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A single impedance formula now covers any multi-ion electrolyte

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2026-08-04 20:42 UTC pith:HUKCSFYS

load-bearing objection The ternary analysis is likely correct and useful, but the general-N derivation has an unflagged orthogonality condition that contradicts the paper's own example, so Eq. (33) is not established as written. the 3 major comments →

arxiv 2608.01799 v1 pith:HUKCSFYS submitted 2026-08-03 cond-mat.soft physics.chem-ph

Impedance of an electric double layer capacitor with a multi-component electrolyte

David Fertig This is my paper
classification cond-mat.soft physics.chem-ph
keywords electrochemical impedance spectroscopyPoisson-Nernst-Planck equationsmulti-component electrolyteelectric double layer capacitorWarburg impedancecharge-salt decompositionternary electrolyteambipolar diffusion
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper derives the impedance of a capacitor made from a simple electrolyte with any number of dissolved ion species, between flat blocking electrodes, within the linearized Poisson–Nernst–Planck model. It shows that the response is fixed by the eigenvalues of a diffusion–migration matrix: when all ions diffuse equally fast, the system looks exactly like a two-ion electrolyte, but when diffusivities differ, charge relaxation couples to neutral concentration modes. For a three-ion electrolyte with two cations and one anion, this coupling produces additional slanted, Warburg-like regions in the impedance spectrum and broadens the transition from resistive to capacitive behavior. The central claim is that mixed electrolytes therefore cannot generally be represented by a binary electrolyte with an averaged diffusion coefficient.

Core claim

The paper's central claim is that, in the linear-response regime, the impedance of an N-component ideal electrolyte between blocking electrodes is given exactly by Z = (α_q S3 η + α_q S2 η L̃)/(p̃ α_q S1 η) (Eq. 33), where every ingredient is constructed from the eigenvectors and eigenvalues of the matrix Λ = M⁻¹(p̃I + B) that governs coupled charge and salt modes. When all diffusivities are equal, the charge mode decouples, the salt modes become degenerate, and the formula reduces to Macdonald's classical binary-electrolyte impedance. When diffusivities differ, one neutral composition mode couples to charge relaxation (and further salt–salt coupling appears if all diffusivities differ), gen

What carries the argument

The charge–salt transformation: instead of tracking each ionic density, the paper changes basis to one charge variable q = Σ z_i ρ_i and N−1 salt variables s_a = Σ ν_ai ρ_i chosen as electroneutral combinations. In this basis the linearized PNP equations become ∂_t w = M ∂²_x w − B w, with M = C ε C⁻¹ encoding ion diffusivities and B encoding electrostatic coupling. Diagonalizing Λ = M⁻¹(p̃I + B) decouples the spatial modes, and the impedance is assembled from the eigenvectors α_i and wavenumbers k_i. For a ternary electrolyte, this yields a closed-form impedance (Eq. 43) whose parameters Ω₁ and Ω₂ quantify the charge–salt coupling; setting them to zero recovers the binary result.

Load-bearing premise

The derivation assumes salt variables can be chosen so that each one has zero net charge weighted by bulk concentrations (Σ_i ν_ai z_i X_i = 0), which makes the blocking-electrode boundary conditions separate cleanly; the paper states a different, unweighted electroneutrality condition that its own ternary example does not satisfy, so for arbitrary stoichiometry this corrected orthogonality must be imposed for Eq. (33) to follow as written.

What would settle it

Numerically solve the full linearized PNP equations for a ternary electrolyte with unequal diffusivities and a composition where the stated salt variables are not weighted-neutral (e.g., an asymmetric divalent mixture), compute the impedance by direct integration, and compare with Eq. (43): a disagreement would show the boundary-condition assumption breaks. Experimentally, impedance spectra of a well-characterized NaCl–KCl mixture with unequal ionic diffusivities should show the predicted two-slope Warburg region and a broader crossover than any binary effective-diffusion model; the claim woul

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • Impedance spectra of electric double layer capacitors with mixed electrolytes will show an extra slanted diffusive region between the high-frequency arc and the low-frequency capacitive branch whenever one ion has a different mobility.
  • The width and slope of that region are set by the ratio of ambipolar diffusion coefficients, so fitting a spectrum can in principle extract individual ion diffusivities.
  • When all ions diffuse at the same rate, the impedance is exactly that of a binary electrolyte regardless of how many species are present—extra salts become invisible to impedance.
  • The general N-component formula provides a numerical recipe for arbitrary mixtures: diagonalize an N×N matrix and assemble Eq. (33).
  • Equivalent-circuit fits to real mixed-electrolyte data will require additional Warburg elements rather than a single averaged diffusion constant.

Where Pith is reading between the lines

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

  • The same charge–salt diagonalization could predict other linear transport properties of mixed electrolytes, such as dielectric relaxation or ac conductivity, giving a unified route to multi-ion response.
  • Because equal-diffusivity salt modes are invisible to impedance, impedance data alone cannot determine salt transport parameters in such mixtures; complementary measurements, like diffusion-weighted NMR or concentration profiling, would be needed.
  • If the predicted two-slope Warburg feature is confirmed experimentally, it could serve as an impedance-spectroscopy fingerprint of how many distinct ion mobilities are present in a mixed electrolyte.
  • Extending the salt-mode coupling idea to Faradaic (non-blocking) electrodes would likely modify the low-frequency charge-transfer response and may produce additional features beyond the blocking-electrode case treated here.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript derives an impedance expression for an ideal N-component electrolyte between blocking planar electrodes, starting from linearized Poisson–Nernst–Planck equations. A charge–salt transformation is introduced, the linear system is diagonalized in Laplace space, and a general impedance formula is written as Eq. (33). The paper then specializes to a monovalent ternary electrolyte, obtains a three-mode impedance expression, and evaluates it numerically for different diffusivity ratios. The equal-diffusivity limit reproduces Macdonald’s binary result, and unequal diffusivities produce additional Warburg-like features that broaden the resistive–capacitive crossover.

Significance. If Eq. (33) is established rigorously, it is a useful compact starting point for multi-component EDLC impedance, and the ternary analysis is a valuable explicit example. The checks against known binary limits are a strength, and the numerical results are clearly presented. However, the general-N derivation currently relies on an incorrectly stated and insufficient orthogonality condition for the salt variables, so the central formula is not established as written. The ternary section appears salvageable, but the paper’s advertised generality needs repair.

major comments (3)
  1. [§3, Eqs. (10)–(11) and (23a)] The boundary condition ∂_x ŝ_a=0 used in Eq. (23a) is not a consequence of Eq. (11). From the blocking condition j_i=0, one obtains ∂_x s_a = -(1/S)(∑_i ν_ai z_i X_i) ∂_x ψ. Hence ∂_x s_a=0 requires ∑_i ν_ai z_i X_i=0, not ∑_i ν_ai z_i=0. These conditions coincide only when all X_i are equal. Moreover, the ternary basis used in §4 violates Eq. (11): for z=(1,1,-1), X=(1,1,2), ν_1=(2,0,1), one has z·ν_1=1, while the weighted condition holds. The same issue appears in the Appendix B salt variables. Consequently, the derivation of Eq. (33) from the stated assumptions is internally inconsistent. The paper should either define the salt basis by the weighted orthogonality ∑ν_ai z_i X_i=0, or carry the extra ∂_x ψ term through the general boundary-value solution.
  2. [§5, 'For a general multi-component electrolyte'] The statement that for a general N-component electrolyte 'the charge relaxation is coupled to one salt-relaxation mode, and the rest of the N−2 salt variables remain decoupled' is asserted without proof for N>3. It is also in tension with the abstract’s 'one or more neutral composition modes'. This matters for the qualitative interpretation, because the paper’s novelty partly rests on predicting how many diffusive modes appear. Either prove this claim from the eigenvalue structure of Λ in Eq. (17), or explicitly restrict it to the ternary cases studied here.
  3. [Abstract and §5, effective-binary claim] The abstract claims that the results explain why mixed-electrolyte spectra 'cannot be interpreted as a simple binary electrolyte with an averaged diffusion coefficient.' The evidence in Fig. 1 is qualitative: no comparison is made against the best-fit binary unequal-diffusivity model, which already has a Warburg-like region. To make this claim load-bearing, the ternary impedance should be fitted with the binary expression (e.g., Eq. (45) with adjusted effective parameters) and the residuals shown. Alternatively, the conclusion should be softened to say that the spectra contain additional features beyond the single-ambipolar-diffusion binary form.
minor comments (5)
  1. [Eq. (37a)] The term written as '− ϵ1 + ϵ2 + 2ϵ3' should be '−(1 + ϵ2 + 2ϵ3)'; the explicit 'ϵ1' is confusing since ϵ1=1.
  2. [Fig. 1 caption] The caption states X1=1/4, X2=1/4, X3=1/2, while §4 uses X1=X2=1, X3=2. These differ by a common scale, but the equivalence should be stated.
  3. [§5 and §6] Typos: 'freuqencies' and 'high freuency' should be corrected.
  4. [Data availability] The data-availability statement contains a placeholder 'link: xx' and should be completed.
  5. [Eqs. (42)–(43)] The abbreviations Ω1 and Ω2 are introduced abruptly; showing their relation to the η vector defined in Eq. (26) would improve readability.

Circularity Check

0 steps flagged

No circularity found: the impedance derivation is an algebraic consequence of the stated PNP equations and matches external benchmarks; the general-N boundary-condition concern is an internal-consistency issue, not a circular one.

full rationale

The paper derives Eq. (33) by transforming the linearized PNP equations into a charge–salt basis, diagonalizing the matrix Λ, and applying blocking-electrode boundary conditions. The derivation is self-contained: no diffusivity or other parameter is fitted to the impedance spectra, and the final formula is a closed-form algebraic expression in the eigenvalues/eigenvectors of the transport matrix. The equal-diffusivity limit reproduces Macdonald's 1953 binary result (Eqs. 43→44), and the unequal-diffusivity binary limit is cross-checked against both the authors' preprint [3] and the independent Balu–Khair derivation [25]; since the binary result is also independently available, the self-citation is not load-bearing. The ternary analysis (Section IV) is likewise derived from the same linearized equations with no fitted inputs. The reader's identified boundary-condition problem—that ∂x s_a=0 follows from j_i=0 only under the X-weighted condition Σν_ai z_i X_i=0, whereas Eq. (11) states the unweighted condition—is a genuine mathematical correctness concern for the claimed general-N validity of Eq. (33), but it is not an instance of a prediction reducing to an input or of a fitted parameter being renamed as a result. Therefore the circularity score is 0; the internal inconsistency should be evaluated as a correctness/rigor issue, not circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 0 invented entities

The central result rests on standard linearized PNP electrodiffusion and on a charge-salt basis choice. No data are fitted; the diffusivity ratios in the figure are illustrative inputs. The main accounting issue is the unstated and partially incorrect orthogonality condition on the salt basis (see weakest_assumption): the worked ternary example satisfies the required condition implicitly even though Eq. (11) is inconsistent with it. No new physical entities are introduced.

free parameters (2)
  • Diffusivity ratios ε2 = D2/D1, ε3 = D3/D1 = Fig. 1 uses (1,1), (1,0.1), (0.1,1), (0.01,0.3), (0.01,0.1)
    Illustrative model inputs chosen by hand for the figure and discussion; not fitted to any data.
  • Dimensionless electrode separation L̃ = L/λ_D = 100 in Fig. 1
    Fixed value for the impedance plots, representing the large-cell limit; a standard choice, not fitted.
axioms (5)
  • domain assumption Poisson-Nernst-Planck with ideal dilute solution, constant diffusivities, linearized around equilibrium
    Section II B. The entire derivation lives in this model; sterics, correlations, adsorption, and Faradaic reactions are explicitly excluded.
  • domain assumption Small applied potential (Ψ̃ = eΨ/kT << 1) justifying linearization
    Section II B. The impedance is defined as the linear-response limit of the system.
  • domain assumption Planar symmetry: blocking electrodes at x = ±L with symmetric boundary data, giving B_i = 0 and C0 = 0
    Section III, after Eq. (21): 'Considering the symmetry of the system, I find B_i = 0 and C0 = 0.'
  • standard math Λ is diagonalizable with distinct eigenvalues for generic parameters; degenerate limits handled by limits
    Section III, Eq. (18); the ternary analysis computes eigenvalues numerically (Section IV), and the equal-diffusivity case is treated through limiting expressions for Ω1, Ω2.
  • ad hoc to paper Blocking boundary conditions decouple in the charge-salt basis as ∂_x ŝ = 0, i.e., salts satisfy Σ ν_ai z_i X_i = 0
    Section III, Eq. (23a). The stated condition in Eq. (11) is Σ ν_ai z_i = 0, which is the wrong orthogonality and is violated by the paper's own ternary example; the condition actually required by the boundary treatment is unstated.

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Cite this review

Pith. "Pith review of Impedance of an electric double layer capacitor with a multi-component electrolyte." pith.science (2026). https://pith.science/paper/HUKCSFYS

@misc{pith2026260801799,
  author       = {Pith},
  title        = {Pith review of: Impedance of an electric double layer capacitor with a multi-component electrolyte},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HUKCSFYS}},
  note         = {Machine review of arXiv:2608.01799}
}
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read the original abstract

I derive the impedance response of an ideal electrolyte containing an arbitrary number of mobile ionic species between blocking planar electrodes, described by the Poisson--Nernst--Planck equations. By transforming the linearized equations to a charge--salt basis, the response is written in terms of a multi-component diffusion--migration matrix and its eigenvalues/eigenvectors. When all diffusivities are equal, the charge mode decouples from the neutral concentration subspace and the classical binary-electrolyte result is recovered. In contrast, unequal diffusivities couple charge relaxation to one or more neutral composition modes. For a ternary electrolyte with two cations and one anion, this coupling produces additional diffusive features and broadens the crossover between resistive and capacitive regimes. The results found in this paper provide a minimal continuum explanation for why mixed electrolytes can display impedance spectra that cannot be interpreted as a simple binary electrolyte with an averaged diffusion coefficient.

Figures

Figures reproduced from arXiv: 2608.01799 by David Fertig.

Figure 1
Figure 1. Figure 1: FIG. 1. Impedance obtained from Eq. (43) for a ternary mix [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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Reference graph

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This paper was first reviewed by deepseek-v4-flash on August 4, 2026.