{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,13]],"date-time":"2026-02-13T13:19:32Z","timestamp":1770988772874,"version":"3.50.1"},"reference-count":48,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2016,3,30]],"date-time":"2016-03-30T00:00:00Z","timestamp":1459296000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Computation"],"abstract":"<jats:p>We model the behavior of an ideal liquid junction, across a porous and possibly charged medium between two ion-containing solutions, by means of the Nernst\u2013Planck equation for the stationary state, in conditions of local electroneutrality. An analytical solution of the equation was found long ago by Planck for the uncharged junction with only ions of valences +1 and \u22121. Other analytical results, which have later been obtained also for more general situations, seem impractical for performing calculations. In this paper, we obtain analytical solutions for systems with up to three valence classes, which can be applied to perform numerical calculations in a straightforward way. Our method provides a much larger amount of information on the behavior of the system than the well-known Henderson\u2019s approximation. At the same time, it is more simple and reliable, and much less demanding in terms of computational effort, than the nowadays commonly employed numerical methods, typically based on discrete integration and trial-and-error numerical inversions. We present some examples of practical applications of our results. We study in particular the uphill transport (i.e., the transport from the lower-concentration to the higher-concentration region) of a divalent cation in a liquid junction containing also other univalent anions and cations.<\/jats:p>","DOI":"10.3390\/computation4020017","type":"journal-article","created":{"date-parts":[[2016,3,30]],"date-time":"2016-03-30T16:47:27Z","timestamp":1459356447000},"page":"17","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Analytical Results on the Behavior of a Liquid Junction across a Porous Diaphragm or a Charged Porous Membrane between Two Solutions According to the Nernst\u2013Planck Equation"],"prefix":"10.3390","volume":"4","author":[{"given":"Massimo","family":"Marino","sequence":"first","affiliation":[{"name":"Universit\u00e0 degli Studi di Milano, Dipartimento di Matematica, via Saldini 50, Milano 20133, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Doriano","family":"Brogioli","sequence":"additional","affiliation":[{"name":"Universit\u00e0 degli Studi di Milano, Dipartimento di Matematica, via Saldini 50, Milano 20133, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2016,3,30]]},"reference":[{"key":"ref_1","unstructured":"MacInnes, D. 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