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Unformatted text preview: V. Electrostatics Lecture 25: Diffuse double layer structure MIT Student Last time we showed that whenever λ D L the electrolyte has a quasi-neutral “bulk” (or “outer”) region at the geometrical scale L , where there is very little mean charge density ρ = i z i ec i compared to the to- tal charge density c , or more precisely | ρ ˜ | = O ( ε 2 ) where ρ ˜ = ρ/ec ,ε = λ D /L 1. In order to satisfy electrostatic boundary conditions, how- ever, diffuse charge exists in thin quasi-equilibrium double layers (which are mathematical “boundary layers”). The ion profiles are approximately in thermal equilibrium ( µ i ≈ constant), even when there is a non-zero current of ﬂuid ﬂow, due to the small scale λ D L . [Note: the double layer can go out of equilibrium if c → at a limiting current, or a very fast transient can occur, e.g. high frequency impedance with ω ∼ D/λ 2 D .] 1 Poisson-Boltzmann Equation We start with an assumption of quasi-equilibrium, so that the chemical po- tential µ i =constant. We separate the electric potential φ into two parts: φ = φ + ψ , where φ is the (approximately) constant bulk potential and ψ is the part due to diffuse charge. From the quasi-equilibrium assumption, the concentrations and charge density are in equilibrium with the spatially vary- ing part ψ , so c i = eq c i ( ψ ) ,ρ = ρ eq ( ψ ). The generalized...
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This note was uploaded on 11/27/2011 for the course CHEMICAL E 20.410j taught by Professor Rogerd.kamm during the Spring '03 term at MIT.
- Spring '03