pith. sign in

arxiv: 2606.18028 · v1 · pith:2OCINMKGnew · submitted 2026-06-16 · ⚛️ physics.chem-ph

Thermodynamically consistent modeling of ion exchange membranes in multi-ionic environments

Noah Lettner , Felix K. Schwab , Birger Horstmann This is my paper

Pith reviewed 2026-06-26 21:54 UTC · model grok-4.3

classification ⚛️ physics.chem-ph
keywords ion exchange membranesthermodynamic consistencymass-action site occupationmean-field electrostaticsmulticomponent electrolytespolymer backbonemembrane modeling
0
0 comments X

The pith

A thermodynamically consistent model for ion exchange membranes arises from combining mass-action site occupation with mean-field electrostatic interactions along the polymer backbone.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper reviews existing models of ion exchange membranes and identifies useful physical contributions before deriving a new one that enforces thermodynamic consistency. It does so by merging mass-action laws for site occupation with mean-field electrostatics along the polymer backbone while explicitly treating multicomponent electrolytes at elevated concentrations. The resulting parameters become interpretable and linked rather than independent. When tested against data, the model reproduces both static and dynamic membrane properties with good accuracy. This supplies a coherent starting point for designing membranes used in desalination, fuel cells, and batteries.

Core claim

The authors derive a thermodynamically consistent model for ion exchange membranes by combining mass-action site occupation with mean-field electrostatic interactions along the polymer backbone, explicitly accounting for multicomponent electrolytes at elevated concentrations. The model parameters relate to those of other models but gain consistency and interpretability through the underlying derivation. Comparison to experimental data shows that both static and dynamic membrane properties are reproduced with good accuracy.

What carries the argument

The combination of mass-action site occupation with mean-field electrostatic interactions along the polymer backbone, which enforces thermodynamic consistency for multicomponent electrolytes.

If this is right

  • Parameters acquire explicit linkages, reducing the number of quantities treated as fully independent.
  • The model supplies a basis for theory-driven optimization of ion exchange membranes.
  • It supports tailored membrane design for water desalination, fuel cells, and aqueous batteries.
  • Both static equilibrium properties and dynamic transport rates match experimental observations with good accuracy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same combination of mass-action and mean-field terms could be tested on other polymer electrolytes with similar site-binding chemistry.
  • The linked parameters may simplify calibration when the model is embedded in device-scale simulations.
  • High-concentration multi-ion cases become more tractable without separate ad-hoc corrections for each new mixture.

Load-bearing premise

That mass-action site occupation combined with mean-field electrostatic interactions along the polymer backbone produces thermodynamic consistency for multicomponent electrolytes at elevated concentrations without additional constraints.

What would settle it

An experiment on membrane ion uptake or conductivity in a multicomponent electrolyte at high concentration that shows clear mismatch with the model's predicted equilibrium or transport values.

Figures

Figures reproduced from arXiv: 2606.18028 by Birger Horstmann, Felix K. Schwab, Noah Lettner.

Figure 1
Figure 1. Figure 1: Schematic representation of the SDE model. Colors indi￾cate the sign of ion charge, the hatched pattern denotes fixed sites and pore walls. Equilibrium partitioning is described by a partitioning factor S, which relates pore (p) and bulk (b) concentra￾tions ci . It accounts for electric (Donnan, do), steric (st) and dielectric (de) contributions: Si = c p i c b i = S st i S de i S do i (1) Expressions for … view at source ↗
Figure 2
Figure 2. Figure 2: Schematic representation of the Donnan-Manning model Note that in multi-ionic environments it can be useful to estimate the condensed concentrations c c i of all counte￾rions. These can be approximated by assuming a propor￾tionality to charge, radius and uncondensed concentration c u i [16,42] , e.g.: c c i =  cX − qeff zX  |zX| c u i P /ri j |zj | c u j /rj ∀ i, j with zi/j zX < 0 (13) 2.2.3. low-T* mod… view at source ↗
Figure 3
Figure 3. Figure 3: Schematic representation of the basic occupational states of fixed charge sites for a monovalent permeating salt[7] . Colors indicate the sign of ion charge, the hatched pattern denotes fixed sites. Based on these assumptions, an analytic expression for the partitioning can be derived. For a monovalent salt this yields the following coion and counterion partitioning co￾efficients SA = c m A /cb A and SM = … view at source ↗
Figure 4
Figure 4. Figure 4: Treatment of the electric potential: Homogeneous in the SDE-model, distinct interactions in the low-T* model The Donnan-Manning model, while also being a mean￾field model, introduces a thermodynamic threshold for the inevitable onset of counterion condensation. This thresh￾old arises from limiting laws[38] and can be interpreted as a non-distinct interaction of fixed sites with mobile ions standing in betw… view at source ↗
Figure 5
Figure 5. Figure 5: Illustration of model variables and indexing. For the depicted system, the occupation fractions θα∈{0,1,2,3} would be 1 4 each. 3.2. Thermodynamics For the bulk electrolyte surrounding the membrane, we write the free energy density g b as the sum of ideal mix￾ing, electrostatic, and remaining linear excess contribu￾tions. Assuming sufficient abundance of neutral solvent to eliminate its entropic contributi… view at source ↗
Figure 6
Figure 6. Figure 6: Water uptake and the subsequent ion exchange capacity cX depending on the bulk salt concentration c b s for a CR61 mem￾brane. 4.3. Association constants The experimental determination of association constants is a complicated matter[69–71] . Since the present model can require multiple association constants (see equation 39), approximation from theory can be useful. To do so, we assume that the association… view at source ↗
Figure 7
Figure 7. Figure 7: Left: The influence of the cavity radius rcav on the dielectric exclusion factor S de for a divalent ion |z| = 2. Right: The influence of the geometry parameter g on the steric exclusion factor S st . But the model uncertainty does not end here. Generally, neither the bulk permittivity, nor the membrane permit￾tivity are constant. The permittivity of the bulk phase 9 [PITH_FULL_IMAGE:figures/full_fig_p009… view at source ↗
Figure 8
Figure 8. Figure 8: Left: The dependence of bulk permittivity on salt concen￾tration c b s according to Mollerup et al. [86] ; Right: The membrane static permittivity depending on the water uptake with either the polymer as the continuous phase (C=pb ) or the solute as the contin￾uous phase (C=s b ) according to equation 58 compared to experimen￾tal data for Nafion 117 and XL-pGMA-z [88–90] ; the polymer relative permittivity… view at source ↗
Figure 9
Figure 9. Figure 9: compares both theories along this premise. It shows that the Manning condensation in the D-M model can be regarded as a discontinuous piecewise approxima￾tion of the IOM. 5.1.2. The role of interaction To isolate the influence of nearest-neighbor interactions, we vary the dimensionless interaction parameter nnU for the CR61 membrane at otherwise fixed parameters. Its 100 101 104 105 0 − z zx cX / mol m3 θ … view at source ↗
Figure 10
Figure 10. Figure 10: The influence of the interaction term nnU on the effec￾tive charge qeff for divalent counterions zct = 2 at constant water uptake. 5.1.3. Sensitivity to association constants To assess the sensitivity of the IOM to association con￾stants, we vary the counterion association constant Kct within a physically reasonable range while keeping all other parameters fixed. As seen from equation 39, Kct enters the o… view at source ↗
Figure 12
Figure 12. Figure 12: compares the concentrations of co- and coun￾terions in a CR61 membrane equilibrated with aqueous NaCl solutions, as predicted by the four models, to ex￾perimental data[68] . It shows, that the experimental data is reproduced with appropriate accuracy by all four mod￾els. Only the low-T* model shows noticeable deviations from the experimental data at high salt concentrations, because the active sites satur… view at source ↗
Figure 13
Figure 13. Figure 13: Co- and counterion concentrations in the CR61 mem￾brane depending on ambient bulk salt concentration (partitioning) for aqueous solutions of CaCl2. 13 [PITH_FULL_IMAGE:figures/full_fig_p013_13.png] view at source ↗
Figure 15
Figure 15. Figure 15: CR61 membrane permeability of NaCl and MgCl2 as predicted by the presented models compared to experimental data from Kamcev et al. [50] 5.3. Model comparison Sections 5.1 and 5.2 analyze the IOM and compare its pre￾dictions with those of the SDE, D-M, and low-T* model. This comparison shows that all presented models repro￾duce the considered experimental data with reasonable accuracy. However, for the SDE… view at source ↗
Figure 16
Figure 16. Figure 16: The effective membrane charge qeff of the CR61 mem￾brane as predicted by the IOM (left) and the D-M model (right) depending on the concentrations of NaCl and MgCl2 in the bulk salt mixture. 101 102 101 103 c b NaCl c b MgCl2 IOM 100 10 c 4 m co / mol m3 101 103 102 103 c b MgCl2 IOM D-M 0.75 1.25 η IOM D-M [PITH_FULL_IMAGE:figures/full_fig_p015_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Left: Membrane coion (Cl−) concentrations as pre￾dicted by the IOM for a CR61 membrane depending on the concen￾trations of NaCl and MgCl2 in the bulk salt mixture. Right: The relative differences η IOM D-M = c m,IOM co /cm,D-M co of the predictions by the IOM and the D-M model. Both models correctly reproduce the uphill transport of the divalent counterion at low divalent salt fractions known from literat… view at source ↗
Figure 18
Figure 18. Figure 18: The permeabilities of Na+ and Mg2+ in a CR61 mem￾brane depending on the concentrations of NaCl and MgCl2 in the bulk salt mixture. 7. Conclusion and outlook Through a consistent derivation from fundamental ther￾modynamics, combining mass-action site occupation with mean-field electrostatic contributions, we obtain a model with broad structural applicability for ion exchange mem￾branes. The main benefit of… view at source ↗
read the original abstract

Ion exchange membranes are useful for a wide range of applications, including water desalination, fuel cells, and aqueous batteries. Accordingly, a variety of models for ion exchange membranes has been proposed, each emphasizing different aspects that govern their static and dynamic properties. By reviewing these models, we identify key physical contributions and beneficial modeling strategies. Based on these insights, we derive a thermodynamically consistent model by combining mass-action site occupation with mean-field electrostatic interactions along the polymer backbone. In this derivation, we explicitly account for multicomponent electrolytes at elevated concentrations. The parameters of the resulting model relate closely to those of other models, but gain consistency and interpretability through the underlying derivation. A discussion of the model parameters highlights redundancies and linkages between quantities that are commonly treated independently. Comparison to experimental data shows that both static and dynamic membrane properties are reproduced with good accuracy by the presented model. This makes it a promising basis for theory-driven membrane optimization and supports the tailored design of ion exchange membranes for various technologies.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper reviews existing models for ion exchange membranes, identifies key physical contributions, and derives a thermodynamically consistent model by combining mass-action site occupation with mean-field electrostatic interactions along the polymer backbone. The derivation explicitly accounts for multicomponent electrolytes at elevated concentrations; model parameters are related to those of prior models but gain consistency and interpretability through the derivation. A discussion of parameter redundancies and linkages is provided, followed by comparison to experimental data claiming good accuracy for both static and dynamic membrane properties.

Significance. If the thermodynamic consistency holds via an explicit construction, the model could serve as a unified, interpretable framework for optimizing ion exchange membranes in desalination, fuel cells, and batteries, with parameters that reduce redundancies common in earlier approaches.

major comments (2)
  1. [Derivation section] Derivation section (around the combination of mass-action and mean-field terms): the central claim that this combination produces thermodynamic consistency for multicomponent electrolytes at elevated concentrations lacks an explicit demonstration (e.g., via variational free-energy construction, verification of chemical-potential equality, or Gibbs-Duhem relation satisfaction across all species). Mean-field electrostatics typically omits short-range correlations dominant above ~0.5 M, so an unstated constraint or correction may be required; this is load-bearing for the consistency assertion.
  2. [Results/comparison section] Results/comparison section: the assertion that 'both static and dynamic membrane properties are reproduced with good accuracy' is not supported by quantitative metrics (e.g., RMSE values, specific datasets, or error bars on figures) in the provided text; without these, the validation cannot be assessed as load-bearing evidence for the model's superiority.
minor comments (2)
  1. The abstract states that parameters 'relate closely to those of other models'; a explicit table or section mapping parameters (e.g., site-binding constants, dielectric constants) to literature equivalents would clarify the claimed interpretability gains.
  2. Notation for multicomponent concentrations and electrostatic potentials should be defined at first use to avoid ambiguity when extending to elevated concentrations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments, which help clarify the presentation of thermodynamic consistency and strengthen the validation. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Derivation section] Derivation section (around the combination of mass-action and mean-field terms): the central claim that this combination produces thermodynamic consistency for multicomponent electrolytes at elevated concentrations lacks an explicit demonstration (e.g., via variational free-energy construction, verification of chemical-potential equality, or Gibbs-Duhem relation satisfaction across all species). Mean-field electrostatics typically omits short-range correlations dominant above ~0.5 M, so an unstated constraint or correction may be required; this is load-bearing for the consistency assertion.

    Authors: We agree that an explicit verification strengthens the central claim. The derivation constructs the free energy from mass-action site occupation (ensuring correct chemical potentials for bound and mobile ions) combined with mean-field electrostatics along the backbone; thermodynamic consistency follows by construction because the resulting chemical potentials derive from a single variational free-energy functional. However, we did not include a dedicated verification step (e.g., explicit Gibbs-Duhem check or chemical-potential equality across all species). In the revised manuscript we will add a short subsection that (i) writes the explicit free-energy functional, (ii) derives the chemical potentials, and (iii) verifies the Gibbs-Duhem relation holds for the multicomponent system. Regarding short-range correlations above ~0.5 M, the model is intended for the mean-field regime typical of many IEM applications; we will note this scope limitation and the possible need for correlation corrections in future extensions. revision: yes

  2. Referee: [Results/comparison section] Results/comparison section: the assertion that 'both static and dynamic membrane properties are reproduced with good accuracy' is not supported by quantitative metrics (e.g., RMSE values, specific datasets, or error bars on figures) in the provided text; without these, the validation cannot be assessed as load-bearing evidence for the model's superiority.

    Authors: We agree that quantitative metrics are necessary for rigorous assessment. The manuscript text states that the model reproduces experimental data with good accuracy, but does not report RMSE values, list the exact datasets, or include error bars. In the revised manuscript we will (i) specify the experimental datasets used for both static (e.g., sorption isotherms) and dynamic (e.g., conductivity, transport numbers) properties, (ii) add RMSE or normalized error values for each comparison, and (iii) include error bars on the relevant figures where experimental uncertainties are available. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation presented as independent combination without self-referential reductions or fitted predictions.

full rationale

The abstract describes deriving thermodynamic consistency by combining mass-action site occupation with mean-field electrostatic interactions, explicitly for multicomponent electrolytes. No equations, self-citations, or fitted parameters are quoted that would reduce a claimed prediction or consistency result to the inputs by construction. The parameters are stated to relate to other models but gain consistency through the derivation, with no indication that the consistency itself is assumed or fitted. Comparison to experimental data is presented as validation rather than a circular fit. Without load-bearing steps that collapse to self-definition or renaming, the chain is treated as self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no specific free parameters, axioms, or invented entities are identifiable from the provided text. The model is described at a high level as combining mass-action site occupation with mean-field electrostatic interactions.

pith-pipeline@v0.9.1-grok · 5705 in / 1115 out tokens · 30222 ms · 2026-06-26T21:54:01.463965+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

128 extracted references

  1. [1]

    W. R. Bowen, J. S. Welfoot,Chemical Engineering Science2002

  2. [2]

    Kamcev, D

    J. Kamcev, D. R. Paul, B. D. Freeman,Macro- molecules2015,48, 8011

  3. [3]

    K. H. Meyer, J.-F. Sievers,Helvetica Chimica Acta 1936,19, 649

    1936

  4. [4]

    Teorell,Proceedings of the National Academy of Sciences1935,21, 152

    T. Teorell,Proceedings of the National Academy of Sciences1935,21, 152

  5. [5]

    F. A. Morrison, J. F. Osterle,The Journal of Chem- ical Physics1965,43, 2111

  6. [6]

    A. Z. Weber, J. Newman,Journal of The Electro- chemical Society2004,151, A311

  7. [7]

    Freger,Advances in Colloid and Interface Science 2020,277, 102107

    V. Freger,Advances in Colloid and Interface Science 2020,277, 102107

    2020

  8. [8]

    Freger,Journal of Membrane Science2025,722, 123795

    V. Freger,Journal of Membrane Science2025,722, 123795

  9. [9]

    T.Luo, S.Abdu, M.Wessling,JournalofMembrane Science2018,555, 429

  10. [10]

    Kitto, J

    D. Kitto, J. Kamcev,Journal of Polymer Science 2022,60, 2929

    2022

  11. [11]

    Yaroshchuk, M

    A. Yaroshchuk, M. L. Bruening, E. Zholkovskiy,Ad- vances in Colloid and Interface Science2019,268, 39

  12. [12]

    S. M. Bannon, G. M. Geise,Journal of Membrane Science2024,694, 122396. 16

  13. [13]

    A. R. Crothers, R. M. Darling, A. Kusoglu, C. J. Radke, A.Z.Weber,JournalofTheElectrochemical Society2020,167, 013547

  14. [14]

    Freger,Advances in Colloid and Interface Science 2023,319, 102972

    V. Freger,Advances in Colloid and Interface Science 2023,319, 102972

    2023

  15. [15]

    Santafé-Moros, J

    A. Santafé-Moros, J. Gozálvez-Zafrilla, J. Lora- García,Desalination2008,221, 268

  16. [16]

    R. Wang, R. Duddu, S. Lin,Journal of Membrane Science2023,681, 121782

  17. [17]

    Chao,SCIENCE ADVANCES2020

    D. Chao,SCIENCE ADVANCES2020

  18. [18]

    Liang, Y

    Y. Liang, Y. Yao,Nature Reviews Materials2022, 8, 109

  19. [19]

    J. O. G. Posada, A. J. Rennie, S. P. Villar, V. L. Martins, J. Marinaccio, A. Barnes, C. F. Glover, D. A. Worsley, P. J. Hall,Renewable and Sustain- able Energy Reviews2017,68, 1174

  20. [20]

    Z. Ju, Q. Zhao, D. Chao, Y. Hou, H. Pan, W. Sun, Z. Yuan, H. Li, T. Ma, D. Su, B. Jia,Advanced Energy Materials2022,12, 2201074

  21. [21]

    Borchers, S

    N. Borchers, S. Clark, B. Horstmann, K. Jayasayee, M. Juel, P. Stevens,Journal of Power Sources 2021,484, 229309

    2021

  22. [22]

    Dembélé, L

    K. Dembélé, L. Chikh, S. Alfonsi, O. Fichet,Poly- mer Degradation and Stability2023,215, 110462

  23. [23]

    M. T. Tsehaye, F. Alloin, C. Iojoiu, R. A. Tufa, D. Aili, P. Fischer, S. Velizarov,Journal of Power Sources2020,475, 228689

  24. [24]

    M. T. Tsehaye, G. Teklay Gebreslassie, N. Heon Choi, D. Milian, V. Martin, P. Fis- cher, J. Tübke, N. El Kissi, M. L. Donten, F. Alloin, C. Iojoiu,Molecules2021,26, 4062

  25. [25]

    D. L. Oatley,Swansea University Prifysgol Abertawe 2004

    2004

  26. [26]

    Söllner,Biochem

    K. Söllner,Biochem. Z1932,244, 390

  27. [27]

    L. Liu, C. Wang, Z. He, R. Das, B. Dong, X. Xie, Z. Guo,Journal of Materials Science & Technology 2021,69, 212

    2021

  28. [28]

    Bowen, A

    W. Bowen, A. Mohammad, N. Hilal,Journal of Membrane Science1997,126, 91

  29. [29]

    Teorell,Progress in Biophysics and Biophysical Chemistry1953,3, 305

    T. Teorell,Progress in Biophysics and Biophysical Chemistry1953,3, 305

  30. [30]

    F. G. Donnan,Zeitschrift für Elektrochemie und angewandte physikalische Chemie1911,17, 572

  31. [31]

    J. W. Gibbs,American Journal of Science1878, s3-16, 441

  32. [32]

    J. R. Pappenheimer, E. M. Renkin, L. M. Bor- rero,American Journal of Physiology-Legacy Con- tent1951,167, 13

  33. [33]

    J. C. Giddings, E. Kucera, C. P. Russell, M. N. My- ers,The Journal of Physical Chemistry1968,72, 4397

  34. [34]

    W. M. Deen,AIChE Journal1987,33, 1409

  35. [35]

    Born,Zeitschrift für Physik1920,1, 45

    M. Born,Zeitschrift für Physik1920,1, 45

  36. [36]

    R. Wang, S. Lin,Journal of Membrane Science 2021,620, 118809

    2021

  37. [37]

    Dechadilok, W

    P. Dechadilok, W. M. Deen,Industrial & Engineer- ing Chemistry Research2006,45, 6953

  38. [38]

    G. S. Manning,The Journal of Chemical Physics 1969,51, 924

    1969

  39. [39]

    G. S. Manning,The Journal of Chemical Physics 1969,51, 934

    1969

  40. [40]

    G. S. Manning,The Journal of Chemical Physics 1969,51, 3249

    1969

  41. [41]

    Mareev, A

    S. Mareev, A. Gorobchenko, D. Ivanov, D. Anokhin, V. Nikonenko,International Journal of Molecular Sciences2022,24, 34

  42. [42]

    Purpura, E

    G. Purpura, E. Papiewska, A. Culcasi, A. Filingeri, A. Tamburini, M. C. Ferrari, G. Micale, A. Cipollina, Journal of Membrane Science2024,700, 122659

  43. [43]

    Kamcev, D

    J. Kamcev, D. R. Paul, G. S. Manning, B. D. Free- man,Macromolecules2018,51, 5519

  44. [44]

    J.-N. Aqua, S. Banerjee, M. E. Fisher,Physical Re- view E2005,72, 041501

  45. [45]

    Lifson, J

    S. Lifson, J. L. Jackson,The Journal of Chemical Physics1962,36, 2410

  46. [46]

    J. L. Jackson, S. R. Coriell,The Journal of Chemical Physics1963,38, 959

  47. [47]

    Einstein,Annalen der Physik1905,322, 549

    A. Einstein,Annalen der Physik1905,322, 549

  48. [48]

    Von Smoluchowski,Annalen der Physik1906, 326, 756

    M. Von Smoluchowski,Annalen der Physik1906, 326, 756

  49. [49]

    J. S. Mackie, P. Meares,Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences1955,232, 510

  50. [50]

    Kamcev, D

    J. Kamcev, D. R. Paul, G. S. Manning, B. D. Free- man,ACS Applied Materials & Interfaces2017,9, 4044

  51. [51]

    Marioni, O

    N. Marioni, O. Nordness, Z. Zhang, R. Sujanani, B. D. Freeman, R. A. Segalman, R. J. Clément, V. Ganesan,ACS Macro Letters2024,13, 341

  52. [52]

    B. I. Shklovskii,Physical Review E1999,60, 5802

  53. [53]

    Deserno, C

    M. Deserno, C. Holm, S. May,Macromolecules 2000,33, 199

    2000

  54. [54]

    R. Wang, P. Biesheuvel, M. Elimelech,Journal of Membrane Science2024,705, 122921

  55. [55]

    Y. Yu, Y. Li, N. Hossain, C.-C. Chen,Fluid Phase Equilibria2019,497, 1

  56. [56]

    Z. Lu, G. Polizos, E. Manias, D. Macdonald,ECS Transactions2010,28, 81

  57. [57]

    Z. Lu, G. Polizos, D. D. Macdonald, E. Manias, Journal of The Electrochemical Society2008

  58. [58]

    G. S. Manning,Accounts of Chemical Research 1979,12, 443

    1979

  59. [59]

    Schurr, B

    J. Schurr, B. S. Fujimoto,Biophysical Chemistry 2002,101–102, 425

    2002

  60. [60]

    K. D. Fong, J. Self, K. M. Diederichsen, B. M. Wood, B. D. McCloskey, K. A. Persson,ACS Cen- tral Science2019,5, 1250

  61. [61]

    Schammer, B

    M. Schammer, B. Horstmann, A. Latz,Journal of The Electrochemical Society2021,168, 026511

  62. [62]

    Kilchert, M

    F. Kilchert, M. Lorenz, M. Schammer, P. Nürn- berg, M. Schönhoff, A. Latz, B. Horstmann,Physi- cal Chemistry Chemical Physics2023,25, 25965

  63. [63]

    A. Latz, J. Zausch,Journal of Power Sources2011, 196, 3296

  64. [64]

    Stamm, A

    J. Stamm, A. Varzi, A. Latz, B. Horstmann,Journal of Power Sources2017,360, 136

  65. [65]

    S. . V. W. T. \. Solutions, Ionics* Ion Exchange Membranes2020. 17

  66. [66]

    Galizia, F

    M. Galizia, F. M. Benedetti, D. R. Paul, B. D. Free- man,Journal of Membrane Science2017,535, 132

  67. [67]

    Y. S. Oren, O. Nir, V. Freger,Journal of Membrane Science2024,690, 122202

  68. [68]

    Galizia, G

    M. Galizia, G. S. Manning, D. R. Paul, B. D. Free- man,Polymer2019,165, 91

  69. [69]

    Marcus, G

    Y. Marcus, G. Hefter,Chemical Reviews2006,106, 4585

  70. [70]

    Hefter,Pure and Applied Chemistry2006,78, 1571

    G. Hefter,Pure and Applied Chemistry2006,78, 1571

  71. [71]

    M. A. Peshkova, A. I. Korobeynikov, K. N. Mikhel- son,Electrochimica Acta2008,53, 5819

  72. [72]

    Bjerrum,Videnskabernes Selskab1926

    N. Bjerrum,Videnskabernes Selskab1926

  73. [73]

    R. M. Fuoss, C. A. Kraus,Journal of the American Chemical Society1933,55, 1019

  74. [74]

    R. M. Fuoss,Journal of the American Chemical So- ciety1958,80, 5059

  75. [75]

    Eigen, K

    M. Eigen, K. Tamm,Zeitschrift für Elektrochemie, Berichte der Bunsengesellschaft für physikalische Chemie1962,66, 93

  76. [76]

    Justice, J.-C

    M.-C. Justice, J.-C. Justice,Journal of Solution Chemistry1976,5, 543

  77. [77]

    Barthel, R

    J. Barthel, R. Wachter, H.-J. Gores, Temperature Dependence of Conductance of Electrolytes in Non- aqueous Solutions, in B. E. Conway, J. O. Bock- ris (Editors),Modern Aspects of Electrochemistry, pages 1–79, Springer US, Boston, MA1979

  78. [78]

    Ebeling, M

    W. Ebeling, M. Grigo,Annalen der Physik1980, 492, 21

  79. [79]

    Krienke, J

    H. Krienke, J. Barthel,Journal of Molecular Liquids 1998,78, 123

    1998

  80. [80]

    Krienke, J

    H. Krienke, J. Barthel,Zeitschrift für Physikalische Chemie1998,204, 71

Showing first 80 references.