thermophoresis

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Aerosol Science and Technology 36: 1099 1117 (2002 )c 2002 American Association for Aerosol ResearchPublished by Taylor and Francis0278-6826 =02=$12.00 C .00DOI: 10.1080/0278682029009216 8

Thermophoresis in Rare ed Gas Flows

M. A. Gallis, D. J. Rader, and J. R. Torczynski Engineering Sciences Center, Sandia National Laboratories , Albuquerque, New Mexico

Numerical calculations are presented for the thermophoreticforce acting on a free-molecular, motionless, spherical particle sus-pended in a rare ed gas ow between parallel plates of unequaltemperature. The rare ed gas ow is calculated with the direct

simulation Monte Carlo (DSMC ) method, which provides a time-averagedapproximation to the localmolecularvelocity distributionat discrete locations between the plates. A force Greens function isused to calculate the thermophoretic force directly from the DSMCsimulations for the molecular velocity distribution, with the under-lying assumption that the particle does not in uence the molecularvelocity distribution. Perfect accommodation of energy and mo-mentum is assumed at all solid/gas boundaries. Earlier work formonatomic gases (helium and argon ) is reviewed, and new calcu-lations for a diatomic gas (nitrogen ) are presented. Gas heat uxand particle thermophoretic forces for argon, helium, and nitro-gen are given for a 0.01 m spacing between plates held at 263 and283 K over a pressure range from 0.1 to 1000 mTorr (0.01333 133.3 Pa ). A simple, approximate expression is introduced that can

be used to correlate the thermophoretic force calculations accu-rately over a wide range of pressures, corresponding to a widerange of Knudsen numbers (ratio of the gas mean free path to theinterplate separation ).

INTRODUCTIONA small particle suspended in a gas with a temperature gra-

dient experiences a force called the thermophoretic force forthe case where energy is transferred to the particle by the mean

Received 13 July 2001; accepted 15 April 2002.The authors gratefully acknowledge many useful technical discus-

sions with Dr. W. L. Hermina, Dr. M. E. Riley, and Dr. R. J. Bussof Sandia National Laboratories. This work was performed at SandiaNational Laboratories. Sandia is a multiprogram laboratory operatedby Sandia Corporation, a Lockheed Martin Company, for the UnitedStates Department of Energy under contract DE-AC04-94AL85000.This work was partially supported by the Of ce of Basic Energy Sci-ences within the Of ce of Science of the United States Department of Energy.

Address correspondence to Daniel J. Rader, Engineering SciencesCenter, Sandia National Laboratories, Albuquerque, NM 87185-0834.

E-mail: [email protected]

thermal motion of the surrounding gas molecules. 1 The thmophoretic force will induce particle motion even in the absence of gas ow, causing the particle to move from warmetoward cooler gas regions. Thermophoresis was rst describedby Tyndall (1870), who observed a dust-free zone surrounding ahotbody immersed ina dustygas.Thermophoresis isof practicalinterest in a wide range of applications, including aerosol collec-tors (thermal precipitators), aerosol manufacture of ber opticsgas cleaning, nuclear reactor safety, and the corrosion/foulingof heat exchangers. Many authors have realized the possibilitof protecting valuable surfaces from particle contaminant deposition by keeping the surface warmer than the surroundinggas; examples include thermophoretic protection of paintingsand other works of art (Nazaroff and Cass 1987) and semiconductor wafers during manufacturing (e.g., Gokoglu and Rosne1986). Thermophoretic protection has been shown to be a po

tentwafer-protection strategy at ambientpressures (e.g., Ye et al.1991; Bae et al. 1995), but questions have emerged about thpotential effectiveness of thermophoresis at the low pressurelikely to be encountered in modern semiconductor manufacturing processes. This concern is well founded, as in the limiof a perfect vacuum the thermophoretic force vanishes alonwith the gas molecules that cause it. Another area of conceris the potential effectiveness of using thermophoresis to protect MicroElectroMechanical Systems (MEMS); although thesedevices may be operated at ambient pressures, the very smalfeature size of such devices may lead to gas rarefaction effectthat could affect the magnitude of the thermophoretic force. The

goal of this work is to explore the behavior of the thermophoreticforce under the small-scale or low-pressure conditions likely tobe encountered in these new applications.

Brock (1967) rst recognized that at least 3 length scales arneeded to quantify the thermophoretic force: a particle size, Ra characteristic system dimension, L, and the molecular mefree path, . A general description of thermophoresis woulinvolve at least 2 dimensionless parameters related to geometrya particleKnudsen number,Kn R D = R p, anda system Knudse

1 Brock (1967 ) separatesthe forcesarising fromenergy transfer tothe particleby gas collisions (thermophoresis ) from thatarising from radiant energy transfe

to the particle (photophoresis ).

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1100 M. A. GALLIS ET AL.

number, Kn L D = L. The continuum regime applies for smallKnudsen numbers, Kn ! 0, while the free-molecular regime isreached in the limit of very large Knudsen numbers, Kn ! 1 .Rarefaction (noncontinuum ) effects can arise either because of small length scales or because of large mean free paths (e.g., inthe limit of low gas pressure as / 1=P ). Extensive literaturehas addressed the case where the gas is restricted to the con-tinuum regime, Kn L ! 0, while the particle Knudsen numbermay lie anywhere in the range 0 < Kn R < 1 . Important reviewsof this case include those by Loyalka (1992 ), Bakanov (1991 ),Fuchs (1982 ), Talbot et al. (1980 ), and Waldmann and Schmitt(1966 ). Of particular importance here is Waldmanns expres-sion (Waldmann 1959; Waldmann and Schmitt 1966 ) for thethermophoretic force on a free-molecular particle (Kn R ! 1 )in a continuum gas described by a rst-order approximationChapman-Enskog molecular velocity distribution function.Waldmann found that the thermophoretic force for this case isproportional to the particle area and the local gas translationalheat ux, is inversely proportional to the mean molecular speed,and is independent of pressure. Experimental (Waldmann andSchmitt 1966; Li and Davis 1995a ) and subsequent theoretical(e.g., Yamamoto and Ishihara 1988; Loyalka 1992; Beresnevand Chernyak 1995 ) studies support the Waldmann result.

Less common are studies of the thermophoretic force for arare ed gas (Kn L > 0); those available all deal with the prob-lem of a free-molecular particle suspended between 2 parallel,in nite plates. The restriction that the particle/gas interactionis free molecular is based on the fact that the particle radius( microns ) is typically many orders of magnitude smallerthan the characteristic system length ( centimeters ), R p L .Thus, the fact that the gas is exhibiting noncontinuum behavioron thesystem length scale usuallyimplies thatthe free-molecularlimit must be achieved on the particles length scale. 2 Brock (1967 ) rst approached the rare ed-gas thermophoresis prob-lem for the case where both the plate separation and particleradius were much smaller than the mean free path. In this casethegas ow iseverywhere free-molecular, with molecules travel-ing back andforth between collisionswith the plates andparticleand never colliding with other molecules. For small temperaturedifferences, Brock found that the thermophoretic force was pro-portional to the temperature difference between the plates, theparticle area, and the pressure. Phillips (1972 ) later pointed outa aw in Brocks analysis and gave the correct limit, which dif-

fers only by a numerical constant that was 50% larger than thatobtained by Brock. In the samework, Phillips (1972 ) used a mo-ment solution to the Boltzmann equation to obtain an expressionfor the thermophoretic force on a particle between parallel platesvalid for all Kn L provided that the particle is much smaller thanthe plate separation, R p L. Phillips expression approachesthe correct free-molecular and continuum limits. Later, Cha and

2However, theperpetual drive toward smallerfeaturesizesin microelectronicand MEMS devices may soon lead to conditions where the particle size is notsmall compared to system length scales.

McCoy (1974 ) also derived a closed-form expression for thethermophoretic force covering all values of the particle and sys-tem Knudsen numbers; notably, their result fails to approach theWaldmann limit. Vestner (1974 ) proposed that when the gas iin the near-continuum slip regime (Kn L not too large), the themophoretic force can be approximated by the continuum-gaslimit wherein the temperature gradient is adjusted to accountfor the temperature jumps that occur at each plate. Vestnersprescription was not intended to be extended beyond the slipregime.

The theoretical predictions from these 3 studies (Phillip1972; Cha and McCoy 1974; and Vestner 1974 ) do not all agreand, unfortunately, the available experimental data are not suf-

cient to distinguish the correct thermophoretic behavior in arare ed gas. Davis and Adair (1975 ) used a microstress gaugto measure the thermophoretic force on a sphere centered be-tween parallel plates as a function of gas pressure. Their mea-surements were made on a nite system (the sphere radius waapproximately 20% of the plate separation ), however, so it is nopossible to directly compare their results to the previous theoret-ical analyses, which all assumed R p L. More recently, Li andDavis (1995a, b ) used an electrodynamic balance to study parti-cle thermophoresis in the particle transition regime for a varietyof particle/gas combinations. For Kn R 3 (particles enterinthe free-molecular regime ), the measured thermophoretic forcepeaked about 10% below the Waldmann limit, then declinedwith increasing Kn R. The authors attributed the decline to gasrarefaction, i.e., by a reduction in the local temperature gradientresulting from temperature jumps at the plates. Li and Davis didreport some success in describing their data by correcting thetemperature gradient for temperature jumps; unfortunately, theirstrategy required an accommodation coef cient of 1

3 for the

air/copper surfaces that is substantially lower than previouslyreported values. The authors conjecture that this relatively lowaccommodation coef cient could result from edge effects intheir chamber or their use of highly oxidized (dirty ) plateThese uncertaintiesmake itdif cult tocomparethe dataof Li andDavis to analytical or numerical solutions. It is important to note,however, that Li and Davis data did approach the Waldmannlimit (within 10% ) for air, helium, and carbon dioxide.

Recently, Gallis et al. (2001a ) provided new theoretical computations forthe thermophoretic force ina rare edgas ow. Theyused a force Greens function (Gallis et al. 2001b ) to compute th

force on a spherical particle directly from the molecular veloc-ity distribution calculated by the direct simulation Monte Carlo(DSMC ) method of Bird (1994 ). The selected geometry was thatof a motionless, spherical particle suspended in a monatomic,quiescent gas lling theregionbetween2 in nite, parallel plates.The particle was assumed to be much smaller than the plate sep-aration, R p L, and the gas/particle interaction was assumedto be free-molecular, Kn R ! 1 . All gas/surface interactionwere assumed to be diffuse and fully accommodated (at thsurface temperature ). Gallis et al. (2001a ) calculated the thermophoretic force in argon and helium over a range of pressures


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THERMOPHORESIS IN RAREFIED GAS FLOWS 110

(0.1 1000 mTorr ) that spanned the transition regime.In the free-molecular and continuum limits, the Gallis et al. calculationsagreed very well with the theoretical free-molecular result of Brock/Phillips and the continuum result of Waldmann, respec-tively. In the transition regime, however, the thermophoreticforce calculated by Gallis et al. (2001a ) showed marked dif-ferences (up to 40% ) from the predictions of Phillips model(1972 ). The DSMC calculations also showed that the magni-tude of the thermophoretic force was nearly constant betweenthe plates, contradicting earlier theoretical studies (Havnes et al.1994; Chen 1999, 2000 ) that predicted signi cant variations inthe thermophoretic force near the walls.

Gallis et al. (2001b ) also used the DSMC/Greens functionmethod to analyze the relation between gas heat ux and thethermophoretic force. Vestner (1974 ) appears to be the rst tohave noticed that the thermophoretic force (in both the free-molecular and continuum limits ) is proportional to the particlecross-sectional area and the local heat ux and inversely pro-portional to the mean molecular speed. Based on theoreticalarguments, Gallis et al. (2001b ) conjecture that the constant of proportionality depends on the form of the molecular velocitydistribution. Even for apparently distinct velocity distributions,such as the Chapman-Enskog (nonequilibrium continuum ) orthe combination of 2 half-range Maxwellians (free molecular ),the proportionality constant differs by only about 10%. Gallisetal. (2001b ) calculated the thermophoretic proportionality con-stant over theentire transition regimeand found thatthe constantvaried smoothly from the continuum to the free-molecular val-ues for particles in argon in a parallel-plate geometry. Thus, theclose relationship between the heat ux and the thermophoreticforce was demonstrated in this geometry.

The present work expands on the previous DSMC/Greensfunction studies of the thermophoretic force on a particle sus-pended in a rare ed gas between parallel plates (Gallis et al.2001a,b ). The earlier work for monatomic gases (helium andargon ) is brie y reviewed, and new calculations for a diatomicgas (nitrogen ) are presented. For the parallel-plate geometry, asimple, approximate expression is introduced that can be used to

Figure 1. Schematic diagram of a particle suspended between parallel plates.

correlate the thermophoretic force calculations accurately over awide range of pressures, corresponding to a wide range of KnAs in theearlierwork, the local molecular velocity distribution iscalculated with the DSMC method (Bird 1994 ). Perfect accommodation of energy and momentum is assumed at all solid/gasboundaries. Calculations for the heat ux between the platesare presented to benchmark the accuracy of the DSMC method.Using the force Greens function (Gallis et al. 2001b ), the themophoretic force is calculated directly from the DSMC simula-tions for the molecular velocity distribution, with the underlyingassumption that the particle does not in uence the molecular ve-locity distribution. Although the force Greens function can beapplied to a variety of gas/particle surface models, only diffusere ections from the particle are considered here. Gas heat uxesand particle thermophoretic forces for argon, helium, and ni-trogen are given for a 0.01 m spacing between plates held at263 and 283 K over a pressure range from 0.1 to 1000 mTorr(0.01333 133.3 Pa ).

THEORYGeometry

The present modeling effort seeks to explore the pressuredependence of the thermophoretic force acting on a particlebetween 2 in nite, parallel plates of unequal temperature. Aschematic diagram ofthis geometry is shown in Figure 1. The topplate is oriented face down and is assumed to be warmer than theplate below it. The 2 plates are separated by a gap, L ; the regiobetween the gaps is lled with a monatomic or diatomic gas.The coordinate system is de ned such that x D 0 corresponds the surface of the bottom plate and x D L corresponds to the suface of the top plate. Temperatures of the top and bottom platesarede ned as T h and T c , where T h > T c andthese 2 temperaturedo not differ by much (i.e., T h T c T c ). A spherical particlof radius R p suspended between the plates experiences a ther-mophoretic force pushing it away from the warmer surface. Thegas is assumed to be quiescent (no mass ow ); consequentlheat transfer between the plates is dominated by conduction


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(radiation is neglected ). Perfect accommodation is assumed atall gas/solid interfaces, i.e., thermal, tangential, and normal ac-commodation coef cients are assumed to be unity at each plateand on the particle surface.

Rare ed Gas Flow At high enough (e.g., atmospheric ) pressures, a gas acts as

a continuum and can be described by the Navier Stokes equa-tions. As the pressure decreases, the gas will eventually begin toexhibit noncontinuum effects in that the discrete molecular na-ture of thegasbecomesapparent. Manifestations of ow rarefac-tion include the well-known temperature-jump and velocity-slipdiscontinuities at walls (Springer 1971 ). The onset of noncon-tinuum gas behavior is typically indicated by a Knudsen numberbased on a characteristic length of the system, here taken as theplate separation L , so that

Kn L D

L; [1]

where is the molecular mean free path. The mean free path isde ned as the average distance traveled by molecules betweencollisions. As the de nition of mean free path is somewhat arbi-trary, a number of de nitions persist in the literature; the de ni-tion of mean free path given by Talbot et al. (1980 ) and Springer(1971 ) is used here:

D2c

; [2]

where and are the gas viscosity and density, c is the mean

molecular speed,

c D r 8k BT m ; [3]k B is Boltzmanns constant, T is the local gas temperature, and mis the gas molecularmass.Thecontinuum limit is achieved whenthe gas mean free path is much smaller than the characteristicsystem length (i.e., Kn L ! 0), while free-molecular (rare ed )

ow is achieved when the gas mean free path is much largerthan L (i.e., Kn L ! 1 ). The so-called transition ow regimelies between these extremes.

The particle radius isa secondlengthscale in the problem,anda particle Knudsen number can be de ned based on the particleradius: Kn R D = R p. In this work, the particle is assumed to bemuch smaller than the plate separation, R p L. Consequently,for the present problem of interest where the gas is exhibitingnoncontinuum behavior, the free-molecular limit is assumed toapply on the particles length scale, Kn R ! 1 . An implicationof the assumptions R p L and Kn R ! 1 is that the localmolecular velocity distribution is not affected by the particlespresence. Although the gas ow in the vicinity of the particle isrestricted to the free-molecular limit, no restriction is placed on

the ow eld, so the free-molecular, transition, and continuumgas ow regimes will each be considered.

Gas Heat Flux and the Thermophoretic Force Vestner (1974 ) rst noted that the gas heat ux and the ther

mophoretic force acting on a particle are closely related quanti-ties in both the continuum and rare ed gas limits; recent workby Gallis et al. (2001b ) supports this association over the entiretransition regime for a monatomic gas. In this section, closed-form expressions are presented for the gas heat ux and the ther-mophoretic force between in nite parallel-plates (Figure 1 ) fthe case of an arbitrary system Knudsen number, 0 < Kn L < 1The particle is assumed to be isothermal and the gas ow aroundit free-molecular. Reliable solutions forboth thegasheat ux andparticle thermophoretic force areavailable inboth thecontinuumand free-molecular limits; an approximate bridging expressionis proposed for each in the transition regime. Unless otherwiseindicated, theuseof the terms continuum andfree molecularin this work refer to the system condition (determined by Kn L

as the particle is always assumed to lie in the free-molecularregime.Continuum Gas. For small system Knudsen number

(Kn L ! 0), the gas acts as a continuum and the solution to theone-dimensional energy equation (for small temperature differences ) gives thewell-known result of a linear temperature pro lebetween the plates, where the temperature of the gas immedi-ately adjacent to each plate equals the plate temperature. In thiscase the continuum heat ux between the plates, qC , is given b

qC D k dT dx

D k T h T c

L; T h T c T c ; [

where k is the gas thermal conductivity. Equation (4) appliequally well for any gas (i.e., monatomic, diatomic ) if the apropriate value of k is used. The temperature dependence ofthe thermal conductivity has been neglected in Equation (4which is reasonable for modest temperature differences. Theone-dimensional geometry requires that the heat ux is constantacross the domain (independent of x). Note that the thermalconductivity is independent of pressure; hence, the heat ux is alsoindependent of pressure as long as the ow is continuum.

For large temperature differences, the temperature depen-dence of the thermal conductivity should be considered. Eucken

rst suggested the following form relating the thermal conduc-tivity to the viscosity (which itself exhibits a temperature depen-dence ) (Reid et al. 1987 ):

k D f tr ctr

RC f int

c int

R k B

m 32 f tr C

2 f int k B

m; [

where c tr and c int are the gas translational-energy and internal-energy speci c heat capacities at constant volume, R is the spci c gas constant ( R D k B=m ), and f tr and f int are parameters characterizing the contributions to the thermal conductivityresulting from translational and internal degrees of freedom,


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respectively (Reid et al. 1987 ). The right-hand expression inEquation (5) makes use of the classical approximations c tr D3 R=2 and c int D R=2, where is the number of molecularinternal degrees of freedom. For a monatomic gas, D 0 and f tr D 5=2, so Equation (5) reduces to k D 15 k B =4m , which isaccurate to a rst approximation according to elementarykinetictheory (Bird 1994, p. 66 ). For polyatomic gases, the accuracyof using Equation (5) depends on the choices for the parameters f tr and f int . Reid et al. (1987 ) report approximate theoreticalexpressions for these parameters thatdepend on thegas Schmidtnumber and acollision number, Z rot , representing the number of collisions required to exchange a quantum of rotational energywith translational energy. Based on comparison to experimentaldata, Reid et al. determineda valueof Z rot D 7 fornitrogen (Reidet al. 1987, p. 497 ); the corresponding values f tr D 2:321 and f int D 1:462 are used for nitrogen here. The translational part of the thermal conductivity is approximated by

k tr D32

f tr k B

m: [6]

Note that for monatomic gases (e.g., helium and argon ), thetranslational andtotal thermalconductivities areequal, k tr D k D15 k B =4m , while for nitrogen k tr D 3:482 k B =m . The presentpartition, Equation (5), between translational and internal con-tributions to the heat ux gives a ratio of k tr =k D 0:704 for ni-trogen, which is in good agreement with the ratio 0.71 reportedby Vestner and Waldmann (1977 ).

The variable soft sphere (VSS ) molecular-interaction modelis used in the following DSMC simulation, which leads to apower-law temperature dependence of the coef cient of viscosity (Bird 1994, p. 68 ):

D ref T T ref

!

: [7]

Experimental data are used to specify the reference viscosity, ref , at reference temperature, T ref D 273 K, and the exponent! results from a best t of experimental viscosity data nearthe reference temperature. In this work, the tabulated exponentvalues given in Appendix A of Bird (1994 ) are used. Givena power-law temperature dependence of viscosity (and hencethermal conductivity ), Equation (4) can be solved to give thefollowingnonlinear expressionforthe one-dimensional heat ux

between parallel plates (Bird 1994, p. 155 ):

qC D k ref T

! C 1h T

! C 1c (! C 1) T !ref L : [8]

The heat ux given by Equation (8) is constant across the in-terplate gap as required for a continuum, motionless gas; thelinearized limit of Equation (4) is retrieved in the constant-conductivity limit (! D 0).

Waldmann (1959 ) derived an expression for the thermo-phoretic force in the free-molecular particle limit (Kn R ! 1 ),

but where the system is continuum (Kn L ! 0). In this derivtion the particle was assumed to be immersed in an in nite bathof gas molecules with a rst-order approximation Chapman-Enskog molecular velocity distribution, and the force on theparticle was calculated by integrating the momentum exchangefrom molecular collisions over the particles surface. Assum-ing complete thermal accommodation at the particle surface,Waldmanns result for the parallel-plate geometry is given as

F th ;C D 3215 m8 k BT

1=2

R2 pk

tr d T dx

D 3215

R2 pc q tr C :

Waldmann and Schmitt (1966 ) advocated the use of the translational part of the thermal conductivity, Equation (6), for polatomic gases; they support this contention by comparison toexperimental data (see also the recent work of Li and Davi(1995a )). The nal equality in Equation (9) states that the the

mophoretic force is proportional to the translational componentof the heat ux, q tr C . Vestner (1974 ) considered thermophoresifor nonspherical particles in a polyatomic gas and arrived at thesame expression as Waldmann for spherical particles. Thus, inthe continuum ow limit, the thermophoretic force is propor-tional to the cross-sectional area of the particle and the transla-tional heat ux and inversely proportional to the mean thermalspeed. Interestingly, thethermophoretic force in this regime is idependent of gas pressure as long as the gas can be described ba continuum, Chapman-Enskog, distribution. Clearly, the ther-mophoretic force and the translational heat ux are closely re-lated in the continuum limit.

Theprevious discussion applies to a continuum gas surround-ing a particle that lies in the free-molecular regime. Extensiveliterature exists on the closely related problem of a continuumgas wherein the suspended particle is not restricted to the free-molecular regime. Talbot et al. (1980 ) provide an excellent review of this literature and give a semiempirical formula thatis useful over the entire range of particle Knudsen numbers(continuum to rare ed ) assuming a continuum gas. Talbots result reduces to Equation (9) in the free-molecular particle limitMore recently, several authors (e.g., Yamamoto and Ishihara1988; Loyalka 1992; Beresnev and Chernyak 1995 ) have solvethe continuum-gas/rare ed-particle problem numerically, typ-

ically using the linearized Boltzmann equation. Beresnev andChernyak (1995 ) caution that the Talbot semiempirical expres-sion agrees satisfactorily with their calculations only for the lim-ited range of particle-gas thermal conductivity ratio near unity.For all of these theoretical studies, the Waldmann limit is ob-tained in the particle free-molecular limit.

3 Talbot et al. (1980 ) extended Waldmanns analysis to allow for incompletethermal accommodation at the particle surface; Beresnev and Chernyak (199used a different method to derive a force expression, including both energy andtangential momentum accommodation coef cients.


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Free-Molecular Gas. In the free-molecular limit (Kn L !1 ), molecules travel back and forth between the plates withoutcolliding with each other; in this case the heat transfer betweenthe plates can be described from a molecular point of view. Thewalls are assumed to re ect molecules diffusely with completethermal accommodation, which means that the molecules re-

ected from a wall possess a half-range Maxwellian molecularvelocity distribution in equilibrium with the wall temperature. Inthe rare ed limit, the space between the walls is characterizedby 2 streams of noncollisional molecules, with higher-energymolecules streaming downward from the hot plate while lower-energy molecules stream upward from the cold plate. For a sta-tionary gas with internal degreesof freedom, Bird (1994, p. 84)has shown that the total heat ux to a surface is increased bya factor of (1 C =4) compared to the translational heat ux.Thus, Birds (1994, p. 280) monatomic-gas result for the free-molecular heat ux between parallel plates can be extended toa polyatomic gas according to

qFM D 8k

Bm

1=2

(P ) 1 C

4 T 1=2h T 1=2c : [10]

Unlike the continuum result, the free-molecular heat ux is di-rectly proportional to the gas pressure, P , so in the limit of vanishing pressure the heat ux approaches zero, as it must ina vacuum. In the free-molecular limit, the translational compo-nent of the heat ux for any gas is obtained with D 0. Forsmall temperature differences, the linearized result for the heat

ux becomes

qFM D 12

8k B T m

1=2 PT

1 C

4

(T h T c );

[11]T h T c T c ;

where T D T 1=2h T 1=2

c is the temperature in the free-molecularlimit and the approximation T 1=2h C T

1=2c 2T 1=2 has been used.

An expression for the thermophoretic force between 2 plateswhen both the system and particle are free-molecular (Kn L !1 and Kn R ! 1 ) has been given by Gallis et al. (2001b):

F th ;FM D 32 R

2 p P"

T 1=2h T 1=2

c

T 1=2h C T 1=2

c #: [12]For small temperature differences, Equation (12) reduces to

F th;FM D 38 R

2 p

PT (T h T c)

D34

R2 pc q tr FM ; T h T c T c ; [13]

where q tr FM is the translational component of the heat ux. Thenal equality given in Equation (13) follows from the linearized

form of the free-molecular heat ux, Equation (11). The lin-earized result agrees with the earlier result of Phillips (1972),

who presented the force in a slightly different form; Vestner(1974) later obtained the identical form given in Equation (13).As in the continuum limit, the free-molecular thermophoreticforce is proportional to the cross-sectional area of the particleand the local translational heat ux and inversely proportionalto the mean thermal speed. Unlike the continuum limit, the free-molecular thermophoretic force is directly proportional to thepressure, so in the limit of vanishing pressure this force on theparticle approaches zero, as it must in a vacuum.

Comparison of Equations (9) and (13) reveals a striking re-sult rst realized by Vestner (1974) and later exploited by Galliset al. (2001a,b): the 2 expressions for the thermophoretic forceare identical within a numerical constant. That is, in both thecontinuum and free-molecular limits the thermophoretic forceis proportional to the particle cross-sectional area and the localtranslational component of the heat ux and inversely propor-tional to the mean molecular thermal speed. Moreover, althoughthese 2 limiting cases derive from substantially different molec-ular velocity distributions (Chapman-Enskog for a continuumgas, 2 half-range Maxwellians at different temperatures for arare ed gas), the resulting numerical constants differ only byabout 10% (i.e., the ratio 45 =128).

Transition Regime Gas. The prediction of the heat ux ithe gas transition region is challenging, ultimately requiring acomplete solution of the Boltzmann equation. Bird (1994) gavean approximate closed-form result for the heat ux based onLees four-moment solution of the Boltzmann equation for amonatomic gas. For 2 parallel plates where the temperature dif-ference between the plates is assumed to be small, the heat uxfor a monatomic gas at all pressures can be approximated by

q D k

L(T h T c )

1C k 2PL

m T h2k B

1=2C

mT c2k B

1=2

;

[1T h T c T c :

Note that Equation (14) correctly reproduces both the contin-uum and free-molecular limits for the heat ux given above.Recovering the free-molecular limit requires noting that

T 1=2h T 1=2

c D(T h T c)

T 1=2

h C T 1=2

c : [1

Springer (1971) has shown that Equation (14) agrees reasonablywell with the limited available experimental data for monatomicand diatomic gases over a wide range of Knudsen numbers.

A simple interpolation formula for heat ux has been sug-gested by Sherman (1963):

qSh DqFM

1 C (qFM =qC ): [1

Equation (16) could be applied to monatomic or polyatomicgases if the heat ux expressions of Equations (8) and (10) areused. Springer (1971) noted that Equation (16) is identical to the


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Greens Function Method for Particle Force Calculations Once the local gas velocity distribution is known, the ther-

mophoretic force on a suspended particle can be calculated.For the present case where the particle is in the free-molecularregime (i.e., the velocity distribution function of gas moleculeshitting the particle is not appreciably altered by the presenceof the particle ), an elementary kinetic approach can be used.Following the classic derivation of Waldmann (1959 ), the netthermophoretic force is found directly by calculating the netrate of momentum transferred to the particle by the imping-ing and re ected molecules. In Waldmanns work, the incomingmolecules were represented by an analytical Chapman-Enskogdistribution, characteristic of a near-equilibrium gas sustain-ing a temperature gradient. Using this analytical distributionfunction and certain re ection models, Waldmann integratedthe pressure and shear stress over the surface of a sphericalparticle to obtain the result for thermophoretic force given byEquation (9).

In principle, Waldmanns momentum transfer method could

be applied for any velocity distribution function, e.g., usingthe numerical approximation to the distribution function suchas can be calculated directly by the DSMC method. Recently,Gallis et al. (2001b ) showed that the calculation of particleforces directly fromDSMC velocity distribution functions couldbe greatly facilitated by using a force Greens function. 5 TheGreens function approach assumes free-molecular sphericalparticles that are suf ciently dilute so that the surrounding gas

ow is not affected by their presence or motion. Brownian uc-tuations are not considered. The Greens function representsthe force on a spherical particle from a delta-function incidentmolecular velocity distribution function. A spherical particle of

radius R p and velocity u p is placed in an incident stream of gasmolecules with mass m , number density n , and velocity u (orvelocity c D u u p with respect to the particle reference frame ).The particle is taken to have a uniform temperature T p , a con-sequence of assuming that the particle thermal conductivity islarge compared to the gas thermal conductivity. Although Galliset al. (2001b ) considered an extended version of the Maxwellgas/surface-interaction model, in this work the molecules are as-sumed to experience only isothermal diffuse re ections fromthe particle surface (the re ected molecules having a half-rangeMaxwellian molecular velocity distribution function at the par-ticle temperature ).

The force Greens function is determined from the rates of delivery and removal of momentum by incident molecules andthe isothermally re ected molecules, respectively. These ratesare integrated over the hemispherical surface of the particle thatis exposed to the incident gas molecules, and the correspondingintegrals are carried out as in Epstein (1924 ) or Baines et al.

5A heat-transfer Greensfunctionwas alsopresented by Gallis et al. (2001b ).In the present work, particles are assumed to be moving at suf ciently lowspeeds relative to the gas so that particle heating can be ignored and the particleequilibrates to the local gas temperature.

(1965 ). For the present case of isothermal diffuse re ection, theforce Greens function (i.e., the force from a delta-function inci-dent molecular velocity distribution function ) derived by Gallet al. (2001b ) for a monatomic gas reduces to

F [c] D mn R2 p c [ jcj C (=6)c p]; [2where c p D (8k BT p=m )1=2 is the mean molecular speed at th

particle temperature. The force on a spherical particle from anarbitrary molecular velocity distribution function f [u ] is detemined by integration:

F [ f ] D Z F[u u p] d u ; [2where s f [u ] d u D 1. These expressions are obtained based onthe assumption that the ow in the vicinity of the particle islocally free-molecular so that linear superposition of the forcefrom molecules having different velocities is allowed. The formof these expressions is particularly convenientboth forobtaining

analytical results and for numerical implementation in DSMCmethods. For a motionless particle in a quiescent gas, theparticle-temperature dependent term in Equation (22) vanishewhen performing the integration in Equation (23).

The previous work of Gallis et al. (2001a,b ) applied the forcGreens function method to the case of monatomic gases. Theirmonatomic-gas implementation also can be applied for poly-atomic gases under the present assumptions: for a motionless,isothermal, diffuse-re ecting particle, the rotational and vibra-tional degrees of freedom of a gas can exchange only energy(not momentum ) with theparticle. Thusalthough internal energymodes can affect the temperature of the particle, these modes

do not contribute to the thermophoretic force that is calculateddirectly from the molecular velocity distribution function usingEquations (22) and (23) (see Gallis et al. 2001b ). In the presework, additional discussion of the monatomic gas results arecombined with new results for a diatomic gas (nitrogen ).

NUMERICAL APPROACHThe present numerical approach for determining the ther-

mophoretic force on a particle has 2 components: computingthe gas ow itself and computing the transport of the particlesby the gas ow. In this work, the DSMC method is used to cal-

culate the local molecular velocity distribution function, and theforce Greens function is used to calculate the resulting ther-mophoretic force on a suspended particle. Brief discussions of the numerical methods follow. For more complete details of thepresent implementation of the DSMC method, see Gallis et al.(2001a ), and for the force Greens function, see Gallis et al.(2001b ).

DSMC Method for Rare ed Gas Flow Calculations The DSMC method of Bird (1976, 1994 ) is a well-established

method for simulating many types of rare ed gas ows (Bart


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enough to ensure that molecules travel no more than 13 of thecell size in a time step, satisfying another constraint indicatedby Bird (1994 ). Typically, 100 molecules per cell are used forstatistical accuracy. Simulations are initialized using a constanttemperature pro le at the arithmetic mean of the 2 wall tem-peratures. To obtain steady results, averaging is initiated aftertransients have decayed. Studies by Gallis et al. (2001a ) are re-ported elsewhere, verifying that this procedure produces resultsthat are essentially independent of the computational mesh, thetime step, the number of molecules per cell, the initial transient,and the averaging duration (to acceptable statistical uncertaintyof about 2% ).

Force Greens Function Implementation with DSMC The numerical implementation of the force Greens function

with the DSMC method is described by Gallis et al. (2001b ).Based on Equations (22) and (23), over each time step 1 t andwithin each mesh cell of volume V , a computational moleculemoving with velocity u and representing N real gas molecules

contributes momentum F[u ]1 t to a spherical particle at rest(u p D 0). The number density corresponding to the computa-tional molecule is averaged over the mesh cell: n D N =V . Thetotal force on a spherical particle within a particular mesh cell iscomputed by summing the contributions from all computationalmolecules within the cell and dividing by the time step (orby theduration over which momentum accumulation occurs if this du-ration is greater than 1 time step ). As indicated earlier, the com-putational molecules arenot affected by collisionswith a particle(i.e., neither their momentanor their energies are altered ), sothisapproach is appropriate only for dilute particle concentrations.

There are 2 other minor details related to computing the force

on a particle with the Greens function method. First, under theconstraint of free-molecular ow around the particle used here,the force is exactlyproportional to the cross-sectional area of the

Figure 2. Temperature pro les for nitrogen for pressures between 0.1 and 1000 mTorr. Solid lines correspond to the translationaltemperature, while symbols correspond to the rotational temperature.

particle, R2 p . Thus it is expedient to compute the force per unicross-sectional area because the values so obtained are appli-cable to any particle satisfying this constraint. Second, becauseodd powers of the quantity jcj D j u u p j are present in F , thcontributions to be summed depend implicitly on the particlevelocity u p . In the present application, however, the particle isassumed to be at rest, so u p D 0 and the resulting force includeonly thermophoretic contributions (no drag ).

RESULTS AND DISCUSSIONThis section presents the temperature pro les and heat ux

between parallel plates calculated with the DSMC method andthe thermophoretic force calculated from the resulting veloc-ity distribution functions using the force Greens function. Theplates are at temperatures of T c D 263 K and T h D 283 and are separated by a distance of L D 0:01 m. The particle motionless at a temperature of T p D 273 K and has a diffusisothermally re ecting surface. The computational meshes andnumerical parameters are as indicated previously. The calcula-tions are performed for argon, helium, and nitrogen for pres-sures in the range of 0.1 to 1000 mTorr (0.01333 133.3 PaThis geometry and pressure range provide calculations from thenear-free-molecular to near-continuum regimes for all 3 gases.

Temperature Pro les and Heat Flux Figure 2 shows temperaturepro les fromDSMC simulations

for nitrogen at 5 pressures that span the desired range (0.11000 mTorr ). Each temperature point on a computed pro le isthe average value of the 10 cells in the y direction at thatposition. At each pressure, the translational temperature pro le

is indicated by a solid curve, while the rotational temperatureis indicated by the symbols (not all data points are plotted ). Aexpected, the temperature pro le at the lowest pressure is nearly


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uniform (approaching the free-molecular limit ), and the tem-perature pro le at the highest pressure almost matches the walltemperatures and varies linearly in between (approaching thecontinuum limit ). For intermediate pressures, the temperaturepro les exhibit temperature jumps at the walls and are curvedover a distance of a few mean freepathsfromthe walls.Althoughthe translational and rotational temperatures are generally ingood agreement, slight systematic differences are observed atintermediate pressures (e.g., 10 mTorr ). This level of agree-ment suggests that the gas translational and rotational modesare nearly in equilibrium, which is not surprising consideringthat only small temperature differences are considered. Tem-perature pro les using the DSMC technique on the present ge-ometry have previously been reported by Gallis et al. (2001a,b )for argon and helium. The results for argon are very similarto the present nitrogen results, with nearly continuum ow ob-served at 1000 mTorr and nearly free-molecular ow observed at0.1 mTorr. Such similarities are not surprising, as the mean freepaths of nitrogen and argon are very close. Because the meanfree pathfor heliumis signi cantly longer (byafactorofabout3 )than for argon or nitrogen, however, temperature pro les for he-lium show more rarefaction thanfor argonor nitrogen at thesamepressure.

Figure 3 shows the gas-phase heat ux for argon, helium, andnitrogencalculated with theDSMC method (solidcircles ), alongwith theoretical heat- ux predictions from both the Sherman/ Lees interpolating function (solid line ) and the Phillips model(dashed line ). The Sherman/Lees curve is generated from Sher-mans interpolating formula, Equation (16), using the contin-uum and free-molecular heat- ux expressions of Equations (8)and (10), respectively. The Phillips curve is generated from

(a)

Figure 3. Heat ux for helium, argon, and nitrogen. (Continued )

Equation (17). In all calculations, the thermal conductivity iscalculated from Equation (5) using the gas properties listed iTable 1. The DSMC calculations and the corresponding Sher-man predictions are also listed in Tables 2 and 3.

The free-molecular and continuum heat- ux limits are alsoplotted in Figure 3 for reference; as expected, the DSMC cal-culations approach these 2 limits in the appropriate regime. Theapproach to the free-molecular limit is particularly good: DSMCcalculations of the heat ux for helium, argon, and nitrogen fallonly 0.5, 1.6, and 1.5% below the free-molecular limit, Equa-tion (10), at the lowest pressure tested (0.1 mTorr ). These smadiscrepancies result from the fact that some collisions are oc-curring even at this low pressure. Helium lies closer to the free-molecular limit than the other 2 gases because its mean free pathis the longest. To demonstrate theaccuracyof thepresentmethodfor heat ux in collisionless ow, DSMC test calculations wereperformed for each gas at 0.1 mTorr in which collisions wereturned off (an option available in the Icarus DSMC method ); tDSMC-calculated heat- ux valuesagreed to within3 signi cant

gures with the free-molecular limit given by Equation (10) (sTables 2 and 3 ).

A direct comparison between the DSMC calculations andthe continuum heat- ux limit is dif cult because at the high-est pressure tested (1000 mTorr ), small temperature jumps athe wall are still present, so the continuum limit has not beencompletely achieved. DSMC simulations become computation-ally prohibitive at high pressure, as the small mean free pathrequires an extremely ne mesh and extraordinarily large num-bers of computational molecules to provide reliable results. Thefairest comparison that can be made is to the Sherman interpo-lation expression, Equation (16), which implicitly accounts fo


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(b)

(c)

Figure 3. (Continued )

temperature jumps while asymptotically approaching the con-tinuum limit. DSMC calculations for the heat ux are 0.4% lowfor argon and 1.2% and 1.5% high for helium and nitrogen, ascompared to the Sherman values. Considering the challenges as-sociated with DSMC simulations of near-continuum ow, thislevel of agreement is representative of the potential accuracy of the DSMC method at higher pressures.

The agreement between the DSMC simulations and theSherman/Lees predictions in the transition regime is acceptable,

although for pressures in the 10 30 mTorr range the DSMCdata lie about 8% below the Sherman/Lees predictions. Thepresent heat- ux results are in very good agreement with previ-ous calculations made by Bird (1994, p. 281 ) using the DSMmethod for argon in the same geometry; Bird noted that hiscalculations fell about 10% below the Lees four-moment solu-tion over most of the transition regime. On the basis of lim-ited available experimental data, Springer (1971 ) claimed ththe Sherman/Lees interpolation formula adequately describes


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Table 2DSMC/Greens function calculations and interpolating formula predictions for heat ux and normalized thermophoretic

force for nitrogen between parallel plates (0.01 m separation, 263 and 283 K )

Heat ux (W/m 2 ) Force/area (N/m2 )

Sherman EquationsPressure (mTorr ) DSMC interpolation DSMC/Greens (16), (20), (21)

0.1 (collisionless ) 0.333 0.333 3.66 10 4 3.66 10 4(10.1 0.328 0.331 3.60 10 4 3.30 10 4

0.3 0.963 0.979 1.06 10 3 9.76 10 4

1.0 2.99 3.12 3.27 10 3 3.12 10 3

3.0 7.77 8.28 8.43 10 3 8.33 10 3

10.0 18.4 19.8 1.98 10 2 2.01 10 2

30.0 31.3 32.7 3.35 10 2 3.38 10 2

100 42.0 42.4 4.40 10 2 4.43 10 2

300 46.9 46.4 4.65 10 2 4.87 10 2

1000 48.6 47.9 4.98 10 2 5.04 10 2

(continuum ) 48.6 5.12 10 2

1 8 The closed-form interpolation scheme embodied by Equations (16), (20), and (21) is known to approach thewrong limit in the free-molecularlimit; for the tabulated collisionless thermophoretic force, the exact free-molecular result of Equation (12) is used instead.

rare ed heat ux between parallel plates; if true, this wouldimply that the present DSMC simulations underpredicttransition-regime heat- ux data by as much as 8%. It is dif cultto determine if this discrepancy is real, however, as the subjectof rare ed heat transfer continues to be debated in the litera-ture. Springer (1971 ) rst acknowledged the apparent paradoxthat the admittedly simple four-moment methods seem to agreewith experimental data better than more sophisticated methods

Table 3DSMC/Greens function calculations and interpolating formula predictions for heat ux and normalized thermophoretic

force for helium and argon between parallel plates (0.01 m separation, 263 and 283 K )

Helium Argon

Heat ux (W/m 2 ) Force/area (N/m 2) Heat ux (W/m 2 ) Force/area (N/m2 )

Sherman DSMC/ Equations Sherman DSMC/ EquationsPressure (mTorr ) DSMC interpol. Greens (16), (20), (21) DSMC interpol. Greens (16), (20), (2

0.1 (collisionless ) 0.587 0.587 3.67 10 4 3.66 10 4(9) 0.186 0.186 3.67 10 4 3.66 10 4

0.1 0.584 0.586 3.64 10 4 3.31 10 4 0.183 0.185 3.61 10 4 3.30 10 4

0.3 1.73 1.75 1.08 10 3 9.89 10 4 0.540 0.548 1.06 10 3 9.79 10 4

1.0 5.66 5.75 3.52 10 3 3.25 10 3 1.69 1.76 3.30 10 3 3.14 10 33.0 16.0 16.6 9.86 10 3 9.38 10 3 4.45 4.77 8.53 10 3 8.52 10 3

10.0 45.8 48.8 2.78 10 2 2.76 10 2 10.9 11.9 2.05 10 2 2.12 10 2

30.0 102 110 6.07 10 2 6.20 10 2 19.6 20.8 3.57 10 2 3.70 10 2

100 187 194 1.08 10 1 1.10 10 1 27.7 28.1 4.89 10 2 5.01 10 2

300 247 249 1.38 10 1 1.41 10 1 30.2 31.2 5.35 10 2 5.57 10 2

1000 280 277 1.54 10 1 1.56 10 1 32.4 32.5 5.69 10 2 5.80 10 2

1 (continuum ) 290 1.64 10 1 33.1 5.90 10 2

9The closed-form interpolation scheme embodied by Equations (16), (20), and (21) is known to approach the wrong limit in the free-moleculalimit; for the tabulated collisionless thermophoretic force, the exact free-molecular result of Equation (12) is used instead.

that purport to solve the same mathematical problem more accu-rately. Ohwada (1996 ) more recently analyzed the parallel-plateproblem using a nite-difference numerical solution of the fullnonlinear Boltzmann equation and also predicted heat- ux val-ues 8% below four-moment method predictions. Ohwada iden-ti ed 1 potential weakness with the existing data: the platesused in the experiments were not baked, so the exact state ofthe surface was not controlled. In addition, Ohwada (199


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Figure 4. Pro le in the x direction of the thermophoretic force (per unit cross-sectional area ) for nitrogen.

expressed concerns about rare ed gas ows (e.g., thermal creepow) induced by nonuniformities in the temperature eld and

called for additional experiments. Indeed, more careful experi-ments are likely the only way to resolve the apparent discrep-

ancies between the predictions and the limited existing data forheat ux.

For both monatomic gases, 6 the Phillips model overpredictsthe heat ux compared to the DSMC calculations and the Sher-man/Lees predictions; the discrepancy is most pronounced near30 mTorr, where Phillips model is about 40% higher than theDSMC results and 30% higher than the four-moment results.Gallis et al. (2001a ) suggested that Phillips assumption of two-half-range Chapman-Enskog molecular velocity distributionsis not a good approximation in the transition regime. In anycase, Phillips model provides an inadequate representation of transition-regime heat ux.

Thermophoretic Force This sectionpresents calculations of the thermophoretic force

acting on a motionless particle suspended between 2 plates.Themolecular velocity distributions for the rare ed gas are calcu-lated with the DSMC method, and the resulting force on theparticle is calculated using the force Greens function. Calcu-lations are presented for argon, helium, and nitrogen over the

6Phillips model was derived for monatomic gases and is not shown fornitrogen.

pressure range 0.1 to 1000 mTorr. Force values are normalizedby the particle cross-sectional area, R2 p .

Thermophoretic Force Pro les. Figure 4 shows pro les othe thermophoretic force per unit particle cross-sectional area

for motionless particles suspended in nitrogen for the samepressures shown in Figure 2. Again, each point on a computedpro leis theaverage value of the 10 cells in the y direction at tha x position. At pressures below 100 mTorr, the thermophoreticforce is seen to increase with increasing pressure. For pressuresabove 100 mTorr, the thermophoretic force asymptotically ap-proaches the continuum limit, which is pressure-independent.For all pressures, the pro les are seen to be fairly uniform acrossthe domain. This observation is consistent with the present au-thors work for helium and argon (Gallis et al. 2001a,b ), bdiffers from previous theoretical analyses that predicted thatthe thermophoretic force would be strongly effected near walls

(Havnes et al. 1994; Chen 1999, 2000 ). For example, Havneet al. (1994 ) predicted a thermophoretic force that is 12 the cotinuum (Waldmann ) limit at the wall, while Chen (1999, 2000predicted 40% deviations in the thermophoretic force near awall. Large near-wall effects are clearly not observed in thepresent DSMC results, although the present levels of statisti-cal scatter (particularly at higher pressures ) could mask smallenear-wall deviations.

ThepresentDSMC/Greens function simulations for the ther-mophoretic force suggest that proximity to the wall has a rela-tively small effect on the thermophoretic force. This numerical


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result is consistent with the present theoretical understanding,embodied in Equation (20), which is that the thermophoreticforce is proportional to the ratio of the local heat ux to themolecular mean velocity, where the constant of proportionalityis only a weak function of the local molecular velocity distri-bution. For the present one-dimensional system, the heat uxmust be constant across the gap and, for the small temperaturedifferences under study, the mean molecular velocity is nearlyconstant. Thus, based on Equation (20), any wall effects wouldbe expected to arise from variations in the thermophoresis pa-rameter, which would beexpected to be < 10% for themolecularvelocity distributions likely to be encountered in the present ge-ometry. The detailed behavior of the thermophoretic force nearwalls will be the subject of future work.

Effect of Gas Rarefaction on Thermophoretic Force. Fig-ure 5 shows plots of the thermophoretic force per unit particlecross-sectional area for a motionless particle as a function of pressure for helium, argon, and nitrogen (solid circles ); the re-sults are also tabulated in Table 2 for nitrogen and Table 3 forargon and helium. To reduce stochastic uctuations, the ther-mophoretic force valuespresented herehave beenaveraged overthe entire domain (i.e., in both the x and y directions ).7 Thefree-molecular and continuum thermophoretic force limits arepresented for reference; as expected, the DSMC/Greens func-tion calculations approach these 2 limits in the correspondingregimes. The approach to the free-molecular limit is particularlygood: helium, argon, and nitrogen fall only 0.6, 1.4, and 1.7%below the free-molecular limit of Equation (12) at the lowestpressure tested (0.1 mTorr ). As discussed in regard to the heat

ux, these small discrepancies result from the fact that somecollisions are occurring even at this low pressure. Helium lies

closest to the free-molecular limit of the3 gases because its meanfreepath is3 times as long asthatofargon or nitrogen. To demon-strate the accuracy of the DSMC/Greens function method forthermophoresis in free-molecular ow, test calculations wereperformed for each gas at 0.1 mTorr in which collisions wereturned off (an option available in the Icarus DSMC method );the calculated thermophoretic force values agreed with the free-molecular limit to within 0.3% (see Tables 2 and 3 ).

Although it is clear in Figure 5 that the DSMC/Greens func-tion calculationsare approaching the continuum thermophoreticforce limit as pressure increases, a direct comparison betweenthe two is dif cult because at the highest pressure tested

(1000 mTorr ), small temperature jumps at the wall are stillpresent, so thecontinuum limit hasnotbeen completely achieved.As discussed above, ow simulations with the DSMC methodbecome computationally prohibitiveat highpressures.Onecom-parison that can be made is between the 1000 mTorr DSMC/ Greens function calculation and the continuum limit based on

7 Averaging across the interplate gap will introduce some systematic uncer-tainty. Although the heat ux across the gap must be constant for the parallel-plate geometry, slight spatial variations in the thermophoretic force would arisefrom variations in the mean thermal speed or .

Waldmanns expression, Equation (9); the present calculationare 6.3%, 3.6%, and 2.7% low for helium, argon, and nitrogen,respectively. That the present results fall below the continuumlimit is notsurprising: temperature jumps at the walls will reducethe temperature gradient between the plates and thereby reducethe predicted thermophoretic force. Even at 1000 mTorr, theDSMC calculations predict temperature jumps of about 0.5 Kfor helium and 0.2 K for nitrogen and argon. A fairer compari-son is between the DSMC/Greens function calculations and theproposed approximation for the thermophoretic force given bythe closed-form approximation resulting from the combinationof Equations (16), (20), and (21). This approach should worwell in the near-continuum regime, as the use of Shermans in-terpolation for the heat ux implicitly accounts for temperature- jump effects and the use of the Waldmann equation with a jump-corrected heat ux has already been suggested by Vestner(1974 ). As seen in Figure 5, the present calculations for the ther-mophoretic force are in very good agreement with the proposedclosed-form expressions (given as solid lines ). At 1000 mTorthe DSMC calculations and the closed-form results agree towithin 2% for all 3 gases, with the DSMC values being smaller.Considering the challenges associated with DSMC simulationsof near-continuum ow, this level of agreement is representa-tive of the potential accuracy of the DSMC method at higherpressures.

A comparison of the present DSMC/Greens functionforce calculations to the proposed closed-form expression of Equations (16), (20), and (21) reveals several additional trendsMost importantly, the overall agreement between the calcula-tions and the closed-form expression in Figure 5 is very good,although some subtleties are masked by the use of a log-logplot. In the free-molecular limit, it is seen that the closed-formexpression for the thermophoretic force falls slightly below butparallel to the presentcalculations.This discrepancy is expected,as the choiceof CE D 0:6791 as the numerical constant inEqua-tion (20) is known to be about 10% low in the free-moleculalimit (for which FM D 0:75). The use of CE appears to bwell justi ed at pressures above 10 mTorr, however, as the max-imum difference between the DSMC/Greens function forcecalculations and the proposed closed-form expression of Equa-tions (16), (20), and (21) is < 4% for the pressure range of 10 t1000 mTorr for all 3 gases. Thus the proposed closed-form ex-pression does an excellent job of correlating the present DSMC/

Greens function calculations for the thermophoretic force forthepresent geometry and gases. Interestingly, fora 0.01 m gap, apressure of 10 mTorr corresponds to a system Knudsen numberof about 1.3 for helium and about 0.5 for argon and nitrogen.Thus, the closed-form expression would be an attractive choicefor similar parallel-plate geometries as long as Kn L < 1:5.

Surprisingly, the agreement between DSMC/Greens func-tion calculations and predictions using Equations (16), (20and (21) for the thermophoretic force is much better than theagreement between the DSMC heat- ux calculations and theSherman/Lees correlation. Thiseffect is likelya fortuitous result


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(a)

(b)

Figure 5. Thermophoretic force (per unit cross-sectional area ) versus pressure for helium, argon, and nitrogen. (Continued )

of offsetting errors. Consider that as the free-molecular regimeis approached, the use of the thermophoretic parameter CE inEquation (20) results in predictions for the thermophoretic forcethat are 10% low; coincidentally, the use of the Sherman/Leesinterpolationfor theheat uxpredicts values that are 8% higherthan the DSMC calculations in the transition regime. These 2effects nearly offset each other, so the proposed closed-formexpression of Equations (16), (20), and (21) ends up providinga very good estimation of the present DSMC/Greens functioncalculations for the thermophoretic force.

Finally, predictions for the thermophoretic force based onPhillips method are shown in Figure 5 for the 2 monatomicgases. As seen, the Phillips result overestimates the thermophor-etic force in the transition regime by as much as 40% comparedto the DSMC calculations. A similar discrepancy was observedpreviously between the Phillips and Sherman model predictionsof the heat ux. As the Sherman/Lees formulation has beenfound to agree well with experimental data (Springer 1971Phillips model provides an inadequate model for transition-regimeheat ux.Basedon theresults shown inFigure5,Phillips


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(c)Figure 5. (Continued )

model is also inadequate for predicting the thermophoretic forcein the transition regime.

CONCLUSIONSNumerical calculations are presented for the thermophoretic

force acting on a motionless, spherical particle suspended in arare ed ow between parallel plates of unequal temperature.The molecular velocity distributions for the rare ed gas arecalculated with the DSMC method, and the resulting force onthe particle is calculated using the force Greens function. Thepresent work expands on the previous DSMC/Greens functionstudies of the thermophoretic force on a particle suspended in arare ed, monatomic gas between parallel plates (Gallis et al.2001a,b ). The earlier work for monatomic gases (helium and ar-gon ) is brie y reviewed, and new calculations for a diatomic gas(nitrogen ) are presented. Calculations are performed for pres-sures in the range of 0.1 to 1000 mTorr (0.01333 133.3 Pa ) tospan the range from near-free molecular to near-continuum forall 3 gases.

To establish the validity of the DSMC method for nitrogen,temperature pro les and heat ux between the plates are calcu-lated. As expected, the temperature pro le at the lowest pressureis nearly uniform (approaching the free-molecular limit ), and thetemperature pro le at the highest pressure almost matches thewall temperatures and varies l inearly in between (approachingthe continuum limit ). For intermediate pressures, the temper-ature pro les exhibit temperature jumps at the walls and arecurved over a distance of a few mean free paths from the walls.Good agreement between calculated translational and rotationaltemperatures suggests that the gas translational and rotational

modes are nearly in equilibrium although slight systematic dif-ferences are observed at intermediate pressures (e.g., 10 mTorrThe temperature pro les calculated for nitrogen are very similarto those previously reported for argon (Gallis et al. 2001a,bwhich is not surprising considering that the mean free paths ofnitrogen and argon are very close.

DSMC calculations for the total gas-phase heat ux are alsopresented for argon, helium, and nitrogen. For each gas, theDSMC calculations approach the free-molecular and continuumlimits in theappropriate regime:excellent agreement is observedin the free-molecular limit, while agreement in the continuumlimit is good although complicated by dif culties in perform-ing DSMC simulations at high enough pressures to rmly es-tablish continuum ow. In the intervening transition regime,the present DSMC data are found to be in acceptable agree-ment with the Sherman/Lees predictions although for pres-sures in the 10 30 mTorr range the DSMC data lie about 8%below the Sherman/Lees predictions. On the basis of limitedavailable experimental data, Springer (1971 ) claimed that th

Sherman/Lees interpolation formula adequately describes rar-e ed heat ux between parallel plates; if true, this would im-ply that the present DSMC simulations underpredict transition-regime heat- ux data by as much as 8%. Similar discrepanciesusing the DSMC method have been reported previously (Bi1994 ). It is dif cult to determine if this discrepancy is real, how-ever, as the subject of rare ed heat transfer continues to bedebated in the literature. More careful experiments are likelythe only way to resolve the apparent discrepancies betweenthe present DSMC predictions and the limited existing heat-

ux data. Phillips expression for transition-regime heat ux


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seriously overpredicts both the Sherman/Lees and the presentDSMC results and thus fails to provide an adequate model forheat ux in the transition regime.

Based on the rare ed-gas molecular velocity distributionscalculated with the DSMC method, a force Greens functionapproach is used to determine the thermophoretic force actingon a motionless particle. For all pressures and gases, the ther-mophoretic force is found to be nearly uniform across the do-main. This observation differs from previous theoretical analy-ses suggesting that the thermophoretic force varies signi cantlynear walls (Havnes et al. 1994; Chen 1999, 2000 ). The DSMCresults are consistent with the theoretical interpretation that thethermophoretic force is proportional to the local heat ux, whichmust be constant across the gap for the present one-dimensionalgeometry.

DSMC/Greens function calculations of the thermophoreticforce as a function of pressure are presented for helium, argon,and nitrogen. As required, the DSMC/Greens function calcula-tions for the thermophoretic force approach the free-molecularand continuum (Waldmann ) limits in the appropriate regimes.Excellent agreement between DSMC calculations and theoret-ical predictions is observed in the free-molecular limit, whileagreement in the continuum limit is good although complicatedby dif culties in performing DSMC simulations at high pres-sures. Correcting for temperature jumps, theDSMC calculationsagree with thecontinuum limit towithin 2% for all 3 gases. Pre-dictions for the thermophoretic force based on Phillips methodare found to overestimate the simulation results by as much as40%, suggesting that Phillips model is inadequateforpredictingthe thermophoretic force in the transition regime.

An approximate, closed-form expression for the thermo-phoretic force is proposed. This expression takes advantageof the close coupling between the thermophoretic force andthe local translational heat ux (which is approximated usingShermans interpolating formula ). Comparisons between theDSMC/Greens function calculations and this closed-form ex-pression are very good, with agreement to better than 4% for thepressure range 10 to 1000 mTorrfor all 3 gases. At low pressures,the present closed-form expression suffers from the assump-tion regarding the constant of proportionality and gives ther-mophoretic forces that fall about 10% below the free-molecularlimit (this defect could be corrected if needed ). Interestingly,for a 0.01 m gap, a pressure of 10 mTorr corresponds to a sys-

tem Knudsen number of about 1.3 for nitrogen and about 0.5for helium and nitrogen. Thus, the proposed closed-form ex-pression would be an attractive choice for similar parallel-plategeometries as long as Kn L < 1:5.

NOMENCLATURE

c mean molecular speed, c D (8k B T =m )1=2, m/sc p mean molecular speed at the particle temperature,

c p D (8k B T p=m )1=2 , m/s

c int internal energy contribution to speci c heat capacity,J/ (K kg)

c tr translational energy contribution to speci c heatcapacity,J/ (K kg)

c gas-molecule velocity relative to the particle, c D u um/s

f [u ] molecular velocity distribution function, s 3 /m3

f int internal energy factor in thermal conductivity expres-sion,

f tr translational energy factor in thermal conductivity ex-pression,

F th thermophoretic force, NF force Greens function, Nk gas thermal conductivity, W/ (m K)k B Boltzmann constant, 1.380658 10 23 J/KKn L system Knudsen number, = L, Kn R particle Knudsen number, = R p, L plate separation distance, mm gas-molecule mass, kg

n gas number density, 1/m3

P gas pressure, Paq gas heat ux, W/m 2

R speci c gas constant, k B=m , J/ (K kg) R p particle radius, mT gas temperature, KT c cold plate temperature, KT h hot plate temperature, KT p particle temperature, Ku gas-molecule velocity, m/su p particle velocity, m/s x distance from the bottom (cold ) plate, m

Z rot number of collisions required to exchange a quantum ofrotational energy with translational energy,

Greek Symbols gas mean free path, m gas viscosity, kg/ (ms) gas density, kg/m 3

! viscosity temperature exponent, thermophoresis parameter, number of molecular internal degrees of freedom,

Superscripts int part of a quantity arising from the internal energy modes

of a gastr part of a quantity arising from the translational energy

modes of a gas

Subscripts C continuum limit of a quantityCE quantity calculated from Chapman-Enskog theoryFM free-molecular limit of a quantityref quantity value at reference conditions T ref D 273 K an

Pref D 101,325 Pa


19/19


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