D. BURGREEN AND F. R. NAKACHE
1084
Acknowledgment. The authors wish to thank Mr. Robert Holmes for his effort in carrying out the experi-
ments, and Mr. M. J. Massa for his aid In making the statistical analysis of the data.
Electrokinetic Flow in Ultrafine Capillary Slits'
by D. Burgreen and F. R. Nakache Deaelopment Diuision, United Nuclear Corporation, White Plaine, New YorB
(Received November 9, 1963)
This paper contains an analytical study of electrokinetic flow in very fine capillary channels of rectangular cross section. It is a natural extension of the general theory of electrokinetic flow which heretofore was limited to channels of large electrokinetic radius or to interfaces exhibiting low source potential. The practical implications of the results of the study are explored.
Nomenclature
U
cross section area of capillary channel
uo
M 2io2~ / K
U,
ion concentration in moles per liter dielectric constant of fluid applied axial voltage streaming potential elliptic integral of second kind elliptic integral of first kind parameter plotted in Fig. 6 half of distance between plates current in external circuit electroosmotic conduction current = KOAY O electroosmotic transport current lumped conductivity (fluid path plus external path) = KO L/2hRo = K O L/rro2R0 specific conductivity of fluid Boltemann constant length of capillary passage mobility = & D / 4 ~ por fD/47rp average number of positive or negative ions per unit volume average number of ions of ith kind per unit volume number of ions of ith kind per unit volume preesure in fluid Reynolds number resistance of external circuit internal or fluid resistance radius of tube wetted perimeter of capillary channel absolute temperature
+
The Journal of Physical Chemistry
+
UP Ur
V VP
V, We WP
X
Y
YO Y* Y," Y 2
ZI
velocity a t a given point in capillary channel velocity a t center of channel electroosmotic velocity pressure-induced velocity retarding flow component volumetric flow volumetric pressure-induced flow volumetric retarding flow electrokinetic pumping energy mechanical pumping energy axial distance along capillary channel axial electric field = dE/dx applied axial electric field = dEo/dx dE,/dx = streaming axial electric field unreduced streaming potential distance measured from capillary wall valence of ions when one kind of salt is present and dissociates into two equal and oppositely charged ions valence of ith type of ion
Greek symbols LY
ai
7
e eo
ionic energy parameter = ez$o/kT ionic energy parameter = ezi$o/kT rectilinear coordinate measured from center of channel in flow between plates sin-' [cosh ( ~ ~ $ , / 2 $ ~ ) / c o(a$/2$0)1 sh sin-' [cosh (a$c/2$o)/cosh (or/2)]
(1) Work performed for Aeronautical Systems Division, AFSC, ASRFS-2, Wright-Patterson AFB.
ELECTROKINETIC FLOW IN ULTRAFINE CAPILLARY SLITS
K
x c1
P 7
$J $0
n w
[cash (~$0/2$0)l-’ Debye length = [D$o/8?r+kzm] ‘la viscosity of fluid net charge per unit volume viscous shear stress electric potential of ions (relative to solution containing an equal number of positive1 and negative ions) potential a t surface of capillary electrokinetic radius = 2wA/e, or 2wh, or WTO reciprocal of Debye length = [8aReza/D$J0]’Iz
Introduction Before proceeding with the analytical development of the theory of electrokinetic flow iin ultrafine capillary elements, it is worth reviewing briefly the historical development and present status of the subject. Electrokinetic phenomena were observed as far back as the beginning of the 19th century. In 1808, Reuss2 discovered that flow through capillary elements can be induced by the application of an electric field. About half a century later, Wiedema,nns performed a number of quantitative experiments and promulgated one of the fundamental theories of electrokinetics. This theory, which has been verified many times, states that the volumetric electroosmotic flow is proportional to the applied current. I n 1859, Quinke4 discovered the phenomenon of streaming potential which is the converse of electroosmosis. His experiments showed that when fluid was forced through a diaphragm, the voltage developed across the diaphragm was proportional to the pressure differential causing the flow. In 1879, Helmholtz6 developed the double layer theory which related analytically the electrical and flow parameters of electrokinetic transport. Although the theory was based on a somewhat intuitive analysis, it has stood the test of time and E;till represents an acceptable formulation of the electroosmotic phenomenon in most capillary materials. Smoluchowski,o in 1903, expanded on Helmholtz’s double layer theory by taking into account the actual. distribution of velocity in the capillary channel. In 1909, Freundlich’ published results of a number of comprehensive experiments dealing with electrokinetic effects, and was the first to use the word electrokinetics to describe the phenomena. He demonstrated that the proportionality constant in electroosmosis, relating volumetric flow to electric current, was identical with the proportionality constant relating streaming potential and applied pressure. A more realistic concept of the potential and charge distribution in the fluid adjacent to the capillary wall was introduced by Gouy* in 1910. He computed the electric charge distribution in a diffuse layer. Debye and H u ~ k e l in , ~ 1923, determined the ionic distribu-
1085
tion in solutions of low ionic energy, by means of a linear simplification of the exponential Boltzmann ion energy distribution. Contributions have been made in more recent times to the technology of electrokinetics by Freundlich,’ Abramson,lO Bikerman, l1 Davies and Rideal,12and others. A comprehensive treatment of the classical theories of electroosmosis and streaming potential is available in a recently published article by the authors.18 Analytical solutions exist13 for electroosmotic flow in slits and capillary tubes when the surface ionic energy eqb0 is small in comparison to the thermal energy kT. The solution of these problems indicates that when the electrokinetic radius, defined as 2wh or wq, (see Nomenclature), becomes moderately large, the electrokinetic effects are confined to the area close to the capillary wall. As a result, it is possible to solve the electroosmotic flow problem for arbitrary crosssection channels applicable to capillaries having any magnitude of ionic energy, provided the electrokinetic radius is sufficiently large. It is difficult to specify an exact range of validity for solutions applicable to large electrokinetic radius since the solution becomes more accurate as the electrokinetic radius increases. For the case of flow in a slit, we find that when wh > 5 the error is less than 1%. In the case of flow in a tube or in an odd-shaped capillary, the nonrectilinear geometry requires that oro or the electrokinetic radius be larger than 40 in order to obtain reasonably accurate results. The requirement of large electrokinetic radius would appear, from a practical point of view, not very restrictive, since only in very dilute solutions and in very fine capillaries will the electrokinetic radius be small. However, it is specifically the results of the present treatment, applicable to small electrokinetic radius, that suggest practical application. (2) F. F. Reuss, Memoires de la Societe Imperiale de Naturalistes de Moscou, Vol. 2, 1809, p. 327. (3) G. Wiedemann, Pogg. Ann., 87, 321 (1852). (4) G. Quinke, ibid., 107, 1 (1859). (5) H. Helmholtz, Ann., 7 , 337 (1879). (6) M. Smoluchowski, Krak. A n a , 182 (1903); also in Graets, “Handbuch der Elektrisitiit und des Magnetismus,” Vol. 2, Barth, Leipsig, 1921, p. 336. (7) H. Freundlich, “Kapillarchemie,” Akademischer Verlag, 1909. (8) L. Gouy, J. Phys., 9, 456 (1910). (9) P. Debye and E. HUckel, Physik. Z.,24, 185, 305 (1923). (10) H. A. Abramson, “Elecktrokinetic Phenomena and Their Application to Biology and Medicine,” Chemical Catalog Co., New York, N. Y., 1934. (11) J. J. Bikerman, “Surface Chemistry,” Academic Press Inc., New York, N. Y., 1958. (12) J. T. Davies and E. K. Rideal, “Interfacial Phenomena,” Academic Press Inc., New York and London, 1961. (13) D. Burgreen and F. R. Nakache, “Electrokinetic Flow in Capillary Elements,” ASD-TDR-63-243, March, 1963.
Volume 68, Number 6
May, 1964
D. BURGREEN AND F. R. NAHACHE
1086
In solving the streaming potential problem, similar limitations were imposed. Solutions of problems are thus available covering the more common ranges of ionic energy and electrokinetic radius. This paper contains the analysis of the streaming potential generation and flow in a capillary slit for an arbitrary ionic energy and electrokinetic radius. It provides a solution for the case of high ionic energy and small electrokinetic radius, which is not contained in the earlier work. In the treatment which follows, electroosmotic flow and streaming potential generation are included as part of the comprehensive problem.
Charge and Potential Distribution The theory of electrokinetic flow is based on the fundamental observation that a t the interface of a dilute solution and nonconducting surface a potential $0 (relative to the fluid far from the wall where the positive and negative ion concentrations are equal) exists whose magnitude is dependent upon the character of the surface and the solution. The question of how or why such a potential is developed need not be considered here. It exists, and its presence produces a redistribution of ions in the vicinity of the surface. As a result of the surface potential C0, the equilibrium ionic distribution is disturbed, the net electric charge a t various points in the solution is not zero, and the number of ions of each type per unit volume in the fluid differs from the average. When the material of the capillary wall is an insulator and the net lateral current is zero, an equilibrium will be obtained between the ionic diffusion and conduction current. This resulting ionic distribution is the Boltzmann distribution which gives the number of ions of a particular kind (ni) which are a t a potential $ above that of a solution containing uniformly distributed ions. This is expressed as 1zi =
aie
-ezl+/kT
(1)
We have introduced in eq. 4 the inverse of the Debye length, w, defined as (5)
Integration of eq. 4 yields
If
$c
is the potential a t the center of the channel
( q = 0) then by means of the boundary condition
eq. 6 becomes
or wq = w ( h
- y)
=
where F(e,K) is the elliptic integral of the first kind, and
K
=
[cash (a$/2$0)]-'
(11)
The subscript c (center) refers to q = 0 agd the subscript 0 to q = h. The quantity $o/$o is defined by the expression
Assuming that we are dealing with a simple electrolyte which ionizes into two equally charged ions of valence z, then the net charge per unit volume, p, is given by p =
fiex(e"$c/il.o-
where a
=
=
-2aez sinh CY$/$^)
(2)
ez$o/kT. The charge density and potential
are also related through the one-dimensional Poisson
equation
9--- 04nP
dy2 When combined with eq. 2 this yields
(3)
(4)
The Journal of Physical Chemietry
From eq. 12 we can determine, for any given value of CY = ez$o/iiT, the variation of wh with $c/$o. The relationship between wh and $e/#o is shown graphically in Fig. 1. The knowledge of the variation of $ c / $ ~ with wh permits us to plot, with the use of eq. 9, the distribution of potential. The variation of potential across the slit is shown in Fig. 2, 3, and 4 for several values of a and wh. This is the undisturbed transverse distribution of potential, which will be maintained as long as there is no lateral mixing. Since laminar-type flow will
ELECTROKINETIC: FLOW IN ULTRAFINE CAPILLARY SLITS
1087
1
D.Q5 0.90 0.85
0.80 0.75
0.70 0.85
0.00 0.55
8 2'
0.50
?)
0.45 0.40 0.35
0
0.30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
0.9
1.0
+/h 0.25
Figure 2. Variation of potential and electrokinetic velocity in a slit; a = 1.
0.20 0.15 0.10
u,/MY 0.05 0
.o
2
4
6
8
10
12
ElectroktneUcAadIus. 2wh
Figure 1. Variation of
+,/+o
with electrokinetic radius.
normally exist in a fine slit, the foregoing computed variation of potential and charge will remain valid.
General Equation of Electrokinetic Flow Consider the case of laminar flow between parallel plates. Figure 5 shows an element of fluid of unit width acted upon by pressure forces, viscous forces, and electric body forces generated by an axial electric field, Y = dE/dy. Under steady conditions, the sum of these elemental forces is zero, and with the use of the relationship for laminar flow r = Mdu/dy, the equation
is obtained. In eq. 13, the charge density, p, is related to the spatial distribution of potential, J/, by. the Poisson equation (eq. 3) so that eq. 13 may be written as -
dP --
dx
,DY d2J/ = 0 + p d2u - +%-
dy2
4.lr dy
(14)
The velocity, u, is composed of two discrete parts. The first, up, is the laminar flow caused by the pressure
A . "
0
0.1
0.2
0.3
0.4
0.5
0.0
0.7
0.8
3/$0
Figure 3. Variation of potential and electrokinetic velocity in a slit; a = 4.
gradient, dP/dx, alone. The second is the electroosmotic velocity, ue. If only pressure and viscous forces were present, the equation of flow would be dP -+ / . -d2up =o ax dY2
If only electric body forces and viscous forces were acting, the equation of flow would be eq. 16. Volume 68,Number 5
M a y , 1964
D. BURGREEN AND F. R. NAKACHE
1088
where 11 = h - y. The mean velocity of flow in a slit is given by the expression G = - - - h2 + - dP
3 p dx
Mylh - i) (1
dq
(19)
h
We define G(a,wh) as
which, by use of eq. 8, becomes G(a,wh) =
u*o Figure 4. Variation of potential and electrokinetic velocity in a slit; CY = 10.
The mean velocity may, therefore, be written as G =
-
h2 d P
-3p
dx
+ M Y [ 1 - G(a,wh)]
(22)
The function G is found by numerical integration.
It is plotted in Fig. 6 for various values of the parameters a and wh.
Figure 5. Forces acting on a capillary fluid element between flat plates.
The Streaming Potential Up to this point, we have computed the potential and charge distribution and also the local and mean velocities in terms of the axial field Y (eq. 18 and 22). The field Y can be an externally applied field YO = Eo/L, as shown in Fig. 7 ; it can be a self-induced field Y,, defined as the streaming potential; or it can be a combination of both. If there were only an externally applied voltage (of zero internal resistance) which was maintained at E,, then the electroosmotic flow could
p d2ue - + - - =DoY d 2 # dyz
4a dy2
The addition of the foregoing equations yields eq. 14, in which u = ue up. The boundary conditions that apply are up = ue = 0, and 1c, = fi0 a t the surface of the slit. Also, due/dy = dfi/dy a t the center. The solution for the velocity will always be of the form
+
or Electrokinetic Radius, 2wh
Figure 6. Variation of G(oc,wA) with electrokinetic radius.
The Journal of Physical Chemistry
ELECTROKINETIC FLOW IN ULTRAFINE CAPILLARY SLITS
4
EQ
II
1089
+-I-
Equation 28 becomes
+-
I
? I
I
Figure 7. Basic electrokinetic circuit.
be computed by means of eq. 22. Practical considerations require, however, that we take into account the external resistance of a,n applied electric field source in addition to the internal resistance of the fluid in the capillary. The problem is treated below in a general way by including the external voltage and both the internal and external resistances, as shown in Fig. 7. Assume that across the capillary element there is placed a voltage source Eo and an internal resistance Ro in series with the voltage source, as shown in Fig. 7. The net voltage across the capillary element is
It is noted that only conduction current is associated with a voltage drop, since Ohm’s law is not applicable to the transport current. The current in the external circuit is I
=
Io
+ It
(24)
The foregoing expression for the net axial field is valid for both electroosmotic flow and streaming potential flow and does not carry any restrictions with regard to ionic energy or electrokinetic radius. Thus, the short circuit condition (Ro = 0) yields the electric field for electroosmotic flow, Y = E o / L , and the open circuit condition (Ro a ) yields the expression for open circuit streaming potential --)r
1
y = 1
It = -
pudA
JA
(25)
The substitution of eq. 23 and 25 into eq. 24 yields
-
JA
KOA
p
(1 -
$)dA
(31)
The function G(a,wh), evaluated previously (eq. 21) and plotted in Fig. 6, appears in the expression for the electric field Y as given by eq. 30. For flow in a slit, the integral term in the numerator becomes
By partial integration, eq. 32 becomes
lh
pup
where
”
I J p u p dA KOA A
d7
M dP K dx [l - G( a,wh)l
= --
(33)
Thus, the same averaging factor, 1 - G(a,wh), appears in both the electroosmotic component of the mean flow and in the expression for streaming potential. The integral term in the denominator of eq. 30
From eq. 26, I is set into eq. 23 and the result is
y
=
L
%’(L
+L
pu
dA - KoAY)
(27)
can be evaluated analytically. The result is
The substitution of u from eq. 17 into eq. 27 gives the net electric field explicitly as
+
: I
y = y
I
p u p dA
RO - .. -I, KOA -- M JA Ro J A . I
+
*UU
n
/
p
k))
(1 - ’ \
dA
(28)
where E($,K)is an elliptic integral of the second kind, and as before
YU
We can lump the internal and external resistances by defining the lumped conductivity K as Volume 68, Number 6
M a y , 1964
1090
D. BURGREEN AND F. R. NAKACHE
The integral term in eq. 33a has a definite physical significance. In simple short circuit (Ro+ 0) electroosmotic flow, given by the expression
from which it follows that the electroosmotic transport current in a slit is
The electroosmotic conduction current, moving in the same direction as the transport current, is
2KhEo I, = L The ratio of electroosmotic transport current to conduction current, It/Io, is thus 0.02
0.04
0.08
o.ia
0.2
0.4
0.8
i.a 1.6
z
4
8
12 16ao
EL'ectrokine?ic Radius, 2wb
Figure 8. Variation of It/BI, with electrokinetic radius.
or 1
- --
BIG
[tanh ( 4 2 ) cot Bo
Wh(o1/2)'K
+
where B = M 2 w 2 p / K . The quantity It/BIGis plotted in Fig. 8 as a function of the surface ionic energy, eqbo/kT, and the electrokinetic radius, 2wh. The parameter B, as will be observed, has a strong influence in electrokinetic systems. The open circuit streaming potential for flow in a slit is thus given by the expression
Y,
=
~r
I f
wh( 4 2 )21(
Ltanh ( ~ 4 2 cot ) Bo
a + 0 and ah >> d, eq. 40 approaches the approximate equations obtained in earlier work13for low ionic potential and/or large electrokinetic radius. The usefulness of the curves in Fig. 9 lies primarily in the determination of the streaming potential in the case of small electrokinetic radius and large surface potential. The necessity of using lumped parameters in the presentation of the results obscures somewhat the role of the basic parameters that influence the streaming potential. For example, the surface ionic potential $o is contained in the ordinate (Le., in M ) as well as in the parameters a and B. It is not clear from the figures what effect mill be produced by alter-
+
0.7
0.6 n0.5
0l l
?!Yo4 \
or
'
m'
0.3 0.2
0.1
00
It is plotted in Fig. 9 as a function of the electrokinetic radius, Zwh, the surface ionic energy, a, and the parameter, B. It can be shown that in the limit, as The Journal of Physical Chemistry
2
4
6
8
Electrokinetic Radius. 2wh
Figure 9. Variation of streaming potential with electrokinetic radius for several values of LY and B.
10
12
ELECTROKINETIC FLOW IN ULTRAFINE CAPILLARY SLITB
1091
ing the magnitude of +o. A broader generalization of the problem, beyond that shown in Fig. 9, is not feasible because of the large number of basic parameters upon which Y, depends. These are: rl.o, D, p, dP/dx, K , z, T , c, and h. The solution of the problem of electrokinetic flow in a slit can be used in estimating the electroosmotic velocity and st reaming potential in capillary elements having other tban slit cross sections. If we assume that the same generalization holds as in the case of low potential flow, then it is possible to substitute 2 w A / s for the electrokinetic radius in capillary elements of arbitrary cross section. The error of this approximation has not been established.
Flow Retardation in Fine Capillary Channels I n mechanicail pumping through fine capillary channels, the net velocity, which is the algebraic sum of the pressure-induced velocity and the retarding velocity, is given by eq. 22 and 41 as fi =
I
M 2d P -~ (1 - G)’ .+ 3p dlL: K dx 1 Ib/’Io h2 dP -
+
(42)
or
Electrokinetic Radius, 2wh
Figure 10. Variation of retarding flow with electrokinetic radius for several values of CY and B.
practical case of a! = 4, B = 4, which corresponds roughly to the CY and B of distilled water in equilibrium with the COZ of the air, one finds that the retarding flow component, can be as much as 68% of pressureinduced flow. This occurs when the electrokinetic radius is 2wh = 1.6. Since in equilibrium water c 2.1 X loeB mole/l., the channel width at which the large retardation occurs is =1
A
=
up - ii,
(43)
The ratio of the electrokinetic counterflow component, Zl, to the pressure-induced component, A,, is
The variation of this ratio with electrokinetic radius is plotted in Fig. 10 for several values of CY and B . These curves show that when the elelctrokinetic radius is greater than 20, the retarding flow will be less than 10% of the pressure-induced flow. However, for the
h=3.26
0.8
x
107d2.1
x
=
1.7 X
cm.
10-6
Thus, when the fineness of the capillary element approaches 10-5 em., it is to be expected that a large resistance to the transmission of fluids, via pumping, will be experienced. When the surface ionic potential +o is sufficiently large, with CY and B correspondingly large, Fig. 10 shows that the retardation approaches 100% of the flow. Under these conditions, the porous material would appear completely impermeable.
Volume 68,Number 6 M a y , 1964
