Treating Electrostatic Shielding at the Surface of Silica as Discrete Siloxide‚Cation Interactions Mathai Mammen, Jeffrey D. Carbeck, Eric E. Simanek, and George M. Whitesides* Contribution from the Department of Chemistry and Chemical Biology, HarVard UniVersity, 12 Oxford Street, Cambridge, Massachusetts 02138 ReceiVed NoVember 4, 1996 X Abstract: This work examines the influence of ions in solution on electroosmosis inside a fused silica capillary using capillary electrophoresis; it thereby examines “shielding” at charged interfaces. Theories are reviewed that model ionic solutions as continuous dielectrics: the nonlinear form of the Poisson-Boltzmann equation gives rise to the simplified, more commonly used Debye-Hu ¨ckel (DH) equation. Capillary electrophoresis (CE) is used to measure the rate of electroosmotic flow as a function of the concentration of different monovalent and divalent cations in aqueous solution. These data are used to test three specific predictions of DH theory: this theory does not describe these data adequately. The central reason behind the inadequacy of DH theory here is its inability to account for details at the level of individual ions other than by mean-field electrostatics: that is, chemical characteristics of ionssfor example, polarizability, hydrated size, energy of hydration, ability to coordinate other ions by chelationsare not accounted for. A model (the “dissociation model”) is described that treats the interactions between cations in solution and negatively charged groups on a surface in terms of discrete association equilibria with characteristic dissociation constants, K d eff . CE is then used as a tool to measure values of K d eff for different cations. These dissociation constants follow patterns that are consistent with ones that are familiar from studies in solution. Introduction We have measured the shielding of siloxide ions (SiO - ) at the surface of a fixed silica capillary as a function of the concentration and nature of ions in solution by measuring electroosmotic (EO) mobility, µ EO . We compare these experi- mental values of µ EO to those predicted using Debye-Hu ¨ckel (DH) theory under conditions where DH theory is a valid approximation to the more complete Poisson-Boltzmann (PB) theory. We organize this paper conceptually into three parts. In the first part, we briefly review electroosmosis and examine the theory that leads to its description using both PB theory and DH theory. In the second part, we describe capillary electrophoresis as a convenient tool to examine three specific predictions of DH theory. We demonstrate that, in many cases, the experimental data are not described well by DH theory. We examine these three predictions over a range of concentrations that is consistent with the assumptions of DH theory, that is, concentrations at which the electrostatic potential at the charged surface of the capillary is less than ∼25 mV. In the third part, we introduce an empirical model for the electrostatic potential that incorporates ionic detail and describes the data well over the entire concentration range examined (0.005-500 mM). This second (dissociation) model treats the siloxide‚cation interactions (SiO - M + ) as discrete, noncovalent association equilibria, with a characteristic dissociation constant, K d eff (much as for an acid- base dissociation). Many specific chemical properties of an ion may influence its interaction with another ion: the value of K d eff is a chemically convenient way of summarizing the collective influence of these interacting properties into a single number. The relative magnitudes of K d eff are consistent with chemical intuition. Theoretical Models for Electroosmosis Electroosmosis. Electroosmosis is the motion of liquid driven by a combination of a charge imbalance at a solid- liquid interface and an electrical field applied to that liquid (Figure 1). The liquid has finite viscosity and resists flow. The velocity of electroosmotic flow, ν EO (m s -1 ) results from a balance between electrical forces and viscous forces. Elec- troosmotic mobility, µ EO (m 2 V -1 s -1 ) is defined experimentally as the value of ν EO divided by the strength of the electrical field E (V m -1 ). This paper examines the influence of ions in solution on the value of µ EO . In this work, the solid is the interior, negatively charged surface of a hollow cylinder (capillary) composed of fused silica and the solution is aqueous and ionic. The charge imbalance at the surface of the capillary gives the solution a net charge, and the charged solution moves by electroosmosis in the uniform electrical field applied within the capillary. We first describe in qualitative terms how we will examine the properties of the solution that affect µ EO (and ν EO ). In the section that follows this one, we will introduce some limited mathematical detail to make our qualitative arguments clear. As we will show later, changes in µ EO are directly proportional to changes in the “effective charge” at the surface of the capillary; because of charge neutrality, the effective charge at the surface is equal to the effective charge of the solution. This charged solution moves in bulk in a uniform electrical field. We examine the properties of the solution that influence this charge imbalance. Methods based on Poisson-Boltzmann (PB) theory do not directly or explicitly predict this charge imbal- ance. 1 Rather, PB theory is used to relate the electrostatic potential at the surface of the capillary (Ψ o , volts) to the concentration of ions in solution ([X ( ] i , mol/L), the temperature (T, K), the dielectric constant (ǫ, unitless), and the unshielded (naked) charge density (σ o , C/m 2 ) at the surface. 2-5 This * Author to whom correspondence should be addressed. X Abstract published in AdVance ACS Abstracts, April 1, 1997. (1) Healy, T. W.; White, L. R. AdV. Colloid Interface Sci. 1978, 9, 303- 345. (2) Smith, P. E.; Pettitt, B. M. J. Phys. Chem. 1994, 98, 9700-9711. (3) Rashin, A. A.; Bukatim, M. A. Biophys. J. 1994, 51, 167-192. (4) Honig, B.; Sharp, K.; Yang, A. S. J. Phys. Chem. 1993, 97, 1101- 1109. (5) Rice, C. L.; Whitehead, R. J. Phys. Chem. 1965, 69, 4017-4024. 3469 J. Am. Chem. 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