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REVIEW 4 major objections 4 minor 23 references

Thermodynamic adsorption potential of superconductors

T0 review · 4 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read A superconductor's adsorption potential is almost proportional to its Tc.

desk verdict Novel Polanyi-adsorption framework for superconductivity, but the central ε–Tc correlation is baked into the formulas and the underlying equation of state misbehaves in the regime used. read the letter →

arxiv 2507.09869 v1 pith:EFNGFGJK submitted 2025-07-14 cond-mat.supr-con quant-ph

Jiu Hui Wu , Jiamin Niu , Kejiang Zhou This is my paper · ORCID
classification cond-mat.supr-conquant-ph PACS 74.20.-z
keywords superconductivityPolanyiadsorptionpotentialhigh-temperaturesuperconductorsCooperpairscatastrophetheoryquantumstateequationtransitiontemperaturethermodynamics
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

This paper argues that high-temperature superconductivity is governed by the same physical adsorption potential that describes how gases condense on solid surfaces, applied here to free electrons near a crystal lattice. The authors define the molar adsorption potential $\varepsilon$ of a superconductor, compute it from a catastrophe-theory quantum state equation, and find that $\varepsilon$ is almost proportional to the transition temperature $T_c$. For high-$T_c$ materials ($T_c$ above roughly 40 K), they conclude that Cooper pairs are formed mainly by this adsorption potential rather than by the electron-phonon interaction alone. The adsorption potential still operates at $T_c$ even though the BCS energy gap vanishes there, and the authors say the theory can explain many anomalies of the normal and superconducting states.

What carries the argument

The load-bearing object is the Polanyi molar adsorption potential $\varepsilon$, adapted from gas-adsorption chemistry to electrons: Eq. (6b) sets $\varepsilon$ as the change in chemical potential when one mole of electrons is reversibly adsorbed from the bulk phase $\mu_b$ to the adsorption phase $\mu_a$ near the lattice surface. To evaluate $\mu_n$ and $\mu_s$, the paper uses the quantum state equation Eq. (7) for Fermi electrons, derived by catastrophe theory in Ref. 21 and simplified to Eq. (8) in the limit $\beta \ll 1$. The phase-transition index $\alpha$ is fixed at $15/11$ below $T_c$ and $15/14$ above it from the specific-heat behavior, and the superconducting-electron fraction $\omega$ is fixed by continuity of entropy at $T_c$. Equation (16b) then expresses the adsorption condition $\varepsilon_p = \varepsilon/A \ge \mu_s - \mu_n$ as a closed expression in $T_c$, the electron concentration $n_c$, and $\omega$.

What would settle it

Take measured heat-capacity data for tin across $T_c$, compute the entropy from Eq. (13), and impose its continuity at $T_c$ to solve for the superconducting-electron fraction $\omega$; if the resulting $\omega$ departs from the value 0.0006223 used in Table 1, the adsorption-potential derivation is internally inconsistent with the measured thermodynamics.

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Extended reading notes

Core claim

The central claim is that the molar adsorption potential $\varepsilon$ of a superconductor, defined as the reversible work to move one mole of electrons from the bulk normal phase to the adsorption phase next to the lattice, is almost proportional to the superconductivity temperature $T_c$. Using Eq. (16b), the authors compute $\varepsilon$ for seven superconductors spanning Ag ($T_c = 0.9$ K) to HgBa$_2$Ca$_2$Cu$_3$O$_{8+\delta}$ ($T_c = 135$ K) and find an almost linear relation, with a higher slope below $T_c = 40$ K. They interpret this as evidence that high-$T_c$ superconductors ($T_c \ge 40$ K) are mainly formed by the molar adsorption potential, while low-$T_c$ superconductors require both the adsorption potential and the BCS energy-gap mechanism. Because the adsorption potential is nonzero at $T = T_c$, it can form Cooper pairs even where the BCS gap is zero.

Load-bearing premise

The whole calculation stands on the quantum state equation Eq. (7) for Fermi electrons, taken from Ref. 21; if that equation is not an accurate description of electrons in a superconductor, the adsorption potential and its claimed proportionality to $T_c$ do not follow.

Editorial extensions

If this is right

  • The claimed $\varepsilon$–$T_c$ proportionality implies that changing a material's lattice composition and structure, which sets the adsorption potential, can raise or lower $T_c$ in a predictable way.
  • Because the adsorption potential operates at $T = T_c$ where the BCS gap vanishes, high-$T_c$ pairing does not depend on a phonon-induced energy gap.
  • The superconducting-electron fraction $\omega$ is much larger in high-$T_c$ than in low-$T_c$ materials, so simple, high-carrier-density metals stay at low $T_c$.
  • The theory explains why the minimal structural unit for copper-oxide superconductivity is an intact cell layer containing a CuO$_2$ bilayer, since $\omega$ reflects composition and structure.
  • It can account for normal-state anomalies and the isotope effect of copper oxides without abandoning the Cooper-pair picture.

Reading between the lines

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

  • If the adsorption mechanism is real, surface and interface engineering that enlarges the adsorption space (nanostructuring, heterostructure stacking) should systematically raise $T_c$, a testable prediction distinct from phonon-mediated pairing.
  • The $\varepsilon \propto T_c$ scaling may apply across unconventional superconductors generally, suggesting that $T_c$ limits are set by an adsorption-energy scale rather than a phonon-energy scale.
  • The derivation hinges entirely on Eq. (7); an independent derivation of that quantum state equation from standard statistical mechanics would either confirm or undermine the entire framework.
  • The Polanyi analogy could be made quantitative by computing the adsorption potential from first principles for a single superconductor, providing a direct numerical check of Table 1.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The manuscript applies Polanyi adsorption theory to electrons near the lattice of a superconductor, using a quantum state equation for Fermi electrons taken from the authors' earlier work (Ref. 21). From that equation the authors derive chemical potentials for the normal and superconducting phases, fix the phase-transition indices so that the specific heat has the conventional T and T^3 behaviors, determine the superconducting electron fraction by entropy continuity, and compute an average adsorption energy ε_p and molar adsorption potential ε for seven superconductors. The central claim is that ε is almost proportional to T_c, and that this reveals a physical adsorption mechanism that, beyond the electron-phonon interaction, forms Cooper pairs in high-T_c superconductors.

Significance. If established, the claim would provide a new thermodynamic route to high-temperature superconductivity and could be a falsifiable alternative to phonon-mediated pairing. The paper does present a concrete formula, Eq. (16b), and a data table that could in principle be tested against new materials. However, the result is not supported in its current form: the central equation of state is neither derived nor validated in the relevant regime, the derivation chain is internally inconsistent, and the reported ε–T_c correlation is substantially built into the input variables. The strengths are the clear claim and the explicit tabulation; the weaknesses are fundamental to the derivation.

major comments (4)
  1. [§3.1, Eq. (8a)] The load-bearing equation of state Eq. (7) is cited only to Ref. 21 and is not derived in this paper. More seriously, in the regime relevant to Table 1 (β = m k_B T_c/(ħ² n_c^{2/3}) between roughly 5×10^-6 and 0.04), Eq. (8a) with the normal-phase index α = 15/14 gives P ∝ n^{5/7} T^{10/7}, which vanishes as T→0. This contradicts the exact low-temperature Fermi-gas behavior P ≈ P_0[1 + O((k_B T/E_F)^2)] with P_0 ∝ n^{5/3}, and underestimates the Fermi degeneracy pressure by many orders of magnitude. Because Eqs. (10)–(16b) inherit this equation of state, the derived chemical potentials and adsorption potentials are not based on a valid low-temperature Fermi-gas description.
  2. [§3.3, Eq. (16b) and Table 1] The reported proportionality between the molar adsorption potential ε and T_c is substantially encoded in the inputs. Equation (16b) has an explicit prefactor k_B T_c, and the relative proportion ω is fixed by the entropy-continuity condition at T = T_c using the same T_c and n_c that appear in the formula. Thus ε ≈ c R T_c with c determined from the fitted ω is partly a restatement of the input data, not an independent discovery about adsorption physics. The manuscript needs a control calculation or a demonstration that the prefactor c varies widely under a null model of random material parameters; without that, the sentence after Table 1 that high-T_c superconductors 'are mainly formed by the molar adsorption potentials' is not supported.
  3. [§3.1, Eqs. (9)–(11) and Eq. (16a)] The derivation chain from Eq. (9) to Eqs. (10)–(11) is internally inconsistent. Differentiating Eq. (9) with respect to N at fixed T and V gives, for the second term, the coefficient 12.88 × (5/3) = 21.47 times (1 − 8α/15), not 4.88 as printed in Eqs. (10) and (11). The values 5.856 and 9.194 in Eq. (16a) correspond to 21.47, so either Eq. (9) is not the correct Gibbs free energy or Eqs. (10)–(11) and Eq. (16a) cannot be reproduced from it. This internal inconsistency prevents the numerical results in Table 1 from being checked against the printed equations.
  4. [§3.2] The phase-transition indices α = 15/14 and α′ = 15/11 are chosen so that the specific heat behaves as C_p ∝ T in the normal phase and C_p ∝ T^3 in the superconducting phase for low-T_c superconductors, and the same indices are then applied universally to all materials in Table 1, including high-T_c cuprates. No evidence is given that these indices are appropriate for strongly correlated cuprate superconductors, and the superconducting fraction ω is determined by entropy continuity, which is a fitting constraint rather than a prediction. The universality of the adsorption-potential claim therefore rests on an unjustified extrapolation.
minor comments (4)
  1. [Table 1] The table lists Ag with T_c = 0.9 K, but bulk silver is not generally considered a superconductor at ambient pressure; the source of this value should be cited or the entry should be removed.
  2. [Throughout] The units of the electron concentration n_c in Table 1 should be stated explicitly (m^-3), and 'Avogadro number' in Eq. (6b) should be 'Avogadro constant'.
  3. [Figures 2 and 3] The manuscript refers to Figures 2 and 3, but the figures are not included in the submitted text; please ensure that the actual plots are present in the final version.
  4. [General presentation] The text contains many spacing and encoding artifacts (e.g., missing spaces between words and formulas, broken ligatures such as 'high-T_c' appearing as 'high- 푇푐'), which make the paper difficult to read and require a thorough copy edit.

Circularity Check

2 steps flagged · score 7.0 of 10

The ε–Tc correlation is computed from Eq. (16b), which has Tc as an explicit prefactor and uses α, ω fixed by known specific-heat laws and entropy continuity; the underlying equation of state is imported from the authors' own Ref. 21 without derivation.

  1. fitted input called prediction [Section 3.3, Eq. (16b); Table 1; Fig. 3]
    "When 푇 = 푇c, there is εp ≥ μs − μn = kB Tc [ 0.4/ω (m kB Tc/(ħ² nc^{2/3}))^{14/11} + 5.856/ω (m kB Tc/(ħ² nc^{2/3}))^{9/11} − 1.256 (m kB Tc/(ħ² nc^{2/3}))^{11/14} − 9.194 (m kB Tc/(ħ² nc^{2/3}))^{3/7} ] ... where ω is determined by the continuity of the entropy S at Tc from Eq.(13). ... According to Eq.(16b), in Table 1 we calculate ... It can be found that the molar adsorption potential ε is almost proportional to the superconductivity temperature Tc."

    Every ε value in Table 1 is obtained by inserting the material's Tc and nc into Eq. (16b). The equation has an explicit kB Tc prefactor, and the remaining bracket is a function of βc = m kB Tc/(ħ² nc^{2/3}) and of ω, with ω itself fixed by imposing entropy continuity at Tc. Thus the reported 'ε ∝ Tc' relation is not an independent prediction: it is a deterministic re-expression of the very Tc (and nc) used as inputs, with parameters α and ω chosen to reproduce standard C_p ∝ T and C_p ∝ T^3 behavior. The approximate proportionality is largely the explicit prefactor, not a newly discovered law.

  2. self citation load bearing [Section 3.1, Eq. (7); used in Eqs. (8)–(16)]
    "According to Ref. 21, we have obtained the correlation among the electron concentration, temperature and pressure of Fermi electrons by the catastrophe theory as: [Eq. (7)]"

    All subsequent equations—the Gibbs free energy (9), the chemical potentials (10)–(11), the entropy and specific heat (13)–(14), and the adsorption potential condition (16)—are derived from Eq. (7). Eq. (7) is not derived or independently validated in this paper; it is simply attributed to Ref. 21, an earlier paper by the same group (Wu, Niu, Liu, Zhou). The only checks offered are limiting cases (β→∞, α=1/2 gives ideal gas; α=0, β→0 gives degeneracy pressure). These limits do not establish the low-β, finite-α form used for Table 1, so the paper's central mechanism rests on an unverified same-author citation chain.

full rationale

Accepting Eq. (7) at face value, the numerical central claim still reduces by construction. Eq. (16b) is of the form ε = A kB Tc × f(m kB Tc/(ħ² nc^{2/3}), ω), with ω determined from Eq. (13) by entropy continuity at Tc; therefore computing ε from Tc,nc and then announcing a correlation between ε and Tc is a restatement of the input data through a fitted formula. The determination of α'=15/11 and α=15/14 in Section 3.2 is itself calibrated to the known empirical specific-heat laws (C_p ∝ T^3 superconducting, C_p ∝ T normal) rather than derived from the adsorption mechanism. Additionally, the printed derivation is internally not reproducible: differentiating Eq. (9) with respect to N gives a coefficient 21.47(1−8α/15), not the printed 4.88 in Eqs. (10)–(11), while the numerical values 9.194 and 5.856 in Eq. (16) correspond to the 21.47 coefficient. These issues make the ε–Tc relation a fitted input/output correlation rather than an independent first-principles prediction. The self-citation of Eq. (7) is load-bearing, but the larger problem is that the main 'finding' is already contained in Eq. (16b) and in the values chosen for α and ω.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claim relies on the catastrophic-theory quantum state equation from Ref. 21 (self-cited), the analogy between molecular adsorption and electron condensation, and the fitted parameters α, α', ω. These are not independently verified in this paper.

free parameters (3)
  • α (phase transition index, normal phase) = 15/14
    Chosen so that Cp ∝ T in the normal phase of low-Tc superconductors (Section 3.2). Assumed to hold for all superconductors, including high-Tc cuprates, without independent verification.
  • α' (phase transition index, superconducting phase) = 15/11
    Chosen so that Cp ∝ T^3 in the superconducting phase of low-Tc superconductors (Section 3.2). Again assumed universal.
  • ω (relative proportion of superconducting electrons) = Material-dependent, e.g., 0.0006223 for Sn, 0.0244192 for HgBa2Ca2Cu3O8
    Determined by imposing continuity of entropy S at Tc (Eq. 13). This is a fitting constraint, not a prediction; the paper uses it to compute the adsorption potential, making the correlation with Tc partly circular.
assumptions (3)
  • domain assumption The quantum state equation (Eq. 7) derived in Ref. 21 via catastrophe theory is correct and applicable to electrons in superconductors.
    The entire derivation of adsorption potential uses this equation as a starting point. The paper provides no derivation or external validation; it relies entirely on a self-cited prior work.
  • ad hoc to paper Polanyi adsorption theory for molecular gases applies to free electrons near a crystal lattice.
    The analogy between gas adsorption on solids and electron condensation near a lattice is asserted in the introduction and Section 2, without quantitative justification.
  • domain assumption The superconducting phase transition is a second-order phase transition describable by the same catastrophe-theory framework.
    The paper assumes the phase-transition index concept from catastrophe theory applies to superconductivity, and uses entropy continuity to fix ω.

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Cite this review

Pith. "Pith review of Thermodynamic adsorption potential of superconductors." pith.science (2026). https://pith.science/paper/EFNGFGJK

@misc{pith2026250709869,
  author       = {Pith},
  title        = {Pith review of: Thermodynamic adsorption potential of superconductors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EFNGFGJK}},
  note         = {Machine review of arXiv:2507.09869}
}
read the original abstract

Based on the general thermodynamic analysis of Polanyi adsorption potential, the adsorption potential condition for superconductors is obtained exactly by using the quantum state equation we presented. Because this adsorption potential results in changes of electron concentration, temperature and pressure in a certain volume (adsorption space) adjacent to the surface of the lattice, the composition and structure of superconductors are of course decisive for the adsorption potential. Then we calculate the molar adsorption potentials for those typical superconductors, and find that it is positively correlated to the superconductivity temperature , which reveals that those high-superconductors are mainly determined by the higher molar adsorption potentials. In addition, the adsorption potential at still works despite the disappearance of the energy gap of the BCS theory. This shows that beyond the electron-phonon interaction mechanism, the Cooper-paired electrons are mainly formed by this physical adsorption potential for high-superconductors. This adsorption potential theory could explain almost all common facts about high-temperature superconductors, including many anomalies of the normal and superconducting states.

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