10.626 Electrochemical Energy Systems Spring 2011 MIT, M. Z. Bazant Problem Set 5 – Electrokinetics and Electrochemical Kinetics Due: Lecture 35 1. General electro-osmotic slip formula. Consider a thin, flat, uniformly charged double layer subject to a tangential electric field �E = Exˆ, which results in an effective electro-osmotic slip velocity, us = ux(y = ∞), outside the double layer. Define the zeta potential ζ as a measure of the slip velocity per field in units of voltage via, εbζ us = − E ηb where εb and ηb are the permittivity and viscosity of the bulk solution. Let ψ0 be the potential of the no-slip plane y = 0 relative to the neutral bulk solution y = ∞. (a) For a quasi-equilibrium double layer, we can express dielectric saturation (reduced pemit-tivity in large fields due to aligning solvent dipoles) as ε(ψ) and the viscoelectric effect (thickening of the fluid in large fields or large charge density) as η(ψ), where ε(0) = εb and η(0) = ηb. Show that εbζ � ψ0 ε = dψ ηb 0 η (b) Derive ζ(ψ0) for the lattice-gas model of homework 4, problem 3 for a z : z electrolyte with ions of excluded volume v, assuming that the viscosity diverges at the maximum charge density, ηb v= 1 |ρ|η −ze with ε = εb =constant. Plot ζ vs. ψ0, and compare it to the Helmholtz-Smoluchowski formula, ζ = ψ0. 2. Electrokinetic energy conversion. Consider a quasi-2D nanochannel of height 2h, width w � h and length L � w, with parallel surfaces at x = ±h of fixed charge density qs per area. Assume the surface charge is small enough to use Debye-Huc¨ kel theory for the potential d2ψ λ2 D = ψdx2 where λD = κ−1 is the screening length. Let ε be the permittivity and η the viscosity (both constant). Let σ be the conductivity of a quasi-neutral reference solution in quasi-equilibrium with the nanochannel. Assume that all ions have the same diffusivity. (a) Derive the hydrodynamic conductance KP . (b) Derive the electrical conductance KE . (c) Derive the quasi-equilibrium potential profile ψ(x). (d) Derive the electro-osmotic conductance, KEO. (e) Derive the maximum efficiency of electrokinetic energy conversion, �max. Assuming �max � 1, obtain approximations for thin (κh � 1) and thick (κh � 1) double layers.3. Frumkin-Butler-Volmer kinetics. Consider the oxidation reaction, R O + e−, where →R is a neutral species and O is the cation in a dilute 1:1 electrolyte. The reaction occurs at the Stern plane within a diffuse double layer described by the Gouy-Chapman-Stern model, where δ = λS /λD. Assume symmetric Butler-Volmer kinetics, φS /2 φS /2j/e = kacRe Δ˜− kccOe−Δ˜where cO and cR are concentrations at the Stern plane and Δ φ˜S = eΔφS /kT is the dimen-sionless Stern-layer voltage. Let Δ φ˜D = eΔφD/kT be the dimensionless diffuse-layer voltage, which relates cO to its bulk value ¯cO outside the double layer. Let Δ φ˜= Δ φ˜D + Δ φ˜S be the dimensionless total voltage across the double layer. (a) What is the equilibrium total voltage, Δ φ˜eq, such that j = 0? (b) Let η˜ = Δ φ˜− Δφ˜eq be the overpotential across the full double layer, and ˜j = j/ekacR. Relate ˜j to η˜, δ and the dimensionless surface charge density q˜s = qs/(eλDc¯O). Show that the double layer acts like a diode in the GC limit δ = 0. (c) Write down (but do not solve) a set of equations to determine, q˜s(δ, Δφ˜).MIT OpenCourseWarehttp://ocw.mit.edu 10.626 / 10.462 Electrochemical Energy Systems Spring 2011 For information about citing these materials or our Terms of Use, visit: