Thermophoretic force on a single particle—I. Numerical solution of the linearized Boltzmann equation

J. AerosolSci.,Vol. 23, No. 3, pp. 291-300, 1992. Printed in Great Britain.

0021-8502/92 $5.00+0.00 © 1992PergamonPrcu Lid

THERMOPHORETIC FORCE ON A SINGLE PARTICLE--I. NUMERICAL SOLUTION OF THE LINEARIZED BOLTZMANN EQUATION S. K. LOYALKA Nuclear Engineering Program and Particulate Systems Research Center, College of Engineering, University of Missouri--Columbia, Columbia, M O 65211, U.S.A. (Received 15 October 1991; and in final form 20 December i991) Ai~tract--The force experienced by a small aerosol particle in the presence of a temperature gradient is known as the thermophoretic force. Motion of particles under such a force is known as thermophoresis. Thermophoresis has important technological applications, but uncertainties continue to exist regarding the experimental measurements as well as the theoretical estimations of the thermophoretic force. We describe, in this work, a prescription for calculation of the force on a sphere based on a numerical solution of the linearizcd Boltzmann equation. Then, we report some explicit results for a rigid sphere gas.

1. I N T R O D U C T I O N

Thermophoretic force is the force experienced by an aerosol particle due to the presence of a temperature gradient (Fuchs, 1964; Davies, 1966; Hidy and Brock, 1971-1973; Friedlander, 1977; Rocsner, 1986). Thermophoresis is the motion of that particle under the influence of the force. Thermophorctic deposition is of importance in many areas such as optical fiber fabrication, nuclear reactor safety, microcontamination control, etc. The boundary layer theory analyses show that in a short tube the thcrmophoretic particle deposition rate (Jthermoohoreti¢) i8 approximately expressed as (Walker et al., 1979; Morse and Cipolla, 1984; Batcbelor and Sben, 1985; Williams and Loyalka, 1991)

Jtbermophoretic ~C/~ Jisothermal 1 + 0 ' '

(1)

where Sc is the Schmidt number, 0* is a measure of the temperature difference a n d / ( is strictly proportional to the thermophoretic velocity. Since Sc~ 104, the thermophoretic deposition rate can be quite dominant. This effect arises from the fact that in convective flows, diffusion boundary layers are very thin and thermophoresis can bring particles near the edges of this layer where they may then deposit by way of diffusion. Tbermophoresis has been studied extensively, both experimentally (Rosenblatt and La Mer, 1946; Saxton and Ranz, 1952; Schadt and Cadle, 1961; Schmitt, 1959; Waldmann and Schmitt, 1966; Jacobsen and Brock, 1965; Davis and Adair, 1975; Talbot et al., 1980) and theoretically (Epstein, 1924; Waldmann, 1959; Brock, 1962, 1967; Williams, 1971; Dwyer, 1967, Sone and Aoki, 1981, 1983; Law, 1986, Yamamoto and Ishihara, 1988), and the subject has been continually reviewed, in the past as well as more recently (Brock, 1980; Talbot, 1981; Fuchs, 1982; Loyalka, 1983, 1986; Bakanov, 1991). But the data and the theory remain controversial. The former due to the difficulties in measuring the relatively small effect because of competing phenomena, and the latter because of the approximations employed in the theoretical efforts. It is our purpose in this series of papers to depart substantially from all previous theoretical work on the problem, and to provide numerical solution of the relevant boundary value problem based on the linearized Boltzmann equation itself. In this first paper we provide a clear statement of the problem, discuss the transformations necessary to numerical solution of the problem, outline the method of solution which follows directly from our recent work on related problems of the kinetic theory (Loyalka, 1989, 1991a-c, 1992; Loyalka and Hamoodi, 1990; Loyalka and Hickey, 1990, 1991; Loyalka et al., 1989a, b), benchmark the method by comparing the 291

292

S.K. LOYALKA

numerical results for the BGK model with those reported previously (Yamamoto and Ishihara, 1988), and then provide initial results for a rigid sphere gas. The latter papers will cover the complete range of results for the rigid sphere gas, and realistic intermolecular force laws for several gas-surface interaction operators. The work we describe here, however, is quite general, and getting additional results is mostly a case of running the computer program for a range of conditions, and rendering it more efficient for some values of the parameters. The computations are intensive, and hence, dictate a series of papers on the problem. We should emphasize, however, that in the present paper we are considering the force on a stationary sphere situated in a monatomic gas. Also, we assume small temperature gradients, a circumstance that permits linearization of the problem. In section 2 we review very briefly the previous theoretical research. We provide a clear statement of the problem next, and then in the section 4 we discuss some transformations that facilitate the numerical solution of the problem. These transformations, and the numerical solution follow our recent work on the solving the Boltzmann equation, and in particular the considerations employed in solving the drag and the torque (Loyalka, 1992) problems. We discuss our results in section 5.

2. PREVIOUS WORK Expressions for the thermophoretic force have been developed by several authors (Epstein, 1924; Waldmann, 1959; Brock, 1962, 1967; Williams, 1971; Dwyer, 1967; Sone and Aoki, 1981, 1983; Law, 1986, Yamamoto and Ishihara, 1988; Brock, 1980; Talbot, 1981; Fuchs, 1982; Loyalka, 1983, 1986). In the free molecular limit (small particles), an expression for the thermophoretic force can be obtained from simple arguments (Waidmann, 1959; Williams, 1971). The other limit, of very large particles (the first order slip limit), is quite complicated, but it has also now been clarified. None of the previous works have attempted to solve the problem in its generality. The work of Brock (1962), however, has found a great deal of use due to its simple nature and success in describing a range of experimental data. This is particularly true when corrections to the slip coefficients are used (Talbot, 1981; Talbot et al., 1980). The works of Dwyer (1967) and Sone and Aoki (1981, 1983) are quite interesting in that, for certain combinations of particle size and thermal conductivity, negative thermal forces (motion of particles up the temperature gradient) have been predicted. These predictions have not been experimentally verified at this time and it is possible that such results are either due to the models or the methods used in the calculations (Dwyer used the questionable moments method while Sone and Aoki carried out a careful asymptotic analysis but used the linearized BGK model). Numerical solutions of the integral transport equation for the BGK model were attempted by Law (1986) and Yamamoto and Ishihara (1988). The results of Yamamoto and Ishihara appear particularly interesting, and these indicate that the BGK model does predict negative thermophoresis. To summarize, there is no reported publication that addresses numerical solution of the linearized Boltzmann equation relevant to this problem. We have recently solved the drag problem (1992), and we had remarked that the work could be applied directly to the thermophoresis problem. This indeed is the case, and our effort here has been in providing a clear statement of the problem, and then casting it into a form so that the computer program constructed for solving the drag problem could be adapted with some minimal changes only. 3. STATEMENT OF THE PROBLEM We begin by considering a sphere in an infinite expanse of gas. Far from the sphere the gas is subjected to a temperature gradient (say in the z-direction). Then the molecular distribution, f, is determined by the boundary value problem:

Thermophoreticforceon a singleparticle

293

of ~.~-=s(ff,) f + (~, ~)=Af-, ~" n~>0, r~cgS lim f(F, ~)=fo(1 + g(r, e)x),

(2)

r-*oo

where J is the Boltzmann operator, A is the gas-surface interaction operator, ~ is the position vector, ~ is the molecular velocity, n, is the unit normal at the surface of the sphere directed in to the gas. Also, fo is the absolute Maxwellian: //

~3/2

m

:o=no~2---n~o:

m -2 exp(-2~oe ),

(3,

where m is the molecular mass, no is the number density, and k and To are, respectively, the Boltzmann constant and the temperature. Also,

a(r,e)=( c2-~ )z +c,,~,(c),

(4)

where ~bt is the Chapman-Enskog solution for heat conductivity, and has been discussed elsewhere (Loyalka, 1989, 1991a). Further, 1 AT ~:= - - - To Az

(5)

is the non-dimensionalized temperature gradient, and the non-dimensionalization is such that

r=~t,

R=~t,

e = 2--~o

~..

(6)

The mean free path is expressed as

:-4~'

I m \,/2

in which 2g is the thermal conductivity of the gas, and Po = nokTo is the scalar pressure. Note that R = Kn- t, where Kn is the Knudsen number. The operator A in the present case is determined by the nature of molecular scattering at the surface and the boundary conditions: f d $ $ ' n , f ( R , $)= Yr ~ =0 1 m(~. ~r)2(~:-vr)f(R, $)

f d~ 5

-

~ c3TP(r)

= - % - - - f f - ~=R'

(8)

where 2p is the thermal conductivity of the particle. The temperature of the sphere also satisfies the Laplace's equation: ¢2 Tp(i) = 0.

(9)

Note that for diffuse reflection:

(

m

f+=nw(R) k,2~kT,,(ft)j

exp

(

2kT,,(---R)J' c-n~>0,

where nw(R) and Tw(R)= Tp(R) then are determined from the equations (8}-(9).

(10)

294

s.K. LOYALKA

For x ~ 1, we set: f(~, ~:)=fo(1 + [0(r, e) + h(r, e)] x).

(11)

The perturbation h is now determined by the linearized boundary value problem: Oh e "3r = etLh(r' C) 1 h+(r, c)= s(r, c) +(1, h)- +~-~ (c 2 - 2)(1,(c 2 - 2)h)-, c" n~>0, re0S, lim h(r, e) = 0,

(12)

e~oo

where the scalar product on the negative half space is defined as: (hi, h2) - =-2 ~

de exp(_ c2)[e • n, lhl(r, e)hz(r, e)

(13)

7~ Jc'n~
and s(R, e) = - 9(R, c) + (1, 9)- + 2 ~ (c2 - 2)(1, (c 2 - 2)0)-

(14)

~o = 1 -~ 5r?/~ ~.p I 4 ).g R"

(15)

with

The constant et has been discussed elsewhere (Loyalka, 1989, 1991a). Note that the Chapman-Enskog solution satisfies the equation: 2 5 (c-~)cz=etL(czc~,(c)).

(16)

It is useful to note that referring to the co-ordinate system of Fig. 1, the equation (14) can now be written, more explicitly as (# = cos 0~. s(R, e) = - [fl(c 2 - 2) + c/a~bt(c)] cosct + c(1 - #2)1/2 ~bt(c)sin0t sin~,

(17)

where 5 ~ 1/2 1

5 n 1/2

fl = R

1-

4

1 2fa

1

1-~

5rex/2 2p 1 4 2B R

The thermophoretic force on the sphere is the quantity of major interest here, and is given by (note nr is the unit normal directed into the gas, and hence the negative sign):

f'z= -m f dg f d~.(~'n,)(~'nr)f.

(19)

/7= _2port2 x ([e. nr] [e.nz3, h) '

(20)

This simplifies to:

where the non-dimensionalized scalar product is defined as:

(hl, h2)=~5~ f dS f deexp(-c2)hdR,e)h2(R,e).

(21)

Note that in the free molecular limit, h(R, c)=0 ,/~<0 s(R, c ) , u > 0

(22)

Thermophoretic force on a single particle

297

For, the numerical solution, we begin by noting that the ansatz: h(r, e) = cos 0th.(r, c, ]/) + sin ~tsin ¢ hb(r, c, ]/)

(37)

allows the problem to be converted to that of solving two coupled equations with associated boundary conditions: [ ]/-~r ~ "r 1 -r~ 0 ~#+a(c) 1 h~(r,c,#)=q.(r,c,#)

h~+(R, c, ]/)= - [c#¢,(c)+~h +(c 2-2)(fl+~/2)] , ]/>0 lim ha(r, c, ]/)=0

(38)

r~ct)

and I

63 1--]/2 63 ] ]/ff-r -t -r 63]/+a(c) hb(r , c, ]/)=qb(r, C, ]/) h~ (R, c, ]/) = c(1 - - ] / 2 ) 1 / 2 ( ~ t ( C ) ,

]/>0

lim hb(r, c, ]/)= 0,

(39)

r--*~

where

nx=4fodcc3exp(-c2)f~ d]/]/h,(R,c,]/) t/2 = ~

de c3(c2 - 2 ) exp(- c2)

;o d]/]/h.(R, c, ]/)

(40)

-!

also, e, v(c) a(c) = - -

(41)

C

and qa and qb are the quantities:

q . ( r , c , ] / ) = \( ec, .~ 2=n0+~

1 P.(]/) f : dc'c '2 exp(- c'2)k.(c, c') f l - 1 d]/' P.(]/')h.(r, c', ]/'))

(1 _#2)1/2 r

qb(r,c,]/)=

fib(r, ¢, #)

(42)

2 n + l pl(]/) 1) (ce . o"" 2n(n+ fo°dc, c,2exp(_c,2)k.(c,c,) ?-1 d# P.(]/ )hb(r,c ,#') r 1

t

i

/

(1 __]/2)1/2

+

r

h.(r, c, ]/).

(43)

The thermophoretic force is now expressed as •~ =

-po~2K~z,

(44)

where Fz is the non-dimensionalized (or the normalized) force: Fz-

16~zl/2 3

x

+

[fo fo

dc -1 d#c4exp(-c2)#2ha(R'c'#)-

f:

dc

;1

d#c4exp(

C2)/~(1

ll2)l/2hb(R,c,g)

]}" (45)

298

S.K. LOYALKA 5. N U M E R I C A L

RESULTS

AND

DISCUSSIONS

We have solved the coupled equations (38)--(39) using the SN method, with finite elements (cubics) on speed, and collocation, in the manner of our recent work (Loyalka, 1989, 1991a--c, 1992; Loyalka and Hamoodi, 1990; Loyalka and Hickey, 1990, 1991; Loyalka et al., 1989a, b). Essentially, we use the computer program already written for the drag problem (Loyalka, 1992), and the needed changes only relate to the boundary condition expressions and the force. Such changes are rather minor. We used 81-point nodal scheme on c with cubic 'cross-section' sets (that is/~,,jj,) appropriate to rigid spheres (these universal sets had been created earlier in our previous work (Loyalka, 1991a~c)); 64 (R<0.25), 32 (R >0.25) quadrature nodes on #; and variable 101 (R <0.25), 201 (R > 0.25) point spatial mesh on r, with n'= 3. Our calculations generally converged monotonically, albeit, slowly for large R, indicating that here use of acceleration techniques would be quite attractive. Because of machine memory limitations (we had 24 MB to work with) we could not go beyond R = 1.0 as here for reasonable accuracy more spatial nodes are needed. Our results for the thermophoretic force are reported in Table 1, and compared with Brock's fit (Brock, 1962) to the data of Schmitt (1959) and Jacobsen and Brock (1965). For the range (R < 1.0) that we have studied, the computed results agree rather well with Jacobsen and Brock's data although small discernible disagreements exist at both R __<0.25 and as R approaches 1.0. For R <0.5 the computations describe Schmitt's data rather well, while for R > 0.5 the data of Schmitt are 10-15% higher than our results. Note that partial accommodation would lead to a smaller computed value, and thus only enhance the disagreement. It is quite possible, however, that use of intermolecular potential for argon in the computations would reduce the disagreements. Overall, the computed results are in good qualitative agreement with the data. We have verified the accuracy of the numerical scheme by considering the BGK model, for which numerical results based on integral transport equation are available. The structure of the BGK model allows use of a 41-quadrature set on speed c, and also n'= 1. Thus the storage requirements are reduced, and also computations per iteration step are reduced by a factor of six. We were, thus, able to carry out calculations for a larger range of R (we used a 401-point mesh on r. Results for a 201-point mesh, for R < 1.0, differed from those with 401-point mesh by less than 2%). The comparison of the present results with those of Yamamoto and Ishihara (1988) is reported in Table 2, which indicates that the present results should be accurate to within about 2-3%, except for large R, where the discrepancies are higher (note that the integral transport results are also not 'exact', and are estimated to be in error by 2-3%). For the BGK model we do, however, also find existence of negative thermophoresis. The accuracy of the present results can be improved further by considering higher order quadratures and more speed and spatial nodes, particularly the last because of the algebraic decay at large r. This aspect can be possibly handled by

Table 1. Thermophoretic force R (Kn - 1) 0.1 0.25 0.50 0.75 1.0

FT

Present 2p/). K= 10.0 ).p/2 R= 100.0 2.2698 2.0259 1.6459 1.3273 1.0811

2.2677 2.0115 1.6150 1.2865 1.0290

for a rigid sphere gas ~.p/2B= 10.0 2.2123 2.0037 1.6989 1.4404 ! .2213

Data* ,,tp/,,[B= 100.0 2.1788 1.9288 1.5743 1.2849 1.0487

* These results are from Brock (1962), who indicated that the data of Schmitt (1959) (M 300 silicone oil droplets in argon) and Jacobsen and Brock (1965) (spherical sodium chloride particles in argon) can be fitted, for R < 5.0, by the expression (in the present nomenclature): Fz = 4~1/2 " - 7 exp

(--~R) z=O.39fOr2p/A.~lO.O;z=O.48for~p/2.~-,lO0.O.

Thermophoretic force on a single particle

295

nZ z

nr

n

I

Fig. 1. The spherical co-ordinate system in curvilinear geometry.

and the thermophoretic force is:

F,=2pod2r ~ - R

[ fl--~-+

(23) /

where fl has been defined earlier, equation (18), and, 1

0=~ fo dcexp(-cZ)cS Ot(c)"

(24)

Since for R = 0,/~ = 0; we have, /~z, fm = 2Po d2K ---~ --R2~8~t/2

-"2 167tl/2

=poR r - - ~ 0

(25)

we note that for Maxwell molecules: 0= -]

1

(26)

and expression (25) reduces to Fz fm=

.

-Po

J~2/¢47t1/2 3 =

32 /~2 AT ~'~ / 8k~o-o~1/2 ,~B~-~ , \-m-m / 1

(27)

which is a result reported by Waldmann (1959). In the slip limit, R>> 1, and following Epstein (1924), as well as Brock (1962) (note that for Knudsen number, Kn=I/R~. 1, Brock's expression is the same as Epstein's) one obtains: I

2~.~ 1 - 1

AT

(28)

where ~'is the thermal creep slip velocity, and # is the viscosity (one should not confuse it with direction cosine as used elsewhere in this paper). We should remark that the recently computed (Loyalka, 1990) values of this slip coefficient (which have some experimental

296

S.K. LOYALKA

et al., 1932) are quite different 2~-=34 # 1 I ( 2kTo']1/2 p ~ ~~k ~ j

verification (Annis, 1972; Weber

from Maxwell's value: (29)

used by Epstein (1924), Brock (1962) and others. We remark that while ~9 is almost independent of intermolecular force laws (-0.25 for Maxwell molecules and -0.2567 for rigid spheres), the thermal creep slip coefficient has a distinct dependence on intermolecular force laws (the factor 1/2 in equation (29) is replaced by values ranging from 0.68 to 0.76). For future reference we note that the macroscopic gaseous density, velocity, and temperature are given as:

,j

r(~)=3~nn da m(~-~)2f(~, ~)

(3O)

whence, n(r)

no =

tl o

1_ -

z +

1 fdcexp,

v(r)=[~-/~fdeexp(-cZ)eh(r,e)Jx T(r)-T°-[z+ 3n2~fdeexp(-cZ)(cZ-~)h(r'e)] 3/z

(31)

4. SOME T R A N S F O R M A T I O N S We begin by noting that the non-dimensional collision operator L may be written as

Lh(r, c) = - v(c)h(r,e) + f d c '

e x p ( - c'2)K(e, e')h(r, c'),

(32)

where v(c) is the 'collision frequency', and K(c, c') is a symmetric kernel. The rotational invariance of K permits us to express it in the form 1 ~ {2n+l\ K(e, e')= ~ . ~ o ~)k,(c,c')P.(#o),

(33)

where /~o is the cosine of angle between e ( c , # = c o s 0 , ~b) and e', P. are the Legendre polynomials, and

k.(c, c')=

f'

d~oP.(#o)K(c, e').

(34)

-1

Note that from the addition theorem of spherical harmonics: .

(n-:)!

P"f#°)=P"(#)P"(ff)+ 2:=,~"(n+:)! P~"(#)Pe"(~')c°s:(¢-dP')'

(35)

where p : are the associated Legendre functions. For the streaming term in the Boltzmann equation, referring to Fig. 1, with n, a unit vector in the direction of the temperature gradient, we consider a spherical co-ordinate system in curvilinear geometry, and have: c'~-r=C

/~ffrr+

r

ap

~_(1-

r

sin4~

-~

sin~ Off

~-cos~bcota

.

(36)

Thermophoretic force on a single particle

299

Table 2. Thermophoretic force F r for the BGK model R (Kn -t) 0.1 0.50 1.0 5.0 10.0

Present 2p/2j = 10.0 ~ / 2 , = 100.0 2.2250 1.6865 1.1674 0.1234 0.04015

2.2200 1.6528 1.1086 - 0.002628 -0.05581

Integral transport* ).p/2, = 10.0 2p/2s = 100.0 2.2160 1.6625 1.1313 0.09899 0.03518

2.2110 1.6343 1.0760 - 0.0085 -0.04480

* Yamamoto and Ishihara (1988).

subtraction of asymptotic solution, while the slow convergence for large R requires investigation of acceleration or multigrid schemes. We thus conclude that the linearized Boltzmann equation corresponding to the thermophoresis problem can be solved accurately, and that the rigid sphere results agree qualitatively with the available data over a range of the Knudsen number (R < 1.0; Kn > 1.0). We hope to report comprehensive and detailed numerical results for the force, as well as the molecular distribution and other macroscopic quantities, for all R and different and thermal conductivity ratios in the near future. General molecular force laws and boundary conditions will be considered as discussed in our recent work (Loyalka, 1991c). We note that the present effort was computationally intensive, and it did require hundreds of hours on a dedicated RISC/6000 workstation. We are presently attempting to get access to a massively parallel computer (reputed to be several orders of magnitude faster than RISC/6000), and depending on our success, the remainder of the task ahead may not be difficult. Lest one should think that the task here was easy, we only quote Brock (1962) writing in the very context of this problem, "A numerical solution of the full Boltzmann equation would be most certainly a prohibitive task". It is really only the advances of the recent years in the numerical approaches and the computational hardware that have now made the task feasible. We should again, emphasize though, that the present work applies to a monatomic gas under the restriction of linearization. Extensions to polyatomic gases under linearization can be carried out in the framework of the technique discussed in this paper, but the computational costs would increase very substantially. Methods for accurate deterministic treatment of the non-linear Boltzmann equation, in geometries such as of this paper, however, still remain to be realized. Acknowledoement--This research was supported by the Exploratory Research Division of the Environmental Protection Agency. REFERENCES Annis, B. K. (1972) J. chem. Phys. 57, 2898. Bakanov, S. P. (1991) Aerosol Sci. Technol. 15, 77. Batchelor, G. K. and Shen, C. (1985) J. Colloid lnterf. Sci. 107, 21. Brock, J. R. (1962) J. Colloid Sci. 17, 768. Brock, J. R. (1967) J. Colloid Inter.:. Sci. 23, 448. Brock, J. R. (1980) in Aerosol Microphysics I (Edited by Marlow, W. H.), Springer-Verlag. Davies, C. N. (Ed.) (1966) Aerosol Science. Academic Press, London. Davis, L. A. and Adair, T. W. (1975) J. Chem. Phys. 62, 2278. Dwyer, H. A. (1967) Phys. Fluids 10, 976. Epstein, P. (1924) Phys. Rev. 23, 710. Friedlander, S. K. (1977) Smoke, Dust and Haze. Wiley, New York. Fuchs, N. A. (1964) The Mechanics of Aerosols. Pergamon Press, Oxford. Fuchs, N. A. (1982) J. Aerosol Sci. 13, 327. Hidy, G. M. and Brock, J. R. (Eds) (1971-1973) Topics in Aerosol Research, Vols. 1-3, Pergamon Press, Oxford. [See in particular, the Vol. II paper by Deryagin, B. V. and Yalamov, Y. I.] Jacobsen, S. and Brock, J. R. (1965) J. Colloid Sci. 20, 544. Law, W. S. (1986) Ph. D. Thesis, University of Missouri-Columbia. Loyalka, S. K. (1983) Proo. Nucl. Eneroy 12, 1. Loyalka, S. K. (1986) in Rarefied Gas Dynamics, XVI Symposium (Edited by Boffi, V. and Cercignani, C.). Teubner, Stuttgart. Loyalka, S. K. (1989) Phys. Fluids 1, 403.

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