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V. Electrostatics Lecture 27a: Diffuse charge at electrodes Notes by MIT Student We have talked about the electric double structures and corresponding models describing the ion and potential distribution in the double layer. Now we start to investigate the effect of double layer on the reaction kinetics, especially how the diffuse ionic charges affect the Faradaic half-cell reaction kinetics. 1. Frumkin Correction to Butler Volmer Kinetics First we assume that Faradaic reaction takes place at a reaction plane, which is roughly the same as stern plane. The charge carriers need to hop over this stern layer barrier (with a thickness of one solvent molecule) and pick up an electron to finish the Faradic reaction. This region is basically a solvation shell of the electrode surface. Remember that the Butler Volmer (B-V) kinetics is a molecular scale equation, thus, in order to introduce the Frumkin correction, we apply the B-V equation only across this compact layer and the reaction rate, where the reaction occurs. Then the B-V equation can be described as: (1 ) / /0100[]:ln( )SSSne kT ne kTROAeqSeq OORR R e ek a aprefactor Roverpotential acrosssternlayerakTne a            Combining all these equations we will get: (1 )( ) ( )0[]OOSSSSne nekT kTROARR a e a e       Note that for every charged species, activities or concentrations are all evaluated locally at the stern plane, which differs from the corresponding values in the neutral bulk electrolyte as we used in BV equation before. Now, in order to relate it to the bulk concentrations, we need a double layer model. For example, if the diffuse part of the double layer satisfies dilute solution approximation, then for ionic species i with charge zieLecture 27a: Diffuse charge at Electrodes 10.626 (2011) Bazant 2 /( 0)( 0)( 0) ( 0)iz e kToi x iixoix bulk D xc c ewherec isconcentrationat stern planec isbulkconcentration      /iz e kTe is the Boltzman factor for charge across the diffuse layer. This makes0,eqSRand thus and Rall depend on zeta potential or surface charge. At surface x=0, ( 0) ( 0)x D x    . This was first noted by Frumkin in the 1930s, but Zeta potential is always related to charge, and or sqare usually fitted and assumed constants. However, as we seen in the previous lecture,  must depend on the current and should be predicted by the model. 2. Complete mathematical model based on Stern boundary condition After we split double layer into Stern/compact layer and diffuse layer, potential boundary condition at the interface of these two layers is always missing in electrochemical system modeling. To complete the mathematical model, we have to introduce a reasonable boundary condition. To an approximation, since no charge is located within the compact layer, the electric field within it can be treated as constant, so if we denote x = 0 to be the Stern plane, our boundary condition can be written in terms of our effective compact layer thickness S as follows based on a linear extrapolation of the potential across the compact layer.  ( 0) ( 0)S e x S x Sn n E             where nis the outward normal pointing from solution to the electrode SShSis the effective thickness of the stern layer This makes reaction kinetics dependent on normal electric field EApply Gauss’s law, we will figure out that it also depends on surface charge densitysq. ()solution metalsolution metal sssolutionsolutionn E E qqnE       Then we relate sqtoS: S Sn ESqssolutionhSqsSLecture 27a: Diffuse charge at Electrodes 10.626 (2011) Bazant 3 A. Helmholtz limit Remember the dimensionless group SDis the ratio of stern layer effective thickness and Debye screening length. When goes to infinity SD      , and we recover the usual BV kinetics using bulk concentrations without any Frumkin correction for diffuse charge. But surface charge is not independent. The model predicts surface charge self-consistently by qsSShS, qseqSSeqhS B. Gouy-Chapman limit When goes to zero, DS      , and the flux boundary condition loses its explicit voltage dependence. Consider the reaction below: iziiis M ne Flux boundary condition is: (1 )( ) ( )ooSSSSne neiokT kTii R OAsRn F s R a e a e          o R r OR k c k c (Chang-Jaffe BC) (1 )ROSOSneokToAneokTroARk r ewhereRk r e  In the derivation, it seems that voltage no longer matters for the reaction kinetics, but actually it does. The current or reaction rate still depends implicitly on the total double layer voltage via the concentrations,ROcc. With,ork k consts this is Chang-Jaffe’ boundary condition (just standard chemical kinetics applied at the stern plane)Lecture 27a: Diffuse charge at Electrodes 10.626 (2011) Bazant 4 Example: Consider a cathode reaction M M e(M=Li, H, etc) Assume dilute solution and quasi-equilibrium, cis the stern plane concentration, and oc is the bulk concentration. ///()De kTooD c bulko e kT e kTO R rc c c eIR k c k c e e is FrumkincorrectioneA            This just looks like BV equation with =1 At equilibrium R=0, we get ln( ) ln( )ooOreq SO R Rk c akT kTe k c e a     This is the Nernst equation. ///0()(1 ) (1 )eqo e kToRo R RoRe kT e kToRkcIR k c k c eeA k cI k c eA e I e           Based on this current expression, we see that in the Gouy-Chapman limit, double layer acts like an ideal semiconductor diode (like p-n junction behavior), and pass current in one direction only. This can be further explained from microscopic point of view in figure 2 and 3. Fig. 1 In Gouy-Chapman limit double layer only allows current in one directionLecture 27a: Diffuse


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MIT 10 626 - Diffuse charge at electrodes

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