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PCCP Paper mechanism responsible for the emf. If our theory is correct then there must be detectable signals of this high-frequency oscillation. Observation of the 0.1–1 GHz electromagnetic radiation, which is strongly absorbed by water, may be experi- mentally challenging. Other possible signals include fast resi- dual modulations of the battery’s output voltage (much faster than those reported in ref. 16 and 19) and the battery’s response when subject to ultrasound signals capable of reso- nantly driving the double layer’s oscillation. Moreover, as noted in Section IV, if the present model were extended to include mechanical elasticity of the capacitor plates and spatial inhomo- geneity of the charging processes, quantitative predictions of slower self-oscillations (in a range of frequency more accessible experimentally) might result. Our model is consistent with the established principles for the optimization of battery design, based on increasing the chemical potential difference at the anode and cathode, and increasing the capacitance of each half-cell. However, our dynamical treatment introduces a new consideration, namely the mechanical properties of the double layer, to which very little consideration has been given in the past. The theory that we have proposed here for battery half-cell as a dynamical engine might be generalizable to other chemically- active surfaces, including the catalytic systems considered in ref. 17 and 18. In those cases the work extracted from the external disequilibrium is used to drive a non-spontaneous chemical reaction, rather than to pump a macroscopic electrical current. The fact that some kinds of catalytic reaction are associated with micro-mechanical self-oscillations of the catalyst is already well established experimentally.17 Conflicts of interest There are no conflicts to declare. Appendix: Double-layer dynamics and self-oscillation In this Appendix we provide the details of the mathematical treatment of the LEC model for the battery, whose results are used in Section III C. It is convenient to introduce dimension- less variables x,y to characterize the deviation of capacitor plates from their equilibrium position and charge separation, respectively, so that X = X0(1 + x) and Q = Q0(1 + y), (A.1) where X0 and Q0 give zero force in eqn (7). We also parametrize not relevant to the battery, in which no macroscopic and time- varying magnetic field is present (as would be required by the Faraday-Maxwell law to give electric circulation).6,13 To date, no microphysical, quantitative description of the generation of the battery’s emf has been worked out. We believe that this theoretical blind spot arises from several factors. Firstly, the emf cannot be directly measured, monitored, or detected in a discharging battery. Experimental studies of batteries rely on discharge curves that relate the potential between the two terminals to the charge over time. Secondly, the emf – by its nature – is not an electrostatic phenomenon, and therefore lies beyond the usual theoretical framework of electrochemistry. That framework has proved powerful by making many useful and accurate predictions, but it neglects the dynamical nature of the operation of electrochemical cells. As the historian and philosopher of science Hasok Chang has noted correctly, ‘‘for anyone wanting a rather mechanical or causal story about how free electrons start getting produced and get moved about, the modern textbook theory is a difficult thing to apply.’’4 This situation is not unusual in physics and chemistry. Classical thermodynamics can be used to predict the efficiency of a steam engine, with no knowledge of the mechanical or dynamical properties of the moving parts. However, as we have argued here, to fully understand the battery’s emf we must consider such off-equilibrium dynamics. Our description of the battery’s emf is based on an active, non-conservative force that pumps the current in the circuit. This leads to an account that requires conceptual tools not ordinarily taught to either chem- istry or physics students, whose training in the dynamics of non-conservative systems is usually limited to stochastic fluc- tuations and the associated dissipation. Our own thinking in this subject has been influenced by recent developments in non-equilibrium thermodynamics, and especially by research in ‘‘quantum thermodynamics’’ seeking to understand the microphysics of work extraction by an open system coupled to an external disequilibrium.48 For a recent theoretical investigation in which the microphysics of the generation an emf by a triboelectric generator is considered in those terms, see ref. 49. In the present case of the battery, however, our treatment has been wholly classical. The observations of slow self-oscillations in certain battery configurations16,19 have clearly established that the electrochemical double layer at the electrode–electrolyte interface is a complex dynamical system, capable of generating an active cycle through the interplay of its chemical, electric, and mechanical properties. Together with the theoretical considerations, based on electro- dynamics and thermodynamics, detailed in this article, we believe that this makes a strong case that the generation of the emf by the battery must be understood as an active periodic process, like other instances of the pumping of a macroscopic current. The recent report of the response of the Li-ion battery to an external acoustic driving with a frequency of 0.1 GHz44 is consistent with our theory and, in our view, points towards the rapid double-layer dynamic that we have identified as the the capacitance as View Article Online 9436 | Phys. Chem. Chem. Phys., 2021, 23, 9428–9439 This journal is © the Owner Societies 2021 CðXÞ1⁄4C X0 1⁄4 C0 : 0X 1þx (A.2) (A.3) eqn (6) then takes the form 1 x€þgx_1⁄4O20 ð1þyÞ21þx; Open Access Article. Published on 23 March 2021. Downloaded on 6/26/2022 1:50:45 PM. 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