|
\begin{eqnarray} |
| 165 |
|
M_c & = & \sum_{i = 1}^{N_c} m_i \\ |
| 166 |
|
M_h & = & \sum_{j = 1}^{N_h} m_j |
| 167 |
|
\end{eqnarray} |
| 168 |
< |
and $\Delta t$ is the interval between two operations. |
| 168 |
> |
and $\Delta t$ is the interval between two separate operations. |
| 169 |
|
|
| 170 |
< |
The above operations conserve the linear momentum of a periodic |
| 171 |
< |
system. To satisfy total energy conservation as well as to impose a |
| 172 |
< |
thermal flux $J_z$, one would have |
| 173 |
< |
[SUPPORT INFO MIGHT BE NECESSARY TO PUT EXTRA MATH IN] |
| 170 |
> |
The above operations already conserve the linear momentum of a |
| 171 |
> |
periodic system. To further satisfy total energy conservation as well |
| 172 |
> |
as to impose the thermal flux $J_z$, the following equations are |
| 173 |
> |
included as well: |
| 174 |
> |
[MAY PUT EXTRA MATH IN SUPPORT INFO OR APPENDIX] |
| 175 |
|
\begin{eqnarray} |
| 176 |
|
K_c - J_z\Delta t & = & c^2 (K_c - \frac{1}{2}M_c \langle\vec{v}_c |
| 177 |
|
\rangle^2) + \frac{1}{2}M_c (\langle \vec{v}_c \rangle + \vec{a}_c)^2 \\ |
| 179 |
|
\rangle^2) + \frac{1}{2}M_h (\langle \vec{v}_h \rangle + \vec{a}_h)^2 |
| 180 |
|
\end{eqnarray} |
| 181 |
|
where $K_c$ and $K_h$ denotes translational kinetic energy of slabs |
| 182 |
< |
$c$ and $h$ respectively before an operation occurs. These |
| 182 |
> |
$c$ and $h$ respectively before an operation is applied. These |
| 183 |
|
translational kinetic energy conservation equations are sufficient to |
| 184 |
< |
ensure total energy conservation, as the operations applied do not |
| 185 |
< |
change the potential energy of a system, given that the potential |
| 184 |
> |
ensure total energy conservation, as the operations applied in our |
| 185 |
> |
method do not change the kinetic energy related to other degrees of |
| 186 |
> |
freedom or the potential energy of a system, given that its potential |
| 187 |
|
energy does not depend on particle velocity. |
| 188 |
|
|
| 189 |
|
The above sets of equations are sufficient to determine the velocity |
| 190 |
|
scaling coefficients ($c$ and $h$) as well as $\vec{a}_c$ and |
| 191 |
< |
$\vec{a}_h$. Note that two roots of $c$ and $h$ exist |
| 192 |
< |
respectively. However, to avoid dramatic perturbations to a system, |
| 193 |
< |
the positive roots (which are closer to 1) are chosen. Figure |
| 194 |
< |
\ref{method} illustrates the implementation of this algorithm in an |
| 195 |
< |
individual step. |
| 191 |
> |
$\vec{a}_h$. Note that there are two roots respectively for $c$ and |
| 192 |
> |
$h$. However, the positive roots (which are closer to 1) are chosen so |
| 193 |
> |
that the perturbations to a system can be reduced to a minimum. Figure |
| 194 |
> |
\ref{method} illustrates the implementation sketch of this algorithm |
| 195 |
> |
in an individual step. |
| 196 |
|
|
| 197 |
|
\begin{figure} |
| 198 |
|
\includegraphics[width=\linewidth]{method} |
| 199 |
|
\caption{Illustration of the implementation of the algorithm in a |
| 200 |
|
single step. Starting from an ideal velocity distribution, the |
| 201 |
< |
transformation is used to apply both thermal and momentum flux from |
| 202 |
< |
the ``c'' slab to the ``h'' slab. As the figure shows, the thermal |
| 203 |
< |
distributions preserve after this operation.} |
| 201 |
> |
transformation is used to apply the effect of both a thermal and a |
| 202 |
> |
momentum flux from the ``c'' slab to the ``h'' slab. As the figure |
| 203 |
> |
shows, thermal distributions can preserve after this operation.} |
| 204 |
|
\label{method} |
| 205 |
|
\end{figure} |
| 206 |
|
|
| 208 |
|
thermal and/or momentum flux can be applied and the corresponding |
| 209 |
|
temperature and/or momentum gradients can be established. |
| 210 |
|
|
| 211 |
< |
This approach is more computationaly efficient compared to the |
| 212 |
< |
previous NIVS method, in that only quadratic equations are involved, |
| 213 |
< |
while the NIVS method needs to solve a quartic equations. Furthermore, |
| 214 |
< |
the method implements isotropic scaling of velocities in respective |
| 215 |
< |
slabs, unlike the NIVS, where an extra criteria function is necessary |
| 216 |
< |
to choose a set of coefficients that performs the most isotropic |
| 217 |
< |
scaling. More importantly, separating the momentum flux imposing from |
| 218 |
< |
velocity scaling avoids the underlying cause that NIVS produced |
| 219 |
< |
thermal anisotropy when applying a momentum flux. |
| 211 |
> |
Compared to the previous NIVS method, this approach is computationally |
| 212 |
> |
more efficient in that only quadratic equations are involved to |
| 213 |
> |
determine a set of scaling coefficients, while the NIVS method needs |
| 214 |
> |
to solve quartic equations. Furthermore, this method implements |
| 215 |
> |
isotropic scaling of velocities in respective slabs, unlike the NIVS, |
| 216 |
> |
where an extra criteria function is necessary to choose a set of |
| 217 |
> |
coefficients that performs a scaling as isotropic as possible. More |
| 218 |
> |
importantly, separating the means of momentum flux imposing from |
| 219 |
> |
velocity scaling avoids the underlying cause to thermal anisotropy in |
| 220 |
> |
NIVS when applying a momentum flux. And later sections will |
| 221 |
> |
demonstrate that this can improve the performance in shear viscosity |
| 222 |
> |
simulations. |
| 223 |
|
|
| 224 |
< |
The advantages of the approach over the original momentum swapping |
| 225 |
< |
approach lies in its nature to preserve a Gaussian |
| 226 |
< |
distribution. Because the momentum swapping tends to render a |
| 227 |
< |
nonthermal distribution, when the imposed flux is relatively large, |
| 228 |
< |
diffusion of the neighboring slabs could no longer remedy this effect, |
| 229 |
< |
and nonthermal distributions would be observed. Results in later |
| 230 |
< |
section will illustrate this effect. |
| 224 |
> |
This approach is advantageous over the original momentum swapping in |
| 225 |
> |
many aspects. In one swapping, the velocity vectors involved are |
| 226 |
> |
usually very different (or the generated flux is trivial to obtain |
| 227 |
> |
gradients), thus the swapping tends to incur perturbations to the |
| 228 |
> |
neighbors of the particles involved. Comparatively, our approach |
| 229 |
> |
disperse the flux to every selected particle in a slab so that |
| 230 |
> |
perturbations in the flux generating region could be |
| 231 |
> |
minimized. Additionally, because the momentum swapping steps tend to |
| 232 |
> |
result in a nonthermal distribution, when an imposed flux is |
| 233 |
> |
relatively large and diffusions from the neighboring slabs could no |
| 234 |
> |
longer remedy this effect, problematic distributions would be |
| 235 |
> |
observed. In comparison, the operations of our approach has the nature |
| 236 |
> |
of preserving the equilibrium velocity distributions (commonly |
| 237 |
> |
Maxwell-Boltzmann), and results in later section will illustrate that |
| 238 |
> |
this is helpful to retain thermal distributions in a simulation. |
| 239 |
|
|
| 240 |
|
\section{Computational Details} |
| 241 |
|
The algorithm has been implemented in our MD simulation code, |
| 242 |
|
OpenMD\cite{Meineke:2005gd,openmd}. We compare the method with |
| 243 |
< |
previous RNEMD methods or equilibrium MD methods in homogeneous fluids |
| 243 |
> |
previous RNEMD methods or equilibrium MD (EMD) methods in homogeneous fluids |
| 244 |
|
(Lennard-Jones and SPC/E water). And taking advantage of the method, |
| 245 |
|
we simulate the interfacial friction of different heterogeneous |
| 246 |
|
interfaces (gold-organic solvent and gold-SPC/E water and ice-liquid |
| 247 |
|
water). |
| 248 |
|
|
| 249 |
|
\subsection{Simulation Protocols} |
| 250 |
< |
The systems to be investigated are set up in a orthorhombic simulation |
| 251 |
< |
cell with periodic boundary conditions in all three dimensions. The |
| 252 |
< |
$z$ axis of these cells were longer and was set as the gradient axis |
| 253 |
< |
of temperature and/or momentum. Thus the cells were divided into $N$ |
| 250 |
> |
The systems to be investigated are set up in orthorhombic simulation |
| 251 |
> |
cells with periodic boundary conditions in all three dimensions. The |
| 252 |
> |
$z$ axis of these cells were longer and set as the temperature and/or |
| 253 |
> |
momentum gradient axis. And the cells were evenly divided into $N$ |
| 254 |
|
slabs along this axis, with various $N$ depending on individual |
| 255 |
< |
system. The $x$ and $y$ axis were usually of the same length in |
| 256 |
< |
homogeneous systems or close to each other where interfaces |
| 257 |
< |
presents. In all cases, before introducing a nonequilibrium method to |
| 258 |
< |
establish steady thermal and/or momentum gradients for further |
| 259 |
< |
measurements and calculations, canonical ensemble with a Nos\'e-Hoover |
| 260 |
< |
thermostat\cite{hoover85} and microcanonical ensemble equilibrations |
| 261 |
< |
were used to prepare systems ready for data |
| 262 |
< |
collections. Isobaric-isothermal equilibrations are performed before |
| 263 |
< |
this for SPC/E water systems to reach normal pressure (1 bar), while |
| 264 |
< |
similar equilibrations are used for interfacial systems to relax the |
| 265 |
< |
surface tensions. |
| 255 |
> |
system. The $x$ and $y$ axis were of the same length in homogeneous |
| 256 |
> |
systems or had length scale close to each other where heterogeneous |
| 257 |
> |
interfaces presents. In all cases, before introducing a nonequilibrium |
| 258 |
> |
method to establish steady thermal and/or momentum gradients for |
| 259 |
> |
further measurements and calculations, canonical ensemble with a |
| 260 |
> |
Nos\'e-Hoover thermostat\cite{hoover85} and microcanonical ensemble |
| 261 |
> |
equilibrations were used before data collections. For SPC/E water |
| 262 |
> |
simulations, isobaric-isothermal equilibrations\cite{melchionna93} are |
| 263 |
> |
performed before the above to reach normal pressure (1 bar); for |
| 264 |
> |
interfacial systems, similar equilibrations are used to relax the |
| 265 |
> |
surface tensions of the $xy$ plane. |
| 266 |
|
|
| 267 |
< |
While homogeneous fluid systems can be set up with random |
| 268 |
< |
configurations, our interfacial systems needs extra steps to ensure |
| 269 |
< |
the interfaces be established properly for computations. The |
| 267 |
> |
While homogeneous fluid systems can be set up with rather random |
| 268 |
> |
configurations, our interfacial systems needs a series of steps to |
| 269 |
> |
ensure the interfaces be established properly for computations. The |
| 270 |
|
preparation and equilibration of butanethiol covered gold (111) |
| 271 |
|
surface and further solvation and equilibration process is described |
| 272 |
< |
as in reference \cite{kuang:AuThl}. |
| 272 |
> |
in details as in reference \cite{kuang:AuThl}. |
| 273 |
|
|
| 274 |
|
As for the ice/liquid water interfaces, the basal surface of ice |
| 275 |
|
lattice was first constructed. Hirsch {\it et |
| 276 |
|
al.}\cite{doi:10.1021/jp048434u} explored the energetics of ice |
| 277 |
< |
lattices with different proton orders. We refer to their results and |
| 278 |
< |
choose the configuration of the lowest energy after geometry |
| 279 |
< |
optimization as the unit cells of our ice lattices. Although |
| 280 |
< |
experimental solid/liquid coexistant temperature near normal pressure |
| 281 |
< |
is 273K, Bryk and Haymet's simulations of ice/liquid water interfaces |
| 282 |
< |
with different models suggest that for SPC/E, the most stable |
| 283 |
< |
interface is observed at 225$\pm$5K. Therefore, all our ice/liquid |
| 284 |
< |
water simulations were carried out under 225K. To have extra |
| 285 |
< |
protection of the ice lattice during initial equilibration (when the |
| 286 |
< |
randomly generated liquid phase configuration could release large |
| 287 |
< |
amount of energy in relaxation), a constraint method (REF?) was |
| 288 |
< |
adopted until the high energy configuration was relaxed. |
| 289 |
< |
[MAY ADD A FIGURE HERE FOR BASAL PLANE, MAY INCLUDE PRISM IF POSSIBLE] |
| 277 |
> |
lattices with all possible proton order configurations. We refer to |
| 278 |
> |
their results and choose the configuration of the lowest energy after |
| 279 |
> |
geometry optimization as the unit cell for our ice lattices. Although |
| 280 |
> |
experimental solid/liquid coexistant temperature under normal pressure |
| 281 |
> |
should be close to 273K, Bryk and Haymet's simulations of ice/liquid |
| 282 |
> |
water interfaces with different models suggest that for SPC/E, the |
| 283 |
> |
most stable interface is observed at 225$\pm$5K.\cite{bryk:10258} |
| 284 |
> |
Therefore, our ice/liquid water simulations were carried out at |
| 285 |
> |
225K. To have extra protection of the ice lattice during initial |
| 286 |
> |
equilibration (when the randomly generated liquid phase configuration |
| 287 |
> |
could release large amount of energy in relaxation), restraints were |
| 288 |
> |
applied to the ice lattice to avoid inadvertent melting by the heat |
| 289 |
> |
dissipated from the high enery configurations. |
| 290 |
> |
[MAY ADD A SNAPSHOT FOR BASAL PLANE] |
| 291 |
|
|
| 292 |
|
\subsection{Force Field Parameters} |
| 293 |
|
For comparison of our new method with previous work, we retain our |
| 294 |
< |
force field parameters consistent with the results we will compare |
| 295 |
< |
with. The Lennard-Jones fluid used here for argon , and reduced unit |
| 296 |
< |
results are reported for direct comparison purpose. |
| 294 |
> |
force field parameters consistent with previous simulations. Argon is |
| 295 |
> |
the Lennard-Jones fluid used here, and its results are reported in |
| 296 |
> |
reduced unit for direct comparison purpose. |
| 297 |
|
|
| 298 |
|
As for our water simulations, SPC/E model is used throughout this work |
| 299 |
|
for consistency. Previous work for transport properties of SPC/E water |
| 308 |
|
Spohr potential was adopted\cite{ISI:000167766600035} to depict |
| 309 |
|
Au-H$_2$O interactions. |
| 310 |
|
|
| 311 |
< |
The small organic molecules included in our simulations are the Au |
| 312 |
< |
surface capping agent butanethiol and liquid hexane and toluene. The |
| 313 |
< |
United-Atom |
| 311 |
> |
For our gold/organic liquid interfaces, the small organic molecules |
| 312 |
> |
included in our simulations are the Au surface capping agent |
| 313 |
> |
butanethiol and liquid hexane and toluene. The United-Atom |
| 314 |
|
models\cite{TraPPE-UA.thiols,TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} |
| 315 |
|
for these components were used in this work for better computational |
| 316 |
|
efficiency, while maintaining good accuracy. We refer readers to our |
| 338 |
|
\end{equation} |
| 339 |
|
where $L_x$ and $L_y$ denotes the dimensions of the plane in a |
| 340 |
|
simulation cell perpendicular to the thermal gradient, and a factor of |
| 341 |
< |
two in the denominator is present for the heat transport occurs in |
| 342 |
< |
both $+z$ and $-z$ directions. The temperature gradient |
| 341 |
> |
two in the denominator is necessary for the heat transport occurs in |
| 342 |
> |
both $+z$ and $-z$ directions. The average temperature gradient |
| 343 |
|
${\langle\partial T/\partial z\rangle}$ can be obtained by a linear |
| 344 |
|
regression of the temperature profile, which is recorded during a |
| 345 |
|
simulation for each slab in a cell. For Lennard-Jones simulations, |
| 348 |
|
|
| 349 |
|
\subsection{Shear viscosities} |
| 350 |
|
Alternatively, the method can carry out shear viscosity calculations |
| 351 |
< |
by switching off $J_z$. One can specify the vector |
| 352 |
< |
$\vec{j}_z(\vec{p})$ by choosing the three components |
| 351 |
> |
by specify a momentum flux. In our algorithm, one can specify the |
| 352 |
> |
three components of the flux vector $\vec{j}_z(\vec{p})$ |
| 353 |
|
respectively. For shear viscosity simulations, $j_z(p_z)$ is usually |
| 354 |
< |
set to zero. Although for isotropic systems, the direction of |
| 355 |
< |
$\vec{j}_z(\vec{p})$ on the $xy$ plane should not matter, the ability |
| 356 |
< |
of arbitarily specifying the vector direction in our method provides |
| 357 |
< |
convenience in anisotropic simulations. |
| 354 |
> |
set to zero. For isotropic systems, the direction of |
| 355 |
> |
$\vec{j}_z(\vec{p})$ 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. |
| 358 |
|
|
| 359 |
< |
Similar to thermal conductivity computations, linear response of the |
| 360 |
< |
momentum gradient with respect to the shear stress is assumed, and the |
| 361 |
< |
shear viscosity ($\eta$) can be obtained with the imposed momentum |
| 362 |
< |
flux (e.g. in $x$ direction) and the measured gradient: |
| 359 |
> |
Similar to thermal conductivity computations, for a homogeneous |
| 360 |
> |
system, linear response of the momentum gradient with respect to the |
| 361 |
> |
shear stress is assumed, and the shear viscosity ($\eta$) can be |
| 362 |
> |
obtained with the imposed momentum flux (e.g. in $x$ direction) and |
| 363 |
> |
the measured gradient: |
| 364 |
|
\begin{equation} |
| 365 |
|
j_z(p_x) = -\eta \frac{\partial v_x}{\partial z} |
| 366 |
|
\end{equation} |
| 369 |
|
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
| 370 |
|
\end{equation} |
| 371 |
|
with $P_x$ being the total non-physical momentum transferred within |
| 372 |
< |
the data collection time. Also, the velocity gradient |
| 372 |
> |
the data collection time. Also, the averaged velocity gradient |
| 373 |
|
${\langle\partial v_x/\partial z\rangle}$ can be obtained using linear |
| 374 |
< |
regression of the $x$ component of the mean velocity, $\langle |
| 375 |
< |
v_x\rangle$, in each of the bins. For Lennard-Jones simulations, shear |
| 376 |
< |
viscosities are reported in reduced units |
| 374 |
> |
regression of the $x$ component of the mean velocity ($\langle |
| 375 |
> |
v_x\rangle$) in each of the bins. For Lennard-Jones simulations, shear |
| 376 |
> |
viscosities are also reported in reduced units |
| 377 |
|
(${\eta^*=\eta\sigma^2(\varepsilon m)^{-1/2}}$). |
| 378 |
+ |
|
| 379 |
+ |
Although $J_z$ may be switched off for shear viscosity simulations at |
| 380 |
+ |
a certain temperature, our method's ability to impose both a thermal |
| 381 |
+ |
and a momentum flux in one simulation allows the combination of a |
| 382 |
+ |
temperature and a velocity gradient. In this case, since viscosity is |
| 383 |
+ |
generally a function of temperature, the local viscosity also depends |
| 384 |
+ |
on the local temperature. Therefore, in one such simulation, viscosity |
| 385 |
+ |
at $z$ (corresponding to a certain $T$) can be computed with the |
| 386 |
+ |
applied shear flux and the local velocity gradient (which can be |
| 387 |
+ |
obtained by finite difference approximation). As a whole, the |
| 388 |
+ |
viscosity can be mapped out as the function of temperature in one |
| 389 |
+ |
single trajectory of simulation. Results for shear viscosity |
| 390 |
+ |
computations of SPC/E water will demonstrate its effectiveness in |
| 391 |
+ |
detail. |
| 392 |
|
|
| 393 |
|
\subsection{Interfacial friction and Slip length} |
| 394 |
|
While the shear stress results in a velocity gradient within bulk |
| 395 |
|
fluid phase, its effect at a solid-liquid interface could vary due to |
| 396 |
|
the interaction strength between the two phases. The interfacial |
| 397 |
|
friction coefficient $\kappa$ is defined to relate the shear stress |
| 398 |
< |
(e.g. along $x$-axis) and the relative fluid velocity tangent to the |
| 398 |
> |
(e.g. along $x$-axis) with the relative fluid velocity tangent to the |
| 399 |
|
interface: |
| 400 |
|
\begin{equation} |
| 401 |
|
j_z(p_x)|_{interface} = \kappa\Delta v_x|_{interface} |
| 402 |
|
\end{equation} |
| 403 |
|
Under ``stick'' boundary condition, $\Delta v_x|_{interface} |
| 404 |
|
\rightarrow 0$, which leads to $\kappa\rightarrow\infty$. However, for |
| 405 |
< |
``slip'' boundary condition at the solid-liquid interface, $\kappa$ |
| 405 |
> |
``slip'' boundary conditions at the solid-liquid interfaces, $\kappa$ |
| 406 |
|
becomes finite. To characterize the interfacial boundary conditions, |
| 407 |
|
slip length ($\delta$) is defined using $\kappa$ and the shear |
| 408 |
|
viscocity of liquid phase ($\eta$): |
| 412 |
|
so that $\delta\rightarrow 0$ in the ``no-slip'' boundary condition, |
| 413 |
|
and depicts how ``slippery'' an interface is. Figure \ref{slipLength} |
| 414 |
|
illustrates how this quantity is defined and computed for a |
| 415 |
< |
solid-liquid interface. [MAY INCLUDE A SNAPSHOT IN FIGURE] |
| 415 |
> |
solid-liquid interface. [MAY INCLUDE SNAPSHOT IN FIGURE] |
| 416 |
|
|
| 417 |
|
\begin{figure} |
| 418 |
|
\includegraphics[width=\linewidth]{defDelta} |
| 419 |
|
\caption{The slip length $\delta$ can be obtained from a velocity |
| 420 |
< |
profile of a solid-liquid interface simulation. An example of |
| 421 |
< |
Au/hexane interfaces is shown. Calculation for the left side is |
| 422 |
< |
illustrated. The right side is similar to the left side.} |
| 420 |
> |
profile of a solid-liquid interface simulation, when a momentum flux |
| 421 |
> |
is applied. An example of Au/hexane interfaces is shown, and the |
| 422 |
> |
calculation for the left side is illustrated. The calculation for |
| 423 |
> |
the right side is similar to the left.} |
| 424 |
|
\label{slipLength} |
| 425 |
|
\end{figure} |
| 426 |
|
|
| 463 |
|
\multicolumn{2}{c}{$\eta^*$} \\ |
| 464 |
|
\hline |
| 465 |
|
Swap Interval $t^*$ & Equivalent $J_z^*$ or $j_z^*(p_x)$ & |
| 466 |
< |
NIVS & This work & Swapping & This work \\ |
| 466 |
> |
NIVS\cite{kuang:164101} & This work & Swapping & This work \\ |
| 467 |
|
\hline |
| 468 |
|
0.116 & 0.16 & 7.30(0.10) & 6.25(0.23) & 3.57(0.06) & 3.53(0.16)\\ |
| 469 |
|
0.232 & 0.09 & 6.95(0.09) & 8.02(0.56) & 3.64(0.05) & 3.43(0.17)\\ |
| 480 |
|
\subsubsection{Thermal conductivity} |
| 481 |
|
Our thermal conductivity calculations with this method yields |
| 482 |
|
comparable results to the previous NIVS algorithm. This indicates that |
| 483 |
< |
the thermal gradients rendered using this method are also close to |
| 483 |
> |
the thermal gradients introduced using this method are also close to |
| 484 |
|
previous RNEMD methods. Simulations with moderately higher thermal |
| 485 |
|
fluxes tend to yield more reliable thermal gradients and thus avoid |
| 486 |
|
large errors, while overly high thermal fluxes could introduce side |
| 489 |
|
|
| 490 |
|
Since the scaling operation is isotropic in this method, one does not |
| 491 |
|
need extra care to ensure temperature isotropy between the $x$, $y$ |
| 492 |
< |
and $z$ axes, while thermal anisotropy might happen if the criteria |
| 493 |
< |
function for choosing scaling coefficients does not perform as |
| 494 |
< |
expected. Furthermore, this method avoids inadvertent concomitant |
| 492 |
> |
and $z$ axes, while for NIVS, thermal anisotropy might happen if the |
| 493 |
> |
criteria function for choosing scaling coefficients does not perform |
| 494 |
> |
as expected. Furthermore, this method avoids inadvertent concomitant |
| 495 |
|
momentum flux when only thermal flux is imposed, which could not be |
| 496 |
|
achieved with swapping or NIVS approaches. The thermal energy exchange |
| 497 |
|
in swapping ($\vec{p}_i$ in slab ``c'' with $\vec{p}_j$ in slab ``h'') |
| 498 |
|
or NIVS (total slab momentum conponemts $P^\alpha$ scaled to $\alpha |
| 499 |
< |
P^\alpha$) would not obtain this result unless thermal flux vanishes |
| 500 |
< |
(i.e. $\vec{p}_i=\vec{p}_j$ or $\alpha=1$ which are trivial to apply a |
| 501 |
< |
thermal flux). In this sense, this method contributes to having |
| 502 |
< |
minimal perturbation to a simulation while imposing thermal flux. |
| 499 |
> |
P^\alpha$) would not achieve this effect unless thermal flux vanishes |
| 500 |
> |
(i.e. $\vec{p}_i=\vec{p}_j$ or $\alpha=1$ which do not contribute to |
| 501 |
> |
applying a thermal flux). In this sense, this method aids to achieve |
| 502 |
> |
minimal perturbation to a simulation while imposing a thermal flux. |
| 503 |
|
|
| 504 |
|
\subsubsection{Shear viscosity} |
| 505 |
< |
Table \ref{LJ} also compares our shear viscosity results with momentum |
| 506 |
< |
swapping approach. Our calculations show that our method predicted |
| 507 |
< |
similar values for shear viscosities to the momentum swapping |
| 505 |
> |
Table \ref{LJ} also compares our shear viscosity results with the |
| 506 |
> |
momentum swapping approach. Our calculations show that our method |
| 507 |
> |
predicted similar values of shear viscosities to the momentum swapping |
| 508 |
|
approach, as well as the velocity gradient profiles. Moderately larger |
| 509 |
|
momentum fluxes are helpful to reduce the errors of measured velocity |
| 510 |
|
gradients and thus the final result. However, it is pointed out that |
| 514 |
|
To examine that temperature isotropy holds in simulations using our |
| 515 |
|
method, we measured the three one-dimensional temperatures in each of |
| 516 |
|
the slabs (Figure \ref{tempXyz}). Note that the $x$-dimensional |
| 517 |
< |
temperatures were calculated after subtracting the effects from bulk |
| 518 |
< |
velocities of the slabs. The one-dimensional temperature profiles |
| 517 |
> |
temperatures were calculated after subtracting the contribution from |
| 518 |
> |
bulk velocities of the slabs. The one-dimensional temperature profiles |
| 519 |
|
showed no observable difference between the three dimensions. This |
| 520 |
|
ensures that isotropic scaling automatically preserves temperature |
| 521 |
|
isotropy and that our method is useful in shear viscosity |
| 538 |
|
distributions against the momentum swapping approach even under large |
| 539 |
|
imposed fluxes. Previous swapping methods tend to deplete particles of |
| 540 |
|
positive velocities in the negative velocity slab (``c'') and vice |
| 541 |
< |
versa in slab ``h'', where the distributions leave a notch. This |
| 541 |
> |
versa in slab ``h'', where the distributions leave notchs. This |
| 542 |
|
problematic profiles become significant when the imposed-flux becomes |
| 543 |
|
larger and diffusions from neighboring slabs could not offset the |
| 544 |
< |
depletion. Simutaneously, abnormal peaks appear corresponding to |
| 545 |
< |
excessive velocity swapped from the other slab. This nonthermal |
| 546 |
< |
distributions limit applications of the swapping approach in shear |
| 547 |
< |
stress simulations. Our method avoids the above problematic |
| 544 |
> |
depletions. Simutaneously, abnormal peaks appear corresponding to |
| 545 |
> |
excessive particles having velocity swapped from the other slab. These |
| 546 |
> |
nonthermal distributions limit applications of the swapping approach |
| 547 |
> |
in shear stress simulations. Our method avoids the above problematic |
| 548 |
|
distributions by altering the means of applying momentum |
| 549 |
|
fluxes. Comparatively, velocity distributions recorded from |
| 550 |
|
simulations with our method is so close to the ideal thermal |
| 551 |
< |
prediction that no observable difference is shown in Figure |
| 552 |
< |
\ref{vDist}. Conclusively, our method avoids problems happened in |
| 551 |
> |
prediction that no obvious difference is shown in Figure |
| 552 |
> |
\ref{vDist}. Conclusively, our method avoids problems that occurs in |
| 553 |
|
previous RNEMD methods and provides a useful means for shear viscosity |
| 554 |
|
computations. |
| 555 |
|
|
| 556 |
|
\begin{figure} |
| 557 |
|
\includegraphics[width=\linewidth]{velDist} |
| 558 |
|
\caption{Velocity distributions that develop under the swapping and |
| 559 |
< |
our methods at high flux. These distributions were obtained from |
| 559 |
> |
our methods at a large flux. These distributions were obtained from |
| 560 |
|
Lennard-Jones simulations with $j_z^*(p_x)\sim 0.4$ (equivalent to a |
| 561 |
|
swapping interval of 20 time steps). This is a relatively large flux |
| 562 |
|
to demonstrate the nonthermal distributions that develop under the |
| 563 |
< |
swapping method. Distributions produced by our method are very close |
| 564 |
< |
to the ideal thermal situations.} |
| 563 |
> |
swapping method. In comparison, distributions produced by our method |
| 564 |
> |
are very close to the ideal thermal situations.} |
| 565 |
|
\label{vDist} |
| 566 |
|
\end{figure} |
| 567 |
|
|
| 568 |
|
\subsection{Bulk SPC/E water} |
| 569 |
< |
Since our method was in good performance of thermal conductivity and |
| 570 |
< |
shear viscosity computations for simple Lennard-Jones fluid, we extend |
| 571 |
< |
our applications of these simulations to complex fluid like SPC/E |
| 572 |
< |
water model. A simulation cell with 1000 molecules was set up in the |
| 573 |
< |
same manner as in \cite{kuang:164101}. For thermal conductivity |
| 574 |
< |
simulations, measurements were taken to compare with previous RNEMD |
| 575 |
< |
methods; for shear viscosity computations, simulations were run under |
| 576 |
< |
a series of temperatures (with corresponding pressure relaxation using |
| 577 |
< |
the isobaric-isothermal ensemble[CITE NIVS REF 32]), and results were |
| 578 |
< |
compared to available data from Equilibrium MD methods[CITATIONS]. |
| 569 |
> |
We extend our applications of thermal conductivity and shear viscosity |
| 570 |
> |
computations to a complex fluid model of SPC/E water. A simulation |
| 571 |
> |
cell with 1000 molecules was set up in the similar manner as in |
| 572 |
> |
\cite{kuang:164101}. For thermal conductivity simulations, |
| 573 |
> |
measurements were taken to compare with previous RNEMD methods; for |
| 574 |
> |
shear viscosity computations, simulations were run under a series of |
| 575 |
> |
temperatures (with corresponding pressure relaxation using the |
| 576 |
> |
isobaric-isothermal ensemble\cite{melchionna93}), and results were |
| 577 |
> |
compared to available data from EMD |
| 578 |
> |
methods\cite{10.1063/1.3330544,Medina2011}. Besides, a simulation with |
| 579 |
> |
both thermal and momentum gradient was carried out to map out shear |
| 580 |
> |
viscosity as a function of temperature to see the effectiveness and |
| 581 |
> |
accuracy our method could reach. |
| 582 |
|
|
| 583 |
|
\subsubsection{Thermal conductivity} |
| 584 |
|
Table \ref{spceThermal} summarizes our thermal conductivity |
| 587 |
|
experimental measurements\cite{WagnerKruse}. Note that no appreciable |
| 588 |
|
drift of total system energy or temperature was observed when our |
| 589 |
|
method is applied, which indicates that our algorithm conserves total |
| 590 |
< |
energy even for systems involving electrostatic interactions. |
| 590 |
> |
energy well for systems involving electrostatic interactions. |
| 591 |
|
|
| 592 |
|
Measurements using our method established similar temperature |
| 593 |
|
gradients to the previous NIVS method. Our simulation results are in |
| 650 |
|
\hline\hline |
| 651 |
|
$T$ & $\partial v_x/\partial z$ & \multicolumn{2}{c} |
| 652 |
|
{$\eta (\mathrm{mPa}\cdot\mathrm{s})$} \\ |
| 653 |
< |
(K) & (10$^{10}$s$^{-1}$) & This work & Previous simulations[CITATION]\\ |
| 653 |
> |
(K) & (10$^{10}$s$^{-1}$) & This work & Previous |
| 654 |
> |
simulations\cite{Medina2011} \\ |
| 655 |
|
\hline |
| 656 |
< |
273 & & 1.218(0.004) & \\ |
| 657 |
< |
& & 1.140(0.012) & \\ |
| 658 |
< |
303 & & 0.646(0.008) & \\ |
| 659 |
< |
318 & & 0.536(0.007) & \\ |
| 660 |
< |
& & 0.510(0.007) & \\ |
| 661 |
< |
& & & \\ |
| 662 |
< |
333 & & 0.428(0.002) & \\ |
| 663 |
< |
363 & & 0.279(0.014) & \\ |
| 664 |
< |
& & 0.306(0.001) & \\ |
| 656 |
> |
273 & 1.12 & 1.218(0.004) & 1.282(0.048) \\ |
| 657 |
> |
& 1.79 & 1.140(0.012) & \\ |
| 658 |
> |
303 & 2.09 & 0.646(0.008) & 0.643(0.019) \\ |
| 659 |
> |
318 & 2.50 & 0.536(0.007) & \\ |
| 660 |
> |
& 5.25 & 0.510(0.007) & \\ |
| 661 |
> |
& 2.82 & 0.474(0.003)\footnote{Simulation with $L_z$ twice |
| 662 |
> |
as long.} & \\ |
| 663 |
> |
333 & 3.10 & 0.428(0.002) & 0.421(0.008) \\ |
| 664 |
> |
363 & 2.34 & 0.279(0.014) & 0.291(0.005) \\ |
| 665 |
> |
& 4.26 & 0.306(0.001) & \\ |
| 666 |
|
\hline\hline |
| 667 |
|
\end{tabular} |
| 668 |
|
\label{spceShear} |
| 670 |
|
\end{minipage} |
| 671 |
|
\end{table*} |
| 672 |
|
|
| 673 |
< |
[MAY COMBINE JZPX AND JZKE TO RUN ONE JOB BUT GET ETA=F(T)] |
| 674 |
< |
[PUT RESULTS AND FIGURE HERE IF IT WORKS] |
| 673 |
> |
A more effective way to map out $\eta$ vs $T$ is to combine a momentum |
| 674 |
> |
flux with a thermal flux. Figure \ref{Tvxdvdz} shows the thermal and |
| 675 |
> |
velocity gradient in one such simulation. At different positions with |
| 676 |
> |
different temperatures, the velocity gradient is not a constant but |
| 677 |
> |
can be computed locally. With the data provided in Figure |
| 678 |
> |
\ref{Tvxdvdz}, a series of $\eta$ is calculated as in Figure |
| 679 |
> |
\ref{etaT} and a linear fit was performed to $\partial v_x/\partial z$ |
| 680 |
> |
vs. $z$ so that the resulted $\eta$ can be present as a curve as |
| 681 |
> |
well. For comparison, other results are also mapped in the figure. |
| 682 |
> |
|
| 683 |
> |
\begin{figure} |
| 684 |
> |
\includegraphics[width=\linewidth]{tvxdvdz} |
| 685 |
> |
\caption{With a combination of a thermal and a momentum flux, a |
| 686 |
> |
simulation can have both a temperature (top) and a velocity (middle) |
| 687 |
> |
gradient. Due to the thermal gradient, $\partial v_x/\partial z$ is |
| 688 |
> |
not constant but can be computed using finite difference |
| 689 |
> |
approximations (lower). These data can be used further to calculate |
| 690 |
> |
$\eta$ vs $T$ (Figure \ref{etaT}).} |
| 691 |
> |
\label{Tvxdvdz} |
| 692 |
> |
\end{figure} |
| 693 |
> |
|
| 694 |
> |
From Figure \ref{etaT}, one can see that the generated curve agrees |
| 695 |
> |
well with the above RNEMD simulations at different temperatures, as |
| 696 |
> |
well as results reported using EMD |
| 697 |
> |
methods\cite{10.1063/1.3330544,Medina2011} in much of the temperature |
| 698 |
> |
range simulated. However, this curve has relatively large error in |
| 699 |
> |
lower temperature regions and has some difference in predicting $\eta$ |
| 700 |
> |
near 273K. Provided that this curve only takes one trajectory to |
| 701 |
> |
generate, these results are of satisfactory efficiency and |
| 702 |
> |
accuracy. Since previous work already pointed out that the SPC/E model |
| 703 |
> |
tends to predict lower viscosity compared to experimental |
| 704 |
> |
data,\cite{Medina2011} experimental comparison are not given here. |
| 705 |
> |
|
| 706 |
> |
\begin{figure} |
| 707 |
> |
\includegraphics[width=\linewidth]{etaT} |
| 708 |
> |
\caption{The curve generated by single simulation with thermal and |
| 709 |
> |
momentum gradient predicts satisfatory values in much of the |
| 710 |
> |
temperature range under test.} |
| 711 |
> |
\label{etaT} |
| 712 |
> |
\end{figure} |
| 713 |
> |
|
| 714 |
|
\subsection{Interfacial frictions and slip lengths} |
| 715 |
< |
An attractive aspect of our method is the ability to apply momentum |
| 716 |
< |
and/or thermal flux in nonhomogeneous systems, where molecules of |
| 717 |
< |
different identities (or phases) are segregated in different |
| 718 |
< |
regions. We have previously studied the interfacial thermal transport |
| 719 |
< |
of a series of metal gold-liquid |
| 720 |
< |
surfaces\cite{kuang:164101,kuang:AuThl}, and attemptions have been |
| 721 |
< |
made to investigate the relationship between this phenomenon and the |
| 715 |
> |
Another attractive aspect of our method is the ability to apply |
| 716 |
> |
momentum and/or thermal flux in nonhomogeneous systems, where |
| 717 |
> |
molecules of different identities (or phases) are segregated in |
| 718 |
> |
different regions. We have previously studied the interfacial thermal |
| 719 |
> |
transport of a series of metal gold-liquid |
| 720 |
> |
surfaces\cite{kuang:164101,kuang:AuThl}, and would like to further |
| 721 |
> |
investigate the relationship between this phenomenon and the |
| 722 |
|
interfacial frictions. |
| 723 |
|
|
| 724 |
|
Table \ref{etaKappaDelta} includes these computations and previous |
| 725 |
|
calculations of corresponding interfacial thermal conductance. For |
| 726 |
|
bare Au(111) surfaces, slip boundary conditions were observed for both |
| 727 |
|
organic and aqueous liquid phases, corresponding to previously |
| 728 |
< |
computed low interfacial thermal conductance. Instead, the butanethiol |
| 729 |
< |
covered Au(111) surface appeared to be sticky to the organic liquid |
| 730 |
< |
molecules in our simulations. We have reported conductance enhancement |
| 731 |
< |
effect for this surface capping agent,\cite{kuang:AuThl} and these |
| 732 |
< |
observations have a qualitative agreement with the thermal conductance |
| 733 |
< |
results. This agreement also supports discussions on the relationship |
| 734 |
< |
between surface wetting and slip effect and thermal conductance of the |
| 735 |
< |
interface.[CITE BARRAT, GARDE] |
| 728 |
> |
computed low interfacial thermal conductance. In comparison, the |
| 729 |
> |
butanethiol covered Au(111) surface appeared to be sticky to the |
| 730 |
> |
organic liquid layers in our simulations. We have reported conductance |
| 731 |
> |
enhancement effect for this surface capping agent,\cite{kuang:AuThl} |
| 732 |
> |
and these observations have a qualitative agreement with the thermal |
| 733 |
> |
conductance results. This agreement also supports discussions on the |
| 734 |
> |
relationship between surface wetting and slip effect and thermal |
| 735 |
> |
conductance of the |
| 736 |
> |
interface.\cite{PhysRevLett.82.4671,doi:10.1080/0026897031000068578,garde:PhysRevLett2009} |
| 737 |
|
|
| 738 |
|
\begin{table*} |
| 739 |
|
\begin{minipage}{\linewidth} |
| 748 |
|
Solid & Liquid & $T$ & $j_z(p_x)$ & $\eta_{liquid}$ & $\kappa$ |
| 749 |
|
& $\delta$ & $G$\footnote{References \cite{kuang:AuThl} and |
| 750 |
|
\cite{kuang:164101}.} \\ |
| 751 |
< |
surface & molecules & K & MPa & mPa$\cdot$s & Pa$\cdot$s/m & nm |
| 752 |
< |
& MW/m$^2$/K \\ |
| 751 |
> |
surface & molecules & K & MPa & mPa$\cdot$s & |
| 752 |
> |
10$^4$Pa$\cdot$s/m & nm & MW/m$^2$/K \\ |
| 753 |
|
\hline |
| 754 |
< |
Au(111) & hexane & 200 & 1.08 & 0.20() & 5.3$\times$10$^4$() & |
| 755 |
< |
3.7 & 46.5 \\ |
| 756 |
< |
& & & 2.15 & 0.14() & 5.3$\times$10$^4$() & |
| 757 |
< |
2.7 & \\ |
| 758 |
< |
Au-SC$_4$H$_9$ & hexane & 200 & 2.16 & 0.29() & $\infty$ & 0 & |
| 759 |
< |
131 \\ |
| 760 |
< |
& & & 5.39 & 0.32() & $\infty$ & 0 & |
| 761 |
< |
\\ |
| 754 |
> |
Au(111) & hexane & 200 & 1.08 & 0.197(0.009) & 5.30(0.36) & |
| 755 |
> |
3.72 & 46.5 \\ |
| 756 |
> |
& & & 2.15 & 0.141(0.002) & 5.31(0.26) & |
| 757 |
> |
2.76 & \\ |
| 758 |
> |
Au-SC$_4$H$_9$ & hexane & 200 & 2.16 & 0.286(0.019) & $\infty$ |
| 759 |
> |
& 0 & 131 \\ |
| 760 |
> |
& & & 5.39 & 0.320(0.006) & $\infty$ |
| 761 |
> |
& 0 & \\ |
| 762 |
|
\hline |
| 763 |
< |
Au(111) & toluene & 200 & 1.08 & 0.72() & 1.?$\times$10$^5$() & |
| 764 |
< |
4.6 & 70.1 \\ |
| 765 |
< |
& & & 2.16 & 0.54() & 1.?$\times$10$^5$() & |
| 766 |
< |
4.9 & \\ |
| 767 |
< |
Au-SC$_4$H$_9$ & toluene & 200 & 5.39 & 0.98() & $\infty$ & 0 |
| 768 |
< |
& 187 \\ |
| 769 |
< |
& & & 10.8 & 0.99() & $\infty$ & 0 |
| 770 |
< |
& \\ |
| 763 |
> |
Au(111) & toluene & 200 & 1.08 & 0.722(0.035) & 15.7(0.7) & |
| 764 |
> |
4.60 & 70.1 \\ |
| 765 |
> |
& & & 2.16 & 0.544(0.030) & 11.2(0.5) & |
| 766 |
> |
4.86 & \\ |
| 767 |
> |
Au-SC$_4$H$_9$ & toluene & 200 & 5.39 & 0.980(0.057) & |
| 768 |
> |
$\infty$ & 0 & 187 \\ |
| 769 |
> |
& & & 10.8 & 0.995(0.005) & |
| 770 |
> |
$\infty$ & 0 & \\ |
| 771 |
|
\hline |
| 772 |
< |
Au(111) & water & 300 & 1.08 & 0.40() & 1.9$\times$10$^4$() & |
| 772 |
> |
Au(111) & water & 300 & 1.08 & 0.399(0.050) & 1.928(0.022) & |
| 773 |
|
20.7 & 1.65 \\ |
| 774 |
< |
& & & 2.16 & 0.79() & 1.9$\times$10$^4$() & |
| 774 |
> |
& & & 2.16 & 0.794(0.255) & 1.895(0.003) & |
| 775 |
|
41.9 & \\ |
| 776 |
|
\hline |
| 777 |
< |
ice(basal) & water & 225 & 19.4 & 15.8() & $\infty$ & 0 & \\ |
| 777 |
> |
ice(basal) & water & 225 & 19.4 & 15.8(0.2) & $\infty$ & 0 & \\ |
| 778 |
|
\hline\hline |
| 779 |
|
\end{tabular} |
| 780 |
|
\label{etaKappaDelta} |
| 784 |
|
|
| 785 |
|
An interesting effect alongside the surface friction change is |
| 786 |
|
observed on the shear viscosity of liquids in the regions close to the |
| 787 |
< |
solid surface. Note that $\eta$ measured near a ``slip'' surface tends |
| 788 |
< |
to be smaller than that near a ``stick'' surface. This suggests that |
| 789 |
< |
an interface could affect the dynamic properties on its neighbor |
| 790 |
< |
regions. It is known that diffusions of solid particles in liquid |
| 791 |
< |
phase is affected by their surface conditions (stick or slip |
| 792 |
< |
boundary).[CITE SCHMIDT AND SKINNER] Our observations could provide |
| 793 |
< |
support to this phenomenon. |
| 787 |
> |
solid surface. In our results, $\eta$ measured near a ``slip'' surface |
| 788 |
> |
tends to be smaller than that near a ``stick'' surface. This may |
| 789 |
> |
suggest the influence from an interface on the dynamic properties of |
| 790 |
> |
liquid within its neighbor regions. It is known that diffusions of |
| 791 |
> |
solid particles in liquid phase is affected by their surface |
| 792 |
> |
conditions (stick or slip boundary).\cite{10.1063/1.1610442} Our |
| 793 |
> |
observations could provide a support to this phenomenon. |
| 794 |
|
|
| 795 |
|
In addition to these previously studied interfaces, we attempt to |
| 796 |
|
construct ice-water interfaces and the basal plane of ice lattice was |
| 797 |
< |
first studied. In contrast to the Au(111)/water interface, where the |
| 798 |
< |
friction coefficient is relatively small and large slip effect |
| 797 |
> |
studied here. In contrast to the Au(111)/water interface, where the |
| 798 |
> |
friction coefficient is substantially small and large slip effect |
| 799 |
|
presents, the ice/liquid water interface demonstrates strong |
| 800 |
< |
interactions and appears to be sticky. The supercooled liquid phase is |
| 801 |
< |
an order of magnitude viscous than measurements in previous |
| 802 |
< |
section. It would be of interst to investigate the effect of different |
| 803 |
< |
ice lattice planes (such as prism surface) on interfacial friction and |
| 804 |
< |
corresponding liquid viscosity. |
| 800 |
> |
solid-liquid interactions and appears to be sticky. The supercooled |
| 801 |
> |
liquid phase is an order of magnitude more viscous than measurements |
| 802 |
> |
in previous section. It would be of interst to investigate the effect |
| 803 |
> |
of different ice lattice planes (such as prism and other surfaces) on |
| 804 |
> |
interfacial friction and the corresponding liquid viscosity. |
| 805 |
|
|
| 806 |
|
\section{Conclusions} |
| 807 |
|
Our simulations demonstrate the validity of our method in RNEMD |
| 813 |
|
applied in various ensembles, so prospective applications to |
| 814 |
|
extended-system methods are possible. |
| 815 |
|
|
| 816 |
< |
Furthermore, using this method, investigations can be carried out to |
| 817 |
< |
characterize interfacial interactions. Our method is capable of |
| 818 |
< |
effectively imposing both thermal and momentum flux accross an |
| 819 |
< |
interface and thus facilitates studies that relates dynamic property |
| 820 |
< |
measurements to the chemical details of an interface. |
| 816 |
> |
Our method is capable of effectively imposing thermal and/or momentum |
| 817 |
> |
flux accross an interface. This facilitates studies that relates |
| 818 |
> |
dynamic property measurements to the chemical details of an |
| 819 |
> |
interface. Therefore, investigations can be carried out to |
| 820 |
> |
characterize interfacial interactions using the method. |
| 821 |
|
|
| 822 |
|
Another attractive feature of our method is the ability of |
| 823 |
< |
simultaneously imposing thermal and momentum flux in a |
| 824 |
< |
system. potential researches that might be benefit include complex |
| 825 |
< |
systems that involve thermal and momentum gradients. For example, the |
| 826 |
< |
Soret effects under a velocity gradient would be of interest to |
| 827 |
< |
purification and separation researches. |
| 823 |
> |
simultaneously introducing thermal and momentum gradients in a |
| 824 |
> |
system. This facilitates us to effectively map out the shear viscosity |
| 825 |
> |
with respect to a range of temperature in single trajectory of |
| 826 |
> |
simulation with satisafactory accuracy. Complex systems that involve |
| 827 |
> |
thermal and momentum gradients might potentially benefit from |
| 828 |
> |
this. For example, the Soret effects under a velocity gradient might |
| 829 |
> |
be models of interest to purification and separation researches. |
| 830 |
|
|
| 831 |
|
\section{Acknowledgments} |
| 832 |
|
Support for this project was provided by the National Science |
