Langmuir 2006, 22, 9765-9769
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Calculation of the Streaming Potential near a Rotating Disk Paul J. Sides,*,† John Newman,‡ James D. Hoggard,† and Dennis C. Prieve† Department of Chemical Engineering, Carnegie Mellon UniVersity, Pittsburgh, PennsylVania 15213, and Department of Chemical Engineering, UniVersity of California, Berkeley, California 94720-1462 ReceiVed April 17, 2006. In Final Form: August 23, 2006 A corrected theory of the streaming potential in the vicinity of a disk-shaped sample rotating in an electrolytic solution is presented. When streaming-potential measurements on a variety of materials were reduced to a ζ potential according to a prior theory, the results exceeded expected values by a factor of approximately 2, even though other aspects of the same experiments seemed to confirm the theory. Investigation of the source of the discrepancy revealed a flaw in the prior theory. The crucial understanding is that the surface current produced by the rotation of the disk emerges from the diffuse layer and enters the bulk solution at the periphery of the disk. The new treatment accounts entirely for the discrepancy between literature data and results based on the prior theory.
Introduction The ζ potential of a surface is defined as the electric potential at the plane of shear relative to the adjacent, electrically neutral, bulk solution.1 ζ potential is usually determined by measuring either the streaming current or the streaming potential arising from the forced convection of the diffuse layer. A new apparatus and a method for obtaining the ζ potential of a flat surface based on the measurement of streaming potential in the vicinity of a rotating disk were described in two prior publications.2,3 A schematic drawing of an apparatus employing this concept appears in Figure 1. A disk-shaped sample is appended to a cylindrical rotating support. Spinning the disk convects the diffuse-layer charge adjacent to the face of the sample, which produces a spatially distributed streaming potential. Reference electrodes, such as Ag/AgCl, sense the streaming potential in their vicinity. When a 2.5 cm disk spins at 4000 rpm in 1 mM salt solution, the voltmeter of Figure 1 registers a streaming potential on the order of 100 µV, depending on the material and pH. Analysis of the streaming potential in the vicinity of a disk rotating in contact with a semi-infinite medium yielded predictions of the dependence of the measured streaming potential on radial and axial position.2,3 Experiments described previously2,3 seemed to validate the analysis with respect to the dependence of measured streaming potential on radial position, axial position, and rotation speed, but the ζ potentials deduced from experiments on a range of samples consistently exceeded results from the literature by a factor of approximately 2. Consideration of all aspects of the problem revealed a flaw in the theory identified by one of the authors of the present work (Newman). Rotation of a disk as shown in Figure 1 generates radial flow with a velocity proportional to distance from the axis.4-6 Radial flow near the disk surface convects mobile ionic charge in the diffuse part of the double layer, thereby creating * Corresponding author. Address: Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213. TEL: 412 268 3846. FAX: 412 268 2183. E-mail:
[email protected]. † Carnegie Mellon University. ‡ University of California. (1) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: London, 1981. (2) Sides, P. J.; Hoggard, J. D. Langmuir 2004, 20, 11493-11498. (3) Hoggard, J. D.; Sides, P. J.; Prieve, D. C. Langmuir 2005, 21, 7433-7438. (4) von Ka´rma´n, Th. Z. Angew. Math. Mech. 1921, 1, 233-252. (5) Cochran, W. G. Proc. Cambridge Philos. Soc. 1934, 30, 365-375. (6) Rogers, M. H.; Lance, G. N. J. Fluid Mech. 1960, 7, 617-631.
Figure 1. A rotating disk of diameter 2a is supported on a cylindrical spindle and immersed in electrolytic solution. The disk has a ζ potential equal to ζ. When the disk is rotated, the voltmeter records a voltage, the streaming potential, between the reference electrodes represented by the dots at the end of the leads to the voltmeter.
Figure 2. This sketch of the immediate vicinity of the disk shows the radial component of the surface current flowing along the disk and the bulk current returning from the periphery of the disk through the electrolyte to the disk’s face.
a sheet of ionic current that flows both concentrically and radially outward along the disk surface. The convected radial current must be conducted through the electrolyte to complete the circuit. The flaw in the previous theory was an incorrect view of the location at which streaming current emerges from the diffuse layer and enters the bulk electrolyte. A sketch of the region near the disk (Figure 2) shows the correct physics; all of the charge convected radially near the disk, that is, the surface current, emerges from the diffuse layer at the periphery of the disk and returns as bulk current through the electrolyte to the disk’s face.
10.1021/la061041x CCC: $33.50 © 2006 American Chemical Society Published on Web 10/07/2006
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We present an analysis reflecting this understanding and confirm by experiment that the results from this apparatus agree with published data based on the flow between parallel plates, the traditional method of measuring ζ potential.
Theory of the Streaming Potential near a Rotating Disk The theory of streaming potential measurement begins with derivation of an expression for the radial surface current jsr ([)] A/m). In the general case, js comprises surface current due to the convective flow of ions in the diffuse layer and current due to surface conductivity.1 For a disk with a radius greater than 1 cm, the effect of surface conductivity is small because the large volume of solution in the disk’s vicinity provides a low resistance path for the return of current. Calculations presented in the appendix demonstrate that surface conductivity is negligible, even for experiments in distilled water. The surface current is the integrated flow of unbalanced charge in the diffuse layer, given by
jsr )
∫0∞VrFedz
(1)
where Vr is the local radial velocity, and Fe is the charge density at any point in space. The radial velocity in the immediate vicinity of a rotating disk is
Vr )
0.51023Ω3/2 rz ) γrz ν1/2
φd(rj,zj)κ )2 γζa
∫0∞
J1(p) J (prj)e-pzjdp p o
(5)
where J1 is the Bessel function of order 1; φd is the potential due to the disk; κ is the solution conductivity; and the overbars on r and z indicate that the coordinate is scaled by the disk radius, a. The surface current leaving the disk is a ring source at radius a. Consider a flat ring of radius r′ and thickness dr′ centered about the axis in the plane of the disk; the ring is a source of current density in flowing into the semi-infinite domain below the disk. The contribution of this ring to the electric potential φr(r,z) is10
dφr(r,z) ) (2)
where γ ≡ 0.51023xΩ3/ν, Ω is the rotation rate in radians per second, and ν is the kinematic viscosity of the liquid. Substituting Fe from Poisson’s equation (neglecting the radial partial derivatives) into eq 1 and integrating by parts2 leads to
jsr ) -γrζ
Deriving an expression for the overall electric potential distribution in the bulk solution is the remaining task. According to the mechanism described in the previous paragraph, the potential at an arbitrary location is the superposition of a potential arising from the uniform flow of current to the disk and from a ring source of current at the periphery of the disk. Nanis and Kesselman9 solved the case for a disk embedded in an infinite insulating plane. The equation expressing the potential on such a disk, when the current density is a constant equal to the result presented in eq 4, is
(3)
2i dr′
∫02π 4πκ[z2 + (r′ cos θ′ -n r)2 + (r′ sin θ′)2]1/2dθ′ ) in(K(m)r′dr′ 2 (6) πκ [z2 + (r + r′)2]1/2 in which K(m) is the complete elliptic integral of the first kind with10
where is the electric permittivity of the fluid, r is the radial position, and ζ is the ζ potential. Both the electric field and the potential vanish at infinity. Thus, the surface current due to convection of diffuse-layer charge is proportional to the radial position and the rotation rate raised to the 3/2 power. The surface current flows radially outward along the disk surface within a few Debye lengths of the surface. The surface current density is positive for a negative ζ potential. Charge must be conducted into the diffuse layer at each r from the neutral bulk to supply the larger radial current at larger r. Conservation of charge requires a current density normal to the disk equal to7,8
The variable φr denotes the potential due to the ring, and the current density on the ring is in at r′. The ring is an infinitesimally thin source of current so that r′ f a and indr′ f -aγζ. Furthermore, m f (4ra)/[z2 + (r + a)2] and [z2 + (r + r′)2]1/2 f [z2 + (r + a)2]1/2. Equation 6 becomes
iz ) -∇s‚js ) 2 γζ
Superimposing eqs 5 and 8, one obtains the overall potential distribution φ in the semi-infinite domain,
(4)
Equation 4 indicates that a uniform current density flows from the bulk electrolyte to the diffuse layer or vice versa. When ζ is negative, the surface current jsr is positive, and the bulk current density to the disk given by eq 4 is negative; in this case, the current in the bulk electrolyte flows toward the disk for 0 < r < a. When ζ is positive, the direction of the current from the face of the disk is opposite. In summary, a total surface current of magnitude -2γζπa2 flows in the bulk solution between the disk’s face and periphery to close the electrical circuit. The streaming potential is the voltage required to drive this current through the electrolyte. (7) Levich, V. G. Physicochemical Hydrodynamics; Prentice Hall: Englewood Cliffs, NJ, 1962. (8) Newman, J.; Thomas-Alyea, K. E. Electrochemical Systems, 3rd ed.; WileyInterscience: Hoboken, NJ, 2004.
m)
4rr′ and K(m) ) z2 + (r + r′)2
∫0π/2 (1 - mdRsin2 R)1/2
κφr(rj,zj) -2 K(m) ) 2 γaζ π [zj + (rj + 1)2]1/2
(7)
(8)
φ(rj,zj)κ φd(rj,zj)κ φr(rj,zj)κ ) + ) γζa γζa γζa 2
∫0∞
J1(p) K(m) 2 J (prj)e-pzjdp (9) p o π [zj2 + (rj + 1)2]1/2
Equation 9 permits calculation of the potential anywhere in the bulk electrolyte subject to several assumptions: (1) The domain is semi-infinite. (2) The spindle supporting and spinning the disk is taken as infinitesimally thin and frictionless with respect to the flow, and the plane z ) 0 is a mirror plane. (3) Laplace’s (9) Nanis, L.; Kesselman, W. J. Electrochem. Soc. 1971, 118, 454-461. (10) Newman, J. Electroanalytical Chemistry, A Series of AdVances; Bard, A., Ed.; Elsevier: New York, 1972; Vol. 6, pp 187-352.
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Figure 3. Streaming potential (solid lines) and lines of current (dotted) in the vicinity of the disk. The equipotentials represent constant values of κφ/aγζ shown at intervals of 0.05. The zero of potential is the surface of rotation extending from 0.908 radii on the surface of the disk toward infinity. Values of κφ/aγζ at radii smaller than the zero of potential are positive, while the same quantity at radii larger than the zero of potential is negative. Four percent of the total current flows between each line of current.
equation is a valid basis for analyzing the electric potential in the bulk. (4) Surface conductivity is negligible. The effects of these assumptions and the methods for exploring their significance are mentioned in the next section, and the assumption about surface conductivity is discussed in the appendix. Approaches other than the one described above lead to the same quantitative predictions of the streaming potential distribution in the bulk solution. One obtains identical numerical results by using an analogous superposition of a solution in rotational elliptical (oblate spheroidal) coordinates and a ring current. One can also solve the entire problem directly with Coulomb’s law. In this approach, the electric field required for conduction to supply the correct normal component of current to the surface of the disk and the radial component from the periphery of the disk is obtained by specifying an apparent charge distributed uniformly over the surface of the disk and an equal but opposite charge distributed uniformly along the perimeter of the disk. This method provides a closed-form result for the streaming potential along the axis relative to the streaming potential at infinity.
φ(0,zj)κ 1 - 2zjx1 + jz2 + 2zj2 ) γζa x1 + jz2
(10)
The dimensionless streaming potential along the axis varies as 1 - 2zj + O(zj2) near the plane of the disk and as 1/(8zj3) + O|1/zj5| as jz f ∞.
Results and Discussion Equipotentials and current lines calculated with the aid of eq 9 appear in Figure 3. The equipotentials correspond to constant values of κφs/aγζ at intervals of 0.05. The zero of potential is the surface of rotation extending from r ) 0.908a on the disk and flaring out to infinity. Potentials at radii smaller than the radius of this surface are positive, while potentials at radii larger than the radius of this surface are negative. The maximum potential is in the plane of the disk at the axis, and the potential diverges to negative infinity in the plane of the disk at r ) a where the ring source emits current to the bulk solution. The equipotentials are practically indistinguishable near the edge of the disk. Any two current lines enclose 4% of the total current. Streaming current emerges from the edge of the disk (r ) a) and flows
Figure 4. The dependence of the streaming potential on position with respect to the disk. (a) Radial position dependence at a constant axial distance in the plane of the disk (z ) 0) for the present calculation (solid line) and the previous calculation1,2 (dotted line). Comparison of the present result with the prior calculation demonstrates the origin of a factor of 2.33 discrepancy on the axis of the disk, even though the profiles both cross zero and go through a minimum near the edge of the disk. (b) Axial position dependence of the streaming potential. This line shows the effect of the position of the reference electrode on the measured streaming potential.
through the bulk solution back to the diffuse layer where it is distributed uniformly between r ) 0 and r ) a. Figure 4 shows the potential distribution plotted along the coordinate axes. The solid black lines in Figure 4 correspond to the potential distribution calculated according to eq 9. The dimensionless potential in the plane of the disk is exactly unity at the axis (Figure 4a); it becomes negative along the disk edge, a necessary requirement since no current flows to infinity. The region of sharply varying potential near the edge of the disk reflects the emergence of the surface current into bulk solution at the edge of the disk. The potential diverges to negative infinity at the edge of the disk and decays back toward zero beyond it. Also appearing in Figure 4a is the potential calculated according to the previous theory.2,3 The ratio of the two results at the axis in Figure 4a, a factor of 2.33, accounts for the discrepancy between literature values of ζ potential and the data of experiments with the rotating disk analyzed according to the prior theory. The dependence of the streaming potential on axial position appears in Figure 4b. The streaming potential decays to less than 5% of its maximum value at the axis within one diameter of the disk. The corrected potential distributions appearing in Figures 3 and 4 lead to the same conclusions about the placement of reference electrodes described previously.2,3 Positioning one reference electrode at the axis near the disk takes advantage of the insensitivity of potential to radial position there. One must, however, know accurately the distance of the reference electrode from the disk for absolute calculation of ζ from measured streaming potentials; eq 10 is helpful in this regard. Placement of the other reference electrode near the disk at r ) a is problematic because of the extreme gradients of potential. The best position
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Figure 6. (b) The ζ potential of muscovite mica acquired with the rotating disk apparatus in a background of approximately 1 mM KCl and (O) the data of Scales et al.11 The agreement is excellent.
Figure 5. Comparison of radial (a) and axial (b) streaming potentials to experimental data reported in Hoggard et al.2 for measurements on silicon. The concentration of KCl was 0.014 mM, the disk was 50 mm in diameter, and the temperature was 23 °C. The data were fit to the theory by calculating the ζ potential of the disk based on the streaming potential measured at the axis at an axial distance of 1.0 mm. The calculated ζ potential is -62.8 mV. The data agree reasonably well with the theory, particularly capturing the maximum of potential at the disk edge.
for the second reference electrode is far from the disk where the potential is zero and not sensitive to position. Experimental data reported previously2,3 appear in Figure 5, showing the streaming potential along the two main coordinate axes. The streaming potential of a 50 mm diameter silicon wafer was measured in solutions of KCl at a concentration of 0.014 mM at a temperature of 23 °C. One reference electrode was placed far from the disk, and the other reference electrode was positioned 1 mm from the disk and scanned across it to obtain Figure 5a. Figure 5b was obtained by positioning one electrode far from the disk while the other was scanned vertically along the axis of the disk. The data were fit to the theory at a single point, 1 mm from the disk and on the axis; the same value of ζ, -62 mV, was used for both graphs. Both the radial and axial variations of streaming potential agree reasonably well with the calculations. Note particularly the agreement between the theory and the experimentally measured streaming potential near the periphery of the disk. Comparison of the measurements of the ζ potential of muscovite mica with the careful measurements of Scales et al.11 demonstrates the effectiveness of the rotating disk for deducing ζ potential from streaming-potential measurements. A 25 mm diameter mica disk was rotated at rates between 0 and 4000 rpm, during which streaming potential was measured as per the schematic drawing of Figure 1. Figure 6 shows data acquired with the rotating disk and data acquired by Scales et al. with a parallel plate method. The mica was titrated in a background of approximately 1 mM KCl. The agreement is excellent. Several assumptions were enumerated in the Theory section. Figure 6 serves as initial empirical evidence that the assumptions (11) Scales, P. J.; Grieser, F.; Healy, T. W. Langmuir 1990, 6, 582-589.
are not too severe. Detailed treatment of the inaccuracies of the present theory for a disk-shaped sample attached to a cylindrical spindle and rotating in a finite cell volume is beyond the scope of the present work, but a few comments are possible. Mandin et al.12 numerically simulated both the inner and outer flows in a cell of finite volume and found good agreement with the classical treatment in the boundary layer of the disk, even when the spinning disk spanned nearly the entire diameter of the cell. The use of the asymptotic expression for radial flow of fluid in the diffuse layer (eq 2), therefore, is not a significant source of inaccuracy. The effect of cell volume on the distribution of electric potential can be addressed in a straightforward manner by solving Laplace’s equation in a cylindrical cell, as demonstrated previously for an electrochemical system.13 Separation of variables yields a Bessel series with zero field at the walls and a specified current distribution on the plane, including the disk, from which the effects of finite cell volume can be estimated if necessary. Since the streaming potential is small for cell dimensions greater than a few radii (see eq 10), the cell must have a dimension less than approximately 3 times the radius for cell volume to be significant. Laplace’s equation is valid for analysis of the potential in the bulk solution because the salt concentration gradients are nil due to stirring by the disk. See the appendix for discussion of the assumption of negligible surface conductivity.
Conclusion This contribution corrects prior theory of the streaming potential in the vicinity of a rotating disk. Rotation of the disk causes mobile charge in the diffuse layer to flow concentrically and radially outward, which constitutes a surface current. The radial component of the surface current emerges from the periphery of the disk and returns to the face of the disk to conserve charge. The theory describing the streaming potential near the disk provides a foundation for use of this apparatus in experiments to determine the value of the ζ potential of the disk. Acknowledgment. The National Science Foundation supported this work under CTS0338089. The authors acknowledge the support of Malvern Instruments, Ltd., and experimental findings provided by Fraser McNeill-Watson. Finally, the authors thank Thomas Saiget and Samuel Aigen for their measurements of the ζ potential of mica. (12) Mandin, Ph.; Pauporte´, Th.; Fanouille`re, Ph.; Lincot, D. J. Electroanal. Chem. 2004, 565, 159-173. (13) Pierini, P.; Newman, J. J. Electrochem. Soc. 1979, 126, 1348-1352.
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Note Added after ASAP Publication. The ordinates of the plots in Figure 4 were inadvertently omitted in the version published ASAP October 7, 2006; the corrected version was posted ASAP October 13, 2006.
(
κs ) -2κλ sinh
js ) κsEr - γζr
(A.1)
where κs is the surface conductivity and Er is the radial component of the electric field near the disk. The radial electric field can be calculated with the aid of eq 9. For the purpose of estimating the error incurred by the neglect of surface conductivity, we take the quotient of the streaming potential on the disk at the axis, where φs ) γζr/κ, and the radius a as a characteristic electric field in the problem. Setting r ) a, normalizing all terms by -γζr, and subtracting unity from both sides, one can express the fractional error in eq 1 as
κs js -1) -γζa aκ
)(
)
λ -2RT Fζ 2 sinh µ Fλ 2RT
[
]
1+
Appendix: The Contribution of Surface Current If the diffuse layer is thin relative to the disk radius, the complete expression for the surface current js is8
Fζ Fζ t - t - tanh + 2RT + 4RT
x
(A.3)
Fζ 1 + sinh 2RT 2
where t is the transference number of the indicated ion, λ is the Debye length, µ is viscosity, and F is Faraday’s constant. The first term arises from the difference between the bulk conductivity and the diffuse-layer conductivity. The second term expresses the effect of electroosmosis driven by the gradient of streaming potential near the surface. Combining eqs A.2 and A.3 yields
js 2λ Fζ 1 2RT Fζ 2 - 1 ≈ sinh + sinh -γζa a 2RT µλκa F 2RT
(
)
(A.4)
which allows estimation of the effect of assuming negligible surface conductivity. The first term on the right side is essentially the ratio of the Debye length to the disk radius. The second term on the right side reflects the contribution of electroosmostic flow in the diffuse layer in response to the average gradient of streaming potential. Both of these terms depend on the reciprocal square root of concentration, so the worst case would be a concentration of 10-7 mol/L, reflecting the background concentration of protons and hydroxyl ions in pure water. If the disk is 1 cm in radius and the ζ potential is 100 mV, the values from eq A.4 are
js - 1 ≈ O(10-3) + O(10-3) -γζa
(A.2)
(A.5)
This calculation shows that neglecting surface conductivity in the disk system is appropriate. If the right side of eq A.2 is small, surface conductivity can be neglected. The surface conductivity κs comprises two terms,8
LA061041X
