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On the coalescence of electrically charged droplets
On the Coalescence of Electrically Charged Droplets I S A I A H G A L L I L Y 1 AND G E D A L I A AILA!V[ ( V O L I N E Z ) Israeli Institute for Biological Research, Ness-Ziona, Israel
Received January 20, 1969; revised April 18, 1969 The feasibility of coalescence of two perfectly conducting, electrically charged droplets is studied from a thermodynamic point of view. It is assumed that the medium is devoid of external fields of force and that the system changes only in its electrostatic and surface free energies. A suitable expression is developed for the electrical energy of the two droplets which make the initial contact. In addition, the difference between the free energy of an assumed, final sphere and the initia:l aggregate is reduced to a dimensionless form depending on a geometric parameter and the ratio a/2 between the electric and surface free energies of the sphere. The behavior of the free-energy function is studied, and it is concluded that, when e < 5.293, coalescence is feasible just as for neutral drops. I. INTRODUCTION
charged droplets, and we shall study the feasibility of coalescence from the thermodynamic, energetic point of view. I n doing so, we shall treat the case of initially spherical particles, which we think to be quite common. Prior to a collision, the single droplets are usually in the shape of a sphere, which is the stable form with respect to small perturbations when the ratio a / 2 between its electrostatic and surface free energies is less t h a n 2 (4, 5) ; likewise, at the m o m e n t of encounter, these droplets will stay essentially spherical as long as their kinetic energy is a negligible fraction of their surface energy. I t is assumed t h a t the system changes only in its electrostatic and surface free energies, t h a t effects caused b y mass transfer between the aggregate and its surroundings can be neglected (6), and t h a t the medium is devoid II. SYSTEM CHOSEN AND FORMULATION of external fields of force. I n addition, it is OF THE PROBLEM assumed t h a t the surface tension of the We shall deal here with an initial aggreliquid is constant. gate composed of two perfectly conducting, As a measure of the feasibility of the i Present address: Section of Lunar and Planeprocess, we shall adopt the difference AF be~ tary Sciences, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Cali- tween the free energy of a final shape, which fornia 91103. is t a k e n to be a sphere, and the free energy Journal of Colloidand InterfaceScience, Vol.30, No. 4~August 1969 537 The shape of the aggregates formed by collisions among aerosol particles is an important factor in their coagulation, in their dynamics, and in processes of heat and mass transfer. Thus, the geometry of the composite body strongly affects the coagulation constant (1), the aerodynamic drag (2), and the interracial transport rates. For electrically neutral droplets, the final product of a collision is a sphere, which is the shape of m i n i m u m energy in a force-free space. However, for charged liquid particles (3), there is still much to be learned with regard to the result of mutual encounters. As electrically charged liquid aerosols occur v e r y frequently, we thought it worth while to investigate some points related to the coalescence of their droplets.
538
GALLILY AND AILAM
of the initial aggregate. This difference is given by
the method of images by Kirchoff and Poisson (9), i.e., p~+l -- [(d 2 -- rl 2 -- r~2 )/rlr2]p~
AF = 4~rR2(r + (q~/2eR) [1] --
w h e r e q - - q l + q 2 , R ~ = rl ~ + r 2 ~ , a n d w i s the electric energy of the two droplets. Here, one should distinguish between the following situations: (1) If the assumed, final sphere is in a stable state (a < 4), then AF indicates the extent of displacement from equilibrium. In this case, AF < 0 implies a thermodynamically possible coalescence. (2) If the sphere is unstable (a > 4), then AF leads to a definite conclusion only when it is negative. Since at a > 4 the actual final shape has a lower energy than that of the corresponding spherical form, AF < 0 shows that coalescence is even more likely to occur. III. ELECTROSTATIC ENERGY OF THE AGGREGATE AND THE REDUCED FREEENERGY EQUATION The electrostatic energy of two perfectly conducting spheres is (7) w = ( ½ ) ( c ~ i v / + 2c~v~v~ + c~2v~),
[2]
where v~ and v2 are the potentials and c~, c22, and c~2 are the two coefficients of capacity and the coefficient of inductance, respectively (8). As at contact U1 = V2 = V, ql =
and
-/- p~-I = 0
[ 4 w ( r l 2 -~- r2 2) -~- w ] ,
I
pn+l
[5] -
[(d ~ -
rl ~
-
2 r~
)/r~r2]p~
!
!
+pn-1 = 0 where e/p~ is the n-th charge induced in the sphere of radius r~, e/p~' is the n'-th charge induced in the sphere of radius r2, and d > rl + r2 is the inter-center distance. In the presently discussed ease d = rl + r2; consequently, we can write (Eq. [5] ) : pn+l - - 2pn "~ pn-1 = O ]
and
t
[3]
[6]
p ~! + l - - 2p~ ! f l - p n! _ l = O,J
which h~ve the general solution p~ = a n - t - b
~ I l p~ = a'n + b'.J
and
[7]
The constants a and b are determined by (9) 5 = po = l/r1 and
a+b
= p~=
1
2 2 I (d~ - r 2 ) / r ~ r 2 .
[8]
Thus, p~ = [(rl -~- r2)/rlr2]n -1- l / r l .
[9]
Likewise (9), a' d- b' = Pl' = -- (rt -t- r2)jrlr2 and
(eli -~- C21)?) )
t
and
/
[10]
2a' + b' = p ( = - 2 ( r l + r2)/rlr~, which yields p~' = --(rl + r~)n/rlr2.
where c12 = c21. Thus, from Eqs. [2] and [3], and the definition of q, we get w = q~/[2(oi + 2c12 -~- c22)].
[4]
To calculate cn, c2~., and c12, let us use the difference-equations obtained according to Journal of Colloid and Interface Bcience, Vol. 30, No. 4, A.ugus~ 1969
[11]
The coefficients cn can be expressed now as cll = e ~
l/p~ [12]
= e ~ r l r z / [ n ( r l + r2) -~ r~], n~0
COALESCENCE OF ELECTRICALLY CHARGED DROPLETS
539
we can replace Eq. [16] by
and, from symmetry,
Z / u ' 3 ( 1 - u ) "~ = (1 -
u ) -:/~
c~2 = e ~ ro'2/[n(rl ~- r2) ~- rl]. [13] n=0
" k {1/[(n -k 1)x -k n] n ~0
Similarly,
+ 1/[n(x + 1) + 1] 2 / [ ( n + 1 ) ( x + 1)]}
-
c12 = e k
l/p,( =~ a o
[14]
= [u-~l~x/(x + 1)] ~o
= --~ ~ rlr2/(n q- 1) (rl q- r2).
"~-~n~01 / ( { n - ~ 1 -- [ 1 / ( x + 1)]}
n=l
• {n + [1/(x ~- 1)]1)
Denoting rl = u*/aR and r2 = (1 - u)l/aR, and by [4] and [12] through [14], we can put the equation for AF in the dimensionless form
-- [2u-1/ax2/(x q- 1 )a] • ~ 1 / ( { n + 1 - [ 1 / ( . + 1)]} r~0
• {n -~ [1/(x + 1)]} (n ~- 1)).
G =- (AF/4~-R=~) = 1 + (a/2) - - [u 2/a q-
(1 -- u) m q- a / 2 Z ] ,
[18]
[15]
where a = q2/4rrRa(*e and
The first term on the right-hand side of Eq. [18] may be written as [u:l~/(1 -- x)] k {1/In q- 1 - 1/(x -k 1)] n~O
Z = u*/~(1 - u) 1/~ •k
(1/{n[ u~/a q- (1
-
= u-1/~x (x q- 1 ) / ( x -- 1 )
[19]
- - u ) 1/3] -}- u I/3}
=0
-- [2u-1/ax/(x -- 1 ) (x q- 1 )]
[16]
-1- 1/{n[u ~la q- (1 -- u) ~/a] Jr (1 -- u) */a} -- 2/(n +
1~In + 1 / ( x + 1)]}
•~
1)[u :/3 ~- (1 -- u):/a]).
[1/(x q- 1)]2},
which, by the identity (10)
IV. BEHAVIOR OF THE REDUCED FREEENERGY DIFFERENCE FUNCTION To study the dependence of the reduced free-energy difference G on the geometryof the initial aggregate, we shall make Eq. [16] more tractable, investigate the derivative dG/du, and find the extremal points of G = G(u, ~). A. The Reduced Capacity of the Aggregate• Designating
ul/a/(1 -- u ) 113 = x,
1~In 2 - -
[17]
k
1/{ n2 -- [1/(x + 1)]2}
= [(x + 1)/2]{x + 1 -- 7r etg[~r/(x + 1)]}, yields
[u-1/'a/(1 -- x)] "k
n~0
-- l/In
{1/In q- 1 -- 1 / ( x -+- 1)]
Jr 1 / ( x ~- 1)]}
[20]
= [~u-1/ax/(x -- 1 )] ctg [~/(x ~- 1 )]. The second term on the right-hand side of Journal of Colloid and Interface Science, Vol, 30~ No. 4, A u g u s t 1969
540
GALLILY AND AILAM
[18] may be expressed as
integer values of u except zero, and that
[2u-l%V (x + 1)31 ~l/.(.+g) •~
= (l/g)
[25]
1 / [ ( n + 1 -- g ) ( n + g ) ( n + 1)]
n=0
= A ~
f01 • 1 - --- : ~~g d~ _-- ~,
[21]
{1/[,ff - g)]
v=l
--
we obtain (Eqs. [18], [20], and [23])
1/[.(.
--
1 +
g)]},
w h e r e g = 1/(x + 1) a n d A
z = u1~3{- [~/(z + 1)]
= [(x -}-1)/
(1 - x)][2u-~%V (x + 1)~].
• ctg [~/(x +
Since
1 )]
[26]
+ 2 -- [2~/(x + 1)2]}.
E 1/[.(. - 1 + g)] v~l
= [1/(g-- 1)] ~ {[1/v -- 1/(v + g)] + [1/(v + g) -- 1/(v -- 1 + g)]}
[22
For computational convenience, we shall transform fl by applying repeatedly Kummer's transformation (11) while using the series ~ . = 1 1/u (i = 2, 3 , . . . k). Thus, we get
ao
= [1/(g-- 1)]~[1/v--
1/(u+
-
-
1/[g(g
(-1)q-(. + 2)g"
= ~
g)]
v~O
v=l
k
-- 1)],
= Z: ( - 1 ) ' ~ ( . + 2)g v
[27]
V~0
one gets + (--
1 )/~-}-ig/Z+l
A { ~ 1/[,(. -- g)] -- 1/[.(v - 1 + g)]}
[23]
- g)
Z = [x/(1 + xS) ~/~]
v=l
-- [(2g -- 1 ) / ( g
• {[~r/(x + 1)] ctg [~rx/(x + 1)]
-- 1)]
+ 2 - 2[~(2)/(x + 1) 2
.I.=~ 1 / . ( . + g)] + 1/g(g-- l)}. -
Thus, noting that (10) oo oo L / u ( . -- g) + ~ i / ( - - . ) ( - - . v=l
-- g)
v=l oo
=
~ ' 1 / . ( . - g)
= 1/g 2 -where ~ '
l/k+2 (, + g),
where ~(z) is Riemann's zeta function. Equation [26] may be written now in the form:
= A { ~ 1/[v(v -- g)
+ ~ 1/(-.)(-~
~ v~l
[24]
Or/g) ctg (vg),
stands for summation over all
Journal of Colloid and Interface Science, Vol. 30, No. 4, A u g u s t 1969
~(3)/(x
[28]
+ 1) ~ + . . . ] / .
It is worth while to mention that for x = 1 one obtains, by evaluating ~ (Eq. [25] ), that Z = 2m In 2. B. The change of G with u. Differentiating G in [15] with respect to u, one gets
dG/du = - { (2/~)[u -1/~ -- (1 - u) -in] -
(a/2Z) (dZ/du)l
[291
COALESCENCE OF ELECTRICALLY CHARGED DROPLETS and according to [26]
Furthermore, expanding the expressions contained within the brackets of [34] in a power series in x, we get
dZ/du = (u-2/3/3) •{ -
[~/(x + 1)] ctg [~/(x + 1)]
G = l - k (a/2)
~- 2 -- [2fl/ (z -~ 1)2]} --[1
~- {u-2r~x/[3(1 -- u)]}
1)2]ctg[Tr/(x
• ([Tr/(x +
[30]
(a/2)[1
-k
x~/3 --k
0@4)]
Hence, in the vicinity of x = 0, [36]
dG/dx ~ - 2x
where
and 1) 2] = --2fl/(x + 1) ~
(d/dx)[2~/(x •
+ [2/(x 4 1 ) 3 ] f01 ~11n29 ~ .
dG/du = (dG/dx) (dx/du) ~ -- 2/3x.
[31]
At u = 1/~ (corresponding to droplets of equal size), dZ/du = 0; so, dG/du = 0,
1 / ( 1 + x ~)2~
-
(~ (1 + x ~)~/<2xI [2 (~ +
[2~/~(x
+
2ul/3/3(1 _ u) Ij3.
l
-oo
dG/du--~ L oo
=
as
u~0;
as
u--~l•
1)]
-- (1 -- u)-ll~]/(dZ/du),
f G I d u 2 = (2/I)[u-4/~ + (1 -
u) -4/3]
we obtain, for the neighborhood of x = 0
-~ (a/2Z 2) ( f Z / d u 2) -- a (dZ/du )2/Z3.
(~ = o ) , G = 1 + (~/2)
To evaluate d2G/du 2 we use -
x~/(1 + x~)~/~
[2 (x + 1)/~r] tan [wx/(x + 1)]
+ [2~/~(x + 1)] • tan
[Trx/(x -k 1)] -k O(x~)}).
[411
fZ/du ~
[a(1 q- x~)113/2x]
= (d/dx)[(dZ/dx) (dx/du)] (dx/du)
• ([(x ~- 1 ) / v ] t a n [ ~ x / ( z ~- 1)] • { 1 --
[40]
and their nature is specified by
•tan [~rx/(x -]- 1 )]
--
[39]
( ~ / ) Z 2 [ u -1/~
[33]
•t a n [ ~ x / ( x + 1)]}))[(x + 1)/~r]
1/(1 + x3) 2/3
[3s]
c• The Extremal Points of G = G (u, a). The points at which dG/du = 0 are determined from (Eq. [29] )
1)/~]
•tan [~x/(x -t- 1 )] 1 -
da/du ~-- 2/3(1/x)
Thus,
G = 1 + (~/2) - x~/(1 + x~)~ -
[37]
For the neighborhood of u = 1, one has, by symmetry (Eq. [37]),
[32]
which could be expected from symmetry• The value of dG/du when u --~ 0 and when u --* 1 can be calculated as follows: Transforming the equation for G ([15], [17], and [26] ) into
--
[35]
• [1 + 0 (x~)] = - x ~ + 0 (x~).
(d/~x)[2~/(z + 1)~]),
-
-
-
-k x ~ -- 2x3/3 -k O(x4)]
+ 1)]
(x + 1)2sin 2 [r/(x + 1)]}
--v"/{
541
[34]
= ( f Z / d x 2) (dz/du) 2
[42]
+ (~Z/du) (~/dx) (dx/du)• From Eq. [26] (with u 1/3 expressed in terms Journal of Colloid and Interface Science, Vo]. 30, No. 4, A u g u s t 1969
542
GALLILY AND ALLAN[ TABLE I
THE I~EDUCED CAPACITY OF TItE Two DROPLETS AT VARIOUS VALUES OF X x
1.0013 1.0088
0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.91 0.93 0.95 0.97 0.98 1
1.0221 1.0385 1.0563 1.0724 1.0851 1.0940 1.0988 1.0991 1.0996 1.0999 1.10008 1.1002
dZ/du
0.1 0.2 0.3
1.384 0.882 0.601
0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.98 1
0.416 0.290 0.200 0.133 0.081 0.037 0.018 0.007 0
E q u a t i o n [44] m a y be written in a more convenient form with the aid of Eq. [27]. F r o m Eq. [17] we get (dx/du) 9 ~nd (d/dx) (dx/du); consequently ([42], [43], and [44]), it becomes possible to calculate d2Z/du 2 and
1.1003
of x), we o b t a i n
d2Z/dx 2 = - [4x2/ (1 + x~) m] • {[~/(x +
1)] ctg [~rx/(x +
d~G/ du 2. Atu
1)]
+ 2 -- [23/(x + 1)~]} + [2/(1 + x~) 4/~]
= ~,
d~/du ~ =
[4(1//)-~/~/9]
+ ~ (d~Z/du~)u~=1/~/2'/~ In ~ 2,
+ 1 )~1 ctg [~x/(x + 1 )]
• (-[~/(x
-
x
Z
0.1 0.2
--7r2/{(x
TABLE II THE DERIVATIVEdZ/du AT VAI~IOUSVALUESOF X
+ 1)Bsin2[~rx/(x + 1)]}
(d/dx)[23/(z + 1)2])
-5 [x/(1 + x~)~/3]([2~r/(x -5 1) 3]
[43]
which shows t h a t the free-energy difference acquires a m i n i m u m or a m a x i m u m value when a is correspondingly smaller or greater t h a n % where ~/ -- -- 2 Im In 2 2/9 (d2Z/du 2)n=1/2
• etg [~rx/(x -5 1 )]
One should observe that, at x = 1, (d/dx) • [2~/(x -5 1 )~] = In 2 -- ~r2/8, and
-5 ~r2/{ (x + 1) 4 sin s [~rx/(x -5 1)]} -5 (~2{3(x -5 1) 2 sin [~rx/(x + 1)]
(d~/dx~) [23/(x + 1 )~]
+ 2~(x + 1) cos [~x/(z + 1)11/
= ~r2/4 -- In 2 -- 14~(3)/16• 2
{ (x + 1) 6 sin 3 [~rx/(x + 1)]}) -
(d/d~
~)[23/(~
Thus,
+ 1)~1),
in which (d/dx)[2~/(x [31], und
-5 1) ~] is given b y
(d2Z/du 2)~=1/~ = - (2va/9)[16 in 2 - 7~" (3)] ~nd -~ = 5.745• This is ~lso the v a l u e of a which character-
i(d~/dZ)[23/(z + ~)~1 = [43/(x + ~)~] -- [ 8 / ( x
-5 1) ~] jo 1 ~o1la-- } }d~
-- [ 2 / ( x
-5 1) ~]
2 Designating ~1/2 = ~, we have (12) : [44]
f / } ~ in ~ ( d~ 1 --
Journal of Colloid and Interface Science, Vol. 30, No. 4, August 1969
l~,/Zln ~} .
=
--8
¢l~aln ~0
£ln2e dO +
811(2a - 1)/2a12!ff(3)}.
543
C O A L E S C E N C E OF E L E C T R I C A L L Y C H A R G E D D R O P L E T S TABLE III
2.0
T H E E X T R E M A L P O I N T S O F G = G ( u , re) A T V A R I O U S VALUES
OF X
1.6
X
0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.8 0.9 0.95 0.98 1
1.2
8.695 6.166 5.458 5.293 5.336 5.451 5.577 5.664 5.728 5.741 5.744 5.745
dZ/du 0.8
0.4
o
'
' 0.2
'
0I 4
'
0 I6
0.8
1.0
x
Fla. 2. The derivative dZ/du vs. x
9 1.10 I
8
1.08
7 ct d
6
1.06
Z
I 1.04
40
0.2 . . . . 0.4
0I6
'
0 I8
' 1.0
x
1.02
F~G. 3. The extremal points of G = G(u, x) VS. o~. I .00 l
l
I
0.2
.I
0 4
[
I
0.6
I _ 018
I
I .0
x
FIG. 1. The reduced capacity of the two drop]ets vs. x.
0.10
izes t h e e x t r e m a l p o i n t a t u = 1/~ ( E q . [40] ), i.e.~
]ima u~l/2
-G
=
0.05
lira { (d/du) ( ~ ) [ u -~/3
u~l[2 --
(1
-
u)-'~]}/(d
2z
/du2 )~1~2 0
V. C O M P U T A T I O N S
The reduced capacity Z and the derivative
dZ/du a r e g i v e n f o r v a r i o u s v a l u e s of x i n
'0',2
'0'.4
'0'.6 x
'
0.8
'
1.0
FIO. 4. The reduced free-energy difference vs. x for a = 3. Journal of Colloid and Interface Science, Vol. 30, No. 4, A u g u s t
1969
544
G A L L I L Y AND A I L A M
Tables I and II, and Figs. 1 and 2. The values of a corresponding to the points at which d G / d u = 0 are shown in Table III and Fig. 3. Finally, the computed freeenergy differences are presented in Tables IV through VI, and Figs. 4 through 6. Here one should observe that since, from symmetry,
0.005
Z(x) = zo/x)
and 0
I
I
~
0.2
I
0.4
I
. ~ - -
01.8
0.6
I
~
1,0
x
FIo. 5. The reduced free-energy difference vs. x f o r ~ = 5.5.
[45]
(x, it is sufficient to carry out calculations for 0
~:~,EDUCED
FREE-ENERGY
VARIOUS VALUES
-G
-0.01[
-0.01
i
i
I __l
0.2
,
0.4
t
. ,
0.6
p
i
0,8
1.0
x
FIG. 6. The reduced free-energy difference vs. x for o~ = 6.
x
-G
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 1
0 0.0057 0.0105 0.0115 0.0110 O.OO9O 0.0082 0.0083 0.0086 0.0091 0.0092(0) 0.0092(3)
FREE-ENERGY
VARIOUS VALUES
AT
5.5
TABLE VI
T A B L E IV REDUCED
DIFFERENCE
O F X AND FOR t~ ~
DIFFERENCE
OF X AND FOR t~ ~
AT
3
THE
l~]~DUCED
FREE-ENERGY
VARIOUS VALUES
DIFFERENCE
OF ~: AND FO R (~ =
x
-G
x
6:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 1
0 0.0074 0.0211 0.0384 0.0574 0.0756 0.0925 0.1064 0.116 0.1215 0.1228 0.1232
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 1
0 --0.0054 --0.0084 --0.0061 --0.0017 0.0044 0.0087 0.0113 0.0128 0.0134 0.0135 0.0136
Journal of Colloid and Interface Science, Vol. SO, No. 4, August 1969
6
AT
COALESCENCE OF ELECTRICALLY CHARGED DROPLETS VI. DISCUSSION AN]) CONCLUSIONS According to Fig. 3 and Eq. [36], one sees that, for a < 5.293, G is always negative and decreases monotonically with x to a minimum at x = 1. Hence, under these conditions, the process of coalescence is thermodynamically feasible. This process can occur even at a = 5.5 (Table V and Fig. 5). In the case of a system composed of two droplets of equal size, G =
i +
(~/2)
-
2 (½)2/~ _
(~/2/~ In 2),
which allows coalescence at a < 5.702. Note again that the adopted measure of feasibility is satisfactory at all values of a for G < 0;forG > 0, however (Table VI and Fig. 6), our analysis will provide a definite result only in the stable case. For most liquid aerosols in nature, the values of the parameter a of the individual droplets are smaller than 1 (13); consequently, these particles may coalesce as do the electrically neutral ones. NOMENCLATURE a, b--constants defined in text. A - - f u n c t i o n of x, defined in text. c,--coefficient of capacity of the i-th droplet. c~j--coeffcient of inductance of the i-th on the j - t h droplet. d--inter-center distance. AF--difference between the free energies of the system in its final and initial states. g = 1/(x+
1).
G = AF/4~R2(T.
p~/e--reciprocal of the n-th charge induced in sphere of radius h • q~-electric charge of the i-th droplet. q--electric charge of the final droplet. r~--radius of the i-th droplet. R - - r a d i u s of the final, spherical drop that may be produced. u = r ~ / R ~.
w--electrostatic potential of the i 4 h droplet.
545
v--e~ectrostatic potential of the initial aggregate. w--electrostatic energy of the initial aggregate. x - - r a t i o between radii of droplets in the initial aggregate. Z - - r e d u c e d capacity of the initial aggregate. Greek Letters
a--dimensionless
parameter
defined
by
a = q2/4~R~e.
fl = [1/(x + 1) 2]
jo1
1 --
~g
d~.
~--constant defined in text. e--dielectric permittivity of external medium. (z)--Riemann's zeta function. a--interfacial tension between liquid and external medium. ACKNOWLEDGMENT The authors wish to thank Mr. E. Fisher for carrying out the numerical calculations.
REFERENCES 1. FUCHS, N. A., "The Mechanics of Aerosols," pp. 302-5. Macmillan, New York, 1964. 2. IBID, pp. 34--42. 3. SCHNEIDER, J. M., LINDBLAD, N. R., AND HENDRICKS, C. D., J. Colloid Sci. 20, 610 (1965), and references cited therein. 4. LORD RAYLEIGH,Phil. Mag. 14, 184 (1882). 5. AILAM (VOLINEZ), G., AND GALLILY, I., The
Physics of Fluids 5, 575 (1962). 6. GALLILY, I., AND AILAM (VoLINEZ), G., J. Chem. Phys. 36, 1781 (1962). 7. JEANS, J., "The Mathematical Theory of Electricity and Magnetism," p. 95. Cambridge University Press, Cambridge, 1960. 8. ImD, pp. 93-4. 9. ImD, pp. 196-7. 10. RIEZHII(, E. M., AND GRADSHTEIN, E. C., "Tables of Integrals, Sums, Series and Products," p. 50 (in Russian). Moscow, 1951. 11. KNO:eI,, K., "Theory and Application of Infinite Series," p. 247. Blackie and Son, London and Glasgow, 1959 12. REF. 10, p. 214. 13. WHITB¥, K. T., AND LIU, B. Y. H., I n C. N. Davies, ed., "Aerosol Science," pp. 63, 65. Academic Press, New York, 1966.
Journal of Colloid and Interface Science, VoI.30, No. 4, August1969
Received January 20, 1969; revised April 18, 1969 The feasibility of coalescence of two perfectly conducting, electrically charged droplets is studied from a thermodynamic point of view. It is assumed that the medium is devoid of external fields of force and that the system changes only in its electrostatic and surface free energies. A suitable expression is developed for the electrical energy of the two droplets which make the initial contact. In addition, the difference between the free energy of an assumed, final sphere and the initia:l aggregate is reduced to a dimensionless form depending on a geometric parameter and the ratio a/2 between the electric and surface free energies of the sphere. The behavior of the free-energy function is studied, and it is concluded that, when e < 5.293, coalescence is feasible just as for neutral drops. I. INTRODUCTION
charged droplets, and we shall study the feasibility of coalescence from the thermodynamic, energetic point of view. I n doing so, we shall treat the case of initially spherical particles, which we think to be quite common. Prior to a collision, the single droplets are usually in the shape of a sphere, which is the stable form with respect to small perturbations when the ratio a / 2 between its electrostatic and surface free energies is less t h a n 2 (4, 5) ; likewise, at the m o m e n t of encounter, these droplets will stay essentially spherical as long as their kinetic energy is a negligible fraction of their surface energy. I t is assumed t h a t the system changes only in its electrostatic and surface free energies, t h a t effects caused b y mass transfer between the aggregate and its surroundings can be neglected (6), and t h a t the medium is devoid II. SYSTEM CHOSEN AND FORMULATION of external fields of force. I n addition, it is OF THE PROBLEM assumed t h a t the surface tension of the We shall deal here with an initial aggreliquid is constant. gate composed of two perfectly conducting, As a measure of the feasibility of the i Present address: Section of Lunar and Planeprocess, we shall adopt the difference AF be~ tary Sciences, Jet Propulsion Laboratory, California Institute of Technology, Pasadena, Cali- tween the free energy of a final shape, which fornia 91103. is t a k e n to be a sphere, and the free energy Journal of Colloidand InterfaceScience, Vol.30, No. 4~August 1969 537 The shape of the aggregates formed by collisions among aerosol particles is an important factor in their coagulation, in their dynamics, and in processes of heat and mass transfer. Thus, the geometry of the composite body strongly affects the coagulation constant (1), the aerodynamic drag (2), and the interracial transport rates. For electrically neutral droplets, the final product of a collision is a sphere, which is the shape of m i n i m u m energy in a force-free space. However, for charged liquid particles (3), there is still much to be learned with regard to the result of mutual encounters. As electrically charged liquid aerosols occur v e r y frequently, we thought it worth while to investigate some points related to the coalescence of their droplets.
538
GALLILY AND AILAM
of the initial aggregate. This difference is given by
the method of images by Kirchoff and Poisson (9), i.e., p~+l -- [(d 2 -- rl 2 -- r~2 )/rlr2]p~
AF = 4~rR2(r + (q~/2eR) [1] --
w h e r e q - - q l + q 2 , R ~ = rl ~ + r 2 ~ , a n d w i s the electric energy of the two droplets. Here, one should distinguish between the following situations: (1) If the assumed, final sphere is in a stable state (a < 4), then AF indicates the extent of displacement from equilibrium. In this case, AF < 0 implies a thermodynamically possible coalescence. (2) If the sphere is unstable (a > 4), then AF leads to a definite conclusion only when it is negative. Since at a > 4 the actual final shape has a lower energy than that of the corresponding spherical form, AF < 0 shows that coalescence is even more likely to occur. III. ELECTROSTATIC ENERGY OF THE AGGREGATE AND THE REDUCED FREEENERGY EQUATION The electrostatic energy of two perfectly conducting spheres is (7) w = ( ½ ) ( c ~ i v / + 2c~v~v~ + c~2v~),
[2]
where v~ and v2 are the potentials and c~, c22, and c~2 are the two coefficients of capacity and the coefficient of inductance, respectively (8). As at contact U1 = V2 = V, ql =
and
-/- p~-I = 0
[ 4 w ( r l 2 -~- r2 2) -~- w ] ,
I
pn+l
[5] -
[(d ~ -
rl ~
-
2 r~
)/r~r2]p~
!
!
+pn-1 = 0 where e/p~ is the n-th charge induced in the sphere of radius r~, e/p~' is the n'-th charge induced in the sphere of radius r2, and d > rl + r2 is the inter-center distance. In the presently discussed ease d = rl + r2; consequently, we can write (Eq. [5] ) : pn+l - - 2pn "~ pn-1 = O ]
and
t
[3]
[6]
p ~! + l - - 2p~ ! f l - p n! _ l = O,J
which h~ve the general solution p~ = a n - t - b
~ I l p~ = a'n + b'.J
and
[7]
The constants a and b are determined by (9) 5 = po = l/r1 and
a+b
= p~=
1
2 2 I (d~ - r 2 ) / r ~ r 2 .
[8]
Thus, p~ = [(rl -~- r2)/rlr2]n -1- l / r l .
[9]
Likewise (9), a' d- b' = Pl' = -- (rt -t- r2)jrlr2 and
(eli -~- C21)?) )
t
and
/
[10]
2a' + b' = p ( = - 2 ( r l + r2)/rlr~, which yields p~' = --(rl + r~)n/rlr2.
where c12 = c21. Thus, from Eqs. [2] and [3], and the definition of q, we get w = q~/[2(oi + 2c12 -~- c22)].
[4]
To calculate cn, c2~., and c12, let us use the difference-equations obtained according to Journal of Colloid and Interface Bcience, Vol. 30, No. 4, A.ugus~ 1969
[11]
The coefficients cn can be expressed now as cll = e ~
l/p~ [12]
= e ~ r l r z / [ n ( r l + r2) -~ r~], n~0
COALESCENCE OF ELECTRICALLY CHARGED DROPLETS
539
we can replace Eq. [16] by
and, from symmetry,
Z / u ' 3 ( 1 - u ) "~ = (1 -
u ) -:/~
c~2 = e ~ ro'2/[n(rl ~- r2) ~- rl]. [13] n=0
" k {1/[(n -k 1)x -k n] n ~0
Similarly,
+ 1/[n(x + 1) + 1] 2 / [ ( n + 1 ) ( x + 1)]}
-
c12 = e k
l/p,( =~ a o
[14]
= [u-~l~x/(x + 1)] ~o
= --~ ~ rlr2/(n q- 1) (rl q- r2).
"~-~n~01 / ( { n - ~ 1 -- [ 1 / ( x + 1)]}
n=l
• {n + [1/(x ~- 1)]1)
Denoting rl = u*/aR and r2 = (1 - u)l/aR, and by [4] and [12] through [14], we can put the equation for AF in the dimensionless form
-- [2u-1/ax2/(x q- 1 )a] • ~ 1 / ( { n + 1 - [ 1 / ( . + 1)]} r~0
• {n -~ [1/(x + 1)]} (n ~- 1)).
G =- (AF/4~-R=~) = 1 + (a/2) - - [u 2/a q-
(1 -- u) m q- a / 2 Z ] ,
[18]
[15]
where a = q2/4rrRa(*e and
The first term on the right-hand side of Eq. [18] may be written as [u:l~/(1 -- x)] k {1/In q- 1 - 1/(x -k 1)] n~O
Z = u*/~(1 - u) 1/~ •k
(1/{n[ u~/a q- (1
-
= u-1/~x (x q- 1 ) / ( x -- 1 )
[19]
- - u ) 1/3] -}- u I/3}
=0
-- [2u-1/ax/(x -- 1 ) (x q- 1 )]
[16]
-1- 1/{n[u ~la q- (1 -- u) ~/a] Jr (1 -- u) */a} -- 2/(n +
1~In + 1 / ( x + 1)]}
•~
1)[u :/3 ~- (1 -- u):/a]).
[1/(x q- 1)]2},
which, by the identity (10)
IV. BEHAVIOR OF THE REDUCED FREEENERGY DIFFERENCE FUNCTION To study the dependence of the reduced free-energy difference G on the geometryof the initial aggregate, we shall make Eq. [16] more tractable, investigate the derivative dG/du, and find the extremal points of G = G(u, ~). A. The Reduced Capacity of the Aggregate• Designating
ul/a/(1 -- u ) 113 = x,
1~In 2 - -
[17]
k
1/{ n2 -- [1/(x + 1)]2}
= [(x + 1)/2]{x + 1 -- 7r etg[~r/(x + 1)]}, yields
[u-1/'a/(1 -- x)] "k
n~0
-- l/In
{1/In q- 1 -- 1 / ( x -+- 1)]
Jr 1 / ( x ~- 1)]}
[20]
= [~u-1/ax/(x -- 1 )] ctg [~/(x ~- 1 )]. The second term on the right-hand side of Journal of Colloid and Interface Science, Vol, 30~ No. 4, A u g u s t 1969
540
GALLILY AND AILAM
[18] may be expressed as
integer values of u except zero, and that
[2u-l%V (x + 1)31 ~l/.(.+g) •~
= (l/g)
[25]
1 / [ ( n + 1 -- g ) ( n + g ) ( n + 1)]
n=0
= A ~
f01 • 1 - --- : ~~g d~ _-- ~,
[21]
{1/[,ff - g)]
v=l
--
we obtain (Eqs. [18], [20], and [23])
1/[.(.
--
1 +
g)]},
w h e r e g = 1/(x + 1) a n d A
z = u1~3{- [~/(z + 1)]
= [(x -}-1)/
(1 - x)][2u-~%V (x + 1)~].
• ctg [~/(x +
Since
1 )]
[26]
+ 2 -- [2~/(x + 1)2]}.
E 1/[.(. - 1 + g)] v~l
= [1/(g-- 1)] ~ {[1/v -- 1/(v + g)] + [1/(v + g) -- 1/(v -- 1 + g)]}
[22
For computational convenience, we shall transform fl by applying repeatedly Kummer's transformation (11) while using the series ~ . = 1 1/u (i = 2, 3 , . . . k). Thus, we get
ao
= [1/(g-- 1)]~[1/v--
1/(u+
-
-
1/[g(g
(-1)q-(. + 2)g"
= ~
g)]
v~O
v=l
k
-- 1)],
= Z: ( - 1 ) ' ~ ( . + 2)g v
[27]
V~0
one gets + (--
1 )/~-}-ig/Z+l
A { ~ 1/[,(. -- g)] -- 1/[.(v - 1 + g)]}
[23]
- g)
Z = [x/(1 + xS) ~/~]
v=l
-- [(2g -- 1 ) / ( g
• {[~r/(x + 1)] ctg [~rx/(x + 1)]
-- 1)]
+ 2 - 2[~(2)/(x + 1) 2
.I.=~ 1 / . ( . + g)] + 1/g(g-- l)}. -
Thus, noting that (10) oo oo L / u ( . -- g) + ~ i / ( - - . ) ( - - . v=l
-- g)
v=l oo
=
~ ' 1 / . ( . - g)
= 1/g 2 -where ~ '
l/k+2 (, + g),
where ~(z) is Riemann's zeta function. Equation [26] may be written now in the form:
= A { ~ 1/[v(v -- g)
+ ~ 1/(-.)(-~
~ v~l
[24]
Or/g) ctg (vg),
stands for summation over all
Journal of Colloid and Interface Science, Vol. 30, No. 4, A u g u s t 1969
~(3)/(x
[28]
+ 1) ~ + . . . ] / .
It is worth while to mention that for x = 1 one obtains, by evaluating ~ (Eq. [25] ), that Z = 2m In 2. B. The change of G with u. Differentiating G in [15] with respect to u, one gets
dG/du = - { (2/~)[u -1/~ -- (1 - u) -in] -
(a/2Z) (dZ/du)l
[291
COALESCENCE OF ELECTRICALLY CHARGED DROPLETS and according to [26]
Furthermore, expanding the expressions contained within the brackets of [34] in a power series in x, we get
dZ/du = (u-2/3/3) •{ -
[~/(x + 1)] ctg [~/(x + 1)]
G = l - k (a/2)
~- 2 -- [2fl/ (z -~ 1)2]} --[1
~- {u-2r~x/[3(1 -- u)]}
1)2]ctg[Tr/(x
• ([Tr/(x +
[30]
(a/2)[1
-k
x~/3 --k
0@4)]
Hence, in the vicinity of x = 0, [36]
dG/dx ~ - 2x
where
and 1) 2] = --2fl/(x + 1) ~
(d/dx)[2~/(x •
+ [2/(x 4 1 ) 3 ] f01 ~11n29 ~ .
dG/du = (dG/dx) (dx/du) ~ -- 2/3x.
[31]
At u = 1/~ (corresponding to droplets of equal size), dZ/du = 0; so, dG/du = 0,
1 / ( 1 + x ~)2~
-
(~ (1 + x ~)~/<2xI [2 (~ +
[2~/~(x
+
2ul/3/3(1 _ u) Ij3.
l
-oo
dG/du--~ L oo
=
as
u~0;
as
u--~l•
1)]
-- (1 -- u)-ll~]/(dZ/du),
f G I d u 2 = (2/I)[u-4/~ + (1 -
u) -4/3]
we obtain, for the neighborhood of x = 0
-~ (a/2Z 2) ( f Z / d u 2) -- a (dZ/du )2/Z3.
(~ = o ) , G = 1 + (~/2)
To evaluate d2G/du 2 we use -
x~/(1 + x~)~/~
[2 (x + 1)/~r] tan [wx/(x + 1)]
+ [2~/~(x + 1)] • tan
[Trx/(x -k 1)] -k O(x~)}).
[411
fZ/du ~
[a(1 q- x~)113/2x]
= (d/dx)[(dZ/dx) (dx/du)] (dx/du)
• ([(x ~- 1 ) / v ] t a n [ ~ x / ( z ~- 1)] • { 1 --
[40]
and their nature is specified by
•tan [~rx/(x -]- 1 )]
--
[39]
( ~ / ) Z 2 [ u -1/~
[33]
•t a n [ ~ x / ( x + 1)]}))[(x + 1)/~r]
1/(1 + x3) 2/3
[3s]
c• The Extremal Points of G = G (u, a). The points at which dG/du = 0 are determined from (Eq. [29] )
1)/~]
•tan [~x/(x -t- 1 )] 1 -
da/du ~-- 2/3(1/x)
Thus,
G = 1 + (~/2) - x~/(1 + x~)~ -
[37]
For the neighborhood of u = 1, one has, by symmetry (Eq. [37]),
[32]
which could be expected from symmetry• The value of dG/du when u --~ 0 and when u --* 1 can be calculated as follows: Transforming the equation for G ([15], [17], and [26] ) into
--
[35]
• [1 + 0 (x~)] = - x ~ + 0 (x~).
(d/~x)[2~/(z + 1)~]),
-
-
-
-k x ~ -- 2x3/3 -k O(x4)]
+ 1)]
(x + 1)2sin 2 [r/(x + 1)]}
--v"/{
541
[34]
= ( f Z / d x 2) (dz/du) 2
[42]
+ (~Z/du) (~/dx) (dx/du)• From Eq. [26] (with u 1/3 expressed in terms Journal of Colloid and Interface Science, Vo]. 30, No. 4, A u g u s t 1969
542
GALLILY AND ALLAN[ TABLE I
THE I~EDUCED CAPACITY OF TItE Two DROPLETS AT VARIOUS VALUES OF X x
1.0013 1.0088
0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.91 0.93 0.95 0.97 0.98 1
1.0221 1.0385 1.0563 1.0724 1.0851 1.0940 1.0988 1.0991 1.0996 1.0999 1.10008 1.1002
dZ/du
0.1 0.2 0.3
1.384 0.882 0.601
0.4 0.5 0.6 0.7 0.8 0.9 0.95 0.98 1
0.416 0.290 0.200 0.133 0.081 0.037 0.018 0.007 0
E q u a t i o n [44] m a y be written in a more convenient form with the aid of Eq. [27]. F r o m Eq. [17] we get (dx/du) 9 ~nd (d/dx) (dx/du); consequently ([42], [43], and [44]), it becomes possible to calculate d2Z/du 2 and
1.1003
of x), we o b t a i n
d2Z/dx 2 = - [4x2/ (1 + x~) m] • {[~/(x +
1)] ctg [~rx/(x +
d~G/ du 2. Atu
1)]
+ 2 -- [23/(x + 1)~]} + [2/(1 + x~) 4/~]
= ~,
d~/du ~ =
[4(1//)-~/~/9]
+ ~ (d~Z/du~)u~=1/~/2'/~ In ~ 2,
+ 1 )~1 ctg [~x/(x + 1 )]
• (-[~/(x
-
x
Z
0.1 0.2
--7r2/{(x
TABLE II THE DERIVATIVEdZ/du AT VAI~IOUSVALUESOF X
+ 1)Bsin2[~rx/(x + 1)]}
(d/dx)[23/(z + 1)2])
-5 [x/(1 + x~)~/3]([2~r/(x -5 1) 3]
[43]
which shows t h a t the free-energy difference acquires a m i n i m u m or a m a x i m u m value when a is correspondingly smaller or greater t h a n % where ~/ -- -- 2 Im In 2 2/9 (d2Z/du 2)n=1/2
• etg [~rx/(x -5 1 )]
One should observe that, at x = 1, (d/dx) • [2~/(x -5 1 )~] = In 2 -- ~r2/8, and
-5 ~r2/{ (x + 1) 4 sin s [~rx/(x -5 1)]} -5 (~2{3(x -5 1) 2 sin [~rx/(x + 1)]
(d~/dx~) [23/(x + 1 )~]
+ 2~(x + 1) cos [~x/(z + 1)11/
= ~r2/4 -- In 2 -- 14~(3)/16• 2
{ (x + 1) 6 sin 3 [~rx/(x + 1)]}) -
(d/d~
~)[23/(~
Thus,
+ 1)~1),
in which (d/dx)[2~/(x [31], und
-5 1) ~] is given b y
(d2Z/du 2)~=1/~ = - (2va/9)[16 in 2 - 7~" (3)] ~nd -~ = 5.745• This is ~lso the v a l u e of a which character-
i(d~/dZ)[23/(z + ~)~1 = [43/(x + ~)~] -- [ 8 / ( x
-5 1) ~] jo 1 ~o1la-- } }d~
-- [ 2 / ( x
-5 1) ~]
2 Designating ~1/2 = ~, we have (12) : [44]
f / } ~ in ~ ( d~ 1 --
Journal of Colloid and Interface Science, Vol. 30, No. 4, August 1969
l~,/Zln ~} .
=
--8
¢l~aln ~0
£ln2e dO +
811(2a - 1)/2a12!ff(3)}.
543
C O A L E S C E N C E OF E L E C T R I C A L L Y C H A R G E D D R O P L E T S TABLE III
2.0
T H E E X T R E M A L P O I N T S O F G = G ( u , re) A T V A R I O U S VALUES
OF X
1.6
X
0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.8 0.9 0.95 0.98 1
1.2
8.695 6.166 5.458 5.293 5.336 5.451 5.577 5.664 5.728 5.741 5.744 5.745
dZ/du 0.8
0.4
o
'
' 0.2
'
0I 4
'
0 I6
0.8
1.0
x
Fla. 2. The derivative dZ/du vs. x
9 1.10 I
8
1.08
7 ct d
6
1.06
Z
I 1.04
40
0.2 . . . . 0.4
0I6
'
0 I8
' 1.0
x
1.02
F~G. 3. The extremal points of G = G(u, x) VS. o~. I .00 l
l
I
0.2
.I
0 4
[
I
0.6
I _ 018
I
I .0
x
FIG. 1. The reduced capacity of the two drop]ets vs. x.
0.10
izes t h e e x t r e m a l p o i n t a t u = 1/~ ( E q . [40] ), i.e.~
]ima u~l/2
-G
=
0.05
lira { (d/du) ( ~ ) [ u -~/3
u~l[2 --
(1
-
u)-'~]}/(d
2z
/du2 )~1~2 0
V. C O M P U T A T I O N S
The reduced capacity Z and the derivative
dZ/du a r e g i v e n f o r v a r i o u s v a l u e s of x i n
'0',2
'0'.4
'0'.6 x
'
0.8
'
1.0
FIO. 4. The reduced free-energy difference vs. x for a = 3. Journal of Colloid and Interface Science, Vol. 30, No. 4, A u g u s t
1969
544
G A L L I L Y AND A I L A M
Tables I and II, and Figs. 1 and 2. The values of a corresponding to the points at which d G / d u = 0 are shown in Table III and Fig. 3. Finally, the computed freeenergy differences are presented in Tables IV through VI, and Figs. 4 through 6. Here one should observe that since, from symmetry,
0.005
Z(x) = zo/x)
and 0
I
I
~
0.2
I
0.4
I
. ~ - -
01.8
0.6
I
~
1,0
x
FIo. 5. The reduced free-energy difference vs. x f o r ~ = 5.5.
[45]
(x, it is sufficient to carry out calculations for 0
~:~,EDUCED
FREE-ENERGY
VARIOUS VALUES
-G
-0.01[
-0.01
i
i
I __l
0.2
,
0.4
t
. ,
0.6
p
i
0,8
1.0
x
FIG. 6. The reduced free-energy difference vs. x for o~ = 6.
x
-G
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 1
0 0.0057 0.0105 0.0115 0.0110 O.OO9O 0.0082 0.0083 0.0086 0.0091 0.0092(0) 0.0092(3)
FREE-ENERGY
VARIOUS VALUES
AT
5.5
TABLE VI
T A B L E IV REDUCED
DIFFERENCE
O F X AND FOR t~ ~
DIFFERENCE
OF X AND FOR t~ ~
AT
3
THE
l~]~DUCED
FREE-ENERGY
VARIOUS VALUES
DIFFERENCE
OF ~: AND FO R (~ =
x
-G
x
6:
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 1
0 0.0074 0.0211 0.0384 0.0574 0.0756 0.0925 0.1064 0.116 0.1215 0.1228 0.1232
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.95 1
0 --0.0054 --0.0084 --0.0061 --0.0017 0.0044 0.0087 0.0113 0.0128 0.0134 0.0135 0.0136
Journal of Colloid and Interface Science, Vol. SO, No. 4, August 1969
6
AT
COALESCENCE OF ELECTRICALLY CHARGED DROPLETS VI. DISCUSSION AN]) CONCLUSIONS According to Fig. 3 and Eq. [36], one sees that, for a < 5.293, G is always negative and decreases monotonically with x to a minimum at x = 1. Hence, under these conditions, the process of coalescence is thermodynamically feasible. This process can occur even at a = 5.5 (Table V and Fig. 5). In the case of a system composed of two droplets of equal size, G =
i +
(~/2)
-
2 (½)2/~ _
(~/2/~ In 2),
which allows coalescence at a < 5.702. Note again that the adopted measure of feasibility is satisfactory at all values of a for G < 0;forG > 0, however (Table VI and Fig. 6), our analysis will provide a definite result only in the stable case. For most liquid aerosols in nature, the values of the parameter a of the individual droplets are smaller than 1 (13); consequently, these particles may coalesce as do the electrically neutral ones. NOMENCLATURE a, b--constants defined in text. A - - f u n c t i o n of x, defined in text. c,--coefficient of capacity of the i-th droplet. c~j--coeffcient of inductance of the i-th on the j - t h droplet. d--inter-center distance. AF--difference between the free energies of the system in its final and initial states. g = 1/(x+
1).
G = AF/4~R2(T.
p~/e--reciprocal of the n-th charge induced in sphere of radius h • q~-electric charge of the i-th droplet. q--electric charge of the final droplet. r~--radius of the i-th droplet. R - - r a d i u s of the final, spherical drop that may be produced. u = r ~ / R ~.
w--electrostatic potential of the i 4 h droplet.
545
v--e~ectrostatic potential of the initial aggregate. w--electrostatic energy of the initial aggregate. x - - r a t i o between radii of droplets in the initial aggregate. Z - - r e d u c e d capacity of the initial aggregate. Greek Letters
a--dimensionless
parameter
defined
by
a = q2/4~R~e.
fl = [1/(x + 1) 2]
jo1
1 --
~g
d~.
~--constant defined in text. e--dielectric permittivity of external medium. (z)--Riemann's zeta function. a--interfacial tension between liquid and external medium. ACKNOWLEDGMENT The authors wish to thank Mr. E. Fisher for carrying out the numerical calculations.
REFERENCES 1. FUCHS, N. A., "The Mechanics of Aerosols," pp. 302-5. Macmillan, New York, 1964. 2. IBID, pp. 34--42. 3. SCHNEIDER, J. M., LINDBLAD, N. R., AND HENDRICKS, C. D., J. Colloid Sci. 20, 610 (1965), and references cited therein. 4. LORD RAYLEIGH,Phil. Mag. 14, 184 (1882). 5. AILAM (VOLINEZ), G., AND GALLILY, I., The
Physics of Fluids 5, 575 (1962). 6. GALLILY, I., AND AILAM (VoLINEZ), G., J. Chem. Phys. 36, 1781 (1962). 7. JEANS, J., "The Mathematical Theory of Electricity and Magnetism," p. 95. Cambridge University Press, Cambridge, 1960. 8. ImD, pp. 93-4. 9. ImD, pp. 196-7. 10. RIEZHII(, E. M., AND GRADSHTEIN, E. C., "Tables of Integrals, Sums, Series and Products," p. 50 (in Russian). Moscow, 1951. 11. KNO:eI,, K., "Theory and Application of Infinite Series," p. 247. Blackie and Son, London and Glasgow, 1959 12. REF. 10, p. 214. 13. WHITB¥, K. T., AND LIU, B. Y. H., I n C. N. Davies, ed., "Aerosol Science," pp. 63, 65. Academic Press, New York, 1966.
Journal of Colloid and Interface Science, VoI.30, No. 4, August1969