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PCCP Paper (i.e., cyclically), at the expense of the external disequilibrium. In many cases this work generation appears as the self-oscillation of a piston.1,26,27 The macroscopic kinetic energy of this self- oscillation can be used to pump a flow against an external potential, or along a closed path. More specifically, a self-oscillator is a system that can generate and maintain a periodic motion at the expense of a power source with no corresponding periodicity.28 Such a process is necessarily dissipative and requires a positive feed- back between the oscillating system and the action upon it of the external power source.1 Here we will describe the pumping of charge within a battery in terms of the self-oscillation of the electrochemical double layer of the half-cell. This is a direct application to a Gouy–Chapman-type model of the ‘‘leaking elastic capacitor’’ (LEC) model that has recently been worked out mathematically in ref. 13. That work, in turn, is closely related to the theory of the ‘‘electron shuttle’’ as an autonomous engine powered by an external disequilibrium in chemical potential.29–31 The passive half-cell can be described by the AC equivalent circuit model shown in Fig. 2.32,33 The capacitance C of the double layer is not fixed, but rather increases as the potential difference between the Helmholtz plane of ions and the elec- trode surface is increased. In the Gouy–Chapman model this is explained as the result of the diffuse layer of ions in the electrolyte becoming more compact as the applied potential is increased. It is well established that such a re-arrangement of charges at the solid–liquid interface can lead to dynamical instabilities.11,12 In this section we shall extend this into a dynamical description that can explain the pumping of charge that generates the battery’s emf. A Leaking elastic capacitor model In order to describe the dynamics of the active half-cell in a battery, we consider the general mathematical description of the LEC model introduced in ref. 13. Fig. 3 shows the equivalent electromechanical circuit for the LEC, consisting of the external source of energy (the voltage V0), a capacitance C, an external resistance Re in series with C, and internal resistance Ri in parallel with C. The capacitance C, as well as the resistances Re,i, Fig. 2 Equivalent AC circuit for the half-cell, in the case in which the impedance is dominated by the electron transfer resistance: Re is the electrolyte resistance, C is the double layer’s capacitance, and Ri is the double layer’s internal resistance. See ref. 32 and 33. emf, which remains almost constant up to the moment when the chemical fuel runs out. The emf in the battery is often equated to the voltage measured at the terminals, but this is a conceptual error, as the authors of ref. 2 underline. In open-circuit conditions, the potential Voc is indeed equal to the emf E, and the relation Voc 1⁄4 E (1) provides an accurate measurement of the emf. However, even the zero-current limit of the battery’s operation cannot be understood electrostatically, and the relation between emf and voltage at the terminals becomes more involved as the battery is operated away from open-circuit conditions. The mathematics of the emf as a form of pumping is discussed in detail in ref. 13. In Section III we propose a model for the physical origin of the non-electrostatic force that gives rise to the battery’s emf. As shown in Fig. 1(b), the current that a charged super- capacitor generates when connected to an external load R is not closed: there is a current I both through the load (given by a flow of electrons) and through the bulk of the electrolyte (given by diffusion of ions), but no current flows through the double layer at each electrode–electrolyte interface. This is equivalent to replacing the supercapacitor by two simple capacitors (each corresponding to one of the double layers) connected in series. On the other hand, as illustrated in Fig. 1(c), for the battery the circuit is closed by a ballistic flow of ions within the electrolyte and through the double layers. Electronics textbooks distinguish between active devices that can amplify the power that they receive from the circuit, and passive devices that cannot. In this classification, the supercapacitor, like the ordinary capacitor, is passive. Horowitz and Hill note that active devices ‘‘are distinguishable by their ability to make oscillators, by feeding from output signal back into the input,’’ i.e., to self-oscillate.25 The self-oscillations reported in ref. 16 and 19 can therefore be interpreted as evidence that the battery (considered as a circuit element, rather than as just an external power source) is active. The distinction between active and passive devices offers an instructive way of framing the key qualitative distinction between the battery and the supercapacitor. A passive device can consume free energy, but it cannot use it to perform sustained work or to pump a flow. An active system, on the other hand, uses some of the free energy that it consumes from an external source in order to generate an active, non- conservative force, which can be used to pump a flow against an external potential or to sustain a circulation.13 The emf corresponds to that non-conservative force per unit charge, integrated over the closed path of the current. We return to this specific issue in Section III E. III. Active double-layer dynamics Macroscopic engines generate an active, non-conservative force, via thermodynamically irreversible processes involving a positive feedback. This allows them to do work persistently View Article Online 9430 | Phys. Chem. Chem. Phys., 2021, 23, 9428–9439 This journal is © the Owner Societies 2021 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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