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PCCP Paper According to the calculations detailed in the Appendix, the necessary condition for self-oscillation of the electrochemical double layer (i.e., for positive feedback between X and Q), is (16) We assume that the concentrations [AB] and [B] in the double layer are kept constant by diffusion in the electrolyte, while [e] is dynamical. The change in [e] is accompanied by the corresponding variation of [A+], so as to maintain the net charge neutrality of the double layer. But the average of [A+], inside the effective volume associated with the reaction of eqn (10), is much larger than the variation of [e], because the double layer is sparse compared to the density of atoms and of ions in solution: an estimate for the Li-ion battery, using the parameters reported in ref. 36, gives a density B103q Å2, where q is the elementary charge. We may therefore approx- imate [A+] as constant in eqn (10). Stability of the equilibrium state for the double layer produced by the redox reaction implies that @ DGðX0; Q0Þ 1⁄4 @ DGðX0; Q0Þ 1⁄4 0: (12) @X @Q In order to obtain the engine dynamics that we shall associate with the pumping of charge inside the battery, we must consider an additional process not contemplated in eqn (10): A leakage of charge is needed to perturb the equili- brium state (X0,Q0) of the double layer and the corresponding potential difference Vd(X0,Q0). This leakage reduces Vd, inducing a chemical reaction that tends to restore equilibrium, in accordance with Le Chˆatelier’s principle. Under specific conditions that we will determine, the system then overshoots the equilibrium configu- ration and the double layer goes into a persistent oscillation. We will show that such a self-oscillation can give the pumping necessary to account for the generation of the battery’s emf. The charge of the double layer is Q 1⁄4 1⁄2e qA‘; (13) where A is the total surface area (an extensive parameter) and c is the effective width of the region containing the double layer in which the reaction takes place (an intensive parameter). Combining the chemical reaction described by eqn (10) with the leakage of the double layer gives us a dynamical equation for Q of a form very similar to eqn (5): 0ob1⁄4rðX;QÞX @lnrþðX;Q0Þ 0 0 0@X rðX;Q0Þ @ 1⁄4 koutX0@X ln kðX0; Q0Þ: X1⁄4X0 View Article Online Positivity of qXlnk follows from the fact that increasing X increases the electrostatic potential difference Vd(X,Q) = Q/C(X). This favors the reverse reaction in eqn (9), in which the positive ion A+ goes from the positively charged Helmholtz layer and into the negatively charged electrode, i.e., down in the potential difference Vd. Larger X therefore increases the reaction rate k in eqn (9). In particular, when b 4 12g; (17) the stationary solution becomes unstable and any small per- turbation will give rise to a self-oscillation, which in the linear regime has angular frequency pffiffiffiffiffiffiVdðX0;Q0Þ O0 1⁄4 ee0 pffiffiffiffiffi 2 (18) and exponentially growing amplitude. Eqn (18) is expressed in terms of experimental parameters characterizing the electro- chemical double layer, namely the equilibrium width of the double layer X0, the potential drop Vd(X,Q) = Q/C(X) at {X0,Q0} over the double layer, which is practically equal to the mea- sured potential (i.e., the average electrostatic potential over a complete oscillation cycle), the density of the electrolyte r, and its permittivity ee0. The condition of eqn (17) corresponds to the ‘‘Hopf bifurca- tion’’ of the dynamical system, at which the equilibrium becomes unstable due to the anti-damping of small oscillations.37 As the amplitude of such an oscillation increases, non-linearities become important so that eventually a limit-cycle regime is reached, giving a regular oscillation with steady amplitude.1,37 D Thermodynamic interpretation Note that, despite the mathematical similarity between eqn (5) and (14), the physics that they describe is different. In the electromechanical model of the LEC, based on eqn (5), all electric currents are driven by the external voltage source V0, and the mechanical energy of the self-oscillation is simply dissipated. On the other hand, in the case of the model for the battery’s half-cell, based on eqn (14), electric current can leave the system and circulate in an external circuit, pumped by the mechanical self-oscillations of the double layer. That is, part of the mechanical energy of the self-oscillation is dissi- pated and part of it is transformed into an electrical work W = EQ% , where E is the emf and Q% is the total charge driven around the closed circuit. with Q: = r(X,Q)Q + qr+(X,Q), rðX; QÞ 1⁄4 kðX; QÞ1⁄2B1⁄2Aþ þ kout; rþðX; QÞ 1⁄4 kþðX; QÞ1⁄2ABA‘: (14) (15) In eqn (15) the term kout describes the leakage of charge from the double layer due to an internal resistance and to consumption of current by an external load connected to the battery’s terminals. This kout is intensive (i.e., it does not depend on the size of the system, given by the area A). C Conditions for self-oscillation Note that the rates r in eqn (14) depend on both Q and X. This is more general than eqn (5) for the purely electromechanical system, but the basic mechanism of feedback-induced self- oscillation is qualitatively similar in both cases. 9432 | Phys. Chem. Chem. Phys., 2021, 23, 9428–9439 This journal is © the Owner Societies 2021 2rX0 Open Access Article. Published on 23 March 2021. Downloaded on 6/26/2022 1:50:45 PM. This article is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Licence.PDF Image | Dynamic theory battery electromotive force
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