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Current-Voltage Relations for Electrochemical Thin Films

Martin Z. Bazant, Kevin T. Chu, B. J. Bayly

2005
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SIAM Journal on Applied Mathematics
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The dc response of an electrochemical thin film, such as the separator in a microbattery, is analyzed by solving the Poisson-Nernst-Planck equations, subject to boundary conditions appropriate for an electrolytic/galvanic cell. The model system consists of a binary electrolyte between parallel-plate electrodes, each possessing a compact Stern layer, which mediates Faradaic reactions with nonlinear Butler-Volmer kinetics. Analytical results are obtained by matched asymptotic expansions in the
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... xpansions in the limit of thin double layers and compared with full numerical solutions. The analysis shows that (i) decreasing the system size relative to the Debye screening length decreases the voltage of the cell and allows currents higher than the classical diffusion-limited current; (ii) finite reaction rates lead to the important possibility of a reaction-limited current; (iii) the Stern-layer capacitance is critical for allowing the cell to achieve currents above the reaction-limited current; and (iv) all polarographic (current-voltage) curves tend to the same limit as reaction kinetics become fast. Dimensional analysis, however, shows that "fast" reactions tend to become "slow" with decreasing system size, so the nonlinear effects of surface polarization may dominate the dc response of thin films. Introduction. Micro-electrochemical systems pose interesting problems for applied mathematics because traditional "macroscopic" approximations of electroneutrality and thermal equilibrium [1], which make the classical transport equations more tractable [2] , break down at small scales, approaching the Debye screening length. Of course, the relative importance of surface phenomena also increases with miniaturization. Micro-electrochemical systems of current interest include ion channels in biological membranes [3, 4, 5] and thin-film batteries [6, 7, 8, 9, 10] , which could revolutionize the design of modern electronics with distributed on-chip power sources. In the latter context, the internal resistance of the battery is related to the nonlinear current-voltage characteristics of the separator, consisting of a thin-film electrolyte (solid, liquid, or gel) sandwiched between flat electrodes and interfacial layers where Faradaic electron-transfer reactions occur [11] . Under such conditions, the internal resistance is unlikely to be simply constant, as is usually assumed. Motivated by the application to thin-film batteries, here we revisit the classical problem of steady conduction between parallel, flat electrodes, studied by Nernst [12] and Brunner [13, 14] a century ago. As in subsequent studies of liquid [15, 16] and solid [17, 18] electrolytes, we do not make Nernst's assumption of bulk electroneutrality and work instead with the Poisson-Nernst-Planck (PNP) equations, allowing for diffuse charge in solution [1, 2] . What distinguishes our analysis from previous work on current-voltage relations (or "polarographic curves") is the use of more realistic nonlinear boundary conditions describing (i) Butler-Volmer reaction kinetics and (ii) the surface capacitance of the compact Stern layer, as in the recent paper of Bonnefont et. al [19] . Such boundary conditions, although complicating mathematical analysis, generally cannot be ignored in micro-electrochemical cells, where interfaces play a crucial role. Diffuse-charge dynamics, which can be important for high-power

doi:10.1137/040609938
fatcat:ojtvawyw6jbulfbipv3xbi775i