ELECTROKINETIC FLOW IN
A
NARROW CYLINDRICAL CAPILLARY
Science Foundation for the award of a Senior Foreign Scientist Fellowship during the tenure of which this work was done. This investigation was supported in
4017
part by Public Health Service Research Grant AM05177 from the National Institute of Arthritis and Rletabolic Diseases.
Electrokinetic Flow in a Narrow Cylindrical Capillary
by C. L. Rice and R. Whitehead Royal College of Advanced Technology, Salforord 6, Lamaahire, England
(Received June 28, 1065)
This paper is a theoretical study of electrokinetic flow in narrow cylindrical capillaries. It is concerned with the dependence of the usual electrokinetic phenomena on the electrokinetic radius. The results obtained for this dependence must, however, be treated with caution for the higher values of the interface potential due to the use of the DebyeHuckel approximation. Of interest is the prediction of a maximum in the electroviscous effect.
Introduction The results of applying pressure and potential gradients across a capillary are well known, the basic relationships involved having been formulated mainly by Smoluchowski. However, all these relationships involve the assumption that the double-layer thickness is small compared with the capillary diameter. I n some investigations, this condition is no longer satisfied and the dependence of electroosmosis, the streaming potential, and the apparent viscosity of the capillary fluid on the electrokinetic radius must be taken into account. It has been shown2 that a bed of fine particles or a porous diaphragm can be regarded in the same manner as a single capillary when the electrokinetic radius is large and probably also when it is not large, though greater care must’be exercised here. Pertinent also is the fact that it is now possible to make a uniform capillary with a radius as small as em. In a recent paper, Burgreen and Nakache3 studied the electrokinetic flow in a very fine capillary channel of rectangular cross section. I n this paper we shall be concerned with a single long uniform circular capillary of radius a, containing an electrolyte, and no assump-
tions will be made as to the magnitude of the electrokinetic radius Ka where K is the reciprocal of the Debye length.
Double-Layer Potential and Net Charge Density Consider a uni-univalent electrolyte of bulk ionic concentration n ions/unit volume, with the wall of the capillary at a potential +bo and a potential $ a t a point distance r from the axis. If the excess charge density at this point is p ( r ) , then the Poisson equation has the form
for, due to symmetry, $ is a function of r only. t is the dielectric constant which is assumed to be uniform throughout the liquid. From the Boltzmann equation (1) M. von Smoluchowski in Graetz, “Handbuch der Elektrizitat und des Magnetismus,” Vol. 11, Leipzig, 1914, p. 366. (2) P. Mazur and J. T. G. Overbeek, Rec. trap. chim., 70, 83 (1951). (3) D. Burgreen and F. R. Nakache, J . Phys. Chem., 68, 1084 (1964).
Volume 69, Number 11
November 1966
4018
C . L. RICEAND R. WHITEHEAD
p(r) =
e* -2ne sinh kT
If e$/kT is small, sinh e+/kT
1 : e#/kT.
1 dP
F, 7
When
$ N 25 mv., e$/kT = 1, so that it would appear that
the above approximation would only be applicable to very small values of #. Fortunately, however, this approximation results in an expression for the potential distribution at a single plane surface which is in good agreement with that obtained from the exact solution for values of $0 up to 50 mv. in the case of a uni-univalent electrolyte. This approximation can be adopted in the case of a cylindrical capillary, with the proviso that any results derived must be viewed with caution when applied to higher values of This is particularly so when KU is small, since it is for these values that the double layers begin to overlap in the center of the capillary. The Poisson-Boltzmann distribution equation then becomes
(8)
with the conditions that p and v, must be finite everywhere, that p is a function of z only, and that
v,(u) =
$1
= 0 r=O
(9)
In the following sections dP -dz
=
P,
where P, is the uniform applied pressure gradient, and the body force is caused by the action of the applied electric field, E,, on the net charge density p ( r ) in the double layer.
Electroosmosis In the presence of an applied potential gradient and an applied pressure gradient, the equation of fluid motion is
(3) where K = (8me2/dcT)”’ is the reciprocal of the double-layer thickness. The solution of (3) which is finite at T = 0 is $ =
BIo(KT)
(4)
where Io is the zero-order modified Bessel function of the first kind. At r = a the condition is $ = #o; therefore (5)
From Poisson’s equation, the net charge density is given by
--*
Substituting expression 6 for p ( r ) in this equation and using the boundary conditions (9), the solution of (10) is found as
a sum of a Poisseuille flow term and an electrokinetic term. Electroosmotic velocity and volume t,ra,nsport are observed under conditions of no applied pressure gradient; i e . , P , = 0 and so
CK2
P(4 =
477
The Equations of Motion The basic equation of motion for the fluid is
+ (v.V)V + V p = F
~VAVAV
(7)
with V . v = 0 as the fluid is incompressible. 7 is the coefficient of viscosity and F is the body force per unit volume. In an infinite cylindrical tube under the influence of an axial applied electric field, v, = v,(T) and v, = v4 = 0, in terms of cylindrical polar coordinates. Thus in this case the inertia terms ( v . V ) v vanish. Also, V p = dp/dze, and F = F,(r)e,. The equation of motion thus reduces to The Journal of Ph.yaical Chemistry
wherefl = C + O / ~ T ~ . For KU >> 1, asymptotic expansions show that I ~ ( K T ) / I O ( K U ) is negligible, except in the double-layer region very close to the wall. Under these conditions (12) reduces to the classical formula
v,(r)
=
-flE,
(13)
The function [l - ( I o ( K T ) ) / (is~ o plotted ( K uin ))] Figure 1 to give velocity profiles for various values of KU. This shows that eq. 13 holds for 0 6 r 6 0 . 9 ~ if KU > 50. For KU = 10, the divergence from (13) becomes marked as the double layers begin to overlap in the center of the capillary. For KU 6 2, the velocity profiles take on a noticeable Poisseuille form. Using
4019
ELECTROKINETIC FLOW IN A NARROW CYLINDRICAL CAPILLARY
V = 2 r 1 v Z ( r ) r d r = -QE,A,
[1 - KaIo(Ka) ~
211(xa)] (15)
As Ka becomes large, the function in brackets tends to 1,giving
0.8-
V
0.6-
=
-QE,A,
(16)
which is the usual classical result. The function [l - (211(~a))/(~aIo(~a))] is plotted against log Ka in Figure 2, which shows clearly that the greatest change of V with Ka is in the region 1 < Ka 10 while V has 90% of its classical value for K a = 20. The electroosmotic counter pressure can be obtained by first integrating eq. 11 for u,(r) over a cross section, obtaining for the volume transport
> 1gives the usual form
P, - 2 4 0 E, Tu2
0.0
0.0
0.2
0.4
0.6
0.8 1.0 Log nu.
1.2
1.4
1.6
1.8
2.0
Figure 2. Volume transported per unit area and unit time for various values of KU.
(19)
If condition 18 is substituted in eq. 11, the expression for the velocity distribution in the case of no total flow is obtained
the series expansion for Ioand neglecting terms of o[(KU)4]and higher for Ka > 1the above expression reduces to
except in the double-layer regions close to the wall. The velocity profiles for a number of values of Ka are plotted in Figure 3. It can then be seen that near the walls the contribution of P, to the flow is not as great as the contribution in the opposite direction caused by the migration of the ions under the applied field E,. Thus if $0 is positive, there is in this region a preponderance of negative ions and the flow is in the negative direction. On the other hand, in regions nearer the center of the capillary the flow in the positive direction due to P, is greater than the now much smaller Volume 69,Number 11
November 1966
4020
C. L. RICEAND R. WHITEHEAD
evaluating the integrals, noting that the last is of the Lommel type, we obtain
If X is the conductivity of the fluid and this is assumed to be uniform throughout the fluid up to the wall of the capillary, then =
XE,A,
(24)
Hence
-0.2
-0.4
The first term in the above expression represents the current for electroosmotic flow, since in this case P, = 0. The ratio of volume flow V to current i is then given by
-0.6
V
- 0.8
-i =
Q --f(Ka,p)
x
wherep = Q2qK2/X and
- 1.0 0.0
0.2
0.4 r/a.
0.6
0.8
1.0
Figure 3. Velocity profiles for the electroosmotic counter pressure condition.
contribution in the negative direction to which E, gives rise.
The Current in Electroosmotic Flow The current, i, flowing can be written as i = il &, where il is the current due to transport of charge by the fluid and iz is the conduction current.
+
il = 2?rSaDv,(r)p(r)rdr
(22)
With both an applied electric field and a pressure gradient, expressions 11 and 6 for v,(T) and p ( r ) , respectively, are used, giving
c.g.s., the range For water, where E = 81, q = of the parameter p is effectively covered by the values 0.1, 4.5, 27.0. The results for the higher values of p corresponding to high values of $0 will be less reliable, but they should nevertheless indicate the general trend and so are worth inclusion. In Figure 4, f(Ka,P) is plotted against log Ka for these three values of p. For large values of Ka, asymptotic expansions indicate that f (xa,p) tends to 1and we have the usual result that
V i
Q x
= --
As the curves indicate, the deviations from this result become larger as Ka becomes progressively smaller and, when 0 = 4.5, they are important for values of Ka less than 5.0. The Journal of P h y s k d Chemistry
ELECTROKINETIC FLOW IN
0.01 0.0
.
. 11.4
Figure 4. Tariation of
.
A
. 0.8 Log
f(KU,@)
4021
NARROW CYLINDRICAL CAPILLARY
.
, 1.2
,
,
1.6
.
1
2 .o
Ka.
with
KU.
Streaming Potential The streaming potential is the steady potenthl which builds up across a capillary in the presence of an applied pressure gradient and which is just sufficient to prevent any net current flow. Putting i = 0 in eq. 25 and rearranging, the following condition is obtained
-0.8 1 0.0
0.2
0.4
0.G
0.8
1.0
r/a.
Figure 5. Variation of current density across the capillary with p = 4.5.
Current Density Distribution The current density j ( r ) at a point in t'he fluid is given by By comparison with (26) it is seen that
V - _ _E i P as Mazur and Overbeek2 have shown it must be, in accordance with the Onsager principle of reciprocity of irreversible phenomena. The variation of the ratio E,/P, can therefore be inferred from Figure 4, and again for large values of KU we have the well-known result that
j ( r ) = vz(r)p(r)
+ EZX
Substituting expressions 11 and 6 for respectively, we find that
j(r> _ p,
-
(a2 - r2) 16a~
€K2#o
IO(K?)
I o (Ka)
(32) z',(T)
and p ( r ) ,
+
I n the streaming potential case with no net flow of current, condition 28 for t,he ratio E,/P, is appropriate and eq. 33 becomes
If the {-potential, where { = #o, is determined experimentally for narrow capillaries from measurements of the streaming potential on the basis of eq. 30, an apparent value la would be found. This value is related to the electrokinetic potentid f by the relation
Thus the apparent {-potential would again be expected to vary with the electrokinetic radius after the manner shown in Figure 4.
With p = 4.5, the function on the right-hand side of expression 34 is plotted in Figures 5 and 6 for various values of KU. In order to show the effect of the parameter p, the function is plotted in Figure 7 with KU = 10 for the three values of p which were used in Figure 4. Volume 69, Number 11
.Vovember 1966
C. L. RICEAND R. WHITEHEAD
4022
1.0
0.0
- 1.0
- 2.0
-3.0
p-, - 4.0 - 5.0 - 6.0 - 7.0 - 8.0 0.0
0.2
0.4
0.6
0.8
1.0
r/a.
0.0
0.2
0.4
0.6
0.8
1.0
r/a.
Figure 6 . Variation of current density across the capillary with p = 4.5.
Figure 7. Variation of the current density across the capillary with KU = 10.
The curves have been obtained by assuming that the conductivity X is uniform throughout the fluid and by neglecting the effect of surface conductance. Due to this there must be some uncertainty as to the behavior of the curves at distances very close to the capillary wall. cm. containing a For a capillary of radius a = uni-univalent electrolyte of concentration 10-' M , KU = 100, and the reversal point of the current pearest to the capillary wall occurs a t a distance of 10 A. from it. For values of Ka less than 100 or for values of the radius greater than low5cm., the reversal point is a t a larger distance than this from the wall. Despite the uncertainty in the behavior of the curves very close to the wall, it is felt that these reversal points are meaningful physically. From Figures 5 and 6, it is seen that there is only one reversal point for each of the values Ka = 1, 2, but for the higher values of Ka there are two. This can be explained if the situation pertaining for the higher values of Ka is considered first. Assuming
that $0 is positive, then p(r) is negative, and neSor the wall the negative ions move in the negative direction, as their mobility under the field E,, which is set up, is in the negative direction and is greater than their transport in the positive direction due to P,, u,(r) being very small. Thus j ( r ) is positive; i.e., E,X > v,(r) P(T). Near the center of the capillary $ N 0 and so p(r) I: 0. j ( r ) then depends on the conductance and is virtually independent of P,. Thus again j ( r ) is positive; Le., E,X > u,(r)p(r). In part of the intervening region, $ and p(r) are small but not negligible, and the transport of net negative charge in the positive direction outweighs the conductance effect in t,he opposite direction. j ( r ) is therefore negative, Le., E,X < v I ( r ) p ( r ) , and there will be two reversal points. For the lower values of Ka, e.g.? Ka = 1, 2, the situation near the center of the capillary is different from the above. Now as r + 0, $ and p(r) are still appreciable and hence there is no region where the conduct-
The J O U Tof~Physical ~ Chemistry
ELECTROKIXETIC FLOW IN
4023
h NARROW CYLINDRICsL CAPILLARY
ance contribution can again overcome the transport contribution, as it does near the wall. So E,x < u,(r)p(r), j ( r ) remains negative, and there will be only one reversal point .
6.0
4
5.0
The Electroviscous Effect When a liquid is forced through a narrow cylindrical capillary under an applied pressure gradient, a streaniing potential gradient is set up, and the volume flow is given by eq. IL7. This reduced rate of flow results in the apparer1.t viscosity qa where (35)
In the streaming potential case we can substitute relationship 28 for E,/P, obtaining
Using the asymptotic expansions, it is found that for very large values of KU
(KU)
1--
3.0 2.0 1.0 0.0
5s
10.0
15.0
Ka.
Figrire 8. Tariation of the apparent viscosity with xu.
Subst'it,utingthis expression in eq. 17 gives
1---
4.0 c
2 7r2u2Xq
Under the conditions E = 81, 9 = lo-? c.g.s., #o = 100 mv., A = 100 ohm-' c m 2 g.-equiv.-I, for a uniA9 with univalent electrolyte of concentration a capillary radius a = cni., then KU = 100 and ~ ~ # ~ ~ / 2 7 rN ~ u4 ~ X Xq Expanding (38) by the binomial theorem, since ~ ~ $ ~ ~ / 2 7 r
