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The prediction of electrokinetic phenomena within multiparticle systems. I. Electrophoresis and electroosmosis
The Prediction of Electrokinetic Phenomena within Multiparticle Systems I. Electrophoresis and Electroosmosis S. L E V I N E 1 AND G R A H A M H. N E A L E Department of Chemical Engineering, The University of British Columbia Vancouver 8, B. C., Canada Received March 30, 1973; accepted April 30, 1973 Henry's classic theory for the electrophoresis of a single isolated sphere is extended analytically to cover the more practical problem of the electrophoresis of a swarm of identical,
dielectric, spherical particles. The effects of interaction of the individual particles and their associated electric fields are taken into explicit account by employing a fundamental cellmodel representation which is known to provide good predictions for the motion of a swarm of spheres within a fluid in the absence of electrical effects. For thin double layers, the electrophorefic velocity of (or the electroosmotic velocity within) a swarm of identical dielectric spheres is shown to be invariant, practically speaking, with respect to the void fraction of the swarm. However, as the double layer thickness increases the electrophoretic (electroosmotic) velocity decreases sharply and becomes strongly dependent on the void fraction. The computed results provide theoretical justification for Smoluchowski's widely quoted approximate result for electroosmotic flow within porous media of arbitrary pore geometry. I. INTRODUCTION An understanding of the flow behavior of electrolytes relative to swarms of solid particles is of considerable importance in m a n y fields of science and engineering, for example electrophoresis, electroosmosis, pressure-induced fluid flow, and gravitational sedimentation. This paper examines analytically those electrokinetic phenomena which occur within particle swarms as a result of externally applied electric fields. These processes can be classified as (i) electrophoresis of suspended particles, and (ii) electroosmosis within rigid beds of particles. For convenience, attention is here confined to swarms composed of identical, spherical particles. 1 Permanent address: Department of Mathematics, University of Manchester, England.
Several theories, of increasing sophistication, have been advanced to predict the electrophoretic velocity of a single isolated sphere [-Smoluchowski (22 ), Huckel (11), H e n r y (8, 9), Hermans (10), Overbeek (15-17), Booth (2, 3), Pickard (18), Dukhin (5)J. However, no serious attempt appears to have been made (3), (21) to extend the validity of these theories to the more practical system of a s w a r m of particles. This problem is much more difficult because it becomes necessary to take into account the effects of the particle interactions on the electric field, and the fluid motions associated with the individual particles. Similarly, for electroosmotic flow within beds of particles it is difficult to account explicitly for the effects of pore geometry, tortuosity, etc. When calculating this electroosmotic flow it is customary to utilize Smoluchowski's
520 Jegrnal of Colloid and ln~erface Science, Vol. 47, No. 2, May 1974
Copyright ~ 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.
ELECTROKINETIC PHENOMENA WITHIN MULTIPARTICLE SYSTEMS
521
dassic result (17), (22), (23), which is valid extension of Henry's classic treatment of the for porous media in general (i.e., independent electrophoresis of a single isolated spherical of the pore geometry) provided that (i) the particle. Throughout the present analysis we assume internal surface of the porous medium is nonconducting, (ii) the thickness of the diffuse negligible surface conductance and small ~" layer is small in comparison with the radius of potentials (in the Debye-Hiickel range), whence relaxation effects may be neglected curvature of the constituent particles. Because the pore geometry of particle (1), (16), and (25). In the main text we confine swarms usually defies description in all but a our attention to large values of Ka, whereas loose statistical sense, it becomes necessary in the Appendix we develop the theory for to introduce certain simplifying assumptions arbitrary values of Ka. For large Ka, it is demonstrated that the in order to permit a quantitative assessment of flow behavior occurring therein. The problem electrophoretic velocity of a swarm of solid of slow viscous flow relative to a swarm of spheres is given by the following simple uncharged spherical particles has been treated expressions : analytically by a number of authors [Cunningham (4), Happel (6), Kuwabara (12), Neale V = EE~a f(~') for the Happel model, [1.12 and Nader (14)] using familar cell-model 47r/~ representations, the principal function of which is to permit a simple, yet reasonably accurate, for the Kuwabara model, [1.2] V = __ assessment of the effects of particle interaction. 4~r~ The most amenable of these models are the so-called "free-surface" model due to Happel, where e denotes the dielectric constant of the and the "zero-vorticity" model due to Kuwa- electrolyte, # its viscosity, E the externally bara, the predictions of which are in close applied electric field, ~a the potential at the agreement with experimental data throughout surface of the spheres (this potential may be the entire range of porosity. However, in good identified with the ~" potential), and ~, the ionizing solvents, like water, the particles porosity of the swarm (porosity = void fracconstituting the swarm would invariably carry tion = 1 -- volume packing fraction of solids). an associated charge and, consequently, be The function f@) can range in value between surrounded by diffuse layers bearing an excess 1.0 and 1.09, indicating that the predictions of oppositely charged ions. In fact, there is of the two models are in satisfactory agreement, considerable experimental evidence (13), (20), practically speaking. Both [1.1] and [1.22 (24) that the effects of these diffuse (double) reduce to Henry's original result for the case layers are often too significant to be neglected. of a single isolated sphere (~, = 1) at large Ka, In other words, the predictions of the above namely, geometric models, when assuming uncharged V = eEpa/47r#, [1.3] particles, should not be applied indiscrimi- as required. nately when dealing with electrolytes. It is The conclusion to be drawn from these therefore desirable, in fact necessary, to be observations is that Henry's result [1.3] for a able to extend the theories of Happel and single isolated sphere remains valid, practically Kuwabara to accommodate these important speaking, for a swarm of spheres of arbitrary effects of the double layer. This can be achieved porosity [this being in complete contrast to the analytically for small ~" potentials (in the case of gravitationally--induced particle moDebye-Htickel range) and negligible surface tion, for which the settling velocity of a swarm conductance, for all values of Ka (where 1/K decreases rapidly with decreasing porosity denotes the double-layer thickness and a the (7)]. The presented theory therefore justifies particle radius), by a logical adaptation and the conventional usage (when Ka is large) of Journal of Colloid and Interface Science, Vol. 47, No. 2, May 1974
522
LEVINE AND NEALE
O,
,:.0.'., ,.m, C1, ,
,
.
.
.
.
',L2 ~
:v-i
0
Z
- O:
FIG. 1. Aqueouselectrolyte mediumin relative motion with a swarm of chargednonconductingspheres (dashes denotes outer extent of diffuselayer). Henry's result [1.3-] for swarms, although it was derived specifically for a single isolated particle. It should be noted that El.i] and [1.2"] are valid also for evaluating the mean electroosmotic velocity V within a rigid bed of identical dielectric spheres. This is because, in the absence of an applied pressure gradient, electrophoresis of, and electroosmosis within, a swarm of spheres constitute two equivalent processes, each involving the relative motion of solid particles and electrolyte. Equations [1.1] and [1.2] can be seen to be completely consistent with Smoluchowski's classic result for electroosmotic flow within a porous medium of arbitrary pore geometry, namely, V = eE~b~/47r#
[-1.43
which is stated to be valid when the doublelayer thicknesses are much smaller than the pore diameters and when surface conductance is negligible. The substantial agreement of [1.1] and El.2] with [-1.4] i s particularly encouraging because it provides further theoretical justification for the implications inherent in Smoluchowski's result, namely that V is independent of porosity, particle size and shape. II. DEFINING EQUATIONS AND BOUNDARY CONDITIONS We consider a uniform three-dimensional distribution of solid nonconducting spherical particles of radius a, in steady relative motion Journal o f Colloid and Interface Scienae, Vol. 47, No. 2, May 1974
with respect to an aqueous electrolyte medium. For convenience, we consider the particles to be stationary and the fluid to be moving parallel to the Z axis with a uniform velocity V (Fig. 1). The particles carry a uniform surface charge density ~ in the equilibrium state of zero flow. Following Happel (6) and Kuwabara (12) we introduce a cell model in which each particle is envisaged to be surrounded by a concentric spherical shell of electrolyte, having an outer radius b such that the cell contains the same volumetric proportion of solid to fluid as exists in the entire assemblage [Fig. 2]. Each particle is surrounded by a diffuse layer, contained within the cell, which carries a net charge equal in magnitude but opposite in sign to that on the particle, such that the net charge within each cell is zero. ELECTRICAL CONSIDERATIONS
In the equilibrium state of zero fluid motion, the potential ~ = 6(r) in the diffuse layer depends only on distance r from the center of the particle, and satisfies Poisson's equation, V~ = -- 4~p/~,
[2.13
where p ~r) denotes the volume charge density and ~ the dielectric constant of the electrolyte medium. Using Gauss' theorem, the boundary conditions on $ read =----, r=a
= 0. ~
r=b
E2.2-]
523
E L E C T R O K I N E T I C P H E N O M E N A W I T H I N MULTIPARTICLE SYSTEMS
FIG. 2. The proposed model for the physical system depicted in Fig. 1.
Adopting the approximation used by Henry (7) in his classical work on electrophoresis, we superimpose on ~b(r) the electrostatic perturbation potential ¢ = ¢ ( r , O) outside each partide. This potential exists as a result of the externally applied electric field, and in the steady state satisfies the Laplace equation
can be incorporated if necessary, as demonstrated by Henry himself (9).
V2¢ = O.
#72~ = Vp + pV(¢ + ~b),
[-2.3]
If E denotes the externally applied potential gradient in the z direction, then q~is stipulated to satisfy the following boundary conditions:
0 where 0 denotes the polar angle measured from the Z axis, about which we have axial symmetry. The first relation in [-2.4] is that used by Henry (8), and follows from the condition of continuity of current flow at the surface of the particle. We assume in the second relation that at the outer surface of the cell (r = b) the local electric field is parallel to the externally applied electric field (this condition is analogous to Happel's hydrodynamic condition in [-2.9]). The solution of I-2.3], subject to [2.4] is ¢(~, 0) -
~+ 1 -- (a/b) 3
cos 0.
[2.5]
2rZ/
Following Henry (8), the effects of surface conductance and relaxation are here being neglected. The latter effect has been shown (1, 16, 25) to be small for large Ka(>~100 for ~b~ ~> 100 mV) or for small 6a(~<50 mV when Kay---10). The effect of surface conductance
HYDRODYNAMIC CONSIDERATIONS
The fluid flow around any typical particle is governed by the Navier-Stokes equation, as simplified for creeping flow,
[2.6]
and the continuity equation for an incompressible fluid,
V.u = O,
[2.7]
where u denotes the fluid velocity vector, u the electrolyte viscosity, and p the fluid pressure referred to a datum plane. We adopt hydrodynamic boundary conditions which are equivalent to those used by Happel (6, 7), namely, ur(,, 0) = ,0(r, 0) = o
at
u,.(r, O) = - V cos 0 r O( u o / r ) r,.o(r, O) = - - # \ Or + r
r = a,
[-2.8]
10ur~
} [-2.9]
00/ =0 at
r = b, [-2.10]
where V denotes the mean fluid velocity in the Z direction, and rr0 the tangential shear stress component. The condition [2.9] is the hydrodynamic counterpart of the second electrical condition in [2.4]. Happel himself considered the complementary problem of spheres falling under gravity through a viscous fluid. Consequently, his conditions may be retrieved from [2.8] and [-2.9] by adding V cos 0 to the value of ur and -- V sin 0 Journal of Colloid and Interface Science,
Vol. 47, No. 2, May 1974
524
LEVINE AND NEALE
to the value of uo [Happel and Brenner (7)]. A satisfactory alternative to [2.10], introduced by Kuwabara (12), is that the azimuthal component of the vorticity w~ (rather than r~o) be zero at the outer envelope of the cell, that is2
where Ao, A1, A2, B1, and ]32 are constants to be determined from the hydrodynamical boundary conditions F2.83-E2.10] or E2.8], [2.97, and [2.117, and where E x =
10(ruo)
1 Our
r
r O0
w~-
=0 Or
[2.153 1 -
at r = b .
(~/b)~
'
E2.11] &k
a3r f r
} = }(r) = dr + - - 2 This representation provides predictions for uncharged spheres which agree closely with those of Happel (7). The general solution of [2.6]-[2.7], which can be obtained from Henry's work (8) and which, in the absence of an electric field, reduces to Lamb's classic solution as used by Happel (7) and Kuwabara (12), reads
fJ b
p = Ao
I
+-
dr
Expressions for fro and w~ are readily obtained from E2.10]-[2.14]. It will become apparent that only the value of B1 is required here.
III. ULTIMATE SOLUTIONS FOR THE CELL MODELS
r2
4~r \ dr Fri = 2~ra2 -Air 2
u,=
B1 +--+A2+-_ 10/~ ~r
_
X sin OdO,
'; )1
6re#
r 3} dr
where r,0 is defined in E2.10] and
--
ra . a
uo =
Air 2
B1
5~
2#r
~X 6~'#
(L
A~+--
'f
[3.13
cos 0, O¢,t,r
[2.133
E
(pr, cos 0 + fro sin 0)r=a
ra
dr -- - -
_
f
Tr
-/0
B2
i
[2.15-]
The total force F acting on any typical particle is the sum of a hydrodynamical force F~ and an electrical force FE. The hydrodynamic force is given by
BI
+
1
L --H v2¢~ dr.
P~,r = --P + 2g
,
E3.2]
Or
B2
and the electrical force by
2r 3
~dr + - r3~dr 2r 3 . . . .
/ dT,\ sin O,
E2.14] 2For problems involving axial symmetry, which prevails here, the radial and tangential components of vorficity wr and wo are automatically zero. Similar remarks apply to the shear stress components. Journal of Colloid and Interface Science, Vol. 47. No. 2, M a y 1974
FE = - , E a 2 l - ) . \ dr/r=a
[3.3]
The calculated expression for the total force F is F = -- 4~rBt = -- 6zc#Va~ + eEaA,
[3.4]
ELECTROKINETIC PHENOMENA WITHIN MULTIPARTICLE SYSTEMS
525
where ~ and A represent the following functions : 6 + 4y ~
[3.5] 6 - - 9 y + 9 y 5 - - 6y 6
according to the Happel model, --
(6 + 4y5)
A=
~ dr + 10y5
(rS/a ~) ~ dr
(1 -- y~)(6 -- 9y + 9y5 -- 6y6)
[3.6]
'
~=
[3.7]
5 -- 9y + 5y 3 -- y6 according to the Kuwabara model. --5
±=
~dr + 5y 3
(rS/aS)~dr
[3.8]
( 1 - - y s ) ( S - - 9 y + S y ~ - y6) '
where [3.9]
y = a/b.
Following Henry (7), we integrate by parts and obtain
Utilizing the expressions given in [3.5]-[3.8] and [3.12] and [3.13] we obtain the following ultimate predictions, valid for Ka>> 1 : V=
f b
f b d~
dr = ~b -- t.ka -- aa
Jo
~E¢,o 47r/~
.((1
)
(1 -- yS) - y3)(1 + ~y~),
for the Happel model
[3.14]
and 3a 5 fab d~ +---, 2 r5 fb r3
Ja F
1
.
.
[3.10] g
.
Ja r ~ rig,
+
+ - -3a 5 fo b d~ --. 5 r5
[3.113
In particular, for thin double layers (i.e., Ka)) 1) it may be further shown (7) that f b
~dr~_]
fb r 3 ~ ~dr~'--~(~
daft,
-- ~bb),
~-~a,
for the Kuwabara model.
[3.15]
From the definition of the cell model, y is related to the porosity 3' by 3" = 1 -
(a/b) 3 = 1 --y3.
[3.12]
[3.13]
[3.16]
It can be seen that [3.14] and [3.15] both reduce to Henry's result [1.3] as 3"---* 1 (i.e., b---+ ~ , y---~O, ~ - - ~ 0), as required. DETERMINATION
since ~ b < < ~ for Ka>> 1 in all practical applications (i.e., 3" > 0.26), as will be demonstrated in the next section. For steady motion, the net force F on any typical particle must vanish (7, 8). Hence, from [3.4] it follows that V = (,E/6~,)(A/a).
-
4~-/~
1 fb
.
=
OF ~b
The potential distribution ~b = ~ (r) within the model cell is given by the solution of Poisson's equation [2.1] subject to [2.2"]. Unfortunately, an analytic solution of this equation seems difficult to obtain except at small potentials. Thus, by making use of the Debye-Hiickel approximation (applicable for ~b < 25 mV in most practical applications), Eq. [2.1] reduces to a form which can be Journal of Colloid and Interface Science, Vol. 47, No. 2, May 1974
526
LEVINE AND NEALE
solved directly (3' 163, namely
~
= ~.
[3.173
The solution of [-3.173 subject to [-2.23 is
¢, =
////{k---~-/\r/\(1 -
K2ab) sinh (xb -- ~a) -- (~b -- ~a) cosh (~b -- Ka)) "
It can be shown from [-3.183 that for Ka >~ 100 the value of 6b will be less than 1% of $, for all porosities exceeding 0.26. This statement assumes that physical credence can be given to the adopted cell model towards the lower end of the porosity range [N.B. the minimum attainable porosity for a swarm of monosized spheres is 0.26 (7)3.
Kuwabara model when extending the analysis to arbitrary values of Ka (0 < ~a < ~). A possible reason for the success of the Kuwabara model is that the contribution to the vorticity which is directly due to the electrical effects and, from Henry's work, is proportional to
2 DISCUSSION AND CONCLUSIONS
One important observation is that the expressions derived for V in this work are equally valid for electroosmotic flow within a rigid bed of spheres as they are for the electrophoresis of a suspension of spheres. As noted earlier this is because both processes involve the relative motion of solid particles and electrolyte under the influence of an externally applied electric field, in the absence of an applied pressure gradient (3,8, 17, 22, 23), For large Ka, the predictions of Happel's model [-3.14-] and Kuwabara's model [-3.15-1 are in numerical agreement to within better than 9% at all porosities (0 _< 3' _< 1). From a practical point of view this can be considered reasonable agreement but from mathematical and physical points of view the prediction of the Happel model [3.14-1 is unsatisfactory since it does not agree with the classic Smoluchowski result [-1.4-1, which is known to be valid for all porosities % at large Ka and negligible surface conductance. (The Smoluchowski result is valid at all potentials, provided Ka>> 1, but this observation need not concern us here.) In contrast, the prediction of the Kuwabara model [-3.15-1 at large Ka is identical with Smoluchowski's result [1.4-]. It is for this reason that we have employed only the Journal of Colloid and Interface Science, Vol. 47, No. 2, May 1974
[-3.183
ra
in fact vanishes at r = b. This is consistent with the condition employed by Kuwabara that the purely hydrodynamic part of the vorticity and hence the total vorticity [-2.11-1 is made to vanish. In the Happel model, the combination of electrical and hydrodynamic conditions seems less consistent. Another important inference to be drawn from this work is that for large Ka the classic result of Henry J-1.33 for the electrophoresis of a single isolated sphere and that of Smoluchowski [-1.4-1 for electroosmosis within both straight capillaries and arbitrary porous media, remain valid, practically speaking, for all particle swarms likely to be encountered in practice. This is particularly significant because it implies in turn that the many other electrokinetic theories based upon single isolated particles, not necessarily spherical, are likely to remain equally valid for swarms at large Ka. It has been shown in the Appendix that as Ka decreases, so the electrophoretic (electroosmotic) velocity V for a swarm decreases sharply from that predicted by [-1.4-1, owing to double-layer interaction within the voids. In a subsequent paper, the theory developed here will be modified and generalized to cover the related, but more complicated, phenomena of streaming potential and sedimentation potential within swarms of spherical particles,
ELECTROKINETIC PHENOMENA WITHIN MULTIPARTICLE SYSTEMS that is, processes in which the electric field E is internally induced, rather than externally applied, and must consequently be predicted. ACKNOWLEDGMENT The authors extend thanks to Professor Norman Epstein of the University of British Columbia for several valuable suggestions and for reviewing this manuscript. Financial support provided by the National Research Council of Canada is gratefully acknowledged. REFERENCES 1. ABRA~rOWlTZ,M., AND STEGUN, I. A., in "Handbook of Mathematical Functions." National Bureau of Standards, U.S. GPO, Washington, D.C., 1964. 2. BooTI% F., Proc. Roy. Soc. London A203, 514 (1950); J. Colloid Sd. 6, 549 (1951). 3. Booa'tI, F., Progr. Biophys. 3, 131 (1953). 4. CUNNmOI~A~t,E., Proc. Roy. Soc. London A83, 357 (1910). 5. DUKmN, S. S., in "Research in Surface Forces" (B. Derjaguin, Ed.), Voh 3, p. 313. Consultants Bureau, New York, 1971. 6. HAI'rEL, J., Amer. Inst. Chem. Eng. J. 4, 197 (1958). 7. HAPPEL, J., AND BRENNER, H., "Low Reynolds
8. 9.
10. 11. 12. 13. 14. 15. 16. 17. 18.
19. 20. 21. 22.
Number Hydrodynamics." Prentice-Hall, Englewood Cliffs, New Jersey, 1965. HENRY, D. C., Proc. Roy. Soc. London A133, 106 (1931). HENRY, D. C., Trans. Faraday Soc. 44, 1021 (1948). HERMAI~S,J. J., Phil. Mag. 26, 650 (1938). HOeKEL,E., Physik Z. 25, 204 (1924). KUWABARA,S., J. Phys. Soc. Japan 14, 527 (1959). MICI~AELS,A. S., ANn LIN, C. S., Ind. Eng. Chem. 47, 1249 (1955). NEALE, G. H., ANn NADAR,W., Amer. Inst. Chem. Eng. J. 19, 112 (1973). OVERBEEK,J. Th. G., Kolloid Beih. 54, 287 (1943). OVERBEEK,J. Th. G., Advan. Colloid Sci., Vol. 3, 97 (1950). OVEI~BEEK,J. Th. G., in "Colloid Science," Vol. 1, p. 194. Elsevier, Amsterdam, 1952. PICKARD,W. F., Kolloid Z. 179, 117 (1961). RICE, C., Axn WmTEI-IEAn,R., J. Phys. Chem. 69, 4017 (1965) SCI~EIDEGGER,A. E., in "The Physics of Flow through Porous Media." Univ. of Toronto, Toronto, 1960. SENGUPTA, M., J. Colloid Interface Sci. 26, 240 (1968). S~oLucllowsI~I, M., Z. Phys. Chem. 93, 129 (1918).
527
23. S~oLvcI~owsI(I, M., in "Handbuch der Electrizitat und des Magnetismus" (L. Graetz, Ed.), Vol. 2. Liepzig, 1914. 24. SWARTZENDRUBER, D., "Flow Through Porous Media" (R. De Wiest, Ed.), p. 215. Academic Press, New York, 1969. 25. WIERSEM~ P. H., LOEB, h. L., AND OVERBEEK, J. Th. G., J. Colloid. Interface Sci. 22, 78 (1966).
APPENDIX In the main text we derived expressions for the electrophoretic (electroosmotic) velocity V within a swarm of identical spheres subject to the restriction that Ka is large. We here seek to extend the analysis of the main text to accommodate all values of Ka (0 < Ka < oo ). Although the predictions for V given in Eqs. [-3.14] and [,3.15] are in satisfactory agreement from a practical viewpoint, it is evident that the K u w a b a r a result [,,3.15] is physically preferable to the Happel result [,3.14], since the latter predicts that V should increase, albeit slightly, as the porosity of the swarm decreases ; such an increase in V has apparently not been observed experimentally. Hence, in this Appendix we shall confine our attention exclusively to the K u w a b a r a model. The velocity V according to the K u w a b a r a model, can be expressed using Eqs. [,3.13], [-3.7], and [.3.8] as follows: ~E¢~ V =
~ f(~a, 7),
4~
[A.1-]
where a -
f(~a,"r) =
-
f
/ a 5 r3 dr + y3
-
a 3 ~dr -
[,A.2-]
~(1 -- y")~o
The function f(Ka, 7) is therefore a correction factor which measures the extent to which V differs from Smoluchowski's prediction [,1.4-]. The integrals in [-A.2] can be determined b y substitution of [,3.18-] directly into [,3.10-] and [,3.11-], followed by partial integration and some straightforward but tedious algebraic manipulation. The final expressions for the Journal of Colloid and Interface Science, Vol. 47, No. 2, May 1974
528
LEVINE AND NEALE I
I
I
I
I.O-~E~ . f(~,,,r)
-
V = 4-T~
-- )% I.O(
~"
'-T-- 0.5
H E N R Y ~
~
5 /
®
~'~
o.~
0 .Ol
.I
I
I
IO
I00
IO00
KCl
FiG. 3. Plot off(tea, 3') vs Ka for various porosities % integrals appearing in [ A . 2 ] can be shown to be
~a b
I dr=--~b~
Q = [1 - Kb tanh (Kb -- ¢a)]/ [ t a n h (~b -- ~a) -- Kb],
K2~2 I + RKa--[ 16 "JI-
K3a3
4
A = R['sinh ~b -- ~b cosh Kb],
[A.7]
B = R [ c o s h Kb -- ~b sinh ~b],
[A.8]
12
I = __ _ _
-~-
K5a 5
96
Pa6~I] /
FA.3]
I b r3
f
,b A cosh t -- B sinh t
-t
dt.
[A.9]
For Ka >~ 10, we write
+C~'
f J,
[A.6]
(A + B) f ' ~ e-~ J, - - d r 2 ,a t
I
- - ~ d r = --~b~ 1 + 3K-2a-2 + 3Q~-la -1
(A - B) f , b e t + ---f - dz [A.10]
a3
2
-t- Ry -3
+
~a
10
Ka
t
where 40
A+B 2e `a
+(£
7o)
1 + [-@b + 1)/(Kb -- 1)]e - 2 ( ' b - ~ )
"°'°,l.. [A.4]
where
1/R
=
[sinh
(Kb - -
Ka) -- Kb cosh (~b
Journal of Colloid and Interface Science,
-- Ka)],
[A.5]
Vol. 47, No. 2, May 1974
[A.11]
A--B 2e-,,a 1 + [(~b - 1)/(~b + i)]~ ~<~-~o) and use a s y m p t o t i c expansions or rational approximations (1) to evaluate the integrals in [-A.10-]. Figure 3 shows the functionf(~a, ~,) plotted against Ka for various porosities %
E L E C T R O K I N E T I C P H E N O M E N A W I T H I N M U L T I P A R T I C L E SYSTEMS
The following aspects of the solution presented here are worthy of mention. (i) For the case of a single isolated sphere (i.e., b--+oo, -r-+ I, y--~ 0) the solution given in EA.1"]-[-A.9"] reduces to Henry's original solution, as quoted by Overbeek (16), viz: EJ~a [-
V = __ 6~rv
K2a2
5K~a3
K4a4
1-~
L
16
48
e Ka
\ 8
KSa5
+--
96 /
96
96
rA.12"]
dt
. ,a -~-
"
I-It should be noted that there is an error in Henry's result for V, although his derivation appears to be in order (8)"].
529
(ii) For Ka--+oo, f(Ka,~,) --+ 1 for all porosities. In other words, Smoluchowski's classic result for electroosmosis ['1.4"] has been justified theoretically as being valid for porous media of arbitrary porosity composed of equal spheres, when based on the Kuwabara model. (iii) For Ka --+ O, f(Ka, "r) --+ 0 for all porosities 7 < 1.0. In other words, as double-layer interaction within the voids increases, the velocity V decreases sharply. This trend is completely consistent with the theoretical predictions of Rice and Whitehead (19) for electroosmosis within circular capillaries. Their results indicate a qualitatively similar dependence on mean electroosmotic velocity V on Ka (here a denotes capillary radius) and also predict that V approaches zero as Ka --+ 0.
Journal of Colloid and Interface Science, Vol. 47, No. 2, May 1974
dielectric, spherical particles. The effects of interaction of the individual particles and their associated electric fields are taken into explicit account by employing a fundamental cellmodel representation which is known to provide good predictions for the motion of a swarm of spheres within a fluid in the absence of electrical effects. For thin double layers, the electrophorefic velocity of (or the electroosmotic velocity within) a swarm of identical dielectric spheres is shown to be invariant, practically speaking, with respect to the void fraction of the swarm. However, as the double layer thickness increases the electrophoretic (electroosmotic) velocity decreases sharply and becomes strongly dependent on the void fraction. The computed results provide theoretical justification for Smoluchowski's widely quoted approximate result for electroosmotic flow within porous media of arbitrary pore geometry. I. INTRODUCTION An understanding of the flow behavior of electrolytes relative to swarms of solid particles is of considerable importance in m a n y fields of science and engineering, for example electrophoresis, electroosmosis, pressure-induced fluid flow, and gravitational sedimentation. This paper examines analytically those electrokinetic phenomena which occur within particle swarms as a result of externally applied electric fields. These processes can be classified as (i) electrophoresis of suspended particles, and (ii) electroosmosis within rigid beds of particles. For convenience, attention is here confined to swarms composed of identical, spherical particles. 1 Permanent address: Department of Mathematics, University of Manchester, England.
Several theories, of increasing sophistication, have been advanced to predict the electrophoretic velocity of a single isolated sphere [-Smoluchowski (22 ), Huckel (11), H e n r y (8, 9), Hermans (10), Overbeek (15-17), Booth (2, 3), Pickard (18), Dukhin (5)J. However, no serious attempt appears to have been made (3), (21) to extend the validity of these theories to the more practical system of a s w a r m of particles. This problem is much more difficult because it becomes necessary to take into account the effects of the particle interactions on the electric field, and the fluid motions associated with the individual particles. Similarly, for electroosmotic flow within beds of particles it is difficult to account explicitly for the effects of pore geometry, tortuosity, etc. When calculating this electroosmotic flow it is customary to utilize Smoluchowski's
520 Jegrnal of Colloid and ln~erface Science, Vol. 47, No. 2, May 1974
Copyright ~ 1974 by Academic Press, Inc. All rights of reproduction in any form reserved.
ELECTROKINETIC PHENOMENA WITHIN MULTIPARTICLE SYSTEMS
521
dassic result (17), (22), (23), which is valid extension of Henry's classic treatment of the for porous media in general (i.e., independent electrophoresis of a single isolated spherical of the pore geometry) provided that (i) the particle. Throughout the present analysis we assume internal surface of the porous medium is nonconducting, (ii) the thickness of the diffuse negligible surface conductance and small ~" layer is small in comparison with the radius of potentials (in the Debye-Hiickel range), whence relaxation effects may be neglected curvature of the constituent particles. Because the pore geometry of particle (1), (16), and (25). In the main text we confine swarms usually defies description in all but a our attention to large values of Ka, whereas loose statistical sense, it becomes necessary in the Appendix we develop the theory for to introduce certain simplifying assumptions arbitrary values of Ka. For large Ka, it is demonstrated that the in order to permit a quantitative assessment of flow behavior occurring therein. The problem electrophoretic velocity of a swarm of solid of slow viscous flow relative to a swarm of spheres is given by the following simple uncharged spherical particles has been treated expressions : analytically by a number of authors [Cunningham (4), Happel (6), Kuwabara (12), Neale V = EE~a f(~') for the Happel model, [1.12 and Nader (14)] using familar cell-model 47r/~ representations, the principal function of which is to permit a simple, yet reasonably accurate, for the Kuwabara model, [1.2] V = __ assessment of the effects of particle interaction. 4~r~ The most amenable of these models are the so-called "free-surface" model due to Happel, where e denotes the dielectric constant of the and the "zero-vorticity" model due to Kuwa- electrolyte, # its viscosity, E the externally bara, the predictions of which are in close applied electric field, ~a the potential at the agreement with experimental data throughout surface of the spheres (this potential may be the entire range of porosity. However, in good identified with the ~" potential), and ~, the ionizing solvents, like water, the particles porosity of the swarm (porosity = void fracconstituting the swarm would invariably carry tion = 1 -- volume packing fraction of solids). an associated charge and, consequently, be The function f@) can range in value between surrounded by diffuse layers bearing an excess 1.0 and 1.09, indicating that the predictions of oppositely charged ions. In fact, there is of the two models are in satisfactory agreement, considerable experimental evidence (13), (20), practically speaking. Both [1.1] and [1.22 (24) that the effects of these diffuse (double) reduce to Henry's original result for the case layers are often too significant to be neglected. of a single isolated sphere (~, = 1) at large Ka, In other words, the predictions of the above namely, geometric models, when assuming uncharged V = eEpa/47r#, [1.3] particles, should not be applied indiscrimi- as required. nately when dealing with electrolytes. It is The conclusion to be drawn from these therefore desirable, in fact necessary, to be observations is that Henry's result [1.3] for a able to extend the theories of Happel and single isolated sphere remains valid, practically Kuwabara to accommodate these important speaking, for a swarm of spheres of arbitrary effects of the double layer. This can be achieved porosity [this being in complete contrast to the analytically for small ~" potentials (in the case of gravitationally--induced particle moDebye-Htickel range) and negligible surface tion, for which the settling velocity of a swarm conductance, for all values of Ka (where 1/K decreases rapidly with decreasing porosity denotes the double-layer thickness and a the (7)]. The presented theory therefore justifies particle radius), by a logical adaptation and the conventional usage (when Ka is large) of Journal of Colloid and Interface Science, Vol. 47, No. 2, May 1974
522
LEVINE AND NEALE
O,
,:.0.'., ,.m, C1, ,
,
.
.
.
.
',L2 ~
:v-i
0
Z
- O:
FIG. 1. Aqueouselectrolyte mediumin relative motion with a swarm of chargednonconductingspheres (dashes denotes outer extent of diffuselayer). Henry's result [1.3-] for swarms, although it was derived specifically for a single isolated particle. It should be noted that El.i] and [1.2"] are valid also for evaluating the mean electroosmotic velocity V within a rigid bed of identical dielectric spheres. This is because, in the absence of an applied pressure gradient, electrophoresis of, and electroosmosis within, a swarm of spheres constitute two equivalent processes, each involving the relative motion of solid particles and electrolyte. Equations [1.1] and [1.2] can be seen to be completely consistent with Smoluchowski's classic result for electroosmotic flow within a porous medium of arbitrary pore geometry, namely, V = eE~b~/47r#
[-1.43
which is stated to be valid when the doublelayer thicknesses are much smaller than the pore diameters and when surface conductance is negligible. The substantial agreement of [1.1] and El.2] with [-1.4] i s particularly encouraging because it provides further theoretical justification for the implications inherent in Smoluchowski's result, namely that V is independent of porosity, particle size and shape. II. DEFINING EQUATIONS AND BOUNDARY CONDITIONS We consider a uniform three-dimensional distribution of solid nonconducting spherical particles of radius a, in steady relative motion Journal o f Colloid and Interface Scienae, Vol. 47, No. 2, May 1974
with respect to an aqueous electrolyte medium. For convenience, we consider the particles to be stationary and the fluid to be moving parallel to the Z axis with a uniform velocity V (Fig. 1). The particles carry a uniform surface charge density ~ in the equilibrium state of zero flow. Following Happel (6) and Kuwabara (12) we introduce a cell model in which each particle is envisaged to be surrounded by a concentric spherical shell of electrolyte, having an outer radius b such that the cell contains the same volumetric proportion of solid to fluid as exists in the entire assemblage [Fig. 2]. Each particle is surrounded by a diffuse layer, contained within the cell, which carries a net charge equal in magnitude but opposite in sign to that on the particle, such that the net charge within each cell is zero. ELECTRICAL CONSIDERATIONS
In the equilibrium state of zero fluid motion, the potential ~ = 6(r) in the diffuse layer depends only on distance r from the center of the particle, and satisfies Poisson's equation, V~ = -- 4~p/~,
[2.13
where p ~r) denotes the volume charge density and ~ the dielectric constant of the electrolyte medium. Using Gauss' theorem, the boundary conditions on $ read =----, r=a
= 0. ~
r=b
E2.2-]
523
E L E C T R O K I N E T I C P H E N O M E N A W I T H I N MULTIPARTICLE SYSTEMS
FIG. 2. The proposed model for the physical system depicted in Fig. 1.
Adopting the approximation used by Henry (7) in his classical work on electrophoresis, we superimpose on ~b(r) the electrostatic perturbation potential ¢ = ¢ ( r , O) outside each partide. This potential exists as a result of the externally applied electric field, and in the steady state satisfies the Laplace equation
can be incorporated if necessary, as demonstrated by Henry himself (9).
V2¢ = O.
#72~ = Vp + pV(¢ + ~b),
[-2.3]
If E denotes the externally applied potential gradient in the z direction, then q~is stipulated to satisfy the following boundary conditions:
0 where 0 denotes the polar angle measured from the Z axis, about which we have axial symmetry. The first relation in [-2.4] is that used by Henry (8), and follows from the condition of continuity of current flow at the surface of the particle. We assume in the second relation that at the outer surface of the cell (r = b) the local electric field is parallel to the externally applied electric field (this condition is analogous to Happel's hydrodynamic condition in [-2.9]). The solution of I-2.3], subject to [2.4] is ¢(~, 0) -
~+ 1 -- (a/b) 3
cos 0.
[2.5]
2rZ/
Following Henry (8), the effects of surface conductance and relaxation are here being neglected. The latter effect has been shown (1, 16, 25) to be small for large Ka(>~100 for ~b~ ~> 100 mV) or for small 6a(~<50 mV when Kay---10). The effect of surface conductance
HYDRODYNAMIC CONSIDERATIONS
The fluid flow around any typical particle is governed by the Navier-Stokes equation, as simplified for creeping flow,
[2.6]
and the continuity equation for an incompressible fluid,
V.u = O,
[2.7]
where u denotes the fluid velocity vector, u the electrolyte viscosity, and p the fluid pressure referred to a datum plane. We adopt hydrodynamic boundary conditions which are equivalent to those used by Happel (6, 7), namely, ur(,, 0) = ,0(r, 0) = o
at
u,.(r, O) = - V cos 0 r O( u o / r ) r,.o(r, O) = - - # \ Or + r
r = a,
[-2.8]
10ur~
} [-2.9]
00/ =0 at
r = b, [-2.10]
where V denotes the mean fluid velocity in the Z direction, and rr0 the tangential shear stress component. The condition [2.9] is the hydrodynamic counterpart of the second electrical condition in [2.4]. Happel himself considered the complementary problem of spheres falling under gravity through a viscous fluid. Consequently, his conditions may be retrieved from [2.8] and [-2.9] by adding V cos 0 to the value of ur and -- V sin 0 Journal of Colloid and Interface Science,
Vol. 47, No. 2, May 1974
524
LEVINE AND NEALE
to the value of uo [Happel and Brenner (7)]. A satisfactory alternative to [2.10], introduced by Kuwabara (12), is that the azimuthal component of the vorticity w~ (rather than r~o) be zero at the outer envelope of the cell, that is2
where Ao, A1, A2, B1, and ]32 are constants to be determined from the hydrodynamical boundary conditions F2.83-E2.10] or E2.8], [2.97, and [2.117, and where E x =
10(ruo)
1 Our
r
r O0
w~-
=0 Or
[2.153 1 -
at r = b .
(~/b)~
'
E2.11] &k
a3r f r
} = }(r) = dr + - - 2 This representation provides predictions for uncharged spheres which agree closely with those of Happel (7). The general solution of [2.6]-[2.7], which can be obtained from Henry's work (8) and which, in the absence of an electric field, reduces to Lamb's classic solution as used by Happel (7) and Kuwabara (12), reads
fJ b
p = Ao
I
+-
dr
Expressions for fro and w~ are readily obtained from E2.10]-[2.14]. It will become apparent that only the value of B1 is required here.
III. ULTIMATE SOLUTIONS FOR THE CELL MODELS
r2
4~r \ dr Fri = 2~ra2 -Air 2
u,=
B1 +--+A2+-_ 10/~ ~r
_
X sin OdO,
'; )1
6re#
r 3} dr
where r,0 is defined in E2.10] and
--
ra . a
uo =
Air 2
B1
5~
2#r
~X 6~'#
(L
A~+--
'f
[3.13
cos 0, O¢,t,r
[2.133
E
(pr, cos 0 + fro sin 0)r=a
ra
dr -- - -
_
f
Tr
-/0
B2
i
[2.15-]
The total force F acting on any typical particle is the sum of a hydrodynamical force F~ and an electrical force FE. The hydrodynamic force is given by
BI
+
1
L --H v2¢~ dr.
P~,r = --P + 2g
,
E3.2]
Or
B2
and the electrical force by
2r 3
~dr + - r3~dr 2r 3 . . . .
/ dT,\ sin O,
E2.14] 2For problems involving axial symmetry, which prevails here, the radial and tangential components of vorficity wr and wo are automatically zero. Similar remarks apply to the shear stress components. Journal of Colloid and Interface Science, Vol. 47. No. 2, M a y 1974
FE = - , E a 2 l - ) . \ dr/r=a
[3.3]
The calculated expression for the total force F is F = -- 4~rBt = -- 6zc#Va~ + eEaA,
[3.4]
ELECTROKINETIC PHENOMENA WITHIN MULTIPARTICLE SYSTEMS
525
where ~ and A represent the following functions : 6 + 4y ~
[3.5] 6 - - 9 y + 9 y 5 - - 6y 6
according to the Happel model, --
(6 + 4y5)
A=
~ dr + 10y5
(rS/a ~) ~ dr
(1 -- y~)(6 -- 9y + 9y5 -- 6y6)
[3.6]
'
~=
[3.7]
5 -- 9y + 5y 3 -- y6 according to the Kuwabara model. --5
±=
~dr + 5y 3
(rS/aS)~dr
[3.8]
( 1 - - y s ) ( S - - 9 y + S y ~ - y6) '
where [3.9]
y = a/b.
Following Henry (7), we integrate by parts and obtain
Utilizing the expressions given in [3.5]-[3.8] and [3.12] and [3.13] we obtain the following ultimate predictions, valid for Ka>> 1 : V=
f b
f b d~
dr = ~b -- t.ka -- aa
Jo
~E¢,o 47r/~
.((1
)
(1 -- yS) - y3)(1 + ~y~),
for the Happel model
[3.14]
and 3a 5 fab d~ +---, 2 r5 fb r3
Ja F
1
.
.
[3.10] g
.
Ja r ~ rig,
+
+ - -3a 5 fo b d~ --. 5 r5
[3.113
In particular, for thin double layers (i.e., Ka)) 1) it may be further shown (7) that f b
~dr~_]
fb r 3 ~ ~dr~'--~(~
daft,
-- ~bb),
~-~a,
for the Kuwabara model.
[3.15]
From the definition of the cell model, y is related to the porosity 3' by 3" = 1 -
(a/b) 3 = 1 --y3.
[3.12]
[3.13]
[3.16]
It can be seen that [3.14] and [3.15] both reduce to Henry's result [1.3] as 3"---* 1 (i.e., b---+ ~ , y---~O, ~ - - ~ 0), as required. DETERMINATION
since ~ b < < ~ for Ka>> 1 in all practical applications (i.e., 3" > 0.26), as will be demonstrated in the next section. For steady motion, the net force F on any typical particle must vanish (7, 8). Hence, from [3.4] it follows that V = (,E/6~,)(A/a).
-
4~-/~
1 fb
.
=
OF ~b
The potential distribution ~b = ~ (r) within the model cell is given by the solution of Poisson's equation [2.1] subject to [2.2"]. Unfortunately, an analytic solution of this equation seems difficult to obtain except at small potentials. Thus, by making use of the Debye-Hiickel approximation (applicable for ~b < 25 mV in most practical applications), Eq. [2.1] reduces to a form which can be Journal of Colloid and Interface Science, Vol. 47, No. 2, May 1974
526
LEVINE AND NEALE
solved directly (3' 163, namely
~
= ~.
[3.173
The solution of [-3.173 subject to [-2.23 is
¢, =
////{k---~-/\r/\(1 -
K2ab) sinh (xb -- ~a) -- (~b -- ~a) cosh (~b -- Ka)) "
It can be shown from [-3.183 that for Ka >~ 100 the value of 6b will be less than 1% of $, for all porosities exceeding 0.26. This statement assumes that physical credence can be given to the adopted cell model towards the lower end of the porosity range [N.B. the minimum attainable porosity for a swarm of monosized spheres is 0.26 (7)3.
Kuwabara model when extending the analysis to arbitrary values of Ka (0 < ~a < ~). A possible reason for the success of the Kuwabara model is that the contribution to the vorticity which is directly due to the electrical effects and, from Henry's work, is proportional to
2 DISCUSSION AND CONCLUSIONS
One important observation is that the expressions derived for V in this work are equally valid for electroosmotic flow within a rigid bed of spheres as they are for the electrophoresis of a suspension of spheres. As noted earlier this is because both processes involve the relative motion of solid particles and electrolyte under the influence of an externally applied electric field, in the absence of an applied pressure gradient (3,8, 17, 22, 23), For large Ka, the predictions of Happel's model [-3.14-] and Kuwabara's model [-3.15-1 are in numerical agreement to within better than 9% at all porosities (0 _< 3' _< 1). From a practical point of view this can be considered reasonable agreement but from mathematical and physical points of view the prediction of the Happel model [3.14-1 is unsatisfactory since it does not agree with the classic Smoluchowski result [-1.4-1, which is known to be valid for all porosities % at large Ka and negligible surface conductance. (The Smoluchowski result is valid at all potentials, provided Ka>> 1, but this observation need not concern us here.) In contrast, the prediction of the Kuwabara model [-3.15-1 at large Ka is identical with Smoluchowski's result [1.4-]. It is for this reason that we have employed only the Journal of Colloid and Interface Science, Vol. 47, No. 2, May 1974
[-3.183
ra
in fact vanishes at r = b. This is consistent with the condition employed by Kuwabara that the purely hydrodynamic part of the vorticity and hence the total vorticity [-2.11-1 is made to vanish. In the Happel model, the combination of electrical and hydrodynamic conditions seems less consistent. Another important inference to be drawn from this work is that for large Ka the classic result of Henry J-1.33 for the electrophoresis of a single isolated sphere and that of Smoluchowski [-1.4-1 for electroosmosis within both straight capillaries and arbitrary porous media, remain valid, practically speaking, for all particle swarms likely to be encountered in practice. This is particularly significant because it implies in turn that the many other electrokinetic theories based upon single isolated particles, not necessarily spherical, are likely to remain equally valid for swarms at large Ka. It has been shown in the Appendix that as Ka decreases, so the electrophoretic (electroosmotic) velocity V for a swarm decreases sharply from that predicted by [-1.4-1, owing to double-layer interaction within the voids. In a subsequent paper, the theory developed here will be modified and generalized to cover the related, but more complicated, phenomena of streaming potential and sedimentation potential within swarms of spherical particles,
ELECTROKINETIC PHENOMENA WITHIN MULTIPARTICLE SYSTEMS that is, processes in which the electric field E is internally induced, rather than externally applied, and must consequently be predicted. ACKNOWLEDGMENT The authors extend thanks to Professor Norman Epstein of the University of British Columbia for several valuable suggestions and for reviewing this manuscript. Financial support provided by the National Research Council of Canada is gratefully acknowledged. REFERENCES 1. ABRA~rOWlTZ,M., AND STEGUN, I. A., in "Handbook of Mathematical Functions." National Bureau of Standards, U.S. GPO, Washington, D.C., 1964. 2. BooTI% F., Proc. Roy. Soc. London A203, 514 (1950); J. Colloid Sd. 6, 549 (1951). 3. Booa'tI, F., Progr. Biophys. 3, 131 (1953). 4. CUNNmOI~A~t,E., Proc. Roy. Soc. London A83, 357 (1910). 5. DUKmN, S. S., in "Research in Surface Forces" (B. Derjaguin, Ed.), Voh 3, p. 313. Consultants Bureau, New York, 1971. 6. HAI'rEL, J., Amer. Inst. Chem. Eng. J. 4, 197 (1958). 7. HAPPEL, J., AND BRENNER, H., "Low Reynolds
8. 9.
10. 11. 12. 13. 14. 15. 16. 17. 18.
19. 20. 21. 22.
Number Hydrodynamics." Prentice-Hall, Englewood Cliffs, New Jersey, 1965. HENRY, D. C., Proc. Roy. Soc. London A133, 106 (1931). HENRY, D. C., Trans. Faraday Soc. 44, 1021 (1948). HERMAI~S,J. J., Phil. Mag. 26, 650 (1938). HOeKEL,E., Physik Z. 25, 204 (1924). KUWABARA,S., J. Phys. Soc. Japan 14, 527 (1959). MICI~AELS,A. S., ANn LIN, C. S., Ind. Eng. Chem. 47, 1249 (1955). NEALE, G. H., ANn NADAR,W., Amer. Inst. Chem. Eng. J. 19, 112 (1973). OVERBEEK,J. Th. G., Kolloid Beih. 54, 287 (1943). OVERBEEK,J. Th. G., Advan. Colloid Sci., Vol. 3, 97 (1950). OVEI~BEEK,J. Th. G., in "Colloid Science," Vol. 1, p. 194. Elsevier, Amsterdam, 1952. PICKARD,W. F., Kolloid Z. 179, 117 (1961). RICE, C., Axn WmTEI-IEAn,R., J. Phys. Chem. 69, 4017 (1965) SCI~EIDEGGER,A. E., in "The Physics of Flow through Porous Media." Univ. of Toronto, Toronto, 1960. SENGUPTA, M., J. Colloid Interface Sci. 26, 240 (1968). S~oLucllowsI~I, M., Z. Phys. Chem. 93, 129 (1918).
527
23. S~oLvcI~owsI(I, M., in "Handbuch der Electrizitat und des Magnetismus" (L. Graetz, Ed.), Vol. 2. Liepzig, 1914. 24. SWARTZENDRUBER, D., "Flow Through Porous Media" (R. De Wiest, Ed.), p. 215. Academic Press, New York, 1969. 25. WIERSEM~ P. H., LOEB, h. L., AND OVERBEEK, J. Th. G., J. Colloid. Interface Sci. 22, 78 (1966).
APPENDIX In the main text we derived expressions for the electrophoretic (electroosmotic) velocity V within a swarm of identical spheres subject to the restriction that Ka is large. We here seek to extend the analysis of the main text to accommodate all values of Ka (0 < Ka < oo ). Although the predictions for V given in Eqs. [-3.14] and [,3.15] are in satisfactory agreement from a practical viewpoint, it is evident that the K u w a b a r a result [,,3.15] is physically preferable to the Happel result [,3.14], since the latter predicts that V should increase, albeit slightly, as the porosity of the swarm decreases ; such an increase in V has apparently not been observed experimentally. Hence, in this Appendix we shall confine our attention exclusively to the K u w a b a r a model. The velocity V according to the K u w a b a r a model, can be expressed using Eqs. [,3.13], [-3.7], and [.3.8] as follows: ~E¢~ V =
~ f(~a, 7),
4~
[A.1-]
where a -
f(~a,"r) =
-
f
/ a 5 r3 dr + y3
-
a 3 ~dr -
[,A.2-]
~(1 -- y")~o
The function f(Ka, 7) is therefore a correction factor which measures the extent to which V differs from Smoluchowski's prediction [,1.4-]. The integrals in [-A.2] can be determined b y substitution of [,3.18-] directly into [,3.10-] and [,3.11-], followed by partial integration and some straightforward but tedious algebraic manipulation. The final expressions for the Journal of Colloid and Interface Science, Vol. 47, No. 2, May 1974
528
LEVINE AND NEALE I
I
I
I
I.O-~E~ . f(~,,,r)
-
V = 4-T~
-- )% I.O(
~"
'-T-- 0.5
H E N R Y ~
~
5 /
®
~'~
o.~
0 .Ol
.I
I
I
IO
I00
IO00
KCl
FiG. 3. Plot off(tea, 3') vs Ka for various porosities % integrals appearing in [ A . 2 ] can be shown to be
~a b
I dr=--~b~
Q = [1 - Kb tanh (Kb -- ¢a)]/ [ t a n h (~b -- ~a) -- Kb],
K2~2 I + RKa--[ 16 "JI-
K3a3
4
A = R['sinh ~b -- ~b cosh Kb],
[A.7]
B = R [ c o s h Kb -- ~b sinh ~b],
[A.8]
12
I = __ _ _
-~-
K5a 5
96
Pa6~I] /
FA.3]
I b r3
f
,b A cosh t -- B sinh t
-t
dt.
[A.9]
For Ka >~ 10, we write
+C~'
f J,
[A.6]
(A + B) f ' ~ e-~ J, - - d r 2 ,a t
I
- - ~ d r = --~b~ 1 + 3K-2a-2 + 3Q~-la -1
(A - B) f , b e t + ---f - dz [A.10]
a3
2
-t- Ry -3
+
~a
10
Ka
t
where 40
A+B 2e `a
+(£
7o)
1 + [-@b + 1)/(Kb -- 1)]e - 2 ( ' b - ~ )
"°'°,l.. [A.4]
where
1/R
=
[sinh
(Kb - -
Ka) -- Kb cosh (~b
Journal of Colloid and Interface Science,
-- Ka)],
[A.5]
Vol. 47, No. 2, May 1974
[A.11]
A--B 2e-,,a 1 + [(~b - 1)/(~b + i)]~ ~<~-~o) and use a s y m p t o t i c expansions or rational approximations (1) to evaluate the integrals in [-A.10-]. Figure 3 shows the functionf(~a, ~,) plotted against Ka for various porosities %
E L E C T R O K I N E T I C P H E N O M E N A W I T H I N M U L T I P A R T I C L E SYSTEMS
The following aspects of the solution presented here are worthy of mention. (i) For the case of a single isolated sphere (i.e., b--+oo, -r-+ I, y--~ 0) the solution given in EA.1"]-[-A.9"] reduces to Henry's original solution, as quoted by Overbeek (16), viz: EJ~a [-
V = __ 6~rv
K2a2
5K~a3
K4a4
1-~
L
16
48
e Ka
\ 8
KSa5
+--
96 /
96
96
rA.12"]
dt
. ,a -~-
"
I-It should be noted that there is an error in Henry's result for V, although his derivation appears to be in order (8)"].
529
(ii) For Ka--+oo, f(Ka,~,) --+ 1 for all porosities. In other words, Smoluchowski's classic result for electroosmosis ['1.4"] has been justified theoretically as being valid for porous media of arbitrary porosity composed of equal spheres, when based on the Kuwabara model. (iii) For Ka --+ O, f(Ka, "r) --+ 0 for all porosities 7 < 1.0. In other words, as double-layer interaction within the voids increases, the velocity V decreases sharply. This trend is completely consistent with the theoretical predictions of Rice and Whitehead (19) for electroosmosis within circular capillaries. Their results indicate a qualitatively similar dependence on mean electroosmotic velocity V on Ka (here a denotes capillary radius) and also predict that V approaches zero as Ka --+ 0.
Journal of Colloid and Interface Science, Vol. 47, No. 2, May 1974