JOURNAL OF COLLOID AND INTERFACE SCIENCE 191, 482–488 (1997)ARTICLE NO. CS974979

Thermophoresis of a Small Evaporating Particlein a High-Temperature Diatomic Gas

Xi Chen1

Department of Engineering Mechanics, Tsinghua University,Beijing 100084, People’s Republic of China

Received March 6, 1997; accepted May 5, 1997

sented in Ref. (5) was restricted to a monatomic gas andKinetic-theory analytical results are presented concerning the for completely diffuse reflection of atoms at the particle

effect of intense evaporation on the thermophoretic force acting surface. The present paper intends to extend the previouson a spherical particle suspended in a high-temperature diatomic analysis (3, 4) to the case of high-temperature diatomicgas for the case of free-molecule regime. Molecule dissociation and

gases in which molecule dissociation in the bulk gas andatom recombination are included in the analysis. It has been shownatom recombination at the particle surface are concerned,that evaporation may substantially enhance the thermophoreticbut the gas temperature is not too high so that gas ionizationforce acting on a particle, especially for the case of the particleis still negligible. Incomplete thermal accommodation and amaterials with low evaporation latent heat and small molecularcombined specular and diffuse reflection of gas atoms orweight and at high gas temperatures. The values of the effective

atomic and molecular thermal-accommodation factors do not af- molecules at the particle surface will be incorporated in thefect the thermophoretic force acting on a nonevaporating particle, analysis. The research results presented here will be usefulbut they affect significantly the evaporation-added thermopho- for many applications such as the synthesis of fine powdersretic force. It has been shown that the recombination fraction of by injecting solute droplets as the raw materials into a high-atoms at the particle surface does not influence the thermo- temperature gas reactor, the electron depletion technique byphoresis. q 1997 Academic Press injecting liquid droplets into the plasma sheath in the space-

Key Words: thermophoresis; evaporating particle; kinetic-theorycraft communication studies, etc.analysis; high-temperature gas.

The study of the thermophoretic force acting on a particlewith intense evaporation represents a combined problem ofmass, heat, and momentum transfer. Here the term ‘‘intense

INTRODUCTION evaporation’’ means that the evaporation of the particle iscontrolled by the heat transfer from the high-temperature gas

A small particle suspended in a gas with an overall tem- to the particle and the surface temperature of the evaporatingperature gradient in the gas will experience a force in the particle is approximately equal to (or slightly lower than)direction opposite to that of the temperature gradient. This the boiling temperature of particle material. It is expectedphysical phenomenon is known as thermophoresis and has that an evaporation-added thermophoretic force will bebeen extensively studied (e.g., see Refs. (1–5)) due to its caused by a nonuniform evaporated-mass flux distributionimportance in the atmospheric and aerosol sciences, the LDV around an intensely evaporating particle, while the localmeasurements and visualization of non-isothermal flow evaporated-mass flux is determined through the local heatfields, the materials processing by using a high-temperature flux to the particle surface divided by the latent heat ofgas reactor coupled with a deposition process and many evaporation of the particle material. Such a problem of si-other applications. Especially for the free-molecule regime multaneous heat, mass and momentum transfer will be stud-and ordinary temperature gases without dissociation or ion- ied here for the case of free-molecule regime with greatization, an analytical expression was derived by Waldmann Knudsen numbers (Kn ú 10). Where the Knudsen number(3) for the thermophoretic force acting on a nonevaporating (Kn) is defined as the ratio of the mean free path lengthspherical particle, while the effect of evaporation on the of gas particles to the diameter of the suspended sphericalthermophoresis was studied in Ref. (4) . Reference (5) ex- particle.tended the analysis of Refs. (3, 4) to the case of high-temperature ionized gases (plasmas), but the analysis pre- ANALYSIS

In the analysis, the following assumptions are employed:1 To whom correspondence should be addressed. (A) the mean free path length of gas particles (molecules

4820021-9797/97 $25.00Copyright q 1997 by Academic PressAll rights of reproduction in any form reserved.

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483THERMOPHORESIS OF AN EVAPORATING PARTICLE

and atoms) is much greater than the diameter of sphericalparticle (free-molecule regime); (B) there exists a tempera-ture gradient within the bulk gas but there is no relative veloc-ity between the bulk gas and the suspended particle; (C)equilibrium atom–molecule composition prevails in the bulkhigh-temperature diatomic gas, while atoms incident to parti-cle surface will partially recombine into molecules at the coldparticle surface; (D) the reflection of gas particles (moleculesor atoms) from the particle surface is partially specular andpartially diffuse; (E) there is no interaction between the inci-dent gas particles and evaporated vapor molecules flowingout from the evaporating particle; and (F) the surface temper-ature of the suspended particle is constant (e.g., equal to theboiling temperature of the particle material).

For the present case, the velocity distribution function forgas molecules ( j Å m) or atoms ( j Å a) incident to theparticle surface can be expressed as (5–7)

f 0j Å f 0oj/ f 0ojFm 2j (n 2

x / n 2y / n 2

z )5njk

3T 3 0 mj

njk2T 2G(vrq*j ) , [1]

where mj and nj are the particle mass and number densityof the j th gas species, k is the Boltzmann constant, T is thetemperature of the bulk gas, and v is the velocity vector ofa gas particle and v Å nxi / nyj / nzk . f 0oj in [1] is theMaxwellian velocity distribution function, i.e.,

FIG. 1. The coordinate system used in this analysis.f 0oj Å

nj

(2pkT /mj)3/2 expS0 n 2

x / n 2y / n 2

z

2kT /mjD . [2]

a)ca for atoms and wm Å (a /2)ca / cm for molecules,where cj is the particle flux of the j th gas species incidentq*j is the heat flux vector due to the j th species ( j Å m forto the particle surface at the point P in Fig. 1 and can bemolecules and j Å a for atoms) within the bulk gas whichcalculated byis related with the temperature gradient ÇT in the bulk gas

by q*j Å 0kjÇT . kj is the gas thermal conductivity due tocj Å *

`

0*

`

0`*

`

0`

nz f 0j dnx dny dnz [4]the j th species. Referring to Fig. 1 for the coordinate systemand angle u used in this analysis, (vrq*j ) can be expressed

After the expression for f 0j given in Eq. [1] is inserted intoas (vrq*j ) Å (nzcos u / nxsin u)q*j , in which q*j is theEq. [4] , one obtains

magnitude of the heat flux vector q*j within the bulk gas.The velocity distribution function for the j th gas particle

cj Å14

njnV j [5]reflected specularly from the particle surface is the same asthat given in Eq. [1] , except that 0nz is used instead of nz ,while the velocity distribution function for the j th gas parti- for atoms ( j Å a) or molecules ( j Å m) , where n

V j Åcle reflected diffusely from the particle surface is (4, 7, 8)

√8kT / (pmj) is the mean thermal speed of the j th gas particle.

For the case of completely diffuse reflection, if we usef /jd0 to denote the velocity distribution function of the j thf /jd Å

wj

2p(kTW/mj)2 expS0 n 2

x / n 2y / n 2

z

2kTW/mjD , [3]

reflected gas particles when the recombination factor a Å 0,the following relation can be obtained:

in which TW is the surface temperature of the suspendedparticle, while wj is the atom or molecule flux leaving from f /md Å Sa2DS na

nmD√

mm

ma

f /md0 / f /md0 ,the particle surface. If complete diffuse-reflection is assumedand the fraction of atoms which recombine into moleculesat the particle surface is denoted by a, we have wa Å (1 0 f /ad Å (1 0 a) f /ad0 . [6]

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484 XI CHEN

For a nonevaporating particle, the surface pressure pj can tj Å (tzx) j

be calculated by (7)Å (1 0 ej) *

`

0*

`

0`*

`

0`

nz(mjnx) f 0j dnx dny dnz

pj Å p0j / ejp/j s / (1 0 ej)p/jd [7]

Å (1 0 ej)q*j sin u

5√2pkT /mj

. [12]Å (1 / ej)p0j / (1 0 ej)p/jd ,

The thermophoretic force FT acting on a spherical non-where ej is the fraction of the specularly reflected gas parti-

evaporating particle is in the direction of q*, and its magni-cles, and the subscript j Å a stands for atoms while j Å m

tude can be calculated by (p Å pa / pm , t Å ta / tm)for molecules. p0j is related to the momentum flux carriedin by the incident particles and can be calculated by

FT Å 2pR 2 *p

0

(p cos u / t sin u)sin u du

p0j Å *`

0*

`

0`*

`

0`

nz(mjnz) f 0j dnx dny dnz , [8]Å (4pR 2)F 4q*a

15√2pkT /ma

/ 4q*m15

√2pkT /mm

G , [13]

p/j s in Eq. [7] denotes the contribution to the surface pressurein which R is the sphere radius. Since q*a Å 0kaÇT and(pj) of reflected gas particles if completely specular reflec-

tion is assumed, and p/j s Å p0j . On the other hand, p/jd q*m Å 0kmÇT , Eq. [13] can also be rewritten through therepresents the contribution to the surface pressure (pj) of temperature gradient ÇT within the bulk gas as follows:reflected gas particles if completely diffuse reflection is as-sumed and can be calculated by

FT Å 0(4pR 2)F 4ka

15√2pkT /ma

/ 4km

15√2pkT /mm

GÇT . [14]

p/jd Å *0

0`*

`

0`*

`

0`

nz(mjnz) f /jd dnx dny dnz . [9]In order to calculate the evaporation-added thermopho-

retic force, one has to calculate the local heat flux from thehigh-temperature gas to the suspended particle, q Å qc 0 qr ,From Eqs. [1] – [3] and [7] – [9] , the following expressionswhere qc is the heat flux due to the energy transport by thefor surface pressure at a non-evaporating particle can beincident and reflected gas particles, qr is the radiative heatobtained:loss from the particle surface, and qr Å csT 4

w , in which cand s are the surface emissivity and the Stefan–Boltzmannconstant. Since there is no energy exchange between the gaspa Å

12

nakT[(1 / ea) / (1 0 ea)(1 0 a)√TW/T]

and the suspended particle when the reflection of gas atomsor molecules at the sphere surface is specular (7) , the localheat flux components due to gas molecules (qc)m and atoms/ (1 / ea)S2

5D q*a cos u√2pkT /ma

[10](qc)a are as follows:

pm Å12

nmkTF(1 / em) / (1 0 em)(qc)m Å (1 0 em)amF*

`

0*

`

0`*

`

0`

nzSmm

2n 2 / i

2kTD

1 S1 / a

2 S na

nmD√

mm

maD√

TW

T G 1 f 0m dnx dny dnz 0 *0

0`*

`

0`*

`

0`

(0nz)

1 Smm

2n 2 / i

2kTWD f /md0 dnx dny dnzG/ (1 / em)S2

5D q*m cos u√2pkT /mm

[11]

No shear stress appears if completely specular reflection is Å (1 0 em)amF14

nmnV mFS(2 / i

2Dk(T 0 TW )Gconcerned. On the other hand, no shear stress is caused bythe diffusely reflected gas particles. Hence, the surface shearstress components tj ( j Å a for atoms and j Å m for mole- / 1

2q*m cos uG , [15]

cules) is calculated by

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485THERMOPHORESIS OF AN EVAPORATING PARTICLE

to evaporation are assumed to satisfy the Maxwellian veloc-(qc)aÅ (10 ea)HaaF*

`

0*

`

0`*

`

0`

nzSma

2n 2D ity distribution function as follows:

fV Å A exp[0B(V 2x / V 2

y / V 2z )] , [17]

1 f 0a dnx dny dnz0*0

0`*

`

0`*

`

0`

(0nz)Sma

2n 2D

where A and B are two constants which are determined bythe following two physical conditions (4, 5): ( i ) The averagekinetic energy of the vapor molecules leaving from the

1 (10 a) f /ad0 dnx dny dnz0*0

0`*

`

0`*

`

0`particle surface corresponds to the surface temperature TW,and (ii) The local flux of the outflowing vapor molecules isequal to the local mass flux m

g

divided by the vapor molecule1 (0nz)Smm

2n 2/ i

2kTWDSa2DS na

nmD mass mV. By using the two physical conditions, one obtains

(4, 5)

A Å 12p(kTW/mV )2 S qc 0 qr

mVLVD , B Å 1

2kTW/mV

. [18]1√

mm

ma

f /md0 dnx dny dnzG/ acaErJConsequently, the velocity distribution function fV in Eq.Å (10 ea)H1

4aananV aF2kT0 S10 a/ Sa2D/ i

4D [17] about the evaporated vapor molecules takes the follow-ing form:

1 2kTW/ S a

aaD ErG/ 1

2aaq*a cos uJ , [16]

fV Å1

2p(kTW/mV )2 S qc 0 csT 4W

mVLVD

where aa and am are the thermal-accommodation factors for1 expS0 V 2

x / V 2y / V 2

z

2kTW/mVD . [19]energy transfer due to atoms and molecules, i is the number

of the internal free degree in a molecule, Er is the recombina-tion energy per recombined atom, and all of the energy

It is easy to show that the additional shear stress due to thereleased in the recombination reaction A / A r M is as-

evaporated vapor molecules t* is zero, while the additionalsumed to be transferred completely to the suspended particle

pressure at the sphere surface due to these vapor moleculesfor the case of diffuse reflection.

isIt is noted that the local heat flux qc Å (qc)a / (qc)m or

q Å qc 0 qr contains two parts. The first part does not dependp* Å *

0

0`*

`

0`*

`

0`

mVV 2z fV dVx dVy dVzon the u position, while the other part involving ajq*j cos u

is u-dependent. Although the second part involving ajq*j cosu is usually very small in comparison with the first part

Å qc 0 csT 4W

mVLV

√pmV kTW

2. [20](their ratio is less than 1% even as the temperature gradient

is as high as 105 K/m), it is the second part that is criticalfor the thermophoresis enhancement. The local heat flux q And the evaporation-added thermophoretic force acting onassumes its maximum at the frontal ‘‘stagnation point’’ (u the evaporating particle is thusÅ 0) and its minimum at the rear ‘‘stagnation point’’ (u Åp) . Since the local evaporated-mass flux m

g

is equal to thelocal heat flux (q) divided by the evaporation latent-heat of F *V Å 2pR 2 *

p

0

(p * cos u / t* sin u)sin u duparticle material (LV ), i.e., m

g

Å q /LV, mg

will also assume asimilar u distribution as q . As a result, the evaporation- Å (4pR 2)[(1 0 ea)aaq*aadded thermophoretic force due to the reaction force of theevaporated-mass flux will also be in the direction of the heat / (1 0 em)amq*m ]

√2pkTW/mV

12LV

. [21]flux vector q* (or that of 0ÇT ) within the bulk gas.Namely, evaporation always enhances the thermophoreticforce. The total thermophoretic force acting on an evaporating

particle can thus be calculated byThe vapor molecules issuing from the particle surface due

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486 XI CHEN

FTV Å FT / F *V Å (4pR 2)HS 415DS q*a√

2pkT /ma

/ q*m√2pkT /mm

D/ [(1 0 ea)aaq*a

/ (1 0 em)amq*m ]

√2pkTW/mV

12LVJ [22]

or

FTV Å 0(4pR 2)FS 415DSka

√ma / km

√mm√

2pkTD

/ aa ,effka / am ,effkm

12LV

√2pkTW

mVGÇT . [23]

FIG. 2. Variation with the gas temperature of the ratio of the thermo-phoretic forces with evaporation to without evaporation (FTV/FT ) . WaterSince aa(1 0 ea) and am(1 0 em) can be treated as effectivedroplet in nitrogen; 105 Pa; TW Å 373 K; LV Å 2.26 1 106 J/kg; effectiveatomic and molecular thermal-accommodation factors, wethermal accommodation factors aa ,eff Å am ,eff .have used aa ,eff and am ,eff to substitute them in Eq. [23],

where aa ,eff Å aa(1 0 ea) while am ,eff Å am(1 0 em) .accommodation factors aa ,eff and am ,eff . (B) The atom recom-bination fraction a at particle surface does not affect theRESULTS AND DISCUSSIONthermophoretic force, although a appears in the expression

Equations [14] and [23] show that the thermophoretic [16] for the atom heat-flux component. (C) The ratio of theforces acting on both non-evaporating and evaporating parti- predicted thermophoretic forces with evaporation to withoutcles for free molecule regime (Kn @ 1) are all directly evaporation (FTV/FT ) does not depend on the temperatureproportional to the temperature gradient within the bulk gas gradient (ÇT ) . (D) The atomic and molecular componentsand to the square of particle radius. Hence, the thermopho- of the thermophoretic force acting on a nonevaporating orretic force reduced to per unit temperature gradient and per evaporating particle are directly proportional to the atom andunit surface area of the particle, i.e., FT/(4pR 2

ÇT ) or FTV/ molecule heat-flux components within the bulk gas (q*a or(4pR 2

ÇT ) , will be independent of the temperature gradient kaÇT and q*m or kmÇT ) , respectively, and thus dependentwithin the bulk gas and independent of the particle radius. on the level of the gas temperature.The thermophoretic force on a spherical particle is in the Some typical calculated results are shown in Figs. 2–direction opposite to that of the temperature gradient. Evapo- 5 for a small water droplet suspended in high-temperatureration always enhances the thermophoretic force acting on diatomic gases within free molecule regime. Calculated vari-a particle, and the evaporation-added part of the total ations with the gas temperature of the ratio of the predictedthermophoretic force is inversely proportional to LV (evapo- thermophoretic forces with evaporation to without evapora-

tion are plotted in Figs. 2 and 3 for nitrogen and oxygen,ration latent-heat) and√mV (square root of the vapor mole-

cule mass) . All of these results are the same as those ob- respectively. It can be seen that evaporation may signifi-cantly enhance the thermophoretic force, especially at hightained before for the case of ordinary temperature gases

without considering molecule dissociation and atom recom- gas temperatures. For the case of aa ,eff Å am ,eff Å 1.0 and TÅ 7000 K, the ratio of the calculated thermophoretic forcesbination (3–5).

Examination of Eqs. [14] and [23] derived for the case with evaporation to without evaporation (FTV/FT ) can be asgreat as 1.67 for both nitrogen and oxygen. The effectiveof high temperature diatomic gases and with a combined

specular and diffuse reflection can also reveal the following thermal accommodation factors for atoms (aa ,eff ) and mole-cules (am ,eff ) affect significantly the ratio of the thermopho-new results: (A) The thermophoretic force acting on a non-

evaporating particle does not depend on the specular-reflec- retic forces with evaporation to without evaporation, sincethe atomic and molecular components of the evaporation-tion fractions (ea and em) and the thermal accommodation

factors (aa and am) , while the evaporation-added part of the added thermophoretic force are directly proportional to theeffective thermal accommodation factors aa ,eff or am ,eff , re-thermophoretic force is dependent on the effective thermal-

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487THERMOPHORESIS OF AN EVAPORATING PARTICLE

FIG. 3. Variation with the gas temperature of the ratio of the thermo-FIG. 5. Variation with the gas temperature of the atomic (continuousphoretic forces with evaporation to without evaporation (FTV/FT ) . Water

lines) and molecular (dotted lines) fractions in the thermophoretic force.droplet in oxygen; 105 Pa; TW Å 373 K; LV Å 2.26 1 106 J/kg; effectiveWater droplet in oxygen and nitrogen; 105 Pa; TW Å 373 K; LV Å 2.26 1thermal accommodation factors aa ,eff Å am ,eff .106 J/kg; effective thermal accommodation factors aa ,eff Å am ,eff Å 1.

spectively. Figures 3 and 4 compare the effect of gas pressurekm will increases with decreasing gas pressure. However,on the thermophoretic force ratio FTV/FT . It is expected thatcomparison of the calculated results in Fig. 3 with those inthe equilibrium state of the dissociation reaction O2 r O /Fig. 4 shows that the effect of gas pressure on the ratioO will shift to the direction with greater mole fraction ofof the calculated thermophoretic forces with evaporation toatom species at lower gas pressure according to the LeCha-without evaporation FTV/FT is just a little. Figure 5 comparestelier’s principle, and thus the thermal conductivity ratio ka /the calculated variations with the gas temperature of theatomic and molecular fractions in the thermophoretic forcefor a water droplet suspended in a nitrogen or oxygen. Thecalculated results for a nonevaporating particle are almostidentical to those for an evaporating particle. As expected,Fig. 5 shows that at low gas temperatures with negligiblemolecule dissociation (e.g., less than 4000 K for nitrogenor less than 2000 K for oxygen), the values of the molecularfraction are almost 1.0 and the thermophoretic force is almostcompletely caused by the molecule species of the diatomicgases. For this case, Eqs. [14] and [23] will reduce to theircounterparts obtained previously for ordinary temperaturegases (3, 4) . On the other hand, at high gas temperaturesassociated with appreciable gas dissociation degree (e.g.,when the gas temperature is higher than 4000 K for oxygenor higher than 7000 K for nitrogen), the thermophoreticforce will be caused mainly by the atom species. Includinggas dissociation in the thermophoresis calculation will beimportant for this case.

CONCLUSIONSFIG. 4. Variation with the gas temperature of the ratio of the thermo-

phoretic forces with evaporation to without evaporation (FTV/FT ) . WaterFor the thermophoretic force acting on a nonevaporatingin oxygen; 104 Pa; TW Å 319 K; LV Å 2.39 1 106 J/kg; effective thermal

accommodation factors aa ,eff Å am ,eff . or evaporating particle suspended in a diatomic gas, the

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488 XI CHEN

present analysis for free-molecule regime obtains the follow- while it can be attributed mainly to atom species at high gastemperatures.ing conclusions: As usual, the thermophoretic force is in the

direction opposite to the temperature gradient within the gas,ACKNOWLEDGMENT

and is proportional to the magnitude of temperature gradientThis work was supported by the National Natural Science Foundation ofand to the square of particle radius. Evaporation may sig-

China through Grants 59376308 and 59676011.nificantly enhance the thermophoretic force, especially forthe case of the particle materials with low evaporation latent- REFERENCESheat and small molecular weight. The ratio of the predicted

1. Brock, J. R., J. Colloid Sci. 17, 768 (1962).thermophoretic forces with evaporation to without evapora- 2. Talbot, L., Cheng, R. K., Schefer, R. W., and Willis, D. R., J. Fluidtion does not depend on the temperature gradient in the bulk Mech. 104, 737 (1980).

3. Waldmann, L., Z. Naturforsch. A 14, 589 (1959).gas. The thermophoretic force acting on a nonevaporating4. Chen, X., in ‘‘Kinetic Theory and Its Applications to Studies of Heatparticle does not depend on the specular-reflection fractions

Transfer and Fluid Flow,’’ Chap. 3. Tsinghua University Press, Beijing,(ea and em) and the thermal accommodation factors (aa and 1996. [In Chinese]am) , while the effective thermal-accommodation factors af- 5. Chen, X., J. Phys. D: Appl. Phys. 30, 826 (1997).

6. Chapman, S., and Cowling, T. G., ‘‘The Mathematical Theory of Non-fect substantially the evaporation-added thermophoreticUniform Gases,’’ 3rd ed., Sec. 15.6. Cambridge University Press, Cam-force. The thermophoretic force does not depend on thebridge, 1970.recombination fraction of atoms at the particle surface. At

7. Bird, G. A., in ‘‘Molecular Gas Dynamics,’’ pp. 73–75, 91–98.low gas temperatures with negligible gas dissociation, the Clarendon Press, Oxford, 1976.

8. Chen, X., and He, P., Plasma Chem. Plasma Process. 6, 313 (1986).thermophoretic force is caused by the molecule species,

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