| 43 |
|
\begin{doublespace} |
| 44 |
|
|
| 45 |
|
\begin{abstract} |
| 46 |
< |
REPLACE ABSTRACT HERE |
| 47 |
< |
With the Non-Isotropic Velocity Scaling (NIVS) approach to Reverse |
| 48 |
< |
Non-Equilibrium Molecular Dynamics (RNEMD) it is possible to impose |
| 49 |
< |
an unphysical thermal flux between different regions of |
| 50 |
< |
inhomogeneous systems such as solid / liquid interfaces. We have |
| 51 |
< |
applied NIVS to compute the interfacial thermal conductance at a |
| 52 |
< |
metal / organic solvent interface that has been chemically capped by |
| 53 |
< |
butanethiol molecules. Our calculations suggest that coupling |
| 54 |
< |
between the metal and liquid phases is enhanced by the capping |
| 55 |
< |
agents, leading to a greatly enhanced conductivity at the interface. |
| 56 |
< |
Specifically, the chemical bond between the metal and the capping |
| 57 |
< |
agent introduces a vibrational overlap that is not present without |
| 58 |
< |
the capping agent, and the overlap between the vibrational spectra |
| 59 |
< |
(metal to cap, cap to solvent) provides a mechanism for rapid |
| 60 |
< |
thermal transport across the interface. Our calculations also |
| 61 |
< |
suggest that this is a non-monotonic function of the fractional |
| 62 |
< |
coverage of the surface, as moderate coverages allow diffusive heat |
| 63 |
< |
transport of solvent molecules that have been in close contact with |
| 64 |
< |
the capping agent. |
| 46 |
> |
We present a new method for introducing stable nonequilibrium |
| 47 |
> |
velocity and temperature gradients in molecular dynamics simulations |
| 48 |
> |
of heterogeneous systems. This method conserves the linear momentum |
| 49 |
> |
and total energy of the system and improves previous Reverse |
| 50 |
> |
Non-Equilibrium Molecular Dynamics (RNEMD) methods and maintains |
| 51 |
> |
thermal velocity distributions. It also avoid thermal anisotropy |
| 52 |
> |
occured in NIVS simulations by using isotropic velocity scaling on |
| 53 |
> |
the molecules in specific regions of a system. To test the method, |
| 54 |
> |
we have computed the thermal conductivity and shear viscosity of |
| 55 |
> |
model liquid systems as well as the interfacial frictions of a |
| 56 |
> |
series of metal/liquid interfaces. |
| 57 |
|
|
| 58 |
|
\end{abstract} |
| 59 |
|
|
| 210 |
|
scaling. More importantly, separating the momentum flux imposing from |
| 211 |
|
velocity scaling avoids the underlying cause that NIVS produced |
| 212 |
|
thermal anisotropy when applying a momentum flux. |
| 221 |
– |
%NEW METHOD DOESN'T CAUSE UNDESIRED CONCOMITENT MOMENTUM FLUX WHEN |
| 222 |
– |
%IMPOSING A THERMAL FLUX |
| 213 |
|
|
| 214 |
|
The advantages of the approach over the original momentum swapping |
| 215 |
|
approach lies in its nature to preserve a Gaussian |
| 218 |
|
diffusion of the neighboring slabs could no longer remedy this effect, |
| 219 |
|
and nonthermal distributions would be observed. Results in later |
| 220 |
|
section will illustrate this effect. |
| 231 |
– |
%NEW METHOD (AND NIVS) HAVE LESS PERTURBATION THAN MOMENTUM SWAPPING |
| 221 |
|
|
| 222 |
|
\section{Computational Details} |
| 223 |
|
The algorithm has been implemented in our MD simulation code, |
| 231 |
|
\subsection{Simulation Protocols} |
| 232 |
|
The systems to be investigated are set up in a orthorhombic simulation |
| 233 |
|
cell with periodic boundary conditions in all three dimensions. The |
| 234 |
< |
$z$ axis of these cells were longer and was used as the gradient axis |
| 234 |
> |
$z$ axis of these cells were longer and was set as the gradient axis |
| 235 |
|
of temperature and/or momentum. Thus the cells were divided into $N$ |
| 236 |
|
slabs along this axis, with various $N$ depending on individual |
| 237 |
|
system. The $x$ and $y$ axis were usually of the same length in |
| 248 |
|
|
| 249 |
|
While homogeneous fluid systems can be set up with random |
| 250 |
|
configurations, our interfacial systems needs extra steps to ensure |
| 251 |
< |
the interfaces be established properly for computations. |
| 252 |
< |
[AU(THIOL)ORGANIC SOLVENTS: REFER TO JPCC] |
| 253 |
< |
[ICE-WATER REFER TO OTHER REF.S] |
| 251 |
> |
the interfaces be established properly for computations. The |
| 252 |
> |
preparation and equilibration of butanethiol covered gold (111) |
| 253 |
> |
surface and further solvation and equilibration process is described |
| 254 |
> |
as in reference \cite{kuang:AuThl}. |
| 255 |
|
|
| 256 |
< |
Metal slabs of 6 or 11 layers of Au atoms were first equilibrated |
| 257 |
< |
under atmospheric pressure (1 atm) and 200K. After equilibration, |
| 258 |
< |
butanethiol capping agents were placed at three-fold hollow sites on |
| 259 |
< |
the Au(111) surfaces. These sites are either {\it fcc} or {\it |
| 260 |
< |
hcp} sites, although Hase {\it et al.} found that they are |
| 261 |
< |
equivalent in a heat transfer process,\cite{hase:2010} so we did not |
| 262 |
< |
distinguish between these sites in our study. The maximum butanethiol |
| 263 |
< |
capacity on Au surface is $1/3$ of the total number of surface Au |
| 264 |
< |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
| 265 |
< |
structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A |
| 266 |
< |
series of lower coverages was also prepared by eliminating |
| 267 |
< |
butanethiols from the higher coverage surface in a regular manner. The |
| 268 |
< |
lower coverages were prepared in order to study the relation between |
| 269 |
< |
coverage and interfacial conductance. |
| 270 |
< |
|
| 271 |
< |
The capping agent molecules were allowed to migrate during the |
| 282 |
< |
simulations. They distributed themselves uniformly and sampled a |
| 283 |
< |
number of three-fold sites throughout out study. Therefore, the |
| 284 |
< |
initial configuration does not noticeably affect the sampling of a |
| 285 |
< |
variety of configurations of the same coverage, and the final |
| 286 |
< |
conductance measurement would be an average effect of these |
| 287 |
< |
configurations explored in the simulations. |
| 288 |
< |
|
| 289 |
< |
After the modified Au-butanethiol surface systems were equilibrated in |
| 290 |
< |
the canonical (NVT) ensemble, organic solvent molecules were packed in |
| 291 |
< |
the previously empty part of the simulation cells.\cite{packmol} Two |
| 292 |
< |
solvents were investigated, one which has little vibrational overlap |
| 293 |
< |
with the alkanethiol and which has a planar shape (toluene), and one |
| 294 |
< |
which has similar vibrational frequencies to the capping agent and |
| 295 |
< |
chain-like shape ({\it n}-hexane). |
| 256 |
> |
As for the ice/liquid water interfaces, the basal surface of ice |
| 257 |
> |
lattice was first constructed. Hirsch {\it et |
| 258 |
> |
al.}\cite{doi:10.1021/jp048434u} explored the energetics of ice |
| 259 |
> |
lattices with different proton orders. We refer to their results and |
| 260 |
> |
choose the configuration of the lowest energy after geometry |
| 261 |
> |
optimization as the unit cells of our ice lattices. Although |
| 262 |
> |
experimental solid/liquid coexistant temperature near normal pressure |
| 263 |
> |
is 273K, Bryk and Haymet's simulations of ice/liquid water interfaces |
| 264 |
> |
with different models suggest that for SPC/E, the most stable |
| 265 |
> |
interface is observed at 225$\pm$5K. Therefore, all our ice/liquid |
| 266 |
> |
water simulations were carried out under 225K. To have extra |
| 267 |
> |
protection of the ice lattice during initial equilibration (when the |
| 268 |
> |
randomly generated liquid phase configuration could release large |
| 269 |
> |
amount of energy in relaxation), a constraint method (REF?) was |
| 270 |
> |
adopted until the high energy configuration was relaxed. |
| 271 |
> |
[MAY ADD A FIGURE HERE FOR BASAL PLANE, MAY INCLUDE PRISM IF POSSIBLE] |
| 272 |
|
|
| 273 |
< |
The simulation cells were not particularly extensive along the |
| 274 |
< |
$z$-axis, as a very long length scale for the thermal gradient may |
| 275 |
< |
cause excessively hot or cold temperatures in the middle of the |
| 276 |
< |
solvent region and lead to undesired phenomena such as solvent boiling |
| 277 |
< |
or freezing when a thermal flux is applied. Conversely, too few |
| 302 |
< |
solvent molecules would change the normal behavior of the liquid |
| 303 |
< |
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
| 304 |
< |
these extreme cases did not happen to our simulations. The spacing |
| 305 |
< |
between periodic images of the gold interfaces is $45 \sim 75$\AA in |
| 306 |
< |
our simulations. |
| 273 |
> |
\subsection{Force Field Parameters} |
| 274 |
> |
For comparison of our new method with previous work, we retain our |
| 275 |
> |
force field parameters consistent with the results we will compare |
| 276 |
> |
with. The Lennard-Jones fluid used here for argon , and reduced unit |
| 277 |
> |
results are reported for direct comparison purpose. |
| 278 |
|
|
| 279 |
< |
The initial configurations generated are further equilibrated with the |
| 280 |
< |
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
| 281 |
< |
change. This is to ensure that the equilibration of liquid phase does |
| 282 |
< |
not affect the metal's crystalline structure. Comparisons were made |
| 312 |
< |
with simulations that allowed changes of $L_x$ and $L_y$ during NPT |
| 313 |
< |
equilibration. No substantial changes in the box geometry were noticed |
| 314 |
< |
in these simulations. After ensuring the liquid phase reaches |
| 315 |
< |
equilibrium at atmospheric pressure (1 atm), further equilibration was |
| 316 |
< |
carried out under canonical (NVT) and microcanonical (NVE) ensembles. |
| 279 |
> |
As for our water simulations, SPC/E model is used throughout this work |
| 280 |
> |
for consistency. Previous work for transport properties of SPC/E water |
| 281 |
> |
model is available\cite{Bedrov:2000,10.1063/1.3330544,Medina2011} so |
| 282 |
> |
that unnecessary repetition of previous methods can be avoided. |
| 283 |
|
|
| 284 |
< |
After the systems reach equilibrium, NIVS was used to impose an |
| 285 |
< |
unphysical thermal flux between the metal and the liquid phases. Most |
| 286 |
< |
of our simulations were done under an average temperature of |
| 287 |
< |
$\sim$200K. Therefore, thermal flux usually came from the metal to the |
| 288 |
< |
liquid so that the liquid has a higher temperature and would not |
| 289 |
< |
freeze due to lowered temperatures. After this induced temperature |
| 290 |
< |
gradient had stabilized, the temperature profile of the simulation cell |
| 291 |
< |
was recorded. To do this, the simulation cell is divided evenly into |
| 292 |
< |
$N$ slabs along the $z$-axis. The average temperatures of each slab |
| 293 |
< |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
| 294 |
< |
the same, the derivatives of $T$ with respect to slab number $n$ can |
| 295 |
< |
be directly used for $G^\prime$ calculations: \begin{equation} |
| 296 |
< |
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 297 |
< |
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
| 298 |
< |
= |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| |
| 299 |
< |
\Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 |
| 300 |
< |
= |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| |
| 301 |
< |
\Big/\left(\frac{\partial T}{\partial n}\right)^2 |
| 302 |
< |
\label{derivativeG2} |
| 284 |
> |
The Au-Au interaction parameters in all simulations are described by |
| 285 |
> |
the quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The |
| 286 |
> |
QSC potentials include zero-point quantum corrections and are |
| 287 |
> |
reparametrized for accurate surface energies compared to the |
| 288 |
> |
Sutton-Chen potentials.\cite{Chen90} For gold/water interfaces, the |
| 289 |
> |
Spohr potential was adopted\cite{ISI:000167766600035} to depict |
| 290 |
> |
Au-H$_2$O interactions. |
| 291 |
> |
|
| 292 |
> |
The small organic molecules included in our simulations are the Au |
| 293 |
> |
surface capping agent butanethiol and liquid hexane and toluene. The |
| 294 |
> |
United-Atom |
| 295 |
> |
models\cite{TraPPE-UA.thiols,TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} |
| 296 |
> |
for these components were used in this work for better computational |
| 297 |
> |
efficiency, while maintaining good accuracy. We refer readers to our |
| 298 |
> |
previous work\cite{kuang:AuThl} for further details of these models, |
| 299 |
> |
as well as the interactions between Au and the above organic molecule |
| 300 |
> |
components. |
| 301 |
> |
|
| 302 |
> |
\subsection{Thermal conductivities} |
| 303 |
> |
When $\vec{j}_z(\vec{p})$ is set to zero and a target $J_z$ is set to |
| 304 |
> |
impose kinetic energy transfer, the method can be used for thermal |
| 305 |
> |
conductivity computations. Similar to previous RNEMD methods, we |
| 306 |
> |
assume linear response of the temperature gradient with respect to the |
| 307 |
> |
thermal flux in general case. And the thermal conductivity ($\lambda$) |
| 308 |
> |
can be obtained with the imposed kinetic energy flux and the measured |
| 309 |
> |
thermal gradient: |
| 310 |
> |
\begin{equation} |
| 311 |
> |
J_z = -\lambda \frac{\partial T}{\partial z} |
| 312 |
|
\end{equation} |
| 313 |
< |
The absolute values in Eq. \ref{derivativeG2} appear because the |
| 314 |
< |
direction of the flux $\vec{J}$ is in an opposing direction on either |
| 315 |
< |
side of the metal slab. |
| 313 |
> |
Like other imposed-flux methods, the energy flux was calculated using |
| 314 |
> |
the total non-physical energy transferred (${E_{total}}$) from slab |
| 315 |
> |
``c'' to slab ``h'', which is recorded throughout a simulation, and |
| 316 |
> |
the time for data collection $t$: |
| 317 |
> |
\begin{equation} |
| 318 |
> |
J_z = \frac{E_{total}}{2 t L_x L_y} |
| 319 |
> |
\end{equation} |
| 320 |
> |
where $L_x$ and $L_y$ denotes the dimensions of the plane in a |
| 321 |
> |
simulation cell perpendicular to the thermal gradient, and a factor of |
| 322 |
> |
two in the denominator is present for the heat transport occurs in |
| 323 |
> |
both $+z$ and $-z$ directions. The temperature gradient |
| 324 |
> |
${\langle\partial T/\partial z\rangle}$ can be obtained by a linear |
| 325 |
> |
regression of the temperature profile, which is recorded during a |
| 326 |
> |
simulation for each slab in a cell. For Lennard-Jones simulations, |
| 327 |
> |
thermal conductivities are reported in reduced units |
| 328 |
> |
(${\lambda^*=\lambda \sigma^2 m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$). |
| 329 |
|
|
| 330 |
< |
All of the above simulation procedures use a time step of 1 fs. Each |
| 331 |
< |
equilibration stage took a minimum of 100 ps, although in some cases, |
| 332 |
< |
longer equilibration stages were utilized. |
| 330 |
> |
\subsection{Shear viscosities} |
| 331 |
> |
Alternatively, the method can carry out shear viscosity calculations |
| 332 |
> |
by switching off $J_z$. One can specify the vector |
| 333 |
> |
$\vec{j}_z(\vec{p})$ by choosing the three components |
| 334 |
> |
respectively. For shear viscosity simulations, $j_z(p_z)$ is usually |
| 335 |
> |
set to zero. Although for isotropic systems, the direction of |
| 336 |
> |
$\vec{j}_z(\vec{p})$ on the $xy$ plane should not matter, the ability |
| 337 |
> |
of arbitarily specifying the vector direction in our method provides |
| 338 |
> |
convenience in anisotropic simulations. |
| 339 |
|
|
| 340 |
< |
\subsection{Force Field Parameters} |
| 341 |
< |
Our simulations include a number of chemically distinct components. |
| 342 |
< |
Figure \ref{demoMol} demonstrates the sites defined for both |
| 343 |
< |
United-Atom and All-Atom models of the organic solvent and capping |
| 344 |
< |
agents in our simulations. Force field parameters are needed for |
| 345 |
< |
interactions both between the same type of particles and between |
| 346 |
< |
particles of different species. |
| 347 |
< |
|
| 348 |
< |
\begin{figure} |
| 349 |
< |
\includegraphics[width=\linewidth]{structures} |
| 350 |
< |
\caption{Structures of the capping agent and solvents utilized in |
| 351 |
< |
these simulations. The chemically-distinct sites (a-e) are expanded |
| 352 |
< |
in terms of constituent atoms for both United Atom (UA) and All Atom |
| 353 |
< |
(AA) force fields. Most parameters are from References |
| 354 |
< |
\protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
| 355 |
< |
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
| 356 |
< |
atoms are given in Table 1 in the supporting information.} |
| 357 |
< |
\label{demoMol} |
| 364 |
< |
\end{figure} |
| 365 |
< |
|
| 366 |
< |
The Au-Au interactions in metal lattice slab is described by the |
| 367 |
< |
quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC |
| 368 |
< |
potentials include zero-point quantum corrections and are |
| 369 |
< |
reparametrized for accurate surface energies compared to the |
| 370 |
< |
Sutton-Chen potentials.\cite{Chen90} |
| 371 |
< |
|
| 372 |
< |
For the two solvent molecules, {\it n}-hexane and toluene, two |
| 373 |
< |
different atomistic models were utilized. Both solvents were modeled |
| 374 |
< |
using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA |
| 375 |
< |
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
| 376 |
< |
for our UA solvent molecules. In these models, sites are located at |
| 377 |
< |
the carbon centers for alkyl groups. Bonding interactions, including |
| 378 |
< |
bond stretches and bends and torsions, were used for intra-molecular |
| 379 |
< |
sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones |
| 380 |
< |
potentials are used. |
| 381 |
< |
|
| 382 |
< |
By eliminating explicit hydrogen atoms, the TraPPE-UA models are |
| 383 |
< |
simple and computationally efficient, while maintaining good accuracy. |
| 384 |
< |
However, the TraPPE-UA model for alkanes is known to predict a slightly |
| 385 |
< |
lower boiling point than experimental values. This is one of the |
| 386 |
< |
reasons we used a lower average temperature (200K) for our |
| 387 |
< |
simulations. If heat is transferred to the liquid phase during the |
| 388 |
< |
NIVS simulation, the liquid in the hot slab can actually be |
| 389 |
< |
substantially warmer than the mean temperature in the simulation. The |
| 390 |
< |
lower mean temperatures therefore prevent solvent boiling. |
| 391 |
< |
|
| 392 |
< |
For UA-toluene, the non-bonded potentials between intermolecular sites |
| 393 |
< |
have a similar Lennard-Jones formulation. The toluene molecules were |
| 394 |
< |
treated as a single rigid body, so there was no need for |
| 395 |
< |
intramolecular interactions (including bonds, bends, or torsions) in |
| 396 |
< |
this solvent model. |
| 397 |
< |
|
| 398 |
< |
Besides the TraPPE-UA models, AA models for both organic solvents are |
| 399 |
< |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
| 400 |
< |
were used. For hexane, additional explicit hydrogen sites were |
| 401 |
< |
included. Besides bonding and non-bonded site-site interactions, |
| 402 |
< |
partial charges and the electrostatic interactions were added to each |
| 403 |
< |
CT and HC site. For toluene, a flexible model for the toluene molecule |
| 404 |
< |
was utilized which included bond, bend, torsion, and inversion |
| 405 |
< |
potentials to enforce ring planarity. |
| 406 |
< |
|
| 407 |
< |
The butanethiol capping agent in our simulations, were also modeled |
| 408 |
< |
with both UA and AA model. The TraPPE-UA force field includes |
| 409 |
< |
parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for |
| 410 |
< |
UA butanethiol model in our simulations. The OPLS-AA also provides |
| 411 |
< |
parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) |
| 412 |
< |
surfaces do not have the hydrogen atom bonded to sulfur. To derive |
| 413 |
< |
suitable parameters for butanethiol adsorbed on Au(111) surfaces, we |
| 414 |
< |
adopt the S parameters from Luedtke and Landman\cite{landman:1998} and |
| 415 |
< |
modify the parameters for the CTS atom to maintain charge neutrality |
| 416 |
< |
in the molecule. Note that the model choice (UA or AA) for the capping |
| 417 |
< |
agent can be different from the solvent. Regardless of model choice, |
| 418 |
< |
the force field parameters for interactions between capping agent and |
| 419 |
< |
solvent can be derived using Lorentz-Berthelot Mixing Rule: |
| 420 |
< |
\begin{eqnarray} |
| 421 |
< |
\sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ |
| 422 |
< |
\epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} |
| 423 |
< |
\end{eqnarray} |
| 424 |
< |
|
| 425 |
< |
To describe the interactions between metal (Au) and non-metal atoms, |
| 426 |
< |
we refer to an adsorption study of alkyl thiols on gold surfaces by |
| 427 |
< |
Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective |
| 428 |
< |
Lennard-Jones form of potential parameters for the interaction between |
| 429 |
< |
Au and pseudo-atoms CH$_x$ and S based on a well-established and |
| 430 |
< |
widely-used effective potential of Hautman and Klein for the Au(111) |
| 431 |
< |
surface.\cite{hautman:4994} As our simulations require the gold slab |
| 432 |
< |
to be flexible to accommodate thermal excitation, the pair-wise form |
| 433 |
< |
of potentials they developed was used for our study. |
| 340 |
> |
Similar to thermal conductivity computations, linear response of the |
| 341 |
> |
momentum gradient with respect to the shear stress is assumed, and the |
| 342 |
> |
shear viscosity ($\eta$) can be obtained with the imposed momentum |
| 343 |
> |
flux (e.g. in $x$ direction) and the measured gradient: |
| 344 |
> |
\begin{equation} |
| 345 |
> |
j_z(p_x) = -\eta \frac{\partial v_x}{\partial z} |
| 346 |
> |
\end{equation} |
| 347 |
> |
where the flux is similarly defined: |
| 348 |
> |
\begin{equation} |
| 349 |
> |
j_z(p_x) = \frac{P_x}{2 t L_x L_y} |
| 350 |
> |
\end{equation} |
| 351 |
> |
with $P_x$ being the total non-physical momentum transferred within |
| 352 |
> |
the data collection time. Also, the velocity gradient |
| 353 |
> |
${\langle\partial v_x/\partial z\rangle}$ can be obtained using linear |
| 354 |
> |
regression of the $x$ component of the mean velocity, $\langle |
| 355 |
> |
v_x\rangle$, in each of the bins. For Lennard-Jones simulations, shear |
| 356 |
> |
viscosities are reported in reduced units |
| 357 |
> |
(${\eta^*=\eta\sigma^2(\varepsilon m)^{-1/2}}$). |
| 358 |
|
|
| 359 |
< |
The potentials developed from {\it ab initio} calculations by Leng |
| 360 |
< |
{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 361 |
< |
interactions between Au and aromatic C/H atoms in toluene. However, |
| 362 |
< |
the Lennard-Jones parameters between Au and other types of particles, |
| 363 |
< |
(e.g. AA alkanes) have not yet been established. For these |
| 364 |
< |
interactions, the Lorentz-Berthelot mixing rule can be used to derive |
| 365 |
< |
effective single-atom LJ parameters for the metal using the fit values |
| 366 |
< |
for toluene. These are then used to construct reasonable mixing |
| 367 |
< |
parameters for the interactions between the gold and other atoms. |
| 368 |
< |
Table 1 in the supporting information summarizes the |
| 369 |
< |
``metal/non-metal'' parameters utilized in our simulations. |
| 359 |
> |
\subsection{Interfacial friction and Slip length} |
| 360 |
> |
While the shear stress results in a velocity gradient within bulk |
| 361 |
> |
fluid phase, its effect at a solid-liquid interface could vary due to |
| 362 |
> |
the interaction strength between the two phases. The interfacial |
| 363 |
> |
friction coefficient $\kappa$ is defined to relate the shear stress |
| 364 |
> |
(e.g. along $x$-axis) and the relative fluid velocity tangent to the |
| 365 |
> |
interface: |
| 366 |
> |
\begin{equation} |
| 367 |
> |
j_z(p_x)|_{interface} = \kappa\Delta v_x|_{interface} |
| 368 |
> |
\end{equation} |
| 369 |
> |
Under ``stick'' boundary condition, $\Delta v_x|_{interface} |
| 370 |
> |
\rightarrow 0$, which leads to $\kappa\rightarrow\infty$. However, for |
| 371 |
> |
``slip'' boundary condition at the solid-liquid interface, $\kappa$ |
| 372 |
> |
becomes finite. To characterize the interfacial boundary conditions, |
| 373 |
> |
slip length ($\delta$) is defined using $\kappa$ and the shear |
| 374 |
> |
viscocity of liquid phase ($\eta$): |
| 375 |
> |
\begin{equation} |
| 376 |
> |
\delta = \frac{\eta}{\kappa} |
| 377 |
> |
\end{equation} |
| 378 |
> |
so that $\delta\rightarrow 0$ in the ``no-slip'' boundary condition, |
| 379 |
> |
and depicts how ``slippery'' an interface is. Figure \ref{slipLength} |
| 380 |
> |
illustrates how this quantity is defined and computed for a |
| 381 |
> |
solid-liquid interface. [MAY INCLUDE A SNAPSHOT IN FIGURE] |
| 382 |
|
|
| 447 |
– |
\section{Results} |
| 448 |
– |
[L-J COMPARED TO RNEMD NIVS; WATER COMPARED TO RNEMD NIVS AND EMD; |
| 449 |
– |
SLIP BOUNDARY VS STICK BOUNDARY; ICE-WATER INTERFACES] |
| 450 |
– |
|
| 451 |
– |
There are many factors contributing to the measured interfacial |
| 452 |
– |
conductance; some of these factors are physically motivated |
| 453 |
– |
(e.g. coverage of the surface by the capping agent coverage and |
| 454 |
– |
solvent identity), while some are governed by parameters of the |
| 455 |
– |
methodology (e.g. applied flux and the formulas used to obtain the |
| 456 |
– |
conductance). In this section we discuss the major physical and |
| 457 |
– |
calculational effects on the computed conductivity. |
| 458 |
– |
|
| 459 |
– |
\subsection{Effects due to capping agent coverage} |
| 460 |
– |
|
| 461 |
– |
A series of different initial conditions with a range of surface |
| 462 |
– |
coverages was prepared and solvated with various with both of the |
| 463 |
– |
solvent molecules. These systems were then equilibrated and their |
| 464 |
– |
interfacial thermal conductivity was measured with the NIVS |
| 465 |
– |
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
| 466 |
– |
with respect to surface coverage. |
| 467 |
– |
|
| 383 |
|
\begin{figure} |
| 384 |
< |
\includegraphics[width=\linewidth]{coverage} |
| 385 |
< |
\caption{The interfacial thermal conductivity ($G$) has a |
| 386 |
< |
non-monotonic dependence on the degree of surface capping. This |
| 387 |
< |
data is for the Au(111) / butanethiol / solvent interface with |
| 388 |
< |
various UA force fields at $\langle T\rangle \sim $200K.} |
| 389 |
< |
\label{coverage} |
| 384 |
> |
\includegraphics[width=\linewidth]{defDelta} |
| 385 |
> |
\caption{The slip length $\delta$ can be obtained from a velocity |
| 386 |
> |
profile of a solid-liquid interface simulation. An example of |
| 387 |
> |
Au/hexane interfaces is shown. Calculation for the left side is |
| 388 |
> |
illustrated. The right side is similar to the left side.} |
| 389 |
> |
\label{slipLength} |
| 390 |
|
\end{figure} |
| 391 |
|
|
| 392 |
< |
In partially covered surfaces, the derivative definition for |
| 393 |
< |
$G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the |
| 394 |
< |
location of maximum change of $\lambda$ becomes washed out. The |
| 395 |
< |
discrete definition (Eq. \ref{discreteG}) is easier to apply, as the |
| 396 |
< |
Gibbs dividing surface is still well-defined. Therefore, $G$ (not |
| 397 |
< |
$G^\prime$) was used in this section. |
| 392 |
> |
In our method, a shear stress can be applied similar to shear |
| 393 |
> |
viscosity computations by applying an unphysical momentum flux |
| 394 |
> |
(e.g. $j_z(p_x)$). A corresponding velocity profile can be obtained as |
| 395 |
> |
shown in Figure \ref{slipLength}, in which the velocity gradients |
| 396 |
> |
within liquid phase and velocity difference at the liquid-solid |
| 397 |
> |
interface can be measured respectively. Further calculations and |
| 398 |
> |
characterizations of the interface can be carried out using these |
| 399 |
> |
data. |
| 400 |
|
|
| 401 |
< |
From Figure \ref{coverage}, one can see the significance of the |
| 402 |
< |
presence of capping agents. When even a small fraction of the Au(111) |
| 403 |
< |
surface sites are covered with butanethiols, the conductivity exhibits |
| 404 |
< |
an enhancement by at least a factor of 3. Capping agents are clearly |
| 405 |
< |
playing a major role in thermal transport at metal / organic solvent |
| 406 |
< |
surfaces. |
| 401 |
> |
\section{Results and Discussions} |
| 402 |
> |
\subsection{Lennard-Jones fluid} |
| 403 |
> |
Our orthorhombic simulation cell of Lennard-Jones fluid has identical |
| 404 |
> |
parameters to our previous work\cite{kuang:164101} to facilitate |
| 405 |
> |
comparison. Thermal conductivitis and shear viscosities were computed |
| 406 |
> |
with the algorithm applied to the simulations. The results of thermal |
| 407 |
> |
conductivity are compared with our previous NIVS algorithm. However, |
| 408 |
> |
since the NIVS algorithm could produce temperature anisotropy for |
| 409 |
> |
shear viscocity computations, these results are instead compared to |
| 410 |
> |
the momentum swapping approaches. Table \ref{LJ} lists these |
| 411 |
> |
calculations with various fluxes in reduced units. |
| 412 |
|
|
| 491 |
– |
We note a non-monotonic behavior in the interfacial conductance as a |
| 492 |
– |
function of surface coverage. The maximum conductance (largest $G$) |
| 493 |
– |
happens when the surfaces are about 75\% covered with butanethiol |
| 494 |
– |
caps. The reason for this behavior is not entirely clear. One |
| 495 |
– |
explanation is that incomplete butanethiol coverage allows small gaps |
| 496 |
– |
between butanethiols to form. These gaps can be filled by transient |
| 497 |
– |
solvent molecules. These solvent molecules couple very strongly with |
| 498 |
– |
the hot capping agent molecules near the surface, and can then carry |
| 499 |
– |
away (diffusively) the excess thermal energy from the surface. |
| 500 |
– |
|
| 501 |
– |
There appears to be a competition between the conduction of the |
| 502 |
– |
thermal energy away from the surface by the capping agents (enhanced |
| 503 |
– |
by greater coverage) and the coupling of the capping agents with the |
| 504 |
– |
solvent (enhanced by interdigitation at lower coverages). This |
| 505 |
– |
competition would lead to the non-monotonic coverage behavior observed |
| 506 |
– |
here. |
| 507 |
– |
|
| 508 |
– |
Results for rigid body toluene solvent, as well as the UA hexane, are |
| 509 |
– |
within the ranges expected from prior experimental |
| 510 |
– |
work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests |
| 511 |
– |
that explicit hydrogen atoms might not be required for modeling |
| 512 |
– |
thermal transport in these systems. C-H vibrational modes do not see |
| 513 |
– |
significant excited state population at low temperatures, and are not |
| 514 |
– |
likely to carry lower frequency excitations from the solid layer into |
| 515 |
– |
the bulk liquid. |
| 516 |
– |
|
| 517 |
– |
The toluene solvent does not exhibit the same behavior as hexane in |
| 518 |
– |
that $G$ remains at approximately the same magnitude when the capping |
| 519 |
– |
coverage increases from 25\% to 75\%. Toluene, as a rigid planar |
| 520 |
– |
molecule, cannot occupy the relatively small gaps between the capping |
| 521 |
– |
agents as easily as the chain-like {\it n}-hexane. The effect of |
| 522 |
– |
solvent coupling to the capping agent is therefore weaker in toluene |
| 523 |
– |
except at the very lowest coverage levels. This effect counters the |
| 524 |
– |
coverage-dependent conduction of heat away from the metal surface, |
| 525 |
– |
leading to a much flatter $G$ vs. coverage trend than is observed in |
| 526 |
– |
{\it n}-hexane. |
| 527 |
– |
|
| 528 |
– |
\subsection{Effects due to Solvent \& Solvent Models} |
| 529 |
– |
In addition to UA solvent and capping agent models, AA models have |
| 530 |
– |
also been included in our simulations. In most of this work, the same |
| 531 |
– |
(UA or AA) model for solvent and capping agent was used, but it is |
| 532 |
– |
also possible to utilize different models for different components. |
| 533 |
– |
We have also included isotopic substitutions (Hydrogen to Deuterium) |
| 534 |
– |
to decrease the explicit vibrational overlap between solvent and |
| 535 |
– |
capping agent. Table \ref{modelTest} summarizes the results of these |
| 536 |
– |
studies. |
| 537 |
– |
|
| 413 |
|
\begin{table*} |
| 414 |
|
\begin{minipage}{\linewidth} |
| 415 |
|
\begin{center} |
| 416 |
+ |
|
| 417 |
+ |
\caption{Thermal conductivity ($\lambda^*$) and shear viscosity |
| 418 |
+ |
($\eta^*$) (in reduced units) of a Lennard-Jones fluid at |
| 419 |
+ |
${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$ computed |
| 420 |
+ |
at various momentum fluxes. The new method yields similar |
| 421 |
+ |
results to previous RNEMD methods. All results are reported in |
| 422 |
+ |
reduced unit. Uncertainties are indicated in parentheses.} |
| 423 |
|
|
| 424 |
< |
\caption{Computed interfacial thermal conductance ($G$ and |
| 543 |
< |
$G^\prime$) values for interfaces using various models for |
| 544 |
< |
solvent and capping agent (or without capping agent) at |
| 545 |
< |
$\langle T\rangle\sim$200K. Here ``D'' stands for deuterated |
| 546 |
< |
solvent or capping agent molecules. Error estimates are |
| 547 |
< |
indicated in parentheses.} |
| 548 |
< |
|
| 549 |
< |
\begin{tabular}{llccc} |
| 424 |
> |
\begin{tabular}{cccccc} |
| 425 |
|
\hline\hline |
| 426 |
< |
Butanethiol model & Solvent & $G$ & $G^\prime$ \\ |
| 427 |
< |
(or bare surface) & model & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 426 |
> |
\multicolumn{2}{c}{Momentum Exchange} & |
| 427 |
> |
\multicolumn{2}{c}{$\lambda^*$} & |
| 428 |
> |
\multicolumn{2}{c}{$\eta^*$} \\ |
| 429 |
|
\hline |
| 430 |
< |
UA & UA hexane & 131(9) & 87(10) \\ |
| 431 |
< |
& UA hexane(D) & 153(5) & 136(13) \\ |
| 556 |
< |
& AA hexane & 131(6) & 122(10) \\ |
| 557 |
< |
& UA toluene & 187(16) & 151(11) \\ |
| 558 |
< |
& AA toluene & 200(36) & 149(53) \\ |
| 430 |
> |
Swap Interval $t^*$ & Equivalent $J_z^*$ or $j_z^*(p_x)$ & |
| 431 |
> |
NIVS & This work & Swapping & This work \\ |
| 432 |
|
\hline |
| 433 |
< |
AA & UA hexane & 116(9) & 129(8) \\ |
| 434 |
< |
& AA hexane & 442(14) & 356(31) \\ |
| 435 |
< |
& AA hexane(D) & 222(12) & 234(54) \\ |
| 436 |
< |
& UA toluene & 125(25) & 97(60) \\ |
| 437 |
< |
& AA toluene & 487(56) & 290(42) \\ |
| 565 |
< |
\hline |
| 566 |
< |
AA(D) & UA hexane & 158(25) & 172(4) \\ |
| 567 |
< |
& AA hexane & 243(29) & 191(11) \\ |
| 568 |
< |
& AA toluene & 364(36) & 322(67) \\ |
| 569 |
< |
\hline |
| 570 |
< |
bare & UA hexane & 46.5(3.2) & 49.4(4.5) \\ |
| 571 |
< |
& UA hexane(D) & 43.9(4.6) & 43.0(2.0) \\ |
| 572 |
< |
& AA hexane & 31.0(1.4) & 29.4(1.3) \\ |
| 573 |
< |
& UA toluene & 70.1(1.3) & 65.8(0.5) \\ |
| 433 |
> |
0.116 & 0.16 & 7.30(0.10) & 6.25(0.23) & 3.57(0.06) & 3.53(0.16)\\ |
| 434 |
> |
0.232 & 0.09 & 6.95(0.09) & 8.02(0.56) & 3.64(0.05) & 3.43(0.17)\\ |
| 435 |
> |
0.463 & 0.047 & 7.19(0.07) & 6.48(0.15) & 3.52(0.16) & 3.51(0.08)\\ |
| 436 |
> |
0.926 & 0.024 & 7.19(0.28) & 7.02(0.13) & 3.72(0.05) & 3.79(0.11)\\ |
| 437 |
> |
1.158 & 0.019 & 7.98(0.33) & 7.37(0.23) & 3.42(0.06) & 3.53(0.06)\\ |
| 438 |
|
\hline\hline |
| 439 |
|
\end{tabular} |
| 440 |
< |
\label{modelTest} |
| 440 |
> |
\label{LJ} |
| 441 |
|
\end{center} |
| 442 |
|
\end{minipage} |
| 443 |
|
\end{table*} |
| 444 |
|
|
| 445 |
< |
To facilitate direct comparison between force fields, systems with the |
| 446 |
< |
same capping agent and solvent were prepared with the same length |
| 447 |
< |
scales for the simulation cells. |
| 445 |
> |
\subsubsection{Thermal conductivity} |
| 446 |
> |
Our thermal conductivity calculations with this method yields |
| 447 |
> |
comparable results to the previous NIVS algorithm. This indicates that |
| 448 |
> |
the thermal gradients rendered using this method are also close to |
| 449 |
> |
previous RNEMD methods. Simulations with moderately higher thermal |
| 450 |
> |
fluxes tend to yield more reliable thermal gradients and thus avoid |
| 451 |
> |
large errors, while overly high thermal fluxes could introduce side |
| 452 |
> |
effects such as non-linear temperature gradient response or |
| 453 |
> |
inadvertent phase transitions. |
| 454 |
|
|
| 455 |
< |
On bare metal / solvent surfaces, different force field models for |
| 456 |
< |
hexane yield similar results for both $G$ and $G^\prime$, and these |
| 457 |
< |
two definitions agree with each other very well. This is primarily an |
| 458 |
< |
indicator of weak interactions between the metal and the solvent. |
| 455 |
> |
Since the scaling operation is isotropic in this method, one does not |
| 456 |
> |
need extra care to ensure temperature isotropy between the $x$, $y$ |
| 457 |
> |
and $z$ axes, while thermal anisotropy might happen if the criteria |
| 458 |
> |
function for choosing scaling coefficients does not perform as |
| 459 |
> |
expected. Furthermore, this method avoids inadvertent concomitant |
| 460 |
> |
momentum flux when only thermal flux is imposed, which could not be |
| 461 |
> |
achieved with swapping or NIVS approaches. The thermal energy exchange |
| 462 |
> |
in swapping ($\vec{p}_i$ in slab ``c'' with $\vec{p}_j$ in slab ``h'') |
| 463 |
> |
or NIVS (total slab momentum conponemts $P^\alpha$ scaled to $\alpha |
| 464 |
> |
P^\alpha$) would not obtain this result unless thermal flux vanishes |
| 465 |
> |
(i.e. $\vec{p}_i=\vec{p}_j$ or $\alpha=1$ which are trivial to apply a |
| 466 |
> |
thermal flux). In this sense, this method contributes to having |
| 467 |
> |
minimal perturbation to a simulation while imposing thermal flux. |
| 468 |
|
|
| 469 |
< |
For the fully-covered surfaces, the choice of force field for the |
| 470 |
< |
capping agent and solvent has a large impact on the calculated values |
| 471 |
< |
of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are |
| 472 |
< |
much larger than their UA to UA counterparts, and these values exceed |
| 473 |
< |
the experimental estimates by a large measure. The AA force field |
| 474 |
< |
allows significant energy to go into C-H (or C-D) stretching modes, |
| 475 |
< |
and since these modes are high frequency, this non-quantum behavior is |
| 476 |
< |
likely responsible for the overestimate of the conductivity. Compared |
| 477 |
< |
to the AA model, the UA model yields more reasonable conductivity |
| 599 |
< |
values with much higher computational efficiency. |
| 469 |
> |
\subsubsection{Shear viscosity} |
| 470 |
> |
Table \ref{LJ} also compares our shear viscosity results with momentum |
| 471 |
> |
swapping approach. Our calculations show that our method predicted |
| 472 |
> |
similar values for shear viscosities to the momentum swapping |
| 473 |
> |
approach, as well as the velocity gradient profiles. Moderately larger |
| 474 |
> |
momentum fluxes are helpful to reduce the errors of measured velocity |
| 475 |
> |
gradients and thus the final result. However, it is pointed out that |
| 476 |
> |
the momentum swapping approach tends to produce nonthermal velocity |
| 477 |
> |
distributions.\cite{Maginn:2010} |
| 478 |
|
|
| 479 |
< |
\subsubsection{Are electronic excitations in the metal important?} |
| 480 |
< |
Because they lack electronic excitations, the QSC and related embedded |
| 481 |
< |
atom method (EAM) models for gold are known to predict unreasonably |
| 482 |
< |
low values for bulk conductivity |
| 483 |
< |
($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the |
| 484 |
< |
conductance between the phases ($G$) is governed primarily by phonon |
| 485 |
< |
excitation (and not electronic degrees of freedom), one would expect a |
| 486 |
< |
classical model to capture most of the interfacial thermal |
| 487 |
< |
conductance. Our results for $G$ and $G^\prime$ indicate that this is |
| 610 |
< |
indeed the case, and suggest that the modeling of interfacial thermal |
| 611 |
< |
transport depends primarily on the description of the interactions |
| 612 |
< |
between the various components at the interface. When the metal is |
| 613 |
< |
chemically capped, the primary barrier to thermal conductivity appears |
| 614 |
< |
to be the interface between the capping agent and the surrounding |
| 615 |
< |
solvent, so the excitations in the metal have little impact on the |
| 616 |
< |
value of $G$. |
| 479 |
> |
To examine that temperature isotropy holds in simulations using our |
| 480 |
> |
method, we measured the three one-dimensional temperatures in each of |
| 481 |
> |
the slabs (Figure \ref{tempXyz}). Note that the $x$-dimensional |
| 482 |
> |
temperatures were calculated after subtracting the effects from bulk |
| 483 |
> |
velocities of the slabs. The one-dimensional temperature profiles |
| 484 |
> |
showed no observable difference between the three dimensions. This |
| 485 |
> |
ensures that isotropic scaling automatically preserves temperature |
| 486 |
> |
isotropy and that our method is useful in shear viscosity |
| 487 |
> |
computations. |
| 488 |
|
|
| 489 |
< |
\subsection{Effects due to methodology and simulation parameters} |
| 489 |
> |
\begin{figure} |
| 490 |
> |
\includegraphics[width=\linewidth]{tempXyz} |
| 491 |
> |
\caption{Unlike the previous NIVS algorithm, the new method does not |
| 492 |
> |
produce a thermal anisotropy. No temperature difference between |
| 493 |
> |
different dimensions were observed beyond the magnitude of the error |
| 494 |
> |
bars. Note that the two ``hotter'' regions are caused by the shear |
| 495 |
> |
stress, as reported by Tenney and Maginn\cite{Maginn:2010}, but not |
| 496 |
> |
an effect that only observed in our methods.} |
| 497 |
> |
\label{tempXyz} |
| 498 |
> |
\end{figure} |
| 499 |
|
|
| 500 |
< |
We have varied the parameters of the simulations in order to |
| 501 |
< |
investigate how these factors would affect the computation of $G$. Of |
| 502 |
< |
particular interest are: 1) the length scale for the applied thermal |
| 503 |
< |
gradient (modified by increasing the amount of solvent in the system), |
| 504 |
< |
2) the sign and magnitude of the applied thermal flux, 3) the average |
| 505 |
< |
temperature of the simulation (which alters the solvent density during |
| 506 |
< |
equilibration), and 4) the definition of the interfacial conductance |
| 507 |
< |
(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the |
| 508 |
< |
calculation. |
| 500 |
> |
Furthermore, the velocity distribution profiles are tested by imposing |
| 501 |
> |
a large shear stress into the simulations. Figure \ref{vDist} |
| 502 |
> |
demonstrates how our method is able to maintain thermal velocity |
| 503 |
> |
distributions against the momentum swapping approach even under large |
| 504 |
> |
imposed fluxes. Previous swapping methods tend to deplete particles of |
| 505 |
> |
positive velocities in the negative velocity slab (``c'') and vice |
| 506 |
> |
versa in slab ``h'', where the distributions leave a notch. This |
| 507 |
> |
problematic profiles become significant when the imposed-flux becomes |
| 508 |
> |
larger and diffusions from neighboring slabs could not offset the |
| 509 |
> |
depletion. Simutaneously, abnormal peaks appear corresponding to |
| 510 |
> |
excessive velocity swapped from the other slab. This nonthermal |
| 511 |
> |
distributions limit applications of the swapping approach in shear |
| 512 |
> |
stress simulations. Our method avoids the above problematic |
| 513 |
> |
distributions by altering the means of applying momentum |
| 514 |
> |
fluxes. Comparatively, velocity distributions recorded from |
| 515 |
> |
simulations with our method is so close to the ideal thermal |
| 516 |
> |
prediction that no observable difference is shown in Figure |
| 517 |
> |
\ref{vDist}. Conclusively, our method avoids problems happened in |
| 518 |
> |
previous RNEMD methods and provides a useful means for shear viscosity |
| 519 |
> |
computations. |
| 520 |
|
|
| 521 |
< |
Systems of different lengths were prepared by altering the number of |
| 522 |
< |
solvent molecules and extending the length of the box along the $z$ |
| 523 |
< |
axis to accomodate the extra solvent. Equilibration at the same |
| 524 |
< |
temperature and pressure conditions led to nearly identical surface |
| 525 |
< |
areas ($L_x$ and $L_y$) available to the metal and capping agent, |
| 526 |
< |
while the extra solvent served mainly to lengthen the axis that was |
| 527 |
< |
used to apply the thermal flux. For a given value of the applied |
| 528 |
< |
flux, the different $z$ length scale has only a weak effect on the |
| 529 |
< |
computed conductivities. |
| 521 |
> |
\begin{figure} |
| 522 |
> |
\includegraphics[width=\linewidth]{velDist} |
| 523 |
> |
\caption{Velocity distributions that develop under the swapping and |
| 524 |
> |
our methods at high flux. These distributions were obtained from |
| 525 |
> |
Lennard-Jones simulations with $j_z^*(p_x)\sim 0.4$ (equivalent to a |
| 526 |
> |
swapping interval of 20 time steps). This is a relatively large flux |
| 527 |
> |
to demonstrate the nonthermal distributions that develop under the |
| 528 |
> |
swapping method. Distributions produced by our method are very close |
| 529 |
> |
to the ideal thermal situations.} |
| 530 |
> |
\label{vDist} |
| 531 |
> |
\end{figure} |
| 532 |
|
|
| 533 |
< |
\subsubsection{Effects of applied flux} |
| 534 |
< |
The NIVS algorithm allows changes in both the sign and magnitude of |
| 535 |
< |
the applied flux. It is possible to reverse the direction of heat |
| 536 |
< |
flow simply by changing the sign of the flux, and thermal gradients |
| 537 |
< |
which would be difficult to obtain experimentally ($5$ K/\AA) can be |
| 538 |
< |
easily simulated. However, the magnitude of the applied flux is not |
| 539 |
< |
arbitrary if one aims to obtain a stable and reliable thermal gradient. |
| 540 |
< |
A temperature gradient can be lost in the noise if $|J_z|$ is too |
| 541 |
< |
small, and excessive $|J_z|$ values can cause phase transitions if the |
| 542 |
< |
extremes of the simulation cell become widely separated in |
| 543 |
< |
temperature. Also, if $|J_z|$ is too large for the bulk conductivity |
| 651 |
< |
of the materials, the thermal gradient will never reach a stable |
| 652 |
< |
state. |
| 533 |
> |
\subsection{Bulk SPC/E water} |
| 534 |
> |
Since our method was in good performance of thermal conductivity and |
| 535 |
> |
shear viscosity computations for simple Lennard-Jones fluid, we extend |
| 536 |
> |
our applications of these simulations to complex fluid like SPC/E |
| 537 |
> |
water model. A simulation cell with 1000 molecules was set up in the |
| 538 |
> |
same manner as in \cite{kuang:164101}. For thermal conductivity |
| 539 |
> |
simulations, measurements were taken to compare with previous RNEMD |
| 540 |
> |
methods; for shear viscosity computations, simulations were run under |
| 541 |
> |
a series of temperatures (with corresponding pressure relaxation using |
| 542 |
> |
the isobaric-isothermal ensemble[CITE NIVS REF 32]), and results were |
| 543 |
> |
compared to available data from Equilibrium MD methods[CITATIONS]. |
| 544 |
|
|
| 545 |
< |
Within a reasonable range of $J_z$ values, we were able to study how |
| 546 |
< |
$G$ changes as a function of this flux. In what follows, we use |
| 547 |
< |
positive $J_z$ values to denote the case where energy is being |
| 548 |
< |
transferred by the method from the metal phase and into the liquid. |
| 549 |
< |
The resulting gradient therefore has a higher temperature in the |
| 550 |
< |
liquid phase. Negative flux values reverse this transfer, and result |
| 551 |
< |
in higher temperature metal phases. The conductance measured under |
| 552 |
< |
different applied $J_z$ values is listed in Tables 2 and 3 in the |
| 662 |
< |
supporting information. These results do not indicate that $G$ depends |
| 663 |
< |
strongly on $J_z$ within this flux range. The linear response of flux |
| 664 |
< |
to thermal gradient simplifies our investigations in that we can rely |
| 665 |
< |
on $G$ measurement with only a small number $J_z$ values. |
| 545 |
> |
\subsubsection{Thermal conductivity} |
| 546 |
> |
Table \ref{spceThermal} summarizes our thermal conductivity |
| 547 |
> |
computations under different temperatures and thermal gradients, in |
| 548 |
> |
comparison to the previous NIVS results\cite{kuang:164101} and |
| 549 |
> |
experimental measurements\cite{WagnerKruse}. Note that no appreciable |
| 550 |
> |
drift of total system energy or temperature was observed when our |
| 551 |
> |
method is applied, which indicates that our algorithm conserves total |
| 552 |
> |
energy even for systems involving electrostatic interactions. |
| 553 |
|
|
| 554 |
< |
The sign of $J_z$ is a different matter, however, as this can alter |
| 555 |
< |
the temperature on the two sides of the interface. The average |
| 556 |
< |
temperature values reported are for the entire system, and not for the |
| 557 |
< |
liquid phase, so at a given $\langle T \rangle$, the system with |
| 558 |
< |
positive $J_z$ has a warmer liquid phase. This means that if the |
| 559 |
< |
liquid carries thermal energy via diffusive transport, {\it positive} |
| 560 |
< |
$J_z$ values will result in increased molecular motion on the liquid |
| 674 |
< |
side of the interface, and this will increase the measured |
| 675 |
< |
conductivity. |
| 554 |
> |
Measurements using our method established similar temperature |
| 555 |
> |
gradients to the previous NIVS method. Our simulation results are in |
| 556 |
> |
good agreement with those from previous simulations. And both methods |
| 557 |
> |
yield values in reasonable agreement with experimental |
| 558 |
> |
values. Simulations using moderately higher thermal gradient or those |
| 559 |
> |
with longer gradient axis ($z$) for measurement seem to have better |
| 560 |
> |
accuracy, from our results. |
| 561 |
|
|
| 562 |
< |
\subsubsection{Effects due to average temperature} |
| 562 |
> |
\begin{table*} |
| 563 |
> |
\begin{minipage}{\linewidth} |
| 564 |
> |
\begin{center} |
| 565 |
> |
|
| 566 |
> |
\caption{Thermal conductivity of SPC/E water under various |
| 567 |
> |
imposed thermal gradients. Uncertainties are indicated in |
| 568 |
> |
parentheses.} |
| 569 |
> |
|
| 570 |
> |
\begin{tabular}{ccccc} |
| 571 |
> |
\hline\hline |
| 572 |
> |
$\langle T\rangle$ & $\langle dT/dz\rangle$ & \multicolumn{3}{c} |
| 573 |
> |
{$\lambda (\mathrm{W m}^{-1} \mathrm{K}^{-1})$} \\ |
| 574 |
> |
(K) & (K/\AA) & This work & Previous NIVS\cite{kuang:164101} & |
| 575 |
> |
Experiment\cite{WagnerKruse} \\ |
| 576 |
> |
\hline |
| 577 |
> |
300 & 0.8 & 0.815(0.027) & 0.770(0.008) & 0.61 \\ |
| 578 |
> |
318 & 0.8 & 0.801(0.024) & 0.750(0.032) & 0.64 \\ |
| 579 |
> |
& 1.6 & 0.766(0.007) & 0.778(0.019) & \\ |
| 580 |
> |
& 0.8 & 0.786(0.009)\footnote{Simulation with $L_z$ |
| 581 |
> |
twice as long.} & & \\ |
| 582 |
> |
\hline\hline |
| 583 |
> |
\end{tabular} |
| 584 |
> |
\label{spceThermal} |
| 585 |
> |
\end{center} |
| 586 |
> |
\end{minipage} |
| 587 |
> |
\end{table*} |
| 588 |
|
|
| 589 |
< |
We also studied the effect of average system temperature on the |
| 590 |
< |
interfacial conductance. The simulations are first equilibrated in |
| 591 |
< |
the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to |
| 592 |
< |
predict a lower boiling point (and liquid state density) than |
| 593 |
< |
experiments. This lower-density liquid phase leads to reduced contact |
| 594 |
< |
between the hexane and butanethiol, and this accounts for our |
| 595 |
< |
observation of lower conductance at higher temperatures. In raising |
| 596 |
< |
the average temperature from 200K to 250K, the density drop of |
| 597 |
< |
$\sim$20\% in the solvent phase leads to a $\sim$40\% drop in the |
| 598 |
< |
conductance. |
| 589 |
> |
\subsubsection{Shear viscosity} |
| 590 |
> |
The improvement our method achieves for shear viscosity computations |
| 591 |
> |
enables us to apply it on SPC/E water models. The series of |
| 592 |
> |
temperatures under which our shear viscosity calculations were carried |
| 593 |
> |
out covers the liquid range under normal pressure. Our simulations |
| 594 |
> |
predict a similar trend of $\eta$ vs. $T$ to EMD results we refer to |
| 595 |
> |
(Table \ref{spceShear}). Considering subtlties such as temperature or |
| 596 |
> |
pressure/density errors in these two series of measurements, our |
| 597 |
> |
results show no significant difference from those with EMD |
| 598 |
> |
methods. Since each value reported using our method takes only one |
| 599 |
> |
single trajectory of simulation, instead of average from many |
| 600 |
> |
trajectories when using EMD, our method provides an effective means |
| 601 |
> |
for shear viscosity computations. |
| 602 |
|
|
| 603 |
< |
Similar behavior is observed in the TraPPE-UA model for toluene, |
| 604 |
< |
although this model has better agreement with the experimental |
| 605 |
< |
densities of toluene. The expansion of the toluene liquid phase is |
| 606 |
< |
not as significant as that of the hexane (8.3\% over 100K), and this |
| 607 |
< |
limits the effect to $\sim$20\% drop in thermal conductivity. |
| 603 |
> |
\begin{table*} |
| 604 |
> |
\begin{minipage}{\linewidth} |
| 605 |
> |
\begin{center} |
| 606 |
> |
|
| 607 |
> |
\caption{Computed shear viscosity of SPC/E water under different |
| 608 |
> |
temperatures. Results are compared to those obtained with EMD |
| 609 |
> |
method[CITATION]. Uncertainties are indicated in parentheses.} |
| 610 |
> |
|
| 611 |
> |
\begin{tabular}{cccc} |
| 612 |
> |
\hline\hline |
| 613 |
> |
$T$ & $\partial v_x/\partial z$ & \multicolumn{2}{c} |
| 614 |
> |
{$\eta (\mathrm{mPa}\cdot\mathrm{s})$} \\ |
| 615 |
> |
(K) & (10$^{10}$s$^{-1}$) & This work & Previous simulations[CITATION]\\ |
| 616 |
> |
\hline |
| 617 |
> |
273 & & 1.218(0.004) & \\ |
| 618 |
> |
& & 1.140(0.012) & \\ |
| 619 |
> |
303 & & 0.646(0.008) & \\ |
| 620 |
> |
318 & & 0.536(0.007) & \\ |
| 621 |
> |
& & 0.510(0.007) & \\ |
| 622 |
> |
& & & \\ |
| 623 |
> |
333 & & 0.428(0.002) & \\ |
| 624 |
> |
363 & & 0.279(0.014) & \\ |
| 625 |
> |
& & 0.306(0.001) & \\ |
| 626 |
> |
\hline\hline |
| 627 |
> |
\end{tabular} |
| 628 |
> |
\label{spceShear} |
| 629 |
> |
\end{center} |
| 630 |
> |
\end{minipage} |
| 631 |
> |
\end{table*} |
| 632 |
|
|
| 633 |
< |
Although we have not mapped out the behavior at a large number of |
| 634 |
< |
temperatures, is clear that there will be a strong temperature |
| 635 |
< |
dependence in the interfacial conductance when the physical properties |
| 636 |
< |
of one side of the interface (notably the density) change rapidly as a |
| 637 |
< |
function of temperature. |
| 633 |
> |
[MAY COMBINE JZPX AND JZKE TO RUN ONE JOB BUT GET ETA=F(T)] |
| 634 |
> |
[PUT RESULTS AND FIGURE HERE IF IT WORKS] |
| 635 |
> |
\subsection{Interfacial frictions and slip lengths} |
| 636 |
> |
An attractive aspect of our method is the ability to apply momentum |
| 637 |
> |
and/or thermal flux in nonhomogeneous systems, where molecules of |
| 638 |
> |
different identities (or phases) are segregated in different |
| 639 |
> |
regions. We have previously studied the interfacial thermal transport |
| 640 |
> |
of a series of metal gold-liquid |
| 641 |
> |
surfaces\cite{kuang:164101,kuang:AuThl}, and attemptions have been |
| 642 |
> |
made to investigate the relationship between this phenomenon and the |
| 643 |
> |
interfacial frictions. |
| 644 |
|
|
| 645 |
< |
Besides the lower interfacial thermal conductance, surfaces at |
| 646 |
< |
relatively high temperatures are susceptible to reconstructions, |
| 647 |
< |
particularly when butanethiols fully cover the Au(111) surface. These |
| 648 |
< |
reconstructions include surface Au atoms which migrate outward to the |
| 649 |
< |
S atom layer, and butanethiol molecules which embed into the surface |
| 650 |
< |
Au layer. The driving force for this behavior is the strong Au-S |
| 651 |
< |
interactions which are modeled here with a deep Lennard-Jones |
| 652 |
< |
potential. This phenomenon agrees with reconstructions that have been |
| 653 |
< |
experimentally |
| 654 |
< |
observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt |
| 655 |
< |
{\it et al.} kept their Au(111) slab rigid so that their simulations |
| 656 |
< |
could reach 300K without surface |
| 714 |
< |
reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions |
| 715 |
< |
blur the interface, the measurement of $G$ becomes more difficult to |
| 716 |
< |
conduct at higher temperatures. For this reason, most of our |
| 717 |
< |
measurements are undertaken at $\langle T\rangle\sim$200K where |
| 718 |
< |
reconstruction is minimized. |
| 719 |
< |
|
| 720 |
< |
However, when the surface is not completely covered by butanethiols, |
| 721 |
< |
the simulated system appears to be more resistent to the |
| 722 |
< |
reconstruction. Our Au / butanethiol / toluene system had the Au(111) |
| 723 |
< |
surfaces 90\% covered by butanethiols, but did not see this above |
| 724 |
< |
phenomena even at $\langle T\rangle\sim$300K. That said, we did |
| 725 |
< |
observe butanethiols migrating to neighboring three-fold sites during |
| 726 |
< |
a simulation. Since the interface persisted in these simulations, we |
| 727 |
< |
were able to obtain $G$'s for these interfaces even at a relatively |
| 728 |
< |
high temperature without being affected by surface reconstructions. |
| 645 |
> |
Table \ref{etaKappaDelta} includes these computations and previous |
| 646 |
> |
calculations of corresponding interfacial thermal conductance. For |
| 647 |
> |
bare Au(111) surfaces, slip boundary conditions were observed for both |
| 648 |
> |
organic and aqueous liquid phases, corresponding to previously |
| 649 |
> |
computed low interfacial thermal conductance. Instead, the butanethiol |
| 650 |
> |
covered Au(111) surface appeared to be sticky to the organic liquid |
| 651 |
> |
molecules in our simulations. We have reported conductance enhancement |
| 652 |
> |
effect for this surface capping agent,\cite{kuang:AuThl} and these |
| 653 |
> |
observations have a qualitative agreement with the thermal conductance |
| 654 |
> |
results. This agreement also supports discussions on the relationship |
| 655 |
> |
between surface wetting and slip effect and thermal conductance of the |
| 656 |
> |
interface.[CITE BARRAT, GARDE] |
| 657 |
|
|
| 658 |
< |
\section{Discussion} |
| 659 |
< |
[COMBINE W. RESULTS] |
| 660 |
< |
The primary result of this work is that the capping agent acts as an |
| 661 |
< |
efficient thermal coupler between solid and solvent phases. One of |
| 662 |
< |
the ways the capping agent can carry out this role is to down-shift |
| 663 |
< |
between the phonon vibrations in the solid (which carry the heat from |
| 664 |
< |
the gold) and the molecular vibrations in the liquid (which carry some |
| 665 |
< |
of the heat in the solvent). |
| 658 |
> |
\begin{table*} |
| 659 |
> |
\begin{minipage}{\linewidth} |
| 660 |
> |
\begin{center} |
| 661 |
> |
|
| 662 |
> |
\caption{Computed interfacial friction coefficient values for |
| 663 |
> |
interfaces with various components for liquid and solid |
| 664 |
> |
phase. Error estimates are indicated in parentheses.} |
| 665 |
> |
|
| 666 |
> |
\begin{tabular}{llcccccc} |
| 667 |
> |
\hline\hline |
| 668 |
> |
Solid & Liquid & $T$ & $j_z(p_x)$ & $\eta_{liquid}$ & $\kappa$ |
| 669 |
> |
& $\delta$ & $G$\footnote{References \cite{kuang:AuThl} and |
| 670 |
> |
\cite{kuang:164101}.} \\ |
| 671 |
> |
surface & molecules & K & MPa & mPa$\cdot$s & Pa$\cdot$s/m & nm |
| 672 |
> |
& MW/m$^2$/K \\ |
| 673 |
> |
\hline |
| 674 |
> |
Au(111) & hexane & 200 & 1.08 & 0.20() & 5.3$\times$10$^4$() & |
| 675 |
> |
3.7 & 46.5 \\ |
| 676 |
> |
& & & 2.15 & 0.14() & 5.3$\times$10$^4$() & |
| 677 |
> |
2.7 & \\ |
| 678 |
> |
Au-SC$_4$H$_9$ & hexane & 200 & 2.16 & 0.29() & $\infty$ & 0 & |
| 679 |
> |
131 \\ |
| 680 |
> |
& & & 5.39 & 0.32() & $\infty$ & 0 & |
| 681 |
> |
\\ |
| 682 |
> |
\hline |
| 683 |
> |
Au(111) & toluene & 200 & 1.08 & 0.72() & 1.?$\times$10$^5$() & |
| 684 |
> |
4.6 & 70.1 \\ |
| 685 |
> |
& & & 2.16 & 0.54() & 1.?$\times$10$^5$() & |
| 686 |
> |
4.9 & \\ |
| 687 |
> |
Au-SC$_4$H$_9$ & toluene & 200 & 5.39 & 0.98() & $\infty$ & 0 |
| 688 |
> |
& 187 \\ |
| 689 |
> |
& & & 10.8 & 0.99() & $\infty$ & 0 |
| 690 |
> |
& \\ |
| 691 |
> |
\hline |
| 692 |
> |
Au(111) & water & 300 & 1.08 & 0.40() & 1.9$\times$10$^4$() & |
| 693 |
> |
20.7 & 1.65 \\ |
| 694 |
> |
& & & 2.16 & 0.79() & 1.9$\times$10$^4$() & |
| 695 |
> |
41.9 & \\ |
| 696 |
> |
\hline |
| 697 |
> |
ice(basal) & water & 225 & 19.4 & 15.8() & $\infty$ & 0 & \\ |
| 698 |
> |
\hline\hline |
| 699 |
> |
\end{tabular} |
| 700 |
> |
\label{etaKappaDelta} |
| 701 |
> |
\end{center} |
| 702 |
> |
\end{minipage} |
| 703 |
> |
\end{table*} |
| 704 |
|
|
| 705 |
< |
To investigate the mechanism of interfacial thermal conductance, the |
| 706 |
< |
vibrational power spectrum was computed. Power spectra were taken for |
| 707 |
< |
individual components in different simulations. To obtain these |
| 708 |
< |
spectra, simulations were run after equilibration in the |
| 709 |
< |
microcanonical (NVE) ensemble and without a thermal |
| 710 |
< |
gradient. Snapshots of configurations were collected at a frequency |
| 711 |
< |
that is higher than that of the fastest vibrations occurring in the |
| 712 |
< |
simulations. With these configurations, the velocity auto-correlation |
| 713 |
< |
functions can be computed: |
| 748 |
< |
\begin{equation} |
| 749 |
< |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
| 750 |
< |
\label{vCorr} |
| 751 |
< |
\end{equation} |
| 752 |
< |
The power spectrum is constructed via a Fourier transform of the |
| 753 |
< |
symmetrized velocity autocorrelation function, |
| 754 |
< |
\begin{equation} |
| 755 |
< |
\hat{f}(\omega) = |
| 756 |
< |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
| 757 |
< |
\label{fourier} |
| 758 |
< |
\end{equation} |
| 705 |
> |
An interesting effect alongside the surface friction change is |
| 706 |
> |
observed on the shear viscosity of liquids in the regions close to the |
| 707 |
> |
solid surface. Note that $\eta$ measured near a ``slip'' surface tends |
| 708 |
> |
to be smaller than that near a ``stick'' surface. This suggests that |
| 709 |
> |
an interface could affect the dynamic properties on its neighbor |
| 710 |
> |
regions. It is known that diffusions of solid particles in liquid |
| 711 |
> |
phase is affected by their surface conditions (stick or slip |
| 712 |
> |
boundary).[CITE SCHMIDT AND SKINNER] Our observations could provide |
| 713 |
> |
support to this phenomenon. |
| 714 |
|
|
| 715 |
< |
\subsection{The role of specific vibrations} |
| 716 |
< |
The vibrational spectra for gold slabs in different environments are |
| 717 |
< |
shown as in Figure \ref{specAu}. Regardless of the presence of |
| 718 |
< |
solvent, the gold surfaces which are covered by butanethiol molecules |
| 719 |
< |
exhibit an additional peak observed at a frequency of |
| 720 |
< |
$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding |
| 721 |
< |
vibration. This vibration enables efficient thermal coupling of the |
| 722 |
< |
surface Au layer to the capping agents. Therefore, in our simulations, |
| 723 |
< |
the Au / S interfaces do not appear to be the primary barrier to |
| 724 |
< |
thermal transport when compared with the butanethiol / solvent |
| 770 |
< |
interfaces. This supports the results of Luo {\it et |
| 771 |
< |
al.}\cite{Luo20101}, who reported $G$ for Au-SAM junctions roughly |
| 772 |
< |
twice as large as what we have computed for the thiol-liquid |
| 773 |
< |
interfaces. |
| 715 |
> |
In addition to these previously studied interfaces, we attempt to |
| 716 |
> |
construct ice-water interfaces and the basal plane of ice lattice was |
| 717 |
> |
first studied. In contrast to the Au(111)/water interface, where the |
| 718 |
> |
friction coefficient is relatively small and large slip effect |
| 719 |
> |
presents, the ice/liquid water interface demonstrates strong |
| 720 |
> |
interactions and appears to be sticky. The supercooled liquid phase is |
| 721 |
> |
an order of magnitude viscous than measurements in previous |
| 722 |
> |
section. It would be of interst to investigate the effect of different |
| 723 |
> |
ice lattice planes (such as prism surface) on interfacial friction and |
| 724 |
> |
corresponding liquid viscosity. |
| 725 |
|
|
| 775 |
– |
\begin{figure} |
| 776 |
– |
\includegraphics[width=\linewidth]{vibration} |
| 777 |
– |
\caption{The vibrational power spectrum for thiol-capped gold has an |
| 778 |
– |
additional vibrational peak at $\sim $165cm$^{-1}$. Bare gold |
| 779 |
– |
surfaces (both with and without a solvent over-layer) are missing |
| 780 |
– |
this peak. A similar peak at $\sim $165cm$^{-1}$ also appears in |
| 781 |
– |
the vibrational power spectrum for the butanethiol capping agents.} |
| 782 |
– |
\label{specAu} |
| 783 |
– |
\end{figure} |
| 784 |
– |
|
| 785 |
– |
Also in this figure, we show the vibrational power spectrum for the |
| 786 |
– |
bound butanethiol molecules, which also exhibits the same |
| 787 |
– |
$\sim$165cm$^{-1}$ peak. |
| 788 |
– |
|
| 789 |
– |
\subsection{Overlap of power spectra} |
| 790 |
– |
A comparison of the results obtained from the two different organic |
| 791 |
– |
solvents can also provide useful information of the interfacial |
| 792 |
– |
thermal transport process. In particular, the vibrational overlap |
| 793 |
– |
between the butanethiol and the organic solvents suggests a highly |
| 794 |
– |
efficient thermal exchange between these components. Very high |
| 795 |
– |
thermal conductivity was observed when AA models were used and C-H |
| 796 |
– |
vibrations were treated classically. The presence of extra degrees of |
| 797 |
– |
freedom in the AA force field yields higher heat exchange rates |
| 798 |
– |
between the two phases and results in a much higher conductivity than |
| 799 |
– |
in the UA force field. The all-atom classical models include high |
| 800 |
– |
frequency modes which should be unpopulated at our relatively low |
| 801 |
– |
temperatures. This artifact is likely the cause of the high thermal |
| 802 |
– |
conductance in all-atom MD simulations. |
| 803 |
– |
|
| 804 |
– |
The similarity in the vibrational modes available to solvent and |
| 805 |
– |
capping agent can be reduced by deuterating one of the two components |
| 806 |
– |
(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols |
| 807 |
– |
are deuterated, one can observe a significantly lower $G$ and |
| 808 |
– |
$G^\prime$ values (Table \ref{modelTest}). |
| 809 |
– |
|
| 810 |
– |
\begin{figure} |
| 811 |
– |
\includegraphics[width=\linewidth]{aahxntln} |
| 812 |
– |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
| 813 |
– |
systems. When butanethiol is deuterated (lower left), its |
| 814 |
– |
vibrational overlap with hexane decreases significantly. Since |
| 815 |
– |
aromatic molecules and the butanethiol are vibrationally dissimilar, |
| 816 |
– |
the change is not as dramatic when toluene is the solvent (right).} |
| 817 |
– |
\label{aahxntln} |
| 818 |
– |
\end{figure} |
| 819 |
– |
|
| 820 |
– |
For the Au / butanethiol / toluene interfaces, having the AA |
| 821 |
– |
butanethiol deuterated did not yield a significant change in the |
| 822 |
– |
measured conductance. Compared to the C-H vibrational overlap between |
| 823 |
– |
hexane and butanethiol, both of which have alkyl chains, the overlap |
| 824 |
– |
between toluene and butanethiol is not as significant and thus does |
| 825 |
– |
not contribute as much to the heat exchange process. |
| 826 |
– |
|
| 827 |
– |
Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate |
| 828 |
– |
that the {\it intra}molecular heat transport due to alkylthiols is |
| 829 |
– |
highly efficient. Combining our observations with those of Zhang {\it |
| 830 |
– |
et al.}, it appears that butanethiol acts as a channel to expedite |
| 831 |
– |
heat flow from the gold surface and into the alkyl chain. The |
| 832 |
– |
vibrational coupling between the metal and the liquid phase can |
| 833 |
– |
therefore be enhanced with the presence of suitable capping agents. |
| 834 |
– |
|
| 835 |
– |
Deuterated models in the UA force field did not decouple the thermal |
| 836 |
– |
transport as well as in the AA force field. The UA models, even |
| 837 |
– |
though they have eliminated the high frequency C-H vibrational |
| 838 |
– |
overlap, still have significant overlap in the lower-frequency |
| 839 |
– |
portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating |
| 840 |
– |
the UA models did not decouple the low frequency region enough to |
| 841 |
– |
produce an observable difference for the results of $G$ (Table |
| 842 |
– |
\ref{modelTest}). |
| 843 |
– |
|
| 844 |
– |
\begin{figure} |
| 845 |
– |
\includegraphics[width=\linewidth]{uahxnua} |
| 846 |
– |
\caption{Vibrational power spectra for UA models for the butanethiol |
| 847 |
– |
and hexane solvent (upper panel) show the high degree of overlap |
| 848 |
– |
between these two molecules, particularly at lower frequencies. |
| 849 |
– |
Deuterating a UA model for the solvent (lower panel) does not |
| 850 |
– |
decouple the two spectra to the same degree as in the AA force |
| 851 |
– |
field (see Fig \ref{aahxntln}).} |
| 852 |
– |
\label{uahxnua} |
| 853 |
– |
\end{figure} |
| 854 |
– |
|
| 726 |
|
\section{Conclusions} |
| 727 |
< |
The NIVS algorithm has been applied to simulations of |
| 728 |
< |
butanethiol-capped Au(111) surfaces in the presence of organic |
| 729 |
< |
solvents. This algorithm allows the application of unphysical thermal |
| 730 |
< |
flux to transfer heat between the metal and the liquid phase. With the |
| 731 |
< |
flux applied, we were able to measure the corresponding thermal |
| 732 |
< |
gradients and to obtain interfacial thermal conductivities. Under |
| 733 |
< |
steady states, 2-3 ns trajectory simulations are sufficient for |
| 734 |
< |
computation of this quantity. |
| 727 |
> |
Our simulations demonstrate the validity of our method in RNEMD |
| 728 |
> |
computations of thermal conductivity and shear viscosity in atomic and |
| 729 |
> |
molecular liquids. Our method maintains thermal velocity distributions |
| 730 |
> |
and avoids thermal anisotropy in previous NIVS shear stress |
| 731 |
> |
simulations, as well as retains attractive features of previous RNEMD |
| 732 |
> |
methods. There is no {\it a priori} restrictions to the method to be |
| 733 |
> |
applied in various ensembles, so prospective applications to |
| 734 |
> |
extended-system methods are possible. |
| 735 |
|
|
| 736 |
< |
Our simulations have seen significant conductance enhancement in the |
| 737 |
< |
presence of capping agent, compared with the bare gold / liquid |
| 738 |
< |
interfaces. The vibrational coupling between the metal and the liquid |
| 739 |
< |
phase is enhanced by a chemically-bonded capping agent. Furthermore, |
| 740 |
< |
the coverage percentage of the capping agent plays an important role |
| 870 |
< |
in the interfacial thermal transport process. Moderately low coverages |
| 871 |
< |
allow higher contact between capping agent and solvent, and thus could |
| 872 |
< |
further enhance the heat transfer process, giving a non-monotonic |
| 873 |
< |
behavior of conductance with increasing coverage. |
| 736 |
> |
Furthermore, using this method, investigations can be carried out to |
| 737 |
> |
characterize interfacial interactions. Our method is capable of |
| 738 |
> |
effectively imposing both thermal and momentum flux accross an |
| 739 |
> |
interface and thus facilitates studies that relates dynamic property |
| 740 |
> |
measurements to the chemical details of an interface. |
| 741 |
|
|
| 742 |
< |
Our results, particularly using the UA models, agree well with |
| 743 |
< |
available experimental data. The AA models tend to overestimate the |
| 744 |
< |
interfacial thermal conductance in that the classically treated C-H |
| 745 |
< |
vibrations become too easily populated. Compared to the AA models, the |
| 746 |
< |
UA models have higher computational efficiency with satisfactory |
| 747 |
< |
accuracy, and thus are preferable in modeling interfacial thermal |
| 881 |
< |
transport. |
| 742 |
> |
Another attractive feature of our method is the ability of |
| 743 |
> |
simultaneously imposing thermal and momentum flux in a |
| 744 |
> |
system. potential researches that might be benefit include complex |
| 745 |
> |
systems that involve thermal and momentum gradients. For example, the |
| 746 |
> |
Soret effects under a velocity gradient would be of interest to |
| 747 |
> |
purification and separation researches. |
| 748 |
|
|
| 883 |
– |
Of the two definitions for $G$, the discrete form |
| 884 |
– |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
| 885 |
– |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
| 886 |
– |
is not as versatile. Although $G^\prime$ gives out comparable results |
| 887 |
– |
and follows similar trend with $G$ when measuring close to fully |
| 888 |
– |
covered or bare surfaces, the spatial resolution of $T$ profile |
| 889 |
– |
required for the use of a derivative form is limited by the number of |
| 890 |
– |
bins and the sampling required to obtain thermal gradient information. |
| 891 |
– |
|
| 892 |
– |
Vlugt {\it et al.} have investigated the surface thiol structures for |
| 893 |
– |
nanocrystalline gold and pointed out that they differ from those of |
| 894 |
– |
the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This |
| 895 |
– |
difference could also cause differences in the interfacial thermal |
| 896 |
– |
transport behavior. To investigate this problem, one would need an |
| 897 |
– |
effective method for applying thermal gradients in non-planar |
| 898 |
– |
(i.e. spherical) geometries. |
| 899 |
– |
|
| 749 |
|
\section{Acknowledgments} |
| 750 |
|
Support for this project was provided by the National Science |
| 751 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
| 758 |
|
|
| 759 |
|
\end{doublespace} |
| 760 |
|
\end{document} |
| 912 |
– |
|
