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Electrical interactions involved in aerosol filtration by a dielectric porous membrane: The case of nuclepore filter
Pergamon
S0021-8502(96)00006-7
J. Aerosol Sci. Vol.27, No. 5, pp. 751 757, 1996
Copyright ~3 1996ElsevierScienceLtd Printed in Great Britain. All rights reserved 0021-8502/96 $15.00 + 0.00
ELECTRICAL INTERACTIONS INVOLVED IN AEROSOL FILTRATION BY A DIELECTRIC POROUS MEMBRANE: THE CASE OF NUCLEPORE FILTER Carlos Manuel Romo-Krrger* and Vicente Diaz t * Departamento de Fisica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile t Departamento de Fisica Universidad Iberoamericana de Ciencia y Tecnologia, Santiago, Chile (First received 1 June 1995; and in final form 1 December 1995)
Abstract--Filtration by a dielectric medium has associated important electrostatic effects, since during the operations (the filtration process, manipulations, etc.) this medium accumulates considerable amounts of charge. In a filter-like Nuclepore, the charge is concentrated mainly at the rims of the pores. The electric field and its derivatives are analytically obtained and numerically evaluated for such a configuration of charge, to determine Coulomb and dipolar forces acting on a particle, supposed to be charged and polarizable. The important finding here is: in the vicinity of the pore rim the dipolar force is very high, exceeding the Coulomb force. This points to the importance of including polarization for a model of particle collection for that kind of filters. It could explain in part the observed accumulation of particles deposited in the pore vicinity in normal filtration situations. Copyright © 1996 Elsevier Science Ltd. 1. I N T R O D U C T I O N
This work intends to give attention to a problem that emerges when particles are being separated from a gas forced to pass through a porous foil or membrane filter. Any filter composed of dielectric material presents a strong tendency to acquire electrical charge, by friction during air passage and manipulations (tribo-electricity). Charges in the order of nano-Coulombs may be found in Nuclepore filters during this process (Zebel, 1974; Romo-KriSger, 1990). Such charge has a significant effect on particles of microscopic sizes, producing an interesting physical situation when they are near the pore entrance. Particle deposition on charged bodies is a very complicated problem (Pich, 1966) and in many cases ignored its importance on particle capture (Heidam, 1981). On the contrary, by the light of measurements, we think that charge effects are important for the particle trapping mechanism. Since the 1970s Nuclepore filter has a considerable use for atmospheric aerosol studies. It is specially well suited for subsequent analyses, like PIXE and XRF high-resolution spectroscopy, and light and electron microscopy. The Nuclepore is a dielectric foil of about 10/~m thickness with straight holes and very well-defined structures (Fig. 1). The tendency of this filter to acquire electrostatic charge" has being studied (Engelbrecht et al., 1980; Feeney et al., 1984; Romo-Kr/Sger, 1990). Also, good intents to describe particle collection in relation to electrical interactions have been done (Zebel, 1974; Fan et al., 1978; Gentry et al., 1985; Romo-Kr~Sger and Morales, 1988). Zebel (1974) has studied in detail how the retention ability of the Nuclepore filter is influenced by the Coulomb interactions on particles. He worked with an equation of motion for a particle that included the drag force due to the gas flowing through pores and a Coulomb force between the particle and the filter. Later, we proposed an equation including a dipolar term (Romo-Krrger and Morales, 1988), accounting for the interaction between a polarizable particle and the non-uniform field generated by the charged filter. The high polarizability observed in aerosol particles---evidenced by the absorption of visible light (Friedlander, 1977; Krishna-Moorthy et al., 1988)---argues in favor to include the polarization term in any study of filter-particle interaction. The dipolar force is FD = I(p.V)EI, 751
(1)
752
C. M.
Romo-Kr/Sgerand V. Diaz
Fig. 1. Scheme of the cross section of a Nuclepore membrane filter.
where p is the dipole moment of the particle and E is the electric field generated by the charged filter. We evaluate FD and the Coulomb force Fc = [qE], to compare them. q is the particle charge. For such evaluation, typical parameters relevant to aerosols must be used. In this case, isotropic particles may be considered (Friedlander, 1977). The dipole moment is expressed in terms of the electric field E and a scalar polarizability :~. That is p = ~E.
(2)
FD = e(E. V)E.
(3)
The dipolar force is, therefore,
The polarizability of a spherical particle of radius r acting as a dipole, given in terms of its dielectric constant c in appropriate units, is (Jackson, 1975; Friedlander, 1977; Fuchs and Claro, 1989) ( ~ - 1)r 3 -
-
-
(~: + 2)
(4)
Friedlander (1977), in the subject "Field charging", presents a formula to evaluate the charge of saturation of a particle when an electrical field is present. Further, this formula is evaluated for parameters approximating those conditions when real particles are collected in electrical precipitators. This leads to the value e = 58, for the dielectric constant of the particle. This is an evaluation using experimental conditions for a real aerosol, thus includes polarization of particle material and the polarization due to the surface conductivity of the particle. It is concordant to the results of K o g a n et al. (1993) on the efficiency of electrostatic deposition of aerosol particles on sorbent beds, when we assume beds composed of 6 x 10- -~m diameter fibers with a charge density of 3.3 A sm - 3. The value e = 58 leads to the value 0.95 for the factor (~:- 1)/(e. + 2) used in equation (4). The same is 1.0 for a conducting particle, 0.97 for water spheres and 0.3 for a dielectric sphere composed of benzene. Water on the surface of particles and inclusions of conducting materials are factors that increase the dielectric constant of the particles (Mier and Wyder, 1972; KrishnaMoorthy, 1988; Claro and Brovers, 1989). The dielectric properties of small particles have been studied mainly in relation to oscillatory electric fields (scattering of electromagnetic radiation, etc.). In this case we must use these properties to determine the interactions between a polarizable particle in a static electric field. 2. M A T H E M A T I C A L F O R M A L I S M Jackson (1975) evaluates fields and charge densities near corners a':d edges of conductor bodies. In particular, a body with an edge formed with the intersection of two planes at right angle has a charge density proportional to 6 1/3 (being 6 the distance from a surface point to the edge). The configuration of charge has a singularity as 6 ~ 0. It implies that the field strength becomes very large at the edge. The very concentrated charge at the edge is a known effect named "point effect". At small scale, the edge of a filter pore is formed by the intersection of two planes at right angle. The first plane is the filter face. The second is a plane tangent to the cylindrical surface of the pore. If we consider the filter as a "conductor" body, is plausible to expect the above charge distribution verified near the pore edge.
Filtration by dielectric porous membrane
753
Migration of charge in Nuclepore filters has been observed (Romo-Kr/Sger, 1990) with typical times for filter charge decay on the order of 100 min. That migration of charge induces us to think the filter charge concentrated mainly at the edges of the pores, by point effect (within the standard operation times of the Nuclepore filter, normally of 24 h). The same consideration has been done by other authors (Zebel, 1974; Fan et al., 1978). Then, we calculate the electric field E as the derivative of a potential arising from a distribution of charge consisting in the uniformly charged ring at the entrance of the pore, with total charge Q. The complete form of the potential is obtained by integrating each charge element dQ, in terms of the pore diameter rp and the coordinates (x, y) of a general point of the space (Fig. 2). In virtue of the cylindrical symmetry, the potential q~ at (x, y) is the same for equivalent points in planes rotated around the pore axis. This reduces the problem to one in two dimensions, with the Cartesian coordinates x and y. Thus,
~(X, y) = Oprp
f
21t (r 2
_[_ X 2 _.[_ y 2
dO.
2xrp c o s 0 ) - 1 / 2
--
o
(5)
Using the symmetry ~b(x, y) = ~b( - x, y) and an integral equivalence (Gradshteyn and Ryzhik, 1965), this may be written as ~?(X, y) = 2crprlp/2X - 1/2 k K (k).
(6)
where crv = Q/2rcrp is the linear charge density of the ring and K(k) is the complete elliptical integral of the first kind (Abramowitz and Stegun, 1965) with the argument
I k_=2 i
rpX p+
11/2 +y23
(7)
.
We find the derivatives of ~b as
a,~
2~.r~/~
(8)
and q~,
02~b 2apt 1/2 [l+k 2 = @k2 - x 1 / 2 k ( l _ k 2 ) ~ E ( k ) - K ( k )
] .
(9)
E(k) is the complete elliptical integral of the second kind (Abramowitz and Stegun, 1965) with the same argument k.
~ Y
°o)
)X
Z Fig. 2. The charged ring at the pore edge centered at the origin of the xyz frame, for calculation of the electrical potential ~b at a general point P on the plane xy.
754
c . M . Romo-Kr6ger and V. Diaz
W i t h these e x p r e s s i o n s the c o m p o n e n t s of the electric field a n d its d e r i v a t i v e s are:
O'~k'~ + ~ ,
a-~
Ex-
E~,-
-
0),
+~k;,
#E x _ EXX -- ( ' ~ X
ff)k'(k'x)2
?Ex Exy = Eyx - -i?y Eyy-
~Ey ~ -
(~'kk '~ +
-
-
-,;
3 d?
--
(lO)
(o'kk'jx -- I 4X 2'
dp'kk ;, _ ,~, k" c~}~k;, + --27-x V'k xy,
~b/(k;,) 2 - qS~k;:y,
w i t h the explicit forms:
G_ & t
_ _
k~, k;~x
[1
~x - k
2x
(?k ?y
yk 12'
?2 ( ? .k~ -
(rp + x)] 12
k I
(rP + gx) + 3 (rp +14 x)2 xl 2
•
(O2k kJy = k'(~ - (9x ~), .. -
~2
k;; - Cy 2
'
YkL
F3(rp + x)
-
_
k[3y2
1]
L I~
1"-
-la
1 4x 2
,
1 | 2xl 2
]
'
l = E(rp + x) 2 -[- y2]1/2.
Y
)i I
Fig. 3. The equipotential lines (weak traces) and the electric field lines (hard traces) for a charge distributed over the pore rim. The top of the pore and the x, y coordinates frame are also drawn.
(11)
Filtration by dielectric porous membrane
755
In Fig. 3 are the e q u i p o t e n t i a l a n d the electric field lines a r o u n d the a n n u l a r d i s t r i b u t i o n of c h a r g e at the p o r e top, they were n u m e r i c a l l y d e t e r m i n e d using the f o r m e r formulas. W i t h the s a m e f o r m a l i s m the m a g n i t u d e of the C o u l o m b force F c = IqE[ a n d the d i p o l a r force FD = I(P" V)EI were n u m e r i c a l l y e v a l u a t e d for different p o i n t s n e a r the p o r e top. Results are p l o t t e d in F i g s 4-6. F o r these e v a l u a t i o n s we used the t a b u l a t e d values ( A b r a m o w i t z a n d Stegun, 1965) for the elliptical integrals K a n d E. O t h e r p a r a m e t e r s were t a k e n from o t h e r a e r o s o l studies (Zebel, 1974; F r i e d l a n d e r , 1977; Romo-Kr~Sger a n d M o r a l e s , 1988). These p h y s i c a l p a r a m e t e r s were: r = 5.0× 1 0 - V m , rp = 2.0× 1 0 - 6 m , ap = 1.27 x 10 - 7 A s m - 1 a n d e = 58. F o r the c a l c u l a t i o n s of F c a n d F D in Figs 4 - 6 we used charges for the p a r t i c l e e q u a l to l e - , 10e- a n d 100e-. e - is the c h a r g e of one electron ( e - = - 1.6 × 10-19 A s). FD results i n d e p e n d e n t of the value of particle charge. F r o m these figures we see t h a t FD has a very increasing b e h a v i o r n e a r the p o r e edge (x = rp, y = 0). At this s u r r o u n d i n g s the d i p o l a r force F o exceeds in extent the C o u l o m b force F c , even for the high value o f p a r t i c l e c h a r g e q = 100e-.
i
i
i
i
-15
N E
I
i
' } '
~
~c3}F°
-20
-25 °
0.0
014
012
0.6
1.0 '
018
x/rp
Fig. 4. Dipolar and Coulomb force intensities evaluated along the x axis (y = 0). Coulomb forces Fcl, Fc2 and Fc3 a r e for particle charges ql = le-, q2 = 10e- and q3 = 100e-, respectively (e- = 1.60 x 10-19 A s is the charge of one electron). The dipolar force FD results independent of the net charge on particle.
-3
~-" I
-6
~D
E 6, 'w" ¢.. m
-9
-12
la. ¢-
Fc -15
Fo I
I
I
I
I
I
I
o.o
0.5
,.o
1.5
2.o
2.5
3.0
y/rp Fig. 5. D i p o l a r f o r c e FD a n d C o u l o m b f o r c e F c intensities as f u n c t i o n s o f the d i s t a n c e f r o m the p o r e
border along a vertical axis (x = rp). Particle charge is q = 100e (e- = 1.60 x 10-19 A s is the charge of one electron).
756
C . M . R o m o - K r S g e r and V. Diaz
I.:37
1.68
0.00 0.2 0.4
-24.7
•
-
" ~nF ~tn K9 "re's-2)
Fig. 6. Each curve is the dipolar force FD as functions of the distance from the pore axis along different horizontal lines (y/r = 0.00, y/r = 0.34, y/r = 0.68, y/r = 1.02, y/r = 1.37, y/r - 1.71).
The values used in our calculations for the particle charge are in concordance with those given in the literature. Turner et al. (1988) predicts the value q = 2.9e- for a monodisperse aerosol particle. For particles exposed to bipolar ions Turner et al. (1988) predicts q ~ 100e- and Fjeld et al. (1983) predicts q ~ 1000e-. According to Burtscher et al. (1986) in combustion processes a particle can reach the value q = 10e .
3. C O N C L U S I O N S The charge on filters and particles may influence the process of filtration in two ways (Pich, 1966): (1) the particles may be attracted (or repelled) to the filter surface from a greater distance; (2) the charge makes the particle "stick" to the filter surface. For the first may be more important the Coulombic force, because it is of longer range than the dipolar force. For the effect of sticking on surfaces, in turn, the dipolar interactions play a principal role. Particles that fall across the filter surface by inertial impaction or by interception (or by influence of the Coulomb force), may be strongly fixed near the pore edges due to the intensive dipolar force there. Dipolar force contributes to avoid the detachment of particles. These conclusions are expectable according to our results. Microscopic analyses of particles on the Nuclepore surface are in agreement with theory, predicting particles mainly trapped by interception at the pore edges and the dipolar force sticking them. Results plotted in Figs 4 6 are important because they show the magnitude of Coulomb and dipolar forces. They also show that dipolar force greatly exceeds the Coulomb force near the pore edge (x = r v, y = 0). For example, for a particle situated on the ring plane and 5 x 10- v m (0.5 #m) from the pore edge the Coulomb force is 6.3 x 10-18 kg m s- 2, while the dipolar force is t.0 x 10-15 kg m s- 2. Inclusion of dipolar effects is fundamental in order to describe the particle collection mechanism on dielectric membrane filters. This can solve the discrepancies between the application of the Zebel model--considering only the Coulomb force--and the experimental measurements of collection efficiency (Fan et al., 1978). According to the considerations presented in this work, it is reasonable to consider a spatial distribution of charge with the highest concentration near the edge of the pore. Consequently, the equipotential and electric field lines around the pore top calculated on the base of the "charged ring model" (Fig. 3) should be very close to the real distribution of charge. The inclusion of other features, as rebound of particles, Van der Waals and diffusion forces, particle size and composition; is advisable to have accuracy in the solution of the trapping mechanism problem.
Filtration by dielectric porous membrane
757
Acknowledgements--Discussions with Dr Luis Moraga and other colleagues were very useful to clarify the concepts. We thank the University of Chile for supporting this publication.
REFERENCES Abramowitz, M. and Stegun, I. A. (eds) (1965) Handbook of Mathematical Functions, pp. 610-611. Dover Publications, New York. Burtscher, H., Reis, A. and Shmidt-Ott, A. (1986) Particle charge in combustion aerosols. J. Aerosol Sci. 17, 4%51. Claro, F. and Brovers, F. (1989) Dielectric anomaly of porous media: the role of multipolar interactions. Phys. Rev. B40, 3261-3265. Engelbrecht, D. R., Cahill, T. A. and Feeney, P. J. (1980) Electrostatical effects on gravimetric analysis of membrane filters. J. Air Pollut. Control Ass. 30, 391-392. Fan, K. C., Leaseburge, C., Hyun, Y. and Gentry, J. (1978) Clogging in Nuclepore filters: cap formation model. Atmos. Envir. 12, 1797-1802. Feeney, P., Cahill, T., Olivera, J. and Guidara, R. (1984) Gravimetric determination of mass on lightly membrane filters. J. Air Pollut. Control Ass. 31,376-378. Fjeld, R. A., Gauntt, R. O. and McFarland, A. R. (1983) Continuum field-diffusion theory for bipolar charging of aerosols. J. Aerosol Sci. 14, 541-556. Friedlander, S. K. (1977) Smoke, Dust and Haze. Wiley, New York. Fuchs, R. and Claro, F. (1989) Spectral representation for polarizability of a collection of dielectric spheres. Phys. Rev. B39, 3875-3878. Gentry, J. W., Spumy, K. R., Boose, C. and Scbrrrnann, J. (1985) Electrical enhancement of filtration in Nuclepore filters--I. Experimental design and measurement with spherical particles. J. Aerosol Sci. 16, 379-389. Gradshteyn, I. S. and Ryzhik, I. M. (1965) Tables of Integrals, Series and Products, pp. 154 and 150-153. Academic Press, New York. Heidam, N. (1981) Review: aerosol fractionation by sequential filtration with Nuclepore filters. Atmos. Envir. 15, 891-904. Jackson, J. D. (1975) Classical Electrodynamics, 2nd Edition. Wiley, New York. Kogan, V., Kuhlman, M. R., Coutant, R. W. and Lewis, R. G. (1993) Aerosol filtration by sorbent beds. JAPCA J. Air Waste Manage. Ass. 43, 1367-1373. Krishna-Moorthy, K., Nair, P. B. and Krishna-Moorthy, B. N. (1988) A study on aerosol optical depth at a coastal station, Trivandrum. Indian J. Radio and Space Phys. 17, 16-22. Meier, F. and Wyder, P. (1972) Polarizability of small metallic particles in a quasi-static electric field. Phys. Lett. 39A, 51-52. Pich, J. (1966) Theory of aerosol filtration by fibrous and membrane filters. In Aerosol Science (Edited by Davies, C. N.). Academic Press, New York. Romo-Krrger, C. M. (1990) Time-dependent charge in a dielectric membrane filter. J. Electrostatics 25, 145-154. Romo-Krrger, C. M. and Morales, J. R. (1988) Charge effect in Nuclepore filter. Phys. Scr. 37, 270-273. Turner, J. R., Fissan, H. J. and Liguras, D. K. (1988) Particle deposition form plane stagnation flow: competition between electrostatic and thermophoretic effects. J. Aerosol Sci. 19, 797-800. Zebel, G. (1974) A simple model for the calculation of particle trajectories approaching Nuclepore filter pores with allowance for electrical forces. J. Aerosol Sci. 5, 473-482.
S0021-8502(96)00006-7
J. Aerosol Sci. Vol.27, No. 5, pp. 751 757, 1996
Copyright ~3 1996ElsevierScienceLtd Printed in Great Britain. All rights reserved 0021-8502/96 $15.00 + 0.00
ELECTRICAL INTERACTIONS INVOLVED IN AEROSOL FILTRATION BY A DIELECTRIC POROUS MEMBRANE: THE CASE OF NUCLEPORE FILTER Carlos Manuel Romo-Krrger* and Vicente Diaz t * Departamento de Fisica, Facultad de Ciencias, Universidad de Chile, Casilla 653, Santiago, Chile t Departamento de Fisica Universidad Iberoamericana de Ciencia y Tecnologia, Santiago, Chile (First received 1 June 1995; and in final form 1 December 1995)
Abstract--Filtration by a dielectric medium has associated important electrostatic effects, since during the operations (the filtration process, manipulations, etc.) this medium accumulates considerable amounts of charge. In a filter-like Nuclepore, the charge is concentrated mainly at the rims of the pores. The electric field and its derivatives are analytically obtained and numerically evaluated for such a configuration of charge, to determine Coulomb and dipolar forces acting on a particle, supposed to be charged and polarizable. The important finding here is: in the vicinity of the pore rim the dipolar force is very high, exceeding the Coulomb force. This points to the importance of including polarization for a model of particle collection for that kind of filters. It could explain in part the observed accumulation of particles deposited in the pore vicinity in normal filtration situations. Copyright © 1996 Elsevier Science Ltd. 1. I N T R O D U C T I O N
This work intends to give attention to a problem that emerges when particles are being separated from a gas forced to pass through a porous foil or membrane filter. Any filter composed of dielectric material presents a strong tendency to acquire electrical charge, by friction during air passage and manipulations (tribo-electricity). Charges in the order of nano-Coulombs may be found in Nuclepore filters during this process (Zebel, 1974; Romo-KriSger, 1990). Such charge has a significant effect on particles of microscopic sizes, producing an interesting physical situation when they are near the pore entrance. Particle deposition on charged bodies is a very complicated problem (Pich, 1966) and in many cases ignored its importance on particle capture (Heidam, 1981). On the contrary, by the light of measurements, we think that charge effects are important for the particle trapping mechanism. Since the 1970s Nuclepore filter has a considerable use for atmospheric aerosol studies. It is specially well suited for subsequent analyses, like PIXE and XRF high-resolution spectroscopy, and light and electron microscopy. The Nuclepore is a dielectric foil of about 10/~m thickness with straight holes and very well-defined structures (Fig. 1). The tendency of this filter to acquire electrostatic charge" has being studied (Engelbrecht et al., 1980; Feeney et al., 1984; Romo-Kr/Sger, 1990). Also, good intents to describe particle collection in relation to electrical interactions have been done (Zebel, 1974; Fan et al., 1978; Gentry et al., 1985; Romo-Kr~Sger and Morales, 1988). Zebel (1974) has studied in detail how the retention ability of the Nuclepore filter is influenced by the Coulomb interactions on particles. He worked with an equation of motion for a particle that included the drag force due to the gas flowing through pores and a Coulomb force between the particle and the filter. Later, we proposed an equation including a dipolar term (Romo-Krrger and Morales, 1988), accounting for the interaction between a polarizable particle and the non-uniform field generated by the charged filter. The high polarizability observed in aerosol particles---evidenced by the absorption of visible light (Friedlander, 1977; Krishna-Moorthy et al., 1988)---argues in favor to include the polarization term in any study of filter-particle interaction. The dipolar force is FD = I(p.V)EI, 751
(1)
752
C. M.
Romo-Kr/Sgerand V. Diaz
Fig. 1. Scheme of the cross section of a Nuclepore membrane filter.
where p is the dipole moment of the particle and E is the electric field generated by the charged filter. We evaluate FD and the Coulomb force Fc = [qE], to compare them. q is the particle charge. For such evaluation, typical parameters relevant to aerosols must be used. In this case, isotropic particles may be considered (Friedlander, 1977). The dipole moment is expressed in terms of the electric field E and a scalar polarizability :~. That is p = ~E.
(2)
FD = e(E. V)E.
(3)
The dipolar force is, therefore,
The polarizability of a spherical particle of radius r acting as a dipole, given in terms of its dielectric constant c in appropriate units, is (Jackson, 1975; Friedlander, 1977; Fuchs and Claro, 1989) ( ~ - 1)r 3 -
-
-
(~: + 2)
(4)
Friedlander (1977), in the subject "Field charging", presents a formula to evaluate the charge of saturation of a particle when an electrical field is present. Further, this formula is evaluated for parameters approximating those conditions when real particles are collected in electrical precipitators. This leads to the value e = 58, for the dielectric constant of the particle. This is an evaluation using experimental conditions for a real aerosol, thus includes polarization of particle material and the polarization due to the surface conductivity of the particle. It is concordant to the results of K o g a n et al. (1993) on the efficiency of electrostatic deposition of aerosol particles on sorbent beds, when we assume beds composed of 6 x 10- -~m diameter fibers with a charge density of 3.3 A sm - 3. The value e = 58 leads to the value 0.95 for the factor (~:- 1)/(e. + 2) used in equation (4). The same is 1.0 for a conducting particle, 0.97 for water spheres and 0.3 for a dielectric sphere composed of benzene. Water on the surface of particles and inclusions of conducting materials are factors that increase the dielectric constant of the particles (Mier and Wyder, 1972; KrishnaMoorthy, 1988; Claro and Brovers, 1989). The dielectric properties of small particles have been studied mainly in relation to oscillatory electric fields (scattering of electromagnetic radiation, etc.). In this case we must use these properties to determine the interactions between a polarizable particle in a static electric field. 2. M A T H E M A T I C A L F O R M A L I S M Jackson (1975) evaluates fields and charge densities near corners a':d edges of conductor bodies. In particular, a body with an edge formed with the intersection of two planes at right angle has a charge density proportional to 6 1/3 (being 6 the distance from a surface point to the edge). The configuration of charge has a singularity as 6 ~ 0. It implies that the field strength becomes very large at the edge. The very concentrated charge at the edge is a known effect named "point effect". At small scale, the edge of a filter pore is formed by the intersection of two planes at right angle. The first plane is the filter face. The second is a plane tangent to the cylindrical surface of the pore. If we consider the filter as a "conductor" body, is plausible to expect the above charge distribution verified near the pore edge.
Filtration by dielectric porous membrane
753
Migration of charge in Nuclepore filters has been observed (Romo-Kr/Sger, 1990) with typical times for filter charge decay on the order of 100 min. That migration of charge induces us to think the filter charge concentrated mainly at the edges of the pores, by point effect (within the standard operation times of the Nuclepore filter, normally of 24 h). The same consideration has been done by other authors (Zebel, 1974; Fan et al., 1978). Then, we calculate the electric field E as the derivative of a potential arising from a distribution of charge consisting in the uniformly charged ring at the entrance of the pore, with total charge Q. The complete form of the potential is obtained by integrating each charge element dQ, in terms of the pore diameter rp and the coordinates (x, y) of a general point of the space (Fig. 2). In virtue of the cylindrical symmetry, the potential q~ at (x, y) is the same for equivalent points in planes rotated around the pore axis. This reduces the problem to one in two dimensions, with the Cartesian coordinates x and y. Thus,
~(X, y) = Oprp
f
21t (r 2
_[_ X 2 _.[_ y 2
dO.
2xrp c o s 0 ) - 1 / 2
--
o
(5)
Using the symmetry ~b(x, y) = ~b( - x, y) and an integral equivalence (Gradshteyn and Ryzhik, 1965), this may be written as ~?(X, y) = 2crprlp/2X - 1/2 k K (k).
(6)
where crv = Q/2rcrp is the linear charge density of the ring and K(k) is the complete elliptical integral of the first kind (Abramowitz and Stegun, 1965) with the argument
I k_=2 i
rpX p+
11/2 +y23
(7)
.
We find the derivatives of ~b as
a,~
2~.r~/~
(8)
and q~,
02~b 2apt 1/2 [l+k 2 = @k2 - x 1 / 2 k ( l _ k 2 ) ~ E ( k ) - K ( k )
] .
(9)
E(k) is the complete elliptical integral of the second kind (Abramowitz and Stegun, 1965) with the same argument k.
~ Y
°o)
)X
Z Fig. 2. The charged ring at the pore edge centered at the origin of the xyz frame, for calculation of the electrical potential ~b at a general point P on the plane xy.
754
c . M . Romo-Kr6ger and V. Diaz
W i t h these e x p r e s s i o n s the c o m p o n e n t s of the electric field a n d its d e r i v a t i v e s are:
O'~k'~ + ~ ,
a-~
Ex-
E~,-
-
0),
+~k;,
#E x _ EXX -- ( ' ~ X
ff)k'(k'x)2
?Ex Exy = Eyx - -i?y Eyy-
~Ey ~ -
(~'kk '~ +
-
-
-,;
3 d?
--
(lO)
(o'kk'jx -- I 4X 2'
dp'kk ;, _ ,~, k" c~}~k;, + --27-x V'k xy,
~b/(k;,) 2 - qS~k;:y,
w i t h the explicit forms:
G_ & t
_ _
k~, k;~x
[1
~x - k
2x
(?k ?y
yk 12'
?2 ( ? .k~ -
(rp + x)] 12
k I
(rP + gx) + 3 (rp +14 x)2 xl 2
•
(O2k kJy = k'(~ - (9x ~), .. -
~2
k;; - Cy 2
'
YkL
F3(rp + x)
-
_
k[3y2
1]
L I~
1"-
-la
1 4x 2
,
1 | 2xl 2
]
'
l = E(rp + x) 2 -[- y2]1/2.
Y
)i I
Fig. 3. The equipotential lines (weak traces) and the electric field lines (hard traces) for a charge distributed over the pore rim. The top of the pore and the x, y coordinates frame are also drawn.
(11)
Filtration by dielectric porous membrane
755
In Fig. 3 are the e q u i p o t e n t i a l a n d the electric field lines a r o u n d the a n n u l a r d i s t r i b u t i o n of c h a r g e at the p o r e top, they were n u m e r i c a l l y d e t e r m i n e d using the f o r m e r formulas. W i t h the s a m e f o r m a l i s m the m a g n i t u d e of the C o u l o m b force F c = IqE[ a n d the d i p o l a r force FD = I(P" V)EI were n u m e r i c a l l y e v a l u a t e d for different p o i n t s n e a r the p o r e top. Results are p l o t t e d in F i g s 4-6. F o r these e v a l u a t i o n s we used the t a b u l a t e d values ( A b r a m o w i t z a n d Stegun, 1965) for the elliptical integrals K a n d E. O t h e r p a r a m e t e r s were t a k e n from o t h e r a e r o s o l studies (Zebel, 1974; F r i e d l a n d e r , 1977; Romo-Kr~Sger a n d M o r a l e s , 1988). These p h y s i c a l p a r a m e t e r s were: r = 5.0× 1 0 - V m , rp = 2.0× 1 0 - 6 m , ap = 1.27 x 10 - 7 A s m - 1 a n d e = 58. F o r the c a l c u l a t i o n s of F c a n d F D in Figs 4 - 6 we used charges for the p a r t i c l e e q u a l to l e - , 10e- a n d 100e-. e - is the c h a r g e of one electron ( e - = - 1.6 × 10-19 A s). FD results i n d e p e n d e n t of the value of particle charge. F r o m these figures we see t h a t FD has a very increasing b e h a v i o r n e a r the p o r e edge (x = rp, y = 0). At this s u r r o u n d i n g s the d i p o l a r force F o exceeds in extent the C o u l o m b force F c , even for the high value o f p a r t i c l e c h a r g e q = 100e-.
i
i
i
i
-15
N E
I
i
' } '
~
~c3}F°
-20
-25 °
0.0
014
012
0.6
1.0 '
018
x/rp
Fig. 4. Dipolar and Coulomb force intensities evaluated along the x axis (y = 0). Coulomb forces Fcl, Fc2 and Fc3 a r e for particle charges ql = le-, q2 = 10e- and q3 = 100e-, respectively (e- = 1.60 x 10-19 A s is the charge of one electron). The dipolar force FD results independent of the net charge on particle.
-3
~-" I
-6
~D
E 6, 'w" ¢.. m
-9
-12
la. ¢-
Fc -15
Fo I
I
I
I
I
I
I
o.o
0.5
,.o
1.5
2.o
2.5
3.0
y/rp Fig. 5. D i p o l a r f o r c e FD a n d C o u l o m b f o r c e F c intensities as f u n c t i o n s o f the d i s t a n c e f r o m the p o r e
border along a vertical axis (x = rp). Particle charge is q = 100e (e- = 1.60 x 10-19 A s is the charge of one electron).
756
C . M . R o m o - K r S g e r and V. Diaz
I.:37
1.68
0.00 0.2 0.4
-24.7
•
-
" ~nF ~tn K9 "re's-2)
Fig. 6. Each curve is the dipolar force FD as functions of the distance from the pore axis along different horizontal lines (y/r = 0.00, y/r = 0.34, y/r = 0.68, y/r = 1.02, y/r = 1.37, y/r - 1.71).
The values used in our calculations for the particle charge are in concordance with those given in the literature. Turner et al. (1988) predicts the value q = 2.9e- for a monodisperse aerosol particle. For particles exposed to bipolar ions Turner et al. (1988) predicts q ~ 100e- and Fjeld et al. (1983) predicts q ~ 1000e-. According to Burtscher et al. (1986) in combustion processes a particle can reach the value q = 10e .
3. C O N C L U S I O N S The charge on filters and particles may influence the process of filtration in two ways (Pich, 1966): (1) the particles may be attracted (or repelled) to the filter surface from a greater distance; (2) the charge makes the particle "stick" to the filter surface. For the first may be more important the Coulombic force, because it is of longer range than the dipolar force. For the effect of sticking on surfaces, in turn, the dipolar interactions play a principal role. Particles that fall across the filter surface by inertial impaction or by interception (or by influence of the Coulomb force), may be strongly fixed near the pore edges due to the intensive dipolar force there. Dipolar force contributes to avoid the detachment of particles. These conclusions are expectable according to our results. Microscopic analyses of particles on the Nuclepore surface are in agreement with theory, predicting particles mainly trapped by interception at the pore edges and the dipolar force sticking them. Results plotted in Figs 4 6 are important because they show the magnitude of Coulomb and dipolar forces. They also show that dipolar force greatly exceeds the Coulomb force near the pore edge (x = r v, y = 0). For example, for a particle situated on the ring plane and 5 x 10- v m (0.5 #m) from the pore edge the Coulomb force is 6.3 x 10-18 kg m s- 2, while the dipolar force is t.0 x 10-15 kg m s- 2. Inclusion of dipolar effects is fundamental in order to describe the particle collection mechanism on dielectric membrane filters. This can solve the discrepancies between the application of the Zebel model--considering only the Coulomb force--and the experimental measurements of collection efficiency (Fan et al., 1978). According to the considerations presented in this work, it is reasonable to consider a spatial distribution of charge with the highest concentration near the edge of the pore. Consequently, the equipotential and electric field lines around the pore top calculated on the base of the "charged ring model" (Fig. 3) should be very close to the real distribution of charge. The inclusion of other features, as rebound of particles, Van der Waals and diffusion forces, particle size and composition; is advisable to have accuracy in the solution of the trapping mechanism problem.
Filtration by dielectric porous membrane
757
Acknowledgements--Discussions with Dr Luis Moraga and other colleagues were very useful to clarify the concepts. We thank the University of Chile for supporting this publication.
REFERENCES Abramowitz, M. and Stegun, I. A. (eds) (1965) Handbook of Mathematical Functions, pp. 610-611. Dover Publications, New York. Burtscher, H., Reis, A. and Shmidt-Ott, A. (1986) Particle charge in combustion aerosols. J. Aerosol Sci. 17, 4%51. Claro, F. and Brovers, F. (1989) Dielectric anomaly of porous media: the role of multipolar interactions. Phys. Rev. B40, 3261-3265. Engelbrecht, D. R., Cahill, T. A. and Feeney, P. J. (1980) Electrostatical effects on gravimetric analysis of membrane filters. J. Air Pollut. Control Ass. 30, 391-392. Fan, K. C., Leaseburge, C., Hyun, Y. and Gentry, J. (1978) Clogging in Nuclepore filters: cap formation model. Atmos. Envir. 12, 1797-1802. Feeney, P., Cahill, T., Olivera, J. and Guidara, R. (1984) Gravimetric determination of mass on lightly membrane filters. J. Air Pollut. Control Ass. 31,376-378. Fjeld, R. A., Gauntt, R. O. and McFarland, A. R. (1983) Continuum field-diffusion theory for bipolar charging of aerosols. J. Aerosol Sci. 14, 541-556. Friedlander, S. K. (1977) Smoke, Dust and Haze. Wiley, New York. Fuchs, R. and Claro, F. (1989) Spectral representation for polarizability of a collection of dielectric spheres. Phys. Rev. B39, 3875-3878. Gentry, J. W., Spumy, K. R., Boose, C. and Scbrrrnann, J. (1985) Electrical enhancement of filtration in Nuclepore filters--I. Experimental design and measurement with spherical particles. J. Aerosol Sci. 16, 379-389. Gradshteyn, I. S. and Ryzhik, I. M. (1965) Tables of Integrals, Series and Products, pp. 154 and 150-153. Academic Press, New York. Heidam, N. (1981) Review: aerosol fractionation by sequential filtration with Nuclepore filters. Atmos. Envir. 15, 891-904. Jackson, J. D. (1975) Classical Electrodynamics, 2nd Edition. Wiley, New York. Kogan, V., Kuhlman, M. R., Coutant, R. W. and Lewis, R. G. (1993) Aerosol filtration by sorbent beds. JAPCA J. Air Waste Manage. Ass. 43, 1367-1373. Krishna-Moorthy, K., Nair, P. B. and Krishna-Moorthy, B. N. (1988) A study on aerosol optical depth at a coastal station, Trivandrum. Indian J. Radio and Space Phys. 17, 16-22. Meier, F. and Wyder, P. (1972) Polarizability of small metallic particles in a quasi-static electric field. Phys. Lett. 39A, 51-52. Pich, J. (1966) Theory of aerosol filtration by fibrous and membrane filters. In Aerosol Science (Edited by Davies, C. N.). Academic Press, New York. Romo-Krrger, C. M. (1990) Time-dependent charge in a dielectric membrane filter. J. Electrostatics 25, 145-154. Romo-Krrger, C. M. and Morales, J. R. (1988) Charge effect in Nuclepore filter. Phys. Scr. 37, 270-273. Turner, J. R., Fissan, H. J. and Liguras, D. K. (1988) Particle deposition form plane stagnation flow: competition between electrostatic and thermophoretic effects. J. Aerosol Sci. 19, 797-800. Zebel, G. (1974) A simple model for the calculation of particle trajectories approaching Nuclepore filter pores with allowance for electrical forces. J. Aerosol Sci. 5, 473-482.