| 131 |
|
properties. Different models were used for both the capping agent and |
| 132 |
|
the solvent force field parameters. Using the NIVS algorithm, the |
| 133 |
|
thermal transport across these interfaces was studied and the |
| 134 |
< |
underlying mechanism for this phenomena was investigated. |
| 134 |
> |
underlying mechanism for the phenomena was investigated. |
| 135 |
|
|
| 136 |
|
[MAY ADD WHY STUDY AU-THIOL SURFACE; CITE SHAOYI JIANG] |
| 137 |
|
|
| 138 |
|
\section{Methodology} |
| 139 |
|
\subsection{Imposd-Flux Methods in MD Simulations} |
| 140 |
+ |
[CF. CAHILL] |
| 141 |
|
For systems with low interfacial conductivity one must have a method |
| 142 |
|
capable of generating relatively small fluxes, compared to those |
| 143 |
|
required for bulk conductivity. This requirement makes the calculation |
| 173 |
|
momenta and energy and does not depend on an external thermostat. |
| 174 |
|
|
| 175 |
|
\subsection{Defining Interfacial Thermal Conductivity $G$} |
| 176 |
< |
For interfaces with a relatively low interfacial conductance, the bulk |
| 177 |
< |
regions on either side of an interface rapidly come to a state in |
| 178 |
< |
which the two phases have relatively homogeneous (but distinct) |
| 179 |
< |
temperatures. The interfacial thermal conductivity $G$ can therefore |
| 180 |
< |
be approximated as: |
| 176 |
> |
Given a system with thermal gradients and the corresponding thermal |
| 177 |
> |
flux, for interfaces with a relatively low interfacial conductance, |
| 178 |
> |
the bulk regions on either side of an interface rapidly come to a |
| 179 |
> |
state in which the two phases have relatively homogeneous (but |
| 180 |
> |
distinct) temperatures. The interfacial thermal conductivity $G$ can |
| 181 |
> |
therefore be approximated as: |
| 182 |
|
\begin{equation} |
| 183 |
|
G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - |
| 184 |
|
\langle T_\mathrm{cold}\rangle \right)} |
| 248 |
|
the unphysical flux, we are able to obtain a temperature profile and |
| 249 |
|
its spatial derivatives. These quantities enable the evaluation of the |
| 250 |
|
interfacial thermal conductance of a surface. Figure \ref{gradT} is an |
| 251 |
< |
example how those applied thermal fluxes can be used to obtain the 1st |
| 251 |
> |
example of how an applied thermal flux can be used to obtain the 1st |
| 252 |
|
and 2nd derivatives of the temperature profile. |
| 253 |
|
|
| 254 |
|
\begin{figure} |
| 264 |
|
\subsection{Simulation Protocol} |
| 265 |
|
The NIVS algorithm has been implemented in our MD simulation code, |
| 266 |
|
OpenMD\cite{Meineke:2005gd,openmd}, and was used for our |
| 267 |
< |
simulations. Different metal slab thickness (layer numbers of Au) were |
| 267 |
> |
simulations. Different metal slab thickness (layer numbers of Au) was |
| 268 |
|
simulated. Metal slabs were first equilibrated under atmospheric |
| 269 |
|
pressure (1 atm) and a desired temperature (e.g. 200K). After |
| 270 |
|
equilibration, butanethiol capping agents were placed at three-fold |
| 271 |
< |
sites on the Au(111) surfaces. The maximum butanethiol capacity on Au |
| 272 |
< |
surface is $1/3$ of the total number of surface Au |
| 273 |
< |
atoms\cite{vlugt:cpc2007154}[CITE CHEM REV]. |
| 274 |
< |
A series of different coverages was |
| 275 |
< |
investigated in order to study the relation between coverage and |
| 276 |
< |
interfacial conductance. |
| 271 |
> |
hollow sites on the Au(111) surfaces. These sites could be either a |
| 272 |
> |
{\it fcc} or {\it hcp} site. However, Hase {\it et al.} found that |
| 273 |
> |
they are equivalent in a heat transfer process\cite{hase:2010}, so |
| 274 |
> |
they are not distinguished in our study. The maximum butanethiol |
| 275 |
> |
capacity on Au surface is $1/3$ of the total number of surface Au |
| 276 |
> |
atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ |
| 277 |
> |
structure[CITE PORTER]. |
| 278 |
> |
A series of different coverages was derived by evenly eliminating |
| 279 |
> |
butanethiols on the surfaces, and was investigated in order to study |
| 280 |
> |
the relation between coverage and interfacial conductance. |
| 281 |
|
|
| 282 |
|
The capping agent molecules were allowed to migrate during the |
| 283 |
|
simulations. They distributed themselves uniformly and sampled a |
| 303 |
|
solvent molecules would change the normal behavior of the liquid |
| 304 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
| 305 |
|
these extreme cases did not happen to our simulations. And the |
| 306 |
< |
corresponding spacing is usually $35[?] \sim 75$\AA. |
| 306 |
> |
corresponding spacing is usually $35[DOUBLE CHECK] \sim 75$\AA. |
| 307 |
|
|
| 308 |
|
The initial configurations generated are further equilibrated with the |
| 309 |
|
$x$ and $y$ dimensions fixed, only allowing length scale change in $z$ |
| 310 |
|
dimension. This is to ensure that the equilibration of liquid phase |
| 311 |
|
does not affect the metal crystal structure in $x$ and $y$ dimensions. |
| 312 |
|
To investigate this effect, comparisons were made with simulations |
| 313 |
< |
allowed to change $L_x$ and $L_y$ during NPT equilibration, and the |
| 314 |
< |
results are shown in later sections. After ensuring the liquid phase |
| 315 |
< |
reaches equilibrium at atmospheric pressure (1 atm), further |
| 313 |
> |
that allow changes of $L_x$ and $L_y$ during NPT equilibration, and |
| 314 |
> |
the results are shown in later sections. After ensuring the liquid |
| 315 |
> |
phase reaches equilibrium at atmospheric pressure (1 atm), further |
| 316 |
|
equilibration are followed under NVT and then NVE ensembles. |
| 317 |
|
|
| 318 |
|
After the systems reach equilibrium, NIVS is implemented to impose a |
| 320 |
|
phase. Most of our simulations are under an average temperature of |
| 321 |
|
$\sim$200K. Therefore, this flux usually comes from the metal to the |
| 322 |
|
liquid so that the liquid has a higher temperature and would not |
| 323 |
< |
freeze due to excessively low temperature. This induced temperature |
| 324 |
< |
gradient is stablized and the simulation cell is devided evenly into |
| 325 |
< |
N slabs along the $z$-axis and the temperatures of each slab are |
| 326 |
< |
recorded. When the slab width $d$ of each slab is the same, the |
| 327 |
< |
derivatives of $T$ with respect to slab number $n$ can be directly |
| 328 |
< |
used for $G^\prime$ calculations: |
| 323 |
> |
freeze due to excessively low temperature. After this induced |
| 324 |
> |
temperature gradient is stablized, the temperature profile of the |
| 325 |
> |
simulation cell is recorded. To do this, the simulation cell is |
| 326 |
> |
devided evenly into $N$ slabs along the $z$-axis and $N$ is maximized |
| 327 |
> |
for highest possible spatial resolution but not too many to have some |
| 328 |
> |
slabs empty most of the time. The average temperatures of each slab |
| 329 |
> |
are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is |
| 330 |
> |
the same, the derivatives of $T$ with respect to slab number $n$ can |
| 331 |
> |
be directly used for $G^\prime$ calculations: |
| 332 |
|
\begin{equation} |
| 333 |
|
G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| |
| 334 |
|
\Big/\left(\frac{\partial T}{\partial z}\right)^2 |
| 339 |
|
\label{derivativeG2} |
| 340 |
|
\end{equation} |
| 341 |
|
|
| 342 |
+ |
All of the above simulation procedures use a time step of 1 fs. And |
| 343 |
+ |
each equilibration / stabilization step usually takes 100 ps, or |
| 344 |
+ |
longer, if necessary. |
| 345 |
+ |
|
| 346 |
|
\subsection{Force Field Parameters} |
| 347 |
|
Our simulations include various components. Figure \ref{demoMol} |
| 348 |
|
demonstrates the sites defined for both United-Atom and All-Atom |
| 377 |
|
the carbon centers for alkyl groups. Bonding interactions, including |
| 378 |
|
bond stretches and bends and torsions, were used for intra-molecular |
| 379 |
|
sites not separated by more than 3 bonds. Otherwise, for non-bonded |
| 380 |
< |
interactions, Lennard-Jones potentials are used. [MORE CITATION?] |
| 380 |
> |
interactions, Lennard-Jones potentials are used. [CHECK CITATION] |
| 381 |
|
|
| 382 |
|
By eliminating explicit hydrogen atoms, these models are simple and |
| 383 |
|
computationally efficient, while maintains good accuracy. However, the |
| 384 |
|
TraPPE-UA for alkanes is known to predict a lower boiling point than |
| 385 |
|
experimental values. Considering that after an unphysical thermal flux |
| 386 |
|
is applied to a system, the temperature of ``hot'' area in the liquid |
| 387 |
< |
phase would be significantly higher than the average, to prevent over |
| 388 |
< |
heating and boiling of the liquid phase, the average temperature in |
| 389 |
< |
our simulations should be much lower than the liquid boiling point. |
| 387 |
> |
phase would be significantly higher than the average of the system, to |
| 388 |
> |
prevent over heating and boiling of the liquid phase, the average |
| 389 |
> |
temperature in our simulations should be much lower than the liquid |
| 390 |
> |
boiling point. |
| 391 |
|
|
| 392 |
|
For UA-toluene model, the non-bonded potentials between |
| 393 |
|
inter-molecular sites have a similar Lennard-Jones formulation. For |
| 406 |
|
adopted. Without the rigid body constraints, bonding interactions were |
| 407 |
|
included. For the aromatic ring, improper torsions (inversions) were |
| 408 |
|
added as an extra potential for maintaining the planar shape. |
| 409 |
< |
[MORE CITATION?] |
| 409 |
> |
[CHECK CITATION] |
| 410 |
|
|
| 411 |
|
The capping agent in our simulations, the butanethiol molecules can |
| 412 |
|
either use UA or AA model. The TraPPE-UA force fields includes |
| 416 |
|
surfaces do not have the hydrogen atom bonded to sulfur. To adapt this |
| 417 |
|
change and derive suitable parameters for butanethiol adsorbed on |
| 418 |
|
Au(111) surfaces, we adopt the S parameters from Luedtke and |
| 419 |
< |
Landman\cite{landman:1998} and modify parameters for its neighbor C |
| 419 |
> |
Landman\cite{landman:1998}[CHECK CITATION] |
| 420 |
> |
and modify parameters for its neighbor C |
| 421 |
|
atom for charge balance in the molecule. Note that the model choice |
| 422 |
|
(UA or AA) of capping agent can be different from the |
| 423 |
|
solvent. Regardless of model choice, the force field parameters for |
| 492 |
|
\end{table*} |
| 493 |
|
|
| 494 |
|
\subsection{Vibrational Spectrum} |
| 495 |
< |
|
| 496 |
< |
[MAY ADD EQN'S] |
| 497 |
< |
To obtain these |
| 498 |
< |
spectra, one first runs a simulation in the NVE ensemble and collects |
| 499 |
< |
snapshots of configurations; these configurations are used to compute |
| 500 |
< |
the velocity auto-correlation functions, which is used to construct a |
| 501 |
< |
power spectrum via a Fourier transform. |
| 495 |
> |
To investigate the mechanism of interfacial thermal conductance, the |
| 496 |
> |
vibrational spectrum is utilized as a complementary tool. Vibrational |
| 497 |
> |
spectra were taken for individual components in different |
| 498 |
> |
simulations. To obtain these spectra, simulations were run after |
| 499 |
> |
equilibration, in the NVE ensemble. Snapshots of configurations were |
| 500 |
> |
collected at a frequency that is higher than that of the fastest |
| 501 |
> |
vibrations occuring in the simulations. With these configurations, the |
| 502 |
> |
velocity auto-correlation functions can be computed: |
| 503 |
> |
\begin{equation} |
| 504 |
> |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
| 505 |
> |
\label{vCorr} |
| 506 |
> |
\end{equation} |
| 507 |
|
|
| 508 |
+ |
Followed by Fourier transforms, the power spectrum can be constructed: |
| 509 |
+ |
\begin{equation} |
| 510 |
+ |
\hat{f}(\omega) = \int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
| 511 |
+ |
\label{fourier} |
| 512 |
+ |
\end{equation} |
| 513 |
+ |
|
| 514 |
|
\section{Results and Discussions} |
| 515 |
< |
[MAY HAVE A BRIEF SUMMARY] |
| 515 |
> |
In what follows, how the parameters and protocol of simulations would |
| 516 |
> |
affect the measurement of $G$'s is first discussed. With a reliable |
| 517 |
> |
protocol and set of parameters, the influence of capping agent |
| 518 |
> |
coverage on thermal conductance is investigated. Besides, different |
| 519 |
> |
force field models for both solvents and selected deuterated models |
| 520 |
> |
were tested and compared. Finally, a summary of the role of capping |
| 521 |
> |
agent in the interfacial thermal transport process is given. |
| 522 |
> |
|
| 523 |
|
\subsection{How Simulation Parameters Affects $G$} |
| 491 |
– |
[MAY NOT PUT AT FIRST] |
| 524 |
|
We have varied our protocol or other parameters of the simulations in |
| 525 |
|
order to investigate how these factors would affect the measurement of |
| 526 |
|
$G$'s. It turned out that while some of these parameters would not |
| 856 |
|
butanethiols are deuterated, one can see a significantly lower $G$ and |
| 857 |
|
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
| 858 |
|
between the solvent and the capping agent is removed. |
| 859 |
< |
[MAY NEED SPECTRA FIGURE] Conclusively, the |
| 859 |
> |
[ NEED SPECTRA FIGURE] Conclusively, the |
| 860 |
|
improperly treated C-H vibration in the AA model produced |
| 861 |
|
over-predicted results accordingly. Compared to the AA model, the UA |
| 862 |
|
model yields more reasonable results with higher computational |
| 881 |
|
occupying the interfaces. |
| 882 |
|
|
| 883 |
|
\subsection{Role of Capping Agent in Interfacial Thermal Conductance} |
| 884 |
< |
To investigate the mechanism of this interfacial thermal conductance, |
| 885 |
< |
the vibrational spectra of various gold systems were obtained and are |
| 886 |
< |
shown as in Fig. \ref{vibration}. |
| 887 |
< |
[MAY RELATE TO HASE'S] |
| 888 |
< |
The gold surfaces covered by butanethiol molecules, compared to bare |
| 889 |
< |
gold surfaces, exhibit an additional peak observed at the frequency of |
| 890 |
< |
$\sim$170cm$^{-1}$, which is attributed to the S-Au bonding |
| 891 |
< |
vibration. This vibration enables efficient thermal transport from |
| 892 |
< |
surface Au layer to the capping agents. |
| 861 |
< |
[MAY PUT IN OTHER SECTION] Simultaneously, as shown in |
| 862 |
< |
the lower panel of Fig. \ref{vibration}, the large overlap of the |
| 863 |
< |
vibration spectra of butanethiol and hexane in the All-Atom model, |
| 864 |
< |
including the C-H vibration, also suggests high thermal exchange |
| 865 |
< |
efficiency. The combination of these two effects produces the drastic |
| 866 |
< |
interfacial thermal conductance enhancement in the All-Atom model. |
| 884 |
> |
The vibrational spectra for gold slabs in different environments are |
| 885 |
> |
shown as in Figure \ref{specAu}. Regardless of the presence of |
| 886 |
> |
solvent, the gold surfaces covered by butanethiol molecules, compared |
| 887 |
> |
to bare gold surfaces, exhibit an additional peak observed at the |
| 888 |
> |
frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au |
| 889 |
> |
bonding vibration. This vibration enables efficient thermal transport |
| 890 |
> |
from surface Au layer to the capping agents. Therefore, in our |
| 891 |
> |
simulations, the Au/S interfaces do not appear major heat barriers |
| 892 |
> |
compared to the butanethiol / solvent interfaces. |
| 893 |
|
|
| 894 |
< |
[NEED SEPARATE FIGURE. MAY NEED TO CONVERT TO JPEG] |
| 894 |
> |
Simultaneously, the vibrational overlap between butanethiol and |
| 895 |
> |
organic solvents suggests higher thermal exchange efficiency between |
| 896 |
> |
these two components. Even exessively high heat transport was observed |
| 897 |
> |
when All-Atom models were used and C-H vibrations were treated |
| 898 |
> |
classically. Compared to metal and organic liquid phase, the heat |
| 899 |
> |
transfer efficiency between butanethiol and organic solvents is closer |
| 900 |
> |
to that within bulk liquid phase. |
| 901 |
> |
|
| 902 |
> |
As a combinational effects of the above two, butanethiol acts as a |
| 903 |
> |
channel to expedite thermal transport process. The acoustic impedance |
| 904 |
> |
mismatch between the metal and the liquid phase can be effectively |
| 905 |
> |
reduced with the presence of suitable capping agents. |
| 906 |
> |
|
| 907 |
|
\begin{figure} |
| 908 |
|
\includegraphics[width=\linewidth]{vibration} |
| 909 |
|
\caption{Vibrational spectra obtained for gold in different |
| 910 |
|
environments.} |
| 911 |
< |
\label{vibration} |
| 911 |
> |
\label{specAu} |
| 912 |
|
\end{figure} |
| 913 |
|
|
| 914 |
< |
[MAY ADD COMPARISON OF G AND G', AU SLAB WIDTHS, ETC] |
| 877 |
< |
% The results show that the two definitions used for $G$ yield |
| 878 |
< |
% comparable values, though $G^\prime$ tends to be smaller. |
| 914 |
> |
[MAY ADD COMPARISON OF AU SLAB WIDTHS] |
| 915 |
|
|
| 916 |
|
\section{Conclusions} |
| 917 |
|
The NIVS algorithm we developed has been applied to simulations of |
| 919 |
|
effective unphysical thermal flux transferred between the metal and |
| 920 |
|
the liquid phase. With the flux applied, we were able to measure the |
| 921 |
|
corresponding thermal gradient and to obtain interfacial thermal |
| 922 |
< |
conductivities. Our simulations have seen significant conductance |
| 923 |
< |
enhancement with the presence of capping agent, compared to the bare |
| 924 |
< |
gold / liquid interfaces. The acoustic impedance mismatch between the |
| 925 |
< |
metal and the liquid phase is effectively eliminated by proper capping |
| 922 |
> |
conductivities. Under steady states, single trajectory simulation |
| 923 |
> |
would be enough for accurate measurement. This would be advantageous |
| 924 |
> |
compared to transient state simulations, which need multiple |
| 925 |
> |
trajectories to produce reliable average results. |
| 926 |
> |
|
| 927 |
> |
Our simulations have seen significant conductance enhancement with the |
| 928 |
> |
presence of capping agent, compared to the bare gold / liquid |
| 929 |
> |
interfaces. The acoustic impedance mismatch between the metal and the |
| 930 |
> |
liquid phase is effectively eliminated by proper capping |
| 931 |
|
agent. Furthermore, the coverage precentage of the capping agent plays |
| 932 |
< |
an important role in the interfacial thermal transport process. |
| 932 |
> |
an important role in the interfacial thermal transport |
| 933 |
> |
process. Moderately lower coverages allow higher contact between |
| 934 |
> |
capping agent and solvent, and thus could further enhance the heat |
| 935 |
> |
transfer process. |
| 936 |
|
|
| 937 |
|
Our measurement results, particularly of the UA models, agree with |
| 938 |
|
available experimental data. This indicates that our force field |
| 942 |
|
vibration would be overly sampled. Compared to the AA models, the UA |
| 943 |
|
models have higher computational efficiency with satisfactory |
| 944 |
|
accuracy, and thus are preferable in interfacial thermal transport |
| 945 |
< |
modelings. |
| 945 |
> |
modelings. Of the two definitions for $G$, the discrete form |
| 946 |
> |
(Eq. \ref{discreteG}) was easier to use and gives out relatively |
| 947 |
> |
consistent results, while the derivative form (Eq. \ref{derivativeG}) |
| 948 |
> |
is not as versatile. Although $G^\prime$ gives out comparable results |
| 949 |
> |
and follows similar trend with $G$ when measuring close to fully |
| 950 |
> |
covered or bare surfaces, the spatial resolution of $T$ profile is |
| 951 |
> |
limited for accurate computation of derivatives data. |
| 952 |
|
|
| 953 |
|
Vlugt {\it et al.} has investigated the surface thiol structures for |
| 954 |
|
nanocrystal gold and pointed out that they differs from those of the |
| 958 |
|
and measure the corresponding thermal gradient is desirable for |
| 959 |
|
simulating structures with spherical symmetry. |
| 960 |
|
|
| 911 |
– |
|
| 961 |
|
\section{Acknowledgments} |
| 962 |
|
Support for this project was provided by the National Science |
| 963 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
