{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,31]],"date-time":"2026-08-31T23:16:12Z","timestamp":1788218172332,"version":"build-2803163510"},"reference-count":28,"publisher":"American Chemical Society (ACS)","issue":"33","license":[{"start":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T00:00:00Z","timestamp":1785888000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100007812","name":"University of Washington","doi-asserted-by":"publisher","award":["NA"],"award-info":[{"award-number":["NA"]}],"id":[{"id":"10.13039\/100007812","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026,8,25]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Low-frequency impedance or admittance experiments of reversible redox couples in well-supported quiescent electrolyte are normally analyzed using a linearized semi-infinite Warburg element with a single lumped parameter. Because each species of a redox couple has a distinct diffusivity, one cannot determine individual diffusivities by fitting a spectrum with one parameter; linearization causes information loss. Weakly nonlinear theory and experiments presented here extend traditional Warburg admittance analysis to the higher harmonic currents generated from single-sine potential modulations at a Pt electrode in well-supported ferri-\/ferro-cyanide electrolyte. Complex harmonic current data from our commercial instrument is processed to enable direct fitting of theory to measurements. We show that complex first and second harmonic currents are measurable for \u0394E \u2265 5 mVrms, whereas a good compromise between experimental signal quality and leading-order model accuracy is found at \u0394E \u2248 10 mVrms. Higher-order corrections to theory are required at larger potential modulation amplitudes to remove amplitude dependency from fits, until \u0394E \u00d7 FRT&amp;gt; 1, when weakly nonlinear theory breaks down. Fitting the first and second harmonic experimental spectra to theory provides the mean diffusivity and a diffusion asymmetry parameter (deviations from mean), respectively, enabling unique determination of the individual ferricyanide and ferrocyanide diffusivities. Theory also shows that even harmonics disappear in the high-symmetry case of equal species diffusivities. In short, analyzing the first and second harmonics generated with moderate amplitude modulations eliminates information loss from linear Warburg analysis of the first harmonic alone.<\/jats:p>","DOI":"10.1021\/acs.analchem.6c01245","type":"journal-article","created":{"date-parts":[[2026,8,5]],"date-time":"2026-08-05T18:02:12Z","timestamp":1785952932000},"page":"24057-24069","source":"Crossref","is-referenced-by-count":0,"title":["Second Harmonic\nNonlinear Warburg Admittance Analysis\nEliminates Information Loss from Linearization in Traditional EIS:\nTheory and Experimental Validation"],"prefix":"10.1021","volume":"98","author":[{"ORCID":"https:\/\/orcid.org\/0009-0003-6296-3689","authenticated-orcid":true,"given":"Lauren A.","family":"Frank","sequence":"first","affiliation":[{"name":"University of Washington , , Box 351750 , , ,","place":["Seattle, Washington, United States, 98195-1750"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jerome T.","family":"Babauta","sequence":"additional","affiliation":[{"name":"Gamry Instruments, Inc. , 734 Louis Drive , , ,","place":["Warminster, Pennsylvania, United States, 18974"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Daniel T.","family":"Schwartz","sequence":"additional","affiliation":[{"name":"University of Washington , , Box 351750 , , ,","place":["Seattle, Washington, United States, 98195-1750"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"316","published-online":{"date-parts":[[2026,8,5]]},"reference":[{"key":"2026083118452305900_cit1","volume-title":"Electrochemical Methods:\nFundamentals and Applications","author":"Bard","year":"2022","edition":"3"},{"issue":"1","key":"2026083118452305900_cit2","doi-asserted-by":"publisher","first-page":"271","DOI":"10.1186\/s13068-019-1602-9","article-title":"Polyphenolic Compounds as Electron\nShuttles for Sustainable Energy Utilization","volume":"12","author":"Hsueh","year":"2019","journal-title":"Biotechnol. Biofuels"},{"issue":"24","key":"2026083118452305900_cit3","doi-asserted-by":"publisher","first-page":"4858","DOI":"10.1021\/ic00349a009","article-title":"Direct Cyclic Voltammetry\nof Three Ruthenium-Modified\nElectron-Transfer Proteins","volume":"29","author":"Armstrong","year":"1990","journal-title":"Inorg. Chem."},{"issue":"19","key":"2026083118452305900_cit4","doi-asserted-by":"publisher","first-page":"2334","DOI":"10.1021\/ac00146a005","article-title":"Direct\nElectron Transfer Reactions of Cytochrome c\nat Silver Electrodes","volume":"59","author":"Reed","year":"1987","journal-title":"Anal. Chem."},{"issue":"4","key":"2026083118452305900_cit5","doi-asserted-by":"publisher","first-page":"102","DOI":"10.1007\/s00894-024-05882-8","article-title":"Exploring\nJahn-Teller Distortions:\nA Local Vibrational Mode Perspective","volume":"30","author":"Quintano","year":"2024","journal-title":"J. Mol.\nModel."},{"issue":"51","key":"2026083118452305900_cit6","doi-asserted-by":"publisher","first-page":"14899","DOI":"10.1021\/jp511391b","article-title":"Structure and Dynamics of Ferrocyanide and Ferricyanide Anions in\nWater and Heavy Water: An Insight by MD Simulations and 2D IR Spectroscopy","volume":"118","author":"Prampolini","year":"2014","journal-title":"J. Phys. Chem. B"},{"issue":"1","key":"2026083118452305900_cit7","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1016\/S0022-0728(00)00327-2","article-title":"Determination of Diffusion\nCoefficients of the Electrode\nReaction Products by the Double Potential Step Chronoamperometry at\nSmall Disk Electrodes","volume":"493","author":"Ikeuchi","year":"2000","journal-title":"J. Electroanal. 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Electrochem."},{"key":"2026083118452305900_cit21","doi-asserted-by":"publisher","first-page":"80","DOI":"10.1039\/tf9575300080","article-title":"Faradaic\nRectification: Theory and Application to the Hg22+\/Hg Electrode","volume":"53","author":"Oldham","year":"1957","journal-title":"Trans. Faraday Soc."},{"issue":"12","key":"2026083118452305900_cit22","doi-asserted-by":"publisher","DOI":"10.1149\/1945-7111\/ad15ca","article-title":"Second-Harmonic Nonlinear Electrochemical Impedance\nSpectroscopy: Part I. Analytical Theory and Equivalent Circuit Representations\nfor Planar and Porous Electrodes","volume":"170","author":"Ji","year":"2023","journal-title":"J. Electrochem.\nSoc."},{"issue":"2","key":"2026083118452305900_cit23","doi-asserted-by":"publisher","DOI":"10.1149\/1945-7111\/ad2596","article-title":"Second-Harmonic Nonlinear Electrochemical Impedance\nSpectroscopy: Part II. 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