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Paper PCCP (A.14) (A.15) In order to find an approximate solution to the linearized eqn (A.7) and (A.10), we assume that the damping rate g and the reaction rates k are small compared to O0. Then we can expect a solution of the form x(t) = emtcos(O1t), y(t) = Aemtcos(O1t + f) (A.16) with O1 C O0 and |m| { O0. Inserting eqn (A.16) into eqn (A.7) and (A.10), and comparing the coefficients of the functions emtcos(O0t) and emtsin(O0t), one obtains four equations in four unknowns: m,O1,A, and f. One can simplify this system of equations by putting O1 = O0 and neglecting small contribu- tions like m/O0,g/O0, etc. This procedure leads to the final approximate expression where eqn (12) implies that qXDG = 0. Thus we have @ lnrþ 1⁄4r@Xk k@Xr 1⁄4kout@Xk X r kr kr 1⁄4 kout@X ln k: r Inserting eqn (A.14) into eqn (A.12) we obtain 2 O0 Q02 1⁄4 2MX 2C (A.4) O02 1⁄4 ee0Vd2ðX0;Q0Þ; 2rX02 (A.6) 00 View Article Online is the square of the angular frequency O0 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 ee0 of the electrolyte: C0 1⁄4 ee0A: (A.5) X0 This leads to a useful expression for the frequency O0 in terms of the potential drop Vd(X,Q) = Q/C(X) and the effective density r 1⁄4 M=ðAX0Þ: b = koutX0qXln k, which corresponds to eqn (16) in the main text. which corresponds to eqn (18) in the main text. The density r 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 per- turbations about equilibrium.37 The linearized version of eqn (A.3), valid for |x|,|y| { 1, is x€ 1⁄4 gx_ O02ðx þ 2yÞ: (A.7) Inserting the lowest order expansions of the rates in eqn (15), r(X,Q) C r + [qXr](X X0) + [qQr](Q Q0), (A.8) m ’ b 2g and the corresponding approximate threshold for the emergence and the equilibrium relation rþ 1⁄4 Q0 r q (A.9) We thank Luuk Wagenaar and Lotte Schaap for fruitful discus- sions and critical questions. AJ also thanks Esteban Avendan ̃o, Diego Gonz ́alez, Mavis Montero, and Roberto Urcuyo for edu- cating him on electrochemical double layers. RA was supported by the International Research Agendas Programme (IRAP) of the Foundation for Polish Science (FNP), with structural funds from the European Union (EU). DG-K was supported by the Gordon and Betty Moore Foundation as a Physics of Living Systems Fellow (grant no. GBMF45130). AJ was supported by the Polish National Agency for Academic Exchange (NAWA)’s Ulam Programme (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 Research Council (NWO). EvH and RA also gratefully acknowledge the support of the Freiburg Institute of Advanced Study (FRIAS)’s visitors’ program during the first stages of this collaboration. References 1 A. Jenkins, Self-Oscillation, Phys. Rep., 2013, 525, 167–222. 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., 1980, 48, 405. (in obvious shorthand notation) into the kinetic eqn (14), we arrive at the linearized equation with and G 1⁄4 rðX0; Q0Þ y:= Gy + bx, ( @ rþðX0;QÞ ) (A.10) (A.11) 1 Q0@Q @ ln rðX0; QÞ rþðX;Q0Þ Q1⁄4Q0 of self-oscillations: b 4 2g: Acknowledgements (A.18) (A.17) b1⁄4rðX0;Q0ÞX0@X lnrðX;Q0Þ (A.12) For r given by eqn (15), and using eqn (11), we have : @X lnrþ 1⁄4@X lnkþ @X lnr 1⁄4@X lnkþ þ@X lnk r kr 1⁄4 @X DG þ @X k @X r : RT k r This journal is © the Owner Societies 2021 Phys. Chem. Chem. Phys., 2021, 23, 9428–9439 | 9437 X1⁄4X0 (A.13) 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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