A Theoretical Study of the Effect of Electric Charges on the
Efficiency with Which Aerosol Particles are
Collected by Ice Crystal Plates
J. J. MARTIN, P. K. WANG, AND H. R. PRUPPACHER
Department of Atmospheric Sciences, University of California, Los Angeles, California 90024
Received November 19, 1979; accepted January 29, 1980
Two theoretical models are presented which allow computing the efficiency with which electrically
charged aerosol particles of radius 0.001 <-- r -< 10 t~m are collected by electrically charged platelike ice crystals of radius 50 -< ac -< 640/xm in air of various relative humidities and of 700 mb
and - 10°C. Particle capture due to thermophoresis, diffusiophoresis, Brownian diffusion, inertial
impaction, and electrostatic forces is considered. It is shown that Brownian diffusion dominates
the capture process by ice crystals if r -< 0.01 /zm, while inertial impaction controls the capture
process if r > 0.1 /zm. For aerosol particles of 0.01 -< r -< 0.1 /zm, the collection efficiency is
controlled by phoretic forces. Electrical forces significantly affect the collection of aerosol particles in the size range of 0.01 -< r -< 5 p~m. Trajectory analysis demonstrates that electrically
charged and uncharged aerosol particles are preferentially captured at the rim of plate-like
ice crystals. Electrically neutral ice crystals of NRe < 50 capture particles only on the underside
of the ice crystal. Ice crystals which are electrically charged collect aerosol particles by rear
capture ifNR~ ~< 0.5.
and ice crystal sizes, both changing in time.
This makes it very difficult to extract from
the observed data collection efficiencies
which apply to a certain ice crystal and
aerosol particle size. In addition, the relative humidities, and thus the phoretic forces
usually vary greatly during a precipitation event.
Laboratory studies, although capable of
determining the efficiency with which ice
crystals of a specific size collect aerosol
particles of a specific size under controlled
conditions, have the drawback that they are
usually confined to sea level pressure, while
the scavenging of aerosol particles by ice
crystals proceeds at all pressure levels
within the troposphere. In addition, the few
experimental studies available (9-11) considered ice crystals and aerosol particles of
a very limited size range, i.e., only ice
crystals of radii ac -> 1 mm, and aerosol
particles of radii 0.5 -< r --- 7 /zm were
considered.
1. INTRODUCTION
Field studies (1-8), as well as the laboratory studies (9-11), strongly suggest that
snow crystals, in particular those of the
planar type (hexagonal plate, hexagonal
dendrites, etc.), play an important role in the
scavenging of aerosol particles from the
atmosphere.
Unfortunately, the field studies mentioned do not allow the derivation of reliable
quantitative data on particle scavenging
by ice crystals. Field studies are plagued by
serious shortcomings which stem from the
fact that they are based on observed changes
in aerosol particle concentrations measured
during a precipitation event. Such particle
concentration changes may be the result of
scavenging as well as the result of air mass
changes which often accompany a precipitation event. In addition, the collision efficiencies derived from field studies usually
apply to a wide spectrum of aerosol particle
44
0021-9797/80/110044-13502.00/0
Copyright © 1980 by Academic Press, Inc.
All rights of reproduction in any form reserved.
Journal of Colloid and Interface Science, Vo|. 78, No. 1, November 1980
AEROSOL
Theoretical studies avoid the limitations
and drawbacks of both the field and laboratory investigations. However, thus far only
one such theoretical study is available (12).
Unfortunately this investigation covers only
a limited aerosol particle size range (0.5
-< r-< 20 /xm), and does not include the
effects of thermophoretic and diffusio~
phoretic forces.
In recognition of the mentioned shortcomings of present studies on the scavenging of aerosol particles by ice crystals, we
formulated a theoretical model capable of
computing the efficiency with which electrically charged aerosol particles of a wide
range of sizes are collected by electrically
charged plate-like ice crystals of a wide
range of sizes in air of various pressure,
temperature, and humidity.
Two models, complementary to each
other, have been used in the present study.
These models have been described in great
detail (13). Certain aspects of these models
and some of the results derived from them
have also been discussed (14). In the present
article we shall describe extensions of these
models to include the effect of electric
charges on the scavenging of aerosol particles by ice crystal plates.
a. The Trajectory Model
The first model (henceforth called Model-
I) computes the efficiency with which
=
--
m
dv
. . . . rag*
dt
67r~ar
(1 + o~NKn)
0.74DwMa VPv
,
(1 + aNKn)Mwp~
[3]
where ka and kp are the thermal conductivities of air and the aerosol particle, respectively, Dva is the diffusivity of water
vapor in air, T is absolute temperature, Pv
+Fwh +FDf + F e .
[1]
Equation [1] applies to the motion of an
aerosol particle of radius r, mass m, and
velocity v moving around an ice crystal of
radius ac, both falling in air of dynamic viscosity ~a under the effect of gravity, hydrodynamic forces, phoretic forces, and electric forces. In Eq. [1] g* = g(pp - Pa)/Pa,
g is the acceleration of gravity, pp is the
bulk density of the aerosol particle, Pa is
the density of air, NKn = hJr is the
Knudsen number, ha is the free path length
of air molecules, u is the velocity field
around the falling collector, a = 1.25
+ 0.44 e x p ( - 1.10NKn-I), FTh is the thermophoretic force given by the relations (see 15)
127r~/ar(ka + 2.5 k, NKn)ka VT
5(1 + 3NKn)(kv + 2ka + 5koNKn) p
and FDf is the diffusiophoretic force given
by the relation (see 15)
For = -6~'r~)a
aerosol particles of 0.1 - r -< 10 /zm are
collected by simple ice crystal plates due to
phoretic forces, inertial forces, and electric
forces caused by electric charges of opposite signs residing on the ice crystals and
aerosol particles. The model, however,
neglects the effects due to Brownian diffusion. The efficiency was computed from
an analysis of the trajectory of the aerosol
particles moving past the ice crystal. Assuming that the flow around the aerosol
particle does not affect the crystal motion
(which is justified considering the smallness
of the aerosol particle in comparison to the
size of the ice crystal) an aerosol particle
trajectory was determined from the equation
× (v-u)
2. PRESENT MODELS
FTh
45
PARTICLES
[2]
is the water vapor density in air, and Mw
and Ma are the molecular weights of water
and air. Equations [2] and [3] were evaluated
from a knowledge of the vapor density distribution and of the temperature distribution
around the falling crystal. These were determined previously (16) from a numerical
evaluation of the steady-state convective
diffusion equation applied to an ice crystal
Journal o f Colloid and Interface Science, Vol. 78, No. 1, November 1980
46
MARTIN, WANG, AND PRUPPACHER
plate idealized by a thin oblate spheroid of
ice of axis ratio b/a~ = 0.05 (b is the minor
semiaxis and ae is the major semiaxis of the
oblate spheroid). The velocity fields u
around falling ice crystals, necessary for
solving both the convective diffusion equation and Eq. [1], were those determined by
Pitter et al. (17) (for Nrte = 2acV=/va <- 20),
and by us (for NR~ = 50), from a numerical
solution of the complete N a v i e r - S t o k e s
equation of motion for steady-state, incompressible flow, where Vo~ is the terminal
velocity of the ice crystal and Va is the kinematic viscosity of air.
The electric force F~ in Eq. [1] was assumed to be determined by the electric
charges residing at the surface of the ice
crystal and aerosol particles. From electrostatic theory, the force on a charged particle immersed in an electric field, E, is
F~ = - Q p E ,
the electric potential satisfies Laplace's
equation
V2qb~ = O.
[61
In oblate spheroidal coordinates, Eq.
[6] has the solution
~e = c~ sin -1 [tanh ~] + c2.
The constants c~ and c2 may be obtained
from the boundary conditions ~: = ~:o, qbe
= qb~.o, and ~: = ~:o~,qbe = 0, leading to
c~ = -q)e,o
- sin -1 (AR)
c2 = qbe,o
- sin -1 (AR
a&o = Qo/C
E =
,
[9]
[10]
ae[1 - (AR)2p/2
c
=
sin -~ {[1 - (AR)2] 1/2}
[111
Therefore, the electric potential on any ~=surface is
7r
-
Qa
Oe - - C
sin -1 [tanh ~]
2
7r
-
[12]
- sin -1 (AR)
2
The electric field strength E = -Vqbe in
oblate spheroidal coordinates is therefore 1
Q a sech
ac s e c h , 0 C [ 2 -
[8]
where according to Pruppacher and Klett
(15) the capacitance C for a thin oblate
spheroid is
-
where qbe is the electric potential around
the crystal. Taking the potential at infinity
to be zero, and assuming no net space charge,
11
where qbe.o is the electric potential at the
surface of the oblate ice spheroid. F r o m
electrostatic theory
[4]
where Qp is the charge on the particle. In
the absence of an external electric field, the
electric fields affecting the interacting
bodies result from the charges on their surfaces. In the present computations it is
assumed that the effect of the electric field
due to the electrically charged aerosol particle has negligible effect on the motion of
the ice crystal. The electric field around the
electrically charged ice crystal satisfies the
condition
E -- -Vovo,
[51
[7]
[13]
sin-1 (aR)][sinh2 ~: + cos2 ~]
Thus, from Eqs. [4] and [11]
QpQa sech ~: sin -1 {[1 - (AR)] ~/2}
Fe
~
[14]
--
a~ sech £ 0 [ 1 - ( A R ) 2 ] l / 2 [ 2 - s i n - l ( A R ) ] [ s i n h 2 £
F r o m a knowledge of the particle trajectory
determined from an evaluation of Eq. [1],
Journal of Colloid and Interface Science, Vol. 78, No. 1, November 1980
_]_ C O S 2 ,1~]1/2
1 It must be mentioned here that the well-known
electric potential (18) ~e = tan-~ (1/~:)/tan-~(1/~0)
AEROSOL PARTICLES
subject to Eqs. [2], [3], and [14], the collision efficiency
E -
7r(ac + r) 2
[15]
was deduced, where y~ is the largest horizontal offset the particle can have and still
collide with the crystal (y~ being m e a s u r e d
perpendicular to the crystal axis aligned
along g and sufficiently far u p s t r e a m f r o m
the crystal. The collision kernel was computed from Eq. [15] using the relation
K = ETr(ac + r)2(V~,a - V ~ ) .
[16]
b. The Flux Model
Model-I discussed in Section 2a obviously
does not apply to particles of r < 0.1 /xm
since the model does not include Brownian
diffusion. In order to determine the efficiency with which electrically charged aerosol particles of r < 0.1 ~ m are captured by
electrically charged ice crystals, we developed Model-II in analogy to the model
formulated (20, 21) for drops. In this model,
the collision efficiency is found from a description of the flux of aerosol particles to
the collector ice crystal.
The total particle current densityjp to the
crystal can be written as jp = nvp = nvdrift
-DpVn, where Vdrift ~---BpFext, n is the
n u m b e r concentration of aerosol particles,
Vdrift is their drift velocity, Dp is their diffusivity, Bp = (1 + aNK.)/67rrna is their
mobility, and Fext is the sum of the external
forces acting on the particle.
F o r particle moving toward an ice crystal
pertains to a coordinate system in which the surface
of an oblate spheroid is given by s¢ = constant. In
contrast, in the coordinate system of Happel and
Brenner (19), followed in this study, the surface of
an oblate spheroid is given by sinh s¢ = const. Thus,
the ~: in the above equation is not equivalent to the
~:-coordinate of the present work. An unfortunate
error due to mixing up the two coordinate systems
entered the work of Pitter (12) whose expression for
E is incorrect, which can be easily verified noting that
V. E is required to be zero which is not the case in the
expression of Pitter (12).
47
at rest as a result of Brownian diffusion,
thermophoresis, diffusiophoresis, gravitational forces, and electric forces, one finds
j , = nBp(mg* + Fxh + FDf + Fe)
- DpVn.
[17]
We shall a s s u m e now steady state, constant
diffusivity, and further assume that the
aerosol particles are small so that the term
mg* m a y be neglected in c o m p a r i s o n to
FTh, Fvf, and DpVn (a numerical evaluation
of these forces shows that for typical conditions the contribution of mg* to the collision kernel is by several orders of magnitude smaller than the contribution of the
phoretic forces and Brownian diffusion).
The condition of particle continuity (On/Ot)
+ V.(nvp) = 0 leads then to V.j = 0, from
which for an ice crystal at rest
Bp(FTh + FDf + Fe)'Vn
-- DvV2n = 0.
[18]
In order to arrive at Eq. [18] we assume that
V'[n(Fwh + FDf + Fe)]
= (FTh + FDf + Fe)'Vn
[19]
which in turn requires that
V'(FTh + FDf + Fe) = 0.
[20]
This is justified for a stationary ice crystal
whose t e m p e r a t u r e , vapor, density, and
electrical potential fields are described by
the solution of the LaPlace equations
V2T = 0
[21a]
VZpv = 0
[21b]
V ~ e = 0.
[21c]
Since from Eqs. [2], [3], and [5], FTh a: VT,
FDf oc Vpv, and F, cc V6, it follows that Eq.
[20] fulfills Eqs. [21a], [21b], and [21c].
H o w e v e r , for an ice crystal falling at
terminal velocity the t e m p e r a t u r e and v a p o r
density distribution, and hence V T and Vpv,
cannot be determined from Eqs. [21] but
must be determined f r o m solutions to the
Journal of Colloid and Interface Science, Vol. 78, No. 1, November 1980
48
MARTIN, WANG, AND PRUPPACHER
convective diffusion equation
u ' V T = KaV2T
[22a]
U ' ~ 7 p v = Dva~72pv
[22b]
where u is the velocity field in air around
the crystal and Ka is the heat diffusivity in
air. From this, in turn, the enhancement of
the heat and vapor flux due to the presence
of an air flow past the crystal can be found.
This enhancement can be expressed in
terms of an angular dependent ventilation
coefficient fv(O) and fh(O), fp(O), for water
vapor, heat, and particle transport, respectively (where 0 is the angle measured from
the forward stagnation point, of the crystal),
or in terms of an overall ventillation coefficientfv ,fh, andfp for vapor, heat, and particle transport, respectively. Values for the
overall ventillation coefficient of ice crystals
have been derived (16, 22). In the present
problem we are considering the overall flux
of particles to a falling crystal. We are
therefore interested only in the overall,
flow-field enhanced, flux of particles to the
crystal. Thus, neglecting the hydrodynamic
drag force on the aerosol particles (which
is very small indeed due to the small fall
velocity of the aerosol particle) we may
write Eq. [18] for a falling ice crystal as
V T = ( T = - Ts) sech ~[ 2 - sin-l (AR) ] -~
X [ac sech ~0(sinh 2 ~ + cos 2 ~1)1/2]-1ke [251
and
Vp~
= (Owe- Pvs)sech ~[ 2 - sin-1 (AR)] -1
B p ( f h F T h + fvFDf + F e ) ' V n
-fpDpVZn = 0.
[23]
In order to arrive at Eq. [23] we assumed,
instead of [20],
V. (fhFTh + fvFof + F~) = 0,
FTh ~ --
where fh and fv are mean quantities over
the body and only a function of NR~ and
Ns~,v = va/Dva (where va is the kinematic
viscosity of air), and NRe and Np~ = Va/K~,
respectively (see 15). Recall again that FTn
cc VT and Fdf cc Vpv, Eq. [24] fulfills Eqs.
[21a], [21b], and [21c].
In order to solve Eq. [23] it is necessary
to have determined F~,h and FDf for a stationary crystal. This requires knowledge of
the temperature and vapor density fields
around a stationary ice crystal. These fields
can be found from a solution of Eqs. [21a1
and [21b] formulated for the case of an ice
crystal plate whose shape is idealized by a
thin oblate spheroid of axis ratio 0.05.
Transforming these equations into oblate
spheroidal coordinates (~,ag) (see Pitter et
al., 17) and solving them subject to the conditions T(~: = ~:0)= Ts, p(~: = C0) = Ps, and
T(sc = ~:~) = T~, p(sc = ~:~) = p~ one finds
[24]
× [ac sech ¢0(sinh 2 ¢ + cos 2 ~1)~12]-1~. [26]
With these solutions the phoretic forces for
a stationary ice crystal can now be expressed as
127r'Oar(ka + 2.5kpNKn)ka(T~ - Ts)G ^
ee
5(1 + 3NKn)(kp + 2ka + 5koNKn)p
FDf = --67rr~%
0.74DvaMa(Pw - Pvs)G ^
ee
(1 + aNx.)Mw p~
[27]
[28]
where
G = sech ~
For a moving ice crystal we shall define
- sin -1 (AR)
x [at sech ~o(sinh2 ~ + cos 2 ~).2]-1
now
[291
with ~:o = tanh -1 (AR).
Journal of Colloid and Interface Science, Vol. 78, No. 1, November 1980
?hFTh = G C T h e ~
[30a]
fvFDf = G C D f O ~
[30b]
AEROSOL
49
PARTICLES
where 0f is the unit vector in sO-direction, and where
121r~/ar(k a + 2.5kpNKn)ka(T~ - Ts)fn
CTh
~
--
CDf
=
--6,n-r,0a
0.74DvaMa(Pv~ - Pvs)f~
where
Fext = (fhFTh
QpO~ sin -1 {[1 - (AR)2] v2}
exp
{[
-~c
- fpDr, V2n = 0
[35]
sin-'tanhsq[2-sin-l(AR)]
[36]
[371
-~}
-exp{[-flC
exp{[-
CGke
and considering Eq. [29].
Writing Eq. [35] in oblate spheroidal coordinates one finds for ~: = ~ , n = n~, and
for ~ = C0, n = 0, and using Eqs. [31], [321,
[341, [371, and [361
We shall now write Eq. [23] in the form
B F e x t • Vn
=
C = --(CTh + CDf + Ce),
[34]
(AR)2] 1/2
-
+ fvFDf + Fe)
with
~¢here G is given by Eq. [29], and where
ac[1
[32]
(1 + aNKn)Mwp~
Recalling Eq. [14] we may write the elecxic force now analogously to the phoretic
Forces as
Fe = GC~O~
[33]
C, =
[31]
5(1 + 3Nxn)(hp + 2ha + 5hpNxn)p
sin -t ( A R ) ] [ 2 -
sin l aR>l' }
[38]
sin 1 Z '1-1}
- exp {[-/3C sin-1 ( A R ) ] [ 2 -
s i n - l ( A R ) J -1}
vhere fl = Bp/Dpfp. Differentiation leads to
n~43C sech ~ : [ 2 -
sin-~ (AR) ] -~ e x p [ - f l C sin -1 (tanh ~:)]Oe
7/,/ = _
[391
ac sech s%(sinh 2 ~ + cos z n) v2 exp - -~- tiC
"he total flux of particles to the crystal is
JP = fs Dpfp(Vn)e=e° "dS.
[40]
~rom Eq. [40] we find for the collision kernel
T = Jp/n
_
4rrBpCa¢ sech ~:o
r
/ BpC ~
x /exp/-----=-_ / - 1
J
L
\ Dpfv /
[41]
- e x p [ - f l C sin -1 (AR)]
t
with so0 = tanh -1 (AR). Using Eq. [41] the
collision efficiency can be determined from
E =
K
~'(ac
+ r)2(V~,a
[42]
-- V ~ , r )
It is reasonable to assume that adhesive
forces ensure that an aerosol particle remains at the surface of an ice crystal once
it has collided with it. This assumption is
particularly justified at temperatures between 0 and -10°C at which water moleJournal of Colloid and Interface Science, Vol. 78, No. I, November 1980
50
MARTIN, WANG, AND PRUPPACHER
cules have an appreciable surface mobility
and behave as if part of a "pseudo-liquid"
layer (see 15). With this assumption, the
collision efficiency E calculated by the
above given procedure is then identical with
the collection efficiency, and the collision
kernel K is identical with the collection
kernel.
3. EVALUATION PROCEDURE
Models-I and -II were evaluated for platelike ice crystals idealized by oblate spheroids of ice with an axis ratio A R = b/ac
= 0.05 (where ac is the major and b is the
minor axis of the oblate spheroid, respectively). The aerosol particle radii considered
were 0.001 -< r -< 10 /xm. The ice crystal
plates had radii (i.e., semimajor axes) of
ac = 50.6, 87.9, 112.8, 146.8,213,289, 404,
and 639 txm, corresponding to Reynolds
numbers NRe (=2a~V~/v) 0.1, 0.5, 1.0, 2.0,
5.0, 10.0, 20, and 50, respectively, at 700 mb
and -10°C. In addition to this pressure
level, Models-I and -II were evaluated also
for the levels 1000 mb, 0°C; 900 mb, -5°C;
and 600 mb, -20°C. Due to the particular
choice of corresponding pressure and temperature, the Reynolds numbers corresponding to the above given crystal sizes
were, with sufficient accuracy, the same at
all pressure-temperature levels considered.
At each of these pressure levels we considered four relative humidities (RH)i (with
respect to ice), namely (RH)i = 100, 95,
751 and 50%. The values chosen for ka(p, T),
L~(p,T), Dva(p,T), and ~a(T) were those
recommended by Pruppacher and Klett
(15). The values for p~ were those given by
the Smithsonian Meteorological Tables.
The bulk densities of the particles considered were pp = 1.0, 1.5, 1.75, 2.0, and
5 g cm-L In the present computations we
also assumed that fh ~ f v ; w h e r e f~ is the
ventilation coefficient for mass transport.
In evaluating fp we assumed that its functional dependence on the Reynolds number and on the Schmidt number Nsc is
the same as that given by Hall and Pruppacher (22) for f~, except that now instead
Journal of Colloid and Interface Science, Vol. 78, No. 1, November 1980
ofNse,v = va/Dv,a we used Nsc,p = va/Dp~,
where Dp~ is the diffusivity of the aerosol
particles in air. Values for Dp,a and justifications for both of the above assumptions
are given by Pruppacher and Klett (15). The
thermal conductivity of the aerosol particle
material was assumed to be kp = 4.19
x 10-1 J cm -1 sec -1 °C -1. For evaluating
the phoretic forces (Eqs. [2] and [3]) a
uniform ice crystal temperature was assumed
considering the thinness of the ice crystals
assumed and considering the relatively high
heat conductivity of ice.
A literature search (23-27) reveals that
little is known about the surface charge, Qa,
on plate-like ice crystals. However, the
scant information available provided bounds
from which it was determined that the surface charge on plate-like ice crystals in strongly
electrified clouds may be represented by
IQal = Iqala~ = 2a~.
[43]
An analogous law was shown (15) to hold
for spherical particles. Thus we assumed
for stronglyelectrified clouds IQrl = [qrl
x r 2 = 2r z.
It also appears from the studies cited
above that plate-like ice crystals are predominantly negatively charged. In order to
test the effect of electric charges on the
scavenging of aerosol particles by ice crystals we considered strongly electrified
clouds, as well as weakly electrified clouds,
and therefore investigated the charge effect
for ]qa] = Iq~[ = 0 . 1 4 , 0.20, 0.40, 1.0,
1.4, and 2.0 esu cm -2, assuming that the
crystals were negatively charged and the
aerosol particles were positively charged
(qa = QJa~ and qr = QJr2). (The electric
charges on the ice crystals considered are
listed in Table 1.)
Obviously, the smallest charge an aerosol
particle can carry is Q~ = 4.8 x 10-1° esu,
which is equal to one electron charge.
Smaller particles carry no charge. Thus, it
appears that our formulation Qr = q~r ~
applies only to aerosol particles of r >- (4.8
x 10-Wqr) v2, i.e., to aerosol particles of
r >~ 0.2/xm, if we assume qr = 2.0 esu cm -z.
AEROSOL PARTICLES
51
I
TABLE I
NRe =20
Oc=404ffm
7 0 0 m b ' -IO°C
[-
Electric Charges Which Reside on the Ice Crystals
Considered, for qa = QJa~ = 2.0
Na~
ae (p.m)
50
20
10
5
2
1
0.5
0.1
639.15
404.0
289.0
213.0
146.8
112.8
87.9
50.6
I0
u
x
10 -~
I0 ~
10 -a
10 -4
10 -4
10 -4
10 -4
x
10 -~
x
x
x
×
x
x
_
>-
Q . (esu)
-8.170
-3.264
- 1.670
-9.074
-4.310
-2.545
- 1.545
-5.121
//]
/'"
_
.
/ /
~,
',
'/
,,'(:::~ ........./
~o
...j-_
u_
u_
uJ
~)
....:;::---
//
.-
iO-3
J
o
10-4.
.......
(RH)i :~00°/o"
- -~
( R H ) i = 5 0 % I q°- q r - U
. . . . . . . . . . (RH) i = 1 0 0 % ~ q o = - 2 . 0 e s u cm -z
...................... R H _r.c,o/ ( q r = + 2 . 0 e s u c r n 4a
However, since Model-I considered only
particles ofr > 0.1/zm while Model-II considered particles of 0.001 -< r -< 0.1/~m the
above restrictions apply only to Model-II.
On the other hand, Model-II did not consider the motion of individual particles but
rather the flux of a whole assembly o f particles some of which carry zero charge while
others carry 1, 2 . . . .
electron charges.
Therefore we assumed that, in the mean,
the electric charge of the whole assembly
Oc = 639ffm
700rob,-IO°C
-td ~ ~
z
to
~a
i I lilllil
IO-3
I [ lillill
10-2
RADIUS
I J llliHI
10- t
OF
I r JIIllll
IO
I
AEROSOL
PARTICLE
(/zm)
FIG. 2. Same as Fig. 1 but for ao = 404 /~m and
Nrte = 20.
of particles
affecting
population
of particles
the scavenging
could
of the
be given
by
Q r = q , r 2.
4. R E S U L T S A N D D I S C U S S I O N
The major results of our study are summarized in Figs. 1 to 8. In these figures the
efficiency is given with which electrically
charged and uncharged aerosol particles of
..y
.
>-
"
10- 5
/
-..
IG 2
/f
I
.q
/
...~:" ,"
...
~--
NRe =10
O c : 289p.rn
}_
700rob,-IO°C
/l
///]
l()'"~,
~oZ
toLL
10- 3 !
~"x<~:"'"'"" " ' /
J
0
0
_¢~
I0 ~-- . . . . . . .
~o
i
(RH)i = IOO%'f~qa=qr=U
(RH]i=50%]
t~_
t~
to
. . . . . . . . . . (RH)i = IOO%~ qo: -2.O esu cm "2
5 ~- .......................(RH)i = 5 0 % ] q r : + 2 " O e s u cm-~
I0- /
L , IIHHI
I I IJIHH
I I IJJHII
I J ~IIHII
10-3
I0 "2
I0-I
I
IO
RADIUS
OF
AEROSOL
PARTICLE
(/zm)
FIG. 1. Efficiency with which aerosol particles collide
with a simple planar ice crystal o f radius ao = 639/xm
and Reynolds n u m b e r NR~ = 50, in air of 700 mb,
- 10°C and of relative humidity (RH)~ (with respect to
ice) of 50, 75, 95, and 100%; for pp = 2 g cm-% and
for qa = qr = 0 and qa = qr = 2.0 esu c m -2, where
qa = Q~/a~ and qr = Q Jr 2.
_z id3I
2
d
o
-4J
io F . . . . . . .
(RH) i =100%7
IRU)i:50%'O°:qr:
0
. . . . . . . . . . (RH) i =lOO%'~qa= -2.O esu cm "~
5 ~ - .......................(RH) i : 50 % ] qr =+2.0
esu cm "2
iO- |
~ t tIHnl
I I ~lHnL
t t titHd
I0 ~3
I0 -2
I0 -I
I
RADIUS
OF
AEROSOL
I I =l~ml
10
PARTICLE
(H_m)
FIG. 3. Same as Fig. 1 but for ae = 289 /xm and
Nrte = 10.
Journal
of Colloid
and Interface Science, VoL 78, N o . 1, November 1980
52
MARTIN, WANG, AND PRUPPACHER
l
_~-
NRe= 5
ac=213~m
700mb,-IO°C
:0- '
~-~:,......
N. ~ - - . . . . . . . .
'x"::..
z
.
.
.
.
iO
.
-
z
;:::
J
tj
,
LL
LL
bJ
\.,
10-2 - -
.,..~-%
",<
"~.
/
u..
"°'I
/
_1
_1
O
o
'--%
^-4[
iu I~" . . . . . . .
(RH)i : IOO%'L . . . . . .
(RH)i = 5 0 % j . . . .
O
O
I
0
io'_: ~;:~o0,21~:o I
( R H l i =100%'1 . . . .
r~
J
t-
. . . . . . . . . . (RH) i =IOO°/o~,% = - 2 . O esu cm-2
}-- .......................(RH) = 5 0 °/0 ~ q r = + 2 " O e s u c m ' 2
J~-5|
I
I I Itlill
lO-~
I
I ~llltll
I I~lll]l
IO-]
IO-z
RADIUS
I
OF
I
I
AEROSOL
10-3
IO
PARTICLE
. . . . . . . . . . (RH) i = I 0 0 % " L % = - 2 . 0 esu cm -~
.......................( RH ) -- 5 0 o/,. ~ q r = + 2 0 e s u c r n "z
10-5
I IlllilJ
(M.m)
IO-2
RADIUS
FIG. 4. S a m e as Fig. 1 but for a~ = 213 /.~m and
OF
10- I
AEROSOL
I
IO
PARTICLE
(/zrn)
FIG. 6. S a m e as Fig. 1 but for a~ = 112.8 /xm and
N p ~ = 5.
N n e = 1.
0.001 - r -< 10 ~m are captured by electrically charged and uncharged ice crystal
plates of various radii in air of various
humidities at -10°C and 700 mb. Our resuits at the other pressure-temperature
levels considered, differed only insignificantly from those at -10°C and 700 mb.
We attributed this finding to our particular choice of pressure-temperature level,
which, in combination, affected r/~, Dva,
p~, and k~ in such a manner that the pressure
and temperature sensitive contributions to
the phoretic and hydrodynamic forces compensated each other. Other combinations
of pressure and temperature may well change
the present curves.
The most significant feature of the curves
in Figs. 1 to 8 is the predominant minimum
I
NRe=2
0c=146.8/zm
~
7 O O m b , -IO°C
Id ~
~
,0-, I~_
~.,,.
'~:~.~""-.-.i.~ '- _.--..........'.~..................
:~:~"
I 0 "2 ~-
id a
"~.
/ ....
~.~/~
ix.
u_
uJ
~
10- 3
iO-3
NRe=0.5
o c = 87.9
_
10-4 _~- . . . . . . .
(RH) i = I00%'1
.(RH)i = 5 0 % ~ q°" q r - u
~(~
Z ..........
(RH) i =LOO%~qo= - 2 . O e s u
-- -...................... (RH) i = 5 0 % ~ q r = ÷ 2 " O e s u
IO -~
i
i illlill
10- 3
R~,DIUS
i
I(3"z
OF
I li ItiLL
i
I(3"i
AEROSOL
IO-4
i tililli
~
I
PARTICLE
(R.)i :1oo%t
_
(RH)i =500/°~ qO: q r - O
. .......
..........
c m "2
crn'2
{RH) i = I00%'~ qo = - 2 . 0 esu cm -2
.......................(RH) i = 5 0 % J qr =+ 2 . 0 e s u c m "z
IO"5
I iL~llld
i E llnlLL
i lllltnl
10-3
10-2
10- I
I
i liHIll
10
RADIUS
(/zm)
FIG. 5. S a m e as Fig. 1 but for ae = 146.8 ~ m and
N~
7OOmb,-IO°C
-J
M
0
(J
Journal of Colloid and Interface Science, Vol. 78, N o . 1, N o v e m b e r
Fie.
7. S a m e
NR~ = 0 . 5 .
= 2.
1980
OF
as Fig.
AEROSOL
J I LLLJJJI
IO
PARTICLE
(/zm)
I b u t f o r a~ = 8 7 . 9 /xm a n d
53
AEROSOL PARTICLES
",...
~..
~:#...
I
/
:,':? ..............
i0 -I
)-
/
/.....
/
.,
...
", ...
/
;:/
......
" xx
~
\\
uJ
7j
IG z
L
LL
Ld
Z
0
I0 3 __
----
NRe~O'I
O C = 50,6/~m
co
S
700mb,
10- 4
-IO°C
( R H ) i = 1 0 0 % )_
_
(RH)i =50%]
qa= q r - O
._ ......
..........
( R H ) i = I 0 0 % ' ~ qa = - 2 . 0 esu c m "z
.......................( R H ) i = 5 0
iO-5
~L
II1~11]
I0 -3
RADIUS
% .J q r = + 2 " 0 esu c m "z
I I Iltllll
10- 2
OF
I ; ~tlHII
I 0 -I
AEROSOL
J I IlJllll
I
PARTICLE
I0
(H. rn)
FIG. 8. Same as Fig. 1 but for a~ = 50.6 /xm and
NR~ = 0.1.
in the collision efficiency E for aerosol particles of radius between r = 0.01 /xm and
r = 0.1 /xm. Analogous to the particle
scavenging behavior of water drops (20) this
result can be explained on the basis of
Brownian diffusion which is increasingly
responsible for particle scavenging as the
particle radius decreases below 0.01 /xm,
while inertial capture is increasingly responsible for particle scavenging as the particle
radius increases above 0.1 txm. However,
it is worth noting that the minimum (termed
by 20, as the Greenfield-Gap) for particle
scavenging by ice crystal plates appears at
aerosol particle radii which are about 1 order
of magnitude smaller than the particle radii
at which the minimum appears for water
drops. This result is caused by the unusual
properties of the ice crystal rim as a trap
for the aerosol particles. In contrast to the
air flow past a spherical drop, air flow past
a thin falling ice plate exhibits strong
horizontal flow components on its lower
side, recurving sharply to become more
or less vertical near the crystal's edge,
the streamlines strongly crowding near the
crystal tip (see 17). This flow behavior
causes aerosol particle trajectories of the
type described in Fig. 9. These demonstrate
that the ice crystal rim is a preferred capture
site for aerosol particles. Thus, the capture
of aerosol particles is controlled by the air
flow past the scavenging body to much
smaller aerosol particle sizes if the scavenging body is a thin ice crystal plate than if it
is a drop. No annular behavior of the type
found by Pitter and Pruppacher (28) for
drops captured by ice crystal plates was
observed by us for aerosol particles captured by such plates.
It is evident from Figs. 1 to 8 that, as in
the case of particle scavenging by drops,
particle scavenging by ice crystal plates is
most strongly affected by phoretic forces in
the Greenfield-Gap. The phoretic effects are
quite small for aerosol particles of r > 1
/xm and of r < 0.01 /xm, but are important
for aerosol particles of 0.01 -< r -< 1 p~m.
Note also from Figs. 1 to 8 that the phoretic
<
rc
-=O.05~m
7 0 0 m b , - IO°C
Yc, FRONT = 0 . 4 0 4
Ye,REAR = 0 . 6 9 5
w
~- ,.o
qp : + 2 . 0 esu cm -2
qa : - 2 . 0 e s u cm-2
cl
o
7,
f~ z.o
Ilo
NON DIMENSIONAL
DISTANCE
FROM
VERTICL
AXIS
FIG. 9. Theoretically computed trajectories of an
aerosol particle or r = 0.05 /xm moving around an
ice crystal plate of a0 = 50.6 /xm (NRe = 0.1), for
q~ = q ~ = 2.0 esu cm -2. Note that the capture of
aerosol particles takes place at the ice crystal rim, and
also may take place on the rear side of the ice crystal.
Journal o f Colloid and Interface Science, Vol. 78, N o . 1, N o v e m b e r 1980
54
MARTIN, WANG, AND PRUPPACHER
effects b e c o m e stronger the smaller the ice
crystal. Obviously, the smaller the ice crystal, the smaller its Reynolds n u m b e r and,
therefore, the smaller the particle-deflecting
effect of the flow field beneath the crystal.
Similar to the phoretic effects, the electrical effects on particle scavenging are negligible for r < 0.01 /xm. H o w e v e r , they are
very p r o n o u n c e d in the particle size range
0.01 <- r <- 10/xm, depending on the size of
the ice crystal. Thus, the collision efficiency
for crystals of a~ = 639, 404, and 289/xm
is raised by as much as 1 order of magnitude
in the range of 0.01 --- r -< 5 /xm if
= Iq" I = 2.0 esu cm -2.
We also note from Figs. 1 t o 8 that the
phoretic effects on the scavenging are less
noticeable if the aerosol particles and ice
crystals are electrically charged. Figures 1
to 8 show further that the smaller the
Reynolds n u m b e r of the ice crystal, i.e., the
smaller the particle-deflecting hydrodynamic
forces beneath the crystal, the stronger is
the collision efficiency enhanced by the
electric charges present.
Some particularly strong electric effects
are noted for particles of r > 1/xm and ice
crystals of ac = 213, 146.8, and 112.8/xm.
If crystals of these sizes are electrically
uncharged their collision efficiency rapidly
decreases to zero as r becomes larger than
1 /zm. In fact, no particles are collected if
r > 2 /xm. Trajectory analysis shows that
the reason for this behavior lies in the fact
Iqol
10-2 __
NRe=IO
Gc =289FLrn
r =o.02/~m
700mb'-lO°C
(FIH)i =100%
Z
lu
5
/
LU
CO
°o ,o-~
0.1
i
I
i
I i i I Ill
.0
i
Iqol=Iq,I ( ~ c~~)
FIG. 10. Variation of the collision efficiency with
electric charge [q~ [ = Iqr[ on the crystal and aerosol
particle, respectively; fora¢ = 289/xm and forr = 0.02
/xrn; w h e r e qa = Qa/a~ and qr = Qr/r 2.
Journal of Colloid and Interface Science, V o ] . 7 8 , N o . 1, N o v e m b e r
1980
I0 "I __
NRe" I0
OC =289/zm
r =0,3/~m
?OOmb, -lO*G
z
LO
E
(RH)I-IO0%
LU
~
o
i0 "z
I
0. I
I
I
i
i ] I 11
1.0
I
Iqol=lq~l (esu cm-=)
FIG. l l. Variation of the collision efficiency with
electric charge Iqo I = Iq,I on the crystal and aerosol
particle, respectively, for ac = 289 txm and for r
= 0.3/xm; where qa = Q,/a~and qr = Qr/r~.
that at these relatively low Reynolds numbers the approach velocity of the aerosol
particle to the ice crystal is sufficiently small
so that the strong horizontal hydrodynamic
deflecting forces beneath the ice crystal
have sufficient time to move any aerosol
particle of r > 2 /xm around the crystal
causing the collision efficiency to be zero.
H o w e v e r , if the ice crystal and aerosol particles are electrically charged with Iqa[
= Iqrl = 2.0 esu cm -2 the collision efficiency becomes finite and in fact quite large,
being raised to a value above 10-2 by the
electric charges.
A further dramatic change in the collision
behavior of ice crystal plates is noted if ice
crystals have Nae < 1 (see Figs. 7 and 8).
We note that at these very low Reynolds
numbers aerosol particles of r > 1 /xm are
again captured. Analysis of the velocity
field around the falling crystal shows that
the reason for this behavior lies in a pronounced decrease of the horizontal, particledeflecting velocity if Nae < 1. In fact, the
deflecting force for Nr~e < 1 decreases to
such a low value that, despite the small approach velocity, the particle cannot escape
colliding with the crystal. Nevertheless, the
increase in E due to the presence of electric
charges is considerable over the whole particle size range. In fact, with decreasing r,
the collision efficiency decreases unexpectedly to a minimum near r ~ 0.5 txm
and subsequently increases with further
decrease in r to a maximum near r ~ 0.05
AEROSOL PARTICLES
f.O~_lqal=lqpt=o esucrn'Z
F lqol= lqPl =o.2 esu cm -2 ....
o~
D
~n
~_ lqo,o lq~.loZ.o . . . . . -z._.,1
o.I
0.01
.../
55
the results of the present theories with the
presently available experimental results for
ice crystals and aerosol particles which are
electrically uncharged. This comparison
shows that, at least for uncharged ice crystals and aerosol particles, the agreement between theory and experiment is satisfactory.
NRe =20
700mb _i00c
0.0011
,, rJ [
~
, ~J,
0. I
1.0
I0
RADIUS OF AEROSOL PARTICLE (p.m)
FIG. 12. Variation of the collision efficiency with
aerosol particle radius for various electric charges on
the ice crystal and aerosol particle; for a~ = 404 ~m.
ACKNOWLEDGMENTS
The present study was supported by the Atmospheric
Sciences Section of the National Science Foundation, under grant ATM 78-10817, by the Environmental
Protection Agency under grant R 806257-02-1, and by
Lawrence Livermore Laboratories under grant PO
7683403.
REFERENCES
p.m. Trajectory analysis illustrated in Fig. 9
shows that this effect is due to capture of
the charged aerosol particles on the rear
side of the charged ice crystal. Figures 7
and 8 show that with even further decrease
in particle size, the collision efficiency
decreases again as the electric forces are
now rapidly decreasing in comparison to the
hydrodynamic forces tending to move the
particle around the crystal. The final increase
in collision efficiency forr < 0.01/~m is due
to the effects of Brownian diffusion.
In Figs. 10 and 11 the variation of the
collision efficiency is plotted as a function of
the charge ]qa] = ]qr[ on the ice crystal
and on the aerosol particle, respectively,
where qa = qa/a~ and qr = Qr/r z. We note
that for both aerosol particle sizes considered and for an ice crystal of radius ac
= 289 p~m electric charges begin to noticeably affect the capture of aerosol particles
if ]qal = Iqr] > 0.4. This charge is considerably below the mean thunderstorm
charge on particles in clouds. Figure 12
shows that for an ice crystal of ac = 404
~m the collision efficiency is significantly
affected even though the charge is as low
as [q~[ = ]qr[ = 0.2 e s u c m - K
Unfortunately, no experiments are available for checking the predictions of the
present theoretical results. However, Martin (13) and Martin et aI. (14) have compared
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MARTIN, WANG, AND PRUPPACHER
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