Article pubs.acs.org/Langmuir
Surface Conductivity and the Streaming Potential near a Rotating Disk-Shaped Sample Paul J. Sides* and Dennis C. Prieve Department of Chemical Engineering, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213, United States ABSTRACT: Surface conductivity can complicate the determination of a sample’s zeta potential from electrokinetic measurements. Correction factors have been derived to mitigate the problem for common systems such as particles, plates, and porous plugs. These factors are functions of the Dukhin number Du given by Ks/Ka where Ks is the surface conductivity, K is the electrolyte conductivity, and a is a length scale appropriate for the sample geometry. Here, the correction factor for the rotating disk geometry, in terms comparable to equations for particles, capillaries, and porous plugs, is shown to be f(Du) ≃ 1 + 1.516Du + 0.135Du2. The reciprocal of the f(Du) equation expresses the factor by which surface conductivity reduces the measured streaming potential for a given sample’s true zeta potential. The theory shows that surface conductivity is negligible in the rotating disk geometry for essentially all ionic strengths because the disk radius is the natural length scale for Du. The correction for surface conductivity for a KCl solution with an ionic strength equivalent to the ionic strength of pure water would be only 1% for a disk 10 mm in diameter. The ohmic resistance to the return of surface current from the disk’s periphery through the bulk liquid to the axis is always much smaller than the resistance to the return of surface current back through the diffuse charge cloud of the double layer. The rotating disk geometry is unusual in this regard. where ε is the solution permittivity, ζ is the zeta potential, μ is the dynamic viscosity, K is the bulk electrolyte conductivity, and Du is the Dukhin number defined in eq 2.
1. INTRODUCTION Surface conductivity can “short” the electrokinetic circuit in streaming potential measurements aimed at determining the zeta potential of particles or macroscopic materials.1 The conducting layer is roughly the Debye length κ−1 in thickness and conformal to the sample geometry. For particles, the ratio of the particle radius to the Debye length, κrp, is important in assessing whether surface conductivity is significant in the determination of electrophoretic mobility. When κrp is large but not infinite, the electrophoretic mobility requires a correction for surface conductivity.1−3 Surface conductivity likewise arises in experiments aimed at determining the zeta potential of planar samples when one employs a flow cell based on a thin gap and measures the streaming potential. The parameter analogous to κrp in this case is κh, where h is the thickness of the gap between the two surfaces defining the major walls of the flow cell. If κh is not infinite, current leaks back through the diffuse layer because of surface conductivity, and the measured streaming potential is less than expected for a given zeta potential of the materials forming the gap. The correction typically is cast as a factor that reduces the streaming potential expected for a given zeta potential. The relationship between the measured streaming potential Δϕm and the pressure difference Δp from end to end in a cylindrical capillary of radius rc is given by Δϕm =
1 εζ Δp 1 + 2Du μK
Du ≡
(2)
Du is the dimensionless ratio of the surface conductivity Ks to a characteristic parallel conductivity provided by the product of the bulk ionic conductivity K and an appropriate length scale, in this case the capillary radius rc.3−5 The corresponding correction factor for particles is (1 + Du) where the significant length scale is the particle radius. The correction for surface conductivity can be appreciable when small particles or small capillary gaps are used with solutions of low ionic strength because the particle radius and capillary gap appear in the denominator of Du. The rotating disk is an alternative geometry for determining the zeta potential of planar samples.6 When a material in the form of a disk and having a nonzero zeta potential is attached to a spindle and rotated in a liquid, shear at the disk/liquid interface produces a radial flow of the unbalanced ionic charge in the diffuse part of the double layer adjacent to the solid. This advected charge, called a surface current, flows to the periphery of the disk where the current becomes ohmic in order to return Received: July 17, 2013 Revised: October 1, 2013
(1) © XXXX American Chemical Society
Ks Krc
A
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direction indicated, and the radial distance is r. The face of the cylinder in contact with the liquid has a zeta potential ζ when 0 ≤ r ≤ a and zero when r > a. Laplace’s equation governs the electric potential ϕ outside the thin counterion cloud in the semi-infinite domain below the disk.
to the disk, thus closing what amounts to an electrokinetic circuit. The streaming potential drives the ohmic return current. For a disk with a negative zeta potential, the perimeter of the disk is the positive “pole” of the circuit and the center of the disk is the negative pole. The potential far from the disk vanishes. One places sensors, for example, Ag/AgCl reference electrodes, at the axis and at infinity to detect the streaming potential most reproducibly. The basic theory of the rotating disk is presented, in a more efficient form than previously published, in the Theory section. The rotating disk method, based on the measurement of the streaming potential, suffers from surface conductivity theoretically, but no correction factor analogous to eq 1 for the rotating disk geometry has been presented; its significance or lack thereof has not been explored. The purpose of this article is to provide that correction and demonstrate the circumstances under which an adjustment for surface conductivity is required for the rotating disk geometry. Three currents flow in any streaming potential experiment: (1) streaming source current arises from the convection of charge inside the thin counterion cloud at the liquid/solid interface; (2) ohmic drain current returns source current through the bulk solution to its origin, which completes the electrokinetic circuit; and (3) leak current returns streaming current via the diffuse layer adjacent to the sample when the diffuse part of the double layer is not infinitesimally thin. In the classic capillary-style geometry that features a large ratio of length to radius, these three currents flow parallel to each other, which simplifies the problem. In the rotating disk geometry, the source and leak currents flow in the plane of the diffuse layer adjacent to the disk and the drain current flows out of the plane. In the solution of the problem for the rotating disk, we combine the drain and leak currents and then superimpose the source current to express the streaming potential. The method used in Sides et al.6 is inflexible; other approaches exclusively using polynomials are possible.7 We first derive a formula, for κa approaching infinity, with orthogonal functions only; this structure forms a convenient basis for the second part wherein we calculate a correction for surface conductivity at the rotating disk for large κa. The surface conductivity turns out to be negligible under practical conditions, an unusual property of the rotating disk geometry.
∇2 ϕ = 0
(3)
subject to lim
r 2 + z 2 →∞
ϕ=0
r = 0 ⇒ ϕ well behaved iz z = 0 = −∇·js
(4)
where js is the surface current density in A/m. Boundary conditions in eq 4 express the asymptotic decay to zero potential at infinity, the symmetry at the axis, and the current density normal to the plane of the disk, respectively. 2.1. Solution for κa → ∞. A constitutive equation for the surface current density, as κa approaches infinity, expresses the advection of the unbalanced charge in the diffuse charge layer proximate to the disk. Both the radial and angular components of this surface current near a rotating disk are nonzero, but only the radial component contributes to the streaming potential. The radial fluid velocity near the disk surface is approximately vr ≅ 0.51023
ω3 rz = γrz ν
(5)
Equation 5 uses the known solution of the fluid mechanics of the rotating disk.8 This equation is the first term of a polynomial expansion for the radial velocity; it becomes exact as z tends toward zero at the plane of shear. One obtains the surface current of eq 4 by integrating the product of the charge density in the diffuse part of the double layer adjacent to the disk with the local velocity given by eq 5. This integral appears in eq 6. After substituting eq 5 for the local velocity, replacing ρe by Poisson’s equation for the charge density, and integrating by parts according to standard technique, one obtains the expression for the surface current density when the Debye length is infinitesimally small.9 jsr (r ) =
2. THEORY The cylindrical coordinate system and geometry appear in Figure 1. A disk of radius a contacts a Newtonian liquid of semi-infinite extent. The disk rotates on its axis at angular velocity ω. The liquid has kinematic viscosity ν, permittivity ε, and ionic conductivity K. Axial distance z is positive in the
∫0
∞
ρe (z) vr(r , z) dz = −γεζr[1 − H(r − a)] (6)
The Heaviside step function in eq 6 passes the surface current on the disk and terminates it at the periphery. The negative of the surface divergence of the surface current (eq 4) gives the normal current density, valid for 0 ≤ r ≤ ∞. iz z = 0 = 2γεζ[1 − H(r − a)] − γεζrδ(r − a)
(7)
Equation 7 is the asymptotically correct expression for the current density flowing from the bulk solution to the disk at any radius when the diffuse part of the double layer is infinitesimally thin. The surface current jsr carried by advection inside the counterion cloud for r < a is transmuted at r = a to conduction through the electrically neutral bulk solution; otherwise, charges would accumulate at r = a. Sides et al.6 treated the source current at the edge of the disk as a ring source in the analysis. Here, we have combined the drain current density, the source current density, and the insulation condition for r > a and z = 0 into one expression, eq 7. The δ function expresses
Figure 1. Cylindrical coordinates of a disk of infinite radius rotating in an infinite medium. The zeta potential in the plane of the interface is nonzero from the axis to radius a and zero at any radius greater than a. B
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⎡ K ∂(rEr) ⎤ iz z = 0 = ⎢2εγζ − s ⎥[1 − H(r − a)] − [γεζr − K sEr][δ(r − a)] ⎣ r ∂r ⎦
the singularity of the normal current density at r = a (Appendix A). Integrating eq 7 over the entire disk,
∫0
∞
2πriz
dr = 2εγζ z=0 ∞
− εγζ
∫0
∫0
(15)
∞
2πr[1 − H(r − a)] dr
for the normal current density, applicable at any radius. The drain and source terms of the current density appear as before. Equation 15 is the equivalent of eq 7 but includes surface conduction; it incorporates the leak current due to surface conductivity into the boundary condition expressing the current density normal to the disk. The leak current on the disk in eq 15 prevents the simple use of orthogonality to solve the problem in a single semianalytical step. In principle, one could use collocation to solve eq 15 numerically in a direct calculation, but the singularity at the periphery of the disk complicates that approach. Breaking the overall problem into two parts, however, makes a hybrid solution possible. The first part, superimposing the shear and leak currents on the disk, admits a solution based on orthogonal collocation. The second part takes advantage of orthogonality as before. By momentarily neglecting the term related to the streaming current emitted at the disk’s periphery, multiplying by r dr, and integrating, one obtains
2
2πr δ(r − a) dr = 0
(8)
proves that the total source current leaving the edge of the disk at r = a (the second term on the right-hand side of eq 7) exactly equals the drain current returning to the disk for r < a (the first term on the right-hand side of eq 7), as it must. The solution to Laplace’s equation that satisfies eq 7 represents the streaming potential outside the counterion cloud. The streaming potential is written conveniently as a function of rotational elliptical coordinates.10 The value ξ = 0 defines the plane of the sample (0 < r < a, z = 0); η = 0 defines the insulating plane (r > a, z = 0); and η = 1 defines the axis (r = 0, z > 0). The origin of the system is a circle in the plane of the disk at (η, ξ) = (0, 0). In these coordinates, the solution of eq 3 satisfying all conditions of eq 4, other than the normal current density on the disk itself, is ∞
ϕ(η , ξ) =
∑ BnP2n(η) M2n(ξ)
(9)
n=0
K
where 2
Using Newman’s expression10 for the normal derivative in the rotational elliptic system, one can write eq 7 (Appendix B) as
Ed , z
r dr = εγζr 2 − K srEd ,r
⎡
∞
⎣
n=0
−εγζa [2η − δ(η)] ∑ BnP2n(η) M2n′(0) = K n=0
z=0
∑ Bd̅ ,n⎢M2n′(0) ∫
∞
z=0
η
1
P2n(u) du + Du
(16)
⎤ (1 − η2) P2n′(η)⎥ η ⎦
= 1 − η2
(10)
(17)
where
where10
Bd̅ , n ≡
−2 (2nn! )4 M 2n′(0) ≡ π [(2n) ! ]2
(11)
1 ⎡ ⎤ n K 4n + 1 ⎢ ( −1) Γ n − 2 ⎥ + Bn = P (0) 2n ⎥ εγζa M 2n′(0) ⎢ 2 π (n + 1)! ⎣ ⎦
(
)
(12)
∞
The streaming potential at the axis of the disk and proximal to it (i.e., at r = z = 0) is given by ϕ(η = 1, ξ = 0) =
εγζa K
∞
∑ n=0
Id = πa 2(2εγζ ) ∑ Bd̅ , n M 2n′(0) n=0
1 ⎡ ⎤ n 4n + 1 ⎢ (− 1) Γ n − 2 ⎥ + P2n(0)⎥ ⎢ M 2n′(0) 2 π (n + 1)! ⎣ ⎦
εγζa K
(
∫1
0
P2n(η) dη
(18)
All terms vanish except the first term.
)
Id = −2πεγζa 2Bd̅ ,0 M 0′(0)
(19)
Because the source current per unit arc length must equal the negative of the net current to the disk, in magnitude but opposite in sign, the boundary condition on the disk for the entire problem becomes
(13)
which agrees with our earlier result.6 2.2. Solution for Large (but Not Infinite) κa. We now deduce a factor that corrects for the finite surface conductivity of a thin layer conformal to the disk. The surface current density is jsr = [−γεζr + K sEr][1 − H(r − a)]
K Bd , n εγζa
This expression is ready for solution by orthogonal collocation, which produces the Bd,n coefficients. The magnitude of the source current flowing from the periphery is less than εζγa, the source current for infinite κa, by the amount that leaks back through the surface conductivity, as one can see from eq 15. We calculate the amount of source current associated with the measured streaming potential by integrating the net current flowing to the disk,
One uses the orthogonality of Legendre polynomials and the definition of the δ function to obtain coefficients Bn.
=
r
The subscript d indicates the electric field associated with the drain and leak currents. In rotational elliptical coordinates, this relationship becomes
2
r = a (1 + ξ )(1 − η ) , z = aξη
Bn̅ ≡
∫0
∞
∑ Bn̅ P2n(η) M2n′(0) n=0 ∞
(14)
=
and one finds
∑ Bd̅ ,nP2n(η) M2n′(0) − [Bd̅ ,0 M0′(0)]δ(η) n=0
C
(20)
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The use of orthogonality is possible at this juncture because the Bd,n coefficients are known from the numerical solution of eq 17. Bn̅ = Bd̅ , n +
2 4n + 1 Bd̅ ,0 P2n(0) π M 2n′(0)
(21)
The streaming potential at the axis and on the disk is ϕ r=0 = z=0
εγζa K
∞
⎡
∑ ⎢Bd̅ ,n + n=1
⎣
⎤ 2 (4n + 1)P2n(0) Bd̅ ,0 ⎥ M 2n′(0) π ⎦
(22)
The required correction factor, in terms comparable to the factor for the thin gap cell, is f (Du) =
1 ∞ ⎡ ∑n = 1 ⎢⎣Bd̅ , n
+
⎤ 2 (4n + 1)P2n(0) Bd̅ ,0 ⎥⎦ M 2n ′ (0) π
Figure 2. Dimensionless streaming potential, relative to a reference electrode at infinity, as a function of radial position on the disk for various values of Du between 0 and 1. Increasing surface conductivity diminishes the streaming potential at the axis.
(23)
3. RESULTS AND DISCUSSION Just as the Dukhin number Du is the main parameter governing the importance of surface conductivity in common electrokinetic systems, Du is the essential parameter for the rotating disk as well. Its role becomes clear when eq 15 is normalized naturally by the current density 2εζγ returning to the disk in the absence of surface conduction. The radial position is normalized naturally by the disk radius, and the streaming potential is normalized by its value when the surface conductivity is absent, eq 13. Equation 15 becomes
from zero, the magnitude of the streaming potential decreases everywhere along the disk because streaming current leaks through the surface conductance offered by the diffuse layer adjacent to the disk. For Du = 1, the streaming potential at the axis of the disk and on the plane of the disk equals approximately 40% of the value expected in the absence of surface conductivity. Figure 3 shows the factor by which the streaming potential that would be measured at the center of the disk is reduced by surface conductivity.
⎡ 1 ∂ρEρ̅ ⎤ ⎥[1 − H(r − a)] − [ρ − DuEρ̅ ]a[δ(r − a)] i ̅ = ⎢1 − Du ρ ∂ρ ⎦ ⎣
(24)
where ρ is the dimensionless radius, Du = Ks/(Ka), and the overbars on the current density and electric field indicate dimensionless quantities. When Du = 0, surface conduction is absent. Du is the only parameter of eq 24 that affects the amount of current returning to the disk through the bulk electrolyte; hence, Du is the single essential parameter of this analysis. The key finding is that the natural length scale of Du for the rotating disk is the disk radius. Because this radius is typically tens of millimeters, as opposed to fractions of a millimeter for a capillary gap or micrometers for particles and porous plugs, Du is quite small for most systems, as shown below. Figure 2 shows the dimensionless streaming potential as a function of radial position on the disk for various Du values. The streaming potential is relative to zero potential at infinity. The value is a maximum at the origin. Orthogonal collocation based on Legendre polynomials at 20 points was used to solve eq 17. The results matched exactly the semianalytical solution for Du = 0 when the latter was terminated at the same number of terms. The value of the potential oscillated by a few percent around the correct value for very large n and small Du. The potentials given by successive truncations, however, were 180° out of phase and could be averaged. The result of averaging the potentials given by truncations at 19 and 20 terms agreed almost exactly with the semianalytical result for Du = 0 as presented in Sides et al.,6 especially at the axis where the streaming potential is measured. The faint oscillations for larger r and Du = 0 reflect the fact that the averaging described above did not completely remove the oscillations due to the finite number of terms of the collocation procedure. These small oscillations were not apparent for Du > 0.1. As Du increases
Figure 3. Fraction by which the streaming potential associated with a given zeta potential would be reduced in the rotating disk system. The streaming potential decreases as the relative importance of surface conductance vs bulk conductance increases.
In a form comparable to the factor 1 + 2Du shown in eq 1 for the capillary, one obtains f (Du) ≅ 1 + 1.516Du + 0.135Du 2
(25)
for the disk where Du is based on the disk radius. Having eq 25, one can explore quantitatively the significance of surface conduction for the rotating disk. Newman8 (p. 258) gives an expression for the surface conductivity in the case of thin diffuse layers and a binary 1:1 electrolyte. A similar equation can be found elsewhere.3 Keq ⎛ D − D− eζ ⎞ K s = 2 dl ⎜ + − tanh ⎟+ 4kT ⎠ κ εkT ⎝ D+ + D−
D
qdl 2 κμ
1+
1+
eqdl
2
( 2κεkT )
(26)
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Here the Di are ionic diffusivities, e is the electron charge, and k is Boltzmann’s constant, and Gouy−Chapman theory gives the double-layer charge qdl. qdl =
−2εκkT eζ sinh e 2kT
(27)
The first term in eq 25 is augmented conduction by the surfaceexcess concentrations of the two ions in the diffuse cloud of the double layer; the second term in eq 25 is the contribution of electroosmosis induced by the streaming potential. Equation 25 was used to calculate Du from eq 2 as a function of the concentration of KCl; the results appear in Figure 4. Du is
Figure 5. The factor by which the measured streaming potential must be increased to compensate for surface conductivity is a function of the ionic strength and system geometry. The gap is the separation between parallel plates forming the cell in a capillary gap design. The disk geometry requires no correction even for very dilute solutions.
geometry is negligible under practical circumstances. This geometry is unusual in this regard.
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APPENDIX A: DEFINITION AND PROPERTIES OF THE δ FUNCTION The δ function δ(s) is an example of a generalized function defined by the following equation
Figure 4. Dukhin number as a function of concentration for 0.1, 1, and 10 mm disk radii. The correction is appreciable only for disks 100 μm in diameter and less, an impractical dimension for reasons other than surface conduction.
∞
∫−∞ f (s) δ(s − s0) ds ≡ f (s0)
(A1)
for any function f(s) that is continuous in the vicinity of s = s0. To satisfy this relationship, δ(s) must be zero everywhere except at s = 0 where it becomes infinite; hence δ(s) is highly singular. In particular, notice that the area under the δ function is unity:
approximately 0.04 for disks 10 mm in radius even at an ionic strength of KCl corresponding to the ionic strength of pure water; according to eq 25, the correction factor would be less than 1% in this case. Indeed it would be quite difficult even to observe the effect of surface conduction for a rotating disk. Surface conduction would reduce the measured streaming potential of a disk 1 mm in radius by a few percent at the ionic strength of pure water; it would become appreciable only at very low ionic strength for a disk 100 μm in diameter. Even if one could make a small enough sensor to get a measurement, say 10 μm radius, one would need to rotate the sample at tens of thousands of rpm to generate a measurable signal to test the theory. Figure 5 emphasizes this point. The correction factor for streaming potential measurements, based on capillary gap cells with two different gaps, is compared to the correction factor for a 10-mm-radius disk. The correction factor is substantial for thin gap cells and dilute solutions but is unity for the rotating disk. Physically, the bulk conductance in this geometry is always much larger than the surface conductance for practical disk diameters and electrolyte conductivities. No correction is required for surface conductivity in the rotating disk system for essentially all realizable systems.
∞
∫−∞ δ(s) ds = 1
(A2)
The δ function is the derivative of the Heaviside step function δ(s) =
d H (s ) ds
(A3)
If s has units, then δ(s) has units of s−1, which implies that the δ function is not the same in all coordinate systems.
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APPENDIX B: DERIVATION OF EQUATION 10 The expression for ohmic current density normal to the disk in the rotational elliptical coordinate system is7 iz(η) =
−K ∂ϕ aη ∂ξ
= ξ= 0
−K aη
∞
∑ BnP2n(η) M2n′(0) n=0
(B1)
Expressing the drain portion of the overall current density in the boundary condition is straightforward because it is a constant. Expressing the singular emission of current at the periphery in rotational elliptic coordinates requires development. We integrate to a nonzero radius and provide a function f(η) that adapts the δ function to the new coordinate system.
4. CONCLUSIONS The geometry of the rotating disk, when used for the determination of the zeta potential, offers a broad avenue for the ohmic return of surface current through the bulk solution from the periphery. The consequence of this low-resistance pathway is that surface conductivity in the rotating disk
∫ r →a 0 lim
ro
∫ r →a 0
2εγζ 2πr dr + lim
o
o
ro
f (η) δη(η − ηo)2πr dr = 0
(B2)
Because r dr = −a η dη, this can be written as 2
E
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∫ η →0 1 lim
ηo
Article
∫ η →0 1
− 4πεγζa2η dη + lim
o
ηo
(10) Newman, J. Current Distribution on a Rotating Disk below the Limiting Current. J. Electrochem. Soc. 1966, 113, 1235−1241.
f (η) δη(η − ηo)2π(− a2ηdη) = 0
o
(B3)
which becomes, after applying the definition of the δ function (note the sign reversal because the limits of integration are reversed from the definition of the δ function) and taking the simple limit on the left-hand side, 2πεγζa 2 + lim [2πa 2ηof (ηo)] = 0
(B4)
ηo → 0
Thus lim ηof (ηo) = −εγζ
(B5)
ηo → 0
As ηo asymptotically approaches zero, f (ηo) →
−εγζ ηo
(B6)
Therefore, f (η) = −εγζ/η around η = 0. Given that the source current enters the bulk solution essentially as a line source at the periphery of the disk, the boundary condition at the electrode can be expressed as −K aη
∞
∑ BnP2n(η) M2n′(0) = 2εγζ − n=0
εγζ δ(η) η
(B7)
One obtains eq 10 after a simple rearrangement of eq B7.
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AUTHOR INFORMATION
Corresponding Author
*E-mail:
[email protected]. Phone: (412) 268-3846. Author Contributions
Both authors contributed to this manuscript and have approved the final version. Notes
The authors declare the following competing financial interest(s): Dr. Sides is the founder of ZetaMetrix Inc. that makes and sells a scientific instrument based on principles presented in this contribution.
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REFERENCES
(1) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: New York, 1981. (2) O’Brien, R. W.; White, L. R. Electrophoretic Mobility of a Spherical Colloidal Particle. J. Chem Soc., Faraday Trans. 1978, 74, 1607−1626. (3) ISO 13099-1:2012. Colloidal Systems − Methods for ZetaPotential Determination. Part 1: Electroacoustic and Electrokinetic Phenomena; 2012 . (4) Dukhin, A. S.; Shilov, V. N.; Borkovskaya, Yu. Dynamic Electrophoretic Mobility in Concentrated Dispersed Systems. Cell Model. Langmuir 1999, 15, 3452−3457. (5) Lyklema, J. Fundamentals of Interface and Colloid Science; Academic Press: London, 1995; Vol. II. (6) Sides, P. J.; Newman, J.; Hoggard, J. D.; Prieve, D. C. Calculation of the Streaming Potential near a Rotating Disk. Langmuir 2006, 22, 9765−9769. (7) F.S. Lameiras, E. H.; Nunes, E. H. M. Calculation of the Streaming Potential near a Rotating Disk with Rotational Elliptic Coordinates. Port. Electrochim. Acta 2008, 26, 369−375. (8) Newman, J.; Thomas-Alyea, K. E. Electrochemical Systems, 3rd ed.; John Wiley & Sons: Hoboken, NJ, 2004. (9) Sides, P. J.; Hoggard, J. D. Measurement of the Zeta Potential of Planar Solid Surfaces by Means of a Rotating Disk. Langmuir 2004, 20, 11493−11498. F
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