| 83 |
|
traditional methods developed for homogeneous systems. |
| 84 |
|
|
| 85 |
|
Experimentally, various interfaces have been investigated for their |
| 86 |
< |
thermal conductance. Wang {\it et al.} studied heat transport through |
| 86 |
> |
thermal conductance. Cahill and coworkers studied nanoscale thermal |
| 87 |
> |
transport from metal nanoparticle/fluid interfaces, to epitaxial |
| 88 |
> |
TiN/single crystal oxides interfaces, to hydrophilic and hydrophobic |
| 89 |
> |
interfaces between water and solids with different self-assembled |
| 90 |
> |
monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} |
| 91 |
> |
Wang {\it et al.} studied heat transport through |
| 92 |
|
long-chain hydrocarbon monolayers on gold substrate at individual |
| 93 |
|
molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the |
| 94 |
|
role of CTAB on thermal transport between gold nanorods and |
| 111 |
|
measurements for heat conductance of interfaces between the capping |
| 112 |
|
monolayer on Au and a solvent phase have yet to be studied with their |
| 113 |
|
approach. The comparatively low thermal flux through interfaces is |
| 114 |
< |
difficult to measure with Equilibrium MD or forward NEMD simulation |
| 114 |
> |
difficult to measure with Equilibrium |
| 115 |
> |
MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation |
| 116 |
|
methods. Therefore, the Reverse NEMD (RNEMD) |
| 117 |
|
methods\cite{MullerPlathe:1997xw,kuang:164101} would have the |
| 118 |
|
advantage of applying this difficult to measure flux (while measuring |
| 196 |
|
temperature of the two separated phases. |
| 197 |
|
|
| 198 |
|
When the interfacial conductance is {\it not} small, there are two |
| 199 |
< |
ways to define $G$. One way is to assume the temperature is discrete |
| 200 |
< |
on the two sides of the interface. $G$ can be calculated using the |
| 201 |
< |
applied thermal flux $J$ and the maximum temperature difference |
| 202 |
< |
measured along the thermal gradient max($\Delta T$), which occurs at |
| 203 |
< |
the Gibbs deviding surface (Figure \ref{demoPic}): \begin{equation} |
| 204 |
< |
G=\frac{J}{\Delta T} \label{discreteG} \end{equation} |
| 199 |
> |
ways to define $G$. One common way is to assume the temperature is |
| 200 |
> |
discrete on the two sides of the interface. $G$ can be calculated |
| 201 |
> |
using the applied thermal flux $J$ and the maximum temperature |
| 202 |
> |
difference measured along the thermal gradient max($\Delta T$), which |
| 203 |
> |
occurs at the Gibbs deviding surface (Figure \ref{demoPic}). This is |
| 204 |
> |
known as the Kapitza conductance, which is the inverse of the Kapitza |
| 205 |
> |
resistance. |
| 206 |
> |
\begin{equation} |
| 207 |
> |
G=\frac{J}{\Delta T} |
| 208 |
> |
\label{discreteG} |
| 209 |
> |
\end{equation} |
| 210 |
|
|
| 211 |
|
\begin{figure} |
| 212 |
|
\includegraphics[width=\linewidth]{method} |
| 307 |
|
solvent molecules would change the normal behavior of the liquid |
| 308 |
|
phase. Therefore, our $N_{solvent}$ values were chosen to ensure that |
| 309 |
|
these extreme cases did not happen to our simulations. The spacing |
| 310 |
< |
between periodic images of the gold interfaces is $35 \sim 75$\AA. |
| 310 |
> |
between periodic images of the gold interfaces is $45 \sim 75$\AA. |
| 311 |
|
|
| 312 |
|
The initial configurations generated are further equilibrated with the |
| 313 |
|
$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to |
| 358 |
|
these simulations. The chemically-distinct sites (a-e) are expanded |
| 359 |
|
in terms of constituent atoms for both United Atom (UA) and All Atom |
| 360 |
|
(AA) force fields. Most parameters are from |
| 361 |
< |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} (UA) and |
| 362 |
< |
\protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given |
| 363 |
< |
in Table \ref{MnM}.} |
| 361 |
> |
Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} |
| 362 |
> |
(UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au |
| 363 |
> |
atoms are given in Table \ref{MnM}.} |
| 364 |
|
\label{demoMol} |
| 365 |
|
\end{figure} |
| 366 |
|
|
| 397 |
|
this solvent model. |
| 398 |
|
|
| 399 |
|
Besides the TraPPE-UA models, AA models for both organic solvents are |
| 400 |
< |
included in our studies as well. For hexane, the OPLS-AA\cite{OPLSAA} |
| 401 |
< |
force field is used, and additional explicit hydrogen sites were |
| 400 |
> |
included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields |
| 401 |
> |
were used. For hexane, additional explicit hydrogen sites were |
| 402 |
|
included. Besides bonding and non-bonded site-site interactions, |
| 403 |
|
partial charges and the electrostatic interactions were added to each |
| 404 |
< |
CT and HC site. For toluene, the United Force Field developed by |
| 405 |
< |
Rapp\'{e} {\it et al.}\cite{doi:10.1021/ja00051a040} was adopted, and |
| 406 |
< |
a flexible model for the toluene molecule was utilized which included |
| 396 |
< |
bond, bend, torsion, and inversion potentials to enforce ring |
| 397 |
< |
planarity. |
| 404 |
> |
CT and HC site. For toluene, a flexible model for the toluene molecule |
| 405 |
> |
was utilized which included bond, bend, torsion, and inversion |
| 406 |
> |
potentials to enforce ring planarity. |
| 407 |
|
|
| 408 |
|
The butanethiol capping agent in our simulations, were also modeled |
| 409 |
|
with both UA and AA model. The TraPPE-UA force field includes |
| 480 |
|
\end{minipage} |
| 481 |
|
\end{table*} |
| 482 |
|
|
| 474 |
– |
\subsection{Vibrational Power Spectrum} |
| 483 |
|
|
| 484 |
< |
To investigate the mechanism of interfacial thermal conductance, the |
| 485 |
< |
vibrational power spectrum was computed. Power spectra were taken for |
| 486 |
< |
individual components in different simulations. To obtain these |
| 487 |
< |
spectra, simulations were run after equilibration, in the NVE |
| 488 |
< |
ensemble, and without a thermal gradient. Snapshots of configurations |
| 489 |
< |
were collected at a frequency that is higher than that of the fastest |
| 490 |
< |
vibrations occuring in the simulations. With these configurations, the |
| 491 |
< |
velocity auto-correlation functions can be computed: |
| 484 |
< |
\begin{equation} |
| 485 |
< |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
| 486 |
< |
\label{vCorr} |
| 487 |
< |
\end{equation} |
| 488 |
< |
The power spectrum is constructed via a Fourier transform of the |
| 489 |
< |
symmetrized velocity autocorrelation function, |
| 490 |
< |
\begin{equation} |
| 491 |
< |
\hat{f}(\omega) = |
| 492 |
< |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
| 493 |
< |
\label{fourier} |
| 494 |
< |
\end{equation} |
| 484 |
> |
\section{Results} |
| 485 |
> |
There are many factors contributing to the measured interfacial |
| 486 |
> |
conductance; some of these factors are physically motivated |
| 487 |
> |
(e.g. coverage of the surface by the capping agent coverage and |
| 488 |
> |
solvent identity), while some are governed by parameters of the |
| 489 |
> |
methodology (e.g. applied flux and the formulas used to obtain the |
| 490 |
> |
conductance). In this section we discuss the major physical and |
| 491 |
> |
calculational effects on the computed conductivity. |
| 492 |
|
|
| 493 |
< |
\section{Results and Discussions} |
| 497 |
< |
In what follows, how the parameters and protocol of simulations would |
| 498 |
< |
affect the measurement of $G$'s is first discussed. With a reliable |
| 499 |
< |
protocol and set of parameters, the influence of capping agent |
| 500 |
< |
coverage on thermal conductance is investigated. Besides, different |
| 501 |
< |
force field models for both solvents and selected deuterated models |
| 502 |
< |
were tested and compared. Finally, a summary of the role of capping |
| 503 |
< |
agent in the interfacial thermal transport process is given. |
| 493 |
> |
\subsection{Effects due to capping agent coverage} |
| 494 |
|
|
| 495 |
< |
\subsection{How Simulation Parameters Affects $G$} |
| 496 |
< |
We have varied our protocol or other parameters of the simulations in |
| 497 |
< |
order to investigate how these factors would affect the measurement of |
| 498 |
< |
$G$'s. It turned out that while some of these parameters would not |
| 499 |
< |
affect the results substantially, some other changes to the |
| 500 |
< |
simulations would have a significant impact on the measurement |
| 511 |
< |
results. |
| 495 |
> |
A series of different initial conditions with a range of surface |
| 496 |
> |
coverages was prepared and solvated with various with both of the |
| 497 |
> |
solvent molecules. These systems were then equilibrated and their |
| 498 |
> |
interfacial thermal conductivity was measured with the NIVS |
| 499 |
> |
algorithm. Figure \ref{coverage} demonstrates the trend of conductance |
| 500 |
> |
with respect to surface coverage. |
| 501 |
|
|
| 502 |
< |
In some of our simulations, we allowed $L_x$ and $L_y$ to change |
| 503 |
< |
during equilibrating the liquid phase. Due to the stiffness of the |
| 504 |
< |
crystalline Au structure, $L_x$ and $L_y$ would not change noticeably |
| 505 |
< |
after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a |
| 506 |
< |
system is fully equilibrated in the NPT ensemble, this fluctuation, as |
| 507 |
< |
well as those of $L_x$ and $L_y$ (which is significantly smaller), |
| 508 |
< |
would not be magnified on the calculated $G$'s, as shown in Table |
| 509 |
< |
\ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows |
| 510 |
< |
reliable measurement of $G$'s without the necessity of extremely |
| 511 |
< |
cautious equilibration process. |
| 502 |
> |
\begin{figure} |
| 503 |
> |
\includegraphics[width=\linewidth]{coverage} |
| 504 |
> |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 505 |
> |
for the Au-butanethiol/solvent interface with various UA models and |
| 506 |
> |
different capping agent coverages at $\langle T\rangle\sim$200K.} |
| 507 |
> |
\label{coverage} |
| 508 |
> |
\end{figure} |
| 509 |
> |
|
| 510 |
> |
In partially covered surfaces, the derivative definition for $G^\prime$ (Eq. \ref{derivativeG}) becomes difficult to apply, as the location of maximum change of $\lambda$ becomes washed out. The discrete definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs dividing surface is still well-defined. Therefore, $G$ (not $G^\prime$) was used in this section. |
| 511 |
> |
|
| 512 |
> |
From Figure \ref{coverage}, one can see the significance of the presence of capping agents. When even a small fraction of the Au(111) surface sites are covered with butanethiols, the conductivity exhibits an enhancement by at least a factor of 3. Cappping agents are clearly playing a major role in thermal transport at metal / organic solvent surfaces. |
| 513 |
|
|
| 514 |
+ |
We note a non-monotonic behavior in the interfacial conductance as a function of surface coverage. The maximum conductance (largest $G$) happens when the surfaces are about 75\% covered with butanethiol caps. The reason for this behavior is not entirely clear. One explanation is that incomplete butanethiol coverage allows small gaps between butanethiols to form. These gaps can be filled by transient solvent molecules. These solvent molecules couple very strongly with the hot capping agent molecules near the surface, and can then carry away (diffusively) the excess thermal energy from the surface. |
| 515 |
+ |
|
| 516 |
+ |
There appears to be a competition between the conduction of the thermal energy away from the surface by the capping agents (enhanced by greater coverage) and the coupling of the capping agents with the solvent (enhanced by interdigitation at lower coverages). This competition would lead to the non-monotonic coverage behavior observed here. |
| 517 |
+ |
|
| 518 |
+ |
Results for rigid body toluene solvent, as well as the UA hexane, are within the ranges expected from prior experimental work.\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406} This suggests that explicit hydrogen atoms might not be required for modeling thermal transport in these systems. C-H vibrational modes do not see significant excited state population at low temperatures, and are not likely to carry lower frequency excitations from the solid layer into the bulk liquid. |
| 519 |
+ |
|
| 520 |
+ |
The toluene solvent does not exhibit the same behavior as hexane in that $G$ remains at approximately the same magnitude when the capping coverage increases from 25\% to 75\%. Toluene, as a rigid planar molecule, cannot occupy the relatively small gaps between the capping agents as easily as the chain-like {\it n}-hexane. The effect of solvent coupling to the capping agent is therefore weaker in toluene except at the very lowest coverage levels. This effect counters the coverage-dependent conduction of heat away from the metal surface, leading to a much flatter $G$ vs. coverage trend than is observed in {\it n}-hexane. |
| 521 |
+ |
|
| 522 |
+ |
\subsection{Effects due to Solvent \& Solvent Models} |
| 523 |
+ |
In addition to UA solvent and capping agent models, AA models have also been included in our simulations. In most of this work, the same (UA or AA) model for solvent and capping agent was used, but it is also possible to utilize different models for different components. We have also included isotopic substitutions (Hydrogen to Deuterium) to decrease the explicit vibrational overlap between solvent and capping agent. Table \ref{modelTest} summarizes the results of these studies. |
| 524 |
+ |
|
| 525 |
+ |
\begin{table*} |
| 526 |
+ |
\begin{minipage}{\linewidth} |
| 527 |
+ |
\begin{center} |
| 528 |
+ |
|
| 529 |
+ |
\caption{Computed interfacial thermal conductance ($G$ and |
| 530 |
+ |
$G^\prime$) values for interfaces using various models for |
| 531 |
+ |
solvent and capping agent (or without capping agent) at |
| 532 |
+ |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 533 |
+ |
or capping agent molecules; ``Avg.'' denotes results that are |
| 534 |
+ |
averages of simulations under different applied thermal flux values $(J_z)$. Error |
| 535 |
+ |
estimates are indicated in parentheses.)} |
| 536 |
+ |
|
| 537 |
+ |
\begin{tabular}{llccc} |
| 538 |
+ |
\hline\hline |
| 539 |
+ |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 540 |
+ |
(or bare surface) & model & (GW/m$^2$) & |
| 541 |
+ |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 542 |
+ |
\hline |
| 543 |
+ |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 544 |
+ |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 545 |
+ |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 546 |
+ |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 547 |
+ |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 548 |
+ |
\hline |
| 549 |
+ |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 550 |
+ |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 551 |
+ |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 552 |
+ |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 553 |
+ |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 554 |
+ |
\hline |
| 555 |
+ |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 556 |
+ |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 557 |
+ |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 558 |
+ |
\hline |
| 559 |
+ |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 560 |
+ |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 561 |
+ |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 562 |
+ |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 563 |
+ |
\hline\hline |
| 564 |
+ |
\end{tabular} |
| 565 |
+ |
\label{modelTest} |
| 566 |
+ |
\end{center} |
| 567 |
+ |
\end{minipage} |
| 568 |
+ |
\end{table*} |
| 569 |
+ |
|
| 570 |
+ |
To facilitate direct comparison between force fields, systems with the same capping agent and solvent were prepared with the same length scales for the simulation cells. |
| 571 |
+ |
|
| 572 |
+ |
On bare metal / solvent surfaces, different force field models for hexane yield similar results for both $G$ and $G^\prime$, and these two definitions agree with each other very well. This is primarily an indicator of weak interactions between the metal and the solvent, and is a typical case for acoustic impedance mismatch between these two phases. |
| 573 |
+ |
|
| 574 |
+ |
For the fully-covered surfaces, the choice of force field for the capping agent and solvent has a large impact on the calulated values of $G$ and $G^\prime$. The AA thiol to AA solvent conductivities are much larger than their UA to UA counterparts, and these values exceed the experimental estimates by a large measure. The AA force field allows significant energy to go into C-H (or C-D) stretching modes, and since these modes are high frequency, this non-quantum behavior is likely responsible for the overestimate of the conductivity. |
| 575 |
+ |
|
| 576 |
+ |
The similarity in the vibrational modes available to solvent and capping agent can be reduced by deuterating one of the two components. Once either the hexanes or the butanethiols are deuterated, one can see a significantly lower $G$ and $G^\prime$ (Figure \ref{aahxntln}). Compared to the AA model, the UA model yields more reasonable conductivity values with much higher computational efficiency. |
| 577 |
+ |
|
| 578 |
+ |
\begin{figure} |
| 579 |
+ |
\includegraphics[width=\linewidth]{aahxntln} |
| 580 |
+ |
\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent |
| 581 |
+ |
systems. When butanethiol is deuterated (lower left), its |
| 582 |
+ |
vibrational overlap with hexane decreases significantly. Since aromatic molecules and the butanethiol are vibrationally dissimilar, the change is not as dramatic when toluene is the solvent (right).} |
| 583 |
+ |
\label{aahxntln} |
| 584 |
+ |
\end{figure} |
| 585 |
+ |
|
| 586 |
+ |
For the Au / butanethiol / toluene interfaces, having the AA butanethiol deuterated did not yield a significant change in the measured conductance. Compared to the C-H vibrational overlap between hexane and butanethiol, both of which have alkyl chains, the overlap between toluene and butanethiol is not as significant and thus does not contribute as much to the heat exchange process. The presence of extra degrees of freedom in the AA force field for toluene yields higher heat exchange rates between the two phases and results in a much higher conductivity than in the UA force field. |
| 587 |
+ |
|
| 588 |
+ |
\subsubsection{Are electronic excitations in the metal important?} |
| 589 |
+ |
Because they lack electronic excitations, the QSC and related embedded atom method (EAM) models for gold are known to predict unreasonably low values for bulk conductivity ($\lambda$).\cite{kuang:164101,ISI:000207079300006,Clancy:1992} If the conductance between the phases ($G$) is governed primarily by phonon excitation (and not electronic degrees of freedom), one would expect a classical model to capture most of the interfacial thermal conductance. Our results for $G$ and $G^\prime$ indicate that this is indeed the case, and suggest that the modeling of interfacial thermal transport depends primarily on the description of the interactions between the various components at the interface. When the metal is chemically capped, the primary barrier to thermal conductivity appears to be the interface between the capping agent and the surrounding solvent, so the excitations in the metal have little impact on the value of $G$. |
| 590 |
+ |
|
| 591 |
+ |
\subsection{Effects due to methodology and simulation parameters} |
| 592 |
+ |
|
| 593 |
+ |
START HERE |
| 594 |
+ |
|
| 595 |
+ |
We have varied our protocol or other parameters of the simulations in order to investigate how these factors would affect the computation of $G$. |
| 596 |
+ |
|
| 597 |
+ |
We allowed $L_x$ and $L_y$ to change during equilibrating the liquid phase. Due to the stiffness of the crystalline Au structure, $L_x$ and $L_y$ would not change noticeably after equilibration. Although $L_z$ could fluctuate $\sim$1\% after a system is fully equilibrated in the NPT ensemble, this fluctuation, as well as those of $L_x$ and $L_y$ (which is significantly smaller), would not be magnified on the calculated $G$'s, as shown in Table \ref{AuThiolHexaneUA}. This insensivity to $L_i$ fluctuations allows reliable measurement of $G$'s without the necessity of extremely cautious equilibration process. |
| 598 |
+ |
|
| 599 |
|
As stated in our computational details, the spacing filled with |
| 600 |
|
solvent molecules can be chosen within a range. This allows some |
| 601 |
|
change of solvent molecule numbers for the same Au-butanethiol |
| 606 |
|
smaller system size would be preferable, given that the liquid phase |
| 607 |
|
structure is not affected. |
| 608 |
|
|
| 609 |
+ |
\subsubsection{Effects of applied flux} |
| 610 |
|
Our NIVS algorithm allows change of unphysical thermal flux both in |
| 611 |
|
direction and in quantity. This feature extends our investigation of |
| 612 |
|
interfacial thermal conductance. However, the magnitude of this |
| 670 |
|
\end{minipage} |
| 671 |
|
\end{table*} |
| 672 |
|
|
| 673 |
+ |
\subsubsection{Effects due to average temperature} |
| 674 |
+ |
|
| 675 |
|
Furthermore, we also attempted to increase system average temperatures |
| 676 |
|
to above 200K. These simulations are first equilibrated in the NPT |
| 677 |
|
ensemble under normal pressure. As stated above, the TraPPE-UA model |
| 756 |
|
interfaces even at a relatively high temperature without being |
| 757 |
|
affected by surface reconstructions. |
| 758 |
|
|
| 681 |
– |
\subsection{Influence of Capping Agent Coverage on $G$} |
| 682 |
– |
To investigate the influence of butanethiol coverage on interfacial |
| 683 |
– |
thermal conductance, a series of different coverage Au-butanethiol |
| 684 |
– |
surfaces is prepared and solvated with various organic |
| 685 |
– |
molecules. These systems are then equilibrated and their interfacial |
| 686 |
– |
thermal conductivity are measured with our NIVS algorithm. Figure |
| 687 |
– |
\ref{coverage} demonstrates the trend of conductance change with |
| 688 |
– |
respect to different coverages of butanethiol. To study the isotope |
| 689 |
– |
effect in interfacial thermal conductance, deuterated UA-hexane is |
| 690 |
– |
included as well. |
| 759 |
|
|
| 760 |
< |
\begin{figure} |
| 693 |
< |
\includegraphics[width=\linewidth]{coverage} |
| 694 |
< |
\caption{Comparison of interfacial thermal conductivity ($G$) values |
| 695 |
< |
for the Au-butanethiol/solvent interface with various UA models and |
| 696 |
< |
different capping agent coverages at $\langle T\rangle\sim$200K |
| 697 |
< |
using certain energy flux respectively.} |
| 698 |
< |
\label{coverage} |
| 699 |
< |
\end{figure} |
| 760 |
> |
\section{Discussion} |
| 761 |
|
|
| 762 |
< |
It turned out that with partial covered butanethiol on the Au(111) |
| 763 |
< |
surface, the derivative definition for $G^\prime$ |
| 764 |
< |
(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty |
| 765 |
< |
in locating the maximum of change of $\lambda$. Instead, the discrete |
| 766 |
< |
definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs |
| 767 |
< |
deviding surface can still be well-defined. Therefore, $G$ (not |
| 768 |
< |
$G^\prime$) was used for this section. |
| 762 |
> |
\subsection{Capping agent acts as a vibrational coupler between solid |
| 763 |
> |
and solvent phases} |
| 764 |
> |
To investigate the mechanism of interfacial thermal conductance, the |
| 765 |
> |
vibrational power spectrum was computed. Power spectra were taken for |
| 766 |
> |
individual components in different simulations. To obtain these |
| 767 |
> |
spectra, simulations were run after equilibration, in the NVE |
| 768 |
> |
ensemble, and without a thermal gradient. Snapshots of configurations |
| 769 |
> |
were collected at a frequency that is higher than that of the fastest |
| 770 |
> |
vibrations occuring in the simulations. With these configurations, the |
| 771 |
> |
velocity auto-correlation functions can be computed: |
| 772 |
> |
\begin{equation} |
| 773 |
> |
C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle |
| 774 |
> |
\label{vCorr} |
| 775 |
> |
\end{equation} |
| 776 |
> |
The power spectrum is constructed via a Fourier transform of the |
| 777 |
> |
symmetrized velocity autocorrelation function, |
| 778 |
> |
\begin{equation} |
| 779 |
> |
\hat{f}(\omega) = |
| 780 |
> |
\int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt |
| 781 |
> |
\label{fourier} |
| 782 |
> |
\end{equation} |
| 783 |
|
|
| 709 |
– |
From Figure \ref{coverage}, one can see the significance of the |
| 710 |
– |
presence of capping agents. Even when a fraction of the Au(111) |
| 711 |
– |
surface sites are covered with butanethiols, the conductivity would |
| 712 |
– |
see an enhancement by at least a factor of 3. This indicates the |
| 713 |
– |
important role cappping agent is playing for thermal transport |
| 714 |
– |
phenomena on metal / organic solvent surfaces. |
| 784 |
|
|
| 785 |
< |
Interestingly, as one could observe from our results, the maximum |
| 717 |
< |
conductance enhancement (largest $G$) happens while the surfaces are |
| 718 |
< |
about 75\% covered with butanethiols. This again indicates that |
| 719 |
< |
solvent-capping agent contact has an important role of the thermal |
| 720 |
< |
transport process. Slightly lower butanethiol coverage allows small |
| 721 |
< |
gaps between butanethiols to form. And these gaps could be filled with |
| 722 |
< |
solvent molecules, which acts like ``heat conductors'' on the |
| 723 |
< |
surface. The higher degree of interaction between these solvent |
| 724 |
< |
molecules and capping agents increases the enhancement effect and thus |
| 725 |
< |
produces a higher $G$ than densely packed butanethiol arrays. However, |
| 726 |
< |
once this maximum conductance enhancement is reached, $G$ decreases |
| 727 |
< |
when butanethiol coverage continues to decrease. Each capping agent |
| 728 |
< |
molecule reaches its maximum capacity for thermal |
| 729 |
< |
conductance. Therefore, even higher solvent-capping agent contact |
| 730 |
< |
would not offset this effect. Eventually, when butanethiol coverage |
| 731 |
< |
continues to decrease, solvent-capping agent contact actually |
| 732 |
< |
decreases with the disappearing of butanethiol molecules. In this |
| 733 |
< |
case, $G$ decrease could not be offset but instead accelerated. [NEED |
| 734 |
< |
SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] |
| 735 |
< |
|
| 736 |
< |
A comparison of the results obtained from differenet organic solvents |
| 737 |
< |
can also provide useful information of the interfacial thermal |
| 738 |
< |
transport process. The deuterated hexane (UA) results do not appear to |
| 739 |
< |
be much different from those of normal hexane (UA), given that |
| 740 |
< |
butanethiol (UA) is non-deuterated for both solvents. These UA model |
| 741 |
< |
studies, even though eliminating C-H vibration samplings, still have |
| 742 |
< |
C-C vibrational frequencies different from each other. However, these |
| 743 |
< |
differences in the infrared range do not seem to produce an observable |
| 744 |
< |
difference for the results of $G$ (Figure \ref{uahxnua}). |
| 745 |
< |
|
| 746 |
< |
\begin{figure} |
| 747 |
< |
\includegraphics[width=\linewidth]{uahxnua} |
| 748 |
< |
\caption{Vibrational spectra obtained for normal (upper) and |
| 749 |
< |
deuterated (lower) hexane in Au-butanethiol/hexane |
| 750 |
< |
systems. Butanethiol spectra are shown as reference. Both hexane and |
| 751 |
< |
butanethiol were using United-Atom models.} |
| 752 |
< |
\label{uahxnua} |
| 753 |
< |
\end{figure} |
| 754 |
< |
|
| 755 |
< |
Furthermore, results for rigid body toluene solvent, as well as other |
| 756 |
< |
UA-hexane solvents, are reasonable within the general experimental |
| 757 |
< |
ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. This |
| 758 |
< |
suggests that explicit hydrogen might not be a required factor for |
| 759 |
< |
modeling thermal transport phenomena of systems such as |
| 760 |
< |
Au-thiol/organic solvent. |
| 761 |
< |
|
| 762 |
< |
However, results for Au-butanethiol/toluene do not show an identical |
| 763 |
< |
trend with those for Au-butanethiol/hexane in that $G$ remains at |
| 764 |
< |
approximately the same magnitue when butanethiol coverage differs from |
| 765 |
< |
25\% to 75\%. This might be rooted in the molecule shape difference |
| 766 |
< |
for planar toluene and chain-like {\it n}-hexane. Due to this |
| 767 |
< |
difference, toluene molecules have more difficulty in occupying |
| 768 |
< |
relatively small gaps among capping agents when their coverage is not |
| 769 |
< |
too low. Therefore, the solvent-capping agent contact may keep |
| 770 |
< |
increasing until the capping agent coverage reaches a relatively low |
| 771 |
< |
level. This becomes an offset for decreasing butanethiol molecules on |
| 772 |
< |
its effect to the process of interfacial thermal transport. Thus, one |
| 773 |
< |
can see a plateau of $G$ vs. butanethiol coverage in our results. |
| 774 |
< |
|
| 775 |
< |
\subsection{Influence of Chosen Molecule Model on $G$} |
| 776 |
< |
In addition to UA solvent/capping agent models, AA models are included |
| 777 |
< |
in our simulations as well. Besides simulations of the same (UA or AA) |
| 778 |
< |
model for solvent and capping agent, different models can be applied |
| 779 |
< |
to different components. Furthermore, regardless of models chosen, |
| 780 |
< |
either the solvent or the capping agent can be deuterated, similar to |
| 781 |
< |
the previous section. Table \ref{modelTest} summarizes the results of |
| 782 |
< |
these studies. |
| 783 |
< |
|
| 784 |
< |
\begin{table*} |
| 785 |
< |
\begin{minipage}{\linewidth} |
| 786 |
< |
\begin{center} |
| 787 |
< |
|
| 788 |
< |
\caption{Computed interfacial thermal conductivity ($G$ and |
| 789 |
< |
$G^\prime$) values for interfaces using various models for |
| 790 |
< |
solvent and capping agent (or without capping agent) at |
| 791 |
< |
$\langle T\rangle\sim$200K. (D stands for deuterated solvent |
| 792 |
< |
or capping agent molecules; ``Avg.'' denotes results that are |
| 793 |
< |
averages of simulations under different $J_z$'s. Error |
| 794 |
< |
estimates indicated in parenthesis.)} |
| 795 |
< |
|
| 796 |
< |
\begin{tabular}{llccc} |
| 797 |
< |
\hline\hline |
| 798 |
< |
Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ |
| 799 |
< |
(or bare surface) & model & (GW/m$^2$) & |
| 800 |
< |
\multicolumn{2}{c}{(MW/m$^2$/K)} \\ |
| 801 |
< |
\hline |
| 802 |
< |
UA & UA hexane & Avg. & 131(9) & 87(10) \\ |
| 803 |
< |
& UA hexane(D) & 1.95 & 153(5) & 136(13) \\ |
| 804 |
< |
& AA hexane & Avg. & 131(6) & 122(10) \\ |
| 805 |
< |
& UA toluene & 1.96 & 187(16) & 151(11) \\ |
| 806 |
< |
& AA toluene & 1.89 & 200(36) & 149(53) \\ |
| 807 |
< |
\hline |
| 808 |
< |
AA & UA hexane & 1.94 & 116(9) & 129(8) \\ |
| 809 |
< |
& AA hexane & Avg. & 442(14) & 356(31) \\ |
| 810 |
< |
& AA hexane(D) & 1.93 & 222(12) & 234(54) \\ |
| 811 |
< |
& UA toluene & 1.98 & 125(25) & 97(60) \\ |
| 812 |
< |
& AA toluene & 3.79 & 487(56) & 290(42) \\ |
| 813 |
< |
\hline |
| 814 |
< |
AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ |
| 815 |
< |
& AA hexane & 1.92 & 243(29) & 191(11) \\ |
| 816 |
< |
& AA toluene & 1.93 & 364(36) & 322(67) \\ |
| 817 |
< |
\hline |
| 818 |
< |
bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ |
| 819 |
< |
& UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ |
| 820 |
< |
& AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ |
| 821 |
< |
& UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ |
| 822 |
< |
\hline\hline |
| 823 |
< |
\end{tabular} |
| 824 |
< |
\label{modelTest} |
| 825 |
< |
\end{center} |
| 826 |
< |
\end{minipage} |
| 827 |
< |
\end{table*} |
| 828 |
< |
|
| 829 |
< |
To facilitate direct comparison, the same system with differnt models |
| 830 |
< |
for different components uses the same length scale for their |
| 831 |
< |
simulation cells. Without the presence of capping agent, using |
| 832 |
< |
different models for hexane yields similar results for both $G$ and |
| 833 |
< |
$G^\prime$, and these two definitions agree with eath other very |
| 834 |
< |
well. This indicates very weak interaction between the metal and the |
| 835 |
< |
solvent, and is a typical case for acoustic impedance mismatch between |
| 836 |
< |
these two phases. |
| 837 |
< |
|
| 838 |
< |
As for Au(111) surfaces completely covered by butanethiols, the choice |
| 839 |
< |
of models for capping agent and solvent could impact the measurement |
| 840 |
< |
of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane |
| 841 |
< |
interfaces, using AA model for both butanethiol and hexane yields |
| 842 |
< |
substantially higher conductivity values than using UA model for at |
| 843 |
< |
least one component of the solvent and capping agent, which exceeds |
| 844 |
< |
the general range of experimental measurement results. This is |
| 845 |
< |
probably due to the classically treated C-H vibrations in the AA |
| 846 |
< |
model, which should not be appreciably populated at normal |
| 847 |
< |
temperatures. In comparison, once either the hexanes or the |
| 848 |
< |
butanethiols are deuterated, one can see a significantly lower $G$ and |
| 849 |
< |
$G^\prime$. In either of these cases, the C-H(D) vibrational overlap |
| 850 |
< |
between the solvent and the capping agent is removed (Figure |
| 851 |
< |
\ref{aahxntln}). Conclusively, the improperly treated C-H vibration in |
| 852 |
< |
the AA model produced over-predicted results accordingly. Compared to |
| 853 |
< |
the AA model, the UA model yields more reasonable results with higher |
| 854 |
< |
computational efficiency. |
| 855 |
< |
|
| 856 |
< |
\begin{figure} |
| 857 |
< |
\includegraphics[width=\linewidth]{aahxntln} |
| 858 |
< |
\caption{Spectra obtained for All-Atom model Au-butanethil/solvent |
| 859 |
< |
systems. When butanethiol is deuterated (lower left), its |
| 860 |
< |
vibrational overlap with hexane would decrease significantly, |
| 861 |
< |
compared with normal butanethiol (upper left). However, this |
| 862 |
< |
dramatic change does not apply to toluene as much (right).} |
| 863 |
< |
\label{aahxntln} |
| 864 |
< |
\end{figure} |
| 865 |
< |
|
| 866 |
< |
However, for Au-butanethiol/toluene interfaces, having the AA |
| 867 |
< |
butanethiol deuterated did not yield a significant change in the |
| 868 |
< |
measurement results. Compared to the C-H vibrational overlap between |
| 869 |
< |
hexane and butanethiol, both of which have alkyl chains, that overlap |
| 870 |
< |
between toluene and butanethiol is not so significant and thus does |
| 871 |
< |
not have as much contribution to the heat exchange |
| 872 |
< |
process. Conversely, extra degrees of freedom such as the C-H |
| 873 |
< |
vibrations could yield higher heat exchange rate between these two |
| 874 |
< |
phases and result in a much higher conductivity. |
| 875 |
< |
|
| 876 |
< |
Although the QSC model for Au is known to predict an overly low value |
| 877 |
< |
for bulk metal gold conductivity\cite{kuang:164101}, our computational |
| 878 |
< |
results for $G$ and $G^\prime$ do not seem to be affected by this |
| 879 |
< |
drawback of the model for metal. Instead, our results suggest that the |
| 880 |
< |
modeling of interfacial thermal transport behavior relies mainly on |
| 881 |
< |
the accuracy of the interaction descriptions between components |
| 882 |
< |
occupying the interfaces. |
| 883 |
< |
|
| 884 |
< |
\subsection{Role of Capping Agent in Interfacial Thermal Conductance} |
| 785 |
> |
\subsubsection{The role of specific vibrations} |
| 786 |
|
The vibrational spectra for gold slabs in different environments are |
| 787 |
|
shown as in Figure \ref{specAu}. Regardless of the presence of |
| 788 |
|
solvent, the gold surfaces covered by butanethiol molecules, compared |
| 793 |
|
simulations, the Au/S interfaces do not appear major heat barriers |
| 794 |
|
compared to the butanethiol / solvent interfaces. |
| 795 |
|
|
| 796 |
+ |
\subsubsection{Overlap of power spectrum} |
| 797 |
|
Simultaneously, the vibrational overlap between butanethiol and |
| 798 |
|
organic solvents suggests higher thermal exchange efficiency between |
| 799 |
|
these two components. Even exessively high heat transport was observed |
| 817 |
|
\label{specAu} |
| 818 |
|
\end{figure} |
| 819 |
|
|
| 820 |
< |
[MAY ADD COMPARISON OF AU SLAB WIDTHS] |
| 820 |
> |
\subsubsection{Isotopic substitution and vibrational overlap} |
| 821 |
> |
A comparison of the results obtained from the two different organic |
| 822 |
> |
solvents can also provide useful information of the interfacial |
| 823 |
> |
thermal transport process. The deuterated hexane (UA) results do not |
| 824 |
> |
appear to be substantially different from those of normal hexane (UA), |
| 825 |
> |
given that butanethiol (UA) is non-deuterated for both solvents. The |
| 826 |
> |
UA models, even though they have eliminated C-H vibrational overlap, |
| 827 |
> |
still have significant overlap in the infrared spectra. Because |
| 828 |
> |
differences in the infrared range do not seem to produce an observable |
| 829 |
> |
difference for the results of $G$ (Figure \ref{uahxnua}). |
| 830 |
|
|
| 831 |
+ |
\begin{figure} |
| 832 |
+ |
\includegraphics[width=\linewidth]{uahxnua} |
| 833 |
+ |
\caption{Vibrational spectra obtained for normal (upper) and |
| 834 |
+ |
deuterated (lower) hexane in Au-butanethiol/hexane |
| 835 |
+ |
systems. Butanethiol spectra are shown as reference. Both hexane and |
| 836 |
+ |
butanethiol were using United-Atom models.} |
| 837 |
+ |
\label{uahxnua} |
| 838 |
+ |
\end{figure} |
| 839 |
+ |
|
| 840 |
|
\section{Conclusions} |
| 841 |
|
The NIVS algorithm we developed has been applied to simulations of |
| 842 |
|
Au-butanethiol surfaces with organic solvents. This algorithm allows |
| 876 |
|
|
| 877 |
|
Vlugt {\it et al.} has investigated the surface thiol structures for |
| 878 |
|
nanocrystal gold and pointed out that they differs from those of the |
| 879 |
< |
Au(111) surface\cite{vlugt:cpc2007154}. This difference might lead to |
| 880 |
< |
change of interfacial thermal transport behavior as well. To |
| 881 |
< |
investigate this problem, an effective means to introduce thermal flux |
| 882 |
< |
and measure the corresponding thermal gradient is desirable for |
| 883 |
< |
simulating structures with spherical symmetry. |
| 879 |
> |
Au(111) surface\cite{landman:1998,vlugt:cpc2007154}. This difference |
| 880 |
> |
might lead to change of interfacial thermal transport behavior as |
| 881 |
> |
well. To investigate this problem, an effective means to introduce |
| 882 |
> |
thermal flux and measure the corresponding thermal gradient is |
| 883 |
> |
desirable for simulating structures with spherical symmetry. |
| 884 |
|
|
| 885 |
|
\section{Acknowledgments} |
| 886 |
|
Support for this project was provided by the National Science |
| 887 |
|
Foundation under grant CHE-0848243. Computational time was provided by |
| 888 |
|
the Center for Research Computing (CRC) at the University of Notre |
| 889 |
< |
Dame. \newpage |
| 889 |
> |
Dame. |
| 890 |
> |
\newpage |
| 891 |
|
|
| 892 |
|
\bibliography{interfacial} |
| 893 |
|
|
