Passage of diffusion-migration current across electrode/membrane/solution system. Part 1: short-time evolution. Binary electrolyte (equal mobilities)

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

Express-method proposed recently for experimental determination of diffusion coefficients of electroactive ions inside a membrane and their distribution coefficients at the membrane/solution boundary (Russ. J. Electrochem., 2022, 58, 1103) is based on comparison of measured non-stationary current for the electrode/membrane/electrolyte solution system after a potential step with theoretical expressions for the current-time dependence. Application of this method for the study of bromide-anion transport across the membrane was performed in the previous work under the condition of the permselectivity of the membrane where the amplitude of the electric field inside its space was suppressed owing to a high concentration of non-electroactive counterions. Then, the coion (bromide anion) transport corresponded to the diffusional mechanism, for which the solution was available in an analytical form. This study considers for the first time a non-stationary electrodiffusional transmembrane transport of two singly charged ions (e.g. background cation М+ as the counterion and electroactive anion X– as the coion) having identical values of their diffusion coefficients where the current passage induces a transient electric field in this spatial region, resulting in a deviation from predictions for the diffusional mechanism. It has been established that within the short time interval after a potential step from the equilibrium state of the membrane to the limiting current regime where the thickness of the non-stationary diffusion layer is significantly smaller than the thickness of the membrane, non-stationary distributions of the ion concentrations and of the electric field strength as a function of two variables (spatial and temporal ones, x and t) can be expressed via a function of one variable, Z(z), where z = x/(4Dt)1/2, the form of which, depending on the ratio of the surface concentration of component X to the fixed charge density inside the membrane (Xm/Cf ) has been found by numerical integration. The limiting current varies with time according to the Cottrell formula (I ~ t–1/2); dependence of the dimensionless current amplitude, i, on the ratio, Xm/Cf , has been found via numerical calculation; approximate analytical formula has also been proposed. In particular, it has been shown that the passing current is close to the diffusion–limited one for a low concentration of coions at the membrane/electrolyte solution boundary with respect to the concentration of immobile charged groups inside the membrane (Xm/Cf≪1), whereas the migrational contribution to the ionic fluxes doubles the limiting current if the opposite condition (Xm/Cf≫1) is fulfilled.

Full Text

Restricted Access

About the authors

M. A. Vorotyntsev

Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences

Author for correspondence.
Email: mivo2010@yandex.com
Russian Federation, Moscow

P. А. Zader

Frumkin Institute of Physical Chemistry and Electrochemistry, Russian Academy of Sciences

Email: mivo2010@yandex.com
Russian Federation, Moscow

References

  1. Perry, M.L., Saraidaridis, J.D., and Darling, R.M., Crossover mitigation strategies for redox–flow batteries, Current Opinion in Electrochem., 2020, vol. 21, p. 311.
  2. Saadi, K., Nanikashvili, P., Tatus–Portnoy, Z., Hardisty, S., Shokhen, V., Zysler, M., and Zitoun, D., Crossover–tolerant coated platinum catalysts in hydrogen/bromine redox flow battery, J. Power Sources, 2019, vol. 422, p. 84.
  3. Darling, R.M., Weber, A.Z., Tucker, M. C., and Perry, M.L., The influence of electric field on crossover in redox–flow batteries, J. Electrochem. Soc., 2015, vol. 163, no. 1, p. A5014.
  4. Oh, K., Weber, A.Z., and Ju, H., Study of bromine species crossover in H2/Br2 redox flow batteries, Intern. J. Hydrogen Energy, 2017, vol. 42, no. 6, p. 3753.
  5. Shi, Y., Wei, Z., Liu, H., and Zhao, J., Dynamic modeling of long–term operations of vanadium/air redox flow battery with different membranes, J. Energy Storage, 2022, vol. 50, p. 104171.
  6. Barton, J.L. and Brushett, F.R., A one-dimensional stack model for redox flow battery analysis and operation, Batteries, 2019, vol. 5, no. 1, p. 25.
  7. Cho, K.T., Albertus, P., Battaglia, V., Kojic, A., Srinivasan, V., and Weber, A.Z., Optimization and analysis of high‐power hydrogen/bromine‐flow batteries for grid‐scale energy storage, Energy Technology, 2013, vol, 1, no. 10, p. 596.
  8. Maurya, S., Shin, S.H., Lee, J.Y., Kim, Y., and Moon, S.H., Amphoteric nanoporous polybenzimidazole membrane with extremely low crossover for a vanadium redox flow battery, RSC advances, 2016, vol. 6, no. 7, p. 5198.
  9. Peng, S., Zhang, L., Zhang, C., Ding, Y., Guo, X., He, G., and Yu, G., Gradient‐Distributed Metal–Organic Framework–Based Porous Membranes for Nonaqueous Redox Flow Batteries, Advanced Energy Mater., 2018, vol. 8, no. 33, p. 1802533.
  10. Gvozdik, N.A., Sanginov, E.A., Abunaeva, L.Z., Konev, D.V., Usenko, A.A., Novikova, K.S., Stevenson, K.J., and Dobrovolsky, Y. A., A Composite Membrane Based on Sulfonated Polystyrene Implanted in a Stretched PTFE Film for Vanadium Flow Batteries, ChemPlusChem, 2020, vol. 85, no. 12, p. 2580.
  11. Leung, P.K., Xu, Q., Zhao, T.S., Zeng, L., and Zhang, C., Preparation of silica nanocomposite anion–exchange membranes with low vanadium–ion crossover for vanadium redox flow batteries, Electrochim. Acta, 2013, vol. 105, p. 584.
  12. Bukola, S., Li, Z., Zack, J., Antunes, C., Korzeniewski, C., Teeter, G., & Pivovar, B., Single–layer graphene as a highly selective barrier for vanadium crossover with high proton selectivity, J. Energy Chem., 2021, vol. 59, p. 419.
  13. Huang, S.L., Yu, H.F., and Lin, Y.S., Modification of Nafion® membrane via a sol–gel route for vanadium redox flow energy storage battery applications, J. Chem., 2017, vol. 2017, p. 1.
  14. Will, F.G., Bromine Diffusion Through Nafion® Perfluorinated Ion Exchange Membranes, J. Electrochem. Soc., 1979, vol. 126, no. 1, p. 36.
  15. Park, J.W., Wycisk, R., and Pintauro, P.N., Nafion/PVDF nanofiber composite membranes for regenerative hydrogen/bromine fuel cells, J. Membrane Sci., 2015, vol. 490, p. 103.
  16. Heintz, A. and Illenberger, C., Diffusion coefficients of Br2 in cation exchange membranes, J. Membrane Sci., 1996, vol. 113, no. 2, p. 175.
  17. Yeo, R. and McBreen, J., Transport properties of Nafion membranes in electrochemically regenerative hydrogen/halogen cells, J. Electrochem. Soc., 1979, vol. 126, p. 1682.
  18. Baldwin, R.S., Electrochemical performance and transport properties of a Nafion membrane in a hydrogen-bromine cell environment, NASA TM (NAS 1.15:89862), 1987, vol. 89862, p. 1.
  19. Kimble, M. and White, R., Estimation of the diffusion coefficient and solubility for a gas diffusing through a membrane, J. Electrochem. Soc., 1990, vol. 137, p. 2510.
  20. Haug, A.T. and White, R.E., Oxygen diffusion coefficient and solubility in a new proton exchange membrane, J. Electrochem. Soc., 2000, vol. 147, p. 980.
  21. White, H.S., Leddy, J., and Bard, A.J., Polymer films on electrodes. 8. Investigation of charge–transport mechanisms in Nafion polymer modified electrodes, J. Am. Chem. Soc., 1982, vol. 104, p. 4811.
  22. Mello, R.M.Q. and Ticianelli, E.A., Kinetic study of the hydrogen oxidation reaction on platinum and Nafion ® covered platinum electrodes, Electrochim. Acta, 1997, vol. 42, no. 6, p. 1031.
  23. Ayad, A., Naimi, Y., Bouet, J., and Fauvarque, J.F., Oxygen reduction on platinum electrode coated with Nafion®, J. Power Source, 2004, vol. 130, p. 50.
  24. Brunetti, B., Desimoni, E., and Casati, P., Determination of Caffeine at a Nafion‐Covered Glassy Carbon Electrode, Electroanalysis, 2007, vol. 19, no. 2–3, p. 385.
  25. Sadok, I., Tyszczuk–Rotko, K., and Nosal–Wierci´nska, A., Bismuth particles Nafion covered boron–doped diamond electrode for simultaneous and individual voltammetric assays of paracetamol and caffeine, Sensors & Actuators, B: Chem., 2016, vol. 235, p. 263.
  26. Конев, Д.В., Истакова, О.И., Карташова, Н.В., Абунаева, Л.З., Пырков, П.В., Локтионов, П.А., Воротынцев, М.А. Электрохимическое измерение коэффициента диффузии коиона через ионообменную мембрану. Электрохимия. 2022. Т. 58. С. 870. [Konev, D.V., Istakova, O.I., Kartashova, N.V., Abunaeva, L.Z., Pyrkov, P.V., Loktionov, P.A., and Vorotyntsev, M.A., Electrochemical Measurement of Co–Ion Diffusion Coefficient in Ion–Exchange Membranes, Russ. J. Electrochem., 2022, vol. 58, p. 1103.]
  27. Konev, D.V., Istakova, O.I., and Vorotyntsev, M.A., Electrochemical measurement of interfacial distribution and diffusion coefficients of electroactive species for ion–exchange membranes, Membranes, 2022, vol. 12, no. 11, p. 1041.
  28. Myland, J.C. and Oldham, K.B., Limiting currents in potentiostatic voltammetry without supporting electrolyte, Electrochem. Commun., 1999, vol. 1, no. 10, p. 467.
  29. Bieniasz, L.K., Analytical formulae for chronoamperometry of a charge neutralisation process under conditions of linear migration and diffusion, Electrochem. Commun., 2002, vol. 4, p. 917.
  30. Lange, R. and Doblhofer, K., The transient response of electrodes coated with membrane–type polymer films under conditions of diffusion and migration of the redox ions, J. Electroanal. Chem. and Interfacial Electrochem., 1987, vol. 237, p. 13.
  31. Britz, D., Digital Simulation in Electrochemistry, Berlin: Springer, 1988. 338 p.
  32. Helfferich, F.G., Ion Exchange, New York: Dover Publications, Inc., 1995. 624 p.
  33. Заболоцкий, В.И., Никоненко, В.В. Перенос ионов в мембранах, М.: Наука, 1996. 392 с. [Zabolotsky, V.I. and Nikonenko, V.V., Ion Transfer in Membranes (in Russian), Moscow: Nauka, 1996. 392 p.]
  34. Newman, J.S., Electrochemical Systems, London – Englewood Cliffs: Prentice–Hall, Inc., 1973. 432 p.
  35. Pfabe, K.A., A problem in nonlinear ion transport, Lincoln: University of Nebraska, PhD Thesis, 1995. 125 p.
  36. Cohn, S., Pfabe, K., Redepenning, J., A similarity solution to a problem in nonlinear ion transport with a nonlocal condition, Math. Mod. Meth. Appl. Sci., 1999, vol. 9, p. 445.
  37. Pfabe, K. and Shores, T.S., Numerical methods for an ion transport problem, Appl. Numer. Math., 2000, vol. 32, p. 175.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download
2. 1. Diagram of the electrode/membrane/binary electrolyte solution system. The concentration profiles of background counterions M (M(1), M(2), and M(3)) and electroactive coions X (X(1), X(2), and X(3)) are shown for three time points after the potential jump: 0 < t(1) < t(2) < t(3).

Download (103KB)
Indexing metadata
3. Fig. 2. (a) Dependence of the dimensionless concentration of the electroactive component X, equal to Z = X/Cf, on the dimensionless combination of spatial (x) and temporal (t) variables: z = x/(4Dt)1/2, formula (23), for a set of values of the dimensionless amplitude of the unsteady current i, defined by the formula (24). (b) The dependence of the normalized concentration of component X: Z/Z(∞) on the z coordinate for a set of values of parameter I. (c) The dependence of the ratio of the dimensionless concentration of component X to the dimensionless amplitude of the unsteady current i(Z/i) on the z coordinate for a set of values of parameter I. The values of parameter i are shown in the table in each figure.

Download (265KB)
Indexing metadata
4. 3. (a) The dependence of the dimensionless amplitude of the unsteady current i on the dimensionless surface concentration of component X at the membrane/solution boundary Xm/Cf in bilogarithmic (decimal) coordinates; dotted straight lines correspond to the formulas i = 2π–1/2Xm/Cf ≅ 1.13Xm/Cf (red) and i = 4π-1/2Xm/Cf ≅ 2.26Xm/Cf (blue). (b) Comparison of the exact solution of Z(z) (see the caption to Fig. 2a; solid lines) with its approximate formula (30) (dotted lines) for a set of parameter values i: 0.3 (black), 1 (red), 3 (blue), 10 (green) and 100 (magenta), which are indicated near the graphs.

Download (186KB)
Indexing metadata
5. 4. (a) Dependence of the normalized concentration of component X (Z = X/Cf) from the spatial coordinate x for three time values after the potential jump t: 1 s (black lines), 4 s (red lines) and 9 s (blue lines) with the parameter value i equal to 0.3 or 3 (indicated near each line). (b) The dependence of the electric field strength E in FE/RT coordinates on the spatial coordinate x, formula (28), for the same values of the time after the potential jump t and the values of parameter i, see Fig. 4a. (c) The dependence of the limiting current I(t) in coordinates: I(t)/(F D Cf) = i/(4D t)1/2, formula (24), on time t for three values of parameter i: 0.3, 1 or 3 (indicated near each line).

Download (209KB)
Indexing metadata
6. 5. Comparison of the dependences: (a) the ratio of the dimensionless current amplitude i to the dimensionless surface concentration Z(∞) = Xm/Cf , i.e. i/[2π–1/2Z(∞)]; (b) the dimensionless current amplitude i from Z(∞) = Xm/Cf , calculated numerically (black dots; according to the data in Fig. 3a) or according to the approximate formula (32) (Fig. 5a) or (33) (Fig. 5b), where the parameters of curves a and b are indicated in the table in the figure. The dotted straight lines in Fig. 5b correspond to the formulas i = 2π–1/2Xm/Cf ≅ 1.13Xm/Cf (blue) and i = 4π–1/2Xm/Cf ≅ 2.26 Xm/Cf (green).

Download (214KB)
Indexing metadata

Copyright (c) 2024 Russian Academy of Sciences

Согласие на обработку персональных данных с помощью сервиса «Яндекс.Метрика»

1. Я (далее – «Пользователь» или «Субъект персональных данных»), осуществляя использование сайта https://journals.rcsi.science/ (далее – «Сайт»), подтверждая свою полную дееспособность даю согласие на обработку персональных данных с использованием средств автоматизации Оператору - федеральному государственному бюджетному учреждению «Российский центр научной информации» (РЦНИ), далее – «Оператор», расположенному по адресу: 119991, г. Москва, Ленинский просп., д.32А, со следующими условиями.

2. Категории обрабатываемых данных: файлы «cookies» (куки-файлы). Файлы «cookie» – это небольшой текстовый файл, который веб-сервер может хранить в браузере Пользователя. Данные файлы веб-сервер загружает на устройство Пользователя при посещении им Сайта. При каждом следующем посещении Пользователем Сайта «cookie» файлы отправляются на Сайт Оператора. Данные файлы позволяют Сайту распознавать устройство Пользователя. Содержимое такого файла может как относиться, так и не относиться к персональным данным, в зависимости от того, содержит ли такой файл персональные данные или содержит обезличенные технические данные.

3. Цель обработки персональных данных: анализ пользовательской активности с помощью сервиса «Яндекс.Метрика».

4. Категории субъектов персональных данных: все Пользователи Сайта, которые дали согласие на обработку файлов «cookie».

5. Способы обработки: сбор, запись, систематизация, накопление, хранение, уточнение (обновление, изменение), извлечение, использование, передача (доступ, предоставление), блокирование, удаление, уничтожение персональных данных.

6. Срок обработки и хранения: до получения от Субъекта персональных данных требования о прекращении обработки/отзыва согласия.

7. Способ отзыва: заявление об отзыве в письменном виде путём его направления на адрес электронной почты Оператора: info@rcsi.science или путем письменного обращения по юридическому адресу: 119991, г. Москва, Ленинский просп., д.32А

8. Субъект персональных данных вправе запретить своему оборудованию прием этих данных или ограничить прием этих данных. При отказе от получения таких данных или при ограничении приема данных некоторые функции Сайта могут работать некорректно. Субъект персональных данных обязуется сам настроить свое оборудование таким способом, чтобы оно обеспечивало адекватный его желаниям режим работы и уровень защиты данных файлов «cookie», Оператор не предоставляет технологических и правовых консультаций на темы подобного характера.

9. Порядок уничтожения персональных данных при достижении цели их обработки или при наступлении иных законных оснований определяется Оператором в соответствии с законодательством Российской Федерации.

10. Я согласен/согласна квалифицировать в качестве своей простой электронной подписи под настоящим Согласием и под Политикой обработки персональных данных выполнение мною следующего действия на сайте: https://journals.rcsi.science/ нажатие мною на интерфейсе с текстом: «Сайт использует сервис «Яндекс.Метрика» (который использует файлы «cookie») на элемент с текстом «Принять и продолжить».