Collection of Aerosol Particles by a Conducting Sphere in an ...

Collection of Aerosol Particles by a Conducting Sphere in an ExternalElectric Field-Continuum Regime ApproximationPAO-KUANWANGDepartment of Meteorology, University of Wisconsin, Madison, Wisconsin 53706Received January 4, 1982; accepted December 13, 1982The collection of charged aerosol particles by a charged conducting sphere in a uniform electricfield is theoretically investigated. Continuum regime is considered and particle concentration is assumedto be described by the convective diffusion equation. Analytical solutions satisfying boundaryconditions for different regions are obtained. These solutions include the simultaneous effects ofBrownian motion, static charges, and external electric field. Some numerical results are presented.It is also shown that these solutions can be reduced to various limiting cases where some effects areabsent.INTRODUCTIONThe collision and coagulation process ofaerosol particles has long been an importantproblem in colloidal science. It has applicationsin both atmospheric sciences and engineering.In the atmosphere, for example,the collection mechanism plays an importantrole in the scavenging of aerosol particles.This mechanism has recently received considerableattention (l-lo). In problems ofchemical engineering particle coagulation isof importance as it determines to a considerableextent the size distribution and thusthe reaction rate of the reactants.In this paper we want to investigate thecollection process of aerosol particles by aspherical collector in the continuum regime.The concentration of particles is assumed tobe described by the convective diffusionequation since the particles involved moveby Brownian diffusion while simultaneouslyunder the influence of convective forces suchas gravity, hydrodynamic forces, and possiblyphoretic and electric forces. In this studywe shall consider the case where the externalforces involved are electric forces.PHYSICS AND MATHEMATICS1. The convective d$ikon of aerosol particlesaround an absorbing sphere in conservativeforce fields. In this section we willbriefly describe the general formulation ofthe convective diffusion of small particles inthe presence of conservative force fieldswhich will be used later. Consider a cloud ofmonodispersed aerosol particles around aperfect absorbing sphere under the influenceof a net conservative force field F. Then bythe definition of conservativeness,orVXF=O[IIF = -VV,PIi.e., it is possible, in this case, to find a scalarpotential function V whose gradient will givethe force.We further assume that this potential functionsatisfies the Laplace equation, i.e.,v2v = 0. [31Equation [3] is satisfied by many force fieldssuch as those involved in the irrotational flowof a perfect fluid, surface waves, electromagneticphenomena, and gravitation.Journal ofcolloid and Inkyface Science, Vol. 94, No. 2, August 19833010021-9797183 $3.00Copyright 0 1983 by Academic Press, Inc.All rights of reproduction in any form reserved

2.302 PAO-KUAN WANGIf we assume steady-state condition, then+++++++the convective diffision equation for parti----cles exposed the above force field is (12)DV2n-BF.Vn=O, [41where n is the concentration, D the diffusivity,and B the mobility of the particles, re-I-:COLLECTORspectively. According to Eq. [2], Eq. [4] can---be rewritten as:v2n+~vv.vn=0.153The general solution of Eq. [5] isAEROSOLFlPARTICLEwhere C1 and C, are arbitrary constants tobe fixed by boundary conditions. One simplyhas to substitute Eq. [6] into Eq. [5] to seethat it is indeed a solution of Eq. [5]. Bysuitable transformation of Eqs. [5] and [6],one can obtain other related equations andtheir solutions (see Appendix A).Note that the above derivations were carriedout by vector operations. Thus they arevalid not only for spherical coordinates butfor other coordinates as well. In fact, the applicationsof Eqs. [5] and [6] had been madeby (13) for cylindrical coordinates and by(14) for oblate spherical coordinates. Notealso that the force fields which satisfy theabove derivations need not to be radiallysymmetrical.2. Distribution of aerosol particles arounda spherical conductor under the influence ofBrownian motions, electric charges, and auniform electric field. We now consider thespecific case where a conducting sphere ofradius cy is surrounded by a cloud of monodispersedaerosol particles with radius r,, (seeFig. 1). Electric charges on the sphere andthe particles are & and q, respectively. Wewill assume g > 0, Q < 0 for the convenienceof discussion although the solution we shallderive is valid for general cases. The Coulombforce due to the static charges is thusattractive. This force is in the radial directioni:FIG. 1. ConfigurationF z&pcr* ’of the problem.where we have neglected the image force (seeAppendix B for justification). In addition,when an external electric field E0 is presenteach particle will feel an additional forceFE = &WIwhere E is the actual electric field due to theperturbation of the sphere. In writing Eq. [S 1,we have assumed that (i) electrically the aerosolparticles behave like point charges, and(ii) the electric field caused by the spacecharges nq is negligible. These assumptionsare justified if the particles are small comparedto the collector sphere and if thecharges residing on aerosol particles aresmall. The total external force acting on anaerosol particle iswhereF = FE + F, = qE + e4 u^r2 ’[9]Journal of Co/hid and Inrer/ace Science. Vol. 94, No. 2, August 1983

304 PAO-KUAN WANGwherezero everywhere in this region except when&Icos 0 is itself infinitely close to zero. In the6 = exp - - D 4,) = exp(-co) -+ 0 WI latter case e is close to one. Thus C, and C,given above are not strictly constants butis a very small number, infinitely close to very nearly so. The solution isexp{$[&(r--$)cosB-y]}-exp(-$$)nxn co [241‘- exp$)cos*+(~-$I}).P51(ii) When cos 13 >, 0, I < r,When cos 0 >, 0, the conditions [ 171 do notdetermine C, and CZ uniquely, and additionalconditions are necessary. For this purpose,we note that the concentration gradientis given by taking the radial derivative of Eq.[16]:$= C1~[Eoq(l +$gcoss+gAt cos 0 = 0, the radial component of theexternal field is zero and we would expectthat the radial concentration gradient wouldbe the same as the case when the externalfield is absent. The expression of &z/aR forthe latter case can be found in (8). Thus,dndy I coss=o = Cl ;[F]exP(-$$)Xexp{g[q&(r--$)cosR-T]}. 1261E-0.85I”““““““11 -1and thereforec, =1 - expno012810 - 1 /REGION-I / REGION-II 1The other boundary condition is iz = 0 at r= a, i.e.,and therefore0 = Cl exp1.0 14 1.8 2.2RADIAL DISTANCE (rlFIG. 2. The electric potential $ as a function of theradial distance r for the case a = 1, Q = - 1, l&l = 0.2,and 0 = 0.c, = - [291The complete solution is therefore

COLLECTION OF AEROSOL PARTICLES 305n=n m [301At r = r, the radial components of the external field for all 8 vanish, and we would expectthat an@- = 0 at Y = r, for all 8. But this requirement is automatically satisfied in view ofEq. [26] since the quantity in the first square brackets vanishes at r = r,.Equation [30] can be written asexp(-z)-exp{$[E,q(r-$)cos*-$11n = n, [311exp -BQ4\ - 1( Dal LBoth the numerator and denominator on theright hand side of Eq. [3 11 are now positive.It is easy to see that the concentration is notuniform at r = r, (remember that r,,, is afunction of 0). The concentration y1,, islargest at cos 0 = 0 since the potential is zeroat r = r,. On the other hand, the smallestn,=, occurs at 0 = 0 since the potential maximumis lowest and therefore the negative ofthe potential maximum is largest (see Fig. 2).Even at a constant r near the sphere, the0”FIG. 3. The surfaces of r = r, calculated from Eq.[20] for different values of h. Note that these surfacesare not equipotential surfaces.concentration n is also smallest along 0 = 0,as can be seen from Eq. [3 11. This is consistentwith the physical consideration becausethis is where the particles meet the strongestrepulsive forces.From Eq. [3 I] we can also see that whenEO increases to an extent that the second termin the numerator equals the first term, theconcentration becomes zero. Further increasein EO would cause the concentrationto become negative. The negative value itselfis unimportant; it simply means that undervery strong external field, particles cannotcome close to the hemisphere because of thevery strong repulsive forces. This is also consistentwith physical reasonings.The solution for the region where cos f3> 0 and r > r, is not obtained yet. On theother hand it should not concern us herebecause the particle flux in this region is basicallydrifting along the field lines and isdirected away from the sphere. Thus, insofaras the collection of aerosol particles by thesphere is concerned, we can ignore this region.3. The calculation of collection kernel. Tocalculate the collection kernel, we need toknow the radial concentration gradients atthe surface of the sphere. Thus by taking theradial derivatives of Eqs. [25] and [31] andset r = a, we obtain.Journal of Co/lord and Interface Science, Vol. 94, No. 2, August 1983

306 PAO-KUAN WANGdnBi% ‘33s I=a &co 3E,,q cos 0 + $= -&D 1~321and= --y1 cc; 3Eoq cos 6’ + %11 - exp[331The collection kernel iszzfDUn.dA=i-=ZiK, + K2B cos 0 +ss 2r ~,2 3Eoq 900 1 - exp1 a2sinBdBd4 +~~2B[3EoqcosB+~]a2sinOdOd~= -(3nEoa2Bq + 2rBQq)1 - exp+ (3aEoa2Bq - 2nBQq). [34]-The two terms in Eq. [34] should be evaluated separately before adding together. Negativevalues of K, or K2 should be regarded as zero since it simply means that no particle iscollected.The above derivation is based on the assumption that the surface r = r, extends outsidethe sphere. In the case when X < 3, this surface does not cover the whole hemisphere butcovers only from 0 = 0, to 0 = n/2 (see Fig. 3). In this case, the lower limit of the innerintegral in K, is not 0 but should be replaced by 13,. The resulted collection kernel is- i rEoa2Bq(l + cos 28,) + 2aBQq cos &1K= + (3xEoa2Bq - 2rBQq), [3511 - expwhereThe above formulation is valid for stationary collectors. For collections moving in aviscous medium the hydrodynamic forces should be considered. The calculations whichinclude hydrodynamic forces are, however, quite involved. Examples of this type of calculationswere given in (6) where the Brownian motion was not considered. Alternativelyan approximated method which takes the hydrodynamic effects into account was given by[361Journal of C&id and Inrerjhre Science, Vol. 94, No. 2, August 1983

COLLECTION OF AEROSOL PARTICLES 307( 10). There a ventilation factor was used to represent the overall enhancement of the particleflux. Thus according to this approximation the collection kernel is given byfor X > 3 and- i rEoa2Bq(l + cos 20,) + 2?rBQq cos &1K= + (3*Eoa2Bq - 2aBQq) [3811 - expfor X < 3. The previous paper ( 10) must be consulted for the formulas of various ventilationfactors.RESULTS AND DISCUSSIONSEquations [34]-[38] were used to calculatethe collection kernels of aerosol particles ofradii 0.00 1 to 1 .O pm by spheres of 10 to 400pm in the presence of external electric fieldsof strength 100 to 3000 V/cm. All computationswere carried out for 900 mbar and10°C. The spheres and particles were assumedto carry electric charges ofQ = 2a2 esuq = 2ri esu,[391respectively, with a and r, in centimeters.This size dependence of electric charges isthat proposed in (17) for the mean thunderstormcharges. Note that for particles of radiismaller than about 0.15 pm the above formulasgive values of charges smaller than oneelectron (-4.8 X 10-l’ esu). The values thusrepresent only the average charges of a cloudof mixed charged and neutral particles. Sincethe charges of small particles are usually verysmall, we assume here that, for the case whenq is smaller than one electron, the aerosolcloud is composed of particles carrying eitherone electron charge or no charge. The fractionof charged particles is therefore given byR, =2rz4.8 X 10-l’[401whereas the fraction of neutral particles isgiven byRg= 1 -R,. [411The neutral particles are assumed to be unaffectedby the electric forces and thus performpure Brownian motions with the collectionkernelKB = 4?rDa.The total kernel is therefore1421K=R,K,+(l -R,)KB, [431where Kg is the collection kernel of chargedparticles given by one of the Eqs. [34]-[38].Figures 4 to 8 show the collection kernelsof aerosol particles by stationary spheres ofvarious sizes in the presence of electric fieldsof 100, 500, 1000, 2000, and 3000 V/cm,respectively. Figures 9 to 13 show the collectionkernels of the same set of particle andsphere sizes, except that now the ventilationeffect is considered. The two sets of figureslook similar to each other except in the magnitudes.The general feature is that the kerneldecreases with aerosol size initially, reachesa minimum in the size range between 0.01and 0.1 pm, and then increases with particlesize. This general feature can be understoodin terms of the relative importance of theBrownian motion and the electric effect. Forvery small particles the Brownian motion isJournal of C&id and he&a= Science, Vol. 94, No. 2, August 1983

308 PAO-KUAN WANG1o .OOl 2. 5. .Ol- 2. 5. .l 2. 5.I - ““I 1llll ““I ’ I’I’, IIll, ““I ’ 1’1’1 IBIB ‘lo1 - - 1lo-‘- - 10-I10-z-10-T- - 10-710-a- -10-8‘“-“.001 2. I , I,I*I 5. I1111 .Ol ,,,.I 2. . I .,,, 5. I,,,, .l .*,, 2. I , ,,,,I 5. ,I,, 1 Jo-9RRDIUS OF PARTICLE (MICRONS1FIG. 4. Collection kernels of aerosol particles captured by spheres of various sizes due to Browniandiffusion and electric forces in air of 900 mbar and 10°C and external electric field E,, = 100 V/cm. (I)a = 10 pm, (2) a = 30 pm, (3) a = 50 pm, (4) a = 70 km, (5) a = 100 pm. No ventilation effect.Q= 2a2 esu, q = 2rz esu, a and r, in centimeters.dominant, the effect being larger for smallerparticles, thus the kernel decreases when sizeincreases. On the other hand, the electric effectis dominant for large particle sizes, theeffect being larger for larger particles due tothe larger charges. Thus the kernel increaseswith size at this end In the middle neitherthe Brownian motion nor the electric effectis strong, the kernel is therefore a minimumhere. A more detailed discussion is givenin (10).Figures 14 and 15 compare the collectionkernels of aerosol particles captured byspheres of radii 30 and 100 pm for the caseswith and without ventilation effect. Clearlythe ventilation effect is stronger for smalleraerosol particles while the larger particles(Y > 0.1 pm) are hardly affected. This can berealized in terms of the larger inertia of theparticles. In addition the ventilation effect isstronger for the larger collector (a = 100 pm)because of larger hydrodynamic forces.Figure 16 compares the kernels obtainedby the present formulation and that of (10)for collectors of radii 30 and 106 pm in theelectric field E0 = 3000 V/cm. As can be seenhere the differences between the two are notvery large, being about lo-20% in most cases.The present formulation gives a lower valuethan that in (10). This is to be expected sincethe formulation in (10) simply adds the fluxdue to Brownian diffusion to that due toelectric effect, thus implicitly assuming thatthe Brownian diffusion is not affected by theelectric field. This is of course less accurate.The present formulation, on the other hand,considers the reduction of the Brownian fluxin the presence of the electric field and istherefore more physically realistic. The smalldifference between the two, however, doesJournal of Colloid and Inlerface Science, Vol. 94, No. 2, August 1983

110-lf10-g ’ a 3 ” ’ ’ ’ ’ ” ’ ’ ’ ’ ’ ’ 8 ’ ” ’ * 1 I I I I I I I I 1 , I , I , I , I I I I I 1.lo-g,001 2. 6. .Ol 2. 5. .l 2. 6.FIG. 5. SameRflOIUS OF PFIRTICLE IMICRONS)as Fig. 4 except for E. = 500 V/cm.rRRDIUS OF PRRTICLE IMICRONSIFIG. 6. Same as Fig. 4 except for E. = 1000 V/cm.309Journal of Colloid and Interface Science, Vol. 94, No. 2, hgmt 1983

1o .OOl 2. 6. .Ol 2. 5. .l 2. 5.I - ‘11’1 ““I ““I - “‘11 ““I “.‘I - ‘,“I “‘1 ‘io1 - - 1lo-‘- - 10-l10-z- 45zI10-Z1 o-3+&1 o-’- 10-510-G- L------y 10-610-T- - 10-710-a- - 10-a‘“-g.oo’ 2. I , ,,,#I 6. lllll .Ol I1.II 2. , 1,1#1 5. lllll .l ,*,.I 2. 1 I1l&I 5. 1111 , Jo-9RRIJIUS OF PflRTICLE [MlCRONSlFIG. 7. Same as Fig. 4 except for E. = 2000 V/cm.1 - - 1lo-‘- - 10-llo-*-10-6- ----- - 10-S10-T- - 10-710-e- - 10-a10-g .OOl ~~~~’ 2. “*‘,““” 5. .Ol ,*s*’ 2. “*‘,““” 5. .l *,a,’ 2. ’ I~(~““’ 5. i!O-’RRDIUS OF PARTICLE (MICRONS)FIG. 8. Same as Fig. 4 except for E0 = 3000 V/cm.Journal of Co/hid and Inlerface Science. Vol. 94, No. 2, August 1983310

COLLECTION OF AEROSOL PARTICLES 311x lj& ,,,( ,,,,/,,,,,,,,,,,,/,, I2. 5. .Ol 2. 5. .I 2. 5.RRDIUS OF PRRTICLE [MICRONSI,,,, (,,,, ,,,~;I:FIG. 9. Collection kernels of aerosol particles captured by spheres of various sizes due to Browniandiffusion and electric forces in air of 900 mbar and 10°C and external electric field Ea = 100 V/cm.(1) a = 10 pm, (2) a = 30 pm, (3) a = 50 pm, (4) a = 70 pm, (5) a = 100 pm, (6) a = 200 pm,(7) a = 300 pm, (8) a = 400 pm. Ventilation effects are included. Q = 2a2 esu, q = 2rz esu, a and ri,in centimeters.testify an old empirical rule that the sum ofthe pure Brownian flux and pure conductioncurrent represents a good approximation forthe total flux in many cases.a. Limiting CasesWe shall now show that in various limitingcases the solutions in Sections 2 and 3 canbe reduced and can represent the proper solutionsfor these limits.(i) When there is no external field (EO= 0). In this case, t = 1 in Eq. [23] insteadof zero, and Eq. [24] becomes exactly[441while Eq. [30] also reduces exactly to thisresult. This is identical with the solution obtainedby (8) where the particle distributionis determined only by the Brownian diffusionand central forces such as that caused bystatic charges.(ii) Pure Brownian dljiision. When bothelectric forces due to the external field andthe electric charges are absent, then the particledistribution is solely determined by theBrownian diffusion. We can obtain this fromthe above solutions.Since there is no external electric field, thestarting equation will be the same as Eq. [44].We then take the limit of Eq. [44] when Qand q are infinitely approaching zero. Thus,by expanding the exponential functions in[44] into power series we obtain

lo-‘-11 o-6F10-G3-l-& ,*,,2., , ,,,,5.t ,,,,.Ol),,,, 2., ,#(,( 5., ,,(,.l, ,*,,8., (,,*I5.(jjRRDIUS OF PRRTICLE (MICRONS1FIG. 10. Same as Fig. 9 except for E0 = 500 V/cm.10.OOl 2. 5. .Ol 2. 5. .l 2.0I10-l 11 o-2 110-3 1.\ r////g..--10-4 1RADIUS OF PARTICLE IMICRONS)FIG.11. Same as Fig. 9 except for I& = 1000 V/cm.Journal ofCoNoid and InterSace Science, Vol. 94, No. 2, August 1983312

II , ,(, , 1,, , , (,,;...OOl 2. 5. .Ol 2. 5. .l 2. 5.RRDIUS OF PRRTICLE (MICRONS1FIG. 12. Same as Fig. 9 except for & = 2000 V/cm.1 -110-l1 o-210-a-4FIG.RRDIUS OF PRRTICLE (MICRONS113. Same as Fig. 9 except for ECJ = 3000 V/cm.313Journal of Co/hid and Interface Science, Vol. 94, No. 2, August 1983

314 PAO-KUAN WANGRROIUS OF PRRTICLE IMICRONSIFIG. 14. Comparison between the collection kernels of aerosol particles captured by spheres of radiusa = 30 pm with (-) and without (- - -) ventilation effects. Curves (l)-(5) are for cases withoutventilation effects and E. = 100, 500, 1000, 2000, and 3000 V/cm, respectively. Curves (6)-( 10) are forcases with ventilation effects and E0 = 100, 500, 1000, 2000, and 3000 V/cm.,.,(-g-,.,(-g)= lim n,Leo Q-0q--‘o 930 1 - exp= lim n,e-0PO1Da---- BQq 1Da 2!(neglect higher order terms)1451Journal of C&id and Interfac Science, Vol. 94, No. 2, August 1983

COLLECTION OF AEROSOL PARTICLES 31510 .OOl a.111 2. I ,‘,‘,,,,,,““, 5. .Ol 2. 1,1,,,,,,,,,, 5. .I I,, 2. ,,,,, 5. ,,,, ‘ioRROIUS OF PRRTICLE [MICRONSIFIG. 15. Same as Fig. 14 except for a = 100 Grn.Equation [45] is the well-known particledistribution around a sphere due to pureBrownian motions (see, e.g. (8)).(iii) Very strong external electric field.When the external field is so strong as topredominate the whole process, the particledistribution is solely determined by the externalfield. The total mass flux toward thesphere is the convective current due to theexternal field. This can be easily obtained bytaking Eoa2 S @I in Eq. [35]. Since &- ?r/2 in this case, we haveK x 3?rEoa2Bq 1461since the first term is a negative number andshould be set to zero. Equation [46] is thewell-known convective current due to a uniformexternal field E0 (see, e.g. (1 S), for thecase of ion transport toward a spherical dropdue to Eo).CONCLUSIONSIn the above discussions we have shownthat the distribution of aerosol particles bya stationary, conducting sphere in the presenceof an external electric field can be describedby the convective diffusion Eq. [13]with solution Eqs. [25] and [3 11. The particleflux is given by Eqs. [34]-[38]. We have alsoproved that in the limiting cases (i) E,, = 0,(ii) pure Brownian diffusion, and (iii) verystrong E,, the above solutions can be reducedto proper solutions for these cases.Although the present paper deals exclusivelywith the electrostatic forces, the formulationis quite general and is valid for anyconservative force fields whose potentials satisfythe Laplace equation. Thus one can easilyadd forces such as thermo- and diffusiophoreticforces.We want to stress, however, that we havenot included the hydrodynamic forces whichJournal ofCo/loid and Inlerjace Scrence, Vol. 94, No. 2, August 1983

316 PAO-KUAN WANG- 10-i \1o-6 I I I 1 I I.OOl .Ol .I I.RRDIUS OF PRRTICLE [MICRONS)FIG. 16. Comparison between the collection kernels calculated from the present formulation and fromRef. (10). Solid curves are the present results. Dashed curves are results from Ref. (10). & = 3000 V/cmfor all cases. Only Brownian diffusion and electric forces are considered, but with ventilation effect.are more complicated. Thus the results obtainedabove should not be compared directlywith the experimental results whichwere obtained in the presence of hydrodynamicforces. We are currently working towardthis direction. The present paper, in themean time, should represent a step forwardin solving the convective diffusion problemin a nonsymmetric force field.APPENDIXIn the following we will provide some convenientformulas for obtaining particular solutionsof convective diffusion processes underthe influence of conservative force fields.To our knowledge these formula have notbeen brought to the attention of many investigators.Consider a vector field F which is conservativeand satisfies Eqs. [ 1 ]-[ 31. Let $ be anyAcontinuous function which is at least twicedifferentiable. Then(i) the equationhas a solution(ii) the equationhas a solution(iii) the equationhas the solutionV*$+FVIC/=O1c/ = C,e” + C,,V*$-F-V+=0c471[481[491I) = C,e-’ + C,, WIV*ic, + F*$ = 0 1511$ = Cle’” + C2e-I* (2 = -I), [52]Journal of Co/bid and Inter/ace Science, Vol. 94, No. 2, August 1983

COLLECTION OF AEROSOL PARTICLES 317and (iv) the equationAPPENDIXBhas a solutionV2t,b - F2$ = 0 [531I) = C,e” + C2eev, [541where F2 = F. F = Vv . Vv, v is the potentialof F, defined in Eq. [2]. These can be verifiedby simply substituting the solutions into thecorresponding equations. Equations [47] and[49] are generally regarded as the diffusionequations while Eqs. [5 I] and [53] are theHelmholtz equations.Equation [ 531 is also mathematically identicalwith the time-independent SchrGdingerequation.It is interesting to note that Eqs. [47], [49],and [53] are mutually transformable. If welet $ = $’ exp(v/2) and substitute into Eq.[47] we will obtainororv2gv - $ (Vv - Vv)$’ = 0V2$’ - (Vu’ - Vv’)l+v = 0v’ = 11( 21V2$’ - F’2$’ = 0 (F’ = -0~‘) [55]which is identical with Eq. [53]. On the otherhand if we substitute + = $’ exp(-v/2) intoEq. [49], we will again obtain Eq. [55]. Thistype of transformation has been introducedby Ftirth (193 1) (cited in Eq. [ 121 for onedimensionalcase). In the above we see thatone can perform the transformation in amore general three-dimensional form if thevector F satisfies Eqs. [ l]-[3].When using the above formula one has tobe sure that F does indeed satisfy Eqs. [l]-[3] and that the form of the equation is indeedthe same as those listed in Eqs. [47],[49], [51], and [53]. When these requirementsare met, the remaining work is to findthe potential function v for F. This last stepcan be found in many standard textbooks ofmathematical physics.The complete electric force between apoint charge and a charged conducting sphereis (see, e.g. (15) or (19))IFI = 5 [Q - “$’ aipl?)] , [56]where the second term in the brackets representsthe image force. Since the aerosolparticles are not really point charges but offinite sizes, they will be captured by the collectorat a distance r = a + r,. At this distancethe electric force isIFI = q:-!&-(a + rJ2qa3[a(a + T-,)~ - a”]- (a + r,)[(a + I-,)~ - a2121J ’[571Since the image force is larger for larger qwhich, in turn, requires a larger particle radiusin our calculation (q a Y;), we take thelargest particle (rp = 1 pm) and the smallestsphere (a = 10 pm, so that Q is smallest) foran estimate of the relative importance of thetwo terms. Putting these values of a and r,into Eq. [57] and assuming that Q = 2a2, q= 2r$ (a, rp in centimeters), we obtain-= Q 2 x 1o-6(a + rp)2 (1.1 X 10-3)2 = 1’65’qa3[2(a + rp)’ - a21(a + r,)[(a + rp)2 - a212= 2 X lo-’ X lo-’ X 1.42 X 1O-61.1 x 1o-3 x (0.21 x 10-6)2= 5.85 X lo-‘.Clearly the image force is negligibly small incomparison with the first term for aerosolparticles captured by larger spheres. Thus inour calculation we can safely neglect the imageforce.Journni of CoNoid and Interface Science. Vol. 94, No. 2, August 1983

318 PAO-KUAN WANGACKNOWLEDGMENTThe author thanks the anonymous reviewers for veryuseful suggestions and Ms. Eva Singer for typing themanuscript. This research is partially supported by theUnited States Environmental Protection Agency underassistance agreement R80937 1-O 1-O. Although the researchdescribed in this article has been funded in partby EPA, it has not been subject to the agency’s requiredpeer and administrative review and, therefore, does notreflect the view of the agency and no official endorsementshould be inferred.REFERENCES1. Hampl, V., Kerker, M., Cooke, D. D., and MatijeviC,E., J. Atmos. Sci. 28, 1221 (1971).2. Slinn, W. G., and Hales, J. M., J. Atmos. Sci. 28,1465 (1971).3. Hampl, V., and Kerker, M., J. Colloid Interface Sci.40, 305 (1972).4. Kerker, M., and Hampl, V., J. Atmos. Sci. 31, 1368(1974).5. Beard, K. V., J. Atmos. Sci. 31, 1595 (1974).6. Grover, S. N., Pruppacher, H. R., and Hamielec,A. E., J. Atmos. Sci. 34, 1655 (1977).7.8.9.Sci. 35, 674 (1978).10. Wang, P. K., and Pruppacher, H. R., J. Colloid InterfaceSci. 75, 286 (1980).11. Fuchs, N. A., “The Mechanics of Aerosols.” Pergamon,Elmsford, N.Y., 1964.12. Jost, W., in “Diffusion in Solids, Liquids, andGases.” Academic Press, New York, 1960.13. Wang, P. K., and Pruppacher, H. R., Pure Appl.14.15.Wang, P. K., and Pruppacher, H. R., J. Atmos. Sci.34, 1664 (1977).Wang, P. K., Grover, S. N., and Pruppacher, H. R.,J. Atmos. Sci. 35, 1973 (1978).Lai, K. Y., Dayan, N., and Kerker, M.,J. Atmos.Geophys. 118, 1090-l 108 (1980).Martin, J. J., Wang, P. K., and Pruppacher, H. R.,J. Atmos. Sci. 37, 1628-1638 (1980).Jackson, J. D., in “Classical Electrodynamics,” 64 1pp. Wiley, New York, 1962.16. Chandrasekhar, S., Rev. Mod. Phys. 15, l-89 (1943).17. Takahasi, T., Rev. Geophys. Space Phys. 11, 903(1973).18. Whipple, F. J. W., and Chalmers, J. A., Quart. J.R. Meteor. Sot. 70, 103-120 (1944).19. Marlow, W. H. (Ed.), “Aerosol Microphysics I: Par-ticle Interactions,” Vol. 16, 160 pp. Springer-Verlag, Berlin/Heidelberg/New York, 1980.Jmmal of Co/bid and Interface Science. Vol. 94, No. 2, hgust 1983