Universal scaling of electrochemical information transfer at solid-liquid interfaces

Ashutosh Rathi; Eikichi Kimura; El Mustapha Mansouri; Keigo Arai; Keisuke Oshimi; Kristi Kanya Saikia; Masazumi Fujiwara; Saki Tsuchiya; Sara Mandi\'{c}

REVIEW 3 major objections 5 minor 45 references

Universal scaling of electrochemical information transfer at solid-liquid interfaces

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read This paper claims that the maximum electrochemical information transmissible across a solid–liquid interface is set by a single dimensionless ratio u, yielding a universal inverse-square attenuation law I*/I0 = 1/(1+u)^2.

desk verdict A plausible, potentially useful design rule—Fisher info ~ 1/(1+u)^2—but the derivation sits in a missing supplement and the experimental check fits the one parameter that matters, so treat it as unverified until the supplement appears. read the letter →

arxiv 2607.22190 v1 pith:5RQ7GFSZ submitted 2026-07-24 physics.app-ph cond-mat.mtrl-sci

Eikichi Kimura , Ashutosh Rathi , Saki Tsuchiya , El Mustapha Mansouri , Sara Mandić , Kristi Kanya Saikia , Keisuke Oshimi , Masazumi Fujiwara

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Keigo Arai

This is my paper · ORCID

classification physics.app-phcond-mat.mtrl-sci

keywords electrochemicalpotentialFisherinformationDebyescreeningsolid-liquidinterfacecharge-statesensingelectrostaticpropagationlengthuniversalscalingtwo-levelsystem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the maximum amount of electrochemical information that can be transferred from a liquid to a probe embedded in a solid is controlled by a single dimensionless number u — the ratio of the electrostatic distance between the liquid and the probe to the electrostatic propagation length in the solid. After optimal tuning of the bias voltage, the Fisher information of a two-level charge-state readout equals I0/(1+u)^2, an inverse-square attenuation law that is independent of the probe's microscopic details. This matters because it identifies electrostatic screening, not the probe's internal physics, as the fundamental constraint on subsurface electrochemical sensing. If correct, it turns sensor optimization into a simple geometry problem: minimize u by placing probes shallower, increasing ionic strength, or lengthening the solid's propagation scale.

What carries the argument

The central object is the dimensionless electrostatic screening parameter u = (x~0 + λ~D)/L~s, built from three electrical length scales: the probe's depth below the interface, the Debye screening length in the liquid, and the propagation length over which the solid transmits electrostatic information. Its role is to absorb all microscopic detail: after the bias is set to its optimal value, the maximum Fisher information collapses to I0/(1+u)^2, a universal inverse-square attenuation. The work of u is to encode whether the interface is 'electrostatically transparent' (u≪1) or 'screened' (u≫1), directly setting the Cramér–Rao precision floor for estimating the electrochemical potential.

What would settle it

Measure the optimally biased Fisher information for two systems with the same u but different microscopic parameters (e.g., dopant density, Fermi-to-transition energy gap); the universal law predicts identical I*/I0, so any significant deviation would falsify it. Alternatively, sweep u over at least two orders of magnitude and test the exact 1/(1+u)^2 functional form.

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Extended reading notes

Core claim

Treating a planar solid–liquid interface with Debye–Hückel screening on the liquid side and a depletion layer in the solid, the authors model a near-surface two-level system whose occupation probability follows Fermi–Dirac statistics in thermal equilibrium. The Fisher information of this binary readout with respect to the liquid's electrochemical potential, maximized over the applied bias voltage, takes the closed form I* = I0/(1+u)^2, with I0 = 1/(2k_B T)^2 and u = (x~0 + λ~D)/L~s. Here x~0 is the electrical depth of the probe, λ~D the Debye screening length, and L~s the electrostatic propagation length in the solid. The paper argues that this reduction holds for any localized probe whose s

Load-bearing premise

The derivation assumes that after tuning the bias voltage to its optimum, the Fisher information is a function only of the ratio u, with no residual dependence on the Fermi energy, the defect's transition level, the dopant concentration, or the shape of the potential; if any of these enter the optimized information, the claimed universality breaks down.

Editorial extensions

If this is right

The best possible precision in estimating the liquid's electrochemical potential is bounded below by σ ≥ 1/√I*, so minimizing u directly improves measurement precision.
In the screened regime u ≫ 1, information dies off as u^{-2}, making deep probes or dilute electrolytes fundamentally worse.
Design rules follow: shallow probe depth, high ionic strength (small λ~D), and a long propagation length (low dopant density, large Fermi-to-transition energy gap) all increase accessible information.
Fitting four published charge-state experiments gives u_fit clustered near 1, indicating those systems already sit at the transparent-to-screened crossover.
The scaling law applies beyond solid-state defects to any localized probe governed by electronic free-energy shifts, so it can guide optimization of diverse sensing platforms.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the 1/(1+u)^2 law holds, it should also describe molecular redox probes and electrochemical transistors whose occupation is Fermi–Dirac-like, an extension beyond the defect case the paper studies.
The same ratio may govern kinetic or nonequilibrium sensing where the occupation probability is modulated by current flow rather than thermal equilibrium; the paper does not address this.
The optimal-bias expression could be inverted: fitting the response curve to the model yields u and thus the propagation length from existing data, giving a non-invasive way to measure L~s.
Because the attenuation is quadratic in distance, burying a probe to avoid surface chemistry costs information steeply, suggesting lateral probe geometries or 2D materials may outperform depth-separated ones.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Short version: the central formula, I* = I0/(1+u)^2, may well be right, but as submitted it cannot be checked. The derivation is deferred to a supplement that has not been uploaded, and the experimental "validation" fits the ionized dopant concentration N_D^+ separately for each dataset. That makes the agreement around u≈1 a fit, not an independent prediction. If the supplement delivers the reduction to u, this becomes a genuinely useful paper.

What is new: the explicit inverse-square scaling in terms of u=(x0+λD)/Ls, and the claim that the optimized Fisher information collapses to a single ratio. That framing is not in the cited literature. The design guidance—shallow probes, high ionic strength liquids, lightly doped solids—is clear and practical.

What the paper does well: the model setup is physically standard (Debye–Hückel liquid, depletion-layer solid, Fermi–Dirac two-level probe), and the information-theoretic measure is appropriate for the binary readout. The literature coverage of charge-state experiments is broad and relevant. The data are promised publicly.

Where it is soft: the load-bearing step is the collapse of the full optimized problem to u alone, and we never see it. Reference [9] literally says "See Supplemental Material that will be uploaded with the next version." That is not acceptable for the central equation. The assertion that E_T0−E_F0, N_D^+, and the shape of φ(x) drop out after optimization is exactly what the derivation must show. The experimental test is also weak: four diamond datasets, all clustered around u≈1, with N_D^+ fitted per dataset. That demonstrates the model can be made to match, not that the u-scaling holds across different regimes. Varying u over a range, or using independently measured N_D^+, would actually test the law.

The model assumptions—Debye–Hückel, depletion approximation, exclusion of the accumulation regime—are reasonable as scope conditions, but they mean the word "universal" is doing work. The paper should state these limits more prominently.

My take: if the supplement is supplied and the reduction to u is valid, this is a solid letter. Even without it, the heuristic is plausible and worth discussing. But I would not cite it as a result until the derivation is public and the scaling has an independent test. Send it to peer review with the condition that the supplement must be available and the experimental section must be honest about what is fitted. This is a paper to engage with, not desk-reject—the question matters, and the framework, if correct, would guide real experiments.

Referee Report

3 major / 5 minor

Summary. The paper claims that the maximum Fisher information about the liquid electrochemical potential obtainable by a solid-embedded two-level probe, after optimizing the bias voltage, obeys a universal inverse-square law I* = I0/(1+u)^2, where u = (x̃0 + λ̃D)/L̃s, with I0 = 1/(2kBT)^2. The model treats a planar Debye–Hückel electrolyte in contact with a solid, a two-level defect probe, and a binomial readout. The authors present information landscapes, propose design guidelines, and report fits of four experimental diamond charge-state datasets, all with u ≈ 1.

Significance. If the derivation of Eq. (1) is correct and the collapse to the single parameter u is genuine, the result would be a valuable and general design principle for subsurface electrochemical sensing, showing that the fundamental limit is set by three electrical length scales. The paper formulates a clean Fisher-information framework and explicitly targets falsifiable predictions. However, the central claim is currently unverifiable because the derivation is deferred to a missing Supplemental Material, and the experimental support is not independent, since the free parameter N_D^+ used to compute u is fitted per dataset. The idea is promising, but the manuscript in its present form does not establish the universal law.

major comments (3)

[Eq. (1), p.2 and p.3] The central universal scaling law is stated without derivation in the main text. The only reference is [9], which reads 'See Supplemental Material that will be uploaded with the next version.' The collapse from the full electrostatic model (Fermi–Dirac occupancy, Debye–Hückel/depletion potential, bias optimization) to I* = I0/(1+u)^2 with no residual dependence on E_T0 − E_F0, N_D^+, or the shape of φ(x̃) must be shown explicitly. This is the load-bearing step; without it, Eq. (1) is an unsupported assertion. The derivation must be supplied in a form the referee can check.
[Table I and §'The charge-state responses...'] The experimental 'validation' is not independent: N_D^+ is fitted separately for each dataset, and u_fit is computed from that fitted N_D^+. Thus the agreement of I*/I0 with the theoretical curve is a consequence of the fit, not a prediction. Additionally, all four datasets cluster around u ≈ 1, so they do not span the predicted (1+u)^{-2} dependence across different regimes. To support the universality claim, the authors should either fix N_D^+ from independent measurements or show a test that varies u over a wider range, or otherwise reframe Table I as a consistency check rather than a validation.
[p.3, relationship L̃s ∝ sqrt((E_F0 − E_T0)/N_D^+)] This relation is used to connect the fitted N_D^+ to u and to the design guidelines, but it is nowhere derived in the main text. It appears to follow from a depletion-layer model, but the derivation is deferred to [9]. Like Eq. (1), this is part of the missing machinery. The expression for φ(x̃), the derivation of V*, and the calculation of the optimized Fisher information must be provided in the main text or a complete Supplemental Material.

minor comments (5)

[p.3, sentence before Eq. (1)] The sentence 'By tuning the bias voltage to the optimal value V*, we maximize Eq. (1)' is confusing: Eq. (1) is the result after maximization. It should say 'we maximize the Fisher information I' or similar.
[Funding statement] There is a typo: 'This study was supported by the JST K Program ... (grant number JPMJKP24F3). and by JSPS' — the period before 'and' should be a comma, and 'JST K Program' is likely an incomplete program name.
[Reference [9]] The reference to the Supplemental Material is a placeholder and should be resolved before submission; a published paper cannot have 'will be uploaded'.
[Units and notation, Fig. 2 caption] The electrical length coordinate x̃ = x/ε uses ε with different values for liquid and solid, but the units (e.g., 'nm/(80ε0)') are unusual. Please define clearly the physical dimension of x̃ and the relation to the permittivity, and check that all expressions are dimensionally consistent.
[Fig. 4(a)] This panel claims 'universal scaling' but shows only the theoretical curve. Overlaying the four data points from Table I, with error bars, would make the agreement or scatter visible.

Circularity Check

1 steps flagged · score 5.0 of 10

Experimental 'validation' of Eq. (1) is a fitted-parameter re-expression: u_fit is computed from a per-dataset fit of N_D^+, and I*/I0 = (1+u_fit)^-2 follows by construction; the central derivation is deferred to missing SM [9].

fitted input called prediction [p.4, Table I paragraph ('The charge-state responses ...')]
"We fitted these experimental data using the present electrostatic model, with the ionized dopant concentration N_D^+ treated as the fitting parameter [9]. The dimensionless parameter u_fit, calculated from the fitted N_D^+, and the corresponding optimal Fisher information I* are summarized in Table I."

N_D^+ is a free parameter fitted to each dataset; u_fit is then computed from that fitted N_D^+, and I*/I0 is taken from Eq. (1) as 1/(1+u_fit)^2. Thus Table I's agreement with the inverse-square law is imposed by construction rather than tested. The datasets are all diamond charge-state measurements clustered near u≈1, so the table neither varies u independently nor checks the predicted u-dependence; the 'analysis revealed' a similar intermediate regime only because the fit parameter was used to compute u.

full rationale

The central analytical claim, Eq. (1), is a claimed Fisher-information optimization result; the derivation is entirely deferred to reference [9], which at the time of writing reads 'See Supplemental Material that will be uploaded with the next version.' That is a serious uncheckability/evidence gap but not in itself a circular reduction. No self-citation chain or imported uniqueness theorem is used to force Eq. (1). The only presented empirical support, however, is circular in the pattern-2 sense: the model is fitted to each experiment through N_D^+, u_fit is derived from the fit, and I*/I0 is then read off the same formula being 'validated.' Nothing in the paper predicts u_fit before fitting or tests the claimed collapse across varied microscopic parameters. Because the theoretical law itself could be independently derived from the model (if the SM is supplied and checked), this is partial circularity in the validation, not a fully forced derivation; score 5, not 8.

Assumptions & free parameters 1 free parameters · 5 assumptions · 1 invented entities

The model's predictions depend on three length scales: λ~D and x~0 are inputs from experimental conditions (chosen in the figures), while L~s is tied to a fitted N_D^+. The universal scaling rests on the Debye–Hückel and depletion approximations, Fermi–Dirac statistics, and the exclusion of the accumulation regime. These are standard but restrict the claimed universality.

free parameters (1)

N_D^+ (ionized dopant concentration) = per dataset (not reported directly; u_fit ≈ 1.06–1.20)
Fitted to reproduce the charge-state response in the Sow, Karaveli, Grotz, and Fulton datasets; u_fit is then computed from this fit, so the comparison is not an independent test of the scaling law.

assumptions (5)

domain assumption Debye–Hückel approximation for the liquid: linear screening with length λ~D.
Used throughout; p.2: 'The liquid is described within the Debye–Hückel approximation.'
domain assumption Depletion-layer approximation in the solid: the potential drop occurs across a layer of width L~s, with L~s ∝ sqrt((E_F0−E_T0)/N_D^+).
p.3: 'Because L~s ∝ sqrt( (E_F0−E_T0)/N_D^+ )' — this identifies L~s with a depletion width and ties it to the fitted N_D^+.
domain assumption The probe is a two-level system whose occupancy follows Fermi–Dirac statistics with E_T(x~) = E_T0 − eφ(x~).
p.2: 'the occupation probability of the two-level system follows the Fermi–Dirac distribution.'
domain assumption The electron-accumulation regime (μ~e > E_F0) is excluded from the model.
p.2: 'The electron-accumulation regime μ~e > E_F0 is excluded from the present model.' This restricts the claimed universality.
standard math Fisher information for a binomial readout: I = S'^2/(S(1−S)).
p.2 Eq. (2); standard definition of Fisher information for a Bernoulli random variable.

invented entities (1)

L~s — 'electrostatic propagation length in the solid'
purpose: A single scalar length used to define u and to quantify attenuation of electrochemical information in the solid; identified with the depletion width via L~s ∝ sqrt((E_F0−E_T0)/N_D^+).
No independent measurement is provided. In the experimental fits, L~s is inferred from the fitted N_D^+, so it is a derived, fitted quantity rather than an independently verified physical length. The new name suggests a fundamental scale, but it is a re-scaled depletion width.

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Cite this review

Pith. "Pith review of Universal scaling of electrochemical information transfer at solid-liquid interfaces." pith.science (2026). https://pith.science/paper/5RQ7GFSZ

@misc{pith2026260722190,
  author       = {Pith},
  title        = {Pith review of: Universal scaling of electrochemical information transfer at solid-liquid interfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5RQ7GFSZ}},
  note         = {Machine review of arXiv:2607.22190}
}

abstract

Electrochemical potentials at solid-liquid interfaces govern diverse chemical and energy conversion processes; however, the extent to which their electrochemical influence extends into the solid remains unclear. This study demonstrates that the maximally accessible electrochemical information is controlled by a single dimensionless factor $u$, defined as the ratio of the effective electrostatic separation between the liquid and probe locations to the electrostatic propagation length in the solid. The resulting universal inverse-square scaling is independent of the microscopic details of the probe. This attenuation law identifies electrostatic screening as a fundamental constraint on information transfer across solid-liquid interfaces, providing quantitative design principles for subsurface electrochemical sensing.

Figures

Figures reproduced from arXiv: 2607.22190 by the authors.

**Figure 2.** FIG. 2. Model of a solid–liquid interface. (a) The liquid [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** FIG. 3. Information landscape of electrochemical potential [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: FIG. 4. Optimal Fisher information. (a) The optimized [PITH_FULL_IMAGE:figures/full_fig_p003_4.png]

Discussion (0). Continue with ORCID to comment.

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