Multiscale molecular simulations of argon vapor condensation onto a cooled substratewith bulk flowKai Gu, Charles B. Watkins, and Joel Koplik

Citation: Physics of Fluids (1994-present) 22, 112002 (2010); doi: 10.1063/1.3517293 View online: http://dx.doi.org/10.1063/1.3517293 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/22/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Molecular dynamics studies to understand the mechanism of heat accommodation in homogeneous condensingflow of carbon dioxide J. Chem. Phys. 135, 064503 (2011); 10.1063/1.3624335 A unimolecular evaporation model for simulating argon condensation flows in direct simulation Monte Carlo Phys. Fluids 21, 036101 (2009); 10.1063/1.3094957 Numerical Simulation of a Vapor Flow with Evaporation and Condensation in the Presence of a Small Amount ofa Noncondensable Gas AIP Conf. Proc. 663, 638 (2003); 10.1063/1.1581604 Two-surface problems of a multicomponent mixture of vapors and noncondensable gases in the continuum limitin the light of kinetic theory Phys. Fluids 11, 2743 (1999); 10.1063/1.870133 Vapor flows caused by evaporation and condensation on two parallel plane surfaces: Effect of the presence of anoncondensable gas Phys. Fluids 10, 1519 (1998); 10.1063/1.869671

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Multiscale molecular simulations of argon vapor condensationonto a cooled substrate with bulk flow

Kai Gu,1 Charles B. Watkins,1,a� and Joel Koplik2

1Department of Mechanical Engineering, The City College of New York,160 Convent Avenue, New York, New York 10031, USA2Department of Physics and Benjamin Levich Institute, The City College of New York,160 Convent Avenue, New York, New York 10031, USA

�Received 10 June 2010; accepted 12 October 2010; published online 18 November 2010�

A hybrid simulation method is employed to study the condensation of saturated argon vapor flowingtangentially across a stationary cooled substrate, at nanoscale resolution. The method combines adirect simulation Monte Carlo treatment of the bulk vapor phase with a nonequilibrium moleculardynamics treatment of the condensed liquid and interphase regions; it provides an efficientsimulation procedure for a heterogeneous system with a large ratio of vapor to liquid length scales.Starting from a bare, crystalline solid wall, the condensation process evolves from a transientunsteady state to a quasisteady state, where interfacial properties and heat and mass transferparameters are analyzed. The Knudsen layer structure from the hybrid simulation is compared withkinetic theory predictions from a modified moment method analysis and from pure DSMCsimulation. The effects of condensation strength and a tangential flow velocity that is on the orderof the condensation velocity are examined. A comparison is made between the nonequilibriumresults and equilibrium results for the interphase transition between liquid and vapor. The resultsreveal the structure of the interphase for such phenomena as inverted temperature, drift flux, andheat transfer. Heat transfer phenomena at the substrate surface are also described. © 2010 AmericanInstitute of Physics. �doi:10.1063/1.3517293�

I. INTRODUCTION

Molecular simulation of interfacial heat and mass trans-port in a multiscale condensing flow system is the focus ofthe present research. The system of interest has length scalesspanning the disparate scales of molecular diameters andmolecular collisions. �For argon at standard conditions, theeffective hard sphere molecular diameter is 0.4 nm, while themean free path is 63 nm.� It involves deposition of a nano-scale liquid film onto a solid substrate from a microscalebulk saturated vapor flow over it. Interfacial transport phe-nomena have not been examined at this resolution in thepresence of bulk flow but are of immense scientific interestand practical relevance for heat and mass transfer involvingphase change in microscale heat exchangers, chemical orphysical vapor deposition in microfabrication and thin-filmcoating processes, dew and frost formation with wind, andpostnucleation cloud and fog microphysics.

In the present system, nanoscale effects dominate in asmall but significant portion of a microscale physical region,as opposed to only at the boundaries. Investigation of solid-liquid-vapor heat and mass transport phenomena in suchmultiscale systems has been hampered by lack of an appro-priate simulation tool. Development of such a tool was calledfor a decade ago in the molecular dynamics heat transferreview paper by Maruyama,1 but no satisfactory toolemerged until our recent development of an atomistic hybridmultiscale method for flow over a gas/solid interface.2

Vapor flow above its dense phase forms a kinetic bound-ary layer, known as the Knudsen layer, within a few meanfree paths of the liquid surface. Kinetic theory, based on theBoltzmann equation, can be applied to study the flow withinthe Knudsen layer. Knudsen layer kinetic theory modelingfor vapor over its dense phase in condensing and evaporatingflow systems was reviewed by Ytrehus.3 The work of Aoki etal.4 on the effect, within the Knudsen layer, of velocity alongthe liquid phase is particularly relevant to the present inves-tigation. Their results were obtained by numerical solution ofthe BGK approximation to the Boltzmann equation for theKnudsen layer. Furthermore, recent developments5–7 usingthe higher-order lattice Boltzmann and moment methodshave improved our understanding of the dynamics of thiscritical region. However, kinetic theory cannot resolve thedetailed nanoscale thermophysics of the interphase regionthat exists between the pure liquid phase of the film and thepure vapor phase of the Knudsen layer. This region is at leasta few molecular diameters in thickness. Furthermore, kinetictheory approaches rely on modeled interfacial boundary con-dition approximations using the concepts of evaporation andcondensation coefficients. They also must rely on other phys-ics to deal with solid and liquid phases present in the inter-facial region.

Analytical solutions to the Boltzmann equation cannotreadily model complex flow geometries or complex transientdynamics. Some computational approaches can overcomethese restrictions. In particular, direct simulation MonteCarlo or DSMC �Ref. 8� is a relatively efficient and adapt-

a�Author to whom correspondence should be addressed. Electronic mail:watkins@ccny.cuny.edu.

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able method for molecular simulation of dilute gas flows inKnudsen layers.

In DSMC atomistic simulation, as in other kinetic theoryapproaches, modeled interface boundary conditions are re-quired. They are necessarily restricted to simplified mechani-cal models based on average surface collision dynamics or tostatistically averaged particle flux distributions based on sim-plified molecular interactions at an interface. This latter factis actually an advantage in devising simple gas reservoirouter boundary treatments in Knudsen layer simulations.9

However, nanoscale multiphase interfacial physics cannot beresolved. Moreover, DSMC simulation is limited to dilutegases and, like other kinetic theory methods, cannot resolvethe interphase region.

When the classical molecular dynamics �MD� method10

is applied to fluids that are not in equilibrium because theyare subjected to imposed mechanical or thermal driving, it isoften referred to as nonequilibrium molecular dynamics�NEMD�. NEMD is appealing for interfacial physics becauseit can treat both dilute and concentrated molecular systems.Therefore, it can be applied to simulate an entire solid-liquid-vapor interfacial system, i.e., the solid substrate, theliquid condensate, and the liquid-vapor interphase, as well asthe Knudsen layer.

NEMD simulations of vapor condensation onto a solidsubstrate without bulk flow were conducted for a closed sys-tem by Yi et al.11 and for an open system by He et al.12 Thesystem investigated by He et al. was similar to the system ofthe present investigation but, in addition to the fact that therewas no tangential flow, the vapor was not saturated and theupper boundary treatment was questionable. The substantialdisadvantage of such NEMD simulations is that they are sohighly computationally intensive that their use is restricted tonanodimensioned systems and correspondingly short simula-tion lifetimes. For example, the simulations of He et al. wereterminated before any evolution to quasisteady condensationcould be verified and the additional simulation needed toextract mean values from thermal fluctuations was not done.In pure dilute gas simulations, larger sizes are possible, butpractical multiphase microscale systems of engineering inter-est are not directly accessible to NEMD.

A multiscale hybrid method with the ability to efficientlysimulate the molecular gas dynamics of the Knudsen layerwhile simultaneously resolving the coupled nanoscale mo-lecular interactions in the solid, liquid, and interphase innerregions of the interface could be invaluable. Multiscale hy-brid schemes that couple MD or DSMC to continuum CFDare abundant, e.g., Refs. 13–16. Among them, Donev et al.14

recently introduced a promising Landau–Lifshitz Navier–Stokes �LLNS�-DSMC hybrid method with the capability ofdealing with a gas-liquid interface in an unsteady flow. Theiralgorithm incorporates some liquid phase molecular effectsvia the continuum LLNS numerical solution and gas phasemolecular effects via DSMC. However, using their method todeal with a moving liquid-vapor phase transition or solid-liquid molecular interaction is beyond the method’s currentstate of development.

The ability of DSMC to handle a Knudsen layer andNEMD to handle a nanoscale multiphase region suggests

that a DSMC-NEMD hybrid, combining the two, would bean effective multiscale method for vapor interaction with aninterface. The atomistic method, for coupling NEMD toDSMC in the gas phase, which we described and applied toa gas-solid interface in Ref. 2, is such a hybrid. It was ap-plied here, with a slight modification, to vapor phase cou-pling in the Knudsen layer of a multiscale solid-liquid-vaporsystem, such that the interphase is simulated with NEMDand most of the vapor phase is simulated with DSMC.

In the present simulations, bulk, saturated argon vaporflows above a cooled substrate in a direction tangential �par-allel� to its surface, creating a condensing flow within a re-gion above the surface with a thickness on the order of sev-eral gas mean free paths. The cooled substrate causes thecondensed vapor to deposit continuously on it, building up athin liquid film, as depicted in Fig. 1. The baseline simula-tion cases were nonequilibrium condensation with the tan-gential flow velocity equal to zero. The imposed tangentialvelocities in the other cases were of the order of the conden-sation velocity normal to the surface. The system was al-lowed to develop from an initial unsteady transient state to aquasisteady state over the simulation period.

II. KINETIC ANALYTICAL SOLUTIONOF THE KNUDSEN LAYER

It is useful to revisit the application of kinetic theoryanalytical modeling to the Knudsen layer, vapor region witha view toward comparing the results of kinetic modeling andsimulation to the results of hybrid simulation. At the outset itis understood that neither kinetic analytical modeling norkinetic computational methods can capture the physics ofmolecular interactions at the liquid-vapor interface, which isintrinsic to our hybrid simulation. Given this limitation ofkinetic theory, the Boltzmann equation for the Knudsen layercan be solved by traditional numerical methods as in theBGK solutions of Ref. 4 or by DSMC simulation. Analyticalapproaches, using simplified modeling, are less able to accu-rately resolve the vapor properties within the layer. However,one such approach, the moment method, was used here todevelop a convenient and consistent set of boundary condi-tions for pure �nonhybrid� DSMC simulations of the Knud-sen layer. The pure DSMC simulation results can be com-pared with the results of the hybrid simulations.

FIG. 1. Illustration of condensation on a cooled substrate with tangentialvapor flow.

112002-2 Gu, Watkins, and Koplik Phys. Fluids 22, 112002 �2010�

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A kinetic theory analytical approach to describe vaporflow evaporating from or condensing to a liquid surface, butwithout a tangential flow component was systematically in-troduced by Ytrehus.3 The vapor flow next to the liquid sur-face was in a steady, nonequilibrium state within a Knudsenlayer, which is one to several mean free paths thick. Themass, momentum, and energy fluxes within the Knudsenlayer were obtained from an approximate solution to theBoltzmann equation by a moment method.

Ytrehus made a number of simplifying assumptions inhis analysis such as an assumption that the condensation andevaporation coefficients are both unity. A scaling method forextending the analysis to nonunity coefficients is presentedin a subsequent section. In his moment method, Ytrehus as-sumed a four-term kinetic distribution function such that theboundary conditions were exactly represented in a certainform. This distribution function cannot fully model the addi-tion of a tangential vapor flow to the Knudsen layer so themethod is restricted in its ability to accurately capture all thedetails of such a flow. Nevertheless, if used as a modelingbasis for flows in which the tangential velocities are of thesame order or less than the condensation velocities, it canprovide a reasonable level of approximation for flow vari-ables other than tangential velocities.

The details of applying the moment method to obtainKnudsen layer solutions with tangential velocity are given inthe Appendix. The development follows directly from Ytre-hus’s work.3 The reason for presenting the complete solutionhere is to correct some apparently typographical errors thatwere discovered in Ref. 3 as well as to add tangential vaporflow.

III. HYBRID DSMC-NEMD SIMULATION PROCEDURES

A. Description of hybrid DSMC-NEMD simulation

The hybrid method applied to the present simulation in-corporates an improvement in the DSMC-NEMD couplingalgorithm over that described in Ref. 2. The NEMD portionof the hybrid method employs the Lennard-Jones �LJ� 12-6potential; this potential can be expressed as

�LJ�rij� = 4��� �

rij�12

− �s� �

rij�6 , �1�

where rij is the distance between molecule i and molecule j,and � and � are the characteristic energy and interactionlengths, respectively. The parameter, �s, is not present forinteractions between gas or liquid molecules but is intro-duced for fluid-solid surface molecular interactions and de-scribes the wetting effect at the fluid-solid interface. In thepresent simulations, it was assigned a value of 1.0, corre-sponding to complete wetting. The shift-force potentialmodification10 to the LJ potential was implemented with acutoff distance, rc, to eliminate weak effects from long-distance pair evaluation.

The DSMC molecular collision model employed in thehybrid and the pure DSMC simulations is the modified gen-eralized soft sphere �MGSS� model we developed in Ref. 2.Its Lennard-Jones potential-based collision cross-section en-sures its compatibility with NEMD molecular interactions

and is dependent on particle collision speed for improvedcomputational efficiency and low-temperature performance.

The algorithm used for coupling between DSMC andNEMD in the hybrid is an optimization of the tradeoff be-tween computational efficiency and accuracy. It is com-pletely particle based. There is no conventional buffer zonebetween the DSMC and NEMD simulator domains. Particlesare freely exchanged between the domains in a transmutationprocess and molecular interactions between the two types ofparticles are modeled in a consistent fashion.

The basic hybrid coupling steps are simple in concept.First, a one-to-one correspondence of DSMC particles tomolecules is established, as opposed to the usual DSMCimplementation where actual molecules are represented by afar fewer number of stochastic simulation particles. After aninitial stabilization period of simulation, DSMC and NEMDmolecules are allowed to exchange across the interdomainboundary. For computational efficiency, the DSMC timesteps are larger than the NEMD time steps �20 times larger inthe present simulations�. The coupling procedure synchro-nizes the DSMC simulator with the NEMD simulator at thebeginning of each DSMC time step. The DSMC and NEMDsimulator computations then proceed in parallel as repre-sented in Fig. 2. Details of the coupling procedure and theDSMC-NEMD molecule transmutation process are given inRef. 2. Our atomistic hybrid approach does not introduce anycomputational artifacts at the interdomain boundary since themean collision time in the vapor is 5000 times larger than theinformation exchange time between the DSMC and MD do-mains and both microscopic flux and state continuity arepreserved.

When DSMC molecules enter the NEMD domain to be-come NEMD molecules, they encounter an insertion proce-dure that avoids an overlap in positions with the NEMDmolecules already inside. The insertion routine has been im-proved from the original version presented in Ref. 2. Al-though the new algorithm requires more processor time than

FIG. 2. Hybrid DSMC-NEMD flow diagram.

112002-3 Multiscale molecular simulations of argon vapor Phys. Fluids 22, 112002 �2010�

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the original version, it reduces the stochastic effects carriedfrom the DSMC simulator to the NEMD simulator.

Any necessary adjustment in the location of newly in-serted NEMD molecules is determined by the locations ofnearby pre-existing NEMD molecules. The distance, rij, be-tween the neighboring molecule, j, and newly insertedNEMD molecule, i, is chosen based on an intermoleculardistance determined from the radial distribution function�RDF�. For an equilibrium system, the probability of oneparticle �i� finding another particle �j� at the distance rij isproportional to ng�rij�drij, in which n is the local mean num-ber density and g�rij� is the RDF at a distance rij. Assuminglocal equilibrium at the boundary between the DSMC andNEMD simulator domains, the location of new NEMD par-ticles can be determined by the RDF. If the distances to theneighboring molecules rij are greater than rc, the new NEMDparticle is successfully inserted without adjustment. In a casewhere rij is smaller than rc, the acceptance-rejection method�see Ref. 8� is triggered to judge the state of the new NEMDmolecules and adjust its location by randomly shifting it inthe x-z plane. The new NEMD particle is accepted at thisparticular location, rij, if

Rf �g�rij�

g�rmax�, �2�

where Rf is the uniformly distributed random number be-tween 0 and 1 corresponding to the random shift in locationand rmax is chosen to maximize g�r�. For a dilute gas, theRDF can be shown to be a function of a two-body potentialas

g�rij� = exp�− ��rij�kBT

� , �3�

in which kB is the Boltzmann constant and T is the tempera-ture of the system. For the LJ potential used in the presentsimulations, the maximum value of g�rij� can be obtained atthe distance, rmax=21/6�, and T is selected to be the localtemperature at the interdomain boundary on the NEMD side.

B. Molecular system model

The molecular system was set up in an elongated box asdepicted in Fig. 3. The dimension normal to the interface �ydirection� and the dimensions transverse to it �x and z direc-tions� were chosen to be 700� �b2 in the figure� and 25�,respectively. At the bottom of box, the solid substrate was afcc crystal structure formed from two layers of harmonicmolecules through tethering large mass molecules to a fixedsite with spring connections.17 Above the substrate, 942 ar-gon molecules were employed to initialize the vapor phase.The lower part of the box, with a normal dimension of 126��b1 in the figure�, included the substrate �1156 solid mol-ecules� and initially a small part of the vapor �160 argonmolecules�. The lower box and the substrate were the NEMDsimulator domain and the upper box was the DSMC simula-tor domain. The partitioning was designed to ensure that theupward-moving interphase would not cross the interdomainboundary into the DSMC domain during the simulation life-time, i.e., the physically fixed spatial boundary between the

NEMD and DSMC domains remains within the vapor phase.The NEMD domain was divided into 100 sampling bins andthe DSMC domain was divided into 40 collision cells.

The cutoff distance, rc, was set to be 4.0� in the NEMDsimulator and the Gear predictor-corrector algorithm was ap-plied to solve the molecular equations of motion with a timestep ���MD� of 0.005m�2 /�, where m is the argon molecu-lar mass. Alternative algorithms which make use of the sym-plectic nature of the equations of motion to improve energyconservation characteristics are available,10 but the simplerones have lower accuracy. Furthermore, because we imposea flow in the gas, energy is not conserved anyway.

The molecular dynamics part of the calculation isNEMD in the sense that although the algorithm which up-dates particle positions, etc., is the same as in any MD cal-culation, the part of the system to which such dynamics areapplied is open to molecule insertion/deletion and has animposed temperature difference across it. It is, therefore, notin any of the standard equilibrium ensembles. Likewise, theonly thermostat used is applied to the �lower-temperature�solid while the liquid and gas between the solid and theDSMC region find their own temperature and adjust it inresponse to the condensation

Utilizing data from equilibrium simulations, the initialstate of the argon vapor was set in a saturated condition at atemperature of 80 K, just above the triple point, and a densityof 0.0024 m /�3. The substrate temperature, TS, was con-trolled by velocity rescaling and initially set at the saturatedvapor temperature. The parameters for the argon and solidmolecules used in the simulations are listed in Table I. Thesolid molecular mass, ms, was chosen so that the ratio, k /ms

�where k is the spring constant�, associated with the stiffsprings of the solid molecules produced about the same os-cillation frequency as the argon LJ potential in the fluid.Therefore, the traditional NEMD time step based on the LJpotential of argon is sufficiently small to also capture thesolid oscillations.

The DSMC vapor molecule particle properties were con-sistent with those of the NEMD molecules. Constrained by

FIG. 3. Molecular system and hybrid solution domains. �MD domain heightis greatly exaggerated.�

112002-4 Gu, Watkins, and Koplik Phys. Fluids 22, 112002 �2010�

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the NEMD time step, the DSMC time step ���DSMC� is sig-nificantly smaller than the mean collision time. The DSMCcollision cells were uniformly distributed along the y direc-tion with cell dimensions smaller than the usual restriction ofone-third of the mean free path both in the y and neutraldirections. Each cell initially contained around 20 particleson average.

The top inflow boundary was implemented as a DSMCmolecular reservoir with a specified pressure �p�� and tem-perature �T��, which were set to be identical to the initialvapor condition. During the simulations, the inflow meanvelocity normal to the boundary, V�, was controlled by amass inflow of reservoir molecules tangentially translatingwith velocity, U�, whose thermal velocities were sampledbased on a Maxwellian distribution. The mean y directionvelocity in the reservoir, V�, was computed by having theinner cell mean y velocity extrapolated to the reservoirboundary. This implicit inflow boundary has been proven tobe an acceptable DSMC treatment for a low-speed gas flowboundary.18 Hence, the continuous condensate depositionprocess was made possible by the molecular flow inducedacross the open DSMC boundary with the reservoir. Themean tangential flow velocity, U�, at the inflow boundarywas fixed and assumed to be uncoupled to V�. The vaporinflow molecular thermal velocities were generated by sam-pling from the thermal equilibrium velocity distribution forthe reservoir. Periodic boundary conditions were applied inthe x and z directions.

After simulations began, the vapor molecules in theNEMD domain were allowed to reach their equilibrium stateat the saturation temperature. Then the hybrid DSMC-NEMD procedure was started and the substrate temperaturewas smoothly decreased from the saturation temperature tothe desired cooled value over the first 5104 time steps andkept constant during the rest of the simulation life. The driv-ing force of a temperature gradient between the substrate andthe vapor phase led nonequilibrium condensation to beginand a thin liquid film, with its thickness increasing with time,built up on the cooled substrate.

As listed in Table II, nine simulation cases were investi-gated with three different cooled substrate temperatures andthree different tangential flow velocities. The physical con-ditions simulated were designed to produce a significantamount of condensation in a reasonable simulation time. Thetangential velocities were selected to be of the same order asthe computed condensation �normal� velocities and the Machnumbers for the tangential and condensation velocities arewithin the ranges of the kinetic theory results of Ytrehus3 andAoki et al.4 For all cases, the simulation lifetime was 2

104m�2 /� or 43.21 ns, which ensured that the condensa-tion process reached a quasisteady state. The results of eachcase were obtained from ensemble averaging over ten inde-pendent runs with different initial random seeds to reduce theeffects of statistical and thermal fluctuations.

In molecular simulation, macroscopic physical quantitiesare obtained from dividing the physical domain into sam-pling bins �NEMD� or sampling/collision cells �DSMC� andaveraging over the individual molecules in them. The binheat flux, enthalpy, pressure tensor, and energy flux can becomputed, respectively, from

q,J =1

VJ��

i=1

NJ

ci,� �1

2mci�

2 + �i� +1

2�i,j

NJ

�Fij • ci��rij, ,

�4�

hJ =5

6

1

NJ�

i

NJ

ci�2 +

1

mNJ��

i

NJ

�i +1

3�i�j

NJ

�Fij • rij� , �5�

��,J =1

VJ�m�

i=1

NJ

ci,� ci,�� + �i�j

Fij,rij,�� , �6�

E,J = c,J1

VJ�i=1

NJ �1

2mci

2 + �i� + q,J + ��,Jc�,J, �7�

in which NJ is the number of molecules in the bin J withvolume, VJ, subscripts and � indicate the x, y, or z direc-tion, c� =c−c is the thermal velocity in the direction, c,J

is the stream velocity of bin J in the direction, Fij, is theinteraction force between molecule i and molecule j in the direction, rij,� is the interaction vector in the � direction, and�i is the potential energy of particle i. The interaction forceand potential energy terms in Eqs. �3�–�7� are not applicable

to the DSMC sampling routine. E,J represents the first-lawconservation flux in the direction at an arbitrary systemcontrol volume surface located at bin J. It includes all therelevant energy transport mode terms plus the pressure tensorwork terms.

TABLE I. Parameters for argon molecules.

Parameter Physical value

Length �=3.410−10 m

Energy �=1.6610−21 J

Molecular mass m=6.6410−26 kg

Solid molecular mass ms=6.6410−24 kg

TABLE II. Simulation cases.

Simulation caseSubstrate temperature Ts

�K�Tangential velocity U�

�m/s�

1 75 0

2 75 10

3 75 20

4 70 0

5 70 10

6 70 20

7 65 0

8 65 10

9 65 20

112002-5 Multiscale molecular simulations of argon vapor Phys. Fluids 22, 112002 �2010�

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C. Moving interphase boundaries

To study the interfacial molecular mass and heat transfer,an accurate definition of the interface is mandatory. It is rela-tively simple to define the interface between solid and fluidphases. In the present system, there is no mass transfer be-tween the solid and fluid phase �initially the fluid is a vapor�and the interface is stationary. This interface is also distin-guishable because a different type of molecule is used for thesolid. The solid-fluid interface physical properties on thefluid side were obtained at the first sampling bin �with a ydimension of 0.7�� next to the solid substrate.

Handling the liquid-vapor interface in the nonequilib-rium system is less straightforward. First, significant masstransfer takes place between the liquid and vapor phases andthe finite-thickness, interphase region over which the transi-tion from liquid to vapor occurs cannot be ignored. The sec-ond complication of the liquid-vapor interface is that theliquid film keeps growing from the cooling substrate, dis-placing the interphase along with it. Some classic strategiesfor maintaining a fixed interphase position during NEMDsimulation of nonequilibrium condensation �e.g., the liquidmolecule removal method in Ref. 19� are not suitable for thepresent system.

To overcome these difficulties, we employed a methodwe recently developed for defining the liquid-vaporinterphase.20 This method has the advantage of capturing themoving interphase region without significantly increasing thecomputational expense. It is based on counting the numberof neighboring interacting molecules, N����, in a speciallyconstructed volume, V�, about each molecule in the simula-tion. The volume, V�, is a thin region sliced from the centerof a sphere about the molecule with radius, rc, by planesparallel to the interface, as depicted in Fig. 4. For the presentsimulations, the y thickness of the region was chosen as 0.2�to provide reasonable spatial resolution while containing asufficient number of molecules to be representative. TheN���� are averaged over a small time interval and then these

averages are sampled over sampling bin, J, to generateNJ����, the bin-averaged number of interacting particles permolecule in the special volume, V�.

The method defines interphase boundaries on the vaporside and on the liquid side of the interphase by traversing theliquid-vapor interface to identify the bin location at whichthe NJ���� transitions to a specific value that characterizeseither the vapor �Cg� or the liquid �Cl� phase. In the presentsimulations, Cg=0.02 and Cl=8.1 were the boundary criteriavalues for the gas and liquid boundaries, respectively. Theywere obtained from liquid-vapor coexistence, equilibriumMD simulations at a temperature equal to the present T� �80K�. Once an interphase boundary bin is identified, a similarprocedure is employed to pinpoint the boundary locationwithin the bin. A numerical smoothing algorithm is appliedto the location coordinate to reduce the effect of fluctuations.

As condensation began in the present simulations, thenumbers of neighboring interacting molecules for the con-densate molecules near the solid-liquid interface weresmaller than the liquid criterion number. As the number ofvapor molecules condensing on the bare substrate increased,the neighboring interacting numbers for molecules near thesolid-vapor interphase increased to the liquid criterion num-ber and the solid-vapor interface transitioned to a solid-liquidinterface. At that point, it was assumed that the liquid-vaporinterphase was formed just above the solid-liquid interface.All subsequent steps to track the positions of the movinginterphase boundaries followed the procedures that havebeen described.

IV. RESULTS AND DISCUSSION

This section presents the nonequilibrium condensationresults obtained from the hybrid DSMC-NEMD simulationsand compares them with the moment method and with pureDSMC, where possible. As shown in the first subsection be-low, over the lifetime of the simulation, the systems passthrough an unsteady period; then they reach and maintain aquasisteady state. The quasisteady results are described inthe next subsection. Finally, two interfacial regions, liquid-vapor and solid-liquid, are discussed in separate subsequentsubsections.

A. From unsteady to quasisteady state

Figure 5 depicts the liquid thin-film growth for simula-tion case 4 as a plot of the liquid and vapor moving inter-phase boundary locations for liquid and gas �yl and yg� as afunction of time, with “instantaneous” mass density � =nm�profiles at selected times superimposed on them. The accu-racy of the locations of the liquid and vapor boundaries areverified through comparing them with the mass density pro-files at the different times. The jagged peak shapes in theliquid density profiles indicate the solid-liquid interaction ef-fect �layering pattern� that is also reflected in the steps shapeof the liquid boundary growth profile. The step heights are ofthe order of the characteristic molecular diameter. As theliquid film thickened, this steps shape tended to attenuatewith time.

FIG. 4. �Color online� Method for determining interphase boundaries�Ref. 20�.

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The liquid film growth obtained for simulation cases 1,4, and 7 are compared in Fig. 6, showing the higher growthrates associated with the larger temperature differentials be-tween bulk vapor and substrate. In confirmation of the roughtrend observed in the shorter simulation time results of He etal.,12 the present results show that growth becomes linear.Growth for the remainder of the cases, which involve tan-gential flow, was not plotted because there are only veryslight effects of the tangential flow on growth. In fact, thereis little or no effect of tangential velocity on most of thethermally related quantities computed from simulation of thetangential flow cases.

The quasisteady state can be distinguished from the ini-tial unsteady state by observing the interphase properties atthe liquid and vapor boundaries as a function of time. Thetemperature, molecular number flux, and heat flux were ob-

tained at the NEMD sampling bins within which the yl andyg interphase boundaries were determined to be located. Fig-ures 7–12 show the time evolution of various interfacialproperties from simulation cases 1, 4, and 7, which did nothave tangential vapor flow. Again, the tangential vapor flowin the other simulated cases did not produce any significantdifferences for the properties presented in the figures andwere not plotted.

Even after reasonable ensemble averaging, the statisticaland thermal fluctuations at the vapor interphase boundary inFigs. 7–12 remain large. The fundamental difficulty is thatthe gas is dilute and the mean flow velocities and the thermalgradients are small.

Figure 7 is a plot of temperature, Tl, at the liquid inter-phase boundary versus simulation time. After the liquid filmappears, Tl for simulation case 1 tends to decrease initiallyfrom the upstream vapor temperature, T�. Different fromcase 1, the results from simulation cases 4 and 7, with colder

FIG. 5. Moving interphase of simulation case 4 overlaid with the densityprofiles at selected system evolution times. �Horizontal dashed line indicatesthe liquid interphase boundary location, on the density profiles.�

FIG. 6. �Color online� Moving interphase boundary heights vs evolutiontime.

FIG. 7. �Color online� Liquid interphase boundary temperature vs evolutiontime.

FIG. 8. �Color online� Vapor interphase boundary temperature vs evolutiontime.

112002-7 Multiscale molecular simulations of argon vapor Phys. Fluids 22, 112002 �2010�

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substrates, show liquid layers initially deposited from thevapor phase that are much colder than T�, especially for case7 with the coldest substrate, but then they generally warmwith time, with case 7 exhibiting a warming peak. All of thecases approach a quasisteady steady state with some apparentperiodic oscillation in the quasisteady portion of the curves.The quasisteady state is reached after time, 1104m�2 /�or 21.6 ns.

Figure 7 also compares the hybrid results with solutionsfrom the kinetic theory, steady flow moment method pre-sented in the Appendix, showing good agreement with themoment method at the quasisteady state. The momentmethod as applied here requires specification of at least oneboundary condition that must come from the hybrid methodsimulation in order to compare results. In this case, the mo-ment method results were obtained by taking the hybridmethod quasisteady result for V� as the known quantity sinceit is relatively free from the fluctuations in the other hybrid

results. Further details regarding the application of the mo-ment method to obtain the comparative results are given inthe next section.

The evolution of the temperature at the vapor boundary,Tg, is shown in Fig. 8 for the same cases. There are substan-tial fluctuations in the temperature but some trends can bediscerned. The initial values are much higher than T�. Areasonable explanation is that this is because of the largelatent heat released from forming the liquid in the suddenphase transition from vapor to liquid and the strong solid-gas-liquid interactions. This higher thermal energy is ab-sorbed after a time by the cooled substrate, acting as a heatsink, and then the temperature decreases with the time. Onaverage, Tg is above T� in the quasisteady state. This in-verted temperature phenomenon indicates that the vaporphase near the interphase is superheated �i.e.,undersaturated�.3

The average temperature across the interphase, TI, wasobtained through sampling the particle thermal kinetic en-ergy in the region between yl and yg. It is shown in Fig. 9 andgenerally followed the trend of the liquid boundary tempera-

FIG. 9. �Color online� Averaged interphase temperature vs evolution time.

FIG. 10. �Color online� Fluid temperature at interface with solid vs systemevolution time.

FIG. 11. �Color online� Number flux at vapor boundary vs evolution time.

FIG. 12. �Color online� Energy flux at vapor boundary and wall heat trans-fer vs evolution time.

112002-8 Gu, Watkins, and Koplik Phys. Fluids 22, 112002 �2010�

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ture with a value that is only slightly greater. At the quasi-steady state, all of the simulation cases show an averageinterphase temperature lower than T� at steady state.

Figure 10 displays the temperature of the liquid obtainedfrom the sampling bin adjacent to the cooling substrate, Tw,normalized with the substrate temperature for plotting pur-poses. In the first portion of the simulation, the argon next tothe substrate is still in the vapor phase �as the dashed lineindicates in the figure�. Due to the sudden formation of theliquid film on the bare substrate, the solid-liquid-vapor inter-action is strong initially and causes the higher Tw value.Later, the wall fluid cools down as the phase transition ap-proaches quasisteady state, with only solid-liquid interactionoccurring near the substrate; finally, Tw stabilizes at a nearconstant value. A significant temperature slip,

�Tw = TS − Tw, �8�

across the solid-liquid interface at quasisteady state is evi-dent in the figure as the normalized temperature tends to avalue greater than one.

The number flux, J, of molecules across the vaporboundary at yg is shown in Fig. 11. The large initial masstransfer rates decline in the early stages of the simulationbefore increasing again to a relatively constant level at qua-sisteady state. The results are compared with the momentmethod solutions and show good agreement with the mo-ment method at quasisteady state.

The energy conservation flux, Ey,g, at the vapor inter-phase boundary is shown in Fig. 12. After reaching the qua-

sisteady state, Ey,g is approximately equal to its upstreamvalue, 1 /2 �V��1 /2V�

2 +5 /2RT��. At the solid substrate sur-face, where the normal velocity vanishes, the only contribu-tion to the energy conservation flux comes from the heatflux, qw, which is also shown in Fig. 12. The difference be-tween the two quantities plotted in the figure is the time rateof change of the energy within the liquid. It is obvious fromthe relative magnitudes of the fluxes that most of the heatremoved from the liquid at the substrate is latent heat re-moved from the vapor as opposed to the removal of sensibleheat and the flow energy and work converted to heat.

B. Quasisteady state

Based on examination of the unsteady results, all ninesimulation cases reach a quasisteady state after time, �=1104m�2 /�. Beyond this point, time averaging can be ap-plied along with the ensemble averaging used for the un-steady results to eliminate some of the fluctuations in thequasisteady data.

1. Liquid-vapor interfacial properties

Results obtained from averaging a number of liquid-vapor interface properties are tabulated in Table III. Thestandard deviations of the results are generally within 6%.

�H /RTI in Table III is the normalized latent heat, where�H=hg−hl �hg and hl are the enthalpies at the liquid andvapor boundary, respectively�. It tends to increase with de-creasing substrate temperature.

The condensation and evaporation coefficients given inTable III can be defined, in terms of averaged number fluxes,respectively, as

�c =�Jcnds �Jcoll

and �e =�Jevap

sp �Jout

, �9�

in which the vapor colliding flux, �Jcoll , and the outgoingflux, �Jout , are the number fluxes of molecules crossing thegas surface that originate in the vapor region or in the inter-phase region, respectively. The condensation flux, �Jcnds , isthe number flux of molecules that originate in the vapor andcross the liquid surface and the mass flux of spontaneouslyevaporating molecules, �Jevap

sp , is the number flux of mol-ecules that originate in the liquid and cross the vapor bound-ary. For convenience, the flux terminology and convention ofIshiyama et al.21 were adopted in the present work.

In the simulations, the evaporation and condensation co-efficients were obtained through tracking and counting theappropriate molecules to determine the various fluxes em-ployed in Eq. �9�. The molecules were labeled according totheir initial phase and during the simulation the labels weresubject to change depending on the paths the molecules took.As the liquid film grew, the liquid-vapor interphase bound-aries �yl and yg� were also tracked. The gas molecules origi-

TABLE III. Quasisteady liquid-vapor interphase properties.

Sim. case

Tg

T�

TI

T�

Tl

T� �e �c �

�Jout �Jcoll

�H

RTI

qg

��RT��3/2

�10−2�

1 1.024 0.979 0.978 0.827 0.849 0.872 0.862 9.208 �4.632

2 1.016 0.981 0.980 0.826 0.847 0.879 0.851 9.168 �5.885

3 1.021 0.983 0.983 0.826 0.847 0.879 0.854 9.245 �6.172

4 1.028 0.968 0.968 0.813 0.848 0.810 0.794 9.431 �7.151

5 1.035 0.971 0.970 0.815 0.851 0.804 0.798 9.461 �7.635

6 1.025 0.972 0.972 0.816 0.850 0.812 0.799 9.436 �8.704

7 1.043 0.961 0.960 0.806 0.856 0.744 0.731 9.711 �8.713

8 1.043 0.963 0.962 0.800 0.851 0.744 0.729 9.799 �10.091

9 1.032 0.966 0.965 0.803 0.853 0.744 0.733 9.792 �12.086

112002-9 Multiscale molecular simulations of argon vapor Phys. Fluids 22, 112002 �2010�

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nally above yg became liquid molecules if they crossed yl

moving downward, while liquid molecules originally belowyl became gas molecules if they crossed yg moving upward.

As expected, �e and �c are not equal as they would be inan equilibrium system. Larger differences are found to existbetween them for the stronger condensation cases �i.e., simu-lation cases 7–9�.

Flux conservation in the interfacial region dictates thatthe upward flux, �Jout , at the vapor boundary is equal to thesum of �Jevap

sp and the reflected-back-to-the-vapor flux �seeFig. 13�, �Jref

+ �i.e., �Jout = �Jevapsp + �Jref

+ �. Analogously, thedownward flux, �Jcoll , at the vapor boundary can be de-scribed as the sum of �Jcnds and the reflected-back-to-the-liquid flux, �Jref

− , which is imposed by �Jcoll �i.e., �Jcoll = �Jcnds + �Jref

− �.In Table III, �Jout / �Jcoll is compared with �, which is

defined as

� =1 − �c

1 − �eor � =

�Jout �Jref−

�Jcoll �Jref+

. �10�

Differences between � and �Jout / �Jcoll are small, but�Jref

− / �Jref+ �i.e., � / ��Jout / �Jcoll �� is slightly greater than

unity and demonstrates the fact that, across the liquid-vaporinterphase, a small portion of the evaporating liquid mol-ecules undergo collision with condensing vapor moleculescarrying larger kinetic energy and are reflected back to theliquid phase region. The fact that �Jref

− / �Jref+ is only slightly

greater than unity implies that the nonequilibrium state in theinterphase is relatively weak. This is consistent with the de-parture from Maxwellian distribution functions due to inter-facial drift velocity observed in Ref. 19. There is also aninterfacial drift velocity in the present results as will be pre-sented in Sec. III �Fig. 26�.

The quasisteady liquid-vapor interfacial properties inTable III are unaffected by the vapor tangential flow condi-tion with the exception of heat flux at the vapor interphaseboundary, qg, which is sensitive to the tangential flow veloc-ity. Larger tangential velocities induce higher heat flux ratesat the vapor boundary.

The interphase gas boundary temperatures, Tg /T�,shown in Table III, are greater than unity due to the invertedtemperature phenomenon. As noted earlier, there is very little

difference in the table between the interphase liquid bound-ary temperature, Tl, and the slightly larger, interphase-averaged temperature, TI.

Two important computed quasisteady averaged systemparameters are shown in Table IV, the inflow condensationvelocity and the wall heat transfer. Both quantities must in-crease with a decrease in the substrate temperature. Again,there is no discernible effect of tangential flow velocity. Theresult that condensation velocity is unaffected is consistentwith kinetic theory results for the Knudsen layer.3,4 The heatconduction from the liquid into the wall is two orders ofmagnitude larger than the heat conduction at the vaporboundary, where a tangential flow effect is observed.

The quasisteady, liquid surface temperature, Tl, hybridresults have a linear relationship with the substrate tempera-ture, TS, as shown in Fig. 14. The quasisteady, hybrid results

for normalized upstream condensation velocity, S�

= V� /2RT�, are plotted against the hybrid results for Tl /T�

in Fig. 15.The hybrid results in Fig. 15 are compared with the ki-

netic moment method solution for the liquid surface tem-

FIG. 13. Illustration of mass flux across the interphase.

TABLE IV. Quasisteady computed system parameters.

Sim. case.V�

2RT�

U�0�U�

�TW

T� − TS

qw

��RT��3/2

1 0.0861 � �0.702 �1.246

2 0.0866 0.007 �0.703 �1.285

3 0.0871 0.011 �0.716 �1.289

4 0.130 � �0.766 �1.931

5 0.132 0.006 �0.771 �2.005

6 0.130 0.012 �0.773 �1.949

7 0.173 � �0.776 �2.703

8 0.173 0.006 �0.787 �2.752

9 0.173 0.018 �0.789 �2.758

FIG. 14. Linear dependency of liquid interphase boundary temperature onsubstrate temperature. �Standard deviations of TI /T� are within 3%.�

112002-10 Gu, Watkins, and Koplik Phys. Fluids 22, 112002 �2010�

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perature, TL /T�, plotted as a function of S�, which is a speci-fied quantity in the present application of the momentmethod �see Appendix�. Sx� �i.e., U� /2RT�� and the nor-malized latent heat of condensation, �H /RTL, must also bespecified for the moment solution. The latent heat of conden-sation parameter, �H /RTL, used in all the moment methodsolutions was taken to be the average of the �H /RTI valuesin Table III obtained from the nine hybrid simulation cases,resulting in a value of 9.5. This is possible because the mo-ment method is insensitive to small changes in the latent heat

parameter. TI is used as a proxy for TL instead of Tl in thecomputation of the latent heat parameter simply for the rea-son that it is conceptually more representative of the phasetransition temperature. It is insignificant to the results since

the values for TI and Tl are very close.There is good agreement between the hybrid and mo-

ment method results in Fig. 15. In the moment method, theKnudsen layer structure is solved based on the fifth powerrepulsive interaction force law of Maxwell molecules �seeEq. �A10��. As suggested by Ytrehus,3 the interaction lawinfluences the spatial structure of the vapor flow, but not thedriving parameters. Figure 15 shows the close agreement be-tween the moment method driving parameter, TL /T�, and itsquasisteady hybrid simulation counterpart for a given S�.The figure confirms that the interaction law and other mod-eling assumptions made in the moment method do not have asignificant impact on the values of TL computed from it. Thisextends to the present modification of the moment method toinclude tangential velocity, as shown by the good agreementbetween the methods for nonzero tangential velocity inFig. 15.

In addition, for comparison purposes, the Knudsen layerwas simulated using DSMC alone �“pure DSMC”� withthese results for S� plotted in Fig. 15. The agreement be-tween the hybrid results and the moment solution is alsogood.

All the present pure DSMC results were obtained withthe DSMC equivalent of a kinetic boundary condition22,23

applied at the liquid surface. TL, the surface temperature, and

ne, the vapor number density at the liquid surface, were ob-tained from the moment method solution, while the conden-sation coefficients used were from the hybrid simulation. V�

was not fixed but determined as part of the simulation.The liquid surface interfacial boundary condition in pure

DSMC was treated, with some modification, as a DSMCopen boundary8 for zero streaming velocity. Moleculesevaporating from the liquid surface, whose velocities weresampled by

Fe+ = 1/�2�RTL�3/2exp�− �c�2/2RTL� , �11�

were generated with an outgoing density equal to �ene. Dueto the nonunity condensation coefficient, unlike the tradi-tional DSMC open boundary, only a portion, �c, of collidingvapor molecules was condensed �removed at the boundary�and the rest of the colliding vapor molecules were diffusivelyreflected back to the vapor phase with the temperature, TL.

The condensing molecules were stochastically selected.A vapor molecule striking the interface was condensed �re-moved�, if a random number Rf ��c, or was diffusively re-flected back to the vapor phase, if the random numberRf ��c. This process ensured that the ratio of mean condens-ing mass flux to mean colliding mass flux was equal to �c.Carey et al.24 used this condensation algorithm to simulatemolecular interaction at a droplet surface by DSMC andshowed it to be an accurate interfacial treatment for conden-sation.

The inverted temperature profile phenomenon has beenobserved previously in molecular simulations of condensa-tion �e.g., Refs. 25 and 26�. Ytrehus3 introduced the invertedtemperature criterion number

�c =32 + 9�

4�+ 8

1 − �e/c

�e/c, �12�

where �e/c is a single evaporation and condensation coeffi-cient based on an assumption that �e=�c. The thermody-namic state of the vapor above the liquid is superheated andan inverted temperature profile must occur if

�c ��H

RTL. �13�

For each case of the nonequilibrium condensation simu-lations, the latent heat in Eq. �13� was taken to be the�H /RTI value given in Table III. �e/c in Eq. �12� was takento be the average value of �e and �c in Table III, giving a �c

of about 5.2 for all cases. Based on these values, the criterionin Eq. �13� is met for all nine simulation cases.

2. Knudsen layer structure

To study the profiles of the vapor Knudsen layer struc-ture and the moving liquid-vapor interfacial region, the mov-ing coordinate, y�, was introduced and its origin was fixed atyg �i.e., y�=y−yg�. The hybrid simulation profiles of tem-perature, density, and normal velocity in the vapor phaseregion were obtained by time averaging the hybrid resultsduring the quasisteady period in the region for which y��0. The Knudsen layer profiles for this region were alsoobtained from the moment method and pure DSMC. Knud-

FIG. 15. Relationship between condensation velocity and liquid surfacetemperature at quasisteady state. �Standard deviations of V� are within 5%.�

112002-11 Multiscale molecular simulations of argon vapor Phys. Fluids 22, 112002 �2010�

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sen layer profiles for simulation case 7 are shown in Figs.16–19 with the y� coordinate plotted in units of the referencemean free path3 defined as

�e =��TL�mne

�

2RTL, �14�

in which � is the vapor viscosity at TL. The vapor viscositymodel used was the LJ spline potential model described inRef. 19.

The unity condensation coefficient solution for the mo-ment method was extended to the more general condition ofnonunity, nonequal condensation and evaporation coeffi-cients by scaling the y� /�e coordinate by the factor,27

� =

�c

�e� ne

n��

1+

1 − �c

�e2�T�

TLS�

� ne

n��

1

, �15�

where �ne /n��1 is the unity evaporation/condensation solu-tion of ne /n� from the moment method. The moment methodplots in Figs. 16–19 have been scaled with Eq. �15� using �e

and �c from the hybrid results.Unlike the hybrid method, the kinetic theory-based mo-

ment and pure DSMC methods for the vapor are not influ-enced by the substrate temperature. They cannot indepen-dently determine TL without specification of another vaporboundary condition, nor can they evaluate �e and �c. Sincethe S� values from the hybrid results were used in the mo-ment method to determine the TL and ne values, which were,in turn, used to generate the pure DSMC results, the kineticresults in Figs. 16–19 are not truly independent of the hybrid

FIG. 16. �Color online� Quasisteady kinetic temperature profiles in the va-por region.

FIG. 17. �Color online� Quasisteady temperature profiles in the vaporregion.

FIG. 18. �Color online� Quasisteady density profiles in the vapor region.

FIG. 19. �Color online� Quasisteady normal velocity profiles in the vaporregion.

112002-12 Gu, Watkins, and Koplik Phys. Fluids 22, 112002 �2010�

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results. Furthermore, the �e and �c from the hybrid were alsoused for the DSMC results through the boundary treatmentand for the moment results through Eq. �15�. As a conse-quence of their dependency on the hybrid results and theirmodeling restrictions, the moment and DSMC results cannotbe used to validate the hybrid method in an absolute sense.However, the hybrid can be used to assess the utility of themoment and DSMC methods for the present system. This isof particular interest for the profiles, which are more sensi-tive to the differences in molecular models than the drivingparameter comparisons in Figs. 12 and 15.

In Figs. 16–19, the hybrid simulation and pure DSMCsimulation results for the vapor profiles show varying de-grees of agreement with each other as well as with the mo-ment method results. The figures reveal that fluctuations arestill prominent in the quasisteady profiles from simulation ofthe vapor region by the hybrid and pure DSMC methodseven after smoothing by time averaging as well as by theensemble averaging used in obtaining the unsteady results.The temperature gradients and flow velocities across the va-por region proved to be too small �i.e., temperature gradientsless than 8% and condensing flow Mach numbers less than0.19� to reduce the level of fluctuations further.

The vertical dashed lines in the figures indicate the ap-proximate location of the boundary between the NEMD andDSMC simulator domains of the hybrid. It is not the exactboundary, due to the fact that y� is a moving coordinate.However, the movement during the quasisteady portion ofthe simulation for which the data is plotted is only about0.1�e.

Figure 16 shows the separate profiles for the kinetic tem-peratures in the x direction and in the y direction, demon-strating that they are both inverted as well as out of equilib-rium with each other. In Fig. 16, the hybrid and DSMCinverted interfacial kinetic temperature profiles differ fromthe corresponding moment method inverted profiles in thatthe distance above the interface at which they begin to in-crease �around 5�e� is greater. Tx and Ty are strongly decou-pled in the region within one �e, of the interface. Ty, themajor contributor to the inverted temperature, rapidly in-creases toward the interface, while Tx slightly decreases. Thethermodynamic temperature, T, results are shown in Fig. 17and also differ from the hybrid results in a similar manner.The pure DSMC results for Tx and Ty shown in Fig. 16 andfor T shown in Fig. 17 follow the trends of the results fromthe hybrid simulations.

For density, , and condensing velocity, V, shown inFigs. 18 and 19, respectively, the hybrid simulations displaygood agreement with the moment method, but pure DSMCsimulations show small deviations within 5�e of the inter-phase. The different profile shapes in the Knudsen layer be-tween the hybrid DSMC-NEMD results and the momentmethod solutions in Figs. 16–19 are most likely primarilydue to the different molecular interaction laws employed ineach method since the DSMC profiles tend to follow thehybrid profiles.

The p-T diagram in Fig. 20 displays the thermodynamicstate of the vapor domain at three different locations, y�=0,y�=1.5��0.03�e�, and y�=10�e. The p-T saturation line was

determined from numerically fitting data from equilibriumsimulations. The vapor is close to the local saturation state atthe vapor boundary, y�=0; however, just above, the thermo-dynamic state of the vapor jumps to superheated. The suddenchange of the thermodynamic state is also reflected in thedensity profiles near y�=0 in Fig. 18.

Hybrid simulation tangential velocity profiles in the va-por for cases 2, 5, and 8 are shown in Fig. 21 and for cases3, 6, and 9 are shown in Fig. 22. Agreement with the pureDSMC results, also shown in the figures, is good. For thestrongest condensation and tangential flow case �i.e., simula-tion case 9�, the slip velocities �as a percentage of the upperboundary tangential velocity� at the liquid-vapor interphaseare larger than in the weaker condensation and tangential

FIG. 20. Quasisteady saturation states at various vapor locations. �Opensymbols indicate the hybrid NEMD/DSMC results at y�=10�e, half blackand half white symbols indicate the hybrid NEMD/DSMC results aty�=1.5�, black symbols indicate the hybrid NEMD/DSMC results at y�=0,and the solid line is the Clausius–Clapeyron relation numerical fit to datafrom equilibrium simulations.� �The standard deviations of T and p arewithin 3.5% and 5%, respectively.�

FIG. 21. �Color online� Quasisteady vapor tangential velocity profiles forlower tangential velocity cases.

112002-13 Multiscale molecular simulations of argon vapor Phys. Fluids 22, 112002 �2010�

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flow cases. The slip velocities range from 0.07% to 1.8% ofthe upper boundary, tangential velocities, as shown inTable IV.

The shape of the tangential velocity profiles is consistentwith those obtained by Aoki et al.4 from their kinetic theorysolution of the Knudsen layer for condensation with tangen-tial flow. They are different than the more linear profilestypical of a Couette shear flow that the tangential velocity inthe reservoir at a fixed height above the interface might sug-gest exists below it. The profiles’ shapes suggest that theprofiles are being compressed and displaced downward bythe condensing flow.

The stronger condensation cases produce flattened pro-files at the upper simulation boundary for the hybrid and theDSMC simulations. This implies that at least these profilescould be independent of the boundary height. The modifiedmoment method in the Appendix employs an asymptoticmaximum upper boundary condition for the tangential veloc-ity. The agreement in Fig. 15 with the moment method issomewhat better for the tangential flow cases with the stron-ger condensation. Moreover, previous kinetic theorystudies3,5 have demonstrated that the flow behavior above theKnudsen layer for condensation with tangential flow isequivalent to an Euler flow with inner slip matched to theKnudsen layer edge flow. Hence, the present results are gen-eral in nature for any condensing bulk flow in the absence ofa tangential pressure gradient and are not restricted to a par-ticular bulk viscous shear profile.

Figure 23 shows the tangential velocity-driven shearstress profiles from the hybrid and DSMC simulations for thelowest and highest condensation cases. For the same uppertangential velocity boundary, the stronger condensation re-sults in a higher shear stress above the liquid-vapor inter-phase. The shear profile follows from the form of the tangen-tial velocity profile and indicates that the vapor flow is not ina significant state of shear except within approximately 2�e

from the interface; therefore, it is nearly an Euler flow.

3. Liquid-vapor interphase

The liquid-vapor interphase, which is on the order of tenmolecular diameters, was resolved by the hybrid within itsNEMD simulator partition. The moving interphase was stud-ied with reference to the coordinate y�. The liquid-vapor in-terphase thickness, d=yg−yl, is subject to the minor periodicoscillations in the liquid surface location noted earlier but isotherwise unvaried during the quasisteady final state of thesystem �see Fig. 5�. Thus, the quasisteady interphase regionis stable with respect to coordinate y�.

In Fig. 24, the density profiles across the interfacial re-gion without tangential flow are compared with the equilib-rium interfacial density profile obtained by a preliminaryequilibrium MD simulation of a liquid-vapor equilibriumsystem at 78 K �0.97T��, which corresponds to the liquidsurface temperature for case 4. The present liquid-vapor in-

FIG. 22. �Color online� Quasisteady vapor tangential velocity profiles forhigher tangential velocity cases.

FIG. 23. �Color online� Quasisteady shear stress profiles for tangential va-por velocity cases.

FIG. 24. Comparison between quasisteady nonequilibrium and equilibriuminterphase density profiles. �The solid line identifies the equilibrium densityprofile from equilibrium MD simulation at 78 K.�

112002-14 Gu, Watkins, and Koplik Phys. Fluids 22, 112002 �2010�

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terphase with a strong net condensation process has a differ-ent density profile when compared with an equilibrium pro-file with the same liquid surface temperature. The presentdensity profile is distorted and has a longer upstream tail.This tail effect is caused by molecules accumulating ahead ofthe moving liquid surface that are not accommodated to thetransition state from vapor to liquid. For the equilibriumliquid-vapor interphase, the density profile is well-known tohave a shape close to a hyperbolic tangent28 so the nonequi-librium distortion means that the present profiles deviatefrom this shape.

The distorted interphase density profiles apparently in-crease the ability of the molecules to cross the interphaseregion. Hence, liquid molecules are freer to move downwardto the liquid phase and vice versa. This effect is shown by thecondensation and evaporation coefficients ��e and �c� inTable III, which both exceed their values at phase equilib-rium �i.e., �e=�c=0.7 at 78 K�. Furthermore, the normalizedlatent heat for cases 4, 5, and, 6 of about 9.4, as shown inTable III, is lower than the latent heat for the equilibriumcase, which is 9.71.

Tx and Ty across the interphase are shown in Fig. 25. Theinverted kinetic temperature, Ty, which is most strongly outof equilibrium in the vapor interfacial region, decreases rap-idly, beginning at the vapor boundary of the interphase, to itsequilibrium state at the liquid boundary and a value smallerthan T�. Tx begins the decline from its inverted state in thevapor and continues to decline across the interphase, equili-brating with Ty at the liquid interphase boundary, yl.

The interphase number flux, �J , and heat flux, q, arepresented in Figs. 26 and 27, respectively. The number fluxis a molecular drift velocity that first increases in magnitude,beginning just above the vapor boundary, and then starts todecline in magnitude just below it, continually declining toits equilibrium value of near zero at the liquid boundary,zero.

Figure 27 presents the heat flux across the interphase.Note that the coordinate convention requires heat flow fromvapor to liquid to be negative. The magnitude of the heat flux

in the interfacial region increases monotonically toward theliquid phase. In Fig. 27, heat fluxes for the highest tangentialvapor flow simulation cases are also shown. Comparing themwith the cases for nontangential flow shows that tangentialflow in the vapor phase increases the magnitude of the heatflux in the region of the interphase, but only slightly.

4. Solid-liquid interface

Figure 28 shows the temperature profiles in the liquidfilm, which were obtained by sampling the liquid phase mol-ecules during the period between 1.8104m�2 /� and2.0104m�2 /� for simulation cases 1, 4, and 7. The lineartrending temperature profiles in the liquid reflect a thermalFourier flow.

The quasisteady temperature slip at the solid wall, �Tw,is listed in Table V, which indicates an increase in slip witha decrease in substrate temperature. The temperature discon-

FIG. 25. Quasisteady interphase nonequilibrium temperature profiles. FIG. 26. Quasisteady interphase number flux profiles.

FIG. 27. Quasisteady interphase heat flux profiles.

112002-15 Multiscale molecular simulations of argon vapor Phys. Fluids 22, 112002 �2010�

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tinuity across the solid-liquid interface can be characterizedby the interfacial thermal resistance or Kapitza resistancedefined as

RK =�Tw

qw, �16�

in which qw is the heat flux at the interface on the liquid side.The computed RK values from the simulations, which arelisted in Table V, increased with condensation strength.

Figure 28 demonstrates that the heat transfer at the solid-liquid boundary approximately follows the Fourier law, qw

=−���T /�y�, where � is the thermal conductivity. The ther-mal resistance length or Kapitza length, LK, is related to thetemperature slip and liquid temperature gradient adjacent tothe substrate through the relation

�Tw = LK� �T

�y�

liquid, �17�

where the temperature gradient is evaluated at the interfaceon the liquid side. LK=RK� is the length of the virtual regionwhere the Fourier law extends to TS and can be estimated bynumerically fitting the temperature profile inside the liquidfilm by a linear function �see Table V�. The statistical error iswithin 10%.

LK for a solid-liquid interface has been reported to bedependent on solid-liquid intermolecular stiffness ratio�squared�, solid-liquid interaction strength, �sl, and substratetemperature.29,30 In the present system, the first two terms arealmost constant and so LK should only be a function of thesubstrate temperature, TS. Figure 29 presents LK as a functionof TS. For liquid argon, the dependency of LK on TS wasreported to be linear.30 This linear relationship is verified inthe figure.

The large temperature slips and thermal resistancelengths in Table V can be attributed partly to the relativelylarge solid crystal to argon molecular mass ratio used for thesimulations. The stiffness ratio in the present simulations isessentially controlled by this large molecular mass ratio.Therefore, the effect of its square on the magnitude of thethermal resistance length is quite large. In addition, the solidsubstrate was modeled with only two layers of solid mol-ecules, which were controlled by the thermostat. It is difficultto thermalize fully the nearby liquid molecules by a smallvolume of temperature-controlled solid molecules because ofinsufficient solid-liquid interaction. This also leads to largertemperature slips and thermal resistance lengths.31

V. SUMMARY AND CONCLUSIONS

The results demonstrate that the atomistic hybridDSMC-NEMD method can efficiently resolve a typical un-steady nonequilibrium, multiphase interfacial system withdisparate length scales. In the present work, nanoscale vaporcondensation onto a cooled substrate with a bulk vapor flowparallel to the substrate was successfully simulated with in-teresting results. The method permitted simulating the sys-tem for an evolutionary time period that was sufficiently longfor quasistudy conditions to develop. A new scheme, withinthe method, for identifying liquid and vapor interphaseboundaries was successfully tested on the nonequilibriumphase change at the moving interface.

The structure of the Knudsen vapor layer from the hy-brid simulation was compared with kinetic theory predictions

FIG. 28. Quasisteady liquid film temperature profiles.

TABLE V. Heat transfer quantities at the solid-liquid interphase.

Simulation case�Tw

�K�RK

�10−7 W K−1 m−1� LK ���

1 �3.50 3.45 �163.49

2 �3.52 3.35 �150.82

3 �3.57 3.40 �144.44

4 �7.61 4.81 �218.33

5 �7.71 4.70 �236.40

6 �7.64 4.82 �216.20

7 �11.70 5.28 �242.47

8 �11.81 5.24 �264.87

9 �11.73 5.21 �264.16

FIG. 29. Thermal resistance length dependency on substrate temperature atquasisteady state. �Solid line indicates the least-squares fitting of data forsimulation cases 1, 4, and 7.�

112002-16 Gu, Watkins, and Koplik Phys. Fluids 22, 112002 �2010�

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from a modified moment method analysis and from pureDSMC simulation. In general, the agreement was good, dem-onstrating the utility of the kinetic methods. The pure DSMCresults were more accurate. Some of the differences betweenthe simulation results and the moment method results ap-peared to be due to the different molecular interaction mod-els employed by the two approaches. The kinetic methodswere not totally independent of the hybrid because they re-quired information from the hybrid for their boundary con-ditions due to their inability to model the solid-liquid inter-action and liquid-vapor molecular exchange.

The growth trend of the liquid film height with timebecame linear, while the thickness of the liquid-vapor inter-phase transition region remained constant. The density pro-file in the transition region was distorted from the equilib-rium profile and the condensation and evaporationcoefficients were higher than the equilibrium case, while thelatent heat was lower.

The inverted temperature profile phenomenon was ob-served in the vapor adjacent to the interface with accompa-nying out-of-equilibrium kinetic temperatures in the direc-tions normal and tangential to the interface. The state ofsuperheat in the vapor leading to the inverted temperatureswas verified.

Tangential flow velocity, which was of the same order asthe condensation-induced vapor velocity, had no observableeffect on most of the liquid-vapor interfacial properties. Onenotable exception was the heat flux at the vapor boundarywith the interphase. Tangential velocity also had a small ef-fect on the relationship between the condensation-inducedvapor velocity and the liquid interfacial temperature. It wasnot a factor in condensation film growth or heat transfer tothe substrate, a result that has important practical implica-tions and is consistent with previous kinetic theory results forthe Knudsen layer.

The tangential flow velocity profiles for the strongestcondensation cases were displaced away from the upperboundary and compressed, suggesting that they were inde-pendent of the height of the upper boundary. This result isalso consistent with previous kinetic results and, in the ab-sence of a tangential pressure gradient, generalizes thepresent results to be independent of a specific bulk tangentialvelocity profile. The shear stress profiles are compatible withthis interpretation and are a further indication of an Eulerflow above the Knudsen layer.

The slip velocities at the liquid surface were less than2% of the tangential velocities at the upper boundary andincreased with stronger condensation. The shear stress at theliquid-vapor boundary also increased with stronger conden-sation.

The major portion of the heat removed from the fluid atthe cooled substrate was associated with the latent heat ofcondensation as opposed to the sensible heat and the flowenergy and work converted to heat. Temperature slip, ther-mal resistance, and Kapitza length at the wall increased withthe condensation strength, with the Kapitza length exhibitingthe expected linear dependence on substrate temperature.Large Kapitza lengths were observed due to the molecularmodels used for the solid substrate.

ACKNOWLEDGMENTS

This work was supported financially by NSF Award No.0934206 to the PREM, City College-Chicago MRSEC Part-nership on the Dynamics of Heterogeneous and ParticulateMaterials.

APPENDIX: MODIFIED YTREHUS MOMENT METHOD

The kinetic theory approach to describe vapor flowevaporating from or condensing to a liquid surface was sys-tematically introduced by Ytrehus.3 The present developmentadds the tangential flow effects into this model for compari-sons with the hybrid DSMC-NEMD results.

The four-mode model used for the distribution function,f , describing the condensing vapor above the liquid surfacewith an asymptotic upper flow condition �i.e., u�

= �U� ,V� ,0�� is of the form

f�y,c� = ae+�y�fe

+�c� + a�+�y�f�

+�c� + a�−�y�f�

−�c�

+ a�−�y�f�

−�c� , �A1�

where c indicates the particle velocity vector, �cx ,cy ,cz�, y isthe coordinate above the liquid surface, and �ae

+ ,a�+ ,a�

− ,a�−�

are y dependent amplitude functions. The four parts of thedistribution function on the right-hand side of Eq. �A1� canbe expressed as

fe+�c� = fe�y = 0,cy � 0� =

ne

�2�RTL�3/2exp�−�c�2

2RTL� ,

�A2a�

f�+�c� = f��y → �,cy � 0�

=n�

�2�RT��3/2exp�−�c − u��2

2RT�� , �A2b�

f�−�c� = f��y → �,cy � 0�

=n�

�2�RT��3/2exp�−�c − u��2

2RT�� , �A2c�

f�−�c� = f��cy � 0� =

n�

�2�RT��3/2exp�−�c − u��2

2RT�� , �A2d�

in which ne is the saturated number density corresponding toliquid surface temperature TL and R is the gas constant. Theparameters in Eq. �A2d� are defined as

n� =ne

2, T� = TL�1 −

2

3�� , and u� = �0,2RTL

�,0� .

�A3�

The space-dependent amplitude functions �ae+ ,a�

+ ,a�− ,a�

−�must satisfy the boundary conditions for unity evaporation/condensation coefficient,

at y = 0:

ae+ = 1

a�+ = 0

a�− = �

a�− =

and as y → �:

ae+ = 0

a�+ = 1

a�− = 1

a�− = 0

. �A4�

112002-17 Multiscale molecular simulations of argon vapor Phys. Fluids 22, 112002 �2010�

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The collision invariants, �= �m ,mcy +mcx ,mc2�, i.e., themass, momentum, and energy fluxes, across the Knudsenlayer are conserved as described by

�

�y� cy��fdc = 0, � = 1,2,3. �A5�

It should be noted that since mcx and mcy are indepen-dent, an additional invariant is available to yield another in-dependent equation if a five-mode distribution function isconstructed. Due to the fact that, in this work, we were seek-ing a slight modification to the Ytrehus four-mode model forcondensation, mcx and mcy. Instead of using mcy alone as inYtrehus’s work, evaluation of mcy +mcx will include the tan-gential vapor flow momentum effects to some approximatedegree without using the available additional invariant, mcx.

With simple algebra and applying the boundary condi-tions in Eq. �A4�, Eq. �A5� can be written in the form ofthree linear equations with six unknowns �ne /n�, TL /T�, S�,Sx�, , and �� as follows:

�1 −1 − 2/�3��

2F�

−� ne

n�

TL

T�

− �F− = 2�S�, �A6a�

�1 +1 − 2/�3��

2G�

−� ne

n�

TL

T�

+ �G− − �2

�F−Sx�

= 4S�2 + 4S�Sx� + 2, �A6b�

�1 −�1 − 2/�3���3/2

2H�

−� ne

n�

TL

T�

TL

T�

− �H−

= �S��S�2 + Sx�

2 +5

2� , �A6c�

in which

S� =V�

2RT�

and Sx� =U�

2RT�

. �A7�

F−, F�−, G−, G�

−, H−, and H�− are the functions which can be

shown in the following forms:

F− = �S��− 1 + erf�S��� + exp�− S�2 � , �A8a�

F�− = �S��− 1 + erf�S��� + exp�− S�

2� , �A8b�

G− = �2S�2 + 1��1 − erf�S��� −

2�

S� exp�− S�2 � , �A8c�

G�− = �2S�

2 + 1��1 − erf�S��� −2

�S� exp�− S�

2� , �A8d�

H− =�S�

2�S�

2 +5

2��− 1 + erf�S���

+1

2�S�

2 + 2�exp�− S�2 � +

Sx�2

2F−, �A8e�

H�− =

�S�

2�S�

2 +5

2��− 1 + erf�S���

+1

2�S�

2 + 2�exp�− S�2� , �A8f�

where S�=1 /�−2 /3.In addition to specifying the condition that the vapor is

saturated in the upstream uniform state, p��T��, theClausius–Clapeyron equation,

p�

pe= exp� �H

RTL�1 −

TL

T�� , �A9�

can be used as the fourth equation to solve �ne /n�, TL /T�,S�, Sx�, , and �� combined with Eq. �A6�. If S� and Sx� areknown, four unknowns �ne /n�, TL /T�, , and �� are solved.

The Knudsen layer structure solution is obtained by in-tegrating the collision variant, mcy

2, which is the noncon-served moment. For simplicity, if the molecular interaction isthe fifth power repulsive interaction force �i.e., Maxwellmolecules�, we will have

�

�y� cy�mcy

2�fdc =�

�eRTL

2�

n

ne�yy� , �A10�

where �yy� is the viscous stress and �e is the reference meanfree path. After some simple algebra, the Knudsen layer canbe solved through the following equation set derived fromEq. �A10� as:

da�−�y�dy

= −Pc

�e�a�

−�y� − 1��a�−�y� − r� , �A11a�

ae+�y� =

a�−�y� − 1

� − 1, �A11b�

a�+�y� = 1 −

a�−�y� − 1

� − 1, �A11c�

a�−�y� =

a�−�y� − 1

� − 1, �A11d�

in which

Pc =�� − 1����1�2 − Sx�

2 �32�

12� e

��2 TL

T��1 −

T�

TL+

1 − 2/�3��2

F�−�T�

TL− 1 +

2

3�� +

T�

TL�1 −

1 − 2/�3��2

F�−�Sx�

2 � , �A12a�

112002-18 Gu, Watkins, and Koplik Phys. Fluids 22, 112002 �2010�

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r = 1 −�2 − 2S�

2 �1 + Sx�2 �1 − 2Sx�

2 �3

�1�2

2−

1

2Sx�

2 �32

, �A12b�

�1 =1

� − 1� ne

n�

− 2 + �1 − erf�S���� +

2�1 − erf�S���

ne

n�� , �A12c�

�2 =1

� − 1�− 2Sx�

2 +ne

n�

TL

T�

− 2 + ��1 + Sx�2 ��1 − erf�S��� +

1 − 2/�3��2

�1 − erf�S���ne

n�

TL

T�� , �A12d�

�3 =1

� − 1�− 2 + �1 − erf�S����� . �A12e�

Errors in Eq. �12�, as it was originally presented in Ref. 3,have been corrected along with the addition of extra termsdue to the tangential flow.

If Pc�0;r�1;��r or Pc�0;r�1;��r, the ordinarydifferential equation, Eq. �10�, can be solved analytically as

a�−�y� − 1

� − 1=

a�−�y� − r

� − rexp�− Pc�1 − r�

y

�e . �A13�

Finally, the physical quantities as a function of distance, y,are obtained, i.e.,

n�y� =� fdc , �A14a�

V�y�V�

=n�

n�y�, �A14b�

U�y� =1

n�y�� cxfdc , �A14c�

Ty�y� =1

n�y�R� �cy − V�y��2fdc , �A14d�

Tx�y� =1

n�y�R� �cx − U�y��2fdc . �A14e�

1S. Maruyama, “Molecular dynamics method for microscale heat transfer,”in Advances in Numerical Heat Transfer, edited by W. J. Minkowycz andE. M. Sparrow �Taylor and Francis, New York, 2000�, Vol. 2, Chap. 6, p.189.

2K. Gu, C. B. Watkins, and J. Koplik, “Atomistic hybrid DSMC/NEMDmethod for nonequilibrium multiscale simulations,” J. Comput. Phys. 229,1381 �2010�.

3T. Ytrehus, “Molecular-flow effects in evaporation and condensation atinterfaces,” Multiphase Sci. Technol. 9, 205 �1997�.

4K. Aoki, K. Nishino, Y. Sone, and H. Sugimoto, “Numerical analysis ofsteady flows of a gas condensing on or evaporating from its plane con-densed phase on the basis of kinetic theory: Effect of gas motion along thecondensed phase,” Phys. Fluids A 3, 2260 �1991�.

5A. S. Ansumali, I. V. Karlin, S. Arcidiacono, and N. I. Prasianakis, “Hy-drodynamics beyond Navier-Stokes: Exact solution to the lattice Boltz-mann hierarchy,” Phys. Rev. Lett. 98, 124502 �2007�.

6P. Taheri, M. Torrilhon, and H. Struchtrup, “Couette and Poiseuille micro-

flows: Analytical solutions for regularized 13-moment equations,” Phys.Fluids 21, 017102 �2009�.

7S. H. Kim, H. Pitsch, and I. D. Boyd, “Accuracy of higher-order latticeBoltzmann methods for microscale flows with finite Knudsen numbers,” J.Comput. Phys. 227, 8655 �2008�.

8G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of GasFlows �Oxford, New York, 1994�.

9A. Garcia and W. Wagner, “Generation of the Maxwellian inflow distri-bution,” J. Comput. Phys. 217, 693 �2006�.

10D. Frenkel and B. Smit, Understanding Molecular Simulation �Academic,San Diego, 1996�.

11P. Yi, D. Poulikakos, J. Walther, and G. Yadigaroglu, “Molecular dynam-ics simulation of vaporization of an ultra-thin liquid argon layer on asurface,” Int. J. Heat Mass Transfer 45, 2087 �2002�.

12Y.-L. He, J. Sun, Y.-S. Li, and W.-Q. Tao, “A molecular dynamics study ongrowth of condensation film on a solid surface,” Prog. Comput. Fluid Dyn.9, 262 �2009�.

13Q. Sun, I. D. Boyd, and G. V. Candler, “A hybrid continuum/particleapproach for modeling subsonic, rarefied gas flows,” J. Comput. Phys.194, 256 �2004�.

14A. Donev, J. B. Bell, A. L. Garcia, and B. J. Alder, “A hybrid particle-continuum method for hydrodynamics of complex fluids,” MultiscaleModel. Simul. 8, 871 �2010�.

15X. B. Nie, S. Y. Chen, W. N. E, and M. O. Robbins, “A continuum andmolecular dynamics hybrid method for micro- and nano-fluid flow,” J.Fluid Mech. 500, 55 �2004�.

16R. Delgado-Buscalioni and P. V. Coveney, “Continuum-particle hybridcoupling for mass, momentum, and energy transfers in unsteady fluidflow,” Phys. Rev. E 67, 046704 �2003�.

17J. Koplik, J. Banavar, and J. F. Willemsen, “Molecular dynamics of Poi-seuille flow and moving contact lines,” Phys. Rev. Lett. 60, 1282 �1988�.

18Y. Fang and W. W. Liou, “Computations of the flow and heat transfer inmicrodevices using DSMC with implicit boundary conditions,” J. HeatTransfer 124, 338 �2002�.

19R. Meland, A. Frezzotti, T. Ytrehus, and B. Hafskjold, “Nonequilibriummolecular-dynamics simulation of net evaporation and net condensation,and evaluation of the gas-kinetic boundary condition at the interphase,”Phys. Fluids 16, 223 �2004�.

20K. Gu, C. B. Watkins, and J. Koplik, “Molecular dynamics simulation ofthe equilibrium liquid–vapor interphase with solidification,” Fluid PhaseEquilib. 297, 77 �2010�.

21T. Ishiyama, T. Yano, and S. Fujkawa, “Molecular dynamics study ofkinetic boundary condition at an interface between argon vapor and itscondensed phase,” Phys. Fluids 16, 2899 �2004�.

22C. Cercignani, Rarefied Gas Dynamics �Cambridge University Press, NewYork, 2000�.

23Y. Sone, Kinetic Theory and Fluid Dynamics �Birkhäuser, Boston, 2002�.24V. P. Carey, G. Chen, C. Grigoropoulos, M. Kaviany, and A. Majumdar,

“A review of heat transfer physics,” Nanoscale Microscale Thermophys.Eng. 12, 1 �2008�.

25R. Meland, “Molecular dynamics simulation of the inverted temperature

112002-19 Multiscale molecular simulations of argon vapor Phys. Fluids 22, 112002 �2010�

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

130.63.180.147 On: Wed, 13 Aug 2014 10:11:53

gradient phenomenon,” Phys. Fluids 15, 3244 �2003�.26S. Kjelstrup, T. Tsuruta, and D. Bedeaux, “The inverted temperature pro-

file across a vapor/liquid surface analyzed by molecular computer simula-tions,” J. Colloid Interface Sci. 256, 451 �2002�.

27R. Meland and T. Ytrehus, “Evaporation and condensation Knudsen layersfor nonunity condensation coefficient,” Phys. Fluids 15, 1348 �2003�.

28J. S. Rowlinson and B. Widom, Molecular Theory of Capillarity �Claren-don, Oxford, 1982�.

29R. J. Stevens, L. V. Zhigilei, and P. M. Norris, “Effects of temperature and

disorder on thermal boundary conductance at solid-solid interfaces: Non-equilibrium molecular dynamics simulations,” Int. J. Heat Mass Transfer50, 3977 �2007�.

30B. H. Kim, A. Beskok, and T. Caigen, “Molecular dynamics simulationsof thermal resistance at the liquid-solid interphase,” J. Chem. Phys. 129,174701 �2008�.

31J. Chen, G. Zhang, and B. Li, Molecular dynamics simulations of heatconduction in nanostructures: Effect of heat bath,” J. Phys. Soc. Jpn. 7,074604 �2010�.

112002-20 Gu, Watkins, and Koplik Phys. Fluids 22, 112002 �2010�

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:

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