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Paper PCCP Q: = k2Q + k1QR R: = (k3B)R k1QR (28) implement the well-known Lotka-Volterra dynamical system, leading to self-oscillating concentrations.46 In this case, slow chemical oscillations are superimposed on the fast self-oscillations of the ‘‘piston’’ described in Section III A. Note that, as the battery’s chemical fuel starts to run out, the concentration of B decreases and the stationary point (R0 = k2/k1, Q0 = k3B/k1) of the kinetic eqn (28) is shifted. The slow chemical oscillations of Q, R may therefore grow, which would be consistent with the behavior reported in ref. 16. To obtain useful work from such a chemical self-oscillation some coupling to a mechanical degree of freedom (analogous to the X of the LEC model) is needed.20 V. Discussion We have proposed a dynamic model of the electrode–electrolyte interface that can explain the pumping that gives rise to the battery’s emf. This required us to move beyond the electrostatic description on which most of the literature in both condensed- matter physics and electrochemistry is based. For this we have considered variations in time of the mechanical state and charge distribution of the double layer. In thermodynamic equilibrium, the fluctuations of the double layer would quickly average out, yielding no macroscopic current. But we have shown that an underlying chemical disequilibrium can, in the presence of a positive feedback between the mechanical deformation and the charging of the double layer, cause a coherent self-oscillation that can pump a current within the cell, charging the terminals and sustaining the current in the external circuit. The relevant feedback mechanism is intro- duced by the dependence of the rates r on X and Q in eqn (14). The general idea that we have advanced here, that the battery’s emf is generated by a rapid oscillatory dynamics at the electrode– electrolyte interface of the half-cell, was already suggested by Sir Humphry Davy (the founder of electrochemistry) in 1812: ‘‘It is very probable that [. . .] the action of the [solvents in the electrolyte] exposes continually new surfaces of metal; and the electrical equilibrium may be conceived in consequence, to be alternately destroyed and restored, the changes taking place in imperceptible portions of time.’’47 Although he lacked the conceptual tools to formalize this insight, which was therefore later lost, Davy understood that the action of the battery was at odds with a description in terms of time-independent potentials, as well as with a simple relaxa- tion to an equilibrium state. The very use of the term electromotive force (as distinct from the potential difference or voltage) points, in the context of the battery, towards an off-equilibrium, dynamical process that irreversibly converts chemical energy into electrical work, equal to the total charge separated across the two terminals times the potential difference between those terminals. In classical electrodynamics, emf is often equated to circulation of the electrical field, but this is pumping of current by the forced oscillation of the double layer, as described by eqn (24). IV. Comparison with experiment To estimate the oscillation frequency O0 we use the data from ref. 36: X0 B1030nm, Vd(Q0,X0) = 150 mV. (26) Putting e = 80 and r = 103 kg m3 into eqn (18) gives O0 C 0.1–1 GHz. This is much faster than the self- oscillations that have been experimentally observed so far, in the range of Hz to kHz. However, we have reason to believe that the double-layer of the battery half-cell can indeed oscillate coherently at this high frequency, as required by our model of charge pumping. It has been reported recently that operation of Li-ion batteries responds strongly to an external acoustic driving at 0.1 GHz.44 The present ability of experimenters to manipulate the electrode–electrolyte interface in Li-ion batteries using sound at these high frequencies also suggests that it might be possible to detect the double layer’s self-oscillation directly, by looking for the acoustic signal that it produces. The low frequency self-oscillations that have been reported experimentally must therefore be of a somewhat different nature to the mechanism described in Section III. A useful analogy is to the relationship between the fast dynamics that gives hydrodynamic pumping and the slow, parasitic self- oscillations that can affect the performance of such pumps. Those parasitic self-oscillations get their power from the fast pumping mechanism, but they have a different dynamics from that of the pumping itself.21 One possible source of the slow electrochemical self- oscillations are mechanical surface waves moving along the Helmholtz plane of the double layer. These may produce mod- ulations of the output voltage that are superimposed on the fast ‘‘pumping oscillations’’. Determining whether this is, in fact, the mechanism of some of the electrochemical self-oscillations that have been observed will require detailed numerical simulation of a model of the double-layer that incorporates a mechanical elasticity for the deformation of the Helmholtz plane away from its flat configuration, as well as spatial inhomogeneity of the charging process described by eqn (14).45 Alternatively, the very slow oscillations (with frequencies 1–104 Hz) could be of purely chemical nature. To illustrate their possible dynamics we can replace eqn (14) by a more complicated kinetic equation involving molecules Q and R with varying concentrations and molecules A and B provided by the chemical baths with fixed concentrations. Consider the following set of irreversible chemical reactions containing two autocatalytic reactions: Q + R-2Q; Q-A; R + B-2R, (27) with the rates k1, k2, k3. The corresponding kinetic equations View Article Online This journal is © the Owner Societies 2021 Phys. Chem. Chem. Phys., 2021, 23, 9428–9439 | 9435 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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