--- stokes/stokes.tex 2011/12/09 05:08:56 3776 +++ stokes/stokes.tex 2011/12/11 03:22:47 3780 @@ -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} @@ -226,7 +218,6 @@ section will illustrate this effect. 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. -%NEW METHOD (AND NIVS) HAVE LESS PERTURBATION THAN MOMENTUM SWAPPING \section{Computational Details} The algorithm has been implemented in our MD simulation code, @@ -392,8 +383,9 @@ solid-liquid interface. [MAY INCLUDE A SNAPSHOT IN FIG \begin{figure} \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.} + profile of a solid-liquid interface simulation. An example of + Au/hexane interfaces is shown. Calculation for the left side is + illustrated. The right side is similar to the left side.} \label{slipLength} \end{figure} @@ -405,8 +397,6 @@ data. interface can be measured respectively. Further calculations and characterizations of the interface can be carried out using these data. -[MENTION IN RESULTS THAT ETA OBTAINED HERE DOES NOT NECESSARILY EQUAL -TO BULK VALUES] \section{Results and Discussions} \subsection{Lennard-Jones fluid} @@ -437,7 +427,7 @@ calculations with various fluxes in reduced units. \multicolumn{2}{c}{$\lambda^*$} & \multicolumn{2}{c}{$\eta^*$} \\ \hline - Swap Interval & Equivalent $J_z^*$ or $j_z^*(p_x)$ & + Swap Interval $t^*$ & Equivalent $J_z^*$ or $j_z^*(p_x)$ & NIVS & This work & Swapping & This work \\ \hline 0.116 & 0.16 & 7.30(0.10) & 6.25(0.23) & 3.57(0.06) & 3.53(0.16)\\ @@ -469,7 +459,7 @@ in swapping ($\vec{p}_i$ in slab ``c'' with $\vec{p}_j expected. Furthermore, this method avoids inadvertent concomitant momentum flux when only thermal flux is imposed, which could not be achieved with swapping or NIVS approaches. The thermal energy exchange -in swapping ($\vec{p}_i$ in slab ``c'' with $\vec{p}_j$ in slab ``j'') +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 @@ -498,7 +488,12 @@ computations. \begin{figure} \includegraphics[width=\linewidth]{tempXyz} -\caption{.} +\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} @@ -525,136 +520,232 @@ computations. \begin{figure} \includegraphics[width=\linewidth]{velDist} -\caption{.} +\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} \subsection{Bulk SPC/E water} -[WATER COMPARED TO RNEMD NIVS AND EMD] +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]. \subsubsection{Thermal conductivity} -[VSIS DOES AS WELL AS NIVS] +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. -\subsubsection{Shear viscosity} -[COMPARE W EMD] +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. -[MAY HAVE A FIRURE FOR DATA] - -[MAY COMBINE JZPX AND JZKE TO RUN ONE JOB BUT GET ETA=F(T)] -[PUT RESULT AND FIGURE HERE IF IT WORKS] -\subsection{Interfacial frictions} -[SLIP BOUNDARY VS STICK BOUNDARY; ICE-WATER INTERFACES] +\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*} -qualitative agreement w interfacial thermal conductance +\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. -[FUTURE WORK HERE OR IN CONCLUSIONS] - - \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. Here ``D'' stands for deuterated - solvent or capping agent molecules. Error estimates are - indicated in parentheses.} + \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}{llccc} + \begin{tabular}{cccc} \hline\hline - Butanethiol model & Solvent & $G$ & $G^\prime$ \\ - (or bare surface) & model & \multicolumn{2}{c}{(MW/m$^2$/K)} \\ + $T$ & $\partial v_x/\partial z$ & \multicolumn{2}{c} + {$\eta (\mathrm{mPa}\cdot\mathrm{s})$} \\ + (K) & (10$^{10}$s$^{-1}$) & This work & Previous simulations[CITATION]\\ \hline - 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) \\ - \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) \\ + 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{modelTest} + \label{spceShear} \end{center} \end{minipage} \end{table*} -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. +[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. -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. +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] -\subsubsection{Effects due to average temperature} +\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 & molecules & 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*} -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. +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. -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. +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. -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. - \section{Conclusions} -[VSIS WORKS! COMBINES NICE FEATURES OF PREVIOUS RNEMD METHODS AND -IMPROVEMENTS TO THEIR PROBLEMS!] +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. -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. +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. -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. +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. -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. - \section{Acknowledgments} Support for this project was provided by the National Science Foundation under grant CHE-0848243. Computational time was provided by