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  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  potential of a surface is defined as the electric potential at the plane of shear relative to the adjacent, electrically neutral, bulk solution. 1  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  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 previously 2,3 seemed to validate the analysis with respect to the dependence of measured streaming potential on radial position, axial position, and rotation speed, but the  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 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. * Corresponding author. Address: Department of Chemical Engineering, Carnegie Mellon University, Pittsburgh, PA 15213. TEL: 412 268 3846. FAX: 412 268 2183. E-mail: ps7r@andrew.cmu.edu. † 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  potential equal to . 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. 9765 Langmuir 2006, 22, 9765-9769 10.1021/la061041x CCC: $33.50 © 2006 American Chemical Society Published on Web 10/07/2006