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Improvement on the Hogg—Healy—Fuerstenau formulas for the interaction of dissimilar double layers
Improvement on the Hogg-Healy-Fuerstenau Formulas for the Interaction of Dissimilar Double Layers I. Second and Third Approximations for Moderate Potentials H I R O Y U K I O H S H I M A , T H O M A S W. HEALY, 1 AND LEE R. W H I T E Department o f Physical Chemistry, University o f Melbourne, Parkville, Victoria, 3052, Australia Received October 21, 198I; accepted February 3, 1982 Corrections to the fourth and sixth powers of surface potentials have been m a d e in the H o g g H e a l y - F u e r s t e n a u formulas for the double layer interaction at constant surface potential between two dissimilar plates and between two dissimilar spheres. A g r e e m e n t with the exact numerical values of the interaction of dissimilar plates given by Devereux and de Bruyn is considerably improved for low and moderate surface potentials.
Htickel linear approximation to the PoissonBoltmann equation, which gives a good approximation to the exact numerical values (4) for low potentials. Further, they obtained a simple formula for the interaction of two dissimilar spheres by applying Derjaguin's method (6). Although the H H F formula is applicable for sufficiently low potentials, it has been widely used in practice, e.g., heterocoagulation studies, because of its simple analytic form (7). In this paper, in an attempt to obtain analytic expressions applicable for low and moderate potentials, we make corrections to the fourth and sixth powers of surface potentials in the H H F formula both for the interaction of two dissimilar plates and for the interaction of two dissimilar spheres. We confine ourselves to the interaction at constant surface potential in a symmetrical electrolyte.
1. I N T R O D U C T I O N
The extension of the Derjaguin-LandauVerwey-Overbeek theory of the interaction between similar double layers (1, 2) to the interaction of dissimilar double layers was derived first by Derjaguin (3) who solved the Poisson-Boltzmann equation for a system of two dissimilar charged plates under the boundary condition that the surface potential of the plates remains constant, independent of plate separation. The relation between the interaction force and plate separation was obtained in terms of elliptic integrals. A more complete treatment was done by Devereux and de Bruyn (4), who also tabulated the exact numerical values of the interaction energy of two dissimilar plates. The formulas obtained by the above authors, however, are rather involved, since they require tedious numerical calculations, and are therefore not easily applicable to practical study. To this end, Hogg, Healy, and Fuerstenau ( H H F ) (5) derived a simplified formula for the interaction of two dissimilar plates at constant surface potential using the Debye-
2. I N T E R A C T I O N O F T W O DISSIMILAR PLATES
Consider two parallel plates 1 and 2 at a separation of h in a symmetrical electrolyte of valence v. We take an x-axis perpendicular to the plate surfaces and put its origin
To whom correspondence should be addressed. 484 0021-9797/82/100484-10502.00/0 Copyright© 1982by AcademicPress, Inc. All rights of reproductionin any form reserved.
Journalof Colloidand InterfaceScience, Vol. 89, No. 2, October 1982
485
DISSIMILAR DOUBLE LAYER INTERACTION
with Plate
1
Prate 2
y(O) = ysl,
[6]
y(Kh) = Ys2.
Now let us expand the right-hand side of [5] in a power series of y: d2y y3 y5 d~ 2 - y + ~ - + ~ - 0 + "'" [7]
0
h
~x
FIG. 1. Geometry of two parallel dissimilar plates 1 and 2 at separation h, each having constant surface potentials ~bsl and ff~2, respectively.
The first term on the right-hand side of [7] corresponds to the Debye-Htickel linear approximation, upon which the H H F formulas are based. According to [7], we express y as a double power series of Ysi and Y~2: y(~) = y(~)(~) + y(2)(~) + y(3)(~) + . . . [81
on the surface of the plate 1 (Fig. 1). The Poisson-Boltzmann equation for the electric potential if(x) relative to the bulk solution is (in SI units)
with y(l)(~) = y~IA1(~) + ys2Bl(~),
y(2)(~) = y3 A2(~) + y32B2(~)
2yen sinh verb
d2~b
Er60
dx 2 --
kr'
lp(0) = ~sl,
~/(h) = ~s2.
[2]
The following dimensionless parameters are introduced: ~=rx,
y=~,
ve Y~l=~b~l,
+ y2 yszA3(~) + Ysly~zB3(~), 2
[1]
where e is the elementary electric charge, n is the electrolyte concentration, er is the relative permittivity of the solution, Go is the permittivity of vacuum, k is the Boltzmann constant, and T is the absolute temperature. We assume that the surface potentials ff~l and ~bs2 of the plates 1 and 2, respectively, are constant, independent of separation h. The boundary conditions at x = 0 a n d x = h are then
[10]
y(3)(~) = Ys51A4(~) + y52B4(~)
+ y4 ys2A (O + 3
2
2
3
+ Y~Iys2A6(~) + y ~1y~2B6(~).
[ 1 1]
The functions A~(~), B~(~) (i = 1, 2 . . . . . 6) remain to be determined as set out below. Substituting [8]-[ 1 1 ] in [7] and equating terms of the same order in Y~I and Ys2 on both sides of [7], we obtain the following simultaneous equations for A~(~) and B,-(~): A'; - A I = 0, A~ - A2
6 '
De
ve Y~2=~ffs2,
A'~ - A3 -
A~B1 2
A~--
A~A2 - 2
[3] A 4 --
where = (2nv2e2] 1/2 r
[9]
kereokT]
[4]
is the Debye-Htickel parameter. Then, [ 1] with [2] reduces to
d2y d~ 2 = sinh y
A~ - A5 -
A~A3
2
A~ 120 '
-'b - -
A41B~ + AIBIA2 + - 24 '
A~B3 A~
-
A 6 -
2
B2A2 +
- -
2 3
[5 ]
2
AIBI + AIB1A3 + - 12 ' Journal o f Colloid and Interface Science,
[12]
VoL 89, No. 2, October 1982
486
OHSHIMA, HEALY, AND WHITE
together with another six equations obtained by the interchange Ai ~ B,- in the above equations, where the double prime denotes the second derivative with respect to ~. The boundary conditions [6] give A,(O) = 1,
Al(rh) = 0,
B'4(0) = - A '4(Kh ) =
B,(O) = 0,
6).
A'1(0) = - B ' l ( ~ h ) = - c o t h Kh,
[ 14]
= cosech Kh,
[151
B'~(O) = - A ' l ( r h )
5 3
+
1024 sinh 3 Kh
+
3 Kh cosh Kh 1024 sinh 5 rh - 256 sinh 4 Kh
--
15 Kh cosh Kh + 5(Kh) 2 1024 sinh 6 Kh 512 sinh 5 Kh 3(rh) 2 +256sinh 7rh'
[13]
As will be seen later, we need only the values of A ~ ( O ) , A ~ ( r h ) , B~(0), and B ~ ( r h ) (the prime denoting the first derivative with respect to ~) which are obtained from the solution of the extended equation set [ 12] and the boundary conditions [13]. After a little algebra, we obtain the results
5
1920 sinh Kh
Bl(O) = & ( r h ) = 0
(i = 2, 3 . . . . .
B'5(Kh)
1
-
B l ( K h ) = 1, Ai(O) = A i ( r h ) = O,
A'5(O)
B'5( O) = - A '5(Kh ) =
A'~(O) _
[20]
B'd,,h)
2
2
3 cosh Kh 512 sinh 3 Kh
15 cosh Kh 1024 sinh 5 Kh
+
Kh 17Kh + 192 sinh 2 Kh 256 sinh 4 r h
+
75rh 1024 sinh 6 rh
(rh) 2 cosh Kh 32 sinh 5 Kh
1
A'2(O) = - B ' 2 ( r h ) = - ~-~ coth Kh
cosh Kh + 16sinh 3 r h
Kh 16sinh 4 r h '
A'3(O) B'2( O) = - A '2(~h ) - - 3 1
16 sinh 3 rh Kh cosh Kh + 16 sinh 4 Kh '
+
[17] +
3Kh 16 sinh 4 Kh ' 1
1920
cosh Kh 512 sinh 3 Kh
1
13 15 + 512sinh 3Kh 512sinh 5Kh
3(Kh) 2
75Kh cosh rh 512 sinh 6 Kh 33(Kh) 2
+
128 sinh 3 rh
256 sinh 5 Kh 15(rh) 2
+
128 sinh 7 rh "
[22]
[18]
coth Kh
3 cosh Kh 1024 sinh 5 Kh
Kh 15rh + 128 sinh 4 ~h 1024 sinh 6 Kh 3(~h) 2 cosh Kh 256 sinh 7 Kh '
[21]
768 sinh Kh
17~h cosh Kh 256 sinh 4 rh
3 cosh Kh 16 sinh 3 Kh
rh 8 sinh: ~h
+
B~(0) = - A ~(xh) =
3
1
A'4(0) = - B ' 4 ( Kh ) -
[16]
B'3(Kh)
48 sinh r h
B'3( O) = - A '3(~h ) -
15(Kh) 2 cosh Kh 256 sinh 7 Kh '
[19]
Journal of Colloid and Interface Science, Vol. 89, No. 2, October 1982
The double layer free energy can be expressed in integral form by a charging process (2). For two identical plates with constant surface potential ~bs, the double layer free energy F per unit area is given by (2) F = -2
fo
a(~b)d4~ -- -27~s
2
~(X)dX,
[23]
DISSIMILAR DOUBLE LAYER INTERACTION where 4~ (=X~b~) and a(X), respectively, are the surface potential and the surface charge density of the plates at stage ), in the charging process (?, varying from 0 to 1). For a system of two dissimilar plates, [23] is generalized to F = -
~k
~rk(X)dX.
[24]
k=l
487
where A~(0), A'e(Kh), B~(O), and B'i(Kh) (i = 1, 2 . . . . ,6) are given by [14]-[22]. Since a~1), a~z), and a~3) (k = 1, 2) are, respectively, first-, third-, and fifth-degree homogeneous functions Ofysl and y~: (or, ~bs~and ~b~2),their values at stage X are proportional to X, X3, and Xs, respectively. Therefore, substituting [27] into [24] and integrating with respect to X, we obtain
The surface charge densities ~rk of the plate k ( k = 1, 2) are given by
F = F °) + F (2) -1- F (3) + • • . ,
[34]
FO ) = _ ~ 1 k=, 2 a(k~)g'~k'
[35]
2 1 F (2) = - Z ~ a~2)~bsk,
[36]
with or1 = --~r~0~ 1
iat.a, I
if2 =
,
[25]
X=0
~-ere0 7 • a N I x= h
[26]
Using [9]-[11], we write [25] and [26] as
k=l
--
I
F (3) = - - ~ g O'(k3),lPsk• + ...
( k = 1,2)
[27]
with
2nve
0-~ 1) :
-- - -
f f l 1) =
-.~ - K
0.~2) _
2nve
2nve
[YslA'~(0) +y~2Bi(0)],
[28]
[y~tA'~(Kh) + y~zB i&h)], [29]
[371
k=l
The potential energy of double layer interaction V(h) per unit area of two plates at separation h is given by the value of F at separation h minus that for infinite separation:
V(h) = F(h) - F(oo). [y~lA ~(0) + y2zBi(0)
K 2 ! + y~lYs2A'3(O) + ys~YszB3(O)],
[30]
2nve ~r(22)= + - [y~IA '2(Kh) + y32B'2(Kh)
[38]
We define the first, second, and third approximations to V(h) as V~, V2, and V3, respectively, which are V]
= F(l)(h)
-
F(~)(~),
[39]
K 2 + ys~ ys:A t3(Kh) + Ysl y2zB'3(r~h)],
0.(31) _
2nve
5 [y~IA
t 4(0)
Vz = { F ° ) ( h ) + F(2)(h)} [31]
5 t -~- Y s 2 B 4 ( 0 )
+ Y4slYs2A'5(0) + Ysl Ys2 4 B ,5(O)
- {F°)(~) [32]
2rive o-~2) = + [y~A'4(Kh ) + y~zB5 '4(Kh) K 4
4
+ F ( 2 ) ( ~ ) + F(3)(~)}.
+ y~lY~EA'5(rh) + y~ly~2Bs(rh) [33]
[41]
Substituting [28]-[33] (with [14]-[22]) into [35]-[37] and introducing x+~ - Y s l _ _Ys2, + 2
r
3 2 2 3 ¢ + y~lY~2A'6(Kh) + ys~ys2B6(Kh)],
[40]
V~ = { F ° ) ( h ) + F(2)(h) + F(3)(h)}
K
3 2 2 3 +y~ys2A'6(O) + YslYs2B6(O)],
-- {F(1)(o0) + F(2)(c~)},
ii_ _ Ysl - Ys2 2
,
[42]
from [39]-[41] we obtain Journal of Colloid and Interface Science, Vol. 89, No. 2, October 1982
488
OHSHIMA, HEALY, AND W H I T E
V1 - 2nkT [y2+(1 _ tanh (rh/2)} - y2{coth (Kh/2) - 1}],
[43]
K
112 = V, +
-
( y 4 + 3yzy2){1 _ tanh (Kh/2)} -
( y 4 + 3y2+y2)
K
X {coth (Kh/2) - 1} - y4+ tanh (Kh/2) y 4 coth (Kh/2) 3--2"cosh 2 (Kh/2) + 32 sinh 2 (rh/2) (Kh/2) I _y2 32 tcosh 2 (rh/2) V3=V2+
2nkT F y2 { y a + ~ y 2 _ ( 7 y 2 i
5760 y4 +
.~21
y2 sinh 2 (Kh/2)) .]'
[44]
+ y2_)}{l_tanh(rh/2)}
L5-76-0
yZ+(7y2 + y2) {coth ( r h / 2 ) - 1) 3
+ 1-~
2 y 2 cosh2 (Kh/2)
1536 y 2 + 2 y2+ sinh 2 (~h/2)
y6+ tanh (Kh/2) _ y 6 coth (rh/2) - 1024 cosh4 (Kh/2) 1024 sinh 4 (Kh/2) (xh/2) ~. y4 + ~ [cosh 4 (Kh/2)
(rh/2)2 ~ + - 256
r 2 [cosh 2 (rh/2)
7(Kh/2) I y2 3072 [cosh 2 (Kh/2)
y2 .~3 sinh 2 (Kh/2))
Y% .~sy2 tanh 2 (Kh/2) - y 2 coth 2 (Kh/2)} sinh4 (Kh/Z)J / + y2 ~21 tanh (rh/2) sinh 2 (rh/2)) ( Y 2 cosh2 (rh/2)
The first approximation V1([43]) agrees with the HHF formula for the interaction of two dissimilar plates (Eq. [6] of Ref. (5)). We also note that for a special case when Y~I = Ys2 (i.e., Y_ = 0), [43]-[45] reduce to an expansion of the elliptic integral expression for the interaction energy of two similar plates in powers of surface potential derived by Levine and Suddaby (Eq. [37b] of Ref. (8)). 3. I N T E R A C T I O N OF T W O D I S S I M I L A R L A R G E S P H E R E S AT S M A L L S E P A R A T I O N S
Following the treatment of HHF, we apply Derjaguin's method (6) to derive an expression for the interaction energy of two dissimilar large spheres at small separations. Consider two spheres, each having radii al, a2 and surface potentials ~bs~, ~b~2, respectively. Let H be the shortest distance between two spheres (Fig. 2) and assume that ~k~l and ~b,2 are constant, independent of H. Journal of Colloid and Interface Science, Vol. 89, No. 2, October 1982
r 2 coth (rh/2)~l sinh 2 (rh/2)JJ
[45]
When the conditions: H ~ al, a2
[46]
Kal>> 1, Ka2>> 1
[47]
and are satisfied, Derjaguin's method (6) can be employed, which makes it possible to obtain the interaction energy Vsp of two spheres by integrating the interaction energy V per unit area of two plates ([38]) using the following formula (5):
FIG. 2. Geometry of two dissimilar spheres of radii al and a2 at a shortest separation of H.
DISSIMILAR
V(h)dh.
27rala2 al + a2
V~p(H)
[48]
Substituting [43]-[45], we obtain first, second, and third approximations to V~p, denoted as V~pl, Vsp2, and Vsp3, respectively:
87ralaznkT f e-KH)+ y2_ logo (1 +~2) LY2+logo (1 +
Vspl = ~
Vsp2 = Vspl '~
87rala2nkT[-
K2(al + a2)
l(y
4+
y4+ 1 - (KH/2) tanh (KH/2) 96 cosh 2 (KH/2)
81rala2nkr[
y2_ + 5 - - ~ { Y4- + ~
[49]
-- 1} y 4 (KH/2) coth (KH/2) -- 17 96 sinh 2 (KH/2) .J '
[50]
r2+ {y4 +l~ y2_(Vy2++ y2)}(gH/2){l_tanh(gH/2) )
5760
Y2+(7y2-+ Y2)}(KH/2){coth (KH/2)
+ y 6 17 + 4(~H/2) tanh (KH/2) _ 46080 cosh2 (KH/2)
y6_.4(rH/2)
+ y4yZ
yE+y4_ (~H/2)
_
-- e-~U)],
3y2+yE_)(rH/2){1 _ tanh (KH/2)}
+ ~1 (y4_ + 3y2y2_)(KH/2){coth (KH/2)
Vsp3 = Vsp2+ K2(aI + a2)
489
DOUBLE LAYER INTERACTION
1 + (rH/2) tanh (KH/2) _ 1024 cosh z (rH/2)
1}
coth (KH/2) + 17 46080 sinh 2 (KH/2) coth (KH/2) + 1 1024 sinh 2 (~H/2)
y 6 1 - I I(KH/2) tanh (rH/2) + y 6 l l(KH/2) coth (KH/2) - 1 15360 cosh4 (KH/2) 15360 sinh4 (KH/2) +
(KH/2)2 I y2+ y2 ~3q 1536 tcosh = (,~H/2) - sinh = (KH/2)J d '
[51]
where Y+ and Y_ are defined by [42]. The first approximation V~pl ([49]) is the HHF formula for the interaction of two dissimilar spheres (Eq. [21] of Ref. (5)). For the case of two similar spheres (~1 = ffs2, (i.e., Y_ = 0) and al = a 2 ) [49]-[51] reduce to a series expression (Eq. [4.5] of Ref. (9)) which Hoskin and Levine derived by applying Derjaguin's method to the series expansion (Eq. [37b] of Ref. (8)) of the interaction energy of two similar plates. 4. D I S C U S S I O N
The HHF formula (5) for the interaction of dissimilar double layers at constant surface potential is based on the Debye-Htickel linear approximation. This approximation retains only the first term of the expansion [7] of the Poisson-Boltzmann equation, giving the interaction energy correct to quadratic order in the surface potentials. The HHF formula is thus applicable for sufficiently low potentials.
In the present study we have truncated the expansion of sinh y at the y5 term in [7] and obtained higher order approximations to the electrostatic potential and interaction energy expressed as a power series in surface potentials. The first approximation ([43] for the plate-plate interaction and [49] for the sphere-sphere interaction) agrees with the HHF formula. The second and third approximations ([44] and [45] for the plateJournal of Colloid and Interface Science, Vol. 89, No. 2, October 1982
490
OHSHIMA, HEALY, AND WHITE TABLE I
Percentage Relative Error Ei (i = 1, 2, 3) (%) in the First (HHF), Second, and Third Approximations V/ (i = 1, 2, 3) ([43]-[45]) to the Interaction Energy of Two Dissimilar Plates as a Function of Y~I and Y~2 for Kh = 0.5, 1, and 2a xh = 1
~h=0.5
Y.I
Yd
3.0
3.0 2.2 1.4 -1.4 -2.2 -3.0
2.6
el
~2
~3
+1.3 -1.4 -13.5 +4.9 +4.4 +5.3
+3.3 +2.1 +2.3 +0.11 +0.13 +0.20
-0.21 -0.038 +0.14 -0.019 -0.011 -0.012
2.6 1.8 1.0 -1.0 -1.8 -2.6
+0.40 -2.1 -28.2 +3.9 +3.3 +3.9
+1.9 +1.2 +2.1 +0.059 +0.072 +0.11
2.2
2.2 1.4 0.6 -0.6 -1.4 -2.2
-0.077 -2.5 +30.8 +3.2 +2.3 +2.8
1.8
1.8 1.0 -1.0 -1.8
-0.26 -3.0 +1.6 +1.8
~1
xh = 2
~2
~3
~1
~2
~3
+19.9 +13.6 +5.4 +9.1 +8.9 +11.0
+0.22 +0.66 +1.7 -0.32 -0.12 -0.084
-1.8 -0.88 -0.52 -0.068 -0.054 -0.078
+40.9 +30.4 +21.2 +15.0 +16.7 +21,4
-15.2 -8.5 -4.2 -1.9 -1.6 -2.3
-0.071 0.000 +0.16 -0.012 -0.006 -0.006
+14.4 +9.1 +0.49 +7.2 +6.5 +8.0
+0.30 +0.51 +1.4 -0.22 -0.077 -0.038
-0.78 -0.34 -0.21 -0.029 -0.029 -0.035
+30.5 +21.6 +13.8 +11.4 +12.0 +15.8
-8.5 -4.3 -1.8 -1.2 -0.89 -1.3
+2.4 +0.86 +0.17 +0.074 +0.015 -0.014
+0,97 +0.58 -0.88 +0.024 +0.034 +0.057
-0.021 +0.005 -0.10 -0.009 -0.004 -0.004
+10.0 +5.5 -8.0 +5.8 +4.5 +5.6
+0.24 +0.34 +1.5 -0.16 -0.044 -0.019
-0.30 -0.11 -0.087 -0.010 -0.011 -0.017
+21.7 +14.4 +7.5 +8.7 +8.3 +11.1
-4.3 -1.9 -0.44 -0.72 -0.45 -0.62
+0.88 +0.25 -0.005 +0.044 +0.007 -0.007
+0.44 +0.26 +0.012 +0.024
-0.006 +0.004 -0.004 -0.002
+6.4 +2,7 +3.0 +3.7
+0.14 +0.19 -0.024 -0.006
-0.090 -0.030 -0.003 -0.006
+14.5 +8.6 +5.4 +7.3
-1.9 -0.70 -0.21 -0.27
+0.27 +0.053 +0.004 -0.002
+5.7 +2.4 +0.74 +0.13 +0.028 -0.025
"e~ > 0 corresponds to overestimation and ~ < 0 to underestimation.
plate i n t e r a c t i o n a n d [50] a n d [51] for the s p h e r e - s p h e r e i n t e r a c t i o n ) correspond, respectively, to corrections to the fourth a n d to the sixth power of surface potentials. For the special case of two similar d o u b l e layers, our series expansions [ 4 3 ] - [ 4 5 ] a n d [ 4 9 ] - [ 5 1 ] , respectively, reduce to t h a t derived b y Levine a n d S u d d a b y (8) for the int e r a c t i o n of two s i m i l a r plates a n d t h a t derived b y H o s k i n a n d Levine (9) for the interaction of two similar spheres; these authors o b t a i n e d their e q u a t i o n s on the basis of the e x p a n s i o n of the i n t e r a c t i o n e n e r g y of two s i m i l a r plates expressed in t e r m s of elliptic integrals. Levine a n d S u d d a b y (8) considered the a p p l i c a b i l i t y of the series e x p a n s i o n for the i n t e r a c t i o n of s i m i l a r plates. H o n i g Journal o f Colloid and Interface Science, Vol. 89, No. 2, October 1982
a n d M u l (10) p r e s e n t e d a m o d i f i c a t i o n of the e q u a t i o n s derived b y Levine a n d Sudd a b y (8) a n d b y H o s k i n a n d Levine (9). Let us now c o m p a r e our a p p r o x i m a t i o n s for the i n t e r a c t i o n e n e r g y of two d i s s i m i l a r plates ( [ 4 3 ] - [ 4 5 ] ) with the exact n u m e r i c a l values o b t a i n e d b y D e v e r e u x a n d de B r u y n (4). W e c a l c u l a t e the p e r c e n t a g e relative error defined b y V~-V V
× lO0 (%)
(i = 1, 2, 3),
[521
where Vi (i = 1, 2, 3) are the values calculated with the first ( [ 4 3 ] ) , second ( [ 4 4 ] ) , a n d t h i r d ( [ 4 5 ] ) a p p r o x i m a t i o n s , respectively,
DISSIMILAR DOUBLE LAYER INTERACTION
potentials are of unlike sign or either of them is zero (7). Let the position and height of the potential maximum be hm and Vroax, respectively, and assume Ysl > Ys2 > 0 without loss of generality. Exact equations for hm and Vmaxcan be derived from Eqs. [2.38], [3.61 ], and [3.68] of Ref. (4):
A~_ Kv,
2nkT" 2nkT ( 1-1.2.3) I
Y,,-3.0, Y . - 1.4
'\
[tanh (Y~l/4)] logo [_tanh ~ l '
[53]
32nkT Vmax -- - sinh 2 (ys2/4). K
[54]
• .\ \
rhm ,
i
491
i
=
The first approximation V~ ([43]), that is, the H H F formula gives (Eqs. [44] and [45] of Ref. (7))
-1
rhm
Ys,'3-O, Ys,'- I-4
=
Vmax 0
1
2
3
4
5
Kh
FIG. 3. Potential curves for the interaction of two dissimilar plates with Y,I = 3.0, Ya = 1.4, and with ys] = 3.0, Ys2= -1.4. - - , exact values V (4), - - -, first approximation (HHF) 1:1, calculated with [43], -----, second approximation I:2,calculated with [44]. Curves for the third approximationcalculated with [45] coincide with the exact curves within the width of lines.
and Vis the exact numerical value (4). Table I shows Ei as a function of ys~ and Ys2 for Kh = 0.5, 1, and 2. We see that in the second and third approximations agreement with the exact values is considerably improved for small and moderate surface potentials (Ys], Y~2 ~< 3). In a few exceptional cases, a lower approximation is better than a higher approximation, because the potential curves for V, V~, V2, and/:3 intersect each other. Figure 3 shows two examples of the potential curves for V, 1:1, IrE, and 1:3. As is seen in Fig. 3, a characteristic feature of the interaction of dissimilar plates at constant surface potential is that for the case of plates with surface potentials of like sign the potential curve exhibitsa maximum; while it is always negative when the surface
logo (y~JYs2),
2nkT K
Ys~.
[55] [56]
We note that [54] and [56] are independent of the higher surface potential Y~I. The second (V2 ([44])) and third (1:3 ([45])) approximations, however, give no simple analytic expressions for hm and Vmax, and the values of hm calculated by these approximations depend (but negligibly) on ys,. Table II compares the exact values of hm ([53]) and Vmax ([54]) with approximate values calculated with 1:1 ([55] and [56]), V:, V3 together with their relative errors (defined similarly to [52]), showing that agreement with exact values is improved in higher approximations. Another approximation involved in the derivation of the H H F formula for the interaction of two dissimilar spheres is the application of Derjaguin's method (6), which is applicable for the interaction of large spheres (Kal, ~a2 >> 1) at small separations ( H ,~ al, a2). In the present work, we followed Derjaguin's method after making corrections to moderate potentials and obtained [49]-[51], which are therefore applicable only for large a~, a2 and small H. Several approximations for the interaction of dissimilar spheres have been obtained without recourse to Derjaguin's method (6). Bell et al. Journal of Colloid and Interface Science, Vol, 89, No. 2, October 1982
492
OHSHIMA, HEALY, AND W H I T E TABLE II
Position (hm) and Height (Vmax) of Potential Maximum Calculated with Exact Equations and the First ( H H F ) (1), Second (2), and Third (3) Approximations as a Function of Y~I and ys2a xVm~/ 2nk T
rhm Y,i
Yl2
Exact
1
2
3
Exact
3.3
2.2
0.2382
0.3102 (+30.2)
0.2473 (+3.8)
0.2384 (+0.090)
5.348
4.840 (-9.5)
5.392 (+0.82)
5.357 (+0.16)
1.4
0.6356
0.7621 (+19.9)
0.6366 (+0.15)
0.6317 (-0.61)
2.041
1.960 (-4.0)
2.092 (+2.5)
2.040 (-0.048)
1.8
0.3038
0.3677 (+21.0)
0.3096 (+1.9)
0.3037 (-0.048)
3.465
3.240 (-6.5)
3.448 (+0.66)
3.467 (+0.071)
1.0
0.8476
0.9555 (+12.7)
0.8425 (-0.60)
0.8455 (-0.25)
1.021
1.000 (-2.1)
1.038 (+1.7)
1.020 (-0.13)
1.4
0.3974
0.4520 (+13.7)
0.4004 (+0.76)
0.3972 (-0.060)
2.041
1.960 (-4.0)
2.051 (+0.47)
2.042 (+0.022)
0.6
1.2125
1.2993 (+7.2)
1.2037 (-0.72)
1.2122 (-0.020)
0.3627
0.3600 (-0.75)
0.3664 (+1.0)
0.3623 (-0.12)
1.0
0.5438
0.5878 (+8.1)
0.5447 (+0.16)
0.5437 (-0.031)
1.021
1.000 (-2.1)
1.024 (+0.28)
1.021 (+0.002)
2.6
2.2
1.8
1
2
3
a Percentage relative error (%) is given in parentheses; positive error corresponds to overestimation and negative error to underestimation.
(11) transformed the linearized form of the Poisson-Boltzmann equation [ 1] into an integral equation and so obtained an approximate expression for the interaction energy at large a~, a2 and all H for small potentials. In the same paper (11 ) Bell et al., using the superposition approximation, derived the general asymptotic result (i.e., valid for all surface potentials and radii al, a2,) Vsp(H) =
87rala2nkT
YIY2
K2
H + al + a2
(KH>> 1),
e_m [57]
where YI and Y2 are the coefficients in the asymptotic form for the dimensionless potential distribution yi(r) around isolated spheres
the exact form for the Yi is not known in general. Elsewhere (12) we have obtained accurate analytic approximations to these coefficients which allows an accurate analytic expression for Vsp(H) when r H ~> 1 for all particle radii and surface potentials. Barouch et al. (13) derived an approximate expression without using the Debye-Htickel linearization and Derjaguin's method, but they still used the linearized form for the double layer free energy (our first approximation F (1) in [35]). In a comment on the work of Barouch et al. (13), Chan and White (14) stated that the next-order corrections on Derjaguin's method should give a correction term of orders in (Kai)-l (i = 1, 2). We shall consider these corrections in Part II of this series.
yi(r) = Y~aie-~('-a')
(i=1,2), [58] r (where r is radial distance from center of sphere i). For small potentials Yi ~ Ysi but Journal of Colloid and Interface Science, Vol. 89, No. 2, October 1982
ACKNOWLEDGMENTS One of us (H.O.) acknowledges a Visiting Research Fellowship from the University of Melbourne. This work
DISSIMILAR DOUBLE LAYER INTERACTION was supported by the Australian Research Grants Committee. REFERENCES 1. Derjaguin, B. V., and Landau, L. D., Acta Physicochirn. 14, 633 (1941). 2. Verwey, E. J. W., and Overbeek, J. Th. G., "Theory of the Stability of Lyophobic Colloids." Elsevier, Amsterdam, 1948. 3. Derjaguin, B. V., Discuss. Faraday Soc. 18, 85 (1954). 4. Devereux, O. F., and de Bruyn, P. L., "Interaction of Plane-Parallel Double Layers." M.I.T. Press, Cambridge, 1963. 5. Hogg, R., Healy, T. W., and Fuerstenau, D. W., Trans. Faraday Soc. 62, 1638 (1966). 6. Derjaguin, B. V., Kolloid Z. 69, 155 (1934). 7. Usui, S., in "Progress in Surface and Membrane
8. 9. 10. 11. 12. 13.
493
Science" (J. F. Danielli, M. D. Rosenberg, and D. A. Cadenhead, Eds.), Vol. 5, p. 223. Academic Press, New York, 1972. Levine, S., and Suddaby, A., Proc. Phys. Soc. A64, 287 (1951). Hoskin, N. E., and Levine, S., Phil Trans. Roy. Soc. London A248, 449 (1956). Honig, E. P., and Mul, P. M., J. Colloid Interface Sci. 36, 258 (1971). Bell, G. M., Levine, S., and McCartney, L. N., J. Colloid Interface Sci. 33, 335 (1970). Ohshima, H., Healy, T. W., and White, L. R., J. Colloid Interface Sci., submitted. Barouch, E., Matijevi6, E., Ring, T. A., and Finlan, J. M., J. Colloid Interface Sci. 67, 1 (1979); 70, 400 (1979).
14. Chan, D. Y. C., and White, L. R., J. Colloid Interface Sci. 74, 303 (1980).
Journal of Colloid and Interface Science, Vol. 89, No. 2, October 1982
Htickel linear approximation to the PoissonBoltmann equation, which gives a good approximation to the exact numerical values (4) for low potentials. Further, they obtained a simple formula for the interaction of two dissimilar spheres by applying Derjaguin's method (6). Although the H H F formula is applicable for sufficiently low potentials, it has been widely used in practice, e.g., heterocoagulation studies, because of its simple analytic form (7). In this paper, in an attempt to obtain analytic expressions applicable for low and moderate potentials, we make corrections to the fourth and sixth powers of surface potentials in the H H F formula both for the interaction of two dissimilar plates and for the interaction of two dissimilar spheres. We confine ourselves to the interaction at constant surface potential in a symmetrical electrolyte.
1. I N T R O D U C T I O N
The extension of the Derjaguin-LandauVerwey-Overbeek theory of the interaction between similar double layers (1, 2) to the interaction of dissimilar double layers was derived first by Derjaguin (3) who solved the Poisson-Boltzmann equation for a system of two dissimilar charged plates under the boundary condition that the surface potential of the plates remains constant, independent of plate separation. The relation between the interaction force and plate separation was obtained in terms of elliptic integrals. A more complete treatment was done by Devereux and de Bruyn (4), who also tabulated the exact numerical values of the interaction energy of two dissimilar plates. The formulas obtained by the above authors, however, are rather involved, since they require tedious numerical calculations, and are therefore not easily applicable to practical study. To this end, Hogg, Healy, and Fuerstenau ( H H F ) (5) derived a simplified formula for the interaction of two dissimilar plates at constant surface potential using the Debye-
2. I N T E R A C T I O N O F T W O DISSIMILAR PLATES
Consider two parallel plates 1 and 2 at a separation of h in a symmetrical electrolyte of valence v. We take an x-axis perpendicular to the plate surfaces and put its origin
To whom correspondence should be addressed. 484 0021-9797/82/100484-10502.00/0 Copyright© 1982by AcademicPress, Inc. All rights of reproductionin any form reserved.
Journalof Colloidand InterfaceScience, Vol. 89, No. 2, October 1982
485
DISSIMILAR DOUBLE LAYER INTERACTION
with Plate
1
Prate 2
y(O) = ysl,
[6]
y(Kh) = Ys2.
Now let us expand the right-hand side of [5] in a power series of y: d2y y3 y5 d~ 2 - y + ~ - + ~ - 0 + "'" [7]
0
h
~x
FIG. 1. Geometry of two parallel dissimilar plates 1 and 2 at separation h, each having constant surface potentials ~bsl and ff~2, respectively.
The first term on the right-hand side of [7] corresponds to the Debye-Htickel linear approximation, upon which the H H F formulas are based. According to [7], we express y as a double power series of Ysi and Y~2: y(~) = y(~)(~) + y(2)(~) + y(3)(~) + . . . [81
on the surface of the plate 1 (Fig. 1). The Poisson-Boltzmann equation for the electric potential if(x) relative to the bulk solution is (in SI units)
with y(l)(~) = y~IA1(~) + ys2Bl(~),
y(2)(~) = y3 A2(~) + y32B2(~)
2yen sinh verb
d2~b
Er60
dx 2 --
kr'
lp(0) = ~sl,
~/(h) = ~s2.
[2]
The following dimensionless parameters are introduced: ~=rx,
y=~,
ve Y~l=~b~l,
+ y2 yszA3(~) + Ysly~zB3(~), 2
[1]
where e is the elementary electric charge, n is the electrolyte concentration, er is the relative permittivity of the solution, Go is the permittivity of vacuum, k is the Boltzmann constant, and T is the absolute temperature. We assume that the surface potentials ff~l and ~bs2 of the plates 1 and 2, respectively, are constant, independent of separation h. The boundary conditions at x = 0 a n d x = h are then
[10]
y(3)(~) = Ys51A4(~) + y52B4(~)
+ y4 ys2A (O + 3
2
2
3
+ Y~Iys2A6(~) + y ~1y~2B6(~).
[ 1 1]
The functions A~(~), B~(~) (i = 1, 2 . . . . . 6) remain to be determined as set out below. Substituting [8]-[ 1 1 ] in [7] and equating terms of the same order in Y~I and Ys2 on both sides of [7], we obtain the following simultaneous equations for A~(~) and B,-(~): A'; - A I = 0, A~ - A2
6 '
De
ve Y~2=~ffs2,
A'~ - A3 -
A~B1 2
A~--
A~A2 - 2
[3] A 4 --
where = (2nv2e2] 1/2 r
[9]
kereokT]
[4]
is the Debye-Htickel parameter. Then, [ 1] with [2] reduces to
d2y d~ 2 = sinh y
A~ - A5 -
A~A3
2
A~ 120 '
-'b - -
A41B~ + AIBIA2 + - 24 '
A~B3 A~
-
A 6 -
2
B2A2 +
- -
2 3
[5 ]
2
AIBI + AIB1A3 + - 12 ' Journal o f Colloid and Interface Science,
[12]
VoL 89, No. 2, October 1982
486
OHSHIMA, HEALY, AND WHITE
together with another six equations obtained by the interchange Ai ~ B,- in the above equations, where the double prime denotes the second derivative with respect to ~. The boundary conditions [6] give A,(O) = 1,
Al(rh) = 0,
B'4(0) = - A '4(Kh ) =
B,(O) = 0,
6).
A'1(0) = - B ' l ( ~ h ) = - c o t h Kh,
[ 14]
= cosech Kh,
[151
B'~(O) = - A ' l ( r h )
5 3
+
1024 sinh 3 Kh
+
3 Kh cosh Kh 1024 sinh 5 rh - 256 sinh 4 Kh
--
15 Kh cosh Kh + 5(Kh) 2 1024 sinh 6 Kh 512 sinh 5 Kh 3(rh) 2 +256sinh 7rh'
[13]
As will be seen later, we need only the values of A ~ ( O ) , A ~ ( r h ) , B~(0), and B ~ ( r h ) (the prime denoting the first derivative with respect to ~) which are obtained from the solution of the extended equation set [ 12] and the boundary conditions [13]. After a little algebra, we obtain the results
5
1920 sinh Kh
Bl(O) = & ( r h ) = 0
(i = 2, 3 . . . . .
B'5(Kh)
1
-
B l ( K h ) = 1, Ai(O) = A i ( r h ) = O,
A'5(O)
B'5( O) = - A '5(Kh ) =
A'~(O) _
[20]
B'd,,h)
2
2
3 cosh Kh 512 sinh 3 Kh
15 cosh Kh 1024 sinh 5 Kh
+
Kh 17Kh + 192 sinh 2 Kh 256 sinh 4 r h
+
75rh 1024 sinh 6 rh
(rh) 2 cosh Kh 32 sinh 5 Kh
1
A'2(O) = - B ' 2 ( r h ) = - ~-~ coth Kh
cosh Kh + 16sinh 3 r h
Kh 16sinh 4 r h '
A'3(O) B'2( O) = - A '2(~h ) - - 3 1
16 sinh 3 rh Kh cosh Kh + 16 sinh 4 Kh '
+
[17] +
3Kh 16 sinh 4 Kh ' 1
1920
cosh Kh 512 sinh 3 Kh
1
13 15 + 512sinh 3Kh 512sinh 5Kh
3(Kh) 2
75Kh cosh rh 512 sinh 6 Kh 33(Kh) 2
+
128 sinh 3 rh
256 sinh 5 Kh 15(rh) 2
+
128 sinh 7 rh "
[22]
[18]
coth Kh
3 cosh Kh 1024 sinh 5 Kh
Kh 15rh + 128 sinh 4 ~h 1024 sinh 6 Kh 3(~h) 2 cosh Kh 256 sinh 7 Kh '
[21]
768 sinh Kh
17~h cosh Kh 256 sinh 4 rh
3 cosh Kh 16 sinh 3 Kh
rh 8 sinh: ~h
+
B~(0) = - A ~(xh) =
3
1
A'4(0) = - B ' 4 ( Kh ) -
[16]
B'3(Kh)
48 sinh r h
B'3( O) = - A '3(~h ) -
15(Kh) 2 cosh Kh 256 sinh 7 Kh '
[19]
Journal of Colloid and Interface Science, Vol. 89, No. 2, October 1982
The double layer free energy can be expressed in integral form by a charging process (2). For two identical plates with constant surface potential ~bs, the double layer free energy F per unit area is given by (2) F = -2
fo
a(~b)d4~ -- -27~s
2
~(X)dX,
[23]
DISSIMILAR DOUBLE LAYER INTERACTION where 4~ (=X~b~) and a(X), respectively, are the surface potential and the surface charge density of the plates at stage ), in the charging process (?, varying from 0 to 1). For a system of two dissimilar plates, [23] is generalized to F = -
~k
~rk(X)dX.
[24]
k=l
487
where A~(0), A'e(Kh), B~(O), and B'i(Kh) (i = 1, 2 . . . . ,6) are given by [14]-[22]. Since a~1), a~z), and a~3) (k = 1, 2) are, respectively, first-, third-, and fifth-degree homogeneous functions Ofysl and y~: (or, ~bs~and ~b~2),their values at stage X are proportional to X, X3, and Xs, respectively. Therefore, substituting [27] into [24] and integrating with respect to X, we obtain
The surface charge densities ~rk of the plate k ( k = 1, 2) are given by
F = F °) + F (2) -1- F (3) + • • . ,
[34]
FO ) = _ ~ 1 k=, 2 a(k~)g'~k'
[35]
2 1 F (2) = - Z ~ a~2)~bsk,
[36]
with or1 = --~r~0~ 1
iat.a, I
if2 =
,
[25]
X=0
~-ere0 7 • a N I x= h
[26]
Using [9]-[11], we write [25] and [26] as
k=l
--
I
F (3) = - - ~ g O'(k3),lPsk• + ...
( k = 1,2)
[27]
with
2nve
0-~ 1) :
-- - -
f f l 1) =
-.~ - K
0.~2) _
2nve
2nve
[YslA'~(0) +y~2Bi(0)],
[28]
[y~tA'~(Kh) + y~zB i&h)], [29]
[371
k=l
The potential energy of double layer interaction V(h) per unit area of two plates at separation h is given by the value of F at separation h minus that for infinite separation:
V(h) = F(h) - F(oo). [y~lA ~(0) + y2zBi(0)
K 2 ! + y~lYs2A'3(O) + ys~YszB3(O)],
[30]
2nve ~r(22)= + - [y~IA '2(Kh) + y32B'2(Kh)
[38]
We define the first, second, and third approximations to V(h) as V~, V2, and V3, respectively, which are V]
= F(l)(h)
-
F(~)(~),
[39]
K 2 + ys~ ys:A t3(Kh) + Ysl y2zB'3(r~h)],
0.(31) _
2nve
5 [y~IA
t 4(0)
Vz = { F ° ) ( h ) + F(2)(h)} [31]
5 t -~- Y s 2 B 4 ( 0 )
+ Y4slYs2A'5(0) + Ysl Ys2 4 B ,5(O)
- {F°)(~) [32]
2rive o-~2) = + [y~A'4(Kh ) + y~zB5 '4(Kh) K 4
4
+ F ( 2 ) ( ~ ) + F(3)(~)}.
+ y~lY~EA'5(rh) + y~ly~2Bs(rh) [33]
[41]
Substituting [28]-[33] (with [14]-[22]) into [35]-[37] and introducing x+~ - Y s l _ _Ys2, + 2
r
3 2 2 3 ¢ + y~lY~2A'6(Kh) + ys~ys2B6(Kh)],
[40]
V~ = { F ° ) ( h ) + F(2)(h) + F(3)(h)}
K
3 2 2 3 +y~ys2A'6(O) + YslYs2B6(O)],
-- {F(1)(o0) + F(2)(c~)},
ii_ _ Ysl - Ys2 2
,
[42]
from [39]-[41] we obtain Journal of Colloid and Interface Science, Vol. 89, No. 2, October 1982
488
OHSHIMA, HEALY, AND W H I T E
V1 - 2nkT [y2+(1 _ tanh (rh/2)} - y2{coth (Kh/2) - 1}],
[43]
K
112 = V, +
-
( y 4 + 3yzy2){1 _ tanh (Kh/2)} -
( y 4 + 3y2+y2)
K
X {coth (Kh/2) - 1} - y4+ tanh (Kh/2) y 4 coth (Kh/2) 3--2"cosh 2 (Kh/2) + 32 sinh 2 (rh/2) (Kh/2) I _y2 32 tcosh 2 (rh/2) V3=V2+
2nkT F y2 { y a + ~ y 2 _ ( 7 y 2 i
5760 y4 +
.~21
y2 sinh 2 (Kh/2)) .]'
[44]
+ y2_)}{l_tanh(rh/2)}
L5-76-0
yZ+(7y2 + y2) {coth ( r h / 2 ) - 1) 3
+ 1-~
2 y 2 cosh2 (Kh/2)
1536 y 2 + 2 y2+ sinh 2 (~h/2)
y6+ tanh (Kh/2) _ y 6 coth (rh/2) - 1024 cosh4 (Kh/2) 1024 sinh 4 (Kh/2) (xh/2) ~. y4 + ~ [cosh 4 (Kh/2)
(rh/2)2 ~ + - 256
r 2 [cosh 2 (rh/2)
7(Kh/2) I y2 3072 [cosh 2 (Kh/2)
y2 .~3 sinh 2 (Kh/2))
Y% .~sy2 tanh 2 (Kh/2) - y 2 coth 2 (Kh/2)} sinh4 (Kh/Z)J / + y2 ~21 tanh (rh/2) sinh 2 (rh/2)) ( Y 2 cosh2 (rh/2)
The first approximation V1([43]) agrees with the HHF formula for the interaction of two dissimilar plates (Eq. [6] of Ref. (5)). We also note that for a special case when Y~I = Ys2 (i.e., Y_ = 0), [43]-[45] reduce to an expansion of the elliptic integral expression for the interaction energy of two similar plates in powers of surface potential derived by Levine and Suddaby (Eq. [37b] of Ref. (8)). 3. I N T E R A C T I O N OF T W O D I S S I M I L A R L A R G E S P H E R E S AT S M A L L S E P A R A T I O N S
Following the treatment of HHF, we apply Derjaguin's method (6) to derive an expression for the interaction energy of two dissimilar large spheres at small separations. Consider two spheres, each having radii al, a2 and surface potentials ~bs~, ~b~2, respectively. Let H be the shortest distance between two spheres (Fig. 2) and assume that ~k~l and ~b,2 are constant, independent of H. Journal of Colloid and Interface Science, Vol. 89, No. 2, October 1982
r 2 coth (rh/2)~l sinh 2 (rh/2)JJ
[45]
When the conditions: H ~ al, a2
[46]
Kal>> 1, Ka2>> 1
[47]
and are satisfied, Derjaguin's method (6) can be employed, which makes it possible to obtain the interaction energy Vsp of two spheres by integrating the interaction energy V per unit area of two plates ([38]) using the following formula (5):
FIG. 2. Geometry of two dissimilar spheres of radii al and a2 at a shortest separation of H.
DISSIMILAR
V(h)dh.
27rala2 al + a2
V~p(H)
[48]
Substituting [43]-[45], we obtain first, second, and third approximations to V~p, denoted as V~pl, Vsp2, and Vsp3, respectively:
87ralaznkT f e-KH)+ y2_ logo (1 +~2) LY2+logo (1 +
Vspl = ~
Vsp2 = Vspl '~
87rala2nkT[-
K2(al + a2)
l(y
4+
y4+ 1 - (KH/2) tanh (KH/2) 96 cosh 2 (KH/2)
81rala2nkr[
y2_ + 5 - - ~ { Y4- + ~
[49]
-- 1} y 4 (KH/2) coth (KH/2) -- 17 96 sinh 2 (KH/2) .J '
[50]
r2+ {y4 +l~ y2_(Vy2++ y2)}(gH/2){l_tanh(gH/2) )
5760
Y2+(7y2-+ Y2)}(KH/2){coth (KH/2)
+ y 6 17 + 4(~H/2) tanh (KH/2) _ 46080 cosh2 (KH/2)
y6_.4(rH/2)
+ y4yZ
yE+y4_ (~H/2)
_
-- e-~U)],
3y2+yE_)(rH/2){1 _ tanh (KH/2)}
+ ~1 (y4_ + 3y2y2_)(KH/2){coth (KH/2)
Vsp3 = Vsp2+ K2(aI + a2)
489
DOUBLE LAYER INTERACTION
1 + (rH/2) tanh (KH/2) _ 1024 cosh z (rH/2)
1}
coth (KH/2) + 17 46080 sinh 2 (KH/2) coth (KH/2) + 1 1024 sinh 2 (~H/2)
y 6 1 - I I(KH/2) tanh (rH/2) + y 6 l l(KH/2) coth (KH/2) - 1 15360 cosh4 (KH/2) 15360 sinh4 (KH/2) +
(KH/2)2 I y2+ y2 ~3q 1536 tcosh = (,~H/2) - sinh = (KH/2)J d '
[51]
where Y+ and Y_ are defined by [42]. The first approximation V~pl ([49]) is the HHF formula for the interaction of two dissimilar spheres (Eq. [21] of Ref. (5)). For the case of two similar spheres (~1 = ffs2, (i.e., Y_ = 0) and al = a 2 ) [49]-[51] reduce to a series expression (Eq. [4.5] of Ref. (9)) which Hoskin and Levine derived by applying Derjaguin's method to the series expansion (Eq. [37b] of Ref. (8)) of the interaction energy of two similar plates. 4. D I S C U S S I O N
The HHF formula (5) for the interaction of dissimilar double layers at constant surface potential is based on the Debye-Htickel linear approximation. This approximation retains only the first term of the expansion [7] of the Poisson-Boltzmann equation, giving the interaction energy correct to quadratic order in the surface potentials. The HHF formula is thus applicable for sufficiently low potentials.
In the present study we have truncated the expansion of sinh y at the y5 term in [7] and obtained higher order approximations to the electrostatic potential and interaction energy expressed as a power series in surface potentials. The first approximation ([43] for the plate-plate interaction and [49] for the sphere-sphere interaction) agrees with the HHF formula. The second and third approximations ([44] and [45] for the plateJournal of Colloid and Interface Science, Vol. 89, No. 2, October 1982
490
OHSHIMA, HEALY, AND WHITE TABLE I
Percentage Relative Error Ei (i = 1, 2, 3) (%) in the First (HHF), Second, and Third Approximations V/ (i = 1, 2, 3) ([43]-[45]) to the Interaction Energy of Two Dissimilar Plates as a Function of Y~I and Y~2 for Kh = 0.5, 1, and 2a xh = 1
~h=0.5
Y.I
Yd
3.0
3.0 2.2 1.4 -1.4 -2.2 -3.0
2.6
el
~2
~3
+1.3 -1.4 -13.5 +4.9 +4.4 +5.3
+3.3 +2.1 +2.3 +0.11 +0.13 +0.20
-0.21 -0.038 +0.14 -0.019 -0.011 -0.012
2.6 1.8 1.0 -1.0 -1.8 -2.6
+0.40 -2.1 -28.2 +3.9 +3.3 +3.9
+1.9 +1.2 +2.1 +0.059 +0.072 +0.11
2.2
2.2 1.4 0.6 -0.6 -1.4 -2.2
-0.077 -2.5 +30.8 +3.2 +2.3 +2.8
1.8
1.8 1.0 -1.0 -1.8
-0.26 -3.0 +1.6 +1.8
~1
xh = 2
~2
~3
~1
~2
~3
+19.9 +13.6 +5.4 +9.1 +8.9 +11.0
+0.22 +0.66 +1.7 -0.32 -0.12 -0.084
-1.8 -0.88 -0.52 -0.068 -0.054 -0.078
+40.9 +30.4 +21.2 +15.0 +16.7 +21,4
-15.2 -8.5 -4.2 -1.9 -1.6 -2.3
-0.071 0.000 +0.16 -0.012 -0.006 -0.006
+14.4 +9.1 +0.49 +7.2 +6.5 +8.0
+0.30 +0.51 +1.4 -0.22 -0.077 -0.038
-0.78 -0.34 -0.21 -0.029 -0.029 -0.035
+30.5 +21.6 +13.8 +11.4 +12.0 +15.8
-8.5 -4.3 -1.8 -1.2 -0.89 -1.3
+2.4 +0.86 +0.17 +0.074 +0.015 -0.014
+0,97 +0.58 -0.88 +0.024 +0.034 +0.057
-0.021 +0.005 -0.10 -0.009 -0.004 -0.004
+10.0 +5.5 -8.0 +5.8 +4.5 +5.6
+0.24 +0.34 +1.5 -0.16 -0.044 -0.019
-0.30 -0.11 -0.087 -0.010 -0.011 -0.017
+21.7 +14.4 +7.5 +8.7 +8.3 +11.1
-4.3 -1.9 -0.44 -0.72 -0.45 -0.62
+0.88 +0.25 -0.005 +0.044 +0.007 -0.007
+0.44 +0.26 +0.012 +0.024
-0.006 +0.004 -0.004 -0.002
+6.4 +2,7 +3.0 +3.7
+0.14 +0.19 -0.024 -0.006
-0.090 -0.030 -0.003 -0.006
+14.5 +8.6 +5.4 +7.3
-1.9 -0.70 -0.21 -0.27
+0.27 +0.053 +0.004 -0.002
+5.7 +2.4 +0.74 +0.13 +0.028 -0.025
"e~ > 0 corresponds to overestimation and ~ < 0 to underestimation.
plate i n t e r a c t i o n a n d [50] a n d [51] for the s p h e r e - s p h e r e i n t e r a c t i o n ) correspond, respectively, to corrections to the fourth a n d to the sixth power of surface potentials. For the special case of two similar d o u b l e layers, our series expansions [ 4 3 ] - [ 4 5 ] a n d [ 4 9 ] - [ 5 1 ] , respectively, reduce to t h a t derived b y Levine a n d S u d d a b y (8) for the int e r a c t i o n of two s i m i l a r plates a n d t h a t derived b y H o s k i n a n d Levine (9) for the interaction of two similar spheres; these authors o b t a i n e d their e q u a t i o n s on the basis of the e x p a n s i o n of the i n t e r a c t i o n e n e r g y of two s i m i l a r plates expressed in t e r m s of elliptic integrals. Levine a n d S u d d a b y (8) considered the a p p l i c a b i l i t y of the series e x p a n s i o n for the i n t e r a c t i o n of s i m i l a r plates. H o n i g Journal o f Colloid and Interface Science, Vol. 89, No. 2, October 1982
a n d M u l (10) p r e s e n t e d a m o d i f i c a t i o n of the e q u a t i o n s derived b y Levine a n d Sudd a b y (8) a n d b y H o s k i n a n d Levine (9). Let us now c o m p a r e our a p p r o x i m a t i o n s for the i n t e r a c t i o n e n e r g y of two d i s s i m i l a r plates ( [ 4 3 ] - [ 4 5 ] ) with the exact n u m e r i c a l values o b t a i n e d b y D e v e r e u x a n d de B r u y n (4). W e c a l c u l a t e the p e r c e n t a g e relative error defined b y V~-V V
× lO0 (%)
(i = 1, 2, 3),
[521
where Vi (i = 1, 2, 3) are the values calculated with the first ( [ 4 3 ] ) , second ( [ 4 4 ] ) , a n d t h i r d ( [ 4 5 ] ) a p p r o x i m a t i o n s , respectively,
DISSIMILAR DOUBLE LAYER INTERACTION
potentials are of unlike sign or either of them is zero (7). Let the position and height of the potential maximum be hm and Vroax, respectively, and assume Ysl > Ys2 > 0 without loss of generality. Exact equations for hm and Vmaxcan be derived from Eqs. [2.38], [3.61 ], and [3.68] of Ref. (4):
A~_ Kv,
2nkT" 2nkT ( 1-1.2.3) I
Y,,-3.0, Y . - 1.4
'\
[tanh (Y~l/4)] logo [_tanh ~ l '
[53]
32nkT Vmax -- - sinh 2 (ys2/4). K
[54]
• .\ \
rhm ,
i
491
i
=
The first approximation V~ ([43]), that is, the H H F formula gives (Eqs. [44] and [45] of Ref. (7))
-1
rhm
Ys,'3-O, Ys,'- I-4
=
Vmax 0
1
2
3
4
5
Kh
FIG. 3. Potential curves for the interaction of two dissimilar plates with Y,I = 3.0, Ya = 1.4, and with ys] = 3.0, Ys2= -1.4. - - , exact values V (4), - - -, first approximation (HHF) 1:1, calculated with [43], -----, second approximation I:2,calculated with [44]. Curves for the third approximationcalculated with [45] coincide with the exact curves within the width of lines.
and Vis the exact numerical value (4). Table I shows Ei as a function of ys~ and Ys2 for Kh = 0.5, 1, and 2. We see that in the second and third approximations agreement with the exact values is considerably improved for small and moderate surface potentials (Ys], Y~2 ~< 3). In a few exceptional cases, a lower approximation is better than a higher approximation, because the potential curves for V, V~, V2, and/:3 intersect each other. Figure 3 shows two examples of the potential curves for V, 1:1, IrE, and 1:3. As is seen in Fig. 3, a characteristic feature of the interaction of dissimilar plates at constant surface potential is that for the case of plates with surface potentials of like sign the potential curve exhibitsa maximum; while it is always negative when the surface
logo (y~JYs2),
2nkT K
Ys~.
[55] [56]
We note that [54] and [56] are independent of the higher surface potential Y~I. The second (V2 ([44])) and third (1:3 ([45])) approximations, however, give no simple analytic expressions for hm and Vmax, and the values of hm calculated by these approximations depend (but negligibly) on ys,. Table II compares the exact values of hm ([53]) and Vmax ([54]) with approximate values calculated with 1:1 ([55] and [56]), V:, V3 together with their relative errors (defined similarly to [52]), showing that agreement with exact values is improved in higher approximations. Another approximation involved in the derivation of the H H F formula for the interaction of two dissimilar spheres is the application of Derjaguin's method (6), which is applicable for the interaction of large spheres (Kal, ~a2 >> 1) at small separations ( H ,~ al, a2). In the present work, we followed Derjaguin's method after making corrections to moderate potentials and obtained [49]-[51], which are therefore applicable only for large a~, a2 and small H. Several approximations for the interaction of dissimilar spheres have been obtained without recourse to Derjaguin's method (6). Bell et al. Journal of Colloid and Interface Science, Vol, 89, No. 2, October 1982
492
OHSHIMA, HEALY, AND W H I T E TABLE II
Position (hm) and Height (Vmax) of Potential Maximum Calculated with Exact Equations and the First ( H H F ) (1), Second (2), and Third (3) Approximations as a Function of Y~I and ys2a xVm~/ 2nk T
rhm Y,i
Yl2
Exact
1
2
3
Exact
3.3
2.2
0.2382
0.3102 (+30.2)
0.2473 (+3.8)
0.2384 (+0.090)
5.348
4.840 (-9.5)
5.392 (+0.82)
5.357 (+0.16)
1.4
0.6356
0.7621 (+19.9)
0.6366 (+0.15)
0.6317 (-0.61)
2.041
1.960 (-4.0)
2.092 (+2.5)
2.040 (-0.048)
1.8
0.3038
0.3677 (+21.0)
0.3096 (+1.9)
0.3037 (-0.048)
3.465
3.240 (-6.5)
3.448 (+0.66)
3.467 (+0.071)
1.0
0.8476
0.9555 (+12.7)
0.8425 (-0.60)
0.8455 (-0.25)
1.021
1.000 (-2.1)
1.038 (+1.7)
1.020 (-0.13)
1.4
0.3974
0.4520 (+13.7)
0.4004 (+0.76)
0.3972 (-0.060)
2.041
1.960 (-4.0)
2.051 (+0.47)
2.042 (+0.022)
0.6
1.2125
1.2993 (+7.2)
1.2037 (-0.72)
1.2122 (-0.020)
0.3627
0.3600 (-0.75)
0.3664 (+1.0)
0.3623 (-0.12)
1.0
0.5438
0.5878 (+8.1)
0.5447 (+0.16)
0.5437 (-0.031)
1.021
1.000 (-2.1)
1.024 (+0.28)
1.021 (+0.002)
2.6
2.2
1.8
1
2
3
a Percentage relative error (%) is given in parentheses; positive error corresponds to overestimation and negative error to underestimation.
(11) transformed the linearized form of the Poisson-Boltzmann equation [ 1] into an integral equation and so obtained an approximate expression for the interaction energy at large a~, a2 and all H for small potentials. In the same paper (11 ) Bell et al., using the superposition approximation, derived the general asymptotic result (i.e., valid for all surface potentials and radii al, a2,) Vsp(H) =
87rala2nkT
YIY2
K2
H + al + a2
(KH>> 1),
e_m [57]
where YI and Y2 are the coefficients in the asymptotic form for the dimensionless potential distribution yi(r) around isolated spheres
the exact form for the Yi is not known in general. Elsewhere (12) we have obtained accurate analytic approximations to these coefficients which allows an accurate analytic expression for Vsp(H) when r H ~> 1 for all particle radii and surface potentials. Barouch et al. (13) derived an approximate expression without using the Debye-Htickel linearization and Derjaguin's method, but they still used the linearized form for the double layer free energy (our first approximation F (1) in [35]). In a comment on the work of Barouch et al. (13), Chan and White (14) stated that the next-order corrections on Derjaguin's method should give a correction term of orders in (Kai)-l (i = 1, 2). We shall consider these corrections in Part II of this series.
yi(r) = Y~aie-~('-a')
(i=1,2), [58] r (where r is radial distance from center of sphere i). For small potentials Yi ~ Ysi but Journal of Colloid and Interface Science, Vol. 89, No. 2, October 1982
ACKNOWLEDGMENTS One of us (H.O.) acknowledges a Visiting Research Fellowship from the University of Melbourne. This work
DISSIMILAR DOUBLE LAYER INTERACTION was supported by the Australian Research Grants Committee. REFERENCES 1. Derjaguin, B. V., and Landau, L. D., Acta Physicochirn. 14, 633 (1941). 2. Verwey, E. J. W., and Overbeek, J. Th. G., "Theory of the Stability of Lyophobic Colloids." Elsevier, Amsterdam, 1948. 3. Derjaguin, B. V., Discuss. Faraday Soc. 18, 85 (1954). 4. Devereux, O. F., and de Bruyn, P. L., "Interaction of Plane-Parallel Double Layers." M.I.T. Press, Cambridge, 1963. 5. Hogg, R., Healy, T. W., and Fuerstenau, D. W., Trans. Faraday Soc. 62, 1638 (1966). 6. Derjaguin, B. V., Kolloid Z. 69, 155 (1934). 7. Usui, S., in "Progress in Surface and Membrane
8. 9. 10. 11. 12. 13.
493
Science" (J. F. Danielli, M. D. Rosenberg, and D. A. Cadenhead, Eds.), Vol. 5, p. 223. Academic Press, New York, 1972. Levine, S., and Suddaby, A., Proc. Phys. Soc. A64, 287 (1951). Hoskin, N. E., and Levine, S., Phil Trans. Roy. Soc. London A248, 449 (1956). Honig, E. P., and Mul, P. M., J. Colloid Interface Sci. 36, 258 (1971). Bell, G. M., Levine, S., and McCartney, L. N., J. Colloid Interface Sci. 33, 335 (1970). Ohshima, H., Healy, T. W., and White, L. R., J. Colloid Interface Sci., submitted. Barouch, E., Matijevi6, E., Ring, T. A., and Finlan, J. M., J. Colloid Interface Sci. 67, 1 (1979); 70, 400 (1979).
14. Chan, D. Y. C., and White, L. R., J. Colloid Interface Sci. 74, 303 (1980).
Journal of Colloid and Interface Science, Vol. 89, No. 2, October 1982