When immersed in a non-uniform electrolyte solution, a rigid charged sphere migrates toward higher or lower concentration of the electrolyte depending on the relative ionic mobilities and the charge borne by the sphere. This motion has a twofold origin: first, a macroscopic electrolyte gradient produces an electric field which acts on the charged sphere (electrophoresis); secondly, the electrolyte gradient polarizes the cloud of counterions surrounding the charged sphere by making the cloud thinner on the high-concentration side (chemiphoresis). In this paper, we compute the terminal velocity of a non-conductive sphere through a slightly non-uniform solution of a symmetrically charged binary electrolyte. The analysis proceeds through an expansion in the small parameter λ (defined as the ratio of the counterion-cloud thickness to the particle radius). Results to O(λ) are presented. The only property of the sphere's surface that affects the velocity is its zeta potential ζ when the electrolyte gradient vanishes; no information concerning the dependence of ζ upon ionic strength is needed. While the chemiphoretic effect always directs the particle toward higher electrolyte concentration, the electrophoretic contribution can move the particle in either direction depending on the sign of βζ, where β is a normalized difference in mobilities between cation and anion of the elecytrolyte; thus particle movement could be directed toward either higher or lower electrolyte concentration depending on the physical properties of the system. With slight algebraic rearrangement, our results are also applicable to conventional electrophoresis (particle motion in an applied electric field) and show excellent agreement with the numerical calculations of O'Brien & White (1978).
An analysis is presented for electrophoretic motion of a charged non-conducting sphere in the proximity of rigid boundaries. An important assumption is that κa → ∞, where a is the particle radius and κ is the Debye screening parameter. Three boundary configurations are considered: single flat wall, two parallel walls (slit), and a long circular tube. The boundary is assumed a perfect electrical insulator except when the applied field is directed perpendicular to a single wall, in which case the wall is assumed to have a uniform potential (perfect conductor). There are three basic effects causing the particle velocity to deviate from the value given by Smoluchowski's classic equation: first, a charge on the boundary causes electro-osmotic flow of the suspending fluid; secondly, the boundary alters the interaction between the particle and applied electric field; and, thirdly, the boundary enhances viscous retardation of the particle as it tries to move in response to the applied field. Using a method of reflections, we determine the particle velocity for a constant applied field in increasing powers of λ up to O(λ6), where λ is the ratio of particle radius to distance from the boundary. Ignoring the O(λ0) electro-osmotic effect, the first effect attributable to proximity of the boundary is O(λ3) for all boundary configurations, and in cases when the applied field is parallel to the boundaries the electrophoretic velocity is proportional to ζp − ζw, the difference in zeta potential between the particle and boundary.
Clustering of latex particles 4−10 μm in diameter during and after electrophoretic deposition of the particles onto flat electrodes has been reported by Böhmer (Langmuir 1996, 12, 5747). The particles interacted over length scales comparable to their size in the formative stages of the clusters. Combinations of two or more clusters already deposited approached each other to form larger agglomerates. A model based on electroosmotic flow about charged particles near surfaces is developed here to explain these observations. A charged, nonconducting particle near or on a flat conducting surface creates flow in the adjacent fluid due to electroosmosis about the particle's surface. Fluid is drawn laterally toward the particle near the electrode and pushed outward from the particle farther away from the electrode above the particle. Another particle near the electrode will be drawn toward the central particle by this convection. We first solve for the flow field about a single particle and then compute the rearrangement of neighboring particles in response to the flows. The clustering times for different initial configurations of sets of particles (e.g., regular versus irregular spacing) are calculated. The average clustering times for irregular configurations are greater than those for regular arrays. The qualitative and quantitative features of the experimental observations are captured by this model if the hindrance effect of the solid wall is taken into account. For example, the model correctly predicts the observed declustering (separation) of particles when the polarity of the electric field is reversed as well as the observed cluster-to-cluster motion.
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