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Unformatted text preview: V. Electrostatics Lecture 26: Compact Part of the Double Layer MIT Student 1 Double-layer Capacitance 1.1 Stern Layer As was discussed in the previous lecture, the Gouy-Chapman model pre dicts unphysically large counter-ion concentrations at large potentials. As an example, at 2.5 V, which is a typical battery electrode potential, counter- ion concentrations predicted by Gouy-Chapman would be on the order of e 100 10 43 , which is clearly impossible. The central problem is that the Boltzmann distribution upon which Guoy-Chapman is based only on the bal ance between electrostatic attraction of the counter-ions towards the charged surface and diffusion of them away from it. This approach does not take into account near range effects due to close packing of ions. For zeta po tentials much larger than the thermal voltage, the electrostatic attraction becomes the dominant force in this model, and the infinitesimally small ions pile up infinitesimally close to the surface. In 1924, Stern came up with a model that addressed this problem. He proposed that a finite molecular size could be introduced into the model by requiring that there was some distance of closest approach to the surface. This distance was defined as h s , and the region contained in that distance is commonly called the compact layer. The physical meaning of this model is not precisely defined, but it has been suggested that it is caused by a sphere of solvated water molecules attached to the ions. At sufficiently high voltages, this solvation layer has been shown to break down, leading to ions adsorbing onto the surface at an inner Helmholtz plane. This structure is illustrated in figure 1. As was discussed in lecture 25, the dielectric constant is known to de crease for aqueous solutions at high electric fields due to the alignment of the 1 Lecture 26: Compact part of the double layer 10.626 (2011) Bazant Courtesy of Wiley. Used with permission. Figure 1: Figure from Newman and Thomas-Alyea. Electrochemical Sys tems, 3rd Edition . 2004. The struction of a Stern Model double layer. The inner and outer Helmholtz plane act as planar capacitors in series with the diffuse layer. water molecule dipoles. Rather than implementing a non-uniform dielectric constant, this model assumes that within the compact layer, the dielectric constant is some constant bulk . This approximation is quite reason able since the compact layer is generally only one or two...
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- Spring '03