Dynamical theory for the battery’s electromotive force Robert Alicki,1, ∗ David Gelbwaser-Klimovsky,2, † Alejandro Jenkins,1, 3, ‡ and Elizabeth von Hauff4, § 1International Centre for Theory of Quantum Technologies (ICTQT), University of Gdańsk, 80-308, Gdańsk, Poland 2Physics of Living Systems, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 3Laboratorio de F́ısica Teórica y Computacional, Escuela de F́ısica, Universidad de Costa Rica, 11501-2060, San José, Costa Rica 4Department of Physics and Astronomy, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam, The Netherlands (Dated: First version, 30 Oct. 2020; last revision, 25 Mar. 2021. Published in Phys. Chem. Chem. Phys. 23, 9428 (2021)) We propose a dynamical theory of how the chemical energy stored in a battery generates the electromotive force (emf). In this picture, the battery’s half-cell acts as an engine, cyclically ex- tracting work from its underlying chemical disequilibrium. We show that the double layer at the electrode-electrolyte interface can exhibit a rapid self-oscillation that pumps an electric current, thus accounting for the persistent conversion of chemical energy into electrical work equal to the emf times the separated charge. We suggest a connection between this mechanism and the slow self-oscillations observed in various electrochemical cells, including batteries, as well as the enhance- ment of the current observed when ultrasound is applied to the half-cell. Finally, we propose more direct experimental tests of the predictions of this dynamical theory. I. INTRODUCTION Macroscopic devices capable of delivering sustained power, such as motors, turbines, generators, and combus- tion engines, have moving parts whose cyclical dynamics govern the transformation of potential energy from an external source into useful work.1 In contrast, the oper- ation of solid-state and electrochemical devices such as photovoltaic cells, thermoelectric generators, and batter- ies, relies on an electromotive force (emf) in order to convert chemical potential into electrical power. Volta introduced the concept of emf in 1798 to refer to the mechanism that causes and maintains the separation of opposite charges in a battery, despite the electrostatic attraction between them.2 Volta’s discoveries triggered intense debates on the seat and mechanism of the bat- tery’s emf. These involved the leading physicists and chemists of the 19th and early 20th centuries, including Biot, Davy, Ohm, Berzelius, A. C. Becquerel, Faraday, Helmholtz, Kelvin, Maxwell, Heaviside, Lodge, Ostwald, Nernst, and Langmuir.3 Rather that reaching a decisive resolution, this scientific controversy petered out with the practical advances in electrochemistry and the shift in focus resulting from the quantum revolution.4,5 We know now that the battery’s emf arises because a chemical reaction “yields more energy than it costs to buck the [general electric] field.”6 However, even this cor- rect observation fails to provide a realistic dynamical ac- count of the origin and nature of the emf responsible for the power output of the device during discharge.2,7 We believe that this is because the theoretical treatment of electrochemical cells is still largely based on electrostat- ics. Such a description can account for the flow of current through the external circuit (from high to low electro- static potential), as in the case of a discharging capaci- tor. But, though it provides the correct energy budget for the conversion of chemical into electrical energy, it cannot describe the dynamics responsible for the flow of current within the battery that generates and maintains the electric potential difference between the terminals. Physicist Peter Heller suggested replacing the term emf by electromotive pump (emp), to describe any un- derlying physical mechanism that promotes the circu- lation of electric current around a closed path.7 It is well understood that the battery’s emf results from the action of “surface pumps” at the electrode-electrolyte interfaces within the device, and that the energy con- sumed by this pumping comes from the redox chemi- cal reactions at those interfaces.7,8 However the often cited explanation of this pumping in terms of an “equiv- alent force” in diffusion theory has been characterized as merely “heuristic”.2 Describing the pumping of charge at the electrode-electrolyte interface using the equations for diffusion with fixed-concentration boundary conditions presents serious difficulties due to the indefiniteness of the velocity of Brownian particles.9 Ultimately, we are still lacking a clear quantitative description of the micro- physics at the electrode-electrolyte interface.7 In this article we propose that the pumping of charge that generates the emf of a battery is associated with dynamics-in-time of the double layer at the electrode- electrolyte interface. This double layer incorporates a mechanical elasticity10 or “squishiness”,11,12 that is com- patible with an electrochemical Gouy–Chapman-type model. The mutual coupling between the mechanical and electrical degrees of freedom can cause a dynam- ical instability leading to a self-oscillation. The self- oscillating double layer acts as an internal piston for the half-cell, thereby pumping current against the average electrostatic potential.13 In our picture, the battery’s emf is analogous to the “head rise” produced by a hydraulic pump. In the me- chanical engineering literature, the head rise times the flow rate is equated to the external power consumed by 2 the pump, minus the pump’s internal dissipative losses. In an ideal hydrodynamic pump (much like in a battery), the pressure is nearly independent of the flow, up to some maximum flow rate beyond which the pump cannot run properly.14 We predict that the pumping of charge within the battery must be associated with high-frequency oscil- lations that actively pump current at the double layer. Much slower self-oscillations have been observed in vari- ous electrochemical systems,15 including modified Li-ion batteries.16 We interpret those slow oscillations as sec- ondary effects of the battery’s pumping dynamics, effects that may become evident towards the end of the discharg- ing process as the battery’s operation becomes diffusion- limited and the steady pumping regime is destabilized. Similar slow self-oscillations have also been reported in the catalytic reactions17,18 and in self-charging electro- chemical cells with a ferroelectric glass electrolyte.19 Some authors have proposed exploiting the mechanical energy of electrochemical self-oscillations to perform use- ful work,20 but the experimentally observed electrochem- ical oscillations have not previously been connected to the pumping dynamic responsible for the battery’s emf. Analogous self-oscillations, with frequencies well below that of the pump’s running, are often seen in hydraulic pumps (where they can pose serious problems for the de- vice’s proper functioning).21 II. CAPACITORS VS. BATTERIES Like a battery, a “supercapacitor” has an electrochem- ical double layer at each of the two electrode-electrolyte interfaces.22 The formation of the double layer is a spon- taneous process that can be understood in terms of an equilibration of the electrochemical potential.23 The gain in chemical energy (which is linear in the number of separated charges) is balanced by the electrostatic en- ergy accumulated (which is quadratic). The presence of this double layer —with the corresponding difference in the electrostatic potential between the electrode and the electrolyte— does not by itself imply an emf.24 Indeed, comparison of the discharge curves for the supercapaci- tor and the battery reveals a clear qualitative difference between the two devices. Figure 1(a) shows a capacitor’s linear relation between voltage and integrated current (blue curve), which can be entirely understood in terms of electrostatics. If the capacitor is charged, and its terminals are subsequently connected to an external load, a current flows between the terminals at the expense of the discharging potential V . The capacitor cannot drive current along a closed path, and therefore has no emf. A rechargeable elec- trochemical battery, on the other hand has a non-linear (and, therefore, non-electrostatic) relation between volt- age and integrated current, shown by the red curve in Fig. 1(a). This reflects the presence of an emf within the battery, which keeps the voltage difference between (a) – – – – – + – + + + + ––– electrolyte double layer double layer R – – + : electron : positive ion : negative ion Key no current – ––– – + + +++ no current II (b) – – – – IR– + – + + + + ––– – ––– – + + +++ I (c) FIG. 1: Image (a) shows the voltage V versus integrated cur- rent ∆Q ≡ ∫ t 0 I(t′)dt′ of an ideal supercapacitor (blue curve) and battery (red curve). The other images schematically show the electrical current I in a circuit connected to: (b) a super- capacitor and (c) a battery. The voltage V is the electrostatic potential difference between the + and − terminals of each device. the terminals nearly constant until the chemical reservoir becomes depleted.22 The charging of a supercapacitor involves only re- versible separation of charges, i.e., the storage of electro- static potential energy. We will argue that the charging of a battery, on the other hand, produces a “chemical fuel”, which is then irreversibly consumed during dis- charge. The battery’s half-cell acts like a chemical en- gine, powering a charge pump at the electrode-electrolyte interface that drives the circulation of current in the ex- ternal circuit. The average “pressure” produced by the “pump’s piston” acting on the electron fluid is the source of the battery’s 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 condi- tions, the potential Voc is indeed equal to the emf E , and 3 the relation Voc = E (1) provides an accurate measurement of the emf. However, even the zero-current limit of the battery’s operation can- not be understood electrostatically, and the relation be- tween emf and voltage at the terminals becomes more in- volved 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 Sec. 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 su- percapacitor 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 re- placing 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 de- vices that can amplify the power that they receive from the circuit, and passive devices that cannot. In this classi- fication, 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 Refs. 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 of- fers an instructive way of framing the key qualitative dis- tinction 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 Sec. 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 (i.e., cyclically), at the expense of the external disequilibrium. In many cases this work generation appears as the self-oscillation of 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 re- sistance. See Refs. 32,33. 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 feedback 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 dou- ble layer of the half-cell. This is a direct application to a Gouy–Chapman-type model of the “leaking elastic ca- pacitor” (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 au- tonomous engine powered by an external disequilibrium in chemical potential.29–31 The passive half-cell can be described by the AC equiv- alent 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 electrode surface is increased. In the Gouy– Chapman model this is explained as the result of the dif- fuse layer of ions in the electrolyte becoming more com- pact as the applied potential is increased. It is well estab- lished 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 de- scription 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- ure 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 4 FIG. 3: Equivalent circuit for the leaking elastic capacitor (LEC). Adapted from Ref. 13. in series with C, and internal resistance Ri in parallel with C. The capacitance C, as well as the resistances Re,i, are presumed to be a functions of the distance X between the capacitor plates. Here V0 represents a simple electrostatic potential. Let Q be the total charge accumulated by the ca- pacitor. According to Kirchhoff’s rules, the current Ie through resistor Re is equal to Ie = Q̇+ Ii, (2) where Ii is the current through resistor Ri, and V0 = IeRe + IiRi. (3) The voltage drop across the capacitor is Q/C = IiRi. (4) Combining Eqs. (2), (3), and (4) yields a dynamical equa- tion for the total charge Q(t) stored in the LEC of Fig. 3: Q̇ = − [ 1 C(X)Re(X) + 1 C(X)Ri(X) ] Q+ V0 Re(X) . (5) Assuming that one of the plates of the capacitor is fixed while the other is free to move, Newton’s second law of motion yields a dynamical equation for X(t) of the form Ẍ + γẊ = 1 M f(X,Q). (6) In Eq. (6) γ is a viscous damping coefficient, M is the mass of the moving capacitor plate, and the force f(X,Q) is the sum of electrostatic attraction and molecular re- pulsion. For simplicity and definiteness, we model the repulsion with a phenomenological parameter σ > 0: f(X,Q) = − Q2 2XC(X) + σ X . (7) The choice of the repulsive force in (7) as ∼ X−1 is mo- tivated by a simple picture of the pressure exerted by a gas of ions confined within the double layer, as in the Gouy-Chapman model.34 Note that it is consistent with the fact that if Q → 0 then the equilibrium separation X diverges, since in that case the Helmholtz layer of ions dissolves away. The phenomenological parameter σ in Eq. (7) can be taken from the value of X0 corresponding to the equilibrium width of the double layer. Our con- clusions are essentially independent of the details of this non-linear repulsive term. For recent work on a detailed modeling of the mechanical properties of the electrochem- ical double layer, see Ref. 35. This simple electromechanical system can display a range of qualitatively different dynamical behaviors, in- cluding fixed-point solutions, regular limit cycles, and chaotic behavior; the details of this are discussed in Ref. 13. The physical origins of self-oscillations in this model can be qualitatively understood without detailed mathematical analysis. A change in X implies changes to the internal and external resistances Ri,e. These mod- ulate the rate with which current Q flows into or out of the capacitor. If the charge Q is modulated in phase with −Ẋ, the electrostatic attraction between the two plates will increase when the LEC is contracting, and decrease when the LEC is expanding. Such a positive feedback between X and Q will anti-damp the mechanical oscil- lation. If this feedback-induced anti-damping exceeds the viscous damping γ, the equilibrium configuration be- comes unstable and small oscillations about the equilib- rium configuration grow exponentially in amplitude until they are limited by the non-linearity in Eq. (7). The damping coefficient can be decomposed as γ = γdiss + γload, (8) where γdiss gives the approximately constant contribu- tion from internal (viscous) dissipation and γload models energy loss of the oscillating double layer due to pump- ing of electric current in the external circuit (load). This distinction will be discussed in detail in Sec. III D. B. Chemical engine To describe the dynamics of the double layer within the battery we must replace Eq. (5), which was based on a purely electromechanical model, by a suitable chemical kinetic equation. For definiteness and simplicity, consider a redox reaction of the form AB B +A+ + e−, (9) as in a Li-ion battery. The forward reaction in Eq. (9) increases the absolute value of Q by injecting electrons into the “electrode plate” and positive ions into the “elec- trolyte plate” (corresponding to the Helmholtz plane of ions). According to the law of mass action, d[e−] dt = k+ [AB]− k− [B][A+][e−], (10) where [ · ] represents the concentration of the correspond- ing species in Eq. (9), within the effective volume near 5 the electrode surface in which the reactions take place. Meanwhile, k± are the “rate constants”, which in this case depend on the state of the double layer. These rates are related to the Gibbs free energy of reaction, ∆G(X,Q), by the formula lnKeq(X,Q) = ln k+(X,Q) k−(X,Q) = −∆G(X,Q) RT + c, (11) where Keq is the equilibrium constant, T is the tempera- ture and R the universal gas constant, and c is a constant depending on the units of concentration. 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 dou- ble layer. But the average of [A+], inside the effective volume associated with the reaction of Eq. (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 pa- rameters reported in Ref. 36, gives a density ∼ 10−3q/Å 2 , where q is the elementary charge. We may therefore ap- proximate [A+] as constant in Eq. (10). Stability of the equilibrium state for the double layer produced by the redox reaction implies that ∂ ∂X ∆G(X0, Q0) = ∂ ∂Q ∆G(X0, Q0) = 0. (12) In order to obtain the engine dynamics that we shall as- sociate with the pumping of charge inside the battery, we must consider an additional process not contemplated in Eq. (10): A leakage of charge is needed to perturb the equilibrium state (X0, Q0) of the double layer and the corresponding potential difference Vd(X0, Q0). This leak- age reduces Vd, inducing a chemical reaction that tends to restore equilibrium, in accordance with Le Châtelier’s principle. Under specific conditions that we will deter- mine, the system then overshoots the equilibrium config- uration and the double layer goes into a persistent oscil- lation. 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 = [e−] · qA`, (13) where A is the total surface area (an extensive parame- ter) and ` 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 Eq. (10) with the leakage of the double layer gives us a dynamical equation for Q of a form very similar to Eq. (5): Q̇ = −r−(X,Q)Q+ qr+(X,Q), (14) with r−(X,Q) = k−(X,Q)[B][A+] + kout, r+(X,Q) = k+(X,Q)[AB]A`. (15) In Eq. (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 Eq. (14) depend on both Q and X. This is more general than Eq. (5) for the purely electromechanical system, but the basic mechanism of feedback-induced self-oscillation is qualitatively similar in both cases. According to the calculations detailed in the Appendix, the necessary condition for self-oscillation of the electro- chemical double layer (i.e., for positive feedback between X and Q) is 0 < b = r−(X0, Q0)X0 ∂ ∂X ln [ r+(X,Q0) r−(X,Q0) ] X=X0 = koutX0 ∂ ∂X ln k−(X0, Q0). (16) Positivity of ∂X ln k− follows from the fact that in- creasing X increases the electrostatic potential difference Vd(X,Q) = Q/C(X). This favors the reverse reaction in Eq. (9), in which the positive ion A+ goes from the pos- itively 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 Eq. (9). In particular, when b > 1 2 γ (17) the stationary solution becomes unstable and any small perturbation will give rise to a self-oscillation, which in the linear regime has angular frequency Ω0 = √ εε0 Vd(X0, Q0)√ 2ρX2 0 (18) and exponentially growing amplitude. Equation (18) is expressed in terms of experimental parameters character- izing the electrochemical double layer, namely the equi- librium 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 measured potential (i.e., the average electrostatic potential over a complete os- cillation cycle), the density of the electrolyte ρ, and its permittivity εε0. 6 The condition of Eq. (17) corresponds to the “Hopf bi- furcation” of the dynamical system, at which the equilib- rium becomes unstable due to the anti-damping of small oscillations.37 As the amplitude of such an oscillation in- creases, non-linearities become important so that even- tually 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 Eqs. (5) and (14), the physics that they describe is differ- ent. In the electromechanical model of the LEC, based on Eq. (5), all electric currents are driven by the exter- nal 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 Eq. (14), electric current can leave the system and cir- culate 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 dissipated and part of it is transformed into an electrical work W = E Q̄, where E is the emf and Q̄ is the total charge driven around the closed circuit. Let us first consider this process of transformation of chemical energy into an emf from the point of view of thermodynamics. The total energy of the LEC can be expressed as E = 1 2 MẊ2 + U(X,Q), f(X,Q) = − ∂ ∂X U(X,Q), (19) with time derivative Ė = −(γdiss + γload)MẊ2 + Vd(Q,X)Q̇. (20) Equation (20) can be interpreted in terms of the first law of thermodynamics. Taking the system of interest to be the battery’s half-cell, we have that dE = δQ− δW + µdN, (21) with internal energy E, heat flow to the environ- ment −Q̇ = γdissMẊ2, and electric power output Ẇ = γloadMẊ2. We identify the electrochemical poten- tial µ in Eq. (21) with qVd(Q,X), and the quantity of matter N with Q/q. It should be stressed that the two damping coefficients γdiss and γload, which appear in Eqs. (8) and (20), are qualitatively different from a thermodynamic point of view: γdiss represents the dissipation of mechanical en- ergy into the disordered motion of the microscopic com- ponents of the environment in which the double layer is immersed, while γload describes the transfer of energy from the self-oscillating double layer into the pumping of a coherent, macroscopic current that carries no en- tropy. The fluctuation-dissipation theorem therefore ap- plies only to γdiss, a point that should be born in mind if FIG. 4: Chemical-engine cycle represented in the (µ,N)- plane. the presentmodel were extended to incorporate thermal noise via a Fokker-Planck equation. The second law of thermodynamics is satisfied because heat is dissipated into the environment (δQ < 0) at con- stant temperature T , so that Ṡ = −Q̇/T > 0. Integrat- ing Eq. (21) over a complete cycle of the system’s ther- modynamic state we obtain that W0 ≡ ∮ δW < ∮ Vd(Q,X) dQ = ∮ µdN, (22) where W0 is the useful work generated by one limit cycle of the self-oscillation. The right-hand side of Eq. (22) is the area contained within a closed trajectory on the (µ,N)-plane, as shown in Fig. 4. This curve is the ther- modynamic cycle of the half-cell considered as a chemical engine. Equation (22) implies that sustained work extraction (W0 > 0) by an open system coupled to an external chem- ical disequilibrium requires the system to change its state in time in such a way that µ varies in phase with N . This is a particular instance of what was called the generalized “Rayleigh-Eddington criterion” (after the physicists who clearly formulated this principle for heat engines) in Ref. 38. A clear example of this principle for a microscopic chemical engine is provided by the electron shuttle.29–31 The work W0 extracted by the cycle can then be used to to pump electric charge Q̄ against the time-averaged electric potential difference at the half-cell, which corre- sponds to the generation of an emf E = W0 Q̄ . (23) Note that the modulation of µ with respect to N , re- quired by the Rayleigh-Eddington criterion, is possible only if the thermodynamic cycle (which in this case cor- responds to the self-oscillation of the double layer) is slow compared to the time-scale of the chemical reactions that control the value of µ at each point within the cycle. For instance, an automobile can run because it takes in pris- tine fuel at high chemical potential and expels burnt fuel at low chemical potential.1 Combustion must therefore 1 The internal combustion engine is usually conceptualized as a 7 proceed quickly compared to the period of the motion of engine’s pistons. If the time taken by the combustion were comparable to the period of the piston, µ would not vary effectively with dN in Eq. (22) and little or no net work could be done by the engine. The self-oscillation of the double layer is also the pump- ing cycle, which converts mechanical into electrical work, as we shall discuss in more detail in Sec. III E. This pump- ing must therefore be slow with respect to the redox reac- tions from which the battery ultimately takes the energy to generate the electrical work. This conceptual distinc- tion between the fast chemical reaction and the slower pumping is, in our view, the key missing ingredient in all previous theoretical treatments of the battery. E. Current pumping Having described the extraction of mechanical work by the double layer’s self-oscillation, we proceed to consider how that work can be used to pump an electrical current, thereby generating the battery’s emf. As shown in Ref. 13, the instantaneous power that an irrotational electric field E(t, r) delivers to a current density J(t, r) contained in a volume V is P = ∫ V E · J d3r = − ∫ V φ ∂ρ ∂t d3r, (24) where φ(t, r) is the potential (such that E = −∇φ) and ρ(t, r) is the charge density (such that ∇ · J = −∂ρ/∂t). In an ordinary capacitor, electrical charge comes out (∂ρ/∂t < 0) of the positively-charged plate at poten- tial φ+, while charge enters (∂ρ/∂t > 0) the negatively- charged plate at potential φ−. The instantaneous power is therefore P = V · I > 0, (25) where V = φ+ − φ− and I is the integral of |∂ρ/∂t| over the complete volume of either plate. A similar analysis applies to the discharging of the two double layers in the supercapacitor. The case of the battery is more subtle, because it can generate P > 0 without appreciable accumulation or de- pletion of charge anywhere in the circuit. Purcell ar- gued that this apparent contradiction is resolved by the fact that the battery’s operation involves chemical re- actions, which must be described in terms of quantum mechanics.6 Indeed, the average flow of matter inside the heat engine, with the air in the cylinder as working substance. But it is also possible to consider it as a chemical engine, with the fuel-air mixture as working substance. Of course, in the latter analysis, Eq. (22) gives only a loose upper bound on the extracted work because of the large amount of heat (and therefore entropy) that the engine dumps into the environment when it expels the burnt fuel. Helmholtz layer negative electrode – + + electrolyte I I – – + – + – – + + + + – – + : electron : positive ion : negative ion Key + – – + + + – – oxidation reaction– – FIG. 5: On the left: during the compression of the double layer, positive ions are squeezed out of the double layer and into the bulk of the electrolyte, giving them ballistic rather than diffusive motion. The Coulomb interaction then drives free electrons away from the interface and into the conduct- ing terminal, maintaining charge balance. On the right: no pumping of current occurs during the expansion of the dou- ble layer. The ions and electrons pumped away during the compression phase are replenished by the oxidation reaction. battery can be described in terms of discharging chemi- cal potentials,39 which represent an underlying quantum physics. But the details of how this chemical energy is converted into electrical work, in a thermodynamically irreversible way and without contradicting the laws of classical electrodynamics, have not been adequately clar- ified in the literature. It is well known that charges can be accelerated by pe- riodic oscillations of φ and ∂ρ/∂t, as is done in a modern particle accelerator.40 As we shall see, the self-oscillation of the double layer described in Sec. III modulates ∂ρ/∂t in phase with −φ, allowing net electrical work to be per- formed over a complete period of the oscillation. When the double layer contracts (i.e., Ẋ < 0), the Helmholtz layer is moving towards the electrode, and therefore along a direction of decreasing potential φ. The value of ∂ρ/∂t is negative in the region that the Helmholtz layer is moving out of (where the voltage is higher) and positive in the region that the Helmholtz layer is moving into (where the voltage is lower). Equa- tion (24) therefore implies that P > 0 during the con- traction phase. This reflects the fact that the ion plane is moving along the internal electric field, and is therefore being accelerated by it. On the other hand, when the double layer expands (i.e., Ẋ > 0), the ion plane is moving into a region that is screened from the internal electric field. Thus, the region where ∂ρ/∂t is negative is at a voltage only slightly lower than the region in which it is positive. This allows the net work (i.e., the integral of P in Eq. (24) over a full period of the double layer’s self-oscillation) to come out positive, which corresponds to a sustained pumping of the current. Intuitively, we may describe the pumping at the neg- atively charged terminal in the following terms: During the contraction phase, the positive ions in the double 8 layer are “squeezed”, giving them a ballistic motion that allows some of them to pass through the Helmholtz layer and escape into the neutral bulk of the electrolyte, thus generating an active discharge current. The current is rectified because the ions can pass through the Helmholtz layer but not penetrate the solid electrode. The genera- tion of a net current by this oscillation is similar to the operation of a “valveless pulsejet” engine.41 The Coulomb interaction between the positive ions in the electrolyte and the electrons in the electrode en- forces charge balance, causing an equivalent current to be drawn in from the external circuit and into the electrode during the squeezing phase. This equilibration of charge happens very quickly compared to the double layer’s self- oscillation, and it directs the flow of electrons in the ex- ternal circuit. During expansion, the interior of the double layer is replenished with positive ions and the electrode is re- plenished with electrons by the oxidation reaction that consumes the battery’s “chemical fuel” in a thermody- namically irreversible way. This sets up the double layer for the next pumping cycle. The full process is illustrated in Fig. 5. Note that if the positive ions are injected by the oxidation reaction when the double layer is expanded (so that Vd, and therefore the chemical potential µ for the ions, is high) and ballistically expelled when the double layer is contracted (so that µ = qVd is low), Eq. (22) im- plies that a net positive work can be done to pump the current I. An analogous pumping may occur in the other half- cell. Note, however, that pumping at only one of the half-cells may be sufficient to produce an emf. Further details on the pumping dynamics probably depend on the specific battery configuration being considered. From a microphysical point of view, this active trans- port of charges in an asymmetric potential subjected to a coherent, periodic modulation, is similar to the math- ematical model of an artificial Brownian motor.42 In the battery, it is the mechanical self-oscillation of the double layer that produces the periodic modulation of the po- tential for the ions. Thermodynamically speaking, that modulation is the external work that drives the pumping and which is converted into the electrostatic energy of battery’s charged terminals, minus some internal dissi- pation. There is ample experimental evidence from “sonoelec- trochemistry” that the application of ultrasound to an electrochemical double layer can significantly enhance the electrical current through that layer.43 This effect has usually been explained as resulting from the ultra- sonic driving causing the Nernst diffusion layer at the electrode-electrolyte interface to become thinner, but in our view it may more plausibly be interpreted as result- ing from the pumping of current by the forced oscillation of the double layer, as described by Eq. (24). IV. COMPARISON WITH EXPERIMENT To estimate the oscillation frequency Ω0 we use the data from Ref. 36: X0 ∼ 10 – 30 nm, Vd(Q0, X0) = 150 mV. (26) Putting ε = 80 and ρ = 103 kg/m3 into Eq. (18) gives Ω0 ' 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 re- sponds strongly to an external acoustic driving at 0.1 GHz.44 The present ability of experimenters to manip- ulate the electrode-electrolyte interface in Li-ion batter- ies 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 re- ported experimentally must therefore be of a somewhat different nature to the mechanism described in Sec. III. A useful analogy is to the relationship between the fast dy- namics 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 pro- duce modulations of the output voltage that are superim- posed 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 Eq. (14).45 Alternatively, the very slow oscillations (with frequen- cies 1 – 10−4 Hz) could be of purely chemical nature. To illustrate their possible dynamics we can replace Eq. (14) by a more complicated kinetic equation involv- ing 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 irre- versible chemical reactions containing two autocatalytic reactions: Q+R→ 2Q, Q→ A, R+B → 2R, (27) with the rates k1, k2, k3. The corresponding kinetic equa- 9 tions Q̇ = −k2Q+ k1QR, Ṙ = (k3B)R− k1QR, (28) implement the well-known Lotka-Volterra dynamical sys- tem, leading to self-oscillating concentrations.46 In this case, slow chemical oscillations are superim- posed on the fast self-oscillations of the “piston” de- scribed in Sec. 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 Eq. (28) is shifted. The slow chemical oscil- lations of Q,R may therefore grow, which would be con- sistent 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 electro- chemistry is based. For this we have considered varia- tions in time of the mechanical state and charge distri- bution of the double layer. In thermodynamic equilib- rium, 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 sus- taining the current in the external circuit. The relevant feedback mechanism is introduced by the dependence of the rates r± on X and Q in Eq. (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 continu- ally new surfaces of metal; and the electri- cal equilibrium may be conceived in conse- quence, to be alternately destroyed and re- stored, the changes taking place in impercep- tible 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 descrip- tion in terms of time-independent potentials, as well as with a simple relaxation to an equilibrium state. The very use of the term electromotive force (as dis- tinct from the potential difference or voltage) points, in the context of the battery, towards an off-equilibrium, dynamical process that irreversibly converts chemical en- ergy into electrical work, equal to the total charge sepa- rated across the two terminals times the potential differ- ence between those terminals. In classical electrodynam- ics, emf is often equated to circulation of the electrical field, but this is 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 mea- sured, 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 be- yond the usual theoretical framework of electrochemistry. That framework has proved powerful by making many useful and accurate predictions, but it neglects the dy- namical nature of the operation of electrochemical cells. As the historian and philosopher of science Hasok Chang has noted correctly, “for anyone wanting a rather me- chanical or causal story about how free electrons start getting produced and get moved about, the modern text- book 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 ef- ficiency of a steam engine, with no knowledge of the me- chanical 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 cur- rent in the circuit. This leads to an account that requires conceptual tools not ordinarily taught to either chemistry or physics students, whose training in the dynamics of non-conservative systems is usually limited to stochastic fluctuations and the associated dissipation. Our own thinking in this subject has been influenced by recent developments in non-equilibrium thermody- namics, and especially by research in “quantum ther- modynamics” 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 bat- tery configurations16,19 have clearly established that the electrochemical double layer at the electrode-electrolyte interface is a complex dynamical system, capable of gen- erating an active cycle through the interplay of its chem- 10 ical, electric, and mechanical properties. Together with the theoretical considerations, based on electrodynamics 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 pe- riodic 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 mechanism responsible for the emf. If our theory is correct then there must be detectable sig- nals of this high-frequency oscillation. Observation of the 0.1 – 1 GHz electromagnetic radiation, which is strongly absorbed by water, may be experimentally challenging. Other possible signals include fast residual modulations of the battery’s output voltage (much faster than those reported in Refs. 16,19) and the battery’s response when subject to ultrasound signals capable of resonantly driv- ing the double layer’s oscillation. Moreover, as noted in Sec. IV, if the present model were extended to in- clude mechanical elasticity of the capacitor plates and spatial inhomogeneity of the charging processes, quanti- tative 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 increas- ing the chemical potential difference at the anode and cathode, and increasing the capacitance of each half-cell. However, our dynamical treatment introduces a new con- sideration, namely the mechanical properties of the dou- ble 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 Refs. 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 Acknowledgments We thank Luuk Wagenaar and Lotte Schaap for fruit- ful discussions and critical questions. AJ also thanks Esteban Avendaño, Diego González, Mavis Montero, and Roberto Urcuyo for educating him on electrochem- ical double layers. RA was supported by the In- ternational Research Agendas Programme (IRAP) of the Foundation for Polish Science (FNP), with struc- tural funds from the European Union (EU). DG-K was supported by the Gordon and Betty Moore Founda- tion as a Physics of Living Systems Fellow (grant no. GBMF45130). AJ was supported by the Polish National Agency for Academic Exchange (NAWA)’s Ulam Pro- gramme (project no. PPN/ULM/2019/1/00284). EvH was supported by the research programme ENW XS (grant no. OCENW.XS.040), financed by the Dutch Re- search Council (NWO). EvH and RA also gratefully ac- knowledge the support of the Freiburg Institute of Ad- vanced Study (FRIAS)’s visitors’ program during the first stages of this collaboration. ∗ Electronic address: robert.alicki@ug.edu.pl † Electronic address: dgelbi@mit.edu ‡ Electronic address: alejandro.jenkins@ucr.ac.cr § Electronic address: e.l.von.hauff@vu.nl 1 See, e.g., A. Jenkins, “Self-oscillation”, Phys. Rep. 525, 167 (2013) [arXiv:1109.6640 [physics.class-ph]], and refer- ences therein. 2 For an excellent historical and conceptual review of this subject, see R. N. Varney and L. H. Fisher, “Electromotive force: Volta’s forgotten concept”, Am. J. Phys. 48, 405 (1980). 3 For a modern historical survey of this extremely long- running debate, see H. Kragh, “Confusion and Contro- versy: Nineteenth-Century Theories of the Voltaic Pile”, in Nuova Voltiana: Studies in Volta and His Times, vol. I, eds. F. Bevilacqua and L. Fragonese (Milan: Hoepli, 2000), pp. 133–167. 4 H. Chang, “How Historical Experiments Can Improve Sci- entific Knowledge and Science Education: The Cases of Boiling Water and Electrochemistry”, Sci. & Educ. 20, 317 (2011) 5 H. Chang, “Who cares about the history of science?”, Notes Rec. 71, 91 (2017) 6 E. Purcell and D. J. Morin, Electricity and Magnetism, 3rd ed., (Cambridge, UK: Cambridge U. P., 2013), p. 127 7 W. M. Saslow, “Voltaic cells for physicists: Two surface pumps and an internal resistance”, Am. J. Phys. 67, 574 (1999) 8 R. Baierlein, “The elusive chemical potential”, Am. J. Phys. 69, 423 (2001) 9 A. Singer, Z. Schuss, B. Nadler and R. S. Eisenberg, “Mem- oryless control of boundary concentrations of diffusing par- ticles”, Phys. Rev. E 70, 061106 (2004) 10 The term “elastic capacitor” was first used, in a biophys- ical context, in J. M. Crowley, “Electrical Breakdown of Bimolecular Lipid Membranes as an Electromechanical In- stability”, Biophys. J. 13, 711 (1973) 11 M. B. Partenskii and P. C. Jordan, “‘Squishy capaci- tor’ model for electrical double layers and the stability of charged interfaces”, Phys. Rev. E 80, 011112 (2009) 12 M. B. Partenskii and P. C. Jordan “Relaxing gap capaci- tor models of electrified interfaces”, Am. J. Phys. 79, 103 mailto:robert.alicki@ug.edu.pl mailto:dgelbi@mit.edu mailto:alejandro.jenkins@ucr.ac.cr mailto:e.l.von.hauff@vu.nl https://doi.org/10.1016/j.physrep.2012.10.007 https://doi.org/10.1016/j.physrep.2012.10.007 https://doi.org/10.1119/1.12115 http://ppp.unipv.it/Collana/Pages/Libri/Saggi/NuovaVoltiana_PDF/sei.pdf https://doi.org/10.1007/s11191-010-9301-8 https://doi.org/10.1007/s11191-010-9301-8 https://doi.org/10.1098/rsnr.2016.0042 https://doi.org/10.1119/1.19327 https://doi.org/10.1119/1.1336839 https://doi.org/10.1103/PhysRevE.70.061106 https://dx.doi.org/10.1016%2FS0006-3495(73)86017-5 https://doi.org/10.1103/PhysRevE.80.011112 https://doi.org/10.1119/1.3490647 11 (2011) 13 R. Alicki, D. Gelbwaser-Klimovsky and A. Jenkins, “Leaking elastic capacitor as model for active matter”, arXiv:2010.05534 [physics.class-ph], to be published in Phys. Rev. E 14 C. E. Brennen, Hydrodynamics of Pumps (Cambridge: Cambridge U. P., 2011), ch. 2 15 M. T. M. Koper, “Non-linear phenomena in electrochem- ical systems”, J. Chem. Soc., Faraday Trans. 94, 1369 (1998) 16 D. Li, Y. Sun, Z. Yang, L. Gu, Y. Chen and H. Zhou, “Electrochemical oscillation in Li-ion batteries”, Joule 2, 1265 (2018) 17 R. Imbihl and G. Ertl, “Oscillatory kinetics in heteroge- neous catalysis”, Chem. Rev. 95, 667 (1995) 18 M. V. F. Delmonde, L. F. Sallum, N. Perini, E. R. Gonza- lez, R. Schlögl and H. Varela, “Electrocatalytic Efficiency of the Oxidation of Small Organic Molecules under Oscil- latory Regime”, J. Phys. Chem. C 120, 22365 (2016) 19 M. H. Braga, J. E. Oliveira, A. J. Murchison and J. B. Goodenough, “Performance of a ferroelectric glass electrolyte in a self-charging electrochemical cell with neg- ative capacitance and resistance”, Appl. Phys. Rev. 7, 011406 (2020) 20 A. Isakova and K. Novakovic, “Oscillatory chemical reac- tions in the quest for rhythmic motion of smart materials”, Eur. Polym. J. 95, 430 (2017) 21 See Ref. 14, ch. 8. 22 B. E. Conway, Electrochemical Supercapacitors, (New York: Kluwer Academic / Plenum Publishers, 1993) 23 For recent theoretical work on this subject, in the specific context of the lithium-ion battery, see, e.g., J. Lück and A. Latz, “Modeling of the electrochemical double layer and its impact on intercalation reactions”, Phys. Chem. Chem. Phys. 20, 27804 (2018). 24 This has caused confusion in the pedagogical literature, especially because the spontaneous generation of the dou- ble layer meets Volta’s original definition of the emf as a mechanism that causes and maintains the separation of op- posite charges. See, e.g., D. Roberts, “How batteries work: A gravitational analog”, Am. J. Phys. 51, 829 (1983). Our point here is that both supercapacitors and batteries de- velop double layers, but only the battery can drive a closed circulation of current. 25 P. Horowitz and W. Hill, The Art of Electronics, 3rd ed. (Cambridge: Cambridge U. P., 2015), p. 71 26 P. Le Corbeiller, “The non-linear theory of the mainte- nance of oscillations,” J. Inst. Electr. Engrs. 79, 361 (1936) 27 P. Le Corbeiller, “Theory of prime movers”, in Non- Linear Mechanics, eds. K. O. Friedrichs, P. Le Corbeiller, N. Levinson and J. J. Stoker, (Providence: Brown U., 1943), pp. 2.1–2.18 28 A. A. Andronov, A. A. Vitt and S. È. Khăıkin, Theory of Oscillators, ed. W. Fishwick, (Mineola, NY: Dover, 1987 [1966]), p. 200 29 L. Y. Gorelik et al., “Shuttle Mechanism for Charge Trans- fer in Coulomb Blockade”, Phys. Rev. Lett. 80, 4526 (1998) [arXiv:cond-mat/9711196] 30 C. W. Wächtler, P. Strasberg, S. H. L. Klapp, G. Schaller and C. Jarzynski, “Stochastic thermodynamics of self- oscillations: the electron shuttle”, New J. Phys. 21, 073009 (2019) [arXiv:1902.08174 [cond-mat.stat-mech]] 31 C. W. Wächtler, P. Strasberg and G. Schaller, “Pro- posal of a Realistic Stochastic Rotor Engine Based on Electron Shuttling”, Phys. Rev. Appl. 12, 024001 (2019) [arXiv:1903.07500 [cond-mat.stat-mech]] 32 C. H. Hamann, A. Hamnett and W. Vielstich, Electrochem- istry, 2nd ed., (Weinheim: Wiley-VCH, 2007), sec. 5.2.2 33 E. von Hauff, “Impedance Spectroscopy for Emerging Pho- tovoltaics”, J. Phys. Chem. C 123, 11329 (2019) 34 S. L. Carnie and G. M. Torrie, “The Statistical Mechanics of the Electrical Double Layer”, Adv. Chem. Phys. 56, 141 (1984) 35 C. W. Monroe, “Mechanics of the Ideal Double-Layer Ca- pacitor”, J. Electrochem. Soc. 167, 013550 (2020) 36 M. Favaro, B. Jeong, P. N. Ross, J. Yano, Z. Hussain, Z. Liu and E. J. Crumlin, “Unravelling the electrochemi- cal double layer by direct probing of the solid/liquid inter- face”, Nat. Comm. 7, 12695 (2016) 37 See, e.g., S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., (Boulder: Westview Press, 2014), sec. 8.2. 38 R. Alicki, D. Gelbwaser-Klimovsky, and A. Jenkins, “A thermodynamic cycle for the solar cell”, Ann. Phys. (NY) 378, 71 (2017) [arXiv:1606.03819 [cond-mat.stat-mech]] 39 For such a treatment, in the particular case of the Li-ion battery, see K. E. Thomas, J. Newman, and R. M. Dar- ling, “Mathematical Modeling of Lithium Batteries”, in Advances in Lithium-Ion Batteries, eds. W. van Schalk- wijk and B. Scrosati (New York: Kluwer Academic, 2002), pp. 345–92. 40 E. O. Lawrence and M. S. Livingston, “The Production of High Speed Light Ions Without the Use of High Voltages”, Phys. Rev. 40, 19 (1932) 41 A. Jenkins, “Irreversibility in an ideal fluid”, Am. J. Phys. 82, 1040 (2014) [arXiv:1301.3915 [physics.flu-dyn]] 42 P. Hänggi and F. Marchesoni, “Artificial Brownian motors: Controlling transport on the nanoscale”, Rev. Mod. Phys. 81, 387 (2009) [arXiv:0807.1283 [cond-mat.stat-mech]] 43 R. G. Compton, J. C. Eklund, and F. Marken, “Sonoelec- trochemical Processes: A Review”, Electroanalysis 9, 509 (1997) 44 A. Huang, H. Liu, O. Manor, P. Liu and J. Friend, “En- abling Rapid Charging Lithium Metal Batteries via Sur- face Acoustic Wave-Driven Electrolyte Flow”, Adv. Mater. 32, 1907516 (2020) 45 On such a description, see Sec. VI in Ref. 13. 46 N. S. Goel, C. Maitra and E. W. Montroll, “On the Volterra and Other Nonlinear Models of Interacting Pop- ulations,” Rev. Mod. Phys. 43, 231 (1971) 47 H. Davy, Elements of Chemical Philosophy, part I, vol. I, (London: J. Johnson and Co., 1812), p. 170. Quoted in Ref. 3. 48 See, e.g., R. Alicki and R. Kosloff, “Introduction to quan- tum thermodynamics: History and prospects”, in Ther- modynamics in the Quantum Regime, eds. F. Binder et al., (Cham: Springer, 2019), pp. 1–33 [arXiv:1801.08314 [quant-ph]], and references therein. 49 R. Alicki and A. Jenkins, “Quantum theory of tri- boelectricity”, Phys. Rev. Lett. 125, 186101 (2020) [arXiv:1904.11997 [cond-mat.mes-hall]] http://brennen.caltech.edu/INTPump/pumbook.pdf https://doi.org/10.1039/A708897C https://doi.org/10.1016/j.joule.2018.03.014 https://doi.org/10.1016/j.joule.2018.03.014 https://doi.org/10.1021/cr00035a010 https://doi.org/10.1021/acs.jpcc.6b06692 https://doi.org/10.1063/1.5132841 https://doi.org/10.1063/1.5132841 https://doi.org/10.1016/j.eurpolymj.2017.08.033 https://doi.org/10.1039/c8cp05113e https://doi.org/10.1119/1.13128 https://doi.org/10.1049/PWS.1936.0030 https://doi.org/10.1103/PhysRevLett.80.4526 https://doi.org/10.1088/1367-2630/ab2727 https://doi.org/10.1103/PhysRevApplied.12.024001 https://pubs.acs.org/doi/10.1021/acs.jpcc.9b00892 https://doi.org/10.1002/9780470142806.ch2 https://doi.org/10.1149/1945-7111/ab6b04 https://doi.org/10.1038/ncomms12695 https://doi.org/10.1016/j.aop.2017.01.003 https://doi.org/10.1007/0-306-47508-1_13 https://doi.org/10.1103/PhysRev.40.19 https://doi.org/10.1119/1.4890941 https://doi.org/10.1103/RevModPhys.81.387 https://doi.org/10.1002/elan.1140090702 https://doi.org/10.1002/adma.201907516 http://dx.doi.org/10.1103/RevModPhys.43.231 https://books.google.com/books?id=d6Y5AAAAcAAJ https://doi.org/10.1007/978-3-319-99046-0_1 https://doi.org/10.1103/PhysRevLett.125.186101 12 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 Sec. III C. It is convenient to introduce dimensionless 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 Eq. (7). We also parametrize the capacitance as C(X) = C0 X0 X = C0 1 + x . (A.2) Equation (6) then takes the form ẍ+ γẋ = −Ω2 0 [ (1 + y)2 − 1 1 + x ] , (A.3) where Ω2 0 = Q2 0 2MX2 0C0 (A.4) is the square of the angular frequency Ω0 for small oscillations about equilibrium (x = 0, y = 0). The capacitance of the LEC at equilibrium can be expressed in terms of surface area A, plate separation X0, and electric permittivity εε0 of the electrolyte: C0 = εε0A X0 . (A.5) This leads to a useful expression for the frequency Ω0 in terms of the potential drop Vd(X,Q) = Q/C(X) and the effective density ρ = M/(AX0): Ω2 0 = εε0V 2 d (X0, Q0) 2ρX4 0 , (A.6) which corresponds to Eq. (18) in the main text. The density ρ should be of the same order of magnitude as the electrolyte density. To determine the condition for the onset of self-oscillations (the “Hopf bifurcation”), it is enough to consider linear perturbations about equilibrium.37 The linearized version of Eq. (A.3), valid for |x|, |y| � 1, is ẍ = −γẋ− Ω2 0(x+ 2y). (A.7) Inserting the lowest order expansions of the rates in Eq. (15), r±(X,Q) ' r± + [∂Xr±](X −X0) + [∂Qr±](Q−Q0), (A.8) and the equilibrium relation r+ r− = Q0 q (A.9) (in obvious shorthand notation) into the kinetic Eq. (14), we arrive at the linearized equation ẏ = −Γy + bx, (A.10) with Γ = r−(X0, Q0) { 1−Q0 ∂ ∂Q [ ln r+(X0, Q) r−(X0, Q) ] Q=Q0 } (A.11) 13 and b = r−(X0, Q0)X0 ∂ ∂X [ ln r+(X,Q0) r−(X,Q0) ] X=X0 . (A.12) For r± given by Eq. (15), and using Eq. (11), we have ∂X ln r+ r− = ∂X ln k+ − ∂X ln r− = ∂X ln k+ k− + ∂X ln k− r− = −∂X∆G RT + ∂Xk− k− − ∂Xr− r− . (A.13) Equation (12) implies that ∂X∆G = 0. Thus we have ∂X ln r+ r− = r−∂Xk− − k−∂Xr− k−r− = kout ∂Xk− k−r− = kout ∂X ln k− r− . (A.14) Inserting Eq. (A.14) into Eq. (A.12) we obtain b = koutX0∂X ln k−, (A.15) which corresponds to Eq. (16) in the main text. In order to find an approximate solution to the linearized Eqs. (A.7) and (A.10), we assume that the damping rate γ and the reaction rates κ± are small compared to Ω0. Then we can expect a solution of the form x(t) = eµt cos(Ω1t), y(t) = Aeµt cos(Ω1t+ φ) (A.16) with Ω1 ' Ω0 and |µ| � Ω0. Inserting Eq. (A.16) into Eqs. (A.7) and (A.10), and comparing the coefficients of the functions eµt cos(Ω0t) and eµt sin(Ω0t), one obtains four equations in four unknowns: µ,Ω1, A, and φ. One can simplify this system of equations by putting Ω1 = Ω0 and neglecting small contributions like µ/Ω0, γ/Ω0, etc. This procedure leads to the final approximate expression µ ' b− γ 2 (A.17) and the corresponding approximate threshold for the emergence of self-oscillations: b > γ 2 . (A.18) Introduction Capacitors vs. batteries Active double-layer dynamics Leaking elastic capacitor model Chemical engine Conditions for self-oscillation Thermodynamic interpretation Current pumping Comparison with experiment Discussion Acknowledgments References Double-layer dynamics and self-oscillation