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Measurement of the Zeta Potential of Planar Solid Surfaces by Means of a Rotating Disk Paul J. Sides* and James D. Hoggard Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Received June 25, 2004. In Final Form: September 29, 2004
A method for measuring the zeta potential of disks is described. Combining the hydrodynamic properties of a rotating disk, the solution of Laplace’s equation for the potential, and the electrokinetic boundary condition, one obtains an equation that relates the zeta potential of the disk to the streaming potential in the disk’s vicinity. Theory predicts a dependence of the streaming potential on the rotation rate raised to the 3/2 power. Theory also shows that placement of one reference electrode on the axis of rotation near the disk surface and the other far from the disk is favorable. Measurement of the streaming potential of silicon oxide and indium tin oxide in contact with a solution of potassium chloride demonstrated the expected 3/2 power dependence on rotation rate. The zeta potentials calculated from the combination of the theory and experimental data agreed with published values.
Introduction The zeta potential of a surface characterizes the amount or the apparent amount of charge associated with a solid surface. The solid might be a particle or fiber, or it might be a planar surface. The zeta potential of particles is an important characteristic for maintaining the stability of a colloid. The zeta potential of planar solids, the subject of this contribution, is also useful information. For example, the zeta potential of silicon can determine whether particles from solution are more or less likely to stick to it. If the silicon wafer has a surface charge and the particles in solution are charged with the same sign, then particles have a lower tendency to stick. Anton-Paar GmBH of Graz, Austria, manufactures an instrument that can be used to determine the zeta potential of planar solids by the method of flow through a narrow channel. One clamps a flow channel to the planar solid. In the case where the sample material differs from the material of the flow channel, however, a correction must be supplied. We suggest a simple apparatus suited for measuring the zeta potential of wafers or other disk-shaped solids and not requiring correction for the presence of another material. The proposed method is to spin a disk sample in the plane normal to its axis, as shown in Figure 1, and measure a streaming potential in its vicinity. The interaction of the axisymmetric flow over the disk with the diffuse layer charge of the disk’s double layer produces a spatially distributed measurable potential. The size of the disk is not tightly constrained, so disk diameters from a few millimeters to hundreds of millimeters are practical. This idea arose in response to a desire to measure the zeta potential of a surface isolated from contributions of other surfaces to the measured voltage. A search of the scientific literature revealed a few previous investigations. Sidorova et al.1 used a spinning disk where the electrolyte was fed to the axis of the disk and reference electrodes were placed at the periphery. A film of liquid flowed over the disk; there was no infinite * Corresponding author. Tel: 412 268 3846. Fax: 412 268 2183. E-mail:
[email protected]. (1) Sidorova, M. P.; Fridrikhsberg, D. A.; Kibirova, N. A. Vestn. Leningr. Univ. 1973, 2, 121-123.
Figure 1. A disk of diameter 2R spins in an infinite medium of electrolyte. The disk surface has a uniform zeta potential ζ. The numerals refer to positions: 0, far from the disk; 1, immediately adjacent to the disk; 2, in the plane of the disk; 3, in the plane of the disk at a larger radius; 4, adjacent to the periphery of the disk.
medium of liquid adjacent to the disk. This work represented improvements on prior work of Levashova and Krotov2 and Levashova.3 The theory used in their analysis showed a dependence of the measured zeta potential on the inverse square of the disk rotation rate. Knodler and co-workers4,5 experimented with a rotating disk spinning in contact with a semi-infinite liquid medium, as depicted schematically in Figure 1. They recognized that the streaming potential in the vicinity of the rotating disk should vary with the 3/2 power of the rotation rate. Using an apparatus where the sample was a ring concentric and coplanar with two reference electrodes, they measured voltages corresponding to the potential difference between positions 2 and 3 in Figure 1.4,5 The measured streaming potential did not exhibit the expected variation with the 3 /2 power of the rotation; exponents of 0.27, 0.7, and 2.1 were obtained. A search for citations of this work did not reveal subsequent investigations. (2) Levashova, L. G.; Krotov, V. V. Vestn. Leningr. Univ. 1969, 16, 139-141. (3) Levashova, L. G. Vestn. Leningr. Univ. 1970, 16, 133-136. (4) Knodler, R.; Kohling, A.; Walter, G. Electroanal. Chem. Interfacial Electrochem. 1974, 56, 315-319. (5) Knodler, R.; Langbein, D. Z. Phys. Chem. Neue Folge 1975, 98, 421-433.
10.1021/la048420f CCC: $27.50 © 2004 American Chemical Society Published on Web 11/24/2004
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Mathematical analysis of the streaming potential in the vicinity of a disk rotating in contact with a semi-infinite medium has led to a simple experiment whereby measurements of voltage between two points, one located very near the surface of the disk immediately above the axis of rotation and the other far from the disk, give a voltage directly related to the zeta potential of the disk. Experimental measurements show near perfect agreement with the theoretically expected 3/2 exponent and good agreement with zeta potential measured by the traditional method. Theory of the Streaming Potential near a Rotating Disk A dielectric disk of radius R rotating in an infinite aqueous solution appears in Figure 1. If the disk is charged, an excess of ions of opposite sign form a diffuse layer of mobile charge that shields the surface charge. Rotation of the disk generates radial flow with a velocity proportional to the radial distance from the axis. Radial convection along the disk surface transports the mobile ionic charge in the diffuse part of the double layer, thereby engendering a sheet of ionic current that flows both concentrically and radially outward along the disk surface. If the disk radius is of centimeter order, the sheet of current is thin with respect to the disk in aqueous solutions because the Debye length is about a micron in pure water and usually of nanometer order in aqueous solutions. The convected ionic current must return ohmically through the electrolyte to complete the circuit and conserve charge because no charge crosses the boundary between the disk and the electrolyte. The infinite medium provides a path for conduction of this return current. The two theoretical tasks are to derive an expression for the convected surface current and to use this current in a boundary condition that constrains the solution of Laplace’s equation for the return current. The convected surface current, js, comprises current due to surface conductivity and current due to flow of charge in response to the imposed flow. We first assume that the current due to the surface conductivity is negligible; i.e., that the Dukhin number,6 the ratio of the surface conductivity to the bulk conductivity, is much less than unity. (For disks of centimeter order in radius in aqueous systems where the zeta potential is order 100 mV, Du is of order 10-2 or smaller. This is not a significant constraint.) The surface current is then given by
jsr )
∫0∞ vrFe dz
(1)
where vr is radial velocity of the fluid and Fe is the local concentration of unbalanced charge in the diffuse layer. The fluid dynamics of the rotating disk is well-known, and the radial velocity is easily obtained.7-10 Using standard electrokinetic arguments (see Appendix 1 for details), one can show that the radial component of the surface current density (eq 2) is proportional to the zeta (6) Lyklema, J. Fundamentals of Interface and Colloid Science. Vol. II: Solid-Liquid Interfaces; Academic Press/Harcourt Brace: New York, 1995. (7) von Karman, Th. Z. Angew. Math. Mech. 1921, 1, 233-252. (8) Cochran, W. G. Proc. Cambridge Philos. Soc. 1934, 30, 365-375. (9) Rogers, M.; Lance, G. J. Fluid Mech. 1960, 7, 617-631. (10) Newman, J. Electrochemical Systems; Prentice Hall: New York, 1973.
potential, the radial position, and the 3/2 power of the rotation rate, i.e.
jsr ) -0γζr
(2)
where is the dielectric constant, 0 is the permittivity of free space, r is radial position, ζ is zeta potential, γ ≡ a(Ω3/ν)1/2, a is 0.51023,10 Ω is the rotation rate in radians per second, and ν is the kinematic viscosity of the liquid. Thus the surface current due to convection of charge in the diffuse layer is proportional to the radial position and to the rotation rate raised to the 3/2 power. Laplace’s equation governs the current that closes the circuit in the bulk solution.
∇2φ ) 0
(3)
Newman11 showed that Laplace’s equation is separable and solvable in the semi-infinite domain where the origin is at the center of the disk. Using rotational elliptical coordinates, he wrote the solution as ∞
φ)
∑ BnP2n(η)M2n(ξ)
(4)
n)0
where φ is electric potential that varies as a function of position in the semi-infinite domain and vanishes far from the disk. P represents Legendre polynomials of order 2n, and M is a Legendre function of complex argument; M takes the value of unity on the disk and zero at infinity. The position variables η and ξ are related to cylindrical coordinates by z ) Rξη and r ) R[(1 + ξ2)(1 - η2)]1/2. η ) 0 is off the disk in the plane that includes the disk. η ) 1 indicates the axis of the disk. ξ ) 0 is on the disk, and ξ ) ∞ is far from the disk. Equation 4 satisfies the boundary conditions at all limits other than on the disk itself: symmetry at the axis, vanishing potential at infinity, and zero flux through the plane of the disk at distances greater than the disk radius. One deduces the coefficients Bn by applying an appropriate boundary condition on the disk. A boundary condition valid for the relationship between current density normal to the disk iz and surface current is10,12
iz ) -∇s‚j˜s
(5)
where js is the surface current density in amps per unit of circumference and iz is normally directed current density evaluated at the disk surface. Equation 5 accounts for current that leaks out of the thin sheet of surface current to the bulk electrolyte. Inserting eq 2 into eq 5, one obtains
iz ) 20γζ
(6)
which shows that the local current density flowing between the bulk electrolyte and the surface is a constant over the surface of the disk. Using Ohm’s law and eq 4, one writes13
∂φ -κ ∞ |z)0 ) BnP2n(η)M′2n(0) ∂z Rηn)0
iz ) -κ
∑
(7)
where κ is the electrolyte conductivity and (11) Newman, J. J. Electrochem. Soc. 1966, 113, 501-502. (12) Levich, B. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962. (13) Newman, J. J. Electrochem. Soc. 1966, 113, 1235-1241.
Zeta Potential of Planar Solid Surfaces
M′2n(0) ) -
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2(2nn!)4
(8)
π[(2n)!]2
Equating the results in eqs 6 and 7, one obtains ∞
∑ BnP2n(η)M′2n(0) ) n)0
-20γRζη
from which orthogonality yields
[
∫01 ηP2n(η) dη ∫01 [P2n(η)]2 dη
-20γRζ 1 Bn ) κ M′2n(0)
(9)
κ
]
(10)
Figure 2 shows the results of a sample calculation of the expected potential (in volts) measured as a function of r near the disk. The potential is a maximum at the axis of the disk; the calculated streaming potential is a few millivolts. Figure 2 reveals, however, that the analysis is incomplete. The potential in Figure 2 is always of the same sign, which means that current flows from the disk to infinity where the potential is zero. Physically, one expects the current generated by the spinning disk to close on itself in order to satisfy an overall condition that no net current flows from the disk. For positive charge in the diffuse layer, there must be a net positive current leaving the disk beyond a certain radius and a net negative current flowing to the disk at radii less than this value. Thus, the derivation presented above is incomplete. The missing solution can be conceived as a “return” current from infinity to the disk; the function expressing this current, which also must satisfy Laplace’s equation, is superimposed on the solution already obtained in order to satisfy the requirement of zero net current. The return current must exactly cancel the current density departing from the disk, given as the result in eq 6 multiplied by the area of the disk. The total current flowing from the disk then becomes zero as required. The question arises, however, how exactly to distribute the return current on the disk. The clues are that the potential associated with this return current must satisfy eq 4 and the total current due to it must exactly match the negative of the result of eq 6 multiplied by the area of the disk. As shown in Appendix 2, the appropriate current distribution is given by
ireturn ) -
0γσ
x
1-
(11) 2
r R2
Thus the correct expression for the current density that forms the boundary condition at the electrode is the sum of eq 6 and eq 11.
(
iz ) 20γζ 1 -
1 2
1
)
x1 - (r/R)2
(12)
Integration of the current distribution expressed by eq 12 over the disk yields a zero net current flowing from the disk. Use of this current distribution means that the coefficients Bn are now calculated according to the
Figure 2. The dependence of the potential on position near the surface of the disk. The highest magnitude of the potential is at the axis. For purposes of calculation, the concentration of electrolyte was 0.1 M KCl and the zeta potential of the disk was taken as -60 mV. The disk rotates at 2500 rpm. The mobilities of KCl were used to calculate the conductivity. Note that the potential is always of the same sign, which means that current “leaks” to infinity. This is nonphysical.
Figure 3. The distribution of potential (V) on the disk when opposing current is supplied from infinity according to the description in the theory section. Calculation parameters were the same as in Figure 2. The potential on the disk surface now crosses zero so the current in solution flows from the outer part of the disk to the inner part. There is a circle of zero potential on the disk that is connected by a surface of rotation at zero potential extending to infinity.
following equation, where eq 12 has been converted to rotational elliptical coordinates.
[
-20γRζ 1 Bn ) κ M′2n(0)
∫01 (η - 21)P2n(η) dη ∫01 [P2n(η)]2 dη
]
(13)
Figure 3 shows the results of the calculation of the potential at the surface of the disk according to eq 4 when the coefficients obey eq 13. The potential now crosses zero so that current flows in solution from the outer region of the disk to the inner region, as one expects on physical grounds. In essence, all potentials of Figure 2 have been shifted upward by an amount required to drive the opposing current, so that the net current from the disk is zero. At a concentration of 0.1 mM potassium chloride with a zeta potential of -60 mV, a potential of approximately 650 µV should appear between electrodes placed at the
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Figure 4. A schematic of the experimental apparatus for proof of principle. Electrolyte (1) was poured into a 2 L beaker. A disk of silicon (2) was prepared, affixed to a spindle (3), and spun at various rates. A lead (5) from the sensor electrode near the disk connected to the positive terminal of a Keithley Instruments electrometer (4). A lead from a reference electrode (6) connected to the negative terminal of the electrometer (4). When the disk was rotated, a nonzero voltage was detected by the electrometer (4). The measured voltage depended on the rotation rate as described in the text.
axis and at infinity. The measured potential should depend on the rotation rate to the 3/2 power. Experiment Experiments were performed to check the theory. The apparatus appears in Figure 4. A Pine Instruments (Grove City, PA) precision motor and spindle rotated a 50 mm diameter silicon wafer submerged in an aqueous KCl solution. The zeta potential of the wafer was measured in solutions with concentrations of 1.0 and 0.0014 mM. One silver/silver chloride reference electrode was placed near the disk on the axis of rotation, and a similar one was placed far from the disk. The results in Figure 3 provide the logic of these choices of position. The edge of the disk is where the measured potential is most sensitive to position; the potential crosses zero not too far from the edge and the termination of the disk disrupts the flow pattern. Thus the edge of the disk (position 4 in Figure 1) is the worst position for the reference electrode. Guided by the theory, one places one reference electrode at the axis of rotation (position 1 in Figure 1) where the dependence on axial position is the least and refers it to an electrode at infinity (position 0 in Figure 1). Ag/AgCl electrodes were made according to the method of Westermann-Clark.14 Twelve gauge diameter (2.06 mm) silver wire was obtained from C.C. Silver & Gold Inc. with a purity of 99.99%. The wire was coated with polyolefin heat shrink tubing. Approximately 2 mm of the silver was exposed at one end of each electrode. The exposed wire was soaked in a concentrated ammonium hydroxide solution for 30 min to remove any chloride on the surface and then rinsed with deionized water. The wire was then placed in concentrated nitric acid for about 10 s to clean and roughen the surface, followed again by a rinse in deionized water. The exposed end was then immersed in 0.1 M HCl and attached to the positive lead of a dc power supply. A copper wire was similarly cleaned in nitric acid and attached to the negative lead of the power supply. A current of 4 mA was passed for 45 min to make the AgCl coating characterized by a dark gray/brown color. The electrodes were immersed in the test solution and connected externally to an electrometer. Satisfactory electrodes registered a potential difference at zero rotation rate of less than 1 mV that fluctuated less than 0.1 mV. Silicon wafers were soaked in chromic acid solution for 20 min, followed by rinsing and soaking in deionized water for 5 min. The surface was then dried and the disk was attached to the spindle using 3M double-coated foam tape. The cylinder was rinsed again and soaked in the test electrolyte solution for 20 min or until the potential measured was constant for a 5 min period while rotating the disk at a constant rate. (14) Braem, A. D. Colloidal and interfacial phenomena in polymer/ surfactant mixtures. Ph.D. Thesis, Carnegie Mellon University, Pittsburgh, PA, 2001.
Sides and Hoggard
Figure 5. Streaming potentials measured in 0.0014 mM KCl. 95% confidence limits are barely larger than the symbols. Disk: 50 mm diameter silicon wafer with silicon oxide surface. Streaming potential measured at the axis, referred to the potential far from the disk. The data are plotted vs the 3/2 power of the rotation rate predicted by theory. The linearity is excellent. Fitting the theory to the data with ζ as the single adjustable parameter gave a zeta of -146 mV. In separate experiments, indium tin oxide films coated on glass and obtained from Bioptechs (Butler, PA) were tested in both the rotating disk system described here and in a flat plate capillary flow system.14 The disk and slides were washed in ethanol for 5 min and then rinsed in deionized water. In the parallel plate capillary, we pumped 500 mL of the 10-6 M KCl solution through before measuring the streaming current.
Results and Discussion Measuring the potential difference between the two reference electrodes based on the recommended positioning, one obtains the results of Figure 5. The data of Figure 5 are plotted against the rotation rate raised to the 3/2 power in order to compare the variation found experimentally to the theoretically expected result. The agreement was excellent when standard values for the density, viscosity, and conductivity were employed and a “fitted” value of -146 mV was used for the zeta potential in a 0.0014 mM KCl solution. Positioning of the reference electrode near the axis and adjacent to the disk exhibits the expected 3/2 power behavior. We then scanned the reference electrode in a horizontal plane to probe the radial dependence of the measured potential. The experimental results of the scanning appear in Figure 6. The overall shape of the measured voltage differences agrees with the theoretically expected shape appearing in Figure 3. The magnitude of the measured voltage is a maximum at the axis and crosses zero near the outer edge. The zeta potential obtained from the measurements on the axis was used in the calculation of the theoretically expected line, and the calculated results also appear in Figure 6. The theory agrees well with the experimental result near the axis but diverges from the theory at the periphery of the disk, as one expects from the comments in the previous section. The sensitivity of the measurement to exact location when near the edge, and even more so the uncontrolled hydrodynamic conditions there, make measurements near the outer edge impractical. Measuring the potential between two positions in the plane of the disk, as previously attempted,4,5 also is not optimal.
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mM; streaming potentials in such solutions are often less than 0.1 mV and in our experience are difficult to distinguish from noise. We are currently investigating methods to overcome the constraint on solution conductivity. Conclusions
Figure 6. This graph is a scan of the spatial distribution of potential just outside the double layer as a function of radius. The rotation rate was 1500 rpm. The solid theoretical line was calculated with the same zeta potential used in Figure 5. The experimental data are shown as solid circles. The data match the theoretical predictions near the axis but deviate near the edge where the assumptions underlying the analysis are the least appropriate. Note, however, the crossing of zero potential in both the theory and the data as shown in the sample calculation of Figure 3. Measuring the potential anywhere but on the axis is not optimal. Measuring the potential at the disk edge gives erroneous results.
Mathematical analysis of the electric potential produced in the vicinity of a charged disk rotating in an infinite liquid medium led to the optimal placement of reference electrodes. Experiments confirmed the expected dependence of the measured potential on the 3/2 power of the rotation rate. The analysis allowed direct conversion of the measured streaming potential to a zeta potential characteristic of the charge on the disk. Possibilities for extension of the approach exist and are being developed. Acknowledgment. This work was supported by the National Science Foundation under CTS 0338089. Appendix 1: Calculation of the Radial Component of the Surface Current The derivation of the surface current begins with eq 1 in the text:
jsr )
∫0∞ vrFe dz
(1)
15
Healy and White reported values of the zeta potential of silica at neutral pH in a 0.1 mM solution to be -118 mV. To compare the zeta potential measured using a rotating disk to this value, the same silicon wafer used in the experiments shown in Figures 5 and 6 was tested again in a KCl solution at the same concentration as Healy and White’s experiment. The zeta potential at this concentration was determined to be -122 ( 7 mV which agrees with the reported zeta potential within the margin of error. We performed the experiments on indium tin oxide films in order to compare the results of the rotating disk measurement to data obtained in our laboratories on a parallel plate capillary system. The sample used on the rotating apparatus was a disk, and the sample used in the capillary flow system had the dimensions of a standard microscope slide. We found a zeta potential of 61 ( 5 mV for the capillary flow experiment and 70 ( 3 mV for the rotating disk measurement. This is satisfactory agreement, within 1 mV of statistical agreement. The rotating disk method for measurement of zeta potential is simple and effective, but the approach described here has limitations. First one must prepare the sample in a disk form. Second, the von Karmann theory of fluid dynamics of the rotating disk does not apply at the edge of the disk. This is probably why the agreement between theory and experiment in Figure 6 deteriorates near the edge of the disk. The disturbance at the disk edge is confined to a small region, however, and does not invalidate the method. This problem again points to the axis of the disk as being the optimum position for measurement of the streaming potential. While introduction of the reference electrode also disturbs the fluid dynamics near the axis, this disturbance is confined to a small region and does not degrade the dependence of the streaming potential on the 3/2 power of the rotation rate. One can minimize the effect by making the reference electrode thin. Probably the most severe constraint, however, is a limitation to solutions less than about 1 (15) Healy, T. W.; White, L. R. Adv. Colloid Interface Sci. 1978, 9, 303.
The axial velocity near a rotating disk in an infinite medium is given by Newman7 as
vz ) -a
x
Ω3 2 z ν
(A.1)
Applying the continuity equation for fluid flow and integrating once with respect to r, one obtains
vr )
aΩ3/2 rz ν1/2
(A.2)
We use this equation, Poisson’s equation, and integration by parts to calculate the integral appearing in eq 4. Only the z component of Poisson’s equation is included because the z variation of the electric field in the diffuse layer dominates the radial variation.
jsr )
( ) [ | |]
∫0∞ vrFe dz ) - ∫0∞ axΩν rz0 ∂∂zφ2 3
∂φ ∂z
-0γr z
∞ 0
-φ
2
∞ 0
dz )
) -0γrσ (A.3)
Here the electric field at infinity and the potential at infinity are both zero. This is the basis of eq 2. Appendix 2. Derivation of the Form of the Distribution of the Return Current Density Here we derive the form of the solution of Laplace’s equation for the return current. The general expression for the current density at the disk surface associated with this potential is the same as eq 7 in the text. Thus we can write the constraint on the solution expressing the return current to the disk as
I)
∫0
R
[
-κ
∞
]
∑ βnP2n(η)M′2n(0) 2πr dr ) -20γσπR2
Rηn ) 0
(A.4)
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where the coefficients βn pertain to the return-current solution of Laplace’s equation. Using the expressions given in the main text for the relation between η and r, we find that this equation becomes ∞
∑ βnM′2n(0)∫0 n)0
1
κ
P2n(η) dη ) 0γσR
Thus we require only the first term of the series expressed above to satisfy the constraint on the potential associated with the return current. The expression for the returncurrent density is
(A.5)
ireturn ) -
All the integrals vanish except for the integral associated with n ) 0. So the result is
β0 )
0γσR 1 κ M′0(0)
0γσ )η
0γσ
x
which is the result shown in eq 11.
(A.6) LA048420F
r2 1- 2 R
(A.7)
