Langmuir 1990, 6, 1585-1590
1585
Moving Boundary Electrophoresis of Concentrated Suspensions and Electroosmosis in Porous Media Matthew W. Kozakt and E. James Davis* Department of Chemical Engineering, BF-10, University of Washington, Seattle, Washington 98195 Received October 5, 1989. I n Final Form: April 19, 1990 This is a report of studies designed to validate the unit-cell theory (or swarm theory) of electroosmosis and electrophoresis proposed by Kozak and Davis. That analysis is compared with a numerical analysis of electroosmosis performed by O'Brien and with electrokinetic data of Van der Put and Bijsterbosch. { potentials extracted from electroosmosis data by using swarm theory show a greater dependence on the electrical double-layer thickness than does the numerical analysis, but for moderately thick and for thin electrical double layers both analyses roughly agree with microelectrophoreticdata of Van der Put and Bijsterbosch. In addition, a composite theory of moving boundary electrophoresis was formulated and used to interpret new data for concentrated suspensions of colloidal particles. The composite theory consists of the previously developed unit-cell theory incorporated into the moving boundary electrophoresisanalysis of M a r m ~ r .Theory ~ and experiment show that the electrophoretic mobility is not a strong function of particle concentration, but the composite analysis fails to predict the characteristics of the leading and following boundaries.
Introduction Electrophoresis and electroosmosis are commonly used methods for determining the potential of surfaces, and these electrokinetic phenomena have numerous scientific and technological applications. Electrophoresis occurs when an electrical field is imposed upon a suspension of colloidal particles. The electrical field induces the particles to move due to the charge that resides a t the particle surface. The bound charges and the layer of counterions in the adjacent fluid constitute the electrical double layer. Electroosmosis occurs if the particles are held fixed, for the electrical field produces a net motion of fluid past the particles. The motion of the particles relative to the fluid is affected by viscous interaction with surrounding water molecules, viscous interaction with ions in the electrical double layer (the retardation effect), and interaction with the double layer that has been distorted from its equilibrium configuration due to convection (the relaxation effect). These effects influence the rate of particle motion in electrophoresis and the rate of fluid motion in electroosmosis. Furthermore, hydrodynamic interactions between neighboring particles should also be taken into account. It is usual to measure the electrophoretic velocity of a particle in dilute suspensions so that an individual particle can be observed as it migrates in the applied electrical field, but in many applications concentrated suspensions are encountered. It is desirable to predict the electrokinetic behavior of such suspensions from information on dilute suspension properties. Sprute and Kelsh5 demonstrated in the laboratory and in field studies the dewatering of mine tailings by combined electrophoresis and electroosmosis. Electrophoresis was used to concentrate the suspension of colloidal particles by applying a vertical electrical field, ~
~~
~~
~~
Now at Sandia National Laboratories, Waste Management Systems, Division 6416,Albuquerque, NM 87185-5800. (1)Kozak, M.W.;Davis, E. J. J . Colloid Interface Sci. 1989,127,497. (2)O'Brien, R. W.Electroosmosis in porous materials. J . Colloid t
Interface Sci. 1986,110, 477. (3)Van der Put, A. G.; Bijsterbosch, B. H. J . Colloid Interface Sci. 1983,92,499. (4)Marmur, A. J . Colloid Interface Sci. 1982,85, 556. (5) Sprute, R. H.; Kelsh, D. J. Bur. Mines Rep. Inuest. RI 8666, U S . Dept. of Interior, 1982.
0743-7463/90/2406-1585$02.50/0
and when a dense matrix of particles formed, the electrical field was reversed. Water was then removed from the porous mass by electroosmosis. Sunderland'j reviewed a number of applications of electrokinetics, including enhancement of settling rates in settling ponds, dewatering concrete, and concentrating rubber latices. Electroosmosis has also been applied to the removal of contaminants from ground water systems and has been proposed for use in hazardous waste site remediation (Renaud and Probstein7). The objective of this study was to compare our earlier electrokinetic theory, which takes into account the retardation and relaxation effects and hydrodynamic interactions between particles, with experimental data for concentrated suspensions. In addition, the unit-cell analysis or swarm theory is compared with available electrokinetic data in porous media and with a previous analysis of electroosmosis.
Background The theory of electrophoresis in dilute suspensions is very well developed, and there is a long history of improvements to the classical theory of Smoluchowski8for dilute suspension electrophoresis (Henry,g Overbeek,'O Booth," O'Brien and White,12O'Brien and Hunter,l3 Ohshima et al.14). A principal purpose of these developments has been to infer the desired surface property of the particle ([potential or surface charge density) from the measured rate of electrophoresis. On the whole, the theory of electrophoresis of dilute suspensions is now satisfactory, although Zukoski and Saville15J6pointed out some areas (6)Sunderland, J. G.J . Appl. Electrochem. 1987,17, 889. (7) Renaud, P. C.; Probstein, R. F. PCHPhysicoChem. Hydrodyn. 1987, 9,345. (8)Smoluchowski, M.Handbuch der Electriztat und des Magnetismus (Graetz);Barth Leipzig, 1921;Vol. 2. (9)Henry, D. C.Proc. R. SOC.London, Ser. A 1931,133,106. (10)Overbeek, J. Th. G. Kolloidchem. Beihefte 1943,54,287. (11)Booth, F.Proc. R. SOC.London, Ser. A 1950,203,514. (12)O'Brien, R. W.; White, L. R. J. Chem. SOC.,Faraday Trans. 2, 1978,74,1607. (13)O'Brien, R. W.;Hunter, R. J. Can. J . Chem. 1981,59,1878. (14)Ohshima. H.: Healv. T. W.: White, L. R. J . Chem. SOC..Faraday Trans. 2 1983,80, 1299. (15)Zukoski, C.F.:Saville, D. A. J . Colloid Interface Sci. 1985,107, 322.
0 1990 American Chemical Society
1586 Langmuir, Vol. 6, No. 10, 1990
Kozak and Davis
that need improvement. There has been considerably less fundamental work done on electroosmosis in porous media and electrophoresis of concentrated suspensions. Levine and Neale17developed a unit-cell model for electrophoresis or electroosmosis for spherical particles in the Henry approximation of low potential, and Kozak and Davis18 extended Levine and Neale’s solution to fibrous particles aligned perpendicular to the direction of the applied electrical field. A somewhat different approach was used by O’Brien who solved the equations of motion numerically for electroosmotic flow through cubic arrays of spheres. His analysis was performed for simple, body-centered, and face-centered cubic arrays. He also presented an interpolation method to estimate the electroosmotic flow through random packings of spheres. Recently, Kozak and Davis’ developed a unit-cell model for electroosmosisand concentrated suspensions of spheres that is applicable to any f potential but is restricted to a z-z electrolyte and thin double layers. For a negatively charged particle, their solution for the ratio of the electrophoretic velocity, u , to the current density, i, in the porous medium is given by
where t o is the permittivity of free space, t, is the dielectric constant, w is the fluid viscosity, t is the absolute value of the { potential, R is the gas constant, F is Faraday’s constant, Tis the absolute temperature, and z is the valence of the z-z electrolyte ( z = z+ = -z-). The parameter K is defined by
Here y is the particle volume fraction, D-is the diffusion coefficient of the anions, and K is the inverse Debye length, which for a z-z electrolyte is given by
(3) where C is the total concentration of electrolyte in the solution. The leading term in eq 1is Smoluchowski’s result, and the second term takes into account the retardation and relaxation phenomena. The parameter K , which is a function of the particle volume fraction, incorporates the effects of particle-particle interactions. The ratio of the conductivity of the colloidal suspension to the solution conductivity, A/Ao, is
Table I. Comparison o f t Potentials Obtained from Microelectrophoresis Measurements and Electroosmosis Experiments ~~~
Ka
FTIRP
FTIRP
FlIRP
30 42
2.7 2.9
3.7
67
3.0 3.1
2.7 2.8 2.8 2.5
2.0
1.8
95 300
3.5
3.1 2.6 1.6
a Calculated from the microelectrophoresis data of Van der Put and Bij~terbosch.~Obtained from the electroosmosis data of Van der Put and Bijsterbosch by using the numerical solution of O’Briem2 Obtained from the electroosmosis data of Van der Put and Bijsterbosch using eq 6.
*
Equation 1is valid to order ( K U ) ~but , Kozak and Davis19 extended the analysis to moderately thick double layers, providing a more complicated solution valid to order ( K U ) - ~ . Kozak20 showed that this solution reduces to all earlier equations for the electrophoretic velocity in appropriate limiting cases. The results calculated by using the more elaborate analysis agreed well with eq 1 except for conditions involving thick double layers and high particle c o n c e n t r a t i o n s ( h i g h volume f r a c t i o n s o l i d s ) . Consequently, eq 1 can be used for most situations of technological importance.
Comparison with Electroosmosis Data and Theory There are few extant data with which the unit-cell analysis can be compared, an exception being the data of Van der Put and Bijsterbosch, who measured the streaming current caused by flow through porous plugs of polystyrene latex spheres. They also measured the electrophoretic mobility of the latex particles used to form the porous plugs. From the latter data, the { potential can be determined by applying the numerical solution of O’Brien and White. As shown by O’Brien,2 the electroosmotic mobility &/i can be calculated from the streaming current by application of the Onsager reciprocal relation (HuntePl). O’Brien also extracted { potentials from the data of Van der Put and Bijsterbosch using a numerical solution for electroosmosis through cubic arrays of spheres. One can use eqs 1 and 4 together with i = h ( E ) to determine {potentials from the data of Van der Put and Bijsterbosch. Here ( E ) is the volume-averaged electric field in the porous medium (O’Brien2). Combining the equations, we obtain
where D+ is the diffusion coefficient of the cations.
Since the particle concentration and double-layer thickness are known for each experiment, a simple iterative procedure can be used to determine the {potential, which predicts the measured electroosmotic mobility. The dimensionless { potentials determined in this way are presented in column 4 of Table I together with the { potentials calculated from the electrophoresis data (column 2) and those calculated by O’Brien using his numerical solution for cubic arrays (column 3). The results are plotted in Figure 1. There should be agreement between the results of electroosmosis experiments and electrophoresis experiments, and Figure 1 indicates that there
(16) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1987, 115, 422. (17) Levine, S.; Neale, G. H. J . Colloid Interface Sci. 1974, 47, 520. (18) Kozak, M. W.; Davis, E. J. J . Colloid Interface Sci. 1986,112,403.
(19) Kozak, M. W.; Davis, E. J . J . ColloidInterface Sci. 1989,129,166, (20) Kozak, M. W. Ph.D. Dissertation, Universityof Washington, 1988. (21) Hunter, R. J. Zeta Potential in Colloid Science: Academic Press: New York, 1981.
where the Peclet numbers for cations and anions, Pe+ and Pe-, are defined by Pe+ = u / K D + ; Pe- = u / K D -
(5)
Electrokinetic Theory for Concentrated Suspensions
To interpret mass-transport electrophoresis data or moving boundary electrophoresis data, a relationship is required between the motion of a single particle and the measurable motion of a swarm of particles. Marmur provided such a result for moving boundary electrophoresis. Consequently,we undertook an experimental program to study moving boundary electrophoresis of concentrated suspensions.
A
DATA (REF. 3) 0 O’BRIEN (REF. 2)
*t 1’
0
”
100
”
200
Langmuir, Vol. 6, No. 10, 1990 1587
‘
300
1
Ka
Figure 1. Comparison among the predictions of O’Brien’s* numerical solution, eq 6, and the microelectrophoresis data of Van der Put and Bijsterbo~ch.~
is reasonably good agreement except that at small values of Ka eq 6 diverges from the predictions based on O’Brien’s solution and from the electrophoretic measurements. The reason for the divergence is that the unit-cell analysis does not account for points of contact between adjacent particles. In the neighborhood of the points of contact, there must be an overlap of the double layers associated with adjacent particles. As Ka decreases and the doublelayer thickness increases, more of the current passes through the region of double-layer overlap near points of particle contact. This increase of current is not adequately predicted by unit-cell theory. Thus, the unit-cell model can be expected to be inaccurate for small values of Ka in porous media. The difficulties associated with predicting conductivities and interpreting conductivity data were pointed out by O’Brien based on the conductivity data of Van der Put and Bijsterbosch. { potentials calculated from the conductivity data are substantially larger than those listed in Table I. This should be true for our analysis as well, and as a result we have not attempted the comparison here. Instead, we compare only electroosmosis theory to electrophoresis theory, which O’BrienZ has shown to be consistent. As O’Brien2 stated, “Until these doubts are removed, or the foundations are modified in the light of experimental results, the entire theoretical structure must be viewed with suspicion.” We note that experimental results that have raised these doubts are all based on data from polystyrene latices. Because of the difficulties related to applying unitcell theory to porous media, we found it desirable to perform experiments to test eq 1under conditions in which double-layer overlap would not influence the results as strongly, that is, by working with concentrated suspensions. For dilute suspensions, the technique of microelectrophoresis can be used and interactions between particles can be ignored. By contrast, concentrated suspensions are opaque, so the motion of individual particles cannot be observed. Alternate experimental methods are (i) moving boundary electrophoresis (Bierz2),(ii) mass-transport electrophoresis (James2?, and (iii) microelectrophoresis with a mixture of transparent particles (ghosts) and tracer particles (Zukoski and Saville16). Each of these methods has particular strengths and weaknesses that have been described elsewhere (Kozakzo), but we note that Zukoski and Saville’s method of using optical ghosts has only been applied to systems having low {potential (red blood cells) and a thin double layer. Thus, their results do not provide an adequate test of the theory. Furthermore, there has not been an adequate analysis of mass-transport electrophoresis to permit that method to be used to test eq 6. (22) Bier, M. Electrophoresis; Academic Press: New York, 1959. (23) James, S. D.J. Colloid Interface Sci. 1978,63,577.
Moving Boundary Electrophoresis T h e simplest and most widely used method for measuring the electrophoretic mobilities of concentrated suspensions is moving boundary electrophoresis. Sharp interfaces between the suspension and an electrolyte solution are formed in a U-tube. An electrical field is then imposed on the U-tube, and the interfaces move in response to the electrical field. A suspension of negatively charged colloidal particles will move toward the anode. The interface in the anode arm of the tube is called the leading boundary, and the second interface is the following boundary. The rate of movement of the boundaries can be measured either by direct visual observation or by the Schlieren technique (Bier22). The primary disadvantageof the use of moving boundary electrophoresis is the presence of “boundary anomalies”. It is experimentallyobserved that the leading and following boundaries move a t different rates. In addition, it is observed that one boundary remains sharp while the other becomes diffuse as the experiment progresses. Earlier attempts a t explaining boundary anomalies were not entirely successful (Ross and Long24and Tison%),but Marmur provided a theoretical basis for the analysis of boundary anomalies. Assuming that the particle velocity dependence on concentration could be described by the cell model of Levine and Neale, Marmur solved the conservationequation for particles in the moving boundary electrophoresis apparatus. The particle conservation equation is
aT + Q(r)g= 0
?Y
(7)
where x is the dimensionless distance along the tube (x = z / L ) , is the distance along the tube, L is the overall length of the tube, 7 is the dimensionless time defined by T = u ( K u , { , O ) ~ /and L , Q is a dimensionless function given by
Here, u ( ~ a , { , yis) the dimensionless particle velocity with respect to the electrolyte solution and u(Ka,{,O) is the particle velocity at infinite dilution. In the treatment by Marmur, u is considered to be independent of { because that is the result of the cell model of Levine and Neale. Here u is a function of as well as of Ka and y. Marmur considered the suspension to be a continuum in the calculation of conductivity and electrical field strength, and he assumed constant physical properties including the electrolyte concentration. This excludes the case of highly conductive particles. The solution to eq 7 is given in terms of a characteristic curve on the X-T plane. In this case, the characteristic is a straight line with slope Q(r).Marmur noted that the dimensionless characteristic velocity is different from the dimensionless particle velocity. As a result, the initially (24) Ross, S.; Long, R. P. Znd. Eng. Chem. 1969,61, 58. (25) Tison, R. P. J. Colloid Interface Sci. 1977, 60, 519.
1588 Langmuir, Vol. 6, No. 10, 1990
Kozak and Davis ’
/
/ ’ /
’/
-
SMOLUCHOWSKI THEORY - -
8 -
----- -- - -- - - - - - -
m
0
THEORY OF OHSHIMA ET AL. FOR = 109 mv
5
r , DIMENSIONLESS TIME
,
-
EXPERIMENTAL OATA
0 I , DIMENSIONLESS TIME
F i g u r e 2. Characteristics of moving boundary electrophoretic boundaries. Table 11. Characteristics of t h e Polystyrene Latex 0.989 pm mean diameter &2.2% coefficient of variation on mean density at 20 OC 1055 kg/m3 refractive index 1.591 at 590 nm surface charge density 7.45 pC/cm2 surface area per charge group 214 A2/charge group milliequivalents per gram 0.004 44 specific surface area 57 505 cm2/g particle concentration 1.59 X 1Ol1 particles/(:m3 Table 111. Microelectrophoresis Data for the Polystyrene Latex 95 % KC1 concn, uWi, confidence moliL Ka wcm/(V.s) limits 1 x 10-4 16.4 4.85 i0.063 5 x 10-4 36.7 5.90 i0.073 1 x 10-3 52.0 6.39 f0.060 116.0 7.51 io.100 5 x 10-3
uniform distribution of particles becomes nonuniform as time increases. Plots of Q(7) on the x--7 plane are shown in Figure 2 for Q > 1and for Q < 1. The dashed lines are characteristics that evolve from clear solution, and their slope is equal to unity. The characteristic lines that evolve from the suspension are shown as solid in the figures. When the characteristics that emanate from the suspension intersect the characteristics that emanate from the particlefree liquid, the particle concentration becomes multivalued on a finite region of the plane. This paradox is resolved by the formation of a shock front, a discontinuity that moves at a velocity determined by a mass balance across the discontinuity. When the characteristics diverge in a region, they enclose a region in which t h e particle concentration changes in a continuous manner. Hence, intersecting characteristics correspond to a sharp interface, and diverging characteristics correspond to a diffuse interface. From Marmur’s analysis, one would expect that for Q > 1 the leading boundary should be sharp, and the following boundary diffuse. For Q < 1, the sharp and diffuse boundaries reverse those positions. An experimental program was designed to evaluate the combination of Marmur’s analysis with eq 1using colloidal particles with a relatively high {potential in suspensions of various concentrations. Experimental Section The experimental program consisted of three parts: (i) determination of the { potential as a function of electrolyte concentration, (ii) conductivity measurements for the electrolyte solution in the moving boundary electrophoresis cell, and (iii) measurement of moving boundary electrophoretic mobilities for various particle concentrations and double-layer thicknesses. The colloidal particles were selected based on three criteria: (i) narrow size distribution to provide a narrow range of values
1
0
I 20
I
I 40
I
l
60 ca
I
l 80
I
I 100
l
120
F i g u r e 3. Experimental dimensionless mobility as a function of KU compared with t h e predictions of Ohshima et al.“ and Smoluchowski.* of Ka for a particular experiment, (ii) high charge to examine the relaxation effect, and (iii) small size so that low values of KU could be attained over a range of electrolyte concentrations. Sulfated polystyrene latices with satisfactory characteristics were obtained from Interfacial Dynamics Corp. (IDC). Selected physical properties for IDC lot number 10-110-13 are given in Table 11. The particle diameter of 0.989 pm was determined by IDC using transmission electron microscopy of 500 randomly selected particles. Microelectrophoresis experiments were performed to determine { potentials using a Pen Kem Inc. Lazer Zee Meter Model 501. Experiments were performed using potassium chloride solutions for which D+ = D-.All solutions used in this work were titrated to p H 7.0 f 0.05 by using KOH as a titrant. The p H was measured with a Radiometer PHM 84 digital p H meter. The microelectrophoresis data are presented in Table 111. Each average mobility reported was calculated from three replicates of 16 measurements each. Each suspension replicate was prepared independently, so the variability in the mobility data reflects r a n d o m n e s s in s o l u t i o n p r e p a r a t i o n , s u s p e n s i o n treatment, and intrinsic variability in particle { potential. {potentials were calculated from the mobility data by using the theory of Ohshima et al.14 A { potential of 109 mV was found to provide a satisfactory fit to the data for all doublelayer thicknesses, as shown in Figure 3. Smoluchowski’s theory is shown as a horizontal dashed line for reference. I t can be concluded from Figure 3 that the { potential of these polystyrene latices is not a function of electrolyte concentration in the range of concentrations examined. Furthermore, the { potential is sufficiently high for relaxation effects to be encountered. The moving boundary electrophoresis apparatus consisted of a vertical glass U-tube shown in Figure 4. The 4-mm4.d. tube was separated into two sections, the upper and low portions being connected by Plexiglas plates which caused the upper and lower parts of the tubes to fit flush with the faces of the plates. Vacuum grease was used between the plates as a sealant and lubricant. At the upper ends of the tube were installed 14/35 female ground glass fittings which connected to 15-cm3Buchner funnels that served to isolate electrode products from the electrophoresis tube. Fine glass frits in the funnels served as barriers to prevent convection of electrode products into t h e electrophoresis cell. The electrodes were made of squares of 0.5-mm-thick platinum foil. The squares were 8 mm by 8 mm, and they were spotwelded to 0.5-mm-diameterplatinum wire which was sealed into 2-mm-diameter glass tubing. The exposed platinum was platinized by the procedure outlined by Van den Hoven.26 The electrodes were held in place in the electrode compartments by rubber stoppers. T h e anode compartment was sealed with respect to the atmosphere, but the cathode compartment was left open to the atmosphere to vent hydrogen that evolved at high current densities. The electrical current was calculated from the potential drop across a 5000-Q precision resistor mounted in series with the ap(26) Van den Hoven, Th. J. J. Ph.D. Dissertation, Agricultural University, Wageningen, The Netherlands, 1984.
Electrokinetic Theory for Concentrated Suspensions
r
ANODE
A
I
CATHODE
F i g u r e 4. Apparatus used for moving boundary electrophoresis experiments.
F i g u r e 5. Overall cell resistance as a function of the electrolyte inverse electrical conductivity. T h e dashed line is the limiting slope for high conductivities. paratus. This resistance was always a minor fraction of the overall circuit resistance, so it did not affect the results appreciably. The rate of movement of the interface was measured by aligning a cathetometer with a scale mounted behind the apparatus. This setup ensured that the boundary displacement measurements were not affected by parallax. Conductivity measurements were made to determine the conductivity cell constant of the electrophoresis apparatus as a function of electrolyte concentration. For the conductivity cell, the total resistance between the electrodes is given by (9) where the cell constant &/A) is the ratio of the distance between electrodes L and electrode area A . In Figure 5, resistance is plotted versus inverse conductivity, A-l, for various electrolyte concentrations in the apparatus. At low solution conductivities, the curve is nonlinear, indicating that t h e cell constant is not constant except a t higher conductivities. For X > 600 ps/cm, the slope approaches the limiting value shown on the figure as a dashed line. I t is clear that there is some additional charge-transfer mechanism occurring a t low electrolyte concentrations that reduces the cell resistance. Two phenomena might explain this effect. First, the electric current flow in
Langmuir, Vol. 6, No. 10, 1990 1589 the apparatus can cause electroosmotic circulation, thereby lowering the overall tube resistance due to convective charge transfer. Second, the electrical double layer associated with the tube wall provides a region of enhanced conductivity. Both mechanisms decrease the overall cell resistance, and both become more pronounced a t low solution conductivity. I t is likely that the surface conductivity of the tube is small, because Ka for the tube is very large. Hence it is concluded that the reduced cell resistance is more likely due to electroosmotic circulation. Since the [potential associated with the tube wall can be expected to be negative, electroosmotic flow near the wall would be toward the cathode. These results suggest that moving boundary electrophoresis experiments are complicated by electroosmotic circulation; the effect is expected to be particularly pronounced at low electrolyte concentrations. But to provide a n adequate test of theory, it was necessary to operate a t low electrolyte concentrations to achieve small KU. Consequently, all of the electrophoresis measurements reported below are likely to be affected by the circulation. As a result, absolute values of moving boundary electrophoretic mobilities cannot be compared with microelectrophoretic mobilities, but it is possible to make relative comparisons between moving boundary data at different particle concentrations to assess the effects of particle-particle interactions. T o make such comparisons, we assume t h a t t h e electroosmotic circulation is independent of particle concentration and that the particle [potential is independent of particle concentration. T h e former assumption is reasonable since the circulation is caused by the tube wall double layer, and the latter assumption is reasonable since the f' potential will normally be expected to depend only on the nature of the electrolyte solution. Considerable care was taken to ensure that the ionic strength was identical for all particle concentrations. Concentrated suspensions of particles can affect significantly the p H and ionic strength of the electrolyte solution by dissociation of surface sulfate groups. T o avoid this effect, all suspensions that were mutually compared were dialyzed together in the same vessel. Dialysis was carried out exhaustively to ensure that the electrolyte concentration was accurately known. The colloidal particles supplied by the manufacturer had a concentration of 8.4 vol % in distilled water. Data are reported here for three different suspension concentrations: 1.0, 8.4, and 20 vol % , but some experiments were performed a t lower concentrations. T h e 1% suspensions were prepared by dilution of the original samples. T h e 20% suspensions were made by centrifuging the original material, decanting the supernatant fluid, and then redispersing the solids by ultrasonication. The suspensions were separated into 2-cm3 samples, and each sample was placed in a separate dialysis tube. All of the samples were then dialyzed together in a 1-L vessel for several hours under mild agitation. The dialysis solution was then changed, and dialysis continued overnight. Each suspension was ultrasonicated just before use in an electrophoresis experiment. For each set of suspension concentrations compared, the applied current density was identical. This permits us to make a direct comparison of boundary velocities for different particle concentrations. At suspension concentrations below 0.5 vol % , the boundary behavior was unsatisfactory, for the sharp leading boundary assumed a parabolic shape. This is consistent with the shape one expects from electroosmotic circulation in a closed cell. Above 0.5% solids, the boundary was sharp and flat, but frequently a spike evolved from the center of the flat surface. I t is possible that this resulted from electroosmotic circulation in the free solution with little or no circulation in the suspension, but there is not direct evidence for this assertion. Moving boundary data are reported in Table IV for three suspension concentrations at four levels of electrolyte concentration ( K O ) . In every case, t h e boundaries showed behavior consistent with Q > 1, that is, a sharp leading boundary and a diffuse following boundary. T h e boundary displacement rate was found to be independent of colloid concentration. Tison showed that the boundary displacements must be corrected for the displacement of solution by the migrating particles. Hence, the electrophoretic velocity, u ,is calculated from the boundary
Kozak and Davis
1590 Langmuir, Vol. 6, No. 10, 1990 Table IV. Measured Moving Boundary Electrophoretic Mobilities Ka
16.4 36.7 52.0 116.0
y
0.01 0.084 0.20 0.01 0.084 0.20 0.01 0.084 0.20 0.01 0.084
uhti, rcmjjV.s) a 1.94 2.16 4.13 4.33 5.19 5.39 5.84 6.49 8.31 10.06
95 7; confidence limits
measd
0.06
1.00*
0.04 0.11 0.07 0.09 0.04 0.09 0.14 0.05 0.16
l.ll*
U(Y)/U(O.Ol)
1.00 1.05 1.26 1.00 1.08 1.20 1.00 1.21
calcd
1.00 1.00 1.01 1.00 1.01 1.02 1.00 1.01 1.03 1.00 1.05
*
Not measurable. The interface was unstable for KQ = 16.4 and y = 0.01. As a result, the velocity ratio reported is u ( y ) / u (0.084). a
where
A((”)=dy A,
3
2(1+ y/2)2
+
the electrophoretic velocity, u, is calculated from the boundary by the equation
velocity,
= ub/(l - 7)
(10)
In Table IV, we have listed the values of K Q , the mobilities together with their 95% confidence limits, and the ratio of the electrokinetic velocity to that for the dilute suspension, u ( y ) / ~(0.01). For K a = 16.4 and y = 0.01, the interface was consistently unstable, so it was not possible to specify the boundary velocity. As a result, we report the ratio u(y)/u(O.O84) in column 5 for the lowest value of K U .
Discussion of Results Table IV shows t h a t the mobility increases as K U increases. The electrophoretic velocity is not a strong function of the suspension concentration. For KQ < 116, the electrophoretic mobilities reported in Table IV are generally lower than the microelectrophoretic mobilities listed in Table 111. This could be due to an enhanced free solution conductivity caused by electroosmotic circulation in the moving boundary electrophoresis experiments. One would expect that as the particle concentration decreases the mobility should approach the value determined by microelectrophoresis, but that certainly is not observed at smaller values of K Q . As we have stated above, a direct comparison between the mobilities reported in Tables I11 and IV should not be made because of electroosmotic effects. However, an examination of the effects of particle concentration on the electrophoretic velocities can be made. It is straightforward to calculate the ratios of electrophoretic velocities from theory. Equation 1 can be used to calculate u ( y ) for { = 109 mV by using Peclet numbers determined from the properties of KCl and KU and y specified by the experiment. The conductivity of the suspension can be calculated by using eq 4. For KCl, the Peclet numbers for cation and anion are approximately equal, so Pe+/Pe- = 1. The last column in Table IV compares theoretical and experimental ratios of the electrophoretic velocities for the conditions of the experiments. Theory predicts a small increase in the electrophoretic velocity as the particle concentration increases. However, the experiments show a somewhat larger increase t h a n predicted. This discrepancy between theory and experiment exists for all values of KU examined. It is possible to make another comparison between theory and experiment based on the qualitative behavior of the boundary anomalies observed. Combining eqs 1, 4, and 7, one obtains
Numerical values were calculated for Q for each experimental run. For each run, eq 11yielded Q < 1,which implies that the leading boundary should be diffuse and t h e following boundary sharp. T h e experimental observations were precisely the opposite, for in every case the leading boundary was sharp and the following boundary diffuse. This puts in question the validity of the composite theory of moving boundary electrophoresis, but it does not help us determine whether the unit-cell theory or Marmur’s analysis are adequate. It is possible that one or more of the assumptions made in the interpretation of data are invalid. The assumption that the electroosmotic circulation was independent of particle concentration is certainly violated a t very low particle concentrations. For y C 0.005, the boundary behaved as though particles were convected by electroosmotic circulation, but a t higher concentrations no such effects were observable. I t is also possible that there is a more fundamental problem with our understanding of the conductivity of concentrated suspensions, in which case eq 4 is not satisfactory. O’Brien2 pointed out the large discrepancy between potentials extracted from electroosmosis experiments and conductivity measurements.
Summary The results of the experimental validation are mixed. The composite theory of moving boundary electrophoresis is neither quantitatively nor qualitatively satisfactory. And yet the unit-cell theory on which eqs 1 and 4 are based is in reasonable agreement with O’Brien’s numerical solution for electroosmosis through cubic arrays of spheres and with electrokinetic data of Van der Put and Bijsterbosch for larger values of K U . For small K U , the unit-cell theory diverges from the experimental data and from O’Brien’s predictions. Additional development of the theory and experimental methodology of moving boundary electrophoresis is needed to resolve the problem of electroosmotic circulation in the apparatus. Accounting for such circulation would also allow one t o compare directly microelectrophoretic mobilities with moving boundary results. Acknowledgment is made to the Bureau of Mines for Grant G1175149-5321 and to the National Institutes of Health for a predoctoral traineeship (NIH Grant HL 07403) for M. W.Kozak.
