--- stokes/stokes.tex 2011/12/02 20:14:03 3770 +++ stokes/stokes.tex 2011/12/10 23:46:08 3779 @@ -43,25 +43,17 @@ Notre Dame, Indiana 46556} \begin{doublespace} \begin{abstract} - REPLACE ABSTRACT HERE - With the Non-Isotropic Velocity Scaling (NIVS) approach to Reverse - Non-Equilibrium Molecular Dynamics (RNEMD) it is possible to impose - an unphysical thermal flux between different regions of - inhomogeneous systems such as solid / liquid interfaces. We have - applied NIVS to compute the interfacial thermal conductance at a - metal / organic solvent interface that has been chemically capped by - butanethiol molecules. Our calculations suggest that coupling - between the metal and liquid phases is enhanced by the capping - agents, leading to a greatly enhanced conductivity at the interface. - Specifically, the chemical bond between the metal and the capping - agent introduces a vibrational overlap that is not present without - the capping agent, and the overlap between the vibrational spectra - (metal to cap, cap to solvent) provides a mechanism for rapid - thermal transport across the interface. Our calculations also - suggest that this is a non-monotonic function of the fractional - coverage of the surface, as moderate coverages allow diffusive heat - transport of solvent molecules that have been in close contact with - the capping agent. + We present a new method for introducing stable nonequilibrium + velocity and temperature gradients in molecular dynamics simulations + of heterogeneous systems. This method conserves the linear momentum + and total energy of the system and improves previous Reverse + Non-Equilibrium Molecular Dynamics (RNEMD) methods and maintains + thermal velocity distributions. It also avoid thermal anisotropy + occured in NIVS simulations by using isotropic velocity scaling on + the molecules in specific regions of a system. To test the method, + we have computed the thermal conductivity and shear viscosity of + model liquid systems as well as the interfacial frictions of a + series of metal/liquid interfaces. \end{abstract} @@ -74,112 +66,67 @@ Notre Dame, Indiana 46556} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} -[DO THIS LATER] +[REFINE LATER, ADD MORE REF.S] +Imposed-flux methods in Molecular Dynamics (MD) +simulations\cite{MullerPlathe:1997xw} can establish steady state +systems with a set applied flux vs a corresponding gradient that can +be measured. These methods does not need many trajectories to provide +information of transport properties of a given system. Thus, they are +utilized in computing thermal and mechanical transfer of homogeneous +or bulk systems as well as heterogeneous systems such as liquid-solid +interfaces.\cite{kuang:AuThl} -[IMPORTANCE OF NANOSCALE TRANSPORT PROPERTIES STUDIES] +The Reverse Non-Equilibrium MD (RNEMD) methods adopt constraints that +satisfy linear momentum and total energy conservation of a system when +imposing fluxes in a simulation. Thus they are compatible with various +ensembles, including the micro-canonical (NVE) ensemble, without the +need of an external thermostat. The original approaches by +M\"{u}ller-Plathe {\it et + al.}\cite{MullerPlathe:1997xw,ISI:000080382700030} utilize simple +momentum swapping for generating energy/momentum fluxes, which is also +compatible with particles of different identities. Although simple to +implement in a simulation, this approach can create nonthermal +velocity distributions, as discovered by Tenney and +Maginn\cite{Maginn:2010}. Furthermore, this approach to kinetic energy +transfer between particles of different identities is less efficient +when the mass difference between the particles becomes significant, +which also limits its application on heterogeneous interfacial +systems. -Due to the importance of heat flow (and heat removal) in -nanotechnology, interfacial thermal conductance has been studied -extensively both experimentally and computationally.\cite{cahill:793} -Nanoscale materials have a significant fraction of their atoms at -interfaces, and the chemical details of these interfaces govern the -thermal transport properties. Furthermore, the interfaces are often -heterogeneous (e.g. solid - liquid), which provides a challenge to -computational methods which have been developed for homogeneous or -bulk systems. +Recently, we developed a different approach, using Non-Isotropic +Velocity Scaling (NIVS) \cite{kuang:164101} algorithm to impose +fluxes. Compared to the momentum swapping move, it scales the velocity +vectors in two separate regions of a simulated system with respective +diagonal scaling matrices. These matrices are determined by solving a +set of equations including linear momentum and kinetic energy +conservation constraints and target flux satisfaction. This method is +able to effectively impose a wide range of kinetic energy fluxes +without obvious perturbation to the velocity distributions of the +simulated systems, regardless of the presence of heterogeneous +interfaces. We have successfully applied this approach in studying the +interfacial thermal conductance at metal-solvent +interfaces.\cite{kuang:AuThl} -Experimentally, the thermal properties of a number of interfaces have -been investigated. Cahill and coworkers studied nanoscale thermal -transport from metal nanoparticle/fluid interfaces, to epitaxial -TiN/single crystal oxides interfaces, and 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 -cetyltrimethylammonium bromide (CTAB) on the thermal transport between -gold nanorods and solvent,\cite{doi:10.1021/jp8051888} and Juv\'e {\it - et al.} studied the cooling dynamics, which is controlled by thermal -interface resistance of glass-embedded metal -nanoparticles.\cite{PhysRevB.80.195406} Although interfaces are -normally considered barriers for heat transport, Alper {\it et al.} -suggested that specific ligands (capping agents) could completely -eliminate this barrier -($G\rightarrow\infty$).\cite{doi:10.1021/la904855s} +However, the NIVS approach limits its application in imposing momentum +fluxes. Temperature anisotropy can happen under high momentum fluxes, +due to the nature of the algorithm. Thus, combining thermal and +momentum flux is also difficult to implement with this +approach. However, such combination may provide a means to simulate +thermal/momentum gradient coupled processes such as freeze +desalination. Therefore, developing novel approaches to extend the +application of imposed-flux method is desired. -The acoustic mismatch model for interfacial conductance utilizes the -acoustic impedance ($Z_a = \rho_a v^s_a$) on both sides of the -interface.\cite{swartz1989} Here, $\rho_a$ and $v^s_a$ are the density -and speed of sound in material $a$. The phonon transmission -probability at the $a-b$ interface is -\begin{equation} -t_{ab} = \frac{4 Z_a Z_b}{\left(Z_a + Z_b \right)^2}, -\end{equation} -and the interfacial conductance can then be approximated as -\begin{equation} -G_{ab} \approx \frac{1}{4} C_D v_D t_{ab} -\end{equation} -where $C_D$ is the Debye heat capacity of the hot side, and $v_D$ is -the Debye phonon velocity ($1/v_D^3 = 1/3v_L^3 + 2/3 v_T^3$) where -$v_L$ and $v_T$ are the longitudinal and transverse speeds of sound, -respectively. For the Au/hexane and Au/toluene interfaces, the -acoustic mismatch model predicts room-temperature $G \approx 87 \mbox{ - and } 129$ MW m$^{-2}$ K$^{-1}$, respectively. However, it is not -clear how to apply the acoustic mismatch model to a -chemically-modified surface, particularly when the acoustic properties -of a monolayer film may not be well characterized. +In this paper, we improve the NIVS method and propose a novel approach +to impose fluxes. This approach separate the means of applying +momentum and thermal flux with operations in one time step and thus is +able to simutaneously impose thermal and momentum flux. Furthermore, +the approach retains desirable features of previous RNEMD approaches +and is simpler to implement compared to the NIVS method. In what +follows, we first present the method to implement the method in a +simulation. Then we compare the method on bulk fluids to previous +methods. Also, interfacial frictions are computed for a series of +interfaces. -[PREVIOUS METHODS INCLUDING NIVS AND THEIR LIMITATIONS] -[DIFFICULTY TO GENERATE JZKE AND JZP SIMUTANEOUSLY] - -More precise computational models have also been used to study the -interfacial thermal transport in order to gain an understanding of -this phenomena at the molecular level. Recently, Hase and coworkers -employed Non-Equilibrium Molecular Dynamics (NEMD) simulations to -study thermal transport from hot Au(111) substrate to a self-assembled -monolayer of alkylthiol with relatively long chain (8-20 carbon -atoms).\cite{hase:2010,hase:2011} However, ensemble averaged -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\cite{doi:10.1080/0026897031000068578} or forward NEMD simulation -methods. Therefore, the Reverse NEMD (RNEMD) -methods\cite{MullerPlathe:1997xw,kuang:164101} would be advantageous -in that they {\it apply} the difficult to measure quantity (flux), -while {\it measuring} the easily-computed quantity (the thermal -gradient). This is particularly true for inhomogeneous interfaces -where it would not be clear how to apply a gradient {\it a priori}. -Garde and coworkers\cite{garde:nl2005,garde:PhysRevLett2009} applied -this approach to various liquid interfaces and studied how thermal -conductance (or resistance) is dependent on chemical details of a -number of hydrophobic and hydrophilic aqueous interfaces. And -recently, Luo {\it et al.} studied the thermal conductance of -Au-SAM-Au junctions using the same approach, comparing to a constant -temperature difference method.\cite{Luo20101} While this latter -approach establishes more ideal Maxwell-Boltzmann distributions than -previous RNEMD methods, it does not guarantee momentum or kinetic -energy conservation. - -Recently, we have developed a Non-Isotropic Velocity Scaling (NIVS) -algorithm for RNEMD simulations\cite{kuang:164101}. This algorithm -retains the desirable features of RNEMD (conservation of linear -momentum and total energy, compatibility with periodic boundary -conditions) while establishing true thermal distributions in each of -the two slabs. Furthermore, it allows effective thermal exchange -between particles of different identities, and thus makes the study of -interfacial conductance much simpler. - -[WHAT IS COVERED IN THIS MANUSCRIPT] -[MAY PUT FIGURE 1 HERE] -The work presented here deals with the Au(111) surface covered to -varying degrees by butanethiol, a capping agent with short carbon -chain, and solvated with organic solvents of different molecular -properties. Different models were used for both the capping agent and -the solvent force field parameters. Using the NIVS algorithm, the -thermal transport across these interfaces was studied and the -underlying mechanism for the phenomena was investigated. - \section{Methodology} Similar to the NIVS methodology,\cite{kuang:164101} we consider a periodic system divided into a series of slabs along a certain axis @@ -218,7 +165,7 @@ thermal flux $J_z$, one would have The above operations conserve the linear momentum of a periodic system. To satisfy total energy conservation as well as to impose a thermal flux $J_z$, one would have -%SUPPORT INFO MIGHT BE NECESSARY TO PUT EXTRA MATH IN +[SUPPORT INFO MIGHT BE NECESSARY TO PUT EXTRA MATH IN] \begin{eqnarray} K_c - J_z\Delta t & = & c^2 (K_c - \frac{1}{2}M_c \langle\vec{v}_c \rangle^2) + \frac{1}{2}M_c (\langle \vec{v}_c \rangle + \vec{a}_c)^2 \\ @@ -236,786 +183,568 @@ the positive roots (which are closer to 1) are chosen. scaling coefficients ($c$ and $h$) as well as $\vec{a}_c$ and $\vec{a}_h$. Note that two roots of $c$ and $h$ exist respectively. However, to avoid dramatic perturbations to a system, -the positive roots (which are closer to 1) are chosen. - +the positive roots (which are closer to 1) are chosen. Figure +\ref{method} illustrates the implementation of this algorithm in an +individual step. + +\begin{figure} +\includegraphics[width=\linewidth]{method} +\caption{Illustration of the implementation of the algorithm in a + single step. Starting from an ideal velocity distribution, the + transformation is used to apply both thermal and momentum flux from + the ``c'' slab to the ``h'' slab. As the figure shows, the thermal + distributions preserve after this operation.} +\label{method} +\end{figure} + By implementing these operations at a certain frequency, a steady thermal and/or momentum flux can be applied and the corresponding temperature and/or momentum gradients can be established. -[REFER TO NIVS PAPER] -[ADVANTAGES] -Steady state MD simulations have an advantage in that not many -trajectories are needed to study the relationship between thermal flux -and thermal gradients. For systems with low interfacial conductance, -one must have a method capable of generating or measuring relatively -small fluxes, compared to those required for bulk conductivity. This -requirement makes the calculation even more difficult for -slowly-converging equilibrium methods.\cite{Viscardy:2007lq} Forward -NEMD methods impose a gradient (and measure a flux), but at interfaces -it is not clear what behavior should be imposed at the boundaries -between materials. Imposed-flux reverse non-equilibrium -methods\cite{MullerPlathe:1997xw} set the flux {\it a priori} and -the thermal response becomes an easy-to-measure quantity. Although -M\"{u}ller-Plathe's original momentum swapping approach can be used -for exchanging energy between particles of different identity, the -kinetic energy transfer efficiency is affected by the mass difference -between the particles, which limits its application on heterogeneous -interfacial systems. +This approach is more computationaly efficient compared to the +previous NIVS method, in that only quadratic equations are involved, +while the NIVS method needs to solve a quartic equations. Furthermore, +the method implements isotropic scaling of velocities in respective +slabs, unlike the NIVS, where an extra criteria function is necessary +to choose a set of coefficients that performs the most isotropic +scaling. More importantly, separating the momentum flux imposing from +velocity scaling avoids the underlying cause that NIVS produced +thermal anisotropy when applying a momentum flux. -The non-isotropic velocity scaling (NIVS) \cite{kuang:164101} approach -to non-equilibrium MD simulations is able to impose a wide range of -kinetic energy fluxes without obvious perturbation to the velocity -distributions of the simulated systems. Furthermore, this approach has -the advantage in heterogeneous interfaces in that kinetic energy flux -can be applied between regions of particles of arbitrary identity, and -the flux will not be restricted by difference in particle mass. +The advantages of the approach over the original momentum swapping +approach lies in its nature to preserve a Gaussian +distribution. Because the momentum swapping tends to render a +nonthermal distribution, when the imposed flux is relatively large, +diffusion of the neighboring slabs could no longer remedy this effect, +and nonthermal distributions would be observed. Results in later +section will illustrate this effect. -The NIVS algorithm scales the velocity vectors in two separate regions -of a simulation system with respective diagonal scaling matrices. To -determine these scaling factors in the matrices, a set of equations -including linear momentum conservation and kinetic energy conservation -constraints and target energy flux satisfaction is solved. With the -scaling operation applied to the system in a set frequency, bulk -temperature gradients can be easily established, and these can be used -for computing thermal conductivities. The NIVS algorithm conserves -momenta and energy and does not depend on an external thermostat. +\section{Computational Details} +The algorithm has been implemented in our MD simulation code, +OpenMD\cite{Meineke:2005gd,openmd}. We compare the method with +previous RNEMD methods or equilibrium MD methods in homogeneous fluids +(Lennard-Jones and SPC/E water). And taking advantage of the method, +we simulate the interfacial friction of different heterogeneous +interfaces (gold-organic solvent and gold-SPC/E water and ice-liquid +water). -\subsection{Defining Interfacial Thermal Conductivity ($G$)} +\subsection{Simulation Protocols} +The systems to be investigated are set up in a orthorhombic simulation +cell with periodic boundary conditions in all three dimensions. The +$z$ axis of these cells were longer and was set as the gradient axis +of temperature and/or momentum. Thus the cells were divided into $N$ +slabs along this axis, with various $N$ depending on individual +system. The $x$ and $y$ axis were usually of the same length in +homogeneous systems or close to each other where interfaces +presents. In all cases, before introducing a nonequilibrium method to +establish steady thermal and/or momentum gradients for further +measurements and calculations, canonical ensemble with a Nos\'e-Hoover +thermostat\cite{hoover85} and microcanonical ensemble equilibrations +were used to prepare systems ready for data +collections. Isobaric-isothermal equilibrations are performed before +this for SPC/E water systems to reach normal pressure (1 bar), while +similar equilibrations are used for interfacial systems to relax the +surface tensions. -For an interface with relatively low interfacial conductance, and a -thermal flux between two distinct bulk regions, the regions on either -side of the interface rapidly come to a state in which the two phases -have relatively homogeneous (but distinct) temperatures. The -interfacial thermal conductivity $G$ can therefore be approximated as: -\begin{equation} - G = \frac{E_{total}}{2 t L_x L_y \left( \langle T_\mathrm{hot}\rangle - - \langle T_\mathrm{cold}\rangle \right)} -\label{lowG} -\end{equation} -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. For an applied flux $J_z$ -operating over a simulation time $t$ on a periodically-replicated slab -of dimensions $L_x \times L_y$, $E_{total} = 2 J_z t L_x L_y$. +While homogeneous fluid systems can be set up with random +configurations, our interfacial systems needs extra steps to ensure +the interfaces be established properly for computations. The +preparation and equilibration of butanethiol covered gold (111) +surface and further solvation and equilibration process is described +as in reference \cite{kuang:AuThl}. -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 dividing 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} -\end{equation} +As for the ice/liquid water interfaces, the basal surface of ice +lattice was first constructed. Hirsch {\it et + al.}\cite{doi:10.1021/jp048434u} explored the energetics of ice +lattices with different proton orders. We refer to their results and +choose the configuration of the lowest energy after geometry +optimization as the unit cells of our ice lattices. Although +experimental solid/liquid coexistant temperature near normal pressure +is 273K, Bryk and Haymet's simulations of ice/liquid water interfaces +with different models suggest that for SPC/E, the most stable +interface is observed at 225$\pm$5K. Therefore, all our ice/liquid +water simulations were carried out under 225K. To have extra +protection of the ice lattice during initial equilibration (when the +randomly generated liquid phase configuration could release large +amount of energy in relaxation), a constraint method (REF?) was +adopted until the high energy configuration was relaxed. +[MAY ADD A FIGURE HERE FOR BASAL PLANE, MAY INCLUDE PRISM IF POSSIBLE] -\begin{figure} -\includegraphics[width=\linewidth]{method} -\caption{Interfacial conductance can be calculated by applying an - (unphysical) kinetic energy flux between two slabs, one located - within the metal and another on the edge of the periodic box. The - system responds by forming a thermal gradient. In bulk liquids, - this gradient typically has a single slope, but in interfacial - systems, there are distinct thermal conductivity domains. The - interfacial conductance, $G$ is found by measuring the temperature - gap at the Gibbs dividing surface, or by using second derivatives of - the thermal profile.} -\label{demoPic} -\end{figure} +\subsection{Force Field Parameters} +For comparison of our new method with previous work, we retain our +force field parameters consistent with the results we will compare +with. The Lennard-Jones fluid used here for argon , and reduced unit +results are reported for direct comparison purpose. -Another approach is to assume that the temperature is continuous and -differentiable throughout the space. Given that $\lambda$ is also -differentiable, $G$ can be defined as its gradient ($\nabla\lambda$) -projected along a vector normal to the interface ($\mathbf{\hat{n}}$) -and evaluated at the interface location ($z_0$). This quantity, -\begin{align} -G^\prime &= \left(\nabla\lambda \cdot \mathbf{\hat{n}}\right)_{z_0} \\ - &= \frac{\partial}{\partial z}\left(-\frac{J_z}{ - \left(\frac{\partial T}{\partial z}\right)}\right)_{z_0} \\ - &= J_z\left(\frac{\partial^2 T}{\partial z^2}\right)_{z_0} \Big/ - \left(\frac{\partial T}{\partial z}\right)_{z_0}^2 \label{derivativeG} -\end{align} -has the same units as the common definition for $G$, and the maximum -of its magnitude denotes where thermal conductivity has the largest -change, i.e. the interface. In the geometry used in this study, the -vector normal to the interface points along the $z$ axis, as do -$\vec{J}$ and the thermal gradient. This yields the simplified -expressions in Eq. \ref{derivativeG}. +As for our water simulations, SPC/E model is used throughout this work +for consistency. Previous work for transport properties of SPC/E water +model is available\cite{Bedrov:2000,10.1063/1.3330544,Medina2011} so +that unnecessary repetition of previous methods can be avoided. -With temperature profiles obtained from simulation, one is able to -approximate the first and second derivatives of $T$ with finite -difference methods and calculate $G^\prime$. In what follows, both -definitions have been used, and are compared in the results. +The Au-Au interaction parameters in all simulations are described by +the quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The +QSC potentials include zero-point quantum corrections and are +reparametrized for accurate surface energies compared to the +Sutton-Chen potentials.\cite{Chen90} For gold/water interfaces, the +Spohr potential was adopted\cite{ISI:000167766600035} to depict +Au-H$_2$O interactions. -To investigate the interfacial conductivity at metal / solvent -interfaces, we have modeled a metal slab with its (111) surfaces -perpendicular to the $z$-axis of our simulation cells. The metal slab -has been prepared both with and without capping agents on the exposed -surface, and has been solvated with simple organic solvents, as -illustrated in Figure \ref{gradT}. +The small organic molecules included in our simulations are the Au +surface capping agent butanethiol and liquid hexane and toluene. The +United-Atom +models\cite{TraPPE-UA.thiols,TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} +for these components were used in this work for better computational +efficiency, while maintaining good accuracy. We refer readers to our +previous work\cite{kuang:AuThl} for further details of these models, +as well as the interactions between Au and the above organic molecule +components. -With the simulation cell described above, we are able to equilibrate -the system and impose an unphysical thermal flux between the liquid -and the metal phase using the NIVS algorithm. By periodically applying -the unphysical flux, we obtained a temperature profile and its spatial -derivatives. Figure \ref{gradT} shows how an applied thermal flux can -be used to obtain the 1st and 2nd derivatives of the temperature -profile. +\subsection{Thermal conductivities} +When $\vec{j}_z(\vec{p})$ is set to zero and a target $J_z$ is set to +impose kinetic energy transfer, the method can be used for thermal +conductivity computations. Similar to previous RNEMD methods, we +assume linear response of the temperature gradient with respect to the +thermal flux in general case. And the thermal conductivity ($\lambda$) +can be obtained with the imposed kinetic energy flux and the measured +thermal gradient: +\begin{equation} +J_z = -\lambda \frac{\partial T}{\partial z} +\end{equation} +Like other imposed-flux methods, the energy flux was calculated using +the total non-physical energy transferred (${E_{total}}$) from slab +``c'' to slab ``h'', which is recorded throughout a simulation, and +the time for data collection $t$: +\begin{equation} +J_z = \frac{E_{total}}{2 t L_x L_y} +\end{equation} +where $L_x$ and $L_y$ denotes the dimensions of the plane in a +simulation cell perpendicular to the thermal gradient, and a factor of +two in the denominator is present for the heat transport occurs in +both $+z$ and $-z$ directions. The temperature gradient +${\langle\partial T/\partial z\rangle}$ can be obtained by a linear +regression of the temperature profile, which is recorded during a +simulation for each slab in a cell. For Lennard-Jones simulations, +thermal conductivities are reported in reduced units +(${\lambda^*=\lambda \sigma^2 m^{1/2} k_B^{-1}\varepsilon^{-1/2}}$). -\begin{figure} -\includegraphics[width=\linewidth]{gradT} -\caption{A sample of Au (111) / butanethiol / hexane interfacial - system with the temperature profile after a kinetic energy flux has - been imposed. Note that the largest temperature jump in the thermal - profile (corresponding to the lowest interfacial conductance) is at - the interface between the butanethiol molecules (blue) and the - solvent (grey). First and second derivatives of the temperature - profile are obtained using a finite difference approximation (lower - panel).} -\label{gradT} -\end{figure} +\subsection{Shear viscosities} +Alternatively, the method can carry out shear viscosity calculations +by switching off $J_z$. One can specify the vector +$\vec{j}_z(\vec{p})$ by choosing the three components +respectively. For shear viscosity simulations, $j_z(p_z)$ is usually +set to zero. Although for isotropic systems, the direction of +$\vec{j}_z(\vec{p})$ on the $xy$ plane should not matter, the ability +of arbitarily specifying the vector direction in our method provides +convenience in anisotropic simulations. -\section{Computational Details} -\subsection{Simulation Protocol} -The NIVS algorithm has been implemented in our MD simulation code, -OpenMD\cite{Meineke:2005gd,openmd}, and was used for our simulations. -Metal slabs of 6 or 11 layers of Au atoms were first equilibrated -under atmospheric pressure (1 atm) and 200K. After equilibration, -butanethiol capping agents were placed at three-fold hollow sites on -the Au(111) surfaces. These sites are either {\it fcc} or {\it - hcp} sites, although Hase {\it et al.} found that they are -equivalent in a heat transfer process,\cite{hase:2010} so we did not -distinguish between these sites in our study. The maximum butanethiol -capacity on Au surface is $1/3$ of the total number of surface Au -atoms, and the packing forms a $(\sqrt{3}\times\sqrt{3})R30^\circ$ -structure\cite{doi:10.1021/ja00008a001,doi:10.1021/cr9801317}. A -series of lower coverages was also prepared by eliminating -butanethiols from the higher coverage surface in a regular manner. The -lower coverages were prepared in order to study the relation between -coverage and interfacial conductance. +Similar to thermal conductivity computations, linear response of the +momentum gradient with respect to the shear stress is assumed, and the +shear viscosity ($\eta$) can be obtained with the imposed momentum +flux (e.g. in $x$ direction) and the measured gradient: +\begin{equation} +j_z(p_x) = -\eta \frac{\partial v_x}{\partial z} +\end{equation} +where the flux is similarly defined: +\begin{equation} +j_z(p_x) = \frac{P_x}{2 t L_x L_y} +\end{equation} +with $P_x$ being the total non-physical momentum transferred within +the data collection time. Also, the velocity gradient +${\langle\partial v_x/\partial z\rangle}$ can be obtained using linear +regression of the $x$ component of the mean velocity, $\langle +v_x\rangle$, in each of the bins. For Lennard-Jones simulations, shear +viscosities are reported in reduced units +(${\eta^*=\eta\sigma^2(\varepsilon m)^{-1/2}}$). -The capping agent molecules were allowed to migrate during the -simulations. They distributed themselves uniformly and sampled a -number of three-fold sites throughout out study. Therefore, the -initial configuration does not noticeably affect the sampling of a -variety of configurations of the same coverage, and the final -conductance measurement would be an average effect of these -configurations explored in the simulations. - -After the modified Au-butanethiol surface systems were equilibrated in -the canonical (NVT) ensemble, organic solvent molecules were packed in -the previously empty part of the simulation cells.\cite{packmol} Two -solvents were investigated, one which has little vibrational overlap -with the alkanethiol and which has a planar shape (toluene), and one -which has similar vibrational frequencies to the capping agent and -chain-like shape ({\it n}-hexane). - -The simulation cells were not particularly extensive along the -$z$-axis, as a very long length scale for the thermal gradient may -cause excessively hot or cold temperatures in the middle of the -solvent region and lead to undesired phenomena such as solvent boiling -or freezing when a thermal flux is applied. Conversely, too few -solvent molecules would change the normal behavior of the liquid -phase. Therefore, our $N_{solvent}$ values were chosen to ensure that -these extreme cases did not happen to our simulations. The spacing -between periodic images of the gold interfaces is $45 \sim 75$\AA in -our simulations. - -The initial configurations generated are further equilibrated with the -$x$ and $y$ dimensions fixed, only allowing the $z$-length scale to -change. This is to ensure that the equilibration of liquid phase does -not affect the metal's crystalline structure. Comparisons were made -with simulations that allowed changes of $L_x$ and $L_y$ during NPT -equilibration. No substantial changes in the box geometry were noticed -in these simulations. After ensuring the liquid phase reaches -equilibrium at atmospheric pressure (1 atm), further equilibration was -carried out under canonical (NVT) and microcanonical (NVE) ensembles. - -After the systems reach equilibrium, NIVS was used to impose an -unphysical thermal flux between the metal and the liquid phases. Most -of our simulations were done under an average temperature of -$\sim$200K. Therefore, thermal flux usually came from the metal to the -liquid so that the liquid has a higher temperature and would not -freeze due to lowered temperatures. After this induced temperature -gradient had stabilized, the temperature profile of the simulation cell -was recorded. To do this, the simulation cell is divided evenly into -$N$ slabs along the $z$-axis. The average temperatures of each slab -are recorded for 1$\sim$2 ns. When the slab width $d$ of each slab is -the same, the derivatives of $T$ with respect to slab number $n$ can -be directly used for $G^\prime$ calculations: \begin{equation} - G^\prime = |J_z|\Big|\frac{\partial^2 T}{\partial z^2}\Big| - \Big/\left(\frac{\partial T}{\partial z}\right)^2 - = |J_z|\Big|\frac{1}{d^2}\frac{\partial^2 T}{\partial n^2}\Big| - \Big/\left(\frac{1}{d}\frac{\partial T}{\partial n}\right)^2 - = |J_z|\Big|\frac{\partial^2 T}{\partial n^2}\Big| - \Big/\left(\frac{\partial T}{\partial n}\right)^2 -\label{derivativeG2} +\subsection{Interfacial friction and Slip length} +While the shear stress results in a velocity gradient within bulk +fluid phase, its effect at a solid-liquid interface could vary due to +the interaction strength between the two phases. The interfacial +friction coefficient $\kappa$ is defined to relate the shear stress +(e.g. along $x$-axis) and the relative fluid velocity tangent to the +interface: +\begin{equation} +j_z(p_x)|_{interface} = \kappa\Delta v_x|_{interface} \end{equation} -The absolute values in Eq. \ref{derivativeG2} appear because the -direction of the flux $\vec{J}$ is in an opposing direction on either -side of the metal slab. +Under ``stick'' boundary condition, $\Delta v_x|_{interface} +\rightarrow 0$, which leads to $\kappa\rightarrow\infty$. However, for +``slip'' boundary condition at the solid-liquid interface, $\kappa$ +becomes finite. To characterize the interfacial boundary conditions, +slip length ($\delta$) is defined using $\kappa$ and the shear +viscocity of liquid phase ($\eta$): +\begin{equation} +\delta = \frac{\eta}{\kappa} +\end{equation} +so that $\delta\rightarrow 0$ in the ``no-slip'' boundary condition, +and depicts how ``slippery'' an interface is. Figure \ref{slipLength} +illustrates how this quantity is defined and computed for a +solid-liquid interface. [MAY INCLUDE A SNAPSHOT IN FIGURE] -All of the above simulation procedures use a time step of 1 fs. Each -equilibration stage took a minimum of 100 ps, although in some cases, -longer equilibration stages were utilized. - -\subsection{Force Field Parameters} -Our simulations include a number of chemically distinct components. -Figure \ref{demoMol} demonstrates the sites defined for both -United-Atom and All-Atom models of the organic solvent and capping -agents in our simulations. Force field parameters are needed for -interactions both between the same type of particles and between -particles of different species. - \begin{figure} -\includegraphics[width=\linewidth]{structures} -\caption{Structures of the capping agent and solvents utilized in - 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 References - \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 1 in the supporting information.} -\label{demoMol} +\includegraphics[width=\linewidth]{defDelta} +\caption{The slip length $\delta$ can be obtained from a velocity + profile of a solid-liquid interface. An example of Au/hexane + interfaces is shown.} +\label{slipLength} \end{figure} -The Au-Au interactions in metal lattice slab is described by the -quantum Sutton-Chen (QSC) formulation.\cite{PhysRevB.59.3527} The QSC -potentials include zero-point quantum corrections and are -reparametrized for accurate surface energies compared to the -Sutton-Chen potentials.\cite{Chen90} - -For the two solvent molecules, {\it n}-hexane and toluene, two -different atomistic models were utilized. Both solvents were modeled -using United-Atom (UA) and All-Atom (AA) models. The TraPPE-UA -parameters\cite{TraPPE-UA.alkanes,TraPPE-UA.alkylbenzenes} are used -for our UA solvent molecules. In these models, sites are located at -the carbon centers for alkyl groups. Bonding interactions, including -bond stretches and bends and torsions, were used for intra-molecular -sites closer than 3 bonds. For non-bonded interactions, Lennard-Jones -potentials are used. - -By eliminating explicit hydrogen atoms, the TraPPE-UA models are -simple and computationally efficient, while maintaining good accuracy. -However, the TraPPE-UA model for alkanes is known to predict a slightly -lower boiling point than experimental values. This is one of the -reasons we used a lower average temperature (200K) for our -simulations. If heat is transferred to the liquid phase during the -NIVS simulation, the liquid in the hot slab can actually be -substantially warmer than the mean temperature in the simulation. The -lower mean temperatures therefore prevent solvent boiling. - -For UA-toluene, the non-bonded potentials between intermolecular sites -have a similar Lennard-Jones formulation. The toluene molecules were -treated as a single rigid body, so there was no need for -intramolecular interactions (including bonds, bends, or torsions) in -this solvent model. - -Besides the TraPPE-UA models, AA models for both organic solvents are -included in our studies as well. The OPLS-AA\cite{OPLSAA} force fields -were used. For hexane, additional explicit hydrogen sites were -included. Besides bonding and non-bonded site-site interactions, -partial charges and the electrostatic interactions were added to each -CT and HC site. For toluene, a flexible model for the toluene molecule -was utilized which included bond, bend, torsion, and inversion -potentials to enforce ring planarity. - -The butanethiol capping agent in our simulations, were also modeled -with both UA and AA model. The TraPPE-UA force field includes -parameters for thiol molecules\cite{TraPPE-UA.thiols} and are used for -UA butanethiol model in our simulations. The OPLS-AA also provides -parameters for alkyl thiols. However, alkyl thiols adsorbed on Au(111) -surfaces do not have the hydrogen atom bonded to sulfur. To derive -suitable parameters for butanethiol adsorbed on Au(111) surfaces, we -adopt the S parameters from Luedtke and Landman\cite{landman:1998} and -modify the parameters for the CTS atom to maintain charge neutrality -in the molecule. Note that the model choice (UA or AA) for the capping -agent can be different from the solvent. Regardless of model choice, -the force field parameters for interactions between capping agent and -solvent can be derived using Lorentz-Berthelot Mixing Rule: -\begin{eqnarray} - \sigma_{ij} & = & \frac{1}{2} \left(\sigma_{ii} + \sigma_{jj}\right) \\ - \epsilon_{ij} & = & \sqrt{\epsilon_{ii}\epsilon_{jj}} -\end{eqnarray} - -To describe the interactions between metal (Au) and non-metal atoms, -we refer to an adsorption study of alkyl thiols on gold surfaces by -Vlugt {\it et al.}\cite{vlugt:cpc2007154} They fitted an effective -Lennard-Jones form of potential parameters for the interaction between -Au and pseudo-atoms CH$_x$ and S based on a well-established and -widely-used effective potential of Hautman and Klein for the Au(111) -surface.\cite{hautman:4994} As our simulations require the gold slab -to be flexible to accommodate thermal excitation, the pair-wise form -of potentials they developed was used for our study. - -The potentials developed from {\it ab initio} calculations by Leng -{\it et al.}\cite{doi:10.1021/jp034405s} are adopted for the -interactions between Au and aromatic C/H atoms in toluene. However, -the Lennard-Jones parameters between Au and other types of particles, -(e.g. AA alkanes) have not yet been established. For these -interactions, the Lorentz-Berthelot mixing rule can be used to derive -effective single-atom LJ parameters for the metal using the fit values -for toluene. These are then used to construct reasonable mixing -parameters for the interactions between the gold and other atoms. -Table 1 in the supporting information summarizes the -``metal/non-metal'' parameters utilized in our simulations. - -\section{Results} -[L-J COMPARED TO RENMD NIVS; WATER COMPARED TO RNEMD NIVS; -SLIP BOUNDARY VS STICK BOUNDARY; ICE-WATER INTERFACES] - -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{Effects due to capping agent coverage} - -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. - -\begin{figure} -\includegraphics[width=\linewidth]{coverage} -\caption{The interfacial thermal conductivity ($G$) has a - non-monotonic dependence on the degree of surface capping. This - data is for the Au(111) / butanethiol / solvent interface with - various UA force fields at $\langle T\rangle \sim $200K.} -\label{coverage} -\end{figure} - -In partially covered surfaces, the derivative definition for -$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. Capping agents are clearly -playing a major role in thermal transport at metal / organic solvent -surfaces. +In our method, a shear stress can be applied similar to shear +viscosity computations by applying an unphysical momentum flux +(e.g. $j_z(p_x)$). A corresponding velocity profile can be obtained as +shown in Figure \ref{slipLength}, in which the velocity gradients +within liquid phase and velocity difference at the liquid-solid +interface can be measured respectively. Further calculations and +characterizations of the interface can be carried out using these +data. -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. +\section{Results and Discussions} +\subsection{Lennard-Jones fluid} +Our orthorhombic simulation cell of Lennard-Jones fluid has identical +parameters to our previous work\cite{kuang:164101} to facilitate +comparison. Thermal conductivitis and shear viscosities were computed +with the algorithm applied to the simulations. The results of thermal +conductivity are compared with our previous NIVS algorithm. However, +since the NIVS algorithm could produce temperature anisotropy for +shear viscocity computations, these results are instead compared to +the momentum swapping approaches. Table \ref{LJ} lists these +calculations with various fluxes in reduced units. -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{Thermal conductivity ($\lambda^*$) and shear viscosity + ($\eta^*$) (in reduced units) of a Lennard-Jones fluid at + ${\langle T^* \rangle = 0.72}$ and ${\rho^* = 0.85}$ computed + at various momentum fluxes. The new method yields similar + results to previous RNEMD methods. All results are reported in + reduced unit. Uncertainties are indicated in parentheses.} - \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. Here ``D'' stands for deuterated - solvent or capping agent molecules. Error estimates are - indicated in parentheses.} - - \begin{tabular}{llccc} + \begin{tabular}{cccccc} \hline\hline - Butanethiol model & Solvent & $G$ & $G^\prime$ \\ - (or bare surface) & model & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ + \multicolumn{2}{c}{Momentum Exchange} & + \multicolumn{2}{c}{$\lambda^*$} & + \multicolumn{2}{c}{$\eta^*$} \\ \hline - UA & UA hexane & 131(9) & 87(10) \\ - & UA hexane(D) & 153(5) & 136(13) \\ - & AA hexane & 131(6) & 122(10) \\ - & UA toluene & 187(16) & 151(11) \\ - & AA toluene & 200(36) & 149(53) \\ + Swap Interval $t^*$ & Equivalent $J_z^*$ or $j_z^*(p_x)$ & + NIVS & This work & Swapping & This work \\ \hline - AA & UA hexane & 116(9) & 129(8) \\ - & AA hexane & 442(14) & 356(31) \\ - & AA hexane(D) & 222(12) & 234(54) \\ - & UA toluene & 125(25) & 97(60) \\ - & AA toluene & 487(56) & 290(42) \\ - \hline - AA(D) & UA hexane & 158(25) & 172(4) \\ - & AA hexane & 243(29) & 191(11) \\ - & AA toluene & 364(36) & 322(67) \\ - \hline - bare & UA hexane & 46.5(3.2) & 49.4(4.5) \\ - & UA hexane(D) & 43.9(4.6) & 43.0(2.0) \\ - & AA hexane & 31.0(1.4) & 29.4(1.3) \\ - & UA toluene & 70.1(1.3) & 65.8(0.5) \\ + 0.116 & 0.16 & 7.30(0.10) & 6.25(0.23) & 3.57(0.06) & 3.53(0.16)\\ + 0.232 & 0.09 & 6.95(0.09) & 8.02(0.56) & 3.64(0.05) & 3.43(0.17)\\ + 0.463 & 0.047 & 7.19(0.07) & 6.48(0.15) & 3.52(0.16) & 3.51(0.08)\\ + 0.926 & 0.024 & 7.19(0.28) & 7.02(0.13) & 3.72(0.05) & 3.79(0.11)\\ + 1.158 & 0.019 & 7.98(0.33) & 7.37(0.23) & 3.42(0.06) & 3.53(0.06)\\ \hline\hline \end{tabular} - \label{modelTest} + \label{LJ} \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. +\subsubsection{Thermal conductivity} +Our thermal conductivity calculations with this method yields +comparable results to the previous NIVS algorithm. This indicates that +the thermal gradients rendered using this method are also close to +previous RNEMD methods. Simulations with moderately higher thermal +fluxes tend to yield more reliable thermal gradients and thus avoid +large errors, while overly high thermal fluxes could introduce side +effects such as non-linear temperature gradient response or +inadvertent phase transitions. -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. +Since the scaling operation is isotropic in this method, one does not +need extra care to ensure temperature isotropy between the $x$, $y$ +and $z$ axes, while thermal anisotropy might happen if the criteria +function for choosing scaling coefficients does not perform as +expected. Furthermore, this method avoids inadvertent concomitant +momentum flux when only thermal flux is imposed, which could not be +achieved with swapping or NIVS approaches. The thermal energy exchange +in swapping ($\vec{p}_i$ in slab ``c'' with $\vec{p}_j$ in slab ``h'') +or NIVS (total slab momentum conponemts $P^\alpha$ scaled to $\alpha +P^\alpha$) would not obtain this result unless thermal flux vanishes +(i.e. $\vec{p}_i=\vec{p}_j$ or $\alpha=1$ which are trivial to apply a +thermal flux). In this sense, this method contributes to having +minimal perturbation to a simulation while imposing thermal flux. -For the fully-covered surfaces, the choice of force field for the -capping agent and solvent has a large impact on the calculated 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. - -\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 -arbitrary 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 2 and 3 in the -supporting information. 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. - -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 diffusive 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. - -\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. In raising -the average temperature from 200K to 250K, the density drop of -$\sim$20\% in the solvent phase leads to a $\sim$40\% 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 $\sim$20\% drop in thermal conductivity. - -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. - -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 been -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 appears to be more resistent to the -reconstruction. 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. That said, we did -observe butanethiols migrating to neighboring three-fold sites during -a simulation. Since the interface persisted in these simulations, we -were able to obtain $G$'s for these interfaces even at a relatively -high temperature without being affected by surface reconstructions. - -\section{Discussion} -[COMBINE W. RESULTS] -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). - -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 occurring 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} - -\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. This supports the results of Luo {\it et - al.}\cite{Luo20101}, who reported $G$ for Au-SAM junctions roughly -twice as large as what we have computed for the thiol-liquid -interfaces. - -\begin{figure} -\includegraphics[width=\linewidth]{vibration} -\caption{The vibrational power spectrum for thiol-capped gold has an - additional vibrational peak at $\sim $165cm$^{-1}$. Bare gold - surfaces (both with and without a solvent over-layer) are missing - this peak. A similar peak at $\sim $165cm$^{-1}$ also appears in - the vibrational power spectrum for the butanethiol capping agents.} -\label{specAu} -\end{figure} - -Also in this figure, we show the vibrational power spectrum for the -bound butanethiol molecules, which also exhibits the same -$\sim$165cm$^{-1}$ peak. - -\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. The all-atom classical models include high -frequency modes which should be unpopulated at our relatively low -temperatures. This artifact is likely the cause of the high thermal -conductance in all-atom MD simulations. +\subsubsection{Shear viscosity} +Table \ref{LJ} also compares our shear viscosity results with momentum +swapping approach. Our calculations show that our method predicted +similar values for shear viscosities to the momentum swapping +approach, as well as the velocity gradient profiles. Moderately larger +momentum fluxes are helpful to reduce the errors of measured velocity +gradients and thus the final result. However, it is pointed out that +the momentum swapping approach tends to produce nonthermal velocity +distributions.\cite{Maginn:2010} -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}). +To examine that temperature isotropy holds in simulations using our +method, we measured the three one-dimensional temperatures in each of +the slabs (Figure \ref{tempXyz}). Note that the $x$-dimensional +temperatures were calculated after subtracting the effects from bulk +velocities of the slabs. The one-dimensional temperature profiles +showed no observable difference between the three dimensions. This +ensures that isotropic scaling automatically preserves temperature +isotropy and that our method is useful in shear viscosity +computations. \begin{figure} -\includegraphics[width=\linewidth]{aahxntln} -\caption{Spectra obtained for all-atom (AA) Au / butanethiol / solvent - systems. When butanethiol is deuterated (lower left), its - 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} +\includegraphics[width=\linewidth]{tempXyz} +\caption{Unlike the previous NIVS algorithm, the new method does not + produce a thermal anisotropy. No temperature difference between + different dimensions were observed beyond the magnitude of the error + bars. Note that the two ``hotter'' regions are caused by the shear + stress, as reported by Tenney and Maginn\cite{Maginn:2010}, but not + an effect that only observed in our methods.} +\label{tempXyz} \end{figure} -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. +Furthermore, the velocity distribution profiles are tested by imposing +a large shear stress into the simulations. Figure \ref{vDist} +demonstrates how our method is able to maintain thermal velocity +distributions against the momentum swapping approach even under large +imposed fluxes. Previous swapping methods tend to deplete particles of +positive velocities in the negative velocity slab (``c'') and vice +versa in slab ``h'', where the distributions leave a notch. This +problematic profiles become significant when the imposed-flux becomes +larger and diffusions from neighboring slabs could not offset the +depletion. Simutaneously, abnormal peaks appear corresponding to +excessive velocity swapped from the other slab. This nonthermal +distributions limit applications of the swapping approach in shear +stress simulations. Our method avoids the above problematic +distributions by altering the means of applying momentum +fluxes. Comparatively, velocity distributions recorded from +simulations with our method is so close to the ideal thermal +prediction that no observable difference is shown in Figure +\ref{vDist}. Conclusively, our method avoids problems happened in +previous RNEMD methods and provides a useful means for shear viscosity +computations. -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 -vibrational coupling between the metal and the liquid phase can -therefore be enhanced with the presence of suitable capping agents. - -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}). - \begin{figure} -\includegraphics[width=\linewidth]{uahxnua} -\caption{Vibrational power spectra for UA models for the butanethiol - and hexane solvent (upper panel) show the high degree of overlap - between these two molecules, particularly at lower frequencies. - Deuterating a UA model for the solvent (lower panel) does not - decouple the two spectra to the same degree as in the AA force - field (see Fig \ref{aahxntln}).} -\label{uahxnua} +\includegraphics[width=\linewidth]{velDist} +\caption{Velocity distributions that develop under the swapping and + our methods at high flux. These distributions were obtained from + Lennard-Jones simulations with $j_z^*(p_x)\sim 0.4$ (equivalent to a + swapping interval of 20 time steps). This is a relatively large flux + to demonstrate the nonthermal distributions that develop under the + swapping method. Distributions produced by our method are very close + to the ideal thermal situations.} +\label{vDist} \end{figure} -\section{Conclusions} -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. +\subsection{Bulk SPC/E water} +Since our method was in good performance of thermal conductivity and +shear viscosity computations for simple Lennard-Jones fluid, we extend +our applications of these simulations to complex fluid like SPC/E +water model. A simulation cell with 1000 molecules was set up in the +same manner as in \cite{kuang:164101}. For thermal conductivity +simulations, measurements were taken to compare with previous RNEMD +methods; for shear viscosity computations, simulations were run under +a series of temperatures (with corresponding pressure relaxation using +the isobaric-isothermal ensemble[CITE NIVS REF 32]), and results were +compared to available data from Equilibrium MD methods[CITATIONS]. -Our simulations have seen significant conductance enhancement in the -presence of capping agent, compared with the bare gold / liquid -interfaces. The vibrational coupling between the metal and the liquid -phase is enhanced by a chemically-bonded capping agent. Furthermore, -the coverage percentage of the capping agent plays an important role -in the interfacial thermal transport 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. +\subsubsection{Thermal conductivity} +Table \ref{spceThermal} summarizes our thermal conductivity +computations under different temperatures and thermal gradients, in +comparison to the previous NIVS results\cite{kuang:164101} and +experimental measurements\cite{WagnerKruse}. Note that no appreciable +drift of total system energy or temperature was observed when our +method is applied, which indicates that our algorithm conserves total +energy even for systems involving electrostatic interactions. -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 -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. +Measurements using our method established similar temperature +gradients to the previous NIVS method. Our simulation results are in +good agreement with those from previous simulations. And both methods +yield values in reasonable agreement with experimental +values. Simulations using moderately higher thermal gradient or those +with longer gradient axis ($z$) for measurement seem to have better +accuracy, from our results. -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 -required for the use of a derivative form is limited by the number of -bins and the sampling required to obtain thermal gradient information. +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + + \caption{Thermal conductivity of SPC/E water under various + imposed thermal gradients. Uncertainties are indicated in + parentheses.} + + \begin{tabular}{ccccc} + \hline\hline + $\langle T\rangle$ & $\langle dT/dz\rangle$ & \multicolumn{3}{c} + {$\lambda (\mathrm{W m}^{-1} \mathrm{K}^{-1})$} \\ + (K) & (K/\AA) & This work & Previous NIVS\cite{kuang:164101} & + Experiment\cite{WagnerKruse} \\ + \hline + 300 & 0.8 & 0.815(0.027) & 0.770(0.008) & 0.61 \\ + 318 & 0.8 & 0.801(0.024) & 0.750(0.032) & 0.64 \\ + & 1.6 & 0.766(0.007) & 0.778(0.019) & \\ + & 0.8 & 0.786(0.009)\footnote{Simulation with $L_z$ + twice as long.} & & \\ + \hline\hline + \end{tabular} + \label{spceThermal} + \end{center} + \end{minipage} +\end{table*} -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. +\subsubsection{Shear viscosity} +The improvement our method achieves for shear viscosity computations +enables us to apply it on SPC/E water models. The series of +temperatures under which our shear viscosity calculations were carried +out covers the liquid range under normal pressure. Our simulations +predict a similar trend of $\eta$ vs. $T$ to EMD results we refer to +(Table \ref{spceShear}). Considering subtlties such as temperature or +pressure/density errors in these two series of measurements, our +results show no significant difference from those with EMD +methods. Since each value reported using our method takes only one +single trajectory of simulation, instead of average from many +trajectories when using EMD, our method provides an effective means +for shear viscosity computations. +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + + \caption{Computed shear viscosity of SPC/E water under different + temperatures. Results are compared to those obtained with EMD + method[CITATION]. Uncertainties are indicated in parentheses.} + + \begin{tabular}{cccc} + \hline\hline + $T$ & $\partial v_x/\partial z$ & \multicolumn{2}{c} + {$\eta (\mathrm{mPa}\cdot\mathrm{s})$} \\ + (K) & (10$^{10}$s$^{-1}$) & This work & Previous simulations[CITATION]\\ + \hline + 273 & & 1.218(0.004) & \\ + & & 1.140(0.012) & \\ + 303 & & 0.646(0.008) & \\ + 318 & & 0.536(0.007) & \\ + & & 0.510(0.007) & \\ + & & & \\ + 333 & & 0.428(0.002) & \\ + 363 & & 0.279(0.014) & \\ + & & 0.306(0.001) & \\ + \hline\hline + \end{tabular} + \label{spceShear} + \end{center} + \end{minipage} +\end{table*} + +[MAY COMBINE JZPX AND JZKE TO RUN ONE JOB BUT GET ETA=F(T)] +[PUT RESULTS AND FIGURE HERE IF IT WORKS] +\subsection{Interfacial frictions and slip lengths} +An attractive aspect of our method is the ability to apply momentum +and/or thermal flux in nonhomogeneous systems, where molecules of +different identities (or phases) are segregated in different +regions. We have previously studied the interfacial thermal transport +of a series of metal gold-liquid +surfaces\cite{kuang:164101,kuang:AuThl}, and attemptions have been +made to investigate the relationship between this phenomenon and the +interfacial frictions. + +Table \ref{etaKappaDelta} includes these computations and previous +calculations of corresponding interfacial thermal conductance. For +bare Au(111) surfaces, slip boundary conditions were observed for both +organic and aqueous liquid phases, corresponding to previously +computed low interfacial thermal conductance. Instead, the butanethiol +covered Au(111) surface appeared to be sticky to the organic liquid +molecules in our simulations. We have reported conductance enhancement +effect for this surface capping agent,\cite{kuang:AuThl} and these +observations have a qualitative agreement with the thermal conductance +results. This agreement also supports discussions on the relationship +between surface wetting and slip effect and thermal conductance of the +interface.[CITE BARRAT, GARDE] + +\begin{table*} + \begin{minipage}{\linewidth} + \begin{center} + + \caption{Computed interfacial friction coefficient values for + interfaces with various components for liquid and solid + phase. Error estimates are indicated in parentheses.} + + \begin{tabular}{llcccccc} + \hline\hline + Solid & Liquid & $T$ & $j_z(p_x)$ & $\eta_{liquid}$ & $\kappa$ + & $\delta$ & $G$\footnote{References \cite{kuang:AuThl} and + \cite{kuang:164101}.} \\ + surface & model & K & MPa & mPa$\cdot$s & Pa$\cdot$s/m & nm & + MW/m$^2$/K \\ + \hline + Au(111) & hexane & 200 & 1.08 & 0.20() & 5.3$\times$10$^4$() & + 3.7 & 46.5 \\ + & & & 2.15 & 0.14() & 5.3$\times$10$^4$() & + 2.7 & \\ + Au-SC$_4$H$_9$ & hexane & 200 & 2.16 & 0.29() & $\infty$ & 0 & + 131 \\ + & & & 5.39 & 0.32() & $\infty$ & 0 & + \\ + \hline + Au(111) & toluene & 200 & 1.08 & 0.72() & 1.?$\times$10$^5$() & + 4.6 & 70.1 \\ + & & & 2.16 & 0.54() & 1.?$\times$10$^5$() & + 4.9 & \\ + Au-SC$_4$H$_9$ & toluene & 200 & 5.39 & 0.98() & $\infty$ & 0 + & 187 \\ + & & & 10.8 & 0.99() & $\infty$ & 0 + & \\ + \hline + Au(111) & water & 300 & 1.08 & 0.40() & 1.9$\times$10$^4$() & + 20.7 & 1.65 \\ + & & & 2.16 & 0.79() & 1.9$\times$10$^4$() & + 41.9 & \\ + \hline + ice(basal) & water & 225 & 19.4 & 15.8() & $\infty$ & 0 & \\ + \hline\hline + \end{tabular} + \label{etaKappaDelta} + \end{center} + \end{minipage} +\end{table*} + +An interesting effect alongside the surface friction change is +observed on the shear viscosity of liquids in the regions close to the +solid surface. Note that $\eta$ measured near a ``slip'' surface tends +to be smaller than that near a ``stick'' surface. This suggests that +an interface could affect the dynamic properties on its neighbor +regions. It is known that diffusions of solid particles in liquid +phase is affected by their surface conditions (stick or slip +boundary).[CITE SCHMIDT AND SKINNER] Our observations could provide +support to this phenomenon. + +In addition to these previously studied interfaces, we attempt to +construct ice-water interfaces and the basal plane of ice lattice was +first studied. In contrast to the Au(111)/water interface, where the +friction coefficient is relatively small and large slip effect +presents, the ice/liquid water interface demonstrates strong +interactions and appears to be sticky. The supercooled liquid phase is +an order of magnitude viscous than measurements in previous +section. It would be of interst to investigate the effect of different +ice lattice planes (such as prism surface) on interfacial friction and +corresponding liquid viscosity. + +\section{Conclusions} +Our simulations demonstrate the validity of our method in RNEMD +computations of thermal conductivity and shear viscosity in atomic and +molecular liquids. Our method maintains thermal velocity distributions +and avoids thermal anisotropy in previous NIVS shear stress +simulations, as well as retains attractive features of previous RNEMD +methods. There is no {\it a priori} restrictions to the method to be +applied in various ensembles, so prospective applications to +extended-system methods are possible. + +Furthermore, using this method, investigations can be carried out to +characterize interfacial interactions. Our method is capable of +effectively imposing both thermal and momentum flux accross an +interface and thus facilitates studies that relates dynamic property +measurements to the chemical details of an interface. + +Another attractive feature of our method is the ability of +simultaneously imposing thermal and momentum flux in a +system. potential researches that might be benefit include complex +systems that involve thermal and momentum gradients. For example, the +Soret effects under a velocity gradient would be of interest to +purification and separation researches. + \section{Acknowledgments} Support for this project was provided by the National Science Foundation under grant CHE-0848243. Computational time was provided by @@ -1028,4 +757,3 @@ Dame. \end{doublespace} \end{document} -