| 222 |
|
|
| 223 |
|
To compare the above definitions ($G$ and $G^\prime$), we have modeled |
| 224 |
|
a metal slab with its (111) surfaces perpendicular to the $z$-axis of |
| 225 |
< |
our simulation cells. Both with and withour capping agents on the |
| 225 |
> |
our simulation cells. Both with and without capping agents on the |
| 226 |
|
surfaces, the metal slab is solvated with simple organic solvents, as |
| 227 |
|
illustrated in Figure \ref{demoPic}. |
| 228 |
|
|
| 325 |
|
\end{equation} |
| 326 |
|
|
| 327 |
|
\subsection{Force Field Parameters} |
| 328 |
< |
Our simulations include various components. Therefore, force field |
| 329 |
< |
parameter descriptions are needed for interactions both between the |
| 330 |
< |
same type of particles and between particles of different species. |
| 331 |
< |
|
| 332 |
< |
The Au-Au interactions in metal lattice slab is described by the |
| 333 |
< |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
| 334 |
< |
potentials include zero-point quantum corrections and are |
| 335 |
< |
reparametrized for accurate surface energies compared to the |
| 336 |
< |
Sutton-Chen potentials\cite{Chen90}. |
| 337 |
< |
|
| 338 |
< |
Figure \ref{demoMol} demonstrates how we name our pseudo-atoms of the |
| 339 |
< |
organic solvent molecules in our simulations. |
| 328 |
> |
Our simulations include various components. Figure \ref{demoMol} |
| 329 |
> |
demonstrates the sites defined for both United-Atom and All-Atom |
| 330 |
> |
models of the organic solvent and capping agent molecules in our |
| 331 |
> |
simulations. Force field parameter descriptions are needed for |
| 332 |
> |
interactions both between the same type of particles and between |
| 333 |
> |
particles of different species. |
| 334 |
|
|
| 335 |
|
\begin{figure} |
| 336 |
|
\includegraphics[width=\linewidth]{structures} |
| 344 |
|
\label{demoMol} |
| 345 |
|
\end{figure} |
| 346 |
|
|
| 347 |
+ |
The Au-Au interactions in metal lattice slab is described by the |
| 348 |
+ |
quantum Sutton-Chen (QSC) formulation\cite{PhysRevB.59.3527}. The QSC |
| 349 |
+ |
potentials include zero-point quantum corrections and are |
| 350 |
+ |
reparametrized for accurate surface energies compared to the |
| 351 |
+ |
Sutton-Chen potentials\cite{Chen90}. |
| 352 |
+ |
|
| 353 |
|
For both solvent molecules, straight chain {\it n}-hexane and aromatic |
| 354 |
|
toluene, United-Atom (UA) and All-Atom (AA) models are used |
| 355 |
|
respectively. The TraPPE-UA |
| 356 |
|
parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used |
| 357 |
< |
for our UA solvent molecules. In these models, pseudo-atoms are |
| 358 |
< |
located at the carbon centers for alkyl groups. By eliminating |
| 359 |
< |
explicit hydrogen atoms, these models are simple and computationally |
| 360 |
< |
efficient, while maintains good accuracy. However, the TraPPE-UA for |
| 361 |
< |
alkanes is known to predict a lower boiling point than experimental |
| 362 |
< |
values. Considering that after an unphysical thermal flux is applied |
| 363 |
< |
to a system, the temperature of ``hot'' area in the liquid phase would be |
| 364 |
< |
significantly higher than the average, to prevent over heating and |
| 365 |
< |
boiling of the liquid phase, the average temperature in our |
| 366 |
< |
simulations should be much lower than the liquid boiling point. [MORE DISCUSSION] |
| 367 |
< |
For UA-toluene model, rigid body constraints are applied, so that the |
| 368 |
< |
benzene ring and the methyl-CRar bond are kept rigid. This would save |
| 369 |
< |
computational time.[MORE DETAILS] |
| 357 |
> |
for our UA solvent molecules. In these models, sites are located at |
| 358 |
> |
the carbon centers for alkyl groups. Bonding interactions, including |
| 359 |
> |
bond stretches and bends and torsions, were used for intra-molecular |
| 360 |
> |
sites not separated by more than 3 bonds. Otherwise, for non-bonded |
| 361 |
> |
interactions, Lennard-Jones potentials are used. [MORE CITATION?] |
| 362 |
> |
|
| 363 |
> |
By eliminating explicit hydrogen atoms, these models are simple and |
| 364 |
> |
computationally efficient, while maintains good accuracy. However, the |
| 365 |
> |
TraPPE-UA for alkanes is known to predict a lower boiling point than |
| 366 |
> |
experimental values. Considering that after an unphysical thermal flux |
| 367 |
> |
is applied to a system, the temperature of ``hot'' area in the liquid |
| 368 |
> |
phase would be significantly higher than the average, to prevent over |
| 369 |
> |
heating and boiling of the liquid phase, the average temperature in |
| 370 |
> |
our simulations should be much lower than the liquid boiling point. |
| 371 |
|
|
| 372 |
+ |
For UA-toluene model, the non-bonded potentials between |
| 373 |
+ |
inter-molecular sites have a similar Lennard-Jones formulation. For |
| 374 |
+ |
intra-molecular interactions, considering the stiffness of the benzene |
| 375 |
+ |
ring, rigid body constraints are applied for further computational |
| 376 |
+ |
efficiency. All bonds in the benzene ring and between the ring and the |
| 377 |
+ |
methyl group remain rigid during the progress of simulations. |
| 378 |
+ |
|
| 379 |
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
| 380 |
|
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
| 381 |
< |
force field is used. [MORE DETAILS] |
| 382 |
< |
For toluene, the United Force Field developed by Rapp\'{e} {\it et |
| 383 |
< |
al.}\cite{doi:10.1021/ja00051a040} is adopted.[MORE DETAILS] |
| 381 |
> |
force field is used. Additional explicit hydrogen sites were |
| 382 |
> |
included. Besides bonding and non-bonded site-site interactions, |
| 383 |
> |
partial charges and the electrostatic interactions were added to each |
| 384 |
> |
CT and HC site. For toluene, the United Force Field developed by |
| 385 |
> |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} is |
| 386 |
> |
adopted. Without the rigid body constraints, bonding interactions were |
| 387 |
> |
included. For the aromatic ring, improper torsions (inversions) were |
| 388 |
> |
added as an extra potential for maintaining the planar shape. |
| 389 |
> |
[MORE CITATIONS?] |
| 390 |
|
|
| 391 |
|
The capping agent in our simulations, the butanethiol molecules can |
| 392 |
|
either use UA or AA model. The TraPPE-UA force fields includes |
| 421 |
|
|
| 422 |
|
Besides, the potentials developed from {\it ab initio} calculations by |
| 423 |
|
Leng {\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the |
| 424 |
< |
interactions between Au and aromatic C/H atoms in toluene.[MORE DETAILS] |
| 424 |
> |
interactions between Au and aromatic C/H atoms in toluene. A set of |
| 425 |
> |
pseudo Lennard-Jones parameters were provided for Au in their force |
| 426 |
> |
fields. By using the Mixing Rule, this can be used to derive pair-wise |
| 427 |
> |
potentials for non-bonded interactions between Au and non-metal sites. |
| 428 |
|
|
| 429 |
|
However, the Lennard-Jones parameters between Au and other types of |
| 430 |
< |
particles in our simulations are not yet well-established. For these |
| 431 |
< |
interactions, we attempt to derive their parameters using the Mixing |
| 432 |
< |
Rule. To do this, the ``Metal-non-Metal'' (MnM) interaction parameters |
| 433 |
< |
for Au is first extracted from the Au-CH$_x$ parameters by applying |
| 434 |
< |
the Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
| 430 |
> |
particles, such as All-Atom normal alkanes in our simulations are not |
| 431 |
> |
yet well-established. For these interactions, we attempt to derive |
| 432 |
> |
their parameters using the Mixing Rule. To do this, Au pseudo |
| 433 |
> |
Lennard-Jones parameters for ``Metal-non-Metal'' (MnM) interactions |
| 434 |
> |
were first extracted from the Au-CH$_x$ parameters by applying the |
| 435 |
> |
Mixing Rule reversely. Table \ref{MnM} summarizes these ``MnM'' |
| 436 |
|
parameters in our simulations. |
| 437 |
|
|
| 438 |
|
\begin{table*} |
| 461 |
|
&CA & 3.55 & 0.070 & -0.115 & 3.173 & 0.0640 \\ |
| 462 |
|
&HA & 2.42 & 0.030 & 0.115 & 2.746 & 0.0414 \\ |
| 463 |
|
\hline |
| 464 |
< |
Both UA and AA & S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
| 464 |
> |
Both UA and AA |
| 465 |
> |
& S & 4.45 & 0.25 & - & 2.40 & 8.465 \\ |
| 466 |
|
\hline\hline |
| 467 |
|
\end{tabular} |
| 468 |
|
\label{MnM} |
| 483 |
|
results. |
| 484 |
|
|
| 485 |
|
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
| 486 |
< |
during equilibrating the liquid phase. Due to the stiffness of the Au |
| 487 |
< |
slab, $L_x$ and $L_y$ would not change noticeably after |
| 488 |
< |
equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system |
| 489 |
< |
is fully equilibrated in the NPT ensemble, this fluctuation, as well |
| 490 |
< |
as those comparably smaller to $L_x$ and $L_y$, would not be magnified |
| 491 |
< |
on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This |
| 492 |
< |
insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s |
| 493 |
< |
without the necessity of extremely cautious equilibration process. |
| 486 |
> |
during equilibrating the liquid phase. Due to the stiffness of the |
| 487 |
> |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
| 488 |
> |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
| 489 |
> |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
| 490 |
> |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
| 491 |
> |
would not be magnified on the calculated $G$'s, as shown in Table |
| 492 |
> |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
| 493 |
> |
reliable measurement of $G$'s without the necessity of extremely |
| 494 |
> |
cautious equilibration process. |
| 495 |
|
|
| 496 |
|
As stated in our computational details, the spacing filled with |
| 497 |
|
solvent molecules can be chosen within a range. This allows some |
| 518 |
|
the thermal flux across the interface. For our simulations, we denote |
| 519 |
|
$J_z$ to be positive when the physical thermal flux is from the liquid |
| 520 |
|
to metal, and negative vice versa. The $G$'s measured under different |
| 521 |
< |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and [REF]. These |
| 522 |
< |
results do not suggest that $G$ is dependent on $J_z$ within this flux |
| 523 |
< |
range. The linear response of flux to thermal gradient simplifies our |
| 524 |
< |
investigations in that we can rely on $G$ measurement with only a |
| 525 |
< |
couple $J_z$'s and do not need to test a large series of fluxes. |
| 521 |
> |
$J_z$ is listed in Table \ref{AuThiolHexaneUA} and |
| 522 |
> |
\ref{AuThiolToluene}. These results do not suggest that $G$ is |
| 523 |
> |
dependent on $J_z$ within this flux range. The linear response of flux |
| 524 |
> |
to thermal gradient simplifies our investigations in that we can rely |
| 525 |
> |
on $G$ measurement with only a couple $J_z$'s and do not need to test |
| 526 |
> |
a large series of fluxes. |
| 527 |
|
|
| 507 |
– |
[LOW FLUX, LARGE ERROR] |
| 528 |
|
\begin{table*} |
| 529 |
|
\begin{minipage}{\linewidth} |
| 530 |
|
\begin{center} |
| 573 |
|
temperature is higher than 250K. Additionally, the equilibrated liquid |
| 574 |
|
hexane density under 250K becomes lower than experimental value. This |
| 575 |
|
expanded liquid phase leads to lower contact between hexane and |
| 576 |
< |
butanethiol as well.[MAY NEED FIGURE] And this reduced contact would |
| 576 |
> |
butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] |
| 577 |
> |
And this reduced contact would |
| 578 |
|
probably be accountable for a lower interfacial thermal conductance, |
| 579 |
|
as shown in Table \ref{AuThiolHexaneUA}. |
| 580 |
|
|
| 640 |
|
|
| 641 |
|
However, when the surface is not completely covered by butanethiols, |
| 642 |
|
the simulated system is more resistent to the reconstruction |
| 643 |
< |
above. Our Au-butanethiol/toluene system did not see this phenomena |
| 644 |
< |
even at $\langle T\rangle\sim$300K. The Au(111) surfaces have a 90\% |
| 645 |
< |
coverage of butanethiols and have empty three-fold sites. These empty |
| 646 |
< |
sites could help prevent surface reconstruction in that they provide |
| 647 |
< |
other means of capping agent relaxation. It is observed that |
| 643 |
> |
above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% |
| 644 |
> |
covered by butanethiols, but did not see this above phenomena even at |
| 645 |
> |
$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by |
| 646 |
> |
capping agents could help prevent surface reconstruction in that they |
| 647 |
> |
provide other means of capping agent relaxation. It is observed that |
| 648 |
|
butanethiols can migrate to their neighbor empty sites during a |
| 649 |
|
simulation. Therefore, we were able to obtain $G$'s for these |
| 650 |
|
interfaces even at a relatively high temperature without being |
| 655 |
|
thermal conductance, a series of different coverage Au-butanethiol |
| 656 |
|
surfaces is prepared and solvated with various organic |
| 657 |
|
molecules. These systems are then equilibrated and their interfacial |
| 658 |
< |
thermal conductivity are measured with our NIVS algorithm. Table |
| 659 |
< |
\ref{tlnUhxnUhxnD} lists these results for direct comparison between |
| 660 |
< |
different coverages of butanethiol. To study the isotope effect in |
| 661 |
< |
interfacial thermal conductance, deuterated UA-hexane is included as |
| 662 |
< |
well. |
| 658 |
> |
thermal conductivity are measured with our NIVS algorithm. Figure |
| 659 |
> |
\ref{coverage} demonstrates the trend of conductance change with |
| 660 |
> |
respect to different coverages of butanethiol. To study the isotope |
| 661 |
> |
effect in interfacial thermal conductance, deuterated UA-hexane is |
| 662 |
> |
included as well. |
| 663 |
|
|
| 664 |
|
It turned out that with partial covered butanethiol on the Au(111) |
| 665 |
< |
surface, the derivative definition for $G$ (Eq. \ref{derivativeG}) has |
| 666 |
< |
difficulty to apply, due to the difficulty in locating the maximum of |
| 667 |
< |
change of $\lambda$. Instead, the discrete definition |
| 668 |
< |
(Eq. \ref{discreteG}) is easier to apply, as max($\Delta T$) can still |
| 669 |
< |
be well-defined. Therefore, $G$'s (not $G^\prime$) are used for this |
| 670 |
< |
section. |
| 665 |
> |
surface, the derivative definition for $G^\prime$ |
| 666 |
> |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
| 667 |
> |
in locating the maximum of change of $\lambda$. Instead, the discrete |
| 668 |
> |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
| 669 |
> |
deviding surface can still be well-defined. Therefore, $G$ (not |
| 670 |
> |
$G^\prime$) was used for this section. |
| 671 |
|
|
| 672 |
< |
From Table \ref{tlnUhxnUhxnD}, one can see the significance of the |
| 672 |
> |
From Figure \ref{coverage}, one can see the significance of the |
| 673 |
|
presence of capping agents. Even when a fraction of the Au(111) |
| 674 |
|
surface sites are covered with butanethiols, the conductivity would |
| 675 |
|
see an enhancement by at least a factor of 3. This indicates the |
| 676 |
|
important role cappping agent is playing for thermal transport |
| 677 |
< |
phenomena on metal/organic solvent surfaces. |
| 677 |
> |
phenomena on metal / organic solvent surfaces. |
| 678 |
|
|
| 679 |
|
Interestingly, as one could observe from our results, the maximum |
| 680 |
|
conductance enhancement (largest $G$) happens while the surfaces are |
| 693 |
|
would not offset this effect. Eventually, when butanethiol coverage |
| 694 |
|
continues to decrease, solvent-capping agent contact actually |
| 695 |
|
decreases with the disappearing of butanethiol molecules. In this |
| 696 |
< |
case, $G$ decrease could not be offset but instead accelerated. |
| 696 |
> |
case, $G$ decrease could not be offset but instead accelerated. [NEED |
| 697 |
> |
SNAPSHOT SHOWING THE PHENOMENA] |
| 698 |
|
|
| 699 |
|
A comparison of the results obtained from differenet organic solvents |
| 700 |
|
can also provide useful information of the interfacial thermal |
| 704 |
|
studies, even though eliminating C-H vibration samplings, still have |
| 705 |
|
C-C vibrational frequencies different from each other. However, these |
| 706 |
|
differences in the infrared range do not seem to produce an observable |
| 707 |
< |
difference for the results of $G$. [MAY NEED FIGURE] |
| 707 |
> |
difference for the results of $G$. [MAY NEED SPECTRA FIGURE] |
| 708 |
|
|
| 709 |
|
Furthermore, results for rigid body toluene solvent, as well as other |
| 710 |
|
UA-hexane solvents, are reasonable within the general experimental |
| 713 |
|
such as Au-thiol/organic solvent. |
| 714 |
|
|
| 715 |
|
However, results for Au-butanethiol/toluene do not show an identical |
| 716 |
< |
trend with those for Au-butanethiol/hexane in that $G$'s remain at |
| 716 |
> |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
| 717 |
|
approximately the same magnitue when butanethiol coverage differs from |
| 718 |
|
25\% to 75\%. This might be rooted in the molecule shape difference |
| 719 |
< |
for plane-like toluene and chain-like {\it n}-hexane. Due to this |
| 719 |
> |
for planar toluene and chain-like {\it n}-hexane. Due to this |
| 720 |
|
difference, toluene molecules have more difficulty in occupying |
| 721 |
|
relatively small gaps among capping agents when their coverage is not |
| 722 |
|
too low. Therefore, the solvent-capping agent contact may keep |
| 805 |
|
interfaces, using AA model for both butanethiol and hexane yields |
| 806 |
|
substantially higher conductivity values than using UA model for at |
| 807 |
|
least one component of the solvent and capping agent, which exceeds |
| 808 |
< |
the upper bond of experimental value range. This is probably due to |
| 809 |
< |
the classically treated C-H vibrations in the AA model, which should |
| 810 |
< |
not be appreciably populated at normal temperatures. In comparison, |
| 811 |
< |
once either the hexanes or the butanethiols are deuterated, one can |
| 812 |
< |
see a significantly lower $G$ and $G^\prime$. In either of these |
| 813 |
< |
cases, the C-H(D) vibrational overlap between the solvent and the |
| 814 |
< |
capping agent is removed. [MAY NEED FIGURE] Conclusively, the |
| 808 |
> |
the general range of experimental measurement results. This is |
| 809 |
> |
probably due to the classically treated C-H vibrations in the AA |
| 810 |
> |
model, which should not be appreciably populated at normal |
| 811 |
> |
temperatures. In comparison, once either the hexanes or the |
| 812 |
> |
butanethiols are deuterated, one can see a significantly lower $G$ and |
| 813 |
> |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
| 814 |
> |
between the solvent and the capping agent is removed. |
| 815 |
> |
[MAY NEED SPECTRA FIGURE] Conclusively, the |
| 816 |
|
improperly treated C-H vibration in the AA model produced |
| 817 |
|
over-predicted results accordingly. Compared to the AA model, the UA |
| 818 |
|
model yields more reasonable results with higher computational |
| 838 |
|
|
| 839 |
|
\subsection{Mechanism of Interfacial Thermal Conductance Enhancement |
| 840 |
|
by Capping Agent} |
| 841 |
< |
%OR\subsection{Vibrational spectrum study on conductance mechanism} |
| 841 |
> |
[OR: Vibrational Spectrum Study on Conductance Mechanism] |
| 842 |
|
|
| 843 |
|
[MAY INTRODUCE PROTOCOL IN METHODOLOGY/COMPUTATIONAL DETAIL, EQN'S] |
| 844 |
|
|
| 854 |
|
The gold surfaces covered by |
| 855 |
|
butanethiol molecules, compared to bare gold surfaces, exhibit an |
| 856 |
|
additional peak observed at a frequency of $\sim$170cm$^{-1}$, which |
| 857 |
< |
is attributed to the vibration of the S-Au bond. This vibration |
| 857 |
> |
is attributed to the vibration of the S-Au bonding. This vibration |
| 858 |
|
enables efficient thermal transport from surface Au atoms to the |
| 859 |
|
capping agents. Simultaneously, as shown in the lower panel of |
| 860 |
|
Fig. \ref{vibration}, the large overlap of the vibration spectra of |
| 872 |
|
\label{vibration} |
| 873 |
|
\end{figure} |
| 874 |
|
|
| 875 |
< |
[COMPARISON OF TWO G'S; AU SLAB WIDTHS; ETC] |
| 875 |
> |
[MAY ADD COMPARISON OF G AND G', AU SLAB WIDTHS, ETC] |
| 876 |
|
% The results show that the two definitions used for $G$ yield |
| 877 |
|
% comparable values, though $G^\prime$ tends to be smaller. |
| 878 |
|
|
| 884 |
|
corresponding thermal gradient and to obtain interfacial thermal |
| 885 |
|
conductivities. Our simulations have seen significant conductance |
| 886 |
|
enhancement with the presence of capping agent, compared to the bare |
| 887 |
< |
gold/liquid interfaces. The acoustic impedance mismatch between the |
| 887 |
> |
gold / liquid interfaces. The acoustic impedance mismatch between the |
| 888 |
|
metal and the liquid phase is effectively eliminated by proper capping |
| 889 |
|
agent. Furthermore, the coverage precentage of the capping agent plays |
| 890 |
|
an important role in the interfacial thermal transport process. |
