--- interfacial/interfacial.tex 2011/07/27 03:27:28 3753 +++ interfacial/interfacial.tex 2011/07/29 21:06:30 3759 @@ -83,7 +83,12 @@ thermal conductance. Wang {\it et al.} studied heat tr traditional methods developed for homogeneous systems. Experimentally, various interfaces have been investigated for their -thermal conductance. Wang {\it et al.} studied heat transport through +thermal conductance. Cahill and coworkers studied nanoscale thermal +transport from metal nanoparticle/fluid interfaces, to epitaxial +TiN/single crystal oxides interfaces, to hydrophilic and hydrophobic +interfaces between water and solids with different self-assembled +monolayers.\cite{Wilson:2002uq,PhysRevB.67.054302,doi:10.1021/jp048375k,PhysRevLett.96.186101} +Wang {\it et al.} studied heat transport through long-chain hydrocarbon monolayers on gold substrate at individual molecular level,\cite{Wang10082007} Schmidt {\it et al.} studied the role of CTAB on thermal transport between gold nanorods and @@ -106,7 +111,8 @@ difficult to measure with Equilibrium MD or forward NE measurements for heat conductance of interfaces between the capping monolayer on Au and a solvent phase have yet to be studied with their approach. The comparatively low thermal flux through interfaces is -difficult to measure with Equilibrium MD or forward NEMD simulation +difficult to measure with Equilibrium +MD\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation methods. Therefore, the Reverse NEMD (RNEMD) methods\cite{MullerPlathe:1997xw,kuang:164101} would have the advantage of applying this difficult to measure flux (while measuring @@ -187,14 +193,18 @@ temperature of the two separated phases. where ${E_{total}}$ is the total imposed non-physical kinetic energy transfer during the simulation and ${\langle T_\mathrm{hot}\rangle}$ and ${\langle T_\mathrm{cold}\rangle}$ are the average observed -temperature of the two separated phases. +temperature of the two separated phases. For an applied flux $J_z$ +operating over a simulation time $t$ on a periodically-replicated slab +of dimensions $L_x \times L_y$, $E_{total} = J_z *(t)*(2 L_x L_y)$. When the interfacial conductance is {\it not} small, there are two ways to define $G$. One common way is to assume the temperature is discrete on the two sides of the interface. $G$ can be calculated using the applied thermal flux $J$ and the maximum temperature difference measured along the thermal gradient max($\Delta T$), which -occurs at the Gibbs deviding surface (Figure \ref{demoPic}): +occurs at the Gibbs deviding surface (Figure \ref{demoPic}). This is +known as the Kapitza conductance, which is the inverse of the Kapitza +resistance. \begin{equation} G=\frac{J}{\Delta T} \label{discreteG} @@ -350,7 +360,9 @@ particles of different species. these simulations. The chemically-distinct sites (a-e) are expanded in terms of constituent atoms for both United Atom (UA) and All Atom (AA) force fields. Most parameters are from - Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} (UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au atoms are given in Table \ref{MnM}.} + Refs. \protect\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes,TraPPE-UA.thiols} + (UA) and \protect\cite{OPLSAA} (AA). Cross-interactions with the Au + atoms are given in Table \ref{MnM}.} \label{demoMol} \end{figure} @@ -470,88 +482,229 @@ our simulations. \end{minipage} \end{table*} -\subsection{Vibrational Power Spectrum} -To investigate the mechanism of interfacial thermal conductance, the -vibrational power spectrum was computed. Power spectra were taken for -individual components in different simulations. To obtain these -spectra, simulations were run after equilibration, in the NVE -ensemble, and without a thermal gradient. Snapshots of configurations -were collected at a frequency that is higher than that of the fastest -vibrations occuring in the simulations. With these configurations, the -velocity auto-correlation functions can be computed: -\begin{equation} -C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle -\label{vCorr} -\end{equation} -The power spectrum is constructed via a Fourier transform of the -symmetrized velocity autocorrelation function, -\begin{equation} - \hat{f}(\omega) = - \int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt -\label{fourier} -\end{equation} - -\section{Results and Discussions} -In what follows, how the parameters and protocol of simulations would -affect the measurement of $G$'s is first discussed. With a reliable -protocol and set of parameters, the influence of capping agent -coverage on thermal conductance is investigated. Besides, different -force field models for both solvents and selected deuterated models -were tested and compared. Finally, a summary of the role of capping -agent in the interfacial thermal transport process is given. +\section{Results} +There are many factors contributing to the measured interfacial +conductance; some of these factors are physically motivated +(e.g. coverage of the surface by the capping agent coverage and +solvent identity), while some are governed by parameters of the +methodology (e.g. applied flux and the formulas used to obtain the +conductance). In this section we discuss the major physical and +calculational effects on the computed conductivity. -\subsection{How Simulation Parameters Affects $G$} -We have varied our protocol or other parameters of the simulations in -order to investigate how these factors would affect the measurement of -$G$'s. It turned out that while some of these parameters would not -affect the results substantially, some other changes to the -simulations would have a significant impact on the measurement -results. +\subsection{Effects due to capping agent coverage} -In some of our simulations, 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. +A series of different initial conditions with a range of surface +coverages was prepared and solvated with various with both of the +solvent molecules. These systems were then equilibrated and their +interfacial thermal conductivity was measured with the NIVS +algorithm. Figure \ref{coverage} demonstrates the trend of conductance +with respect to surface coverage. -As stated in our computational details, the spacing filled with -solvent molecules can be chosen within a range. This allows some -change of solvent molecule numbers for the same Au-butanethiol -surfaces. We did this study on our Au-butanethiol/hexane -simulations. Nevertheless, the results obtained from systems of -different $N_{hexane}$ did not indicate that the measurement of $G$ is -susceptible to this parameter. For computational efficiency concern, -smaller system size would be preferable, given that the liquid phase -structure is not affected. +\begin{figure} +\includegraphics[width=\linewidth]{coverage} +\caption{Comparison of interfacial thermal conductivity ($G$) values + for the Au-butanethiol/solvent interface with various UA models and + different capping agent coverages at $\langle T\rangle\sim$200K.} +\label{coverage} +\end{figure} -Our NIVS algorithm allows change of unphysical thermal flux both in -direction and in quantity. This feature extends our investigation of -interfacial thermal conductance. However, the magnitude of this -thermal flux is not arbitary if one aims to obtain a stable and -reliable thermal gradient. A temperature profile would be -substantially affected by noise when $|J_z|$ has a much too low -magnitude; while an excessively large $|J_z|$ that overwhelms the -conductance capacity of the interface would prevent a thermal gradient -to reach a stablized steady state. NIVS has the advantage of allowing -$J$ to vary in a wide range such that the optimal flux range for $G$ -measurement can generally be simulated by the algorithm. Within the -optimal range, we were able to study how $G$ would change according to -the thermal flux across the interface. For our simulations, we denote -$J_z$ to be positive when the physical thermal flux is from the liquid -to metal, and negative vice versa. The $G$'s measured under different -$J_z$ is listed in Table \ref{AuThiolHexaneUA} and -\ref{AuThiolToluene}. These results do not suggest that $G$ is -dependent on $J_z$ within this flux range. The linear response of flux -to thermal gradient simplifies our investigations in that we can rely -on $G$ measurement with only a couple $J_z$'s and do not need to test -a large series of fluxes. +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. + +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. + +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. + +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. + +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. + +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. + +\subsection{Effects due to Solvent \& Solvent Models} +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. + +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + + \caption{Computed interfacial thermal conductance ($G$ and + $G^\prime$) values for interfaces using various models for + solvent and capping agent (or without capping agent) at + $\langle T\rangle\sim$200K. (D stands for deuterated solvent + or capping agent molecules; ``Avg.'' denotes results that are + averages of simulations under different applied thermal flux + values $(J_z)$. Error estimates are indicated in + parentheses.)} + + \begin{tabular}{llccc} + \hline\hline + Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ + (or bare surface) & model & (GW/m$^2$) & + \multicolumn{2}{c}{(MW/m$^2$/K)} \\ + \hline + UA & UA hexane & Avg. & 131(9) & 87(10) \\ + & UA hexane(D) & 1.95 & 153(5) & 136(13) \\ + & AA hexane & Avg. & 131(6) & 122(10) \\ + & UA toluene & 1.96 & 187(16) & 151(11) \\ + & AA toluene & 1.89 & 200(36) & 149(53) \\ + \hline + AA & UA hexane & 1.94 & 116(9) & 129(8) \\ + & AA hexane & Avg. & 442(14) & 356(31) \\ + & AA hexane(D) & 1.93 & 222(12) & 234(54) \\ + & UA toluene & 1.98 & 125(25) & 97(60) \\ + & AA toluene & 3.79 & 487(56) & 290(42) \\ + \hline + AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ + & AA hexane & 1.92 & 243(29) & 191(11) \\ + & AA toluene & 1.93 & 364(36) & 322(67) \\ + \hline + bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ + & UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ + & AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ + & UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ + \hline\hline + \end{tabular} + \label{modelTest} + \end{center} + \end{minipage} +\end{table*} +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. + +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. + +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. Compared +to the AA model, the UA model yields more reasonable conductivity +values with much higher computational efficiency. + +\subsubsection{Are electronic excitations in the metal important?} +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 $G$. + +\subsection{Effects due to methodology and simulation parameters} + +We have varied the parameters of the simulations in order to +investigate how these factors would affect the computation of $G$. Of +particular interest are: 1) the length scale for the applied thermal +gradient (modified by increasing the amount of solvent in the system), +2) the sign and magnitude of the applied thermal flux, 3) the average +temperature of the simulation (which alters the solvent density during +equilibration), and 4) the definition of the interfacial conductance +(Eqs. (\ref{discreteG}) or (\ref{derivativeG})) used in the +calculation. + +Systems of different lengths were prepared by altering the number of +solvent molecules and extending the length of the box along the $z$ +axis to accomodate the extra solvent. Equilibration at the same +temperature and pressure conditions led to nearly identical surface +areas ($L_x$ and $L_y$) available to the metal and capping agent, +while the extra solvent served mainly to lengthen the axis that was +used to apply the thermal flux. For a given value of the applied +flux, the different $z$ length scale has only a weak effect on the +computed conductivities (Table \ref{AuThiolHexaneUA}). + +\subsubsection{Effects of applied flux} +The NIVS algorithm allows changes in both the sign and magnitude of +the applied flux. It is possible to reverse the direction of heat +flow simply by changing the sign of the flux, and thermal gradients +which would be difficult to obtain experimentally ($5$ K/\AA) can be +easily simulated. However, the magnitude of the applied flux is not +arbitary if one aims to obtain a stable and reliable thermal gradient. +A temperature gradient can be lost in the noise if $|J_z|$ is too +small, and excessive $|J_z|$ values can cause phase transitions if the +extremes of the simulation cell become widely separated in +temperature. Also, if $|J_z|$ is too large for the bulk conductivity +of the materials, the thermal gradient will never reach a stable +state. + +Within a reasonable range of $J_z$ values, we were able to study how +$G$ changes as a function of this flux. In what follows, we use +positive $J_z$ values to denote the case where energy is being +transferred by the method from the metal phase and into the liquid. +The resulting gradient therefore has a higher temperature in the +liquid phase. Negative flux values reverse this transfer, and result +in higher temperature metal phases. The conductance measured under +different applied $J_z$ values is listed in Tables +\ref{AuThiolHexaneUA} and \ref{AuThiolToluene}. These results do not +indicate that $G$ depends strongly on $J_z$ within this flux +range. The linear response of flux to thermal gradient simplifies our +investigations in that we can rely on $G$ measurement with only a +small number $J_z$ values. + \begin{table*} \begin{minipage}{\linewidth} \begin{center} @@ -563,29 +716,24 @@ a large series of fluxes. \begin{tabular}{ccccccc} \hline\hline - $\langle T\rangle$ & $N_{hexane}$ & Fixed & $\rho_{hexane}$ & + $\langle T\rangle$ & $N_{hexane}$ & $\rho_{hexane}$ & $J_z$ & $G$ & $G^\prime$ \\ - (K) & & $L_x$ \& $L_y$? & (g/cm$^3$) & (GW/m$^2$) & + (K) & & (g/cm$^3$) & (GW/m$^2$) & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ \hline - 200 & 266 & No & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ - & 200 & Yes & 0.694 & 1.92 & 129(11) & 87.3(0.3) \\ - & & Yes & 0.672 & 1.93 & 131(16) & 78(13) \\ - & & No & 0.688 & 0.96 & 125(16) & 90.2(15) \\ - & & & & 1.91 & 139(10) & 101(10) \\ - & & & & 2.83 & 141(6) & 89.9(9.8) \\ - & 166 & Yes & 0.679 & 0.97 & 115(19) & 69(18) \\ - & & & & 1.94 & 125(9) & 87.1(0.2) \\ - & & No & 0.681 & 0.97 & 141(30) & 78(22) \\ - & & & & 1.92 & 138(4) & 98.9(9.5) \\ + 200 & 266 & 0.672 & -0.96 & 102(3) & 80.0(0.8) \\ + & 200 & 0.688 & 0.96 & 125(16) & 90.2(15) \\ + & & & 1.91 & 139(10) & 101(10) \\ + & & & 2.83 & 141(6) & 89.9(9.8) \\ + & 166 & 0.681 & 0.97 & 141(30) & 78(22) \\ + & & & 1.92 & 138(4) & 98.9(9.5) \\ \hline - 250 & 200 & No & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ - & & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ - & 166 & Yes & 0.570 & 0.98 & 79.0(3.5) & 62.9(3.0) \\ - & & No & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ - & & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ - & & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ - & & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ + 250 & 200 & 0.560 & 0.96 & 75(10) & 61.8(7.3) \\ + & & & -0.95 & 49.4(0.3) & 45.7(2.1) \\ + & 166 & 0.569 & 0.97 & 80.3(0.6) & 67(11) \\ + & & & 1.44 & 76.2(5.0) & 64.8(3.8) \\ + & & & -0.95 & 56.4(2.5) & 54.4(1.1) \\ + & & & -1.85 & 47.8(1.1) & 53.5(1.5) \\ \hline\hline \end{tabular} \label{AuThiolHexaneUA} @@ -593,30 +741,42 @@ Furthermore, we also attempted to increase system aver \end{minipage} \end{table*} -Furthermore, we also attempted to increase system average temperatures -to above 200K. These simulations are first equilibrated in the NPT -ensemble under normal pressure. As stated above, the TraPPE-UA model -for hexane tends to predict a lower boiling point. In our simulations, -hexane had diffculty to remain in liquid phase when NPT equilibration -temperature is higher than 250K. Additionally, the equilibrated liquid -hexane density under 250K becomes lower than experimental value. This -expanded liquid phase leads to lower contact between hexane and -butanethiol as well.[MAY NEED SLAB DENSITY FIGURE] -And this reduced contact would -probably be accountable for a lower interfacial thermal conductance, -as shown in Table \ref{AuThiolHexaneUA}. +The sign of $J_z$ is a different matter, however, as this can alter +the temperature on the two sides of the interface. The average +temperature values reported are for the entire system, and not for the +liquid phase, so at a given $\langle T \rangle$, the system with +positive $J_z$ has a warmer liquid phase. This means that if the +liquid carries thermal energy via convective transport, {\it positive} +$J_z$ values will result in increased molecular motion on the liquid +side of the interface, and this will increase the measured +conductivity. -A similar study for TraPPE-UA toluene agrees with the above result as -well. Having a higher boiling point, toluene tends to remain liquid in -our simulations even equilibrated under 300K in NPT -ensembles. Furthermore, the expansion of the toluene liquid phase is -not as significant as that of the hexane. This prevents severe -decrease of liquid-capping agent contact and the results (Table -\ref{AuThiolToluene}) show only a slightly decreased interface -conductance. Therefore, solvent-capping agent contact should play an -important role in the thermal transport process across the interface -in that higher degree of contact could yield increased conductance. +\subsubsection{Effects due to average temperature} +We also studied the effect of average system temperature on the +interfacial conductance. The simulations are first equilibrated in +the NPT ensemble at 1 atm. The TraPPE-UA model for hexane tends to +predict a lower boiling point (and liquid state density) than +experiments. This lower-density liquid phase leads to reduced contact +between the hexane and butanethiol, and this accounts for our +observation of lower conductance at higher temperatures as shown in +Table \ref{AuThiolHexaneUA}. In raising the average temperature from +200K to 250K, the density drop of ~20\% in the solvent phase leads to +a ~65\% drop in the conductance. + +Similar behavior is observed in the TraPPE-UA model for toluene, +although this model has better agreement with the experimental +densities of toluene. The expansion of the toluene liquid phase is +not as significant as that of the hexane (8.3\% over 100K), and this +limits the effect to ~20\% drop in thermal conductivity (Table +\ref{AuThiolToluene}). + +Although we have not mapped out the behavior at a large number of +temperatures, is clear that there will be a strong temperature +dependence in the interfacial conductance when the physical properties +of one side of the interface (notably the density) change rapidly as a +function of temperature. + \begin{table*} \begin{minipage}{\linewidth} \begin{center} @@ -643,329 +803,203 @@ Besides lower interfacial thermal conductance, surface \end{minipage} \end{table*} -Besides lower interfacial thermal conductance, surfaces in relatively -high temperatures are susceptible to reconstructions, when -butanethiols have a full coverage on the Au(111) surface. These -reconstructions include surface Au atoms migrated outward to the S -atom layer, and butanethiol molecules embedded into the original -surface Au layer. The driving force for this behavior is the strong -Au-S interactions in our simulations. And these reconstructions lead -to higher ratio of Au-S attraction and thus is energetically -favorable. Furthermore, this phenomenon agrees with experimental -results\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt -{\it et al.} had kept their Au(111) slab rigid so that their -simulations can reach 300K without surface reconstructions. Without -this practice, simulating 100\% thiol covered interfaces under higher -temperatures could hardly avoid surface reconstructions. However, our -measurement is based on assuming homogeneity on $x$ and $y$ dimensions -so that measurement of $T$ at particular $z$ would be an effective -average of the particles of the same type. Since surface -reconstructions could eliminate the original $x$ and $y$ dimensional -homogeneity, measurement of $G$ is more difficult to conduct under -higher temperatures. Therefore, most of our measurements are -undertaken at $\langle T\rangle\sim$200K. +Besides the lower interfacial thermal conductance, surfaces at +relatively high temperatures are susceptible to reconstructions, +particularly when butanethiols fully cover the Au(111) surface. These +reconstructions include surface Au atoms which migrate outward to the +S atom layer, and butanethiol molecules which embed into the surface +Au layer. The driving force for this behavior is the strong Au-S +interactions which are modeled here with a deep Lennard-Jones +potential. This phenomenon agrees with reconstructions that have beeen +experimentally +observed.\cite{doi:10.1021/j100035a033,doi:10.1021/la026493y}. Vlugt +{\it et al.} kept their Au(111) slab rigid so that their simulations +could reach 300K without surface +reconstructions.\cite{vlugt:cpc2007154} Since surface reconstructions +blur the interface, the measurement of $G$ becomes more difficult to +conduct at higher temperatures. For this reason, most of our +measurements are undertaken at $\langle T\rangle\sim$200K where +reconstruction is minimized. However, when the surface is not completely covered by butanethiols, -the simulated system is more resistent to the reconstruction -above. Our Au-butanethiol/toluene system had the Au(111) surfaces 90\% -covered by butanethiols, but did not see this above phenomena even at -$\langle T\rangle\sim$300K. The empty three-fold sites not occupied by -capping agents could help prevent surface reconstruction in that they -provide other means of capping agent relaxation. It is observed that -butanethiols can migrate to their neighbor empty sites during a -simulation. Therefore, we were able to obtain $G$'s for these -interfaces even at a relatively high temperature without being -affected by surface reconstructions. +the simulated system appears to be more resistent to the +reconstruction. O ur Au / butanethiol / toluene system had the Au(111) +surfaces 90\% covered by butanethiols, but did not see this above +phenomena even at $\langle T\rangle\sim$300K. That said, we did +observe butanethiols migrating to neighboring three-fold sites during +a simulation. Since the interface persisted in these simulations, +were able to obtain $G$'s for these interfaces even at a relatively +high temperature without being affected by surface reconstructions. -\subsection{Influence of Capping Agent Coverage on $G$} -To investigate the influence of butanethiol coverage on interfacial -thermal conductance, a series of different coverage Au-butanethiol -surfaces is prepared and solvated with various organic -molecules. These systems are then equilibrated and their interfacial -thermal conductivity are measured with our NIVS algorithm. Figure -\ref{coverage} demonstrates the trend of conductance change with -respect to different coverages of butanethiol. To study the isotope -effect in interfacial thermal conductance, deuterated UA-hexane is -included as well. +\section{Discussion} -\begin{figure} -\includegraphics[width=\linewidth]{coverage} -\caption{Comparison of interfacial thermal conductivity ($G$) values - for the Au-butanethiol/solvent interface with various UA models and - different capping agent coverages at $\langle T\rangle\sim$200K - using certain energy flux respectively.} -\label{coverage} -\end{figure} +The primary result of this work is that the capping agent acts as an +efficient thermal coupler between solid and solvent phases. One of +the ways the capping agent can carry out this role is to down-shift +between the phonon vibrations in the solid (which carry the heat from +the gold) and the molecular vibrations in the liquid (which carry some +of the heat in the solvent). -It turned out that with partial covered butanethiol on the Au(111) -surface, the derivative definition for $G^\prime$ -(Eq. \ref{derivativeG}) was difficult to apply, due to the difficulty -in locating the maximum of change of $\lambda$. Instead, the discrete -definition (Eq. \ref{discreteG}) is easier to apply, as the Gibbs -deviding surface can still be well-defined. Therefore, $G$ (not -$G^\prime$) was used for this section. +To investigate the mechanism of interfacial thermal conductance, the +vibrational power spectrum was computed. Power spectra were taken for +individual components in different simulations. To obtain these +spectra, simulations were run after equilibration in the +microcanonical (NVE) ensemble and without a thermal +gradient. Snapshots of configurations were collected at a frequency +that is higher than that of the fastest vibrations occuring in the +simulations. With these configurations, the velocity auto-correlation +functions can be computed: +\begin{equation} +C_A (t) = \langle\vec{v}_A (t)\cdot\vec{v}_A (0)\rangle +\label{vCorr} +\end{equation} +The power spectrum is constructed via a Fourier transform of the +symmetrized velocity autocorrelation function, +\begin{equation} + \hat{f}(\omega) = + \int_{-\infty}^{\infty} C_A (t) e^{-2\pi it\omega}\,dt +\label{fourier} +\end{equation} -From Figure \ref{coverage}, one can see the significance of the -presence of capping agents. Even when a fraction of the Au(111) -surface sites are covered with butanethiols, the conductivity would -see an enhancement by at least a factor of 3. This indicates the -important role cappping agent is playing for thermal transport -phenomena on metal / organic solvent surfaces. +\subsection{The role of specific vibrations} +The vibrational spectra for gold slabs in different environments are +shown as in Figure \ref{specAu}. Regardless of the presence of +solvent, the gold surfaces which are covered by butanethiol molecules +exhibit an additional peak observed at a frequency of +$\sim$165cm$^{-1}$. We attribute this peak to the S-Au bonding +vibration. This vibration enables efficient thermal coupling of the +surface Au layer to the capping agents. Therefore, in our simulations, +the Au / S interfaces do not appear to be the primary barrier to +thermal transport when compared with the butanethiol / solvent +interfaces. -Interestingly, as one could observe from our results, the maximum -conductance enhancement (largest $G$) happens while the surfaces are -about 75\% covered with butanethiols. This again indicates that -solvent-capping agent contact has an important role of the thermal -transport process. Slightly lower butanethiol coverage allows small -gaps between butanethiols to form. And these gaps could be filled with -solvent molecules, which acts like ``heat conductors'' on the -surface. The higher degree of interaction between these solvent -molecules and capping agents increases the enhancement effect and thus -produces a higher $G$ than densely packed butanethiol arrays. However, -once this maximum conductance enhancement is reached, $G$ decreases -when butanethiol coverage continues to decrease. Each capping agent -molecule reaches its maximum capacity for thermal -conductance. Therefore, even higher solvent-capping agent contact -would not offset this effect. Eventually, when butanethiol coverage -continues to decrease, solvent-capping agent contact actually -decreases with the disappearing of butanethiol molecules. In this -case, $G$ decrease could not be offset but instead accelerated. [MAY NEED -SNAPSHOT SHOWING THE PHENOMENA / SLAB DENSITY ANALYSIS] - -A comparison of the results obtained from differenet organic solvents -can also provide useful information of the interfacial thermal -transport process. The deuterated hexane (UA) results do not appear to -be much different from those of normal hexane (UA), given that -butanethiol (UA) is non-deuterated for both solvents. These UA model -studies, even though eliminating C-H vibration samplings, still have -C-C vibrational frequencies different from each other. However, these -differences in the infrared range do not seem to produce an observable -difference for the results of $G$ (Figure \ref{uahxnua}). - \begin{figure} -\includegraphics[width=\linewidth]{uahxnua} -\caption{Vibrational spectra obtained for normal (upper) and - deuterated (lower) hexane in Au-butanethiol/hexane - systems. Butanethiol spectra are shown as reference. Both hexane and - butanethiol were using United-Atom models.} -\label{uahxnua} +\includegraphics[width=\linewidth]{vibration} +\caption{Vibrational power spectra for gold in different solvent + environments. The presence of the butanethiol capping molecules + adds a vibrational peak at $\sim$165cm$^{-1}$. The butanethiol + spectra exhibit a corresponding peak.} +\label{specAu} \end{figure} -Furthermore, results for rigid body toluene solvent, as well as other -UA-hexane solvents, are reasonable within the general experimental -ranges\cite{Wilson:2002uq,cahill:793,PhysRevB.80.195406}. This -suggests that explicit hydrogen might not be a required factor for -modeling thermal transport phenomena of systems such as -Au-thiol/organic solvent. +Also in this figure, we show the vibrational power spectrum for the +bound butanethiol molecules, which also exhibits the same +$\sim$165cm$^{-1}$ peak. -However, results for Au-butanethiol/toluene do not show an identical -trend with those for Au-butanethiol/hexane in that $G$ remains at -approximately the same magnitue when butanethiol coverage differs from -25\% to 75\%. This might be rooted in the molecule shape difference -for planar toluene and chain-like {\it n}-hexane. Due to this -difference, toluene molecules have more difficulty in occupying -relatively small gaps among capping agents when their coverage is not -too low. Therefore, the solvent-capping agent contact may keep -increasing until the capping agent coverage reaches a relatively low -level. This becomes an offset for decreasing butanethiol molecules on -its effect to the process of interfacial thermal transport. Thus, one -can see a plateau of $G$ vs. butanethiol coverage in our results. +\subsection{Overlap of power spectra} +A comparison of the results obtained from the two different organic +solvents can also provide useful information of the interfacial +thermal transport process. In particular, the vibrational overlap +between the butanethiol and the organic solvents suggests a highly +efficient thermal exchange between these components. Very high +thermal conductivity was observed when AA models were used and C-H +vibrations were treated classically. The presence of extra degrees of +freedom in the AA force field yields higher heat exchange rates +between the two phases and results in a much higher conductivity than +in the UA force field. -\subsection{Influence of Chosen Molecule Model on $G$} -In addition to UA solvent/capping agent models, AA models are included -in our simulations as well. Besides simulations of the same (UA or AA) -model for solvent and capping agent, different models can be applied -to different components. Furthermore, regardless of models chosen, -either the solvent or the capping agent can be deuterated, similar to -the previous section. Table \ref{modelTest} summarizes the results of -these studies. +The similarity in the vibrational modes available to solvent and +capping agent can be reduced by deuterating one of the two components +(Fig. \ref{aahxntln}). Once either the hexanes or the butanethiols +are deuterated, one can observe a significantly lower $G$ and +$G^\prime$ values (Table \ref{modelTest}). -\begin{table*} - \begin{minipage}{\linewidth} - \begin{center} - - \caption{Computed interfacial thermal conductivity ($G$ and - $G^\prime$) values for interfaces using various models for - solvent and capping agent (or without capping agent) at - $\langle T\rangle\sim$200K. (D stands for deuterated solvent - or capping agent molecules; ``Avg.'' denotes results that are - averages of simulations under different $J_z$'s. Error - estimates indicated in parenthesis.)} - - \begin{tabular}{llccc} - \hline\hline - Butanethiol model & Solvent & $J_z$ & $G$ & $G^\prime$ \\ - (or bare surface) & model & (GW/m$^2$) & - \multicolumn{2}{c}{(MW/m$^2$/K)} \\ - \hline - UA & UA hexane & Avg. & 131(9) & 87(10) \\ - & UA hexane(D) & 1.95 & 153(5) & 136(13) \\ - & AA hexane & Avg. & 131(6) & 122(10) \\ - & UA toluene & 1.96 & 187(16) & 151(11) \\ - & AA toluene & 1.89 & 200(36) & 149(53) \\ - \hline - AA & UA hexane & 1.94 & 116(9) & 129(8) \\ - & AA hexane & Avg. & 442(14) & 356(31) \\ - & AA hexane(D) & 1.93 & 222(12) & 234(54) \\ - & UA toluene & 1.98 & 125(25) & 97(60) \\ - & AA toluene & 3.79 & 487(56) & 290(42) \\ - \hline - AA(D) & UA hexane & 1.94 & 158(25) & 172(4) \\ - & AA hexane & 1.92 & 243(29) & 191(11) \\ - & AA toluene & 1.93 & 364(36) & 322(67) \\ - \hline - bare & UA hexane & Avg. & 46.5(3.2) & 49.4(4.5) \\ - & UA hexane(D) & 0.98 & 43.9(4.6) & 43.0(2.0) \\ - & AA hexane & 0.96 & 31.0(1.4) & 29.4(1.3) \\ - & UA toluene & 1.99 & 70.1(1.3) & 65.8(0.5) \\ - \hline\hline - \end{tabular} - \label{modelTest} - \end{center} - \end{minipage} -\end{table*} - -To facilitate direct comparison, the same system with differnt models -for different components uses the same length scale for their -simulation cells. Without the presence of capping agent, using -different models for hexane yields similar results for both $G$ and -$G^\prime$, and these two definitions agree with eath other very -well. This indicates very weak interaction between the metal and the -solvent, and is a typical case for acoustic impedance mismatch between -these two phases. - -As for Au(111) surfaces completely covered by butanethiols, the choice -of models for capping agent and solvent could impact the measurement -of $G$ and $G^\prime$ quite significantly. For Au-butanethiol/hexane -interfaces, using AA model for both butanethiol and hexane yields -substantially higher conductivity values than using UA model for at -least one component of the solvent and capping agent, which exceeds -the general range of experimental measurement results. This is -probably due to the classically treated C-H vibrations in the AA -model, which should not be appreciably populated at normal -temperatures. In comparison, once either the hexanes or the -butanethiols are deuterated, one can see a significantly lower $G$ and -$G^\prime$. In either of these cases, the C-H(D) vibrational overlap -between the solvent and the capping agent is removed (Figure -\ref{aahxntln}). Conclusively, the improperly treated C-H vibration in -the AA model produced over-predicted results accordingly. Compared to -the AA model, the UA model yields more reasonable results with higher -computational efficiency. - \begin{figure} \includegraphics[width=\linewidth]{aahxntln} -\caption{Spectra obtained for All-Atom model Au-butanethil/solvent +\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent systems. When butanethiol is deuterated (lower left), its - vibrational overlap with hexane would decrease significantly, - compared with normal butanethiol (upper left). However, this - dramatic change does not apply to toluene as much (right).} + 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).} \label{aahxntln} \end{figure} -However, for Au-butanethiol/toluene interfaces, having the AA +For the Au / butanethiol / toluene interfaces, having the AA butanethiol deuterated did not yield a significant change in the -measurement results. Compared to the C-H vibrational overlap between -hexane and butanethiol, both of which have alkyl chains, that overlap -between toluene and butanethiol is not so significant and thus does -not have as much contribution to the heat exchange -process. Conversely, extra degrees of freedom such as the C-H -vibrations could yield higher heat exchange rate between these two -phases and result in a much higher conductivity. +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. -Although the QSC model for Au is known to predict an overly low value -for bulk metal gold conductivity\cite{kuang:164101}, our computational -results for $G$ and $G^\prime$ do not seem to be affected by this -drawback of the model for metal. Instead, our results suggest that the -modeling of interfacial thermal transport behavior relies mainly on -the accuracy of the interaction descriptions between components -occupying the interfaces. +Previous observations of Zhang {\it et al.}\cite{hase:2010} indicate +that the {\it intra}molecular heat transport due to alkylthiols is +highly efficient. Combining our observations with those of Zhang {\it + et al.}, it appears that butanethiol acts as a channel to expedite +heat flow from the gold surface and into the alkyl chain. The +acoustic impedance mismatch between the metal and the liquid phase can +therefore be effectively reduced with the presence of suitable capping +agents. -\subsection{Role of Capping Agent in Interfacial Thermal Conductance} -The vibrational spectra for gold slabs in different environments are -shown as in Figure \ref{specAu}. Regardless of the presence of -solvent, the gold surfaces covered by butanethiol molecules, compared -to bare gold surfaces, exhibit an additional peak observed at the -frequency of $\sim$170cm$^{-1}$, which is attributed to the S-Au -bonding vibration. This vibration enables efficient thermal transport -from surface Au layer to the capping agents. Therefore, in our -simulations, the Au/S interfaces do not appear major heat barriers -compared to the butanethiol / solvent interfaces. +Deuterated models in the UA force field did not decouple the thermal +transport as well as in the AA force field. The UA models, even +though they have eliminated the high frequency C-H vibrational +overlap, still have significant overlap in the lower-frequency +portions of the infrared spectra (Figure \ref{uahxnua}). Deuterating +the UA models did not decouple the low frequency region enough to +produce an observable difference for the results of $G$ (Table +\ref{modelTest}). -Simultaneously, the vibrational overlap between butanethiol and -organic solvents suggests higher thermal exchange efficiency between -these two components. Even exessively high heat transport was observed -when All-Atom models were used and C-H vibrations were treated -classically. Compared to metal and organic liquid phase, the heat -transfer efficiency between butanethiol and organic solvents is closer -to that within bulk liquid phase. - -Furthermore, our observation validated previous -results\cite{hase:2010} that the intramolecular heat transport of -alkylthiols is highly effecient. As a combinational effects of these -phenomena, butanethiol acts as a channel to expedite thermal transport -process. The acoustic impedance mismatch between the metal and the -liquid phase can be effectively reduced with the presence of suitable -capping agents. - \begin{figure} -\includegraphics[width=\linewidth]{vibration} -\caption{Vibrational spectra obtained for gold in different - environments.} -\label{specAu} +\includegraphics[width=\linewidth]{uahxnua} +\caption{Vibrational spectra obtained for normal (upper) and + deuterated (lower) hexane in Au-butanethiol/hexane + systems. Butanethiol spectra are shown as reference. Both hexane and + butanethiol were using United-Atom models.} +\label{uahxnua} \end{figure} -[MAY ADD COMPARISON OF AU SLAB WIDTHS BUT NOT MUCH TO TALK ABOUT...] - \section{Conclusions} -The NIVS algorithm we developed has been applied to simulations of -Au-butanethiol surfaces with organic solvents. This algorithm allows -effective unphysical thermal flux transferred between the metal and -the liquid phase. With the flux applied, we were able to measure the -corresponding thermal gradient and to obtain interfacial thermal -conductivities. Under steady states, single trajectory simulation -would be enough for accurate measurement. This would be advantageous -compared to transient state simulations, which need multiple -trajectories to produce reliable average results. +The NIVS algorithm has been applied to simulations of +butanethiol-capped Au(111) surfaces in the presence of organic +solvents. This algorithm allows the application of unphysical thermal +flux to transfer heat between the metal and the liquid phase. With the +flux applied, we were able to measure the corresponding thermal +gradients and to obtain interfacial thermal conductivities. Under +steady states, 2-3 ns trajectory simulations are sufficient for +computation of this quantity. -Our simulations have seen significant conductance enhancement with the -presence of capping agent, compared to the bare gold / liquid +Our simulations have seen significant conductance enhancement in the +presence of capping agent, compared with the bare gold / liquid interfaces. The acoustic impedance mismatch between the metal and the -liquid phase is effectively eliminated by proper capping +liquid phase is effectively eliminated by a chemically-bonded capping agent. Furthermore, the coverage precentage of the capping agent plays an important role in the interfacial thermal transport -process. Moderately lower coverages allow higher contact between -capping agent and solvent, and thus could further enhance the heat -transfer process. +process. Moderately low coverages allow higher contact between capping +agent and solvent, and thus could further enhance the heat transfer +process, giving a non-monotonic behavior of conductance with +increasing coverage. -Our measurement results, particularly of the UA models, agree with -available experimental data. This indicates that our force field -parameters have a nice description of the interactions between the -particles at the interfaces. AA models tend to overestimate the +Our results, particularly using the UA models, agree well with +available experimental data. The AA models tend to overestimate the interfacial thermal conductance in that the classically treated C-H -vibration would be overly sampled. Compared to the AA models, the UA -models have higher computational efficiency with satisfactory -accuracy, and thus are preferable in interfacial thermal transport -modelings. Of the two definitions for $G$, the discrete form +vibrations become too easily populated. Compared to the AA models, the +UA models have higher computational efficiency with satisfactory +accuracy, and thus are preferable in modeling interfacial thermal +transport. + +Of the two definitions for $G$, the discrete form (Eq. \ref{discreteG}) was easier to use and gives out relatively consistent results, while the derivative form (Eq. \ref{derivativeG}) is not as versatile. Although $G^\prime$ gives out comparable results and follows similar trend with $G$ when measuring close to fully -covered or bare surfaces, the spatial resolution of $T$ profile is -limited for accurate computation of derivatives data. +covered or bare surfaces, the spatial resolution of $T$ profile +required for the use of a derivative form is limited by the number of +bins and the sampling required to obtain thermal gradient information. -Vlugt {\it et al.} has investigated the surface thiol structures for -nanocrystal gold and pointed out that they differs from those of the -Au(111) surface\cite{landman:1998,vlugt:cpc2007154}. This difference -might lead to change of interfacial thermal transport behavior as -well. To investigate this problem, an effective means to introduce -thermal flux and measure the corresponding thermal gradient is -desirable for simulating structures with spherical symmetry. +Vlugt {\it et al.} have investigated the surface thiol structures for +nanocrystalline gold and pointed out that they differ from those of +the Au(111) surface.\cite{landman:1998,vlugt:cpc2007154} This +difference could also cause differences in the interfacial thermal +transport behavior. To investigate this problem, one would need an +effective method for applying thermal gradients in non-planar +(i.e. spherical) geometries. \section{Acknowledgments} Support for this project was provided by the National Science Foundation under grant CHE-0848243. Computational time was provided by the Center for Research Computing (CRC) at the University of Notre -Dame. \newpage +Dame. +\newpage \bibliography{interfacial}